WORST_CASE(?,O(n^3))
* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1} / {n__0/0,n__plus/2,n__s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0,U11,U12,U13,U14,U15,U16,U21,U22,U23,U31,U32,U41,U51,U52
,U61,U62,U63,U64,activate,isNat,isNatKind,plus,s} and constructors {n__0,n__plus,n__s,tt}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
0#() -> c_1()
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U16#(tt()) -> c_7()
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U23#(tt()) -> c_10()
U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U32#(tt()) -> c_12()
U41#(tt()) -> c_13()
U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
U52#(tt(),N) -> c_15(activate#(N))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
activate#(X) -> c_20()
activate#(n__0()) -> c_21(0#())
activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X))
isNat#(n__0()) -> c_24()
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
isNatKind#(n__0()) -> c_27()
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N))
plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M))
plus#(X1,X2) -> c_32()
s#(X) -> c_33()
Weak DPs
and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
0#() -> c_1()
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U16#(tt()) -> c_7()
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U23#(tt()) -> c_10()
U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U32#(tt()) -> c_12()
U41#(tt()) -> c_13()
U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
U52#(tt(),N) -> c_15(activate#(N))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
activate#(X) -> c_20()
activate#(n__0()) -> c_21(0#())
activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X))
isNat#(n__0()) -> c_24()
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
isNatKind#(n__0()) -> c_27()
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N))
plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M))
plus#(X1,X2) -> c_32()
s#(X) -> c_33()
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4
,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,7,10,12,13,20,24,27,30,31,32,33}
by application of
Pre({1,7,10,12,13,20,24,27,30,31,32,33}) = {2,3,4,5,6,8,9,11,14,15,16,17,18,19,21,22,23,25,26,28,29}.
Here rules are labelled as follows:
1: 0#() -> c_1()
2: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
3: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
4: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
5: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
6: U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
7: U16#(tt()) -> c_7()
8: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
9: U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
10: U23#(tt()) -> c_10()
11: U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
12: U32#(tt()) -> c_12()
13: U41#(tt()) -> c_13()
14: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
15: U52#(tt(),N) -> c_15(activate#(N))
16: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
17: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
18: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
19: U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
20: activate#(X) -> c_20()
21: activate#(n__0()) -> c_21(0#())
22: activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
23: activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X))
24: isNat#(n__0()) -> c_24()
25: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
26: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
27: isNatKind#(n__0()) -> c_27()
28: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
29: isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
30: plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N))
31: plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M))
32: plus#(X1,X2) -> c_32()
33: s#(X) -> c_33()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
U52#(tt(),N) -> c_15(activate#(N))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
activate#(n__0()) -> c_21(0#())
activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
- Weak DPs:
0#() -> c_1()
U16#(tt()) -> c_7()
U23#(tt()) -> c_10()
U32#(tt()) -> c_12()
U41#(tt()) -> c_13()
activate#(X) -> c_20()
isNat#(n__0()) -> c_24()
isNatKind#(n__0()) -> c_27()
plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N))
plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M))
plus#(X1,X2) -> c_32()
s#(X) -> c_33()
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4
,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{15}
by application of
Pre({15}) = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,16,17,18,19,20,21}.
Here rules are labelled as follows:
1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
5: U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
7: U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
8: U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
9: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
10: U52#(tt(),N) -> c_15(activate#(N))
11: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
12: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
13: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
14: U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
15: activate#(n__0()) -> c_21(0#())
16: activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
17: activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X))
18: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
19: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
20: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
21: isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
22: 0#() -> c_1()
23: U16#(tt()) -> c_7()
24: U23#(tt()) -> c_10()
25: U32#(tt()) -> c_12()
26: U41#(tt()) -> c_13()
27: activate#(X) -> c_20()
28: isNat#(n__0()) -> c_24()
29: isNatKind#(n__0()) -> c_27()
30: plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N))
31: plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M))
32: plus#(X1,X2) -> c_32()
33: s#(X) -> c_33()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
U52#(tt(),N) -> c_15(activate#(N))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
- Weak DPs:
0#() -> c_1()
U16#(tt()) -> c_7()
U23#(tt()) -> c_10()
U32#(tt()) -> c_12()
U41#(tt()) -> c_13()
activate#(X) -> c_20()
activate#(n__0()) -> c_21(0#())
isNat#(n__0()) -> c_24()
isNatKind#(n__0()) -> c_27()
plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N))
plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M))
plus#(X1,X2) -> c_32()
s#(X) -> c_33()
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4
,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
-->_5 activate#(n__0()) -> c_21(0#()):27
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2)):2
-->_2 isNatKind#(n__0()) -> c_27():29
-->_5 activate#(X) -> c_20():26
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
-->_5 activate#(n__0()) -> c_21(0#()):27
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2)):3
-->_2 isNatKind#(n__0()) -> c_27():29
-->_5 activate#(X) -> c_20():26
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
-->_5 activate#(n__0()) -> c_21(0#()):27
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):4
-->_2 isNatKind#(n__0()) -> c_27():29
-->_5 activate#(X) -> c_20():26
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):18
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):17
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):5
-->_2 isNat#(n__0()) -> c_24():28
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
5:S:U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):18
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):17
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_2 isNat#(n__0()) -> c_24():28
-->_3 activate#(X) -> c_20():26
-->_1 U16#(tt()) -> c_7():22
6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):7
-->_2 isNatKind#(n__0()) -> c_27():29
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
7:S:U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):18
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):17
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_2 isNat#(n__0()) -> c_24():28
-->_3 activate#(X) -> c_20():26
-->_1 U23#(tt()) -> c_10():23
8:S:U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_2 isNatKind#(n__0()) -> c_27():29
-->_3 activate#(X) -> c_20():26
-->_1 U32#(tt()) -> c_12():24
9:S:U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U52#(tt(),N) -> c_15(activate#(N)):10
-->_2 isNatKind#(n__0()) -> c_27():29
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
10:S:U52#(tt(),N) -> c_15(activate#(N))
-->_1 activate#(n__0()) -> c_21(0#()):27
-->_1 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_1 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 activate#(X) -> c_20():26
11:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
-->_5 activate#(n__0()) -> c_21(0#()):27
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N)):12
-->_2 isNatKind#(n__0()) -> c_27():29
-->_5 activate#(X) -> c_20():26
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
12:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
-->_5 activate#(n__0()) -> c_21(0#()):27
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):18
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):17
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N)):13
-->_2 isNat#(n__0()) -> c_24():28
-->_5 activate#(X) -> c_20():26
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
13:S:U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
-->_5 activate#(n__0()) -> c_21(0#()):27
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M)):14
-->_2 isNatKind#(n__0()) -> c_27():29
-->_5 activate#(X) -> c_20():26
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
14:S:U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 s#(X) -> c_33():33
-->_2 plus#(X1,X2) -> c_32():32
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
15:S:activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_1 plus#(X1,X2) -> c_32():32
-->_3 activate#(X) -> c_20():26
-->_2 activate#(X) -> c_20():26
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
16:S:activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X))
-->_2 activate#(n__0()) -> c_21(0#()):27
-->_1 s#(X) -> c_33():33
-->_2 activate#(X) -> c_20():26
-->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
17:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
-->_5 activate#(n__0()) -> c_21(0#()):27
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_2 isNatKind#(n__0()) -> c_27():29
-->_5 activate#(X) -> c_20():26
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):1
18:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_2 isNatKind#(n__0()) -> c_27():29
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):6
19:S:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_4 activate#(n__0()) -> c_21(0#()):27
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__0()) -> c_27():29
-->_4 activate#(X) -> c_20():26
-->_3 activate#(X) -> c_20():26
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):8
20:S:isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
-->_3 activate#(n__0()) -> c_21(0#()):27
-->_2 isNatKind#(n__0()) -> c_27():29
-->_3 activate#(X) -> c_20():26
-->_1 U41#(tt()) -> c_13():25
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
21:W:0#() -> c_1()
22:W:U16#(tt()) -> c_7()
23:W:U23#(tt()) -> c_10()
24:W:U32#(tt()) -> c_12()
25:W:U41#(tt()) -> c_13()
26:W:activate#(X) -> c_20()
27:W:activate#(n__0()) -> c_21(0#())
-->_1 0#() -> c_1():21
28:W:isNat#(n__0()) -> c_24()
29:W:isNatKind#(n__0()) -> c_27()
30:W:plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N))
31:W:plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M))
32:W:plus#(X1,X2) -> c_32()
33:W:s#(X) -> c_33()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
31: plus#(N,s(M)) -> c_31(U61#(isNat(M),M,N),isNat#(M))
30: plus#(N,0()) -> c_30(U51#(isNat(N),N),isNat#(N))
22: U16#(tt()) -> c_7()
23: U23#(tt()) -> c_10()
28: isNat#(n__0()) -> c_24()
24: U32#(tt()) -> c_12()
32: plus#(X1,X2) -> c_32()
33: s#(X) -> c_33()
25: U41#(tt()) -> c_13()
26: activate#(X) -> c_20()
29: isNatKind#(n__0()) -> c_27()
27: activate#(n__0()) -> c_21(0#())
21: 0#() -> c_1()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
U52#(tt(),N) -> c_15(activate#(N))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/3,c_7/0,c_8/4
,c_9/3,c_10/0,c_11/3,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/4,c_20/0,c_21/1,c_22/3,c_23/2
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/3,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2)):2
2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2)):3
3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):4
4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):18
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):17
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2)):5
5:S:U15#(tt(),V2) -> c_6(U16#(isNat(activate(V2))),isNat#(activate(V2)),activate#(V2))
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):18
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):17
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1)):7
7:S:U22#(tt(),V1) -> c_9(U23#(isNat(activate(V1))),isNat#(activate(V1)),activate#(V1))
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):18
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):17
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
8:S:U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
9:S:U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U52#(tt(),N) -> c_15(activate#(N)):10
10:S:U52#(tt(),N) -> c_15(activate#(N))
-->_1 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_1 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
11:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N)):12
12:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):18
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):17
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N)):13
13:S:U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M)):14
14:S:U64#(tt(),M,N) -> c_19(s#(plus(activate(N),activate(M)))
,plus#(activate(N),activate(M))
,activate#(N)
,activate#(M))
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
15:S:activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2))
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
16:S:activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X))
-->_2 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_2 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
17:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_5 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_5 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):1
18:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):6
19:S:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_4 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_4 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
-->_1 U31#(tt(),V2) -> c_11(U32#(isNatKind(activate(V2))),isNatKind#(activate(V2)),activate#(V2)):8
20:S:isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1))),isNatKind#(activate(V1)),activate#(V1))
-->_2 isNatKind#(n__s(V1)) -> c_29(U41#(isNatKind(activate(V1)))
,isNatKind#(activate(V1))
,activate#(V1)):20
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):19
-->_3 activate#(n__s(X)) -> c_23(s#(activate(X)),activate#(X)):16
-->_3 activate#(n__plus(X1,X2)) -> c_22(plus#(activate(X1),activate(X2)),activate#(X1),activate#(X2)):15
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1))
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2))
U64#(tt(),M,N) -> c_19(activate#(N),activate#(M))
activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(activate#(X))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1))
* Step 6: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1))
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2))
U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
U52#(tt(),N) -> c_15(activate#(N))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U64#(tt(),M,N) -> c_19(activate#(N),activate#(M))
activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(activate#(X))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1))
U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
U52#(tt(),N) -> c_15(activate#(N))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U64#(tt(),M,N) -> c_19(activate#(N),activate#(M))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
and a lower component
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2))
activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(activate#(X))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1))
Further, following extension rules are added to the lower component.
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> activate#(N)
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> activate#(M)
U61#(tt(),M,N) -> activate#(N)
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
** Step 6.a:1: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1))
U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
U52#(tt(),N) -> c_15(activate#(N))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U64#(tt(),M,N) -> c_19(activate#(N),activate#(M))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{9,13}
by application of
Pre({9,13}) = {8,12}.
Here rules are labelled as follows:
1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
5: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2))
6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
7: U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1))
8: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
9: U52#(tt(),N) -> c_15(activate#(N))
10: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
11: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
12: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
13: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M))
14: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
15: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
** Step 6.a:2: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1))
U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
- Weak DPs:
U52#(tt(),N) -> c_15(activate#(N))
U64#(tt(),M,N) -> c_19(activate#(N),activate#(M))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{8,11}
by application of
Pre({8,11}) = {10}.
Here rules are labelled as follows:
1: U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
2: U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
3: U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
4: U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
5: U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2))
6: U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
7: U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1))
8: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
9: U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
10: U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
11: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
12: isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
13: isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
14: U52#(tt(),N) -> c_15(activate#(N))
15: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M))
** Step 6.a:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
- Weak DPs:
U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
U52#(tt(),N) -> c_15(activate#(N))
U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
U64#(tt(),M,N) -> c_19(activate#(N),activate#(M))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
-->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2)):2
2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
-->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2)):3
3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
-->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):4
4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):11
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):10
