+++ /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 ***************************************)
-
-set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/ac_defs".
-
-include "coq.ma".
-
-(* 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
- domain of a propositional function (ie a predicative subset) as a
- quantification domain and therefore as a category, but there is no
- type in CIC representing the domain of a propositional function
- - We set up a single equality predicate, parametric on the category,
- defined as the reflexive, symmetic, transitive and compatible closure
- of the aceq predicate given inside the category. Then we prove the
- properties of the equality that usually are axiomatized inside the
- category structure. This makes categories easier to use
-*)
-
-record AC: Type \def {
- ac: Type;
- acin: ac \to Prop;
- aceq: ac \to ac \to Prop
-}.
-
-coercion ac.
-
-inductive eq (A:AC) (a:A): A \to Prop \def
- | eq_refl: acin ? a \to eq ? a a
- | eq_sing_r: \forall b,c.
- eq ? a b \to acin ? c \to aceq ? b c \to eq ? a c
- | eq_sing_l: \forall b,c.
- eq ? a b \to acin ? c \to aceq ? c b \to eq ? a c.
-
-theorem eq_cl: \forall A,a,b. eq ? a b \to acin A a \land acin A b.
-intros; elim H; clear H; clear b;
- [ auto | decompose H2; auto | decompose H2; auto ].
-qed.
-
-theorem eq_trans: \forall A,b,a,c.
- eq A b c \to eq ? a b \to eq ? a c.
-intros 5; elim H; clear H; clear c;
- [ auto
- | apply eq_sing_r; [||| apply H4 ]; auto
- | apply eq_sing_l; [||| apply H4 ]; auto
- ].
-qed.
-
-theorem eq_conf_rev: \forall A,b,a,c.
- eq A c b \to eq ? a b \to eq ? a c.
-intros 5; elim H; clear H; clear b;
- [ auto
- | lapply eq_cl; [ decompose Hletin |||| apply H1 ].
- apply H2; apply eq_sing_l; [||| apply H4 ]; auto
- | lapply eq_cl; [ decompose Hletin |||| apply H1 ].
- apply H2; apply eq_sing_r; [||| apply H4 ]; auto
- ].
-qed.
-
-theorem eq_sym: \forall A,a,b. eq A a b \to eq A b a.
-intros;
-lapply eq_cl; [ decompose Hletin |||| apply H ].
-auto.
-qed.
-
-theorem eq_conf: \forall A,b,a,c.
- eq A a b \to eq ? a c \to eq ? b c.
-intros.
-lapply eq_sym; [|||| apply H ].
-apply eq_trans; [| auto | auto ].
-qed.
--- /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 ***************************************)
+
+set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/class_defs".
+
+include "../../library/logic/connectives.ma".
+
+(* 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
+ domain of a propositional function (ie a predicative subset) as a
+ quantification domain and therefore as a category, but there is no
+ type in CIC representing the domain of a propositional function
+ - We set up a single equality predicate, parametric on the category,
+ defined as the reflexive, symmetic, transitive and compatible closure
+ of the csub1 predicate given inside the category. Then we prove the
+ properties of the equality that usually are axiomatized inside the
+ category structure. This makes categories easier to use
+*)
+
+record Class: Type \def {
+ class: Type;
+ cin : class \to Prop;
+ csub1: class \to class \to Prop
+}.
+
+coercion class.
+
+inductive eq (C:Class) (c1:C): C \to Prop \def
+ | eq_refl: cin ? c1 \to eq ? c1 c1
+ | eq_sing_r: \forall c2,c3.
+ eq ? c1 c2 \to cin ? c3 \to csub1 ? c2 c3 \to eq ? c1 c3
+ | eq_sing_l: \forall c2,c3.
+ eq ? c1 c2 \to cin ? c3 \to csub1 ? c3 c2 \to eq ? c1 c3.
+
+theorem eq_cl: \forall C,c1,c2. eq ? c1 c2 \to cin C c1 \land cin C c2.
+intros; elim H; clear H; clear c2;
+ [ auto | decompose H2; auto | decompose H2; auto ].
+qed.
+
+theorem eq_trans: \forall C,c2,c1,c3.
+ eq C c2 c3 \to eq ? c1 c2 \to eq ? c1 c3.
+intros 5; elim H; clear H; clear c3;
+ [ auto
+ | apply eq_sing_r; [||| apply H4 ]; auto
+ | apply eq_sing_l; [||| apply H4 ]; auto
+ ].
+qed.
+
+theorem eq_conf_rev: \forall C,c2,c1,c3.
