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Diffstat (limited to 'theories/Logic/Eqdep_dec.v')
| -rw-r--r-- | theories/Logic/Eqdep_dec.v | 32 |
1 files changed, 16 insertions, 16 deletions
diff --git a/theories/Logic/Eqdep_dec.v b/theories/Logic/Eqdep_dec.v index 59088aa7..3a6f6a23 100644 --- a/theories/Logic/Eqdep_dec.v +++ b/theories/Logic/Eqdep_dec.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -9,7 +9,7 @@ (* Created by Bruno Barras, Jan 1998 *) (* Made a module instance for EqdepFacts by Hugo Herbelin, Mar 2006 *) -(** We prove that there is only one proof of [x=x], i.e [refl_equal x]. +(** We prove that there is only one proof of [x=x], i.e [eq_refl x]. This holds if the equality upon the set of [x] is decidable. A corollary of this theorem is the equality of the right projections of two equal dependent pairs. @@ -43,7 +43,7 @@ Section EqdepDec. Let comp (x y y':A) (eq1:x = y) (eq2:x = y') : y = y' := eq_ind _ (fun a => a = y') eq2 _ eq1. - Remark trans_sym_eq : forall (x y:A) (u:x = y), comp u u = refl_equal y. + Remark trans_sym_eq : forall (x y:A) (u:x = y), comp u u = eq_refl y. Proof. intros. case u; trivial. @@ -61,7 +61,7 @@ Section EqdepDec. Let nu_constant : forall (y:A) (u v:x = y), nu u = nu v. intros. - unfold nu in |- *. + unfold nu. case (eq_dec x y); intros. reflexivity. @@ -69,13 +69,13 @@ Section EqdepDec. Qed. - Let nu_inv (y:A) (v:x = y) : x = y := comp (nu (refl_equal x)) v. + Let nu_inv (y:A) (v:x = y) : x = y := comp (nu (eq_refl x)) v. Remark nu_left_inv : forall (y:A) (u:x = y), nu_inv (nu u) = u. Proof. intros. - case u; unfold nu_inv in |- *. + case u; unfold nu_inv. apply trans_sym_eq. Qed. @@ -90,10 +90,10 @@ Section EqdepDec. Qed. Theorem K_dec : - forall P:x = x -> Prop, P (refl_equal x) -> forall p:x = x, P p. + forall P:x = x -> Prop, P (eq_refl x) -> forall p:x = x, P p. Proof. intros. - elim eq_proofs_unicity with x (refl_equal x) p. + elim eq_proofs_unicity with x (eq_refl x) p. trivial. Qed. @@ -115,7 +115,7 @@ Section EqdepDec. Proof. intros. cut (proj (ex_intro P x y) y = proj (ex_intro P x y') y). - simpl in |- *. + simpl. case (eq_dec x x). intro e. elim e using K_dec; trivial. @@ -135,7 +135,7 @@ Require Import EqdepFacts. Theorem K_dec_type : forall A:Type, (forall x y:A, {x = y} + {x <> y}) -> - forall (x:A) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p. + forall (x:A) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p. Proof. intros A eq_dec x P H p. elim p using K_dec; intros. @@ -146,7 +146,7 @@ Qed. Theorem K_dec_set : forall A:Set, (forall x y:A, {x = y} + {x <> y}) -> - forall (x:A) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p. + forall (x:A) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p. Proof fun A => K_dec_type (A:=A). (** We deduce the [eq_rect_eq] axiom for (decidable) types *) @@ -212,13 +212,13 @@ Module DecidableEqDep (M:DecidableType). (** Uniqueness of Reflexive Identity Proofs *) - Lemma UIP_refl : forall (x:U) (p:x = x), p = refl_equal x. + Lemma UIP_refl : forall (x:U) (p:x = x), p = eq_refl x. Proof (UIP__UIP_refl U UIP). (** Streicher's axiom K *) Lemma Streicher_K : - forall (x:U) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p. + forall (x:U) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p. Proof (K_dec_type eq_dec). (** Injectivity of equality on dependent pairs in [Type] *) @@ -281,13 +281,13 @@ Module DecidableEqDepSet (M:DecidableSet). (** Uniqueness of Reflexive Identity Proofs *) - Lemma UIP_refl : forall (x:U) (p:x = x), p = refl_equal x. + Lemma UIP_refl : forall (x:U) (p:x = x), p = eq_refl x. Proof N.UIP_refl. (** Streicher's axiom K *) Lemma Streicher_K : - forall (x:U) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p. + forall (x:U) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p. Proof N.Streicher_K. (** Proof-irrelevance on subsets of decidable sets *) @@ -301,7 +301,7 @@ Module DecidableEqDepSet (M:DecidableSet). Lemma inj_pair2 : forall (P:U -> Type) (p:U) (x y:P p), - existS P p x = existS P p y -> x = y. + existT P p x = existT P p y -> x = y. Proof eq_dep_eq__inj_pair2 U N.eq_dep_eq. (** Injectivity of equality on dependent pairs with second component |