YES(O(1), O(n^1)) 13.32/3.85 YES(O(1), O(n^1)) 13.70/3.98 13.70/3.98 13.70/3.98 13.70/3.98 13.70/3.98 13.70/3.98 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 13.70/3.98 13.70/3.98 13.70/3.98
13.70/3.98
13.70/3.98
13.70/3.98
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
13.70/3.98 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
13.70/3.98 
↳2 CdtProblem
13.70/3.98 
↳3 CdtUnreachableProof (⇔)
13.70/3.98 
↳4 CdtProblem
13.70/3.98 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
13.70/3.98 
↳6 CdtProblem
13.70/3.98 
↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
13.70/3.98 
↳8 CdtProblem
13.70/3.98 
↳9 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
13.70/3.98 
↳10 CdtProblem
13.70/3.98 
↳11 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
13.70/3.98 
↳12 CdtProblem
13.70/3.98 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
13.70/3.98 
↳14 CdtProblem
13.70/3.98 
↳15 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
13.70/3.98 
↳16 CdtProblem
13.70/3.98 
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
13.70/3.98 
↳18 CdtProblem
13.70/3.98 
↳19 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
13.70/3.98 
↳20 CdtProblem
13.70/3.98 
↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
13.70/3.98 
↳22 CdtProblem
13.70/3.98 
↳23 CdtKnowledgeProof (⇔)
13.70/3.98 
↳24 BOUNDS(O(1), O(1))
13.70/3.98
13.70/3.98 13.70/3.98
13.70/3.98
13.70/3.98

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2))) 13.70/3.98
U12(tt) → tt 13.70/3.98
U21(tt) → tt 13.70/3.98
U31(tt, V2) → U32(isNat(activate(V2))) 13.70/3.98
U32(tt) → tt 13.70/3.98
U41(tt, N) → activate(N) 13.70/3.98
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N)) 13.70/3.98
U52(tt, M, N) → s(plus(activate(N), activate(M))) 13.70/3.98
U61(tt) → 0 13.70/3.98
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N)) 13.70/3.98
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N)) 13.70/3.98
isNat(n__0) → tt 13.70/3.98
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2)) 13.70/3.98
isNat(n__s(V1)) → U21(isNat(activate(V1))) 13.70/3.98
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2)) 13.70/3.98
plus(N, 0) → U41(isNat(N), N) 13.70/3.98
plus(N, s(M)) → U51(isNat(M), M, N) 13.70/3.98
x(N, 0) → U61(isNat(N)) 13.70/3.98
x(N, s(M)) → U71(isNat(M), M, N) 13.70/3.98
0 → n__0 13.70/3.98
plus(X1, X2) → n__plus(X1, X2) 13.70/3.98
s(X) → n__s(X) 13.70/3.98
x(X1, X2) → n__x(X1, X2) 13.70/3.98
activate(n__0) → 0 13.70/3.98
activate(n__plus(X1, X2)) → plus(X1, X2) 13.70/3.98
activate(n__s(X)) → s(X) 13.70/3.98
activate(n__x(X1, X2)) → x(X1, X2) 13.70/3.98
activate(X) → X

