The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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)
add(0, x) → x
add(s(x), y) → s(add(x, y))

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative 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)
add(0', x) → x
add(s(x), y) → s(add(x, y))

S is empty.
Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Innermost TRS:
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)
add(0', x) → x
add(s(x), y) → s(add(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_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
cond, gr, add

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

(6) Obligation:

Innermost TRS:
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)
add(0', x) → x
add(s(x), y) → s(add(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_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
gr, cond, add

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

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

Induction Base:
gr(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
false

Induction Step:
gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) →_R^Ω(1)
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) →_IH
false

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
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)
add(0', x) → x
add(s(x), y) → s(add(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_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
add, cond

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

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
add(gen_0':s4_0(n265_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n265_0, b)), rt ∈ Ω(1 + n265₀)

Induction Base:
add(gen_0':s4_0(0), gen_0':s4_0(b)) →_R^Ω(1)
gen_0':s4_0(b)

Induction Step:
add(gen_0':s4_0(+(n265_0, 1)), gen_0':s4_0(b)) →_R^Ω(1)
s(add(gen_0':s4_0(n265_0), gen_0':s4_0(b))) →_IH
s(gen_0':s4_0(+(b, c266_0)))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
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)
add(0', x) → x
add(s(x), y) → s(add(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_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)
add(gen_0':s4_0(n265_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n265_0, b)), rt ∈ Ω(1 + n265₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
cond

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

Innermost TRS:
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)
add(0', x) → x
add(s(x), y) → s(add(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_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)
add(gen_0':s4_0(n265_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n265_0, b)), rt ∈ Ω(1 + n265₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
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)
add(0', x) → x
add(s(x), y) → s(add(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_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)
add(gen_0':s4_0(n265_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n265_0, b)), rt ∈ Ω(1 + n265₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
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)
add(0', x) → x
add(s(x), y) → s(add(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_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n6₀)

(22) BOUNDS(n^1, INF)