hoelzl@30437
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(* Author: Johannes Hoelzl <hoelzl@in.tum.de> 2008 / 2009 *)
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wenzelm@30122
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wenzelm@30886
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header {* Prove Real Valued Inequalities by Computation *}
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wenzelm@30122
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hoelzl@29742
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theory Approximation
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haftmann@29760
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imports Complex_Main Float Reflection Dense_Linear_Order Efficient_Nat
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hoelzl@29742
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begin
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hoelzl@29742
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hoelzl@29742
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section "Horner Scheme"
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hoelzl@29742
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hoelzl@29742
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subsection {* Define auxiliary helper @{text horner} function *}
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hoelzl@29742
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hoelzl@31098
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primrec 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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hoelzl@29742
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"horner F G 0 i k x = 0" |
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hoelzl@29742
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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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hoelzl@29742
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hoelzl@29742
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lemma horner_schema': fixes x :: real and a :: "nat \<Rightarrow> real"
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hoelzl@29742
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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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hoelzl@29742
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proof -
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hoelzl@29742
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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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huffman@36770
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show ?thesis unfolding setsum_right_distrib shift_pow diff_def setsum_negf[symmetric] setsum_head_upt_Suc[OF zero_less_Suc]
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hoelzl@29742
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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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hoelzl@29742
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qed
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hoelzl@29742
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hoelzl@29742
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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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haftmann@30971
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assumes f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
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haftmann@30971
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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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hoelzl@29742
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proof (induct n arbitrary: i k j')
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hoelzl@29742
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case (Suc n)
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hoelzl@29742
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hoelzl@29742
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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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hoelzl@29742
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using horner_schema'[of "\<lambda> j. 1 / real (f (j' + j))"] by auto
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hoelzl@29742
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qed auto
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hoelzl@29742
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hoelzl@29742
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lemma horner_bounds':
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hoelzl@31098
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assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
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hoelzl@29742
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and lb_0: "\<And> i k x. lb 0 i k x = 0"
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hoelzl@29742
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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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hoelzl@29742
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and ub_0: "\<And> i k x. ub 0 i k x = 0"
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hoelzl@29742
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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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hoelzl@31809
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shows "real (lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') (real x) \<and>
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hoelzl@31098
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horner F G n ((F ^^ j') s) (f j') (real x) \<le> real (ub n ((F ^^ j') s) (f j') x)"
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hoelzl@29742
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(is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'")
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hoelzl@29742
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proof (induct n arbitrary: j')
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hoelzl@29742
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case 0 thus ?case unfolding lb_0 ub_0 horner.simps by auto
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hoelzl@29742
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next
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hoelzl@29742
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case (Suc n)
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hoelzl@31098
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have "?lb (Suc n) j' \<le> ?horner (Suc n) j'" unfolding lb_Suc ub_Suc horner.simps real_of_float_sub diff_def
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hoelzl@29742
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proof (rule add_mono)
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hoelzl@31098
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show "real (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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hoelzl@31098
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from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct2] `0 \<le> real x`
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hoelzl@31098
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show "- real (x * ub n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) x) \<le> - (real x * horner F G n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) (real x))"
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hoelzl@31098
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unfolding real_of_float_mult neg_le_iff_le by (rule mult_left_mono)
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hoelzl@29742
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qed
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hoelzl@31098
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moreover have "?horner (Suc n) j' \<le> ?ub (Suc n) j'" unfolding ub_Suc ub_Suc horner.simps real_of_float_sub diff_def
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hoelzl@29742
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proof (rule add_mono)
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hoelzl@31098
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show "1 / real (f j') \<le> real (rapprox_rat prec 1 (int (f j')))" using rapprox_rat[of 1 "int (f j')" prec] by auto
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hoelzl@31098
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from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct1] `0 \<le> real x`
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hoelzl@31809
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show "- (real x * horner F G n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) (real x)) \<le>
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hoelzl@31098
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- real (x * lb n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) x)"
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hoelzl@31098
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unfolding real_of_float_mult neg_le_iff_le by (rule mult_left_mono)
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hoelzl@29742
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qed
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hoelzl@29742
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ultimately show ?case by blast
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hoelzl@29742
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qed
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hoelzl@29742
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hoelzl@29742
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subsection "Theorems for floating point functions implementing the horner scheme"
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hoelzl@29742
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hoelzl@29742
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text {*
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hoelzl@29742
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hoelzl@29742
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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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hoelzl@29742
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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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hoelzl@29742
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hoelzl@29742
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*}
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hoelzl@29742
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hoelzl@29742
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lemma horner_bounds: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
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hoelzl@31098
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assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
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hoelzl@29742
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and lb_0: "\<And> i k x. lb 0 i k x = 0"
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hoelzl@29742
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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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hoelzl@29742
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and ub_0: "\<And> i k x. ub 0 i k x = 0"
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hoelzl@29742
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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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hoelzl@31809
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shows "real (lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. -1 ^ j * (1 / real (f (j' + j))) * real x ^ j)" (is "?lb") and
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hoelzl@31098
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"(\<Sum>j=0..<n. -1 ^ j * (1 / real (f (j' + j))) * (real x ^ j)) \<le> real (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
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hoelzl@29742
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proof -
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hoelzl@31809
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have "?lb \<and> ?ub"
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hoelzl@31098
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using horner_bounds'[where lb=lb, OF `0 \<le> real x` f_Suc lb_0 lb_Suc ub_0 ub_Suc]
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hoelzl@29742
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unfolding horner_schema[where f=f, OF f_Suc] .
