From 6dac171410066642dd976b5cb88f83da9dfa70d8 Mon Sep 17 00:00:00 2001
From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Fri, 26 Dec 2008 17:17:19 +0000
Subject: [PATCH] Some notes by Giovanni.

---
 .../formal_topology/overlap/o-algebra.ma         | 16 +++++++++++++---
 1 file changed, 13 insertions(+), 3 deletions(-)

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
index 0cb3fe117..cb2113414 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
@@ -45,9 +45,13 @@ for @{ 'comprehension_by $s (λ${ident i}:$_. $p) $by}.
 interpretation "unary morphism comprehension with proof" 'comprehension_by s \eta.f p = 
   (mk_unary_morphism s _ f p).
 
+(* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete
+   lattices, Definizione 0.9 *)
+(* USARE L'ESISTENZIALE DEBOLE *)
+(* Far salire SET usando setoidi1 *)
 record OAlgebra : Type := {
   oa_P :> SET;
-  oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
+  oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1, CPROP importante che sia small *)
   oa_overlap: binary_morphism1 oa_P oa_P CPROP;
   oa_meet: ∀I:SET.unary_morphism (arrows1 SET I oa_P) oa_P;
   oa_join: ∀I:SET.unary_morphism (arrows1 SET I oa_P) oa_P;
@@ -57,16 +61,21 @@ record OAlgebra : Type := {
   oa_leq_antisym: ∀a,b:oa_P.oa_leq a b → oa_leq b a → a = b;
   oa_leq_trans: ∀a,b,c:oa_P.oa_leq a b → oa_leq b c → oa_leq a c;
   oa_overlap_sym: ∀a,b:oa_P.oa_overlap a b → oa_overlap b a;
+  (* Errore: = in oa_meet_inf e oa_join_sup *)
   oa_meet_inf: ∀I.∀p_i.∀p:oa_P.oa_leq p (oa_meet I p_i) → ∀i:I.oa_leq p (p_i i);
   oa_join_sup: ∀I.∀p_i.∀p:oa_P.oa_leq (oa_join I p_i) p → ∀i:I.oa_leq (p_i i) p;
   oa_zero_bot: ∀p:oa_P.oa_leq oa_zero p;
   oa_one_top: ∀p:oa_P.oa_leq p oa_one;
+  (* preservers!! (typo) *)
   oa_overlap_preservers_meet_: 
       ∀p,q.oa_overlap p q → oa_overlap p 
        (oa_meet ? { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p oa_P p q });
+  (* ⇔ deve essere =, l'esiste debole *)
   oa_join_split:
       ∀I:SET.∀p.∀q:arrows1 SET I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
   (*oa_base : setoid;
+  1) enum non e' il nome giusto perche' non e' suriettiva
+  2) manca (vedere altro capitolo) la "suriettivita'" come immagine di insiemi di oa_base
   oa_enum : ums oa_base oa_P;
   oa_density: ∀p,q.(∀i.oa_overlap p (oa_enum i) → oa_overlap q (oa_enum i)) → oa_leq p q
   *)
@@ -152,6 +161,8 @@ intros (P Q);
 constructor 1;
 [ apply (ORelation P Q);
 | constructor 1;
+   (* tenere solo una uguaglianza e usare la proposizione 9.9 per
+      le altre (unicita' degli aggiunti e del simmetrico) *)
    [ apply (λp,q. And4 (eq1 ? (or_f_minus_star_ ?? p) (or_f_minus_star_ ?? q)) 
              (eq1 ? (or_f_minus_ ?? p) (or_f_minus_ ?? q)) 
              (eq1 ? (or_f_ ?? p) (or_f_ ?? q)) 
@@ -254,8 +265,7 @@ intros;
 constructor 1;
 [ intros (F G);
   constructor 1;
-  [ lapply (G ∘ F);
-    apply (G ∘ F);
+  [ apply (G ∘ F);
   | apply (G⎻* ∘ F⎻* );
   | apply (F* ∘ G* );
   | apply (F⎻ ∘ G⎻);
-- 
2.39.2