author | blanchet |
Wed, 04 Mar 2009 10:45:52 +0100 | |
changeset 30240 | 5b25fee0362c |
parent 29823 | 0ab754d13ccd |
child 30273 | ecd6f0ca62ea |
permissions | -rw-r--r-- |
30240 | 1 |
(* Title: HOL/Reflection/Approximation.thy |
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Author: Johannes Hoelzl <hoelzl@in.tum.de> 2008 / 2009 |
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*) |
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header {* Prove unequations about real numbers by computation *} |
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theory Approximation |
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0ab754d13ccd
session Reflecion renamed to Decision_Procs, moved Dense_Linear_Order there
haftmann
parents:
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diff
changeset
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imports Complex_Main Float Reflection Dense_Linear_Order Efficient_Nat |
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begin |
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section "Horner Scheme" |
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subsection {* Define auxiliary helper @{text horner} function *} |
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fun horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where |
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"horner F G 0 i k x = 0" | |
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"horner F G (Suc n) i k x = 1 / real k - x * horner F G n (F i) (G i k) x" |
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lemma horner_schema': fixes x :: real and a :: "nat \<Rightarrow> real" |
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shows "a 0 - x * (\<Sum> i=0..<n. (-1)^i * a (Suc i) * x^i) = (\<Sum> i=0..<Suc n. (-1)^i * a i * x^i)" |
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proof - |
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have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)" by auto |
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show ?thesis unfolding setsum_right_distrib shift_pow real_diff_def setsum_negf[symmetric] setsum_head_upt_Suc[OF zero_less_Suc] |
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setsum_reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n *a n * x^n"] by auto |
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qed |
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lemma horner_schema: fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat" |
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assumes f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)" |
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shows "horner F G n ((F^j') s) (f j') x = (\<Sum> j = 0..< n. -1^j * (1 / real (f (j' + j))) * x^j)" |
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proof (induct n arbitrary: i k j') |
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case (Suc n) |
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show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc] |
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using horner_schema'[of "\<lambda> j. 1 / real (f (j' + j))"] by auto |
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qed auto |
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lemma horner_bounds': |
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assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)" |
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and lb_0: "\<And> i k x. lb 0 i k x = 0" |
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and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)" |
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and ub_0: "\<And> i k x. ub 0 i k x = 0" |
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and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)" |
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shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> horner F G n ((F^j') s) (f j') (Ifloat x) \<and> |
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horner F G n ((F^j') s) (f j') (Ifloat x) \<le> Ifloat (ub n ((F^j') s) (f j') x)" |
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(is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'") |
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proof (induct n arbitrary: j') |
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case 0 thus ?case unfolding lb_0 ub_0 horner.simps by auto |
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next |
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case (Suc n) |
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have "?lb (Suc n) j' \<le> ?horner (Suc n) j'" unfolding lb_Suc ub_Suc horner.simps Ifloat_sub diff_def |
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proof (rule add_mono) |
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show "Ifloat (lapprox_rat prec 1 (int (f j'))) \<le> 1 / real (f j')" using lapprox_rat[of prec 1 "int (f j')"] by auto |
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from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct2] `0 \<le> Ifloat x` |
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show "- Ifloat (x * ub n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) x) \<le> - (Ifloat x * horner F G n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) (Ifloat x))" |
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unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono) |
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qed |
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moreover have "?horner (Suc n) j' \<le> ?ub (Suc n) j'" unfolding ub_Suc ub_Suc horner.simps Ifloat_sub diff_def |
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proof (rule add_mono) |
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show "1 / real (f j') \<le> Ifloat (rapprox_rat prec 1 (int (f j')))" using rapprox_rat[of 1 "int (f j')" prec] by auto |
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from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct1] `0 \<le> Ifloat x` |
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show "- (Ifloat x * horner F G n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) (Ifloat x)) \<le> |
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- Ifloat (x * lb n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) x)" |
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unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono) |
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qed |
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ultimately show ?case by blast |
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qed |
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subsection "Theorems for floating point functions implementing the horner scheme" |
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text {* |
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Here @{term_type "f :: nat \<Rightarrow> nat"} is the sequence defining the Taylor series, the coefficients are |
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all alternating and reciprocs. We use @{term G} and @{term F} to describe the computation of @{term f}. |
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*} |
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lemma horner_bounds: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" |
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assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)" |
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and lb_0: "\<And> i k x. lb 0 i k x = 0" |
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and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)" |
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and ub_0: "\<And> i k x. ub 0 i k x = 0" |
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and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)" |
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shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> (\<Sum>j=0..<n. -1^j * (1 / real (f (j' + j))) * (Ifloat x)^j)" (is "?lb") and |
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"(\<Sum>j=0..<n. -1^j * (1 / real (f (j' + j))) * (Ifloat x)^j) \<le> Ifloat (ub n ((F^j') s) (f j') x)" (is "?ub") |
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proof - |
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have "?lb \<and> ?ub" |
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using horner_bounds'[where lb=lb, OF `0 \<le> Ifloat x` f_Suc lb_0 lb_Suc ub_0 ub_Suc] |
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unfolding horner_schema[where f=f, OF f_Suc] . |
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thus "?lb" and "?ub" by auto |
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qed |
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lemma horner_bounds_nonpos: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" |
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assumes "Ifloat x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)" |
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and lb_0: "\<And> i k x. lb 0 i k x = 0" |
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and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) + x * (ub n (F i) (G i k) x)" |
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and ub_0: "\<And> i k x. ub 0 i k x = 0" |
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and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) + x * (lb n (F i) (G i k) x)" |
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shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j)" (is "?lb") and |
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"(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j) \<le> Ifloat (ub n ((F^j') s) (f j') x)" (is "?ub") |
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proof - |
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{ fix x y z :: float have "x - y * z = x + - y * z" |
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by (cases x, cases y, cases z, simp add: plus_float.simps minus_float.simps uminus_float.simps times_float.simps algebra_simps) |
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} note diff_mult_minus = this |
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{ fix x :: float have "- (- x) = x" by (cases x, auto simp add: uminus_float.simps) } note minus_minus = this |
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have move_minus: "Ifloat (-x) = -1 * Ifloat x" by auto |
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have sum_eq: "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j) = |
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(\<Sum>j = 0..<n. -1 ^ j * (1 / real (f (j' + j))) * Ifloat (- x) ^ j)" |
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proof (rule setsum_cong, simp) |
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fix j assume "j \<in> {0 ..< n}" |
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show "1 / real (f (j' + j)) * Ifloat x ^ j = -1 ^ j * (1 / real (f (j' + j))) * Ifloat (- x) ^ j" |
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unfolding move_minus power_mult_distrib real_mult_assoc[symmetric] |
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unfolding real_mult_commute unfolding real_mult_assoc[of "-1^j", symmetric] power_mult_distrib[symmetric] |
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by auto |
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qed |
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have "0 \<le> Ifloat (-x)" using assms by auto |
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from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec |
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and lb="\<lambda> n i k x. lb n i k (-x)" and ub="\<lambda> n i k x. ub n i k (-x)", unfolded lb_Suc ub_Suc diff_mult_minus, |
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OF this f_Suc lb_0 refl ub_0 refl] |
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show "?lb" and "?ub" unfolding minus_minus sum_eq |
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by auto |
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qed |
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subsection {* Selectors for next even or odd number *} |
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text {* |
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The horner scheme computes alternating series. To get the upper and lower bounds we need to |
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guarantee to access a even or odd member. To do this we use @{term get_odd} and @{term get_even}. |
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*} |
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definition get_odd :: "nat \<Rightarrow> nat" where |
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"get_odd n = (if odd n then n else (Suc n))" |
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definition get_even :: "nat \<Rightarrow> nat" where |
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"get_even n = (if even n then n else (Suc n))" |
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lemma get_odd[simp]: "odd (get_odd n)" unfolding get_odd_def by (cases "odd n", auto) |
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lemma get_even[simp]: "even (get_even n)" unfolding get_even_def by (cases "even n", auto) |
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lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)" |
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proof (cases "odd n") |
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case True hence "0 < n" by (rule odd_pos) |
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from gr0_implies_Suc[OF this] obtain k where "Suc k = n" by auto |
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thus ?thesis unfolding get_odd_def if_P[OF True] using True[unfolded `Suc k = n`[symmetric]] by blast |
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next |
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case False hence "odd (Suc n)" by auto |
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thus ?thesis unfolding get_odd_def if_not_P[OF False] by blast |
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qed |
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lemma get_even_double: "\<exists>i. get_even n = 2 * i" using get_even[unfolded even_mult_two_ex] . |
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lemma get_odd_double: "\<exists>i. get_odd n = 2 * i + 1" using get_odd[unfolded odd_Suc_mult_two_ex] by auto |
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section "Power function" |
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definition float_power_bnds :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where |
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"float_power_bnds n l u = (if odd n \<or> 0 < l then (l ^ n, u ^ n) |
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else if u < 0 then (u ^ n, l ^ n) |
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else (0, (max (-l) u) ^ n))" |
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lemma float_power_bnds: assumes "(l1, u1) = float_power_bnds n l u" and "x \<in> {Ifloat l .. Ifloat u}" |
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shows "x^n \<in> {Ifloat l1..Ifloat u1}" |
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proof (cases "even n") |
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case True |
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show ?thesis |
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proof (cases "0 < l") |
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case True hence "odd n \<or> 0 < l" and "0 \<le> Ifloat l" unfolding less_float_def by auto |
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have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \<or> 0 < l`] by auto |
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have "Ifloat l^n \<le> x^n" and "x^n \<le> Ifloat u^n " using `0 \<le> Ifloat l` and assms unfolding atLeastAtMost_iff using power_mono[of "Ifloat l" x] power_mono[of x "Ifloat u"] by auto |
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thus ?thesis using assms `0 < l` unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto |
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next |
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case False hence P: "\<not> (odd n \<or> 0 < l)" using `even n` by auto |
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show ?thesis |
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proof (cases "u < 0") |
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case True hence "0 \<le> - Ifloat u" and "- Ifloat u \<le> - x" and "0 \<le> - x" and "-x \<le> - Ifloat l" using assms unfolding less_float_def by auto |
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hence "Ifloat u^n \<le> x^n" and "x^n \<le> Ifloat l^n" using power_mono[of "-x" "-Ifloat l" n] power_mono[of "-Ifloat u" "-x" n] |
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unfolding power_minus_even[OF `even n`] by auto |
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moreover have u1: "u1 = l ^ n" and l1: "l1 = u ^ n" using assms unfolding float_power_bnds_def if_not_P[OF P] if_P[OF True] by auto |
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ultimately show ?thesis using float_power by auto |
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next |
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case False |
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have "\<bar>x\<bar> \<le> Ifloat (max (-l) u)" |
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proof (cases "-l \<le> u") |
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case True thus ?thesis unfolding max_def if_P[OF True] using assms unfolding le_float_def by auto |
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next |
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case False thus ?thesis unfolding max_def if_not_P[OF False] using assms unfolding le_float_def by auto |
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qed |
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hence x_abs: "\<bar>x\<bar> \<le> \<bar>Ifloat (max (-l) u)\<bar>" by auto |
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have u1: "u1 = (max (-l) u) ^ n" and l1: "l1 = 0" using assms unfolding float_power_bnds_def if_not_P[OF P] if_not_P[OF False] by auto |
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show ?thesis unfolding atLeastAtMost_iff l1 u1 float_power using zero_le_even_power[OF `even n`] power_mono_even[OF `even n` x_abs] by auto |
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qed |
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qed |
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next |
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case False hence "odd n \<or> 0 < l" by auto |
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have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \<or> 0 < l`] by auto |
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have "Ifloat l^n \<le> x^n" and "x^n \<le> Ifloat u^n " using assms unfolding atLeastAtMost_iff using power_mono_odd[OF False] by auto |
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thus ?thesis unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto |
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qed |
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lemma bnds_power: "\<forall> x l u. (l1, u1) = float_power_bnds n l u \<and> x \<in> {Ifloat l .. Ifloat u} \<longrightarrow> Ifloat l1 \<le> x^n \<and> x^n \<le> Ifloat u1" |
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using float_power_bnds by auto |
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section "Square root" |