Rewrite Strategy: INNERMOST
13.70/3.98
13.70/3.98

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT
13.70/3.98
13.70/3.98

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 13.70/3.98
U12(tt) → tt 13.70/3.98
U21(tt) → tt 13.70/3.98
U31(tt, z0) → U32(isNat(activate(z0))) 13.70/3.98
U32(tt) → tt 13.70/3.98
U41(tt, z0) → activate(z0) 13.70/3.98
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 13.70/3.98
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 13.70/3.98
U61(tt) → 0 13.70/3.98
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 13.70/3.98
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 13.70/3.98
isNat(n__0) → tt 13.70/3.98
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 13.70/3.98
isNat(n__s(z0)) → U21(isNat(activate(z0))) 13.70/3.98
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 13.70/3.98
plus(z0, 0) → U41(isNat(z0), z0) 13.70/3.98
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 13.70/3.98
plus(z0, z1) → n__plus(z0, z1) 13.70/3.98
x(z0, 0) → U61(isNat(z0)) 13.70/3.98
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 13.70/3.98
x(z0, z1) → n__x(z0, z1) 13.70/3.98
0 → n__0 13.70/3.98
s(z0) → n__s(z0) 13.70/3.98
activate(n__0) → 0 13.70/3.98
activate(n__plus(z0, z1)) → plus(z0, z1) 13.70/3.98
activate(n__s(z0)) → s(z0) 13.70/3.98
activate(n__x(z0, z1)) → x(z0, z1) 13.70/3.98
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 13.70/3.98
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 13.70/3.98
U41'(tt, z0) → c5(ACTIVATE(z0)) 13.70/3.98
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 13.70/3.98
U61'(tt) → c8(0') 13.70/3.98
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 13.70/3.98
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0)) 13.70/3.98
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1)) 13.70/3.98
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0)) 13.70/3.98
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1)) 13.70/3.98
ACTIVATE(n__0) → c23(0') 13.70/3.98
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(z0, z1)) 13.70/3.98
ACTIVATE(n__s(z0)) → c25(S(z0)) 13.70/3.98
ACTIVATE(n__x(z0, z1)) → c26(X(z0, z1))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 13.70/3.98
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 13.70/3.98
U41'(tt, z0) → c5(ACTIVATE(z0)) 13.70/3.98
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 13.70/3.98
U61'(tt) → c8(0') 13.70/3.98
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 13.70/3.98
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0)) 13.70/3.98
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1)) 13.70/3.98
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0)) 13.70/3.98
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1)) 13.70/3.98
ACTIVATE(n__0) → c23(0') 13.70/3.98
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(z0, z1)) 13.70/3.98
ACTIVATE(n__s(z0)) → c25(S(z0)) 13.70/3.98
ACTIVATE(n__x(z0, z1)) → c26(X(z0, z1))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U11', U31', U41', U51', U52', U61', U71', U72', ISNAT, PLUS, X, ACTIVATE

Compound Symbols:

c, c3, c5, c6, c7, c8, c9, c10, c12, c13, c14, c15, c16, c18, c19, c23, c24, c25, c26

13.70/3.98
13.70/3.98

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0)) 13.70/3.98
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1)) 13.70/3.98
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0)) 13.70/3.98
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1))
13.70/3.98
13.70/3.98

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 13.70/3.98
U12(tt) → tt 13.70/3.98
U21(tt) → tt 13.70/3.98
U31(tt, z0) → U32(isNat(activate(z0))) 13.70/3.98
U32(tt) → tt 13.70/3.98
U41(tt, z0) → activate(z0) 13.70/3.98
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 13.70/3.98
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 13.70/3.98
U61(tt) → 0 13.70/3.98
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 13.70/3.98
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 13.70/3.98
isNat(n__0) → tt 13.70/3.98
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 13.70/3.98
isNat(n__s(z0)) → U21(isNat(activate(z0))) 13.70/3.98
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 13.70/3.98
plus(z0, 0) → U41(isNat(z0), z0) 13.70/3.98
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 13.70/3.98
plus(z0, z1) → n__plus(z0, z1) 13.70/3.98
x(z0, 0) → U61(isNat(z0)) 13.70/3.98
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 13.70/3.98
x(z0, z1) → n__x(z0, z1) 13.70/3.98
0 → n__0 13.70/3.98
s(z0) → n__s(z0) 13.70/3.98
activate(n__0) → 0 13.70/3.98
activate(n__plus(z0, z1)) → plus(z0, z1) 13.70/3.98
activate(n__s(z0)) → s(z0) 13.70/3.98
activate(n__x(z0, z1)) → x(z0, z1) 13.70/3.98
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 13.70/3.98
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 13.70/3.98
U41'(tt, z0) → c5(ACTIVATE(z0)) 13.70/3.98
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 13.70/3.98
U61'(tt) → c8(0') 13.70/3.98
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 13.70/3.98
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 14.09/4.09
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.09/4.09
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 14.09/4.09
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.09/4.09
ACTIVATE(n__0) → c23(0') 14.09/4.09
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(z0, z1)) 14.09/4.09
ACTIVATE(n__s(z0)) → c25(S(z0)) 14.09/4.09
ACTIVATE(n__x(z0, z1)) → c26(X(z0, z1))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 14.09/4.09
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 14.09/4.09
U41'(tt, z0) → c5(ACTIVATE(z0)) 14.09/4.09
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 14.09/4.09
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0)) 14.09/4.09
U61'(tt) → c8(0') 14.09/4.09
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 14.09/4.09
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 14.09/4.09
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.09/4.09
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0)) 14.09/4.09
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.09/4.09
ACTIVATE(n__0) → c23(0') 14.09/4.09
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(z0, z1)) 14.09/4.09
ACTIVATE(n__s(z0)) → c25(S(z0)) 14.09/4.09
ACTIVATE(n__x(z0, z1)) → c26(X(z0, z1))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U11', U31', U41', U51', U52', U61', U71', U72', ISNAT, ACTIVATE

