### (0) Obligation:

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

gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
divides(x, y) → div(x, y, y)
div(0, 0, z) → true
div(0, s(x), z) → false
div(s(x), 0, s(z)) → div(s(x), s(z), s(z))
div(s(x), s(y), z) → div(x, y, z)
prime(x) → test(x, s(s(0)))
test(x, y) → if1(gt(x, y), x, y)
if1(true, x, y) → if2(divides(x, y), x, y)
if1(false, x, y) → true
if2(true, x, y) → false
if2(false, x, y) → test(x, s(y))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

gt(s(x), 0') → true
gt(0', y) → false
gt(s(x), s(y)) → gt(x, y)
divides(x, y) → div(x, y, y)
div(0', 0', z) → true
div(0', s(x), z) → false
div(s(x), 0', s(z)) → div(s(x), s(z), s(z))
div(s(x), s(y), z) → div(x, y, z)
prime(x) → test(x, s(s(0')))
test(x, y) → if1(gt(x, y), x, y)
if1(true, x, y) → if2(divides(x, y), x, y)
if1(false, x, y) → true
if2(true, x, y) → false
if2(false, x, y) → test(x, s(y))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
gt(s(x), 0') → true
gt(0', y) → false
gt(s(x), s(y)) → gt(x, y)
divides(x, y) → div(x, y, y)
div(0', 0', z) → true
div(0', s(x), z) → false
div(s(x), 0', s(z)) → div(s(x), s(z), s(z))
div(s(x), s(y), z) → div(x, y, z)
prime(x) → test(x, s(s(0')))
test(x, y) → if1(gt(x, y), x, y)
if1(true, x, y) → if2(divides(x, y), x, y)
if1(false, x, y) → true
if2(true, x, y) → false
if2(false, x, y) → test(x, s(y))

Types:
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0' → true:false
prime :: s:0' → true:false
test :: s:0' → s:0' → true:false
if1 :: true:false → s:0' → s:0' → true:false
if2 :: true:false → s:0' → s:0' → true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
gt, div, test

They will be analysed ascendingly in the following order:
gt < test

### (8) Obligation:

TRS:
Rules:
gt(s(x), 0') → true
gt(0', y) → false
gt(s(x), s(y)) → gt(x, y)
divides(x, y) → div(x, y, y)
div(0', 0', z) → true
div(0', s(x), z) → false
div(s(x), 0', s(z)) → div(s(x), s(z), s(z))
div(s(x), s(y), z) → div(x, y, z)
prime(x) → test(x, s(s(0')))
test(x, y) → if1(gt(x, y), x, y)
if1(true, x, y) → if2(divides(x, y), x, y)
if1(false, x, y) → true
if2(true, x, y) → false
if2(false, x, y) → test(x, s(y))

Types:
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0' → true:false
prime :: s:0' → true:false
test :: s:0' → s:0' → true:false
if1 :: true:false → s:0' → s:0' → true:false
if2 :: true:false → s:0' → s:0' → true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
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:
gt, div, test

They will be analysed ascendingly in the following order:
gt < test

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

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

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

Induction Step:
gt(gen_s:0'3_0(+(1, +(n5_0, 1))), gen_s:0'3_0(+(n5_0, 1))) →RΩ(1)
gt(gen_s:0'3_0(+(1, 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:
gt(s(x), 0') → true
gt(0', y) → false
gt(s(x), s(y)) → gt(x, y)
divides(x, y) → div(x, y, y)
div(0', 0', z) → true
div(0', s(x), z) → false
div(s(x), 0', s(z)) → div(s(x), s(z), s(z))
div(s(x), s(y), z) → div(x, y, z)
prime(x) → test(x, s(s(0')))
test(x, y) → if1(gt(x, y), x, y)
if1(true, x, y) → if2(divides(x, y), x, y)
if1(false, x, y) → true
if2(true, x, y) → false
if2(false, x, y) → test(x, s(y))

Types:
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0' → true:false
prime :: s:0' → true:false
test :: s:0' → s:0' → true:false
if1 :: true:false → s:0' → s:0' → true:false
if2 :: true:false → s:0' → s:0' → true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'3_0(+(1, 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:
div, test

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

Proved the following rewrite lemma:
div(gen_s:0'3_0(n300_0), gen_s:0'3_0(n300_0), gen_s:0'3_0(c)) → true, rt ∈ Ω(1 + n3000)

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

Induction Step:
div(gen_s:0'3_0(+(n300_0, 1)), gen_s:0'3_0(+(n300_0, 1)), gen_s:0'3_0(c)) →RΩ(1)
div(gen_s:0'3_0(n300_0), gen_s:0'3_0(n300_0), gen_s:0'3_0(c)) →IH
true

