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:

+(0, y) → y
+(s(x), y) → s(+(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0) → x
-(0, s(y)) → 0
-(s(x), s(y)) → -(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

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:

+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

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

Heuristically decided to analyse the following defined symbols:
+', ++, sum, -, quot, length

They will be analysed ascendingly in the following order:
+' < sum
++ < sum
- < quot

(6) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

The following defined symbols remain to be analysed:
+', ++, sum, -, quot, length

They will be analysed ascendingly in the following order:
+' < sum
++ < sum
- < quot

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

Proved the following rewrite lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Induction Base:
+'(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
gen_0':s3_0(b)

Induction Step:
+'(gen_0':s3_0(+(n6_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
s(+'(gen_0':s3_0(n6_0), gen_0':s3_0(b))) →_IH
s(gen_0':s3_0(+(b, c7_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:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

The following defined symbols remain to be analysed:
++, sum, -, quot, length

They will be analysed ascendingly in the following order:
++ < sum
- < quot

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

Proved the following rewrite lemma:
++(gen_nil::4_0(n833_0), gen_nil::4_0(b)) → gen_nil::4_0(+(n833_0, b)), rt ∈ Ω(1 + n833₀)

Induction Base:
++(gen_nil::4_0(0), gen_nil::4_0(b)) →_R^Ω(1)
gen_nil::4_0(b)

Induction Step:
++(gen_nil::4_0(+(n833_0, 1)), gen_nil::4_0(b)) →_R^Ω(1)
:(0', ++(gen_nil::4_0(n833_0), gen_nil::4_0(b))) →_IH
:(0', gen_nil::4_0(+(b, c834_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:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
++(gen_nil::4_0(n833_0), gen_nil::4_0(b)) → gen_nil::4_0(+(n833_0, b)), rt ∈ Ω(1 + n833₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

The following defined symbols remain to be analysed:
sum, -, quot, length

They will be analysed ascendingly in the following order:
- < quot

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

Proved the following rewrite lemma:
sum(gen_nil::4_0(+(1, n1736_0))) → gen_nil::4_0(1), rt ∈ Ω(1 + n1736₀)

Induction Base:
sum(gen_nil::4_0(+(1, 0))) →_R^Ω(1)
:(0', nil)

Induction Step:
sum(gen_nil::4_0(+(1, +(n1736_0, 1)))) →_R^Ω(1)
sum(:(+'(0', 0'), gen_nil::4_0(n1736_0))) →_L^Ω(1)
sum(:(gen_0':s3_0(+(0, 0)), gen_nil::4_0(n1736_0))) →_IH
gen_nil::4_0(1)

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:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
++(gen_nil::4_0(n833_0), gen_nil::4_0(b)) → gen_nil::4_0(+(n833_0, b)), rt ∈ Ω(1 + n833₀)
sum(gen_nil::4_0(+(1, n1736_0))) → gen_nil::4_0(1), rt ∈ Ω(1 + n1736₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

The following defined symbols remain to be analysed:
-, quot, length

They will be analysed ascendingly in the following order:
- < quot

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

Proved the following rewrite lemma:
-(gen_0':s3_0(n2212_0), gen_0':s3_0(n2212_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n2212₀)

Induction Base:
-(gen_0':s3_0(0), gen_0':s3_0(0)) →_R^Ω(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n2212_0, 1)), gen_0':s3_0(+(n2212_0, 1))) →_R^Ω(1)
-(gen_0':s3_0(n2212_0), gen_0':s3_0(n2212_0)) →_IH
gen_0':s3_0(0)

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:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
++(gen_nil::4_0(n833_0), gen_nil::4_0(b)) → gen_nil::4_0(+(n833_0, b)), rt ∈ Ω(1 + n833₀)
sum(gen_nil::4_0(+(1, n1736_0))) → gen_nil::4_0(1), rt ∈ Ω(1 + n1736₀)
-(gen_0':s3_0(n2212_0), gen_0':s3_0(n2212_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n2212₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

The following defined symbols remain to be analysed:
quot, length

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
++(gen_nil::4_0(n833_0), gen_nil::4_0(b)) → gen_nil::4_0(+(n833_0, b)), rt ∈ Ω(1 + n833₀)
sum(gen_nil::4_0(+(1, n1736_0))) → gen_nil::4_0(1), rt ∈ Ω(1 + n1736₀)
-(gen_0':s3_0(n2212_0), gen_0':s3_0(n2212_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n2212₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