Compound Symbols:

c, c3, c5, c6, c7, c8, c9, c10, c12, c13, c14, c23, c24, c25, c26

14.09/4.09
14.09/4.09

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 12 trailing tuple parts
14.09/4.09
14.09/4.09

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 14.09/4.09
U12(tt) → tt 14.09/4.09
U21(tt) → tt 14.09/4.09
U31(tt, z0) → U32(isNat(activate(z0))) 14.09/4.09
U32(tt) → tt 14.09/4.09
U41(tt, z0) → activate(z0) 14.09/4.09
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 14.09/4.09
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 14.09/4.09
U61(tt) → 0 14.09/4.09
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 14.09/4.09
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 14.09/4.09
isNat(n__0) → tt 14.09/4.09
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.09/4.09
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.09/4.09
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.09/4.09
plus(z0, 0) → U41(isNat(z0), z0) 14.09/4.09
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 14.48/4.13
plus(z0, z1) → n__plus(z0, z1) 14.48/4.13
x(z0, 0) → U61(isNat(z0)) 14.48/4.13
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 14.48/4.13
x(z0, z1) → n__x(z0, z1) 14.48/4.13
0 → n__0 14.48/4.13
s(z0) → n__s(z0) 14.48/4.13
activate(n__0) → 0 14.48/4.13
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.13
activate(n__s(z0)) → s(z0) 14.48/4.13
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.13
activate(z0) → z0
Tuples:

U41'(tt, z0) → c5(ACTIVATE(z0)) 14.48/4.13
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U52'(tt, z0, z1) → c7(ACTIVATE(z1), ACTIVATE(z0)) 14.48/4.13
U61'(tt) → c8 14.48/4.13
U72'(tt, z0, z1) → c10(ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26
S tuples:

U41'(tt, z0) → c5(ACTIVATE(z0)) 14.48/4.13
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U52'(tt, z0, z1) → c7(ACTIVATE(z1), ACTIVATE(z0)) 14.48/4.13
U61'(tt) → c8 14.48/4.13
U72'(tt, z0, z1) → c10(ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U41', U51', U71', ISNAT, U11', U31', U52', U61', U72', ACTIVATE

Compound Symbols:

c5, c6, c9, c12, c14, c, c3, c7, c8, c10, c13, c23, c24, c25, c26

14.48/4.13
14.48/4.13

(7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC
14.48/4.13
14.48/4.13

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 14.48/4.13
U12(tt) → tt 14.48/4.13
U21(tt) → tt 14.48/4.13
U31(tt, z0) → U32(isNat(activate(z0))) 14.48/4.13
U32(tt) → tt 14.48/4.13
U41(tt, z0) → activate(z0) 14.48/4.13
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 14.48/4.13
U61(tt) → 0 14.48/4.13
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 14.48/4.13
isNat(n__0) → tt 14.48/4.13
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.48/4.13
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.48/4.13
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.48/4.13
plus(z0, 0) → U41(isNat(z0), z0) 14.48/4.13
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 14.48/4.13
plus(z0, z1) → n__plus(z0, z1) 14.48/4.13
x(z0, 0) → U61(isNat(z0)) 14.48/4.13
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 14.48/4.13
x(z0, z1) → n__x(z0, z1) 14.48/4.13
0 → n__0 14.48/4.13
s(z0) → n__s(z0) 14.48/4.13
activate(n__0) → 0 14.48/4.13
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.13
activate(n__s(z0)) → s(z0) 14.48/4.13
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.13
activate(z0) → z0
Tuples:

