The Runtime Complexity (innermost) 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:

min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

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:

min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
min, len, sum, le, take, if

They will be analysed ascendingly in the following order:
min < if
len < if
sum < if
le < if
take < if

(6) Obligation:

Innermost TRS:
Rules:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
min, len, sum, le, take, if

They will be analysed ascendingly in the following order:
min < if
len < if
sum < if
le < if
take < if

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

Proved the following rewrite lemma:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

Induction Base:
min(gen_0':s4_0(0), gen_0':s4_0(0)) →_R^Ω(1)
0'

Induction Step:
min(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →_R^Ω(1)
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) →_IH
gen_0':s4_0(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:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), 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_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
len, sum, le, take, if

They will be analysed ascendingly in the following order:
len < if
sum < if
le < if
take < if

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

Proved the following rewrite lemma:
len(gen_nil:cons5_0(n503_0)) → gen_0':s4_0(n503_0), rt ∈ Ω(1 + n503₀)

Induction Base:
len(gen_nil:cons5_0(0)) →_R^Ω(1)
0'

Induction Step:
len(gen_nil:cons5_0(+(n503_0, 1))) →_R^Ω(1)
s(len(gen_nil:cons5_0(n503_0))) →_IH
s(gen_0':s4_0(c504_0))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
len(gen_nil:cons5_0(n503_0)) → gen_0':s4_0(n503_0), rt ∈ Ω(1 + n503₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
sum, le, take, if

They will be analysed ascendingly in the following order:
sum < if
le < if
take < if

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

Proved the following rewrite lemma:
sum(gen_0':s4_0(a), gen_0':s4_0(n757_0)) → gen_0':s4_0(+(n757_0, a)), rt ∈ Ω(1 + n757₀)

Induction Base:
sum(gen_0':s4_0(a), gen_0':s4_0(0)) →_R^Ω(1)
gen_0':s4_0(a)

Induction Step:
sum(gen_0':s4_0(a), gen_0':s4_0(+(n757_0, 1))) →_R^Ω(1)
s(sum(gen_0':s4_0(a), gen_0':s4_0(n757_0))) →_IH
s(gen_0':s4_0(+(a, c758_0)))

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:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
len(gen_nil:cons5_0(n503_0)) → gen_0':s4_0(n503_0), rt ∈ Ω(1 + n503₀)
sum(gen_0':s4_0(a), gen_0':s4_0(n757_0)) → gen_0':s4_0(+(n757_0, a)), rt ∈ Ω(1 + n757₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
le, take, if

They will be analysed ascendingly in the following order:
le < if
take < if

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
le(gen_0':s4_0(n1582_0), gen_0':s4_0(n1582_0)) → true, rt ∈ Ω(1 + n1582₀)

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

Induction Step:
le(gen_0':s4_0(+(n1582_0, 1)), gen_0':s4_0(+(n1582_0, 1))) →_R^Ω(1)
le(gen_0':s4_0(n1582_0), gen_0':s4_0(n1582_0)) →_IH
true

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

(17) Complex Obligation (BEST)

(18) Obligation:

Innermost TRS:
Rules:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
len(gen_nil:cons5_0(n503_0)) → gen_0':s4_0(n503_0), rt ∈ Ω(1 + n503₀)
sum(gen_0':s4_0(a), gen_0':s4_0(n757_0)) → gen_0':s4_0(+(n757_0, a)), rt ∈ Ω(1 + n757₀)
le(gen_0':s4_0(n1582_0), gen_0':s4_0(n1582_0)) → true, rt ∈ Ω(1 + n1582₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
take, if

They will be analysed ascendingly in the following order:
take < if

(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
take(gen_0':s4_0(n1959_0), gen_nil:cons5_0(+(1, n1959_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n1959₀)

Induction Base:
take(gen_0':s4_0(0), gen_nil:cons5_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
take(gen_0':s4_0(+(n1959_0, 1)), gen_nil:cons5_0(+(1, +(n1959_0, 1)))) →_R^Ω(1)
take(gen_0':s4_0(n1959_0), gen_nil:cons5_0(+(1, n1959_0))) →_IH
gen_0':s4_0(0)

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

(20) Complex Obligation (BEST)

