(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C. *) From mathcomp Require Import all_ssreflect ssralg ssrint ssrnum matrix. From mathcomp Require Import interval rat. Require Import mathcomp_extra boolp reals ereal nsatz_realtype classical_sets. Require Import signed functions topology normedtype landau sequences derive. Require Import realfun exp. (******************************************************************************) (* Theory of trigonometric functions *) (* *) (* This file provides the definitions of basic trigonometric functions and *) (* develops their theories. *) (* *) (* periodic f T == f is a periodic function of period T *) (* alternating f T == f is an alternating function of period T *) (* sin_coeff x == the sequence of coefficients of sin x *) (* sin x == the sine function, i.e., lim (series (sin_coeff x)) *) (* sin_coeff' x == the sequence of odd coefficients of sin x *) (* cos_coeff x == the sequence of coefficients of cos x *) (* cos x == the cosine function, i.e., lim (series (cos_coeff x)) *) (* cos_coeff' x == the sequence of even coefficients of cos x *) (* pi == pi *) (* tan x == the tangent function *) (* acos x == the arccos function *) (* asin x == the arcsin function *) (* atan x == the arctangent function *) (* *) (* Acknowledgments: the proof of cos 2 < 0 is inspired from HOL-light, some *) (* proofs of trigonometric relations are taken from *) (* https://github.com/affeldt-aist/coq-robot. *) (* *) (******************************************************************************) Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. Import Order.TTheory GRing.Theory Num.Def Num.Theory. Import numFieldNormedType.Exports. Local Open Scope classical_set_scope. Local Open Scope ring_scope. (* NB: backport to mathcomp in progress *) Lemma sqrtrV (R : rcfType) (x : R) : 0 <= x -> Num.sqrt (x^-1) = (Num.sqrt x)^-1. Proof. move=> x_ge0. case: (x =P 0) => [->|/eqP xD0]; first by rewrite invr0 sqrtr0 invr0. rewrite -[LHS]mul1r -(mulVf (_ : Num.sqrt x != 0)); last first. by rewrite sqrtr_eq0 -ltNge; case: ltrgt0P x_ge0 xD0. by rewrite -mulrA -sqrtrM // divff // sqrtr1 mulr1. Qed. Lemma eqr_div (R : numFieldType) (x y z t : R): y != 0 -> t != 0 -> (x / y == z / t) = (x * t == z * y). Proof. move=> yD0 tD0. rewrite -[x in RHS](divfK yD0) -[z in RHS](divfK tD0) mulrAC. by apply/eqP/eqP=> [->//|xyty]; exact/(mulIf tD0)/(mulIf yD0). Qed. Lemma big_nat_mul (R : zmodType) (f : R ^nat) (n k : nat) : \sum_(0 <= i < n * k) f i = \sum_(0 <= i < n) \sum_(i * k <= j < i.+1 * k) f j. Proof. elim: n => [|n ih]; first by rewrite mul0n 2!big_nil. rewrite [in RHS]big_nat_recr//= -ih mulSn addnC [in LHS]/index_iota subn0 iotaD. rewrite big_cat /= [in X in _ = X _]/index_iota subn0; congr (_ + _). by rewrite add0n /index_iota (addnC _ k) addnK. Qed. (* /NB: backport to mathcomp in progress *) Lemma cvg_series_cvg_series_group (R : realFieldType) (f : R ^nat) k : cvg (series f) -> (0 < k)%N -> [series \sum_(n * k <= i < n.+1 * k) f i]_n --> lim (series f). Proof. move=> /cvg_ballP cf k0; apply/cvg_ballP => _/posnumP[e]. have := !! cf _ (gt0 e); rewrite near_map => -[n _ nl]. rewrite near_map; near=> m. rewrite /ball /= [in X in `|_ - X|]/series [in X in `|_ - X|]/= -big_nat_mul. have /nl : (n <= m * k)%N. by near: m; exists n.+1 => //= p /ltnW /leq_trans /(_ (leq_pmulr _ k0)). by rewrite /ball /= distrC. Unshelve. all: by end_near. Qed. Lemma lt_sum_lim_series (R : realFieldType) (f : R ^nat) n : cvg (series f) -> (forall d, 0 < f (n + d.*2)%N + f (n + d.*2.+1)%N) -> \sum_(0 <= i < n) f i < lim (series f). Proof. move=> /cvg_ballP cf fn. have fn0 : 0 < f n + f n.+1 by have := fn 0%N; rewrite double0 addn0 addn1. rewrite ltNge; apply: contraPN cf => ffn /(_ _ fn0). rewrite near_map /ball /=. have nf_ub N : \sum_(0 <= i < n.+2) f i <= \sum_(0 <= i < N.+1.*2 + n) f i. elim: N => // N /le_trans ->//; rewrite -(addn1 (N.+1)) doubleD addnAC. rewrite [in leRHS]/index_iota subn0 iotaD big_cat. rewrite -[in X in _ <= X + _](subn0 (N.+1.*2 + n)%N) ler_addl /= add0n. by rewrite 2!big_cons big_nil addr0 -(addnC n) ltW// -addnS fn. case=> N _ Nfn; have /Nfn/ltr_distlC_addr : (N.+1.*2 + n >= N)%N. by rewrite doubleS -addn2 -addnn -2!addnA leq_addr. rewrite addrA => ffnfn. have : lim (series f) + f n + f n.+1 <= \sum_(0 <= i < N.+1.*2 + n) f i. apply: (le_trans _ (nf_ub N)). by do 2 rewrite big_nat_recr //=; by rewrite -2!addrA ler_add2r. by move/(lt_le_trans ffnfn); rewrite ltxx. Qed. Section periodic. Variables U V : zmodType. Implicit Type f : U -> V. Definition periodic f (T : U) := forall u, f (u + T) = f u. Lemma periodicn f (T : U) : periodic f T -> forall n a, f (a + T *+ n) = f a. Proof. by move=> fT; elim=> [|n ih] a;[rewrite mulr0n addr0|rewrite mulrS addrA ih fT]. Qed. End periodic. Section alternating. Variables (U : zmodType) (V : ringType). Implicit Type f : U -> V. Definition alternating f (T : U) := forall x, f (x + T) = - f x. Lemma alternatingn f (T : U) : alternating f T -> forall n a, f (a + T *+ n) = (- 1) ^+ n * f a. Proof. move=> fT; elim => [a|n ih a]; first by rewrite mulr0n expr0 addr0 mul1r. by rewrite mulrS addrA ih fT exprS mulrN mulN1r mulNr. Qed. End alternating. Section CosSin. Variable R : realType. Implicit Types x y : R. Definition sin_coeff x := [sequence (odd n)%:R * (-1) ^+ n.-1./2 * x ^+ n / n`!%:R]_n. Lemma sin_coeffE x : sin_coeff x = (fun n => (fun n => (odd n)%:R * (-1) ^+ n.-1./2 * (n`!%:R)^-1) n * x ^+ n). Proof. by apply/funext => i; rewrite /sin_coeff /= -!mulrA [_ / _]mulrC. Qed. Lemma sin_coeff_even n x : sin_coeff x n.