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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Specif.v,v 1.25.2.1 2004/07/16 19:31:04 herbelin Exp $ i*)
Set Implicit Arguments.
(** Basic specifications : Sets containing logical information *)
Require Import Notations.
Require Import Datatypes.
Require Import Logic.
(** Subsets *)
(** [(sig A P)], or more suggestively [{x:A | (P x)}], denotes the subset
of elements of the Set [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]. *)
Inductive sig (A:Set) (P:A -> Prop) : Set :=
exist : forall x:A, P x -> sig (A:=A) P.
Inductive sig2 (A:Set) (P Q:A -> Prop) : Set :=
exist2 : forall x:A, P x -> Q x -> sig2 (A:=A) P Q.
(** [(sigS A P)], or more suggestively [{x:A & (P x)}], is a subtle 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.
Inductive sigS2 (A:Set) (P Q:A -> Set) : Set :=
existS2 : forall x:A, P x -> Q x -> sigS2 (A:=A) P Q.
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].
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)) :
type_scope.
Add Printing Let sig.
Add Printing Let sig2.
Add Printing Let sigS.
Add Printing Let sigS2.
(** Projections of sig *)
Section Subset_projections.
Variable A : Set.
Variable P : A -> Prop.
Definition proj1_sig (e:sig P) := match e with
| exist a b => a
end.
Definition proj2_sig (e:sig P) :=
match e return P (proj1_sig e) with
| exist a b => b
end.
End Subset_projections.
(** Projections of sigS *)
Section Projections.
Variable A : Set.
Variable P : A -> Set.
(** An element [y] of a subset [{x:A & (P x)}] is the pair of an [a] of
type [A] and of a proof [h] that [a] satisfies [P].
Then [(projS1 y)] is the witness [a]
and [(projS2 y)] is the proof of [(P a)] *)
Definition projS1 (x:sigS P) : A := match x with
| existS a _ => a
end.
Definition projS2 (x:sigS P) : P (projS1 x) :=
match x return P (projS1 x) with
| existS _ h => h
end.
End Projections.
(** Extended_booleans *)
Inductive sumbool (A B:Prop) : Set :=
| left : A -> {A} + {B}
| right : B -> {A} + {B}
where "{ A } + { B }" := (sumbool A B) : type_scope.
Add Printing If sumbool.
Inductive sumor (A:Set) (B:Prop) : Set :=
| inleft : A -> A + {B}
| inright : B -> A + {B}
where "A + { B }" := (sumor A B) : type_scope.
Add Printing If sumor.
(** Choice *)
Section Choice_lemmas.
(** The following lemmas state various forms of the axiom of choice *)
Variables S S' : Set.
Variable R : S -> S' -> Prop.
Variable R' : S -> S' -> Set.
Variables R1 R2 : S -> Prop.
Lemma Choice :
(forall x:S, sig (fun y:S' => R x y)) ->
sig (fun f:S -> S' => forall z:S, R z (f z)).
Proof.
intro H.
exists (fun z:S => match H z with
| exist y _ => y
end).
intro z; destruct (H z); trivial.
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)).
Proof.
intro H.
exists (fun z:S => match H z with
| existS y _ => y
end).
intro z; destruct (H z); trivial.
Qed.
Lemma bool_choice :
(forall x:S, {R1 x} + {R2 x}) ->
sig
(fun f:S -> bool =>
forall x:S, f x = true /\ R1 x \/ f x = false /\ R2 x).
Proof.
intro H.
exists
(fun z:S => match H z with
| left _ => true
| right _ => false
end).
intro z; destruct (H z); auto.
Qed.
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)]
it is implemented using the option type. *)
Definition Exc := option.
Definition value := Some.
Definition error := @None.
Implicit Arguments error [A].
Definition except := False_rec. (* for compatibility with previous versions *)
Implicit Arguments except [P].
Theorem absurd_set : forall (A:Prop) (C:Set), A -> ~ A -> C.
Proof.
intros A C h1 h2.
apply False_rec.
apply (h2 h1).
Qed.
Hint Resolve left right inleft inright: core v62.
(** Sigma Type at Type level [sigT] *)
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.
|