(21) Obligation:

Innermost TRS:
Rules:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
len(gen_nil:cons5_0(n503_0)) → gen_0':s4_0(n503_0), rt ∈ Ω(1 + n503₀)
sum(gen_0':s4_0(a), gen_0':s4_0(n757_0)) → gen_0':s4_0(+(n757_0, a)), rt ∈ Ω(1 + n757₀)
le(gen_0':s4_0(n1582_0), gen_0':s4_0(n1582_0)) → true, rt ∈ Ω(1 + n1582₀)
take(gen_0':s4_0(n1959_0), gen_nil:cons5_0(+(1, n1959_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n1959₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

The following defined symbols remain to be analysed:
if

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(23) Obligation:

Innermost TRS:
Rules:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
len(gen_nil:cons5_0(n503_0)) → gen_0':s4_0(n503_0), rt ∈ Ω(1 + n503₀)
sum(gen_0':s4_0(a), gen_0':s4_0(n757_0)) → gen_0':s4_0(+(n757_0, a)), rt ∈ Ω(1 + n757₀)
le(gen_0':s4_0(n1582_0), gen_0':s4_0(n1582_0)) → true, rt ∈ Ω(1 + n1582₀)
take(gen_0':s4_0(n1959_0), gen_nil:cons5_0(+(1, n1959_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n1959₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
len(gen_nil:cons5_0(n503_0)) → gen_0':s4_0(n503_0), rt ∈ Ω(1 + n503₀)
sum(gen_0':s4_0(a), gen_0':s4_0(n757_0)) → gen_0':s4_0(+(n757_0, a)), rt ∈ Ω(1 + n757₀)
le(gen_0':s4_0(n1582_0), gen_0':s4_0(n1582_0)) → true, rt ∈ Ω(1 + n1582₀)
take(gen_0':s4_0(n1959_0), gen_nil:cons5_0(+(1, n1959_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n1959₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

(28) BOUNDS(n^1, INF)

(29) Obligation:

Innermost TRS:
Rules:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
len(gen_nil:cons5_0(n503_0)) → gen_0':s4_0(n503_0), rt ∈ Ω(1 + n503₀)
sum(gen_0':s4_0(a), gen_0':s4_0(n757_0)) → gen_0':s4_0(+(n757_0, a)), rt ∈ Ω(1 + n757₀)
le(gen_0':s4_0(n1582_0), gen_0':s4_0(n1582_0)) → true, rt ∈ Ω(1 + n1582₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

(31) BOUNDS(n^1, INF)

(32) Obligation:

Innermost TRS:
Rules:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
len(gen_nil:cons5_0(n503_0)) → gen_0':s4_0(n503_0), rt ∈ Ω(1 + n503₀)
sum(gen_0':s4_0(a), gen_0':s4_0(n757_0)) → gen_0':s4_0(+(n757_0, a)), rt ∈ Ω(1 + n757₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

(34) BOUNDS(n^1, INF)

(35) Obligation:

Innermost TRS:
Rules:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)
len(gen_nil:cons5_0(n503_0)) → gen_0':s4_0(n503_0), rt ∈ Ω(1 + n503₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

(37) BOUNDS(n^1, INF)

(38) Obligation:

Innermost TRS:
Rules:
min(0', y) → 0'
min(s(x), 0') → 0'
min(s(x), s(y)) → min(x, y)
len(nil) → 0'
len(cons(x, xs)) → s(len(xs))
sum(x, 0') → x
sum(x, s(y)) → s(sum(x, y))
le(0', x) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
take(0', cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0', min(len(x), len(y))), 0', x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))

Types:
min :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
len :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
take :: 0':s → nil:cons → 0':s
addList :: nil:cons → nil:cons → nil:cons
if :: true:false → 0':s → nil:cons → nil:cons → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat → 0':s
gen_nil:cons5_0 :: Nat → nil:cons

Lemmas:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), 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_nil:cons5_0(0) ⇔ nil
gen_nil:cons5_0(+(x, 1)) ⇔ cons(0', gen_nil:cons5_0(x))

No more defined symbols left to analyse.

(39) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
min(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n7₀)

(40) BOUNDS(n^1, INF)