### (0) Obligation:

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

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

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(x)) →+ f(x)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / 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(f(x)) → f(x)
f(s(x)) → f(x)
g(s(0')) → g(f(s(0')))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
g :: s:0' → g
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_g2_0 :: g
gen_s:0'3_0 :: Nat → s:0'

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

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

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

### (8) Obligation:

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

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
g :: s:0' → g
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_g2_0 :: g
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

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

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

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

Proved the following rewrite lemma:
f(gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

Induction Base:
f(gen_s:0'3_0(+(1, 0)))

Induction Step:
f(gen_s:0'3_0(+(1, +(n5_0, 1)))) →RΩ(1)
f(gen_s:0'3_0(+(1, n5_0))) →IH
*4_0

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

### (11) Obligation:

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

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
g :: s:0' → g
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_g2_0 :: g
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
f(gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
g

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

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

### (13) Obligation:

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

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
g :: s:0' → g
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_g2_0 :: g
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
f(gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_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:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

### (16) Obligation:

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

Types:
f :: s:0' → s:0'
s :: s:0' → s:0'
g :: s:0' → g
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_g2_0 :: g
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
f(gen_s:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_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:0'3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)