Question

如何证明集合的补集是对合的？

  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)允许firstorder完全证明引理。

Answer 1

complement_involutive正好是~ ~ A x -> A x，众所周知，它等同于被排除的中间，在这种情况下是Type，因此在Coq中无法证明它不是公理。请参阅此回答https://math.stackexchange.com/questions/1370805/why-cant-you-prove-the-law-of-the-excluded-middle-in-intuitionistic-logic-for

Coq证明补充是对合的

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