### (0) Obligation:

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

f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(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(g(x)) →+ g(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 / g(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(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))

Types:
f :: g:h → g:h
g :: g:h → g:h
h :: g:h → g:h
f' :: s → s → s → f'
s :: s → s
hole_g:h1_0 :: g:h
hole_f'2_0 :: f'
hole_s3_0 :: s
gen_g:h4_0 :: Nat → g:h
gen_s5_0 :: Nat → s

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

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

### (8) Obligation:

TRS:
Rules:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))

Types:
f :: g:h → g:h
g :: g:h → g:h
h :: g:h → g:h
f' :: s → s → s → f'
s :: s → s
hole_g:h1_0 :: g:h
hole_f'2_0 :: f'
hole_s3_0 :: s
gen_g:h4_0 :: Nat → g:h
gen_s5_0 :: Nat → s

Generator Equations:
gen_g:h4_0(0) ⇔ hole_g:h1_0
gen_g:h4_0(+(x, 1)) ⇔ g(gen_g:h4_0(x))
gen_s5_0(0) ⇔ hole_s3_0
gen_s5_0(+(x, 1)) ⇔ s(gen_s5_0(x))

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

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

Proved the following rewrite lemma:
f(gen_g:h4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n70)

Induction Base:
f(gen_g:h4_0(+(1, 0)))

Induction Step:
f(gen_g:h4_0(+(1, +(n7_0, 1)))) →RΩ(1)
g(f(f(gen_g:h4_0(+(1, n7_0))))) →IH
g(f(*6_0))

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

### (11) Obligation:

TRS:
Rules:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))

Types:
f :: g:h → g:h
g :: g:h → g:h
h :: g:h → g:h
f' :: s → s → s → f'
s :: s → s
hole_g:h1_0 :: g:h
hole_f'2_0 :: f'
hole_s3_0 :: s
gen_g:h4_0 :: Nat → g:h
gen_s5_0 :: Nat → s

Lemmas:
f(gen_g:h4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n70)

Generator Equations:
gen_g:h4_0(0) ⇔ hole_g:h1_0
gen_g:h4_0(+(x, 1)) ⇔ g(gen_g:h4_0(x))
gen_s5_0(0) ⇔ hole_s3_0
gen_s5_0(+(x, 1)) ⇔ s(gen_s5_0(x))

The following defined symbols remain to be analysed:
f'

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

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

### (13) Obligation:

TRS:
Rules:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))

Types:
f :: g:h → g:h
g :: g:h → g:h
h :: g:h → g:h
f' :: s → s → s → f'
s :: s → s
hole_g:h1_0 :: g:h
hole_f'2_0 :: f'
hole_s3_0 :: s
gen_g:h4_0 :: Nat → g:h
gen_s5_0 :: Nat → s

Lemmas:
f(gen_g:h4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n70)

Generator Equations:
gen_g:h4_0(0) ⇔ hole_g:h1_0
gen_g:h4_0(+(x, 1)) ⇔ g(gen_g:h4_0(x))
gen_s5_0(0) ⇔ hole_s3_0
gen_s5_0(+(x, 1)) ⇔ s(gen_s5_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_g:h4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n70)

### (16) Obligation:

TRS:
Rules:
f(g(x)) → g(f(f(x)))
f(h(x)) → h(g(x))
f'(s(x), y, y) → f'(y, x, s(x))

Types:
f :: g:h → g:h
g :: g:h → g:h
h :: g:h → g:h
f' :: s → s → s → f'
s :: s → s
hole_g:h1_0 :: g:h
hole_f'2_0 :: f'
hole_s3_0 :: s
gen_g:h4_0 :: Nat → g:h
gen_s5_0 :: Nat → s

Lemmas:
f(gen_g:h4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n70)

Generator Equations:
gen_g:h4_0(0) ⇔ hole_g:h1_0
gen_g:h4_0(+(x, 1)) ⇔ g(gen_g:h4_0(x))
gen_s5_0(0) ⇔ hole_s3_0
gen_s5_0(+(x, 1)) ⇔ s(gen_s5_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_g:h4_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n70)