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

Renamed function symbols to avoid clashes with predefined symbol.

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

Sliced the following arguments:
+'/0

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

Rewrite Strategy: INNERMOST

Infered types.

Rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), 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':+'
_hole_0':s':+'1 :: 0':s':+'
_gen_0':s':+'2 :: Nat → 0':s':+'

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

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

Rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), 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':+'
_hole_0':s':+'1 :: 0':s':+'
_gen_0':s':+'2 :: Nat → 0':s':+'

Generator Equations:
_gen_0':s':+'2(0) ⇔ 0'
_gen_0':s':+'2(+(x, 1)) ⇔ s'(_gen_0':s':+'2(x))

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

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

Proved the following rewrite lemma:
*'(_gen_0':s':+'2(+(1, _n4)), _gen_0':s':+'2(b)) → _*3, rt ∈ Ω(n4)

Induction Base:
*'(_gen_0':s':+'2(+(1, 0)), _gen_0':s':+'2(b))

Induction Step:
*'(_gen_0':s':+'2(+(1, +(_\$n5, 1))), _gen_0':s':+'2(_b711)) →RΩ(1)
+'(*'(_gen_0':s':+'2(+(1, _\$n5)), _gen_0':s':+'2(_b711))) →IH
+'(_*3)

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

Rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), 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':+'
_hole_0':s':+'1 :: 0':s':+'
_gen_0':s':+'2 :: Nat → 0':s':+'

Lemmas:
*'(_gen_0':s':+'2(+(1, _n4)), _gen_0':s':+'2(b)) → _*3, rt ∈ Ω(n4)

Generator Equations:
_gen_0':s':+'2(0) ⇔ 0'
_gen_0':s':+'2(+(x, 1)) ⇔ s'(_gen_0':s':+'2(x))

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

Proved the following rewrite lemma:
exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(_n1018)) → _*3, rt ∈ Ω(n1018)

Induction Base:
exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(0))

Induction Step:
exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(+(_\$n1019, 1))) →RΩ(1)
*'(_gen_0':s':+'2(0), exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(_\$n1019))) →IH
*'(_gen_0':s':+'2(0), _*3)

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

Rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), 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':+'
_hole_0':s':+'1 :: 0':s':+'
_gen_0':s':+'2 :: Nat → 0':s':+'

Lemmas:
*'(_gen_0':s':+'2(+(1, _n4)), _gen_0':s':+'2(b)) → _*3, rt ∈ Ω(n4)
exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(_n1018)) → _*3, rt ∈ Ω(n1018)

Generator Equations:
_gen_0':s':+'2(0) ⇔ 0'
_gen_0':s':+'2(+(x, 1)) ⇔ s'(_gen_0':s':+'2(x))

The following defined symbols remain to be analysed:
-'

Proved the following rewrite lemma:
-'(_gen_0':s':+'2(_n2763), _gen_0':s':+'2(_n2763)) → _gen_0':s':+'2(0), rt ∈ Ω(1 + n2763)

Induction Base:
-'(_gen_0':s':+'2(0), _gen_0':s':+'2(0)) →RΩ(1)
0'

Induction Step:
-'(_gen_0':s':+'2(+(_\$n2764, 1)), _gen_0':s':+'2(+(_\$n2764, 1))) →RΩ(1)
-'(_gen_0':s':+'2(_\$n2764), _gen_0':s':+'2(_\$n2764)) →IH
_gen_0':s':+'2(0)

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

Rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), 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':+'
_hole_0':s':+'1 :: 0':s':+'
_gen_0':s':+'2 :: Nat → 0':s':+'

Lemmas:
*'(_gen_0':s':+'2(+(1, _n4)), _gen_0':s':+'2(b)) → _*3, rt ∈ Ω(n4)
exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(_n1018)) → _*3, rt ∈ Ω(n1018)
-'(_gen_0':s':+'2(_n2763), _gen_0':s':+'2(_n2763)) → _gen_0':s':+'2(0), rt ∈ Ω(1 + n2763)

Generator Equations:
_gen_0':s':+'2(0) ⇔ 0'
_gen_0':s':+'2(+(x, 1)) ⇔ s'(_gen_0':s':+'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
*'(_gen_0':s':+'2(+(1, _n4)), _gen_0':s':+'2(b)) → _*3, rt ∈ Ω(n4)