author paulson Mon, 29 Apr 2019 00:36:54 +0100 changeset 70211 2388e0d2827b parent 70207 511352b4d5d3 (current diff) parent 70210 1ececb77b27a (diff) child 70213 ff8386fc191d child 70221 bca14283e1a8
merged
--- a/src/HOL/Nonstandard_Analysis/HTranscendental.thy	Sun Apr 28 22:22:29 2019 +0200
+++ b/src/HOL/Nonstandard_Analysis/HTranscendental.thy	Mon Apr 29 00:36:54 2019 +0100
@@ -12,602 +12,524 @@
begin

definition
-  exphr :: "real => hypreal" where
+  exphr :: "real \<Rightarrow> hypreal" where
\<comment> \<open>define exponential function using standard part\<close>
-  "exphr x =  st(sumhr (0, whn, %n. inverse (fact n) * (x ^ n)))"
+  "exphr x \<equiv>  st(sumhr (0, whn, \<lambda>n. inverse (fact n) * (x ^ n)))"

definition
-  sinhr :: "real => hypreal" where
-  "sinhr x = st(sumhr (0, whn, %n. sin_coeff n * x ^ n))"
+  sinhr :: "real \<Rightarrow> hypreal" where
+  "sinhr x \<equiv> st(sumhr (0, whn, \<lambda>n. sin_coeff n * x ^ n))"

definition
-  coshr :: "real => hypreal" where
-  "coshr x = st(sumhr (0, whn, %n. cos_coeff n * x ^ n))"
+  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)"
-apply (cases x)
-apply (auto simp add: star_n_le star_n_zero_num starfun hrealpow star_n_eq_iff
-            simp del: hpowr_Suc power_Suc)
-done
-
-lemma hypreal_sqrt_gt_zero_pow2: "!!x. 0 < x ==> ( *f* sqrt) (x) ^ 2 = x"
-by (transfer, simp)
+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_pow2_gt_zero: "0 < x ==> 0 < ( *f* sqrt) (x) ^ 2"
-by (frule hypreal_sqrt_gt_zero_pow2, auto)
+lemma hypreal_sqrt_gt_zero_pow2: "\<And>x. 0 < x \<Longrightarrow> ( *f* sqrt) (x) ^ 2 = x"
+  by transfer simp

-lemma hypreal_sqrt_not_zero: "0 < x ==> ( *f* sqrt) (x) \<noteq> 0"
-apply (frule hypreal_sqrt_pow2_gt_zero)
-done
+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 ==> inverse (( *f* sqrt)(x)) ^ 2 = inverse x"
-apply (cut_tac n = 2 and a = "( *f* sqrt) x" in power_inverse [symmetric])
-apply (auto dest: hypreal_sqrt_gt_zero_pow2)
-done
+     "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:
-    "!!x y. [|0 < x; 0 <y |] ==>
+    "\<And>x y. \<lbrakk>0 < x; 0 <y\<rbrakk> \<Longrightarrow>
( *f* sqrt)(x*y) = ( *f* sqrt)(x) * ( *f* sqrt)(y)"
-apply transfer
-apply (auto intro: real_sqrt_mult)
-done
+  by transfer (auto intro: real_sqrt_mult)

lemma hypreal_sqrt_mult_distrib2:
-     "[|0\<le>x; 0\<le>y |] ==>
-     ( *f* sqrt)(x*y) =  ( *f* sqrt)(x) * ( *f* sqrt)(y)"
+     "\<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]:
-     "0 < x ==> (( *f* sqrt)(x) \<approx> 0) = (x \<approx> 0)"
-apply (auto simp add: mem_infmal_iff [symmetric])
-apply (rule hypreal_sqrt_gt_zero_pow2 [THEN subst])
-apply (auto intro: Infinitesimal_mult
-            dest!: hypreal_sqrt_gt_zero_pow2 [THEN ssubst]
-done
+  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 ==> (( *f* sqrt)(x) \<approx> 0) = (x \<approx> 0)"
+  "0 \<le> x \<Longrightarrow> (( *f* sqrt)(x) \<approx> 0) = (x \<approx> 0)"
+  by (auto simp add: order_le_less)

