YES(O(1), O(n^2)) 22.68/6.38 YES(O(1), O(n^2)) 23.42/6.42 23.42/6.42 23.42/6.42
23.42/6.42 23.42/6.420 CpxTRS23.42/6.42
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))23.42/6.42
↳2 CdtProblem23.42/6.42
↳3 CdtUnreachableProof (⇔)23.42/6.42
↳4 CdtProblem23.42/6.42
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))23.42/6.42
↳6 CdtProblem23.42/6.42
↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))23.42/6.42
↳8 CdtProblem23.42/6.42
↳9 CdtLeafRemovalProof (ComplexityIfPolyImplication)23.42/6.42
↳10 CdtProblem23.42/6.42
↳11 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))23.42/6.42
↳12 CdtProblem23.42/6.42
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))23.42/6.42
↳14 CdtProblem23.42/6.42
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))23.42/6.42
↳16 CdtProblem23.42/6.42
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))23.42/6.42
↳18 CdtProblem23.42/6.42
↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))23.42/6.42
↳20 CdtProblem23.42/6.42
↳21 CdtKnowledgeProof (⇔)23.42/6.42
↳22 BOUNDS(O(1), O(1))23.42/6.42
U11(tt, V2) → U12(isNat(activate(V2))) 23.42/6.42
U12(tt) → tt 23.42/6.42
U21(tt) → tt 23.42/6.42
U31(tt, V2) → U32(isNat(activate(V2))) 23.42/6.42
U32(tt) → tt 23.42/6.42
U41(tt, N) → activate(N) 23.42/6.42
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N)) 23.42/6.42
U52(tt, M, N) → s(plus(activate(N), activate(M))) 23.42/6.42
U61(tt) → 0 23.42/6.42
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N)) 23.42/6.42
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N)) 23.42/6.42
isNat(n__0) → tt 23.42/6.42
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2)) 23.42/6.42
isNat(n__s(V1)) → U21(isNat(activate(V1))) 23.42/6.42
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2)) 23.42/6.42
plus(N, 0) → U41(isNat(N), N) 23.42/6.42
plus(N, s(M)) → U51(isNat(M), M, N) 23.42/6.42
x(N, 0) → U61(isNat(N)) 23.42/6.42
x(N, s(M)) → U71(isNat(M), M, N) 23.42/6.42
0 → n__0 23.42/6.42
plus(X1, X2) → n__plus(X1, X2) 23.42/6.42
s(X) → n__s(X) 23.42/6.42
x(X1, X2) → n__x(X1, X2) 23.42/6.42
activate(n__0) → 0 23.42/6.42
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2)) 23.42/6.42
activate(n__s(X)) → s(activate(X)) 23.42/6.42
activate(n__x(X1, X2)) → x(activate(X1), activate(X2)) 23.42/6.42
activate(X) → X
Tuples:
U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.42
U12(tt) → tt 23.42/6.42
U21(tt) → tt 23.42/6.42
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.42
U32(tt) → tt 23.42/6.42
U41(tt, z0) → activate(z0) 23.42/6.42
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.42
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.42
U61(tt) → 0 23.42/6.42
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.42
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.42
isNat(n__0) → tt 23.42/6.42
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.42
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.42
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.42
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.42
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.42
plus(z0, z1) → n__plus(z0, z1) 23.42/6.42
x(z0, 0) → U61(isNat(z0)) 23.42/6.42
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.42
x(z0, z1) → n__x(z0, z1) 23.42/6.42
0 → n__0 23.42/6.42
s(z0) → n__s(z0) 23.42/6.42
activate(n__0) → 0 23.42/6.42
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.42
activate(n__s(z0)) → s(activate(z0)) 23.42/6.42
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.42
activate(z0) → z0
S tuples:
U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.42
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.42
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.42
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.42
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.42
U61'(tt) → c8(0') 23.42/6.42
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.42
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.42
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.42
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.42
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.42
PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0)) 23.42/6.42
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1)) 23.42/6.44
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0)) 23.42/6.44
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1)) 23.42/6.44
ACTIVATE(n__0) → c23(0') 23.42/6.44
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0)) 23.42/6.44
