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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** Basic specifications : sets that may contain logical information *)
Set Implicit Arguments.
Require Import Notations.
Require Import Datatypes.
Require Import Logic.
(** Subsets and Sigma-types *)
(** [(sig A P)], or more suggestively [{x:A | P x}], denotes the subset
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 type [A] which satisfy both [P] and [Q]. *)
Inductive sig (A:Type) (P:A -> Prop) : Type :=
exist : forall x:A, P x -> sig P.
Inductive sig2 (A:Type) (P Q:A -> Prop) : Type :=
exist2 : forall x:A, P x -> Q x -> sig2 P Q.
(** [(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 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 sig (A P)%type.
Arguments sig2 (A P Q)%type.
Arguments sigT (A P)%type.
Arguments sigT2 (A P Q)%type.
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 }" := (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 sigT.
Add Printing Let sigT2.
(** Projections of [sig]
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
[(proj1_sig y)] is the witness [a] and [(proj2_sig y)] is the
proof of [(P a)] *)
Section Subset_projections.
Variable A : Type.
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 [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,
[(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 : Type.
Variable P : A -> Type.
Definition projT1 (x:sigT P) : A := match x with
| existT a _ => a
end.
Definition projT2 (x:sigT P) : P (projT1 x) :=
match x return P (projT1 x) with
| existT _ h => h
end.
End Projections.
(** [sigT] of a predicate is equivalent to [sig] *)
Lemma sig_of_sigT : forall (A:Type) (P:A->Prop), sigT P -> sig P.
Proof. destruct 1 as (x,H); exists x; trivial. Defined.
Lemma sigT_of_sig : forall (A:Type) (P:A->Prop), sig P -> sigT P.
Proof. destruct 1 as (x,H); exists x; trivial. Defined.
Coercion sigT_of_sig : sig >-> sigT.
Coercion sig_of_sigT : sigT >-> sig.
(** [sumbool] is a boolean type equipped with the justification of
their value *)
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.
Arguments left {A B} _, [A] B _.
Arguments right {A B} _ , A [B] _.
(** [sumor] is an option type equipped with the justification of why
it may not be a regular value *)
Inductive sumor (A:Type) (B:Prop) : Type :=
| inleft : A -> A + {B}
| inright : B -> A + {B}
where "A + { B }" := (sumor A B) : type_scope.
Add Printing If sumor.
Arguments inleft {A B} _ , [A] B _.
Arguments inright {A B} _ , A [B] _.
(** Various forms of the axiom of choice for specifications *)
Section Choice_lemmas.
Variables S S' : Set.
Variable R : S -> S' -> Prop.
Variable R' : S -> S' -> Set.
Variables R1 R2 : S -> Prop.
Lemma Choice :
(forall x:S, {y:S' | R x y}) -> {f:S -> S' | forall z:S, R z (f z)}.
Proof.
intro H.
exists (fun z => proj1_sig (H z)).
intro z; destruct (H z); assumption.
Defined.
Lemma Choice2 :
(forall x:S, {y:S' & R' x y}) -> {f:S -> S' & forall z:S, R' z (f z)}.
Proof.
intro H.
exists (fun z => projT1 (H z)).
intro z; destruct (H z); assumption.
Defined.
Lemma bool_choice :
(forall x:S, {R1 x} + {R2 x}) ->
{f:S -> bool | forall x:S, f x = true /\ R1 x \/ f x = false /\ R2 x}.
Proof.
intro H.
exists (fun z:S => if H z then true else false).
intro z; destruct (H z); auto.
Defined.
End Choice_lemmas.
Section Dependent_choice_lemmas.
Variables X : Set.
Variable R : X -> X -> Prop.
Lemma dependent_choice :
(forall x:X, {y | R x y}) ->
forall x0, {f : nat -> X | f O = x0 /\ forall n, R (f n) (f (S n))}.
Proof.
intros H x0.
set (f:=fix f n := match n with O => x0 | S n' => proj1_sig (H (f n')) end).
exists f.
split. reflexivity.
induction n; simpl; apply proj2_sig.
Defined.
End Dependent_choice_lemmas.
(** A result of type [(Exc A)] is either a normal value of type [A] or
an [error] :
[Inductive Exc [A:Type] : Type := value : A->(Exc A) | error : (Exc A)].
It is implemented using the option type. *)
Definition Exc := option.
Definition value := Some.
Definition error := @None.
Arguments error [A].
Definition except := False_rec. (* for compatibility with previous versions *)
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).
Defined.
Hint Resolve left right inleft inright: core v62.
Hint Resolve exist exist2 existT existT2: core.
(* Compatibility *)
Notation sigS := sigT (compat "8.2").
Notation existS := existT (compat "8.2").
Notation sigS_rect := sigT_rect (compat "8.2").
Notation sigS_rec := sigT_rec (compat "8.2").
Notation sigS_ind := sigT_ind (compat "8.2").
Notation projS1 := projT1 (compat "8.2").
Notation projS2 := projT2 (compat "8.2").
Notation sigS2 := sigT2 (compat "8.2").
Notation existS2 := existT2 (compat "8.2").
Notation sigS2_rect := sigT2_rect (compat "8.2").
Notation sigS2_rec := sigT2_rec (compat "8.2").
Notation sigS2_ind := sigT2_ind (compat "8.2").
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