From b47c73ebbc1a6e8cc930a6c8c99f90d4fe2f19b8 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Thu, 29 Jan 2009 13:00:42 +0000
Subject: [PATCH] more polishing

---
 .../o-basic_pairs_to_o-basic_topologies.ma    | 12 +--
 .../overlap/o-basic_topologies.ma             | 96 +++++++++----------
 .../overlap/r-o-basic_pairs.ma                |  9 +-
 3 files changed, 60 insertions(+), 57 deletions(-)

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma
index b2dfffd02..1bf31881c 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-basic_pairs_to_o-basic_topologies.ma
@@ -19,7 +19,7 @@ include "o-basic_topologies.ma".
 alias symbol "eq" = "setoid1 eq".
 
 (* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
-definition o_basic_topology_of_o_basic_pair: OBP → BTop.
+definition o_basic_topology_of_o_basic_pair: OBP → OBTop.
  intro t;
  constructor 1;
   [ apply (Oform t);
@@ -60,7 +60,7 @@ qed.
 
 definition o_continuous_relation_of_o_relation_pair:
  ∀BP1,BP2.arrows2 OBP BP1 BP2 →
-  arrows2 BTop (o_basic_topology_of_o_basic_pair BP1) (o_basic_topology_of_o_basic_pair BP2).
+  arrows2 OBTop (o_basic_topology_of_o_basic_pair BP1) (o_basic_topology_of_o_basic_pair BP2).
  intros (BP1 BP2 t);
  constructor 1;
   [ apply (t \sub \f);
@@ -93,15 +93,15 @@ definition o_continuous_relation_of_o_relation_pair:
 qed.
 
 
-definition OR : carr3 (arrows3 CAT2 OBP BTop).
+definition OR : carr3 (arrows3 CAT2 OBP OBTop).
 constructor 1;
 [ apply o_basic_topology_of_o_basic_pair;
 | intros; constructor 1;
   [ apply o_continuous_relation_of_o_relation_pair;
   | apply hide; 
     intros; whd; unfold o_continuous_relation_of_o_relation_pair; simplify;;
-    change with ((a \sub \f ⎻* ∘ A (o_basic_topology_of_o_basic_pair S)) =
-                 (a' \sub \f ⎻*∘A (o_basic_topology_of_o_basic_pair S)));
+    change with ((a \sub \f ⎻* ∘ oA (o_basic_topology_of_o_basic_pair S)) =
+                 (a' \sub \f ⎻*∘ oA (o_basic_topology_of_o_basic_pair S)));
     whd in e; cases e; clear e e2 e3 e4;
     change in ⊢ (? ? ? (? ? ? ? ? % ?) ?) with ((⊩\sub S)⎻* ∘ (⊩\sub S)⎻);
     apply (.= (comp_assoc2 ? ???? ?? a\sub\f⎻* ));
@@ -113,7 +113,7 @@ constructor 1;
     change in ⊢ (? ? ? (? ? ? ? ? ? %) ?) with (a'\sub\f⎻* ∘ (⊩\sub S)⎻* );    
     apply (.= (comp_assoc2 ? ???? ?? a'\sub\f⎻* )^-1);
     apply refl2;]
-| intros 2 (o a); apply rule #;
+| intros 2 (o a); apply refl1;
 | intros 6; apply refl1;]
 qed.
 
diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-basic_topologies.ma b/helm/software/matita/contribs/formal_topology/overlap/o-basic_topologies.ma
index 30a8b476c..f2ce654ba 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-basic_topologies.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-basic_topologies.ma
@@ -15,37 +15,37 @@
 include "o-algebra.ma".
 include "o-saturations.ma".
 
-record basic_topology: Type2 ≝
- { carrbt:> OA;
-   A: carrbt ⇒ carrbt;
-   J: carrbt ⇒ carrbt;
-   A_is_saturation: is_o_saturation ? A;
-   J_is_reduction: is_o_reduction ? J;
-   compatibility: ∀U,V. (A U >< J V) = (U >< J V)
+record Obasic_topology: Type2 ≝
+ { Ocarrbt:> OA;
+   oA: Ocarrbt ⇒ Ocarrbt;
+   oJ: Ocarrbt ⇒ Ocarrbt;
+   oA_is_saturation: is_o_saturation ? oA;
+   oJ_is_reduction: is_o_reduction ? oJ;
+   Ocompatibility: ∀U,V. (oA U >< oJ V) = (U >< oJ V)
  }.
 
-record continuous_relation (S,T: basic_topology) : Type2 ≝
- { cont_rel:> arrows2 OA S T;
+record Ocontinuous_relation (S,T: Obasic_topology) : Type2 ≝
+ { Ocont_rel:> arrows2 OA S T;
    (* reduces uses eq1, saturated uses eq!!! *)
-   reduced: ∀U. U = J ? U → cont_rel U = J ? (cont_rel U);
-   saturated: ∀U. U = A ? U → cont_rel⎻* U = A ? (cont_rel⎻* U)
+   Oreduced: ∀U. U = oJ ? U → Ocont_rel U = oJ ? (Ocont_rel U);
+   Osaturated: ∀U. U = oA ? U → Ocont_rel⎻* U = oA ? (Ocont_rel⎻* U)
  }. 
 
