src/HOL/Decision_Procs/Approximation.thy
author wenzelm
Thu Feb 26 20:56:59 2009 +0100 (2009-02-26)
changeset 30122 1c912a9d8200
parent 29823 0ab754d13ccd
child 30273 ecd6f0ca62ea
permissions -rw-r--r--
standard headers;
eliminated non-ASCII chars, which are fragile in the age of unicode;
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(*  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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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"
hoelzl@29805
   229
proof -
hoelzl@29805
   230
  from assms have "0 < (b - sqrt x) ^ 2 " by simp
hoelzl@29805
   231
  also have "\<dots> = b ^ 2 - 2 * b * sqrt x + (sqrt x) ^ 2" by algebra
hoelzl@29805
   232
  also have "\<dots> = b ^ 2 - 2 * b * sqrt x + x" using assms by (simp add: real_sqrt_pow2)
hoelzl@29805
   233
  finally have "0 < b ^ 2 - 2 * b * sqrt x + x" by assumption
hoelzl@29805
   234
  hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms
hoelzl@29805
   235
    by (simp add: field_simps power2_eq_square)
hoelzl@29805
   236
  thus ?thesis by (simp add: field_simps)
hoelzl@29805
   237
qed
hoelzl@29805
   238
hoelzl@29805
   239
lemma sqrt_iteration_bound: assumes "0 < Ifloat x"
hoelzl@29805
   240
  shows "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec n x)"
hoelzl@29805
   241
proof (induct n)
hoelzl@29805
   242
  case 0
hoelzl@29805
   243
  show ?case
hoelzl@29805
   244
  proof (cases x)
hoelzl@29805
   245
    case (Float m e)
hoelzl@29805
   246
    hence "0 < m" using float_pos_m_pos[unfolded less_float_def] assms by auto
hoelzl@29805
   247
    hence "0 < sqrt (real m)" by auto
hoelzl@29805
   248
hoelzl@29805
   249
    have int_nat_bl: "int (nat (bitlen m)) = bitlen m" using bitlen_ge0 by auto
hoelzl@29805
   250
hoelzl@29805
   251
    have "Ifloat x = (real m / 2^nat (bitlen m)) * pow2 (e + int (nat (bitlen m)))"
hoelzl@29805
   252
      unfolding pow2_add pow2_int Float Ifloat.simps by auto
hoelzl@29805
   253
    also have "\<dots> < 1 * pow2 (e + int (nat (bitlen m)))"
hoelzl@29805
   254
    proof (rule mult_strict_right_mono, auto)
hoelzl@29805
   255
      show "real m < 2^nat (bitlen m)" using bitlen_bounds[OF `0 < m`, THEN conjunct2] 
hoelzl@29805
   256
	unfolding real_of_int_less_iff[of m, symmetric] by auto
hoelzl@29805
   257
    qed
hoelzl@29805
   258
    finally have "sqrt (Ifloat x) < sqrt (pow2 (e + bitlen m))" unfolding int_nat_bl by auto
hoelzl@29805
   259
    also have "\<dots> \<le> pow2 ((e + bitlen m) div 2 + 1)"
hoelzl@29805
   260
    proof -
hoelzl@29805
   261
      let ?E = "e + bitlen m"
hoelzl@29805
   262
      have E_mod_pow: "pow2 (?E mod 2) < 4"
hoelzl@29805
   263
      proof (cases "?E mod 2 = 1")
hoelzl@29805
   264
	case True thus ?thesis by auto
hoelzl@29805
   265
      next
hoelzl@29805
   266
	case False 
hoelzl@29805
   267
	have "0 \<le> ?E mod 2" by auto 
hoelzl@29805
   268
	have "?E mod 2 < 2" by auto
hoelzl@29805
   269
	from this[THEN zless_imp_add1_zle]
hoelzl@29805
   270
	have "?E mod 2 \<le> 0" using False by auto
hoelzl@29805
   271
	from xt1(5)[OF `0 \<le> ?E mod 2` this]
hoelzl@29805
   272
	show ?thesis by auto
hoelzl@29805
   273
      qed
hoelzl@29805
   274
      hence "sqrt (pow2 (?E mod 2)) < sqrt (2 * 2)" by auto
hoelzl@29805
   275
      hence E_mod_pow: "sqrt (pow2 (?E mod 2)) < 2" unfolding real_sqrt_abs2 by auto
hoelzl@29805
   276
hoelzl@29805
   277
      have E_eq: "pow2 ?E = pow2 (?E div 2 + ?E div 2 + ?E mod 2)" by auto
hoelzl@29805
   278
      have "sqrt (pow2 ?E) = sqrt (pow2 (?E div 2) * pow2 (?E div 2) * pow2 (?E mod 2))"
hoelzl@29805
   279
	unfolding E_eq unfolding pow2_add ..
hoelzl@29805
   280
      also have "\<dots> = pow2 (?E div 2) * sqrt (pow2 (?E mod 2))"
hoelzl@29805
   281
	unfolding real_sqrt_mult[of _ "pow2 (?E mod 2)"] real_sqrt_abs2 by auto
hoelzl@29805
   282
      also have "\<dots> < pow2 (?E div 2) * 2" 
hoelzl@29805
   283
	by (rule mult_strict_left_mono, auto intro: E_mod_pow)
hoelzl@29805
   284
      also have "\<dots> = pow2 (?E div 2 + 1)" unfolding zadd_commute[of _ 1] pow2_add1 by auto
hoelzl@29805
   285
      finally show ?thesis by auto
hoelzl@29805
   286
    qed
hoelzl@29805
   287
    finally show ?thesis 
hoelzl@29805
   288
      unfolding Float sqrt_iteration.simps Ifloat.simps by auto
hoelzl@29805
   289
  qed
hoelzl@29805
   290
next
hoelzl@29805
   291
  case (Suc n)
hoelzl@29805
   292
  let ?b = "sqrt_iteration prec n x"
hoelzl@29805
   293
  have "0 < sqrt (Ifloat x)" using `0 < Ifloat x` by auto
hoelzl@29805
   294
  also have "\<dots> < Ifloat ?b" using Suc .
hoelzl@29805
   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
hoelzl@29805
   296
  also have "\<dots> \<le> (Ifloat ?b + Ifloat (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr)
hoelzl@29805
   297
  also have "\<dots> = Ifloat (Float 1 -1) * (Ifloat ?b + Ifloat (float_divr prec x ?b))" by auto
hoelzl@29805
   298
  finally show ?case unfolding sqrt_iteration.simps Let_def Ifloat_mult Ifloat_add right_distrib .
hoelzl@29805
   299
qed
hoelzl@29805
   300
hoelzl@29805
   301
lemma sqrt_iteration_lower_bound: assumes "0 < Ifloat x"
hoelzl@29805
   302
  shows "0 < Ifloat (sqrt_iteration prec n x)" (is "0 < ?sqrt")
hoelzl@29805
   303
proof -
hoelzl@29805
   304
  have "0 < sqrt (Ifloat x)" using assms by auto
hoelzl@29805
   305
  also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
hoelzl@29805
   306
  finally show ?thesis .
hoelzl@29805
   307
qed
hoelzl@29805
   308
hoelzl@29805
   309
lemma lb_sqrt_lower_bound: assumes "0 \<le> Ifloat x"
hoelzl@29805
   310
  shows "0 \<le> Ifloat (the (lb_sqrt prec x))"
hoelzl@29805
   311
proof (cases "0 < x")
hoelzl@29805
   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
hoelzl@29805
   313
  hence "0 < sqrt_iteration prec prec x" unfolding less_float_def using sqrt_iteration_lower_bound by auto 
hoelzl@29805
   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
hoelzl@29805
   315
  thus ?thesis unfolding lb_sqrt_def using True by auto
hoelzl@29805
   316
next
hoelzl@29805
   317
  case False with `0 \<le> Ifloat x` have "Ifloat x = 0" unfolding less_float_def by auto
hoelzl@29805
   318
  thus ?thesis unfolding lb_sqrt_def less_float_def by auto
hoelzl@29805
   319
qed
hoelzl@29805
   320
hoelzl@29805
   321
lemma lb_sqrt_upper_bound: assumes "0 \<le> Ifloat x"
hoelzl@29805
   322
  shows "Ifloat (the (lb_sqrt prec x)) \<le> sqrt (Ifloat x)"
hoelzl@29805
   323
proof (cases "0 < x")
hoelzl@29805
   324
  case True hence "0 < Ifloat x" and "0 \<le> Ifloat x" unfolding less_float_def by auto
hoelzl@29805
   325
  hence sqrt_gt0: "0 < sqrt (Ifloat x)" by auto
hoelzl@29805
   326
  hence sqrt_ub: "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
hoelzl@29805
   327
  
hoelzl@29805
   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)
hoelzl@29805
   329
  also have "\<dots> < Ifloat x / sqrt (Ifloat x)" 
hoelzl@29805
   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]])
hoelzl@29805
   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
hoelzl@29805
   332
  finally show ?thesis unfolding lb_sqrt_def if_P[OF `0 < x`] by auto
hoelzl@29805
   333
next
hoelzl@29805
   334
  case False with `0 \<le> Ifloat x`
hoelzl@29805
   335
  have "\<not> x < 0" unfolding less_float_def le_float_def by auto
hoelzl@29805
   336
  show ?thesis unfolding lb_sqrt_def if_not_P[OF False] if_not_P[OF `\<not> x < 0`] using assms by auto
hoelzl@29805
   337
qed
hoelzl@29805
   338
hoelzl@29805
   339
lemma lb_sqrt: assumes "Some y = lb_sqrt prec x"
hoelzl@29805
   340
  shows "Ifloat y \<le> sqrt (Ifloat x)" and "0 \<le> Ifloat x"
hoelzl@29805
   341
proof -
hoelzl@29805
   342
  show "0 \<le> Ifloat x"
hoelzl@29805
   343
  proof (rule ccontr)
hoelzl@29805
   344
    assume "\<not> 0 \<le> Ifloat x"
hoelzl@29805
   345
    hence "lb_sqrt prec x = None" unfolding lb_sqrt_def less_float_def by auto
hoelzl@29805
   346
    thus False using assms by auto
hoelzl@29805
   347
  qed
hoelzl@29805
   348
  from lb_sqrt_upper_bound[OF this, of prec]
hoelzl@29805
   349
  show "Ifloat y \<le> sqrt (Ifloat x)" unfolding assms[symmetric] by auto
hoelzl@29805
   350
qed
hoelzl@29805
   351
hoelzl@29805
   352
lemma ub_sqrt_lower_bound: assumes "0 \<le> Ifloat x"
hoelzl@29805
   353
  shows "sqrt (Ifloat x) \<le> Ifloat (the (ub_sqrt prec x))"
hoelzl@29805
   354
proof (cases "0 < x")
hoelzl@29805
   355
  case True hence "0 < Ifloat x" unfolding less_float_def by auto
hoelzl@29805
   356
  hence "0 < sqrt (Ifloat x)" by auto
hoelzl@29805
   357
  hence "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
hoelzl@29805
   358
  thus ?thesis unfolding ub_sqrt_def if_P[OF `0 < x`] by auto
hoelzl@29805
   359
next
hoelzl@29805
   360
  case False with `0 \<le> Ifloat x`
hoelzl@29805
   361
  have "Ifloat x = 0" unfolding less_float_def le_float_def by auto
hoelzl@29805
   362
  thus ?thesis unfolding ub_sqrt_def less_float_def le_float_def by auto
hoelzl@29805
   363
qed
hoelzl@29805
   364
hoelzl@29805
   365
lemma ub_sqrt: assumes "Some y = ub_sqrt prec x"
hoelzl@29805
   366
  shows "sqrt (Ifloat x) \<le> Ifloat y" and "0 \<le> Ifloat x"
hoelzl@29805
   367
proof -
hoelzl@29805
   368
  show "0 \<le> Ifloat x"
hoelzl@29805
   369
  proof (rule ccontr)
hoelzl@29805
   370
    assume "\<not> 0 \<le> Ifloat x"
hoelzl@29805
   371
    hence "ub_sqrt prec x = None" unfolding ub_sqrt_def less_float_def by auto
hoelzl@29805
   372
    thus False using assms by auto
hoelzl@29805
   373
  qed
hoelzl@29805
   374
  from ub_sqrt_lower_bound[OF this, of prec]
hoelzl@29805
   375
  show "sqrt (Ifloat x) \<le> Ifloat y" unfolding assms[symmetric] by auto
hoelzl@29805
   376
qed
hoelzl@29805
   377
hoelzl@29805
   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"
hoelzl@29805
   379
proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@29805
   380
  fix x lx ux
hoelzl@29805
   381
  assume "(Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
hoelzl@29805
   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
hoelzl@29805
   383
  
hoelzl@29805
   384
  have "Ifloat lx \<le> x" and "x \<le> Ifloat ux" using x by auto
hoelzl@29805
   385
hoelzl@29805
   386
  from lb_sqrt(1)[OF l] real_sqrt_le_mono[OF `Ifloat lx \<le> x`]
hoelzl@29805
   387
  have "Ifloat l \<le> sqrt x" by (rule order_trans)
hoelzl@29805
   388
  moreover
hoelzl@29805
   389
  from real_sqrt_le_mono[OF `x \<le> Ifloat ux`] ub_sqrt(1)[OF u]
hoelzl@29805
   390
  have "sqrt x \<le> Ifloat u" by (rule order_trans)
hoelzl@29805
   391
  ultimately show "Ifloat l \<le> sqrt x \<and> sqrt x \<le> Ifloat u" ..
hoelzl@29805
   392
qed
hoelzl@29805
   393
hoelzl@29805
   394
section "Arcus tangens and \<pi>"
hoelzl@29805
   395
hoelzl@29805
   396
subsection "Compute arcus tangens series"
hoelzl@29805
   397
hoelzl@29805
   398
text {*
hoelzl@29805
   399
hoelzl@29805
   400
As first step we implement the computation of the arcus tangens series. This is only valid in the range
hoelzl@29805
   401
@{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens.
hoelzl@29805
   402
hoelzl@29805
   403
*}
hoelzl@29805
   404
hoelzl@29805
   405
fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
hoelzl@29805
   406
and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
   407
  "ub_arctan_horner prec 0 k x = 0"
hoelzl@29805
   408
| "ub_arctan_horner prec (Suc n) k x = 
hoelzl@29805
   409
    (rapprox_rat prec 1 (int k)) - x * (lb_arctan_horner prec n (k + 2) x)"
hoelzl@29805
   410
| "lb_arctan_horner prec 0 k x = 0"
hoelzl@29805
   411
| "lb_arctan_horner prec (Suc n) k x = 
hoelzl@29805
   412
    (lapprox_rat prec 1 (int k)) - x * (ub_arctan_horner prec n (k + 2) x)"
hoelzl@29805
   413
hoelzl@29805
   414
lemma arctan_0_1_bounds': assumes "0 \<le> Ifloat x" "Ifloat x \<le> 1" and "even n"
hoelzl@29805
   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))}"
hoelzl@29805
   416
proof -
hoelzl@29805
   417
  let "?c i" = "-1^i * (1 / real (i * 2 + 1) * Ifloat x ^ (i * 2 + 1))"
hoelzl@29805
   418
  let "?S n" = "\<Sum> i=0..<n. ?c i"
hoelzl@29805
   419
hoelzl@29805
   420
  have "0 \<le> Ifloat (x * x)" by auto
hoelzl@29805
   421
  from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
hoelzl@29805
   422
  
hoelzl@29805
   423
  have "arctan (Ifloat x) \<in> { ?S n .. ?S (Suc n) }"
hoelzl@29805
   424
  proof (cases "Ifloat x = 0")
hoelzl@29805
   425
    case False
hoelzl@29805
   426
    hence "0 < Ifloat x" using `0 \<le> Ifloat x` by auto
hoelzl@29805
   427
    hence prem: "0 < 1 / real (0 * 2 + (1::nat)) * Ifloat x ^ (0 * 2 + 1)" by auto 
hoelzl@29805
   428
hoelzl@29805
   429
    have "\<bar> Ifloat x \<bar> \<le> 1"  using `0 \<le> Ifloat x` `Ifloat x \<le> 1` by auto
hoelzl@29805
   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`]
hoelzl@29805
   431
    show ?thesis unfolding arctan_series[OF `\<bar> Ifloat x \<bar> \<le> 1`] Suc_plus1  .
hoelzl@29805
   432
  qed auto
hoelzl@29805
   433
  note arctan_bounds = this[unfolded atLeastAtMost_iff]
hoelzl@29805
   434
hoelzl@29805
   435
  have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
hoelzl@29805
   436
hoelzl@29805
   437
  note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0 
hoelzl@29805
   438
    and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
hoelzl@29805
   439
    and ub="\<lambda>n i k x. ub_arctan_horner prec n k x", 
hoelzl@29805
   440
    OF `0 \<le> Ifloat (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps]
hoelzl@29805
   441
hoelzl@29805
   442
  { have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n"
hoelzl@29805
   443
      using bounds(1) `0 \<le> Ifloat x`
hoelzl@29805
   444
      unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
hoelzl@29805
   445
      unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"]
hoelzl@29805
   446
      by (auto intro!: mult_left_mono)
hoelzl@29805
   447
    also have "\<dots> \<le> arctan (Ifloat x)" using arctan_bounds ..
hoelzl@29805
   448
    finally have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan (Ifloat x)" . }
hoelzl@29805
   449
  moreover
hoelzl@29805
   450
  { have "arctan (Ifloat x) \<le> ?S (Suc n)" using arctan_bounds ..
hoelzl@29805
   451
    also have "\<dots> \<le> Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))"
hoelzl@29805
   452
      using bounds(2)[of "Suc n"] `0 \<le> Ifloat x`
hoelzl@29805
   453
      unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
hoelzl@29805
   454
      unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"]
hoelzl@29805
   455
      by (auto intro!: mult_left_mono)
hoelzl@29805
   456
    finally have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . }
hoelzl@29805
   457
  ultimately show ?thesis by auto
hoelzl@29805
   458
qed
hoelzl@29805
   459
hoelzl@29805
   460
lemma arctan_0_1_bounds: assumes "0 \<le> Ifloat x" "Ifloat x \<le> 1"
hoelzl@29805
   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))}"
hoelzl@29805
   462
proof (cases "even n")
hoelzl@29805
   463
  case True
hoelzl@29805
   464
  obtain n' where "Suc n' = get_odd n" and "odd (Suc n')" using get_odd_ex by auto
hoelzl@29805
   465
  hence "even n'" unfolding even_nat_Suc by auto
hoelzl@29805
   466
  have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
hoelzl@29805
   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
hoelzl@29805
   468
  moreover
hoelzl@29805
   469
  have "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (Ifloat x)"
hoelzl@29805
   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
hoelzl@29805
   471
  ultimately show ?thesis by auto
hoelzl@29805
   472
next
hoelzl@29805
   473
  case False hence "0 < n" by (rule odd_pos)
hoelzl@29805
   474
  from gr0_implies_Suc[OF this] obtain n' where "n = Suc n'" ..
hoelzl@29805
   475
  from False[unfolded this even_nat_Suc]
hoelzl@29805
   476
  have "even n'" and "even (Suc (Suc n'))" by auto
hoelzl@29805
   477
  have "get_odd n = Suc n'" unfolding get_odd_def if_P[OF False] using `n = Suc n'` .
hoelzl@29805
   478
hoelzl@29805
   479
  have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
hoelzl@29805
   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
hoelzl@29805
   481
  moreover
hoelzl@29805
   482
  have "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (Ifloat x)"
hoelzl@29805
   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
hoelzl@29805
   484
  ultimately show ?thesis by auto
hoelzl@29805
   485
qed
hoelzl@29805
   486
hoelzl@29805
   487
subsection "Compute \<pi>"
hoelzl@29805
   488
hoelzl@29805
   489
definition ub_pi :: "nat \<Rightarrow> float" where
hoelzl@29805
   490
  "ub_pi prec = (let A = rapprox_rat prec 1 5 ; 
hoelzl@29805
   491
                     B = lapprox_rat prec 1 239
hoelzl@29805
   492
                 in ((Float 1 2) * ((Float 1 2) * A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (A * A)) - 
hoelzl@29805
   493
                                                  B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 (B * B)))))"
hoelzl@29805
   494
hoelzl@29805
   495
definition lb_pi :: "nat \<Rightarrow> float" where
hoelzl@29805
   496
  "lb_pi prec = (let A = lapprox_rat prec 1 5 ; 
hoelzl@29805
   497
                     B = rapprox_rat prec 1 239
hoelzl@29805
   498
                 in ((Float 1 2) * ((Float 1 2) * A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (A * A)) - 
hoelzl@29805
   499
                                                  B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (B * B)))))"
hoelzl@29805
   500
hoelzl@29805
   501
lemma pi_boundaries: "pi \<in> {Ifloat (lb_pi n) .. Ifloat (ub_pi n)}"
hoelzl@29805
   502
proof -
hoelzl@29805
   503
  have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))" unfolding machin[symmetric] by auto
hoelzl@29805
   504
hoelzl@29805
   505
  { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto
hoelzl@29805
   506
    let ?k = "rapprox_rat prec 1 k"
hoelzl@29805
   507
    have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
hoelzl@29805
   508
      
