(*  Title       : HTranscendental.thy

     Author      : Jacques D. Fleuriot

     Copyright   : 2001 University of Edinburgh

 Converted to Isar and polished by lcp

 *)

 header{*Nonstandard Extensions of Transcendental Functions*}

 theory HTranscendental

 imports Transcendental Integration

 begin

 text{*really belongs in Transcendental*}

 lemma sqrt_divide_self_eq:

   assumes nneg: "0 \<le> x"

   shows "sqrt x / x = inverse (sqrt x)"

 proof cases

   assume "x=0" thus ?thesis by simp

 next

   assume nz: "x\<noteq>0" 

   hence pos: "0<x" using nneg by arith

   show ?thesis

   proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 

     show "sqrt x / x \<noteq> 0" by (simp add: divide_inverse nneg nz) 

     show "inverse (sqrt x) / (sqrt x / x) = 1"

       by (simp add: divide_inverse mult_assoc [symmetric] 

                   power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 

   qed

 qed

 definition

   exphr :: "real => hypreal"

     --{*define exponential function using standard part *}

   "exphr x =  st(sumhr (0, whn, %n. inverse(real (fact n)) * (x ^ n)))"

   sinhr :: "real => hypreal"

   "sinhr x = st(sumhr (0, whn, %n. (if even(n) then 0 else

              ((-1) ^ ((n - 1) div 2))/(real (fact n))) * (x ^ n)))"

   coshr :: "real => hypreal"

   "coshr x = st(sumhr (0, whn, %n. (if even(n) then

             ((-1) ^ (n div 2))/(real (fact n)) else 0) * x ^ n))"

 subsection{*Nonstandard Extension of Square Root Function*}

 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 realpow_Suc)

 done

 lemma hypreal_sqrt_gt_zero_pow2: "!!x. 0 < x ==> ( *f* sqrt) (x) ^ 2 = x"

 by (transfer, simp)

 lemma hypreal_sqrt_pow2_gt_zero: "0 < x ==> 0 < ( *f* sqrt) (x) ^ 2"

 by (frule hypreal_sqrt_gt_zero_pow2, auto)

 lemma hypreal_sqrt_not_zero: "0 < x ==> ( *f* sqrt) (x) \<noteq> 0"

 apply (frule hypreal_sqrt_pow2_gt_zero)

 apply (auto simp add: numeral_2_eq_2)

 done

 lemma hypreal_inverse_sqrt_pow2:

      "0 < x ==> inverse (( *f* sqrt)(x)) ^ 2 = inverse x"

 apply (cut_tac n1 = 2 and a1 = "( *f* sqrt) x" in power_inverse [symmetric])

 apply (auto dest: hypreal_sqrt_gt_zero_pow2)

 done

 lemma hypreal_sqrt_mult_distrib: 

     "!!x y. [|0 < x; 0 <y |] ==>

       ( *f* sqrt)(x*y) = ( *f* sqrt)(x) * ( *f* sqrt)(y)"

 apply transfer

 apply (auto intro: real_sqrt_mult_distrib) 

 done

 lemma hypreal_sqrt_mult_distrib2:

      "[|0\<le>x; 0\<le>y |] ==>  

      ( *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) @= 0) = (x @= 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] 

             simp add: numeral_2_eq_2)

 done

 lemma hypreal_sqrt_approx_zero2 [simp]:

      "0 \<le> x ==> (( *f* sqrt)(x) @= 0) = (x @= 0)"

 by (auto simp add: order_le_less)

 lemma hypreal_sqrt_sum_squares [simp]:

      "(( *f* sqrt)(x*x + y*y + z*z) @= 0) = (x*x + y*y + z*z @= 0)"

 apply (rule hypreal_sqrt_approx_zero2)

 apply (rule add_nonneg_nonneg)+

 apply (auto simp add: zero_le_square)

 done

 lemma hypreal_sqrt_sum_squares2 [simp]:

