diff options
Diffstat (limited to 'theories/Logic/ClassicalFacts.v')
| -rw-r--r-- | theories/Logic/ClassicalFacts.v | 38 |
1 files changed, 19 insertions, 19 deletions
diff --git a/theories/Logic/ClassicalFacts.v b/theories/Logic/ClassicalFacts.v index bcec657a..34ae1cd5 100644 --- a/theories/Logic/ClassicalFacts.v +++ b/theories/Logic/ClassicalFacts.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* 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 *) @@ -117,7 +117,7 @@ Qed. *) -Notation Local inhabited A := A (only parsing). +Local Notation inhabited A := A (only parsing). Lemma prop_ext_A_eq_A_imp_A : prop_extensionality -> forall A:Prop, inhabited A -> (A -> A) = A. @@ -148,7 +148,7 @@ Proof. case (prop_ext_retract_A_A_imp_A Ext A a); intros g1 g2 g1_o_g2. exists (fun f => (fun x:A => f (g1 x x)) (g2 (fun x => f (g1 x x)))). intro f. - pattern (g1 (g2 (fun x:A => f (g1 x x)))) at 1 in |- *. + pattern (g1 (g2 (fun x:A => f (g1 x x)))) at 1. rewrite (g1_o_g2 (fun x:A => f (g1 x x))). reflexivity. Qed. @@ -191,13 +191,13 @@ Section Proof_irrelevance_gen. intros Ext Ind. case (ext_prop_fixpoint Ext bool true); intros G Gfix. set (neg := fun b:bool => bool_elim bool false true b). - generalize (refl_equal (G neg)). - pattern (G neg) at 1 in |- *. + generalize (eq_refl (G neg)). + pattern (G neg) at 1. apply Ind with (b := G neg); intro Heq. rewrite (bool_elim_redl bool false true). - change (true = neg true) in |- *; rewrite Heq; apply Gfix. + change (true = neg true); rewrite Heq; apply Gfix. rewrite (bool_elim_redr bool false true). - change (neg false = false) in |- *; rewrite Heq; symmetry in |- *; + change (neg false = false); rewrite Heq; symmetry ; apply Gfix. Qed. @@ -207,9 +207,9 @@ Section Proof_irrelevance_gen. intros Ext Ind A a1 a2. set (f := fun b:bool => bool_elim A a1 a2 b). rewrite (bool_elim_redl A a1 a2). - change (f true = a2) in |- *. + change (f true = a2). rewrite (bool_elim_redr A a1 a2). - change (f true = f false) in |- *. + change (f true = f false). rewrite (aux Ext Ind). reflexivity. Qed. @@ -228,9 +228,9 @@ Section Proof_irrelevance_Prop_Ext_CC. Definition FalseP : BoolP := fun C c1 c2 => c2. Definition BoolP_elim C c1 c2 (b:BoolP) := b C c1 c2. Definition BoolP_elim_redl (C:Prop) (c1 c2:C) : - c1 = BoolP_elim C c1 c2 TrueP := refl_equal c1. + c1 = BoolP_elim C c1 c2 TrueP := eq_refl c1. Definition BoolP_elim_redr (C:Prop) (c1 c2:C) : - c2 = BoolP_elim C c1 c2 FalseP := refl_equal c2. + c2 = BoolP_elim C c1 c2 FalseP := eq_refl c2. Definition BoolP_dep_induction := forall P:BoolP -> Prop, P TrueP -> P FalseP -> forall b:BoolP, P b. @@ -263,9 +263,9 @@ Section Proof_irrelevance_CIC. | trueP : boolP | falseP : boolP. Definition boolP_elim_redl (C:Prop) (c1 c2:C) : - c1 = boolP_ind C c1 c2 trueP := refl_equal c1. + c1 = boolP_ind C c1 c2 trueP := eq_refl c1. Definition boolP_elim_redr (C:Prop) (c1 c2:C) : - c2 = boolP_ind C c1 c2 falseP := refl_equal c2. + c2 = boolP_ind C c1 c2 falseP := eq_refl c2. Scheme boolP_indd := Induction for boolP Sort Prop. Lemma ext_prop_dep_proof_irrel_cic : prop_extensionality -> proof_irrelevance. @@ -344,8 +344,8 @@ Section Proof_irrelevance_EM_CC. Lemma p2p1 : forall A:Prop, A -> b2p (p2b A). Proof. - unfold p2b in |- *; intro A; apply or_dep_elim with (b := em A); - unfold b2p in |- *; intros. + unfold p2b; intro A; apply or_dep_elim with (b := em A); + unfold b2p; intros. apply (or_elim_redl A (~ A) B (fun _ => b1) (fun _ => b2)). destruct (b H). Qed. @@ -353,8 +353,8 @@ Section Proof_irrelevance_EM_CC. Lemma p2p2 : b1 <> b2 -> forall A:Prop, b2p (p2b A) -> A. Proof. intro not_eq_b1_b2. - unfold p2b in |- *; intro A; apply or_dep_elim with (b := em A); - unfold b2p in |- *; intros. + unfold p2b; intro A; apply or_dep_elim with (b := em A); + unfold b2p; intros. assumption. destruct not_eq_b1_b2. rewrite <- (or_elim_redr A (~ A) B (fun _ => b1) (fun _ => b2)) in H. @@ -392,9 +392,9 @@ Section Proof_irrelevance_CCI. Hypothesis em : forall A:Prop, A \/ ~ A. Definition or_elim_redl (A B C:Prop) (f:A -> C) (g:B -> C) - (a:A) : f a = or_ind f g (or_introl B a) := refl_equal (f a). + (a:A) : f a = or_ind f g (or_introl B a) := eq_refl (f a). Definition or_elim_redr (A B C:Prop) (f:A -> C) (g:B -> C) - (b:B) : g b = or_ind f g (or_intror A b) := refl_equal (g b). + (b:B) : g b = or_ind f g (or_intror A b) := eq_refl (g b). Scheme or_indd := Induction for or Sort Prop. Theorem proof_irrelevance_cci : forall (B:Prop) (b1 b2:B), b1 = b2. |