hoelzl@29805
   509
    have "0 \<le> Ifloat ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \<le> k`)
hoelzl@29805
   510
    have "Ifloat ?k \<le> 1" unfolding rapprox_rat.simps(2)[OF zero_le_one `0 < k`]
hoelzl@29805
   511
      by (rule rapprox_posrat_le1, auto simp add: `0 < k` `1 \<le> k`)
hoelzl@29805
   512
hoelzl@29805
   513
    have "1 / real k \<le> Ifloat ?k" using rapprox_rat[where x=1 and y=k] by auto
hoelzl@29805
   514
    hence "arctan (1 / real k) \<le> arctan (Ifloat ?k)" by (rule arctan_monotone')
hoelzl@29805
   515
    also have "\<dots> \<le> Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))"
hoelzl@29805
   516
      using arctan_0_1_bounds[OF `0 \<le> Ifloat ?k` `Ifloat ?k \<le> 1`] by auto
hoelzl@29805
   517
    finally have "arctan (1 / (real k)) \<le> Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" .
hoelzl@29805
   518
  } note ub_arctan = this
hoelzl@29805
   519
hoelzl@29805
   520
  { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
hoelzl@29805
   521
    let ?k = "lapprox_rat prec 1 k"
hoelzl@29805
   522
    have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
hoelzl@29805
   523
    have "1 / real k \<le> 1" using `1 < k` by auto
hoelzl@29805
   524
hoelzl@29805
   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`)
hoelzl@29805
   526
    have "\<And>n. Ifloat ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / real k \<le> 1`)
hoelzl@29805
   527
hoelzl@29805
   528
    have "Ifloat ?k \<le> 1 / real k" using lapprox_rat[where x=1 and y=k] by auto
hoelzl@29805
   529
hoelzl@29805
   530
    have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (Ifloat ?k)"
hoelzl@29805
   531
      using arctan_0_1_bounds[OF `0 \<le> Ifloat ?k` `Ifloat ?k \<le> 1`] by auto
hoelzl@29805
   532
    also have "\<dots> \<le> arctan (1 / real k)" using `Ifloat ?k \<le> 1 / real k` by (rule arctan_monotone')
hoelzl@29805
   533
    finally have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (1 / (real k))" .
hoelzl@29805
   534
  } note lb_arctan = this
hoelzl@29805
   535
hoelzl@29805
   536
  have "pi \<le> Ifloat (ub_pi n)"
hoelzl@29805
   537
    unfolding ub_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub unfolding Float_num
hoelzl@29805
   538
    using lb_arctan[of 239] ub_arctan[of 5]
hoelzl@29805
   539
    by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
hoelzl@29805
   540
  moreover
hoelzl@29805
   541
  have "Ifloat (lb_pi n) \<le> pi"
hoelzl@29805
   542
    unfolding lb_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub Float_num
hoelzl@29805
   543
    using lb_arctan[of 5] ub_arctan[of 239]
hoelzl@29805
   544
    by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
hoelzl@29805
   545
  ultimately show ?thesis by auto
hoelzl@29805
   546
qed
hoelzl@29805
   547
hoelzl@29805
   548
subsection "Compute arcus tangens in the entire domain"
hoelzl@29805
   549
hoelzl@29805
   550
function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where 
hoelzl@29805
   551
  "lb_arctan prec x = (let ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) ;
hoelzl@29805
   552
                           lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)
hoelzl@29805
   553
    in (if x < 0          then - ub_arctan prec (-x) else
hoelzl@29805
   554
        if x \<le> Float 1 -1 then lb_horner x else
hoelzl@29805
   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))))
hoelzl@29805
   556
                          else (let inv = float_divr prec 1 x 
hoelzl@29805
   557
                                in if inv > 1 then 0 
hoelzl@29805
   558
                                              else lb_pi prec * Float 1 -1 - ub_horner inv)))"
hoelzl@29805
   559
hoelzl@29805
   560
| "ub_arctan prec x = (let lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) ;
hoelzl@29805
   561
                           ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)
hoelzl@29805
   562
    in (if x < 0          then - lb_arctan prec (-x) else
hoelzl@29805
   563
        if x \<le> Float 1 -1 then ub_horner x else
hoelzl@29805
   564
        if x \<le> Float 1 1  then let y = float_divr prec x (1 + the (lb_sqrt prec (1 + x * x)))
hoelzl@29805
   565
                               in if y > 1 then ub_pi prec * Float 1 -1 
hoelzl@29805
   566
                                           else Float 1 1 * ub_horner y 
hoelzl@29805
   567
                          else ub_pi prec * Float 1 -1 - lb_horner (float_divl prec 1 x)))"
hoelzl@29805
   568
by pat_completeness auto
hoelzl@29805
   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)
hoelzl@29805
   570
hoelzl@29805
   571
declare ub_arctan_horner.simps[simp del]
hoelzl@29805
   572
declare lb_arctan_horner.simps[simp del]
hoelzl@29805
   573
hoelzl@29805
   574
lemma lb_arctan_bound': assumes "0 \<le> Ifloat x"
hoelzl@29805
   575
  shows "Ifloat (lb_arctan prec x) \<le> arctan (Ifloat x)"
hoelzl@29805
   576
proof -
hoelzl@29805
   577
  have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> Ifloat x` by auto
hoelzl@29805
   578
  let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805
   579
    and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805
   580
hoelzl@29805
   581
  show ?thesis
hoelzl@29805
   582
  proof (cases "x \<le> Float 1 -1")
hoelzl@29805
   583
    case True hence "Ifloat x \<le> 1" unfolding le_float_def Float_num by auto
hoelzl@29805
   584
    show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
hoelzl@29805
   585
      using arctan_0_1_bounds[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1`] by auto
hoelzl@29805
   586
  next
hoelzl@29805
   587
    case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto
hoelzl@29805
   588
    let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)"
hoelzl@29805
   589
    let ?fR = "1 + the (ub_sqrt prec (1 + x * x))"
hoelzl@29805
   590
    let ?DIV = "float_divl prec x ?fR"
hoelzl@29805
   591
    
hoelzl@29805
   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
hoelzl@29805
   593
    hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
hoelzl@29805
   594
hoelzl@29805
   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)
hoelzl@29805
   596
    hence "?R \<le> Ifloat ?fR" by auto
hoelzl@29805
   597
    hence "0 < ?fR" and "0 < Ifloat ?fR" unfolding less_float_def using `0 < ?R` by auto
hoelzl@29805
   598
hoelzl@29805
   599
    have monotone: "Ifloat (float_divl prec x ?fR) \<le> Ifloat x / ?R"
hoelzl@29805
   600
    proof -
hoelzl@29805
   601
      have "Ifloat ?DIV \<le> Ifloat x / Ifloat ?fR" by (rule float_divl)
hoelzl@29805
   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]])
hoelzl@29805
   603
      finally show ?thesis .
hoelzl@29805
   604
    qed
hoelzl@29805
   605
hoelzl@29805
   606
    show ?thesis
hoelzl@29805
   607
    proof (cases "x \<le> Float 1 1")
hoelzl@29805
   608
      case True
hoelzl@29805
   609
      
hoelzl@29805
   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
hoelzl@29805
   611
      also have "\<dots> \<le> Ifloat (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0)
hoelzl@29805
   612
      finally have "Ifloat x \<le> Ifloat ?fR" by auto
hoelzl@29805
   613
      moreover have "Ifloat ?DIV \<le> Ifloat x / Ifloat ?fR" by (rule float_divl)
hoelzl@29805
   614
      ultimately have "Ifloat ?DIV \<le> 1" unfolding divide_le_eq_1_pos[OF `0 < Ifloat ?fR`, symmetric] by auto
hoelzl@29805
   615
hoelzl@29805
   616
      have "0 \<le> Ifloat ?DIV" using float_divl_lower_bound[OF `0 \<le> x` `0 < ?fR`] unfolding le_float_def by auto
hoelzl@29805
   617
hoelzl@29805
   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
hoelzl@29805
   619
	using arctan_0_1_bounds[OF `0 \<le> Ifloat ?DIV` `Ifloat ?DIV \<le> 1`] by auto
hoelzl@29805
   620
      also have "\<dots> \<le> 2 * arctan (Ifloat x / ?R)"
hoelzl@29805
   621
	using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
hoelzl@29805
   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 . 
hoelzl@29805
   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] .
hoelzl@29805
   624
    next
hoelzl@29805
   625
      case False
hoelzl@29805
   626
      hence "2 < Ifloat x" unfolding le_float_def Float_num by auto
hoelzl@29805
   627
      hence "1 \<le> Ifloat x" by auto
hoelzl@29805
   628
hoelzl@29805
   629
      let "?invx" = "float_divr prec 1 x"
hoelzl@29805
   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
hoelzl@29805
   631
hoelzl@29805
   632
      show ?thesis
hoelzl@29805
   633
      proof (cases "1 < ?invx")
hoelzl@29805
   634
	case True
hoelzl@29805
   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] 
hoelzl@29805
   636
	  using `0 \<le> arctan (Ifloat x)` by auto
hoelzl@29805
   637
      next
hoelzl@29805
   638
	case False
hoelzl@29805
   639
	hence "Ifloat ?invx \<le> 1" unfolding less_float_def by auto
hoelzl@29805
   640
	have "0 \<le> Ifloat ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> Ifloat x`)
hoelzl@29805
   641
hoelzl@29805
   642
	have "1 / Ifloat x \<noteq> 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto
hoelzl@29805
   643
	
hoelzl@29805
   644
	have "arctan (1 / Ifloat x) \<le> arctan (Ifloat ?invx)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divr)
hoelzl@29805
   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
hoelzl@29805
   646
	finally have "pi / 2 - Ifloat (?ub_horner ?invx) \<le> arctan (Ifloat x)" 
hoelzl@29805
   647
	  using `0 \<le> arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \<noteq> 0`] 
hoelzl@29805
   648
	  unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto
hoelzl@29805
   649
	moreover
hoelzl@29805
   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
hoelzl@29805
   651
	ultimately
hoelzl@29805
   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]
hoelzl@29805
   653
	  by auto
hoelzl@29805
   654
      qed
hoelzl@29805
   655
    qed
hoelzl@29805
   656
  qed
hoelzl@29805
   657
qed
hoelzl@29805
   658
hoelzl@29805
   659
lemma ub_arctan_bound': assumes "0 \<le> Ifloat x"
hoelzl@29805
   660
  shows "arctan (Ifloat x) \<le> Ifloat (ub_arctan prec x)"
hoelzl@29805
   661
proof -
hoelzl@29805
   662
  have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> Ifloat x` by auto
hoelzl@29805
   663
hoelzl@29805
   664
  let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805
   665
    and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805
   666
hoelzl@29805
   667
  show ?thesis
hoelzl@29805
   668
  proof (cases "x \<le> Float 1 -1")
hoelzl@29805
   669
    case True hence "Ifloat x \<le> 1" unfolding le_float_def Float_num by auto
hoelzl@29805
   670
    show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
hoelzl@29805
   671
      using arctan_0_1_bounds[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1`] by auto
hoelzl@29805
   672
  next
hoelzl@29805
   673
    case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto
hoelzl@29805
   674
    let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)"
hoelzl@29805
   675
    let ?fR = "1 + the (lb_sqrt prec (1 + x * x))"
hoelzl@29805
   676
    let ?DIV = "float_divr prec x ?fR"
hoelzl@29805
   677
    
hoelzl@29805
   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
hoelzl@29805
   679
    hence "0 \<le> Ifloat (1 + x*x)" by auto
hoelzl@29805
   680
    
hoelzl@29805
   681
    hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
hoelzl@29805
   682
hoelzl@29805
   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)
hoelzl@29805
   684
    hence "Ifloat ?fR \<le> ?R" by auto
hoelzl@29805
   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)`])
hoelzl@29805
   686
hoelzl@29805
   687
    have monotone: "Ifloat x / ?R \<le> Ifloat (float_divr prec x ?fR)"
hoelzl@29805
   688
    proof -
hoelzl@29805
   689
      from divide_left_mono[OF `Ifloat ?fR \<le> ?R` `0 \<le> Ifloat x` mult_pos_pos[OF divisor_gt0 `0 < Ifloat ?fR`]]
hoelzl@29805
   690
      have "Ifloat x / ?R \<le> Ifloat x / Ifloat ?fR" .
hoelzl@29805
   691
      also have "\<dots> \<le> Ifloat ?DIV" by (rule float_divr)
hoelzl@29805
   692
      finally show ?thesis .
hoelzl@29805
   693
    qed
hoelzl@29805
   694
hoelzl@29805
   695
    show ?thesis
hoelzl@29805
   696
    proof (cases "x \<le> Float 1 1")
hoelzl@29805
   697
      case True
hoelzl@29805
   698
      show ?thesis
hoelzl@29805
   699
      proof (cases "?DIV > 1")
hoelzl@29805
   700
	case True
hoelzl@29805
   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
hoelzl@29805
   702
	from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le]
hoelzl@29805
   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] .
hoelzl@29805
   704
      next
hoelzl@29805
   705
	case False
hoelzl@29805
   706
	hence "Ifloat ?DIV \<le> 1" unfolding less_float_def by auto
hoelzl@29805
   707
      
hoelzl@29805
   708
	have "0 \<le> Ifloat x / ?R" using `0 \<le> Ifloat x` `0 < ?R` unfolding real_0_le_divide_iff by auto
hoelzl@29805
   709
	hence "0 \<le> Ifloat ?DIV" using monotone by (rule order_trans)
hoelzl@29805
   710
hoelzl@29805
   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 .
hoelzl@29805
   712
	also have "\<dots> \<le> 2 * arctan (Ifloat ?DIV)"
hoelzl@29805
   713
	  using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
hoelzl@29805
   714
	also have "\<dots> \<le> Ifloat (Float 1 1 * ?ub_horner ?DIV)" unfolding Ifloat_mult[of "Float 1 1"] Float_num
hoelzl@29805
   715
	  using arctan_0_1_bounds[OF `0 \<le> Ifloat ?DIV` `Ifloat ?DIV \<le> 1`] by auto
hoelzl@29805
   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] .
hoelzl@29805
   717
      qed
hoelzl@29805
   718
    next
hoelzl@29805
   719
      case False
hoelzl@29805
   720
      hence "2 < Ifloat x" unfolding le_float_def Float_num by auto
hoelzl@29805
   721
      hence "1 \<le> Ifloat x" by auto
hoelzl@29805
   722
      hence "0 < Ifloat x" by auto
hoelzl@29805
   723
      hence "0 < x" unfolding less_float_def by auto
hoelzl@29805
   724
hoelzl@29805
   725
      let "?invx" = "float_divl prec 1 x"
hoelzl@29805
   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
hoelzl@29805
   727
hoelzl@29805
   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`])
hoelzl@29805
   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`)
hoelzl@29805
   730
	
hoelzl@29805
   731
      have "1 / Ifloat x \<noteq> 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto
hoelzl@29805
   732
      
hoelzl@29805
   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
hoelzl@29805
   734
      also have "\<dots> \<le> arctan (1 / Ifloat x)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divl)
hoelzl@29805
   735
      finally have "arctan (Ifloat x) \<le> pi / 2 - Ifloat (?lb_horner ?invx)"
hoelzl@29805
   736
	using `0 \<le> arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \<noteq> 0`] 
hoelzl@29805
   737
	unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto
hoelzl@29805
   738
      moreover
hoelzl@29805
   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
hoelzl@29805
   740
      ultimately
hoelzl@29805
   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]
hoelzl@29805
   742
	by auto
hoelzl@29805
   743
    qed
hoelzl@29805
   744
  qed
hoelzl@29805
   745
qed
hoelzl@29805
   746
hoelzl@29805
   747
lemma arctan_boundaries:
hoelzl@29805
   748
  "arctan (Ifloat x) \<in> {Ifloat (lb_arctan prec x) .. Ifloat (ub_arctan prec x)}"
hoelzl@29805
   749
proof (cases "0 \<le> x")
hoelzl@29805
   750
  case True hence "0 \<le> Ifloat x" unfolding le_float_def by auto
hoelzl@29805
   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
hoelzl@29805
   752
next
hoelzl@29805
   753
  let ?mx = "-x"
hoelzl@29805
   754
  case False hence "x < 0" and "0 \<le> Ifloat ?mx" unfolding le_float_def less_float_def by auto
hoelzl@29805
   755
  hence bounds: "Ifloat (lb_arctan prec ?mx) \<le> arctan (Ifloat ?mx) \<and> arctan (Ifloat ?mx) \<le> Ifloat (ub_arctan prec ?mx)"
hoelzl@29805
   756
    using ub_arctan_bound'[OF `0 \<le> Ifloat ?mx`] lb_arctan_bound'[OF `0 \<le> Ifloat ?mx`] by auto
hoelzl@29805
   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`]
hoelzl@29805
   758
    unfolding atLeastAtMost_iff using bounds[unfolded Ifloat_minus arctan_minus] by auto
hoelzl@29805
   759
qed
hoelzl@29805
   760
hoelzl@29805
   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"
hoelzl@29805
   762
proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@29805
   763
  fix x lx ux
hoelzl@29805
   764
  assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
hoelzl@29805
   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
hoelzl@29805
   766
hoelzl@29805
   767
  { from arctan_boundaries[of lx prec, unfolded l]
hoelzl@29805
   768
    have "Ifloat l \<le> arctan (Ifloat lx)" by (auto simp del: lb_arctan.simps)
hoelzl@29805
   769
    also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
hoelzl@29805
   770
    finally have "Ifloat l \<le> arctan x" .
hoelzl@29805
   771
  } moreover
hoelzl@29805
   772
  { have "arctan x \<le> arctan (Ifloat ux)" using x by (auto intro: arctan_monotone')
hoelzl@29805
   773
    also have "\<dots> \<le> Ifloat u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
hoelzl@29805
   774
    finally have "arctan x \<le> Ifloat u" .
hoelzl@29805
   775
  } ultimately show "Ifloat l \<le> arctan x \<and> arctan x \<le> Ifloat u" ..
hoelzl@29805
   776
qed
hoelzl@29805
   777
hoelzl@29805
   778
section "Sinus and Cosinus"
hoelzl@29805
   779
hoelzl@29805
   780
subsection "Compute the cosinus and sinus series"
hoelzl@29805
   781
hoelzl@29805
   782
fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
hoelzl@29805
   783
and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
   784
  "ub_sin_cos_aux prec 0 i k x = 0"
hoelzl@29805
   785
| "ub_sin_cos_aux prec (Suc n) i k x = 
hoelzl@29805
   786
    (rapprox_rat prec 1 (int k)) - x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
hoelzl@29805
   787
| "lb_sin_cos_aux prec 0 i k x = 0"
hoelzl@29805
   788
| "lb_sin_cos_aux prec (Suc n) i k x = 
hoelzl@29805
   789
    (lapprox_rat prec 1 (int k)) - x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
hoelzl@29805
   790
hoelzl@29805
   791
lemma cos_aux:
hoelzl@29805
   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")
hoelzl@29805
   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")
hoelzl@29805
   794
proof -
hoelzl@29805
   795
  have "0 \<le> Ifloat (x * x)" unfolding Ifloat_mult by auto
hoelzl@29805
   796
  let "?f n" = "fact (2 * n)"
hoelzl@29805
   797
hoelzl@29805
   798
  { fix n 
hoelzl@29805
   799
    have F: "\<And>m. ((\<lambda>i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
hoelzl@29805
   800
    have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^ n) 1 * (((\<lambda>i. i + 2) ^ n) 1 + 1)"
hoelzl@29805
   801
      unfolding F by auto } note f_eq = this
hoelzl@29805
   802
    
