The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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

prod(xs) → prodIter(xs, s(0))
prodIter(xs, x) → ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) → x
ifProd(false, xs, x) → prodIter(tail(xs), times(x, head(xs)))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0, 0)
timesIter(x, y, z, u) → ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, plus(y, z), s(u))
isempty(nil) → true
isempty(cons(x, xs)) → false
head(nil) → error
head(cons(x, xs)) → x
tail(nil) → nil
tail(cons(x, xs)) → xs
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
a → b
a → c

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
prodIter(cons(x47455_1, xs47456_1), x) →⁺ prodIter(xs47456_1, times(x, head(cons(x47455_1, xs47456_1))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [xs47456_1 / cons(x47455_1, xs47456_1)].
The result substitution is [x / times(x, head(cons(x47455_1, xs47456_1)))].

(2) BOUNDS(n^1, INF)

(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:

prod(xs) → prodIter(xs, s(0'))
prodIter(xs, x) → ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) → x
ifProd(false, xs, x) → prodIter(tail(xs), times(x, head(xs)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0', 0')
timesIter(x, y, z, u) → ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, plus(y, z), s(u))
isempty(nil) → true
isempty(cons(x, xs)) → false
head(nil) → error
head(cons(x, xs)) → x
tail(nil) → nil
tail(cons(x, xs)) → xs
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
a → b
a → c

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
prod(xs) → prodIter(xs, s(0'))
prodIter(xs, x) → ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) → x
ifProd(false, xs, x) → prodIter(tail(xs), times(x, head(xs)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0', 0')
timesIter(x, y, z, u) → ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, plus(y, z), s(u))
isempty(nil) → true
isempty(cons(x, xs)) → false
head(nil) → error
head(cons(x, xs)) → x
tail(nil) → nil
tail(cons(x, xs)) → xs
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
a → b
a → c

Types:
prod :: nil:cons → 0':s:error
prodIter :: nil:cons → 0':s:error → 0':s:error
s :: 0':s:error → 0':s:error
0' :: 0':s:error
ifProd :: true:false → nil:cons → 0':s:error → 0':s:error
isempty :: nil:cons → true:false
true :: true:false
false :: true:false
tail :: nil:cons → nil:cons
times :: 0':s:error → 0':s:error → 0':s:error
head :: nil:cons → 0':s:error
plus :: 0':s:error → 0':s:error → 0':s:error
timesIter :: 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ifTimes :: true:false → 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ge :: 0':s:error → 0':s:error → true:false
nil :: nil:cons
cons :: 0':s:error → nil:cons → nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat → 0':s:error
gen_nil:cons6_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
prodIter, plus, timesIter, ge

They will be analysed ascendingly in the following order:
plus < timesIter
ge < timesIter

(8) Obligation:

TRS:
Rules:
prod(xs) → prodIter(xs, s(0'))
prodIter(xs, x) → ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) → x
ifProd(false, xs, x) → prodIter(tail(xs), times(x, head(xs)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0', 0')
timesIter(x, y, z, u) → ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, plus(y, z), s(u))
isempty(nil) → true
isempty(cons(x, xs)) → false
head(nil) → error
head(cons(x, xs)) → x
tail(nil) → nil
tail(cons(x, xs)) → xs
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
a → b
a → c

Types:
prod :: nil:cons → 0':s:error
prodIter :: nil:cons → 0':s:error → 0':s:error
s :: 0':s:error → 0':s:error
0' :: 0':s:error
ifProd :: true:false → nil:cons → 0':s:error → 0':s:error
isempty :: nil:cons → true:false
true :: true:false
false :: true:false
tail :: nil:cons → nil:cons
times :: 0':s:error → 0':s:error → 0':s:error
head :: nil:cons → 0':s:error
plus :: 0':s:error → 0':s:error → 0':s:error
timesIter :: 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ifTimes :: true:false → 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ge :: 0':s:error → 0':s:error → true:false
nil :: nil:cons
cons :: 0':s:error → nil:cons → nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat → 0':s:error
gen_nil:cons6_0 :: Nat → nil:cons

