### (0) Obligation:

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

cond1(true, x, y) → cond2(gr(x, 0), x, y)
cond2(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), p(x), y)
cond2(false, x, y) → cond3(gr(y, 0), x, y)
cond3(true, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, p(y))
cond3(false, x, y) → cond1(or(gr(x, 0), gr(y, 0)), x, y)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
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
cond1(true, s(x29_4), y) →+ cond1(true, x29_4, y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x29_4 / s(x29_4)].
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:

cond1(true, x, y) → cond2(gr(x, 0'), x, y)
cond2(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) → cond3(gr(y, 0'), x, y)
cond3(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
p(0') → 0'
p(s(x)) → x

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
cond1(true, x, y) → cond2(gr(x, 0'), x, y)
cond2(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) → cond3(gr(y, 0'), x, y)
cond3(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
or :: true:false → true:false → true:false
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
cond1, cond2, gr, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
cond2 = cond3
gr < cond3

### (8) Obligation:

TRS:
Rules:
cond1(true, x, y) → cond2(gr(x, 0'), x, y)
cond2(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) → cond3(gr(y, 0'), x, y)
cond3(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
or :: true:false → true:false → true:false
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
gr, cond1, cond2, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
cond2 = cond3
gr < cond3

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

Proved the following rewrite lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

Induction Base:
gr(gen_0':s4_0(0), gen_0':s4_0(0)) →RΩ(1)
false

Induction Step:
gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) →RΩ(1)
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) →IH
false

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

### (11) Obligation:

TRS:
Rules:
cond1(true, x, y) → cond2(gr(x, 0'), x, y)
cond2(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) → cond3(gr(y, 0'), x, y)
cond3(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
or :: true:false → true:false → true:false
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
cond2, cond1, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
cond1 = cond3
cond2 = cond3

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

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

### (13) Obligation:

TRS:
Rules:
cond1(true, x, y) → cond2(gr(x, 0'), x, y)
cond2(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) → cond3(gr(y, 0'), x, y)
cond3(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
or :: true:false → true:false → true:false
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
cond1, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
cond1 = cond3
cond2 = cond3

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

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

### (15) Obligation:

TRS:
Rules:
cond1(true, x, y) → cond2(gr(x, 0'), x, y)
cond2(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) → cond3(gr(y, 0'), x, y)
cond3(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
or :: true:false → true:false → true:false
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
cond1 = cond3
cond2 = cond3

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

Could not prove a rewrite lemma for the defined symbol cond3.

### (17) Obligation:

TRS:
Rules:
cond1(true, x, y) → cond2(gr(x, 0'), x, y)
cond2(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) → cond3(gr(y, 0'), x, y)
cond3(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
or :: true:false → true:false → true:false
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

### (20) Obligation:

TRS:
Rules:
cond1(true, x, y) → cond2(gr(x, 0'), x, y)
cond2(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), p(x), y)
cond2(false, x, y) → cond3(gr(y, 0'), x, y)
cond3(true, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, p(y))
cond3(false, x, y) → cond1(or(gr(x, 0'), gr(y, 0')), x, y)
gr(0', x) → false
gr(s(x), 0') → true
gr(s(x), s(y)) → gr(x, y)
or(false, false) → false
or(true, x) → true
or(x, true) → true
p(0') → 0'
p(s(x)) → x

Types:
cond1 :: true:false → 0':s → 0':s → cond1:cond2:cond3
true :: true:false
cond2 :: true:false → 0':s → 0':s → cond1:cond2:cond3
gr :: 0':s → 0':s → true:false
0' :: 0':s
or :: true:false → true:false → true:false
p :: 0':s → 0':s
false :: true:false
cond3 :: true:false → 0':s → 0':s → cond1:cond2:cond3
s :: 0':s → 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → false, rt ∈ Ω(1 + n60)