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:

exp(x, 0) → s(0)
exp(x, s(y)) → *(x, exp(x, y))
*(0, y) → 0
*(s(x), y) → +(y, *(x, y))
-(0, y) → 0
-(x, 0) → x
-(s(x), s(y)) → -(x, y)

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:

exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(y, *'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(y, *'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

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

Heuristically decided to analyse the following defined symbols:
exp, *', -

They will be analysed ascendingly in the following order:
*' < exp

(6) Obligation:

Innermost TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(y, *'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

The following defined symbols remain to be analysed:
*', exp, -

They will be analysed ascendingly in the following order:
*' < exp

(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol *'.

(8) Obligation:

Innermost TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(y, *'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

The following defined symbols remain to be analysed:
exp, -

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

Innermost TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(y, *'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

The following defined symbols remain to be analysed:
-

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
-(gen_0':s:+'2_0(n59498_0), gen_0':s:+'2_0(n59498_0)) → gen_0':s:+'2_0(0), rt ∈ Ω(1 + n59498₀)

Induction Base:
-(gen_0':s:+'2_0(0), gen_0':s:+'2_0(0)) →_R^Ω(1)
0'

Induction Step:
-(gen_0':s:+'2_0(+(n59498_0, 1)), gen_0':s:+'2_0(+(n59498_0, 1))) →_R^Ω(1)
-(gen_0':s:+'2_0(n59498_0), gen_0':s:+'2_0(n59498_0)) →_IH
gen_0':s:+'2_0(0)

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

(12) Complex Obligation (BEST)

(13) Obligation:

Innermost TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(y, *'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
-(gen_0':s:+'2_0(n59498_0), gen_0':s:+'2_0(n59498_0)) → gen_0':s:+'2_0(0), rt ∈ Ω(1 + n59498₀)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
-(gen_0':s:+'2_0(n59498_0), gen_0':s:+'2_0(n59498_0)) → gen_0':s:+'2_0(0), rt ∈ Ω(1 + n59498₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
exp(x, 0') → s(0')
exp(x, s(y)) → *'(x, exp(x, y))
*'(0', y) → 0'
*'(s(x), y) → +'(y, *'(x, y))
-(0', y) → 0'
-(x, 0') → x
-(s(x), s(y)) → -(x, y)

Types:
exp :: 0':s:+' → 0':s:+' → 0':s:+'
0' :: 0':s:+'
s :: 0':s:+' → 0':s:+'
*' :: 0':s:+' → 0':s:+' → 0':s:+'
+' :: 0':s:+' → 0':s:+' → 0':s:+'
- :: 0':s:+' → 0':s:+' → 0':s:+'
hole_0':s:+'1_0 :: 0':s:+'
gen_0':s:+'2_0 :: Nat → 0':s:+'

Lemmas:
-(gen_0':s:+'2_0(n59498_0), gen_0':s:+'2_0(n59498_0)) → gen_0':s:+'2_0(0), rt ∈ Ω(1 + n59498₀)

Generator Equations:
gen_0':s:+'2_0(0) ⇔ 0'
gen_0':s:+'2_0(+(x, 1)) ⇔ s(gen_0':s:+'2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
-(gen_0':s:+'2_0(n59498_0), gen_0':s:+'2_0(n59498_0)) → gen_0':s:+'2_0(0), rt ∈ Ω(1 + n59498₀)

(18) BOUNDS(n^1, INF)