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:

rev1(0, nil) → 0
rev1(s(X), nil) → s(X)
rev1(X, cons(Y, L)) → rev1(Y, L)
rev(nil) → nil
rev(cons(X, L)) → cons(rev1(X, L), rev2(X, L))
rev2(X, nil) → nil
rev2(X, cons(Y, L)) → rev(cons(X, rev(rev2(Y, L))))

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:

rev1(0', nil) → 0'
rev1(s(X), nil) → s(X)
rev1(X, cons(Y, L)) → rev1(Y, L)
rev(nil) → nil
rev(cons(X, L)) → cons(rev1(X, L), rev2(X, L))
rev2(X, nil) → nil
rev2(X, cons(Y, L)) → rev(cons(X, rev(rev2(Y, L))))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
rev1(0', nil) → 0'
rev1(s(X), nil) → s(X)
rev1(X, cons(Y, L)) → rev1(Y, L)
rev(nil) → nil
rev(cons(X, L)) → cons(rev1(X, L), rev2(X, L))
rev2(X, nil) → nil
rev2(X, cons(Y, L)) → rev(cons(X, rev(rev2(Y, L))))

Types:
rev1 :: 0':s → nil:cons → 0':s
0' :: 0':s
nil :: nil:cons
s :: a → 0':s
cons :: 0':s → nil:cons → nil:cons
rev :: nil:cons → nil:cons
rev2 :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_a3_0 :: a
gen_nil:cons4_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
rev1, rev, rev2

They will be analysed ascendingly in the following order:
rev1 < rev
rev = rev2

(6) Obligation:

Innermost TRS:
Rules:
rev1(0', nil) → 0'
rev1(s(X), nil) → s(X)
rev1(X, cons(Y, L)) → rev1(Y, L)
rev(nil) → nil
rev(cons(X, L)) → cons(rev1(X, L), rev2(X, L))
rev2(X, nil) → nil
rev2(X, cons(Y, L)) → rev(cons(X, rev(rev2(Y, L))))

Types:
rev1 :: 0':s → nil:cons → 0':s
0' :: 0':s
nil :: nil:cons
s :: a → 0':s
cons :: 0':s → nil:cons → nil:cons
rev :: nil:cons → nil:cons
rev2 :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_a3_0 :: a
gen_nil:cons4_0 :: Nat → nil:cons

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
rev1, rev, rev2

They will be analysed ascendingly in the following order:
rev1 < rev
rev = rev2

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

Proved the following rewrite lemma:
rev1(0', gen_nil:cons4_0(n6_0)) → 0', rt ∈ Ω(1 + n6₀)

Induction Base:
rev1(0', gen_nil:cons4_0(0)) →_R^Ω(1)
0'

Induction Step:
rev1(0', gen_nil:cons4_0(+(n6_0, 1))) →_R^Ω(1)
rev1(0', gen_nil:cons4_0(n6_0)) →_IH
0'

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:
rev1(0', nil) → 0'
rev1(s(X), nil) → s(X)
rev1(X, cons(Y, L)) → rev1(Y, L)
rev(nil) → nil
rev(cons(X, L)) → cons(rev1(X, L), rev2(X, L))
rev2(X, nil) → nil
rev2(X, cons(Y, L)) → rev(cons(X, rev(rev2(Y, L))))

Types:
rev1 :: 0':s → nil:cons → 0':s
0' :: 0':s
nil :: nil:cons
s :: a → 0':s
cons :: 0':s → nil:cons → nil:cons
rev :: nil:cons → nil:cons
rev2 :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_a3_0 :: a
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
rev1(0', gen_nil:cons4_0(n6_0)) → 0', rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
rev2, rev

They will be analysed ascendingly in the following order:
rev = rev2

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol rev2.

(11) Obligation:

Innermost TRS:
Rules:
rev1(0', nil) → 0'
rev1(s(X), nil) → s(X)
rev1(X, cons(Y, L)) → rev1(Y, L)
rev(nil) → nil
rev(cons(X, L)) → cons(rev1(X, L), rev2(X, L))
rev2(X, nil) → nil
rev2(X, cons(Y, L)) → rev(cons(X, rev(rev2(Y, L))))

Types:
rev1 :: 0':s → nil:cons → 0':s
0' :: 0':s
nil :: nil:cons
s :: a → 0':s
cons :: 0':s → nil:cons → nil:cons
rev :: nil:cons → nil:cons
rev2 :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_a3_0 :: a
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
rev1(0', gen_nil:cons4_0(n6_0)) → 0', rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
rev

They will be analysed ascendingly in the following order:
rev = rev2

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

Innermost TRS:
Rules:
rev1(0', nil) → 0'
rev1(s(X), nil) → s(X)
rev1(X, cons(Y, L)) → rev1(Y, L)
rev(nil) → nil
rev(cons(X, L)) → cons(rev1(X, L), rev2(X, L))
rev2(X, nil) → nil
rev2(X, cons(Y, L)) → rev(cons(X, rev(rev2(Y, L))))

Types:
rev1 :: 0':s → nil:cons → 0':s
0' :: 0':s
nil :: nil:cons
s :: a → 0':s
cons :: 0':s → nil:cons → nil:cons
rev :: nil:cons → nil:cons
rev2 :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_a3_0 :: a
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
rev1(0', gen_nil:cons4_0(n6_0)) → 0', rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
rev1(0', gen_nil:cons4_0(n6_0)) → 0', rt ∈ Ω(1 + n6₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
rev1(0', nil) → 0'
rev1(s(X), nil) → s(X)
rev1(X, cons(Y, L)) → rev1(Y, L)
rev(nil) → nil
rev(cons(X, L)) → cons(rev1(X, L), rev2(X, L))
rev2(X, nil) → nil
rev2(X, cons(Y, L)) → rev(cons(X, rev(rev2(Y, L))))

Types:
rev1 :: 0':s → nil:cons → 0':s
0' :: 0':s
nil :: nil:cons
s :: a → 0':s
cons :: 0':s → nil:cons → nil:cons
rev :: nil:cons → nil:cons
rev2 :: 0':s → nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_a3_0 :: a
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
rev1(0', gen_nil:cons4_0(n6_0)) → 0', rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
rev1(0', gen_nil:cons4_0(n6_0)) → 0', rt ∈ Ω(1 + n6₀)

(18) BOUNDS(n^1, INF)