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.44
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.44
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.44
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.44
U61'(tt) → c8(0') 23.42/6.44
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.44
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0)) 23.42/6.44
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1)) 23.42/6.44
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0)) 23.42/6.44
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1)) 23.42/6.44
ACTIVATE(n__0) → c23(0') 23.42/6.44
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0)) 23.42/6.44
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate
U11', U31', U41', U51', U52', U61', U71', U72', ISNAT, PLUS, X, ACTIVATE
c, c3, c5, c6, c7, c8, c9, c10, c12, c13, c14, c15, c16, c18, c19, c23, c24, c25, c26
PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0)) 23.42/6.44
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1)) 23.42/6.44
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0)) 23.42/6.44
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1))
Tuples:
U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.44
U12(tt) → tt 23.42/6.44
U21(tt) → tt 23.42/6.44
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.44
U32(tt) → tt 23.42/6.44
U41(tt, z0) → activate(z0) 23.42/6.44
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.44
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.44
U61(tt) → 0 23.42/6.44
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.44
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.44
isNat(n__0) → tt 23.42/6.44
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.44
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.44
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.44
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.44
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.44
plus(z0, z1) → n__plus(z0, z1) 23.42/6.44
x(z0, 0) → U61(isNat(z0)) 23.42/6.44
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.44
x(z0, z1) → n__x(z0, z1) 23.42/6.44
0 → n__0 23.42/6.44
s(z0) → n__s(z0) 23.42/6.44
activate(n__0) → 0 23.42/6.44
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.44
activate(n__s(z0)) → s(activate(z0)) 23.42/6.44
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.44
activate(z0) → z0
S tuples:
U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.44
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.44
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.44
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.44
U61'(tt) → c8(0') 23.42/6.44
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.44
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.45
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ACTIVATE(n__0) → c23(0') 23.42/6.45
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0)) 23.42/6.45
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.45
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.45
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.45
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.45
U61'(tt) → c8(0') 23.42/6.45
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.45
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ACTIVATE(n__0) → c23(0') 23.42/6.45
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0)) 23.42/6.45
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate
U11', U31', U41', U51', U52', U61', U71', U72', ISNAT, ACTIVATE
c, c3, c5, c6, c7, c8, c9, c10, c12, c13, c14, c23, c24, c25, c26
Tuples:
U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.45
U12(tt) → tt 23.42/6.45
U21(tt) → tt 23.42/6.45
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.45
U32(tt) → tt 23.42/6.45
U41(tt, z0) → activate(z0) 23.42/6.45
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.45
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.45
U61(tt) → 0 23.42/6.45
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.45
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.45
isNat(n__0) → tt 23.42/6.45
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.45
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.45
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.45
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.45
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.45
plus(z0, z1) → n__plus(z0, z1) 23.42/6.45
x(z0, 0) → U61(isNat(z0)) 23.42/6.45