hoelzl@29805
   803
  from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0, 
hoelzl@29805
   804
    OF `0 \<le> Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
hoelzl@29805
   805
  show "?lb" and "?ub" by (auto simp add: power_mult power2_eq_square[of "Ifloat x"])
hoelzl@29805
   806
qed
hoelzl@29805
   807
hoelzl@29805
   808
lemma cos_boundaries: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
hoelzl@29805
   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))}"
hoelzl@29805
   810
proof (cases "Ifloat x = 0")
hoelzl@29805
   811
  case False hence "Ifloat x \<noteq> 0" by auto
hoelzl@29805
   812
  hence "0 < x" and "0 < Ifloat x" using `0 \<le> Ifloat x` unfolding less_float_def by auto
hoelzl@29805
   813
  have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0
hoelzl@29805
   814
    using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto
hoelzl@29805
   815
hoelzl@29805
   816
  { fix x n have "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x^(2 * i))
hoelzl@29805
   817
    = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum")
hoelzl@29805
   818
  proof -
hoelzl@29805
   819
    have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
hoelzl@29805
   820
    also have "\<dots> = 
hoelzl@29805
   821
      (\<Sum> j = 0 ..< n. -1 ^ ((2 * j) div 2) / (real (fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
hoelzl@29805
   822
    also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then -1 ^ (i div 2) / (real (fact i)) * x ^ i else 0)"
hoelzl@29805
   823
      unfolding sum_split_even_odd ..
hoelzl@29805
   824
    also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then -1 ^ (i div 2) / (real (fact i)) else 0) * x ^ i)"
hoelzl@29805
   825
      by (rule setsum_cong2) auto
hoelzl@29805
   826
    finally show ?thesis by assumption
hoelzl@29805
   827
  qed } note morph_to_if_power = this
hoelzl@29805
   828
hoelzl@29805
   829
hoelzl@29805
   830
  { fix n :: nat assume "0 < n"
hoelzl@29805
   831
    hence "0 < 2 * n" by auto
hoelzl@29805
   832
    obtain t where "0 < t" and "t < Ifloat x" and
hoelzl@29805
   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) 
hoelzl@29805
   834
      + (cos (t + 1/2 * real (2 * n) * pi) / real (fact (2*n))) * (Ifloat x)^(2*n)" 
hoelzl@29805
   835
      (is "_ = ?SUM + ?rest / ?fact * ?pow")
hoelzl@29805
   836
      using Maclaurin_cos_expansion2[OF `0 < Ifloat x` `0 < 2 * n`] by auto
hoelzl@29805
   837
hoelzl@29805
   838
    have "cos t * -1^n = cos t * cos (real n * pi) + sin t * sin (real n * pi)" by auto
hoelzl@29805
   839
    also have "\<dots> = cos (t + real n * pi)"  using cos_add by auto
hoelzl@29805
   840
    also have "\<dots> = ?rest" by auto
hoelzl@29805
   841
    finally have "cos t * -1^n = ?rest" .
hoelzl@29805
   842
    moreover
hoelzl@29805
   843
    have "t \<le> pi / 2" using `t < Ifloat x` and `Ifloat x \<le> pi / 2` by auto
hoelzl@29805
   844
    hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
hoelzl@29805
   845
    ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
hoelzl@29805
   846
hoelzl@29805
   847
    have "0 < ?fact" by auto
hoelzl@29805
   848
    have "0 < ?pow" using `0 < Ifloat x` by auto
hoelzl@29805
   849
hoelzl@29805
   850
    {
hoelzl@29805
   851
      assume "even n"
hoelzl@29805
   852
      have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
hoelzl@29805
   853
	unfolding morph_to_if_power[symmetric] using cos_aux by auto 
hoelzl@29805
   854
      also have "\<dots> \<le> cos (Ifloat x)"
hoelzl@29805
   855
      proof -
hoelzl@29805
   856
	from even[OF `even n`] `0 < ?fact` `0 < ?pow`
hoelzl@29805
   857
	have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
hoelzl@29805
   858
	thus ?thesis unfolding cos_eq by auto
hoelzl@29805
   859
      qed
hoelzl@29805
   860
      finally have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos (Ifloat x)" .
hoelzl@29805
   861
    } note lb = this
hoelzl@29805
   862
hoelzl@29805
   863
    {
hoelzl@29805
   864
      assume "odd n"
hoelzl@29805
   865
      have "cos (Ifloat x) \<le> ?SUM"
hoelzl@29805
   866
      proof -
hoelzl@29805
   867
	from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
hoelzl@29805
   868
	have "0 \<le> (- ?rest) / ?fact * ?pow"
hoelzl@29805
   869
	  by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
hoelzl@29805
   870
	thus ?thesis unfolding cos_eq by auto
hoelzl@29805
   871
      qed
hoelzl@29805
   872
      also have "\<dots> \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))"
hoelzl@29805
   873
	unfolding morph_to_if_power[symmetric] using cos_aux by auto
hoelzl@29805
   874
      finally have "cos (Ifloat x) \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" .
hoelzl@29805
   875
    } note ub = this and lb
hoelzl@29805
   876
  } note ub = this(1) and lb = this(2)
hoelzl@29805
   877
hoelzl@29805
   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] .
hoelzl@29805
   879
  moreover have "Ifloat (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos (Ifloat x)" 
hoelzl@29805
   880
  proof (cases "0 < get_even n")
hoelzl@29805
   881
    case True show ?thesis using lb[OF True get_even] .
hoelzl@29805
   882
  next
hoelzl@29805
   883
    case False
hoelzl@29805
   884
    hence "get_even n = 0" by auto
hoelzl@29805
   885
    have "- (pi / 2) \<le> Ifloat x" by (rule order_trans[OF _ `0 < Ifloat x`[THEN less_imp_le]], auto)
hoelzl@29805
   886
    with `Ifloat x \<le> pi / 2`
hoelzl@29805
   887
    show ?thesis unfolding `get_even n = 0` lb_sin_cos_aux.simps Ifloat_minus Ifloat_0 using cos_ge_zero by auto
hoelzl@29805
   888
  qed
hoelzl@29805
   889
  ultimately show ?thesis by auto
hoelzl@29805
   890
next
hoelzl@29805
   891
  case True
hoelzl@29805
   892
  show ?thesis
hoelzl@29805
   893
  proof (cases "n = 0")
hoelzl@29805
   894
    case True 
hoelzl@29805
   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
hoelzl@29805
   896
  next
hoelzl@29805
   897
    case False with not0_implies_Suc obtain m where "n = Suc m" by blast
hoelzl@29805
   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)
hoelzl@29805
   899
  qed
hoelzl@29805
   900
qed
hoelzl@29805
   901
hoelzl@29805
   902
lemma sin_aux: assumes "0 \<le> Ifloat x"
hoelzl@29805
   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")
hoelzl@29805
   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")
hoelzl@29805
   905
proof -
hoelzl@29805
   906
  have "0 \<le> Ifloat (x * x)" unfolding Ifloat_mult by auto
hoelzl@29805
   907
  let "?f n" = "fact (2 * n + 1)"
hoelzl@29805
   908
hoelzl@29805
   909
  { fix n 
hoelzl@29805
   910
    have F: "\<And>m. ((\<lambda>i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
hoelzl@29805
   911
    have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^ n) 2 * (((\<lambda>i. i + 2) ^ n) 2 + 1)"
hoelzl@29805
   912
      unfolding F by auto } note f_eq = this
hoelzl@29805
   913
    
hoelzl@29805
   914
  from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
hoelzl@29805
   915
    OF `0 \<le> Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
hoelzl@29805
   916
  show "?lb" and "?ub" using `0 \<le> Ifloat x` unfolding Ifloat_mult
hoelzl@29805
   917
    unfolding power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
hoelzl@29805
   918
    unfolding real_mult_commute
hoelzl@29805
   919
    by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "Ifloat x"])
hoelzl@29805
   920
qed
hoelzl@29805
   921
hoelzl@29805
   922
lemma sin_boundaries: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
hoelzl@29805
   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))}"
hoelzl@29805
   924
proof (cases "Ifloat x = 0")
hoelzl@29805
   925
  case False hence "Ifloat x \<noteq> 0" by auto
hoelzl@29805
   926
  hence "0 < x" and "0 < Ifloat x" using `0 \<le> Ifloat x` unfolding less_float_def by auto
hoelzl@29805
   927
  have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0
hoelzl@29805
   928
    using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto
hoelzl@29805
   929
hoelzl@29805
   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))
hoelzl@29805
   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 = _")
hoelzl@29805
   932
    proof -
hoelzl@29805
   933
      have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto
hoelzl@29805
   934
      have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" by auto
hoelzl@29805
   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)"
hoelzl@29805
   936
	unfolding sum_split_even_odd ..
hoelzl@29805
   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)"
hoelzl@29805
   938
	by (rule setsum_cong2) auto
hoelzl@29805
   939
      finally show ?thesis by assumption
hoelzl@29805
   940
    qed } note setsum_morph = this
hoelzl@29805
   941
hoelzl@29805
   942
  { fix n :: nat assume "0 < n"
hoelzl@29805
   943
    hence "0 < 2 * n + 1" by auto
hoelzl@29805
   944
    obtain t where "0 < t" and "t < Ifloat x" and
hoelzl@29805
   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) 
hoelzl@29805
   946
      + (sin (t + 1/2 * real (2 * n + 1) * pi) / real (fact (2*n + 1))) * (Ifloat x)^(2*n + 1)" 
hoelzl@29805
   947
      (is "_ = ?SUM + ?rest / ?fact * ?pow")
hoelzl@29805
   948
      using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < Ifloat x`] by auto
hoelzl@29805
   949
hoelzl@29805
   950
    have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add left_distrib right_distrib by auto
hoelzl@29805
   951
    moreover
hoelzl@29805
   952
    have "t \<le> pi / 2" using `t < Ifloat x` and `Ifloat x \<le> pi / 2` by auto
hoelzl@29805
   953
    hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
hoelzl@29805
   954
    ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
hoelzl@29805
   955
hoelzl@29805
   956
    have "0 < ?fact" by (rule real_of_nat_fact_gt_zero)
hoelzl@29805
   957
    have "0 < ?pow" using `0 < Ifloat x` by (rule zero_less_power)
hoelzl@29805
   958
hoelzl@29805
   959
    {
hoelzl@29805
   960
      assume "even n"
hoelzl@29805
   961
      have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> 
hoelzl@29805
   962
            (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)"
hoelzl@29805
   963
	using sin_aux[OF `0 \<le> Ifloat x`] unfolding setsum_morph[symmetric] by auto
hoelzl@29805
   964
      also have "\<dots> \<le> ?SUM" by auto
hoelzl@29805
   965
      also have "\<dots> \<le> sin (Ifloat x)"
hoelzl@29805
   966
      proof -
hoelzl@29805
   967
	from even[OF `even n`] `0 < ?fact` `0 < ?pow`
hoelzl@29805
   968
	have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
hoelzl@29805
   969
	thus ?thesis unfolding sin_eq by auto
hoelzl@29805
   970
      qed
hoelzl@29805
   971
      finally have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin (Ifloat x)" .
hoelzl@29805
   972
    } note lb = this
hoelzl@29805
   973
hoelzl@29805
   974
    {
hoelzl@29805
   975
      assume "odd n"
hoelzl@29805
   976
      have "sin (Ifloat x) \<le> ?SUM"
hoelzl@29805
   977
      proof -
hoelzl@29805
   978
	from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
hoelzl@29805
   979
	have "0 \<le> (- ?rest) / ?fact * ?pow"
hoelzl@29805
   980
	  by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
hoelzl@29805
   981
	thus ?thesis unfolding sin_eq by auto
hoelzl@29805
   982
      qed
hoelzl@29805
   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)"
hoelzl@29805
   984
	 by auto
hoelzl@29805
   985
      also have "\<dots> \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" 
hoelzl@29805
   986
	using sin_aux[OF `0 \<le> Ifloat x`] unfolding setsum_morph[symmetric] by auto
hoelzl@29805
   987
      finally have "sin (Ifloat x) \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
hoelzl@29805
   988
    } note ub = this and lb
hoelzl@29805
   989
  } note ub = this(1) and lb = this(2)
hoelzl@29805
   990
hoelzl@29805
   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] .
hoelzl@29805
   992
  moreover have "Ifloat (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin (Ifloat x)" 
hoelzl@29805
   993
  proof (cases "0 < get_even n")
hoelzl@29805
   994
    case True show ?thesis using lb[OF True get_even] .
hoelzl@29805
   995
  next
hoelzl@29805
   996
    case False
hoelzl@29805
   997
    hence "get_even n = 0" by auto
hoelzl@29805
   998
    with `Ifloat x \<le> pi / 2` `0 \<le> Ifloat x`
hoelzl@29805
   999
    show ?thesis unfolding `get_even n = 0` ub_sin_cos_aux.simps Ifloat_minus Ifloat_0 using sin_ge_zero by auto
hoelzl@29805
  1000
  qed
hoelzl@29805
  1001
  ultimately show ?thesis by auto
hoelzl@29805
  1002
next
hoelzl@29805
  1003
  case True
hoelzl@29805
  1004
  show ?thesis
hoelzl@29805
  1005
  proof (cases "n = 0")
hoelzl@29805
  1006
    case True 
hoelzl@29805
  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
hoelzl@29805
  1008
  next
hoelzl@29805
  1009
    case False with not0_implies_Suc obtain m where "n = Suc m" by blast
hoelzl@29805
  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)
hoelzl@29805
  1011
  qed
hoelzl@29805
  1012
qed
hoelzl@29805
  1013
hoelzl@29805
  1014
subsection "Compute the cosinus in the entire domain"
hoelzl@29805
  1015
hoelzl@29805
  1016
definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
  1017
"lb_cos prec x = (let
hoelzl@29805
  1018
    horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ;
hoelzl@29805
  1019
    half = \<lambda> x. if x < 0 then - 1 else Float 1 1 * x * x - 1
hoelzl@29805
  1020
  in if x < Float 1 -1 then horner x
hoelzl@29805
  1021
else if x < 1          then half (horner (x * Float 1 -1))
hoelzl@29805
  1022
                       else half (half (horner (x * Float 1 -2))))"
hoelzl@29805
  1023
hoelzl@29805
  1024
definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
  1025
"ub_cos prec x = (let
hoelzl@29805
  1026
    horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ;
hoelzl@29805
  1027
    half = \<lambda> x. Float 1 1 * x * x - 1
hoelzl@29805
  1028
  in if x < Float 1 -1 then horner x
hoelzl@29805
  1029
else if x < 1          then half (horner (x * Float 1 -1))
hoelzl@29805
  1030
                       else half (half (horner (x * Float 1 -2))))"
hoelzl@29805
  1031
hoelzl@29805
  1032
definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
hoelzl@29805
  1033
"bnds_cos prec lx ux = (let  lpi = lb_pi prec
hoelzl@29805
  1034
  in   if lx < -lpi \<or> ux > lpi   then (Float -1 0, Float 1 0)
hoelzl@29805
  1035
  else if ux \<le> 0                 then (lb_cos prec (-lx), ub_cos prec (-ux))
hoelzl@29805
  1036
  else if 0 \<le> lx                 then (lb_cos prec ux, ub_cos prec lx)
hoelzl@29805
  1037
                                 else (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0))"
hoelzl@29805
  1038
hoelzl@29805
  1039
lemma lb_cos: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi" 
hoelzl@29805
  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) }")
hoelzl@29805
  1041
proof -
hoelzl@29805
  1042
  { fix x :: real
hoelzl@29805
  1043
    have "cos x = cos (x / 2 + x / 2)" by auto
hoelzl@29805
  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"
hoelzl@29805
  1045
      unfolding cos_add by auto
hoelzl@29805
  1046
    also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1" by algebra
hoelzl@29805
  1047
    finally have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" .
hoelzl@29805
  1048
  } note x_half = this[symmetric]
hoelzl@29805
  1049
hoelzl@29805
  1050
  have "\<not> x < 0" using `0 \<le> Ifloat x` unfolding less_float_def by auto
hoelzl@29805
  1051
  let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
hoelzl@29805
  1052
  let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
hoelzl@29805
  1053
  let "?ub_half x" = "Float 1 1 * x * x - 1"
hoelzl@29805
  1054
  let "?lb_half x" = "if x < 0 then - 1 else Float 1 1 * x * x - 1"
hoelzl@29805
  1055
hoelzl@29805
  1056
  show ?thesis
hoelzl@29805
  1057
  proof (cases "x < Float 1 -1")
hoelzl@29805
  1058
    case True hence "Ifloat x \<le> pi / 2" unfolding less_float_def using pi_ge_two by auto
hoelzl@29805
  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
hoelzl@29805
  1060
      using cos_boundaries[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi / 2`] .
hoelzl@29805
  1061
  next
hoelzl@29805
  1062
    case False
hoelzl@29805
  1063
    
hoelzl@29805
  1064
    { fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)"
hoelzl@29805
  1065
      assume "Ifloat y \<le> cos ?x2" and "-pi \<le> Ifloat x" and "Ifloat x \<le> pi"
hoelzl@29805
  1066
      hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto
hoelzl@29805
  1067
      hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
hoelzl@29805
  1068
      
hoelzl@29805
  1069
      have "Ifloat (?lb_half y) \<le> cos (Ifloat x)"
hoelzl@29805
  1070
      proof (cases "y < 0")
hoelzl@29805
  1071
	case True show ?thesis using cos_ge_minus_one unfolding if_P[OF True] by auto
hoelzl@29805
  1072
      next
hoelzl@29805
  1073
	case False
hoelzl@29805
  1074
	hence "0 \<le> Ifloat y" unfolding less_float_def by auto
hoelzl@29805
  1075
	from mult_mono[OF `Ifloat y \<le> cos ?x2` `Ifloat y \<le> cos ?x2` `0 \<le> cos ?x2` this]
hoelzl@29805
  1076
	have "Ifloat y * Ifloat y \<le> cos ?x2 * cos ?x2" .
hoelzl@29805
  1077
	hence "2 * Ifloat y * Ifloat y \<le> 2 * cos ?x2 * cos ?x2" by auto
hoelzl@29805
  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
hoelzl@29805
  1079
	thus ?thesis unfolding if_not_P[OF False] x_half Float_num Ifloat_mult Ifloat_sub by auto
hoelzl@29805
  1080
      qed
hoelzl@29805
  1081
    } note lb_half = this
hoelzl@29805
  1082
    
hoelzl@29805
  1083
    { fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)"
hoelzl@29805
  1084
      assume ub: "cos ?x2 \<le> Ifloat y" and "- pi \<le> Ifloat x" and "Ifloat x \<le> pi"
hoelzl@29805
  1085
      hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto
hoelzl@29805
  1086
      hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
hoelzl@29805
  1087
      
hoelzl@29805
  1088
      have "cos (Ifloat x) \<le> Ifloat (?ub_half y)"
hoelzl@29805
  1089
      proof -
hoelzl@29805
  1090
	have "0 \<le> Ifloat y" using `0 \<le> cos ?x2` ub by (rule order_trans)
hoelzl@29805
  1091
	from mult_mono[OF ub ub this `0 \<le> cos ?x2`]
hoelzl@29805
  1092
	have "cos ?x2 * cos ?x2 \<le> Ifloat y * Ifloat y" .
hoelzl@29805
  1093
	hence "2 * cos ?x2 * cos ?x2 \<le> 2 * Ifloat y * Ifloat y" by auto
hoelzl@29805
  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
hoelzl@29805
  1095
	thus ?thesis unfolding x_half Ifloat_mult Float_num Ifloat_sub by auto
hoelzl@29805
  1096
      qed
hoelzl@29805
  1097
    } note ub_half = this
hoelzl@29805
  1098
    
hoelzl@29805
  1099
    let ?x2 = "x * Float 1 -1"
hoelzl@29805
  1100
    let ?x4 = "x * Float 1 -1 * Float 1 -1"
hoelzl@29805
  1101
    
hoelzl@29805
  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)
hoelzl@29805
  1103
    
hoelzl@29805
  1104
    show ?thesis
hoelzl@29805
  1105
    proof (cases "x < 1")
hoelzl@29805
  1106
      case True hence "Ifloat x \<le> 1" unfolding less_float_def by auto
hoelzl@29805
  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
hoelzl@29805
  1108
      from cos_boundaries[OF this]
hoelzl@29805
  1109
      have lb: "Ifloat (?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> Ifloat (?ub_horner ?x2)" by auto
hoelzl@29805
  1110
      
