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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "datatypes/categories.ma".
16 include "logic/cprop_connectives.ma".
18 inductive bool : Type := true : bool | false : bool.
20 lemma BOOL : objs1 SET.
21 constructor 1; [apply bool] constructor 1;
22 [ intros (x y); apply (match x with [ true ⇒ match y with [ true ⇒ True | _ ⇒ False] | false ⇒ match y with [ true ⇒ False | false ⇒ True ]]);
23 | whd; simplify; intros; cases x; apply I;
24 | whd; simplify; intros 2; cases x; cases y; simplify; intros; assumption;
25 | whd; simplify; intros 3; cases x; cases y; cases z; simplify; intros; try assumption; apply I]
28 definition hint: objs1 SET → setoid.
34 lemma IF_THEN_ELSE_p :
35 ∀S:setoid.∀a,b:S.∀x,y:BOOL.x = y →
36 (λm.match m with [ true ⇒ a | false ⇒ b ]) x =
37 (λm.match m with [ true ⇒ a | false ⇒ b ]) y.
38 intros; cases x in H; cases y; simplify; intros; try apply refl; whd in H; cases H;
42 interpretation "unary morphism comprehension with no proof" 'comprehension T P =
43 (mk_unary_morphism T _ P _).
45 notation > "hvbox({ ident i ∈ s | term 19 p | by })" with precedence 90
46 for @{ 'comprehension_by $s (λ${ident i}. $p) $by}.
47 notation < "hvbox({ ident i ∈ s | term 19 p })" with precedence 90
48 for @{ 'comprehension_by $s (λ${ident i}:$_. $p) $by}.
50 interpretation "unary morphism comprehension with proof" 'comprehension_by s \eta.f p =
51 (mk_unary_morphism s _ f p).
54 record OAlgebra : Type := {
56 oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
57 oa_overlap: binary_morphism1 oa_P oa_P CPROP;
58 oa_meet: ∀I:SET.unary_morphism (arrows1 SET I oa_P) oa_P;
59 oa_join: ∀I:SET.unary_morphism (arrows1 SET I oa_P) oa_P;
62 oa_leq_refl: ∀a:oa_P. oa_leq a a;
63 oa_leq_antisym: ∀a,b:oa_P.oa_leq a b → oa_leq b a → a = b;
64 oa_leq_trans: ∀a,b,c:oa_P.oa_leq a b → oa_leq b c → oa_leq a c;
65 oa_overlap_sym: ∀a,b:oa_P.oa_overlap a b → oa_overlap b a;
66 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);
67 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;
68 oa_zero_bot: ∀p:oa_P.oa_leq oa_zero p;
69 oa_one_top: ∀p:oa_P.oa_leq p oa_one;
70 oa_overlap_preservers_meet:
71 ∀p,q.oa_overlap p q → oa_overlap p
72 (oa_meet ? { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p oa_P p q });
73 (*(oa_meet BOOL (if_then_else oa_P p q));*)
74 oa_join_split: (* ha I → oa_P da castare a funX (ums A oa_P) *)
75 ∀I:SET.∀p.∀q:arrows1 SET I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
77 oa_enum : ums oa_base oa_P;
78 oa_density: ∀p,q.(∀i.oa_overlap p (oa_enum i) → oa_overlap q (oa_enum i)) → oa_leq p q
81 ∀p,q.(∀r.oa_overlap p r → oa_overlap q r) → oa_leq p q
84 interpretation "o-algebra leq" 'leq a b = (fun1 ___ (oa_leq _) a b).
86 notation "hovbox(a mpadded width -150% (>)< b)" non associative with precedence 45
87 for @{ 'overlap $a $b}.
88 interpretation "o-algebra overlap" 'overlap a b = (fun1 ___ (oa_overlap _) a b).
90 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (\emsp) \nbsp term 90 p)"
91 non associative with precedence 50 for @{ 'oa_meet $p }.
92 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (ident i ∈ I) break term 90 p)"
93 non associative with precedence 50 for @{ 'oa_meet_mk (λ${ident i}:$I.$p) }.
94 notation < "hovbox(a ∧ b)" left associative with precedence 35
95 for @{ 'oa_meet_mk (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
97 notation > "hovbox(∧ f)" non associative with precedence 60
99 notation > "hovbox(a ∧ b)" left associative with precedence 50
100 for @{ 'oa_meet (mk_unary_morphism BOOL ? (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) (IF_THEN_ELSE_p ? $a $b)) }.
102 interpretation "o-algebra meet" 'oa_meet f =
103 (fun_1 __ (oa_meet __) f).
104 interpretation "o-algebra meet with explicit function" 'oa_meet_mk f =
105 (fun_1 __ (oa_meet __) (mk_unary_morphism _ _ f _)).