      "(( *f* sqrt)(x*x + y*y) @= 0) = (x*x + y*y @= 0)"

 apply (rule hypreal_sqrt_approx_zero2)

 apply (rule add_nonneg_nonneg)

 apply (auto simp add: zero_le_square)

 done

 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_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 ^ 2) = abs(x)"

 by (transfer, simp)

 lemma hypreal_sqrt_hrabs2 [simp]: "!!x. ( *f* sqrt)(x*x) = abs(x)"

 by (transfer, simp)

 lemma hypreal_sqrt_hyperpow_hrabs [simp]:

      "!!x. ( *f* sqrt)(x pow (hypnat_of_nat 2)) = abs(x)"

 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

 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

 lemma hypreal_sqrt_sum_squares_ge1 [simp]: "!!x y. x \<le> ( *f* sqrt)(x ^ 2 + y ^ 2)"

 by (transfer, simp)

 lemma HFinite_hypreal_sqrt:

      "[| 0 \<le> x; x \<in> HFinite |] ==> ( *f* sqrt) x \<in> HFinite"

 apply (auto simp add: order_le_less)

 apply (rule HFinite_square_iff [THEN iffD1])

 apply (drule hypreal_sqrt_gt_zero_pow2)

 apply (simp add: numeral_2_eq_2)

 done

 lemma HFinite_hypreal_sqrt_imp_HFinite:

      "[| 0 \<le> x; ( *f* sqrt) x \<in> HFinite |] ==> x \<in> HFinite"

 apply (auto simp add: order_le_less)

 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

 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 (rule add_nonneg_nonneg)

 apply (auto simp add: zero_le_square)

 done

 lemma Infinitesimal_hypreal_sqrt:

      "[| 0 \<le> x; x \<in> Infinitesimal |] ==> ( *f* sqrt) x \<in> Infinitesimal"

 apply (auto simp add: order_le_less)

 apply (rule Infinitesimal_square_iff [THEN iffD2])

 apply (drule hypreal_sqrt_gt_zero_pow2)

 apply (simp add: numeral_2_eq_2)

 done

 lemma Infinitesimal_hypreal_sqrt_imp_Infinitesimal:

      "[| 0 \<le> x; ( *f* sqrt) x \<in> Infinitesimal |] ==> x \<in> Infinitesimal"

 apply (auto simp add: order_le_less)

 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

 lemma Infinitesimal_hypreal_sqrt_iff [simp]:

      "0 \<le> x ==> (( *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 (rule add_nonneg_nonneg)

 apply (auto simp add: zero_le_square)

 done

 lemma HInfinite_hypreal_sqrt:

      "[| 0 \<le> x; x \<in> HInfinite |] ==> ( *f* sqrt) x \<in> HInfinite"

 apply (auto simp add: order_le_less)

 apply (rule HInfinite_square_iff [THEN iffD1])

 apply (drule hypreal_sqrt_gt_zero_pow2)

 apply (simp add: numeral_2_eq_2)

 done

 lemma HInfinite_hypreal_sqrt_imp_HInfinite:

      "[| 0 \<le> x; ( *f* sqrt) x \<in> HInfinite |] ==> x \<in> HInfinite"

 apply (auto simp add: order_le_less)

 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

 lemma HInfinite_hypreal_sqrt_iff [simp]:

      "0 \<le> x ==> (( *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 (rule add_nonneg_nonneg)

 apply (auto simp add: zero_le_square)

 done

 lemma HFinite_exp [simp]:

      "sumhr (0, whn, %n. inverse (real (fact n)) * x ^ n) \<in> HFinite"

 by (auto intro!: NSBseq_HFinite_hypreal NSconvergent_NSBseq 

          simp add: starfunNat_sumr [symmetric] starfun hypnat_omega_def

                    convergent_NSconvergent_iff [symmetric] 

                    summable_convergent_sumr_iff [symmetric] summable_exp)

 lemma exphr_zero [simp]: "exphr 0 = 1"

 apply (simp add: exphr_def sumhr_split_add

                    [OF hypnat_one_less_hypnat_omega, symmetric]) 

 apply (simp add: sumhr star_n_zero_num starfun star_n_one_num star_n_add

                  hypnat_omega_def

             del: OrderedGroup.add_0)

