Skip to content

Commit d2fecce

Browse files
affeldt-aistaffeldt-aist
committed
mv lemmas to derive
1 parent 7e816dc commit d2fecce

File tree

3 files changed

+236
-245
lines changed

3 files changed

+236
-245
lines changed

‎CHANGELOG_UNRELEASED.md‎

Lines changed: 3 additions & 4 deletions
Original file line numberDiff line numberDiff line change
@@ -30,10 +30,9 @@
3030
- in `classical_sets.v`:
3131
+ lemmas `xsectionE`, `ysectionE`
3232

33-
- in file `realfun.v`,
34-
+ new lemmas `cauchy_MVT`,
35-
`lhopital_right`, and
36-
`lhopital_left`.
33+
- in `derive.v`:
34+
+ lemmas `differentiable_subr_neq0`, `cauchy_MVT`,
35+
`lhopital_right`, `lhopital_left`
3736

3837
### Changed
3938

‎theories/derive.v‎

Lines changed: 233 additions & 1 deletion
Original file line numberDiff line numberDiff line change
@@ -11,7 +11,8 @@ From mathcomp Require Import normedtype landau forms poly.
1111
(* This file provides a theory of differentiation. It includes the standard *)
1212
(* rules of differentiation (differential of a sum, of a product, of *)
1313
(* exponentiation, of the inverse, etc.) as well as standard theorems (the *)
14-
(* Extreme Value Theorem, Rolle's theorem, the Mean Value Theorem). *)
14+
(* Extreme Value Theorem, Rolle's theorem, the Mean Value Theorem, Cauchy's *)
15+
(* mean value theorem, L'Hopital's rule). *)
1516
(* *)
1617
(* Parsable notations (in all of the following, f is not supposed to be *)
1718
(* differentiable): *)
@@ -1602,6 +1603,237 @@ have [c cab D] := MVT altb fdrvbl fcont.
16021603
by exists c => //; rewrite in_itv /= ltW (itvP cab).
16031604
Qed.
16041605

