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
+ of the cle1 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
*)
+definition true_f \def \lambda (X:Type). \lambda (_:X). True.
+
+definition false_f \def \lambda (X:Type). \lambda (_:X). False.
+
record Class: Type \def {
class: Type;
cin : class \to Prop;
- csub1: class \to class \to Prop
+ cle1 : 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.
+inductive ceq (C:Class) (c1:C): C \to Prop \def
+ | ceq_refl: cin ? c1 \to ceq ? c1 c1
+ | ceq_sing_r: \forall c2,c3.
+ ceq ? c1 c2 \to cin ? c3 \to cle1 ? c2 c3 \to ceq ? c1 c3
+ | ceq_sing_l: \forall c2,c3.
+ ceq ? c1 c2 \to cin ? c3 \to cle1 ? c3 c2 \to ceq ? c1 c3.
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.
+(* 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.
+(*
+(* notations **************************************************************)
+
(*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).
@{\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).
for @{ 'iexist ${default
@{\lambda ${ident i} : $ty. $p)}
@{\lambda ${ident i} . $p}}}.
+*)
\ No newline at end of file
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 membership (epsilon) *)
+definition sin : \forall D. Subset D \to D \to Prop \def
+ \lambda (D:Domain). \lambda U,d. cin D d \and U d.
+
+(* subset top (full subset) *)
+definition stop \def \lambda (D:Domain). true_f D.
+
+(* subset bottom (empty subset) *)
+definition sbot \def \lambda (D:Domain). false_f D.
+
+(* subset and (binary intersection) *)
+definition sand: \forall D. Subset D \to Subset D \to Subset D \def
+ \lambda D,U1,U2,d. U1 d \land U2 d.
+(* subset or (binary union) *)
+definition sor: \forall D. Subset D \to Subset D \to Subset D \def
+ \lambda D,U1,U2,d. U1 d \lor U2 d.
+(* subset less or equal (inclusion) *)
+definition sle: \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.
+*)
+
+(* 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 domain built upon a subset *)
+definition domain_of_subset: \forall D. (Subset D) \to Domain \def
+ \lambda (D:Domain). \lambda U.
+ mk_Domain (mk_Class D (sin D U) (cle1 D)).
-(* full subset: "subset top" *)
-definition stop \def \lambda (D:Domain). \lambda (_:D). True.
+coercion domain_of_subset.
+(* the full subset of a domain *)
coercion stop.
-(* empty subset: "subset bottom" *)
-definition sbot \def \lambda (D:Domain). \lambda (_:D). False.
\ No newline at end of file
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