hoelzl@29805
  1111
      have "Ifloat (?lb x) \<le> ?cos x"
hoelzl@29805
  1112
      proof -
hoelzl@29805
  1113
	from lb_half[OF lb `-pi \<le> Ifloat x` `Ifloat x \<le> pi`]
hoelzl@29805
  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
hoelzl@29805
  1115
      qed
hoelzl@29805
  1116
      moreover have "?cos x \<le> Ifloat (?ub x)"
hoelzl@29805
  1117
      proof -
hoelzl@29805
  1118
	from ub_half[OF ub `-pi \<le> Ifloat x` `Ifloat x \<le> pi`]
hoelzl@29805
  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 
hoelzl@29805
  1120
      qed
hoelzl@29805
  1121
      ultimately show ?thesis by auto
hoelzl@29805
  1122
    next
hoelzl@29805
  1123
      case False
hoelzl@29805
  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
hoelzl@29805
  1125
      from cos_boundaries[OF this]
hoelzl@29805
  1126
      have lb: "Ifloat (?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> Ifloat (?ub_horner ?x4)" by auto
hoelzl@29805
  1127
      
hoelzl@29805
  1128
      have eq_4: "?x2 * Float 1 -1 = x * Float 1 -2" by (cases x, auto simp add: times_float.simps)
hoelzl@29805
  1129
      
hoelzl@29805
  1130
      have "Ifloat (?lb x) \<le> ?cos x"
hoelzl@29805
  1131
      proof -
hoelzl@29805
  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
hoelzl@29805
  1133
	from lb_half[OF lb_half[OF lb this] `-pi \<le> Ifloat x` `Ifloat x \<le> pi`, unfolded eq_4]
hoelzl@29805
  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 .
hoelzl@29805
  1135
      qed
hoelzl@29805
  1136
      moreover have "?cos x \<le> Ifloat (?ub x)"
hoelzl@29805
  1137
      proof -
hoelzl@29805
  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
hoelzl@29805
  1139
	from ub_half[OF ub_half[OF ub this] `-pi \<le> Ifloat x` `Ifloat x \<le> pi`, unfolded eq_4]
hoelzl@29805
  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 .
hoelzl@29805
  1141
      qed
hoelzl@29805
  1142
      ultimately show ?thesis by auto
hoelzl@29805
  1143
    qed
hoelzl@29805
  1144
  qed
hoelzl@29805
  1145
qed
hoelzl@29805
  1146
hoelzl@29805
  1147
lemma lb_cos_minus: assumes "-pi \<le> Ifloat x" and "Ifloat x \<le> 0" 
hoelzl@29805
  1148
  shows "cos (Ifloat (-x)) \<in> {Ifloat (lb_cos prec (-x)) .. Ifloat (ub_cos prec (-x))}"
hoelzl@29805
  1149
proof -
hoelzl@29805
  1150
  have "0 \<le> Ifloat (-x)" and "Ifloat (-x) \<le> pi" using `-pi \<le> Ifloat x` `Ifloat x \<le> 0` by auto
hoelzl@29805
  1151
  from lb_cos[OF this] show ?thesis .
hoelzl@29805
  1152
qed
hoelzl@29805
  1153
hoelzl@29805
  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"
hoelzl@29805
  1155
proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@29805
  1156
  fix x lx ux
hoelzl@29805
  1157
  assume "(l, u) = bnds_cos prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux}"
hoelzl@29805
  1158
  hence bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
hoelzl@29805
  1159
hoelzl@29805
  1160
  let ?lpi = "lb_pi prec"  
hoelzl@29805
  1161
  have [intro!]: "Ifloat lx \<le> Ifloat ux" using x by auto
hoelzl@29805
  1162
  hence "lx \<le> ux" unfolding le_float_def .
hoelzl@29805
  1163
hoelzl@29805
  1164
  show "Ifloat l \<le> cos x \<and> cos x \<le> Ifloat u"
hoelzl@29805
  1165
  proof (cases "lx < -?lpi \<or> ux > ?lpi")
hoelzl@29805
  1166
    case True
hoelzl@29805
  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
hoelzl@29805
  1168
  next
hoelzl@29805
  1169
    case False note not_out = this
hoelzl@29805
  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
hoelzl@29805
  1171
hoelzl@29805
  1172
    from pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1, THEN le_imp_neg_le] lpi_lx
hoelzl@29805
  1173
    have "- pi \<le> Ifloat lx" by (rule order_trans)
hoelzl@29805
  1174
    hence "- pi \<le> x" and "- pi \<le> Ifloat ux" and "x \<le> Ifloat ux" using x by auto
hoelzl@29805
  1175
    
hoelzl@29805
  1176
    from lpi_ux pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1]
hoelzl@29805
  1177
    have "Ifloat ux \<le> pi" by (rule order_trans)
hoelzl@29805
  1178
    hence "x \<le> pi" and "Ifloat lx \<le> pi" and "Ifloat lx \<le> x" using x by auto
hoelzl@29805
  1179
hoelzl@29805
  1180
    note lb_cos_minus_bottom = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct1]
hoelzl@29805
  1181
    note lb_cos_minus_top = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct2]
hoelzl@29805
  1182
    note lb_cos_bottom = lb_cos[unfolded atLeastAtMost_iff, THEN conjunct1]
hoelzl@29805
  1183
    note lb_cos_top = lb_cos[unfolded atLeastAtMost_iff, THEN conjunct2]
hoelzl@29805
  1184
hoelzl@29805
  1185
    show ?thesis
hoelzl@29805
  1186
    proof (cases "ux \<le> 0")
hoelzl@29805
  1187
      case True hence "Ifloat ux \<le> 0" unfolding le_float_def by auto
hoelzl@29805
  1188
      hence "x \<le> 0" and "Ifloat lx \<le> 0" using x by auto
hoelzl@29805
  1189
      
hoelzl@29805
  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`] .
hoelzl@29805
  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`] .
hoelzl@29805
  1192
	finally have "Ifloat (lb_cos prec (-lx)) \<le> cos x" . }
hoelzl@29805
  1193
      moreover
hoelzl@29805
  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`] .
hoelzl@29805
  1195
	also have "\<dots> \<le> Ifloat (ub_cos prec (-ux))" using lb_cos_minus_top[OF `-pi \<le> Ifloat ux` `Ifloat ux \<le> 0`] .
hoelzl@29805
  1196
	finally have "cos x \<le> Ifloat (ub_cos prec (-ux))" . }
hoelzl@29805
  1197
      ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_P[OF True] by auto
hoelzl@29805
  1198
    next
hoelzl@29805
  1199
      case False note not_ux = this
hoelzl@29805
  1200
      
hoelzl@29805
  1201
      show ?thesis
hoelzl@29805
  1202
      proof (cases "0 \<le> lx")
hoelzl@29805
  1203
	case True hence "0 \<le> Ifloat lx" unfolding le_float_def by auto
hoelzl@29805
  1204
	hence "0 \<le> x" and "0 \<le> Ifloat ux" using x by auto
hoelzl@29805
  1205
      
hoelzl@29805
  1206
	{ have "Ifloat (lb_cos prec ux) \<le> cos (Ifloat ux)" using lb_cos_bottom[OF `0 \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
hoelzl@29805
  1207
	  also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
hoelzl@29805
  1208
	  finally have "Ifloat (lb_cos prec ux) \<le> cos x" . }
hoelzl@29805
  1209
	moreover
hoelzl@29805
  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`] .
hoelzl@29805
  1211
	  also have "\<dots> \<le> Ifloat (ub_cos prec lx)" using lb_cos_top[OF `0 \<le> Ifloat lx` `Ifloat lx \<le> pi`] .
hoelzl@29805
  1212
	  finally have "cos x \<le> Ifloat (ub_cos prec lx)" . }
hoelzl@29805
  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
hoelzl@29805
  1214
      next
hoelzl@29805
  1215
	case False with not_ux
hoelzl@29805
  1216
	have "Ifloat lx \<le> 0" and "0 \<le> Ifloat ux" unfolding le_float_def by auto
hoelzl@29805
  1217
hoelzl@29805
  1218
	have "Ifloat (min (lb_cos prec (-lx)) (lb_cos prec ux)) \<le> cos x"
hoelzl@29805
  1219
	proof (cases "x \<le> 0")
hoelzl@29805
  1220
	  case True
hoelzl@29805
  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`] .
hoelzl@29805
  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`] .
hoelzl@29805
  1223
	  finally show ?thesis unfolding Ifloat_min by auto
hoelzl@29805
  1224
	next
hoelzl@29805
  1225
	  case False hence "0 \<le> x" by auto
hoelzl@29805
  1226
	  have "Ifloat (lb_cos prec ux) \<le> cos (Ifloat ux)" using lb_cos_bottom[OF `0 \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
hoelzl@29805
  1227
	  also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
hoelzl@29805
  1228
	  finally show ?thesis unfolding Ifloat_min by auto
hoelzl@29805
  1229
	qed
hoelzl@29805
  1230
	moreover have "cos x \<le> Ifloat (Float 1 0)" by auto
hoelzl@29805
  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
hoelzl@29805
  1232
      qed
hoelzl@29805
  1233
    qed
hoelzl@29805
  1234
  qed
hoelzl@29805
  1235
qed
hoelzl@29805
  1236
hoelzl@29805
  1237
subsection "Compute the sinus in the entire domain"
hoelzl@29805
  1238
hoelzl@29805
  1239
function lb_sin :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_sin :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
  1240
"lb_sin prec x = (let sqr_diff = \<lambda> x. if x > 1 then 0 else 1 - x * x 
hoelzl@29805
  1241
  in if x < 0           then - ub_sin prec (- x)
hoelzl@29805
  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)
hoelzl@29805
  1243
                        else the (lb_sqrt prec (sqr_diff (ub_cos prec x))))" |
hoelzl@29805
  1244
hoelzl@29805
  1245
"ub_sin prec x = (let sqr_diff = \<lambda> x. if x < 0 then 1 else 1 - x * x
hoelzl@29805
  1246
  in if x < 0           then - lb_sin prec (- x)
hoelzl@29805
  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)
hoelzl@29805
  1248
                        else the (ub_sqrt prec (sqr_diff (lb_cos prec x))))"
hoelzl@29805
  1249
by pat_completeness auto
hoelzl@29805
  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)
hoelzl@29805
  1251
hoelzl@29805
  1252
definition bnds_sin :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
hoelzl@29805
  1253
"bnds_sin prec lx ux = (let 
hoelzl@29805
  1254
    lpi = lb_pi prec ;
hoelzl@29805
  1255
    half_pi = lpi * Float 1 -1
hoelzl@29805
  1256
  in if lx \<le> - half_pi \<or> half_pi \<le> ux then (Float -1 0, Float 1 0)
hoelzl@29805
  1257
                                       else (lb_sin prec lx, ub_sin prec ux))"
hoelzl@29805
  1258
hoelzl@29805
  1259
lemma lb_sin: assumes "- (pi / 2) \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
hoelzl@29805
  1260
  shows "sin (Ifloat x) \<in> { Ifloat (lb_sin prec x) .. Ifloat (ub_sin prec x) }" (is "?sin x \<in> { ?lb x .. ?ub x}")
hoelzl@29805
  1261
proof -
hoelzl@29805
  1262
  { fix x :: float assume "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
hoelzl@29805
  1263
    hence "\<not> (x < 0)" and "- (pi / 2) \<le> Ifloat x" unfolding less_float_def using pi_ge_two by auto
hoelzl@29805
  1264
hoelzl@29805
  1265
    have "Ifloat x \<le> pi" using `Ifloat x \<le> pi / 2` using pi_ge_two by auto
hoelzl@29805
  1266
hoelzl@29805
  1267
    have "?sin x \<in> { ?lb x .. ?ub x}"
hoelzl@29805
  1268
    proof (cases "x \<le> Float 1 -1")
hoelzl@29805
  1269
      case True from sin_boundaries[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi / 2`]
hoelzl@29805
  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 .
hoelzl@29805
  1271
    next
hoelzl@29805
  1272
      case False
hoelzl@29805
  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
hoelzl@29805
  1274
      have "0 \<le> sin (Ifloat x)" using `0 \<le> Ifloat x` and `Ifloat x \<le> pi / 2` using sin_ge_zero by auto
hoelzl@29805
  1275
      
hoelzl@29805
  1276
      have "?sin x \<le> ?ub x"
hoelzl@29805
  1277
      proof (cases "lb_cos prec x < 0")
hoelzl@29805
  1278
	case True
hoelzl@29805
  1279
	have "?sin x \<le> 1" using sin_le_one .
hoelzl@29805
  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
hoelzl@29805
  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 .
hoelzl@29805
  1282
      next
hoelzl@29805
  1283
	case False hence "0 \<le> Ifloat (lb_cos prec x)" unfolding less_float_def by auto
hoelzl@29805
  1284
	
hoelzl@29805
  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
hoelzl@29805
  1286
	also have "\<dots> \<le> sqrt (Ifloat (1 - lb_cos prec x * lb_cos prec x))" 
hoelzl@29805
  1287
	proof (rule real_sqrt_le_mono)
hoelzl@29805
  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
hoelzl@29805
  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)
hoelzl@29805
  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
hoelzl@29805
  1291
	qed
hoelzl@29805
  1292
	also have "\<dots> \<le> Ifloat (the (ub_sqrt prec (1 - lb_cos prec x * lb_cos prec x)))"
hoelzl@29805
  1293
	proof (rule ub_sqrt_lower_bound)
hoelzl@29805
  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
hoelzl@29805
  1295
	  from mult_mono[OF order_trans[OF this cos_le_one] order_trans[OF this cos_le_one]]
hoelzl@29805
  1296
	  have "Ifloat (lb_cos prec x) * Ifloat (lb_cos prec x) \<le> 1" using `0 \<le> Ifloat (lb_cos prec x)` by auto
hoelzl@29805
  1297
	  thus "0 \<le> Ifloat (1 - lb_cos prec x * lb_cos prec x)" by auto
hoelzl@29805
  1298
	qed
hoelzl@29805
  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 .
hoelzl@29805
  1300
      qed
hoelzl@29805
  1301
      moreover
hoelzl@29805
  1302
      have "?lb x \<le> ?sin x"
hoelzl@29805
  1303
      proof (cases "1 < ub_cos prec x")
hoelzl@29805
  1304
	case True
hoelzl@29805
  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 
hoelzl@29805
  1306
	  by (rule order_trans[OF _ sin_ge_zero[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`]]) 
hoelzl@29805
  1307
        (auto simp add: lb_sqrt_upper_bound[where prec=prec and x=0, unfolded Ifloat_0 real_sqrt_zero])
hoelzl@29805
  1308
      next
hoelzl@29805
  1309
	case False hence "Ifloat (ub_cos prec x) \<le> 1" unfolding less_float_def by auto
hoelzl@29805
  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
hoelzl@29805
  1311
	
hoelzl@29805
  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))"
hoelzl@29805
  1313
	proof (rule lb_sqrt_upper_bound)
hoelzl@29805
  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)`
hoelzl@29805
  1315
	  have "Ifloat (ub_cos prec x) * Ifloat (ub_cos prec x) \<le> 1" by auto
hoelzl@29805
  1316
	  thus "0 \<le> Ifloat (1 - ub_cos prec x * ub_cos prec x)" by auto
hoelzl@29805
  1317
	qed
hoelzl@29805
  1318
	also have "\<dots> \<le> sqrt (1 - cos (Ifloat x) ^ 2)"
hoelzl@29805
  1319
	proof (rule real_sqrt_le_mono)
hoelzl@29805
  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
hoelzl@29805
  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)
hoelzl@29805
  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
hoelzl@29805
  1323
	qed
hoelzl@29805
  1324
	also have "\<dots> = sin (Ifloat x)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \<le> sin (Ifloat x)` by auto
hoelzl@29805
  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 .
hoelzl@29805
  1326
      qed
hoelzl@29805
  1327
      ultimately show ?thesis by auto
hoelzl@29805
  1328
    qed
hoelzl@29805
  1329
  } note for_pos = this
hoelzl@29805
  1330
hoelzl@29805
  1331
  show ?thesis
hoelzl@29805
  1332
  proof (cases "x < 0")
hoelzl@29805
  1333
    case True 
hoelzl@29805
  1334
    hence "0 \<le> Ifloat (-x)" and "Ifloat (- x) \<le> pi / 2" using `-(pi/2) \<le> Ifloat x` unfolding less_float_def by auto
hoelzl@29805
  1335
    from for_pos[OF this]
hoelzl@29805
  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
hoelzl@29805
  1337
  next
hoelzl@29805
  1338
    case False hence "0 \<le> Ifloat x" unfolding less_float_def by auto
hoelzl@29805
  1339
    from for_pos[OF this `Ifloat x \<le> pi /2`]
hoelzl@29805
  1340
    show ?thesis .
hoelzl@29805
  1341
  qed
hoelzl@29805
  1342
qed
hoelzl@29805
  1343
hoelzl@29805
  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"
hoelzl@29805
  1345
proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@29805
  1346
  fix x lx ux
hoelzl@29805
  1347
  assume "(l, u) = bnds_sin prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux}"
hoelzl@29805
  1348
  hence bnds: "(l, u) = bnds_sin prec lx ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
hoelzl@29805
  1349
  show "Ifloat l \<le> sin x \<and> sin x \<le> Ifloat u"
hoelzl@29805
  1350
  proof (cases "lx \<le> - (lb_pi prec * Float 1 -1) \<or> lb_pi prec * Float 1 -1 \<le> ux")
hoelzl@29805
  1351
    case True show ?thesis using bnds unfolding bnds_sin_def if_P[OF True] Let_def by auto
hoelzl@29805
  1352
  next
hoelzl@29805
  1353
    case False
hoelzl@29805
  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
hoelzl@29805
  1355
    moreover have "Ifloat (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding Ifloat_mult using pi_boundaries by auto
hoelzl@29805
  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
hoelzl@29805
  1357
    hence "- (pi / 2) \<le> Ifloat ux" and "Ifloat lx \<le> pi / 2" by auto
hoelzl@29805
  1358
    
hoelzl@29805
  1359
    have "- (pi / 2) \<le> x""x \<le> pi / 2" using `Ifloat ux \<le> pi / 2` `- (pi /2) \<le> Ifloat lx` x by auto
hoelzl@29805
  1360
    
hoelzl@29805
  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
hoelzl@29805
  1362
      also have "\<dots> \<le> sin x" using sin_monotone_2pi' `- (pi / 2) \<le> Ifloat lx` x `x \<le> pi / 2` by auto
hoelzl@29805
  1363
      finally have "Ifloat (lb_sin prec lx) \<le> sin x" . }
hoelzl@29805
  1364
    moreover
hoelzl@29805
  1365
    { have "sin x \<le> sin (Ifloat ux)" using sin_monotone_2pi' `- (pi / 2) \<le> x` x `Ifloat ux \<le> pi / 2` by auto
hoelzl@29805
  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
hoelzl@29805
  1367
      finally have "sin x \<le> Ifloat (ub_sin prec ux)" . }
hoelzl@29805
  1368
    ultimately
hoelzl@29805
  1369
    show ?thesis using bnds unfolding bnds_sin_def if_not_P[OF False] Let_def by auto
hoelzl@29805
  1370
  qed
hoelzl@29805
  1371
qed
hoelzl@29805
  1372
hoelzl@29805
  1373
section "Exponential function"
hoelzl@29805
  1374
hoelzl@29805
  1375
subsection "Compute the series of the exponential function"
hoelzl@29805
  1376
hoelzl@29805
  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
hoelzl@29805
  1378
"ub_exp_horner prec 0 i k x       = 0" |
hoelzl@29805
  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" |
hoelzl@29805
  1380
"lb_exp_horner prec 0 i k x       = 0" |
hoelzl@29805
  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"
hoelzl@29805
  1382
hoelzl@29805
  1383
lemma bnds_exp_horner: assumes "Ifloat x \<le> 0"
hoelzl@29805
  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) }"
hoelzl@29805
  1385
proof -
hoelzl@29805
  1386
  { fix n
hoelzl@29805
  1387
    have F: "\<And> m. ((\<lambda>i. i + 1) ^ n) m = n + m" by (induct n, auto)
hoelzl@29805
  1388
    have "fact (Suc n) = fact n * ((\<lambda>i. i + 1) ^ n) 1" unfolding F by auto } note f_eq = this
hoelzl@29805
  1389
    
