The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

0 CpxTRS
↳1 RenamingProof (⇔, 0 ms)
↳2 CpxRelTRS
↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 typed CpxTrs
↳5 OrderProof (LOWER BOUND(ID), 0 ms)
↳6 typed CpxTrs
↳7 RewriteLemmaProof (LOWER BOUND(ID), 793 ms)
↳8 BEST
    ↳9 typed CpxTrs
        ↳10 RewriteLemmaProof (LOWER BOUND(ID), 87 ms)
        ↳11 BEST
            ↳12 typed CpxTrs
                ↳13 LowerBoundsProof (⇔, 0 ms)
                ↳14 BOUNDS(n^1, INF)
            ↳15 typed CpxTrs
                ↳16 LowerBoundsProof (⇔, 0 ms)
                ↳17 BOUNDS(n^1, INF)
    ↳18 typed CpxTrs
        ↳19 LowerBoundsProof (⇔, 0 ms)
        ↳20 BOUNDS(n^1, INF)

(0) Obligation:

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

plus_x#1(0, x8) → x8
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14))
map#2(plus_x(x2), Nil) → Nil
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
main(x5, x12) → map#2(plus_x(x12), x5)

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:

plus_x#1(0', x8) → x8
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14))
map#2(plus_x(x2), Nil) → Nil
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
main(x5, x12) → map#2(plus_x(x12), x5)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
plus_x#1(0', x8) → x8
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14))
map#2(plus_x(x2), Nil) → Nil
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
main(x5, x12) → map#2(plus_x(x12), x5)

Types:
plus_x#1 :: 0':S → 0':S → 0':S
0' :: 0':S
S :: 0':S → 0':S
map#2 :: plus_x → Nil:Cons → Nil:Cons
plus_x :: 0':S → plus_x
Nil :: Nil:Cons
Cons :: 0':S → Nil:Cons → Nil:Cons
main :: Nil:Cons → 0':S → Nil:Cons
hole_0':S1_0 :: 0':S
hole_Nil:Cons2_0 :: Nil:Cons
hole_plus_x3_0 :: plus_x
gen_0':S4_0 :: Nat → 0':S
gen_Nil:Cons5_0 :: Nat → Nil:Cons

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

Heuristically decided to analyse the following defined symbols:
plus_x#1, map#2

They will be analysed ascendingly in the following order:
plus_x#1 < map#2

(6) Obligation:

Innermost TRS:
Rules:
plus_x#1(0', x8) → x8
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14))
map#2(plus_x(x2), Nil) → Nil
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
main(x5, x12) → map#2(plus_x(x12), x5)

Types:
plus_x#1 :: 0':S → 0':S → 0':S
0' :: 0':S
S :: 0':S → 0':S
map#2 :: plus_x → Nil:Cons → Nil:Cons
plus_x :: 0':S → plus_x
Nil :: Nil:Cons
Cons :: 0':S → Nil:Cons → Nil:Cons
main :: Nil:Cons → 0':S → Nil:Cons
hole_0':S1_0 :: 0':S
hole_Nil:Cons2_0 :: Nil:Cons
hole_plus_x3_0 :: plus_x
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:
plus_x#1, map#2

They will be analysed ascendingly in the following order:
plus_x#1 < map#2

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

Proved the following rewrite lemma:
plus_x#1(gen_0':S4_0(n7_0), gen_0':S4_0(b)) → gen_0':S4_0(+(n7_0, b)), rt ∈ Ω(1 + n70)

Induction Base:
plus_x#1(gen_0':S4_0(0), gen_0':S4_0(b)) →RΩ(1)
gen_0':S4_0(b)

Induction Step:
plus_x#1(gen_0':S4_0(+(n7_0, 1)), gen_0':S4_0(b)) →RΩ(1)
S(plus_x#1(gen_0':S4_0(n7_0), gen_0':S4_0(b))) →IH
S(gen_0':S4_0(+(b, c8_0)))

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
plus_x#1(0', x8) → x8
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14))
map#2(plus_x(x2), Nil) → Nil
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
main(x5, x12) → map#2(plus_x(x12), x5)

Types:
plus_x#1 :: 0':S → 0':S → 0':S
0' :: 0':S
S :: 0':S → 0':S
map#2 :: plus_x → Nil:Cons → Nil:Cons
plus_x :: 0':S → plus_x
Nil :: Nil:Cons
Cons :: 0':S → Nil:Cons → Nil:Cons
main :: Nil:Cons → 0':S → Nil:Cons
hole_0':S1_0 :: 0':S
hole_Nil:Cons2_0 :: Nil:Cons
hole_plus_x3_0 :: plus_x
gen_0':S4_0 :: Nat → 0':S
gen_Nil:Cons5_0 :: Nat → Nil:Cons

Lemmas:
plus_x#1(gen_0':S4_0(n7_0), gen_0':S4_0(b)) → gen_0':S4_0(+(n7_0, b)), 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:
map#2

