### (0) Obligation:

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

minus(0, y) → 0
minus(s(x), 0) → s(x)
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
if(true, x, y) → x
if(false, x, y) → y
perfectp(0) → false
perfectp(s(x)) → f(x, s(0), s(x), s(x))
f(0, y, 0, u) → true
f(0, y, s(z), u) → false
f(s(x), 0, z, u) → f(x, u, minus(z, s(x)), u)
f(s(x), s(y), z, u) → if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
f(s(s(x116272_1)), s(y), z, s(0)) →+ if(le(s(x116272_1), y), f(s(s(x116272_1)), minus(y, s(x116272_1)), z, s(0)), if(le(x116272_1, 0), f(x116272_1, s(0), minus(z, s(x116272_1)), s(0)), f(x116272_1, s(0), z, s(0))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
The pumping substitution is [x116272_1 / s(s(x116272_1))].
The result substitution is [y / 0, z / minus(z, s(x116272_1))].

The rewrite sequence
f(s(s(x116272_1)), s(y), z, s(0)) →+ if(le(s(x116272_1), y), f(s(s(x116272_1)), minus(y, s(x116272_1)), z, s(0)), if(le(x116272_1, 0), f(x116272_1, s(0), minus(z, s(x116272_1)), s(0)), f(x116272_1, s(0), z, s(0))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [2,2].
The pumping substitution is [x116272_1 / s(s(x116272_1))].
The result substitution is [y / 0].

### (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:

minus(0', y) → 0'
minus(s(x), 0') → s(x)
minus(s(x), s(y)) → minus(x, y)
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
if(true, x, y) → x
if(false, x, y) → y
perfectp(0') → false
perfectp(s(x)) → f(x, s(0'), s(x), s(x))
f(0', y, 0', u) → true
f(0', y, s(z), u) → false
f(s(x), 0', z, u) → f(x, u, minus(z, s(x)), u)
f(s(x), s(y), z, u) → if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
minus(0', y) → 0'
minus(s(x), 0') → s(x)
minus(s(x), s(y)) → minus(x, y)
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
if(true, x, y) → x
if(false, x, y) → y
perfectp(0') → false
perfectp(s(x)) → f(x, s(0'), s(x), s(x))
f(0', y, 0', u) → true
f(0', y, s(z), u) → false
f(s(x), 0', z, u) → f(x, u, minus(z, s(x)), u)
f(s(x), s(y), z, u) → if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → true:false → true:false → true:false
perfectp :: 0':s → true:false
f :: 0':s → 0':s → 0':s → 0':s → true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
minus, le, f

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

### (8) Obligation:

TRS:
Rules:
minus(0', y) → 0'
minus(s(x), 0') → s(x)
minus(s(x), s(y)) → minus(x, y)
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
if(true, x, y) → x
if(false, x, y) → y
perfectp(0') → false
perfectp(s(x)) → f(x, s(0'), s(x), s(x))
f(0', y, 0', u) → true
f(0', y, s(z), u) → false
f(s(x), 0', z, u) → f(x, u, minus(z, s(x)), u)
f(s(x), s(y), z, u) → if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → true:false → true:false → true:false
perfectp :: 0':s → true:false
f :: 0':s → 0':s → 0':s → 0':s → true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
minus, le, f

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

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

Proved the following rewrite lemma:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)

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

Induction Step:
minus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →RΩ(1)
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) →IH
gen_0':s3_0(0)

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

### (11) Obligation:

TRS:
Rules:
minus(0', y) → 0'
minus(s(x), 0') → s(x)
minus(s(x), s(y)) → minus(x, y)
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
if(true, x, y) → x
if(false, x, y) → y
perfectp(0') → false
perfectp(s(x)) → f(x, s(0'), s(x), s(x))
f(0', y, 0', u) → true
f(0', y, s(z), u) → false
f(s(x), 0', z, u) → f(x, u, minus(z, s(x)), u)
f(s(x), s(y), z, u) → if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → true:false → true:false → true:false
perfectp :: 0':s → true:false
f :: 0':s → 0':s → 0':s → 0':s → true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(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:
le, f

