Module mathcomp.analysis.exp

From mathcomp Require Import boot order ssralg ssrint ssrnum matrix.
From mathcomp Require Import interval rat interval_inference.
#[warning="-warn-library-file-internal-analysis"]
From mathcomp Require Import unstable.
From mathcomp Require Import boolp classical_sets functions mathcomp_extra.
From mathcomp Require Import reals topology ereal tvs normedtype landau.
From mathcomp Require Import sequences derive realfun convex.

# Theory of exponential/logarithm functions This file defines exponential and logarithm functions and develops their theory. ## Differentiability of series (Section PseriesDiff) This formalization is inspired by HOL-Light (transc.ml). This part is temporary: it should be subsumed by a proper theory of power series. ``` pseries f x == [series f n * x ^ n]_n pseries_diffs f i == (i + 1) * f (i + 1) expeR x == extended real number-valued exponential function ln x == the natural logarithm, in ring_scope lne x == the natural logarithm, in ereal_scope s `^ r == power function, in ring_scope (assumes s >= 0) e `^ r == power function, in ereal_scope (assumes e >= 0) riemannR a == sequence n |-> 1 / (n.+1) `^ a where a has a type of type realType e `^?(r +? s) == validity condition for the distributivity of the power of the addition, in ereal_scope ```

Unset SsrOldRewriteGoalsOrder.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import Order.TTheory GRing.Theory Num.Theory.
Import numFieldNormedType.Exports.

Local Open Scope classical_set_scope.
Local Open Scope ring_scope.

Reserved Notation "x '`^?' ( r +? s )"
(format "x '`^?' ( r +? s )", r at next level, at level 11) .

Lemma ( : numDomainType) ( : R) : `|x| \is Num.nneg.

Proof.

by rewrite qualifE/=. Qed.

#[global] Hint Extern 0 (is_true (@Num.norm _ _ _ \is Num.nneg)) =>
  solve [apply: normr_nneg] : core.

Section PseriesDiff.

Variable : realType.

Definition ( : R) := [series f i * x ^+ i]_.

Fact ( : R) :
    cvgn (pseries f x) -> `|z| < `|x| ->
  cvgn (pseries (fun => `|f i|) z).

Proof.

move=> Cx zLx; have [K [Kreal Kf]] := cvg_series_bounded Cx.
have Kzxn n : 0 <= `|K + 1| * `|z ^+ n| / `|x ^+ n| by rewrite !mulr_ge0.
apply: normed_cvg.
apply: series_le_cvg Kzxn _ _ => [//=| /= n|].
  rewrite (_ : `|_ * _| = `|f n * x ^+ n| * `|z ^+ n| / `|x ^+ n|).
    rewrite !normrM normr_id mulrAC mulfK // normr_eq0 expf_eq0 andbC.
    by case: ltrgt0P zLx; rewrite //= normr_lt0.
  do! (apply: ler_pM || apply: mulr_ge0 || rewrite invr_ge0) => //.
  by apply Kf => //; rewrite (lt_le_trans _ (ler_norm _))// ltrDl.
have F : `|z / x| < 1.
  by rewrite normrM normfV ltr_pdivrMr ?mul1r // (le_lt_trans _ zLx).
rewrite (_ : (fun _ => _) = geometric `|K + 1| `|z / x|).
  by apply/funext => i /=; rewrite normrM exprMn mulrA normfV !normrX exprVn.
by apply: is_cvg_geometric_series; rewrite normr_id.
Qed.

Fact ( : R) :
cvgn (pseries f x) -> `|z| < `|x| -> cvgn (pseries f z).

Proof.

move=> Cx zLx.
apply: normed_cvg; rewrite /normed_series_of /=.
rewrite (_ : (fun _ => _) = (fun => `|f i| * `|z| ^+ i)).
by apply/funext => i; rewrite normrM normrX.
by apply: is_cvg_pseries_inside_norm Cx _; rewrite normr_id.
Qed.

Definition ( : nat -> R) := i.+1%:R * f i.+1.

Lemma ( : nat -> R) : pseries_diffs (- f) = -(pseries_diffs f).

Proof.

by apply/funext => i; rewrite /pseries_diffs /= -mulrN. Qed.

Lemma :
pseries_diffs (fun => (n`!%:R)^-1) = (fun => (n`!%:R)^-1 : R).

Proof.

by apply/funext => i; rewrite /pseries_diffs factS natrM invfM mulVKf.
Qed.

Lemma :
\sum_(0 <= < n) pseries_diffs f i * x ^+ i =
(\sum_(0 <= < n) i%:R * f i * x ^+ i.-1) + n%:R * f n * x ^+ n.-1.

Proof.

case: n => [|n]; first by rewrite !big_nil !mul0r add0r.
under eq_bigr do unfold pseries_diffs.
by rewrite big_nat_recr //= big_nat_recl //= !mul0r add0r.
Qed.

Lemma :
  let := i%:R * f i * x ^+ i.-1 in
  cvgn (pseries (pseries_diffs f) x) ->
  series s @ \oo --> limn (pseries (pseries_diffs f) x).

Proof.

move=> s Cx; rewrite -[lim _]subr0.
rewrite /pseries/= [X in X @ \oo --> _]/series /=.
rewrite [X in X @ \oo --> _](_ : _ = (fun => \sum_(0 <= < n)
pseries_diffs f i * x ^+ i - n%:R * f n * x ^+ n.-1)).
by rewrite funeqE => n; rewrite pseries_diffs_sumE addrK.
by apply: cvgB => //; rewrite -cvg_shiftS; exact: cvg_series_cvg_0.
Qed.

Lemma :
cvgn (pseries (pseries_diffs f) x) ->
cvgn ([series i%:R * f i * x ^+ i.-1]_).

Proof.

by by move=> Cx; have := pseries_diffs_equiv Cx; move/(cvg_lim _) => -> //.
Qed.

Let ( : R) :
\sum_(0 <= < m) ((h + z) ^+ (m - i) * z ^+ i - z ^+ m) =
\sum_(0 <= < m) z ^+ i * ((h + z) ^+ (m - i) - z ^+ (m - i)).

Proof.

rewrite !big_mkord; apply: eq_bigr => i _.
by rewrite mulrDr mulrN -exprD mulrC addnC subnK // ltnW.
Qed.

Let ( : R) :
  h != 0 ->
  ((h + z) ^+ n - (z ^+ n)) / h - n%:R * z ^+ n.-1 =
  h * \sum_(0 <= < n.-1) z ^+ i *
      \sum_(0 <= < n.-1 - i) (h + z) ^+ j * z ^+ (n.-2 - i - j).

Proof.

move=> hNZ; apply: (mulfI hNZ).
rewrite mulrBr mulrC divfK //.
case: n => [|n]; first by rewrite !expr0 !(mul0r, mulr0, subrr, big_geq).
rewrite subrXX addrK -mulrBr; congr (_ * _).
rewrite -(big_mkord xpredT (fun => (h + z) ^+ (n - i) * z ^+ i)).
rewrite big_nat_recr //= subnn expr0 -addrA -mulrBl -nat1r opprD addNKr mulNr.
rewrite mulr_natl -[in X in _ *+ X](subn0 n) -sumr_const_nat -sumrB.
rewrite pseries_diffs_P1 mulr_sumr !big_mkord; apply: eq_bigr => i _.
rewrite mulrCA; congr (_ * _).
rewrite subrXX addrK big_nat_rev /= big_mkord; congr (_ * _).
by apply: eq_bigr => k _; rewrite -!predn_sub subKn // -subnS.
Qed.

Let ( : R) :
  h != 0 -> `|z| <= K -> `|h + z| <= K ->
    `|((h +z) ^+ n - z ^+ n) / h - n%:R * z ^+ n.-1|
      <= n%:R * n.-1%:R * K ^+ n.-2 * `|h|.

Proof.

move=> hNZ zLK zhLk.
rewrite pseries_diffs_P2// normrM mulrC.
rewrite ler_pM2r ?normr_gt0//.
rewrite (le_trans (ler_norm_sum _ _ _))//.
rewrite -mulrA mulrC -mulrA mulr_natl -[X in _ *+ X]subn0 -sumr_const_nat.
apply ler_sum_nat => i /=.
case: n => //= n ni.
rewrite normrM.
pose d := (n.-1 - i)%N.
rewrite -[(n - i)%N]prednK ?subn_gt0// predn_sub -/d.
rewrite -(subnK (_ : i <= n.-1)%N) -/d.
  by rewrite -ltnS prednK// (leq_ltn_trans _ ni).
rewrite addnC exprD mulrAC -mulrA.
apply: ler_pM => //.
  by rewrite normrX lerXn2r// qualifE/= (le_trans _ zLK).
apply: le_trans (_ : d.+1%:R * K ^+ d <= _); last first.
  rewrite ler_wpM2r //; first by rewrite exprn_ge0 // (le_trans _ zLK).
  by rewrite ler_nat ltnS /d -subn1 -subnDA leq_subr.
rewrite (le_trans (ler_norm_sum _ _ _))//.
rewrite mulr_natl -[X in _ *+ X]subn0 -sumr_const_nat ler_sum_nat//= => j jd1.
rewrite -[in leRHS](subnK (_ : j <= d)%N) -1?ltnS // addnC exprD normrM.
by rewrite ler_pM// normrX lerXn2r// qualifE/= (le_trans _ zLK).
Qed.

