inductive bool : Type := true : bool | false : bool.
-<<<<<<< .mine
-lemma BOOL : setoid.
-=======
lemma BOOL : objs1 SET.
->>>>>>> .r9407
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.
+definition setoid_OF_SET: objs1 SET → setoid.
+ intros; apply o; qed.
-coercion hint.
+coercion setoid_OF_SET.
lemma IF_THEN_ELSE_p :
∀S:setoid.∀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.
-<<<<<<< .mine
interpretation "unary morphism comprehension with no proof" 'comprehension T P =
-=======
-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);]
-qed.
-
-interpretation "mk " 'comprehension T P =
->>>>>>> .r9407
(mk_unary_morphism T _ P _).
notation > "hvbox({ ident i ∈ s | term 19 p | by })" with precedence 90
interpretation "unary morphism comprehension with proof" 'comprehension_by s \eta.f p =
(mk_unary_morphism s _ f p).
-<<<<<<< .mine
-=======
-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.
-
->>>>>>> .r9407
record OAlgebra : Type := {
oa_P :> SET;
oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
oa_overlap: binary_morphism1 oa_P oa_P CPROP;
-<<<<<<< .mine
- oa_meet: ∀I:setoid.unary_morphism (unary_morphism_setoid I oa_P) oa_P;
- oa_join: ∀I:setoid.unary_morphism (unary_morphism_setoid I oa_P) oa_P;
-=======
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;
->>>>>>> .r9407
oa_one: oa_P;
oa_zero: oa_P;
oa_leq_refl: ∀a:oa_P. oa_leq a a;
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:
+ oa_overlap_preservers_meet_:
∀p,q.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 });
-<<<<<<< .mine
oa_join_split:
- ∀I:setoid.∀p.∀q:I ⇒ oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
- (*
- oa_base : setoid;
-=======
- (*(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_base : setoid;
->>>>>>> .r9407
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
*)
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 =
(fun_1 __ (oa_meet __) f).
interpretation "o-algebra meet with explicit function" 'oa_meet_mk f =
(fun_1 __ (oa_meet __) (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; apply (prop_1 ?? (oa_meet O BOOL)); 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).
+
+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) # ?);
+[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)"
(fun_1 __ (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;
- 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)
+ 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)
}.
-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.
definition ORelation_setoid : OAlgebra → OAlgebra → setoid1.
intros (P Q);
constructor 1;
[ apply (ORelation P Q);
| constructor 1;
-<<<<<<< .mine
- [ alias symbol "and" = "constructive and".
- apply (λp,q. And4 (∀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;
-=======
- [ apply (λp,q. eq1 ? p⎻* q⎻* ∧ eq1 ? p⎻ q⎻ ∧ eq1 ? p q ∧ eq1 ? p* q* );
+ [ 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;
->>>>>>> .r9407
-<<<<<<< .mine
| 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 (.= (H4 a)); apply H8;
| apply (.= (H5 a)); apply H9;]]]
qed.
-=======
- | 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
- ]]]
+
+definition or_f_minus_star: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET P Q.
+ intros; constructor 1;
+ [ apply or_f_minus_star_;
+ | intros; cases H; assumption]
qed.
->>>>>>> .r9407
-<<<<<<< .mine
-definition ORelation_composition : ∀P,Q,R.
-=======
-lemma hint1 : ∀P,Q. ORelation_setoid P Q → arrows1 SET P Q. intros; apply (or_f ?? c);qed.
-coercion hint1.
+definition or_f: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET P Q.
+ intros; constructor 1;
+ [ apply or_f_;
+ | intros; cases H; assumption]
+qed.
+
+coercion or_f.
+
+definition or_f_minus: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET Q P.
+ intros; constructor 1;
+ [ apply or_f_minus_;
+ | intros; cases H; assumption]
+qed.
+
+definition or_f_star: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET Q P.
+ intros; constructor 1;
+ [ apply or_f_star_;
+ | intros; cases H; assumption]
+qed.
+
+lemma arrows1_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → arrows1 SET P Q.
+intros; apply (or_f ?? c);
+qed.
+
+coercion arrows1_OF_ORelation_setoid nocomposites.
+
+lemma umorphism_OF_ORelation_setoid : ∀P,Q. ORelation_setoid P Q → P ⇒ Q.
+intros; apply (or_f ?? c);
+qed.
+
+coercion umorphism_OF_ORelation_setoid.
+
+
+lemma uncurry_arrows : ∀B,C. arrows1 SET B C → B → C.
+intros; apply ((fun_1 ?? c) t);
+qed.
+
+coercion uncurry_arrows 1.
lemma hint3 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply c;qed.
-coercion hint3.
+coercion hint3 nocomposites.
+(*
lemma hint2: OAlgebra → setoid. intros; apply (oa_P o). qed.
-coercion hint2.
+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}.
+
+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}.
-definition composition : ∀P,Q,R.
->>>>>>> .r9407
+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).
+
+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_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
intros;
constructor 1;
[ intros (F G);
constructor 1;
-<<<<<<< .mine
- [ apply {x ∈ P | G (F x)}; intros; simplify; apply (†(†H));
- | apply {x ∈ P | G⎻* (F⎻* x)}; intros; simplify; apply (†(†H));
- | apply {x ∈ R | F* (G* x)}; intros; simplify; apply (†(†H));
- | apply {x ∈ R | F⎻ (G⎻ x)}; intros; simplify; apply (†(†H));
- | intros; simplify;
- lapply (or_prop1 ?? G (F p) q) as H1; lapply (or_prop1 ?? F p (G* q)) as H2;
- split; intro H;
- [ apply (if1 ?? H2); apply (if1 ?? H1); apply H;
- | apply (fi1 ?? H1); apply (fi1 ?? H2); apply H;]
- | intros; simplify;
- lapply (or_prop2 ?? G p (F⎻* q)) as H1; lapply (or_prop2 ?? F (G⎻ p) q) as H2;
- split; intro H;
- [ apply (if1 ?? H1); apply (if1 ?? H2); apply H;
- | apply (fi1 ?? H2); apply (fi1 ?? H1); apply H;]
- | intros; simplify;
- lapply (or_prop3 ?? F p (G⎻ q)) as H1; lapply (or_prop3 ?? G (F p) q) as H2;
- split; intro H;
- [ apply (if1 ?? H1); apply (if1 ?? H2); apply H;
- | apply (fi1 ?? H2); apply (fi1 ?? H1); apply H;]]
-| intros; simplify; split; simplify; intros; elim DAEMON;]
-=======
- [ apply (G ∘ F);
+ [ lapply (G ∘ F);
+ apply (G ∘ F);
| apply (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;
-*)]
->>>>>>> .r9407
+| intros; split; simplify;
+ [1,3: unfold arrows1_OF_ORelation_setoid; apply ((†H)‡(†H1));
+ |2,4: apply ((†H1)‡(†H));]]
qed.
definition OA : category1.
[ apply (OAlgebra);
| intros; apply (ORelation_setoid o o1);
| intro O; split;
-<<<<<<< .mine
- [1,2,3,4: constructor 1; [1,3,5,7:apply (λx.x);|*:intros;assumption]
- |5,6,7:intros;split;intros; assumption;]
-|4: apply ORelation_composition;
-|*: elim DAEMON;]
-qed.
-
-
-
-=======
[1,2,3,4: apply id1;
|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;]
-qed.>>>>>>> .r9407
+| 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
projects / helm.git / blobdiff