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(* Title : HTranscendental.thy
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Author : Jacques D. Fleuriot
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Copyright : 2001 University of Edinburgh
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Converted to Isar and polished by lcp
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*)
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header{*Nonstandard Extensions of Transcendental Functions*}
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theory HTranscendental
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imports Transcendental HSeries HDeriv
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begin
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definition
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exphr :: "real => hypreal" where
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--{*define exponential function using standard part *}
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"exphr x = st(sumhr (0, whn, %n. inverse(real (fact n)) * (x ^ n)))"
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definition
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sinhr :: "real => hypreal" where
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"sinhr x = st(sumhr (0, whn, %n. (if even(n) then 0 else
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((-1) ^ ((n - 1) div 2))/(real (fact n))) * (x ^ n)))"
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definition
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coshr :: "real => hypreal" where
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"coshr x = st(sumhr (0, whn, %n. (if even(n) then
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((-1) ^ (n div 2))/(real (fact n)) else 0) * x ^ n))"
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subsection{*Nonstandard Extension of Square Root Function*}
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lemma STAR_sqrt_zero [simp]: "( *f* sqrt) 0 = 0"
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by (simp add: starfun star_n_zero_num)
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lemma STAR_sqrt_one [simp]: "( *f* sqrt) 1 = 1"
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by (simp add: starfun star_n_one_num)
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lemma hypreal_sqrt_pow2_iff: "(( *f* sqrt)(x) ^ 2 = x) = (0 \<le> x)"
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apply (cases x)
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apply (auto simp add: star_n_le star_n_zero_num starfun hrealpow star_n_eq_iff
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simp del: hpowr_Suc realpow_Suc)
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done
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lemma hypreal_sqrt_gt_zero_pow2: "!!x. 0 < x ==> ( *f* sqrt) (x) ^ 2 = x"
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by (transfer, simp)
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lemma hypreal_sqrt_pow2_gt_zero: "0 < x ==> 0 < ( *f* sqrt) (x) ^ 2"
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by (frule hypreal_sqrt_gt_zero_pow2, auto)
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lemma hypreal_sqrt_not_zero: "0 < x ==> ( *f* sqrt) (x) \<noteq> 0"
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apply (frule hypreal_sqrt_pow2_gt_zero)
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apply (auto simp add: numeral_2_eq_2)
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done
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lemma hypreal_inverse_sqrt_pow2:
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"0 < x ==> inverse (( *f* sqrt)(x)) ^ 2 = inverse x"
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apply (cut_tac n = 2 and a = "( *f* sqrt) x" in power_inverse [symmetric])
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apply (auto dest: hypreal_sqrt_gt_zero_pow2)
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done
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lemma hypreal_sqrt_mult_distrib:
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"!!x y. [|0 < x; 0 <y |] ==>
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( *f* sqrt)(x*y) = ( *f* sqrt)(x) * ( *f* sqrt)(y)"
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apply transfer
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apply (auto intro: real_sqrt_mult_distrib)
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done
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lemma hypreal_sqrt_mult_distrib2:
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"[|0\<le>x; 0\<le>y |] ==>
