### (0) Obligation:

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

plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
times(0, y) → 0
times(x, 0) → 0
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0)) → 0
fac(s(x)) → times(fac(p(s(x))), 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
plus(x, s(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 [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:

plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
plus, times, p, fac

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

### (8) Obligation:

TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
plus, times, p, fac

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

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

Proved the following rewrite lemma:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)

Induction Base:
plus(gen_0':s2_0(a), gen_0':s2_0(0)) →RΩ(1)
gen_0':s2_0(a)

Induction Step:
plus(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) →RΩ(1)
s(plus(gen_0':s2_0(a), gen_0':s2_0(n4_0))) →IH
s(gen_0':s2_0(+(a, c5_0)))

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

### (11) Obligation:

TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
times, p, fac

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

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

Proved the following rewrite lemma:
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n4750 + n4750)

Induction Base:
times(gen_0':s2_0(0), gen_0':s2_0(b)) →RΩ(1)
0'

Induction Step:
times(gen_0':s2_0(+(n475_0, 1)), gen_0':s2_0(b)) →RΩ(1)
plus(times(gen_0':s2_0(n475_0), gen_0':s2_0(b)), gen_0':s2_0(b)) →IH
plus(gen_0':s2_0(*(c476_0, b)), gen_0':s2_0(b)) →LΩ(1 + b)
gen_0':s2_0(+(b, *(n475_0, b)))

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

### (14) Obligation:

TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n4750 + n4750)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
p, fac

They will be analysed ascendingly in the following order:
p < fac

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

Proved the following rewrite lemma:
p(gen_0':s2_0(+(1, n1092_0))) → gen_0':s2_0(n1092_0), rt ∈ Ω(1 + n10920)

Induction Base:
p(gen_0':s2_0(+(1, 0))) →RΩ(1)
0'

Induction Step:
p(gen_0':s2_0(+(1, +(n1092_0, 1)))) →RΩ(1)
s(p(s(gen_0':s2_0(n1092_0)))) →IH
s(gen_0':s2_0(c1093_0))

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

### (17) Obligation:

TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n4750 + n4750)
p(gen_0':s2_0(+(1, n1092_0))) → gen_0':s2_0(n1092_0), rt ∈ Ω(1 + n10920)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
fac

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

Proved the following rewrite lemma:
fac(gen_0':s2_0(+(1, n1276_0))) → *3_0, rt ∈ Ω(n12760 + n127602)

Induction Base:
fac(gen_0':s2_0(+(1, 0)))

Induction Step:
fac(gen_0':s2_0(+(1, +(n1276_0, 1)))) →RΩ(1)
times(fac(p(s(gen_0':s2_0(+(1, n1276_0))))), s(gen_0':s2_0(+(1, n1276_0)))) →LΩ(2 + n12760)
times(fac(gen_0':s2_0(+(1, n1276_0))), s(gen_0':s2_0(+(1, n1276_0)))) →IH
times(*3_0, s(gen_0':s2_0(+(1, n1276_0))))

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

### (20) Obligation:

TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n4750 + n4750)
p(gen_0':s2_0(+(1, n1092_0))) → gen_0':s2_0(n1092_0), rt ∈ Ω(1 + n10920)
fac(gen_0':s2_0(+(1, n1276_0))) → *3_0, rt ∈ Ω(n12760 + n127602)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n4750 + n4750)

### (23) Obligation:

TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n4750 + n4750)
p(gen_0':s2_0(+(1, n1092_0))) → gen_0':s2_0(n1092_0), rt ∈ Ω(1 + n10920)
fac(gen_0':s2_0(+(1, n1276_0))) → *3_0, rt ∈ Ω(n12760 + n127602)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n4750 + n4750)

### (26) Obligation:

TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n4750 + n4750)
p(gen_0':s2_0(+(1, n1092_0))) → gen_0':s2_0(n1092_0), rt ∈ Ω(1 + n10920)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n4750 + n4750)

### (29) Obligation:

TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n4750 + n4750)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

### (30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n4750 + n4750)

### (32) Obligation:

TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

### (33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)