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:

empty(nil) → true
empty(cons(x, y)) → false
tail(nil) → nil
tail(cons(x, y)) → y
head(cons(x, y)) → x
zero(0) → true
zero(s(x)) → false
p(0) → 0
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
intlist(x) → if_intlist(empty(x), x)
if_intlist(true, x) → nil
if_intlist(false, x) → cons(s(head(x)), intlist(tail(x)))
int(x, y) → if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) → if1(b, x, y)
if_int(false, b, x, y) → if2(b, x, y)
if1(true, x, y) → cons(0, nil)
if1(false, x, y) → cons(0, int(s(0), y))
if2(true, x, y) → nil
if2(false, x, y) → intlist(int(p(x), p(y)))

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:

empty(nil) → true
empty(cons(x, y)) → false
tail(nil) → nil
tail(cons(x, y)) → y
head(cons(x, y)) → x
zero(0') → true
zero(s(x)) → false
p(0') → 0'
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
intlist(x) → if_intlist(empty(x), x)
if_intlist(true, x) → nil
if_intlist(false, x) → cons(s(head(x)), intlist(tail(x)))
int(x, y) → if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) → if1(b, x, y)
if_int(false, b, x, y) → if2(b, x, y)
if1(true, x, y) → cons(0', nil)
if1(false, x, y) → cons(0', int(s(0'), y))
if2(true, x, y) → nil
if2(false, x, y) → intlist(int(p(x), p(y)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
empty(nil) → true
empty(cons(x, y)) → false
tail(nil) → nil
tail(cons(x, y)) → y
head(cons(x, y)) → x
zero(0') → true
zero(s(x)) → false
p(0') → 0'
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
intlist(x) → if_intlist(empty(x), x)
if_intlist(true, x) → nil
if_intlist(false, x) → cons(s(head(x)), intlist(tail(x)))
int(x, y) → if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) → if1(b, x, y)
if_int(false, b, x, y) → if2(b, x, y)
if1(true, x, y) → cons(0', nil)
if1(false, x, y) → cons(0', int(s(0'), y))
if2(true, x, y) → nil
if2(false, x, y) → intlist(int(p(x), p(y)))

Types:
empty :: nil:cons → true:false
nil :: nil:cons
true :: true:false
cons :: 0':s → nil:cons → nil:cons
false :: true:false
tail :: nil:cons → nil:cons
head :: nil:cons → 0':s
zero :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
intlist :: nil:cons → nil:cons
if_intlist :: true:false → nil:cons → nil:cons
int :: 0':s → 0':s → nil:cons
if_int :: true:false → true:false → 0':s → 0':s → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat → nil:cons
gen_0':s5_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
p, intlist, int

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

(6) Obligation:

Innermost TRS:
Rules:
empty(nil) → true
empty(cons(x, y)) → false
tail(nil) → nil
tail(cons(x, y)) → y
head(cons(x, y)) → x
zero(0') → true
zero(s(x)) → false
p(0') → 0'
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
intlist(x) → if_intlist(empty(x), x)
if_intlist(true, x) → nil
if_intlist(false, x) → cons(s(head(x)), intlist(tail(x)))
int(x, y) → if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) → if1(b, x, y)
if_int(false, b, x, y) → if2(b, x, y)
if1(true, x, y) → cons(0', nil)
if1(false, x, y) → cons(0', int(s(0'), y))
if2(true, x, y) → nil
if2(false, x, y) → intlist(int(p(x), p(y)))

Types:
empty :: nil:cons → true:false
nil :: nil:cons
true :: true:false
cons :: 0':s → nil:cons → nil:cons
false :: true:false
tail :: nil:cons → nil:cons
head :: nil:cons → 0':s
zero :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
intlist :: nil:cons → nil:cons
if_intlist :: true:false → nil:cons → nil:cons
int :: 0':s → 0':s → nil:cons
if_int :: true:false → true:false → 0':s → 0':s → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat → nil:cons
gen_0':s5_0 :: Nat → 0':s

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))

