(STRATEGY
    INNERMOST)

(VAR
    M N V1 V2 X X1 X2 X3)
(DATATYPES
    A = µX.< tt, mark(X), 0, ok(X) >)
(SIGNATURES
    active :: [A] -> A
    U101 :: [A x A x A] -> A
    U102 :: [A x A x A] -> A
    U103 :: [A x A x A] -> A
    U104 :: [A x A x A] -> A
    plus :: [A x A] -> A
    x :: [A x A] -> A
    U11 :: [A x A x A] -> A
    U12 :: [A x A x A] -> A
    U13 :: [A x A x A] -> A
    U14 :: [A x A x A] -> A
    U15 :: [A x A] -> A
    U16 :: [A] -> A
    U21 :: [A x A] -> A
    U22 :: [A x A] -> A
    U23 :: [A] -> A
    U31 :: [A x A x A] -> A
    U32 :: [A x A x A] -> A
    U33 :: [A x A x A] -> A
    U34 :: [A x A x A] -> A
    U35 :: [A x A] -> A
    U36 :: [A] -> A
    U41 :: [A x A] -> A
    U42 :: [A] -> A
    U51 :: [A] -> A
    U61 :: [A x A] -> A
    U62 :: [A] -> A
    U71 :: [A x A] -> A
    U72 :: [A x A] -> A
    U81 :: [A x A x A] -> A
    U82 :: [A x A x A] -> A
    U83 :: [A x A x A] -> A
    U84 :: [A x A x A] -> A
    s :: [A] -> A
    U91 :: [A x A] -> A
    U92 :: [A] -> A
    proper :: [A] -> A
    isNatKind :: [A] -> A
    isNat :: [A] -> A
    top :: [A] -> A)
(RULES
    active(U101(tt(),M,N)) ->
      mark(U102(isNatKind(M),M,N))
    active(U102(tt(),M,N)) ->
      mark(U103(isNat(N),M,N))
    active(U103(tt(),M,N)) ->
      mark(U104(isNatKind(N),M,N))
    active(U104(tt(),M,N)) ->
      mark(plus(x(N,M),N))
    active(U11(tt(),V1,V2)) ->
      mark(U12(isNatKind(V1),V1,V2))
    active(U12(tt(),V1,V2)) ->
      mark(U13(isNatKind(V2),V1,V2))
    active(U13(tt(),V1,V2)) ->
      mark(U14(isNatKind(V2),V1,V2))
    active(U14(tt(),V1,V2)) ->
      mark(U15(isNat(V1),V2))
    active(U15(tt(),V2)) ->
      mark(U16(isNat(V2)))
    active(U16(tt())) -> mark(tt())
    active(U21(tt(),V1)) ->
      mark(U22(isNatKind(V1),V1))
    active(U22(tt(),V1)) ->
      mark(U23(isNat(V1)))
    active(U23(tt())) -> mark(tt())
    active(U31(tt(),V1,V2)) ->
      mark(U32(isNatKind(V1),V1,V2))
    active(U32(tt(),V1,V2)) ->
      mark(U33(isNatKind(V2),V1,V2))
    active(U33(tt(),V1,V2)) ->
      mark(U34(isNatKind(V2),V1,V2))
    active(U34(tt(),V1,V2)) ->
      mark(U35(isNat(V1),V2))
    active(U35(tt(),V2)) ->
      mark(U36(isNat(V2)))
    active(U36(tt())) -> mark(tt())
    active(U41(tt(),V2)) ->
      mark(U42(isNatKind(V2)))
    active(U42(tt())) -> mark(tt())
    active(U51(tt())) -> mark(tt())
    active(U61(tt(),V2)) ->
      mark(U62(isNatKind(V2)))
    active(U62(tt())) -> mark(tt())
    active(U71(tt(),N)) ->
      mark(U72(isNatKind(N),N))
    active(U72(tt(),N)) -> mark(N)
    active(U81(tt(),M,N)) ->
      mark(U82(isNatKind(M),M,N))
    active(U82(tt(),M,N)) ->
      mark(U83(isNat(N),M,N))
    active(U83(tt(),M,N)) ->
      mark(U84(isNatKind(N),M,N))
    active(U84(tt(),M,N)) ->
      mark(s(plus(N,M)))
    active(U91(tt(),N)) ->
      mark(U92(isNatKind(N)))
    active(U92(tt())) -> mark(0())
    active(isNat(0())) -> mark(tt())
    active(isNat(plus(V1,V2))) ->
      mark(U11(isNatKind(V1),V1,V2))
    active(isNat(s(V1))) ->
