better notation for oalgebra

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 806859a45dd0ae4e3d70d265843b9a54bdeaf0a4..0a2e84880771c0a30b199f0198f7b55b4bff15b3 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-algebra.ma
@@ -51,40 +51,47 @@ interpretation "unary morphism1 comprehension with proof" 'comprehension_by s \e
 (* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete
    lattices, Definizione 0.9 *)
 (* USARE L'ESISTENZIALE DEBOLE *)
+
+
+notation > "A × B ⇉2,1 C" non associative with precedence 70 for @{binary_morphism1 $A $B $C}.
+notation > "A × B ⇉2,2 C" non associative with precedence 70 for @{binary_morphism2 $A $B $C}.
+notation > "B ⇉1,1 C" non associative with precedence 80 for @{arrows1 SET $B $C}.
+notation > "B ⇉1,2 C" non associative with precedence 80 for @{unary_morphism2 $B $C}.
+notation > "hvbox(a break ≤ b)" non associative with precedence 45 for @{oa_leq $a $b}.
+notation > "a >< b" non associative with precedence 45 for @{oa_overlap $a $b}.
+notation > "⋁ p" non associative with precedence 45 for @{oa_join ? $p}.
+notation > "⋀ p" non associative with precedence 45 for @{oa_meet ? $p}.
 record OAlgebra : Type2 := {
   oa_P :> SET1;
-  oa_leq : binary_morphism1 oa_P oa_P CPROP;
-  oa_overlap: binary_morphism1 oa_P oa_P CPROP;
-  oa_meet: ∀I:SET.unary_morphism2 (I ⇒ oa_P) oa_P;
-  oa_join: ∀I:SET.unary_morphism2 (I ⇒ oa_P) oa_P;
+  oa_leq : oa_P × oa_P ⇉2,1 CPROP;
+  oa_overlap: oa_P × oa_P ⇉2,1 CPROP;
+  oa_meet: ∀I:SET.(I ⇒ oa_P) ⇉1,2 oa_P;
+  oa_join: ∀I:SET.(I ⇒ oa_P) ⇉1,2 oa_P;
   oa_one: oa_P;
   oa_zero: oa_P;
-  oa_leq_refl: ∀a:oa_P. oa_leq a a; 
-  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;
-  oa_meet_inf: 
-    ∀I:SET.∀p_i:I ⇒ oa_P.∀p:oa_P.
-      oa_leq p (oa_meet I p_i) = (∀i:I.oa_leq p (p_i i));
-  oa_join_sup: ∀I:SET.∀p_i:I ⇒ oa_P.∀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;
+  oa_leq_refl: ∀a:oa_P. a ≤ a; 
+  oa_leq_antisym: ∀a,b:oa_P.a ≤ b → b ≤ a → a = b;
+  oa_leq_trans: ∀a,b,c:oa_P.a ≤ b → b ≤ c → a ≤ c;
+  oa_overlap_sym: ∀a,b:oa_P.a >< b → b >< a;
+  oa_meet_inf: ∀I:SET.∀p_i:I ⇒ oa_P.∀p:oa_P.p ≤ (⋀ p_i) = (∀i:I.p ≤ (p_i i));
+  oa_join_sup: ∀I:SET.∀p_i:I ⇒ oa_P.∀p:oa_P.(⋁ p_i) ≤ p = (∀i:I.p_i i ≤ p);
+  oa_zero_bot: ∀p:oa_P.oa_zero ≤ p;
+  oa_one_top: ∀p:oa_P.p ≤ oa_one;
   oa_overlap_preserves_meet_: 
-      ∀p,q:oa_P.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 });
-  oa_join_split:
-      ∀I:SET.∀p.∀q:I ⇒ oa_P.
-       oa_overlap p (oa_join I q) = (∃i:I.oa_overlap p (q i));
+      ∀p,q:oa_P.p >< q → 
+        p >< (⋀ { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p oa_P p q });
+  oa_join_split: ∀I:SET.∀p.∀q:I ⇒ oa_P.p >< (⋁ q) = (∃i:I.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
   *)
-  oa_density: 
-      ∀p,q.(∀r.oa_overlap p r → oa_overlap q r) → oa_leq p q
+  oa_density: ∀p,q.(∀r.p >< r → q >< r) → p ≤ q
 }.
 
+notation "hvbox(a break ≤ b)" non associative with precedence 45 for @{ 'leq $a $b }.
+
 interpretation "o-algebra leq" 'leq a b = (fun21 ??? (oa_leq ?) a b).
 
 notation "hovbox(a mpadded width -150% (>)< b)" non associative with precedence 45