hoelzl@29805
  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,
hoelzl@29805
  1391
    OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]
hoelzl@29805
  1392
hoelzl@29805
  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)"
hoelzl@29805
  1394
      using bounds(1) by auto
hoelzl@29805
  1395
    also have "\<dots> \<le> exp (Ifloat x)"
hoelzl@29805
  1396
    proof -
hoelzl@29805
  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)"
hoelzl@29805
  1398
	using Maclaurin_exp_le by blast
hoelzl@29805
  1399
      moreover have "0 \<le> exp t / real (fact (get_even n)) * (Ifloat x) ^ (get_even n)"
hoelzl@29805
  1400
	by (auto intro!: mult_nonneg_nonneg divide_nonneg_pos simp add: get_even zero_le_even_power exp_gt_zero)
hoelzl@29805
  1401
      ultimately show ?thesis
hoelzl@29805
  1402
	using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
hoelzl@29805
  1403
    qed
hoelzl@29805
  1404
    finally have "Ifloat (lb_exp_horner prec (get_even n) 1 1 x) \<le> exp (Ifloat x)" .
hoelzl@29805
  1405
  } moreover
hoelzl@29805
  1406
  { 
hoelzl@29805
  1407
    have x_less_zero: "Ifloat x ^ get_odd n \<le> 0"
hoelzl@29805
  1408
    proof (cases "Ifloat x = 0")
hoelzl@29805
  1409
      case True
hoelzl@29805
  1410
      have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
hoelzl@29805
  1411
      thus ?thesis unfolding True power_0_left by auto
hoelzl@29805
  1412
    next
hoelzl@29805
  1413
      case False hence "Ifloat x < 0" using `Ifloat x \<le> 0` by auto
hoelzl@29805
  1414
      show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq get_odd `Ifloat x < 0`)
hoelzl@29805
  1415
    qed
hoelzl@29805
  1416
hoelzl@29805
  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)"
hoelzl@29805
  1418
      using Maclaurin_exp_le by blast
hoelzl@29805
  1419
    moreover have "exp t / real (fact (get_odd n)) * (Ifloat x) ^ (get_odd n) \<le> 0"
hoelzl@29805
  1420
      by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero exp_gt_zero)
hoelzl@29805
  1421
    ultimately have "exp (Ifloat x) \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * Ifloat x ^ j)"
hoelzl@29805
  1422
      using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
hoelzl@29805
  1423
    also have "\<dots> \<le> Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)"
hoelzl@29805
  1424
      using bounds(2) by auto
hoelzl@29805
  1425
    finally have "exp (Ifloat x) \<le> Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)" .
hoelzl@29805
  1426
  } ultimately show ?thesis by auto
hoelzl@29805
  1427
qed
hoelzl@29805
  1428
hoelzl@29805
  1429
subsection "Compute the exponential function on the entire domain"
hoelzl@29805
  1430
hoelzl@29805
  1431
function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
  1432
"lb_exp prec x = (if 0 < x then float_divl prec 1 (ub_exp prec (-x))
hoelzl@29805
  1433
             else let 
hoelzl@29805
  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)
hoelzl@29805
  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))
hoelzl@29805
  1436
                           else horner x)" |
hoelzl@29805
  1437
"ub_exp prec x = (if 0 < x    then float_divr prec 1 (lb_exp prec (-x))
hoelzl@29805
  1438
             else if x < - 1  then (case floor_fl x of (Float m e) \<Rightarrow> 
hoelzl@29805
  1439
                                    (ub_exp_horner prec (get_odd (prec + 2)) 1 1 (float_divr prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e))
hoelzl@29805
  1440
                              else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
hoelzl@29805
  1441
by pat_completeness auto
hoelzl@29805
  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)
hoelzl@29805
  1443
hoelzl@29805
  1444
lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)"
hoelzl@29805
  1445
proof -
hoelzl@29805
  1446
  have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
hoelzl@29805
  1447
hoelzl@29805
  1448
  have "1 / 4 = Ifloat (Float 1 -2)" unfolding Float_num by auto
hoelzl@29805
  1449
  also have "\<dots> \<le> Ifloat (lb_exp_horner 1 (get_even 4) 1 1 (- 1))"
hoelzl@29805
  1450
    unfolding get_even_def eq4 
hoelzl@29805
  1451
    by (auto simp add: lapprox_posrat_def rapprox_posrat_def normfloat.simps)
hoelzl@29805
  1452
  also have "\<dots> \<le> exp (Ifloat (- 1))" using bnds_exp_horner[where x="- 1"] by auto
hoelzl@29805
  1453
  finally show ?thesis unfolding Ifloat_minus Ifloat_1 . 
hoelzl@29805
  1454
qed
hoelzl@29805
  1455
hoelzl@29805
  1456
lemma lb_exp_pos: assumes "\<not> 0 < x" shows "0 < lb_exp prec x"
hoelzl@29805
  1457
proof -
hoelzl@29805
  1458
  let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
hoelzl@29805
  1459
  let "?horner x" = "let  y = ?lb_horner x  in if y \<le> 0 then Float 1 -2 else y"
hoelzl@29805
  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)
hoelzl@29805
  1461
  moreover { fix x :: float fix num :: nat
hoelzl@29805
  1462
    have "0 < Ifloat (?horner x) ^ num" using `0 < ?horner x`[unfolded less_float_def Ifloat_0] by (rule zero_less_power)
hoelzl@29805
  1463
    also have "\<dots> = Ifloat ((?horner x) ^ num)" using float_power by auto
hoelzl@29805
  1464
    finally have "0 < Ifloat ((?horner x) ^ num)" .
hoelzl@29805
  1465
  }
hoelzl@29805
  1466
  ultimately show ?thesis
hoelzl@29805
  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) 
hoelzl@29805
  1468
qed
hoelzl@29805
  1469
hoelzl@29805
  1470
lemma exp_boundaries': assumes "x \<le> 0"
hoelzl@29805
  1471
  shows "exp (Ifloat x) \<in> { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}"
hoelzl@29805
  1472
proof -
hoelzl@29805
  1473
  let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
hoelzl@29805
  1474
  let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"
hoelzl@29805
  1475
hoelzl@29805
  1476
  have "Ifloat x \<le> 0" and "\<not> x > 0" using `x \<le> 0` unfolding le_float_def less_float_def by auto
hoelzl@29805
  1477
  show ?thesis
hoelzl@29805
  1478
  proof (cases "x < - 1")
hoelzl@29805
  1479
    case False hence "- 1 \<le> Ifloat x" unfolding less_float_def by auto
hoelzl@29805
  1480
    show ?thesis
hoelzl@29805
  1481
    proof (cases "?lb_exp_horner x \<le> 0")
hoelzl@29805
  1482
      from `\<not> x < - 1` have "- 1 \<le> Ifloat x" unfolding less_float_def by auto
hoelzl@29805
  1483
      hence "exp (- 1) \<le> exp (Ifloat x)" unfolding exp_le_cancel_iff .
hoelzl@29805
  1484
      from order_trans[OF exp_m1_ge_quarter this]
hoelzl@29805
  1485
      have "Ifloat (Float 1 -2) \<le> exp (Ifloat x)" unfolding Float_num .
hoelzl@29805
  1486
      moreover case True
hoelzl@29805
  1487
      ultimately show ?thesis using bnds_exp_horner `Ifloat x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by auto
hoelzl@29805
  1488
    next
hoelzl@29805
  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)
hoelzl@29805
  1490
    qed
hoelzl@29805
  1491
  next
hoelzl@29805
  1492
    case True
hoelzl@29805
  1493
    
hoelzl@29805
  1494
    obtain m e where Float_floor: "floor_fl x = Float m e" by (cases "floor_fl x", auto)
hoelzl@29805
  1495
    let ?num = "nat (- m) * 2 ^ nat e"
hoelzl@29805
  1496
    
hoelzl@29805
  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)
hoelzl@29805
  1498
    hence "Ifloat (floor_fl x) < 0" unfolding Float_floor Ifloat.simps using zero_less_pow2[of xe] by auto
hoelzl@29805
  1499
    hence "m < 0"
hoelzl@29805
  1500
      unfolding less_float_def Ifloat_0 Float_floor Ifloat.simps
hoelzl@29805
  1501
      unfolding pos_prod_lt[OF zero_less_pow2[of e], unfolded real_mult_commute] by auto
hoelzl@29805
  1502
    hence "1 \<le> - m" by auto
hoelzl@29805
  1503
    hence "0 < nat (- m)" by auto
hoelzl@29805
  1504
    moreover
hoelzl@29805
  1505
    have "0 \<le> e" using floor_pos_exp Float_floor[symmetric] by auto
hoelzl@29805
  1506
    hence "(0::nat) < 2 ^ nat e" by auto
hoelzl@29805
  1507
    ultimately have "0 < ?num"  by auto
hoelzl@29805
  1508
    hence "real ?num \<noteq> 0" by auto
hoelzl@29805
  1509
    have e_nat: "int (nat e) = e" using `0 \<le> e` by auto
hoelzl@29805
  1510
    have num_eq: "real ?num = Ifloat (- floor_fl x)" using `0 < nat (- m)`
hoelzl@29805
  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
hoelzl@29805
  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 .
hoelzl@29805
  1513
    hence "Ifloat (floor_fl x) < 0" unfolding less_float_def by auto
hoelzl@29805
  1514
    
hoelzl@29805
  1515
    have "exp (Ifloat x) \<le> Ifloat (ub_exp prec x)"
hoelzl@29805
  1516
    proof -
hoelzl@29805
  1517
      have div_less_zero: "Ifloat (float_divr prec x (- floor_fl x)) \<le> 0" 
hoelzl@29805
  1518
	using float_divr_nonpos_pos_upper_bound[OF `x \<le> 0` `0 < - floor_fl x`] unfolding le_float_def Ifloat_0 .
hoelzl@29805
  1519
      
hoelzl@29805
  1520
      have "exp (Ifloat x) = exp (real ?num * (Ifloat x / real ?num))" using `real ?num \<noteq> 0` by auto
hoelzl@29805
  1521
      also have "\<dots> = exp (Ifloat x / real ?num) ^ ?num" unfolding exp_real_of_nat_mult ..
hoelzl@29805
  1522
      also have "\<dots> \<le> exp (Ifloat (float_divr prec x (- floor_fl x))) ^ ?num" unfolding num_eq
hoelzl@29805
  1523
	by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
hoelzl@29805
  1524
      also have "\<dots> \<le> Ifloat ((?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num)" unfolding float_power
hoelzl@29805
  1525
	by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
hoelzl@29805
  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 .
hoelzl@29805
  1527
    qed
hoelzl@29805
  1528
    moreover 
hoelzl@29805
  1529
    have "Ifloat (lb_exp prec x) \<le> exp (Ifloat x)"
hoelzl@29805
  1530
    proof -
hoelzl@29805
  1531
      let ?divl = "float_divl prec x (- Float m e)"
hoelzl@29805
  1532
      let ?horner = "?lb_exp_horner ?divl"
hoelzl@29805
  1533
      
hoelzl@29805
  1534
      show ?thesis
hoelzl@29805
  1535
      proof (cases "?horner \<le> 0")
hoelzl@29805
  1536
	case False hence "0 \<le> Ifloat ?horner" unfolding le_float_def by auto
hoelzl@29805
  1537
	
hoelzl@29805
  1538
	have div_less_zero: "Ifloat (float_divl prec x (- floor_fl x)) \<le> 0"
hoelzl@29805
  1539
	  using `Ifloat (floor_fl x) < 0` `Ifloat x \<le> 0` by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
hoelzl@29805
  1540
	
hoelzl@29805
  1541
	have "Ifloat ((?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num) \<le>  
hoelzl@29805
  1542
          exp (Ifloat (float_divl prec x (- floor_fl x))) ^ ?num" unfolding float_power 
hoelzl@29805
  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)
hoelzl@29805
  1544
	also have "\<dots> \<le> exp (Ifloat x / real ?num) ^ ?num" unfolding num_eq
hoelzl@29805
  1545
	  using float_divl by (auto intro!: power_mono simp del: Ifloat_minus)
hoelzl@29805
  1546
	also have "\<dots> = exp (real ?num * (Ifloat x / real ?num))" unfolding exp_real_of_nat_mult ..
hoelzl@29805
  1547
	also have "\<dots> = exp (Ifloat x)" using `real ?num \<noteq> 0` by auto
hoelzl@29805
  1548
	finally show ?thesis
hoelzl@29805
  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
hoelzl@29805
  1550
      next
hoelzl@29805
  1551
	case True
hoelzl@29805
  1552
	have "Ifloat (floor_fl x) \<noteq> 0" and "Ifloat (floor_fl x) \<le> 0" using `Ifloat (floor_fl x) < 0` by auto
hoelzl@29805
  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`]]
hoelzl@29805
  1554
	have "- 1 \<le> Ifloat x / Ifloat (- floor_fl x)" unfolding Ifloat_minus by auto
hoelzl@29805
  1555
	from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
hoelzl@29805
  1556
	have "Ifloat (Float 1 -2) \<le> exp (Ifloat x / Ifloat (- floor_fl x))" unfolding Float_num .
hoelzl@29805
  1557
	hence "Ifloat (Float 1 -2) ^ ?num \<le> exp (Ifloat x / Ifloat (- floor_fl x)) ^ ?num"
hoelzl@29805
  1558
	  by (auto intro!: power_mono simp add: Float_num)
hoelzl@29805
  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
hoelzl@29805
  1560
	finally show ?thesis
hoelzl@29805
  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 .
hoelzl@29805
  1562
      qed
hoelzl@29805
  1563
    qed
hoelzl@29805
  1564
    ultimately show ?thesis by auto
hoelzl@29805
  1565
  qed
hoelzl@29805
  1566
qed
hoelzl@29805
  1567
hoelzl@29805
  1568
lemma exp_boundaries: "exp (Ifloat x) \<in> { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}"
hoelzl@29805
  1569
proof -
hoelzl@29805
  1570
  show ?thesis
hoelzl@29805
  1571
  proof (cases "0 < x")
hoelzl@29805
  1572
    case False hence "x \<le> 0" unfolding less_float_def le_float_def by auto 
hoelzl@29805
  1573
    from exp_boundaries'[OF this] show ?thesis .
hoelzl@29805
  1574
  next
hoelzl@29805
  1575
    case True hence "-x \<le> 0" unfolding less_float_def le_float_def by auto
hoelzl@29805
  1576
    
hoelzl@29805
  1577
    have "Ifloat (lb_exp prec x) \<le> exp (Ifloat x)"
hoelzl@29805
  1578
    proof -
hoelzl@29805
  1579
      from exp_boundaries'[OF `-x \<le> 0`]
hoelzl@29805
  1580
      have ub_exp: "exp (- Ifloat x) \<le> Ifloat (ub_exp prec (-x))" unfolding atLeastAtMost_iff Ifloat_minus by auto
hoelzl@29805
  1581
      
hoelzl@29805
  1582
      have "Ifloat (float_divl prec 1 (ub_exp prec (-x))) \<le> Ifloat 1 / Ifloat (ub_exp prec (-x))" using float_divl .
hoelzl@29805
  1583
      also have "Ifloat 1 / Ifloat (ub_exp prec (-x)) \<le> exp (Ifloat x)"
hoelzl@29805
  1584
	using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] exp_gt_zero, symmetric]]
hoelzl@29805
  1585
	unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide by auto
hoelzl@29805
  1586
      finally show ?thesis unfolding lb_exp.simps if_P[OF True] .
hoelzl@29805
  1587
    qed
hoelzl@29805
  1588
    moreover
hoelzl@29805
  1589
    have "exp (Ifloat x) \<le> Ifloat (ub_exp prec x)"
hoelzl@29805
  1590
    proof -
hoelzl@29805
  1591
      have "\<not> 0 < -x" using `0 < x` unfolding less_float_def by auto
hoelzl@29805
  1592
      
hoelzl@29805
  1593
      from exp_boundaries'[OF `-x \<le> 0`]
hoelzl@29805
  1594
      have lb_exp: "Ifloat (lb_exp prec (-x)) \<le> exp (- Ifloat x)" unfolding atLeastAtMost_iff Ifloat_minus by auto
hoelzl@29805
  1595
      