1606+
Section Cauchy_MVT.
1607+
Context {R : realType}.
1608+
Variables (f df g dg : R -> R) (a b c : R).
1609+
Hypothesis ab : a < b.
1610+
Hypotheses (cf : {within `[a, b], continuous f})
1611+
(cg : {within `[a, b], continuous g}).
1612+
Hypotheses (fdf : forall x, x \in `]a, b[%R -> is_derive x 1 f (df x))
1613+
(gdg : forall x, x \in `]a, b[%R -> is_derive x 1 g (dg x)).
1614+
Hypotheses (dg0 : forall x, x \in `]a, b[%R -> dg x != 0).
1615+
1616+
Lemma differentiable_subr_neq0 : g b - g a != 0.
1617+
Proof.
1618+
have [r] := MVT ab gdg cg; rewrite in_itv/= => /andP[ar rb] dgg.
1619+
by rewrite dgg mulf_neq0 ?dg0 ?in_itv/= ?ar//; rewrite subr_eq0 gt_eqF.
1620+
Qed.
1621+
1622+
Lemma cauchy_MVT :
1623+
exists2 c, c \in `]a, b[%R & df c / dg c = (f b - f a) / (g b - g a).
1624+
Proof.
1625+
pose h x := f x - ((f b - f a) / (g b - g a)) * g x.
1626+
have hder x : x \in `]a, b[%R -> derivable h x 1.
1627+
move=> xab; apply: derivableB => /=.
1628+
exact: (@ex_derive _ _ _ _ _ _ _ (fdf xab)).
1629+
by apply: derivableM => //; exact: (@ex_derive _ _ _ _ _ _ _ (gdg xab)).
1630+
have ch : {within `[a, b], continuous h}.
1631+
rewrite continuous_subspace_in => x xab.
1632+
by apply: cvgB; [exact: cf|apply: cvgM; [exact: cvg_cst|exact: cg]].
1633+
have /(Rolle ab hder ch)[x xab derh] : h a = h b.
1634+
rewrite /h; apply/eqP; rewrite subr_eq eq_sym -addrA eq_sym addrC -subr_eq.
1635+
rewrite -mulrN -mulrDr -(addrC (g a)) -[X in _ * X]opprB mulrN -mulrA.
1636+
by rewrite mulVf ?differentiable_subr_neq0// mulr1 opprB.
1637+
pose dh x := df x - (f b - f a) / (g b - g a) * dg x.
1638+
have his_der y : y \in `]a, b[%R -> is_derive x 1 h (dh x).
1639+
by move=> yab; apply: is_deriveB; [exact: fdf|apply: is_deriveZ; exact: gdg].
1640+
exists x => //.
1641+
have := @derive_val _ R _ _ _ _ _ (his_der _ xab).
1642+
rewrite (@derive_val _ R _ _ _ _ _ derh) => /esym/eqP.
1643+
by rewrite subr_eq add0r => /eqP ->; rewrite -mulrA divff ?mulr1//; exact: dg0.
1644+
Qed.
1645+
1646+
End Cauchy_MVT.
1647+
1648+
Section lhopital.
1649+
Context {R : realType}.
1650+
Variables (f df g dg : R -> R) (a : R) (U : set R) (Ua : nbhs a U).
1651+
Hypotheses (fdf : forall x, x \in U -> is_derive x 1 f (df x))
1652+
(gdg : forall x, x \in U -> is_derive x 1 g (dg x)).
1653+
Hypotheses (fa0 : f a = 0) (ga0 : g a = 0)
1654+
(cdg : \forall x \near a^', dg x != 0).
1655+
1656+
Lemma lhopital_right (l : R) :
1657+
df x / dg x @[x --> a^'+] --> l -> f x / g x @[x --> a^'+] --> l.
1658+
Proof.
1659+
case: cdg => r/= r0 cdg'.
1660+
move: Ua; rewrite filter_of_nearI => -[D /= D0 aDU].
1661+
near a^'+ => b.
1662+
have abf x : x \in `]a, b[%R -> {within `[a, x], continuous f}.
1663+
rewrite in_itv/= => /andP[ax xb].
1664+
apply: derivable_within_continuous => y; rewrite in_itv/= => /andP[ay yx].
1665+
apply: ex_derive.
1666+
apply: fdf.
1667+
rewrite inE; apply: aDU => /=.
1668+
rewrite ler0_norm? subr_le0//.
1669+
rewrite opprD opprK addrC ltrBlDr (le_lt_trans yx)// (lt_trans xb)//.
1670+
near: b; apply: nbhs_right_lt.
1671+
by rewrite ltrDr.
1672+
have abg x : x \in `]a, b[%R -> {within `[a, x], continuous g}.
1673+
rewrite in_itv/= => /andP[ax xb].
1674+
apply: derivable_within_continuous => y; rewrite in_itv/= => /andP[ay yx].
1675+
apply: ex_derive.
1676+
apply: gdg.
1677+
rewrite inE; apply: aDU => /=.
1678+
rewrite ler0_norm? subr_le0//.
1679+
rewrite opprD opprK addrC ltrBlDr (le_lt_trans yx)// (lt_trans xb)//.
1680+
near: b; apply: nbhs_right_lt.