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.45
x(z0, z1) → n__x(z0, z1) 23.42/6.45
0 → n__0 23.42/6.45
s(z0) → n__s(z0) 23.42/6.45
activate(n__0) → 0 23.42/6.45
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.45
activate(n__s(z0)) → s(activate(z0)) 23.42/6.45
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.45
activate(z0) → z0
S tuples:
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.45
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.45
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U52'(tt, z0, z1) → c7(ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.46
U61'(tt) → c8 23.42/6.46
U72'(tt, z0, z1) → c10(ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.46
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U52'(tt, z0, z1) → c7(ACTIVATE(z1), ACTIVATE(z0)) 23.42/6.46
U61'(tt) → c8 23.42/6.46
U72'(tt, z0, z1) → c10(ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate
U41', U51', U71', ISNAT, U11', U31', U52', U61', U72', ACTIVATE
c5, c6, c9, c12, c14, c, c3, c7, c8, c10, c13, c23, c24, c25, c26
Tuples:
U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.46
U12(tt) → tt 23.42/6.46
U21(tt) → tt 23.42/6.46
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.46
U32(tt) → tt 23.42/6.46
U41(tt, z0) → activate(z0) 23.42/6.46
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.46
U61(tt) → 0 23.42/6.46
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.46
isNat(n__0) → tt 23.42/6.46
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.46
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.46
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.46
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.46
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.46
plus(z0, z1) → n__plus(z0, z1) 23.42/6.46
x(z0, 0) → U61(isNat(z0)) 23.42/6.46
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.46
x(z0, z1) → n__x(z0, z1) 23.42/6.46
0 → n__0 23.42/6.46
s(z0) → n__s(z0) 23.42/6.46
activate(n__0) → 0 23.42/6.46
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.46
activate(n__s(z0)) → s(activate(z0)) 23.42/6.46
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.46
activate(z0) → z0
S tuples:
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.46
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U61'(tt) → c8 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:none
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.46
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U61'(tt) → c8 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate
U41', ISNAT, U11', U31', U61', ACTIVATE, U51', U71', U52', U72'
c5, c12, c14, c, c3, c8, c13, c23, c24, c25, c26, c1
Removed 2 trailing nodes:
U41'(tt, z0) → c5(ACTIVATE(z0)) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U61'(tt) → c8 23.42/6.46
ACTIVATE(n__0) → c23
Tuples:
U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.46
U12(tt) → tt 23.42/6.46
U21(tt) → tt 23.42/6.46
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.46
U32(tt) → tt 23.42/6.46
U41(tt, z0) → activate(z0) 23.42/6.46
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.46
U61(tt) → 0 23.42/6.46
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.46
isNat(n__0) → tt 23.42/6.46
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.46
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.46
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.46
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.46
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.46
plus(z0, z1) → n__plus(z0, z1) 23.42/6.46
x(z0, 0) → U61(isNat(z0)) 23.42/6.46
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.46
x(z0, z1) → n__x(z0, z1) 23.42/6.46
0 → n__0 23.42/6.46
s(z0) → n__s(z0) 23.42/6.46
activate(n__0) → 0 23.42/6.46
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.46
activate(n__s(z0)) → s(activate(z0)) 23.42/6.46
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.46
activate(z0) → z0
S tuples:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:none
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__0) → c23 23.42/6.46
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.46
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.46
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate
ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'
c12, c14, c, c3, c13, c23, c24, c25, c26, c1
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.46
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.46
