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:

rev(ls) → r1(ls, empty)
r1(empty, a) → a
r1(cons(x, k), a) → r1(k, cons(x, a))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
r1(cons(x, k), a) →⁺ r1(k, cons(x, a))
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:

rev(ls) → r1(ls, empty)
r1(empty, a) → a
r1(cons(x, k), a) → r1(k, cons(x, a))

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:

rev(ls) → r1(ls, empty)
r1(empty, a) → a
r1(cons(k), a) → r1(k, cons(a))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
rev(ls) → r1(ls, empty)
r1(empty, a) → a
r1(cons(k), a) → r1(k, cons(a))

Types:
rev :: empty:cons → empty:cons
r1 :: empty:cons → empty:cons → empty:cons
empty :: 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:
r1

(10) Obligation:

TRS:
Rules:
rev(ls) → r1(ls, empty)
r1(empty, a) → a
r1(cons(k), a) → r1(k, cons(a))

Types:
rev :: empty:cons → empty:cons
r1 :: empty:cons → empty:cons → empty:cons
empty :: 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:
r1

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

Proved the following rewrite lemma:
r1(gen_empty:cons2_0(n4_0), gen_empty:cons2_0(b)) → gen_empty:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

Induction Base:
r1(gen_empty:cons2_0(0), gen_empty:cons2_0(b)) →_R^Ω(1)
gen_empty:cons2_0(b)

Induction Step:
r1(gen_empty:cons2_0(+(n4_0, 1)), gen_empty:cons2_0(b)) →_R^Ω(1)
r1(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:
rev(ls) → r1(ls, empty)
r1(empty, a) → a
r1(cons(k), a) → r1(k, cons(a))

Types:
rev :: empty:cons → empty:cons
r1 :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
cons :: empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
gen_empty:cons2_0 :: Nat → empty:cons

Lemmas:
r1(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.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
r1(gen_empty:cons2_0(n4_0), gen_empty:cons2_0(b)) → gen_empty:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

TRS:
Rules:
rev(ls) → r1(ls, empty)
r1(empty, a) → a
r1(cons(k), a) → r1(k, cons(a))

Types:
rev :: empty:cons → empty:cons
r1 :: empty:cons → empty:cons → empty:cons
empty :: empty:cons
cons :: empty:cons → empty:cons
hole_empty:cons1_0 :: empty:cons
gen_empty:cons2_0 :: Nat → empty:cons

Lemmas:
r1(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.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
r1(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)