*2 = 0. Proof. by rewrite /sin_coeff /= odd_double /= !mul0r. Qed. Lemma is_cvg_series_sin_coeff x : cvg (series (sin_coeff x)). Proof. apply: normed_cvg. apply: series_le_cvg; last exact: (@is_cvg_series_exp_coeff _ `|x|). - by move=> n; rewrite normr_ge0. - by move=> n; rewrite divr_ge0. - move=> n /=; rewrite /exp_coeff /sin_coeff /=. rewrite !normrM normfV !normr_nat !normrX normrN normr1 expr1n mulr1. by case: odd; [rewrite mul1r| rewrite !mul0r]. Qed. Definition sin x : R := lim (series (sin_coeff x)). Lemma sinE : sin = fun x => lim (pseries (fun n => (odd n)%:R * (-1) ^+ n.-1./2 * (n`!%:R)^-1) x). Proof. by apply/funext => x; rewrite /pseries -sin_coeffE. Qed. Definition sin_coeff' x (n : nat) := (-1)^n * x ^+ n.*2.+1 / n.*2.+1`!%:R. Lemma sin_coeff'E x n : sin_coeff' x n = sin_coeff x n.*2.+1. Proof. by rewrite /sin_coeff' /sin_coeff /= odd_double mul1r -2!mulrA doubleK. Qed. Lemma cvg_sin_coeff' x : series (sin_coeff' x) --> sin x. Proof. have /(@cvg_series_cvg_series_group _ _ 2) := @is_cvg_series_sin_coeff x. move=> /(_ isT); apply: cvg_trans. rewrite [X in _ --> series X](_ : _ = (fun n => sin_coeff x n.*2.+1)). rewrite [X in series X --> _](_ : _ = (fun n => sin_coeff x n.*2.+1)) //. by rewrite funeqE => n; exact: sin_coeff'E. rewrite funeqE=> n; rewrite /= 2!muln2 big_nat_recl //= sin_coeff_even add0r. by rewrite big_nat_recl // big_geq // addr0. Qed. Lemma diffs_sin : pseries_diffs (fun n => (odd n)%:R * (-1) ^+ n.-1./2 * (n`!%:R)^-1) = (fun n => (~~(odd n))%:R * (-1) ^+ n./2 * (n`!%:R)^-1 : R). Proof. apply/funext => i; rewrite /pseries_diffs /= factS natrM invfM. by rewrite [_.+1%:R * _]mulrC -!mulrA [_.+1%:R^-1 * _]mulrC mulfK. Qed. Lemma series_sin_coeff0 n : series (sin_coeff 0) n.+1 = 0. Proof. rewrite /series /= big_nat_recl //= /sin_coeff /= expr0n divr1 !mulr1. by rewrite big1 ?addr0 // => i _; rewrite expr0n !(mul0r, mulr0). Qed. Lemma sin0 : sin 0 = 0. Proof. apply: lim_near_cst => //; near=> m; rewrite -[m]prednK; last by near: m. rewrite -addn1 series_addn series_sin_coeff0 big_add1 big1 ?addr0//. by move=> i _; rewrite /sin_coeff /= expr0n !(mulr0, mul0r). Unshelve. all: by end_near. Qed. Definition cos_coeff x := [sequence (~~ odd n)%:R * (-1)^n./2 * x ^+ n / n`!%:R]_n. Lemma cos_coeff_odd n x : cos_coeff x n.*2.+1 = 0. Proof. by rewrite /cos_coeff /= odd_double /= !mul0r. Qed. Lemma cos_coeff_2_0 : cos_coeff 2 0%N = 1 :> R. Proof. by rewrite /cos_coeff /= mul1r expr0 mulr1 expr0z divff. Qed. Lemma cos_coeff_2_2 : cos_coeff 2 2%N = - 2%:R :> R. Proof. by rewrite /cos_coeff /= mul1r expr1z mulN1r expr2 mulNr -mulrA divff// mulr1. Qed. Lemma cos_coeff_2_4 : cos_coeff 2 4%N = 2%:R / 3%:R :> R. Proof. rewrite /cos_coeff /= mul1r -exprnP sqrrN expr1n mul1r 2!factS mulnCA mulnC. by rewrite 3!exprS expr1 2!mulrA natrM -mulf_div -2!natrM divff// mul1r. Qed. Lemma cos_coeffE x : cos_coeff x = (fun n => (fun n => (~~(odd n))%:R * (-1) ^+ n./2 * (n`!%:R)^-1) n * x ^+ n). Proof. by apply/funext => i; rewrite /cos_coeff /= -!mulrA [_ / _]mulrC. Qed. Lemma is_cvg_series_cos_coeff x : cvg (series (cos_coeff x)). Proof. apply: normed_cvg. apply: series_le_cvg; last exact: (@is_cvg_series_exp_coeff _ `|x|). - by move=> n; rewrite normr_ge0. - by move=> n; rewrite divr_ge0. - move=> n /=; rewrite /exp_coeff /cos_coeff /=. rewrite !normrM normfV !normr_nat !normrX normrN normr1 expr1n mulr1. by case: odd; [rewrite !mul0r | rewrite mul1r]. Qed. Definition cos x : R := lim (series (cos_coeff x)). Lemma cosE : cos = fun x => lim (series (fun n => (fun n => (~~(odd n))%:R * (-1)^+ n./2 * (n`!%:R)^-1) n * x ^+ n)). Proof. by apply/funext => x; rewrite -cos_coeffE. Qed. Definition cos_coeff' x (n : nat) := (-1)^n * x ^+ n.*2 / n.*2`!%:R. Lemma cos_coeff'E x n : cos_coeff' x n = cos_coeff x n.*2. Proof. rewrite /cos_coeff' /cos_coeff /= odd_double /= mul1r -2!mulrA; congr (_ * _). by rewrite (half_bit_double n false). Qed. Lemma cvg_cos_coeff' x : series (cos_coeff' x) --> cos x. Proof. have /(@cvg_series_cvg_series_group _ _ 2) := @is_cvg_series_cos_coeff x. move=> /(_ isT); apply: cvg_trans. rewrite [X in _ --> series X](_ : _ = (fun n => cos_coeff x n.*2)); last first. rewrite funeqE=> n; rewrite /= 2!muln2 big_nat_recr //= cos_coeff_odd addr0. by rewrite big_nat_recl//= /index_iota subnn big_nil addr0. rewrite [X in series X --> _](_ : _ = (fun n => cos_coeff x n.*2)) //. by rewrite funeqE => n; exact: cos_coeff'E. Qed. Lemma diffs_cos : pseries_diffs (fun n => (~~(odd n))%:R * (-1) ^+ n./2 * (n`!%:R)^-1) = (fun n => - ((odd n)%:R * (-1) ^+ n.-1./2 * (n`!%:R)^-1): R). Proof. apply/funext => [] [|i] /=. by rewrite /pseries_diffs /= !mul0r mulr0 oppr0. rewrite /pseries_diffs /= negbK exprS mulN1r !(mulNr, mulrN). rewrite factS natrM invfM. by rewrite [_.+1%:R * _]mulrC -!mulrA [_.+1%:R^-1 * _]mulrC mulfK. Qed. Lemma series_cos_coeff0 n : series (cos_coeff 0) n.+1 = 1. Proof. rewrite /series /= big_nat_recl //= /cos_coeff /= expr0n divr1 !mulr1. by rewrite big1 ?addr0 // => i _; rewrite expr0n !(mul0r, mulr0). Qed. Lemma cos0 : cos 0 = 1. Proof. apply: lim_near_cst => //; near=> m; rewrite -[m]prednK; last by near: m. rewrite -addn1 series_addn series_cos_coeff0 big_add1 big1 ?addr0//. by move=> i _; rewrite /cos_coeff /= expr0n !(mulr0, mul0r). Unshelve. all: by end_near. Qed. Global Instance is_derive_sin x : is_derive x 1 sin (cos x). Proof. rewrite sinE /=. pose s : R^nat := fun n => (odd n)%:R * (-1) ^+ (n.-1)./2 / n`!%:R. pose s1 n := pseries_diffs s n * x ^+ n. rewrite cosE /= /pseries (_ : (fun _ => _) = s1); last first. by apply/funext => i; rewrite /s1 diffs_sin. apply: (@pseries_snd_diffs _ _ (`|x| + 1)); rewrite /pseries. - by rewrite -sin_coeffE; apply: is_cvg_series_sin_coeff. - rewrite (_ : (fun _ => _) = cos_coeff (`|x| + 1)). exact: is_cvg_series_cos_coeff. by apply/funext => i; rewrite diffs_sin cos_coeffE. - rewrite /pseries (_ : (fun _ => _) = - sin_coeff (`|x| + 1)). by rewrite is_cvg_seriesN; exact: is_cvg_series_sin_coeff. by apply/funext => i; rewrite diffs_sin diffs_cos sin_coeffE !fctE !mulNr. - by rewrite [ltRHS]ger0_norm// addrC -subr_gt0 addrK. Qed. Lemma derivable_sin x : derivable sin x 1. Proof. by apply: ex_derive; apply: is_derive_sin. Qed. Lemma continuous_sin : continuous sin. Proof. by move=> x; apply/differentiable_continuous/derivable1_diffP/derivable_sin. Qed. Global Instance is_derive_cos x : is_derive x 1 cos (- (sin x)). Proof. rewrite cosE /=. pose s : R^nat := fun n => (~~ odd n)%:R * (-1) ^+ n./2 / n`!