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hoelzl@29742
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thus "?lb" and "?ub" by auto
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hoelzl@29742
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qed
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hoelzl@29742
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hoelzl@29742
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lemma horner_bounds_nonpos: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
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hoelzl@31098
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assumes "real x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
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hoelzl@29742
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and lb_0: "\<And> i k x. lb 0 i k x = 0"
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hoelzl@29742
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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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hoelzl@29742
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and ub_0: "\<And> i k x. ub 0 i k x = 0"
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hoelzl@29742
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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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hoelzl@31809
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shows "real (lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / real (f (j' + j))) * real x ^ j)" (is "?lb") and
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hoelzl@31098
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"(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (real x ^ j)) \<le> real (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
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hoelzl@29742
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proof -
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hoelzl@29742
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{ fix x y z :: float have "x - y * z = x + - y * z"
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haftmann@30968
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by (cases x, cases y, cases z, simp add: plus_float.simps minus_float_def uminus_float.simps times_float.simps algebra_simps)
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hoelzl@29742
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} note diff_mult_minus = this
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hoelzl@29742
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hoelzl@29742
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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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hoelzl@29742
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hoelzl@31098
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have move_minus: "real (-x) = -1 * real x" by auto
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hoelzl@29742
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hoelzl@31809
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have sum_eq: "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * real x ^ j) =
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hoelzl@31098
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(\<Sum>j = 0..<n. -1 ^ j * (1 / real (f (j' + j))) * real (- x) ^ j)"
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hoelzl@29742
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proof (rule setsum_cong, simp)
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hoelzl@29742
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fix j assume "j \<in> {0 ..< n}"
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hoelzl@31098
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show "1 / real (f (j' + j)) * real x ^ j = -1 ^ j * (1 / real (f (j' + j))) * real (- x) ^ j"
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huffman@36770
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unfolding move_minus power_mult_distrib mult_assoc[symmetric]
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huffman@36770
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unfolding mult_commute unfolding mult_assoc[of "-1 ^ j", symmetric] power_mult_distrib[symmetric]
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hoelzl@29742
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by auto
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hoelzl@29742
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qed
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hoelzl@29742
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hoelzl@31098
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have "0 \<le> real (-x)" using assms by auto
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hoelzl@29742
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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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hoelzl@29742
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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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hoelzl@29742
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OF this f_Suc lb_0 refl ub_0 refl]
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hoelzl@29742
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show "?lb" and "?ub" unfolding minus_minus sum_eq
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hoelzl@29742
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by auto
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hoelzl@29742
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qed
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hoelzl@29742
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hoelzl@29742
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subsection {* Selectors for next even or odd number *}
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hoelzl@29742
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hoelzl@29742
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text {*
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hoelzl@29742
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hoelzl@29742
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The horner scheme computes alternating series. To get the upper and lower bounds we need to
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hoelzl@29742
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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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hoelzl@29742
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hoelzl@29742
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*}
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hoelzl@29742
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hoelzl@29742
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definition get_odd :: "nat \<Rightarrow> nat" where
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hoelzl@29742
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"get_odd n = (if odd n then n else (Suc n))"
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hoelzl@29742
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hoelzl@29742
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definition get_even :: "nat \<Rightarrow> nat" where
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hoelzl@29742
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"get_even n = (if even n then n else (Suc n))"
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hoelzl@29742
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hoelzl@29742
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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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hoelzl@29742
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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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hoelzl@29742
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lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)"
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hoelzl@29742
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proof (cases "odd n")
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hoelzl@29742
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case True hence "0 < n" by (rule odd_pos)
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hoelzl@31467
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from gr0_implies_Suc[OF this] obtain k where "Suc k = n" by auto
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hoelzl@29742
