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

cond(true, x, y) → cond(gr(x, y), x, add(x, y))
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(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:

cond'(true', x, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
cond'(true', x, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)

Types:
cond' :: true':false' → 0':s' → 0':s' → cond'
true' :: true':false'
gr' :: 0':s' → 0':s' → true':false'
add' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_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:

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

Rules:
cond'(true', x, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)

Types:
cond' :: true':false' → 0':s' → 0':s' → cond'
true' :: true':false'
gr' :: 0':s' → 0':s' → true':false'
add' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_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:

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

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:
cond'(true', x, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)

Types:
cond' :: true':false' → 0':s' → 0':s' → cond'
true' :: true':false'
gr' :: 0':s' → 0':s' → true':false'
add' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_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:

They will be analysed ascendingly in the following order:

Proved the following rewrite lemma:
add'(_gen_0':s'4(_n354), _gen_0':s'4(b)) → _gen_0':s'4(+(_n354, b)), rt ∈ Ω(1 + n354)

Induction Base:
_gen_0':s'4(b)

Induction Step:
s'(_gen_0':s'4(+(_\$n355, _b487)))

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

Rules:
cond'(true', x, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)

Types:
cond' :: true':false' → 0':s' → 0':s' → cond'
true' :: true':false'
gr' :: 0':s' → 0':s' → true':false'
add' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_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)
add'(_gen_0':s'4(_n354), _gen_0':s'4(b)) → _gen_0':s'4(+(_n354, b)), rt ∈ Ω(1 + n354)

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:
cond'

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

Rules:
cond'(true', x, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)

Types:
cond' :: true':false' → 0':s' → 0':s' → cond'
true' :: true':false'
gr' :: 0':s' → 0':s' → true':false'
add' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_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)
add'(_gen_0':s'4(_n354), _gen_0':s'4(b)) → _gen_0':s'4(+(_n354, b)), rt ∈ Ω(1 + n354)

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)