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( *f* sqrt)(x*y) = ( *f* sqrt)(x) * ( *f* sqrt)(y)"
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by (auto intro: hypreal_sqrt_mult_distrib simp add: order_le_less)
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lemma hypreal_sqrt_approx_zero [simp]:
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"0 < x ==> (( *f* sqrt)(x) @= 0) = (x @= 0)"
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apply (auto simp add: mem_infmal_iff [symmetric])
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apply (rule hypreal_sqrt_gt_zero_pow2 [THEN subst])
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apply (auto intro: Infinitesimal_mult
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dest!: hypreal_sqrt_gt_zero_pow2 [THEN ssubst]
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simp add: numeral_2_eq_2)
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done
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lemma hypreal_sqrt_approx_zero2 [simp]:
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"0 \<le> x ==> (( *f* sqrt)(x) @= 0) = (x @= 0)"
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by (auto simp add: order_le_less)
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lemma hypreal_sqrt_sum_squares [simp]:
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"(( *f* sqrt)(x*x + y*y + z*z) @= 0) = (x*x + y*y + z*z @= 0)"
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apply (rule hypreal_sqrt_approx_zero2)
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apply (rule add_nonneg_nonneg)+
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apply (auto)
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done
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lemma hypreal_sqrt_sum_squares2 [simp]:
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"(( *f* sqrt)(x*x + y*y) @= 0) = (x*x + y*y @= 0)"
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apply (rule hypreal_sqrt_approx_zero2)
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apply (rule add_nonneg_nonneg)
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apply (auto)
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done
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lemma hypreal_sqrt_gt_zero: "!!x. 0 < x ==> 0 < ( *f* sqrt)(x)"
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apply transfer
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apply (auto intro: real_sqrt_gt_zero)
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done
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lemma hypreal_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> ( *f* sqrt)(x)"
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by (auto intro: hypreal_sqrt_gt_zero simp add: order_le_less)
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lemma hypreal_sqrt_hrabs [simp]: "!!x. ( *f* sqrt)(x ^ 2) = abs(x)"
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by (transfer, simp)
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lemma hypreal_sqrt_hrabs2 [simp]: "!!x. ( *f* sqrt)(x*x) = abs(x)"
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by (transfer, simp)
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lemma hypreal_sqrt_hyperpow_hrabs [simp]:
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"!!x. ( *f* sqrt)(x pow (hypnat_of_nat 2)) = abs(x)"
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by (transfer, simp)
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lemma star_sqrt_HFinite: "\<lbrakk>x \<in> HFinite; 0 \<le> x\<rbrakk> \<Longrightarrow> ( *f* sqrt) x \<in> HFinite"
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apply (rule HFinite_square_iff [THEN iffD1])
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apply (simp only: hypreal_sqrt_mult_distrib2 [symmetric], simp)
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done
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lemma st_hypreal_sqrt:
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"[| x \<in> HFinite; 0 \<le> x |] ==> st(( *f* sqrt) x) = ( *f* sqrt)(st x)"
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apply (rule power_inject_base [where n=1])
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apply (auto intro!: st_zero_le hypreal_sqrt_ge_zero)
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apply (rule st_mult [THEN subst])
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apply (rule_tac [3] hypreal_sqrt_mult_distrib2 [THEN subst])
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apply (rule_tac [5] hypreal_sqrt_mult_distrib2 [THEN subst])
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apply (auto simp add: st_hrabs st_zero_le star_sqrt_HFinite)