-lemma hypreal_sqrt_sum_squares [simp]:
-     "(( *f* sqrt)(x*x + y*y + z*z) \<approx> 0) = (x*x + y*y + z*z \<approx> 0)"
-apply (rule hypreal_sqrt_approx_zero2)
-apply (auto)
-done
+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_sum_squares2 [simp]:
-     "(( *f* sqrt)(x*x + y*y) \<approx> 0) = (x*x + y*y \<approx> 0)"
-apply (rule hypreal_sqrt_approx_zero2)
-apply (auto)
-done
+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_gt_zero: "!!x. 0 < x ==> 0 < ( *f* sqrt)(x)"
-apply transfer
-apply (auto intro: real_sqrt_gt_zero)
-done
+lemma hypreal_sqrt_hrabs [simp]: "\<And>x. ( *f* sqrt)(x\<^sup>2) = \<bar>x\<bar>"
+  by transfer simp

-lemma hypreal_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> ( *f* sqrt)(x)"
-by (auto intro: hypreal_sqrt_gt_zero simp add: order_le_less)
-
-lemma hypreal_sqrt_hrabs [simp]: "!!x. ( *f* sqrt)(x\<^sup>2) = \<bar>x\<bar>"
-by (transfer, simp)
-
-lemma hypreal_sqrt_hrabs2 [simp]: "!!x. ( *f* sqrt)(x*x) = \<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]:
-     "!!x. ( *f* sqrt)(x pow (hypnat_of_nat 2)) = \<bar>x\<bar>"
-by (transfer, 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"
-apply (rule HFinite_square_iff [THEN iffD1])
-apply (simp only: hypreal_sqrt_mult_distrib2 [symmetric], simp)
-done
+  by (metis HFinite_square_iff hypreal_sqrt_pow2_iff power2_eq_square)

lemma st_hypreal_sqrt:
-     "[| x \<in> HFinite; 0 \<le> x |] ==> st(( *f* sqrt) x) = ( *f* sqrt)(st x)"
-apply (rule power_inject_base [where n=1])
-apply (auto intro!: st_zero_le hypreal_sqrt_ge_zero)
-apply (rule st_mult [THEN subst])
-apply (rule_tac [3] hypreal_sqrt_mult_distrib2 [THEN subst])
-apply (rule_tac [5] hypreal_sqrt_mult_distrib2 [THEN subst])
-apply (auto simp add: st_hrabs st_zero_le star_sqrt_HFinite)
-done
+  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]: "!!x y. x \<le> ( *f* sqrt)(x\<^sup>2 + y\<^sup>2)"
-by transfer (rule real_sqrt_sum_squares_ge1)
-
-lemma HFinite_hypreal_sqrt:
-     "[| 0 \<le> x; x \<in> HFinite |] ==> ( *f* sqrt) x \<in> HFinite"
-apply (rule HFinite_square_iff [THEN iffD1])
-apply (drule hypreal_sqrt_gt_zero_pow2)
-done
+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:
-     "[| 0 \<le> x; ( *f* sqrt) x \<in> HFinite |] ==> x \<in> HFinite"
-apply (drule HFinite_square_iff [THEN iffD2])
-apply (drule hypreal_sqrt_gt_zero_pow2)
-apply (simp add: numeral_2_eq_2 del: HFinite_square_iff)
-done
+  "\<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 ==> (( *f* sqrt) x \<in> HFinite) = (x \<in> HFinite)"
-by (blast intro: HFinite_hypreal_sqrt HFinite_hypreal_sqrt_imp_HFinite)
-
-lemma HFinite_sqrt_sum_squares [simp]:
-     "(( *f* sqrt)(x*x + y*y) \<in> HFinite) = (x*x + y*y \<in> HFinite)"
-apply (rule HFinite_hypreal_sqrt_iff)
-apply (auto)
-done
+  "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:
-     "[| 0 \<le> x; x \<in> Infinitesimal |] ==> ( *f* sqrt) x \<in> Infinitesimal"
-apply (rule Infinitesimal_square_iff [THEN iffD2])
-apply (drule hypreal_sqrt_gt_zero_pow2)
-done
+     "\<lbrakk>0 \<le> x; x \<in> Infinitesimal\<rbrakk> \<Longrightarrow> ( *f* sqrt) x \<in> Infinitesimal"