-->_1 U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)):5
5:S:U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2))
-->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):11
-->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):10
6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
-->_1 U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)):7
7:S:U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1))
-->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):11
-->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):10
8:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
-->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N)):9
9:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
-->_1 U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N)):14
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):11
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):10
10:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
-->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):1
11:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
-->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):6
12:W:U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
-->_1 U52#(tt(),N) -> c_15(activate#(N)):13
13:W:U52#(tt(),N) -> c_15(activate#(N))
14:W:U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
-->_1 U64#(tt(),M,N) -> c_19(activate#(N),activate#(M)):15
15:W:U64#(tt(),M,N) -> c_19(activate#(N),activate#(M))
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
12: U51#(tt(),N) -> c_14(U52#(isNatKind(activate(N)),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(N))
13: U52#(tt(),N) -> c_15(activate#(N))
14: U63#(tt(),M,N) -> c_18(U64#(isNatKind(activate(N)),activate(M),activate(N))
,isNatKind#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
15: U64#(tt(),M,N) -> c_19(activate#(N),activate#(M))
** Step 6.a:4: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
-->_1 U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2)):2
2:S:U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
-->_1 U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2)):3
3:S:U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
,isNatKind#(activate(V2))
,activate#(V2)
,activate#(V1)
,activate#(V2))
-->_1 U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2)):4
4:S:U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2))
,isNat#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):11
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):10
-->_1 U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2)):5
5:S:U15#(tt(),V2) -> c_6(isNat#(activate(V2)),activate#(V2))
-->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):11
-->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):10
6:S:U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
-->_1 U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1)):7
7:S:U22#(tt(),V1) -> c_9(isNat#(activate(V1)),activate#(V1))
-->_1 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):11
-->_1 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):10
8:S:U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N))
,isNatKind#(activate(M))
,activate#(M)
,activate#(M)
,activate#(N))
-->_1 U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N)):9
9:S:U62#(tt(),M,N) -> c_17(U63#(isNat(activate(N)),activate(M),activate(N))
,isNat#(activate(N))
,activate#(N)
,activate#(M)
,activate#(N))
-->_2 isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):11
-->_2 isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):10
10:S:isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2))
-->_1 U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)
,activate#(V2)):1
11:S:isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1))
-->_1 U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V1)):6
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
** Step 6.a:5: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1
,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_25) = {1},
uargs(c_26) = {1}
Following symbols are considered usable:
{0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#
,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [4]
p(U12) = [4] x2 + [1] x3 + [7]
p(U13) = [4] x1 + [2] x3 + [3]
p(U14) = [1] x2 + [5]
p(U15) = [1] x1 + [2] x2 + [4]
p(U16) = [6]
p(U21) = [0]
p(U22) = [1] x1 + [1] x2 + [0]
p(U23) = [1]
p(U31) = [4]
p(U32) = [5] x1 + [0]
p(U41) = [3]
p(U51) = [3] x1 + [1] x2 + [1]
p(U52) = [6] x1 + [0]
p(U61) = [3] x1 + [5] x2 + [1] x3 + [4]
p(U62) = [1] x2 + [0]
p(U63) = [2] x1 + [1] x2 + [0]
p(U64) = [2] x3 + [0]
p(activate) = [0]
p(isNat) = [1] x1 + [1]
p(isNatKind) = [1]
p(n__0) = [0]
p(n__plus) = [7]
p(n__s) = [1] x1 + [7]
p(plus) = [4] x2 + [4]
p(s) = [1]
p(tt) = [0]
p(0#) = [2]
p(U11#) = [0]
p(U12#) = [0]
p(U13#) = [0]
p(U14#) = [0]
p(U15#) = [0]
p(U16#) = [4] x1 + [0]
p(U21#) = [0]
p(U22#) = [0]
p(U23#) = [2]
p(U31#) = [0]
p(U32#) = [1]
p(U41#) = [1] x1 + [0]
p(U51#) = [1] x1 + [2] x2 + [1]
p(U52#) = [0]
p(U61#) = [4] x1 + [4] x2 + [4] x3 + [5]
p(U62#) = [1]
p(U63#) = [1] x1 + [0]
p(U64#) = [1] x1 + [2] x2 + [2] x3 + [0]
p(activate#) = [0]
p(isNat#) = [0]
p(isNatKind#) = [4] x1 + [0]
p(plus#) = [1]
p(s#) = [1] x1 + [2]
p(c_1) = [0]
p(c_2) = [4] x1 + [0]
p(c_3) = [4] x1 + [0]
p(c_4) = [4] x1 + [0]
p(c_5) = [4] x1 + [1] x2 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [0]
p(c_8) = [4] x1 + [0]
p(c_9) = [4] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x2 + [0]
p(c_12) = [1]
p(c_13) = [1]
p(c_14) = [1] x1 + [1] x2 + [1]
p(c_15) = [2] x1 + [4]
p(c_16) = [1] x1 + [0]
p(c_17) = [1] x1 + [1]
p(c_18) = [1] x1 + [0]
p(c_19) = [1] x1 + [1]
p(c_20) = [4]
p(c_21) = [4]
p(c_22) = [4] x2 + [1]
p(c_23) = [0]
p(c_24) = [0]
p(c_25) = [1] x1 + [0]
p(c_26) = [4] x1 + [0]
p(c_27) = [0]
p(c_28) = [1] x2 + [1] x3 + [1] x4 + [2]
p(c_29) = [1] x1 + [4] x2 + [4]
p(c_30) = [1] x1 + [1] x2 + [1]
p(c_31) = [0]
p(c_32) = [0]
p(c_33) = [1]
Following rules are strictly oriented:
U61#(tt(),M,N) = [4] M + [4] N + [5]
> [1]
= c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [0]
>= [0]
= c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) = [0]
>= [0]
= c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) = [0]
>= [0]
= c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) = [0]
>= [0]
= c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) = [0]
>= [0]
= c_6(isNat#(activate(V2)))
U21#(tt(),V1) = [0]
>= [0]
= c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) = [0]
>= [0]
= c_9(isNat#(activate(V1)))
U62#(tt(),M,N) = [1]
>= [1]
= c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [0]
>= [0]
= c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) = [0]
>= [0]
= c_26(U21#(isNatKind(activate(V1)),activate(V1)))
** Step 6.a:6: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
- Weak DPs:
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1
,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_25) = {1},
uargs(c_26) = {1}
Following symbols are considered usable:
{0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#
,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [1]
p(U11) = [0]
p(U12) = [4] x2 + [1]
p(U13) = [2] x2 + [6]
p(U14) = [1] x1 + [2]
p(U15) = [1] x1 + [2]
p(U16) = [1] x1 + [6]
p(U21) = [1] x1 + [1] x2 + [2]
p(U22) = [2] x2 + [4]
p(U23) = [4] x1 + [2]
p(U31) = [2]
p(U32) = [1] x1 + [2]
p(U41) = [0]
p(U51) = [6]
p(U52) = [0]
p(U61) = [1] x1 + [6]
p(U62) = [2] x1 + [4] x2 + [4] x3 + [4]
p(U63) = [1] x1 + [1] x2 + [4]
p(U64) = [1] x1 + [1] x2 + [1] x3 + [0]
p(activate) = [0]
p(isNat) = [0]
p(isNatKind) = [0]
p(n__0) = [2]
p(n__plus) = [1]
p(n__s) = [1] x1 + [0]
p(plus) = [4] x2 + [4]
p(s) = [2] x1 + [2]
p(tt) = [0]
p(0#) = [1]
p(U11#) = [0]
p(U12#) = [0]
p(U13#) = [0]
p(U14#) = [0]
p(U15#) = [0]
p(U16#) = [1] x1 + [0]
p(U21#) = [0]
p(U22#) = [0]
p(U23#) = [1] x1 + [1]
p(U31#) = [1] x1 + [0]
p(U32#) = [4]
p(U41#) = [1] x1 + [1]
p(U51#) = [0]
p(U52#) = [1]
p(U61#) = [1] x3 + [7]
p(U62#) = [2]
p(U63#) = [1] x3 + [1]
p(U64#) = [1]
p(activate#) = [1] x1 + [4]
p(isNat#) = [0]
p(isNatKind#) = [0]
p(plus#) = [1] x2 + [1]
p(s#) = [1]
p(c_1) = [2]
p(c_2) = [1] x1 + [0]
p(c_3) = [4] x1 + [0]
p(c_4) = [4] x1 + [0]
p(c_5) = [1] x1 + [2] x2 + [0]
p(c_6) = [4] x1 + [0]
p(c_7) = [1]
p(c_8) = [4] x1 + [0]
p(c_9) = [4] x1 + [0]
p(c_10) = [0]
p(c_11) = [2] x1 + [4] x2 + [1]
p(c_12) = [1]
p(c_13) = [0]
p(c_14) = [1] x4 + [1]
p(c_15) = [1]
p(c_16) = [2] x1 + [3]
p(c_17) = [1] x1 + [0]
p(c_18) = [1] x2 + [1] x4 + [0]
p(c_19) = [1]
p(c_20) = [2]
p(c_21) = [4]
p(c_22) = [1] x1 + [1] x2 + [2]
p(c_23) = [4] x1 + [0]
p(c_24) = [1]
p(c_25) = [4] x1 + [0]
p(c_26) = [2] x1 + [0]
p(c_27) = [2]
p(c_28) = [1] x2 + [4] x3 + [0]
p(c_29) = [0]
p(c_30) = [4] x1 + [0]
p(c_31) = [1] x1 + [1] x2 + [2]
p(c_32) = [0]
p(c_33) = [0]
Following rules are strictly oriented:
U62#(tt(),M,N) = [2]
> [0]
= c_17(isNat#(activate(N)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [0]
>= [0]
= c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) = [0]
>= [0]
= c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) = [0]
>= [0]
= c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) = [0]
>= [0]
= c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) = [0]
>= [0]
= c_6(isNat#(activate(V2)))
U21#(tt(),V1) = [0]
>= [0]
= c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) = [0]
>= [0]
= c_9(isNat#(activate(V1)))
U61#(tt(),M,N) = [1] N + [7]
>= [7]
= c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
isNat#(n__plus(V1,V2)) = [0]
>= [0]
= c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) = [0]
>= [0]
= c_26(U21#(isNatKind(activate(V1)),activate(V1)))
** Step 6.a:7: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
- Weak DPs:
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1
,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_25) = {1},
uargs(c_26) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x1 + [2]
p(U12) = [4] x1 + [4]
p(U13) = [4] x1 + [4] x3 + [6]
p(U14) = [4] x3 + [0]
p(U15) = [2] x2 + [5]
p(U16) = [3] x1 + [5]
p(U21) = [6] x2 + [1]
p(U22) = [1] x1 + [2] x2 + [3]
p(U23) = [2]
p(U31) = [4]
p(U32) = [0]
p(U41) = [0]
p(U51) = [1] x2 + [0]
p(U52) = [1] x2 + [0]
p(U61) = [1] x2 + [1] x3 + [1]
p(U62) = [1] x2 + [1] x3 + [1]
p(U63) = [1] x2 + [1] x3 + [1]
p(U64) = [1] x2 + [1] x3 + [1]
p(activate) = [1] x1 + [0]
p(isNat) = [2]
p(isNatKind) = [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [1]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(0#) = [4]
p(U11#) = [1] x2 + [1] x3 + [0]
p(U12#) = [1] x2 + [1] x3 + [0]
p(U13#) = [1] x2 + [1] x3 + [0]
p(U14#) = [1] x2 + [1] x3 + [0]
p(U15#) = [1] x2 + [0]
p(U16#) = [1] x1 + [0]
p(U21#) = [1] x2 + [0]
p(U22#) = [1] x2 + [0]
p(U23#) = [2]
p(U31#) = [0]
p(U32#) = [1]
p(U41#) = [4] x1 + [0]
p(U51#) = [1] x2 + [0]
p(U52#) = [1] x1 + [2] x2 + [4]
p(U61#) = [4] x2 + [4] x3 + [3]
p(U62#) = [4] x3 + [2]
p(U63#) = [2] x3 + [4]
p(U64#) = [1]
p(activate#) = [1] x1 + [0]
p(isNat#) = [1] x1 + [0]
p(isNatKind#) = [1] x1 + [1]
p(plus#) = [1]
p(s#) = [2] x1 + [1]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [1] x2 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [1]
p(c_11) = [1] x1 + [0]
p(c_12) = [0]
p(c_13) = [2]
p(c_14) = [2] x2 + [1] x4 + [1]
p(c_15) = [2] x1 + [1]
p(c_16) = [1] x1 + [1]
p(c_17) = [1] x1 + [2]
p(c_18) = [1] x1 + [1] x2 + [2] x3 + [2] x4 + [1]
p(c_19) = [1] x1 + [1] x2 + [2]
p(c_20) = [1]
p(c_21) = [0]
p(c_22) = [1] x2 + [4]
p(c_23) = [1] x1 + [0]
p(c_24) = [1]
p(c_25) = [1] x1 + [0]
p(c_26) = [1] x1 + [0]
p(c_27) = [0]
p(c_28) = [1] x1 + [1] x3 + [2]
p(c_29) = [4] x2 + [2]
p(c_30) = [4] x1 + [1] x2 + [1]
p(c_31) = [4] x1 + [1] x2 + [2]
p(c_32) = [1]
p(c_33) = [1]
Following rules are strictly oriented:
isNat#(n__s(V1)) = [1] V1 + [1]
> [1] V1 + [0]
= c_26(U21#(isNatKind(activate(V1)),activate(V1)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) = [1] V2 + [0]
>= [1] V2 + [0]
= c_6(isNat#(activate(V2)))
U21#(tt(),V1) = [1] V1 + [0]
>= [1] V1 + [0]
= c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) = [1] V1 + [0]
>= [1] V1 + [0]
= c_9(isNat#(activate(V1)))
U61#(tt(),M,N) = [4] M + [4] N + [3]
>= [4] N + [3]
= c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) = [4] N + [2]
>= [1] N + [2]
= c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
0() = [0]
>= [0]
= n__0()
U51(tt(),N) = [1] N + [0]
>= [1] N + [0]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [1]
>= [1] X + [1]
= s(activate(X))
plus(N,0()) = [1] N + [0]
>= [1] N + [0]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
** Step 6.a:8: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
- Weak DPs:
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1
,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_25) = {1},
uargs(c_26) = {1}
Following symbols are considered usable:
{0,U11,U12,U13,U14,U15,U16,U21,U22,U23,U51,U52,U61,U62,U63,U64,activate,isNat,plus,s,0#,U11#,U12#,U13#
,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#
,plus#,s#}
TcT has computed the following interpretation:
p(0) = [1]
p(U11) = [0]
p(U12) = [0]
p(U13) = [0]
p(U14) = [0]
p(U15) = [0]
p(U16) = [0]
p(U21) = [0]
p(U22) = [0]
p(U23) = [0]
p(U31) = [1]
p(U32) = [1] x1 + [5]
p(U41) = [3]
p(U51) = [1] x1 + [1] x2 + [0]
p(U52) = [1] x2 + [0]
p(U61) = [1] x2 + [1] x3 + [2]
p(U62) = [1] x2 + [1] x3 + [2]
p(U63) = [1] x2 + [1] x3 + [2]
p(U64) = [1] x2 + [1] x3 + [2]
p(activate) = [1] x1 + [0]
p(isNat) = [1]
p(isNatKind) = [0]
p(n__0) = [1]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [2]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [2]
p(tt) = [0]
p(0#) = [0]
p(U11#) = [6] x2 + [6] x3 + [0]
p(U12#) = [6] x2 + [6] x3 + [0]
p(U13#) = [6] x2 + [6] x3 + [0]
p(U14#) = [6] x2 + [6] x3 + [0]
p(U15#) = [6] x2 + [0]
p(U16#) = [1] x1 + [0]
p(U21#) = [6] x2 + [5]
p(U22#) = [6] x2 + [4]
p(U23#) = [0]
p(U31#) = [4] x1 + [0]
p(U32#) = [0]
p(U41#) = [1]
p(U51#) = [1]
p(U52#) = [1] x1 + [0]
p(U61#) = [1] x1 + [1] x2 + [6] x3 + [6]
p(U62#) = [1] x2 + [6] x3 + [1]
p(U63#) = [1] x2 + [0]
p(U64#) = [1] x1 + [0]
p(activate#) = [1] x1 + [1]
p(isNat#) = [6] x1 + [0]
p(isNatKind#) = [2] x1 + [0]
p(plus#) = [4] x1 + [0]
p(s#) = [1] x1 + [1]
p(c_1) = [2]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [1] x2 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1]
p(c_8) = [1] x1 + [0]
p(c_9) = [1] x1 + [4]
p(c_10) = [1]
p(c_11) = [1] x1 + [0]
p(c_12) = [4]
p(c_13) = [0]
p(c_14) = [4] x3 + [1] x4 + [4]
p(c_15) = [1]
p(c_16) = [1] x1 + [5]
p(c_17) = [1] x1 + [1]
p(c_18) = [1] x1 + [1] x3 + [1] x5 + [2]
p(c_19) = [2] x1 + [4]
p(c_20) = [0]
p(c_21) = [2] x1 + [1]
p(c_22) = [4] x1 + [1]
p(c_23) = [4] x1 + [1]
p(c_24) = [0]
p(c_25) = [1] x1 + [0]
p(c_26) = [1] x1 + [5]
p(c_27) = [0]
p(c_28) = [2] x1 + [1] x2 + [2] x3 + [2]
p(c_29) = [4]
p(c_30) = [4]
p(c_31) = [4]
p(c_32) = [0]
p(c_33) = [0]
Following rules are strictly oriented:
U21#(tt(),V1) = [6] V1 + [5]
> [6] V1 + [4]
= c_8(U22#(isNatKind(activate(V1)),activate(V1)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [6] V2 + [0]
= c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [6] V2 + [0]
= c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [6] V2 + [0]
= c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [6] V2 + [0]
= c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) = [6] V2 + [0]
>= [6] V2 + [0]
= c_6(isNat#(activate(V2)))
U22#(tt(),V1) = [6] V1 + [4]
>= [6] V1 + [4]
= c_9(isNat#(activate(V1)))
U61#(tt(),M,N) = [1] M + [6] N + [6]
>= [1] M + [6] N + [6]
= c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) = [1] M + [6] N + [1]
>= [6] N + [1]
= c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [6] V2 + [0]
= c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) = [6] V1 + [12]
>= [6] V1 + [10]
= c_26(U21#(isNatKind(activate(V1)),activate(V1)))
0() = [1]
>= [1]
= n__0()
U11(tt(),V1,V2) = [0]
>= [0]
= U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) = [0]
>= [0]
= U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) = [0]
>= [0]
= U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) = [0]
>= [0]
= U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) = [0]
>= [0]
= U16(isNat(activate(V2)))
U16(tt()) = [0]
>= [0]
= tt()
U21(tt(),V1) = [0]
>= [0]
= U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) = [0]
>= [0]
= U23(isNat(activate(V1)))
U23(tt()) = [0]
>= [0]
= tt()
U51(tt(),N) = [1] N + [0]
>= [1] N + [0]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [1]
>= [1]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [2]
>= [1] X + [2]
= s(activate(X))
isNat(n__0()) = [1]
>= [0]
= tt()
isNat(n__plus(V1,V2)) = [1]
>= [0]
= U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) = [1]
>= [0]
= U21(isNatKind(activate(V1)),activate(V1))
plus(N,0()) = [1] N + [1]
>= [1] N + [1]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [2]
>= [1] X + [2]
= n__s(X)
** Step 6.a:9: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
- Weak DPs:
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1
,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_25) = {1},
uargs(c_26) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [4] x2 + [0]
p(U12) = [2] x1 + [7] x2 + [6]
p(U13) = [4] x3 + [0]
p(U14) = [1] x1 + [1] x3 + [1]
p(U15) = [4] x1 + [4] x2 + [2]
p(U16) = [1] x1 + [7]
p(U21) = [4] x1 + [4] x2 + [1]
p(U22) = [1] x1 + [4] x2 + [4]
p(U23) = [3] x1 + [2]
p(U31) = [3]
p(U32) = [1]
p(U41) = [4] x1 + [0]
p(U51) = [1] x2 + [2]
p(U52) = [1] x2 + [1]
p(U61) = [1] x2 + [1] x3 + [6]
p(U62) = [1] x2 + [1] x3 + [6]
p(U63) = [1] x2 + [1] x3 + [6]
p(U64) = [1] x2 + [1] x3 + [6]
p(activate) = [1] x1 + [0]
p(isNat) = [2]
p(isNatKind) = [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [2]
p(n__s) = [1] x1 + [4]
p(plus) = [1] x1 + [1] x2 + [2]
p(s) = [1] x1 + [4]
p(tt) = [0]
p(0#) = [1]
p(U11#) = [2] x2 + [2] x3 + [4]
p(U12#) = [2] x2 + [2] x3 + [2]
p(U13#) = [2] x2 + [2] x3 + [2]
p(U14#) = [2] x2 + [2] x3 + [0]
p(U15#) = [2] x2 + [0]
p(U16#) = [1] x1 + [1]
p(U21#) = [2] x2 + [4]
p(U22#) = [2] x2 + [0]
p(U23#) = [0]
p(U31#) = [1] x1 + [1] x2 + [1]
p(U32#) = [2]
p(U41#) = [4]
p(U51#) = [1] x2 + [0]
p(U52#) = [0]
p(U61#) = [1] x2 + [7] x3 + [0]
p(U62#) = [1] x2 + [7] x3 + [0]
p(U63#) = [0]
p(U64#) = [1] x3 + [0]
p(activate#) = [1]
p(isNat#) = [2] x1 + [0]
p(isNatKind#) = [1] x1 + [1]
p(plus#) = [4] x1 + [0]
p(s#) = [4] x1 + [1]
p(c_1) = [2]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [2]
p(c_5) = [1] x1 + [1] x2 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [2]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [4] x1 + [2] x2 + [0]
p(c_12) = [4]
p(c_13) = [0]
p(c_14) = [2]
p(c_15) = [2]
p(c_16) = [1] x1 + [0]
p(c_17) = [1] x1 + [0]
p(c_18) = [1] x4 + [1]
p(c_19) = [1] x2 + [2]
p(c_20) = [0]
p(c_21) = [1]
p(c_22) = [0]
p(c_23) = [1] x1 + [1]
p(c_24) = [1]
p(c_25) = [1] x1 + [0]
p(c_26) = [1] x1 + [4]
p(c_27) = [1]
p(c_28) = [1] x3 + [1] x4 + [4]
p(c_29) = [1] x1 + [1] x2 + [0]
p(c_30) = [1]
p(c_31) = [1] x1 + [0]
p(c_32) = [0]
p(c_33) = [2]
Following rules are strictly oriented:
U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [4]
> [2] V1 + [2] V2 + [2]
= c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Following rules are (at-least) weakly oriented:
U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [2]
>= [2] V1 + [2] V2 + [2]
= c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [2]
>= [2] V1 + [2] V2 + [2]
= c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [0]
>= [2] V1 + [2] V2 + [0]
= c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) = [2] V2 + [0]
>= [2] V2 + [0]
= c_6(isNat#(activate(V2)))
U21#(tt(),V1) = [2] V1 + [4]
>= [2] V1 + [2]
= c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) = [2] V1 + [0]
>= [2] V1 + [0]
= c_9(isNat#(activate(V1)))
U61#(tt(),M,N) = [1] M + [7] N + [0]
>= [1] M + [7] N + [0]
= c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) = [1] M + [7] N + [0]
>= [2] N + [0]
= c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [4]
>= [2] V1 + [2] V2 + [4]
