(**************************************************************************)
include "categories.ma".
-include "logic/cprop_connectives.ma".
inductive bool : Type0 := true : bool | false : bool.
qed.
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 proof" 'comprehension_by s \eta.f p =
(mk_unary_morphism1 s _ f p).
-definition Type_of_SET: SET → Type0 := λS.carr S.
-coercion Type_of_SET.
-
(* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete
lattices, Definizione 0.9 *)
(* USARE L'ESISTENZIALE DEBOLE *)
-(* Far salire SET usando setoidi1 *)
-alias symbol "comprehension_by" = "unary morphism comprehension with proof".
-record OAlgebra : Type1 := {
- oa_P :> SET;
- oa_leq : binary_morphism1 (setoid1_of_setoid oa_P) (setoid1_of_setoid oa_P) CPROP; (* CPROP is setoid1, CPROP importante che sia small *)
- oa_overlap: binary_morphism1 (setoid1_of_setoid oa_P) (setoid1_of_setoid oa_P) CPROP;
- oa_meet: ∀I:SET.unary_morphism1 (arrows1 SET I oa_P) (setoid1_of_setoid oa_P);
- oa_join: ∀I:SET.unary_morphism1 (arrows1 SET I oa_P) (setoid1_of_setoid oa_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_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;
- (* preservers!! (typo) *)
- oa_overlap_preservers_meet_:
+ 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: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
∀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 $p }.
notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (ident i ∈ I) break term 90 p)"
non associative with precedence 50 for @{ 'oa_meet_mk (λ${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 =
- (fun11 __ (oa_meet __) f).
+ (fun12 __ (oa_meet __) f).
interpretation "o-algebra meet with explicit function" 'oa_meet_mk f =
- (fun11 __ (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).
+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).
-intros; lapply (oa_overlap_preservers_meet_ O p q f);
-lapply (prop1 O O CPROP (oa_overlap O) p 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;
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 _)).
-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;
+definition hint5: OAlgebra → objs2 SET1.
+ intro; apply (oa_P o);
+qed.
+coercion hint5.
+
+record ORelation (P,Q : OAlgebra) : Type2 ≝ {
+ 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;
(* tenere solo una uguaglianza e usare la proposizione 9.9 per
le altre (unicita' degli aggiunti e del simmetrico) *)
- [ 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;
- | whd; simplify; intros; cases H; clear H; split;
- intro a; apply sym; 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.
-
-definition or_f_minus_star: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET P 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 a; clear a; split;
+ intro a; apply sym1; generalize in match a;assumption;
+ | 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;
+ | apply (.= (e3 a)); apply e7;]]]
+qed.
+
+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);
+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 uncurry_arrows : ∀B,C. arrows1 SET B C → B → C.
-intros; apply ((fun_1 ?? c) t);
+lemma umorphism_setoid_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → unary_morphism1_setoid1 P Q.
+intros; apply (or_f ?? t);
qed.
-coercion uncurry_arrows 1.
+coercion umorphism_setoid_OF_ORelation_setoid.
-lemma hint3 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply c;qed.
-coercion hint3 nocomposites.
+lemma uncurry_arrows : ∀B,C. ORelation_setoid B C → B → C.
+intros; apply ((fun11 ?? t) t1);
+qed.
-(*
-lemma hint2: OAlgebra → setoid. intros; apply (oa_P o). qed.
-coercion hint2 nocomposites.
-*)
+coercion uncurry_arrows 1.
+lemma hint6: ∀P,Q. Type_OF_setoid2 (hint5 P ⇒ hint5 Q) → unary_morphism1 P Q.
+ intros; apply t;
+qed.
+coercion hint6.
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}.
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).
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;]
+ 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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