How to solve this nonstandard system?

Namely, I mean

solve({y >= 4*x^4+4*x^2*y+1/2, sqrt((1/2)*(x-y)^2-(x-y)^4) = -2*x^2+y^2}, [x, y]);
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The answer (no solution) is not correct in view of 

eval({y >= 4*x^4+4*x^2*y+1/2, sqrt((1/2)*(x-y)^2-(x-y)^4) = -2*x^2+y^2}, [x = 0, y = 1/2]);
      {(1/8)*sqrt(4) = 1/4, 1/2 <= 1/2}
eval({y >= 4*x^4+4*x^2*y+1/2, sqrt((1/2)*(x-y)^2-(x-y)^4) = -2*x^2+y^2}, [x = -1, y = -3/2]);
      {(1/8)*sqrt(4) = 1/4, -3/2 <= -3/2}