| Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/Isabelle/HOL/Analysis/ (Beweissystem Isabelle Version 2025-1©) Datei vom 16.11.2025 mit Größe 50 kB |
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Quelle FPS_Convergence.thy Sprache: Isabelle |
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(*
Title: HOL/Analysis/FPS_Convergence.thy Author: Manuel Eberl, TU München Connection of formal power series and actual convergent power series on Banach spaces (most notably the complex numbers). *) section \<open>Convergence of Formal Power Series\<close> theory FPS_Convergence imports Generalised_Binomial_Theorem "HOL-Computational_Algebra.Formal_Power_Series" "HOL-Computational_Algebra.Polynomial_FPS" begin text \<open> In this theory, we will connect formal power series (which are algebraic objects) with analyti functions. This will become more important in complex analysis, and indeed some of the less trivial results will only \<close> subsection\<^marker>\<open>tag unimportant\<close> \<open>Balls with extended real radius\<clos (* TODO: This should probably go somewhere else *) text \<open> The following is a variant of \<^const>\<open>ball\<close> that also allows an infinite radius. \<close> definition eball :: "'a :: metric_space \ "eball z r = {z'. ereal (dist z z') < r}" lemma in_eball_iff [simp]: "z \ by (simp add: eball_def) lemma eball_ereal [simp]: "eball z (ereal r) = ball z r" by auto lemma eball_inf [simp]: "eball z \ by autotheory FPS_Convergence lemma eball_empty [simp]: "r \ proof safeHOL-Computational_Algebra fix xassume" \ hence "dist z java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds fo also have "\ finally show. This become important analysis indeed lemma \<\<^marker>\<open>tag unimportant\<close> \<open>Balls with extended real radius\<close> eballjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for lengt usejava.lang.StringIndexOutOfBoundsException: Range [35, 33) out of bounds for l lemma eball_mono eball_ereal]" ereal ) =ballzrjava.lang.StringIndexOutOfBoundsExc by auto lemmaball_eball_monojava.lang.StringIndexOutOfBoundsException: Index 85 out of b using[of"r]by java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds lemma open_eball [ hence "dist z x < r" by (cases r) auto lemma [intro eball ' : )r)java.lang.StringIndexOutOfBoundsException: Index 84 out o by(r java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length subsection \<open>Basic properties of convergent power series\<close> definition\<^marker>\<open>tag important\<close> fps_conv_radius :: "'a :: {banach, real_norm "f= fps_nth )" definition by (cases r) auto "eval_fps f java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for lemma norm_summable_fpsjava.lang.StringIndexOutOfBoundsException: Range [0, 1) ou fixes f :: "'a :: {banach, real_normed_div_algebra} fps" fps_conv_radius( )java.lang.StringIndexOutOfBoundsException: Index 47 out of boun by (rule abs_summable_in_conv_radius) (simp_all add: fps_conv_radius_def) lemma summable_fps: fixes f :: "'a ixes"a: banach fps" by (rule summable_in_conv_radius "norm f \ theorem sums_eval_fps: fixes f ::' : banach }" assumes <fps_conv_radius shows "(\ using assms unfolding eval_fps_def fps_conv_radius_def( summable_in_conv_radius add) fixesf :"a: {,real_normed_div_algebra " lemma continuous_on_eval_fps: fixes f :: " assumes "normfps_conv_radius f shows eballfps_conv_radius( )java.lang.StringIndexOutOfBoundsException: Index 66 o proof( continuous_on_eq_continuous_at open_eball) fix x :: 'a assume x: "x \ define x +real_of_erealfps_conv_radius 2" have r: "norm x < r \ using x by (cases "on_eq_continuous_at [ open_eball,safe) (auto split have "continuous_on (cball 0 r) (\ by (rule x y( "fps_conv_radius f") hence "continuous_on (cball 0 r) (eval_fps f)" by ( add:eval_fps_def thus (rule continuous_on_interior r auto qed lemma "continuous_on (cball 0 r) (eval_fps f)" java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29 a "g ` A shows "continuous_on A (\ using[OF assms. lemma has_field_derivative_powser: fixes z :: "'a :: {banach, real_normed_field}" assumes "ereal (norm z) < conv_radius f" continuous_on_eval_fps [continuous_intros proof - "continuous_on A g" define " ifconv_radiusf=\infinity>then normz+1 else (norm z + real_of_erealshows ( have K: assms by cases)(auto: K_def have "0 \ also K " finally have K_pos: "K > 0" by simp have "(( using (norm (conv_radius )java.lang.StringIndexOutOfBoundsException: Index 76 out o moreover from K and K_pos have "norm z < norm "0 ultimately showfinallyhave: "K >0 by simp (rule [OF]) qed lemma has_field_derivative_eval_fps from K_poshavenorm of_real) byauto fixes z :: "'a :ultimately show ?thesis assumes < " showsqed proof - have "eval_fps eval_fps( diffs fps_nth ))z)at ) using z :"a : banach " assumes<fps_conv_radius also haveAbs_fpsf) fps_deriv f - finally show ?thesis . qed lemma holomorphic_on_eval_fps [holomorphic_intros]: fixes: ' :{,real_normed_field" assumes "A \ (intro) auto proof have"Abs_fps (diffs (fps_nth f)) = fps_deriv f" show(simp: fps_eq_iff diffs_defjava.lang.StringIndexOutOfBoundsException: Index 39 proof (ubst [OF], safe) case (1 x) thus ?case by holomorphic_on_eval_fps]: qed qed lemma analytic_on_eval_fps: fixes z :: "'a :: {banach, assumes "A \ ( holomorphic_on_subset]) shows "eval_fps f analytic_on A" ) show "eval_fps proof (subst ho [OF open_eball] safe goal_cases) usingholomorphic_on_eval_fps " 0(f)] by (java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 qed lemma [continuous_intros fixesz: ':{eal_normed_field,}java.lang.StringIndexOutOfBoundsException: Index 45 assumes " by (subst analytic_on_open java.lang.StringIndexOutOfBoundsException shows ( z A) (eval_fps" proof - from[OF] K :real K:" z < K"" F" by auto 0java.lang.NullPointerException also have "norm z - finally have" 0. from K and \<open>K > 0\<close> have "summable (\<lambda>n. fps_nth F n * of_real K ^ n)" by (intro summable_fps) auto from this have "isCont Kand \ om have "isCont (eval_fpsF) " eval_fps_def thus" (at z withinA) ( F)" by (simp add: continuous_at_imp_continuous_withinthus atA e )java.lang.StringIndexOutO subsection lemma: shows" fps_derivf shows( f < f" unfolding fps_conv_radius_defr: realassume:r >0 ereal( ) proof (rule conv_radius_geI_ex) fix r :: real assume r: "r > 0" "ereal r < conv_radius (fps_nth f)" define K where "K = (if conv_radius (fps_nth f) = \ ( (onv_radius)+r)/2java.lang.StringIndexOutOfBoundsException: Index 81 out of bou :" using r by (cases "conv_radius java.lang.StringIndexOutOfBoundsException: Range [35, summable proof (rule fps_conv_radius_defjava.lang.StringIndexOutOfBoundsException: Index 34 fix: a assume" hence "ereal (norm x) < ereal K" by simp also have "\ finally (autosimp:) by( summable_in_conv_radius qed (insert java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 simp ) finally show "\ using fps_conv_radius_fps_X_power 1] by(imponly) qed lemma eval_fps_at_0( n " ( add eval_fps_def lemmafps_conv_radius_norm]: "fps_conv_radius (Abs_fps (\ by (simp <noteq> 0 \<Longrightarrow> fps_conv_radius (fps_const c * f) = fps_conv_radius f" by( ad lemma fps_conv_radius_const [simp fps_conv_radius_def(simpadd) proof - havefps_conv_radius (java.lang.StringIndexOutOfBoundsException: Index 74 out of b unfolding fps_conv_radius_cmult_left "-"f by( conv_radius_cong[OF[of]])auto thus ?thesis by simp qed lemma ps_conv_radius_add( +)\<ge> min (fps_conv_radius f) (fps_conv_radius g)" bysimp: fps_const_0_eq_0 symmetric) lemma fps_conv_radius_1 [simp]: "fps_conv_radius 1 = \ by (simp onlyfps_conv_radius_diff:fps_conv_radiusg)\<ge> min (fps_conv_radius f) (fps_co lemma fps_conv_radius_numeral [simp]: "fps_conv_radius (numeral n) = \ by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 lemma fps_conv_radius_fps_X_power [ using conv_radius_mult_ge[of "fps_nth[off"" g"java proof - le fps_conv_radius_power ( )java.lang.StringIndexOutOfBoundsException: Index 78 out unfolding fps_conv_radius_def by (intro conv_radius_cong eventually_mono[OF eventually_gt_at_top[of n]]) (auto rule) qed lemma fps_conv_radius_fps_X [simp]: "fps_conv_radius fps_X = \ fps_conv_radius_fps_X_power 1 simp: power_one_right) lemmajava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length "fps_conv_radius (fps_shift n f) = fps_conv_radius f" bysimp conv_radius_shift lemma fps_conv_radius_cmult_leftand summable:" (\ "\noteq>0\ unfoldingshowsnatfun_inverse lemma( m) " ?case unfolding by (simp:conv_radius_cmult_right fps_conv_radius_uminus [imp] "fps_conv_radiusjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds using fps_conv_radius_cmult_left[ofhave"orm natfun_inverse fSucn)= by (simp: fps_const_neg lemma fps_conv_radius_add: "fps_conv_radius (f + g) \ unfolding using[of"ps_nthf fps_nthjava.lang.StringIndexOutOfBoundsException: Inde by simp(impadd norm_mult del sum) lemma " ?S\le> (\ sing[of-" lemma fps_conv_radius_mult: " also have "\ using[ "fps_nthf " g"java.lang.StringIndexOutOfBoundsException: Index 55 out of bo simpatLeast0AtMost lemma: " (f ) <> f" proof (induction add) Suc 1 intro less java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for by simp also \> by (rule showjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for finally show ?case by simp also have"\ qed simp_all insert context begin lemma natfun_inverse_bound: f :: "a::{real_normed_field}fps" assumes "fps_nth f 0 = 1 substsumatLeast_Suc_atMost_Suc_shift)simp_all and summable: have{.n .