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

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

Proved the following rewrite lemma:
le(gen_0':s3_0(n435_0), gen_0':s3_0(n435_0)) → true, rt ∈ Ω(1 + n4350)

Induction Base:
le(gen_0':s3_0(0), gen_0':s3_0(0)) →RΩ(1)
true

Induction Step:
le(gen_0':s3_0(+(n435_0, 1)), gen_0':s3_0(+(n435_0, 1))) →RΩ(1)
le(gen_0':s3_0(n435_0), gen_0':s3_0(n435_0)) →IH
true

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

### (14) Obligation:

TRS:
Rules:
minus(0', y) → 0'
minus(s(x), 0') → s(x)
minus(s(x), s(y)) → minus(x, y)
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
if(true, x, y) → x
if(false, x, y) → y
perfectp(0') → false
perfectp(s(x)) → f(x, s(0'), s(x), s(x))
f(0', y, 0', u) → true
f(0', y, s(z), u) → false
f(s(x), 0', z, u) → f(x, u, minus(z, s(x)), u)
f(s(x), s(y), z, u) → if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → true:false → true:false → true:false
perfectp :: 0':s → true:false
f :: 0':s → 0':s → 0':s → 0':s → true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)
le(gen_0':s3_0(n435_0), gen_0':s3_0(n435_0)) → true, rt ∈ Ω(1 + n4350)

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:
f

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

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

### (16) Obligation:

TRS:
Rules:
minus(0', y) → 0'
minus(s(x), 0') → s(x)
minus(s(x), s(y)) → minus(x, y)
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
if(true, x, y) → x
if(false, x, y) → y
perfectp(0') → false
perfectp(s(x)) → f(x, s(0'), s(x), s(x))
f(0', y, 0', u) → true
f(0', y, s(z), u) → false
f(s(x), 0', z, u) → f(x, u, minus(z, s(x)), u)
f(s(x), s(y), z, u) → if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → true:false → true:false → true:false
perfectp :: 0':s → true:false
f :: 0':s → 0':s → 0':s → 0':s → true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)
le(gen_0':s3_0(n435_0), gen_0':s3_0(n435_0)) → true, rt ∈ Ω(1 + n4350)

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.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)

### (19) Obligation:

TRS:
Rules:
minus(0', y) → 0'
minus(s(x), 0') → s(x)
minus(s(x), s(y)) → minus(x, y)
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
if(true, x, y) → x
if(false, x, y) → y
perfectp(0') → false
perfectp(s(x)) → f(x, s(0'), s(x), s(x))
f(0', y, 0', u) → true
f(0', y, s(z), u) → false
f(s(x), 0', z, u) → f(x, u, minus(z, s(x)), u)
f(s(x), s(y), z, u) → if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → true:false → true:false → true:false
perfectp :: 0':s → true:false
f :: 0':s → 0':s → 0':s → 0':s → true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)
le(gen_0':s3_0(n435_0), gen_0':s3_0(n435_0)) → true, rt ∈ Ω(1 + n4350)

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.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)

### (22) Obligation:

TRS:
Rules:
minus(0', y) → 0'
minus(s(x), 0') → s(x)
minus(s(x), s(y)) → minus(x, y)
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
if(true, x, y) → x
if(false, x, y) → y
perfectp(0') → false
perfectp(s(x)) → f(x, s(0'), s(x), s(x))
f(0', y, 0', u) → true
f(0', y, s(z), u) → false
f(s(x), 0', z, u) → f(x, u, minus(z, s(x)), u)
f(s(x), s(y), z, u) → if(le(x, y), f(s(x), minus(y, x), z, u), f(x, u, z, u))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if :: true:false → true:false → true:false → true:false
perfectp :: 0':s → true:false
f :: 0':s → 0':s → 0':s → 0':s → true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(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.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n50)