author | huffman |
Wed, 27 May 2009 16:39:22 -0700 | |
changeset 31271 | 0237e5e40b71 |
parent 30273 | ecd6f0ca62ea |
child 37887 | 2ae085b07f2f |
permissions | -rw-r--r-- |
27468 | 1 |
(* 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. sin_coeff 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. cos_coeff n * 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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30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents:
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diff
changeset
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simp del: hpowr_Suc power_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 add: cos_coeff_def) |
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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" |
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by transfer (rule ln_gt_zero) |
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lemma starfun_ln_not_eq_zero [simp]: "!!x. [| 0 < x; x \<noteq> 1 |] ==> ( *f* ln) x \<noteq> 0" |
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by transfer simp |
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lemma starfun_ln_HFinite: "[| x \<in> HFinite; 1 \<le> x |] ==> ( *f* ln) x \<in> HFinite" |
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apply (rule HFinite_bounded) |
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apply assumption |
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apply (simp_all add: starfun_ln_less_self order_less_imp_le) |
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done |
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lemma starfun_ln_inverse: "!!x. 0 < x ==> ( *f* ln) (inverse x) = -( *f* ln) x" |
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by transfer (rule ln_inverse) |
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lemma starfun_abs_exp_cancel: "\<And>x. \<bar>( *f* exp) (x::hypreal)\<bar> = ( *f* exp) x" |
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by transfer (rule abs_exp_cancel) |
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lemma starfun_exp_less_mono: "\<And>x y::hypreal. x < y \<Longrightarrow> ( *f* exp) x < ( *f* exp) y" |
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by transfer (rule exp_less_mono) |
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lemma starfun_exp_HFinite: "x \<in> HFinite ==> ( *f* exp) (x::hypreal) \<in> HFinite" |
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apply (auto simp add: HFinite_def, rename_tac u) |
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apply (rule_tac x="( *f* exp) u" in rev_bexI) |
|
346 |
apply (simp add: Reals_eq_Standard) |
|
347 |
apply (simp add: starfun_abs_exp_cancel) |
|
348 |
apply (simp add: starfun_exp_less_mono) |
|
349 |
done |
|
350 |
||
351 |
lemma starfun_exp_add_HFinite_Infinitesimal_approx: |
|
352 |
"[|x \<in> Infinitesimal; z \<in> HFinite |] ==> ( *f* exp) (z + x::hypreal) @= ( *f* exp) z" |
|
353 |
apply (simp add: STAR_exp_add) |
|
354 |
apply (frule STAR_exp_Infinitesimal) |
|
355 |
apply (drule approx_mult2) |
|
356 |
apply (auto intro: starfun_exp_HFinite) |
|
357 |
done |
|
358 |
||
359 |
(* using previous result to get to result *) |
|
360 |
lemma starfun_ln_HInfinite: |
|
361 |
"[| x \<in> HInfinite; 0 < x |] ==> ( *f* ln) x \<in> HInfinite" |
|
362 |
apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2]) |
|
363 |
apply (drule starfun_exp_HFinite) |
|
364 |
apply (simp add: starfun_exp_ln_iff [THEN iffD2] HFinite_HInfinite_iff) |
|
