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:

power(x', Cons(x, xs)) → mult(x', power(x', xs))
mult(x', Cons(x, xs)) → add0(x', mult(x', xs))
add0(x', Cons(x, xs)) → Cons(Cons(Nil, Nil), add0(x', xs))
power(x, Nil) → Cons(Nil, Nil)
mult(x, Nil) → Nil
add0(x, Nil) → x
goal(x, y) → power(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
power(x', Cons(x, xs)) →⁺ mult(x', power(x', xs))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [xs / Cons(x, xs)].
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:

power(x', Cons(x, xs)) → mult(x', power(x', xs))
mult(x', Cons(x, xs)) → add0(x', mult(x', xs))
add0(x', Cons(x, xs)) → Cons(Cons(Nil, Nil), add0(x', xs))
power(x, Nil) → Cons(Nil, Nil)
mult(x, Nil) → Nil
add0(x, Nil) → x
goal(x, y) → power(x, y)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
power(x', Cons(x, xs)) → mult(x', power(x', xs))
mult(x', Cons(x, xs)) → add0(x', mult(x', xs))
add0(x', Cons(x, xs)) → Cons(Cons(Nil, Nil), add0(x', xs))
power(x, Nil) → Cons(Nil, Nil)
mult(x, Nil) → Nil
add0(x, Nil) → x
goal(x, y) → power(x, y)

Types:
power :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
mult :: Cons:Nil → Cons:Nil → Cons:Nil
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat → Cons:Nil

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

Heuristically decided to analyse the following defined symbols:
power, mult, add0

They will be analysed ascendingly in the following order:
mult < power
add0 < mult

(8) Obligation:

Innermost TRS:
Rules:
power(x', Cons(x, xs)) → mult(x', power(x', xs))
mult(x', Cons(x, xs)) → add0(x', mult(x', xs))
add0(x', Cons(x, xs)) → Cons(Cons(Nil, Nil), add0(x', xs))
power(x, Nil) → Cons(Nil, Nil)
mult(x, Nil) → Nil
add0(x, Nil) → x
goal(x, y) → power(x, y)

Types:
power :: Cons:Nil → Cons:Nil → Cons:Nil
Cons :: Cons:Nil → Cons:Nil → Cons:Nil
mult :: Cons:Nil → Cons:Nil → Cons:Nil
add0 :: Cons:Nil → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil → Cons:Nil → Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat → Cons:Nil

Generator Equations:
gen_Cons:Nil2_1(0) ⇔ Nil
gen_Cons:Nil2_1(+(x, 1)) ⇔ Cons(Nil, gen_Cons:Nil2_1(x))

The following defined symbols remain to be analysed:
add0, power, mult

They will be analysed ascendingly in the following order:
mult < power
add0 < mult