The Runtime Complexity (full) 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:

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0), div2(x))
cond2(false, x) → cond1(neq(x, 0), p(x))
neq(0, 0) → false
neq(0, s(x)) → true
neq(s(x), 0) → true
neq(s(x), s(y)) → neq(x, y)
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
div2(0) → 0
div2(s(0)) → 0
div2(s(s(x))) → s(div2(x))
p(0) → 0
p(s(x)) → x

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)

(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:

cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0'), div2(x))
cond2(false, x) → cond1(neq(x, 0'), p(x))
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
div2(0') → 0'
div2(s(0')) → 0'
div2(s(s(x))) → s(div2(x))
p(0') → 0'
p(s(x)) → x

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0'), div2(x))
cond2(false, x) → cond1(neq(x, 0'), p(x))
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
div2(0') → 0'
div2(s(0')) → 0'
div2(s(s(x))) → s(div2(x))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s:y → cond1:cond2
true :: true:false
cond2 :: true:false → 0':s:y → cond1:cond2
even :: 0':s:y → true:false
neq :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
div2 :: 0':s:y → 0':s:y
false :: true:false
p :: 0':s:y → 0':s:y
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat → 0':s:y

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

Heuristically decided to analyse the following defined symbols:
cond1, cond2, even, neq, div2

They will be analysed ascendingly in the following order:
cond1 = cond2
even < cond1
neq < cond2
div2 < cond2

(8) Obligation:

TRS:
Rules:
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0'), div2(x))
cond2(false, x) → cond1(neq(x, 0'), p(x))
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
div2(0') → 0'
div2(s(0')) → 0'
div2(s(s(x))) → s(div2(x))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s:y → cond1:cond2
true :: true:false
cond2 :: true:false → 0':s:y → cond1:cond2
even :: 0':s:y → true:false
neq :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
div2 :: 0':s:y → 0':s:y
false :: true:false
p :: 0':s:y → 0':s:y
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat → 0':s:y

Generator Equations:
gen_0':s:y4_0(0) ⇔ 0'
gen_0':s:y4_0(+(x, 1)) ⇔ s(gen_0':s:y4_0(x))

The following defined symbols remain to be analysed:
even, cond1, cond2, neq, div2

They will be analysed ascendingly in the following order:
cond1 = cond2
even < cond1
neq < cond2
div2 < cond2

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

Proved the following rewrite lemma:
even(gen_0':s:y4_0(*(2, n6_0))) → true, rt ∈ Ω(1 + n6₀)

Induction Base:
even(gen_0':s:y4_0(*(2, 0))) →_R^Ω(1)
true

Induction Step:
even(gen_0':s:y4_0(*(2, +(n6_0, 1)))) →_R^Ω(1)
even(gen_0':s:y4_0(*(2, n6_0))) →_IH
true

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0'), div2(x))
cond2(false, x) → cond1(neq(x, 0'), p(x))
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
div2(0') → 0'
div2(s(0')) → 0'
div2(s(s(x))) → s(div2(x))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s:y → cond1:cond2
true :: true:false
cond2 :: true:false → 0':s:y → cond1:cond2
even :: 0':s:y → true:false
neq :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
div2 :: 0':s:y → 0':s:y
false :: true:false
p :: 0':s:y → 0':s:y
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat → 0':s:y

Lemmas:
even(gen_0':s:y4_0(*(2, n6_0))) → true, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s:y4_0(0) ⇔ 0'
gen_0':s:y4_0(+(x, 1)) ⇔ s(gen_0':s:y4_0(x))

The following defined symbols remain to be analysed:
neq, cond1, cond2, div2

They will be analysed ascendingly in the following order:
cond1 = cond2
neq < cond2
div2 < cond2

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0'), div2(x))
cond2(false, x) → cond1(neq(x, 0'), p(x))
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
div2(0') → 0'
div2(s(0')) → 0'
div2(s(s(x))) → s(div2(x))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s:y → cond1:cond2
true :: true:false
cond2 :: true:false → 0':s:y → cond1:cond2
even :: 0':s:y → true:false
neq :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
div2 :: 0':s:y → 0':s:y
false :: true:false
p :: 0':s:y → 0':s:y
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat → 0':s:y

Lemmas:
even(gen_0':s:y4_0(*(2, n6_0))) → true, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s:y4_0(0) ⇔ 0'
gen_0':s:y4_0(+(x, 1)) ⇔ s(gen_0':s:y4_0(x))

The following defined symbols remain to be analysed:
div2, cond1, cond2

They will be analysed ascendingly in the following order:
cond1 = cond2
div2 < cond2

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
div2(gen_0':s:y4_0(*(2, n198_0))) → gen_0':s:y4_0(n198_0), rt ∈ Ω(1 + n198₀)

Induction Base:
div2(gen_0':s:y4_0(*(2, 0))) →_R^Ω(1)
0'

Induction Step:
div2(gen_0':s:y4_0(*(2, +(n198_0, 1)))) →_R^Ω(1)
s(div2(gen_0':s:y4_0(*(2, n198_0)))) →_IH
s(gen_0':s:y4_0(c199_0))

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

(15) Complex Obligation (BEST)

(16) Obligation:

TRS:
Rules:
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0'), div2(x))
cond2(false, x) → cond1(neq(x, 0'), p(x))
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
div2(0') → 0'
div2(s(0')) → 0'
div2(s(s(x))) → s(div2(x))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s:y → cond1:cond2
true :: true:false
cond2 :: true:false → 0':s:y → cond1:cond2
even :: 0':s:y → true:false
neq :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
div2 :: 0':s:y → 0':s:y
false :: true:false
p :: 0':s:y → 0':s:y
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat → 0':s:y

Lemmas:
even(gen_0':s:y4_0(*(2, n6_0))) → true, rt ∈ Ω(1 + n6₀)
div2(gen_0':s:y4_0(*(2, n198_0))) → gen_0':s:y4_0(n198_0), rt ∈ Ω(1 + n198₀)

Generator Equations:
gen_0':s:y4_0(0) ⇔ 0'
gen_0':s:y4_0(+(x, 1)) ⇔ s(gen_0':s:y4_0(x))

The following defined symbols remain to be analysed:
cond2, cond1

They will be analysed ascendingly in the following order:
cond1 = cond2

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol cond2.

