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proof of L'Hopital rule #1371

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proof of L'Hopital rule #1371

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c8df1b0
lhopital + cauchy mvt
ndslusarz Oct 28, 2024
17ae5ef
Merge branch 'math-comp:master' into lhopital
ndslusarz Oct 28, 2024
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headings
ndslusarz Oct 28, 2024
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Merge branch 'lhopital' of https://github.com/ndslusarz/analysis into…
ndslusarz Oct 28, 2024
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updated changelog
ndslusarz Oct 29, 2024
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deduplicating into lemma
ndslusarz Nov 15, 2024
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5 changes: 5 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Expand Up @@ -57,6 +57,11 @@
`locally_compact_completely_regular`, and
`completely_regular_regular`.

- in file `realfun.v`,
+ new lemmas `cauchy_MVT`,
`lhopital_right`, and
`lhopital_left`.

### Changed

- in file `normedtype.v`,
Expand Down
239 changes: 239 additions & 0 deletions theories/realfun.v
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Expand Up @@ -41,6 +41,9 @@ From HB Require Import structures.
(* lime_sup f a/lime_inf f a == limit sup/inferior of the extended real- *)
(* valued function f at point a *)
(* ``` *)
(* cauchy_MVT == Cauchy's mean value theorem *)
(* lhopital_right == L'Hopital rule (limit taken on the right) *)
(* lhopital_left == L'Hopital rule (limit taken on the left) *)
(* *)
(******************************************************************************)

Expand Down Expand Up @@ -2544,3 +2547,239 @@ apply/continuous_within_itvP => //; split.
Qed.

End variation_continuity.

Section Cauchy_MVT.
Context {R : realType}.
Variables (f df g dg : R -> R) (a b c : R).
Hypothesis ab : a < b.
Hypotheses (cf : {within `[a, b], continuous f})
(cg : {within `[a, b], continuous g}).
Hypotheses (fdf : forall x, x \in `]a, b[%R -> is_derive x 1 f (df x))
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Should we extend the predicate derivable_oo_continuous_bnd to include an option for an explicit derivative as an argument (E.G. derivable_oo_continuous_bnd_with f df x y)?

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Gave it a try, but couldn't extend quickly, so perhaps something to extend separately for a different PR?

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Yeah, I have no problem with that. Happy to deal with it later.

(gdg : forall x, x \in `]a, b[%R -> is_derive x 1 g (dg x)).
Hypotheses (dg0 : forall x, x \in `]a, b[%R -> dg x != 0).

Lemma differentiable_subr_neq0 :
g b - g a != 0.
Proof.
have [r] := MVT ab gdg cg; rewrite in_itv/= => /andP[ar rb] dgg.
by rewrite dgg mulf_neq0 ?dg0 ?in_itv/= ?ar//; rewrite subr_eq0 gt_eqF.
Qed.

Lemma cauchy_MVT :
exists2 c, c \in `]a, b[%R & df c / dg c = (f b - f a) / (g b - g a).
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Seems like callers will need to know that g b - g a != 0. We might as well deduplicate that a bit. I would recommend either an auxiliary lemma that for any g, a and b, {in ]a,b[, dg x != 0 -> g b - g a != 0. Or maybe put a g b - g a != 0 as an extra clause in the result of cauchy_MVT

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Done, as a separate lemma instead of inside the proof, but unsure if you perhaps meant outside of the MVT Cauchy section (so without all of its assumptions?)

Also I'm a bit unsure of what the naming convention would be for that one, so happy to rename it to something proper.

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This makes sense to me. I don't have any idea for naming here, so I'd say it's fine for now.

