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

-(x, 0) → x
-(s(x), s(y)) → -(x, y)
<=(0, y) → true
<=(s(x), 0) → false
<=(s(x), s(y)) → <=(x, y)
if(true, x, y) → x
if(false, x, y) → y
perfectp(0) → false
perfectp(s(x)) → f(x, s(0), s(x), s(x))
f(0, y, 0, u) → true
f(0, y, s(z), u) → false
f(s(x), 0, z, u) → f(x, u, -(z, s(x)), u)
f(s(x), s(y), z, u) → if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u))

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) → true'
<='(s'(x), 0') → false'
<='(s'(x), s'(y)) → <='(x, y)
if'(true', x, y) → x
if'(false', x, y) → y
perfectp'(0') → false'
perfectp'(s'(x)) → f'(x, s'(0'), s'(x), s'(x))
f'(0', y, 0', u) → true'
f'(0', y, s'(z), u) → false'
f'(s'(x), 0', z, u) → f'(x, u, -'(z, s'(x)), u)
f'(s'(x), s'(y), z, u) → if'(<='(x, y), f'(s'(x), -'(y, x), z, u), f'(x, u, z, u))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
<='(0', y) → true'
<='(s'(x), 0') → false'
<='(s'(x), s'(y)) → <='(x, y)
if'(true', x, y) → x
if'(false', x, y) → y
perfectp'(0') → false'
perfectp'(s'(x)) → f'(x, s'(0'), s'(x), s'(x))
f'(0', y, 0', u) → true'
f'(0', y, s'(z), u) → false'
f'(s'(x), 0', z, u) → f'(x, u, -'(z, s'(x)), u)
f'(s'(x), s'(y), z, u) → if'(<='(x, y), f'(s'(x), -'(y, x), z, u), f'(x, u, z, u))

Types:
-' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
<=' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
if' :: true':false' → true':false' → true':false' → true':false'
perfectp' :: 0':s' → true':false'
f' :: 0':s' → 0':s' → 0':s' → 0':s' → true':false'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_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'

Rules:
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
<='(0', y) → true'
<='(s'(x), 0') → false'
<='(s'(x), s'(y)) → <='(x, y)
if'(true', x, y) → x
if'(false', x, y) → y
perfectp'(0') → false'
perfectp'(s'(x)) → f'(x, s'(0'), s'(x), s'(x))
f'(0', y, 0', u) → true'
f'(0', y, s'(z), u) → false'
f'(s'(x), 0', z, u) → f'(x, u, -'(z, s'(x)), u)
f'(s'(x), s'(y), z, u) → if'(<='(x, y), f'(s'(x), -'(y, x), z, u), f'(x, u, z, u))

Types:
-' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
<=' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
if' :: true':false' → true':false' → true':false' → true':false'
perfectp' :: 0':s' → true':false'
f' :: 0':s' → 0':s' → 0':s' → 0':s' → true':false'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_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'

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) → true'
<='(s'(x), 0') → false'
<='(s'(x), s'(y)) → <='(x, y)
if'(true', x, y) → x
if'(false', x, y) → y
perfectp'(0') → false'
perfectp'(s'(x)) → f'(x, s'(0'), s'(x), s'(x))
f'(0', y, 0', u) → true'
f'(0', y, s'(z), u) → false'
f'(s'(x), 0', z, u) → f'(x, u, -'(z, s'(x)), u)
f'(s'(x), s'(y), z, u) → if'(<='(x, y), f'(s'(x), -'(y, x), z, u), f'(x, u, z, u))

Types:
-' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
<=' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
if' :: true':false' → true':false' → true':false' → true':false'
perfectp' :: 0':s' → true':false'
f' :: 0':s' → 0':s' → 0':s' → 0':s' → true':false'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_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'

Proved the following rewrite lemma:
<='(_gen_0':s'3(_n788), _gen_0':s'3(_n788)) → true', rt ∈ Ω(1 + n788)

Induction Base:
<='(_gen_0':s'3(0), _gen_0':s'3(0)) →RΩ(1)
true'

Induction Step:
<='(_gen_0':s'3(+(_\$n789, 1)), _gen_0':s'3(+(_\$n789, 1))) →RΩ(1)
<='(_gen_0':s'3(_\$n789), _gen_0':s'3(_\$n789)) →IH
true'

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) → true'
<='(s'(x), 0') → false'
<='(s'(x), s'(y)) → <='(x, y)
if'(true', x, y) → x
if'(false', x, y) → y
perfectp'(0') → false'
perfectp'(s'(x)) → f'(x, s'(0'), s'(x), s'(x))
f'(0', y, 0', u) → true'
f'(0', y, s'(z), u) → false'
f'(s'(x), 0', z, u) → f'(x, u, -'(z, s'(x)), u)
f'(s'(x), s'(y), z, u) → if'(<='(x, y), f'(s'(x), -'(y, x), z, u), f'(x, u, z, u))

Types:
-' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
<=' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
if' :: true':false' → true':false' → true':false' → true':false'
perfectp' :: 0':s' → true':false'
f' :: 0':s' → 0':s' → 0':s' → 0':s' → true':false'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_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(_n788), _gen_0':s'3(_n788)) → true', rt ∈ Ω(1 + n788)

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) → true'
<='(s'(x), 0') → false'
<='(s'(x), s'(y)) → <='(x, y)
if'(true', x, y) → x
if'(false', x, y) → y
perfectp'(0') → false'
perfectp'(s'(x)) → f'(x, s'(0'), s'(x), s'(x))
f'(0', y, 0', u) → true'
f'(0', y, s'(z), u) → false'
f'(s'(x), 0', z, u) → f'(x, u, -'(z, s'(x)), u)
f'(s'(x), s'(y), z, u) → if'(<='(x, y), f'(s'(x), -'(y, x), z, u), f'(x, u, z, u))

Types:
-' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
<=' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
if' :: true':false' → true':false' → true':false' → true':false'
perfectp' :: 0':s' → true':false'
f' :: 0':s' → 0':s' → 0':s' → 0':s' → true':false'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_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(_n788), _gen_0':s'3(_n788)) → true', rt ∈ Ω(1 + n788)

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)