      mark(U21(isNatKind(V1),V1))
    active(isNat(x(V1,V2))) ->
      mark(U31(isNatKind(V1),V1,V2))
    active(isNatKind(0())) ->
      mark(tt())
    active(isNatKind(plus(V1
                         ,V2))) -> mark(U41(isNatKind(V1)
                                           ,V2))
    active(isNatKind(s(V1))) ->
      mark(U51(isNatKind(V1)))
    active(isNatKind(x(V1,V2))) ->
      mark(U61(isNatKind(V1),V2))
    active(plus(N,0())) ->
      mark(U71(isNat(N),N))
    active(plus(N,s(M))) ->
      mark(U81(isNat(M),M,N))
    active(x(N,0())) ->
      mark(U91(isNat(N),N))
    active(x(N,s(M))) ->
      mark(U101(isNat(M),M,N))
    active(U101(X1,X2,X3)) ->
      U101(active(X1),X2,X3)
    active(U102(X1,X2,X3)) ->
      U102(active(X1),X2,X3)
    active(U103(X1,X2,X3)) ->
      U103(active(X1),X2,X3)
    active(U104(X1,X2,X3)) ->
      U104(active(X1),X2,X3)
    active(plus(X1,X2)) ->
      plus(active(X1),X2)
    active(plus(X1,X2)) -> plus(X1
                               ,active(X2))
    active(x(X1,X2)) -> x(active(X1)
                         ,X2)
    active(x(X1,X2)) -> x(X1
                         ,active(X2))
    active(U11(X1,X2,X3)) ->
      U11(active(X1),X2,X3)
    active(U12(X1,X2,X3)) ->
      U12(active(X1),X2,X3)
    active(U13(X1,X2,X3)) ->
      U13(active(X1),X2,X3)
    active(U14(X1,X2,X3)) ->
      U14(active(X1),X2,X3)
    active(U15(X1,X2)) ->
      U15(active(X1),X2)
    active(U16(X)) -> U16(active(X))
    active(U21(X1,X2)) ->
      U21(active(X1),X2)
    active(U22(X1,X2)) ->
      U22(active(X1),X2)
    active(U23(X)) -> U23(active(X))
    active(U31(X1,X2,X3)) ->
      U31(active(X1),X2,X3)
    active(U32(X1,X2,X3)) ->
      U32(active(X1),X2,X3)
    active(U33(X1,X2,X3)) ->
      U33(active(X1),X2,X3)
    active(U34(X1,X2,X3)) ->
      U34(active(X1),X2,X3)
    active(U35(X1,X2)) ->
      U35(active(X1),X2)
    active(U36(X)) -> U36(active(X))
    active(U41(X1,X2)) ->
      U41(active(X1),X2)
    active(U42(X)) -> U42(active(X))
    active(U51(X)) -> U51(active(X))
    active(U61(X1,X2)) ->
      U61(active(X1),X2)
    active(U62(X)) -> U62(active(X))
    active(U71(X1,X2)) ->
      U71(active(X1),X2)
    active(U72(X1,X2)) ->
      U72(active(X1),X2)
    active(U81(X1,X2,X3)) ->
      U81(active(X1),X2,X3)
    active(U82(X1,X2,X3)) ->
      U82(active(X1),X2,X3)
    active(U83(X1,X2,X3)) ->
      U83(active(X1),X2,X3)
    active(U84(X1,X2,X3)) ->
      U84(active(X1),X2,X3)
    active(s(X)) -> s(active(X))
    active(U91(X1,X2)) ->
      U91(active(X1),X2)
    active(U92(X)) -> U92(active(X))
    U101(mark(X1),X2,X3) ->
      mark(U101(X1,X2,X3))
    U102(mark(X1),X2,X3) ->
      mark(U102(X1,X2,X3))
    U103(mark(X1),X2,X3) ->
      mark(U103(X1,X2,X3))
    U104(mark(X1),X2,X3) ->
      mark(U104(X1,X2,X3))
    plus(mark(X1),X2) ->
      mark(plus(X1,X2))
    plus(X1,mark(X2)) ->
      mark(plus(X1,X2))
    x(mark(X1),X2) -> mark(x(X1,X2))
    x(X1,mark(X2)) -> mark(x(X1,X2))
    U11(mark(X1),X2,X3) ->
      mark(U11(X1,X2,X3))
