(**************************************************************************)
include "categories.ma".
-include "logic/cprop_connectives.ma".
inductive bool : Type0 := true : bool | false : bool.
interpretation "unary morphism1 comprehension with proof" 'comprehension_by s \eta.f p =
(mk_unary_morphism1 s _ f p).
-definition hint: Type_OF_category2 SET1 → setoid2.
- intro; apply (setoid2_of_setoid1 t); qed.
-coercion hint.
-
-definition hint2: Type_OF_category1 SET → objs2 SET1.
- intro; apply (setoid1_of_setoid t); qed.
-coercion hint2.
-
(* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete
lattices, Definizione 0.9 *)
(* USARE L'ESISTENZIALE DEBOLE *)
(* ⇔ 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);
+ oa_overlap p (oa_join I q) ⇔ ∃i:carr 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
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.
+
lemma oa_overlap_preservers_meet: ∀O:OAlgebra.∀p,q:O.p >< q → p >< (p ∧ q).
+(* next change to avoid universe inconsistency *)
+change in ⊢ (?→%→%→?) with (Type1_OF_OAlgebra O);
intros; lapply (oa_overlap_preserves_meet_ O p q f);
lapply (prop21 O O CPROP (oa_overlap O) p p ? (p ∧ q) # ?);
[3: apply (if ?? (Hletin1)); apply Hletin;|skip] apply refl1;
| 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;
coercion arrows1_OF_ORelation_setoid.
-lemma umorphism_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → P ⇒ Q.
+lemma umorphism_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → unary_morphism1 P Q.
intros; apply (or_f ?? t);
qed.
coercion umorphism_OF_ORelation_setoid.
+lemma umorphism_setoid_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → unary_morphism1_setoid1 P Q.
+intros; apply (or_f ?? t);
+qed.
+
+coercion umorphism_setoid_OF_ORelation_setoid.
-lemma uncurry_arrows : ∀B,C. arrows1 SET B C → B → C.
-intros; apply ((fun1 ?? t) t1);
+lemma uncurry_arrows : ∀B,C. ORelation_setoid B C → B → C.
+intros; apply ((fun11 ?? t) t1);
qed.
coercion uncurry_arrows 1.
-lemma hint6 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply t;qed.
+lemma hint6: ∀P,Q. Type_OF_setoid2 (hint5 P ⇒ hint5 Q) → unary_morphism1 P Q.
+ intros; apply t;
+qed.
coercion hint6.
-(*
-lemma hint2: OAlgebra → setoid. intros; apply (oa_P o). qed.
-coercion hint2 nocomposites.
-*)
-
-
notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
qed.
definition ORelation_composition : ∀P,Q,R.
- binary_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
+ binary_morphism2 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
intros;
constructor 1;
[ intros (F G);
constructor 1;
[ apply (G ∘ F);
- | apply (G⎻* ∘ F⎻* );
+ | apply rule (G⎻* ∘ F⎻* );
| apply (F* ∘ G* );
| apply (F⎻ ∘ G⎻);
| intros;
apply or_prop3;
]
| intros; split; simplify;
- [1,3: unfold arrows1_OF_ORelation_setoid; apply ((†H)‡(†H1));
- |2,4: apply ((†H1)‡(†H));]]
+ [1,3: unfold umorphism_setoid_OF_ORelation_setoid; unfold arrows1_OF_ORelation_setoid; apply ((†e)‡(†e1));
+ |2,4: apply ((†e1)‡(†e));]]
qed.
-definition OA : category1.
+definition OA : category2.
split;
[ apply (OAlgebra);
| intros; apply (ORelation_setoid o o1);
| intro O; split;
- [1,2,3,4: apply id1;
+ [1,2,3,4: apply id2;
|5,6,7:intros; apply refl1;]
| apply ORelation_composition;
| intros (P Q R S F G H); split;
[ change with (H⎻* ∘ G⎻* ∘ F⎻* = H⎻* ∘ (G⎻* ∘ F⎻* ));
- apply (comp_assoc1 ????? (F⎻* ) (G⎻* ) (H⎻* ));
- | apply ((comp_assoc1 ????? (H⎻) (G⎻) (F⎻))^-1);
- | apply ((comp_assoc1 ????? F G H)^-1);
- | apply ((comp_assoc1 ????? H* G* F* ));]
-| intros; split; unfold ORelation_composition; simplify; apply id_neutral_left1;
-| intros; split; unfold ORelation_composition; simplify; apply id_neutral_right1;]
-qed.
\ No newline at end of file
+ apply (comp_assoc2 ????? (F⎻* ) (G⎻* ) (H⎻* ));
+ | apply ((comp_assoc2 ????? (H⎻) (G⎻) (F⎻))^-1);
+ | apply ((comp_assoc2 ????? F G H)^-1);
+ | apply ((comp_assoc2 ????? H* G* F* ));]
+| intros; split; unfold ORelation_composition; simplify; apply id_neutral_left2;
+| intros; split; unfold ORelation_composition; simplify; apply id_neutral_right2;]
+qed.
+
+lemma setoid1_of_OA: OA → setoid1.
+ 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.
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