The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

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

sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0, x), y) → sum(x, y)
sum(nil, y) → y
empty(nil) → true
empty(cons(n, x)) → false
tail(nil) → nil
tail(cons(n, x)) → x
head(cons(n, x)) → n
weight(x) → if(empty(x), empty(tail(x)), x)
if(true, b, x) → weight_undefined_error
if(false, b, x) → if2(b, x)
if2(true, x) → head(x)
if2(false, x) → weight(sum(x, cons(0, tail(tail(x)))))

Rewrite Strategy: FULL

(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:

sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
empty(nil) → true
empty(cons(n, x)) → false
tail(nil) → nil
tail(cons(n, x)) → x
head(cons(n, x)) → n
weight(x) → if(empty(x), empty(tail(x)), x)
if(true, b, x) → weight_undefined_error
if(false, b, x) → if2(b, x)
if2(true, x) → head(x)
if2(false, x) → weight(sum(x, cons(0', tail(tail(x)))))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(4) Obligation:

TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
empty(nil) → true
empty(cons(n, x)) → false
tail(nil) → nil
tail(cons(n, x)) → x
head(cons(n, x)) → n
weight(x) → if(empty(x), empty(tail(x)), x)
if(true, b, x) → weight_undefined_error
if(false, b, x) → if2(b, x)
if2(true, x) → head(x)
if2(false, x) → weight(sum(x, cons(0', tail(tail(x)))))

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0':weight_undefined_error → cons:nil → cons:nil
s :: s:0':weight_undefined_error → s:0':weight_undefined_error
0' :: s:0':weight_undefined_error
nil :: cons:nil
empty :: cons:nil → true:false
true :: true:false
false :: true:false
tail :: cons:nil → cons:nil
head :: cons:nil → s:0':weight_undefined_error
weight :: cons:nil → s:0':weight_undefined_error
if :: true:false → true:false → cons:nil → s:0':weight_undefined_error
weight_undefined_error :: s:0':weight_undefined_error
if2 :: true:false → cons:nil → s:0':weight_undefined_error
hole_cons:nil1_0 :: cons:nil
hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat → cons:nil
gen_s:0':weight_undefined_error5_0 :: Nat → s:0':weight_undefined_error

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

Heuristically decided to analyse the following defined symbols:
sum, weight

They will be analysed ascendingly in the following order:
sum < weight

(6) Obligation:

TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
empty(nil) → true
empty(cons(n, x)) → false
tail(nil) → nil
tail(cons(n, x)) → x
head(cons(n, x)) → n
weight(x) → if(empty(x), empty(tail(x)), x)
if(true, b, x) → weight_undefined_error
if(false, b, x) → if2(b, x)
if2(true, x) → head(x)
if2(false, x) → weight(sum(x, cons(0', tail(tail(x)))))

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0':weight_undefined_error → cons:nil → cons:nil
s :: s:0':weight_undefined_error → s:0':weight_undefined_error
0' :: s:0':weight_undefined_error
nil :: cons:nil
empty :: cons:nil → true:false
true :: true:false
false :: true:false
tail :: cons:nil → cons:nil
head :: cons:nil → s:0':weight_undefined_error
weight :: cons:nil → s:0':weight_undefined_error
if :: true:false → true:false → cons:nil → s:0':weight_undefined_error
weight_undefined_error :: s:0':weight_undefined_error
if2 :: true:false → cons:nil → s:0':weight_undefined_error
hole_cons:nil1_0 :: cons:nil
hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat → cons:nil
gen_s:0':weight_undefined_error5_0 :: Nat → s:0':weight_undefined_error

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(0', gen_cons:nil4_0(x))
gen_s:0':weight_undefined_error5_0(0) ⇔ 0'
gen_s:0':weight_undefined_error5_0(+(x, 1)) ⇔ s(gen_s:0':weight_undefined_error5_0(x))

The following defined symbols remain to be analysed:
sum, weight

They will be analysed ascendingly in the following order:
sum < weight

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

Proved the following rewrite lemma:
sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) → gen_cons:nil4_0(b), rt ∈ Ω(1 + n7₀)

Induction Base:
sum(gen_cons:nil4_0(0), gen_cons:nil4_0(b)) →_R^Ω(1)
gen_cons:nil4_0(b)

Induction Step:
sum(gen_cons:nil4_0(+(n7_0, 1)), gen_cons:nil4_0(b)) →_R^Ω(1)
sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) →_IH
gen_cons:nil4_0(b)

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

(8) Complex Obligation (BEST)

(9) Obligation:

TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
empty(nil) → true
empty(cons(n, x)) → false
tail(nil) → nil
tail(cons(n, x)) → x
head(cons(n, x)) → n
weight(x) → if(empty(x), empty(tail(x)), x)
if(true, b, x) → weight_undefined_error
if(false, b, x) → if2(b, x)
if2(true, x) → head(x)
if2(false, x) → weight(sum(x, cons(0', tail(tail(x)))))