    U12(mark(X1),X2,X3) ->
      mark(U12(X1,X2,X3))
    U13(mark(X1),X2,X3) ->
      mark(U13(X1,X2,X3))
    U14(mark(X1),X2,X3) ->
      mark(U14(X1,X2,X3))
    U15(mark(X1),X2) -> mark(U15(X1
                                ,X2))
    U16(mark(X)) -> mark(U16(X))
    U21(mark(X1),X2) -> mark(U21(X1
                                ,X2))
    U22(mark(X1),X2) -> mark(U22(X1
                                ,X2))
    U23(mark(X)) -> mark(U23(X))
    U31(mark(X1),X2,X3) ->
      mark(U31(X1,X2,X3))
    U32(mark(X1),X2,X3) ->
      mark(U32(X1,X2,X3))
    U33(mark(X1),X2,X3) ->
      mark(U33(X1,X2,X3))
    U34(mark(X1),X2,X3) ->
      mark(U34(X1,X2,X3))
    U35(mark(X1),X2) -> mark(U35(X1
                                ,X2))
    U36(mark(X)) -> mark(U36(X))
    U41(mark(X1),X2) -> mark(U41(X1
                                ,X2))
    U42(mark(X)) -> mark(U42(X))
    U51(mark(X)) -> mark(U51(X))
    U61(mark(X1),X2) -> mark(U61(X1
                                ,X2))
    U62(mark(X)) -> mark(U62(X))
    U71(mark(X1),X2) -> mark(U71(X1
                                ,X2))
    U72(mark(X1),X2) -> mark(U72(X1
                                ,X2))
    U81(mark(X1),X2,X3) ->
      mark(U81(X1,X2,X3))
    U82(mark(X1),X2,X3) ->
      mark(U82(X1,X2,X3))
    U83(mark(X1),X2,X3) ->
      mark(U83(X1,X2,X3))
    U84(mark(X1),X2,X3) ->
      mark(U84(X1,X2,X3))
    s(mark(X)) -> mark(s(X))
    U91(mark(X1),X2) -> mark(U91(X1
                                ,X2))
    U92(mark(X)) -> mark(U92(X))
    proper(U101(X1,X2,X3)) ->
      U101(proper(X1)
          ,proper(X2)
          ,proper(X3))
    proper(tt()) -> ok(tt())
    proper(U102(X1,X2,X3)) ->
      U102(proper(X1)
          ,proper(X2)
          ,proper(X3))
    proper(isNatKind(X)) ->
      isNatKind(proper(X))
    proper(U103(X1,X2,X3)) ->
      U103(proper(X1)
          ,proper(X2)
          ,proper(X3))
    proper(isNat(X)) ->
      isNat(proper(X))
    proper(U104(X1,X2,X3)) ->
      U104(proper(X1)
          ,proper(X2)
          ,proper(X3))
    proper(plus(X1,X2)) ->
      plus(proper(X1),proper(X2))
    proper(x(X1,X2)) -> x(proper(X1)
                         ,proper(X2))
    proper(U11(X1,X2,X3)) ->
      U11(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(U12(X1,X2,X3)) ->
      U12(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(U13(X1,X2,X3)) ->
      U13(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(U14(X1,X2,X3)) ->
      U14(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(U15(X1,X2)) ->
      U15(proper(X1),proper(X2))
    proper(U16(X)) -> U16(proper(X))
    proper(U21(X1,X2)) ->
      U21(proper(X1),proper(X2))
    proper(U22(X1,X2)) ->
      U22(proper(X1),proper(X2))
    proper(U23(X)) -> U23(proper(X))
    proper(U31(X1,X2,X3)) ->
      U31(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(U32(X1,X2,X3)) ->
      U32(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(U33(X1,X2,X3)) ->
      U33(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(U34(X1,X2,X3)) ->
