X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fformal_topology%2Frelations.ma;h=f81e19eeccb87b438dc2215b5f2b3b8744322618;hb=c96d1f2066d37b84a34412f7c49fb3e4f54bd9a2;hp=e67b2ce7c5ea8792fd4b715efb3befe79dcaa2c5;hpb=7fad6f9727bb6f054c0198cf10354be4b355baef;p=helm.git

diff --git a/helm/software/matita/library/formal_topology/relations.ma b/helm/software/matita/library/formal_topology/relations.ma
index e67b2ce7c..f81e19eec 100644
--- a/helm/software/matita/library/formal_topology/relations.ma
+++ b/helm/software/matita/library/formal_topology/relations.ma
@@ -14,132 +14,234 @@
 
 include "datatypes/subsets.ma".
 
-record ssigma (A:Type) (S: powerset A) : Type ≝
- { witness:> A;
-   proof:> witness ∈ S
- }.
-
-coercion ssigma.
-
-record binary_relation (A,B: Type) (U: Ω \sup A) (V: Ω \sup B) : Type ≝
- { satisfy:2> U → V → CProp }.
+record binary_relation (A,B: setoid) : Type ≝
+ { 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 = (fun1 ___ (satisfy __ r) x y).
+
+definition binary_relation_setoid: setoid → setoid → 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 ?? (H ??)) | apply (if ?? (H ??))] assumption
+     | simplify;  intros 7; split; intro;
+        [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
+        [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
+       assumption]]
+qed.
 
 definition composition:
- ∀A,B,C.∀U1: Ω \sup A.∀U2: Ω \sup B.∀U3: Ω \sup C.
-  binary_relation ?? U1 U2 → binary_relation ?? U2 U3 →
-   binary_relation ?? U1 U3.
- intros (A B C U1 U2 U3 R12 R23);
+ ∀A,B,C.
+  binary_morphism1 (binary_relation_setoid A B) (binary_relation_setoid B C) (binary_relation_setoid A C).
+ intros;
  constructor 1;
- intros (s1 s3);
- apply (∃s2. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
+  [ intros (R12 R23);
+    constructor 1;
+    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]]
+  | 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) ]
+    exists; try assumption;
+    split; assumption]
+qed.
+
+definition REL: category1.
+ constructor 1;
+  [ apply setoid
+  | 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)]]
+  | 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 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]]]
+qed.
+
+definition full_subset: ∀s:REL. Ω \sup s.
+ apply (λs.{x | True});
+ intros; simplify; split; intro; assumption.
 qed.
 
-interpretation "binary relation composition" 'compose x y = (composition ______ x y).
+coercion full_subset.
 
-definition equal_relations ≝
- λA,B,U,V.λr,r': binary_relation A B U V.
-  ∀x,y. r x y ↔ r' x y.
+definition setoid1_of_REL: REL → setoid ≝ λS. S.
 
-interpretation "equal relation" 'eq x y = (equal_relations ____ x y).
+coercion setoid1_of_REL.
 
-lemma refl_equal_relations: ∀A,B,U,V. reflexive ? (equal_relations A B U V).
- intros 5; intros 2; split; intro; assumption.
+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.
 qed.
 
-lemma sym_equal_relations: ∀A,B,U,V. symmetric ? (equal_relations A B U V).
- intros 7; intros 2; split; intro;
-  [ apply (fi ?? (H ??)) | apply (if ?? (H ??))] assumption.
+interpretation "subset comprehension" 'comprehension s p =
+ (comprehension s (mk_unary_morphism __ p _)).
+
+definition ext: ∀X,S:REL. binary_morphism1 (arrows1 ? X S) S (Ω \sup X).
+ apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 ? X S.λf:S.{x ∈ X | x ♮r f}) ?);
+  [ intros; simplify; apply (.= (H‡#)); apply refl1
+  | intros; simplify; split; intros; simplify; intros; cases f; split; try assumption;
+     [ apply (. (#‡H1)); whd in H; apply (if ?? (H ??)); assumption
+     | apply (. (#‡H1\sup -1)); whd in H; apply (fi ?? (H ??));assumption]]
 qed.
 
-lemma trans_equal_relations: ∀A,B,U,V. transitive ? (equal_relations A B U V).
- intros 9; intros 2; split; intro;
-  [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
-  [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
-  assumption.
+definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup 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
+       |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;
+     [1,3: assumption
+     |2,4: exists; [1,3: apply w]
+      [ apply (. (#‡H)‡#); assumption
+      | apply (. (#‡H\sup -1)‡#); assumption]]]
 qed.
 