%:R. pose s1 n := pseries_diffs s n * x ^+ n. rewrite sinE /= /pseries. rewrite (_ : (fun _ => _) = - s1); last first. by apply/funext => i; rewrite /s1 diffs_cos !fctE mulNr opprK. rewrite lim_seriesN ?opprK; last first. rewrite (_ : s1 = - sin_coeff x). by rewrite is_cvg_seriesN; exact: is_cvg_series_sin_coeff. by apply/funext => i; rewrite /s1 diffs_cos sin_coeffE !fctE mulNr. apply: (@pseries_snd_diffs _ _ (`|x| + 1)). - by rewrite /pseries -cos_coeffE; apply: is_cvg_series_cos_coeff. - rewrite /pseries (_ : (fun _ => _) = - sin_coeff (`|x| + 1)). by rewrite is_cvg_seriesN; exact: is_cvg_series_sin_coeff. by apply/funext => i; rewrite diffs_cos sin_coeffE !fctE mulNr. - rewrite /pseries (_ : (fun _=> _) = - cos_coeff (`|x| + 1)). by rewrite is_cvg_seriesN; exact: is_cvg_series_cos_coeff. apply/funext => i; rewrite diffs_cos pseries_diffsN. by rewrite diffs_sin cos_coeffE mulNr. - by rewrite [ltRHS]ger0_norm// addrC -subr_gt0 addrK. Qed. Lemma derivable_cos x : derivable cos x 1. Proof. by apply: ex_derive; apply: is_derive_cos. Qed. Lemma continuous_cos : continuous cos. Proof. by move=> x; exact/differentiable_continuous/derivable1_diffP/derivable_cos. Qed. Lemma cos2Dsin2 x : (cos x) ^+ 2 + (sin x) ^+ 2 = 1. Proof. set v := LHS; pattern x in v; move: @v; set f := (X in let _ := X x in _) => /=. apply: (@eq_trans _ _ (f 0)); last by rewrite /f sin0 cos0 expr1n expr0n addr0. apply: is_derive_0_is_cst => {}x. apply: trigger_derive; rewrite /GRing.scale /=. by rewrite mulrN ![sin x * _]mulrC -opprD addrC subrr. Qed. Lemma cos_max x : `| cos x | <= 1. Proof. rewrite -(expr_le1 (_ : 0 < 2)%nat) // -normrX ger0_norm ?exprn_even_ge0 //. by rewrite -(cos2Dsin2 x) ler_addl ?sqr_ge0. Qed. Lemma cos_geN1 x : -1 <= cos x. Proof. by rewrite ler_oppl; have /ler_normlP[] := cos_max x. Qed. Lemma cos_le1 x : cos x <= 1. Proof. by have /ler_normlP[] := cos_max x. Qed. Lemma sin_max x : `| sin x | <= 1. Proof. rewrite -(expr_le1 (_ : 0 < 2)%nat) // -normrX ger0_norm ?exprn_even_ge0 //. by rewrite -(cos2Dsin2 x) ler_addr ?sqr_ge0. Qed. Lemma sin_geN1 x : -1 <= sin x. Proof. by rewrite ler_oppl; have /ler_normlP[] := sin_max x. Qed. Lemma sin_le1 x : sin x <= 1. Proof. by have /ler_normlP[] := sin_max x. Qed. Fact sinD_cosD x y : (sin (x + y) - (sin x * cos y + cos x * sin y)) ^+ 2 + (cos (x + y) - (cos x * cos y - sin x * sin y)) ^+ 2 = 0. Proof. set v := LHS; pattern x in v; move: @v; set f := (X in let _ := X x in _) => /=. apply: (@eq_trans _ _ (f 0)); last first. by rewrite /f cos0 sin0 !(mul1r, mul0r, add0r, subr0, subrr, expr0n). apply: is_derive_0_is_cst => {}x. by apply: trigger_derive; rewrite /GRing.scale /=; nsatz. Qed. Lemma sinD x y : sin (x + y) = sin x * cos y + cos x * sin y. Proof. have /eqP := sinD_cosD x y. rewrite paddr_eq0 => [/andP[]||]; try exact: sqr_ge0. by rewrite sqrf_eq0 subr_eq0 => /eqP. Qed. Lemma cosD x y : cos (x + y) = cos x * cos y - sin x * sin y. Proof. have /eqP := sinD_cosD x y. rewrite paddr_eq0 => [/andP[_]||]; try exact: sqr_ge0. by rewrite sqrf_eq0 subr_eq0 => /eqP. Qed. Lemma sin2cos2 x : sin x ^+ 2 = 1 - cos x ^+ 2. Proof. by move/eqP: (cos2Dsin2 x); rewrite eq_sym addrC -subr_eq => /eqP. Qed. Lemma cos2sin2 x : cos x ^+ 2 = 1 - sin x ^+ 2. Proof. by move/eqP: (cos2Dsin2 x); rewrite eq_sym -subr_eq => /eqP. Qed. Lemma sin_mulr2n x : sin (x *+ 2) = (cos x * sin x) *+ 2. Proof. by rewrite mulr2n sinD mulrC -mulr2n. Qed. Lemma cos_mulr2n x : cos (x *+ 2) = cos x ^+2 *+ 2 - 1. Proof. by rewrite mulr2n cosD -!expr2 sin2cos2 opprB addrA mulr2n. Qed. Fact sinN_cosN x : (sin (- x) + sin x) ^+ 2 + (cos (- x) - cos x) ^+ 2 = 0. Proof. set v := LHS; pattern x in v; move: @v; set f := (X in let _ := X x in _) => /=. apply: (@eq_trans _ _ (f 0)); last first. by rewrite /f oppr0 cos0 sin0 !(addr0, subrr, expr0n). apply: is_derive_0_is_cst => {}x. by apply: trigger_derive; rewrite /GRing.scale /=; nsatz. Qed. Lemma sinN x : sin (- x) = - sin x. Proof. have /eqP := sinN_cosN x. rewrite paddr_eq0 => [/andP[]||]; try exact: sqr_ge0. by rewrite sqrf_eq0 addr_eq0 => /eqP. Qed. Lemma cosN x : cos (- x) = cos x. Proof. have /eqP := sinN_cosN x. rewrite paddr_eq0 => [/andP[_]||]; try exact: sqr_ge0. by rewrite sqrf_eq0 subr_eq0 => /eqP. Qed. Lemma sin_sg x y : sin (Num.sg x * y) = Num.sg x * sin y. Proof. by case: sgrP; rewrite ?mul1r ?mulN1r ?sinN // !mul0r sin0. Qed. Lemma cos_sg x y : x != 0 -> cos (Num.sg x * y) = cos y. Proof. by case: sgrP; rewrite ?mul1r ?mulN1r ?cosN. Qed. Lemma cosB x y : cos (x - y) = cos x * cos y + sin x * sin y. Proof. by rewrite cosD cosN sinN mulrN opprK. Qed. Lemma sinB x y : sin (x - y) = sin x * cos y - cos x * sin y. Proof. by rewrite sinD cosN sinN mulrN. Qed. Lemma norm_cos_eq1 x : (`|cos x| == 1) = (sin x == 0). Proof. rewrite -sqrf_eq0 -sqrp_eq1 // -normrX ger0_norm ?exprn_even_ge0 //. by rewrite [X in _ = (X == _)]sin2cos2 subr_eq0 eq_sym. Qed. Lemma norm_sin_eq1 x : (`|sin x| == 1) = (cos x == 0). Proof. rewrite -sqrf_eq0 -sqrp_eq1 // -normrX ger0_norm ?exprn_even_ge0 //. by rewrite [X in _ = (X == _)]cos2sin2 subr_eq0 eq_sym. Qed. Lemma cos1sin0 x : `|cos x| = 1 -> sin x = 0. Proof. by move/eqP; rewrite norm_cos_eq1 => /eqP. Qed. Lemma sin1cos0 x : `|sin x| = 1 -> cos x = 0. Proof. by move/eqP; rewrite norm_sin_eq1 => /eqP. Qed. Lemma sin0cos1 x : sin x = 0 -> `|cos x| = 1. Proof. by