= c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) = [2] V1 + [8]
>= [2] V1 + [8]
= c_26(U21#(isNatKind(activate(V1)),activate(V1)))
0() = [0]
>= [0]
= n__0()
U51(tt(),N) = [1] N + [2]
>= [1] N + [1]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [1]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [6]
>= [1] M + [1] N + [6]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [6]
>= [1] M + [1] N + [6]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [6]
>= [1] M + [1] N + [6]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [6]
>= [1] M + [1] N + [6]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [4]
>= [1] X + [4]
= s(activate(X))
plus(N,0()) = [1] N + [2]
>= [1] N + [2]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [6]
>= [1] M + [1] N + [6]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__plus(X1,X2)
s(X) = [1] X + [4]
>= [1] X + [4]
= n__s(X)
** Step 6.a:10: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
- Weak DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1
,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_25) = {1},
uargs(c_26) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [0]
p(U12) = [4] x1 + [1] x2 + [0]
p(U13) = [1] x2 + [1]
p(U14) = [1] x1 + [1] x2 + [6] x3 + [0]
p(U15) = [4] x1 + [4] x2 + [0]
p(U16) = [0]
p(U21) = [0]
p(U22) = [1] x2 + [2]
p(U23) = [1] x1 + [5]
p(U31) = [6]
p(U32) = [2] x1 + [1]
p(U41) = [2] x1 + [0]
p(U51) = [1] x2 + [0]
p(U52) = [1] x2 + [0]
p(U61) = [1] x2 + [1] x3 + [2]
p(U62) = [1] x2 + [1] x3 + [2]
p(U63) = [1] x2 + [1] x3 + [2]
p(U64) = [1] x2 + [1] x3 + [2]
p(activate) = [1] x1 + [0]
p(isNat) = [0]
p(isNatKind) = [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [2]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [2]
p(tt) = [0]
p(0#) = [2]
p(U11#) = [1] x2 + [1] x3 + [0]
p(U12#) = [1] x2 + [1] x3 + [0]
p(U13#) = [1] x2 + [1] x3 + [0]
p(U14#) = [1] x2 + [1] x3 + [0]
p(U15#) = [1] x2 + [0]
p(U16#) = [1] x1 + [1]
p(U21#) = [1] x2 + [1]
p(U22#) = [1] x2 + [1]
p(U23#) = [4]
p(U31#) = [1] x1 + [1]
p(U32#) = [1] x1 + [0]
p(U41#) = [0]
p(U51#) = [1] x1 + [4] x2 + [0]
p(U52#) = [1] x1 + [1] x2 + [4]
p(U61#) = [2] x3 + [6]
p(U62#) = [2] x3 + [2]
p(U63#) = [1] x2 + [2]
p(U64#) = [2] x3 + [0]
p(activate#) = [1] x1 + [1]
p(isNat#) = [1] x1 + [0]
p(isNatKind#) = [1]
p(plus#) = [1] x1 + [1] x2 + [1]
p(s#) = [4] x1 + [0]
p(c_1) = [2]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [1] x2 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1]
p(c_8) = [1] x1 + [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [4]
p(c_11) = [1] x1 + [2] x2 + [1]
p(c_12) = [2]
p(c_13) = [1]
p(c_14) = [2] x1 + [1] x3 + [1] x4 + [0]
p(c_15) = [1] x1 + [4]
p(c_16) = [1] x1 + [3]
p(c_17) = [2] x1 + [2]
p(c_18) = [1] x5 + [0]
p(c_19) = [1] x2 + [1]
p(c_20) = [1]
p(c_21) = [1]
p(c_22) = [1] x1 + [0]
p(c_23) = [1]
p(c_24) = [4]
p(c_25) = [1] x1 + [0]
p(c_26) = [1] x1 + [1]
p(c_27) = [2]
p(c_28) = [4] x1 + [1] x3 + [1] x4 + [0]
p(c_29) = [1] x2 + [4]
p(c_30) = [1] x1 + [1]
p(c_31) = [0]
p(c_32) = [0]
p(c_33) = [0]
Following rules are strictly oriented:
U22#(tt(),V1) = [1] V1 + [1]
> [1] V1 + [0]
= c_9(isNat#(activate(V1)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) = [1] V2 + [0]
>= [1] V2 + [0]
= c_6(isNat#(activate(V2)))
U21#(tt(),V1) = [1] V1 + [1]
>= [1] V1 + [1]
= c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U61#(tt(),M,N) = [2] N + [6]
>= [2] N + [5]
= c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) = [2] N + [2]
>= [2] N + [2]
= c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [0]
>= [1] V1 + [1] V2 + [0]
= c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) = [1] V1 + [2]
>= [1] V1 + [2]
= c_26(U21#(isNatKind(activate(V1)),activate(V1)))
0() = [0]
>= [0]
= n__0()
U51(tt(),N) = [1] N + [0]
>= [1] N + [0]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [2]
>= [1] X + [2]
= s(activate(X))
plus(N,0()) = [1] N + [0]
>= [1] N + [0]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [2]
>= [1] X + [2]
= n__s(X)
** Step 6.a:11: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
- Weak DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1
,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_25) = {1},
uargs(c_26) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [5]
p(U11) = [6]
p(U12) = [2] x2 + [0]
p(U13) = [4] x2 + [0]
p(U14) = [2] x2 + [1] x3 + [0]
p(U15) = [4] x2 + [3]
p(U16) = [2]
p(U21) = [0]
p(U22) = [4] x1 + [2] x2 + [4]
p(U23) = [0]
p(U31) = [1] x1 + [1] x2 + [4]
p(U32) = [2] x1 + [6]
p(U41) = [1]
p(U51) = [1] x2 + [3]
p(U52) = [1] x2 + [0]
p(U61) = [1] x2 + [1] x3 + [6]
p(U62) = [1] x2 + [1] x3 + [6]
p(U63) = [1] x2 + [1] x3 + [6]
p(U64) = [1] x2 + [1] x3 + [6]
p(activate) = [1] x1 + [0]
p(isNat) = [4]
p(isNatKind) = [0]
p(n__0) = [5]
p(n__plus) = [1] x1 + [1] x2 + [4]
p(n__s) = [1] x1 + [2]
p(plus) = [1] x1 + [1] x2 + [4]
p(s) = [1] x1 + [2]
p(tt) = [0]
p(0#) = [2]
p(U11#) = [2] x2 + [2] x3 + [6]
p(U12#) = [2] x2 + [2] x3 + [6]
p(U13#) = [2] x2 + [2] x3 + [6]
p(U14#) = [2] x2 + [2] x3 + [6]
p(U15#) = [2] x2 + [3]
p(U16#) = [1]
p(U21#) = [2] x2 + [3]
p(U22#) = [2] x2 + [3]
p(U23#) = [1] x1 + [0]
p(U31#) = [4] x1 + [1]
p(U32#) = [1] x1 + [1]
p(U41#) = [1] x1 + [0]
p(U51#) = [4]
p(U52#) = [1]
p(U61#) = [4] x2 + [4] x3 + [7]
p(U62#) = [4] x3 + [3]
p(U63#) = [2] x2 + [4] x3 + [1]
p(U64#) = [2] x2 + [2]
p(activate#) = [1]
p(isNat#) = [2] x1 + [3]
p(isNatKind#) = [2]
p(plus#) = [4] x1 + [1]
p(s#) = [1] x1 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [1] x2 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1]
p(c_8) = [1] x1 + [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [1]
p(c_11) = [2] x1 + [1]
p(c_12) = [0]
p(c_13) = [2]
p(c_14) = [2] x4 + [0]
p(c_15) = [2] x1 + [4]
p(c_16) = [1] x1 + [4]
p(c_17) = [1] x1 + [0]
p(c_18) = [1] x3 + [1] x4 + [0]
p(c_19) = [2]
p(c_20) = [4]
p(c_21) = [0]
p(c_22) = [2] x1 + [0]
p(c_23) = [1] x1 + [0]
p(c_24) = [1]
p(c_25) = [1] x1 + [4]
p(c_26) = [1] x1 + [1]
p(c_27) = [1]
p(c_28) = [1] x1 + [0]
p(c_29) = [1] x1 + [1] x2 + [4]
p(c_30) = [0]
p(c_31) = [2] x1 + [1] x2 + [0]
p(c_32) = [2]
p(c_33) = [1]
Following rules are strictly oriented:
isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [11]
> [2] V1 + [2] V2 + [10]
= c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [6]
>= [2] V1 + [2] V2 + [6]
= c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [6]
>= [2] V1 + [2] V2 + [6]
= c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [6]
>= [2] V1 + [2] V2 + [6]
= c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [6]
>= [2] V1 + [2] V2 + [6]
= c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) = [2] V2 + [3]
>= [2] V2 + [3]
= c_6(isNat#(activate(V2)))
U21#(tt(),V1) = [2] V1 + [3]
>= [2] V1 + [3]
= c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) = [2] V1 + [3]
>= [2] V1 + [3]
= c_9(isNat#(activate(V1)))
U61#(tt(),M,N) = [4] M + [4] N + [7]
>= [4] N + [7]
= c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) = [4] N + [3]
>= [2] N + [3]
= c_17(isNat#(activate(N)))
isNat#(n__s(V1)) = [2] V1 + [7]
>= [2] V1 + [4]
= c_26(U21#(isNatKind(activate(V1)),activate(V1)))
0() = [5]
>= [5]
= n__0()
U51(tt(),N) = [1] N + [3]
>= [1] N + [0]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [6]
>= [1] M + [1] N + [6]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [6]
>= [1] M + [1] N + [6]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [6]
>= [1] M + [1] N + [6]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [6]
>= [1] M + [1] N + [6]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [5]
>= [5]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [2]
>= [1] X + [2]
= s(activate(X))
plus(N,0()) = [1] N + [9]
>= [1] N + [3]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [6]
>= [1] M + [1] N + [6]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [4]
>= [1] X1 + [1] X2 + [4]
= n__plus(X1,X2)
s(X) = [1] X + [2]
>= [1] X + [2]
= n__s(X)
** Step 6.a:12: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
- Weak DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1
,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_25) = {1},
uargs(c_26) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x1 + [4] x2 + [1] x3 + [2]
p(U12) = [1] x2 + [2] x3 + [0]
p(U13) = [1] x1 + [7]
p(U14) = [2] x3 + [4]
p(U15) = [1] x1 + [1] x2 + [4]
p(U16) = [4] x1 + [4]
p(U21) = [2] x1 + [2] x2 + [4]
p(U22) = [2] x1 + [3]
p(U23) = [4] x1 + [3]
p(U31) = [1] x1 + [5]
p(U32) = [0]
p(U41) = [4] x1 + [0]
p(U51) = [1] x2 + [0]
p(U52) = [1] x2 + [0]
p(U61) = [1] x2 + [1] x3 + [3]
p(U62) = [1] x2 + [1] x3 + [3]
p(U63) = [1] x2 + [1] x3 + [3]
p(U64) = [1] x2 + [1] x3 + [3]
p(activate) = [1] x1 + [0]
p(isNat) = [2] x1 + [0]
p(isNatKind) = [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [2]
p(n__s) = [1] x1 + [1]
p(plus) = [1] x1 + [1] x2 + [2]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(0#) = [0]
p(U11#) = [2] x2 + [2] x3 + [2]
p(U12#) = [2] x2 + [2] x3 + [2]
p(U13#) = [2] x2 + [2] x3 + [2]
p(U14#) = [2] x2 + [2] x3 + [2]
p(U15#) = [2] x2 + [1]
p(U16#) = [2]
p(U21#) = [2] x2 + [2]
p(U22#) = [2] x2 + [2]
p(U23#) = [1]
p(U31#) = [1] x1 + [1] x2 + [0]
p(U32#) = [4] x1 + [1]
p(U41#) = [2]
p(U51#) = [1] x1 + [1]
p(U52#) = [1] x2 + [0]
p(U61#) = [1] x2 + [4] x3 + [5]
p(U62#) = [4] x3 + [1]
p(U63#) = [1] x1 + [1]
p(U64#) = [2] x1 + [1] x2 + [0]
p(activate#) = [2] x1 + [0]
p(isNat#) = [2] x1 + [0]
p(isNatKind#) = [2]
p(plus#) = [1]
p(s#) = [1] x1 + [4]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [1] x2 + [1]
p(c_6) = [1] x1 + [0]
p(c_7) = [2]
p(c_8) = [1] x1 + [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [2]
p(c_12) = [0]
p(c_13) = [1]
p(c_14) = [1] x2 + [1] x3 + [2]
p(c_15) = [0]
p(c_16) = [1] x1 + [0]
p(c_17) = [2] x1 + [0]
p(c_18) = [2] x1 + [0]
p(c_19) = [1] x1 + [0]
p(c_20) = [1]
p(c_21) = [1]
p(c_22) = [4] x1 + [2] x2 + [1]
p(c_23) = [1] x1 + [1]
p(c_24) = [4]
p(c_25) = [1] x1 + [2]
p(c_26) = [1] x1 + [0]
p(c_27) = [1]
p(c_28) = [1] x1 + [2] x2 + [1] x3 + [1] x4 + [0]
p(c_29) = [1]
p(c_30) = [2] x2 + [0]
p(c_31) = [1]
p(c_32) = [0]
p(c_33) = [1]
Following rules are strictly oriented:
U15#(tt(),V2) = [2] V2 + [1]
> [2] V2 + [0]
= c_6(isNat#(activate(V2)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [2]
>= [2] V1 + [2] V2 + [2]
= c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [2]
>= [2] V1 + [2] V2 + [2]
= c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [2]
>= [2] V1 + [2] V2 + [2]
= c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [2]
>= [2] V1 + [2] V2 + [2]
= c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U21#(tt(),V1) = [2] V1 + [2]
>= [2] V1 + [2]
= c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) = [2] V1 + [2]
>= [2] V1 + [0]
= c_9(isNat#(activate(V1)))
U61#(tt(),M,N) = [1] M + [4] N + [5]
>= [4] N + [1]
= c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) = [4] N + [1]
>= [4] N + [0]
= c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [4]
>= [2] V1 + [2] V2 + [4]
= c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) = [2] V1 + [2]
>= [2] V1 + [2]
= c_26(U21#(isNatKind(activate(V1)),activate(V1)))
0() = [0]
>= [0]
= n__0()
U51(tt(),N) = [1] N + [0]
>= [1] N + [0]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [1]
>= [1] X + [1]
= s(activate(X))
plus(N,0()) = [1] N + [2]
>= [1] N + [0]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__plus(X1,X2)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
** Step 6.a:13: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
- Weak DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1
,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_25) = {1},
uargs(c_26) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [1] x1 + [1] x2 + [4] x3 + [0]
p(U12) = [1] x2 + [1] x3 + [0]
p(U13) = [2] x2 + [1] x3 + [0]
p(U14) = [0]
p(U15) = [4] x1 + [0]
p(U16) = [1] x1 + [7]
p(U21) = [4] x1 + [4] x2 + [0]
p(U22) = [2] x1 + [5]
p(U23) = [5] x1 + [0]
p(U31) = [6] x2 + [0]
p(U32) = [1] x1 + [6]
p(U41) = [1]
p(U51) = [1] x2 + [2]
p(U52) = [1] x2 + [2]
p(U61) = [1] x2 + [1] x3 + [3]
p(U62) = [1] x2 + [1] x3 + [3]
p(U63) = [1] x2 + [1] x3 + [3]
p(U64) = [1] x2 + [1] x3 + [3]
p(activate) = [1] x1 + [0]
p(isNat) = [1] x1 + [3]
p(isNatKind) = [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [2]
p(n__s) = [1] x1 + [1]
p(plus) = [1] x1 + [1] x2 + [2]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(0#) = [1]
p(U11#) = [4] x2 + [4] x3 + [6]
p(U12#) = [4] x2 + [4] x3 + [6]
p(U13#) = [4] x2 + [4] x3 + [6]
p(U14#) = [4] x2 + [4] x3 + [4]
p(U15#) = [4] x2 + [2]
p(U16#) = [1] x1 + [0]
p(U21#) = [4] x2 + [1]
p(U22#) = [4] x2 + [1]
p(U23#) = [1]
p(U31#) = [4] x1 + [1]
p(U32#) = [0]
p(U41#) = [1] x1 + [1]
p(U51#) = [2] x1 + [0]
p(U52#) = [1] x1 + [1] x2 + [4]
p(U61#) = [1] x1 + [4] x3 + [0]
p(U62#) = [4] x3 + [0]
p(U63#) = [4] x1 + [1]
p(U64#) = [1] x3 + [1]
p(activate#) = [1]
p(isNat#) = [4] x1 + [0]
p(isNatKind#) = [1]
p(plus#) = [4] x1 + [1] x2 + [1]
p(s#) = [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [1] x2 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [4]
p(c_8) = [1] x1 + [0]
p(c_9) = [1] x1 + [1]
p(c_10) = [1]
p(c_11) = [1]
p(c_12) = [0]
p(c_13) = [4]
p(c_14) = [1] x2 + [4] x3 + [4] x4 + [0]
p(c_15) = [1] x1 + [1]
p(c_16) = [1] x1 + [0]
p(c_17) = [1] x1 + [0]
p(c_18) = [1] x3 + [1] x4 + [1] x5 + [1]
p(c_19) = [4] x1 + [1]
p(c_20) = [0]
p(c_21) = [4] x1 + [1]
p(c_22) = [1] x1 + [1]
p(c_23) = [1] x1 + [4]
p(c_24) = [1]
p(c_25) = [1] x1 + [2]
p(c_26) = [1] x1 + [0]
p(c_27) = [0]
p(c_28) = [2] x3 + [1] x4 + [1]
p(c_29) = [2] x1 + [2]
p(c_30) = [2] x2 + [0]
p(c_31) = [2]
p(c_32) = [1]
p(c_33) = [1]
Following rules are strictly oriented:
U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [6]
> [4] V1 + [4] V2 + [4]
= c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [4]
> [4] V1 + [4] V2 + [2]
= c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [6]
>= [4] V1 + [4] V2 + [6]
= c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [6]
>= [4] V1 + [4] V2 + [6]
= c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U15#(tt(),V2) = [4] V2 + [2]
>= [4] V2 + [0]
= c_6(isNat#(activate(V2)))
U21#(tt(),V1) = [4] V1 + [1]
>= [4] V1 + [1]
= c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) = [4] V1 + [1]
>= [4] V1 + [1]
= c_9(isNat#(activate(V1)))
U61#(tt(),M,N) = [4] N + [0]
>= [4] N + [0]
= c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) = [4] N + [0]
>= [4] N + [0]
= c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [4] V2 + [8]
= c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [1]
= c_26(U21#(isNatKind(activate(V1)),activate(V1)))
0() = [0]
>= [0]
= n__0()
U51(tt(),N) = [1] N + [2]
>= [1] N + [2]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [2]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [1]
>= [1] X + [1]
= s(activate(X))
plus(N,0()) = [1] N + [2]
>= [1] N + [2]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__plus(X1,X2)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
** Step 6.a:14: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
- Weak DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1
,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1,2},
uargs(c_6) = {1},
uargs(c_8) = {1},
uargs(c_9) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_25) = {1},
uargs(c_26) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [1]
p(U11) = [3] x3 + [4]
p(U12) = [4] x1 + [1] x2 + [1] x3 + [0]
p(U13) = [2] x2 + [1] x3 + [3]
p(U14) = [2] x3 + [0]
p(U15) = [2] x2 + [4]
p(U16) = [2]
p(U21) = [0]
p(U22) = [2] x1 + [1]
p(U23) = [5]
p(U31) = [2]
p(U32) = [0]
p(U41) = [0]
p(U51) = [1] x2 + [2]
p(U52) = [1] x2 + [2]
p(U61) = [1] x2 + [1] x3 + [2]
p(U62) = [1] x2 + [1] x3 + [2]
p(U63) = [1] x2 + [1] x3 + [2]
p(U64) = [1] x2 + [1] x3 + [2]
p(activate) = [1] x1 + [0]
p(isNat) = [0]
p(isNatKind) = [0]
p(n__0) = [1]
p(n__plus) = [1] x1 + [1] x2 + [2]
p(n__s) = [1] x1 + [0]
p(plus) = [1] x1 + [1] x2 + [2]
p(s) = [1] x1 + [0]
p(tt) = [0]
p(0#) = [4]
p(U11#) = [4] x2 + [4] x3 + [1]
p(U12#) = [4] x2 + [4] x3 + [1]
p(U13#) = [4] x2 + [4] x3 + [0]
p(U14#) = [4] x2 + [4] x3 + [0]
p(U15#) = [4] x2 + [0]
p(U16#) = [1] x1 + [0]
p(U21#) = [4] x2 + [0]
p(U22#) = [4] x2 + [0]
p(U23#) = [0]
p(U31#) = [1] x2 + [4]
p(U32#) = [4]
p(U41#) = [4]
p(U51#) = [1] x2 + [1]
p(U52#) = [0]
p(U61#) = [4] x2 + [4] x3 + [5]
p(U62#) = [2] x2 + [4] x3 + [3]
p(U63#) = [1] x2 + [1]
p(U64#) = [1] x3 + [1]
p(activate#) = [4] x1 + [1]
p(isNat#) = [4] x1 + [0]
p(isNatKind#) = [1]
p(plus#) = [0]
p(s#) = [1]
p(c_1) = [2]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [1] x1 + [1] x2 + [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [1]
p(c_11) = [2]
p(c_12) = [0]
p(c_13) = [0]
p(c_14) = [2] x1 + [1] x4 + [1]
p(c_15) = [0]
p(c_16) = [1] x1 + [1]
p(c_17) = [1] x1 + [3]
p(c_18) = [1] x1 + [4]
p(c_19) = [1] x1 + [1] x2 + [0]
p(c_20) = [1]
p(c_21) = [0]
p(c_22) = [2]
p(c_23) = [4]
p(c_24) = [2]
p(c_25) = [1] x1 + [7]
p(c_26) = [1] x1 + [0]
p(c_27) = [0]
p(c_28) = [1]
p(c_29) = [1]
p(c_30) = [0]
p(c_31) = [2] x1 + [4]
p(c_32) = [1]
p(c_33) = [4]
Following rules are strictly oriented:
U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [1]
> [4] V1 + [4] V2 + [0]
= c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [1]
>= [4] V1 + [4] V2 + [1]
= c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= c_6(isNat#(activate(V2)))
U21#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= c_9(isNat#(activate(V1)))
U61#(tt(),M,N) = [4] M + [4] N + [5]
>= [2] M + [4] N + [4]
= c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) = [2] M + [4] N + [3]
>= [4] N + [3]
= c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [4] V2 + [8]
= c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) = [4] V1 + [0]
>= [4] V1 + [0]
= c_26(U21#(isNatKind(activate(V1)),activate(V1)))
0() = [1]
>= [1]
= n__0()
U51(tt(),N) = [1] N + [2]
>= [1] N + [2]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [2]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [1]
>= [1]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(activate(X))
plus(N,0()) = [1] N + [3]
>= [1] N + [2]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__plus(X1,X2)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
** Step 6.a:15: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
U11#(tt(),V1,V2) -> c_2(U12#(isNatKind(activate(V1)),activate(V1),activate(V2)))
U12#(tt(),V1,V2) -> c_3(U13#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U13#(tt(),V1,V2) -> c_4(U14#(isNatKind(activate(V2)),activate(V1),activate(V2)))
U14#(tt(),V1,V2) -> c_5(U15#(isNat(activate(V1)),activate(V2)),isNat#(activate(V1)))
U15#(tt(),V2) -> c_6(isNat#(activate(V2)))
U21#(tt(),V1) -> c_8(U22#(isNatKind(activate(V1)),activate(V1)))
U22#(tt(),V1) -> c_9(isNat#(activate(V1)))
U61#(tt(),M,N) -> c_16(U62#(isNatKind(activate(M)),activate(M),activate(N)))
U62#(tt(),M,N) -> c_17(isNat#(activate(N)))
isNat#(n__plus(V1,V2)) -> c_25(U11#(isNatKind(activate(V1)),activate(V1),activate(V2)))
isNat#(n__s(V1)) -> c_26(U21#(isNatKind(activate(V1)),activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/1,c_3/1,c_4/1,c_5/2,c_6/1,c_7/0,c_8/1
,c_9/1,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/1,c_17/1,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/1,c_26/1,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 6.b:1: DecomposeDG WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2))
activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(activate#(X))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> activate#(N)
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> activate#(M)
U61#(tt(),M,N) -> activate#(N)
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> activate#(N)
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> activate#(M)
U61#(tt(),M,N) -> activate#(N)
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1))
and a lower component
activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(activate#(X))
Further, following extension rules are added to the lower component.