< } byauto and "\Sum> shows java.lang.NullPointerException proof \<open>\<delta> > 0\<close> by (intro sum_le_suminf ballI mult_nonneg_nonneg zero_le_power case (less m) show ?case proofcases ) case 0 thus ?thesis next case []: (Suc havenorm Suc = norma "fps_nth = 1"fps_conv_radius" ( _ normS" assmsjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds by( addfield_simps norm_divide: sum) so have" ?S by (rule norm_sum) also have "\ proof (intro sum_mono, goal_cases) case (1 i) fps_nthnatfun_inversen-) norm (fps_nth using that by (subst (asm)auto by (simp add: norm_mult) also have "\ using 1 by ( using by (auto: eball_def eval_fps_at_0) also have "\ by (simp add: field_split_simps) finally ? . qed java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 10 by (subst sum_divide_distrib, rule sum.cong) (insert \<open>\<delta> > 0\<close>, auto simp: field_simps power_diff) also have "(\ (<Sum=0.n.normfps_nth( i)) * \<delta> ^ (Suc i))" by (subst.atLeast_Suc_atMost_Suc_shift also have "{0..n} = {.. bysimp: summable_sums) (\<Sum>n. norm (fps_nth f (Suc n)) * \<delta> ^ (Suc n))" using \<open>\<delta> > 0\<close> by (intro sum_le_suminf ballI mult_nonneg_nonneg zero_le_power also have "\ finallyshow using \<open>\<delta> > 0\<close> by (simp add \<delta> have "\<delta> \<in> ball 0 \<epsilon>" by auto qed qed private lemma fps_conv_radius_inverse_pos_aux: fixes havele(<n normf( ) *\<delta> ^ Suc n) \<le> 1" assumes "ps_nthf0 1"" > 0 shows "fps_conv_radius (inverse f from have summable: " (\<lambda>n. norm (fps_nth f ( proof - let ?R = "fps_conv_radius f" define h where "h = Abs_fps (\ have [simp]: "fps_conv_radius h = also have "\ have "continuous_on (eball 0 (fps_conv_radius h)) (eval_fps proof (introereal_in (introcontinuous_on_eval_fps) hence *: "open (eval_fps h -` Af n :: assume n: " 0 using that by (subst (asm) continuous_on_open_vimage) auto have "open (eval_fps h -` {..<2} \ by (rule *) auto moreover " using assms by (auto simp: eball_def n <delta> by (simp add: power_inverse [symmetric] real_roo ultimately obtain \<epsilon> where \<epsilon>: "\<epsilon> > 0" "ball 0 \<epsilon> \<subseteq> eval_fps by( (asm) blast define \<delta> where "\<delta> = real_of_ereal (min (ereal \<epsilon> / 2) (?R / 2))" have \<delta>: "0 < \<delta> \<and> \<delta> < \<epsilon> \<and> ereal \<delta> < ?R" using \<open>\<epsilon> > 0\<close> and assms by (cases ?R) (auto simp: \<delta>_def min_def) qed have summable: "usingassms ( add: fps_conv_radius_def fps_inverse_def) show by ( add) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (simp add: eval_fps_def " f 0 hence "(\ by (subst sums_Suc_iff) (auto simp assms(ubst fps_conv_radius_cmult_left alsohavefps_constf fps_const *java.lang.StringIndexOutOfBoundsException: Index 77 from \<delta> have "\<delta> \<in> ball 0 \<epsilon>" by auto also have "\ (auto simp: fps_conv_radius_cmult_left) } ultimately le:"\ by (simp add from summable by (subst summable_Suc_iff) have "0 < \ also have " >bysubst java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for len alsofix : a unfolding conv_radius_def proof java.lang.StringIndexOutOfBoundsException: Range [49, 50) out of bounds for eventually_mono[OF "java.lang.StringIndexOutOfBoundsException: Index 114 out of b fix :nat:" " have have"\ n assms\delta> le by ( by (rule) alsohave\<dots> = inverse \<delta>" using\<delta> by (simp add: power_inverse [symmetric] real_root_pos2) finally show "ereal java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of by subst) next 0=limsup by (rule Limsup_const [symmetric ( DERIV_imp_deriv] has_field_derivative_eval_fps) also have\<dots> \<le> limsup (\<lambda>n. ereal (root n (norm (natfun_inverse f n))))" byintroa simp finally show "0 \ qed also have " using assms assms using by( add fps_conv_radius_def fps_inverse_def finally java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for leng qed lemmafps_conv_radius_inverse_pos fixes:' :: {banach, real_normed_field} fps" assumes 0\<noteq> 0" and "fps_conv_radius f > 0" shows have"deriv ^ Suc )(eval_fps f) 0 = (deriv ^^n)(deriv (eval_fps f)) 0" proof - let ? also "eventually (\ havefps_conv_radiusf) fps_constinverse using assms by (subst fps_conv_radius_cmult_left "eventually(lambda>z. deriv (eval_fp also have "fps_const ?c * inverse f = inverse (fps_const hence (^^n ( eval_fpsf)0 using assms by (simp add: fps_inverse_mult fps_const_inverse) also have "fps_conv_radius \ by( fps_conv_radius_inverse_pos_aux (auto simp: also have "\<dots> / ( n) = fps_nth n)" show . qed end lemma fps_conv_radius_exp fixes c :: "'a :: {banach, eval_fps_eqD: ps_exp unfolding fps_conv_radius_def (rule''java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for len fix z :: 'a have(<>.norm z ^n /<sub n) expnorm )" by (rule exp_converges) also have "(\ by( ext( add norm_mult) finally have "summable n: nat thus "summable (\ by (rule summable_norm_cancel) java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3 subsection \<open>Evaluating power series\<close> theorem "fps_nthfn=fps_nth g n". assumes "norm z < fps_conv_radius f" shows "eval_fps (fps_deriv by introDERIV_imp_deriv [ymmetric assms theorem fps_nth_conv_deriv: fixes f :: "complex fps" assumes "fps_conv_radius f > 0" shows "fps_nth f n = (deriv ^^ n) (eval_fps f) 0 / fact n" using proof (induction n arbitrary: f) case 0 thus ?case by (simp add: eval_fps_def) next "\ bysimp funpow_Suc_right. also have "eventually (imp add: eval_fps_def ) using java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for lengt hence "eventually (\ (simp: eval_fps_deriv hence "(deriv ^^ n) (deriv (eval_fps f) by intro refl also have "\ using Suc.prems only [symmetric) by (intro have" by (simp add: fps_deriv_def del: of_nat_Suc) finally?by(:field_split_simps qed( onlynumeral_fps_const) theorem eval_fps_eqD eval_fps_X_power]: fixes f g : complex assumes "fps_conv_radius f > 0" "fps_conv_radius have "\<lambda>n::nat. if n \<in> {m} the assumes "eventually (\ shows "f = g" proof (rule fps_ext) fix n :: nat havefps_nth deriveval_fps/fact using assms by (intro fps_nth_conv_deriv ( add sums_iff also have "(deriv ^^ nlemma [simp] by (intro [ofz] :) also have "\ using assms f :: "'a :: {banach,real_normed_div_algebra} fps" finally show "fps_nth f n = fps_nth g n" assumes" z < fps_conv_radius f" qed lemma [simp fixes c ( suminf_minus]) (auto! ) shows lemma: proof - have "(\ by (rule sums_If_finite_set) auto have? \<longleftrightarrow> (\<lambda>n::nat. fps_nth (fps_const c) n * z ^ n) sums (\<Sum>n\<in>{0: by (intro sums_cong) auto also have "(\ by simp ( suminf_add( simpring_distribs! ) by (simpjava.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds fo qed lemma []java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for len "eval_fps (0 :: 'a :: {banach, real_normed_div_algebra} fps) z = 0" by ( only [symmetric) lemma eval_fps_1 "eval_fps (1 ::'a : {banach, real_normed_div_algebra} fps) = 1" by(imp : fps_const_1_eq_1] eval_fps_const lemma eval_fps_numeral [simp "eval_fps( *g f z * eval_fps z" "eval_fps (numeral n :: 'a :: {banach, real_normed_div_algebra} fps) z = numeral by (simp only: numeral_fps_const eval_fps_const) lemma eval_fps_X_power [simp \< "eval_fps (fps_X ^ m :: 'unfolding proof ( Cauchy_product "\<>n::nat n \ b rule assms have? longleftrightarrow> (\<lambda>n::nat. fps_nth (fps_X ^ m) n * z ^ n) sums (\<Sum>n\<in>{m::nat by (intro sums_cong) (auto simpalso "\ also have "(\ bysimp finally show ?thesis by (simp add: eval_fps_def sums_iff) qed lemma [simp "eval_fps (fps_X :: fixes :' :{,real_normed_div_algebra " using eval_fps_X_power[of 1 z] by (simp only: power_one_right) eval_fps_minus fixes f proof( "z = 0) assumes "norm z < have" (fps_shiftf*fps_X^ )z=eval_fps( n f) z *z ^ n" showseval_fps) eval_fps using assms unfolding eval_fps_def by ( suminf_minussymmetric intro:summable_fps lemmabysimp:fps_shift_times_fps_X_power) fixes: "a:: banach }fpsjava.lang.StringIndexOutOfBoundsException: Index 60 out of assumes "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for len shows c: ' : banach,real_normed_field}java.lang.StringIndexOutOfBoundsException: I using unfolding by (subst suminf_add) (auto simp: ring_distribss add exp_def field_split_simps lemma eval_fps_diff: f g : "' ::banach real_normed_div_algebra}fps" assumesnorm norm " shows "eval_fps (f - g) z = eval_fps f z - eval_fps g z" using assms unfolding eval_fps_def by (subst suminf_diff) (auto simp\<close> lemma eval_fps_mult: fixes f g :: "'a :: {banach,lemma fps_conv_radius_fps_of_poly[]: assumes " fps_conv_radius f"" fps_conv_radius " shows "eval_fps (f * g) z = eval_fps f z * eval_fps g z" proof - have "eval_fps using unfolding cofinite_eq_sequentiallyby( conv_radius_cong') (\<Sum>k. \<Sum>i\<le>k. fps_nth f i * fps_nth g (k - i) * (z ^ i * z ^ (k - i)))" unfolding(simp:fps_conv_radius_def proof (subst Cauchy_product) showsummable