Coq 证明补码是内合的
Coq proof that complement is involutive
如何证明集合的补集是内合的？
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) 使 firstorder 能够完全证明引理。
complement_involutive 恰好是 ~ ~ A x -> A x，即 well-known 等同于排中律，在本例中是 Type，因此如果不假设它是，则无法在 Coq 中证明一个公理。看到这个答案 https://math.stackexchange.com/questions/1370805/why-cant-you-prove-the-law-of-the-excluded-middle-in-intuitionistic-logic-for
如何证明集合的补集是内合的？
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) 使 firstorder 能够完全证明引理。
complement_involutive 恰好是 ~ ~ A x -> A x，即 well-known 等同于排中律，在本例中是 Type，因此如果不假设它是，则无法在 Coq 中证明一个公理。看到这个答案 https://math.stackexchange.com/questions/1370805/why-cant-you-prove-the-law-of-the-excluded-middle-in-intuitionistic-logic-for