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNatKind#(activate(V2))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> activate#(N)
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> activate#(M)
U61#(tt(),M,N) -> activate#(N)
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
isNatKind#(n__plus(V1,V2)) -> activate#(V1)
isNatKind#(n__plus(V1,V2)) -> activate#(V2)
isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNatKind#(n__s(V1)) -> activate#(V1)
isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
*** Step 6.b:1.a:1: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U15#(tt(),V2) -> activate#(V2)
U21#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> activate#(V1)
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2))
U51#(tt(),N) -> activate#(N)
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> activate#(M)
U61#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__s(V1)) -> activate#(V1)
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> isNat#(activate(V1))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> isNatKind#(activate(N))
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{13,15,16}
by application of
Pre({13,15,16}) = {}.
Here rules are labelled as follows:
1: U11#(tt(),V1,V2) -> activate#(V1)
2: U11#(tt(),V1,V2) -> activate#(V2)
3: U12#(tt(),V1,V2) -> activate#(V1)
4: U12#(tt(),V1,V2) -> activate#(V2)
5: U13#(tt(),V1,V2) -> activate#(V1)
6: U13#(tt(),V1,V2) -> activate#(V2)
7: U14#(tt(),V1,V2) -> activate#(V1)
8: U14#(tt(),V1,V2) -> activate#(V2)
9: U15#(tt(),V2) -> activate#(V2)
10: U21#(tt(),V1) -> activate#(V1)
11: U22#(tt(),V1) -> activate#(V1)
12: U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2))
13: U51#(tt(),N) -> activate#(N)
14: U52#(tt(),N) -> activate#(N)
15: U61#(tt(),M,N) -> activate#(M)
16: U61#(tt(),M,N) -> activate#(N)
17: U62#(tt(),M,N) -> activate#(M)
18: U62#(tt(),M,N) -> activate#(N)
19: U63#(tt(),M,N) -> activate#(M)
20: U63#(tt(),M,N) -> activate#(N)
21: U64#(tt(),M,N) -> activate#(M)
22: U64#(tt(),M,N) -> activate#(N)
23: isNat#(n__plus(V1,V2)) -> activate#(V1)
24: isNat#(n__plus(V1,V2)) -> activate#(V2)
25: isNat#(n__s(V1)) -> activate#(V1)
26: isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
27: isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1))
28: U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
29: U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
30: U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
31: U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
32: U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
33: U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
34: U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
35: U14#(tt(),V1,V2) -> isNat#(activate(V1))
36: U15#(tt(),V2) -> isNat#(activate(V2))
37: U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
38: U21#(tt(),V1) -> isNatKind#(activate(V1))
39: U22#(tt(),V1) -> isNat#(activate(V1))
40: U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
41: U51#(tt(),N) -> isNatKind#(activate(N))
42: U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
43: U61#(tt(),M,N) -> isNatKind#(activate(M))
44: U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
45: U62#(tt(),M,N) -> isNat#(activate(N))
46: U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
47: U63#(tt(),M,N) -> isNatKind#(activate(N))
48: isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
49: isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
50: isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
51: isNat#(n__s(V1)) -> isNatKind#(activate(V1))
*** Step 6.b:1.a:2: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U15#(tt(),V2) -> activate#(V2)
U21#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> activate#(V1)
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2))
U52#(tt(),N) -> activate#(N)
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__s(V1)) -> activate#(V1)
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> isNat#(activate(V1))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> activate#(N)
U51#(tt(),N) -> isNatKind#(activate(N))
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> activate#(M)
U61#(tt(),M,N) -> activate#(N)
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> isNatKind#(activate(N))
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:U11#(tt(),V1,V2) -> activate#(V1)
2:S:U11#(tt(),V1,V2) -> activate#(V2)
3:S:U12#(tt(),V1,V2) -> activate#(V1)
4:S:U12#(tt(),V1,V2) -> activate#(V2)
5:S:U13#(tt(),V1,V2) -> activate#(V1)
6:S:U13#(tt(),V1,V2) -> activate#(V2)
7:S:U14#(tt(),V1,V2) -> activate#(V1)
8:S:U14#(tt(),V1,V2) -> activate#(V2)
9:S:U15#(tt(),V2) -> activate#(V2)
10:S:U21#(tt(),V1) -> activate#(V1)
11:S:U22#(tt(),V1) -> activate#(V1)
12:S:U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
13:S:U52#(tt(),N) -> activate#(N)
14:S:U62#(tt(),M,N) -> activate#(M)
15:S:U62#(tt(),M,N) -> activate#(N)
16:S:U63#(tt(),M,N) -> activate#(M)
17:S:U63#(tt(),M,N) -> activate#(N)
18:S:U64#(tt(),M,N) -> activate#(M)
19:S:U64#(tt(),M,N) -> activate#(N)
20:S:isNat#(n__plus(V1,V2)) -> activate#(V1)
21:S:isNat#(n__plus(V1,V2)) -> activate#(V2)
22:S:isNat#(n__s(V1)) -> activate#(V1)
23:S:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
-->_1 U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)):12
24:S:isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
25:W:U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
-->_1 U12#(tt(),V1,V2) -> isNatKind#(activate(V2)):28
-->_1 U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)):27
-->_1 U12#(tt(),V1,V2) -> activate#(V2):4
-->_1 U12#(tt(),V1,V2) -> activate#(V1):3
26:W:U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
27:W:U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
-->_1 U13#(tt(),V1,V2) -> isNatKind#(activate(V2)):30
-->_1 U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)):29
-->_1 U13#(tt(),V1,V2) -> activate#(V2):6
-->_1 U13#(tt(),V1,V2) -> activate#(V1):5
28:W:U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
29:W:U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
-->_1 U14#(tt(),V1,V2) -> isNat#(activate(V1)):32
-->_1 U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)):31
-->_1 U14#(tt(),V1,V2) -> activate#(V2):8
-->_1 U14#(tt(),V1,V2) -> activate#(V1):7
30:W:U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
31:W:U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
-->_1 U15#(tt(),V2) -> isNat#(activate(V2)):33
-->_1 U15#(tt(),V2) -> activate#(V2):9
32:W:U14#(tt(),V1,V2) -> isNat#(activate(V1))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48
-->_1 isNat#(n__s(V1)) -> activate#(V1):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20
33:W:U15#(tt(),V2) -> isNat#(activate(V2))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48
-->_1 isNat#(n__s(V1)) -> activate#(V1):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20
34:W:U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
-->_1 U22#(tt(),V1) -> isNat#(activate(V1)):36
-->_1 U22#(tt(),V1) -> activate#(V1):11
35:W:U21#(tt(),V1) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
36:W:U22#(tt(),V1) -> isNat#(activate(V1))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48
-->_1 isNat#(n__s(V1)) -> activate#(V1):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20
37:W:U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
-->_1 U52#(tt(),N) -> activate#(N):13
38:W:U51#(tt(),N) -> activate#(N)
39:W:U51#(tt(),N) -> isNatKind#(activate(N))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
40:W:U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
-->_1 U62#(tt(),M,N) -> isNat#(activate(N)):45
-->_1 U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)):44
-->_1 U62#(tt(),M,N) -> activate#(N):15
-->_1 U62#(tt(),M,N) -> activate#(M):14
41:W:U61#(tt(),M,N) -> activate#(M)
42:W:U61#(tt(),M,N) -> activate#(N)
43:W:U61#(tt(),M,N) -> isNatKind#(activate(M))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
44:W:U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
-->_1 U63#(tt(),M,N) -> isNatKind#(activate(N)):47
-->_1 U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)):46
-->_1 U63#(tt(),M,N) -> activate#(N):17
-->_1 U63#(tt(),M,N) -> activate#(M):16
45:W:U62#(tt(),M,N) -> isNat#(activate(N))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48
-->_1 isNat#(n__s(V1)) -> activate#(V1):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20
46:W:U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
-->_1 U64#(tt(),M,N) -> activate#(N):19
-->_1 U64#(tt(),M,N) -> activate#(M):18
47:W:U63#(tt(),M,N) -> isNatKind#(activate(N))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
48:W:isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
-->_1 U11#(tt(),V1,V2) -> isNatKind#(activate(V1)):26
-->_1 U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)):25
-->_1 U11#(tt(),V1,V2) -> activate#(V2):2
-->_1 U11#(tt(),V1,V2) -> activate#(V1):1
49:W:isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
50:W:isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
-->_1 U21#(tt(),V1) -> isNatKind#(activate(V1)):35
-->_1 U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)):34
-->_1 U21#(tt(),V1) -> activate#(V1):10
51:W:isNat#(n__s(V1)) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
42: U61#(tt(),M,N) -> activate#(N)
41: U61#(tt(),M,N) -> activate#(M)
38: U51#(tt(),N) -> activate#(N)
*** Step 6.b:1.a:3: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U15#(tt(),V2) -> activate#(V2)
U21#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> activate#(V1)
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2))
U52#(tt(),N) -> activate#(N)
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__s(V1)) -> activate#(V1)
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> isNat#(activate(V1))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> isNatKind#(activate(N))
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:U11#(tt(),V1,V2) -> activate#(V1)
2:S:U11#(tt(),V1,V2) -> activate#(V2)
3:S:U12#(tt(),V1,V2) -> activate#(V1)
4:S:U12#(tt(),V1,V2) -> activate#(V2)
5:S:U13#(tt(),V1,V2) -> activate#(V1)
6:S:U13#(tt(),V1,V2) -> activate#(V2)
7:S:U14#(tt(),V1,V2) -> activate#(V1)
8:S:U14#(tt(),V1,V2) -> activate#(V2)
9:S:U15#(tt(),V2) -> activate#(V2)
10:S:U21#(tt(),V1) -> activate#(V1)
11:S:U22#(tt(),V1) -> activate#(V1)
12:S:U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
13:S:U52#(tt(),N) -> activate#(N)
14:S:U62#(tt(),M,N) -> activate#(M)
15:S:U62#(tt(),M,N) -> activate#(N)
16:S:U63#(tt(),M,N) -> activate#(M)
17:S:U63#(tt(),M,N) -> activate#(N)
18:S:U64#(tt(),M,N) -> activate#(M)
19:S:U64#(tt(),M,N) -> activate#(N)
20:S:isNat#(n__plus(V1,V2)) -> activate#(V1)
21:S:isNat#(n__plus(V1,V2)) -> activate#(V2)
22:S:isNat#(n__s(V1)) -> activate#(V1)
23:S:isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2))
-->_2 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_2 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
-->_1 U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)),activate#(V2)):12
24:S:isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
25:W:U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
-->_1 U12#(tt(),V1,V2) -> isNatKind#(activate(V2)):28
-->_1 U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)):27
-->_1 U12#(tt(),V1,V2) -> activate#(V2):4
-->_1 U12#(tt(),V1,V2) -> activate#(V1):3
26:W:U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
27:W:U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
-->_1 U13#(tt(),V1,V2) -> isNatKind#(activate(V2)):30
-->_1 U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)):29
-->_1 U13#(tt(),V1,V2) -> activate#(V2):6
-->_1 U13#(tt(),V1,V2) -> activate#(V1):5
28:W:U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
29:W:U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
-->_1 U14#(tt(),V1,V2) -> isNat#(activate(V1)):32
-->_1 U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)):31
-->_1 U14#(tt(),V1,V2) -> activate#(V2):8
-->_1 U14#(tt(),V1,V2) -> activate#(V1):7
30:W:U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
31:W:U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
-->_1 U15#(tt(),V2) -> isNat#(activate(V2)):33
-->_1 U15#(tt(),V2) -> activate#(V2):9
32:W:U14#(tt(),V1,V2) -> isNat#(activate(V1))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48
-->_1 isNat#(n__s(V1)) -> activate#(V1):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20
33:W:U15#(tt(),V2) -> isNat#(activate(V2))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48
-->_1 isNat#(n__s(V1)) -> activate#(V1):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20
34:W:U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
-->_1 U22#(tt(),V1) -> isNat#(activate(V1)):36
-->_1 U22#(tt(),V1) -> activate#(V1):11
35:W:U21#(tt(),V1) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
36:W:U22#(tt(),V1) -> isNat#(activate(V1))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48
-->_1 isNat#(n__s(V1)) -> activate#(V1):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20
37:W:U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
-->_1 U52#(tt(),N) -> activate#(N):13
39:W:U51#(tt(),N) -> isNatKind#(activate(N))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
40:W:U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
-->_1 U62#(tt(),M,N) -> isNat#(activate(N)):45
-->_1 U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)):44
-->_1 U62#(tt(),M,N) -> activate#(N):15
-->_1 U62#(tt(),M,N) -> activate#(M):14
43:W:U61#(tt(),M,N) -> isNatKind#(activate(M))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
44:W:U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
-->_1 U63#(tt(),M,N) -> isNatKind#(activate(N)):47
-->_1 U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)):46
-->_1 U63#(tt(),M,N) -> activate#(N):17
-->_1 U63#(tt(),M,N) -> activate#(M):16
45:W:U62#(tt(),M,N) -> isNat#(activate(N))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):48
-->_1 isNat#(n__s(V1)) -> activate#(V1):22
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):21
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):20
46:W:U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
-->_1 U64#(tt(),M,N) -> activate#(N):19
-->_1 U64#(tt(),M,N) -> activate#(M):18
47:W:U63#(tt(),M,N) -> isNatKind#(activate(N))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
48:W:isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
-->_1 U11#(tt(),V1,V2) -> isNatKind#(activate(V1)):26
-->_1 U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)):25
-->_1 U11#(tt(),V1,V2) -> activate#(V2):2
-->_1 U11#(tt(),V1,V2) -> activate#(V1):1
49:W:isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
50:W:isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
-->_1 U21#(tt(),V1) -> isNatKind#(activate(V1)):35
-->_1 U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)):34
-->_1 U21#(tt(),V1) -> activate#(V1):10
51:W:isNat#(n__s(V1)) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)),activate#(V1)):24
-->_1 isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2))
,isNatKind#(activate(V1))
,activate#(V1)
,activate#(V2)):23
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)))
*** Step 6.b:1.a:4: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U15#(tt(),V2) -> activate#(V2)
U21#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> activate#(V1)
