(* *)
(**************************************************************************)
-include "datatypes/categories.ma".
-include "logic/cprop_connectives.ma".
+include "categories.ma".
-inductive bool : Type := true : bool | false : bool.
+inductive bool : Type0 := true : bool | false : bool.
-lemma ums : setoid → setoid → setoid.
-intros (S T);
-constructor 1;
-[ apply (unary_morphism S T);
-| constructor 1;
- [ intros (f1 f2); apply (∀a,b:S.eq1 ? a b → eq1 ? (f1 a) (f2 b));
- | whd; simplify; intros; apply (.= (†H)); apply refl1;
- | whd; simplify; intros; apply (.= (†H1)); apply sym1; apply H; apply refl1;
- | whd; simplify; intros; apply (.= (†H2)); apply (.= (H ?? #)); apply (.= (H1 ?? #)); apply rule #;]]
-qed.
-
-lemma BOOL : setoid.
+lemma BOOL : objs1 SET.
constructor 1; [apply bool] constructor 1;
[ intros (x y); apply (match x with [ true ⇒ match y with [ true ⇒ True | _ ⇒ False] | false ⇒ match y with [ true ⇒ False | false ⇒ True ]]);
| whd; simplify; intros; cases x; apply I;
| whd; simplify; intros 2; cases x; cases y; simplify; intros; assumption;
-| whd; simplify; intros 3; cases x; cases y; cases z; simplify; intros; try assumption; apply I]
+| whd; simplify; intros 3; cases x; cases y; cases z; simplify; intros;
+ try assumption; apply I]
qed.
lemma IF_THEN_ELSE_p :
- ∀S:setoid.∀a,b:S.∀x,y:BOOL.x = y →
- let f ≝ λm.match m with [ true ⇒ a | false ⇒ b ] in f x = f y.
-intros; cases x in H; cases y; simplify; intros; try apply refl; whd in H; cases H;
-qed.
-
-lemma if_then_else : ∀T:setoid. ∀a,b:T. ums BOOL T.
-intros; constructor 1; intros;
-[ apply (match c2 with [ true ⇒ c | false ⇒ c1 ]);
-| apply (IF_THEN_ELSE_p T c c1 a a' H);]
+ ∀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 e; cases y; simplify; intros; try apply refl1; whd in e; cases e;
qed.
-record OAlgebra : Type := {
- oa_P :> setoid;
- oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
+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}.
+notation < "hvbox({ ident i ∈ s | term 19 p })" 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).
+interpretation "unary morphism1 comprehension with proof" 'comprehension_by s \eta.f p =
+ (mk_unary_morphism1 s _ f p).
+
+(* per il set-indexing vedere capitolo BPTools (foundational tools), Sect. 0.3.4 complete
+ lattices, Definizione 0.9 *)
+(* USARE L'ESISTENZIALE DEBOLE *)
+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:setoid.unary_morphism (ums I oa_P) oa_P;
- oa_join: ∀I:setoid.unary_morphism (ums 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;
- 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_preservers_meet:
- ∀p,q.oa_overlap p q → oa_overlap p
- (oa_meet BOOL (if_then_else oa_P p q));
- oa_join_split: (* ha I → oa_P da castare a funX (ums A oa_P) *)
- ∀I:setoid.∀p.∀q:ums I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
+ 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 });
+ oa_join_split:
+ ∀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
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*)
+ oa_density: ∀p,q.(∀i.oa_overlap p (oa_enum i) → oa_overlap q (oa_enum i)) → oa_leq p q
+ *)
oa_density:
∀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(∧ f)" non associative with precedence 60
for @{ 'oa_meet $f }.
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (ident i ∈ I) break term 90 p)" non associative with precedence 50
-for @{ 'oa_meet (λ${ident i}:$I.$p) }.
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (\emsp) \nbsp term 90 p)" non associative with precedence 50
-for @{ 'oa_meet (λ${ident i}.($p $_)) }.
-notation < "hovbox(a ∧ b)" left associative with precedence 50
-for @{ 'oa_meet2 $a $b }.
+interpretation "o-algebra meet" 'oa_meet f =
+ (fun12 __ (oa_meet __) f).
+interpretation "o-algebra meet with explicit function" 'oa_meet_mk f =
+ (fun12 __ (oa_meet __) (mk_unary_morphism _ _ f _)).
-interpretation "o-algebra meet" 'oa_meet \eta.f = (fun_1 __ (oa_meet __) f).
-interpretation "o-algebra binary meet" 'and x y = (fun_1 __ (oa_meet _ BOOL) (if_then_else _ x y)).
+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; 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.
+
+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;
+qed.
+
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)"
+non associative with precedence 49 for @{ 'oa_join $p }.
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (ident i ∈ I) break term 90 p)"
+non associative with precedence 49 for @{ 'oa_join_mk (λ${ident i}:$I.$p) }.
+notation < "hovbox(a ∨ b)" left associative with precedence 49
+for @{ 'oa_join_mk (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
-(*
-notation > "hovbox(a ∨ b)" left associative with precedence 49
-for @{ 'oa_join (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) }.
notation > "hovbox(∨ f)" non associative with precedence 59
for @{ 'oa_join $f }.
