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

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

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
-' :: 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'
f' :: 0':s' → 0':s' → f'
_hole_0':s'1 :: 0':s'
_hole_f'2 :: f'
_gen_0':s'3 :: Nat → 0':s'

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

They will be analysed ascendingly in the following order:
-' < f'
+' < *'
*' < f'

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

Types:
-' :: 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'
f' :: 0':s' → 0':s' → f'
_hole_0':s'1 :: 0':s'
_hole_f'2 :: f'
_gen_0':s'3 :: Nat → 0':s'

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

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

They will be analysed ascendingly in the following order:
-' < f'
+' < *'
*' < f'

Proved the following rewrite lemma:
-'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)

Induction Base:
-'(_gen_0':s'3(0), _gen_0':s'3(0)) →RΩ(1)
_gen_0':s'3(0)

Induction Step:
-'(_gen_0':s'3(+(_$n6, 1)), _gen_0':s'3(+(_$n6, 1))) →RΩ(1)
-'(_gen_0':s'3(_$n6), _gen_0':s'3(_$n6)) →IH
_gen_0':s'3(0)

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

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

Types:
-' :: 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'
f' :: 0':s' → 0':s' → f'
_hole_0':s'1 :: 0':s'
_hole_f'2 :: f'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
-'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

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

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

Proved the following rewrite lemma:
+'(_gen_0':s'3(_n473), _gen_0':s'3(b)) → _gen_0':s'3(+(_n473, b)), rt ∈ Ω(1 + n473)

Induction Base:
+'(_gen_0':s'3(0), _gen_0':s'3(b)) →RΩ(1)
_gen_0':s'3(b)

Induction Step:
+'(_gen_0':s'3(+(_$n474, 1)), _gen_0':s'3(_b606)) →RΩ(1)
s'(+'(_gen_0':s'3(_$n474), _gen_0':s'3(_b606))) →IH
s'(_gen_0':s'3(+(_$n474, _b606)))

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

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

Types:
-' :: 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'
f' :: 0':s' → 0':s' → f'
_hole_0':s'1 :: 0':s'
_hole_f'2 :: f'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
-'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
+'(_gen_0':s'3(_n473), _gen_0':s'3(b)) → _gen_0':s'3(+(_n473, b)), rt ∈ Ω(1 + n473)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

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

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

Proved the following rewrite lemma:
*'(_gen_0':s'3(a), _gen_0':s'3(_n1040)) → _gen_0':s'3(*(_n1040, a)), rt ∈ Ω(1 + a1300·n1040 + n1040)

Induction Base:
*'(_gen_0':s'3(a), _gen_0':s'3(0)) →RΩ(1)
0'

Induction Step:
*'(_gen_0':s'3(_a1300), _gen_0':s'3(+(_$n1041, 1))) →RΩ(1)
+'(_gen_0':s'3(_a1300), *'(_gen_0':s'3(_a1300), _gen_0':s'3(_$n1041))) →IH
+'(_gen_0':s'3(_a1300), _gen_0':s'3(*(_$n1041, _a1300))) →LΩ(1 + a1300)
_gen_0':s'3(+(_a1300, *(_$n1041, _a1300)))

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

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

Types:
-' :: 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'
f' :: 0':s' → 0':s' → f'
_hole_0':s'1 :: 0':s'
_hole_f'2 :: f'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
-'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
+'(_gen_0':s'3(_n473), _gen_0':s'3(b)) → _gen_0':s'3(+(_n473, b)), rt ∈ Ω(1 + n473)
*'(_gen_0':s'3(a), _gen_0':s'3(_n1040)) → _gen_0':s'3(*(_n1040, a)), rt ∈ Ω(1 + a1300·n1040 + n1040)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
f'

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

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

Types:
-' :: 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'
f' :: 0':s' → 0':s' → f'
_hole_0':s'1 :: 0':s'
_hole_f'2 :: f'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
-'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
+'(_gen_0':s'3(_n473), _gen_0':s'3(b)) → _gen_0':s'3(+(_n473, b)), rt ∈ Ω(1 + n473)
*'(_gen_0':s'3(a), _gen_0':s'3(_n1040)) → _gen_0':s'3(*(_n1040, a)), rt ∈ Ω(1 + a1300·n1040 + n1040)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n2) was proven with the following lemma:
*'(_gen_0':s'3(a), _gen_0':s'3(_n1040)) → _gen_0':s'3(*(_n1040, a)), rt ∈ Ω(1 + a1300·n1040 + n1040)