### (0) Obligation:

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

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0), s(0))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 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:
lt, times, plus, loop

They will be analysed ascendingly in the following order:
lt < loop
plus < times
times < loop

### (8) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 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:
lt, times, plus, loop

They will be analysed ascendingly in the following order:
lt < loop
plus < times
times < loop

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

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

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

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

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

### (11) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 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:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, 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:
plus, times, loop

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

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

Proved the following rewrite lemma:
plus(gen_0':s3_0(n282_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n282_0, b)), rt ∈ Ω(1 + n2820)

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(+(n282_0, 1)), gen_0':s3_0(b)) →RΩ(1)
s(plus(gen_0':s3_0(n282_0), gen_0':s3_0(b))) →IH
s(gen_0':s3_0(+(b, c283_0)))

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

### (14) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 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:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)
plus(gen_0':s3_0(n282_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n282_0, b)), rt ∈ Ω(1 + n2820)

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:
times, loop

They will be analysed ascendingly in the following order:
times < loop

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

Proved the following rewrite lemma:
times(gen_0':s3_0(n825_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n825_0, b)), rt ∈ Ω(1 + b·n8250 + n8250)

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

Induction Step:
times(gen_0':s3_0(+(n825_0, 1)), gen_0':s3_0(b)) →RΩ(1)
plus(gen_0':s3_0(b), times(gen_0':s3_0(n825_0), gen_0':s3_0(b))) →IH
plus(gen_0':s3_0(b), gen_0':s3_0(*(c826_0, b))) →LΩ(1 + b)
gen_0':s3_0(+(b, *(n825_0, b)))

We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).

### (17) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 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:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)
plus(gen_0':s3_0(n282_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n282_0, b)), rt ∈ Ω(1 + n2820)
times(gen_0':s3_0(n825_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n825_0, b)), rt ∈ Ω(1 + b·n8250 + n8250)

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:
loop

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

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

### (19) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 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:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)
plus(gen_0':s3_0(n282_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n282_0, b)), rt ∈ Ω(1 + n2820)
times(gen_0':s3_0(n825_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n825_0, b)), rt ∈ Ω(1 + b·n8250 + n8250)

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(n2) was proven with the following lemma:
times(gen_0':s3_0(n825_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n825_0, b)), rt ∈ Ω(1 + b·n8250 + n8250)

### (22) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 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:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)
plus(gen_0':s3_0(n282_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n282_0, b)), rt ∈ Ω(1 + n2820)
times(gen_0':s3_0(n825_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n825_0, b)), rt ∈ Ω(1 + b·n8250 + n8250)

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(n2) was proven with the following lemma:
times(gen_0':s3_0(n825_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n825_0, b)), rt ∈ Ω(1 + b·n8250 + n8250)

### (25) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 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:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n50)
plus(gen_0':s3_0(n282_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n282_0, b)), rt ∈ Ω(1 + n2820)

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.

### (26) LowerBoundsProof (EQUIVALENT transformation)

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

### (28) Obligation:

TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 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:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, 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.

### (29) LowerBoundsProof (EQUIVALENT transformation)

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