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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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hoelzl@29742
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next
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hoelzl@29742
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case False hence "odd (Suc n)" by auto
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hoelzl@29742
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thus ?thesis unfolding get_odd_def if_not_P[OF False] by blast
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hoelzl@29742
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qed
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hoelzl@29742
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hoelzl@29742
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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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hoelzl@29742
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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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hoelzl@29742
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hoelzl@29742
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section "Power function"
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hoelzl@29742
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hoelzl@29742
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definition float_power_bnds :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
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hoelzl@29742
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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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hoelzl@29742
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else if u < 0 then (u ^ n, l ^ n)
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hoelzl@29742
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else (0, (max (-l) u) ^ n))"
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hoelzl@29742
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hoelzl@31098
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lemma float_power_bnds: assumes "(l1, u1) = float_power_bnds n l u" and "x \<in> {real l .. real u}"
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hoelzl@31098
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shows "x ^ n \<in> {real l1..real u1}"
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hoelzl@29742
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proof (cases "even n")
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hoelzl@31467
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case True
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hoelzl@29742
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show ?thesis
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hoelzl@29742
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proof (cases "0 < l")
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hoelzl@31098
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case True hence "odd n \<or> 0 < l" and "0 \<le> real l" unfolding less_float_def by auto
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hoelzl@29742
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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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hoelzl@31098
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have "real l ^ n \<le> x ^ n" and "x ^ n \<le> real u ^ n " using `0 \<le> real l` and assms unfolding atLeastAtMost_iff using power_mono[of "real l" x] power_mono[of x "real u"] by auto
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hoelzl@29742
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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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hoelzl@29742
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next
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hoelzl@29742
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case False hence P: "\<not> (odd n \<or> 0 < l)" using `even n` by auto
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hoelzl@29742
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show ?thesis
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hoelzl@29742
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proof (cases "u < 0")
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hoelzl@31098
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case True hence "0 \<le> - real u" and "- real u \<le> - x" and "0 \<le> - x" and "-x \<le> - real l" using assms unfolding less_float_def by auto
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hoelzl@31809
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hence "real u ^ n \<le> x ^ n" and "x ^ n \<le> real l ^ n" using power_mono[of "-x" "-real l" n] power_mono[of "-real u" "-x" n]
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wenzelm@32962
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unfolding power_minus_even[OF `even n`] by auto
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hoelzl@29742
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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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hoelzl@29742
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ultimately show ?thesis using float_power by auto
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hoelzl@29742
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next
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hoelzl@31467
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case False
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hoelzl@31098
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have "\<bar>x\<bar> \<le> real (max (-l) u)"
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hoelzl@29742
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proof (cases "-l \<le> u")
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wenzelm@32962
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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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hoelzl@29742
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next
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wenzelm@32962
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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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hoelzl@29742
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qed
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hoelzl@31098
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hence x_abs: "\<bar>x\<bar> \<le> \<bar>real (max (-l) u)\<bar>" by auto
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hoelzl@29742
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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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hoelzl@29742
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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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hoelzl@29742
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qed
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hoelzl@29742
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qed
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hoelzl@29742
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next
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hoelzl@29742
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case False hence "odd n \<or> 0 < l" by auto
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hoelzl@29742
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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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hoelzl@31098
|
197 |
have "real l ^ n \<le> x ^ n" and "x ^ n \<le> real u ^ n " using assms unfolding atLeastAtMost_iff using power_mono_odd[OF False] by auto
|
hoelzl@29742
|
198 |
thus ?thesis unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto
|
hoelzl@29742
|
199 |
qed
|
hoelzl@29742
|
200 |
|
hoelzl@31098
|
201 |
lemma bnds_power: "\<forall> x l u. (l1, u1) = float_power_bnds n l u \<and> x \<in> {real l .. real u} \<longrightarrow> real l1 \<le> x ^ n \<and> x ^ n \<le> real u1"
|
hoelzl@29742
|
202 |
using float_power_bnds by auto
|
hoelzl@29742
|
203 |
|
hoelzl@29742
|
204 |
section "Square root"
|
hoelzl@29742
|
205 |
|
hoelzl@29742
|
206 |
text {*
|
hoelzl@29742
|
207 |
|
hoelzl@29742
|
208 |
The square root computation is implemented as newton iteration. As first first step we use the
|
hoelzl@29742
|
209 |
nearest power of two greater than the square root.