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text {* |
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The square root computation is implemented as newton iteration. As first first step we use the |
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nearest power of two greater than the square root. |
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*} |
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fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
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"sqrt_iteration prec 0 (Float m e) = Float 1 ((e + bitlen m) div 2 + 1)" | |
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"sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x |
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in Float 1 -1 * (y + float_divr prec x y))" |
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definition ub_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float option" where |
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"ub_sqrt prec x = (if 0 < x then Some (sqrt_iteration prec prec x) else if x < 0 then None else Some 0)" |
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definition lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float option" where |
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"lb_sqrt prec x = (if 0 < x then Some (float_divl prec x (sqrt_iteration prec prec x)) else if x < 0 then None else Some 0)" |
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lemma sqrt_ub_pos_pos_1: |
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assumes "sqrt x < b" and "0 < b" and "0 < x" |
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shows "sqrt x < (b + x / b)/2" |
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proof - |
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from assms have "0 < (b - sqrt x) ^ 2 " by simp |
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also have "\<dots> = b ^ 2 - 2 * b * sqrt x + (sqrt x) ^ 2" by algebra |
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also have "\<dots> = b ^ 2 - 2 * b * sqrt x + x" using assms by (simp add: real_sqrt_pow2) |
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finally have "0 < b ^ 2 - 2 * b * sqrt x + x" by assumption |
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hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms |
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by (simp add: field_simps power2_eq_square) |
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thus ?thesis by (simp add: field_simps) |
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qed |
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lemma sqrt_iteration_bound: assumes "0 < Ifloat x" |
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shows "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec n x)" |
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proof (induct n) |
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case 0 |
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show ?case |
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proof (cases x) |
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case (Float m e) |
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hence "0 < m" using float_pos_m_pos[unfolded less_float_def] assms by auto |
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hence "0 < sqrt (real m)" by auto |
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have int_nat_bl: "int (nat (bitlen m)) = bitlen m" using bitlen_ge0 by auto |
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have "Ifloat x = (real m / 2^nat (bitlen m)) * pow2 (e + int (nat (bitlen m)))" |
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unfolding pow2_add pow2_int Float Ifloat.simps by auto |
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also have "\<dots> < 1 * pow2 (e + int (nat (bitlen m)))" |
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proof (rule mult_strict_right_mono, auto) |
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show "real m < 2^nat (bitlen m)" using bitlen_bounds[OF `0 < m`, THEN conjunct2] |
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unfolding real_of_int_less_iff[of m, symmetric] by auto |
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qed |
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finally have "sqrt (Ifloat x) < sqrt (pow2 (e + bitlen m))" unfolding int_nat_bl by auto |
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also have "\<dots> \<le> pow2 ((e + bitlen m) div 2 + 1)" |
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proof - |
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let ?E = "e + bitlen m" |
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have E_mod_pow: "pow2 (?E mod 2) < 4" |
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proof (cases "?E mod 2 = 1") |
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case True thus ?thesis by auto |
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next |
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case False |
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have "0 \<le> ?E mod 2" by auto |
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have "?E mod 2 < 2" by auto |
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from this[THEN zless_imp_add1_zle] |
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have "?E mod 2 \<le> 0" using False by auto |
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from xt1(5)[OF `0 \<le> ?E mod 2` this] |
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show ?thesis by auto |
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qed |
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hence "sqrt (pow2 (?E mod 2)) < sqrt (2 * 2)" by auto |
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hence E_mod_pow: "sqrt (pow2 (?E mod 2)) < 2" unfolding real_sqrt_abs2 by auto |
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have E_eq: "pow2 ?E = pow2 (?E div 2 + ?E div 2 + ?E mod 2)" by auto |
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have "sqrt (pow2 ?E) = sqrt (pow2 (?E div 2) * pow2 (?E div 2) * pow2 (?E mod 2))" |
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unfolding E_eq unfolding pow2_add .. |
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also have "\<dots> = pow2 (?E div 2) * sqrt (pow2 (?E mod 2))" |
|
281 |
unfolding real_sqrt_mult[of _ "pow2 (?E mod 2)"] real_sqrt_abs2 by auto |
|
282 |
also have "\<dots> < pow2 (?E div 2) * 2" |
|
283 |
by (rule mult_strict_left_mono, auto intro: E_mod_pow) |
|
284 |
also have "\<dots> = pow2 (?E div 2 + 1)" unfolding zadd_commute[of _ 1] pow2_add1 by auto |
|
285 |
finally show ?thesis by auto |
|
286 |
qed |
|
287 |
finally show ?thesis |
|
288 |
unfolding Float sqrt_iteration.simps Ifloat.simps by auto |
|
289 |
qed |
|
290 |
next |
|
291 |
case (Suc n) |
|
292 |
let ?b = "sqrt_iteration prec n x" |
|
293 |
have "0 < sqrt (Ifloat x)" using `0 < Ifloat x` by auto |
|
294 |
also have "\<dots> < Ifloat ?b" using Suc . |
|
295 |
finally have "sqrt (Ifloat x) < (Ifloat ?b + Ifloat x / Ifloat ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ `0 < Ifloat x`] by auto |
|
296 |
also have "\<dots> \<le> (Ifloat ?b + Ifloat (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr) |
|
297 |
also have "\<dots> = Ifloat (Float 1 -1) * (Ifloat ?b + Ifloat (float_divr prec x ?b))" by auto |
|
298 |
finally show ?case unfolding sqrt_iteration.simps Let_def Ifloat_mult Ifloat_add right_distrib . |
|
299 |
qed |
|
300 |
||
301 |
lemma sqrt_iteration_lower_bound: assumes "0 < Ifloat x" |
|
302 |
shows "0 < Ifloat (sqrt_iteration prec n x)" (is "0 < ?sqrt") |
|
303 |
proof - |
|
304 |
have "0 < sqrt (Ifloat x)" using assms by auto |
|
305 |
also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] . |
|
306 |
finally show ?thesis . |
|
307 |
qed |
|
308 |
||
309 |
lemma lb_sqrt_lower_bound: assumes "0 \<le> Ifloat x" |
|
310 |
shows "0 \<le> Ifloat (the (lb_sqrt prec x))" |
|
311 |
proof (cases "0 < x") |
|
312 |
case True hence "0 < Ifloat x" and "0 \<le> x" using `0 \<le> Ifloat x` unfolding less_float_def le_float_def by auto |
|
313 |
hence "0 < sqrt_iteration prec prec x" unfolding less_float_def using sqrt_iteration_lower_bound by auto |
|
314 |
hence "0 \<le> Ifloat (float_divl prec x (sqrt_iteration prec prec x))" using float_divl_lower_bound[OF `0 \<le> x`] unfolding le_float_def by auto |
|
315 |
thus ?thesis unfolding lb_sqrt_def using True by auto |
|
316 |
next |
|
317 |
case False with `0 \<le> Ifloat x` have "Ifloat x = 0" unfolding less_float_def by auto |
|
318 |
thus ?thesis unfolding lb_sqrt_def less_float_def by auto |
|
319 |
qed |
|
320 |
||
321 |
lemma lb_sqrt_upper_bound: assumes "0 \<le> Ifloat x" |
|
322 |
shows "Ifloat (the (lb_sqrt prec x)) \<le> sqrt (Ifloat x)" |
|
323 |
proof (cases "0 < x") |
|
324 |
case True hence "0 < Ifloat x" and "0 \<le> Ifloat x" unfolding less_float_def by auto |
|
325 |
hence sqrt_gt0: "0 < sqrt (Ifloat x)" by auto |
|
326 |
hence sqrt_ub: "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto |
|
327 |
||
328 |
have "Ifloat (float_divl prec x (sqrt_iteration prec prec x)) \<le> Ifloat x / Ifloat (sqrt_iteration prec prec x)" by (rule float_divl) |
|
329 |
also have "\<dots> < Ifloat x / sqrt (Ifloat x)" |
|
330 |
by (rule divide_strict_left_mono[OF sqrt_ub `0 < Ifloat x` mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]]) |
|
331 |
also have "\<dots> = sqrt (Ifloat x)" unfolding inverse_eq_iff_eq[of _ "sqrt (Ifloat x)", symmetric] sqrt_divide_self_eq[OF `0 \<le> Ifloat x`, symmetric] by auto |
|
332 |
finally show ?thesis unfolding lb_sqrt_def if_P[OF `0 < x`] by auto |
|
333 |
next |
|
334 |
case False with `0 \<le> Ifloat x` |
|
335 |
have "\<not> x < 0" unfolding less_float_def le_float_def by auto |
|
336 |
show ?thesis unfolding lb_sqrt_def if_not_P[OF False] if_not_P[OF `\<not> x < 0`] using assms by auto |
|
337 |
qed |
|
338 |
||
339 |
lemma lb_sqrt: assumes "Some y = lb_sqrt prec x" |
|
340 |
shows "Ifloat y \<le> sqrt (Ifloat x)" and "0 \<le> Ifloat x" |
|
341 |
proof - |
|
342 |
show "0 \<le> Ifloat x" |
|
343 |
proof (rule ccontr) |
|
344 |
assume "\<not> 0 \<le> Ifloat x" |
|
345 |
hence "lb_sqrt prec x = None" unfolding lb_sqrt_def less_float_def by auto |
|
346 |
thus False using assms by auto |
|
347 |
qed |
|
348 |
from lb_sqrt_upper_bound[OF this, of prec] |
|
349 |
show "Ifloat y \<le> sqrt (Ifloat x)" unfolding assms[symmetric] by auto |
|
350 |
qed |
|
351 |
||
352 |
lemma ub_sqrt_lower_bound: assumes "0 \<le> Ifloat x" |
|
353 |
shows "sqrt (Ifloat x) \<le> Ifloat (the (ub_sqrt prec x))" |
|
354 |
proof (cases "0 < x") |
|
355 |
case True hence "0 < Ifloat x" unfolding less_float_def by auto |
|
356 |
hence "0 < sqrt (Ifloat x)" by auto |
|
357 |
hence "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto |
|
358 |
thus ?thesis unfolding ub_sqrt_def if_P[OF `0 < x`] by auto |
|
359 |
next |
|
360 |
case False with `0 \<le> Ifloat x` |
|
361 |
have "Ifloat x = 0" unfolding less_float_def le_float_def by auto |
|
362 |
thus ?thesis unfolding ub_sqrt_def less_float_def le_float_def by auto |
|
363 |
qed |
|
364 |
||
365 |
lemma ub_sqrt: assumes "Some y = ub_sqrt prec x" |
|
366 |
shows "sqrt (Ifloat x) \<le> Ifloat y" and "0 \<le> Ifloat x" |
|
367 |
proof - |
|
368 |
show "0 \<le> Ifloat x" |
|
369 |
proof (rule ccontr) |
|
370 |
assume "\<not> 0 \<le> Ifloat x" |
|
371 |
hence "ub_sqrt prec x = None" unfolding ub_sqrt_def less_float_def by auto |
|
372 |
thus False using assms by auto |
|
373 |
qed |
|
374 |
from ub_sqrt_lower_bound[OF this, of prec] |
|
375 |
show "sqrt (Ifloat x) \<le> Ifloat y" unfolding assms[symmetric] by auto |
|
376 |
qed |
|
377 |
||
378 |
lemma bnds_sqrt: "\<forall> x lx ux. (Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> sqrt x \<and> sqrt x \<le> Ifloat u" |
|
379 |
proof (rule allI, rule allI, rule allI, rule impI) |
|
380 |
fix x lx ux |
|
381 |
assume "(Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}" |
|
382 |
hence l: "Some l = lb_sqrt prec lx " and u: "Some u = ub_sqrt prec ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto |
|
383 |
||
384 |
have "Ifloat lx \<le> x" and "x \<le> Ifloat ux" using x by auto |
|
385 |
||
386 |
from lb_sqrt(1)[OF l] real_sqrt_le_mono[OF `Ifloat lx \<le> x`] |
|
387 |
have "Ifloat l \<le> sqrt x" by (rule order_trans) |
|
388 |
moreover |
|
389 |
from real_sqrt_le_mono[OF `x \<le> Ifloat ux`] ub_sqrt(1)[OF u] |
|
390 |
have "sqrt x \<le> Ifloat u" by (rule order_trans) |
|
391 |
ultimately show "Ifloat l \<le> sqrt x \<and> sqrt x \<le> Ifloat u" .. |
|
392 |
qed |
|
393 |
||
394 |
section "Arcus tangens and \<pi>" |
|
395 |
||
396 |
subsection "Compute arcus tangens series" |
|
397 |
||
398 |
text {* |
|
399 |
||
400 |
As first step we implement the computation of the arcus tangens series. This is only valid in the range |
|
401 |
@{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens. |
|
402 |
||
403 |
*} |
|
404 |
||
405 |
fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
|
406 |
and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
|
407 |
"ub_arctan_horner prec 0 k x = 0" |
|
408 |
| "ub_arctan_horner prec (Suc n) k x = |
|
409 |
(rapprox_rat prec 1 (int k)) - x * (lb_arctan_horner prec n (k + 2) x)" |
|
410 |
| "lb_arctan_horner prec 0 k x = 0" |
|
411 |
| "lb_arctan_horner prec (Suc n) k x = |
|
412 |
(lapprox_rat prec 1 (int k)) - x * (ub_arctan_horner prec n (k + 2) x)" |
|
413 |
||
414 |
lemma arctan_0_1_bounds': assumes "0 \<le> Ifloat x" "Ifloat x \<le> 1" and "even n" |
|
415 |
shows "arctan (Ifloat x) \<in> {Ifloat (x * lb_arctan_horner prec n 1 (x * x)) .. Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x * x))}" |
|
416 |
proof - |
|
417 |
let "?c i" = "-1^i * (1 / real (i * 2 + 1) * Ifloat x ^ (i * 2 + 1))" |
|
418 |
let "?S n" = "\<Sum> i=0..<n. ?c i" |
|
419 |
||
420 |
have "0 \<le> Ifloat (x * x)" by auto |
|
421 |
from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto |
|
422 |
||
423 |
have "arctan (Ifloat x) \<in> { ?S n .. ?S (Suc n) }" |
|
424 |
proof (cases "Ifloat x = 0") |
|
425 |
case False |
|
426 |
hence "0 < Ifloat x" using `0 \<le> Ifloat x` by auto |
|
427 |
hence prem: "0 < 1 / real (0 * 2 + (1::nat)) * Ifloat x ^ (0 * 2 + 1)" by auto |
|
428 |
||
429 |
have "\<bar> Ifloat x \<bar> \<le> 1" using `0 \<le> Ifloat x` `Ifloat x \<le> 1` by auto |
|
430 |
from mp[OF summable_Leibniz(2)[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] prem, THEN spec, of m, unfolded `2 * m = n`] |
|
431 |
show ?thesis unfolding arctan_series[OF `\<bar> Ifloat x \<bar> \<le> 1`] Suc_plus1 . |
|
432 |
qed auto |
|
433 |
note arctan_bounds = this[unfolded atLeastAtMost_iff] |
|
434 |
||
435 |
have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto |
|
436 |
||
437 |
note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0 |
|
438 |
and lb="\<lambda>n i k x. lb_arctan_horner prec n k x" |
|
439 |
and ub="\<lambda>n i k x. ub_arctan_horner prec n k x", |
|
440 |
OF `0 \<le> Ifloat (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps] |
|
441 |
||
442 |
{ have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n" |
|
443 |
using bounds(1) `0 \<le> Ifloat x` |
|
444 |
unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric] |
|
445 |
unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"] |
|
446 |
by (auto intro!: mult_left_mono) |
|
447 |
also have "\<dots> \<le> arctan (Ifloat x)" using arctan_bounds .. |
|
448 |
finally have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan (Ifloat x)" . } |
|
449 |
moreover |
|
450 |
{ have "arctan (Ifloat x) \<le> ?S (Suc n)" using arctan_bounds .. |
|
451 |
also have "\<dots> \<le> Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))" |
|
452 |
using bounds(2)[of "Suc n"] `0 \<le> Ifloat x` |
|
453 |
unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric] |
|
454 |
unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"] |
|
455 |
by (auto intro!: mult_left_mono) |
|
456 |
finally have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . } |
|
457 |
ultimately show ?thesis by auto |
|
458 |
qed |
|
459 |
||
460 |
lemma arctan_0_1_bounds: assumes "0 \<le> Ifloat x" "Ifloat x \<le> 1" |
|
461 |
shows "arctan (Ifloat x) \<in> {Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) .. Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))}" |
|
462 |
proof (cases "even n") |
|
463 |
case True |
|
464 |
obtain n' where "Suc n' = get_odd n" and "odd (Suc n')" using get_odd_ex by auto |
|
465 |
hence "even n'" unfolding even_nat_Suc by auto |
|
466 |
have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))" |
|
467 |
unfolding `Suc n' = get_odd n`[symmetric] using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even n'`] by auto |
|
468 |
moreover |
|
469 |
have "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (Ifloat x)" |
|
470 |
unfolding get_even_def if_P[OF True] using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even n`] by auto |
|
471 |
ultimately show ?thesis by auto |
|
472 |
next |
|
473 |
case False hence "0 < n" by (rule odd_pos) |
|
474 |
from gr0_implies_Suc[OF this] obtain n' where "n = Suc n'" .. |
|
475 |
from False[unfolded this even_nat_Suc] |
|
476 |
have "even n'" and "even (Suc (Suc n'))" by auto |
|
477 |
have "get_odd n = Suc n'" unfolding get_odd_def if_P[OF False] using `n = Suc n'` . |
|
478 |
||
479 |
have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))" |
|
480 |
unfolding `get_odd n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even n'`] by auto |
|
481 |
moreover |
|
482 |
have "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (Ifloat x)" |
|
483 |
unfolding get_even_def if_not_P[OF False] unfolding `n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even (Suc (Suc n'))`] by auto |
|
484 |
ultimately show ?thesis by auto |
|
485 |
qed |
|
486 |
||
487 |
subsection "Compute \<pi>" |
|
488 |
||
489 |
definition ub_pi :: "nat \<Rightarrow> float" where |
|
490 |
"ub_pi prec = (let A = rapprox_rat prec 1 5 ; |
|
491 |
B = lapprox_rat prec 1 239 |
|
492 |
in ((Float 1 2) * ((Float 1 2) * A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (A * A)) - |
|
493 |
B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 (B * B)))))" |
|
494 |
||
495 |
definition lb_pi :: "nat \<Rightarrow> float" where |
|
496 |
"lb_pi prec = (let A = lapprox_rat prec 1 5 ; |
|
497 |
B = rapprox_rat prec 1 239 |
|
498 |
in ((Float 1 2) * ((Float 1 2) * A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (A * A)) - |
|
499 |
B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (B * B)))))" |
|
500 |
||
501 |
lemma pi_boundaries: "pi \<in> {Ifloat (lb_pi n) .. Ifloat (ub_pi n)}" |
|
502 |
proof - |
|
503 |
have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))" unfolding machin[symmetric] by auto |
|
504 |
||
505 |
{ fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto |
|
506 |
let ?k = "rapprox_rat prec 1 k" |
|
507 |
have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto |
|
508 |
||
509 |
have "0 \<le> Ifloat ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \<le> k`) |
|
510 |
have "Ifloat ?k \<le> 1" unfolding rapprox_rat.simps(2)[OF zero_le_one `0 < k`] |
|
511 |
by (rule rapprox_posrat_le1, auto simp add: `0 < k` `1 \<le> k`) |
|
512 |
||
513 |
have "1 / real k \<le> Ifloat ?k" using rapprox_rat[where x=1 and y=k] by auto |
|
514 |
hence "arctan (1 / real k) \<le> arctan (Ifloat ?k)" by (rule arctan_monotone') |
|
515 |
also have "\<dots> \<le> Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" |
|
516 |
using arctan_0_1_bounds[OF `0 \<le> Ifloat ?k` `Ifloat ?k \<le> 1`] by auto |
|
517 |
finally have "arctan (1 / (real k)) \<le> Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" . |
|
518 |
} note ub_arctan = this |
|
519 |
||
520 |
{ fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" by auto |
|
521 |
let ?k = "lapprox_rat prec 1 k" |
|
522 |