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done
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lemma hypreal_sqrt_sum_squares_ge1 [simp]: "!!x y. x \<le> ( *f* sqrt)(x ^ 2 + y ^ 2)"
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by transfer (rule real_sqrt_sum_squares_ge1)
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lemma HFinite_hypreal_sqrt:
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"[| 0 \<le> x; x \<in> HFinite |] ==> ( *f* sqrt) x \<in> HFinite"
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apply (auto simp add: order_le_less)
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apply (rule HFinite_square_iff [THEN iffD1])
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apply (drule hypreal_sqrt_gt_zero_pow2)
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apply (simp add: numeral_2_eq_2)
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done
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lemma HFinite_hypreal_sqrt_imp_HFinite:
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"[| 0 \<le> x; ( *f* sqrt) x \<in> HFinite |] ==> x \<in> HFinite"
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apply (auto simp add: order_le_less)
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apply (drule HFinite_square_iff [THEN iffD2])
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apply (drule hypreal_sqrt_gt_zero_pow2)
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apply (simp add: numeral_2_eq_2 del: HFinite_square_iff)
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done
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lemma HFinite_hypreal_sqrt_iff [simp]:
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"0 \<le> x ==> (( *f* sqrt) x \<in> HFinite) = (x \<in> HFinite)"
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by (blast intro: HFinite_hypreal_sqrt HFinite_hypreal_sqrt_imp_HFinite)
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lemma HFinite_sqrt_sum_squares [simp]:
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"(( *f* sqrt)(x*x + y*y) \<in> HFinite) = (x*x + y*y \<in> HFinite)"
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apply (rule HFinite_hypreal_sqrt_iff)
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apply (rule add_nonneg_nonneg)
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apply (auto)
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done
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lemma Infinitesimal_hypreal_sqrt:
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"[| 0 \<le> x; x \<in> Infinitesimal |] ==> ( *f* sqrt) x \<in> Infinitesimal"
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apply (auto simp add: order_le_less)
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apply (rule Infinitesimal_square_iff [THEN iffD2])
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apply (drule hypreal_sqrt_gt_zero_pow2)
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apply (simp add: numeral_2_eq_2)
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done
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lemma Infinitesimal_hypreal_sqrt_imp_Infinitesimal:
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"[| 0 \<le> x; ( *f* sqrt) x \<in> Infinitesimal |] ==> x \<in> Infinitesimal"
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apply (auto simp add: order_le_less)
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apply (drule Infinitesimal_square_iff [THEN iffD1])
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apply (drule hypreal_sqrt_gt_zero_pow2)
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apply (simp add: numeral_2_eq_2 del: Infinitesimal_square_iff [symmetric])
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done
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lemma Infinitesimal_hypreal_sqrt_iff [simp]:
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"0 \<le> x ==> (( *f* sqrt) x \<in> Infinitesimal) = (x \<in> Infinitesimal)"
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by (blast intro: Infinitesimal_hypreal_sqrt_imp_Infinitesimal Infinitesimal_hypreal_sqrt)
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lemma Infinitesimal_sqrt_sum_squares [simp]:
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"(( *f* sqrt)(x*x + y*y) \<in> Infinitesimal) = (x*x + y*y \<in> Infinitesimal)"
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apply (rule Infinitesimal_hypreal_sqrt_iff)
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apply (rule add_nonneg_nonneg)
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apply (auto)
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done
