The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

f(0) → 1
f(s(x)) → g(x, s(x))
g(0, y) → y
g(s(x), y) → g(x, +(y, s(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
g(s(x), y) → g(x, s(+(y, x)))

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(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
f(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

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

Heuristically decided to analyse the following defined symbols:
g, +'

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

(6) Obligation:

Innermost TRS:
Rules:
f(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

Generator Equations:
gen_0':1':s2_0(0) ⇔ 0'
gen_0':1':s2_0(+(x, 1)) ⇔ s(gen_0':1':s2_0(x))

The following defined symbols remain to be analysed:
+', g

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

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

Proved the following rewrite lemma:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) → gen_0':1':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Induction Base:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(0)) →_R^Ω(1)
gen_0':1':s2_0(a)

Induction Step:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(+(n4_0, 1))) →_R^Ω(1)
s(+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0))) →_IH
s(gen_0':1':s2_0(+(a, c5_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(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

Lemmas:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) → gen_0':1':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':1':s2_0(0) ⇔ 0'
gen_0':1':s2_0(+(x, 1)) ⇔ s(gen_0':1':s2_0(x))

The following defined symbols remain to be analysed:
g

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
g(gen_0':1':s2_0(n527_0), gen_0':1':s2_0(b)) → *3_0, rt ∈ Ω(n527₀ + n527₀²)

Induction Base:
g(gen_0':1':s2_0(0), gen_0':1':s2_0(b))

Induction Step:
g(gen_0':1':s2_0(+(n527_0, 1)), gen_0':1':s2_0(b)) →_R^Ω(1)
g(gen_0':1':s2_0(n527_0), +'(gen_0':1':s2_0(b), s(gen_0':1':s2_0(n527_0)))) →_L^{Ω(2 + n527₀)}
g(gen_0':1':s2_0(n527_0), gen_0':1':s2_0(+(+(n527_0, 1), b))) →_IH
*3_0

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
f(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

Lemmas:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) → gen_0':1':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
g(gen_0':1':s2_0(n527_0), gen_0':1':s2_0(b)) → *3_0, rt ∈ Ω(n527₀ + n527₀²)

Generator Equations:
gen_0':1':s2_0(0) ⇔ 0'
gen_0':1':s2_0(+(x, 1)) ⇔ s(gen_0':1':s2_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
g(gen_0':1':s2_0(n527_0), gen_0':1':s2_0(b)) → *3_0, rt ∈ Ω(n527₀ + n527₀²)

(14) BOUNDS(n^2, INF)

(15) Obligation:

Innermost TRS:
Rules:
f(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

Lemmas:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) → gen_0':1':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
g(gen_0':1':s2_0(n527_0), gen_0':1':s2_0(b)) → *3_0, rt ∈ Ω(n527₀ + n527₀²)

Generator Equations:
gen_0':1':s2_0(0) ⇔ 0'
gen_0':1':s2_0(+(x, 1)) ⇔ s(gen_0':1':s2_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
g(gen_0':1':s2_0(n527_0), gen_0':1':s2_0(b)) → *3_0, rt ∈ Ω(n527₀ + n527₀²)

(17) BOUNDS(n^2, INF)

(18) Obligation:

Innermost TRS:
Rules:
f(0') → 1'
f(s(x)) → g(x, s(x))
g(0', y) → y
g(s(x), y) → g(x, +'(y, s(x)))
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
g(s(x), y) → g(x, s(+'(y, x)))

Types:
f :: 0':1':s → 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s → 0':1':s
g :: 0':1':s → 0':1':s → 0':1':s
+' :: 0':1':s → 0':1':s → 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat → 0':1':s

Lemmas:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) → gen_0':1':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':1':s2_0(0) ⇔ 0'
gen_0':1':s2_0(+(x, 1)) ⇔ s(gen_0':1':s2_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) → gen_0':1':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

(20) BOUNDS(n^1, INF)