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:

car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0) → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0) → 0
shorter(nil, y) → true
shorter(cons(x, l), 0) → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0), shorter(l, s(0)), l)
if(true, b, l) → s(0)
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))

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:

car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0') → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0'
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0') → 0'
shorter(nil, y) → true
shorter(cons(x, l), 0') → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) → s(0')
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0') → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0'
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0') → 0'
shorter(nil, y) → true
shorter(cons(x, l), 0') → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) → s(0')
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil → 0':s
cons :: 0':s → cons:nil → cons:nil
cddr :: cons:nil → cons:nil
nil :: cons:nil
cadr :: cons:nil → 0':s
isZero :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
plus :: 0':s → 0':s → 0':s
ifplus :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
iftimes :: true:false → 0':s → 0':s → 0':s
shorter :: cons:nil → 0':s → true:false
prod :: cons:nil → 0':s
if :: true:false → true:false → cons:nil → 0':s
if2 :: true:false → cons:nil → 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

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

Heuristically decided to analyse the following defined symbols:
plus, times, shorter, prod

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

(6) Obligation:

Innermost TRS:
Rules:
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0') → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0'
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0') → 0'
shorter(nil, y) → true
shorter(cons(x, l), 0') → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) → s(0')
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil → 0':s
cons :: 0':s → cons:nil → cons:nil
cddr :: cons:nil → cons:nil
nil :: cons:nil
cadr :: cons:nil → 0':s
isZero :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
plus :: 0':s → 0':s → 0':s
ifplus :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
iftimes :: true:false → 0':s → 0':s → 0':s
shorter :: cons:nil → 0':s → true:false
prod :: cons:nil → 0':s
if :: true:false → true:false → cons:nil → 0':s
if2 :: true:false → cons:nil → 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
plus, times, shorter, prod

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

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

Proved the following rewrite lemma:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

Induction Base:
plus(gen_0':s4_0(0), gen_0':s4_0(b)) →_R^Ω(1)
ifplus(isZero(gen_0':s4_0(0)), gen_0':s4_0(0), gen_0':s4_0(b)) →_R^Ω(1)
ifplus(true, gen_0':s4_0(0), gen_0':s4_0(b)) →_R^Ω(1)
gen_0':s4_0(b)

Induction Step:
plus(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(b)) →_R^Ω(1)
ifplus(isZero(gen_0':s4_0(+(n7_0, 1))), gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(b)) →_R^Ω(1)
ifplus(false, gen_0':s4_0(+(1, n7_0)), gen_0':s4_0(b)) →_R^Ω(1)
s(plus(p(gen_0':s4_0(+(1, n7_0))), gen_0':s4_0(b))) →_R^Ω(1)
s(plus(gen_0':s4_0(n7_0), gen_0':s4_0(b))) →_IH
s(gen_0':s4_0(+(b, 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:
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0') → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0'
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0') → 0'
shorter(nil, y) → true
shorter(cons(x, l), 0') → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) → s(0')
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil → 0':s
cons :: 0':s → cons:nil → cons:nil
cddr :: cons:nil → cons:nil
nil :: cons:nil
cadr :: cons:nil → 0':s
isZero :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
plus :: 0':s → 0':s → 0':s
ifplus :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
iftimes :: true:false → 0':s → 0':s → 0':s
shorter :: cons:nil → 0':s → true:false
prod :: cons:nil → 0':s
if :: true:false → true:false → cons:nil → 0':s
if2 :: true:false → cons:nil → 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
times, shorter, prod

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

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

Proved the following rewrite lemma:
times(gen_0':s4_0(n1404_0), gen_0':s4_0(b)) → gen_0':s4_0(*(n1404_0, b)), rt ∈ Ω(1 + b·n1404₀ + n1404₀)

Induction Base:
times(gen_0':s4_0(0), gen_0':s4_0(b)) →_R^Ω(1)
iftimes(isZero(gen_0':s4_0(0)), gen_0':s4_0(0), gen_0':s4_0(b)) →_R^Ω(1)
iftimes(true, gen_0':s4_0(0), gen_0':s4_0(b)) →_R^Ω(1)
0'

