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/categories.ma".
16 include "logic/cprop_connectives.ma".
18 inductive bool : Type := true : bool | false : bool.
20 lemma ums : setoid → setoid → setoid.
23 [ apply (unary_morphism S T);
25 [ intros (f1 f2); apply (∀a,b:S.eq1 ? a b → eq1 ? (f1 a) (f2 b));
26 | whd; simplify; intros; apply (.= (†H)); apply refl1;
27 | whd; simplify; intros; apply (.= (†H1)); apply sym1; apply H; apply refl1;
28 | whd; simplify; intros; apply (.= (†H2)); apply (.= (H ?? #)); apply (.= (H1 ?? #)); apply rule #;]]
32 constructor 1; [apply bool] constructor 1;
33 [ intros (x y); apply (match x with [ true ⇒ match y with [ true ⇒ True | _ ⇒ False] | false ⇒ match y with [ true ⇒ False | false ⇒ True ]]);
34 | whd; simplify; intros; cases x; apply I;
35 | whd; simplify; intros 2; cases x; cases y; simplify; intros; assumption;
36 | whd; simplify; intros 3; cases x; cases y; cases z; simplify; intros; try assumption; apply I]
39 lemma IF_THEN_ELSE_p :
40 ∀S:setoid.∀a,b:S.∀x,y:BOOL.x = y →
41 let f ≝ λm.match m with [ true ⇒ a | false ⇒ b ] in f x = f y.
42 intros; cases x in H; cases y; simplify; intros; try apply refl; whd in H; cases H;
45 lemma if_then_else : ∀T:setoid. ∀a,b:T. ums BOOL T.
46 intros; constructor 1; intros;
47 [ apply (match c2 with [ true ⇒ c | false ⇒ c1 ]);
48 | apply (IF_THEN_ELSE_p T c c1 a a' H);]
51 interpretation "mk " 'comprehension T P =
52 (mk_unary_morphism T _ P _).
54 notation > "hvbox({ ident i ∈ s | term 19 p | by })" with precedence 90
55 for @{ 'comprehension_by $s (\lambda ${ident i}. $p) $by}.
57 interpretation "unary morphism comprehension with proof" 'comprehension_by s f p =
58 (mk_unary_morphism s _ f p).
60 definition A : ∀S:setoid.∀a,b:S.ums BOOL S.
61 apply (λS,a,b.{ x ∈ BOOL | match x with [ true ⇒ a | false ⇒ b] | IF_THEN_ELSE_p S a b}).
64 record OAlgebra : Type := {
66 oa_leq : binary_morphism1 oa_P oa_P CPROP; (* CPROP is setoid1 *)
67 oa_overlap: binary_morphism1 oa_P oa_P CPROP;
68 oa_meet: ∀I:setoid.unary_morphism (ums I oa_P) oa_P;
69 oa_join: ∀I:setoid.unary_morphism (ums I oa_P) oa_P;
72 oa_leq_refl: ∀a:oa_P. oa_leq a a;
73 oa_leq_antisym: ∀a,b:oa_P.oa_leq a b → oa_leq b a → a = b;
74 oa_leq_trans: ∀a,b,c:oa_P.oa_leq a b → oa_leq b c → oa_leq a c;
75 oa_overlap_sym: ∀a,b:oa_P.oa_overlap a b → oa_overlap b a;
76 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);
77 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;
78 oa_zero_bot: ∀p:oa_P.oa_leq oa_zero p;
79 oa_one_top: ∀p:oa_P.oa_leq p oa_one;
80 oa_overlap_preservers_meet:
81 ∀p,q.oa_overlap p q → oa_overlap p
82 (oa_meet ? { x ∈ BOOL | match x with [ true ⇒ p | false ⇒ q ] | IF_THEN_ELSE_p oa_P p q });
83 (*(oa_meet BOOL (if_then_else oa_P p q));*)
84 oa_join_split: (* ha I → oa_P da castare a funX (ums A oa_P) *)
85 ∀I:setoid.∀p.∀q:ums I oa_P.oa_overlap p (oa_join I q) ⇔ ∃i:I.oa_overlap p (q i);
87 oa_enum : ums oa_base oa_P;
88 oa_density: ∀p,q.(∀i.oa_overlap p (oa_enum i) → oa_overlap q (oa_enum i)) → oa_leq p q*)
90 ∀p,q.(∀r.oa_overlap p r → oa_overlap q r) → oa_leq p q
93 interpretation "o-algebra leq" 'leq a b = (fun1 ___ (oa_leq _) a b).