365 |
done |
|
366 |
||
367 |
lemma starfun_exp_HInfinite_Infinitesimal_disj: |
|
368 |
"x \<in> HInfinite ==> ( *f* exp) x \<in> HInfinite | ( *f* exp) (x::hypreal) \<in> Infinitesimal" |
|
369 |
apply (insert linorder_linear [of x 0]) |
|
370 |
apply (auto intro: starfun_exp_HInfinite starfun_exp_Infinitesimal) |
|
371 |
done |
|
372 |
||
373 |
(* check out this proof!!! *) |
|
374 |
lemma starfun_ln_HFinite_not_Infinitesimal: |
|
375 |
"[| x \<in> HFinite - Infinitesimal; 0 < x |] ==> ( *f* ln) x \<in> HFinite" |
|
376 |
apply (rule ccontr, drule HInfinite_HFinite_iff [THEN iffD2]) |
|
377 |
apply (drule starfun_exp_HInfinite_Infinitesimal_disj) |
|
378 |
apply (simp add: starfun_exp_ln_iff [symmetric] HInfinite_HFinite_iff |
|
379 |
del: starfun_exp_ln_iff) |
|
380 |
done |
|
381 |
||
382 |
(* we do proof by considering ln of 1/x *) |
|
383 |
lemma starfun_ln_Infinitesimal_HInfinite: |
|
384 |
"[| x \<in> Infinitesimal; 0 < x |] ==> ( *f* ln) x \<in> HInfinite" |
|
385 |
apply (drule Infinitesimal_inverse_HInfinite) |
|
386 |
apply (frule positive_imp_inverse_positive) |
|
387 |
apply (drule_tac [2] starfun_ln_HInfinite) |
|
388 |
apply (auto simp add: starfun_ln_inverse HInfinite_minus_iff) |
|
389 |
done |
|
390 |
||
391 |
lemma starfun_ln_less_zero: "!!x. [| 0 < x; x < 1 |] ==> ( *f* ln) x < 0" |
|
392 |
by transfer (rule ln_less_zero) |
|
393 |
||
394 |
lemma starfun_ln_Infinitesimal_less_zero: |
|
395 |
"[| x \<in> Infinitesimal; 0 < x |] ==> ( *f* ln) x < 0" |
|
396 |
by (auto intro!: starfun_ln_less_zero simp add: Infinitesimal_def) |
|
397 |
||
398 |
lemma starfun_ln_HInfinite_gt_zero: |
|
399 |
"[| x \<in> HInfinite; 0 < x |] ==> 0 < ( *f* ln) x" |
|
400 |
by (auto intro!: starfun_ln_gt_zero simp add: HInfinite_def) |
|
401 |
||
402 |
||
403 |
(* |
|
404 |
Goalw [NSLIM_def] "(%h. ((x powr h) - 1) / h) -- 0 --NS> ln x" |
|
405 |
*) |
|
406 |
||
31271 | 407 |
lemma HFinite_sin [simp]: "sumhr (0, whn, %n. sin_coeff n * x ^ n) \<in> HFinite" |
27468 | 408 |
unfolding sumhr_app |
409 |
apply (simp only: star_zero_def starfun2_star_of) |
|
410 |
apply (rule NSBseqD2) |
|
411 |
apply (rule NSconvergent_NSBseq) |
|
412 |
apply (rule convergent_NSconvergent_iff [THEN iffD1]) |
|
413 |
apply (rule summable_convergent_sumr_iff [THEN iffD1]) |
|
31271 | 414 |
apply (rule summable_sin) |
27468 | 415 |
done |
416 |
||
417 |
lemma STAR_sin_zero [simp]: "( *f* sin) 0 = 0" |
|
418 |
by transfer (rule sin_zero) |
|
419 |
||
420 |
lemma STAR_sin_Infinitesimal [simp]: "x \<in> Infinitesimal ==> ( *f* sin) x @= x" |
|
421 |
apply (case_tac "x = 0") |
|
422 |
apply (cut_tac [2] x = 0 in DERIV_sin) |
|
423 |
apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def) |
|
424 |
apply (drule bspec [where x = x], auto) |
|
425 |
apply (drule approx_mult1 [where c = x]) |
|
426 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] |
|
427 |
simp add: mult_assoc) |
|
428 |
done |
|
429 |
||
31271 | 430 |
lemma HFinite_cos [simp]: "sumhr (0, whn, %n. cos_coeff n * x ^ n) \<in> HFinite" |
27468 | 431 |
unfolding sumhr_app |
432 |
apply (simp only: star_zero_def starfun2_star_of) |
|
433 |
apply (rule NSBseqD2) |
|
434 |
apply (rule NSconvergent_NSBseq) |
|
435 |
apply (rule convergent_NSconvergent_iff [THEN iffD1]) |
|
436 |
apply (rule summable_convergent_sumr_iff [THEN iffD1]) |
|
437 |
apply (rule summable_cos) |
|
438 |
done |
|
439 |
||
440 |
lemma STAR_cos_zero [simp]: "( *f* cos) 0 = 1" |
|
441 |
by transfer (rule cos_zero) |
|
442 |
||
443 |
lemma STAR_cos_Infinitesimal [simp]: "x \<in> Infinitesimal ==> ( *f* cos) x @= 1" |
