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:

g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0, x2) → x2
f(x1, 0) → g(x1, 0)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(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:

g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0', x2) → x2
f(x1, 0') → g(x1, 0')

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0', x2) → x2
f(x1, 0') → g(x1, 0')

Types:
g :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
f :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

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

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

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

(8) Obligation:

Innermost TRS:
Rules:
g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0', x2) → x2
f(x1, 0') → g(x1, 0')

Types:
g :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
f :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

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

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

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

Proved the following rewrite lemma:
g(gen_S:0'2_0(n4_0), gen_S:0'2_0(b)) → gen_S:0'2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Induction Base:
g(gen_S:0'2_0(0), gen_S:0'2_0(b)) →_R^Ω(1)
gen_S:0'2_0(b)

Induction Step:
g(gen_S:0'2_0(+(n4_0, 1)), gen_S:0'2_0(b)) →_R^Ω(1)
g(gen_S:0'2_0(n4_0), S(gen_S:0'2_0(b))) →_IH
gen_S:0'2_0(+(+(b, 1), c5_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:
g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0', x2) → x2
f(x1, 0') → g(x1, 0')

Types:
g :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
f :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
g(gen_S:0'2_0(n4_0), gen_S:0'2_0(b)) → gen_S:0'2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

The following defined symbols remain to be analysed:
f

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_S:0'2_0(a), gen_S:0'2_0(n431_0)) → gen_S:0'2_0(+(n431_0, a)), rt ∈ Ω(1 + a + n431₀)

Induction Base:
f(gen_S:0'2_0(a), gen_S:0'2_0(0)) →_R^Ω(1)
g(gen_S:0'2_0(a), 0') →_L^{Ω(1 + a)}
gen_S:0'2_0(+(a, 0))

Induction Step:
f(gen_S:0'2_0(a), gen_S:0'2_0(+(n431_0, 1))) →_R^Ω(1)
f(S(gen_S:0'2_0(a)), gen_S:0'2_0(n431_0)) →_IH
gen_S:0'2_0(+(+(a, 1), c432_0))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0', x2) → x2
f(x1, 0') → g(x1, 0')

Types:
g :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
f :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
g(gen_S:0'2_0(n4_0), gen_S:0'2_0(b)) → gen_S:0'2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
f(gen_S:0'2_0(a), gen_S:0'2_0(n431_0)) → gen_S:0'2_0(+(n431_0, a)), rt ∈ Ω(1 + a + n431₀)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_S:0'2_0(n4_0), gen_S:0'2_0(b)) → gen_S:0'2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0', x2) → x2
f(x1, 0') → g(x1, 0')

Types:
g :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
f :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
g(gen_S:0'2_0(n4_0), gen_S:0'2_0(b)) → gen_S:0'2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
f(gen_S:0'2_0(a), gen_S:0'2_0(n431_0)) → gen_S:0'2_0(+(n431_0, a)), rt ∈ Ω(1 + a + n431₀)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_S:0'2_0(n4_0), gen_S:0'2_0(b)) → gen_S:0'2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
g(S(x), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0', x2) → x2
f(x1, 0') → g(x1, 0')

Types:
g :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
f :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
g(gen_S:0'2_0(n4_0), gen_S:0'2_0(b)) → gen_S:0'2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_S:0'2_0(n4_0), gen_S:0'2_0(b)) → gen_S:0'2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(22) BOUNDS(n^1, INF)