The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

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

fact(X) → if(zero(X), s(0), prod(X, fact(p(X))))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
prod(0, X) → 0
prod(s(X), Y) → add(Y, prod(X, Y))
if(true, X, Y) → X
if(false, X, Y) → Y
zero(0) → true
zero(s(X)) → false
p(s(X)) → X

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

fact(X) → if(zero(X), s(0'), prod(X, fact(p(X))))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
prod(0', X) → 0'
prod(s(X), Y) → add(Y, prod(X, Y))
if(true, X, Y) → X
if(false, X, Y) → Y
zero(0') → true
zero(s(X)) → false
p(s(X)) → X

S is empty.
Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
fact(X) → if(zero(X), s(0'), prod(X, fact(p(X))))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
prod(0', X) → 0'
prod(s(X), Y) → add(Y, prod(X, Y))
if(true, X, Y) → X
if(false, X, Y) → Y
zero(0') → true
zero(s(X)) → false
p(s(X)) → X

Types:
fact :: 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
zero :: 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
prod :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
fact, prod, add

They will be analysed ascendingly in the following order:
prod < fact
add < prod

(6) Obligation:

Innermost TRS:
Rules:
fact(X) → if(zero(X), s(0'), prod(X, fact(p(X))))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
prod(0', X) → 0'
prod(s(X), Y) → add(Y, prod(X, Y))
if(true, X, Y) → X
if(false, X, Y) → Y
zero(0') → true
zero(s(X)) → false
p(s(X)) → X

Types:
fact :: 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
zero :: 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
prod :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
add, fact, prod

They will be analysed ascendingly in the following order:
prod < fact
add < prod

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

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

Induction Base:
add(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
gen_0':s3_0(b)

Induction Step:
add(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
s(add(gen_0':s3_0(n5_0), gen_0':s3_0(b))) →_IH
s(gen_0':s3_0(+(b, c6_0)))

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
fact(X) → if(zero(X), s(0'), prod(X, fact(p(X))))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
prod(0', X) → 0'
prod(s(X), Y) → add(Y, prod(X, Y))
if(true, X, Y) → X
if(false, X, Y) → Y
zero(0') → true
zero(s(X)) → false
p(s(X)) → X

Types:
fact :: 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
zero :: 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
prod :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
add(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), 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:
prod, fact

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

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
prod(gen_0':s3_0(n592_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n592_0, b)), rt ∈ Ω(1 + b·n592₀ + n592₀)

Induction Base:
prod(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
0'

Induction Step:
prod(gen_0':s3_0(+(n592_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
add(gen_0':s3_0(b), prod(gen_0':s3_0(n592_0), gen_0':s3_0(b))) →_IH
add(gen_0':s3_0(b), gen_0':s3_0(*(c593_0, b))) →_L^{Ω(1 + b)}
gen_0':s3_0(+(b, *(n592_0, b)))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
fact(X) → if(zero(X), s(0'), prod(X, fact(p(X))))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
prod(0', X) → 0'
prod(s(X), Y) → add(Y, prod(X, Y))
if(true, X, Y) → X
if(false, X, Y) → Y
zero(0') → true
zero(s(X)) → false
p(s(X)) → X

Types:
fact :: 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
zero :: 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
prod :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
add(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
prod(gen_0':s3_0(n592_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n592_0, b)), rt ∈ Ω(1 + b·n592₀ + n592₀)

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:
fact

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

Innermost TRS:
Rules:
fact(X) → if(zero(X), s(0'), prod(X, fact(p(X))))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
prod(0', X) → 0'
prod(s(X), Y) → add(Y, prod(X, Y))
if(true, X, Y) → X
if(false, X, Y) → Y
zero(0') → true
zero(s(X)) → false
p(s(X)) → X

Types:
fact :: 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
zero :: 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
prod :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
add(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
prod(gen_0':s3_0(n592_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n592_0, b)), rt ∈ Ω(1 + b·n592₀ + n592₀)

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.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
prod(gen_0':s3_0(n592_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n592_0, b)), rt ∈ Ω(1 + b·n592₀ + n592₀)

(16) BOUNDS(n^2, INF)

(17) Obligation:

Innermost TRS:
Rules:
fact(X) → if(zero(X), s(0'), prod(X, fact(p(X))))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
prod(0', X) → 0'
prod(s(X), Y) → add(Y, prod(X, Y))
if(true, X, Y) → X
if(false, X, Y) → Y
zero(0') → true
zero(s(X)) → false
p(s(X)) → X

Types:
fact :: 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
zero :: 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
prod :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
add(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
prod(gen_0':s3_0(n592_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n592_0, b)), rt ∈ Ω(1 + b·n592₀ + n592₀)

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.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
prod(gen_0':s3_0(n592_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n592_0, b)), rt ∈ Ω(1 + b·n592₀ + n592₀)

(19) BOUNDS(n^2, INF)

(20) Obligation:

Innermost TRS:
Rules:
fact(X) → if(zero(X), s(0'), prod(X, fact(p(X))))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
prod(0', X) → 0'
prod(s(X), Y) → add(Y, prod(X, Y))
if(true, X, Y) → X
if(false, X, Y) → Y
zero(0') → true
zero(s(X)) → false
p(s(X)) → X

Types:
fact :: 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s
zero :: 0':s → true:false
s :: 0':s → 0':s
0' :: 0':s
prod :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
add :: 0':s → 0':s → 0':s
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
add(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), 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.

(21) LowerBoundsProof (EQUIVALENT transformation)

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

(22) BOUNDS(n^1, INF)