### (0) Obligation:

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

h(X, Z) → f(X, s(X), Z)
f(X, Y, g(X, Y)) → h(0, g(X, Y))
g(0, Y) → 0
g(X, s(Y)) → g(X, 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(X, s(Y)) →+ g(X, Y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [Y / s(Y)].
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:

h(X, Z) → f(X, s(X), Z)
f(X, Y, g(X, Y)) → h(0', g(X, Y))
g(0', Y) → 0'
g(X, s(Y)) → g(X, Y)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
h(X, Z) → f(X, s(X), Z)
f(X, Y, g(X, Y)) → h(0', g(X, Y))
g(0', Y) → 0'
g(X, s(Y)) → g(X, Y)

Types:
h :: s:0' → s:0' → h:f
f :: s:0' → s:0' → s:0' → h:f
s :: s:0' → s:0'
g :: s:0' → s:0' → s:0'
0' :: s:0'
hole_h:f1_0 :: h:f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

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

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

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

### (8) Obligation:

TRS:
Rules:
h(X, Z) → f(X, s(X), Z)
f(X, Y, g(X, Y)) → h(0', g(X, Y))
g(0', Y) → 0'
g(X, s(Y)) → g(X, Y)

Types:
h :: s:0' → s:0' → h:f
f :: s:0' → s:0' → s:0' → h:f
s :: s:0' → s:0'
g :: s:0' → s:0' → s:0'
0' :: s:0'
hole_h:f1_0 :: h:f
hole_s:0'2_0 :: s:0'
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:
g, h

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

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

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

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

Induction Step:
g(gen_s:0'3_0(a), gen_s:0'3_0(+(1, +(n5_0, 1)))) →RΩ(1)
g(gen_s:0'3_0(a), 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:
h(X, Z) → f(X, s(X), Z)
f(X, Y, g(X, Y)) → h(0', g(X, Y))
g(0', Y) → 0'
g(X, s(Y)) → g(X, Y)

Types:
h :: s:0' → s:0' → h:f
f :: s:0' → s:0' → s:0' → h:f
s :: s:0' → s:0'
g :: s:0' → s:0' → s:0'
0' :: s:0'
hole_h:f1_0 :: h:f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
g(gen_s:0'3_0(a), 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:
h

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

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

### (13) Obligation:

TRS:
Rules:
h(X, Z) → f(X, s(X), Z)
f(X, Y, g(X, Y)) → h(0', g(X, Y))
g(0', Y) → 0'
g(X, s(Y)) → g(X, Y)

Types:
h :: s:0' → s:0' → h:f
f :: s:0' → s:0' → s:0' → h:f
s :: s:0' → s:0'
g :: s:0' → s:0' → s:0'
0' :: s:0'
hole_h:f1_0 :: h:f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

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

### (16) Obligation:

TRS:
Rules:
h(X, Z) → f(X, s(X), Z)
f(X, Y, g(X, Y)) → h(0', g(X, Y))
g(0', Y) → 0'
g(X, s(Y)) → g(X, Y)

Types:
h :: s:0' → s:0' → h:f
f :: s:0' → s:0' → s:0' → h:f
s :: s:0' → s:0'
g :: s:0' → s:0' → s:0'
0' :: s:0'
hole_h:f1_0 :: h:f
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

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