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:

f(s(x)) → f(id_inc(c(x, x)))
f(c(s(x), y)) → g(c(x, y))
g(c(s(x), y)) → g(c(y, x))
g(c(x, s(y))) → g(c(y, x))
g(c(x, x)) → f(x)
id_inc(c(x, y)) → c(id_inc(x), id_inc(y))
id_inc(s(x)) → s(id_inc(x))
id_inc(0) → 0
id_inc(0) → s(0)

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:

f(s(x)) → f(id_inc(c(x, x)))
f(c(s(x), y)) → g(c(x, y))
g(c(s(x), y)) → g(c(y, x))
g(c(x, s(y))) → g(c(y, x))
g(c(x, x)) → f(x)
id_inc(c(x, y)) → c(id_inc(x), id_inc(y))
id_inc(s(x)) → s(id_inc(x))
id_inc(0') → 0'
id_inc(0') → s(0')

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
f(s(x)) → f(id_inc(c(x, x)))
f(c(s(x), y)) → g(c(x, y))
g(c(s(x), y)) → g(c(y, x))
g(c(x, s(y))) → g(c(y, x))
g(c(x, x)) → f(x)
id_inc(c(x, y)) → c(id_inc(x), id_inc(y))
id_inc(s(x)) → s(id_inc(x))
id_inc(0') → 0'
id_inc(0') → s(0')

Types:
f :: s:c:0' → f:g
s :: s:c:0' → s:c:0'
id_inc :: s:c:0' → s:c:0'
c :: s:c:0' → s:c:0' → s:c:0'
g :: s:c:0' → f:g
0' :: s:c:0'
hole_f:g1_0 :: f:g
hole_s:c:0'2_0 :: s:c:0'
gen_s:c:0'3_0 :: Nat → s:c:0'

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

Heuristically decided to analyse the following defined symbols:
f, id_inc, g

They will be analysed ascendingly in the following order:
id_inc < f
f = g

(6) Obligation:

Innermost TRS:
Rules:
f(s(x)) → f(id_inc(c(x, x)))
f(c(s(x), y)) → g(c(x, y))
g(c(s(x), y)) → g(c(y, x))
g(c(x, s(y))) → g(c(y, x))
g(c(x, x)) → f(x)
id_inc(c(x, y)) → c(id_inc(x), id_inc(y))
id_inc(s(x)) → s(id_inc(x))
id_inc(0') → 0'
id_inc(0') → s(0')

Types:
f :: s:c:0' → f:g
s :: s:c:0' → s:c:0'
id_inc :: s:c:0' → s:c:0'
c :: s:c:0' → s:c:0' → s:c:0'
g :: s:c:0' → f:g
0' :: s:c:0'
hole_f:g1_0 :: f:g
hole_s:c:0'2_0 :: s:c:0'
gen_s:c:0'3_0 :: Nat → s:c:0'

Generator Equations:
gen_s:c:0'3_0(0) ⇔ 0'
gen_s:c:0'3_0(+(x, 1)) ⇔ s(gen_s:c:0'3_0(x))

The following defined symbols remain to be analysed:
id_inc, f, g

They will be analysed ascendingly in the following order:
id_inc < f
f = g

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

Proved the following rewrite lemma:
id_inc(gen_s:c:0'3_0(n5_0)) → gen_s:c:0'3_0(n5_0), rt ∈ Ω(1 + n5₀)

Induction Base:
id_inc(gen_s:c:0'3_0(0)) →_R^Ω(1)
0'

Induction Step:
id_inc(gen_s:c:0'3_0(+(n5_0, 1))) →_R^Ω(1)
s(id_inc(gen_s:c:0'3_0(n5_0))) →_IH
s(gen_s:c:0'3_0(c6_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:
f(s(x)) → f(id_inc(c(x, x)))
f(c(s(x), y)) → g(c(x, y))
g(c(s(x), y)) → g(c(y, x))
g(c(x, s(y))) → g(c(y, x))
g(c(x, x)) → f(x)
id_inc(c(x, y)) → c(id_inc(x), id_inc(y))
id_inc(s(x)) → s(id_inc(x))
id_inc(0') → 0'
id_inc(0') → s(0')

Types:
f :: s:c:0' → f:g
s :: s:c:0' → s:c:0'
id_inc :: s:c:0' → s:c:0'
c :: s:c:0' → s:c:0' → s:c:0'
g :: s:c:0' → f:g
0' :: s:c:0'
hole_f:g1_0 :: f:g
hole_s:c:0'2_0 :: s:c:0'
gen_s:c:0'3_0 :: Nat → s:c:0'