lemma Infinitesimal_hypreal_sqrt_imp_Infinitesimal:
-     "[| 0 \<le> x; ( *f* sqrt) x \<in> Infinitesimal |] ==> x \<in> Infinitesimal"
-apply (drule Infinitesimal_square_iff [THEN iffD1])
-apply (drule hypreal_sqrt_gt_zero_pow2)
-apply (simp add: numeral_2_eq_2 del: Infinitesimal_square_iff [symmetric])
-done
+     "\<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 ==> (( *f* sqrt) x \<in> Infinitesimal) = (x \<in> Infinitesimal)"
+     "0 \<le> x \<Longrightarrow> (( *f* sqrt) x \<in> Infinitesimal) = (x \<in> Infinitesimal)"
by (blast intro: Infinitesimal_hypreal_sqrt_imp_Infinitesimal Infinitesimal_hypreal_sqrt)

-lemma Infinitesimal_sqrt_sum_squares [simp]:
-     "(( *f* sqrt)(x*x + y*y) \<in> Infinitesimal) = (x*x + y*y \<in> Infinitesimal)"
-apply (rule Infinitesimal_hypreal_sqrt_iff)
-apply (auto)
-done
-
lemma HInfinite_hypreal_sqrt:
-     "[| 0 \<le> x; x \<in> HInfinite |] ==> ( *f* sqrt) x \<in> HInfinite"
-apply (rule HInfinite_square_iff [THEN iffD1])
-apply (drule hypreal_sqrt_gt_zero_pow2)
-done
+     "\<lbrakk>0 \<le> x; x \<in> HInfinite\<rbrakk> \<Longrightarrow> ( *f* sqrt) x \<in> HInfinite"

lemma HInfinite_hypreal_sqrt_imp_HInfinite:
-     "[| 0 \<le> x; ( *f* sqrt) x \<in> HInfinite |] ==> x \<in> HInfinite"
-apply (drule HInfinite_square_iff [THEN iffD2])
-apply (drule hypreal_sqrt_gt_zero_pow2)
-apply (simp add: numeral_2_eq_2 del: HInfinite_square_iff)
-done
+     "\<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 ==> (( *f* sqrt) x \<in> HInfinite) = (x \<in> HInfinite)"
+     "0 \<le> x \<Longrightarrow> (( *f* sqrt) x \<in> HInfinite) = (x \<in> HInfinite)"
by (blast intro: HInfinite_hypreal_sqrt HInfinite_hypreal_sqrt_imp_HInfinite)

-lemma HInfinite_sqrt_sum_squares [simp]:
-     "(( *f* sqrt)(x*x + y*y) \<in> HInfinite) = (x*x + y*y \<in> HInfinite)"
-apply (rule HInfinite_hypreal_sqrt_iff)
-apply (auto)
-done
-
lemma HFinite_exp [simp]:
-     "sumhr (0, whn, %n. inverse (fact 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])
-apply (rule summable_exp)
-done
+  "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"
-apply (rule st_unique, simp)
-apply (rule subst [where P="\<lambda>x. 1 \<approx> x", OF _ approx_refl])
-apply (rule rev_mp [OF hypnat_one_less_hypnat_omega])
-apply (rule_tac x="whn" in spec)
-apply (unfold sumhr_app, transfer, simp add: power_0_left)
-done
+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"
-                   [OF hypnat_one_less_hypnat_omega, symmetric])
-apply (rule st_unique, simp)
-apply (rule subst [where P="\<lambda>x. 1 \<approx> x", OF _ approx_refl])
-apply (rule rev_mp [OF hypnat_one_less_hypnat_omega])
-apply (rule_tac x="whn" in spec)
-apply (unfold sumhr_app, transfer, simp add: cos_coeff_def power_0_left)
-done
+  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"
-apply (subgoal_tac "( *f* exp) (0::hypreal) = 1", simp)
-apply (transfer, simp)
-done
+  proof -
+  have "(*f* exp) (0::real star) = 1"
+    by transfer simp
+  then show ?thesis
+    by auto
+qed

-lemma STAR_exp_Infinitesimal: "x \<in> Infinitesimal ==> ( *f* exp) (x::hypreal) \<approx> 1"
-apply (case_tac "x = 0")
-apply (cut_tac [2] x = 0 in DERIV_exp)
-apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
-apply (drule_tac x = x in bspec, auto)
-apply (drule_tac c = x in approx_mult1)
-apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
-apply (rule approx_sym [THEN [2] approx_trans2])
-done
+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)
+  by (auto intro: STAR_exp_Infinitesimal)