Lemma ( : R^nat) :
  cvgn (pseries c K) ->
  cvgn (pseries (pseries_diffs c) K) ->
  cvgn (pseries (pseries_diffs (pseries_diffs c)) K) ->
  `|x| < `|K| ->
  is_derive x (1 : R)
    (fun => limn (pseries c x))
    (limn (pseries (pseries_diffs c) x)).

Proof.

move=> Ck CdK CddK xLK; rewrite /pseries.
set s := (fun : nat => _); set (f := fun => _).
suff hfxs : h^-1 *: (f (h + x) - f x) @[ --> 0^'] --> limn (series s).
  have F : f^`() x = limn (series s) by apply: cvg_lim hfxs.
  have Df : derivable f x 1.
    move: hfxs; rewrite /derivable [X in X @ 0^'](_ : _ =
        (fun => h^-1 *: (f (h%:A + x) - f x))) /=.
      by apply/funext => i //=; rewrite [i%:A]mulr1.
    by move=> /(cvg_lim _) -> //.
  by constructor; [exact: Df|rewrite -derive1E].
pose sx := fun : nat => c n * x ^+ n.
have Csx : cvgn (pseries c x) by apply: is_cvg_pseries_inside Ck _.
pose shx := fun ( : nat) => c n * (h + x) ^+ n.
suff Cc : limn (h^-1 *: (series (shx h - sx))) @[ --> 0^'] --> limn (series s).
  apply: cvg_sub0 Cc.
  apply/cvgrPdist_lt => eps eps_gt0 /=.
  near=> h; rewrite sub0r normrN /=.
  rewrite (le_lt_trans _ eps_gt0)//.
  rewrite normr_le0 subr_eq0 -/sx -/(shx _); apply/eqP.
  suff Cshx' : cvgn (series (shx h)).
    rewrite limZl_tmp; first exact/is_cvg_seriesB/Csx.
    by rewrite lim_seriesB; [|exact: Csx|].
  apply: is_cvg_pseries_inside Ck _.
  rewrite (le_lt_trans (ler_normD _ _))// -(subrK `|x| `|K|) ltrD2r.
  near: h.
  apply/nbhs_ballP => /=; exists ((`|K| - `|x|) /2%:R) => /=.
    by rewrite divr_gt0 // subr_gt0.
  move=> t; rewrite /ball /= sub0r normrN => H tNZ.
  rewrite (lt_le_trans H)// ler_pdivrMr // mulr2n mulrDr mulr1.
  by rewrite ler_wpDr // subr_ge0 ltW.
apply: cvg_zero => /=.
suff Cc : limn
    (series (fun => c n * (((h + x) ^+ n - x ^+ n) / h - n%:R * x ^+ n.-1)))
    @[ --> 0^'] --> 0.
  apply: cvg_sub0 Cc.
  apply/cvgrPdist_lt => eps eps_gt0 /=.
  near=> h; rewrite sub0r normrN /=.
  rewrite (le_lt_trans _ eps_gt0)// normr_le0 subr_eq0; apply/eqP.
  have Cs : cvgn (series s) by apply: is_cvg_pseries_inside CdK _.
  have Cs1 := is_cvg_pseries_diffs_equiv Cs.
  have Fs1 := pseries_diffs_equiv Cs.
  set s1 := (fun => _) in Cs1.
  have Cshx : cvgn (series (shx h)).
    apply: is_cvg_pseries_inside Ck _.
    rewrite (le_lt_trans (ler_normD _ _))// -(subrK `|x| `|K|) ltrD2r.
    near: h.
    apply/nbhs_ballP => /=; exists ((`|K| - `|x|) / 2%:R) => /=.
      by rewrite divr_gt0 // subr_gt0.
    move=> t; rewrite /ball /= sub0r normrN => H tNZ.
    rewrite (lt_le_trans H)// ler_pdivrMr // mulr2n mulrDr mulr1.
    by rewrite ler_wpDr // subr_ge0 ltW.
  have C1 := is_cvg_seriesB Cshx Csx.
  have Ckf := @is_cvg_seriesZ _ _ h^-1 C1.
  have Cu : (series (h^-1 *: (shx h - sx)) - series s1) x0 @[ --> \oo] -->
      limn (series (h^-1 *: (shx h - sx))) - limn (series s).
    exact: cvgB Ckf Fs1.
  set w := (fun : nat => _ in RHS).
  have -> : w = h^-1 *: (shx h - sx) - s1.
    apply: funext => i; rewrite !fctE.
    rewrite /w /shx /sx /s1 /= mulrBr; congr (_ - _); last first.
      by rewrite mulrCA !mulrA.
    by rewrite -mulrBr [RHS]mulrCA [_^-1 * _]mulrC.
  rewrite [X in X h = _]/+%R /= [X in _ + X h = _]/-%R /=.
  have -> : series (h^-1 *: (shx h - sx) - s1) =
           series (h^-1 *: (shx h - sx)) - (series s1).
    by apply/funext => i; rewrite /series /= sumrB.
  have -> : h^-1 *: series (shx h - sx) = series (h^-1 *: (shx h - sx)).
    by apply/funext => i; rewrite /series /= -scaler_sumr.
  exact/esym/cvg_lim.
pose r := (`|x| + `|K|) / 2.
have xLr : `|x| < r by rewrite ltr_pdivlMr // mulrDr mulr1 ltrD2l.
have rLx : r < `|K| by rewrite ltr_pdivrMr // mulrDr mulr1 ltrD2r.
have r_gt0 : 0 < r by apply: le_lt_trans xLr.
have rNZ : r != 0by case: ltrgt0P r_gt0.
apply: (@lim_cvg_to_0_linear _
  (fun => `|c n| * n%:R * (n.-1)%:R * r ^+ n.-2)
  (fun => c n * (((h + x) ^+ n - x ^+ n) / h - n%:R * x ^+ n.-1))
  (r - `|x|)); first by rewrite subr_gt0.
- have : cvgn ([series `|pseries_diffs (pseries_diffs c) n| * r ^+ n]_).
    apply: is_cvg_pseries_inside_norm CddK _.
    by rewrite ger0_norm // ltW // (le_lt_trans _ xLr).
  have -> : (fun => `|pseries_diffs (pseries_diffs c) n| * r ^+ n) =
            (fun => pseries_diffs (pseries_diffs
                                      (fun => `|c m|)) n * r ^+ n).
    apply/funext => i.
    by rewrite /pseries_diffs !normrM !mulrA ger0_norm // ger0_norm.
  move=> /is_cvg_pseries_diffs_equiv.
  rewrite /pseries_diffs.
  have -> : (fun => n%:R * ((n.+1)%:R * `|c n.+1|) * r ^+ n.-1) =
           (fun => pseries_diffs
             (fun => (m.-1)%:R * `|c m| * r^-1) n * r ^+ n).
    apply/funext => n.
    rewrite /pseries_diffs /= mulrA.
    case: n => [|n /=]; first by rewrite !(mul0r, mulr0).
    by rewrite [_%:R *_]mulrC !mulrA -[RHS]mulrA exprS mulKf.
  move/is_cvg_pseries_diffs_equiv.
  have ->// : (fun => n%:R * (n.-1%:R * `|c n| / r) * r ^+ n.-1) =
              (fun => `|c n| * n%:R * n.-1%:R * r ^+ n.-2).
  apply/funext => [] [|[|i]]; rewrite ?(mul0r, mulr0) //=.
  rewrite mulrA -mulrA exprS mulKf// !mulrA; congr (_ * _).
  by rewrite mulrC mulrA.
- move=> h /andP[h_gt0 hLrBx] n.
  rewrite normrM -!mulrA ler_wpM2l //.
  rewrite (le_trans (pseries_diffs_P3 _ _ (ltW xLr) _))// ?mulrA -?normr_gt0//.
  by rewrite (le_trans (ler_normD _ _))// -(subrK `|x| r) lerD2r ltW.
Unshelve. all: by end_near.
Qed.

End PseriesDiff.

Section expR.
Variable : realType.
Implicit Types x : R.

Lemma : expR 0 = 1 :> R.

Proof.

apply: lim_near_cst => //.
near=> m; rewrite -[m]prednK; first by near: m.
rewrite -addn1 series_addn series_exp_coeff0 big_add1 big1 ?addr0//.
by move=> i _; rewrite /exp_coeff /= expr0n mul0r.
Unshelve. all: by end_near. Qed.

Lemma : 0 <= x -> 1 + x ^+ n.+1 / n.+1`!%:R <= expR x.

