(* Title: HOL/Nonstandard_Analysis/HTranscendental.thy
Author: Jacques D. Fleuriot
Copyright: 2001 University of Edinburgh
Converted to Isar and polished by lcp
*)
section\<open>Nonstandard Extensions of Transcendental Functions\<close>
theory HTranscendental
imports Complex_Main HSeries HDeriv Sketch_and_Explore
begin
sledgehammer_params [timeout = 90]
definition
exphr :: "real \<Rightarrow> hypreal" where
\<comment> \<open>define exponential function using standard part\<close>
"exphr x \<equiv> st(sumhr (0, whn, \<lambda>n. inverse (fact n) * (x ^ n)))"
definition
sinhr :: "real \<Rightarrow> hypreal" where
"sinhr x \<equiv> st(sumhr (0, whn, \<lambda>n. sin_coeff n * x ^ n))"
definition
coshr :: "real \<Rightarrow> hypreal" where
"coshr x \<equiv> st(sumhr (0, whn, \<lambda>n. cos_coeff n * x ^ n))"
subsection\<open>Nonstandard Extension of Square Root Function\<close>
lemma STAR_sqrt_zero [simp]: "( *f* sqrt) 0 = 0"
by (simp add: starfun star_n_zero_num)
lemma STAR_sqrt_one [simp]: "( *f* sqrt) 1 = 1"
by (simp add: starfun star_n_one_num)
lemma hypreal_sqrt_pow2_iff: "(( *f* sqrt)(x) ^ 2 = x) = (0 \<le> x)"
proof (cases x)
case (star_n X)
then show ?thesis
by (simp add: star_n_le star_n_zero_num starfun hrealpow star_n_eq_iff del: hpowr_Suc power_Suc)
qed
lemma hypreal_sqrt_gt_zero_pow2: "\<And>x. 0 < x \<Longrightarrow> ( *f* sqrt) (x) ^ 2 = x"
by transfer simp
lemma hypreal_sqrt_pow2_gt_zero: "0 < x \<Longrightarrow> 0 < ( *f* sqrt) (x) ^ 2"
by (frule hypreal_sqrt_gt_zero_pow2, auto)
lemma hypreal_sqrt_not_zero: "0 < x \<Longrightarrow> ( *f* sqrt) (x) \<noteq> 0"
using hypreal_sqrt_gt_zero_pow2 by fastforce
lemma hypreal_inverse_sqrt_pow2:
"0 < x \<Longrightarrow> inverse (( *f* sqrt)(x)) ^ 2 = inverse x"
by (simp add: hypreal_sqrt_gt_zero_pow2 power_inverse)
lemma hypreal_sqrt_mult_distrib:
"\<And>x y. \<lbrakk>0 < x; 0 <y\<rbrakk> \<Longrightarrow>
( *f* sqrt)(x*y) = ( *f* sqrt)(x) * ( *f* sqrt)(y)"
by transfer (auto intro: real_sqrt_mult)
lemma hypreal_sqrt_mult_distrib2:
"\<lbrakk>0\<le>x; 0\<le>y\<rbrakk> \<Longrightarrow> ( *f* sqrt)(x*y) = ( *f* sqrt)(x) * ( *f* sqrt)(y)"
by (auto intro: hypreal_sqrt_mult_distrib simp add: order_le_less)
lemma hypreal_sqrt_approx_zero [simp]:
assumes "0 < x"
shows "(( *f* sqrt) x \<approx> 0) \<longleftrightarrow> (x \<approx> 0)"
proof -
have "( *f* sqrt) x \<in> Infinitesimal \<longleftrightarrow> ((*f* sqrt) x)\<^sup>2 \<in> Infinitesimal"
by (metis Infinitesimal_hrealpow pos2 power2_eq_square Infinitesimal_square_iff)
also have "... \<longleftrightarrow> x \<in> Infinitesimal"
by (simp add: assms hypreal_sqrt_gt_zero_pow2)
finally show ?thesis
using mem_infmal_iff by blast
qed
lemma hypreal_sqrt_approx_zero2 [simp]:
"0 \<le> x \<Longrightarrow> (( *f* sqrt)(x) \<approx> 0) = (x \<approx> 0)"