Proof.
have [r] := MVT ab gdg cg; rewrite in_itv/= => /andP[ar rb] dgg.
pose h (x : R) := f x - ((f b - f a) / (g b - g a)) * g x.
have hder x : x \in `]a, b[%R -> derivable h x 1.
move=> xab; apply: derivableB => /=.
exact: (@ex_derive _ _ _ _ _ _ _ (fdf xab)).
by apply: derivableM => //; exact: (@ex_derive _ _ _ _ _ _ _ (gdg xab)).
have ch : {within `[a, b], continuous h}.
rewrite continuous_subspace_in => x xab.
by apply: cvgB; [exact: cf|apply: cvgM; [exact: cvg_cst|exact: cg]].
have /(Rolle ab hder ch)[x xab derh] : h a = h b.
rewrite /h; apply/eqP; rewrite subr_eq eq_sym -addrA eq_sym addrC -subr_eq.
rewrite -mulrN -mulrDr -(addrC (g a)) -[X in _ * X]opprB mulrN -mulrA.
rewrite mulVf//.
by rewrite mulr1 opprB.
by rewrite differentiable_subr_neq0.
pose dh (x : R) := df x - (f b - f a) / (g b - g a) * dg x.
have his_der y : y \in `]a, b[%R -> is_derive x 1 h (dh x).
by move=> yab; apply: is_deriveB; [exact: fdf|apply: is_deriveZ; exact: gdg].
exists x => //.
have := @derive_val _ R _ _ _ _ _ (his_der _ xab).
have -> := @derive_val _ R _ _ _ _ _ derh.
move=> /eqP; rewrite eq_sym subr_eq add0r => /eqP ->.
by rewrite -mulrA divff ?mulr1//; exact: dg0.
Qed.

End Cauchy_MVT.

Section lhopital.
Context {R : realType}.
Variables (f df g dg : R -> R) (a : R) (U : set R) (Ua : nbhs a U).
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As usual, a question about boundary conditions. I'm a bit surprised to see we require f to be derivable in a full neighborhood of a. But then only take the right/left limit. Instead I would expect to see either a^'- U or a^'+U depending on the direction of the limit. Will the theorem still go through with that weakening?

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Good catch - I think this should go through, but changes are a bit more time-consuming than I expected so still in progress, should be done sometime next week.

Hypotheses (fdf : forall x, x \in U -> is_derive x 1 f (df x))
(gdg : forall x, x \in U -> is_derive x 1 g (dg x)).
Hypotheses (fa0 : f a = 0) (ga0 : g a = 0)
(cdg : \forall x \near a^', dg x != 0).