Proof.

move=> x_ge0; rewrite /expR.
pose f ( : R) := (i == 0)%:R + x ^+ n / n`!%:R *+ (i == n).
have F m : (n.+1 < m)%N ->
    \sum_(0 <= < m) (f n.+1 x i) = 1 + x ^+ n.+1 / n.+1`!%:R.
  move=> n1m.
  rewrite (@big_cat_nat _ _ _ n.+2)//= big_nat_recr// big_nat_recl//=.
  rewrite big_nat_cond big1 ?addr0.
    by move=> i /[!andbT] /[!leq0n]/= ni; rewrite /f/= lt_eqF//= add0r.
  rewrite big_nat_cond big1 ?addr0.
    move=> i /[!andbT] /andP[ni mi]; rewrite /f !gtn_eqF//= ?add0r//.
    exact: ltn_trans ni.
  by rewrite /f/= add0r mulr0n addr0 eqxx 2!mulr1n.
rewrite [leLHS](_ : _ = limn (series (f n.+1 x))).
  by apply/esym/(@lim_near_cst R^o) => //; near=> k; apply: F; near: k.
apply: ler_lim; first by apply: is_cvg_near_cst; near=> k; apply: F; near: k.
  exact: is_cvg_series_exp_coeff.
near=> k; apply: ler_sum => -[|[|i]] _; rewrite /f /exp_coeff/= ?add0r.
- by rewrite !(mulr0n, expr0, addr0, divr1).
- case: n F; first by rewrite !(mulr0n, mulr1n, expr0, addr0, add0r, divr1).
  by move=> n F; rewrite mulr0n expr1 divr1.
- rewrite eqSS; case: eqP => [->|_]; rewrite ?mulr1n//.
  by rewrite mulr0n divr_ge0// exprn_ge0.
Unshelve. all: by end_near. Qed.

Lemma : exp_coeff x = fun => (fun => (n`!%:R)^-1) n * x ^+ n.

Proof.

by apply/funext => i; rewrite /exp_coeff /= mulrC. Qed.

Import GRing.Theory.
Local Open Scope ring_scope.

Lemma :
expR = fun => limn (pseries (fun => (fun => (n`!%:R)^-1) n) x).

Proof.

by apply/funext => x; rewrite /pseries -exp_coeffE. Qed.

Global Instance : is_derive x 1 expR (expR x).

Proof.

pose s1 := pseries_diffs (fun => n`!%:R^-1) n * x ^+ n.
rewrite expRE /= /pseries (_ : (fun _ => _) = s1).
  by apply/funext => i; rewrite /s1 pseries_diffs_inv_fact.
apply: (@pseries_snd_diffs _ _ (`|x| + 1)); rewrite /pseries.
- by rewrite -exp_coeffE; apply: is_cvg_series_exp_coeff.
- rewrite (_ : (fun _ => _) = exp_coeff (`|x| + 1)).
    by apply/funext => i; rewrite pseries_diffs_inv_fact exp_coeffE.
  exact: is_cvg_series_exp_coeff.
- rewrite (_ : (fun _ => _) = exp_coeff (`|x| + 1)).
    by apply/funext => i; rewrite !pseries_diffs_inv_fact exp_coeffE.
  exact: is_cvg_series_exp_coeff.
by rewrite [ltRHS]ger0_norm// addrC -subr_gt0 addrK.
Qed.

Lemma : derivable expR x 1.

Proof.

by apply: ex_derive; apply: is_derive_exp. Qed.

Lemma : 'D_1 expR = expR :> (R -> R).

Proof.

by apply/funext => r /=; rewrite derive_val. Qed.

Lemma : continuous (@expR R).

Proof.

by move=> x; exact/differentiable_continuous/derivable1_diffP/derivable_expR.
Qed.

Lemma : expR (x + y) * expR (- x) = expR y.

Proof.

set v := LHS; pattern x in v; move: @v; set f := (X in let _ := X x in _) => /=.
apply: etrans (_ : f x = f 0) _; last by rewrite /f add0r oppr0 expR0 mulr1.
apply: is_derive_0_is_cst => x1.
apply: trigger_derive.
by rewrite /GRing.scale/= addr0 2!mulrN 2!mulr1 mulrC addNr.
Qed.

Lemma : expR x * expR (- x) = 1.

Proof.

by rewrite -[X in _ X * _ = _]addr0 expRxDyMexpx expR0. Qed.

Lemma : 0 < x -> 1 < expR x.

Proof.

move=> x0; rewrite (lt_le_trans _ (expR_ge1Dxn 0%N (ltW x0)))// ltrDl.
by rewrite expr1 divr1.
Qed.

Lemma : 0 < expR x.

Proof.

case: (ltrgt0P x) => [x_gt0|x_gt0|->]; last by rewrite expR0.
- exact: lt_trans (pexpR_gt1 x_gt0).
- have F : 0 < expR (- x) by rewrite (lt_trans _ (pexpR_gt1 _))// oppr_gt0.
by rewrite -(pmulr_lgt0 _ F) expRxMexpNx_1.
Qed.

Lemma : 0 <= expR x

Proof.

by rewrite ltW// expR_gt0. Qed.

Lemma : Num.norm \o expR = (expR : R -> R).

Proof.

by apply/funext => x /=; rewrite ger0_norm ?expR_ge0. Qed.

Lemma : (expR x == 0) = false.

Proof.

by rewrite gt_eqF ?expR_gt0. Qed.

Lemma : expR (- x) = (expR x)^-1.

Proof.

apply: (mulfI (lt0r_neq0 (expR_gt0 x))).
by rewrite expRxMexpNx_1 mulfV // (lt0r_neq0 (expR_gt0 x)).
Qed.

Lemma : expR (x + y) = expR x * expR y.

Proof.

apply: (mulIf (lt0r_neq0 (expR_gt0 (- x)))).
rewrite expRxDyMexpx expRN [_ * expR y]mulrC mulfK //.
by case: ltrgt0P (expR_gt0 x).
Qed.

Lemma ( : pred T) ( : T -> R) :
expR (\sum_( <- s | P i) f i) = \prod_( <- s | P i) expR (f i).

Proof.

by elim/big_ind2 : _ => [|? ? ? ? <- <-|]; rewrite ?expR0// expRD. Qed.

Lemma : expR (n%:R * x) = expR x ^+ n.

Proof.

elim: n x => [x|n IH x] /=; first by rewrite mul0r expr0 expR0.
by rewrite exprS -nat1r mulrDl mul1r expRD IH.
Qed.

Lemma : expR (x * n%:R) = expR x ^+ n.

Proof.

by rewrite mulrC expRM_natl. Qed.

Lemma : (1 < expR x) = (0 < x).

Proof.

case: ltrgt0P => [x_gt0| xN|->]; last by rewrite expR0.
- by rewrite (pexpR_gt1 x_gt0).
- apply/idP/negP.
rewrite -[x]opprK expRN -leNgt invf_cp1 ?expR_gt0 //.
by rewrite ltW // pexpR_gt1 // lterNE.
Qed.

Lemma : (expR x < 1) = (x < 0).

Proof.

case: ltrgt0P => [x_gt0|xN|->]; last by rewrite expR0.
- by apply/idP/negP; rewrite -leNgt ltW // expR_gt1.
- by rewrite -[x]opprK expRN invf_cp1 ?expR_gt0 // expR_gt1 lterNE.
Qed.

Lemma : (expR x <= 1) = (x <= 0).

Proof.

case: ltrgt0P => [||->]; last by rewrite expR0 lexx.
- by rewrite -expR_gt1 ltNge => /negbTE.
- by rewrite -expR_lt1 => /ltW.
Qed.

Lemma : expR (x - y) = expR x / expR y.

Proof.

by rewrite expRD expRN. Qed.

Lemma : {mono (@expR R) : / x < y}.

Proof.

move=> x y.
by rewrite -[in LHS](subrK x y) expRD ltr_pMl ?expR_gt0 // expR_gt1 subr_gt0.
Qed.

Lemma : {mono (@expR R) : / x <= y}.

Proof.

move=> x y.
case: (ltrgtP x y) => [xLy|yLx|<-]; last by rewrite lexx.
- by rewrite ltW // ltr_expR.
- by rewrite leNgt ltr_expR yLx.
Qed.

Lemma : x != 0 -> 1 + x < expR x.

Proof.

rewrite neq_lt => /orP[x0|x0].
- have [?|_] := leP x (-1).
    by rewrite (@le_lt_trans _ _ 0) ?expR_gt0// -lerBrDl sub0r.
  have [] := @MVT R expR expR _ _ x0 (fun _ => is_derive_expR x).
    exact/continuous_subspaceT/continuous_expR.
  move=> c; rewrite in_itv/= => /andP[xc c0].
  rewrite expR0 sub0r => /eqP; rewrite subr_eq addrC -subr_eq => /eqP <-.
  by rewrite mulrN opprK ltrD2l ltr_nMl// -expR0 ltr_expR.
- have [] := @MVT R expR expR _ _ x0 (fun _ => is_derive_expR x).
    exact/continuous_subspaceT/continuous_expR.
  move=> c; rewrite in_itv/= => /andP[c0 cx].
  rewrite subr0 expR0 => /eqP /[!subr_eq] /eqP ->.
  by rewrite addrC ltrD2r ltr_pMl// -expR0 ltr_expR.
Qed.

Lemma : 1 + x <= expR x.

Proof.

by have [->|/expR_gt1Dx/ltW//] := eqVneq x 0; rewrite expR0 addr0.
Qed.

Lemma : @expR R (- x) @[ --> +oo] --> 0.

Proof.

apply/cvgrPdist_le => e e0; near=> x.
rewrite sub0r normrN ger0_norm; first exact: expR_ge0.
rewrite expRN -[leRHS]invrK lef_pV2 ?posrE ?expR_gt0 ?invr_gt0//.
rewrite (le_trans _ (expR_ge1Dx _))// -lerBlDl.
by near: x; apply: nbhs_pinfty_ge; exact: num_real.
Unshelve. end_near. Qed.

Lemma : @expR R (- n%:R) @[ --> \oo] --> 0%R.