Generator Equations:
gen_0':s:error5_0(0) ⇔ 0'
gen_0':s:error5_0(+(x, 1)) ⇔ s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
prodIter, plus, timesIter, ge

They will be analysed ascendingly in the following order:
plus < timesIter
ge < timesIter

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

Proved the following rewrite lemma:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)

Induction Base:
prodIter(gen_nil:cons6_0(0), gen_0':s:error5_0(0)) →_R^Ω(1)
ifProd(isempty(gen_nil:cons6_0(0)), gen_nil:cons6_0(0), gen_0':s:error5_0(0)) →_R^Ω(1)
ifProd(true, gen_nil:cons6_0(0), gen_0':s:error5_0(0)) →_R^Ω(1)
gen_0':s:error5_0(0)

Induction Step:
prodIter(gen_nil:cons6_0(+(n8_0, 1)), gen_0':s:error5_0(0)) →_R^Ω(1)
ifProd(isempty(gen_nil:cons6_0(+(n8_0, 1))), gen_nil:cons6_0(+(n8_0, 1)), gen_0':s:error5_0(0)) →_R^Ω(1)
ifProd(false, gen_nil:cons6_0(+(1, n8_0)), gen_0':s:error5_0(0)) →_R^Ω(1)
prodIter(tail(gen_nil:cons6_0(+(1, n8_0))), times(gen_0':s:error5_0(0), head(gen_nil:cons6_0(+(1, n8_0))))) →_R^Ω(1)
prodIter(gen_nil:cons6_0(n8_0), times(gen_0':s:error5_0(0), head(gen_nil:cons6_0(+(1, n8_0))))) →_R^Ω(1)
prodIter(gen_nil:cons6_0(n8_0), times(gen_0':s:error5_0(0), 0')) →_R^Ω(1)
prodIter(gen_nil:cons6_0(n8_0), timesIter(gen_0':s:error5_0(0), 0', 0', 0')) →_R^Ω(1)
prodIter(gen_nil:cons6_0(n8_0), ifTimes(ge(0', gen_0':s:error5_0(0)), gen_0':s:error5_0(0), 0', 0', 0')) →_R^Ω(1)
prodIter(gen_nil:cons6_0(n8_0), ifTimes(true, gen_0':s:error5_0(0), 0', 0', 0')) →_R^Ω(1)
prodIter(gen_nil:cons6_0(n8_0), 0') →_IH
gen_0':s:error5_0(0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
prod(xs) → prodIter(xs, s(0'))
prodIter(xs, x) → ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) → x
ifProd(false, xs, x) → prodIter(tail(xs), times(x, head(xs)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0', 0')
timesIter(x, y, z, u) → ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, plus(y, z), s(u))
isempty(nil) → true
isempty(cons(x, xs)) → false
head(nil) → error
head(cons(x, xs)) → x
tail(nil) → nil
tail(cons(x, xs)) → xs
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
a → b
a → c

Types:
prod :: nil:cons → 0':s:error
prodIter :: nil:cons → 0':s:error → 0':s:error
s :: 0':s:error → 0':s:error
0' :: 0':s:error
ifProd :: true:false → nil:cons → 0':s:error → 0':s:error
isempty :: nil:cons → true:false
true :: true:false
false :: true:false
tail :: nil:cons → nil:cons
times :: 0':s:error → 0':s:error → 0':s:error
head :: nil:cons → 0':s:error
plus :: 0':s:error → 0':s:error → 0':s:error
timesIter :: 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ifTimes :: true:false → 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ge :: 0':s:error → 0':s:error → true:false
nil :: nil:cons
cons :: 0':s:error → nil:cons → nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat → 0':s:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)

Generator Equations:
gen_0':s:error5_0(0) ⇔ 0'
gen_0':s:error5_0(+(x, 1)) ⇔ s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
plus, timesIter, ge

They will be analysed ascendingly in the following order:
plus < timesIter
ge < timesIter

