Runtime Complexity (innermost) proof of /tmp/tmp4xq0SY/MYNAT_nokinds-noand

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳4 CdtProblem

↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 16 ms)

↳8 CdtProblem

↳9 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳10 CdtProblem

↳11 CdtKnowledgeProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CdtProblem

↳13 CdtUsableRulesProof (⇔, 0 ms)

↳14 CdtProblem

↳15 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 1161 ms)

↳16 CdtProblem

↳17 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 1249 ms)

↳18 CdtProblem

↳19 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 1024 ms)

↳20 CdtProblem

↳21 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 545 ms)

↳22 CdtProblem

↳23 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))), 1020 ms)

↳24 CdtProblem

↳25 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

↳26 BOUNDS(O(1), O(1))

(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U12'(tt) → c1
U21'(tt) → c2
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U32'(tt) → c4
U41'(tt, z0) → c5(ACTIVATE(z0))
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U61'(tt) → c8(0')
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__0) → c11
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, z1) → c17
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0))
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1))
X(z0, z1) → c20
0' → c21
S(z0) → c22
ACTIVATE(n__0) → c23(0')
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c27

S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U12'(tt) → c1
U21'(tt) → c2
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U32'(tt) → c4
U41'(tt, z0) → c5(ACTIVATE(z0))
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U61'(tt) → c8(0')
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__0) → c11
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, z1) → c17
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0))
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1))
X(z0, z1) → c20
0' → c21
S(z0) → c22
ACTIVATE(n__0) → c23(0')
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c27

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', U12', U21', U31', U32', U41', U51', U52', U61', U71', U72', ISNAT, PLUS, X, 0', S, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 5 leading nodes:

PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1))
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0))
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1))
U41'(tt, z0) → c5(ACTIVATE(z0))

Removed 11 trailing nodes:

ACTIVATE(z0) → c27
U32'(tt) → c4
ACTIVATE(n__0) → c23(0')
PLUS(z0, z1) → c17
X(z0, z1) → c20
S(z0) → c22
0' → c21
U12'(tt) → c1
U21'(tt) → c2
U61'(tt) → c8(0')
ISNAT(n__0) → c11

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))

S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(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', U51', U52', U71', U72', ISNAT, ACTIVATE

Compound Symbols:

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

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

Removed 10 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
U52'(tt, z0, z1) → c7(ACTIVATE(z1), ACTIVATE(z0))
U72'(tt, z0, z1) → c10(ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))

S tuples:

U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
U52'(tt, z0, z1) → c7(ACTIVATE(z1), ACTIVATE(z0))
U72'(tt, z0, z1) → c10(ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(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:

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

Compound Symbols:

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

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
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))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
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, c24, c25, c26, c1

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 4 leading nodes:

U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
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))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
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, c24, c25, c26, c1

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

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
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))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))

K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
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, c24, c25, c26, c1

(13) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
x(z0, z1) → n__x(z0, z1)
U11(tt, z0) → U12(isNat(activate(z0)))
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U12(tt) → tt

Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
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))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))

K tuples:

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

Defined Rule Symbols:

isNat, activate, 0, plus, s, x, U11, U21, U31, U32, U12

Defined Pair Symbols:

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

Compound Symbols:

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

(15) CdtRuleRemovalProof (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))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))

We considered the (Usable) Rules:

activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
plus(z0, z1) → n__plus(z0, z1)
activate(n__0) → 0
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)

And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

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

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
x(z0, z1) → n__x(z0, z1)
U11(tt, z0) → U12(isNat(activate(z0)))
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U12(tt) → tt

Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
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))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))

K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))

Defined Rule Symbols:

isNat, activate, 0, plus, s, x, U11, U21, U31, U32, U12

Defined Pair Symbols:

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

Compound Symbols:

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

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

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

U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))

We considered the (Usable) Rules:

activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(z0) → z0
isNat(n__0) → tt
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
plus(z0, z1) → n__plus(z0, z1)
U21(tt) → tt
isNat(n__s(z0)) → U21(isNat(activate(z0)))
U32(tt) → tt
U12(tt) → tt
0 → n__0
s(z0) → n__s(z0)
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
activate(n__0) → 0
x(z0, z1) → n__x(z0, z1)
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
U31(tt, z0) → U32(isNat(activate(z0)))
U11(tt, z0) → U12(isNat(activate(z0)))

