(* *)
(**************************************************************************)
-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 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.
-definition hint: objs1 SET → setoid.
- intros; apply o;
-qed.
-
-coercion hint.
-
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:SET. ∀a,b:T. arrows1 SET BOOL T.
-intros; constructor 1; intros;
-[ apply (match c with [ true ⇒ t | false ⇒ t1 ]);
-| apply (IF_THEN_ELSE_p T t t1 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.
-interpretation "mk " 'comprehension T P =
- (mk_unary_morphism T _ P _).
+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 (\lambda ${ident i}. $p) $by}.
-
-interpretation "unary morphism comprehension with proof" 'comprehension_by s f p =
- (mk_unary_morphism s _ f p).
+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}.
-definition A : ∀S:SET.∀a,b:S.arrows1 SET BOOL S.
-apply (λS,a,b.{ x ∈ BOOL | match x with [ true ⇒ a | false ⇒ b] | IF_THEN_ELSE_p S a b}).
-qed.
+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).
-record OAlgebra : Type := {
- oa_P :> SET;
- oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
+(* 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: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 (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_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_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:SET.∀p.∀q:arrows1 SET I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
+ 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 $_)) }.
-notation < "hovbox(a ∧ b)" left associative with precedence 50
-for @{ 'oa_meet
- ($foo $bar $baz
- (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ])
- $res) }.
-
-interpretation "o-algebra meet" 'oa_meet f = (fun_1 __ (oa_meet __) f).
-(*interpretation "o-algebra binary meet" 'and x y = (fun_1 __ (oa_meet _ BOOL) (if_then_else _ x y)).*)
-
-(*
-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
+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 ?)).
+
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)"
+non associative with precedence 50 for @{ 'oa_join $p }.
+notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (ident i ∈ I) break term 90 p)"
+non associative with precedence 50 for @{ 'oa_join_mk (λ${ident i}:$I.$p) }.
+
+notation > "hovbox(∨ f)" non associative with precedence 60
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 $_)) }.
+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 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).
+
+prefer coercion Type1_OF_OAlgebra.
+
+definition binary_join : ∀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_join 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 join" 'or a b =
+ (fun21 ??? (binary_join ?) a b).
+
+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 (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
-
-interpretation "o-algebra join" 'oa_join \eta.f = (oa_join _ _ f).
-*)
-
-record ORelation (P,Q : OAlgebra) : Type ≝ {
- or_f :> arrows1 SET P Q;
- or_f_minus_star : arrows1 SET P Q;
- or_f_star : arrows1 SET Q P;
- or_f_minus : arrows1 SET Q P;
- or_prop1 : ∀p,q. (or_f p ≤ q) = (p ≤ or_f_star q);
- or_prop2 : ∀p,q. (or_f_minus p ≤ q) = (p ≤ or_f_minus_star q);
- or_prop3 : ∀p,q. (or_f p >< q) = (p >< or_f_minus q)
+for @{ 'oa_join_mk (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
+
+notation > "hovbox(∨ f)" non associative with precedence 59
+for @{ 'oa_join $f }.
+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 ?)).
+
+record ORelation (P,Q : OAlgebra) : Type2 ≝ {
+ or_f_ : carr2 (P ⇒ Q);
+ or_f_minus_star_ : carr2(P ⇒ Q);
+ or_f_star_ : carr2(Q ⇒ P);
+ or_f_minus_ : carr2(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
+ (or_f_minus_star_ ?? p = or_f_minus_star_ ?? q)
+ (or_f_minus_ ?? p = or_f_minus_ ?? q)
+ (or_f_ ?? p = or_f_ ?? q)
+ (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 ORelation_of_ORelation_setoid :
+ ∀P,Q.ORelation_setoid P Q → ORelation P Q ≝ λP,Q,x.x.
+coercion ORelation_of_ORelation_setoid.
+
+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.
+
+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 ?? c);
+qed.
+coercion arrows1_of_ORelation_setoid.