hoelzl@29805
  1596
      have "exp (Ifloat x) \<le> Ifloat 1 / Ifloat (lb_exp prec (-x))"
hoelzl@29805
  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]]
hoelzl@29805
  1598
	unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide Ifloat_1 by auto
hoelzl@29805
  1599
      also have "\<dots> \<le> Ifloat (float_divr prec 1 (lb_exp prec (-x)))" using float_divr .
hoelzl@29805
  1600
      finally show ?thesis unfolding ub_exp.simps if_P[OF True] .
hoelzl@29805
  1601
    qed
hoelzl@29805
  1602
    ultimately show ?thesis by auto
hoelzl@29805
  1603
  qed
hoelzl@29805
  1604
qed
hoelzl@29805
  1605
hoelzl@29805
  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"
hoelzl@29805
  1607
proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@29805
  1608
  fix x lx ux
hoelzl@29805
  1609
  assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
hoelzl@29805
  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
hoelzl@29805
  1611
hoelzl@29805
  1612
  { from exp_boundaries[of lx prec, unfolded l]
hoelzl@29805
  1613
    have "Ifloat l \<le> exp (Ifloat lx)" by (auto simp del: lb_exp.simps)
hoelzl@29805
  1614
    also have "\<dots> \<le> exp x" using x by auto
hoelzl@29805
  1615
    finally have "Ifloat l \<le> exp x" .
hoelzl@29805
  1616
  } moreover
hoelzl@29805
  1617
  { have "exp x \<le> exp (Ifloat ux)" using x by auto
hoelzl@29805
  1618
    also have "\<dots> \<le> Ifloat u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
hoelzl@29805
  1619
    finally have "exp x \<le> Ifloat u" .
hoelzl@29805
  1620
  } ultimately show "Ifloat l \<le> exp x \<and> exp x \<le> Ifloat u" ..
hoelzl@29805
  1621
qed
hoelzl@29805
  1622
hoelzl@29805
  1623
section "Logarithm"
hoelzl@29805
  1624
hoelzl@29805
  1625
subsection "Compute the logarithm series"
hoelzl@29805
  1626
hoelzl@29805
  1627
fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" 
hoelzl@29805
  1628
and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805
  1629
"ub_ln_horner prec 0 i x       = 0" |
hoelzl@29805
  1630
"ub_ln_horner prec (Suc n) i x = rapprox_rat prec 1 (int i) - x * lb_ln_horner prec n (Suc i) x" |
hoelzl@29805
  1631
"lb_ln_horner prec 0 i x       = 0" |
hoelzl@29805
  1632
"lb_ln_horner prec (Suc n) i x = lapprox_rat prec 1 (int i) - x * ub_ln_horner prec n (Suc i) x"
hoelzl@29805
  1633
hoelzl@29805
  1634
lemma ln_bounds:
hoelzl@29805
  1635
  assumes "0 \<le> x" and "x < 1"
hoelzl@29805
  1636
  shows "(\<Sum>i=0..<2*n. -1^i * (1 / real (i + 1)) * x^(Suc i)) \<le> ln (x + 1)" (is "?lb")
hoelzl@29805
  1637
  and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. -1^i * (1 / real (i + 1)) * x^(Suc i))" (is "?ub")
hoelzl@29805
  1638
proof -
hoelzl@29805
  1639
  let "?a n" = "(1/real (n +1)) * x^(Suc n)"
hoelzl@29805
  1640
hoelzl@29805
  1641
  have ln_eq: "(\<Sum> i. -1^i * ?a i) = ln (x + 1)"
hoelzl@29805
  1642
    using ln_series[of "x + 1"] `0 \<le> x` `x < 1` by auto
hoelzl@29805
  1643
hoelzl@29805
  1644
  have "norm x < 1" using assms by auto
hoelzl@29805
  1645
  have "?a ----> 0" unfolding Suc_plus1[symmetric] inverse_eq_divide[symmetric] 
hoelzl@29805
  1646
    using LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF `norm x < 1`]]] by auto
hoelzl@29805
  1647
  { fix n have "0 \<le> ?a n" by (rule mult_nonneg_nonneg, auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`) }
hoelzl@29805
  1648
  { fix n have "?a (Suc n) \<le> ?a n" unfolding inverse_eq_divide[symmetric]
hoelzl@29805
  1649
    proof (rule mult_mono)
hoelzl@29805
  1650
      show "0 \<le> x ^ Suc (Suc n)" by (auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
hoelzl@29805
  1651
      have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 real_mult_assoc[symmetric] 
hoelzl@29805
  1652
	by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
hoelzl@29805
  1653
      thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
hoelzl@29805
  1654
    qed auto }
hoelzl@29805
  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]
hoelzl@29805
  1656
  show "?lb" and "?ub" by auto
hoelzl@29805
  1657
qed
hoelzl@29805
  1658
hoelzl@29805
  1659
lemma ln_float_bounds: 
hoelzl@29805
  1660
  assumes "0 \<le> Ifloat x" and "Ifloat x < 1"
hoelzl@29805
  1661
  shows "Ifloat (x * lb_ln_horner prec (get_even n) 1 x) \<le> ln (Ifloat x + 1)" (is "?lb \<le> ?ln")
hoelzl@29805
  1662
  and "ln (Ifloat x + 1) \<le> Ifloat (x * ub_ln_horner prec (get_odd n) 1 x)" (is "?ln \<le> ?ub")
hoelzl@29805
  1663
proof -
hoelzl@29805
  1664
  obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
hoelzl@29805
  1665
  obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..
hoelzl@29805
  1666
hoelzl@29805
  1667
  let "?s n" = "-1^n * (1 / real (1 + n)) * (Ifloat x)^(Suc n)"
hoelzl@29805
  1668
hoelzl@29805
  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
hoelzl@29805
  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",
hoelzl@29805
  1671
      OF `0 \<le> Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> Ifloat x`
hoelzl@29805
  1672
    by (rule mult_right_mono)
hoelzl@29805
  1673
  also have "\<dots> \<le> ?ln" using ln_bounds(1)[OF `0 \<le> Ifloat x` `Ifloat x < 1`] by auto
hoelzl@29805
  1674
  finally show "?lb \<le> ?ln" . 
hoelzl@29805
  1675
hoelzl@29805
  1676
  have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> Ifloat x` `Ifloat x < 1`] by auto
hoelzl@29805
  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
hoelzl@29805
  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",
hoelzl@29805
  1679
      OF `0 \<le> Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> Ifloat x`
hoelzl@29805
  1680
    by (rule mult_right_mono)
hoelzl@29805
  1681
  finally show "?ln \<le> ?ub" . 
hoelzl@29805
  1682
qed
hoelzl@29805
  1683
hoelzl@29805
  1684
lemma ln_add: assumes "0 < x" and "0 < y" shows "ln (x + y) = ln x + ln (1 + y / x)"
hoelzl@29805
  1685
proof -
hoelzl@29805
  1686
  have "x \<noteq> 0" using assms by auto
hoelzl@29805
  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
hoelzl@29805
  1688
  moreover 
hoelzl@29805
  1689
  have "0 < y / x" using assms divide_pos_pos by auto
hoelzl@29805
  1690
  hence "0 < 1 + y / x" by auto
hoelzl@29805
  1691
  ultimately show ?thesis using ln_mult assms by auto
hoelzl@29805
  1692
qed
hoelzl@29805
  1693
hoelzl@29805
  1694
subsection "Compute the logarithm of 2"
hoelzl@29805
  1695
hoelzl@29805
  1696
definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3 
hoelzl@29805
  1697
                                        in (Float 1 -1 * ub_ln_horner prec (get_odd prec) 1 (Float 1 -1)) + 
hoelzl@29805
  1698
                                           (third * ub_ln_horner prec (get_odd prec) 1 third))"
hoelzl@29805
  1699
definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3 
hoelzl@29805
  1700
                                        in (Float 1 -1 * lb_ln_horner prec (get_even prec) 1 (Float 1 -1)) + 
hoelzl@29805
  1701
                                           (third * lb_ln_horner prec (get_even prec) 1 third))"
hoelzl@29805
  1702
hoelzl@29805
  1703
lemma ub_ln2: "ln 2 \<le> Ifloat (ub_ln2 prec)" (is "?ub_ln2")
hoelzl@29805
  1704
  and lb_ln2: "Ifloat (lb_ln2 prec) \<le> ln 2" (is "?lb_ln2")
hoelzl@29805
  1705
proof -
hoelzl@29805
  1706
  let ?uthird = "rapprox_rat (max prec 1) 1 3"
hoelzl@29805
  1707
  let ?lthird = "lapprox_rat prec 1 3"
hoelzl@29805
  1708
hoelzl@29805
  1709
  have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1)"
hoelzl@29805
  1710
    using ln_add[of "3 / 2" "1 / 2"] by auto
hoelzl@29805
  1711
  have lb3: "Ifloat ?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
hoelzl@29805
  1712
  hence lb3_ub: "Ifloat ?lthird < 1" by auto
hoelzl@29805
  1713
  have lb3_lb: "0 \<le> Ifloat ?lthird" using lapprox_rat_bottom[of 1 3] by auto
hoelzl@29805
  1714
  have ub3: "1 / 3 \<le> Ifloat ?uthird" using rapprox_rat[of 1 3] by auto
hoelzl@29805
  1715
  hence ub3_lb: "0 \<le> Ifloat ?uthird" by auto
hoelzl@29805
  1716
hoelzl@29805
  1717
  have lb2: "0 \<le> Ifloat (Float 1 -1)" and ub2: "Ifloat (Float 1 -1) < 1" unfolding Float_num by auto
hoelzl@29805
  1718
hoelzl@29805
  1719
  have "0 \<le> (1::int)" and "0 < (3::int)" by auto
hoelzl@29805
  1720
  have ub3_ub: "Ifloat ?uthird < 1" unfolding rapprox_rat.simps(2)[OF `0 \<le> 1` `0 < 3`]
hoelzl@29805
  1721
    by (rule rapprox_posrat_less1, auto)
hoelzl@29805
  1722
hoelzl@29805
  1723
  have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
hoelzl@29805
  1724
  have uthird_gt0: "0 < Ifloat ?uthird + 1" using ub3_lb by auto
hoelzl@29805
  1725
  have lthird_gt0: "0 < Ifloat ?lthird + 1" using lb3_lb by auto
hoelzl@29805
  1726
hoelzl@29805
  1727
  show ?ub_ln2 unfolding ub_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric]
hoelzl@29805
  1728
  proof (rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
hoelzl@29805
  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
hoelzl@29805
  1730
    also have "\<dots> \<le> Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)"
hoelzl@29805
  1731
      using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
hoelzl@29805
  1732
    finally show "ln (1 / 3 + 1) \<le> Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
hoelzl@29805
  1733
  qed
hoelzl@29805
  1734
  show ?lb_ln2 unfolding lb_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric]
hoelzl@29805
  1735
  proof (rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
hoelzl@29805
  1736
    have "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (Ifloat ?lthird + 1)"
hoelzl@29805
  1737
      using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
hoelzl@29805
  1738
    also have "\<dots> \<le> ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto
hoelzl@29805
  1739
    finally show "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (1 / 3 + 1)" .
hoelzl@29805
  1740
  qed
hoelzl@29805
  1741
qed
hoelzl@29805
  1742
hoelzl@29805
  1743
subsection "Compute the logarithm in the entire domain"
hoelzl@29805
  1744
hoelzl@29805
  1745
function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where
hoelzl@29805
  1746
"ub_ln prec x = (if x \<le> 0         then None
hoelzl@29805
  1747
            else if x < 1         then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
hoelzl@29805
  1748
            else let horner = \<lambda>x. (x - 1) * ub_ln_horner prec (get_odd prec) 1 (x - 1) in
hoelzl@29805
  1749
                 if x < Float 1 1 then Some (horner x)
hoelzl@29805
  1750
                                  else let l = bitlen (mantissa x) - 1 in 
hoelzl@29805
  1751
                                       Some (ub_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l))))" |
hoelzl@29805
  1752
"lb_ln prec x = (if x \<le> 0         then None
hoelzl@29805
  1753
            else if x < 1         then Some (- the (ub_ln prec (float_divr prec 1 x)))
hoelzl@29805
  1754
            else let horner = \<lambda>x. (x - 1) * lb_ln_horner prec (get_even prec) 1 (x - 1) in
hoelzl@29805
  1755
                 if x < Float 1 1 then Some (horner x)
hoelzl@29805
  1756
                                  else let l = bitlen (mantissa x) - 1 in 
hoelzl@29805
  1757
                                       Some (lb_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l))))"
hoelzl@29805
  1758
by pat_completeness auto
hoelzl@29805
  1759
hoelzl@29805
  1760
termination proof (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 1 then 1 else 0))", auto)
hoelzl@29805
  1761
  fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divl (max prec (Suc 0)) 1 x < 1"
hoelzl@29805
  1762
  hence "0 < x" and "0 < max prec (Suc 0)" unfolding less_float_def le_float_def by auto
hoelzl@29805
  1763
  from float_divl_pos_less1_bound[OF `0 < x` `x < 1` `0 < max prec (Suc 0)`]
hoelzl@29805
  1764
  show False using `float_divl (max prec (Suc 0)) 1 x < 1` unfolding less_float_def le_float_def by auto
hoelzl@29805
  1765
next
hoelzl@29805
  1766
  fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divr prec 1 x < 1"
hoelzl@29805
  1767
  hence "0 < x" unfolding less_float_def le_float_def by auto
hoelzl@29805
  1768
  from float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`, of prec]
hoelzl@29805
  1769
  show False using `float_divr prec 1 x < 1` unfolding less_float_def le_float_def by auto
hoelzl@29805
  1770
qed
hoelzl@29805
  1771
hoelzl@29805
  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))))"
hoelzl@29805
  1773
proof -
hoelzl@29805
  1774
  let ?B = "2^nat (bitlen m - 1)"
hoelzl@29805
  1775
  have "0 < real m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0" using assms by auto
hoelzl@29805
  1776
  hence "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto
hoelzl@29805
  1777
  show ?thesis 
hoelzl@29805
  1778
  proof (cases "0 \<le> e")
hoelzl@29805
  1779
    case True
hoelzl@29805
  1780
    show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
hoelzl@29805
  1781
      unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`] 
hoelzl@29805
  1782
      unfolding Ifloat_ge0_exp[OF True] ln_mult[OF `0 < real m` `0 < 2^nat e`] 
hoelzl@29805
  1783
      ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` True by auto
hoelzl@29805
  1784
  next
hoelzl@29805
  1785
    case False hence "0 < -e" by auto
hoelzl@29805
  1786
    hence pow_gt0: "(0::real) < 2^nat (-e)" by auto
hoelzl@29805
  1787
    hence inv_gt0: "(0::real) < inverse (2^nat (-e))" by auto
hoelzl@29805
  1788
    show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
hoelzl@29805
  1789
      unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`] 
hoelzl@29805
  1790
      unfolding Ifloat_nge0_exp[OF False] ln_mult[OF `0 < real m` inv_gt0] ln_inverse[OF pow_gt0]
hoelzl@29805
  1791
      ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` False by auto
hoelzl@29805
  1792
  qed
hoelzl@29805
  1793
qed
hoelzl@29805
  1794
hoelzl@29805
  1795
lemma ub_ln_lb_ln_bounds': assumes "1 \<le> x"
hoelzl@29805
  1796
  shows "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x) \<and> ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))"
hoelzl@29805
  1797
  (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
hoelzl@29805
  1798
proof (cases "x < Float 1 1")
hoelzl@29805
  1799
  case True hence "Ifloat (x - 1) < 1" unfolding less_float_def Float_num by auto
hoelzl@29805
  1800
  have "\<not> x \<le> 0" and "\<not> x < 1" using `1 \<le> x` unfolding less_float_def le_float_def by auto
hoelzl@29805
  1801
  hence "0 \<le> Ifloat (x - 1)" using `1 \<le> x` unfolding less_float_def Float_num by auto
hoelzl@29805
  1802
  show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
hoelzl@29805
  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
hoelzl@29805
  1804
next
hoelzl@29805
  1805
  case False
hoelzl@29805
  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
hoelzl@29805
  1807
  show ?thesis
hoelzl@29805
  1808
  proof (cases x)
hoelzl@29805
  1809
    case (Float m e)
hoelzl@29805
  1810
    let ?s = "Float (e + (bitlen m - 1)) 0"
hoelzl@29805
  1811
    let ?x = "Float m (- (bitlen m - 1))"
hoelzl@29805
  1812
hoelzl@29805
  1813
    have "0 < m" and "m \<noteq> 0" using float_pos_m_pos `0 < x` Float by auto
hoelzl@29805
  1814
hoelzl@29805
  1815
    {
hoelzl@29805
  1816
      have "Ifloat (lb_ln2 prec * ?s) \<le> ln 2 * real (e + (bitlen m - 1))" (is "?lb2 \<le> _")
hoelzl@29805
  1817
	unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 realpow_0 mult_1_right
hoelzl@29805
  1818
	using lb_ln2[of prec]
hoelzl@29805
  1819
      proof (rule mult_right_mono)
hoelzl@29805
  1820
	have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
hoelzl@29805
  1821
	from float_gt1_scale[OF this]
hoelzl@29805
  1822
	show "0 \<le> real (e + (bitlen m - 1))" by auto
hoelzl@29805
  1823
      qed
hoelzl@29805
  1824
      moreover
hoelzl@29805
  1825
      from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
hoelzl@29805
  1826
      have "0 \<le> Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto
hoelzl@29805
  1827
      from ln_float_bounds(1)[OF this]
hoelzl@29805
  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
hoelzl@29805
  1829
      ultimately have "?lb2 + ?lb_horner \<le> ln (Ifloat x)"
hoelzl@29805
  1830
	unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
hoelzl@29805
  1831
    } 
hoelzl@29805
  1832
    moreover
hoelzl@29805
  1833
    {
hoelzl@29805
  1834
      from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
hoelzl@29805
  1835
      have "0 \<le> Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto
hoelzl@29805
  1836
      from ln_float_bounds(2)[OF this]
hoelzl@29805
  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
hoelzl@29805
  1838
      moreover
hoelzl@29805
  1839
      have "ln 2 * real (e + (bitlen m - 1)) \<le> Ifloat (ub_ln2 prec * ?s)" (is "_ \<le> ?ub2")
hoelzl@29805
  1840
	unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 realpow_0 mult_1_right
hoelzl@29805
  1841
	using ub_ln2[of prec] 
hoelzl@29805
  1842
      proof (rule mult_right_mono)
hoelzl@29805
  1843
	have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
hoelzl@29805
  1844
	from float_gt1_scale[OF this]
hoelzl@29805
  1845
	show "0 \<le> real (e + (bitlen m - 1))" by auto
hoelzl@29805
  1846
      qed
hoelzl@29805
  1847
      ultimately have "ln (Ifloat x) \<le> ?ub2 + ?ub_horner"
hoelzl@29805
  1848
	unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
hoelzl@29805
  1849
    }
hoelzl@29805
  1850
    ultimately show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps
hoelzl@29805
  1851
      unfolding if_not_P[OF `\<not> x \<le> 0`] if_not_P[OF `\<not> x < 1`] if_not_P[OF False] Let_def
hoelzl@29805
  1852
      unfolding scale.simps[of m e, unfolded Float[symmetric]] mantissa.simps[of m e, unfolded Float[symmetric]] Ifloat_add by auto
hoelzl@29805
  1853
  qed
hoelzl@29805
  1854
qed
hoelzl@29805
  1855
hoelzl@29805
  1856
lemma ub_ln_lb_ln_bounds: assumes "0 < x"
hoelzl@29805
  1857
  shows "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x) \<and> ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))"
hoelzl@29805
  1858
  (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
hoelzl@29805
  1859
proof (cases "x < 1")
hoelzl@29805
  1860
  case False hence "1 \<le> x" unfolding less_float_def le_float_def by auto
hoelzl@29805
  1861
  show ?thesis using ub_ln_lb_ln_bounds'[OF `1 \<le> x`] .
hoelzl@29805
  1862
next
hoelzl@29805
  1863
  case True have "\<not> x \<le> 0" using `0 < x` unfolding less_float_def le_float_def by auto
hoelzl@29805
  1864
hoelzl@29805
  1865
  have "0 < Ifloat x" and "Ifloat x \<noteq> 0" using `0 < x` unfolding less_float_def by auto
hoelzl@29805
  1866
  hence A: "0 < 1 / Ifloat x" by auto
hoelzl@29805
  1867
hoelzl@29805
  1868
  {
hoelzl@29805
  1869
    let ?divl = "float_divl (max prec 1) 1 x"
hoelzl@29805
  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
hoelzl@29805
  1871
    hence B: "0 < Ifloat ?divl" unfolding le_float_def by auto
hoelzl@29805
  1872
    
hoelzl@29805
  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
hoelzl@29805
  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
hoelzl@29805
  1875
    from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le] 
hoelzl@29805
  1876
    have "?ln \<le> Ifloat (- the (lb_ln prec ?divl))" unfolding Ifloat_minus by (rule order_trans)
hoelzl@29805
  1877
  } moreover
hoelzl@29805
  1878
  {
hoelzl@29805
  1879
    let ?divr = "float_divr prec 1 x"
hoelzl@29805
  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
hoelzl@29805
  1881
    hence B: "0 < Ifloat ?divr" unfolding le_float_def by auto
hoelzl@29805
  1882
    
