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authorGravatar Stephane Glondu <steph@glondu.net>2012-01-12 16:02:20 +0100
committerGravatar Stephane Glondu <steph@glondu.net>2012-01-12 16:02:20 +0100
commit97fefe1fcca363a1317e066e7f4b99b9c1e9987b (patch)
tree97ec6b7d831cc5fb66328b0c63a11db1cbb2f158 /theories/Numbers/NatInt/NZDiv.v
parent300293c119981054c95182a90c829058530a6b6f (diff)
Imported Upstream version 8.4~betaupstream/8.4_beta
Diffstat (limited to 'theories/Numbers/NatInt/NZDiv.v')
-rw-r--r--theories/Numbers/NatInt/NZDiv.v112
1 files changed, 58 insertions, 54 deletions
diff --git a/theories/Numbers/NatInt/NZDiv.v b/theories/Numbers/NatInt/NZDiv.v
index ba1c171e..bc109ace 100644
--- a/theories/Numbers/NatInt/NZDiv.v
+++ b/theories/Numbers/NatInt/NZDiv.v
@@ -1,6 +1,6 @@
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
-(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *)
+(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
@@ -12,44 +12,36 @@ Require Import NZAxioms NZMulOrder.
(** The first signatures will be common to all divisions over NZ, N and Z *)
-Module Type DivMod (Import T:Typ).
+Module Type DivMod (Import A : Typ).
Parameters Inline div modulo : t -> t -> t.
End DivMod.
-Module Type DivModNotation (T:Typ)(Import NZ:DivMod T).
+Module Type DivModNotation (A : Typ)(Import B : DivMod A).
Infix "/" := div.
Infix "mod" := modulo (at level 40, no associativity).
End DivModNotation.
-Module Type DivMod' (T:Typ) := DivMod T <+ DivModNotation T.
+Module Type DivMod' (A : Typ) := DivMod A <+ DivModNotation A.
-Module Type NZDivCommon (Import NZ : NZAxiomsSig')(Import DM : DivMod' NZ).
+Module Type NZDivSpec (Import A : NZOrdAxiomsSig')(Import B : DivMod' A).
Declare Instance div_wd : Proper (eq==>eq==>eq) div.
Declare Instance mod_wd : Proper (eq==>eq==>eq) modulo.
Axiom div_mod : forall a b, b ~= 0 -> a == b*(a/b) + (a mod b).
-End NZDivCommon.
+ Axiom mod_bound_pos : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b.
+End NZDivSpec.
(** The different divisions will only differ in the conditions
- they impose on [modulo]. For NZ, we only describe behavior
- on positive numbers.
-
- NB: This axiom would also be true for N and Z, but redundant.
+ they impose on [modulo]. For NZ, we have only described the
+ behavior on positive numbers.
*)
-Module Type NZDivSpecific (Import NZ : NZOrdAxiomsSig')(Import DM : DivMod' NZ).
- Axiom mod_bound : forall a b, 0<=a -> 0<b -> 0 <= a mod b < b.
-End NZDivSpecific.
-
-Module Type NZDiv (NZ:NZOrdAxiomsSig)
- := DivMod NZ <+ NZDivCommon NZ <+ NZDivSpecific NZ.
+Module Type NZDiv (A : NZOrdAxiomsSig) := DivMod A <+ NZDivSpec A.
+Module Type NZDiv' (A : NZOrdAxiomsSig) := NZDiv A <+ DivModNotation A.
-Module Type NZDiv' (NZ:NZOrdAxiomsSig) := NZDiv NZ <+ DivModNotation NZ.
-
-Module NZDivPropFunct
- (Import NZ : NZOrdAxiomsSig')
- (Import NZP : NZMulOrderPropSig NZ)
- (Import NZD : NZDiv' NZ)
-.
+Module Type NZDivProp
+ (Import A : NZOrdAxiomsSig')
+ (Import B : NZDiv' A)
+ (Import C : NZMulOrderProp A).
(** Uniqueness theorems *)
@@ -84,7 +76,7 @@ Theorem div_unique:
Proof.
intros a b q r Ha (Hb,Hr) EQ.
destruct (div_mod_unique b q (a/b) r (a mod b)); auto.
-apply mod_bound; order.
+apply mod_bound_pos; order.
rewrite <- div_mod; order.
Qed.
@@ -94,18 +86,21 @@ Theorem mod_unique:
Proof.
intros a b q r Ha (Hb,Hr) EQ.
destruct (div_mod_unique b q (a/b) r (a mod b)); auto.
-apply mod_bound; order.
+apply mod_bound_pos; order.
rewrite <- div_mod; order.
Qed.
+Theorem div_unique_exact a b q:
+ 0<=a -> 0<b -> a == b*q -> q == a/b.
+Proof.
+ intros Ha Hb H. apply div_unique with 0; nzsimpl; now try split.
+Qed.
(** A division by itself returns 1 *)
Lemma div_same : forall a, 0<a -> a/a == 1.
