From 05085e80668a4d1dedc522c6af343168870cc648 Mon Sep 17 00:00:00 2001 From: pboutill Date: Fri, 10 Dec 2010 13:22:29 +0000 Subject: First release of Vector library. To avoid names¬ations clashs with list, Vector shouldn't be "Import"ed but one can "Import Vector.VectorNotations." to have notations. SetoidVector at least remains to do. git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@13702 85f007b7-540e-0410-9357-904b9bb8a0f7 --- theories/Arith/Bool_nat.v | 2 +- theories/Arith/Compare.v | 2 +- theories/Arith/Min.v | 2 +- theories/Arith/Peano_dec.v | 22 +++++++++++++++++++++- theories/Arith/Plus.v | 25 ++++++------------------- 5 files changed, 30 insertions(+), 23 deletions(-) (limited to 'theories/Arith') diff --git a/theories/Arith/Bool_nat.v b/theories/Arith/Bool_nat.v index 08a090325..f384e1488 100644 --- a/theories/Arith/Bool_nat.v +++ b/theories/Arith/Bool_nat.v @@ -34,4 +34,4 @@ Definition nat_noteq_bool x y := bool_of_sumbool (sumbool_not _ _ (eq_nat_dec x y)). Definition zerop_bool x := bool_of_sumbool (zerop x). -Definition notzerop_bool x := bool_of_sumbool (notzerop x). \ No newline at end of file +Definition notzerop_bool x := bool_of_sumbool (notzerop x). diff --git a/theories/Arith/Compare.v b/theories/Arith/Compare.v index c6bf35bb0..c9e6d3cf3 100644 --- a/theories/Arith/Compare.v +++ b/theories/Arith/Compare.v @@ -50,4 +50,4 @@ Qed. Require Export Wf_nat. -Require Export Min Max. \ No newline at end of file +Require Export Min Max. diff --git a/theories/Arith/Min.v b/theories/Arith/Min.v index 2b2cf860d..bcfbe0efe 100644 --- a/theories/Arith/Min.v +++ b/theories/Arith/Min.v @@ -39,4 +39,4 @@ Definition min_glb := Nat.min_glb. (* Compatibility *) Notation min_case2 := min_case (only parsing). Notation min_SS := Nat.succ_min_distr (only parsing). -(* end hide *) \ No newline at end of file +(* end hide *) diff --git a/theories/Arith/Peano_dec.v b/theories/Arith/Peano_dec.v index 9e3823d7a..6eb667c11 100644 --- a/theories/Arith/Peano_dec.v +++ b/theories/Arith/Peano_dec.v @@ -7,7 +7,8 @@ (************************************************************************) Require Import Decidable. - +Require Eqdep_dec. +Require Import Le Lt. Open Local Scope nat_scope. Implicit Types m n x y : nat. @@ -30,3 +31,22 @@ Hint Resolve O_or_S eq_nat_dec: arith. Theorem dec_eq_nat : forall n m, decidable (n = m). intros x y; unfold decidable in |- *; elim (eq_nat_dec x y); auto with arith. Defined. + +Definition UIP_nat:= Eqdep_dec.UIP_dec eq_nat_dec. + +Lemma le_unique: forall m n (h1 h2: m <= n), h1 = h2. +Proof. +fix 3. +refine (fun m _ h1 => match h1 as h' in _ <= k return forall hh: m <= k, h' = hh + with le_n => _ |le_S i H => _ end). +refine (fun hh => match hh as h' in _ <= k return forall eq: m = k, + le_n m = match eq in _ = p return m <= p -> m <= m with |eq_refl => fun bli => bli end h' with + |le_n => fun eq => _ |le_S j H' => fun eq => _ end eq_refl). +rewrite (UIP_nat _ _ eq eq_refl). reflexivity. +subst m. destruct (Lt.lt_irrefl j H'). +refine (fun hh => match hh as h' in _ <= k return match k as k' return m <= k' -> Prop + with |0 => fun _ => True |S i' => fun h'' => forall H':m <= i', le_S m i' H' = h'' end h' + with |le_n => _ |le_S j H2 => fun H' => _ end H). +destruct m. exact I. intros; destruct (Lt.lt_irrefl m H'). +f_equal. apply le_unique. +Qed. diff --git a/theories/Arith/Plus.v b/theories/Arith/Plus.v index 48d730319..eb2d4df4c 100644 --- a/theories/Arith/Plus.v +++ b/theories/Arith/Plus.v @@ -24,17 +24,10 @@ Open Local Scope nat_scope. Implicit Types m n p q : nat. -(** * Zero is neutral *) - -Lemma plus_0_l : forall n, 0 + n = n. -Proof. - reflexivity. -Qed. - -Lemma plus_0_r : forall n, n + 0 = n. -Proof. - intro; symmetry in |- *; apply plus_n_O. -Qed. +(** * Zero is neutral +Deprecated : Already in Init/Peano.v *) +Definition plus_0_l n := eq_sym (plus_O_n n). +Definition plus_0_r n := eq_sym (plus_n_O n). (** * Commutativity *) @@ -47,14 +40,8 @@ Hint Immediate plus_comm: arith v62. (** * Associativity *) -Lemma plus_Snm_nSm : forall n m, S n + m = n + S m. -Proof. - intros. - simpl in |- *. - rewrite (plus_comm n m). - rewrite (plus_comm n (S m)). - trivial with arith. -Qed. +Definition plus_Snm_nSm : forall n m, S n + m = n + S m:= + plus_n_Sm. Lemma plus_assoc : forall n m p, n + (m + p) = n + m + p. Proof. -- cgit v1.2.3