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lemma HInfinite_hypreal_sqrt:
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"[| 0 \<le> x; x \<in> HInfinite |] ==> ( *f* sqrt) x \<in> HInfinite"
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apply (auto simp add: order_le_less)
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apply (rule HInfinite_square_iff [THEN iffD1])
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apply (drule hypreal_sqrt_gt_zero_pow2)
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apply (simp add: numeral_2_eq_2)
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done
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lemma HInfinite_hypreal_sqrt_imp_HInfinite:
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"[| 0 \<le> x; ( *f* sqrt) x \<in> HInfinite |] ==> x \<in> HInfinite"
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apply (auto simp add: order_le_less)
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apply (drule HInfinite_square_iff [THEN iffD2])
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apply (drule hypreal_sqrt_gt_zero_pow2)
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apply (simp add: numeral_2_eq_2 del: HInfinite_square_iff)
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done
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lemma HInfinite_hypreal_sqrt_iff [simp]:
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"0 \<le> x ==> (( *f* sqrt) x \<in> HInfinite) = (x \<in> HInfinite)"
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by (blast intro: HInfinite_hypreal_sqrt HInfinite_hypreal_sqrt_imp_HInfinite)
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lemma HInfinite_sqrt_sum_squares [simp]:
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"(( *f* sqrt)(x*x + y*y) \<in> HInfinite) = (x*x + y*y \<in> HInfinite)"
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apply (rule HInfinite_hypreal_sqrt_iff)
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apply (rule add_nonneg_nonneg)
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apply (auto)
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done
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lemma HFinite_exp [simp]:
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"sumhr (0, whn, %n. inverse (real (fact n)) * x ^ n) \<in> HFinite"
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unfolding sumhr_app
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apply (simp only: star_zero_def starfun2_star_of)
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apply (rule NSBseqD2)
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apply (rule NSconvergent_NSBseq)
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apply (rule convergent_NSconvergent_iff [THEN iffD1])
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apply (rule summable_convergent_sumr_iff [THEN iffD1])
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apply (rule summable_exp)
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done
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lemma exphr_zero [simp]: "exphr 0 = 1"
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apply (simp add: exphr_def sumhr_split_add
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[OF hypnat_one_less_hypnat_omega, symmetric])
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apply (rule st_unique, simp)
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apply (rule subst [where P="\<lambda>x. 1 \<approx> x", OF _ approx_refl])
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apply (rule rev_mp [OF hypnat_one_less_hypnat_omega])
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apply (rule_tac x="whn" in spec)
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apply (unfold sumhr_app, transfer, simp)
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done
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lemma coshr_zero [simp]: "coshr 0 = 1"
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apply (simp add: coshr_def sumhr_split_add
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[OF hypnat_one_less_hypnat_omega, symmetric])
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apply (rule st_unique, simp)
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apply (rule subst [where P="\<lambda>x. 1 \<approx> x", OF _ approx_refl])
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apply (rule rev_mp [OF hypnat_one_less_hypnat_omega])
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apply (rule_tac x="whn" in spec)
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apply (unfold sumhr_app, transfer, simp)
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done