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0':weight_undefined_error → cons:nil → cons:nil
s :: s:0':weight_undefined_error → s:0':weight_undefined_error
0' :: s:0':weight_undefined_error
nil :: cons:nil
empty :: cons:nil → true:false
true :: true:false
false :: true:false
tail :: cons:nil → cons:nil
head :: cons:nil → s:0':weight_undefined_error
weight :: cons:nil → s:0':weight_undefined_error
if :: true:false → true:false → cons:nil → s:0':weight_undefined_error
weight_undefined_error :: s:0':weight_undefined_error
if2 :: true:false → cons:nil → s:0':weight_undefined_error
hole_cons:nil1_0 :: cons:nil
hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat → cons:nil
gen_s:0':weight_undefined_error5_0 :: Nat → s:0':weight_undefined_error

Lemmas:
sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) → gen_cons:nil4_0(b), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(0', gen_cons:nil4_0(x))
gen_s:0':weight_undefined_error5_0(0) ⇔ 0'
gen_s:0':weight_undefined_error5_0(+(x, 1)) ⇔ s(gen_s:0':weight_undefined_error5_0(x))

The following defined symbols remain to be analysed:
weight

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

Proved the following rewrite lemma:
weight(gen_cons:nil4_0(+(1, n664_0))) → gen_s:0':weight_undefined_error5_0(0), rt ∈ Ω(1 + n664₀ + n664₀²)

Induction Base:
weight(gen_cons:nil4_0(+(1, 0))) →_R^Ω(1)
if(empty(gen_cons:nil4_0(+(1, 0))), empty(tail(gen_cons:nil4_0(+(1, 0)))), gen_cons:nil4_0(+(1, 0))) →_R^Ω(1)
if(false, empty(tail(gen_cons:nil4_0(1))), gen_cons:nil4_0(1)) →_R^Ω(1)
if(false, empty(gen_cons:nil4_0(0)), gen_cons:nil4_0(1)) →_R^Ω(1)
if(false, true, gen_cons:nil4_0(1)) →_R^Ω(1)
if2(true, gen_cons:nil4_0(1)) →_R^Ω(1)
head(gen_cons:nil4_0(1)) →_R^Ω(1)
0'

Induction Step:
weight(gen_cons:nil4_0(+(1, +(n664_0, 1)))) →_R^Ω(1)
if(empty(gen_cons:nil4_0(+(1, +(n664_0, 1)))), empty(tail(gen_cons:nil4_0(+(1, +(n664_0, 1))))), gen_cons:nil4_0(+(1, +(n664_0, 1)))) →_R^Ω(1)
if(false, empty(tail(gen_cons:nil4_0(+(2, n664_0)))), gen_cons:nil4_0(+(2, n664_0))) →_R^Ω(1)
if(false, empty(gen_cons:nil4_0(+(1, n664_0))), gen_cons:nil4_0(+(2, n664_0))) →_R^Ω(1)
if(false, false, gen_cons:nil4_0(+(2, n664_0))) →_R^Ω(1)
if2(false, gen_cons:nil4_0(+(2, n664_0))) →_R^Ω(1)
weight(sum(gen_cons:nil4_0(+(2, n664_0)), cons(0', tail(tail(gen_cons:nil4_0(+(2, n664_0))))))) →_R^Ω(1)
weight(sum(gen_cons:nil4_0(+(2, n664_0)), cons(0', tail(gen_cons:nil4_0(+(1, n664_0)))))) →_R^Ω(1)
weight(sum(gen_cons:nil4_0(+(2, n664_0)), cons(0', gen_cons:nil4_0(n664_0)))) →_L^{Ω(3 + n664₀)}
weight(gen_cons:nil4_0(+(n664_0, 1))) →_IH
gen_s:0':weight_undefined_error5_0(0)

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(11) Complex Obligation (BEST)

(12) Obligation:

TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
empty(nil) → true
empty(cons(n, x)) → false
tail(nil) → nil
tail(cons(n, x)) → x
head(cons(n, x)) → n
weight(x) → if(empty(x), empty(tail(x)), x)
if(true, b, x) → weight_undefined_error
if(false, b, x) → if2(b, x)
if2(true, x) → head(x)
if2(false, x) → weight(sum(x, cons(0', tail(tail(x)))))

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0':weight_undefined_error → cons:nil → cons:nil
s :: s:0':weight_undefined_error → s:0':weight_undefined_error
0' :: s:0':weight_undefined_error
nil :: cons:nil
empty :: cons:nil → true:false
true :: true:false
false :: true:false
tail :: cons:nil → cons:nil
head :: cons:nil → s:0':weight_undefined_error
weight :: cons:nil → s:0':weight_undefined_error
if :: true:false → true:false → cons:nil → s:0':weight_undefined_error
weight_undefined_error :: s:0':weight_undefined_error
if2 :: true:false → cons:nil → s:0':weight_undefined_error
hole_cons:nil1_0 :: cons:nil
hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat → cons:nil
gen_s:0':weight_undefined_error5_0 :: Nat → s:0':weight_undefined_error

