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

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
twice(0) → 0
twice(s(x)) → s(s(twice(x)))
f(s(x), s(y)) → f(-(y, min(x, y)), s(twice(min(x, y))))
f(s(x), s(y)) → f(-(x, min(x, y)), s(twice(min(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:

-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
twice'(0') → 0'
twice'(s'(x)) → s'(s'(twice'(x)))
f'(s'(x), s'(y)) → f'(-'(y, min'(x, y)), s'(twice'(min'(x, y))))
f'(s'(x), s'(y)) → f'(-'(x, min'(x, y)), s'(twice'(min'(x, y))))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
twice'(0') → 0'
twice'(s'(x)) → s'(s'(twice'(x)))
f'(s'(x), s'(y)) → f'(-'(y, min'(x, y)), s'(twice'(min'(x, y))))
f'(s'(x), s'(y)) → f'(-'(x, min'(x, y)), s'(twice'(min'(x, y))))

Types:
-' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
min' :: 0':s' → 0':s' → 0':s'
twice' :: 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:
-', min', twice', f'

They will be analysed ascendingly in the following order:
-' < f'
min' < f'
twice' < f'

Rules:
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
twice'(0') → 0'
twice'(s'(x)) → s'(s'(twice'(x)))
f'(s'(x), s'(y)) → f'(-'(y, min'(x, y)), s'(twice'(min'(x, y))))
f'(s'(x), s'(y)) → f'(-'(x, min'(x, y)), s'(twice'(min'(x, y))))

Types:
-' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
min' :: 0':s' → 0':s' → 0':s'
twice' :: 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:
-', min', twice', f'

They will be analysed ascendingly in the following order:
-' < f'
min' < f'
twice' < 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)
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
twice'(0') → 0'
twice'(s'(x)) → s'(s'(twice'(x)))
f'(s'(x), s'(y)) → f'(-'(y, min'(x, y)), s'(twice'(min'(x, y))))
f'(s'(x), s'(y)) → f'(-'(x, min'(x, y)), s'(twice'(min'(x, y))))

Types:
-' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
min' :: 0':s' → 0':s' → 0':s'
twice' :: 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:
min', twice', f'

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

Proved the following rewrite lemma:
min'(_gen_0':s'3(_n520), _gen_0':s'3(_n520)) → _gen_0':s'3(_n520), rt ∈ Ω(1 + n520)

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

Induction Step:
min'(_gen_0':s'3(+(_\$n521, 1)), _gen_0':s'3(+(_\$n521, 1))) →RΩ(1)
s'(min'(_gen_0':s'3(_\$n521), _gen_0':s'3(_\$n521))) →IH
s'(_gen_0':s'3(_\$n521))

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)
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
twice'(0') → 0'
twice'(s'(x)) → s'(s'(twice'(x)))
f'(s'(x), s'(y)) → f'(-'(y, min'(x, y)), s'(twice'(min'(x, y))))
f'(s'(x), s'(y)) → f'(-'(x, min'(x, y)), s'(twice'(min'(x, y))))

Types:
-' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
min' :: 0':s' → 0':s' → 0':s'
twice' :: 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)
min'(_gen_0':s'3(_n520), _gen_0':s'3(_n520)) → _gen_0':s'3(_n520), rt ∈ Ω(1 + n520)

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:
twice', f'

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

Proved the following rewrite lemma:
twice'(_gen_0':s'3(_n1075)) → _gen_0':s'3(*(2, _n1075)), rt ∈ Ω(1 + n1075)

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

Induction Step:
twice'(_gen_0':s'3(+(_\$n1076, 1))) →RΩ(1)
s'(s'(twice'(_gen_0':s'3(_\$n1076)))) →IH
s'(s'(_gen_0':s'3(*(2, _\$n1076))))

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)
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
twice'(0') → 0'
twice'(s'(x)) → s'(s'(twice'(x)))
f'(s'(x), s'(y)) → f'(-'(y, min'(x, y)), s'(twice'(min'(x, y))))
f'(s'(x), s'(y)) → f'(-'(x, min'(x, y)), s'(twice'(min'(x, y))))

Types:
-' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
min' :: 0':s' → 0':s' → 0':s'
twice' :: 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)
min'(_gen_0':s'3(_n520), _gen_0':s'3(_n520)) → _gen_0':s'3(_n520), rt ∈ Ω(1 + n520)
twice'(_gen_0':s'3(_n1075)) → _gen_0':s'3(*(2, _n1075)), rt ∈ Ω(1 + n1075)

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)
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
twice'(0') → 0'
twice'(s'(x)) → s'(s'(twice'(x)))
f'(s'(x), s'(y)) → f'(-'(y, min'(x, y)), s'(twice'(min'(x, y))))
f'(s'(x), s'(y)) → f'(-'(x, min'(x, y)), s'(twice'(min'(x, y))))

Types:
-' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
min' :: 0':s' → 0':s' → 0':s'
twice' :: 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)
min'(_gen_0':s'3(_n520), _gen_0':s'3(_n520)) → _gen_0':s'3(_n520), rt ∈ Ω(1 + n520)
twice'(_gen_0':s'3(_n1075)) → _gen_0':s'3(*(2, _n1075)), rt ∈ Ω(1 + n1075)

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(n) was proven with the following lemma:
-'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)