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lemma STAR_exp_zero_approx_one [simp]: "( *f* exp) (0::hypreal) @= 1"
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apply (subgoal_tac "( *f* exp) (0::hypreal) = 1", simp)
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apply (transfer, simp)
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done
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lemma STAR_exp_Infinitesimal: "x \<in> Infinitesimal ==> ( *f* exp) (x::hypreal) @= 1"
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apply (case_tac "x = 0")
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apply (cut_tac [2] x = 0 in DERIV_exp)
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apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
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apply (drule_tac x = x in bspec, auto)
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apply (drule_tac c = x in approx_mult1)
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apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
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simp add: mult_assoc)
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apply (rule approx_add_right_cancel [where d="-1"])
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apply (rule approx_sym [THEN [2] approx_trans2])
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apply (auto simp add: diff_def mem_infmal_iff)
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done
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lemma STAR_exp_epsilon [simp]: "( *f* exp) epsilon @= 1"
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by (auto intro: STAR_exp_Infinitesimal)
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lemma STAR_exp_add: "!!x y. ( *f* exp)(x + y) = ( *f* exp) x * ( *f* exp) y"
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by transfer (rule exp_add)
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lemma exphr_hypreal_of_real_exp_eq: "exphr x = hypreal_of_real (exp x)"
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apply (simp add: exphr_def)
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apply (rule st_unique, simp)
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apply (subst starfunNat_sumr [symmetric])
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apply (rule NSLIMSEQ_D [THEN approx_sym])
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apply (rule LIMSEQ_NSLIMSEQ)
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apply (subst sums_def [symmetric])
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apply (cut_tac exp_converges [where x=x], simp)
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apply (rule HNatInfinite_whn)
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done
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lemma starfun_exp_ge_add_one_self [simp]: "!!x::hypreal. 0 \<le> x ==> (1 + x) \<le> ( *f* exp) x"
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by transfer (rule exp_ge_add_one_self_aux)
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(* exp (oo) is infinite *)
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lemma starfun_exp_HInfinite:
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"[| x \<in> HInfinite; 0 \<le> x |] ==> ( *f* exp) (x::hypreal) \<in> HInfinite"
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apply (frule starfun_exp_ge_add_one_self)
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apply (rule HInfinite_ge_HInfinite, assumption)
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apply (rule order_trans [of _ "1+x"], auto)
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done
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lemma starfun_exp_minus: "!!x. ( *f* exp) (-x) = inverse(( *f* exp) x)"
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by transfer (rule exp_minus)
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(* exp (-oo) is infinitesimal *)
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lemma starfun_exp_Infinitesimal:
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"[| x \<in> HInfinite; x \<le> 0 |] ==> ( *f* exp) (x::hypreal) \<in> Infinitesimal"
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apply (subgoal_tac "\<exists>y. x = - y")
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apply (rule_tac [2] x = "- x" in exI)
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apply (auto intro!: HInfinite_inverse_Infinitesimal starfun_exp_HInfinite
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simp add: starfun_exp_minus HInfinite_minus_iff)
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done
