COQ证明补体是参与的

Question

我如何证明集合的补充是涉及的？

  Require Import Ensembles. Arguments In {_}. Arguments Complement {_}.
  Variables (T:Type) (A:Ensemble T).
  Axiom set_eq: forall (E1 E2:Ensemble T), (forall x, E1 x <-> E2 x) -> E1 = E2.
  Lemma complement_involutive: 
      forall x, In (Complement (Complement A)) x -> In A x.

编辑：假设decidable (In A x)使firstorder完全证明了引理。

Answer 1

complement_involutive恰好是 ~ ~ A x -> A x，众所周知，在 Type中，在这种情况下等效于中间，因此在COQ中无法证明，而不假设它是公理。请参阅此答案https://math.stackexchange.com/questions/1370805/why-cant-you-prove-theaw-the-thear-of-the-middle-middle-in-intuitionistic in-intuitionistic logic-forc-forfor