use named types to force some constraints asap

[helm.git] / helm / software / matita / library / formal_topology / basic_pairs.ma

diff --git a/helm/software/matita/library/formal_topology/basic_pairs.ma b/helm/software/matita/library/formal_topology/basic_pairs.ma
index 3ee8649374e598918cfb24477a439f09089c62fd..0d51724de7114ad3eb7c2d3936856a13933ed313 100644 (file)
--- a/helm/software/matita/library/formal_topology/basic_pairs.ma
+++ b/helm/software/matita/library/formal_topology/basic_pairs.ma
@@ -30,7 +30,7 @@ interpretation "basic pair relation (non applied)" 'Vdash = (rel _).
 record relation_pair (BP1,BP2: basic_pair): Type ≝
  { concr_rel: arrows1 ? (concr BP1) (concr BP2);
    form_rel: arrows1 ? (form BP1) (form BP2);
-   commute: concr_rel ∘ ⊩ = ⊩ ∘ form_rel
+   commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
  }.
 
 notation "hvbox (r \sub \c)"  with precedence 90 for @{'concr_rel $r}.
@@ -43,7 +43,7 @@ definition relation_pair_equality:
  ∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
  intros;
  constructor 1;
-  [ apply (λr,r'. r \sub\c ∘ ⊩ = r' \sub\c ∘ ⊩);
+  [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
   | simplify;
     intros;
     apply refl1;
@@ -64,7 +64,7 @@ definition relation_pair_setoid: basic_pair → basic_pair → setoid1.
   ]
 qed.
 
-lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → ⊩ \circ r \sub\f = ⊩ \circ r'\sub\f.
+lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
  intros 7 (o1 o2 r r' H c1 f2);
  split; intro H1;
   [ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
@@ -76,12 +76,12 @@ lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → ⊩ \cir
   ]
 qed.
 
-definition id: ∀o:basic_pair. relation_pair o o.
+definition id_relation_pair: ∀o:basic_pair. relation_pair o o.
  intro;
  constructor 1;
   [1,2: apply id1;
-  | lapply (id_neutral_left1 ? (concr o) ? (⊩)) as H;
-    lapply (id_neutral_right1 ?? (form o) (⊩)) as H1;
+  | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
+    lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
     apply (.= H);
     apply (H1 \sup -1);]
 qed.
@@ -92,8 +92,8 @@ definition relation_pair_composition:
  constructor 1;
   [ intros (r r1);
     constructor 1;
-     [ apply (r \sub\c ∘ r1 \sub\c) 
-     | apply (r \sub\f ∘ r1 \sub\f)
+     [ apply (r1 \sub\c ∘ r \sub\c) 
+     | apply (r1 \sub\f ∘ r \sub\f)
      | lapply (commute ?? r) as H;
        lapply (commute ?? r1) as H1;
        apply (.= ASSOC1);
@@ -102,9 +102,9 @@ definition relation_pair_composition:
        apply (.= H‡#);
        apply ASSOC1]
   | intros;
-    change with (a\sub\c ∘ b\sub\c ∘ ⊩ = a'\sub\c ∘ b'\sub\c ∘ ⊩);  
-    change in H with (a \sub\c ∘ ⊩ = a' \sub\c ∘ ⊩);
-    change in H1 with (b \sub\c ∘ ⊩ = b' \sub\c ∘ ⊩);
+    change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));  
+    change in H with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
+    change in H1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
     apply (.= ASSOC1);
     apply (.= #‡H1);
     apply (.= #‡(commute ?? b'));
@@ -119,17 +119,61 @@ definition BP: category1.
  constructor 1;
   [ apply basic_pair
   | apply relation_pair_setoid
-  | apply id
+  | apply id_relation_pair
   | apply relation_pair_composition
   | intros;
-    change with (a12\sub\c ∘ a23\sub\c ∘ a34\sub\c ∘ ⊩ =
-                 (a12\sub\c ∘ (a23\sub\c ∘ a34\sub\c) ∘ ⊩));
+    change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
+                 ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
     apply (ASSOC1‡#);
   | intros;
-    change with ((id o1)\sub\c ∘ a\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
-    apply ((id_neutral_left1 ????)‡#);
-  | intros;
-    change with (a\sub\c ∘ (id o2)\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
+    change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
     apply ((id_neutral_right1 ????)‡#);
-  ]
-qed.
\ No newline at end of file
+  | intros;
+    change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
+    apply ((id_neutral_left1 ????)‡#);]
+qed.
+
+definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o).
+ intros; constructor 1;
+  [ apply (ext ? ? (rel o));
+  | intros;
+    apply (.= #‡H);
+    apply refl1]
+qed.
+
+definition BPextS: ∀o: BP. Ω \sup (form o) ⇒ Ω \sup (concr o) ≝
+ λo.extS ?? (rel o).
+
+definition fintersects: ∀o: BP. binary_morphism1 (form o) (form o) (Ω \sup (form o)).
+ intros (o); constructor 1;
+  [ apply (λa,b: form o.{c | BPext o c ⊆ BPext o a ∩ BPext o b });
+    intros; simplify; apply (.= (†H)‡#); apply refl1
+  | intros; split; simplify; intros;
+     [ apply (. #‡((†H)‡(†H1))); assumption
+     | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+qed.
+
+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)).
+ intros (o); constructor 1;
+  [ apply (λo: basic_pair.λa,b: Ω \sup (form o).{c | BPext o c ⊆ BPextS o a ∩ BPextS o b });
+    intros; simplify; apply (.= (†H)‡#); apply refl1
+  | intros; split; simplify; intros;
+     [ apply (. #‡((†H)‡(†H1))); assumption
+     | apply (. #‡((†H\sup -1)‡(†H1\sup -1))); assumption]]
+qed.
+
+interpretation "fintersectsS" 'fintersects U V = (fun1 ___ (fintersectsS _) U V).
+
+definition relS: ∀o: BP. binary_morphism1 (concr o) (Ω \sup (form o)) CPROP.
+ intros (o); constructor 1;
+  [ apply (λx:concr o.λS: Ω \sup (form o).∃y: form o.y ∈ S ∧ x ⊩ y);
+  | intros; split; intros; cases H2; exists [1,3: apply w]
+     [ apply (. (#‡H1)‡(H‡#)); assumption
+     | 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 _)).