some notation for map_arrows2

[helm.git] / helm / software / matita / contribs / formal_topology / overlap / basic_topologies.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/basic_topologies.ma b/helm/software/matita/contribs/formal_topology/overlap/basic_topologies.ma
index 013ddb94d5a15db81551249caca71610974733a2..5cb82832a167a8bafa07910d59b8c5e60e973a93 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/basic_topologies.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_topologies.ma
@@ -17,11 +17,11 @@ include "saturations.ma".
 
 record basic_topology: Type1 ≝
  { carrbt:> REL;
-   A: unary_morphism1 (Ω \sup carrbt) (Ω \sup carrbt);
-   J: unary_morphism1 (Ω \sup carrbt) (Ω \sup carrbt);
+   A: Ω^carrbt ⇒_1 Ω^carrbt;
+   J: Ω^carrbt ⇒_1 Ω^carrbt;
    A_is_saturation: is_saturation ? A;
    J_is_reduction: is_reduction ? J;
-   compatibility: ∀U,V. (A U ≬ J V) = (U ≬ J V)
+   compatibility: ∀U,V. (A U ≬ J V) =_1 (U ≬ J V)
  }.
 
 record continuous_relation (S,T: basic_topology) : Type1 ≝
@@ -29,20 +29,25 @@ record continuous_relation (S,T: basic_topology) : Type1 ≝
    reduced: ∀U. U = J ? U → image ?? cont_rel U = J ? (image ?? cont_rel U);
    saturated: ∀U. U = A ? U → minus_star_image ?? cont_rel U = A ? (minus_star_image ?? cont_rel U)
  }. 
-(*
+
 definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
  intros (S T); constructor 1;
   [ apply (continuous_relation S T)
   | constructor 1;
      [ apply (λr,s:continuous_relation S T.∀b. A ? (ext ?? r b) = A ? (ext ?? s b));
      | simplify; intros; apply refl1;
-     | simplify; intros; apply sym1; apply H
-     | simplify; intros; apply trans1; [2: apply H |3: apply H1; |1: skip]]]
+     | simplify; intros (x y H); apply sym1; apply H
+     | simplify; intros; apply trans1; [2: apply f |3: apply f1; |1: skip]]]
 qed.
 
-theorem continuous_relation_eq':
+definition continuos_relation_of_continuous_relation_setoid :
+ ∀P,Q. continuous_relation_setoid P Q → continuous_relation P Q ≝ λP,Q,x.x.
+coercion continuos_relation_of_continuous_relation_setoid.
+
+axiom continuous_relation_eq':
  ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
   a = a' → ∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X).
+(*
  intros; split; intro; unfold minus_star_image; simplify; intros;
   [ cut (ext ?? a a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;]
     lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut;
@@ -56,12 +61,12 @@ theorem continuous_relation_eq':
     lapply (fi ?? (A_is_saturation ???) Hcut);
     apply (Hletin1 x); change with (x ∈ ext ?? a a1); split; simplify;
      [ apply I | assumption ]]
-qed.
+qed.*)
 
-theorem continuous_relation_eq_inv':
+axiom continuous_relation_eq_inv':
  ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2.
   (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → a=a'.
- intros 6;
+(* intros 6;
  cut (∀a,a': continuous_relation_setoid o1 o2.
   (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → 
    ∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V));
@@ -86,6 +91,7 @@ theorem continuous_relation_eq_inv':
       assumption;]
  split; apply Hcut; [2: assumption | intro; apply sym1; apply H]
 qed.
+*)
 
 definition continuous_relation_comp:
  ∀o1,o2,o3.
@@ -131,16 +137,17 @@ definition BTop: category1.
   | intros; constructor 1;
      [ apply continuous_relation_comp;
      | intros; simplify; intro x; simplify;
-       lapply depth=0 (continuous_relation_eq' ???? H) as H';
-       lapply depth=0 (continuous_relation_eq' ???? H1) as H1';
+       lapply depth=0 (continuous_relation_eq' ???? e) as H';
+       lapply depth=0 (continuous_relation_eq' ???? e1) as H1';
        letin K ≝ (λX.H1' (minus_star_image ?? a (A ? X))); clearbody K;
        cut (∀X:Ω \sup o1.
               minus_star_image o2 o3 b (A o2 (minus_star_image o1 o2 a (A o1 X)))
             = minus_star_image o2 o3 b' (A o2 (minus_star_image o1 o2 a' (A o1 X))));
         [2: intro; apply sym1; apply (.= #‡(†((H' ?)\sup -1))); apply sym1; apply (K X);]
        clear K H' H1';
-       cut (∀X:Ω \sup o1.
-              minus_star_image o1 o3 (b ∘ a) (A o1 X) = minus_star_image o1 o3 (b'∘a') (A o1 X));
+alias symbol "compose" (instance 1) = "category1 composition".
+cut (∀X:Ω^o1.
+              minus_star_image ?? (b ∘ a) (A o1 X) =_1 minus_star_image ?? (b'∘a') (A o1 X));
         [2: intro;
             apply (.= (minus_star_image_comp ??????));
             apply (.= #‡(saturated ?????));
@@ -154,7 +161,12 @@ definition BTop: category1.
        apply (continuous_relation_eq_inv');
        apply Hcut1;]
   | intros; simplify; intro; do 2 (unfold continuous_relation_comp); simplify;
-    apply (.= †(ASSOC1‡#));
+    alias symbol "trans" (instance 1) = "trans1".
+alias symbol "refl" (instance 5) = "refl1".
+alias symbol "prop2" (instance 3) = "prop21".
+alias symbol "prop1" (instance 2) = "prop11".
+alias symbol "assoc" (instance 4) = "category1 assoc".
+apply (.= †(ASSOC‡#));
     apply refl1
   | intros; simplify; intro; unfold continuous_relation_comp; simplify;
     apply (.= †((id_neutral_right1 ????)‡#));
@@ -190,4 +202,3 @@ theorem continuous_relation_eqS:
  apply Hcut2; assumption.
 qed.
 *)
-*)
\ No newline at end of file