+
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;
- [ apply (λp,q. eq1 ? p⎻* q⎻* ∧ eq1 ? p⎻ q⎻ ∧ eq1 ? p q ∧ eq1 ? p* q* );
- | whd; simplify; intros; repeat split; intros; apply refl1;
- | whd; simplify; intros; cases H; cases H1; cases H3; clear H H3 H1;
- repeat split; intros; apply sym1; assumption;
- | whd; simplify; intros; cases H; cases H1; cases H2; cases H4; cases H6; cases H8;
- repeat split; intros; clear H H1 H2 H4 H6 H8; apply trans1;
- [2: apply H10;
- |5: apply H11;
- |8: apply H7;
- |11: apply H3;
- |1,4,7,10: skip
- |*: assumption
- ]]]
-qed.
-
-lemma hint1 : ∀P,Q. ORelation_setoid P Q → arrows1 SET P Q. intros; apply (or_f ?? c);qed.
-coercion hint1.
-
-lemma hint3 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply c;qed.
-coercion hint3.
-
-lemma hint2: OAlgebra → setoid. intros; apply (oa_P o). qed.
-coercion hint2.
-
-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;
[ apply (G ∘ F);
- | 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 ??? (F p) ?);
- apply (.= or_prop1 ??? p ?);
- apply refl1
- | intros; change with ((F⎻ (G⎻ p) ≤ q) = (p ≤ (G⎻* (F⎻* q))));
- apply (.= or_prop2 ??? (G⎻ p) ?);
- apply (.= or_prop2 ??? p ?);
- apply refl1;
+ | 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 ??? (F p) ?);
- apply (.= or_prop3 ??? p ?);
- apply refl1
+ apply (.= (or_prop3 :?));
+ apply or_prop3;
]
-| intros; repeat split; simplify; cases DAEMON (*
- [ apply trans1; [2: apply prop1; [3: apply rule #; | skip | 4:
- apply rule (†?);
-
- lapply (.= ((†H1)‡#)); [8: apply Hletin;
- [ apply trans1; [2: lapply (prop1); [apply Hletin;
-*)]
+| intros; split; simplify;
+ [3: unfold arrows1_of_ORelation_setoid; apply ((†e)‡(†e1));
+ |1: 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 composition;
-| intros; repeat split; unfold composition; simplify;
- [1,3: apply (comp_assoc1); | 2,4: apply ((comp_assoc1 ????????) \sup -1);]
-| intros; repeat split; unfold composition; simplify; apply id_neutral_left1;
-| intros; repeat split; unfold composition; simplify; apply id_neutral_right1;]
+| 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.
+
+definition OAlgebra_of_objs2_OA: objs2 OA → OAlgebra ≝ λx.x.
+coercion OAlgebra_of_objs2_OA.
+
+definition ORelation_setoid_of_arrows2_OA:
+ ∀P,Q. arrows2 OA P Q → ORelation_setoid P Q ≝ λP,Q,c.c.
+coercion ORelation_setoid_of_arrows2_OA.
+
+prefer coercion Type_OF_objs2.
+
+(* alias symbol "eq" = "setoid1 eq". *)
+
+(* qui la notazione non va *)
+lemma leq_to_eq_join: ∀S:OA.∀p,q:S. p ≤ q → q = (binary_join ? p q).
+ intros;
+ apply oa_leq_antisym;
+ [ apply oa_density; intros;
+ apply oa_overlap_sym;
+ unfold binary_join; simplify;
+ apply (. (oa_join_split : ?));
+ exists; [ apply false ]
+ apply oa_overlap_sym;
+ assumption
+ | unfold binary_join; simplify;
+ apply (. (oa_join_sup : ?)); intro;
+ cases i; whd in ⊢ (? ? ? ? ? % ?);
+ [ assumption | apply oa_leq_refl ]]
+qed.
+
+lemma overlap_monotone_left: ∀S:OA.∀p,q,r:S. p ≤ q → p >< r → q >< r.
+ intros;
+ apply (. (leq_to_eq_join : ?)‡#);
+ [ apply f;
+ | skip
+ | apply oa_overlap_sym;
+ unfold binary_join; simplify;
+ apply (. (oa_join_split : ?));
+ exists [ apply true ]
+ apply oa_overlap_sym;
+ assumption; ]
+qed.