-definition continuous_relation_setoid: basic_topology → basic_topology → setoid2.
+definition Ocontinuous_relation_setoid: Obasic_topology → Obasic_topology → setoid2.
  intros (S T); constructor 1;
-  [ apply (continuous_relation S T)
+  [ apply (Ocontinuous_relation S T)
   | constructor 1;
      [ alias symbol "eq" = "setoid2 eq".
        alias symbol "compose" = "category2 composition".
-       apply (λr,s:continuous_relation S T. (r⎻* ) ∘ (A S) = (s⎻* ∘ (A ?)));
+       apply (λr,s:Ocontinuous_relation S T. (r⎻* ) ∘ (oA S) = (s⎻* ∘ (oA ?)));
      | simplify; intros; apply refl2;
      | simplify; intros; apply sym2; apply e
      | simplify; intros; apply trans2; [2: apply e |3: apply e1; |1: skip]]]
 qed.
 
-definition continuous_relation_of_continuous_relation_setoid: 
-  ∀P,Q. continuous_relation_setoid P Q → continuous_relation P Q ≝ λP,Q,c.c.
-coercion continuous_relation_of_continuous_relation_setoid.
+definition Ocontinuous_relation_of_Ocontinuous_relation_setoid: 
+  ∀P,Q. Ocontinuous_relation_setoid P Q → Ocontinuous_relation P Q ≝ λP,Q,c.c.
+coercion Ocontinuous_relation_of_Ocontinuous_relation_setoid.
 
 (*
 theorem continuous_relation_eq':
@@ -96,69 +96,69 @@ theorem continuous_relation_eq_inv':
 qed.
 *)
 
-definition continuous_relation_comp:
+definition Ocontinuous_relation_comp:
  ∀o1,o2,o3.
-  continuous_relation_setoid o1 o2 →
-   continuous_relation_setoid o2 o3 →
-    continuous_relation_setoid o1 o3.
+  Ocontinuous_relation_setoid o1 o2 →
+   Ocontinuous_relation_setoid o2 o3 →
+    Ocontinuous_relation_setoid o1 o3.
  intros (o1 o2 o3 r s); constructor 1;
   [ apply (s ∘ r);
   | intros;
     apply sym1; 
     change in match ((s ∘ r) U) with (s (r U));
-    apply (.= (reduced : ?)\sup -1);
-     [ apply (.= (reduced :?)); [ assumption | apply refl1 ]
+    apply (.= (Oreduced : ?)\sup -1);
+     [ apply (.= (Oreduced :?)); [ assumption | apply refl1 ]
      | apply refl1]
   | intros;
     apply sym1;
     change in match ((s ∘ r)⎻* U) with (s⎻* (r⎻* U));
-    apply (.= (saturated : ?)\sup -1);
-     [ apply (.= (saturated : ?)); [ assumption | apply refl1 ]
+    apply (.= (Osaturated : ?)\sup -1);
+     [ apply (.= (Osaturated : ?)); [ assumption | apply refl1 ]
      | apply refl1]]
 qed.
 
