Fixing universe levels for saturations and (partially) basic_topologies.

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

diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations.ma b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
index eb60386fe3d1abc0831f7f747d7a5093062845c9..c9685f2207f4cf0e93db0d23f2d89fde0fbd2bf7 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/relations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations.ma
@@ -103,11 +103,18 @@ definition full_subset: ∀s:REL. Ω \sup s.
 qed.
 
 coercion full_subset.
+*)
 
 definition setoid1_of_REL: REL → setoid ≝ λS. S.
-
 coercion setoid1_of_REL.
 
+lemma Type_OF_setoid1_of_REL: ∀o1:Type_OF_category1 REL. Type_OF_objs1 o1 → Type_OF_setoid1 ?(*(setoid1_of_SET o1)*).
+ [ apply (setoid1_of_SET o1);
+ | intros; apply t;]
+qed.
+coercion Type_OF_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.
@@ -175,12 +182,6 @@ lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (e
 qed.
 *)
 
-lemma hint: ∀U. carr U → Type_OF_setoid1 ?(*(setoid1_of_SET U)*).
- [ apply setoid1_of_SET; apply U
- | intros; apply c;]
-qed.
-coercion hint.
-
 (* 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;
@@ -291,7 +292,7 @@ qed.
 
 include "o-algebra.ma".
 
-definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → ORelation (SUBSETS o1) (SUBSETS o2).
+definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → arrows2 OA (SUBSETS o1) (SUBSETS o2).
  intros;
  constructor 1;
   [ constructor 1; 
@@ -327,4 +328,82 @@ definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → ORelation (S
      | cases x1; cases x2; clear x1 x2; exists; [apply w1]
         [ exists; [apply w] split; assumption;
         | assumption; ]]]
-qed. (*sistemare anche l'hint da un'altra parte *)
+qed.
+
+lemma orelation_of_relation_preserves_equality:
+ ∀o1,o2:REL.∀t,t': arrows1 ? o1 o2. eq1 ? t t' → orelation_of_relation ?? t = orelation_of_relation ?? t'.
+ intros; split; unfold orelation_of_relation; simplify; intro; split; intro;
+ simplify; whd in o1 o2;
+  [ change with (a1 ∈ minus_star_image ?? t a → a1 ∈ minus_star_image ?? t' a);
+    apply (. #‡(e‡#));
+  | change with (a1 ∈ minus_star_image ?? t' a → a1 ∈ minus_star_image ?? t a);
+    apply (. #‡(e ^ -1‡#));
+  | change with (a1 ∈ minus_image ?? t a → a1 ∈ minus_image ?? t' a);
+    apply (. #‡(e‡#));
+  | change with (a1 ∈ minus_image ?? t' a → a1 ∈ minus_image ?? t a);
+    apply (. #‡(e ^ -1‡#));
+  | change with (a1 ∈ image ?? t a → a1 ∈ image ?? t' a);
+    apply (. #‡(e‡#));
+  | change with (a1 ∈ image ?? t' a → a1 ∈ image ?? t a);
+    apply (. #‡(e ^ -1‡#));
+  | change with (a1 ∈ star_image ?? t a → a1 ∈ star_image ?? t' a);
+    apply (. #‡(e‡#));
+  | change with (a1 ∈ star_image ?? t' a → a1 ∈ star_image ?? t a);
+    apply (. #‡(e ^ -1‡#)); ]
+qed.
+
+lemma hint: ∀o1,o2:OA. Type_OF_setoid2 (arrows2 ? o1 o2) → carr2 (arrows2 OA o1 o2).
+ intros; apply t;
+qed.
+coercion hint.
+
+lemma orelation_of_relation_preserves_identity:
+ ∀o1:REL. orelation_of_relation ?? (id1 ? o1) = id2 OA (SUBSETS o1).
+ intros; split; intro; split; whd; intro; 
+  [ change with ((∀x. x ♮(id1 REL o1) a1→x∈a) → a1 ∈ a); intros;
+    apply (f a1); change with (a1 = a1); apply refl1;
+  | change with (a1 ∈ a → ∀x. x ♮(id1 REL o1) a1→x∈a); intros;
+    change in f1 with (x = a1); apply (. f1 ^ -1‡#); apply f;
+  | alias symbol "and" = "and_morphism".
+    change with ((∃y: carr o1.a1 ♮(id1 REL o1) y ∧ y∈a) → a1 ∈ a);
+    intro; cases e; clear e; cases x; clear x; change in f with (a1=w);
+    apply (. f^-1‡#); apply f1;
+  | change with (a1 ∈ a → ∃y: carr o1.a1 ♮(id1 REL o1) y ∧ y∈a);
+    intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
+  | change with ((∃x: carr o1.x ♮(id1 REL o1) a1∧x∈a) → a1 ∈ a);
+    intro; cases e; clear e; cases x; clear x; change in f with (w=a1);
+    apply (. f‡#); apply f1;
+  | change with (a1 ∈ a → ∃x: carr o1.x ♮(id1 REL o1) a1∧x∈a);
+    intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
+  | change with ((∀y.a1 ♮(id1 REL o1) y→y∈a) → a1 ∈ a); intros;
+    apply (f a1); change with (a1 = a1); apply refl1;
+  | change with (a1 ∈ a → ∀y.a1 ♮(id1 REL o1) y→y∈a); intros;
+    change in f1 with (a1 = y); apply (. f1‡#); apply f;]
+qed.
+
+lemma hint2: ∀S,T. carr2 (arrows2 OA S T) → Type_OF_setoid2 (arrows2 OA S T).
+ intros; apply c;
+qed.
+coercion hint2.
+
+(* CSC: ???? forse un uncertain mancato *)
+lemma orelation_of_relation_preserves_composition:
+ ∀o1,o2,o3:REL.∀F: arrows1 ? o1 o2.∀G: arrows1 ? o2 o3.
+  orelation_of_relation ?? (G ∘ F) =
+   comp2 OA (SUBSETS o1) (SUBSETS o2) (SUBSETS o3)
+    ?? (*(orelation_of_relation ?? F) (orelation_of_relation ?? G)*).
+ [ apply (orelation_of_relation ?? F); | apply (orelation_of_relation ?? G); ]
+ intros; split; intro; split; whd; intro; whd in ⊢ (% → %); intros;
+  [ whd; intros; apply f; exists; [ apply x] split; assumption; 
+  | cases f1; clear f1; cases x1; clear x1; apply (f w); assumption;
+  | cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
+    split; [ assumption | exists; [apply w] split; assumption ]
+  | cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
+    split; [ exists; [apply w] split; assumption | assumption ]
+  | cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
+    split; [ assumption | exists; [apply w] split; assumption ]
+  | cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
+    split; [ exists; [apply w] split; assumption | assumption ]
+  | whd; intros; apply f; exists; [ apply y] split; assumption;
+  | cases f1; clear f1; cases x; clear x; apply (f w); assumption;]
+qed.
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