hoelzl@29805
  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
hoelzl@29805
  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
hoelzl@29805
  1885
    from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
hoelzl@29805
  1886
    have "Ifloat (- the (ub_ln prec ?divr)) \<le> ?ln" unfolding Ifloat_minus by (rule order_trans)
hoelzl@29805
  1887
  }
hoelzl@29805
  1888
  ultimately show ?thesis unfolding lb_ln.simps[where x=x]  ub_ln.simps[where x=x]
hoelzl@29805
  1889
    unfolding if_not_P[OF `\<not> x \<le> 0`] if_P[OF True] by auto
hoelzl@29805
  1890
qed
hoelzl@29805
  1891
hoelzl@29805
  1892
lemma lb_ln: assumes "Some y = lb_ln prec x"
hoelzl@29805
  1893
  shows "Ifloat y \<le> ln (Ifloat x)" and "0 < Ifloat x"
hoelzl@29805
  1894
proof -
hoelzl@29805
  1895
  have "0 < x"
hoelzl@29805
  1896
  proof (rule ccontr)
hoelzl@29805
  1897
    assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
hoelzl@29805
  1898
    thus False using assms by auto
hoelzl@29805
  1899
  qed
hoelzl@29805
  1900
  thus "0 < Ifloat x" unfolding less_float_def by auto
hoelzl@29805
  1901
  have "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x)" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
hoelzl@29805
  1902
  thus "Ifloat y \<le> ln (Ifloat x)" unfolding assms[symmetric] by auto
hoelzl@29805
  1903
qed
hoelzl@29805
  1904
hoelzl@29805
  1905
lemma ub_ln: assumes "Some y = ub_ln prec x"
hoelzl@29805
  1906
  shows "ln (Ifloat x) \<le> Ifloat y" and "0 < Ifloat x"
hoelzl@29805
  1907
proof -
hoelzl@29805
  1908
  have "0 < x"
hoelzl@29805
  1909
  proof (rule ccontr)
hoelzl@29805
  1910
    assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
hoelzl@29805
  1911
    thus False using assms by auto
hoelzl@29805
  1912
  qed
hoelzl@29805
  1913
  thus "0 < Ifloat x" unfolding less_float_def by auto
hoelzl@29805
  1914
  have "ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
hoelzl@29805
  1915
  thus "ln (Ifloat x) \<le> Ifloat y" unfolding assms[symmetric] by auto
hoelzl@29805
  1916
qed
hoelzl@29805
  1917
hoelzl@29805
  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"
hoelzl@29805
  1919
proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@29805
  1920
  fix x lx ux
hoelzl@29805
  1921
  assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
hoelzl@29805
  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
hoelzl@29805
  1923
hoelzl@29805
  1924
  have "ln (Ifloat ux) \<le> Ifloat u" and "0 < Ifloat ux" using ub_ln u by auto
hoelzl@29805
  1925
  have "Ifloat l \<le> ln (Ifloat lx)" and "0 < Ifloat lx" and "0 < x" using lb_ln[OF l] x by auto
hoelzl@29805
  1926
hoelzl@29805
  1927
  from ln_le_cancel_iff[OF `0 < Ifloat lx` `0 < x`] `Ifloat l \<le> ln (Ifloat lx)` 
hoelzl@29805
  1928
  have "Ifloat l \<le> ln x" using x unfolding atLeastAtMost_iff by auto
hoelzl@29805
  1929
  moreover
hoelzl@29805
  1930
  from ln_le_cancel_iff[OF `0 < x` `0 < Ifloat ux`] `ln (Ifloat ux) \<le> Ifloat u` 
hoelzl@29805
  1931
  have "ln x \<le> Ifloat u" using x unfolding atLeastAtMost_iff by auto
hoelzl@29805
  1932
  ultimately show "Ifloat l \<le> ln x \<and> ln x \<le> Ifloat u" ..
hoelzl@29805
  1933
qed
hoelzl@29805
  1934
hoelzl@29805
  1935
hoelzl@29805
  1936
section "Implement floatarith"
hoelzl@29805
  1937
hoelzl@29805
  1938
subsection "Define syntax and semantics"
hoelzl@29805
  1939
hoelzl@29805
  1940
datatype floatarith
hoelzl@29805
  1941
  = Add floatarith floatarith
hoelzl@29805
  1942
  | Minus floatarith
hoelzl@29805
  1943
  | Mult floatarith floatarith
hoelzl@29805
  1944
  | Inverse floatarith
hoelzl@29805
  1945
  | Sin floatarith
hoelzl@29805
  1946
  | Cos floatarith
hoelzl@29805
  1947
  | Arctan floatarith
hoelzl@29805
  1948
  | Abs floatarith
hoelzl@29805
  1949
  | Max floatarith floatarith
hoelzl@29805
  1950
  | Min floatarith floatarith
hoelzl@29805
  1951
  | Pi
hoelzl@29805
  1952
  | Sqrt floatarith
hoelzl@29805
  1953
  | Exp floatarith
hoelzl@29805
  1954
  | Ln floatarith
hoelzl@29805
  1955
  | Power floatarith nat
hoelzl@29805
  1956
  | Atom nat
hoelzl@29805
  1957
  | Num float
hoelzl@29805
  1958
hoelzl@29805
  1959
fun Ifloatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real"
hoelzl@29805
  1960
where
hoelzl@29805
  1961
"Ifloatarith (Add a b) vs   = (Ifloatarith a vs) + (Ifloatarith b vs)" |
hoelzl@29805
  1962
"Ifloatarith (Minus a) vs    = - (Ifloatarith a vs)" |
hoelzl@29805
  1963
"Ifloatarith (Mult a b) vs   = (Ifloatarith a vs) * (Ifloatarith b vs)" |
hoelzl@29805
  1964
"Ifloatarith (Inverse a) vs  = inverse (Ifloatarith a vs)" |
hoelzl@29805
  1965
"Ifloatarith (Sin a) vs      = sin (Ifloatarith a vs)" |
hoelzl@29805
  1966
"Ifloatarith (Cos a) vs      = cos (Ifloatarith a vs)" |
hoelzl@29805
  1967
"Ifloatarith (Arctan a) vs   = arctan (Ifloatarith a vs)" |
hoelzl@29805
  1968
"Ifloatarith (Min a b) vs    = min (Ifloatarith a vs) (Ifloatarith b vs)" |
hoelzl@29805
  1969
"Ifloatarith (Max a b) vs    = max (Ifloatarith a vs) (Ifloatarith b vs)" |
hoelzl@29805
  1970
"Ifloatarith (Abs a) vs      = abs (Ifloatarith a vs)" |
hoelzl@29805
  1971
"Ifloatarith Pi vs           = pi" |
hoelzl@29805
  1972
"Ifloatarith (Sqrt a) vs     = sqrt (Ifloatarith a vs)" |
hoelzl@29805
  1973
"Ifloatarith (Exp a) vs      = exp (Ifloatarith a vs)" |
hoelzl@29805
  1974
"Ifloatarith (Ln a) vs       = ln (Ifloatarith a vs)" |
hoelzl@29805
  1975
"Ifloatarith (Power a n) vs  = (Ifloatarith a vs)^n" |
hoelzl@29805
  1976
"Ifloatarith (Num f) vs      = Ifloat f" |
hoelzl@29805
  1977
"Ifloatarith (Atom n) vs     = vs ! n"
hoelzl@29805
  1978
hoelzl@29805
  1979
subsection "Implement approximation function"
hoelzl@29805
  1980
hoelzl@29805
  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
hoelzl@29805
  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)
hoelzl@29805
  1983
                                                                     | t \<Rightarrow> None)" |
hoelzl@29805
  1984
"lift_bin a b f = None"
hoelzl@29805
  1985
hoelzl@29805
  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
hoelzl@29805
  1987
"lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" |
hoelzl@29805
  1988
"lift_bin' a b f = None"
hoelzl@29805
  1989
hoelzl@29805
  1990
fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where
hoelzl@29805
  1991
"lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u)
hoelzl@29805
  1992
                                             | t \<Rightarrow> None)" |
hoelzl@29805
  1993
"lift_un b f = None"
hoelzl@29805
  1994
hoelzl@29805
  1995
fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
hoelzl@29805
  1996
"lift_un' (Some (l1, u1)) f = Some (f l1 u1)" |
hoelzl@29805
  1997
"lift_un' b f = None"
hoelzl@29805
  1998
hoelzl@29805
  1999
fun bounded_by :: "real list \<Rightarrow> (float * float) list \<Rightarrow> bool " where
hoelzl@29805
  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)" |
hoelzl@29805
  2001
bounded_by_Nil: "bounded_by [] [] = True" |
hoelzl@29805
  2002
"bounded_by _ _ = False"
hoelzl@29805
  2003
hoelzl@29805
  2004
lemma bounded_by: assumes "bounded_by vs bs" and "i < length bs"
hoelzl@29805
  2005
  shows "Ifloat (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> Ifloat (snd (bs ! i))"
hoelzl@29805
  2006
  using `bounded_by vs bs` and `i < length bs`
hoelzl@29805
  2007
proof (induct arbitrary: i rule: bounded_by.induct)
hoelzl@29805
  2008
  fix v :: real and vs :: "real list" and l u :: float and bs :: "(float * float) list" and i :: nat
hoelzl@29805
  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))"
hoelzl@29805
  2010
  assume bounded: "bounded_by (v # vs) ((l, u) # bs)" and length: "i < length ((l, u) # bs)"
hoelzl@29805
  2011
  show "Ifloat (fst (((l, u) # bs) ! i)) \<le> (v # vs) ! i \<and> (v # vs) ! i \<le> Ifloat (snd (((l, u) # bs) ! i))"
hoelzl@29805
  2012
  proof (cases i)
hoelzl@29805
  2013
    case 0
hoelzl@29805
  2014
    show ?thesis using bounded unfolding 0 nth_Cons_0 fst_conv snd_conv bounded_by.simps ..
hoelzl@29805
  2015
  next
hoelzl@29805
  2016
    case (Suc i) with length have "i < length bs" by auto
hoelzl@29805
  2017
    show ?thesis unfolding Suc nth_Cons_Suc bounded_by.simps
hoelzl@29805
  2018
      using hyp[OF bounded[unfolded bounded_by.simps, THEN conjunct2] `i < length bs`] .
hoelzl@29805
  2019
  qed
hoelzl@29805
  2020
qed auto
hoelzl@29805
  2021
hoelzl@29805
  2022
fun approx approx' :: "nat \<Rightarrow> floatarith \<Rightarrow> (float * float) list \<Rightarrow> (float * float) option" where
hoelzl@29805
  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)" |
hoelzl@29805
  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))" | 
hoelzl@29805
  2025
"approx prec (Minus a) bs   = lift_un' (approx' prec a bs) (\<lambda> l u. (-u, -l))" |
hoelzl@29805
  2026
"approx prec (Mult a b) bs  = lift_bin' (approx' prec a bs) (approx' prec b bs)
hoelzl@29805
  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, 
hoelzl@29805
  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))" |
hoelzl@29805
  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))" |
hoelzl@29805
  2030
"approx prec (Sin a) bs     = lift_un' (approx' prec a bs) (bnds_sin prec)" |
hoelzl@29805
  2031
"approx prec (Cos a) bs     = lift_un' (approx' prec a bs) (bnds_cos prec)" |
hoelzl@29805
  2032
"approx prec Pi bs          = Some (lb_pi prec, ub_pi prec)" |
hoelzl@29805
  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))" |
hoelzl@29805
  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))" |
hoelzl@29805
  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>))" |
hoelzl@29805
  2036
"approx prec (Arctan a) bs  = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_arctan prec l, ub_arctan prec u))" |
hoelzl@29805
  2037
"approx prec (Sqrt a) bs    = lift_un (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" |
hoelzl@29805
  2038
"approx prec (Exp a) bs     = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" |
hoelzl@29805
  2039
"approx prec (Ln a) bs      = lift_un (approx' prec a bs) (\<lambda> l u. (lb_ln prec l, ub_ln prec u))" |
hoelzl@29805
  2040
"approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds n)" |
hoelzl@29805
  2041
"approx prec (Num f) bs     = Some (f, f)" |
hoelzl@29805
  2042
"approx prec (Atom i) bs    = (if i < length bs then Some (bs ! i) else None)"
hoelzl@29805
  2043
hoelzl@29805
  2044
lemma lift_bin'_ex:
hoelzl@29805
  2045
  assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f"
hoelzl@29805
  2046
  shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b"
hoelzl@29805
  2047
proof (cases a)
hoelzl@29805
  2048
  case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
hoelzl@29805
  2049
  thus ?thesis using lift_bin'_Some by auto
hoelzl@29805
  2050
next
hoelzl@29805
  2051
  case (Some a')
hoelzl@29805
  2052
  show ?thesis
hoelzl@29805
  2053
  proof (cases b)
hoelzl@29805
  2054
    case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
hoelzl@29805
  2055
    thus ?thesis using lift_bin'_Some by auto
hoelzl@29805
  2056
  next
hoelzl@29805
  2057
    case (Some b')
hoelzl@29805
  2058
    obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
hoelzl@29805
  2059
    obtain lb ub where b': "b' = (lb, ub)" by (cases b', auto)
hoelzl@29805
  2060
    thus ?thesis unfolding `a = Some a'` `b = Some b'` a' b' by auto
hoelzl@29805
  2061
  qed
hoelzl@29805
  2062
qed
hoelzl@29805
  2063
hoelzl@29805
  2064
lemma lift_bin'_f:
hoelzl@29805
  2065
  assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f"
hoelzl@29805
  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"
hoelzl@29805
  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)"
hoelzl@29805
  2068
proof -
hoelzl@29805
  2069
  obtain l1 u1 l2 u2
hoelzl@29805
  2070
    where Sa: "Some (l1, u1) = g a" and Sb: "Some (l2, u2) = g b" using lift_bin'_ex[OF assms(1)] by auto
hoelzl@29805
  2071
  have lu: "(l, u) = f l1 u1 l2 u2" using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto 
hoelzl@29805
  2072
  have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" unfolding lu[symmetric] by auto
hoelzl@29805
  2073
  thus ?thesis using Pa[OF Sa] Pb[OF Sb] by auto 
hoelzl@29805
  2074
qed
hoelzl@29805
  2075
hoelzl@29805
  2076
lemma approx_approx':
hoelzl@29805
  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"
hoelzl@29805
  2078
  and approx': "Some (l, u) = approx' prec a vs"
hoelzl@29805
  2079
  shows "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
hoelzl@29805
  2080
proof -
hoelzl@29805
  2081
  obtain l' u' where S: "Some (l', u') = approx prec a vs"
hoelzl@29805
  2082
    using approx' unfolding approx'.simps by (cases "approx prec a vs", auto)
hoelzl@29805
  2083
  have l': "l = round_down prec l'" and u': "u = round_up prec u'"
hoelzl@29805
  2084
    using approx' unfolding approx'.simps S[symmetric] by auto
hoelzl@29805
  2085
  show ?thesis unfolding l' u' 
hoelzl@29805
  2086
    using order_trans[OF Pa[OF S, THEN conjunct2] round_up[of u']]
hoelzl@29805
  2087
    using order_trans[OF round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto
hoelzl@29805
  2088
qed
hoelzl@29805
  2089
hoelzl@29805
  2090
lemma lift_bin':
hoelzl@29805
  2091
  assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f"
hoelzl@29805
  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")
hoelzl@29805
  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"
hoelzl@29805
  2094
  shows "\<exists> l1 u1 l2 u2. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> 
hoelzl@29805
  2095
                        (Ifloat l2 \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u2) \<and> 
hoelzl@29805
  2096
                        l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
hoelzl@29805
  2097
proof -
hoelzl@29805
  2098
  { fix l u assume "Some (l, u) = approx' prec a bs"
hoelzl@29805
  2099
    with approx_approx'[of prec a bs, OF _ this] Pa
hoelzl@29805
  2100
    have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
hoelzl@29805
  2101
  { fix l u assume "Some (l, u) = approx' prec b bs"
hoelzl@29805
  2102
    with approx_approx'[of prec b bs, OF _ this] Pb
hoelzl@29805
  2103
    have "Ifloat l \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u" by auto } note Pb = this
hoelzl@29805
  2104
hoelzl@29805
  2105
  from lift_bin'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb]
hoelzl@29805
  2106
  show ?thesis by auto
hoelzl@29805
  2107
qed
hoelzl@29805
  2108
hoelzl@29805
  2109
lemma lift_un'_ex:
hoelzl@29805
  2110
  assumes lift_un'_Some: "Some (l, u) = lift_un' a f"
hoelzl@29805
  2111
  shows "\<exists> l u. Some (l, u) = a"
hoelzl@29805
  2112
proof (cases a)
hoelzl@29805
  2113
  case None hence "None = lift_un' a f" unfolding None lift_un'.simps ..
hoelzl@29805
  2114
  thus ?thesis using lift_un'_Some by auto
hoelzl@29805
  2115
next
hoelzl@29805
  2116
  case (Some a')
hoelzl@29805
  2117
  obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
hoelzl@29805
  2118
  thus ?thesis unfolding `a = Some a'` a' by auto
hoelzl@29805
  2119
qed
hoelzl@29805
  2120
hoelzl@29805
  2121
lemma lift_un'_f:
hoelzl@29805
  2122
  assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f"
hoelzl@29805
  2123
  and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
hoelzl@29805
  2124
  shows "\<exists> l1 u1. P l1 u1 a \<and> l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
hoelzl@29805
  2125
proof -
hoelzl@29805
  2126
  obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un'_ex[OF assms(1)] by auto
hoelzl@29805
  2127
  have lu: "(l, u) = f l1 u1" using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto
hoelzl@29805
  2128
  have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" unfolding lu[symmetric] by auto
hoelzl@29805
  2129
  thus ?thesis using Pa[OF Sa] by auto
hoelzl@29805
  2130
qed
hoelzl@29805
  2131
hoelzl@29805
  2132
lemma lift_un':
hoelzl@29805
  2133
  assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
hoelzl@29805
  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")
hoelzl@29805
  2135
  shows "\<exists> l1 u1. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> 
hoelzl@29805
  2136
                        l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
hoelzl@29805
  2137
proof -
hoelzl@29805
  2138
  { fix l u assume "Some (l, u) = approx' prec a bs"
hoelzl@29805
  2139
    with approx_approx'[of prec a bs, OF _ this] Pa
hoelzl@29805
  2140
    have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
hoelzl@29805
  2141
  from lift_un'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa]
hoelzl@29805
  2142
  show ?thesis by auto
hoelzl@29805
  2143
qed
hoelzl@29805
  2144
hoelzl@29805
  2145
lemma lift_un'_bnds:
hoelzl@29805
  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"
hoelzl@29805
  2147
  and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
hoelzl@29805
  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"
hoelzl@29805
  2149
  shows "Ifloat l \<le> f' (Ifloatarith a xs) \<and> f' (Ifloatarith a xs) \<le> Ifloat u"
hoelzl@29805
  2150
proof -
hoelzl@29805
  2151
  from lift_un'[OF lift_un'_Some Pa]
hoelzl@29805
  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
hoelzl@29805
  2153
  hence "(l, u) = f l1 u1" and "Ifloatarith a xs \<in> {Ifloat l1 .. Ifloat u1}" by auto
hoelzl@29805
  2154
  thus ?thesis using bnds by auto
hoelzl@29805
  2155
qed
hoelzl@29805
  2156
hoelzl@29805
  2157
lemma lift_un_ex:
hoelzl@29805
  2158
  assumes lift_un_Some: "Some (l, u) = lift_un a f"
hoelzl@29805
  2159
  shows "\<exists> l u. Some (l, u) = a"
hoelzl@29805
  2160
proof (cases a)
hoelzl@29805
  2161
  case None hence "None = lift_un a f" unfolding None lift_un.simps ..
hoelzl@29805
  2162
  thus ?thesis using lift_un_Some by auto
hoelzl@29805
  2163
next
hoelzl@29805
  2164
  case (Some a')
hoelzl@29805
  2165
  obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
hoelzl@29805
  2166
  thus ?thesis unfolding `a = Some a'` a' by auto
hoelzl@29805
  2167
qed
hoelzl@29805
  2168
hoelzl@29805
  2169
lemma lift_un_f:
hoelzl@29805
  2170
  assumes lift_un_Some: "Some (l, u) = lift_un (g a) f"
hoelzl@29805
  2171
  and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
hoelzl@29805
  2172
  shows "\<exists> l1 u1. P l1 u1 a \<and> Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
hoelzl@29805
  2173
proof -
hoelzl@29805
  2174
  obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un_ex[OF assms(1)] by auto
hoelzl@29805
  2175
  have "fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None"
hoelzl@29805
  2176
  proof (rule ccontr)
hoelzl@29805
  2177
    assume "\<not> (fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None)"
hoelzl@29805
  2178
    hence or: "fst (f l1 u1) = None \<or> snd (f l1 u1) = None" by auto
hoelzl@29805
  2179
    hence "lift_un (g a) f = None" 
hoelzl@29805
  2180
    proof (cases "fst (f l1 u1) = None")
hoelzl@29805
  2181
      case True
hoelzl@29805
  2182
      then obtain b where b: "f l1 u1 = (None, b)" by (cases "f l1 u1", auto)
hoelzl@29805
  2183
      thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
hoelzl@29805
  2184
    next
hoelzl@29805
  2185
      case False hence "snd (f l1 u1) = None" using or by auto
hoelzl@29805
  2186
      with False obtain b where b: "f l1 u1 = (Some b, None)" by (cases "f l1 u1", auto)
hoelzl@29805
  2187
      thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
hoelzl@29805
  2188
    qed
hoelzl@29805
  2189
    thus False using lift_un_Some by auto
hoelzl@29805
  2190
  qed
hoelzl@29805
  2191
  then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" by (cases "f l1 u1", auto)
hoelzl@29805
  2192
  from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f]
hoelzl@29805
  2193
  have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" unfolding f by auto
hoelzl@29805
  2194
  thus ?thesis unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto
hoelzl@29805
  2195
qed
hoelzl@29805
  2196
hoelzl@29805
  2197
lemma lift_un:
hoelzl@29805
  2198
  assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
hoelzl@29805
  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")
hoelzl@29805
  2200
  shows "\<exists> l1 u1. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> 
hoelzl@29805
  2201
                  Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
hoelzl@29805
  2202
proof -
hoelzl@29805
  2203
  { fix l u assume "Some (l, u) = approx' prec a bs"
hoelzl@29805
  2204
    with approx_approx'[of prec a bs, OF _ this] Pa
hoelzl@29805
  2205
    have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
hoelzl@29805
  2206
  from lift_un_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa]
hoelzl@29805
  2207
  show ?thesis by auto
hoelzl@29805
  2208
qed
hoelzl@29805
  2209
hoelzl@29805
  2210
lemma lift_un_bnds:
hoelzl@29805
  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"
hoelzl@29805
  2212
  and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
hoelzl@29805
  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"
hoelzl@29805
  2214
  shows "Ifloat l \<le> f' (Ifloatarith a xs) \<and> f' (Ifloatarith a xs) \<le> Ifloat u"
hoelzl@29805
  2215
proof -
hoelzl@29805
  2216
  from lift_un[OF lift_un_Some Pa]
hoelzl@29805
  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
hoelzl@29805
  2218
  hence "(Some l, Some u) = f l1 u1" and "Ifloatarith a xs \<in> {Ifloat l1 .. Ifloat u1}" by auto
hoelzl@29805
  2219
  thus ?thesis using bnds by auto
hoelzl@29805
  2220
qed
hoelzl@29805
  2221
hoelzl@29805
  2222
lemma approx:
hoelzl@29805
  2223
  assumes "bounded_by xs vs"
hoelzl@29805
  2224
  and "Some (l, u) = approx prec arith vs" (is "_ = ?g arith")
hoelzl@29805
  2225
  shows "Ifloat l \<le> Ifloatarith arith xs \<and> Ifloatarith arith xs \<le> Ifloat u" (is "?P l u arith")
hoelzl@29805
  2226
  using `Some (l, u) = approx prec arith vs` 
hoelzl@29805
  2227
proof (induct arith arbitrary: l u x)
hoelzl@29805
  2228
  case (Add a b)
hoelzl@29805
  2229
  from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps
hoelzl@29805
  2230
  obtain l1 u1 l2 u2 where "l = l1 + l2" and "u = u1 + u2"
hoelzl@29805
  2231
    "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1"
hoelzl@29805
  2232
    "Ifloat l2 \<le> Ifloatarith b xs" and "Ifloatarith b xs \<le> Ifloat u2" unfolding fst_conv snd_conv by blast
hoelzl@29805
  2233
  thus ?case unfolding Ifloatarith.simps by auto
hoelzl@29805
  2234
next
hoelzl@29805
  2235
  case (Minus a)
hoelzl@29805
  2236
  from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps
hoelzl@29805
  2237
  obtain l1 u1 where "l = -u1" and "u = -l1"
hoelzl@29805
  2238
    "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" unfolding fst_conv snd_conv by blast
hoelzl@29805
  2239
  thus ?case unfolding Ifloatarith.simps using Ifloat_minus by auto
hoelzl@29805
  2240
next
hoelzl@29805
  2241
  case (Mult a b)
hoelzl@29805
  2242
  from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps
hoelzl@29805
  2243
  obtain l1 u1 l2 u2 
hoelzl@29805
  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"
hoelzl@29805
  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"
hoelzl@29805
  2246
    and "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1"
hoelzl@29805
  2247
    and "Ifloat l2 \<le> Ifloatarith b xs" and "Ifloatarith b xs \<le> Ifloat u2" unfolding fst_conv snd_conv by blast
hoelzl@29805
  2248
  thus ?case unfolding Ifloatarith.simps l u Ifloat_add Ifloat_mult Ifloat_nprt Ifloat_pprt 
hoelzl@29805
  2249
    using mult_le_prts mult_ge_prts by auto
hoelzl@29805
  2250
next
hoelzl@29805
  2251
  case (Inverse a)
hoelzl@29805
  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
hoelzl@29805
  2253
  obtain l1 u1 where l': "Some l = (if 0 < l1 \<or> u1 < 0 then Some (float_divl prec 1 u1) else None)" 
hoelzl@29805
  2254
    and u': "Some u = (if 0 < l1 \<or> u1 < 0 then Some (float_divr prec 1 l1) else None)"
hoelzl@29805
  2255
    and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1" by blast
hoelzl@29805
  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
hoelzl@29805
  2257
  moreover have l1_le_u1: "Ifloat l1 \<le> Ifloat u1" using l1 u1 by auto
hoelzl@29805
  2258
  ultimately have "Ifloat l1 \<noteq> 0" and "Ifloat u1 \<noteq> 0" unfolding less_float_def by auto
hoelzl@29805
  2259
hoelzl@29805
  2260
  have inv: "inverse (Ifloat u1) \<le> inverse (Ifloatarith a xs)
hoelzl@29805
  2261
           \<and> inverse (Ifloatarith a xs) \<le> inverse (Ifloat l1)"
hoelzl@29805
  2262
  proof (cases "0 < l1")
hoelzl@29805
  2263
    case True hence "0 < Ifloat u1" and "0 < Ifloat l1" "0 < Ifloatarith a xs" 
hoelzl@29805
  2264
      unfolding less_float_def using l1_le_u1 l1 by auto
hoelzl@29805
  2265
    show ?thesis
hoelzl@29805
  2266
      unfolding inverse_le_iff_le[OF `0 < Ifloat u1` `0 < Ifloatarith a xs`]
hoelzl@29805
  2267
	inverse_le_iff_le[OF `0 < Ifloatarith a xs` `0 < Ifloat l1`]
hoelzl@29805
  2268
      using l1 u1 by auto
hoelzl@29805
  2269
  next
hoelzl@29805
  2270
    case False hence "u1 < 0" using either by blast
hoelzl@29805
  2271
    hence "Ifloat u1 < 0" and "Ifloat l1 < 0" "Ifloatarith a xs < 0" 
hoelzl@29805
  2272
      unfolding less_float_def using l1_le_u1 u1 by auto
hoelzl@29805
  2273
    show ?thesis
hoelzl@29805
  2274
      unfolding inverse_le_iff_le_neg[OF `Ifloat u1 < 0` `Ifloatarith a xs < 0`]
hoelzl@29805
  2275
	inverse_le_iff_le_neg[OF `Ifloatarith a xs < 0` `Ifloat l1 < 0`]
hoelzl@29805
  2276
      using l1 u1 by auto
hoelzl@29805
  2277
  qed
hoelzl@29805
  2278
    