Lemmas:
id_inc(gen_s:c:0'3_0(n5_0)) → gen_s:c:0'3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:c:0'3_0(0) ⇔ 0'
gen_s:c:0'3_0(+(x, 1)) ⇔ s(gen_s:c:0'3_0(x))

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

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

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

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

(11) Obligation:

Innermost TRS:
Rules:
f(s(x)) → f(id_inc(c(x, x)))
f(c(s(x), y)) → g(c(x, y))
g(c(s(x), y)) → g(c(y, x))
g(c(x, s(y))) → g(c(y, x))
g(c(x, x)) → f(x)
id_inc(c(x, y)) → c(id_inc(x), id_inc(y))
id_inc(s(x)) → s(id_inc(x))
id_inc(0') → 0'
id_inc(0') → s(0')

Types:
f :: s:c:0' → f:g
s :: s:c:0' → s:c:0'
id_inc :: s:c:0' → s:c:0'
c :: s:c:0' → s:c:0' → s:c:0'
g :: s:c:0' → f:g
0' :: s:c:0'
hole_f:g1_0 :: f:g
hole_s:c:0'2_0 :: s:c:0'
gen_s:c:0'3_0 :: Nat → s:c:0'

Lemmas:
id_inc(gen_s:c:0'3_0(n5_0)) → gen_s:c:0'3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:c:0'3_0(0) ⇔ 0'
gen_s:c:0'3_0(+(x, 1)) ⇔ s(gen_s:c:0'3_0(x))

The following defined symbols remain to be analysed:
f

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

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

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

(13) Obligation:

Innermost TRS:
Rules:
f(s(x)) → f(id_inc(c(x, x)))
f(c(s(x), y)) → g(c(x, y))
g(c(s(x), y)) → g(c(y, x))
g(c(x, s(y))) → g(c(y, x))
g(c(x, x)) → f(x)
id_inc(c(x, y)) → c(id_inc(x), id_inc(y))
id_inc(s(x)) → s(id_inc(x))
id_inc(0') → 0'
id_inc(0') → s(0')

Types:
f :: s:c:0' → f:g
s :: s:c:0' → s:c:0'
id_inc :: s:c:0' → s:c:0'
c :: s:c:0' → s:c:0' → s:c:0'
g :: s:c:0' → f:g
0' :: s:c:0'
hole_f:g1_0 :: f:g
hole_s:c:0'2_0 :: s:c:0'
gen_s:c:0'3_0 :: Nat → s:c:0'

Lemmas:
id_inc(gen_s:c:0'3_0(n5_0)) → gen_s:c:0'3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:c:0'3_0(0) ⇔ 0'
gen_s:c:0'3_0(+(x, 1)) ⇔ s(gen_s:c: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:
id_inc(gen_s:c:0'3_0(n5_0)) → gen_s:c:0'3_0(n5_0), rt ∈ Ω(1 + n5₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
f(s(x)) → f(id_inc(c(x, x)))
f(c(s(x), y)) → g(c(x, y))
g(c(s(x), y)) → g(c(y, x))
g(c(x, s(y))) → g(c(y, x))
g(c(x, x)) → f(x)
id_inc(c(x, y)) → c(id_inc(x), id_inc(y))
id_inc(s(x)) → s(id_inc(x))
id_inc(0') → 0'
id_inc(0') → s(0')

Types:
f :: s:c:0' → f:g
s :: s:c:0' → s:c:0'
id_inc :: s:c:0' → s:c:0'
c :: s:c:0' → s:c:0' → s:c:0'
g :: s:c:0' → f:g
0' :: s:c:0'
hole_f:g1_0 :: f:g
hole_s:c:0'2_0 :: s:c:0'
gen_s:c:0'3_0 :: Nat → s:c:0'

Lemmas:
id_inc(gen_s:c:0'3_0(n5_0)) → gen_s:c:0'3_0(n5_0), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:c:0'3_0(0) ⇔ 0'
gen_s:c:0'3_0(+(x, 1)) ⇔ s(gen_s:c: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:
id_inc(gen_s:c:0'3_0(n5_0)) → gen_s:c:0'3_0(n5_0), rt ∈ Ω(1 + n5₀)

(18) BOUNDS(n^1, INF)