move/eqP; rewrite -norm_cos_eq1 => /eqP. Qed. Lemma cos_norm x : cos `|x| = cos x. Proof. by case: (ler0P x); rewrite ?cosN. Qed. End CosSin. Arguments sin {R}. Arguments cos {R}. Section Pi. Variable R : realType. Implicit Types (x y : R) (n k : nat). Definition pi : R := get [set x | 0 <= x <= 2 /\ cos x = 0] *+ 2. Lemma pihalfE : pi / 2 = get [set x | 0 <= x <= 2 /\ cos x = 0]. Proof. by rewrite /pi -(mulr_natr (get _)) -mulrA divff ?mulr1. Qed. Lemma cos2_lt0 : cos 2 < 0 :> R. Proof. rewrite -(opprK (cos _)) oppr_lt0; have /cvgN h := @cvg_cos_coeff' R 2. rewrite -(cvg_lim (@Rhausdorff R) h). apply: (@lt_trans _ _ (\sum_(0 <= i < 3) - cos_coeff' 2 i)). do 3 rewrite big_nat_recl//; rewrite big_nil addr0 3!cos_coeff'E double0. rewrite cos_coeff_2_0 cos_coeff_2_2 -muln2 cos_coeff_2_4 addrA -(opprD 1). rewrite opprB -(@natrB _ 2 1)// subn1/= -[in X in X - _](@divff _ 3%:R)//. by rewrite -mulrBl divr_gt0// -natrB// -[(_ - _)%N]/_.+1. rewrite -seriesN lt_sum_lim_series //. by move/cvgP in h; by rewrite seriesN. move=> d. rewrite /cos_coeff' 2!exprzD_nat (exprSz _ d.*2) -[in (-1) ^ d.*2](muln2 d). rewrite -(exprnP _ (d * 2)) (exprM (-1)) sqrr_sign 2!mulr1 -exprSzr. rewrite (_ : 4 = 2 * 2)%N // -(exprnP _ (2 * 2)) (exprM (-1)) sqrr_sign. rewrite mul1r [(-1) ^ 3](_ : _ = -1) ?mulN1r ?mulNr ?opprK; last first. by rewrite -exprnP 2!exprS expr1 mulrN1 opprK mulr1. rewrite subr_gt0. rewrite addnS doubleS -[X in 2 ^+ X]addn2 exprD -mulrA ltr_pmul2l//. rewrite factS factS 2!natrM mulrA invfM !mulrA. rewrite ltr_pdivr_mulr ?ltr0n ?fact_gt0// mulVf ?pnatr_eq0 ?gtn_eqF ?fact_gt0//. rewrite ltr_pdivr_mulr ?mul1r //. by rewrite expr2 -!natrM ltr_nat !mulSn !add2n mul0n !addnS. Qed. Lemma sin2_gt0 x : 0 < x < 2 -> 0 < sin x. Proof. move=> /andP[x_gt0 x_lt2]. have sinx := @cvg_sin_coeff' _ x. rewrite -(cvg_lim (@Rhausdorff R) sinx). rewrite [ltLHS](_ : 0 = \sum_(0 <= i < 0) sin_coeff' x i :> R); last first. by rewrite big_nil. rewrite lt_sum_lim_series //; first by move/cvgP in sinx. move=> d. rewrite /sin_coeff' 2!exprzD_nat (exprSz _ d.*2) -[in (-1) ^ d.*2](muln2 d). rewrite -(exprnP _ (d * 2)) (exprM (-1)) sqrr_sign 2!mulr1 -exprSzr. rewrite !add0n!mul1r mulN1r -[d.*2.+1]addn1 doubleD -addSn exprD. rewrite -(ffact_fact (leq_addl _ _)) addnK. rewrite mulNr -!mulrA -mulrBr mulr_gt0 ?exprn_gt0 //. set u := _.+1. rewrite natrM invfM. rewrite -[X in _ < X - _]mul1r !mulrA -mulrBl divr_gt0 //; last first. by rewrite (ltr_nat _ 0) fact_gt0. rewrite subr_gt0. set v := _ ^_ _; rewrite -[ltRHS](divff (_ : v%:R != 0)); last first. by rewrite lt0r_neq0 // (ltr_nat _ 0) ffact_gt0 leq_addl. rewrite ltr_pmul2r; last by rewrite invr_gt0 (ltr_nat _ 0) ffact_gt0 leq_addl. rewrite {}/v !addnS addn0 !ffactnS ffactn0 muln1 /= natrM. by rewrite (ltr_pmul (ltW _ ) (ltW _)) // (lt_le_trans x_lt2) // ler_nat. Qed. Lemma cos1_gt0 : cos 1 > 0 :> R. Proof. have h := @cvg_cos_coeff' R 1; rewrite -(cvg_lim (@Rhausdorff R) h). apply: (@lt_trans _ _ (\sum_(0 <= i < 2) cos_coeff' 1 i)). rewrite big_nat_recr//= big_nat_recr//= big_nil add0r. rewrite /cos_coeff' expr0z expr1n fact0 !mul1r expr1n expr1z. by rewrite !mulNr subr_gt0 mul1r div1r ltf_pinv ?posrE ?ltr0n// ltr_nat. rewrite lt_sum_lim_series //; [by move/cvgP in h|move=> d]. rewrite /cos_coeff' !(expr1n,mulr1). rewrite -muln2 -mulSn muln2 -exprnP -signr_odd odd_double expr0. rewrite -exprnP -signr_odd oddD/= muln2 odd_double/= expr1 add2n. rewrite mulNr subr_gt0 2!div1r ltf_pinv ?posrE ?ltr0n ?fact_gt0//. by rewrite ltr_nat ltn_pfact//ltn_double doubleS. Qed. Lemma cos_exists : exists2 pih : R, 1 <= pih <= 2 & cos pih = 0. Proof. have /IVT[] : minr (cos 1) (cos 2) <= (0 : R) <= maxr (cos 1) (cos 2). - rewrite /minr /maxr ltNge (ltW (lt_trans cos2_lt0 cos1_gt0))/=. by rewrite (ltW cos2_lt0)/= (ltW cos1_gt0). - by rewrite ler1n. - by move=> *; apply/continuous_subspaceT=> ? _; exact: continuous_cos. by move=> pih /itvP pihI chpi_eq0; exists pih; rewrite ?pihI. Qed. Lemma cos_02_uniq x y : 0 <= x <= 2 -> cos x = 0 -> 0 <= y <= 2 -> cos y = 0 -> x = y. Proof. wlog xLy : x y / x <= y => [H xB cx0 yB cy0|]. by case: (lerP x y) => [/H //| /ltW /H H1]; [exact|exact/esym/H1]. move=> /andP[x_ge0 x_le2] cx0 /andP[y_ge0 y_le2] cy0. case: (x =P y) => // /eqP xDy. have xLLs : x < y by rewrite le_eqVlt (negPf xDy) in xLy. have /(Rolle xLLs)[x1 _|x1|x1 x1I [_ x1D]] : cos x = cos y by rewrite cy0. - exact: derivable_cos. - by apply/continuous_subspaceT=> ? _; exact: continuous_cos. - have [_ /esym/eqP] := is_derive_cos x1; rewrite x1D oppr_eq0 => /eqP Hs. suff : 0 < sin x1 by rewrite Hs ltxx. apply/sin2_gt0/andP; split. + by rewrite (le_lt_trans x_ge0)// (itvP x1I). + by rewrite (lt_le_trans _ y_le2)// (itvP x1I). Qed. Lemma pihalf_02_cos_pihalf : 0 <= pi / 2 <= 2 /\ cos (pi / 2) = 0. Proof. have [x /andP[x1 x2] cs0] := cos_exists; rewrite pihalfE. case: xgetP => [_->[]//|/(_ x)/=]. by rewrite cs0 (le_trans _ x1)// x2 => /not_andP[]. Qed. #[deprecated(note="Use pihalf_ge1 and pihalf_lt2 instead")] Lemma pihalf_02 : 0 < pi / 2 < 2. Proof. have [pih02 cpih] := pihalf_02_cos_pihalf. rewrite 2!lt_neqAle andbCA -andbA pih02 andbT; apply/andP; split. by apply/eqP => pih2; have := cos2_lt0; rewrite -pih2 cpih ltxx. apply/eqP => pih0; have := @cos0 R. by rewrite pih0 cpih; apply/eqP; rewrite eq_sym oner_eq0. Qed. Let pihalf_12 : 1 <= pi / 2 < 2. Proof. have [/andP[pih0 pih2] cpih] := pihalf_02_cos_pihalf. rewrite lt_neqAle andbA andbAC pih2 andbT; apply/andP; split; last first. by apply/eqP => hpi2; have := cos2_lt0; rewrite -hpi2 cpih ltxx. rewrite leNgt; apply/negP => hpi1; have [x /andP[x1 x2] cs0] := cos_exists. have := @cos_02_uniq (pi / 2) x. rewrite pih0 pih2 cpih (le_trans _ x1)// x2 cs0 => /(_ erefl erefl erefl erefl). by move=> pih; move: hpi1; rewrite pih => /lt_le_trans/(_ x1); rewrite ltxx. Qed. Lemma