diff options
Diffstat (limited to 'mathcomp')
| -rw-r--r-- | mathcomp/algebra/ssrnum.v | 78 |
1 files changed, 73 insertions, 5 deletions
diff --git a/mathcomp/algebra/ssrnum.v b/mathcomp/algebra/ssrnum.v index e1e5992..95c40cd 100644 --- a/mathcomp/algebra/ssrnum.v +++ b/mathcomp/algebra/ssrnum.v @@ -2894,7 +2894,19 @@ rewrite real_ltNge ?real_ler_norml // negb_and -?real_ltNge ?realN //. by rewrite orbC ltr_oppr. Qed. -Definition real_lter_normr := (real_ler_normr, real_ltr_normr). +Definition real_lter_normr := (real_ler_normr, real_ltr_normr). + +Lemma real_ltr_normlW x y : x \is real -> `|x| < y -> x < y. +Proof. by move=> ?; case/real_ltr_normlP. Qed. + +Lemma real_ltrNnormlW x y : x \is real -> `|x| < y -> - y < x. +Proof. by move=> ?; case/real_ltr_normlP => //; rewrite ltr_oppl. Qed. + +Lemma real_ler_normlW x y : x \is real -> `|x| <= y -> x <= y. +Proof. by move=> ?; case/real_ler_normlP. Qed. + +Lemma real_lerNnormlW x y : x \is real -> `|x| <= y -> - y <= x. +Proof. by move=> ?; case/real_ler_normlP => //; rewrite ler_oppl. Qed. Lemma real_ler_distl x y e : x - y \is real -> (`|x - y| <= e) = (y - e <= x <= y + e). @@ -2906,6 +2918,30 @@ Proof. by move=> Rxy; rewrite real_lter_norml // !lter_sub_addl. Qed. Definition real_lter_distl := (real_ler_distl, real_ltr_distl). +Lemma real_ltr_distl_addr x y e : x - y \is real -> `|x - y| < e -> x < y + e. +Proof. by move=> ?; rewrite real_ltr_distl // => /andP[]. Qed. + +Lemma real_ler_distl_addr x y e : x - y \is real -> `|x - y| <= e -> x <= y + e. +Proof. by move=> ?; rewrite real_ler_distl // => /andP[]. Qed. + +Lemma real_ltr_distlC_addr x y e : x - y \is real -> `|x - y| < e -> y < x + e. +Proof. by rewrite realBC (distrC x) => ? /real_ltr_distl_addr; apply. Qed. + +Lemma real_ler_distlC_addr x y e : x - y \is real -> `|x - y| <= e -> y <= x + e. +Proof. by rewrite realBC distrC => ? /real_ler_distl_addr; apply. Qed. + +Lemma real_ltr_distl_subl x y e : x - y \is real -> `|x - y| < e -> x - e < y. +Proof. by move/real_ltr_distl_addr; rewrite ltr_sub_addr; apply. Qed. + +Lemma real_ler_distl_subl x y e : x - y \is real -> `|x - y| <= e -> x - e <= y. +Proof. by move/real_ler_distl_addr; rewrite ler_sub_addr; apply. Qed. + +Lemma real_ltr_distlC_subl x y e : x - y \is real -> `|x - y| < e -> y - e < x. +Proof. by rewrite realBC distrC => ? /real_ltr_distl_subl; apply. Qed. + +Lemma real_ler_distlC_subl x y e : x - y \is real -> `|x - y| <= e -> y - e <= x. +Proof. by rewrite realBC distrC => ? /real_ler_distl_subl; apply. Qed. + (* GG: pointless duplication }-( *) Lemma eqr_norm_id x : (`|x| == x) = (0 <= x). Proof. by rewrite ger0_def. Qed. Lemma eqr_normN x : (`|x| == - x) = (x <= 0). Proof. by rewrite ler0_def. Qed. @@ -3738,22 +3774,54 @@ Lemma ltr_normlP x y : reflect ((-x < y) * (x < y)) (`|x| < y). Proof. exact: real_ltr_normlP. Qed. Arguments ltr_normlP {x y}. +Lemma ltr_normlW x y : `|x| < y -> x < y. Proof. exact: real_ltr_normlW. Qed. + +Lemma ltrNnormlW x y : `|x| < y -> - y < x. Proof. exact: real_ltrNnormlW. Qed. + +Lemma ler_normlW x y : `|x| <= y -> x <= y. Proof. exact: real_ler_normlW. Qed. + +Lemma lerNnormlW x y : `|x| <= y -> - y <= x. Proof. exact: real_lerNnormlW. Qed. + Lemma ler_normr x y : (x <= `|y|) = (x <= y) || (x <= - y). -Proof. by rewrite leNgt ltr_norml negb_and -!leNgt orbC ler_oppr. Qed. +Proof. exact: real_ler_normr. Qed. Lemma ltr_normr x y : (x < `|y|) = (x < y) || (x < - y). -Proof. by rewrite ltNge ler_norml negb_and -!ltNge orbC ltr_oppr. Qed. +Proof. exact: real_ltr_normr. Qed. Definition lter_normr := (ler_normr, ltr_normr). Lemma ler_distl x y e : (`|x - y| <= e) = (y - e <= x <= y + e). -Proof. by rewrite lter_norml !lter_sub_addl. Qed. +Proof. exact: real_ler_distl. Qed. Lemma ltr_distl x y e : (`|x - y| < e) = (y - e < x < y + e). -Proof. by rewrite lter_norml !lter_sub_addl. Qed. +Proof. exact: real_ltr_distl. Qed. Definition lter_distl := (ler_distl, ltr_distl). +Lemma ltr_distl_addr x y e : `|x - y| < e -> x < y + e. +Proof. exact: real_ltr_distl_addr. Qed. + +Lemma ler_distl_addr x y e : `|x - y| <= e -> x <= y + e. +Proof. exact: real_ler_distl_addr. Qed. + +Lemma ltr_distlC_addr x y e : `|x - y| < e -> y < x + e. +Proof. exact: real_ltr_distlC_addr. Qed. + +Lemma ler_distlC_addr x y e : `|x - y| <= e -> y <= x + e. +Proof. exact: real_ler_distlC_addr. Qed. + +Lemma ltr_distl_subl x y e : `|x - y| < e -> x - e < y. +Proof. exact: real_ltr_distl_subl. Qed. + +Lemma ler_distl_subl x y e : `|x - y| <= e -> x - e <= y. +Proof. exact: real_ler_distl_subl. Qed. + +Lemma ltr_distlC_subl x y e : `|x - y| < e -> y - e < x. +Proof. exact: real_ltr_distlC_subl. Qed. + +Lemma ler_distlC_subr x y e : `|x - y| <= e -> y - e <= x. +Proof. exact: real_ler_distlC_subl. Qed. + Lemma exprn_even_ge0 n x : ~~ odd n -> 0 <= x ^+ n. Proof. by move=> even_n; rewrite real_exprn_even_ge0 ?num_real. Qed. |