X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fformal_topology%2Frelations.ma;h=789f312cf0260da24d333203d898af38a3abca3f;hb=e880d6eab5e1700f4a625ddcd7d0fa8f0cce2dcc;hp=033788e928fb193b877330b806c89a8cf92281cc;hpb=42e87a2db149c53be4001e0f3c46ad2fa0a9579e;p=helm.git

diff --git a/helm/software/matita/library/formal_topology/relations.ma b/helm/software/matita/library/formal_topology/relations.ma
index 033788e92..789f312cf 100644
--- a/helm/software/matita/library/formal_topology/relations.ma
+++ b/helm/software/matita/library/formal_topology/relations.ma
@@ -12,114 +12,315 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "datatypes/subsets.ma".
+include "formal_topology/subsets.ma".
 
-record binary_relation (A,B: Type) : Type ≝
- { satisfy:2> A → B → CProp }.
+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 = (satisfy __ r x y).
+interpretation "relation applied" 'satisfy r x y = (fun21 ??? (satisfy ?? r) x y).
+
+definition binary_relation_setoid: SET → SET → setoid1.
+ intros (A B);
+ constructor 1;
+  [ apply (binary_relation A B)
+  | constructor 1;
+     [ 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 ?? (f ??)) | apply (if ?? (f ??))] assumption
+     | simplify;  intros 7; split; intro;
+        [ 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_relation A B → binary_relation B C →
-   binary_relation A C.
- intros (A B C R12 R23);
+  (binary_relation_setoid A B) × (binary_relation_setoid B C) ⇒_1 (binary_relation_setoid A C).
+ intros;
+ constructor 1;
+  [ intros (R12 R23);
+    constructor 1;
+    constructor 1;
+     [ apply (λs1:A.λs3:C.∃s2:B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
+     | intros;
+       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 ?? (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.
+
+definition REL: category1.
  constructor 1;
- intros (s1 s3);
- apply (∃s2. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
+  [ apply setoid
+  | intros (T T1); apply (binary_relation_setoid T T1)
+  | intros; constructor 1;
+    constructor 1; unfold setoid1_of_setoid; simplify;
+     [ (* 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;
+    cases f (w H); clear f; cases H; clear H;
+    [cases f (w1 H); clear f | cases f1 (w1 H); clear f1]
+    cases H; clear H;
+    exists; try assumption;
+    split; try assumption;
+    exists; try assumption;
+    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 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.
 
-interpretation "binary relation composition" 'compose x y = (composition ___ x y).
+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.
 
-definition equal_relations ≝
- λA,B.λr,r': binary_relation A B.
-  ∀x,y. r x y ↔ r' x y.
 
-interpretation "equal relation" 'eq x y = (equal_relations __ x y).
+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).
 
-lemma refl_equal_relations: ∀A,B. reflexive ? (equal_relations A B).
- intros 3; intros 2; split; intro; assumption.
+
+definition full_subset: ∀s:REL. Ω^s.
+ apply (λs.{x | True});
+ intros; simplify; split; intro; assumption.
 qed.
 
-lemma sym_equal_relations: ∀A,B. symmetric ? (equal_relations A B).
- intros 5; intros 2; split; intro;
-  [ apply (fi ?? (H ??)) | apply (if ?? (H ??))] assumption.
+coercion full_subset.
+
+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.
 
-lemma trans_equal_relations: ∀A,B. transitive ? (equal_relations A B).
- intros 7; intros 2; split; intro;
-  [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
-  [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
-  assumption.
+interpretation "subset comprehension" 'comprehension s p =
+ (comprehension s (mk_unary_morphism1 ?? p ?)).
+
+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.
 
-lemma associative_composition:
- ∀A,B,C,D.
-  ∀r1:binary_relation A B.
-   ∀r2:binary_relation B C.
-    ∀r3:binary_relation C D.
-     (r1 ∘ r2) ∘ r3 = r1 ∘ (r2 ∘ r3).
- intros 9;
- split; intro;
- cases H; clear H; cases H1; clear H1;
- [cases H; clear H | cases H2; clear H2]
- cases H1; clear H1;
- exists; try assumption;
- split; try assumption;
- exists; try assumption;
- split; assumption.
+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 (. (e^-1‡#)); assumption
+       |3: apply (. (e‡#)); assumption
+       |2,4: cases H2 (w H3); exists; [1,3: apply w]
+         [ 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 (. (#‡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;
+ split; simplify;
+  [ intros 2; change with (a ∈ X);
+    cases f; clear f;
+    cases H; clear H;
+    cases x; clear x;
+    change in f2 with (eq1 ? a w);
+    apply (. (f2\sup -1‡#));
+    assumption
+  | intros 2; change in f with (a ∈ X);
+    split;
+     [ whd; exact I 
+     | exists; [ apply a ]
+       split;
+        [ assumption
+        | change with (a = a); apply refl]]]
 qed.
 
-lemma composition_morphism:
- ∀A,B,C.
-  ∀r1,r1':binary_relation A B.
-   ∀r2,r2':binary_relation B C.
-    r1 = r1' → r2 = r2' → r1 ∘ r2 = r1' ∘ r2'.
- intros 11; split; intro;
- cases H2; clear H2; cases H3; 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) ]
- exists; try assumption;
- split; assumption.
-qed.
-
-definition binary_relation_setoid: Type → Type → setoid.
- intros (A B);
- constructor 1;
-  [ apply (binary_relation A B)
-  | constructor 1;
-     [ apply equal_relations
-     | apply refl_equal_relations
-     | apply sym_equal_relations
-     | apply trans_equal_relations
-     ]]
+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;
+    exists; [apply w1] split [2: assumption] constructor 1; [assumption]
+    exists; [apply w] split; assumption
+  | cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
+    cases H3 (H4 H5); cases H4 (w1 H6); clear H3 H4; cases H6 (w2 H7); clear H6;
+    cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
+    assumption]
 qed.
+*)
 
-definition REL: category.
- constructor 1;
-  [ apply Type
-  | intros; apply (binary_relation_setoid T T1)
-  | intros; constructor 1; intros; apply (eq ? o1 o2);
-  | intros; constructor 1;
-     [ apply composition
-     | apply composition_morphism
-     ]
-  | intros; unfold mk_binary_morphism; simplify;
-    apply associative_composition
-  |6,7: intros 5; simplify; split; intro;
-     [1,3: cases H; clear H; cases H1; clear H1;
-       [ alias id "eq_elim_r''" = "cic:/matita/logic/equality/eq_elim_r''.con".
-         apply (eq_elim_r'' ? w ?? x H); assumption
-       | alias id "eq_rect" = "cic:/matita/logic/equality/eq_rect.con".
-         apply (eq_rect ? w ?? y H2); assumption ]
-       assumption
-     |*: exists; try assumption; split;
-         alias id "refl_eq" = "cic:/matita/logic/equality/eq.ind#xpointer(1/1/1)".
-         first [ apply refl_eq | assumption ]]]
-qed.
-
-definition full_subset: ∀s:REL. Ω \sup s ≝ λs.{x | True}.
-
-coercion full_subset.
\ No newline at end of file
+(* 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