by (auto simp add: order_le_less)
lemma hypreal_sqrt_gt_zero: "\<And>x. 0 < x \<Longrightarrow> 0 < ( *f* sqrt)(x)"
by transfer (simp add: real_sqrt_gt_zero)
lemma hypreal_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> ( *f* sqrt)(x)"
by (auto intro: hypreal_sqrt_gt_zero simp add: order_le_less)
lemma hypreal_sqrt_hrabs [simp]: "\<And>x. ( *f* sqrt)(x\<^sup>2) = \<bar>x\<bar>"
by transfer simp
lemma hypreal_sqrt_hrabs2 [simp]: "\<And>x. ( *f* sqrt)(x*x) = \<bar>x\<bar>"
by transfer simp
lemma hypreal_sqrt_hyperpow_hrabs [simp]:
"\<And>x. ( *f* sqrt)(x pow (hypnat_of_nat 2)) = \<bar>x\<bar>"
by transfer simp
lemma star_sqrt_HFinite: "\<lbrakk>x \<in> HFinite; 0 \<le> x\<rbrakk> \<Longrightarrow> ( *f* sqrt) x \<in> HFinite"
by (metis HFinite_square_iff hypreal_sqrt_pow2_iff power2_eq_square)
lemma st_hypreal_sqrt:
assumes "x \<in> HFinite" "0 \<le> x"
shows "st(( *f* sqrt) x) = ( *f* sqrt)(st x)"
proof (rule power_inject_base)
show "st ((*f* sqrt) x) ^ Suc 1 = (*f* sqrt) (st x) ^ Suc 1"
using assms hypreal_sqrt_pow2_iff [of x]
by (metis HFinite_square_iff hypreal_sqrt_hrabs2 power2_eq_square st_hrabs st_mult)
show "0 \<le> st ((*f* sqrt) x)"
by (simp add: assms hypreal_sqrt_ge_zero st_zero_le star_sqrt_HFinite)
show "0 \<le> (*f* sqrt) (st x)"
by (simp add: assms hypreal_sqrt_ge_zero st_zero_le)
qed
lemma hypreal_sqrt_sum_squares_ge1 [simp]: "\<And>x y. x \<le> ( *f* sqrt)(x\<^sup>2 + y\<^sup>2)"
by transfer (rule real_sqrt_sum_squares_ge1)
lemma HFinite_hypreal_sqrt_imp_HFinite:
"\<lbrakk>0 \<le> x; ( *f* sqrt) x \<in> HFinite\<rbrakk> \<Longrightarrow> x \<in> HFinite"
by (metis HFinite_mult hrealpow_two hypreal_sqrt_pow2_iff numeral_2_eq_2)
lemma HFinite_hypreal_sqrt_iff [simp]:
"0 \<le> x \<Longrightarrow> (( *f* sqrt) x \<in> HFinite) = (x \<in> HFinite)"
by (blast intro: star_sqrt_HFinite HFinite_hypreal_sqrt_imp_HFinite)
lemma Infinitesimal_hypreal_sqrt:
"\<lbrakk>0 \<le> x; x \<in> Infinitesimal\<rbrakk> \<Longrightarrow> ( *f* sqrt) x \<in> Infinitesimal"
by (simp add: mem_infmal_iff)
lemma Infinitesimal_hypreal_sqrt_imp_Infinitesimal:
"\<lbrakk>0 \<le> x; ( *f* sqrt) x \<in> Infinitesimal\<rbrakk> \<Longrightarrow> x \<in> Infinitesimal"
using hypreal_sqrt_approx_zero2 mem_infmal_iff by blast
lemma Infinitesimal_hypreal_sqrt_iff [simp]:
"0 \<le> x \<Longrightarrow> (( *f* sqrt) x \<in> Infinitesimal) = (x \<in> Infinitesimal)"
by (blast intro: Infinitesimal_hypreal_sqrt_imp_Infinitesimal Infinitesimal_hypreal_sqrt)
lemma HInfinite_hypreal_sqrt:
"\<lbrakk>0 \<le> x; x \<in> HInfinite\<rbrakk> \<Longrightarrow> ( *f* sqrt) x \<in> HInfinite"
by (simp add: HInfinite_HFinite_iff)
lemma HInfinite_hypreal_sqrt_imp_HInfinite:
"\<lbrakk>0 \<le> x; ( *f* sqrt) x \<in> HInfinite\<rbrakk> \<Longrightarrow> x \<in> HInfinite"
using HFinite_hypreal_sqrt_iff HInfinite_HFinite_iff by blast
lemma HInfinite_hypreal_sqrt_iff [simp]:
"0 \<le> x \<Longrightarrow> (( *f* sqrt) x \<in> HInfinite) = (x \<in> HInfinite)"
by (blast intro: HInfinite_hypreal_sqrt HInfinite_hypreal_sqrt_imp_HInfinite)
lemma HFinite_exp [simp]:
"sumhr (0, whn, \<lambda>n. inverse (fact n) * x ^ n) \<in> HFinite"
unfolding sumhr_app star_zero_def starfun2_star_of atLeast0LessThan
by (metis NSBseqD2 NSconvergent_NSBseq convergent_NSconvergent_iff summable_iff_convergent summable_exp)
lemma exphr_zero [simp]: "exphr 0 = 1"
proof -
have "\<forall>x>1. 1 = sumhr (0, 1, \<lambda>n. inverse (fact n) * 0 ^ n) + sumhr (1, x, \<lambda>n. inverse (fact n) * 0 ^ n)"
unfolding sumhr_app by transfer (simp add: power_0_left)
then have "sumhr (0, 1, \<lambda>n. inverse (fact n) * 0 ^ n) + sumhr (1, whn, \<lambda>n. inverse (fact n) * 0 ^ n) \<approx> 1"
by auto
then show ?thesis
unfolding exphr_def
using sumhr_split_add [OF hypnat_one_less_hypnat_omega] st_unique by auto
qed
lemma coshr_zero [simp]: "coshr 0 = 1"
proof -
have "\<forall>x>1. 1 = sumhr (0, 1, \<lambda>n. cos_coeff n * 0 ^ n) + sumhr (1, x, \<lambda>n. cos_coeff n * 0 ^ n)"
unfolding sumhr_app by transfer (simp add: power_0_left)
then have "sumhr (0, 1, \<lambda>n. cos_coeff n * 0 ^ n) + sumhr (1, whn, \<lambda>n. cos_coeff n * 0 ^ n) \<approx> 1"
by auto
then show ?thesis
unfolding coshr_def
using sumhr_split_add [OF hypnat_one_less_hypnat_omega] st_unique by auto
qed
lemma STAR_exp_zero_approx_one [simp]: "( *f* exp) (0::hypreal) \<approx> 1"
proof -
have "(*f* exp) (0::real star) = 1"
by transfer simp
then show ?thesis
by auto
qed
lemma STAR_exp_Infinitesimal:
assumes "x \<in> Infinitesimal" shows "( *f* exp) (x::hypreal) \<approx> 1"
proof (cases "x = 0")
case False
have "NSDERIV exp 0 :> 1"
by (metis DERIV_exp NSDERIV_DERIV_iff exp_zero)
then have "((*f* exp) x - 1) / x \<approx> 1"
using nsderiv_def False NSDERIVD2 assms by fastforce
then have "(*f* exp) x - 1 \<approx> x"
using NSDERIVD4 \<open>NSDERIV exp 0 :> 1\<close> assms by fastforce
then show ?thesis
by (meson Infinitesimal_approx approx_minus_iff approx_trans2 assms not_Infinitesimal_not_zero)
qed auto
lemma STAR_exp_epsilon [simp]: "( *f* exp) \<epsilon> \<approx> 1"
by (auto intro: STAR_exp_Infinitesimal)
lemma STAR_exp_add:
"\<And>(x::'a:: {banach,real_normed_field} star) y. ( *f* exp)(x + y) = ( *f* exp) x * ( *f* exp) y"
by transfer (rule exp_add)
lemma exphr_hypreal_of_real_exp_eq: "exphr x = hypreal_of_real (exp x)"
proof -
have "(\<lambda>n. inverse (fact n) * x ^ n) sums exp x"
using exp_converges [of x] by simp
then have "(\<lambda>n. \<Sum>n<n. inverse (fact n) * x ^ n) \<longlonglongrightarrow>\<^sub>N\<^sub>S exp x"