-  "!!(x::'a:: {banach,real_normed_field} star) y. ( *f* exp)(x + y) = ( *f* exp) x * ( *f* exp) y"
+  "\<And>(x::'a:: {banach,real_normed_field} star) y. ( *f* exp)(x + y) = ( *f* exp) x * ( *f* exp) y"

lemma exphr_hypreal_of_real_exp_eq: "exphr x = hypreal_of_real (exp x)"
-apply (rule st_unique, simp)
-apply (subst starfunNat_sumr [symmetric])
-unfolding atLeast0LessThan
-apply (rule NSLIMSEQ_D [THEN approx_sym])
-apply (rule LIMSEQ_NSLIMSEQ)
-apply (subst sums_def [symmetric])
-apply (cut_tac exp_converges [where x=x], simp)
-apply (rule HNatInfinite_whn)
-done
+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]: "!!x::hypreal. 0 \<le> x ==> (1 + x) \<le> ( *f* exp) x"
+lemma starfun_exp_ge_add_one_self [simp]: "\<And>x::hypreal. 0 \<le> x \<Longrightarrow> (1 + x) \<le> ( *f* exp) x"

-(* exp (oo) is infinite *)
+text\<open>exp maps infinities to infinities\<close>
lemma starfun_exp_HInfinite:
-     "[| x \<in> HInfinite; 0 \<le> x |] ==> ( *f* exp) (x::hypreal) \<in> HInfinite"
-apply (rule HInfinite_ge_HInfinite, assumption)
-apply (rule order_trans [of _ "1+x"], auto)
-done
+  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:
-  "!!x::'a:: {banach,real_normed_field} star. ( *f* exp) (-x) = inverse(( *f* exp) x)"
-by transfer (rule exp_minus)
+  "\<And>x::'a:: {banach,real_normed_field} star. ( *f* exp) (-x) = inverse(( *f* exp) x)"
+  by transfer (rule exp_minus)

-(* exp (-oo) is infinitesimal *)
+text\<open>exp maps infinitesimals to infinitesimals\<close>
lemma starfun_exp_Infinitesimal:
-     "[| x \<in> HInfinite; x \<le> 0 |] ==> ( *f* exp) (x::hypreal) \<in> Infinitesimal"
-apply (subgoal_tac "\<exists>y. x = - y")
-apply (rule_tac [2] x = "- x" in exI)
-apply (auto intro!: HInfinite_inverse_Infinitesimal starfun_exp_HInfinite
-done
+  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]: "!!x::hypreal. 0 < x ==> 1 < ( *f* exp) x"
-by transfer (rule exp_gt_one)
+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]: "!!x. ( *f* real_ln) (( *f* exp) x) = x"
-by transfer (rule ln_exp)
+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]: "!!x. (( *f* exp)(( *f* real_ln) x) = x) = (0 < x)"
-by transfer (rule exp_ln_iff)
+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: "!!u x. ( *f* exp) u = x ==> ( *f* real_ln) x = u"
-by transfer (rule ln_unique)
+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]: "!!x. 0 < x ==> ( *f* real_ln) x < x"
-by transfer (rule ln_less_self)
-
-lemma starfun_ln_ge_zero [simp]: "!!x. 1 \<le> x ==> 0 \<le> ( *f* real_ln) x"
-by transfer (rule ln_ge_zero)
+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_gt_zero [simp]: "!!x .1 < x ==> 0 < ( *f* real_ln) x"
-by transfer (rule ln_gt_zero)
+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_not_eq_zero [simp]: "!!x. [| 0 < x; x \<noteq> 1 |] ==> ( *f* real_ln) x \<noteq> 0"
-by transfer simp
+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_HFinite: "[| x \<in> HFinite; 1 \<le> x |] ==> ( *f* real_ln) x \<in> HFinite"
-apply (rule HFinite_bounded)
-apply assumption
-done
+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_inverse: "!!x. 0 < x ==> ( *f* real_ln) (inverse x) = -( *f* ln) x"
-by transfer (rule ln_inverse)
+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)
+  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)
+  by transfer (rule exp_less_mono)

