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left/right lim prop for monotonic fun #1265

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Aug 6, 2024
Merged

left/right lim prop for monotonic fun #1265

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13800f6
left/right lim prop for monotonic fun
affeldt-aist Jul 22, 2024
ff2c2e4
rm redundant lemma
affeldt-aist Aug 5, 2024
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3 changes: 3 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -122,6 +122,9 @@
+ lemmas `antisymmetric_wo_chain`, `antisymmetric_well_order`, `wo_chainW`, `wo_chain_reflexive`,
`wo_chain_antisymmetric`, `Zorn's_lemma`, `Hausdorff_maximal_principle`, `well_ordering_principle`

- in `realfun.v`:
+ lemmas `nondecreasing_at_left_at_right`, `nonincreasing_at_left_at_right`

### Changed

- in `topology.v`:
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55 changes: 55 additions & 0 deletions theories/realfun.v
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Expand Up @@ -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_itvW xrs; exact/ltW.
+ by apply: subset_itvW 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_itvW 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_itvW xrs; exact/ltW.
+ by apply: subset_itvW yrs; exact/ltW.
- near=> x; exists (f r) => _ /= [s srx <-]; rewrite ndf//.
+ by apply: subset_itvW 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.

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