have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto |
|
523 |
have "1 / real k \<le> 1" using `1 < k` by auto |
|
524 |
||
525 |
have "\<And>n. 0 \<le> Ifloat ?k" using lapprox_rat_bottom[where x=1 and y=k, OF zero_le_one `0 < k`] by (auto simp add: `1 div k = 0`) |
|
526 |
have "\<And>n. Ifloat ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / real k \<le> 1`) |
|
527 |
||
528 |
have "Ifloat ?k \<le> 1 / real k" using lapprox_rat[where x=1 and y=k] by auto |
|
529 |
||
530 |
have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (Ifloat ?k)" |
|
531 |
using arctan_0_1_bounds[OF `0 \<le> Ifloat ?k` `Ifloat ?k \<le> 1`] by auto |
|
532 |
also have "\<dots> \<le> arctan (1 / real k)" using `Ifloat ?k \<le> 1 / real k` by (rule arctan_monotone') |
|
533 |
finally have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (1 / (real k))" . |
|
534 |
} note lb_arctan = this |
|
535 |
||
536 |
have "pi \<le> Ifloat (ub_pi n)" |
|
537 |
unfolding ub_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub unfolding Float_num |
|
538 |
using lb_arctan[of 239] ub_arctan[of 5] |
|
539 |
by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps) |
|
540 |
moreover |
|
541 |
have "Ifloat (lb_pi n) \<le> pi" |
|
542 |
unfolding lb_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub Float_num |
|
543 |
using lb_arctan[of 5] ub_arctan[of 239] |
|
544 |
by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps) |
|
545 |
ultimately show ?thesis by auto |
|
546 |
qed |
|
547 |
||
548 |
subsection "Compute arcus tangens in the entire domain" |
|
549 |
||
550 |
function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where |
|
551 |
"lb_arctan prec x = (let ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) ; |
|
552 |
lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) |
|
553 |
in (if x < 0 then - ub_arctan prec (-x) else |
|
554 |
if x \<le> Float 1 -1 then lb_horner x else |
|
555 |
if x \<le> Float 1 1 then Float 1 1 * lb_horner (float_divl prec x (1 + the (ub_sqrt prec (1 + x * x)))) |
|
556 |
else (let inv = float_divr prec 1 x |
|
557 |
in if inv > 1 then 0 |
|
558 |
else lb_pi prec * Float 1 -1 - ub_horner inv)))" |
|
559 |
||
560 |
| "ub_arctan prec x = (let lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) ; |
|
561 |
ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) |
|
562 |
in (if x < 0 then - lb_arctan prec (-x) else |
|
563 |
if x \<le> Float 1 -1 then ub_horner x else |
|
564 |
if x \<le> Float 1 1 then let y = float_divr prec x (1 + the (lb_sqrt prec (1 + x * x))) |
|
565 |
in if y > 1 then ub_pi prec * Float 1 -1 |
|
566 |
else Float 1 1 * ub_horner y |
|
567 |
else ub_pi prec * Float 1 -1 - lb_horner (float_divl prec 1 x)))" |
|
568 |
by pat_completeness auto |
|
569 |
termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 0 then 1 else 0))", auto simp add: less_float_def) |
|
570 |
||
571 |
declare ub_arctan_horner.simps[simp del] |
|
572 |
declare lb_arctan_horner.simps[simp del] |
|
573 |
||
574 |
lemma lb_arctan_bound': assumes "0 \<le> Ifloat x" |
|
575 |
shows "Ifloat (lb_arctan prec x) \<le> arctan (Ifloat x)" |
|
576 |
proof - |
|
577 |
have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> Ifloat x` by auto |
|
578 |
let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)" |
|
579 |
and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)" |
|
580 |
||
581 |
show ?thesis |
|
582 |
proof (cases "x \<le> Float 1 -1") |
|
583 |
case True hence "Ifloat x \<le> 1" unfolding le_float_def Float_num by auto |
|
584 |
show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True] |
|
585 |
using arctan_0_1_bounds[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1`] by auto |
|
586 |
next |
|
587 |
case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto |
|
588 |
let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)" |
|
589 |
let ?fR = "1 + the (ub_sqrt prec (1 + x * x))" |
|
590 |
let ?DIV = "float_divl prec x ?fR" |
|
591 |
||
592 |
have sqr_ge0: "0 \<le> 1 + Ifloat x * Ifloat x" using sum_power2_ge_zero[of 1 "Ifloat x", unfolded numeral_2_eq_2] by auto |
|
593 |
hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg) |
|
594 |
||
595 |
have "sqrt (Ifloat (1 + x * x)) \<le> Ifloat (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0) |
|
596 |
hence "?R \<le> Ifloat ?fR" by auto |
|
597 |
hence "0 < ?fR" and "0 < Ifloat ?fR" unfolding less_float_def using `0 < ?R` by auto |
|
598 |
||
599 |
have monotone: "Ifloat (float_divl prec x ?fR) \<le> Ifloat x / ?R" |
|
600 |
proof - |
|
601 |
have "Ifloat ?DIV \<le> Ifloat x / Ifloat ?fR" by (rule float_divl) |
|
602 |
also have "\<dots> \<le> Ifloat x / ?R" by (rule divide_left_mono[OF `?R \<le> Ifloat ?fR` `0 \<le> Ifloat x` mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 `?R \<le> Ifloat ?fR`] divisor_gt0]]) |
|
603 |
finally show ?thesis . |
|
604 |
qed |
|
605 |
||
606 |
show ?thesis |
|
607 |
proof (cases "x \<le> Float 1 1") |
|
608 |
case True |
|
609 |
||
610 |
have "Ifloat x \<le> sqrt (Ifloat (1 + x * x))" using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto |
|
611 |
also have "\<dots> \<le> Ifloat (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0) |
|
612 |
finally have "Ifloat x \<le> Ifloat ?fR" by auto |
|
613 |
moreover have "Ifloat ?DIV \<le> Ifloat x / Ifloat ?fR" by (rule float_divl) |
|
614 |
ultimately have "Ifloat ?DIV \<le> 1" unfolding divide_le_eq_1_pos[OF `0 < Ifloat ?fR`, symmetric] by auto |
|
615 |
||
616 |
have "0 \<le> Ifloat ?DIV" using float_divl_lower_bound[OF `0 \<le> x` `0 < ?fR`] unfolding le_float_def by auto |
|
617 |
||
618 |
have "Ifloat (Float 1 1 * ?lb_horner ?DIV) \<le> 2 * arctan (Ifloat (float_divl prec x ?fR))" unfolding Ifloat_mult[of "Float 1 1"] Float_num |
|
619 |
using arctan_0_1_bounds[OF `0 \<le> Ifloat ?DIV` `Ifloat ?DIV \<le> 1`] by auto |
|
620 |
also have "\<dots> \<le> 2 * arctan (Ifloat x / ?R)" |
|
621 |
using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono) |
|
622 |
also have "2 * arctan (Ifloat x / ?R) = arctan (Ifloat x)" using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 realpow_0 real_mult_1 . |
|
623 |
finally show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] . |
|
624 |
next |
|
625 |
case False |
|
626 |
hence "2 < Ifloat x" unfolding le_float_def Float_num by auto |
|
627 |
hence "1 \<le> Ifloat x" by auto |
|
628 |
||
629 |
let "?invx" = "float_divr prec 1 x" |
|
630 |
have "0 \<le> arctan (Ifloat x)" using arctan_monotone'[OF `0 \<le> Ifloat x`] using arctan_tan[of 0, unfolded tan_zero] by auto |
|
631 |
||
632 |
show ?thesis |
|
633 |
proof (cases "1 < ?invx") |
|
634 |
case True |
|
635 |
show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] if_P[OF True] |
|
636 |
using `0 \<le> arctan (Ifloat x)` by auto |
|
637 |
next |
|
638 |
case False |
|
639 |
hence "Ifloat ?invx \<le> 1" unfolding less_float_def by auto |
|
640 |
have "0 \<le> Ifloat ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> Ifloat x`) |
|
641 |
||
642 |
have "1 / Ifloat x \<noteq> 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto |
|
643 |
||
644 |
have "arctan (1 / Ifloat x) \<le> arctan (Ifloat ?invx)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divr) |
|
645 |
also have "\<dots> \<le> Ifloat (?ub_horner ?invx)" using arctan_0_1_bounds[OF `0 \<le> Ifloat ?invx` `Ifloat ?invx \<le> 1`] by auto |
|
646 |
finally have "pi / 2 - Ifloat (?ub_horner ?invx) \<le> arctan (Ifloat x)" |
|
647 |
using `0 \<le> arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \<noteq> 0`] |
|
648 |
unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto |
|
649 |
moreover |
|
650 |
have "Ifloat (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding Ifloat_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto |
|
651 |
ultimately |
|
652 |
show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF `\<not> x \<le> Float 1 1`] if_not_P[OF False] |
|
653 |
by auto |
|
654 |
qed |
|
655 |
qed |
|
656 |
qed |
|
657 |
qed |
|
658 |
||
659 |
lemma ub_arctan_bound': assumes "0 \<le> Ifloat x" |
|
660 |
shows "arctan (Ifloat x) \<le> Ifloat (ub_arctan prec x)" |
|
661 |
proof - |
|
662 |
have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> Ifloat x` by auto |
|
663 |
||
664 |
let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)" |
|
665 |
and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)" |
|
666 |
||
667 |
show ?thesis |
|
668 |
proof (cases "x \<le> Float 1 -1") |
|
669 |
case True hence "Ifloat x \<le> 1" unfolding le_float_def Float_num by auto |
|
670 |
show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True] |
|
671 |
using arctan_0_1_bounds[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1`] by auto |
|
672 |
next |
|
673 |
case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto |
|
674 |
let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)" |
|
675 |
let ?fR = "1 + the (lb_sqrt prec (1 + x * x))" |
|
676 |
let ?DIV = "float_divr prec x ?fR" |
|
677 |
||
678 |
have sqr_ge0: "0 \<le> 1 + Ifloat x * Ifloat x" using sum_power2_ge_zero[of 1 "Ifloat x", unfolded numeral_2_eq_2] by auto |
|
679 |
hence "0 \<le> Ifloat (1 + x*x)" by auto |
|
680 |
||
681 |
hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg) |
|
682 |
||
683 |
have "Ifloat (the (lb_sqrt prec (1 + x * x))) \<le> sqrt (Ifloat (1 + x * x))" by (rule lb_sqrt_upper_bound, auto simp add: sqr_ge0) |
|
684 |
hence "Ifloat ?fR \<le> ?R" by auto |
|
685 |
have "0 < Ifloat ?fR" unfolding Ifloat_add Ifloat_1 by (rule order_less_le_trans[OF zero_less_one], auto simp add: lb_sqrt_lower_bound[OF `0 \<le> Ifloat (1 + x*x)`]) |
|
686 |
||
687 |
have monotone: "Ifloat x / ?R \<le> Ifloat (float_divr prec x ?fR)" |
|
688 |
proof - |
|
689 |
from divide_left_mono[OF `Ifloat ?fR \<le> ?R` `0 \<le> Ifloat x` mult_pos_pos[OF divisor_gt0 `0 < Ifloat ?fR`]] |
|
690 |
have "Ifloat x / ?R \<le> Ifloat x / Ifloat ?fR" . |
|
691 |
also have "\<dots> \<le> Ifloat ?DIV" by (rule float_divr) |
|
692 |
finally show ?thesis . |
|
693 |
qed |
|
694 |
||
695 |
show ?thesis |
|
696 |
proof (cases "x \<le> Float 1 1") |
|
697 |
case True |
|
698 |
show ?thesis |
|
699 |
proof (cases "?DIV > 1") |
|
700 |
case True |
|
701 |
have "pi / 2 \<le> Ifloat (ub_pi prec * Float 1 -1)" unfolding Ifloat_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto |
|
702 |
from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le] |
|
703 |
show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF `x \<le> Float 1 1`] if_P[OF True] . |
|
704 |
next |
|
705 |
case False |
|
706 |
hence "Ifloat ?DIV \<le> 1" unfolding less_float_def by auto |
|
707 |
||
708 |
have "0 \<le> Ifloat x / ?R" using `0 \<le> Ifloat x` `0 < ?R` unfolding real_0_le_divide_iff by auto |
|
709 |
hence "0 \<le> Ifloat ?DIV" using monotone by (rule order_trans) |
|
710 |
||
711 |
have "arctan (Ifloat x) = 2 * arctan (Ifloat x / ?R)" using arctan_half unfolding numeral_2_eq_2 power_Suc2 realpow_0 real_mult_1 . |
|
712 |
also have "\<dots> \<le> 2 * arctan (Ifloat ?DIV)" |
|
713 |
using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono) |
|
714 |
also have "\<dots> \<le> Ifloat (Float 1 1 * ?ub_horner ?DIV)" unfolding Ifloat_mult[of "Float 1 1"] Float_num |
|
715 |
using arctan_0_1_bounds[OF `0 \<le> Ifloat ?DIV` `Ifloat ?DIV \<le> 1`] by auto |
|
716 |
finally show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF `x \<le> Float 1 1`] if_not_P[OF False] . |
|
717 |
qed |
|
718 |
next |
|
719 |
case False |
|
720 |
hence "2 < Ifloat x" unfolding le_float_def Float_num by auto |
|
721 |
hence "1 \<le> Ifloat x" by auto |
|
722 |
hence "0 < Ifloat x" by auto |
|
723 |
hence "0 < x" unfolding less_float_def by auto |
|
724 |
||
725 |
let "?invx" = "float_divl prec 1 x" |
|
726 |
have "0 \<le> arctan (Ifloat x)" using arctan_monotone'[OF `0 \<le> Ifloat x`] using arctan_tan[of 0, unfolded tan_zero] by auto |
|
727 |
||
728 |
have "Ifloat ?invx \<le> 1" unfolding less_float_def by (rule order_trans[OF float_divl], auto simp add: `1 \<le> Ifloat x` divide_le_eq_1_pos[OF `0 < Ifloat x`]) |
|
729 |
have "0 \<le> Ifloat ?invx" unfolding Ifloat_0[symmetric] by (rule float_divl_lower_bound[unfolded le_float_def], auto simp add: `0 < x`) |
|
730 |
||
731 |
have "1 / Ifloat x \<noteq> 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto |
|
732 |
||
733 |
have "Ifloat (?lb_horner ?invx) \<le> arctan (Ifloat ?invx)" using arctan_0_1_bounds[OF `0 \<le> Ifloat ?invx` `Ifloat ?invx \<le> 1`] by auto |
|
734 |
also have "\<dots> \<le> arctan (1 / Ifloat x)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divl) |
|
735 |
finally have "arctan (Ifloat x) \<le> pi / 2 - Ifloat (?lb_horner ?invx)" |
|
736 |
using `0 \<le> arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \<noteq> 0`] |
|
737 |
unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto |
|
738 |
moreover |
|
739 |
have "pi / 2 \<le> Ifloat (ub_pi prec * Float 1 -1)" unfolding Ifloat_mult Float_num times_divide_eq_right mult_1_right using pi_boundaries by auto |
|
740 |
ultimately |
|
741 |
show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF `\<not> x \<le> Float 1 1`] if_not_P[OF False] |
|
742 |
by auto |
|
743 |
qed |
|
744 |
qed |
|
745 |
qed |
|
746 |
||
747 |
lemma arctan_boundaries: |
|
748 |
"arctan (Ifloat x) \<in> {Ifloat (lb_arctan prec x) .. Ifloat (ub_arctan prec x)}" |
|
749 |
proof (cases "0 \<le> x") |
|
750 |
case True hence "0 \<le> Ifloat x" unfolding le_float_def by auto |
|
751 |
show ?thesis using ub_arctan_bound'[OF `0 \<le> Ifloat x`] lb_arctan_bound'[OF `0 \<le> Ifloat x`] unfolding atLeastAtMost_iff by auto |
|
752 |
next |
|
753 |
let ?mx = "-x" |
|
754 |
case False hence "x < 0" and "0 \<le> Ifloat ?mx" unfolding le_float_def less_float_def by auto |
|
755 |
hence bounds: "Ifloat (lb_arctan prec ?mx) \<le> arctan (Ifloat ?mx) \<and> arctan (Ifloat ?mx) \<le> Ifloat (ub_arctan prec ?mx)" |
|
756 |
using ub_arctan_bound'[OF `0 \<le> Ifloat ?mx`] lb_arctan_bound'[OF `0 \<le> Ifloat ?mx`] by auto |
|
757 |
show ?thesis unfolding Ifloat_minus arctan_minus lb_arctan.simps[where x=x] ub_arctan.simps[where x=x] Let_def if_P[OF `x < 0`] |
|
758 |
unfolding atLeastAtMost_iff using bounds[unfolded Ifloat_minus arctan_minus] by auto |
|
759 |
qed |
|
760 |
||
761 |
lemma bnds_arctan: "\<forall> x lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> arctan x \<and> arctan x \<le> Ifloat u" |
|
762 |
proof (rule allI, rule allI, rule allI, rule impI) |
|
763 |
fix x lx ux |
|
764 |
assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}" |
|
765 |
hence l: "lb_arctan prec lx = l " and u: "ub_arctan prec ux = u" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto |
|
766 |
||
767 |
{ from arctan_boundaries[of lx prec, unfolded l] |
|
768 |
have "Ifloat l \<le> arctan (Ifloat lx)" by (auto simp del: lb_arctan.simps) |
|
769 |
also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone') |
|
770 |
finally have "Ifloat l \<le> arctan x" . |
|
771 |
} moreover |
|
772 |
{ have "arctan x \<le> arctan (Ifloat ux)" using x by (auto intro: arctan_monotone') |
|
773 |
also have "\<dots> \<le> Ifloat u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps) |
|
774 |
finally have "arctan x \<le> Ifloat u" . |
|
775 |
} ultimately show "Ifloat l \<le> arctan x \<and> arctan x \<le> Ifloat u" .. |
|
776 |
qed |
|
777 |
||
778 |
section "Sinus and Cosinus" |
|
779 |
||
780 |
subsection "Compute the cosinus and sinus series" |
|
781 |
||
782 |
fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
|
783 |
and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
|
784 |
"ub_sin_cos_aux prec 0 i k x = 0" |
|
785 |
| "ub_sin_cos_aux prec (Suc n) i k x = |
|
786 |
(rapprox_rat prec 1 (int k)) - x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)" |
|
787 |
| "lb_sin_cos_aux prec 0 i k x = 0" |
|
788 |
| "lb_sin_cos_aux prec (Suc n) i k x = |
|
789 |
(lapprox_rat prec 1 (int k)) - x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)" |
|
790 |
||
791 |
lemma cos_aux: |
|
792 |
shows "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (Ifloat x)^(2 * i))" (is "?lb") |
|
793 |
and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (Ifloat x)^(2 * i)) \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub") |
|
794 |
proof - |
|
795 |
have "0 \<le> Ifloat (x * x)" unfolding Ifloat_mult by auto |
|
796 |
let "?f n" = "fact (2 * n)" |
|
797 |
||
798 |
{ fix n |
|
799 |
have F: "\<And>m. ((\<lambda>i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto) |
|
800 |
have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^ n) 1 * (((\<lambda>i. i + 2) ^ n) 1 + 1)" |
|
801 |
unfolding F by auto } note f_eq = this |
|
802 |
||
803 |
from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0, |
|
804 |
OF `0 \<le> Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps] |
|
805 |
show "?lb" and "?ub" by (auto simp add: power_mult power2_eq_square[of "Ifloat x"]) |
|
806 |
qed |
|
807 |
||
808 |
lemma cos_boundaries: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2" |
|
809 |
shows "cos (Ifloat x) \<in> {Ifloat (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. Ifloat (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}" |
|
810 |
proof (cases "Ifloat x = 0") |
|
811 |
case False hence "Ifloat x \<noteq> 0" by auto |
|
812 |
hence "0 < x" and "0 < Ifloat x" using `0 \<le> Ifloat x` unfolding less_float_def by auto |
|
813 |
have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0 |
|
814 |
using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto |
|
815 |
||
816 |
{ fix x n have "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x^(2 * i)) |
|
817 |
= (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum") |
|
818 |
proof - |
|
819 |
have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto |
|
820 |
also have "\<dots> = |
|
821 |
(\<Sum> j = 0 ..< n. -1 ^ ((2 * j) div 2) / (real (fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto |
|
822 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then -1 ^ (i div 2) / (real (fact i)) * x ^ i else 0)" |
|
823 |
unfolding sum_split_even_odd .. |
|
824 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then -1 ^ (i div 2) / (real (fact i)) else 0) * x ^ i)" |
|
825 |
by (rule setsum_cong2) auto |
|
826 |
finally show ?thesis by assumption |
|
827 |
qed } note morph_to_if_power = this |
|
828 |
||
829 |
||
830 |
{ fix n :: nat assume "0 < n" |
|
831 |
hence "0 < 2 * n" by auto |
|
832 |
obtain t where "0 < t" and "t < Ifloat x" and |
|
833 |
cos_eq: "cos (Ifloat x) = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * (Ifloat x) ^ i) |
|
834 |
+ (cos (t + 1/2 * real (2 * n) * pi) / real (fact (2*n))) * (Ifloat x)^(2*n)" |
|
835 |
(is "_ = ?SUM + ?rest / ?fact * ?pow") |
|
836 |
using Maclaurin_cos_expansion2[OF `0 < Ifloat x` `0 < 2 * n`] by auto |
|
837 |
||
838 |
have "cos t * -1^n = cos t * cos (real n * pi) + sin t * sin (real n * pi)" by auto |
|
839 |
also have "\<dots> = cos (t + real n * pi)" using cos_add by auto |
|
840 |
also have "\<dots> = ?rest" by auto |
|
841 |
finally have "cos t * -1^n = ?rest" . |
|
842 |
moreover |
|
843 |
have "t \<le> pi / 2" using `t < Ifloat x` and `Ifloat x \<le> pi / 2` by auto |
|
844 |
hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto |
|
845 |
ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto |
|
846 |
||
847 |
have "0 < ?fact" by auto |
|
848 |
have "0 < ?pow" using `0 < Ifloat x` by auto |
|
849 |
||
850 |
{ |
|
851 |
assume "even n" |
|
852 |
have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM" |
|
853 |
unfolding morph_to_if_power[symmetric] using cos_aux by auto |
|
854 |
also have "\<dots> \<le> cos (Ifloat x)" |
|
855 |
proof - |
|
856 |
from even[OF `even n`] `0 < ?fact` `0 < ?pow` |
|
857 |
have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le) |
|
858 |
thus ?thesis unfolding cos_eq by auto |
|
859 |
qed |
|
860 |
finally have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos (Ifloat x)" . |
|
861 |
} note lb = this |
|
862 |
||
863 |
{ |
|
864 |
assume "odd n" |
|
865 |
have "cos (Ifloat x) \<le> ?SUM" |
