(0) Obligation:

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

f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
p(s(X)) → X
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(z0)) → z0
activate(n__f(z0)) → f(z0)
activate(z0) → z0
Tuples:

F(s(0)) → c1(F(p(s(0))), P(s(0)))
ACTIVATE(n__f(z0)) → c4(F(z0))
S tuples:

F(s(0)) → c1(F(p(s(0))), P(s(0)))
ACTIVATE(n__f(z0)) → c4(F(z0))
K tuples:none
Defined Rule Symbols:

f, p, activate

Defined Pair Symbols:

F, ACTIVATE

Compound Symbols:

c1, c4

(3) CdtGraphRemoveDanglingProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 of 2 dangling nodes:

ACTIVATE(n__f(z0)) → c4(F(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(z0)) → z0
activate(n__f(z0)) → f(z0)
activate(z0) → z0
Tuples:

F(s(0)) → c1(F(p(s(0))), P(s(0)))
S tuples:

F(s(0)) → c1(F(p(s(0))), P(s(0)))
K tuples:none
Defined Rule Symbols:

f, p, activate

Defined Pair Symbols:

F

Compound Symbols:

c1

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(z0)) → z0
activate(n__f(z0)) → f(z0)
activate(z0) → z0
Tuples:

F(s(0)) → c1(F(p(s(0))))
S tuples:

F(s(0)) → c1(F(p(s(0))))
K tuples:none
Defined Rule Symbols:

f, p, activate

Defined Pair Symbols:

F

Compound Symbols:

c1

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

Use narrowing to replace F(s(0)) → c1(F(p(s(0)))) by

F(s(0)) → c1(F(0))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(z0)) → z0
activate(n__f(z0)) → f(z0)
activate(z0) → z0
Tuples:

F(s(0)) → c1(F(0))
S tuples:

F(s(0)) → c1(F(0))
K tuples:none
Defined Rule Symbols:

f, p, activate

Defined Pair Symbols:

F

Compound Symbols:

c1

(9) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 1 dangling nodes:

F(s(0)) → c1(F(0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(s(0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(z0)) → z0
activate(n__f(z0)) → f(z0)
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

f, p, activate

Defined Pair Symbols:none

Compound Symbols:none

(11) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(12) BOUNDS(O(1), O(1))