|
444 |
apply (case_tac "x = 0") |
|
445 |
apply (cut_tac [2] x = 0 in DERIV_cos) |
|
446 |
apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def) |
|
447 |
apply (drule bspec [where x = x]) |
|
448 |
apply auto |
|
449 |
apply (drule approx_mult1 [where c = x]) |
|
450 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] |
|
451 |
simp add: mult_assoc) |
|
452 |
apply (rule approx_add_right_cancel [where d = "-1"]) |
|
453 |
apply (simp add: diff_def) |
|
454 |
done |
|
455 |
||
456 |
lemma STAR_tan_zero [simp]: "( *f* tan) 0 = 0" |
|
457 |
by transfer (rule tan_zero) |
|
458 |
||
459 |
lemma STAR_tan_Infinitesimal: "x \<in> Infinitesimal ==> ( *f* tan) x @= x" |
|
460 |
apply (case_tac "x = 0") |
|
461 |
apply (cut_tac [2] x = 0 in DERIV_tan) |
|
462 |
apply (auto simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def) |
|
463 |
apply (drule bspec [where x = x], auto) |
|
464 |
apply (drule approx_mult1 [where c = x]) |
|
465 |
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] |
|
466 |
simp add: mult_assoc) |
|
467 |
done |
|
468 |
||
469 |
lemma STAR_sin_cos_Infinitesimal_mult: |
|
470 |
"x \<in> Infinitesimal ==> ( *f* sin) x * ( *f* cos) x @= x" |
|
471 |
apply (insert approx_mult_HFinite [of "( *f* sin) x" _ "( *f* cos) x" 1]) |
|
472 |
apply (simp add: Infinitesimal_subset_HFinite [THEN subsetD]) |
|
473 |
done |
|
474 |
||
475 |
lemma HFinite_pi: "hypreal_of_real pi \<in> HFinite" |
|
476 |
by simp |
|
477 |
||
478 |
(* lemmas *) |
|
479 |
||
480 |
lemma lemma_split_hypreal_of_real: |
|
481 |
"N \<in> HNatInfinite |
|
482 |
==> hypreal_of_real a = |
|
483 |
hypreal_of_hypnat N * (inverse(hypreal_of_hypnat N) * hypreal_of_real a)" |
|
484 |
by (simp add: mult_assoc [symmetric] zero_less_HNatInfinite) |
|
485 |
||
486 |
lemma STAR_sin_Infinitesimal_divide: |
|
487 |
"[|x \<in> Infinitesimal; x \<noteq> 0 |] ==> ( *f* sin) x/x @= 1" |
|
488 |
apply (cut_tac x = 0 in DERIV_sin) |
|
489 |
apply (simp add: NSDERIV_DERIV_iff [symmetric] nsderiv_def) |
|
490 |
done |
|
491 |
||
492 |
(*------------------------------------------------------------------------*) |
|
493 |
(* sin* (1/n) * 1/(1/n) @= 1 for n = oo *) |
|
494 |
(*------------------------------------------------------------------------*) |
|
495 |
||
496 |
lemma lemma_sin_pi: |
|
497 |
"n \<in> HNatInfinite |
|
498 |
==> ( *f* sin) (inverse (hypreal_of_hypnat n))/(inverse (hypreal_of_hypnat n)) @= 1" |
|
499 |
apply (rule STAR_sin_Infinitesimal_divide) |
|
500 |
apply (auto simp add: zero_less_HNatInfinite) |
|
501 |
done |
|
502 |
||
503 |
lemma STAR_sin_inverse_HNatInfinite: |
|
504 |
"n \<in> HNatInfinite |
|
505 |
==> ( *f* sin) (inverse (hypreal_of_hypnat n)) * hypreal_of_hypnat n @= 1" |
|
506 |
apply (frule lemma_sin_pi) |
|
507 |
apply (simp add: divide_inverse) |
|
508 |
done |
|
509 |
||
510 |
lemma Infinitesimal_pi_divide_HNatInfinite: |
|
511 |
"N \<in> HNatInfinite |
|
512 |
==> hypreal_of_real pi/(hypreal_of_hypnat N) \<in> Infinitesimal" |
|
513 |
apply (simp add: divide_inverse) |
|
514 |
apply (auto intro: Infinitesimal_HFinite_mult2) |
|
515 |
done |
|
516 |
||
517 |
lemma pi_divide_HNatInfinite_not_zero [simp]: |
|
518 |
"N \<in> HNatInfinite ==> hypreal_of_real pi/(hypreal_of_hypnat N) \<noteq> 0" |
|
519 |
by (simp add: zero_less_HNatInfinite) |
|
520 |
||
521 |
lemma STAR_sin_pi_divide_HNatInfinite_approx_pi: |
|
522 |
"n \<in> HNatInfinite |
|
523 |
==> ( *f* sin) (hypreal_of_real pi/(hypreal_of_hypnat n)) * hypreal_of_hypnat n |
|
524 |
@= hypreal_of_real pi" |
|
525 |
