Merge pull request #514 from affeldt-aist/lemmas_from_analysis_20200521

Lemma addition to ssrnum

3 files changed, 85 insertions, 8 deletions

diff --git a/CHANGELOG_UNRELEASED.md b/CHANGELOG_UNRELEASED.md
index 0693252..eb39d33 100644
--- a/CHANGELOG_UNRELEASED.md
+++ b/CHANGELOG_UNRELEASED.md
@@ -20,6 +20,11 @@ The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/).
 
 ### Added
 
+- in `ssrnum.v`, new lemmas:
+  + `(real_)ltr_normlW`, `(real_)ltrNnormlW`, `(real_)ler_normlW`, `(real_)lerNnormlW`
+  + `(real_)ltr_distl_addr`, `(real_)ler_distl_addr`, `(real_)ltr_distlC_addr`, `(real_)ler_distlC_addr`,
+    `(real_)ltr_distl_subl`, `(real_)ler_distl_subl`, `(real_)ltr_distlC_subl`, `(real_)ler_distlC_subl`
+
 ### Changed
 
 ### Renamed
diff --git a/CONTRIBUTING.md b/CONTRIBUTING.md
index 6313d01..19ab0e5 100644
--- a/CONTRIBUTING.md
+++ b/CONTRIBUTING.md
@@ -124,7 +124,9 @@ Abbreviations are in the header of the file which introduces them. We list here
   - `g` -- a group argument.
   - `I` -- left/right injectivity, as in `addbI : right_injective addb.`
         -- alternatively predicate or set intersection, as in `predI.`
-  - `l` -- the left-hand of an operation, as in `andb_orl : left_distributive andb orb.`
+  - `l` -- the left-hand of an operation, as in
+    + `andb_orl : left_distributive andb orb.`
+    + ``ltr_norml x y : (`|x| < y) = (- y < x < y).``
   - `L` -- the left-hand of a relation, as in `ltn_subrL : n - m < n = (0 < m) && (0 < n).`
   - `LR` -- moving an operator from the left-hand to the right-hand of an relation, as in `leq_subLR : (m - n <= p) = (m <= n + p).`
   - `N` or `n` -- boolean negation, as in `andbN : a && (~~ a) = false.`
@@ -132,8 +134,10 @@ Abbreviations are in the header of the file which introduces them. We list here
   - `N` -- alternatively ring negation, as in `mulNr : (- x) * y = - (x * y).`
   - `P` -- a characteristic property, often a reflection lemma, as in
      `andP : reflect (a /\ b) (a && b)`.
-  - `r` -- a right-hand operation, as `orb_andr : right_distributive orb andb.`
-      -- alternatively, it is a ring argument.
+  - `r` -- a right-hand operation, as in
+    + `orb_andr : right_distributive orb andb.`
+    + ``ler_normr x y : (x <= `|y|) = (x <= y) || (x <= - y).``
+    + alternatively, it is a ring argument.
   - `R` -- the right-hand of a relation, as in `ltn_subrR : n < n - m = false`.
   - `RL` -- moving an operator from the right-hand to the left-hand of an relation, as in `ltn_subRL : (n < p - m) = (m + n < p).`
   - `T` or `t` -- boolean truth, as in `andbT: right_id true andb.`
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.