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)))
U52#(tt(),N) -> activate#(N)
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__s(V1)) -> activate#(V1)
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> isNat#(activate(V1))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> isNatKind#(activate(N))
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_11) = {1},
uargs(c_28) = {1,2},
uargs(c_29) = {1}
Following symbols are considered usable:
{0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#
,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [4] x2 + [0]
p(U12) = [0]
p(U13) = [0]
p(U14) = [0]
p(U15) = [0]
p(U16) = [0]
p(U21) = [1] x2 + [1]
p(U22) = [0]
p(U23) = [1] x1 + [0]
p(U31) = [0]
p(U32) = [3]
p(U41) = [0]
p(U51) = [2] x1 + [0]
p(U52) = [0]
p(U61) = [5]
p(U62) = [0]
p(U63) = [4] x3 + [0]
p(U64) = [0]
p(activate) = [1]
p(isNat) = [2] x1 + [6]
p(isNatKind) = [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1]
p(plus) = [2]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(0#) = [0]
p(U11#) = [0]
p(U12#) = [0]
p(U13#) = [0]
p(U14#) = [0]
p(U15#) = [0]
p(U16#) = [0]
p(U21#) = [0]
p(U22#) = [0]
p(U23#) = [0]
p(U31#) = [0]
p(U32#) = [0]
p(U41#) = [0]
p(U51#) = [0]
p(U52#) = [0]
p(U61#) = [5]
p(U62#) = [4]
p(U63#) = [4]
p(U64#) = [0]
p(activate#) = [0]
p(isNat#) = [0]
p(isNatKind#) = [0]
p(plus#) = [0]
p(s#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [2]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [0]
p(c_11) = [2] x1 + [0]
p(c_12) = [0]
p(c_13) = [0]
p(c_14) = [0]
p(c_15) = [0]
p(c_16) = [0]
p(c_17) = [0]
p(c_18) = [0]
p(c_19) = [0]
p(c_20) = [0]
p(c_21) = [0]
p(c_22) = [0]
p(c_23) = [0]
p(c_24) = [0]
p(c_25) = [0]
p(c_26) = [1] x4 + [0]
p(c_27) = [0]
p(c_28) = [4] x1 + [1] x2 + [0]
p(c_29) = [1] x1 + [0]
p(c_30) = [1] x1 + [0]
p(c_31) = [2] x2 + [0]
p(c_32) = [1]
p(c_33) = [0]
Following rules are strictly oriented:
U62#(tt(),M,N) = [4]
> [0]
= activate#(M)
U62#(tt(),M,N) = [4]
> [0]
= activate#(N)
U63#(tt(),M,N) = [4]
> [0]
= activate#(M)
U63#(tt(),M,N) = [4]
> [0]
= activate#(N)
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [0]
>= [0]
= U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) = [0]
>= [0]
= activate#(V1)
U11#(tt(),V1,V2) = [0]
>= [0]
= activate#(V2)
U11#(tt(),V1,V2) = [0]
>= [0]
= isNatKind#(activate(V1))
U12#(tt(),V1,V2) = [0]
>= [0]
= U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) = [0]
>= [0]
= activate#(V1)
U12#(tt(),V1,V2) = [0]
>= [0]
= activate#(V2)
U12#(tt(),V1,V2) = [0]
>= [0]
= isNatKind#(activate(V2))
U13#(tt(),V1,V2) = [0]
>= [0]
= U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) = [0]
>= [0]
= activate#(V1)
U13#(tt(),V1,V2) = [0]
>= [0]
= activate#(V2)
U13#(tt(),V1,V2) = [0]
>= [0]
= isNatKind#(activate(V2))
U14#(tt(),V1,V2) = [0]
>= [0]
= U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) = [0]
>= [0]
= activate#(V1)
U14#(tt(),V1,V2) = [0]
>= [0]
= activate#(V2)
U14#(tt(),V1,V2) = [0]
>= [0]
= isNat#(activate(V1))
U15#(tt(),V2) = [0]
>= [0]
= activate#(V2)
U15#(tt(),V2) = [0]
>= [0]
= isNat#(activate(V2))
U21#(tt(),V1) = [0]
>= [0]
= U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) = [0]
>= [0]
= activate#(V1)
U21#(tt(),V1) = [0]
>= [0]
= isNatKind#(activate(V1))
U22#(tt(),V1) = [0]
>= [0]
= activate#(V1)
U22#(tt(),V1) = [0]
>= [0]
= isNat#(activate(V1))
U31#(tt(),V2) = [0]
>= [0]
= c_11(isNatKind#(activate(V2)))
U51#(tt(),N) = [0]
>= [0]
= U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) = [0]
>= [0]
= isNatKind#(activate(N))
U52#(tt(),N) = [0]
>= [0]
= activate#(N)
U61#(tt(),M,N) = [5]
>= [4]
= U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) = [5]
>= [0]
= isNatKind#(activate(M))
U62#(tt(),M,N) = [4]
>= [4]
= U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) = [4]
>= [0]
= isNat#(activate(N))
U63#(tt(),M,N) = [4]
>= [0]
= U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) = [4]
>= [0]
= isNatKind#(activate(N))
U64#(tt(),M,N) = [0]
>= [0]
= activate#(M)
U64#(tt(),M,N) = [0]
>= [0]
= activate#(N)
isNat#(n__plus(V1,V2)) = [0]
>= [0]
= U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) = [0]
>= [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [0]
>= [0]
= activate#(V2)
isNat#(n__plus(V1,V2)) = [0]
>= [0]
= isNatKind#(activate(V1))
isNat#(n__s(V1)) = [0]
>= [0]
= U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) = [0]
>= [0]
= activate#(V1)
isNat#(n__s(V1)) = [0]
>= [0]
= isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) = [0]
>= [0]
= c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) = [0]
>= [0]
= c_29(isNatKind#(activate(V1)))
*** Step 6.b:1.a:5: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U15#(tt(),V2) -> activate#(V2)
U21#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> activate#(V1)
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)))
U52#(tt(),N) -> activate#(N)
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__s(V1)) -> activate#(V1)
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> isNat#(activate(V1))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_11) = {1},
uargs(c_28) = {1,2},
uargs(c_29) = {1}
Following symbols are considered usable:
{0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#
,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [0]
p(U12) = [0]
p(U13) = [3]
p(U14) = [0]
p(U15) = [0]
p(U16) = [0]
p(U21) = [1] x2 + [0]
p(U22) = [0]
p(U23) = [0]
p(U31) = [0]
p(U32) = [0]
p(U41) = [3]
p(U51) = [0]
p(U52) = [2] x1 + [0]
p(U61) = [0]
p(U62) = [0]
p(U63) = [0]
p(U64) = [0]
p(activate) = [4]
p(isNat) = [5]
p(isNatKind) = [1] x1 + [2]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [0]
p(plus) = [0]
p(s) = [0]
p(tt) = [0]
p(0#) = [0]
p(U11#) = [0]
p(U12#) = [0]
p(U13#) = [0]
p(U14#) = [0]
p(U15#) = [0]
p(U16#) = [0]
p(U21#) = [0]
p(U22#) = [0]
p(U23#) = [0]
p(U31#) = [0]
p(U32#) = [0]
p(U41#) = [0]
p(U51#) = [1]
p(U52#) = [1]
p(U61#) = [1] x1 + [3]
p(U62#) = [2]
p(U63#) = [2]
p(U64#) = [2]
p(activate#) = [0]
p(isNat#) = [0]
p(isNatKind#) = [0]
p(plus#) = [0]
p(s#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [1] x2 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [0]
p(c_13) = [0]
p(c_14) = [0]
p(c_15) = [0]
p(c_16) = [0]
p(c_17) = [0]
p(c_18) = [2] x1 + [0]
p(c_19) = [0]
p(c_20) = [0]
p(c_21) = [0]
p(c_22) = [0]
p(c_23) = [0]
p(c_24) = [0]
p(c_25) = [0]
p(c_26) = [0]
p(c_27) = [0]
p(c_28) = [4] x1 + [1] x2 + [0]
p(c_29) = [1] x1 + [0]
p(c_30) = [1] x1 + [1] x2 + [0]
p(c_31) = [1] x1 + [0]
p(c_32) = [0]
p(c_33) = [0]
Following rules are strictly oriented:
U52#(tt(),N) = [1]
> [0]
= activate#(N)
U64#(tt(),M,N) = [2]
> [0]
= activate#(M)
U64#(tt(),M,N) = [2]
> [0]
= activate#(N)
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [0]
>= [0]
= U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) = [0]
>= [0]
= activate#(V1)
U11#(tt(),V1,V2) = [0]
>= [0]
= activate#(V2)
U11#(tt(),V1,V2) = [0]
>= [0]
= isNatKind#(activate(V1))
U12#(tt(),V1,V2) = [0]
>= [0]
= U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) = [0]
>= [0]
= activate#(V1)
U12#(tt(),V1,V2) = [0]
>= [0]
= activate#(V2)
U12#(tt(),V1,V2) = [0]
>= [0]
= isNatKind#(activate(V2))
U13#(tt(),V1,V2) = [0]
>= [0]
= U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) = [0]
>= [0]
= activate#(V1)
U13#(tt(),V1,V2) = [0]
>= [0]
= activate#(V2)
U13#(tt(),V1,V2) = [0]
>= [0]
= isNatKind#(activate(V2))
U14#(tt(),V1,V2) = [0]
>= [0]
= U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) = [0]
>= [0]
= activate#(V1)
U14#(tt(),V1,V2) = [0]
>= [0]
= activate#(V2)
U14#(tt(),V1,V2) = [0]
>= [0]
= isNat#(activate(V1))
U15#(tt(),V2) = [0]
>= [0]
= activate#(V2)
U15#(tt(),V2) = [0]
>= [0]
= isNat#(activate(V2))
U21#(tt(),V1) = [0]
>= [0]
= U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) = [0]
>= [0]
= activate#(V1)
U21#(tt(),V1) = [0]
>= [0]
= isNatKind#(activate(V1))
U22#(tt(),V1) = [0]
>= [0]
= activate#(V1)
U22#(tt(),V1) = [0]
>= [0]
= isNat#(activate(V1))
U31#(tt(),V2) = [0]
>= [0]
= c_11(isNatKind#(activate(V2)))
U51#(tt(),N) = [1]
>= [1]
= U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) = [1]
>= [0]
= isNatKind#(activate(N))
U61#(tt(),M,N) = [3]
>= [2]
= U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) = [3]
>= [0]
= isNatKind#(activate(M))
U62#(tt(),M,N) = [2]
>= [2]
= U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) = [2]
>= [0]
= activate#(M)
U62#(tt(),M,N) = [2]
>= [0]
= activate#(N)
U62#(tt(),M,N) = [2]
>= [0]
= isNat#(activate(N))
U63#(tt(),M,N) = [2]
>= [2]
= U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) = [2]
>= [0]
= activate#(M)
U63#(tt(),M,N) = [2]
>= [0]
= activate#(N)
U63#(tt(),M,N) = [2]
>= [0]
= isNatKind#(activate(N))
isNat#(n__plus(V1,V2)) = [0]
>= [0]
= U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) = [0]
>= [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [0]
>= [0]
= activate#(V2)
isNat#(n__plus(V1,V2)) = [0]
>= [0]
= isNatKind#(activate(V1))
isNat#(n__s(V1)) = [0]
>= [0]
= U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) = [0]
>= [0]
= activate#(V1)
isNat#(n__s(V1)) = [0]
>= [0]
= isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) = [0]
>= [0]
= c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) = [0]
>= [0]
= c_29(isNatKind#(activate(V1)))
*** Step 6.b:1.a:6: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U15#(tt(),V2) -> activate#(V2)
U21#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> activate#(V1)
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__s(V1)) -> activate#(V1)
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> isNat#(activate(V1))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_11) = {1},
uargs(c_28) = {1,2},
uargs(c_29) = {1}
Following symbols are considered usable:
{0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#
,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [0]
p(U12) = [0]
p(U13) = [0]
p(U14) = [0]
p(U15) = [0]
p(U16) = [0]
p(U21) = [0]
p(U22) = [0]
p(U23) = [0]
p(U31) = [0]
p(U32) = [0]
p(U41) = [0]
p(U51) = [0]
p(U52) = [0]
p(U61) = [0]
p(U62) = [0]
p(U63) = [0]
p(U64) = [7]
p(activate) = [0]
p(isNat) = [0]
p(isNatKind) = [0]
p(n__0) = [0]
p(n__plus) = [1] x2 + [0]
p(n__s) = [0]
p(plus) = [0]
p(s) = [0]
p(tt) = [0]
p(0#) = [0]
p(U11#) = [4]
p(U12#) = [4]
p(U13#) = [4]
p(U14#) = [4]
p(U15#) = [4]
p(U16#) = [0]
p(U21#) = [4]
p(U22#) = [4]
p(U23#) = [0]
p(U31#) = [0]
p(U32#) = [0]
p(U41#) = [0]
p(U51#) = [4]
p(U52#) = [3]
p(U61#) = [1] x2 + [7]
p(U62#) = [7]
p(U63#) = [0]
p(U64#) = [0]
p(activate#) = [0]
p(isNat#) = [4]
p(isNatKind#) = [0]
p(plus#) = [0]
p(s#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [1] x3 + [1]
p(c_6) = [1]
p(c_7) = [2]
p(c_8) = [4] x3 + [1]
p(c_9) = [4] x1 + [4]
p(c_10) = [1]
p(c_11) = [1] x1 + [0]
p(c_12) = [4]
p(c_13) = [1]
p(c_14) = [2] x1 + [2] x2 + [4] x3 + [1] x4 + [0]
p(c_15) = [1]
p(c_16) = [1] x2 + [1] x5 + [4]
p(c_17) = [1] x1 + [1] x3 + [2]
p(c_18) = [2] x5 + [2]
p(c_19) = [1] x1 + [1]
p(c_20) = [0]
p(c_21) = [0]
p(c_22) = [1] x1 + [0]
p(c_23) = [1]
p(c_24) = [1]
p(c_25) = [1] x3 + [1] x4 + [0]
p(c_26) = [1] x1 + [1]
p(c_27) = [1]
p(c_28) = [2] x1 + [4] x2 + [0]
p(c_29) = [4] x1 + [0]
p(c_30) = [1] x1 + [4] x2 + [4]
p(c_31) = [1] x2 + [0]
p(c_32) = [1]
p(c_33) = [1]
Following rules are strictly oriented:
U11#(tt(),V1,V2) = [4]
> [0]
= activate#(V1)
U11#(tt(),V1,V2) = [4]
> [0]
= activate#(V2)
U12#(tt(),V1,V2) = [4]
> [0]
= activate#(V1)
U12#(tt(),V1,V2) = [4]
> [0]
= activate#(V2)
U13#(tt(),V1,V2) = [4]
> [0]
= activate#(V1)
U13#(tt(),V1,V2) = [4]
> [0]
= activate#(V2)
U14#(tt(),V1,V2) = [4]
> [0]
= activate#(V1)
U14#(tt(),V1,V2) = [4]
> [0]
= activate#(V2)
U15#(tt(),V2) = [4]
> [0]
= activate#(V2)
U21#(tt(),V1) = [4]
> [0]
= activate#(V1)
U22#(tt(),V1) = [4]
> [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [4]
> [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [4]
> [0]
= activate#(V2)
isNat#(n__s(V1)) = [4]
> [0]
= activate#(V1)
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [4]
>= [4]
= U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) = [4]
>= [0]
= isNatKind#(activate(V1))
U12#(tt(),V1,V2) = [4]
>= [4]
= U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) = [4]
>= [0]
= isNatKind#(activate(V2))
U13#(tt(),V1,V2) = [4]
>= [4]
= U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) = [4]
>= [0]
= isNatKind#(activate(V2))
U14#(tt(),V1,V2) = [4]
>= [4]
= U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) = [4]
>= [4]
= isNat#(activate(V1))
U15#(tt(),V2) = [4]
>= [4]
= isNat#(activate(V2))
U21#(tt(),V1) = [4]
>= [4]
= U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) = [4]
>= [0]
= isNatKind#(activate(V1))
U22#(tt(),V1) = [4]
>= [4]
= isNat#(activate(V1))
U31#(tt(),V2) = [0]
>= [0]
= c_11(isNatKind#(activate(V2)))
U51#(tt(),N) = [4]
>= [3]
= U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) = [4]
>= [0]
= isNatKind#(activate(N))
U52#(tt(),N) = [3]
>= [0]
= activate#(N)
U61#(tt(),M,N) = [1] M + [7]
>= [7]
= U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) = [1] M + [7]
>= [0]
= isNatKind#(activate(M))
U62#(tt(),M,N) = [7]
>= [0]
= U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) = [7]
>= [0]
= activate#(M)
U62#(tt(),M,N) = [7]
>= [0]
= activate#(N)
U62#(tt(),M,N) = [7]
>= [4]
= isNat#(activate(N))
U63#(tt(),M,N) = [0]
>= [0]
= U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) = [0]
>= [0]
= activate#(M)
U63#(tt(),M,N) = [0]
>= [0]
= activate#(N)
U63#(tt(),M,N) = [0]
>= [0]
= isNatKind#(activate(N))
U64#(tt(),M,N) = [0]
>= [0]
= activate#(M)
U64#(tt(),M,N) = [0]
>= [0]
= activate#(N)
isNat#(n__plus(V1,V2)) = [4]
>= [4]
= U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) = [4]
>= [0]
= isNatKind#(activate(V1))
isNat#(n__s(V1)) = [4]
>= [4]
= U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) = [4]
>= [0]
= isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) = [0]
>= [0]
= c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) = [0]
>= [0]
= c_29(isNatKind#(activate(V1)))
*** Step 6.b:1.a:7: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_11) = {1},
uargs(c_28) = {1,2},
uargs(c_29) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [1]
p(U11) = [1]
p(U12) = [2]
p(U13) = [2] x1 + [2] x3 + [0]
p(U14) = [0]
p(U15) = [2] x2 + [4]
p(U16) = [4] x1 + [4]
p(U21) = [4] x2 + [0]
p(U22) = [1] x1 + [7]
p(U23) = [0]
p(U31) = [0]
p(U32) = [5] x1 + [5]
p(U41) = [4] x1 + [0]
p(U51) = [1] x2 + [1]
p(U52) = [1] x2 + [0]
p(U61) = [1] x2 + [1] x3 + [1]
p(U62) = [1] x2 + [1] x3 + [1]
p(U63) = [1] x2 + [1] x3 + [1]
p(U64) = [1] x2 + [1] x3 + [1]
p(activate) = [1] x1 + [0]
p(isNat) = [1] x1 + [0]
p(isNatKind) = [1]
p(n__0) = [1]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [1]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(0#) = [1]
p(U11#) = [6] x2 + [6] x3 + [0]
p(U12#) = [6] x2 + [6] x3 + [0]
p(U13#) = [6] x2 + [6] x3 + [0]
p(U14#) = [6] x2 + [6] x3 + [0]
p(U15#) = [6] x2 + [0]
p(U16#) = [1] x1 + [0]
p(U21#) = [6] x2 + [6]
p(U22#) = [6] x2 + [5]
p(U23#) = [4]
p(U31#) = [6] x2 + [0]
p(U32#) = [0]
p(U41#) = [1] x1 + [0]
p(U51#) = [7] x2 + [2]
p(U52#) = [4] x2 + [2]
p(U61#) = [1] x1 + [7] x2 + [7] x3 + [1]
p(U62#) = [7] x2 + [6] x3 + [1]
p(U63#) = [4] x2 + [6] x3 + [1]
p(U64#) = [4] x2 + [1]
p(activate#) = [0]
p(isNat#) = [6] x1 + [0]