|
hoelzl@29742
|
210 |
|
hoelzl@29742
|
211 |
*}
|
hoelzl@29742
|
212 |
|
hoelzl@29742
|
213 |
fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
|
hoelzl@29742
|
214 |
"sqrt_iteration prec 0 (Float m e) = Float 1 ((e + bitlen m) div 2 + 1)" |
|
hoelzl@31467
|
215 |
"sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x
|
hoelzl@29742
|
216 |
in Float 1 -1 * (y + float_divr prec x y))"
|
hoelzl@29742
|
217 |
|
hoelzl@31467
|
218 |
function ub_sqrt lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float" where
|
hoelzl@31467
|
219 |
"ub_sqrt prec x = (if 0 < x then (sqrt_iteration prec prec x)
|
hoelzl@31467
|
220 |
else if x < 0 then - lb_sqrt prec (- x)
|
hoelzl@31467
|
221 |
else 0)" |
|
hoelzl@31467
|
222 |
"lb_sqrt prec x = (if 0 < x then (float_divl prec x (sqrt_iteration prec prec x))
|
hoelzl@31467
|
223 |
else if x < 0 then - ub_sqrt prec (- x)
|
hoelzl@31467
|
224 |
else 0)"
|
hoelzl@31467
|
225 |
by pat_completeness auto
|
hoelzl@31467
|
226 |
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@29742
|
227 |
|
hoelzl@31467
|
228 |
declare lb_sqrt.simps[simp del]
|
hoelzl@31467
|
229 |
declare ub_sqrt.simps[simp del]
|
hoelzl@29742
|
230 |
|
hoelzl@29742
|
231 |
lemma sqrt_ub_pos_pos_1:
|
hoelzl@29742
|
232 |
assumes "sqrt x < b" and "0 < b" and "0 < x"
|
hoelzl@29742
|
233 |
shows "sqrt x < (b + x / b)/2"
|
hoelzl@29742
|
234 |
proof -
|
hoelzl@29742
|
235 |
from assms have "0 < (b - sqrt x) ^ 2 " by simp
|
hoelzl@29742
|
236 |
also have "\<dots> = b ^ 2 - 2 * b * sqrt x + (sqrt x) ^ 2" by algebra
|
hoelzl@29742
|
237 |
also have "\<dots> = b ^ 2 - 2 * b * sqrt x + x" using assms by (simp add: real_sqrt_pow2)
|
hoelzl@29742
|
238 |
finally have "0 < b ^ 2 - 2 * b * sqrt x + x" by assumption
|
hoelzl@29742
|
239 |
hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms
|
hoelzl@29742
|
240 |
by (simp add: field_simps power2_eq_square)
|
hoelzl@29742
|
241 |
thus ?thesis by (simp add: field_simps)
|
hoelzl@29742
|
242 |
qed
|
hoelzl@29742
|
243 |
|
hoelzl@31098
|
244 |
lemma sqrt_iteration_bound: assumes "0 < real x"
|
hoelzl@31098
|
245 |
shows "sqrt (real x) < real (sqrt_iteration prec n x)"
|
hoelzl@29742
|
246 |
proof (induct n)
|
hoelzl@29742
|
247 |
case 0
|
hoelzl@29742
|
248 |
show ?case
|
hoelzl@29742
|
249 |
proof (cases x)
|
hoelzl@29742
|
250 |
case (Float m e)
|
hoelzl@29742
|
251 |
hence "0 < m" using float_pos_m_pos[unfolded less_float_def] assms by auto
|
hoelzl@29742
|
252 |
hence "0 < sqrt (real m)" by auto
|
hoelzl@29742
|
253 |
|
hoelzl@29742
|
254 |
have int_nat_bl: "int (nat (bitlen m)) = bitlen m" using bitlen_ge0 by auto
|
hoelzl@29742
|
255 |
|
hoelzl@31098
|
256 |
have "real x = (real m / 2^nat (bitlen m)) * pow2 (e + int (nat (bitlen m)))"