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (ident i ∈ I) break term 90 p)" non associative with precedence 49
-for @{ 'oa_join (λ${ident i}:$I.$p) }.
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)" non associative with precedence 49
-for @{ 'oa_join (λ${ident i}.($p $_)) }.
-notation < "hovbox(a ∨ b)" left associative with precedence 49
-for @{ 'oa_join (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
-
-interpretation "o-algebra join" 'oa_join \eta.f = (oa_join _ _ f).
-*)
-
-record ORelation (P,Q : OAlgebra) : Type ≝ {
- 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
+notation > "hovbox(a ∨ b)" left associative with precedence 49
+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 =
+ (fun12 __ (oa_join __) f).
+interpretation "o-algebra join with explicit function" 'oa_join_mk 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) : 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 → 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. 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 e; assumption]
+qed.
+
+definition or_f: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (P ⇒ Q).
+ intros; constructor 1;
+ [ apply or_f_;
+ | intros; cases e; assumption]
+qed.
+
+coercion or_f.
+
+definition or_f_minus: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (Q ⇒ P).
+ intros; constructor 1;
+ [ apply or_f_minus_;
+ | intros; cases e; assumption]
+qed.
+
+definition or_f_star: ∀P,Q:OAlgebra.unary_morphism2 (ORelation_setoid P Q) (Q ⇒ P).
+ intros; constructor 1;
+ [ apply or_f_star_;
+ | intros; cases e; assumption]
+qed.
+
+lemma arrows1_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → (P ⇒ Q).
+intros; apply (or_f ?? t);
+qed.
+
+coercion arrows1_OF_ORelation_setoid.
+
+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. ORelation_setoid B C → B → C.
+intros; apply ((fun11 ?? t) t1);
+qed.
+
+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}.
-interpretation "o-relation f*" 'OR_f_star r = (or_f_star _ _ r).
notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
-interpretation "o-relation f⎻*" 'OR_f_minus_star r = (or_f_minus_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 r = (or_f_minus _ _ r).
-axiom DAEMON: False.
+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 ORelation_setoid : OAlgebra → OAlgebra → setoid1.
-intros (P Q);
-constructor 1;
-[ apply (ORelation P Q);
-| constructor 1;
- [
- alias symbol "and" = "constructive and".
- apply (λp,q.
- (∀a.p⎻* a = q⎻* a) ∧
- (∀a.p⎻ a = q⎻ a) ∧
- (∀a.p a = q a) ∧
- (∀a.p* a = q* a));
- | whd; simplify; intros; repeat split; intros; apply refl;
- | whd; simplify; intros; cases H; cases H1; cases H3; clear H H3 H1;
- repeat split; intros; apply sym; generalize in match a;assumption;
- | whd; simplify; intros; elim DAEMON;]]
-qed.
-
-lemma hint : ∀P,Q. ORelation_setoid P Q → P ⇒ Q. intros; apply (or_f ?? c);qed.
-coercion hint.
-
-definition composition : ∀P,Q,R.
- binary_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
+definition or_prop1 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
+ (F p ≤ q) = (p ≤ F* q).
+intros; apply (or_prop1_ ?? F p q);
+qed.
+
+definition or_prop2 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
+ (F⎻ p ≤ q) = (p ≤ F⎻* q).
+intros; apply (or_prop2_ ?? F p q);
+qed.
+
+definition or_prop3 : ∀P,Q:OAlgebra.∀F:ORelation_setoid P Q.∀p,q.
+ (F p >< q) = (p >< F⎻ q).
+intros; apply (or_prop3_ ?? F p q);
+qed.
+
+definition ORelation_composition : ∀P,Q,R.
+ binary_morphism2 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
intros;
constructor 1;
[ intros (F G);
constructor 1;
- [ constructor 1; [apply (λx. G (F x)); | intros; apply (†(†H));]
- |2,3,4,5,6,7: cases DAEMON;]
-| intros; cases DAEMON;]
+ [ apply (G ∘ F);
+ | apply rule (G⎻* ∘ F⎻* );
+ | apply (F* ∘ G* );
+ | apply (F⎻ ∘ G⎻);
+ | intros;
+ change with ((G (F p) ≤ q) = (p ≤ (F* (G* q))));
+ apply (.= (or_prop1 :?));
+ apply (or_prop1 :?);
+ | intros;
+ change with ((F⎻ (G⎻ p) ≤ q) = (p ≤ (G⎻* (F⎻* q))));
+ apply (.= (or_prop2 :?));
+ apply or_prop2 ;
+ | intros; change with ((G (F (p)) >< q) = (p >< (F⎻ (G⎻ q))));
+ apply (.= (or_prop3 :?));
+ apply or_prop3;
+ ]
+| intros; split; simplify;
+ [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. (* category2 *)
+definition OA : category2.
split;
[ apply (OAlgebra);
| intros; apply (ORelation_setoid o o1);
| intro O; split;
- [1,2,3,4: constructor 1; [1,3,5,7:apply (λx.x);|*:intros;assumption]
- |5,6,7:intros;split;intros; assumption; ]
-|4: apply composition;
-|*: elim DAEMON;]
-qed.
+ [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_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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