|
866 |
proof - |
|
867 |
from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`] |
|
868 |
have "0 \<le> (- ?rest) / ?fact * ?pow" |
|
869 |
by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le) |
|
870 |
thus ?thesis unfolding cos_eq by auto |
|
871 |
qed |
|
872 |
also have "\<dots> \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" |
|
873 |
unfolding morph_to_if_power[symmetric] using cos_aux by auto |
|
874 |
finally have "cos (Ifloat x) \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" . |
|
875 |
} note ub = this and lb |
|
876 |
} note ub = this(1) and lb = this(2) |
|
877 |
||
878 |
have "cos (Ifloat x) \<le> Ifloat (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] . |
|
879 |
moreover have "Ifloat (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos (Ifloat x)" |
|
880 |
proof (cases "0 < get_even n") |
|
881 |
case True show ?thesis using lb[OF True get_even] . |
|
882 |
next |
|
883 |
case False |
|
884 |
hence "get_even n = 0" by auto |
|
885 |
have "- (pi / 2) \<le> Ifloat x" by (rule order_trans[OF _ `0 < Ifloat x`[THEN less_imp_le]], auto) |
|
886 |
with `Ifloat x \<le> pi / 2` |
|
887 |
show ?thesis unfolding `get_even n = 0` lb_sin_cos_aux.simps Ifloat_minus Ifloat_0 using cos_ge_zero by auto |
|
888 |
qed |
|
889 |
ultimately show ?thesis by auto |
|
890 |
next |
|
891 |
case True |
|
892 |
show ?thesis |
|
893 |
proof (cases "n = 0") |
|
894 |
case True |
|
895 |
thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `Ifloat x = 0` lapprox_rat[where x="-1" and y=1] by auto |
|
896 |
next |
|
897 |
case False with not0_implies_Suc obtain m where "n = Suc m" by blast |
|
898 |
thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `Ifloat x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto) |
|
899 |
qed |
|
900 |
qed |
|
901 |
||
902 |
lemma sin_aux: assumes "0 \<le> Ifloat x" |
|
903 |
shows "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (Ifloat x)^(2 * i + 1))" (is "?lb") |
|
904 |
and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (Ifloat x)^(2 * i + 1)) \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub") |
|
905 |
proof - |
|
906 |
have "0 \<le> Ifloat (x * x)" unfolding Ifloat_mult by auto |
|
907 |
let "?f n" = "fact (2 * n + 1)" |
|
908 |
||
909 |
{ fix n |
|
910 |
have F: "\<And>m. ((\<lambda>i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto) |
|
911 |
have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^ n) 2 * (((\<lambda>i. i + 2) ^ n) 2 + 1)" |
|
912 |
unfolding F by auto } note f_eq = this |
|
913 |
||
914 |
from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0, |
|
915 |
OF `0 \<le> Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps] |
|
916 |
show "?lb" and "?ub" using `0 \<le> Ifloat x` unfolding Ifloat_mult |
|
917 |
unfolding power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric] |
|
918 |
unfolding real_mult_commute |
|
919 |
by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "Ifloat x"]) |
|
920 |
qed |
|
921 |
||
922 |
lemma sin_boundaries: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2" |
|
923 |
shows "sin (Ifloat x) \<in> {Ifloat (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. Ifloat (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}" |
|
924 |
proof (cases "Ifloat x = 0") |
|
925 |
case False hence "Ifloat x \<noteq> 0" by auto |
|
926 |
hence "0 < x" and "0 < Ifloat x" using `0 \<le> Ifloat x` unfolding less_float_def by auto |
|
927 |
have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0 |
|
928 |
using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto |
|
929 |
||
930 |
{ fix x n have "(\<Sum> j = 0 ..< n. -1 ^ (((2 * j + 1) - Suc 0) div 2) / (real (fact (2 * j + 1))) * x ^(2 * j + 1)) |
|
931 |
= (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * x ^ i)" (is "?SUM = _") |
|
932 |
proof - |
|
933 |
have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto |
|
934 |
have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" by auto |
|
935 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then 0 else -1 ^ ((i - Suc 0) div 2) / (real (fact i)) * x ^ i)" |
|
936 |
unfolding sum_split_even_odd .. |
|
937 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then 0 else -1 ^ ((i - Suc 0) div 2) / (real (fact i))) * x ^ i)" |
|
938 |
by (rule setsum_cong2) auto |
|
939 |
finally show ?thesis by assumption |
|
940 |
qed } note setsum_morph = this |
|
941 |
||
942 |
{ fix n :: nat assume "0 < n" |
|
943 |
hence "0 < 2 * n + 1" by auto |
|
944 |
obtain t where "0 < t" and "t < Ifloat x" and |
|
945 |
sin_eq: "sin (Ifloat x) = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i) |
|
946 |
+ (sin (t + 1/2 * real (2 * n + 1) * pi) / real (fact (2*n + 1))) * (Ifloat x)^(2*n + 1)" |
|
947 |
(is "_ = ?SUM + ?rest / ?fact * ?pow") |
|
948 |
using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < Ifloat x`] by auto |
|
949 |
||
950 |
have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add left_distrib right_distrib by auto |
|
951 |
moreover |
|
952 |
have "t \<le> pi / 2" using `t < Ifloat x` and `Ifloat x \<le> pi / 2` by auto |
|
953 |
hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto |
|
954 |
ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto |
|
955 |
||
956 |
have "0 < ?fact" by (rule real_of_nat_fact_gt_zero) |
|
957 |
have "0 < ?pow" using `0 < Ifloat x` by (rule zero_less_power) |
|
958 |
||
959 |
{ |
|
960 |
assume "even n" |
|
961 |
have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> |
|
962 |
(\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)" |
|
963 |
using sin_aux[OF `0 \<le> Ifloat x`] unfolding setsum_morph[symmetric] by auto |
|
964 |
also have "\<dots> \<le> ?SUM" by auto |
|
965 |
also have "\<dots> \<le> sin (Ifloat x)" |
|
966 |
proof - |
|
967 |
from even[OF `even n`] `0 < ?fact` `0 < ?pow` |
|
968 |
have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le) |
|
969 |
thus ?thesis unfolding sin_eq by auto |
|
970 |
qed |
|
971 |
finally have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin (Ifloat x)" . |
|
972 |
} note lb = this |
|
973 |
||
974 |
{ |
|
975 |
assume "odd n" |
|
976 |
have "sin (Ifloat x) \<le> ?SUM" |
|
977 |
proof - |
|
978 |
from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`] |
|
979 |
have "0 \<le> (- ?rest) / ?fact * ?pow" |
|
980 |
by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le) |
|
981 |
thus ?thesis unfolding sin_eq by auto |
|
982 |
qed |
|
983 |
also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)" |
|
984 |
by auto |
|
985 |
also have "\<dots> \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" |
|
986 |
using sin_aux[OF `0 \<le> Ifloat x`] unfolding setsum_morph[symmetric] by auto |
|
987 |
finally have "sin (Ifloat x) \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" . |
|
988 |
} note ub = this and lb |
|
989 |
} note ub = this(1) and lb = this(2) |
|
990 |
||
991 |
have "sin (Ifloat x) \<le> Ifloat (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] . |
|
992 |
moreover have "Ifloat (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin (Ifloat x)" |
|
993 |
proof (cases "0 < get_even n") |
|
994 |
case True show ?thesis using lb[OF True get_even] . |
|
995 |
next |
|
996 |
case False |
|
997 |
hence "get_even n = 0" by auto |
|
998 |
with `Ifloat x \<le> pi / 2` `0 \<le> Ifloat x` |
|
999 |
show ?thesis unfolding `get_even n = 0` ub_sin_cos_aux.simps Ifloat_minus Ifloat_0 using sin_ge_zero by auto |
|
1000 |
qed |
|
1001 |
ultimately show ?thesis by auto |
|
1002 |
next |
|
1003 |
case True |
|
1004 |
show ?thesis |
|
1005 |
proof (cases "n = 0") |
|
1006 |
case True |
|
1007 |
thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `Ifloat x = 0` lapprox_rat[where x="-1" and y=1] by auto |
|
1008 |
next |
|
1009 |
case False with not0_implies_Suc obtain m where "n = Suc m" by blast |
|
1010 |
thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `Ifloat x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto) |
|
1011 |
qed |
|
1012 |
qed |
|
1013 |
||
1014 |
subsection "Compute the cosinus in the entire domain" |
|
1015 |
||
1016 |
definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where |
|
1017 |
"lb_cos prec x = (let |
|
1018 |
horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ; |
|
1019 |
half = \<lambda> x. if x < 0 then - 1 else Float 1 1 * x * x - 1 |
|
1020 |
in if x < Float 1 -1 then horner x |
|
1021 |
else if x < 1 then half (horner (x * Float 1 -1)) |
|
1022 |
else half (half (horner (x * Float 1 -2))))" |
|
1023 |
||
1024 |
definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where |
|
1025 |
"ub_cos prec x = (let |
|
1026 |
horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ; |
|
1027 |
half = \<lambda> x. Float 1 1 * x * x - 1 |
|
1028 |
in if x < Float 1 -1 then horner x |
|
1029 |
else if x < 1 then half (horner (x * Float 1 -1)) |
|
1030 |
else half (half (horner (x * Float 1 -2))))" |
|
1031 |
||
1032 |
definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where |
|
1033 |
"bnds_cos prec lx ux = (let lpi = lb_pi prec |
|
1034 |
in if lx < -lpi \<or> ux > lpi then (Float -1 0, Float 1 0) |
|
1035 |
else if ux \<le> 0 then (lb_cos prec (-lx), ub_cos prec (-ux)) |
|
1036 |
else if 0 \<le> lx then (lb_cos prec ux, ub_cos prec lx) |
|
1037 |
else (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0))" |
|
1038 |
||
1039 |
lemma lb_cos: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi" |
|
1040 |
shows "cos (Ifloat x) \<in> {Ifloat (lb_cos prec x) .. Ifloat (ub_cos prec x)}" (is "?cos x \<in> { Ifloat (?lb x) .. Ifloat (?ub x) }") |
|
1041 |
proof - |
|
1042 |
{ fix x :: real |
|
1043 |
have "cos x = cos (x / 2 + x / 2)" by auto |
|
1044 |
also have "\<dots> = cos (x / 2) * cos (x / 2) + sin (x / 2) * sin (x / 2) - sin (x / 2) * sin (x / 2) + cos (x / 2) * cos (x / 2) - 1" |
|
1045 |
unfolding cos_add by auto |
|
1046 |
also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1" by algebra |
|
1047 |
finally have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" . |
|
1048 |
} note x_half = this[symmetric] |
|
1049 |
||
1050 |
have "\<not> x < 0" using `0 \<le> Ifloat x` unfolding less_float_def by auto |
|
1051 |
let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)" |
|
1052 |
let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)" |
|
1053 |
let "?ub_half x" = "Float 1 1 * x * x - 1" |
|
1054 |
let "?lb_half x" = "if x < 0 then - 1 else Float 1 1 * x * x - 1" |
|
1055 |
||
1056 |
show ?thesis |
|
1057 |
proof (cases "x < Float 1 -1") |
|
1058 |
case True hence "Ifloat x \<le> pi / 2" unfolding less_float_def using pi_ge_two by auto |
|
1059 |
show ?thesis unfolding lb_cos_def[where x=x] ub_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_P[OF `x < Float 1 -1`] Let_def |
|
1060 |
using cos_boundaries[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi / 2`] . |
|
1061 |
next |
|
1062 |
case False |
|
1063 |
||
1064 |
{ fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)" |
|
1065 |
assume "Ifloat y \<le> cos ?x2" and "-pi \<le> Ifloat x" and "Ifloat x \<le> pi" |
|
1066 |
hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto |
|
1067 |
hence "0 \<le> cos ?x2" by (rule cos_ge_zero) |
|
1068 |
||
1069 |
have "Ifloat (?lb_half y) \<le> cos (Ifloat x)" |
|
1070 |
proof (cases "y < 0") |
|
1071 |
case True show ?thesis using cos_ge_minus_one unfolding if_P[OF True] by auto |
|
1072 |
next |
|
1073 |
case False |
|
1074 |
hence "0 \<le> Ifloat y" unfolding less_float_def by auto |
|
1075 |
from mult_mono[OF `Ifloat y \<le> cos ?x2` `Ifloat y \<le> cos ?x2` `0 \<le> cos ?x2` this] |
|
1076 |
have "Ifloat y * Ifloat y \<le> cos ?x2 * cos ?x2" . |
|
1077 |
hence "2 * Ifloat y * Ifloat y \<le> 2 * cos ?x2 * cos ?x2" by auto |
|
1078 |
hence "2 * Ifloat y * Ifloat y - 1 \<le> 2 * cos (Ifloat x / 2) * cos (Ifloat x / 2) - 1" unfolding Float_num Ifloat_mult by auto |
|
1079 |
thus ?thesis unfolding if_not_P[OF False] x_half Float_num Ifloat_mult Ifloat_sub by auto |
|
1080 |
qed |
|
1081 |
} note lb_half = this |
|
1082 |
||
1083 |
{ fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)" |
|
1084 |
assume ub: "cos ?x2 \<le> Ifloat y" and "- pi \<le> Ifloat x" and "Ifloat x \<le> pi" |
|
1085 |
hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto |
|
1086 |
hence "0 \<le> cos ?x2" by (rule cos_ge_zero) |
|
1087 |
||
1088 |
have "cos (Ifloat x) \<le> Ifloat (?ub_half y)" |
|
1089 |
proof - |
|
1090 |
have "0 \<le> Ifloat y" using `0 \<le> cos ?x2` ub by (rule order_trans) |
|
1091 |
from mult_mono[OF ub ub this `0 \<le> cos ?x2`] |
|
1092 |
have "cos ?x2 * cos ?x2 \<le> Ifloat y * Ifloat y" . |
|
1093 |
hence "2 * cos ?x2 * cos ?x2 \<le> 2 * Ifloat y * Ifloat y" by auto |
|
1094 |
hence "2 * cos (Ifloat x / 2) * cos (Ifloat x / 2) - 1 \<le> 2 * Ifloat y * Ifloat y - 1" unfolding Float_num Ifloat_mult by auto |
|
1095 |
thus ?thesis unfolding x_half Ifloat_mult Float_num Ifloat_sub by auto |
|
1096 |
qed |
|
1097 |
} note ub_half = this |
|
1098 |
||
1099 |
let ?x2 = "x * Float 1 -1" |
|
1100 |
let ?x4 = "x * Float 1 -1 * Float 1 -1" |
|
1101 |
||
1102 |
have "-pi \<le> Ifloat x" using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] `0 \<le> Ifloat x` by (rule order_trans) |
|
1103 |
||
1104 |
show ?thesis |
|
1105 |
proof (cases "x < 1") |
|
1106 |
case True hence "Ifloat x \<le> 1" unfolding less_float_def by auto |
|
1107 |
have "0 \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi / 2" using pi_ge_two `0 \<le> Ifloat x` unfolding Ifloat_mult Float_num using assms by auto |
|
1108 |
from cos_boundaries[OF this] |
|
1109 |
have lb: "Ifloat (?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> Ifloat (?ub_horner ?x2)" by auto |
|
1110 |
||
1111 |
have "Ifloat (?lb x) \<le> ?cos x" |
|
1112 |
proof - |
|
1113 |
from lb_half[OF lb `-pi \<le> Ifloat x` `Ifloat x \<le> pi`] |
|
1114 |
show ?thesis unfolding lb_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto |
|
1115 |
qed |
|
1116 |
moreover have "?cos x \<le> Ifloat (?ub x)" |
|
1117 |
proof - |
|
1118 |
from ub_half[OF ub `-pi \<le> Ifloat x` `Ifloat x \<le> pi`] |
|
1119 |
show ?thesis unfolding ub_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto |
|
1120 |
qed |
|
1121 |
ultimately show ?thesis by auto |
|
1122 |
next |
|
1123 |
case False |
|
1124 |
have "0 \<le> Ifloat ?x4" and "Ifloat ?x4 \<le> pi / 2" using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` unfolding Ifloat_mult Float_num by auto |
|
1125 |
from cos_boundaries[OF this] |
|
1126 |
have lb: "Ifloat (?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> Ifloat (?ub_horner ?x4)" by auto |
|
1127 |
||
1128 |
have eq_4: "?x2 * Float 1 -1 = x * Float 1 -2" by (cases x, auto simp add: times_float.simps) |
|
1129 |
||
1130 |
have "Ifloat (?lb x) \<le> ?cos x" |
|
1131 |
proof - |
|
1132 |
have "-pi \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi" unfolding Ifloat_mult Float_num using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto |
|
1133 |
from lb_half[OF lb_half[OF lb this] `-pi \<le> Ifloat x` `Ifloat x \<le> pi`, unfolded eq_4] |
|
1134 |
show ?thesis unfolding lb_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x < Float 1 -1`] if_not_P[OF `\<not> x < 1`] Let_def . |
|
1135 |
qed |
|
1136 |
moreover have "?cos x \<le> Ifloat (?ub x)" |
|
1137 |
proof - |
|
1138 |
have "-pi \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi" unfolding Ifloat_mult Float_num using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto |
|
1139 |
from ub_half[OF ub_half[OF ub this] `-pi \<le> Ifloat x` `Ifloat x \<le> pi`, unfolded eq_4] |
|
1140 |
show ?thesis unfolding ub_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x < Float 1 -1`] if_not_P[OF `\<not> x < 1`] Let_def . |
|
1141 |
qed |
|
1142 |
ultimately show ?thesis by auto |
|
1143 |
qed |
|
1144 |
qed |
|
1145 |
qed |
|
1146 |
||
1147 |
lemma lb_cos_minus: assumes "-pi \<le> Ifloat x" and "Ifloat x \<le> 0" |
|
1148 |
shows "cos (Ifloat (-x)) \<in> {Ifloat (lb_cos prec (-x)) .. Ifloat (ub_cos prec (-x))}" |
|
1149 |
proof - |
|
1150 |
have "0 \<le> Ifloat (-x)" and "Ifloat (-x) \<le> pi" using `-pi \<le> Ifloat x` `Ifloat x \<le> 0` by auto |
|
1151 |
from lb_cos[OF this] show ?thesis . |
|
1152 |
qed |
|
1153 |
||
1154 |
lemma bnds_cos: "\<forall> x lx ux. (l, u) = bnds_cos prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> cos x \<and> cos x \<le> Ifloat u" |
|
1155 |
proof (rule allI, rule allI, rule allI, rule impI) |
|
1156 |
fix x lx ux |
|
1157 |
assume "(l, u) = bnds_cos prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux}" |
|
1158 |
hence bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto |
|
1159 |
||
1160 |
let ?lpi = "lb_pi prec" |
|
1161 |
have [intro!]: "Ifloat lx \<le> Ifloat ux" using x by auto |
|
1162 |
hence "lx \<le> ux" unfolding le_float_def . |
|
1163 |
||
1164 |
show "Ifloat l \<le> cos x \<and> cos x \<le> Ifloat u" |
|
1165 |
proof (cases "lx < -?lpi \<or> ux > ?lpi") |
|
1166 |
case True |
|
1167 |
show ?thesis using bnds unfolding bnds_cos_def if_P[OF True] Let_def using cos_le_one cos_ge_minus_one by auto |
|
1168 |
next |
|
1169 |
case False note not_out = this |
|
1170 |
hence lpi_lx: "- Ifloat ?lpi \<le> Ifloat lx" and lpi_ux: "Ifloat ux \<le> Ifloat ?lpi" unfolding le_float_def less_float_def by auto |
|
1171 |
||
1172 |
from pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1, THEN le_imp_neg_le] lpi_lx |
|
1173 |
have "- pi \<le> Ifloat lx" by (rule order_trans) |
|
1174 |
hence "- pi \<le> x" and "- pi \<le> Ifloat ux" and "x \<le> Ifloat ux" using x by auto |
|
1175 |
||
1176 |
from lpi_ux pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1] |
|
1177 |
have "Ifloat ux \<le> pi" by (rule order_trans) |
|
1178 |
hence "x \<le> pi" and "Ifloat lx \<le> pi" and "Ifloat lx \<le> x" using x by auto |
|
1179 |
||
1180 |
note lb_cos_minus_bottom = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct1] |
|
1181 |
note lb_cos_minus_top = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct2] |
|
1182 |
note lb_cos_bottom = lb_cos[unfolded atLeastAtMost_iff, THEN conjunct1] |
|
1183 |
note lb_cos_top = lb_cos[unfolded atLeastAtMost_iff, THEN conjunct2] |
|
1184 |
||
1185 |
show ?thesis |
|
1186 |
proof (cases "ux \<le> 0") |
|
1187 |
case True hence "Ifloat ux \<le> 0" unfolding le_float_def by auto |
|
1188 |
hence "x \<le> 0" and "Ifloat lx \<le> 0" using x by auto |
|
1189 |
||
1190 |
{ have "Ifloat (lb_cos prec (-lx)) \<le> cos (Ifloat (-lx))" using lb_cos_minus_bottom[OF `-pi \<le> Ifloat lx` `Ifloat lx \<le> 0`] . |
|
1191 |
also have "\<dots> \<le> cos x" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> 0`] . |
|
1192 |
finally have "Ifloat (lb_cos prec (-lx)) \<le> cos x" . } |
|
1193 |
moreover |
|
1194 |
{ have "cos x \<le> cos (Ifloat (-ux))" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> 0`] . |
|
1195 |
also have "\<dots> \<le> Ifloat (ub_cos prec (-ux))" using lb_cos_minus_top[OF `-pi \<le> Ifloat ux` `Ifloat ux \<le> 0`] . |
|
1196 |
finally have "cos x \<le> Ifloat (ub_cos prec (-ux))" . } |
|
1197 |
ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_P[OF True] by auto |
|
1198 |
next |
|
1199 |
case False note not_ux = this |
|
1200 |
||
1201 |
show ?thesis |
|
1202 |
proof (cases "0 \<le> lx") |
|
1203 |
case True hence "0 \<le> Ifloat lx" unfolding le_float_def by auto |
|
1204 |
hence "0 \<le> x" and "0 \<le> Ifloat ux" using x by auto |
|
1205 |
||
1206 |
{ have "Ifloat (lb_cos prec ux) \<le> cos (Ifloat ux)" using lb_cos_bottom[OF `0 \<le> Ifloat ux` `Ifloat ux \<le> pi`] . |
|
1207 |
also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> pi`] . |
|
1208 |
finally have "Ifloat (lb_cos prec ux) \<le> cos x" . } |
|
1209 |
moreover |
|
1210 |
{ have "cos x \<le> cos (Ifloat lx)" using cos_monotone_0_pi'[OF `0 \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> pi`] . |
|
1211 |
also have "\<dots> \<le> Ifloat (ub_cos prec lx)" using lb_cos_top[OF `0 \<le> Ifloat lx` `Ifloat lx \<le> pi`] . |
|
1212 |
finally have "cos x \<le> Ifloat (ub_cos prec lx)" . } |
|
1213 |
ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_not_P[OF not_ux] if_P[OF True] by auto |
|
1214 |
next |
|
1215 |
case False with not_ux |
|
1216 |
have "Ifloat lx \<le> 0" and "0 \<le> Ifloat ux" unfolding le_float_def by auto |
|
1217 |
||
1218 |
have "Ifloat (min (lb_cos prec (-lx)) (lb_cos prec ux)) \<le> cos x" |
|
1219 |
proof (cases "x \<le> 0") |
|
1220 |
case True |
|
1221 |
have "Ifloat (lb_cos prec (-lx)) \<le> cos (Ifloat (-lx))" using lb_cos_minus_bottom[OF `-pi \<le> Ifloat lx` `Ifloat lx \<le> 0`] . |
|
1222 |
also have "\<dots> \<le> cos x" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> 0`] . |
|
1223 |
finally show ?thesis unfolding Ifloat_min by auto |
|
1224 |
next |
|
1225 |
case False hence "0 \<le> x" by auto |
|
1226 |
have "Ifloat (lb_cos prec ux) \<le> cos (Ifloat ux)" using lb_cos_bottom[OF `0 \<le> Ifloat ux` `Ifloat ux \<le> pi`] . |
|
1227 |
also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> pi`] . |
|
1228 |
finally show ?thesis unfolding Ifloat_min by auto |
|
1229 |
qed |
|
1230 |
moreover have "cos x \<le> Ifloat (Float 1 0)" by auto |
|
1231 |
ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_not_P[OF not_ux] if_not_P[OF False] by auto |
|
1232 |
qed |
|
1233 |
qed |
|
1234 |
qed |
|
1235 |
qed |
|
1236 |
||
1237 |
subsection "Compute the sinus in the entire domain" |
|
1238 |
||
1239 |
function lb_sin :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_sin :: "nat \<Rightarrow> float \<Rightarrow> float" where |
|
1240 |
"lb_sin prec x = (let sqr_diff = \<lambda> x. if x > 1 then 0 else 1 - x * x |
|
1241 |
in if x < 0 then - ub_sin prec (- x) |
|
1242 |
else if x \<le> Float 1 -1 then x * lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 2 1 (x * x) |
|
1243 |
else the (lb_sqrt prec (sqr_diff (ub_cos prec x))))" | |
|
1244 |
||
1245 |
"ub_sin prec x = (let sqr_diff = \<lambda> x. if x < 0 then 1 else 1 - x * x |
|
1246 |
in if x < 0 then - lb_sin prec (- x) |
|
1247 |
else if x \<le> Float 1 -1 then x * ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 2 1 (x * x) |
|
1248 |
else the (ub_sqrt prec (sqr_diff (lb_cos prec x))))" |
|
1249 |
by pat_completeness auto |
|
1250 |
termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 0 then 1 else 0))", auto simp add: less_float_def) |
|
1251 |
||
1252 |
definition bnds_sin :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where |
|
1253 |
"bnds_sin prec lx ux = (let |
|
1254 |
lpi = lb_pi prec ; |
|
1255 |
half_pi = lpi * Float 1 -1 |
|
1256 |
in if lx \<le> - half_pi \<or> half_pi \<le> ux then (Float -1 0, Float 1 0) |
|
1257 |
else (lb_sin prec lx, ub_sin prec ux))" |
|
1258 |
||
1259 |
lemma lb_sin: assumes "- (pi / 2) \<le> Ifloat x" and "Ifloat x \<le> pi / 2" |
|
1260 |
shows "sin (Ifloat x) \<in> { Ifloat (lb_sin prec x) .. Ifloat (ub_sin prec x) }" (is "?sin x \<in> { ?lb x .. ?ub x}") |
|
1261 |
proof - |
|
1262 |
{ fix x :: float assume "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2" |
|
1263 |
hence "\<not> (x < 0)" and "- (pi / 2) \<le> Ifloat x" unfolding less_float_def using pi_ge_two by auto |
|
1264 |
||
1265 |
have "Ifloat x \<le> pi" using `Ifloat x \<le> pi / 2` using pi_ge_two by auto |
|
1266 |
||
1267 |
have "?sin x \<in> { ?lb x .. ?ub x}" |
|
1268 |
proof (cases "x \<le> Float 1 -1") |
|
1269 |
case True from sin_boundaries[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi / 2`] |
|
1270 |
show ?thesis unfolding lb_sin.simps[of prec x] ub_sin.simps[of prec x] if_not_P[OF `\<not> (x < 0)`] if_P[OF True] Let_def . |
|
1271 |
next |
|
1272 |
case False |
|
1273 |
have "0 \<le> cos (Ifloat x)" using cos_ge_zero[OF _ `Ifloat x \<le> pi /2`] `0 \<le> Ifloat x` pi_ge_two by auto |
|
1274 |
have "0 \<le> sin (Ifloat x)" using `0 \<le> Ifloat x` and `Ifloat x \<le> pi / 2` using sin_ge_zero by auto |
|
1275 |
||
1276 |
have "?sin x \<le> ?ub x" |
|
1277 |
proof (cases "lb_cos prec x < 0") |
|
1278 |
case True |
|
1279 |
have "?sin x \<le> 1" using sin_le_one . |
|
1280 |
also have "\<dots> \<le> Ifloat (the (ub_sqrt prec 1))" using ub_sqrt_lower_bound[where prec=prec and x=1] unfolding Ifloat_1 by auto |
|
1281 |
finally show ?thesis unfolding ub_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] Let_def . |
|
1282 |
next |
|
1283 |
case False hence "0 \<le> Ifloat (lb_cos prec x)" unfolding less_float_def by auto |
|
1284 |
||
1285 |
have "sin (Ifloat x) = sqrt (1 - cos (Ifloat x) ^ 2)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \<le> sin (Ifloat x)` by auto |
|
1286 |
also have "\<dots> \<le> sqrt (Ifloat (1 - lb_cos prec x * lb_cos prec x))" |
|
1287 |
proof (rule real_sqrt_le_mono) |
|
1288 |
have "Ifloat (lb_cos prec x * lb_cos prec x) \<le> cos (Ifloat x) ^ 2" unfolding numeral_2_eq_2 power_Suc2 realpow_0 Ifloat_mult |
|
1289 |
using `0 \<le> Ifloat (lb_cos prec x)` lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] `0 \<le> cos (Ifloat x)` by(auto intro!: mult_mono) |
|
1290 |
thus "1 - cos (Ifloat x) ^ 2 \<le> Ifloat (1 - lb_cos prec x * lb_cos prec x)" unfolding Ifloat_sub Ifloat_1 by auto |
|
1291 |
qed |
|
1292 |
also have "\<dots> \<le> Ifloat (the (ub_sqrt prec (1 - lb_cos prec x * lb_cos prec x)))" |
|
1293 |
proof (rule ub_sqrt_lower_bound) |
|
1294 |
have "Ifloat (lb_cos prec x) \<le> cos (Ifloat x)" using lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] by auto |
|
1295 |
from mult_mono[OF order_trans[OF this cos_le_one] order_trans[OF this cos_le_one]] |
|
1296 |
have "Ifloat (lb_cos prec x) * Ifloat (lb_cos prec x) \<le> 1" using `0 \<le> Ifloat (lb_cos prec x)` by auto |
|
1297 |
thus "0 \<le> Ifloat (1 - lb_cos prec x * lb_cos prec x)" by auto |
|
1298 |
qed |
|
1299 |
finally show ?thesis unfolding ub_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] Let_def . |
|
1300 |
qed |
|
1301 |
moreover |
|
1302 |
have "?lb x \<le> ?sin x" |
|
1303 |
proof (cases "1 < ub_cos prec x") |
|
1304 |
case True |
|
1305 |
show ?thesis unfolding lb_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] Let_def |
|
1306 |
by (rule order_trans[OF _ sin_ge_zero[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`]]) |
|
1307 |
(auto simp add: lb_sqrt_upper_bound[where prec=prec and x=0, unfolded Ifloat_0 real_sqrt_zero]) |
|
1308 |
next |
|
1309 |
case False hence "Ifloat (ub_cos prec x) \<le> 1" unfolding less_float_def by auto |
|
1310 |
have "0 \<le> Ifloat (ub_cos prec x)" using order_trans[OF `0 \<le> cos (Ifloat x)`] lb_cos `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto |
|
1311 |
||
1312 |
have "Ifloat (the (lb_sqrt prec (1 - ub_cos prec x * ub_cos prec x))) \<le> sqrt (Ifloat (1 - ub_cos prec x * ub_cos prec x))" |
|
1313 |
proof (rule lb_sqrt_upper_bound) |
|
1314 |
from mult_mono[OF `Ifloat (ub_cos prec x) \<le> 1` `Ifloat (ub_cos prec x) \<le> 1`] `0 \<le> Ifloat (ub_cos prec x)` |
|
1315 |
have "Ifloat (ub_cos prec x) * Ifloat (ub_cos prec x) \<le> 1" by auto |
|
1316 |
thus "0 \<le> Ifloat (1 - ub_cos prec x * ub_cos prec x)" by auto |
|
1317 |
qed |
|
1318 |
also have "\<dots> \<le> sqrt (1 - cos (Ifloat x) ^ 2)" |
|
1319 |
proof (rule real_sqrt_le_mono) |
|
1320 |
have "cos (Ifloat x) ^ 2 \<le> Ifloat (ub_cos prec x * ub_cos prec x)" unfolding numeral_2_eq_2 power_Suc2 realpow_0 Ifloat_mult |
|
1321 |
using `0 \<le> Ifloat (ub_cos prec x)` lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] `0 \<le> cos (Ifloat x)` by(auto intro!: mult_mono) |
|
1322 |
thus "Ifloat (1 - ub_cos prec x * ub_cos prec x) \<le> 1 - cos (Ifloat x) ^ 2" unfolding Ifloat_sub Ifloat_1 by auto |
|
1323 |
qed |
|
1324 |
also have "\<dots> = sin (Ifloat x)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \<le> sin (Ifloat x)` by auto |
|
1325 |
finally show ?thesis unfolding lb_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] Let_def . |
|
1326 |
qed |
|
1327 |
ultimately show ?thesis by auto |
|
1328 |
qed |
|
1329 |
} note for_pos = this |
|
1330 |
||
1331 |
show ?thesis |
|
1332 |
proof (cases "x < 0") |
|
1333 |
case True |
|
1334 |
hence "0 \<le> Ifloat (-x)" and "Ifloat (- x) \<le> pi / 2" using `-(pi/2) \<le> Ifloat x` unfolding less_float_def by auto |
|
1335 |
from for_pos[OF this] |
|
1336 |
show ?thesis unfolding Ifloat_minus sin_minus lb_sin.simps[of prec x] ub_sin.simps[of prec x] if_P[OF True] Let_def atLeastAtMost_iff by auto |
|
1337 |
next |
|
1338 |
case False hence "0 \<le> Ifloat x" unfolding less_float_def by auto |
|
1339 |
from for_pos[OF this `Ifloat x \<le> pi /2`] |
|
1340 |
show ?thesis . |
|
1341 |
qed |
|
1342 |
qed |
|
1343 |
||
1344 |
lemma bnds_sin: "\<forall> x lx ux. (l, u) = bnds_sin prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> sin x \<and> sin x \<le> Ifloat u" |
|
1345 |
proof (rule allI, rule allI, rule allI, rule impI) |
|
1346 |
fix x lx ux |
|
1347 |
assume "(l, u) = bnds_sin prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux}" |
|
1348 |
hence bnds: "(l, u) = bnds_sin prec lx ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto |
|
1349 |
show "Ifloat l \<le> sin x \<and> sin x \<le> Ifloat u" |
|
1350 |
proof (cases "lx \<le> - (lb_pi prec * Float 1 -1) \<or> lb_pi prec * Float 1 -1 \<le> ux") |
|
1351 |
case True show ?thesis using bnds unfolding bnds_sin_def if_P[OF True] Let_def by auto |
|
1352 |
next |
|
1353 |
case False |
|
1354 |
hence "- lb_pi prec * Float 1 -1 \<le> lx" and "ux \<le> lb_pi prec * Float 1 -1" unfolding le_float_def by auto |
|
1355 |
moreover have "Ifloat (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding Ifloat_mult using pi_boundaries by auto |
|
1356 |
ultimately have "- (pi / 2) \<le> Ifloat lx" and "Ifloat ux \<le> pi / 2" and "Ifloat lx \<le> Ifloat ux" unfolding le_float_def using x by auto |
|
1357 |
hence "- (pi / 2) \<le> Ifloat ux" and "Ifloat lx \<le> pi / 2" by auto |
|
1358 |
||
1359 |
have "- (pi / 2) \<le> x""x \<le> pi / 2" using `Ifloat ux \<le> pi / 2` `- (pi /2) \<le> Ifloat lx` x by auto |
|
1360 |
||
1361 |
{ have "Ifloat (lb_sin prec lx) \<le> sin (Ifloat lx)" using lb_sin[OF `- (pi / 2) \<le> Ifloat lx` `Ifloat lx \<le> pi / 2`] unfolding atLeastAtMost_iff by auto |
|
1362 |
also have "\<dots> \<le> sin x" using sin_monotone_2pi' `- (pi / 2) \<le> Ifloat lx` x `x \<le> pi / 2` by auto |
|
1363 |
finally have "Ifloat (lb_sin prec lx) \<le> sin x" . } |
|
1364 |
moreover |
|
1365 |
{ have "sin x \<le> sin (Ifloat ux)" using sin_monotone_2pi' `- (pi / 2) \<le> x` x `Ifloat ux \<le> pi / 2` by auto |
|
1366 |
also have "\<dots> \<le> Ifloat (ub_sin prec ux)" using lb_sin[OF `- (pi / 2) \<le> Ifloat ux` `Ifloat ux \<le> pi / 2`] unfolding atLeastAtMost_iff by auto |
|
1367 |
finally have "sin x \<le> Ifloat (ub_sin prec ux)" . } |
|
1368 |
ultimately |
|
1369 |
show ?thesis using bnds unfolding bnds_sin_def if_not_P[OF False] Let_def by auto |
|
1370 |
qed |
|
1371 |
qed |
|
1372 |
||
1373 |
section "Exponential function" |
|
1374 |
||
1375 |
subsection "Compute the series of the exponential function" |
|
1376 |
||
1377 |
fun ub_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" and lb_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
|
1378 |
"ub_exp_horner prec 0 i k x = 0" | |
|
1379 |
"ub_exp_horner prec (Suc n) i k x = rapprox_rat prec 1 (int k) + x * lb_exp_horner prec n (i + 1) (k * i) x" | |
|
1380 |
"lb_exp_horner prec 0 i k x = 0" | |
|
1381 |
"lb_exp_horner prec (Suc n) i k x = lapprox_rat prec 1 (int k) + x * ub_exp_horner prec n (i + 1) (k * i) x" |
|
1382 |
||
1383 |
lemma bnds_exp_horner: assumes "Ifloat x \<le> 0" |
|
1384 |
shows "exp (Ifloat x) \<in> { Ifloat (lb_exp_horner prec (get_even n) 1 1 x) .. Ifloat (ub_exp_horner prec (get_odd n) 1 1 x) }" |
|
1385 |
proof - |
|
1386 |
{ fix n |
|
1387 |
have F: "\<And> m. ((\<lambda>i. i + 1) ^ n) m = n + m" by (induct n, auto) |
|
1388 |
have "fact (Suc n) = fact n * ((\<lambda>i. i + 1) ^ n) 1" unfolding F by auto } note f_eq = this |
|
1389 |
||
1390 |
note bounds = horner_bounds_nonpos[where f="fact" and lb="lb_exp_horner prec" and ub="ub_exp_horner prec" and j'=0 and s=1, |
|
1391 |
OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps] |
|
1392 |
||
1393 |
{ have "Ifloat (lb_exp_horner prec (get_even n) 1 1 x) \<le> (\<Sum>j = 0..<get_even n. 1 / real (fact j) * Ifloat x ^ j)" |
|
1394 |
using bounds(1) by auto |
|
1395 |
also have "\<dots> \<le> exp (Ifloat x)" |
|
1396 |
proof - |
|
1397 |
obtain t where "\<bar>t\<bar> \<le> \<bar>Ifloat x\<bar>" and "exp (Ifloat x) = (\<Sum>m = 0..<get_even n. (Ifloat x) ^ m / real (fact m)) + exp t / real (fact (get_even n)) * (Ifloat x) ^ (get_even n)" |
|
1398 |
using Maclaurin_exp_le by blast |
|
1399 |
moreover have "0 \<le> exp t / real (fact (get_even n)) * (Ifloat x) ^ (get_even n)" |
|
1400 |
by (auto intro!: mult_nonneg_nonneg divide_nonneg_pos simp add: get_even zero_le_even_power exp_gt_zero) |
|
1401 |
ultimately show ?thesis |
|
1402 |
using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg) |
|
1403 |
qed |
|
1404 |
finally have "Ifloat (lb_exp_horner prec (get_even n) 1 1 x) \<le> exp (Ifloat x)" . |
|
1405 |
} moreover |
|
1406 |
{ |
|
1407 |
have x_less_zero: "Ifloat x ^ get_odd n \<le> 0" |
|
1408 |
proof (cases "Ifloat x = 0") |
|
1409 |
case True |
|
1410 |
have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto |
|
1411 |
thus ?thesis unfolding True power_0_left by auto |
|
1412 |
next |
|
1413 |
case False hence "Ifloat x < 0" using `Ifloat x \<le> 0` by auto |
|
1414 |
show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq get_odd `Ifloat x < 0`) |
|
1415 |
qed |
|
1416 |
||
1417 |
obtain t where "\<bar>t\<bar> \<le> \<bar>Ifloat x\<bar>" and "exp (Ifloat x) = (\<Sum>m = 0..<get_odd n. (Ifloat x) ^ m / real (fact m)) + exp t / real (fact (get_odd n)) * (Ifloat x) ^ (get_odd n)" |
|
1418 |
using Maclaurin_exp_le by blast |
|
1419 |
moreover have "exp t / real (fact (get_odd n)) * (Ifloat x) ^ (get_odd n) \<le> 0" |
|
1420 |
by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero exp_gt_zero) |
|
1421 |
ultimately have "exp (Ifloat x) \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * Ifloat x ^ j)" |
|
1422 |
using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg) |
|
1423 |
also have "\<dots> \<le> Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)" |
|
1424 |
using bounds(2) by auto |
|
1425 |
finally have "exp (Ifloat x) \<le> Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)" . |
|
1426 |
} ultimately show ?thesis by auto |
|
1427 |
qed |
|
1428 |
||
1429 |
subsection "Compute the exponential function on the entire domain" |
|
1430 |
||
1431 |
function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where |
|
1432 |
"lb_exp prec x = (if 0 < x then float_divl prec 1 (ub_exp prec (-x)) |
|
1433 |
else let |
|
1434 |
horner = (\<lambda> x. let y = lb_exp_horner prec (get_even (prec + 2)) 1 1 x in if y \<le> 0 then Float 1 -2 else y) |
|
1435 |
in if x < - 1 then (case floor_fl x of (Float m e) \<Rightarrow> (horner (float_divl prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e)) |
|
1436 |
else horner x)" | |
|
1437 |
"ub_exp prec x = (if 0 < x then float_divr prec 1 (lb_exp prec (-x)) |
|
1438 |
else if x < - 1 then (case floor_fl x of (Float m e) \<Rightarrow> |
|
1439 |
(ub_exp_horner prec (get_odd (prec + 2)) 1 1 (float_divr prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e)) |
|
1440 |
else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)" |
|
1441 |
by pat_completeness auto |
|
1442 |
termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if 0 < x then 1 else 0))", auto simp add: less_float_def) |
|
1443 |
||
1444 |
lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)" |
|
1445 |
proof - |
|
1446 |
have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto |
|
1447 |
||
1448 |
have "1 / 4 = Ifloat (Float 1 -2)" unfolding Float_num by auto |
|
1449 |
also have "\<dots> \<le> Ifloat (lb_exp_horner 1 (get_even 4) 1 1 (- 1))" |
|
1450 |
unfolding get_even_def eq4 |
|
1451 |
by (auto simp add: lapprox_posrat_def rapprox_posrat_def normfloat.simps) |
|
1452 |
also have "\<dots> \<le> exp (Ifloat (- 1))" using bnds_exp_horner[where x="- 1"] by auto |
|
1453 |
finally show ?thesis unfolding Ifloat_minus Ifloat_1 . |
|
1454 |
qed |
|
1455 |
||
1456 |
lemma lb_exp_pos: assumes "\<not> 0 < x" shows "0 < lb_exp prec x" |
|
1457 |
proof - |
|
1458 |
let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x" |
|
1459 |
let "?horner x" = "let y = ?lb_horner x in if y \<le> 0 then Float 1 -2 else y" |
|
1460 |
have pos_horner: "\<And> x. 0 < ?horner x" unfolding Let_def by (cases "?lb_horner x \<le> 0", auto simp add: le_float_def less_float_def) |
|
1461 |
moreover { fix x :: float fix num :: nat |
|
1462 |
have "0 < Ifloat (?horner x) ^ num" using `0 < ?horner x`[unfolded less_float_def Ifloat_0] by (rule zero_less_power) |
|
1463 |
also have "\<dots> = Ifloat ((?horner x) ^ num)" using float_power by auto |
|
1464 |
finally have "0 < Ifloat ((?horner x) ^ num)" . |
|
1465 |
} |
|
1466 |
ultimately show ?thesis |
|
1467 |
unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def by (cases "floor_fl x", cases "x < - 1", auto simp add: le_float_def less_float_def normfloat) |
|
1468 |
qed |
|
1469 |
||
1470 |
lemma exp_boundaries': assumes "x \<le> 0" |
|
1471 |
shows "exp (Ifloat x) \<in> { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}" |
|
1472 |
proof - |
|
1473 |
let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x" |
|
1474 |
let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x" |
|
1475 |
||
1476 |
have "Ifloat x \<le> 0" and "\<not> x > 0" using `x \<le> 0` unfolding le_float_def less_float_def by auto |
|
1477 |
show ?thesis |
|
1478 |
proof (cases "x < - 1") |
|
1479 |
case False hence "- 1 \<le> Ifloat x" unfolding less_float_def by auto |
|
1480 |
show ?thesis |
|
1481 |
proof (cases "?lb_exp_horner x \<le> 0") |
|
1482 |
from `\<not> x < - 1` have "- 1 \<le> Ifloat x" unfolding less_float_def by auto |
|
1483 |
hence "exp (- 1) \<le> exp (Ifloat x)" unfolding exp_le_cancel_iff . |
|
1484 |
from order_trans[OF exp_m1_ge_quarter this] |
|
1485 |
have "Ifloat (Float 1 -2) \<le> exp (Ifloat x)" unfolding Float_num . |
|
1486 |
moreover case True |
|
1487 |
ultimately show ?thesis using bnds_exp_horner `Ifloat x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by auto |
|
1488 |
next |
|
1489 |
case False thus ?thesis using bnds_exp_horner `Ifloat x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by (auto simp add: Let_def) |
|
1490 |
qed |
|
1491 |
next |
|
1492 |
case True |
|
1493 |
||
1494 |
obtain m e where Float_floor: "floor_fl x = Float m e" by (cases "floor_fl x", auto) |
|
1495 |
let ?num = "nat (- m) * 2 ^ nat e" |
|
1496 |
||
1497 |
have "Ifloat (floor_fl x) < - 1" using floor_fl `x < - 1` unfolding le_float_def less_float_def Ifloat_minus Ifloat_1 by (rule order_le_less_trans) |
|
1498 |
hence "Ifloat (floor_fl x) < 0" unfolding Float_floor Ifloat.simps using zero_less_pow2[of xe] by auto |
|
1499 |
hence "m < 0" |
|
1500 |
unfolding less_float_def Ifloat_0 Float_floor Ifloat.simps |
|
1501 |
unfolding pos_prod_lt[OF zero_less_pow2[of e], unfolded real_mult_commute] by auto |
|
1502 |
hence "1 \<le> - m" by auto |
|
1503 |
hence "0 < nat (- m)" by auto |
|
1504 |
moreover |
|
1505 |
have "0 \<le> e" using floor_pos_exp Float_floor[symmetric] by auto |
|
1506 |
hence "(0::nat) < 2 ^ nat e" by auto |
|
1507 |
ultimately have "0 < ?num" by auto |
|
1508 |
hence "real ?num \<noteq> 0" by auto |
|
1509 |
have e_nat: "int (nat e) = e" using `0 \<le> e` by auto |
|
1510 |
have num_eq: "real ?num = Ifloat (- floor_fl x)" using `0 < nat (- m)` |
|
1511 |
unfolding Float_floor Ifloat_minus Ifloat.simps real_of_nat_mult pow2_int[of "nat e", unfolded e_nat] realpow_real_of_nat[symmetric] by auto |
|
1512 |
have "0 < - floor_fl x" using `0 < ?num`[unfolded real_of_nat_less_iff[symmetric]] unfolding less_float_def num_eq[symmetric] Ifloat_0 real_of_nat_zero . |
|
1513 |
hence "Ifloat (floor_fl x) < 0" unfolding less_float_def by auto |
|
1514 |
||
1515 |
have "exp (Ifloat x) \<le> Ifloat (ub_exp prec x)" |
|
1516 |
proof - |
|
