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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: Logic_Type.v,v 1.3.2.1 2004/07/16 19:31:26 herbelin Exp $ i*)
Set Implicit Arguments.
V7only [Unset Implicit Arguments.].
(** This module defines quantification on the world [Type]
([Logic.v] was defining it on the world [Set]) *)
Require Datatypes.
Require Export Logic.
V7only [
(*
(** [allT A P], or simply [(ALLT x | P(x))], stands for [(x:A)(P x)]
when [A] is of type [Type] *)
Definition allT := [A:Type][P:A->Prop](x:A)(P x).
*)
Notation allT := all (only parsing).
Notation inst := Logic.inst (only parsing).
Notation gen := Logic.gen (only parsing).
(* Order is important to give printing priority to fully typed ALL and EX *)
Notation AllT := (all ?).
Notation "'ALLT' x | p" := (all ? [x]p) (at level 10, p at level 8).
Notation "'ALLT' x : t | p" := (all ? [x:t]p) (at level 10, p at level 8).
(*
Section universal_quantification.
Variable A : Type.
Variable P : A->Prop.
Theorem inst : (x:A)(allT ? [x](P x))->(P x).
Proof.
Unfold all; Auto.
Qed.
Theorem gen : (B:Prop)(f:(y:A)B->(P y))B->(allT A P).
Proof.
Red; Auto.
Qed.
End universal_quantification.
*)
(*
(** * Existential Quantification *)
(** [exT A P], or simply [(EXT x | P(x))], stands for the existential
quantification on the predicate [P] when [A] is of type [Type] *)
(** [exT2 A P Q], or simply [(EXT x | P(x) & Q(x))], stands for the
existential quantification on both [P] and [Q] when [A] is of
type [Type] *)
Inductive exT [A:Type;P:A->Prop] : Prop
:= exT_intro : (x:A)(P x)->(exT A P).
*)
Notation exT := ex (only parsing).
Notation exT_intro := ex_intro (only parsing).
Notation exT_ind := ex_ind (only parsing).
Notation ExT := (ex ?).
Notation "'EXT' x | p" := (ex ? [x]p)
(at level 10, p at level 8, only parsing).
Notation "'EXT' x : t | p" := (ex ? [x:t]p)
(at level 10, p at level 8, only parsing).
(*
Inductive exT2 [A:Type;P,Q:A->Prop] : Prop
:= exT_intro2 : (x:A)(P x)->(Q x)->(exT2 A P Q).
*)
Notation exT2 := ex2 (only parsing).
Notation exT_intro2 := ex_intro2 (only parsing).
Notation exT2_ind := ex2_ind (only parsing).
Notation ExT2 := (ex2 ?).
Notation "'EXT' x | p & q" := (ex2 ? [x]p [x]q)
(at level 10, p, q at level 8).
Notation "'EXT' x : t | p & q" := (ex2 ? [x:t]p [x:t]q)
(at level 10, p, q at level 8).
(*
(** Leibniz equality : [A:Type][x,y:A] (P:A->Prop)(P x)->(P y)
[eqT A x y], or simply [x==y], is Leibniz' equality when [A] is of
type [Type]. This equality satisfies reflexivity (by definition),
symmetry, transitivity and stability by congruence *)
Inductive eqT [A:Type;x:A] : A -> Prop
:= refl_eqT : (eqT A x x).
Hints Resolve refl_eqT (* exT_intro2 exT_intro *) : core v62.
*)
Notation eqT := eq (only parsing).
Notation refl_eqT := refl_equal (only parsing).
Notation eqT_ind := eq_ind (only parsing).
Notation eqT_rect := eq_rect (only parsing).
Notation eqT_rec := eq_rec (only parsing).
Notation "x == y" := (eq ? x y) (at level 5, no associativity, only parsing).
(** Parsing only of things in [Logic_type.v] *)
Notation "< A > x == y" := (eq A x y)
(A annot, at level 1, x at level 0, only parsing).
(*
Section Equality_is_a_congruence.
Variables A,B : Type.
Variable f : A->B.
Variable x,y,z : A.
Lemma sym_eqT : (eqT ? x y) -> (eqT ? y x).
Proof.
NewDestruct 1; Trivial.
Qed.
Lemma trans_eqT : (eqT ? x y) -> (eqT ? y z) -> (eqT ? x z).
Proof.
NewDestruct 2; Trivial.
Qed.
Lemma congr_eqT : (eqT ? x y)->(eqT ? (f x) (f y)).
Proof.
NewDestruct 1; Trivial.
Qed.
Lemma sym_not_eqT : ~(eqT ? x y) -> ~(eqT ? y x).
Proof.
Red; Intros H H'; Apply H; NewDestruct H'; Trivial.
Qed.
End Equality_is_a_congruence.
*)
Notation sym_eqT := sym_eq (only parsing).
Notation trans_eqT := trans_eq (only parsing).
Notation congr_eqT := f_equal (only parsing).