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

Proved the following rewrite lemma:
plus(gen_0':s:error5_0(n1635_0), gen_0':s:error5_0(b)) → gen_0':s:error5_0(+(n1635_0, b)), rt ∈ Ω(1 + n1635₀)

Induction Base:
plus(gen_0':s:error5_0(0), gen_0':s:error5_0(b)) →_R^Ω(1)
gen_0':s:error5_0(b)

Induction Step:
plus(gen_0':s:error5_0(+(n1635_0, 1)), gen_0':s:error5_0(b)) →_R^Ω(1)
s(plus(gen_0':s:error5_0(n1635_0), gen_0':s:error5_0(b))) →_IH
s(gen_0':s:error5_0(+(b, c1636_0)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
prod(xs) → prodIter(xs, s(0'))
prodIter(xs, x) → ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) → x
ifProd(false, xs, x) → prodIter(tail(xs), times(x, head(xs)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0', 0')
timesIter(x, y, z, u) → ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, plus(y, z), s(u))
isempty(nil) → true
isempty(cons(x, xs)) → false
head(nil) → error
head(cons(x, xs)) → x
tail(nil) → nil
tail(cons(x, xs)) → xs
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
a → b
a → c

Types:
prod :: nil:cons → 0':s:error
prodIter :: nil:cons → 0':s:error → 0':s:error
s :: 0':s:error → 0':s:error
0' :: 0':s:error
ifProd :: true:false → nil:cons → 0':s:error → 0':s:error
isempty :: nil:cons → true:false
true :: true:false
false :: true:false
tail :: nil:cons → nil:cons
times :: 0':s:error → 0':s:error → 0':s:error
head :: nil:cons → 0':s:error
plus :: 0':s:error → 0':s:error → 0':s:error
timesIter :: 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ifTimes :: true:false → 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ge :: 0':s:error → 0':s:error → true:false
nil :: nil:cons
cons :: 0':s:error → nil:cons → nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat → 0':s:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)
plus(gen_0':s:error5_0(n1635_0), gen_0':s:error5_0(b)) → gen_0':s:error5_0(+(n1635_0, b)), rt ∈ Ω(1 + n1635₀)

Generator Equations:
gen_0':s:error5_0(0) ⇔ 0'
gen_0':s:error5_0(+(x, 1)) ⇔ s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
ge, timesIter

They will be analysed ascendingly in the following order:
ge < timesIter

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

Proved the following rewrite lemma:
ge(gen_0':s:error5_0(n2460_0), gen_0':s:error5_0(n2460_0)) → true, rt ∈ Ω(1 + n2460₀)

Induction Base:
ge(gen_0':s:error5_0(0), gen_0':s:error5_0(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_0':s:error5_0(+(n2460_0, 1)), gen_0':s:error5_0(+(n2460_0, 1))) →_R^Ω(1)
ge(gen_0':s:error5_0(n2460_0), gen_0':s:error5_0(n2460_0)) →_IH
true

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
prod(xs) → prodIter(xs, s(0'))
prodIter(xs, x) → ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) → x
ifProd(false, xs, x) → prodIter(tail(xs), times(x, head(xs)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0', 0')
timesIter(x, y, z, u) → ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, plus(y, z), s(u))
isempty(nil) → true
isempty(cons(x, xs)) → false
head(nil) → error
head(cons(x, xs)) → x
tail(nil) → nil
tail(cons(x, xs)) → xs
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
a → b
a → c

Types:
prod :: nil:cons → 0':s:error
prodIter :: nil:cons → 0':s:error → 0':s:error
s :: 0':s:error → 0':s:error
0' :: 0':s:error
ifProd :: true:false → nil:cons → 0':s:error → 0':s:error
isempty :: nil:cons → true:false
true :: true:false
false :: true:false
tail :: nil:cons → nil:cons
times :: 0':s:error → 0':s:error → 0':s:error
head :: nil:cons → 0':s:error
plus :: 0':s:error → 0':s:error → 0':s:error
timesIter :: 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ifTimes :: true:false → 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ge :: 0':s:error → 0':s:error → true:false
nil :: nil:cons
cons :: 0':s:error → nil:cons → nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat → 0':s:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)
plus(gen_0':s:error5_0(n1635_0), gen_0':s:error5_0(b)) → gen_0':s:error5_0(+(n1635_0, b)), rt ∈ Ω(1 + n1635₀)
ge(gen_0':s:error5_0(n2460_0), gen_0':s:error5_0(n2460_0)) → true, rt ∈ Ω(1 + n2460₀)

