X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fformal_topology%2Frelations.ma;h=789f312cf0260da24d333203d898af38a3abca3f;hb=e880d6eab5e1700f4a625ddcd7d0fa8f0cce2dcc;hp=72e12aa9dc500009518aebf7fb7038f629821771;hpb=dd7f52dfd7f80b80368661fce5b58b644c102d7b;p=helm.git

diff --git a/helm/software/matita/library/formal_topology/relations.ma b/helm/software/matita/library/formal_topology/relations.ma
index 72e12aa9d..789f312cf 100644
--- a/helm/software/matita/library/formal_topology/relations.ma
+++ b/helm/software/matita/library/formal_topology/relations.ma
@@ -12,16 +12,16 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "datatypes/subsets.ma".
+include "formal_topology/subsets.ma".
 
-record binary_relation (A,B: setoid) : Type ≝
+record binary_relation (A,B: SET) : Type1 ≝
  { satisfy:> binary_morphism1 A B CPROP }.
 
 notation < "hvbox (x \nbsp \natur term 90 r \nbsp y)"  with precedence 45 for @{'satisfy $r $x $y}.
 notation > "hvbox (x \natur term 90 r y)"  with precedence 45 for @{'satisfy $r $x $y}.
-interpretation "relation applied" 'satisfy r x y = (fun1 ___ (satisfy __ r) x y).
+interpretation "relation applied" 'satisfy r x y = (fun21 ??? (satisfy ?? r) x y).
 
-definition binary_relation_setoid: setoid → setoid → setoid1.
+definition binary_relation_setoid: SET → SET → setoid1.
  intros (A B);
  constructor 1;
   [ apply (binary_relation A B)
@@ -29,16 +29,20 @@ definition binary_relation_setoid: setoid → setoid → setoid1.
      [ apply (λA,B.λr,r': binary_relation A B. ∀x,y. r x y ↔ r' x y)
      | simplify; intros 3; split; intro; assumption
      | simplify; intros 5; split; intro;
-       [ apply (fi ?? (H ??)) | apply (if ?? (H ??))] assumption
+       [ apply (fi ?? (f ??)) | apply (if ?? (f ??))] assumption
      | simplify;  intros 7; split; intro;
-        [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
-        [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
+        [ apply (if ?? (f1 ??)) | apply (fi ?? (f ??)) ]
+        [ apply (if ?? (f ??)) | apply (fi ?? (f1 ??)) ]
        assumption]]
 qed.
 
