YES(O(1),O(n^3))
752.91/207.95	YES(O(1),O(n^3))
752.91/207.95	
752.91/207.95	We are left with following problem, upon which TcT provides the
752.91/207.95	certificate YES(O(1),O(n^3)).
752.91/207.95	
752.91/207.95	Strict Trs:
752.91/207.95	  { U11(tt(), V1, V2) ->
752.91/207.95	    U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.95	  , U12(tt(), V1, V2) ->
752.91/207.95	    U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.95	  , isNatKind(n__0()) -> tt()
752.91/207.95	  , isNatKind(n__plus(V1, V2)) ->
752.91/207.95	    U31(isNatKind(activate(V1)), activate(V2))
752.91/207.95	  , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.95	  , activate(X) -> X
752.91/207.95	  , activate(n__0()) -> 0()
752.91/207.95	  , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.95	  , activate(n__s(X)) -> s(activate(X))
752.91/207.95	  , U13(tt(), V1, V2) ->
752.91/207.95	    U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.95	  , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.95	  , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.95	  , isNat(n__0()) -> tt()
752.91/207.95	  , isNat(n__plus(V1, V2)) ->
752.91/207.95	    U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.95	  , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.95	  , U16(tt()) -> tt()
752.91/207.95	  , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.95	  , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.95	  , U23(tt()) -> tt()
752.91/207.95	  , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.95	  , U32(tt()) -> tt()
752.91/207.95	  , U41(tt()) -> tt()
752.91/207.95	  , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N))
752.91/207.95	  , U52(tt(), N) -> activate(N)
752.91/207.95	  , U61(tt(), M, N) ->
752.91/207.95	    U62(isNatKind(activate(M)), activate(M), activate(N))
752.91/207.95	  , U62(tt(), M, N) ->
752.91/207.95	    U63(isNat(activate(N)), activate(M), activate(N))
752.91/207.95	  , U63(tt(), M, N) ->
752.91/207.95	    U64(isNatKind(activate(N)), activate(M), activate(N))
752.91/207.95	  , U64(tt(), M, N) -> s(plus(activate(N), activate(M)))
752.91/207.95	  , s(X) -> n__s(X)
752.91/207.95	  , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.95	  , plus(N, s(M)) -> U61(isNat(M), M, N)
752.91/207.95	  , plus(N, 0()) -> U51(isNat(N), N)
752.91/207.95	  , 0() -> n__0() }
752.91/207.95	Obligation:
752.91/207.95	  innermost runtime complexity
752.91/207.95	Answer:
752.91/207.95	  YES(O(1),O(n^3))
752.91/207.95	
752.91/207.95	Arguments of following rules are not normal-forms:
752.91/207.95	
752.91/207.95	{ plus(N, s(M)) -> U61(isNat(M), M, N)
752.91/207.95	, plus(N, 0()) -> U51(isNat(N), N) }
752.91/207.95	
752.91/207.95	All above mentioned rules can be savely removed.
752.91/207.95	
752.91/207.95	We are left with following problem, upon which TcT provides the
752.91/207.95	certificate YES(O(1),O(n^3)).
752.91/207.95	
752.91/207.95	Strict Trs:
752.91/207.95	  { U11(tt(), V1, V2) ->
752.91/207.95	    U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.95	  , U12(tt(), V1, V2) ->
752.91/207.95	    U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.95	  , isNatKind(n__0()) -> tt()
752.91/207.95	  , isNatKind(n__plus(V1, V2)) ->
752.91/207.95	    U31(isNatKind(activate(V1)), activate(V2))
752.91/207.95	  , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.95	  , activate(X) -> X
752.91/207.95	  , activate(n__0()) -> 0()
752.91/207.95	  , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.95	  , activate(n__s(X)) -> s(activate(X))
752.91/207.95	  , U13(tt(), V1, V2) ->
752.91/207.95	    U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.95	  , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.95	  , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.95	  , isNat(n__0()) -> tt()
752.91/207.95	  , isNat(n__plus(V1, V2)) ->
752.91/207.95	    U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.95	  , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.95	  , U16(tt()) -> tt()
752.91/207.95	  , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.95	  , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.95	  , U23(tt()) -> tt()
752.91/207.95	  , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.95	  , U32(tt()) -> tt()
752.91/207.95	  , U41(tt()) -> tt()
752.91/207.95	  , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N))
752.91/207.95	  , U52(tt(), N) -> activate(N)
752.91/207.95	  , U61(tt(), M, N) ->
752.91/207.95	    U62(isNatKind(activate(M)), activate(M), activate(N))
752.91/207.95	  , U62(tt(), M, N) ->
752.91/207.95	    U63(isNat(activate(N)), activate(M), activate(N))
752.91/207.95	  , U63(tt(), M, N) ->
752.91/207.95	    U64(isNatKind(activate(N)), activate(M), activate(N))
752.91/207.95	  , U64(tt(), M, N) -> s(plus(activate(N), activate(M)))
752.91/207.95	  , s(X) -> n__s(X)
752.91/207.95	  , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.95	  , 0() -> n__0() }
752.91/207.95	Obligation:
752.91/207.95	  innermost runtime complexity
752.91/207.95	Answer:
752.91/207.95	  YES(O(1),O(n^3))
752.91/207.95	
752.91/207.95	We add the following dependency tuples:
752.91/207.95	
752.91/207.95	Strict DPs:
752.91/207.95	  { U11^#(tt(), V1, V2) ->
752.91/207.95	    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.95	        isNatKind^#(activate(V1)),
752.91/207.95	        activate^#(V1),
752.91/207.95	        activate^#(V1),
752.91/207.95	        activate^#(V2))
752.91/207.95	  , U12^#(tt(), V1, V2) ->
752.91/207.95	    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.95	        isNatKind^#(activate(V2)),
752.91/207.95	        activate^#(V2),
752.91/207.95	        activate^#(V1),
752.91/207.95	        activate^#(V2))
752.91/207.95	  , isNatKind^#(n__0()) -> c_3()
752.91/207.95	  , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.95	    c_4(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.95	        isNatKind^#(activate(V1)),
752.91/207.95	        activate^#(V1),
752.91/207.95	        activate^#(V2))
752.91/207.95	  , isNatKind^#(n__s(V1)) ->
752.91/207.95	    c_5(U41^#(isNatKind(activate(V1))),
752.91/207.95	        isNatKind^#(activate(V1)),
752.91/207.95	        activate^#(V1))
752.91/207.95	  , activate^#(X) -> c_6()
752.91/207.95	  , activate^#(n__0()) -> c_7(0^#())
752.91/207.95	  , activate^#(n__plus(X1, X2)) ->
752.91/207.95	    c_8(plus^#(activate(X1), activate(X2)),
752.91/207.95	        activate^#(X1),
752.91/207.95	        activate^#(X2))
752.91/207.95	  , activate^#(n__s(X)) -> c_9(s^#(activate(X)), activate^#(X))
752.91/207.95	  , U13^#(tt(), V1, V2) ->
752.91/207.95	    c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.95	         isNatKind^#(activate(V2)),
752.91/207.95	         activate^#(V2),
752.91/207.95	         activate^#(V1),
752.91/207.95	         activate^#(V2))
752.91/207.95	  , U31^#(tt(), V2) ->
752.91/207.95	    c_20(U32^#(isNatKind(activate(V2))),
752.91/207.95	         isNatKind^#(activate(V2)),
752.91/207.95	         activate^#(V2))
752.91/207.95	  , U41^#(tt()) -> c_22()
752.91/207.95	  , 0^#() -> c_31()
752.91/207.95	  , plus^#(X1, X2) -> c_30()
752.91/207.95	  , s^#(X) -> c_29()
752.91/207.95	  , U14^#(tt(), V1, V2) ->
752.91/207.95	    c_11(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.95	         isNat^#(activate(V1)),
752.91/207.95	         activate^#(V1),
752.91/207.95	         activate^#(V2))
752.91/207.95	  , U15^#(tt(), V2) ->
752.91/207.95	    c_12(U16^#(isNat(activate(V2))),
752.91/207.95	         isNat^#(activate(V2)),
752.91/207.95	         activate^#(V2))
752.91/207.95	  , isNat^#(n__0()) -> c_13()
752.91/207.95	  , isNat^#(n__plus(V1, V2)) ->
752.91/207.95	    c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.95	         isNatKind^#(activate(V1)),
752.91/207.95	         activate^#(V1),
752.91/207.95	         activate^#(V1),
752.91/207.95	         activate^#(V2))
752.91/207.95	  , isNat^#(n__s(V1)) ->
752.91/207.95	    c_15(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.95	         isNatKind^#(activate(V1)),
752.91/207.95	         activate^#(V1),
752.91/207.95	         activate^#(V1))
752.91/207.95	  , U16^#(tt()) -> c_16()
752.91/207.95	  , U21^#(tt(), V1) ->
752.91/207.95	    c_17(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.95	         isNatKind^#(activate(V1)),
752.91/207.95	         activate^#(V1),
752.91/207.95	         activate^#(V1))
752.91/207.95	  , U22^#(tt(), V1) ->
752.91/207.95	    c_18(U23^#(isNat(activate(V1))),
752.91/207.95	         isNat^#(activate(V1)),
752.91/207.95	         activate^#(V1))
752.91/207.95	  , U23^#(tt()) -> c_19()
752.91/207.95	  , U32^#(tt()) -> c_21()
752.91/207.95	  , U51^#(tt(), N) ->
752.91/207.95	    c_23(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.95	         isNatKind^#(activate(N)),
752.91/207.95	         activate^#(N),
752.91/207.95	         activate^#(N))
752.91/207.95	  , U52^#(tt(), N) -> c_24(activate^#(N))
752.91/207.95	  , U61^#(tt(), M, N) ->
752.91/207.95	    c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.95	         isNatKind^#(activate(M)),
752.91/207.95	         activate^#(M),
752.91/207.95	         activate^#(M),
752.91/207.95	         activate^#(N))
752.91/207.95	  , U62^#(tt(), M, N) ->
752.91/207.95	    c_26(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.95	         isNat^#(activate(N)),
752.91/207.95	         activate^#(N),
752.91/207.95	         activate^#(M),
752.91/207.95	         activate^#(N))
752.91/207.95	  , U63^#(tt(), M, N) ->
752.91/207.95	    c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.95	         isNatKind^#(activate(N)),
752.91/207.95	         activate^#(N),
752.91/207.95	         activate^#(M),
752.91/207.95	         activate^#(N))
752.91/207.95	  , U64^#(tt(), M, N) ->
752.91/207.95	    c_28(s^#(plus(activate(N), activate(M))),
752.91/207.95	         plus^#(activate(N), activate(M)),
752.91/207.95	         activate^#(N),
752.91/207.95	         activate^#(M)) }
752.91/207.95	
752.91/207.95	and mark the set of starting terms.
752.91/207.95	
752.91/207.95	We are left with following problem, upon which TcT provides the
752.91/207.95	certificate YES(O(1),O(n^3)).
752.91/207.95	
752.91/207.95	Strict DPs:
752.91/207.95	  { U11^#(tt(), V1, V2) ->
752.91/207.95	    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.95	        isNatKind^#(activate(V1)),
752.91/207.95	        activate^#(V1),
752.91/207.95	        activate^#(V1),
752.91/207.95	        activate^#(V2))
752.91/207.95	  , U12^#(tt(), V1, V2) ->
752.91/207.95	    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.95	        isNatKind^#(activate(V2)),
752.91/207.95	        activate^#(V2),
752.91/207.95	        activate^#(V1),
752.91/207.95	        activate^#(V2))
752.91/207.95	  , isNatKind^#(n__0()) -> c_3()
752.91/207.95	  , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.95	    c_4(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.95	        isNatKind^#(activate(V1)),
752.91/207.95	        activate^#(V1),
752.91/207.95	        activate^#(V2))
752.91/207.95	  , isNatKind^#(n__s(V1)) ->
752.91/207.95	    c_5(U41^#(isNatKind(activate(V1))),
752.91/207.95	        isNatKind^#(activate(V1)),
752.91/207.96	        activate^#(V1))
752.91/207.96	  , activate^#(X) -> c_6()
752.91/207.96	  , activate^#(n__0()) -> c_7(0^#())
752.91/207.96	  , activate^#(n__plus(X1, X2)) ->
752.91/207.96	    c_8(plus^#(activate(X1), activate(X2)),
752.91/207.96	        activate^#(X1),
752.91/207.96	        activate^#(X2))
752.91/207.96	  , activate^#(n__s(X)) -> c_9(s^#(activate(X)), activate^#(X))
752.91/207.96	  , U13^#(tt(), V1, V2) ->
752.91/207.96	    c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.96	         isNatKind^#(activate(V2)),
752.91/207.96	         activate^#(V2),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V2))
752.91/207.96	  , U31^#(tt(), V2) ->
752.91/207.96	    c_20(U32^#(isNatKind(activate(V2))),
752.91/207.96	         isNatKind^#(activate(V2)),
752.91/207.96	         activate^#(V2))
752.91/207.96	  , U41^#(tt()) -> c_22()
752.91/207.96	  , 0^#() -> c_31()
752.91/207.96	  , plus^#(X1, X2) -> c_30()
752.91/207.96	  , s^#(X) -> c_29()
752.91/207.96	  , U14^#(tt(), V1, V2) ->
752.91/207.96	    c_11(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.96	         isNat^#(activate(V1)),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V2))
752.91/207.96	  , U15^#(tt(), V2) ->
752.91/207.96	    c_12(U16^#(isNat(activate(V2))),
752.91/207.96	         isNat^#(activate(V2)),
752.91/207.96	         activate^#(V2))
752.91/207.96	  , isNat^#(n__0()) -> c_13()
752.91/207.96	  , isNat^#(n__plus(V1, V2)) ->
752.91/207.96	    c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.96	         isNatKind^#(activate(V1)),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V2))
752.91/207.96	  , isNat^#(n__s(V1)) ->
752.91/207.96	    c_15(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.96	         isNatKind^#(activate(V1)),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V1))
752.91/207.96	  , U16^#(tt()) -> c_16()
752.91/207.96	  , U21^#(tt(), V1) ->
752.91/207.96	    c_17(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.96	         isNatKind^#(activate(V1)),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V1))
752.91/207.96	  , U22^#(tt(), V1) ->
752.91/207.96	    c_18(U23^#(isNat(activate(V1))),
752.91/207.96	         isNat^#(activate(V1)),
752.91/207.96	         activate^#(V1))
752.91/207.96	  , U23^#(tt()) -> c_19()
752.91/207.96	  , U32^#(tt()) -> c_21()
752.91/207.96	  , U51^#(tt(), N) ->
752.91/207.96	    c_23(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.96	         isNatKind^#(activate(N)),
752.91/207.96	         activate^#(N),
752.91/207.96	         activate^#(N))
752.91/207.96	  , U52^#(tt(), N) -> c_24(activate^#(N))
752.91/207.96	  , U61^#(tt(), M, N) ->
752.91/207.96	    c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.96	         isNatKind^#(activate(M)),
752.91/207.96	         activate^#(M),
752.91/207.96	         activate^#(M),
752.91/207.96	         activate^#(N))
752.91/207.96	  , U62^#(tt(), M, N) ->
752.91/207.96	    c_26(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.96	         isNat^#(activate(N)),
752.91/207.96	         activate^#(N),
752.91/207.96	         activate^#(M),
752.91/207.96	         activate^#(N))
752.91/207.96	  , U63^#(tt(), M, N) ->
752.91/207.96	    c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.96	         isNatKind^#(activate(N)),
752.91/207.96	         activate^#(N),
752.91/207.96	         activate^#(M),
752.91/207.96	         activate^#(N))
752.91/207.96	  , U64^#(tt(), M, N) ->
752.91/207.96	    c_28(s^#(plus(activate(N), activate(M))),
752.91/207.96	         plus^#(activate(N), activate(M)),
752.91/207.96	         activate^#(N),
752.91/207.96	         activate^#(M)) }
752.91/207.96	Weak Trs:
752.91/207.96	  { U11(tt(), V1, V2) ->
752.91/207.96	    U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.96	  , U12(tt(), V1, V2) ->
752.91/207.96	    U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.96	  , isNatKind(n__0()) -> tt()
752.91/207.96	  , isNatKind(n__plus(V1, V2)) ->
752.91/207.96	    U31(isNatKind(activate(V1)), activate(V2))
752.91/207.96	  , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.96	  , activate(X) -> X
752.91/207.96	  , activate(n__0()) -> 0()
752.91/207.96	  , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.96	  , activate(n__s(X)) -> s(activate(X))
752.91/207.96	  , U13(tt(), V1, V2) ->
752.91/207.96	    U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.96	  , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.96	  , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.96	  , isNat(n__0()) -> tt()
752.91/207.96	  , isNat(n__plus(V1, V2)) ->
752.91/207.96	    U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.96	  , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.96	  , U16(tt()) -> tt()
752.91/207.96	  , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.96	  , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.96	  , U23(tt()) -> tt()
752.91/207.96	  , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.96	  , U32(tt()) -> tt()
752.91/207.96	  , U41(tt()) -> tt()
752.91/207.96	  , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N))
752.91/207.96	  , U52(tt(), N) -> activate(N)
752.91/207.96	  , U61(tt(), M, N) ->
752.91/207.96	    U62(isNatKind(activate(M)), activate(M), activate(N))
752.91/207.96	  , U62(tt(), M, N) ->
752.91/207.96	    U63(isNat(activate(N)), activate(M), activate(N))
752.91/207.96	  , U63(tt(), M, N) ->
752.91/207.96	    U64(isNatKind(activate(N)), activate(M), activate(N))
752.91/207.96	  , U64(tt(), M, N) -> s(plus(activate(N), activate(M)))
752.91/207.96	  , s(X) -> n__s(X)
752.91/207.96	  , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.96	  , 0() -> n__0() }
752.91/207.96	Obligation:
752.91/207.96	  innermost runtime complexity
752.91/207.96	Answer:
752.91/207.96	  YES(O(1),O(n^3))
752.91/207.96	
752.91/207.96	We estimate the number of application of
752.91/207.96	{3,6,12,13,14,15,18,21,24,25} by applications of
752.91/207.96	Pre({3,6,12,13,14,15,18,21,24,25}) =
752.91/207.96	{1,2,4,5,7,8,9,10,11,16,17,19,20,22,23,26,27,28,29,30,31}. Here
752.91/207.96	rules are labeled as follows:
752.91/207.96	
752.91/207.96	  DPs:
752.91/207.96	    { 1: U11^#(tt(), V1, V2) ->
752.91/207.96	         c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.96	             isNatKind^#(activate(V1)),
752.91/207.96	             activate^#(V1),
752.91/207.96	             activate^#(V1),
752.91/207.96	             activate^#(V2))
752.91/207.96	    , 2: U12^#(tt(), V1, V2) ->
752.91/207.96	         c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.96	             isNatKind^#(activate(V2)),
752.91/207.96	             activate^#(V2),
752.91/207.96	             activate^#(V1),
752.91/207.96	             activate^#(V2))
752.91/207.96	    , 3: isNatKind^#(n__0()) -> c_3()
752.91/207.96	    , 4: isNatKind^#(n__plus(V1, V2)) ->
752.91/207.96	         c_4(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.96	             isNatKind^#(activate(V1)),
752.91/207.96	             activate^#(V1),
752.91/207.96	             activate^#(V2))
752.91/207.96	    , 5: isNatKind^#(n__s(V1)) ->
752.91/207.96	         c_5(U41^#(isNatKind(activate(V1))),
752.91/207.96	             isNatKind^#(activate(V1)),
752.91/207.96	             activate^#(V1))
752.91/207.96	    , 6: activate^#(X) -> c_6()
752.91/207.96	    , 7: activate^#(n__0()) -> c_7(0^#())
752.91/207.96	    , 8: activate^#(n__plus(X1, X2)) ->
752.91/207.96	         c_8(plus^#(activate(X1), activate(X2)),
752.91/207.96	             activate^#(X1),
752.91/207.96	             activate^#(X2))
752.91/207.96	    , 9: activate^#(n__s(X)) -> c_9(s^#(activate(X)), activate^#(X))
752.91/207.96	    , 10: U13^#(tt(), V1, V2) ->
752.91/207.96	          c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.96	               isNatKind^#(activate(V2)),
752.91/207.96	               activate^#(V2),
752.91/207.96	               activate^#(V1),
752.91/207.96	               activate^#(V2))
752.91/207.96	    , 11: U31^#(tt(), V2) ->
752.91/207.96	          c_20(U32^#(isNatKind(activate(V2))),
752.91/207.96	               isNatKind^#(activate(V2)),
752.91/207.96	               activate^#(V2))
752.91/207.96	    , 12: U41^#(tt()) -> c_22()
752.91/207.96	    , 13: 0^#() -> c_31()
752.91/207.96	    , 14: plus^#(X1, X2) -> c_30()
752.91/207.96	    , 15: s^#(X) -> c_29()
752.91/207.96	    , 16: U14^#(tt(), V1, V2) ->
752.91/207.96	          c_11(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.96	               isNat^#(activate(V1)),
752.91/207.96	               activate^#(V1),
752.91/207.96	               activate^#(V2))
752.91/207.96	    , 17: U15^#(tt(), V2) ->
752.91/207.96	          c_12(U16^#(isNat(activate(V2))),
752.91/207.96	               isNat^#(activate(V2)),
752.91/207.96	               activate^#(V2))
752.91/207.96	    , 18: isNat^#(n__0()) -> c_13()
752.91/207.96	    , 19: isNat^#(n__plus(V1, V2)) ->
752.91/207.96	          c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.96	               isNatKind^#(activate(V1)),
752.91/207.96	               activate^#(V1),
752.91/207.96	               activate^#(V1),
752.91/207.96	               activate^#(V2))
752.91/207.96	    , 20: isNat^#(n__s(V1)) ->
752.91/207.96	          c_15(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.96	               isNatKind^#(activate(V1)),
752.91/207.96	               activate^#(V1),
752.91/207.96	               activate^#(V1))
752.91/207.96	    , 21: U16^#(tt()) -> c_16()
752.91/207.96	    , 22: U21^#(tt(), V1) ->
752.91/207.96	          c_17(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.96	               isNatKind^#(activate(V1)),
752.91/207.96	               activate^#(V1),
752.91/207.96	               activate^#(V1))
752.91/207.96	    , 23: U22^#(tt(), V1) ->
752.91/207.96	          c_18(U23^#(isNat(activate(V1))),
752.91/207.96	               isNat^#(activate(V1)),
752.91/207.96	               activate^#(V1))
752.91/207.96	    , 24: U23^#(tt()) -> c_19()
752.91/207.96	    , 25: U32^#(tt()) -> c_21()
752.91/207.96	    , 26: U51^#(tt(), N) ->
752.91/207.96	          c_23(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.96	               isNatKind^#(activate(N)),
752.91/207.96	               activate^#(N),
752.91/207.96	               activate^#(N))
752.91/207.96	    , 27: U52^#(tt(), N) -> c_24(activate^#(N))
752.91/207.96	    , 28: U61^#(tt(), M, N) ->
752.91/207.96	          c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.96	               isNatKind^#(activate(M)),
752.91/207.96	               activate^#(M),
752.91/207.96	               activate^#(M),
752.91/207.96	               activate^#(N))
752.91/207.96	    , 29: U62^#(tt(), M, N) ->
752.91/207.96	          c_26(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.96	               isNat^#(activate(N)),
752.91/207.96	               activate^#(N),
752.91/207.96	               activate^#(M),
752.91/207.96	               activate^#(N))
752.91/207.96	    , 30: U63^#(tt(), M, N) ->
752.91/207.96	          c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.96	               isNatKind^#(activate(N)),
752.91/207.96	               activate^#(N),
752.91/207.96	               activate^#(M),
752.91/207.96	               activate^#(N))
752.91/207.96	    , 31: U64^#(tt(), M, N) ->
752.91/207.96	          c_28(s^#(plus(activate(N), activate(M))),
752.91/207.96	               plus^#(activate(N), activate(M)),
752.91/207.96	               activate^#(N),
752.91/207.96	               activate^#(M)) }
752.91/207.96	
752.91/207.96	We are left with following problem, upon which TcT provides the
752.91/207.96	certificate YES(O(1),O(n^3)).
752.91/207.96	
752.91/207.96	Strict DPs:
752.91/207.96	  { U11^#(tt(), V1, V2) ->
752.91/207.96	    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.96	        isNatKind^#(activate(V1)),
752.91/207.96	        activate^#(V1),
752.91/207.96	        activate^#(V1),
752.91/207.96	        activate^#(V2))
752.91/207.96	  , U12^#(tt(), V1, V2) ->
752.91/207.96	    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.96	        isNatKind^#(activate(V2)),
752.91/207.96	        activate^#(V2),
752.91/207.96	        activate^#(V1),
752.91/207.96	        activate^#(V2))
752.91/207.96	  , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.96	    c_4(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.96	        isNatKind^#(activate(V1)),
752.91/207.96	        activate^#(V1),
752.91/207.96	        activate^#(V2))
752.91/207.96	  , isNatKind^#(n__s(V1)) ->
752.91/207.96	    c_5(U41^#(isNatKind(activate(V1))),
752.91/207.96	        isNatKind^#(activate(V1)),
752.91/207.96	        activate^#(V1))
752.91/207.96	  , activate^#(n__0()) -> c_7(0^#())
752.91/207.96	  , activate^#(n__plus(X1, X2)) ->
752.91/207.96	    c_8(plus^#(activate(X1), activate(X2)),
752.91/207.96	        activate^#(X1),
752.91/207.96	        activate^#(X2))
752.91/207.96	  , activate^#(n__s(X)) -> c_9(s^#(activate(X)), activate^#(X))
752.91/207.96	  , U13^#(tt(), V1, V2) ->
752.91/207.96	    c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.96	         isNatKind^#(activate(V2)),
752.91/207.96	         activate^#(V2),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V2))
752.91/207.96	  , U31^#(tt(), V2) ->
752.91/207.96	    c_20(U32^#(isNatKind(activate(V2))),
752.91/207.96	         isNatKind^#(activate(V2)),
752.91/207.96	         activate^#(V2))
752.91/207.96	  , U14^#(tt(), V1, V2) ->
752.91/207.96	    c_11(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.96	         isNat^#(activate(V1)),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V2))
752.91/207.96	  , U15^#(tt(), V2) ->
752.91/207.96	    c_12(U16^#(isNat(activate(V2))),
752.91/207.96	         isNat^#(activate(V2)),
752.91/207.96	         activate^#(V2))
752.91/207.96	  , isNat^#(n__plus(V1, V2)) ->
752.91/207.96	    c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.96	         isNatKind^#(activate(V1)),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V2))
752.91/207.96	  , isNat^#(n__s(V1)) ->
752.91/207.96	    c_15(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.96	         isNatKind^#(activate(V1)),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V1))
752.91/207.96	  , U21^#(tt(), V1) ->
752.91/207.96	    c_17(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.96	         isNatKind^#(activate(V1)),
752.91/207.96	         activate^#(V1),
752.91/207.96	         activate^#(V1))
752.91/207.96	  , U22^#(tt(), V1) ->
752.91/207.96	    c_18(U23^#(isNat(activate(V1))),
752.91/207.96	         isNat^#(activate(V1)),
752.91/207.96	         activate^#(V1))
752.91/207.96	  , U51^#(tt(), N) ->
752.91/207.96	    c_23(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.96	         isNatKind^#(activate(N)),
752.91/207.96	         activate^#(N),
752.91/207.96	         activate^#(N))
752.91/207.96	  , U52^#(tt(), N) -> c_24(activate^#(N))
752.91/207.96	  , U61^#(tt(), M, N) ->
752.91/207.96	    c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.96	         isNatKind^#(activate(M)),
752.91/207.96	         activate^#(M),
752.91/207.96	         activate^#(M),
752.91/207.96	         activate^#(N))
752.91/207.96	  , U62^#(tt(), M, N) ->
752.91/207.96	    c_26(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.96	         isNat^#(activate(N)),
752.91/207.96	         activate^#(N),
752.91/207.96	         activate^#(M),
752.91/207.96	         activate^#(N))
752.91/207.96	  , U63^#(tt(), M, N) ->
752.91/207.96	    c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.96	         isNatKind^#(activate(N)),
752.91/207.96	         activate^#(N),
752.91/207.96	         activate^#(M),
752.91/207.96	         activate^#(N))
752.91/207.96	  , U64^#(tt(), M, N) ->
752.91/207.96	    c_28(s^#(plus(activate(N), activate(M))),
752.91/207.96	         plus^#(activate(N), activate(M)),
752.91/207.96	         activate^#(N),
752.91/207.96	         activate^#(M)) }
752.91/207.96	Weak DPs:
752.91/207.96	  { isNatKind^#(n__0()) -> c_3()
752.91/207.96	  , activate^#(X) -> c_6()
752.91/207.96	  , U41^#(tt()) -> c_22()
752.91/207.96	  , 0^#() -> c_31()
752.91/207.96	  , plus^#(X1, X2) -> c_30()
752.91/207.96	  , s^#(X) -> c_29()
752.91/207.96	  , isNat^#(n__0()) -> c_13()
752.91/207.96	  , U16^#(tt()) -> c_16()
752.91/207.96	  , U23^#(tt()) -> c_19()
752.91/207.96	  , U32^#(tt()) -> c_21() }
752.91/207.96	Weak Trs:
752.91/207.96	  { U11(tt(), V1, V2) ->
752.91/207.96	    U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.96	  , U12(tt(), V1, V2) ->
752.91/207.96	    U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.96	  , isNatKind(n__0()) -> tt()
752.91/207.96	  , isNatKind(n__plus(V1, V2)) ->
752.91/207.96	    U31(isNatKind(activate(V1)), activate(V2))
752.91/207.96	  , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.96	  , activate(X) -> X
752.91/207.96	  , activate(n__0()) -> 0()
752.91/207.96	  , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.96	  , activate(n__s(X)) -> s(activate(X))
752.91/207.96	  , U13(tt(), V1, V2) ->
752.91/207.96	    U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.96	  , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.96	  , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.96	  , isNat(n__0()) -> tt()
752.91/207.96	  , isNat(n__plus(V1, V2)) ->
752.91/207.96	    U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.96	  , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.96	  , U16(tt()) -> tt()
752.91/207.96	  , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.96	  , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.96	  , U23(tt()) -> tt()
752.91/207.96	  , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.96	  , U32(tt()) -> tt()
752.91/207.96	  , U41(tt()) -> tt()
752.91/207.96	  , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N))
752.91/207.96	  , U52(tt(), N) -> activate(N)
752.91/207.96	  , U61(tt(), M, N) ->
752.91/207.96	    U62(isNatKind(activate(M)), activate(M), activate(N))
752.91/207.96	  , U62(tt(), M, N) ->
752.91/207.96	    U63(isNat(activate(N)), activate(M), activate(N))
752.91/207.96	  , U63(tt(), M, N) ->
752.91/207.96	    U64(isNatKind(activate(N)), activate(M), activate(N))
752.91/207.96	  , U64(tt(), M, N) -> s(plus(activate(N), activate(M)))
752.91/207.96	  , s(X) -> n__s(X)
752.91/207.96	  , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.96	  , 0() -> n__0() }
752.91/207.96	Obligation:
752.91/207.96	  innermost runtime complexity
752.91/207.96	Answer:
752.91/207.96	  YES(O(1),O(n^3))
752.91/207.96	
752.91/207.96	We estimate the number of application of {5} by applications of
752.91/207.96	Pre({5}) = {1,2,3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21}.
752.91/207.96	Here rules are labeled as follows:
752.91/207.96	
752.91/207.96	  DPs:
752.91/207.96	    { 1: U11^#(tt(), V1, V2) ->
752.91/207.96	         c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.96	             isNatKind^#(activate(V1)),
752.91/207.96	             activate^#(V1),
752.91/207.96	             activate^#(V1),
752.91/207.96	             activate^#(V2))
752.91/207.96	    , 2: U12^#(tt(), V1, V2) ->
752.91/207.96	         c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.96	             isNatKind^#(activate(V2)),
752.91/207.96	             activate^#(V2),
752.91/207.96	             activate^#(V1),
752.91/207.96	             activate^#(V2))
752.91/207.96	    , 3: isNatKind^#(n__plus(V1, V2)) ->
752.91/207.96	         c_4(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.96	             isNatKind^#(activate(V1)),
752.91/207.96	             activate^#(V1),
752.91/207.96	             activate^#(V2))
752.91/207.96	    , 4: isNatKind^#(n__s(V1)) ->
752.91/207.96	         c_5(U41^#(isNatKind(activate(V1))),
752.91/207.96	             isNatKind^#(activate(V1)),
752.91/207.96	             activate^#(V1))
752.91/207.96	    , 5: activate^#(n__0()) -> c_7(0^#())
752.91/207.96	    , 6: activate^#(n__plus(X1, X2)) ->
752.91/207.96	         c_8(plus^#(activate(X1), activate(X2)),
752.91/207.96	             activate^#(X1),
752.91/207.96	             activate^#(X2))
752.91/207.96	    , 7: activate^#(n__s(X)) -> c_9(s^#(activate(X)), activate^#(X))
752.91/207.96	    , 8: U13^#(tt(), V1, V2) ->
752.91/207.96	         c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.96	              isNatKind^#(activate(V2)),
752.91/207.96	              activate^#(V2),
752.91/207.96	              activate^#(V1),
752.91/207.96	              activate^#(V2))
752.91/207.96	    , 9: U31^#(tt(), V2) ->
752.91/207.96	         c_20(U32^#(isNatKind(activate(V2))),
752.91/207.96	              isNatKind^#(activate(V2)),
752.91/207.96	              activate^#(V2))
752.91/207.96	    , 10: U14^#(tt(), V1, V2) ->
752.91/207.96	          c_11(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.96	               isNat^#(activate(V1)),
752.91/207.96	               activate^#(V1),
752.91/207.96	               activate^#(V2))
752.91/207.96	    , 11: U15^#(tt(), V2) ->
752.91/207.96	          c_12(U16^#(isNat(activate(V2))),
752.91/207.96	               isNat^#(activate(V2)),
752.91/207.96	               activate^#(V2))
752.91/207.96	    , 12: isNat^#(n__plus(V1, V2)) ->
752.91/207.96	          c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.96	               isNatKind^#(activate(V1)),
752.91/207.96	               activate^#(V1),
752.91/207.96	               activate^#(V1),
752.91/207.96	               activate^#(V2))
752.91/207.96	    , 13: isNat^#(n__s(V1)) ->
752.91/207.96	          c_15(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.96	               isNatKind^#(activate(V1)),
752.91/207.96	               activate^#(V1),
752.91/207.97	               activate^#(V1))
752.91/207.97	    , 14: U21^#(tt(), V1) ->
752.91/207.97	          c_17(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.97	               isNatKind^#(activate(V1)),
752.91/207.97	               activate^#(V1),
752.91/207.97	               activate^#(V1))
752.91/207.97	    , 15: U22^#(tt(), V1) ->
752.91/207.97	          c_18(U23^#(isNat(activate(V1))),
752.91/207.97	               isNat^#(activate(V1)),
752.91/207.97	               activate^#(V1))
752.91/207.97	    , 16: U51^#(tt(), N) ->
752.91/207.97	          c_23(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.97	               isNatKind^#(activate(N)),
752.91/207.97	               activate^#(N),
752.91/207.97	               activate^#(N))
752.91/207.97	    , 17: U52^#(tt(), N) -> c_24(activate^#(N))
752.91/207.97	    , 18: U61^#(tt(), M, N) ->
752.91/207.97	          c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.97	               isNatKind^#(activate(M)),
752.91/207.97	               activate^#(M),
752.91/207.97	               activate^#(M),
752.91/207.97	               activate^#(N))
752.91/207.97	    , 19: U62^#(tt(), M, N) ->
752.91/207.97	          c_26(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.97	               isNat^#(activate(N)),
752.91/207.97	               activate^#(N),
752.91/207.97	               activate^#(M),
752.91/207.97	               activate^#(N))
752.91/207.97	    , 20: U63^#(tt(), M, N) ->
752.91/207.97	          c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.97	               isNatKind^#(activate(N)),
752.91/207.97	               activate^#(N),
752.91/207.97	               activate^#(M),
752.91/207.97	               activate^#(N))
752.91/207.97	    , 21: U64^#(tt(), M, N) ->
752.91/207.97	          c_28(s^#(plus(activate(N), activate(M))),
752.91/207.97	               plus^#(activate(N), activate(M)),
752.91/207.97	               activate^#(N),
752.91/207.97	               activate^#(M))
752.91/207.97	    , 22: isNatKind^#(n__0()) -> c_3()
752.91/207.97	    , 23: activate^#(X) -> c_6()
752.91/207.97	    , 24: U41^#(tt()) -> c_22()
752.91/207.97	    , 25: 0^#() -> c_31()
752.91/207.97	    , 26: plus^#(X1, X2) -> c_30()
752.91/207.97	    , 27: s^#(X) -> c_29()
752.91/207.97	    , 28: isNat^#(n__0()) -> c_13()
752.91/207.97	    , 29: U16^#(tt()) -> c_16()
752.91/207.97	    , 30: U23^#(tt()) -> c_19()
752.91/207.97	    , 31: U32^#(tt()) -> c_21() }
752.91/207.97	
752.91/207.97	We are left with following problem, upon which TcT provides the
752.91/207.97	certificate YES(O(1),O(n^3)).
752.91/207.97	
752.91/207.97	Strict DPs:
752.91/207.97	  { U11^#(tt(), V1, V2) ->
752.91/207.97	    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.97	        isNatKind^#(activate(V1)),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V2))
752.91/207.97	  , U12^#(tt(), V1, V2) ->
752.91/207.97	    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.97	        isNatKind^#(activate(V2)),
752.91/207.97	        activate^#(V2),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V2))
752.91/207.97	  , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.97	    c_4(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.97	        isNatKind^#(activate(V1)),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V2))
752.91/207.97	  , isNatKind^#(n__s(V1)) ->
752.91/207.97	    c_5(U41^#(isNatKind(activate(V1))),
752.91/207.97	        isNatKind^#(activate(V1)),
752.91/207.97	        activate^#(V1))
752.91/207.97	  , activate^#(n__plus(X1, X2)) ->
752.91/207.97	    c_8(plus^#(activate(X1), activate(X2)),
752.91/207.97	        activate^#(X1),
752.91/207.97	        activate^#(X2))
752.91/207.97	  , activate^#(n__s(X)) -> c_9(s^#(activate(X)), activate^#(X))
752.91/207.97	  , U13^#(tt(), V1, V2) ->
752.91/207.97	    c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.97	         isNatKind^#(activate(V2)),
752.91/207.97	         activate^#(V2),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , U31^#(tt(), V2) ->
752.91/207.97	    c_20(U32^#(isNatKind(activate(V2))),
752.91/207.97	         isNatKind^#(activate(V2)),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , U14^#(tt(), V1, V2) ->
752.91/207.97	    c_11(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.97	         isNat^#(activate(V1)),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , U15^#(tt(), V2) ->
752.91/207.97	    c_12(U16^#(isNat(activate(V2))),
752.91/207.97	         isNat^#(activate(V2)),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , isNat^#(n__plus(V1, V2)) ->
752.91/207.97	    c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.97	         isNatKind^#(activate(V1)),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , isNat^#(n__s(V1)) ->
752.91/207.97	    c_15(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.97	         isNatKind^#(activate(V1)),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V1))
752.91/207.97	  , U21^#(tt(), V1) ->
752.91/207.97	    c_17(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.97	         isNatKind^#(activate(V1)),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V1))
752.91/207.97	  , U22^#(tt(), V1) ->
752.91/207.97	    c_18(U23^#(isNat(activate(V1))),
752.91/207.97	         isNat^#(activate(V1)),
752.91/207.97	         activate^#(V1))
752.91/207.97	  , U51^#(tt(), N) ->
752.91/207.97	    c_23(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.97	         isNatKind^#(activate(N)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U52^#(tt(), N) -> c_24(activate^#(N))
752.91/207.97	  , U61^#(tt(), M, N) ->
752.91/207.97	    c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.97	         isNatKind^#(activate(M)),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U62^#(tt(), M, N) ->
752.91/207.97	    c_26(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.97	         isNat^#(activate(N)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U63^#(tt(), M, N) ->
752.91/207.97	    c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.97	         isNatKind^#(activate(N)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U64^#(tt(), M, N) ->
752.91/207.97	    c_28(s^#(plus(activate(N), activate(M))),
752.91/207.97	         plus^#(activate(N), activate(M)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(M)) }
752.91/207.97	Weak DPs:
752.91/207.97	  { isNatKind^#(n__0()) -> c_3()
752.91/207.97	  , activate^#(X) -> c_6()
752.91/207.97	  , activate^#(n__0()) -> c_7(0^#())
752.91/207.97	  , U41^#(tt()) -> c_22()
752.91/207.97	  , 0^#() -> c_31()
752.91/207.97	  , plus^#(X1, X2) -> c_30()
752.91/207.97	  , s^#(X) -> c_29()
752.91/207.97	  , isNat^#(n__0()) -> c_13()
752.91/207.97	  , U16^#(tt()) -> c_16()
752.91/207.97	  , U23^#(tt()) -> c_19()
752.91/207.97	  , U32^#(tt()) -> c_21() }
752.91/207.97	Weak Trs:
752.91/207.97	  { U11(tt(), V1, V2) ->
752.91/207.97	    U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.97	  , U12(tt(), V1, V2) ->
752.91/207.97	    U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.97	  , isNatKind(n__0()) -> tt()
752.91/207.97	  , isNatKind(n__plus(V1, V2)) ->
752.91/207.97	    U31(isNatKind(activate(V1)), activate(V2))
752.91/207.97	  , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.97	  , activate(X) -> X
752.91/207.97	  , activate(n__0()) -> 0()
752.91/207.97	  , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.97	  , activate(n__s(X)) -> s(activate(X))
752.91/207.97	  , U13(tt(), V1, V2) ->
752.91/207.97	    U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.97	  , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.97	  , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.97	  , isNat(n__0()) -> tt()
752.91/207.97	  , isNat(n__plus(V1, V2)) ->
752.91/207.97	    U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.97	  , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.97	  , U16(tt()) -> tt()
752.91/207.97	  , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.97	  , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.97	  , U23(tt()) -> tt()
752.91/207.97	  , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.97	  , U32(tt()) -> tt()
752.91/207.97	  , U41(tt()) -> tt()
752.91/207.97	  , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N))
752.91/207.97	  , U52(tt(), N) -> activate(N)
752.91/207.97	  , U61(tt(), M, N) ->
752.91/207.97	    U62(isNatKind(activate(M)), activate(M), activate(N))
752.91/207.97	  , U62(tt(), M, N) ->
752.91/207.97	    U63(isNat(activate(N)), activate(M), activate(N))
752.91/207.97	  , U63(tt(), M, N) ->
752.91/207.97	    U64(isNatKind(activate(N)), activate(M), activate(N))
752.91/207.97	  , U64(tt(), M, N) -> s(plus(activate(N), activate(M)))
752.91/207.97	  , s(X) -> n__s(X)
752.91/207.97	  , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.97	  , 0() -> n__0() }
752.91/207.97	Obligation:
752.91/207.97	  innermost runtime complexity
752.91/207.97	Answer:
752.91/207.97	  YES(O(1),O(n^3))
752.91/207.97	
752.91/207.97	The following weak DPs constitute a sub-graph of the DG that is
752.91/207.97	closed under successors. The DPs are removed.
752.91/207.97	
752.91/207.97	{ isNatKind^#(n__0()) -> c_3()
752.91/207.97	, activate^#(X) -> c_6()
752.91/207.97	, activate^#(n__0()) -> c_7(0^#())
752.91/207.97	, U41^#(tt()) -> c_22()
752.91/207.97	, 0^#() -> c_31()
752.91/207.97	, plus^#(X1, X2) -> c_30()
752.91/207.97	, s^#(X) -> c_29()
752.91/207.97	, isNat^#(n__0()) -> c_13()
752.91/207.97	, U16^#(tt()) -> c_16()
752.91/207.97	, U23^#(tt()) -> c_19()
752.91/207.97	, U32^#(tt()) -> c_21() }
752.91/207.97	
752.91/207.97	We are left with following problem, upon which TcT provides the
752.91/207.97	certificate YES(O(1),O(n^3)).
752.91/207.97	
752.91/207.97	Strict DPs:
752.91/207.97	  { U11^#(tt(), V1, V2) ->
752.91/207.97	    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.97	        isNatKind^#(activate(V1)),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V2))
752.91/207.97	  , U12^#(tt(), V1, V2) ->
752.91/207.97	    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.97	        isNatKind^#(activate(V2)),
752.91/207.97	        activate^#(V2),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V2))
752.91/207.97	  , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.97	    c_4(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.97	        isNatKind^#(activate(V1)),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V2))
752.91/207.97	  , isNatKind^#(n__s(V1)) ->
752.91/207.97	    c_5(U41^#(isNatKind(activate(V1))),
752.91/207.97	        isNatKind^#(activate(V1)),
752.91/207.97	        activate^#(V1))
752.91/207.97	  , activate^#(n__plus(X1, X2)) ->
752.91/207.97	    c_8(plus^#(activate(X1), activate(X2)),
752.91/207.97	        activate^#(X1),
752.91/207.97	        activate^#(X2))
752.91/207.97	  , activate^#(n__s(X)) -> c_9(s^#(activate(X)), activate^#(X))
752.91/207.97	  , U13^#(tt(), V1, V2) ->
752.91/207.97	    c_10(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.97	         isNatKind^#(activate(V2)),
752.91/207.97	         activate^#(V2),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , U31^#(tt(), V2) ->
752.91/207.97	    c_20(U32^#(isNatKind(activate(V2))),
752.91/207.97	         isNatKind^#(activate(V2)),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , U14^#(tt(), V1, V2) ->
752.91/207.97	    c_11(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.97	         isNat^#(activate(V1)),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , U15^#(tt(), V2) ->
752.91/207.97	    c_12(U16^#(isNat(activate(V2))),
752.91/207.97	         isNat^#(activate(V2)),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , isNat^#(n__plus(V1, V2)) ->
752.91/207.97	    c_14(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.97	         isNatKind^#(activate(V1)),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , isNat^#(n__s(V1)) ->
752.91/207.97	    c_15(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.97	         isNatKind^#(activate(V1)),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V1))
752.91/207.97	  , U21^#(tt(), V1) ->
752.91/207.97	    c_17(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.97	         isNatKind^#(activate(V1)),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V1))
752.91/207.97	  , U22^#(tt(), V1) ->
752.91/207.97	    c_18(U23^#(isNat(activate(V1))),
752.91/207.97	         isNat^#(activate(V1)),
752.91/207.97	         activate^#(V1))
752.91/207.97	  , U51^#(tt(), N) ->
752.91/207.97	    c_23(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.97	         isNatKind^#(activate(N)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U52^#(tt(), N) -> c_24(activate^#(N))
752.91/207.97	  , U61^#(tt(), M, N) ->
752.91/207.97	    c_25(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.97	         isNatKind^#(activate(M)),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U62^#(tt(), M, N) ->
752.91/207.97	    c_26(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.97	         isNat^#(activate(N)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U63^#(tt(), M, N) ->
752.91/207.97	    c_27(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.97	         isNatKind^#(activate(N)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U64^#(tt(), M, N) ->
752.91/207.97	    c_28(s^#(plus(activate(N), activate(M))),
752.91/207.97	         plus^#(activate(N), activate(M)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(M)) }
752.91/207.97	Weak Trs:
752.91/207.97	  { U11(tt(), V1, V2) ->
752.91/207.97	    U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.97	  , U12(tt(), V1, V2) ->
752.91/207.97	    U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.97	  , isNatKind(n__0()) -> tt()
752.91/207.97	  , isNatKind(n__plus(V1, V2)) ->
752.91/207.97	    U31(isNatKind(activate(V1)), activate(V2))
752.91/207.97	  , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.97	  , activate(X) -> X
752.91/207.97	  , activate(n__0()) -> 0()
752.91/207.97	  , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.97	  , activate(n__s(X)) -> s(activate(X))
752.91/207.97	  , U13(tt(), V1, V2) ->
752.91/207.97	    U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.97	  , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.97	  , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.97	  , isNat(n__0()) -> tt()
752.91/207.97	  , isNat(n__plus(V1, V2)) ->
752.91/207.97	    U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.97	  , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.97	  , U16(tt()) -> tt()
752.91/207.97	  , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.97	  , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.97	  , U23(tt()) -> tt()
752.91/207.97	  , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.97	  , U32(tt()) -> tt()
752.91/207.97	  , U41(tt()) -> tt()
752.91/207.97	  , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N))
752.91/207.97	  , U52(tt(), N) -> activate(N)
752.91/207.97	  , U61(tt(), M, N) ->
752.91/207.97	    U62(isNatKind(activate(M)), activate(M), activate(N))
752.91/207.97	  , U62(tt(), M, N) ->
752.91/207.97	    U63(isNat(activate(N)), activate(M), activate(N))
752.91/207.97	  , U63(tt(), M, N) ->
752.91/207.97	    U64(isNatKind(activate(N)), activate(M), activate(N))
752.91/207.97	  , U64(tt(), M, N) -> s(plus(activate(N), activate(M)))
752.91/207.97	  , s(X) -> n__s(X)
752.91/207.97	  , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.97	  , 0() -> n__0() }
752.91/207.97	Obligation:
752.91/207.97	  innermost runtime complexity
752.91/207.97	Answer:
752.91/207.97	  YES(O(1),O(n^3))
752.91/207.97	
752.91/207.97	Due to missing edges in the dependency-graph, the right-hand sides
752.91/207.97	of following rules could be simplified:
752.91/207.97	
752.91/207.97	  { isNatKind^#(n__s(V1)) ->
752.91/207.97	    c_5(U41^#(isNatKind(activate(V1))),
752.91/207.97	        isNatKind^#(activate(V1)),
752.91/207.97	        activate^#(V1))
752.91/207.97	  , activate^#(n__plus(X1, X2)) ->
752.91/207.97	    c_8(plus^#(activate(X1), activate(X2)),
752.91/207.97	        activate^#(X1),
752.91/207.97	        activate^#(X2))
752.91/207.97	  , activate^#(n__s(X)) -> c_9(s^#(activate(X)), activate^#(X))
752.91/207.97	  , U31^#(tt(), V2) ->
752.91/207.97	    c_20(U32^#(isNatKind(activate(V2))),
752.91/207.97	         isNatKind^#(activate(V2)),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , U15^#(tt(), V2) ->
752.91/207.97	    c_12(U16^#(isNat(activate(V2))),
752.91/207.97	         isNat^#(activate(V2)),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , U22^#(tt(), V1) ->
752.91/207.97	    c_18(U23^#(isNat(activate(V1))),
752.91/207.97	         isNat^#(activate(V1)),
752.91/207.97	         activate^#(V1))
752.91/207.97	  , U64^#(tt(), M, N) ->
752.91/207.97	    c_28(s^#(plus(activate(N), activate(M))),
752.91/207.97	         plus^#(activate(N), activate(M)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(M)) }
752.91/207.97	
752.91/207.97	We are left with following problem, upon which TcT provides the
752.91/207.97	certificate YES(O(1),O(n^3)).
752.91/207.97	
752.91/207.97	Strict DPs:
752.91/207.97	  { U11^#(tt(), V1, V2) ->
752.91/207.97	    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.97	        isNatKind^#(activate(V1)),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V2))
752.91/207.97	  , U12^#(tt(), V1, V2) ->
752.91/207.97	    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.97	        isNatKind^#(activate(V2)),
752.91/207.97	        activate^#(V2),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V2))
752.91/207.97	  , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.97	    c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.97	        isNatKind^#(activate(V1)),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V2))
752.91/207.97	  , isNatKind^#(n__s(V1)) ->
752.91/207.97	    c_4(isNatKind^#(activate(V1)), activate^#(V1))
752.91/207.97	  , activate^#(n__plus(X1, X2)) ->
752.91/207.97	    c_5(activate^#(X1), activate^#(X2))
752.91/207.97	  , activate^#(n__s(X)) -> c_6(activate^#(X))
752.91/207.97	  , U13^#(tt(), V1, V2) ->
752.91/207.97	    c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.97	        isNatKind^#(activate(V2)),
752.91/207.97	        activate^#(V2),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V2))
752.91/207.97	  , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
752.91/207.97	  , U14^#(tt(), V1, V2) ->
752.91/207.97	    c_9(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.97	        isNat^#(activate(V1)),
752.91/207.97	        activate^#(V1),
752.91/207.97	        activate^#(V2))
752.91/207.97	  , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
752.91/207.97	  , isNat^#(n__plus(V1, V2)) ->
752.91/207.97	    c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.97	         isNatKind^#(activate(V1)),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V2))
752.91/207.97	  , isNat^#(n__s(V1)) ->
752.91/207.97	    c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.97	         isNatKind^#(activate(V1)),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V1))
752.91/207.97	  , U21^#(tt(), V1) ->
752.91/207.97	    c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.97	         isNatKind^#(activate(V1)),
752.91/207.97	         activate^#(V1),
752.91/207.97	         activate^#(V1))
752.91/207.97	  , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
752.91/207.97	  , U51^#(tt(), N) ->
752.91/207.97	    c_15(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.97	         isNatKind^#(activate(N)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U52^#(tt(), N) -> c_16(activate^#(N))
752.91/207.97	  , U61^#(tt(), M, N) ->
752.91/207.97	    c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.97	         isNatKind^#(activate(M)),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U62^#(tt(), M, N) ->
752.91/207.97	    c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.97	         isNat^#(activate(N)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U63^#(tt(), M, N) ->
752.91/207.97	    c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.97	         isNatKind^#(activate(N)),
752.91/207.97	         activate^#(N),
752.91/207.97	         activate^#(M),
752.91/207.97	         activate^#(N))
752.91/207.97	  , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
752.91/207.97	Weak Trs:
752.91/207.97	  { U11(tt(), V1, V2) ->
752.91/207.97	    U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.97	  , U12(tt(), V1, V2) ->
752.91/207.97	    U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.97	  , isNatKind(n__0()) -> tt()
752.91/207.97	  , isNatKind(n__plus(V1, V2)) ->
752.91/207.97	    U31(isNatKind(activate(V1)), activate(V2))
752.91/207.97	  , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.97	  , activate(X) -> X
752.91/207.97	  , activate(n__0()) -> 0()
752.91/207.97	  , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.97	  , activate(n__s(X)) -> s(activate(X))
752.91/207.97	  , U13(tt(), V1, V2) ->
752.91/207.97	    U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.97	  , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.97	  , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.97	  , isNat(n__0()) -> tt()
752.91/207.97	  , isNat(n__plus(V1, V2)) ->
752.91/207.97	    U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.97	  , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.97	  , U16(tt()) -> tt()
752.91/207.97	  , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.97	  , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.97	  , U23(tt()) -> tt()
752.91/207.97	  , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.97	  , U32(tt()) -> tt()
752.91/207.97	  , U41(tt()) -> tt()
752.91/207.97	  , U51(tt(), N) -> U52(isNatKind(activate(N)), activate(N))
752.91/207.97	  , U52(tt(), N) -> activate(N)
752.91/207.97	  , U61(tt(), M, N) ->
752.91/207.97	    U62(isNatKind(activate(M)), activate(M), activate(N))
752.91/207.97	  , U62(tt(), M, N) ->
752.91/207.97	    U63(isNat(activate(N)), activate(M), activate(N))
752.91/207.97	  , U63(tt(), M, N) ->
752.91/207.97	    U64(isNatKind(activate(N)), activate(M), activate(N))
752.91/207.97	  , U64(tt(), M, N) -> s(plus(activate(N), activate(M)))
752.91/207.97	  , s(X) -> n__s(X)
752.91/207.97	  , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.97	  , 0() -> n__0() }
752.91/207.97	Obligation:
752.91/207.97	  innermost runtime complexity
752.91/207.97	Answer:
752.91/207.97	  YES(O(1),O(n^3))
752.91/207.97	
752.91/207.97	We replace rewrite rules by usable rules:
752.91/207.97	
752.91/207.97	  Weak Usable Rules:
752.91/207.97	    { U11(tt(), V1, V2) ->
752.91/207.97	      U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.97	    , U12(tt(), V1, V2) ->
752.91/207.97	      U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.97	    , isNatKind(n__0()) -> tt()
752.91/207.97	    , isNatKind(n__plus(V1, V2)) ->
752.91/207.97	      U31(isNatKind(activate(V1)), activate(V2))
752.91/207.97	    , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.97	    , activate(X) -> X
752.91/207.97	    , activate(n__0()) -> 0()
752.91/207.98	    , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.98	    , activate(n__s(X)) -> s(activate(X))
752.91/207.98	    , U13(tt(), V1, V2) ->
752.91/207.98	      U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.98	    , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.98	    , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.98	    , isNat(n__0()) -> tt()
752.91/207.98	    , isNat(n__plus(V1, V2)) ->
752.91/207.98	      U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.98	    , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.98	    , U16(tt()) -> tt()
752.91/207.98	    , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.98	    , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.98	    , U23(tt()) -> tt()
752.91/207.98	    , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.98	    , U32(tt()) -> tt()
752.91/207.98	    , U41(tt()) -> tt()
752.91/207.98	    , s(X) -> n__s(X)
752.91/207.98	    , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.98	    , 0() -> n__0() }
752.91/207.98	
752.91/207.98	We are left with following problem, upon which TcT provides the
752.91/207.98	certificate YES(O(1),O(n^3)).
752.91/207.98	
752.91/207.98	Strict DPs:
752.91/207.98	  { U11^#(tt(), V1, V2) ->
752.91/207.98	    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.98	        isNatKind^#(activate(V1)),
752.91/207.98	        activate^#(V1),
752.91/207.98	        activate^#(V1),
752.91/207.98	        activate^#(V2))
752.91/207.98	  , U12^#(tt(), V1, V2) ->
752.91/207.98	    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.98	        isNatKind^#(activate(V2)),
752.91/207.98	        activate^#(V2),
752.91/207.98	        activate^#(V1),
752.91/207.98	        activate^#(V2))
752.91/207.98	  , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.98	    c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.98	        isNatKind^#(activate(V1)),
752.91/207.98	        activate^#(V1),
752.91/207.98	        activate^#(V2))
752.91/207.98	  , isNatKind^#(n__s(V1)) ->
752.91/207.98	    c_4(isNatKind^#(activate(V1)), activate^#(V1))
752.91/207.98	  , activate^#(n__plus(X1, X2)) ->
752.91/207.98	    c_5(activate^#(X1), activate^#(X2))
752.91/207.98	  , activate^#(n__s(X)) -> c_6(activate^#(X))
752.91/207.98	  , U13^#(tt(), V1, V2) ->
752.91/207.98	    c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.98	        isNatKind^#(activate(V2)),
752.91/207.98	        activate^#(V2),
752.91/207.98	        activate^#(V1),
752.91/207.98	        activate^#(V2))
752.91/207.98	  , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
752.91/207.98	  , U14^#(tt(), V1, V2) ->
752.91/207.98	    c_9(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.98	        isNat^#(activate(V1)),
752.91/207.98	        activate^#(V1),
752.91/207.98	        activate^#(V2))
752.91/207.98	  , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
752.91/207.98	  , isNat^#(n__plus(V1, V2)) ->
752.91/207.98	    c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.98	         isNatKind^#(activate(V1)),
752.91/207.98	         activate^#(V1),
752.91/207.98	         activate^#(V1),
752.91/207.98	         activate^#(V2))
752.91/207.98	  , isNat^#(n__s(V1)) ->
752.91/207.98	    c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.98	         isNatKind^#(activate(V1)),
752.91/207.98	         activate^#(V1),
752.91/207.98	         activate^#(V1))
752.91/207.98	  , U21^#(tt(), V1) ->
752.91/207.98	    c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.98	         isNatKind^#(activate(V1)),
752.91/207.98	         activate^#(V1),
752.91/207.98	         activate^#(V1))
752.91/207.98	  , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
752.91/207.98	  , U51^#(tt(), N) ->
752.91/207.98	    c_15(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.98	         isNatKind^#(activate(N)),
752.91/207.98	         activate^#(N),
752.91/207.98	         activate^#(N))
752.91/207.98	  , U52^#(tt(), N) -> c_16(activate^#(N))
752.91/207.98	  , U61^#(tt(), M, N) ->
752.91/207.98	    c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.98	         isNatKind^#(activate(M)),
752.91/207.98	         activate^#(M),
752.91/207.98	         activate^#(M),
752.91/207.98	         activate^#(N))
752.91/207.98	  , U62^#(tt(), M, N) ->
752.91/207.98	    c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.98	         isNat^#(activate(N)),
752.91/207.98	         activate^#(N),
752.91/207.98	         activate^#(M),
752.91/207.98	         activate^#(N))
752.91/207.98	  , U63^#(tt(), M, N) ->
752.91/207.98	    c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.98	         isNatKind^#(activate(N)),
752.91/207.98	         activate^#(N),
752.91/207.98	         activate^#(M),
752.91/207.98	         activate^#(N))
752.91/207.98	  , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
752.91/207.98	Weak Trs:
752.91/207.98	  { U11(tt(), V1, V2) ->
752.91/207.98	    U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.98	  , U12(tt(), V1, V2) ->
752.91/207.98	    U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.98	  , isNatKind(n__0()) -> tt()
752.91/207.98	  , isNatKind(n__plus(V1, V2)) ->
752.91/207.98	    U31(isNatKind(activate(V1)), activate(V2))
752.91/207.98	  , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.98	  , activate(X) -> X
752.91/207.98	  , activate(n__0()) -> 0()
752.91/207.98	  , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.98	  , activate(n__s(X)) -> s(activate(X))
752.91/207.98	  , U13(tt(), V1, V2) ->
752.91/207.98	    U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.98	  , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.98	  , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.98	  , isNat(n__0()) -> tt()
752.91/207.98	  , isNat(n__plus(V1, V2)) ->
752.91/207.98	    U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.98	  , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.98	  , U16(tt()) -> tt()
752.91/207.98	  , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.98	  , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.98	  , U23(tt()) -> tt()
752.91/207.98	  , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.98	  , U32(tt()) -> tt()
752.91/207.98	  , U41(tt()) -> tt()
752.91/207.98	  , s(X) -> n__s(X)
752.91/207.98	  , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.98	  , 0() -> n__0() }
752.91/207.98	Obligation:
752.91/207.98	  innermost runtime complexity
752.91/207.98	Answer:
752.91/207.98	  YES(O(1),O(n^3))
752.91/207.98	
752.91/207.98	We analyse the complexity of following sub-problems (R) and (S).