|
hoelzl@31098
|
257 |
unfolding pow2_add pow2_int Float real_of_float_simp by auto
|
hoelzl@29742
|
258 |
also have "\<dots> < 1 * pow2 (e + int (nat (bitlen m)))"
|
hoelzl@29742
|
259 |
proof (rule mult_strict_right_mono, auto)
|
hoelzl@31467
|
260 |
show "real m < 2^nat (bitlen m)" using bitlen_bounds[OF `0 < m`, THEN conjunct2]
|
wenzelm@32962
|
261 |
unfolding real_of_int_less_iff[of m, symmetric] by auto
|
hoelzl@29742
|
262 |
qed
|
hoelzl@31098
|
263 |
finally have "sqrt (real x) < sqrt (pow2 (e + bitlen m))" unfolding int_nat_bl by auto
|
hoelzl@29742
|
264 |
also have "\<dots> \<le> pow2 ((e + bitlen m) div 2 + 1)"
|
hoelzl@29742
|
265 |
proof -
|
hoelzl@29742
|
266 |
let ?E = "e + bitlen m"
|
hoelzl@29742
|
267 |
have E_mod_pow: "pow2 (?E mod 2) < 4"
|
hoelzl@29742
|
268 |
proof (cases "?E mod 2 = 1")
|
wenzelm@32962
|
269 |
case True thus ?thesis by auto
|
hoelzl@29742
|
270 |
next
|
wenzelm@32962
|
271 |
case False
|
wenzelm@32962
|
272 |
have "0 \<le> ?E mod 2" by auto
|
wenzelm@32962
|
273 |
have "?E mod 2 < 2" by auto
|
wenzelm@32962
|
274 |
from this[THEN zless_imp_add1_zle]
|
wenzelm@32962
|
275 |
have "?E mod 2 \<le> 0" using False by auto
|
wenzelm@32962
|
276 |
from xt1(5)[OF `0 \<le> ?E mod 2` this]
|
wenzelm@32962
|
277 |
show ?thesis by auto
|
hoelzl@29742
|
278 |
qed
|
hoelzl@29742
|
279 |
hence "sqrt (pow2 (?E mod 2)) < sqrt (2 * 2)" by auto
|
hoelzl@29742
|
280 |
hence E_mod_pow: "sqrt (pow2 (?E mod 2)) < 2" unfolding real_sqrt_abs2 by auto
|
hoelzl@29742
|
281 |
|
hoelzl@29742
|
282 |
have E_eq: "pow2 ?E = pow2 (?E div 2 + ?E div 2 + ?E mod 2)" by auto
|
hoelzl@29742
|
283 |
have "sqrt (pow2 ?E) = sqrt (pow2 (?E div 2) * pow2 (?E div 2) * pow2 (?E mod 2))"
|
wenzelm@32962
|
284 |
unfolding E_eq unfolding pow2_add ..
|
hoelzl@29742
|
285 |
also have "\<dots> = pow2 (?E div 2) * sqrt (pow2 (?E mod 2))"
|
wenzelm@32962
|
286 |
unfolding real_sqrt_mult[of _ "pow2 (?E mod 2)"] real_sqrt_abs2 by auto
|
hoelzl@31467
|
287 |
also have "\<dots> < pow2 (?E div 2) * 2"
|
wenzelm@32962
|
288 |
by (rule mult_strict_left_mono, auto intro: E_mod_pow)
|
hoelzl@29742
|
289 |
also have "\<dots> = pow2 (?E div 2 + 1)" unfolding zadd_commute[of _ 1] pow2_add1 by auto
|
hoelzl@29742
|
290 |
finally show ?thesis by auto
|
hoelzl@29742
|
291 |
qed
|
hoelzl@31467
|
292 |
finally show ?thesis
|
hoelzl@31098
|
293 |
unfolding Float sqrt_iteration.simps real_of_float_simp by auto
|
hoelzl@29742
|
294 |
qed
|
hoelzl@29742
|
295 |
next
|
hoelzl@29742
|
296 |
case (Suc n)
|
hoelzl@29742
|
297 |
let ?b = "sqrt_iteration prec n x"
|
hoelzl@31098
|
298 |
have "0 < sqrt (real x)" using `0 < real x` by auto
|
hoelzl@31098
|
299 |
also have "\<dots> < real ?b" using Suc .