      U34(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(U35(X1,X2)) ->
      U35(proper(X1),proper(X2))
    proper(U36(X)) -> U36(proper(X))
    proper(U41(X1,X2)) ->
      U41(proper(X1),proper(X2))
    proper(U42(X)) -> U42(proper(X))
    proper(U51(X)) -> U51(proper(X))
    proper(U61(X1,X2)) ->
      U61(proper(X1),proper(X2))
    proper(U62(X)) -> U62(proper(X))
    proper(U71(X1,X2)) ->
      U71(proper(X1),proper(X2))
    proper(U72(X1,X2)) ->
      U72(proper(X1),proper(X2))
    proper(U81(X1,X2,X3)) ->
      U81(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(U82(X1,X2,X3)) ->
      U82(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(U83(X1,X2,X3)) ->
      U83(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(U84(X1,X2,X3)) ->
      U84(proper(X1)
         ,proper(X2)
         ,proper(X3))
    proper(s(X)) -> s(proper(X))
    proper(U91(X1,X2)) ->
      U91(proper(X1),proper(X2))
    proper(U92(X)) -> U92(proper(X))
    proper(0()) -> ok(0())
    U101(ok(X1),ok(X2),ok(X3)) ->
      ok(U101(X1,X2,X3))
    U102(ok(X1),ok(X2),ok(X3)) ->
      ok(U102(X1,X2,X3))
    isNatKind(ok(X)) ->
      ok(isNatKind(X))
    U103(ok(X1),ok(X2),ok(X3)) ->
      ok(U103(X1,X2,X3))
    isNat(ok(X)) -> ok(isNat(X))
    U104(ok(X1),ok(X2),ok(X3)) ->
      ok(U104(X1,X2,X3))
    plus(ok(X1),ok(X2)) ->
      ok(plus(X1,X2))
    x(ok(X1),ok(X2)) -> ok(x(X1,X2))
    U11(ok(X1),ok(X2),ok(X3)) ->
      ok(U11(X1,X2,X3))
    U12(ok(X1),ok(X2),ok(X3)) ->
      ok(U12(X1,X2,X3))
    U13(ok(X1),ok(X2),ok(X3)) ->
      ok(U13(X1,X2,X3))
    U14(ok(X1),ok(X2),ok(X3)) ->
      ok(U14(X1,X2,X3))
    U15(ok(X1),ok(X2)) -> ok(U15(X1
                                ,X2))
    U16(ok(X)) -> ok(U16(X))
    U21(ok(X1),ok(X2)) -> ok(U21(X1
                                ,X2))
    U22(ok(X1),ok(X2)) -> ok(U22(X1
                                ,X2))
    U23(ok(X)) -> ok(U23(X))
    U31(ok(X1),ok(X2),ok(X3)) ->
      ok(U31(X1,X2,X3))
    U32(ok(X1),ok(X2),ok(X3)) ->
      ok(U32(X1,X2,X3))
    U33(ok(X1),ok(X2),ok(X3)) ->
      ok(U33(X1,X2,X3))
    U34(ok(X1),ok(X2),ok(X3)) ->
      ok(U34(X1,X2,X3))
    U35(ok(X1),ok(X2)) -> ok(U35(X1
                                ,X2))
    U36(ok(X)) -> ok(U36(X))
    U41(ok(X1),ok(X2)) -> ok(U41(X1
                                ,X2))
    U42(ok(X)) -> ok(U42(X))
    U51(ok(X)) -> ok(U51(X))
    U61(ok(X1),ok(X2)) -> ok(U61(X1
                                ,X2))
    U62(ok(X)) -> ok(U62(X))
    U71(ok(X1),ok(X2)) -> ok(U71(X1
                                ,X2))
    U72(ok(X1),ok(X2)) -> ok(U72(X1
                                ,X2))
    U81(ok(X1),ok(X2),ok(X3)) ->
      ok(U81(X1,X2,X3))
    U82(ok(X1),ok(X2),ok(X3)) ->
      ok(U82(X1,X2,X3))
    U83(ok(X1),ok(X2),ok(X3)) ->
      ok(U83(X1,X2,X3))
    U84(ok(X1),ok(X2),ok(X3)) ->
      ok(U84(X1,X2,X3))
    s(ok(X)) -> ok(s(X))
    U91(ok(X1),ok(X2)) -> ok(U91(X1
                                ,X2))
    U92(ok(X)) -> ok(U92(X))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X)))