+definition binary_relation_of_binary_relation_setoid : 
+  ∀A,B.binary_relation_setoid A B → binary_relation A B ≝ λA,B,c.c.
+coercion binary_relation_of_binary_relation_setoid.
+
 definition composition:
  ∀A,B,C.
-  binary_morphism1 (binary_relation_setoid A B) (binary_relation_setoid B C) (binary_relation_setoid A C).
+  (binary_relation_setoid A B) × (binary_relation_setoid B C) ⇒_1 (binary_relation_setoid A C).
  intros;
  constructor 1;
   [ intros (R12 R23);
@@ -46,13 +50,13 @@ definition composition:
     constructor 1;
      [ apply (λs1:A.λs3:C.∃s2:B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
      | intros;
-       split; intro; cases H2 (w H3); clear H2; exists; [1,3: apply w ]
-        [ apply (. (H‡#)‡(#‡H1)); assumption
-        | apply (. ((H \sup -1)‡#)‡(#‡(H1 \sup -1))); assumption]]
+       split; intro; cases e2 (w H3); clear e2; exists; [1,3: apply w ]
+        [ apply (. (e^-1‡#)‡(#‡e1^-1)); assumption
+        | apply (. (e‡#)‡(#‡e1)); assumption]]
   | intros 8; split; intro H2; simplify in H2 ⊢ %;
     cases H2 (w H3); clear H2; exists [1,3: apply w] cases H3 (H2 H4); clear H3;
-    [ lapply (if ?? (H x w) H2) | lapply (fi ?? (H x w) H2) ]
-    [ lapply (if ?? (H1 w y) H4)| lapply (fi ?? (H1 w y) H4) ]
+    [ lapply (if ?? (e x w) H2) | lapply (fi ?? (e x w) H2) ]
+    [ lapply (if ?? (e1 w y) H4)| lapply (fi ?? (e1 w y) H4) ]
     exists; try assumption;
     split; assumption]
 qed.
@@ -63,14 +67,16 @@ definition REL: category1.
   | intros (T T1); apply (binary_relation_setoid T T1)
   | intros; constructor 1;
     constructor 1; unfold setoid1_of_setoid; simplify;
-     [ intros; apply (c = c1)
-     | intros; split; intro;
-        [ apply (trans ????? (H \sup -1));
-          apply (trans ????? H2);
-          apply H1
-        | apply (trans ????? H);
-          apply (trans ????? H2);
-          apply (H1 \sup -1)]]
+     [ (* changes required to avoid universe inconsistency *)
+       change with (carr o → carr o → CProp); intros; apply (eq ? c c1)
+     | intros; split; intro; change in a a' b b' with (carr o);
+       change in e with (eq ? a a'); change in e1 with (eq ? b b');
+        [ apply (.= (e ^ -1));
+          apply (.= e2);
+          apply e1
+        | apply (.= e);
+          apply (.= e2);
+          apply (e1 ^ -1)]]
   | apply composition
   | intros 9;
     split; intro;
@@ -83,57 +89,75 @@ definition REL: category1.
     split; assumption
   |6,7: intros 5; unfold composition; simplify; split; intro;
         unfold setoid1_of_setoid in x y; simplify in x y;
-        [1,3: cases H (w H1); clear H; cases H1; clear H1; unfold;
-          [ apply (. (H \sup -1 : eq1 ? w x)‡#); assumption
-          | apply (. #‡(H : eq1 ? w y)); assumption]
-        |2,4: exists; try assumption; split; first [apply refl | assumption]]]
+        [1,3: cases e (w H1); clear e; cases H1; clear H1; unfold;
+          [ apply (. (e : eq1 ? x w)‡#); assumption
+          | apply (. #‡(e : eq1 ? w y)^-1); assumption]
+        |2,4: exists; try assumption; split;
+          (* change required to avoid universe inconsistency *)
+          change in x with (carr o1); change in y with (carr o2);
+          first [apply refl | assumption]]]
 qed.
 
-definition full_subset: ∀s:REL. Ω \sup s.
+definition setoid_of_REL : objs1 REL → setoid ≝ λx.x.
+coercion setoid_of_REL.
+
+definition binary_relation_setoid_of_arrow1_REL : 
+  ∀P,Q. arrows1 REL P Q → binary_relation_setoid P Q ≝ λP,Q,x.x.
+coercion binary_relation_setoid_of_arrow1_REL.
+
+
+notation > "B ⇒_\r1 C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
+notation "B ⇒\sub (\r 1) C" right associative with precedence 72 for @{'arrows1_REL $B $C}.
+interpretation "'arrows1_REL" 'arrows1_REL A B = (arrows1 REL A B).
+notation > "B ⇒_\r2 C" right associative with precedence 72 for @{'arrows2_REL $B $C}.
+notation "B ⇒\sub (\r 2) C" right associative with precedence 72 for @{'arrows2_REL $B $C}.
+interpretation "'arrows2_REL" 'arrows2_REL A B = (arrows2 (category2_of_category1 REL) A B).
+
+
+definition full_subset: ∀s:REL. Ω^s.
  apply (λs.{x | True});
  intros; simplify; split; intro; assumption.
 qed.
 
 coercion full_subset.
 
