The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

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

rev(xs) → revtl(xs, nil)
revtl(nil, ys) → ys
revtl(cons(x, xs), ys) → revtl(xs, cons(x, ys))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

rev(z0) → revtl(z0, nil)
revtl(nil, z0) → z0
revtl(cons(z0, z1), z2) → revtl(z1, cons(z0, z2))

Tuples:

REV(z0) → c(REVTL(z0, nil))
REVTL(cons(z0, z1), z2) → c2(REVTL(z1, cons(z0, z2)))

S tuples:

REV(z0) → c(REVTL(z0, nil))
REVTL(cons(z0, z1), z2) → c2(REVTL(z1, cons(z0, z2)))

K tuples:none
Defined Rule Symbols:

rev, revtl

Defined Pair Symbols:

REV, REVTL

Compound Symbols:

c, c2

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

Removed 1 of 2 dangling nodes:

REV(z0) → c(REVTL(z0, nil))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

rev(z0) → revtl(z0, nil)
revtl(nil, z0) → z0
revtl(cons(z0, z1), z2) → revtl(z1, cons(z0, z2))

Tuples:

REVTL(cons(z0, z1), z2) → c2(REVTL(z1, cons(z0, z2)))

S tuples:

REVTL(cons(z0, z1), z2) → c2(REVTL(z1, cons(z0, z2)))

K tuples:none
Defined Rule Symbols:

rev, revtl

Defined Pair Symbols:

REVTL

Compound Symbols:

c2

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

REVTL(cons(z0, z1), z2) → c2(REVTL(z1, cons(z0, z2)))

We considered the (Usable) Rules:none
And the Tuples:

REVTL(cons(z0, z1), z2) → c2(REVTL(z1, cons(z0, z2)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(REVTL(x₁, x₂)) = [4]x₁ + [3]x₂   
POL(c2(x₁)) = x₁   
POL(cons(x₁, x₂)) = [5] + x₁ + x₂   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

rev(z0) → revtl(z0, nil)
revtl(nil, z0) → z0
revtl(cons(z0, z1), z2) → revtl(z1, cons(z0, z2))

Tuples:

REVTL(cons(z0, z1), z2) → c2(REVTL(z1, cons(z0, z2)))

S tuples:none
K tuples:

REVTL(cons(z0, z1), z2) → c2(REVTL(z1, cons(z0, z2)))

Defined Rule Symbols:

rev, revtl

Defined Pair Symbols:

REVTL

Compound Symbols:

c2

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))