U41'(tt, z0) → c5(ACTIVATE(z0)) 14.48/4.13
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U61'(tt) → c8 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.13
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U51'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U71'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

U41'(tt, z0) → c5(ACTIVATE(z0)) 14.48/4.13
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U61'(tt) → c8 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.13
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U51'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U71'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U41', ISNAT, U11', U31', U61', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c5, c12, c14, c, c3, c8, c13, c23, c24, c25, c26, c1

14.48/4.13
14.48/4.13

(9) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 16 trailing nodes:

ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.13
U71'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U61'(tt) → c8 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U51'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
U71'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U41'(tt, z0) → c5(ACTIVATE(z0))
14.48/4.13
14.48/4.13

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 14.48/4.13
U12(tt) → tt 14.48/4.13
U21(tt) → tt 14.48/4.13
U31(tt, z0) → U32(isNat(activate(z0))) 14.48/4.13
U32(tt) → tt 14.48/4.13
U41(tt, z0) → activate(z0) 14.48/4.13
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 14.48/4.13
U61(tt) → 0 14.48/4.13
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 14.48/4.13
isNat(n__0) → tt 14.48/4.13
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.48/4.13
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.48/4.13
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.48/4.13
plus(z0, 0) → U41(isNat(z0), z0) 14.48/4.13
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 14.48/4.13
plus(z0, z1) → n__plus(z0, z1) 14.48/4.13
x(z0, 0) → U61(isNat(z0)) 14.48/4.13
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 14.48/4.13
x(z0, z1) → n__x(z0, z1) 14.48/4.13
0 → n__0 14.48/4.13
s(z0) → n__s(z0) 14.48/4.13
activate(n__0) → 0 14.48/4.13
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.13
activate(n__s(z0)) → s(z0) 14.48/4.13
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.13
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.13
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.13
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

14.48/4.13
14.48/4.13

(11) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
14.48/4.13
14.48/4.13

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 14.48/4.13
U12(tt) → tt 14.48/4.13
U21(tt) → tt 14.48/4.13
U31(tt, z0) → U32(isNat(activate(z0))) 14.48/4.13
U32(tt) → tt 14.48/4.13
U41(tt, z0) → activate(z0) 14.48/4.13
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 14.48/4.13
U61(tt) → 0 14.48/4.13
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 14.48/4.13
isNat(n__0) → tt 14.48/4.13
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.48/4.13
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.48/4.13
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.48/4.13
plus(z0, 0) → U41(isNat(z0), z0) 14.48/4.13
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 14.48/4.13
plus(z0, z1) → n__plus(z0, z1) 14.48/4.13
x(z0, 0) → U61(isNat(z0)) 14.48/4.13
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 14.48/4.13
x(z0, z1) → n__x(z0, z1) 14.48/4.13
0 → n__0 14.48/4.13
s(z0) → n__s(z0) 14.48/4.13
activate(n__0) → 0 14.48/4.13
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.13
activate(n__s(z0)) → s(z0) 14.48/4.13
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.13
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.13
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

14.48/4.13
14.48/4.13

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:

activate(n__0) → 0 14.48/4.13
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.13
activate(n__s(z0)) → s(z0) 14.48/4.13
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.13
activate(z0) → z0 14.48/4.13
x(z0, z1) → n__x(z0, z1) 14.48/4.13
s(z0) → n__s(z0) 14.48/4.13
plus(z0, z1) → n__plus(z0, z1) 14.48/4.13
0 → n__0 14.48/4.13
isNat(n__0) → tt 14.48/4.13
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.48/4.13
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.48/4.13
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.48/4.13
U31(tt, z0) → U32(isNat(activate(z0))) 14.48/4.13
U32(tt) → tt 14.48/4.13
U21(tt) → tt 14.48/4.13
U11(tt, z0) → U12(isNat(activate(z0))) 14.48/4.13
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.13
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 14.48/4.13