pihalf_ge1 : 1 <= pi / 2. Proof. by have /andP[] := pihalf_12. Qed. Lemma pihalf_lt2 : pi / 2 < 2. Proof. by have /andP[] := pihalf_12. Qed. Lemma pi_ge2 : 2 <= pi. Proof. by have := pihalf_ge1; rewrite ler_pdivl_mulr// mul1r. Qed. Lemma pi_gt0 : 0 < pi. Proof. by rewrite (lt_le_trans _ pi_ge2). Qed. Lemma pi_ge0 : 0 <= pi. Proof. exact: (ltW pi_gt0). Qed. Lemma sin_gt0_pihalf x : 0 < x < pi / 2 -> 0 < sin x. Proof. move=> /andP[x_gt0 xLpi]; apply: sin2_gt0; rewrite x_gt0 /=. by apply: lt_trans xLpi _; exact: pihalf_lt2. Qed. Lemma cos_gt0_pihalf x : -(pi / 2) < x < pi / 2 -> 0 < cos x. Proof. wlog : x / 0 <= x => [Hw|x_ge0]. case: (leP 0 x) => [/Hw//| x_lt_0]. rewrite -{-1}[x]opprK ltr_oppl andbC [-- _ < _]ltr_oppl cosN. by apply: Hw => //; rewrite oppr_cp0 ltW. move=> /andP[x_gt0 xLpi2]; case: (ler0P (cos x)) => // cx_le0. have /IVT[]// : minr (cos 0) (cos x) <= 0 <= maxr (cos 0) (cos x). by rewrite cos0 /minr /maxr !ifN ?cx_le0 //= -leNgt (le_trans cx_le0). - by move=> *; apply/continuous_subspaceT=> ? _; apply: continuous_cos. move=> x1 /itvP Hx1 cx1_eq0. suff x1E : x1 = pi/2. have : x1 < pi / 2 by apply: le_lt_trans xLpi2; rewrite Hx1. by rewrite x1E ltxx. apply: cos_02_uniq=> //; last by case pihalf_02_cos_pihalf => _ ->. by rewrite Hx1 ltW // (lt_trans _ pihalf_lt2) // (le_lt_trans _ xLpi2) // Hx1. by rewrite divr_ge0 ?(ltW pihalf_lt2)// pi_ge0. Qed. Lemma cos_pihalf : cos (pi / 2) = 0. Proof. exact: pihalf_02_cos_pihalf.2. Qed. Lemma sin_pihalf : sin (pi / 2) = 1. Proof. have := cos2Dsin2 (pi / 2); rewrite cos_pihalf expr0n add0r. rewrite -[in X in _ = X -> _](expr1n _ 2%N) => /eqP; rewrite -subr_eq0 subr_sqr. rewrite mulf_eq0=> /orP[|]; first by rewrite subr_eq0=> /eqP. rewrite addr_eq0 => /eqP spi21; have /sin2_gt0: 0 < pi / 2 < 2. by rewrite pihalf_lt2 andbT (lt_le_trans _ pihalf_ge1). by rewrite spi21 ltr0N1. Qed. Lemma cos_ge0_pihalf x : -(pi / 2) <= x <= pi / 2 -> 0 <= cos x. Proof. rewrite le_eqVlt; case: (_ =P x) => /= [<-|_]. by rewrite cosN cos_pihalf. rewrite le_eqVlt; case: (x =P _) => /= [->|_ H]; first by rewrite cos_pihalf. by rewrite ltW //; apply: cos_gt0_pihalf. Qed. Lemma cospi : cos pi = - 1. Proof. by rewrite /pi mulr2n cosD -pihalfE sin_pihalf mulr1 cos_pihalf mulr0 add0r. Qed. Lemma sinpi : sin pi = 0. Proof. have := sinD (pi / 2) (pi / 2); rewrite cos_pihalf mulr0 mul0r. by rewrite -mulrDl -mulr2n -mulr_natr -mulrA divff// mulr1 addr0. Qed. Lemma cos2pi : cos (pi *+ 2) = 1. Proof. by rewrite mulr2n cosD cospi sinpi !mulrN1 mulr0 subr0 opprK. Qed. Lemma sin2pi : sin (pi *+ 2) = 0. Proof. by rewrite mulr2n sinD sinpi cospi !mulrN1 mulr0 oppr0 addr0. Qed. Lemma sinDpi : alternating sin pi. Proof. by move=> a; rewrite sinD cospi mulrN1 sinpi mulr0 addr0. Qed. Lemma cosDpi : alternating cos pi. Proof. by move=> a; rewrite cosD cospi mulrN1 sinpi mulr0 subr0. Qed. Lemma sinD2pi : periodic sin (pi *+ 2). Proof. by move=> a; rewrite sinD cos2pi sin2pi mulr0 mulr1 addr0. Qed. Lemma cosD2pi : periodic cos (pi *+ 2). Proof. by move=> a; rewrite cosD cos2pi mulr1 sin2pi mulr0 subr0. Qed. Lemma cosDpihalf a : cos (a + pi / 2) = - sin a. Proof. by rewrite cosD cos_pihalf mulr0 add0r sin_pihalf mulr1. Qed. Lemma cosBpihalf a : cos (a - pi / 2) = sin a. Proof. by rewrite cosB cos_pihalf mulr0 add0r sin_pihalf mulr1. Qed. Lemma sinDpihalf a : sin (a + pi / 2) = cos a. Proof. by rewrite sinD cos_pihalf mulr0 add0r sin_pihalf mulr1. Qed. Lemma sinBpihalf a : sin (a - pi / 2) = - cos a. Proof. by rewrite sinB cos_pihalf mulr0 add0r sin_pihalf mulr1. Qed. Lemma sin_ge0_pi x : 0 <= x <= pi -> 0 <= sin x. Proof. move=> xI; rewrite -cosBpihalf cos_ge0_pihalf //. by rewrite ler_subr_addl subrr ler_sub_addr -mulr2n -[_ *+ 2]mulr_natr divfK. Qed. Lemma sin_gt0_pi x : 0 < x < pi -> 0 < sin x. Proof. move=> xI; rewrite -cosBpihalf cos_gt0_pihalf //. by rewrite ltr_subr_addl subrr ltr_sub_addr -mulr2n -[_ *+ 2]mulr_natr divfK. Qed. Lemma ltr_cos : {in `[0, pi] &, {mono cos : x y /~ y < x}}. Proof. move=> x y; rewrite !in_itv/= le_eqVlt; case: eqP => [<- _|_] /=. rewrite cos0 le_eqVlt; case: eqP => /= [<- _|_ /andP[y_gt0 gLpi]]. by rewrite cos0 !ltxx. rewrite y_gt0; apply/idP. suff : cos y != 1 by case: ltrgtP (cos_le1 y). rewrite -cos0 eq_sym; apply/eqP => /Rolle [||x1|x1 /itvP x1I [_ x1D]] //. by apply/continuous_subspaceT=> ? _; exact: continuous_cos. case: (is_derive_cos x1) => _ /eqP; rewrite x1D eq_sym oppr_eq0 => /eqP s_eq0. suff : 0 < sin x1 by rewrite s_eq0 ltxx. by apply: sin_gt0_pi; rewrite x1I /= (lt_le_trans (_ : _ < y)) ?x1I // yI. rewrite le_eqVlt; case: eqP => [-> _ /andP[y_ge0]|/= _ /andP[x_gt0 x_ltpi]] /=. rewrite cospi le_eqVlt; case: eqP => /= [-> _|/eqP yDpi y_ltpi]. by rewrite cospi ltxx. by rewrite ltNge cos_geN1 ltNge ltW. rewrite le_eqVlt; case: eqP => [<- _|_] /=. rewrite cos0 [_ < 0]ltNge ltW //=. by apply/idP/negP; rewrite -leNgt cos_le1. rewrite le_eqVlt; case: eqP => /= [-> _ | _ /andP[y_gt0 y_ltpi]]. rewrite cospi x_ltpi; apply/idP. suff : cos x != -1 by case: ltrgtP (cos_geN1 x). rewrite -cospi; apply/eqP => /Rolle [||x1|x1 /itvP x1I [_ x1D]] //. by apply/continuous_subspaceT=> ? _; exact: continuous_cos. case: (is_derive_cos x1) => _ /eqP; rewrite x1D eq_sym oppr_eq0 => /eqP s_eq0. suff : 0 < sin x1 by rewrite s_eq0 ltxx. by apply: sin_gt0_pi; rewrite x1I /= (lt_le_trans (_ : _ < x)) ?x1I. wlog xLy : x y x_gt0 x_ltpi y_gt0 y_ltpi / x <= y => [H | ]. case: (lerP x y) => [/H //->//|yLx]. by rewrite !ltNge ltW ?