left/right lim prop for monotonic fun

Co-authored-by: IshiguroYoshihiro <[email protected]>

CHANGELOG_UNRELEASED.md

@@ -39,6 +39,12 @@
 - in `realfun.v`:
   + lemma `nondecreasing_at_left_is_cvgr`
 
+- in `set_interval.v`:
+  + lemma `subset_itv_oo_ccW`
+
+- in `realfun.v`:
+  + lemmas `nondecreasing_at_left_at_right`, `nonincreasing_at_left_at_right`
+
 ### Changed
 
 - in `topology.v`:

classical/set_interval.v

@@ -156,6 +156,13 @@ move=> /andP[]; rewrite lt_neqAle => /andP[xz zx ->].
 by rewrite andbT; split => //; exact/nesym/eqP.
 Qed.
 
+Lemma subset_itv_oo_ccW x y u v : (u <= x)%O -> (y <= v)%O ->
+  `]x, y[ `<=` `[u, v].
+Proof.
+move=> ux yv; apply: (subset_trans _ (@subset_itv_oo_cc _ _ _ _)) => z/=.
+by rewrite !in_itv/= => /andP[? zy]; rewrite (le_lt_trans ux) ?(lt_le_trans zy).
+Qed.
+
 End set_itv_porderType.
 Arguments neitv {d T} _.
 

theories/realfun.v

@@ -381,6 +381,61 @@ apply: @nonincreasing_at_right_is_cvgr.
   by rewrite oppr_itvoo.
 Qed.
 
+Let nondecreasing_at_right_is_cvgrW f a b r : a < r -> r < b ->
+  {in `[a, b] &, nondecreasing_fun f} -> cvg (f x @[x --> r^'+]).
+Proof.
+move=> ar rb ndf H; apply: nondecreasing_at_right_is_cvgr.
+- near=> s => x y xrs yrs xy; rewrite ndf//.
+  + by apply: subset_itv_oo_ccW xrs; exact/ltW.
+  + by apply: subset_itv_oo_ccW yrs; exact/ltW.
+- near=> x; exists (f r) => _ /= [s srx <-]; rewrite ndf//.
+  + by apply: subset_itv_oo_cc; rewrite in_itv/= ar.
+  + by apply: subset_itv_oo_ccW srx; exact/ltW.
+  + by move: srx; rewrite in_itv/= => /andP[/ltW].
+Unshelve. all: by end_near. Qed.
+
+Let nondecreasing_at_left_is_cvgrW f a b r : a < r -> r < b ->
+  {in `[a, b] &, nondecreasing_fun f} -> cvg (f x @[x --> r^'-]).
+Proof.
+move=> ar rb ndf H; apply: nondecreasing_at_left_is_cvgr.
+- near=> s => x y xrs yrs xy; rewrite ndf//.
+  + by apply: subset_itv_oo_ccW xrs; exact/ltW.
+  + by apply: subset_itv_oo_ccW yrs; exact/ltW.
+- near=> x; exists (f r) => _ /= [s srx <-]; rewrite ndf//.
+  + by apply: subset_itv_oo_ccW srx; exact/ltW.
+  + by apply: subset_itv_oo_cc; rewrite in_itv/= ar.
+  + by move: srx; rewrite in_itv/= => /andP[_ /ltW].
+Unshelve. all: by end_near. Qed.
+
+Lemma nondecreasing_at_left_at_right f a b :
+  {in `[a, b] &, nondecreasing_fun f} ->
+  {in `]a, b[, forall r, lim (f x @[x --> r^'-]) <= lim (f x @[x --> r^'+])}.
+Proof.
+move=> ndf r; rewrite in_itv/= => /andP[ar rb]; apply: limr_ge.
+  exact: nondecreasing_at_right_is_cvgrW ndf.
+near=> x; apply: limr_le; first exact: nondecreasing_at_left_is_cvgrW ndf.
+near=> y; rewrite ndf// ?in_itv/=.
+- apply/andP; split; first by near: y; apply: nbhs_left_ge.
+  exact: (le_trans _ (ltW rb)).
+- by rewrite (le_trans (ltW ar))/= ltW.
+- exact: (@le_trans _ _ r).
+Unshelve. all: by end_near. Qed.
+
+Lemma nonincreasing_at_left_at_right f a b :
+  {in `[a, b] &, nonincreasing_fun f} ->
+  {in `]a, b[, forall r, lim (f x @[x --> r^'-]) >= lim (f x @[x --> r^'+])}.
+Proof.
+move=> nif; have ndNf : {in `[a, b] &, nondecreasing_fun (-%R \o f)}.
+  by move=> x y xab yab xy /=; rewrite lerNl opprK nif.
+move/nondecreasing_at_left_at_right : (ndNf) => H x.
+rewrite in_itv/= => /andP[ax xb]; rewrite -[leLHS]opprK lerNl -!limN//.
+- by apply: H; rewrite !in_itv/= ax.
+- rewrite -(opprK f); apply: is_cvgN.
+  exact: nondecreasing_at_right_is_cvgrW ndNf.
+- rewrite -(opprK f);apply: is_cvgN.
+  exact: nondecreasing_at_left_is_cvgrW ndNf.
+Qed.
+
 End fun_cvg_realType.
 Arguments nondecreasing_at_right_cvgr {R f a} b.