-definition BTop: category2.
+definition OBTop: category2.
  constructor 1;
-  [ apply basic_topology
-  | apply continuous_relation_setoid
+  [ apply Obasic_topology
+  | apply Ocontinuous_relation_setoid
   | intro; constructor 1;
      [ apply id2
      | intros; apply e;
      | intros; apply e;]
   | intros; constructor 1;
-     [ apply continuous_relation_comp;
+     [ apply Ocontinuous_relation_comp;
      | intros; simplify;
-       change with ((b⎻* ∘ a⎻* ) ∘ A o1 = ((b'⎻* ∘ a'⎻* ) ∘ A o1)); 
-       change with (b⎻* ∘ (a⎻* ∘ A o1) = b'⎻* ∘ (a'⎻* ∘ A o1));
-       change in e with (a⎻* ∘ A o1 = a'⎻* ∘ A o1);
-       change in e1 with (b⎻* ∘ A o2 = b'⎻* ∘ A o2);
+       change with ((b⎻* ∘ a⎻* ) ∘ oA o1 = ((b'⎻* ∘ a'⎻* ) ∘ oA o1)); 
+       change with (b⎻* ∘ (a⎻* ∘ oA o1) = b'⎻* ∘ (a'⎻* ∘ oA o1));
+       change in e with (a⎻* ∘ oA o1 = a'⎻* ∘ oA o1);
+       change in e1 with (b⎻* ∘ oA o2 = b'⎻* ∘ oA o2);
        apply (.= e‡#);
        intro x;          
-       change with (eq1 ? (b⎻* (a'⎻* (A o1 x))) (b'⎻*(a'⎻* (A o1 x)))); 
-       apply (.= †(saturated o1 o2 a' (A o1 x) ?)); [
-         apply ((o_saturation_idempotent ?? (A_is_saturation o1) x)^-1);]
-       apply (.= (e1 (a'⎻* (A o1 x))));
-       change with (eq1 ? (b'⎻* (A o2 (a'⎻* (A o1 x)))) (b'⎻*(a'⎻* (A o1 x))));   
-       apply (.= †(saturated o1 o2 a' (A o1 x):?)^-1); [
-         apply ((o_saturation_idempotent ?? (A_is_saturation o1) x)^-1);]
+       change with (eq1 ? (b⎻* (a'⎻* (oA o1 x))) (b'⎻*(a'⎻* (oA o1 x)))); 
+       apply (.= †(Osaturated o1 o2 a' (oA o1 x) ?)); [
+         apply ((o_saturation_idempotent ?? (oA_is_saturation o1) x)^-1);]
+       apply (.= (e1 (a'⎻* (oA o1 x))));
+       change with (eq1 ? (b'⎻* (oA o2 (a'⎻* (oA o1 x)))) (b'⎻*(a'⎻* (oA o1 x))));   
+       apply (.= †(Osaturated o1 o2 a' (oA o1 x):?)^-1); [
+         apply ((o_saturation_idempotent ?? (oA_is_saturation o1) x)^-1);]
        apply rule #;]
   | intros; simplify;
-    change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ A o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ A o1));
+    change with (((a34⎻* ∘ a23⎻* ) ∘ a12⎻* ) ∘ oA o1 = ((a34⎻* ∘ (a23⎻* ∘ a12⎻* )) ∘ oA o1));
     apply rule (#‡ASSOC ^ -1);
   | intros; simplify;
-    change with ((a⎻* ∘ (id2 ? o1)⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+    change with ((a⎻* ∘ (id2 ? o1)⎻* ) ∘ oA o1 = a⎻* ∘ oA o1);
     apply (#‡(id_neutral_right2 : ?));
   | intros; simplify;
-    change with (((id2 ? o2)⎻* ∘ a⎻* ) ∘ A o1 = a⎻* ∘ A o1);
+    change with (((id2 ? o2)⎻* ∘ a⎻* ) ∘ oA o1 = a⎻* ∘ oA o1);
     apply (#‡(id_neutral_left2 : ?));]
 qed.
 
-definition basic_topology_of_BTop: objs2 BTop → basic_topology ≝ λx.x.
-coercion basic_topology_of_BTop.
+definition Obasic_topology_of_OBTop: objs2 OBTop → Obasic_topology ≝ λx.x.
+coercion Obasic_topology_of_OBTop.
 
-definition continuous_relation_setoid_of_arrows2_BTop : 
-  ∀P,Q. arrows2 BTop P Q → continuous_relation_setoid P Q ≝ λP,Q,x.x.
-coercion continuous_relation_setoid_of_arrows2_BTop.
+definition Ocontinuous_relation_setoid_of_arrows2_OBTop : 
+  ∀P,Q. arrows2 OBTop P Q → Ocontinuous_relation_setoid P Q ≝ λP,Q,x.x.
+coercion Ocontinuous_relation_setoid_of_arrows2_OBTop.
 
 (*
 (*CSC: unused! *)
diff --git a/helm/software/matita/contribs/formal_topology/overlap/r-o-basic_pairs.ma b/helm/software/matita/contribs/formal_topology/overlap/r-o-basic_pairs.ma
index 1dcc1b91f..f90b8bbba 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/r-o-basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/r-o-basic_pairs.ma
@@ -20,10 +20,13 @@ definition rOBP ≝ Apply ?? BP_to_OBP.
 include "o-basic_pairs_to_o-basic_topologies.ma".
 
 lemma rOR_full : 
-  ∀rS,rT:rOBP.∀f:arrows2 BTop (OR (ℱ_2 rS)) (OR (ℱ_2 rT)).
-    exT22 ? (λg:arrows2 rOBP rS rT.
+  ∀s,t:rOBP.∀f:arrows2 OBTop (OR (ℱ_2 s)) (OR (ℱ_2 t)).
+    exT22 ? (λg:arrows2 rOBP s t.
        map_arrows2 ?? OR ?? (ℳ_2 g) = f).
-intros; cases f (c H1 H2); simplify;
+intros; cases f (h H1 H2); clear f; simplify;
+change in ⊢ (? ? (λg:?.? ? ? (? ? ? %) ?)) with 
+  (o_continuous_relation_of_o_relation_pair ?? (ℳ_2 g)).
+
 STOP;
  
 
-- 
2.39.2