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

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

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

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

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

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' → 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'
*' < f'

Proved the following rewrite lemma:
+'(_gen_0':s'3(_n450), _gen_0':s'3(b)) → _gen_0':s'3(+(_n450, b)), rt ∈ Ω(1 + n450)

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(+(_$n451, 1)), _gen_0':s'3(_b583)) →RΩ(1)
s'(+'(_gen_0':s'3(_$n451), _gen_0':s'3(_b583))) →IH
s'(_gen_0':s'3(+(_$n451, _b583)))

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

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' → 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(_n450), _gen_0':s'3(b)) → _gen_0':s'3(+(_n450, b)), rt ∈ Ω(1 + n450)

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(_n995)) → _gen_0':s'3(*(_n995, a)), rt ∈ Ω(1 + a1255·n995 + n995)

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

Induction Step:
*'(_gen_0':s'3(_a1255), _gen_0':s'3(+(_$n996, 1))) →RΩ(1)
+'(_gen_0':s'3(_a1255), *'(_gen_0':s'3(_a1255), _gen_0':s'3(_$n996))) →IH
+'(_gen_0':s'3(_a1255), _gen_0':s'3(*(_$n996, _a1255))) →LΩ(1 + a1255)
_gen_0':s'3(+(_a1255, *(_$n996, _a1255)))

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

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' → 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(_n450), _gen_0':s'3(b)) → _gen_0':s'3(+(_n450, b)), rt ∈ Ω(1 + n450)
*'(_gen_0':s'3(a), _gen_0':s'3(_n995)) → _gen_0':s'3(*(_n995, a)), rt ∈ Ω(1 + a1255·n995 + n995)

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

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' → 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(_n450), _gen_0':s'3(b)) → _gen_0':s'3(+(_n450, b)), rt ∈ Ω(1 + n450)
*'(_gen_0':s'3(a), _gen_0':s'3(_n995)) → _gen_0':s'3(*(_n995, a)), rt ∈ Ω(1 + a1255·n995 + n995)

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(_n995)) → _gen_0':s'3(*(_n995, a)), rt ∈ Ω(1 + a1255·n995 + n995)