using NSsums_def sums_NSsums_iff by blast
then have "hypreal_of_real (exp x) \<approx> sumhr (0, whn, \<lambda>n. inverse (fact n) * x ^ n)"
unfolding starfunNat_sumr [symmetric] atLeast0LessThan
using HNatInfinite_whn NSLIMSEQ_iff approx_sym by blast
then show ?thesis
unfolding exphr_def using st_eq_approx_iff by auto
qed
lemma starfun_exp_ge_add_one_self [simp]: "\<And>x::hypreal. 0 \<le> x \<Longrightarrow> (1 + x) \<le> ( *f* exp) x"
by transfer (rule exp_ge_add_one_self_aux)
text\<open>exp maps infinities to infinities\<close>
lemma starfun_exp_HInfinite:
fixes x :: hypreal
assumes "x \<in> HInfinite" "0 \<le> x"
shows "( *f* exp) x \<in> HInfinite"
proof -
have "x \<le> 1 + x"
by simp
also have "\<dots> \<le> (*f* exp) x"
by (simp add: \<open>0 \<le> x\<close>)
finally show ?thesis
using HInfinite_ge_HInfinite assms by blast
qed
lemma starfun_exp_minus:
"\<And>x::'a:: {banach,real_normed_field} star. ( *f* exp) (-x) = inverse(( *f* exp) x)"
by transfer (rule exp_minus)
text\<open>exp maps infinitesimals to infinitesimals\<close>
lemma starfun_exp_Infinitesimal:
fixes x :: hypreal
assumes "x \<in> HInfinite" "x \<le> 0"
shows "( *f* exp) x \<in> Infinitesimal"
proof -
obtain y where "x = -y" "y \<ge> 0"
by (metis abs_of_nonpos assms(2) eq_abs_iff')
then have "( *f* exp) y \<in> HInfinite"
using HInfinite_minus_iff assms(1) starfun_exp_HInfinite by blast
then show ?thesis
by (simp add: HInfinite_inverse_Infinitesimal \<open>x = - y\<close> starfun_exp_minus)
qed
lemma starfun_exp_gt_one [simp]: "\<And>x::hypreal. 0 < x \<Longrightarrow> 1 < ( *f* exp) x"
by transfer (rule exp_gt_one)
abbreviation real_ln :: "real \<Rightarrow> real" where
"real_ln \<equiv> ln"
lemma starfun_ln_exp [simp]: "\<And>x. ( *f* real_ln) (( *f* exp) x) = x"
by transfer (rule ln_exp)
lemma starfun_exp_ln_iff [simp]: "\<And>x. (( *f* exp)(( *f* real_ln) x) = x) = (0 < x)"
by transfer (rule exp_ln_iff)
lemma starfun_exp_ln_eq: "\<And>u x. ( *f* exp) u = x \<Longrightarrow> ( *f* real_ln) x = u"
by transfer (rule ln_unique)
lemma starfun_ln_less_self [simp]: "\<And>x. 0 < x \<Longrightarrow> ( *f* real_ln) x < x"
by transfer (rule ln_less_self)
lemma starfun_ln_ge_zero [simp]: "\<And>x. 1 \<le> x \<Longrightarrow> 0 \<le> ( *f* real_ln) x"
by transfer (rule ln_ge_zero)
lemma starfun_ln_gt_zero [simp]: "\<And>x .1 < x \<Longrightarrow> 0 < ( *f* real_ln) x"
by transfer (rule ln_gt_zero)
lemma starfun_ln_not_eq_zero [simp]: "\<And>x. \<lbrakk>0 < x; x \<noteq> 1\<rbrakk> \<Longrightarrow> ( *f* real_ln) x \<noteq> 0"
by transfer simp
lemma starfun_ln_HFinite: "\<lbrakk>x \<in> HFinite; 1 \<le> x\<rbrakk> \<Longrightarrow> ( *f* real_ln) x \<in> HFinite"
by (metis HFinite_HInfinite_iff less_le_trans starfun_exp_HInfinite starfun_exp_ln_iff starfun_ln_ge_zero zero_less_one)