(18) Obligation:

TRS:
Rules:
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0'), div2(x))
cond2(false, x) → cond1(neq(x, 0'), p(x))
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
div2(0') → 0'
div2(s(0')) → 0'
div2(s(s(x))) → s(div2(x))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s:y → cond1:cond2
true :: true:false
cond2 :: true:false → 0':s:y → cond1:cond2
even :: 0':s:y → true:false
neq :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
div2 :: 0':s:y → 0':s:y
false :: true:false
p :: 0':s:y → 0':s:y
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat → 0':s:y

Lemmas:
even(gen_0':s:y4_0(*(2, n6_0))) → true, rt ∈ Ω(1 + n6₀)
div2(gen_0':s:y4_0(*(2, n198_0))) → gen_0':s:y4_0(n198_0), rt ∈ Ω(1 + n198₀)

Generator Equations:
gen_0':s:y4_0(0) ⇔ 0'
gen_0':s:y4_0(+(x, 1)) ⇔ s(gen_0':s:y4_0(x))

The following defined symbols remain to be analysed:
cond1

They will be analysed ascendingly in the following order:
cond1 = cond2

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol cond1.

(20) Obligation:

TRS:
Rules:
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0'), div2(x))
cond2(false, x) → cond1(neq(x, 0'), p(x))
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
div2(0') → 0'
div2(s(0')) → 0'
div2(s(s(x))) → s(div2(x))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s:y → cond1:cond2
true :: true:false
cond2 :: true:false → 0':s:y → cond1:cond2
even :: 0':s:y → true:false
neq :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
div2 :: 0':s:y → 0':s:y
false :: true:false
p :: 0':s:y → 0':s:y
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat → 0':s:y

Lemmas:
even(gen_0':s:y4_0(*(2, n6_0))) → true, rt ∈ Ω(1 + n6₀)
div2(gen_0':s:y4_0(*(2, n198_0))) → gen_0':s:y4_0(n198_0), rt ∈ Ω(1 + n198₀)

Generator Equations:
gen_0':s:y4_0(0) ⇔ 0'
gen_0':s:y4_0(+(x, 1)) ⇔ s(gen_0':s:y4_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
even(gen_0':s:y4_0(*(2, n6_0))) → true, rt ∈ Ω(1 + n6₀)

(22) BOUNDS(n^1, INF)

(23) Obligation:

TRS:
Rules:
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0'), div2(x))
cond2(false, x) → cond1(neq(x, 0'), p(x))
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
div2(0') → 0'
div2(s(0')) → 0'
div2(s(s(x))) → s(div2(x))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s:y → cond1:cond2
true :: true:false
cond2 :: true:false → 0':s:y → cond1:cond2
even :: 0':s:y → true:false
neq :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
div2 :: 0':s:y → 0':s:y
false :: true:false
p :: 0':s:y → 0':s:y
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat → 0':s:y

Lemmas:
even(gen_0':s:y4_0(*(2, n6_0))) → true, rt ∈ Ω(1 + n6₀)
div2(gen_0':s:y4_0(*(2, n198_0))) → gen_0':s:y4_0(n198_0), rt ∈ Ω(1 + n198₀)

Generator Equations:
gen_0':s:y4_0(0) ⇔ 0'
gen_0':s:y4_0(+(x, 1)) ⇔ s(gen_0':s:y4_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
even(gen_0':s:y4_0(*(2, n6_0))) → true, rt ∈ Ω(1 + n6₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

TRS:
Rules:
cond1(true, x) → cond2(even(x), x)
cond2(true, x) → cond1(neq(x, 0'), div2(x))
cond2(false, x) → cond1(neq(x, 0'), p(x))
neq(0', 0') → false
neq(0', s(x)) → true
neq(s(x), 0') → true
neq(s(x), s(y)) → neq(x, y)
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
div2(0') → 0'
div2(s(0')) → 0'
div2(s(s(x))) → s(div2(x))
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s:y → cond1:cond2
true :: true:false
cond2 :: true:false → 0':s:y → cond1:cond2
even :: 0':s:y → true:false
neq :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
div2 :: 0':s:y → 0':s:y
false :: true:false
p :: 0':s:y → 0':s:y
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s:y3_0 :: 0':s:y
gen_0':s:y4_0 :: Nat → 0':s:y

Lemmas:
even(gen_0':s:y4_0(*(2, n6_0))) → true, rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s:y4_0(0) ⇔ 0'
gen_0':s:y4_0(+(x, 1)) ⇔ s(gen_0':s:y4_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
even(gen_0':s:y4_0(*(2, n6_0))) → true, rt ∈ Ω(1 + n6₀)

(28) BOUNDS(n^1, INF)