Library Stdlib.Sets.Powerset_Classical_facts

Require Export Ensembles.
Require Export Constructive_sets.
Require Export Relations_1.
Require Export Relations_1_facts.
Require Export Partial_Order.
Require Export Cpo.
Require Export Powerset.
Require Export Powerset_facts.
Require Export Classical.
Require Export Classical_sets.

Section Sets_as_an_algebra.

  Variable U : Type.

  Lemma sincl_add_x :
    forall (A B:Ensemble U) (x:U),
      ~ In U A x ->
      Strict_Included U (Add U A x) (Add U B x) -> Strict_Included U A B.

  Lemma incl_soustr_in :
    forall (X:Ensemble U) (x:U), In U X x -> Included U (Subtract U X x) X.

  Lemma incl_soustr :
    forall (X Y:Ensemble U) (x:U),
      Included U X Y -> Included U (Subtract U X x) (Subtract U Y x).

  Lemma incl_soustr_add_l :
    forall (X:Ensemble U) (x:U), Included U (Subtract U (Add U X x) x) X.

  Lemma incl_soustr_add_r :
    forall (X:Ensemble U) (x:U),
      ~ In U X x -> Included U X (Subtract U (Add U X x) x).
  #[local]
  Hint Resolve incl_soustr_add_r: sets.

  Lemma add_soustr_2 :
    forall (X:Ensemble U) (x:U),
      In U X x -> Included U X (Add U (Subtract U X x) x).

  Lemma add_soustr_1 :
    forall (X:Ensemble U) (x:U),
      In U X x -> Included U (Add U (Subtract U X x) x) X.

  Lemma add_soustr_xy :
    forall (X:Ensemble U) (x y:U),
      x <> y -> Subtract U (Add U X x) y = Add U (Subtract U X y) x.

  Lemma incl_st_add_soustr :
    forall (X Y:Ensemble U) (x:U),
      ~ In U X x ->
      Strict_Included U (Add U X x) Y -> Strict_Included U X (Subtract U Y x).

  Lemma Sub_Add_new :
    forall (X:Ensemble U) (x:U), ~ In U X x -> X = Subtract U (Add U X x) x.

  Lemma Simplify_add :
    forall (X X0:Ensemble U) (x:U),
      ~ In U X x -> ~ In U X0 x -> Add U X x = Add U X0 x -> X = X0.

  Lemma Included_Add :
    forall (X A:Ensemble U) (x:U),
      Included U X (Add U A x) ->
      Included U X A \/ (exists A' : _, X = Add U A' x /\ Included U A' A).

  Lemma setcover_inv :
    forall A x y:Ensemble U,
      covers (Ensemble U) (Power_set_PO U A) y x ->
      Strict_Included U x y /\
      (forall z:Ensemble U, Included U x z -> Included U z y -> x = z \/ z = y).

  Theorem Add_covers :
    forall A a:Ensemble U,
      Included U a A ->
      forall x:U,
        In U A x ->
        ~ In U a x -> covers (Ensemble U) (Power_set_PO U A) (Add U a x) a.

  Theorem covers_Add :
    forall A a a':Ensemble U,
      Included U a A ->
      Included U a' A ->
      covers (Ensemble U) (Power_set_PO U A) a' a ->
      exists x : _, a' = Add U a x /\ In U A x /\ ~ In U a x.

  Theorem covers_is_Add :
    forall A a a':Ensemble U,
      Included U a A ->
      Included U a' A ->
      (covers (Ensemble U) (Power_set_PO U A) a' a <->
        (exists x : _, a' = Add U a x /\ In U A x /\ ~ In U a x)).

  Theorem Singleton_atomic :
    forall (x:U) (A:Ensemble U),
      In U A x ->
      covers (Ensemble U) (Power_set_PO U A) (Singleton U x) (Empty_set U).

  Lemma less_than_singleton :
    forall (X:Ensemble U) (x:U),
      Strict_Included U X (Singleton U x) -> X = Empty_set U.

End Sets_as_an_algebra.

#[global]
Hint Resolve incl_soustr_in: sets.
#[global]
Hint Resolve incl_soustr: sets.
#[global]
Hint Resolve incl_soustr_add_l: sets.
#[global]
Hint Resolve incl_soustr_add_r: sets.
#[global]
Hint Resolve add_soustr_1 add_soustr_2: sets.
#[global]
Hint Resolve add_soustr_xy: sets.