### (0) Obligation:

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

cond(true, x) → cond(and(even(x), gr(x, 0)), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
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 [ ].

### (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:

cond(true, x) → cond(and(even(x), gr(x, 0')), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
p(0') → 0'
p(s(x)) → x

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
cond(true, x) → cond(and(even(x), gr(x, 0')), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond :: true:false → 0':s:y → cond
true :: true:false
and :: true:false → true:false → true:false
even :: 0':s:y → true:false
gr :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
p :: 0':s:y → 0':s:y
false :: true:false
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1_0 :: cond
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:
cond, even, gr

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

### (8) Obligation:

TRS:
Rules:
cond(true, x) → cond(and(even(x), gr(x, 0')), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond :: true:false → 0':s:y → cond
true :: true:false
and :: true:false → true:false → true:false
even :: 0':s:y → true:false
gr :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
p :: 0':s:y → 0':s:y
false :: true:false
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1_0 :: cond
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, cond, gr

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

### (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))) →RΩ(1)
true

Induction Step:
even(gen_0':s:y4_0(*(2, +(n6_0, 1)))) →RΩ(1)
even(gen_0':s:y4_0(*(2, n6_0))) →IH
true

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

### (11) Obligation:

TRS:
Rules:
cond(true, x) → cond(and(even(x), gr(x, 0')), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond :: true:false → 0':s:y → cond
true :: true:false
and :: true:false → true:false → true:false
even :: 0':s:y → true:false
gr :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
p :: 0':s:y → 0':s:y
false :: true:false
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1_0 :: cond
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:
gr, cond

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

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

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

### (13) Obligation:

TRS:
Rules:
cond(true, x) → cond(and(even(x), gr(x, 0')), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond :: true:false → 0':s:y → cond
true :: true:false
and :: true:false → true:false → true:false
even :: 0':s:y → true:false
gr :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
p :: 0':s:y → 0':s:y
false :: true:false
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1_0 :: cond
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:
cond

### (14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (15) Obligation:

TRS:
Rules:
cond(true, x) → cond(and(even(x), gr(x, 0')), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond :: true:false → 0':s:y → cond
true :: true:false
and :: true:false → true:false → true:false
even :: 0':s:y → true:false
gr :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
p :: 0':s:y → 0':s:y
false :: true:false
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1_0 :: cond
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.

### (16) 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)

### (18) Obligation:

TRS:
Rules:
cond(true, x) → cond(and(even(x), gr(x, 0')), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0') → true
even(s(0')) → false
even(s(s(x))) → even(x)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
p(0') → 0'
p(s(x)) → x

Types:
cond :: true:false → 0':s:y → cond
true :: true:false
and :: true:false → true:false → true:false
even :: 0':s:y → true:false
gr :: 0':s:y → 0':s:y → true:false
0' :: 0':s:y
p :: 0':s:y → 0':s:y
false :: true:false
s :: 0':s:y → 0':s:y
y :: 0':s:y
hole_cond1_0 :: cond
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.

### (19) 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)