apply (frule STAR_sin_Infinitesimal_divide |
|
526 |
[OF Infinitesimal_pi_divide_HNatInfinite |
|
527 |
pi_divide_HNatInfinite_not_zero]) |
|
528 |
apply (auto) |
|
529 |
apply (rule approx_SReal_mult_cancel [of "inverse (hypreal_of_real pi)"]) |
|
530 |
apply (auto intro: Reals_inverse simp add: divide_inverse mult_ac) |
|
531 |
done |
|
532 |
||
533 |
lemma STAR_sin_pi_divide_HNatInfinite_approx_pi2: |
|
534 |
"n \<in> HNatInfinite |
|
535 |
==> hypreal_of_hypnat n * |
|
536 |
( *f* sin) (hypreal_of_real pi/(hypreal_of_hypnat n)) |
|
537 |
@= hypreal_of_real pi" |
|
538 |
apply (rule mult_commute [THEN subst]) |
|
539 |
apply (erule STAR_sin_pi_divide_HNatInfinite_approx_pi) |
|
540 |
done |
|
541 |
||
542 |
lemma starfunNat_pi_divide_n_Infinitesimal: |
|
543 |
"N \<in> HNatInfinite ==> ( *f* (%x. pi / real x)) N \<in> Infinitesimal" |
|
544 |
by (auto intro!: Infinitesimal_HFinite_mult2 |
|
545 |
simp add: starfun_mult [symmetric] divide_inverse |
|
546 |
starfun_inverse [symmetric] starfunNat_real_of_nat) |
|
547 |
||
548 |
lemma STAR_sin_pi_divide_n_approx: |
|
549 |
"N \<in> HNatInfinite ==> |
|
550 |
( *f* sin) (( *f* (%x. pi / real x)) N) @= |
|
551 |
hypreal_of_real pi/(hypreal_of_hypnat N)" |
|
552 |
apply (simp add: starfunNat_real_of_nat [symmetric]) |
|
553 |
apply (rule STAR_sin_Infinitesimal) |
|
554 |
apply (simp add: divide_inverse) |
|
555 |
apply (rule Infinitesimal_HFinite_mult2) |
|
556 |
apply (subst starfun_inverse) |
|
557 |
apply (erule starfunNat_inverse_real_of_nat_Infinitesimal) |
|
558 |
apply simp |
|
559 |
done |
|
560 |
||
561 |
lemma NSLIMSEQ_sin_pi: "(%n. real n * sin (pi / real n)) ----NS> pi" |
|
562 |
apply (auto simp add: NSLIMSEQ_def starfun_mult [symmetric] starfunNat_real_of_nat) |
|
563 |
apply (rule_tac f1 = sin in starfun_o2 [THEN subst]) |
|
564 |
apply (auto simp add: starfun_mult [symmetric] starfunNat_real_of_nat divide_inverse) |
|
565 |
apply (rule_tac f1 = inverse in starfun_o2 [THEN subst]) |
|
566 |
apply (auto dest: STAR_sin_pi_divide_HNatInfinite_approx_pi |
|
567 |
simp add: starfunNat_real_of_nat mult_commute divide_inverse) |
|
568 |
done |
|
569 |
||
570 |
lemma NSLIMSEQ_cos_one: "(%n. cos (pi / real n))----NS> 1" |
|
571 |
apply (simp add: NSLIMSEQ_def, auto) |
|
572 |
apply (rule_tac f1 = cos in starfun_o2 [THEN subst]) |
|
573 |
apply (rule STAR_cos_Infinitesimal) |
|
574 |
apply (auto intro!: Infinitesimal_HFinite_mult2 |
|
575 |
simp add: starfun_mult [symmetric] divide_inverse |
|
576 |
starfun_inverse [symmetric] starfunNat_real_of_nat) |
|
577 |
done |
|
578 |
||
579 |
lemma NSLIMSEQ_sin_cos_pi: |
|
580 |
"(%n. real n * sin (pi / real n) * cos (pi / real n)) ----NS> pi" |
|
581 |
by (insert NSLIMSEQ_mult [OF NSLIMSEQ_sin_pi NSLIMSEQ_cos_one], simp) |
|
582 |
||
583 |
||
584 |
text{*A familiar approximation to @{term "cos x"} when @{term x} is small*} |
|
585 |
||
586 |
lemma STAR_cos_Infinitesimal_approx: |
|
587 |
"x \<in> Infinitesimal ==> ( *f* cos) x @= 1 - x ^ 2" |
|
588 |
apply (rule STAR_cos_Infinitesimal [THEN approx_trans]) |
|
589 |
apply (auto simp add: Infinitesimal_approx_minus [symmetric] |
|
590 |
diff_minus add_assoc [symmetric] numeral_2_eq_2) |
|
591 |
done |
|
592 |
||
593 |
lemma STAR_cos_Infinitesimal_approx2: |
|
594 |
"x \<in> Infinitesimal ==> ( *f* cos) x @= 1 - (x ^ 2)/2" |
|
595 |
apply (rule STAR_cos_Infinitesimal [THEN approx_trans]) |
|
596 |
apply (auto intro: Infinitesimal_SReal_divide |
|
597 |
simp add: Infinitesimal_approx_minus [symmetric] numeral_2_eq_2) |
|
598 |
done |
|
599 |
||
600 |
end |