### (0) Obligation:

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

lcm(x, y) → lcmIter(x, y, 0, times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0, x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0, x), x, y)
ifTimes(true, x, y) → 0
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0) → s(s(0))
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0, s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0) → divisible(x, y)
div(0, y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
plus(s(x), y) →+ s(plus(x, y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x / s(x)].
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:

lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Types:
lcm :: 0':s → 0':s → 0':s
lcmIter :: 0':s → 0':s → 0':s → 0':s → 0':s
0' :: 0':s
times :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
or :: true:false → true:false → true:false
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if2 :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
divisible :: 0':s → 0':s → true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
ifTimes :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s → true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
lcmIter, times, ge, divisible, plus, div

They will be analysed ascendingly in the following order:
ge < lcmIter
divisible < lcmIter
plus < lcmIter
ge < times
plus < times
divisible = div

### (8) Obligation:

TRS:
Rules:
lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Types:
lcm :: 0':s → 0':s → 0':s
lcmIter :: 0':s → 0':s → 0':s → 0':s → 0':s
0' :: 0':s
times :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
or :: true:false → true:false → true:false
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if2 :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
divisible :: 0':s → 0':s → true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
ifTimes :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s → true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
ge, lcmIter, times, divisible, plus, div

They will be analysed ascendingly in the following order:
ge < lcmIter
divisible < lcmIter
plus < lcmIter
ge < times
plus < times
divisible = div

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

Proved the following rewrite lemma:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)

Induction Base:
ge(gen_0':s4_0(0), gen_0':s4_0(0)) →RΩ(1)
true

Induction Step:
ge(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) →RΩ(1)
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) →IH
true

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

### (11) Obligation:

TRS:
Rules:
lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Types:
lcm :: 0':s → 0':s → 0':s
lcmIter :: 0':s → 0':s → 0':s → 0':s → 0':s
0' :: 0':s
times :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
or :: true:false → true:false → true:false
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if2 :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
divisible :: 0':s → 0':s → true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
ifTimes :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s → true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat → 0':s

Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
plus, lcmIter, times, divisible, div

They will be analysed ascendingly in the following order:
divisible < lcmIter
plus < lcmIter
plus < times
divisible = div

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

Proved the following rewrite lemma:
plus(gen_0':s4_0(n367_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n367_0, b)), rt ∈ Ω(1 + n3670)

Induction Base:
plus(gen_0':s4_0(0), gen_0':s4_0(b)) →RΩ(1)
gen_0':s4_0(b)

Induction Step:
plus(gen_0':s4_0(+(n367_0, 1)), gen_0':s4_0(b)) →RΩ(1)
s(plus(gen_0':s4_0(n367_0), gen_0':s4_0(b))) →IH
s(gen_0':s4_0(+(b, c368_0)))

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

### (14) Obligation:

TRS:
Rules:
lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Types:
lcm :: 0':s → 0':s → 0':s
lcmIter :: 0':s → 0':s → 0':s → 0':s → 0':s
0' :: 0':s
times :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
or :: true:false → true:false → true:false
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if2 :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
divisible :: 0':s → 0':s → true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
ifTimes :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s → true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat → 0':s

Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(n367_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n367_0, b)), rt ∈ Ω(1 + n3670)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
times, lcmIter, divisible, div

They will be analysed ascendingly in the following order:
divisible < lcmIter
divisible = div

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

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

### (16) Obligation:

TRS:
Rules:
lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Types:
lcm :: 0':s → 0':s → 0':s
lcmIter :: 0':s → 0':s → 0':s → 0':s → 0':s
0' :: 0':s
times :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
or :: true:false → true:false → true:false
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if2 :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
divisible :: 0':s → 0':s → true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
ifTimes :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s → true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat → 0':s

Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(n367_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n367_0, b)), rt ∈ Ω(1 + n3670)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
div, lcmIter, divisible

They will be analysed ascendingly in the following order:
divisible < lcmIter
divisible = div

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

Proved the following rewrite lemma:
div(gen_0':s4_0(n1228_0), gen_0':s4_0(b), gen_0':s4_0(+(1, n1228_0))) → false, rt ∈ Ω(1 + n12280)

Induction Base:
div(gen_0':s4_0(0), gen_0':s4_0(b), gen_0':s4_0(+(1, 0))) →RΩ(1)
false

Induction Step:
div(gen_0':s4_0(+(n1228_0, 1)), gen_0':s4_0(b), gen_0':s4_0(+(1, +(n1228_0, 1)))) →RΩ(1)
div(gen_0':s4_0(n1228_0), gen_0':s4_0(b), gen_0':s4_0(+(1, n1228_0))) →IH
false