Proof.

under eq_cvg do rewrite -mulNrn -mulr_natr expRM_natr; apply: cvg_expr.
by rewrite ger0_norm ?expR_ge0// expRN invf_lt1 ?expR_gt1// expR_gt0.
Qed.

Lemma : injective (@expR R).

Proof.

move=> x y exE.
by have [] := (ltr_expR x y, ltr_expR y x); rewrite exE ltxx; case: ltrgtP.
Qed.

Lemma :
1 <= x -> exists , [/\ 0 <= y, 1 + y <= x & expR y = x].

Proof.

move=> x_ge1; have x_ge0 : 0 <= x by apply: le_trans x_ge1.
have [x1 x1Ix| |x1 _ /eqP] := @IVT _ (fun => expR y - x) _ _ 0 x_ge0.
- apply: continuousB => // y1; last exact: cst_continuous.
  by apply/continuous_subspaceT=> ?; exact: continuous_expR.
- rewrite expR0; have [_| |] := ltrgtP (1 - x) (expR x - x).
  + by rewrite subr_le0 x_ge1 subr_ge0 (le_trans _ (expR_ge1Dx _)) ?lerDr.
  + by rewrite ltrD2r expR_lt1 ltNge x_ge0.
  + rewrite subr_le0 x_ge1 => -> /=; rewrite subr_ge0.
    by rewrite (le_trans _ (expR_ge1Dx _)) ?lerDr.
- rewrite subr_eq0 => /eqP x1_x; exists x1; split => //.
  + by rewrite -ler_expR expR0 x1_x.
  + by rewrite -x1_x expR_ge1Dx // -ler_expR x1_x expR0.
Qed.

Lemma : 0 < x -> exists , expR y = x.

Proof.

case: (lerP 1 x) => [/expR_total_gt1[y [_ _ Hy]]|x_lt1 x_gt0].
by exists y.
have /expR_total_gt1[y [H1y H2y H3y]] : 1 <= x^-1 by rewrite ltW // !invf_cp1.
by exists (-y); rewrite expRN H3y invrK.
Qed.

Local Open Scope convex_scope.
Lemma ( : {i01 R}) ( : R^o) :
expR (a <| t |> b) <= (expR a : R^o) <| t |> (expR b : R^o).

Proof.

have [ab|/ltW ba] := leP a b.
- apply: second_derivative_convex => //.
  + by move=> x axb; rewrite derive_expR derive_val expR_ge0.
  + exact/cvg_at_left_filter/continuous_expR.
  + exact/cvg_at_right_filter/continuous_expR.
  + by move=> z zab; rewrite derive_expR; exact: derivable_expR.
- rewrite convC [leRHS]convC; apply: second_derivative_convex => //.
  + by move=> x axb; rewrite derive_expR derive_val expR_ge0.
  + exact/cvg_at_left_filter/continuous_expR.
  + exact/cvg_at_right_filter/continuous_expR.
  + by move=> z zab; rewrite derive_expR; exact: derivable_expR.
Qed.

Local Close Scope convex_scope.

Definition :=
  Order.max (BRight 0%Z) (IntItv.add_boundl b (BLeft 1)).

Definition :=
  match b with
  | BSide _ (Negz _) => BLeft 1%Z
  | BSide b 0%Z => BSide b 1%Z
  | _ => +oo%O
  end.

Definition :=
  match i with
  | Itv.Top => Itv.Real `[0%Z, +oo[
  | Itv.Real (Interval l u) =>
      Itv.Real (Interval (expR_itv_boundl l) (expR_itv_boundr u))
  end.

Lemma ( : Itv.t) ( : Itv.def (@Itv.num_sem R) i)
    ( := expR_itv i) :
  Itv.spec (@Itv.num_sem R) r (expR x%:num).

Proof.

rewrite {}/r; case: i x => [|[l u]] x /=.
  by apply/and3P; rewrite ?num_real// bnd_simp expR_ge0.
case: x => [x /=/and3P[xr lx xu]]; apply/and3P; split; [exact: num_real| | ].
- rewrite Instances.num_itv_bound_max maxEge.
  case: ifP; rewrite ?bnd_simp ?expR_gt0// => _.
  apply: le_trans (Instances.num_itv_add_boundl lx _) _; first exact: lexx.
  by rewrite bnd_simp addrC expR_ge1Dx.
- case: u xu => [[] [[|//] | u] |//]; rewrite !bnd_simp.
  + by rewrite expR_lt1.
  + by rewrite expR_lt1 => /lt_trans; apply.
  + by rewrite expR_le1.
  + by rewrite expR_lt1 => /le_lt_trans; apply.
Qed.

Canonical ( : Itv.t) ( : Itv.def (@Itv.num_sem R) i) :=
Itv.mk (num_spec_expR x).

End expR.

Section expeR.
Context { : realType}.
Implicit Types (x y : \bar R) (r s : R).

Local Open Scope ereal_scope.

Definition :=
match x with | r%:E => (expR r)%:E | +oo => +oo | -oo => 0 end.

Lemma : expeR 0 = 1

Proof.

by rewrite /= expR0. Qed.

Lemma : 0 <= expeR x.

Proof.

by case: x => //= r; rewrite lee_fin expR_ge0. Qed.

Lemma : -oo < x -> 0 < expeR x.

Proof.

by case: x => //= r; rewrite lte_fin expR_gt0. Qed.

Lemma : (expeR x == 0) = (x == -oo).

Proof.

by case: x => //= [r|]; rewrite ?eqxx// eqe expR_eq0. Qed.

Lemma : (expeR x == +oo) = (x == +oo).

Proof.

by case: x. Qed.

Lemma : expeR (x + y) = expeR x * expeR y.

Proof.

case: x => /= [r| |]; last by rewrite mul0e.
- case: y => /= [s| |]; last by rewrite mule0.
  + by rewrite expRD EFinM.
  + by rewrite mulry gtr0_sg ?mul1e// expR_gt0.
- case: y => /= [s| |]; last by rewrite mule0.
  + by rewrite mulyr gtr0_sg ?mul1e// expR_gt0.
  + by rewrite mulyy.
Qed.

Lemma : 1 + x <= expeR x.

Proof.

case : x => //= [r|]; last by rewrite addeNy.
by rewrite -EFinD !lee_fin; exact: expR_ge1Dx.
Qed.

Lemma : {mono expeR : / x < y}.

Proof.

move=> [r| |] [s| |]//=; rewrite ?ltry//.
- by rewrite !lte_fin ltr_expR.
- by rewrite !ltNge lee_fin expR_ge0 leNye.
- by rewrite lte_fin expR_gt0 ltNye.
Qed.

Lemma : {mono expeR : / x <= y}.

Proof.

move=> [r| |] [s| |]//=; rewrite ?leey ?lexx//.
- by rewrite !lee_fin ler_expR.
- by rewrite !leNgt lte_fin expR_gt0 ltNye.
- by rewrite lee_fin expR_ge0 leNye.
Qed.

Lemma : injective expeR.

Proof.

move=> [r| |] [s| |] => //=.
- by move=> [] /expR_inj ->.
- by case => /eqP; rewrite expR_eq0.
- by case => /esym/eqP; rewrite expR_eq0.
Qed.

Lemma : 0 <= x -> exists , expeR y = x.

Proof.

move: x => [r|_|//]; last by exists +oo.
rewrite le_eqVlt => /predU1P[<-|]; first by exists -oo.
by rewrite lte_fin => /expR_total[y <-]; exists y%:E.
Qed.

End expeR.
#[deprecated(since="mathcomp-analysis 1.13.0", note="renamed `lte_expeR`")]
Notation := lte_expeR (only parsing).
#[deprecated(since="mathcomp-analysis 1.13.0", note="renamed `lee_expeR`")]
Notation := lee_expeR (only parsing).

Section Ln.
Variable : realType.
Implicit Types x : R.

Notation := (@expR R).

Definition : R := [get | exp y == x ].

Fact : x <= 0 -> ln x = 0.

Proof.

rewrite /ln; case: xgetP => //= y _ /eqP yx x0.
by have := expR_gt0 y; rewrite yx => /(le_lt_trans x0); rewrite ltxx.
Qed.

Lemma : cancel exp ln.

Proof.

by move=> x; rewrite /ln; case: xgetP => [x1 _ /eqP/expR_inj //|/(_ x)[]/=].
Qed.

Lemma : {in Num.pos, cancel ln exp}.

Proof.

move=> x; rewrite qualifE => x_gt0.
rewrite /ln; case: xgetP=> [x1 _ /eqP// |H].
by case: (expR_total x_gt0) => y /eqP Hy; case: (H y).
Qed.

Lemma : (exp (ln x) == x) = (0 < x).

Proof.

by apply/eqP/idP=> [<-|x0]; [exact: expR_gt0|rewrite lnK// in_itv/= x0].
Qed.

Lemma : ln 1 = 0.

Proof.

by apply/expR_inj; rewrite lnK// ?expR0// qualifE/=. Qed.

Lemma : {in Num.pos &, {morph ln : / x * y >-> x + y}}.

Proof.

move=> x y x0 y0; apply: expR_inj; rewrite expRD !lnK//.
by move: x0 y0; rewrite !qualifE; exact: mulr_gt0.
Qed.

Lemma : {in Num.pos &, injective ln}.

Proof.

by move=> x y /lnK {2}<- /lnK {2}<- ->. Qed.

Lemma : {in Num.pos, {morph ln : / x ^-1 >-> - x}}.