Notation sym_not_eqT := sym_not_eq (only parsing).
(*
Hints Immediate sym_eqT sym_not_eqT : core v62.
*)
(** This states the replacement of equals by equals *)
(*
Definition eqT_ind_r : (A:Type)(x:A)(P:A->Prop)(P x)->(y:A)(eqT ? y x)->(P y).
Intros A x P H y H0; Case sym_eqT with 1:=H0; Trivial.
Defined.
Definition eqT_rec_r : (A:Type)(x:A)(P:A->Set)(P x)->(y:A)(eqT ? y x)->(P y).
Intros A x P H y H0; Case sym_eqT with 1:=H0; Trivial.
Defined.
Definition eqT_rect_r : (A:Type)(x:A)(P:A->Type)(P x)->(y:A)(eqT ? y x)->(P y).
Intros A x P H y H0; Case sym_eqT with 1:=H0; Trivial.
Defined.
*)
Notation eqT_ind_r := eq_ind_r (only parsing).
Notation eqT_rec_r := eq_rec_r (only parsing).
Notation eqT_rect_r := eq_rect_r (only parsing).
(** Some datatypes at the [Type] level *)
(*
Inductive EmptyT: Type :=.
Inductive UnitT : Type := IT : UnitT.
*)
Notation EmptyT := False (only parsing).
Notation UnitT := unit (only parsing).
Notation IT := tt.
].
Definition notT := [A:Type] A->EmptyT.
V7only [
(** Have you an idea of what means [identityT A a b]? No matter! *)
(*
Inductive identityT [A:Type; a:A] : A -> Type :=
refl_identityT : (identityT A a a).
*)
Notation identityT := identity (only parsing).
Notation refl_identityT := refl_identity (only parsing).
Notation "< A > x === y" := (!identityT A x y)
(A annot, at level 1, x at level 0, only parsing) : type_scope.
Notation "x === y" := (identityT ? x y)
(at level 5, no associativity, only parsing) : type_scope.
(*
Hints Resolve refl_identityT : core v62.
*)
].
Section identity_is_a_congruence.
Variables A,B : Type.
Variable f : A->B.
Variable x,y,z : A.
Lemma sym_id : (identityT ? x y) -> (identityT ? y x).
Proof.
NewDestruct 1; Trivial.
Qed.
Lemma trans_id : (identityT ? x y) -> (identityT ? y z) -> (identityT ? x z).
Proof.
NewDestruct 2; Trivial.
Qed.
Lemma congr_id : (identityT ? x y)->(identityT ? (f x) (f y)).
Proof.
NewDestruct 1; Trivial.
Qed.
Lemma sym_not_id : (notT (identityT ? x y)) -> (notT (identityT ? y x)).
Proof.
Red; Intros H H'; Apply H; NewDestruct H'; Trivial.
Qed.
End identity_is_a_congruence.
Definition identity_ind_r :
(A:Type)
(a:A)
(P:A->Prop)
(P a)->(y:A)(identityT ? y a)->(P y).
Intros A x P H y H0; Case sym_id with 1:= H0; Trivial.
Defined.
Definition identity_rec_r :
(A:Type)
(a:A)
(P:A->Set)
(P a)->(y:A)(identityT ? y a)->(P y).
Intros A x P H y H0; Case sym_id with 1:= H0; Trivial.
Defined.
Definition identity_rect_r :
(A:Type)
(a:A)
(P:A->Type)
(P a)->(y:A)(identityT ? y a)->(P y).
Intros A x P H y H0; Case sym_id with 1:= H0; Trivial.
Defined.
V7only [
Notation sym_idT := sym_id (only parsing).
Notation trans_idT := trans_id (only parsing).
Notation congr_idT := congr_id (only parsing).
Notation sym_not_idT := sym_not_id (only parsing).
Notation identityT_ind_r := identity_ind_r (only parsing).
Notation identityT_rec_r := identity_rec_r (only parsing).
Notation identityT_rect_r := identity_rect_r (only parsing).
].
Inductive prodT [A,B:Type] : Type := pairT : A -> B -> (prodT A B).
Section prodT_proj.
Variables A, B : Type.
Definition fstT := [H:(prodT A B)]Cases H of (pairT x _) => x end.
Definition sndT := [H:(prodT A B)]Cases H of (pairT _ y) => y end.
End prodT_proj.
Definition prodT_uncurry : (A,B,C:Type)((prodT A B)->C)->A->B->C :=
[A,B,C:Type; f:((prodT A B)->C); x:A; y:B]
(f (pairT A B x y)).
Definition prodT_curry : (A,B,C:Type)(A->B->C)->(prodT A B)->C :=
[A,B,C:Type; f:(A->B->C); p:(prodT A B)]
Cases p of
| (pairT x y) => (f x y)
end.
Hints Immediate sym_id sym_not_id : core v62.
V7only [
Implicits fstT [1 2].
Implicits sndT [1 2].
Implicits pairT [1 2].
].
|