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lemma starfun_exp_gt_one [simp]: "!!x::hypreal. 0 < x ==> 1 < ( *f* exp) x"
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by transfer (rule exp_gt_one)
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lemma starfun_ln_exp [simp]: "!!x. ( *f* ln) (( *f* exp) x) = x"
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by transfer (rule ln_exp)
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lemma starfun_exp_ln_iff [simp]: "!!x. (( *f* exp)(( *f* ln) x) = x) = (0 < x)"
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by transfer (rule exp_ln_iff)
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lemma starfun_exp_ln_eq: "!!u x. ( *f* exp) u = x ==> ( *f* ln) x = u"
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by transfer (rule exp_ln_eq)
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lemma starfun_ln_less_self [simp]: "!!x. 0 < x ==> ( *f* ln) x < x"
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by transfer (rule ln_less_self)
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lemma starfun_ln_ge_zero [simp]: "!!x. 1 \<le> x ==> 0 \<le> ( *f* ln) x"
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by transfer (rule ln_ge_zero)
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lemma starfun_ln_gt_zero [simp]: "!!x .1 < x ==> 0 < ( *f* ln) x"
|
|
325 |
by transfer (rule ln_gt_zero)
|
|
326 |
|
|
327 |
lemma starfun_ln_not_eq_zero [simp]: "!!x. [| 0 < x; x \<noteq> 1 |] ==> ( *f* ln) x \<noteq> 0"
|
|
328 |
by transfer simp
|
|
329 |
|
|
330 |
lemma starfun_ln_HFinite: "[| x \<in> HFinite; 1 \<le> x |] ==> ( *f* ln) x \<in> HFinite"
|
|
331 |
apply (rule HFinite_bounded)
|
|
332 |
apply assumption
|
|
333 |
apply (simp_all add: starfun_ln_less_self order_less_imp_le)
|
|
334 |
done
|
|
335 |
|
|
336 |
lemma starfun_ln_inverse: "!!x. 0 < x ==> ( *f* ln) (inverse x) = -( *f* ln) x"
|
|
337 |
by transfer (rule ln_inverse)
|
|
338 |
|
|
339 |
lemma starfun_abs_exp_cancel: "\<And>x. \<bar>( *f* exp) (x::hypreal)\<bar> = ( *f* exp) x"
|
|
340 |
by transfer (rule abs_exp_cancel)
|
|
341 |
|
|
342 |
lemma starfun_exp_less_mono: "\<And>x y::hypreal. x < y \<Longrightarrow> ( *f* exp) x < ( *f* exp) y"
|
|
343 |
by transfer (rule exp_less_mono)
|
|
344 |
|
|
345 |
lemma starfun_exp_HFinite: "x \<in> HFinite ==> ( *f* exp) (x::hypreal) \<in> HFinite"
|
|
346 |
apply (auto simp add: HFinite_def, rename_tac u)
|
|
347 |
apply (rule_tac x="( *f* exp) u" in rev_bexI)
|
|
348 |
apply (simp add: Reals_eq_Standard)
|
|
349 |
apply (simp add: starfun_abs_exp_cancel)
|
|
350 |
apply (simp add: starfun_exp_less_mono)
|
|
351 |
done
|
|
352 |
|
|
353 |
lemma starfun_exp_add_HFinite_Infinitesimal_approx:
|
|
354 |
"[|x \<in> Infinitesimal; z \<in> HFinite |] ==> ( *f* exp) (z + x::hypreal) @= ( *f* exp) z"
|
|
355 |
apply (simp add: STAR_exp_add)
|
|
356 |
apply (frule STAR_exp_Infinitesimal)
|
|
357 |
apply (drule approx_mult2)
|
|
358 |
apply (auto intro: starfun_exp_HFinite)
|
|
359 |
done
|
|
360 |
|
|
361 |
(* using previous result to get to result *)
|
|
362 |
lemma starfun_ln_HInfinite:
|
|
363 |
"[| x \<in> HInfinite; 0 < x |] ==> ( *f* ln) x \<in> HInfinite"
|
|
364 |
apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
|
|
365 |
apply (drule starfun_exp_HFinite)
|
|
366 |
apply (simp add: starfun_exp_ln_iff [THEN iffD2] HFinite_HInfinite_iff)
|
|
367 |
done
|
|
368 |
|
|
369 |
lemma starfun_exp_HInfinite_Infinitesimal_disj:
|
|
370 |
"x \<in> HInfinite ==> ( *f* exp) x \<in> HInfinite | ( *f* exp) (x::hypreal) \<in> Infinitesimal"
|
|
371 |
apply (insert linorder_linear [of x 0])
|
|
372 |
apply (auto intro: starfun_exp_HInfinite starfun_exp_Infinitesimal)
|
|
373 |
done
|
|
374 |
|
|
375 |
(* check out this proof!!! *)
|
|
376 |
lemma starfun_ln_HFinite_not_Infinitesimal:
|
|
377 |
"[| x \<in> HFinite - Infinitesimal; 0 < x |] ==> ( *f* ln) x \<in> HFinite"
|
|
378 |
apply (rule ccontr, drule HInfinite_HFinite_iff [THEN iffD2])
|
|
379 |
apply (drule starfun_exp_HInfinite_Infinitesimal_disj)
|
|
380 |
apply (simp add: starfun_exp_ln_iff [symmetric] HInfinite_HFinite_iff
|
|
381 |
del: starfun_exp_ln_iff)
|
|
382 |
done
|
|
383 |
|
|
384 |
(* we do proof by considering ln of 1/x *)
|
|
385 |
lemma starfun_ln_Infinitesimal_HInfinite:
|
|
386 |
"[| x \<in> Infinitesimal; 0 < x |] ==> ( *f* ln) x \<in> HInfinite"
|
|
387 |
apply (drule Infinitesimal_inverse_HInfinite)
|
|
388 |
apply (frule positive_imp_inverse_positive)
|
|
389 |
apply (drule_tac [2] starfun_ln_HInfinite)
|
|
390 |
apply (auto simp add: starfun_ln_inverse HInfinite_minus_iff)
|
|
391 |
done
|
|
392 |
|
|
393 |
lemma starfun_ln_less_zero: "!!x. [| 0 < x; x < 1 |] ==> ( *f* ln) x < 0"
|
|
394 |
by transfer (rule ln_less_zero)