Proof.

move=> x x0; apply: expR_inj; rewrite lnK// ?expRN ?lnK//.
by move: x0; rewrite !qualifE/= invr_gt0.
Qed.

Lemma : {in Num.pos &, {morph ln : / x / y >-> x - y}}.

Proof.

move=> x y x0 y0; rewrite (lnM x0) ?lnV//.
by move: y0; rewrite !qualifE/= invr_gt0.
Qed.

Lemma : {in Num.pos &, {mono ln : / x < y}}.

Proof.

by move=> x y x_gt0 y_gt0; rewrite -ltr_expR !lnK. Qed.

Lemma : {in Num.pos &, {mono ln : / x <= y}}.

Proof.

by move=> x y x_gt0 y_gt0; rewrite -ler_expR !lnK. Qed.

Lemma : 0 < x -> ln (x ^+ n) = ln x *+ n.

Proof.

move=> x_gt0; elim: n => [|n ih] /=; first by rewrite expr0 ln1 mulr0n.
by rewrite !exprS lnM ?qualifE//= ?exprn_gt0// mulrS ih.
Qed.

Lemma : -1 < x -> ln (1 + x) <= x.

Proof.

move=> N1x; rewrite -ler_expR lnK ?expR_ge1Dx //.
by rewrite posrE addrC -ltrBlDr sub0r.
Qed.

Lemma : 0 < x -> ln x < x.

Proof.

move=> x_gt0; apply: lt_le_trans (_ : ln (1 + x) <= _).
by rewrite -ltr_expR !lnK ?qualifE/= ?addr_gt0 // ltrDr.
by rewrite -ler_expR lnK ?qualifE/= ?addr_gt0// expR_ge1Dx // ltW.
Qed.

Lemma : 1 <= x -> 0 <= ln x.

Proof.

by move=> x_ge1; rewrite -ler_expR expR0 lnK// qualifE/= (lt_le_trans _ x_ge1).
Qed.

Lemma : 0 < x < 1 -> ln x < 0.

Proof.

by move=> /andP[x_gt0 x_lt1]; rewrite -ltr_expR expR0 lnK. Qed.

Lemma : 1 < x -> 0 < ln x.

Proof.

by move=> x_gt1; rewrite -ltr_expR expR0 lnK // qualifE/= (lt_trans _ x_gt1).
Qed.

Lemma : x <= 1 -> ln x <= 0.

Proof.

have [x0|x0 x1] := leP x 0; first by rewrite ln0.
by rewrite -ler_expR expR0 lnK.
Qed.

Lemma : 0 < x -> (ln x == 0) = (x == 1).

Proof.

move=> x0; apply/idP/idP => [/eqP lnx0|/eqP ->]; last by rewrite ln1.
have [| |//] := ltgtP x 1; last by move/ln_gt0; rewrite lnx0 ltxx.
by move/(conj x0)/andP/ln_lt0; rewrite lnx0 ltxx.
Qed.

Lemma : 0 < x -> {for x, continuous ln}.

Proof.

move=> x_gt0; rewrite -[x]lnK//.
apply: nbhs_singleton (near_can_continuous _ _); near=> z; first exact: expRK.
by apply: continuous_expR.
Unshelve. all: by end_near. Qed.

Global Instance ( : R) : 0 < x -> is_derive x 1 ln x^-1.

Proof.

move=> x_gt0; rewrite -[x]lnK//.
apply: (@is_derive_inverse R expR); first by near=> z; apply: expRK.
by near=>z; apply: continuous_expR.
by rewrite lnK // lt0r_neq0.
Unshelve. all: by end_near. Qed.

Local Open Scope convex_scope.
Lemma ( : {i01 R}) ( : R^o) : 0 < a -> 0 < b ->
(ln a : R^o) <| t |> (ln b : R^o) <= ln (a <| t |> b).

Proof.

move=> a0 b0; have := convex_expR t (ln a) (ln b).
by rewrite !lnK// -(@ler_ln) ?posrE ?expR_gt0 ?convR_gt0// expRK.
Qed.

Local Close Scope convex_scope.

Lemma : ln x @[ --> 0^'+] --> -oo.

Proof.

apply/cvgrNyPle => y; near=> x; rewrite -ler_expR lnK; first by rewrite posrE.
by near: x; apply: nbhs_right_le; exact: expR_gt0.
Unshelve. end_near. Qed.

End Ln.

Section PowR.
Variable : realType.
Implicit Types a x y z r : R.

Definition := if a == 0 then (x == 0)%:R else expR (x * ln a).

Local Notation := (powR a x).

Lemma : expR (x * y) = (expR x) `^ y.

Proof.

by rewrite /powR gt_eqF ?expR_gt0// expRK mulrC. Qed.

Lemma : 0 <= a `^ x.

Proof.

by rewrite /powR; case: ifPn => // _; exact: expR_ge0. Qed.

Lemma : 0 < a -> 0 < a `^ x.

Proof.

by move=> a0; rewrite /powR gt_eqF// expR_gt0. Qed.

Lemma : 0 < x -> 0 <= a -> 0 < a `^ x -> 0 < a.

Proof.

move=> x0 a0; rewrite /powR; case: ifPn => [_|a_neq0 _].
by rewrite gt_eqF//= ltxx.
by rewrite lt_neqAle a0 andbT eq_sym.
Qed.

Lemma : x != 0 -> 0 `^ x = 0.

Proof.

by move=> x0; rewrite /powR eqxx (negbTE x0). Qed.

Lemma : 0 <= a -> a `^ 1 = a.

Proof.

rewrite le_eqVlt => /predU1P[<-|a0]; first by rewrite powR0// oner_eq0.
by rewrite /powR gt_eqF// mul1r lnK// posrE.
Qed.

Lemma : a `^ 0 = 1.

Proof.

by rewrite /powR; case: ifPn; rewrite ?eqxx// mul0r expR0. Qed.

Lemma : powR 1 = fun=> 1.

Proof.

by apply/funext => x; rewrite /powR oner_eq0 ln1 mulr0 expR0. Qed.

Lemma : (x `^ p == 0) = (x == 0) && (p != 0).

Proof.

rewrite /powR; have [_|x_neq0] := eqVneq x 0 => //.
by case: (p == 0); rewrite (oner_eq0, eqxx).
by rewrite expR_eq0.
Qed.

Lemma : x `^ p = 0 -> x = 0.

Proof.

by move=> /eqP; rewrite powR_eq0 => /andP[/eqP]. Qed.

Lemma : 0 < a <= 1 -> {homo powR a : /~ y <= x}.

Proof.

move=> /andP[a0 a1] x y xy.
by rewrite /powR gt_eqF// ler_expR ler_wnM2r// ln_le0.
Qed.

Lemma : 1 <= a -> {homo powR a : / x <= y}.

Proof.

move=> a1 x y xy.
by rewrite /powR gt_eqF ?(lt_le_trans _ a1)// ler_expR ler_wpM2r ?ln_ge0.
Qed.

Lemma : 0 < r -> {in Num.nneg &, injective (powR ^~ r)}.

Proof.

move=> r0 x y x0 y0; rewrite /powR; case: ifPn => [/eqP ->|xneq0].
by case: ifPn => [/eqP ->//|_ /eqP]; rewrite (gt_eqF r0) eq_sym expR_eq0.
case: ifPn => [/eqP -> /eqP|yneq0]; first by rewrite (gt_eqF r0) expR_eq0.
by move/expR_inj/mulfI => /(_ (negbT (gt_eqF r0))); apply: ln_inj;
rewrite posrE lt_neqAle eq_sym (xneq0,yneq0).
Qed.

Lemma : 1 <= a -> r <= 1 -> a >= a `^ r.

Proof.

by move=> a1 r1; rewrite (le_trans (ler_powR _ r1)) ?powRr1// (le_trans _ a1).
Qed.

Lemma : 1 <= a -> 1 <= r -> a <= a `^ r.

Proof.

by move=> a1 r1; rewrite (le_trans _ (ler_powR _ r1)) ?powRr1// (le_trans _ a1).
Qed.

Lemma : 0 < a <= 1 -> r <= 1 -> a <= a `^ r.

Proof.

move=> /andP[a0 _a1] r1.
by rewrite (le_trans _ (ger_powR _ r1)) ?powRr1 ?a0// ltW.
Qed.

Lemma : 0 < a <= 1 -> 1 <= r -> a >= a `^ r.

Proof.

move=> /andP[a0 a1] r1.
by rewrite (le_trans (ger_powR _ r1)) ?powRr1 ?a0// ltW.
Qed.

Lemma : 0 <= r ->
{in Num.nneg &, {homo powR ^~ r : / x <= y >-> x <= y}}.

Proof.

rewrite le_eqVlt => /predU1P[<- x y _ _ _|]; first by rewrite !powRr0.
move=> a0 x y; rewrite 2!nnegrE !le_eqVlt => /predU1P[<-|x0].
move=> /predU1P[<- _|y0 _]; first by rewrite eqxx.
by rewrite !powR0 ?(gt_eqF a0)// powR_gt0 ?orbT.
move=> /predU1P[<-|y0]; first by rewrite gt_eqF//= ltNge (ltW x0).
move=> /predU1P[->//|xy]; first by rewrite eqxx.
by apply/orP; right; rewrite /powR !gt_eqF// ltr_expR ltr_pM2l// ltr_ln.
Qed.

