### (0) Obligation:

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

minus(0, y) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
zero(s(x)) → false
zero(0) → true
p(s(x)) → x
div(x, y) → quot(x, y, 0)
quot(x, y, z) → if(zero(x), x, y, plus(z, s(0)))
if(true, x, y, z) → p(z)
if(false, x, s(y), z) → quot(minus(x, s(y)), s(y), z)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
minus(s(x), s(y)) →+ minus(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:

minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
div(x, y) → quot(x, y, 0')
quot(x, y, z) → if(zero(x), x, y, plus(z, s(0')))
if(true, x, y, z) → p(z)
if(false, x, s(y), z) → quot(minus(x, s(y)), s(y), z)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
div(x, y) → quot(x, y, 0')
quot(x, y, z) → if(zero(x), x, y, plus(z, s(0')))
if(true, x, y, z) → p(z)
if(false, x, s(y), z) → quot(minus(x, s(y)), s(y), z)

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
minus, plus, quot

They will be analysed ascendingly in the following order:
minus < quot
plus < quot

### (8) Obligation:

TRS:
Rules:
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
div(x, y) → quot(x, y, 0')
quot(x, y, z) → if(zero(x), x, y, plus(z, s(0')))
if(true, x, y, z) → p(z)
if(false, x, s(y), z) → quot(minus(x, s(y)), s(y), z)

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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, plus, quot

They will be analysed ascendingly in the following order:
minus < quot
plus < quot

### (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(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
div(x, y) → quot(x, y, 0')
quot(x, y, z) → if(zero(x), x, y, plus(z, s(0')))
if(true, x, y, z) → p(z)
if(false, x, s(y), z) → quot(minus(x, s(y)), s(y), z)

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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:
plus, quot

They will be analysed ascendingly in the following order:
plus < quot

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

Proved the following rewrite lemma:
plus(gen_0':s3_0(n275_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n275_0, b)), rt ∈ Ω(1 + n2750)

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

Induction Step:
plus(gen_0':s3_0(+(n275_0, 1)), gen_0':s3_0(b)) →RΩ(1)
plus(gen_0':s3_0(n275_0), s(gen_0':s3_0(b))) →IH
gen_0':s3_0(+(+(b, 1), c276_0))

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

### (14) Obligation:

TRS:
Rules:
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
div(x, y) → quot(x, y, 0')
quot(x, y, z) → if(zero(x), x, y, plus(z, s(0')))
if(true, x, y, z) → p(z)
if(false, x, s(y), z) → quot(minus(x, s(y)), s(y), z)

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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)
plus(gen_0':s3_0(n275_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n275_0, b)), rt ∈ Ω(1 + n2750)

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:
quot

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

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

### (16) Obligation:

TRS:
Rules:
minus(0', y) → 0'
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
div(x, y) → quot(x, y, 0')
quot(x, y, z) → if(zero(x), x, y, plus(z, s(0')))
if(true, x, y, z) → p(z)
if(false, x, s(y), z) → quot(minus(x, s(y)), s(y), z)

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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)
plus(gen_0':s3_0(n275_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n275_0, b)), rt ∈ Ω(1 + n2750)

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(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
div(x, y) → quot(x, y, 0')
quot(x, y, z) → if(zero(x), x, y, plus(z, s(0')))
if(true, x, y, z) → p(z)
if(false, x, s(y), z) → quot(minus(x, s(y)), s(y), z)

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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)
plus(gen_0':s3_0(n275_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n275_0, b)), rt ∈ Ω(1 + n2750)

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(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
plus(0', y) → y
plus(s(x), y) → plus(x, s(y))
zero(s(x)) → false
zero(0') → true
p(s(x)) → x
div(x, y) → quot(x, y, 0')
quot(x, y, z) → if(zero(x), x, y, plus(z, s(0')))
if(true, x, y, z) → p(z)
if(false, x, s(y), z) → quot(minus(x, s(y)), s(y), z)

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
plus :: 0':s → 0':s → 0':s
zero :: 0':s → false:true
false :: false:true
true :: false:true
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s → 0':s
if :: false:true → 0':s → 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_false:true2_0 :: false:true
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)