The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, 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) 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:

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

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

(9) 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

(10) Obligation:

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

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

Proved the following rewrite lemma:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) → gen_Cons:Nil2_1(+(n4_1, a)), rt ∈ Ω(1 + n4₁)

Induction Base:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(0)) →_R^Ω(1)
gen_Cons:Nil2_1(a)

Induction Step:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(n4_1, 1))) →_R^Ω(1)
Cons(add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1))) →_IH
Cons(gen_Cons:Nil2_1(+(a, c5_1)))

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

(12) Complex Obligation (BEST)

(13) Obligation:

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

Lemmas:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) → gen_Cons:Nil2_1(+(n4_1, a)), rt ∈ Ω(1 + n4₁)

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

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

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

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

Proved the following rewrite lemma:
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n518_1)) → gen_Cons:Nil2_1(*(n518_1, a)), rt ∈ Ω(1 + a·n518₁² + n518₁)

Induction Base:
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(0)) →_R^Ω(1)
Nil

Induction Step:
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(n518_1, 1))) →_R^Ω(1)
add0(gen_Cons:Nil2_1(a), mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n518_1))) →_IH
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(*(c519_1, a))) →_L^{Ω(1 + a·n518₁)}
gen_Cons:Nil2_1(+(*(n518_1, a), a))

We have rt ∈ Ω(n³) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n³).

(15) Complex Obligation (BEST)

(16) Obligation:

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

Lemmas:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) → gen_Cons:Nil2_1(+(n4_1, a)), rt ∈ Ω(1 + n4₁)
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n518_1)) → gen_Cons:Nil2_1(*(n518_1, a)), rt ∈ Ω(1 + a·n518₁² + n518₁)

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

The following defined symbols remain to be analysed:
power

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
power(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(1, n1170_1))) → *3_1, rt ∈ Ω(n1170₁)

Induction Base:
power(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(1, 0)))

Induction Step:
power(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(1, +(n1170_1, 1)))) →_R^Ω(1)
mult(gen_Cons:Nil2_1(a), power(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(1, n1170_1)))) →_IH
mult(gen_Cons:Nil2_1(a), *3_1)

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

(18) Complex Obligation (BEST)

(19) Obligation:

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

Lemmas:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) → gen_Cons:Nil2_1(+(n4_1, a)), rt ∈ Ω(1 + n4₁)
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n518_1)) → gen_Cons:Nil2_1(*(n518_1, a)), rt ∈ Ω(1 + a·n518₁² + n518₁)
power(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(1, n1170_1))) → *3_1, rt ∈ Ω(n1170₁)

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

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n518_1)) → gen_Cons:Nil2_1(*(n518_1, a)), rt ∈ Ω(1 + a·n518₁² + n518₁)

(21) BOUNDS(n^3, INF)

(22) Obligation:

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

Lemmas:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) → gen_Cons:Nil2_1(+(n4_1, a)), rt ∈ Ω(1 + n4₁)
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n518_1)) → gen_Cons:Nil2_1(*(n518_1, a)), rt ∈ Ω(1 + a·n518₁² + n518₁)
power(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(+(1, n1170_1))) → *3_1, rt ∈ Ω(n1170₁)

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

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n518_1)) → gen_Cons:Nil2_1(*(n518_1, a)), rt ∈ Ω(1 + a·n518₁² + n518₁)

(24) BOUNDS(n^3, INF)

(25) Obligation:

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

Lemmas:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) → gen_Cons:Nil2_1(+(n4_1, a)), rt ∈ Ω(1 + n4₁)
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n518_1)) → gen_Cons:Nil2_1(*(n518_1, a)), rt ∈ Ω(1 + a·n518₁² + n518₁)

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

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
mult(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n518_1)) → gen_Cons:Nil2_1(*(n518_1, a)), rt ∈ Ω(1 + a·n518₁² + n518₁)

(27) BOUNDS(n^3, INF)

(28) Obligation:

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

Lemmas:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) → gen_Cons:Nil2_1(+(n4_1, a)), rt ∈ Ω(1 + n4₁)

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

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
add0(gen_Cons:Nil2_1(a), gen_Cons:Nil2_1(n4_1)) → gen_Cons:Nil2_1(+(n4_1, a)), rt ∈ Ω(1 + n4₁)

(30) BOUNDS(n^1, INF)