Standard Library | The Coq Proof Assistant

The Coq Proof Assistant

# Library Coq.Sets.Classical_sets

Require Export Ensembles.
Require Export Constructive_sets.
Require Export Classical.
Section Ensembles_classical.
Variable U : Type.
Lemma not_included_empty_Inhabited :
forall A:Ensemble U, ~ Included U A (Empty_set U) -> Inhabited U A.
Lemma not_empty_Inhabited :
forall A:Ensemble U, A <> Empty_set U -> Inhabited U A.
Lemma Inhabited_Setminus :
forall X Y:Ensemble U,
Included U X Y -> ~ Included U Y X -> Inhabited U (Setminus U Y X).
Lemma Strict_super_set_contains_new_element :
forall X Y:Ensemble U,
Included U X Y -> X <> Y -> Inhabited U (Setminus U Y X).
Lemma Subtract_intro :
forall (A:Ensemble U) (x y:U), In U A y -> x <> y -> In U (Subtract U A x) y.
Hint Resolve Subtract_intro : sets.
Lemma Subtract_inv :
forall (A:Ensemble U) (x y:U), In U (Subtract U A x) y -> In U A y /\ x <> y.
Lemma Included_Strict_Included :
forall X Y:Ensemble U, Included U X Y -> Strict_Included U X Y \/ X = Y.
Lemma Strict_Included_inv :
forall X Y:Ensemble U,
Strict_Included U X Y -> Included U X Y /\ Inhabited U (Setminus U Y X).
Lemma not_SIncl_empty :
forall X:Ensemble U, ~ Strict_Included U X (Empty_set U).
Lemma Complement_Complement :
forall A:Ensemble U, Complement U (Complement U A) = A.
End Ensembles_classical.
Hint Resolve Strict_super_set_contains_new_element Subtract_intro
not_SIncl_empty: sets.