### (0) Obligation:

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

f(x) → s(x)
f(s(s(x))) → s(f(f(x)))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
f(x) → s(x)
f(s(s(x))) → s(f(f(x)))

Types:
f :: s → s
s :: s → s
hole_s1_0 :: s
gen_s2_0 :: Nat → s

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
f

### (8) Obligation:

TRS:
Rules:
f(x) → s(x)
f(s(s(x))) → s(f(f(x)))

Types:
f :: s → s
s :: s → s
hole_s1_0 :: s
gen_s2_0 :: Nat → s

Generator Equations:
gen_s2_0(0) ⇔ hole_s1_0
gen_s2_0(+(x, 1)) ⇔ s(gen_s2_0(x))

The following defined symbols remain to be analysed:
f

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

Proved the following rewrite lemma:
f(gen_s2_0(+(2, *(2, n4_0)))) → *3_0, rt ∈ Ω(n40)

Induction Base:
f(gen_s2_0(+(2, *(2, 0))))

Induction Step:
f(gen_s2_0(+(2, *(2, +(n4_0, 1))))) →RΩ(1)
s(f(f(gen_s2_0(+(2, *(2, n4_0)))))) →IH
s(f(*3_0))

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

### (11) Obligation:

TRS:
Rules:
f(x) → s(x)
f(s(s(x))) → s(f(f(x)))

Types:
f :: s → s
s :: s → s
hole_s1_0 :: s
gen_s2_0 :: Nat → s

Lemmas:
f(gen_s2_0(+(2, *(2, n4_0)))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_s2_0(0) ⇔ hole_s1_0
gen_s2_0(+(x, 1)) ⇔ s(gen_s2_0(x))

No more defined symbols left to analyse.

### (12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_s2_0(+(2, *(2, n4_0)))) → *3_0, rt ∈ Ω(n40)

### (14) Obligation:

TRS:
Rules:
f(x) → s(x)
f(s(s(x))) → s(f(f(x)))

Types:
f :: s → s
s :: s → s
hole_s1_0 :: s
gen_s2_0 :: Nat → s

Lemmas:
f(gen_s2_0(+(2, *(2, n4_0)))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_s2_0(0) ⇔ hole_s1_0
gen_s2_0(+(x, 1)) ⇔ s(gen_s2_0(x))

No more defined symbols left to analyse.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_s2_0(+(2, *(2, n4_0)))) → *3_0, rt ∈ Ω(n40)