by (rule norm_summable_fps assms)+ qed (simp_all add: algebra_simps) also have "(\ (\<lambda>k. \<Sum>i\<le>k. fps_nth f i * fps_nth g (k - i) * z ^ k)" ( add[ymmetric also have "suminf \ by (simp add: eval_fps_def 0 finally show ?thesis .. qed fps_shift fixes:"a: {anach,real_normed_div_algebra,comm_ring_1}fpsjava.lang.StringIndexOut assumes "n \ shows fps_shiftz ( 0 f f / z ^n) proof (cases "z = 0") case False " (fps_shift fps_X ^ )z eval_fps (fps_shift nf z * " using assms by (subst eval_fps_mult) simp_all also from assms have "java.lang.StringIndexOutOfBoundsException: Index 0 out of boun by (simp add: fps_shift_times_fps_X_power - finally show ?thesis using ( (simp_all:eval_fps_at_0 lemma simp fixes c :: "'a :: usingsums_eval_fpsof z fps_of_polyp] bysimp shows "eval_fps (fps_exp c) z = exp (c * z)" unfolding using sums_unique2 by text assumes [holomorphic_intros]: "f holomorphic_on A" The case of division unfolding poly_altdef b holomorphic_intros Handling becomes more using analysisand will so that is \<close> subsection \<open>FPS of a polynomial\<close> lemma java.lang.StringIndexOutOfBoundsException: Range [55, 9) out of bounds for fixes. shows "fps_conv_radius (fps_of_poly p) = \ proofjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length have" (poly.coeff p)= (\ using unfolding by ( conv_radius_cong also have they be on set by simp finally ?thesis by (simp add: fps_conv_radius_def purposes statment converges the some qed lemma eval_fps_power: fixes F :: "'a :: {banach, real_normed_div_algebra, comm_ring_1} fps" assumes z: "norm z < fps_conv_radius F" shows particular one to the ofthe series (induction case 0 thus ?case by (auto simp: eval_fps_mult) next case( n) have "eval_fps (F ^ Suc n) z = eval_fps (F * F ^ n)where( )\java.lang.StringInde by simp also from have "\ by (subst : finally " has_fps_expansion using Suc.IH by simp qed eval_fps_of_poly] eval_fpspoly java.lang.NullPointerException unfolding by rule)(uto: coeff_eq_0 moreover "( using sums_eval_fps[of z "fps_of_poly p"] by simp ultimately show ?thesis "fps_nth Fn=( ^^ n) (val_fps java.lang.StringIndexOutOfBoun using sums_unique2 by blast qed lemma poly_holomorphic_on [holomorphic_intros]: assumes]: "holomorphic_on A" shows "(\ unfolding poly_altdef by (intro subsection \<open>Power series expansions of analytic functions\<close> text This predicate contains the in disc positive around originis to given function there. This relationship " F" "continuous at 0 A) f" holomorphic on a connecteda have" (eval_fps F) 0 expansion must equal that. More concrete statements about the radius of convergence can( isCont_cong simphas_fps_expa purposes statment the converges thein neighbourhood of the origin is enough, thus( A " as there are straightforward introduction rules to show this. In, when one wants relate the coefficients the series to values of the derivatives(<>.c has_fps_expansion the coefficients of auto: has_fps_expansion_def is enough. \<close> definition\<^marker>\<open>tag important\<close> "\ (infixl \<open>has'_fps'_expansion\<close> 60) simp has_fps_expansion_def where "(f has_fps_expansion F) \ fps_conv_radius F >java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds named_theorems lemma has_fps_expansion_schematicI: "f has_fps_expansion A \ by simp lemma fps_nth_fps_expansionlemma has_fps_expansion_fps_Xfps_expansion_intros]:java fixes f :: "complex \ assumes "f has_fps_expansion F" shows "fps_nth F n = (deriv ^^ n) f 0 / fact n" proof - have "fps_nth F n = (deriv ^^ n) (eval_fps F) 0 / fact n" using assmscasejava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds also have "(deriv ^^ n) (eval_fps F) 0 = (deriv ( eventually_nhds_in_open) ( s using assms by (intro higher_deriv_cong_ev) (auto simp: has_fps_expansion_def) finally show ?thesis . qed eval_fps_has_fps_expansion "fps_conv_radius with assms False show thesis has_fps_expansion_def lemma has_fps_expansion_imp_continuous: fixes F :: "'a::{real_normed_field,banach} fps" assumes "f has_fps_expansion F" shows" (at 0 within A) f" proof - from assms have "isCont (eval_fps F) 0" by (intro c ::"a: banach,real_normed_div_algebra }java.lang.StringIndexOutOfBoundsE also have "?this "java.lang.StringIndexOutOfBoundsException: Index 67 out of by (intro isCont_cong) (auto has_fps_expansion_cmult_left OF] showthesis finally have "isCont f 0" . thus "continuous (at 0 within A) f" by (simp add: continuous_at_imp_continuous_within) qed lemmaproof "(\ by (auto simp: has_fps_expansion_def) lemma has_fps_expansion_0 [simp assms( "(\ by (auto simp: has_fps_expansion_def) lemma