How to make Mathematica try harder to perform symbolic comparisons?

Question

(I suspect this question is a duplicate, but I didn't find a sufficiently similar question with an answer to it.)

I'm having trouble with comparisons of symbolic Reals that are equal, but which Mathematica has trouble recognising them as equal, apparently because it uses N to compare these (apparently after some simple symbolic manipulation). Typically these issues creep up in conditions like one (greatly simplified) below:

x > y /. {x -> Sqrt[(2 - Sqrt[3])^2 + (-3 + 2*Sqrt[3])^2], y -> 4 - 2*Sqrt[3]}

N::meprec: "Internal precision limit $MaxExtraPrecision = 50.` reached while evaluating -4+2\ Sqrt[3]+Sqrt[(2-Sqrt[3])^2+(-3+2 Sqrt[3])^2]."

Sqrt[(2 - Sqrt[3])^2 + (-3 + 2*Sqrt[3])^2] > 4 - 2*Sqrt[3]

This is in no way undocumented feature; it is discussed under Possible Issues section of $MaxExtraPrecision.

What I really want is Mathematica to try a bit harder on solving these (in)equalities numerically in a block of code. How do I accomplish this? As a workaround for specific problem above, this does work (while Simplify on inequality part doesn't):

x > y /. Simplify @ {x -> Sqrt[(2 - Sqrt[3])^2 + (-3 + 2*Sqrt[3])^2], y -> 4 - 2*Sqrt[3]}

False

I would be happy to see a solution that could wrap up around a block of code, and which could possibly work also on non-algebraics that FullSimplify can handle.

EDIT:

To clarify my question: I want first example to evaluate like the expression below, and in general Greater to evaluate for NumericQ argument list similarly inside a code block where I want this feature to be used:

  Simplify[Sqrt[(2 - Sqrt[3])^2 + (-3 + 2*Sqrt[3])^2]] > Simplify[4 - 2*Sqrt[3]]

False

Answer 1

This solution uses trick to redefine internal functions presented in this earlier answer by Leonid Shifrin.

(* Code run as an argument of WithFullySimplifiedComparisons tries
   harder to symbolically resolve equality and inequality tests. *)

ClearAll[WithFullySimplifiedComparisons];
WithFullySimplifiedComparisons =
  Module[
   {withReplacedFunctions, fullSimplifiedTest, testMappings, 
    splitInequality, wrapper},

   (* Runs code with functions in { old -> new, ... } mappings list
      executing new function in place of old, but old definitions
      working inside new implementations. *)

   SetAttributes[withReplacedFunctions, HoldRest];
   withReplacedFunctions[mappings_List, code_] :=
    Module[{outer = True},
     Internal`InheritedBlock[Evaluate@mappings[[All, 1]],
      (Unprotect[#1];
         #1[args___] :=
          Block[{outer = False}, #2[args]] /; outer;
         Protect[#1];) & @@@ mappings;
      code]];

   (* Returns a version of comparison function fun that attempts to
      simplify numeric comparisons in domain dom with FullSimplify. *)

   fullySimplifiedTest[fun_, dom_] :=
    Module[{test},
     test[] := True;
     test[_] := True;
     test[a_, b_, r___] :=
      With[{bn = FullSimplify[b]},
        If[fun[0, FullSimplify[bn - a]],
          test[bn, r], False, test[a, bn] && test[bn, r]]] /; 
       NumericQ[a] && NumericQ[b] && (a | b) \[Element] dom;
     test[a_, b_, r___] := fun[a, b] && test[b, r];
     test];

   (* Mappings for normal comparison functions. *)

   testMappings = 
    #1 -> fullySimplifiedTest[##] & @@@ 
     Join[{#, Complexes} & /@ {Equal, Unequal},
       {#, Reals} & /@ {Greater, GreaterEqual, Less, LessEqual}];

   (* Conversion function from Inequality to chain of simple tests. *)

   splitInequality[_] := True;
   splitInequality[a_, test_, b_, r___] := test[a, b] && splitInequality[b, r];

   (* Actual wrapper to be used.
      splitInequality uses new comparison functions. *)

   SetAttributes[wrapper, HoldFirst];
   wrapper[code_] :=
     withReplacedFunctions[testMappings,
       withReplacedFunctions[{Inequality -> splitInequality},
         code]];

   wrapper];

Now earlier problematic tests can run purely symbolically:

WithFullySimplifiedComparisons[
  x > y /. {x -> Sqrt[(2 - Sqrt[3])^2 + (-3 + 2*Sqrt[3])^2], y -> 4 - 2*Sqrt[3]}]

False