The following defined symbols remain to be analysed:
p, intlist, int

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

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

Proved the following rewrite lemma:
p(gen_0':s5_0(+(1, n7_0))) → gen_0':s5_0(n7_0), rt ∈ Ω(1 + n7₀)

Induction Base:
p(gen_0':s5_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
p(gen_0':s5_0(+(1, +(n7_0, 1)))) →_R^Ω(1)
s(p(s(gen_0':s5_0(n7_0)))) →_IH
s(gen_0':s5_0(c8_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:
empty(nil) → true
empty(cons(x, y)) → false
tail(nil) → nil
tail(cons(x, y)) → y
head(cons(x, y)) → x
zero(0') → true
zero(s(x)) → false
p(0') → 0'
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
intlist(x) → if_intlist(empty(x), x)
if_intlist(true, x) → nil
if_intlist(false, x) → cons(s(head(x)), intlist(tail(x)))
int(x, y) → if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) → if1(b, x, y)
if_int(false, b, x, y) → if2(b, x, y)
if1(true, x, y) → cons(0', nil)
if1(false, x, y) → cons(0', int(s(0'), y))
if2(true, x, y) → nil
if2(false, x, y) → intlist(int(p(x), p(y)))

Types:
empty :: nil:cons → true:false
nil :: nil:cons
true :: true:false
cons :: 0':s → nil:cons → nil:cons
false :: true:false
tail :: nil:cons → nil:cons
head :: nil:cons → 0':s
zero :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
intlist :: nil:cons → nil:cons
if_intlist :: true:false → nil:cons → nil:cons
int :: 0':s → 0':s → nil:cons
if_int :: true:false → true:false → 0':s → 0':s → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat → nil:cons
gen_0':s5_0 :: Nat → 0':s

Lemmas:
p(gen_0':s5_0(+(1, n7_0))) → gen_0':s5_0(n7_0), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))

The following defined symbols remain to be analysed:
intlist, int

They will be analysed ascendingly in the following order:
intlist < int

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
empty(nil) → true
empty(cons(x, y)) → false
tail(nil) → nil
tail(cons(x, y)) → y
head(cons(x, y)) → x
zero(0') → true
zero(s(x)) → false
p(0') → 0'
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
intlist(x) → if_intlist(empty(x), x)
if_intlist(true, x) → nil
if_intlist(false, x) → cons(s(head(x)), intlist(tail(x)))
int(x, y) → if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) → if1(b, x, y)
if_int(false, b, x, y) → if2(b, x, y)
if1(true, x, y) → cons(0', nil)
if1(false, x, y) → cons(0', int(s(0'), y))
if2(true, x, y) → nil
if2(false, x, y) → intlist(int(p(x), p(y)))

Types:
empty :: nil:cons → true:false
nil :: nil:cons
true :: true:false
cons :: 0':s → nil:cons → nil:cons
false :: true:false
tail :: nil:cons → nil:cons
head :: nil:cons → 0':s
zero :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
intlist :: nil:cons → nil:cons
if_intlist :: true:false → nil:cons → nil:cons
int :: 0':s → 0':s → nil:cons
if_int :: true:false → true:false → 0':s → 0':s → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat → nil:cons
gen_0':s5_0 :: Nat → 0':s

Lemmas:
p(gen_0':s5_0(+(1, n7_0))) → gen_0':s5_0(n7_0), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))

The following defined symbols remain to be analysed:
int

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

Proved the following rewrite lemma:
int(gen_0':s5_0(+(1, n37645_0)), gen_0':s5_0(n37645_0)) → gen_nil:cons4_0(0), rt ∈ Ω(1 + n37645₀ + n37645₀²)

Induction Base:
int(gen_0':s5_0(+(1, 0)), gen_0':s5_0(0)) →_R^Ω(1)
if_int(zero(gen_0':s5_0(+(1, 0))), zero(gen_0':s5_0(0)), gen_0':s5_0(+(1, 0)), gen_0':s5_0(0)) →_R^Ω(1)
if_int(false, zero(gen_0':s5_0(0)), gen_0':s5_0(1), gen_0':s5_0(0)) →_R^Ω(1)
if_int(false, true, gen_0':s5_0(1), gen_0':s5_0(0)) →_R^Ω(1)
if2(true, gen_0':s5_0(1), gen_0':s5_0(0)) →_R^Ω(1)
nil