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
Tuples:
U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.46
U12(tt) → tt 23.42/6.46
U21(tt) → tt 23.42/6.46
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.46
U32(tt) → tt 23.42/6.46
U41(tt, z0) → activate(z0) 23.42/6.46
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.46
U61(tt) → 0 23.42/6.46
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.46
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.46
isNat(n__0) → tt 23.42/6.46
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.46
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.46
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.46
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.46
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.42/6.46
plus(z0, z1) → n__plus(z0, z1) 23.42/6.46
x(z0, 0) → U61(isNat(z0)) 23.42/6.46
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.42/6.46
x(z0, z1) → n__x(z0, z1) 23.42/6.46
0 → n__0 23.42/6.46
s(z0) → n__s(z0) 23.42/6.46
activate(n__0) → 0 23.42/6.46
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.46
activate(n__s(z0)) → s(activate(z0)) 23.42/6.46
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.46
activate(z0) → z0
S tuples:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__0) → c23 23.42/6.49
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__0) → c23 23.42/6.49
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate
ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'
c12, c14, c, c3, c13, c23, c24, c25, c26, c1
We considered the (Usable) Rules:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
And the Tuples:
activate(n__0) → 0 23.42/6.49
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.42/6.49
activate(n__s(z0)) → s(activate(z0)) 23.42/6.49
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.42/6.49
activate(z0) → z0 23.42/6.49
x(z0, z1) → n__x(z0, z1) 23.42/6.49
s(z0) → n__s(z0) 23.42/6.49
plus(z0, z1) → n__plus(z0, z1) 23.42/6.49
0 → n__0 23.42/6.49
isNat(n__0) → tt 23.42/6.49
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.49
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.49
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.49
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.49
U32(tt) → tt 23.42/6.49
U21(tt) → tt 23.42/6.49
U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.49
U12(tt) → tt
The order we found is given by the following interpretation:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__0) → c23 23.42/6.49
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.42/6.49
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.42/6.49
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.42/6.49
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.42/6.49
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
POL(0) = 0 23.42/6.49
POL(ACTIVATE(x1)) = 0 23.42/6.49
POL(ISNAT(x1)) = [4]x1 23.42/6.49
POL(U11(x1, x2)) = [3] 23.42/6.49
POL(U11'(x1, x2)) = [3] + [4]x2 23.42/6.49
POL(U12(x1)) = [3] 23.42/6.49
POL(U21(x1)) = [3] 23.42/6.49
POL(U31(x1, x2)) = [3] 23.42/6.49
POL(U31'(x1, x2)) = [4]x2 23.42/6.49
POL(U32(x1)) = [3] + [3]x1 23.42/6.49
POL(U51'(x1, x2, x3)) = x1 + [5]x2 + [4]x3 23.42/6.49
POL(U52'(x1, x2, x3)) = 0 23.42/6.49
POL(U71'(x1, x2, x3)) = x1 + [3]x2 + [4]x3 23.42/6.49
POL(U72'(x1, x2, x3)) = 0 23.42/6.49
POL(activate(x1)) = x1 23.42/6.49
POL(c(x1, x2)) = x1 + x2 23.42/6.49
POL(c1(x1)) = x1 23.42/6.49
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4 23.42/6.49
POL(c13(x1, x2)) = x1 + x2 23.42/6.49
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4 23.42/6.49
POL(c23) = 0 23.42/6.49
POL(c24(x1, x2)) = x1 + x2 23.42/6.49
POL(c25(x1)) = x1 23.42/6.49
POL(c26(x1, x2)) = x1 + x2 23.42/6.49
POL(c3(x1, x2)) = x1 + x2 23.42/6.49
POL(isNat(x1)) = 0 23.42/6.49
POL(n__0) = 0 23.42/6.49
POL(n__plus(x1, x2)) = [1] + x1 + x2 23.42/6.49
POL(n__s(x1)) = [4] + x1 23.42/6.49
POL(n__x(x1, x2)) = x1 + x2 23.42/6.49
POL(plus(x1, x2)) = [1] + x1 + x2 23.42/6.49
POL(s(x1)) = [4] + x1 23.42/6.49
POL(tt) = 0 23.42/6.49
POL(x(x1, x2)) = x1 + x2
Tuples:
U11(tt, z0) → U12(isNat(activate(z0))) 23.42/6.49
U12(tt) → tt 23.42/6.49
U21(tt) → tt 23.42/6.49
U31(tt, z0) → U32(isNat(activate(z0))) 23.42/6.49
U32(tt) → tt 23.42/6.49
U41(tt, z0) → activate(z0) 23.42/6.49
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.49
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.42/6.49
U61(tt) → 0 23.42/6.49
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.42/6.49