And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

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

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
x(z0, z1) → n__x(z0, z1)
U11(tt, z0) → U12(isNat(activate(z0)))
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U12(tt) → tt

Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
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))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))

K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))

Defined Rule Symbols:

isNat, activate, 0, plus, s, x, U11, U21, U31, U32, U12

Defined Pair Symbols:

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

Compound Symbols:

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

(19) CdtRuleRemovalProof (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__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
plus(z0, z1) → n__plus(z0, z1)
activate(n__0) → 0
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)

And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

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

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
x(z0, z1) → n__x(z0, z1)
U11(tt, z0) → U12(isNat(activate(z0)))
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U12(tt) → tt

Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
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))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))

K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))

Defined Rule Symbols:

isNat, activate, 0, plus, s, x, U11, U21, U31, U32, U12

Defined Pair Symbols:

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

Compound Symbols:

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

(21) CdtRuleRemovalProof (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__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
plus(z0, z1) → n__plus(z0, z1)
activate(n__0) → 0
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)

And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

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

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
x(z0, z1) → n__x(z0, z1)
U11(tt, z0) → U12(isNat(activate(z0)))
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U12(tt) → tt

Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))

S tuples:

ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))

K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))

Defined Rule Symbols:

isNat, activate, 0, plus, s, x, U11, U21, U31, U32, U12

Defined Pair Symbols:

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

Compound Symbols:

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

(23) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))

We considered the (Usable) Rules:

activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
plus(z0, z1) → n__plus(z0, z1)
activate(n__0) → 0
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)

And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]
POL(ACTIVATE(x₁)) = x₁
POL(ISNAT(x₁)) = x₁²
POL(U11(x₁, x₂)) = 0
POL(U11'(x₁, x₂)) = [2]x₂ + x₂²
POL(U12(x₁)) = 0
POL(U21(x₁)) = 0
POL(U31(x₁, x₂)) = 0
POL(U31'(x₁, x₂)) = x₂ + x₂²
POL(U32(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = [3] + [2]x₂ + [3]x₃ + [2]x₃² + [2]x₂·x₃ + [3]x₂²
POL(U52'(x₁, x₂, x₃)) = [2]x₂ + [3]x₃ + x₃²
POL(U71'(x₁, x₂, x₃)) = [3] + [2]x₂ + x₃ + [3]x₃² + [3]x₂·x₃ + [3]x₂²
POL(U72'(x₁, x₂, x₃)) = [2]x₂ + x₃
POL(activate(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1(x₁)) = x₁
POL(c12(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c13(x₁, x₂)) = x₁ + x₂
POL(c14(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c24(x₁, x₂)) = x₁ + x₂
POL(c25(x₁)) = x₁
POL(c26(x₁, x₂)) = x₁ + x₂
POL(c3(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(n__0) = [3]
POL(n__plus(x₁, x₂)) = [2] + x₁ + x₂
POL(n__s(x₁)) = [1] + x₁
POL(n__x(x₁, x₂)) = [1] + x₁ + x₂
POL(plus(x₁, x₂)) = [2] + x₁ + x₂
POL(s(x₁)) = [1] + x₁
POL(tt) = 0
POL(x(x₁, x₂)) = [1] + x₁ + x₂

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
0 → n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
x(z0, z1) → n__x(z0, z1)
U11(tt, z0) → U12(isNat(activate(z0)))
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U12(tt) → tt

Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))

S tuples:none
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))

Defined Rule Symbols:

isNat, activate, 0, plus, s, x, U11, U21, U31, U32, U12

Defined Pair Symbols:

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

Compound Symbols:

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

(25) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

(4) Obligation:

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

(6) Obligation:

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

(8) Obligation:

(9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

(10) Obligation:

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

(12) Obligation:

(13) CdtUsableRulesProof (EQUIVALENT transformation)

(14) Obligation:

(15) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(16) Obligation:

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

(18) Obligation:

(19) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

(20) Obligation:

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

(22) Obligation:

(23) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^2))) transformation)

(24) Obligation:

(25) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

(26) BOUNDS(O(1), O(1))