p(isNatKind#) = [6] x1 + [0]
p(plus#) = [2]
p(s#) = [1]
p(c_1) = [0]
p(c_2) = [4] x1 + [1] x2 + [2] x3 + [1] x4 + [0]
p(c_3) = [4] x1 + [1] x3 + [1]
p(c_4) = [1] x4 + [2] x5 + [1]
p(c_5) = [4] x1 + [1] x2 + [1] x4 + [0]
p(c_6) = [4] x1 + [1] x2 + [0]
p(c_7) = [1]
p(c_8) = [1] x4 + [0]
p(c_9) = [1] x2 + [1]
p(c_10) = [1]
p(c_11) = [1] x1 + [0]
p(c_12) = [1]
p(c_13) = [2]
p(c_14) = [1] x1 + [4] x2 + [4] x4 + [0]
p(c_15) = [1] x1 + [0]
p(c_16) = [4] x2 + [1] x5 + [0]
p(c_17) = [1] x2 + [1] x3 + [2] x5 + [0]
p(c_18) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1]
p(c_19) = [0]
p(c_20) = [0]
p(c_21) = [1]
p(c_22) = [2] x1 + [2]
p(c_23) = [1] x1 + [2]
p(c_24) = [0]
p(c_25) = [4] x3 + [4] x5 + [0]
p(c_26) = [0]
p(c_27) = [0]
p(c_28) = [1] x1 + [1] x2 + [0]
p(c_29) = [1] x1 + [4]
p(c_30) = [1] x1 + [2] x2 + [1]
p(c_31) = [0]
p(c_32) = [1]
p(c_33) = [1]
Following rules are strictly oriented:
isNatKind#(n__s(V1)) = [6] V1 + [6]
> [6] V1 + [4]
= c_29(isNatKind#(activate(V1)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [6] V2 + [0]
= U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [0]
= activate#(V1)
U11#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [0]
= activate#(V2)
U11#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [0]
= isNatKind#(activate(V1))
U12#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [6] V2 + [0]
= U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [0]
= activate#(V1)
U12#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [0]
= activate#(V2)
U12#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V2 + [0]
= isNatKind#(activate(V2))
U13#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [6] V2 + [0]
= U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [0]
= activate#(V1)
U13#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [0]
= activate#(V2)
U13#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V2 + [0]
= isNatKind#(activate(V2))
U14#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V2 + [0]
= U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [0]
= activate#(V1)
U14#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [0]
= activate#(V2)
U14#(tt(),V1,V2) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [0]
= isNat#(activate(V1))
U15#(tt(),V2) = [6] V2 + [0]
>= [0]
= activate#(V2)
U15#(tt(),V2) = [6] V2 + [0]
>= [6] V2 + [0]
= isNat#(activate(V2))
U21#(tt(),V1) = [6] V1 + [6]
>= [6] V1 + [5]
= U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) = [6] V1 + [6]
>= [0]
= activate#(V1)
U21#(tt(),V1) = [6] V1 + [6]
>= [6] V1 + [0]
= isNatKind#(activate(V1))
U22#(tt(),V1) = [6] V1 + [5]
>= [0]
= activate#(V1)
U22#(tt(),V1) = [6] V1 + [5]
>= [6] V1 + [0]
= isNat#(activate(V1))
U31#(tt(),V2) = [6] V2 + [0]
>= [6] V2 + [0]
= c_11(isNatKind#(activate(V2)))
U51#(tt(),N) = [7] N + [2]
>= [4] N + [2]
= U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) = [7] N + [2]
>= [6] N + [0]
= isNatKind#(activate(N))
U52#(tt(),N) = [4] N + [2]
>= [0]
= activate#(N)
U61#(tt(),M,N) = [7] M + [7] N + [1]
>= [7] M + [6] N + [1]
= U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) = [7] M + [7] N + [1]
>= [6] M + [0]
= isNatKind#(activate(M))
U62#(tt(),M,N) = [7] M + [6] N + [1]
>= [4] M + [6] N + [1]
= U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) = [7] M + [6] N + [1]
>= [0]
= activate#(M)
U62#(tt(),M,N) = [7] M + [6] N + [1]
>= [0]
= activate#(N)
U62#(tt(),M,N) = [7] M + [6] N + [1]
>= [6] N + [0]
= isNat#(activate(N))
U63#(tt(),M,N) = [4] M + [6] N + [1]
>= [4] M + [1]
= U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) = [4] M + [6] N + [1]
>= [0]
= activate#(M)
U63#(tt(),M,N) = [4] M + [6] N + [1]
>= [0]
= activate#(N)
U63#(tt(),M,N) = [4] M + [6] N + [1]
>= [6] N + [0]
= isNatKind#(activate(N))
U64#(tt(),M,N) = [4] M + [1]
>= [0]
= activate#(M)
U64#(tt(),M,N) = [4] M + [1]
>= [0]
= activate#(N)
isNat#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [6] V2 + [0]
= U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0]
>= [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0]
>= [0]
= activate#(V2)
isNat#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [0]
= isNatKind#(activate(V1))
isNat#(n__s(V1)) = [6] V1 + [6]
>= [6] V1 + [6]
= U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) = [6] V1 + [6]
>= [0]
= activate#(V1)
isNat#(n__s(V1)) = [6] V1 + [6]
>= [6] V1 + [0]
= isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) = [6] V1 + [6] V2 + [0]
>= [6] V1 + [6] V2 + [0]
= c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
0() = [1]
>= [1]
= n__0()
U51(tt(),N) = [1] N + [1]
>= [1] N + [0]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [1]
>= [1]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [1]
>= [1] X + [1]
= s(activate(X))
plus(N,0()) = [1] N + [1]
>= [1] N + [1]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
*** Step 6.b:1.a:8: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_11) = {1},
uargs(c_28) = {1,2},
uargs(c_29) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [2] x1 + [3] x2 + [4]
p(U12) = [2] x1 + [7]
p(U13) = [1]
p(U14) = [2] x1 + [1] x2 + [4] x3 + [6]
p(U15) = [4] x2 + [6]
p(U16) = [0]
p(U21) = [3] x2 + [4]
p(U22) = [2] x1 + [1] x2 + [0]
p(U23) = [5] x1 + [7]
p(U31) = [2] x1 + [2]
p(U32) = [5]
p(U41) = [2]
p(U51) = [1] x2 + [0]
p(U52) = [1] x2 + [0]
p(U61) = [1] x2 + [1] x3 + [3]
p(U62) = [1] x2 + [1] x3 + [3]
p(U63) = [1] x2 + [1] x3 + [3]
p(U64) = [1] x2 + [1] x3 + [3]
p(activate) = [1] x1 + [0]
p(isNat) = [1]
p(isNatKind) = [1]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [3]
p(n__s) = [1] x1 + [0]
p(plus) = [1] x1 + [1] x2 + [3]
p(s) = [1] x1 + [0]
p(tt) = [0]
p(0#) = [1]
p(U11#) = [1] x2 + [1] x3 + [4]
p(U12#) = [1] x2 + [1] x3 + [4]
p(U13#) = [1] x2 + [1] x3 + [4]
p(U14#) = [1] x2 + [1] x3 + [3]
p(U15#) = [1] x2 + [3]
p(U16#) = [0]
p(U21#) = [1] x2 + [2]
p(U22#) = [1] x2 + [2]
p(U23#) = [1] x1 + [0]
p(U31#) = [1] x2 + [0]
p(U32#) = [4] x1 + [0]
p(U41#) = [0]
p(U51#) = [4] x2 + [2]
p(U52#) = [3] x2 + [0]
p(U61#) = [4] x2 + [4] x3 + [2]
p(U62#) = [4] x2 + [4] x3 + [2]
p(U63#) = [1] x2 + [4] x3 + [2]
p(U64#) = [1] x3 + [0]
p(activate#) = [0]
p(isNat#) = [1] x1 + [2]
p(isNatKind#) = [1] x1 + [0]
p(plus#) = [4] x1 + [1]
p(s#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [1] x3 + [1] x4 + [1]
p(c_3) = [1] x1 + [2] x5 + [0]
p(c_4) = [4] x1 + [4] x2 + [1] x4 + [1]
p(c_5) = [1] x4 + [2]
p(c_6) = [2] x1 + [2]
p(c_7) = [2]
p(c_8) = [1] x3 + [1] x4 + [0]
p(c_9) = [2] x1 + [1]
p(c_10) = [4]
p(c_11) = [1] x1 + [0]
p(c_12) = [1]
p(c_13) = [1]
p(c_14) = [4] x1 + [4] x3 + [1]
p(c_15) = [2] x1 + [2]
p(c_16) = [4] x2 + [4] x4 + [1]
p(c_17) = [1] x1 + [2] x5 + [0]
p(c_18) = [1] x5 + [0]
p(c_19) = [1] x1 + [1]
p(c_20) = [4]
p(c_21) = [1] x1 + [0]
p(c_22) = [4] x1 + [2]
p(c_23) = [1]
p(c_24) = [1]
p(c_25) = [0]
p(c_26) = [1] x2 + [4] x3 + [0]
p(c_27) = [1]
p(c_28) = [1] x1 + [1] x2 + [0]
p(c_29) = [1] x1 + [0]
p(c_30) = [1] x1 + [1] x2 + [0]
p(c_31) = [4]
p(c_32) = [2]
p(c_33) = [0]
Following rules are strictly oriented:
isNatKind#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [3]
> [1] V1 + [1] V2 + [0]
= c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [1] V1 + [1] V2 + [4]
= U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [0]
= activate#(V1)
U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [0]
= activate#(V2)
U11#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [1] V1 + [0]
= isNatKind#(activate(V1))
U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [1] V1 + [1] V2 + [4]
= U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [0]
= activate#(V1)
U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [0]
= activate#(V2)
U12#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [1] V2 + [0]
= isNatKind#(activate(V2))
U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [1] V1 + [1] V2 + [3]
= U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [0]
= activate#(V1)
U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [0]
= activate#(V2)
U13#(tt(),V1,V2) = [1] V1 + [1] V2 + [4]
>= [1] V2 + [0]
= isNatKind#(activate(V2))
U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [3]
>= [1] V2 + [3]
= U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [3]
>= [0]
= activate#(V1)
U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [3]
>= [0]
= activate#(V2)
U14#(tt(),V1,V2) = [1] V1 + [1] V2 + [3]
>= [1] V1 + [2]
= isNat#(activate(V1))
U15#(tt(),V2) = [1] V2 + [3]
>= [0]
= activate#(V2)
U15#(tt(),V2) = [1] V2 + [3]
>= [1] V2 + [2]
= isNat#(activate(V2))
U21#(tt(),V1) = [1] V1 + [2]
>= [1] V1 + [2]
= U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) = [1] V1 + [2]
>= [0]
= activate#(V1)
U21#(tt(),V1) = [1] V1 + [2]
>= [1] V1 + [0]
= isNatKind#(activate(V1))
U22#(tt(),V1) = [1] V1 + [2]
>= [0]
= activate#(V1)
U22#(tt(),V1) = [1] V1 + [2]
>= [1] V1 + [2]
= isNat#(activate(V1))
U31#(tt(),V2) = [1] V2 + [0]
>= [1] V2 + [0]
= c_11(isNatKind#(activate(V2)))
U51#(tt(),N) = [4] N + [2]
>= [3] N + [0]
= U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) = [4] N + [2]
>= [1] N + [0]
= isNatKind#(activate(N))
U52#(tt(),N) = [3] N + [0]
>= [0]
= activate#(N)
U61#(tt(),M,N) = [4] M + [4] N + [2]
>= [4] M + [4] N + [2]
= U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) = [4] M + [4] N + [2]
>= [1] M + [0]
= isNatKind#(activate(M))
U62#(tt(),M,N) = [4] M + [4] N + [2]
>= [1] M + [4] N + [2]
= U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) = [4] M + [4] N + [2]
>= [0]
= activate#(M)
U62#(tt(),M,N) = [4] M + [4] N + [2]
>= [0]
= activate#(N)
U62#(tt(),M,N) = [4] M + [4] N + [2]
>= [1] N + [2]
= isNat#(activate(N))
U63#(tt(),M,N) = [1] M + [4] N + [2]
>= [1] N + [0]
= U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) = [1] M + [4] N + [2]
>= [0]
= activate#(M)
U63#(tt(),M,N) = [1] M + [4] N + [2]
>= [0]
= activate#(N)
U63#(tt(),M,N) = [1] M + [4] N + [2]
>= [1] N + [0]
= isNatKind#(activate(N))
U64#(tt(),M,N) = [1] N + [0]
>= [0]
= activate#(M)
U64#(tt(),M,N) = [1] N + [0]
>= [0]
= activate#(N)
isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5]
>= [1] V1 + [1] V2 + [4]
= U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5]
>= [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5]
>= [0]
= activate#(V2)
isNat#(n__plus(V1,V2)) = [1] V1 + [1] V2 + [5]
>= [1] V1 + [0]
= isNatKind#(activate(V1))
isNat#(n__s(V1)) = [1] V1 + [2]
>= [1] V1 + [2]
= U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) = [1] V1 + [2]
>= [0]
= activate#(V1)
isNat#(n__s(V1)) = [1] V1 + [2]
>= [1] V1 + [0]
= isNatKind#(activate(V1))
isNatKind#(n__s(V1)) = [1] V1 + [0]
>= [1] V1 + [0]
= c_29(isNatKind#(activate(V1)))
0() = [0]
>= [0]
= n__0()
U51(tt(),N) = [1] N + [0]
>= [1] N + [0]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [3]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(activate(X))
plus(N,0()) = [1] N + [3]
>= [1] N + [0]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [3]
>= [1] X1 + [1] X2 + [3]
= n__plus(X1,X2)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
*** Step 6.b:1.a:9: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_11) = {1},
uargs(c_28) = {1,2},
uargs(c_29) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [6] x1 + [2] x3 + [4]
p(U12) = [4] x1 + [4] x3 + [4]
p(U13) = [2]
p(U14) = [4] x3 + [1]
p(U15) = [1] x1 + [2] x2 + [0]
p(U16) = [0]
p(U21) = [2] x1 + [2] x2 + [2]
p(U22) = [4] x1 + [4] x2 + [4]
p(U23) = [2]
p(U31) = [6]
p(U32) = [5]
p(U41) = [5] x1 + [3]
p(U51) = [1] x2 + [2]
p(U52) = [1] x2 + [2]
p(U61) = [1] x2 + [1] x3 + [2]
p(U62) = [1] x2 + [1] x3 + [2]
p(U63) = [1] x2 + [1] x3 + [2]
p(U64) = [1] x2 + [1] x3 + [2]
p(activate) = [1] x1 + [0]
p(isNat) = [0]
p(isNatKind) = [1]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [2]
p(n__s) = [1] x1 + [0]
p(plus) = [1] x1 + [1] x2 + [2]
p(s) = [1] x1 + [0]
p(tt) = [0]
p(0#) = [0]
p(U11#) = [4] x2 + [4] x3 + [0]
p(U12#) = [4] x2 + [4] x3 + [0]
p(U13#) = [4] x2 + [4] x3 + [0]
p(U14#) = [4] x2 + [4] x3 + [0]
p(U15#) = [4] x2 + [0]
p(U16#) = [0]
p(U21#) = [4] x2 + [0]
p(U22#) = [4] x2 + [0]
p(U23#) = [1]
p(U31#) = [4] x2 + [2]
p(U32#) = [1] x1 + [0]
p(U41#) = [1]
p(U51#) = [6] x2 + [0]
p(U52#) = [4] x2 + [0]
p(U61#) = [1] x1 + [5] x2 + [4] x3 + [1]
p(U62#) = [1] x2 + [4] x3 + [1]
p(U63#) = [4] x3 + [0]
p(U64#) = [0]
p(activate#) = [0]
p(isNat#) = [4] x1 + [0]
p(isNatKind#) = [4] x1 + [0]
p(plus#) = [1] x1 + [1] x2 + [0]
p(s#) = [1]
p(c_1) = [0]
p(c_2) = [1] x1 + [4] x2 + [1] x3 + [1]
p(c_3) = [4] x2 + [2] x3 + [1] x5 + [0]
p(c_4) = [2] x2 + [1] x3 + [1] x4 + [0]
p(c_5) = [1] x1 + [1] x2 + [1] x4 + [2]
p(c_6) = [1] x1 + [0]
p(c_7) = [4]
p(c_8) = [1] x2 + [1]
p(c_9) = [1] x1 + [1] x2 + [1]
p(c_10) = [2]
p(c_11) = [1] x1 + [1]
p(c_12) = [1]
p(c_13) = [1]
p(c_14) = [4] x1 + [1] x2 + [4] x4 + [0]
p(c_15) = [1]
p(c_16) = [1] x2 + [2] x4 + [2] x5 + [1]
p(c_17) = [2] x1 + [2] x3 + [1] x4 + [4]
p(c_18) = [1] x1 + [1] x3 + [4] x5 + [1]
p(c_19) = [2] x1 + [4]
p(c_20) = [0]
p(c_21) = [2] x1 + [0]
p(c_22) = [2] x2 + [0]
p(c_23) = [2] x1 + [4]
p(c_24) = [1]
p(c_25) = [1] x3 + [0]
p(c_26) = [0]
p(c_27) = [1]
p(c_28) = [1] x1 + [1] x2 + [6]
p(c_29) = [1] x1 + [0]
p(c_30) = [1] x2 + [0]
p(c_31) = [1] x1 + [2] x2 + [1]
p(c_32) = [4]
p(c_33) = [1]
Following rules are strictly oriented:
U31#(tt(),V2) = [4] V2 + [2]
> [4] V2 + [1]
= c_11(isNatKind#(activate(V2)))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [0]
= activate#(V1)
U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [0]
= activate#(V2)
U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= isNatKind#(activate(V1))
U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [0]
= activate#(V1)
U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [0]
= activate#(V2)
U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= isNatKind#(activate(V2))
U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [0]
= activate#(V1)
U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [0]
= activate#(V2)
U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= isNatKind#(activate(V2))
U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [0]
= activate#(V1)
U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [0]
= activate#(V2)
U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= isNat#(activate(V1))
U15#(tt(),V2) = [4] V2 + [0]
>= [0]
= activate#(V2)
U15#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= isNat#(activate(V2))
U21#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) = [4] V1 + [0]
>= [0]
= activate#(V1)
U21#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= isNatKind#(activate(V1))
U22#(tt(),V1) = [4] V1 + [0]
>= [0]
= activate#(V1)
U22#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= isNat#(activate(V1))
U51#(tt(),N) = [6] N + [0]
>= [4] N + [0]
= U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) = [6] N + [0]
>= [4] N + [0]
= isNatKind#(activate(N))
U52#(tt(),N) = [4] N + [0]
>= [0]
= activate#(N)
U61#(tt(),M,N) = [5] M + [4] N + [1]
>= [1] M + [4] N + [1]
= U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) = [5] M + [4] N + [1]
>= [4] M + [0]
= isNatKind#(activate(M))
U62#(tt(),M,N) = [1] M + [4] N + [1]
>= [4] N + [0]
= U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) = [1] M + [4] N + [1]
>= [0]
= activate#(M)
U62#(tt(),M,N) = [1] M + [4] N + [1]
>= [0]
= activate#(N)
U62#(tt(),M,N) = [1] M + [4] N + [1]
