### (0) Obligation:

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

nonZero(0) → false
nonZero(s(x)) → true
p(s(0)) → 0
p(s(s(x))) → s(p(s(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: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
p(s(s(x))) →+ s(p(s(x)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / s(x)].
The result substitution is [ ].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

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

nonZero(0') → false
nonZero(s(x)) → true
p(s(0')) → 0'
p(s(s(x))) → s(p(s(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: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
nonZero(0') → false
nonZero(s(x)) → true
p(s(0')) → 0'
p(s(s(x))) → s(p(s(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

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
p, rand

They will be analysed ascendingly in the following order:
p < rand

### (8) Obligation:

TRS:
Rules:
nonZero(0') → false
nonZero(s(x)) → true
p(s(0')) → 0'
p(s(s(x))) → s(p(s(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:
p, rand

They will be analysed ascendingly in the following order:
p < rand

### (9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

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

Induction Base:
p(gen_0':s3_0(+(1, 0))) →RΩ(1)
0'

Induction Step:
p(gen_0':s3_0(+(1, +(n5_0, 1)))) →RΩ(1)
s(p(s(gen_0':s3_0(n5_0)))) →IH
s(gen_0':s3_0(c6_0))

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

### (11) Obligation:

TRS:
Rules:
nonZero(0') → false
nonZero(s(x)) → true
p(s(0')) → 0'
p(s(s(x))) → s(p(s(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:
p(gen_0':s3_0(+(1, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)

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

### (12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
rand(gen_0':s3_0(n198_0), gen_0':s3_0(b)) → gen_0':s3_0(b), rt ∈ Ω(1 + n1980 + n19802)

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

Induction Step:
rand(gen_0':s3_0(+(n198_0, 1)), gen_0':s3_0(b)) →RΩ(1)
if(nonZero(gen_0':s3_0(+(n198_0, 1))), gen_0':s3_0(+(n198_0, 1)), gen_0':s3_0(b)) →RΩ(1)
if(true, gen_0':s3_0(+(1, n198_0)), gen_0':s3_0(b)) →RΩ(1)
rand(p(gen_0':s3_0(+(1, n198_0))), id_inc(gen_0':s3_0(b))) →LΩ(1 + n1980)
rand(gen_0':s3_0(n198_0), id_inc(gen_0':s3_0(b))) →RΩ(1)
rand(gen_0':s3_0(n198_0), gen_0':s3_0(b)) →IH
gen_0':s3_0(b)

We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).

### (14) Obligation:

TRS:
Rules:
nonZero(0') → false
nonZero(s(x)) → true
p(s(0')) → 0'
p(s(s(x))) → s(p(s(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:
p(gen_0':s3_0(+(1, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
rand(gen_0':s3_0(n198_0), gen_0':s3_0(b)) → gen_0':s3_0(b), rt ∈ Ω(1 + n1980 + n19802)

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.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
rand(gen_0':s3_0(n198_0), gen_0':s3_0(b)) → gen_0':s3_0(b), rt ∈ Ω(1 + n1980 + n19802)

### (17) Obligation:

TRS:
Rules:
nonZero(0') → false
nonZero(s(x)) → true
p(s(0')) → 0'
p(s(s(x))) → s(p(s(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:
p(gen_0':s3_0(+(1, n5_0))) → gen_0':s3_0(n5_0), rt ∈ Ω(1 + n50)
rand(gen_0':s3_0(n198_0), gen_0':s3_0(b)) → gen_0':s3_0(b), rt ∈ Ω(1 + n1980 + n19802)

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.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
rand(gen_0':s3_0(n198_0), gen_0':s3_0(b)) → gen_0':s3_0(b), rt ∈ Ω(1 + n1980 + n19802)

### (20) Obligation:

TRS:
Rules:
nonZero(0') → false
nonZero(s(x)) → true
p(s(0')) → 0'
p(s(s(x))) → s(p(s(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:
p(gen_0':s3_0(+(1, n5_0))) → gen_0':s3_0(n5_0), 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.

### (21) LowerBoundsProof (EQUIVALENT transformation)

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