-lemma associative_composition:
- ∀A,B,C,D.∀U1,U2,U3,U4.
-  ∀r1:binary_relation A B U1 U2.
-   ∀r2:binary_relation B C U2 U3.
-    ∀r3:binary_relation C D U3 U4.
-     (r1 ∘ r2) ∘ r3 = r1 ∘ (r2 ∘ r3).
- intros 13;
- 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.
+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.∀U1,U2,U3.
-  ∀r1,r1':binary_relation A B U1 U2.
-   ∀r2,r2':binary_relation B C U2 U3.
-    r1 = r1' → r2 = r2' → r1 ∘ r2 = r1' ∘ r2'.
- intros 14; 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.
+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 binary_relation_setoid: ∀A,B. Ω \sup A → Ω \sup B → setoid.
- intros (A B U V);
- constructor 1;
-  [ apply (binary_relation ?? U V)
-  | constructor 1;
-     [ apply equal_relations
-     | apply refl_equal_relations
-     | apply sym_equal_relations
-     | apply trans_equal_relations
-     ]]
+(* the same as ⋄ for a basic pair *)
+definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
+ intros; constructor 1;
+  [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S});
+    intros; simplify; split; intro; cases H1; exists [1,3: apply w]
+     [ apply (. (#‡H)‡#); assumption
+     | apply (. (#‡H \sup -1)‡#); assumption]
+  | intros; split; simplify; intros; cases H2; exists [1,3: apply w]
+     [ apply (. #‡(#‡H1)); cases x; split; try assumption;
+       apply (if ?? (H ??)); assumption
+     | apply (. #‡(#‡H1 \sup -1)); cases x; split; try assumption;
+       apply (if ?? (H \sup -1 ??)); assumption]]
 qed.
 
-record sigma (A:Type) (P: A → Type) : Type ≝
- { s_witness:> A;
-   s_proof:> P s_witness
- }.
+(* the same as □ for a basic pair *)
+definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
+ intros; constructor 1;
+  [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
+    intros; simplify; split; intros; apply H1;
+     [ apply (. #‡H \sup -1); assumption
+     | apply (. #‡H); assumption]
+  | intros; split; simplify; intros; [ apply (. #‡H1); | apply (. #‡H1 \sup -1)]
+    apply H2; [ apply (if ?? (H \sup -1 ??)); | apply (if ?? (H ??)) ] assumption]
+qed.
 
-interpretation "sigma" 'sigma \eta.x = (sigma _ x).
+(* minus_image is the same as ext *)
 
-definition REL: category.
- constructor 1;
-  [ apply (ΣA:Type.Ω \sup A)
-  | intros; apply (binary_relation_setoid ?? (s_proof ?? s) (s_proof ?? s1))
-  | intros; constructor 1; intros; apply (s=s1) 
-  | 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;
-       [ rewrite > H | rewrite < H2 ]
-       assumption
-     |*: exists; try assumption; split; first [ reflexivity | assumption ]]]
+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
+  | change in f with (a ∈ U);
+    exists; [apply a] split; [ change with (a = a); apply refl | assumption]]
 qed.
 
-definition elements: objs REL → Type ≝
- λb:ΣA.Ω\sup A.ssigma (s_witness ?? b) (s_proof ?? b).
-
-coercion elements.
+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]
+qed.
 
-definition carrier: objs REL → Type ≝
- λb:ΣA.Ω\sup A.s_witness ?? b.
+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;
+ 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.
+qed.
 
-interpretation "REL carrier" 'card c = (carrier c).
+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]
+qed.
 
-definition subset: ∀b:objs REL. Ω \sup (carrier b) ≝
- λb:ΣA.Ω\sup A.s_proof ?? b.
+(*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.
 
-coercion subset.
+theorem extS_singleton:
+ ∀o1,o2.∀a:arrows1 ? o1 o2.∀x.extS o1 o2 a (singleton o2 x) = ext o1 o2 a x.
+ intros; unfold extS; unfold ext; unfold singleton; simplify;
+ split; intros 2; simplify; cases f; split; try assumption;
+  [ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
+    assumption
+  | exists; try assumption; split; try assumption; change with (x = x); apply refl]
+qed.
\ No newline at end of file