### (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, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
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) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
0n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
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(ACTIVATE(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
ISNAT(n__0) → c6
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, z1) → c11
0'c12
S(z0) → c13
ACTIVATE(n__0) → c14(0')
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(z0, z1))
ACTIVATE(n__s(z0)) → c16(S(z0))
ACTIVATE(z0) → c17
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(ACTIVATE(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
ISNAT(n__0) → c6
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, z1) → c11
0'c12
S(z0) → c13
ACTIVATE(n__0) → c14(0')
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(z0, z1))
ACTIVATE(n__s(z0)) → c16(S(z0))
ACTIVATE(z0) → c17
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', ISNAT, PLUS, 0', S, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17

### (3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
Removed 12 trailing nodes:

PLUS(z0, z1) → c11
ACTIVATE(z0) → c17
ACTIVATE(n__0) → c14(0')
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(z0, z1))
U12'(tt) → c1
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U21'(tt) → c2
U31'(tt, z0) → c3(ACTIVATE(z0))
0'c12
ISNAT(n__0) → c6
S(z0) → c13
ACTIVATE(n__s(z0)) → c16(S(z0))

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
0n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U11', U41', ISNAT

Compound Symbols:

c, c4, c7, c8

### (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) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
0n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
S tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U11', U41', ISNAT

Compound Symbols:

c, c4, c7, c8

### (7) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

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

U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
0n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
S tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
K tuples:

U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U11', U41', ISNAT

Compound Symbols:

c, c4, c7, c8

### (9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

U31(tt, z0) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
0n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
U11(tt, z0) → U12(isNat(activate(z0)))
U21(tt) → tt
U12(tt) → tt
Tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
S tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
K tuples:

U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
Defined Rule Symbols:

activate, 0, plus, s, isNat, U11, U21, U12

Defined Pair Symbols:

U11', U41', ISNAT

Compound Symbols:

c, c4, c7, c8

### (11) 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)) → c8(ISNAT(activate(z0)))
We considered the (Usable) Rules:

activate(z0) → z0
activate(n__s(z0)) → s(z0)
plus(z0, z1) → n__plus(z0, z1)
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__0) → 0
s(z0) → n__s(z0)
0n__0
And the Tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]
POL(ISNAT(x1)) = [4]x1
POL(U11(x1, x2)) = [3] + [2]x1 + [2]x2
POL(U11'(x1, x2)) = [4]x2
POL(U12(x1)) = [3] + [4]x1
POL(U21(x1)) = [3] + [3]x1
POL(U41'(x1, x2, x3)) = [3] + [4]x1 + [4]x3
POL(activate(x1)) = x1
POL(c(x1)) = x1
POL(c4(x1)) = x1
POL(c7(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(isNat(x1)) = [3]
POL(n__0) = [3]
POL(n__plus(x1, x2)) = x1 + x2
POL(n__s(x1)) = [1] + x1
POL(plus(x1, x2)) = x1 + x2
POL(s(x1)) = [1] + x1
POL(tt) = 0

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
0n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
U11(tt, z0) → U12(isNat(activate(z0)))
U21(tt) → tt
U12(tt) → tt
Tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
S tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
K tuples:

U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
Defined Rule Symbols:

activate, 0, plus, s, isNat, U11, U21, U12

Defined Pair Symbols:

U11', U41', ISNAT

Compound Symbols:

c, c4, c7, c8

### (13) 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)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
We considered the (Usable) Rules:

activate(z0) → z0
activate(n__s(z0)) → s(z0)
plus(z0, z1) → n__plus(z0, z1)
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__0) → 0
s(z0) → n__s(z0)
0n__0
And the Tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [4]
POL(ISNAT(x1)) = [4]x1
POL(U11(x1, x2)) = [2] + [3]x1 + [3]x2
POL(U11'(x1, x2)) = [4]x2
POL(U12(x1)) = [2] + [2]x1
POL(U21(x1)) = [3] + [2]x1
POL(U41'(x1, x2, x3)) = [2] + [4]x1 + [4]x3
POL(activate(x1)) = x1
POL(c(x1)) = x1
POL(c4(x1)) = x1
POL(c7(x1, x2)) = x1 + x2
POL(c8(x1)) = x1
POL(isNat(x1)) = 0
POL(n__0) = [4]
POL(n__plus(x1, x2)) = [2] + x1 + x2
POL(n__s(x1)) = x1
POL(plus(x1, x2)) = [2] + x1 + x2
POL(s(x1)) = x1
POL(tt) = [2]

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(z0) → z0
0n__0
plus(z0, z1) → n__plus(z0, z1)
s(z0) → n__s(z0)
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
U11(tt, z0) → U12(isNat(activate(z0)))
U21(tt) → tt
U12(tt) → tt
Tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
S tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)))
K tuples:

U41'(tt, z0, z1) → c4(ISNAT(activate(z1)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
Defined Rule Symbols:

activate, 0, plus, s, isNat, U11, U21, U12

Defined Pair Symbols:

U11', U41', ISNAT

Compound Symbols:

c, c4, c7, c8

### (15) CdtKnowledgeProof (EQUIVALENT transformation)

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

U11'(tt, z0) → c(ISNAT(activate(z0)))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)))
Now S is empty