set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/class_eq".
-include "class_defs.ma".
+include "class_le.ma".
theorem ceq_cl: \forall C,c1,c2. ceq ? c1 c2 \to cin C c1 \land cin C c2.
-intros; elim H; clear H; clear c2;
- [ auto | decompose H2; auto | decompose H2; auto ].
+intros; elim H; clear H.
+lapply cle_cl to H1 using H; clear H1; decompose H;
+lapply cle_cl to H2 using H; clear H2; decompose H.
+auto.
qed.
-theorem ceq_trans: \forall C,c2,c1,c3.
- ceq C c2 c3 \to ceq ? c1 c2 \to ceq ? c1 c3.
-intros 5; elim H; clear H; clear c3;
- [ auto
- | apply ceq_sing_r; [||| apply H4 ]; auto
- | apply ceq_sing_l; [||| apply H4 ]; auto
- ].
+theorem ceq_refl: \forall C,c. cin C c \to ceq ? c c.
+intros; apply ceq_intro; auto.
qed.
-theorem ceq_conf_rev: \forall C,c2,c1,c3.
- ceq C c3 c2 \to ceq ? c1 c2 \to ceq ? c1 c3.
-intros 5; elim H; clear H; clear c2;
- [ auto
- | lapply ceq_cl; [ decompose Hletin |||| apply H1 ].
- apply H2; apply ceq_sing_l; [||| apply H4 ]; auto
- | lapply ceq_cl; [ decompose Hletin |||| apply H1 ].
- apply H2; apply ceq_sing_r; [||| apply H4 ]; auto
- ].
+theorem ceq_trans: \forall C,c2,c1,c3.
+ ceq C c2 c3 \to ceq ? c1 c2 \to ceq ? c1 c3.
+intros; elim H; elim H1; clear H; clear H1.
+apply ceq_intro; apply cle_trans; [|auto|auto||auto|auto].
qed.
theorem ceq_sym: \forall C,c1,c2. ceq C c1 c2 \to ceq C c2 c1.
-intros;
-lapply ceq_cl; [ decompose Hletin |||| apply H ].
-auto.
-qed.
-
-theorem ceq_conf: \forall C,c2,c1,c3.
- ceq C c1 c2 \to ceq ? c1 c3 \to ceq ? c2 c3.
-intros.
-lapply ceq_sym; [|||| apply H ].
-apply ceq_trans; [| auto | auto ].
+intros; elim H; clear H.; auto.
qed.