95 notation "hovbox(a mpadded width -150% (>)< b)" non associative with precedence 45
96 for @{ 'overlap $a $b}.
97 interpretation "o-algebra overlap" 'overlap a b = (fun1 ___ (oa_overlap _) a b).
99 notation > "hovbox(∧ f)" non associative with precedence 60
100 for @{ 'oa_meet $f }.
101 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (ident i ∈ I) break term 90 p)" non associative with precedence 50
102 for @{ 'oa_meet (λ${ident i}:$I.$p) }.
103 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∧) \below (\emsp) \nbsp term 90 p)" non associative with precedence 50
104 for @{ 'oa_meet (λ${ident i}.($p $_)) }.
105 notation < "hovbox(a ∧ b)" left associative with precedence 50
108 (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ])
111 interpretation "o-algebra meet" 'oa_meet f = (fun_1 __ (oa_meet __) f).
112 (*interpretation "o-algebra binary meet" 'and x y = (fun_1 __ (oa_meet _ BOOL) (if_then_else _ x y)).*)
115 notation > "hovbox(a ∨ b)" left associative with precedence 49
116 for @{ 'oa_join (λx__:bool.match x__ with [ true ⇒ $a | false ⇒ $b ]) }.
117 notation > "hovbox(∨ f)" non associative with precedence 59
118 for @{ 'oa_join $f }.
119 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (ident i ∈ I) break term 90 p)" non associative with precedence 49
120 for @{ 'oa_join (λ${ident i}:$I.$p) }.
121 notation < "hovbox(mstyle scriptlevel 1 scriptsizemultiplier 1.7 (∨) \below (\emsp) \nbsp term 90 p)" non associative with precedence 49
122 for @{ 'oa_join (λ${ident i}.($p $_)) }.
123 notation < "hovbox(a ∨ b)" left associative with precedence 49
124 for @{ 'oa_join (λ${ident i}:$_.match $i with [ true ⇒ $a | false ⇒ $b ]) }.
126 interpretation "o-algebra join" 'oa_join \eta.f = (oa_join _ _ f).
129 record ORelation (P,Q : OAlgebra) : Type ≝ {
131 or_f_minus_star : P ⇒ Q;
134 or_prop1 : ∀p,q. or_f p ≤ q ⇔ p ≤ or_f_star q;
135 or_prop2 : ∀p,q. or_f_minus p ≤ q ⇔ p ≤ or_f_minus_star q;
136 or_prop3 : ∀p,q. or_f p >< q ⇔ p >< or_f_minus q
139 notation "r \sup *" non associative with precedence 90 for @{'OR_f_star $r}.
140 notation > "r *" non associative with precedence 90 for @{'OR_f_star $r}.
141 interpretation "o-relation f*" 'OR_f_star r = (or_f_star _ _ r).
143 notation "r \sup (⎻* )" non associative with precedence 90 for @{'OR_f_minus_star $r}.
144 notation > "r⎻*" non associative with precedence 90 for @{'OR_f_minus_star $r}.
145 interpretation "o-relation f⎻*" 'OR_f_minus_star r = (or_f_minus_star _ _ r).
147 notation "r \sup ⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
148 notation > "r⎻" non associative with precedence 90 for @{'OR_f_minus $r}.
149 interpretation "o-relation f⎻" 'OR_f_minus r = (or_f_minus _ _ r).
153 definition ORelation_setoid : OAlgebra → OAlgebra → setoid1.
156 [ apply (ORelation P Q);
159 alias symbol "and" = "constructive and".
165 | whd; simplify; intros; repeat split; intros; apply refl;
166 | whd; simplify; intros; cases H; cases H1; cases H3; clear H H3 H1;
167 repeat split; intros; apply sym; generalize in match a;assumption;
168 | whd; simplify; intros; elim DAEMON;]]
171 lemma hint : ∀P,Q. ORelation_setoid P Q → P ⇒ Q. intros; apply (or_f ?? c);qed.
174 definition composition : ∀P,Q,R.
175 binary_morphism1 (ORelation_setoid P Q) (ORelation_setoid Q R) (ORelation_setoid P R).
180 [ constructor 1; [apply (λx. G (F x)); | intros; apply (†(†H));]
181 |2,3,4,5,6,7: cases DAEMON;]
182 | intros; cases DAEMON;]
185 definition OA : category1. (* category2 *)
188 | intros; apply (ORelation_setoid o o1);
190 [1,2,3,4: constructor 1; [1,3,5,7:apply (λx.x);|*:intros;assumption]
191 |5,6,7:intros;split;intros; assumption; ]
192 |4: apply composition;