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:

div(x, s(y)) → d(x, s(y), 0)
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0
ge(u, 0) → true
ge(0, s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0) → n
plus(n, s(m)) → s(plus(n, m))

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:

div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
d, ge, plus

They will be analysed ascendingly in the following order:
ge < d
plus < d

(6) Obligation:

Innermost TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
ge, d, plus

They will be analysed ascendingly in the following order:
ge < d
plus < d

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

Proved the following rewrite lemma:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
ge(gen_s:0'3_0(0), gen_s:0'3_0(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) →_R^Ω(1)
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) →_IH
true

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:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
plus, d

They will be analysed ascendingly in the following order:
plus < d

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

Proved the following rewrite lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n306_0)) → gen_s:0'3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

Induction Base:
plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) →_R^Ω(1)
gen_s:0'3_0(a)

Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n306_0, 1))) →_R^Ω(1)
s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n306_0))) →_IH
s(gen_s:0'3_0(+(a, c307_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:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n306_0)) → gen_s:0'3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
d

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

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

(14) Obligation:

Innermost TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n306_0)) → gen_s:0'3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
plus(gen_s:0'3_0(a), gen_s:0'3_0(n306_0)) → gen_s:0'3_0(+(n306_0, a)), rt ∈ Ω(1 + n306₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
div(x, s(y)) → d(x, s(y), 0')
d(x, s(y), z) → cond(ge(x, z), x, y, z)
cond(true, x, y, z) → s(d(x, s(y), plus(s(y), z)))
cond(false, x, y, z) → 0'
ge(u, 0') → true
ge(0', s(v)) → false
ge(s(u), s(v)) → ge(u, v)
plus(n, 0') → n
plus(n, s(m)) → s(plus(n, m))

Types:
div :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
d :: s:0' → s:0' → s:0' → s:0'
0' :: s:0'
cond :: true:false → s:0' → s:0' → s:0' → s:0'
ge :: s:0' → s:0' → true:false
true :: true:false
plus :: s:0' → s:0' → s:0'
false :: true:false
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(22) BOUNDS(n^1, INF)