include "preamble.ma".
-(* ACZEL CATEGORIES:
+(* ACZEL CATEGORIES
- We use typoids with a compatible membership relation
- The category is intended to be the domain of the membership relation
- The membership relation is necessary because we need to regard the
| ceq_trans: ∀c1,c,c2. cin ? c1 → ces ? c1 c → ceq ? c c2 → ceq ? c1 c2
| ceq_conf : ∀c1,c,c2. cin ? c1 → ces ? c c1 → ceq ? c c2 → ceq ? c1 c2
.
-(*
-(* external universal quantification *)
-inductive call (C:Class) (P:C \to Prop) : Prop ≝
- | call_intro: (\forall c. cin ? c \to P c) \to call C P.
-inductive call2 (C1,C2:Class) (P:C1 \to C2 \to Prop) : Prop \def
- | call2_intro:
- (\forall c1,c2. cin ? c1 \to cin ? c2 \to P c1 c2) \to call2 C1 C2 P.
-*)
-(* notations **************************************************************)
-(*
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "external for all" 'xforall \eta.x =
- (cic:/matita/limits/Class/defs/call.ind#xpointer(1/1) _ x).
-
-notation < "hvbox(\xforall ident i opt (: ty) break . p)"
- right associative with precedence 20
-for @{ 'xforall ${default
- @{\lambda ${ident i} : $ty. $p}
- @{\lambda ${ident i} . $p}}}.
-
-notation > "\forall list1 ident x sep , opt (: T). term 19 Px"
- with precedence 20
- for ${ default
- @{ ${ fold right @{$Px} rec acc @{'xforall (λ${ident x}:$T.$acc)} } }
- @{ ${ fold right @{$Px} rec acc @{'xforall (λ${ident x}.$acc)} } }
- }.
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "external for all 2" 'xforall2 \eta.x \eta.y =
- (cic:/matita/limits/Class/defs/call2.ind#xpointer(1/1) _ _ x y).
-
-notation > "hvbox(\xforall ident i1 opt (: ty1) ident i2 opt (: ty2) break . p)"
- with precedence 20
-for @{ 'xforall2 ${default
- @{\lambda ${ident i1} : $ty1. \lambda ${ident i2} : $ty2. $p}
- @{\lambda ${ident i1}, ${ident i2}. $p}}}.
-*)
(* internal universal quantification *)
inductive dall (D:Domain) (P:D → Prop): Prop ≝
- | dall_intro: (∀d. cin ? d → P d) → dall D P.
+ | dall_intro: (∀d. cin ? d → P d) → dall D P
+.
+
+interpretation "internal for all" 'iforall η.x = (dall _ x).
(* internal existential quantification *)
inductive dex (D:Domain) (P:D → Prop): Prop ≝
- | dex_intro: ∀d. cin D d → P d → dex D P.
-
-(* notations **************************************************************)
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "internal for all" 'iforall η.x =
- (cic:/matita/limits/Domain/defs/dall.ind#xpointer(1/1) _ x).
+ | dex_intro: ∀d. cin D d → P d → dex D P
+.
-notation < "hvbox(\iforall ident i opt (: ty) break . p)"
- right associative with precedence 20
-for @{ 'iforall ${default
- @{\lambda ${ident i} : $ty. $p}
- @{\lambda ${ident i} . $p}}}.
-
-notation > "\iforall list1 ident x sep , opt (: T). term 19 Px"
- with precedence 20
-for ${ default
- @{ ${ fold right @{$Px} rec acc @{'iforall (λ${ident x}:$T.$acc)} } }
- @{ ${ fold right @{$Px} rec acc @{'iforall (λ${ident x}.$acc)} } }
-}.
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "internal exists" 'iexists η.x =
- (cic:/matita/limits/Domain/defs/dex.ind#xpointer(1/1) _ x).
-
-notation < "hvbox(\iexists ident i opt (: ty) break . p)"
- right associative with precedence 20
-for @{ 'iexists ${default
- @{\lambda ${ident i} : $ty. $p}
- @{\lambda ${ident i} . $p}}}.
-
-notation > "\iexists list1 ident x sep , opt (: T). term 19 Px"
- with precedence 20
-for ${ default
- @{ ${ fold right @{$Px} rec acc @{'iexists (λ${ident x}:$T.$acc)} } }
- @{ ${ fold right @{$Px} rec acc @{'iexists (λ${ident x}.$acc)} } }
-}.
+interpretation "internal exists" 'iexists η.x = (dex _ x).
(* subset top (full subset) *)
definition stop ≝ λD:Domain. true_f D.
+coercion stop.
+
(* subset bottom (empty subset) *)
definition sbot ≝ λD:Domain. false_f D.
definition sover: ∀D. Subset D → Subset D → Prop ≝
λD,U1,U2. \iexists d. U1 d ∧ U2 d.
-(* coercions **************************************************************)
-
-(*
-(* the class of the subsets of a domain (not an implicit coercion) *)
-definition class_of_subsets_of \def
- \lambda D. mk_Class (Subset D) (true_f ?) (sle ?).
-*)
+(* the class of the subsets of a domain *)
+definition subsets_of_domain ≝
+ λD. mk_Class (Subset D) (true_f ?) (sle ?).
-(* the domain built upon a subset (not an implicit coercion) *)
-definition domain_of_subset: ∀D. Subset D \to Domain ≝
+(* the domain built upon a subset *)
+definition domain_of_subset: ∀D. Subset D → Domain ≝
λD:Domain. λU. mk_Domain (mk_Class D (sin D U) (ces D)).
-
-(* the full subset of a domain *)
-coercion stop.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* Project started Wed Oct 12, 2005 ***************************************)
+(* Project taken over by "formal_topology" and restarted Mon Apr 6, 2009 **)
+
+include "logic/connectives.ma".
+include "datatypes/constructors.ma".
+include "datatypes/bool.ma".
+
+(* notations **************************************************************)
+
+notation < "hvbox(\iforall ident i opt (: ty) break . p)"
+ right associative with precedence 20
+for @{ 'iforall ${ default
+ @{λ ${ident i}: $ty. $p}
+ @{λ ${ident i}. $p}
+}}.
+
+notation > "\iforall list1 ident x sep , opt (: T). term 19 Px"
+ with precedence 20
+for ${ default
+ @{ ${ fold right @{$Px} rec acc @{'iforall (λ ${ident x} :$T . $acc)} } }
+ @{ ${ fold right @{$Px} rec acc @{'iforall (λ ${ident x} . $acc)} } }
+}.
+
+notation < "hvbox(\iexists ident i opt (: ty) break . p)"
+ right associative with precedence 20
+for @{ 'iexists ${ default
+ @{λ ${ident i}: $ty. $p}
+ @{λ ${ident i}. $p}
+}}.
+
+notation > "\iexists list1 ident x sep , opt (: T). term 19 Px"
+ with precedence 20
+for ${ default
+ @{ ${ fold right @{$Px} rec acc @{'iexists (λ ${ident x}: $T. $acc)} } }
+ @{ ${ fold right @{$Px} rec acc @{'iexists (λ ${ident x}. $acc)} } }
+}.
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