### (0) Obligation:

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

leq(0, y) → true
leq(s(x), 0) → false
leq(s(x), s(y)) → leq(x, y)
if(true, x, y) → x
if(false, x, y) → y
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
mod(0, y) → 0
mod(s(x), 0) → 0
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

leq(0', y) → true
leq(s(x), 0') → false
leq(s(x), s(y)) → leq(x, y)
if(true, x, y) → x
if(false, x, y) → y
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
mod(0', y) → 0'
mod(s(x), 0') → 0'
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
leq(0', y) → true
leq(s(x), 0') → false
leq(s(x), s(y)) → leq(x, y)
if(true, x, y) → x
if(false, x, y) → y
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
mod(0', y) → 0'
mod(s(x), 0') → 0'
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
leq, -, mod

They will be analysed ascendingly in the following order:
leq < mod
- < mod

### (8) Obligation:

TRS:
Rules:
leq(0', y) → true
leq(s(x), 0') → false
leq(s(x), s(y)) → leq(x, y)
if(true, x, y) → x
if(false, x, y) → y
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
mod(0', y) → 0'
mod(s(x), 0') → 0'
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
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:
leq, -, mod

They will be analysed ascendingly in the following order:
leq < mod
- < mod

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

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

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

Induction Step:
leq(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →RΩ(1)
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) →IH
true

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

### (11) Obligation:

TRS:
Rules:
leq(0', y) → true
leq(s(x), 0') → false
leq(s(x), s(y)) → leq(x, y)
if(true, x, y) → x
if(false, x, y) → y
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
mod(0', y) → 0'
mod(s(x), 0') → 0'
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, 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:
-, mod

They will be analysed ascendingly in the following order:
- < mod

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

Proved the following rewrite lemma:
-(gen_0':s3_0(n276_0), gen_0':s3_0(n276_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n2760)

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

Induction Step:
-(gen_0':s3_0(+(n276_0, 1)), gen_0':s3_0(+(n276_0, 1))) →RΩ(1)
-(gen_0':s3_0(n276_0), gen_0':s3_0(n276_0)) →IH
gen_0':s3_0(0)

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

### (14) Obligation:

TRS:
Rules:
leq(0', y) → true
leq(s(x), 0') → false
leq(s(x), s(y)) → leq(x, y)
if(true, x, y) → x
if(false, x, y) → y
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
mod(0', y) → 0'
mod(s(x), 0') → 0'
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
-(gen_0':s3_0(n276_0), gen_0':s3_0(n276_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n2760)

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:
mod

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

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

### (16) Obligation:

TRS:
Rules:
leq(0', y) → true
leq(s(x), 0') → false
leq(s(x), s(y)) → leq(x, y)
if(true, x, y) → x
if(false, x, y) → y
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
mod(0', y) → 0'
mod(s(x), 0') → 0'
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
-(gen_0':s3_0(n276_0), gen_0':s3_0(n276_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n2760)

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:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

### (19) Obligation:

TRS:
Rules:
leq(0', y) → true
leq(s(x), 0') → false
leq(s(x), s(y)) → leq(x, y)
if(true, x, y) → x
if(false, x, y) → y
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
mod(0', y) → 0'
mod(s(x), 0') → 0'
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
-(gen_0':s3_0(n276_0), gen_0':s3_0(n276_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n2760)

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:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)

### (22) Obligation:

TRS:
Rules:
leq(0', y) → true
leq(s(x), 0') → false
leq(s(x), s(y)) → leq(x, y)
if(true, x, y) → x
if(false, x, y) → y
-(x, 0') → x
-(s(x), s(y)) → -(x, y)
mod(0', y) → 0'
mod(s(x), 0') → 0'
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(x))

Types:
leq :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
if :: true:false → 0':s → 0':s → 0':s
- :: 0':s → 0':s → 0':s
mod :: 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, 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:
leq(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)