The Runtime Complexity (full) 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(z, e(x)) → h(c(z), d(z, x))
d(z, g(0, 0)) → e(0)
d(z, g(x, y)) → g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) → g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) → e(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(e(x), e(y)) →⁺ e(g(x, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / e(x), y / e(y)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

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

Types:
h :: c → e:0' → h
e :: e:0' → e:0'
c :: c → c
d :: c → e:0' → e:0'
g :: e:0' → e:0' → e:0'
0' :: e:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_e:0'3_0 :: e:0'
gen_c4_0 :: Nat → c
gen_e:0'5_0 :: Nat → e:0'

(7) 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

(8) Obligation:

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

Types:
h :: c → e:0' → h
e :: e:0' → e:0'
c :: c → c
d :: c → e:0' → e:0'
g :: e:0' → e:0' → e:0'
0' :: e:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_e:0'3_0 :: e:0'
gen_c4_0 :: Nat → c
gen_e:0'5_0 :: Nat → e:0'

Generator Equations:
gen_c4_0(0) ⇔ hole_c2_0
gen_c4_0(+(x, 1)) ⇔ c(gen_c4_0(x))
gen_e:0'5_0(0) ⇔ 0'
gen_e:0'5_0(+(x, 1)) ⇔ e(gen_e:0'5_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

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

Proved the following rewrite lemma:
g(gen_e:0'5_0(+(1, n7_0)), gen_e:0'5_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Induction Base:
g(gen_e:0'5_0(+(1, 0)), gen_e:0'5_0(+(1, 0)))

Induction Step:
g(gen_e:0'5_0(+(1, +(n7_0, 1))), gen_e:0'5_0(+(1, +(n7_0, 1)))) →_R^Ω(1)
e(g(gen_e:0'5_0(+(1, n7_0)), gen_e:0'5_0(+(1, n7_0)))) →_IH
e(*6_0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

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

Types:
h :: c → e:0' → h
e :: e:0' → e:0'
c :: c → c
d :: c → e:0' → e:0'
g :: e:0' → e:0' → e:0'
0' :: e:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_e:0'3_0 :: e:0'
gen_c4_0 :: Nat → c
gen_e:0'5_0 :: Nat → e:0'

Lemmas:
g(gen_e:0'5_0(+(1, n7_0)), gen_e:0'5_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Generator Equations:
gen_c4_0(0) ⇔ hole_c2_0
gen_c4_0(+(x, 1)) ⇔ c(gen_c4_0(x))
gen_e:0'5_0(0) ⇔ 0'
gen_e:0'5_0(+(x, 1)) ⇔ e(gen_e:0'5_0(x))

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

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

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

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

(13) Obligation:

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

Types:
h :: c → e:0' → h
e :: e:0' → e:0'
c :: c → c
d :: c → e:0' → e:0'
g :: e:0' → e:0' → e:0'
0' :: e:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_e:0'3_0 :: e:0'
gen_c4_0 :: Nat → c
gen_e:0'5_0 :: Nat → e:0'

Lemmas:
g(gen_e:0'5_0(+(1, n7_0)), gen_e:0'5_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Generator Equations:
gen_c4_0(0) ⇔ hole_c2_0
gen_c4_0(+(x, 1)) ⇔ c(gen_c4_0(x))
gen_e:0'5_0(0) ⇔ 0'
gen_e:0'5_0(+(x, 1)) ⇔ e(gen_e:0'5_0(x))

The following defined symbols remain to be analysed:
h

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

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

Types:
h :: c → e:0' → h
e :: e:0' → e:0'
c :: c → c
d :: c → e:0' → e:0'
g :: e:0' → e:0' → e:0'
0' :: e:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_e:0'3_0 :: e:0'
gen_c4_0 :: Nat → c
gen_e:0'5_0 :: Nat → e:0'

Lemmas:
g(gen_e:0'5_0(+(1, n7_0)), gen_e:0'5_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Generator Equations:
gen_c4_0(0) ⇔ hole_c2_0
gen_c4_0(+(x, 1)) ⇔ c(gen_c4_0(x))
gen_e:0'5_0(0) ⇔ 0'
gen_e:0'5_0(+(x, 1)) ⇔ e(gen_e:0'5_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_e:0'5_0(+(1, n7_0)), gen_e:0'5_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

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

Types:
h :: c → e:0' → h
e :: e:0' → e:0'
c :: c → c
d :: c → e:0' → e:0'
g :: e:0' → e:0' → e:0'
0' :: e:0'
hole_h1_0 :: h
hole_c2_0 :: c
hole_e:0'3_0 :: e:0'
gen_c4_0 :: Nat → c
gen_e:0'5_0 :: Nat → e:0'

Lemmas:
g(gen_e:0'5_0(+(1, n7_0)), gen_e:0'5_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

Generator Equations:
gen_c4_0(0) ⇔ hole_c2_0
gen_c4_0(+(x, 1)) ⇔ c(gen_c4_0(x))
gen_e:0'5_0(0) ⇔ 0'
gen_e:0'5_0(+(x, 1)) ⇔ e(gen_e:0'5_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_e:0'5_0(+(1, n7_0)), gen_e:0'5_0(+(1, n7_0))) → *6_0, rt ∈ Ω(n7₀)

(20) BOUNDS(n^1, INF)