POL(0) = 0    14.48/4.13
POL(ACTIVATE(x1)) = 0    14.48/4.13
POL(ISNAT(x1)) = x1    14.48/4.13
POL(U11(x1, x2)) = [3]    14.48/4.13
POL(U11'(x1, x2)) = x2    14.48/4.13
POL(U12(x1)) = [3]    14.48/4.13
POL(U21(x1)) = [3]    14.48/4.13
POL(U31(x1, x2)) = [3]    14.48/4.13
POL(U31'(x1, x2)) = x2    14.48/4.13
POL(U32(x1)) = [3] + [3]x1    14.48/4.13
POL(U51'(x1, x2, x3)) = x1 + [3]x3    14.48/4.13
POL(U52'(x1, x2, x3)) = 0    14.48/4.13
POL(U71'(x1, x2, x3)) = [1] + x1 + [2]x3    14.48/4.13
POL(U72'(x1, x2, x3)) = 0    14.48/4.13
POL(activate(x1)) = x1    14.48/4.13
POL(c(x1, x2)) = x1 + x2    14.48/4.13
POL(c1(x1)) = x1    14.48/4.13
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    14.48/4.13
POL(c13(x1, x2)) = x1 + x2    14.48/4.13
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    14.48/4.13
POL(c23) = 0    14.48/4.13
POL(c24) = 0    14.48/4.13
POL(c25) = 0    14.48/4.13
POL(c26) = 0    14.48/4.13
POL(c3(x1, x2)) = x1 + x2    14.48/4.13
POL(isNat(x1)) = 0    14.48/4.13
POL(n__0) = 0    14.48/4.13
POL(n__plus(x1, x2)) = [2] + x1 + x2    14.48/4.13
POL(n__s(x1)) = x1    14.48/4.13
POL(n__x(x1, x2)) = x1 + x2    14.48/4.13
POL(plus(x1, x2)) = [2] + x1 + x2    14.48/4.13
POL(s(x1)) = x1    14.48/4.13
POL(tt) = 0    14.48/4.13
POL(x(x1, x2)) = x1 + x2   
14.48/4.13
14.48/4.13

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 14.48/4.13
U12(tt) → tt 14.48/4.13
U21(tt) → tt 14.48/4.13
U31(tt, z0) → U32(isNat(activate(z0))) 14.48/4.13
U32(tt) → tt 14.48/4.13
U41(tt, z0) → activate(z0) 14.48/4.13
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 14.48/4.13
U61(tt) → 0 14.48/4.13
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 14.48/4.13
isNat(n__0) → tt 14.48/4.13
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.48/4.13
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.48/4.13
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.48/4.13
plus(z0, 0) → U41(isNat(z0), z0) 14.48/4.13
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 14.48/4.13
plus(z0, z1) → n__plus(z0, z1) 14.48/4.13
x(z0, 0) → U61(isNat(z0)) 14.48/4.13
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 14.48/4.13
x(z0, z1) → n__x(z0, z1) 14.48/4.13
0 → n__0 14.48/4.13
s(z0) → n__s(z0) 14.48/4.13
activate(n__0) → 0 14.48/4.13
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.13
activate(n__s(z0)) → s(z0) 14.48/4.13
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.13
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.13
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

14.48/4.13
14.48/4.13

(15) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
14.48/4.13
14.48/4.13

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 14.48/4.13
U12(tt) → tt 14.48/4.13
U21(tt) → tt 14.48/4.13
U31(tt, z0) → U32(isNat(activate(z0))) 14.48/4.13
U32(tt) → tt 14.48/4.13
U41(tt, z0) → activate(z0) 14.48/4.13
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 14.48/4.13
U61(tt) → 0 14.48/4.13
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 14.48/4.13
isNat(n__0) → tt 14.48/4.13
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.48/4.13
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.48/4.13
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.48/4.13
plus(z0, 0) → U41(isNat(z0), z0) 14.48/4.13
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 14.48/4.13
plus(z0, z1) → n__plus(z0, z1) 14.48/4.13
x(z0, 0) → U61(isNat(z0)) 14.48/4.13
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 14.48/4.13
x(z0, z1) → n__x(z0, z1) 14.48/4.13
0 → n__0 14.48/4.13
s(z0) → n__s(z0) 14.48/4.13
activate(n__0) → 0 14.48/4.13
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.13
activate(n__s(z0)) → s(z0) 14.48/4.13
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.13
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.13
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