Generator Equations:
gen_0':s:error5_0(0) ⇔ 0'
gen_0':s:error5_0(+(x, 1)) ⇔ s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
timesIter

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

TRS:
Rules:
prod(xs) → prodIter(xs, s(0'))
prodIter(xs, x) → ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) → x
ifProd(false, xs, x) → prodIter(tail(xs), times(x, head(xs)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0', 0')
timesIter(x, y, z, u) → ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, plus(y, z), s(u))
isempty(nil) → true
isempty(cons(x, xs)) → false
head(nil) → error
head(cons(x, xs)) → x
tail(nil) → nil
tail(cons(x, xs)) → xs
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
a → b
a → c

Types:
prod :: nil:cons → 0':s:error
prodIter :: nil:cons → 0':s:error → 0':s:error
s :: 0':s:error → 0':s:error
0' :: 0':s:error
ifProd :: true:false → nil:cons → 0':s:error → 0':s:error
isempty :: nil:cons → true:false
true :: true:false
false :: true:false
tail :: nil:cons → nil:cons
times :: 0':s:error → 0':s:error → 0':s:error
head :: nil:cons → 0':s:error
plus :: 0':s:error → 0':s:error → 0':s:error
timesIter :: 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ifTimes :: true:false → 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ge :: 0':s:error → 0':s:error → true:false
nil :: nil:cons
cons :: 0':s:error → nil:cons → nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat → 0':s:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)
plus(gen_0':s:error5_0(n1635_0), gen_0':s:error5_0(b)) → gen_0':s:error5_0(+(n1635_0, b)), rt ∈ Ω(1 + n1635₀)
ge(gen_0':s:error5_0(n2460_0), gen_0':s:error5_0(n2460_0)) → true, rt ∈ Ω(1 + n2460₀)

Generator Equations:
gen_0':s:error5_0(0) ⇔ 0'
gen_0':s:error5_0(+(x, 1)) ⇔ s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
prod(xs) → prodIter(xs, s(0'))
prodIter(xs, x) → ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) → x
ifProd(false, xs, x) → prodIter(tail(xs), times(x, head(xs)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0', 0')
timesIter(x, y, z, u) → ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, plus(y, z), s(u))
isempty(nil) → true
isempty(cons(x, xs)) → false
head(nil) → error
head(cons(x, xs)) → x
tail(nil) → nil
tail(cons(x, xs)) → xs
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
a → b
a → c

Types:
prod :: nil:cons → 0':s:error
prodIter :: nil:cons → 0':s:error → 0':s:error
s :: 0':s:error → 0':s:error
0' :: 0':s:error
ifProd :: true:false → nil:cons → 0':s:error → 0':s:error
isempty :: nil:cons → true:false
true :: true:false
false :: true:false
tail :: nil:cons → nil:cons
times :: 0':s:error → 0':s:error → 0':s:error
head :: nil:cons → 0':s:error
plus :: 0':s:error → 0':s:error → 0':s:error
timesIter :: 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ifTimes :: true:false → 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ge :: 0':s:error → 0':s:error → true:false
nil :: nil:cons
cons :: 0':s:error → nil:cons → nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat → 0':s:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)
plus(gen_0':s:error5_0(n1635_0), gen_0':s:error5_0(b)) → gen_0':s:error5_0(+(n1635_0, b)), rt ∈ Ω(1 + n1635₀)
ge(gen_0':s:error5_0(n2460_0), gen_0':s:error5_0(n2460_0)) → true, rt ∈ Ω(1 + n2460₀)

Generator Equations:
gen_0':s:error5_0(0) ⇔ 0'
gen_0':s:error5_0(+(x, 1)) ⇔ s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)