 apply (simp add: star_n_one_num [symmetric])

 done

 lemma coshr_zero [simp]: "coshr 0 = 1"

 apply (simp add: coshr_def sumhr_split_add

                    [OF hypnat_one_less_hypnat_omega, symmetric]) 

 apply (simp add: sumhr star_n_zero_num star_n_one_num hypnat_omega_def)

 apply (simp add: star_n_one_num [symmetric] star_n_zero_num [symmetric])

 done

 lemma STAR_exp_zero_approx_one [simp]: "( *f* exp) 0 @= 1"

 by (simp add: star_n_zero_num star_n_one_num starfun)

 lemma STAR_exp_Infinitesimal: "x \<in> Infinitesimal ==> ( *f* exp) x @= 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] 

             simp add: mult_assoc)

 apply (rule approx_add_right_cancel [where d="-1"])

 apply (rule approx_sym [THEN [2] approx_trans2])

 apply (auto simp add: mem_infmal_iff)

 done

 lemma STAR_exp_epsilon [simp]: "( *f* exp) epsilon @= 1"

 by (auto intro: STAR_exp_Infinitesimal)

 lemma STAR_exp_add: "!!x 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)"

 apply (simp add: exphr_def)

 apply (rule st_hypreal_of_real [THEN subst])

 apply (rule approx_st_eq, auto)

 apply (rule approx_minus_iff [THEN iffD2])

 apply (simp only: mem_infmal_iff [symmetric])

 apply (auto simp add: mem_infmal_iff [symmetric] star_of_def star_n_zero_num hypnat_omega_def sumhr star_n_minus star_n_add)

 apply (rule NSLIMSEQ_zero_Infinitesimal_hypreal)

 apply (insert exp_converges [of x]) 

 apply (simp add: sums_def) 

 apply (drule LIMSEQ_const [THEN [2] LIMSEQ_add, where b = "- exp x"])

 apply (simp add: LIMSEQ_NSLIMSEQ_iff)

 done

 lemma starfun_exp_ge_add_one_self [simp]: "!!x. 0 \<le> x ==> (1 + x) \<le> ( *f* exp) x"

 by (transfer, rule exp_ge_add_one_self_aux)

 (* exp (oo) is infinite *)

 lemma starfun_exp_HInfinite:

      "[| x \<in> HInfinite; 0 \<le> x |] ==> ( *f* exp) x \<in> HInfinite"

 apply (frule starfun_exp_ge_add_one_self)

 apply (rule HInfinite_ge_HInfinite, assumption)

 apply (rule order_trans [of _ "1+x"], auto) 

 done

 lemma starfun_exp_minus: "!!x. ( *f* exp) (-x) = inverse(( *f* exp) x)"

 by (transfer, rule exp_minus)

 (* exp (-oo) is infinitesimal *)

 lemma starfun_exp_Infinitesimal:

      "[| x \<in> HInfinite; x \<le> 0 |] ==> ( *f* exp) x \<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

             simp add: starfun_exp_minus HInfinite_minus_iff)

 done

 lemma starfun_exp_gt_one [simp]: "!!x. 0 < x ==> 1 < ( *f* exp) x"

 by (transfer, simp)

 (* needs derivative of inverse function

    TRY a NS one today!!!

 Goal "x @= 1 ==> ( *f* ln) x @= 1"

 by (res_inst_tac [("z","x")] eq_Abs_star 1);

 by (auto_tac (claset(),simpset() addsimps [hypreal_one_def]));

 Goalw [nsderiv_def] "0r < x ==> NSDERIV ln x :> inverse x";