+ eq C c3 c2 \to eq ? c1 c2 \to eq ? c1 c3.
+intros 5; elim H; clear H; clear c2;
+ [ auto
+ | lapply eq_cl; [ decompose Hletin |||| apply H1 ].
+ apply H2; apply eq_sing_l; [||| apply H4 ]; auto
+ | lapply eq_cl; [ decompose Hletin |||| apply H1 ].
+ apply H2; apply eq_sing_r; [||| apply H4 ]; auto
+ ].
+qed.
+
+theorem eq_sym: \forall C,c1,c2. eq C c1 c2 \to eq C c2 c1.
+intros;
+lapply eq_cl; [ decompose Hletin |||| apply H ].
+auto.
+qed.
+
+theorem eq_conf: \forall C,c2,c1,c3.
+ eq C c1 c2 \to eq ? c1 c3 \to eq ? c2 c3.
+intros.
+lapply eq_sym; [|||| apply H ].
+apply eq_trans; [| auto | auto ].
+qed.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/domain_defs".
+
+include "class_defs.ma".
+
+(* QUANTIFICATION DOMAINS
+ - These are the categories on which we allow quantification
+ - We set up single quantifiers, parametric on the domain, so they
+ already have the properties that usually are axiomatized inside the
+ domain structure. This makes domains easier to use
+*)
+
+
+
+record Domain: Type \def {
+ qd: Class
+}.
+
+coercion qd.
+
+
+
+(* internal universal quantification *)
+inductive iall (D:Domain) (P:D \to Prop) : Prop \def
+ | iall_intro: (\forall d:D. cin D d \to P d) \to iall D P.
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "internal for all" 'iall \eta.x =
+ (cic:/matita/PREDICATIVE-TOPOLOGY/domain_defs/iall.ind#xpointer(1/1) _ x).
+
+notation < "hvbox(\iall ident i opt (: ty) break . p)"
+ right associative with precedence 20
+for @{ 'iall ${default
+ @{\lambda ${ident i} : $ty. $p)}
+ @{\lambda ${ident i} . $p}}}.
+
+
+
+(* internal existential quantification *)
+inductive iex (D:Domain) (P:D \to Prop) : Prop \def
+ | iex_intro: \forall d:D. cin D d \land P d \to iex D P.
+
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "internal exist" 'iexist \eta.x =
+ (cic:/matita/PREDICATIVE-TOPOLOGY/domain_defs/iex.ind#xpointer(1/1) _ x).
+
+notation < "hvbox(\iexist ident i opt (: ty) break . p)"
+ right associative with precedence 20
+for @{ 'iexist ${default
+ @{\lambda ${ident i} : $ty. $p)}
+ @{\lambda ${ident i} . $p}}}.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/qd_defs".
-
-include "ac_defs.ma".
-
-(* QUANTIFICATION DOMAINS
-*)
-
-record QD: Type \def {
- qd: AC
-}.
-
-coercion qd.
-
-(* se togli il D da acin D a non ce la fa ancora *)
-inductive iall (D:QD) (P:D \to Prop) : Prop \def
- | iall_intro: (\forall a:D. acin D a \to P a) \to iall D P.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/subset_defs".
+
+include "domain_defs.ma".
+
+(* SUBSETS
+ - We use predicative subsets coded as propositional functions
+ according to G.Sambin and S.Valentini "Toolbox"
+*)
+
+definition Subset \def \lambda (D:Domain). D \to Prop.
+
+(* subset inclusion *)
+definition ssub: \forall D. Subset D \to Subset D \to Prop \def
+ \lambda D,U1,U2. \forall d. U1 d \to U2 d.
+
+
+(* subset overlap *)
+definition sover: \forall D. Subset D \to Subset D \to Prop \def
+ \lambda D,U1,U2. \forall d. U1 d \to U2 d.
+
+
+
+(* full subset: "subset top" *)
+definition stop \def \lambda (D:Domain). \lambda (_:D). True.
+
+coercion stop.
+
+(* empty subset: "subset bottom" *)
+definition sbot \def \lambda (D:Domain). \lambda (_:D). False.
\ No newline at end of file
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/PREDICATIVE-TOPOLOGY/subsets_defs".
-
-include "qd_defs.ma".
-
-(* SUBSETS
- - We use predicative subsets coded as propositional functions
- according to G.Sambin and S.Valentini "Toolbox"
-*)
-
-definition Subset \def \forall (D:QD). D \to Prop.
-
-alias id "True" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1)".
-
-definition stop \def \lambda (D:QD). \lambda (a:D). True.
projects / helm.git / commitdiff