### (0) Obligation:

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

g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
g(s(s(x2_1))) →+ s(s(g(x2_1)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [x2_1 / s(s(x2_1))].
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:

g(s(x)) → f(x)
f(0') → s(0')
f(s(x)) → s(s(g(x)))
g(0') → 0'

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
g(s(x)) → f(x)
f(0') → s(0')
f(s(x)) → s(s(g(x)))
g(0') → 0'

Types:
g :: s:0' → s:0'
s :: s:0' → s:0'
f :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

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

They will be analysed ascendingly in the following order:
g = f

### (8) Obligation:

TRS:
Rules:
g(s(x)) → f(x)
f(0') → s(0')
f(s(x)) → s(s(g(x)))
g(0') → 0'

Types:
g :: s:0' → s:0'
s :: s:0' → s:0'
f :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

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

They will be analysed ascendingly in the following order:
g = f

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

Proved the following rewrite lemma:
f(gen_s:0'2_0(*(2, n4_0))) → gen_s:0'2_0(+(1, *(2, n4_0))), rt ∈ Ω(1 + n40)

Induction Base:
f(gen_s:0'2_0(*(2, 0))) →RΩ(1)
s(0')

Induction Step:
f(gen_s:0'2_0(*(2, +(n4_0, 1)))) →RΩ(1)
s(s(g(gen_s:0'2_0(+(1, *(2, n4_0)))))) →RΩ(1)
s(s(f(gen_s:0'2_0(*(2, n4_0))))) →IH
s(s(gen_s:0'2_0(+(1, *(2, c5_0)))))

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

### (11) Obligation:

TRS:
Rules:
g(s(x)) → f(x)
f(0') → s(0')
f(s(x)) → s(s(g(x)))
g(0') → 0'

Types:
g :: s:0' → s:0'
s :: s:0' → s:0'
f :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
f(gen_s:0'2_0(*(2, n4_0))) → gen_s:0'2_0(+(1, *(2, n4_0))), rt ∈ Ω(1 + n40)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
g

They will be analysed ascendingly in the following order:
g = f

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

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

### (13) Obligation:

TRS:
Rules:
g(s(x)) → f(x)
f(0') → s(0')
f(s(x)) → s(s(g(x)))
g(0') → 0'

Types:
g :: s:0' → s:0'
s :: s:0' → s:0'
f :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
f(gen_s:0'2_0(*(2, n4_0))) → gen_s:0'2_0(+(1, *(2, n4_0))), rt ∈ Ω(1 + n40)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_s:0'2_0(*(2, n4_0))) → gen_s:0'2_0(+(1, *(2, n4_0))), rt ∈ Ω(1 + n40)

### (16) Obligation:

TRS:
Rules:
g(s(x)) → f(x)
f(0') → s(0')
f(s(x)) → s(s(g(x)))
g(0') → 0'

Types:
g :: s:0' → s:0'
s :: s:0' → s:0'
f :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
f(gen_s:0'2_0(*(2, n4_0))) → gen_s:0'2_0(+(1, *(2, n4_0))), rt ∈ Ω(1 + n40)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_s:0'2_0(*(2, n4_0))) → gen_s:0'2_0(+(1, *(2, n4_0))), rt ∈ Ω(1 + n40)