lemma starfun_ln_inverse: "\<And>x. 0 < x \<Longrightarrow> ( *f* real_ln) (inverse x) = -( *f* ln) x"
by transfer (rule ln_inverse)
lemma starfun_abs_exp_cancel: "\<And>x. \<bar>( *f* exp) (x::hypreal)\<bar> = ( *f* exp) x"
by transfer (rule abs_exp_cancel)
lemma starfun_exp_less_mono: "\<And>x y::hypreal. x < y \<Longrightarrow> ( *f* exp) x < ( *f* exp) y"
by transfer (rule exp_less_mono)
lemma starfun_exp_HFinite:
fixes x :: hypreal
assumes "x \<in> HFinite"
shows "( *f* exp) x \<in> HFinite"
proof -
obtain u where "u \<in> \<real>" "\<bar>x\<bar> < u"
using HFiniteD assms by force
with assms have "\<bar>(*f* exp) x\<bar> < (*f* exp) u"
using starfun_abs_exp_cancel starfun_exp_less_mono by auto
with \<open>u \<in> \<real>\<close> show ?thesis
by (force simp: HFinite_def Reals_eq_Standard)
qed
lemma starfun_exp_add_HFinite_Infinitesimal_approx:
"\<lbrakk>x \<in> Infinitesimal; z \<in> HFinite\<rbrakk> \<Longrightarrow> ( *f* exp) (z + x::hypreal) \<approx> ( *f* exp) z"
apply (simp add: STAR_exp_add)
apply (frule STAR_exp_Infinitesimal)
apply (drule approx_mult2)
apply (auto intro: starfun_exp_HFinite)
done
(* using previous result to get to result *)
lemma starfun_ln_HInfinite:
"\<lbrakk>x \<in> HInfinite; 0 < x\<rbrakk> \<Longrightarrow> ( *f* real_ln) x \<in> HInfinite"
apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
apply (drule starfun_exp_HFinite)
apply (simp add: starfun_exp_ln_iff [THEN iffD2] HFinite_HInfinite_iff)
done
lemma starfun_exp_HInfinite_Infinitesimal_disj:
"x \<in> HInfinite \<Longrightarrow> ( *f* exp) x \<in> HInfinite \<or> ( *f* exp) (x::hypreal) \<in> Infinitesimal"
apply (insert linorder_linear [of x 0])
apply (auto intro: starfun_exp_HInfinite starfun_exp_Infinitesimal)
done
(* check out this proof\<And>! *)
lemma starfun_ln_HFinite_not_Infinitesimal:
"\<lbrakk>x \<in> HFinite - Infinitesimal; 0 < x\<rbrakk> \<Longrightarrow> ( *f* real_ln) x \<in> HFinite"
apply (rule ccontr, drule HInfinite_HFinite_iff [THEN iffD2])
apply (drule starfun_exp_HInfinite_Infinitesimal_disj)
apply (simp add: starfun_exp_ln_iff [symmetric] HInfinite_HFinite_iff
del: starfun_exp_ln_iff)
done
(* we do proof by considering ln of 1/x *)
lemma starfun_ln_Infinitesimal_HInfinite:
"\<lbrakk>x \<in> Infinitesimal; 0 < x\<rbrakk> \<Longrightarrow> ( *f* real_ln) x \<in> HInfinite"
apply (drule Infinitesimal_inverse_HInfinite)
apply (frule positive_imp_inverse_positive)
apply (drule_tac [2] starfun_ln_HInfinite)
apply (auto simp add: starfun_ln_inverse HInfinite_minus_iff)
done
lemma starfun_ln_less_zero: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ( *f* real_ln) x < 0"
by transfer (rule ln_less_zero)
lemma starfun_ln_Infinitesimal_less_zero:
"\<lbrakk>x \<in> Infinitesimal; 0 < x\<rbrakk> \<Longrightarrow> ( *f* real_ln) x < 0"
by (auto intro!: starfun_ln_less_zero simp add: Infinitesimal_def)
lemma starfun_ln_HInfinite_gt_zero:
"\<lbrakk>x \<in> HInfinite; 0 < x\<rbrakk> \<Longrightarrow> 0 < ( *f* real_ln) x"
by (auto intro!: starfun_ln_gt_zero simp add: HInfinite_def)
(*
Goalw [NSLIM_def] "(\<lambda>h. ((x powr h) - 1) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S ln x"
*)
lemma HFinite_sin [simp]: "sumhr (0, whn, \<lambda>n. sin_coeff n * x ^ n) \<in> HFinite"
unfolding sumhr_app
apply (simp only: star_zero_def starfun2_star_of atLeast0LessThan)
apply (rule NSBseqD2)
apply (rule NSconvergent_NSBseq)
apply (rule convergent_NSconvergent_iff [THEN iffD1])
apply (rule summable_iff_convergent [THEN iffD1])
using summable_norm_sin [of x]
apply (simp add: summable_rabs_cancel)
done
lemma STAR_sin_zero [simp]: "( *f* sin) 0 = 0"
by transfer (rule sin_zero)
lemma STAR_sin_Infinitesimal [simp]:
fixes x :: "'a::{real_normed_field,banach} star"
shows "x \<in> Infinitesimal \<Longrightarrow> ( *f* sin) x \<approx> x"
apply (case_tac "x = 0")
apply (cut_tac [2] x = 0 in DERIV_sin)
apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
apply (drule bspec [where x = x], auto)
apply (drule approx_mult1 [where c = x])
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
simp add: mult.assoc)
done
lemma HFinite_cos [simp]: "sumhr (0, whn, \<lambda>n. cos_coeff n * x ^ n) \<in> HFinite"
unfolding sumhr_app
apply (simp only: star_zero_def starfun2_star_of atLeast0LessThan)
apply (rule NSBseqD2)
apply (rule NSconvergent_NSBseq)
apply (rule convergent_NSconvergent_iff [THEN iffD1])
apply (rule summable_iff_convergent [THEN iffD1])
using summable_norm_cos [of x]
apply (simp add: summable_rabs_cancel)
done
lemma STAR_cos_zero [simp]: "( *f* cos) 0 = 1"
by transfer (rule cos_zero)
lemma STAR_cos_Infinitesimal [simp]:
fixes x :: "'a::{real_normed_field,banach} star"
shows "x \<in> Infinitesimal \<Longrightarrow> ( *f* cos) x \<approx> 1"
apply (case_tac "x = 0")
apply (cut_tac [2] x = 0 in DERIV_cos)
apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
apply (drule bspec [where x = x])
apply auto
apply (drule approx_mult1 [where c = x])
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
simp add: mult.assoc)
apply (rule approx_add_right_cancel [where d = "-1"])
apply simp
done
lemma STAR_tan_zero [simp]: "( *f* tan) 0 = 0"
by transfer (rule tan_zero)
lemma STAR_tan_Infinitesimal: "x \<in> Infinitesimal \<Longrightarrow> ( *f* tan) x \<approx> x"
apply (case_tac "x = 0")
apply (cut_tac [2] x = 0 in DERIV_tan)
apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
apply (drule bspec [where x = x], auto)
apply (drule approx_mult1 [where c = x])