The following defined symbols remain to be analysed:
length

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

Proved the following rewrite lemma:
length(gen_nil::4_0(n2871_0)) → gen_0':s3_0(n2871_0), rt ∈ Ω(1 + n2871₀)

Induction Base:
length(gen_nil::4_0(0)) →_R^Ω(1)
0'

Induction Step:
length(gen_nil::4_0(+(n2871_0, 1))) →_R^Ω(1)
s(length(gen_nil::4_0(n2871_0))) →_IH
s(gen_0':s3_0(c2872_0))

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

(22) Complex Obligation (BEST)

(23) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
++(gen_nil::4_0(n833_0), gen_nil::4_0(b)) → gen_nil::4_0(+(n833_0, b)), rt ∈ Ω(1 + n833₀)
sum(gen_nil::4_0(+(1, n1736_0))) → gen_nil::4_0(1), rt ∈ Ω(1 + n1736₀)
-(gen_0':s3_0(n2212_0), gen_0':s3_0(n2212_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n2212₀)
length(gen_nil::4_0(n2871_0)) → gen_0':s3_0(n2871_0), rt ∈ Ω(1 + n2871₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
++(gen_nil::4_0(n833_0), gen_nil::4_0(b)) → gen_nil::4_0(+(n833_0, b)), rt ∈ Ω(1 + n833₀)
sum(gen_nil::4_0(+(1, n1736_0))) → gen_nil::4_0(1), rt ∈ Ω(1 + n1736₀)
-(gen_0':s3_0(n2212_0), gen_0':s3_0(n2212_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n2212₀)
length(gen_nil::4_0(n2871_0)) → gen_0':s3_0(n2871_0), rt ∈ Ω(1 + n2871₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(28) BOUNDS(n^1, INF)

(29) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
++(gen_nil::4_0(n833_0), gen_nil::4_0(b)) → gen_nil::4_0(+(n833_0, b)), rt ∈ Ω(1 + n833₀)
sum(gen_nil::4_0(+(1, n1736_0))) → gen_nil::4_0(1), rt ∈ Ω(1 + n1736₀)
-(gen_0':s3_0(n2212_0), gen_0':s3_0(n2212_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n2212₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(31) BOUNDS(n^1, INF)

(32) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
++(gen_nil::4_0(n833_0), gen_nil::4_0(b)) → gen_nil::4_0(+(n833_0, b)), rt ∈ Ω(1 + n833₀)
sum(gen_nil::4_0(+(1, n1736_0))) → gen_nil::4_0(1), rt ∈ Ω(1 + n1736₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(34) BOUNDS(n^1, INF)

(35) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
++(gen_nil::4_0(n833_0), gen_nil::4_0(b)) → gen_nil::4_0(+(n833_0, b)), rt ∈ Ω(1 + n833₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

No more defined symbols left to analyse.

(36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(37) BOUNDS(n^1, INF)

(38) Obligation:

Innermost TRS:
Rules:
+'(0', y) → y
+'(s(x), y) → s(+'(x, y))
++(nil, ys) → ys
++(:(x, xs), ys) → :(x, ++(xs, ys))
sum(:(x, nil)) → :(x, nil)
sum(:(x, :(y, xs))) → sum(:(+'(x, y), xs))
sum(++(xs, :(x, :(y, ys)))) → sum(++(xs, sum(:(x, :(y, ys)))))
-(x, 0') → x
-(0', s(y)) → 0'
-(s(x), s(y)) → -(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(-(x, y), s(y)))
length(nil) → 0'
length(:(x, xs)) → s(length(xs))
hd(:(x, xs)) → x
avg(xs) → quot(hd(sum(xs)), length(xs))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
++ :: nil:: → nil:: → nil::
nil :: nil::
: :: 0':s → nil:: → nil::
sum :: nil:: → nil::
- :: 0':s → 0':s → 0':s
quot :: 0':s → 0':s → 0':s
length :: nil:: → 0':s
hd :: nil:: → 0':s
avg :: nil:: → 0':s
hole_0':s1_0 :: 0':s
hole_nil::2_0 :: nil::
gen_0':s3_0 :: Nat → 0':s
gen_nil::4_0 :: Nat → nil::

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil::4_0(0) ⇔ nil
gen_nil::4_0(+(x, 1)) ⇔ :(0', gen_nil::4_0(x))

No more defined symbols left to analyse.

(39) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(40) BOUNDS(n^1, INF)