The Runtime Complexity (innermost) 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:

+(x, 0) → x
+(0, x) → x
+(s(x), s(y)) → s(s(+(x, y)))
*(x, 0) → 0
*(0, x) → 0
*(s(x), s(y)) → s(+(*(x, y), +(x, y)))
sum(nil) → 0
sum(cons(x, l)) → +(x, sum(l))
prod(nil) → s(0)
prod(cons(x, l)) → *(x, prod(l))

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:

+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
+', *', sum, prod

They will be analysed ascendingly in the following order:
+' < *'
+' < sum
*' < prod

(6) Obligation:

Innermost TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
+', *', sum, prod

They will be analysed ascendingly in the following order:
+' < *'
+' < sum
*' < prod

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

Proved the following rewrite lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)

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(+(n6_0, 1)), gen_0':s3_0(+(n6_0, 1))) →_R^Ω(1)
s(s(+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)))) →_IH
s(s(gen_0':s3_0(*(2, 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:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), 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:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
*', sum, prod

They will be analysed ascendingly in the following order:
*' < prod

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

Proved the following rewrite lemma:
*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)) → *5_0, rt ∈ Ω(n566₀ + n566₀²)

Induction Base:
*'(gen_0':s3_0(0), gen_0':s3_0(0))

Induction Step:
*'(gen_0':s3_0(+(n566_0, 1)), gen_0':s3_0(+(n566_0, 1))) →_R^Ω(1)
s(+'(*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)), +'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)))) →_IH
s(+'(*5_0, +'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)))) →_L^{Ω(1 + n566₀)}
s(+'(*5_0, gen_0':s3_0(*(2, n566_0))))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)) → *5_0, rt ∈ Ω(n566₀ + n566₀²)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

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

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

Proved the following rewrite lemma:
sum(gen_nil:cons4_0(n10521_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n10521₀)

Induction Base:
sum(gen_nil:cons4_0(0)) →_R^Ω(1)
0'

Induction Step:
sum(gen_nil:cons4_0(+(n10521_0, 1))) →_R^Ω(1)
+'(0', sum(gen_nil:cons4_0(n10521_0))) →_IH
+'(0', gen_0':s3_0(0)) →_L^Ω(1)
gen_0':s3_0(*(2, 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:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)) → *5_0, rt ∈ Ω(n566₀ + n566₀²)
sum(gen_nil:cons4_0(n10521_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n10521₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
prod

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

Proved the following rewrite lemma:
prod(gen_nil:cons4_0(n10985_0)) → *5_0, rt ∈ Ω(n10985₀)

Induction Base:
prod(gen_nil:cons4_0(0))

Induction Step:
prod(gen_nil:cons4_0(+(n10985_0, 1))) →_R^Ω(1)
*'(0', prod(gen_nil:cons4_0(n10985_0))) →_IH
*'(0', *5_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:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)) → *5_0, rt ∈ Ω(n566₀ + n566₀²)
sum(gen_nil:cons4_0(n10521_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n10521₀)
prod(gen_nil:cons4_0(n10985_0)) → *5_0, rt ∈ Ω(n10985₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)) → *5_0, rt ∈ Ω(n566₀ + n566₀²)

(20) BOUNDS(n^2, INF)

(21) Obligation:

Innermost TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)) → *5_0, rt ∈ Ω(n566₀ + n566₀²)
sum(gen_nil:cons4_0(n10521_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n10521₀)
prod(gen_nil:cons4_0(n10985_0)) → *5_0, rt ∈ Ω(n10985₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)) → *5_0, rt ∈ Ω(n566₀ + n566₀²)

(23) BOUNDS(n^2, INF)

(24) Obligation:

Innermost TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)) → *5_0, rt ∈ Ω(n566₀ + n566₀²)
sum(gen_nil:cons4_0(n10521_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n10521₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)) → *5_0, rt ∈ Ω(n566₀ + n566₀²)

(26) BOUNDS(n^2, INF)

(27) Obligation:

Innermost TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)
*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)) → *5_0, rt ∈ Ω(n566₀ + n566₀²)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s3_0(n566_0), gen_0':s3_0(n566_0)) → *5_0, rt ∈ Ω(n566₀ + n566₀²)

(29) BOUNDS(n^2, INF)

(30) Obligation:

Innermost TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
sum(nil) → 0'
sum(cons(x, l)) → +'(x, sum(l))
prod(nil) → s(0')
prod(cons(x, l)) → *'(x, prod(l))

Types:
+' :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
sum :: nil:cons → 0':s
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
prod :: nil:cons → 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), 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:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_0':s3_0(n6_0), gen_0':s3_0(n6_0)) → gen_0':s3_0(*(2, n6_0)), rt ∈ Ω(1 + n6₀)

(32) BOUNDS(n^1, INF)