Solving a simple geometric puzzle in CLPQ/R (Prolog)

Consider the following square:

You are given three constraints:

1. All rectangles (A, B, C, D and E) have the same area;
2. Their geometric layout constitutes a square; and
3. The height of A is 2.

Now, I know this is very simple to solve by hand, but I thought it would be a very good example to show off the capabilities of CLP(Q/R) with Prolog:

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SPOILER ALERT: If you want to first solve the puzzle by yourself, do not continue to read this, as there are constraints that will give away the solution.

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Anyway, here's my attempt at defining (with I think include redundant constraints) this puzzle with CLP(Q/R):

:- use_module(library(clpr)).

solve(Eh) :-
  A = B, B = C, C = D, D = E,

  { A  >= 1, B  >= 1, C  >= 1, D  >= 1, E  >= 1,
    Aw >= 1, Bw >= 1, Cw >= 1, Dw >= 1, Ew >= 1 },
  
  { Ah = 2 },
  
  { A = Ah * Aw,
    B = Bh * Bw,
    C = Ch * Cw,
    D = Dh * Dw,
    E = Eh * Ew },

  { (Bw + Cw) = Aw,
     Dw = Cw,
    (Ah + Bh) = Eh,
    (Ch + Dh) = Bh,
    (Aw + Ew) = Eh },

  minimize(Eh).

Which when queried:

?- solve(Eh).
false.

...makes me sad. Such a beautiful example for a constraint solver... Anyone cares to undo my sadness?

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Addendum: I used Mathematica and the FindMinimum function to check for my constraints. It seems to be working:

domain = a >= 1 && b >= 1 && c >= 1 && d >= 1 && e >= 1 && ah == 2.0 && a == b == c == d == e && aw >= 1 && bw >= 1 && cw >= 1 && dw >= 1 && ew >= 1
rectangles = (a == ah*aw && b == bh*bw && c == ch*cw && d == dh*dw && e == eh*ew)

FindMinimum[{eh, 
  domain && rectangles &&
  ((bw + cw ) == aw && dw == cw && (ah + bh) == eh && (ch + dh) == bh && (aw + ew) == eh)}, 
  {a, b, c, d, e, ah, aw, bh, bw, ch, cw, dh, dw, eh, ew}]

Answers:

{8., {a -> 12.8, b -> 12.8, c -> 12.8, d -> 12.8, e -> 12.8, 
      ah -> 2., aw -> 6.4, bh -> 6., bw -> 2.13333, ch -> 3., 
      cw -> 4.26667, dh -> 3., dw -> 4.26667, 
      eh -> 8., ew -> 1.6}}

There is a old/new entry in CLP, clpBNR. You can install it in a recent version of SWI-Prolog.

I think it would require to group equations together into a single {}.

?- pack_install(clpBNR).

:- use_module(library(clpBNR)).

solve_(Eh) :-
  Vs = [A,B,C,D,E, Aw,Bw,Cw,Dw,Ew, Ah,Bh,Ch,Dh,Eh],
  Vs::real(1,100),

  { Ah == 2,

    A is Ah * Aw,
    B is Bh * Bw,
    C is Ch * Cw,
    D is Dh * Dw,
    E is Eh * Ew,

    A == B,
    B == C,
    C == D,
    D == E,

    (Bw + Cw) == Aw,
     Dw == Cw,
    (Ah + Bh) == Eh,
    (Ch + Dh) == Bh,
    (Aw + Ew) == Eh
  },

  solve(Vs).

?- solve_(Eh).
::(Eh, ...( 8.000000)) .