Proof.
-intros. symmetry.
-apply div_unique with 0; intuition; try order.
-now nzsimpl.
+intros. symmetry. apply div_unique_exact; nzsimpl; order.
Qed.
Lemma mod_same : forall a, 0<a -> a mod a == 0.
@@ -147,9 +142,7 @@ Qed.
Lemma div_1_r: forall a, 0<=a -> a/1 == a.
Proof.
-intros. symmetry.
-apply div_unique with 0; try split; try order; try apply lt_0_1.
-now nzsimpl.
+intros. symmetry. apply div_unique_exact; nzsimpl; order'.
Qed.
Lemma mod_1_r: forall a, 0<=a -> a mod 1 == 0.
@@ -161,20 +154,19 @@ Qed.
Lemma div_1_l: forall a, 1<a -> 1/a == 0.
Proof.
-intros; apply div_small; split; auto. apply le_succ_diag_r.
+intros; apply div_small; split; auto. apply le_0_1.
Qed.
Lemma mod_1_l: forall a, 1<a -> 1 mod a == 1.
Proof.
-intros; apply mod_small; split; auto. apply le_succ_diag_r.
+intros; apply mod_small; split; auto. apply le_0_1.
Qed.
Lemma div_mul : forall a b, 0<=a -> 0<b -> (a*b)/b == a.
Proof.
-intros; symmetry.
-apply div_unique with 0; try split; try order.
+intros; symmetry. apply div_unique_exact; trivial.
apply mul_nonneg_nonneg; order.
-nzsimpl; apply mul_comm.
+apply mul_comm.
Qed.
Lemma mod_mul : forall a b, 0<=a -> 0<b -> (a*b) mod b == 0.
@@ -194,7 +186,7 @@ Theorem mod_le: forall a b, 0<=a -> 0<b -> a mod b <= a.
Proof.
intros. destruct (le_gt_cases b a).
apply le_trans with b; auto.
-apply lt_le_incl. destruct (mod_bound a b); auto.
+apply lt_le_incl. destruct (mod_bound_pos a b); auto.
rewrite lt_eq_cases; right.
apply mod_small; auto.
Qed.
@@ -216,7 +208,7 @@ Lemma div_str_pos : forall a b, 0<b<=a -> 0 < a/b.
Proof.
intros a b (Hb,Hab).
assert (LE : 0 <= a/b) by (apply div_pos; order).
-assert (MOD : a mod b < b) by (destruct (mod_bound a b); order).
+assert (MOD : a mod b < b) by (destruct (mod_bound_pos a b); order).
rewrite lt_eq_cases in LE; destruct LE as [LT|EQ]; auto.
exfalso; revert Hab.
rewrite (div_mod a b), <-EQ; nzsimpl; order.
@@ -263,7 +255,7 @@ rewrite <- (mul_1_l (a/b)) at 1.
rewrite <- mul_lt_mono_pos_r; auto.
apply div_str_pos; auto.
rewrite <- (add_0_r (b*(a/b))) at 1.
-rewrite <- add_le_mono_l. destruct (mod_bound a b); order.
+rewrite <- add_le_mono_l. destruct (mod_bound_pos a b); order.
Qed.
(** [le] is compatible with a positive division. *)
@@ -282,8 +274,8 @@ apply lt_le_trans with b; auto.
rewrite (div_mod b c) at 1 by order.
rewrite <- add_assoc, <- add_le_mono_l.
apply le_trans with (c+0).
-nzsimpl; destruct (mod_bound b c); order.
-rewrite <- add_le_mono_l. destruct (mod_bound a c); order.
+nzsimpl; destruct (mod_bound_pos b c); order.
+rewrite <- add_le_mono_l. destruct (mod_bound_pos a c); order.
Qed.
(** The following two properties could be used as specification of div *)
@@ -293,7 +285,7 @@ Proof.
intros.
rewrite (add_le_mono_r _ _ (a mod b)), <- div_mod by order.
rewrite <- (add_0_r a) at 1.
-rewrite <- add_le_mono_l. destruct (mod_bound a b); order.
+rewrite <- add_le_mono_l. destruct (mod_bound_pos a b); order.
Qed.
Lemma mul_succ_div_gt : forall a b, 0<=a -> 0<b -> a < b*(S (a/b)).
@@ -302,7 +294,7 @@ intros.
rewrite (div_mod a b) at 1 by order.
rewrite (mul_succ_r).
rewrite <- add_lt_mono_l.
-destruct (mod_bound a b); auto.
+destruct (mod_bound_pos a b); auto.
Qed.
@@ -359,7 +351,7 @@ Proof.
apply mul_le_mono_nonneg_r; try order.
apply div_pos; order.
rewrite <- (add_0_r (r*(p/r))) at 1.
- rewrite <- add_le_mono_l. destruct (mod_bound p r); order.
+ rewrite <- add_le_mono_l. destruct (mod_bound_pos p r); order.