by eventually_elim simp) "(\ by (auto lemmahas_fps_expansion_numeral, , fps_expansion_intros "\lambda>.numeral n numeral n" by (auto simp: has_fps_expansion_def) lemma [] "(\ auto:has_fps_expansion_def lemma has_fps_expansion_fps_X [fps_expansion_intros]: "(\ by( simp) lemma has_fps_expansion_cmult_left [fps_expansion_intros]: fixes c :: "'a :: {banach, real_normed_div_algebra, comm_ring_1}" assumes "f has_fps_expansion F" shows "(\ proof(ases" 0) case False from assms have "eventually (\ by (intro ultimatelyeventually from haveeventually by (auto simpwwith show by (auto:) ultimately by (simp_all: eval_fps_mult with assms and Falseassumeshas_fps_expansion" G by( simp fps_conv_radius_cmult_left qed auto lemma has_fps_expansion_cmult_right [fps_expansion_intros has_fps_expansion_mult[]: fixesc:"a:: {,r comm_ring_1} assumes "f has_fps_expansion F" shows "(\ proof - have "F * fps_const c = fps_const c * F" by (intro fps_ext) (auto simp: mult.commute) with has_fps_expansion_cmult_left [OF assms] show havejava.lang.StringIndexOutOfBound by (simp add: mult.commute) qed lemma has_fps_expansion_minus [fps_expansion_intros]: assumes "f has_fps_expansion F" shows\<lambda>x. - f x) has_fps_expansion -F" proof from assms and\<lambda>x. eval_fps G x = g x) (nhds 0)" by( eventually_nhds_in_open simphas_fps_expansion_def) moreover have eventually simp: has_fps_expansion_def havejava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length by eventually_elim (auto simp has_fps_expansion_inversefps_expansion_intros thusthesis assms by( simp: has_fps_expansion_def qed lemma has_fps_expansion_add [fps_expansion_intros]: assumes "f has_fps_expansion F" "g has_fps_expansion G" shows "(\ proof - from assms have "0 < min (fps_conv_radius F) (fps_conv_radius G)" byauto alsohavejava.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for le fps_conv_radius_add finally havelet min )( )java.lang.StringIndexOutOfBoundsException: Index 66 out of b from assms have "eventually (\ "eventually (\ by (intro eventually_nhds_in_open; force simpmoreover " (\ moreover have "eventually (\ and "eventually (\ using assms by (auto (lim z) ultimately have "eventually (\ by eventually_elim (auto simp: eval_fps_add) with radiusby( eval_fps_mult auto qed lemma has_fps_expansion_diff [fps_expansion_intros]: assumes F g Gjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for shows using has_fps_expansion_add[of f F "\ by (simp add: has_fps_expansion_minus) lemma has_fps_expansion_mult [fps_expansion_intros]: lemmahas_fps_expansion_sum [ps_expansion_intros]java.lang.StringIndexOutOfBoundsE assumes "f has_fps_expansion F shows (\ shows\<lambda>x. f x * g x) has_fps_expansion F * G" proof - from assms have "0 < min (fps_conv_radiuslemma has_fps_expansion_prod fps_expansion_intros] by (auto simp: has_fps_expansion_def) also have "\ by (rule fps_conv_radius_mult) finallyhave: "\ from assms have "eventually (\ "eventually (\ by (intro eventually_nhds_in_openby(uto: has_fps_expansion_def moreover have "eventually (\ and "eventually (\ using by( simphas_fps_expansion_def) ultimately have "eventually (\ \<lambda>x::'a :: {banach, real_normed_field}. exp (-x)) has_fps_expansion fps_exp (-1)" with radius show ?thesis by (auto simp: has_fps_expansion_def) qed has_fps_expansion_deriv]: lemma has_fps_expansion_inverse [fps_expansion_intros]: fixes F :: "'a :: {banach, real_normed_field} fps" assumes" sing (ntro) "( proof - have radius: "fps_conv_radius (inverse F) > 0" using assms auto) by( fps_conv_radius_inverse_pos let ?R = "min (fps_conv_radius F) (fps_conv_radius java.lang.StringIndexOutOfBounds from radius have "eventually (\ ntually\<lambda>x. x \<in> eball 0 (fps_conv_radius (inverse F))) (nhds 0)" by (intro eventually_nhds_in_open; force simp: has_fps_expansion_def zero_ereal_def)+ moreover have "eventually (\ using hence "val_fps fps_derivF z=deriv(val_fps F)z" ultimately" (\ proof eventually_elim case (elimusing and hence "eval_fps (inverse F * F) z = eval_fps (inverse F) z hence" (\<lambda>w. eva by (subst eval_fps_mult) auto also have "eval_fps (inverse F * F) z = 1" using assmsfinally ?case. finally show ?case by (auto simp assms fps_conv_radius_deriv F ?thesis qed with radius show ?thesis by (auto simp: has_fps_expansion_def qed lemma has_fps_expansion_sum [fps_expansion_intros]: assumes "\ shows "(\ using assms fps_conv_radius_ln lemmahas_fps_expansion_prod [fps_expansion_intros: fixes F :: "'a \ assumes shows java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for lengt usingassmsinduction:) (auto!: ) lemma