The Runtime Complexity (full) 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:

f(a, empty) → g(a, empty)
f(a, cons(x, k)) → f(cons(x, a), k)
g(empty, d) → d
g(cons(x, k), d) → g(k, cons(x, d))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(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:

f(a, empty) → g(a, empty)
f(a, cons(x, k)) → f(cons(x, a), k)
g(empty, d) → d
g(cons(x, k), d) → g(k, cons(x, d))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
f(a, empty) → g(a, empty)
f(a, cons(x, k)) → f(cons(x, a), k)
g(empty, d) → d
g(cons(x, k), d) → g(k, cons(x, d))

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: a → empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
hole_a2_0 :: a
gen_empty:cons3_0 :: Nat → empty:cons

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

Heuristically decided to analyse the following defined symbols:
f, g

They will be analysed ascendingly in the following order:
g < f

(8) Obligation:

TRS:
Rules:
f(a, empty) → g(a, empty)
f(a, cons(x, k)) → f(cons(x, a), k)
g(empty, d) → d
g(cons(x, k), d) → g(k, cons(x, d))

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: a → empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
hole_a2_0 :: a
gen_empty:cons3_0 :: Nat → empty:cons

Generator Equations:
gen_empty:cons3_0(0) ⇔ empty
gen_empty:cons3_0(+(x, 1)) ⇔ cons(hole_a2_0, gen_empty:cons3_0(x))

The following defined symbols remain to be analysed:
g, f

They will be analysed ascendingly in the following order:
g < f

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
g(gen_empty:cons3_0(n5_0), gen_empty:cons3_0(b)) → gen_empty:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Induction Base:
g(gen_empty:cons3_0(0), gen_empty:cons3_0(b)) →_R^Ω(1)
gen_empty:cons3_0(b)

Induction Step:
g(gen_empty:cons3_0(+(n5_0, 1)), gen_empty:cons3_0(b)) →_R^Ω(1)
g(gen_empty:cons3_0(n5_0), cons(hole_a2_0, gen_empty:cons3_0(b))) →_IH
gen_empty:cons3_0(+(+(b, 1), c6_0))

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
f(a, empty) → g(a, empty)
f(a, cons(x, k)) → f(cons(x, a), k)
g(empty, d) → d
g(cons(x, k), d) → g(k, cons(x, d))

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: a → empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
hole_a2_0 :: a
gen_empty:cons3_0 :: Nat → empty:cons

Lemmas:
g(gen_empty:cons3_0(n5_0), gen_empty:cons3_0(b)) → gen_empty:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_empty:cons3_0(0) ⇔ empty
gen_empty:cons3_0(+(x, 1)) ⇔ cons(hole_a2_0, gen_empty:cons3_0(x))

The following defined symbols remain to be analysed:
f

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_empty:cons3_0(a), gen_empty:cons3_0(n426_0)) → gen_empty:cons3_0(+(n426_0, a)), rt ∈ Ω(1 + a + n426₀)

Induction Base:
f(gen_empty:cons3_0(a), gen_empty:cons3_0(0)) →_R^Ω(1)
g(gen_empty:cons3_0(a), empty) →_L^{Ω(1 + a)}
gen_empty:cons3_0(+(a, 0))

Induction Step:
f(gen_empty:cons3_0(a), gen_empty:cons3_0(+(n426_0, 1))) →_R^Ω(1)
f(cons(hole_a2_0, gen_empty:cons3_0(a)), gen_empty:cons3_0(n426_0)) →_IH
gen_empty:cons3_0(+(+(a, 1), c427_0))

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
f(a, empty) → g(a, empty)
f(a, cons(x, k)) → f(cons(x, a), k)
g(empty, d) → d
g(cons(x, k), d) → g(k, cons(x, d))

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: a → empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
hole_a2_0 :: a
gen_empty:cons3_0 :: Nat → empty:cons

Lemmas:
g(gen_empty:cons3_0(n5_0), gen_empty:cons3_0(b)) → gen_empty:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
f(gen_empty:cons3_0(a), gen_empty:cons3_0(n426_0)) → gen_empty:cons3_0(+(n426_0, a)), rt ∈ Ω(1 + a + n426₀)

Generator Equations:
gen_empty:cons3_0(0) ⇔ empty
gen_empty:cons3_0(+(x, 1)) ⇔ cons(hole_a2_0, gen_empty:cons3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_empty:cons3_0(n5_0), gen_empty:cons3_0(b)) → gen_empty:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

TRS:
Rules:
f(a, empty) → g(a, empty)
f(a, cons(x, k)) → f(cons(x, a), k)
g(empty, d) → d
g(cons(x, k), d) → g(k, cons(x, d))

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: a → empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
hole_a2_0 :: a
gen_empty:cons3_0 :: Nat → empty:cons

Lemmas:
g(gen_empty:cons3_0(n5_0), gen_empty:cons3_0(b)) → gen_empty:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
f(gen_empty:cons3_0(a), gen_empty:cons3_0(n426_0)) → gen_empty:cons3_0(+(n426_0, a)), rt ∈ Ω(1 + a + n426₀)

Generator Equations:
gen_empty:cons3_0(0) ⇔ empty
gen_empty:cons3_0(+(x, 1)) ⇔ cons(hole_a2_0, gen_empty:cons3_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_empty:cons3_0(n5_0), gen_empty:cons3_0(b)) → gen_empty:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

TRS:
Rules:
f(a, empty) → g(a, empty)
f(a, cons(x, k)) → f(cons(x, a), k)
g(empty, d) → d
g(cons(x, k), d) → g(k, cons(x, d))

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: a → empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
hole_a2_0 :: a
gen_empty:cons3_0 :: Nat → empty:cons

Lemmas:
g(gen_empty:cons3_0(n5_0), gen_empty:cons3_0(b)) → gen_empty:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_empty:cons3_0(0) ⇔ empty
gen_empty:cons3_0(+(x, 1)) ⇔ cons(hole_a2_0, gen_empty:cons3_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_empty:cons3_0(n5_0), gen_empty:cons3_0(b)) → gen_empty:cons3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)

(22) BOUNDS(n^1, INF)