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[helm.git] / helm / software / matita / contribs / formal_topology / overlap / o-basic_pairs.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma
index b6d6e1b3dd23cb088149fb1c509e25a0fd050fc8..3cbb70058ca879290b8f92a8f41d7996fb46e02a 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs.ma
@@ -22,8 +22,8 @@ record Obasic_pair: Type2 ≝
  }.
 
 (* FIX *)
-interpretation "basic pair relation indexed" 'Vdash2 x y c = (Orel c x y).
-interpretation "basic pair relation (non applied)" 'Vdash c = (Orel c).
+interpretation "o-basic pair relation indexed" 'Vdash2 x y c = (Orel c x y).
+interpretation "o-basic pair relation (non applied)" 'Vdash c = (Orel c).
 
 alias symbol "eq" = "setoid1 eq".
 alias symbol "compose" = "category1 composition".
@@ -38,8 +38,8 @@ record Orelation_pair (BP1,BP2: Obasic_pair): Type2 ≝
  }.
  
 (* FIX *)
-interpretation "concrete relation" 'concr_rel r = (Oconcr_rel __ r). 
-interpretation "formal relation" 'form_rel r = (Oform_rel __ r). 
+interpretation "o-concrete relation" 'concr_rel r = (Oconcr_rel ?? r). 
+interpretation "o-formal relation" 'form_rel r = (Oform_rel ?? r). 
 
 definition Orelation_pair_equality:
  ∀o1,o2. equivalence_relation2 (Orelation_pair o1 o2).
@@ -87,14 +87,14 @@ definition Oid_relation_pair: ∀o:Obasic_pair. Orelation_pair o o.
   | lapply (id_neutral_right2 ? (Oconcr o) ? (⊩)) as H;
     lapply (id_neutral_left2 ?? (Oform o) (⊩)) as H1;
     apply (.= H);
-    apply (H1 \sup -1);]
+    apply (H1^-1);]
 qed.
 
-definition Orelation_pair_composition:
- ∀o1,o2,o3. binary_morphism2 (Orelation_pair_setoid o1 o2) (Orelation_pair_setoid o2 o3) (Orelation_pair_setoid o1 o3).
- intros;
- constructor 1;
-  [ intros (r r1);
+lemma Orelation_pair_composition:
+ ∀o1,o2,o3:Obasic_pair.
+ Orelation_pair_setoid o1 o2 → Orelation_pair_setoid o2 o3→Orelation_pair_setoid o1 o3.
+intros 3 (o1 o2 o3);
+   intros (r r1);
     constructor 1;
      [ apply (r1 \sub\c ∘ r \sub\c) 
      | apply (r1 \sub\f ∘ r \sub\f)
@@ -105,36 +105,80 @@ definition Orelation_pair_composition:
        apply rule (.= ASSOC ^ -1);
        apply (.= H‡#);
        apply rule ASSOC]
-  | intros;
+qed.
+
+
+lemma Orelation_pair_composition_is_morphism:
+  ∀o1,o2,o3:Obasic_pair.
+  Πa,a':Orelation_pair_setoid o1 o2.Πb,b':Orelation_pair_setoid o2 o3.
+   a=a' →b=b' →
+      Orelation_pair_composition o1 o2 o3 a b
+      = Orelation_pair_composition o1 o2 o3 a' b'.
+intros;
     change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));  
     change in e with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
     change in e1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
     apply rule (.= ASSOC);
     apply (.= #‡e1);
     apply (.= #‡(Ocommute ?? b'));
-    apply rule (.= ASSOC \sup -1);
+    apply rule (.= ASSOC^-1);
     apply (.= e‡#);
     apply rule (.= ASSOC);
-    apply (.= #‡(Ocommute ?? b')\sup -1);
-    apply rule (ASSOC \sup -1)]
+    apply (.= #‡(Ocommute ?? b')^-1);
+    apply rule (ASSOC^-1);
 qed.
-    
-definition OBP: category2.
- constructor 1;
-  [ apply Obasic_pair
-  | apply Orelation_pair_setoid
-  | apply Oid_relation_pair
-  | apply Orelation_pair_composition
-  | intros;
+
+definition Orelation_pair_composition_morphism:
+ ∀o1,o2,o3. binary_morphism2 (Orelation_pair_setoid o1 o2) (Orelation_pair_setoid o2 o3) (Orelation_pair_setoid o1 o3).
+intros; constructor 1;
+[ apply Orelation_pair_composition;
+| apply Orelation_pair_composition_is_morphism;]
+qed.
+
+lemma Orelation_pair_composition_morphism_assoc:
+∀o1,o2,o3,o4:Obasic_pair
+   .Πa12:Orelation_pair_setoid o1 o2
+    .Πa23:Orelation_pair_setoid o2 o3
+     .Πa34:Orelation_pair_setoid o3 o4
+      .Orelation_pair_composition_morphism o1 o3 o4
+       (Orelation_pair_composition_morphism o1 o2 o3 a12 a23) a34
+       =Orelation_pair_composition_morphism o1 o2 o4 a12
+        (Orelation_pair_composition_morphism o2 o3 o4 a23 a34).  
+   intros;
     change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
                  ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
     apply rule (ASSOC‡#);
-  | intros;
+qed.
+
+lemma Orelation_pair_composition_morphism_respects_id:
+Πo1:Obasic_pair
+.Πo2:Obasic_pair
+ .Πa:Orelation_pair_setoid o1 o2
+  .Orelation_pair_composition_morphism o1 o1 o2 (Oid_relation_pair o1) a=a.
+   intros;
     change with (⊩ ∘ (a\sub\c ∘ (Oid_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
     apply ((id_neutral_right2 ????)‡#);
-  | intros;
+qed.
+
+lemma Orelation_pair_composition_morphism_respects_id_r:
+Πo1:Obasic_pair
+.Πo2:Obasic_pair
+ .Πa:Orelation_pair_setoid o1 o2
+  .Orelation_pair_composition_morphism o1 o2 o2 a (Oid_relation_pair o2)=a.
+intros;
     change with (⊩ ∘ ((Oid_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
-    apply ((id_neutral_left2 ????)‡#);]
+    apply ((id_neutral_left2 ????)‡#);
+qed.
+
+definition OBP: category2.
+ constructor 1;
+  [ apply Obasic_pair
+  | apply Orelation_pair_setoid
+  | apply Oid_relation_pair
+  | apply Orelation_pair_composition_morphism
+  | apply Orelation_pair_composition_morphism_assoc;
+  | apply Orelation_pair_composition_morphism_respects_id;
+  | apply Orelation_pair_composition_morphism_respects_id_r;]
 qed.
 