hoelzl@29805
  2279
  from l' have "l = float_divl prec 1 u1" by (cases "0 < l1 \<or> u1 < 0", auto)
hoelzl@29805
  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
hoelzl@29805
  2281
  also have "\<dots> \<le> inverse (Ifloatarith a xs)" using inv by auto
hoelzl@29805
  2282
  finally have "Ifloat l \<le> inverse (Ifloatarith a xs)" .
hoelzl@29805
  2283
  moreover
hoelzl@29805
  2284
  from u' have "u = float_divr prec 1 l1" by (cases "0 < l1 \<or> u1 < 0", auto)
hoelzl@29805
  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
hoelzl@29805
  2286
  hence "inverse (Ifloatarith a xs) \<le> Ifloat u" by (rule order_trans[OF inv[THEN conjunct2]])
hoelzl@29805
  2287
  ultimately show ?case unfolding Ifloatarith.simps using l1 u1 by auto
hoelzl@29805
  2288
next
hoelzl@29805
  2289
  case (Abs x)
hoelzl@29805
  2290
  from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps
hoelzl@29805
  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>"
hoelzl@29805
  2292
    and l1: "Ifloat l1 \<le> Ifloatarith x xs" and u1: "Ifloatarith x xs \<le> Ifloat u1" by blast
hoelzl@29805
  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)
hoelzl@29805
  2294
next
hoelzl@29805
  2295
  case (Min a b)
hoelzl@29805
  2296
  from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps
hoelzl@29805
  2297
  obtain l1 u1 l2 u2 where l': "l = min l1 l2" and u': "u = min u1 u2"
hoelzl@29805
  2298
    and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1"
hoelzl@29805
  2299
    and l1: "Ifloat l2 \<le> Ifloatarith b xs" and u1: "Ifloatarith b xs \<le> Ifloat u2" by blast
hoelzl@29805
  2300
  thus ?case unfolding l' u' by (auto simp add: Ifloat_min)
hoelzl@29805
  2301
next
hoelzl@29805
  2302
  case (Max a b)
hoelzl@29805
  2303
  from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps
hoelzl@29805
  2304
  obtain l1 u1 l2 u2 where l': "l = max l1 l2" and u': "u = max u1 u2"
hoelzl@29805
  2305
    and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1"
hoelzl@29805
  2306
    and l1: "Ifloat l2 \<le> Ifloatarith b xs" and u1: "Ifloatarith b xs \<le> Ifloat u2" by blast
hoelzl@29805
  2307
  thus ?case unfolding l' u' by (auto simp add: Ifloat_max)
hoelzl@29805
  2308
next case (Sin a) with lift_un'_bnds[OF bnds_sin] show ?case by auto
hoelzl@29805
  2309
next case (Cos a) with lift_un'_bnds[OF bnds_cos] show ?case by auto
hoelzl@29805
  2310
next case (Arctan a) with lift_un'_bnds[OF bnds_arctan] show ?case by auto
hoelzl@29805
  2311
next case Pi with pi_boundaries show ?case by auto
hoelzl@29805
  2312
next case (Sqrt a) with lift_un_bnds[OF bnds_sqrt] show ?case by auto
hoelzl@29805
  2313
next case (Exp a) with lift_un'_bnds[OF bnds_exp] show ?case by auto
hoelzl@29805
  2314
next case (Ln a) with lift_un_bnds[OF bnds_ln] show ?case by auto
hoelzl@29805
  2315
next case (Power a n) with lift_un'_bnds[OF bnds_power] show ?case by auto
hoelzl@29805
  2316
next case (Num f) thus ?case by auto
hoelzl@29805
  2317
next
hoelzl@29805
  2318
  case (Atom n) 
hoelzl@29805
  2319
  show ?case
hoelzl@29805
  2320
  proof (cases "n < length vs")
hoelzl@29805
  2321
    case True
hoelzl@29805
  2322
    with Atom have "vs ! n = (l, u)" by auto
hoelzl@29805
  2323
    thus ?thesis using bounded_by[OF assms(1) True] by auto
hoelzl@29805
  2324
  next
hoelzl@29805
  2325
    case False thus ?thesis using Atom by auto
hoelzl@29805
  2326
  qed
hoelzl@29805
  2327
qed
hoelzl@29805
  2328
hoelzl@29805
  2329
datatype ApproxEq = Less floatarith floatarith 
hoelzl@29805
  2330
                  | LessEqual floatarith floatarith 
hoelzl@29805
  2331
hoelzl@29805
  2332
fun uneq :: "ApproxEq \<Rightarrow> real list \<Rightarrow> bool" where 
hoelzl@29805
  2333
"uneq (Less a b) vs                   = (Ifloatarith a vs < Ifloatarith b vs)" |
hoelzl@29805
  2334
"uneq (LessEqual a b) vs              = (Ifloatarith a vs \<le> Ifloatarith b vs)"
hoelzl@29805
  2335
hoelzl@29805
  2336
fun uneq' :: "nat \<Rightarrow> ApproxEq \<Rightarrow> (float * float) list \<Rightarrow> bool" where 
hoelzl@29805
  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)" |
hoelzl@29805
  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)"
hoelzl@29805
  2339
hoelzl@29805
  2340
lemma uneq_approx: fixes m :: nat assumes "bounded_by vs bs" and "uneq' prec eq bs"
hoelzl@29805
  2341
  shows "uneq eq vs"
hoelzl@29805
  2342
proof (cases eq)
hoelzl@29805
  2343
  case (Less a b)
hoelzl@29805
  2344
  show ?thesis
hoelzl@29805
  2345
  proof (cases "\<exists> u l u' l'. approx prec a bs = Some (l, u) \<and> 
hoelzl@29805
  2346
                             approx prec b bs = Some (l', u')")
hoelzl@29805
  2347
    case True
hoelzl@29805
  2348
    then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)"
hoelzl@29805
  2349
      and b_approx: "approx prec b bs = Some (l', u') " by auto
hoelzl@29805
  2350
    with `uneq' prec eq bs` have "Ifloat u < Ifloat l'"
hoelzl@29805
  2351
      unfolding Less uneq'.simps less_float_def by auto
hoelzl@29805
  2352
    moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs`
hoelzl@29805
  2353
    have "Ifloatarith a vs \<le> Ifloat u" and "Ifloat l' \<le> Ifloatarith b vs"
hoelzl@29805
  2354
      using approx by auto
hoelzl@29805
  2355
    ultimately show ?thesis unfolding uneq.simps Less by auto
hoelzl@29805
  2356
  next
hoelzl@29805
  2357
    case False
hoelzl@29805
  2358
    hence "approx prec a bs = None \<or> approx prec b bs = None"
hoelzl@29805
  2359
      unfolding not_Some_eq[symmetric] by auto
hoelzl@29805
  2360
    hence "\<not> uneq' prec eq bs" unfolding Less uneq'.simps 
hoelzl@29805
  2361
      by (cases "approx prec a bs = None", auto)
hoelzl@29805
  2362
    thus ?thesis using assms by auto
hoelzl@29805
  2363
  qed
hoelzl@29805
  2364
next
hoelzl@29805
  2365
  case (LessEqual a b)
hoelzl@29805
  2366
  show ?thesis
hoelzl@29805
  2367
  proof (cases "\<exists> u l u' l'. approx prec a bs = Some (l, u) \<and> 
hoelzl@29805
  2368
                             approx prec b bs = Some (l', u')")
hoelzl@29805
  2369
    case True
hoelzl@29805
  2370
    then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)"
hoelzl@29805
  2371
      and b_approx: "approx prec b bs = Some (l', u') " by auto
hoelzl@29805
  2372
    with `uneq' prec eq bs` have "Ifloat u \<le> Ifloat l'"
hoelzl@29805
  2373
      unfolding LessEqual uneq'.simps le_float_def by auto
hoelzl@29805
  2374
    moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs`
hoelzl@29805
  2375
    have "Ifloatarith a vs \<le> Ifloat u" and "Ifloat l' \<le> Ifloatarith b vs"
hoelzl@29805
  2376
      using approx by auto
hoelzl@29805
  2377
    ultimately show ?thesis unfolding uneq.simps LessEqual by auto
hoelzl@29805
  2378
  next
hoelzl@29805
  2379
    case False
hoelzl@29805
  2380
    hence "approx prec a bs = None \<or> approx prec b bs = None"
hoelzl@29805
  2381
      unfolding not_Some_eq[symmetric] by auto
hoelzl@29805
  2382
    hence "\<not> uneq' prec eq bs" unfolding LessEqual uneq'.simps 
hoelzl@29805
  2383
      by (cases "approx prec a bs = None", auto)
hoelzl@29805
  2384
    thus ?thesis using assms by auto
hoelzl@29805
  2385
  qed
hoelzl@29805
  2386
qed
hoelzl@29805
  2387
hoelzl@29805
  2388
lemma Ifloatarith_divide: "Ifloatarith (Mult a (Inverse b)) vs = (Ifloatarith a vs) / (Ifloatarith b vs)"
hoelzl@29805
  2389
  unfolding real_divide_def Ifloatarith.simps ..
hoelzl@29805
  2390
hoelzl@29805
  2391
lemma Ifloatarith_diff: "Ifloatarith (Add a (Minus b)) vs = (Ifloatarith a vs) - (Ifloatarith b vs)"
hoelzl@29805
  2392
  unfolding real_diff_def Ifloatarith.simps ..
hoelzl@29805
  2393
hoelzl@29805
  2394
lemma Ifloatarith_tan: "Ifloatarith (Mult (Sin a) (Inverse (Cos a))) vs = tan (Ifloatarith a vs)"
hoelzl@29805
  2395
  unfolding tan_def Ifloatarith.simps real_divide_def ..
hoelzl@29805
  2396
hoelzl@29805
  2397
lemma Ifloatarith_powr: "Ifloatarith (Exp (Mult b (Ln a))) vs = (Ifloatarith a vs) powr (Ifloatarith b vs)"
hoelzl@29805
  2398
  unfolding powr_def Ifloatarith.simps ..
hoelzl@29805
  2399
hoelzl@29805
  2400
lemma Ifloatarith_log: "Ifloatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (Ifloatarith b vs) (Ifloatarith x vs)"
hoelzl@29805
  2401
  unfolding log_def Ifloatarith.simps real_divide_def ..
hoelzl@29805
  2402
hoelzl@29805
  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
hoelzl@29805
  2404
hoelzl@29805
  2405
subsection {* Implement proof method \texttt{approximation} *}
hoelzl@29805
  2406
hoelzl@29805
  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)
hoelzl@29805
  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)
hoelzl@29805
  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)"
hoelzl@29805
  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)"
hoelzl@29805
  2411
  by (auto simp add: Ifloat.simps pow2_def)
hoelzl@29805
  2412
hoelzl@29805
  2413
lemmas bounded_by_equations = bounded_by_Cons bounded_by_Nil float_power bounded_divl bounded_divr bounded_num HOL.simp_thms
hoelzl@29805
  2414
lemmas uneq_equations = uneq.simps Ifloatarith.simps Ifloatarith_num Ifloatarith_divide Ifloatarith_diff Ifloatarith_tan Ifloatarith_powr Ifloatarith_log
hoelzl@29805
  2415
hoelzl@29805
  2416
lemma "x div (0::int) = 0" by auto -- "What happens in the zero case for div"
hoelzl@29805
  2417
lemma "x mod (0::int) = x" by auto -- "What happens in the zero case for mod"
hoelzl@29805
  2418
hoelzl@29805
  2419
text {* The following equations must hold for div & mod 
hoelzl@29805
  2420
        -- see "The Definition of Standard ML" by R. Milner, M. Tofte and R. Harper (pg. 79) *}
hoelzl@29805
  2421
lemma "d * (i div d) + i mod d = (i::int)" by auto
hoelzl@29805
  2422
lemma "0 < (d :: int) \<Longrightarrow> 0 \<le> i mod d \<and> i mod d < d" by auto
hoelzl@29805
  2423
lemma "(d :: int) < 0 \<Longrightarrow> d < i mod d \<and> i mod d \<le> 0" by auto
hoelzl@29805
  2424
hoelzl@29805
  2425
code_const "op div :: int \<Rightarrow> int \<Rightarrow> int" (SML "(fn i => fn d => if d = 0 then 0 else i div d)")
hoelzl@29805
  2426
code_const "op mod :: int \<Rightarrow> int \<Rightarrow> int" (SML "(fn i => fn d => if d = 0 then i else i mod d)")
hoelzl@29805
  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))")
hoelzl@29805
  2428
hoelzl@29805
  2429
ML {*
hoelzl@29805
  2430
  val uneq_equations = PureThy.get_thms @{theory} "uneq_equations";
hoelzl@29805
  2431
  val bounded_by_equations = PureThy.get_thms @{theory} "bounded_by_equations";
hoelzl@29805
  2432
  val bounded_by_simpset = (HOL_basic_ss addsimps bounded_by_equations)
hoelzl@29805
  2433
hoelzl@29805
  2434
  fun reify_uneq ctxt i = (fn st =>
hoelzl@29805
  2435
    let
hoelzl@29805
  2436
      val to = HOLogic.dest_Trueprop (Logic.strip_imp_concl (List.nth (prems_of st, i - 1)))
hoelzl@29805
  2437
    in (Reflection.genreify_tac ctxt uneq_equations (SOME to) i) st
hoelzl@29805
  2438
    end)
hoelzl@29805
  2439
hoelzl@29805
  2440
  fun rule_uneq ctxt prec i thm = let
hoelzl@29805
  2441
    fun conv_num typ = HOLogic.dest_number #> snd #> HOLogic.mk_number typ
hoelzl@29805
  2442
    val to_natc = conv_num @{typ "nat"} #> Thm.cterm_of (ProofContext.theory_of ctxt)
hoelzl@29805
  2443
    val to_nat = conv_num @{typ "nat"}
hoelzl@29805
  2444
    val to_int = conv_num @{typ "int"}
hoelzl@29805
  2445
hoelzl@29805
  2446
    val prec' = to_nat prec
hoelzl@29805
  2447
hoelzl@29805
  2448
    fun bot_float (Const (@{const_name "times"}, _) $ mantisse $ (Const (@{const_name "pow2"}, _) $ exp))
hoelzl@29805
  2449
                   = @{term "Float"} $ to_int mantisse $ to_int exp
hoelzl@29805
  2450
      | bot_float (Const (@{const_name "divide"}, _) $ mantisse $ (Const (@{const_name "power"}, _) $ ten $ exp))
hoelzl@29805
  2451
                   = @{term "float_divl"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ (@{term "power_float (Float 5 1)"} $ to_nat exp)
hoelzl@29805
  2452
      | bot_float (Const (@{const_name "divide"}, _) $ mantisse $ ten)
hoelzl@29805
  2453
                   = @{term "float_divl"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ @{term "Float 5 1"}
hoelzl@29805
  2454
      | bot_float mantisse = @{term "Float"} $ to_int mantisse $ @{term "0 :: int"}
hoelzl@29805
  2455
hoelzl@29805
  2456
    fun top_float (Const (@{const_name "times"}, _) $ mantisse $ (Const (@{const_name "pow2"}, _) $ exp))
hoelzl@29805
  2457
                   = @{term "Float"} $ to_int mantisse $ to_int exp
hoelzl@29805
  2458
      | top_float (Const (@{const_name "divide"}, _) $ mantisse $ (Const (@{const_name "power"}, _) $ ten $ exp))