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

min(x, 0) → 0
min(0, y) → 0
min(s(x), s(y)) → s(min(x, y))
max(x, 0) → x
max(0, y) → y
max(s(x), s(y)) → s(max(x, y))
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
gcd(s(x), s(y)) → gcd(-(s(max(x, y)), s(min(x, y))), s(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'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
max'(x, 0') → x
max'(0', y) → y
max'(s'(x), s'(y)) → s'(max'(x, y))
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
gcd'(s'(x), s'(y)) → gcd'(-'(s'(max'(x, y)), s'(min'(x, y))), s'(min'(x, y)))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
max'(x, 0') → x
max'(0', y) → y
max'(s'(x), s'(y)) → s'(max'(x, y))
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
gcd'(s'(x), s'(y)) → gcd'(-'(s'(max'(x, y)), s'(min'(x, y))), s'(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'
-' :: 0':s' → 0':s' → 0':s'
gcd' :: 0':s' → 0':s' → gcd'
_hole_0':s'1 :: 0':s'
_hole_gcd'2 :: gcd'
_gen_0':s'3 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
min', max', -', gcd'

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

Rules:
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
max'(x, 0') → x
max'(0', y) → y
max'(s'(x), s'(y)) → s'(max'(x, y))
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
gcd'(s'(x), s'(y)) → gcd'(-'(s'(max'(x, y)), s'(min'(x, y))), s'(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'
-' :: 0':s' → 0':s' → 0':s'
gcd' :: 0':s' → 0':s' → gcd'
_hole_0':s'1 :: 0':s'
_hole_gcd'2 :: gcd'
_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', -', gcd'

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

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'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
max'(x, 0') → x
max'(0', y) → y
max'(s'(x), s'(y)) → s'(max'(x, y))
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
gcd'(s'(x), s'(y)) → gcd'(-'(s'(max'(x, y)), s'(min'(x, y))), s'(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'
-' :: 0':s' → 0':s' → 0':s'
gcd' :: 0':s' → 0':s' → gcd'
_hole_0':s'1 :: 0':s'
_hole_gcd'2 :: gcd'
_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', -', gcd'

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

Proved the following rewrite lemma:
max'(_gen_0':s'3(_n560), _gen_0':s'3(_n560)) → _gen_0':s'3(_n560), rt ∈ Ω(1 + n560)

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(+(_\$n561, 1)), _gen_0':s'3(+(_\$n561, 1))) →RΩ(1)
s'(max'(_gen_0':s'3(_\$n561), _gen_0':s'3(_\$n561))) →IH
s'(_gen_0':s'3(_\$n561))

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

Rules:
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
max'(x, 0') → x
max'(0', y) → y
max'(s'(x), s'(y)) → s'(max'(x, y))
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
gcd'(s'(x), s'(y)) → gcd'(-'(s'(max'(x, y)), s'(min'(x, y))), s'(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'
-' :: 0':s' → 0':s' → 0':s'
gcd' :: 0':s' → 0':s' → gcd'
_hole_0':s'1 :: 0':s'
_hole_gcd'2 :: gcd'
_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(_n560), _gen_0':s'3(_n560)) → _gen_0':s'3(_n560), rt ∈ Ω(1 + n560)

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:
-', gcd'

They will be analysed ascendingly in the following order:
-' < gcd'

Proved the following rewrite lemma:
-'(_gen_0':s'3(_n1249), _gen_0':s'3(_n1249)) → _gen_0':s'3(0), rt ∈ Ω(1 + n1249)

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(+(_\$n1250, 1)), _gen_0':s'3(+(_\$n1250, 1))) →RΩ(1)
-'(_gen_0':s'3(_\$n1250), _gen_0':s'3(_\$n1250)) →IH
_gen_0':s'3(0)

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

Rules:
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
max'(x, 0') → x
max'(0', y) → y
max'(s'(x), s'(y)) → s'(max'(x, y))
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
gcd'(s'(x), s'(y)) → gcd'(-'(s'(max'(x, y)), s'(min'(x, y))), s'(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'
-' :: 0':s' → 0':s' → 0':s'
gcd' :: 0':s' → 0':s' → gcd'
_hole_0':s'1 :: 0':s'
_hole_gcd'2 :: gcd'
_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(_n560), _gen_0':s'3(_n560)) → _gen_0':s'3(_n560), rt ∈ Ω(1 + n560)
-'(_gen_0':s'3(_n1249), _gen_0':s'3(_n1249)) → _gen_0':s'3(0), rt ∈ Ω(1 + n1249)

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:
gcd'

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

Rules:
min'(x, 0') → 0'
min'(0', y) → 0'
min'(s'(x), s'(y)) → s'(min'(x, y))
max'(x, 0') → x
max'(0', y) → y
max'(s'(x), s'(y)) → s'(max'(x, y))
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
gcd'(s'(x), s'(y)) → gcd'(-'(s'(max'(x, y)), s'(min'(x, y))), s'(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'
-' :: 0':s' → 0':s' → 0':s'
gcd' :: 0':s' → 0':s' → gcd'
_hole_0':s'1 :: 0':s'
_hole_gcd'2 :: gcd'
_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(_n560), _gen_0':s'3(_n560)) → _gen_0':s'3(_n560), rt ∈ Ω(1 + n560)
-'(_gen_0':s'3(_n1249), _gen_0':s'3(_n1249)) → _gen_0':s'3(0), rt ∈ Ω(1 + n1249)

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)