### (0) Obligation:

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

p(s(x)) → x
p(0) → 0
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
average(x, y) → if(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y)
if(true, b1, b2, b3, x, y) → if2(b1, b2, b3, x, y)
if(false, b1, b2, b3, x, y) → average(p(x), s(y))
if2(true, b2, b3, x, y) → 0
if2(false, b2, b3, x, y) → if3(b2, b3, x, y)
if3(true, b3, x, y) → 0
if3(false, b3, x, y) → if4(b3, x, y)
if4(true, x, y) → s(0)
if4(false, x, y) → average(s(x), p(p(y)))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

p(s(x)) → x
p(0') → 0'
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
average(x, y) → if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y)
if(true, b1, b2, b3, x, y) → if2(b1, b2, b3, x, y)
if(false, b1, b2, b3, x, y) → average(p(x), s(y))
if2(true, b2, b3, x, y) → 0'
if2(false, b2, b3, x, y) → if3(b2, b3, x, y)
if3(true, b3, x, y) → 0'
if3(false, b3, x, y) → if4(b3, x, y)
if4(true, x, y) → s(0')
if4(false, x, y) → average(s(x), p(p(y)))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
p(s(x)) → x
p(0') → 0'
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
average(x, y) → if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y)
if(true, b1, b2, b3, x, y) → if2(b1, b2, b3, x, y)
if(false, b1, b2, b3, x, y) → average(p(x), s(y))
if2(true, b2, b3, x, y) → 0'
if2(false, b2, b3, x, y) → if3(b2, b3, x, y)
if3(true, b3, x, y) → 0'
if3(false, b3, x, y) → if4(b3, x, y)
if4(true, x, y) → s(0')
if4(false, x, y) → average(s(x), p(p(y)))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
le :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
average :: s:0' → s:0' → s:0'
if :: true:false → true:false → true:false → true:false → s:0' → s:0' → s:0'
if2 :: true:false → true:false → true:false → s:0' → s:0' → s:0'
if3 :: true:false → true:false → s:0' → s:0' → s:0'
if4 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
le, average

They will be analysed ascendingly in the following order:
le < average

### (8) Obligation:

TRS:
Rules:
p(s(x)) → x
p(0') → 0'
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
average(x, y) → if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y)
if(true, b1, b2, b3, x, y) → if2(b1, b2, b3, x, y)
if(false, b1, b2, b3, x, y) → average(p(x), s(y))
if2(true, b2, b3, x, y) → 0'
if2(false, b2, b3, x, y) → if3(b2, b3, x, y)
if3(true, b3, x, y) → 0'
if3(false, b3, x, y) → if4(b3, x, y)
if4(true, x, y) → s(0')
if4(false, x, y) → average(s(x), p(p(y)))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
le :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
average :: s:0' → s:0' → s:0'
if :: true:false → true:false → true:false → true:false → s:0' → s:0' → s:0'
if2 :: true:false → true:false → true:false → s:0' → s:0' → s:0'
if3 :: true:false → true:false → s:0' → s:0' → s:0'
if4 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
le, average

They will be analysed ascendingly in the following order:
le < average

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

Proved the following rewrite lemma:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Induction Base:
le(gen_s:0'3_0(0), gen_s:0'3_0(0)) →RΩ(1)
true

Induction Step:
le(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) →RΩ(1)
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) →IH
true

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

### (11) Obligation:

TRS:
Rules:
p(s(x)) → x
p(0') → 0'
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
average(x, y) → if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y)
if(true, b1, b2, b3, x, y) → if2(b1, b2, b3, x, y)
if(false, b1, b2, b3, x, y) → average(p(x), s(y))
if2(true, b2, b3, x, y) → 0'
if2(false, b2, b3, x, y) → if3(b2, b3, x, y)
if3(true, b3, x, y) → 0'
if3(false, b3, x, y) → if4(b3, x, y)
if4(true, x, y) → s(0')
if4(false, x, y) → average(s(x), p(p(y)))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
le :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
average :: s:0' → s:0' → s:0'
if :: true:false → true:false → true:false → true:false → s:0' → s:0' → s:0'
if2 :: true:false → true:false → true:false → s:0' → s:0' → s:0'
if3 :: true:false → true:false → s:0' → s:0' → s:0'
if4 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
average

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

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

### (13) Obligation:

TRS:
Rules:
p(s(x)) → x
p(0') → 0'
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
average(x, y) → if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y)
if(true, b1, b2, b3, x, y) → if2(b1, b2, b3, x, y)
if(false, b1, b2, b3, x, y) → average(p(x), s(y))
if2(true, b2, b3, x, y) → 0'
if2(false, b2, b3, x, y) → if3(b2, b3, x, y)
if3(true, b3, x, y) → 0'
if3(false, b3, x, y) → if4(b3, x, y)
if4(true, x, y) → s(0')
if4(false, x, y) → average(s(x), p(p(y)))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
le :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
average :: s:0' → s:0' → s:0'
if :: true:false → true:false → true:false → true:false → s:0' → s:0' → s:0'
if2 :: true:false → true:false → true:false → s:0' → s:0' → s:0'
if3 :: true:false → true:false → s:0' → s:0' → s:0'
if4 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

### (16) Obligation:

TRS:
Rules:
p(s(x)) → x
p(0') → 0'
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
average(x, y) → if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y)
if(true, b1, b2, b3, x, y) → if2(b1, b2, b3, x, y)
if(false, b1, b2, b3, x, y) → average(p(x), s(y))
if2(true, b2, b3, x, y) → 0'
if2(false, b2, b3, x, y) → if3(b2, b3, x, y)
if3(true, b3, x, y) → 0'
if3(false, b3, x, y) → if4(b3, x, y)
if4(true, x, y) → s(0')
if4(false, x, y) → average(s(x), p(p(y)))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
le :: s:0' → s:0' → true:false
true :: true:false
false :: true:false
average :: s:0' → s:0' → s:0'
if :: true:false → true:false → true:false → true:false → s:0' → s:0' → s:0'
if2 :: true:false → true:false → true:false → s:0' → s:0' → s:0'
if3 :: true:false → true:false → s:0' → s:0' → s:0'
if4 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)