Lemma : 0 < r ->
{in Num.nneg &, {homo powR ^~ r : / x < y >-> x < y}}.

Proof.

move=> r0 x y x0 y0 xy; have := ge0_ler_powR (ltW r0) x0 y0 (ltW xy).
rewrite le_eqVlt => /orP[/eqP/(powR_injective r0 x0 y0)/eqP|//].
by rewrite lt_eqF.
Qed.

Lemma : 0 <= x -> 0 <= y -> (x * y) `^ r = x `^ r * y `^ r.

Proof.

rewrite /powR mulf_eq0.
case: (ltgtP x 0) => // x0 _; case: (ltgtP y 0) => //= y0 _; do ?
by case: eqVneq => [r0|]; rewrite ?r0 ?mul0r ?expR0 ?mulr0 ?mul1r.
by rewrite lnM// mulrDr expRD.
Qed.

Lemma : 0 < x <= 1 -> 0 <= y -> 1 <= r ->
(x * y) `^ r <= x * (y `^ r).

Proof.

move=> /andP[x0 x1] y0 r1.
by rewrite (powRM _ (ltW _))// ler_wpM2r ?powR_ge0// ge1r_powR// x0.
Qed.

Lemma : x >= 1 -> 0 <= y -> 1 <= r ->
(x * y) `^ r >= x * (y `^ r).

Proof.

move=> x1 y0 r1.
by rewrite (powRM _ (le_trans _ x1))// ler_wpM2r ?powR_ge0// le1r_powR// x0.
Qed.

Lemma : x `^ (y * z) = (x `^ y) `^ z.

Proof.

rewrite /powR mulf_eq0; have [_|xN0] := eqVneq x 0.
by case: (y == 0); rewrite ?eqxx//= oner_eq0 ln1 mulr0 expR0.
by rewrite expR_eq0 expRK mulrCA mulrA.
Qed.

Lemma : (x `^ y) `^ z = (x `^ z) `^ y.

Proof.

by rewrite -!powRrM mulrC. Qed.

Lemma : (r + s == 0) ==> (x != 0) -> x `^ (r + s) = x `^ r * x `^ s.

Proof.

rewrite /powR; case: (eqVneq x 0) => //= [_|x_neq0 _];
last by rewrite mulrDl expRD.
have [->|] := eqVneq r 0; first by rewrite mul1r add0r.
by rewrite implybF mul0r => _ /negPf ->.
Qed.

Lemma : 0 <= x -> 0 < p -> x * x `^ (p - 1) = x `^ p.

Proof.

rewrite le_eqVlt => /predU1P[<- p0|x0 p0]; first by rewrite mul0r powR0 ?gt_eqF.
by rewrite -{1}(powRr1 (ltW x0))// -powRD subrKC// gt_eqF.
Qed.

Lemma : x `^ (- r) = (x `^ r)^-1.

Proof.

have [r0|r0] := eqVneq r 0%R; first by rewrite r0 oppr0 powRr0 invr1.
have [->|xN0] := eqVneq x 0; first by rewrite !powR0 ?oppr_eq0// invr0.
rewrite -div1r; apply: (canRL (mulfK _)); first by rewrite powR_eq0 (negPf xN0).
by rewrite -powRD ?addNr ?powRr0// xN0 eqxx.
Qed.

Lemma : (r == s) ==> (x != 0) -> x `^ (r - s) = x `^ r / x `^ s.

Proof.

by move=> ?; rewrite powRD ?subr_eq0// powRN. Qed.

Lemma : 0 <= a -> a `^ n%:R = a ^+ n.

Proof.

move=> a_ge0; elim: n => [|n IHn]; first by rewrite powRr0 expr0.
by rewrite -natr1 powRD ?natr1 ?pnatr_eq0// powRr1// IHn exprSr.
Qed.

Lemma : 0 <= a -> a `^ (-1) = a ^-1.

Proof.

by move=> a_ge0; rewrite powRN powRr1. Qed.

Lemma : 0 <= a -> a `^ (- n%:R) = a ^- n.

Proof.

by move=> a_ge0; rewrite powRN powR_mulrn. Qed.

Lemma ( : int) : 0 <= a -> a `^ z%:~R = a ^ z.

Proof.

by move=> a0; case: z => n; [exact: powR_mulrn | exact: powR_invn]. Qed.

Lemma : ln (a `^ x) = x * ln a.

Proof.

have [->|x0] := eqVneq x 0; first by rewrite powRr0 ln1// mul0r.
have [->|a0] := eqVneq a 0; first by rewrite powR0// ln0// mulr0.
by rewrite /powR (negbTE a0) expRK.
Qed.

Lemma : 0 <= a -> a `^ (2^-1) = Num.sqrt a.

Proof.

rewrite le_eqVlt => /predU1P[<-|a0].
  by rewrite powR0 ?invr_eq0 ?pnatr_eq0// sqrtr0.
have /eqP : (a `^ (2^-1)) ^+ 2 = (Num.sqrt a) ^+ 2.
  rewrite sqr_sqrtr; first exact: ltW.
  by rewrite /powR gt_eqF// -expRM_natl mulVKf// lnK.
rewrite eqf_sqr => /predU1P[//|/eqP h].
have : 0 < a `^ 2^-1 by exact: powR_gt0.
by rewrite h oppr_gt0 ltNge sqrtr_ge0.
Qed.

Lemma : 0 <= a -> `|a `^ x| = `|a| `^ x.

Proof.

move=> a0; rewrite /powR; case: ifPn => [/eqP ->|].
by rewrite normr0 eqxx normr_nat.
rewrite neq_lt ltNge a0/= => {}a0.
by rewrite gtr0_norm ?expR_gt0// gtr0_norm// gt_eqF.
Qed.

Lemma : a < 0 -> `|a `^ x| = 1.

Proof.

move=> a0; rewrite /powR lt_eqF// gtr0_norm ?expR_gt0//.
by rewrite ln0 ?mulr0 ?expR0// ltW.
Qed.

Lemma : 0 <= a -> 0 <= b ->
0 < p -> 0 < q -> p^-1 + q^-1 = 1 ->
a * b <= a `^ p / p + b `^ q / q.

Proof.

rewrite le_eqVlt => /predU1P[<- b0 p0 q0 _|a0].
by rewrite mul0r powR0 ?gt_eqF// mul0r add0r divr_ge0 ?powR_ge0 ?ltW.
rewrite le_eqVlt => /predU1P[<-|b0] p0 q0 pq.
by rewrite mulr0 powR0 ?gt_eqF// mul0r addr0 divr_ge0 ?powR_ge0 ?ltW.
have iq1 : q^-1 <= 1 by rewrite -pq ler_wpDl// invr_ge0 ltW.
have ap0 : (0 < a `^ p)%R by rewrite powR_gt0.
have bq0 : (0 < b `^ q)%R by rewrite powR_gt0.
have pq' : (p^-1 = 1 - q^-1)%R by rewrite -pq addrK.
have qp' : (q^-1 = 1 - p^-1)%R by rewrite -pq -addrA subrKC.
have := @concave_ln _ (Itv01 (eqbRL (invr_ge0 _) (ltW q0)) iq1) _ _ bq0 ap0.
rewrite 2!(convC (Itv01 _ _)) !convRE/= /onem -pq' -[_ <= ln _]ler_expR expRD.
rewrite 2!ln_powR mulrCA mulVKf ?gt_eqF// -qp' mulrCA mulVKf ?gt_eqF//.
by rewrite 2![_^-1 * _]mulrC 3?lnK ?posrE// addr_gt0// mulr_gt0// ?invr_gt0.
Qed.

Definition :=
  match i with
  | Itv.Top => Itv.Real `]-oo, +oo[
  | Itv.Real (Interval l u) =>
      Itv.Real (Interval (IntItv.keep_pos_bound l) +oo%O)
  end.

Lemma ( : Itv.t) ( : Itv.def (@Itv.num_sem R) i)
    ( := powR_itv i) :
  Itv.spec (@Itv.num_sem R) r (powR x%:num p).

Proof.

rewrite {}/r; case: i x => [|[l u]] x /=; [by apply/and3P; rewrite ?num_real|].
case: x => [x /=/and3P[xr lx xu]]; apply/and3P; split; [exact: num_real| |by[]].
case: l lx => [[] [[|l] |//] |//]; rewrite !bnd_simp => lx.
- by rewrite powR_ge0.
- by apply: powR_gt0; apply: lt_le_trans lx.
- by apply: powR_gt0; apply: le_lt_trans lx.
- by apply: powR_gt0; apply: le_lt_trans lx.
Qed.

Canonical ( : Itv.t) ( : Itv.def (@Itv.num_sem R) i) :=
Itv.mk (num_spec_powR x p).

Lemma ( : R) : 0 < x -> a `^ x @[ --> 0^'+] --> 0.

Proof.

move=> x0; apply: (@cvg_trans _ ((expR (ln a * x)) @[ --> 0^'+])).
by apply: near_eq_cvg; near=> a; rewrite /powR gt_eqF 1?mulrC.
apply: (@cvg_comp _ _ _ _ _ _ (@ninfty_nbhs R)).
by apply: gt0_cvgMlNy; rewrite ?x0//; exact: lnNy.
exact/cvgNy_compNP/cvgr_expR.
Unshelve. end_near. Qed.