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

Proved the following rewrite lemma:
map#2(plus_x(0'), gen_Nil:Cons5_0(n546_0)) → gen_Nil:Cons5_0(n546_0), rt ∈ Ω(1 + n5460)

Induction Base:
map#2(plus_x(0'), gen_Nil:Cons5_0(0)) →RΩ(1)
Nil

Induction Step:
map#2(plus_x(0'), gen_Nil:Cons5_0(+(n546_0, 1))) →RΩ(1)
Cons(plus_x#1(0', 0'), map#2(plus_x(0'), gen_Nil:Cons5_0(n546_0))) →LΩ(1)
Cons(gen_0':S4_0(+(0, 0)), map#2(plus_x(0'), gen_Nil:Cons5_0(n546_0))) →IH
Cons(gen_0':S4_0(0), gen_Nil:Cons5_0(c547_0))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
plus_x#1(0', x8) → x8
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14))
map#2(plus_x(x2), Nil) → Nil
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
main(x5, x12) → map#2(plus_x(x12), x5)

Types:
plus_x#1 :: 0':S → 0':S → 0':S
0' :: 0':S
S :: 0':S → 0':S
map#2 :: plus_x → Nil:Cons → Nil:Cons
plus_x :: 0':S → plus_x
Nil :: Nil:Cons
Cons :: 0':S → Nil:Cons → Nil:Cons
main :: Nil:Cons → 0':S → Nil:Cons
hole_0':S1_0 :: 0':S
hole_Nil:Cons2_0 :: Nil:Cons
hole_plus_x3_0 :: plus_x
gen_0':S4_0 :: Nat → 0':S
gen_Nil:Cons5_0 :: Nat → Nil:Cons

Lemmas:
plus_x#1(gen_0':S4_0(n7_0), gen_0':S4_0(b)) → gen_0':S4_0(+(n7_0, b)), rt ∈ Ω(1 + n70)
map#2(plus_x(0'), gen_Nil:Cons5_0(n546_0)) → gen_Nil:Cons5_0(n546_0), rt ∈ Ω(1 + n5460)

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.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
plus_x#1(gen_0':S4_0(n7_0), gen_0':S4_0(b)) → gen_0':S4_0(+(n7_0, b)), rt ∈ Ω(1 + n70)

(14) BOUNDS(n^1, INF)

(15) Obligation:

Innermost TRS:
Rules:
plus_x#1(0', x8) → x8
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14))
map#2(plus_x(x2), Nil) → Nil
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
main(x5, x12) → map#2(plus_x(x12), x5)

Types:
plus_x#1 :: 0':S → 0':S → 0':S
0' :: 0':S
S :: 0':S → 0':S
map#2 :: plus_x → Nil:Cons → Nil:Cons
plus_x :: 0':S → plus_x
Nil :: Nil:Cons
Cons :: 0':S → Nil:Cons → Nil:Cons
main :: Nil:Cons → 0':S → Nil:Cons
hole_0':S1_0 :: 0':S
hole_Nil:Cons2_0 :: Nil:Cons
hole_plus_x3_0 :: plus_x
gen_0':S4_0 :: Nat → 0':S
gen_Nil:Cons5_0 :: Nat → Nil:Cons

Lemmas:
plus_x#1(gen_0':S4_0(n7_0), gen_0':S4_0(b)) → gen_0':S4_0(+(n7_0, b)), rt ∈ Ω(1 + n70)
map#2(plus_x(0'), gen_Nil:Cons5_0(n546_0)) → gen_Nil:Cons5_0(n546_0), rt ∈ Ω(1 + n5460)

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.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
plus_x#1(gen_0':S4_0(n7_0), gen_0':S4_0(b)) → gen_0':S4_0(+(n7_0, b)), rt ∈ Ω(1 + n70)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
Rules:
plus_x#1(0', x8) → x8
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14))
map#2(plus_x(x2), Nil) → Nil
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
main(x5, x12) → map#2(plus_x(x12), x5)

Types:
plus_x#1 :: 0':S → 0':S → 0':S
0' :: 0':S
S :: 0':S → 0':S
map#2 :: plus_x → Nil:Cons → Nil:Cons
plus_x :: 0':S → plus_x
Nil :: Nil:Cons
Cons :: 0':S → Nil:Cons → Nil:Cons
main :: Nil:Cons → 0':S → Nil:Cons
hole_0':S1_0 :: 0':S
hole_Nil:Cons2_0 :: Nil:Cons
hole_plus_x3_0 :: plus_x
gen_0':S4_0 :: Nat → 0':S
gen_Nil:Cons5_0 :: Nat → Nil:Cons

Lemmas:
plus_x#1(gen_0':S4_0(n7_0), gen_0':S4_0(b)) → gen_0':S4_0(+(n7_0, b)), 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.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
plus_x#1(gen_0':S4_0(n7_0), gen_0':S4_0(b)) → gen_0':S4_0(+(n7_0, b)), rt ∈ Ω(1 + n70)

(20) BOUNDS(n^1, INF)