752.91/207.98	Problem (S) is obtained from the input problem by shifting strict
752.91/207.98	rules from (R) into the weak component:
752.91/207.98	
752.91/207.98	Problem (R):
752.91/207.98	------------
752.91/207.98	  Strict DPs:
752.91/207.98	    { activate^#(n__plus(X1, X2)) ->
752.91/207.98	      c_5(activate^#(X1), activate^#(X2))
752.91/207.98	    , activate^#(n__s(X)) -> c_6(activate^#(X))
752.91/207.98	    , U51^#(tt(), N) ->
752.91/207.98	      c_15(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.98	           isNatKind^#(activate(N)),
752.91/207.98	           activate^#(N),
752.91/207.98	           activate^#(N))
752.91/207.98	    , U52^#(tt(), N) -> c_16(activate^#(N)) }
752.91/207.98	  Weak DPs:
752.91/207.98	    { U11^#(tt(), V1, V2) ->
752.91/207.98	      c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.98	          isNatKind^#(activate(V1)),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V2))
752.91/207.98	    , U12^#(tt(), V1, V2) ->
752.91/207.98	      c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.98	          isNatKind^#(activate(V2)),
752.91/207.98	          activate^#(V2),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V2))
752.91/207.98	    , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.98	      c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.98	          isNatKind^#(activate(V1)),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V2))
752.91/207.98	    , isNatKind^#(n__s(V1)) ->
752.91/207.98	      c_4(isNatKind^#(activate(V1)), activate^#(V1))
752.91/207.98	    , U13^#(tt(), V1, V2) ->
752.91/207.98	      c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.98	          isNatKind^#(activate(V2)),
752.91/207.98	          activate^#(V2),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V2))
752.91/207.98	    , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
752.91/207.98	    , U14^#(tt(), V1, V2) ->
752.91/207.98	      c_9(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.98	          isNat^#(activate(V1)),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V2))
752.91/207.98	    , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
752.91/207.98	    , isNat^#(n__plus(V1, V2)) ->
752.91/207.98	      c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.98	           isNatKind^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	    , isNat^#(n__s(V1)) ->
752.91/207.98	      c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.98	           isNatKind^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V1))
752.91/207.98	    , U21^#(tt(), V1) ->
752.91/207.98	      c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.98	           isNatKind^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V1))
752.91/207.98	    , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
752.91/207.98	    , U61^#(tt(), M, N) ->
752.91/207.98	      c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.98	           isNatKind^#(activate(M)),
752.91/207.98	           activate^#(M),
752.91/207.98	           activate^#(M),
752.91/207.98	           activate^#(N))
752.91/207.98	    , U62^#(tt(), M, N) ->
752.91/207.98	      c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.98	           isNat^#(activate(N)),
752.91/207.98	           activate^#(N),
752.91/207.98	           activate^#(M),
752.91/207.98	           activate^#(N))
752.91/207.98	    , U63^#(tt(), M, N) ->
752.91/207.98	      c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.98	           isNatKind^#(activate(N)),
752.91/207.98	           activate^#(N),
752.91/207.98	           activate^#(M),
752.91/207.98	           activate^#(N))
752.91/207.98	    , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
752.91/207.98	  Weak Trs:
752.91/207.98	    { U11(tt(), V1, V2) ->
752.91/207.98	      U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.98	    , U12(tt(), V1, V2) ->
752.91/207.98	      U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.98	    , isNatKind(n__0()) -> tt()
752.91/207.98	    , isNatKind(n__plus(V1, V2)) ->
752.91/207.98	      U31(isNatKind(activate(V1)), activate(V2))
752.91/207.98	    , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.98	    , activate(X) -> X
752.91/207.98	    , activate(n__0()) -> 0()
752.91/207.98	    , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.98	    , activate(n__s(X)) -> s(activate(X))
752.91/207.98	    , U13(tt(), V1, V2) ->
752.91/207.98	      U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.98	    , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.98	    , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.98	    , isNat(n__0()) -> tt()
752.91/207.98	    , isNat(n__plus(V1, V2)) ->
752.91/207.98	      U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.98	    , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.98	    , U16(tt()) -> tt()
752.91/207.98	    , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.98	    , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.98	    , U23(tt()) -> tt()
752.91/207.98	    , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.98	    , U32(tt()) -> tt()
752.91/207.98	    , U41(tt()) -> tt()
752.91/207.98	    , s(X) -> n__s(X)
752.91/207.98	    , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.98	    , 0() -> n__0() }
752.91/207.98	  StartTerms: basic terms
752.91/207.98	  Strategy: innermost
752.91/207.98	
752.91/207.98	Problem (S):
752.91/207.98	------------
752.91/207.98	  Strict DPs:
752.91/207.98	    { U11^#(tt(), V1, V2) ->
752.91/207.98	      c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.98	          isNatKind^#(activate(V1)),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V2))
752.91/207.98	    , U12^#(tt(), V1, V2) ->
752.91/207.98	      c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.98	          isNatKind^#(activate(V2)),
752.91/207.98	          activate^#(V2),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V2))
752.91/207.98	    , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.98	      c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.98	          isNatKind^#(activate(V1)),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V2))
752.91/207.98	    , isNatKind^#(n__s(V1)) ->
752.91/207.98	      c_4(isNatKind^#(activate(V1)), activate^#(V1))
752.91/207.98	    , U13^#(tt(), V1, V2) ->
752.91/207.98	      c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.98	          isNatKind^#(activate(V2)),
752.91/207.98	          activate^#(V2),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V2))
752.91/207.98	    , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
752.91/207.98	    , U14^#(tt(), V1, V2) ->
752.91/207.98	      c_9(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.98	          isNat^#(activate(V1)),
752.91/207.98	          activate^#(V1),
752.91/207.98	          activate^#(V2))
752.91/207.98	    , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
752.91/207.98	    , isNat^#(n__plus(V1, V2)) ->
752.91/207.98	      c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.98	           isNatKind^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	    , isNat^#(n__s(V1)) ->
752.91/207.98	      c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.98	           isNatKind^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V1))
752.91/207.98	    , U21^#(tt(), V1) ->
752.91/207.98	      c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.98	           isNatKind^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V1))
752.91/207.98	    , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
752.91/207.98	    , U61^#(tt(), M, N) ->
752.91/207.98	      c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.98	           isNatKind^#(activate(M)),
752.91/207.98	           activate^#(M),
752.91/207.98	           activate^#(M),
752.91/207.98	           activate^#(N))
752.91/207.98	    , U62^#(tt(), M, N) ->
752.91/207.98	      c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.98	           isNat^#(activate(N)),
752.91/207.98	           activate^#(N),
752.91/207.98	           activate^#(M),
752.91/207.98	           activate^#(N))
752.91/207.98	    , U63^#(tt(), M, N) ->
752.91/207.98	      c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.98	           isNatKind^#(activate(N)),
752.91/207.98	           activate^#(N),
752.91/207.98	           activate^#(M),
752.91/207.98	           activate^#(N))
752.91/207.98	    , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
752.91/207.98	  Weak DPs:
752.91/207.98	    { activate^#(n__plus(X1, X2)) ->
752.91/207.98	      c_5(activate^#(X1), activate^#(X2))
752.91/207.98	    , activate^#(n__s(X)) -> c_6(activate^#(X))
752.91/207.98	    , U51^#(tt(), N) ->
752.91/207.98	      c_15(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.98	           isNatKind^#(activate(N)),
752.91/207.98	           activate^#(N),
752.91/207.98	           activate^#(N))
752.91/207.98	    , U52^#(tt(), N) -> c_16(activate^#(N)) }
752.91/207.98	  Weak Trs:
752.91/207.98	    { U11(tt(), V1, V2) ->
752.91/207.98	      U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.98	    , U12(tt(), V1, V2) ->
752.91/207.98	      U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.98	    , isNatKind(n__0()) -> tt()
752.91/207.98	    , isNatKind(n__plus(V1, V2)) ->
752.91/207.98	      U31(isNatKind(activate(V1)), activate(V2))
752.91/207.98	    , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.98	    , activate(X) -> X
752.91/207.98	    , activate(n__0()) -> 0()
752.91/207.98	    , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.98	    , activate(n__s(X)) -> s(activate(X))
752.91/207.98	    , U13(tt(), V1, V2) ->
752.91/207.98	      U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.98	    , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.98	    , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.98	    , isNat(n__0()) -> tt()
752.91/207.98	    , isNat(n__plus(V1, V2)) ->
752.91/207.98	      U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.98	    , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.98	    , U16(tt()) -> tt()
752.91/207.98	    , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.98	    , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.98	    , U23(tt()) -> tt()
752.91/207.98	    , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.98	    , U32(tt()) -> tt()
752.91/207.98	    , U41(tt()) -> tt()
752.91/207.98	    , s(X) -> n__s(X)
752.91/207.98	    , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.98	    , 0() -> n__0() }
752.91/207.98	  StartTerms: basic terms
752.91/207.98	  Strategy: innermost
752.91/207.98	
752.91/207.98	Overall, the transformation results in the following sub-problem(s):
752.91/207.98	
752.91/207.98	Generated new problems:
752.91/207.98	-----------------------
752.91/207.98	R) Strict DPs:
752.91/207.98	     { activate^#(n__plus(X1, X2)) ->
752.91/207.98	       c_5(activate^#(X1), activate^#(X2))
752.91/207.98	     , activate^#(n__s(X)) -> c_6(activate^#(X))
752.91/207.98	     , U51^#(tt(), N) ->
752.91/207.98	       c_15(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.98	            isNatKind^#(activate(N)),
752.91/207.98	            activate^#(N),
752.91/207.98	            activate^#(N))
752.91/207.98	     , U52^#(tt(), N) -> c_16(activate^#(N)) }
752.91/207.98	   Weak DPs:
752.91/207.98	     { U11^#(tt(), V1, V2) ->
752.91/207.98	       c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.98	           isNatKind^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	     , U12^#(tt(), V1, V2) ->
752.91/207.98	       c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.98	           isNatKind^#(activate(V2)),
752.91/207.98	           activate^#(V2),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	     , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.98	       c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.98	           isNatKind^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	     , isNatKind^#(n__s(V1)) ->
752.91/207.98	       c_4(isNatKind^#(activate(V1)), activate^#(V1))
752.91/207.98	     , U13^#(tt(), V1, V2) ->
752.91/207.98	       c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.98	           isNatKind^#(activate(V2)),
752.91/207.98	           activate^#(V2),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	     , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
752.91/207.98	     , U14^#(tt(), V1, V2) ->
752.91/207.98	       c_9(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.98	           isNat^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	     , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
752.91/207.98	     , isNat^#(n__plus(V1, V2)) ->
752.91/207.98	       c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.98	            isNatKind^#(activate(V1)),
752.91/207.98	            activate^#(V1),
752.91/207.98	            activate^#(V1),
752.91/207.98	            activate^#(V2))
752.91/207.98	     , isNat^#(n__s(V1)) ->
752.91/207.98	       c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.98	            isNatKind^#(activate(V1)),
752.91/207.98	            activate^#(V1),
752.91/207.98	            activate^#(V1))
752.91/207.98	     , U21^#(tt(), V1) ->
752.91/207.98	       c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.98	            isNatKind^#(activate(V1)),
752.91/207.98	            activate^#(V1),
752.91/207.98	            activate^#(V1))
752.91/207.98	     , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
752.91/207.98	     , U61^#(tt(), M, N) ->
752.91/207.98	       c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.98	            isNatKind^#(activate(M)),
752.91/207.98	            activate^#(M),
752.91/207.98	            activate^#(M),
752.91/207.98	            activate^#(N))
752.91/207.98	     , U62^#(tt(), M, N) ->
752.91/207.98	       c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.98	            isNat^#(activate(N)),
752.91/207.98	            activate^#(N),
752.91/207.98	            activate^#(M),
752.91/207.98	            activate^#(N))
752.91/207.98	     , U63^#(tt(), M, N) ->
752.91/207.98	       c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.98	            isNatKind^#(activate(N)),
752.91/207.98	            activate^#(N),
752.91/207.98	            activate^#(M),
752.91/207.98	            activate^#(N))
752.91/207.98	     , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
752.91/207.98	   Weak Trs:
752.91/207.98	     { U11(tt(), V1, V2) ->
752.91/207.98	       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.98	     , U12(tt(), V1, V2) ->
752.91/207.98	       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.98	     , isNatKind(n__0()) -> tt()
752.91/207.98	     , isNatKind(n__plus(V1, V2)) ->
752.91/207.98	       U31(isNatKind(activate(V1)), activate(V2))
752.91/207.98	     , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.98	     , activate(X) -> X
752.91/207.98	     , activate(n__0()) -> 0()
752.91/207.98	     , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.98	     , activate(n__s(X)) -> s(activate(X))
752.91/207.98	     , U13(tt(), V1, V2) ->
752.91/207.98	       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.98	     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.98	     , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.98	     , isNat(n__0()) -> tt()
752.91/207.98	     , isNat(n__plus(V1, V2)) ->
752.91/207.98	       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.98	     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.98	     , U16(tt()) -> tt()
752.91/207.98	     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.98	     , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.98	     , U23(tt()) -> tt()
752.91/207.98	     , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.98	     , U32(tt()) -> tt()
752.91/207.98	     , U41(tt()) -> tt()
752.91/207.98	     , s(X) -> n__s(X)
752.91/207.98	     , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.98	     , 0() -> n__0() }
752.91/207.98	   StartTerms: basic terms
752.91/207.98	   Strategy: innermost
752.91/207.98	   
752.91/207.98	   This problem was proven YES(O(1),O(n^3)).
752.91/207.98	
752.91/207.98	S) Strict DPs:
752.91/207.98	     { U11^#(tt(), V1, V2) ->
752.91/207.98	       c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.98	           isNatKind^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	     , U12^#(tt(), V1, V2) ->
752.91/207.98	       c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.98	           isNatKind^#(activate(V2)),
752.91/207.98	           activate^#(V2),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	     , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.98	       c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.98	           isNatKind^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	     , isNatKind^#(n__s(V1)) ->
752.91/207.98	       c_4(isNatKind^#(activate(V1)), activate^#(V1))
752.91/207.98	     , U13^#(tt(), V1, V2) ->
752.91/207.98	       c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.98	           isNatKind^#(activate(V2)),
752.91/207.98	           activate^#(V2),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	     , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
752.91/207.98	     , U14^#(tt(), V1, V2) ->
752.91/207.98	       c_9(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.98	           isNat^#(activate(V1)),
752.91/207.98	           activate^#(V1),
752.91/207.98	           activate^#(V2))
752.91/207.98	     , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
752.91/207.98	     , isNat^#(n__plus(V1, V2)) ->
752.91/207.98	       c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.98	            isNatKind^#(activate(V1)),
752.91/207.98	            activate^#(V1),
752.91/207.98	            activate^#(V1),
752.91/207.98	            activate^#(V2))
752.91/207.98	     , isNat^#(n__s(V1)) ->
752.91/207.98	       c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.98	            isNatKind^#(activate(V1)),
752.91/207.98	            activate^#(V1),
752.91/207.98	            activate^#(V1))
752.91/207.98	     , U21^#(tt(), V1) ->
752.91/207.98	       c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.98	            isNatKind^#(activate(V1)),
752.91/207.98	            activate^#(V1),
752.91/207.98	            activate^#(V1))
752.91/207.98	     , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
752.91/207.98	     , U61^#(tt(), M, N) ->
752.91/207.98	       c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.98	            isNatKind^#(activate(M)),
752.91/207.98	            activate^#(M),
752.91/207.98	            activate^#(M),
752.91/207.98	            activate^#(N))
752.91/207.98	     , U62^#(tt(), M, N) ->
752.91/207.98	       c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.98	            isNat^#(activate(N)),
752.91/207.98	            activate^#(N),
752.91/207.98	            activate^#(M),
752.91/207.98	            activate^#(N))
752.91/207.98	     , U63^#(tt(), M, N) ->
752.91/207.98	       c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.98	            isNatKind^#(activate(N)),
752.91/207.98	            activate^#(N),
752.91/207.98	            activate^#(M),
752.91/207.98	            activate^#(N))
752.91/207.98	     , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
752.91/207.98	   Weak DPs:
752.91/207.98	     { activate^#(n__plus(X1, X2)) ->
752.91/207.98	       c_5(activate^#(X1), activate^#(X2))
752.91/207.98	     , activate^#(n__s(X)) -> c_6(activate^#(X))
752.91/207.98	     , U51^#(tt(), N) ->
752.91/207.98	       c_15(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.98	            isNatKind^#(activate(N)),
752.91/207.98	            activate^#(N),
752.91/207.98	            activate^#(N))
752.91/207.98	     , U52^#(tt(), N) -> c_16(activate^#(N)) }
752.91/207.98	   Weak Trs:
752.91/207.98	     { U11(tt(), V1, V2) ->
752.91/207.98	       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.98	     , U12(tt(), V1, V2) ->
752.91/207.98	       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.98	     , isNatKind(n__0()) -> tt()
752.91/207.98	     , isNatKind(n__plus(V1, V2)) ->
752.91/207.98	       U31(isNatKind(activate(V1)), activate(V2))
752.91/207.98	     , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.98	     , activate(X) -> X
752.91/207.98	     , activate(n__0()) -> 0()
752.91/207.98	     , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.98	     , activate(n__s(X)) -> s(activate(X))
752.91/207.98	     , U13(tt(), V1, V2) ->
752.91/207.98	       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.98	     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.98	     , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.98	     , isNat(n__0()) -> tt()
752.91/207.98	     , isNat(n__plus(V1, V2)) ->
752.91/207.98	       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.98	     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.98	     , U16(tt()) -> tt()
752.91/207.98	     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.98	     , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.99	     , U23(tt()) -> tt()
752.91/207.99	     , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.99	     , U32(tt()) -> tt()
752.91/207.99	     , U41(tt()) -> tt()
752.91/207.99	     , s(X) -> n__s(X)
752.91/207.99	     , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.99	     , 0() -> n__0() }
752.91/207.99	   StartTerms: basic terms
752.91/207.99	   Strategy: innermost
752.91/207.99	   
752.91/207.99	   This problem was proven YES(O(1),O(n^2)).
752.91/207.99	
752.91/207.99	
752.91/207.99	Proofs for generated problems:
752.91/207.99	------------------------------
752.91/207.99	R) We are left with following problem, upon which TcT provides the
752.91/207.99	   certificate YES(O(1),O(n^3)).
752.91/207.99	   
752.91/207.99	   Strict DPs:
752.91/207.99	     { activate^#(n__plus(X1, X2)) ->
752.91/207.99	       c_5(activate^#(X1), activate^#(X2))
752.91/207.99	     , activate^#(n__s(X)) -> c_6(activate^#(X))
752.91/207.99	     , U51^#(tt(), N) ->
752.91/207.99	       c_15(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.99	            isNatKind^#(activate(N)),
752.91/207.99	            activate^#(N),
752.91/207.99	            activate^#(N))
752.91/207.99	     , U52^#(tt(), N) -> c_16(activate^#(N)) }
752.91/207.99	   Weak DPs:
752.91/207.99	     { U11^#(tt(), V1, V2) ->
752.91/207.99	       c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.99	           isNatKind^#(activate(V1)),
752.91/207.99	           activate^#(V1),
752.91/207.99	           activate^#(V1),
752.91/207.99	           activate^#(V2))
752.91/207.99	     , U12^#(tt(), V1, V2) ->
752.91/207.99	       c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.99	           isNatKind^#(activate(V2)),
752.91/207.99	           activate^#(V2),
752.91/207.99	           activate^#(V1),
752.91/207.99	           activate^#(V2))
752.91/207.99	     , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.99	       c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.99	           isNatKind^#(activate(V1)),
752.91/207.99	           activate^#(V1),
752.91/207.99	           activate^#(V2))
752.91/207.99	     , isNatKind^#(n__s(V1)) ->
752.91/207.99	       c_4(isNatKind^#(activate(V1)), activate^#(V1))
752.91/207.99	     , U13^#(tt(), V1, V2) ->
752.91/207.99	       c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.99	           isNatKind^#(activate(V2)),
752.91/207.99	           activate^#(V2),
752.91/207.99	           activate^#(V1),
752.91/207.99	           activate^#(V2))
752.91/207.99	     , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
752.91/207.99	     , U14^#(tt(), V1, V2) ->
752.91/207.99	       c_9(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.99	           isNat^#(activate(V1)),
752.91/207.99	           activate^#(V1),
752.91/207.99	           activate^#(V2))
752.91/207.99	     , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
752.91/207.99	     , isNat^#(n__plus(V1, V2)) ->
752.91/207.99	       c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.99	            isNatKind^#(activate(V1)),
752.91/207.99	            activate^#(V1),
752.91/207.99	            activate^#(V1),
752.91/207.99	            activate^#(V2))
752.91/207.99	     , isNat^#(n__s(V1)) ->
752.91/207.99	       c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.99	            isNatKind^#(activate(V1)),
752.91/207.99	            activate^#(V1),
752.91/207.99	            activate^#(V1))
752.91/207.99	     , U21^#(tt(), V1) ->
752.91/207.99	       c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.99	            isNatKind^#(activate(V1)),
752.91/207.99	            activate^#(V1),
752.91/207.99	            activate^#(V1))
752.91/207.99	     , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
752.91/207.99	     , U61^#(tt(), M, N) ->
752.91/207.99	       c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.99	            isNatKind^#(activate(M)),
752.91/207.99	            activate^#(M),
752.91/207.99	            activate^#(M),
752.91/207.99	            activate^#(N))
752.91/207.99	     , U62^#(tt(), M, N) ->
752.91/207.99	       c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.99	            isNat^#(activate(N)),
752.91/207.99	            activate^#(N),
752.91/207.99	            activate^#(M),
752.91/207.99	            activate^#(N))
752.91/207.99	     , U63^#(tt(), M, N) ->
752.91/207.99	       c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.99	            isNatKind^#(activate(N)),
752.91/207.99	            activate^#(N),
752.91/207.99	            activate^#(M),
752.91/207.99	            activate^#(N))
752.91/207.99	     , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
752.91/207.99	   Weak Trs:
752.91/207.99	     { U11(tt(), V1, V2) ->
752.91/207.99	       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.99	     , U12(tt(), V1, V2) ->
752.91/207.99	       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.99	     , isNatKind(n__0()) -> tt()
752.91/207.99	     , isNatKind(n__plus(V1, V2)) ->
752.91/207.99	       U31(isNatKind(activate(V1)), activate(V2))
752.91/207.99	     , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.99	     , activate(X) -> X
752.91/207.99	     , activate(n__0()) -> 0()
752.91/207.99	     , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.99	     , activate(n__s(X)) -> s(activate(X))
752.91/207.99	     , U13(tt(), V1, V2) ->
752.91/207.99	       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.99	     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.99	     , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.99	     , isNat(n__0()) -> tt()
752.91/207.99	     , isNat(n__plus(V1, V2)) ->
752.91/207.99	       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.99	     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.99	     , U16(tt()) -> tt()
752.91/207.99	     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.99	     , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.99	     , U23(tt()) -> tt()
752.91/207.99	     , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.99	     , U32(tt()) -> tt()
752.91/207.99	     , U41(tt()) -> tt()
752.91/207.99	     , s(X) -> n__s(X)
752.91/207.99	     , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.99	     , 0() -> n__0() }
752.91/207.99	   Obligation:
752.91/207.99	     innermost runtime complexity
752.91/207.99	   Answer:
752.91/207.99	     YES(O(1),O(n^3))
752.91/207.99	   
752.91/207.99	   We analyse the complexity of following sub-problems (R) and (S).
752.91/207.99	   Problem (S) is obtained from the input problem by shifting strict
752.91/207.99	   rules from (R) into the weak component:
752.91/207.99	   
752.91/207.99	   Problem (R):
752.91/207.99	   ------------
752.91/207.99	     Strict DPs:
752.91/207.99	       { activate^#(n__plus(X1, X2)) ->
752.91/207.99	         c_5(activate^#(X1), activate^#(X2))
752.91/207.99	       , activate^#(n__s(X)) -> c_6(activate^#(X))
752.91/207.99	       , U51^#(tt(), N) ->
752.91/207.99	         c_15(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.99	              isNatKind^#(activate(N)),
752.91/207.99	              activate^#(N),
752.91/207.99	              activate^#(N)) }
752.91/207.99	     Weak DPs:
752.91/207.99	       { U11^#(tt(), V1, V2) ->
752.91/207.99	         c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.99	             isNatKind^#(activate(V1)),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V2))
752.91/207.99	       , U12^#(tt(), V1, V2) ->
752.91/207.99	         c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.99	             isNatKind^#(activate(V2)),
752.91/207.99	             activate^#(V2),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V2))
752.91/207.99	       , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.99	         c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.99	             isNatKind^#(activate(V1)),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V2))
752.91/207.99	       , isNatKind^#(n__s(V1)) ->
752.91/207.99	         c_4(isNatKind^#(activate(V1)), activate^#(V1))
752.91/207.99	       , U13^#(tt(), V1, V2) ->
752.91/207.99	         c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.99	             isNatKind^#(activate(V2)),
752.91/207.99	             activate^#(V2),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V2))
752.91/207.99	       , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
752.91/207.99	       , U14^#(tt(), V1, V2) ->
752.91/207.99	         c_9(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.99	             isNat^#(activate(V1)),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V2))
752.91/207.99	       , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
752.91/207.99	       , isNat^#(n__plus(V1, V2)) ->
752.91/207.99	         c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.99	              isNatKind^#(activate(V1)),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V2))
752.91/207.99	       , isNat^#(n__s(V1)) ->
752.91/207.99	         c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.99	              isNatKind^#(activate(V1)),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V1))
752.91/207.99	       , U21^#(tt(), V1) ->
752.91/207.99	         c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.99	              isNatKind^#(activate(V1)),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V1))
752.91/207.99	       , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
752.91/207.99	       , U52^#(tt(), N) -> c_16(activate^#(N))
752.91/207.99	       , U61^#(tt(), M, N) ->
752.91/207.99	         c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.99	              isNatKind^#(activate(M)),
752.91/207.99	              activate^#(M),
752.91/207.99	              activate^#(M),
752.91/207.99	              activate^#(N))
752.91/207.99	       , U62^#(tt(), M, N) ->
752.91/207.99	         c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.99	              isNat^#(activate(N)),
752.91/207.99	              activate^#(N),
752.91/207.99	              activate^#(M),
752.91/207.99	              activate^#(N))
752.91/207.99	       , U63^#(tt(), M, N) ->
752.91/207.99	         c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.99	              isNatKind^#(activate(N)),
752.91/207.99	              activate^#(N),
752.91/207.99	              activate^#(M),
752.91/207.99	              activate^#(N))
752.91/207.99	       , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
752.91/207.99	     Weak Trs:
752.91/207.99	       { U11(tt(), V1, V2) ->
752.91/207.99	         U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.99	       , U12(tt(), V1, V2) ->
752.91/207.99	         U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.99	       , isNatKind(n__0()) -> tt()
752.91/207.99	       , isNatKind(n__plus(V1, V2)) ->
752.91/207.99	         U31(isNatKind(activate(V1)), activate(V2))
752.91/207.99	       , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.99	       , activate(X) -> X
752.91/207.99	       , activate(n__0()) -> 0()
752.91/207.99	       , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.99	       , activate(n__s(X)) -> s(activate(X))
752.91/207.99	       , U13(tt(), V1, V2) ->
752.91/207.99	         U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.99	       , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.99	       , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.99	       , isNat(n__0()) -> tt()
752.91/207.99	       , isNat(n__plus(V1, V2)) ->
752.91/207.99	         U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.99	       , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.99	       , U16(tt()) -> tt()
752.91/207.99	       , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.99	       , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.99	       , U23(tt()) -> tt()
752.91/207.99	       , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.99	       , U32(tt()) -> tt()
752.91/207.99	       , U41(tt()) -> tt()
752.91/207.99	       , s(X) -> n__s(X)
752.91/207.99	       , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.99	       , 0() -> n__0() }
752.91/207.99	     StartTerms: basic terms
752.91/207.99	     Strategy: innermost
752.91/207.99	   
752.91/207.99	   Problem (S):
752.91/207.99	   ------------
752.91/207.99	     Strict DPs: { U52^#(tt(), N) -> c_16(activate^#(N)) }
752.91/207.99	     Weak DPs:
752.91/207.99	       { U11^#(tt(), V1, V2) ->
752.91/207.99	         c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.99	             isNatKind^#(activate(V1)),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V2))
752.91/207.99	       , U12^#(tt(), V1, V2) ->
752.91/207.99	         c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.99	             isNatKind^#(activate(V2)),
752.91/207.99	             activate^#(V2),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V2))
752.91/207.99	       , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.99	         c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.99	             isNatKind^#(activate(V1)),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V2))
752.91/207.99	       , isNatKind^#(n__s(V1)) ->
752.91/207.99	         c_4(isNatKind^#(activate(V1)), activate^#(V1))
752.91/207.99	       , activate^#(n__plus(X1, X2)) ->
752.91/207.99	         c_5(activate^#(X1), activate^#(X2))
752.91/207.99	       , activate^#(n__s(X)) -> c_6(activate^#(X))
752.91/207.99	       , U13^#(tt(), V1, V2) ->
752.91/207.99	         c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.99	             isNatKind^#(activate(V2)),
752.91/207.99	             activate^#(V2),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V2))
752.91/207.99	       , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
752.91/207.99	       , U14^#(tt(), V1, V2) ->
752.91/207.99	         c_9(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.99	             isNat^#(activate(V1)),
752.91/207.99	             activate^#(V1),
752.91/207.99	             activate^#(V2))
752.91/207.99	       , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
752.91/207.99	       , isNat^#(n__plus(V1, V2)) ->
752.91/207.99	         c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.99	              isNatKind^#(activate(V1)),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V2))
752.91/207.99	       , isNat^#(n__s(V1)) ->
752.91/207.99	         c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.99	              isNatKind^#(activate(V1)),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V1))
752.91/207.99	       , U21^#(tt(), V1) ->
752.91/207.99	         c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
752.91/207.99	              isNatKind^#(activate(V1)),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V1))
752.91/207.99	       , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
752.91/207.99	       , U51^#(tt(), N) ->
752.91/207.99	         c_15(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.99	              isNatKind^#(activate(N)),
752.91/207.99	              activate^#(N),
752.91/207.99	              activate^#(N))
752.91/207.99	       , U61^#(tt(), M, N) ->
752.91/207.99	         c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
752.91/207.99	              isNatKind^#(activate(M)),
752.91/207.99	              activate^#(M),
752.91/207.99	              activate^#(M),
752.91/207.99	              activate^#(N))
752.91/207.99	       , U62^#(tt(), M, N) ->
752.91/207.99	         c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
752.91/207.99	              isNat^#(activate(N)),
752.91/207.99	              activate^#(N),
752.91/207.99	              activate^#(M),
752.91/207.99	              activate^#(N))
752.91/207.99	       , U63^#(tt(), M, N) ->
752.91/207.99	         c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
752.91/207.99	              isNatKind^#(activate(N)),
752.91/207.99	              activate^#(N),
752.91/207.99	              activate^#(M),
752.91/207.99	              activate^#(N))
752.91/207.99	       , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
752.91/207.99	     Weak Trs:
752.91/207.99	       { U11(tt(), V1, V2) ->
752.91/207.99	         U12(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.99	       , U12(tt(), V1, V2) ->
752.91/207.99	         U13(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.99	       , isNatKind(n__0()) -> tt()
752.91/207.99	       , isNatKind(n__plus(V1, V2)) ->
752.91/207.99	         U31(isNatKind(activate(V1)), activate(V2))
752.91/207.99	       , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
752.91/207.99	       , activate(X) -> X
752.91/207.99	       , activate(n__0()) -> 0()
752.91/207.99	       , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
752.91/207.99	       , activate(n__s(X)) -> s(activate(X))
752.91/207.99	       , U13(tt(), V1, V2) ->
752.91/207.99	         U14(isNatKind(activate(V2)), activate(V1), activate(V2))
752.91/207.99	       , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
752.91/207.99	       , U15(tt(), V2) -> U16(isNat(activate(V2)))
752.91/207.99	       , isNat(n__0()) -> tt()
752.91/207.99	       , isNat(n__plus(V1, V2)) ->
752.91/207.99	         U11(isNatKind(activate(V1)), activate(V1), activate(V2))
752.91/207.99	       , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
752.91/207.99	       , U16(tt()) -> tt()
752.91/207.99	       , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
752.91/207.99	       , U22(tt(), V1) -> U23(isNat(activate(V1)))
752.91/207.99	       , U23(tt()) -> tt()
752.91/207.99	       , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
752.91/207.99	       , U32(tt()) -> tt()
752.91/207.99	       , U41(tt()) -> tt()
752.91/207.99	       , s(X) -> n__s(X)
752.91/207.99	       , plus(X1, X2) -> n__plus(X1, X2)
752.91/207.99	       , 0() -> n__0() }
752.91/207.99	     StartTerms: basic terms
752.91/207.99	     Strategy: innermost
752.91/207.99	   
752.91/207.99	   Overall, the transformation results in the following sub-problem(s):
752.91/207.99	   
752.91/207.99	   Generated new problems:
752.91/207.99	   -----------------------
752.91/207.99	   R) Strict DPs:
752.91/207.99	        { activate^#(n__plus(X1, X2)) ->
752.91/207.99	          c_5(activate^#(X1), activate^#(X2))
752.91/207.99	        , activate^#(n__s(X)) -> c_6(activate^#(X))
752.91/207.99	        , U51^#(tt(), N) ->
752.91/207.99	          c_15(U52^#(isNatKind(activate(N)), activate(N)),
752.91/207.99	               isNatKind^#(activate(N)),
752.91/207.99	               activate^#(N),
752.91/207.99	               activate^#(N)) }
752.91/207.99	      Weak DPs:
752.91/207.99	        { U11^#(tt(), V1, V2) ->
752.91/207.99	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.99	              isNatKind^#(activate(V1)),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V2))
752.91/207.99	        , U12^#(tt(), V1, V2) ->
752.91/207.99	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.99	              isNatKind^#(activate(V2)),
752.91/207.99	              activate^#(V2),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V2))
752.91/207.99	        , isNatKind^#(n__plus(V1, V2)) ->
752.91/207.99	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
752.91/207.99	              isNatKind^#(activate(V1)),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V2))
752.91/207.99	        , isNatKind^#(n__s(V1)) ->
752.91/207.99	          c_4(isNatKind^#(activate(V1)), activate^#(V1))
752.91/207.99	        , U13^#(tt(), V1, V2) ->
752.91/207.99	          c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
752.91/207.99	              isNatKind^#(activate(V2)),
752.91/207.99	              activate^#(V2),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V2))
752.91/207.99	        , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
752.91/207.99	        , U14^#(tt(), V1, V2) ->
752.91/207.99	          c_9(U15^#(isNat(activate(V1)), activate(V2)),
752.91/207.99	              isNat^#(activate(V1)),
752.91/207.99	              activate^#(V1),
752.91/207.99	              activate^#(V2))
752.91/207.99	        , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
752.91/207.99	        , isNat^#(n__plus(V1, V2)) ->
752.91/207.99	          c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
752.91/207.99	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V2))
753.12/208.00	        , isNat^#(n__s(V1)) ->
753.12/208.00	          c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.00	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1))
753.12/208.00	        , U21^#(tt(), V1) ->
753.12/208.00	          c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.00	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1))
753.12/208.00	        , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.00	        , U52^#(tt(), N) -> c_16(activate^#(N))
753.12/208.00	        , U61^#(tt(), M, N) ->
753.12/208.00	          c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.00	               isNatKind^#(activate(M)),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U62^#(tt(), M, N) ->
753.12/208.00	          c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.00	               isNat^#(activate(N)),
753.12/208.00	               activate^#(N),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U63^#(tt(), M, N) ->
753.12/208.00	          c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.00	               isNatKind^#(activate(N)),
753.12/208.00	               activate^#(N),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.00	      Weak Trs:
753.12/208.00	        { U11(tt(), V1, V2) ->
753.12/208.00	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.00	        , U12(tt(), V1, V2) ->
753.12/208.00	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.00	        , isNatKind(n__0()) -> tt()
753.12/208.00	        , isNatKind(n__plus(V1, V2)) ->
753.12/208.00	          U31(isNatKind(activate(V1)), activate(V2))
753.12/208.00	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.00	        , activate(X) -> X
753.12/208.00	        , activate(n__0()) -> 0()
753.12/208.00	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.00	        , activate(n__s(X)) -> s(activate(X))
753.12/208.00	        , U13(tt(), V1, V2) ->
753.12/208.00	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.00	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.00	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.00	        , isNat(n__0()) -> tt()
753.12/208.00	        , isNat(n__plus(V1, V2)) ->
753.12/208.00	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.00	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.00	        , U16(tt()) -> tt()
753.12/208.00	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.00	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.00	        , U23(tt()) -> tt()
753.12/208.00	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.00	        , U32(tt()) -> tt()
753.12/208.00	        , U41(tt()) -> tt()
753.12/208.00	        , s(X) -> n__s(X)
753.12/208.00	        , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.00	        , 0() -> n__0() }
753.12/208.00	      StartTerms: basic terms
753.12/208.00	      Strategy: innermost
753.12/208.00	      
753.12/208.00	      This problem was proven YES(O(1),O(n^3)).
753.12/208.00	   
753.12/208.00	   S) Strict DPs: { U52^#(tt(), N) -> c_16(activate^#(N)) }
753.12/208.00	      Weak DPs:
753.12/208.00	        { U11^#(tt(), V1, V2) ->
753.12/208.00	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V1)),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , U12^#(tt(), V1, V2) ->
753.12/208.00	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V2)),
753.12/208.00	              activate^#(V2),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , isNatKind^#(n__plus(V1, V2)) ->
753.12/208.00	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V1)),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , isNatKind^#(n__s(V1)) ->
753.12/208.00	          c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.12/208.00	        , activate^#(n__plus(X1, X2)) ->
753.12/208.00	          c_5(activate^#(X1), activate^#(X2))
753.12/208.00	        , activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.00	        , U13^#(tt(), V1, V2) ->
753.12/208.00	          c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V2)),
753.12/208.00	              activate^#(V2),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.12/208.00	        , U14^#(tt(), V1, V2) ->
753.12/208.00	          c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.00	              isNat^#(activate(V1)),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.00	        , isNat^#(n__plus(V1, V2)) ->
753.12/208.00	          c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.00	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V2))
753.12/208.00	        , isNat^#(n__s(V1)) ->
753.12/208.00	          c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.00	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1))
753.12/208.00	        , U21^#(tt(), V1) ->
753.12/208.00	          c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.00	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1))
753.12/208.00	        , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.00	        , U51^#(tt(), N) ->
753.12/208.00	          c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.00	               isNatKind^#(activate(N)),
753.12/208.00	               activate^#(N),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U61^#(tt(), M, N) ->
753.12/208.00	          c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.00	               isNatKind^#(activate(M)),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U62^#(tt(), M, N) ->
753.12/208.00	          c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.00	               isNat^#(activate(N)),
753.12/208.00	               activate^#(N),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U63^#(tt(), M, N) ->
753.12/208.00	          c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.00	               isNatKind^#(activate(N)),
753.12/208.00	               activate^#(N),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.00	      Weak Trs:
753.12/208.00	        { U11(tt(), V1, V2) ->
753.12/208.00	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.00	        , U12(tt(), V1, V2) ->
753.12/208.00	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.00	        , isNatKind(n__0()) -> tt()
753.12/208.00	        , isNatKind(n__plus(V1, V2)) ->
753.12/208.00	          U31(isNatKind(activate(V1)), activate(V2))
753.12/208.00	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.00	        , activate(X) -> X
753.12/208.00	        , activate(n__0()) -> 0()
753.12/208.00	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.00	        , activate(n__s(X)) -> s(activate(X))
753.12/208.00	        , U13(tt(), V1, V2) ->
753.12/208.00	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.00	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.00	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.00	        , isNat(n__0()) -> tt()
753.12/208.00	        , isNat(n__plus(V1, V2)) ->
753.12/208.00	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.00	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.00	        , U16(tt()) -> tt()
753.12/208.00	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.00	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.00	        , U23(tt()) -> tt()
753.12/208.00	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.00	        , U32(tt()) -> tt()
753.12/208.00	        , U41(tt()) -> tt()
753.12/208.00	        , s(X) -> n__s(X)
753.12/208.00	        , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.00	        , 0() -> n__0() }
753.12/208.00	      StartTerms: basic terms
753.12/208.00	      Strategy: innermost
753.12/208.00	      
753.12/208.00	      This problem was proven YES(O(1),O(1)).
753.12/208.00	   
753.12/208.00	   
753.12/208.00	   Proofs for generated problems:
753.12/208.00	   ------------------------------
753.12/208.00	   R) We are left with following problem, upon which TcT provides the
753.12/208.00	      certificate YES(O(1),O(n^3)).
753.12/208.00	      
753.12/208.00	      Strict DPs:
753.12/208.00	        { activate^#(n__plus(X1, X2)) ->
753.12/208.00	          c_5(activate^#(X1), activate^#(X2))
753.12/208.00	        , activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.00	        , U51^#(tt(), N) ->
753.12/208.00	          c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.00	               isNatKind^#(activate(N)),
753.12/208.00	               activate^#(N),
753.12/208.00	               activate^#(N)) }
753.12/208.00	      Weak DPs:
753.12/208.00	        { U11^#(tt(), V1, V2) ->
753.12/208.00	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V1)),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , U12^#(tt(), V1, V2) ->
753.12/208.00	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V2)),
753.12/208.00	              activate^#(V2),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , isNatKind^#(n__plus(V1, V2)) ->
753.12/208.00	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V1)),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , isNatKind^#(n__s(V1)) ->
753.12/208.00	          c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.12/208.00	        , U13^#(tt(), V1, V2) ->
753.12/208.00	          c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V2)),
753.12/208.00	              activate^#(V2),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.12/208.00	        , U14^#(tt(), V1, V2) ->
753.12/208.00	          c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.00	              isNat^#(activate(V1)),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.00	        , isNat^#(n__plus(V1, V2)) ->
753.12/208.00	          c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.00	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V2))
753.12/208.00	        , isNat^#(n__s(V1)) ->
753.12/208.00	          c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.00	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1))
753.12/208.00	        , U21^#(tt(), V1) ->
753.12/208.00	          c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.00	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1))
753.12/208.00	        , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.00	        , U52^#(tt(), N) -> c_16(activate^#(N))
753.12/208.00	        , U61^#(tt(), M, N) ->
753.12/208.00	          c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.00	               isNatKind^#(activate(M)),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U62^#(tt(), M, N) ->
753.12/208.00	          c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.00	               isNat^#(activate(N)),
753.12/208.00	               activate^#(N),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U63^#(tt(), M, N) ->
753.12/208.00	          c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.00	               isNatKind^#(activate(N)),
753.12/208.00	               activate^#(N),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.00	      Weak Trs:
753.12/208.00	        { U11(tt(), V1, V2) ->
753.12/208.00	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.00	        , U12(tt(), V1, V2) ->
753.12/208.00	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.00	        , isNatKind(n__0()) -> tt()
753.12/208.00	        , isNatKind(n__plus(V1, V2)) ->
753.12/208.00	          U31(isNatKind(activate(V1)), activate(V2))
753.12/208.00	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.00	        , activate(X) -> X
753.12/208.00	        , activate(n__0()) -> 0()
753.12/208.00	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.00	        , activate(n__s(X)) -> s(activate(X))
753.12/208.00	        , U13(tt(), V1, V2) ->
753.12/208.00	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.00	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.00	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.00	        , isNat(n__0()) -> tt()
753.12/208.00	        , isNat(n__plus(V1, V2)) ->
753.12/208.00	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.00	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.00	        , U16(tt()) -> tt()
753.12/208.00	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.00	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.00	        , U23(tt()) -> tt()
753.12/208.00	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.00	        , U32(tt()) -> tt()
753.12/208.00	        , U41(tt()) -> tt()
753.12/208.00	        , s(X) -> n__s(X)
753.12/208.00	        , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.00	        , 0() -> n__0() }
753.12/208.00	      Obligation:
753.12/208.00	        innermost runtime complexity
753.12/208.00	      Answer:
753.12/208.00	        YES(O(1),O(n^3))
753.12/208.00	      
753.12/208.00	      We decompose the input problem according to the dependency graph
753.12/208.00	      into the upper component
753.12/208.00	      
753.12/208.00	        { U11^#(tt(), V1, V2) ->
753.12/208.00	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V1)),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , U12^#(tt(), V1, V2) ->
753.12/208.00	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V2)),
753.12/208.00	              activate^#(V2),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , U13^#(tt(), V1, V2) ->
753.12/208.00	          c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V2)),
753.12/208.00	              activate^#(V2),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , U14^#(tt(), V1, V2) ->
753.12/208.00	          c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.00	              isNat^#(activate(V1)),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.00	        , isNat^#(n__plus(V1, V2)) ->
753.12/208.00	          c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.00	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V2))
753.12/208.00	        , isNat^#(n__s(V1)) ->
753.12/208.00	          c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.00	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1))
753.12/208.00	        , U21^#(tt(), V1) ->
753.12/208.00	          c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.00	               isNatKind^#(activate(V1)),
753.12/208.00	               activate^#(V1),
753.12/208.00	               activate^#(V1))
753.12/208.00	        , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.00	        , U51^#(tt(), N) ->
753.12/208.00	          c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.00	               isNatKind^#(activate(N)),
753.12/208.00	               activate^#(N),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U52^#(tt(), N) -> c_16(activate^#(N))
753.12/208.00	        , U61^#(tt(), M, N) ->
753.12/208.00	          c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.00	               isNatKind^#(activate(M)),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U62^#(tt(), M, N) ->
753.12/208.00	          c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.00	               isNat^#(activate(N)),
753.12/208.00	               activate^#(N),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U63^#(tt(), M, N) ->
753.12/208.00	          c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.00	               isNatKind^#(activate(N)),
753.12/208.00	               activate^#(N),
753.12/208.00	               activate^#(M),
753.12/208.00	               activate^#(N))
753.12/208.00	        , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.00	      
753.12/208.00	      and lower component
753.12/208.00	      
753.12/208.00	        { isNatKind^#(n__plus(V1, V2)) ->
753.12/208.00	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.12/208.00	              isNatKind^#(activate(V1)),
753.12/208.00	              activate^#(V1),
753.12/208.00	              activate^#(V2))
753.12/208.00	        , isNatKind^#(n__s(V1)) ->
753.12/208.00	          c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.12/208.00	        , activate^#(n__plus(X1, X2)) ->
753.12/208.00	          c_5(activate^#(X1), activate^#(X2))
753.12/208.00	        , activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.00	        , U31^#(tt(), V2) ->
753.12/208.00	          c_8(isNatKind^#(activate(V2)), activate^#(V2)) }
753.12/208.00	      
753.12/208.00	      Further, following extension rules are added to the lower
753.12/208.00	      component.
753.12/208.00	      
753.12/208.00	      { U11^#(tt(), V1, V2) ->
753.12/208.00	        U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.00	      , U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.12/208.00	      , U11^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.01	      , U11^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.01	      , U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.01	      , U12^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.01	      , U12^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.01	      , U12^#(tt(), V1, V2) ->
753.12/208.01	        U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.01	      , U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.01	      , U13^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.01	      , U13^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.01	      , U13^#(tt(), V1, V2) ->
753.12/208.01	        U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.01	      , U14^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.01	      , U14^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.01	      , U14^#(tt(), V1, V2) -> U15^#(isNat(activate(V1)), activate(V2))
753.12/208.01	      , U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.12/208.01	      , U15^#(tt(), V2) -> activate^#(V2)
753.12/208.01	      , U15^#(tt(), V2) -> isNat^#(activate(V2))
753.12/208.01	      , isNat^#(n__plus(V1, V2)) ->
753.12/208.01	        U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.01	      , isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.12/208.01	      , isNat^#(n__plus(V1, V2)) -> activate^#(V1)
753.12/208.01	      , isNat^#(n__plus(V1, V2)) -> activate^#(V2)
753.12/208.01	      , isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.12/208.01	      , isNat^#(n__s(V1)) -> activate^#(V1)
753.12/208.01	      , isNat^#(n__s(V1)) -> U21^#(isNatKind(activate(V1)), activate(V1))
753.12/208.01	      , U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.12/208.01	      , U21^#(tt(), V1) -> activate^#(V1)
753.12/208.01	      , U21^#(tt(), V1) -> U22^#(isNatKind(activate(V1)), activate(V1))
753.12/208.01	      , U22^#(tt(), V1) -> activate^#(V1)
753.12/208.01	      , U22^#(tt(), V1) -> isNat^#(activate(V1))
753.12/208.01	      , U51^#(tt(), N) -> isNatKind^#(activate(N))
753.12/208.01	      , U51^#(tt(), N) -> activate^#(N)
753.12/208.01	      , U51^#(tt(), N) -> U52^#(isNatKind(activate(N)), activate(N))
753.12/208.01	      , U52^#(tt(), N) -> activate^#(N)
753.12/208.01	      , U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.12/208.01	      , U61^#(tt(), M, N) -> activate^#(M)
753.12/208.01	      , U61^#(tt(), M, N) -> activate^#(N)
753.12/208.01	      , U61^#(tt(), M, N) ->
753.12/208.01	        U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.12/208.01	      , U62^#(tt(), M, N) -> activate^#(M)
753.12/208.01	      , U62^#(tt(), M, N) -> activate^#(N)
753.12/208.01	      , U62^#(tt(), M, N) -> isNat^#(activate(N))
753.12/208.01	      , U62^#(tt(), M, N) ->
753.12/208.01	        U63^#(isNat(activate(N)), activate(M), activate(N))
753.12/208.01	      , U63^#(tt(), M, N) -> isNatKind^#(activate(N))
753.12/208.01	      , U63^#(tt(), M, N) -> activate^#(M)
753.12/208.01	      , U63^#(tt(), M, N) -> activate^#(N)
753.12/208.01	      , U63^#(tt(), M, N) ->
753.12/208.01	        U64^#(isNatKind(activate(N)), activate(M), activate(N))
753.12/208.01	      , U64^#(tt(), M, N) -> activate^#(M)
753.12/208.01	      , U64^#(tt(), M, N) -> activate^#(N) }
753.12/208.01	      
753.12/208.01	      TcT solves the upper component with certificate YES(O(1),O(n^1)).
753.12/208.01	      
753.12/208.01	      Sub-proof:
753.12/208.01	      ----------
753.12/208.01	        We are left with following problem, upon which TcT provides the
753.12/208.01	        certificate YES(O(1),O(n^1)).
753.12/208.01	        
753.12/208.01	        Strict DPs:
753.12/208.01	          { U11^#(tt(), V1, V2) ->
753.12/208.01	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.01	                isNatKind^#(activate(V1)),
753.12/208.01	                activate^#(V1),
753.12/208.01	                activate^#(V1),
753.12/208.01	                activate^#(V2))
753.12/208.01	          , U12^#(tt(), V1, V2) ->
753.12/208.01	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.01	                isNatKind^#(activate(V2)),
753.12/208.01	                activate^#(V2),
753.12/208.01	                activate^#(V1),
753.12/208.01	                activate^#(V2))
753.12/208.01	          , U13^#(tt(), V1, V2) ->
753.12/208.01	            c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.01	                isNatKind^#(activate(V2)),
753.12/208.01	                activate^#(V2),
753.12/208.01	                activate^#(V1),
753.12/208.01	                activate^#(V2))
753.12/208.01	          , U14^#(tt(), V1, V2) ->
753.12/208.01	            c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.01	                isNat^#(activate(V1)),
753.12/208.01	                activate^#(V1),
753.12/208.01	                activate^#(V2))
753.12/208.01	          , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.01	          , isNat^#(n__plus(V1, V2)) ->
753.12/208.01	            c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.01	                 isNatKind^#(activate(V1)),
753.12/208.01	                 activate^#(V1),
753.12/208.01	                 activate^#(V1),
753.12/208.01	                 activate^#(V2))
753.12/208.01	          , isNat^#(n__s(V1)) ->
753.12/208.01	            c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.01	                 isNatKind^#(activate(V1)),
753.12/208.01	                 activate^#(V1),
753.12/208.01	                 activate^#(V1))
753.12/208.01	          , U21^#(tt(), V1) ->
753.12/208.01	            c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.01	                 isNatKind^#(activate(V1)),
753.12/208.01	                 activate^#(V1),
753.12/208.01	                 activate^#(V1))
753.12/208.01	          , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.01	          , U51^#(tt(), N) ->
753.12/208.01	            c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.01	                 isNatKind^#(activate(N)),
753.12/208.01	                 activate^#(N),
753.12/208.01	                 activate^#(N))
753.12/208.01	          , U52^#(tt(), N) -> c_16(activate^#(N))
753.12/208.01	          , U61^#(tt(), M, N) ->
753.12/208.01	            c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.01	                 isNatKind^#(activate(M)),
753.12/208.01	                 activate^#(M),
753.12/208.01	                 activate^#(M),
753.12/208.01	                 activate^#(N))
753.12/208.01	          , U62^#(tt(), M, N) ->
753.12/208.01	            c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.01	                 isNat^#(activate(N)),
753.12/208.01	                 activate^#(N),
753.12/208.01	                 activate^#(M),
753.12/208.01	                 activate^#(N))
753.12/208.01	          , U63^#(tt(), M, N) ->
753.12/208.01	            c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.01	                 isNatKind^#(activate(N)),
753.12/208.01	                 activate^#(N),
753.12/208.01	                 activate^#(M),
753.12/208.01	                 activate^#(N))
753.12/208.01	          , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.01	        Weak Trs:
753.12/208.01	          { U11(tt(), V1, V2) ->
753.12/208.01	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.01	          , U12(tt(), V1, V2) ->
753.12/208.01	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.01	          , isNatKind(n__0()) -> tt()
753.12/208.01	          , isNatKind(n__plus(V1, V2)) ->
753.12/208.01	            U31(isNatKind(activate(V1)), activate(V2))
753.12/208.01	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.01	          , activate(X) -> X
753.12/208.01	          , activate(n__0()) -> 0()
753.12/208.01	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.01	          , activate(n__s(X)) -> s(activate(X))
753.12/208.01	          , U13(tt(), V1, V2) ->
753.12/208.01	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.01	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.01	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.01	          , isNat(n__0()) -> tt()
753.12/208.01	          , isNat(n__plus(V1, V2)) ->
753.12/208.01	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.01	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.01	          , U16(tt()) -> tt()
753.12/208.01	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.01	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.01	          , U23(tt()) -> tt()
753.12/208.01	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.01	          , U32(tt()) -> tt()
753.12/208.01	          , U41(tt()) -> tt()
753.12/208.01	          , s(X) -> n__s(X)
753.12/208.01	          , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.01	          , 0() -> n__0() }
753.12/208.01	        Obligation:
753.12/208.01	          innermost runtime complexity
753.12/208.01	        Answer:
753.12/208.01	          YES(O(1),O(n^1))
753.12/208.01	        
753.12/208.01	        We estimate the number of application of {11,15} by applications of
753.12/208.01	        Pre({11,15}) = {10,14}. Here rules are labeled as follows:
753.12/208.01	        
753.12/208.01	          DPs:
753.12/208.01	            { 1: U11^#(tt(), V1, V2) ->
753.12/208.01	                 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.01	                     isNatKind^#(activate(V1)),
753.12/208.01	                     activate^#(V1),
753.12/208.01	                     activate^#(V1),
753.12/208.01	                     activate^#(V2))
753.12/208.01	            , 2: U12^#(tt(), V1, V2) ->
753.12/208.01	                 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.01	                     isNatKind^#(activate(V2)),
753.12/208.01	                     activate^#(V2),
753.12/208.01	                     activate^#(V1),
753.12/208.01	                     activate^#(V2))
753.12/208.01	            , 3: U13^#(tt(), V1, V2) ->
753.12/208.01	                 c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.01	                     isNatKind^#(activate(V2)),
753.12/208.01	                     activate^#(V2),
753.12/208.01	                     activate^#(V1),
753.12/208.01	                     activate^#(V2))
753.12/208.01	            , 4: U14^#(tt(), V1, V2) ->
753.12/208.01	                 c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.01	                     isNat^#(activate(V1)),
753.12/208.01	                     activate^#(V1),
753.12/208.01	                     activate^#(V2))
753.12/208.01	            , 5: U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.01	            , 6: isNat^#(n__plus(V1, V2)) ->
753.12/208.01	                 c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.01	                      isNatKind^#(activate(V1)),
753.12/208.01	                      activate^#(V1),
753.12/208.01	                      activate^#(V1),
753.12/208.01	                      activate^#(V2))
753.12/208.01	            , 7: isNat^#(n__s(V1)) ->
753.12/208.01	                 c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.01	                      isNatKind^#(activate(V1)),
753.12/208.01	                      activate^#(V1),
753.12/208.01	                      activate^#(V1))
753.12/208.01	            , 8: U21^#(tt(), V1) ->
753.12/208.01	                 c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.01	                      isNatKind^#(activate(V1)),
753.12/208.01	                      activate^#(V1),
753.12/208.01	                      activate^#(V1))
753.12/208.01	            , 9: U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.01	            , 10: U51^#(tt(), N) ->
753.12/208.01	                  c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.01	                       isNatKind^#(activate(N)),
753.12/208.01	                       activate^#(N),
753.12/208.01	                       activate^#(N))
753.12/208.01	            , 11: U52^#(tt(), N) -> c_16(activate^#(N))
753.12/208.01	            , 12: U61^#(tt(), M, N) ->
753.12/208.01	                  c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.01	                       isNatKind^#(activate(M)),
753.12/208.01	                       activate^#(M),
753.12/208.01	                       activate^#(M),
753.12/208.01	                       activate^#(N))
753.12/208.01	            , 13: U62^#(tt(), M, N) ->
753.12/208.01	                  c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.01	                       isNat^#(activate(N)),
753.12/208.01	                       activate^#(N),
753.12/208.01	                       activate^#(M),
753.12/208.01	                       activate^#(N))
753.12/208.01	            , 14: U63^#(tt(), M, N) ->
753.12/208.01	                  c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.01	                       isNatKind^#(activate(N)),
753.12/208.01	                       activate^#(N),
753.12/208.01	                       activate^#(M),
753.12/208.01	                       activate^#(N))
753.12/208.01	            , 15: U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.01	        
753.12/208.01	        We are left with following problem, upon which TcT provides the
753.12/208.01	        certificate YES(O(1),O(n^1)).
753.12/208.01	        
753.12/208.01	        Strict DPs:
753.12/208.01	          { U11^#(tt(), V1, V2) ->
753.12/208.01	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.01	                isNatKind^#(activate(V1)),
753.12/208.01	                activate^#(V1),
753.12/208.01	                activate^#(V1),
753.12/208.01	                activate^#(V2))
753.12/208.01	          , U12^#(tt(), V1, V2) ->
753.12/208.01	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.01	                isNatKind^#(activate(V2)),
753.12/208.01	                activate^#(V2),
753.12/208.01	                activate^#(V1),
753.12/208.01	                activate^#(V2))
753.12/208.01	          , U13^#(tt(), V1, V2) ->
753.12/208.01	            c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.01	                isNatKind^#(activate(V2)),
753.12/208.01	                activate^#(V2),
753.12/208.01	                activate^#(V1),
753.12/208.01	                activate^#(V2))
753.12/208.01	          , U14^#(tt(), V1, V2) ->
753.12/208.01	            c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.01	                isNat^#(activate(V1)),
753.12/208.01	                activate^#(V1),
753.12/208.01	                activate^#(V2))
753.12/208.01	          , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.01	          , isNat^#(n__plus(V1, V2)) ->
753.12/208.01	            c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.01	                 isNatKind^#(activate(V1)),
753.12/208.01	                 activate^#(V1),
753.12/208.01	                 activate^#(V1),
753.12/208.01	                 activate^#(V2))
753.12/208.01	          , isNat^#(n__s(V1)) ->
753.12/208.01	            c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.01	                 isNatKind^#(activate(V1)),
753.12/208.01	                 activate^#(V1),
753.12/208.01	                 activate^#(V1))
753.12/208.01	          , U21^#(tt(), V1) ->
753.12/208.01	            c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.01	                 isNatKind^#(activate(V1)),
753.12/208.01	                 activate^#(V1),
753.12/208.01	                 activate^#(V1))
753.12/208.01	          , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.01	          , U51^#(tt(), N) ->
753.12/208.01	            c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.01	                 isNatKind^#(activate(N)),
753.12/208.01	                 activate^#(N),
753.12/208.01	                 activate^#(N))
753.12/208.01	          , U61^#(tt(), M, N) ->
753.12/208.01	            c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.01	                 isNatKind^#(activate(M)),
753.12/208.01	                 activate^#(M),
753.12/208.01	                 activate^#(M),
753.12/208.01	                 activate^#(N))
753.12/208.01	          , U62^#(tt(), M, N) ->
753.12/208.01	            c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.01	                 isNat^#(activate(N)),
753.12/208.01	                 activate^#(N),
753.12/208.01	                 activate^#(M),
753.12/208.01	                 activate^#(N))
753.12/208.01	          , U63^#(tt(), M, N) ->
753.12/208.01	            c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.01	                 isNatKind^#(activate(N)),
753.12/208.01	                 activate^#(N),
753.12/208.01	                 activate^#(M),
753.12/208.01	                 activate^#(N)) }
753.12/208.01	        Weak DPs:
753.12/208.01	          { U52^#(tt(), N) -> c_16(activate^#(N))
753.12/208.01	          , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.01	        Weak Trs:
753.12/208.01	          { U11(tt(), V1, V2) ->
753.12/208.01	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.01	          , U12(tt(), V1, V2) ->
753.12/208.01	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.01	          , isNatKind(n__0()) -> tt()
753.12/208.01	          , isNatKind(n__plus(V1, V2)) ->
753.12/208.01	            U31(isNatKind(activate(V1)), activate(V2))
753.12/208.01	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.01	          , activate(X) -> X
753.12/208.01	          , activate(n__0()) -> 0()
753.12/208.01	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.01	          , activate(n__s(X)) -> s(activate(X))
753.12/208.01	          , U13(tt(), V1, V2) ->
753.12/208.01	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.01	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.01	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.01	          , isNat(n__0()) -> tt()
753.12/208.01	          , isNat(n__plus(V1, V2)) ->
753.12/208.01	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.01	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.01	          , U16(tt()) -> tt()
753.12/208.01	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.01	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.01	          , U23(tt()) -> tt()
753.12/208.01	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.01	          , U32(tt()) -> tt()
753.12/208.01	          , U41(tt()) -> tt()
753.12/208.01	          , s(X) -> n__s(X)
753.12/208.01	          , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.01	          , 0() -> n__0() }
753.12/208.01	        Obligation:
753.12/208.01	          innermost runtime complexity
753.12/208.01	        Answer:
753.12/208.01	          YES(O(1),O(n^1))
753.12/208.01	        
753.12/208.01	        We estimate the number of application of {10,13} by applications of
753.12/208.01	        Pre({10,13}) = {12}. Here rules are labeled as follows:
753.12/208.01	        
753.12/208.01	          DPs:
753.12/208.01	            { 1: U11^#(tt(), V1, V2) ->
753.12/208.01	                 c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.01	                     isNatKind^#(activate(V1)),
753.12/208.01	                     activate^#(V1),
753.12/208.01	                     activate^#(V1),
753.12/208.01	                     activate^#(V2))
753.12/208.01	            , 2: U12^#(tt(), V1, V2) ->
753.12/208.01	                 c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.01	                     isNatKind^#(activate(V2)),
753.12/208.01	                     activate^#(V2),
753.12/208.01	                     activate^#(V1),
753.12/208.01	                     activate^#(V2))
753.12/208.01	            , 3: U13^#(tt(), V1, V2) ->
753.12/208.01	                 c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.01	                     isNatKind^#(activate(V2)),
753.12/208.01	                     activate^#(V2),
753.12/208.01	                     activate^#(V1),
753.12/208.01	                     activate^#(V2))
753.12/208.01	            , 4: U14^#(tt(), V1, V2) ->
753.12/208.01	                 c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.01	                     isNat^#(activate(V1)),
753.12/208.02	                     activate^#(V1),
753.12/208.02	                     activate^#(V2))
753.12/208.02	            , 5: U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.02	            , 6: isNat^#(n__plus(V1, V2)) ->
753.12/208.02	                 c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.02	                      isNatKind^#(activate(V1)),
753.12/208.02	                      activate^#(V1),
753.12/208.02	                      activate^#(V1),
753.12/208.02	                      activate^#(V2))
753.12/208.02	            , 7: isNat^#(n__s(V1)) ->
753.12/208.02	                 c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.02	                      isNatKind^#(activate(V1)),
753.12/208.02	                      activate^#(V1),
753.12/208.02	                      activate^#(V1))
753.12/208.02	            , 8: U21^#(tt(), V1) ->
753.12/208.02	                 c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.02	                      isNatKind^#(activate(V1)),
753.12/208.02	                      activate^#(V1),
753.12/208.02	                      activate^#(V1))
753.12/208.02	            , 9: U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.02	            , 10: U51^#(tt(), N) ->
753.12/208.02	                  c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.02	                       isNatKind^#(activate(N)),
753.12/208.02	                       activate^#(N),
753.12/208.02	                       activate^#(N))
753.12/208.02	            , 11: U61^#(tt(), M, N) ->
753.12/208.02	                  c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.02	                       isNatKind^#(activate(M)),
753.12/208.02	                       activate^#(M),
753.12/208.02	                       activate^#(M),
753.12/208.02	                       activate^#(N))
753.12/208.02	            , 12: U62^#(tt(), M, N) ->
753.12/208.02	                  c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.02	                       isNat^#(activate(N)),
753.12/208.02	                       activate^#(N),
753.12/208.02	                       activate^#(M),
753.12/208.02	                       activate^#(N))
753.12/208.02	            , 13: U63^#(tt(), M, N) ->
753.12/208.02	                  c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.02	                       isNatKind^#(activate(N)),
753.12/208.02	                       activate^#(N),
753.12/208.02	                       activate^#(M),
753.12/208.02	                       activate^#(N))
753.12/208.02	            , 14: U52^#(tt(), N) -> c_16(activate^#(N))
753.12/208.02	            , 15: U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.02	        
753.12/208.02	        We are left with following problem, upon which TcT provides the
753.12/208.02	        certificate YES(O(1),O(n^1)).
753.12/208.02	        
753.12/208.02	        Strict DPs:
753.12/208.02	          { U11^#(tt(), V1, V2) ->
753.12/208.02	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.02	                isNatKind^#(activate(V1)),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U12^#(tt(), V1, V2) ->
753.12/208.02	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.02	                isNatKind^#(activate(V2)),
753.12/208.02	                activate^#(V2),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U13^#(tt(), V1, V2) ->
753.12/208.02	            c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.02	                isNatKind^#(activate(V2)),
753.12/208.02	                activate^#(V2),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U14^#(tt(), V1, V2) ->
753.12/208.02	            c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.02	                isNat^#(activate(V1)),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.02	          , isNat^#(n__plus(V1, V2)) ->
753.12/208.02	            c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.02	                 isNatKind^#(activate(V1)),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V2))
753.12/208.02	          , isNat^#(n__s(V1)) ->
753.12/208.02	            c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.02	                 isNatKind^#(activate(V1)),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V1))
753.12/208.02	          , U21^#(tt(), V1) ->
753.12/208.02	            c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.02	                 isNatKind^#(activate(V1)),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V1))
753.12/208.02	          , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.02	          , U61^#(tt(), M, N) ->
753.12/208.02	            c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.02	                 isNatKind^#(activate(M)),
753.12/208.02	                 activate^#(M),
753.12/208.02	                 activate^#(M),
753.12/208.02	                 activate^#(N))
753.12/208.02	          , U62^#(tt(), M, N) ->
753.12/208.02	            c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.02	                 isNat^#(activate(N)),
753.12/208.02	                 activate^#(N),
753.12/208.02	                 activate^#(M),
753.12/208.02	                 activate^#(N)) }
753.12/208.02	        Weak DPs:
753.12/208.02	          { U51^#(tt(), N) ->
753.12/208.02	            c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.02	                 isNatKind^#(activate(N)),
753.12/208.02	                 activate^#(N),
753.12/208.02	                 activate^#(N))
753.12/208.02	          , U52^#(tt(), N) -> c_16(activate^#(N))
753.12/208.02	          , U63^#(tt(), M, N) ->
753.12/208.02	            c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.02	                 isNatKind^#(activate(N)),
753.12/208.02	                 activate^#(N),
753.12/208.02	                 activate^#(M),
753.12/208.02	                 activate^#(N))
753.12/208.02	          , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.02	        Weak Trs:
753.12/208.02	          { U11(tt(), V1, V2) ->
753.12/208.02	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.02	          , U12(tt(), V1, V2) ->
753.12/208.02	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.02	          , isNatKind(n__0()) -> tt()
753.12/208.02	          , isNatKind(n__plus(V1, V2)) ->
753.12/208.02	            U31(isNatKind(activate(V1)), activate(V2))
753.12/208.02	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.02	          , activate(X) -> X
753.12/208.02	          , activate(n__0()) -> 0()
753.12/208.02	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.02	          , activate(n__s(X)) -> s(activate(X))
753.12/208.02	          , U13(tt(), V1, V2) ->
753.12/208.02	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.02	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.02	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.02	          , isNat(n__0()) -> tt()
753.12/208.02	          , isNat(n__plus(V1, V2)) ->
753.12/208.02	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.02	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.02	          , U16(tt()) -> tt()
753.12/208.02	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.02	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.02	          , U23(tt()) -> tt()
753.12/208.02	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.02	          , U32(tt()) -> tt()
753.12/208.02	          , U41(tt()) -> tt()
753.12/208.02	          , s(X) -> n__s(X)
753.12/208.02	          , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.02	          , 0() -> n__0() }
753.12/208.02	        Obligation:
753.12/208.02	          innermost runtime complexity
753.12/208.02	        Answer:
753.12/208.02	          YES(O(1),O(n^1))
753.12/208.02	        
753.12/208.02	        The following weak DPs constitute a sub-graph of the DG that is
753.12/208.02	        closed under successors. The DPs are removed.
753.12/208.02	        
753.12/208.02	        { U51^#(tt(), N) ->
753.12/208.02	          c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.02	               isNatKind^#(activate(N)),
753.12/208.02	               activate^#(N),
753.12/208.02	               activate^#(N))
753.12/208.02	        , U52^#(tt(), N) -> c_16(activate^#(N))
753.12/208.02	        , U63^#(tt(), M, N) ->
753.12/208.02	          c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.02	               isNatKind^#(activate(N)),
753.12/208.02	               activate^#(N),
753.12/208.02	               activate^#(M),
753.12/208.02	               activate^#(N))
753.12/208.02	        , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.02	        
753.12/208.02	        We are left with following problem, upon which TcT provides the
753.12/208.02	        certificate YES(O(1),O(n^1)).
753.12/208.02	        
753.12/208.02	        Strict DPs:
753.12/208.02	          { U11^#(tt(), V1, V2) ->
753.12/208.02	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.02	                isNatKind^#(activate(V1)),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U12^#(tt(), V1, V2) ->
753.12/208.02	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.02	                isNatKind^#(activate(V2)),
753.12/208.02	                activate^#(V2),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U13^#(tt(), V1, V2) ->
753.12/208.02	            c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.02	                isNatKind^#(activate(V2)),
753.12/208.02	                activate^#(V2),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U14^#(tt(), V1, V2) ->
753.12/208.02	            c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.02	                isNat^#(activate(V1)),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.02	          , isNat^#(n__plus(V1, V2)) ->
753.12/208.02	            c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.02	                 isNatKind^#(activate(V1)),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V2))
753.12/208.02	          , isNat^#(n__s(V1)) ->
753.12/208.02	            c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.02	                 isNatKind^#(activate(V1)),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V1))
753.12/208.02	          , U21^#(tt(), V1) ->
753.12/208.02	            c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.02	                 isNatKind^#(activate(V1)),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V1))
753.12/208.02	          , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.02	          , U61^#(tt(), M, N) ->
753.12/208.02	            c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.02	                 isNatKind^#(activate(M)),
753.12/208.02	                 activate^#(M),
753.12/208.02	                 activate^#(M),
753.12/208.02	                 activate^#(N))
753.12/208.02	          , U62^#(tt(), M, N) ->
753.12/208.02	            c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.02	                 isNat^#(activate(N)),
753.12/208.02	                 activate^#(N),
753.12/208.02	                 activate^#(M),
753.12/208.02	                 activate^#(N)) }
753.12/208.02	        Weak Trs:
753.12/208.02	          { U11(tt(), V1, V2) ->
753.12/208.02	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.02	          , U12(tt(), V1, V2) ->
753.12/208.02	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.02	          , isNatKind(n__0()) -> tt()
753.12/208.02	          , isNatKind(n__plus(V1, V2)) ->
753.12/208.02	            U31(isNatKind(activate(V1)), activate(V2))
753.12/208.02	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.02	          , activate(X) -> X
753.12/208.02	          , activate(n__0()) -> 0()
753.12/208.02	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.02	          , activate(n__s(X)) -> s(activate(X))
753.12/208.02	          , U13(tt(), V1, V2) ->
753.12/208.02	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.02	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.02	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.02	          , isNat(n__0()) -> tt()
753.12/208.02	          , isNat(n__plus(V1, V2)) ->
753.12/208.02	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.02	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.02	          , U16(tt()) -> tt()
753.12/208.02	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.02	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.02	          , U23(tt()) -> tt()
753.12/208.02	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.02	          , U32(tt()) -> tt()
753.12/208.02	          , U41(tt()) -> tt()
753.12/208.02	          , s(X) -> n__s(X)
753.12/208.02	          , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.02	          , 0() -> n__0() }
753.12/208.02	        Obligation:
753.12/208.02	          innermost runtime complexity
753.12/208.02	        Answer:
753.12/208.02	          YES(O(1),O(n^1))
753.12/208.02	        
753.12/208.02	        Due to missing edges in the dependency-graph, the right-hand sides
753.12/208.02	        of following rules could be simplified:
753.12/208.02	        
753.12/208.02	          { U11^#(tt(), V1, V2) ->
753.12/208.02	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.02	                isNatKind^#(activate(V1)),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U12^#(tt(), V1, V2) ->
753.12/208.02	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.02	                isNatKind^#(activate(V2)),
753.12/208.02	                activate^#(V2),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U13^#(tt(), V1, V2) ->
753.12/208.02	            c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.02	                isNatKind^#(activate(V2)),
753.12/208.02	                activate^#(V2),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U14^#(tt(), V1, V2) ->
753.12/208.02	            c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.02	                isNat^#(activate(V1)),
753.12/208.02	                activate^#(V1),
753.12/208.02	                activate^#(V2))
753.12/208.02	          , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.02	          , isNat^#(n__plus(V1, V2)) ->
753.12/208.02	            c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.02	                 isNatKind^#(activate(V1)),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V2))
753.12/208.02	          , isNat^#(n__s(V1)) ->
753.12/208.02	            c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.02	                 isNatKind^#(activate(V1)),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V1))
753.12/208.02	          , U21^#(tt(), V1) ->
753.12/208.02	            c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.02	                 isNatKind^#(activate(V1)),
753.12/208.02	                 activate^#(V1),
753.12/208.02	                 activate^#(V1))
753.12/208.02	          , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.02	          , U61^#(tt(), M, N) ->
753.12/208.02	            c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.02	                 isNatKind^#(activate(M)),
753.12/208.02	                 activate^#(M),
753.12/208.02	                 activate^#(M),
753.12/208.02	                 activate^#(N))
753.12/208.02	          , U62^#(tt(), M, N) ->
753.12/208.02	            c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.02	                 isNat^#(activate(N)),
753.12/208.02	                 activate^#(N),
753.12/208.02	                 activate^#(M),
753.12/208.02	                 activate^#(N)) }
753.12/208.02	        
753.12/208.02	        We are left with following problem, upon which TcT provides the
753.12/208.02	        certificate YES(O(1),O(n^1)).
