inductive bool : Type := 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 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.
+ (λm.match m with [ true ⇒ a | false ⇒ b ]) x =
+ (λm.match m with [ true ⇒ a | false ⇒ b ]) 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);]
-qed.
-interpretation "mk " 'comprehension T P =
+interpretation "unary morphism comprehension with no proof" 'comprehension T P =
(mk_unary_morphism T _ P _).
notation > "hvbox({ ident i ∈ s | term 19 p | by })" with precedence 90
-for @{ 'comprehension_by $s (\lambda ${ident i}. $p) $by}.
+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 f p =
+interpretation "unary morphism comprehension with proof" 'comprehension_by s \eta.f p =
(mk_unary_morphism s _ f p).
-definition A : ∀S:setoid.∀a,b:S.ums BOOL S.
-apply (λS,a,b.{ x ∈ BOOL | match x with [ true ⇒ a | false ⇒ b] | IF_THEN_ELSE_p S a b}).
-qed.
record OAlgebra : Type := {
- oa_P :> setoid;
+ oa_P :> SET;
oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
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_morphism (arrows1 SET I oa_P) oa_P;
+ oa_join: ∀I:SET.unary_morphism (arrows1 SET I oa_P) oa_P;
oa_one: oa_P;
oa_zero: oa_P;
oa_leq_refl: ∀a:oa_P. oa_leq a a;
(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:setoid.∀p.∀q:ums I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
+ ∀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;
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
}.
for @{ 'overlap $a $b}.
interpretation "o-algebra overlap" 'overlap a b = (fun1 ___ (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 $f }.
-notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (ident i ∈ I) break term 90 p)" non associative with precedence 50
-for @{ 'oa_meet (λ${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(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 _)).
+
+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(∨ 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 ]) }.
+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 \eta.f = (oa_join _ _ f).
-*)
+interpretation "o-algebra join" 'oa_join f =
+ (fun_1 __ (oa_join __) f).
+interpretation "o-algebra join with explicit function" 'oa_join_mk f =
+ (fun_1 __ (oa_join __) (mk_unary_morphism _ _ 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
+ 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_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;
- [
- 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;]]
+ [ apply (λp,q. And4 (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; 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.
-lemma hint : ∀P,Q. ORelation_setoid P Q → P ⇒ Q. intros; apply (or_f ?? c);qed.
-coercion hint.
+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 or_f_minus_star2: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET P Q.
+ intros; constructor 1;
+ [ apply or_f_minus_star;
+ | intros; cases H; assumption]
+qed.
+
+definition or_f2: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET P Q.
+ intros; constructor 1;
+ [ apply or_f;
+ | intros; cases H; assumption]
+qed.
+
+definition or_f_minus2: ∀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_star2: ∀P,Q:OAlgebra.ORelation_setoid P Q ⇒ arrows1 SET Q P.
+ intros; constructor 1;
+ [ apply or_f_star;
+ | intros; cases H; assumption]
+qed.
+
+interpretation "o-relation f⎻* 2" 'OR_f_minus_star r = (fun_1 __ (or_f_minus_star2 _ _) r).
+interpretation "o-relation f⎻ 2" 'OR_f_minus r = (fun_1 __ (or_f_minus2 _ _) r).
+interpretation "o-relation f* 2" 'OR_f_star r = (fun_1 __ (or_f_star2 _ _) r).
+coercion or_f2.
-definition composition : ∀P,Q,R.
+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;
- [ constructor 1; [apply (λx. G (F x)); | intros; apply (†(†H));]
- |2,3,4,5,6,7: cases DAEMON;]
-| intros; cases DAEMON;]
+ [ apply (or_f2 ?? G ∘ or_f2 ?? F);
+ | alias symbol "compose" = "category1 composition".
+ apply (G⎻* ∘ F⎻* );
+ | apply (F* ∘ G* );
+ | apply (F⎻ ∘ G⎻);
+ | intros;
+ alias symbol "eq" = "setoid1 eq".
+ change with ((G (F p) ≤ q) = (p ≤ (F* (G* q))));
+ apply (.= or_prop1 ??? (F p) ?);
+ apply (.= or_prop1 ??? p ?);
+ apply refl1
+ | intros; alias symbol "eq" = "setoid1 eq".
+ change with ((F⎻ (G⎻ p) ≤ q) = (p ≤ (G⎻* (F⎻* q))));
+ alias symbol "trans" = "trans1".
+ apply (.= or_prop2 ?? F ??);
+ apply (.= or_prop2 ?? G ??);
+ apply refl1;
+ | intros; change with ((G (F (p)) >< q) = (p >< (F⎻ (G⎻ q))));
+ apply (.= or_prop3 ??? (F p) ?);
+ apply (.= or_prop3 ??? p ?);
+ apply refl1
+ ]
+| intros; split; simplify; [1,3: apply ((†H)‡(†H1)); | 2,4: apply ((†H1)‡(†H));]]
qed.
-definition OA : category1. (* category2 *)
+definition OA : category1.
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 id1;
+ |5,6,7:intros; apply refl1;]
+| apply ORelation_composition;
+| intros; split;
+ [ apply (comp_assoc1 ????? (a12⎻* ) (a23⎻* ) (a34⎻* ));
+ | alias symbol "invert" = "setoid1 symmetry".
+ apply ((comp_assoc1 ????? (a34⎻) (a23⎻) (a12⎻)) \sup -1);
+ | apply (comp_assoc1 ????? a12 a23 a34);
+ | apply ((comp_assoc1 ????? (a34* ) (a23* ) (a12* )) \sup -1);]
+| 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