### (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: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

### (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:

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: FULL

Infered types.

### (6) Obligation:

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

### (7) 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

### (8) Obligation:

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

### (9) 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)) →RΩ(1)
0'

Induction Step:
min(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →RΩ(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 ircR ∈ Ω(n).

### (11) Obligation:

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

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

Proved the following rewrite lemma:
len(gen_nil:cons5_0(n440_0)) → gen_0':s4_0(n440_0), rt ∈ Ω(1 + n4400)

Induction Base:
len(gen_nil:cons5_0(0)) →RΩ(1)
0'

Induction Step:
len(gen_nil:cons5_0(+(n440_0, 1))) →RΩ(1)
s(len(gen_nil:cons5_0(n440_0))) →IH
s(gen_0':s4_0(c441_0))

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

### (14) Obligation:

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(n440_0)) → gen_0':s4_0(n440_0), rt ∈ Ω(1 + n4400)

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

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

Proved the following rewrite lemma:
sum(gen_0':s4_0(a), gen_0':s4_0(n676_0)) → gen_0':s4_0(+(n676_0, a)), rt ∈ Ω(1 + n6760)

Induction Base:
sum(gen_0':s4_0(a), gen_0':s4_0(0)) →RΩ(1)
gen_0':s4_0(a)

Induction Step:
sum(gen_0':s4_0(a), gen_0':s4_0(+(n676_0, 1))) →RΩ(1)
s(sum(gen_0':s4_0(a), gen_0':s4_0(n676_0))) →IH
s(gen_0':s4_0(+(a, c677_0)))

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

### (17) Obligation:

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(n440_0)) → gen_0':s4_0(n440_0), rt ∈ Ω(1 + n4400)
sum(gen_0':s4_0(a), gen_0':s4_0(n676_0)) → gen_0':s4_0(+(n676_0, a)), rt ∈ Ω(1 + n6760)

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

### (18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
le(gen_0':s4_0(n1411_0), gen_0':s4_0(n1411_0)) → true, rt ∈ Ω(1 + n14110)

Induction Base:
le(gen_0':s4_0(0), gen_0':s4_0(0)) →RΩ(1)
true

Induction Step:
le(gen_0':s4_0(+(n1411_0, 1)), gen_0':s4_0(+(n1411_0, 1))) →RΩ(1)
le(gen_0':s4_0(n1411_0), gen_0':s4_0(n1411_0)) →IH
true

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

### (20) Obligation:

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(n440_0)) → gen_0':s4_0(n440_0), rt ∈ Ω(1 + n4400)
sum(gen_0':s4_0(a), gen_0':s4_0(n676_0)) → gen_0':s4_0(+(n676_0, a)), rt ∈ Ω(1 + n6760)
le(gen_0':s4_0(n1411_0), gen_0':s4_0(n1411_0)) → true, rt ∈ Ω(1 + n14110)

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

### (21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
take(gen_0':s4_0(n1734_0), gen_nil:cons5_0(+(1, n1734_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n17340)

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

Induction Step:
take(gen_0':s4_0(+(n1734_0, 1)), gen_nil:cons5_0(+(1, +(n1734_0, 1)))) →RΩ(1)
take(gen_0':s4_0(n1734_0), gen_nil:cons5_0(+(1, n1734_0))) →IH
gen_0':s4_0(0)

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

### (23) Obligation:

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(n440_0)) → gen_0':s4_0(n440_0), rt ∈ Ω(1 + n4400)
sum(gen_0':s4_0(a), gen_0':s4_0(n676_0)) → gen_0':s4_0(+(n676_0, a)), rt ∈ Ω(1 + n6760)
le(gen_0':s4_0(n1411_0), gen_0':s4_0(n1411_0)) → true, rt ∈ Ω(1 + n14110)
take(gen_0':s4_0(n1734_0), gen_nil:cons5_0(+(1, n1734_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n17340)

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

### (24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (25) Obligation:

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(n440_0)) → gen_0':s4_0(n440_0), rt ∈ Ω(1 + n4400)
sum(gen_0':s4_0(a), gen_0':s4_0(n676_0)) → gen_0':s4_0(+(n676_0, a)), rt ∈ Ω(1 + n6760)
le(gen_0':s4_0(n1411_0), gen_0':s4_0(n1411_0)) → true, rt ∈ Ω(1 + n14110)
take(gen_0':s4_0(n1734_0), gen_nil:cons5_0(+(1, n1734_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n17340)

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.

### (26) 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) Obligation:

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(n440_0)) → gen_0':s4_0(n440_0), rt ∈ Ω(1 + n4400)
sum(gen_0':s4_0(a), gen_0':s4_0(n676_0)) → gen_0':s4_0(+(n676_0, a)), rt ∈ Ω(1 + n6760)
le(gen_0':s4_0(n1411_0), gen_0':s4_0(n1411_0)) → true, rt ∈ Ω(1 + n14110)
take(gen_0':s4_0(n1734_0), gen_nil:cons5_0(+(1, n1734_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n17340)

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.

### (29) 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) Obligation:

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(n440_0)) → gen_0':s4_0(n440_0), rt ∈ Ω(1 + n4400)
sum(gen_0':s4_0(a), gen_0':s4_0(n676_0)) → gen_0':s4_0(+(n676_0, a)), rt ∈ Ω(1 + n6760)
le(gen_0':s4_0(n1411_0), gen_0':s4_0(n1411_0)) → true, rt ∈ Ω(1 + n14110)

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.

### (32) 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) Obligation:

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(n440_0)) → gen_0':s4_0(n440_0), rt ∈ Ω(1 + n4400)
sum(gen_0':s4_0(a), gen_0':s4_0(n676_0)) → gen_0':s4_0(+(n676_0, a)), rt ∈ Ω(1 + n6760)

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.

### (35) 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) Obligation:

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(n440_0)) → gen_0':s4_0(n440_0), rt ∈ Ω(1 + n4400)

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.

### (38) 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) Obligation:

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.

### (41) 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)