diff options
Diffstat (limited to 'mathcomp/algebra/interval.v')
| -rw-r--r-- | mathcomp/algebra/interval.v | 60 |
1 files changed, 29 insertions, 31 deletions
diff --git a/mathcomp/algebra/interval.v b/mathcomp/algebra/interval.v index eb0785f..48d5254 100644 --- a/mathcomp/algebra/interval.v +++ b/mathcomp/algebra/interval.v @@ -1,7 +1,8 @@ (* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *) (* Distributed under the terms of CeCILL-B. *) From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq choice. -From mathcomp Require Import div fintype bigop ssralg finset fingroup ssrnum. +From mathcomp Require Import div fintype bigop order ssralg finset fingroup. +From mathcomp Require Import ssrnum. (*****************************************************************************) (* This file provide support for intervals in numerical and real domains. *) @@ -39,13 +40,14 @@ Unset Strict Implicit. Unset Printing Implicit Defensive. Local Open Scope ring_scope. -Import GRing.Theory Num.Theory. +Import Order.Theory Order.Syntax GRing.Theory Num.Theory. Local Notation mid x y := ((x + y) / 2%:R). Section LersifPo. Variable R : numDomainType. +Implicit Types (b : bool) (x y z : R). Definition lersif (x y : R) b := if b then x < y else x <= y. @@ -65,7 +67,7 @@ Lemma lersif_trans x y z b1 b2 : x <= y ?< if b1 -> y <= z ?< if b2 -> x <= z ?< if b1 || b2. Proof. by case: b1 b2 => [] []; - apply (ltr_trans, ltr_le_trans, ler_lt_trans, ler_trans). + [apply: lt_trans | apply: lt_le_trans | apply: le_lt_trans | apply: le_trans]. Qed. Lemma lersif01 b : 0 <= 1 ?< if b. @@ -74,16 +76,16 @@ Proof. by case: b; apply lter01. Qed. Lemma lersif_anti b1 b2 x y : (x <= y ?< if b1) && (y <= x ?< if b2) = if b1 || b2 then false else x == y. -Proof. by case: b1 b2 => [] []; rewrite lter_anti. Qed. +Proof. by case: b1 b2 => [] []; rewrite lte_anti. Qed. Lemma lersifxx x b : (x <= x ?< if b) = ~~ b. -Proof. by case: b; rewrite /= lterr. Qed. +Proof. by case: b; rewrite /= ltexx. Qed. Lemma lersifNF x y b : y <= x ?< if ~~ b -> x <= y ?< if b = false. -Proof. by case: b => /= [/ler_gtF|/ltr_geF]. Qed. +Proof. by case: b => /= [/le_gtF|/lt_geF]. Qed. Lemma lersifS x y b : x < y -> x <= y ?< if b. -Proof. by case: b => //= /ltrW. Qed. +Proof. by case: b => //= /ltW. Qed. Lemma lersifT x y : x <= y ?< if true = (x < y). Proof. by []. Qed. @@ -132,25 +134,25 @@ Definition lersif_sub_addl := (lersif_subl_addl, lersif_subr_addl). Lemma lersif_andb x y : {morph lersif x y : p q / p || q >-> p && q}. Proof. -by case=> [] [] /=; rewrite ?ler_eqVlt; +by case=> [] [] /=; rewrite ?le_eqVlt; case: (_ < _)%R; rewrite ?(orbT, orbF, andbF, andbb). Qed. Lemma lersif_orb x y : {morph lersif x y : p q / p && q >-> p || q}. Proof. -by case=> [] [] /=; rewrite ?ler_eqVlt; +by case=> [] [] /=; rewrite ?le_eqVlt; case: (_ < _)%R; rewrite ?(orbT, orbF, orbb). Qed. Lemma lersif_imply b1 b2 r1 r2 : b2 ==> b1 -> r1 <= r2 ?< if b1 -> r1 <= r2 ?< if b2. -Proof. by case: b1 b2 => [] [] //= _ /ltrW. Qed. +Proof. by case: b1 b2 => [] [] //= _ /ltW. Qed. Lemma lersifW b x y : x <= y ?< if b -> x <= y. -Proof. by case: b => // /ltrW. Qed. +Proof. by case: b => // /ltW. Qed. Lemma ltrW_lersif b x y : x < y -> x <= y ?< if b. -Proof. by case: b => // /ltrW. Qed. +Proof. by case: b => // /ltW. Qed. Lemma lersif_pmul2l b x : 0 < x -> {mono *%R x : y z / y <= z ?