Some important proofs/definitions were (and are still) commented out and

do not compile.

Added the notion of functor1 to state the main theorem.

diff --git a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
index 44f8e9f213aa2753789d4b1b83b727cdc32a4da3..8e5421c1f0b39617a9896c045a7070b64c606061 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs.ma
@@ -172,6 +172,7 @@ definition relation_pair_setoid_of_arrows1_BP :
   ∀P,Q. arrows1 BP P Q → relation_pair_setoid P Q ≝ λP,Q,x.x.
 coercion relation_pair_setoid_of_arrows1_BP.
 
+(*
 definition BPext: ∀o: BP. (form o) ⇒_1 Ω^(concr o).
  intros; constructor 1;
   [ apply (ext ? ? (rel o));
@@ -219,3 +220,4 @@ qed.
 
 interpretation "basic pair relation for subsets" 'Vdash2 x y c = (fun21 (concr ?) ?? (relS c) x y).
 interpretation "basic pair relation for subsets (non applied)" 'Vdash c = (fun21 ??? (relS c)).
+*)

diff --git a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_basic_topologies.ma b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_basic_topologies.ma
index 5a98136bc889dc6ef7b5e8ee246580cf9078cab4..b557bedae9bcd025243eabccbde33c70615a004d 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_basic_topologies.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_pairs_to_basic_topologies.ma
@@ -40,4 +40,23 @@ definition continuous_relation_of_relation_pair:
   [ apply (rp \sub \f);
   | apply (Oreduced ?? ocr);
   | apply (Osaturated ?? ocr); ]
+qed.
+
+alias symbol "compose" (instance 3) = "category1 composition".
+alias symbol "compose" (instance 3) = "category1 composition".
+record functor1 (C1: category1) (C2: category1) : Type2 ≝
+ { map_objs1:1> C1 → C2;
+   map_arrows1: ∀S,T. unary_morphism1 (arrows1 ? S T) (arrows1 ? (map_objs1 S) (map_objs1 T));
+   respects_id1: ∀o:C1. map_arrows1 ?? (id1 ? o) = id1 ? (map_objs1 o);
+   respects_comp1:
+     ∀o1,o2,o3.∀f1:arrows1 ? o1 o2.∀f2:arrows1 ? o2 o3.
+     map_arrows1 ?? (f2 ∘ f1) = map_arrows1 ?? f2 ∘ map_arrows1 ?? f1}.
+
+definition BTop_of_BP: functor1 BP BTop.
+ lapply OR as F;
+ constructor 1;
+  [ apply basic_topology_of_basic_pair
+  | intros; constructor 1 [ apply continuous_relation_of_relation_pair; ]
+  | simplify; intro;
+  ]
 qed.
\ No newline at end of file

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 a48aae41c5c598e23c3420fb90e1eddf5402077c..b24a730cb05381873fc80103b8491ab1288e6f8e 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/basic_topologies.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/basic_topologies.ma
@@ -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.
@@ -131,15 +137,16 @@ 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.
+       alias symbol "compose" (instance 2) = "category1 composition".
+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));
         [2: intro;
             apply (.= (minus_star_image_comp ??????));
@@ -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

diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations.ma b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
index f7827939d2a5f7d950c276d40ec9b2031f636d0a..b1589a827fbc50717a08be48ec3fc2f2adf3eae2 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/relations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
@@ -240,25 +240,25 @@ definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) × Ω^V ⇒_1 Ω^U.
        apply (if ?? (e ^ -1 ??)); assumption]]
 qed.
 
-(*
 (* minus_image is the same as ext *)
 
 theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
  intros; unfold image; simplify; split; simplify; intros;
   [ change with (a ∈ U);
-    cases H; cases x; change in f with (eq1 ? w a); apply (. f‡#); assumption
+    cases e; cases x; change in f with (eq1 ? w a); apply (. f^-1‡#); assumption
   | change in f with (a ∈ U);
-    exists; [apply a] split; [ change with (a = a); apply refl | assumption]]
+    exists; [apply a] split; [ change with (a = a); apply refl1 | assumption]]
 qed.
 
 theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
  intros; unfold minus_star_image; simplify; split; simplify; intros;
-  [ change with (a ∈ U); apply H; change with (a=a); apply refl
-  | change in f1 with (eq1 ? x a); apply (. f1 \sup -1‡#); apply f]
+  [ change with (a ∈ U); apply f; change with (a=a); apply refl1
+  | change in f1 with (eq1 ? x a); apply (. f1‡#); apply f]
 qed.
 
+alias symbol "compose" (instance 2) = "category1 composition".
 theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
- intros; unfold image; simplify; split; simplify; intros; cases H; clear H; cases x;
+ intros; unfold image; simplify; split; simplify; intros; cases e; clear e; cases x;
  clear x; [ cases f; clear f; | cases f1; clear f1 ]
  exists; try assumption; cases x; clear x; split; try assumption;
  exists; try assumption; split; assumption.
@@ -268,10 +268,11 @@ theorem minus_star_image_comp:
  ∀A,B,C,r,s,X.
   minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
  intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
-  [ apply H; exists; try assumption; split; assumption
-  | change with (x ∈ X); cases f; cases x1; apply H; assumption]
+  [ apply f; exists; try assumption; split; assumption
+  | change with (x ∈ X); cases f1; cases x1; apply f; assumption]
 qed.
 
+(*
 (*CSC: unused! *)
 theorem ext_comp:
  ∀o1,o2,o3: REL.