-definition setoid1_of_REL: REL → setoid ≝ λS. S.
-
-coercion setoid1_of_REL.
-
-definition comprehension: ∀b:REL. (b ⇒ CPROP) → Ω \sup b.
- apply (λb:REL. λP: b ⇒ CPROP. {x | x ∈ b ∧ P x});
- intros; simplify; apply (.= (H‡#)‡(†H)); apply refl1.
+definition comprehension: ∀b:REL. (b ⇒_1. CPROP) → Ω^b.
+ apply (λb:REL. λP: b ⇒_1 CPROP. {x | P x});
+ intros; simplify;
+ apply (.= †e); apply refl1.
 qed.
 
 interpretation "subset comprehension" 'comprehension s p =
- (comprehension s (mk_unary_morphism __ p _)).
+ (comprehension s (mk_unary_morphism1 ?? p ?)).
 
-definition ext: ∀X,S:REL. ∀r: arrows1 ? X S. S ⇒ Ω \sup X.
- apply (λX,S,r.mk_unary_morphism ?? (λf.{x ∈ X | x ♮r f}) ?);
-  [ intros; simplify; apply (.= (H‡#)); apply refl1
-  | intros; simplify; split; intros; simplify; intros;
-     [ apply (. #‡(#‡H)); assumption
-     | apply (. #‡(#‡H\sup -1)); assumption]]
+definition ext: ∀X,S:REL. (X ⇒_\r1 S) × S ⇒_1 (Ω^X).
+ intros (X S); constructor 1; 
+  [ apply (λr:X ⇒_\r1 S.λf:S.{x ∈ X | x ♮r f}); intros; simplify; apply (.= (e‡#)); apply refl1
+  | intros; simplify; split; intros; simplify;
+     [ change with (∀x. x ♮a b → x ♮a' b'); intros;
+       apply (. (#‡e1^-1)); whd in e; apply (if ?? (e ??)); assumption
+     | change with (∀x. x ♮a' b' → x ♮a b); intros;
+       apply (. (#‡e1)); whd in e; apply (fi ?? (e ??));assumption]]
 qed.
 
-definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup X.
+definition extS: ∀X,S:REL. ∀r:X ⇒_\r1 S. Ω^S ⇒_1 Ω^X.
  (* ∃ is not yet a morphism apply (λX,S,r,F.{x ∈ X | ∃a. a ∈ F ∧ x ♮r a});*)
  intros (X S r); constructor 1;
   [ intro F; constructor 1; constructor 1;
     [ apply (λx. x ∈ X ∧ ∃a:S. a ∈ F ∧ x ♮r a);
     | intros; split; intro; cases f (H1 H2); clear f; split;
-       [ apply (. (H‡#)); assumption
-       |3: apply (. (H\sup -1‡#)); assumption
+       [ apply (. (e^-1‡#)); assumption
+       |3: apply (. (e‡#)); assumption
        |2,4: cases H2 (w H3); exists; [1,3: apply w]
-         [ apply (. (#‡(H‡#))); assumption
-         | apply (. (#‡(H \sup -1‡#))); assumption]]]
-  | intros; split; simplify; intros; cases f; cases H1; split;
+         [ apply (. (#‡(e^-1‡#))); assumption
+         | apply (. (#‡(e‡#))); assumption]]]
+  | intros; split; simplify; intros; cases f; cases e1; split;
      [1,3: assumption
      |2,4: exists; [1,3: apply w]
-      [ apply (. (#‡H)‡#); assumption
-      | apply (. (#‡H\sup -1)‡#); assumption]]]
+      [ apply (. (#‡e^-1)‡#); assumption
+      | apply (. (#‡e)‡#); assumption]]]
 qed.
-
+(*
 lemma equalset_extS_id_X_X: ∀o:REL.∀X.extS ?? (id1 ? o) X = X.
  intros;
  unfold extS; simplify;
@@ -154,7 +178,7 @@ lemma equalset_extS_id_X_X: ∀o:REL.∀X.extS ?? (id1 ? o) X = X.
         | change with (a = a); apply refl]]]
 qed.
 
-lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c1 ∘ c2) S = extS o1 o2 c1 (extS o2 o3 c2 S).
+lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (extS o2 o3 c2 S).
  intros; unfold extS; simplify; split; intros; simplify; intros;
   [ cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
     cases H3 (H4 H5); cases H5 (w1 H6); clear H3 H5; cases H6 (H7 H8); clear H6;
@@ -165,4 +189,138 @@ lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c1 ∘ c2) S = extS o1 o2 c1 (e
     cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
     assumption]
 qed.
+*)
+
+(* the same as ⋄ for a basic pair *)
+definition image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^U ⇒_2 Ω^V).
+ intros; constructor 1;
+ [ intro r; constructor 1;
+   [ apply (λS: Ω^U. {y | ∃x:U. x ♮r y ∧ x ∈ S });
+     intros; simplify; split; intro; cases e1; exists [1,3: apply w]
+     [ apply (. (#‡e^-1)‡#); assumption
+     | apply (. (#‡e)‡#); assumption]
+   | intros; split; 
+     [ intro y; simplify; intro yA; cases yA; exists; [ apply w ];
+       apply (. #‡(#‡e^-1)); assumption;
+     | intro y; simplify; intro yA; cases yA; exists; [ apply w ];
+       apply (. #‡(#‡e)); assumption;]]
+ | simplify; intros; intro y; simplify; split; simplify; intros (b H); cases H;
+   exists; [1,3: apply w]; cases x; split; try assumption;
+   [ apply (if ?? (e ??)); | apply (fi ?? (e ??)); ] assumption;]
+qed.
+
+(* the same as □ for a basic pair *)
+definition minus_star_image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^U ⇒_2 Ω^V).
+ intros; constructor 1; intros;
+  [ constructor 1;
+    [ apply (λS: Ω^U. {y | ∀x:U. x ♮c y → x ∈ S});
+      intros; simplify; split; intros; apply f;
+      [ apply (. #‡e); | apply (. #‡e ^ -1)] assumption;
+    | intros; split; intro; simplify; intros;
+      [ apply (. #‡e^-1);| apply (. #‡e); ] apply f; assumption;]
+  | intros; intro; simplify; split; simplify; intros; apply f;
+    [ apply (. (e x a2)); assumption | apply (. (e^-1 x a2)); assumption]]
+qed.
+
+(* the same as Rest for a basic pair *)
+definition star_image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^V ⇒_2 Ω^U).
+ intros; constructor 1;
+  [ intro r; constructor 1; 
+    [ apply (λS: Ω \sup V. {x | ∀y:V. x ♮r y → y ∈ S});
+      intros; simplify; split; intros; apply f;
+      [ apply (. e ‡#);| apply (. e^ -1‡#);] assumption;
+    | intros; split; simplify; intros;
+      [ apply (. #‡e^-1);| apply (. #‡e); ] apply f; assumption;]
+  | intros; intro; simplify; split; simplify; intros; apply f; 
+    [ apply (. e a2 y); | apply (. e^-1 a2 y)] assumption;]
+qed.
+
+(* the same as Ext for a basic pair *)
+definition minus_image: ∀U,V:REL. (U ⇒_\r1 V) ⇒_2 (Ω^V ⇒_2 Ω^U).
+ intros; constructor 1;
+  [ intro r; constructor 1; 
+    [ apply (λS: Ω^V. {x | ∃y:V. x ♮r y ∧ y ∈ S }).
+      intros; simplify; split; intros; cases e1; cases x; exists; [1,3: apply w]
+      split; try assumption; [ apply (. (e^-1‡#)); | apply (. (e‡#));] assumption;
+    | intros; simplify; split; simplify; intros; cases e1; cases x; 
+      exists [1,3: apply w] split; try assumption;
+      [ apply (. (#‡e^-1)); | apply (. (#‡e));] assumption]
