### (0) Obligation:

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

g(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

### (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(f(x), y) → f(h(x, y))
h(x, y) → g(x, f(y))

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
g/1
h/1

### (6) Obligation:

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

g(f(x)) → f(h(x))
h(x) → g(x)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
g(f(x)) → f(h(x))
h(x) → g(x)

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

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

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

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

### (10) Obligation:

TRS:
Rules:
g(f(x)) → f(h(x))
h(x) → g(x)

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

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

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

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

Proved the following rewrite lemma:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Induction Base:
h(gen_f2_0(0))

Induction Step:
h(gen_f2_0(+(n4_0, 1))) →RΩ(1)
g(gen_f2_0(+(n4_0, 1))) →RΩ(1)
f(h(gen_f2_0(n4_0))) →IH
f(*3_0)

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

### (13) Obligation:

TRS:
Rules:
g(f(x)) → f(h(x))
h(x) → g(x)

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

The following defined symbols remain to be analysed:
g

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

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

Proved the following rewrite lemma:
g(gen_f2_0(+(1, n112_0))) → *3_0, rt ∈ Ω(n1120)

Induction Base:
g(gen_f2_0(+(1, 0)))

Induction Step:
g(gen_f2_0(+(1, +(n112_0, 1)))) →RΩ(1)
f(h(gen_f2_0(+(1, n112_0)))) →RΩ(1)
f(g(gen_f2_0(+(1, n112_0)))) →IH
f(*3_0)

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

### (16) Obligation:

TRS:
Rules:
g(f(x)) → f(h(x))
h(x) → g(x)

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n40)
g(gen_f2_0(+(1, n112_0))) → *3_0, rt ∈ Ω(n1120)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

The following defined symbols remain to be analysed:
h

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

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

Proved the following rewrite lemma:
h(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n3350)

Induction Base:
h(gen_f2_0(0))

Induction Step:
h(gen_f2_0(+(n335_0, 1))) →RΩ(1)
g(gen_f2_0(+(n335_0, 1))) →RΩ(1)
f(h(gen_f2_0(n335_0))) →IH
f(*3_0)

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

### (19) Obligation:

TRS:
Rules:
g(f(x)) → f(h(x))
h(x) → g(x)

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n3350)
g(gen_f2_0(+(1, n112_0))) → *3_0, rt ∈ Ω(n1120)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
h(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n3350)

### (22) Obligation:

TRS:
Rules:
g(f(x)) → f(h(x))
h(x) → g(x)

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n3350)
g(gen_f2_0(+(1, n112_0))) → *3_0, rt ∈ Ω(n1120)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
h(gen_f2_0(n335_0)) → *3_0, rt ∈ Ω(n3350)

### (25) Obligation:

TRS:
Rules:
g(f(x)) → f(h(x))
h(x) → g(x)

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n40)
g(gen_f2_0(+(1, n112_0))) → *3_0, rt ∈ Ω(n1120)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

No more defined symbols left to analyse.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

### (28) Obligation:

TRS:
Rules:
g(f(x)) → f(h(x))
h(x) → g(x)

Types:
g :: f → f
f :: f → f
h :: f → f
hole_f1_0 :: f
gen_f2_0 :: Nat → f

Lemmas:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_f2_0(0) ⇔ hole_f1_0
gen_f2_0(+(x, 1)) ⇔ f(gen_f2_0(x))

No more defined symbols left to analyse.

### (29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
h(gen_f2_0(n4_0)) → *3_0, rt ∈ Ω(n40)