753.12/208.02	        
753.12/208.02	        Strict DPs:
753.12/208.02	          { U11^#(tt(), V1, V2) ->
753.12/208.02	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.02	          , U12^#(tt(), V1, V2) ->
753.12/208.02	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.02	          , U13^#(tt(), V1, V2) ->
753.12/208.02	            c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.02	          , U14^#(tt(), V1, V2) ->
753.12/208.02	            c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.02	                isNat^#(activate(V1)))
753.12/208.02	          , U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.12/208.02	          , isNat^#(n__plus(V1, V2)) ->
753.12/208.02	            c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.02	          , isNat^#(n__s(V1)) ->
753.12/208.02	            c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.02	          , U21^#(tt(), V1) ->
753.12/208.02	            c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.02	          , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.12/208.02	          , U61^#(tt(), M, N) ->
753.12/208.02	            c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.12/208.02	          , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.12/208.02	        Weak Trs:
753.12/208.02	          { U11(tt(), V1, V2) ->
753.12/208.02	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.02	          , U12(tt(), V1, V2) ->
753.12/208.02	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.02	          , isNatKind(n__0()) -> tt()
753.12/208.02	          , isNatKind(n__plus(V1, V2)) ->
753.12/208.02	            U31(isNatKind(activate(V1)), activate(V2))
753.12/208.02	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.02	          , activate(X) -> X
753.12/208.02	          , activate(n__0()) -> 0()
753.12/208.02	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.02	          , activate(n__s(X)) -> s(activate(X))
753.12/208.02	          , U13(tt(), V1, V2) ->
753.12/208.02	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.02	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.02	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.02	          , isNat(n__0()) -> tt()
753.12/208.02	          , isNat(n__plus(V1, V2)) ->
753.12/208.02	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.02	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.02	          , U16(tt()) -> tt()
753.12/208.02	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.02	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.02	          , U23(tt()) -> tt()
753.12/208.02	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.02	          , U32(tt()) -> tt()
753.12/208.02	          , U41(tt()) -> tt()
753.12/208.02	          , s(X) -> n__s(X)
753.12/208.02	          , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.02	          , 0() -> n__0() }
753.12/208.02	        Obligation:
753.12/208.02	          innermost runtime complexity
753.12/208.02	        Answer:
753.12/208.02	          YES(O(1),O(n^1))
753.12/208.02	        
753.12/208.02	        We use the processor 'matrix interpretation of dimension 1' to
753.12/208.02	        orient following rules strictly.
753.12/208.02	        
753.12/208.02	        DPs:
753.12/208.02	          { 1: U11^#(tt(), V1, V2) ->
753.12/208.02	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.02	          , 10: U61^#(tt(), M, N) ->
753.12/208.02	                c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N))) }
753.12/208.02	        
753.12/208.02	        Sub-proof:
753.12/208.02	        ----------
753.12/208.02	          The following argument positions are usable:
753.12/208.02	            Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1},
753.12/208.02	            Uargs(c_4) = {1, 2}, Uargs(c_5) = {1}, Uargs(c_6) = {1},
753.12/208.02	            Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1},
753.12/208.02	            Uargs(c_10) = {1}, Uargs(c_11) = {1}
753.12/208.02	          
753.12/208.02	          TcT has computed the following constructor-based matrix
753.12/208.02	          interpretation satisfying not(EDA).
753.12/208.02	          
753.12/208.02	                     [U11](x1, x2, x3) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                                  [tt] = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                     [U12](x1, x2, x3) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                       [isNatKind](x1) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                        [activate](x1) = [1] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                     [U13](x1, x2, x3) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                     [U14](x1, x2, x3) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                         [U15](x1, x2) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                           [isNat](x1) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                             [U16](x1) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                         [U21](x1, x2) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                         [U22](x1, x2) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                             [U23](x1) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                         [U31](x1, x2) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                             [U32](x1) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                             [U41](x1) = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                               [s](x1) = [1] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                        [plus](x1, x2) = [1] x1 + [1] x2 + [2]                           
753.12/208.02	                                                                                         
753.12/208.02	                                [n__0] = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                     [n__plus](x1, x2) = [1] x1 + [1] x2 + [2]                           
753.12/208.02	                                                                                         
753.12/208.02	                            [n__s](x1) = [1] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                                   [0] = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                   [U11^#](x1, x2, x3) = [2] x2 + [2] x3 + [4]                           
753.12/208.02	                                                                                         
753.12/208.02	                   [U12^#](x1, x2, x3) = [2] x2 + [2] x3 + [0]                           
753.12/208.02	                                                                                         
753.12/208.02	                     [isNatKind^#](x1) = [7] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                      [activate^#](x1) = [7] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                   [U13^#](x1, x2, x3) = [2] x2 + [2] x3 + [0]                           
753.12/208.02	                                                                                         
753.12/208.02	                   [U14^#](x1, x2, x3) = [2] x2 + [2] x3 + [0]                           
753.12/208.02	                                                                                         
753.12/208.02	                       [U15^#](x1, x2) = [2] x2 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                         [isNat^#](x1) = [2] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                       [U21^#](x1, x2) = [2] x2 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                       [U22^#](x1, x2) = [2] x2 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                       [U51^#](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.02	                                                                                         
753.12/208.02	                       [U52^#](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.02	                                                                                         
753.12/208.02	                   [U61^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [7]                  
753.12/208.02	                                                                                         
753.12/208.02	                   [U62^#](x1, x2, x3) = [4] x3 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                   [U63^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0]                  
753.12/208.02	                                                                                         
753.12/208.02	                   [U64^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0]                  
753.12/208.02	                                                                                         
753.12/208.02	             [c_1](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.02	                                                                                         
753.12/208.02	             [c_2](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.02	                                                                                         
753.12/208.02	             [c_7](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.02	                                                                                         
753.12/208.02	                 [c_9](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.02	                                                                                         
753.12/208.02	                        [c_10](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.02	                                                                                         
753.12/208.02	            [c_11](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.02	                                                                                         
753.12/208.02	                [c_12](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.02	                                                                                         
753.12/208.02	                [c_13](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.02	                                                                                         
753.12/208.02	                        [c_14](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.02	                                                                                         
753.12/208.02	                [c_15](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.02	                                                                                         
753.12/208.02	                            [c_16](x1) = [7] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	            [c_17](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.02	                                                                                         
753.12/208.02	            [c_18](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.02	                                                                                         
753.12/208.02	            [c_19](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.02	                                                                                         
753.12/208.02	                        [c_20](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.02	                                                                                         
753.12/208.02	                                   [c] = [0]                                             
753.12/208.02	                                                                                         
753.12/208.02	                             [c_1](x1) = [1] x1 + [1]                                    
753.12/208.02	                                                                                         
753.12/208.02	                             [c_2](x1) = [1] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                             [c_3](x1) = [1] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                         [c_4](x1, x2) = [1] x1 + [1] x2 + [0]                           
753.12/208.02	                                                                                         
753.12/208.02	                             [c_5](x1) = [1] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                             [c_6](x1) = [1] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                             [c_7](x1) = [1] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                             [c_8](x1) = [1] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                             [c_9](x1) = [1] x1 + [0]                                    
753.12/208.02	                                                                                         
753.12/208.02	                            [c_10](x1) = [1] x1 + [3]                                    
753.12/208.02	                                                                                         
753.12/208.02	                            [c_11](x1) = [1] x1 + [0]                                    
753.12/208.02	          
753.12/208.02	          The order satisfies the following ordering constraints:
753.12/208.02	          
753.12/208.02	                     [U11(tt(), V1, V2)] =  [0]                                                              
753.12/208.02	                                         >= [0]                                                              
753.12/208.02	                                         =  [U12(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.12/208.02	                                                                                                             
753.12/208.02	                     [U12(tt(), V1, V2)] =  [0]                                                              
753.12/208.02	                                         >= [0]                                                              
753.12/208.02	                                         =  [U13(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.12/208.02	                                                                                                             
753.12/208.02	                     [isNatKind(n__0())] =  [0]                                                              
753.12/208.02	                                         >= [0]                                                              
753.12/208.02	                                         =  [tt()]                                                           
753.12/208.02	                                                                                                             
753.12/208.02	            [isNatKind(n__plus(V1, V2))] =  [0]                                                              
753.12/208.02	                                         >= [0]                                                              
753.12/208.02	                                         =  [U31(isNatKind(activate(V1)), activate(V2))]                     
753.12/208.02	                                                                                                             
753.12/208.02	                   [isNatKind(n__s(V1))] =  [0]                                                              
753.12/208.02	                                         >= [0]                                                              
753.12/208.02	                                         =  [U41(isNatKind(activate(V1)))]                                   
753.12/208.02	                                                                                                             
753.12/208.02	                           [activate(X)] =  [1] X + [0]                                                      
753.12/208.02	                                         >= [1] X + [0]                                                      
753.12/208.02	                                         =  [X]                                                              
753.12/208.02	                                                                                                             
753.12/208.02	                      [activate(n__0())] =  [0]                                                              
753.12/208.02	                                         >= [0]                                                              
753.12/208.02	                                         =  [0()]                                                            
753.12/208.02	                                                                                                             
753.12/208.02	             [activate(n__plus(X1, X2))] =  [1] X1 + [1] X2 + [2]                                            
753.12/208.02	                                         >= [1] X1 + [1] X2 + [2]                                            
753.12/208.03	                                         =  [plus(activate(X1), activate(X2))]                               
753.12/208.03	                                                                                                             
753.12/208.03	                     [activate(n__s(X))] =  [1] X + [0]                                                      
753.12/208.03	                                         >= [1] X + [0]                                                      
753.12/208.03	                                         =  [s(activate(X))]                                                 
753.12/208.03	                                                                                                             
753.12/208.03	                     [U13(tt(), V1, V2)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U14(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.12/208.03	                                                                                                             
753.12/208.03	                     [U14(tt(), V1, V2)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U15(isNat(activate(V1)), activate(V2))]                         
753.12/208.03	                                                                                                             
753.12/208.03	                         [U15(tt(), V2)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U16(isNat(activate(V2)))]                                       
753.12/208.03	                                                                                                             
753.12/208.03	                         [isNat(n__0())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [tt()]                                                           
753.12/208.03	                                                                                                             
753.12/208.03	                [isNat(n__plus(V1, V2))] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U11(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.12/208.03	                                                                                                             
753.12/208.03	                       [isNat(n__s(V1))] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U21(isNatKind(activate(V1)), activate(V1))]                     
753.12/208.03	                                                                                                             
753.12/208.03	                             [U16(tt())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [tt()]                                                           
753.12/208.03	                                                                                                             
753.12/208.03	                         [U21(tt(), V1)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U22(isNatKind(activate(V1)), activate(V1))]                     
753.12/208.03	                                                                                                             
753.12/208.03	                         [U22(tt(), V1)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U23(isNat(activate(V1)))]                                       
753.12/208.03	                                                                                                             
753.12/208.03	                             [U23(tt())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [tt()]                                                           
753.12/208.03	                                                                                                             
753.12/208.03	                         [U31(tt(), V2)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U32(isNatKind(activate(V2)))]                                   
753.12/208.03	                                                                                                             
753.12/208.03	                             [U32(tt())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [tt()]                                                           
753.12/208.03	                                                                                                             
753.12/208.03	                             [U41(tt())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [tt()]                                                           
753.12/208.03	                                                                                                             
753.12/208.03	                                  [s(X)] =  [1] X + [0]                                                      
753.12/208.03	                                         >= [1] X + [0]                                                      
753.12/208.03	                                         =  [n__s(X)]                                                        
753.12/208.03	                                                                                                             
753.12/208.03	                          [plus(X1, X2)] =  [1] X1 + [1] X2 + [2]                                            
753.12/208.03	                                         >= [1] X1 + [1] X2 + [2]                                            
753.12/208.03	                                         =  [n__plus(X1, X2)]                                                
753.12/208.03	                                                                                                             
753.12/208.03	                                   [0()] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [n__0()]                                                         
753.12/208.03	                                                                                                             
753.12/208.03	                   [U11^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [4]                                            
753.12/208.03	                                         >  [2] V1 + [2] V2 + [1]                                            
753.12/208.03	                                         =  [c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.12/208.03	                                                                                                             
753.12/208.03	                   [U12^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                            
753.12/208.03	                                         >= [2] V1 + [2] V2 + [0]                                            
753.12/208.03	                                         =  [c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.12/208.03	                                                                                                             
753.12/208.03	                   [U13^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                            
753.12/208.03	                                         >= [2] V1 + [2] V2 + [0]                                            
753.12/208.03	                                         =  [c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.12/208.03	                                                                                                             
753.12/208.03	                   [U14^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                            
753.12/208.03	                                         >= [2] V1 + [2] V2 + [0]                                            
753.12/208.03	                                         =  [c_4(U15^#(isNat(activate(V1)), activate(V2)),                   
753.12/208.03	                                                 isNat^#(activate(V1)))]                                     
753.12/208.03	                                                                                                             
753.12/208.03	                       [U15^#(tt(), V2)] =  [2] V2 + [0]                                                     
753.12/208.03	                                         >= [2] V2 + [0]                                                     
753.12/208.03	                                         =  [c_5(isNat^#(activate(V2)))]                                     
753.12/208.03	                                                                                                             
753.12/208.03	              [isNat^#(n__plus(V1, V2))] =  [2] V1 + [2] V2 + [4]                                            
753.12/208.03	                                         >= [2] V1 + [2] V2 + [4]                                            
753.12/208.03	                                         =  [c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.12/208.03	                                                                                                             
753.12/208.03	                     [isNat^#(n__s(V1))] =  [2] V1 + [0]                                                     
753.12/208.03	                                         >= [2] V1 + [0]                                                     
753.12/208.03	                                         =  [c_7(U21^#(isNatKind(activate(V1)), activate(V1)))]              
753.12/208.03	                                                                                                             
753.12/208.03	                       [U21^#(tt(), V1)] =  [2] V1 + [0]                                                     
753.12/208.03	                                         >= [2] V1 + [0]                                                     
753.12/208.03	                                         =  [c_8(U22^#(isNatKind(activate(V1)), activate(V1)))]              
753.12/208.03	                                                                                                             
753.12/208.03	                       [U22^#(tt(), V1)] =  [2] V1 + [0]                                                     
753.12/208.03	                                         >= [2] V1 + [0]                                                     
753.12/208.03	                                         =  [c_9(isNat^#(activate(V1)))]                                     
753.12/208.03	                                                                                                             
753.12/208.03	                     [U61^#(tt(), M, N)] =  [7] N + [7] M + [7]                                              
753.12/208.03	                                         >  [4] N + [3]                                                      
753.12/208.03	                                         =  [c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))]  
753.12/208.03	                                                                                                             
753.12/208.03	                     [U62^#(tt(), M, N)] =  [4] N + [0]                                                      
753.12/208.03	                                         >= [2] N + [0]                                                      
753.12/208.03	                                         =  [c_11(isNat^#(activate(N)))]                                     
753.12/208.03	                                                                                                             
753.12/208.03	        
753.12/208.03	        We return to the main proof. Consider the set of all dependency
753.12/208.03	        pairs
753.12/208.03	        
753.12/208.03	        :
753.12/208.03	          { 1: U11^#(tt(), V1, V2) ->
753.12/208.03	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.03	          , 2: U12^#(tt(), V1, V2) ->
753.12/208.03	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.03	          , 3: U13^#(tt(), V1, V2) ->
753.12/208.03	               c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.03	          , 4: U14^#(tt(), V1, V2) ->
753.12/208.03	               c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.03	                   isNat^#(activate(V1)))
753.12/208.03	          , 5: U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.12/208.03	          , 6: isNat^#(n__plus(V1, V2)) ->
753.12/208.03	               c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.03	          , 7: isNat^#(n__s(V1)) ->
753.12/208.03	               c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.03	          , 8: U21^#(tt(), V1) ->
753.12/208.03	               c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.03	          , 9: U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.12/208.03	          , 10: U61^#(tt(), M, N) ->
753.12/208.03	                c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.12/208.03	          , 11: U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.12/208.03	        
753.12/208.03	        Processor 'matrix interpretation of dimension 1' induces the
753.12/208.03	        complexity certificate YES(?,O(n^1)) on application of dependency
753.12/208.03	        pairs {1,10}. These cover all (indirect) predecessors of dependency
753.12/208.03	        pairs {1,2,3,4,5,10,11}, their number of application is equally
753.12/208.03	        bounded. The dependency pairs are shifted into the weak component.
753.12/208.03	        
753.12/208.03	        We are left with following problem, upon which TcT provides the
753.12/208.03	        certificate YES(O(1),O(n^1)).
753.12/208.03	        
753.12/208.03	        Strict DPs:
753.12/208.03	          { isNat^#(n__plus(V1, V2)) ->
753.12/208.03	            c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.03	          , isNat^#(n__s(V1)) ->
753.12/208.03	            c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.03	          , U21^#(tt(), V1) ->
753.12/208.03	            c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.03	          , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1))) }
753.12/208.03	        Weak DPs:
753.12/208.03	          { U11^#(tt(), V1, V2) ->
753.12/208.03	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.03	          , U12^#(tt(), V1, V2) ->
753.12/208.03	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.03	          , U13^#(tt(), V1, V2) ->
753.12/208.03	            c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.03	          , U14^#(tt(), V1, V2) ->
753.12/208.03	            c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.03	                isNat^#(activate(V1)))
753.12/208.03	          , U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.12/208.03	          , U61^#(tt(), M, N) ->
753.12/208.03	            c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.12/208.03	          , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.12/208.03	        Weak Trs:
753.12/208.03	          { U11(tt(), V1, V2) ->
753.12/208.03	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.03	          , U12(tt(), V1, V2) ->
753.12/208.03	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.03	          , isNatKind(n__0()) -> tt()
753.12/208.03	          , isNatKind(n__plus(V1, V2)) ->
753.12/208.03	            U31(isNatKind(activate(V1)), activate(V2))
753.12/208.03	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.03	          , activate(X) -> X
753.12/208.03	          , activate(n__0()) -> 0()
753.12/208.03	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.03	          , activate(n__s(X)) -> s(activate(X))
753.12/208.03	          , U13(tt(), V1, V2) ->
753.12/208.03	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.03	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.03	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.03	          , isNat(n__0()) -> tt()
753.12/208.03	          , isNat(n__plus(V1, V2)) ->
753.12/208.03	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.03	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.03	          , U16(tt()) -> tt()
753.12/208.03	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.03	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.03	          , U23(tt()) -> tt()
753.12/208.03	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.03	          , U32(tt()) -> tt()
753.12/208.03	          , U41(tt()) -> tt()
753.12/208.03	          , s(X) -> n__s(X)
753.12/208.03	          , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.03	          , 0() -> n__0() }
753.12/208.03	        Obligation:
753.12/208.03	          innermost runtime complexity
753.12/208.03	        Answer:
753.12/208.03	          YES(O(1),O(n^1))
753.12/208.03	        
753.12/208.03	        We use the processor 'matrix interpretation of dimension 1' to
753.12/208.03	        orient following rules strictly.
753.12/208.03	        
753.12/208.03	        DPs:
753.12/208.03	          { 1: isNat^#(n__plus(V1, V2)) ->
753.12/208.03	               c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.03	          , 10: U61^#(tt(), M, N) ->
753.12/208.03	                c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N))) }
753.12/208.03	        
753.12/208.03	        Sub-proof:
753.12/208.03	        ----------
753.12/208.03	          The following argument positions are usable:
753.12/208.03	            Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1},
753.12/208.03	            Uargs(c_4) = {1, 2}, Uargs(c_5) = {1}, Uargs(c_6) = {1},
753.12/208.03	            Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1},
753.12/208.03	            Uargs(c_10) = {1}, Uargs(c_11) = {1}
753.12/208.03	          
753.12/208.03	          TcT has computed the following constructor-based matrix
753.12/208.03	          interpretation satisfying not(EDA).
753.12/208.03	          
753.12/208.03	                     [U11](x1, x2, x3) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                                  [tt] = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                     [U12](x1, x2, x3) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                       [isNatKind](x1) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                        [activate](x1) = [1] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                     [U13](x1, x2, x3) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                     [U14](x1, x2, x3) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                         [U15](x1, x2) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                           [isNat](x1) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                             [U16](x1) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                         [U21](x1, x2) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                         [U22](x1, x2) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                             [U23](x1) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                         [U31](x1, x2) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                             [U32](x1) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                             [U41](x1) = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                               [s](x1) = [1] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                        [plus](x1, x2) = [1] x1 + [1] x2 + [2]                           
753.12/208.03	                                                                                         
753.12/208.03	                                [n__0] = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                     [n__plus](x1, x2) = [1] x1 + [1] x2 + [2]                           
753.12/208.03	                                                                                         
753.12/208.03	                            [n__s](x1) = [1] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                                   [0] = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                   [U11^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]                           
753.12/208.03	                                                                                         
753.12/208.03	                   [U12^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]                           
753.12/208.03	                                                                                         
753.12/208.03	                     [isNatKind^#](x1) = [7] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                      [activate^#](x1) = [7] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                   [U13^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]                           
753.12/208.03	                                                                                         
753.12/208.03	                   [U14^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]                           
753.12/208.03	                                                                                         
753.12/208.03	                       [U15^#](x1, x2) = [1] x2 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                         [isNat^#](x1) = [1] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                       [U21^#](x1, x2) = [1] x2 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                       [U22^#](x1, x2) = [1] x2 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                       [U51^#](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.03	                                                                                         
753.12/208.03	                       [U52^#](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.03	                                                                                         
753.12/208.03	                   [U61^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [7]                  
753.12/208.03	                                                                                         
753.12/208.03	                   [U62^#](x1, x2, x3) = [1] x3 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                   [U63^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0]                  
753.12/208.03	                                                                                         
753.12/208.03	                   [U64^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0]                  
753.12/208.03	                                                                                         
753.12/208.03	             [c_1](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.03	                                                                                         
753.12/208.03	             [c_2](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.03	                                                                                         
753.12/208.03	             [c_7](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.03	                                                                                         
753.12/208.03	                 [c_9](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.03	                                                                                         
753.12/208.03	                        [c_10](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.03	                                                                                         
753.12/208.03	            [c_11](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.03	                                                                                         
753.12/208.03	                [c_12](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.03	                                                                                         
753.12/208.03	                [c_13](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.03	                                                                                         
753.12/208.03	                        [c_14](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.03	                                                                                         
753.12/208.03	                [c_15](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.03	                                                                                         
753.12/208.03	                            [c_16](x1) = [7] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	            [c_17](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.03	                                                                                         
753.12/208.03	            [c_18](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.03	                                                                                         
753.12/208.03	            [c_19](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.03	                                                                                         
753.12/208.03	                        [c_20](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.03	                                                                                         
753.12/208.03	                                   [c] = [0]                                             
753.12/208.03	                                                                                         
753.12/208.03	                             [c_1](x1) = [1] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                             [c_2](x1) = [1] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                             [c_3](x1) = [1] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                         [c_4](x1, x2) = [1] x1 + [1] x2 + [0]                           
753.12/208.03	                                                                                         
753.12/208.03	                             [c_5](x1) = [1] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                             [c_6](x1) = [1] x1 + [1]                                    
753.12/208.03	                                                                                         
753.12/208.03	                             [c_7](x1) = [1] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                             [c_8](x1) = [1] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                             [c_9](x1) = [1] x1 + [0]                                    
753.12/208.03	                                                                                         
753.12/208.03	                            [c_10](x1) = [4] x1 + [3]                                    
753.12/208.03	                                                                                         
753.12/208.03	                            [c_11](x1) = [1] x1 + [0]                                    
753.12/208.03	          
753.12/208.03	          The order satisfies the following ordering constraints:
753.12/208.03	          
753.12/208.03	                     [U11(tt(), V1, V2)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U12(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.12/208.03	                                                                                                             
753.12/208.03	                     [U12(tt(), V1, V2)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U13(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.12/208.03	                                                                                                             
753.12/208.03	                     [isNatKind(n__0())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [tt()]                                                           
753.12/208.03	                                                                                                             
753.12/208.03	            [isNatKind(n__plus(V1, V2))] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U31(isNatKind(activate(V1)), activate(V2))]                     
753.12/208.03	                                                                                                             
753.12/208.03	                   [isNatKind(n__s(V1))] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U41(isNatKind(activate(V1)))]                                   
753.12/208.03	                                                                                                             
753.12/208.03	                           [activate(X)] =  [1] X + [0]                                                      
753.12/208.03	                                         >= [1] X + [0]                                                      
753.12/208.03	                                         =  [X]                                                              
753.12/208.03	                                                                                                             
753.12/208.03	                      [activate(n__0())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [0()]                                                            
753.12/208.03	                                                                                                             
753.12/208.03	             [activate(n__plus(X1, X2))] =  [1] X1 + [1] X2 + [2]                                            
753.12/208.03	                                         >= [1] X1 + [1] X2 + [2]                                            
753.12/208.03	                                         =  [plus(activate(X1), activate(X2))]                               
753.12/208.03	                                                                                                             
753.12/208.03	                     [activate(n__s(X))] =  [1] X + [0]                                                      
753.12/208.03	                                         >= [1] X + [0]                                                      
753.12/208.03	                                         =  [s(activate(X))]                                                 
753.12/208.03	                                                                                                             
753.12/208.03	                     [U13(tt(), V1, V2)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U14(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.12/208.03	                                                                                                             
753.12/208.03	                     [U14(tt(), V1, V2)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U15(isNat(activate(V1)), activate(V2))]                         
753.12/208.03	                                                                                                             
753.12/208.03	                         [U15(tt(), V2)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U16(isNat(activate(V2)))]                                       
753.12/208.03	                                                                                                             
753.12/208.03	                         [isNat(n__0())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [tt()]                                                           
753.12/208.03	                                                                                                             
753.12/208.03	                [isNat(n__plus(V1, V2))] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U11(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.12/208.03	                                                                                                             
753.12/208.03	                       [isNat(n__s(V1))] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U21(isNatKind(activate(V1)), activate(V1))]                     
753.12/208.03	                                                                                                             
753.12/208.03	                             [U16(tt())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [tt()]                                                           
753.12/208.03	                                                                                                             
753.12/208.03	                         [U21(tt(), V1)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U22(isNatKind(activate(V1)), activate(V1))]                     
753.12/208.03	                                                                                                             
753.12/208.03	                         [U22(tt(), V1)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U23(isNat(activate(V1)))]                                       
753.12/208.03	                                                                                                             
753.12/208.03	                             [U23(tt())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [tt()]                                                           
753.12/208.03	                                                                                                             
753.12/208.03	                         [U31(tt(), V2)] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [U32(isNatKind(activate(V2)))]                                   
753.12/208.03	                                                                                                             
753.12/208.03	                             [U32(tt())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [tt()]                                                           
753.12/208.03	                                                                                                             
753.12/208.03	                             [U41(tt())] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [tt()]                                                           
753.12/208.03	                                                                                                             
753.12/208.03	                                  [s(X)] =  [1] X + [0]                                                      
753.12/208.03	                                         >= [1] X + [0]                                                      
753.12/208.03	                                         =  [n__s(X)]                                                        
753.12/208.03	                                                                                                             
753.12/208.03	                          [plus(X1, X2)] =  [1] X1 + [1] X2 + [2]                                            
753.12/208.03	                                         >= [1] X1 + [1] X2 + [2]                                            
753.12/208.03	                                         =  [n__plus(X1, X2)]                                                
753.12/208.03	                                                                                                             
753.12/208.03	                                   [0()] =  [0]                                                              
753.12/208.03	                                         >= [0]                                                              
753.12/208.03	                                         =  [n__0()]                                                         
753.12/208.03	                                                                                                             
753.12/208.03	                   [U11^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                            
753.12/208.03	                                         >= [1] V1 + [1] V2 + [0]                                            
753.12/208.03	                                         =  [c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.12/208.03	                                                                                                             
753.12/208.03	                   [U12^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                            
753.12/208.03	                                         >= [1] V1 + [1] V2 + [0]                                            
753.12/208.03	                                         =  [c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.12/208.03	                                                                                                             
753.12/208.03	                   [U13^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                            
753.12/208.03	                                         >= [1] V1 + [1] V2 + [0]                                            
753.12/208.03	                                         =  [c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.12/208.03	                                                                                                             
753.12/208.03	                   [U14^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                            
753.12/208.03	                                         >= [1] V1 + [1] V2 + [0]                                            
753.12/208.03	                                         =  [c_4(U15^#(isNat(activate(V1)), activate(V2)),                   
753.12/208.03	                                                 isNat^#(activate(V1)))]                                     
753.12/208.03	                                                                                                             
753.12/208.03	                       [U15^#(tt(), V2)] =  [1] V2 + [0]                                                     
753.12/208.04	                                         >= [1] V2 + [0]                                                     
753.12/208.04	                                         =  [c_5(isNat^#(activate(V2)))]                                     
753.12/208.04	                                                                                                             
753.12/208.04	              [isNat^#(n__plus(V1, V2))] =  [1] V1 + [1] V2 + [2]                                            
753.12/208.04	                                         >  [1] V1 + [1] V2 + [1]                                            
753.12/208.04	                                         =  [c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.12/208.04	                                                                                                             
753.12/208.04	                     [isNat^#(n__s(V1))] =  [1] V1 + [0]                                                     
753.12/208.04	                                         >= [1] V1 + [0]                                                     
753.12/208.04	                                         =  [c_7(U21^#(isNatKind(activate(V1)), activate(V1)))]              
753.12/208.04	                                                                                                             
753.12/208.04	                       [U21^#(tt(), V1)] =  [1] V1 + [0]                                                     
753.12/208.04	                                         >= [1] V1 + [0]                                                     
753.12/208.04	                                         =  [c_8(U22^#(isNatKind(activate(V1)), activate(V1)))]              
753.12/208.04	                                                                                                             
753.12/208.04	                       [U22^#(tt(), V1)] =  [1] V1 + [0]                                                     
753.12/208.04	                                         >= [1] V1 + [0]                                                     
753.12/208.04	                                         =  [c_9(isNat^#(activate(V1)))]                                     
753.12/208.04	                                                                                                             
753.12/208.04	                     [U61^#(tt(), M, N)] =  [7] N + [7] M + [7]                                              
753.12/208.04	                                         >  [4] N + [3]                                                      
753.12/208.04	                                         =  [c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))]  
753.12/208.04	                                                                                                             
753.12/208.04	                     [U62^#(tt(), M, N)] =  [1] N + [0]                                                      
753.12/208.04	                                         >= [1] N + [0]                                                      
753.12/208.04	                                         =  [c_11(isNat^#(activate(N)))]                                     
753.12/208.04	                                                                                                             
753.12/208.04	        
753.12/208.04	        We return to the main proof. Consider the set of all dependency
753.12/208.04	        pairs
753.12/208.04	        
753.12/208.04	        :
753.12/208.04	          { 1: isNat^#(n__plus(V1, V2)) ->
753.12/208.04	               c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.04	          , 2: isNat^#(n__s(V1)) ->
753.12/208.04	               c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.04	          , 3: U21^#(tt(), V1) ->
753.12/208.04	               c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.04	          , 4: U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.12/208.04	          , 5: U11^#(tt(), V1, V2) ->
753.12/208.04	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.04	          , 6: U12^#(tt(), V1, V2) ->
753.12/208.04	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.04	          , 7: U13^#(tt(), V1, V2) ->
753.12/208.04	               c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.04	          , 8: U14^#(tt(), V1, V2) ->
753.12/208.04	               c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.04	                   isNat^#(activate(V1)))
753.12/208.04	          , 9: U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.12/208.04	          , 10: U61^#(tt(), M, N) ->
753.12/208.04	                c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.12/208.04	          , 11: U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.12/208.04	        
753.12/208.04	        Processor 'matrix interpretation of dimension 1' induces the
753.12/208.04	        complexity certificate YES(?,O(n^1)) on application of dependency
753.12/208.04	        pairs {1,10}. These cover all (indirect) predecessors of dependency
753.12/208.04	        pairs {1,5,6,7,8,9,10,11}, their number of application is equally
753.12/208.04	        bounded. The dependency pairs are shifted into the weak component.
753.12/208.04	        
753.12/208.04	        We are left with following problem, upon which TcT provides the
753.12/208.04	        certificate YES(O(1),O(n^1)).
753.12/208.04	        
753.12/208.04	        Strict DPs:
753.12/208.04	          { isNat^#(n__s(V1)) ->
753.12/208.04	            c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.04	          , U21^#(tt(), V1) ->
753.12/208.04	            c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.04	          , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1))) }
753.12/208.04	        Weak DPs:
753.12/208.04	          { U11^#(tt(), V1, V2) ->
753.12/208.04	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.04	          , U12^#(tt(), V1, V2) ->
753.12/208.04	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.04	          , U13^#(tt(), V1, V2) ->
753.12/208.04	            c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.04	          , U14^#(tt(), V1, V2) ->
753.12/208.04	            c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.04	                isNat^#(activate(V1)))
753.12/208.04	          , U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.12/208.04	          , isNat^#(n__plus(V1, V2)) ->
753.12/208.04	            c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.04	          , U61^#(tt(), M, N) ->
753.12/208.04	            c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.12/208.04	          , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.12/208.04	        Weak Trs:
753.12/208.04	          { U11(tt(), V1, V2) ->
753.12/208.04	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.04	          , U12(tt(), V1, V2) ->
753.12/208.04	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.04	          , isNatKind(n__0()) -> tt()
753.12/208.04	          , isNatKind(n__plus(V1, V2)) ->
753.12/208.04	            U31(isNatKind(activate(V1)), activate(V2))
753.12/208.04	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.04	          , activate(X) -> X
753.12/208.04	          , activate(n__0()) -> 0()
753.12/208.04	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.04	          , activate(n__s(X)) -> s(activate(X))
753.12/208.04	          , U13(tt(), V1, V2) ->
753.12/208.04	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.04	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.04	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.04	          , isNat(n__0()) -> tt()
753.12/208.04	          , isNat(n__plus(V1, V2)) ->
753.12/208.04	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.04	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.04	          , U16(tt()) -> tt()
753.12/208.04	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.04	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.04	          , U23(tt()) -> tt()
753.12/208.04	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.04	          , U32(tt()) -> tt()
753.12/208.04	          , U41(tt()) -> tt()
753.12/208.04	          , s(X) -> n__s(X)
753.12/208.04	          , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.04	          , 0() -> n__0() }
753.12/208.04	        Obligation:
753.12/208.04	          innermost runtime complexity
753.12/208.04	        Answer:
753.12/208.04	          YES(O(1),O(n^1))
753.12/208.04	        
753.12/208.04	        We use the processor 'matrix interpretation of dimension 1' to
753.12/208.04	        orient following rules strictly.
753.12/208.04	        
753.12/208.04	        DPs:
753.12/208.04	          { 2: U21^#(tt(), V1) ->
753.12/208.04	               c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.04	          , 10: U61^#(tt(), M, N) ->
753.12/208.04	                c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N))) }
753.12/208.04	        
753.12/208.04	        Sub-proof:
753.12/208.04	        ----------
753.12/208.04	          The following argument positions are usable:
753.12/208.04	            Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1},
753.12/208.04	            Uargs(c_4) = {1, 2}, Uargs(c_5) = {1}, Uargs(c_6) = {1},
753.12/208.04	            Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1},
753.12/208.04	            Uargs(c_10) = {1}, Uargs(c_11) = {1}
753.12/208.04	          
753.12/208.04	          TcT has computed the following constructor-based matrix
753.12/208.04	          interpretation satisfying not(EDA).
753.12/208.04	          
753.12/208.04	                     [U11](x1, x2, x3) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                                  [tt] = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                     [U12](x1, x2, x3) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                       [isNatKind](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                        [activate](x1) = [1] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                     [U13](x1, x2, x3) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                     [U14](x1, x2, x3) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                         [U15](x1, x2) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                           [isNat](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                             [U16](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                         [U21](x1, x2) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                         [U22](x1, x2) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                             [U23](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                         [U31](x1, x2) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                             [U32](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                             [U41](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                               [s](x1) = [1] x1 + [1]                                    
753.12/208.04	                                                                                         
753.12/208.04	                        [plus](x1, x2) = [1] x1 + [1] x2 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                                [n__0] = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                     [n__plus](x1, x2) = [1] x1 + [1] x2 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                            [n__s](x1) = [1] x1 + [1]                                    
753.12/208.04	                                                                                         
753.12/208.04	                                   [0] = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                   [U11^#](x1, x2, x3) = [4] x2 + [4] x3 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                   [U12^#](x1, x2, x3) = [4] x2 + [4] x3 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                     [isNatKind^#](x1) = [7] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                      [activate^#](x1) = [7] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                   [U13^#](x1, x2, x3) = [4] x2 + [4] x3 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                   [U14^#](x1, x2, x3) = [4] x2 + [4] x3 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                       [U15^#](x1, x2) = [4] x2 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                         [isNat^#](x1) = [4] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                       [U21^#](x1, x2) = [4] x2 + [4]                                    
753.12/208.04	                                                                                         
753.12/208.04	                       [U22^#](x1, x2) = [4] x2 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                       [U51^#](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                       [U52^#](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                   [U61^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [7]                  
753.12/208.04	                                                                                         
753.12/208.04	                   [U62^#](x1, x2, x3) = [4] x3 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                   [U63^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0]                  
753.12/208.04	                                                                                         
753.12/208.04	                   [U64^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0]                  
753.12/208.04	                                                                                         
753.12/208.04	             [c_1](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.04	                                                                                         
753.12/208.04	             [c_2](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.04	                                                                                         
753.12/208.04	             [c_7](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.04	                                                                                         
753.12/208.04	                 [c_9](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.04	                                                                                         
753.12/208.04	                        [c_10](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	            [c_11](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.04	                                                                                         
753.12/208.04	                [c_12](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.04	                                                                                         
753.12/208.04	                [c_13](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.04	                                                                                         
753.12/208.04	                        [c_14](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                [c_15](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.04	                                                                                         
753.12/208.04	                            [c_16](x1) = [7] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	            [c_17](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.04	                                                                                         
753.12/208.04	            [c_18](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.04	                                                                                         
753.12/208.04	            [c_19](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.04	                                                                                         
753.12/208.04	                        [c_20](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                                   [c] = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                             [c_1](x1) = [1] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                             [c_2](x1) = [1] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                             [c_3](x1) = [1] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                         [c_4](x1, x2) = [1] x1 + [1] x2 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                             [c_5](x1) = [1] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                             [c_6](x1) = [1] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                             [c_7](x1) = [1] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                             [c_8](x1) = [1] x1 + [1]                                    
753.12/208.04	                                                                                         
753.12/208.04	                             [c_9](x1) = [1] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                            [c_10](x1) = [1] x1 + [3]                                    
753.12/208.04	                                                                                         
753.12/208.04	                            [c_11](x1) = [1] x1 + [0]                                    
753.12/208.04	          
753.12/208.04	          The order satisfies the following ordering constraints:
753.12/208.04	          
753.12/208.04	                     [U11(tt(), V1, V2)] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U12(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.12/208.04	                                                                                                             
753.12/208.04	                     [U12(tt(), V1, V2)] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U13(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.12/208.04	                                                                                                             
753.12/208.04	                     [isNatKind(n__0())] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [tt()]                                                           
753.12/208.04	                                                                                                             
753.12/208.04	            [isNatKind(n__plus(V1, V2))] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U31(isNatKind(activate(V1)), activate(V2))]                     
753.12/208.04	                                                                                                             
753.12/208.04	                   [isNatKind(n__s(V1))] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U41(isNatKind(activate(V1)))]                                   
753.12/208.04	                                                                                                             
753.12/208.04	                           [activate(X)] =  [1] X + [0]                                                      
753.12/208.04	                                         >= [1] X + [0]                                                      
753.12/208.04	                                         =  [X]                                                              
753.12/208.04	                                                                                                             
753.12/208.04	                      [activate(n__0())] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [0()]                                                            
753.12/208.04	                                                                                                             
753.12/208.04	             [activate(n__plus(X1, X2))] =  [1] X1 + [1] X2 + [0]                                            
753.12/208.04	                                         >= [1] X1 + [1] X2 + [0]                                            
753.12/208.04	                                         =  [plus(activate(X1), activate(X2))]                               
753.12/208.04	                                                                                                             
753.12/208.04	                     [activate(n__s(X))] =  [1] X + [1]                                                      
753.12/208.04	                                         >= [1] X + [1]                                                      
753.12/208.04	                                         =  [s(activate(X))]                                                 
753.12/208.04	                                                                                                             
753.12/208.04	                     [U13(tt(), V1, V2)] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U14(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.12/208.04	                                                                                                             
753.12/208.04	                     [U14(tt(), V1, V2)] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U15(isNat(activate(V1)), activate(V2))]                         
753.12/208.04	                                                                                                             
753.12/208.04	                         [U15(tt(), V2)] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U16(isNat(activate(V2)))]                                       
753.12/208.04	                                                                                                             
753.12/208.04	                         [isNat(n__0())] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [tt()]                                                           
753.12/208.04	                                                                                                             
753.12/208.04	                [isNat(n__plus(V1, V2))] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U11(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.12/208.04	                                                                                                             
753.12/208.04	                       [isNat(n__s(V1))] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U21(isNatKind(activate(V1)), activate(V1))]                     
753.12/208.04	                                                                                                             
753.12/208.04	                             [U16(tt())] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [tt()]                                                           
753.12/208.04	                                                                                                             
753.12/208.04	                         [U21(tt(), V1)] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U22(isNatKind(activate(V1)), activate(V1))]                     
753.12/208.04	                                                                                                             
753.12/208.04	                         [U22(tt(), V1)] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U23(isNat(activate(V1)))]                                       
753.12/208.04	                                                                                                             
753.12/208.04	                             [U23(tt())] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [tt()]                                                           
753.12/208.04	                                                                                                             
753.12/208.04	                         [U31(tt(), V2)] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [U32(isNatKind(activate(V2)))]                                   
753.12/208.04	                                                                                                             
753.12/208.04	                             [U32(tt())] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [tt()]                                                           
753.12/208.04	                                                                                                             
753.12/208.04	                             [U41(tt())] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [tt()]                                                           
753.12/208.04	                                                                                                             
753.12/208.04	                                  [s(X)] =  [1] X + [1]                                                      
753.12/208.04	                                         >= [1] X + [1]                                                      
753.12/208.04	                                         =  [n__s(X)]                                                        
753.12/208.04	                                                                                                             
753.12/208.04	                          [plus(X1, X2)] =  [1] X1 + [1] X2 + [0]                                            
753.12/208.04	                                         >= [1] X1 + [1] X2 + [0]                                            
753.12/208.04	                                         =  [n__plus(X1, X2)]                                                
753.12/208.04	                                                                                                             
753.12/208.04	                                   [0()] =  [0]                                                              
753.12/208.04	                                         >= [0]                                                              
753.12/208.04	                                         =  [n__0()]                                                         
753.12/208.04	                                                                                                             
753.12/208.04	                   [U11^#(tt(), V1, V2)] =  [4] V1 + [4] V2 + [0]                                            
753.12/208.04	                                         >= [4] V1 + [4] V2 + [0]                                            
753.12/208.04	                                         =  [c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.12/208.04	                                                                                                             
753.12/208.04	                   [U12^#(tt(), V1, V2)] =  [4] V1 + [4] V2 + [0]                                            
753.12/208.04	                                         >= [4] V1 + [4] V2 + [0]                                            
753.12/208.04	                                         =  [c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.12/208.04	                                                                                                             
753.12/208.04	                   [U13^#(tt(), V1, V2)] =  [4] V1 + [4] V2 + [0]                                            
753.12/208.04	                                         >= [4] V1 + [4] V2 + [0]                                            
753.12/208.04	                                         =  [c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.12/208.04	                                                                                                             
753.12/208.04	                   [U14^#(tt(), V1, V2)] =  [4] V1 + [4] V2 + [0]                                            
753.12/208.04	                                         >= [4] V1 + [4] V2 + [0]                                            
753.12/208.04	                                         =  [c_4(U15^#(isNat(activate(V1)), activate(V2)),                   
753.12/208.04	                                                 isNat^#(activate(V1)))]                                     
753.12/208.04	                                                                                                             
753.12/208.04	                       [U15^#(tt(), V2)] =  [4] V2 + [0]                                                     
753.12/208.04	                                         >= [4] V2 + [0]                                                     
753.12/208.04	                                         =  [c_5(isNat^#(activate(V2)))]                                     
753.12/208.04	                                                                                                             
753.12/208.04	              [isNat^#(n__plus(V1, V2))] =  [4] V1 + [4] V2 + [0]                                            
753.12/208.04	                                         >= [4] V1 + [4] V2 + [0]                                            
753.12/208.04	                                         =  [c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.12/208.04	                                                                                                             
753.12/208.04	                     [isNat^#(n__s(V1))] =  [4] V1 + [4]                                                     
753.12/208.04	                                         >= [4] V1 + [4]                                                     
753.12/208.04	                                         =  [c_7(U21^#(isNatKind(activate(V1)), activate(V1)))]              
753.12/208.04	                                                                                                             
753.12/208.04	                       [U21^#(tt(), V1)] =  [4] V1 + [4]                                                     
753.12/208.04	                                         >  [4] V1 + [1]                                                     
753.12/208.04	                                         =  [c_8(U22^#(isNatKind(activate(V1)), activate(V1)))]              
753.12/208.04	                                                                                                             
753.12/208.04	                       [U22^#(tt(), V1)] =  [4] V1 + [0]                                                     
753.12/208.04	                                         >= [4] V1 + [0]                                                     
753.12/208.04	                                         =  [c_9(isNat^#(activate(V1)))]                                     
753.12/208.04	                                                                                                             
753.12/208.04	                     [U61^#(tt(), M, N)] =  [7] N + [7] M + [7]                                              
753.12/208.04	                                         >  [4] N + [3]                                                      
753.12/208.04	                                         =  [c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))]  
753.12/208.04	                                                                                                             
753.12/208.04	                     [U62^#(tt(), M, N)] =  [4] N + [0]                                                      
753.12/208.04	                                         >= [4] N + [0]                                                      
753.12/208.04	                                         =  [c_11(isNat^#(activate(N)))]                                     
753.12/208.04	                                                                                                             
753.12/208.04	        
753.12/208.04	        We return to the main proof. Consider the set of all dependency
753.12/208.04	        pairs
753.12/208.04	        
753.12/208.04	        :
753.12/208.04	          { 1: isNat^#(n__s(V1)) ->
753.12/208.04	               c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.04	          , 2: U21^#(tt(), V1) ->
753.12/208.04	               c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.04	          , 3: U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.12/208.04	          , 4: U11^#(tt(), V1, V2) ->
753.12/208.04	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.04	          , 5: U12^#(tt(), V1, V2) ->
753.12/208.04	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.04	          , 6: U13^#(tt(), V1, V2) ->
753.12/208.04	               c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.04	          , 7: U14^#(tt(), V1, V2) ->
753.12/208.04	               c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.04	                   isNat^#(activate(V1)))
753.12/208.04	          , 8: U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.12/208.04	          , 9: isNat^#(n__plus(V1, V2)) ->
753.12/208.04	               c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.04	          , 10: U61^#(tt(), M, N) ->
753.12/208.04	                c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.12/208.04	          , 11: U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.12/208.04	        
753.12/208.04	        Processor 'matrix interpretation of dimension 1' induces the
753.12/208.04	        complexity certificate YES(?,O(n^1)) on application of dependency
753.12/208.04	        pairs {2,10}. These cover all (indirect) predecessors of dependency
753.12/208.04	        pairs {2,3,10,11}, their number of application is equally bounded.
753.12/208.04	        The dependency pairs are shifted into the weak component.
753.12/208.04	        
753.12/208.04	        We are left with following problem, upon which TcT provides the
753.12/208.04	        certificate YES(O(1),O(n^1)).
753.12/208.04	        
753.12/208.04	        Strict DPs:
753.12/208.04	          { isNat^#(n__s(V1)) ->
753.12/208.04	            c_7(U21^#(isNatKind(activate(V1)), activate(V1))) }
753.12/208.04	        Weak DPs:
753.12/208.04	          { U11^#(tt(), V1, V2) ->
753.12/208.04	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.04	          , U12^#(tt(), V1, V2) ->
753.12/208.04	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.04	          , U13^#(tt(), V1, V2) ->
753.12/208.04	            c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.04	          , U14^#(tt(), V1, V2) ->
753.12/208.04	            c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.04	                isNat^#(activate(V1)))
753.12/208.04	          , U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.12/208.04	          , isNat^#(n__plus(V1, V2)) ->
753.12/208.04	            c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.04	          , U21^#(tt(), V1) ->
753.12/208.04	            c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.04	          , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.12/208.04	          , U61^#(tt(), M, N) ->
753.12/208.04	            c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.12/208.04	          , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.12/208.04	        Weak Trs:
753.12/208.04	          { U11(tt(), V1, V2) ->
753.12/208.04	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.04	          , U12(tt(), V1, V2) ->
753.12/208.04	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.04	          , isNatKind(n__0()) -> tt()
753.12/208.04	          , isNatKind(n__plus(V1, V2)) ->
753.12/208.04	            U31(isNatKind(activate(V1)), activate(V2))
753.12/208.04	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.04	          , activate(X) -> X
753.12/208.04	          , activate(n__0()) -> 0()
753.12/208.04	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.04	          , activate(n__s(X)) -> s(activate(X))
753.12/208.04	          , U13(tt(), V1, V2) ->
753.12/208.04	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.04	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.04	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.04	          , isNat(n__0()) -> tt()
753.12/208.04	          , isNat(n__plus(V1, V2)) ->
753.12/208.04	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.04	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.04	          , U16(tt()) -> tt()
753.12/208.04	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.04	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.04	          , U23(tt()) -> tt()
753.12/208.04	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.04	          , U32(tt()) -> tt()
753.12/208.04	          , U41(tt()) -> tt()
753.12/208.04	          , s(X) -> n__s(X)
753.12/208.04	          , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.04	          , 0() -> n__0() }
753.12/208.04	        Obligation:
753.12/208.04	          innermost runtime complexity
753.12/208.04	        Answer:
753.12/208.04	          YES(O(1),O(n^1))
753.12/208.04	        
753.12/208.04	        We use the processor 'matrix interpretation of dimension 1' to
753.12/208.04	        orient following rules strictly.
753.12/208.04	        
753.12/208.04	        DPs:
753.12/208.04	          { 1: isNat^#(n__s(V1)) ->
753.12/208.04	               c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.04	          , 10: U61^#(tt(), M, N) ->
753.12/208.04	                c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N))) }
753.12/208.04	        
753.12/208.04	        Sub-proof:
753.12/208.04	        ----------
753.12/208.04	          The following argument positions are usable:
753.12/208.04	            Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1},
753.12/208.04	            Uargs(c_4) = {1, 2}, Uargs(c_5) = {1}, Uargs(c_6) = {1},
753.12/208.04	            Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1},
753.12/208.04	            Uargs(c_10) = {1}, Uargs(c_11) = {1}
753.12/208.04	          
753.12/208.04	          TcT has computed the following constructor-based matrix
753.12/208.04	          interpretation satisfying not(EDA).
753.12/208.04	          
753.12/208.04	                     [U11](x1, x2, x3) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                                  [tt] = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                     [U12](x1, x2, x3) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                       [isNatKind](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                        [activate](x1) = [1] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                     [U13](x1, x2, x3) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                     [U14](x1, x2, x3) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                         [U15](x1, x2) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                           [isNat](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                             [U16](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                         [U21](x1, x2) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                         [U22](x1, x2) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                             [U23](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                         [U31](x1, x2) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                             [U32](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                             [U41](x1) = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                               [s](x1) = [1] x1 + [1]                                    
753.12/208.04	                                                                                         
753.12/208.04	                        [plus](x1, x2) = [1] x1 + [1] x2 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                                [n__0] = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                     [n__plus](x1, x2) = [1] x1 + [1] x2 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                            [n__s](x1) = [1] x1 + [1]                                    
753.12/208.04	                                                                                         
753.12/208.04	                                   [0] = [0]                                             
753.12/208.04	                                                                                         
753.12/208.04	                   [U11^#](x1, x2, x3) = [2] x2 + [2] x3 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                   [U12^#](x1, x2, x3) = [2] x2 + [2] x3 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                     [isNatKind^#](x1) = [7] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                      [activate^#](x1) = [7] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                   [U13^#](x1, x2, x3) = [2] x2 + [2] x3 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                   [U14^#](x1, x2, x3) = [2] x2 + [2] x3 + [0]                           
753.12/208.04	                                                                                         
753.12/208.04	                       [U15^#](x1, x2) = [2] x2 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.04	                         [isNat^#](x1) = [2] x1 + [0]                                    
753.12/208.04	                                                                                         
753.12/208.05	                       [U21^#](x1, x2) = [2] x2 + [0]                                    
753.12/208.05	                                                                                         
753.12/208.05	                       [U22^#](x1, x2) = [2] x2 + [0]                                    
753.12/208.05	                                                                                         
753.12/208.05	                       [U51^#](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.05	                                                                                         
753.12/208.05	                       [U52^#](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.05	                                                                                         
753.12/208.05	                   [U61^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [7]                  
753.12/208.05	                                                                                         
753.12/208.05	                   [U62^#](x1, x2, x3) = [2] x3 + [0]                                    
753.12/208.05	                                                                                         
753.12/208.05	                   [U63^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0]                  
753.12/208.05	                                                                                         
753.12/208.05	                   [U64^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0]                  
753.12/208.05	                                                                                         
753.12/208.05	             [c_1](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.05	                                                                                         
753.12/208.05	             [c_2](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.05	                                                                                         
753.12/208.05	             [c_7](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.05	                                                                                         
753.12/208.05	                 [c_9](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.05	                                                                                         
753.12/208.05	                        [c_10](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.05	                                                                                         
753.12/208.05	            [c_11](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.05	                                                                                         
753.12/208.05	                [c_12](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.05	                                                                                         
753.12/208.05	                [c_13](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.05	                                                                                         
753.12/208.05	                        [c_14](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.05	                                                                                         
753.12/208.05	                [c_15](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0]         
753.12/208.05	                                                                                         
753.12/208.05	                            [c_16](x1) = [7] x1 + [0]                                    
753.12/208.05	                                                                                         
753.12/208.05	            [c_17](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.05	                                                                                         
753.12/208.05	            [c_18](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.05	                                                                                         
753.12/208.05	            [c_19](x1, x2, x3, x4, x5) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [0]
753.12/208.05	                                                                                         
753.12/208.05	                        [c_20](x1, x2) = [7] x1 + [7] x2 + [0]                           
753.12/208.05	                                                                                         
753.12/208.05	                                   [c] = [0]                                             
753.12/208.05	                                                                                         
753.12/208.05	                             [c_1](x1) = [1] x1 + [0]                                    
753.12/208.05	                                                                                         
753.12/208.05	                             [c_2](x1) = [1] x1 + [0]                                    
753.12/208.05	                                                                                         
753.12/208.05	                             [c_3](x1) = [1] x1 + [0]                                    
753.12/208.05	                                                                                         
753.12/208.05	                         [c_4](x1, x2) = [1] x1 + [1] x2 + [0]                           
753.12/208.05	                                                                                         
753.12/208.05	                             [c_5](x1) = [1] x1 + [0]                                    
753.12/208.05	                                                                                         
753.12/208.05	                             [c_6](x1) = [1] x1 + [0]                                    
753.12/208.05	                                                                                         
753.12/208.05	                             [c_7](x1) = [1] x1 + [1]                                    
753.12/208.05	                                                                                         
753.12/208.05	                             [c_8](x1) = [1] x1 + [0]                                    
753.12/208.05	                                                                                         
753.12/208.05	                             [c_9](x1) = [1] x1 + [0]                                    
753.12/208.05	                                                                                         
753.12/208.05	                            [c_10](x1) = [1] x1 + [3]                                    
753.12/208.05	                                                                                         
753.12/208.05	                            [c_11](x1) = [1] x1 + [0]                                    
753.12/208.05	          
753.12/208.05	          The order satisfies the following ordering constraints:
753.12/208.05	          
753.12/208.05	                     [U11(tt(), V1, V2)] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U12(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.12/208.05	                                                                                                             
753.12/208.05	                     [U12(tt(), V1, V2)] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U13(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.12/208.05	                                                                                                             
753.12/208.05	                     [isNatKind(n__0())] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [tt()]                                                           
753.12/208.05	                                                                                                             
753.12/208.05	            [isNatKind(n__plus(V1, V2))] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U31(isNatKind(activate(V1)), activate(V2))]                     
753.12/208.05	                                                                                                             
753.12/208.05	                   [isNatKind(n__s(V1))] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U41(isNatKind(activate(V1)))]                                   
753.12/208.05	                                                                                                             
753.12/208.05	                           [activate(X)] =  [1] X + [0]                                                      
753.12/208.05	                                         >= [1] X + [0]                                                      
753.12/208.05	                                         =  [X]                                                              
753.12/208.05	                                                                                                             
753.12/208.05	                      [activate(n__0())] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [0()]                                                            
753.12/208.05	                                                                                                             
753.12/208.05	             [activate(n__plus(X1, X2))] =  [1] X1 + [1] X2 + [0]                                            
753.12/208.05	                                         >= [1] X1 + [1] X2 + [0]                                            
753.12/208.05	                                         =  [plus(activate(X1), activate(X2))]                               
753.12/208.05	                                                                                                             
753.12/208.05	                     [activate(n__s(X))] =  [1] X + [1]                                                      
753.12/208.05	                                         >= [1] X + [1]                                                      
753.12/208.05	                                         =  [s(activate(X))]                                                 
753.12/208.05	                                                                                                             
753.12/208.05	                     [U13(tt(), V1, V2)] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U14(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.12/208.05	                                                                                                             
753.12/208.05	                     [U14(tt(), V1, V2)] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U15(isNat(activate(V1)), activate(V2))]                         
753.12/208.05	                                                                                                             
753.12/208.05	                         [U15(tt(), V2)] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U16(isNat(activate(V2)))]                                       
753.12/208.05	                                                                                                             
753.12/208.05	                         [isNat(n__0())] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [tt()]                                                           
753.12/208.05	                                                                                                             
753.12/208.05	                [isNat(n__plus(V1, V2))] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U11(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.12/208.05	                                                                                                             
753.12/208.05	                       [isNat(n__s(V1))] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U21(isNatKind(activate(V1)), activate(V1))]                     
753.12/208.05	                                                                                                             
753.12/208.05	                             [U16(tt())] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [tt()]                                                           
753.12/208.05	                                                                                                             
753.12/208.05	                         [U21(tt(), V1)] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U22(isNatKind(activate(V1)), activate(V1))]                     
753.12/208.05	                                                                                                             
753.12/208.05	                         [U22(tt(), V1)] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U23(isNat(activate(V1)))]                                       
753.12/208.05	                                                                                                             
753.12/208.05	                             [U23(tt())] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [tt()]                                                           
753.12/208.05	                                                                                                             
753.12/208.05	                         [U31(tt(), V2)] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [U32(isNatKind(activate(V2)))]                                   
753.12/208.05	                                                                                                             
753.12/208.05	                             [U32(tt())] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [tt()]                                                           
753.12/208.05	                                                                                                             
753.12/208.05	                             [U41(tt())] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [tt()]                                                           
753.12/208.05	                                                                                                             
753.12/208.05	                                  [s(X)] =  [1] X + [1]                                                      
753.12/208.05	                                         >= [1] X + [1]                                                      
753.12/208.05	                                         =  [n__s(X)]                                                        
753.12/208.05	                                                                                                             
753.12/208.05	                          [plus(X1, X2)] =  [1] X1 + [1] X2 + [0]                                            
753.12/208.05	                                         >= [1] X1 + [1] X2 + [0]                                            
753.12/208.05	                                         =  [n__plus(X1, X2)]                                                
753.12/208.05	                                                                                                             
753.12/208.05	                                   [0()] =  [0]                                                              
753.12/208.05	                                         >= [0]                                                              
753.12/208.05	                                         =  [n__0()]                                                         
753.12/208.05	                                                                                                             
753.12/208.05	                   [U11^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                            
753.12/208.05	                                         >= [2] V1 + [2] V2 + [0]                                            
753.12/208.05	                                         =  [c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.12/208.05	                                                                                                             
753.12/208.05	                   [U12^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                            
753.12/208.05	                                         >= [2] V1 + [2] V2 + [0]                                            
753.12/208.05	                                         =  [c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.12/208.05	                                                                                                             
753.12/208.05	                   [U13^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                            
753.12/208.05	                                         >= [2] V1 + [2] V2 + [0]                                            
753.12/208.05	                                         =  [c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.12/208.05	                                                                                                             
753.12/208.05	                   [U14^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                            
753.12/208.05	                                         >= [2] V1 + [2] V2 + [0]                                            
753.12/208.05	                                         =  [c_4(U15^#(isNat(activate(V1)), activate(V2)),                   
753.12/208.05	                                                 isNat^#(activate(V1)))]                                     
753.12/208.05	                                                                                                             
753.12/208.05	                       [U15^#(tt(), V2)] =  [2] V2 + [0]                                                     
753.12/208.05	                                         >= [2] V2 + [0]                                                     
753.12/208.05	                                         =  [c_5(isNat^#(activate(V2)))]                                     
753.12/208.05	                                                                                                             
753.12/208.05	              [isNat^#(n__plus(V1, V2))] =  [2] V1 + [2] V2 + [0]                                            
753.12/208.05	                                         >= [2] V1 + [2] V2 + [0]                                            
753.12/208.05	                                         =  [c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.12/208.05	                                                                                                             
753.12/208.05	                     [isNat^#(n__s(V1))] =  [2] V1 + [2]                                                     
753.12/208.05	                                         >  [2] V1 + [1]                                                     
753.12/208.05	                                         =  [c_7(U21^#(isNatKind(activate(V1)), activate(V1)))]              
753.12/208.05	                                                                                                             
753.12/208.05	                       [U21^#(tt(), V1)] =  [2] V1 + [0]                                                     
753.12/208.05	                                         >= [2] V1 + [0]                                                     
753.12/208.05	                                         =  [c_8(U22^#(isNatKind(activate(V1)), activate(V1)))]              
753.12/208.05	                                                                                                             
753.12/208.05	                       [U22^#(tt(), V1)] =  [2] V1 + [0]                                                     
753.12/208.05	                                         >= [2] V1 + [0]                                                     
753.12/208.05	                                         =  [c_9(isNat^#(activate(V1)))]                                     
753.12/208.05	                                                                                                             
753.12/208.05	                     [U61^#(tt(), M, N)] =  [7] N + [7] M + [7]                                              
753.12/208.05	                                         >  [2] N + [3]                                                      
753.12/208.05	                                         =  [c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))]  
753.12/208.05	                                                                                                             
753.12/208.05	                     [U62^#(tt(), M, N)] =  [2] N + [0]                                                      
753.12/208.05	                                         >= [2] N + [0]                                                      
753.12/208.05	                                         =  [c_11(isNat^#(activate(N)))]                                     
753.12/208.05	                                                                                                             
753.12/208.05	        
753.12/208.05	        We return to the main proof. Consider the set of all dependency
753.12/208.05	        pairs
753.12/208.05	        
753.12/208.05	        :
753.12/208.05	          { 1: isNat^#(n__s(V1)) ->
753.12/208.05	               c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.05	          , 2: U11^#(tt(), V1, V2) ->
753.12/208.05	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.05	          , 3: U12^#(tt(), V1, V2) ->
753.12/208.05	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.05	          , 4: U13^#(tt(), V1, V2) ->
753.12/208.05	               c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.05	          , 5: U14^#(tt(), V1, V2) ->
753.12/208.05	               c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.05	                   isNat^#(activate(V1)))
753.12/208.05	          , 6: U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.12/208.05	          , 7: isNat^#(n__plus(V1, V2)) ->
753.12/208.05	               c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.05	          , 8: U21^#(tt(), V1) ->
753.12/208.05	               c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.05	          , 9: U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.12/208.05	          , 10: U61^#(tt(), M, N) ->
753.12/208.05	                c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.12/208.05	          , 11: U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.12/208.05	        
753.12/208.05	        Processor 'matrix interpretation of dimension 1' induces the
753.12/208.05	        complexity certificate YES(?,O(n^1)) on application of dependency
753.12/208.05	        pairs {1,10}. These cover all (indirect) predecessors of dependency
753.12/208.05	        pairs {1,8,9,10,11}, their number of application is equally
753.12/208.05	        bounded. The dependency pairs are shifted into the weak component.
753.12/208.05	        
753.12/208.05	        We are left with following problem, upon which TcT provides the
753.12/208.05	        certificate YES(O(1),O(1)).
753.12/208.05	        
753.12/208.05	        Weak DPs:
753.12/208.05	          { U11^#(tt(), V1, V2) ->
753.12/208.05	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.05	          , U12^#(tt(), V1, V2) ->
753.12/208.05	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.05	          , U13^#(tt(), V1, V2) ->
753.12/208.05	            c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.05	          , U14^#(tt(), V1, V2) ->
753.12/208.05	            c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.05	                isNat^#(activate(V1)))
753.12/208.05	          , U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.12/208.05	          , isNat^#(n__plus(V1, V2)) ->
753.12/208.05	            c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.05	          , isNat^#(n__s(V1)) ->
753.12/208.05	            c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.05	          , U21^#(tt(), V1) ->
753.12/208.05	            c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.05	          , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.12/208.05	          , U61^#(tt(), M, N) ->
753.12/208.05	            c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.12/208.05	          , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.12/208.05	        Weak Trs:
753.12/208.05	          { U11(tt(), V1, V2) ->
753.12/208.05	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.05	          , U12(tt(), V1, V2) ->
753.12/208.05	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.05	          , isNatKind(n__0()) -> tt()
753.12/208.05	          , isNatKind(n__plus(V1, V2)) ->
753.12/208.05	            U31(isNatKind(activate(V1)), activate(V2))
753.12/208.05	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.05	          , activate(X) -> X
753.12/208.05	          , activate(n__0()) -> 0()
753.12/208.05	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.05	          , activate(n__s(X)) -> s(activate(X))
753.12/208.05	          , U13(tt(), V1, V2) ->
753.12/208.05	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.05	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.05	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.05	          , isNat(n__0()) -> tt()
753.12/208.05	          , isNat(n__plus(V1, V2)) ->
753.12/208.05	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.05	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.05	          , U16(tt()) -> tt()
753.12/208.05	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.05	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.05	          , U23(tt()) -> tt()
753.12/208.05	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.05	          , U32(tt()) -> tt()
753.12/208.05	          , U41(tt()) -> tt()
753.12/208.05	          , s(X) -> n__s(X)
753.12/208.05	          , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.05	          , 0() -> n__0() }
753.12/208.05	        Obligation:
753.12/208.05	          innermost runtime complexity
753.12/208.05	        Answer:
753.12/208.05	          YES(O(1),O(1))
753.12/208.05	        
753.12/208.05	        The following weak DPs constitute a sub-graph of the DG that is
753.12/208.05	        closed under successors. The DPs are removed.