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
simp add: mult.assoc)
done
lemma STAR_sin_cos_Infinitesimal_mult:
fixes x :: "'a::{real_normed_field,banach} star"
shows "x \<in> Infinitesimal \<Longrightarrow> ( *f* sin) x * ( *f* cos) x \<approx> x"
using approx_mult_HFinite [of "( *f* sin) x" _ "( *f* cos) x" 1]
by (simp add: Infinitesimal_subset_HFinite [THEN subsetD])
lemma HFinite_pi: "hypreal_of_real pi \<in> HFinite"
by simp
(* lemmas *)
lemma lemma_split_hypreal_of_real:
"N \<in> HNatInfinite
\<Longrightarrow> hypreal_of_real a =
hypreal_of_hypnat N * (inverse(hypreal_of_hypnat N) * hypreal_of_real a)"
by (simp add: mult.assoc [symmetric] zero_less_HNatInfinite)
lemma STAR_sin_Infinitesimal_divide:
fixes x :: "'a::{real_normed_field,banach} star"
shows "\<lbrakk>x \<in> Infinitesimal; x \<noteq> 0\<rbrakk> \<Longrightarrow> ( *f* sin) x/x \<approx> 1"
using DERIV_sin [of "0::'a"]
by (simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
(*------------------------------------------------------------------------*)
(* sin* (1/n) * 1/(1/n) \<approx> 1 for n = oo *)
(*------------------------------------------------------------------------*)
lemma lemma_sin_pi:
"n \<in> HNatInfinite
\<Longrightarrow> ( *f* sin) (inverse (hypreal_of_hypnat n))/(inverse (hypreal_of_hypnat n)) \<approx> 1"
apply (rule STAR_sin_Infinitesimal_divide)
apply (auto simp add: zero_less_HNatInfinite)
done
lemma STAR_sin_inverse_HNatInfinite:
"n \<in> HNatInfinite
\<Longrightarrow> ( *f* sin) (inverse (hypreal_of_hypnat n)) * hypreal_of_hypnat n \<approx> 1"
apply (frule lemma_sin_pi)
apply (simp add: divide_inverse)
done
lemma Infinitesimal_pi_divide_HNatInfinite:
"N \<in> HNatInfinite
\<Longrightarrow> hypreal_of_real pi/(hypreal_of_hypnat N) \<in> Infinitesimal"
apply (simp add: divide_inverse)
apply (auto intro: Infinitesimal_HFinite_mult2)
done
lemma pi_divide_HNatInfinite_not_zero [simp]:
"N \<in> HNatInfinite \<Longrightarrow> hypreal_of_real pi/(hypreal_of_hypnat N) \<noteq> 0"
by (simp add: zero_less_HNatInfinite)
lemma STAR_sin_pi_divide_HNatInfinite_approx_pi:
"n \<in> HNatInfinite
\<Longrightarrow> ( *f* sin) (hypreal_of_real pi/(hypreal_of_hypnat n)) * hypreal_of_hypnat n
\<approx> hypreal_of_real pi"
apply (frule STAR_sin_Infinitesimal_divide
[OF Infinitesimal_pi_divide_HNatInfinite
pi_divide_HNatInfinite_not_zero])
apply (auto)
apply (rule approx_SReal_mult_cancel [of "inverse (hypreal_of_real pi)"])
apply (auto intro: Reals_inverse simp add: divide_inverse ac_simps)
done
lemma STAR_sin_pi_divide_HNatInfinite_approx_pi2:
"n \<in> HNatInfinite
\<Longrightarrow> hypreal_of_hypnat n *
( *f* sin) (hypreal_of_real pi/(hypreal_of_hypnat n))
\<approx> hypreal_of_real pi"
apply (rule mult.commute [THEN subst])
