The Runtime Complexity (full) 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:

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

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
+(s(x), s(y)) →⁺ s(s(+(x, y)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [x / s(x), y / s(y)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(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:

+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) → *'(x, *'(y, z))
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: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) → *'(x, *'(y, z))
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

(7) 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

(8) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) → *'(x, *'(y, z))
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

(9) 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).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) → *'(x, *'(y, z))
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

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol *'.

(13) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) → *'(x, *'(y, z))
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

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sum(gen_nil:cons4_0(n13244_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n13244₀)

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

Induction Step:
sum(gen_nil:cons4_0(+(n13244_0, 1))) →_R^Ω(1)
+'(0', sum(gen_nil:cons4_0(n13244_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).

(15) Complex Obligation (BEST)

(16) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) → *'(x, *'(y, z))
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)
sum(gen_nil:cons4_0(n13244_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n13244₀)

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

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) → *'(x, *'(y, z))
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)
sum(gen_nil:cons4_0(n13244_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n13244₀)

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(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)

(20) BOUNDS(n^1, INF)

(21) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) → *'(x, *'(y, z))
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)
sum(gen_nil:cons4_0(n13244_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n13244₀)

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(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)

(23) BOUNDS(n^1, INF)

(24) Obligation:

TRS:
Rules:
+'(x, 0') → x
+'(0', x) → x
+'(s(x), s(y)) → s(s(+'(x, y)))
+'(+'(x, y), z) → +'(x, +'(y, z))
*'(x, 0') → 0'
*'(0', x) → 0'
*'(s(x), s(y)) → s(+'(*'(x, y), +'(x, y)))
*'(*'(x, y), z) → *'(x, *'(y, z))
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.

(25) 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)

(26) BOUNDS(n^1, INF)