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

(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(X)) → X
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: FULL

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative 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(X)) → X
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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

TRS:
Rules:
f(0') → cons(0', n__f(n__s(n__0)))
f(s(0')) → f(p(s(0')))
p(s(X)) → X
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

Types:
f :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
0' :: n__0:n__s:n__f:cons
cons :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__f :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__s :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__0 :: n__0:n__s:n__f:cons
s :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
p :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
activate :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
hole_n__0:n__s:n__f:cons1_1 :: n__0:n__s:n__f:cons
gen_n__0:n__s:n__f:cons2_1 :: Nat → n__0:n__s:n__f:cons

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
f, activate

They will be analysed ascendingly in the following order:
f < activate

(6) Obligation:

TRS:
Rules:
f(0') → cons(0', n__f(n__s(n__0)))
f(s(0')) → f(p(s(0')))
p(s(X)) → X
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

Types:
f :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
0' :: n__0:n__s:n__f:cons
cons :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__f :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__s :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__0 :: n__0:n__s:n__f:cons
s :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
p :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
activate :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
hole_n__0:n__s:n__f:cons1_1 :: n__0:n__s:n__f:cons
gen_n__0:n__s:n__f:cons2_1 :: Nat → n__0:n__s:n__f:cons

Generator Equations:
gen_n__0:n__s:n__f:cons2_1(0) ⇔ n__0
gen_n__0:n__s:n__f:cons2_1(+(x, 1)) ⇔ n__f(gen_n__0:n__s:n__f:cons2_1(x))

The following defined symbols remain to be analysed:
f, activate

They will be analysed ascendingly in the following order:
f < activate

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f.

(8) Obligation:

TRS:
Rules:
f(0') → cons(0', n__f(n__s(n__0)))
f(s(0')) → f(p(s(0')))
p(s(X)) → X
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

Types:
f :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
0' :: n__0:n__s:n__f:cons
cons :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__f :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__s :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__0 :: n__0:n__s:n__f:cons
s :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
p :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
activate :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
hole_n__0:n__s:n__f:cons1_1 :: n__0:n__s:n__f:cons
gen_n__0:n__s:n__f:cons2_1 :: Nat → n__0:n__s:n__f:cons

Generator Equations:
gen_n__0:n__s:n__f:cons2_1(0) ⇔ n__0
gen_n__0:n__s:n__f:cons2_1(+(x, 1)) ⇔ n__f(gen_n__0:n__s:n__f:cons2_1(x))

The following defined symbols remain to be analysed:
activate

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_n__0:n__s:n__f:cons2_1(n12_1)) → gen_n__0:n__s:n__f:cons2_1(n12_1), rt ∈ Ω(1 + n12₁)

Induction Base:
activate(gen_n__0:n__s:n__f:cons2_1(0)) →_R^Ω(1)
gen_n__0:n__s:n__f:cons2_1(0)

Induction Step:
activate(gen_n__0:n__s:n__f:cons2_1(+(n12_1, 1))) →_R^Ω(1)
f(activate(gen_n__0:n__s:n__f:cons2_1(n12_1))) →_IH
f(gen_n__0:n__s:n__f:cons2_1(c13_1)) →_R^Ω(1)
n__f(gen_n__0:n__s:n__f:cons2_1(n12_1))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
f(0') → cons(0', n__f(n__s(n__0)))
f(s(0')) → f(p(s(0')))
p(s(X)) → X
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

Types:
f :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
0' :: n__0:n__s:n__f:cons
cons :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__f :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__s :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__0 :: n__0:n__s:n__f:cons
s :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
p :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
activate :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
hole_n__0:n__s:n__f:cons1_1 :: n__0:n__s:n__f:cons
gen_n__0:n__s:n__f:cons2_1 :: Nat → n__0:n__s:n__f:cons

Lemmas:
activate(gen_n__0:n__s:n__f:cons2_1(n12_1)) → gen_n__0:n__s:n__f:cons2_1(n12_1), rt ∈ Ω(1 + n12₁)

Generator Equations:
gen_n__0:n__s:n__f:cons2_1(0) ⇔ n__0
gen_n__0:n__s:n__f:cons2_1(+(x, 1)) ⇔ n__f(gen_n__0:n__s:n__f:cons2_1(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_n__0:n__s:n__f:cons2_1(n12_1)) → gen_n__0:n__s:n__f:cons2_1(n12_1), rt ∈ Ω(1 + n12₁)

(13) BOUNDS(n^1, INF)

(14) Obligation:

TRS:
Rules:
f(0') → cons(0', n__f(n__s(n__0)))
f(s(0')) → f(p(s(0')))
p(s(X)) → X
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

Types:
f :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
0' :: n__0:n__s:n__f:cons
cons :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__f :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__s :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
n__0 :: n__0:n__s:n__f:cons
s :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
p :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
activate :: n__0:n__s:n__f:cons → n__0:n__s:n__f:cons
hole_n__0:n__s:n__f:cons1_1 :: n__0:n__s:n__f:cons
gen_n__0:n__s:n__f:cons2_1 :: Nat → n__0:n__s:n__f:cons

Lemmas:
activate(gen_n__0:n__s:n__f:cons2_1(n12_1)) → gen_n__0:n__s:n__f:cons2_1(n12_1), rt ∈ Ω(1 + n12₁)

Generator Equations:
gen_n__0:n__s:n__f:cons2_1(0) ⇔ n__0
gen_n__0:n__s:n__f:cons2_1(+(x, 1)) ⇔ n__f(gen_n__0:n__s:n__f:cons2_1(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_n__0:n__s:n__f:cons2_1(n12_1)) → gen_n__0:n__s:n__f:cons2_1(n12_1), rt ∈ Ω(1 + n12₁)

(16) BOUNDS(n^1, INF)