Qed.
@@ -371,7 +363,7 @@ Proof.
intros.
symmetry.
apply mod_unique with (a/c+b); auto.
- apply mod_bound; auto.
+ apply mod_bound_pos; auto.
rewrite mul_add_distr_l, add_shuffle0, <- div_mod by order.
now rewrite mul_comm.
Qed.
@@ -404,8 +396,8 @@ Proof.
apply div_unique with ((a mod b)*c).
apply mul_nonneg_nonneg; order.
split.
- apply mul_nonneg_nonneg; destruct (mod_bound a b); order.
- rewrite <- mul_lt_mono_pos_r; auto. destruct (mod_bound a b); auto.
+ apply mul_nonneg_nonneg; destruct (mod_bound_pos a b); order.
+ rewrite <- mul_lt_mono_pos_r; auto. destruct (mod_bound_pos a b); auto.
rewrite (div_mod a b) at 1 by order.
rewrite mul_add_distr_r.
rewrite add_cancel_r.
@@ -441,7 +433,7 @@ Qed.
Theorem mod_mod: forall a n, 0<=a -> 0<n ->
(a mod n) mod n == a mod n.
Proof.
- intros. destruct (mod_bound a n); auto. now rewrite mod_small_iff.
+ intros. destruct (mod_bound_pos a n); auto. now rewrite mod_small_iff.
Qed.
Lemma mul_mod_idemp_l : forall a b n, 0<=a -> 0<=b -> 0<n ->
@@ -454,7 +446,7 @@ Proof.
rewrite mul_add_distr_l, mul_assoc.
intros. rewrite mod_add; auto.
now rewrite mul_comm.
- apply mul_nonneg_nonneg; destruct (mod_bound a n); auto.
+ apply mul_nonneg_nonneg; destruct (mod_bound_pos a n); auto.
Qed.
Lemma mul_mod_idemp_r : forall a b n, 0<=a -> 0<=b -> 0<n ->
@@ -467,7 +459,7 @@ Theorem mul_mod: forall a b n, 0<=a -> 0<=b -> 0<n ->
(a * b) mod n == ((a mod n) * (b mod n)) mod n.
Proof.
intros. rewrite mul_mod_idemp_l, mul_mod_idemp_r; trivial. reflexivity.
- now destruct (mod_bound b n).
+ now destruct (mod_bound_pos b n).
Qed.
Lemma add_mod_idemp_l : forall a b n, 0<=a -> 0<=b -> 0<n ->
@@ -478,7 +470,7 @@ Proof.
rewrite (div_mod a n) at 1 2 by order.
rewrite <- add_assoc, add_comm, mul_comm.
intros. rewrite mod_add; trivial. reflexivity.
- apply add_nonneg_nonneg; auto. destruct (mod_bound a n); auto.
+ apply add_nonneg_nonneg; auto. destruct (mod_bound_pos a n); auto.
Qed.
Lemma add_mod_idemp_r : forall a b n, 0<=a -> 0<=b -> 0<n ->
@@ -491,7 +483,7 @@ Theorem add_mod: forall a b n, 0<=a -> 0<=b -> 0<n ->
(a+b) mod n == (a mod n + b mod n) mod n.
Proof.
intros. rewrite add_mod_idemp_l, add_mod_idemp_r; trivial. reflexivity.
- now destruct (mod_bound b n).
+ now destruct (mod_bound_pos b n).
Qed.
Lemma div_div : forall a b c, 0<=a -> 0<b -> 0<c ->
@@ -500,7 +492,7 @@ Proof.
intros a b c Ha Hb Hc.
apply div_unique with (b*((a/b) mod c) + a mod b); trivial.
(* begin 0<= ... <b*c *)
- destruct (mod_bound (a/b) c), (mod_bound a b); auto using div_pos.
+ destruct (mod_bound_pos (a/b) c), (mod_bound_pos a b); auto using div_pos.
split.
apply add_nonneg_nonneg; auto.
apply mul_nonneg_nonneg; order.
@@ -514,6 +506,18 @@ Proof.
apply div_mod; order.
Qed.
+Lemma mod_mul_r : forall a b c, 0<=a -> 0<b -> 0<c ->
+ a mod (b*c) == a mod b + b*((a/b) mod c).
+Proof.
+ intros a b c Ha Hb Hc.
+ apply add_cancel_l with (b*c*(a/(b*c))).
+ rewrite <- div_mod by (apply neq_mul_0; split; order).
+ rewrite <- div_div by trivial.
+ rewrite add_assoc, add_shuffle0, <- mul_assoc, <- mul_add_distr_l.
+ rewrite <- div_mod by order.
+ apply div_mod; order.
+Qed.
+
(** A last inequality: *)
Theorem div_mul_le:
@@ -538,5 +542,5 @@ Proof.
rewrite (mul_le_mono_pos_l _ _ b); auto. nzsimpl. order.
Qed.
-End NZDivPropFunct.
+End NZDivProp.