Lemma : {in `]0, +oo[, forall , derivable (powR ^~ x) a v}.

Proof.

have [-> y _|] := eqVneq v 0; first exact/derivable0.
move=> v0 a; rewrite in_itv/= andbT => a0.
apply: (@near_eq_derivable _ _ _ (fun => expR (x * ln a'))) => //.
by near=> b; rewrite /powR gt_eqF//; near: b; exact: lt_nbhsr.
apply: diff_derivable; apply: differentiable_comp; last exact/derivable1_diffP.
apply: differentiableM => //; apply/derivable1_diffP.
by apply: ex_derive; exact: is_derive1_ln.
Unshelve. end_near. Qed.

Lemma : {in `]0, +oo[%R,
(powR ^~ a) ^`()%classic =1 (fun => a * x `^ (a - 1))}.

Proof.

move=> x; rewrite in_itv/= andbT => x0.
rewrite derive1E.
rewrite (@near_eq_derive _ _ _ _ (fun => expR (a * ln x)))//.
by near=> z; rewrite /powR gt_eqF//; near: z; exact: lt_nbhsr.
rewrite -derive1E.
rewrite derive1_comp//=.
by apply: derivableM => //; apply: ex_derive; exact: is_derive1_ln.
rewrite 2!derive1E.
rewrite deriveM//; first by apply: ex_derive; exact: is_derive1_ln.
rewrite !derive_val scaler0 addr0.
have [_ ->] := (is_derive1_ln x0).
rewrite mulrC powRB; first by rewrite (@gt_eqF _ _ x)//; apply: implybT.
by rewrite powRr1 ?ltW// mulrA [in RHS]mulrAC /powR gt_eqF.
Unshelve. end_near. Qed.

Global Instance ( : R) : 0 < x ->
is_derive x 1 (powR ^~ a) (a * x `^ (a - 1))%R.

Proof.

move=> x_gt0; split.
by apply: derivable_powR; rewrite ?in_itv/= ?andbT.
by rewrite -derive1E powR_derive1// in_itv andbT.
Qed.

Lemma : x < 0 -> x `^ p = 1.

Proof.

by move=> x0; rewrite /powR lt_eqF// (ln0 (ltW x0)) mulr0 expR0.
Qed.

Lemma : (x `^ p == 1) = [|| (x == 1), (x < 0) | (p == 0)].

Proof.

have [->/=|x1] := eqVneq x 1; first by rewrite powR1 eqxx.
have [->|p0]/= := eqVneq p 0; first by rewrite powRr0 eqxx orbT.
have [x0|x0|->]/= := ltgtP x 0; first by rewrite lt0_powR1// eqxx.
- rewrite /powR (gt_eqF x0)// -expR0; apply/negP => /eqP/expR_inj/eqP.
rewrite mulf_eq0 (negbTE p0)/= -ln1 => /eqP/ln_inj => /(_ _ _)/eqP.
by rewrite (negbTE x1) falseE; apply => //; rewrite posrE.
- by rewrite powR0// eq_sym oner_eq0.
Qed.

End PowR.
Notation := (powR a x) : ring_scope.

Section Lne.
Variable : realType.
Implicit Types (x : \bar R) (r : R).
Local Open Scope ereal_scope.

Definition :=
  match x with
  | r%:E => if (r <= 0)%R then -oo else (ln r)%:E
  | +oo => +oo
  | -oo => -oo
  end.

Lemma : x <= 0 -> lne x = -oo.

Proof.

by move: x => [r| |]//=; case: ifPn => //; rewrite lee_fin => /negbTE ->.
Qed.

Lemma : (0 < r)%R -> lne (r%:E) = (ln r)%:E.

Proof.

by move=> /=; case: ifPn => //; rewrite leNgt => /negbTE ->. Qed.

Lemma : cancel expeR lne.

Proof.

by case=> //=[x|]; rewrite ?lexx// leNgt expR_gt0/= expRK. Qed.

Lemma : {in `[0, +oo], cancel lne (@expeR R)}.

Proof.

case=> [r|//|//] /[!in_itv]/=/andP[]; rewrite le_eqVlt eqe lte_fin.
by move=> /predU1P[<- _|r0 _]; rewrite ?lexx// leNgt r0/= lnK.
Qed.

Lemma : (expeR (lne x) == x) = (0 <= x).

Proof.

move: x => [r| |]//=; last by rewrite eqxx leey.
case: ifPn => [r0|]/=; first by rewrite eq_le !lee_fin r0 andbT.
by rewrite -ltNge => r0; rewrite eqe lnK_eq lee_fin le_eqVlt r0 orbT.
Qed.

Lemma : lne 1 = 0

Proof.

by rewrite lne_EFin//= ln1. Qed.

Lemma : {in `]0, +oo] &, {morph lne : / x * y >-> x + y}}.

Proof.

move=> x y /[!in_itv]/= /[!@leey R]/= /[!andbT] x0 y0.
by apply: expeR_inj; rewrite expeRD !lneK// in_itv/= leey ltW ?mule_gt0.
Qed.

Lemma : {in `[0, +oo] &, injective lne}.

Proof.

by move=> x y /lneK {2}<- /lneK {2}<- ->. Qed.

Lemma : (0 < r)%R -> lne r^-1%:E = - lne (r%:E).

Proof.

by move=> r0; rewrite !lne_EFin ?gt_eqF ?invr_gt0// lnV. Qed.

Lemma : {in `]0, +oo] &, {morph lne : / x / y >-> x - y}}.

Proof.

move=> [x| |] [y| |]//; rewrite !in_itv/= !leey !andbT ?lte_fin => x0 y0.
- rewrite inver// !gt_eqF//= pmulr_rle0// invr_le0 leNgt y0/=.
by rewrite leNgt x0/= ln_div.
- by rewrite mulr0 lexx leNgt x0/=.
- by rewrite inver !gt_eqF// leNgt y0/= gt0_mulye// lte_fin invr_gt0.
- by rewrite invey mule0/= lexx.
Qed.

Lemma : {in `[0, +oo] &, {mono lne : / x < y}}.

Proof.

by move => x y x0 y0; rewrite -lte_expeR !lneK. Qed.

Lemma : {in `[0, +oo] &, {mono lne : / x <= y}}.

Proof.

by move=> x y x_gt0 y_gt0; rewrite -lee_expeR !lneK. Qed.

Lemma : 0 < x -> lne (x ^+ n) = lne x *+ n.

Proof.

elim: n x => [x x0|n ih x x0]; first by rewrite expe0 mule0n lne1.
by rewrite expeS lneM ?ih ?muleS// in_itv/= ?expe_gt0// ?x0/= leey.
Qed.

Lemma : - 1%E <= x -> lne (1 + x) <= x.

Proof.

move=> ?; rewrite -lee_expeR lneK ?expeR_ge1Dx //.
by rewrite in_itv //= leey andbT addrC -(oppeK 1) sube_ge0.
Qed.

Lemma : x \is a fin_num -> lne x < x.

Proof.

move=> xfin; have [x0|x0] := leP x 0.
by rewrite le0_lneNy// ltNye_eq xfin.
have ? : (0 < fine x)%R by rewrite fine_gt0// x0/= ltey_eq xfin.
by rewrite -(fineK xfin) lne_EFin// lte_fin ln_sublinear.
Qed.

Lemma : (0 <= lne x) = (1 <= x).

Proof.

apply/idP/idP.
  case: x => [r||]//=; last by move=> _; rewrite leey.
  case: ifPn => //; rewrite -ltNge lee_fin => r0.
  by rewrite -ln1 ler_ln// posrE.
case: x => [r||]//; rewrite ?leey// lee_fin => r1.
by rewrite lne_EFin// ?(lt_le_trans _ r1)// lee_fin ln_ge0.
Qed.

Lemma : (lne x < 0) = (x < 1).

Proof.

by rewrite 2!ltNge lne_ge0. Qed.

Lemma : (lne x == 0) = (x == 1).

Proof.

rewrite /lne; move: x => [r| |] //; case: ifPn => r0.
by apply/esym; rewrite lt_eqF// lte_fin (le_lt_trans r0).
by rewrite !eqe ln_eq0// ltNge.
Qed.

Lemma : (0 < lne x) = (1 < x).

Proof.

by rewrite 2!lt_neqAle eq_sym lne_eq0 eq_sym lne_ge0. Qed.

Lemma : (lne x <= 0) = (x <= 1).

Proof.

by rewrite 2!leNgt lne_gt0. Qed.

End Lne.

Section poweR.
Local Open Scope ereal_scope.
Context { : realType}.
Implicit Types (s r : R) (x y : \bar R).

Definition :=
  match x with
  | x'%:E => (x' `^ r)%:E
  | +oo => if r == 0%R then 1%E else +oo
  | -oo => if r == 0%R then 1%E else 0%E
  end.

Local Notation := (poweR x r).

Lemma : s%:E `^ r = (s `^ r)%:E.

Proof.

by []. Qed.

Lemma : r != 0%R -> +oo `^ r = +oo.

Proof.

by move/negbTE => /= ->. Qed.

Lemma : x `^ 0 = 1.

Proof.

by move: x => [x'| |]/=; rewrite ?powRr0// eqxx. Qed.

Lemma : 0 <= x -> x `^ 1 = x.

Proof.

by move: x => [x'| |]//= x0; rewrite ?powRr1// (negbTE (oner_neq0 _)).
Qed.