< if b}. Proof. by case: b; apply lter_pmul2l. Qed. @@ -166,7 +168,7 @@ Proof. by case: b; apply lter_nmul2r. Qed. Lemma real_lersifN x y b : x \is Num.real -> y \is Num.real -> x <= y ?< if ~~b = ~~ (y <= x ?< if b). -Proof. by case: b => [] xR yR /=; case: real_ltrgtP. Qed. +Proof. by case: b => [] xR yR /=; case: real_ltgtP. Qed. Lemma real_lersif_norml b x y : x \is Num.real -> @@ -207,20 +209,20 @@ Lemma lersif_distl : Proof. by case: b; apply lter_distl. Qed. Lemma lersif_minr : - (x <= Num.min y z ?< if b) = (x <= y ?< if b) && (x <= z ?< if b). -Proof. by case: b; apply lter_minr. Qed. + (x <= y `&` z ?< if b) = (x <= y ?< if b) && (x <= z ?< if b). +Proof. by case: b; rewrite /= ltexI. Qed. Lemma lersif_minl : - (Num.min y z <= x ?< if b) = (y <= x ?< if b) || (z <= x ?< if b). -Proof. by case: b; apply lter_minl. Qed. + (y `&` z <= x ?< if b) = (y <= x ?< if b) || (z <= x ?< if b). +Proof. by case: b; rewrite /= lteIx. Qed. Lemma lersif_maxr : - (x <= Num.max y z ?< if b) = (x <= y ?< if b) || (x <= z ?< if b). -Proof. by case: b; apply lter_maxr. Qed. + (x <= y `|` z ?< if b) = (x <= y ?< if b) || (x <= z ?< if b). +Proof. by case: b; rewrite /= ltexU. Qed. Lemma lersif_maxl : - (Num.max y z <= x ?< if b) = (y <= x ?< if b) && (z <= x ?< if b). -Proof. by case: b; apply lter_maxl. Qed. + (y `|` z <= x ?< if b) = (y <= x ?< if b) && (z <= x ?< if b). +Proof. by case: b; rewrite /= lteUx. Qed. End LersifOrdered. @@ -459,13 +461,9 @@ Definition bound_in_itv := (boundl_in_itv, boundr_in_itv). Lemma itvP : forall (x : R) (i : interval R), x \in i -> itv_rewrite i x. Proof. -move=> x [[[] a|] [[] b|]] /itv_dec // [? ?]; - do ?split => //; rewrite ?bound_in_itv /le_boundl /le_boundr //=; - do 1?[apply/negbTE; rewrite (ler_gtF, ltr_geF) //]; - by [ rewrite ltrW - | rewrite (@ler_trans _ x) // 1?ltrW - | rewrite (@ltr_le_trans _ x) - | rewrite (@ler_lt_trans _ x) // 1?ltrW ]. +move=> x [[[] a|] [[] b|]] /itv_dec [ha hb]; do !split; + rewrite ?bound_in_itv //=; do 1?[apply/negbTE; rewrite (le_gtF, lt_geF)]; + by [ | apply: ltW | move: (lersif_trans ha hb) => //=; exact: ltW ]. Qed. Arguments itvP [x i]. @@ -484,7 +482,7 @@ Definition itv_intersectioni1 : right_id `]-oo, +oo[ itv_intersection. Proof. by case=> [[lb lr |] [ub ur |]]. Qed. Lemma itv_intersectionii : idempotent itv_intersection. -Proof. by case=> [[[] lr |] [[] ur |]] //=; rewrite !lerr. Qed. +Proof. by case=> [[[] lr |] [[] ur |]] //=; rewrite !lexx. Qed. Definition subitv (i1 i2 : interval R) := match i1, i2 with @@ -558,7 +556,7 @@ Lemma le_boundl_total : total (@le_boundl R). Proof. by move=> [[] l |] [[] r |] //=; case: (ltrgtP l r). Qed. Lemma le_boundr_total : total (@le_boundr R). -Proof. by move=> [[] l |] [[] r |] //=; case (ltrgtP l r). Qed. +Proof. by move=> [[] l |] [[] r |] //=; case: (ltrgtP l r). Qed. Lemma itv_splitU (xc : R) bc a b : xc \in Interval a b -> forall y, y \in Interval a b = @@ -626,7 +624,7 @@ Variable R : realFieldType. Lemma mid_in_itv : forall ba bb (xa xb : R), xa <= xb ?< if ba || bb -> mid xa xb \in Interval (BOpen_if ba xa) (BOpen_if bb xb). Proof. -by move=> [] [] xa xb /= ?; apply/itv_dec=> /=; rewrite ?midf_lte // ?ltrW. +by move=> [] [] xa xb /= ?; apply/itv_dec=> /=; rewrite ?midf_lte // ?ltW. Qed. Lemma mid_in_itvoo : forall (xa xb : R), xa < xb -> mid xa xb \in `]xa, xb[. |