apply (erule STAR_sin_pi_divide_HNatInfinite_approx_pi)
done
lemma starfunNat_pi_divide_n_Infinitesimal:
"N \<in> HNatInfinite \<Longrightarrow> ( *f* (\<lambda>x. pi / real x)) N \<in> Infinitesimal"
by (auto intro!: Infinitesimal_HFinite_mult2
simp add: starfun_mult [symmetric] divide_inverse
starfun_inverse [symmetric] starfunNat_real_of_nat)
lemma STAR_sin_pi_divide_n_approx:
"N \<in> HNatInfinite \<Longrightarrow>
( *f* sin) (( *f* (\<lambda>x. pi / real x)) N) \<approx>
hypreal_of_real pi/(hypreal_of_hypnat N)"
apply (simp add: starfunNat_real_of_nat [symmetric])
apply (rule STAR_sin_Infinitesimal)
apply (simp add: divide_inverse)
apply (rule Infinitesimal_HFinite_mult2)
apply (subst starfun_inverse)
apply (erule starfunNat_inverse_real_of_nat_Infinitesimal)
apply simp
done
lemma NSLIMSEQ_sin_pi: "(\<lambda>n. real n * sin (pi / real n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S pi"
apply (auto simp add: NSLIMSEQ_def starfun_mult [symmetric] starfunNat_real_of_nat)
apply (rule_tac f1 = sin in starfun_o2 [THEN subst])
apply (auto simp add: starfun_mult [symmetric] starfunNat_real_of_nat divide_inverse)
apply (rule_tac f1 = inverse in starfun_o2 [THEN subst])
apply (auto dest: STAR_sin_pi_divide_HNatInfinite_approx_pi
simp add: starfunNat_real_of_nat mult.commute divide_inverse)
done
lemma NSLIMSEQ_cos_one: "(\<lambda>n. cos (pi / real n))\<longlonglongrightarrow>\<^sub>N\<^sub>S 1"
apply (simp add: NSLIMSEQ_def, auto)
apply (rule_tac f1 = cos in starfun_o2 [THEN subst])
apply (rule STAR_cos_Infinitesimal)
apply (auto intro!: Infinitesimal_HFinite_mult2
simp add: starfun_mult [symmetric] divide_inverse
starfun_inverse [symmetric] starfunNat_real_of_nat)
done
lemma NSLIMSEQ_sin_cos_pi:
"(\<lambda>n. real n * sin (pi / real n) * cos (pi / real n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S pi"
by (insert NSLIMSEQ_mult [OF NSLIMSEQ_sin_pi NSLIMSEQ_cos_one], simp)
text\<open>A familiar approximation to \<^term>\<open>cos x\<close> when \<^term>\<open>x\<close> is small\<close>
lemma STAR_cos_Infinitesimal_approx:
fixes x :: "'a::{real_normed_field,banach} star"
shows "x \<in> Infinitesimal \<Longrightarrow> ( *f* cos) x \<approx> 1 - x\<^sup>2"
apply (rule STAR_cos_Infinitesimal [THEN approx_trans])
apply (auto simp add: Infinitesimal_approx_minus [symmetric]
add.assoc [symmetric] numeral_2_eq_2)
done
lemma STAR_cos_Infinitesimal_approx2:
fixes x :: hypreal \<comment> \<open>perhaps could be generalised, like many other hypreal results\<close>
shows "x \<in> Infinitesimal \<Longrightarrow> ( *f* cos) x \<approx> 1 - (x\<^sup>2)/2"
apply (rule STAR_cos_Infinitesimal [THEN approx_trans])
apply (auto intro: Infinitesimal_SReal_divide Infinitesimal_mult
simp add: Infinitesimal_approx_minus [symmetric] numeral_2_eq_2)
done
end