 definition Obasic_pair_of_objs2_OBP: objs2 OBP → Obasic_pair ≝ λx.x.
@@ -167,7 +211,7 @@ definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (f
      | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
 qed.
 
-interpretation "fintersects" 'fintersects U V = (fun1 ___ (fintersects _) U V).
+interpretation "fintersects" 'fintersects U V = (fun1 ??? (fintersects ?) U V).
 
 definition fintersectsS:
  ∀o:BP. binary_morphism1 (Ω \sup (form o)) (Ω \sup (form o)) (Ω \sup (form o)).
@@ -179,7 +223,7 @@ definition fintersectsS:
      | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
 qed.
 
-interpretation "fintersectsS" 'fintersects U V = (fun1 ___ (fintersectsS _) U V).
+interpretation "fintersectsS" 'fintersects U V = (fun1 ??? (fintersectsS ?) U V).
 *)
 
 (*
@@ -191,22 +235,22 @@ definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
      | apply (. (#‡H1 \sup -1)‡(H \sup -1‡#)); assumption]]
 qed.
 
-interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr _) __ (relS _) x y).
-interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ___ (relS _)).
+interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr ?) ?? (relS ?) x y).
+interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ??? (relS ?)).
 *)
 
 notation "□ \sub b" non associative with precedence 90 for @{'box $b}.
-notation > "□_term 90 b" non associative with precedence 90 for @{'box $b}.
-interpretation "Universal image ⊩⎻*" 'box x = (fun12 _ _ (or_f_minus_star _ _) (Orel x)).
+notation > "□⎽term 90 b" non associative with precedence 90 for @{'box $b}.
+interpretation "Universal image ⊩⎻*" 'box x = (fun12 ? ? (or_f_minus_star ? ?) (Orel x)).
  
 notation "◊ \sub b" non associative with precedence 90 for @{'diamond $b}.
-notation > "◊_term 90 b" non associative with precedence 90 for @{'diamond $b}.
-interpretation "Existential image ⊩" 'diamond x = (fun12 _ _ (or_f _ _) (Orel x)).
+notation > "◊⎽term 90 b" non associative with precedence 90 for @{'diamond $b}.
+interpretation "Existential image ⊩" 'diamond x = (fun12 ? ? (or_f ? ?) (Orel x)).
 
 notation "'Rest' \sub b" non associative with precedence 90 for @{'rest $b}.
 notation > "'Rest'⎽term 90 b" non associative with precedence 90 for @{'rest $b}.
-interpretation "Universal pre-image ⊩*" 'rest x = (fun12 _ _ (or_f_star _ _) (Orel x)).
+interpretation "Universal pre-image ⊩*" 'rest x = (fun12 ? ? (or_f_star ? ?) (Orel x)).
 
 notation "'Ext' \sub b" non associative with precedence 90 for @{'ext $b}.
 notation > "'Ext'⎽term 90 b" non associative with precedence 90 for @{'ext $b}.
-interpretation "Existential pre-image ⊩⎻" 'ext x = (fun12 _ _ (or_f_minus _ _) (Orel x)).
\ No newline at end of file
+interpretation "Existential pre-image ⊩⎻" 'ext x = (fun12 ? ? (or_f_minus ? ?) (Orel x)).
\ No newline at end of file