1681+
by rewrite ltrDr.
1682+
have fdf' y : y \in `]a, b[%R -> is_derive y 1 f (df y).
1683+
rewrite in_itv/= => /andP[ay yb]; apply: fdf.
1684+
rewrite inE; apply: aDU => /=.
1685+
rewrite ltr0_norm ?subr_lt0//.
1686+
rewrite opprD opprK addrC ltrBlDr (lt_le_trans yb)//.
1687+
near: b; apply: nbhs_right_le.
1688+
by rewrite ltrDr.
1689+
have gdg' y : y \in `]a, b[%R -> is_derive y 1 g (dg y).
1690+
rewrite in_itv/= => /andP[ay yb]; apply: gdg.
1691+
rewrite inE; apply: aDU => /=.
1692+
rewrite ltr0_norm ?subr_lt0//.
1693+
rewrite opprD opprK addrC ltrBlDr (lt_le_trans yb)//.
1694+
near: b; apply: nbhs_right_le.
1695+
by rewrite ltrDr.
1696+
have {}dg0 y : y \in `]a, b[%R -> dg y != 0.
1697+
rewrite in_itv/= => /andP[ay yb].
1698+
apply: (cdg' y) => /=; last by rewrite gt_eqF.
1699+
rewrite ltr0_norm; last by rewrite subr_lt0.
1700+
rewrite opprB ltrBlDl (lt_trans yb)//.
1701+
near: b; apply: nbhs_right_lt.
1702+
by rewrite ltrDl.
1703+
move=> fgal.
1704+
have L : \forall x \near a^'+,
1705+
exists2 c, c \in `]a, x[%R & df c / dg c = f x / g x.
1706+
near=> x.
1707+
have /andP[ax xb] : a < x < b by exact/andP.
1708+
have {}fdf' y : y \in `]a, x[%R -> is_derive y 1 f (df y).
1709+
rewrite !in_itv/= => /andP[ay yx].
1710+
by apply: fdf'; rewrite in_itv/= ay/= (lt_trans yx).
1711+
have {}gdg' y : y \in `]a, x[%R -> is_derive y 1 g (dg y).
1712+
rewrite !in_itv/= => /andP[ay yx].
1713+
by apply: gdg'; rewrite in_itv/= ay/= (lt_trans yx).
1714+
have {}dg0 y : y \in `]a, x[%R -> dg y != 0.
1715+
rewrite in_itv/= => /andP[ya yx].
1716+
by apply: dg0; rewrite in_itv/= ya/= (lt_trans yx).
1717+
have {}axf : {within `[a, x], continuous f}.
1718+
rewrite continuous_subspace_in => y; rewrite inE/= in_itv/= => /andP[ay yx].
1719+
by apply: abf; rewrite in_itv/= xb andbT.
1720+
have {}axg : {within `[a, x], continuous g}.
1721+
rewrite continuous_subspace_in => y; rewrite inE/= in_itv/= => /andP[ay yx].
1722+
by apply: abg; rewrite in_itv/= xb andbT.
1723+
have := @cauchy_MVT _ f df g dg _ _ ax axf axg fdf' gdg' dg0.
1724+
by rewrite fa0 ga0 2!subr0.
1725+
apply/cvgrPdist_le => /= e e0.
1726+
move/cvgrPdist_le : fgal.
1727+
move=> /(_ _ e0)[r'/= r'0 fagl].
1728+
case: L => d /= d0 L.
1729+
near=> t.
1730+
have /= := L t.
1731+
have atd : `|a - t| < d.
1732+
rewrite ltr0_norm; last by rewrite subr_lt0.
1733+
rewrite opprB ltrBlDl.
1734+
near: t; apply: nbhs_right_lt.
1735+
by rewrite ltrDl.
1736+
have at_ : a < t by [].
1737+
move=> /(_ atd)/(_ at_)[c]; rewrite in_itv/= => /andP[ac ct <-].
1738+
apply: fagl => //=.
1739+
rewrite ltr0_norm; last by rewrite subr_lt0.
1740+
rewrite opprB ltrBlDl (lt_trans ct)//.
1741+
near: t; apply: nbhs_right_lt.
1742+
by rewrite ltrDl.
1743+
Unshelve. all: by end_near. Qed.
1744+
1745+
Lemma lhopital_left (l : R) :
1746+
df x / dg x @[x --> a^'-] --> l -> f x / g x @[x --> a^'-] --> l.
1747+
Proof.
1748+
case: cdg => r/= r0 cdg'.
1749+
move: Ua; rewrite filter_of_nearI => -[D /= D0 aDU].
1750+
near a^'- => b.
1751+
have baf x : x \in `]b, a[%R -> {within `[x, a], continuous f}.
1752+
rewrite in_itv/= => /andP[bx xa].
1753+
apply: derivable_within_continuous => y; rewrite in_itv/= => /andP[xy ya].
1754+
apply: ex_derive.
1755+
apply: fdf.
1756+
rewrite inE; apply: aDU => /=.
1757+
rewrite ger0_norm ?subr_ge0//.
1758+
rewrite ltrBlDr -ltrBlDl (lt_le_trans _ xy)// (le_lt_trans _ bx)//.
1759+
near: b; apply: nbhs_left_ge.
1760+
by rewrite ltrBlDl ltrDr.
1761+
