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-rw-r--r--theories/Init/Specif.v113
1 files changed, 55 insertions, 58 deletions
diff --git a/theories/Init/Specif.v b/theories/Init/Specif.v
index e7fc1ac4..dd2f7697 100644
--- a/theories/Init/Specif.v
+++ b/theories/Init/Specif.v
@@ -6,7 +6,7 @@
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
-(*i $Id: Specif.v 8642 2006-03-17 10:09:02Z notin $ i*)
+(*i $Id: Specif.v 8866 2006-05-28 16:21:04Z herbelin $ i*)
(** Basic specifications : sets that may contain logical information *)
@@ -19,42 +19,45 @@ Require Import Logic.
(** Subsets and Sigma-types *)
(** [(sig A P)], or more suggestively [{x:A | P x}], denotes the subset
- of elements of the Set [A] which satisfy the predicate [P].
+ of elements of the type [A] which satisfy the predicate [P].
Similarly [(sig2 A P Q)], or [{x:A | P x & Q x}], denotes the subset
- of elements of the Set [A] which satisfy both [P] and [Q]. *)
+ of elements of the type [A] which satisfy both [P] and [Q]. *)
-Inductive sig (A:Set) (P:A -> Prop) : Set :=
- exist : forall x:A, P x -> sig (A:=A) P.
+Inductive sig (A:Type) (P:A -> Prop) : Type :=
+ exist : forall x:A, P x -> sig P.
-Inductive sig2 (A:Set) (P Q:A -> Prop) : Set :=
- exist2 : forall x:A, P x -> Q x -> sig2 (A:=A) P Q.
+Inductive sig2 (A:Type) (P Q:A -> Prop) : Type :=
+ exist2 : forall x:A, P x -> Q x -> sig2 P Q.
-(** [(sigS A P)], or more suggestively [{x:A & (P x)}] is a Sigma-type.
- It is a variant of subset where [P] is now of type [Set].
- Similarly for [(sigS2 A P Q)], also written [{x:A & (P x) & (Q x)}]. *)
-
-Inductive sigS (A:Set) (P:A -> Set) : Set :=
- existS : forall x:A, P x -> sigS (A:=A) P.
+(** [(sigT A P)], or more suggestively [{x:A & (P x)}] is a Sigma-type.
+ Similarly for [(sigT2 A P Q)], also written [{x:A & (P x) & (Q x)}]. *)
-Inductive sigS2 (A:Set) (P Q:A -> Set) : Set :=
- existS2 : forall x:A, P x -> Q x -> sigS2 (A:=A) P Q.
+Inductive sigT (A:Type) (P:A -> Type) : Type :=
+ existT : forall x:A, P x -> sigT P.
+
+Inductive sigT2 (A:Type) (P Q:A -> Type) : Type :=
+ existT2 : forall x:A, P x -> Q x -> sigT2 P Q.
+
+(* Notations *)
Arguments Scope sig [type_scope type_scope].
Arguments Scope sig2 [type_scope type_scope type_scope].
-Arguments Scope sigS [type_scope type_scope].
-Arguments Scope sigS2 [type_scope type_scope type_scope].
+Arguments Scope sigT [type_scope type_scope].
+Arguments Scope sigT2 [type_scope type_scope type_scope].
+Notation "{ x | P }" := (sig (fun x => P)) : type_scope.
+Notation "{ x | P & Q }" := (sig2 (fun x => P) (fun x => Q)) : type_scope.
Notation "{ x : A | P }" := (sig (fun x:A => P)) : type_scope.
Notation "{ x : A | P & Q }" := (sig2 (fun x:A => P) (fun x:A => Q)) :
type_scope.
-Notation "{ x : A & P }" := (sigS (fun x:A => P)) : type_scope.
-Notation "{ x : A & P & Q }" := (sigS2 (fun x:A => P) (fun x:A => Q)) :
+Notation "{ x : A & P }" := (sigT (fun x:A => P)) : type_scope.
+Notation "{ x : A & P & Q }" := (sigT2 (fun x:A => P) (fun x:A => Q)) :
type_scope.
Add Printing Let sig.
Add Printing Let sig2.
-Add Printing Let sigS.
-Add Printing Let sigS2.
+Add Printing Let sigT.
+Add Printing Let sigT2.
