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) 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:

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

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

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

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
gen_empty:cons2_0 :: Nat → empty:cons

(9) 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

(10) Obligation:

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

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
gen_empty:cons2_0 :: Nat → empty:cons

Generator Equations:
gen_empty:cons2_0(0) ⇔ empty
gen_empty:cons2_0(+(x, 1)) ⇔ cons(gen_empty:cons2_0(x))

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

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

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

Proved the following rewrite lemma:
g(gen_empty:cons2_0(n4_0), gen_empty:cons2_0(b)) → gen_empty:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Induction Base:
g(gen_empty:cons2_0(0), gen_empty:cons2_0(b)) →_R^Ω(1)
gen_empty:cons2_0(b)

Induction Step:
g(gen_empty:cons2_0(+(n4_0, 1)), gen_empty:cons2_0(b)) →_R^Ω(1)
g(gen_empty:cons2_0(n4_0), cons(gen_empty:cons2_0(b))) →_IH
gen_empty:cons2_0(+(+(b, 1), c5_0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

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

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
gen_empty:cons2_0 :: Nat → empty:cons

Lemmas:
g(gen_empty:cons2_0(n4_0), gen_empty:cons2_0(b)) → gen_empty:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_empty:cons2_0(0) ⇔ empty
gen_empty:cons2_0(+(x, 1)) ⇔ cons(gen_empty:cons2_0(x))

The following defined symbols remain to be analysed:
f

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

Proved the following rewrite lemma:
f(gen_empty:cons2_0(a), gen_empty:cons2_0(n407_0)) → gen_empty:cons2_0(+(n407_0, a)), rt ∈ Ω(1 + a + n407₀)

Induction Base:
f(gen_empty:cons2_0(a), gen_empty:cons2_0(0)) →_R^Ω(1)
g(gen_empty:cons2_0(a), empty) →_L^{Ω(1 + a)}
gen_empty:cons2_0(+(a, 0))

Induction Step:
f(gen_empty:cons2_0(a), gen_empty:cons2_0(+(n407_0, 1))) →_R^Ω(1)
f(cons(gen_empty:cons2_0(a)), gen_empty:cons2_0(n407_0)) →_IH
gen_empty:cons2_0(+(+(a, 1), c408_0))

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

(15) Complex Obligation (BEST)

(16) Obligation:

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

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
gen_empty:cons2_0 :: Nat → empty:cons

Lemmas:
g(gen_empty:cons2_0(n4_0), gen_empty:cons2_0(b)) → gen_empty:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
f(gen_empty:cons2_0(a), gen_empty:cons2_0(n407_0)) → gen_empty:cons2_0(+(n407_0, a)), rt ∈ Ω(1 + a + n407₀)

Generator Equations:
gen_empty:cons2_0(0) ⇔ empty
gen_empty:cons2_0(+(x, 1)) ⇔ cons(gen_empty:cons2_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_empty:cons2_0(n4_0), gen_empty:cons2_0(b)) → gen_empty:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)

(19) Obligation:

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

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
gen_empty:cons2_0 :: Nat → empty:cons

Lemmas:
g(gen_empty:cons2_0(n4_0), gen_empty:cons2_0(b)) → gen_empty:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)
f(gen_empty:cons2_0(a), gen_empty:cons2_0(n407_0)) → gen_empty:cons2_0(+(n407_0, a)), rt ∈ Ω(1 + a + n407₀)

Generator Equations:
gen_empty:cons2_0(0) ⇔ empty
gen_empty:cons2_0(+(x, 1)) ⇔ cons(gen_empty:cons2_0(x))

No more defined symbols left to analyse.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_empty:cons2_0(n4_0), gen_empty:cons2_0(b)) → gen_empty:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

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

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
g :: empty:cons → empty:cons → empty:cons
cons :: empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
gen_empty:cons2_0 :: Nat → empty:cons

Lemmas:
g(gen_empty:cons2_0(n4_0), gen_empty:cons2_0(b)) → gen_empty:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_empty:cons2_0(0) ⇔ empty
gen_empty:cons2_0(+(x, 1)) ⇔ cons(gen_empty:cons2_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
g(gen_empty:cons2_0(n4_0), gen_empty:cons2_0(b)) → gen_empty:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(24) BOUNDS(n^1, INF)