|
|
395 |
|
|
396 |
lemma starfun_ln_Infinitesimal_less_zero:
|
|
397 |
"[| x \<in> Infinitesimal; 0 < x |] ==> ( *f* ln) x < 0"
|
|
398 |
by (auto intro!: starfun_ln_less_zero simp add: Infinitesimal_def)
|
|
399 |
|
|
400 |
lemma starfun_ln_HInfinite_gt_zero:
|
|
401 |
"[| x \<in> HInfinite; 0 < x |] ==> 0 < ( *f* ln) x"
|
|
402 |
by (auto intro!: starfun_ln_gt_zero simp add: HInfinite_def)
|
|
403 |
|
|
404 |
|
|
405 |
(*
|
|
406 |
Goalw [NSLIM_def] "(%h. ((x powr h) - 1) / h) -- 0 --NS> ln x"
|
|
407 |
*)
|
|
408 |
|
|
409 |
lemma HFinite_sin [simp]:
|
|
410 |
"sumhr (0, whn, %n. (if even(n) then 0 else
|
|
411 |
(-1 ^ ((n - 1) div 2))/(real (fact n))) * x ^ n)
|
|
412 |
\<in> HFinite"
|
|
413 |
unfolding sumhr_app
|
|
414 |
apply (simp only: star_zero_def starfun2_star_of)
|
|
415 |
apply (rule NSBseqD2)
|
|
416 |
apply (rule NSconvergent_NSBseq)
|
|
417 |
apply (rule convergent_NSconvergent_iff [THEN iffD1])
|
|
418 |
apply (rule summable_convergent_sumr_iff [THEN iffD1])
|
|
419 |
apply (simp only: One_nat_def summable_sin)
|
|
420 |
done
|
|
421 |
|
|
422 |
lemma STAR_sin_zero [simp]: "( *f* sin) 0 = 0"
|
|
423 |
by transfer (rule sin_zero)
|
|
424 |
|
|
425 |
lemma STAR_sin_Infinitesimal [simp]: "x \<in> Infinitesimal ==> ( *f* sin) x @= x"
|
|
426 |
apply (case_tac "x = 0")
|
|
427 |
apply (cut_tac [2] x = 0 in DERIV_sin)
|
|
428 |
apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
|
|
429 |
apply (drule bspec [where x = x], auto)
|
|
430 |
apply (drule approx_mult1 [where c = x])
|
|
431 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
|
|
432 |
simp add: mult_assoc)
|
|
433 |
done
|
|
434 |
|
|
435 |
lemma HFinite_cos [simp]:
|
|
436 |
"sumhr (0, whn, %n. (if even(n) then
|
|
437 |
(-1 ^ (n div 2))/(real (fact n)) else
|
|
438 |
0) * x ^ n) \<in> HFinite"
|
|
439 |
unfolding sumhr_app
|
|
440 |
apply (simp only: star_zero_def starfun2_star_of)
|
|
441 |
apply (rule NSBseqD2)
|
|
442 |
apply (rule NSconvergent_NSBseq)
|
|
443 |
apply (rule convergent_NSconvergent_iff [THEN iffD1])
|
|
444 |
apply (rule summable_convergent_sumr_iff [THEN iffD1])
|
|
445 |
apply (rule summable_cos)
|
|
446 |
done
|
|
447 |
|
|
448 |
lemma STAR_cos_zero [simp]: "( *f* cos) 0 = 1"
|
|
449 |
by transfer (rule cos_zero)
|
|
450 |
|
|
451 |
lemma STAR_cos_Infinitesimal [simp]: "x \<in> Infinitesimal ==> ( *f* cos) x @= 1"
|
|
452 |
apply (case_tac "x = 0")
|
|
453 |
apply (cut_tac [2] x = 0 in DERIV_cos)
|
|
454 |
apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
|
|
455 |
apply (drule bspec [where x = x])
|
|
456 |
apply auto
|
|
457 |
apply (drule approx_mult1 [where c = x])
|
|
458 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
|
|
459 |
simp add: mult_assoc)
|
|
460 |
apply (rule approx_add_right_cancel [where d = "-1"])
|
|
461 |
apply (simp add: diff_def)
|
|
462 |
done
|
|
463 |
|
|
464 |
lemma STAR_tan_zero [simp]: "( *f* tan) 0 = 0"
|
|
465 |
by transfer (rule tan_zero)
|
|
466 |
|
|
467 |
lemma STAR_tan_Infinitesimal: "x \<in> Infinitesimal ==> ( *f* tan) x @= x"
|
|
468 |
apply (case_tac "x = 0")
|
|
469 |
apply (cut_tac [2] x = 0 in DERIV_tan)
|
|
470 |
apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
|
|
471 |
apply (drule bspec [where x = x], auto)
|
|
472 |
apply (drule approx_mult1 [where c = x])
|
|
473 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
|
|
474 |
simp add: mult_assoc)
|
|
475 |
done
|
|
476 |
|
|
477 |
lemma STAR_sin_cos_Infinitesimal_mult:
|
|
478 |
"x \<in> Infinitesimal ==> ( *f* sin) x * ( *f* cos) x @= x"
|
|
479 |
apply (insert approx_mult_HFinite [of "( *f* sin) x" _ "( *f* cos) x" 1])
|
|
480 |
apply (simp add: Infinitesimal_subset_HFinite [THEN subsetD])
|
|
481 |
done
|
|
482 |
|
|
483 |
lemma HFinite_pi: "hypreal_of_real pi \<in> HFinite"
|
|
484 |
by simp
|
|
485 |
|
|
486 |
(* lemmas *)
|
|
487 |
|
|
488 |
lemma lemma_split_hypreal_of_real:
|
|
489 |
"N \<in> HNatInfinite
|
|
490 |
==> hypreal_of_real a =
|
|
491 |
hypreal_of_hypnat N * (inverse(hypreal_of_hypnat N) * hypreal_of_real a)"
|
|
492 |
by (simp add: mult_assoc [symmetric] zero_less_HNatInfinite)
|
|
493 |
|
|
494 |
lemma STAR_sin_Infinitesimal_divide:
|
|
495 |
"[|x \<in> Infinitesimal; x \<noteq> 0 |] ==> ( *f* sin) x/x @= 1"
|
|
496 |
apply (cut_tac x = 0 in DERIV_sin)
|
|
497 |
apply (simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def)
|
|
498 |
done
|
|
499 |
|
|
500 |
(*------------------------------------------------------------------------*)
|
|
501 |
(* sin* (1/n) * 1/(1/n) @= 1 for n = oo *)
|
|
502 |
(*------------------------------------------------------------------------*)
|
|
503 |
|
|
504 |
lemma lemma_sin_pi:
|
|
505 |
"n \<in> HNatInfinite
|
|
506 |
