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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtUnreachableProof (⇔)
↳4 CdtProblem
↳5 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳6 CdtProblem
↳7 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳10 CdtProblem
↳11 SIsEmptyProof (⇔)
↳12 BOUNDS(O(1), O(1))

(0) Obligation:

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

f(0) → cons(0, n__f(n__s(n__0)))
f(s(0)) → f(p(s(0)))
p(s(0)) → 0
f(X) → n__f(X)
s(X) → n__s(X)
0 → n__0
activate(n__f(X)) → f(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__0) → 0
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(n__s(n__0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(0)) → 0
s(z0) → n__s(z0)
0 → n__0
activate(n__f(z0)) → f(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__0) → 0
activate(z0) → z0
Tuples:

F(0) → c(0')
F(s(0)) → c1(F(p(s(0))), P(s(0)), S(0), 0')
P(s(0)) → c3(0')
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__0) → c8(0')
S tuples:

F(0) → c(0')
F(s(0)) → c1(F(p(s(0))), P(s(0)), S(0), 0')
P(s(0)) → c3(0')
ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__0) → c8(0')
K tuples:none
Defined Rule Symbols:

f, p, s, 0, activate

Defined Pair Symbols:

F, P, ACTIVATE

Compound Symbols:

c, c1, c3, c6, c7, c8

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

F(0) → c(0')
F(s(0)) → c1(F(p(s(0))), P(s(0)), S(0), 0')
P(s(0)) → c3(0')

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(n__s(n__0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(0)) → 0
s(z0) → n__s(z0)
0 → n__0
activate(n__f(z0)) → f(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__0) → 0
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__0) → c8(0')
S tuples:

ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__0) → c8(0')
K tuples:none
Defined Rule Symbols:

f, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7, c8

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 3 dangling nodes:

ACTIVATE(n__0) → c8(0')

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(n__s(n__0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(0)) → 0
s(z0) → n__s(z0)
0 → n__0
activate(n__f(z0)) → f(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__0) → 0
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c6(F(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7

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

Removed 2 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(n__s(n__0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(0)) → 0
s(z0) → n__s(z0)
0 → n__0
activate(n__f(z0)) → f(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__0) → 0
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
S tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

f, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7

(9) 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.

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [4]x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(n__f(x1)) = [1] + x1   
POL(n__s(x1)) = [4] + x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f(0) → cons(0, n__f(n__s(n__0)))
f(s(0)) → f(p(s(0)))
f(z0) → n__f(z0)
p(s(0)) → 0
s(z0) → n__s(z0)
0 → n__0
activate(n__f(z0)) → f(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__0) → 0
activate(z0) → z0
Tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__f(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
Defined Rule Symbols:

f, p, s, 0, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7

(11) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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