+  | intros; intro; simplify; split; simplify; intros; cases e1; exists [1,3: apply w]
+    cases x; split; try assumption;
+    [ apply (. e^-1 a2 w); | apply (. e a2 w)] assumption;]
+qed.
+
+definition foo : ∀o1,o2:REL.carr1 (o1 ⇒_\r1 o2) → carr2 (setoid2_of_setoid1 (o1 ⇒_\r1 o2)) ≝ λo1,o2,x.x.
+
+interpretation "relation f⎻*" 'OR_f_minus_star r = (fun12 ?? (minus_star_image ? ?) (foo ?? r)).
+interpretation "relation f⎻" 'OR_f_minus r = (fun12 ?? (minus_image ? ?) (foo ?? r)).
+interpretation "relation f*" 'OR_f_star r = (fun12 ?? (star_image ? ?) (foo ?? r)).
+
+definition image_coercion: ∀U,V:REL. (U ⇒_\r1 V) → Ω^U ⇒_2 Ω^V.
+intros (U V r Us); apply (image U V r); qed.
+coercion image_coercion.
+
+(* minus_image is the same as ext *)
+
+theorem image_id: ∀o. (id1 REL o : carr2 (Ω^o ⇒_2 Ω^o)) =_1 (id2 SET1 Ω^o).
+ intros; unfold image_coercion; unfold image; simplify;
+ whd in match (?:carr2 ?);
+  intro U; simplify; split; simplify; intros;
+  [ change with (a ∈ U);
+    cases e; cases x; change in e1 with (w =_1 a); apply (. e1^-1‡#); assumption
+  | change in f with (a ∈ U);
+    exists; [apply a] split; [ change with (a = a); apply refl1 | assumption]]
+qed.
+
+theorem minus_star_image_id: ∀o:REL. 
+  ((id1 REL o)⎻* : carr2 (Ω^o ⇒_2 Ω^o)) =_1 (id2 SET1 Ω^o).
+ intros; unfold minus_star_image; simplify; intro U; simplify; 
+ split; simplify; intros;
+  [ 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 5) = "category2 composition".
+alias symbol "compose" (instance 4) = "category1 composition".
+theorem image_comp: ∀A,B,C.∀r:B ⇒_\r1 C.∀s:A ⇒_\r1 B. 
+  ((r ∘ s) : carr2 (Ω^A ⇒_2 Ω^C)) =_1 r ∘ s.
+ intros; intro U; split; intro x; (unfold image; unfold SET1; simplify);
+ intro H; cases H; 
+ cases x1; [cases f|cases f1]; exists; [1,3: apply w1] cases x2; split; try assumption;
+   exists; try assumption; split; assumption;
+qed.
+
+theorem minus_star_image_comp:
+ ∀A,B,C.∀r:B ⇒_\r1 C.∀s:A ⇒_\r1 B.
+  minus_star_image A C (r ∘ s) =_1 minus_star_image B C r ∘ (minus_star_image A B s).
+ intros; unfold minus_star_image; intro X; simplify; split; simplify; intros;
+ [ whd; intros; simplify; whd; intros; apply f; exists; try assumption; split; assumption;
+ | cases f1; cases x1; apply f; assumption]
+qed.
+
+
+(*
+(*CSC: unused! *)
+theorem ext_comp:
+ ∀o1,o2,o3: REL.
+  ∀a: arrows1 ? o1 o2.
+   ∀b: arrows1 ? o2 o3.
+    ∀x. ext ?? (b∘a) x = extS ?? a (ext ?? b x).
+ intros;
+ unfold ext; unfold extS; simplify; split; intro; simplify; intros;
+ cases f; clear f; split; try assumption;
+  [ cases f2; clear f2; cases x1; clear x1; exists; [apply w] split;
+     [1: split] assumption;
+  | cases H; clear H; cases x1; clear x1; exists [apply w]; split;
+     [2: cases f] assumption]
+qed.
+*)
+
+axiom daemon : False.
 
+theorem extS_singleton:
+ ∀o1,o2.∀a.∀x.extS o1 o2 a {(x)} = ext o1 o2 a x. 
+ intros; unfold extS; unfold ext; unfold singleton; simplify;
+ split; intros 2; simplify; simplify in f; 
+ [ cases f; cases e; cases x1; change in f2 with (x =_1 w); apply (. #‡f2); assumption;
+ | split; try apply I; exists [apply x] split; try assumption; change with (x = x); apply rule #;] qed.
\ No newline at end of file