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

minus(x, y) → cond(equal(min(x, y), y), x, y)
cond(true, x, y) → s(minus(x, s(y)))
min(0, v) → 0
min(u, 0) → 0
min(s(u), s(v)) → s(min(u, v))
equal(0, 0) → true
equal(s(x), 0) → false
equal(0, s(y)) → false
equal(s(x), s(y)) → equal(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:

minus'(x, y) → cond'(equal'(min'(x, y), y), x, y)
cond'(true', x, y) → s'(minus'(x, s'(y)))
min'(0', v) → 0'
min'(u, 0') → 0'
min'(s'(u), s'(v)) → s'(min'(u, v))
equal'(0', 0') → true'
equal'(s'(x), 0') → false'
equal'(0', s'(y)) → false'
equal'(s'(x), s'(y)) → equal'(x, y)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
minus'(x, y) → cond'(equal'(min'(x, y), y), x, y)
cond'(true', x, y) → s'(minus'(x, s'(y)))
min'(0', v) → 0'
min'(u, 0') → 0'
min'(s'(u), s'(v)) → s'(min'(u, v))
equal'(0', 0') → true'
equal'(s'(x), 0') → false'
equal'(0', s'(y)) → false'
equal'(s'(x), s'(y)) → equal'(x, y)

Types:
minus' :: s':0' → s':0' → s':0'
cond' :: true':false' → s':0' → s':0' → s':0'
equal' :: s':0' → s':0' → true':false'
min' :: s':0' → s':0' → s':0'
true' :: true':false'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_s':0'1 :: s':0'
_hole_true':false'2 :: true':false'
_gen_s':0'3 :: Nat → s':0'

Heuristically decided to analyse the following defined symbols:
minus', equal', min'

They will be analysed ascendingly in the following order:
equal' < minus'
min' < minus'

Rules:
minus'(x, y) → cond'(equal'(min'(x, y), y), x, y)
cond'(true', x, y) → s'(minus'(x, s'(y)))
min'(0', v) → 0'
min'(u, 0') → 0'
min'(s'(u), s'(v)) → s'(min'(u, v))
equal'(0', 0') → true'
equal'(s'(x), 0') → false'
equal'(0', s'(y)) → false'
equal'(s'(x), s'(y)) → equal'(x, y)

Types:
minus' :: s':0' → s':0' → s':0'
cond' :: true':false' → s':0' → s':0' → s':0'
equal' :: s':0' → s':0' → true':false'
min' :: s':0' → s':0' → s':0'
true' :: true':false'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_s':0'1 :: s':0'
_hole_true':false'2 :: true':false'
_gen_s':0'3 :: Nat → s':0'

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

The following defined symbols remain to be analysed:
equal', minus', min'

They will be analysed ascendingly in the following order:
equal' < minus'
min' < minus'

Proved the following rewrite lemma:
equal'(_gen_s':0'3(_n5), _gen_s':0'3(_n5)) → true', rt ∈ Ω(1 + n5)

Induction Base:
equal'(_gen_s':0'3(0), _gen_s':0'3(0)) →RΩ(1)
true'

Induction Step:
equal'(_gen_s':0'3(+(_\$n6, 1)), _gen_s':0'3(+(_\$n6, 1))) →RΩ(1)
equal'(_gen_s':0'3(_\$n6), _gen_s':0'3(_\$n6)) →IH
true'

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

Rules:
minus'(x, y) → cond'(equal'(min'(x, y), y), x, y)
cond'(true', x, y) → s'(minus'(x, s'(y)))
min'(0', v) → 0'
min'(u, 0') → 0'
min'(s'(u), s'(v)) → s'(min'(u, v))
equal'(0', 0') → true'
equal'(s'(x), 0') → false'
equal'(0', s'(y)) → false'
equal'(s'(x), s'(y)) → equal'(x, y)

Types:
minus' :: s':0' → s':0' → s':0'
cond' :: true':false' → s':0' → s':0' → s':0'
equal' :: s':0' → s':0' → true':false'
min' :: s':0' → s':0' → s':0'
true' :: true':false'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_s':0'1 :: s':0'
_hole_true':false'2 :: true':false'
_gen_s':0'3 :: Nat → s':0'

Lemmas:
equal'(_gen_s':0'3(_n5), _gen_s':0'3(_n5)) → true', rt ∈ Ω(1 + n5)

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

The following defined symbols remain to be analysed:
min', minus'

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

Proved the following rewrite lemma:
min'(_gen_s':0'3(_n561), _gen_s':0'3(_n561)) → _gen_s':0'3(_n561), rt ∈ Ω(1 + n561)

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

Induction Step:
min'(_gen_s':0'3(+(_\$n562, 1)), _gen_s':0'3(+(_\$n562, 1))) →RΩ(1)
s'(min'(_gen_s':0'3(_\$n562), _gen_s':0'3(_\$n562))) →IH
s'(_gen_s':0'3(_\$n562))

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

Rules:
minus'(x, y) → cond'(equal'(min'(x, y), y), x, y)
cond'(true', x, y) → s'(minus'(x, s'(y)))
min'(0', v) → 0'
min'(u, 0') → 0'
min'(s'(u), s'(v)) → s'(min'(u, v))
equal'(0', 0') → true'
equal'(s'(x), 0') → false'
equal'(0', s'(y)) → false'
equal'(s'(x), s'(y)) → equal'(x, y)

Types:
minus' :: s':0' → s':0' → s':0'
cond' :: true':false' → s':0' → s':0' → s':0'
equal' :: s':0' → s':0' → true':false'
min' :: s':0' → s':0' → s':0'
true' :: true':false'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_s':0'1 :: s':0'
_hole_true':false'2 :: true':false'
_gen_s':0'3 :: Nat → s':0'

Lemmas:
equal'(_gen_s':0'3(_n5), _gen_s':0'3(_n5)) → true', rt ∈ Ω(1 + n5)
min'(_gen_s':0'3(_n561), _gen_s':0'3(_n561)) → _gen_s':0'3(_n561), rt ∈ Ω(1 + n561)

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

The following defined symbols remain to be analysed:
minus'

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

Rules:
minus'(x, y) → cond'(equal'(min'(x, y), y), x, y)
cond'(true', x, y) → s'(minus'(x, s'(y)))
min'(0', v) → 0'
min'(u, 0') → 0'
min'(s'(u), s'(v)) → s'(min'(u, v))
equal'(0', 0') → true'
equal'(s'(x), 0') → false'
equal'(0', s'(y)) → false'
equal'(s'(x), s'(y)) → equal'(x, y)

Types:
minus' :: s':0' → s':0' → s':0'
cond' :: true':false' → s':0' → s':0' → s':0'
equal' :: s':0' → s':0' → true':false'
min' :: s':0' → s':0' → s':0'
true' :: true':false'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_s':0'1 :: s':0'
_hole_true':false'2 :: true':false'
_gen_s':0'3 :: Nat → s':0'

Lemmas:
equal'(_gen_s':0'3(_n5), _gen_s':0'3(_n5)) → true', rt ∈ Ω(1 + n5)
min'(_gen_s':0'3(_n561), _gen_s':0'3(_n561)) → _gen_s':0'3(_n561), rt ∈ Ω(1 + n561)

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
equal'(_gen_s':0'3(_n5), _gen_s':0'3(_n5)) → true', rt ∈ Ω(1 + n5)