1517 |
have div_less_zero: "Ifloat (float_divr prec x (- floor_fl x)) \<le> 0" |
|
1518 |
using float_divr_nonpos_pos_upper_bound[OF `x \<le> 0` `0 < - floor_fl x`] unfolding le_float_def Ifloat_0 . |
|
1519 |
||
1520 |
have "exp (Ifloat x) = exp (real ?num * (Ifloat x / real ?num))" using `real ?num \<noteq> 0` by auto |
|
1521 |
also have "\<dots> = exp (Ifloat x / real ?num) ^ ?num" unfolding exp_real_of_nat_mult .. |
|
1522 |
also have "\<dots> \<le> exp (Ifloat (float_divr prec x (- floor_fl x))) ^ ?num" unfolding num_eq |
|
1523 |
by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto |
|
1524 |
also have "\<dots> \<le> Ifloat ((?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num)" unfolding float_power |
|
1525 |
by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto) |
|
1526 |
finally show ?thesis unfolding ub_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def . |
|
1527 |
qed |
|
1528 |
moreover |
|
1529 |
have "Ifloat (lb_exp prec x) \<le> exp (Ifloat x)" |
|
1530 |
proof - |
|
1531 |
let ?divl = "float_divl prec x (- Float m e)" |
|
1532 |
let ?horner = "?lb_exp_horner ?divl" |
|
1533 |
||
1534 |
show ?thesis |
|
1535 |
proof (cases "?horner \<le> 0") |
|
1536 |
case False hence "0 \<le> Ifloat ?horner" unfolding le_float_def by auto |
|
1537 |
||
1538 |
have div_less_zero: "Ifloat (float_divl prec x (- floor_fl x)) \<le> 0" |
|
1539 |
using `Ifloat (floor_fl x) < 0` `Ifloat x \<le> 0` by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg) |
|
1540 |
||
1541 |
have "Ifloat ((?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num) \<le> |
|
1542 |
exp (Ifloat (float_divl prec x (- floor_fl x))) ^ ?num" unfolding float_power |
|
1543 |
using `0 \<le> Ifloat ?horner`[unfolded Float_floor[symmetric]] bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1] by (auto intro!: power_mono) |
|
1544 |
also have "\<dots> \<le> exp (Ifloat x / real ?num) ^ ?num" unfolding num_eq |
|
1545 |
using float_divl by (auto intro!: power_mono simp del: Ifloat_minus) |
|
1546 |
also have "\<dots> = exp (real ?num * (Ifloat x / real ?num))" unfolding exp_real_of_nat_mult .. |
|
1547 |
also have "\<dots> = exp (Ifloat x)" using `real ?num \<noteq> 0` by auto |
|
1548 |
finally show ?thesis |
|
1549 |
unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def if_not_P[OF False] by auto |
|
1550 |
next |
|
1551 |
case True |
|
1552 |
have "Ifloat (floor_fl x) \<noteq> 0" and "Ifloat (floor_fl x) \<le> 0" using `Ifloat (floor_fl x) < 0` by auto |
|
1553 |
from divide_right_mono_neg[OF floor_fl[of x] `Ifloat (floor_fl x) \<le> 0`, unfolded divide_self[OF `Ifloat (floor_fl x) \<noteq> 0`]] |
|
1554 |
have "- 1 \<le> Ifloat x / Ifloat (- floor_fl x)" unfolding Ifloat_minus by auto |
|
1555 |
from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]] |
|
1556 |
have "Ifloat (Float 1 -2) \<le> exp (Ifloat x / Ifloat (- floor_fl x))" unfolding Float_num . |
|
1557 |
hence "Ifloat (Float 1 -2) ^ ?num \<le> exp (Ifloat x / Ifloat (- floor_fl x)) ^ ?num" |
|
1558 |
by (auto intro!: power_mono simp add: Float_num) |
|
1559 |
also have "\<dots> = exp (Ifloat x)" unfolding num_eq exp_real_of_nat_mult[symmetric] using `Ifloat (floor_fl x) \<noteq> 0` by auto |
|
1560 |
finally show ?thesis |
|
1561 |
unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def if_P[OF True] float_power . |
|
1562 |
qed |
|
1563 |
qed |
|
1564 |
ultimately show ?thesis by auto |
|
1565 |
qed |
|
1566 |
qed |
|
1567 |
||
1568 |
lemma exp_boundaries: "exp (Ifloat x) \<in> { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}" |
|
1569 |
proof - |
|
1570 |
show ?thesis |
|
1571 |
proof (cases "0 < x") |
|
1572 |
case False hence "x \<le> 0" unfolding less_float_def le_float_def by auto |
|
1573 |
from exp_boundaries'[OF this] show ?thesis . |
|
1574 |
next |
|
1575 |
case True hence "-x \<le> 0" unfolding less_float_def le_float_def by auto |
|
1576 |
||
1577 |
have "Ifloat (lb_exp prec x) \<le> exp (Ifloat x)" |
|
1578 |
proof - |
|
1579 |
from exp_boundaries'[OF `-x \<le> 0`] |
|
1580 |
have ub_exp: "exp (- Ifloat x) \<le> Ifloat (ub_exp prec (-x))" unfolding atLeastAtMost_iff Ifloat_minus by auto |
|
1581 |
||
1582 |
have "Ifloat (float_divl prec 1 (ub_exp prec (-x))) \<le> Ifloat 1 / Ifloat (ub_exp prec (-x))" using float_divl . |
|
1583 |
also have "Ifloat 1 / Ifloat (ub_exp prec (-x)) \<le> exp (Ifloat x)" |
|
1584 |
using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] exp_gt_zero, symmetric]] |
|
1585 |
unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide by auto |
|
1586 |
finally show ?thesis unfolding lb_exp.simps if_P[OF True] . |
|
1587 |
qed |
|
1588 |
moreover |
|
1589 |
have "exp (Ifloat x) \<le> Ifloat (ub_exp prec x)" |
|
1590 |
proof - |
|
1591 |
have "\<not> 0 < -x" using `0 < x` unfolding less_float_def by auto |
|
1592 |
||
1593 |
from exp_boundaries'[OF `-x \<le> 0`] |
|
1594 |
have lb_exp: "Ifloat (lb_exp prec (-x)) \<le> exp (- Ifloat x)" unfolding atLeastAtMost_iff Ifloat_minus by auto |
|
1595 |
||
1596 |
have "exp (Ifloat x) \<le> Ifloat 1 / Ifloat (lb_exp prec (-x))" |
|
1597 |
using lb_exp[unfolded inverse_le_iff_le[OF exp_gt_zero lb_exp_pos[OF `\<not> 0 < -x`, unfolded less_float_def Ifloat_0], symmetric]] |
|
1598 |
unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide Ifloat_1 by auto |
|
1599 |
also have "\<dots> \<le> Ifloat (float_divr prec 1 (lb_exp prec (-x)))" using float_divr . |
|
1600 |
finally show ?thesis unfolding ub_exp.simps if_P[OF True] . |
|
1601 |
qed |
|
1602 |
ultimately show ?thesis by auto |
|
1603 |
qed |
|
1604 |
qed |
|
1605 |
||
1606 |
lemma bnds_exp: "\<forall> x lx ux. (l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> exp x \<and> exp x \<le> Ifloat u" |
|
1607 |
proof (rule allI, rule allI, rule allI, rule impI) |
|
1608 |
fix x lx ux |
|
1609 |
assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}" |
|
1610 |
hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto |
|
1611 |
||
1612 |
{ from exp_boundaries[of lx prec, unfolded l] |
|
1613 |
have "Ifloat l \<le> exp (Ifloat lx)" by (auto simp del: lb_exp.simps) |
|
1614 |
also have "\<dots> \<le> exp x" using x by auto |
|
1615 |
finally have "Ifloat l \<le> exp x" . |
|
1616 |
} moreover |
|
1617 |
{ have "exp x \<le> exp (Ifloat ux)" using x by auto |
|
1618 |
also have "\<dots> \<le> Ifloat u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps) |
|
1619 |
finally have "exp x \<le> Ifloat u" . |
|
1620 |
} ultimately show "Ifloat l \<le> exp x \<and> exp x \<le> Ifloat u" .. |
|
1621 |
qed |
|
1622 |
||
1623 |
section "Logarithm" |
|
1624 |
||
1625 |
subsection "Compute the logarithm series" |
|
1626 |
||
1627 |
fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" |
|
1628 |
and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where |
|
1629 |
"ub_ln_horner prec 0 i x = 0" | |
|
1630 |
"ub_ln_horner prec (Suc n) i x = rapprox_rat prec 1 (int i) - x * lb_ln_horner prec n (Suc i) x" | |
|
1631 |
"lb_ln_horner prec 0 i x = 0" | |
|
1632 |
"lb_ln_horner prec (Suc n) i x = lapprox_rat prec 1 (int i) - x * ub_ln_horner prec n (Suc i) x" |
|
1633 |
||
1634 |
lemma ln_bounds: |
|
1635 |
assumes "0 \<le> x" and "x < 1" |
|
1636 |
shows "(\<Sum>i=0..<2*n. -1^i * (1 / real (i + 1)) * x^(Suc i)) \<le> ln (x + 1)" (is "?lb") |
|
1637 |
and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. -1^i * (1 / real (i + 1)) * x^(Suc i))" (is "?ub") |
|
1638 |
proof - |
|
1639 |
let "?a n" = "(1/real (n +1)) * x^(Suc n)" |
|
1640 |
||
1641 |
have ln_eq: "(\<Sum> i. -1^i * ?a i) = ln (x + 1)" |
|
1642 |
using ln_series[of "x + 1"] `0 \<le> x` `x < 1` by auto |
|
1643 |
||
1644 |
have "norm x < 1" using assms by auto |
|
1645 |
have "?a ----> 0" unfolding Suc_plus1[symmetric] inverse_eq_divide[symmetric] |
|
1646 |
using LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF `norm x < 1`]]] by auto |
|
1647 |
{ fix n have "0 \<le> ?a n" by (rule mult_nonneg_nonneg, auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`) } |
|
1648 |
{ fix n have "?a (Suc n) \<le> ?a n" unfolding inverse_eq_divide[symmetric] |
|
1649 |
proof (rule mult_mono) |
|
1650 |
show "0 \<le> x ^ Suc (Suc n)" by (auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`) |
|
1651 |
have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 real_mult_assoc[symmetric] |
|
1652 |
by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`) |
|
1653 |
thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto |
|
1654 |
qed auto } |
|
1655 |
from summable_Leibniz'(2,4)[OF `?a ----> 0` `\<And>n. 0 \<le> ?a n`, OF `\<And>n. ?a (Suc n) \<le> ?a n`, unfolded ln_eq] |
|
1656 |
show "?lb" and "?ub" by auto |
|
1657 |
qed |
|
1658 |
||
1659 |
lemma ln_float_bounds: |
|
1660 |
assumes "0 \<le> Ifloat x" and "Ifloat x < 1" |
|
1661 |
shows "Ifloat (x * lb_ln_horner prec (get_even n) 1 x) \<le> ln (Ifloat x + 1)" (is "?lb \<le> ?ln") |
|
1662 |
and "ln (Ifloat x + 1) \<le> Ifloat (x * ub_ln_horner prec (get_odd n) 1 x)" (is "?ln \<le> ?ub") |
|
1663 |
proof - |
|
1664 |
obtain ev where ev: "get_even n = 2 * ev" using get_even_double .. |
|
1665 |
obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double .. |
|
1666 |
||
1667 |
let "?s n" = "-1^n * (1 / real (1 + n)) * (Ifloat x)^(Suc n)" |
|
1668 |
||
1669 |
have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 real_mult_assoc[symmetric] Ifloat_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "Ifloat x"] ev |
|
1670 |
using horner_bounds(1)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*ev", |
|
1671 |
OF `0 \<le> Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> Ifloat x` |
|
1672 |
by (rule mult_right_mono) |
|
1673 |
also have "\<dots> \<le> ?ln" using ln_bounds(1)[OF `0 \<le> Ifloat x` `Ifloat x < 1`] by auto |
|
1674 |
finally show "?lb \<le> ?ln" . |
|
1675 |
||
1676 |
have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> Ifloat x` `Ifloat x < 1`] by auto |
|
1677 |
also have "\<dots> \<le> ?ub" unfolding power_Suc2 real_mult_assoc[symmetric] Ifloat_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "Ifloat x"] od |
|
1678 |
using horner_bounds(2)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*od+1", |
|
1679 |
OF `0 \<le> Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> Ifloat x` |
|
1680 |
by (rule mult_right_mono) |
|
1681 |
finally show "?ln \<le> ?ub" . |
|
1682 |
qed |
|
1683 |
||
1684 |
lemma ln_add: assumes "0 < x" and "0 < y" shows "ln (x + y) = ln x + ln (1 + y / x)" |
|
1685 |
proof - |
|
1686 |
have "x \<noteq> 0" using assms by auto |
|
1687 |
have "x + y = x * (1 + y / x)" unfolding right_distrib times_divide_eq_right nonzero_mult_divide_cancel_left[OF `x \<noteq> 0`] by auto |
|
1688 |
moreover |
|
1689 |
have "0 < y / x" using assms divide_pos_pos by auto |
|
1690 |
hence "0 < 1 + y / x" by auto |
|
1691 |
ultimately show ?thesis using ln_mult assms by auto |
|
1692 |
qed |
|
1693 |
||
1694 |
subsection "Compute the logarithm of 2" |
|
1695 |
||
1696 |
definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3 |
|
1697 |
in (Float 1 -1 * ub_ln_horner prec (get_odd prec) 1 (Float 1 -1)) + |
|
1698 |
(third * ub_ln_horner prec (get_odd prec) 1 third))" |
|
1699 |
definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3 |
|
1700 |
in (Float 1 -1 * lb_ln_horner prec (get_even prec) 1 (Float 1 -1)) + |
|
1701 |
(third * lb_ln_horner prec (get_even prec) 1 third))" |
|
1702 |
||
1703 |
lemma ub_ln2: "ln 2 \<le> Ifloat (ub_ln2 prec)" (is "?ub_ln2") |
|
1704 |
and lb_ln2: "Ifloat (lb_ln2 prec) \<le> ln 2" (is "?lb_ln2") |
|
1705 |
proof - |
|
1706 |
let ?uthird = "rapprox_rat (max prec 1) 1 3" |
|
1707 |
let ?lthird = "lapprox_rat prec 1 3" |
|
1708 |
||
1709 |
have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1)" |
|
1710 |
using ln_add[of "3 / 2" "1 / 2"] by auto |
|
1711 |
have lb3: "Ifloat ?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto |
|
1712 |
hence lb3_ub: "Ifloat ?lthird < 1" by auto |
|
1713 |
have lb3_lb: "0 \<le> Ifloat ?lthird" using lapprox_rat_bottom[of 1 3] by auto |
|
1714 |
have ub3: "1 / 3 \<le> Ifloat ?uthird" using rapprox_rat[of 1 3] by auto |
|
1715 |
hence ub3_lb: "0 \<le> Ifloat ?uthird" by auto |
|
1716 |
||
1717 |
have lb2: "0 \<le> Ifloat (Float 1 -1)" and ub2: "Ifloat (Float 1 -1) < 1" unfolding Float_num by auto |
|
1718 |
||
1719 |
have "0 \<le> (1::int)" and "0 < (3::int)" by auto |
|
1720 |
have ub3_ub: "Ifloat ?uthird < 1" unfolding rapprox_rat.simps(2)[OF `0 \<le> 1` `0 < 3`] |
|
1721 |
by (rule rapprox_posrat_less1, auto) |
|
1722 |
||
1723 |
have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto |
|
1724 |
have uthird_gt0: "0 < Ifloat ?uthird + 1" using ub3_lb by auto |
|
1725 |
have lthird_gt0: "0 < Ifloat ?lthird + 1" using lb3_lb by auto |
|
1726 |
||
1727 |
show ?ub_ln2 unfolding ub_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric] |
|
1728 |
proof (rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2]) |
|
1729 |
have "ln (1 / 3 + 1) \<le> ln (Ifloat ?uthird + 1)" unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto |
|
1730 |
also have "\<dots> \<le> Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" |
|
1731 |
using ln_float_bounds(2)[OF ub3_lb ub3_ub] . |
|
1732 |
finally show "ln (1 / 3 + 1) \<le> Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" . |
|
1733 |
qed |
|
1734 |
show ?lb_ln2 unfolding lb_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric] |
|
1735 |
proof (rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2]) |
|
1736 |
have "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (Ifloat ?lthird + 1)" |
|
1737 |
using ln_float_bounds(1)[OF lb3_lb lb3_ub] . |
|
1738 |
also have "\<dots> \<le> ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto |
|
1739 |
finally show "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (1 / 3 + 1)" . |
|
1740 |
qed |
|
1741 |
qed |
|
1742 |
||
1743 |
subsection "Compute the logarithm in the entire domain" |
|
1744 |
||
1745 |
function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where |
|
1746 |
"ub_ln prec x = (if x \<le> 0 then None |
|
1747 |
else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x))) |
|
1748 |
else let horner = \<lambda>x. (x - 1) * ub_ln_horner prec (get_odd prec) 1 (x - 1) in |
|
1749 |
if x < Float 1 1 then Some (horner x) |
|
1750 |
else let l = bitlen (mantissa x) - 1 in |
|
1751 |
Some (ub_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l))))" | |
|
1752 |
"lb_ln prec x = (if x \<le> 0 then None |
|
1753 |
else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x))) |
|
1754 |
else let horner = \<lambda>x. (x - 1) * lb_ln_horner prec (get_even prec) 1 (x - 1) in |
|
1755 |
if x < Float 1 1 then Some (horner x) |
|
1756 |
else let l = bitlen (mantissa x) - 1 in |
|
1757 |
Some (lb_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l))))" |
|
1758 |
by pat_completeness auto |
|
1759 |
||
1760 |
termination proof (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 1 then 1 else 0))", auto) |
|
1761 |
fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divl (max prec (Suc 0)) 1 x < 1" |
|
1762 |
hence "0 < x" and "0 < max prec (Suc 0)" unfolding less_float_def le_float_def by auto |
|
1763 |
from float_divl_pos_less1_bound[OF `0 < x` `x < 1` `0 < max prec (Suc 0)`] |
|
1764 |
show False using `float_divl (max prec (Suc 0)) 1 x < 1` unfolding less_float_def le_float_def by auto |
|
1765 |
next |
|
1766 |
fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divr prec 1 x < 1" |
|
1767 |
hence "0 < x" unfolding less_float_def le_float_def by auto |
|
1768 |
from float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`, of prec] |
|
1769 |
show False using `float_divr prec 1 x < 1` unfolding less_float_def le_float_def by auto |
|
1770 |
qed |
|
1771 |
||
1772 |
lemma ln_shifted_float: assumes "0 < m" shows "ln (Ifloat (Float m e)) = ln 2 * real (e + (bitlen m - 1)) + ln (Ifloat (Float m (- (bitlen m - 1))))" |
|
1773 |
proof - |
|
1774 |
let ?B = "2^nat (bitlen m - 1)" |
|
1775 |
have "0 < real m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0" using assms by auto |
|
1776 |
hence "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto |
|
1777 |
show ?thesis |
|
1778 |
proof (cases "0 \<le> e") |
|
1779 |
case True |
|
1780 |
show ?thesis unfolding normalized_float[OF `m \<noteq> 0`] |
|
1781 |
unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`] |
|
1782 |
unfolding Ifloat_ge0_exp[OF True] ln_mult[OF `0 < real m` `0 < 2^nat e`] |
|
1783 |
ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` True by auto |
|
1784 |
next |
|
1785 |
case False hence "0 < -e" by auto |
|
1786 |
hence pow_gt0: "(0::real) < 2^nat (-e)" by auto |
|
1787 |
hence inv_gt0: "(0::real) < inverse (2^nat (-e))" by auto |
|
1788 |
show ?thesis unfolding normalized_float[OF `m \<noteq> 0`] |
|
1789 |
unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`] |
|
1790 |
unfolding Ifloat_nge0_exp[OF False] ln_mult[OF `0 < real m` inv_gt0] ln_inverse[OF pow_gt0] |
|
1791 |
ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` False by auto |
|
1792 |
qed |
|
1793 |
qed |
|
1794 |
||
1795 |
lemma ub_ln_lb_ln_bounds': assumes "1 \<le> x" |
|
1796 |
shows "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x) \<and> ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))" |
|
1797 |
(is "?lb \<le> ?ln \<and> ?ln \<le> ?ub") |
|
1798 |
proof (cases "x < Float 1 1") |
|
1799 |
case True hence "Ifloat (x - 1) < 1" unfolding less_float_def Float_num by auto |
|
1800 |
have "\<not> x \<le> 0" and "\<not> x < 1" using `1 \<le> x` unfolding less_float_def le_float_def by auto |
|
1801 |
hence "0 \<le> Ifloat (x - 1)" using `1 \<le> x` unfolding less_float_def Float_num by auto |
|
1802 |
show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def |
|
1803 |
using ln_float_bounds[OF `0 \<le> Ifloat (x - 1)` `Ifloat (x - 1) < 1`] `\<not> x \<le> 0` `\<not> x < 1` True by auto |
|
1804 |
next |
|
1805 |
case False |
|
1806 |
have "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" using `1 \<le> x` unfolding less_float_def le_float_def by auto |
|
1807 |
show ?thesis |
|
1808 |
proof (cases x) |
|
1809 |
case (Float m e) |
|
1810 |
let ?s = "Float (e + (bitlen m - 1)) 0" |
|
1811 |
let ?x = "Float m (- (bitlen m - 1))" |
|
1812 |
||
1813 |
have "0 < m" and "m \<noteq> 0" using float_pos_m_pos `0 < x` Float by auto |
|
1814 |
||
1815 |
{ |
|
1816 |
have "Ifloat (lb_ln2 prec * ?s) \<le> ln 2 * real (e + (bitlen m - 1))" (is "?lb2 \<le> _") |
|
1817 |
unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 realpow_0 mult_1_right |
|
1818 |
using lb_ln2[of prec] |
|
1819 |
proof (rule mult_right_mono) |
|
1820 |
have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto |
|
1821 |
from float_gt1_scale[OF this] |
|
1822 |
show "0 \<le> real (e + (bitlen m - 1))" by auto |
|
1823 |
qed |
|
1824 |
moreover |
|
1825 |
from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]] |
|
1826 |
have "0 \<le> Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto |
|
1827 |
from ln_float_bounds(1)[OF this] |
|
1828 |
have "Ifloat ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln (Ifloat ?x)" (is "?lb_horner \<le> _") by auto |
|
1829 |
ultimately have "?lb2 + ?lb_horner \<le> ln (Ifloat x)" |
|
1830 |
unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto |
|
1831 |
} |
|
1832 |
moreover |
|
1833 |
{ |
|
1834 |
from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]] |
|
1835 |
have "0 \<le> Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto |
|
1836 |
from ln_float_bounds(2)[OF this] |
|
1837 |
have "ln (Ifloat ?x) \<le> Ifloat ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))" (is "_ \<le> ?ub_horner") by auto |
|
1838 |
moreover |
|
1839 |
have "ln 2 * real (e + (bitlen m - 1)) \<le> Ifloat (ub_ln2 prec * ?s)" (is "_ \<le> ?ub2") |
|
1840 |
unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 realpow_0 mult_1_right |
|
1841 |
using ub_ln2[of prec] |
|
1842 |
proof (rule mult_right_mono) |
|
1843 |
have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto |
|
1844 |
from float_gt1_scale[OF this] |
|
