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:

eq0(S(x'), S(x)) → eq0(x', x)
eq0(S(x), 0) → 0
eq0(0, S(x)) → 0
eq0(0, 0) → S(0)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
eq0(S(x'), S(x)) →⁺ eq0(x', x)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x' / S(x'), x / S(x)].
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:

eq0(S(x'), S(x)) → eq0(x', x)
eq0(S(x), 0') → 0'
eq0(0', S(x)) → 0'
eq0(0', 0') → S(0')

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
eq0(S(x'), S(x)) → eq0(x', x)
eq0(S(x), 0') → 0'
eq0(0', S(x)) → 0'
eq0(0', 0') → S(0')

Types:
eq0 :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

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

Heuristically decided to analyse the following defined symbols:
eq0

(8) Obligation:

Innermost TRS:
Rules:
eq0(S(x'), S(x)) → eq0(x', x)
eq0(S(x), 0') → 0'
eq0(0', S(x)) → 0'
eq0(0', 0') → S(0')

Types:
eq0 :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

The following defined symbols remain to be analysed:
eq0

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

Proved the following rewrite lemma:
eq0(gen_S:0'2_1(+(1, n4_1)), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(0), rt ∈ Ω(1 + n4₁)

Induction Base:
eq0(gen_S:0'2_1(+(1, 0)), gen_S:0'2_1(0)) →_R^Ω(1)
0'

Induction Step:
eq0(gen_S:0'2_1(+(1, +(n4_1, 1))), gen_S:0'2_1(+(n4_1, 1))) →_R^Ω(1)
eq0(gen_S:0'2_1(+(1, n4_1)), gen_S:0'2_1(n4_1)) →_IH
gen_S:0'2_1(0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
eq0(S(x'), S(x)) → eq0(x', x)
eq0(S(x), 0') → 0'
eq0(0', S(x)) → 0'
eq0(0', 0') → S(0')

Types:
eq0 :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Lemmas:
eq0(gen_S:0'2_1(+(1, n4_1)), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(0), rt ∈ Ω(1 + n4₁)

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq0(gen_S:0'2_1(+(1, n4_1)), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(0), rt ∈ Ω(1 + n4₁)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
eq0(S(x'), S(x)) → eq0(x', x)
eq0(S(x), 0') → 0'
eq0(0', S(x)) → 0'
eq0(0', 0') → S(0')

Types:
eq0 :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
0' :: S:0'
hole_S:0'1_1 :: S:0'
gen_S:0'2_1 :: Nat → S:0'

Lemmas:
eq0(gen_S:0'2_1(+(1, n4_1)), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(0), rt ∈ Ω(1 + n4₁)

Generator Equations:
gen_S:0'2_1(0) ⇔ 0'
gen_S:0'2_1(+(x, 1)) ⇔ S(gen_S:0'2_1(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq0(gen_S:0'2_1(+(1, n4_1)), gen_S:0'2_1(n4_1)) → gen_S:0'2_1(0), rt ∈ Ω(1 + n4₁)

(16) BOUNDS(n^1, INF)