have bag x : x \in `]b, a[%R -> {within `[x, a], continuous g}.
1762+
rewrite in_itv/= => /andP[bx xa].
1763+
apply: derivable_within_continuous => y; rewrite in_itv/= => /andP[xy ya].
1764+
apply: ex_derive.
1765+
apply: gdg.
1766+
rewrite inE; apply: aDU => /=.
1767+
rewrite ger0_norm ?subr_ge0//.
1768+
rewrite ltrBlDr -ltrBlDl (lt_le_trans _ xy)// (lt_trans _ bx)//.
1769+
near: b; apply: nbhs_left_gt.
1770+
by rewrite ltrBlDl ltrDr.
1771+
have fdf' y : y \in `]b, a[%R -> is_derive y 1 f (df y).
1772+
rewrite in_itv/= => /andP[by_ ya]; apply: fdf.
1773+
rewrite inE.
1774+
apply: aDU => /=.
1775+
rewrite gtr0_norm ?subr_gt0//.
1776+
rewrite ltrBlDl -ltrBlDr (le_lt_trans _ by_)//.
1777+
near: b; apply: nbhs_left_ge.
1778+
by rewrite ltrBlDr ltrDl.
1779+
have gdg' y : y \in `]b, a[%R -> is_derive y 1 g (dg y).
1780+
rewrite in_itv/= => /andP[by_ ya]; apply: gdg.
1781+
rewrite inE; apply: aDU => /=.
1782+
rewrite gtr0_norm ?subr_gt0//.
1783+
rewrite ltrBlDl -ltrBlDr (le_lt_trans _ by_)//.
1784+
near: b; apply: nbhs_left_ge.
1785+
by rewrite ltrBlDr ltrDl.
1786+
have {}dg0 y : y \in `]b, a[%R -> dg y != 0.
1787+
rewrite in_itv/= => /andP[by_ ya].
1788+
apply: (cdg' y) => /=; last by rewrite lt_eqF.
1789+
rewrite gtr0_norm; last by rewrite subr_gt0.
1790+
rewrite ltrBlDr -ltrBlDl (lt_trans _ by_)//.
1791+
near: b; apply: nbhs_left_gt.
1792+
by rewrite ltrBlDl ltrDr.
1793+
move=> fgal.
1794+
have L : \forall x \near a^'-,
1795+
exists2 c, c \in `]x, a[%R & df c / dg c = f x / g x.
1796+
near=> x.
1797+
have /andP[bx xa] : b < x < a by exact/andP.
1798+
have {}fdf' y : y \in `]x, a[%R -> is_derive y 1 f (df y).
1799+
rewrite in_itv/= => /andP[xy ya].
1800+
by apply: fdf'; rewrite in_itv/= ya andbT (lt_trans bx).
1801+
have {}gdg' y : y \in `]x, a[%R -> is_derive y 1 g (dg y).
1802+
rewrite in_itv/= => /andP[xy ya].
1803+
by apply: gdg'; rewrite in_itv/= ya andbT (lt_trans _ xy).
1804+
have {}dg0 y : y \in `]x, a[%R -> dg y != 0.
1805+
rewrite in_itv/= => /andP[xy ya].
1806+
by apply: dg0; rewrite in_itv/= ya andbT (lt_trans bx).
1807+
have {}xaf : {within `[x, a], continuous f}.
1808+
rewrite continuous_subspace_in => y; rewrite inE/= in_itv/= => /andP[xy ya].
1809+
by apply: baf; rewrite in_itv/= bx.
1810+
have {}xag : {within `[x, a], continuous g}.
1811+
rewrite continuous_subspace_in => y; rewrite inE/= in_itv/= => /andP[xy ya].
1812+
by apply: bag; rewrite in_itv/= bx.
1813+
have := @cauchy_MVT _ f df g dg _ _ xa xaf xag fdf' gdg' dg0.
1814+
by rewrite fa0 ga0 !sub0r divrN mulNr opprK.
1815+
apply/cvgrPdist_le => /= e e0.
1816+
move/cvgrPdist_le : fgal.
1817+
move=> /(_ _ e0)[r'/= r'0 fagl].
1818+
case: L => d /= d0 L.
1819+
near=> t.
1820+
have /= := L t.
1821+
have atd : `|a - t| < d.
1822+
rewrite gtr0_norm; last by rewrite subr_gt0.
1823+
rewrite ltrBlDr -ltrBlDl.
1824+
near: t; apply: nbhs_left_gt.
1825+
by rewrite ltrBlDl ltrDr.
1826+
have ta : t < a by [].
1827+
move=> /(_ atd)/(_ ta)[c]; rewrite in_itv/= => /andP[tc ca <-].
1828+
apply: fagl => //=.
1829+
rewrite gtr0_norm; last by rewrite subr_gt0.
1830+
rewrite ltrBlDr -ltrBlDl (lt_trans _ tc)//.
1831+
near: t; apply: nbhs_left_gt.
1832+
by rewrite ltrBlDl ltrDr.
1833+
Unshelve. all: by end_near. Qed.
1834+
1835+
End lhopital.
1836+
16051837
Lemma ler0_derive1_nincr (R : realType) (f : R -> R) (a b : R) :
16061838
(forall x, x \in `]a, b[%R -> derivable f x 1) ->
16071839
(forall x, x \in `]a, b[%R -> f^`() x <= 0) ->

0 commit comments

Comments
 (0)