(** Projections of [sig]
@@ -67,7 +70,7 @@ Add Printing Let sigS2.
Section Subset_projections.
- Variable A : Set.
+ Variable A : Type.
Variable P : A -> Prop.
Definition proj1_sig (e:sig P) := match e with
@@ -82,24 +85,24 @@ Section Subset_projections.
End Subset_projections.
-(** Projections of [sigS]
+(** Projections of [sigT]
An element [x] of a sigma-type [{y:A & P y}] is a dependent pair
made of an [a] of type [A] and an [h] of type [P a]. Then,
- [(projS1 x)] is the first projection and [(projS2 x)] is the
- second projection, the type of which depends on the [projS1]. *)
+ [(projT1 x)] is the first projection and [(projT2 x)] is the
+ second projection, the type of which depends on the [projT1]. *)
Section Projections.
- Variable A : Set.
- Variable P : A -> Set.
+ Variable A : Type.
+ Variable P : A -> Type.
- Definition projS1 (x:sigS P) : A := match x with
- | existS a _ => a
+ Definition projT1 (x:sigT P) : A := match x with
+ | existT a _ => a
end.
- Definition projS2 (x:sigS P) : P (projS1 x) :=
- match x return P (projS1 x) with
- | existS _ h => h
+ Definition projT2 (x:sigT P) : P (projT1 x) :=
+ match x return P (projT1 x) with
+ | existT _ h => h
end.
End Projections.
@@ -118,7 +121,7 @@ Add Printing If sumbool.
(** [sumor] is an option type equipped with the justification of why
it may not be a regular value *)
-Inductive sumor (A:Set) (B:Prop) : Set :=
+Inductive sumor (A:Type) (B:Prop) : Type :=
| inleft : A -> A + {B}
| inright : B -> A + {B}
where "A + { B }" := (sumor A B) : type_scope.
@@ -146,12 +149,12 @@ Section Choice_lemmas.
Qed.
Lemma Choice2 :
- (forall x:S, sigS (fun y:S' => R' x y)) ->
- sigS (fun f:S -> S' => forall z:S, R' z (f z)).
+ (forall x:S, sigT (fun y:S' => R' x y)) ->
+ sigT (fun f:S -> S' => forall z:S, R' z (f z)).
Proof.
intro H.
exists (fun z:S => match H z with
- | existS y _ => y
+ | existT y _ => y
end).
intro z; destruct (H z); trivial.
Qed.
@@ -176,7 +179,7 @@ End Choice_lemmas.
(** A result of type [(Exc A)] is either a normal value of type [A] or
an [error] :
- [Inductive Exc [A:Set] : Set := value : A->(Exc A) | error : (Exc A)].
+ [Inductive Exc [A:Type] : Type := value : A->(Exc A) | error : (Exc A)].
It is implemented using the option type. *)
@@ -199,24 +202,18 @@ Qed.
Hint Resolve left right inleft inright: core v62.
-(** Sigma-type for types in [Type] *)
-
-Inductive sigT (A:Type) (P:A -> Type) : Type :=
- existT : forall x:A, P x -> sigT (A:=A) P.
-
-Section projections_sigT.
-
- Variable A : Type.
- Variable P : A -> Type.
-
- Definition projT1 (H:sigT P) : A := match H with
- | existT x _ => x
- end.
-
- Definition projT2 : forall x:sigT P, P (projT1 x) :=
- fun H:sigT P => match H return P (projT1 H) with
- | existT x h => h
- end.
-
-End projections_sigT.
-
+(* Compatibility *)
+
+Notation sigS := sigT (only parsing).
+Notation existS := existT (only parsing).
+Notation sigS_rect := sigT_rect (only parsing).
+Notation sigS_rec := sigT_rec (only parsing).
+Notation sigS_ind := sigT_ind (only parsing).
+Notation projS1 := projT1 (only parsing).
+Notation projS2 := projT2 (only parsing).
+
+Notation sigS2 := sigT2 (only parsing).
+Notation existS2 := existT2 (only parsing).
+Notation sigS2_rect := sigT2_rect (only parsing).
+Notation sigS2_rec := sigT2_rec (only parsing).
+Notation sigS2_ind := sigT2_ind (only parsing).
generated by cgit v1.2.3 (git 2.39.1) at 2025-01-14 07:03:57 +0000