>= [4] N + [0]
= isNat#(activate(N))
U63#(tt(),M,N) = [4] N + [0]
>= [0]
= U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) = [4] N + [0]
>= [0]
= activate#(M)
U63#(tt(),M,N) = [4] N + [0]
>= [0]
= activate#(N)
U63#(tt(),M,N) = [4] N + [0]
>= [4] N + [0]
= isNatKind#(activate(N))
U64#(tt(),M,N) = [0]
>= [0]
= activate#(M)
U64#(tt(),M,N) = [0]
>= [0]
= activate#(N)
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [4] V2 + [0]
= U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [0]
= activate#(V2)
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [0]
= isNatKind#(activate(V1))
isNat#(n__s(V1)) = [4] V1 + [0]
>= [4] V1 + [0]
= U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) = [4] V1 + [0]
>= [0]
= activate#(V1)
isNat#(n__s(V1)) = [4] V1 + [0]
>= [4] V1 + [0]
= isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [4] V2 + [8]
= c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) = [4] V1 + [0]
>= [4] V1 + [0]
= c_29(isNatKind#(activate(V1)))
0() = [0]
>= [0]
= n__0()
U51(tt(),N) = [1] N + [2]
>= [1] N + [2]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [2]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [0]
>= [1] X + [0]
= s(activate(X))
plus(N,0()) = [1] N + [2]
>= [1] N + [2]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [2]
>= [1] M + [1] N + [2]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__plus(X1,X2)
s(X) = [1] X + [0]
>= [1] X + [0]
= n__s(X)
*** Step 6.b:1.a:10: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U31#(tt(),V2) -> c_11(isNatKind#(activate(V2)))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) -> c_28(U31#(isNatKind(activate(V1)),activate(V2)),isNatKind#(activate(V1)))
isNatKind#(n__s(V1)) -> c_29(isNatKind#(activate(V1)))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/1,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/2,c_29/1,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 6.b:1.b:1: RemoveHeads WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(activate#(X))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNatKind#(activate(V2))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> activate#(N)
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> activate#(M)
U61#(tt(),M,N) -> activate#(N)
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
isNatKind#(n__plus(V1,V2)) -> activate#(V1)
isNatKind#(n__plus(V1,V2)) -> activate#(V2)
isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNatKind#(n__s(V1)) -> activate#(V1)
isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2))
-->_2 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_2 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
2:S:activate#(n__s(X)) -> c_23(activate#(X))
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
3:W:U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
-->_1 U12#(tt(),V1,V2) -> isNatKind#(activate(V2)):10
-->_1 U12#(tt(),V1,V2) -> activate#(V2):9
-->_1 U12#(tt(),V1,V2) -> activate#(V1):8
-->_1 U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2)):7
4:W:U11#(tt(),V1,V2) -> activate#(V1)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
5:W:U11#(tt(),V1,V2) -> activate#(V2)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
6:W:U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
7:W:U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
-->_1 U13#(tt(),V1,V2) -> isNatKind#(activate(V2)):14
-->_1 U13#(tt(),V1,V2) -> activate#(V2):13
-->_1 U13#(tt(),V1,V2) -> activate#(V1):12
-->_1 U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2)):11
8:W:U12#(tt(),V1,V2) -> activate#(V1)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
9:W:U12#(tt(),V1,V2) -> activate#(V2)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
10:W:U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
11:W:U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
-->_1 U14#(tt(),V1,V2) -> isNat#(activate(V1)):18
-->_1 U14#(tt(),V1,V2) -> activate#(V2):17
-->_1 U14#(tt(),V1,V2) -> activate#(V1):16
-->_1 U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2)):15
12:W:U13#(tt(),V1,V2) -> activate#(V1)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
13:W:U13#(tt(),V1,V2) -> activate#(V2)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
14:W:U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
15:W:U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
-->_1 U15#(tt(),V2) -> isNat#(activate(V2)):20
-->_1 U15#(tt(),V2) -> activate#(V2):19
16:W:U14#(tt(),V1,V2) -> activate#(V1)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
17:W:U14#(tt(),V1,V2) -> activate#(V2)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
18:W:U14#(tt(),V1,V2) -> isNat#(activate(V1))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):52
-->_1 isNat#(n__s(V1)) -> activate#(V1):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):48
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):47
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):46
19:W:U15#(tt(),V2) -> activate#(V2)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
20:W:U15#(tt(),V2) -> isNat#(activate(V2))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):52
-->_1 isNat#(n__s(V1)) -> activate#(V1):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):48
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):47
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):46
21:W:U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
-->_1 U22#(tt(),V1) -> isNat#(activate(V1)):25
-->_1 U22#(tt(),V1) -> activate#(V1):24
22:W:U21#(tt(),V1) -> activate#(V1)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
23:W:U21#(tt(),V1) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
24:W:U22#(tt(),V1) -> activate#(V1)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
25:W:U22#(tt(),V1) -> isNat#(activate(V1))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):52
-->_1 isNat#(n__s(V1)) -> activate#(V1):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):48
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):47
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):46
26:W:U31#(tt(),V2) -> activate#(V2)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
27:W:U31#(tt(),V2) -> isNatKind#(activate(V2))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
28:W:U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
-->_1 U52#(tt(),N) -> activate#(N):31
29:W:U51#(tt(),N) -> activate#(N)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
30:W:U51#(tt(),N) -> isNatKind#(activate(N))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
31:W:U52#(tt(),N) -> activate#(N)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
32:W:U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
-->_1 U62#(tt(),M,N) -> isNat#(activate(N)):39
-->_1 U62#(tt(),M,N) -> activate#(N):38
-->_1 U62#(tt(),M,N) -> activate#(M):37
-->_1 U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N)):36
33:W:U61#(tt(),M,N) -> activate#(M)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
34:W:U61#(tt(),M,N) -> activate#(N)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
35:W:U61#(tt(),M,N) -> isNatKind#(activate(M))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
36:W:U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
-->_1 U63#(tt(),M,N) -> isNatKind#(activate(N)):43
-->_1 U63#(tt(),M,N) -> activate#(N):42
-->_1 U63#(tt(),M,N) -> activate#(M):41
-->_1 U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N)):40
37:W:U62#(tt(),M,N) -> activate#(M)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
38:W:U62#(tt(),M,N) -> activate#(N)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
39:W:U62#(tt(),M,N) -> isNat#(activate(N))
-->_1 isNat#(n__s(V1)) -> isNatKind#(activate(V1)):52
-->_1 isNat#(n__s(V1)) -> activate#(V1):51
-->_1 isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1)):50
-->_1 isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):49
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V2):48
-->_1 isNat#(n__plus(V1,V2)) -> activate#(V1):47
-->_1 isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2)):46
40:W:U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
-->_1 U64#(tt(),M,N) -> activate#(N):45
-->_1 U64#(tt(),M,N) -> activate#(M):44
41:W:U63#(tt(),M,N) -> activate#(M)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
42:W:U63#(tt(),M,N) -> activate#(N)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
43:W:U63#(tt(),M,N) -> isNatKind#(activate(N))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
44:W:U64#(tt(),M,N) -> activate#(M)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
45:W:U64#(tt(),M,N) -> activate#(N)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
46:W:isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
-->_1 U11#(tt(),V1,V2) -> isNatKind#(activate(V1)):6
-->_1 U11#(tt(),V1,V2) -> activate#(V2):5
-->_1 U11#(tt(),V1,V2) -> activate#(V1):4
-->_1 U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2)):3
47:W:isNat#(n__plus(V1,V2)) -> activate#(V1)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
48:W:isNat#(n__plus(V1,V2)) -> activate#(V2)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
49:W:isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
50:W:isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
-->_1 U21#(tt(),V1) -> isNatKind#(activate(V1)):23
-->_1 U21#(tt(),V1) -> activate#(V1):22
-->_1 U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1)):21
51:W:isNat#(n__s(V1)) -> activate#(V1)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
52:W:isNat#(n__s(V1)) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
53:W:isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
-->_1 U31#(tt(),V2) -> isNatKind#(activate(V2)):27
-->_1 U31#(tt(),V2) -> activate#(V2):26
54:W:isNatKind#(n__plus(V1,V2)) -> activate#(V1)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
55:W:isNatKind#(n__plus(V1,V2)) -> activate#(V2)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
56:W:isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
57:W:isNatKind#(n__s(V1)) -> activate#(V1)
-->_1 activate#(n__s(X)) -> c_23(activate#(X)):2
-->_1 activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2)):1
58:W:isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
-->_1 isNatKind#(n__s(V1)) -> isNatKind#(activate(V1)):58
-->_1 isNatKind#(n__s(V1)) -> activate#(V1):57
-->_1 isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1)):56
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V2):55
-->_1 isNatKind#(n__plus(V1,V2)) -> activate#(V1):54
-->_1 isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2)):53
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(29,U51#(tt(),N) -> activate#(N)),(33,U61#(tt(),M,N) -> activate#(M)),(34,U61#(tt(),M,N) -> activate#(N))]
*** Step 6.b:1.b:2: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(activate#(X))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNatKind#(activate(V2))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
isNatKind#(n__plus(V1,V2)) -> activate#(V1)
isNatKind#(n__plus(V1,V2)) -> activate#(V2)
isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNatKind#(n__s(V1)) -> activate#(V1)
isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_22) = {1,2},
uargs(c_23) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [2] x1 + [1] x3 + [6]
p(U12) = [1] x3 + [0]
p(U13) = [5] x1 + [2] x3 + [1]
p(U14) = [7] x2 + [4]
p(U15) = [2] x1 + [0]
p(U16) = [0]
p(U21) = [4] x2 + [0]
p(U22) = [2] x1 + [0]
p(U23) = [1] x1 + [7]
p(U31) = [1] x1 + [2]
p(U32) = [0]
p(U41) = [1]
p(U51) = [1] x2 + [0]
p(U52) = [1] x2 + [0]
p(U61) = [1] x2 + [1] x3 + [1]
p(U62) = [1] x2 + [1] x3 + [1]
p(U63) = [1] x2 + [1] x3 + [1]
p(U64) = [1] x2 + [1] x3 + [1]
p(activate) = [1] x1 + [0]
p(isNat) = [0]
p(isNatKind) = [2] x1 + [1]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [0]
p(n__s) = [1] x1 + [1]
p(plus) = [1] x1 + [1] x2 + [0]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(0#) = [2]
p(U11#) = [2] x2 + [2] x3 + [1]
p(U12#) = [2] x2 + [2] x3 + [1]
p(U13#) = [2] x2 + [2] x3 + [1]
p(U14#) = [2] x2 + [2] x3 + [1]
p(U15#) = [2] x2 + [1]
p(U16#) = [2] x1 + [1]
p(U21#) = [2] x2 + [1]
p(U22#) = [2] x2 + [1]
p(U23#) = [0]
p(U31#) = [2] x2 + [0]
p(U32#) = [1] x1 + [1]
p(U41#) = [4] x1 + [1]
p(U51#) = [1] x1 + [4] x2 + [0]
p(U52#) = [2] x2 + [0]
p(U61#) = [3] x2 + [4] x3 + [1]
p(U62#) = [2] x2 + [2] x3 + [1]
p(U63#) = [2] x2 + [2] x3 + [0]
p(U64#) = [2] x2 + [2] x3 + [0]
p(activate#) = [2] x1 + [0]
p(isNat#) = [2] x1 + [1]
p(isNatKind#) = [2] x1 + [0]
p(plus#) = [1] x2 + [4]
p(s#) = [0]
p(c_1) = [1]
p(c_2) = [4] x2 + [1] x5 + [1]
p(c_3) = [1] x1 + [1] x4 + [1] x5 + [0]
p(c_4) = [1] x4 + [1] x5 + [1]
p(c_5) = [1] x1 + [2] x3 + [1]
p(c_6) = [2] x1 + [1] x2 + [1]
p(c_7) = [0]
p(c_8) = [1] x2 + [4]
p(c_9) = [1]
p(c_10) = [4]
p(c_11) = [4] x1 + [0]
p(c_12) = [4]
p(c_13) = [0]
p(c_14) = [1] x1 + [1] x2 + [1] x3 + [1]
p(c_15) = [1] x1 + [2]
p(c_16) = [1] x1 + [1] x2 + [2] x5 + [0]
p(c_17) = [2] x1 + [1] x2 + [1] x3 + [0]
p(c_18) = [1] x1 + [2] x2 + [1] x3 + [1] x4 + [0]
p(c_19) = [0]
p(c_20) = [0]
p(c_21) = [2]
p(c_22) = [1] x1 + [1] x2 + [0]
p(c_23) = [1] x1 + [0]
p(c_24) = [2]
p(c_25) = [1] x3 + [0]
p(c_26) = [1] x1 + [1] x2 + [2] x3 + [1] x4 + [1]
p(c_27) = [1]
p(c_28) = [1] x1 + [2] x2 + [4] x3 + [2] x4 + [1]
p(c_29) = [1] x1 + [0]
p(c_30) = [2] x1 + [4] x2 + [1]
p(c_31) = [4] x1 + [1]
p(c_32) = [1]
p(c_33) = [0]
Following rules are strictly oriented:
activate#(n__s(X)) = [2] X + [2]
> [2] X + [0]
= c_23(activate#(X))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [2] V2 + [1]
= U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [0]
= activate#(V1)
U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V2 + [0]
= activate#(V2)
U11#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [0]
= isNatKind#(activate(V1))
U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [2] V2 + [1]
= U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [0]
= activate#(V1)
U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V2 + [0]
= activate#(V2)
U12#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V2 + [0]
= isNatKind#(activate(V2))
U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [2] V2 + [1]
= U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [0]
= activate#(V1)
U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V2 + [0]
= activate#(V2)
U13#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V2 + [0]
= isNatKind#(activate(V2))
U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V2 + [1]
= U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [0]
= activate#(V1)
U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V2 + [0]
= activate#(V2)
U14#(tt(),V1,V2) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [1]
= isNat#(activate(V1))
U15#(tt(),V2) = [2] V2 + [1]
>= [2] V2 + [0]
= activate#(V2)
U15#(tt(),V2) = [2] V2 + [1]
>= [2] V2 + [1]
= isNat#(activate(V2))
U21#(tt(),V1) = [2] V1 + [1]
>= [2] V1 + [1]
= U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) = [2] V1 + [1]
>= [2] V1 + [0]
= activate#(V1)
U21#(tt(),V1) = [2] V1 + [1]
>= [2] V1 + [0]
= isNatKind#(activate(V1))
U22#(tt(),V1) = [2] V1 + [1]
>= [2] V1 + [0]
= activate#(V1)
U22#(tt(),V1) = [2] V1 + [1]
>= [2] V1 + [1]
= isNat#(activate(V1))
U31#(tt(),V2) = [2] V2 + [0]
>= [2] V2 + [0]
= activate#(V2)
U31#(tt(),V2) = [2] V2 + [0]
>= [2] V2 + [0]