Lemmas:
sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) → gen_cons:nil4_0(b), rt ∈ Ω(1 + n7₀)
weight(gen_cons:nil4_0(+(1, n664_0))) → gen_s:0':weight_undefined_error5_0(0), rt ∈ Ω(1 + n664₀ + n664₀²)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(0', gen_cons:nil4_0(x))
gen_s:0':weight_undefined_error5_0(0) ⇔ 0'
gen_s:0':weight_undefined_error5_0(+(x, 1)) ⇔ s(gen_s:0':weight_undefined_error5_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
weight(gen_cons:nil4_0(+(1, n664_0))) → gen_s:0':weight_undefined_error5_0(0), rt ∈ Ω(1 + n664₀ + n664₀²)

(14) BOUNDS(n^2, INF)

(15) Obligation:

TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
empty(nil) → true
empty(cons(n, x)) → false
tail(nil) → nil
tail(cons(n, x)) → x
head(cons(n, x)) → n
weight(x) → if(empty(x), empty(tail(x)), x)
if(true, b, x) → weight_undefined_error
if(false, b, x) → if2(b, x)
if2(true, x) → head(x)
if2(false, x) → weight(sum(x, cons(0', tail(tail(x)))))

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0':weight_undefined_error → cons:nil → cons:nil
s :: s:0':weight_undefined_error → s:0':weight_undefined_error
0' :: s:0':weight_undefined_error
nil :: cons:nil
empty :: cons:nil → true:false
true :: true:false
false :: true:false
tail :: cons:nil → cons:nil
head :: cons:nil → s:0':weight_undefined_error
weight :: cons:nil → s:0':weight_undefined_error
if :: true:false → true:false → cons:nil → s:0':weight_undefined_error
weight_undefined_error :: s:0':weight_undefined_error
if2 :: true:false → cons:nil → s:0':weight_undefined_error
hole_cons:nil1_0 :: cons:nil
hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat → cons:nil
gen_s:0':weight_undefined_error5_0 :: Nat → s:0':weight_undefined_error

Lemmas:
sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) → gen_cons:nil4_0(b), rt ∈ Ω(1 + n7₀)
weight(gen_cons:nil4_0(+(1, n664_0))) → gen_s:0':weight_undefined_error5_0(0), rt ∈ Ω(1 + n664₀ + n664₀²)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(0', gen_cons:nil4_0(x))
gen_s:0':weight_undefined_error5_0(0) ⇔ 0'
gen_s:0':weight_undefined_error5_0(+(x, 1)) ⇔ s(gen_s:0':weight_undefined_error5_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
weight(gen_cons:nil4_0(+(1, n664_0))) → gen_s:0':weight_undefined_error5_0(0), rt ∈ Ω(1 + n664₀ + n664₀²)

(17) BOUNDS(n^2, INF)

(18) Obligation:

TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) → sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) → sum(x, y)
sum(nil, y) → y
empty(nil) → true
empty(cons(n, x)) → false
tail(nil) → nil
tail(cons(n, x)) → x
head(cons(n, x)) → n
weight(x) → if(empty(x), empty(tail(x)), x)
if(true, b, x) → weight_undefined_error
if(false, b, x) → if2(b, x)
if2(true, x) → head(x)
if2(false, x) → weight(sum(x, cons(0', tail(tail(x)))))

Types:
sum :: cons:nil → cons:nil → cons:nil
cons :: s:0':weight_undefined_error → cons:nil → cons:nil
s :: s:0':weight_undefined_error → s:0':weight_undefined_error
0' :: s:0':weight_undefined_error
nil :: cons:nil
empty :: cons:nil → true:false
true :: true:false
false :: true:false
tail :: cons:nil → cons:nil
head :: cons:nil → s:0':weight_undefined_error
weight :: cons:nil → s:0':weight_undefined_error
if :: true:false → true:false → cons:nil → s:0':weight_undefined_error
weight_undefined_error :: s:0':weight_undefined_error
if2 :: true:false → cons:nil → s:0':weight_undefined_error
hole_cons:nil1_0 :: cons:nil
hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat → cons:nil
gen_s:0':weight_undefined_error5_0 :: Nat → s:0':weight_undefined_error

Lemmas:
sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) → gen_cons:nil4_0(b), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(0', gen_cons:nil4_0(x))
gen_s:0':weight_undefined_error5_0(0) ⇔ 0'
gen_s:0':weight_undefined_error5_0(+(x, 1)) ⇔ s(gen_s:0':weight_undefined_error5_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) → gen_cons:nil4_0(b), rt ∈ Ω(1 + n7₀)

(20) BOUNDS(n^1, INF)