 *)

 lemma starfun_ln_exp [simp]: "!!x. ( *f* ln) (( *f* exp) x) = x"

 by (transfer, simp)

 lemma starfun_exp_ln_iff [simp]: "!!x. (( *f* exp)(( *f* ln) x) = x) = (0 < x)"

 by (transfer, simp)

 lemma starfun_exp_ln_eq: "( *f* exp) u = x ==> ( *f* ln) x = u"

 by auto

 lemma starfun_ln_less_self [simp]: "!!x. 0 < x ==> ( *f* ln) x < x"

 by (transfer, simp)

 lemma starfun_ln_ge_zero [simp]: "!!x. 1 \<le> x ==> 0 \<le> ( *f* ln) x"

 by (transfer, simp)

 lemma starfun_ln_gt_zero [simp]: "!!x .1 < x ==> 0 < ( *f* ln) x"

 by (transfer, simp)

 lemma starfun_ln_not_eq_zero [simp]: "!!x. [| 0 < x; x \<noteq> 1 |] ==> ( *f* ln) x \<noteq> 0"

 by (transfer, simp)

 lemma starfun_ln_HFinite: "[| x \<in> HFinite; 1 \<le> x |] ==> ( *f* ln) x \<in> HFinite"

 apply (rule HFinite_bounded)

 apply assumption 

 apply (simp_all add: starfun_ln_less_self order_less_imp_le)

 done

 lemma starfun_ln_inverse: "!!x. 0 < x ==> ( *f* ln) (inverse x) = -( *f* ln) x"

 by (transfer, rule ln_inverse)

 lemma starfun_exp_HFinite: "x \<in> HFinite ==> ( *f* exp) x \<in> HFinite"

 apply (cases x)

 apply (auto simp add: starfun HFinite_FreeUltrafilterNat_iff)

 apply (rule bexI [OF _ Rep_star_star_n], auto)

 apply (rule_tac x = "exp u" in exI)

 apply ultra

 done

 lemma starfun_exp_add_HFinite_Infinitesimal_approx:

      "[|x \<in> Infinitesimal; z \<in> HFinite |] ==> ( *f* exp) (z + x) @= ( *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:

      "[| x \<in> HInfinite; 0 < x |] ==> ( *f* 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 ==> ( *f* exp) x \<in> HInfinite | ( *f* exp) x \<in> Infinitesimal"

 apply (insert linorder_linear [of x 0]) 

 apply (auto intro: starfun_exp_HInfinite starfun_exp_Infinitesimal)

 done

 (* check out this proof!!! *)

 lemma starfun_ln_HFinite_not_Infinitesimal:

      "[| x \<in> HFinite - Infinitesimal; 0 < x |] ==> ( *f* 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:

      "[| x \<in> Infinitesimal; 0 < x |] ==> ( *f* 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: "!!x. [| 0 < x; x < 1 |] ==> ( *f* ln) x < 0"

 by (transfer, simp)

 lemma starfun_ln_Infinitesimal_less_zero:

      "[| x \<in> Infinitesimal; 0 < x |] ==> ( *f* 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* ln) x"

 by (auto intro!: starfun_ln_gt_zero simp add: HInfinite_def)

 (*

 Goalw [NSLIM_def] "(%h. ((x powr h) - 1) / h) -- 0 --NS> ln x"

 *)

 lemma HFinite_sin [simp]:

      "sumhr (0, whn, %n. (if even(n) then 0 else  

               ((- 1) ^ ((n - 1) div 2))/(real (fact n))) * x ^ n)  

               \<in> HFinite"

 apply (auto intro!: NSBseq_HFinite_hypreal NSconvergent_NSBseq 

             simp add: starfunNat_sumr [symmetric] starfun hypnat_omega_def

                       convergent_NSconvergent_iff [symmetric] 

                       summable_convergent_sumr_iff [symmetric])

 apply (simp only: One_nat_def summable_sin)

 done

 lemma STAR_sin_zero [simp]: "( *f* sin) 0 = 0"

 by (transfer, simp)

 lemma STAR_sin_Infinitesimal [simp]: "x \<in> Infinitesimal ==> ( *f* sin) x @= 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, %n. (if even(n) then  