14.48/4.13
14.48/4.13

(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:

activate(n__0) → 0 14.48/4.13
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.13
activate(n__s(z0)) → s(z0) 14.48/4.13
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.13
activate(z0) → z0 14.48/4.13
x(z0, z1) → n__x(z0, z1) 14.48/4.13
s(z0) → n__s(z0) 14.48/4.13
plus(z0, z1) → n__plus(z0, z1) 14.48/4.13
0 → n__0 14.48/4.13
isNat(n__0) → tt 14.48/4.13
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.48/4.13
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.48/4.13
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.48/4.13
U31(tt, z0) → U32(isNat(activate(z0))) 14.48/4.13
U32(tt) → tt 14.48/4.13
U21(tt) → tt 14.48/4.13
U11(tt, z0) → U12(isNat(activate(z0))) 14.48/4.13
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.13
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.13
ACTIVATE(n__0) → c23 14.48/4.13
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.13
ACTIVATE(n__s(z0)) → c25 14.48/4.13
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.13
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.13
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.13
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 14.48/4.13

POL(0) = 0    14.48/4.13
POL(ACTIVATE(x1)) = 0    14.48/4.13
POL(ISNAT(x1)) = x1    14.48/4.13
POL(U11(x1, x2)) = [3]    14.48/4.13
POL(U11'(x1, x2)) = x2    14.48/4.13
POL(U12(x1)) = [3]    14.48/4.13
POL(U21(x1)) = [3]    14.48/4.13
POL(U31(x1, x2)) = [3]    14.48/4.13
POL(U31'(x1, x2)) = x2    14.48/4.13
POL(U32(x1)) = [3] + [3]x1    14.48/4.13
POL(U51'(x1, x2, x3)) = [3] + x1 + [4]x2 + [2]x3    14.48/4.13
POL(U52'(x1, x2, x3)) = [2] + [2]x3    14.48/4.13
POL(U71'(x1, x2, x3)) = x1 + [4]x2 + [4]x3    14.48/4.13
POL(U72'(x1, x2, x3)) = [2]x2    14.48/4.13
POL(activate(x1)) = x1    14.48/4.13
POL(c(x1, x2)) = x1 + x2    14.48/4.13
POL(c1(x1)) = x1    14.48/4.13
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    14.48/4.13
POL(c13(x1, x2)) = x1 + x2    14.48/4.13
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    14.48/4.13
POL(c23) = 0    14.48/4.13
POL(c24) = 0    14.48/4.13
POL(c25) = 0    14.48/4.13
POL(c26) = 0    14.48/4.13
POL(c3(x1, x2)) = x1 + x2    14.48/4.13
POL(isNat(x1)) = 0    14.48/4.13
POL(n__0) = 0    14.48/4.13
POL(n__plus(x1, x2)) = x1 + x2    14.48/4.13
POL(n__s(x1)) = x1    14.48/4.13
POL(n__x(x1, x2)) = [1] + x1 + x2    14.48/4.13
POL(plus(x1, x2)) = x1 + x2    14.48/4.13
POL(s(x1)) = x1    14.48/4.13
POL(tt) = 0    14.48/4.13
POL(x(x1, x2)) = [1] + x1 + x2   
14.48/4.13
14.48/4.13

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 14.48/4.13
U12(tt) → tt 14.48/4.13
U21(tt) → tt 14.48/4.13
U31(tt, z0) → U32(isNat(activate(z0))) 14.48/4.13
U32(tt) → tt 14.48/4.13
U41(tt, z0) → activate(z0) 14.48/4.13
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 14.48/4.13
U61(tt) → 0 14.48/4.13
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.13
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 14.48/4.13
isNat(n__0) → tt 14.48/4.13
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.48/4.13
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.48/4.13
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.48/4.13
plus(z0, 0) → U41(isNat(z0), z0) 14.48/4.13
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 14.48/4.13
plus(z0, z1) → n__plus(z0, z1) 14.48/4.13
x(z0, 0) → U61(isNat(z0)) 14.48/4.13
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 14.48/4.13
x(z0, z1) → n__x(z0, z1) 14.48/4.13
0 → n__0 14.48/4.13
s(z0) → n__s(z0) 14.48/4.15
activate(n__0) → 0 14.48/4.15
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.15
activate(n__s(z0)) → s(z0) 14.48/4.15
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.15
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ACTIVATE(n__0) → c23 14.48/4.15
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.15
ACTIVATE(n__s(z0)) → c25 14.48/4.15
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.15
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ACTIVATE(n__0) → c23 14.48/4.15
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.15
ACTIVATE(n__s(z0)) → c25 14.48/4.15
ACTIVATE(n__x(z0, z1)) → c26
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.15
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