==> ( *f* sin) (inverse (hypreal_of_hypnat n))/(inverse (hypreal_of_hypnat n)) @= 1"
|
|
507 |
apply (rule STAR_sin_Infinitesimal_divide)
|
|
508 |
apply (auto simp add: zero_less_HNatInfinite)
|
|
509 |
done
|
|
510 |
|
|
511 |
lemma STAR_sin_inverse_HNatInfinite:
|
|
512 |
"n \<in> HNatInfinite
|
|
513 |
==> ( *f* sin) (inverse (hypreal_of_hypnat n)) * hypreal_of_hypnat n @= 1"
|
|
514 |
apply (frule lemma_sin_pi)
|
|
515 |
apply (simp add: divide_inverse)
|
|
516 |
done
|
|
517 |
|
|
518 |
lemma Infinitesimal_pi_divide_HNatInfinite:
|
|
519 |
"N \<in> HNatInfinite
|
|
520 |
==> hypreal_of_real pi/(hypreal_of_hypnat N) \<in> Infinitesimal"
|
|
521 |
apply (simp add: divide_inverse)
|
|
522 |
apply (auto intro: Infinitesimal_HFinite_mult2)
|
|
523 |
done
|
|
524 |
|
|
525 |
lemma pi_divide_HNatInfinite_not_zero [simp]:
|
|
526 |
"N \<in> HNatInfinite ==> hypreal_of_real pi/(hypreal_of_hypnat N) \<noteq> 0"
|
|
527 |
by (simp add: zero_less_HNatInfinite)
|
|
528 |
|
|
529 |
lemma STAR_sin_pi_divide_HNatInfinite_approx_pi:
|
|
530 |
"n \<in> HNatInfinite
|
|
531 |
==> ( *f* sin) (hypreal_of_real pi/(hypreal_of_hypnat n)) * hypreal_of_hypnat n
|
|
532 |
@= hypreal_of_real pi"
|
|
533 |
apply (frule STAR_sin_Infinitesimal_divide
|
|
534 |
[OF Infinitesimal_pi_divide_HNatInfinite
|
|
535 |
pi_divide_HNatInfinite_not_zero])
|
|
536 |
apply (auto)
|
|
537 |
apply (rule approx_SReal_mult_cancel [of "inverse (hypreal_of_real pi)"])
|
|
538 |
apply (auto intro: Reals_inverse simp add: divide_inverse mult_ac)
|
|
539 |
done
|
|
540 |
|
|
541 |
lemma STAR_sin_pi_divide_HNatInfinite_approx_pi2:
|
|
542 |
"n \<in> HNatInfinite
|
|
543 |
==> hypreal_of_hypnat n *
|
|
544 |
( *f* sin) (hypreal_of_real pi/(hypreal_of_hypnat n))
|
|
545 |
@= hypreal_of_real pi"
|
|
546 |
apply (rule mult_commute [THEN subst])
|
|
547 |
apply (erule STAR_sin_pi_divide_HNatInfinite_approx_pi)
|
|
548 |
done
|
|
549 |
|
|
550 |
lemma starfunNat_pi_divide_n_Infinitesimal:
|
|
551 |
"N \<in> HNatInfinite ==> ( *f* (%x. pi / real x)) N \<in> Infinitesimal"
|
|
552 |
by (auto intro!: Infinitesimal_HFinite_mult2
|
|
553 |
simp add: starfun_mult [symmetric] divide_inverse
|
|
554 |
starfun_inverse [symmetric] starfunNat_real_of_nat)
|
|
555 |
|
|
556 |
lemma STAR_sin_pi_divide_n_approx:
|
|
557 |
"N \<in> HNatInfinite ==>
|
|
558 |
( *f* sin) (( *f* (%x. pi / real x)) N) @=
|
|
559 |
hypreal_of_real pi/(hypreal_of_hypnat N)"
|
|
560 |
apply (simp add: starfunNat_real_of_nat [symmetric])
|
|
561 |
apply (rule STAR_sin_Infinitesimal)
|
|
562 |
apply (simp add: divide_inverse)
|
|
563 |
apply (rule Infinitesimal_HFinite_mult2)
|
|
564 |
apply (subst starfun_inverse)
|
|
565 |
apply (erule starfunNat_inverse_real_of_nat_Infinitesimal)
|
|
566 |
apply simp
|
|
567 |
done
|
|
568 |
|
|
569 |
lemma NSLIMSEQ_sin_pi: "(%n. real n * sin (pi / real n)) ----NS> pi"
|
|
570 |
apply (auto simp add: NSLIMSEQ_def starfun_mult [symmetric] starfunNat_real_of_nat)
|
|
571 |
apply (rule_tac f1 = sin in starfun_o2 [THEN subst])
|
|
572 |
apply (auto simp add: starfun_mult [symmetric] starfunNat_real_of_nat divide_inverse)
|
|
573 |
apply (rule_tac f1 = inverse in starfun_o2 [THEN subst])
|
|
574 |
apply (auto dest: STAR_sin_pi_divide_HNatInfinite_approx_pi
|
|
575 |
simp add: starfunNat_real_of_nat mult_commute divide_inverse)
|
|
576 |
done
|
|
577 |
|
|
578 |
lemma NSLIMSEQ_cos_one: "(%n. cos (pi / real n))----NS> 1"
|
|
579 |
apply (simp add: NSLIMSEQ_def, auto)
|
|
580 |
apply (rule_tac f1 = cos in starfun_o2 [THEN subst])
|
|
581 |
apply (rule STAR_cos_Infinitesimal)
|
|
582 |
apply (auto intro!: Infinitesimal_HFinite_mult2
|
|
583 |
simp add: starfun_mult [symmetric] divide_inverse
|
|
584 |
starfun_inverse [symmetric] starfunNat_real_of_nat)
|
|
585 |
done
|
|
586 |
|
|
587 |
lemma NSLIMSEQ_sin_cos_pi:
|
|
588 |
"(%n. real n * sin (pi / real n) * cos (pi / real n)) ----NS> pi"
|
|
589 |
by (insert NSLIMSEQ_mult [OF NSLIMSEQ_sin_pi NSLIMSEQ_cos_one], simp)
|
|
590 |
|
|
591 |
|
|
592 |
text{*A familiar approximation to @{term "cos x"} when @{term x} is small*}
|
|
593 |
|
|
594 |
lemma STAR_cos_Infinitesimal_approx:
|
|
595 |
"x \<in> Infinitesimal ==> ( *f* cos) x @= 1 - x ^ 2"
|
|
596 |
apply (rule STAR_cos_Infinitesimal [THEN approx_trans])
|
|
597 |
apply (auto simp add: Infinitesimal_approx_minus [symmetric]
|
|
598 |
diff_minus add_assoc [symmetric] numeral_2_eq_2)
|
|
599 |
done
|
|
600 |
|
|
601 |
lemma STAR_cos_Infinitesimal_approx2:
|
|
602 |
"x \<in> Infinitesimal ==> ( *f* cos) x @= 1 - (x ^ 2)/2"
|
|
603 |
apply (rule STAR_cos_Infinitesimal [THEN approx_trans])
|
|
604 |
apply (auto intro: Infinitesimal_SReal_divide
|
|
605 |
simp add: Infinitesimal_approx_minus [symmetric] numeral_2_eq_2)
|
|
606 |
done
|
|
607 |
|
|
608 |
end
|