(**************************************************************************)
include "categories.ma".
-include "logic/cprop_connectives.ma".
inductive bool : Type0 := true : bool | false : bool.
(* 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: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 =
intro x; simplify; cases x; simplify; assumption;]
qed.
-notation "hovbox(a ∧ b)" left associative with precedence 35
-for @{ 'oa_meet_bin $a $b }.
-interpretation "o-algebra binary meet" 'oa_meet_bin a b =
+interpretation "o-algebra binary meet" 'and a b =
(fun21 ___ (binary_meet _) a b).
coercion Type1_OF_OAlgebra nocomposites.
qed.
coercion hint5.
-record ORelation (P,Q : OAlgebra) : Type ≝ {
+record ORelation (P,Q : OAlgebra) : Type2 ≝ {
or_f_ : P ⇒ Q;
or_f_minus_star_ : P ⇒ Q;
or_f_star_ : Q ⇒ P;
| 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 (eq2 ? (or_f_minus_star_ ?? p) (or_f_minus_star_ ?? q))
+ [ apply (λp,q. And42 (eq2 ? (or_f_minus_star_ ?? p) (or_f_minus_star_ ?? q))
(eq2 ? (or_f_minus_ ?? p) (or_f_minus_ ?? q))
(eq2 ? (or_f_ ?? p) (or_f_ ?? q))
(eq2 ? (or_f_star_ ?? p) (or_f_star_ ?? q)));
| whd; simplify; intros; repeat split; intros; apply refl2;
- | whd; simplify; intros; cases H; clear H; split;
+ | whd; simplify; intros; cases a; clear a; split;
intro a; apply sym1; generalize in match a;assumption;
- | whd; simplify; intros; cases H; cases H1; clear H H1; split; intro a;
+ | whd; simplify; intros; cases a; cases a1; clear a a1; split; intro a;
[ apply (.= (e a)); apply e4;
| apply (.= (e1 a)); apply e5;
| apply (.= (e2 a)); apply e6;
intro; apply (oa_P t);
qed.
coercion setoid1_of_OA.
+
+lemma SET1_of_OA: OA → SET1.
+ intro; whd; apply (setoid1_of_OA t);
+qed.
+coercion SET1_of_OA.
+
+lemma objs2_SET1_OF_OA: OA → objs2 SET1.
+ intro; whd; apply (setoid1_of_OA t);
+qed.
+coercion objs2_SET1_OF_OA.
+
+lemma Type_OF_category2_OF_SET1_OF_OA: OA → Type_OF_category2 SET1.
+ intro; apply (oa_P t);
+qed.
+coercion Type_OF_category2_OF_SET1_OF_OA.