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

cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), p(x), p(y), z)
cond2(false, x, y, z) → cond1(and(eq(x, y), gr(x, z)), x, y, z)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
eq(0, 0) → true
eq(s(x), 0) → false
eq(0, s(x)) → false
eq(s(x), s(y)) → eq(x, y)
and(true, true) → true
and(false, x) → false
and(x, false) → false

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, z) → cond2'(gr'(y, z), x, y, z)
cond2'(true', x, y, z) → cond2'(gr'(y, z), p'(x), p'(y), z)
cond2'(false', x, y, z) → cond1'(and'(eq'(x, y), gr'(x, z)), x, y, z)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
p'(0') → 0'
p'(s'(x)) → x
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(x)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
and'(true', true') → true'
and'(false', x) → false'
and'(x, false') → false'

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
cond1' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
gr' :: 0':s' → 0':s' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
and' :: true':false' → true':false' → true':false'
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_cond1':cond2'1 :: cond1':cond2'
_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', eq'

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

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

Types:
cond1' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
gr' :: 0':s' → 0':s' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
and' :: true':false' → true':false' → true':false'
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_cond1':cond2'1 :: cond1':cond2'
_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', eq'

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

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, z) → cond2'(gr'(y, z), x, y, z)
cond2'(true', x, y, z) → cond2'(gr'(y, z), p'(x), p'(y), z)
cond2'(false', x, y, z) → cond1'(and'(eq'(x, y), gr'(x, z)), x, y, z)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
p'(0') → 0'
p'(s'(x)) → x
eq'(0', 0') → true'
eq'(s'(x), 0') → false'
eq'(0', s'(x)) → false'
eq'(s'(x), s'(y)) → eq'(x, y)
and'(true', true') → true'
and'(false', x) → false'
and'(x, false') → false'

Types:
cond1' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
gr' :: 0':s' → 0':s' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
and' :: true':false' → true':false' → true':false'
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_cond1':cond2'1 :: cond1':cond2'
_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:
eq', cond1', cond2'

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

Proved the following rewrite lemma:
eq'(_gen_0':s'4(_n639), _gen_0':s'4(_n639)) → true', rt ∈ Ω(1 + n639)

Induction Base:
eq'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
true'

Induction Step:
eq'(_gen_0':s'4(+(_\$n640, 1)), _gen_0':s'4(+(_\$n640, 1))) →RΩ(1)
eq'(_gen_0':s'4(_\$n640), _gen_0':s'4(_\$n640)) →IH
true'

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

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

Types:
cond1' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
gr' :: 0':s' → 0':s' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
and' :: true':false' → true':false' → true':false'
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_cond1':cond2'1 :: cond1':cond2'
_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)
eq'(_gen_0':s'4(_n639), _gen_0':s'4(_n639)) → true', rt ∈ Ω(1 + n639)

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'

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

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

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

Types:
cond1' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
gr' :: 0':s' → 0':s' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
and' :: true':false' → true':false' → true':false'
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_cond1':cond2'1 :: cond1':cond2'
_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)
eq'(_gen_0':s'4(_n639), _gen_0':s'4(_n639)) → true', rt ∈ Ω(1 + n639)

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'

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

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

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

Types:
cond1' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
true' :: true':false'
cond2' :: true':false' → 0':s' → 0':s' → 0':s' → cond1':cond2'
gr' :: 0':s' → 0':s' → true':false'
p' :: 0':s' → 0':s'
false' :: true':false'
and' :: true':false' → true':false' → true':false'
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_cond1':cond2'1 :: cond1':cond2'
_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)
eq'(_gen_0':s'4(_n639), _gen_0':s'4(_n639)) → true', rt ∈ Ω(1 + n639)

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)