Lemma lhopital_right (l : R) :
df x / dg x @[x --> a^'+] --> l -> f x / g x @[x --> a^'+] --> l.
Proof.
case: cdg => r/= r0 cdg'.
move: Ua; rewrite filter_of_nearI => -[D /= D0 aDU].
near a^'+ => b.
have abf x : x \in `]a, b[%R -> {within `[a, x], continuous f}.
rewrite in_itv/= => /andP[ax xb].
apply: derivable_within_continuous => y; rewrite in_itv/= => /andP[ay yx].
apply: ex_derive.
apply: fdf.
rewrite inE; apply: aDU => /=.
rewrite ler0_norm? subr_le0//.
rewrite opprD opprK addrC ltrBlDr (le_lt_trans yx)// (lt_trans xb)//.
near: b; apply: nbhs_right_lt.
by rewrite ltrDr.
have abg x : x \in `]a, b[%R -> {within `[a, x], continuous g}.
rewrite in_itv/= => /andP[ax xb].
apply: derivable_within_continuous => y; rewrite in_itv/= => /andP[ay yx].
apply: ex_derive.
apply: gdg.
rewrite inE; apply: aDU => /=.
rewrite ler0_norm? subr_le0//.
rewrite opprD opprK addrC ltrBlDr (le_lt_trans yx)// (lt_trans xb)//.
near: b; apply: nbhs_right_lt.
by rewrite ltrDr.
have fdf' y : y \in `]a, b[%R -> is_derive y 1 f (df y).
rewrite in_itv/= => /andP[ay yb]; apply: fdf.
rewrite inE; apply: aDU => /=.
rewrite ltr0_norm ?subr_lt0//.
rewrite opprD opprK addrC ltrBlDr (lt_le_trans yb)//.
near: b; apply: nbhs_right_le.
by rewrite ltrDr.
have gdg' y : y \in `]a, b[%R -> is_derive y 1 g (dg y).
rewrite in_itv/= => /andP[ay yb]; apply: gdg.
rewrite inE; apply: aDU => /=.
rewrite ltr0_norm ?subr_lt0//.
rewrite opprD opprK addrC ltrBlDr (lt_le_trans yb)//.
near: b; apply: nbhs_right_le.
by rewrite ltrDr.
have {}dg0 y : y \in `]a, b[%R -> dg y != 0.
rewrite in_itv/= => /andP[ay yb].
apply: (cdg' y) => /=; last by rewrite gt_eqF.
rewrite ltr0_norm; last by rewrite subr_lt0.
rewrite opprB ltrBlDl (lt_trans yb)//.
near: b; apply: nbhs_right_lt.
by rewrite ltrDl.
move=> fgal.
have L : \forall x \near a^'+,
exists2 c, c \in `]a, x[%R & df c / dg c = f x / g x.
near=> x.
have /andP[ax xb] : a < x < b by exact/andP.
have {}fdf' y : y \in `]a, x[%R -> is_derive y 1 f (df y).
rewrite !in_itv/= => /andP[ay yx].
by apply: fdf'; rewrite in_itv/= ay/= (lt_trans yx).
have {}gdg' y : y \in `]a, x[%R -> is_derive y 1 g (dg y).
rewrite !in_itv/= => /andP[ay yx].
by apply: gdg'; rewrite in_itv/= ay/= (lt_trans yx).
have {}dg0 y : y \in `]a, x[%R -> dg y != 0.
rewrite in_itv/= => /andP[ya yx].
by apply: dg0; rewrite in_itv/= ya/= (lt_trans yx).
have {}axf : {within `[a, x], continuous f}.
rewrite continuous_subspace_in => y; rewrite inE/= in_itv/= => /andP[ay yx].
by apply: abf; rewrite in_itv/= xb andbT.
have {}axg : {within `[a, x], continuous g}.
rewrite continuous_subspace_in => y; rewrite inE/= in_itv/= => /andP[ay yx].
by apply: abg; rewrite in_itv/= xb andbT.
have := @cauchy_MVT _ f df g dg _ _ ax axf axg fdf' gdg' dg0.
by rewrite fa0 ga0 2!subr0.
apply/cvgrPdist_le => /= e e0.
move/cvgrPdist_le : fgal.
move=> /(_ _ e0)[r'/= r'0 fagl].
case: L => d /= d0 L.
near=> t.
have /= := L t.
have atd : `|a - t| < d.
rewrite ltr0_norm; last by rewrite subr_lt0.
rewrite opprB ltrBlDl.
near: t; apply: nbhs_right_lt.
by rewrite ltrDl.
have at_ : a < t by [].
move=> /(_ atd)/(_ at_)[c]; rewrite in_itv/= => /andP[ac ct <-].
apply: fagl => //=.
rewrite ltr0_norm; last by rewrite subr_lt0.
rewrite opprB ltrBlDl (lt_trans ct)//.
near: t; apply: nbhs_right_lt.
by rewrite ltrDl.
Unshelve. all: by end_near. Qed.