14.48/4.15
14.48/4.15

(19) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

The following tuples could be moved from S to K by knowledge propagation:

U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
14.48/4.15
14.48/4.15

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 14.48/4.15
U12(tt) → tt 14.48/4.15
U21(tt) → tt 14.48/4.15
U31(tt, z0) → U32(isNat(activate(z0))) 14.48/4.15
U32(tt) → tt 14.48/4.15
U41(tt, z0) → activate(z0) 14.48/4.15
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.15
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 14.48/4.15
U61(tt) → 0 14.48/4.15
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.15
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 14.48/4.15
isNat(n__0) → tt 14.48/4.15
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.48/4.15
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.48/4.15
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.48/4.15
plus(z0, 0) → U41(isNat(z0), z0) 14.48/4.15
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 14.48/4.15
plus(z0, z1) → n__plus(z0, z1) 14.48/4.15
x(z0, 0) → U61(isNat(z0)) 14.48/4.15
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 14.48/4.15
x(z0, z1) → n__x(z0, z1) 14.48/4.15
0 → n__0 14.48/4.15
s(z0) → n__s(z0) 14.48/4.15
activate(n__0) → 0 14.48/4.15
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.15
activate(n__s(z0)) → s(z0) 14.48/4.15
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.15
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ACTIVATE(n__0) → c23 14.48/4.15
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.15
ACTIVATE(n__s(z0)) → c25 14.48/4.15
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.15
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ACTIVATE(n__0) → c23 14.48/4.15
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.15
ACTIVATE(n__s(z0)) → c25 14.48/4.15
ACTIVATE(n__x(z0, z1)) → c26
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.15
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

14.48/4.15
14.48/4.15

(21) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0 14.48/4.15
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.15
activate(n__s(z0)) → s(z0) 14.48/4.15
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.15
activate(z0) → z0 14.48/4.15
x(z0, z1) → n__x(z0, z1) 14.48/4.15
s(z0) → n__s(z0) 14.48/4.15
plus(z0, z1) → n__plus(z0, z1) 14.48/4.15
0 → n__0 14.48/4.15
isNat(n__0) → tt 14.48/4.15
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.48/4.15
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.48/4.15
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.48/4.15
U31(tt, z0) → U32(isNat(activate(z0))) 14.48/4.15
U32(tt) → tt 14.48/4.15
U21(tt) → tt 14.48/4.15
U11(tt, z0) → U12(isNat(activate(z0))) 14.48/4.15
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ACTIVATE(n__0) → c23 14.48/4.15
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.15
ACTIVATE(n__s(z0)) → c25 14.48/4.15
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.15
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 14.48/4.15