+
+(* Part of proposition 9.9 *)
+lemma f_minus_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R⎻ p ≤ R⎻ q.
+ intros;
+ apply (. (or_prop2 : ?));
+ apply oa_leq_trans; [2: apply f; | skip | apply (. (or_prop2 : ?)^ -1); apply oa_leq_refl;]
+qed.
+
+(* Part of proposition 9.9 *)
+lemma f_minus_star_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R⎻* p ≤ R⎻* q.
+ intros;
+ apply (. (or_prop2 : ?)^ -1);
+ apply oa_leq_trans; [3: apply f; | skip | apply (. (or_prop2 : ?)); apply oa_leq_refl;]
+qed.
+
+(* Part of proposition 9.9 *)
+lemma f_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R p ≤ R q.
+ intros;
+ apply (. (or_prop1 : ?));
+ apply oa_leq_trans; [2: apply f; | skip | apply (. (or_prop1 : ?)^ -1); apply oa_leq_refl;]
+qed.
+
+(* Part of proposition 9.9 *)
+lemma f_star_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R* p ≤ R* q.
+ intros;
+ apply (. (or_prop1 : ?)^ -1);
+ apply oa_leq_trans; [3: apply f; | skip | apply (. (or_prop1 : ?)); apply oa_leq_refl;]
+qed.
+
+lemma lemma_10_2_a: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R⎻* (R⎻ p).
+ intros;
+ apply (. (or_prop2 : ?)^-1);
+ apply oa_leq_refl.
+qed.
+
+lemma lemma_10_2_b: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻ (R⎻* p) ≤ p.
+ intros;
+ apply (. (or_prop2 : ?));
+ apply oa_leq_refl.
+qed.
+
+lemma lemma_10_2_c: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R* (R p).
+ intros;
+ apply (. (or_prop1 : ?)^-1);
+ apply oa_leq_refl.
+qed.
+
+lemma lemma_10_2_d: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* p) ≤ p.
+ intros;
+ apply (. (or_prop1 : ?));
+ apply oa_leq_refl.
+qed.
+
+lemma lemma_10_3_a: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻ (R⎻* (R⎻ p)) = R⎻ p.
+ intros; apply oa_leq_antisym;
+ [ apply lemma_10_2_b;
+ | apply f_minus_image_monotone;
+ apply lemma_10_2_a; ]
+qed.
+
+lemma lemma_10_3_b: ∀S,T.∀R:arrows2 OA S T.∀p. R* (R (R* p)) = R* p.
+ intros; apply oa_leq_antisym;
+ [ apply f_star_image_monotone;
+ apply (lemma_10_2_d ?? R p);
+ | apply lemma_10_2_c; ]
+qed.
+
+lemma lemma_10_3_c: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* (R p)) = R p.
+ intros; apply oa_leq_antisym;
+ [ apply lemma_10_2_d;
+ | apply f_image_monotone;
+ apply (lemma_10_2_c ?? R p); ]
+qed.
+
+lemma lemma_10_3_d: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻* (R⎻ (R⎻* p)) = R⎻* p.
+ intros; apply oa_leq_antisym;
+ [ apply f_minus_star_image_monotone;
+ apply (lemma_10_2_b ?? R p);
+ | apply lemma_10_2_a; ]
+qed.
+
+lemma lemma_10_4_a: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻* (R⎻ (R⎻* (R⎻ p))) = R⎻* (R⎻ p).
+ intros; apply (†(lemma_10_3_a ?? R p));
+qed.
+
+lemma lemma_10_4_b: ∀S,T.∀R:arrows2 OA S T.∀p. R (R* (R (R* p))) = R (R* p).
+intros; unfold in ⊢ (? ? ? % %); apply (†(lemma_10_3_b ?? R p));
+qed.
+
+lemma oa_overlap_sym': ∀o:OA.∀U,V:o. (U >< V) = (V >< U).
+ intros; split; intro; apply oa_overlap_sym; assumption.
qed.
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projects / helm.git / blobdiff