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(s(s(s(s(s(s(s(x)))))))), y, y) → f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
id(s(x)) → s(id(x))
id(0) → 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(s(s(s(s(s(s(s(x)))))))), y, y) → f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
id(s(x)) → s(id(x))
id(0') → 0'

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
f(s(s(s(s(s(s(s(s(x)))))))), y, y) → f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
id(s(x)) → s(id(x))
id(0') → 0'

Types:
f :: s:0' → a → a → f
s :: s:0' → s:0'
id :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
hole_a3_0 :: a
gen_s:0'4_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
f, id

They will be analysed ascendingly in the following order:
id < f

(6) Obligation:

Innermost TRS:
Rules:
f(s(s(s(s(s(s(s(s(x)))))))), y, y) → f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
id(s(x)) → s(id(x))
id(0') → 0'

Types:
f :: s:0' → a → a → f
s :: s:0' → s:0'
id :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
hole_a3_0 :: a
gen_s:0'4_0 :: Nat → s:0'

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

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

They will be analysed ascendingly in the following order:
id < f

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

Proved the following rewrite lemma:
id(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(n6_0), rt ∈ Ω(1 + n6₀)

Induction Base:
id(gen_s:0'4_0(0)) →_R^Ω(1)
0'

Induction Step:
id(gen_s:0'4_0(+(n6_0, 1))) →_R^Ω(1)
s(id(gen_s:0'4_0(n6_0))) →_IH
s(gen_s:0'4_0(c7_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(s(s(s(s(s(s(s(x)))))))), y, y) → f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
id(s(x)) → s(id(x))
id(0') → 0'

Types:
f :: s:0' → a → a → f
s :: s:0' → s:0'
id :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
hole_a3_0 :: a
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
id(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(n6_0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

The following defined symbols remain to be analysed:
f

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

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

(11) Obligation:

Innermost TRS:
Rules:
f(s(s(s(s(s(s(s(s(x)))))))), y, y) → f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
id(s(x)) → s(id(x))
id(0') → 0'

Types:
f :: s:0' → a → a → f
s :: s:0' → s:0'
id :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
hole_a3_0 :: a
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
id(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(n6_0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
id(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(n6_0), rt ∈ Ω(1 + n6₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
f(s(s(s(s(s(s(s(s(x)))))))), y, y) → f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
id(s(x)) → s(id(x))
id(0') → 0'

Types:
f :: s:0' → a → a → f
s :: s:0' → s:0'
id :: s:0' → s:0'
0' :: s:0'
hole_f1_0 :: f
hole_s:0'2_0 :: s:0'
hole_a3_0 :: a
gen_s:0'4_0 :: Nat → s:0'

Lemmas:
id(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(n6_0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
id(gen_s:0'4_0(n6_0)) → gen_s:0'4_0(n6_0), rt ∈ Ω(1 + n6₀)

(16) BOUNDS(n^1, INF)