(ltW yLx) // H // ltW. case: (x =P y) => [->| /eqP xDy]; first by rewrite ltxx. have xLLs : x < y by rewrite le_eqVlt (negPf xDy) in xLy. rewrite xLLs -subr_gt0 -opprB; rewrite -subr_gt0 in xLLs; apply/idP. have [x1|z /itvP zI ->] := @MVT_segment _ cos (-sin) _ _ xLy. by apply/continuous_subspaceT=> ? _; exact: continuous_cos. rewrite -mulNr opprK mulr_gt0 //; apply: sin_gt0_pi. by rewrite (lt_le_trans x_gt0) ?zI //= (le_lt_trans _ y_ltpi) ?zI. Qed. Lemma ltr_sin : {in `[ (- (pi/2)), pi/2] &, {mono sin : x y / x < y}}. Proof. move=> x y /itvP xpi /itvP ypi; rewrite -[sin x]opprK ltr_oppl. rewrite -!cosDpihalf -[x < y](ltr_add2r (pi /2)) ltr_cos// !in_itv/=. - by rewrite -ler_subl_addr sub0r xpi/= [leRHS]splitr ler_add2r xpi. - by rewrite -ler_subl_addr sub0r ypi/= [leRHS]splitr ler_add2r ypi. Qed. Lemma cos_inj : {in `[0,pi] &, injective (@cos R)}. Proof. move=> x y x0pi y0pi xy; apply/eqP; rewrite eq_le; apply/andP; split. - by have := ltr_cos y0pi x0pi; rewrite xy ltxx => /esym/negbT; rewrite -leNgt. - by have := ltr_cos x0pi y0pi; rewrite xy ltxx => /esym/negbT; rewrite -leNgt. Qed. Lemma sin_inj : {in `[(- (pi/2)), (pi/2)] &, injective sin}. Proof. move=> x y /itvP xpi /itvP ypi sinE; have : - sin x = - sin y by rewrite sinE. rewrite -!cosDpihalf => /cos_inj h; apply/(addIr (pi/2))/h; rewrite !in_itv/=. - by rewrite -ler_subl_addr sub0r xpi/= [leRHS]splitr ler_add2r xpi. - by rewrite -ler_subl_addr sub0r ypi/= [leRHS]splitr ler_add2r ypi. Qed. End Pi. Arguments pi {R}. Section Tan. Variable R : realType. Definition tan (x : R) := sin x / cos x. Lemma tan0 : tan 0 = 0 :> R. Proof. by rewrite /tan sin0 cos0 mul0r. Qed. Lemma tanpi : tan pi = 0. Proof. by rewrite /tan sinpi mul0r. Qed. Lemma tanN x : tan (- x) = - tan x. Proof. by rewrite /tan sinN cosN mulNr. Qed. Lemma tanD x y : cos x != 0 -> cos y != 0 -> tan (x + y) = (tan x + tan y) / (1 - tan x * tan y). Proof. move=> cxNZ cyNZ. rewrite /tan sinD cosD !addf_div // [sin y * cos x]mulrC -!mulrA -invfM. congr (_ / _). rewrite mulrBr mulr1 !mulrA. rewrite -[_ * _ * sin x]mulrA [cos x * (_ * _)]mulrC mulfK //. by rewrite -[_ * _ * sin y]mulrA [cos y * (_ * _)]mulrC mulfK. Qed. Lemma tan_mulr2n x : cos x != 0 -> tan (x *+ 2) = tan x *+ 2 / (1 - tan x ^+ 2). Proof. move=> cxNZ. rewrite /tan cos_mulr2n sin_mulr2n. rewrite !mulr2n exprMn exprVn -[in RHS](divff (_ : 1 != 0)) //. rewrite -mulNr !addf_div ?sqrf_eq0 //. rewrite mul1r mulr1 -!mulrA -invfM -expr2; congr (_ / _). by rewrite [cos x * _]mulrC. rewrite mulrCA mulrA mulfK ?sqrf_eq0 // [X in _ = _ - X]sin2cos2. by rewrite opprB addrA. Qed. Lemma cos2_tan2 x : cos x != 0 -> (cos x) ^- 2 = 1 + (tan x) ^+ 2. Proof. move=> cosx. rewrite /tan exprMn [X in _ = 1 + X * _]sin2cos2 mulrBl -exprMn divff //. by rewrite expr1n addrCA subrr addr0 mul1r exprVn. Qed. Lemma tan_pihalf : tan (pi / 2) = 0. Proof. by rewrite /tan cos_pihalf invr0 mulr0. Qed. Lemma tan_piquarter : tan (pi / 4%:R) = 1. Proof. rewrite /tan -cosBpihalf (splitr (pi / 2)) opprD addrA -mulrA -invfM -natrM. rewrite subrr sub0r cosN divff// gt_eqF// cos_gt0_pihalf//. rewrite ltr_pmul2l ?pi_gt0// ltf_pinv ?qualifE// ltr_nat andbT. by rewrite (@lt_trans _ _ 0)// ?oppr_lt0 ?divr_gt0 ?pi_gt0. Qed. Lemma tanDpi x : tan (x + pi) = tan x. Proof. by rewrite /tan cosDpi sinDpi mulNr invrN mulrN opprK. Qed. Lemma continuous_tan x : cos x != 0 -> {for x, continuous tan}. Proof. move=> cxNZ. apply: continuousM; first exact: continuous_sin. exact/(continuousV cxNZ)/continuous_cos. Qed. Lemma is_derive_tan x : cos x != 0 -> is_derive x 1 tan ((cos x)^-2). Proof. move=> cxNZ; apply: trigger_derive. rewrite /= ![_ *: - _]mulrN mulNr mulrN opprK [_^-1 *: _]mulVf //. rewrite mulrCA -expr2 [X in _ * X + _ = _]sin2cos2. by rewrite mulrBr mulr1 mulVf ?sqrf_eq0 // subrK. Qed. Lemma derivable_tan x : cos x != 0 -> derivable tan x 1. Proof. by move=> /is_derive_tan[]. Qed. Lemma ltr_tan : {in `](- (pi/2)), (pi/2)[ &, {mono tan : x y / x < y}}. Proof. move=> x y. wlog xLy : x y / x <= y => [H | ] xB yB. case: (lerP x y) => [/H //->//|yLx]. by rewrite !ltNge ltW ?(ltW yLx) // H // ltW. case: (x =P y) => [->| /eqP xDy]; first by rewrite ltxx. have xLLs : x < y by rewrite le_eqVlt (negPf xDy) in xLy. rewrite -subr_gt0 xLLs; rewrite -subr_gt0 in xLLs; apply/idP. have [x1 /itvP x1I|z |] := @MVT_segment _ tan (fun x => (cos x) ^-2) _ _ xLy. - apply: is_derive_tan. rewrite gt_eqF // cos_gt0_pihalf // (@lt_le_trans _ _ x) ?x1I ?(itvP xB)//=. by rewrite (@le_lt_trans _ _ y) ?x1I ?(itvP yB). - apply/continuous_subspaceT=> ? inI; apply: continuous_tan. rewrite /= inE /<=%O/= in inI; move/andP: inI => /= [? ?]. rewrite gt_eqF // cos_gt0_pihalf // (@lt_le_trans _ _ x) ?zI ?(itvP xB)//=. rewrite (@le_lt_trans _ _ y) ?zI ?(itvP yB) //. - move=> x1 /itvP x1I ->. rewrite mulr_gt0 // invr_gt0 // exprn_gte0 // cos_gt0_pihalf //. rewrite (@lt_le_trans _ _ x) ?x1I ?(itvP xB)//=. by rewrite (@le_lt_trans _ _ y) ?x1I ?(itvP yB). Qed. Lemma tan_inj : {in `](- (pi/2)), (pi/2)[ &, injective tan}. Proof. move=> x y xB yB tanE. by case: (ltrgtP x y); rewrite // -ltr_tan ?tanE ?ltxx. Qed. End Tan. Arguments tan {R}. #[global] Hint Extern 0 (is_derive _ _ tan _) => (eapply is_derive_tan; first by []) : typeclass_instances. Section Acos. Variable R : realType. Definition acos (x : R) : R := get [set y | 0 <= y <= pi /\ cos y = x]. Lemma acos_def x : -1 <= x <= 1 -> 0 <= acos x <= pi /\ cos (acos x) = x. Proof. move=> xB; rewrite /acos; case: xgetP => //= He. pose f y := cos y - x. have /(IVT (@pi_ge0 _))[] // : minr (f 0) (f pi) <= 0 <= maxr (f 0) (f pi). rewrite /f cos0 cospi /minr /maxr ltr_add2r -subr_lt0 opprK (_ : 1 + 1 = 2)//. by rewrite ltrn0 subr_le0 subr_ge0. - move=> y y0pi. by apply: continuousB; apply/continuous_subspaceT=> ? ?; [exact: continuous_cos|exact: cst_continuous]. - rewrite /f => x1 /itvP x1I /eqP; rewrite subr_eq0 => /eqP cosx1E. by case: (He x1); rewrite !x1I. Qed. Lemma acos_ge0 x : -1 <= x <= 1 -> 0 <= acos x. Proof. by move=> /acos_def[/andP[]]. Qed. Lemma acos_lepi x : -1 <= x <= 1 -> acos x <= pi. Proof. by move=> /acos_def[/andP[]]. Qed. Lemma acosK : {in `[(-1),1], cancel acos cos}. Proof. by move=> x; rewrite in_itv/==> /acos_def[/andP[]]. Qed. Lemma acos_gt0 x : -1 <= x < 1 -> 0 < acos x. Proof. move=> /andP[x_geN1 x_lt1]; move: (x_lt1). have : 0 <= acos x by rewrite acos_ge0 // x_geN1 ltW. have : cos (acos x) = x by rewrite acosK// in_itv/= x_geN1/= ltW. by case: ltrgt0P => // ->; rewrite cos0 => ->; rewrite ltxx. Qed. Lemma acos_ltpi x : -1 < x <= 1 -> acos x < pi. Proof. move=> /andP[x_gtN1 x_le1]; move: (x_gtN1). have : acos x <= pi by rewrite acos_lepi // x_le1 ltW. have : cos (acos x) = x by rewrite acosK// in_itv/= x_le1 ltW. by case: (ltrgtP (acos x) pi) => // ->; rewrite cospi => ->; rewrite ltxx. Qed. Lemma cosK : {in `[0, pi], cancel cos acos}. Proof. move=> x xB; apply: cos_inj => //; rewrite ?acosK//; last first. by move: xB; rewrite !in_itv/= => /andP[? ?];rewrite cos_geN1 cos_le1. move: xB; rewrite !in_itv/= => /andP[? ?]