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.42/6.49
isNat(n__0) → tt 23.42/6.49
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.42/6.49
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.42/6.49
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.42/6.49
plus(z0, 0) → U41(isNat(z0), z0) 23.42/6.49
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
x(z0, 0) → U61(isNat(z0)) 23.80/6.52
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
0 → n__0 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0
S tuples:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate
ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'
c12, c14, c, c3, c13, c23, c24, c25, c26, c1
We considered the (Usable) Rules:
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
And the Tuples:
activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
0 → n__0 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt
The order we found is given by the following interpretation:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
POL(0) = 0 23.80/6.52
POL(ACTIVATE(x1)) = 0 23.80/6.52
POL(ISNAT(x1)) = [3] + [4]x1 23.80/6.52
POL(U11(x1, x2)) = [2] 23.80/6.52
POL(U11'(x1, x2)) = [3] + x1 + [4]x2 23.80/6.52
POL(U12(x1)) = [1] 23.80/6.52
POL(U21(x1)) = [4] 23.80/6.52
POL(U31(x1, x2)) = [4] 23.80/6.52
POL(U31'(x1, x2)) = [4] + [4]x2 23.80/6.52
POL(U32(x1)) = 0 23.80/6.52
POL(U51'(x1, x2, x3)) = [3] + x1 + [5]x2 + [4]x3 23.80/6.52
POL(U52'(x1, x2, x3)) = [2]x2 23.80/6.52
POL(U71'(x1, x2, x3)) = [3] + x1 + [4]x3 23.80/6.52
POL(U72'(x1, x2, x3)) = 0 23.80/6.52
POL(activate(x1)) = x1 23.80/6.52
POL(c(x1, x2)) = x1 + x2 23.80/6.52
POL(c1(x1)) = x1 23.80/6.52
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4 23.80/6.52
POL(c13(x1, x2)) = x1 + x2 23.80/6.52
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4 23.80/6.52
POL(c23) = 0 23.80/6.52
POL(c24(x1, x2)) = x1 + x2 23.80/6.52
POL(c25(x1)) = x1 23.80/6.52
POL(c26(x1, x2)) = x1 + x2 23.80/6.52
POL(c3(x1, x2)) = x1 + x2 23.80/6.52
POL(isNat(x1)) = [4] 23.80/6.52
POL(n__0) = 0 23.80/6.52
POL(n__plus(x1, x2)) = [4] + x1 + x2 23.80/6.52
POL(n__s(x1)) = x1 23.80/6.52
POL(n__x(x1, x2)) = [1] + x1 + x2 23.80/6.52
POL(plus(x1, x2)) = [4] + x1 + x2 23.80/6.52
POL(s(x1)) = x1 23.80/6.52
POL(tt) = 0 23.80/6.52
POL(x(x1, x2)) = [1] + x1 + x2
Tuples:
U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U41(tt, z0) → activate(z0) 23.80/6.52
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.80/6.52
U61(tt) → 0 23.80/6.52
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
plus(z0, 0) → U41(isNat(z0), z0) 23.80/6.52
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
x(z0, 0) → U61(isNat(z0)) 23.80/6.52
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
0 → n__0 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0
S tuples:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate
ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'
c12, c14, c, c3, c13, c23, c24, c25, c26, c1
We considered the (Usable) Rules:
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
And the Tuples:
activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
0 → n__0 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt
The order we found is given by the following interpretation:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
POL(0) = 0 23.80/6.52
POL(ACTIVATE(x1)) = 0 23.80/6.52
POL(ISNAT(x1)) = x1 23.80/6.52
POL(U11(x1, x2)) = [3] 23.80/6.52
POL(U11'(x1, x2)) = x2 23.80/6.52
POL(U12(x1)) = [3] 23.80/6.52
POL(U21(x1)) = [3] 23.80/6.52
POL(U31(x1, x2)) = [3] 23.80/6.52
POL(U31'(x1, x2)) = [3] + x2 23.80/6.52
POL(U32(x1)) = [3] + [3]x1 23.80/6.52
POL(U51'(x1, x2, x3)) = [5]x2 + [3]x3 23.80/6.52
POL(U52'(x1, x2, x3)) = 0 23.80/6.52
POL(U71'(x1, x2, x3)) = [2]x1 + [5]x2 + [2]x3 23.80/6.52
POL(U72'(x1, x2, x3)) = 0 23.80/6.52
POL(activate(x1)) = x1 23.80/6.52
POL(c(x1, x2)) = x1 + x2 23.80/6.52
POL(c1(x1)) = x1 23.80/6.52
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4 23.80/6.52
POL(c13(x1, x2)) = x1 + x2 23.80/6.52
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4 23.80/6.52
POL(c23) = 0 23.80/6.52
POL(c24(x1, x2)) = x1 + x2 23.80/6.52
POL(c25(x1)) = x1 23.80/6.52
POL(c26(x1, x2)) = x1 + x2 23.80/6.52
POL(c3(x1, x2)) = x1 + x2 23.80/6.52
POL(isNat(x1)) = 0 23.80/6.52
POL(n__0) = 0 23.80/6.52
POL(n__plus(x1, x2)) = x1 + x2 23.80/6.52
POL(n__s(x1)) = x1 23.80/6.52
POL(n__x(x1, x2)) = [4] + x1 + x2 23.80/6.52
POL(plus(x1, x2)) = x1 + x2 23.80/6.52
POL(s(x1)) = x1 23.80/6.52