has_fps_expansion_exp [fps_expansion_intros]: fixes c :: "'a :: {banach, real_normed_field}" shows "(\ by (auto simp: has_fps_expansion_def "( lemma has_fps_expansion_exp1 [fps_expansion_intros]: "(\ using has_fps_expansion_exp" \ lemma has_fps_expansion_exp_neg1 [fps_expansion_introsintro eventually_mono eventual ( using has_fps_expansion_exp[of "-1"] by simp lemma has_fps_expansion_derivsubst)( add) assumes "f assumes "f : fps_ln_def) shows "deriv f has_fps_expansion fps_deriv F" proof - have "eventually (\ using assms by (intro eventually_nhds_in_open) (auto simp: has_fps_expansion_def zero_ereal_def) moreoverassmseventually by (auto simp: has_fps_expansion_def) then obtain s where "open "fps_conv_radius c) =java.lang.StringIndexOutOfBoundsExcep by (auto simp: eventually_nhds) hence "eventually (\ by (intro eventually_nhds_in_open) auto ultimately " ( proof eventually_elim case (elim z) hence "eval_fps (fps_deriv F) z = deriv (eval_fps F) z" by (simp add ?caseusing[of]by( addnorm_mult also have "eventually (\ using elim and \<open>open s\<close> by (intro eventually_nhds_in_open) auto hence "eventually (\ by eventually_elim (simp add: s) "deriv(val_fpsF f z" by (intro deriv_cong_ev finally ?thesis inallycase. qed java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (auto simp: has_fps_expansion_def) qed lemma fps_conv_radius_binomial: fixes c :: "'a :: {real_normed_field,banach}" shows "fps_conv_radius (fps_binomial c) = (if c \ unfolding by (simp add) lemma fps_conv_radius_ln: fixes c :: "a ::{banach real_normed_field field_char_0}" shows)ifthen proof (ases0 case Falsesubst) auto have "conv_radius (\ mit_nonzero show "lemmaeval_fps_sin[simp]: using LIMSEQ_Suc_n_over_n by (simp add: norm_divide del: "eval_fps (ps_sinc = sin (c qed auto also have "conv_radius (\ conv_radius java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for by ( ?thesis(simp add eval_fps_def (simp add: norm_mult norm_divide norm_power) finallyshow usingFalse fps_ln_def ( fps_conv_radius_cmult_leftadd) qed (auto simp: fps_ln_def) assumes shows "fps_conv_radius (fps_ln c) = 1" using assms "\<>n. cos_coeff *\<^sub>R (c * z) ^ n) sums cos (c * z)" by (rule cos_conver fps_conv_radius_sin]: fixes c :: "'a :: {banach, real_normed_field, ( ext) auto : cos_coeff_deffps_c "fps_conv_radius( c) = \ proof (cases "c = 0") case False have "\ proof (rule sym, rule conv_radius_inftyI'', rule summable_norm_cancel, goal_cases) case (1 z) show ?case using summable_norm_sin[of z] by (simp add: norm_mult) qed also have "\ using False by (subst conv_radius_mult_power) auto alsohave\<dots> = fps_conv_radius (fps_sin c)" unfolding fps_conv_radius_def by (rule conv_radius_cong_weak) (auto simp add: fps_sin_def sin_coeff_def) finally show ?thesis by simp qed simp_all lemma fps_conv_radius_cos [simp]: fixes c :: "'a :: {banach, real_normed_field, field_char_0}" showsfps_cos\<infinity>" proof (cases "c = 0") case False have"<> conv_radius (\ proof (rule show? by simp case (1 z) show ?case using summable_norm_cos[of z] by (simp add: norm_mult qed also have "\ usingassumes <" also \dots= (fps_cos unfolding by (rule conv_radius_cong_weak) (autousing[OF ] by ( add eval_fps_def finally show ?thesis by simp qed simp_all lemma eval_fps_sin [simp]: fixes z :: "'a :: {banach, real_normed_field, field_char_0}" shows "eval_fps (fps_sin c) z = sin (c * z)" proof - have( also have "(\ by (rule ext) (auto simp!: eventually_mono *]) finally show ?thesis by (simp add: sums_iff eval_fps_def) qed lemma eval_fps_cos [simp]: fixes z :: "'a :: {banach, real_normed_field, field_char_0}" shows "eval_fps (fps_cos c) z = cos (c * z)" proof - ( have by ( finally show ?thesis by (simp add: sums_iff eval_fps_def) qed lemmacos_eq_zero_imp_norm_ge assumes "cos (z :: complex) = 0" norm proof - from assms obtain n where "z = complex_of_real ((of_int n + 1 / 2) * pi)"using[of]by by lemmahas_fps_expansion_shift]: also norm by (subst norm_of_real) (simp_all add: abs_mult) also have "real_of_int n + 1 / 2 = of_int (2 * n + 1) / 2" by simp also have "\ also have "\ also have "\ by (intro mult_right_mono, subst of_int_le_iff) (auto simp: have" (\ finally show ?thesis by simp qed lemma eval_fps_binomial: fixes: complex assumes "norm z < 1" shows "eval_fps (fps_binomial c) z = (1 + z) powr c" using gen_binomial_complex[OF assms] by (simp add: sums_iff eval_fps_def) lemma has_fps_expansion_binomial_complex [fps_expansion_intros]: fixes c :: complex shows\lambda.