The Runtime Complexity (innermost) 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:

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

Infered types.

(6) Obligation:

Innermost TRS:
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)

Types:
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
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

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
add0

(8) Obligation:

Innermost TRS:
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)

Types:
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
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(Nil, gen_Cons:Nil3_1(x))

The following defined symbols remain to be analysed:
add0