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:

nonZero(0) → false
nonZero(s(x)) → true
p(0) → 0
p(s(x)) → x
id_inc(x) → x
id_inc(x) → s(x)
random(x) → rand(x, 0)
rand(x, y) → if(nonZero(x), x, y)
if(false, x, y) → y
if(true, x, y) → rand(p(x), id_inc(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:

nonZero(0') → false
nonZero(s(x)) → true
p(0') → 0'
p(s(x)) → x
id_inc(x) → x
id_inc(x) → s(x)
random(x) → rand(x, 0')
rand(x, y) → if(nonZero(x), x, y)
if(false, x, y) → y
if(true, x, y) → rand(p(x), id_inc(y))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
nonZero(0') → false
nonZero(s(x)) → true
p(0') → 0'
p(s(x)) → x
id_inc(x) → x
id_inc(x) → s(x)
random(x) → rand(x, 0')
rand(x, y) → if(nonZero(x), x, y)
if(false, x, y) → y
if(true, x, y) → rand(p(x), id_inc(y))

Types:
nonZero :: 0':s → false:true
0' :: 0':s
false :: false:true
s :: 0':s → 0':s
true :: false:true
p :: 0':s → 0':s
id_inc :: 0':s → 0':s
random :: 0':s → 0':s
rand :: 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
rand

(6) Obligation:

Innermost TRS:
Rules:
nonZero(0') → false
nonZero(s(x)) → true
p(0') → 0'
p(s(x)) → x
id_inc(x) → x
id_inc(x) → s(x)
random(x) → rand(x, 0')
rand(x, y) → if(nonZero(x), x, y)
if(false, x, y) → y
if(true, x, y) → rand(p(x), id_inc(y))

Types:
nonZero :: 0':s → false:true
0' :: 0':s
false :: false:true
s :: 0':s → 0':s
true :: false:true
p :: 0':s → 0':s
id_inc :: 0':s → 0':s
random :: 0':s → 0':s
rand :: 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
rand

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

Proved the following rewrite lemma:
rand(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(b), rt ∈ Ω(1 + n5₀)

Induction Base:
rand(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
if(nonZero(gen_0':s3_0(0)), gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
if(false, gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
gen_0':s3_0(b)

Induction Step:
rand(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
if(nonZero(gen_0':s3_0(+(n5_0, 1))), gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
if(true, gen_0':s3_0(+(1, n5_0)), gen_0':s3_0(b)) →_R^Ω(1)
rand(p(gen_0':s3_0(+(1, n5_0))), id_inc(gen_0':s3_0(b))) →_R^Ω(1)
rand(gen_0':s3_0(n5_0), id_inc(gen_0':s3_0(b))) →_R^Ω(1)
rand(gen_0':s3_0(n5_0), gen_0':s3_0(b)) →_IH
gen_0':s3_0(b)

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:
nonZero(0') → false
nonZero(s(x)) → true
p(0') → 0'
p(s(x)) → x
id_inc(x) → x
id_inc(x) → s(x)
random(x) → rand(x, 0')
rand(x, y) → if(nonZero(x), x, y)
if(false, x, y) → y
if(true, x, y) → rand(p(x), id_inc(y))

Types:
nonZero :: 0':s → false:true
0' :: 0':s
false :: false:true
s :: 0':s → 0':s
true :: false:true
p :: 0':s → 0':s
id_inc :: 0':s → 0':s
random :: 0':s → 0':s
rand :: 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
rand(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(b), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
rand(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(b), rt ∈ Ω(1 + n5₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

Innermost TRS:
Rules:
nonZero(0') → false
nonZero(s(x)) → true
p(0') → 0'
p(s(x)) → x
id_inc(x) → x
id_inc(x) → s(x)
random(x) → rand(x, 0')
rand(x, y) → if(nonZero(x), x, y)
if(false, x, y) → y
if(true, x, y) → rand(p(x), id_inc(y))

Types:
nonZero :: 0':s → false:true
0' :: 0':s
false :: false:true
s :: 0':s → 0':s
true :: false:true
p :: 0':s → 0':s
id_inc :: 0':s → 0':s
random :: 0':s → 0':s
rand :: 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s
hole_false:true1_0 :: false:true
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
rand(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(b), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
rand(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(b), rt ∈ Ω(1 + n5₀)

(14) BOUNDS(n^1, INF)