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

### (19) Obligation:

TRS:
Rules:
lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Types:
lcm :: 0':s → 0':s → 0':s
lcmIter :: 0':s → 0':s → 0':s → 0':s → 0':s
0' :: 0':s
times :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
or :: true:false → true:false → true:false
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if2 :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
divisible :: 0':s → 0':s → true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
ifTimes :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s → true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat → 0':s

Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(n367_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n367_0, b)), rt ∈ Ω(1 + n3670)
div(gen_0':s4_0(n1228_0), gen_0':s4_0(b), gen_0':s4_0(+(1, n1228_0))) → false, rt ∈ Ω(1 + n12280)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
divisible, lcmIter

They will be analysed ascendingly in the following order:
divisible < lcmIter
divisible = div

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

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

### (21) Obligation:

TRS:
Rules:
lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Types:
lcm :: 0':s → 0':s → 0':s
lcmIter :: 0':s → 0':s → 0':s → 0':s → 0':s
0' :: 0':s
times :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
or :: true:false → true:false → true:false
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if2 :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
divisible :: 0':s → 0':s → true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
ifTimes :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s → true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat → 0':s

Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(n367_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n367_0, b)), rt ∈ Ω(1 + n3670)
div(gen_0':s4_0(n1228_0), gen_0':s4_0(b), gen_0':s4_0(+(1, n1228_0))) → false, rt ∈ Ω(1 + n12280)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
lcmIter

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

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

### (23) Obligation:

TRS:
Rules:
lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Types:
lcm :: 0':s → 0':s → 0':s
lcmIter :: 0':s → 0':s → 0':s → 0':s → 0':s
0' :: 0':s
times :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
or :: true:false → true:false → true:false
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if2 :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
divisible :: 0':s → 0':s → true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
ifTimes :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s → true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat → 0':s

Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(n367_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n367_0, b)), rt ∈ Ω(1 + n3670)
div(gen_0':s4_0(n1228_0), gen_0':s4_0(b), gen_0':s4_0(+(1, n1228_0))) → false, rt ∈ Ω(1 + n12280)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)

### (26) Obligation:

TRS:
Rules:
lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Types:
lcm :: 0':s → 0':s → 0':s
lcmIter :: 0':s → 0':s → 0':s → 0':s → 0':s
0' :: 0':s
times :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
or :: true:false → true:false → true:false
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if2 :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
divisible :: 0':s → 0':s → true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
ifTimes :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s → true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat → 0':s

Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(n367_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n367_0, b)), rt ∈ Ω(1 + n3670)
div(gen_0':s4_0(n1228_0), gen_0':s4_0(b), gen_0':s4_0(+(1, n1228_0))) → false, rt ∈ Ω(1 + n12280)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)

### (29) Obligation:

TRS:
Rules:
lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Types:
lcm :: 0':s → 0':s → 0':s
lcmIter :: 0':s → 0':s → 0':s → 0':s → 0':s
0' :: 0':s
times :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
or :: true:false → true:false → true:false
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if2 :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
divisible :: 0':s → 0':s → true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
ifTimes :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s → true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat → 0':s

Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)
plus(gen_0':s4_0(n367_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n367_0, b)), rt ∈ Ω(1 + n3670)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)

### (32) Obligation:

TRS:
Rules:
lcm(x, y) → lcmIter(x, y, 0', times(x, y))
lcmIter(x, y, z, u) → if(or(ge(0', x), ge(z, u)), x, y, z, u)
if(true, x, y, z, u) → z
if(false, x, y, z, u) → if2(divisible(z, y), x, y, z, u)
if2(true, x, y, z, u) → z
if2(false, x, y, z, u) → lcmIter(x, y, plus(x, z), u)
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → ifTimes(ge(0', x), x, y)
ifTimes(true, x, y) → 0'
ifTimes(false, x, y) → plus(y, times(y, p(x)))
p(s(x)) → x
p(0') → s(s(0'))
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
or(true, y) → true
or(false, y) → y
divisible(0', s(y)) → true
divisible(s(x), s(y)) → div(s(x), s(y), s(y))
div(x, y, 0') → divisible(x, y)
div(0', y, s(z)) → false
div(s(x), y, s(z)) → div(x, y, z)
ab
ac

Types:
lcm :: 0':s → 0':s → 0':s
lcmIter :: 0':s → 0':s → 0':s → 0':s → 0':s
0' :: 0':s
times :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
or :: true:false → true:false → true:false
ge :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
if2 :: true:false → 0':s → 0':s → 0':s → 0':s → 0':s
divisible :: 0':s → 0':s → true:false
plus :: 0':s → 0':s → 0':s
s :: 0':s → 0':s
ifTimes :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
div :: 0':s → 0':s → 0':s → true:false
a :: b:c
b :: b:c
c :: b:c
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_b:c3_0 :: b:c
gen_0':s4_0 :: Nat → 0':s

Lemmas:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
ge(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) → true, rt ∈ Ω(1 + n60)