### (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)
0n__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) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
activate(n__f(X)) →+ f(activate(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__f(X)].
The result substitution is [ ].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) 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(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

S is empty.
Rewrite Strategy: FULL

### (5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
cons/1

### (6) Obligation:

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

f(0') → cons(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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
f(0') → cons(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

Types:
f :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
0' :: cons:n__f:n__s:n__0
cons :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
s :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
p :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__f :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__s :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__0 :: cons:n__f:n__s:n__0
activate :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
hole_cons:n__f:n__s:n__01_1 :: cons:n__f:n__s:n__0
gen_cons:n__f:n__s:n__02_1 :: Nat → cons:n__f:n__s:n__0

### (9) 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

### (10) Obligation:

TRS:
Rules:
f(0') → cons(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

Types:
f :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
0' :: cons:n__f:n__s:n__0
cons :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
s :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
p :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__f :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__s :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__0 :: cons:n__f:n__s:n__0
activate :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
hole_cons:n__f:n__s:n__01_1 :: cons:n__f:n__s:n__0
gen_cons:n__f:n__s:n__02_1 :: Nat → cons:n__f:n__s:n__0

Generator Equations:
gen_cons:n__f:n__s:n__02_1(0) ⇔ n__0
gen_cons:n__f:n__s:n__02_1(+(x, 1)) ⇔ n__f(gen_cons:n__f:n__s:n__02_1(x))

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

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

### (11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (12) Obligation:

TRS:
Rules:
f(0') → cons(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

Types:
f :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
0' :: cons:n__f:n__s:n__0
cons :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
s :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
p :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__f :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__s :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__0 :: cons:n__f:n__s:n__0
activate :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
hole_cons:n__f:n__s:n__01_1 :: cons:n__f:n__s:n__0
gen_cons:n__f:n__s:n__02_1 :: Nat → cons:n__f:n__s:n__0

Generator Equations:
gen_cons:n__f:n__s:n__02_1(0) ⇔ n__0
gen_cons:n__f:n__s:n__02_1(+(x, 1)) ⇔ n__f(gen_cons:n__f:n__s:n__02_1(x))

The following defined symbols remain to be analysed:
activate

### (13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_cons:n__f:n__s:n__02_1(n12_1)) → gen_cons:n__f:n__s:n__02_1(n12_1), rt ∈ Ω(1 + n121)

Induction Base:
activate(gen_cons:n__f:n__s:n__02_1(0)) →RΩ(1)
gen_cons:n__f:n__s:n__02_1(0)

Induction Step:
activate(gen_cons:n__f:n__s:n__02_1(+(n12_1, 1))) →RΩ(1)
f(activate(gen_cons:n__f:n__s:n__02_1(n12_1))) →IH
f(gen_cons:n__f:n__s:n__02_1(c13_1)) →RΩ(1)
n__f(gen_cons:n__f:n__s:n__02_1(n12_1))

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

### (15) Obligation:

TRS:
Rules:
f(0') → cons(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

Types:
f :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
0' :: cons:n__f:n__s:n__0
cons :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
s :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
p :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__f :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__s :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__0 :: cons:n__f:n__s:n__0
activate :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
hole_cons:n__f:n__s:n__01_1 :: cons:n__f:n__s:n__0
gen_cons:n__f:n__s:n__02_1 :: Nat → cons:n__f:n__s:n__0

Lemmas:
activate(gen_cons:n__f:n__s:n__02_1(n12_1)) → gen_cons:n__f:n__s:n__02_1(n12_1), rt ∈ Ω(1 + n121)

Generator Equations:
gen_cons:n__f:n__s:n__02_1(0) ⇔ n__0
gen_cons:n__f:n__s:n__02_1(+(x, 1)) ⇔ n__f(gen_cons:n__f:n__s:n__02_1(x))

No more defined symbols left to analyse.

### (16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_cons:n__f:n__s:n__02_1(n12_1)) → gen_cons:n__f:n__s:n__02_1(n12_1), rt ∈ Ω(1 + n121)

### (18) Obligation:

TRS:
Rules:
f(0') → cons(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

Types:
f :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
0' :: cons:n__f:n__s:n__0
cons :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
s :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
p :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__f :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__s :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
n__0 :: cons:n__f:n__s:n__0
activate :: cons:n__f:n__s:n__0 → cons:n__f:n__s:n__0
hole_cons:n__f:n__s:n__01_1 :: cons:n__f:n__s:n__0
gen_cons:n__f:n__s:n__02_1 :: Nat → cons:n__f:n__s:n__0

Lemmas:
activate(gen_cons:n__f:n__s:n__02_1(n12_1)) → gen_cons:n__f:n__s:n__02_1(n12_1), rt ∈ Ω(1 + n121)

Generator Equations:
gen_cons:n__f:n__s:n__02_1(0) ⇔ n__0
gen_cons:n__f:n__s:n__02_1(+(x, 1)) ⇔ n__f(gen_cons:n__f:n__s:n__02_1(x))

No more defined symbols left to analyse.

### (19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_cons:n__f:n__s:n__02_1(n12_1)) → gen_cons:n__f:n__s:n__02_1(n12_1), rt ∈ Ω(1 + n121)