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

0 CpxTRS
↳1 DecreasingLoopProof (⇔, 293 ms)
↳2 BOUNDS(n^1, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 SlicingProof (LOWER BOUND(ID), 0 ms)
    ↳6 CpxRelTRS
    ↳7 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳10 typed CpxTrs
    ↳11 RewriteLemmaProof (LOWER BOUND(ID), 864 ms)
    ↳12 BEST
        ↳13 typed CpxTrs
            ↳14 LowerBoundsProof (⇔, 0 ms)
            ↳15 BOUNDS(n^1, INF)
        ↳16 typed CpxTrs
            ↳17 LowerBoundsProof (⇔, 0 ms)
            ↳18 BOUNDS(n^1, INF)

(0) Obligation:

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

add0(x', Cons(x, xs)) → add0(Cons(Cons(Nil, Nil), x'), xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
add0(x, Nil) → x
goal(x, y) → add0(x, y)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
add0(x', Cons(x, xs)) →+ add0(Cons(Cons(Nil, Nil), x'), xs)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [xs / Cons(x, xs)].
The result substitution is [x' / Cons(Cons(Nil, Nil), x')].

(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:

add0(x', Cons(x, xs)) → add0(Cons(Cons(Nil, Nil), x'), xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
add0(x, Nil) → x
goal(x, y) → add0(x, y)

S is empty.
Rewrite Strategy: INNERMOST

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
Cons/0

(6) Obligation:

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

add0(x', Cons(xs)) → add0(Cons(x'), xs)
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
add0(x, Nil) → x
goal(x, y) → add0(x, y)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

Innermost TRS:
Rules:
add0(x', Cons(xs)) → add0(Cons(x'), xs)
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
add0(x, Nil) → x
goal(x, y) → add0(x, y)

Types:
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
Nil :: Cons:Nil
False :: True:False
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
hole_True:False2_1 :: True:False
gen_Cons:Nil3_1 :: Nat → Cons:Nil

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
add0

(10) Obligation:

Innermost TRS:
Rules:
add0(x', Cons(xs)) → add0(Cons(x'), xs)
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
add0(x, Nil) → x
goal(x, y) → add0(x, y)

Types:
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
Nil :: Cons:Nil
False :: True:False
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
hole_True:False2_1 :: True:False
gen_Cons:Nil3_1 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil3_1(0) ⇔ Nil
gen_Cons:Nil3_1(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_1(x))

The following defined symbols remain to be analysed:
add0

(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
add0(gen_Cons:Nil3_1(a), gen_Cons:Nil3_1(n5_1)) → gen_Cons:Nil3_1(+(n5_1, a)), rt ∈ Ω(1 + n51)

Induction Base:
add0(gen_Cons:Nil3_1(a), gen_Cons:Nil3_1(0)) →RΩ(1)
gen_Cons:Nil3_1(a)

Induction Step:
add0(gen_Cons:Nil3_1(a), gen_Cons:Nil3_1(+(n5_1, 1))) →RΩ(1)
add0(Cons(gen_Cons:Nil3_1(a)), gen_Cons:Nil3_1(n5_1)) →IH
gen_Cons:Nil3_1(+(+(a, 1), c6_1))

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

(12) Complex Obligation (BEST)

(13) Obligation:

Innermost TRS:
Rules:
add0(x', Cons(xs)) → add0(Cons(x'), xs)
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
add0(x, Nil) → x
goal(x, y) → add0(x, y)

Types:
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
Nil :: Cons:Nil
False :: True:False
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
hole_True:False2_1 :: True:False
gen_Cons:Nil3_1 :: Nat → Cons:Nil

Lemmas:
add0(gen_Cons:Nil3_1(a), gen_Cons:Nil3_1(n5_1)) → gen_Cons:Nil3_1(+(n5_1, a)), rt ∈ Ω(1 + n51)

Generator Equations:
gen_Cons:Nil3_1(0) ⇔ Nil
gen_Cons:Nil3_1(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_1(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
add0(gen_Cons:Nil3_1(a), gen_Cons:Nil3_1(n5_1)) → gen_Cons:Nil3_1(+(n5_1, a)), rt ∈ Ω(1 + n51)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
add0(x', Cons(xs)) → add0(Cons(x'), xs)
notEmpty(Cons(xs)) → True
notEmpty(Nil) → False
add0(x, Nil) → x
goal(x, y) → add0(x, y)

Types:
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
Nil :: Cons:Nil
False :: True:False
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
hole_True:False2_1 :: True:False
gen_Cons:Nil3_1 :: Nat → Cons:Nil

Lemmas:
add0(gen_Cons:Nil3_1(a), gen_Cons:Nil3_1(n5_1)) → gen_Cons:Nil3_1(+(n5_1, a)), rt ∈ Ω(1 + n51)

Generator Equations:
gen_Cons:Nil3_1(0) ⇔ Nil
gen_Cons:Nil3_1(+(x, 1)) ⇔ Cons(gen_Cons:Nil3_1(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
add0(gen_Cons:Nil3_1(a), gen_Cons:Nil3_1(n5_1)) → gen_Cons:Nil3_1(+(n5_1, a)), rt ∈ Ω(1 + n51)

(18) BOUNDS(n^1, INF)