POL(tt) = [5] 23.80/6.52
POL(x(x1, x2)) = [4] + x1 + x2
Tuples:
U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U41(tt, z0) → activate(z0) 23.80/6.52
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.80/6.52
U61(tt) → 0 23.80/6.52
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
plus(z0, 0) → U41(isNat(z0), z0) 23.80/6.52
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
x(z0, 0) → U61(isNat(z0)) 23.80/6.52
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
0 → n__0 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0
S tuples:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate
ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'
c12, c14, c, c3, c13, c23, c24, c25, c26, c1
We considered the (Usable) Rules:
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
And the Tuples:
activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
0 → n__0 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt
The order we found is given by the following interpretation:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
POL(0) = 0 23.80/6.52
POL(ACTIVATE(x1)) = [2]x1 23.80/6.52
POL(ISNAT(x1)) = x12 23.80/6.52
POL(U11(x1, x2)) = [3] + [3]x1 23.80/6.52
POL(U11'(x1, x2)) = [2]x2 + x22 23.80/6.52
POL(U12(x1)) = [3] 23.80/6.52
POL(U21(x1)) = [3] + [3]x1 23.80/6.52
POL(U31(x1, x2)) = [3] + [3]x1 23.80/6.52
POL(U31'(x1, x2)) = [2] + [2]x2 + x22 23.80/6.52
POL(U32(x1)) = [3] + [3]x1 23.80/6.52
POL(U51'(x1, x2, x3)) = [2] + [3]x1 + [3]x2 + [3]x3 + [3]x32 + [2]x2·x3 + [3]x1·x3 + [3]x12 + [3]x1·x2 23.80/6.52
POL(U52'(x1, x2, x3)) = [2]x2 + [2]x3 + [2]x2·x3 23.80/6.52
POL(U71'(x1, x2, x3)) = [2] + [3]x1 + [3]x2 + [3]x3 + [3]x32 + [3]x1·x3 + [3]x12 + [3]x1·x2 + x22 23.80/6.52
POL(U72'(x1, x2, x3)) = [2]x2 + [2]x3 + [2]x32 23.80/6.52
POL(activate(x1)) = x1 23.80/6.52
POL(c(x1, x2)) = x1 + x2 23.80/6.52
POL(c1(x1)) = x1 23.80/6.52
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4 23.80/6.52
POL(c13(x1, x2)) = x1 + x2 23.80/6.52
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4 23.80/6.52
POL(c23) = 0 23.80/6.52
POL(c24(x1, x2)) = x1 + x2 23.80/6.52
POL(c25(x1)) = x1 23.80/6.52
POL(c26(x1, x2)) = x1 + x2 23.80/6.52
POL(c3(x1, x2)) = x1 + x2 23.80/6.52
POL(isNat(x1)) = 0 23.80/6.52
POL(n__0) = 0 23.80/6.52
POL(n__plus(x1, x2)) = [3] + x1 + x2 23.80/6.52
POL(n__s(x1)) = [1] + x1 23.80/6.52
POL(n__x(x1, x2)) = [2] + x1 + x2 23.80/6.52
POL(plus(x1, x2)) = [3] + x1 + x2 23.80/6.52
POL(s(x1)) = [1] + x1 23.80/6.52
POL(tt) = 0 23.80/6.52
POL(x(x1, x2)) = [2] + x1 + x2
Tuples:
U11(tt, z0) → U12(isNat(activate(z0))) 23.80/6.52
U12(tt) → tt 23.80/6.52
U21(tt) → tt 23.80/6.52
U31(tt, z0) → U32(isNat(activate(z0))) 23.80/6.52
U32(tt) → tt 23.80/6.52
U41(tt, z0) → activate(z0) 23.80/6.52
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 23.80/6.52
U61(tt) → 0 23.80/6.52
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 23.80/6.52
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 23.80/6.52
isNat(n__0) → tt 23.80/6.52
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 23.80/6.52
isNat(n__s(z0)) → U21(isNat(activate(z0))) 23.80/6.52
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 23.80/6.52
plus(z0, 0) → U41(isNat(z0), z0) 23.80/6.52
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 23.80/6.52
plus(z0, z1) → n__plus(z0, z1) 23.80/6.52
x(z0, 0) → U61(isNat(z0)) 23.80/6.52
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 23.80/6.52
x(z0, z1) → n__x(z0, z1) 23.80/6.52
0 → n__0 23.80/6.52
s(z0) → n__s(z0) 23.80/6.52
activate(n__0) → 0 23.80/6.52
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1)) 23.80/6.52
activate(n__s(z0)) → s(activate(z0)) 23.80/6.52
activate(n__x(z0, z1)) → x(activate(z0), activate(z1)) 23.80/6.52
activate(z0) → z0
S tuples:
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__0) → c23 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:
ACTIVATE(n__0) → c23
Defined Rule Symbols:
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 23.80/6.52
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 23.80/6.52
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 23.80/6.52
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 23.80/6.52
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1)) 23.80/6.52
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0)) 23.80/6.52
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate
ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'
c12, c14, c, c3, c13, c23, c24, c25, c26, c1
Now S is empty
ACTIVATE(n__0) → c23