753.12/208.05	        
753.12/208.05	        { U11^#(tt(), V1, V2) ->
753.12/208.05	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.05	        , U12^#(tt(), V1, V2) ->
753.12/208.05	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.05	        , U13^#(tt(), V1, V2) ->
753.12/208.05	          c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.12/208.05	        , U14^#(tt(), V1, V2) ->
753.12/208.05	          c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.05	              isNat^#(activate(V1)))
753.12/208.05	        , U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.12/208.05	        , isNat^#(n__plus(V1, V2)) ->
753.12/208.05	          c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.12/208.05	        , isNat^#(n__s(V1)) ->
753.12/208.05	          c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.05	        , U21^#(tt(), V1) ->
753.12/208.05	          c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.12/208.05	        , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.12/208.05	        , U61^#(tt(), M, N) ->
753.12/208.05	          c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.12/208.05	        , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.12/208.05	        
753.12/208.05	        We are left with following problem, upon which TcT provides the
753.12/208.05	        certificate YES(O(1),O(1)).
753.12/208.05	        
753.12/208.05	        Weak Trs:
753.12/208.05	          { U11(tt(), V1, V2) ->
753.12/208.05	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.05	          , U12(tt(), V1, V2) ->
753.12/208.05	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.05	          , isNatKind(n__0()) -> tt()
753.12/208.05	          , isNatKind(n__plus(V1, V2)) ->
753.12/208.05	            U31(isNatKind(activate(V1)), activate(V2))
753.12/208.05	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.05	          , activate(X) -> X
753.12/208.05	          , activate(n__0()) -> 0()
753.12/208.05	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.05	          , activate(n__s(X)) -> s(activate(X))
753.12/208.05	          , U13(tt(), V1, V2) ->
753.12/208.05	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.05	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.05	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.05	          , isNat(n__0()) -> tt()
753.12/208.05	          , isNat(n__plus(V1, V2)) ->
753.12/208.05	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.05	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.05	          , U16(tt()) -> tt()
753.12/208.05	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.05	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.05	          , U23(tt()) -> tt()
753.12/208.05	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.05	          , U32(tt()) -> tt()
753.12/208.05	          , U41(tt()) -> tt()
753.12/208.05	          , s(X) -> n__s(X)
753.12/208.05	          , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.05	          , 0() -> n__0() }
753.12/208.05	        Obligation:
753.12/208.05	          innermost runtime complexity
753.12/208.05	        Answer:
753.12/208.05	          YES(O(1),O(1))
753.12/208.05	        
753.12/208.05	        No rule is usable, rules are removed from the input problem.
753.12/208.05	        
753.12/208.05	        We are left with following problem, upon which TcT provides the
753.12/208.05	        certificate YES(O(1),O(1)).
753.12/208.05	        
753.12/208.05	        Rules: Empty
753.12/208.05	        Obligation:
753.12/208.05	          innermost runtime complexity
753.12/208.05	        Answer:
753.12/208.05	          YES(O(1),O(1))
753.12/208.05	        
753.12/208.05	        Empty rules are trivially bounded
753.12/208.05	      
753.12/208.05	      We return to the main proof.
753.12/208.05	      
753.12/208.05	      We are left with following problem, upon which TcT provides the
753.12/208.05	      certificate YES(O(1),O(n^2)).
753.12/208.05	      
753.12/208.05	      Strict DPs:
753.12/208.05	        { activate^#(n__plus(X1, X2)) ->
753.12/208.05	          c_5(activate^#(X1), activate^#(X2))
753.12/208.05	        , activate^#(n__s(X)) -> c_6(activate^#(X)) }
753.12/208.05	      Weak DPs:
753.12/208.05	        { U11^#(tt(), V1, V2) ->
753.12/208.05	          U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.05	        , U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.12/208.05	        , U11^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.05	        , U11^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.05	        , U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.05	        , U12^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.05	        , U12^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.05	        , U12^#(tt(), V1, V2) ->
753.12/208.05	          U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.05	        , isNatKind^#(n__plus(V1, V2)) ->
753.12/208.05	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.12/208.05	              isNatKind^#(activate(V1)),
753.12/208.05	              activate^#(V1),
753.12/208.05	              activate^#(V2))
753.12/208.05	        , isNatKind^#(n__s(V1)) ->
753.12/208.05	          c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.12/208.05	        , U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.05	        , U13^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.05	        , U13^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.05	        , U13^#(tt(), V1, V2) ->
753.12/208.05	          U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.05	        , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.12/208.05	        , U14^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.05	        , U14^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.05	        , U14^#(tt(), V1, V2) -> U15^#(isNat(activate(V1)), activate(V2))
753.12/208.05	        , U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.12/208.05	        , U15^#(tt(), V2) -> activate^#(V2)
753.12/208.05	        , U15^#(tt(), V2) -> isNat^#(activate(V2))
753.12/208.05	        , isNat^#(n__plus(V1, V2)) ->
753.12/208.05	          U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.05	        , isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.12/208.05	        , isNat^#(n__plus(V1, V2)) -> activate^#(V1)
753.12/208.05	        , isNat^#(n__plus(V1, V2)) -> activate^#(V2)
753.12/208.05	        , isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.12/208.05	        , isNat^#(n__s(V1)) -> activate^#(V1)
753.12/208.05	        , isNat^#(n__s(V1)) -> U21^#(isNatKind(activate(V1)), activate(V1))
753.12/208.05	        , U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.12/208.05	        , U21^#(tt(), V1) -> activate^#(V1)
753.12/208.05	        , U21^#(tt(), V1) -> U22^#(isNatKind(activate(V1)), activate(V1))
753.12/208.05	        , U22^#(tt(), V1) -> activate^#(V1)
753.12/208.05	        , U22^#(tt(), V1) -> isNat^#(activate(V1))
753.12/208.05	        , U51^#(tt(), N) -> isNatKind^#(activate(N))
753.12/208.06	        , U51^#(tt(), N) -> activate^#(N)
753.12/208.06	        , U51^#(tt(), N) -> U52^#(isNatKind(activate(N)), activate(N))
753.12/208.06	        , U52^#(tt(), N) -> activate^#(N)
753.12/208.06	        , U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.12/208.06	        , U61^#(tt(), M, N) -> activate^#(M)
753.12/208.06	        , U61^#(tt(), M, N) -> activate^#(N)
753.12/208.06	        , U61^#(tt(), M, N) ->
753.12/208.06	          U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.12/208.06	        , U62^#(tt(), M, N) -> activate^#(M)
753.12/208.06	        , U62^#(tt(), M, N) -> activate^#(N)
753.12/208.06	        , U62^#(tt(), M, N) -> isNat^#(activate(N))
753.12/208.06	        , U62^#(tt(), M, N) ->
753.12/208.06	          U63^#(isNat(activate(N)), activate(M), activate(N))
753.12/208.06	        , U63^#(tt(), M, N) -> isNatKind^#(activate(N))
753.12/208.06	        , U63^#(tt(), M, N) -> activate^#(M)
753.12/208.06	        , U63^#(tt(), M, N) -> activate^#(N)
753.12/208.06	        , U63^#(tt(), M, N) ->
753.12/208.06	          U64^#(isNatKind(activate(N)), activate(M), activate(N))
753.12/208.06	        , U64^#(tt(), M, N) -> activate^#(M)
753.12/208.06	        , U64^#(tt(), M, N) -> activate^#(N) }
753.12/208.06	      Weak Trs:
753.12/208.06	        { U11(tt(), V1, V2) ->
753.12/208.06	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.06	        , U12(tt(), V1, V2) ->
753.12/208.06	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.06	        , isNatKind(n__0()) -> tt()
753.12/208.06	        , isNatKind(n__plus(V1, V2)) ->
753.12/208.06	          U31(isNatKind(activate(V1)), activate(V2))
753.12/208.06	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.06	        , activate(X) -> X
753.12/208.06	        , activate(n__0()) -> 0()
753.12/208.06	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.06	        , activate(n__s(X)) -> s(activate(X))
753.12/208.06	        , U13(tt(), V1, V2) ->
753.12/208.06	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.06	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.06	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.06	        , isNat(n__0()) -> tt()
753.12/208.06	        , isNat(n__plus(V1, V2)) ->
753.12/208.06	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.06	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.06	        , U16(tt()) -> tt()
753.12/208.06	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.06	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.06	        , U23(tt()) -> tt()
753.12/208.06	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.06	        , U32(tt()) -> tt()
753.12/208.06	        , U41(tt()) -> tt()
753.12/208.06	        , s(X) -> n__s(X)
753.12/208.06	        , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.06	        , 0() -> n__0() }
753.12/208.06	      Obligation:
753.12/208.06	        innermost runtime complexity
753.12/208.06	      Answer:
753.12/208.06	        YES(O(1),O(n^2))
753.12/208.06	      
753.12/208.06	      We use the processor 'matrix interpretation of dimension 2' to
753.12/208.06	      orient following rules strictly.
753.12/208.06	      
753.12/208.06	      DPs:
753.12/208.06	        { 2: activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.06	        , 36: U51^#(tt(), N) -> isNatKind^#(activate(N))
753.12/208.06	        , 37: U51^#(tt(), N) -> activate^#(N)
753.12/208.06	        , 38: U51^#(tt(), N) -> U52^#(isNatKind(activate(N)), activate(N))
753.12/208.06	        , 40: U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.12/208.06	        , 41: U61^#(tt(), M, N) -> activate^#(M)
753.12/208.06	        , 42: U61^#(tt(), M, N) -> activate^#(N)
753.12/208.06	        , 43: U61^#(tt(), M, N) ->
753.12/208.06	              U62^#(isNatKind(activate(M)), activate(M), activate(N)) }
753.12/208.06	      
753.12/208.06	      Sub-proof:
753.12/208.06	      ----------
753.12/208.06	        The following argument positions are usable:
753.12/208.06	          Uargs(c_3) = {1, 2, 3, 4}, Uargs(c_4) = {1, 2},
753.12/208.06	          Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}, Uargs(c_8) = {1, 2}
753.12/208.06	        
753.12/208.06	        TcT has computed the following constructor-based matrix
753.12/208.06	        interpretation satisfying not(EDA).
753.12/208.06	        
753.12/208.06	              [U11](x1, x2, x3) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                           [tt] = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	              [U12](x1, x2, x3) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                [isNatKind](x1) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                 [activate](x1) = [1 0] x1 + [0]                               
753.12/208.06	                                  [0 1]      [0]                               
753.12/208.06	                                                                               
753.12/208.06	              [U13](x1, x2, x3) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	              [U14](x1, x2, x3) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                  [U15](x1, x2) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                    [isNat](x1) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                      [U16](x1) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                  [U21](x1, x2) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                  [U22](x1, x2) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                      [U23](x1) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                  [U31](x1, x2) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                      [U32](x1) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                      [U41](x1) = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	                        [s](x1) = [1 0] x1 + [2]                               
753.12/208.06	                                  [1 1]      [0]                               
753.12/208.06	                                                                               
753.12/208.06	                 [plus](x1, x2) = [1 0] x1 + [1 0] x2 + [0]                    
753.12/208.06	                                  [1 1]      [1 1]      [0]                    
753.12/208.06	                                                                               
753.12/208.06	                         [n__0] = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	              [n__plus](x1, x2) = [1 0] x1 + [1 0] x2 + [0]                    
753.12/208.06	                                  [1 1]      [1 1]      [0]                    
753.12/208.06	                                                                               
753.12/208.06	                     [n__s](x1) = [1 0] x1 + [2]                               
753.12/208.06	                                  [1 1]      [0]                               
753.12/208.06	                                                                               
753.12/208.06	                            [0] = [0]                                          
753.12/208.06	                                  [0]                                          
753.12/208.06	                                                                               
753.12/208.06	            [U11^#](x1, x2, x3) = [1 4] x2 + [1 4] x3 + [0]                    
753.12/208.06	                                  [2 0]      [2 0]      [2]                    
753.12/208.06	                                                                               
753.12/208.06	            [U12^#](x1, x2, x3) = [1 4] x2 + [1 4] x3 + [0]                    
753.12/208.06	                                  [2 0]      [2 0]      [2]                    
753.12/208.06	                                                                               
753.12/208.06	              [isNatKind^#](x1) = [0 4] x1 + [0]                               
753.12/208.06	                                  [1 0]      [2]                               
753.12/208.06	                                                                               
753.12/208.06	               [activate^#](x1) = [1 0] x1 + [0]                               
753.12/208.06	                                  [0 0]      [0]                               
753.12/208.06	                                                                               
753.12/208.06	            [U13^#](x1, x2, x3) = [1 4] x2 + [1 4] x3 + [0]                    
753.12/208.06	                                  [2 0]      [2 0]      [2]                    
753.12/208.06	                                                                               
753.12/208.06	                [U31^#](x1, x2) = [1 4] x2 + [0]                               
753.12/208.06	                                  [0 0]      [4]                               
753.12/208.06	                                                                               
753.12/208.06	            [U14^#](x1, x2, x3) = [1 4] x2 + [1 4] x3 + [0]                    
753.12/208.06	                                  [2 0]      [2 0]      [2]                    
753.12/208.06	                                                                               
753.12/208.06	                [U15^#](x1, x2) = [1 4] x2 + [0]                               
753.12/208.06	                                  [2 0]      [2]                               
753.12/208.06	                                                                               
753.12/208.06	                  [isNat^#](x1) = [0 4] x1 + [0]                               
753.12/208.06	                                  [2 0]      [2]                               
753.12/208.06	                                                                               
753.12/208.06	                [U21^#](x1, x2) = [4 4] x2 + [0]                               
753.12/208.06	                                  [2 0]      [4]                               
753.12/208.06	                                                                               
753.12/208.06	                [U22^#](x1, x2) = [1 4] x2 + [0]                               
753.12/208.06	                                  [2 0]      [4]                               
753.12/208.06	                                                                               
753.12/208.06	                [U51^#](x1, x2) = [5 4] x1 + [7 7] x2 + [7]                    
753.12/208.06	                                  [5 5]      [7 7]      [3]                    
753.12/208.06	                                                                               
753.12/208.06	                [U52^#](x1, x2) = [4 4] x2 + [0]                               
753.12/208.06	                                  [0 0]      [0]                               
753.12/208.06	                                                                               
753.12/208.06	            [U61^#](x1, x2, x3) = [5 4] x1 + [7 7] x2 + [7 7] x3 + [7]         
753.12/208.06	                                  [5 5]      [7 7]      [7 7]      [7]         
753.12/208.06	                                                                               
753.12/208.06	            [U62^#](x1, x2, x3) = [1 0] x2 + [4 4] x3 + [0]                    
753.12/208.06	                                  [4 0]      [2 0]      [2]                    
753.12/208.06	                                                                               
753.12/208.06	            [U63^#](x1, x2, x3) = [1 0] x2 + [4 4] x3 + [0]                    
753.12/208.06	                                  [0 0]      [1 0]      [2]                    
753.12/208.06	                                                                               
753.12/208.06	            [U64^#](x1, x2, x3) = [1 0] x2 + [2 0] x3 + [0]                    
753.12/208.06	                                  [0 0]      [0 0]      [0]                    
753.12/208.06	                                                                               
753.12/208.06	          [c_3](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1          
753.12/208.06	                                                                    0] x4 + [0]
753.12/208.06	                                  [0 0]      [0 0]      [0 0]      [0          
753.12/208.06	                                                                    0]      [2]
753.12/208.06	                                                                               
753.12/208.06	                  [c_4](x1, x2) = [1 0] x1 + [1 0] x2 + [0]                    
753.12/208.06	                                  [0 0]      [0 0]      [3]                    
753.12/208.06	                                                                               
753.12/208.06	                  [c_5](x1, x2) = [1 0] x1 + [1 0] x2 + [0]                    
753.12/208.06	                                  [0 0]      [0 0]      [0]                    
753.12/208.06	                                                                               
753.12/208.06	                      [c_6](x1) = [1 0] x1 + [1]                               
753.12/208.06	                                  [0 0]      [0]                               
753.12/208.06	                                                                               
753.12/208.06	                  [c_8](x1, x2) = [1 0] x1 + [1 0] x2 + [0]                    
753.12/208.06	                                  [0 0]      [0 0]      [3]                    
753.12/208.06	        
753.12/208.06	        The order satisfies the following ordering constraints:
753.12/208.06	        
753.12/208.06	                     [U11(tt(), V1, V2)] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U12(isNatKind(activate(V1)), activate(V1), activate(V2))]  
753.12/208.06	                                                                                                        
753.12/208.06	                     [U12(tt(), V1, V2)] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U13(isNatKind(activate(V2)), activate(V1), activate(V2))]  
753.12/208.06	                                                                                                        
753.12/208.06	                     [isNatKind(n__0())] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [tt()]                                                      
753.12/208.06	                                                                                                        
753.12/208.06	            [isNatKind(n__plus(V1, V2))] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U31(isNatKind(activate(V1)), activate(V2))]                
753.12/208.06	                                                                                                        
753.12/208.06	                   [isNatKind(n__s(V1))] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U41(isNatKind(activate(V1)))]                              
753.12/208.06	                                                                                                        
753.12/208.06	                           [activate(X)] =  [1 0] X + [0]                                               
753.12/208.06	                                            [0 1]     [0]                                               
753.12/208.06	                                         >= [1 0] X + [0]                                               
753.12/208.06	                                            [0 1]     [0]                                               
753.12/208.06	                                         =  [X]                                                         
753.12/208.06	                                                                                                        
753.12/208.06	                      [activate(n__0())] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [0()]                                                       
753.12/208.06	                                                                                                        
753.12/208.06	             [activate(n__plus(X1, X2))] =  [1 0] X1 + [1 0] X2 + [0]                                   
753.12/208.06	                                            [1 1]      [1 1]      [0]                                   
753.12/208.06	                                         >= [1 0] X1 + [1 0] X2 + [0]                                   
753.12/208.06	                                            [1 1]      [1 1]      [0]                                   
753.12/208.06	                                         =  [plus(activate(X1), activate(X2))]                          
753.12/208.06	                                                                                                        
753.12/208.06	                     [activate(n__s(X))] =  [1 0] X + [2]                                               
753.12/208.06	                                            [1 1]     [0]                                               
753.12/208.06	                                         >= [1 0] X + [2]                                               
753.12/208.06	                                            [1 1]     [0]                                               
753.12/208.06	                                         =  [s(activate(X))]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                     [U13(tt(), V1, V2)] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U14(isNatKind(activate(V2)), activate(V1), activate(V2))]  
753.12/208.06	                                                                                                        
753.12/208.06	                     [U14(tt(), V1, V2)] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U15(isNat(activate(V1)), activate(V2))]                    
753.12/208.06	                                                                                                        
753.12/208.06	                         [U15(tt(), V2)] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U16(isNat(activate(V2)))]                                  
753.12/208.06	                                                                                                        
753.12/208.06	                         [isNat(n__0())] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [tt()]                                                      
753.12/208.06	                                                                                                        
753.12/208.06	                [isNat(n__plus(V1, V2))] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U11(isNatKind(activate(V1)), activate(V1), activate(V2))]  
753.12/208.06	                                                                                                        
753.12/208.06	                       [isNat(n__s(V1))] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U21(isNatKind(activate(V1)), activate(V1))]                
753.12/208.06	                                                                                                        
753.12/208.06	                             [U16(tt())] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [tt()]                                                      
753.12/208.06	                                                                                                        
753.12/208.06	                         [U21(tt(), V1)] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U22(isNatKind(activate(V1)), activate(V1))]                
753.12/208.06	                                                                                                        
753.12/208.06	                         [U22(tt(), V1)] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U23(isNat(activate(V1)))]                                  
753.12/208.06	                                                                                                        
753.12/208.06	                             [U23(tt())] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [tt()]                                                      
753.12/208.06	                                                                                                        
753.12/208.06	                         [U31(tt(), V2)] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [U32(isNatKind(activate(V2)))]                              
753.12/208.06	                                                                                                        
753.12/208.06	                             [U32(tt())] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [tt()]                                                      
753.12/208.06	                                                                                                        
753.12/208.06	                             [U41(tt())] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [tt()]                                                      
753.12/208.06	                                                                                                        
753.12/208.06	                                  [s(X)] =  [1 0] X + [2]                                               
753.12/208.06	                                            [1 1]     [0]                                               
753.12/208.06	                                         >= [1 0] X + [2]                                               
753.12/208.06	                                            [1 1]     [0]                                               
753.12/208.06	                                         =  [n__s(X)]                                                   
753.12/208.06	                                                                                                        
753.12/208.06	                          [plus(X1, X2)] =  [1 0] X1 + [1 0] X2 + [0]                                   
753.12/208.06	                                            [1 1]      [1 1]      [0]                                   
753.12/208.06	                                         >= [1 0] X1 + [1 0] X2 + [0]                                   
753.12/208.06	                                            [1 1]      [1 1]      [0]                                   
753.12/208.06	                                         =  [n__plus(X1, X2)]                                           
753.12/208.06	                                                                                                        
753.12/208.06	                                   [0()] =  [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         >= [0]                                                         
753.12/208.06	                                            [0]                                                         
753.12/208.06	                                         =  [n__0()]                                                    
753.12/208.06	                                                                                                        
753.12/208.06	                   [U11^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         =  [U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))]
753.12/208.06	                                                                                                        
753.12/208.06	                   [U11^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [0 4] V1 + [0]                                              
753.12/208.06	                                            [1 0]      [2]                                              
753.12/208.06	                                         =  [isNatKind^#(activate(V1))]                                 
753.12/208.06	                                                                                                        
753.12/208.06	                   [U11^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 0] V1 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V1)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                   [U11^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 0] V2 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V2)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                   [U12^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [0 4] V2 + [0]                                              
753.12/208.06	                                            [1 0]      [2]                                              
753.12/208.06	                                         =  [isNatKind^#(activate(V2))]                                 
753.12/208.06	                                                                                                        
753.12/208.06	                   [U12^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 0] V1 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V1)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                   [U12^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 0] V2 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V2)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                   [U12^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         =  [U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))]
753.12/208.06	                                                                                                        
753.12/208.06	          [isNatKind^#(n__plus(V1, V2))] =  [4 4] V1 + [4 4] V2 + [0]                                   
753.12/208.06	                                            [1 0]      [1 0]      [2]                                   
753.12/208.06	                                         >= [1 4] V1 + [2 4] V2 + [0]                                   
753.12/208.06	                                            [0 0]      [0 0]      [2]                                   
753.12/208.06	                                         =  [c_3(U31^#(isNatKind(activate(V1)), activate(V2)),          
753.12/208.06	                                                 isNatKind^#(activate(V1)),                             
753.12/208.06	                                                 activate^#(V1),                                        
753.12/208.06	                                                 activate^#(V2))]                                       
753.12/208.06	                                                                                                        
753.12/208.06	                 [isNatKind^#(n__s(V1))] =  [4 4] V1 + [0]                                              
753.12/208.06	                                            [1 0]      [4]                                              
753.12/208.06	                                         >= [1 4] V1 + [0]                                              
753.12/208.06	                                            [0 0]      [3]                                              
753.12/208.06	                                         =  [c_4(isNatKind^#(activate(V1)), activate^#(V1))]            
753.12/208.06	                                                                                                        
753.12/208.06	           [activate^#(n__plus(X1, X2))] =  [1 0] X1 + [1 0] X2 + [0]                                   
753.12/208.06	                                            [0 0]      [0 0]      [0]                                   
753.12/208.06	                                         >= [1 0] X1 + [1 0] X2 + [0]                                   
753.12/208.06	                                            [0 0]      [0 0]      [0]                                   
753.12/208.06	                                         =  [c_5(activate^#(X1), activate^#(X2))]                       
753.12/208.06	                                                                                                        
753.12/208.06	                   [activate^#(n__s(X))] =  [1 0] X + [2]                                               
753.12/208.06	                                            [0 0]     [0]                                               
753.12/208.06	                                         >  [1 0] X + [1]                                               
753.12/208.06	                                            [0 0]     [0]                                               
753.12/208.06	                                         =  [c_6(activate^#(X))]                                        
753.12/208.06	                                                                                                        
753.12/208.06	                   [U13^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [0 4] V2 + [0]                                              
753.12/208.06	                                            [1 0]      [2]                                              
753.12/208.06	                                         =  [isNatKind^#(activate(V2))]                                 
753.12/208.06	                                                                                                        
753.12/208.06	                   [U13^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 0] V1 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V1)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                   [U13^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 0] V2 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V2)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                   [U13^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         =  [U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))]
753.12/208.06	                                                                                                        
753.12/208.06	                       [U31^#(tt(), V2)] =  [1 4] V2 + [0]                                              
753.12/208.06	                                            [0 0]      [4]                                              
753.12/208.06	                                         >= [1 4] V2 + [0]                                              
753.12/208.06	                                            [0 0]      [3]                                              
753.12/208.06	                                         =  [c_8(isNatKind^#(activate(V2)), activate^#(V2))]            
753.12/208.06	                                                                                                        
753.12/208.06	                   [U14^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 0] V1 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V1)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                   [U14^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 0] V2 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V2)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                   [U14^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 4] V2 + [0]                                              
753.12/208.06	                                            [2 0]      [2]                                              
753.12/208.06	                                         =  [U15^#(isNat(activate(V1)), activate(V2))]                  
753.12/208.06	                                                                                                        
753.12/208.06	                   [U14^#(tt(), V1, V2)] =  [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [0 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [2]                                              
753.12/208.06	                                         =  [isNat^#(activate(V1))]                                     
753.12/208.06	                                                                                                        
753.12/208.06	                       [U15^#(tt(), V2)] =  [1 4] V2 + [0]                                              
753.12/208.06	                                            [2 0]      [2]                                              
753.12/208.06	                                         >= [1 0] V2 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V2)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                       [U15^#(tt(), V2)] =  [1 4] V2 + [0]                                              
753.12/208.06	                                            [2 0]      [2]                                              
753.12/208.06	                                         >= [0 4] V2 + [0]                                              
753.12/208.06	                                            [2 0]      [2]                                              
753.12/208.06	                                         =  [isNat^#(activate(V2))]                                     
753.12/208.06	                                                                                                        
753.12/208.06	              [isNat^#(n__plus(V1, V2))] =  [4 4] V1 + [4 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 4] V1 + [1 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         =  [U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))]
753.12/208.06	                                                                                                        
753.12/208.06	              [isNat^#(n__plus(V1, V2))] =  [4 4] V1 + [4 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [0 4] V1 + [0]                                              
753.12/208.06	                                            [1 0]      [2]                                              
753.12/208.06	                                         =  [isNatKind^#(activate(V1))]                                 
753.12/208.06	                                                                                                        
753.12/208.06	              [isNat^#(n__plus(V1, V2))] =  [4 4] V1 + [4 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 0] V1 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V1)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	              [isNat^#(n__plus(V1, V2))] =  [4 4] V1 + [4 4] V2 + [0]                                   
753.12/208.06	                                            [2 0]      [2 0]      [2]                                   
753.12/208.06	                                         >= [1 0] V2 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V2)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                     [isNat^#(n__s(V1))] =  [4 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [6]                                              
753.12/208.06	                                         >= [0 4] V1 + [0]                                              
753.12/208.06	                                            [1 0]      [2]                                              
753.12/208.06	                                         =  [isNatKind^#(activate(V1))]                                 
753.12/208.06	                                                                                                        
753.12/208.06	                     [isNat^#(n__s(V1))] =  [4 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [6]                                              
753.12/208.06	                                         >= [1 0] V1 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V1)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                     [isNat^#(n__s(V1))] =  [4 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [6]                                              
753.12/208.06	                                         >= [4 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [4]                                              
753.12/208.06	                                         =  [U21^#(isNatKind(activate(V1)), activate(V1))]              
753.12/208.06	                                                                                                        
753.12/208.06	                       [U21^#(tt(), V1)] =  [4 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [4]                                              
753.12/208.06	                                         >= [0 4] V1 + [0]                                              
753.12/208.06	                                            [1 0]      [2]                                              
753.12/208.06	                                         =  [isNatKind^#(activate(V1))]                                 
753.12/208.06	                                                                                                        
753.12/208.06	                       [U21^#(tt(), V1)] =  [4 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [4]                                              
753.12/208.06	                                         >= [1 0] V1 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V1)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                       [U21^#(tt(), V1)] =  [4 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [4]                                              
753.12/208.06	                                         >= [1 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [4]                                              
753.12/208.06	                                         =  [U22^#(isNatKind(activate(V1)), activate(V1))]              
753.12/208.06	                                                                                                        
753.12/208.06	                       [U22^#(tt(), V1)] =  [1 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [4]                                              
753.12/208.06	                                         >= [1 0] V1 + [0]                                              
753.12/208.06	                                            [0 0]      [0]                                              
753.12/208.06	                                         =  [activate^#(V1)]                                            
753.12/208.06	                                                                                                        
753.12/208.06	                       [U22^#(tt(), V1)] =  [1 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [4]                                              
753.12/208.06	                                         >= [0 4] V1 + [0]                                              
753.12/208.06	                                            [2 0]      [2]                                              
753.12/208.06	                                         =  [isNat^#(activate(V1))]                                     
753.12/208.06	                                                                                                        
753.12/208.06	                        [U51^#(tt(), N)] =  [7 7] N + [7]                                               
753.12/208.06	                                            [7 7]     [3]                                               
753.12/208.06	                                         >  [0 4] N + [0]                                               
753.12/208.06	                                            [1 0]     [2]                                               
753.12/208.06	                                         =  [isNatKind^#(activate(N))]                                  
753.12/208.06	                                                                                                        
753.12/208.06	                        [U51^#(tt(), N)] =  [7 7] N + [7]                                               
753.12/208.06	                                            [7 7]     [3]                                               
753.12/208.06	                                         >  [1 0] N + [0]                                               
753.12/208.06	                                            [0 0]     [0]                                               
753.12/208.06	                                         =  [activate^#(N)]                                             
753.12/208.06	                                                                                                        
753.12/208.06	                        [U51^#(tt(), N)] =  [7 7] N + [7]                                               
753.12/208.06	                                            [7 7]     [3]                                               
753.12/208.06	                                         >  [4 4] N + [0]                                               
753.12/208.06	                                            [0 0]     [0]                                               
753.12/208.06	                                         =  [U52^#(isNatKind(activate(N)), activate(N))]                
753.12/208.06	                                                                                                        
753.12/208.06	                        [U52^#(tt(), N)] =  [4 4] N + [0]                                               
753.12/208.06	                                            [0 0]     [0]                                               
753.12/208.06	                                         >= [1 0] N + [0]                                               
753.12/208.06	                                            [0 0]     [0]                                               
753.12/208.06	                                         =  [activate^#(N)]                                             
753.12/208.06	                                                                                                        
753.12/208.06	                     [U61^#(tt(), M, N)] =  [7 7] N + [7 7] M + [7]                                     
753.12/208.06	                                            [7 7]     [7 7]     [7]                                     
753.12/208.06	                                         >  [0 4] M + [0]                                               
753.12/208.06	                                            [1 0]     [2]                                               
753.12/208.06	                                         =  [isNatKind^#(activate(M))]                                  
753.12/208.06	                                                                                                        
753.12/208.06	                     [U61^#(tt(), M, N)] =  [7 7] N + [7 7] M + [7]                                     
753.12/208.06	                                            [7 7]     [7 7]     [7]                                     
753.12/208.06	                                         >  [1 0] M + [0]                                               
753.12/208.06	                                            [0 0]     [0]                                               
753.12/208.06	                                         =  [activate^#(M)]                                             
753.12/208.06	                                                                                                        
753.12/208.06	                     [U61^#(tt(), M, N)] =  [7 7] N + [7 7] M + [7]                                     
753.12/208.06	                                            [7 7]     [7 7]     [7]                                     
753.12/208.06	                                         >  [1 0] N + [0]                                               
753.12/208.06	                                            [0 0]     [0]                                               
753.12/208.06	                                         =  [activate^#(N)]                                             
753.12/208.06	                                                                                                        
753.12/208.07	                     [U61^#(tt(), M, N)] =  [7 7] N + [7 7] M + [7]                                     
753.12/208.07	                                            [7 7]     [7 7]     [7]                                     
753.12/208.07	                                         >  [4 4] N + [1 0] M + [0]                                     
753.12/208.07	                                            [2 0]     [4 0]     [2]                                     
753.12/208.07	                                         =  [U62^#(isNatKind(activate(M)), activate(M), activate(N))]   
753.12/208.07	                                                                                                        
753.12/208.07	                     [U62^#(tt(), M, N)] =  [4 4] N + [1 0] M + [0]                                     
753.12/208.07	                                            [2 0]     [4 0]     [2]                                     
753.12/208.07	                                         >= [1 0] M + [0]                                               
753.12/208.07	                                            [0 0]     [0]                                               
753.12/208.07	                                         =  [activate^#(M)]                                             
753.12/208.07	                                                                                                        
753.12/208.07	                     [U62^#(tt(), M, N)] =  [4 4] N + [1 0] M + [0]                                     
753.12/208.07	                                            [2 0]     [4 0]     [2]                                     
753.12/208.07	                                         >= [1 0] N + [0]                                               
753.12/208.07	                                            [0 0]     [0]                                               
753.12/208.07	                                         =  [activate^#(N)]                                             
753.12/208.07	                                                                                                        
753.12/208.07	                     [U62^#(tt(), M, N)] =  [4 4] N + [1 0] M + [0]                                     
753.12/208.07	                                            [2 0]     [4 0]     [2]                                     
753.12/208.07	                                         >= [0 4] N + [0]                                               
753.12/208.07	                                            [2 0]     [2]                                               
753.12/208.07	                                         =  [isNat^#(activate(N))]                                      
753.12/208.07	                                                                                                        
753.12/208.07	                     [U62^#(tt(), M, N)] =  [4 4] N + [1 0] M + [0]                                     
753.12/208.07	                                            [2 0]     [4 0]     [2]                                     
753.12/208.07	                                         >= [4 4] N + [1 0] M + [0]                                     
753.12/208.07	                                            [1 0]     [0 0]     [2]                                     
753.12/208.07	                                         =  [U63^#(isNat(activate(N)), activate(M), activate(N))]       
753.12/208.07	                                                                                                        
753.12/208.07	                     [U63^#(tt(), M, N)] =  [4 4] N + [1 0] M + [0]                                     
753.12/208.07	                                            [1 0]     [0 0]     [2]                                     
753.12/208.07	                                         >= [0 4] N + [0]                                               
753.12/208.07	                                            [1 0]     [2]                                               
753.12/208.07	                                         =  [isNatKind^#(activate(N))]                                  
753.12/208.07	                                                                                                        
753.12/208.07	                     [U63^#(tt(), M, N)] =  [4 4] N + [1 0] M + [0]                                     
753.12/208.07	                                            [1 0]     [0 0]     [2]                                     
753.12/208.07	                                         >= [1 0] M + [0]                                               
753.12/208.07	                                            [0 0]     [0]                                               
753.12/208.07	                                         =  [activate^#(M)]                                             
753.12/208.07	                                                                                                        
753.12/208.07	                     [U63^#(tt(), M, N)] =  [4 4] N + [1 0] M + [0]                                     
753.12/208.07	                                            [1 0]     [0 0]     [2]                                     
753.12/208.07	                                         >= [1 0] N + [0]                                               
753.12/208.07	                                            [0 0]     [0]                                               
753.12/208.07	                                         =  [activate^#(N)]                                             
753.12/208.07	                                                                                                        
753.12/208.07	                     [U63^#(tt(), M, N)] =  [4 4] N + [1 0] M + [0]                                     
753.12/208.07	                                            [1 0]     [0 0]     [2]                                     
753.12/208.07	                                         >= [2 0] N + [1 0] M + [0]                                     
753.12/208.07	                                            [0 0]     [0 0]     [0]                                     
753.12/208.07	                                         =  [U64^#(isNatKind(activate(N)), activate(M), activate(N))]   
753.12/208.07	                                                                                                        
753.12/208.07	                     [U64^#(tt(), M, N)] =  [2 0] N + [1 0] M + [0]                                     
753.12/208.07	                                            [0 0]     [0 0]     [0]                                     
753.12/208.07	                                         >= [1 0] M + [0]                                               
753.12/208.07	                                            [0 0]     [0]                                               
753.12/208.07	                                         =  [activate^#(M)]                                             
753.12/208.07	                                                                                                        
753.12/208.07	                     [U64^#(tt(), M, N)] =  [2 0] N + [1 0] M + [0]                                     
753.12/208.07	                                            [0 0]     [0 0]     [0]                                     
753.12/208.07	                                         >= [1 0] N + [0]                                               
753.12/208.07	                                            [0 0]     [0]                                               
753.12/208.07	                                         =  [activate^#(N)]                                             
753.12/208.07	                                                                                                        
753.12/208.07	      
753.12/208.07	      We return to the main proof. Consider the set of all dependency
753.12/208.07	      pairs
753.12/208.07	      
753.12/208.07	      :
753.12/208.07	        { 1: activate^#(n__plus(X1, X2)) ->
753.12/208.07	             c_5(activate^#(X1), activate^#(X2))
753.12/208.07	        , 2: activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.07	        , 3: U11^#(tt(), V1, V2) ->
753.12/208.07	             U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.07	        , 4: U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.12/208.07	        , 5: U11^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.07	        , 6: U11^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.07	        , 7: U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.07	        , 8: U12^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.07	        , 9: U12^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.07	        , 10: U12^#(tt(), V1, V2) ->
753.12/208.07	              U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.07	        , 11: isNatKind^#(n__plus(V1, V2)) ->
753.12/208.07	              c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.12/208.07	                  isNatKind^#(activate(V1)),
753.12/208.07	                  activate^#(V1),
753.12/208.07	                  activate^#(V2))
753.12/208.07	        , 12: isNatKind^#(n__s(V1)) ->
753.12/208.07	              c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.12/208.07	        , 13: U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.07	        , 14: U13^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.07	        , 15: U13^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.07	        , 16: U13^#(tt(), V1, V2) ->
753.12/208.07	              U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.07	        , 17: U31^#(tt(), V2) ->
753.12/208.07	              c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.12/208.07	        , 18: U14^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.07	        , 19: U14^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.07	        , 20: U14^#(tt(), V1, V2) ->
753.12/208.07	              U15^#(isNat(activate(V1)), activate(V2))
753.12/208.07	        , 21: U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.12/208.07	        , 22: U15^#(tt(), V2) -> activate^#(V2)
753.12/208.07	        , 23: U15^#(tt(), V2) -> isNat^#(activate(V2))
753.12/208.07	        , 24: isNat^#(n__plus(V1, V2)) ->
753.12/208.07	              U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.07	        , 25: isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.12/208.07	        , 26: isNat^#(n__plus(V1, V2)) -> activate^#(V1)
753.12/208.07	        , 27: isNat^#(n__plus(V1, V2)) -> activate^#(V2)
753.12/208.07	        , 28: isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.12/208.07	        , 29: isNat^#(n__s(V1)) -> activate^#(V1)
753.12/208.07	        , 30: isNat^#(n__s(V1)) ->
753.12/208.07	              U21^#(isNatKind(activate(V1)), activate(V1))
753.12/208.07	        , 31: U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.12/208.07	        , 32: U21^#(tt(), V1) -> activate^#(V1)
753.12/208.07	        , 33: U21^#(tt(), V1) ->
753.12/208.07	              U22^#(isNatKind(activate(V1)), activate(V1))
753.12/208.07	        , 34: U22^#(tt(), V1) -> activate^#(V1)
753.12/208.07	        , 35: U22^#(tt(), V1) -> isNat^#(activate(V1))
753.12/208.07	        , 36: U51^#(tt(), N) -> isNatKind^#(activate(N))
753.12/208.07	        , 37: U51^#(tt(), N) -> activate^#(N)
753.12/208.07	        , 38: U51^#(tt(), N) -> U52^#(isNatKind(activate(N)), activate(N))
753.12/208.07	        , 39: U52^#(tt(), N) -> activate^#(N)
753.12/208.07	        , 40: U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.12/208.07	        , 41: U61^#(tt(), M, N) -> activate^#(M)
753.12/208.07	        , 42: U61^#(tt(), M, N) -> activate^#(N)
753.12/208.07	        , 43: U61^#(tt(), M, N) ->
753.12/208.07	              U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.12/208.07	        , 44: U62^#(tt(), M, N) -> activate^#(M)
753.12/208.07	        , 45: U62^#(tt(), M, N) -> activate^#(N)
753.12/208.07	        , 46: U62^#(tt(), M, N) -> isNat^#(activate(N))
753.12/208.07	        , 47: U62^#(tt(), M, N) ->
753.12/208.07	              U63^#(isNat(activate(N)), activate(M), activate(N))
753.12/208.07	        , 48: U63^#(tt(), M, N) -> isNatKind^#(activate(N))
753.12/208.07	        , 49: U63^#(tt(), M, N) -> activate^#(M)
753.12/208.07	        , 50: U63^#(tt(), M, N) -> activate^#(N)
753.12/208.07	        , 51: U63^#(tt(), M, N) ->
753.12/208.07	              U64^#(isNatKind(activate(N)), activate(M), activate(N))
753.12/208.07	        , 52: U64^#(tt(), M, N) -> activate^#(M)
753.12/208.07	        , 53: U64^#(tt(), M, N) -> activate^#(N) }
753.12/208.07	      
753.12/208.07	      Processor 'matrix interpretation of dimension 2' induces the
753.12/208.07	      complexity certificate YES(?,O(n^2)) on application of dependency
753.12/208.07	      pairs {2,36,37,38,40,41,42,43}. These cover all (indirect)
753.12/208.07	      predecessors of dependency pairs
753.12/208.07	      {2,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53}, their
753.12/208.07	      number of application is equally bounded. The dependency pairs are
753.12/208.07	      shifted into the weak component.
753.12/208.07	      
753.12/208.07	      We are left with following problem, upon which TcT provides the
753.12/208.07	      certificate YES(O(1),O(n^2)).
753.12/208.07	      
753.12/208.07	      Strict DPs:
753.12/208.07	        { activate^#(n__plus(X1, X2)) ->
753.12/208.07	          c_5(activate^#(X1), activate^#(X2)) }
753.12/208.07	      Weak DPs:
753.12/208.07	        { U11^#(tt(), V1, V2) ->
753.12/208.07	          U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.07	        , U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.12/208.07	        , U11^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.07	        , U11^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.07	        , U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.07	        , U12^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.07	        , U12^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.07	        , U12^#(tt(), V1, V2) ->
753.12/208.07	          U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.07	        , isNatKind^#(n__plus(V1, V2)) ->
753.12/208.07	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.12/208.07	              isNatKind^#(activate(V1)),
753.12/208.07	              activate^#(V1),
753.12/208.07	              activate^#(V2))
753.12/208.07	        , isNatKind^#(n__s(V1)) ->
753.12/208.07	          c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.12/208.07	        , activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.07	        , U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.07	        , U13^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.07	        , U13^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.07	        , U13^#(tt(), V1, V2) ->
753.12/208.07	          U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.07	        , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.12/208.07	        , U14^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.07	        , U14^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.07	        , U14^#(tt(), V1, V2) -> U15^#(isNat(activate(V1)), activate(V2))
753.12/208.07	        , U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.12/208.07	        , U15^#(tt(), V2) -> activate^#(V2)
753.12/208.07	        , U15^#(tt(), V2) -> isNat^#(activate(V2))
753.12/208.07	        , isNat^#(n__plus(V1, V2)) ->
753.12/208.07	          U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.07	        , isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.12/208.07	        , isNat^#(n__plus(V1, V2)) -> activate^#(V1)
753.12/208.07	        , isNat^#(n__plus(V1, V2)) -> activate^#(V2)
753.12/208.07	        , isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.12/208.07	        , isNat^#(n__s(V1)) -> activate^#(V1)
753.12/208.07	        , isNat^#(n__s(V1)) -> U21^#(isNatKind(activate(V1)), activate(V1))
753.12/208.07	        , U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.12/208.07	        , U21^#(tt(), V1) -> activate^#(V1)
753.12/208.07	        , U21^#(tt(), V1) -> U22^#(isNatKind(activate(V1)), activate(V1))
753.12/208.07	        , U22^#(tt(), V1) -> activate^#(V1)
753.12/208.07	        , U22^#(tt(), V1) -> isNat^#(activate(V1))
753.12/208.07	        , U51^#(tt(), N) -> isNatKind^#(activate(N))
753.12/208.07	        , U51^#(tt(), N) -> activate^#(N)
753.12/208.07	        , U51^#(tt(), N) -> U52^#(isNatKind(activate(N)), activate(N))
753.12/208.07	        , U52^#(tt(), N) -> activate^#(N)
753.12/208.07	        , U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.12/208.07	        , U61^#(tt(), M, N) -> activate^#(M)
753.12/208.07	        , U61^#(tt(), M, N) -> activate^#(N)
753.12/208.07	        , U61^#(tt(), M, N) ->
753.12/208.07	          U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.12/208.07	        , U62^#(tt(), M, N) -> activate^#(M)
753.12/208.07	        , U62^#(tt(), M, N) -> activate^#(N)
753.12/208.07	        , U62^#(tt(), M, N) -> isNat^#(activate(N))
753.12/208.07	        , U62^#(tt(), M, N) ->
753.12/208.07	          U63^#(isNat(activate(N)), activate(M), activate(N))
753.12/208.07	        , U63^#(tt(), M, N) -> isNatKind^#(activate(N))
753.12/208.07	        , U63^#(tt(), M, N) -> activate^#(M)
753.12/208.07	        , U63^#(tt(), M, N) -> activate^#(N)
753.12/208.07	        , U63^#(tt(), M, N) ->
753.12/208.07	          U64^#(isNatKind(activate(N)), activate(M), activate(N))
753.12/208.07	        , U64^#(tt(), M, N) -> activate^#(M)
753.12/208.07	        , U64^#(tt(), M, N) -> activate^#(N) }
753.12/208.07	      Weak Trs:
753.12/208.07	        { U11(tt(), V1, V2) ->
753.12/208.07	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.07	        , U12(tt(), V1, V2) ->
753.12/208.07	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.07	        , isNatKind(n__0()) -> tt()
753.12/208.07	        , isNatKind(n__plus(V1, V2)) ->
753.12/208.07	          U31(isNatKind(activate(V1)), activate(V2))
753.12/208.07	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.07	        , activate(X) -> X
753.12/208.07	        , activate(n__0()) -> 0()
753.12/208.07	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.07	        , activate(n__s(X)) -> s(activate(X))
753.12/208.07	        , U13(tt(), V1, V2) ->
753.12/208.07	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.07	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.07	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.07	        , isNat(n__0()) -> tt()
753.12/208.07	        , isNat(n__plus(V1, V2)) ->
753.12/208.07	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.07	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.07	        , U16(tt()) -> tt()
753.12/208.07	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.07	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.07	        , U23(tt()) -> tt()
753.12/208.07	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.07	        , U32(tt()) -> tt()
753.12/208.07	        , U41(tt()) -> tt()
753.12/208.07	        , s(X) -> n__s(X)
753.12/208.07	        , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.07	        , 0() -> n__0() }
753.12/208.07	      Obligation:
753.12/208.07	        innermost runtime complexity
753.12/208.07	      Answer:
753.12/208.07	        YES(O(1),O(n^2))
753.12/208.07	      
753.12/208.07	      We use the processor 'matrix interpretation of dimension 2' to
753.12/208.07	      orient following rules strictly.
753.12/208.07	      
753.12/208.07	      DPs:
753.12/208.07	        { 1: activate^#(n__plus(X1, X2)) ->
753.12/208.07	             c_5(activate^#(X1), activate^#(X2))
753.12/208.07	        , 12: activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.07	        , 28: isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.12/208.07	        , 29: isNat^#(n__s(V1)) -> activate^#(V1)
753.12/208.07	        , 30: isNat^#(n__s(V1)) ->
753.12/208.07	              U21^#(isNatKind(activate(V1)), activate(V1))
753.12/208.07	        , 36: U51^#(tt(), N) -> isNatKind^#(activate(N))
753.12/208.07	        , 37: U51^#(tt(), N) -> activate^#(N)
753.12/208.07	        , 38: U51^#(tt(), N) -> U52^#(isNatKind(activate(N)), activate(N))
753.12/208.07	        , 40: U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.12/208.07	        , 41: U61^#(tt(), M, N) -> activate^#(M)
753.12/208.07	        , 42: U61^#(tt(), M, N) -> activate^#(N)
753.12/208.07	        , 43: U61^#(tt(), M, N) ->
753.12/208.07	              U62^#(isNatKind(activate(M)), activate(M), activate(N)) }
753.12/208.07	      
753.12/208.07	      Sub-proof:
753.12/208.07	      ----------
753.12/208.07	        The following argument positions are usable:
753.12/208.07	          Uargs(c_3) = {1, 2, 3, 4}, Uargs(c_4) = {1, 2},
753.12/208.07	          Uargs(c_5) = {1, 2}, Uargs(c_6) = {1}, Uargs(c_8) = {1, 2}
753.12/208.07	        
753.12/208.07	        TcT has computed the following constructor-based matrix
753.12/208.07	        interpretation satisfying not(EDA).
753.12/208.07	        
753.12/208.07	              [U11](x1, x2, x3) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                           [tt] = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	              [U12](x1, x2, x3) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                [isNatKind](x1) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                 [activate](x1) = [1 0] x1 + [0]                               
753.12/208.07	                                  [0 1]      [0]                               
753.12/208.07	                                                                               
753.12/208.07	              [U13](x1, x2, x3) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	              [U14](x1, x2, x3) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                  [U15](x1, x2) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                    [isNat](x1) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                      [U16](x1) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                  [U21](x1, x2) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                  [U22](x1, x2) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                      [U23](x1) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                  [U31](x1, x2) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                      [U32](x1) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                      [U41](x1) = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	                        [s](x1) = [1 2] x1 + [1]                               
753.12/208.07	                                  [0 1]      [4]                               
753.12/208.07	                                                                               
753.12/208.07	                 [plus](x1, x2) = [1 2] x1 + [1 2] x2 + [0]                    
753.12/208.07	                                  [0 1]      [0 1]      [2]                    
753.12/208.07	                                                                               
753.12/208.07	                         [n__0] = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	              [n__plus](x1, x2) = [1 2] x1 + [1 2] x2 + [0]                    
753.12/208.07	                                  [0 1]      [0 1]      [2]                    
753.12/208.07	                                                                               
753.12/208.07	                     [n__s](x1) = [1 2] x1 + [1]                               
753.12/208.07	                                  [0 1]      [4]                               
753.12/208.07	                                                                               
753.12/208.07	                            [0] = [0]                                          
753.12/208.07	                                  [0]                                          
753.12/208.07	                                                                               
753.12/208.07	            [U11^#](x1, x2, x3) = [1 2] x2 + [1 2] x3 + [0]                    
753.12/208.07	                                  [4 1]      [4 1]      [4]                    
753.12/208.07	                                                                               
753.12/208.07	            [U12^#](x1, x2, x3) = [1 1] x2 + [1 2] x3 + [0]                    
753.12/208.07	                                  [4 1]      [4 1]      [4]                    
753.12/208.07	                                                                               
753.12/208.07	              [isNatKind^#](x1) = [1 0] x1 + [0]                               
753.12/208.07	                                  [0 0]      [0]                               
753.12/208.07	                                                                               
753.12/208.07	               [activate^#](x1) = [0 1] x1 + [0]                               
753.12/208.07	                                  [0 0]      [0]                               
753.12/208.07	                                                                               
753.12/208.07	            [U13^#](x1, x2, x3) = [1 1] x2 + [1 1] x3 + [0]                    
753.12/208.07	                                  [4 1]      [4 1]      [2]                    
753.12/208.07	                                                                               
753.12/208.07	                [U31^#](x1, x2) = [1 1] x2 + [0]                               
753.12/208.07	                                  [4 4]      [4]                               
753.12/208.07	                                                                               
753.12/208.07	            [U14^#](x1, x2, x3) = [1 1] x2 + [1 1] x3 + [0]                    
753.12/208.07	                                  [4 1]      [4 1]      [2]                    
753.12/208.07	                                                                               
753.12/208.07	                [U15^#](x1, x2) = [1 1] x2 + [0]                               
753.12/208.07	                                  [4 1]      [2]                               
753.12/208.07	                                                                               
753.12/208.07	                  [isNat^#](x1) = [1 0] x1 + [0]                               
753.12/208.07	                                  [4 1]      [2]                               
753.12/208.07	                                                                               
753.12/208.07	                [U21^#](x1, x2) = [1 2] x2 + [0]                               
753.12/208.07	                                  [4 1]      [2]                               
753.12/208.07	                                                                               
753.12/208.07	                [U22^#](x1, x2) = [1 2] x2 + [0]                               
753.12/208.07	                                  [4 1]      [2]                               
753.12/208.07	                                                                               
753.12/208.07	                [U51^#](x1, x2) = [7 4] x1 + [7 7] x2 + [7]                    
753.12/208.07	                                  [5 7]      [7 7]      [3]                    
753.12/208.07	                                                                               
753.12/208.07	                [U52^#](x1, x2) = [0 4] x2 + [0]                               
753.12/208.07	                                  [0 0]      [0]                               
753.12/208.07	                                                                               
753.12/208.07	            [U61^#](x1, x2, x3) = [7 4] x1 + [7 7] x2 + [7 7] x3 + [7]         
753.12/208.07	                                  [7 7]      [7 7]      [7 7]      [7]         
753.12/208.07	                                                                               
753.12/208.07	            [U62^#](x1, x2, x3) = [0 4] x2 + [2 2] x3 + [0]                    
753.12/208.07	                                  [0 0]      [4 1]      [2]                    
753.12/208.07	                                                                               
753.12/208.07	            [U63^#](x1, x2, x3) = [0 4] x2 + [1 1] x3 + [0]                    
753.12/208.07	                                  [0 0]      [0 0]      [0]                    
753.12/208.07	                                                                               
753.12/208.07	            [U64^#](x1, x2, x3) = [0 4] x2 + [0 1] x3 + [0]                    
753.12/208.07	                                  [0 0]      [0 0]      [0]                    
753.12/208.07	                                                                               
753.12/208.07	          [c_3](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [2 0] x3 + [1          
753.12/208.07	                                                                    0] x4 + [0]
753.12/208.07	                                  [0 0]      [0 0]      [0 0]      [0          
753.12/208.07	                                                                    0]      [0]
753.12/208.07	                                                                               
753.12/208.07	                  [c_4](x1, x2) = [1 0] x1 + [2 0] x2 + [1]                    
753.12/208.07	                                  [0 0]      [0 0]      [0]                    
753.12/208.07	                                                                               
753.12/208.07	                  [c_5](x1, x2) = [1 0] x1 + [1 0] x2 + [1]                    
753.12/208.07	                                  [0 0]      [0 0]      [0]                    
753.12/208.07	                                                                               
753.12/208.07	                      [c_6](x1) = [1 0] x1 + [3]                               
753.12/208.07	                                  [0 0]      [0]                               
753.12/208.07	                                                                               
753.12/208.07	                  [c_8](x1, x2) = [1 0] x1 + [1 0] x2 + [0]                    
753.12/208.07	                                  [0 0]      [0 0]      [3]                    
753.12/208.07	        
753.12/208.07	        The order satisfies the following ordering constraints:
753.12/208.07	        
753.12/208.07	                     [U11(tt(), V1, V2)] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [U12(isNatKind(activate(V1)), activate(V1), activate(V2))]  
753.12/208.07	                                                                                                        
753.12/208.07	                     [U12(tt(), V1, V2)] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [U13(isNatKind(activate(V2)), activate(V1), activate(V2))]  
753.12/208.07	                                                                                                        
753.12/208.07	                     [isNatKind(n__0())] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [tt()]                                                      
753.12/208.07	                                                                                                        
753.12/208.07	            [isNatKind(n__plus(V1, V2))] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [U31(isNatKind(activate(V1)), activate(V2))]                
753.12/208.07	                                                                                                        
753.12/208.07	                   [isNatKind(n__s(V1))] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [U41(isNatKind(activate(V1)))]                              
753.12/208.07	                                                                                                        
753.12/208.07	                           [activate(X)] =  [1 0] X + [0]                                               
753.12/208.07	                                            [0 1]     [0]                                               
753.12/208.07	                                         >= [1 0] X + [0]                                               
753.12/208.07	                                            [0 1]     [0]                                               
753.12/208.07	                                         =  [X]                                                         
753.12/208.07	                                                                                                        
753.12/208.07	                      [activate(n__0())] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [0()]                                                       
753.12/208.07	                                                                                                        
753.12/208.07	             [activate(n__plus(X1, X2))] =  [1 2] X1 + [1 2] X2 + [0]                                   
753.12/208.07	                                            [0 1]      [0 1]      [2]                                   
753.12/208.07	                                         >= [1 2] X1 + [1 2] X2 + [0]                                   
753.12/208.07	                                            [0 1]      [0 1]      [2]                                   
753.12/208.07	                                         =  [plus(activate(X1), activate(X2))]                          
753.12/208.07	                                                                                                        
753.12/208.07	                     [activate(n__s(X))] =  [1 2] X + [1]                                               
753.12/208.07	                                            [0 1]     [4]                                               
753.12/208.07	                                         >= [1 2] X + [1]                                               
753.12/208.07	                                            [0 1]     [4]                                               
753.12/208.07	                                         =  [s(activate(X))]                                            
753.12/208.07	                                                                                                        
753.12/208.07	                     [U13(tt(), V1, V2)] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [U14(isNatKind(activate(V2)), activate(V1), activate(V2))]  
753.12/208.07	                                                                                                        
753.12/208.07	                     [U14(tt(), V1, V2)] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [U15(isNat(activate(V1)), activate(V2))]                    
753.12/208.07	                                                                                                        
753.12/208.07	                         [U15(tt(), V2)] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [U16(isNat(activate(V2)))]                                  
753.12/208.07	                                                                                                        
753.12/208.07	                         [isNat(n__0())] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [tt()]                                                      
753.12/208.07	                                                                                                        
753.12/208.07	                [isNat(n__plus(V1, V2))] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [U11(isNatKind(activate(V1)), activate(V1), activate(V2))]  
753.12/208.07	                                                                                                        
753.12/208.07	                       [isNat(n__s(V1))] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [U21(isNatKind(activate(V1)), activate(V1))]                
753.12/208.07	                                                                                                        
753.12/208.07	                             [U16(tt())] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [tt()]                                                      
753.12/208.07	                                                                                                        
753.12/208.07	                         [U21(tt(), V1)] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [U22(isNatKind(activate(V1)), activate(V1))]                
753.12/208.07	                                                                                                        
753.12/208.07	                         [U22(tt(), V1)] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [U23(isNat(activate(V1)))]                                  
753.12/208.07	                                                                                                        
753.12/208.07	                             [U23(tt())] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         =  [tt()]                                                      
753.12/208.07	                                                                                                        
753.12/208.07	                         [U31(tt(), V2)] =  [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.07	                                         >= [0]                                                         
753.12/208.07	                                            [0]                                                         
753.12/208.08	                                         =  [U32(isNatKind(activate(V2)))]                              
753.12/208.08	                                                                                                        
753.12/208.08	                             [U32(tt())] =  [0]                                                         
753.12/208.08	                                            [0]                                                         
753.12/208.08	                                         >= [0]                                                         
753.12/208.08	                                            [0]                                                         
753.12/208.08	                                         =  [tt()]                                                      
753.12/208.08	                                                                                                        
753.12/208.08	                             [U41(tt())] =  [0]                                                         
753.12/208.08	                                            [0]                                                         
753.12/208.08	                                         >= [0]                                                         
753.12/208.08	                                            [0]                                                         
753.12/208.08	                                         =  [tt()]                                                      
753.12/208.08	                                                                                                        
753.12/208.08	                                  [s(X)] =  [1 2] X + [1]                                               
753.12/208.08	                                            [0 1]     [4]                                               
753.12/208.08	                                         >= [1 2] X + [1]                                               
753.12/208.08	                                            [0 1]     [4]                                               
753.12/208.08	                                         =  [n__s(X)]                                                   
753.12/208.08	                                                                                                        
753.12/208.08	                          [plus(X1, X2)] =  [1 2] X1 + [1 2] X2 + [0]                                   
753.12/208.08	                                            [0 1]      [0 1]      [2]                                   
753.12/208.08	                                         >= [1 2] X1 + [1 2] X2 + [0]                                   
753.12/208.08	                                            [0 1]      [0 1]      [2]                                   
753.12/208.08	                                         =  [n__plus(X1, X2)]                                           
753.12/208.08	                                                                                                        
753.12/208.08	                                   [0()] =  [0]                                                         
753.12/208.08	                                            [0]                                                         
753.12/208.08	                                         >= [0]                                                         
753.12/208.08	                                            [0]                                                         
753.12/208.08	                                         =  [n__0()]                                                    
753.12/208.08	                                                                                                        
753.12/208.08	                   [U11^#(tt(), V1, V2)] =  [1 2] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [4]                                   
753.12/208.08	                                         >= [1 1] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [4]                                   
753.12/208.08	                                         =  [U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))]
753.12/208.08	                                                                                                        
753.12/208.08	                   [U11^#(tt(), V1, V2)] =  [1 2] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [4]                                   
753.12/208.08	                                         >= [1 0] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [isNatKind^#(activate(V1))]                                 
753.12/208.08	                                                                                                        
753.12/208.08	                   [U11^#(tt(), V1, V2)] =  [1 2] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [4]                                   
753.12/208.08	                                         >= [0 1] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V1)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                   [U11^#(tt(), V1, V2)] =  [1 2] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [4]                                   
753.12/208.08	                                         >= [0 1] V2 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V2)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                   [U12^#(tt(), V1, V2)] =  [1 1] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [4]                                   
753.12/208.08	                                         >= [1 0] V2 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [isNatKind^#(activate(V2))]                                 
753.12/208.08	                                                                                                        
753.12/208.08	                   [U12^#(tt(), V1, V2)] =  [1 1] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [4]                                   
753.12/208.08	                                         >= [0 1] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V1)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                   [U12^#(tt(), V1, V2)] =  [1 1] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [4]                                   
753.12/208.08	                                         >= [0 1] V2 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V2)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                   [U12^#(tt(), V1, V2)] =  [1 1] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [4]                                   
753.12/208.08	                                         >= [1 1] V1 + [1 1] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [2]                                   
753.12/208.08	                                         =  [U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))]
753.12/208.08	                                                                                                        
753.12/208.08	          [isNatKind^#(n__plus(V1, V2))] =  [1 2] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [0 0]      [0 0]      [0]                                   
753.12/208.08	                                         >= [1 2] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [0 0]      [0 0]      [0]                                   
753.12/208.08	                                         =  [c_3(U31^#(isNatKind(activate(V1)), activate(V2)),          
753.12/208.08	                                                 isNatKind^#(activate(V1)),                             
753.12/208.08	                                                 activate^#(V1),                                        
753.12/208.08	                                                 activate^#(V2))]                                       
753.12/208.08	                                                                                                        
753.12/208.08	                 [isNatKind^#(n__s(V1))] =  [1 2] V1 + [1]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         >= [1 2] V1 + [1]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [c_4(isNatKind^#(activate(V1)), activate^#(V1))]            
753.12/208.08	                                                                                                        
753.12/208.08	           [activate^#(n__plus(X1, X2))] =  [0 1] X1 + [0 1] X2 + [2]                                   
753.12/208.08	                                            [0 0]      [0 0]      [0]                                   
753.12/208.08	                                         >  [0 1] X1 + [0 1] X2 + [1]                                   
753.12/208.08	                                            [0 0]      [0 0]      [0]                                   
753.12/208.08	                                         =  [c_5(activate^#(X1), activate^#(X2))]                       
753.12/208.08	                                                                                                        
753.12/208.08	                   [activate^#(n__s(X))] =  [0 1] X + [4]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         >  [0 1] X + [3]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [c_6(activate^#(X))]                                        
753.12/208.08	                                                                                                        
753.12/208.08	                   [U13^#(tt(), V1, V2)] =  [1 1] V1 + [1 1] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [2]                                   
753.12/208.08	                                         >= [1 0] V2 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [isNatKind^#(activate(V2))]                                 
753.12/208.08	                                                                                                        
753.12/208.08	                   [U13^#(tt(), V1, V2)] =  [1 1] V1 + [1 1] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [2]                                   
753.12/208.08	                                         >= [0 1] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V1)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                   [U13^#(tt(), V1, V2)] =  [1 1] V1 + [1 1] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [2]                                   
753.12/208.08	                                         >= [0 1] V2 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V2)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                   [U13^#(tt(), V1, V2)] =  [1 1] V1 + [1 1] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [2]                                   
753.12/208.08	                                         >= [1 1] V1 + [1 1] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [2]                                   
753.12/208.08	                                         =  [U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))]
753.12/208.08	                                                                                                        
753.12/208.08	                       [U31^#(tt(), V2)] =  [1 1] V2 + [0]                                              
753.12/208.08	                                            [4 4]      [4]                                              
753.12/208.08	                                         >= [1 1] V2 + [0]                                              
753.12/208.08	                                            [0 0]      [3]                                              
753.12/208.08	                                         =  [c_8(isNatKind^#(activate(V2)), activate^#(V2))]            
753.12/208.08	                                                                                                        
753.12/208.08	                   [U14^#(tt(), V1, V2)] =  [1 1] V1 + [1 1] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [2]                                   
753.12/208.08	                                         >= [0 1] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V1)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                   [U14^#(tt(), V1, V2)] =  [1 1] V1 + [1 1] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [2]                                   
753.12/208.08	                                         >= [0 1] V2 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V2)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                   [U14^#(tt(), V1, V2)] =  [1 1] V1 + [1 1] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [2]                                   
753.12/208.08	                                         >= [1 1] V2 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         =  [U15^#(isNat(activate(V1)), activate(V2))]                  
753.12/208.08	                                                                                                        
753.12/208.08	                   [U14^#(tt(), V1, V2)] =  [1 1] V1 + [1 1] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [2]                                   
753.12/208.08	                                         >= [1 0] V1 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         =  [isNat^#(activate(V1))]                                     
753.12/208.08	                                                                                                        
753.12/208.08	                       [U15^#(tt(), V2)] =  [1 1] V2 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         >= [0 1] V2 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V2)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                       [U15^#(tt(), V2)] =  [1 1] V2 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         >= [1 0] V2 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         =  [isNat^#(activate(V2))]                                     
753.12/208.08	                                                                                                        
753.12/208.08	              [isNat^#(n__plus(V1, V2))] =  [1 2] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 9]      [4 9]      [4]                                   
753.12/208.08	                                         >= [1 2] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 1]      [4 1]      [4]                                   
753.12/208.08	                                         =  [U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))]
753.12/208.08	                                                                                                        
753.12/208.08	              [isNat^#(n__plus(V1, V2))] =  [1 2] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 9]      [4 9]      [4]                                   
753.12/208.08	                                         >= [1 0] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [isNatKind^#(activate(V1))]                                 
753.12/208.08	                                                                                                        
753.12/208.08	              [isNat^#(n__plus(V1, V2))] =  [1 2] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 9]      [4 9]      [4]                                   
753.12/208.08	                                         >= [0 1] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V1)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	              [isNat^#(n__plus(V1, V2))] =  [1 2] V1 + [1 2] V2 + [0]                                   
753.12/208.08	                                            [4 9]      [4 9]      [4]                                   
753.12/208.08	                                         >= [0 1] V2 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V2)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                     [isNat^#(n__s(V1))] =  [1 2] V1 + [1]                                              
753.12/208.08	                                            [4 9]      [10]                                             
753.12/208.08	                                         >  [1 0] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [isNatKind^#(activate(V1))]                                 
753.12/208.08	                                                                                                        
753.12/208.08	                     [isNat^#(n__s(V1))] =  [1 2] V1 + [1]                                              
753.12/208.08	                                            [4 9]      [10]                                             
753.12/208.08	                                         >  [0 1] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V1)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                     [isNat^#(n__s(V1))] =  [1 2] V1 + [1]                                              
753.12/208.08	                                            [4 9]      [10]                                             
753.12/208.08	                                         >  [1 2] V1 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         =  [U21^#(isNatKind(activate(V1)), activate(V1))]              
753.12/208.08	                                                                                                        
753.12/208.08	                       [U21^#(tt(), V1)] =  [1 2] V1 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         >= [1 0] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [isNatKind^#(activate(V1))]                                 
753.12/208.08	                                                                                                        
753.12/208.08	                       [U21^#(tt(), V1)] =  [1 2] V1 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         >= [0 1] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V1)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                       [U21^#(tt(), V1)] =  [1 2] V1 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         >= [1 2] V1 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         =  [U22^#(isNatKind(activate(V1)), activate(V1))]              
753.12/208.08	                                                                                                        
753.12/208.08	                       [U22^#(tt(), V1)] =  [1 2] V1 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         >= [0 1] V1 + [0]                                              
753.12/208.08	                                            [0 0]      [0]                                              
753.12/208.08	                                         =  [activate^#(V1)]                                            
753.12/208.08	                                                                                                        
753.12/208.08	                       [U22^#(tt(), V1)] =  [1 2] V1 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         >= [1 0] V1 + [0]                                              
753.12/208.08	                                            [4 1]      [2]                                              
753.12/208.08	                                         =  [isNat^#(activate(V1))]                                     
753.12/208.08	                                                                                                        
753.12/208.08	                        [U51^#(tt(), N)] =  [7 7] N + [7]                                               
753.12/208.08	                                            [7 7]     [3]                                               
753.12/208.08	                                         >  [1 0] N + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [isNatKind^#(activate(N))]                                  
753.12/208.08	                                                                                                        
753.12/208.08	                        [U51^#(tt(), N)] =  [7 7] N + [7]                                               
753.12/208.08	                                            [7 7]     [3]                                               
753.12/208.08	                                         >  [0 1] N + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [activate^#(N)]                                             
753.12/208.08	                                                                                                        
753.12/208.08	                        [U51^#(tt(), N)] =  [7 7] N + [7]                                               
753.12/208.08	                                            [7 7]     [3]                                               
753.12/208.08	                                         >  [0 4] N + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [U52^#(isNatKind(activate(N)), activate(N))]                
753.12/208.08	                                                                                                        
753.12/208.08	                        [U52^#(tt(), N)] =  [0 4] N + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         >= [0 1] N + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [activate^#(N)]                                             
753.12/208.08	                                                                                                        
753.12/208.08	                     [U61^#(tt(), M, N)] =  [7 7] N + [7 7] M + [7]                                     
753.12/208.08	                                            [7 7]     [7 7]     [7]                                     
753.12/208.08	                                         >  [1 0] M + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [isNatKind^#(activate(M))]                                  
753.12/208.08	                                                                                                        
753.12/208.08	                     [U61^#(tt(), M, N)] =  [7 7] N + [7 7] M + [7]                                     
753.12/208.08	                                            [7 7]     [7 7]     [7]                                     
753.12/208.08	                                         >  [0 1] M + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [activate^#(M)]                                             
753.12/208.08	                                                                                                        
753.12/208.08	                     [U61^#(tt(), M, N)] =  [7 7] N + [7 7] M + [7]                                     
753.12/208.08	                                            [7 7]     [7 7]     [7]                                     
753.12/208.08	                                         >  [0 1] N + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [activate^#(N)]                                             
753.12/208.08	                                                                                                        
753.12/208.08	                     [U61^#(tt(), M, N)] =  [7 7] N + [7 7] M + [7]                                     
753.12/208.08	                                            [7 7]     [7 7]     [7]                                     
753.12/208.08	                                         >  [2 2] N + [0 4] M + [0]                                     
753.12/208.08	                                            [4 1]     [0 0]     [2]                                     
753.12/208.08	                                         =  [U62^#(isNatKind(activate(M)), activate(M), activate(N))]   
753.12/208.08	                                                                                                        
753.12/208.08	                     [U62^#(tt(), M, N)] =  [2 2] N + [0 4] M + [0]                                     
753.12/208.08	                                            [4 1]     [0 0]     [2]                                     
753.12/208.08	                                         >= [0 1] M + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [activate^#(M)]                                             
753.12/208.08	                                                                                                        
753.12/208.08	                     [U62^#(tt(), M, N)] =  [2 2] N + [0 4] M + [0]                                     
753.12/208.08	                                            [4 1]     [0 0]     [2]                                     
753.12/208.08	                                         >= [0 1] N + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [activate^#(N)]                                             
753.12/208.08	                                                                                                        
753.12/208.08	                     [U62^#(tt(), M, N)] =  [2 2] N + [0 4] M + [0]                                     
753.12/208.08	                                            [4 1]     [0 0]     [2]                                     
753.12/208.08	                                         >= [1 0] N + [0]                                               
753.12/208.08	                                            [4 1]     [2]                                               
753.12/208.08	                                         =  [isNat^#(activate(N))]                                      
753.12/208.08	                                                                                                        
753.12/208.08	                     [U62^#(tt(), M, N)] =  [2 2] N + [0 4] M + [0]                                     
753.12/208.08	                                            [4 1]     [0 0]     [2]                                     
753.12/208.08	                                         >= [1 1] N + [0 4] M + [0]                                     
753.12/208.08	                                            [0 0]     [0 0]     [0]                                     
753.12/208.08	                                         =  [U63^#(isNat(activate(N)), activate(M), activate(N))]       
753.12/208.08	                                                                                                        
753.12/208.08	                     [U63^#(tt(), M, N)] =  [1 1] N + [0 4] M + [0]                                     
753.12/208.08	                                            [0 0]     [0 0]     [0]                                     
753.12/208.08	                                         >= [1 0] N + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [isNatKind^#(activate(N))]                                  
753.12/208.08	                                                                                                        
753.12/208.08	                     [U63^#(tt(), M, N)] =  [1 1] N + [0 4] M + [0]                                     
753.12/208.08	                                            [0 0]     [0 0]     [0]                                     
753.12/208.08	                                         >= [0 1] M + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [activate^#(M)]                                             
753.12/208.08	                                                                                                        
753.12/208.08	                     [U63^#(tt(), M, N)] =  [1 1] N + [0 4] M + [0]                                     
753.12/208.08	                                            [0 0]     [0 0]     [0]                                     
753.12/208.08	                                         >= [0 1] N + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [activate^#(N)]                                             
753.12/208.08	                                                                                                        
753.12/208.08	                     [U63^#(tt(), M, N)] =  [1 1] N + [0 4] M + [0]                                     
753.12/208.08	                                            [0 0]     [0 0]     [0]                                     
753.12/208.08	                                         >= [0 1] N + [0 4] M + [0]                                     
753.12/208.08	                                            [0 0]     [0 0]     [0]                                     
753.12/208.08	                                         =  [U64^#(isNatKind(activate(N)), activate(M), activate(N))]   
753.12/208.08	                                                                                                        
753.12/208.08	                     [U64^#(tt(), M, N)] =  [0 1] N + [0 4] M + [0]                                     
753.12/208.08	                                            [0 0]     [0 0]     [0]                                     
753.12/208.08	                                         >= [0 1] M + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [activate^#(M)]                                             
753.12/208.08	                                                                                                        
753.12/208.08	                     [U64^#(tt(), M, N)] =  [0 1] N + [0 4] M + [0]                                     
753.12/208.08	                                            [0 0]     [0 0]     [0]                                     
753.12/208.08	                                         >= [0 1] N + [0]                                               
753.12/208.08	                                            [0 0]     [0]                                               
753.12/208.08	                                         =  [activate^#(N)]                                             
753.12/208.08	                                                                                                        
753.12/208.08	      
753.12/208.08	      We return to the main proof. Consider the set of all dependency
753.12/208.08	      pairs
753.12/208.08	      
753.12/208.08	      :
753.12/208.08	        { 1: activate^#(n__plus(X1, X2)) ->
753.12/208.08	             c_5(activate^#(X1), activate^#(X2))
753.12/208.08	        , 2: U11^#(tt(), V1, V2) ->
753.12/208.08	             U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.08	        , 3: U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.12/208.08	        , 4: U11^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.08	        , 5: U11^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.08	        , 6: U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.08	        , 7: U12^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.08	        , 8: U12^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.08	        , 9: U12^#(tt(), V1, V2) ->
753.12/208.08	             U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.08	        , 10: isNatKind^#(n__plus(V1, V2)) ->
753.12/208.08	              c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.12/208.08	                  isNatKind^#(activate(V1)),
753.12/208.08	                  activate^#(V1),
753.12/208.08	                  activate^#(V2))
753.12/208.08	        , 11: isNatKind^#(n__s(V1)) ->
753.12/208.08	              c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.12/208.08	        , 12: activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.08	        , 13: U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.08	        , 14: U13^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.08	        , 15: U13^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.08	        , 16: U13^#(tt(), V1, V2) ->
753.12/208.08	              U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.08	        , 17: U31^#(tt(), V2) ->
753.12/208.08	              c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.12/208.08	        , 18: U14^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.08	        , 19: U14^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.08	        , 20: U14^#(tt(), V1, V2) ->
753.12/208.08	              U15^#(isNat(activate(V1)), activate(V2))
753.12/208.08	        , 21: U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.12/208.08	        , 22: U15^#(tt(), V2) -> activate^#(V2)
753.12/208.08	        , 23: U15^#(tt(), V2) -> isNat^#(activate(V2))
753.12/208.08	        , 24: isNat^#(n__plus(V1, V2)) ->
753.12/208.08	              U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.08	        , 25: isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.12/208.08	        , 26: isNat^#(n__plus(V1, V2)) -> activate^#(V1)
753.12/208.08	        , 27: isNat^#(n__plus(V1, V2)) -> activate^#(V2)
753.12/208.08	        , 28: isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.12/208.08	        , 29: isNat^#(n__s(V1)) -> activate^#(V1)
753.12/208.08	        , 30: isNat^#(n__s(V1)) ->
753.12/208.08	              U21^#(isNatKind(activate(V1)), activate(V1))
753.12/208.08	        , 31: U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.12/208.08	        , 32: U21^#(tt(), V1) -> activate^#(V1)
753.12/208.08	        , 33: U21^#(tt(), V1) ->
753.12/208.08	              U22^#(isNatKind(activate(V1)), activate(V1))
753.12/208.08	        , 34: U22^#(tt(), V1) -> activate^#(V1)
753.12/208.08	        , 35: U22^#(tt(), V1) -> isNat^#(activate(V1))
753.12/208.08	        , 36: U51^#(tt(), N) -> isNatKind^#(activate(N))
753.12/208.08	        , 37: U51^#(tt(), N) -> activate^#(N)
753.12/208.08	        , 38: U51^#(tt(), N) -> U52^#(isNatKind(activate(N)), activate(N))
753.12/208.08	        , 39: U52^#(tt(), N) -> activate^#(N)
753.12/208.08	        , 40: U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.12/208.08	        , 41: U61^#(tt(), M, N) -> activate^#(M)
753.12/208.08	        , 42: U61^#(tt(), M, N) -> activate^#(N)
753.12/208.08	        , 43: U61^#(tt(), M, N) ->
753.12/208.08	              U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.12/208.08	        , 44: U62^#(tt(), M, N) -> activate^#(M)
753.12/208.08	        , 45: U62^#(tt(), M, N) -> activate^#(N)
753.12/208.08	        , 46: U62^#(tt(), M, N) -> isNat^#(activate(N))
753.12/208.08	        , 47: U62^#(tt(), M, N) ->
753.12/208.08	              U63^#(isNat(activate(N)), activate(M), activate(N))
753.12/208.09	        , 48: U63^#(tt(), M, N) -> isNatKind^#(activate(N))
753.12/208.09	        , 49: U63^#(tt(), M, N) -> activate^#(M)
753.12/208.09	        , 50: U63^#(tt(), M, N) -> activate^#(N)
753.12/208.09	        , 51: U63^#(tt(), M, N) ->
753.12/208.09	              U64^#(isNatKind(activate(N)), activate(M), activate(N))
753.12/208.09	        , 52: U64^#(tt(), M, N) -> activate^#(M)
753.12/208.09	        , 53: U64^#(tt(), M, N) -> activate^#(N) }
753.12/208.09	      
753.12/208.09	      Processor 'matrix interpretation of dimension 2' induces the
753.12/208.09	      complexity certificate YES(?,O(n^2)) on application of dependency
753.12/208.09	      pairs {1,12,28,29,30,36,37,38,40,41,42,43}. These cover all
753.12/208.09	      (indirect) predecessors of dependency pairs
753.12/208.09	      {1,12,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53},
753.12/208.09	      their number of application is equally bounded. The dependency
753.12/208.09	      pairs are shifted into the weak component.