|
hoelzl@31098
|
300 |
finally have "sqrt (real x) < (real ?b + real x / real ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ `0 < real x`] by auto
|
hoelzl@31098
|
301 |
also have "\<dots> \<le> (real ?b + real (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr)
|
hoelzl@31098
|
302 |
also have "\<dots> = real (Float 1 -1) * (real ?b + real (float_divr prec x ?b))" by auto
|
hoelzl@31098
|
303 |
finally show ?case unfolding sqrt_iteration.simps Let_def real_of_float_mult real_of_float_add right_distrib .
|
hoelzl@29742
|
304 |
qed
|
hoelzl@29742
|
305 |
|
hoelzl@31098
|
306 |
lemma sqrt_iteration_lower_bound: assumes "0 < real x"
|
hoelzl@31098
|
307 |
shows "0 < real (sqrt_iteration prec n x)" (is "0 < ?sqrt")
|
hoelzl@29742
|
308 |
proof -
|
hoelzl@31098
|
309 |
have "0 < sqrt (real x)" using assms by auto
|
hoelzl@29742
|
310 |
also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
|
hoelzl@29742
|
311 |
finally show ?thesis .
|
hoelzl@29742
|
312 |
qed
|
hoelzl@29742
|
313 |
|
hoelzl@31098
|
314 |
lemma lb_sqrt_lower_bound: assumes "0 \<le> real x"
|
hoelzl@31467
|
315 |
shows "0 \<le> real (lb_sqrt prec x)"
|
hoelzl@29742
|
316 |
proof (cases "0 < x")
|
hoelzl@31098
|
317 |
case True hence "0 < real x" and "0 \<le> x" using `0 \<le> real x` unfolding less_float_def le_float_def by auto
|
hoelzl@31809
|
318 |
hence "0 < sqrt_iteration prec prec x" unfolding less_float_def using sqrt_iteration_lower_bound by auto
|
hoelzl@31098
|
319 |
hence "0 \<le> real (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@31467
|
320 |
thus ?thesis unfolding lb_sqrt.simps using True by auto
|
hoelzl@29742
|
321 |
next
|
hoelzl@31098
|
322 |
case False with `0 \<le> real x` have "real x = 0" unfolding less_float_def by auto
|
hoelzl@31467
|
323 |
thus ?thesis unfolding lb_sqrt.simps less_float_def by auto
|
hoelzl@29742
|
324 |
qed
|
hoelzl@29742
|
325 |
|
hoelzl@31467
|
326 |
lemma bnds_sqrt':
|
hoelzl@31467
|
327 |
shows "sqrt (real x) \<in> { real (lb_sqrt prec x) .. real (ub_sqrt prec x) }"
|
hoelzl@31467
|
328 |
proof -
|
hoelzl@31467
|
329 |
{ fix x :: float assume "0 < x"
|
hoelzl@31467
|
330 |
hence "0 < real x" and "0 \<le> real x" unfolding less_float_def by auto
|
hoelzl@31467
|
331 |
hence sqrt_gt0: "0 < sqrt (real x)" by auto
|
hoelzl@31467
|
332 |
hence sqrt_ub: "sqrt (real x) < real (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
|
hoelzl@31467
|
333 |
|
hoelzl@31467
|
334 |
have "real (float_divl prec x (sqrt_iteration prec prec x)) \<le>
|
hoelzl@31467
|
335 |
real x / real (sqrt_iteration prec prec x)" by (rule float_divl)
|
hoelzl@31467
|
336 |
also have "\<dots> < real x / sqrt (real x)"
|
hoelzl@31467
|
337 |
by (rule divide_strict_left_mono[OF sqrt_ub `0 < real x`
|
hoelzl@31467
|
338 |
mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
|
hoelzl@31809
|
339 |
also have "\<dots> = sqrt (real x)"
|
hoelzl@31467
|
340 |
unfolding inverse_eq_iff_eq[of _ "sqrt (real x)", symmetric]
|
wenzelm@32962
|
341 |
sqrt_divide_self_eq[OF `0 \<le> real x`, symmetric] by auto
|
hoelzl@31467
