### (0) Obligation:

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

p(s(x)) → x
fact(0) → s(0)
fact(s(x)) → *(s(x), fact(p(s(x))))
*(0, y) → 0
*(s(x), y) → +(*(x, y), y)
+(x, 0) → x
+(x, s(y)) → s(+(x, y))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
fact(s(x)) →+ +(*(x, fact(x)), fact(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
The pumping substitution is [x / s(x)].
The result substitution is [ ].

The rewrite sequence
fact(s(x)) →+ +(*(x, fact(x)), fact(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
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:

p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
fact, *', +'

They will be analysed ascendingly in the following order:
*' < fact
+' < *'

### (8) Obligation:

TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
+', fact, *'

They will be analysed ascendingly in the following order:
*' < fact
+' < *'

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

Proved the following rewrite lemma:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)

Induction Base:
+'(gen_s:0'2_0(a), gen_s:0'2_0(0)) →RΩ(1)
gen_s:0'2_0(a)

Induction Step:
+'(gen_s:0'2_0(a), gen_s:0'2_0(+(n4_0, 1))) →RΩ(1)
s(+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0))) →IH
s(gen_s:0'2_0(+(a, c5_0)))

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

### (11) Obligation:

TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
*', fact

They will be analysed ascendingly in the following order:
*' < fact

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

Proved the following rewrite lemma:
*'(gen_s:0'2_0(n457_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n457_0, b)), rt ∈ Ω(1 + b·n4570 + n4570)

Induction Base:
*'(gen_s:0'2_0(0), gen_s:0'2_0(b)) →RΩ(1)
0'

Induction Step:
*'(gen_s:0'2_0(+(n457_0, 1)), gen_s:0'2_0(b)) →RΩ(1)
+'(*'(gen_s:0'2_0(n457_0), gen_s:0'2_0(b)), gen_s:0'2_0(b)) →IH
+'(gen_s:0'2_0(*(c458_0, b)), gen_s:0'2_0(b)) →LΩ(1 + b)
gen_s:0'2_0(+(b, *(n457_0, b)))

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

### (14) Obligation:

TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
*'(gen_s:0'2_0(n457_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n457_0, b)), rt ∈ Ω(1 + b·n4570 + n4570)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
fact

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

Proved the following rewrite lemma:
fact(gen_s:0'2_0(n1007_0)) → *3_0, rt ∈ Ω(n10070)

Induction Base:
fact(gen_s:0'2_0(0))

Induction Step:
fact(gen_s:0'2_0(+(n1007_0, 1))) →RΩ(1)
*'(s(gen_s:0'2_0(n1007_0)), fact(p(s(gen_s:0'2_0(n1007_0))))) →RΩ(1)
*'(s(gen_s:0'2_0(n1007_0)), fact(gen_s:0'2_0(n1007_0))) →IH
*'(s(gen_s:0'2_0(n1007_0)), *3_0)

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

### (17) Obligation:

TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
*'(gen_s:0'2_0(n457_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n457_0, b)), rt ∈ Ω(1 + b·n4570 + n4570)
fact(gen_s:0'2_0(n1007_0)) → *3_0, rt ∈ Ω(n10070)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
*'(gen_s:0'2_0(n457_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n457_0, b)), rt ∈ Ω(1 + b·n4570 + n4570)

### (20) Obligation:

TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
*'(gen_s:0'2_0(n457_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n457_0, b)), rt ∈ Ω(1 + b·n4570 + n4570)
fact(gen_s:0'2_0(n1007_0)) → *3_0, rt ∈ Ω(n10070)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
*'(gen_s:0'2_0(n457_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n457_0, b)), rt ∈ Ω(1 + b·n4570 + n4570)

### (23) Obligation:

TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)
*'(gen_s:0'2_0(n457_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n457_0, b)), rt ∈ Ω(1 + b·n4570 + n4570)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
*'(gen_s:0'2_0(n457_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n457_0, b)), rt ∈ Ω(1 + b·n4570 + n4570)

### (26) Obligation:

TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n40)