753.12/208.09	      
753.12/208.09	      We are left with following problem, upon which TcT provides the
753.12/208.09	      certificate YES(O(1),O(1)).
753.12/208.09	      
753.12/208.09	      Weak DPs:
753.12/208.09	        { U11^#(tt(), V1, V2) ->
753.12/208.09	          U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.09	        , U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.12/208.09	        , U11^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.09	        , U11^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.09	        , U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.09	        , U12^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.09	        , U12^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.09	        , U12^#(tt(), V1, V2) ->
753.12/208.09	          U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.09	        , isNatKind^#(n__plus(V1, V2)) ->
753.12/208.09	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.12/208.09	              isNatKind^#(activate(V1)),
753.12/208.09	              activate^#(V1),
753.12/208.09	              activate^#(V2))
753.12/208.09	        , isNatKind^#(n__s(V1)) ->
753.12/208.09	          c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.12/208.09	        , activate^#(n__plus(X1, X2)) ->
753.12/208.09	          c_5(activate^#(X1), activate^#(X2))
753.12/208.09	        , activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.09	        , U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.09	        , U13^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.09	        , U13^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.09	        , U13^#(tt(), V1, V2) ->
753.12/208.09	          U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.09	        , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.12/208.09	        , U14^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.09	        , U14^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.09	        , U14^#(tt(), V1, V2) -> U15^#(isNat(activate(V1)), activate(V2))
753.12/208.09	        , U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.12/208.09	        , U15^#(tt(), V2) -> activate^#(V2)
753.12/208.09	        , U15^#(tt(), V2) -> isNat^#(activate(V2))
753.12/208.09	        , isNat^#(n__plus(V1, V2)) ->
753.12/208.09	          U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.09	        , isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.12/208.09	        , isNat^#(n__plus(V1, V2)) -> activate^#(V1)
753.12/208.09	        , isNat^#(n__plus(V1, V2)) -> activate^#(V2)
753.12/208.09	        , isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.12/208.09	        , isNat^#(n__s(V1)) -> activate^#(V1)
753.12/208.09	        , isNat^#(n__s(V1)) -> U21^#(isNatKind(activate(V1)), activate(V1))
753.12/208.09	        , U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.12/208.09	        , U21^#(tt(), V1) -> activate^#(V1)
753.12/208.09	        , U21^#(tt(), V1) -> U22^#(isNatKind(activate(V1)), activate(V1))
753.12/208.09	        , U22^#(tt(), V1) -> activate^#(V1)
753.12/208.09	        , U22^#(tt(), V1) -> isNat^#(activate(V1))
753.12/208.09	        , U51^#(tt(), N) -> isNatKind^#(activate(N))
753.12/208.09	        , U51^#(tt(), N) -> activate^#(N)
753.12/208.09	        , U51^#(tt(), N) -> U52^#(isNatKind(activate(N)), activate(N))
753.12/208.09	        , U52^#(tt(), N) -> activate^#(N)
753.12/208.09	        , U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.12/208.09	        , U61^#(tt(), M, N) -> activate^#(M)
753.12/208.09	        , U61^#(tt(), M, N) -> activate^#(N)
753.12/208.09	        , U61^#(tt(), M, N) ->
753.12/208.09	          U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.12/208.09	        , U62^#(tt(), M, N) -> activate^#(M)
753.12/208.09	        , U62^#(tt(), M, N) -> activate^#(N)
753.12/208.09	        , U62^#(tt(), M, N) -> isNat^#(activate(N))
753.12/208.09	        , U62^#(tt(), M, N) ->
753.12/208.09	          U63^#(isNat(activate(N)), activate(M), activate(N))
753.12/208.09	        , U63^#(tt(), M, N) -> isNatKind^#(activate(N))
753.12/208.09	        , U63^#(tt(), M, N) -> activate^#(M)
753.12/208.09	        , U63^#(tt(), M, N) -> activate^#(N)
753.12/208.09	        , U63^#(tt(), M, N) ->
753.12/208.09	          U64^#(isNatKind(activate(N)), activate(M), activate(N))
753.12/208.09	        , U64^#(tt(), M, N) -> activate^#(M)
753.12/208.09	        , U64^#(tt(), M, N) -> activate^#(N) }
753.12/208.09	      Weak Trs:
753.12/208.09	        { U11(tt(), V1, V2) ->
753.12/208.09	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.09	        , U12(tt(), V1, V2) ->
753.12/208.09	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.09	        , isNatKind(n__0()) -> tt()
753.12/208.09	        , isNatKind(n__plus(V1, V2)) ->
753.12/208.09	          U31(isNatKind(activate(V1)), activate(V2))
753.12/208.09	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.09	        , activate(X) -> X
753.12/208.09	        , activate(n__0()) -> 0()
753.12/208.09	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.09	        , activate(n__s(X)) -> s(activate(X))
753.12/208.09	        , U13(tt(), V1, V2) ->
753.12/208.09	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.09	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.09	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.09	        , isNat(n__0()) -> tt()
753.12/208.09	        , isNat(n__plus(V1, V2)) ->
753.12/208.09	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.09	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.09	        , U16(tt()) -> tt()
753.12/208.09	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.09	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.09	        , U23(tt()) -> tt()
753.12/208.09	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.09	        , U32(tt()) -> tt()
753.12/208.09	        , U41(tt()) -> tt()
753.12/208.09	        , s(X) -> n__s(X)
753.12/208.09	        , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.09	        , 0() -> n__0() }
753.12/208.09	      Obligation:
753.12/208.09	        innermost runtime complexity
753.12/208.09	      Answer:
753.12/208.09	        YES(O(1),O(1))
753.12/208.09	      
753.12/208.09	      The following weak DPs constitute a sub-graph of the DG that is
753.12/208.09	      closed under successors. The DPs are removed.
753.12/208.09	      
753.12/208.09	      { U11^#(tt(), V1, V2) ->
753.12/208.09	        U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.09	      , U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.12/208.09	      , U11^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.09	      , U11^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.09	      , U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.09	      , U12^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.09	      , U12^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.09	      , U12^#(tt(), V1, V2) ->
753.12/208.09	        U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.09	      , isNatKind^#(n__plus(V1, V2)) ->
753.12/208.09	        c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.12/208.09	            isNatKind^#(activate(V1)),
753.12/208.09	            activate^#(V1),
753.12/208.09	            activate^#(V2))
753.12/208.09	      , isNatKind^#(n__s(V1)) ->
753.12/208.09	        c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.12/208.09	      , activate^#(n__plus(X1, X2)) ->
753.12/208.09	        c_5(activate^#(X1), activate^#(X2))
753.12/208.09	      , activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.09	      , U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.12/208.09	      , U13^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.09	      , U13^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.09	      , U13^#(tt(), V1, V2) ->
753.12/208.09	        U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.09	      , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.12/208.09	      , U14^#(tt(), V1, V2) -> activate^#(V1)
753.12/208.09	      , U14^#(tt(), V1, V2) -> activate^#(V2)
753.12/208.09	      , U14^#(tt(), V1, V2) -> U15^#(isNat(activate(V1)), activate(V2))
753.12/208.09	      , U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.12/208.09	      , U15^#(tt(), V2) -> activate^#(V2)
753.12/208.09	      , U15^#(tt(), V2) -> isNat^#(activate(V2))
753.12/208.09	      , isNat^#(n__plus(V1, V2)) ->
753.12/208.09	        U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.09	      , isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.12/208.09	      , isNat^#(n__plus(V1, V2)) -> activate^#(V1)
753.12/208.09	      , isNat^#(n__plus(V1, V2)) -> activate^#(V2)
753.12/208.09	      , isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.12/208.09	      , isNat^#(n__s(V1)) -> activate^#(V1)
753.12/208.09	      , isNat^#(n__s(V1)) -> U21^#(isNatKind(activate(V1)), activate(V1))
753.12/208.09	      , U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.12/208.09	      , U21^#(tt(), V1) -> activate^#(V1)
753.12/208.09	      , U21^#(tt(), V1) -> U22^#(isNatKind(activate(V1)), activate(V1))
753.12/208.09	      , U22^#(tt(), V1) -> activate^#(V1)
753.12/208.09	      , U22^#(tt(), V1) -> isNat^#(activate(V1))
753.12/208.09	      , U51^#(tt(), N) -> isNatKind^#(activate(N))
753.12/208.09	      , U51^#(tt(), N) -> activate^#(N)
753.12/208.09	      , U51^#(tt(), N) -> U52^#(isNatKind(activate(N)), activate(N))
753.12/208.09	      , U52^#(tt(), N) -> activate^#(N)
753.12/208.09	      , U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.12/208.09	      , U61^#(tt(), M, N) -> activate^#(M)
753.12/208.09	      , U61^#(tt(), M, N) -> activate^#(N)
753.12/208.09	      , U61^#(tt(), M, N) ->
753.12/208.09	        U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.12/208.09	      , U62^#(tt(), M, N) -> activate^#(M)
753.12/208.09	      , U62^#(tt(), M, N) -> activate^#(N)
753.12/208.09	      , U62^#(tt(), M, N) -> isNat^#(activate(N))
753.12/208.09	      , U62^#(tt(), M, N) ->
753.12/208.09	        U63^#(isNat(activate(N)), activate(M), activate(N))
753.12/208.09	      , U63^#(tt(), M, N) -> isNatKind^#(activate(N))
753.12/208.09	      , U63^#(tt(), M, N) -> activate^#(M)
753.12/208.09	      , U63^#(tt(), M, N) -> activate^#(N)
753.12/208.09	      , U63^#(tt(), M, N) ->
753.12/208.09	        U64^#(isNatKind(activate(N)), activate(M), activate(N))
753.12/208.09	      , U64^#(tt(), M, N) -> activate^#(M)
753.12/208.09	      , U64^#(tt(), M, N) -> activate^#(N) }
753.12/208.09	      
753.12/208.09	      We are left with following problem, upon which TcT provides the
753.12/208.09	      certificate YES(O(1),O(1)).
753.12/208.09	      
753.12/208.09	      Weak Trs:
753.12/208.09	        { U11(tt(), V1, V2) ->
753.12/208.09	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.09	        , U12(tt(), V1, V2) ->
753.12/208.09	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.09	        , isNatKind(n__0()) -> tt()
753.12/208.09	        , isNatKind(n__plus(V1, V2)) ->
753.12/208.09	          U31(isNatKind(activate(V1)), activate(V2))
753.12/208.09	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.09	        , activate(X) -> X
753.12/208.09	        , activate(n__0()) -> 0()
753.12/208.09	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.09	        , activate(n__s(X)) -> s(activate(X))
753.12/208.09	        , U13(tt(), V1, V2) ->
753.12/208.09	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.09	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.09	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.09	        , isNat(n__0()) -> tt()
753.12/208.09	        , isNat(n__plus(V1, V2)) ->
753.12/208.09	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.09	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.09	        , U16(tt()) -> tt()
753.12/208.09	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.09	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.09	        , U23(tt()) -> tt()
753.12/208.09	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.09	        , U32(tt()) -> tt()
753.12/208.09	        , U41(tt()) -> tt()
753.12/208.09	        , s(X) -> n__s(X)
753.12/208.09	        , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.09	        , 0() -> n__0() }
753.12/208.09	      Obligation:
753.12/208.09	        innermost runtime complexity
753.12/208.09	      Answer:
753.12/208.09	        YES(O(1),O(1))
753.12/208.09	      
753.12/208.09	      No rule is usable, rules are removed from the input problem.
753.12/208.09	      
753.12/208.09	      We are left with following problem, upon which TcT provides the
753.12/208.09	      certificate YES(O(1),O(1)).
753.12/208.09	      
753.12/208.09	      Rules: Empty
753.12/208.09	      Obligation:
753.12/208.09	        innermost runtime complexity
753.12/208.09	      Answer:
753.12/208.09	        YES(O(1),O(1))
753.12/208.09	      
753.12/208.09	      Empty rules are trivially bounded
753.12/208.09	   
753.12/208.09	   S) We are left with following problem, upon which TcT provides the
753.12/208.09	      certificate YES(O(1),O(1)).
753.12/208.09	      
753.12/208.09	      Strict DPs: { U52^#(tt(), N) -> c_16(activate^#(N)) }
753.12/208.09	      Weak DPs:
753.12/208.09	        { U11^#(tt(), V1, V2) ->
753.12/208.09	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.09	              isNatKind^#(activate(V1)),
753.12/208.09	              activate^#(V1),
753.12/208.09	              activate^#(V1),
753.12/208.09	              activate^#(V2))
753.12/208.09	        , U12^#(tt(), V1, V2) ->
753.12/208.09	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.09	              isNatKind^#(activate(V2)),
753.12/208.09	              activate^#(V2),
753.12/208.09	              activate^#(V1),
753.12/208.09	              activate^#(V2))
753.12/208.09	        , isNatKind^#(n__plus(V1, V2)) ->
753.12/208.09	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.12/208.09	              isNatKind^#(activate(V1)),
753.12/208.09	              activate^#(V1),
753.12/208.09	              activate^#(V2))
753.12/208.09	        , isNatKind^#(n__s(V1)) ->
753.12/208.09	          c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.12/208.09	        , activate^#(n__plus(X1, X2)) ->
753.12/208.09	          c_5(activate^#(X1), activate^#(X2))
753.12/208.09	        , activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.09	        , U13^#(tt(), V1, V2) ->
753.12/208.09	          c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.09	              isNatKind^#(activate(V2)),
753.12/208.09	              activate^#(V2),
753.12/208.09	              activate^#(V1),
753.12/208.09	              activate^#(V2))
753.12/208.09	        , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.12/208.09	        , U14^#(tt(), V1, V2) ->
753.12/208.09	          c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.09	              isNat^#(activate(V1)),
753.12/208.09	              activate^#(V1),
753.12/208.09	              activate^#(V2))
753.12/208.09	        , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.09	        , isNat^#(n__plus(V1, V2)) ->
753.12/208.09	          c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.09	               isNatKind^#(activate(V1)),
753.12/208.09	               activate^#(V1),
753.12/208.09	               activate^#(V1),
753.12/208.09	               activate^#(V2))
753.12/208.09	        , isNat^#(n__s(V1)) ->
753.12/208.09	          c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.09	               isNatKind^#(activate(V1)),
753.12/208.09	               activate^#(V1),
753.12/208.09	               activate^#(V1))
753.12/208.09	        , U21^#(tt(), V1) ->
753.12/208.09	          c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.09	               isNatKind^#(activate(V1)),
753.12/208.09	               activate^#(V1),
753.12/208.09	               activate^#(V1))
753.12/208.09	        , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.09	        , U51^#(tt(), N) ->
753.12/208.09	          c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.09	               isNatKind^#(activate(N)),
753.12/208.09	               activate^#(N),
753.12/208.09	               activate^#(N))
753.12/208.09	        , U61^#(tt(), M, N) ->
753.12/208.09	          c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.09	               isNatKind^#(activate(M)),
753.12/208.09	               activate^#(M),
753.12/208.09	               activate^#(M),
753.12/208.09	               activate^#(N))
753.12/208.09	        , U62^#(tt(), M, N) ->
753.12/208.09	          c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.09	               isNat^#(activate(N)),
753.12/208.09	               activate^#(N),
753.12/208.09	               activate^#(M),
753.12/208.09	               activate^#(N))
753.12/208.09	        , U63^#(tt(), M, N) ->
753.12/208.09	          c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.09	               isNatKind^#(activate(N)),
753.12/208.09	               activate^#(N),
753.12/208.09	               activate^#(M),
753.12/208.09	               activate^#(N))
753.12/208.09	        , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.09	      Weak Trs:
753.12/208.09	        { U11(tt(), V1, V2) ->
753.12/208.09	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.09	        , U12(tt(), V1, V2) ->
753.12/208.09	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.09	        , isNatKind(n__0()) -> tt()
753.12/208.09	        , isNatKind(n__plus(V1, V2)) ->
753.12/208.09	          U31(isNatKind(activate(V1)), activate(V2))
753.12/208.09	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.09	        , activate(X) -> X
753.12/208.09	        , activate(n__0()) -> 0()
753.12/208.09	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.09	        , activate(n__s(X)) -> s(activate(X))
753.12/208.09	        , U13(tt(), V1, V2) ->
753.12/208.09	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.09	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.09	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.09	        , isNat(n__0()) -> tt()
753.12/208.09	        , isNat(n__plus(V1, V2)) ->
753.12/208.09	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.09	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.09	        , U16(tt()) -> tt()
753.12/208.09	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.09	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.09	        , U23(tt()) -> tt()
753.12/208.09	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.09	        , U32(tt()) -> tt()
753.12/208.09	        , U41(tt()) -> tt()
753.12/208.09	        , s(X) -> n__s(X)
753.12/208.09	        , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.09	        , 0() -> n__0() }
753.12/208.09	      Obligation:
753.12/208.09	        innermost runtime complexity
753.12/208.09	      Answer:
753.12/208.09	        YES(O(1),O(1))
753.12/208.09	      
753.12/208.09	      The following weak DPs constitute a sub-graph of the DG that is
753.12/208.09	      closed under successors. The DPs are removed.
753.12/208.09	      
753.12/208.09	      { U11^#(tt(), V1, V2) ->
753.12/208.09	        c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.09	            isNatKind^#(activate(V1)),
753.12/208.09	            activate^#(V1),
753.12/208.09	            activate^#(V1),
753.12/208.09	            activate^#(V2))
753.12/208.09	      , U12^#(tt(), V1, V2) ->
753.12/208.09	        c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.09	            isNatKind^#(activate(V2)),
753.12/208.09	            activate^#(V2),
753.12/208.09	            activate^#(V1),
753.12/208.09	            activate^#(V2))
753.12/208.09	      , isNatKind^#(n__plus(V1, V2)) ->
753.12/208.09	        c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.12/208.09	            isNatKind^#(activate(V1)),
753.12/208.09	            activate^#(V1),
753.12/208.09	            activate^#(V2))
753.12/208.09	      , isNatKind^#(n__s(V1)) ->
753.12/208.09	        c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.12/208.09	      , activate^#(n__plus(X1, X2)) ->
753.12/208.09	        c_5(activate^#(X1), activate^#(X2))
753.12/208.09	      , activate^#(n__s(X)) -> c_6(activate^#(X))
753.12/208.09	      , U13^#(tt(), V1, V2) ->
753.12/208.09	        c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.12/208.09	            isNatKind^#(activate(V2)),
753.12/208.09	            activate^#(V2),
753.12/208.09	            activate^#(V1),
753.12/208.09	            activate^#(V2))
753.12/208.09	      , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.12/208.09	      , U14^#(tt(), V1, V2) ->
753.12/208.09	        c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.12/208.09	            isNat^#(activate(V1)),
753.12/208.09	            activate^#(V1),
753.12/208.09	            activate^#(V2))
753.12/208.09	      , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.12/208.09	      , isNat^#(n__plus(V1, V2)) ->
753.12/208.09	        c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.12/208.09	             isNatKind^#(activate(V1)),
753.12/208.09	             activate^#(V1),
753.12/208.09	             activate^#(V1),
753.12/208.09	             activate^#(V2))
753.12/208.09	      , isNat^#(n__s(V1)) ->
753.12/208.09	        c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.09	             isNatKind^#(activate(V1)),
753.12/208.09	             activate^#(V1),
753.12/208.09	             activate^#(V1))
753.12/208.09	      , U21^#(tt(), V1) ->
753.12/208.09	        c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.12/208.09	             isNatKind^#(activate(V1)),
753.12/208.09	             activate^#(V1),
753.12/208.09	             activate^#(V1))
753.12/208.09	      , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.12/208.09	      , U61^#(tt(), M, N) ->
753.12/208.09	        c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.12/208.10	             isNatKind^#(activate(M)),
753.12/208.10	             activate^#(M),
753.12/208.10	             activate^#(M),
753.12/208.10	             activate^#(N))
753.12/208.10	      , U62^#(tt(), M, N) ->
753.12/208.10	        c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.12/208.10	             isNat^#(activate(N)),
753.12/208.10	             activate^#(N),
753.12/208.10	             activate^#(M),
753.12/208.10	             activate^#(N))
753.12/208.10	      , U63^#(tt(), M, N) ->
753.12/208.10	        c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.12/208.10	             isNatKind^#(activate(N)),
753.12/208.10	             activate^#(N),
753.12/208.10	             activate^#(M),
753.12/208.10	             activate^#(N))
753.12/208.10	      , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.12/208.10	      
753.12/208.10	      We are left with following problem, upon which TcT provides the
753.12/208.10	      certificate YES(O(1),O(1)).
753.12/208.10	      
753.12/208.10	      Strict DPs: { U52^#(tt(), N) -> c_16(activate^#(N)) }
753.12/208.10	      Weak DPs:
753.12/208.10	        { U51^#(tt(), N) ->
753.12/208.10	          c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.10	               isNatKind^#(activate(N)),
753.12/208.10	               activate^#(N),
753.12/208.10	               activate^#(N)) }
753.12/208.10	      Weak Trs:
753.12/208.10	        { U11(tt(), V1, V2) ->
753.12/208.10	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.10	        , U12(tt(), V1, V2) ->
753.12/208.10	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.10	        , isNatKind(n__0()) -> tt()
753.12/208.10	        , isNatKind(n__plus(V1, V2)) ->
753.12/208.10	          U31(isNatKind(activate(V1)), activate(V2))
753.12/208.10	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.10	        , activate(X) -> X
753.12/208.10	        , activate(n__0()) -> 0()
753.12/208.10	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.10	        , activate(n__s(X)) -> s(activate(X))
753.12/208.10	        , U13(tt(), V1, V2) ->
753.12/208.10	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.10	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.10	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.10	        , isNat(n__0()) -> tt()
753.12/208.10	        , isNat(n__plus(V1, V2)) ->
753.12/208.10	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.10	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.10	        , U16(tt()) -> tt()
753.12/208.10	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.10	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.10	        , U23(tt()) -> tt()
753.12/208.10	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.10	        , U32(tt()) -> tt()
753.12/208.10	        , U41(tt()) -> tt()
753.12/208.10	        , s(X) -> n__s(X)
753.12/208.10	        , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.10	        , 0() -> n__0() }
753.12/208.10	      Obligation:
753.12/208.10	        innermost runtime complexity
753.12/208.10	      Answer:
753.12/208.10	        YES(O(1),O(1))
753.12/208.10	      
753.12/208.10	      Due to missing edges in the dependency-graph, the right-hand sides
753.12/208.10	      of following rules could be simplified:
753.12/208.10	      
753.12/208.10	        { U51^#(tt(), N) ->
753.12/208.10	          c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.12/208.10	               isNatKind^#(activate(N)),
753.12/208.10	               activate^#(N),
753.12/208.10	               activate^#(N))
753.12/208.10	        , U52^#(tt(), N) -> c_16(activate^#(N)) }
753.12/208.10	      
753.12/208.10	      We are left with following problem, upon which TcT provides the
753.12/208.10	      certificate YES(O(1),O(1)).
753.12/208.10	      
753.12/208.10	      Strict DPs: { U52^#(tt(), N) -> c_1() }
753.12/208.10	      Weak DPs:
753.12/208.10	        { U51^#(tt(), N) ->
753.12/208.10	          c_2(U52^#(isNatKind(activate(N)), activate(N))) }
753.12/208.10	      Weak Trs:
753.12/208.10	        { U11(tt(), V1, V2) ->
753.12/208.10	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.10	        , U12(tt(), V1, V2) ->
753.12/208.10	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.10	        , isNatKind(n__0()) -> tt()
753.12/208.10	        , isNatKind(n__plus(V1, V2)) ->
753.12/208.10	          U31(isNatKind(activate(V1)), activate(V2))
753.12/208.10	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.10	        , activate(X) -> X
753.12/208.10	        , activate(n__0()) -> 0()
753.12/208.10	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.10	        , activate(n__s(X)) -> s(activate(X))
753.12/208.10	        , U13(tt(), V1, V2) ->
753.12/208.10	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.12/208.10	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.12/208.10	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.12/208.10	        , isNat(n__0()) -> tt()
753.12/208.10	        , isNat(n__plus(V1, V2)) ->
753.12/208.10	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.12/208.10	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.12/208.10	        , U16(tt()) -> tt()
753.12/208.10	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.12/208.10	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.12/208.10	        , U23(tt()) -> tt()
753.12/208.10	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.10	        , U32(tt()) -> tt()
753.12/208.10	        , U41(tt()) -> tt()
753.12/208.10	        , s(X) -> n__s(X)
753.12/208.10	        , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.10	        , 0() -> n__0() }
753.12/208.10	      Obligation:
753.12/208.10	        innermost runtime complexity
753.12/208.10	      Answer:
753.12/208.10	        YES(O(1),O(1))
753.12/208.10	      
753.12/208.10	      We replace rewrite rules by usable rules:
753.12/208.10	      
753.12/208.10	        Weak Usable Rules:
753.12/208.10	          { isNatKind(n__0()) -> tt()
753.12/208.10	          , isNatKind(n__plus(V1, V2)) ->
753.12/208.10	            U31(isNatKind(activate(V1)), activate(V2))
753.12/208.10	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.10	          , activate(X) -> X
753.12/208.10	          , activate(n__0()) -> 0()
753.12/208.10	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.10	          , activate(n__s(X)) -> s(activate(X))
753.12/208.10	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.12/208.10	          , U32(tt()) -> tt()
753.12/208.10	          , U41(tt()) -> tt()
753.12/208.10	          , s(X) -> n__s(X)
753.12/208.10	          , plus(X1, X2) -> n__plus(X1, X2)
753.12/208.10	          , 0() -> n__0() }
753.12/208.10	      
753.12/208.10	      We are left with following problem, upon which TcT provides the
753.12/208.10	      certificate YES(O(1),O(1)).
753.12/208.10	      
753.12/208.10	      Strict DPs: { U52^#(tt(), N) -> c_1() }
753.12/208.10	      Weak DPs:
753.12/208.10	        { U51^#(tt(), N) ->
753.12/208.10	          c_2(U52^#(isNatKind(activate(N)), activate(N))) }
753.12/208.10	      Weak Trs:
753.12/208.10	        { isNatKind(n__0()) -> tt()
753.12/208.10	        , isNatKind(n__plus(V1, V2)) ->
753.12/208.10	          U31(isNatKind(activate(V1)), activate(V2))
753.12/208.10	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.12/208.10	        , activate(X) -> X
753.12/208.10	        , activate(n__0()) -> 0()
753.12/208.10	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.12/208.10	        , activate(n__s(X)) -> s(activate(X))
753.12/208.10	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.10	        , U32(tt()) -> tt()
753.37/208.10	        , U41(tt()) -> tt()
753.37/208.10	        , s(X) -> n__s(X)
753.37/208.10	        , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.10	        , 0() -> n__0() }
753.37/208.10	      Obligation:
753.37/208.10	        innermost runtime complexity
753.37/208.10	      Answer:
753.37/208.10	        YES(O(1),O(1))
753.37/208.10	      
753.37/208.10	      The dependency graph contains no loops, we remove all dependency
753.37/208.10	      pairs.
753.37/208.10	      
753.37/208.10	      We are left with following problem, upon which TcT provides the
753.37/208.10	      certificate YES(O(1),O(1)).
753.37/208.10	      
753.37/208.10	      Weak Trs:
753.37/208.10	        { isNatKind(n__0()) -> tt()
753.37/208.10	        , isNatKind(n__plus(V1, V2)) ->
753.37/208.10	          U31(isNatKind(activate(V1)), activate(V2))
753.37/208.10	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.10	        , activate(X) -> X
753.37/208.10	        , activate(n__0()) -> 0()
753.37/208.10	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.10	        , activate(n__s(X)) -> s(activate(X))
753.37/208.10	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.10	        , U32(tt()) -> tt()
753.37/208.10	        , U41(tt()) -> tt()
753.37/208.10	        , s(X) -> n__s(X)
753.37/208.10	        , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.10	        , 0() -> n__0() }
753.37/208.10	      Obligation:
753.37/208.10	        innermost runtime complexity
753.37/208.10	      Answer:
753.37/208.10	        YES(O(1),O(1))
753.37/208.10	      
753.37/208.10	      No rule is usable, rules are removed from the input problem.
753.37/208.10	      
753.37/208.10	      We are left with following problem, upon which TcT provides the
753.37/208.10	      certificate YES(O(1),O(1)).
753.37/208.10	      
753.37/208.10	      Rules: Empty
753.37/208.10	      Obligation:
753.37/208.10	        innermost runtime complexity
753.37/208.10	      Answer:
753.37/208.10	        YES(O(1),O(1))
753.37/208.10	      
753.37/208.10	      Empty rules are trivially bounded
753.37/208.10	   
753.37/208.10	
753.37/208.10	S) We are left with following problem, upon which TcT provides the
753.37/208.10	   certificate YES(O(1),O(n^2)).
753.37/208.10	   
753.37/208.10	   Strict DPs:
753.37/208.10	     { U11^#(tt(), V1, V2) ->
753.37/208.10	       c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.10	           isNatKind^#(activate(V1)),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , U12^#(tt(), V1, V2) ->
753.37/208.10	       c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.10	           isNatKind^#(activate(V2)),
753.37/208.10	           activate^#(V2),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , isNatKind^#(n__plus(V1, V2)) ->
753.37/208.10	       c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.10	           isNatKind^#(activate(V1)),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , isNatKind^#(n__s(V1)) ->
753.37/208.10	       c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.37/208.10	     , U13^#(tt(), V1, V2) ->
753.37/208.10	       c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.10	           isNatKind^#(activate(V2)),
753.37/208.10	           activate^#(V2),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.37/208.10	     , U14^#(tt(), V1, V2) ->
753.37/208.10	       c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.10	           isNat^#(activate(V1)),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.37/208.10	     , isNat^#(n__plus(V1, V2)) ->
753.37/208.10	       c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.10	            isNatKind^#(activate(V1)),
753.37/208.10	            activate^#(V1),
753.37/208.10	            activate^#(V1),
753.37/208.10	            activate^#(V2))
753.37/208.10	     , isNat^#(n__s(V1)) ->
753.37/208.10	       c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.10	            isNatKind^#(activate(V1)),
753.37/208.10	            activate^#(V1),
753.37/208.10	            activate^#(V1))
753.37/208.10	     , U21^#(tt(), V1) ->
753.37/208.10	       c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.10	            isNatKind^#(activate(V1)),
753.37/208.10	            activate^#(V1),
753.37/208.10	            activate^#(V1))
753.37/208.10	     , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.37/208.10	     , U61^#(tt(), M, N) ->
753.37/208.10	       c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.10	            isNatKind^#(activate(M)),
753.37/208.10	            activate^#(M),
753.37/208.10	            activate^#(M),
753.37/208.10	            activate^#(N))
753.37/208.10	     , U62^#(tt(), M, N) ->
753.37/208.10	       c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.10	            isNat^#(activate(N)),
753.37/208.10	            activate^#(N),
753.37/208.10	            activate^#(M),
753.37/208.10	            activate^#(N))
753.37/208.10	     , U63^#(tt(), M, N) ->
753.37/208.10	       c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.37/208.10	            isNatKind^#(activate(N)),
753.37/208.10	            activate^#(N),
753.37/208.10	            activate^#(M),
753.37/208.10	            activate^#(N))
753.37/208.10	     , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.37/208.10	   Weak DPs:
753.37/208.10	     { activate^#(n__plus(X1, X2)) ->
753.37/208.10	       c_5(activate^#(X1), activate^#(X2))
753.37/208.10	     , activate^#(n__s(X)) -> c_6(activate^#(X))
753.37/208.10	     , U51^#(tt(), N) ->
753.37/208.10	       c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.37/208.10	            isNatKind^#(activate(N)),
753.37/208.10	            activate^#(N),
753.37/208.10	            activate^#(N))
753.37/208.10	     , U52^#(tt(), N) -> c_16(activate^#(N)) }
753.37/208.10	   Weak Trs:
753.37/208.10	     { U11(tt(), V1, V2) ->
753.37/208.10	       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.10	     , U12(tt(), V1, V2) ->
753.37/208.10	       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.10	     , isNatKind(n__0()) -> tt()
753.37/208.10	     , isNatKind(n__plus(V1, V2)) ->
753.37/208.10	       U31(isNatKind(activate(V1)), activate(V2))
753.37/208.10	     , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.10	     , activate(X) -> X
753.37/208.10	     , activate(n__0()) -> 0()
753.37/208.10	     , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.10	     , activate(n__s(X)) -> s(activate(X))
753.37/208.10	     , U13(tt(), V1, V2) ->
753.37/208.10	       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.10	     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.10	     , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.10	     , isNat(n__0()) -> tt()
753.37/208.10	     , isNat(n__plus(V1, V2)) ->
753.37/208.10	       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.10	     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.10	     , U16(tt()) -> tt()
753.37/208.10	     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.10	     , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.10	     , U23(tt()) -> tt()
753.37/208.10	     , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.10	     , U32(tt()) -> tt()
753.37/208.10	     , U41(tt()) -> tt()
753.37/208.10	     , s(X) -> n__s(X)
753.37/208.10	     , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.10	     , 0() -> n__0() }
753.37/208.10	   Obligation:
753.37/208.10	     innermost runtime complexity
753.37/208.10	   Answer:
753.37/208.10	     YES(O(1),O(n^2))
753.37/208.10	   
753.37/208.10	   We estimate the number of application of {16} by applications of
753.37/208.10	   Pre({16}) = {15}. Here rules are labeled as follows:
753.37/208.10	   
753.37/208.10	     DPs:
753.37/208.10	       { 1: U11^#(tt(), V1, V2) ->
753.37/208.10	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.10	                isNatKind^#(activate(V1)),
753.37/208.10	                activate^#(V1),
753.37/208.10	                activate^#(V1),
753.37/208.10	                activate^#(V2))
753.37/208.10	       , 2: U12^#(tt(), V1, V2) ->
753.37/208.10	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.10	                isNatKind^#(activate(V2)),
753.37/208.10	                activate^#(V2),
753.37/208.10	                activate^#(V1),
753.37/208.10	                activate^#(V2))
753.37/208.10	       , 3: isNatKind^#(n__plus(V1, V2)) ->
753.37/208.10	            c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.10	                isNatKind^#(activate(V1)),
753.37/208.10	                activate^#(V1),
753.37/208.10	                activate^#(V2))
753.37/208.10	       , 4: isNatKind^#(n__s(V1)) ->
753.37/208.10	            c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.37/208.10	       , 5: U13^#(tt(), V1, V2) ->
753.37/208.10	            c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.10	                isNatKind^#(activate(V2)),
753.37/208.10	                activate^#(V2),
753.37/208.10	                activate^#(V1),
753.37/208.10	                activate^#(V2))
753.37/208.10	       , 6: U31^#(tt(), V2) ->
753.37/208.10	            c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.37/208.10	       , 7: U14^#(tt(), V1, V2) ->
753.37/208.10	            c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.10	                isNat^#(activate(V1)),
753.37/208.10	                activate^#(V1),
753.37/208.10	                activate^#(V2))
753.37/208.10	       , 8: U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.37/208.10	       , 9: isNat^#(n__plus(V1, V2)) ->
753.37/208.10	            c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.10	                 isNatKind^#(activate(V1)),
753.37/208.10	                 activate^#(V1),
753.37/208.10	                 activate^#(V1),
753.37/208.10	                 activate^#(V2))
753.37/208.10	       , 10: isNat^#(n__s(V1)) ->
753.37/208.10	             c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.10	                  isNatKind^#(activate(V1)),
753.37/208.10	                  activate^#(V1),
753.37/208.10	                  activate^#(V1))
753.37/208.10	       , 11: U21^#(tt(), V1) ->
753.37/208.10	             c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.10	                  isNatKind^#(activate(V1)),
753.37/208.10	                  activate^#(V1),
753.37/208.10	                  activate^#(V1))
753.37/208.10	       , 12: U22^#(tt(), V1) ->
753.37/208.10	             c_14(isNat^#(activate(V1)), activate^#(V1))
753.37/208.10	       , 13: U61^#(tt(), M, N) ->
753.37/208.10	             c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.10	                  isNatKind^#(activate(M)),
753.37/208.10	                  activate^#(M),
753.37/208.10	                  activate^#(M),
753.37/208.10	                  activate^#(N))
753.37/208.10	       , 14: U62^#(tt(), M, N) ->
753.37/208.10	             c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.10	                  isNat^#(activate(N)),
753.37/208.10	                  activate^#(N),
753.37/208.10	                  activate^#(M),
753.37/208.10	                  activate^#(N))
753.37/208.10	       , 15: U63^#(tt(), M, N) ->
753.37/208.10	             c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.37/208.10	                  isNatKind^#(activate(N)),
753.37/208.10	                  activate^#(N),
753.37/208.10	                  activate^#(M),
753.37/208.10	                  activate^#(N))
753.37/208.10	       , 16: U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M))
753.37/208.10	       , 17: activate^#(n__plus(X1, X2)) ->
753.37/208.10	             c_5(activate^#(X1), activate^#(X2))
753.37/208.10	       , 18: activate^#(n__s(X)) -> c_6(activate^#(X))
753.37/208.10	       , 19: U51^#(tt(), N) ->
753.37/208.10	             c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.37/208.10	                  isNatKind^#(activate(N)),
753.37/208.10	                  activate^#(N),
753.37/208.10	                  activate^#(N))
753.37/208.10	       , 20: U52^#(tt(), N) -> c_16(activate^#(N)) }
753.37/208.10	   
753.37/208.10	   We are left with following problem, upon which TcT provides the
753.37/208.10	   certificate YES(O(1),O(n^2)).
753.37/208.10	   
753.37/208.10	   Strict DPs:
753.37/208.10	     { U11^#(tt(), V1, V2) ->
753.37/208.10	       c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.10	           isNatKind^#(activate(V1)),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , U12^#(tt(), V1, V2) ->
753.37/208.10	       c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.10	           isNatKind^#(activate(V2)),
753.37/208.10	           activate^#(V2),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , isNatKind^#(n__plus(V1, V2)) ->
753.37/208.10	       c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.10	           isNatKind^#(activate(V1)),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , isNatKind^#(n__s(V1)) ->
753.37/208.10	       c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.37/208.10	     , U13^#(tt(), V1, V2) ->
753.37/208.10	       c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.10	           isNatKind^#(activate(V2)),
753.37/208.10	           activate^#(V2),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.37/208.10	     , U14^#(tt(), V1, V2) ->
753.37/208.10	       c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.10	           isNat^#(activate(V1)),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.37/208.10	     , isNat^#(n__plus(V1, V2)) ->
753.37/208.10	       c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.10	            isNatKind^#(activate(V1)),
753.37/208.10	            activate^#(V1),
753.37/208.10	            activate^#(V1),
753.37/208.10	            activate^#(V2))
753.37/208.10	     , isNat^#(n__s(V1)) ->
753.37/208.10	       c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.10	            isNatKind^#(activate(V1)),
753.37/208.10	            activate^#(V1),
753.37/208.10	            activate^#(V1))
753.37/208.10	     , U21^#(tt(), V1) ->
753.37/208.10	       c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.10	            isNatKind^#(activate(V1)),
753.37/208.10	            activate^#(V1),
753.37/208.10	            activate^#(V1))
753.37/208.10	     , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.37/208.10	     , U61^#(tt(), M, N) ->
753.37/208.10	       c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.10	            isNatKind^#(activate(M)),
753.37/208.10	            activate^#(M),
753.37/208.10	            activate^#(M),
753.37/208.10	            activate^#(N))
753.37/208.10	     , U62^#(tt(), M, N) ->
753.37/208.10	       c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.10	            isNat^#(activate(N)),
753.37/208.10	            activate^#(N),
753.37/208.10	            activate^#(M),
753.37/208.10	            activate^#(N))
753.37/208.10	     , U63^#(tt(), M, N) ->
753.37/208.10	       c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.37/208.10	            isNatKind^#(activate(N)),
753.37/208.10	            activate^#(N),
753.37/208.10	            activate^#(M),
753.37/208.10	            activate^#(N)) }
753.37/208.10	   Weak DPs:
753.37/208.10	     { activate^#(n__plus(X1, X2)) ->
753.37/208.10	       c_5(activate^#(X1), activate^#(X2))
753.37/208.10	     , activate^#(n__s(X)) -> c_6(activate^#(X))
753.37/208.10	     , U51^#(tt(), N) ->
753.37/208.10	       c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.37/208.10	            isNatKind^#(activate(N)),
753.37/208.10	            activate^#(N),
753.37/208.10	            activate^#(N))
753.37/208.10	     , U52^#(tt(), N) -> c_16(activate^#(N))
753.37/208.10	     , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.37/208.10	   Weak Trs:
753.37/208.10	     { U11(tt(), V1, V2) ->
753.37/208.10	       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.10	     , U12(tt(), V1, V2) ->
753.37/208.10	       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.10	     , isNatKind(n__0()) -> tt()
753.37/208.10	     , isNatKind(n__plus(V1, V2)) ->
753.37/208.10	       U31(isNatKind(activate(V1)), activate(V2))
753.37/208.10	     , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.10	     , activate(X) -> X
753.37/208.10	     , activate(n__0()) -> 0()
753.37/208.10	     , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.10	     , activate(n__s(X)) -> s(activate(X))
753.37/208.10	     , U13(tt(), V1, V2) ->
753.37/208.10	       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.10	     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.10	     , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.10	     , isNat(n__0()) -> tt()
753.37/208.10	     , isNat(n__plus(V1, V2)) ->
753.37/208.10	       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.10	     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.10	     , U16(tt()) -> tt()
753.37/208.10	     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.10	     , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.10	     , U23(tt()) -> tt()
753.37/208.10	     , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.10	     , U32(tt()) -> tt()
753.37/208.10	     , U41(tt()) -> tt()
753.37/208.10	     , s(X) -> n__s(X)
753.37/208.10	     , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.10	     , 0() -> n__0() }
753.37/208.10	   Obligation:
753.37/208.10	     innermost runtime complexity
753.37/208.10	   Answer:
753.37/208.10	     YES(O(1),O(n^2))
753.37/208.10	   
753.37/208.10	   The following weak DPs constitute a sub-graph of the DG that is
753.37/208.10	   closed under successors. The DPs are removed.
753.37/208.10	   
753.37/208.10	   { activate^#(n__plus(X1, X2)) ->
753.37/208.10	     c_5(activate^#(X1), activate^#(X2))
753.37/208.10	   , activate^#(n__s(X)) -> c_6(activate^#(X))
753.37/208.10	   , U52^#(tt(), N) -> c_16(activate^#(N))
753.37/208.10	   , U64^#(tt(), M, N) -> c_20(activate^#(N), activate^#(M)) }
753.37/208.10	   
753.37/208.10	   We are left with following problem, upon which TcT provides the
753.37/208.10	   certificate YES(O(1),O(n^2)).
753.37/208.10	   
753.37/208.10	   Strict DPs:
753.37/208.10	     { U11^#(tt(), V1, V2) ->
753.37/208.10	       c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.10	           isNatKind^#(activate(V1)),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , U12^#(tt(), V1, V2) ->
753.37/208.10	       c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.10	           isNatKind^#(activate(V2)),
753.37/208.10	           activate^#(V2),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , isNatKind^#(n__plus(V1, V2)) ->
753.37/208.10	       c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.10	           isNatKind^#(activate(V1)),
753.37/208.10	           activate^#(V1),
753.37/208.10	           activate^#(V2))
753.37/208.10	     , isNatKind^#(n__s(V1)) ->
753.37/208.10	       c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.37/208.10	     , U13^#(tt(), V1, V2) ->
753.37/208.10	       c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.11	           isNatKind^#(activate(V2)),
753.37/208.11	           activate^#(V2),
753.37/208.11	           activate^#(V1),
753.37/208.11	           activate^#(V2))
753.37/208.11	     , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.37/208.11	     , U14^#(tt(), V1, V2) ->
753.37/208.11	       c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.11	           isNat^#(activate(V1)),
753.37/208.11	           activate^#(V1),
753.37/208.11	           activate^#(V2))
753.37/208.11	     , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.37/208.11	     , isNat^#(n__plus(V1, V2)) ->
753.37/208.11	       c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.11	            isNatKind^#(activate(V1)),
753.37/208.11	            activate^#(V1),
753.37/208.11	            activate^#(V1),
753.37/208.11	            activate^#(V2))
753.37/208.11	     , isNat^#(n__s(V1)) ->
753.37/208.11	       c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.11	            isNatKind^#(activate(V1)),
753.37/208.11	            activate^#(V1),
753.37/208.11	            activate^#(V1))
753.37/208.11	     , U21^#(tt(), V1) ->
753.37/208.11	       c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.11	            isNatKind^#(activate(V1)),
753.37/208.11	            activate^#(V1),
753.37/208.11	            activate^#(V1))
753.37/208.11	     , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.37/208.11	     , U61^#(tt(), M, N) ->
753.37/208.11	       c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.11	            isNatKind^#(activate(M)),
753.37/208.11	            activate^#(M),
753.37/208.11	            activate^#(M),
753.37/208.11	            activate^#(N))
753.37/208.11	     , U62^#(tt(), M, N) ->
753.37/208.11	       c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.11	            isNat^#(activate(N)),
753.37/208.11	            activate^#(N),
753.37/208.11	            activate^#(M),
753.37/208.11	            activate^#(N))
753.37/208.11	     , U63^#(tt(), M, N) ->
753.37/208.11	       c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.37/208.11	            isNatKind^#(activate(N)),
753.37/208.11	            activate^#(N),
753.37/208.11	            activate^#(M),
753.37/208.11	            activate^#(N)) }
753.37/208.11	   Weak DPs:
753.37/208.11	     { U51^#(tt(), N) ->
753.37/208.11	       c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.37/208.11	            isNatKind^#(activate(N)),
753.37/208.11	            activate^#(N),
753.37/208.11	            activate^#(N)) }
753.37/208.11	   Weak Trs:
753.37/208.11	     { U11(tt(), V1, V2) ->
753.37/208.11	       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.11	     , U12(tt(), V1, V2) ->
753.37/208.11	       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.11	     , isNatKind(n__0()) -> tt()
753.37/208.11	     , isNatKind(n__plus(V1, V2)) ->
753.37/208.11	       U31(isNatKind(activate(V1)), activate(V2))
753.37/208.11	     , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.11	     , activate(X) -> X
753.37/208.11	     , activate(n__0()) -> 0()
753.37/208.11	     , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.11	     , activate(n__s(X)) -> s(activate(X))
753.37/208.11	     , U13(tt(), V1, V2) ->
753.37/208.11	       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.11	     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.11	     , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.11	     , isNat(n__0()) -> tt()
753.37/208.11	     , isNat(n__plus(V1, V2)) ->
753.37/208.11	       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.11	     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.11	     , U16(tt()) -> tt()
753.37/208.11	     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.11	     , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.11	     , U23(tt()) -> tt()
753.37/208.11	     , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.11	     , U32(tt()) -> tt()
753.37/208.11	     , U41(tt()) -> tt()
753.37/208.11	     , s(X) -> n__s(X)
753.37/208.11	     , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.11	     , 0() -> n__0() }
753.37/208.11	   Obligation:
753.37/208.11	     innermost runtime complexity
753.37/208.11	   Answer:
753.37/208.11	     YES(O(1),O(n^2))
753.37/208.11	   
753.37/208.11	   Due to missing edges in the dependency-graph, the right-hand sides
753.37/208.11	   of following rules could be simplified:
753.37/208.11	   
753.37/208.11	     { U11^#(tt(), V1, V2) ->
753.37/208.11	       c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.11	           isNatKind^#(activate(V1)),
753.37/208.11	           activate^#(V1),
753.37/208.11	           activate^#(V1),
753.37/208.11	           activate^#(V2))
753.37/208.11	     , U12^#(tt(), V1, V2) ->
753.37/208.11	       c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.11	           isNatKind^#(activate(V2)),
753.37/208.11	           activate^#(V2),
753.37/208.11	           activate^#(V1),
753.37/208.11	           activate^#(V2))
753.37/208.11	     , isNatKind^#(n__plus(V1, V2)) ->
753.37/208.11	       c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.11	           isNatKind^#(activate(V1)),
753.37/208.11	           activate^#(V1),
753.37/208.11	           activate^#(V2))
753.37/208.11	     , isNatKind^#(n__s(V1)) ->
753.37/208.11	       c_4(isNatKind^#(activate(V1)), activate^#(V1))
753.37/208.11	     , U13^#(tt(), V1, V2) ->
753.37/208.11	       c_7(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.11	           isNatKind^#(activate(V2)),
753.37/208.11	           activate^#(V2),
753.37/208.11	           activate^#(V1),
753.37/208.11	           activate^#(V2))
753.37/208.11	     , U31^#(tt(), V2) -> c_8(isNatKind^#(activate(V2)), activate^#(V2))
753.37/208.11	     , U14^#(tt(), V1, V2) ->
753.37/208.11	       c_9(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.11	           isNat^#(activate(V1)),
753.37/208.11	           activate^#(V1),
753.37/208.11	           activate^#(V2))
753.37/208.11	     , U15^#(tt(), V2) -> c_10(isNat^#(activate(V2)), activate^#(V2))
753.37/208.11	     , isNat^#(n__plus(V1, V2)) ->
753.37/208.11	       c_11(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.11	            isNatKind^#(activate(V1)),
753.37/208.11	            activate^#(V1),
753.37/208.11	            activate^#(V1),
753.37/208.11	            activate^#(V2))
753.37/208.11	     , isNat^#(n__s(V1)) ->
753.37/208.11	       c_12(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.11	            isNatKind^#(activate(V1)),
753.37/208.11	            activate^#(V1),
753.37/208.11	            activate^#(V1))
753.37/208.11	     , U21^#(tt(), V1) ->
753.37/208.11	       c_13(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.11	            isNatKind^#(activate(V1)),
753.37/208.11	            activate^#(V1),
753.37/208.11	            activate^#(V1))
753.37/208.11	     , U22^#(tt(), V1) -> c_14(isNat^#(activate(V1)), activate^#(V1))
753.37/208.11	     , U51^#(tt(), N) ->
753.37/208.11	       c_15(U52^#(isNatKind(activate(N)), activate(N)),
753.37/208.11	            isNatKind^#(activate(N)),
753.37/208.11	            activate^#(N),
753.37/208.11	            activate^#(N))
753.37/208.11	     , U61^#(tt(), M, N) ->
753.37/208.11	       c_17(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.11	            isNatKind^#(activate(M)),
753.37/208.11	            activate^#(M),
753.37/208.11	            activate^#(M),
753.37/208.11	            activate^#(N))
753.37/208.11	     , U62^#(tt(), M, N) ->
753.37/208.11	       c_18(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.11	            isNat^#(activate(N)),
753.37/208.11	            activate^#(N),
753.37/208.11	            activate^#(M),
753.37/208.11	            activate^#(N))
753.37/208.11	     , U63^#(tt(), M, N) ->
753.37/208.11	       c_19(U64^#(isNatKind(activate(N)), activate(M), activate(N)),
753.37/208.11	            isNatKind^#(activate(N)),
753.37/208.11	            activate^#(N),
753.37/208.11	            activate^#(M),
753.37/208.11	            activate^#(N)) }
753.37/208.11	   
753.37/208.11	   We are left with following problem, upon which TcT provides the
753.37/208.11	   certificate YES(O(1),O(n^2)).
753.37/208.11	   
753.37/208.11	   Strict DPs:
753.37/208.11	     { U11^#(tt(), V1, V2) ->
753.37/208.11	       c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.11	           isNatKind^#(activate(V1)))
753.37/208.11	     , U12^#(tt(), V1, V2) ->
753.37/208.11	       c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.11	           isNatKind^#(activate(V2)))
753.37/208.11	     , isNatKind^#(n__plus(V1, V2)) ->
753.37/208.11	       c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.11	           isNatKind^#(activate(V1)))
753.37/208.11	     , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.11	     , U13^#(tt(), V1, V2) ->
753.37/208.11	       c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.11	           isNatKind^#(activate(V2)))
753.37/208.11	     , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.11	     , U14^#(tt(), V1, V2) ->
753.37/208.11	       c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.11	           isNat^#(activate(V1)))
753.37/208.11	     , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.11	     , isNat^#(n__plus(V1, V2)) ->
753.37/208.11	       c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.11	           isNatKind^#(activate(V1)))
753.37/208.11	     , isNat^#(n__s(V1)) ->
753.37/208.11	       c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.11	            isNatKind^#(activate(V1)))
753.37/208.11	     , U21^#(tt(), V1) ->
753.37/208.11	       c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.11	            isNatKind^#(activate(V1)))
753.37/208.11	     , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.11	     , U61^#(tt(), M, N) ->
753.37/208.11	       c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.11	            isNatKind^#(activate(M)))
753.37/208.11	     , U62^#(tt(), M, N) ->
753.37/208.11	       c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.11	            isNat^#(activate(N)))
753.37/208.11	     , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.11	   Weak DPs: { U51^#(tt(), N) -> c_16(isNatKind^#(activate(N))) }
753.37/208.11	   Weak Trs:
753.37/208.11	     { U11(tt(), V1, V2) ->
753.37/208.11	       U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.11	     , U12(tt(), V1, V2) ->
753.37/208.11	       U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.11	     , isNatKind(n__0()) -> tt()
753.37/208.11	     , isNatKind(n__plus(V1, V2)) ->
753.37/208.11	       U31(isNatKind(activate(V1)), activate(V2))
753.37/208.11	     , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.11	     , activate(X) -> X
753.37/208.11	     , activate(n__0()) -> 0()
753.37/208.11	     , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.11	     , activate(n__s(X)) -> s(activate(X))
753.37/208.11	     , U13(tt(), V1, V2) ->
753.37/208.11	       U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.11	     , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.11	     , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.11	     , isNat(n__0()) -> tt()
753.37/208.11	     , isNat(n__plus(V1, V2)) ->
753.37/208.11	       U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.11	     , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.11	     , U16(tt()) -> tt()
753.37/208.11	     , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.11	     , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.11	     , U23(tt()) -> tt()
753.37/208.11	     , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.11	     , U32(tt()) -> tt()
753.37/208.11	     , U41(tt()) -> tt()
753.37/208.11	     , s(X) -> n__s(X)
753.37/208.11	     , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.11	     , 0() -> n__0() }
753.37/208.11	   Obligation:
753.37/208.11	     innermost runtime complexity
753.37/208.11	   Answer:
753.37/208.11	     YES(O(1),O(n^2))
753.37/208.11	   
753.37/208.11	   We analyse the complexity of following sub-problems (R) and (S).