-lemma starfun_exp_HFinite: "x \<in> HFinite ==> ( *f* exp) (x::hypreal) \<in> HFinite"
-apply (auto simp add: HFinite_def, rename_tac u)
-apply (rule_tac x="( *f* exp) u" in rev_bexI)
-done
+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

-     "[|x \<in> Infinitesimal; z \<in> HFinite |] ==> ( *f* exp) (z + x::hypreal) \<approx> ( *f* exp) z"
-apply (frule STAR_exp_Infinitesimal)
-apply (drule approx_mult2)
-apply (auto intro: starfun_exp_HFinite)
-done
+  fixes x :: hypreal
+  shows "\<lbrakk>x \<in> Infinitesimal; z \<in> HFinite\<rbrakk> \<Longrightarrow> ( *f* exp) (z + x::hypreal) \<approx> ( *f* exp) z"

-(* using previous result to get to result *)
lemma starfun_ln_HInfinite:
-     "[| x \<in> HInfinite; 0 < x |] ==> ( *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
+  "\<lbrakk>x \<in> HInfinite; 0 < x\<rbrakk> \<Longrightarrow> ( *f* real_ln) x \<in> HInfinite"
+  by (metis HInfinite_HFinite_iff starfun_exp_HFinite starfun_exp_ln_iff)

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
+  fixes x :: hypreal
+  shows "x \<in> HInfinite \<Longrightarrow> ( *f* exp) x \<in> HInfinite \<or> ( *f* exp) (x::hypreal) \<in> Infinitesimal"
+  by (meson linear starfun_exp_HInfinite starfun_exp_Infinitesimal)

-(* check out this proof!!! *)
lemma starfun_ln_HFinite_not_Infinitesimal:
-     "[| x \<in> HFinite - Infinitesimal; 0 < x |] ==> ( *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
+     "\<lbrakk>x \<in> HFinite - Infinitesimal; 0 < x\<rbrakk> \<Longrightarrow> ( *f* real_ln) x \<in> HFinite"
+  by (metis DiffD1 DiffD2 HInfinite_HFinite_iff starfun_exp_HInfinite_Infinitesimal_disj starfun_exp_ln_iff)

(* we do proof by considering ln of 1/x *)
lemma starfun_ln_Infinitesimal_HInfinite:
-     "[| x \<in> Infinitesimal; 0 < x |] ==> ( *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
+  assumes "x \<in> Infinitesimal" "0 < x"
+  shows "( *f* real_ln) x \<in> HInfinite"
+proof -
+  have "inverse x \<in> HInfinite"
+    using Infinitesimal_inverse_HInfinite assms by blast
+  then show ?thesis
+    using HInfinite_minus_iff assms(2) starfun_ln_HInfinite starfun_ln_inverse by fastforce
+qed

-lemma starfun_ln_less_zero: "!!x. [| 0 < x; x < 1 |] ==> ( *f* real_ln) x < 0"
-by transfer (rule ln_less_zero)
+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:
-     "[| x \<in> Infinitesimal; 0 < x |] ==> ( *f* real_ln) x < 0"
-by (auto intro!: starfun_ln_less_zero simp add: Infinitesimal_def)
+  "\<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:
-     "[| x \<in> HInfinite; 0 < x |] ==> 0 < ( *f* real_ln) x"
-by (auto intro!: starfun_ln_gt_zero simp add: HInfinite_def)
+  "\<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] "(%h. ((x powr h) - 1) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S ln x"
-*)
-
-lemma HFinite_sin [simp]: "sumhr (0, whn, %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]
-done
+lemma HFinite_sin [simp]: "sumhr (0, whn, \<lambda>n. sin_coeff n * x ^ n) \<in> HFinite"
+proof -
+  have "summable (\<lambda>i. sin_coeff i * x ^ i)"
+    using summable_norm_sin [of x] by (simp add: summable_rabs_cancel)
+  then have "(*f* (\<lambda>n. \<Sum>n<n. sin_coeff n * x ^ n)) whn \<in> HFinite"
+    unfolding summable_sums_iff sums_NSsums_iff NSsums_def NSLIMSEQ_iff
+    using HFinite_star_of HNatInfinite_whn approx_HFinite approx_sym by blast
+  then show ?thesis
+    unfolding sumhr_app
+    by (simp only: star_zero_def starfun2_star_of atLeast0LessThan)
+qed