|
342 |
finally have "real (lb_sqrt prec x) \<le> sqrt (real x)"
|
hoelzl@31467
|
343 |
unfolding lb_sqrt.simps if_P[OF `0 < x`] by auto }
|
hoelzl@31467
|
344 |
note lb = this
|
hoelzl@31467
|
345 |
|
hoelzl@31467
|
346 |
{ fix x :: float assume "0 < x"
|
hoelzl@31467
|
347 |
hence "0 < real x" unfolding less_float_def by auto
|
hoelzl@31467
|
348 |
hence "0 < sqrt (real x)" by auto
|
hoelzl@31467
|
349 |
hence "sqrt (real x) < real (sqrt_iteration prec prec x)"
|
hoelzl@31467
|
350 |
using sqrt_iteration_bound by auto
|
hoelzl@31467
|
351 |
hence "sqrt (real x) \<le> real (ub_sqrt prec x)"
|
hoelzl@31467
|
352 |
unfolding ub_sqrt.simps if_P[OF `0 < x`] by auto }
|
hoelzl@31467
|
353 |
note ub = this
|
hoelzl@31467
|
354 |
|
hoelzl@31467
|
355 |
show ?thesis
|
hoelzl@31467
|
356 |
proof (cases "0 < x")
|
hoelzl@31467
|
357 |
case True with lb ub show ?thesis by auto
|
hoelzl@31467
|
358 |
next case False show ?thesis
|
hoelzl@31467
|
359 |
proof (cases "real x = 0")
|
hoelzl@31809
|
360 |
case True thus ?thesis
|
hoelzl@31467
|
361 |
by (auto simp add: less_float_def lb_sqrt.simps ub_sqrt.simps)
|
hoelzl@31467
|
362 |
next
|
hoelzl@31467
|
363 |
case False with `\<not> 0 < x` have "x < 0" and "0 < -x"
|
hoelzl@31467
|
364 |
by (auto simp add: less_float_def)
|
hoelzl@31467
|
365 |
|
hoelzl@31467
|
366 |
with `\<not> 0 < x`
|
hoelzl@31467
|
367 |
show ?thesis using lb[OF `0 < -x`] ub[OF `0 < -x`]
|
hoelzl@31467
|
368 |
by (auto simp add: real_sqrt_minus lb_sqrt.simps ub_sqrt.simps)
|
hoelzl@31467
|
369 |
qed qed
|
hoelzl@29742
|
370 |
qed
|
hoelzl@29742
|
371 |
|
hoelzl@31467
|
372 |
lemma bnds_sqrt: "\<forall> x lx ux. (l, u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> sqrt x \<and> sqrt x \<le> real u"
|
hoelzl@31467
|
373 |
proof ((rule allI) +, rule impI, erule conjE, rule conjI)
|
hoelzl@31467
|
374 |
fix x lx ux
|
hoelzl@31467
|
375 |
assume "(l, u) = (lb_sqrt prec lx, ub_sqrt prec ux)"
|
hoelzl@31467
|
376 |
and x: "x \<in> {real lx .. real ux}"
|
hoelzl@31467
|
377 |
hence l: "l = lb_sqrt prec lx " and u: "u = ub_sqrt prec ux" by auto
|
hoelzl@29742
|
378 |
|
hoelzl@31467
|
379 |
have "sqrt (real lx) \<le> sqrt x" using x by auto
|
hoelzl@31467
|
380 |
from order_trans[OF _ this]
|
hoelzl@31467
|
381 |
show "real l \<le> sqrt x" unfolding l using bnds_sqrt'[of lx prec] by auto
|
hoelzl@29742
|
382 |
|
hoelzl@31467
|
383 |
have "sqrt x \<le> sqrt (real ux)" using x by auto
|
hoelzl@31467
|
384 |
from order_trans[OF this]
|
hoelzl@31467
|
385 |
show "sqrt x \<le> real u" unfolding u using bnds_sqrt'[of ux prec] by auto
|
hoelzl@29742
|
386 |
qed
|
hoelzl@29742
|
387 |
|
hoelzl@29742
|
388 |
section "Arcus tangens and \<pi>"
|
hoelzl@29742
|
389 |
|
hoelzl@29742
|
390 |
subsection "Compute arcus tangens series"
|
hoelzl@29742
|
391 |
|
hoelzl@29742
|
392 |
text {*
|
hoelzl@29742
|
393 |
|
hoelzl@29742
|
394 |
As first step we implement the computation of the arcus tangens series. This is only valid in the range
|
hoelzl@29742
|
395 |
@{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens.