= isNatKind#(activate(V2))
U51#(tt(),N) = [4] N + [0]
>= [2] N + [0]
= U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) = [4] N + [0]
>= [2] N + [0]
= isNatKind#(activate(N))
U52#(tt(),N) = [2] N + [0]
>= [2] N + [0]
= activate#(N)
U61#(tt(),M,N) = [3] M + [4] N + [1]
>= [2] M + [2] N + [1]
= U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) = [3] M + [4] N + [1]
>= [2] M + [0]
= isNatKind#(activate(M))
U62#(tt(),M,N) = [2] M + [2] N + [1]
>= [2] M + [2] N + [0]
= U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) = [2] M + [2] N + [1]
>= [2] M + [0]
= activate#(M)
U62#(tt(),M,N) = [2] M + [2] N + [1]
>= [2] N + [0]
= activate#(N)
U62#(tt(),M,N) = [2] M + [2] N + [1]
>= [2] N + [1]
= isNat#(activate(N))
U63#(tt(),M,N) = [2] M + [2] N + [0]
>= [2] M + [2] N + [0]
= U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) = [2] M + [2] N + [0]
>= [2] M + [0]
= activate#(M)
U63#(tt(),M,N) = [2] M + [2] N + [0]
>= [2] N + [0]
= activate#(N)
U63#(tt(),M,N) = [2] M + [2] N + [0]
>= [2] N + [0]
= isNatKind#(activate(N))
U64#(tt(),M,N) = [2] M + [2] N + [0]
>= [2] M + [0]
= activate#(M)
U64#(tt(),M,N) = [2] M + [2] N + [0]
>= [2] N + [0]
= activate#(N)
activate#(n__plus(X1,X2)) = [2] X1 + [2] X2 + [0]
>= [2] X1 + [2] X2 + [0]
= c_22(activate#(X1),activate#(X2))
isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [2] V2 + [1]
= U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1]
>= [2] V2 + [0]
= activate#(V2)
isNat#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [1]
>= [2] V1 + [0]
= isNatKind#(activate(V1))
isNat#(n__s(V1)) = [2] V1 + [3]
>= [2] V1 + [1]
= U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) = [2] V1 + [3]
>= [2] V1 + [0]
= activate#(V1)
isNat#(n__s(V1)) = [2] V1 + [3]
>= [2] V1 + [0]
= isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [0]
>= [2] V2 + [0]
= U31#(isNatKind(activate(V1)),activate(V2))
isNatKind#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [0]
>= [2] V1 + [0]
= activate#(V1)
isNatKind#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [0]
>= [2] V2 + [0]
= activate#(V2)
isNatKind#(n__plus(V1,V2)) = [2] V1 + [2] V2 + [0]
>= [2] V1 + [0]
= isNatKind#(activate(V1))
isNatKind#(n__s(V1)) = [2] V1 + [2]
>= [2] V1 + [0]
= activate#(V1)
isNatKind#(n__s(V1)) = [2] V1 + [2]
>= [2] V1 + [0]
= isNatKind#(activate(V1))
0() = [0]
>= [0]
= n__0()
U51(tt(),N) = [1] N + [0]
>= [1] N + [0]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [1]
>= [1] X + [1]
= s(activate(X))
plus(N,0()) = [1] N + [0]
>= [1] N + [0]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [1]
>= [1] M + [1] N + [1]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [0]
>= [1] X1 + [1] X2 + [0]
= n__plus(X1,X2)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
*** Step 6.b:1.b:3: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2))
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNatKind#(activate(V2))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
activate#(n__s(X)) -> c_23(activate#(X))
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
isNatKind#(n__plus(V1,V2)) -> activate#(V1)
isNatKind#(n__plus(V1,V2)) -> activate#(V2)
isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNatKind#(n__s(V1)) -> activate#(V1)
isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_22) = {1,2},
uargs(c_23) = {1}
Following symbols are considered usable:
{0,U51,U52,U61,U62,U63,U64,activate,plus,s,0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#,U41#
,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#}
TcT has computed the following interpretation:
p(0) = [0]
p(U11) = [2]
p(U12) = [1] x1 + [4] x2 + [4] x3 + [4]
p(U13) = [4] x3 + [0]
p(U14) = [0]
p(U15) = [0]
p(U16) = [2] x1 + [0]
p(U21) = [0]
p(U22) = [1] x1 + [4]
p(U23) = [1] x1 + [2]
p(U31) = [1]
p(U32) = [0]
p(U41) = [0]
p(U51) = [1] x2 + [0]
p(U52) = [1] x2 + [0]
p(U61) = [1] x2 + [1] x3 + [3]
p(U62) = [1] x2 + [1] x3 + [3]
p(U63) = [1] x2 + [1] x3 + [3]
p(U64) = [1] x2 + [1] x3 + [3]
p(activate) = [1] x1 + [0]
p(isNat) = [0]
p(isNatKind) = [4] x1 + [0]
p(n__0) = [0]
p(n__plus) = [1] x1 + [1] x2 + [2]
p(n__s) = [1] x1 + [1]
p(plus) = [1] x1 + [1] x2 + [2]
p(s) = [1] x1 + [1]
p(tt) = [0]
p(0#) = [0]
p(U11#) = [4] x2 + [4] x3 + [0]
p(U12#) = [4] x2 + [4] x3 + [0]
p(U13#) = [4] x2 + [4] x3 + [0]
p(U14#) = [4] x2 + [4] x3 + [0]
p(U15#) = [4] x2 + [0]
p(U16#) = [2] x1 + [1]
p(U21#) = [4] x2 + [0]
p(U22#) = [4] x2 + [0]
p(U23#) = [2] x1 + [1]
p(U31#) = [4] x2 + [0]
p(U32#) = [1] x1 + [1]
p(U41#) = [4]
p(U51#) = [1] x1 + [5] x2 + [4]
p(U52#) = [4] x2 + [2]
p(U61#) = [7] x2 + [4] x3 + [4]
p(U62#) = [4] x2 + [4] x3 + [4]
p(U63#) = [4] x2 + [4] x3 + [0]
p(U64#) = [4] x2 + [4] x3 + [0]
p(activate#) = [4] x1 + [0]
p(isNat#) = [4] x1 + [0]
p(isNatKind#) = [4] x1 + [0]
p(plus#) = [4]
p(s#) = [2] x1 + [0]
p(c_1) = [2]
p(c_2) = [1] x3 + [1] x4 + [0]
p(c_3) = [2] x1 + [4] x2 + [1] x3 + [2]
p(c_4) = [2] x3 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [1] x2 + [1]
p(c_7) = [1]
p(c_8) = [1] x1 + [2] x3 + [0]
p(c_9) = [1] x1 + [4] x2 + [0]
p(c_10) = [0]
p(c_11) = [1] x1 + [1]
p(c_12) = [1]
p(c_13) = [0]
p(c_14) = [2] x2 + [0]
p(c_15) = [0]
p(c_16) = [1] x1 + [1] x2 + [2] x4 + [2] x5 + [1]
p(c_17) = [1] x5 + [4]
p(c_18) = [2] x1 + [4] x4 + [1]
p(c_19) = [4] x2 + [4]
p(c_20) = [2]
p(c_21) = [1] x1 + [1]
p(c_22) = [1] x1 + [1] x2 + [0]
p(c_23) = [1] x1 + [4]
p(c_24) = [1]
p(c_25) = [4]
p(c_26) = [4] x3 + [0]
p(c_27) = [1]
p(c_28) = [1] x1 + [2] x2 + [0]
p(c_29) = [1] x2 + [4]
p(c_30) = [4] x1 + [0]
p(c_31) = [4]
p(c_32) = [0]
p(c_33) = [1]
Following rules are strictly oriented:
activate#(n__plus(X1,X2)) = [4] X1 + [4] X2 + [8]
> [4] X1 + [4] X2 + [0]
= c_22(activate#(X1),activate#(X2))
Following rules are (at-least) weakly oriented:
U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= activate#(V1)
U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= activate#(V2)
U11#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= isNatKind#(activate(V1))
U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= activate#(V1)
U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= activate#(V2)
U12#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= isNatKind#(activate(V2))
U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [4] V2 + [0]
= U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= activate#(V1)
U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= activate#(V2)
U13#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= isNatKind#(activate(V2))
U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= activate#(V1)
U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V2 + [0]
= activate#(V2)
U14#(tt(),V1,V2) = [4] V1 + [4] V2 + [0]
>= [4] V1 + [0]
= isNat#(activate(V1))
U15#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= activate#(V2)
U15#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= isNat#(activate(V2))
U21#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= activate#(V1)
U21#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= isNatKind#(activate(V1))
U22#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= activate#(V1)
U22#(tt(),V1) = [4] V1 + [0]
>= [4] V1 + [0]
= isNat#(activate(V1))
U31#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= activate#(V2)
U31#(tt(),V2) = [4] V2 + [0]
>= [4] V2 + [0]
= isNatKind#(activate(V2))
U51#(tt(),N) = [5] N + [4]
>= [4] N + [2]
= U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) = [5] N + [4]
>= [4] N + [0]
= isNatKind#(activate(N))
U52#(tt(),N) = [4] N + [2]
>= [4] N + [0]
= activate#(N)
U61#(tt(),M,N) = [7] M + [4] N + [4]
>= [4] M + [4] N + [4]
= U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) = [7] M + [4] N + [4]
>= [4] M + [0]
= isNatKind#(activate(M))
U62#(tt(),M,N) = [4] M + [4] N + [4]
>= [4] M + [4] N + [0]
= U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) = [4] M + [4] N + [4]
>= [4] M + [0]
= activate#(M)
U62#(tt(),M,N) = [4] M + [4] N + [4]
>= [4] N + [0]
= activate#(N)
U62#(tt(),M,N) = [4] M + [4] N + [4]
>= [4] N + [0]
= isNat#(activate(N))
U63#(tt(),M,N) = [4] M + [4] N + [0]
>= [4] M + [4] N + [0]
= U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) = [4] M + [4] N + [0]
>= [4] M + [0]
= activate#(M)
U63#(tt(),M,N) = [4] M + [4] N + [0]
>= [4] N + [0]
= activate#(N)
U63#(tt(),M,N) = [4] M + [4] N + [0]
>= [4] N + [0]
= isNatKind#(activate(N))
U64#(tt(),M,N) = [4] M + [4] N + [0]
>= [4] M + [0]
= activate#(M)
U64#(tt(),M,N) = [4] M + [4] N + [0]
>= [4] N + [0]
= activate#(N)
activate#(n__s(X)) = [4] X + [4]
>= [4] X + [4]
= c_23(activate#(X))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [4] V2 + [0]
= U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [0]
= activate#(V1)
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V2 + [0]
= activate#(V2)
isNat#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [0]
= isNatKind#(activate(V1))
isNat#(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [0]
= U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [0]
= activate#(V1)
isNat#(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [0]
= isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V2 + [0]
= U31#(isNatKind(activate(V1)),activate(V2))
isNatKind#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [0]
= activate#(V1)
isNatKind#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V2 + [0]
= activate#(V2)
isNatKind#(n__plus(V1,V2)) = [4] V1 + [4] V2 + [8]
>= [4] V1 + [0]
= isNatKind#(activate(V1))
isNatKind#(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [0]
= activate#(V1)
isNatKind#(n__s(V1)) = [4] V1 + [4]
>= [4] V1 + [0]
= isNatKind#(activate(V1))
0() = [0]
>= [0]
= n__0()
U51(tt(),N) = [1] N + [0]
>= [1] N + [0]
= U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) = [1] N + [0]
>= [1] N + [0]
= activate(N)
U61(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= s(plus(activate(N),activate(M)))
activate(X) = [1] X + [0]
>= [1] X + [0]
= X
activate(n__0()) = [0]
>= [0]
= 0()
activate(n__plus(X1,X2)) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= plus(activate(X1),activate(X2))
activate(n__s(X)) = [1] X + [1]
>= [1] X + [1]
= s(activate(X))
plus(N,0()) = [1] N + [2]
>= [1] N + [0]
= U51(isNat(N),N)
plus(N,s(M)) = [1] M + [1] N + [3]
>= [1] M + [1] N + [3]
= U61(isNat(M),M,N)
plus(X1,X2) = [1] X1 + [1] X2 + [2]
>= [1] X1 + [1] X2 + [2]
= n__plus(X1,X2)
s(X) = [1] X + [1]
>= [1] X + [1]
= n__s(X)
*** Step 6.b:1.b:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
U11#(tt(),V1,V2) -> U12#(isNatKind(activate(V1)),activate(V1),activate(V2))
U11#(tt(),V1,V2) -> activate#(V1)
U11#(tt(),V1,V2) -> activate#(V2)
U11#(tt(),V1,V2) -> isNatKind#(activate(V1))
U12#(tt(),V1,V2) -> U13#(isNatKind(activate(V2)),activate(V1),activate(V2))
U12#(tt(),V1,V2) -> activate#(V1)
U12#(tt(),V1,V2) -> activate#(V2)
U12#(tt(),V1,V2) -> isNatKind#(activate(V2))
U13#(tt(),V1,V2) -> U14#(isNatKind(activate(V2)),activate(V1),activate(V2))
U13#(tt(),V1,V2) -> activate#(V1)
U13#(tt(),V1,V2) -> activate#(V2)
U13#(tt(),V1,V2) -> isNatKind#(activate(V2))
U14#(tt(),V1,V2) -> U15#(isNat(activate(V1)),activate(V2))
U14#(tt(),V1,V2) -> activate#(V1)
U14#(tt(),V1,V2) -> activate#(V2)
U14#(tt(),V1,V2) -> isNat#(activate(V1))
U15#(tt(),V2) -> activate#(V2)
U15#(tt(),V2) -> isNat#(activate(V2))
U21#(tt(),V1) -> U22#(isNatKind(activate(V1)),activate(V1))
U21#(tt(),V1) -> activate#(V1)
U21#(tt(),V1) -> isNatKind#(activate(V1))
U22#(tt(),V1) -> activate#(V1)
U22#(tt(),V1) -> isNat#(activate(V1))
U31#(tt(),V2) -> activate#(V2)
U31#(tt(),V2) -> isNatKind#(activate(V2))
U51#(tt(),N) -> U52#(isNatKind(activate(N)),activate(N))
U51#(tt(),N) -> isNatKind#(activate(N))
U52#(tt(),N) -> activate#(N)
U61#(tt(),M,N) -> U62#(isNatKind(activate(M)),activate(M),activate(N))
U61#(tt(),M,N) -> isNatKind#(activate(M))
U62#(tt(),M,N) -> U63#(isNat(activate(N)),activate(M),activate(N))
U62#(tt(),M,N) -> activate#(M)
U62#(tt(),M,N) -> activate#(N)
U62#(tt(),M,N) -> isNat#(activate(N))
U63#(tt(),M,N) -> U64#(isNatKind(activate(N)),activate(M),activate(N))
U63#(tt(),M,N) -> activate#(M)
U63#(tt(),M,N) -> activate#(N)
U63#(tt(),M,N) -> isNatKind#(activate(N))
U64#(tt(),M,N) -> activate#(M)
U64#(tt(),M,N) -> activate#(N)
activate#(n__plus(X1,X2)) -> c_22(activate#(X1),activate#(X2))
activate#(n__s(X)) -> c_23(activate#(X))
isNat#(n__plus(V1,V2)) -> U11#(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat#(n__plus(V1,V2)) -> activate#(V1)
isNat#(n__plus(V1,V2)) -> activate#(V2)
isNat#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNat#(n__s(V1)) -> U21#(isNatKind(activate(V1)),activate(V1))
isNat#(n__s(V1)) -> activate#(V1)
isNat#(n__s(V1)) -> isNatKind#(activate(V1))
isNatKind#(n__plus(V1,V2)) -> U31#(isNatKind(activate(V1)),activate(V2))
isNatKind#(n__plus(V1,V2)) -> activate#(V1)
isNatKind#(n__plus(V1,V2)) -> activate#(V2)
isNatKind#(n__plus(V1,V2)) -> isNatKind#(activate(V1))
isNatKind#(n__s(V1)) -> activate#(V1)
isNatKind#(n__s(V1)) -> isNatKind#(activate(V1))
- Weak TRS:
0() -> n__0()
U11(tt(),V1,V2) -> U12(isNatKind(activate(V1)),activate(V1),activate(V2))
U12(tt(),V1,V2) -> U13(isNatKind(activate(V2)),activate(V1),activate(V2))
U13(tt(),V1,V2) -> U14(isNatKind(activate(V2)),activate(V1),activate(V2))
U14(tt(),V1,V2) -> U15(isNat(activate(V1)),activate(V2))
U15(tt(),V2) -> U16(isNat(activate(V2)))
U16(tt()) -> tt()
U21(tt(),V1) -> U22(isNatKind(activate(V1)),activate(V1))
U22(tt(),V1) -> U23(isNat(activate(V1)))
U23(tt()) -> tt()
U31(tt(),V2) -> U32(isNatKind(activate(V2)))
U32(tt()) -> tt()
U41(tt()) -> tt()
U51(tt(),N) -> U52(isNatKind(activate(N)),activate(N))
U52(tt(),N) -> activate(N)
U61(tt(),M,N) -> U62(isNatKind(activate(M)),activate(M),activate(N))
U62(tt(),M,N) -> U63(isNat(activate(N)),activate(M),activate(N))
U63(tt(),M,N) -> U64(isNatKind(activate(N)),activate(M),activate(N))
U64(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
activate(n__0()) -> 0()
activate(n__plus(X1,X2)) -> plus(activate(X1),activate(X2))
activate(n__s(X)) -> s(activate(X))
isNat(n__0()) -> tt()
isNat(n__plus(V1,V2)) -> U11(isNatKind(activate(V1)),activate(V1),activate(V2))
isNat(n__s(V1)) -> U21(isNatKind(activate(V1)),activate(V1))
isNatKind(n__0()) -> tt()
isNatKind(n__plus(V1,V2)) -> U31(isNatKind(activate(V1)),activate(V2))
isNatKind(n__s(V1)) -> U41(isNatKind(activate(V1)))
plus(N,0()) -> U51(isNat(N),N)
plus(N,s(M)) -> U61(isNat(M),M,N)
plus(X1,X2) -> n__plus(X1,X2)
s(X) -> n__s(X)
- Signature:
{0/0,U11/3,U12/3,U13/3,U14/3,U15/2,U16/1,U21/2,U22/2,U23/1,U31/2,U32/1,U41/1,U51/2,U52/2,U61/3,U62/3,U63/3
,U64/3,activate/1,isNat/1,isNatKind/1,plus/2,s/1,0#/0,U11#/3,U12#/3,U13#/3,U14#/3,U15#/2,U16#/1,U21#/2
,U22#/2,U23#/1,U31#/2,U32#/1,U41#/1,U51#/2,U52#/2,U61#/3,U62#/3,U63#/3,U64#/3,activate#/1,isNat#/1
,isNatKind#/1,plus#/2,s#/1} / {n__0/0,n__plus/2,n__s/1,tt/0,c_1/0,c_2/5,c_3/5,c_4/5,c_5/4,c_6/2,c_7/0,c_8/4
,c_9/2,c_10/0,c_11/2,c_12/0,c_13/0,c_14/4,c_15/1,c_16/5,c_17/5,c_18/5,c_19/2,c_20/0,c_21/1,c_22/2,c_23/1
,c_24/0,c_25/5,c_26/4,c_27/0,c_28/4,c_29/2,c_30/2,c_31/2,c_32/0,c_33/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {0#,U11#,U12#,U13#,U14#,U15#,U16#,U21#,U22#,U23#,U31#,U32#
,U41#,U51#,U52#,U61#,U62#,U63#,U64#,activate#,isNat#,isNatKind#,plus#,s#} and constructors {n__0,n__plus
,n__s,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^3))