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(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
f(cons(x, k), l) →⁺ f(k, cons(l, cons(x, 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 [l / cons(l, cons(x, k))].

(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(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
f(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))

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

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

Heuristically decided to analyse the following defined symbols:
f

(8) Obligation:

TRS:
Rules:
f(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
cons :: empty:cons → empty:cons → empty:cons
g :: empty:cons → empty: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(empty, gen_empty:cons2_0(x))

The following defined symbols remain to be analysed:
f

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol f.

(10) Obligation:

TRS:
Rules:
f(empty, l) → l
f(cons(x, k), l) → g(k, l, cons(x, k))
g(a, b, c) → f(a, cons(b, c))

Types:
f :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
cons :: empty:cons → empty:cons → empty:cons
g :: empty:cons → empty: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(empty, gen_empty:cons2_0(x))

No more defined symbols left to analyse.