Induction Step:
int(gen_0':s5_0(+(1, +(n37645_0, 1))), gen_0':s5_0(+(n37645_0, 1))) →_R^Ω(1)
if_int(zero(gen_0':s5_0(+(1, +(n37645_0, 1)))), zero(gen_0':s5_0(+(n37645_0, 1))), gen_0':s5_0(+(1, +(n37645_0, 1))), gen_0':s5_0(+(n37645_0, 1))) →_R^Ω(1)
if_int(false, zero(gen_0':s5_0(+(1, n37645_0))), gen_0':s5_0(+(2, n37645_0)), gen_0':s5_0(+(1, n37645_0))) →_R^Ω(1)
if_int(false, false, gen_0':s5_0(+(2, n37645_0)), gen_0':s5_0(+(1, n37645_0))) →_R^Ω(1)
if2(false, gen_0':s5_0(+(2, n37645_0)), gen_0':s5_0(+(1, n37645_0))) →_R^Ω(1)
intlist(int(p(gen_0':s5_0(+(2, n37645_0))), p(gen_0':s5_0(+(1, n37645_0))))) →_L^{Ω(2 + n37645₀)}
intlist(int(gen_0':s5_0(+(1, n37645_0)), p(gen_0':s5_0(+(1, n37645_0))))) →_L^{Ω(1 + n37645₀)}
intlist(int(gen_0':s5_0(+(1, n37645_0)), gen_0':s5_0(n37645_0))) →_IH
intlist(gen_nil:cons4_0(0)) →_R^Ω(1)
if_intlist(empty(gen_nil:cons4_0(0)), gen_nil:cons4_0(0)) →_R^Ω(1)
if_intlist(true, gen_nil:cons4_0(0)) →_R^Ω(1)
nil

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
empty(nil) → true
empty(cons(x, y)) → false
tail(nil) → nil
tail(cons(x, y)) → y
head(cons(x, y)) → x
zero(0') → true
zero(s(x)) → false
p(0') → 0'
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
intlist(x) → if_intlist(empty(x), x)
if_intlist(true, x) → nil
if_intlist(false, x) → cons(s(head(x)), intlist(tail(x)))
int(x, y) → if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) → if1(b, x, y)
if_int(false, b, x, y) → if2(b, x, y)
if1(true, x, y) → cons(0', nil)
if1(false, x, y) → cons(0', int(s(0'), y))
if2(true, x, y) → nil
if2(false, x, y) → intlist(int(p(x), p(y)))

Types:
empty :: nil:cons → true:false
nil :: nil:cons
true :: true:false
cons :: 0':s → nil:cons → nil:cons
false :: true:false
tail :: nil:cons → nil:cons
head :: nil:cons → 0':s
zero :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
intlist :: nil:cons → nil:cons
if_intlist :: true:false → nil:cons → nil:cons
int :: 0':s → 0':s → nil:cons
if_int :: true:false → true:false → 0':s → 0':s → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat → nil:cons
gen_0':s5_0 :: Nat → 0':s

Lemmas:
p(gen_0':s5_0(+(1, n7_0))) → gen_0':s5_0(n7_0), rt ∈ Ω(1 + n7₀)
int(gen_0':s5_0(+(1, n37645_0)), gen_0':s5_0(n37645_0)) → gen_nil:cons4_0(0), rt ∈ Ω(1 + n37645₀ + n37645₀²)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
int(gen_0':s5_0(+(1, n37645_0)), gen_0':s5_0(n37645_0)) → gen_nil:cons4_0(0), rt ∈ Ω(1 + n37645₀ + n37645₀²)

(16) BOUNDS(n^2, INF)

(17) Obligation:

Innermost TRS:
Rules:
empty(nil) → true
empty(cons(x, y)) → false
tail(nil) → nil
tail(cons(x, y)) → y
head(cons(x, y)) → x
zero(0') → true
zero(s(x)) → false
p(0') → 0'
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
intlist(x) → if_intlist(empty(x), x)
if_intlist(true, x) → nil
if_intlist(false, x) → cons(s(head(x)), intlist(tail(x)))
int(x, y) → if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) → if1(b, x, y)
if_int(false, b, x, y) → if2(b, x, y)
if1(true, x, y) → cons(0', nil)
if1(false, x, y) → cons(0', int(s(0'), y))
if2(true, x, y) → nil
if2(false, x, y) → intlist(int(p(x), p(y)))

Types:
empty :: nil:cons → true:false
nil :: nil:cons
true :: true:false
cons :: 0':s → nil:cons → nil:cons
false :: true:false
tail :: nil:cons → nil:cons
head :: nil:cons → 0':s
zero :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
intlist :: nil:cons → nil:cons
if_intlist :: true:false → nil:cons → nil:cons
int :: 0':s → 0':s → nil:cons
if_int :: true:false → true:false → 0':s → 0':s → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat → nil:cons
gen_0':s5_0 :: Nat → 0':s

Lemmas:
p(gen_0':s5_0(+(1, n7_0))) → gen_0':s5_0(n7_0), rt ∈ Ω(1 + n7₀)
int(gen_0':s5_0(+(1, n37645_0)), gen_0':s5_0(n37645_0)) → gen_nil:cons4_0(0), rt ∈ Ω(1 + n37645₀ + n37645₀²)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
int(gen_0':s5_0(+(1, n37645_0)), gen_0':s5_0(n37645_0)) → gen_nil:cons4_0(0), rt ∈ Ω(1 + n37645₀ + n37645₀²)

(19) BOUNDS(n^2, INF)

(20) Obligation:

Innermost TRS:
Rules:
empty(nil) → true
empty(cons(x, y)) → false
tail(nil) → nil
tail(cons(x, y)) → y
head(cons(x, y)) → x
zero(0') → true
zero(s(x)) → false
p(0') → 0'
p(s(0')) → 0'
p(s(s(x))) → s(p(s(x)))
intlist(x) → if_intlist(empty(x), x)
if_intlist(true, x) → nil
if_intlist(false, x) → cons(s(head(x)), intlist(tail(x)))
int(x, y) → if_int(zero(x), zero(y), x, y)
if_int(true, b, x, y) → if1(b, x, y)
if_int(false, b, x, y) → if2(b, x, y)
if1(true, x, y) → cons(0', nil)
if1(false, x, y) → cons(0', int(s(0'), y))
if2(true, x, y) → nil
if2(false, x, y) → intlist(int(p(x), p(y)))

Types:
empty :: nil:cons → true:false
nil :: nil:cons
true :: true:false
cons :: 0':s → nil:cons → nil:cons
false :: true:false
tail :: nil:cons → nil:cons
head :: nil:cons → 0':s
zero :: 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
intlist :: nil:cons → nil:cons
if_intlist :: true:false → nil:cons → nil:cons
int :: 0':s → 0':s → nil:cons
if_int :: true:false → true:false → 0':s → 0':s → nil:cons
if1 :: true:false → 0':s → 0':s → nil:cons
if2 :: true:false → 0':s → 0':s → nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_0':s3_0 :: 0':s
gen_nil:cons4_0 :: Nat → nil:cons
gen_0':s5_0 :: Nat → 0':s

Lemmas:
p(gen_0':s5_0(+(1, n7_0))) → gen_0':s5_0(n7_0), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))
gen_0':s5_0(0) ⇔ 0'
gen_0':s5_0(+(x, 1)) ⇔ s(gen_0':s5_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
p(gen_0':s5_0(+(1, n7_0))) → gen_0':s5_0(n7_0), rt ∈ Ω(1 + n7₀)

(22) BOUNDS(n^1, INF)