Induction Step:
times(gen_0':s4_0(+(n1404_0, 1)), gen_0':s4_0(b)) →_R^Ω(1)
iftimes(isZero(gen_0':s4_0(+(n1404_0, 1))), gen_0':s4_0(+(n1404_0, 1)), gen_0':s4_0(b)) →_R^Ω(1)
iftimes(false, gen_0':s4_0(+(1, n1404_0)), gen_0':s4_0(b)) →_R^Ω(1)
plus(gen_0':s4_0(b), times(p(gen_0':s4_0(+(1, n1404_0))), gen_0':s4_0(b))) →_R^Ω(1)
plus(gen_0':s4_0(b), times(gen_0':s4_0(n1404_0), gen_0':s4_0(b))) →_IH
plus(gen_0':s4_0(b), gen_0':s4_0(*(c1405_0, b))) →_L^{Ω(1 + b)}
gen_0':s4_0(+(b, *(n1404_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:
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0') → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0'
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0') → 0'
shorter(nil, y) → true
shorter(cons(x, l), 0') → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) → s(0')
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil → 0':s
cons :: 0':s → cons:nil → cons:nil
cddr :: cons:nil → cons:nil
nil :: cons:nil
cadr :: cons:nil → 0':s
isZero :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
plus :: 0':s → 0':s → 0':s
ifplus :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
iftimes :: true:false → 0':s → 0':s → 0':s
shorter :: cons:nil → 0':s → true:false
prod :: cons:nil → 0':s
if :: true:false → true:false → cons:nil → 0':s
if2 :: true:false → cons:nil → 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
times(gen_0':s4_0(n1404_0), gen_0':s4_0(b)) → gen_0':s4_0(*(n1404_0, b)), rt ∈ Ω(1 + b·n1404₀ + n1404₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
shorter, prod

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

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
shorter(gen_cons:nil5_0(n3435_0), gen_0':s4_0(n3435_0)) → true, rt ∈ Ω(1 + n3435₀)

Induction Base:
shorter(gen_cons:nil5_0(0), gen_0':s4_0(0)) →_R^Ω(1)
true

Induction Step:
shorter(gen_cons:nil5_0(+(n3435_0, 1)), gen_0':s4_0(+(n3435_0, 1))) →_R^Ω(1)
shorter(gen_cons:nil5_0(n3435_0), gen_0':s4_0(n3435_0)) →_IH
true

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0') → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0'
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0') → 0'
shorter(nil, y) → true
shorter(cons(x, l), 0') → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) → s(0')
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil → 0':s
cons :: 0':s → cons:nil → cons:nil
cddr :: cons:nil → cons:nil
nil :: cons:nil
cadr :: cons:nil → 0':s
isZero :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
plus :: 0':s → 0':s → 0':s
ifplus :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
iftimes :: true:false → 0':s → 0':s → 0':s
shorter :: cons:nil → 0':s → true:false
prod :: cons:nil → 0':s
if :: true:false → true:false → cons:nil → 0':s
if2 :: true:false → cons:nil → 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
times(gen_0':s4_0(n1404_0), gen_0':s4_0(b)) → gen_0':s4_0(*(n1404_0, b)), rt ∈ Ω(1 + b·n1404₀ + n1404₀)
shorter(gen_cons:nil5_0(n3435_0), gen_0':s4_0(n3435_0)) → true, rt ∈ Ω(1 + n3435₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
prod

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

Innermost TRS:
Rules:
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0') → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0'
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0') → 0'
shorter(nil, y) → true
shorter(cons(x, l), 0') → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) → s(0')
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil → 0':s
cons :: 0':s → cons:nil → cons:nil
cddr :: cons:nil → cons:nil
nil :: cons:nil
cadr :: cons:nil → 0':s
isZero :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
plus :: 0':s → 0':s → 0':s
ifplus :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
iftimes :: true:false → 0':s → 0':s → 0':s
shorter :: cons:nil → 0':s → true:false
prod :: cons:nil → 0':s
if :: true:false → true:false → cons:nil → 0':s
if2 :: true:false → cons:nil → 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
times(gen_0':s4_0(n1404_0), gen_0':s4_0(b)) → gen_0':s4_0(*(n1404_0, b)), rt ∈ Ω(1 + b·n1404₀ + n1404₀)
shorter(gen_cons:nil5_0(n3435_0), gen_0':s4_0(n3435_0)) → true, rt ∈ Ω(1 + n3435₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s4_0(n1404_0), gen_0':s4_0(b)) → gen_0':s4_0(*(n1404_0, b)), rt ∈ Ω(1 + b·n1404₀ + n1404₀)

(19) BOUNDS(n^2, INF)