lemma STAR_sin_zero [simp]: "( *f* sin) 0 = 0"
-by transfer (rule sin_zero)
+  by transfer (rule sin_zero)

lemma STAR_sin_Infinitesimal [simp]:
fixes x :: "'a::{real_normed_field,banach} star"
-  shows "x \<in> Infinitesimal ==> ( *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]
-done
+  assumes "x \<in> Infinitesimal"
+  shows "( *f* sin) x \<approx> x"
+proof (cases "x = 0")
+  case False
+  have "NSDERIV sin 0 :> 1"
+    by (metis DERIV_sin NSDERIV_DERIV_iff cos_zero)
+  then have "(*f* sin) x / x \<approx> 1"
+    using False NSDERIVD2 assms by fastforce
+  with assms show ?thesis
+    unfolding star_one_def
+    by (metis False Infinitesimal_approx Infinitesimal_ratio approx_star_of_HFinite)
+qed auto

-lemma HFinite_cos [simp]: "sumhr (0, whn, %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]
-done
+lemma HFinite_cos [simp]: "sumhr (0, whn, \<lambda>n. cos_coeff n * x ^ n) \<in> HFinite"
+proof -
+  have "summable (\<lambda>i. cos_coeff i * x ^ i)"
+    using summable_norm_cos [of x] by (simp add: summable_rabs_cancel)
+  then have "(*f* (\<lambda>n. \<Sum>n<n. cos_coeff n * x ^ n)) whn \<in> HFinite"
+    unfolding summable_sums_iff sums_NSsums_iff NSsums_def NSLIMSEQ_iff
+    using HFinite_star_of HNatInfinite_whn approx_HFinite approx_sym by blast
+  then show ?thesis
+    unfolding sumhr_app
+    by (simp only: star_zero_def starfun2_star_of atLeast0LessThan)
+qed

lemma STAR_cos_zero [simp]: "( *f* cos) 0 = 1"
-by transfer (rule cos_zero)
+  by transfer (rule cos_zero)

lemma STAR_cos_Infinitesimal [simp]:
fixes x :: "'a::{real_normed_field,banach} star"
-  shows "x \<in> Infinitesimal ==> ( *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]
-apply (rule approx_add_right_cancel [where d = "-1"])
-apply simp
-done
+  assumes "x \<in> Infinitesimal"
+  shows "( *f* cos) x \<approx> 1"
+proof (cases "x = 0")
+  case False
+  have "NSDERIV cos 0 :> 0"
+    by (metis DERIV_cos NSDERIV_DERIV_iff add.inverse_neutral sin_zero)
+  then have "(*f* cos) x - 1 \<approx> 0"
+    using NSDERIV_approx assms by fastforce
+  with assms show ?thesis
+    using approx_minus_iff by blast
+qed auto

lemma STAR_tan_zero [simp]: "( *f* tan) 0 = 0"
-by transfer (rule tan_zero)
+  by transfer (rule tan_zero)

-lemma STAR_tan_Infinitesimal: "x \<in> Infinitesimal ==> ( *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]
-done
+lemma STAR_tan_Infinitesimal [simp]:
+  assumes "x \<in> Infinitesimal"
+  shows "( *f* tan) x \<approx> x"
+proof (cases "x = 0")
+  case False
+  have "NSDERIV tan 0 :> 1"
+    using DERIV_tan [of 0] by (simp add: NSDERIV_DERIV_iff)
+  then have "(*f* tan) x / x \<approx> 1"
+    using False NSDERIVD2 assms by fastforce
+  with assms show ?thesis
+    unfolding star_one_def
+    by (metis False Infinitesimal_approx Infinitesimal_ratio approx_star_of_HFinite)
+qed auto

lemma STAR_sin_cos_Infinitesimal_mult:
fixes x :: "'a::{real_normed_field,banach} star"
-  shows "x \<in> Infinitesimal ==> ( *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])
+  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 *)
+  by simp

