The Runtime Complexity (innermost) 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(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Types:
rev :: a:b:++ → a:b:++
a :: a:b:++
b :: a:b:++
++ :: a:b:++ → a:b:++ → a:b:++
hole_a:b:++1_0 :: a:b:++
gen_a:b:++2_0 :: Nat → a:b:++

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
rev

(6) Obligation:

Innermost TRS:
Rules:
rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Types:
rev :: a:b:++ → a:b:++
a :: a:b:++
b :: a:b:++
++ :: a:b:++ → a:b:++ → a:b:++
hole_a:b:++1_0 :: a:b:++
gen_a:b:++2_0 :: Nat → a:b:++

Generator Equations:
gen_a:b:++2_0(0) ⇔ a
gen_a:b:++2_0(+(x, 1)) ⇔ ++(a, gen_a:b:++2_0(x))

The following defined symbols remain to be analysed:
rev

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
rev(gen_a:b:++2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Induction Base:
rev(gen_a:b:++2_0(+(1, 0)))

Induction Step:
rev(gen_a:b:++2_0(+(1, +(n4_0, 1)))) →_R^Ω(1)
++(rev(gen_a:b:++2_0(+(1, n4_0))), rev(a)) →_IH
++(*3_0, rev(a)) →_R^Ω(1)
++(*3_0, a)

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Types:
rev :: a:b:++ → a:b:++
a :: a:b:++
b :: a:b:++
++ :: a:b:++ → a:b:++ → a:b:++
hole_a:b:++1_0 :: a:b:++
gen_a:b:++2_0 :: Nat → a:b:++

Lemmas:
rev(gen_a:b:++2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_a:b:++2_0(0) ⇔ a
gen_a:b:++2_0(+(x, 1)) ⇔ ++(a, gen_a:b:++2_0(x))

No more defined symbols left to analyse.

(10) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
rev(gen_a:b:++2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(11) BOUNDS(n^1, INF)

(12) Obligation:

Innermost TRS:
Rules:
rev(a) → a
rev(b) → b
rev(++(x, y)) → ++(rev(y), rev(x))
rev(++(x, x)) → rev(x)

Types:
rev :: a:b:++ → a:b:++
a :: a:b:++
b :: a:b:++
++ :: a:b:++ → a:b:++ → a:b:++
hole_a:b:++1_0 :: a:b:++
gen_a:b:++2_0 :: Nat → a:b:++

Lemmas:
rev(gen_a:b:++2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

Generator Equations:
gen_a:b:++2_0(0) ⇔ a
gen_a:b:++2_0(+(x, 1)) ⇔ ++(a, gen_a:b:++2_0(x))

No more defined symbols left to analyse.

(13) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
rev(gen_a:b:++2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n4₀)

(14) BOUNDS(n^1, INF)