Lemma : x \is a fin_num -> x `^ (- r) = (fine x `^ r)^-1%:E.

Proof.

by case: x => // x xf; rewrite poweR_EFin powRN. Qed.

Lemma : r != 0%R -> -oo `^ r = 0.

Proof.

by move=> r0 /=; rewrite (negbTE r0). Qed.

Lemma :
  poweR x r = if r == 0%R then
                (if x \is a fin_num then fine x `^ r else 1)%:E
               else if x == +oo then +oo
                    else if x == -oo then 0
                    else (fine x `^ r)%:E.

Proof.

case: ifPn => [/eqP r0|r0].
case: ifPn => finx; last by rewrite r0 poweRe0.
by rewrite -poweR_EFin; case: x finx.
case: ifPn => [/eqP ->|xy]; first by rewrite poweRyr.
case: ifPn => [/eqP ->|xNy]; first by rewrite poweRNyr.
by rewrite -poweR_EFin// fineK// fin_numE xNy.
Qed.

Lemma : x `^ r = +oo -> x = +oo.

Proof.

by case: x => [x| |] //=; case: ifP. Qed.

Lemma : (0 < r)%R -> x = +oo -> x `^ r = +oo.

Proof.

by move: x => [| |]//= r0 _; rewrite gt_eqF. Qed.

Lemma : x < +oo -> x `^ r < +oo.

Proof.

by move: x => [x| |]//=; rewrite ?ltry//; case: ifPn => // _; rewrite ltry.
Qed.

Lemma : r != 0%R -> x `^ r < +oo -> x < +oo.

Proof.

by move=> r0; move: x => [x| | _]//=; rewrite ?ltry// (negbTE r0).
Qed.

Lemma : r != 0%R -> 0 `^ r = 0.

Proof.

by move=> r0; rewrite poweR_EFin powR0. Qed.

Lemma : 1 `^ r = 1

Proof.

by rewrite poweR_EFin powR1. Qed.

Lemma : fine (x `^ r) = ((fine x) `^ r)%R.

Proof.

by move: x => [x| |]//=; case: ifPn => [/eqP ->|?]; rewrite ?powRr0 ?powR0.
Qed.

Lemma : 0 <= x `^ r.

Proof.

by move: x => [x| |]/=; rewrite ?lee_fin ?powR_ge0//; case: ifPn. Qed.

Lemma : 0 < x -> 0 < x `^ r.

Proof.

by move: x => [x|_|]//=; [rewrite lte_fin; exact: powR_gt0|case: ifPn].
Qed.

Lemma : (0 < r)%R -> 0 <= x -> 0 < x `^ r -> 0 < x.

Proof.

move=> r0; move: x => [x|//|]; rewrite ?leeNe_eq// lee_fin !lte_fin.
exact: gt0_powR.
Qed.

Lemma : 0 <= x -> (x `^ r == 0) = ((x == 0) && (r != 0%R)).

Proof.

move: x => [x _|_/=|//]; first by rewrite poweR_EFin eqe powR_eq0.
by case: ifP => //; rewrite onee_eq0.
Qed.

Lemma : 0 <= x -> x `^ r = 0 -> x = 0.

Proof.

by move=> + /eqP => /poweR_eq0-> /andP[/eqP]. Qed.

Lemma : (0 <= r)%R ->
{in `[0, +oo] &, {homo poweR ^~ r : / x <= y >-> x <= y}}.

Proof.

move=> r0 + y; case=> //= [x /[1!in_itv]/= /andP[xint _]| _ _].
- case: y => //= [y /[1!in_itv]/= /andP[yint _] xy| _ _].
+ by rewrite !lee_fin ge0_ler_powR.
+ by case: eqP => [->|]; rewrite ?powRr0 ?leey.
- by rewrite leye_eq => /eqP ->.
Qed.

Lemma : x \is a fin_num -> x `^ r \is a fin_num.

Proof.

by move=> xfin; rewrite ge0_fin_numE ?poweR_lty ?ltey_eq ?xfin// poweR_ge0.
Qed.

Lemma : 0 <= x -> 0 <= y -> (x * y) `^ r = x `^ r * y `^ r.

Proof.

have [->|rN0] := eqVneq r 0%R; first by rewrite !poweRe0 mule1.
have powyrM s : (0 <= s)%R -> (+oo * s%:E) `^ r = +oo `^ r * s%:E `^ r.
  case: ltgtP => // [s_gt0 _|<- _]; last first.
    by rewrite mule0 poweRyr// !poweR0r// mule0.
  by rewrite gt0_mulye// poweRyr// gt0_mulye// poweR_gt0.
case: x y => [x| |] [y| |]// x0 y0; first by rewrite /= -EFinM powRM.
- by rewrite muleC powyrM// muleC.
- by rewrite powyrM.
- by rewrite mulyy !poweRyr// mulyy.
Qed.

Lemma : x `^ (r * s) = (x `^ r) `^ s.

Proof.

have [->|s0] := eqVneq s 0%R; first by rewrite mulr0 !poweRe0.
have [->|r0] := eqVneq r 0%R; first by rewrite mul0r poweRe0 poweR1r.
case: x => [x| |]//; first by rewrite /= powRrM.
by rewrite !poweRyr// mulf_neq0.
by rewrite !poweRNyr ?poweR0r ?(negPf s0)// mulf_neq0.
Qed.

Lemma : (x `^ r) `^ s = (x `^ s) `^ r.

Proof.

by rewrite -!poweRrM mulrC. Qed.

Definition := ((r + s == 0)%R ==>
  ((x != 0) && ((x \isn't a fin_num) ==> (r == 0%R) && (s == 0%R)))).
Notation := (poweRD_def x r s) : ereal_scope.

Lemma :
  x `^?(r +? s) = ((r + s == 0)%R ==>
  ((x != 0) && ((x \isn't a fin_num) ==> (r == 0%R) && (s == 0%R)))).

Proof.

by []. Qed.

Lemma :
x `^?(r +? - s) = ((r == s)%R ==>
((x != 0) && ((x \isn't a fin_num) ==> (r == 0%R) && (s == 0%R)))).

Proof.

by rewrite poweRD_defE subr_eq0 oppr_eq0. Qed.

Lemma : (r + s != 0)%R -> x `^?(r +? s).

Proof.

by rewrite poweRD_defE => /negPf->. Qed.

Lemma : (r != s)%R -> x `^?(r +? - s).

Proof.

by rewrite poweRB_defE => /negPf->. Qed.

Lemma : x != 0 -> (r >= 0)%R -> (s >= 0)%R ->
x `^?(r +? s).

Proof.

move=> xN0 rge0 sge0; rewrite /poweRD_def xN0/=.
by case: ltgtP rge0 => // [r_gt0|<-]; case: ltgtP sge0 => // [s_gt0|<-]//=;
rewrite ?addr0 ?add0r ?implybT// gt_eqF//= ?addr_gt0.
Qed.

Lemma : x != 0 -> (r >= 0)%R -> (s <= 0)%R ->
x `^?(r +? - s).

Proof.

by move=> *; rewrite nneg_neq0_poweRD_def// oppr_ge0. Qed.

Lemma : x `^?(r +? s) -> x `^ (r + s) = x `^ r * x `^ s.

Proof.

rewrite /poweRD_def.
have [->|r0]/= := eqVneq r 0%R; first by rewrite add0r poweRe0 mul1e.
have [->|s0]/= := eqVneq s 0%R; first by rewrite addr0 poweRe0 mule1.
case: x => // [t|/=|/=]; rewrite ?(negPf r0, negPf s0, implybF); last 2 first.
- by move=> /negPf->; rewrite mulyy.
- by move=> /negPf->; rewrite mule0.
rewrite !poweR_EFin eqe => /implyP/(_ _)/andP cnd.
by rewrite powRD//; apply/implyP => /cnd[].
Qed.

Lemma : x `^?(r +? - s) -> x `^ (r - s) = x `^ r * x `^ (- s).

Proof.

by move=> rs; rewrite poweRD. Qed.

Lemma : 0 <= x -> x `^ 2^-1 = sqrte x.

Proof.

move: x => [x|_|//]; last by rewrite poweRyr.
by rewrite lee_fin => x0 /=; rewrite powR12_sqrt.
Qed.

End poweR.
Notation := (poweR a x) : ereal_scope.

Section riemannR_series.
Variable : realType.
Implicit Types a : R.
Local Open Scope real_scope.

Definition : R ^nat := fun => (n.+1%:R `^ a)^-1.
Arguments riemannR a n /.

Lemma : 0 <= a -> 0 < riemannR a i.

Proof.

by move=> ?; rewrite /riemannR invr_gt0 powR_gt0. Qed.

Lemma : 0 <= a <= 1 -> ~ cvgn (series (riemannR a)).

Proof.

move=> /andP[a0 a1].
have : forall , harmonic n <= riemannR a n.
  move=> [/=|n]; first by rewrite powR1 invr1.
  rewrite -[leRHS]div1r ler_pdivlMr ?powR_gt0// mulrC ler_pdivrMr//.
  by rewrite mul1r -[leRHS]powRr1// ler_powR// ler1n.
move/(series_le_cvg harmonic_ge0 (fun => ltW (riemannR_gt0 i a0))).
by move/contra_not; apply; exact: dvg_harmonic.
Qed.

End riemannR_series.

Files

Module mathcomp.analysis.exp