The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

h(e(x), y) → h(d(x, y), s(y))
d(g(g(0, x), y), s(z)) → g(e(x), d(g(g(0, x), y), z))
d(g(g(0, x), y), 0) → e(y)
d(g(0, x), y) → e(x)
d(g(x, y), z) → g(d(x, z), e(y))
g(e(x), e(y)) → e(g(x, y))

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

h(e(x), y) → h(d(x, y), s(y))
d(g(g(0', x), y), s(z)) → g(e(x), d(g(g(0', x), y), z))
d(g(g(0', x), y), 0') → e(y)
d(g(0', x), y) → e(x)
d(g(x, y), z) → g(d(x, z), e(y))
g(e(x), e(y)) → e(g(x, y))

S is empty.
Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
h(e(x), y) → h(d(x, y), s(y))
d(g(g(0', x), y), s(z)) → g(e(x), d(g(g(0', x), y), z))
d(g(g(0', x), y), 0') → e(y)
d(g(0', x), y) → e(x)
d(g(x, y), z) → g(d(x, z), e(y))
g(e(x), e(y)) → e(g(x, y))

Types:
h :: e:s:0' → e:s:0' → h
e :: e:s:0' → e:s:0'
d :: e:s:0' → e:s:0' → e:s:0'
s :: e:s:0' → e:s:0'
g :: e:s:0' → e:s:0' → e:s:0'
0' :: e:s:0'
hole_h1_0 :: h
hole_e:s:0'2_0 :: e:s:0'
gen_e:s:0'3_0 :: Nat → e:s:0'

(5) OrderProof (LOWER BOUND(ID) transformation)

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

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

(6) Obligation:

Innermost TRS:
Rules:
h(e(x), y) → h(d(x, y), s(y))
d(g(g(0', x), y), s(z)) → g(e(x), d(g(g(0', x), y), z))
d(g(g(0', x), y), 0') → e(y)
d(g(0', x), y) → e(x)
d(g(x, y), z) → g(d(x, z), e(y))
g(e(x), e(y)) → e(g(x, y))

Types:
h :: e:s:0' → e:s:0' → h
e :: e:s:0' → e:s:0'
d :: e:s:0' → e:s:0' → e:s:0'
s :: e:s:0' → e:s:0'
g :: e:s:0' → e:s:0' → e:s:0'
0' :: e:s:0'
hole_h1_0 :: h
hole_e:s:0'2_0 :: e:s:0'
gen_e:s:0'3_0 :: Nat → e:s:0'

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

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

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

Induction Base:
g(gen_e:s:0'3_0(+(1, 0)), gen_e:s:0'3_0(+(1, 0)))

Induction Step:
g(gen_e:s:0'3_0(+(1, +(n5_0, 1))), gen_e:s:0'3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
e(g(gen_e:s:0'3_0(+(1, n5_0)), gen_e:s:0'3_0(+(1, n5_0)))) →_IH
e(*4_0)

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
h(e(x), y) → h(d(x, y), s(y))
d(g(g(0', x), y), s(z)) → g(e(x), d(g(g(0', x), y), z))
d(g(g(0', x), y), 0') → e(y)
d(g(0', x), y) → e(x)
d(g(x, y), z) → g(d(x, z), e(y))
g(e(x), e(y)) → e(g(x, y))

Types:
h :: e:s:0' → e:s:0' → h
e :: e:s:0' → e:s:0'
d :: e:s:0' → e:s:0' → e:s:0'
s :: e:s:0' → e:s:0'
g :: e:s:0' → e:s:0' → e:s:0'
0' :: e:s:0'
hole_h1_0 :: h
hole_e:s:0'2_0 :: e:s:0'
gen_e:s:0'3_0 :: Nat → e:s:0'

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

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

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

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

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
h(e(x), y) → h(d(x, y), s(y))
d(g(g(0', x), y), s(z)) → g(e(x), d(g(g(0', x), y), z))
d(g(g(0', x), y), 0') → e(y)
d(g(0', x), y) → e(x)
d(g(x, y), z) → g(d(x, z), e(y))
g(e(x), e(y)) → e(g(x, y))

Types:
h :: e:s:0' → e:s:0' → h
e :: e:s:0' → e:s:0'
d :: e:s:0' → e:s:0' → e:s:0'
s :: e:s:0' → e:s:0'
g :: e:s:0' → e:s:0' → e:s:0'
0' :: e:s:0'
hole_h1_0 :: h
hole_e:s:0'2_0 :: e:s:0'
gen_e:s:0'3_0 :: Nat → e:s:0'

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

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

Innermost TRS:
Rules:
h(e(x), y) → h(d(x, y), s(y))
d(g(g(0', x), y), s(z)) → g(e(x), d(g(g(0', x), y), z))
d(g(g(0', x), y), 0') → e(y)
d(g(0', x), y) → e(x)
d(g(x, y), z) → g(d(x, z), e(y))
g(e(x), e(y)) → e(g(x, y))

Types:
h :: e:s:0' → e:s:0' → h
e :: e:s:0' → e:s:0'
d :: e:s:0' → e:s:0' → e:s:0'
s :: e:s:0' → e:s:0'
g :: e:s:0' → e:s:0' → e:s:0'
0' :: e:s:0'
hole_h1_0 :: h
hole_e:s:0'2_0 :: e:s:0'
gen_e:s:0'3_0 :: Nat → e:s:0'

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

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

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
h(e(x), y) → h(d(x, y), s(y))
d(g(g(0', x), y), s(z)) → g(e(x), d(g(g(0', x), y), z))
d(g(g(0', x), y), 0') → e(y)
d(g(0', x), y) → e(x)
d(g(x, y), z) → g(d(x, z), e(y))
g(e(x), e(y)) → e(g(x, y))

Types:
h :: e:s:0' → e:s:0' → h
e :: e:s:0' → e:s:0'
d :: e:s:0' → e:s:0' → e:s:0'
s :: e:s:0' → e:s:0'
g :: e:s:0' → e:s:0' → e:s:0'
0' :: e:s:0'
hole_h1_0 :: h
hole_e:s:0'2_0 :: e:s:0'
gen_e:s:0'3_0 :: Nat → e:s:0'

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

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

(18) BOUNDS(n^1, INF)