753.37/208.11	   Problem (S) is obtained from the input problem by shifting strict
753.37/208.11	   rules from (R) into the weak component:
753.37/208.11	   
753.37/208.11	   Problem (R):
753.37/208.11	   ------------
753.37/208.11	     Strict DPs:
753.37/208.11	       { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.11	         c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.11	             isNatKind^#(activate(V1)))
753.37/208.11	       , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.11	       , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.11	       , U61^#(tt(), M, N) ->
753.37/208.11	         c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.11	              isNatKind^#(activate(M)))
753.37/208.11	       , U62^#(tt(), M, N) ->
753.37/208.11	         c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.11	              isNat^#(activate(N)))
753.37/208.11	       , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.11	     Weak DPs:
753.37/208.11	       { U11^#(tt(), V1, V2) ->
753.37/208.11	         c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.11	             isNatKind^#(activate(V1)))
753.37/208.11	       , U12^#(tt(), V1, V2) ->
753.37/208.11	         c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.11	             isNatKind^#(activate(V2)))
753.37/208.11	       , U13^#(tt(), V1, V2) ->
753.37/208.11	         c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.11	             isNatKind^#(activate(V2)))
753.37/208.11	       , U14^#(tt(), V1, V2) ->
753.37/208.11	         c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.11	             isNat^#(activate(V1)))
753.37/208.11	       , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.11	       , isNat^#(n__plus(V1, V2)) ->
753.37/208.11	         c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.11	             isNatKind^#(activate(V1)))
753.37/208.11	       , isNat^#(n__s(V1)) ->
753.37/208.11	         c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.11	              isNatKind^#(activate(V1)))
753.37/208.11	       , U21^#(tt(), V1) ->
753.37/208.11	         c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.11	              isNatKind^#(activate(V1)))
753.37/208.11	       , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.11	       , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N))) }
753.37/208.11	     Weak Trs:
753.37/208.11	       { U11(tt(), V1, V2) ->
753.37/208.11	         U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.11	       , U12(tt(), V1, V2) ->
753.37/208.11	         U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.11	       , isNatKind(n__0()) -> tt()
753.37/208.11	       , isNatKind(n__plus(V1, V2)) ->
753.37/208.11	         U31(isNatKind(activate(V1)), activate(V2))
753.37/208.11	       , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.11	       , activate(X) -> X
753.37/208.11	       , activate(n__0()) -> 0()
753.37/208.11	       , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.11	       , activate(n__s(X)) -> s(activate(X))
753.37/208.11	       , U13(tt(), V1, V2) ->
753.37/208.11	         U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.11	       , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.11	       , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.11	       , isNat(n__0()) -> tt()
753.37/208.11	       , isNat(n__plus(V1, V2)) ->
753.37/208.11	         U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.11	       , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.11	       , U16(tt()) -> tt()
753.37/208.11	       , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.11	       , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.11	       , U23(tt()) -> tt()
753.37/208.11	       , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.11	       , U32(tt()) -> tt()
753.37/208.11	       , U41(tt()) -> tt()
753.37/208.11	       , s(X) -> n__s(X)
753.37/208.11	       , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.11	       , 0() -> n__0() }
753.37/208.11	     StartTerms: basic terms
753.37/208.11	     Strategy: innermost
753.37/208.11	   
753.37/208.11	   Problem (S):
753.37/208.11	   ------------
753.37/208.11	     Strict DPs:
753.37/208.11	       { U11^#(tt(), V1, V2) ->
753.37/208.11	         c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.11	             isNatKind^#(activate(V1)))
753.37/208.11	       , U12^#(tt(), V1, V2) ->
753.37/208.11	         c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.11	             isNatKind^#(activate(V2)))
753.37/208.11	       , U13^#(tt(), V1, V2) ->
753.37/208.11	         c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.11	             isNatKind^#(activate(V2)))
753.37/208.11	       , U14^#(tt(), V1, V2) ->
753.37/208.11	         c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.11	             isNat^#(activate(V1)))
753.37/208.11	       , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.11	       , isNat^#(n__plus(V1, V2)) ->
753.37/208.11	         c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.11	             isNatKind^#(activate(V1)))
753.37/208.11	       , isNat^#(n__s(V1)) ->
753.37/208.11	         c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.11	              isNatKind^#(activate(V1)))
753.37/208.11	       , U21^#(tt(), V1) ->
753.37/208.11	         c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.11	              isNatKind^#(activate(V1)))
753.37/208.11	       , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1))) }
753.37/208.11	     Weak DPs:
753.37/208.11	       { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.11	         c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.11	             isNatKind^#(activate(V1)))
753.37/208.11	       , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.11	       , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.11	       , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.11	       , U61^#(tt(), M, N) ->
753.37/208.11	         c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.11	              isNatKind^#(activate(M)))
753.37/208.11	       , U62^#(tt(), M, N) ->
753.37/208.11	         c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.11	              isNat^#(activate(N)))
753.37/208.11	       , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.11	     Weak Trs:
753.37/208.11	       { U11(tt(), V1, V2) ->
753.37/208.11	         U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.11	       , U12(tt(), V1, V2) ->
753.37/208.11	         U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.11	       , isNatKind(n__0()) -> tt()
753.37/208.11	       , isNatKind(n__plus(V1, V2)) ->
753.37/208.11	         U31(isNatKind(activate(V1)), activate(V2))
753.37/208.11	       , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.11	       , activate(X) -> X
753.37/208.11	       , activate(n__0()) -> 0()
753.37/208.11	       , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.11	       , activate(n__s(X)) -> s(activate(X))
753.37/208.11	       , U13(tt(), V1, V2) ->
753.37/208.11	         U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.11	       , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.11	       , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.11	       , isNat(n__0()) -> tt()
753.37/208.11	       , isNat(n__plus(V1, V2)) ->
753.37/208.11	         U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.11	       , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.11	       , U16(tt()) -> tt()
753.37/208.11	       , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.11	       , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.11	       , U23(tt()) -> tt()
753.37/208.11	       , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.11	       , U32(tt()) -> tt()
753.37/208.11	       , U41(tt()) -> tt()
753.37/208.11	       , s(X) -> n__s(X)
753.37/208.11	       , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.11	       , 0() -> n__0() }
753.37/208.11	     StartTerms: basic terms
753.37/208.11	     Strategy: innermost
753.37/208.11	   
753.37/208.11	   Overall, the transformation results in the following sub-problem(s):
753.37/208.11	   
753.37/208.11	   Generated new problems:
753.37/208.11	   -----------------------
753.37/208.11	   R) Strict DPs:
753.37/208.11	        { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.11	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.11	              isNatKind^#(activate(V1)))
753.37/208.11	        , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.11	        , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.11	        , U61^#(tt(), M, N) ->
753.37/208.11	          c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.11	               isNatKind^#(activate(M)))
753.37/208.11	        , U62^#(tt(), M, N) ->
753.37/208.11	          c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.11	               isNat^#(activate(N)))
753.37/208.11	        , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.11	      Weak DPs:
753.37/208.11	        { U11^#(tt(), V1, V2) ->
753.37/208.11	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.11	              isNatKind^#(activate(V1)))
753.37/208.11	        , U12^#(tt(), V1, V2) ->
753.37/208.11	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.11	              isNatKind^#(activate(V2)))
753.37/208.11	        , U13^#(tt(), V1, V2) ->
753.37/208.11	          c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V2)))
753.37/208.12	        , U14^#(tt(), V1, V2) ->
753.37/208.12	          c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.12	              isNat^#(activate(V1)))
753.37/208.12	        , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.12	        , isNat^#(n__plus(V1, V2)) ->
753.37/208.12	          c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V1)))
753.37/208.12	        , isNat^#(n__s(V1)) ->
753.37/208.12	          c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.12	               isNatKind^#(activate(V1)))
753.37/208.12	        , U21^#(tt(), V1) ->
753.37/208.12	          c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.12	               isNatKind^#(activate(V1)))
753.37/208.12	        , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.12	        , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N))) }
753.37/208.12	      Weak Trs:
753.37/208.12	        { U11(tt(), V1, V2) ->
753.37/208.12	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.12	        , U12(tt(), V1, V2) ->
753.37/208.12	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.12	        , isNatKind(n__0()) -> tt()
753.37/208.12	        , isNatKind(n__plus(V1, V2)) ->
753.37/208.12	          U31(isNatKind(activate(V1)), activate(V2))
753.37/208.12	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.12	        , activate(X) -> X
753.37/208.12	        , activate(n__0()) -> 0()
753.37/208.12	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.12	        , activate(n__s(X)) -> s(activate(X))
753.37/208.12	        , U13(tt(), V1, V2) ->
753.37/208.12	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.12	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.12	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.12	        , isNat(n__0()) -> tt()
753.37/208.12	        , isNat(n__plus(V1, V2)) ->
753.37/208.12	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.12	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.12	        , U16(tt()) -> tt()
753.37/208.12	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.12	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.12	        , U23(tt()) -> tt()
753.37/208.12	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.12	        , U32(tt()) -> tt()
753.37/208.12	        , U41(tt()) -> tt()
753.37/208.12	        , s(X) -> n__s(X)
753.37/208.12	        , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.12	        , 0() -> n__0() }
753.37/208.12	      StartTerms: basic terms
753.37/208.12	      Strategy: innermost
753.37/208.12	      
753.37/208.12	      This problem was proven YES(O(1),O(n^2)).
753.37/208.12	   
753.37/208.12	   S) Strict DPs:
753.37/208.12	        { U11^#(tt(), V1, V2) ->
753.37/208.12	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V1)))
753.37/208.12	        , U12^#(tt(), V1, V2) ->
753.37/208.12	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V2)))
753.37/208.12	        , U13^#(tt(), V1, V2) ->
753.37/208.12	          c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V2)))
753.37/208.12	        , U14^#(tt(), V1, V2) ->
753.37/208.12	          c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.12	              isNat^#(activate(V1)))
753.37/208.12	        , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.12	        , isNat^#(n__plus(V1, V2)) ->
753.37/208.12	          c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V1)))
753.37/208.12	        , isNat^#(n__s(V1)) ->
753.37/208.12	          c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.12	               isNatKind^#(activate(V1)))
753.37/208.12	        , U21^#(tt(), V1) ->
753.37/208.12	          c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.12	               isNatKind^#(activate(V1)))
753.37/208.12	        , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1))) }
753.37/208.12	      Weak DPs:
753.37/208.12	        { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.12	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V1)))
753.37/208.12	        , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.12	        , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.12	        , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.12	        , U61^#(tt(), M, N) ->
753.37/208.12	          c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.12	               isNatKind^#(activate(M)))
753.37/208.12	        , U62^#(tt(), M, N) ->
753.37/208.12	          c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.12	               isNat^#(activate(N)))
753.37/208.12	        , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.12	      Weak Trs:
753.37/208.12	        { U11(tt(), V1, V2) ->
753.37/208.12	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.12	        , U12(tt(), V1, V2) ->
753.37/208.12	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.12	        , isNatKind(n__0()) -> tt()
753.37/208.12	        , isNatKind(n__plus(V1, V2)) ->
753.37/208.12	          U31(isNatKind(activate(V1)), activate(V2))
753.37/208.12	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.12	        , activate(X) -> X
753.37/208.12	        , activate(n__0()) -> 0()
753.37/208.12	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.12	        , activate(n__s(X)) -> s(activate(X))
753.37/208.12	        , U13(tt(), V1, V2) ->
753.37/208.12	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.12	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.12	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.12	        , isNat(n__0()) -> tt()
753.37/208.12	        , isNat(n__plus(V1, V2)) ->
753.37/208.12	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.12	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.12	        , U16(tt()) -> tt()
753.37/208.12	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.12	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.12	        , U23(tt()) -> tt()
753.37/208.12	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.12	        , U32(tt()) -> tt()
753.37/208.12	        , U41(tt()) -> tt()
753.37/208.12	        , s(X) -> n__s(X)
753.37/208.12	        , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.12	        , 0() -> n__0() }
753.37/208.12	      StartTerms: basic terms
753.37/208.12	      Strategy: innermost
753.37/208.12	      
753.37/208.12	      This problem was proven YES(O(1),O(n^1)).
753.37/208.12	   
753.37/208.12	   
753.37/208.12	   Proofs for generated problems:
753.37/208.12	   ------------------------------
753.37/208.12	   R) We are left with following problem, upon which TcT provides the
753.37/208.12	      certificate YES(O(1),O(n^2)).
753.37/208.12	      
753.37/208.12	      Strict DPs:
753.37/208.12	        { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.12	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V1)))
753.37/208.12	        , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.12	        , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.12	        , U61^#(tt(), M, N) ->
753.37/208.12	          c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.12	               isNatKind^#(activate(M)))
753.37/208.12	        , U62^#(tt(), M, N) ->
753.37/208.12	          c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.12	               isNat^#(activate(N)))
753.37/208.12	        , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.12	      Weak DPs:
753.37/208.12	        { U11^#(tt(), V1, V2) ->
753.37/208.12	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V1)))
753.37/208.12	        , U12^#(tt(), V1, V2) ->
753.37/208.12	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V2)))
753.37/208.12	        , U13^#(tt(), V1, V2) ->
753.37/208.12	          c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V2)))
753.37/208.12	        , U14^#(tt(), V1, V2) ->
753.37/208.12	          c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.12	              isNat^#(activate(V1)))
753.37/208.12	        , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.12	        , isNat^#(n__plus(V1, V2)) ->
753.37/208.12	          c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.12	              isNatKind^#(activate(V1)))
753.37/208.12	        , isNat^#(n__s(V1)) ->
753.37/208.12	          c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.12	               isNatKind^#(activate(V1)))
753.37/208.12	        , U21^#(tt(), V1) ->
753.37/208.12	          c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.12	               isNatKind^#(activate(V1)))
753.37/208.12	        , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.12	        , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N))) }
753.37/208.12	      Weak Trs:
753.37/208.12	        { U11(tt(), V1, V2) ->
753.37/208.12	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.12	        , U12(tt(), V1, V2) ->
753.37/208.12	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.12	        , isNatKind(n__0()) -> tt()
753.37/208.12	        , isNatKind(n__plus(V1, V2)) ->
753.37/208.12	          U31(isNatKind(activate(V1)), activate(V2))
753.37/208.12	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.12	        , activate(X) -> X
753.37/208.12	        , activate(n__0()) -> 0()
753.37/208.12	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.12	        , activate(n__s(X)) -> s(activate(X))
753.37/208.12	        , U13(tt(), V1, V2) ->
753.37/208.12	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.12	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.12	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.12	        , isNat(n__0()) -> tt()
753.37/208.12	        , isNat(n__plus(V1, V2)) ->
753.37/208.12	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.12	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.12	        , U16(tt()) -> tt()
753.37/208.12	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.12	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.12	        , U23(tt()) -> tt()
753.37/208.12	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.12	        , U32(tt()) -> tt()
753.37/208.12	        , U41(tt()) -> tt()
753.37/208.12	        , s(X) -> n__s(X)
753.37/208.12	        , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.12	        , 0() -> n__0() }
753.37/208.12	      Obligation:
753.37/208.12	        innermost runtime complexity
753.37/208.12	      Answer:
753.37/208.12	        YES(O(1),O(n^2))
753.37/208.12	      
753.37/208.12	      We analyse the complexity of following sub-problems (R) and (S).
753.37/208.12	      Problem (S) is obtained from the input problem by shifting strict
753.37/208.12	      rules from (R) into the weak component:
753.37/208.12	      
753.37/208.12	      Problem (R):
753.37/208.12	      ------------
753.37/208.12	        Strict DPs:
753.37/208.12	          { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.12	            c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.12	                isNatKind^#(activate(V1)))
753.37/208.12	          , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.12	          , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.12	          , U61^#(tt(), M, N) ->
753.37/208.12	            c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.12	                 isNatKind^#(activate(M)))
753.37/208.12	          , U62^#(tt(), M, N) ->
753.37/208.12	            c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.12	                 isNat^#(activate(N))) }
753.37/208.12	        Weak DPs:
753.37/208.12	          { U11^#(tt(), V1, V2) ->
753.37/208.12	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.12	                isNatKind^#(activate(V1)))
753.37/208.12	          , U12^#(tt(), V1, V2) ->
753.37/208.12	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.12	                isNatKind^#(activate(V2)))
753.37/208.12	          , U13^#(tt(), V1, V2) ->
753.37/208.12	            c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.12	                isNatKind^#(activate(V2)))
753.37/208.12	          , U14^#(tt(), V1, V2) ->
753.37/208.12	            c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.12	                isNat^#(activate(V1)))
753.37/208.12	          , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.12	          , isNat^#(n__plus(V1, V2)) ->
753.37/208.12	            c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.12	                isNatKind^#(activate(V1)))
753.37/208.12	          , isNat^#(n__s(V1)) ->
753.37/208.12	            c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.12	                 isNatKind^#(activate(V1)))
753.37/208.12	          , U21^#(tt(), V1) ->
753.37/208.12	            c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.12	                 isNatKind^#(activate(V1)))
753.37/208.12	          , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.12	          , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.12	          , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.12	        Weak Trs:
753.37/208.12	          { U11(tt(), V1, V2) ->
753.37/208.12	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.12	          , U12(tt(), V1, V2) ->
753.37/208.12	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.12	          , isNatKind(n__0()) -> tt()
753.37/208.12	          , isNatKind(n__plus(V1, V2)) ->
753.37/208.12	            U31(isNatKind(activate(V1)), activate(V2))
753.37/208.12	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.12	          , activate(X) -> X
753.37/208.12	          , activate(n__0()) -> 0()
753.37/208.12	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.12	          , activate(n__s(X)) -> s(activate(X))
753.37/208.12	          , U13(tt(), V1, V2) ->
753.37/208.12	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.12	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.12	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.12	          , isNat(n__0()) -> tt()
753.37/208.12	          , isNat(n__plus(V1, V2)) ->
753.37/208.12	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.12	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.12	          , U16(tt()) -> tt()
753.37/208.12	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.12	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.12	          , U23(tt()) -> tt()
753.37/208.12	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.12	          , U32(tt()) -> tt()
753.37/208.12	          , U41(tt()) -> tt()
753.37/208.12	          , s(X) -> n__s(X)
753.37/208.12	          , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.12	          , 0() -> n__0() }
753.37/208.12	        StartTerms: basic terms
753.37/208.12	        Strategy: innermost
753.37/208.12	      
753.37/208.12	      Problem (S):
753.37/208.12	      ------------
753.37/208.12	        Strict DPs: { U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.12	        Weak DPs:
753.37/208.12	          { U11^#(tt(), V1, V2) ->
753.37/208.12	            c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.12	                isNatKind^#(activate(V1)))
753.37/208.12	          , U12^#(tt(), V1, V2) ->
753.37/208.12	            c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.12	                isNatKind^#(activate(V2)))
753.37/208.12	          , isNatKind^#(n__plus(V1, V2)) ->
753.37/208.12	            c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.12	                isNatKind^#(activate(V1)))
753.37/208.12	          , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.12	          , U13^#(tt(), V1, V2) ->
753.37/208.12	            c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.12	                isNatKind^#(activate(V2)))
753.37/208.12	          , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.12	          , U14^#(tt(), V1, V2) ->
753.37/208.12	            c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.12	                isNat^#(activate(V1)))
753.37/208.12	          , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.12	          , isNat^#(n__plus(V1, V2)) ->
753.37/208.12	            c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.12	                isNatKind^#(activate(V1)))
753.37/208.12	          , isNat^#(n__s(V1)) ->
753.37/208.12	            c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.12	                 isNatKind^#(activate(V1)))
753.37/208.12	          , U21^#(tt(), V1) ->
753.37/208.12	            c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.12	                 isNatKind^#(activate(V1)))
753.37/208.12	          , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.12	          , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.12	          , U61^#(tt(), M, N) ->
753.37/208.12	            c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.12	                 isNatKind^#(activate(M)))
753.37/208.12	          , U62^#(tt(), M, N) ->
753.37/208.12	            c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.12	                 isNat^#(activate(N))) }
753.37/208.12	        Weak Trs:
753.37/208.12	          { U11(tt(), V1, V2) ->
753.37/208.12	            U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.12	          , U12(tt(), V1, V2) ->
753.37/208.12	            U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.12	          , isNatKind(n__0()) -> tt()
753.37/208.12	          , isNatKind(n__plus(V1, V2)) ->
753.37/208.12	            U31(isNatKind(activate(V1)), activate(V2))
753.37/208.12	          , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.12	          , activate(X) -> X
753.37/208.12	          , activate(n__0()) -> 0()
753.37/208.12	          , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.12	          , activate(n__s(X)) -> s(activate(X))
753.37/208.12	          , U13(tt(), V1, V2) ->
753.37/208.12	            U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.12	          , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.12	          , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.12	          , isNat(n__0()) -> tt()
753.37/208.12	          , isNat(n__plus(V1, V2)) ->
753.37/208.12	            U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.12	          , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.12	          , U16(tt()) -> tt()
753.37/208.12	          , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.12	          , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.12	          , U23(tt()) -> tt()
753.37/208.12	          , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.12	          , U32(tt()) -> tt()
753.37/208.12	          , U41(tt()) -> tt()
753.37/208.12	          , s(X) -> n__s(X)
753.37/208.12	          , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.12	          , 0() -> n__0() }
753.37/208.12	        StartTerms: basic terms
753.37/208.12	        Strategy: innermost
753.37/208.13	      
753.37/208.13	      Overall, the transformation results in the following sub-problem(s):
753.37/208.13	      
753.37/208.13	      Generated new problems:
753.37/208.13	      -----------------------
753.37/208.13	      R) Strict DPs:
753.37/208.13	           { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.13	             c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.13	           , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.13	           , U61^#(tt(), M, N) ->
753.37/208.13	             c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.13	                  isNatKind^#(activate(M)))
753.37/208.13	           , U62^#(tt(), M, N) ->
753.37/208.13	             c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.13	                  isNat^#(activate(N))) }
753.37/208.13	         Weak DPs:
753.37/208.13	           { U11^#(tt(), V1, V2) ->
753.37/208.13	             c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , U12^#(tt(), V1, V2) ->
753.37/208.13	             c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V2)))
753.37/208.13	           , U13^#(tt(), V1, V2) ->
753.37/208.13	             c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V2)))
753.37/208.13	           , U14^#(tt(), V1, V2) ->
753.37/208.13	             c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.13	                 isNat^#(activate(V1)))
753.37/208.13	           , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.13	           , isNat^#(n__plus(V1, V2)) ->
753.37/208.13	             c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , isNat^#(n__s(V1)) ->
753.37/208.13	             c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.13	                  isNatKind^#(activate(V1)))
753.37/208.13	           , U21^#(tt(), V1) ->
753.37/208.13	             c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.13	                  isNatKind^#(activate(V1)))
753.37/208.13	           , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.13	           , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.13	           , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.13	         Weak Trs:
753.37/208.13	           { U11(tt(), V1, V2) ->
753.37/208.13	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.13	           , U12(tt(), V1, V2) ->
753.37/208.13	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.13	           , isNatKind(n__0()) -> tt()
753.37/208.13	           , isNatKind(n__plus(V1, V2)) ->
753.37/208.13	             U31(isNatKind(activate(V1)), activate(V2))
753.37/208.13	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.13	           , activate(X) -> X
753.37/208.13	           , activate(n__0()) -> 0()
753.37/208.13	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.13	           , activate(n__s(X)) -> s(activate(X))
753.37/208.13	           , U13(tt(), V1, V2) ->
753.37/208.13	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.13	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.13	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.13	           , isNat(n__0()) -> tt()
753.37/208.13	           , isNat(n__plus(V1, V2)) ->
753.37/208.13	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.13	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.13	           , U16(tt()) -> tt()
753.37/208.13	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.13	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.13	           , U23(tt()) -> tt()
753.37/208.13	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.13	           , U32(tt()) -> tt()
753.37/208.13	           , U41(tt()) -> tt()
753.37/208.13	           , s(X) -> n__s(X)
753.37/208.13	           , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.13	           , 0() -> n__0() }
753.37/208.13	         StartTerms: basic terms
753.37/208.13	         Strategy: innermost
753.37/208.13	         
753.37/208.13	         This problem was proven YES(O(1),O(n^2)).
753.37/208.13	      
753.37/208.13	      S) Strict DPs:
753.37/208.13	           { U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.13	         Weak DPs:
753.37/208.13	           { U11^#(tt(), V1, V2) ->
753.37/208.13	             c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , U12^#(tt(), V1, V2) ->
753.37/208.13	             c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V2)))
753.37/208.13	           , isNatKind^#(n__plus(V1, V2)) ->
753.37/208.13	             c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.13	           , U13^#(tt(), V1, V2) ->
753.37/208.13	             c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V2)))
753.37/208.13	           , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.13	           , U14^#(tt(), V1, V2) ->
753.37/208.13	             c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.13	                 isNat^#(activate(V1)))
753.37/208.13	           , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.13	           , isNat^#(n__plus(V1, V2)) ->
753.37/208.13	             c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , isNat^#(n__s(V1)) ->
753.37/208.13	             c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.13	                  isNatKind^#(activate(V1)))
753.37/208.13	           , U21^#(tt(), V1) ->
753.37/208.13	             c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.13	                  isNatKind^#(activate(V1)))
753.37/208.13	           , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.13	           , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.13	           , U61^#(tt(), M, N) ->
753.37/208.13	             c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.13	                  isNatKind^#(activate(M)))
753.37/208.13	           , U62^#(tt(), M, N) ->
753.37/208.13	             c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.13	                  isNat^#(activate(N))) }
753.37/208.13	         Weak Trs:
753.37/208.13	           { U11(tt(), V1, V2) ->
753.37/208.13	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.13	           , U12(tt(), V1, V2) ->
753.37/208.13	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.13	           , isNatKind(n__0()) -> tt()
753.37/208.13	           , isNatKind(n__plus(V1, V2)) ->
753.37/208.13	             U31(isNatKind(activate(V1)), activate(V2))
753.37/208.13	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.13	           , activate(X) -> X
753.37/208.13	           , activate(n__0()) -> 0()
753.37/208.13	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.13	           , activate(n__s(X)) -> s(activate(X))
753.37/208.13	           , U13(tt(), V1, V2) ->
753.37/208.13	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.13	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.13	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.13	           , isNat(n__0()) -> tt()
753.37/208.13	           , isNat(n__plus(V1, V2)) ->
753.37/208.13	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.13	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.13	           , U16(tt()) -> tt()
753.37/208.13	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.13	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.13	           , U23(tt()) -> tt()
753.37/208.13	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.13	           , U32(tt()) -> tt()
753.37/208.13	           , U41(tt()) -> tt()
753.37/208.13	           , s(X) -> n__s(X)
753.37/208.13	           , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.13	           , 0() -> n__0() }
753.37/208.13	         StartTerms: basic terms
753.37/208.13	         Strategy: innermost
753.37/208.13	         
753.37/208.13	         This problem was proven YES(O(1),O(1)).
753.37/208.13	      
753.37/208.13	      
753.37/208.13	      Proofs for generated problems:
753.37/208.13	      ------------------------------
753.37/208.13	      R) We are left with following problem, upon which TcT provides the
753.37/208.13	         certificate YES(O(1),O(n^2)).
753.37/208.13	         
753.37/208.13	         Strict DPs:
753.37/208.13	           { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.13	             c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.13	           , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.13	           , U61^#(tt(), M, N) ->
753.37/208.13	             c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.13	                  isNatKind^#(activate(M)))
753.37/208.13	           , U62^#(tt(), M, N) ->
753.37/208.13	             c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.13	                  isNat^#(activate(N))) }
753.37/208.13	         Weak DPs:
753.37/208.13	           { U11^#(tt(), V1, V2) ->
753.37/208.13	             c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , U12^#(tt(), V1, V2) ->
753.37/208.13	             c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V2)))
753.37/208.13	           , U13^#(tt(), V1, V2) ->
753.37/208.13	             c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V2)))
753.37/208.13	           , U14^#(tt(), V1, V2) ->
753.37/208.13	             c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.13	                 isNat^#(activate(V1)))
753.37/208.13	           , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.13	           , isNat^#(n__plus(V1, V2)) ->
753.37/208.13	             c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , isNat^#(n__s(V1)) ->
753.37/208.13	             c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.13	                  isNatKind^#(activate(V1)))
753.37/208.13	           , U21^#(tt(), V1) ->
753.37/208.13	             c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.13	                  isNatKind^#(activate(V1)))
753.37/208.13	           , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.13	           , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.13	           , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.13	         Weak Trs:
753.37/208.13	           { U11(tt(), V1, V2) ->
753.37/208.13	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.13	           , U12(tt(), V1, V2) ->
753.37/208.13	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.13	           , isNatKind(n__0()) -> tt()
753.37/208.13	           , isNatKind(n__plus(V1, V2)) ->
753.37/208.13	             U31(isNatKind(activate(V1)), activate(V2))
753.37/208.13	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.13	           , activate(X) -> X
753.37/208.13	           , activate(n__0()) -> 0()
753.37/208.13	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.13	           , activate(n__s(X)) -> s(activate(X))
753.37/208.13	           , U13(tt(), V1, V2) ->
753.37/208.13	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.13	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.13	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.13	           , isNat(n__0()) -> tt()
753.37/208.13	           , isNat(n__plus(V1, V2)) ->
753.37/208.13	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.13	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.13	           , U16(tt()) -> tt()
753.37/208.13	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.13	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.13	           , U23(tt()) -> tt()
753.37/208.13	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.13	           , U32(tt()) -> tt()
753.37/208.13	           , U41(tt()) -> tt()
753.37/208.13	           , s(X) -> n__s(X)
753.37/208.13	           , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.13	           , 0() -> n__0() }
753.37/208.13	         Obligation:
753.37/208.13	           innermost runtime complexity
753.37/208.13	         Answer:
753.37/208.13	           YES(O(1),O(n^2))
753.37/208.13	         
753.37/208.13	         We estimate the number of application of {5} by applications of
753.37/208.13	         Pre({5}) = {4}. Here rules are labeled as follows:
753.37/208.13	         
753.37/208.13	           DPs:
753.37/208.13	             { 1: isNatKind^#(n__plus(V1, V2)) ->
753.37/208.13	                  c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.13	                      isNatKind^#(activate(V1)))
753.37/208.13	             , 2: isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.13	             , 3: U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.13	             , 4: U61^#(tt(), M, N) ->
753.37/208.13	                  c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.13	                       isNatKind^#(activate(M)))
753.37/208.13	             , 5: U62^#(tt(), M, N) ->
753.37/208.13	                  c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.13	                       isNat^#(activate(N)))
753.37/208.13	             , 6: U11^#(tt(), V1, V2) ->
753.37/208.13	                  c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.13	                      isNatKind^#(activate(V1)))
753.37/208.13	             , 7: U12^#(tt(), V1, V2) ->
753.37/208.13	                  c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.13	                      isNatKind^#(activate(V2)))
753.37/208.13	             , 8: U13^#(tt(), V1, V2) ->
753.37/208.13	                  c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.13	                      isNatKind^#(activate(V2)))
753.37/208.13	             , 9: U14^#(tt(), V1, V2) ->
753.37/208.13	                  c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.13	                      isNat^#(activate(V1)))
753.37/208.13	             , 10: U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.13	             , 11: isNat^#(n__plus(V1, V2)) ->
753.37/208.13	                   c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.13	                       isNatKind^#(activate(V1)))
753.37/208.13	             , 12: isNat^#(n__s(V1)) ->
753.37/208.13	                   c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.13	                        isNatKind^#(activate(V1)))
753.37/208.13	             , 13: U21^#(tt(), V1) ->
753.37/208.13	                   c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.13	                        isNatKind^#(activate(V1)))
753.37/208.13	             , 14: U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.13	             , 15: U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.13	             , 16: U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.13	         
753.37/208.13	         We are left with following problem, upon which TcT provides the
753.37/208.13	         certificate YES(O(1),O(n^2)).
753.37/208.13	         
753.37/208.13	         Strict DPs:
753.37/208.13	           { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.13	             c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.13	           , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.13	           , U61^#(tt(), M, N) ->
753.37/208.13	             c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.13	                  isNatKind^#(activate(M))) }
753.37/208.13	         Weak DPs:
753.37/208.13	           { U11^#(tt(), V1, V2) ->
753.37/208.13	             c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , U12^#(tt(), V1, V2) ->
753.37/208.13	             c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V2)))
753.37/208.13	           , U13^#(tt(), V1, V2) ->
753.37/208.13	             c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V2)))
753.37/208.13	           , U14^#(tt(), V1, V2) ->
753.37/208.13	             c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.13	                 isNat^#(activate(V1)))
753.37/208.13	           , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.13	           , isNat^#(n__plus(V1, V2)) ->
753.37/208.13	             c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.13	                 isNatKind^#(activate(V1)))
753.37/208.13	           , isNat^#(n__s(V1)) ->
753.37/208.13	             c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.13	                  isNatKind^#(activate(V1)))
753.37/208.13	           , U21^#(tt(), V1) ->
753.37/208.13	             c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.13	                  isNatKind^#(activate(V1)))
753.37/208.13	           , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.13	           , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.13	           , U62^#(tt(), M, N) ->
753.37/208.13	             c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.13	                  isNat^#(activate(N)))
753.37/208.13	           , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.13	         Weak Trs:
753.37/208.13	           { U11(tt(), V1, V2) ->
753.37/208.13	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.13	           , U12(tt(), V1, V2) ->
753.37/208.13	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.13	           , isNatKind(n__0()) -> tt()
753.37/208.13	           , isNatKind(n__plus(V1, V2)) ->
753.37/208.13	             U31(isNatKind(activate(V1)), activate(V2))
753.37/208.13	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.13	           , activate(X) -> X
753.37/208.13	           , activate(n__0()) -> 0()
753.37/208.13	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.13	           , activate(n__s(X)) -> s(activate(X))
753.37/208.13	           , U13(tt(), V1, V2) ->
753.37/208.13	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.13	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.13	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.13	           , isNat(n__0()) -> tt()
753.37/208.13	           , isNat(n__plus(V1, V2)) ->
753.37/208.13	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.14	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.14	           , U16(tt()) -> tt()
753.37/208.14	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.14	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.14	           , U23(tt()) -> tt()
753.37/208.14	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.14	           , U32(tt()) -> tt()
753.37/208.14	           , U41(tt()) -> tt()
753.37/208.14	           , s(X) -> n__s(X)
753.37/208.14	           , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.14	           , 0() -> n__0() }
753.37/208.14	         Obligation:
753.37/208.14	           innermost runtime complexity
753.37/208.14	         Answer:
753.37/208.14	           YES(O(1),O(n^2))
753.37/208.14	         
753.37/208.14	         We decompose the input problem according to the dependency graph
753.37/208.14	         into the upper component
753.37/208.14	         
753.37/208.14	           { U11^#(tt(), V1, V2) ->
753.37/208.14	             c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                 isNatKind^#(activate(V1)))
753.37/208.14	           , U12^#(tt(), V1, V2) ->
753.37/208.14	             c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.14	                 isNatKind^#(activate(V2)))
753.37/208.14	           , U13^#(tt(), V1, V2) ->
753.37/208.14	             c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.14	                 isNatKind^#(activate(V2)))
753.37/208.14	           , U14^#(tt(), V1, V2) ->
753.37/208.14	             c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.14	                 isNat^#(activate(V1)))
753.37/208.14	           , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.14	           , isNat^#(n__plus(V1, V2)) ->
753.37/208.14	             c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                 isNatKind^#(activate(V1)))
753.37/208.14	           , isNat^#(n__s(V1)) ->
753.37/208.14	             c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.14	                  isNatKind^#(activate(V1)))
753.37/208.14	           , U21^#(tt(), V1) ->
753.37/208.14	             c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.14	                  isNatKind^#(activate(V1)))
753.37/208.14	           , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.14	           , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.14	           , U61^#(tt(), M, N) ->
753.37/208.14	             c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.14	                  isNatKind^#(activate(M)))
753.37/208.14	           , U62^#(tt(), M, N) ->
753.37/208.14	             c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.14	                  isNat^#(activate(N)))
753.37/208.14	           , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.14	         
753.37/208.14	         and lower component
753.37/208.14	         
753.37/208.14	           { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.14	             c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.14	                 isNatKind^#(activate(V1)))
753.37/208.14	           , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.14	           , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2))) }
753.37/208.14	         
753.37/208.14	         Further, following extension rules are added to the lower
753.37/208.14	         component.
753.37/208.14	         
753.37/208.14	         { U11^#(tt(), V1, V2) ->
753.37/208.14	           U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.14	         , U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.37/208.14	         , U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.37/208.14	         , U12^#(tt(), V1, V2) ->
753.37/208.14	           U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.14	         , U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.37/208.14	         , U13^#(tt(), V1, V2) ->
753.37/208.14	           U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.14	         , U14^#(tt(), V1, V2) -> U15^#(isNat(activate(V1)), activate(V2))
753.37/208.14	         , U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.37/208.14	         , U15^#(tt(), V2) -> isNat^#(activate(V2))
753.37/208.14	         , isNat^#(n__plus(V1, V2)) ->
753.37/208.14	           U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.14	         , isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.37/208.14	         , isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.37/208.14	         , isNat^#(n__s(V1)) -> U21^#(isNatKind(activate(V1)), activate(V1))
753.37/208.14	         , U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.37/208.14	         , U21^#(tt(), V1) -> U22^#(isNatKind(activate(V1)), activate(V1))
753.37/208.14	         , U22^#(tt(), V1) -> isNat^#(activate(V1))
753.37/208.14	         , U51^#(tt(), N) -> isNatKind^#(activate(N))
753.37/208.14	         , U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.37/208.14	         , U61^#(tt(), M, N) ->
753.37/208.14	           U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.37/208.14	         , U62^#(tt(), M, N) -> isNat^#(activate(N))
753.37/208.14	         , U62^#(tt(), M, N) ->
753.37/208.14	           U63^#(isNat(activate(N)), activate(M), activate(N))
753.37/208.14	         , U63^#(tt(), M, N) -> isNatKind^#(activate(N)) }
753.37/208.14	         
753.37/208.14	         TcT solves the upper component with certificate YES(O(1),O(n^1)).
753.37/208.14	         
753.37/208.14	         Sub-proof:
753.37/208.14	         ----------
753.37/208.14	           We are left with following problem, upon which TcT provides the
753.37/208.14	           certificate YES(O(1),O(n^1)).
753.37/208.14	           
753.37/208.14	           Strict DPs:
753.37/208.14	             { U11^#(tt(), V1, V2) ->
753.37/208.14	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                   isNatKind^#(activate(V1)))
753.37/208.14	             , U12^#(tt(), V1, V2) ->
753.37/208.14	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.14	                   isNatKind^#(activate(V2)))
753.37/208.14	             , U13^#(tt(), V1, V2) ->
753.37/208.14	               c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.14	                   isNatKind^#(activate(V2)))
753.37/208.14	             , isNat^#(n__plus(V1, V2)) ->
753.37/208.14	               c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                   isNatKind^#(activate(V1)))
753.37/208.14	             , isNat^#(n__s(V1)) ->
753.37/208.14	               c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.14	                    isNatKind^#(activate(V1)))
753.37/208.14	             , U21^#(tt(), V1) ->
753.37/208.14	               c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.14	                    isNatKind^#(activate(V1)))
753.37/208.14	             , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.14	             , U61^#(tt(), M, N) ->
753.37/208.14	               c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.14	                    isNatKind^#(activate(M)))
753.37/208.14	             , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.14	           Weak DPs:
753.37/208.14	             { U14^#(tt(), V1, V2) ->
753.37/208.14	               c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.14	                   isNat^#(activate(V1)))
753.37/208.14	             , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.14	             , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.14	             , U62^#(tt(), M, N) ->
753.37/208.14	               c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.14	                    isNat^#(activate(N))) }
753.37/208.14	           Weak Trs:
753.37/208.14	             { U11(tt(), V1, V2) ->
753.37/208.14	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.14	             , U12(tt(), V1, V2) ->
753.37/208.14	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.14	             , isNatKind(n__0()) -> tt()
753.37/208.14	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.14	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.14	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.14	             , activate(X) -> X
753.37/208.14	             , activate(n__0()) -> 0()
753.37/208.14	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.14	             , activate(n__s(X)) -> s(activate(X))
753.37/208.14	             , U13(tt(), V1, V2) ->
753.37/208.14	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.14	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.14	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.14	             , isNat(n__0()) -> tt()
753.37/208.14	             , isNat(n__plus(V1, V2)) ->
753.37/208.14	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.14	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.14	             , U16(tt()) -> tt()
753.37/208.14	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.14	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.14	             , U23(tt()) -> tt()
753.37/208.14	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.14	             , U32(tt()) -> tt()
753.37/208.14	             , U41(tt()) -> tt()
753.37/208.14	             , s(X) -> n__s(X)
753.37/208.14	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.14	             , 0() -> n__0() }
753.37/208.14	           Obligation:
753.37/208.14	             innermost runtime complexity
753.37/208.14	           Answer:
753.37/208.14	             YES(O(1),O(n^1))
753.37/208.14	           
753.37/208.14	           We estimate the number of application of {6,7} by applications of
753.37/208.14	           Pre({6,7}) = {5}. Here rules are labeled as follows:
753.37/208.14	           
753.37/208.14	             DPs:
753.37/208.14	               { 1: U11^#(tt(), V1, V2) ->
753.37/208.14	                    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                        isNatKind^#(activate(V1)))
753.37/208.14	               , 2: U12^#(tt(), V1, V2) ->
753.37/208.14	                    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.14	                        isNatKind^#(activate(V2)))
753.37/208.14	               , 3: U13^#(tt(), V1, V2) ->
753.37/208.14	                    c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.14	                        isNatKind^#(activate(V2)))
753.37/208.14	               , 4: isNat^#(n__plus(V1, V2)) ->
753.37/208.14	                    c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                        isNatKind^#(activate(V1)))
753.37/208.14	               , 5: isNat^#(n__s(V1)) ->
753.37/208.14	                    c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.14	                         isNatKind^#(activate(V1)))
753.37/208.14	               , 6: U21^#(tt(), V1) ->
753.37/208.14	                    c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.14	                         isNatKind^#(activate(V1)))
753.37/208.14	               , 7: U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.14	               , 8: U61^#(tt(), M, N) ->
753.37/208.14	                    c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.14	                         isNatKind^#(activate(M)))
753.37/208.14	               , 9: U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N)))
753.37/208.14	               , 10: U14^#(tt(), V1, V2) ->
753.37/208.14	                     c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.14	                         isNat^#(activate(V1)))
753.37/208.14	               , 11: U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.14	               , 12: U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.14	               , 13: U62^#(tt(), M, N) ->
753.37/208.14	                     c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.14	                          isNat^#(activate(N))) }
753.37/208.14	           
753.37/208.14	           We are left with following problem, upon which TcT provides the
753.37/208.14	           certificate YES(O(1),O(n^1)).
753.37/208.14	           
753.37/208.14	           Strict DPs:
753.37/208.14	             { U11^#(tt(), V1, V2) ->
753.37/208.14	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                   isNatKind^#(activate(V1)))
753.37/208.14	             , U12^#(tt(), V1, V2) ->
753.37/208.14	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.14	                   isNatKind^#(activate(V2)))
753.37/208.14	             , U13^#(tt(), V1, V2) ->
753.37/208.14	               c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.14	                   isNatKind^#(activate(V2)))
753.37/208.14	             , isNat^#(n__plus(V1, V2)) ->
753.37/208.14	               c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                   isNatKind^#(activate(V1)))
753.37/208.14	             , isNat^#(n__s(V1)) ->
753.37/208.14	               c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.14	                    isNatKind^#(activate(V1)))
753.37/208.14	             , U61^#(tt(), M, N) ->
753.37/208.14	               c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.14	                    isNatKind^#(activate(M)))
753.37/208.14	             , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.14	           Weak DPs:
753.37/208.14	             { U14^#(tt(), V1, V2) ->
753.37/208.14	               c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.14	                   isNat^#(activate(V1)))
753.37/208.14	             , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.14	             , U21^#(tt(), V1) ->
753.37/208.14	               c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.14	                    isNatKind^#(activate(V1)))
753.37/208.14	             , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.14	             , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.14	             , U62^#(tt(), M, N) ->
753.37/208.14	               c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.14	                    isNat^#(activate(N))) }
753.37/208.14	           Weak Trs:
753.37/208.14	             { U11(tt(), V1, V2) ->
753.37/208.14	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.14	             , U12(tt(), V1, V2) ->
753.37/208.14	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.14	             , isNatKind(n__0()) -> tt()
753.37/208.14	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.14	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.14	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.14	             , activate(X) -> X
753.37/208.14	             , activate(n__0()) -> 0()
753.37/208.14	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.14	             , activate(n__s(X)) -> s(activate(X))
753.37/208.14	             , U13(tt(), V1, V2) ->
753.37/208.14	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.14	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.14	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.14	             , isNat(n__0()) -> tt()
753.37/208.14	             , isNat(n__plus(V1, V2)) ->
753.37/208.14	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.14	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.14	             , U16(tt()) -> tt()
753.37/208.14	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.14	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.14	             , U23(tt()) -> tt()
753.37/208.14	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.14	             , U32(tt()) -> tt()
753.37/208.14	             , U41(tt()) -> tt()
753.37/208.14	             , s(X) -> n__s(X)
753.37/208.14	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.14	             , 0() -> n__0() }
753.37/208.14	           Obligation:
753.37/208.14	             innermost runtime complexity
753.37/208.14	           Answer:
753.37/208.14	             YES(O(1),O(n^1))
753.37/208.14	           
753.37/208.14	           We estimate the number of application of {3} by applications of
753.37/208.14	           Pre({3}) = {2}. Here rules are labeled as follows:
753.37/208.14	           
753.37/208.14	             DPs:
753.37/208.14	               { 1: U11^#(tt(), V1, V2) ->
753.37/208.14	                    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                        isNatKind^#(activate(V1)))
753.37/208.14	               , 2: U12^#(tt(), V1, V2) ->
753.37/208.14	                    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.14	                        isNatKind^#(activate(V2)))
753.37/208.14	               , 3: U13^#(tt(), V1, V2) ->
753.37/208.14	                    c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.14	                        isNatKind^#(activate(V2)))
753.37/208.14	               , 4: isNat^#(n__plus(V1, V2)) ->
753.37/208.14	                    c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                        isNatKind^#(activate(V1)))
753.37/208.14	               , 5: isNat^#(n__s(V1)) ->
753.37/208.14	                    c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.14	                         isNatKind^#(activate(V1)))
753.37/208.14	               , 6: U61^#(tt(), M, N) ->
753.37/208.14	                    c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.14	                         isNatKind^#(activate(M)))
753.37/208.14	               , 7: U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N)))
753.37/208.14	               , 8: U14^#(tt(), V1, V2) ->
753.37/208.14	                    c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.14	                        isNat^#(activate(V1)))
753.37/208.14	               , 9: U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.14	               , 10: U21^#(tt(), V1) ->
753.37/208.14	                     c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.14	                          isNatKind^#(activate(V1)))
753.37/208.14	               , 11: U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.14	               , 12: U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.14	               , 13: U62^#(tt(), M, N) ->
753.37/208.14	                     c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.14	                          isNat^#(activate(N))) }
753.37/208.14	           
753.37/208.14	           We are left with following problem, upon which TcT provides the
753.37/208.14	           certificate YES(O(1),O(n^1)).
753.37/208.14	           
753.37/208.14	           Strict DPs:
753.37/208.14	             { U11^#(tt(), V1, V2) ->
753.37/208.14	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                   isNatKind^#(activate(V1)))
753.37/208.14	             , U12^#(tt(), V1, V2) ->
753.37/208.14	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.14	                   isNatKind^#(activate(V2)))
753.37/208.14	             , isNat^#(n__plus(V1, V2)) ->
753.37/208.14	               c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.14	                   isNatKind^#(activate(V1)))
753.37/208.15	             , isNat^#(n__s(V1)) ->
753.37/208.15	               c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                    isNatKind^#(activate(V1)))
753.37/208.15	             , U61^#(tt(), M, N) ->
753.37/208.15	               c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.15	                    isNatKind^#(activate(M)))
753.37/208.15	             , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.15	           Weak DPs:
753.37/208.15	             { U13^#(tt(), V1, V2) ->
753.37/208.15	               c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.15	                   isNatKind^#(activate(V2)))
753.37/208.15	             , U14^#(tt(), V1, V2) ->
753.37/208.15	               c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.15	                   isNat^#(activate(V1)))
753.37/208.15	             , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.15	             , U21^#(tt(), V1) ->
753.37/208.15	               c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                    isNatKind^#(activate(V1)))
753.37/208.15	             , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.15	             , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.15	             , U62^#(tt(), M, N) ->
753.37/208.15	               c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.15	                    isNat^#(activate(N))) }
753.37/208.15	           Weak Trs:
753.37/208.15	             { U11(tt(), V1, V2) ->
753.37/208.15	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.15	             , U12(tt(), V1, V2) ->
753.37/208.15	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.15	             , isNatKind(n__0()) -> tt()
753.37/208.15	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.15	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.15	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.15	             , activate(X) -> X
753.37/208.15	             , activate(n__0()) -> 0()
753.37/208.15	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.15	             , activate(n__s(X)) -> s(activate(X))
753.37/208.15	             , U13(tt(), V1, V2) ->
753.37/208.15	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.15	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.15	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.15	             , isNat(n__0()) -> tt()
753.37/208.15	             , isNat(n__plus(V1, V2)) ->
753.37/208.15	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.15	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.15	             , U16(tt()) -> tt()
753.37/208.15	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.15	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.15	             , U23(tt()) -> tt()
753.37/208.15	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.15	             , U32(tt()) -> tt()
753.37/208.15	             , U41(tt()) -> tt()
753.37/208.15	             , s(X) -> n__s(X)
753.37/208.15	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.15	             , 0() -> n__0() }
753.37/208.15	           Obligation:
753.37/208.15	             innermost runtime complexity
753.37/208.15	           Answer:
753.37/208.15	             YES(O(1),O(n^1))
753.37/208.15	           
753.37/208.15	           We estimate the number of application of {2} by applications of
753.37/208.15	           Pre({2}) = {1}. Here rules are labeled as follows:
753.37/208.15	           
753.37/208.15	             DPs:
753.37/208.15	               { 1: U11^#(tt(), V1, V2) ->
753.37/208.15	                    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V1)))
753.37/208.15	               , 2: U12^#(tt(), V1, V2) ->
753.37/208.15	                    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V2)))
753.37/208.15	               , 3: isNat^#(n__plus(V1, V2)) ->
753.37/208.15	                    c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V1)))
753.37/208.15	               , 4: isNat^#(n__s(V1)) ->
753.37/208.15	                    c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                         isNatKind^#(activate(V1)))
753.37/208.15	               , 5: U61^#(tt(), M, N) ->
753.37/208.15	                    c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.15	                         isNatKind^#(activate(M)))
753.37/208.15	               , 6: U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N)))
753.37/208.15	               , 7: U13^#(tt(), V1, V2) ->
753.37/208.15	                    c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V2)))
753.37/208.15	               , 8: U14^#(tt(), V1, V2) ->
753.37/208.15	                    c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.15	                        isNat^#(activate(V1)))
753.37/208.15	               , 9: U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.15	               , 10: U21^#(tt(), V1) ->
753.37/208.15	                     c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                          isNatKind^#(activate(V1)))
753.37/208.15	               , 11: U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.15	               , 12: U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.15	               , 13: U62^#(tt(), M, N) ->
753.37/208.15	                     c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.15	                          isNat^#(activate(N))) }
753.37/208.15	           
753.37/208.15	           We are left with following problem, upon which TcT provides the
753.37/208.15	           certificate YES(O(1),O(n^1)).
753.37/208.15	           
753.37/208.15	           Strict DPs:
753.37/208.15	             { U11^#(tt(), V1, V2) ->
753.37/208.15	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.15	                   isNatKind^#(activate(V1)))
753.37/208.15	             , isNat^#(n__plus(V1, V2)) ->
753.37/208.15	               c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.15	                   isNatKind^#(activate(V1)))
753.37/208.15	             , isNat^#(n__s(V1)) ->
753.37/208.15	               c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                    isNatKind^#(activate(V1)))
753.37/208.15	             , U61^#(tt(), M, N) ->
753.37/208.15	               c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.15	                    isNatKind^#(activate(M)))
753.37/208.15	             , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.15	           Weak DPs:
753.37/208.15	             { U12^#(tt(), V1, V2) ->
753.37/208.15	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.15	                   isNatKind^#(activate(V2)))
753.37/208.15	             , U13^#(tt(), V1, V2) ->
753.37/208.15	               c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.15	                   isNatKind^#(activate(V2)))
753.37/208.15	             , U14^#(tt(), V1, V2) ->
753.37/208.15	               c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.15	                   isNat^#(activate(V1)))
753.37/208.15	             , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.15	             , U21^#(tt(), V1) ->
753.37/208.15	               c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                    isNatKind^#(activate(V1)))
753.37/208.15	             , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.15	             , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.15	             , U62^#(tt(), M, N) ->
753.37/208.15	               c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.15	                    isNat^#(activate(N))) }
753.37/208.15	           Weak Trs:
753.37/208.15	             { U11(tt(), V1, V2) ->
753.37/208.15	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.15	             , U12(tt(), V1, V2) ->
753.37/208.15	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.15	             , isNatKind(n__0()) -> tt()
753.37/208.15	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.15	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.15	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.15	             , activate(X) -> X
753.37/208.15	             , activate(n__0()) -> 0()
753.37/208.15	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.15	             , activate(n__s(X)) -> s(activate(X))
753.37/208.15	             , U13(tt(), V1, V2) ->
753.37/208.15	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.15	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.15	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.15	             , isNat(n__0()) -> tt()
753.37/208.15	             , isNat(n__plus(V1, V2)) ->
753.37/208.15	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.15	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.15	             , U16(tt()) -> tt()
753.37/208.15	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.15	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.15	             , U23(tt()) -> tt()
753.37/208.15	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.15	             , U32(tt()) -> tt()
753.37/208.15	             , U41(tt()) -> tt()
753.37/208.15	             , s(X) -> n__s(X)
753.37/208.15	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.15	             , 0() -> n__0() }
753.37/208.15	           Obligation:
753.37/208.15	             innermost runtime complexity
753.37/208.15	           Answer:
753.37/208.15	             YES(O(1),O(n^1))
753.37/208.15	           
753.37/208.15	           We estimate the number of application of {1} by applications of
753.37/208.15	           Pre({1}) = {2}. Here rules are labeled as follows:
753.37/208.15	           
753.37/208.15	             DPs:
753.37/208.15	               { 1: U11^#(tt(), V1, V2) ->
753.37/208.15	                    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V1)))
753.37/208.15	               , 2: isNat^#(n__plus(V1, V2)) ->
753.37/208.15	                    c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V1)))
753.37/208.15	               , 3: isNat^#(n__s(V1)) ->
753.37/208.15	                    c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                         isNatKind^#(activate(V1)))
753.37/208.15	               , 4: U61^#(tt(), M, N) ->
753.37/208.15	                    c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.15	                         isNatKind^#(activate(M)))
753.37/208.15	               , 5: U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N)))
753.37/208.15	               , 6: U12^#(tt(), V1, V2) ->
753.37/208.15	                    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V2)))
753.37/208.15	               , 7: U13^#(tt(), V1, V2) ->
753.37/208.15	                    c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V2)))
753.37/208.15	               , 8: U14^#(tt(), V1, V2) ->
753.37/208.15	                    c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.15	                        isNat^#(activate(V1)))
753.37/208.15	               , 9: U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.15	               , 10: U21^#(tt(), V1) ->
753.37/208.15	                     c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                          isNatKind^#(activate(V1)))
753.37/208.15	               , 11: U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.15	               , 12: U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.15	               , 13: U62^#(tt(), M, N) ->
753.37/208.15	                     c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.15	                          isNat^#(activate(N))) }
753.37/208.15	           
753.37/208.15	           We are left with following problem, upon which TcT provides the
753.37/208.15	           certificate YES(O(1),O(n^1)).
753.37/208.15	           
753.37/208.15	           Strict DPs:
753.37/208.15	             { isNat^#(n__plus(V1, V2)) ->
753.37/208.15	               c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.15	                   isNatKind^#(activate(V1)))
753.37/208.15	             , isNat^#(n__s(V1)) ->
753.37/208.15	               c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                    isNatKind^#(activate(V1)))
753.37/208.15	             , U61^#(tt(), M, N) ->
753.37/208.15	               c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.15	                    isNatKind^#(activate(M)))
753.37/208.15	             , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.15	           Weak DPs:
753.37/208.15	             { U11^#(tt(), V1, V2) ->
753.37/208.15	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.15	                   isNatKind^#(activate(V1)))
753.37/208.15	             , U12^#(tt(), V1, V2) ->
753.37/208.15	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.15	                   isNatKind^#(activate(V2)))
753.37/208.15	             , U13^#(tt(), V1, V2) ->
753.37/208.15	               c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.15	                   isNatKind^#(activate(V2)))
753.37/208.15	             , U14^#(tt(), V1, V2) ->
753.37/208.15	               c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.15	                   isNat^#(activate(V1)))
753.37/208.15	             , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.15	             , U21^#(tt(), V1) ->
753.37/208.15	               c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                    isNatKind^#(activate(V1)))
753.37/208.15	             , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.15	             , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.15	             , U62^#(tt(), M, N) ->
753.37/208.15	               c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.15	                    isNat^#(activate(N))) }
753.37/208.15	           Weak Trs:
753.37/208.15	             { U11(tt(), V1, V2) ->
753.37/208.15	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.15	             , U12(tt(), V1, V2) ->
753.37/208.15	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.15	             , isNatKind(n__0()) -> tt()
753.37/208.15	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.15	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.15	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.15	             , activate(X) -> X
753.37/208.15	             , activate(n__0()) -> 0()
753.37/208.15	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.15	             , activate(n__s(X)) -> s(activate(X))
753.37/208.15	             , U13(tt(), V1, V2) ->
753.37/208.15	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.15	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.15	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.15	             , isNat(n__0()) -> tt()
753.37/208.15	             , isNat(n__plus(V1, V2)) ->
753.37/208.15	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.15	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.15	             , U16(tt()) -> tt()
753.37/208.15	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.15	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.15	             , U23(tt()) -> tt()
753.37/208.15	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.15	             , U32(tt()) -> tt()
753.37/208.15	             , U41(tt()) -> tt()
753.37/208.15	             , s(X) -> n__s(X)
753.37/208.15	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.15	             , 0() -> n__0() }
753.37/208.15	           Obligation:
753.37/208.15	             innermost runtime complexity
753.37/208.15	           Answer:
753.37/208.15	             YES(O(1),O(n^1))
753.37/208.15	           
753.37/208.15	           We estimate the number of application of {3} by applications of
753.37/208.15	           Pre({3}) = {}. Here rules are labeled as follows:
753.37/208.15	           
753.37/208.15	             DPs:
753.37/208.15	               { 1: isNat^#(n__plus(V1, V2)) ->
753.37/208.15	                    c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V1)))
753.37/208.15	               , 2: isNat^#(n__s(V1)) ->
753.37/208.15	                    c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                         isNatKind^#(activate(V1)))
753.37/208.15	               , 3: U61^#(tt(), M, N) ->
753.37/208.15	                    c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.15	                         isNatKind^#(activate(M)))
753.37/208.15	               , 4: U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N)))
753.37/208.15	               , 5: U11^#(tt(), V1, V2) ->
753.37/208.15	                    c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V1)))
753.37/208.15	               , 6: U12^#(tt(), V1, V2) ->
753.37/208.15	                    c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V2)))
753.37/208.15	               , 7: U13^#(tt(), V1, V2) ->
753.37/208.15	                    c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.15	                        isNatKind^#(activate(V2)))
753.37/208.15	               , 8: U14^#(tt(), V1, V2) ->
753.37/208.15	                    c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.15	                        isNat^#(activate(V1)))
753.37/208.15	               , 9: U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.15	               , 10: U21^#(tt(), V1) ->
753.37/208.15	                     c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.15	                          isNatKind^#(activate(V1)))
753.37/208.15	               , 11: U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.15	               , 12: U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.15	               , 13: U62^#(tt(), M, N) ->
753.37/208.15	                     c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.15	                          isNat^#(activate(N))) }
753.37/208.15	           
753.37/208.15	           We are left with following problem, upon which TcT provides the
753.37/208.15	           certificate YES(O(1),O(n^1)).
753.37/208.15	           
753.37/208.15	           Strict DPs:
753.37/208.15	             { isNat^#(n__plus(V1, V2)) ->
753.37/208.15	               c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.15	                   isNatKind^#(activate(V1)))
753.37/208.16	             , isNat^#(n__s(V1)) ->
753.37/208.16	               c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.16	                    isNatKind^#(activate(V1)))
753.37/208.16	             , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.16	           Weak DPs:
753.37/208.16	             { U11^#(tt(), V1, V2) ->
753.37/208.16	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.16	                   isNatKind^#(activate(V1)))
753.37/208.16	             , U12^#(tt(), V1, V2) ->
753.37/208.16	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.16	                   isNatKind^#(activate(V2)))
753.37/208.16	             , U13^#(tt(), V1, V2) ->
753.37/208.16	               c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.16	                   isNatKind^#(activate(V2)))
753.37/208.16	             , U14^#(tt(), V1, V2) ->
753.37/208.16	               c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.16	                   isNat^#(activate(V1)))
753.37/208.16	             , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.16	             , U21^#(tt(), V1) ->
753.37/208.16	               c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.16	                    isNatKind^#(activate(V1)))
753.37/208.16	             , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.16	             , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.37/208.16	             , U61^#(tt(), M, N) ->
753.37/208.16	               c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.16	                    isNatKind^#(activate(M)))
753.37/208.16	             , U62^#(tt(), M, N) ->
753.37/208.16	               c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.16	                    isNat^#(activate(N))) }
753.37/208.16	           Weak Trs:
753.37/208.16	             { U11(tt(), V1, V2) ->
753.37/208.16	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.16	             , U12(tt(), V1, V2) ->
753.37/208.16	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.16	             , isNatKind(n__0()) -> tt()
753.37/208.16	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.16	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.16	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.16	             , activate(X) -> X
753.37/208.16	             , activate(n__0()) -> 0()
753.37/208.16	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.16	             , activate(n__s(X)) -> s(activate(X))
753.37/208.16	             , U13(tt(), V1, V2) ->
753.37/208.16	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.16	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.16	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.16	             , isNat(n__0()) -> tt()
753.37/208.16	             , isNat(n__plus(V1, V2)) ->
753.37/208.16	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.16	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.16	             , U16(tt()) -> tt()
753.37/208.16	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.16	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.16	             , U23(tt()) -> tt()
753.37/208.16	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.16	             , U32(tt()) -> tt()
753.37/208.16	             , U41(tt()) -> tt()
753.37/208.16	             , s(X) -> n__s(X)
753.37/208.16	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.16	             , 0() -> n__0() }
753.37/208.16	           Obligation:
753.37/208.16	             innermost runtime complexity
753.37/208.16	           Answer:
753.37/208.16	             YES(O(1),O(n^1))
753.37/208.16	           
753.37/208.16	           The following weak DPs constitute a sub-graph of the DG that is
753.37/208.16	           closed under successors. The DPs are removed.
753.37/208.16	           
753.37/208.16	           { U51^#(tt(), N) -> c_16(isNatKind^#(activate(N))) }
753.37/208.16	           
753.37/208.16	           We are left with following problem, upon which TcT provides the
753.37/208.16	           certificate YES(O(1),O(n^1)).
753.37/208.16	           
753.37/208.16	           Strict DPs:
753.37/208.16	             { isNat^#(n__plus(V1, V2)) ->
753.37/208.16	               c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.16	                   isNatKind^#(activate(V1)))
753.37/208.16	             , isNat^#(n__s(V1)) ->
753.37/208.16	               c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.16	                    isNatKind^#(activate(V1)))
753.37/208.16	             , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.16	           Weak DPs:
753.37/208.16	             { U11^#(tt(), V1, V2) ->
753.37/208.16	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.16	                   isNatKind^#(activate(V1)))
753.37/208.16	             , U12^#(tt(), V1, V2) ->
753.37/208.16	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.16	                   isNatKind^#(activate(V2)))
753.37/208.16	             , U13^#(tt(), V1, V2) ->
753.37/208.16	               c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.16	                   isNatKind^#(activate(V2)))
753.37/208.16	             , U14^#(tt(), V1, V2) ->
753.37/208.16	               c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.16	                   isNat^#(activate(V1)))
753.37/208.16	             , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.16	             , U21^#(tt(), V1) ->
753.37/208.16	               c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.16	                    isNatKind^#(activate(V1)))
753.37/208.16	             , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.37/208.16	             , U61^#(tt(), M, N) ->
753.37/208.16	               c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.16	                    isNatKind^#(activate(M)))
753.37/208.16	             , U62^#(tt(), M, N) ->
753.37/208.16	               c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.16	                    isNat^#(activate(N))) }
753.37/208.16	           Weak Trs:
753.37/208.16	             { U11(tt(), V1, V2) ->
753.37/208.16	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.16	             , U12(tt(), V1, V2) ->
753.37/208.16	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.16	             , isNatKind(n__0()) -> tt()
753.37/208.16	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.16	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.16	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.16	             , activate(X) -> X
753.37/208.16	             , activate(n__0()) -> 0()
753.37/208.16	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.16	             , activate(n__s(X)) -> s(activate(X))
753.37/208.16	             , U13(tt(), V1, V2) ->
753.37/208.16	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.16	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.16	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.16	             , isNat(n__0()) -> tt()
753.37/208.16	             , isNat(n__plus(V1, V2)) ->
753.37/208.16	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.16	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.16	             , U16(tt()) -> tt()
753.37/208.16	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.16	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.16	             , U23(tt()) -> tt()
753.37/208.16	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.16	             , U32(tt()) -> tt()
753.37/208.16	             , U41(tt()) -> tt()
753.37/208.16	             , s(X) -> n__s(X)
753.37/208.16	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.16	             , 0() -> n__0() }
753.37/208.16	           Obligation:
753.37/208.16	             innermost runtime complexity
753.37/208.16	           Answer:
753.37/208.16	             YES(O(1),O(n^1))
753.37/208.16	           
753.37/208.16	           Due to missing edges in the dependency-graph, the right-hand sides
753.37/208.16	           of following rules could be simplified:
753.37/208.16	           
753.37/208.16	             { U11^#(tt(), V1, V2) ->
753.37/208.16	               c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.16	                   isNatKind^#(activate(V1)))
753.37/208.16	             , U12^#(tt(), V1, V2) ->
753.37/208.16	               c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.16	                   isNatKind^#(activate(V2)))
753.37/208.16	             , U13^#(tt(), V1, V2) ->
753.37/208.16	               c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.37/208.16	                   isNatKind^#(activate(V2)))
753.37/208.16	             , isNat^#(n__plus(V1, V2)) ->
753.37/208.16	               c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.37/208.16	                   isNatKind^#(activate(V1)))
753.37/208.16	             , isNat^#(n__s(V1)) ->
753.37/208.16	               c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.16	                    isNatKind^#(activate(V1)))
753.37/208.16	             , U21^#(tt(), V1) ->
753.37/208.16	               c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.37/208.16	                    isNatKind^#(activate(V1)))
753.37/208.16	             , U61^#(tt(), M, N) ->
753.37/208.16	               c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.37/208.16	                    isNatKind^#(activate(M)))
753.37/208.16	             , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.37/208.16	           
753.37/208.16	           We are left with following problem, upon which TcT provides the
753.37/208.16	           certificate YES(O(1),O(n^1)).
753.37/208.16	           
753.37/208.16	           Strict DPs:
753.37/208.16	             { isNat^#(n__plus(V1, V2)) ->
753.37/208.16	               c_1(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.16	             , isNat^#(n__s(V1)) ->
753.37/208.16	               c_2(U21^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.16	             , U63^#(tt(), M, N) -> c_3() }
753.37/208.16	           Weak DPs:
753.37/208.16	             { U11^#(tt(), V1, V2) ->
753.37/208.16	               c_4(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.16	             , U12^#(tt(), V1, V2) ->
753.37/208.16	               c_5(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.16	             , U13^#(tt(), V1, V2) ->
753.37/208.16	               c_6(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.16	             , U14^#(tt(), V1, V2) ->
753.37/208.16	               c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.16	                   isNat^#(activate(V1)))
753.37/208.16	             , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.16	             , U21^#(tt(), V1) ->
753.37/208.16	               c_9(U22^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.16	             , U22^#(tt(), V1) -> c_10(isNat^#(activate(V1)))
753.37/208.16	             , U61^#(tt(), M, N) ->
753.37/208.16	               c_11(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.37/208.16	             , U62^#(tt(), M, N) ->
753.37/208.16	               c_12(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.16	                    isNat^#(activate(N))) }
753.37/208.16	           Weak Trs:
753.37/208.16	             { U11(tt(), V1, V2) ->
753.37/208.16	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.16	             , U12(tt(), V1, V2) ->
753.37/208.16	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.16	             , isNatKind(n__0()) -> tt()
753.37/208.16	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.16	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.16	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.16	             , activate(X) -> X
753.37/208.16	             , activate(n__0()) -> 0()
753.37/208.16	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.16	             , activate(n__s(X)) -> s(activate(X))
753.37/208.16	             , U13(tt(), V1, V2) ->
753.37/208.16	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.16	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.16	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.16	             , isNat(n__0()) -> tt()
753.37/208.16	             , isNat(n__plus(V1, V2)) ->
753.37/208.16	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.16	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.16	             , U16(tt()) -> tt()
753.37/208.16	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.16	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.16	             , U23(tt()) -> tt()
753.37/208.16	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.16	             , U32(tt()) -> tt()
753.37/208.16	             , U41(tt()) -> tt()
753.37/208.16	             , s(X) -> n__s(X)
753.37/208.16	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.16	             , 0() -> n__0() }
753.37/208.16	           Obligation:
753.37/208.16	             innermost runtime complexity
753.37/208.16	           Answer:
753.37/208.16	             YES(O(1),O(n^1))
753.37/208.16	           
753.37/208.16	           We use the processor 'matrix interpretation of dimension 1' to
753.37/208.16	           orient following rules strictly.