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

### (14) Obligation:

TRS:
Rules:
gt(s(x), 0') → true
gt(0', y) → false
gt(s(x), s(y)) → gt(x, y)
divides(x, y) → div(x, y, y)
div(0', 0', z) → true
div(0', s(x), z) → false
div(s(x), 0', s(z)) → div(s(x), s(z), s(z))
div(s(x), s(y), z) → div(x, y, z)
prime(x) → test(x, s(s(0')))
test(x, y) → if1(gt(x, y), x, y)
if1(true, x, y) → if2(divides(x, y), x, y)
if1(false, x, y) → true
if2(true, x, y) → false
if2(false, x, y) → test(x, s(y))

Types:
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0' → true:false
prime :: s:0' → true:false
test :: s:0' → s:0' → true:false
if1 :: true:false → s:0' → s:0' → true:false
if2 :: true:false → s:0' → s:0' → true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'3_0(+(1, n5_0)), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
div(gen_s:0'3_0(n300_0), gen_s:0'3_0(n300_0), gen_s:0'3_0(c)) → true, rt ∈ Ω(1 + n3000)

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:
test

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

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

### (16) Obligation:

TRS:
Rules:
gt(s(x), 0') → true
gt(0', y) → false
gt(s(x), s(y)) → gt(x, y)
divides(x, y) → div(x, y, y)
div(0', 0', z) → true
div(0', s(x), z) → false
div(s(x), 0', s(z)) → div(s(x), s(z), s(z))
div(s(x), s(y), z) → div(x, y, z)
prime(x) → test(x, s(s(0')))
test(x, y) → if1(gt(x, y), x, y)
if1(true, x, y) → if2(divides(x, y), x, y)
if1(false, x, y) → true
if2(true, x, y) → false
if2(false, x, y) → test(x, s(y))

Types:
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0' → true:false
prime :: s:0' → true:false
test :: s:0' → s:0' → true:false
if1 :: true:false → s:0' → s:0' → true:false
if2 :: true:false → s:0' → s:0' → true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'3_0(+(1, n5_0)), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
div(gen_s:0'3_0(n300_0), gen_s:0'3_0(n300_0), gen_s:0'3_0(c)) → true, rt ∈ Ω(1 + n3000)

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

### (19) Obligation:

TRS:
Rules:
gt(s(x), 0') → true
gt(0', y) → false
gt(s(x), s(y)) → gt(x, y)
divides(x, y) → div(x, y, y)
div(0', 0', z) → true
div(0', s(x), z) → false
div(s(x), 0', s(z)) → div(s(x), s(z), s(z))
div(s(x), s(y), z) → div(x, y, z)
prime(x) → test(x, s(s(0')))
test(x, y) → if1(gt(x, y), x, y)
if1(true, x, y) → if2(divides(x, y), x, y)
if1(false, x, y) → true
if2(true, x, y) → false
if2(false, x, y) → test(x, s(y))

Types:
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0' → true:false
prime :: s:0' → true:false
test :: s:0' → s:0' → true:false
if1 :: true:false → s:0' → s:0' → true:false
if2 :: true:false → s:0' → s:0' → true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'3_0(+(1, n5_0)), gen_s:0'3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
div(gen_s:0'3_0(n300_0), gen_s:0'3_0(n300_0), gen_s:0'3_0(c)) → true, rt ∈ Ω(1 + n3000)

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.

### (20) LowerBoundsProof (EQUIVALENT transformation)

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

### (22) Obligation:

TRS:
Rules:
gt(s(x), 0') → true
gt(0', y) → false
gt(s(x), s(y)) → gt(x, y)
divides(x, y) → div(x, y, y)
div(0', 0', z) → true
div(0', s(x), z) → false
div(s(x), 0', s(z)) → div(s(x), s(z), s(z))
div(s(x), s(y), z) → div(x, y, z)
prime(x) → test(x, s(s(0')))
test(x, y) → if1(gt(x, y), x, y)
if1(true, x, y) → if2(divides(x, y), x, y)
if1(false, x, y) → true
if2(true, x, y) → false
if2(false, x, y) → test(x, s(y))

Types:
gt :: s:0' → s:0' → true:false
s :: s:0' → s:0'
0' :: s:0'
true :: true:false
false :: true:false
divides :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0' → true:false
prime :: s:0' → true:false
test :: s:0' → s:0' → true:false
if1 :: true:false → s:0' → s:0' → true:false
if2 :: true:false → s:0' → s:0' → true:false
hole_true:false1_0 :: true:false
hole_s:0'2_0 :: s:0'
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'3_0(+(1, 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.

### (23) LowerBoundsProof (EQUIVALENT transformation)

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