RegionIntersection not returning what I expected

Question

Consider the following implicit region defined on one variable

reg = {x > 0 && x < 0}

If I Reduce this system, I obviously get

Reduce[reg]

False

However, If I use

RegionIntersection[ImplicitRegion[reg, {x}]]

I expect to get EmptyRegion[], instead I get the output

ImplicitRegion[x > 0 && x < 0, {x}]

Why does RegionIntersection not return EmptyRegion[]?

Answer 1

Region-combination functions such as RegionIntersection call BooleanRegion to compute the result. For instance, RegionIntersection[reg1, reg2,…] is equivalent to BooleanRegion[And, {reg1, reg2,…}]

In turn, BooleanRegion seems to apply some basic logic to eliminate unnecessary computation. The following and their equivalent RegionIntersection calls return region without inspecting, simplifying, or otherwise altering region:

BooleanRegion[And, {region}]
BooleanRegion[And, {region, region}]  (* DeleteDuplicates[] is used to remove copies *)
BooleanRegion[And, {region, FullRegion[n]}]  (* where n is the dimension of region *)

Possible workarounds include intersecting region with a region distinct from region and FullRegion[n] that covers region. Simply specifying a full region as an ImplicitRegion or changing the variables in region suffice. Unfortunately Simplify[ImplicitRegion[..]] does nothing. In this case, if we apply Simplify or Reduce to the first argument gets around this.

ireg = ImplicitRegion[x < 0 && x > 0, {x}]
yreg = ireg /. x -> y  (* change variable *)
fullreg = ImplicitRegion[-Infinity < x < Infinity, {x}] (* a disguised full region *)
(*
  ImplicitRegion[x < 0 && x > 0, {x}]
  ImplicitRegion[y < 0 && y > 0, {y}]
  ImplicitRegion[-∞ < x < ∞, {x}]
*)

RegionIntersection[ireg, yreg]
RegionIntersection[ireg, fullreg]
(*
  EmptyRegion[1]
  EmptyRegion[1]
*)

Simplification:

MapAt[Simplify, ireg, 1]
MapAt[Reduce, ireg, 1]
(*
  EmptyRegion[1]
  EmptyRegion[1]
*)

It seems Mathematica is missing a RegionSimplify or RegionReduce function. At least, I didn't find one.

Answer 2

$\underline{\mathbf{UPDATE}}$

Q = ImplicitRegion[x > 0 && x < 0, {x}];
Q = If[RegionDimension[Q] == -\[Infinity], EmptyRegion[1]]

returns:

EmptyRegion[1]