             ((- 1) ^ (n div 2))/(real (fact n)) else  

             0) * x ^ n) \<in> HFinite"

 by (auto intro!: NSBseq_HFinite_hypreal NSconvergent_NSBseq 

          simp add: starfunNat_sumr [symmetric] starfun hypnat_omega_def

                    convergent_NSconvergent_iff [symmetric] 

                    summable_convergent_sumr_iff [symmetric] summable_cos)

 lemma STAR_cos_zero [simp]: "( *f* cos) 0 = 1"

 by (simp add: starfun star_n_zero_num star_n_one_num)

 lemma STAR_cos_Infinitesimal [simp]: "x \<in> Infinitesimal ==> ( *f* cos) x @= 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"], auto)

 done

 lemma STAR_tan_zero [simp]: "( *f* tan) 0 = 0"

 by (simp add: starfun star_n_zero_num)

 lemma STAR_tan_Infinitesimal: "x \<in> Infinitesimal ==> ( *f* tan) x @= 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:

      "x \<in> Infinitesimal ==> ( *f* sin) x * ( *f* cos) x @= x"

 apply (insert approx_mult_HFinite [of "( *f* sin) x" _ "( *f* cos) x" 1]) 

 apply (simp add: Infinitesimal_subset_HFinite [THEN subsetD])

 done

 lemma HFinite_pi: "hypreal_of_real pi \<in> HFinite"

 by simp

 (* lemmas *)

 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] HNatInfinite_not_eq_zero)

 lemma STAR_sin_Infinitesimal_divide:

      "[|x \<in> Infinitesimal; x \<noteq> 0 |] ==> ( *f* sin) x/x @= 1"

 apply (cut_tac x = 0 in DERIV_sin)

 apply (simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)

 done

 (*------------------------------------------------------------------------*) 

 (* sin* (1/n) * 1/(1/n) @= 1 for n = oo                                   *)

 (*------------------------------------------------------------------------*)

 lemma lemma_sin_pi:

      "n \<in> HNatInfinite  

       ==> ( *f* sin) (inverse (hypreal_of_hypnat n))/(inverse (hypreal_of_hypnat n)) @= 1"

 apply (rule STAR_sin_Infinitesimal_divide)

 apply (auto simp add: HNatInfinite_not_eq_zero)

 done

 lemma STAR_sin_inverse_HNatInfinite:

      "n \<in> HNatInfinite  

       ==> ( *f* sin) (inverse (hypreal_of_hypnat n)) * hypreal_of_hypnat n @= 1"

 apply (frule lemma_sin_pi)

 apply (simp add: divide_inverse)

 done

 lemma Infinitesimal_pi_divide_HNatInfinite: 

      "N \<in> HNatInfinite  

       ==> 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 ==> hypreal_of_real pi/(hypreal_of_hypnat N) \<noteq> 0"

 by (simp add: HNatInfinite_not_eq_zero)

 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  

           @= 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: SReal_inverse simp add: divide_inverse mult_ac)

 done

 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))  

           @= 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 ==> ( *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)

 lemma STAR_sin_pi_divide_n_approx:

      "N \<in> HNatInfinite ==>  

       ( *f* sin) (( *f* (%x. pi / real x)) N) @=  

       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: "(%n. real n * sin (pi / real n)) ----NS> 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: "(%n. cos (pi / real n))----NS> 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:

      "(%n. real n * sin (pi / real n) * cos (pi / real n)) ----NS> pi"

 by (insert NSLIMSEQ_mult [OF NSLIMSEQ_sin_pi NSLIMSEQ_cos_one], simp)

 text{*A familiar approximation to @{term "cos x"} when @{term x} is small*}

 lemma STAR_cos_Infinitesimal_approx:

      "x \<in> Infinitesimal ==> ( *f* cos) x @= 1 - x ^ 2"

 apply (rule STAR_cos_Infinitesimal [THEN approx_trans])

 apply (auto simp add: Infinitesimal_approx_minus [symmetric] 

             diff_minus add_assoc [symmetric] numeral_2_eq_2)

 done

 lemma STAR_cos_Infinitesimal_approx2:

      "x \<in> Infinitesimal ==> ( *f* cos) x @= 1 - (x ^ 2)/2"

 apply (rule STAR_cos_Infinitesimal [THEN approx_trans])

 apply (auto intro: Infinitesimal_SReal_divide 

             simp add: Infinitesimal_approx_minus [symmetric] numeral_2_eq_2)

 done

 end