(* *)
(**************************************************************************)
-include "datatypes/categories.ma".
+include "categories.ma".
include "logic/cprop_connectives.ma".
-inductive bool : Type := true : bool | false : bool.
+inductive bool : Type0 := true : bool | false : bool.
lemma BOOL : objs1 SET.
constructor 1; [apply bool] constructor 1;
try assumption; apply I]
qed.
-definition setoid_OF_SET: objs1 SET → setoid.
- intros; apply o; qed.
-
-coercion setoid_OF_SET.
-
lemma IF_THEN_ELSE_p :
- ∀S:setoid.∀a,b:S.∀x,y:BOOL.x = y →
+ ∀S:setoid1.∀a,b:S.∀x,y:BOOL.x = y →
(λm.match m with [ true ⇒ a | false ⇒ b ]) x =
(λm.match m with [ true ⇒ a | false ⇒ b ]) y.
whd in ⊢ (?→?→?→%→?);
-intros; cases x in H; cases y; simplify; intros; try apply refl; whd in H; cases H;
-qed.
+intros; cases x in e; cases y; simplify; intros; try apply refl1; whd in e; cases e;
+qed.
interpretation "unary morphism comprehension with no proof" 'comprehension T P =
(mk_unary_morphism T _ P _).
+interpretation "unary morphism1 comprehension with no proof" 'comprehension T P =
+ (mk_unary_morphism1 T _ P _).
notation > "hvbox({ ident i ∈ s | term 19 p | by })" with precedence 90
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).
-
-record OAlgebra : Type := {
- oa_P :> SET;
- oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
+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 *)
+(*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_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;
+ 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_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_zero_bot: ∀p:oa_P.oa_leq oa_zero p;
oa_one_top: ∀p:oa_P.oa_leq p oa_one;
- oa_overlap_preservers_meet_:
- ∀p,q.oa_overlap p q → oa_overlap p
+ 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:arrows1 SET I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
+ ∀I:SET.∀p.∀q:arrows2 SET1 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
*)
∀p,q.(∀r.oa_overlap p r → oa_overlap q r) → oa_leq p q
}.
-interpretation "o-algebra leq" 'leq a b = (fun1 ___ (oa_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
for @{ 'overlap $a $b}.
-interpretation "o-algebra overlap" 'overlap a b = (fun1 ___ (oa_overlap _) a b).
+interpretation "o-algebra overlap" 'overlap a b = (fun21 ___ (oa_overlap _) a b).
notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (\emsp) \nbsp term 90 p)"
non associative with precedence 50 for @{ 'oa_meet $p }.
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 =
- (fun_1 __ (oa_meet __) f).
+ (fun12 __ (oa_meet __) f).
interpretation "o-algebra meet with explicit function" 'oa_meet_mk f =
- (fun_1 __ (oa_meet __) (mk_unary_morphism _ _ f _)).
+ (fun12 __ (oa_meet __) (mk_unary_morphism _ _ f _)).
+
+definition hint3: OAlgebra → setoid1.
+ intro; apply (oa_P o);
+qed.
+coercion hint3.
+
+definition hint4: ∀A. setoid2_OF_OAlgebra A → hint3 A.
+ intros; apply t;
+qed.
+coercion hint4.
definition binary_meet : ∀O:OAlgebra. binary_morphism1 O O O.
intros; split;
[ intros (p q);
apply (∧ { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p ? p q });
-| intros; apply (prop_1 ?? (oa_meet O BOOL)); intro x; simplify;
- cases x; simplify; assumption;]
+| intros; lapply (prop12 ? O (oa_meet O BOOL));
+ [2: apply ({ x ∈ BOOL | match x with [ true ⇒ a | false ⇒ b ] | IF_THEN_ELSE_p ? a b });
+ |3: apply ({ x ∈ BOOL | match x with [ true ⇒ a' | false ⇒ b' ] | IF_THEN_ELSE_p ? a' b' });
+ | apply Hletin;]
+ 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 =
- (fun1 ___ (binary_meet _) a b).
+ (fun21 ___ (binary_meet _) a b).
lemma oa_overlap_preservers_meet: ∀O:OAlgebra.∀p,q:O.p >< q → p >< (p ∧ q).
-intros; lapply (oa_overlap_preservers_meet_ O p q f);
-lapply (prop1 O O CPROP (oa_overlap O) p p ? (p ∧ q) # ?);
+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;
qed.
for @{ 'oa_join (mk_unary_morphism BOOL ? (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) (IF_THEN_ELSE_p ? $a $b)) }.
interpretation "o-algebra join" 'oa_join f =
- (fun_1 __ (oa_join __) f).
+ (fun12 __ (oa_join __) f).
interpretation "o-algebra join with explicit function" 'oa_join_mk f =
- (fun_1 __ (oa_join __) (mk_unary_morphism _ _ f _)).
+ (fun12 __ (oa_join __) (mk_unary_morphism _ _ f _)).
