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

cond1(true, x, y) → cond2(gr(x, 0), x, y)
cond2(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), p(x), y)
cond2(false, x, y) → cond3(gr(y, 0), x, y)
cond3(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, p(y))
cond3(false, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, y)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
p(0) → 0
p(s(x)) → x

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

cond1'(true', x, y) → cond2'(gr'(x, 0'), x, y)
cond2'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), p'(x), y)
cond2'(false', x, y) → cond3'(gr'(y, 0'), x, y)
cond3'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, p'(y))
cond3'(false', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, y)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
or'(false', false') → false'
or'(true', x) → true'
or'(x, true') → true'
p'(0') → 0'
p'(s'(x)) → x

Rewrite Strategy: INNERMOST

Infered types.

Rules:
cond1'(true', x, y) → cond2'(gr'(x, 0'), x, y)
cond2'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), p'(x), y)
cond2'(false', x, y) → cond3'(gr'(y, 0'), x, y)
cond3'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, p'(y))
cond3'(false', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, y)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
or'(false', false') → false'
or'(true', x) → true'
or'(x, true') → true'
p'(0') → 0'
p'(s'(x)) → x

Types:
cond1' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
gr' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
or' :: true':false' → true':false' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
cond3' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
s' :: 0':s' → 0':s'
_hole_cond1':cond2':cond3'1 :: cond1':cond2':cond3'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
cond1', cond2', gr', cond3'

They will be analysed ascendingly in the following order:
cond1' = cond2'
gr' < cond1'
cond1' = cond3'
gr' < cond2'
cond2' = cond3'
gr' < cond3'

Rules:
cond1'(true', x, y) → cond2'(gr'(x, 0'), x, y)
cond2'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), p'(x), y)
cond2'(false', x, y) → cond3'(gr'(y, 0'), x, y)
cond3'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, p'(y))
cond3'(false', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, y)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
or'(false', false') → false'
or'(true', x) → true'
or'(x, true') → true'
p'(0') → 0'
p'(s'(x)) → x

Types:
cond1' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
gr' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
or' :: true':false' → true':false' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
cond3' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
s' :: 0':s' → 0':s'
_hole_cond1':cond2':cond3'1 :: cond1':cond2':cond3'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
gr', cond1', cond2', cond3'

They will be analysed ascendingly in the following order:
cond1' = cond2'
gr' < cond1'
cond1' = cond3'
gr' < cond2'
cond2' = cond3'
gr' < cond3'

Proved the following rewrite lemma:
gr'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)

Induction Base:
gr'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
false'

Induction Step:
gr'(_gen_0':s'4(+(_\$n7, 1)), _gen_0':s'4(+(_\$n7, 1))) →RΩ(1)
gr'(_gen_0':s'4(_\$n7), _gen_0':s'4(_\$n7)) →IH
false'

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

Rules:
cond1'(true', x, y) → cond2'(gr'(x, 0'), x, y)
cond2'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), p'(x), y)
cond2'(false', x, y) → cond3'(gr'(y, 0'), x, y)
cond3'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, p'(y))
cond3'(false', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, y)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
or'(false', false') → false'
or'(true', x) → true'
or'(x, true') → true'
p'(0') → 0'
p'(s'(x)) → x

Types:
cond1' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
gr' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
or' :: true':false' → true':false' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
cond3' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
s' :: 0':s' → 0':s'
_hole_cond1':cond2':cond3'1 :: cond1':cond2':cond3'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
gr'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
cond2', cond1', cond3'

They will be analysed ascendingly in the following order:
cond1' = cond2'
cond1' = cond3'
cond2' = cond3'

Could not prove a rewrite lemma for the defined symbol cond2'.

Rules:
cond1'(true', x, y) → cond2'(gr'(x, 0'), x, y)
cond2'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), p'(x), y)
cond2'(false', x, y) → cond3'(gr'(y, 0'), x, y)
cond3'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, p'(y))
cond3'(false', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, y)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
or'(false', false') → false'
or'(true', x) → true'
or'(x, true') → true'
p'(0') → 0'
p'(s'(x)) → x

Types:
cond1' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
gr' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
or' :: true':false' → true':false' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
cond3' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
s' :: 0':s' → 0':s'
_hole_cond1':cond2':cond3'1 :: cond1':cond2':cond3'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
gr'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
cond1', cond3'

They will be analysed ascendingly in the following order:
cond1' = cond2'
cond1' = cond3'
cond2' = cond3'

Could not prove a rewrite lemma for the defined symbol cond1'.

Rules:
cond1'(true', x, y) → cond2'(gr'(x, 0'), x, y)
cond2'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), p'(x), y)
cond2'(false', x, y) → cond3'(gr'(y, 0'), x, y)
cond3'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, p'(y))
cond3'(false', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, y)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
or'(false', false') → false'
or'(true', x) → true'
or'(x, true') → true'
p'(0') → 0'
p'(s'(x)) → x

Types:
cond1' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
gr' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
or' :: true':false' → true':false' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
cond3' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
s' :: 0':s' → 0':s'
_hole_cond1':cond2':cond3'1 :: cond1':cond2':cond3'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
gr'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
cond3'

They will be analysed ascendingly in the following order:
cond1' = cond2'
cond1' = cond3'
cond2' = cond3'

Could not prove a rewrite lemma for the defined symbol cond3'.

Rules:
cond1'(true', x, y) → cond2'(gr'(x, 0'), x, y)
cond2'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), p'(x), y)
cond2'(false', x, y) → cond3'(gr'(y, 0'), x, y)
cond3'(true', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, p'(y))
cond3'(false', x, y) → cond1'(or'(gr'(x, 0'), gr'(y, 0')), x, y)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
or'(false', false') → false'
or'(true', x) → true'
or'(x, true') → true'
p'(0') → 0'
p'(s'(x)) → x

Types:
cond1' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
gr' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
or' :: true':false' → true':false' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
cond3' :: true':false' → 0':s' → 0':s' → cond1':cond2':cond3'
s' :: 0':s' → 0':s'
_hole_cond1':cond2':cond3'1 :: cond1':cond2':cond3'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
gr'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
gr'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)