(* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete
lattices, Definizione 0.9 *)
(* USARE L'ESISTENZIALE DEBOLE *)
-(*alias symbol "comprehension_by" = "unary morphism comprehension with proof".*)
record OAlgebra : Type2 := {
oa_P :> SET1;
- oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1, CPROP importante che sia small *)
+ oa_leq : binary_morphism1 oa_P oa_P CPROP;
oa_overlap: binary_morphism1 oa_P oa_P CPROP;
- oa_meet: ∀I:SET.unary_morphism2 (arrows2 SET1 I oa_P) oa_P;
- oa_join: ∀I:SET.unary_morphism2 (arrows2 SET1 I oa_P) oa_P;
+ oa_meet: ∀I:SET.unary_morphism2 (I ⇒ oa_P) oa_P;
+ oa_join: ∀I:SET.unary_morphism2 (I ⇒ oa_P) 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;
- (* 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_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_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 });
- (* ⇔ deve essere =, l'esiste debole *)
oa_join_split:
- ∀I:SET.∀p.∀q:arrows2 SET1 I oa_P.
- oa_overlap p (oa_join I q) ⇔ ∃i:carr I.oa_overlap p (q i);
+ ∀I:SET.∀p.∀q: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
notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (ident i ∈ I) break term 90 p)"
non associative with precedence 50 for @{ 'oa_meet_mk (λ${ident i}:$I.$p) }.
-(*
-notation < "hovbox(a ∧ b)" left associative with precedence 35
-for @{ 'oa_meet_mk (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
-*)
notation > "hovbox(∧ f)" non associative with precedence 60
for @{ 'oa_meet $f }.
-(*
-notation > "hovbox(a ∧ b)" left associative with precedence 50
-for @{ 'oa_meet (mk_unary_morphism BOOL ? (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) (IF_THEN_ELSE_p ? $a $b)) }.
-*)
interpretation "o-algebra meet" 'oa_meet f =
(fun12 __ (oa_meet __) f).
interpretation "o-algebra meet with explicit function" 'oa_meet_mk f =