107 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)"
108 non associative with precedence 49 for @{ 'oa_join $p }.
109 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (ident i ∈ I) break term 90 p)"
110 non associative with precedence 49 for @{ 'oa_join_mk (λ${ident i}:$I.$p) }.
111 notation < "hovbox(a ∨ b)" left associative with precedence 49
112 for @{ 'oa_join_mk (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
114 notation > "hovbox(∨ f)" non associative with precedence 59
115 for @{ 'oa_join $f }.
116 notation > "hovbox(a ∨ b)" left associative with precedence 49
117 for @{ 'oa_join (mk_unary_morphism BOOL ? (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) (IF_THEN_ELSE_p ? $a $b)) }.
119 interpretation "o-algebra join" 'oa_join f =
120 (fun_1 __ (oa_join __) f).
121 interpretation "o-algebra join with explicit function" 'oa_join_mk f =
122 (fun_1 __ (oa_join __) (mk_unary_morphism _ _ f _)).
124 record ORelation (P,Q : OAlgebra) : Type ≝ {
125 or_f :> arrows1 SET P Q;
126 or_f_minus_star : arrows1 SET P Q;
127 or_f_star : arrows1 SET Q P;
128 or_f_minus : arrows1 SET Q P;
129 or_prop1 : ∀p,q. (or_f p ≤ q) = (p ≤ or_f_star q);
130 or_prop2 : ∀p,q. (or_f_minus p ≤ q) = (p ≤ or_f_minus_star q);
131 or_prop3 : ∀p,q. (or_f p >< q) = (p >< or_f_minus q)
134 notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
135 notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
136 interpretation "o-relation f*" 'OR_f_star r = (or_f_star _ _ r).
138 notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
139 notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
140 interpretation "o-relation f⎻*" 'OR_f_minus_star r = (or_f_minus_star _ _ r).
142 notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
143 notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
144 interpretation "o-relation f⎻" 'OR_f_minus r = (or_f_minus _ _ r).
148 definition ORelation_setoid : OAlgebra → OAlgebra → setoid1.
151 [ apply (ORelation P Q);
153 [ apply (λp,q. And4 (eq1 ? p⎻* q⎻* ) (eq1 ? p⎻ q⎻) (eq1 ? p q) (eq1 ? p* q* ));
154 | whd; simplify; intros; repeat split; intros; apply refl1;
155 | whd; simplify; intros; cases H; clear H; split;
156 intro a; apply sym; generalize in match a;assumption;
157 | whd; simplify; intros; cases H; cases H1; clear H H1; split; intro a;
158 [ apply (.= (H2 a)); apply H6;
159 | apply (.= (H3 a)); apply H7;
160 | apply (.= (H4 a)); apply H8;
161 | apply (.= (H5 a)); apply H9;]]]
164 lemma hint1 : ∀P,Q. ORelation_setoid P Q → arrows1 SET P Q. intros; apply (or_f ?? c);qed.
167 lemma hint3 : ∀P,Q. arrows1 SET P Q → P ⇒ Q. intros; apply c;qed.
170 lemma hint2: OAlgebra → setoid. intros; apply (oa_P o). qed.
173 definition ORelation_composition : ∀P,Q,R.
174 binary_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
180 | apply (G⎻* ∘ F⎻* );
183 | intros; change with ((G (F p) ≤ q) = (p ≤ (F* (G* q))));
184 apply (.= or_prop1 ??? (F p) ?);
185 apply (.= or_prop1 ??? p ?);
187 | intros; change with ((F⎻ (G⎻ p) ≤ q) = (p ≤ (G⎻* (F⎻* q))));
188 apply (.= or_prop2 ??? (G⎻ p) ?);
189 apply (.= or_prop2 ??? p ?);
191 | intros; change with ((G (F (p)) >< q) = (p >< (F⎻ (G⎻ q))));
192 apply (.= or_prop3 ??? (F p) ?);
193 apply (.= or_prop3 ??? p ?);
196 | intros; repeat split; simplify; cases DAEMON (*
197 [ apply trans1; [2: apply prop1; [3: apply rule #; | skip | 4:
200 lapply (.= ((†H1)‡#)); [8: apply Hletin;
201 [ apply trans1; [2: lapply (prop1); [apply Hletin;
205 definition OA : category1.
208 | intros; apply (ORelation_setoid o o1);
211 |5,6,7:intros; apply refl1;]
212 | apply ORelation_composition;
213 | intros; repeat split; unfold ORelation_composition; simplify;
214 [1,3: apply (comp_assoc1); | 2,4: apply ((comp_assoc1 :?) ^ -1);]
215 | intros; repeat split; unfold ORelation_composition; simplify; apply id_neutral_left1;
216 | intros; repeat split; unfold ORelation_composition; simplify; apply id_neutral_right1;]