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Commit de7dd977 authored by François Clément's avatar François Clément
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Renaming (Inter_with_empty -> Inter_wEmpty).

Add results about Inter and Union when adding {empty,full}set.
parent c86706a3
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......@@ -210,8 +210,8 @@ Proof.
intros H A B _ HA HB; now apply H.
Qed.
Lemma Inter_with_empty :
Inter P -> Inter (fun A => P A \/ A = emptyset).
Lemma Inter_wEmpty :
Inter P -> Inter (add P emptyset).
Proof.
intros H A B [HA | HA] [HB | HB].
left; now apply H.
......@@ -219,6 +219,16 @@ right; rewrite HB; apply inter_emptyset_r.
1,2: right; rewrite HA; apply inter_emptyset_l.
Qed.
Lemma Inter_wFull :
Inter P -> Inter (add P fullset).
Proof.
intros H A B [HA | HA] [HB | HB].
left; now apply H.
left; now rewrite HB, inter_fullset_r.
left; now rewrite HA, inter_fullset_l.
right; now rewrite HA, HB, inter_fullset_l.
Qed.
Lemma Inter_Diff :
Compl P -> Inter P -> Diff P.
Proof.
......@@ -263,6 +273,25 @@ now apply Inter_Diff.
apply Diff_Inter.
Qed.
Lemma Union_wEmpty :
Union P -> Union (add P emptyset).
Proof.
intros H A B [HA | HA] [HB | HB].
left; now apply H.
left; now rewrite HB, union_emptyset_r.
left; now rewrite HA, union_emptyset_l.
right; now rewrite HA, HB, union_emptyset_l.
Qed.
Lemma Union_wFull :
Union P -> Union (add P fullset).
Proof.
intros H A B [HA | HA] [HB | HB].
left; now apply H.
right; rewrite HB; apply union_fullset_r.
1,2: right; rewrite HA; apply union_fullset_l.
Qed.
Lemma Union_disj_Union :
Diff P -> Union_disj P -> Union P.
Proof.
......
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