(24) BOUNDS(n^1, INF)

(25) Obligation:

TRS:
Rules:
prod(xs) → prodIter(xs, s(0'))
prodIter(xs, x) → ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) → x
ifProd(false, xs, x) → prodIter(tail(xs), times(x, head(xs)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0', 0')
timesIter(x, y, z, u) → ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, plus(y, z), s(u))
isempty(nil) → true
isempty(cons(x, xs)) → false
head(nil) → error
head(cons(x, xs)) → x
tail(nil) → nil
tail(cons(x, xs)) → xs
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
a → b
a → c

Types:
prod :: nil:cons → 0':s:error
prodIter :: nil:cons → 0':s:error → 0':s:error
s :: 0':s:error → 0':s:error
0' :: 0':s:error
ifProd :: true:false → nil:cons → 0':s:error → 0':s:error
isempty :: nil:cons → true:false
true :: true:false
false :: true:false
tail :: nil:cons → nil:cons
times :: 0':s:error → 0':s:error → 0':s:error
head :: nil:cons → 0':s:error
plus :: 0':s:error → 0':s:error → 0':s:error
timesIter :: 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ifTimes :: true:false → 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ge :: 0':s:error → 0':s:error → true:false
nil :: nil:cons
cons :: 0':s:error → nil:cons → nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat → 0':s:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)
plus(gen_0':s:error5_0(n1635_0), gen_0':s:error5_0(b)) → gen_0':s:error5_0(+(n1635_0, b)), rt ∈ Ω(1 + n1635₀)

Generator Equations:
gen_0':s:error5_0(0) ⇔ 0'
gen_0':s:error5_0(+(x, 1)) ⇔ s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)

(27) BOUNDS(n^1, INF)

(28) Obligation:

TRS:
Rules:
prod(xs) → prodIter(xs, s(0'))
prodIter(xs, x) → ifProd(isempty(xs), xs, x)
ifProd(true, xs, x) → x
ifProd(false, xs, x) → prodIter(tail(xs), times(x, head(xs)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
times(x, y) → timesIter(x, y, 0', 0')
timesIter(x, y, z, u) → ifTimes(ge(u, x), x, y, z, u)
ifTimes(true, x, y, z, u) → z
ifTimes(false, x, y, z, u) → timesIter(x, y, plus(y, z), s(u))
isempty(nil) → true
isempty(cons(x, xs)) → false
head(nil) → error
head(cons(x, xs)) → x
tail(nil) → nil
tail(cons(x, xs)) → xs
ge(x, 0') → true
ge(0', s(y)) → false
ge(s(x), s(y)) → ge(x, y)
a → b
a → c

Types:
prod :: nil:cons → 0':s:error
prodIter :: nil:cons → 0':s:error → 0':s:error
s :: 0':s:error → 0':s:error
0' :: 0':s:error
ifProd :: true:false → nil:cons → 0':s:error → 0':s:error
isempty :: nil:cons → true:false
true :: true:false
false :: true:false
tail :: nil:cons → nil:cons
times :: 0':s:error → 0':s:error → 0':s:error
head :: nil:cons → 0':s:error
plus :: 0':s:error → 0':s:error → 0':s:error
timesIter :: 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ifTimes :: true:false → 0':s:error → 0':s:error → 0':s:error → 0':s:error → 0':s:error
ge :: 0':s:error → 0':s:error → true:false
nil :: nil:cons
cons :: 0':s:error → nil:cons → nil:cons
error :: 0':s:error
a :: b:c
b :: b:c
c :: b:c
hole_0':s:error1_0 :: 0':s:error
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s:error5_0 :: Nat → 0':s:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)

Generator Equations:
gen_0':s:error5_0(0) ⇔ 0'
gen_0':s:error5_0(+(x, 1)) ⇔ s(gen_0':s:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
prodIter(gen_nil:cons6_0(n8_0), gen_0':s:error5_0(0)) → gen_0':s:error5_0(0), rt ∈ Ω(1 + n8₀)

(30) BOUNDS(n^1, INF)