+
+definition hint5: OAlgebra → objs2 SET1.
+ intro; apply (oa_P o);
+qed.
+coercion hint5.
record ORelation (P,Q : OAlgebra) : Type ≝ {
- or_f_ : arrows1 SET P Q;
- or_f_minus_star_ : arrows1 SET P Q;
- or_f_star_ : arrows1 SET Q P;
- or_f_minus_ : arrows1 SET Q P;
+ or_f_ : P ⇒ Q;
+ or_f_minus_star_ : P ⇒ Q;
+ or_f_star_ : Q ⇒ P;
+ or_f_minus_ : Q ⇒ P;
or_prop1_ : ∀p,q. (or_f_ p ≤ q) = (p ≤ or_f_star_ q);
or_prop2_ : ∀p,q. (or_f_minus_ p ≤ q) = (p ≤ or_f_minus_star_ q);
or_prop3_ : ∀p,q. (or_f_ p >< q) = (p >< or_f_minus_ q)
}.
-
-definition ORelation_setoid : OAlgebra → OAlgebra → setoid1.
+definition ORelation_setoid : OAlgebra → OAlgebra → setoid2.
intros (P Q);
constructor 1;
[ apply (ORelation P Q);
| constructor 1;
- [ 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))
- (eq1 ? (or_f_star_ ?? p) (or_f_star_ ?? q)));
- | whd; simplify; intros; repeat split; intros; apply refl1;
+ (* 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))
+ (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;
- intro a; apply sym; generalize in match a;assumption;
+ intro a; apply sym1; generalize in match a;assumption;
| whd; simplify; intros; cases H; cases H1; clear H H1; split; intro a;
- [ apply (.= (H2 a)); apply H6;
- | apply (.= (H3 a)); apply H7;
- | apply (.= (H4 a)); apply H8;
- | apply (.= (H5 a)); apply H9;]]]
-qed.
+ [ apply (.= (e a)); apply e4;
+ | apply (.= (e1 a)); apply e5;
+ | apply (.= (e2 a)); apply e6;
+ | apply (.= (e3 a)); apply e7;]]]
+qed.
-definition or_f_minus_star: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET P Q.
+definition or_f_minus_star:
+ ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (P ⇒ Q).
intros; constructor 1;
[ apply or_f_minus_star_;
- | intros; cases H; assumption]
+ | intros; cases e; assumption]
qed.
-definition or_f: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET P Q.
+definition or_f: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (P ⇒ Q).
intros; constructor 1;
[ apply or_f_;
- | intros; cases H; assumption]
+ | intros; cases e; assumption]
qed.
coercion or_f.
-definition or_f_minus: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET Q P.
+definition or_f_minus: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (Q ⇒ P).
intros; constructor 1;
[ apply or_f_minus_;
- | intros; cases H; assumption]
+ | intros; cases e; assumption]
qed.
-definition or_f_star: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET Q P.
+definition or_f_star: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (Q ⇒ P).
intros; constructor 1;
[ apply or_f_star_;
- | intros; cases H; assumption]
+ | intros; cases e; assumption]
qed.
-lemma arrows1_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → arrows1 SET P Q.
-intros; apply (or_f ?? c);
+lemma arrows1_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → (P ⇒ Q).
+intros; apply (or_f ?? t);
qed.
-coercion arrows1_OF_ORelation_setoid nocomposites.
+coercion arrows1_OF_ORelation_setoid.
lemma umorphism_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → P ⇒ Q.
-intros; apply (or_f ?? c);
+intros; apply (or_f ?? t);
qed.
coercion umorphism_OF_ORelation_setoid.
lemma uncurry_arrows : ∀B,C. arrows1 SET B C → B → C.
-intros; apply ((fun_1 ?? c) t);
+intros; apply ((fun1 ?? t) t1);
qed.
coercion uncurry_arrows 1.
-lemma hint3 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply c;qed.
-coercion hint3 nocomposites.
+lemma hint6 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply t;qed.
+coercion hint6.
(*
lemma hint2: OAlgebra → setoid. intros; apply (oa_P o). qed.
notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
-interpretation "o-relation f⎻*" 'OR_f_minus_star r = (fun_1 __ (or_f_minus_star _ _) r).
-interpretation "o-relation f⎻" 'OR_f_minus r = (fun_1 __ (or_f_minus _ _) r).
-interpretation "o-relation f*" 'OR_f_star r = (fun_1 __ (or_f_star _ _) r).
+interpretation "o-relation f⎻*" 'OR_f_minus_star r = (fun12 __ (or_f_minus_star _ _) r).
+interpretation "o-relation f⎻" 'OR_f_minus r = (fun12 __ (or_f_minus _ _) r).
+interpretation "o-relation f*" 'OR_f_star r = (fun12 __ (or_f_star _ _) r).
definition or_prop1 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
(F p ≤ q) = (p ≤ F* q).
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⎻);
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