753.37/208.16	           
753.37/208.16	           DPs:
753.37/208.16	             { 11: U61^#(tt(), M, N) ->
753.37/208.16	                   c_11(U62^#(isNatKind(activate(M)), activate(M), activate(N))) }
753.37/208.16	           Trs: { 0() -> n__0() }
753.37/208.16	           
753.37/208.16	           Sub-proof:
753.37/208.16	           ----------
753.37/208.16	             The following argument positions are usable:
753.37/208.16	               Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_4) = {1},
753.37/208.16	               Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1, 2},
753.37/208.16	               Uargs(c_8) = {1}, Uargs(c_9) = {1}, Uargs(c_10) = {1},
753.37/208.16	               Uargs(c_11) = {1}, Uargs(c_12) = {1, 2}
753.37/208.16	             
753.37/208.16	             TcT has computed the following constructor-restricted matrix
753.37/208.16	             interpretation. Note that the diagonal of the component-wise maxima
753.37/208.16	             of interpretation-entries (of constructors) contains no more than 0
753.37/208.16	             non-zero entries.
753.37/208.16	             
753.37/208.16	                 [U11](x1, x2, x3) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                              [tt] = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                 [U12](x1, x2, x3) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                   [isNatKind](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                    [activate](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                 [U13](x1, x2, x3) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                 [U14](x1, x2, x3) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                     [U15](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                       [isNat](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                         [U16](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                     [U21](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                     [U22](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                         [U23](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                     [U31](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                         [U32](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                         [U41](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                           [s](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                    [plus](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                            [n__0] = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                 [n__plus](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                        [n__s](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                               [0] = [7]                           
753.37/208.16	                                                                   
753.37/208.16	               [U11^#](x1, x2, x3) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	               [U12^#](x1, x2, x3) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                 [isNatKind^#](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	               [U13^#](x1, x2, x3) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	               [U14^#](x1, x2, x3) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                   [U15^#](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                     [isNat^#](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                   [U21^#](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                   [U22^#](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                   [U51^#](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	               [U61^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [7]
753.37/208.16	                                                                   
753.37/208.16	               [U62^#](x1, x2, x3) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	               [U63^#](x1, x2, x3) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                     [c_1](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                     [c_2](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                     [c_5](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                     [c_7](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                         [c_8](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                     [c_9](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                    [c_10](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                    [c_11](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                        [c_12](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                    [c_13](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                    [c_14](x1, x2) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                        [c_15](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                        [c_16](x1) = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                               [c] = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                         [c_1](x1) = [4] x1 + [0]                  
753.37/208.16	                                                                   
753.37/208.16	                         [c_2](x1) = [4] x1 + [0]                  
753.37/208.16	                                                                   
753.37/208.16	                             [c_3] = [0]                           
753.37/208.16	                                                                   
753.37/208.16	                         [c_4](x1) = [2] x1 + [0]                  
753.37/208.16	                                                                   
753.37/208.16	                         [c_5](x1) = [4] x1 + [0]                  
753.37/208.16	                                                                   
753.37/208.16	                         [c_6](x1) = [4] x1 + [0]                  
753.37/208.16	                                                                   
753.37/208.16	                     [c_7](x1, x2) = [1] x1 + [4] x2 + [0]         
753.37/208.16	                                                                   
753.37/208.16	                         [c_8](x1) = [4] x1 + [0]                  
753.37/208.16	                                                                   
753.37/208.16	                         [c_9](x1) = [4] x1 + [0]                  
753.37/208.16	                                                                   
753.37/208.16	                        [c_10](x1) = [4] x1 + [0]                  
753.37/208.16	                                                                   
753.37/208.16	                        [c_11](x1) = [2] x1 + [3]                  
753.37/208.16	                                                                   
753.37/208.16	                    [c_12](x1, x2) = [4] x1 + [4] x2 + [0]         
753.37/208.16	             
753.37/208.16	             The order satisfies the following ordering constraints:
753.37/208.16	             
753.37/208.16	                        [U11(tt(), V1, V2)] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U12(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.37/208.16	                                                                                                                
753.37/208.16	                        [U12(tt(), V1, V2)] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U13(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.37/208.16	                                                                                                                
753.37/208.16	                        [isNatKind(n__0())] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [tt()]                                                           
753.37/208.16	                                                                                                                
753.37/208.16	               [isNatKind(n__plus(V1, V2))] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U31(isNatKind(activate(V1)), activate(V2))]                     
753.37/208.16	                                                                                                                
753.37/208.16	                      [isNatKind(n__s(V1))] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U41(isNatKind(activate(V1)))]                                   
753.37/208.16	                                                                                                                
753.37/208.16	                              [activate(X)] =  [0]                                                              
753.37/208.16	                                            ?  [1] X + [0]                                                      
753.37/208.16	                                            =  [X]                                                              
753.37/208.16	                                                                                                                
753.37/208.16	                         [activate(n__0())] =  [0]                                                              
753.37/208.16	                                            ?  [7]                                                              
753.37/208.16	                                            =  [0()]                                                            
753.37/208.16	                                                                                                                
753.37/208.16	                [activate(n__plus(X1, X2))] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [plus(activate(X1), activate(X2))]                               
753.37/208.16	                                                                                                                
753.37/208.16	                        [activate(n__s(X))] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [s(activate(X))]                                                 
753.37/208.16	                                                                                                                
753.37/208.16	                        [U13(tt(), V1, V2)] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U14(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.37/208.16	                                                                                                                
753.37/208.16	                        [U14(tt(), V1, V2)] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U15(isNat(activate(V1)), activate(V2))]                         
753.37/208.16	                                                                                                                
753.37/208.16	                            [U15(tt(), V2)] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U16(isNat(activate(V2)))]                                       
753.37/208.16	                                                                                                                
753.37/208.16	                            [isNat(n__0())] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [tt()]                                                           
753.37/208.16	                                                                                                                
753.37/208.16	                   [isNat(n__plus(V1, V2))] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U11(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.37/208.16	                                                                                                                
753.37/208.16	                          [isNat(n__s(V1))] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U21(isNatKind(activate(V1)), activate(V1))]                     
753.37/208.16	                                                                                                                
753.37/208.16	                                [U16(tt())] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [tt()]                                                           
753.37/208.16	                                                                                                                
753.37/208.16	                            [U21(tt(), V1)] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U22(isNatKind(activate(V1)), activate(V1))]                     
753.37/208.16	                                                                                                                
753.37/208.16	                            [U22(tt(), V1)] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U23(isNat(activate(V1)))]                                       
753.37/208.16	                                                                                                                
753.37/208.16	                                [U23(tt())] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [tt()]                                                           
753.37/208.16	                                                                                                                
753.37/208.16	                            [U31(tt(), V2)] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [U32(isNatKind(activate(V2)))]                                   
753.37/208.16	                                                                                                                
753.37/208.16	                                [U32(tt())] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [tt()]                                                           
753.37/208.16	                                                                                                                
753.37/208.16	                                [U41(tt())] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [tt()]                                                           
753.37/208.16	                                                                                                                
753.37/208.16	                                     [s(X)] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [n__s(X)]                                                        
753.37/208.16	                                                                                                                
753.37/208.16	                             [plus(X1, X2)] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [n__plus(X1, X2)]                                                
753.37/208.16	                                                                                                                
753.37/208.16	                                      [0()] =  [7]                                                              
753.37/208.16	                                            >  [0]                                                              
753.37/208.16	                                            =  [n__0()]                                                         
753.37/208.16	                                                                                                                
753.37/208.16	                      [U11^#(tt(), V1, V2)] =  [0]                                                              
753.37/208.16	                                            >= [0]                                                              
753.37/208.16	                                            =  [c_4(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.37/208.16	                                                                                                                
753.37/208.16	                      [U12^#(tt(), V1, V2)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [c_5(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.37/208.17	                                                                                                                
753.37/208.17	                      [U13^#(tt(), V1, V2)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [c_6(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.37/208.17	                                                                                                                
753.37/208.17	                      [U14^#(tt(), V1, V2)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [c_7(U15^#(isNat(activate(V1)), activate(V2)),                   
753.37/208.17	                                                    isNat^#(activate(V1)))]                                     
753.37/208.17	                                                                                                                
753.37/208.17	                          [U15^#(tt(), V2)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [c_8(isNat^#(activate(V2)))]                                     
753.37/208.17	                                                                                                                
753.37/208.17	                 [isNat^#(n__plus(V1, V2))] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [c_1(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.37/208.17	                                                                                                                
753.37/208.17	                        [isNat^#(n__s(V1))] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [c_2(U21^#(isNatKind(activate(V1)), activate(V1)))]              
753.37/208.17	                                                                                                                
753.37/208.17	                          [U21^#(tt(), V1)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [c_9(U22^#(isNatKind(activate(V1)), activate(V1)))]              
753.37/208.17	                                                                                                                
753.37/208.17	                          [U22^#(tt(), V1)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [c_10(isNat^#(activate(V1)))]                                    
753.37/208.17	                                                                                                                
753.37/208.17	                        [U61^#(tt(), M, N)] =  [7] N + [7] M + [7]                                              
753.37/208.17	                                            >  [3]                                                              
753.37/208.17	                                            =  [c_11(U62^#(isNatKind(activate(M)), activate(M), activate(N)))]  
753.37/208.17	                                                                                                                
753.37/208.17	                        [U62^#(tt(), M, N)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [c_12(U63^#(isNat(activate(N)), activate(M), activate(N)),       
753.37/208.17	                                                     isNat^#(activate(N)))]                                     
753.37/208.17	                                                                                                                
753.37/208.17	                        [U63^#(tt(), M, N)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [c_3()]                                                          
753.37/208.17	                                                                                                                
753.37/208.17	           
753.37/208.17	           We return to the main proof. Consider the set of all dependency
753.37/208.17	           pairs
753.37/208.17	           
753.37/208.17	           :
753.37/208.17	             { 1: isNat^#(n__plus(V1, V2)) ->
753.37/208.17	                  c_1(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.17	             , 2: isNat^#(n__s(V1)) ->
753.37/208.17	                  c_2(U21^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.17	             , 3: U63^#(tt(), M, N) -> c_3()
753.37/208.17	             , 4: U11^#(tt(), V1, V2) ->
753.37/208.17	                  c_4(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.17	             , 5: U12^#(tt(), V1, V2) ->
753.37/208.17	                  c_5(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.17	             , 6: U13^#(tt(), V1, V2) ->
753.37/208.17	                  c_6(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.17	             , 7: U14^#(tt(), V1, V2) ->
753.37/208.17	                  c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.17	                      isNat^#(activate(V1)))
753.37/208.17	             , 8: U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.17	             , 9: U21^#(tt(), V1) ->
753.37/208.17	                  c_9(U22^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.17	             , 10: U22^#(tt(), V1) -> c_10(isNat^#(activate(V1)))
753.37/208.17	             , 11: U61^#(tt(), M, N) ->
753.37/208.17	                   c_11(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.37/208.17	             , 12: U62^#(tt(), M, N) ->
753.37/208.17	                   c_12(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.17	                        isNat^#(activate(N))) }
753.37/208.17	           
753.37/208.17	           Processor 'matrix interpretation of dimension 1' induces the
753.37/208.17	           complexity certificate YES(?,O(1)) on application of dependency
753.37/208.17	           pairs {11}. These cover all (indirect) predecessors of dependency
753.37/208.17	           pairs {3,11,12}, their number of application is equally bounded.
753.37/208.17	           The dependency pairs are shifted into the weak component.
753.37/208.17	           
753.37/208.17	           We are left with following problem, upon which TcT provides the
753.37/208.17	           certificate YES(O(1),O(n^1)).
753.37/208.17	           
753.37/208.17	           Strict DPs:
753.37/208.17	             { isNat^#(n__plus(V1, V2)) ->
753.37/208.17	               c_1(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.17	             , isNat^#(n__s(V1)) ->
753.37/208.17	               c_2(U21^#(isNatKind(activate(V1)), activate(V1))) }
753.37/208.17	           Weak DPs:
753.37/208.17	             { U11^#(tt(), V1, V2) ->
753.37/208.17	               c_4(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.17	             , U12^#(tt(), V1, V2) ->
753.37/208.17	               c_5(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.17	             , U13^#(tt(), V1, V2) ->
753.37/208.17	               c_6(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.17	             , U14^#(tt(), V1, V2) ->
753.37/208.17	               c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.17	                   isNat^#(activate(V1)))
753.37/208.17	             , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.17	             , U21^#(tt(), V1) ->
753.37/208.17	               c_9(U22^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.17	             , U22^#(tt(), V1) -> c_10(isNat^#(activate(V1)))
753.37/208.17	             , U61^#(tt(), M, N) ->
753.37/208.17	               c_11(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.37/208.17	             , U62^#(tt(), M, N) ->
753.37/208.17	               c_12(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.17	                    isNat^#(activate(N)))
753.37/208.17	             , U63^#(tt(), M, N) -> c_3() }
753.37/208.17	           Weak Trs:
753.37/208.17	             { U11(tt(), V1, V2) ->
753.37/208.17	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.17	             , U12(tt(), V1, V2) ->
753.37/208.17	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.17	             , isNatKind(n__0()) -> tt()
753.37/208.17	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.17	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.17	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.17	             , activate(X) -> X
753.37/208.17	             , activate(n__0()) -> 0()
753.37/208.17	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.17	             , activate(n__s(X)) -> s(activate(X))
753.37/208.17	             , U13(tt(), V1, V2) ->
753.37/208.17	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.17	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.17	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.17	             , isNat(n__0()) -> tt()
753.37/208.17	             , isNat(n__plus(V1, V2)) ->
753.37/208.17	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.17	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.17	             , U16(tt()) -> tt()
753.37/208.17	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.17	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.17	             , U23(tt()) -> tt()
753.37/208.17	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.17	             , U32(tt()) -> tt()
753.37/208.17	             , U41(tt()) -> tt()
753.37/208.17	             , s(X) -> n__s(X)
753.37/208.17	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.17	             , 0() -> n__0() }
753.37/208.17	           Obligation:
753.37/208.17	             innermost runtime complexity
753.37/208.17	           Answer:
753.37/208.17	             YES(O(1),O(n^1))
753.37/208.17	           
753.37/208.17	           The following weak DPs constitute a sub-graph of the DG that is
753.37/208.17	           closed under successors. The DPs are removed.
753.37/208.17	           
753.37/208.17	           { U63^#(tt(), M, N) -> c_3() }
753.37/208.17	           
753.37/208.17	           We are left with following problem, upon which TcT provides the
753.37/208.17	           certificate YES(O(1),O(n^1)).
753.37/208.17	           
753.37/208.17	           Strict DPs:
753.37/208.17	             { isNat^#(n__plus(V1, V2)) ->
753.37/208.17	               c_1(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.17	             , isNat^#(n__s(V1)) ->
753.37/208.17	               c_2(U21^#(isNatKind(activate(V1)), activate(V1))) }
753.37/208.17	           Weak DPs:
753.37/208.17	             { U11^#(tt(), V1, V2) ->
753.37/208.17	               c_4(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.17	             , U12^#(tt(), V1, V2) ->
753.37/208.17	               c_5(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.17	             , U13^#(tt(), V1, V2) ->
753.37/208.17	               c_6(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.17	             , U14^#(tt(), V1, V2) ->
753.37/208.17	               c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.17	                   isNat^#(activate(V1)))
753.37/208.17	             , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.37/208.17	             , U21^#(tt(), V1) ->
753.37/208.17	               c_9(U22^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.17	             , U22^#(tt(), V1) -> c_10(isNat^#(activate(V1)))
753.37/208.17	             , U61^#(tt(), M, N) ->
753.37/208.17	               c_11(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.37/208.17	             , U62^#(tt(), M, N) ->
753.37/208.17	               c_12(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.17	                    isNat^#(activate(N))) }
753.37/208.17	           Weak Trs:
753.37/208.17	             { U11(tt(), V1, V2) ->
753.37/208.17	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.17	             , U12(tt(), V1, V2) ->
753.37/208.17	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.17	             , isNatKind(n__0()) -> tt()
753.37/208.17	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.17	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.17	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.17	             , activate(X) -> X
753.37/208.17	             , activate(n__0()) -> 0()
753.37/208.17	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.17	             , activate(n__s(X)) -> s(activate(X))
753.37/208.17	             , U13(tt(), V1, V2) ->
753.37/208.17	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.17	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.17	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.17	             , isNat(n__0()) -> tt()
753.37/208.17	             , isNat(n__plus(V1, V2)) ->
753.37/208.17	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.17	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.17	             , U16(tt()) -> tt()
753.37/208.17	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.17	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.17	             , U23(tt()) -> tt()
753.37/208.17	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.17	             , U32(tt()) -> tt()
753.37/208.17	             , U41(tt()) -> tt()
753.37/208.17	             , s(X) -> n__s(X)
753.37/208.17	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.17	             , 0() -> n__0() }
753.37/208.17	           Obligation:
753.37/208.17	             innermost runtime complexity
753.37/208.17	           Answer:
753.37/208.17	             YES(O(1),O(n^1))
753.37/208.17	           
753.37/208.17	           Due to missing edges in the dependency-graph, the right-hand sides
753.37/208.17	           of following rules could be simplified:
753.37/208.17	           
753.37/208.17	             { U62^#(tt(), M, N) ->
753.37/208.17	               c_12(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.37/208.17	                    isNat^#(activate(N))) }
753.37/208.17	           
753.37/208.17	           We are left with following problem, upon which TcT provides the
753.37/208.17	           certificate YES(O(1),O(n^1)).
753.37/208.17	           
753.37/208.17	           Strict DPs:
753.37/208.17	             { isNat^#(n__plus(V1, V2)) ->
753.37/208.17	               c_1(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.17	             , isNat^#(n__s(V1)) ->
753.37/208.17	               c_2(U21^#(isNatKind(activate(V1)), activate(V1))) }
753.37/208.17	           Weak DPs:
753.37/208.17	             { U11^#(tt(), V1, V2) ->
753.37/208.17	               c_3(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.17	             , U12^#(tt(), V1, V2) ->
753.37/208.17	               c_4(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.17	             , U13^#(tt(), V1, V2) ->
753.37/208.17	               c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.17	             , U14^#(tt(), V1, V2) ->
753.37/208.17	               c_6(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.17	                   isNat^#(activate(V1)))
753.37/208.17	             , U15^#(tt(), V2) -> c_7(isNat^#(activate(V2)))
753.37/208.17	             , U21^#(tt(), V1) ->
753.37/208.17	               c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.17	             , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.37/208.17	             , U61^#(tt(), M, N) ->
753.37/208.17	               c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.37/208.17	             , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.37/208.17	           Weak Trs:
753.37/208.17	             { U11(tt(), V1, V2) ->
753.37/208.17	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.17	             , U12(tt(), V1, V2) ->
753.37/208.17	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.17	             , isNatKind(n__0()) -> tt()
753.37/208.17	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.17	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.17	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.17	             , activate(X) -> X
753.37/208.17	             , activate(n__0()) -> 0()
753.37/208.17	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.17	             , activate(n__s(X)) -> s(activate(X))
753.37/208.17	             , U13(tt(), V1, V2) ->
753.37/208.17	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.17	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.17	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.17	             , isNat(n__0()) -> tt()
753.37/208.17	             , isNat(n__plus(V1, V2)) ->
753.37/208.17	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.17	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.17	             , U16(tt()) -> tt()
753.37/208.17	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.17	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.17	             , U23(tt()) -> tt()
753.37/208.17	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.17	             , U32(tt()) -> tt()
753.37/208.17	             , U41(tt()) -> tt()
753.37/208.17	             , s(X) -> n__s(X)
753.37/208.17	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.17	             , 0() -> n__0() }
753.37/208.17	           Obligation:
753.37/208.17	             innermost runtime complexity
753.37/208.17	           Answer:
753.37/208.17	             YES(O(1),O(n^1))
753.37/208.17	           
753.37/208.17	           We use the processor 'matrix interpretation of dimension 1' to
753.37/208.17	           orient following rules strictly.
753.37/208.17	           
753.37/208.17	           DPs:
753.37/208.17	             { 1: isNat^#(n__plus(V1, V2)) ->
753.37/208.17	                  c_1(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.17	             , 2: isNat^#(n__s(V1)) ->
753.37/208.17	                  c_2(U21^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.17	             , 10: U61^#(tt(), M, N) ->
753.37/208.17	                   c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N))) }
753.37/208.17	           
753.37/208.17	           Sub-proof:
753.37/208.17	           ----------
753.37/208.17	             The following argument positions are usable:
753.37/208.17	               Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1},
753.37/208.17	               Uargs(c_4) = {1}, Uargs(c_5) = {1}, Uargs(c_6) = {1, 2},
753.37/208.17	               Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1},
753.37/208.17	               Uargs(c_10) = {1}, Uargs(c_11) = {1}
753.37/208.17	             
753.37/208.17	             TcT has computed the following constructor-based matrix
753.37/208.17	             interpretation satisfying not(EDA).
753.37/208.17	             
753.37/208.17	                 [U11](x1, x2, x3) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                              [tt] = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                 [U12](x1, x2, x3) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                   [isNatKind](x1) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                    [activate](x1) = [1] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                 [U13](x1, x2, x3) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                 [U14](x1, x2, x3) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                     [U15](x1, x2) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                       [isNat](x1) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                         [U16](x1) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                     [U21](x1, x2) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                     [U22](x1, x2) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                         [U23](x1) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                     [U31](x1, x2) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                         [U32](x1) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                         [U41](x1) = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                           [s](x1) = [1] x1 + [4]                  
753.37/208.17	                                                                   
753.37/208.17	                    [plus](x1, x2) = [1] x1 + [1] x2 + [4]         
753.37/208.17	                                                                   
753.37/208.17	                            [n__0] = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                 [n__plus](x1, x2) = [1] x1 + [1] x2 + [4]         
753.37/208.17	                                                                   
753.37/208.17	                        [n__s](x1) = [1] x1 + [4]                  
753.37/208.17	                                                                   
753.37/208.17	                               [0] = [0]                           
753.37/208.17	                                                                   
753.37/208.17	               [U11^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]         
753.37/208.17	                                                                   
753.37/208.17	               [U12^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                 [isNatKind^#](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	               [U13^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]         
753.37/208.17	                                                                   
753.37/208.17	               [U14^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                   [U15^#](x1, x2) = [1] x2 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                     [isNat^#](x1) = [1] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                   [U21^#](x1, x2) = [1] x2 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                   [U22^#](x1, x2) = [1] x2 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                   [U51^#](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	               [U61^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [7]
753.37/208.17	                                                                   
753.37/208.17	               [U62^#](x1, x2, x3) = [1] x3 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	               [U63^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0]
753.37/208.17	                                                                   
753.37/208.17	                     [c_1](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                     [c_2](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                     [c_5](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                     [c_7](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                         [c_8](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                     [c_9](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                    [c_10](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                    [c_11](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                        [c_12](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                    [c_13](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                    [c_14](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                        [c_15](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                        [c_16](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                               [c] = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                         [c_1](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                         [c_2](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                             [c_3] = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                         [c_4](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                         [c_5](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                         [c_6](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                     [c_7](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                         [c_8](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                         [c_9](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                        [c_10](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                        [c_11](x1) = [7] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                    [c_12](x1, x2) = [7] x1 + [7] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                               [c] = [0]                           
753.37/208.17	                                                                   
753.37/208.17	                         [c_1](x1) = [1] x1 + [1]                  
753.37/208.17	                                                                   
753.37/208.17	                         [c_2](x1) = [1] x1 + [3]                  
753.37/208.17	                                                                   
753.37/208.17	                         [c_3](x1) = [1] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                         [c_4](x1) = [1] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                         [c_5](x1) = [1] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                     [c_6](x1, x2) = [1] x1 + [1] x2 + [0]         
753.37/208.17	                                                                   
753.37/208.17	                         [c_7](x1) = [1] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                         [c_8](x1) = [1] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                         [c_9](x1) = [1] x1 + [0]                  
753.37/208.17	                                                                   
753.37/208.17	                        [c_10](x1) = [1] x1 + [3]                  
753.37/208.17	                                                                   
753.37/208.17	                        [c_11](x1) = [1] x1 + [0]                  
753.37/208.17	             
753.37/208.17	             The order satisfies the following ordering constraints:
753.37/208.17	             
753.37/208.17	                        [U11(tt(), V1, V2)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [U12(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.37/208.17	                                                                                                                
753.37/208.17	                        [U12(tt(), V1, V2)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [U13(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.37/208.17	                                                                                                                
753.37/208.17	                        [isNatKind(n__0())] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [tt()]                                                           
753.37/208.17	                                                                                                                
753.37/208.17	               [isNatKind(n__plus(V1, V2))] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [U31(isNatKind(activate(V1)), activate(V2))]                     
753.37/208.17	                                                                                                                
753.37/208.17	                      [isNatKind(n__s(V1))] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [U41(isNatKind(activate(V1)))]                                   
753.37/208.17	                                                                                                                
753.37/208.17	                              [activate(X)] =  [1] X + [0]                                                      
753.37/208.17	                                            >= [1] X + [0]                                                      
753.37/208.17	                                            =  [X]                                                              
753.37/208.17	                                                                                                                
753.37/208.17	                         [activate(n__0())] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [0()]                                                            
753.37/208.17	                                                                                                                
753.37/208.17	                [activate(n__plus(X1, X2))] =  [1] X1 + [1] X2 + [4]                                            
753.37/208.17	                                            >= [1] X1 + [1] X2 + [4]                                            
753.37/208.17	                                            =  [plus(activate(X1), activate(X2))]                               
753.37/208.17	                                                                                                                
753.37/208.17	                        [activate(n__s(X))] =  [1] X + [4]                                                      
753.37/208.17	                                            >= [1] X + [4]                                                      
753.37/208.17	                                            =  [s(activate(X))]                                                 
753.37/208.17	                                                                                                                
753.37/208.17	                        [U13(tt(), V1, V2)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [U14(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.37/208.17	                                                                                                                
753.37/208.17	                        [U14(tt(), V1, V2)] =  [0]                                                              
753.37/208.17	                                            >= [0]                                                              
753.37/208.17	                                            =  [U15(isNat(activate(V1)), activate(V2))]                         
753.37/208.18	                                                                                                                
753.37/208.18	                            [U15(tt(), V2)] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [U16(isNat(activate(V2)))]                                       
753.37/208.18	                                                                                                                
753.37/208.18	                            [isNat(n__0())] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [tt()]                                                           
753.37/208.18	                                                                                                                
753.37/208.18	                   [isNat(n__plus(V1, V2))] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [U11(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.37/208.18	                                                                                                                
753.37/208.18	                          [isNat(n__s(V1))] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [U21(isNatKind(activate(V1)), activate(V1))]                     
753.37/208.18	                                                                                                                
753.37/208.18	                                [U16(tt())] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [tt()]                                                           
753.37/208.18	                                                                                                                
753.37/208.18	                            [U21(tt(), V1)] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [U22(isNatKind(activate(V1)), activate(V1))]                     
753.37/208.18	                                                                                                                
753.37/208.18	                            [U22(tt(), V1)] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [U23(isNat(activate(V1)))]                                       
753.37/208.18	                                                                                                                
753.37/208.18	                                [U23(tt())] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [tt()]                                                           
753.37/208.18	                                                                                                                
753.37/208.18	                            [U31(tt(), V2)] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [U32(isNatKind(activate(V2)))]                                   
753.37/208.18	                                                                                                                
753.37/208.18	                                [U32(tt())] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [tt()]                                                           
753.37/208.18	                                                                                                                
753.37/208.18	                                [U41(tt())] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [tt()]                                                           
753.37/208.18	                                                                                                                
753.37/208.18	                                     [s(X)] =  [1] X + [4]                                                      
753.37/208.18	                                            >= [1] X + [4]                                                      
753.37/208.18	                                            =  [n__s(X)]                                                        
753.37/208.18	                                                                                                                
753.37/208.18	                             [plus(X1, X2)] =  [1] X1 + [1] X2 + [4]                                            
753.37/208.18	                                            >= [1] X1 + [1] X2 + [4]                                            
753.37/208.18	                                            =  [n__plus(X1, X2)]                                                
753.37/208.18	                                                                                                                
753.37/208.18	                                      [0()] =  [0]                                                              
753.37/208.18	                                            >= [0]                                                              
753.37/208.18	                                            =  [n__0()]                                                         
753.37/208.18	                                                                                                                
753.37/208.18	                      [U11^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                            
753.37/208.18	                                            >= [1] V1 + [1] V2 + [0]                                            
753.37/208.18	                                            =  [c_3(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.37/208.18	                                                                                                                
753.37/208.18	                      [U12^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                            
753.37/208.18	                                            >= [1] V1 + [1] V2 + [0]                                            
753.37/208.18	                                            =  [c_4(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.37/208.18	                                                                                                                
753.37/208.18	                      [U13^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                            
753.37/208.18	                                            >= [1] V1 + [1] V2 + [0]                                            
753.37/208.18	                                            =  [c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.37/208.18	                                                                                                                
753.37/208.18	                      [U14^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                            
753.37/208.18	                                            >= [1] V1 + [1] V2 + [0]                                            
753.37/208.18	                                            =  [c_6(U15^#(isNat(activate(V1)), activate(V2)),                   
753.37/208.18	                                                    isNat^#(activate(V1)))]                                     
753.37/208.18	                                                                                                                
753.37/208.18	                          [U15^#(tt(), V2)] =  [1] V2 + [0]                                                     
753.37/208.18	                                            >= [1] V2 + [0]                                                     
753.37/208.18	                                            =  [c_7(isNat^#(activate(V2)))]                                     
753.37/208.18	                                                                                                                
753.37/208.18	                 [isNat^#(n__plus(V1, V2))] =  [1] V1 + [1] V2 + [4]                                            
753.37/208.18	                                            >  [1] V1 + [1] V2 + [1]                                            
753.37/208.18	                                            =  [c_1(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.37/208.18	                                                                                                                
753.37/208.18	                        [isNat^#(n__s(V1))] =  [1] V1 + [4]                                                     
753.37/208.18	                                            >  [1] V1 + [3]                                                     
753.37/208.18	                                            =  [c_2(U21^#(isNatKind(activate(V1)), activate(V1)))]              
753.37/208.18	                                                                                                                
753.37/208.18	                          [U21^#(tt(), V1)] =  [1] V1 + [0]                                                     
753.37/208.18	                                            >= [1] V1 + [0]                                                     
753.37/208.18	                                            =  [c_8(U22^#(isNatKind(activate(V1)), activate(V1)))]              
753.37/208.18	                                                                                                                
753.37/208.18	                          [U22^#(tt(), V1)] =  [1] V1 + [0]                                                     
753.37/208.18	                                            >= [1] V1 + [0]                                                     
753.37/208.18	                                            =  [c_9(isNat^#(activate(V1)))]                                     
753.37/208.18	                                                                                                                
753.37/208.18	                        [U61^#(tt(), M, N)] =  [7] N + [7] M + [7]                                              
753.37/208.18	                                            >  [1] N + [3]                                                      
753.37/208.18	                                            =  [c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))]  
753.37/208.18	                                                                                                                
753.37/208.18	                        [U62^#(tt(), M, N)] =  [1] N + [0]                                                      
753.37/208.18	                                            >= [1] N + [0]                                                      
753.37/208.18	                                            =  [c_11(isNat^#(activate(N)))]                                     
753.37/208.18	                                                                                                                
753.37/208.18	           
753.37/208.18	           We return to the main proof. Consider the set of all dependency
753.37/208.18	           pairs
753.37/208.18	           
753.37/208.18	           :
753.37/208.18	             { 1: isNat^#(n__plus(V1, V2)) ->
753.37/208.18	                  c_1(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.18	             , 2: isNat^#(n__s(V1)) ->
753.37/208.18	                  c_2(U21^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.18	             , 3: U11^#(tt(), V1, V2) ->
753.37/208.18	                  c_3(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.18	             , 4: U12^#(tt(), V1, V2) ->
753.37/208.18	                  c_4(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.18	             , 5: U13^#(tt(), V1, V2) ->
753.37/208.18	                  c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.18	             , 6: U14^#(tt(), V1, V2) ->
753.37/208.18	                  c_6(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.18	                      isNat^#(activate(V1)))
753.37/208.18	             , 7: U15^#(tt(), V2) -> c_7(isNat^#(activate(V2)))
753.37/208.18	             , 8: U21^#(tt(), V1) ->
753.37/208.18	                  c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.18	             , 9: U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.37/208.18	             , 10: U61^#(tt(), M, N) ->
753.37/208.18	                   c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.37/208.18	             , 11: U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.37/208.18	           
753.37/208.18	           Processor 'matrix interpretation of dimension 1' induces the
753.37/208.18	           complexity certificate YES(?,O(n^1)) on application of dependency
753.37/208.18	           pairs {1,2,10}. These cover all (indirect) predecessors of
753.37/208.18	           dependency pairs {1,2,3,4,5,6,7,8,9,10,11}, their number of
753.37/208.18	           application is equally bounded. The dependency pairs are shifted
753.37/208.18	           into the weak component.
753.37/208.18	           
753.37/208.18	           We are left with following problem, upon which TcT provides the
753.37/208.18	           certificate YES(O(1),O(1)).
753.37/208.18	           
753.37/208.18	           Weak DPs:
753.37/208.18	             { U11^#(tt(), V1, V2) ->
753.37/208.18	               c_3(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.18	             , U12^#(tt(), V1, V2) ->
753.37/208.18	               c_4(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.18	             , U13^#(tt(), V1, V2) ->
753.37/208.18	               c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.18	             , U14^#(tt(), V1, V2) ->
753.37/208.18	               c_6(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.18	                   isNat^#(activate(V1)))
753.37/208.18	             , U15^#(tt(), V2) -> c_7(isNat^#(activate(V2)))
753.37/208.18	             , isNat^#(n__plus(V1, V2)) ->
753.37/208.18	               c_1(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.18	             , isNat^#(n__s(V1)) ->
753.37/208.18	               c_2(U21^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.18	             , U21^#(tt(), V1) ->
753.37/208.18	               c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.18	             , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.37/208.18	             , U61^#(tt(), M, N) ->
753.37/208.18	               c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.37/208.18	             , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.37/208.18	           Weak Trs:
753.37/208.18	             { U11(tt(), V1, V2) ->
753.37/208.18	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.18	             , U12(tt(), V1, V2) ->
753.37/208.18	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.18	             , isNatKind(n__0()) -> tt()
753.37/208.18	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.18	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.18	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.18	             , activate(X) -> X
753.37/208.18	             , activate(n__0()) -> 0()
753.37/208.18	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.18	             , activate(n__s(X)) -> s(activate(X))
753.37/208.18	             , U13(tt(), V1, V2) ->
753.37/208.18	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.18	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.18	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.18	             , isNat(n__0()) -> tt()
753.37/208.18	             , isNat(n__plus(V1, V2)) ->
753.37/208.18	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.18	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.18	             , U16(tt()) -> tt()
753.37/208.18	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.18	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.18	             , U23(tt()) -> tt()
753.37/208.18	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.18	             , U32(tt()) -> tt()
753.37/208.18	             , U41(tt()) -> tt()
753.37/208.18	             , s(X) -> n__s(X)
753.37/208.18	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.18	             , 0() -> n__0() }
753.37/208.18	           Obligation:
753.37/208.18	             innermost runtime complexity
753.37/208.18	           Answer:
753.37/208.18	             YES(O(1),O(1))
753.37/208.18	           
753.37/208.18	           The following weak DPs constitute a sub-graph of the DG that is
753.37/208.18	           closed under successors. The DPs are removed.
753.37/208.18	           
753.37/208.18	           { U11^#(tt(), V1, V2) ->
753.37/208.18	             c_3(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.18	           , U12^#(tt(), V1, V2) ->
753.37/208.18	             c_4(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.18	           , U13^#(tt(), V1, V2) ->
753.37/208.18	             c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.37/208.18	           , U14^#(tt(), V1, V2) ->
753.37/208.18	             c_6(U15^#(isNat(activate(V1)), activate(V2)),
753.37/208.18	                 isNat^#(activate(V1)))
753.37/208.18	           , U15^#(tt(), V2) -> c_7(isNat^#(activate(V2)))
753.37/208.18	           , isNat^#(n__plus(V1, V2)) ->
753.37/208.18	             c_1(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.37/208.18	           , isNat^#(n__s(V1)) ->
753.37/208.18	             c_2(U21^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.18	           , U21^#(tt(), V1) ->
753.37/208.18	             c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.37/208.18	           , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.37/208.18	           , U61^#(tt(), M, N) ->
753.37/208.18	             c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.37/208.18	           , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.37/208.18	           
753.37/208.18	           We are left with following problem, upon which TcT provides the
753.37/208.18	           certificate YES(O(1),O(1)).
753.37/208.18	           
753.37/208.18	           Weak Trs:
753.37/208.18	             { U11(tt(), V1, V2) ->
753.37/208.18	               U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.18	             , U12(tt(), V1, V2) ->
753.37/208.18	               U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.18	             , isNatKind(n__0()) -> tt()
753.37/208.18	             , isNatKind(n__plus(V1, V2)) ->
753.37/208.18	               U31(isNatKind(activate(V1)), activate(V2))
753.37/208.18	             , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.18	             , activate(X) -> X
753.37/208.18	             , activate(n__0()) -> 0()
753.37/208.18	             , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.18	             , activate(n__s(X)) -> s(activate(X))
753.37/208.18	             , U13(tt(), V1, V2) ->
753.37/208.18	               U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.18	             , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.18	             , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.18	             , isNat(n__0()) -> tt()
753.37/208.18	             , isNat(n__plus(V1, V2)) ->
753.37/208.18	               U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.18	             , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.18	             , U16(tt()) -> tt()
753.37/208.18	             , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.18	             , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.18	             , U23(tt()) -> tt()
753.37/208.18	             , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.18	             , U32(tt()) -> tt()
753.37/208.18	             , U41(tt()) -> tt()
753.37/208.18	             , s(X) -> n__s(X)
753.37/208.18	             , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.18	             , 0() -> n__0() }
753.37/208.18	           Obligation:
753.37/208.18	             innermost runtime complexity
753.37/208.18	           Answer:
753.37/208.18	             YES(O(1),O(1))
753.37/208.18	           
753.37/208.18	           No rule is usable, rules are removed from the input problem.
753.37/208.18	           
753.37/208.18	           We are left with following problem, upon which TcT provides the
753.37/208.18	           certificate YES(O(1),O(1)).
753.37/208.18	           
753.37/208.18	           Rules: Empty
753.37/208.18	           Obligation:
753.37/208.18	             innermost runtime complexity
753.37/208.18	           Answer:
753.37/208.18	             YES(O(1),O(1))
753.37/208.18	           
753.37/208.18	           Empty rules are trivially bounded
753.37/208.18	         
753.37/208.18	         We return to the main proof.
753.37/208.18	         
753.37/208.18	         We are left with following problem, upon which TcT provides the
753.37/208.18	         certificate YES(O(1),O(n^1)).
753.37/208.18	         
753.37/208.18	         Strict DPs:
753.37/208.18	           { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.18	             c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.18	                 isNatKind^#(activate(V1)))
753.37/208.18	           , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.18	           , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2))) }
753.37/208.18	         Weak DPs:
753.37/208.18	           { U11^#(tt(), V1, V2) ->
753.37/208.18	             U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.18	           , U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.37/208.18	           , U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.37/208.18	           , U12^#(tt(), V1, V2) ->
753.37/208.18	             U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.18	           , U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.37/208.18	           , U13^#(tt(), V1, V2) ->
753.37/208.18	             U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.18	           , U14^#(tt(), V1, V2) -> U15^#(isNat(activate(V1)), activate(V2))
753.37/208.18	           , U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.37/208.18	           , U15^#(tt(), V2) -> isNat^#(activate(V2))
753.37/208.18	           , isNat^#(n__plus(V1, V2)) ->
753.37/208.18	             U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.18	           , isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.37/208.18	           , isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.37/208.18	           , isNat^#(n__s(V1)) -> U21^#(isNatKind(activate(V1)), activate(V1))
753.37/208.18	           , U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.37/208.18	           , U21^#(tt(), V1) -> U22^#(isNatKind(activate(V1)), activate(V1))
753.37/208.18	           , U22^#(tt(), V1) -> isNat^#(activate(V1))
753.37/208.18	           , U51^#(tt(), N) -> isNatKind^#(activate(N))
753.37/208.18	           , U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.37/208.18	           , U61^#(tt(), M, N) ->
753.37/208.18	             U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.37/208.18	           , U62^#(tt(), M, N) -> isNat^#(activate(N))
753.37/208.18	           , U62^#(tt(), M, N) ->
753.37/208.18	             U63^#(isNat(activate(N)), activate(M), activate(N))
753.37/208.18	           , U63^#(tt(), M, N) -> isNatKind^#(activate(N)) }
753.37/208.18	         Weak Trs:
753.37/208.18	           { U11(tt(), V1, V2) ->
753.37/208.18	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.18	           , U12(tt(), V1, V2) ->
753.37/208.18	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.18	           , isNatKind(n__0()) -> tt()
753.37/208.18	           , isNatKind(n__plus(V1, V2)) ->
753.37/208.18	             U31(isNatKind(activate(V1)), activate(V2))
753.37/208.18	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.18	           , activate(X) -> X
753.37/208.18	           , activate(n__0()) -> 0()
753.37/208.18	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.18	           , activate(n__s(X)) -> s(activate(X))
753.37/208.18	           , U13(tt(), V1, V2) ->
753.37/208.18	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.18	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.18	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.18	           , isNat(n__0()) -> tt()
753.37/208.18	           , isNat(n__plus(V1, V2)) ->
753.37/208.18	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.18	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.18	           , U16(tt()) -> tt()
753.37/208.18	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.18	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.18	           , U23(tt()) -> tt()
753.37/208.18	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.18	           , U32(tt()) -> tt()
753.37/208.18	           , U41(tt()) -> tt()
753.37/208.18	           , s(X) -> n__s(X)
753.37/208.18	           , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.18	           , 0() -> n__0() }
753.37/208.18	         Obligation:
753.37/208.18	           innermost runtime complexity
753.37/208.18	         Answer:
753.37/208.18	           YES(O(1),O(n^1))
753.37/208.18	         
753.37/208.18	         We use the processor 'matrix interpretation of dimension 1' to
753.37/208.18	         orient following rules strictly.
753.37/208.18	         
753.37/208.18	         DPs:
753.37/208.18	           { 2: isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.18	           , 15: isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.37/208.18	           , 16: isNat^#(n__s(V1)) ->
753.37/208.18	                 U21^#(isNatKind(activate(V1)), activate(V1))
753.37/208.18	           , 20: U51^#(tt(), N) -> isNatKind^#(activate(N))
753.37/208.18	           , 21: U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.37/208.18	           , 22: U61^#(tt(), M, N) ->
753.37/208.18	                 U62^#(isNatKind(activate(M)), activate(M), activate(N)) }
753.37/208.18	         
753.37/208.18	         Sub-proof:
753.37/208.18	         ----------
753.37/208.18	           The following argument positions are usable:
753.37/208.18	             Uargs(c_3) = {1, 2}, Uargs(c_4) = {1}, Uargs(c_6) = {1}
753.37/208.18	           
753.37/208.18	           TcT has computed the following constructor-based matrix
753.37/208.18	           interpretation satisfying not(EDA).
753.37/208.18	           
753.37/208.18	               [U11](x1, x2, x3) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                            [tt] = [0]                           
753.37/208.18	                                                                 
753.37/208.18	               [U12](x1, x2, x3) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                 [isNatKind](x1) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                  [activate](x1) = [1] x1 + [0]                  
753.37/208.18	                                                                 
753.37/208.18	               [U13](x1, x2, x3) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	               [U14](x1, x2, x3) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                   [U15](x1, x2) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                     [isNat](x1) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                       [U16](x1) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                   [U21](x1, x2) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                   [U22](x1, x2) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                       [U23](x1) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                   [U31](x1, x2) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                       [U32](x1) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                       [U41](x1) = [0]                           
753.37/208.18	                                                                 
753.37/208.18	                         [s](x1) = [1] x1 + [1]                  
753.37/208.18	                                                                 
753.37/208.18	                  [plus](x1, x2) = [1] x1 + [1] x2 + [0]         
753.37/208.18	                                                                 
753.37/208.18	                          [n__0] = [0]                           
753.37/208.18	                                                                 
753.37/208.18	               [n__plus](x1, x2) = [1] x1 + [1] x2 + [0]         
753.37/208.18	                                                                 
753.37/208.18	                      [n__s](x1) = [1] x1 + [1]                  
753.37/208.18	                                                                 
753.37/208.18	                             [0] = [0]                           
753.37/208.18	                                                                 
753.37/208.18	             [U11^#](x1, x2, x3) = [2] x2 + [2] x3 + [0]         
753.37/208.18	                                                                 
753.37/208.18	             [U12^#](x1, x2, x3) = [2] x2 + [2] x3 + [0]         
753.37/208.18	                                                                 
753.37/208.18	               [isNatKind^#](x1) = [2] x1 + [0]                  
753.37/208.18	                                                                 
753.37/208.18	             [U13^#](x1, x2, x3) = [2] x2 + [2] x3 + [0]         
753.37/208.18	                                                                 
753.37/208.18	                 [U31^#](x1, x2) = [2] x2 + [0]                  
753.37/208.18	                                                                 
753.37/208.18	             [U14^#](x1, x2, x3) = [2] x2 + [2] x3 + [0]         
753.37/208.18	                                                                 
753.37/208.18	                 [U15^#](x1, x2) = [2] x2 + [0]                  
753.37/208.18	                                                                 
753.37/208.18	                   [isNat^#](x1) = [2] x1 + [0]                  
753.37/208.18	                                                                 
753.37/208.18	                 [U21^#](x1, x2) = [2] x2 + [0]                  
753.37/208.18	                                                                 
753.37/208.18	                 [U22^#](x1, x2) = [2] x2 + [0]                  
753.37/208.18	                                                                 
753.37/208.18	                 [U51^#](x1, x2) = [7] x1 + [7] x2 + [7]         
753.37/208.18	                                                                 
753.37/208.18	             [U61^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [1]
753.37/208.18	                                                                 
753.37/208.18	             [U62^#](x1, x2, x3) = [4] x2 + [4] x3 + [0]         
753.37/208.18	                                                                 
753.37/208.18	             [U63^#](x1, x2, x3) = [2] x3 + [0]                  
753.37/208.18	                                                                 
753.37/208.18	                   [c_3](x1, x2) = [1] x1 + [1] x2 + [0]         
753.37/208.18	                                                                 
753.37/208.18	                       [c_4](x1) = [1] x1 + [1]                  
753.37/208.18	                                                                 
753.37/208.18	                       [c_6](x1) = [1] x1 + [0]                  
753.37/208.18	           
753.37/208.18	           The order satisfies the following ordering constraints:
753.37/208.18	           
753.37/208.18	                        [U11(tt(), V1, V2)] =  [0]                                                         
753.37/208.18	                                            >= [0]                                                         
753.37/208.18	                                            =  [U12(isNatKind(activate(V1)), activate(V1), activate(V2))]  
753.37/208.18	                                                                                                           
753.37/208.18	                        [U12(tt(), V1, V2)] =  [0]                                                         
753.37/208.18	                                            >= [0]                                                         
753.37/208.18	                                            =  [U13(isNatKind(activate(V2)), activate(V1), activate(V2))]  
753.37/208.18	                                                                                                           
753.37/208.18	                        [isNatKind(n__0())] =  [0]                                                         
753.37/208.18	                                            >= [0]                                                         
753.37/208.18	                                            =  [tt()]                                                      
753.37/208.18	                                                                                                           
753.37/208.18	               [isNatKind(n__plus(V1, V2))] =  [0]                                                         
753.37/208.18	                                            >= [0]                                                         
753.37/208.18	                                            =  [U31(isNatKind(activate(V1)), activate(V2))]                
753.37/208.18	                                                                                                           
753.37/208.18	                      [isNatKind(n__s(V1))] =  [0]                                                         
753.37/208.18	                                            >= [0]                                                         
753.37/208.18	                                            =  [U41(isNatKind(activate(V1)))]                              
753.37/208.18	                                                                                                           
753.37/208.18	                              [activate(X)] =  [1] X + [0]                                                 
753.37/208.18	                                            >= [1] X + [0]                                                 
753.37/208.18	                                            =  [X]                                                         
753.37/208.18	                                                                                                           
753.37/208.18	                         [activate(n__0())] =  [0]                                                         
753.37/208.18	                                            >= [0]                                                         
753.37/208.18	                                            =  [0()]                                                       
753.37/208.19	                                                                                                           
753.37/208.19	                [activate(n__plus(X1, X2))] =  [1] X1 + [1] X2 + [0]                                       
753.37/208.19	                                            >= [1] X1 + [1] X2 + [0]                                       
753.37/208.19	                                            =  [plus(activate(X1), activate(X2))]                          
753.37/208.19	                                                                                                           
753.37/208.19	                        [activate(n__s(X))] =  [1] X + [1]                                                 
753.37/208.19	                                            >= [1] X + [1]                                                 
753.37/208.19	                                            =  [s(activate(X))]                                            
753.37/208.19	                                                                                                           
753.37/208.19	                        [U13(tt(), V1, V2)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U14(isNatKind(activate(V2)), activate(V1), activate(V2))]  
753.37/208.19	                                                                                                           
753.37/208.19	                        [U14(tt(), V1, V2)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U15(isNat(activate(V1)), activate(V2))]                    
753.37/208.19	                                                                                                           
753.37/208.19	                            [U15(tt(), V2)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U16(isNat(activate(V2)))]                                  
753.37/208.19	                                                                                                           
753.37/208.19	                            [isNat(n__0())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [tt()]                                                      
753.37/208.19	                                                                                                           
753.37/208.19	                   [isNat(n__plus(V1, V2))] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U11(isNatKind(activate(V1)), activate(V1), activate(V2))]  
753.37/208.19	                                                                                                           
753.37/208.19	                          [isNat(n__s(V1))] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U21(isNatKind(activate(V1)), activate(V1))]                
753.37/208.19	                                                                                                           
753.37/208.19	                                [U16(tt())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [tt()]                                                      
753.37/208.19	                                                                                                           
753.37/208.19	                            [U21(tt(), V1)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U22(isNatKind(activate(V1)), activate(V1))]                
753.37/208.19	                                                                                                           
753.37/208.19	                            [U22(tt(), V1)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U23(isNat(activate(V1)))]                                  
753.37/208.19	                                                                                                           
753.37/208.19	                                [U23(tt())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [tt()]                                                      
753.37/208.19	                                                                                                           
753.37/208.19	                            [U31(tt(), V2)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U32(isNatKind(activate(V2)))]                              
753.37/208.19	                                                                                                           
753.37/208.19	                                [U32(tt())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [tt()]                                                      
753.37/208.19	                                                                                                           
753.37/208.19	                                [U41(tt())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [tt()]                                                      
753.37/208.19	                                                                                                           
753.37/208.19	                                     [s(X)] =  [1] X + [1]                                                 
753.37/208.19	                                            >= [1] X + [1]                                                 
753.37/208.19	                                            =  [n__s(X)]                                                   
753.37/208.19	                                                                                                           
753.37/208.19	                             [plus(X1, X2)] =  [1] X1 + [1] X2 + [0]                                       
753.37/208.19	                                            >= [1] X1 + [1] X2 + [0]                                       
753.37/208.19	                                            =  [n__plus(X1, X2)]                                           
753.37/208.19	                                                                                                           
753.37/208.19	                                      [0()] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [n__0()]                                                    
753.37/208.19	                                                                                                           
753.37/208.19	                      [U11^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            >= [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            =  [U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))]
753.37/208.19	                                                                                                           
753.37/208.19	                      [U11^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            >= [2] V1 + [0]                                                
753.37/208.19	                                            =  [isNatKind^#(activate(V1))]                                 
753.37/208.19	                                                                                                           
753.37/208.19	                      [U12^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            >= [2] V2 + [0]                                                
753.37/208.19	                                            =  [isNatKind^#(activate(V2))]                                 
753.37/208.19	                                                                                                           
753.37/208.19	                      [U12^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            >= [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            =  [U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))]
753.37/208.19	                                                                                                           
753.37/208.19	             [isNatKind^#(n__plus(V1, V2))] =  [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            >= [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            =  [c_3(U31^#(isNatKind(activate(V1)), activate(V2)),          
753.37/208.19	                                                    isNatKind^#(activate(V1)))]                            
753.37/208.19	                                                                                                           
753.37/208.19	                    [isNatKind^#(n__s(V1))] =  [2] V1 + [2]                                                
753.37/208.19	                                            >  [2] V1 + [1]                                                
753.37/208.19	                                            =  [c_4(isNatKind^#(activate(V1)))]                            
753.37/208.19	                                                                                                           
753.37/208.19	                      [U13^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            >= [2] V2 + [0]                                                
753.37/208.19	                                            =  [isNatKind^#(activate(V2))]                                 
753.37/208.19	                                                                                                           
753.37/208.19	                      [U13^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            >= [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            =  [U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))]
753.37/208.19	                                                                                                           
753.37/208.19	                          [U31^#(tt(), V2)] =  [2] V2 + [0]                                                
753.37/208.19	                                            >= [2] V2 + [0]                                                
753.37/208.19	                                            =  [c_6(isNatKind^#(activate(V2)))]                            
753.37/208.19	                                                                                                           
753.37/208.19	                      [U14^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            >= [2] V2 + [0]                                                
753.37/208.19	                                            =  [U15^#(isNat(activate(V1)), activate(V2))]                  
753.37/208.19	                                                                                                           
753.37/208.19	                      [U14^#(tt(), V1, V2)] =  [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            >= [2] V1 + [0]                                                
753.37/208.19	                                            =  [isNat^#(activate(V1))]                                     
753.37/208.19	                                                                                                           
753.37/208.19	                          [U15^#(tt(), V2)] =  [2] V2 + [0]                                                
753.37/208.19	                                            >= [2] V2 + [0]                                                
753.37/208.19	                                            =  [isNat^#(activate(V2))]                                     
753.37/208.19	                                                                                                           
753.37/208.19	                 [isNat^#(n__plus(V1, V2))] =  [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            >= [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            =  [U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))]
753.37/208.19	                                                                                                           
753.37/208.19	                 [isNat^#(n__plus(V1, V2))] =  [2] V1 + [2] V2 + [0]                                       
753.37/208.19	                                            >= [2] V1 + [0]                                                
753.37/208.19	                                            =  [isNatKind^#(activate(V1))]                                 
753.37/208.19	                                                                                                           
753.37/208.19	                        [isNat^#(n__s(V1))] =  [2] V1 + [2]                                                
753.37/208.19	                                            >  [2] V1 + [0]                                                
753.37/208.19	                                            =  [isNatKind^#(activate(V1))]                                 
753.37/208.19	                                                                                                           
753.37/208.19	                        [isNat^#(n__s(V1))] =  [2] V1 + [2]                                                
753.37/208.19	                                            >  [2] V1 + [0]                                                
753.37/208.19	                                            =  [U21^#(isNatKind(activate(V1)), activate(V1))]              
753.37/208.19	                                                                                                           
753.37/208.19	                          [U21^#(tt(), V1)] =  [2] V1 + [0]                                                
753.37/208.19	                                            >= [2] V1 + [0]                                                
753.37/208.19	                                            =  [isNatKind^#(activate(V1))]                                 
753.37/208.19	                                                                                                           
753.37/208.19	                          [U21^#(tt(), V1)] =  [2] V1 + [0]                                                
753.37/208.19	                                            >= [2] V1 + [0]                                                
753.37/208.19	                                            =  [U22^#(isNatKind(activate(V1)), activate(V1))]              
753.37/208.19	                                                                                                           
753.37/208.19	                          [U22^#(tt(), V1)] =  [2] V1 + [0]                                                
753.37/208.19	                                            >= [2] V1 + [0]                                                
753.37/208.19	                                            =  [isNat^#(activate(V1))]                                     
753.37/208.19	                                                                                                           
753.37/208.19	                           [U51^#(tt(), N)] =  [7] N + [7]                                                 
753.37/208.19	                                            >  [2] N + [0]                                                 
753.37/208.19	                                            =  [isNatKind^#(activate(N))]                                  
753.37/208.19	                                                                                                           
753.37/208.19	                        [U61^#(tt(), M, N)] =  [7] N + [7] M + [1]                                         
753.37/208.19	                                            >  [2] M + [0]                                                 
753.37/208.19	                                            =  [isNatKind^#(activate(M))]                                  
753.37/208.19	                                                                                                           
753.37/208.19	                        [U61^#(tt(), M, N)] =  [7] N + [7] M + [1]                                         
753.37/208.19	                                            >  [4] N + [4] M + [0]                                         
753.37/208.19	                                            =  [U62^#(isNatKind(activate(M)), activate(M), activate(N))]   
753.37/208.19	                                                                                                           
753.37/208.19	                        [U62^#(tt(), M, N)] =  [4] N + [4] M + [0]                                         
753.37/208.19	                                            >= [2] N + [0]                                                 
753.37/208.19	                                            =  [isNat^#(activate(N))]                                      
753.37/208.19	                                                                                                           
753.37/208.19	                        [U62^#(tt(), M, N)] =  [4] N + [4] M + [0]                                         
753.37/208.19	                                            >= [2] N + [0]                                                 
753.37/208.19	                                            =  [U63^#(isNat(activate(N)), activate(M), activate(N))]       
753.37/208.19	                                                                                                           
753.37/208.19	                        [U63^#(tt(), M, N)] =  [2] N + [0]                                                 
753.37/208.19	                                            >= [2] N + [0]                                                 
753.37/208.19	                                            =  [isNatKind^#(activate(N))]                                  
753.37/208.19	                                                                                                           
753.37/208.19	         
753.37/208.19	         We return to the main proof. Consider the set of all dependency
753.37/208.19	         pairs
753.37/208.19	         
753.37/208.19	         :
753.37/208.19	           { 1: isNatKind^#(n__plus(V1, V2)) ->
753.37/208.19	                c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.19	                    isNatKind^#(activate(V1)))
753.37/208.19	           , 2: isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.19	           , 3: U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.37/208.19	           , 4: U11^#(tt(), V1, V2) ->
753.37/208.19	                U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.19	           , 5: U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.37/208.19	           , 6: U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.37/208.19	           , 7: U12^#(tt(), V1, V2) ->
753.37/208.19	                U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.19	           , 8: U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.37/208.19	           , 9: U13^#(tt(), V1, V2) ->
753.37/208.19	                U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.19	           , 10: U14^#(tt(), V1, V2) ->
753.37/208.19	                 U15^#(isNat(activate(V1)), activate(V2))
753.37/208.19	           , 11: U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.37/208.19	           , 12: U15^#(tt(), V2) -> isNat^#(activate(V2))
753.37/208.19	           , 13: isNat^#(n__plus(V1, V2)) ->
753.37/208.19	                 U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.19	           , 14: isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.37/208.19	           , 15: isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.37/208.19	           , 16: isNat^#(n__s(V1)) ->
753.37/208.19	                 U21^#(isNatKind(activate(V1)), activate(V1))
753.37/208.19	           , 17: U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.37/208.19	           , 18: U21^#(tt(), V1) ->
753.37/208.19	                 U22^#(isNatKind(activate(V1)), activate(V1))
753.37/208.19	           , 19: U22^#(tt(), V1) -> isNat^#(activate(V1))
753.37/208.19	           , 20: U51^#(tt(), N) -> isNatKind^#(activate(N))
753.37/208.19	           , 21: U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.37/208.19	           , 22: U61^#(tt(), M, N) ->
753.37/208.19	                 U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.37/208.19	           , 23: U62^#(tt(), M, N) -> isNat^#(activate(N))
753.37/208.19	           , 24: U62^#(tt(), M, N) ->
753.37/208.19	                 U63^#(isNat(activate(N)), activate(M), activate(N))
753.37/208.19	           , 25: U63^#(tt(), M, N) -> isNatKind^#(activate(N)) }
753.37/208.19	         
753.37/208.19	         Processor 'matrix interpretation of dimension 1' induces the
753.37/208.19	         complexity certificate YES(?,O(n^1)) on application of dependency
753.37/208.19	         pairs {2,15,16,20,21,22}. These cover all (indirect) predecessors
753.37/208.19	         of dependency pairs {2,15,16,17,18,19,20,21,22,23,24,25}, their
753.37/208.19	         number of application is equally bounded. The dependency pairs are
753.37/208.19	         shifted into the weak component.
753.37/208.19	         
753.37/208.19	         We are left with following problem, upon which TcT provides the
753.37/208.19	         certificate YES(O(1),O(n^1)).
753.37/208.19	         
753.37/208.19	         Strict DPs:
753.37/208.19	           { isNatKind^#(n__plus(V1, V2)) ->
753.37/208.19	             c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.19	                 isNatKind^#(activate(V1)))
753.37/208.19	           , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2))) }
753.37/208.19	         Weak DPs:
753.37/208.19	           { U11^#(tt(), V1, V2) ->
753.37/208.19	             U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.19	           , U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.37/208.19	           , U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.37/208.19	           , U12^#(tt(), V1, V2) ->
753.37/208.19	             U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.19	           , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.19	           , U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.37/208.19	           , U13^#(tt(), V1, V2) ->
753.37/208.19	             U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.19	           , U14^#(tt(), V1, V2) -> U15^#(isNat(activate(V1)), activate(V2))
753.37/208.19	           , U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.37/208.19	           , U15^#(tt(), V2) -> isNat^#(activate(V2))
753.37/208.19	           , isNat^#(n__plus(V1, V2)) ->
753.37/208.19	             U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.19	           , isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.37/208.19	           , isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.37/208.19	           , isNat^#(n__s(V1)) -> U21^#(isNatKind(activate(V1)), activate(V1))
753.37/208.19	           , U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.37/208.19	           , U21^#(tt(), V1) -> U22^#(isNatKind(activate(V1)), activate(V1))
753.37/208.19	           , U22^#(tt(), V1) -> isNat^#(activate(V1))
753.37/208.19	           , U51^#(tt(), N) -> isNatKind^#(activate(N))
753.37/208.19	           , U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.37/208.19	           , U61^#(tt(), M, N) ->
753.37/208.19	             U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.37/208.19	           , U62^#(tt(), M, N) -> isNat^#(activate(N))
753.37/208.19	           , U62^#(tt(), M, N) ->
753.37/208.19	             U63^#(isNat(activate(N)), activate(M), activate(N))
753.37/208.19	           , U63^#(tt(), M, N) -> isNatKind^#(activate(N)) }
753.37/208.19	         Weak Trs:
753.37/208.19	           { U11(tt(), V1, V2) ->
753.37/208.19	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.19	           , U12(tt(), V1, V2) ->
753.37/208.19	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.19	           , isNatKind(n__0()) -> tt()
753.37/208.19	           , isNatKind(n__plus(V1, V2)) ->
753.37/208.19	             U31(isNatKind(activate(V1)), activate(V2))
753.37/208.19	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.37/208.19	           , activate(X) -> X
753.37/208.19	           , activate(n__0()) -> 0()
753.37/208.19	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.37/208.19	           , activate(n__s(X)) -> s(activate(X))
753.37/208.19	           , U13(tt(), V1, V2) ->
753.37/208.19	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.37/208.19	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.37/208.19	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.37/208.19	           , isNat(n__0()) -> tt()
753.37/208.19	           , isNat(n__plus(V1, V2)) ->
753.37/208.19	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.19	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.37/208.19	           , U16(tt()) -> tt()
753.37/208.19	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.37/208.19	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.37/208.19	           , U23(tt()) -> tt()
753.37/208.19	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.37/208.19	           , U32(tt()) -> tt()
753.37/208.19	           , U41(tt()) -> tt()
753.37/208.19	           , s(X) -> n__s(X)
753.37/208.19	           , plus(X1, X2) -> n__plus(X1, X2)
753.37/208.19	           , 0() -> n__0() }
753.37/208.19	         Obligation:
753.37/208.19	           innermost runtime complexity
753.37/208.19	         Answer:
753.37/208.19	           YES(O(1),O(n^1))
753.37/208.19	         
753.37/208.19	         We use the processor 'matrix interpretation of dimension 1' to
753.37/208.19	         orient following rules strictly.
753.37/208.19	         
753.37/208.19	         DPs:
753.37/208.19	           { 1: isNatKind^#(n__plus(V1, V2)) ->
753.37/208.19	                c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.37/208.19	                    isNatKind^#(activate(V1)))
753.37/208.19	           , 7: isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.37/208.19	           , 13: isNat^#(n__plus(V1, V2)) ->
753.37/208.19	                 U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.37/208.19	           , 14: isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.37/208.19	           , 15: isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.37/208.19	           , 17: U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.37/208.19	           , 19: U22^#(tt(), V1) -> isNat^#(activate(V1))
753.37/208.19	           , 20: U51^#(tt(), N) -> isNatKind^#(activate(N))
753.37/208.19	           , 21: U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.37/208.19	           , 22: U61^#(tt(), M, N) ->
753.37/208.19	                 U62^#(isNatKind(activate(M)), activate(M), activate(N)) }
753.37/208.19	         
753.37/208.19	         Sub-proof:
753.37/208.19	         ----------
753.37/208.19	           The following argument positions are usable:
753.37/208.19	             Uargs(c_3) = {1, 2}, Uargs(c_4) = {1}, Uargs(c_6) = {1}
753.37/208.19	           
753.37/208.19	           TcT has computed the following constructor-based matrix
753.37/208.19	           interpretation satisfying not(EDA).