-lemma lemma_split_hypreal_of_real:
-     "N \<in> HNatInfinite
-      ==> 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 "[|x \<in> Infinitesimal; x \<noteq> 0 |] ==> ( *f* sin) x/x \<approx> 1"
-using DERIV_sin [of "0::'a"]
-by (simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
+  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                                   *)
-(*------------------------------------------------------------------------*)
+subsection \<open>Proving $\sin* (1/n) \times 1/(1/n) \approx 1$ for $n = \infty$ \<close>

lemma lemma_sin_pi:
-     "n \<in> HNatInfinite
-      ==> ( *f* sin) (inverse (hypreal_of_hypnat n))/(inverse (hypreal_of_hypnat n)) \<approx> 1"
-apply (rule STAR_sin_Infinitesimal_divide)
-done
+  "n \<in> HNatInfinite
+      \<Longrightarrow> ( *f* sin) (inverse (hypreal_of_hypnat n))/(inverse (hypreal_of_hypnat n)) \<approx> 1"
+  by (simp add: STAR_sin_Infinitesimal_divide zero_less_HNatInfinite)

lemma STAR_sin_inverse_HNatInfinite:
"n \<in> HNatInfinite
-      ==> ( *f* sin) (inverse (hypreal_of_hypnat n)) * hypreal_of_hypnat n \<approx> 1"
-apply (frule lemma_sin_pi)
-done
+      \<Longrightarrow> ( *f* sin) (inverse (hypreal_of_hypnat n)) * hypreal_of_hypnat n \<approx> 1"
+  by (metis field_class.field_divide_inverse inverse_inverse_eq lemma_sin_pi)

lemma Infinitesimal_pi_divide_HNatInfinite:
"N \<in> HNatInfinite
-      ==> hypreal_of_real pi/(hypreal_of_hypnat N) \<in> Infinitesimal"
-apply (auto intro: Infinitesimal_HFinite_mult2)
-done
+      \<Longrightarrow> hypreal_of_real pi/(hypreal_of_hypnat N) \<in> Infinitesimal"
+  by (simp add: Infinitesimal_HFinite_mult2 field_class.field_divide_inverse)

lemma pi_divide_HNatInfinite_not_zero [simp]:
-     "N \<in> HNatInfinite ==> hypreal_of_real pi/(hypreal_of_hypnat N) \<noteq> 0"
+  "N \<in> HNatInfinite \<Longrightarrow> hypreal_of_real pi/(hypreal_of_hypnat N) \<noteq> 0"

lemma STAR_sin_pi_divide_HNatInfinite_approx_pi:
-     "n \<in> HNatInfinite
-      ==> ( *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
+  assumes "n \<in> HNatInfinite"
+  shows "(*f* sin) (hypreal_of_real pi / hypreal_of_hypnat n) * hypreal_of_hypnat n \<approx>
+         hypreal_of_real pi"
+proof -
+  have "(*f* sin) (hypreal_of_real pi / hypreal_of_hypnat n) / (hypreal_of_real pi / hypreal_of_hypnat n) \<approx> 1"
+    using Infinitesimal_pi_divide_HNatInfinite STAR_sin_Infinitesimal_divide assms pi_divide_HNatInfinite_not_zero by blast
+  then have "hypreal_of_hypnat n * star_of sin \<star> (hypreal_of_real pi / hypreal_of_hypnat n) / hypreal_of_real pi \<approx> 1"
+    by (simp add: mult.commute starfun_def)
+  then show ?thesis
+    apply (simp add: starfun_def field_simps)
+    by (metis (no_types, lifting) approx_mult_subst_star_of approx_refl mult_cancel_right1 nonzero_eq_divide_eq pi_neq_zero star_of_eq_0)
+qed

lemma STAR_sin_pi_divide_HNatInfinite_approx_pi2:
"n \<in> HNatInfinite
-      ==> 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
+      \<Longrightarrow> hypreal_of_hypnat n * ( *f* sin) (hypreal_of_real pi/(hypreal_of_hypnat n)) \<approx> hypreal_of_real pi"
+  by (metis STAR_sin_pi_divide_HNatInfinite_approx_pi mult.commute)

lemma starfunNat_pi_divide_n_Infinitesimal:
-     "N \<in> HNatInfinite ==> ( *f* (%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)
+     "N \<in> HNatInfinite \<Longrightarrow> ( *f* (\<lambda>x. pi / real x)) N \<in> Infinitesimal"
+  by (simp add: Infinitesimal_HFinite_mult2 divide_inverse starfunNat_real_of_nat)