Lemma lhopital_left (l : R) :
df x / dg x @[x --> a^'-] --> l -> f x / g x @[x --> a^'-] --> l.
Proof.
case: cdg => r/= r0 cdg'.
move: Ua; rewrite filter_of_nearI => -[D /= D0 aDU].
near a^'- => b.
have baf x : x \in `]b, a[%R -> {within `[x, a], continuous f}.
rewrite in_itv/= => /andP[bx xa].
apply: derivable_within_continuous => y; rewrite in_itv/= => /andP[xy ya].
apply: ex_derive.
apply: fdf.
rewrite inE; apply: aDU => /=.
rewrite ger0_norm ?subr_ge0//.
rewrite ltrBlDr -ltrBlDl (lt_le_trans _ xy)// (le_lt_trans _ bx)//.
near: b; apply: nbhs_left_ge.
by rewrite ltrBlDl ltrDr.
have bag x : x \in `]b, a[%R -> {within `[x, a], continuous g}.
rewrite in_itv/= => /andP[bx xa].
apply: derivable_within_continuous => y; rewrite in_itv/= => /andP[xy ya].
apply: ex_derive.
apply: gdg.
rewrite inE; apply: aDU => /=.
rewrite ger0_norm ?subr_ge0//.
rewrite ltrBlDr -ltrBlDl (lt_le_trans _ xy)// (lt_trans _ bx)//.
near: b; apply: nbhs_left_gt.
by rewrite ltrBlDl ltrDr.
have fdf' y : y \in `]b, a[%R -> is_derive y 1 f (df y).
rewrite in_itv/= => /andP[by_ ya]; apply: fdf.
rewrite inE.
apply: aDU => /=.
rewrite gtr0_norm ?subr_gt0//.
rewrite ltrBlDl -ltrBlDr (le_lt_trans _ by_)//.
near: b; apply: nbhs_left_ge.
by rewrite ltrBlDr ltrDl.
have gdg' y : y \in `]b, a[%R -> is_derive y 1 g (dg y).
rewrite in_itv/= => /andP[by_ ya]; apply: gdg.
rewrite inE; apply: aDU => /=.
rewrite gtr0_norm ?subr_gt0//.
rewrite ltrBlDl -ltrBlDr (le_lt_trans _ by_)//.
near: b; apply: nbhs_left_ge.
by rewrite ltrBlDr ltrDl.
have {}dg0 y : y \in `]b, a[%R -> dg y != 0.
rewrite in_itv/= => /andP[by_ ya].
apply: (cdg' y) => /=; last by rewrite lt_eqF.
rewrite gtr0_norm; last by rewrite subr_gt0.
rewrite ltrBlDr -ltrBlDl (lt_trans _ by_)//.
near: b; apply: nbhs_left_gt.
by rewrite ltrBlDl ltrDr.
move=> fgal.
have L : \forall x \near a^'-,
exists2 c, c \in `]x, a[%R & df c / dg c = f x / g x.
near=> x.
have /andP[bx xa] : b < x < a by exact/andP.
have {}fdf' y : y \in `]x, a[%R -> is_derive y 1 f (df y).
rewrite in_itv/= => /andP[xy ya].
by apply: fdf'; rewrite in_itv/= ya andbT (lt_trans bx).
have {}gdg' y : y \in `]x, a[%R -> is_derive y 1 g (dg y).
rewrite in_itv/= => /andP[xy ya].
by apply: gdg'; rewrite in_itv/= ya andbT (lt_trans _ xy).
have {}dg0 y : y \in `]x, a[%R -> dg y != 0.
rewrite in_itv/= => /andP[xy ya].
by apply: dg0; rewrite in_itv/= ya andbT (lt_trans bx).
have {}xaf : {within `[x, a], continuous f}.
rewrite continuous_subspace_in => y; rewrite inE/= in_itv/= => /andP[xy ya].
by apply: baf; rewrite in_itv/= bx.
have {}xag : {within `[x, a], continuous g}.
rewrite continuous_subspace_in => y; rewrite inE/= in_itv/= => /andP[xy ya].
by apply: bag; rewrite in_itv/= bx.
have := @cauchy_MVT _ f df g dg _ _ xa xaf xag fdf' gdg' dg0.
by rewrite fa0 ga0 !sub0r divrN mulNr opprK.
apply/cvgrPdist_le => /= e e0.
move/cvgrPdist_le : fgal.
move=> /(_ _ e0)[r'/= r'0 fagl].
case: L => d /= d0 L.
near=> t.
have /= := L t.
have atd : `|a - t| < d.
rewrite gtr0_norm; last by rewrite subr_gt0.
rewrite ltrBlDr -ltrBlDl.
near: t; apply: nbhs_left_gt.
by rewrite ltrBlDl ltrDr.
have ta : t < a by [].
move=> /(_ atd)/(_ ta)[c]; rewrite in_itv/= => /andP[tc ca <-].
apply: fagl => //=.
rewrite gtr0_norm; last by rewrite subr_gt0.
rewrite ltrBlDr -ltrBlDl (lt_trans _ tc)//.
near: t; apply: nbhs_left_gt.
by rewrite ltrBlDl ltrDr.
Unshelve. all: by end_near. Qed.

End lhopital.