1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "datatypes/bool.ma".
16 include "datatypes/categories.ma".
17 include "logic/cprop_connectives.ma".
19 lemma ums : setoid → setoid → setoid.
22 [ apply (unary_morphism S T);
24 [ intros (f1 f2); apply (∀a,b:S.eq1 ? a b → eq1 ? (f1 a) (f2 b));
25 | whd; simplify; intros; apply (.= (†H)); apply refl1;
26 | whd; simplify; intros; apply (.= (†H1)); apply sym1; apply H; apply refl1;
27 | whd; simplify; intros; apply (.= (†H2)); apply (.= (H ?? #)); apply (.= (H1 ?? #)); apply rule #;]]
31 constructor 1; [apply bool] constructor 1;
32 [ intros (x y); apply (match x with [ true ⇒ match y with [ true ⇒ True | _ ⇒ False] | false ⇒ match y with [ true ⇒ False | false ⇒ True ]]);
33 | whd; simplify; intros; cases x; apply I;
34 | whd; simplify; intros 2; cases x; cases y; simplify; intros; assumption;
35 | whd; simplify; intros 3; cases x; cases y; cases z; simplify; intros; try assumption; apply I]
38 lemma IF_THEN_ELSE_p :
39 ∀S:setoid.∀a,b:S.∀x,y:BOOL.x = y →
40 let f ≝ λm.match m with [ true ⇒ a | false ⇒ b ] in f x = f y.
41 intros; cases x in H; cases y; simplify; intros; try apply refl; whd in H; cases H;
44 lemma if_then_else : ∀T:setoid. ∀a,b:T. ums BOOL T.
45 intros; constructor 1; intros;
46 [ apply (match c2 with [ true ⇒ c | false ⇒ c1 ]);
47 | apply (IF_THEN_ELSE_p T c c1 a a' H);]
50 record OAlgebra : Type := {
52 oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
53 oa_overlap: binary_morphism1 oa_P oa_P CPROP;
54 oa_meet: ∀I:setoid.unary_morphism (ums I oa_P) oa_P;
55 oa_join: ∀I:setoid.unary_morphism (ums I oa_P) oa_P;
58 oa_leq_refl: ∀a:oa_P. oa_leq a a;
59 oa_leq_antisym: ∀a,b:oa_P.oa_leq a b → oa_leq b a → a = b;
60 oa_leq_trans: ∀a,b,c:oa_P.oa_leq a b → oa_leq b c → oa_leq a c;
61 oa_overlap_sym: ∀a,b:oa_P.oa_overlap a b → oa_overlap b a;
62 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);
63 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;
64 oa_zero_bot: ∀p:oa_P.oa_leq oa_zero p;
65 oa_one_top: ∀p:oa_P.oa_leq p oa_one;
66 oa_overlap_preservers_meet:
67 ∀p,q.oa_overlap p q → oa_overlap p
68 (oa_meet BOOL (if_then_else oa_P p q));
69 oa_join_split: (* ha I → oa_P da castare a funX (ums A oa_P) *)
70 ∀I:setoid.∀p.∀q:ums I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
72 oa_enum : ums oa_base oa_P;
73 oa_density: ∀p,q.(∀i.oa_overlap p (oa_enum i) → oa_overlap q (oa_enum i)) → oa_leq p q*)
75 ∀p,q.(∀r.oa_overlap p r → oa_overlap q r) → oa_leq p q
80 axiom x : carr (oa_P Al).
81 definition wwww := (oa_density Al x x).
82 definition X := ((λx:Type.λa:x.True) ? wwww).
85 interpretation "o-algebra leq" 'leq a b = (fun1 ___ (oa_leq _) a b).
87 notation "hovbox(a mpadded width -150% (>)< b)" non associative with precedence 45
88 for @{ 'overlap $a $b}.
89 interpretation "o-algebra overlap" 'overlap a b = (fun1 ___ (oa_overlap _) a b).
91 notation > "hovbox(a ∧ b)" left associative with precedence 50
92 for @{ 'oa_meet2 $a $b }.
93 notation > "hovbox(∧ f)" non associative with precedence 60
95 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (ident i ∈ I) break term 90 p)" non associative with precedence 50
96 for @{ 'oa_meet (λ${ident i}:$I.$p) }.
97 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (\emsp) \nbsp term 90 p)" non associative with precedence 50
98 for @{ 'oa_meet (λ${ident i}.($p $_)) }.
99 notation < "hovbox(a ∧ b)" left associative with precedence 50
100 for @{ 'oa_meet2 $a $b }.
102 interpretation "o-algebra meet" 'oa_meet \eta.f = (fun_1 __ (oa_meet __) f).
103 interpretation "o-algebra binary meet" 'oa_meet2 x y = (fun_1 __ (oa_meet _ BOOL) (if_then_else _ x y)).
105 notation > "hovbox(a ∨ b)" left associative with precedence 49
106 for @{ 'oa_join (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) }.
107 notation > "hovbox(∨ f)" non associative with precedence 59
108 for @{ 'oa_join $f }.
109 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (ident i ∈ I) break term 90 p)" non associative with precedence 49
110 for @{ 'oa_join (λ${ident i}:$I.$p) }.
111 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)" non associative with precedence 49
112 for @{ 'oa_join (λ${ident i}.($p $_)) }.
113 notation < "hovbox(a ∨ b)" left associative with precedence 49
114 for @{ 'oa_join (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
116 interpretation "o-algebra join" 'oa_join \eta.f = (oa_join _ _ f).
119 record ORelation (P,Q : OAlgebra) : Type ≝ {
121 or_f_minus_star : P → Q;
124 or_prop1 : ∀p,q. or_f p ≤ q ⇔ p ≤ or_f_star q;
125 or_prop2 : ∀p,q. or_f_minus p ≤ q ⇔ p ≤ or_f_minus_star q;
126 or_prop3 : ∀p,q. or_f p >< q ⇔ p >< or_f_minus q
129 notation < "⨍ \sub (term 90 r)" non associative with precedence 90 for @{'OR_f $r}.
130 notation < "⨍ \sub (term 90 r) term 90 a" non associative with precedence 70 for @{'OR_f_app1 $r $a}.
131 notation > "⨍_(term 90 r)" non associative with precedence 90 for @{'OR_f $r}.
132 interpretation "o-relation f" 'OR_f r = (or_f _ _ r).
133 interpretation "o-relation f x" 'OR_f_app1 r a = (or_f _ _ r a).
136 definition ORelation_setoid : OAlgebra → OAlgebra → setoid1.
139 [ apply (ORelation P Q);
141 [ apply (λp,q. ∀a.⨍_p a = ⨍_q a (* ∧ f^-1 a = .... *));
142 | whd; simplify; intros; apply refl;
143 | whd; simplify; intros; apply (H ? \sup -1);
144 | whd; simplify; intros; apply trans; [2: apply H;|3: apply H1]]]
149 definition composition : ∀P,Q,R.
150 binary_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
155 [ apply (λx.⨍_G (⨍_F x));
156 |2,3,4,5,6,7: cases DAEMON;]
157 | intros; cases DAEMON;]
160 definition OA : category1. (* category2 *)
163 | intros; apply (ORelation_setoid o o1);
165 [1,2,3,4: apply (λx.x);
166 |*:intros;split;intros; assumption; ]