753.37/208.19	           
753.37/208.19	               [U11](x1, x2, x3) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                            [tt] = [0]                           
753.37/208.19	                                                                 
753.37/208.19	               [U12](x1, x2, x3) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                 [isNatKind](x1) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                  [activate](x1) = [1] x1 + [0]                  
753.37/208.19	                                                                 
753.37/208.19	               [U13](x1, x2, x3) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	               [U14](x1, x2, x3) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                   [U15](x1, x2) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                     [isNat](x1) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                       [U16](x1) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                   [U21](x1, x2) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                   [U22](x1, x2) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                       [U23](x1) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                   [U31](x1, x2) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                       [U32](x1) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                       [U41](x1) = [0]                           
753.37/208.19	                                                                 
753.37/208.19	                         [s](x1) = [1] x1 + [4]                  
753.37/208.19	                                                                 
753.37/208.19	                  [plus](x1, x2) = [1] x1 + [1] x2 + [1]         
753.37/208.19	                                                                 
753.37/208.19	                          [n__0] = [0]                           
753.37/208.19	                                                                 
753.37/208.19	               [n__plus](x1, x2) = [1] x1 + [1] x2 + [1]         
753.37/208.19	                                                                 
753.37/208.19	                      [n__s](x1) = [1] x1 + [4]                  
753.37/208.19	                                                                 
753.37/208.19	                             [0] = [0]                           
753.37/208.19	                                                                 
753.37/208.19	             [U11^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]         
753.37/208.19	                                                                 
753.37/208.19	             [U12^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]         
753.37/208.19	                                                                 
753.37/208.19	               [isNatKind^#](x1) = [1] x1 + [0]                  
753.37/208.19	                                                                 
753.37/208.19	             [U13^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]         
753.37/208.19	                                                                 
753.37/208.19	                 [U31^#](x1, x2) = [1] x2 + [0]                  
753.37/208.19	                                                                 
753.37/208.19	             [U14^#](x1, x2, x3) = [1] x2 + [1] x3 + [0]         
753.37/208.19	                                                                 
753.37/208.19	                 [U15^#](x1, x2) = [1] x2 + [0]                  
753.37/208.19	                                                                 
753.37/208.19	                   [isNat^#](x1) = [1] x1 + [0]                  
753.37/208.19	                                                                 
753.37/208.19	                 [U21^#](x1, x2) = [1] x2 + [4]                  
753.37/208.19	                                                                 
753.37/208.19	                 [U22^#](x1, x2) = [1] x2 + [4]                  
753.37/208.19	                                                                 
753.37/208.19	                 [U51^#](x1, x2) = [7] x1 + [7] x2 + [7]         
753.37/208.19	                                                                 
753.37/208.19	             [U61^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [1]
753.37/208.19	                                                                 
753.37/208.19	             [U62^#](x1, x2, x3) = [4] x3 + [0]                  
753.37/208.19	                                                                 
753.37/208.19	             [U63^#](x1, x2, x3) = [1] x3 + [0]                  
753.37/208.19	                                                                 
753.37/208.19	                   [c_3](x1, x2) = [1] x1 + [1] x2 + [0]         
753.37/208.19	                                                                 
753.37/208.19	                       [c_4](x1) = [1] x1 + [3]                  
753.37/208.19	                                                                 
753.37/208.19	                       [c_6](x1) = [1] x1 + [0]                  
753.37/208.19	           
753.37/208.19	           The order satisfies the following ordering constraints:
753.37/208.19	           
753.37/208.19	                        [U11(tt(), V1, V2)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U12(isNatKind(activate(V1)), activate(V1), activate(V2))]  
753.37/208.19	                                                                                                           
753.37/208.19	                        [U12(tt(), V1, V2)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U13(isNatKind(activate(V2)), activate(V1), activate(V2))]  
753.37/208.19	                                                                                                           
753.37/208.19	                        [isNatKind(n__0())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [tt()]                                                      
753.37/208.19	                                                                                                           
753.37/208.19	               [isNatKind(n__plus(V1, V2))] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U31(isNatKind(activate(V1)), activate(V2))]                
753.37/208.19	                                                                                                           
753.37/208.19	                      [isNatKind(n__s(V1))] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U41(isNatKind(activate(V1)))]                              
753.37/208.19	                                                                                                           
753.37/208.19	                              [activate(X)] =  [1] X + [0]                                                 
753.37/208.19	                                            >= [1] X + [0]                                                 
753.37/208.19	                                            =  [X]                                                         
753.37/208.19	                                                                                                           
753.37/208.19	                         [activate(n__0())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [0()]                                                       
753.37/208.19	                                                                                                           
753.37/208.19	                [activate(n__plus(X1, X2))] =  [1] X1 + [1] X2 + [1]                                       
753.37/208.19	                                            >= [1] X1 + [1] X2 + [1]                                       
753.37/208.19	                                            =  [plus(activate(X1), activate(X2))]                          
753.37/208.19	                                                                                                           
753.37/208.19	                        [activate(n__s(X))] =  [1] X + [4]                                                 
753.37/208.19	                                            >= [1] X + [4]                                                 
753.37/208.19	                                            =  [s(activate(X))]                                            
753.37/208.19	                                                                                                           
753.37/208.19	                        [U13(tt(), V1, V2)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U14(isNatKind(activate(V2)), activate(V1), activate(V2))]  
753.37/208.19	                                                                                                           
753.37/208.19	                        [U14(tt(), V1, V2)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U15(isNat(activate(V1)), activate(V2))]                    
753.37/208.19	                                                                                                           
753.37/208.19	                            [U15(tt(), V2)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U16(isNat(activate(V2)))]                                  
753.37/208.19	                                                                                                           
753.37/208.19	                            [isNat(n__0())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [tt()]                                                      
753.37/208.19	                                                                                                           
753.37/208.19	                   [isNat(n__plus(V1, V2))] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U11(isNatKind(activate(V1)), activate(V1), activate(V2))]  
753.37/208.19	                                                                                                           
753.37/208.19	                          [isNat(n__s(V1))] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U21(isNatKind(activate(V1)), activate(V1))]                
753.37/208.19	                                                                                                           
753.37/208.19	                                [U16(tt())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [tt()]                                                      
753.37/208.19	                                                                                                           
753.37/208.19	                            [U21(tt(), V1)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U22(isNatKind(activate(V1)), activate(V1))]                
753.37/208.19	                                                                                                           
753.37/208.19	                            [U22(tt(), V1)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U23(isNat(activate(V1)))]                                  
753.37/208.19	                                                                                                           
753.37/208.19	                                [U23(tt())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [tt()]                                                      
753.37/208.19	                                                                                                           
753.37/208.19	                            [U31(tt(), V2)] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [U32(isNatKind(activate(V2)))]                              
753.37/208.19	                                                                                                           
753.37/208.19	                                [U32(tt())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [tt()]                                                      
753.37/208.19	                                                                                                           
753.37/208.19	                                [U41(tt())] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [tt()]                                                      
753.37/208.19	                                                                                                           
753.37/208.19	                                     [s(X)] =  [1] X + [4]                                                 
753.37/208.19	                                            >= [1] X + [4]                                                 
753.37/208.19	                                            =  [n__s(X)]                                                   
753.37/208.19	                                                                                                           
753.37/208.19	                             [plus(X1, X2)] =  [1] X1 + [1] X2 + [1]                                       
753.37/208.19	                                            >= [1] X1 + [1] X2 + [1]                                       
753.37/208.19	                                            =  [n__plus(X1, X2)]                                           
753.37/208.19	                                                                                                           
753.37/208.19	                                      [0()] =  [0]                                                         
753.37/208.19	                                            >= [0]                                                         
753.37/208.19	                                            =  [n__0()]                                                    
753.37/208.19	                                                                                                           
753.37/208.19	                      [U11^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            >= [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            =  [U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))]
753.37/208.19	                                                                                                           
753.37/208.19	                      [U11^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            >= [1] V1 + [0]                                                
753.37/208.19	                                            =  [isNatKind^#(activate(V1))]                                 
753.37/208.19	                                                                                                           
753.37/208.19	                      [U12^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            >= [1] V2 + [0]                                                
753.37/208.19	                                            =  [isNatKind^#(activate(V2))]                                 
753.37/208.19	                                                                                                           
753.37/208.19	                      [U12^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            >= [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            =  [U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))]
753.37/208.19	                                                                                                           
753.37/208.19	             [isNatKind^#(n__plus(V1, V2))] =  [1] V1 + [1] V2 + [1]                                       
753.37/208.19	                                            >  [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            =  [c_3(U31^#(isNatKind(activate(V1)), activate(V2)),          
753.37/208.19	                                                    isNatKind^#(activate(V1)))]                            
753.37/208.19	                                                                                                           
753.37/208.19	                    [isNatKind^#(n__s(V1))] =  [1] V1 + [4]                                                
753.37/208.19	                                            >  [1] V1 + [3]                                                
753.37/208.19	                                            =  [c_4(isNatKind^#(activate(V1)))]                            
753.37/208.19	                                                                                                           
753.37/208.19	                      [U13^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            >= [1] V2 + [0]                                                
753.37/208.19	                                            =  [isNatKind^#(activate(V2))]                                 
753.37/208.19	                                                                                                           
753.37/208.19	                      [U13^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            >= [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            =  [U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))]
753.37/208.19	                                                                                                           
753.37/208.19	                          [U31^#(tt(), V2)] =  [1] V2 + [0]                                                
753.37/208.19	                                            >= [1] V2 + [0]                                                
753.37/208.19	                                            =  [c_6(isNatKind^#(activate(V2)))]                            
753.37/208.19	                                                                                                           
753.37/208.19	                      [U14^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            >= [1] V2 + [0]                                                
753.37/208.19	                                            =  [U15^#(isNat(activate(V1)), activate(V2))]                  
753.37/208.19	                                                                                                           
753.37/208.19	                      [U14^#(tt(), V1, V2)] =  [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            >= [1] V1 + [0]                                                
753.37/208.19	                                            =  [isNat^#(activate(V1))]                                     
753.37/208.19	                                                                                                           
753.37/208.19	                          [U15^#(tt(), V2)] =  [1] V2 + [0]                                                
753.37/208.19	                                            >= [1] V2 + [0]                                                
753.37/208.19	                                            =  [isNat^#(activate(V2))]                                     
753.37/208.19	                                                                                                           
753.37/208.19	                 [isNat^#(n__plus(V1, V2))] =  [1] V1 + [1] V2 + [1]                                       
753.37/208.19	                                            >  [1] V1 + [1] V2 + [0]                                       
753.37/208.19	                                            =  [U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))]
753.37/208.19	                                                                                                           
753.37/208.19	                 [isNat^#(n__plus(V1, V2))] =  [1] V1 + [1] V2 + [1]                                       
753.37/208.19	                                            >  [1] V1 + [0]                                                
753.37/208.19	                                            =  [isNatKind^#(activate(V1))]                                 
753.37/208.19	                                                                                                           
753.64/208.20	                        [isNat^#(n__s(V1))] =  [1] V1 + [4]                                                
753.64/208.20	                                            >  [1] V1 + [0]                                                
753.64/208.20	                                            =  [isNatKind^#(activate(V1))]                                 
753.64/208.20	                                                                                                           
753.64/208.20	                        [isNat^#(n__s(V1))] =  [1] V1 + [4]                                                
753.64/208.20	                                            >= [1] V1 + [4]                                                
753.64/208.20	                                            =  [U21^#(isNatKind(activate(V1)), activate(V1))]              
753.64/208.20	                                                                                                           
753.64/208.20	                          [U21^#(tt(), V1)] =  [1] V1 + [4]                                                
753.64/208.20	                                            >  [1] V1 + [0]                                                
753.64/208.20	                                            =  [isNatKind^#(activate(V1))]                                 
753.64/208.20	                                                                                                           
753.64/208.20	                          [U21^#(tt(), V1)] =  [1] V1 + [4]                                                
753.64/208.20	                                            >= [1] V1 + [4]                                                
753.64/208.20	                                            =  [U22^#(isNatKind(activate(V1)), activate(V1))]              
753.64/208.20	                                                                                                           
753.64/208.20	                          [U22^#(tt(), V1)] =  [1] V1 + [4]                                                
753.64/208.20	                                            >  [1] V1 + [0]                                                
753.64/208.20	                                            =  [isNat^#(activate(V1))]                                     
753.64/208.20	                                                                                                           
753.64/208.20	                           [U51^#(tt(), N)] =  [7] N + [7]                                                 
753.64/208.20	                                            >  [1] N + [0]                                                 
753.64/208.20	                                            =  [isNatKind^#(activate(N))]                                  
753.64/208.20	                                                                                                           
753.64/208.20	                        [U61^#(tt(), M, N)] =  [7] N + [7] M + [1]                                         
753.64/208.20	                                            >  [1] M + [0]                                                 
753.64/208.20	                                            =  [isNatKind^#(activate(M))]                                  
753.64/208.20	                                                                                                           
753.64/208.20	                        [U61^#(tt(), M, N)] =  [7] N + [7] M + [1]                                         
753.64/208.20	                                            >  [4] N + [0]                                                 
753.64/208.20	                                            =  [U62^#(isNatKind(activate(M)), activate(M), activate(N))]   
753.64/208.20	                                                                                                           
753.64/208.20	                        [U62^#(tt(), M, N)] =  [4] N + [0]                                                 
753.64/208.20	                                            >= [1] N + [0]                                                 
753.64/208.20	                                            =  [isNat^#(activate(N))]                                      
753.64/208.20	                                                                                                           
753.64/208.20	                        [U62^#(tt(), M, N)] =  [4] N + [0]                                                 
753.64/208.20	                                            >= [1] N + [0]                                                 
753.64/208.20	                                            =  [U63^#(isNat(activate(N)), activate(M), activate(N))]       
753.64/208.20	                                                                                                           
753.64/208.20	                        [U63^#(tt(), M, N)] =  [1] N + [0]                                                 
753.64/208.20	                                            >= [1] N + [0]                                                 
753.64/208.20	                                            =  [isNatKind^#(activate(N))]                                  
753.64/208.20	                                                                                                           
753.64/208.20	         
753.64/208.20	         We return to the main proof. Consider the set of all dependency
753.64/208.20	         pairs
753.64/208.20	         
753.64/208.20	         :
753.64/208.20	           { 1: isNatKind^#(n__plus(V1, V2)) ->
753.64/208.20	                c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.64/208.20	                    isNatKind^#(activate(V1)))
753.64/208.20	           , 2: U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.64/208.20	           , 3: U11^#(tt(), V1, V2) ->
753.64/208.20	                U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.20	           , 4: U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.64/208.20	           , 5: U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.64/208.20	           , 6: U12^#(tt(), V1, V2) ->
753.64/208.20	                U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.20	           , 7: isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.64/208.20	           , 8: U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.64/208.20	           , 9: U13^#(tt(), V1, V2) ->
753.64/208.20	                U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.20	           , 10: U14^#(tt(), V1, V2) ->
753.64/208.20	                 U15^#(isNat(activate(V1)), activate(V2))
753.64/208.20	           , 11: U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.64/208.20	           , 12: U15^#(tt(), V2) -> isNat^#(activate(V2))
753.64/208.20	           , 13: isNat^#(n__plus(V1, V2)) ->
753.64/208.20	                 U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.20	           , 14: isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.64/208.20	           , 15: isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.64/208.20	           , 16: isNat^#(n__s(V1)) ->
753.64/208.20	                 U21^#(isNatKind(activate(V1)), activate(V1))
753.64/208.20	           , 17: U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.64/208.20	           , 18: U21^#(tt(), V1) ->
753.64/208.20	                 U22^#(isNatKind(activate(V1)), activate(V1))
753.64/208.20	           , 19: U22^#(tt(), V1) -> isNat^#(activate(V1))
753.64/208.20	           , 20: U51^#(tt(), N) -> isNatKind^#(activate(N))
753.64/208.20	           , 21: U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.64/208.20	           , 22: U61^#(tt(), M, N) ->
753.64/208.20	                 U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.64/208.20	           , 23: U62^#(tt(), M, N) -> isNat^#(activate(N))
753.64/208.20	           , 24: U62^#(tt(), M, N) ->
753.64/208.20	                 U63^#(isNat(activate(N)), activate(M), activate(N))
753.64/208.20	           , 25: U63^#(tt(), M, N) -> isNatKind^#(activate(N)) }
753.64/208.20	         
753.64/208.20	         Processor 'matrix interpretation of dimension 1' induces the
753.64/208.20	         complexity certificate YES(?,O(n^1)) on application of dependency
753.64/208.20	         pairs {1,7,13,14,15,17,19,20,21,22}. These cover all (indirect)
753.64/208.20	         predecessors of dependency pairs
753.64/208.20	         {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25},
753.64/208.20	         their number of application is equally bounded. The dependency
753.64/208.20	         pairs are shifted into the weak component.
753.64/208.20	         
753.64/208.20	         We are left with following problem, upon which TcT provides the
753.64/208.20	         certificate YES(O(1),O(1)).
753.64/208.20	         
753.64/208.20	         Weak DPs:
753.64/208.20	           { U11^#(tt(), V1, V2) ->
753.64/208.20	             U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.20	           , U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.64/208.20	           , U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.64/208.20	           , U12^#(tt(), V1, V2) ->
753.64/208.20	             U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.20	           , isNatKind^#(n__plus(V1, V2)) ->
753.64/208.20	             c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.64/208.20	                 isNatKind^#(activate(V1)))
753.64/208.20	           , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.64/208.20	           , U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.64/208.20	           , U13^#(tt(), V1, V2) ->
753.64/208.20	             U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.20	           , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.64/208.20	           , U14^#(tt(), V1, V2) -> U15^#(isNat(activate(V1)), activate(V2))
753.64/208.20	           , U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.64/208.20	           , U15^#(tt(), V2) -> isNat^#(activate(V2))
753.64/208.20	           , isNat^#(n__plus(V1, V2)) ->
753.64/208.20	             U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.20	           , isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.64/208.20	           , isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.64/208.20	           , isNat^#(n__s(V1)) -> U21^#(isNatKind(activate(V1)), activate(V1))
753.64/208.20	           , U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.64/208.20	           , U21^#(tt(), V1) -> U22^#(isNatKind(activate(V1)), activate(V1))
753.64/208.20	           , U22^#(tt(), V1) -> isNat^#(activate(V1))
753.64/208.20	           , U51^#(tt(), N) -> isNatKind^#(activate(N))
753.64/208.20	           , U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.64/208.20	           , U61^#(tt(), M, N) ->
753.64/208.20	             U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.64/208.20	           , U62^#(tt(), M, N) -> isNat^#(activate(N))
753.64/208.20	           , U62^#(tt(), M, N) ->
753.64/208.20	             U63^#(isNat(activate(N)), activate(M), activate(N))
753.64/208.20	           , U63^#(tt(), M, N) -> isNatKind^#(activate(N)) }
753.64/208.20	         Weak Trs:
753.64/208.20	           { U11(tt(), V1, V2) ->
753.64/208.20	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.20	           , U12(tt(), V1, V2) ->
753.64/208.20	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.20	           , isNatKind(n__0()) -> tt()
753.64/208.20	           , isNatKind(n__plus(V1, V2)) ->
753.64/208.20	             U31(isNatKind(activate(V1)), activate(V2))
753.64/208.20	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.64/208.20	           , activate(X) -> X
753.64/208.20	           , activate(n__0()) -> 0()
753.64/208.20	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.64/208.20	           , activate(n__s(X)) -> s(activate(X))
753.64/208.20	           , U13(tt(), V1, V2) ->
753.64/208.20	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.20	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.64/208.20	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.64/208.20	           , isNat(n__0()) -> tt()
753.64/208.20	           , isNat(n__plus(V1, V2)) ->
753.64/208.20	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.20	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.64/208.20	           , U16(tt()) -> tt()
753.64/208.20	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.64/208.20	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.64/208.20	           , U23(tt()) -> tt()
753.64/208.20	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.64/208.20	           , U32(tt()) -> tt()
753.64/208.20	           , U41(tt()) -> tt()
753.64/208.20	           , s(X) -> n__s(X)
753.64/208.20	           , plus(X1, X2) -> n__plus(X1, X2)
753.64/208.20	           , 0() -> n__0() }
753.64/208.20	         Obligation:
753.64/208.20	           innermost runtime complexity
753.64/208.20	         Answer:
753.64/208.20	           YES(O(1),O(1))
753.64/208.20	         
753.64/208.20	         The following weak DPs constitute a sub-graph of the DG that is
753.64/208.20	         closed under successors. The DPs are removed.
753.64/208.20	         
753.64/208.20	         { U11^#(tt(), V1, V2) ->
753.64/208.20	           U12^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.20	         , U11^#(tt(), V1, V2) -> isNatKind^#(activate(V1))
753.64/208.20	         , U12^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.64/208.20	         , U12^#(tt(), V1, V2) ->
753.64/208.20	           U13^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.20	         , isNatKind^#(n__plus(V1, V2)) ->
753.64/208.20	           c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.64/208.20	               isNatKind^#(activate(V1)))
753.64/208.20	         , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.64/208.20	         , U13^#(tt(), V1, V2) -> isNatKind^#(activate(V2))
753.64/208.20	         , U13^#(tt(), V1, V2) ->
753.64/208.20	           U14^#(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.20	         , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.64/208.20	         , U14^#(tt(), V1, V2) -> U15^#(isNat(activate(V1)), activate(V2))
753.64/208.20	         , U14^#(tt(), V1, V2) -> isNat^#(activate(V1))
753.64/208.20	         , U15^#(tt(), V2) -> isNat^#(activate(V2))
753.64/208.20	         , isNat^#(n__plus(V1, V2)) ->
753.64/208.20	           U11^#(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.20	         , isNat^#(n__plus(V1, V2)) -> isNatKind^#(activate(V1))
753.64/208.20	         , isNat^#(n__s(V1)) -> isNatKind^#(activate(V1))
753.64/208.20	         , isNat^#(n__s(V1)) -> U21^#(isNatKind(activate(V1)), activate(V1))
753.64/208.20	         , U21^#(tt(), V1) -> isNatKind^#(activate(V1))
753.64/208.20	         , U21^#(tt(), V1) -> U22^#(isNatKind(activate(V1)), activate(V1))
753.64/208.20	         , U22^#(tt(), V1) -> isNat^#(activate(V1))
753.64/208.20	         , U51^#(tt(), N) -> isNatKind^#(activate(N))
753.64/208.20	         , U61^#(tt(), M, N) -> isNatKind^#(activate(M))
753.64/208.20	         , U61^#(tt(), M, N) ->
753.64/208.20	           U62^#(isNatKind(activate(M)), activate(M), activate(N))
753.64/208.20	         , U62^#(tt(), M, N) -> isNat^#(activate(N))
753.64/208.20	         , U62^#(tt(), M, N) ->
753.64/208.20	           U63^#(isNat(activate(N)), activate(M), activate(N))
753.64/208.20	         , U63^#(tt(), M, N) -> isNatKind^#(activate(N)) }
753.64/208.20	         
753.64/208.20	         We are left with following problem, upon which TcT provides the
753.64/208.20	         certificate YES(O(1),O(1)).
753.64/208.20	         
753.64/208.20	         Weak Trs:
753.64/208.20	           { U11(tt(), V1, V2) ->
753.64/208.20	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.20	           , U12(tt(), V1, V2) ->
753.64/208.20	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.20	           , isNatKind(n__0()) -> tt()
753.64/208.20	           , isNatKind(n__plus(V1, V2)) ->
753.64/208.20	             U31(isNatKind(activate(V1)), activate(V2))
753.64/208.20	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.64/208.20	           , activate(X) -> X
753.64/208.20	           , activate(n__0()) -> 0()
753.64/208.20	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.64/208.20	           , activate(n__s(X)) -> s(activate(X))
753.64/208.20	           , U13(tt(), V1, V2) ->
753.64/208.20	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.20	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.64/208.20	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.64/208.20	           , isNat(n__0()) -> tt()
753.64/208.20	           , isNat(n__plus(V1, V2)) ->
753.64/208.20	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.20	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.64/208.20	           , U16(tt()) -> tt()
753.64/208.20	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.64/208.20	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.64/208.20	           , U23(tt()) -> tt()
753.64/208.20	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.64/208.20	           , U32(tt()) -> tt()
753.64/208.20	           , U41(tt()) -> tt()
753.64/208.20	           , s(X) -> n__s(X)
753.64/208.20	           , plus(X1, X2) -> n__plus(X1, X2)
753.64/208.20	           , 0() -> n__0() }
753.64/208.20	         Obligation:
753.64/208.20	           innermost runtime complexity
753.64/208.20	         Answer:
753.64/208.20	           YES(O(1),O(1))
753.64/208.20	         
753.64/208.20	         No rule is usable, rules are removed from the input problem.
753.64/208.20	         
753.64/208.20	         We are left with following problem, upon which TcT provides the
753.64/208.20	         certificate YES(O(1),O(1)).
753.64/208.20	         
753.64/208.20	         Rules: Empty
753.64/208.20	         Obligation:
753.64/208.20	           innermost runtime complexity
753.64/208.20	         Answer:
753.64/208.20	           YES(O(1),O(1))
753.64/208.20	         
753.64/208.20	         Empty rules are trivially bounded
753.64/208.20	      
753.64/208.20	      S) We are left with following problem, upon which TcT provides the
753.64/208.20	         certificate YES(O(1),O(1)).
753.64/208.20	         
753.64/208.20	         Strict DPs: { U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.64/208.20	         Weak DPs:
753.64/208.20	           { U11^#(tt(), V1, V2) ->
753.64/208.20	             c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.64/208.20	                 isNatKind^#(activate(V1)))
753.64/208.20	           , U12^#(tt(), V1, V2) ->
753.64/208.20	             c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.64/208.20	                 isNatKind^#(activate(V2)))
753.64/208.20	           , isNatKind^#(n__plus(V1, V2)) ->
753.64/208.20	             c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.64/208.20	                 isNatKind^#(activate(V1)))
753.64/208.20	           , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.64/208.20	           , U13^#(tt(), V1, V2) ->
753.64/208.20	             c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.64/208.20	                 isNatKind^#(activate(V2)))
753.64/208.20	           , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.64/208.21	           , U14^#(tt(), V1, V2) ->
753.64/208.21	             c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.64/208.21	                 isNat^#(activate(V1)))
753.64/208.21	           , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.64/208.21	           , isNat^#(n__plus(V1, V2)) ->
753.64/208.21	             c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.64/208.21	                 isNatKind^#(activate(V1)))
753.64/208.21	           , isNat^#(n__s(V1)) ->
753.64/208.21	             c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.64/208.21	                  isNatKind^#(activate(V1)))
753.64/208.21	           , U21^#(tt(), V1) ->
753.64/208.21	             c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.64/208.21	                  isNatKind^#(activate(V1)))
753.64/208.21	           , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.64/208.21	           , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.64/208.21	           , U61^#(tt(), M, N) ->
753.64/208.21	             c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.64/208.21	                  isNatKind^#(activate(M)))
753.64/208.21	           , U62^#(tt(), M, N) ->
753.64/208.21	             c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.64/208.21	                  isNat^#(activate(N))) }
753.64/208.21	         Weak Trs:
753.64/208.21	           { U11(tt(), V1, V2) ->
753.64/208.21	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.21	           , U12(tt(), V1, V2) ->
753.64/208.21	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.21	           , isNatKind(n__0()) -> tt()
753.64/208.21	           , isNatKind(n__plus(V1, V2)) ->
753.64/208.21	             U31(isNatKind(activate(V1)), activate(V2))
753.64/208.21	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.64/208.21	           , activate(X) -> X
753.64/208.21	           , activate(n__0()) -> 0()
753.64/208.21	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.64/208.21	           , activate(n__s(X)) -> s(activate(X))
753.64/208.21	           , U13(tt(), V1, V2) ->
753.64/208.21	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.21	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.64/208.21	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.64/208.21	           , isNat(n__0()) -> tt()
753.64/208.21	           , isNat(n__plus(V1, V2)) ->
753.64/208.21	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.21	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.64/208.21	           , U16(tt()) -> tt()
753.64/208.21	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.64/208.21	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.64/208.21	           , U23(tt()) -> tt()
753.64/208.21	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.64/208.21	           , U32(tt()) -> tt()
753.64/208.21	           , U41(tt()) -> tt()
753.64/208.21	           , s(X) -> n__s(X)
753.64/208.21	           , plus(X1, X2) -> n__plus(X1, X2)
753.64/208.21	           , 0() -> n__0() }
753.64/208.21	         Obligation:
753.64/208.21	           innermost runtime complexity
753.64/208.21	         Answer:
753.64/208.21	           YES(O(1),O(1))
753.64/208.21	         
753.64/208.21	         The following weak DPs constitute a sub-graph of the DG that is
753.64/208.21	         closed under successors. The DPs are removed.
753.64/208.21	         
753.64/208.21	         { U11^#(tt(), V1, V2) ->
753.64/208.21	           c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.64/208.21	               isNatKind^#(activate(V1)))
753.64/208.21	         , U12^#(tt(), V1, V2) ->
753.64/208.21	           c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.64/208.21	               isNatKind^#(activate(V2)))
753.64/208.21	         , isNatKind^#(n__plus(V1, V2)) ->
753.64/208.21	           c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.64/208.21	               isNatKind^#(activate(V1)))
753.64/208.21	         , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.64/208.21	         , U13^#(tt(), V1, V2) ->
753.64/208.21	           c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.64/208.21	               isNatKind^#(activate(V2)))
753.64/208.21	         , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.64/208.21	         , U14^#(tt(), V1, V2) ->
753.64/208.21	           c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.64/208.21	               isNat^#(activate(V1)))
753.64/208.21	         , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.64/208.21	         , isNat^#(n__plus(V1, V2)) ->
753.64/208.21	           c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.64/208.21	               isNatKind^#(activate(V1)))
753.64/208.21	         , isNat^#(n__s(V1)) ->
753.64/208.21	           c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.64/208.21	                isNatKind^#(activate(V1)))
753.64/208.21	         , U21^#(tt(), V1) ->
753.64/208.21	           c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.64/208.21	                isNatKind^#(activate(V1)))
753.64/208.21	         , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1)))
753.64/208.21	         , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N))) }
753.64/208.21	         
753.64/208.21	         We are left with following problem, upon which TcT provides the
753.64/208.21	         certificate YES(O(1),O(1)).
753.64/208.21	         
753.64/208.21	         Strict DPs: { U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.64/208.21	         Weak DPs:
753.64/208.21	           { U61^#(tt(), M, N) ->
753.64/208.21	             c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.64/208.21	                  isNatKind^#(activate(M)))
753.64/208.21	           , U62^#(tt(), M, N) ->
753.64/208.21	             c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.64/208.21	                  isNat^#(activate(N))) }
753.64/208.21	         Weak Trs:
753.64/208.21	           { U11(tt(), V1, V2) ->
753.64/208.21	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.21	           , U12(tt(), V1, V2) ->
753.64/208.21	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.21	           , isNatKind(n__0()) -> tt()
753.64/208.21	           , isNatKind(n__plus(V1, V2)) ->
753.64/208.21	             U31(isNatKind(activate(V1)), activate(V2))
753.64/208.21	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.64/208.21	           , activate(X) -> X
753.64/208.21	           , activate(n__0()) -> 0()
753.64/208.21	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.64/208.21	           , activate(n__s(X)) -> s(activate(X))
753.64/208.21	           , U13(tt(), V1, V2) ->
753.64/208.21	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.21	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.64/208.21	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.64/208.21	           , isNat(n__0()) -> tt()
753.64/208.21	           , isNat(n__plus(V1, V2)) ->
753.64/208.21	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.21	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.64/208.21	           , U16(tt()) -> tt()
753.64/208.21	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.64/208.21	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.64/208.21	           , U23(tt()) -> tt()
753.64/208.21	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.64/208.21	           , U32(tt()) -> tt()
753.64/208.21	           , U41(tt()) -> tt()
753.64/208.21	           , s(X) -> n__s(X)
753.64/208.21	           , plus(X1, X2) -> n__plus(X1, X2)
753.64/208.21	           , 0() -> n__0() }
753.64/208.21	         Obligation:
753.64/208.21	           innermost runtime complexity
753.64/208.21	         Answer:
753.64/208.21	           YES(O(1),O(1))
753.64/208.21	         
753.64/208.21	         Due to missing edges in the dependency-graph, the right-hand sides
753.64/208.21	         of following rules could be simplified:
753.64/208.21	         
753.64/208.21	           { U61^#(tt(), M, N) ->
753.64/208.21	             c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.64/208.21	                  isNatKind^#(activate(M)))
753.64/208.21	           , U62^#(tt(), M, N) ->
753.64/208.21	             c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.64/208.21	                  isNat^#(activate(N)))
753.64/208.21	           , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.64/208.21	         
753.64/208.21	         We are left with following problem, upon which TcT provides the
753.64/208.21	         certificate YES(O(1),O(1)).
753.64/208.21	         
753.64/208.21	         Strict DPs: { U63^#(tt(), M, N) -> c_1() }
753.64/208.21	         Weak DPs:
753.64/208.21	           { U61^#(tt(), M, N) ->
753.64/208.21	             c_2(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.64/208.21	           , U62^#(tt(), M, N) ->
753.64/208.21	             c_3(U63^#(isNat(activate(N)), activate(M), activate(N))) }
753.64/208.21	         Weak Trs:
753.64/208.21	           { U11(tt(), V1, V2) ->
753.64/208.21	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.21	           , U12(tt(), V1, V2) ->
753.64/208.21	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.21	           , isNatKind(n__0()) -> tt()
753.64/208.21	           , isNatKind(n__plus(V1, V2)) ->
753.64/208.21	             U31(isNatKind(activate(V1)), activate(V2))
753.64/208.21	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.64/208.21	           , activate(X) -> X
753.64/208.21	           , activate(n__0()) -> 0()
753.64/208.21	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.64/208.21	           , activate(n__s(X)) -> s(activate(X))
753.64/208.21	           , U13(tt(), V1, V2) ->
753.64/208.21	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.21	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.64/208.21	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.64/208.21	           , isNat(n__0()) -> tt()
753.64/208.21	           , isNat(n__plus(V1, V2)) ->
753.64/208.21	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.21	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.64/208.21	           , U16(tt()) -> tt()
753.64/208.21	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.64/208.21	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.64/208.21	           , U23(tt()) -> tt()
753.64/208.21	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.64/208.21	           , U32(tt()) -> tt()
753.64/208.21	           , U41(tt()) -> tt()
753.64/208.21	           , s(X) -> n__s(X)
753.64/208.21	           , plus(X1, X2) -> n__plus(X1, X2)
753.64/208.21	           , 0() -> n__0() }
753.64/208.21	         Obligation:
753.64/208.21	           innermost runtime complexity
753.64/208.21	         Answer:
753.64/208.21	           YES(O(1),O(1))
753.64/208.21	         
753.64/208.21	         The dependency graph contains no loops, we remove all dependency
753.64/208.21	         pairs.
753.64/208.21	         
753.64/208.21	         We are left with following problem, upon which TcT provides the
753.64/208.21	         certificate YES(O(1),O(1)).
753.64/208.21	         
753.64/208.21	         Weak Trs:
753.64/208.21	           { U11(tt(), V1, V2) ->
753.64/208.21	             U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.21	           , U12(tt(), V1, V2) ->
753.64/208.21	             U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.21	           , isNatKind(n__0()) -> tt()
753.64/208.21	           , isNatKind(n__plus(V1, V2)) ->
753.64/208.21	             U31(isNatKind(activate(V1)), activate(V2))
753.64/208.21	           , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.64/208.21	           , activate(X) -> X
753.64/208.21	           , activate(n__0()) -> 0()
753.64/208.21	           , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.64/208.21	           , activate(n__s(X)) -> s(activate(X))
753.64/208.21	           , U13(tt(), V1, V2) ->
753.64/208.21	             U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.21	           , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.64/208.21	           , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.64/208.21	           , isNat(n__0()) -> tt()
753.64/208.21	           , isNat(n__plus(V1, V2)) ->
753.64/208.21	             U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.21	           , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.64/208.21	           , U16(tt()) -> tt()
753.64/208.21	           , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.64/208.21	           , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.64/208.21	           , U23(tt()) -> tt()
753.64/208.21	           , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.64/208.21	           , U32(tt()) -> tt()
753.64/208.21	           , U41(tt()) -> tt()
753.64/208.21	           , s(X) -> n__s(X)
753.64/208.21	           , plus(X1, X2) -> n__plus(X1, X2)
753.64/208.21	           , 0() -> n__0() }
753.64/208.21	         Obligation:
753.64/208.21	           innermost runtime complexity
753.64/208.21	         Answer:
753.64/208.21	           YES(O(1),O(1))
753.64/208.21	         
753.64/208.21	         No rule is usable, rules are removed from the input problem.
753.64/208.21	         
753.64/208.21	         We are left with following problem, upon which TcT provides the
753.64/208.21	         certificate YES(O(1),O(1)).
753.64/208.21	         
753.64/208.21	         Rules: Empty
753.64/208.21	         Obligation:
753.64/208.21	           innermost runtime complexity
753.64/208.21	         Answer:
753.64/208.21	           YES(O(1),O(1))
753.64/208.21	         
753.64/208.21	         Empty rules are trivially bounded
753.64/208.21	      
753.64/208.21	   
753.64/208.21	   S) We are left with following problem, upon which TcT provides the
753.64/208.21	      certificate YES(O(1),O(n^1)).
753.64/208.21	      
753.64/208.21	      Strict DPs:
753.64/208.21	        { U11^#(tt(), V1, V2) ->
753.64/208.21	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.64/208.21	              isNatKind^#(activate(V1)))
753.64/208.21	        , U12^#(tt(), V1, V2) ->
753.64/208.21	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.64/208.21	              isNatKind^#(activate(V2)))
753.64/208.21	        , U13^#(tt(), V1, V2) ->
753.64/208.21	          c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.64/208.21	              isNatKind^#(activate(V2)))
753.64/208.21	        , U14^#(tt(), V1, V2) ->
753.64/208.21	          c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.64/208.21	              isNat^#(activate(V1)))
753.64/208.21	        , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.64/208.21	        , isNat^#(n__plus(V1, V2)) ->
753.64/208.21	          c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.64/208.21	              isNatKind^#(activate(V1)))
753.64/208.21	        , isNat^#(n__s(V1)) ->
753.64/208.21	          c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.64/208.21	               isNatKind^#(activate(V1)))
753.64/208.21	        , U21^#(tt(), V1) ->
753.64/208.21	          c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.64/208.21	               isNatKind^#(activate(V1)))
753.64/208.21	        , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1))) }
753.64/208.21	      Weak DPs:
753.64/208.21	        { isNatKind^#(n__plus(V1, V2)) ->
753.64/208.21	          c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.64/208.21	              isNatKind^#(activate(V1)))
753.64/208.21	        , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.64/208.21	        , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.64/208.21	        , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.64/208.21	        , U61^#(tt(), M, N) ->
753.64/208.21	          c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.64/208.21	               isNatKind^#(activate(M)))
753.64/208.21	        , U62^#(tt(), M, N) ->
753.64/208.21	          c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.64/208.21	               isNat^#(activate(N)))
753.64/208.21	        , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.64/208.21	      Weak Trs:
753.64/208.21	        { U11(tt(), V1, V2) ->
753.64/208.21	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.21	        , U12(tt(), V1, V2) ->
753.64/208.21	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.21	        , isNatKind(n__0()) -> tt()
753.64/208.21	        , isNatKind(n__plus(V1, V2)) ->
753.64/208.21	          U31(isNatKind(activate(V1)), activate(V2))
753.64/208.21	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.64/208.21	        , activate(X) -> X
753.64/208.21	        , activate(n__0()) -> 0()
753.64/208.21	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.64/208.21	        , activate(n__s(X)) -> s(activate(X))
753.64/208.21	        , U13(tt(), V1, V2) ->
753.64/208.21	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.21	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.64/208.21	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.64/208.21	        , isNat(n__0()) -> tt()
753.64/208.21	        , isNat(n__plus(V1, V2)) ->
753.64/208.21	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.21	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.64/208.21	        , U16(tt()) -> tt()
753.64/208.21	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.64/208.21	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.64/208.21	        , U23(tt()) -> tt()
753.64/208.21	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.64/208.21	        , U32(tt()) -> tt()
753.64/208.21	        , U41(tt()) -> tt()
753.64/208.21	        , s(X) -> n__s(X)
753.64/208.21	        , plus(X1, X2) -> n__plus(X1, X2)
753.64/208.21	        , 0() -> n__0() }
753.64/208.21	      Obligation:
753.64/208.21	        innermost runtime complexity
753.64/208.21	      Answer:
753.64/208.21	        YES(O(1),O(n^1))
753.64/208.21	      
753.64/208.21	      The following weak DPs constitute a sub-graph of the DG that is
753.64/208.21	      closed under successors. The DPs are removed.
753.64/208.21	      
753.64/208.21	      { isNatKind^#(n__plus(V1, V2)) ->
753.64/208.21	        c_3(U31^#(isNatKind(activate(V1)), activate(V2)),
753.64/208.21	            isNatKind^#(activate(V1)))
753.64/208.21	      , isNatKind^#(n__s(V1)) -> c_4(isNatKind^#(activate(V1)))
753.64/208.21	      , U31^#(tt(), V2) -> c_6(isNatKind^#(activate(V2)))
753.64/208.21	      , U51^#(tt(), N) -> c_16(isNatKind^#(activate(N)))
753.64/208.21	      , U63^#(tt(), M, N) -> c_15(isNatKind^#(activate(N))) }
753.64/208.21	      
753.64/208.21	      We are left with following problem, upon which TcT provides the
753.64/208.21	      certificate YES(O(1),O(n^1)).
753.64/208.21	      
753.64/208.21	      Strict DPs:
753.64/208.21	        { U11^#(tt(), V1, V2) ->
753.64/208.21	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.64/208.21	              isNatKind^#(activate(V1)))
753.64/208.21	        , U12^#(tt(), V1, V2) ->
753.64/208.21	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.64/208.21	              isNatKind^#(activate(V2)))
753.64/208.21	        , U13^#(tt(), V1, V2) ->
753.64/208.21	          c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.64/208.21	              isNatKind^#(activate(V2)))
753.64/208.21	        , U14^#(tt(), V1, V2) ->
753.64/208.21	          c_7(U15^#(isNat(activate(V1)), activate(V2)),
753.64/208.21	              isNat^#(activate(V1)))
753.64/208.21	        , U15^#(tt(), V2) -> c_8(isNat^#(activate(V2)))
753.64/208.21	        , isNat^#(n__plus(V1, V2)) ->
753.64/208.21	          c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.64/208.21	              isNatKind^#(activate(V1)))
753.64/208.21	        , isNat^#(n__s(V1)) ->
753.64/208.21	          c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.64/208.21	               isNatKind^#(activate(V1)))
753.64/208.21	        , U21^#(tt(), V1) ->
753.64/208.21	          c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.64/208.22	               isNatKind^#(activate(V1)))
753.64/208.22	        , U22^#(tt(), V1) -> c_12(isNat^#(activate(V1))) }
753.64/208.22	      Weak DPs:
753.64/208.22	        { U61^#(tt(), M, N) ->
753.64/208.22	          c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.64/208.22	               isNatKind^#(activate(M)))
753.64/208.22	        , U62^#(tt(), M, N) ->
753.64/208.22	          c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.64/208.22	               isNat^#(activate(N))) }
753.64/208.22	      Weak Trs:
753.64/208.22	        { U11(tt(), V1, V2) ->
753.64/208.22	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.22	        , U12(tt(), V1, V2) ->
753.64/208.22	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.22	        , isNatKind(n__0()) -> tt()
753.64/208.22	        , isNatKind(n__plus(V1, V2)) ->
753.64/208.22	          U31(isNatKind(activate(V1)), activate(V2))
753.64/208.22	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.64/208.22	        , activate(X) -> X
753.64/208.22	        , activate(n__0()) -> 0()
753.64/208.22	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.64/208.22	        , activate(n__s(X)) -> s(activate(X))
753.64/208.22	        , U13(tt(), V1, V2) ->
753.64/208.22	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.22	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.64/208.22	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.64/208.22	        , isNat(n__0()) -> tt()
753.64/208.22	        , isNat(n__plus(V1, V2)) ->
753.64/208.22	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.22	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.64/208.22	        , U16(tt()) -> tt()
753.64/208.22	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.64/208.22	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.64/208.22	        , U23(tt()) -> tt()
753.64/208.22	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.64/208.22	        , U32(tt()) -> tt()
753.64/208.22	        , U41(tt()) -> tt()
753.64/208.22	        , s(X) -> n__s(X)
753.64/208.22	        , plus(X1, X2) -> n__plus(X1, X2)
753.64/208.22	        , 0() -> n__0() }
753.64/208.22	      Obligation:
753.64/208.22	        innermost runtime complexity
753.64/208.22	      Answer:
753.64/208.22	        YES(O(1),O(n^1))
753.64/208.22	      
753.64/208.22	      Due to missing edges in the dependency-graph, the right-hand sides
753.64/208.22	      of following rules could be simplified:
753.64/208.22	      
753.64/208.22	        { U11^#(tt(), V1, V2) ->
753.64/208.22	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.64/208.22	              isNatKind^#(activate(V1)))
753.64/208.22	        , U12^#(tt(), V1, V2) ->
753.64/208.22	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.64/208.22	              isNatKind^#(activate(V2)))
753.64/208.22	        , U13^#(tt(), V1, V2) ->
753.64/208.22	          c_5(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)),
753.64/208.22	              isNatKind^#(activate(V2)))
753.64/208.22	        , isNat^#(n__plus(V1, V2)) ->
753.64/208.22	          c_9(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)),
753.64/208.22	              isNatKind^#(activate(V1)))
753.64/208.22	        , isNat^#(n__s(V1)) ->
753.64/208.22	          c_10(U21^#(isNatKind(activate(V1)), activate(V1)),
753.64/208.22	               isNatKind^#(activate(V1)))
753.64/208.22	        , U21^#(tt(), V1) ->
753.64/208.22	          c_11(U22^#(isNatKind(activate(V1)), activate(V1)),
753.64/208.22	               isNatKind^#(activate(V1)))
753.64/208.22	        , U61^#(tt(), M, N) ->
753.64/208.22	          c_13(U62^#(isNatKind(activate(M)), activate(M), activate(N)),
753.64/208.22	               isNatKind^#(activate(M)))
753.64/208.22	        , U62^#(tt(), M, N) ->
753.64/208.22	          c_14(U63^#(isNat(activate(N)), activate(M), activate(N)),
753.64/208.22	               isNat^#(activate(N))) }
753.64/208.22	      
753.64/208.22	      We are left with following problem, upon which TcT provides the
753.64/208.22	      certificate YES(O(1),O(n^1)).
753.64/208.22	      
753.64/208.22	      Strict DPs:
753.64/208.22	        { U11^#(tt(), V1, V2) ->
753.64/208.22	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.64/208.22	        , U12^#(tt(), V1, V2) ->
753.64/208.22	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.64/208.22	        , U13^#(tt(), V1, V2) ->
753.64/208.22	          c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.64/208.22	        , U14^#(tt(), V1, V2) ->
753.64/208.22	          c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.64/208.22	              isNat^#(activate(V1)))
753.64/208.22	        , U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.64/208.22	        , isNat^#(n__plus(V1, V2)) ->
753.64/208.22	          c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.64/208.22	        , isNat^#(n__s(V1)) ->
753.64/208.22	          c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.64/208.22	        , U21^#(tt(), V1) ->
753.64/208.22	          c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.64/208.22	        , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1))) }
753.64/208.22	      Weak DPs:
753.64/208.22	        { U61^#(tt(), M, N) ->
753.64/208.22	          c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.64/208.22	        , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.64/208.22	      Weak Trs:
753.64/208.22	        { U11(tt(), V1, V2) ->
753.64/208.22	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.22	        , U12(tt(), V1, V2) ->
753.64/208.22	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.22	        , isNatKind(n__0()) -> tt()
753.64/208.22	        , isNatKind(n__plus(V1, V2)) ->
753.64/208.22	          U31(isNatKind(activate(V1)), activate(V2))
753.64/208.22	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.64/208.22	        , activate(X) -> X
753.64/208.22	        , activate(n__0()) -> 0()
753.64/208.22	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.64/208.22	        , activate(n__s(X)) -> s(activate(X))
753.64/208.22	        , U13(tt(), V1, V2) ->
753.64/208.22	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.22	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.64/208.22	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.64/208.22	        , isNat(n__0()) -> tt()
753.64/208.22	        , isNat(n__plus(V1, V2)) ->
753.64/208.22	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.22	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.64/208.22	        , U16(tt()) -> tt()
753.64/208.22	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.64/208.22	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.64/208.22	        , U23(tt()) -> tt()
753.64/208.22	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.64/208.22	        , U32(tt()) -> tt()
753.64/208.22	        , U41(tt()) -> tt()
753.64/208.22	        , s(X) -> n__s(X)
753.64/208.22	        , plus(X1, X2) -> n__plus(X1, X2)
753.64/208.22	        , 0() -> n__0() }
753.64/208.22	      Obligation:
753.64/208.22	        innermost runtime complexity
753.64/208.22	      Answer:
753.64/208.22	        YES(O(1),O(n^1))
753.64/208.22	      
753.64/208.22	      We use the processor 'matrix interpretation of dimension 1' to
753.64/208.22	      orient following rules strictly.
753.64/208.22	      
753.64/208.22	      DPs:
753.64/208.22	        { 1: U11^#(tt(), V1, V2) ->
753.64/208.22	             c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.64/208.22	        , 6: isNat^#(n__plus(V1, V2)) ->
753.64/208.22	             c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.64/208.22	        , 7: isNat^#(n__s(V1)) ->
753.64/208.22	             c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.64/208.22	        , 8: U21^#(tt(), V1) ->
753.64/208.22	             c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.64/208.22	        , 10: U61^#(tt(), M, N) ->
753.64/208.22	              c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N))) }
753.64/208.22	      
753.64/208.22	      Sub-proof:
753.64/208.22	      ----------
753.64/208.22	        The following argument positions are usable:
753.64/208.22	          Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1},
753.64/208.22	          Uargs(c_4) = {1, 2}, Uargs(c_5) = {1}, Uargs(c_6) = {1},
753.64/208.22	          Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1},
753.64/208.22	          Uargs(c_10) = {1}, Uargs(c_11) = {1}
753.64/208.22	        
753.64/208.22	        TcT has computed the following constructor-based matrix
753.64/208.22	        interpretation satisfying not(EDA).
753.64/208.22	        
753.64/208.22	            [U11](x1, x2, x3) = [0]                           
753.64/208.22	                                                              
753.64/208.22	                         [tt] = [0]                           
753.64/208.22	                                                              
753.64/208.22	            [U12](x1, x2, x3) = [0]                           
753.64/208.22	                                                              
753.64/208.22	              [isNatKind](x1) = [0]                           
753.64/208.22	                                                              
753.64/208.22	               [activate](x1) = [1] x1 + [0]                  
753.64/208.22	                                                              
753.64/208.22	            [U13](x1, x2, x3) = [0]                           
753.64/208.22	                                                              
753.64/208.22	            [U14](x1, x2, x3) = [0]                           
753.64/208.22	                                                              
753.64/208.22	                [U15](x1, x2) = [0]                           
753.64/208.22	                                                              
753.64/208.22	                  [isNat](x1) = [0]                           
753.64/208.22	                                                              
753.64/208.22	                    [U16](x1) = [0]                           
753.64/208.22	                                                              
753.64/208.22	                [U21](x1, x2) = [0]                           
753.64/208.22	                                                              
753.64/208.22	                [U22](x1, x2) = [0]                           
753.64/208.22	                                                              
753.64/208.22	                    [U23](x1) = [0]                           
753.64/208.22	                                                              
753.64/208.22	                [U31](x1, x2) = [0]                           
753.64/208.22	                                                              
753.64/208.22	                    [U32](x1) = [0]                           
753.64/208.22	                                                              
753.64/208.22	                    [U41](x1) = [0]                           
753.64/208.22	                                                              
753.64/208.22	                      [s](x1) = [1] x1 + [2]                  
753.64/208.22	                                                              
753.64/208.22	               [plus](x1, x2) = [1] x1 + [1] x2 + [2]         
753.64/208.22	                                                              
753.64/208.22	                       [n__0] = [0]                           
753.64/208.22	                                                              
753.64/208.22	            [n__plus](x1, x2) = [1] x1 + [1] x2 + [2]         
753.64/208.22	                                                              
753.64/208.22	                   [n__s](x1) = [1] x1 + [2]                  
753.64/208.22	                                                              
753.64/208.22	                          [0] = [0]                           
753.64/208.22	                                                              
753.64/208.22	          [U11^#](x1, x2, x3) = [4] x2 + [4] x3 + [4]         
753.64/208.22	                                                              
753.64/208.22	          [U12^#](x1, x2, x3) = [4] x2 + [4] x3 + [0]         
753.64/208.22	                                                              
753.64/208.22	          [U13^#](x1, x2, x3) = [4] x2 + [4] x3 + [0]         
753.64/208.22	                                                              
753.64/208.22	          [U14^#](x1, x2, x3) = [4] x2 + [4] x3 + [0]         
753.64/208.22	                                                              
753.64/208.22	              [U15^#](x1, x2) = [4] x2 + [0]                  
753.64/208.22	                                                              
753.64/208.22	                [isNat^#](x1) = [4] x1 + [0]                  
753.64/208.22	                                                              
753.64/208.22	              [U21^#](x1, x2) = [4] x2 + [4]                  
753.64/208.22	                                                              
753.64/208.22	              [U22^#](x1, x2) = [4] x2 + [0]                  
753.64/208.22	                                                              
753.64/208.22	          [U61^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [7]
753.64/208.22	                                                              
753.64/208.22	          [U62^#](x1, x2, x3) = [4] x3 + [0]                  
753.64/208.22	                                                              
753.64/208.22	                    [c_1](x1) = [1] x1 + [3]                  
753.64/208.22	                                                              
753.64/208.22	                    [c_2](x1) = [1] x1 + [0]                  
753.64/208.22	                                                              
753.64/208.22	                    [c_3](x1) = [1] x1 + [0]                  
753.64/208.22	                                                              
753.64/208.22	                [c_4](x1, x2) = [1] x1 + [1] x2 + [0]         
753.64/208.22	                                                              
753.64/208.22	                    [c_5](x1) = [1] x1 + [0]                  
753.64/208.22	                                                              
753.64/208.22	                    [c_6](x1) = [1] x1 + [3]                  
753.64/208.22	                                                              
753.64/208.22	                    [c_7](x1) = [1] x1 + [3]                  
753.64/208.22	                                                              
753.64/208.22	                    [c_8](x1) = [1] x1 + [1]                  
753.64/208.22	                                                              
753.64/208.22	                    [c_9](x1) = [1] x1 + [0]                  
753.64/208.22	                                                              
753.64/208.22	                   [c_10](x1) = [1] x1 + [3]                  
753.64/208.22	                                                              
753.64/208.22	                   [c_11](x1) = [1] x1 + [0]                  
753.64/208.22	        
753.64/208.22	        The order satisfies the following ordering constraints:
753.64/208.22	        
753.64/208.22	                   [U11(tt(), V1, V2)] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U12(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.64/208.22	                                                                                                           
753.64/208.22	                   [U12(tt(), V1, V2)] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U13(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.64/208.22	                                                                                                           
753.64/208.22	                   [isNatKind(n__0())] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [tt()]                                                           
753.64/208.22	                                                                                                           
753.64/208.22	          [isNatKind(n__plus(V1, V2))] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U31(isNatKind(activate(V1)), activate(V2))]                     
753.64/208.22	                                                                                                           
753.64/208.22	                 [isNatKind(n__s(V1))] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U41(isNatKind(activate(V1)))]                                   
753.64/208.22	                                                                                                           
753.64/208.22	                         [activate(X)] =  [1] X + [0]                                                      
753.64/208.22	                                       >= [1] X + [0]                                                      
753.64/208.22	                                       =  [X]                                                              
753.64/208.22	                                                                                                           
753.64/208.22	                    [activate(n__0())] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [0()]                                                            
753.64/208.22	                                                                                                           
753.64/208.22	           [activate(n__plus(X1, X2))] =  [1] X1 + [1] X2 + [2]                                            
753.64/208.22	                                       >= [1] X1 + [1] X2 + [2]                                            
753.64/208.22	                                       =  [plus(activate(X1), activate(X2))]                               
753.64/208.22	                                                                                                           
753.64/208.22	                   [activate(n__s(X))] =  [1] X + [2]                                                      
753.64/208.22	                                       >= [1] X + [2]                                                      
753.64/208.22	                                       =  [s(activate(X))]                                                 
753.64/208.22	                                                                                                           
753.64/208.22	                   [U13(tt(), V1, V2)] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U14(isNatKind(activate(V2)), activate(V1), activate(V2))]       
753.64/208.22	                                                                                                           
753.64/208.22	                   [U14(tt(), V1, V2)] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U15(isNat(activate(V1)), activate(V2))]                         
753.64/208.22	                                                                                                           
753.64/208.22	                       [U15(tt(), V2)] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U16(isNat(activate(V2)))]                                       
753.64/208.22	                                                                                                           
753.64/208.22	                       [isNat(n__0())] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [tt()]                                                           
753.64/208.22	                                                                                                           
753.64/208.22	              [isNat(n__plus(V1, V2))] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U11(isNatKind(activate(V1)), activate(V1), activate(V2))]       
753.64/208.22	                                                                                                           
753.64/208.22	                     [isNat(n__s(V1))] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U21(isNatKind(activate(V1)), activate(V1))]                     
753.64/208.22	                                                                                                           
753.64/208.22	                           [U16(tt())] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [tt()]                                                           
753.64/208.22	                                                                                                           
753.64/208.22	                       [U21(tt(), V1)] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U22(isNatKind(activate(V1)), activate(V1))]                     
753.64/208.22	                                                                                                           
753.64/208.22	                       [U22(tt(), V1)] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U23(isNat(activate(V1)))]                                       
753.64/208.22	                                                                                                           
753.64/208.22	                           [U23(tt())] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [tt()]                                                           
753.64/208.22	                                                                                                           
753.64/208.22	                       [U31(tt(), V2)] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [U32(isNatKind(activate(V2)))]                                   
753.64/208.22	                                                                                                           
753.64/208.22	                           [U32(tt())] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [tt()]                                                           
753.64/208.22	                                                                                                           
753.64/208.22	                           [U41(tt())] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [tt()]                                                           
753.64/208.22	                                                                                                           
753.64/208.22	                                [s(X)] =  [1] X + [2]                                                      
753.64/208.22	                                       >= [1] X + [2]                                                      
753.64/208.22	                                       =  [n__s(X)]                                                        
753.64/208.22	                                                                                                           
753.64/208.22	                        [plus(X1, X2)] =  [1] X1 + [1] X2 + [2]                                            
753.64/208.22	                                       >= [1] X1 + [1] X2 + [2]                                            
753.64/208.22	                                       =  [n__plus(X1, X2)]                                                
753.64/208.22	                                                                                                           
753.64/208.22	                                 [0()] =  [0]                                                              
753.64/208.22	                                       >= [0]                                                              
753.64/208.22	                                       =  [n__0()]                                                         
753.64/208.22	                                                                                                           
753.64/208.22	                 [U11^#(tt(), V1, V2)] =  [4] V1 + [4] V2 + [4]                                            
753.64/208.22	                                       >  [4] V1 + [4] V2 + [3]                                            
753.64/208.22	                                       =  [c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.64/208.22	                                                                                                           
753.64/208.22	                 [U12^#(tt(), V1, V2)] =  [4] V1 + [4] V2 + [0]                                            
753.64/208.22	                                       >= [4] V1 + [4] V2 + [0]                                            
753.64/208.22	                                       =  [c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.64/208.22	                                                                                                           
753.64/208.22	                 [U13^#(tt(), V1, V2)] =  [4] V1 + [4] V2 + [0]                                            
753.64/208.22	                                       >= [4] V1 + [4] V2 + [0]                                            
753.64/208.22	                                       =  [c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))]
753.64/208.22	                                                                                                           
753.64/208.22	                 [U14^#(tt(), V1, V2)] =  [4] V1 + [4] V2 + [0]                                            
753.64/208.22	                                       >= [4] V1 + [4] V2 + [0]                                            
753.64/208.22	                                       =  [c_4(U15^#(isNat(activate(V1)), activate(V2)),                   
753.64/208.22	                                               isNat^#(activate(V1)))]                                     
753.64/208.22	                                                                                                           
753.64/208.22	                     [U15^#(tt(), V2)] =  [4] V2 + [0]                                                     
753.64/208.22	                                       >= [4] V2 + [0]                                                     
753.64/208.22	                                       =  [c_5(isNat^#(activate(V2)))]                                     
753.64/208.22	                                                                                                           
753.64/208.22	            [isNat^#(n__plus(V1, V2))] =  [4] V1 + [4] V2 + [8]                                            
753.64/208.22	                                       >  [4] V1 + [4] V2 + [7]                                            
753.64/208.22	                                       =  [c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))]
753.64/208.22	                                                                                                           
753.64/208.22	                   [isNat^#(n__s(V1))] =  [4] V1 + [8]                                                     
753.64/208.22	                                       >  [4] V1 + [7]                                                     
753.64/208.22	                                       =  [c_7(U21^#(isNatKind(activate(V1)), activate(V1)))]              
753.64/208.22	                                                                                                           
753.64/208.22	                     [U21^#(tt(), V1)] =  [4] V1 + [4]                                                     
753.64/208.22	                                       >  [4] V1 + [1]                                                     
753.64/208.22	                                       =  [c_8(U22^#(isNatKind(activate(V1)), activate(V1)))]              
753.64/208.22	                                                                                                           
753.64/208.22	                     [U22^#(tt(), V1)] =  [4] V1 + [0]                                                     
753.64/208.22	                                       >= [4] V1 + [0]                                                     
753.64/208.22	                                       =  [c_9(isNat^#(activate(V1)))]                                     
753.64/208.22	                                                                                                           
753.64/208.22	                   [U61^#(tt(), M, N)] =  [7] N + [7] M + [7]                                              
753.64/208.22	                                       >  [4] N + [3]                                                      
753.64/208.22	                                       =  [c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))]  
753.64/208.22	                                                                                                           
753.64/208.22	                   [U62^#(tt(), M, N)] =  [4] N + [0]                                                      
753.64/208.22	                                       >= [4] N + [0]                                                      
753.64/208.22	                                       =  [c_11(isNat^#(activate(N)))]                                     
753.64/208.22	                                                                                                           
753.64/208.22	      
753.64/208.22	      We return to the main proof. Consider the set of all dependency
753.64/208.22	      pairs
753.64/208.22	      
753.64/208.22	      :
753.64/208.22	        { 1: U11^#(tt(), V1, V2) ->
753.64/208.22	             c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.64/208.22	        , 2: U12^#(tt(), V1, V2) ->
753.64/208.22	             c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.64/208.22	        , 3: U13^#(tt(), V1, V2) ->
753.64/208.22	             c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.64/208.22	        , 4: U14^#(tt(), V1, V2) ->
753.64/208.22	             c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.64/208.22	                 isNat^#(activate(V1)))
753.64/208.22	        , 5: U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.64/208.22	        , 6: isNat^#(n__plus(V1, V2)) ->
753.64/208.22	             c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.64/208.22	        , 7: isNat^#(n__s(V1)) ->
753.64/208.22	             c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.64/208.22	        , 8: U21^#(tt(), V1) ->
753.64/208.22	             c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.64/208.22	        , 9: U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.64/208.22	        , 10: U61^#(tt(), M, N) ->
753.64/208.22	              c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.64/208.22	        , 11: U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.64/208.22	      
753.64/208.22	      Processor 'matrix interpretation of dimension 1' induces the
753.64/208.22	      complexity certificate YES(?,O(n^1)) on application of dependency
753.64/208.22	      pairs {1,6,7,8,10}. These cover all (indirect) predecessors of
753.64/208.22	      dependency pairs {1,2,3,4,5,6,7,8,9,10,11}, their number of
753.64/208.22	      application is equally bounded. The dependency pairs are shifted
753.64/208.22	      into the weak component.
753.64/208.22	      
753.64/208.22	      We are left with following problem, upon which TcT provides the
753.64/208.22	      certificate YES(O(1),O(1)).
753.64/208.22	      
753.64/208.22	      Weak DPs:
753.64/208.22	        { U11^#(tt(), V1, V2) ->
753.64/208.22	          c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.64/208.22	        , U12^#(tt(), V1, V2) ->
753.64/208.22	          c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.64/208.22	        , U13^#(tt(), V1, V2) ->
753.64/208.22	          c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.64/208.22	        , U14^#(tt(), V1, V2) ->
753.64/208.22	          c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.64/208.22	              isNat^#(activate(V1)))
753.64/208.22	        , U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.64/208.22	        , isNat^#(n__plus(V1, V2)) ->
753.64/208.22	          c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.64/208.22	        , isNat^#(n__s(V1)) ->
753.64/208.22	          c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.64/208.22	        , U21^#(tt(), V1) ->
753.64/208.22	          c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.64/208.22	        , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.64/208.22	        , U61^#(tt(), M, N) ->
753.64/208.22	          c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.64/208.22	        , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.64/208.22	      Weak Trs:
753.64/208.22	        { U11(tt(), V1, V2) ->
753.64/208.22	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.22	        , U12(tt(), V1, V2) ->
753.64/208.22	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.22	        , isNatKind(n__0()) -> tt()
753.64/208.22	        , isNatKind(n__plus(V1, V2)) ->
753.64/208.22	          U31(isNatKind(activate(V1)), activate(V2))
753.64/208.22	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.64/208.22	        , activate(X) -> X
753.64/208.22	        , activate(n__0()) -> 0()
753.64/208.22	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.64/208.22	        , activate(n__s(X)) -> s(activate(X))
753.64/208.22	        , U13(tt(), V1, V2) ->
753.64/208.22	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.22	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.64/208.22	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.64/208.22	        , isNat(n__0()) -> tt()
753.64/208.22	        , isNat(n__plus(V1, V2)) ->
753.64/208.22	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.22	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.64/208.22	        , U16(tt()) -> tt()
753.64/208.22	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.64/208.22	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.64/208.22	        , U23(tt()) -> tt()
753.64/208.22	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.64/208.22	        , U32(tt()) -> tt()
753.64/208.22	        , U41(tt()) -> tt()
753.64/208.22	        , s(X) -> n__s(X)
753.64/208.22	        , plus(X1, X2) -> n__plus(X1, X2)
753.64/208.22	        , 0() -> n__0() }
753.64/208.22	      Obligation:
753.64/208.22	        innermost runtime complexity
753.64/208.22	      Answer:
753.64/208.22	        YES(O(1),O(1))
753.64/208.22	      
753.64/208.22	      The following weak DPs constitute a sub-graph of the DG that is
753.64/208.22	      closed under successors. The DPs are removed.
753.64/208.22	      
753.64/208.22	      { U11^#(tt(), V1, V2) ->
753.64/208.22	        c_1(U12^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.64/208.22	      , U12^#(tt(), V1, V2) ->
753.64/208.22	        c_2(U13^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.64/208.22	      , U13^#(tt(), V1, V2) ->
753.64/208.22	        c_3(U14^#(isNatKind(activate(V2)), activate(V1), activate(V2)))
753.64/208.22	      , U14^#(tt(), V1, V2) ->
753.64/208.22	        c_4(U15^#(isNat(activate(V1)), activate(V2)),
753.64/208.22	            isNat^#(activate(V1)))
753.64/208.22	      , U15^#(tt(), V2) -> c_5(isNat^#(activate(V2)))
753.64/208.22	      , isNat^#(n__plus(V1, V2)) ->
753.64/208.22	        c_6(U11^#(isNatKind(activate(V1)), activate(V1), activate(V2)))
753.64/208.22	      , isNat^#(n__s(V1)) ->
753.64/208.22	        c_7(U21^#(isNatKind(activate(V1)), activate(V1)))
753.64/208.22	      , U21^#(tt(), V1) ->
753.64/208.22	        c_8(U22^#(isNatKind(activate(V1)), activate(V1)))
753.64/208.22	      , U22^#(tt(), V1) -> c_9(isNat^#(activate(V1)))
753.64/208.22	      , U61^#(tt(), M, N) ->
753.64/208.22	        c_10(U62^#(isNatKind(activate(M)), activate(M), activate(N)))
753.64/208.22	      , U62^#(tt(), M, N) -> c_11(isNat^#(activate(N))) }
753.64/208.22	      
753.64/208.22	      We are left with following problem, upon which TcT provides the
753.64/208.22	      certificate YES(O(1),O(1)).
753.64/208.22	      
753.64/208.22	      Weak Trs:
753.64/208.22	        { U11(tt(), V1, V2) ->
753.64/208.22	          U12(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.22	        , U12(tt(), V1, V2) ->
753.64/208.22	          U13(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.22	        , isNatKind(n__0()) -> tt()
753.64/208.22	        , isNatKind(n__plus(V1, V2)) ->
753.64/208.22	          U31(isNatKind(activate(V1)), activate(V2))
753.64/208.22	        , isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
753.64/208.22	        , activate(X) -> X
753.64/208.22	        , activate(n__0()) -> 0()
753.64/208.22	        , activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2))
753.64/208.22	        , activate(n__s(X)) -> s(activate(X))
753.64/208.22	        , U13(tt(), V1, V2) ->
753.64/208.22	          U14(isNatKind(activate(V2)), activate(V1), activate(V2))
753.64/208.22	        , U14(tt(), V1, V2) -> U15(isNat(activate(V1)), activate(V2))
753.64/208.22	        , U15(tt(), V2) -> U16(isNat(activate(V2)))
753.64/208.22	        , isNat(n__0()) -> tt()
753.64/208.22	        , isNat(n__plus(V1, V2)) ->
753.64/208.22	          U11(isNatKind(activate(V1)), activate(V1), activate(V2))
753.64/208.22	        , isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1))
753.64/208.22	        , U16(tt()) -> tt()
753.64/208.22	        , U21(tt(), V1) -> U22(isNatKind(activate(V1)), activate(V1))
753.64/208.22	        , U22(tt(), V1) -> U23(isNat(activate(V1)))
753.64/208.22	        , U23(tt()) -> tt()
753.64/208.22	        , U31(tt(), V2) -> U32(isNatKind(activate(V2)))
753.64/208.22	        , U32(tt()) -> tt()
753.64/208.22	        , U41(tt()) -> tt()
753.64/208.22	        , s(X) -> n__s(X)
753.64/208.22	        , plus(X1, X2) -> n__plus(X1, X2)
753.64/208.22	        , 0() -> n__0() }
753.64/208.22	      Obligation:
753.64/208.22	        innermost runtime complexity
753.64/208.22	      Answer:
753.64/208.22	        YES(O(1),O(1))
753.64/208.22	      
753.64/208.22	      No rule is usable, rules are removed from the input problem.
753.64/208.22	      
753.64/208.22	      We are left with following problem, upon which TcT provides the
753.64/208.22	      certificate YES(O(1),O(1)).
753.64/208.22	      
753.64/208.22	      Rules: Empty
753.64/208.22	      Obligation:
753.64/208.22	        innermost runtime complexity
753.64/208.22	      Answer:
753.64/208.22	        YES(O(1),O(1))
753.64/208.22	      
753.64/208.22	      Empty rules are trivially bounded
753.64/208.22	   
753.64/208.22	
753.64/208.22	
753.64/208.22	Hurray, we answered YES(O(1),O(n^3))
753.78/208.32	EOF