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

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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
cond1' :: s':0' → s':0' → cond1':cond2'
s' :: s':0' → s':0'
cond2' :: true':false' → s':0' → s':0' → cond1':cond2'
gr' :: s':0' → s':0' → true':false'
true' :: true':false'
false' :: true':false'
p' :: s':0' → s':0'
0' :: s':0'
neq' :: s':0' → s':0' → true':false'
_hole_cond1':cond2'1 :: cond1':cond2'
_hole_s':0'2 :: s':0'
_hole_true':false'3 :: true':false'
_gen_s':0'4 :: Nat → s':0'

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

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

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

Types:
cond1' :: s':0' → s':0' → cond1':cond2'
s' :: s':0' → s':0'
cond2' :: true':false' → s':0' → s':0' → cond1':cond2'
gr' :: s':0' → s':0' → true':false'
true' :: true':false'
false' :: true':false'
p' :: s':0' → s':0'
0' :: s':0'
neq' :: s':0' → s':0' → true':false'
_hole_cond1':cond2'1 :: cond1':cond2'
_hole_s':0'2 :: s':0'
_hole_true':false'3 :: true':false'
_gen_s':0'4 :: Nat → s':0'

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

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

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

Proved the following rewrite lemma:
gr'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)

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

Induction Step:
gr'(_gen_s':0'4(+(_\$n7, 1)), _gen_s':0'4(+(_\$n7, 1))) →RΩ(1)
gr'(_gen_s':0'4(_\$n7), _gen_s':0'4(_\$n7)) →IH
false'

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

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

Types:
cond1' :: s':0' → s':0' → cond1':cond2'
s' :: s':0' → s':0'
cond2' :: true':false' → s':0' → s':0' → cond1':cond2'
gr' :: s':0' → s':0' → true':false'
true' :: true':false'
false' :: true':false'
p' :: s':0' → s':0'
0' :: s':0'
neq' :: s':0' → s':0' → true':false'
_hole_cond1':cond2'1 :: cond1':cond2'
_hole_s':0'2 :: s':0'
_hole_true':false'3 :: true':false'
_gen_s':0'4 :: Nat → s':0'

Lemmas:
gr'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)

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

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

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

The following conjecture could not be proven:

cond1'(_gen_s':0'4(+(1, _n516)), _gen_s':0'4(+(1, _n516))) →? cond1'(_gen_s':0'4(0), _gen_s':0'4(+(1, _n516)))

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

Types:
cond1' :: s':0' → s':0' → cond1':cond2'
s' :: s':0' → s':0'
cond2' :: true':false' → s':0' → s':0' → cond1':cond2'
gr' :: s':0' → s':0' → true':false'
true' :: true':false'
false' :: true':false'
p' :: s':0' → s':0'
0' :: s':0'
neq' :: s':0' → s':0' → true':false'
_hole_cond1':cond2'1 :: cond1':cond2'
_hole_s':0'2 :: s':0'
_hole_true':false'3 :: true':false'
_gen_s':0'4 :: Nat → s':0'

Lemmas:
gr'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)

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

The following defined symbols remain to be analysed:
neq'

Proved the following rewrite lemma:
neq'(_gen_s':0'4(_n19478), _gen_s':0'4(_n19478)) → false', rt ∈ Ω(1 + n19478)

Induction Base:
neq'(_gen_s':0'4(0), _gen_s':0'4(0)) →RΩ(1)
false'

Induction Step:
neq'(_gen_s':0'4(+(_\$n19479, 1)), _gen_s':0'4(+(_\$n19479, 1))) →RΩ(1)
neq'(_gen_s':0'4(_\$n19479), _gen_s':0'4(_\$n19479)) →IH
false'

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

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

Types:
cond1' :: s':0' → s':0' → cond1':cond2'
s' :: s':0' → s':0'
cond2' :: true':false' → s':0' → s':0' → cond1':cond2'
gr' :: s':0' → s':0' → true':false'
true' :: true':false'
false' :: true':false'
p' :: s':0' → s':0'
0' :: s':0'
neq' :: s':0' → s':0' → true':false'
_hole_cond1':cond2'1 :: cond1':cond2'
_hole_s':0'2 :: s':0'
_hole_true':false'3 :: true':false'
_gen_s':0'4 :: Nat → s':0'

Lemmas:
gr'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)
neq'(_gen_s':0'4(_n19478), _gen_s':0'4(_n19478)) → false', rt ∈ Ω(1 + n19478)

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
gr'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)