1845 |
show "0 \<le> real (e + (bitlen m - 1))" by auto |
|
1846 |
qed |
|
1847 |
ultimately have "ln (Ifloat x) \<le> ?ub2 + ?ub_horner" |
|
1848 |
unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto |
|
1849 |
} |
|
1850 |
ultimately show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps |
|
1851 |
unfolding if_not_P[OF `\<not> x \<le> 0`] if_not_P[OF `\<not> x < 1`] if_not_P[OF False] Let_def |
|
1852 |
unfolding scale.simps[of m e, unfolded Float[symmetric]] mantissa.simps[of m e, unfolded Float[symmetric]] Ifloat_add by auto |
|
1853 |
qed |
|
1854 |
qed |
|
1855 |
||
1856 |
lemma ub_ln_lb_ln_bounds: assumes "0 < x" |
|
1857 |
shows "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x) \<and> ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))" |
|
1858 |
(is "?lb \<le> ?ln \<and> ?ln \<le> ?ub") |
|
1859 |
proof (cases "x < 1") |
|
1860 |
case False hence "1 \<le> x" unfolding less_float_def le_float_def by auto |
|
1861 |
show ?thesis using ub_ln_lb_ln_bounds'[OF `1 \<le> x`] . |
|
1862 |
next |
|
1863 |
case True have "\<not> x \<le> 0" using `0 < x` unfolding less_float_def le_float_def by auto |
|
1864 |
||
1865 |
have "0 < Ifloat x" and "Ifloat x \<noteq> 0" using `0 < x` unfolding less_float_def by auto |
|
1866 |
hence A: "0 < 1 / Ifloat x" by auto |
|
1867 |
||
1868 |
{ |
|
1869 |
let ?divl = "float_divl (max prec 1) 1 x" |
|
1870 |
have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF `0 < x` `x < 1`] unfolding le_float_def less_float_def by auto |
|
1871 |
hence B: "0 < Ifloat ?divl" unfolding le_float_def by auto |
|
1872 |
||
1873 |
have "ln (Ifloat ?divl) \<le> ln (1 / Ifloat x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto |
|
1874 |
hence "ln (Ifloat x) \<le> - ln (Ifloat ?divl)" unfolding nonzero_inverse_eq_divide[OF `Ifloat x \<noteq> 0`, symmetric] ln_inverse[OF `0 < Ifloat x`] by auto |
|
1875 |
from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le] |
|
1876 |
have "?ln \<le> Ifloat (- the (lb_ln prec ?divl))" unfolding Ifloat_minus by (rule order_trans) |
|
1877 |
} moreover |
|
1878 |
{ |
|
1879 |
let ?divr = "float_divr prec 1 x" |
|
1880 |
have A': "1 \<le> ?divr" using float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`] unfolding le_float_def less_float_def by auto |
|
1881 |
hence B: "0 < Ifloat ?divr" unfolding le_float_def by auto |
|
1882 |
||
1883 |
have "ln (1 / Ifloat x) \<le> ln (Ifloat ?divr)" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto |
|
1884 |
hence "- ln (Ifloat ?divr) \<le> ln (Ifloat x)" unfolding nonzero_inverse_eq_divide[OF `Ifloat x \<noteq> 0`, symmetric] ln_inverse[OF `0 < Ifloat x`] by auto |
|
1885 |
from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this |
|
1886 |
have "Ifloat (- the (ub_ln prec ?divr)) \<le> ?ln" unfolding Ifloat_minus by (rule order_trans) |
|
1887 |
} |
|
1888 |
ultimately show ?thesis unfolding lb_ln.simps[where x=x] ub_ln.simps[where x=x] |
|
1889 |
unfolding if_not_P[OF `\<not> x \<le> 0`] if_P[OF True] by auto |
|
1890 |
qed |
|
1891 |
||
1892 |
lemma lb_ln: assumes "Some y = lb_ln prec x" |
|
1893 |
shows "Ifloat y \<le> ln (Ifloat x)" and "0 < Ifloat x" |
|
1894 |
proof - |
|
1895 |
have "0 < x" |
|
1896 |
proof (rule ccontr) |
|
1897 |
assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto |
|
1898 |
thus False using assms by auto |
|
1899 |
qed |
|
1900 |
thus "0 < Ifloat x" unfolding less_float_def by auto |
|
1901 |
have "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x)" using ub_ln_lb_ln_bounds[OF `0 < x`] .. |
|
1902 |
thus "Ifloat y \<le> ln (Ifloat x)" unfolding assms[symmetric] by auto |
|
1903 |
qed |
|
1904 |
||
1905 |
lemma ub_ln: assumes "Some y = ub_ln prec x" |
|
1906 |
shows "ln (Ifloat x) \<le> Ifloat y" and "0 < Ifloat x" |
|
1907 |
proof - |
|
1908 |
have "0 < x" |
|
1909 |
proof (rule ccontr) |
|
1910 |
assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto |
|
1911 |
thus False using assms by auto |
|
1912 |
qed |
|
1913 |
thus "0 < Ifloat x" unfolding less_float_def by auto |
|
1914 |
have "ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))" using ub_ln_lb_ln_bounds[OF `0 < x`] .. |
|
1915 |
thus "ln (Ifloat x) \<le> Ifloat y" unfolding assms[symmetric] by auto |
|
1916 |
qed |
|
1917 |
||
1918 |
lemma bnds_ln: "\<forall> x lx ux. (Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> ln x \<and> ln x \<le> Ifloat u" |
|
1919 |
proof (rule allI, rule allI, rule allI, rule impI) |
|
1920 |
fix x lx ux |
|
1921 |
assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}" |
|
1922 |
hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto |
|
1923 |
||
1924 |
have "ln (Ifloat ux) \<le> Ifloat u" and "0 < Ifloat ux" using ub_ln u by auto |
|
1925 |
have "Ifloat l \<le> ln (Ifloat lx)" and "0 < Ifloat lx" and "0 < x" using lb_ln[OF l] x by auto |
|
1926 |
||
1927 |
from ln_le_cancel_iff[OF `0 < Ifloat lx` `0 < x`] `Ifloat l \<le> ln (Ifloat lx)` |
|
1928 |
have "Ifloat l \<le> ln x" using x unfolding atLeastAtMost_iff by auto |
|
1929 |
moreover |
|
1930 |
from ln_le_cancel_iff[OF `0 < x` `0 < Ifloat ux`] `ln (Ifloat ux) \<le> Ifloat u` |
|
1931 |
have "ln x \<le> Ifloat u" using x unfolding atLeastAtMost_iff by auto |
|
1932 |
ultimately show "Ifloat l \<le> ln x \<and> ln x \<le> Ifloat u" .. |
|
1933 |
qed |
|
1934 |
||
1935 |
||
1936 |
section "Implement floatarith" |
|
1937 |
||
1938 |
subsection "Define syntax and semantics" |
|
1939 |
||
1940 |
datatype floatarith |
|
1941 |
= Add floatarith floatarith |
|
1942 |
| Minus floatarith |
|
1943 |
| Mult floatarith floatarith |
|
1944 |
| Inverse floatarith |
|
1945 |
| Sin floatarith |
|
1946 |
| Cos floatarith |
|
1947 |
| Arctan floatarith |
|
1948 |
| Abs floatarith |
|
1949 |
| Max floatarith floatarith |
|
1950 |
| Min floatarith floatarith |
|
1951 |
| Pi |
|
1952 |
| Sqrt floatarith |
|
1953 |
| Exp floatarith |
|
1954 |
| Ln floatarith |
|
1955 |
| Power floatarith nat |
|
1956 |
| Atom nat |
|
1957 |
| Num float |
|
1958 |
||
1959 |
fun Ifloatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real" |
|
1960 |
where |
|
1961 |
"Ifloatarith (Add a b) vs = (Ifloatarith a vs) + (Ifloatarith b vs)" | |
|
1962 |
"Ifloatarith (Minus a) vs = - (Ifloatarith a vs)" | |
|
1963 |
"Ifloatarith (Mult a b) vs = (Ifloatarith a vs) * (Ifloatarith b vs)" | |
|
1964 |
"Ifloatarith (Inverse a) vs = inverse (Ifloatarith a vs)" | |
|
1965 |
"Ifloatarith (Sin a) vs = sin (Ifloatarith a vs)" | |
|
1966 |
"Ifloatarith (Cos a) vs = cos (Ifloatarith a vs)" | |
|
1967 |
"Ifloatarith (Arctan a) vs = arctan (Ifloatarith a vs)" | |
|
1968 |
"Ifloatarith (Min a b) vs = min (Ifloatarith a vs) (Ifloatarith b vs)" | |
|
1969 |
"Ifloatarith (Max a b) vs = max (Ifloatarith a vs) (Ifloatarith b vs)" | |
|
1970 |
"Ifloatarith (Abs a) vs = abs (Ifloatarith a vs)" | |
|
1971 |
"Ifloatarith Pi vs = pi" | |
|
1972 |
"Ifloatarith (Sqrt a) vs = sqrt (Ifloatarith a vs)" | |
|
1973 |
"Ifloatarith (Exp a) vs = exp (Ifloatarith a vs)" | |
|
1974 |
"Ifloatarith (Ln a) vs = ln (Ifloatarith a vs)" | |
|
1975 |
"Ifloatarith (Power a n) vs = (Ifloatarith a vs)^n" | |
|
1976 |
"Ifloatarith (Num f) vs = Ifloat f" | |
|
1977 |
"Ifloatarith (Atom n) vs = vs ! n" |
|
1978 |
||
1979 |
subsection "Implement approximation function" |
|
1980 |
||
1981 |
fun lift_bin :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float option * float option)) \<Rightarrow> (float * float) option" where |
|
1982 |
"lift_bin (Some (l1, u1)) (Some (l2, u2)) f = (case (f l1 u1 l2 u2) of (Some l, Some u) \<Rightarrow> Some (l, u) |
|
1983 |
| t \<Rightarrow> None)" | |
|
1984 |
"lift_bin a b f = None" |
|
1985 |
||
1986 |
fun lift_bin' :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where |
|
1987 |
"lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" | |
|
1988 |
"lift_bin' a b f = None" |
|
1989 |
||
1990 |
fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where |
|
1991 |
"lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u) |
|
1992 |
| t \<Rightarrow> None)" | |
|
1993 |
"lift_un b f = None" |
|
1994 |
||
1995 |
fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where |
|
1996 |
"lift_un' (Some (l1, u1)) f = Some (f l1 u1)" | |
|
1997 |
"lift_un' b f = None" |
|
1998 |
||
1999 |
fun bounded_by :: "real list \<Rightarrow> (float * float) list \<Rightarrow> bool " where |
|
2000 |
bounded_by_Cons: "bounded_by (v#vs) ((l, u)#bs) = ((Ifloat l \<le> v \<and> v \<le> Ifloat u) \<and> bounded_by vs bs)" | |
|
2001 |
bounded_by_Nil: "bounded_by [] [] = True" | |
|
2002 |
"bounded_by _ _ = False" |
|
2003 |
||
2004 |
lemma bounded_by: assumes "bounded_by vs bs" and "i < length bs" |
|
2005 |
shows "Ifloat (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> Ifloat (snd (bs ! i))" |
|
2006 |
using `bounded_by vs bs` and `i < length bs` |
|
2007 |
proof (induct arbitrary: i rule: bounded_by.induct) |
|
2008 |
fix v :: real and vs :: "real list" and l u :: float and bs :: "(float * float) list" and i :: nat |
|
2009 |
assume hyp: "\<And>i. \<lbrakk>bounded_by vs bs; i < length bs\<rbrakk> \<Longrightarrow> Ifloat (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> Ifloat (snd (bs ! i))" |
|
2010 |
assume bounded: "bounded_by (v # vs) ((l, u) # bs)" and length: "i < length ((l, u) # bs)" |
|
2011 |
show "Ifloat (fst (((l, u) # bs) ! i)) \<le> (v # vs) ! i \<and> (v # vs) ! i \<le> Ifloat (snd (((l, u) # bs) ! i))" |
|
2012 |
proof (cases i) |
|
2013 |
case 0 |
|
2014 |
show ?thesis using bounded unfolding 0 nth_Cons_0 fst_conv snd_conv bounded_by.simps .. |
|
2015 |
next |
|
2016 |
case (Suc i) with length have "i < length bs" by auto |
|
2017 |
show ?thesis unfolding Suc nth_Cons_Suc bounded_by.simps |
|
2018 |
using hyp[OF bounded[unfolded bounded_by.simps, THEN conjunct2] `i < length bs`] . |
|
2019 |
qed |
|
2020 |
qed auto |
|
2021 |
||
2022 |
fun approx approx' :: "nat \<Rightarrow> floatarith \<Rightarrow> (float * float) list \<Rightarrow> (float * float) option" where |
|
2023 |
"approx' prec a bs = (case (approx prec a bs) of Some (l, u) \<Rightarrow> Some (round_down prec l, round_up prec u) | None \<Rightarrow> None)" | |
|
2024 |
"approx prec (Add a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (l1 + l2, u1 + u2))" | |
|
2025 |
"approx prec (Minus a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (-u, -l))" | |
|
2026 |
"approx prec (Mult a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) |
|
2027 |
(\<lambda> a1 a2 b1 b2. (float_nprt a1 * float_pprt b2 + float_nprt a2 * float_nprt b2 + float_pprt a1 * float_pprt b1 + float_pprt a2 * float_nprt b1, |
|
2028 |
float_pprt a2 * float_pprt b2 + float_pprt a1 * float_nprt b2 + float_nprt a2 * float_pprt b1 + float_nprt a1 * float_nprt b1))" | |
|
2029 |
"approx prec (Inverse a) bs = lift_un (approx' prec a bs) (\<lambda> l u. if (0 < l \<or> u < 0) then (Some (float_divl prec 1 u), Some (float_divr prec 1 l)) else (None, None))" | |
|
2030 |
"approx prec (Sin a) bs = lift_un' (approx' prec a bs) (bnds_sin prec)" | |
|
2031 |
"approx prec (Cos a) bs = lift_un' (approx' prec a bs) (bnds_cos prec)" | |
|
2032 |
"approx prec Pi bs = Some (lb_pi prec, ub_pi prec)" | |
|
2033 |
"approx prec (Min a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (min l1 l2, min u1 u2))" | |
|
2034 |
"approx prec (Max a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (max l1 l2, max u1 u2))" | |
|
2035 |
"approx prec (Abs a) bs = lift_un' (approx' prec a bs) (\<lambda>l u. (if l < 0 \<and> 0 < u then 0 else min \<bar>l\<bar> \<bar>u\<bar>, max \<bar>l\<bar> \<bar>u\<bar>))" | |
|
2036 |
"approx prec (Arctan a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_arctan prec l, ub_arctan prec u))" | |
|
2037 |
"approx prec (Sqrt a) bs = lift_un (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" | |
|
2038 |
"approx prec (Exp a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" | |
|
2039 |
"approx prec (Ln a) bs = lift_un (approx' prec a bs) (\<lambda> l u. (lb_ln prec l, ub_ln prec u))" | |
|
2040 |
"approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds n)" | |
|
2041 |
"approx prec (Num f) bs = Some (f, f)" | |
|
2042 |
"approx prec (Atom i) bs = (if i < length bs then Some (bs ! i) else None)" |
|
2043 |
||
2044 |
lemma lift_bin'_ex: |
|
2045 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f" |
|
2046 |
shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b" |
|
2047 |
proof (cases a) |
|
2048 |
case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps .. |
|
2049 |
thus ?thesis using lift_bin'_Some by auto |
|
2050 |
next |
|
2051 |
case (Some a') |
|
2052 |
show ?thesis |
|
2053 |
proof (cases b) |
|
2054 |
case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps .. |
|
2055 |
thus ?thesis using lift_bin'_Some by auto |
|
2056 |
next |
|
2057 |
case (Some b') |
|
2058 |
obtain la ua where a': "a' = (la, ua)" by (cases a', auto) |
|
2059 |
obtain lb ub where b': "b' = (lb, ub)" by (cases b', auto) |
|
2060 |
thus ?thesis unfolding `a = Some a'` `b = Some b'` a' b' by auto |
|
2061 |
qed |
|
2062 |
qed |
|
2063 |
||
2064 |
lemma lift_bin'_f: |
|
2065 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f" |
|
2066 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" and Pb: "\<And>l u. Some (l, u) = g b \<Longrightarrow> P l u b" |
|
2067 |
shows "\<exists> l1 u1 l2 u2. P l1 u1 a \<and> P l2 u2 b \<and> l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)" |
|
2068 |
proof - |
|
2069 |
obtain l1 u1 l2 u2 |
|
2070 |
where Sa: "Some (l1, u1) = g a" and Sb: "Some (l2, u2) = g b" using lift_bin'_ex[OF assms(1)] by auto |
|
2071 |
have lu: "(l, u) = f l1 u1 l2 u2" using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto |
|
2072 |
have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" unfolding lu[symmetric] by auto |
|
2073 |
thus ?thesis using Pa[OF Sa] Pb[OF Sb] by auto |
|
2074 |
qed |
|
2075 |
||
2076 |
lemma approx_approx': |
|
2077 |
assumes Pa: "\<And>l u. Some (l, u) = approx prec a vs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" |
|
2078 |
and approx': "Some (l, u) = approx' prec a vs" |
|
2079 |
shows "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" |
|
2080 |
proof - |
|
2081 |
obtain l' u' where S: "Some (l', u') = approx prec a vs" |
|
2082 |
using approx' unfolding approx'.simps by (cases "approx prec a vs", auto) |
|
2083 |
have l': "l = round_down prec l'" and u': "u = round_up prec u'" |
|
2084 |
using approx' unfolding approx'.simps S[symmetric] by auto |
|
2085 |
show ?thesis unfolding l' u' |
|
2086 |
using order_trans[OF Pa[OF S, THEN conjunct2] round_up[of u']] |
|
2087 |
using order_trans[OF round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto |
|
2088 |
qed |
|
2089 |
||
2090 |
lemma lift_bin': |
|
2091 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f" |
|
2092 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a") |
|
2093 |
and Pb: "\<And>l u. Some (l, u) = approx prec b bs \<Longrightarrow> Ifloat l \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u" |
|
2094 |
shows "\<exists> l1 u1 l2 u2. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> |
|
2095 |
(Ifloat l2 \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u2) \<and> |
|
2096 |
l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)" |
|
2097 |
proof - |
|
2098 |
{ fix l u assume "Some (l, u) = approx' prec a bs" |
|
2099 |
with approx_approx'[of prec a bs, OF _ this] Pa |
|
2100 |
have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this |
|
2101 |
{ fix l u assume "Some (l, u) = approx' prec b bs" |
|
2102 |
with approx_approx'[of prec b bs, OF _ this] Pb |
|
2103 |
have "Ifloat l \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u" by auto } note Pb = this |
|
2104 |
||
2105 |
from lift_bin'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb] |
|
2106 |
show ?thesis by auto |
|
2107 |
qed |
|
2108 |
||
2109 |
lemma lift_un'_ex: |
|
2110 |
assumes lift_un'_Some: "Some (l, u) = lift_un' a f" |
|
2111 |
shows "\<exists> l u. Some (l, u) = a" |
|
2112 |
proof (cases a) |
|
2113 |
case None hence "None = lift_un' a f" unfolding None lift_un'.simps .. |
|
2114 |
thus ?thesis using lift_un'_Some by auto |
|
2115 |
next |
|
2116 |
case (Some a') |
|
2117 |
obtain la ua where a': "a' = (la, ua)" by (cases a', auto) |
|
2118 |
thus ?thesis unfolding `a = Some a'` a' by auto |
|
2119 |
qed |
|
2120 |
||
2121 |
lemma lift_un'_f: |
|
2122 |
assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f" |
|
2123 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" |
|
2124 |
shows "\<exists> l1 u1. P l1 u1 a \<and> l = fst (f l1 u1) \<and> u = snd (f l1 u1)" |
|
2125 |
proof - |
|
2126 |
obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un'_ex[OF assms(1)] by auto |
|
2127 |
have lu: "(l, u) = f l1 u1" using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto |
|
2128 |
have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" unfolding lu[symmetric] by auto |
|
2129 |
thus ?thesis using Pa[OF Sa] by auto |
|
2130 |
qed |
|
2131 |
||
2132 |
lemma lift_un': |
|
2133 |
assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f" |
|
2134 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a") |
|
2135 |
shows "\<exists> l1 u1. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> |
|
2136 |
l = fst (f l1 u1) \<and> u = snd (f l1 u1)" |
|
2137 |
proof - |
|
2138 |
{ fix l u assume "Some (l, u) = approx' prec a bs" |
|
2139 |
with approx_approx'[of prec a bs, OF _ this] Pa |
|
2140 |
have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this |
|
2141 |
from lift_un'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa] |
|
2142 |
show ?thesis by auto |
|
2143 |
qed |
|
2144 |
||
2145 |
lemma lift_un'_bnds: |
|
2146 |
assumes bnds: "\<forall> x lx ux. (l, u) = f lx ux \<and> x \<in> { Ifloat lx .. Ifloat ux } \<longrightarrow> Ifloat l \<le> f' x \<and> f' x \<le> Ifloat u" |
|
2147 |
and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f" |
|
2148 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" |
|
2149 |
shows "Ifloat l \<le> f' (Ifloatarith a xs) \<and> f' (Ifloatarith a xs) \<le> Ifloat u" |
|
2150 |
proof - |
|
2151 |
from lift_un'[OF lift_un'_Some Pa] |
|
2152 |
obtain l1 u1 where "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" and "l = fst (f l1 u1)" and "u = snd (f l1 u1)" by blast |
|
2153 |
hence "(l, u) = f l1 u1" and "Ifloatarith a xs \<in> {Ifloat l1 .. Ifloat u1}" by auto |
|
2154 |
thus ?thesis using bnds by auto |
|
2155 |
qed |
|
2156 |
||
2157 |
lemma lift_un_ex: |
|
2158 |
assumes lift_un_Some: "Some (l, u) = lift_un a f" |
|
2159 |
shows "\<exists> l u. Some (l, u) = a" |
|
2160 |
proof (cases a) |
|
2161 |
case None hence "None = lift_un a f" unfolding None lift_un.simps .. |
|
2162 |
thus ?thesis using lift_un_Some by auto |
|
2163 |
next |
|
2164 |
case (Some a') |
|
2165 |
obtain la ua where a': "a' = (la, ua)" by (cases a', auto) |
|
2166 |
thus ?thesis unfolding `a = Some a'` a' by auto |
|
2167 |
qed |
|
2168 |
||
2169 |
lemma lift_un_f: |
|
2170 |
assumes lift_un_Some: "Some (l, u) = lift_un (g a) f" |
|
2171 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" |
|
2172 |
shows "\<exists> l1 u1. P l1 u1 a \<and> Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)" |
|
2173 |
proof - |
|
2174 |
obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un_ex[OF assms(1)] by auto |
|
2175 |
have "fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None" |
|
2176 |
proof (rule ccontr) |
|
2177 |
assume "\<not> (fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None)" |
|
2178 |
hence or: "fst (f l1 u1) = None \<or> snd (f l1 u1) = None" by auto |
|
2179 |
hence "lift_un (g a) f = None" |
|
2180 |
proof (cases "fst (f l1 u1) = None") |
|
2181 |
case True |
|
2182 |
then obtain b where b: "f l1 u1 = (None, b)" by (cases "f l1 u1", auto) |
|
2183 |
thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto |
|
2184 |
next |
|
2185 |
case False hence "snd (f l1 u1) = None" using or by auto |
|
2186 |
with False obtain b where b: "f l1 u1 = (Some b, None)" by (cases "f l1 u1", auto) |
|
2187 |
thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto |
|
2188 |
qed |
|
2189 |
thus False using lift_un_Some by auto |
|
2190 |
qed |
|
2191 |
then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" by (cases "f l1 u1", auto) |
|
2192 |
from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f] |
|
2193 |
have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" unfolding f by auto |
|
2194 |
thus ?thesis unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto |
|
2195 |
qed |
|
2196 |
||
2197 |
lemma lift_un: |
|
2198 |
assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f" |
|
2199 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a") |
|
2200 |
shows "\<exists> l1 u1. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> |
|
2201 |
Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)" |
|
2202 |
proof - |
|
2203 |
{ fix l u assume "Some (l, u) = approx' prec a bs" |
|
2204 |
with approx_approx'[of prec a bs, OF _ this] Pa |
|
2205 |
have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this |
|
2206 |
from lift_un_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa] |
|
2207 |
show ?thesis by auto |
|
2208 |
qed |
|
2209 |
||
2210 |
lemma lift_un_bnds: |
|
2211 |
assumes bnds: "\<forall> x lx ux. (Some l, Some u) = f lx ux \<and> x \<in> { Ifloat lx .. Ifloat ux } \<longrightarrow> Ifloat l \<le> f' x \<and> f' x \<le> Ifloat u" |
|
2212 |
and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f" |
|
2213 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" |
|
2214 |
shows "Ifloat l \<le> f' (Ifloatarith a xs) \<and> f' (Ifloatarith a xs) \<le> Ifloat u" |
|
2215 |
proof - |
|
2216 |
from lift_un[OF lift_un_Some Pa] |
|
2217 |
obtain l1 u1 where "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" and "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" by blast |
|
2218 |
hence "(Some l, Some u) = f l1 u1" and "Ifloatarith a xs \<in> {Ifloat l1 .. Ifloat u1}" by auto |
|
2219 |
thus ?thesis using bnds by auto |
|
2220 |
qed |
|
2221 |
||
2222 |
lemma approx: |
|
2223 |
assumes "bounded_by xs vs" |
|
2224 |
and "Some (l, u) = approx prec arith vs" (is "_ = ?g arith") |
|
2225 |
shows "Ifloat l \<le> Ifloatarith arith xs \<and> Ifloatarith arith xs \<le> Ifloat u" (is "?P l u arith") |
|
2226 |
using `Some (l, u) = approx prec arith vs` |
|
2227 |
proof (induct arith arbitrary: l u x) |
|
2228 |
case (Add a b) |
|
2229 |
from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps |
|
2230 |
obtain l1 u1 l2 u2 where "l = l1 + l2" and "u = u1 + u2" |
|
2231 |
"Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" |
|
2232 |
"Ifloat l2 \<le> Ifloatarith b xs" and "Ifloatarith b xs \<le> Ifloat u2" unfolding fst_conv snd_conv by blast |
|
2233 |
thus ?case unfolding Ifloatarith.simps by auto |
|
2234 |
next |
|
2235 |
case (Minus a) |
|
2236 |
from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps |
|
2237 |
obtain l1 u1 where "l = -u1" and "u = -l1" |
|
2238 |
"Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" unfolding fst_conv snd_conv by blast |
|
2239 |
thus ?case unfolding Ifloatarith.simps using Ifloat_minus by auto |
|
2240 |
next |
|
2241 |
case (Mult a b) |
|
2242 |
from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps |
|
2243 |
obtain l1 u1 l2 u2 |
|
2244 |
where l: "l = float_nprt l1 * float_pprt u2 + float_nprt u1 * float_nprt u2 + float_pprt l1 * float_pprt l2 + float_pprt u1 * float_nprt l2" |
|
2245 |
and u: "u = float_pprt u1 * float_pprt u2 + float_pprt l1 * float_nprt u2 + float_nprt u1 * float_pprt l2 + float_nprt l1 * float_nprt l2" |
|
2246 |
and "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" |
|
2247 |
and "Ifloat l2 \<le> Ifloatarith b xs" and "Ifloatarith b xs \<le> Ifloat u2" unfolding fst_conv snd_conv by blast |
|
2248 |
thus ?case unfolding Ifloatarith.simps l u Ifloat_add Ifloat_mult Ifloat_nprt Ifloat_pprt |
|
2249 |
using mult_le_prts mult_ge_prts by auto |
|
2250 |
next |
|
2251 |
case (Inverse a) |
|
2252 |
from lift_un[OF Inverse.prems[unfolded approx.simps], unfolded if_distrib[of fst] if_distrib[of snd] fst_conv snd_conv] Inverse.hyps |
|
2253 |
obtain l1 u1 where l': "Some l = (if 0 < l1 \<or> u1 < 0 then Some (float_divl prec 1 u1) else None)" |
|
2254 |
and u': "Some u = (if 0 < l1 \<or> u1 < 0 then Some (float_divr prec 1 l1) else None)" |
|
2255 |
and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1" by blast |
|
2256 |
have either: "0 < l1 \<or> u1 < 0" proof (rule ccontr) assume P: "\<not> (0 < l1 \<or> u1 < 0)" show False using l' unfolding if_not_P[OF P] by auto qed |
|
2257 |
moreover have l1_le_u1: "Ifloat l1 \<le> Ifloat u1" using l1 u1 by auto |
|
2258 |
ultimately have "Ifloat l1 \<noteq> 0" and "Ifloat u1 \<noteq> 0" unfolding less_float_def by auto |
|
2259 |
||
2260 |
have inv: "inverse (Ifloat u1) \<le> inverse (Ifloatarith a xs) |
|
2261 |
\<and> inverse (Ifloatarith a xs) \<le> inverse (Ifloat l1)" |
|
2262 |
proof (cases "0 < l1") |
|
2263 |
case True hence "0 < Ifloat u1" and "0 < Ifloat l1" "0 < Ifloatarith a xs" |
|
2264 |
unfolding less_float_def using l1_le_u1 l1 by auto |
|
2265 |
show ?thesis |
|
2266 |
unfolding inverse_le_iff_le[OF `0 < Ifloat u1` `0 < Ifloatarith a xs`] |
|
2267 |
inverse_le_iff_le[OF `0 < Ifloatarith a xs` `0 < Ifloat l1`] |
|
2268 |
using l1 u1 by auto |
|
2269 |
next |
|
2270 |
case False hence "u1 < 0" using either by blast |
|
2271 |
hence "Ifloat u1 < 0" and "Ifloat l1 < 0" "Ifloatarith a xs < 0" |
|
2272 |
unfolding less_float_def using l1_le_u1 u1 by auto |
|
2273 |
show ?thesis |
|
2274 |
unfolding inverse_le_iff_le_neg[OF `Ifloat u1 < 0` `Ifloatarith a xs < 0`] |
|
2275 |
inverse_le_iff_le_neg[OF `Ifloatarith a xs < 0` `Ifloat l1 < 0`] |
|
2276 |
using l1 u1 by auto |
|
2277 |
qed |
|
2278 |
||
2279 |
from l' have "l = float_divl prec 1 u1" by (cases "0 < l1 \<or> u1 < 0", auto) |
|
2280 |
hence "Ifloat l \<le> inverse (Ifloat u1)" unfolding nonzero_inverse_eq_divide[OF `Ifloat u1 \<noteq> 0`] using float_divl[of prec 1 u1] by auto |
|
2281 |
also have "\<dots> \<le> inverse (Ifloatarith a xs)" using inv by auto |
|
2282 |
finally have "Ifloat l \<le> inverse (Ifloatarith a xs)" . |
|
2283 |
moreover |
|
2284 |
from u' have "u = float_divr prec 1 l1" by (cases "0 < l1 \<or> u1 < 0", auto) |
|
2285 |
hence "inverse (Ifloat l1) \<le> Ifloat u" unfolding nonzero_inverse_eq_divide[OF `Ifloat l1 \<noteq> 0`] using float_divr[of 1 l1 prec] by auto |
|
2286 |
hence "inverse (Ifloatarith a xs) \<le> Ifloat u" by (rule order_trans[OF inv[THEN conjunct2]]) |
|
2287 |
ultimately show ?case unfolding Ifloatarith.simps using l1 u1 by auto |
|
2288 |
next |
|
2289 |
case (Abs x) |
|
2290 |
from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps |
|
2291 |
obtain l1 u1 where l': "l = (if l1 < 0 \<and> 0 < u1 then 0 else min \<bar>l1\<bar> \<bar>u1\<bar>)" and u': "u = max \<bar>l1\<bar> \<bar>u1\<bar>" |
|
2292 |
and l1: "Ifloat l1 \<le> Ifloatarith x xs" and u1: "Ifloatarith x xs \<le> Ifloat u1" by blast |
|
2293 |
thus ?case unfolding l' u' by (cases "l1 < 0 \<and> 0 < u1", auto simp add: Ifloat_min Ifloat_max Ifloat_abs less_float_def) |
|
2294 |
next |
|
2295 |
case (Min a b) |
|
2296 |
from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps |
|
2297 |
obtain l1 u1 l2 u2 where l': "l = min l1 l2" and u': "u = min u1 u2" |
|
2298 |
and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1" |
|
2299 |
and l1: "Ifloat l2 \<le> Ifloatarith b xs" and u1: "Ifloatarith b xs \<le> Ifloat u2" by blast |
|
2300 |
thus ?case unfolding l' u' by (auto simp add: Ifloat_min) |
|
2301 |
next |
|
2302 |
case (Max a b) |
|
2303 |
from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps |
|
2304 |
obtain l1 u1 l2 u2 where l': "l = max l1 l2" and u': "u = max u1 u2" |
|
2305 |
and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1" |
|
2306 |
and l1: "Ifloat l2 \<le> Ifloatarith b xs" and u1: "Ifloatarith b xs \<le> Ifloat u2" by blast |
|
2307 |
thus ?case unfolding l' u' by (auto simp add: Ifloat_max) |
|
2308 |
next case (Sin a) with lift_un'_bnds[OF bnds_sin] show ?case by auto |
|
2309 |
next case (Cos a) with lift_un'_bnds[OF bnds_cos] show ?case by auto |
|
2310 |
next case (Arctan a) with lift_un'_bnds[OF bnds_arctan] show ?case by auto |
|
2311 |
next case Pi with pi_boundaries show ?case by auto |
|
2312 |
next case (Sqrt a) with lift_un_bnds[OF bnds_sqrt] show ?case by auto |
|
2313 |
next case (Exp a) with lift_un'_bnds[OF bnds_exp] show ?case by auto |
|
2314 |
next case (Ln a) with lift_un_bnds[OF bnds_ln] show ?case by auto |
|
2315 |
next case (Power a n) with lift_un'_bnds[OF bnds_power] show ?case by auto |
|
2316 |
next case (Num f) thus ?case by auto |
|
2317 |
next |
|
2318 |
case (Atom n) |
|
2319 |
show ?case |
|
2320 |
proof (cases "n < length vs") |
|
2321 |
case True |
|
2322 |
with Atom have "vs ! n = (l, u)" by auto |
|
2323 |
thus ?thesis using bounded_by[OF assms(1) True] by auto |
|
2324 |
next |
|
2325 |
case False thus ?thesis using Atom by auto |
|
2326 |
qed |
|
2327 |
qed |
|
2328 |
||
2329 |
datatype ApproxEq = Less floatarith floatarith |
|
2330 |
| LessEqual floatarith floatarith |
|
2331 |
||
2332 |
fun uneq :: "ApproxEq \<Rightarrow> real list \<Rightarrow> bool" where |
|
2333 |
"uneq (Less a b) vs = (Ifloatarith a vs < Ifloatarith b vs)" | |
|
2334 |
"uneq (LessEqual a b) vs = (Ifloatarith a vs \<le> Ifloatarith b vs)" |
|
2335 |
||
2336 |
fun uneq' :: "nat \<Rightarrow> ApproxEq \<Rightarrow> (float * float) list \<Rightarrow> bool" where |
|
2337 |
"uneq' prec (Less a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u < l' | _ \<Rightarrow> False)" | |
|
2338 |
"uneq' prec (LessEqual a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u \<le> l' | _ \<Rightarrow> False)" |
|
2339 |
||
2340 |
lemma uneq_approx: fixes m :: nat assumes "bounded_by vs bs" and "uneq' prec eq bs" |
|
2341 |
shows "uneq eq vs" |
|
2342 |
proof (cases eq) |
|
2343 |
case (Less a b) |
|
2344 |
show ?thesis |
|
2345 |
proof (cases "\<exists> u l u' l'. approx prec a bs = Some (l, u) \<and> |
|
2346 |
approx prec b bs = Some (l', u')") |
|
2347 |
case True |
|
2348 |
then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)" |
|
2349 |
and b_approx: "approx prec b bs = Some (l', u') " by auto |
|
2350 |
with `uneq' prec eq bs` have "Ifloat u < Ifloat l'" |
|
2351 |
unfolding Less uneq'.simps less_float_def by auto |
|
2352 |
moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs` |
|
2353 |
have "Ifloatarith a vs \<le> Ifloat u" and "Ifloat l' \<le> Ifloatarith b vs" |
|
2354 |
using approx by auto |
|
2355 |
ultimately show ?thesis unfolding uneq.simps Less by auto |
|
2356 |
next |
|
2357 |
case False |
|
2358 |
hence "approx prec a bs = None \<or> approx prec b bs = None" |
|
2359 |
unfolding not_Some_eq[symmetric] by auto |
|
2360 |
hence "\<not> uneq' prec eq bs" unfolding Less uneq'.simps |
|
2361 |
by (cases "approx prec a bs = None", auto) |
|
2362 |
thus ?thesis using assms by auto |
|
2363 |
qed |
|
2364 |
next |
|
2365 |
case (LessEqual a b) |
|
2366 |
show ?thesis |
|
2367 |
proof (cases "\<exists> u l u' l'. approx prec a bs = Some (l, u) \<and> |
|
2368 |
approx prec b bs = Some (l', u')") |
|
2369 |
case True |
|
2370 |
then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)" |
|
2371 |
and b_approx: "approx prec b bs = Some (l', u') " by auto |
|
2372 |
with `uneq' prec eq bs` have "Ifloat u \<le> Ifloat l'" |
|
2373 |
unfolding LessEqual uneq'.simps le_float_def by auto |
|
2374 |
moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs` |
|
2375 |
have "Ifloatarith a vs \<le> Ifloat u" and "Ifloat l' \<le> Ifloatarith b vs" |
|
2376 |
using approx by auto |
|
2377 |
ultimately show ?thesis unfolding uneq.simps LessEqual by auto |
|
2378 |
next |
|
2379 |
case False |
|
2380 |
hence "approx prec a bs = None \<or> approx prec b bs = None" |
|
2381 |
unfolding not_Some_eq[symmetric] by auto |
|
2382 |
hence "\<not> uneq' prec eq bs" unfolding LessEqual uneq'.simps |
|
2383 |
by (cases "approx prec a bs = None", auto) |
|
2384 |
thus ?thesis using assms by auto |
|
2385 |
qed |
|
2386 |
qed |
|
2387 |
||
2388 |
lemma Ifloatarith_divide: "Ifloatarith (Mult a (Inverse b)) vs = (Ifloatarith a vs) / (Ifloatarith b vs)" |
|
2389 |
unfolding real_divide_def Ifloatarith.simps .. |
|
2390 |
||
2391 |
lemma Ifloatarith_diff: "Ifloatarith (Add a (Minus b)) vs = (Ifloatarith a vs) - (Ifloatarith b vs)" |
|
2392 |
unfolding real_diff_def Ifloatarith.simps .. |
|
2393 |
||
2394 |
lemma Ifloatarith_tan: "Ifloatarith (Mult (Sin a) (Inverse (Cos a))) vs = tan (Ifloatarith a vs)" |
|
2395 |
unfolding tan_def Ifloatarith.simps real_divide_def .. |
|
2396 |
||
2397 |
lemma Ifloatarith_powr: "Ifloatarith (Exp (Mult b (Ln a))) vs = (Ifloatarith a vs) powr (Ifloatarith b vs)" |
|
2398 |
unfolding powr_def Ifloatarith.simps .. |
|
2399 |
||
2400 |
lemma Ifloatarith_log: "Ifloatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (Ifloatarith b vs) (Ifloatarith x vs)" |
|
2401 |
unfolding log_def Ifloatarith.simps real_divide_def .. |
|
2402 |
||
2403 |
lemma Ifloatarith_num: shows "Ifloatarith (Num (Float 0 0)) vs = 0" and "Ifloatarith (Num (Float 1 0)) vs = 1" and "Ifloatarith (Num (Float (number_of a) 0)) vs = number_of a" by auto |
|
2404 |
||
2405 |
subsection {* Implement proof method \texttt{approximation} *} |
|
2406 |
||
2407 |
lemma bounded_divl: assumes "Ifloat a / Ifloat b \<le> x" shows "Ifloat (float_divl p a b) \<le> x" by (rule order_trans[OF _ assms], rule float_divl) |
|
2408 |
lemma bounded_divr: assumes "x \<le> Ifloat a / Ifloat b" shows "x \<le> Ifloat (float_divr p a b)" by (rule order_trans[OF assms _], rule float_divr) |
|
2409 |
lemma bounded_num: shows "Ifloat (Float 5 1) = 10" and "Ifloat (Float 0 0) = 0" and "Ifloat (Float 1 0) = 1" and "Ifloat (Float (number_of n) 0) = (number_of n)" |
|
2410 |
and "0 * pow2 e = Ifloat (Float 0 e)" and "1 * pow2 e = Ifloat (Float 1 e)" and "number_of m * pow2 e = Ifloat (Float (number_of m) e)" |
|
2411 |
by (auto simp add: Ifloat.simps pow2_def) |
|
2412 |
||
2413 |
lemmas bounded_by_equations = bounded_by_Cons bounded_by_Nil float_power bounded_divl bounded_divr bounded_num HOL.simp_thms |
|
2414 |
lemmas uneq_equations = uneq.simps Ifloatarith.simps Ifloatarith_num Ifloatarith_divide Ifloatarith_diff Ifloatarith_tan Ifloatarith_powr Ifloatarith_log |
|
2415 |
||
2416 |
lemma "x div (0::int) = 0" by auto -- "What happens in the zero case for div" |
|
2417 |
lemma "x mod (0::int) = x" by auto -- "What happens in the zero case for mod" |
|
2418 |
||
2419 |
text {* The following equations must hold for div & mod |
|
2420 |
-- see "The Definition of Standard ML" by R. Milner, M. Tofte and R. Harper (pg. 79) *} |
|
2421 |
lemma "d * (i div d) + i mod d = (i::int)" by auto |
|
2422 |
lemma "0 < (d :: int) \<Longrightarrow> 0 \<le> i mod d \<and> i mod d < d" by auto |
|
2423 |
lemma "(d :: int) < 0 \<Longrightarrow> d < i mod d \<and> i mod d \<le> 0" by auto |
|
2424 |
||
2425 |
code_const "op div :: int \<Rightarrow> int \<Rightarrow> int" (SML "(fn i => fn d => if d = 0 then 0 else i div d)") |
|
2426 |
code_const "op mod :: int \<Rightarrow> int \<Rightarrow> int" (SML "(fn i => fn d => if d = 0 then i else i mod d)") |
|
2427 |
code_const "divmod :: int \<Rightarrow> int \<Rightarrow> (int * int)" (SML "(fn i => fn d => if d = 0 then (0, i) else IntInf.divMod (i, d))") |
|
2428 |
||
2429 |
ML {* |
|
2430 |
val uneq_equations = PureThy.get_thms @{theory} "uneq_equations"; |
|
2431 |
val bounded_by_equations = PureThy.get_thms @{theory} "bounded_by_equations"; |
|
2432 |
val bounded_by_simpset = (HOL_basic_ss addsimps bounded_by_equations) |
|
2433 |
||
2434 |
fun reify_uneq ctxt i = (fn st => |
|
2435 |
let |
|
2436 |
val to = HOLogic.dest_Trueprop (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))) |
|
2437 |
in (Reflection.genreify_tac ctxt uneq_equations (SOME to) i) st |
|
2438 |
end) |
|
2439 |
||
2440 |
fun rule_uneq ctxt prec i thm = let |
|
2441 |
fun conv_num typ = HOLogic.dest_number #> snd #> HOLogic.mk_number typ |
|
2442 |
val to_natc = conv_num @{typ "nat"} #> Thm.cterm_of (ProofContext.theory_of ctxt) |
|
2443 |
val to_nat = conv_num @{typ "nat"} |
|
2444 |
val to_int = conv_num @{typ "int"} |
|
2445 |
||
2446 |
val prec' = to_nat prec |
|
2447 |
||
2448 |
fun bot_float (Const (@{const_name "times"}, _) $ mantisse $ (Const (@{const_name "pow2"}, _) $ exp)) |
|
2449 |
= @{term "Float"} $ to_int mantisse $ to_int exp |
|
2450 |
| bot_float (Const (@{const_name "divide"}, _) $ mantisse $ (Const (@{const_name "power"}, _) $ ten $ exp)) |
|
2451 |
= @{term "float_divl"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ (@{term "power_float (Float 5 1)"} $ to_nat exp) |
|
2452 |
| bot_float (Const (@{const_name "divide"}, _) $ mantisse $ ten) |
|
2453 |
= @{term "float_divl"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ @{term "Float 5 1"} |
|
2454 |
| bot_float mantisse = @{term "Float"} $ to_int mantisse $ @{term "0 :: int"} |
|
2455 |
||
2456 |
fun top_float (Const (@{const_name "times"}, _) $ mantisse $ (Const (@{const_name "pow2"}, _) $ exp)) |
|
2457 |
= @{term "Float"} $ to_int mantisse $ to_int exp |
|
2458 |
| top_float (Const (@{const_name "divide"}, _) $ mantisse $ (Const (@{const_name "power"}, _) $ ten $ exp)) |
|
2459 |
= @{term "float_divr"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ (@{term "power_float (Float 5 1)"} $ to_nat exp) |
|
2460 |
| top_float (Const (@{const_name "divide"}, _) $ mantisse $ ten) |
|
2461 |
= @{term "float_divr"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ @{term "Float 5 1"} |
|
2462 |
| top_float mantisse = @{term "Float"} $ to_int mantisse $ @{term "0 :: int"} |
|
2463 |
||
2464 |
val goal' : term = List.nth (prems_of thm, i - 1) |
|
2465 |
||
2466 |
fun lift_bnd (t as (Const (@{const_name "op &"}, _) $ |
|
2467 |
(Const (@{const_name "less_eq"}, _) $ |
|
2468 |
bottom $ (Free (name, _))) $ |
|
2469 |
(Const (@{const_name "less_eq"}, _) $ _ $ top))) |
|
2470 |
= ((name, HOLogic.mk_prod (bot_float bottom, top_float top)) |
|
2471 |
handle TERM (txt, ts) => raise TERM ("Premisse needs format '<num> <= <var> & <var> <= <num>', but found " ^ |
|
2472 |
(Syntax.string_of_term ctxt t), [t])) |
|
2473 |
| lift_bnd t = raise TERM ("Premisse needs format '<num> <= <var> & <var> <= <num>', but found " ^ |
|
2474 |
(Syntax.string_of_term ctxt t), [t]) |
|
2475 |
val bound_eqs = map (HOLogic.dest_Trueprop #> lift_bnd) (Logic.strip_imp_prems goal') |
|
2476 |
||
2477 |
fun lift_var (Free (varname, _)) = (case AList.lookup (op =) bound_eqs varname of |
|
2478 |
SOME bound => bound |
|
2479 |
| NONE => raise TERM ("No bound equations found for " ^ varname, [])) |
|
2480 |
| lift_var t = raise TERM ("Can not convert expression " ^ |
|
2481 |
(Syntax.string_of_term ctxt t), [t]) |
|
2482 |
||
2483 |
val _ $ vs = HOLogic.dest_Trueprop (Logic.strip_imp_concl goal') |
|
2484 |
||
2485 |
val bs = (HOLogic.dest_list #> map lift_var #> HOLogic.mk_list @{typ "float * float"}) vs |
|
2486 |
val map = [(@{cpat "?prec::nat"}, to_natc prec), |
|
2487 |
(@{cpat "?bs::(float * float) list"}, Thm.cterm_of (ProofContext.theory_of ctxt) bs)] |
|
2488 |
in rtac (Thm.instantiate ([], map) @{thm "uneq_approx"}) i thm end |
|
2489 |
||
2490 |
val eval_tac = CSUBGOAL (fn (ct, i) => rtac (eval_oracle ct) i) |
|
2491 |
||
2492 |
fun gen_eval_tac conv ctxt = CONVERSION (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt) |
|
2493 |
THEN' rtac TrueI |
|
2494 |
||
2495 |
*} |
|
2496 |
||
2497 |
method_setup approximation = {* fn src => |
|
2498 |
Method.syntax Args.term src #> |
|
2499 |
(fn (prec, ctxt) => let |
|
2500 |
in Method.SIMPLE_METHOD' (fn i => |
|
2501 |
(DETERM (reify_uneq ctxt i) |
|
2502 |
THEN rule_uneq ctxt prec i |
|
2503 |
THEN Simplifier.asm_full_simp_tac bounded_by_simpset i |
|
2504 |
THEN (TRY (filter_prems_tac (fn t => false) i)) |
|
2505 |
THEN (gen_eval_tac eval_oracle ctxt) i)) |
|
2506 |
end) |
|
2507 |
*} "real number approximation" |
|
2508 |
||
2509 |
end |