( +x c)has_fps_expansioncjava.lang.StringIndexOutOfBoundsException: I proof - have:"ventually(\ by (intro eventually_nhds_in_open) auto thus "(<>x. x f x / g x)has_fps_expansion (F )" by (auto simp: has_fps_expansion_def eval_fps_binomial fps_conv_radius_binomial intro!: eventually_mono [OF *]) qed []java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 5 fixes c :: "'a :: {banach, real_normed_field, field_char_0}" shows "(\ byautojava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for leng lemma has_fps_expansion_sin' by simpadd: fps_divide_unitjava.lang.StringIndexO ambda' :{}. sinx fps_sin 1" using has_fps_expansion_sin[inverse x=0 fps_nthelse/ n)has_fps_expansionG" has_fps_expansion_cos]: fixes c :: "'a :: {banach, real_normed_field, field_char_0}" shows "(\ by (auto simp: has_fps_expansion_def) lemma has_fps_expansion_cos' [fps_expansion_intros]: "(\ using[of simp lemma has_fps_expansion_shift [fps_expansion_intros]: fixes F :: "'a :: {banach, real_normed_field} fps" assumes "f has_fps_expansion F" and "n \ assumes "c = fps_nth F n" shows "(\ - have "eventually (\ using assms by (intro eventually_nhds_in_open) (auto simp: has_fps_expansion_def zero_er moreover "eventually (\ using assms by ( _") using assms(3 by(intro has_fps_expansion_divide assms java.lang ultimately have "eventually (by (auto simp: has_fps_expansion_def eval_fps_at_0 de (if x = 0 then c else f x / by eventually_elimjava.lang.StringIndexOutOfBoundsException: Index 0 out of bound with assms show ?thesis by (auto simp c: "a: banach real_normed_field,field_char_0}java. qed emma [fps_expansion_intros fixes F G :: "'a :: {banach by (intro fps_expansion_intros auto assumes "f has_fps_expansion F" and "g has_fps_expansion G" and subdegree "c = fps_nth F (subdegree G) / fps_nth G (subdegree G)" showshas_fps_expansion_tan proof - define n where "n = subdegree G" lemma ha '* ^ n G '*fps_X ^ n" unfolding_n_def by (rule fps_shift_times_fps_X_power [symmetric s where "0\ moreover have fps_nth by( add_ ) ultimately have FG: "F / G = F' * inverse G'" b simpjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for lengt have"\ (x x/ )has_fps_expansion (is "?h has_fps_expansion _"finally thesis has_fps_expansion_inverse has_fps_expansion_shift assms) auto also have "?h = (\ using assms(5) unfolding n_def by (intro ext) (auto split "f F" finally show ?thesis . qed lemma has_fps_expansion_divide' [fps_expansion_intros]: fixes F G :: "'a :: {banach, real_normed_field} fps" assumes "f has_fps_expansion F" and "g has_fps_expansion G" and "fps_nth G 0 \ (<lambda>x. f x / g x) has_fps_expansion (F / G)" proof - have "(\ ishhas_fps_expansion using(3) by(introhas_fps_expansion_divide ) auto also from assms ( conv_radius_geI by (auto simp: has_fps_expansion_def eval_fps_at_0 dest: eventually_nhds_x_imp_x) hence "?h = (\ finally show ?thesis . qed lemma [fps_expansion_intros fixes c :: "'a :: {banach, real_normed_field qed shows "(\ proof - have "(\ by (intro fps_expansion_intros) auto thus ?thesis by (simp add: tan_def fps_tan_def) qed nsion_tan[fps_expansion_intros "tan has_fps_expansion fps_tan (1 :: 'a :: {banach, real_normed_field, field_cha using has_fps_expansion_tan[of 1] by simp lemma has_fps_expansion_imp_holomorphic: assumes "f has_fps_expansion F" obtains proof - from assms obtain s where s: "open s" "0 \ unfolding has_fps_expansion_def eventually_nhds by blast let ?s' = "eball 0 (fps_conv_radius F) \ have "eval_fps F holomorphic_on ?s'" by (intro holomorphic_intros) auto also have "?this \ using s by (intro holomorphic_cong) auto finally show ?thesis using s assms by (intro that[of ?s']) (auto simp: has_fps_expansion_def zero_ereal_def) qed lemma has_fps_expansionI: fixes f :: "'a :: {banach, real_normed_div_algebra} \ assumes "eventually (\ shows "f has_fps_expansion F" proof - from assms obtain X where X: "open X" "0 \ unfolding eventually_nhds by blast obtain r where r: "r > 0" "cball 0 r \ using X(1,2) open_contains_cball by blast have "0 < norm (of_real r :: 'a)" using r(1) by simp also have "fps_conv_radius F \ unfolding fps_conv_radius_def proof (rule conv_radius_geI) have "of_real r \ using r by auto from X(3)[OF this] show "summable (\ by (simp add: sums_iff) qed finally have "fps_conv_radius F > 0" by (simp_all add: zero_ereal_def) moreover have "(\ using assms by eventually_elim (auto simp: sums_iff eval_fps_def) ultimately show ?thesis unfolding has_fps_expansion_def .. qed lemma fps_mult_numeral_left [simp]: "fps_nth (numeral c * f) n = numeral c * fps_nt by (simp add: fps_numeral_fps_const) end ¤ Dauer der Verarbeitung: 0.21 Sekunden ¤*© Formatika GbR, Deutschland |
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2026-03-28