. by rewrite acos_ge0 ?acos_lepi ?cos_geN1 ?cos_le1. Qed. Lemma acos1 : acos (1 : R) = 0. Proof. by have := @cosK 0; rewrite cos0 => -> //; rewrite in_itv //= lexx pi_ge0. Qed. Lemma acos0 : acos (0 : R) = pi / 2%:R. Proof. have := @cosK (pi / 2%:R). rewrite cos_pihalf => -> //; rewrite in_itv//= divr_ge0 ?ler0n ?pi_ge0//=. by rewrite ler_pdivr_mulr ?ltr0n// ler_pemulr ?pi_ge0// ler1n. Qed. Lemma acosN a : -1 <= a <= 1 -> acos (- a) = pi - acos a. Proof. move=> a1; have ? : -1 <= - a <= 1 by rewrite ler_oppl opprK ler_oppl andbC. apply: cos_inj; first by rewrite in_itv/= acos_ge0//= acos_lepi. - by rewrite in_itv/= subr_ge0 acos_lepi//= ler_subl_addl ler_addr acos_ge0. - by rewrite addrC cosDpi cosN !acosK. Qed. Lemma acosN1 : acos (- 1) = (pi : R). Proof. by rewrite acosN ?acos1 ?subr0 ?lexx// -subr_ge0 opprK addr_ge0. Qed. Lemma cosKN a : - pi <= a <= 0 -> acos (cos a) = - a. Proof. by move=> pia0; rewrite -(cosN a) cosK// in_itv/= ler_oppr oppr0 ler_oppl andbC. Qed. Lemma sin_acos x : -1 <= x <= 1 -> sin (acos x) = Num.sqrt (1 - x^+2). Proof. move=> xB. rewrite -[LHS]ger0_norm; last by rewrite sin_ge0_pi // acos_ge0 ?acos_lepi. by rewrite -sqrtr_sqr sin2cos2 acosK. Qed. Lemma continuous_acos x : -1 < x < 1 -> {for x, continuous acos}. Proof. move=> /andP[x_gtN1 x_lt1]; rewrite -[x]acosK; first last. by have : -1 <= x <= 1 by rewrite !ltW //; case/andP: xB. apply: nbhs_singleton (near_can_continuous _ _); last first. by near=> z; apply: continuous_cos. have /near_in_itv aI : acos x \in `]0, pi[. suff : 0 < acos x < pi by []. by rewrite acos_gt0 ?ltW //= acos_ltpi // ltW ?andbT. near=> z; apply: cosK. suff /itvP zI : z \in `]0, pi[ by have : 0 <= z <= pi by rewrite ltW ?zI. by near: z. Unshelve. all: by end_near. Qed. Lemma is_derive1_acos (x : R) : -1 < x < 1 -> is_derive x 1 acos (- (Num.sqrt (1 - x ^+ 2))^-1). Proof. move=> /andP[x_gtN1 x_lt1]; rewrite -sin_acos ?ltW // -invrN. rewrite -{1}[x]acosK; last by have : -1 <= x <= 1 by rewrite ltW // ltW. have /near_in_itv aI : acos x \in `]0, pi[. suff : 0 < acos x < pi by []. by rewrite acos_gt0 ?ltW //= acos_ltpi // ltW ?andbT. apply: (@is_derive_inverse R cos). - near=> z; apply: cosK. suff /itvP zI : z \in `]0, pi[ by have : 0 <= z <= pi by rewrite ltW ?zI. by near: z. - by near=> z; apply: continuous_cos. - rewrite oppr_eq0 sin_acos ?ltW // sqrtr_eq0 // -ltNge subr_gt0. rewrite -real_normK ?qualifE; last by case: ltrgt0P. by rewrite exprn_cp1 // ltr_norml x_gtN1. Unshelve. all: by end_near. Qed. End Acos. #[global] Hint Extern 0 (is_derive _ 1 (@acos _) _) => (eapply is_derive1_acos; first by []) : typeclass_instances. Section Asin. Variable R : realType. Definition asin (x : R) : R := get [set y | -(pi / 2) <= y <= pi / 2 /\ sin y = x]. Lemma asin_def x : -1 <= x <= 1 -> -(pi / 2) <= asin x <= pi / 2 /\ sin (asin x) = x. Proof. move=> xB; rewrite /asin; case: xgetP => //= He. pose f y := sin y - x. have /IVT[] // : minr (f (-(pi/2))) (f (pi/2)) <= 0 <= maxr (f (-(pi/2))) (f (pi/2)). rewrite /f sinN sin_pihalf /minr /maxr ltr_add2r -subr_gt0 opprK. by rewrite (_ : 1 + 1 = 2)// ltr0n/= subr_le0 subr_ge0. - by rewrite -subr_ge0 opprK -splitr pi_ge0. - by move=> *; apply: continuousB; apply/continuous_subspaceT=> ? ?; [exact: continuous_sin| exact: cst_continuous]. - rewrite /f => x1 /itvP x1I /eqP; rewrite subr_eq0 => /eqP sinx1E. by case: (He x1); rewrite !x1I. Qed. Lemma asin_geNpi2 x : -1 <= x <= 1 -> -(pi / 2) <= asin x. Proof. by move=> /asin_def[/andP[]]. Qed. Lemma asin_lepi2 x : -1 <= x <= 1 -> asin x <= pi / 2. Proof. by move=> /asin_def[/andP[]]. Qed. Lemma asinK : {in `[(-1),1], cancel asin sin}. Proof. by move=> x; rewrite in_itv/= => /asin_def[/andP[]]. Qed. Lemma asin_ltpi2 x : -1 <= x < 1 -> asin x < pi/2. Proof. move=> /andP[x_geN1 x_lt1]; move: (x_lt1). have : asin x <= pi / 2 by rewrite asin_lepi2 // x_geN1 ltW. have : sin (asin x) = x by rewrite asinK// in_itv/= x_geN1 ltW. case: (ltrgtP _ ((pi / 2))) => // ->. by rewrite sin_pihalf => <-; rewrite ltxx. Qed. Lemma asin_gtNpi2 x : -1 < x <= 1 -> - (pi / 2) < asin x. Proof. move=> /andP[x_gtN1 x_le1]; move: (x_gtN1). have : - (pi / 2) <= asin x by rewrite asin_geNpi2 // x_le1 ltW. have : sin (asin x) = x by rewrite asinK// in_itv/= x_le1 ltW. by case: (ltrgtP (asin x)) => //->; rewrite sinN sin_pihalf => <-; rewrite ltxx. Qed. Lemma sinK : {in `[(- (pi / 2)), pi / 2], cancel sin asin}. Proof. move=> x; rewrite !in_itv/= => xB ; apply: sin_inj => //; last first. by rewrite asinK// in_itv/= sin_geN1 sin_le1. by rewrite in_itv/= asin_geNpi2/= ?asin_lepi2 ?sin_geN1 ?sin_le1. Qed. Lemma cos_asin x : -1 <= x <= 1 -> cos (asin x) = Num.sqrt (1 - x^+2). Proof. move=> xB; rewrite -[LHS]ger0_norm; first by rewrite -sqrtr_sqr cos2sin2 asinK. by apply: cos_ge0_pihalf; rewrite asin_lepi2 // asin_geNpi2. Qed. Lemma continuous_asin x : -1 < x < 1 -> {for x, continuous asin}. Proof. move=> /andP[x_gtN1 x_lt1]; rewrite -[x]asinK; first