lemma STAR_sin_pi_divide_n_approx:
-     "N \<in> HNatInfinite ==>
-      ( *f* sin) (( *f* (%x. pi / real x)) N) \<approx>
-      hypreal_of_real pi/(hypreal_of_hypnat N)"
-apply (rule STAR_sin_Infinitesimal)
-apply (rule Infinitesimal_HFinite_mult2)
-apply (subst starfun_inverse)
-apply (erule starfunNat_inverse_real_of_nat_Infinitesimal)
-apply simp
-done
+  assumes "N \<in> HNatInfinite"
+  shows "( *f* sin) (( *f* (\<lambda>x. pi / real x)) N) \<approx> hypreal_of_real pi/(hypreal_of_hypnat N)"
+proof -
+  have "\<exists>s. (*f* sin) ((*f* (\<lambda>n. pi / real n)) N) \<approx> s \<and> hypreal_of_real pi / hypreal_of_hypnat N \<approx> s"
+    by (metis (lifting) Infinitesimal_approx Infinitesimal_pi_divide_HNatInfinite STAR_sin_Infinitesimal assms starfunNat_pi_divide_n_Infinitesimal)
+  then show ?thesis
+    by (meson approx_trans2)
+qed

-lemma NSLIMSEQ_sin_pi: "(%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_sin_pi: "(\<lambda>n. real n * sin (pi / real n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S pi"
+proof -
+  have *: "hypreal_of_hypnat N * (*f* sin) ((*f* (\<lambda>x. pi / real x)) N) \<approx> hypreal_of_real pi"
+    if "N \<in> HNatInfinite"
+    for N :: "nat star"
+    using that
+    by simp (metis STAR_sin_pi_divide_HNatInfinite_approx_pi2 starfunNat_real_of_nat)
+  show ?thesis
+    by (simp add: NSLIMSEQ_def starfunNat_real_of_nat) (metis * starfun_o2)
+qed

-lemma NSLIMSEQ_cos_one: "(%n. cos (pi / real n))\<longlonglongrightarrow>\<^sub>N\<^sub>S 1"
-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_cos_one: "(\<lambda>n. cos (pi / real n))\<longlonglongrightarrow>\<^sub>N\<^sub>S 1"
+proof -
+  have "(*f* cos) ((*f* (\<lambda>x. pi / real x)) N) \<approx> 1"
+    if "N \<in> HNatInfinite" for N
+    using that STAR_cos_Infinitesimal starfunNat_pi_divide_n_Infinitesimal by blast
+  then show ?thesis
+    by (simp add: NSLIMSEQ_def) (metis STAR_cos_Infinitesimal starfunNat_pi_divide_n_Infinitesimal starfun_o2)
+qed

lemma NSLIMSEQ_sin_cos_pi:
-     "(%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)
+  "(\<lambda>n. real n * sin (pi / real n) * cos (pi / real n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S pi"
+  using NSLIMSEQ_cos_one NSLIMSEQ_mult NSLIMSEQ_sin_pi by force

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 ==> ( *f* cos) x \<approx> 1 - x\<^sup>2"
-apply (rule STAR_cos_Infinitesimal [THEN approx_trans])
-apply (auto simp add: Infinitesimal_approx_minus [symmetric]
-done
+  shows "x \<in> Infinitesimal \<Longrightarrow> ( *f* cos) x \<approx> 1 - x\<^sup>2"
+  by (metis Infinitesimal_square_iff STAR_cos_Infinitesimal approx_diff approx_sym diff_zero mem_infmal_iff power2_eq_square)

lemma STAR_cos_Infinitesimal_approx2:
fixes x :: hypreal  \<comment> \<open>perhaps could be generalised, like many other hypreal results\<close>
-  shows "x \<in> Infinitesimal ==> ( *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
+  assumes "x \<in> Infinitesimal"
+  shows "( *f* cos) x \<approx> 1 - (x\<^sup>2)/2"
+proof -
+  have "1 \<approx> 1 - x\<^sup>2 / 2"
+    using assms
+    by (auto intro: Infinitesimal_SReal_divide simp add: Infinitesimal_approx_minus [symmetric] numeral_2_eq_2)
+  then show ?thesis
+    using STAR_cos_Infinitesimal approx_trans assms by blast
+qed

end