(20) Obligation:

Innermost TRS:
Rules:
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0') → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0'
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0') → 0'
shorter(nil, y) → true
shorter(cons(x, l), 0') → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) → s(0')
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil → 0':s
cons :: 0':s → cons:nil → cons:nil
cddr :: cons:nil → cons:nil
nil :: cons:nil
cadr :: cons:nil → 0':s
isZero :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
plus :: 0':s → 0':s → 0':s
ifplus :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
iftimes :: true:false → 0':s → 0':s → 0':s
shorter :: cons:nil → 0':s → true:false
prod :: cons:nil → 0':s
if :: true:false → true:false → cons:nil → 0':s
if2 :: true:false → cons:nil → 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
times(gen_0':s4_0(n1404_0), gen_0':s4_0(b)) → gen_0':s4_0(*(n1404_0, b)), rt ∈ Ω(1 + b·n1404₀ + n1404₀)
shorter(gen_cons:nil5_0(n3435_0), gen_0':s4_0(n3435_0)) → true, rt ∈ Ω(1 + n3435₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_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':s4_0(n1404_0), gen_0':s4_0(b)) → gen_0':s4_0(*(n1404_0, b)), rt ∈ Ω(1 + b·n1404₀ + n1404₀)

(22) BOUNDS(n^2, INF)

(23) Obligation:

Innermost TRS:
Rules:
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0') → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0'
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0') → 0'
shorter(nil, y) → true
shorter(cons(x, l), 0') → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) → s(0')
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil → 0':s
cons :: 0':s → cons:nil → cons:nil
cddr :: cons:nil → cons:nil
nil :: cons:nil
cadr :: cons:nil → 0':s
isZero :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
plus :: 0':s → 0':s → 0':s
ifplus :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
iftimes :: true:false → 0':s → 0':s → 0':s
shorter :: cons:nil → 0':s → true:false
prod :: cons:nil → 0':s
if :: true:false → true:false → cons:nil → 0':s
if2 :: true:false → cons:nil → 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
times(gen_0':s4_0(n1404_0), gen_0':s4_0(b)) → gen_0':s4_0(*(n1404_0, b)), rt ∈ Ω(1 + b·n1404₀ + n1404₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_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':s4_0(n1404_0), gen_0':s4_0(b)) → gen_0':s4_0(*(n1404_0, b)), rt ∈ Ω(1 + b·n1404₀ + n1404₀)

(25) BOUNDS(n^2, INF)

(26) Obligation:

Innermost TRS:
Rules:
car(cons(x, l)) → x
cddr(nil) → nil
cddr(cons(x, nil)) → nil
cddr(cons(x, cons(y, l))) → l
cadr(cons(x, cons(y, l))) → y
isZero(0') → true
isZero(s(x)) → false
plus(x, y) → ifplus(isZero(x), x, y)
ifplus(true, x, y) → y
ifplus(false, x, y) → s(plus(p(x), y))
times(x, y) → iftimes(isZero(x), x, y)
iftimes(true, x, y) → 0'
iftimes(false, x, y) → plus(y, times(p(x), y))
p(s(x)) → x
p(0') → 0'
shorter(nil, y) → true
shorter(cons(x, l), 0') → false
shorter(cons(x, l), s(y)) → shorter(l, y)
prod(l) → if(shorter(l, 0'), shorter(l, s(0')), l)
if(true, b, l) → s(0')
if(false, b, l) → if2(b, l)
if2(true, l) → car(l)
if2(false, l) → prod(cons(times(car(l), cadr(l)), cddr(l)))

Types:
car :: cons:nil → 0':s
cons :: 0':s → cons:nil → cons:nil
cddr :: cons:nil → cons:nil
nil :: cons:nil
cadr :: cons:nil → 0':s
isZero :: 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
plus :: 0':s → 0':s → 0':s
ifplus :: true:false → 0':s → 0':s → 0':s
p :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
iftimes :: true:false → 0':s → 0':s → 0':s
shorter :: cons:nil → 0':s → true:false
prod :: cons:nil → 0':s
if :: true:false → true:false → cons:nil → 0':s
if2 :: true:false → cons:nil → 0':s
hole_0':s1_0 :: 0':s
hole_cons:nil2_0 :: cons:nil
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s4_0(n7_0), gen_0':s4_0(b)) → gen_0':s4_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

(28) BOUNDS(n^1, INF)