last. by have : -1 <= x <= 1 by rewrite !ltW //; case/andP: xB. apply: nbhs_singleton (near_can_continuous _ _); last first. by near=> z; apply: continuous_sin. have /near_in_itv aI : asin x \in `](-(pi/2)), (pi/2)[. suff : - (pi / 2) < asin x < pi / 2 by []. by rewrite asin_gtNpi2 ?ltW ?andbT //= asin_ltpi2 // ltW. near=> z; apply: sinK. suff /itvP zI : z \in `](-(pi/2)), (pi/2)[. by have : - (pi / 2) <= z <= pi / 2 by rewrite ltW ?zI. by near: z. Unshelve. all: by end_near. Qed. Lemma is_derive1_asin (x : R) : -1 < x < 1 -> is_derive x 1 asin ((Num.sqrt (1 - x ^+ 2))^-1). Proof. move=> /andP[x_gtN1 x_lt1]; rewrite -cos_asin ?ltW //. rewrite -{1}[x]asinK; last by have : -1 <= x <= 1 by rewrite ltW // ltW. have /near_in_itv aI : asin x \in `](-(pi/2)), (pi/2)[. suff : -(pi/2) < asin x < pi/2 by []. by rewrite asin_gtNpi2 ?ltW ?andbT //= asin_ltpi2 // ltW. apply: (@is_derive_inverse R sin). - near=> z; apply: sinK. suff /itvP zI : z \in `](-(pi/2)), (pi/2)[. by have : - (pi / 2) <= z <= pi / 2 by rewrite ltW ?zI. by near: z. - by near=> z; exact: continuous_sin. - rewrite cos_asin ?ltW // sqrtr_eq0 // -ltNge subr_gt0. rewrite -real_normK ?qualifE; last by case: ltrgt0P. by rewrite exprn_cp1 // ltr_norml x_gtN1. Unshelve. all: by end_near. Qed. End Asin. #[global] Hint Extern 0 (is_derive _ 1 (@asin _) _) => (eapply is_derive1_asin; first by []) : typeclass_instances. Section Atan. Variable R : realType. Definition atan (x : R) : R := get [set y | -(pi / 2) < y < pi / 2 /\ tan y = x]. (* Did not see how to use ITV like in the other *) Lemma atan_def x : -(pi / 2) < atan x < pi / 2 /\ tan (atan x) = x. Proof. rewrite /atan; case: xgetP => //= He. pose x1 := Num.sqrt (1 + x^+ 2) ^-1. have ox2_gt0 : 0 < 1 + x^2. by apply: lt_le_trans (_ : 1 <= _); rewrite ?ler_addl ?sqr_ge0. have ox2_ge0 : 0 <= 1 + x^2 by rewrite ltW. have x1B : -1 <= x1 <= 1. rewrite -ler_norml /x1 ger0_norm ?sqrtr_ge0 // -[leRHS]sqrtr1. by rewrite ler_psqrt ?qualifE ?invr_gte0 //= invf_cp1 // ler_addl sqr_ge0. case: (He (Num.sg x * acos x1)); split; last first. case: (x =P 0) => [->|/eqP xD0]; first by rewrite /tan sgr0 mul0r sin0 mul0r. rewrite /tan sin_sg cos_sg // acosK ?sin_acos //. rewrite /x1 sqr_sqrtr// ?invr_ge0 //. rewrite -{1}[_^-1 in X in X / _ = _]mul1r. rewrite -{1}[X in X - _](divff (_: 1 != 0)) //. rewrite -mulNr addf_div ?lt0r_neq0 //. rewrite mul1r mulr1 [X in X - 1]addrC addrK // sqrtrM ?sqr_ge0 //. rewrite sqrtrV // invrK // mulrA divfK //; last by rewrite sqrtr_eq0 -ltNge. by rewrite sqrtr_sqr mulr_sg_norm. rewrite -ltr_norml normrM. have pi2 : 0 < pi / 2 :> R by rewrite divr_gt0 // pi_gt0. case: (x =P 0) => [->|/eqP xD0]; first by rewrite sgr0 normr0 mul0r. rewrite normr_sg xD0 mul1r ltr_norml. rewrite (@lt_le_trans _ _ 0) ?acos_ge0 ?oppr_cp0 //=. rewrite -ltr_cos ?in_itv/= ?acos_ge0/= ?acos_lepi//; last first. by rewrite divr_ge0 ?pi_ge0//= ler_pdivr_mulr// ler_pmulr ?pi_gt0// ler1n. by rewrite cos_pihalf acosK // ?sqrtr_gt0 ?invr_gt0. Qed. Lemma atan_gtNpi2 x : - (pi / 2) < atan x. Proof. by case: (atan_def x) => [] /andP[]. Qed. Lemma atan_ltpi2 x : atan x < pi / 2. Proof. by case: (atan_def x) => [] /andP[]. Qed. Lemma atanK : cancel atan tan. Proof. by move=> x; case: (atan_def x). Qed. Lemma atan0 : atan 0 = 0 :> R. Proof. apply: tan_inj; last by rewrite atanK tan0. - by rewrite in_itv/= atan_gtNpi2 atan_ltpi2. - by rewrite in_itv/= oppr_cp0 divr_gt0 ?pi_gt0. Qed. Lemma atan1 : atan 1 = pi / 4%:R :> R. Proof. apply: tan_inj; first 2 last. by rewrite atanK tan_piquarter. by rewrite in_itv/= atan_gtNpi2 atan_ltpi2. rewrite in_itv/= -mulNr (lt_trans _ (_ : 0 < _ )) /=; last 2 first. by rewrite mulNr oppr_cp0 divr_gt0 // pi_gt0. by rewrite divr_gt0 ?pi_gt0 // ltr0n. rewrite ltr_pdivr_mulr// -mulrA ltr_pmulr// ?pi_gt0//. by rewrite (natrM _ 2 2) mulrA mulVf// mul1r ltr1n. Qed. Lemma atanN x : atan (- x) = - atan x. Proof. apply: tan_inj; first by rewrite in_itv/= atan_ltpi2 atan_gtNpi2. - by rewrite in_itv/= ltr_oppl opprK ltr_oppl andbC atan_ltpi2 atan_gtNpi2. - by rewrite tanN !atanK. Qed. Lemma tanK : {in `](- (pi / 2)), (pi / 2)[ , cancel tan atan}. Proof. move=> x xB; apply tan_inj => //; rewrite ?atanK//. by rewrite in_itv/= atan_gtNpi2 atan_ltpi2. Qed. Lemma continuous_atan x : {for x, continuous atan}. Proof. rewrite -[x]atanK. have /near_in_itv aI : atan x \in `](-(pi / 2)), (pi / 2)[. suff : - (pi / 2) < atan x < pi / 2 by []. by rewrite atan_gtNpi2 atan_ltpi2. apply: nbhs_singleton (near_can_continuous _ _); last first. by near=> z; apply/continuous_tan/lt0r_neq0/cos_gt0_pihalf; near: z. by near=> z; apply: tanK; near: z. Unshelve. all: by end_near. Qed. Lemma cos_atan x : cos (atan x) = (Num.sqrt (1 + x ^+ 2)) ^-1. Proof. have cos_gt0 : 0 < cos (atan x). by apply: cos_gt0_pihalf; rewrite atan_gtNpi2 atan_ltpi2. have cosD0 : cos (atan x) != 0 by apply: lt0r_neq0. have /eqP : cos (atan x) ^+2 = (Num.sqrt (1 + x ^+ 2))^-2. by rewrite -[LHS]invrK cos2_tan2 // atanK sqr_sqrtr // addr_ge0 // sqr_ge0. rewrite -exprVn eqf_sqr => /orP[] /eqP // cosE. move: cos_gt0; rewrite cosE ltNge; case/negP. by rewrite oppr_le0 invr_ge0 sqrtr_ge0. Qed. Global Instance is_derive1_atan (x : R) : is_derive x 1 atan (1 + x ^+ 2)^-1. Proof. rewrite -{1}[x]atanK. have cosD0 : cos (atan x) != 0. by apply/lt0r_neq0/cos_gt0_pihalf; rewrite atan_gtNpi2 atan_ltpi2. have /near_in_itv aI : atan x \in `](-(pi/2)), (pi/2)[. suff : - (pi / 2) < atan x < pi / 2 by []. by rewrite atan_gtNpi2 atan_ltpi2. apply: (@is_derive_inverse R tan). - by near=> z; apply: tanK; near: z. - by near=> z; apply/continuous_tan/lt0r_neq0/cos_gt0_pihalf; near: z. - by rewrite -[X in 1 + X ^+ 2]atanK -cos2_tan2 //; exact: is_derive_tan. by apply/lt0r_neq0/(@lt_le_trans _ _ 1) => //; rewrite ler_addl sqr_ge0. Unshelve. all: by end_near. Qed. End Atan.