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

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

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

Rewrite Strategy: INNERMOST

Infered types.

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

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

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

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

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

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

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

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

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

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

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

Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
max' :: 0':s' → 0':s' → 0':s'
twice' :: 0':s' → 0':s'
-' :: 0':s' → 0':s' → 0':s'
p' :: 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:
min'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(_n5), 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:
max', twice', -', f'

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

Proved the following rewrite lemma:
max'(_gen_0':s'3(_n650), _gen_0':s'3(_n650)) → _gen_0':s'3(_n650), rt ∈ Ω(1 + n650)

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

Induction Step:
max'(_gen_0':s'3(+(_\$n651, 1)), _gen_0':s'3(+(_\$n651, 1))) →RΩ(1)
s'(max'(_gen_0':s'3(_\$n651), _gen_0':s'3(_\$n651))) →IH
s'(_gen_0':s'3(_\$n651))

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

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

Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
max' :: 0':s' → 0':s' → 0':s'
twice' :: 0':s' → 0':s'
-' :: 0':s' → 0':s' → 0':s'
p' :: 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:
min'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)
max'(_gen_0':s'3(_n650), _gen_0':s'3(_n650)) → _gen_0':s'3(_n650), rt ∈ Ω(1 + n650)

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'
-' < f'

Proved the following rewrite lemma:
twice'(_gen_0':s'3(_n1445)) → _gen_0':s'3(*(2, _n1445)), rt ∈ Ω(1 + n1445)

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

Induction Step:
twice'(_gen_0':s'3(+(_\$n1446, 1))) →RΩ(1)
s'(s'(twice'(_gen_0':s'3(_\$n1446)))) →IH
s'(s'(_gen_0':s'3(*(2, _\$n1446))))

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

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

Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
max' :: 0':s' → 0':s' → 0':s'
twice' :: 0':s' → 0':s'
-' :: 0':s' → 0':s' → 0':s'
p' :: 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:
min'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)
max'(_gen_0':s'3(_n650), _gen_0':s'3(_n650)) → _gen_0':s'3(_n650), rt ∈ Ω(1 + n650)
twice'(_gen_0':s'3(_n1445)) → _gen_0':s'3(*(2, _n1445)), rt ∈ Ω(1 + n1445)

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(_n1947), _gen_0':s'3(_n1947)) → _gen_0':s'3(0), rt ∈ Ω(1 + n1947)

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(+(_\$n1948, 1)), _gen_0':s'3(+(_\$n1948, 1))) →RΩ(1)
-'(_gen_0':s'3(_\$n1948), _gen_0':s'3(_\$n1948)) →IH
_gen_0':s'3(0)

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

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

Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
max' :: 0':s' → 0':s' → 0':s'
twice' :: 0':s' → 0':s'
-' :: 0':s' → 0':s' → 0':s'
p' :: 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:
min'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)
max'(_gen_0':s'3(_n650), _gen_0':s'3(_n650)) → _gen_0':s'3(_n650), rt ∈ Ω(1 + n650)
twice'(_gen_0':s'3(_n1445)) → _gen_0':s'3(*(2, _n1445)), rt ∈ Ω(1 + n1445)
-'(_gen_0':s'3(_n1947), _gen_0':s'3(_n1947)) → _gen_0':s'3(0), rt ∈ Ω(1 + n1947)

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

Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
max' :: 0':s' → 0':s' → 0':s'
twice' :: 0':s' → 0':s'
-' :: 0':s' → 0':s' → 0':s'
p' :: 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:
min'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(_n5), rt ∈ Ω(1 + n5)
max'(_gen_0':s'3(_n650), _gen_0':s'3(_n650)) → _gen_0':s'3(_n650), rt ∈ Ω(1 + n650)
twice'(_gen_0':s'3(_n1445)) → _gen_0':s'3(*(2, _n1445)), rt ∈ Ω(1 + n1445)
-'(_gen_0':s'3(_n1947), _gen_0':s'3(_n1947)) → _gen_0':s'3(0), rt ∈ Ω(1 + n1947)

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