diff options
Diffstat (limited to 'theories/Logic/Eqdep_dec.v')
| -rw-r--r-- | theories/Logic/Eqdep_dec.v | 32 |
1 files changed, 19 insertions, 13 deletions
diff --git a/theories/Logic/Eqdep_dec.v b/theories/Logic/Eqdep_dec.v index 0281916e..fc1c4a97 100644 --- a/theories/Logic/Eqdep_dec.v +++ b/theories/Logic/Eqdep_dec.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Eqdep_dec.v 10144 2007-09-26 15:12:17Z vsiles $ i*) +(*i $Id$ i*) (** We prove that there is only one proof of [x=x], i.e [refl_equal x]. This holds if the equality upon the set of [x] is decidable. @@ -38,7 +38,7 @@ Set Implicit Arguments. Section EqdepDec. Variable A : Type. - + Let comp (x y y':A) (eq1:x = y) (eq2:x = y') : y = y' := eq_ind _ (fun a => a = y') eq2 _ eq1. @@ -49,7 +49,7 @@ Section EqdepDec. Qed. Variable eq_dec : forall x y:A, x = y \/ x <> y. - + Variable x : A. Let nu (y:A) (u:x = y) : x = y := @@ -63,13 +63,13 @@ Section EqdepDec. unfold nu in |- *. case (eq_dec x y); intros. reflexivity. - + case n; trivial. Qed. Let nu_inv (y:A) (v:x = y) : x = y := comp (nu (refl_equal x)) v. - + Remark nu_left_inv : forall (y:A) (u:x = y), nu_inv (nu u) = u. Proof. @@ -88,7 +88,7 @@ Section EqdepDec. reflexivity. Qed. - Theorem K_dec : + Theorem K_dec : forall P:x = x -> Prop, P (refl_equal x) -> forall p:x = x, P p. Proof. intros. @@ -118,10 +118,10 @@ Section EqdepDec. case (eq_dec x x). intro e. elim e using K_dec; trivial. - + intros. case n; trivial. - + case H. reflexivity. Qed. @@ -165,6 +165,12 @@ Theorem eq_dep_eq_dec : forall (P:A->Type) (p:A) (x y:P p), eq_dep A P p x p y -> x = y. Proof (fun A eq_dec => eq_rect_eq__eq_dep_eq A (eq_rect_eq_dec eq_dec)). +Theorem UIP_dec : + forall (A:Type), + (forall x y:A, {x = y} + {x <> y}) -> + forall (x y:A) (p1 p2:x = y), p1 = p2. +Proof (fun A eq_dec => eq_dep_eq__UIP A (eq_dep_eq_dec eq_dec)). + Unset Implicit Arguments. (************************************************************************) @@ -173,13 +179,13 @@ Unset Implicit Arguments. (** The signature of decidable sets in [Type] *) Module Type DecidableType. - + Parameter U:Type. Axiom eq_dec : forall x y:U, {x = y} + {x <> y}. End DecidableType. -(** The module [DecidableEqDep] collects equality properties for decidable +(** The module [DecidableEqDep] collects equality properties for decidable set in [Type] *) Module DecidableEqDep (M:DecidableType). @@ -247,7 +253,7 @@ Module Type DecidableSet. End DecidableSet. -(** The module [DecidableEqDepSet] collects equality properties for decidable +(** The module [DecidableEqDepSet] collects equality properties for decidable set in [Set] *) Module DecidableEqDepSet (M:DecidableSet). @@ -307,11 +313,11 @@ End DecidableEqDepSet. (** From decidability to inj_pair2 **) Lemma inj_pair2_eq_dec : forall A:Type, (forall x y:A, {x=y}+{x<>y}) -> ( forall (P:A -> Type) (p:A) (x y:P p), existT P p x = existT P p y -> x = y ). -Proof. +Proof. intros A eq_dec. apply eq_dep_eq__inj_pair2. apply eq_rect_eq__eq_dep_eq. - unfold Eq_rect_eq. + unfold Eq_rect_eq. apply eq_rect_eq_dec. apply eq_dec. Qed. |