POL(0) = 0    14.48/4.15
POL(ACTIVATE(x1)) = 0    14.48/4.15
POL(ISNAT(x1)) = x1    14.48/4.15
POL(U11(x1, x2)) = [3]    14.48/4.15
POL(U11'(x1, x2)) = x2    14.48/4.15
POL(U12(x1)) = [3]    14.48/4.15
POL(U21(x1)) = [3]    14.48/4.15
POL(U31(x1, x2)) = [3]    14.48/4.15
POL(U31'(x1, x2)) = [2] + x2    14.48/4.15
POL(U32(x1)) = [3] + [3]x1    14.48/4.15
POL(U51'(x1, x2, x3)) = x1 + [3]x2 + [5]x3    14.48/4.15
POL(U52'(x1, x2, x3)) = x3    14.48/4.15
POL(U71'(x1, x2, x3)) = [1] + x1 + [5]x2 + [5]x3    14.48/4.15
POL(U72'(x1, x2, x3)) = x2 + [4]x3    14.48/4.15
POL(activate(x1)) = x1    14.48/4.15
POL(c(x1, x2)) = x1 + x2    14.48/4.15
POL(c1(x1)) = x1    14.48/4.15
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    14.48/4.15
POL(c13(x1, x2)) = x1 + x2    14.48/4.15
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4    14.48/4.15
POL(c23) = 0    14.48/4.15
POL(c24) = 0    14.48/4.15
POL(c25) = 0    14.48/4.15
POL(c26) = 0    14.48/4.15
POL(c3(x1, x2)) = x1 + x2    14.48/4.15
POL(isNat(x1)) = 0    14.48/4.15
POL(n__0) = 0    14.48/4.15
POL(n__plus(x1, x2)) = x1 + x2    14.48/4.15
POL(n__s(x1)) = [1] + x1    14.48/4.15
POL(n__x(x1, x2)) = [2] + x1 + x2    14.48/4.15
POL(plus(x1, x2)) = x1 + x2    14.48/4.15
POL(s(x1)) = [1] + x1    14.48/4.15
POL(tt) = 0    14.48/4.15
POL(x(x1, x2)) = [2] + x1 + x2   
14.48/4.15
14.48/4.15

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0))) 14.48/4.15
U12(tt) → tt 14.48/4.15
U21(tt) → tt 14.48/4.15
U31(tt, z0) → U32(isNat(activate(z0))) 14.48/4.15
U32(tt) → tt 14.48/4.15
U41(tt, z0) → activate(z0) 14.48/4.15
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.15
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0))) 14.48/4.15
U61(tt) → 0 14.48/4.15
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1)) 14.48/4.15
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1)) 14.48/4.15
isNat(n__0) → tt 14.48/4.15
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1)) 14.48/4.15
isNat(n__s(z0)) → U21(isNat(activate(z0))) 14.48/4.15
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1)) 14.48/4.15
plus(z0, 0) → U41(isNat(z0), z0) 14.48/4.15
plus(z0, s(z1)) → U51(isNat(z1), z1, z0) 14.48/4.15
plus(z0, z1) → n__plus(z0, z1) 14.48/4.15
x(z0, 0) → U61(isNat(z0)) 14.48/4.15
x(z0, s(z1)) → U71(isNat(z1), z1, z0) 14.48/4.15
x(z0, z1) → n__x(z0, z1) 14.48/4.15
0 → n__0 14.48/4.15
s(z0) → n__s(z0) 14.48/4.15
activate(n__0) → 0 14.48/4.15
activate(n__plus(z0, z1)) → plus(z0, z1) 14.48/4.15
activate(n__s(z0)) → s(z0) 14.48/4.15
activate(n__x(z0, z1)) → x(z0, z1) 14.48/4.15
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ACTIVATE(n__0) → c23 14.48/4.15
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.15
ACTIVATE(n__s(z0)) → c25 14.48/4.15
ACTIVATE(n__x(z0, z1)) → c26 14.48/4.15
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

ACTIVATE(n__0) → c23 14.48/4.15
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.15
ACTIVATE(n__s(z0)) → c25 14.48/4.15
ACTIVATE(n__x(z0, z1)) → c26
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U51'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1))) 14.48/4.15
U71'(tt, z0, z1) → c1(ISNAT(activate(z1))) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U52'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z1)) 14.48/4.15
U72'(tt, z0, z1) → c1(ACTIVATE(z0)) 14.48/4.15
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1)) 14.48/4.15
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0)) 14.48/4.15
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U11', U31', ACTIVATE, U51', U71', U52', U72'

Compound Symbols:

c12, c14, c, c3, c13, c23, c24, c25, c26, c1

14.48/4.15
14.48/4.15

(23) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

ACTIVATE(n__0) → c23 14.48/4.15
ACTIVATE(n__plus(z0, z1)) → c24 14.48/4.15
ACTIVATE(n__s(z0)) → c25 14.48/4.15
ACTIVATE(n__x(z0, z1)) → c26
Now S is empty
14.48/4.15
14.48/4.15

(24) BOUNDS(O(1), O(1))

14.48/4.15
14.48/4.15
14.88/4.22 EOF