|
hoelzl@29742
|
396 |
|
hoelzl@29742
|
397 |
*}
|
hoelzl@29742
|
398 |
|
hoelzl@29742
|
399 |
fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
|
hoelzl@29742
|
400 |
and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
|
hoelzl@29742
|
401 |
"ub_arctan_horner prec 0 k x = 0"
|
hoelzl@31809
|
402 |
| "ub_arctan_horner prec (Suc n) k x =
|
hoelzl@29742
|
403 |
(rapprox_rat prec 1 (int k)) - x * (lb_arctan_horner prec n (k + 2) x)"
|
hoelzl@29742
|
404 |
| "lb_arctan_horner prec 0 k x = 0"
|
hoelzl@31809
|
405 |
| "lb_arctan_horner prec (Suc n) k x =
|
hoelzl@29742
|
406 |
(lapprox_rat prec 1 (int k)) - x * (ub_arctan_horner prec n (k + 2) x)"
|
hoelzl@29742
|
407 |
|
hoelzl@31098
|
408 |
lemma arctan_0_1_bounds': assumes "0 \<le> real x" "real x \<le> 1" and "even n"
|
hoelzl@31098
|
409 |
shows "arctan (real x) \<in> {real (x * lb_arctan_horner prec n 1 (x * x)) .. real (x * ub_arctan_horner prec (Suc n) 1 (x * x))}"
|
hoelzl@29742
|
410 |
proof -
|
hoelzl@31098
|
411 |
let "?c i" = "-1^i * (1 / real (i * 2 + 1) * real x ^ (i * 2 + 1))"
|
hoelzl@29742
|
412 |
let "?S n" = "\<Sum> i=0..<n. ?c i"
|
hoelzl@29742
|
413 |
|
hoelzl@31098
|
414 |
have "0 \<le> real (x * x)" by auto
|
hoelzl@29742
|
415 |
from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
|
hoelzl@31809
|
416 |
|
hoelzl@31098
|
417 |
have "arctan (real x) \<in> { ?S n .. ?S (Suc n) }"
|
hoelzl@31098
|
418 |
proof (cases "real x = 0")
|
hoelzl@29742
|
419 |
case False
|
hoelzl@31098
|
420 |
hence "0 < real x" using `0 \<le> real x` by auto
|
hoelzl@31809
|
421 |
hence prem: "0 < 1 / real (0 * 2 + (1::nat)) * real x ^ (0 * 2 + 1)" by auto
|
hoelzl@29742
|
422 |
|
hoelzl@31098
|
423 |
have "\<bar> real x \<bar> \<le> 1" using `0 \<le> real x` `real x \<le> 1` by auto
|
hoelzl@29742
|
424 |
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`]
|
nipkow@31790
|
425 |
show ?thesis unfolding arctan_series[OF `\<bar> real x \<bar> \<le> 1`] Suc_eq_plus1 .
|
hoelzl@29742
|
426 |
qed auto
|
hoelzl@29742
|
427 |
note arctan_bounds = this[unfolded atLeastAtMost_iff]
|
hoelzl@29742
|
428 |
|
hoelzl@29742
|
429 |
have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
|
hoelzl@29742
|
430 |
|
hoelzl@31809
|
431 |
note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0
|
hoelzl@29742
|
432 |
and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
|
hoelzl@31809
|
433 |
and ub="\<lambda>n i k x. ub_arctan_horner prec n k x",
|
hoelzl@31098
|
434 |
OF `0 \<le> real (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps]
|
hoelzl@29742
|
435 |
|
hoelzl@31098
|
436 |
{ have "real (x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n"
|
hoelzl@31098
|
437 |
using bounds(1) `0 \<le> real x`
|
huffman@36770
|
438 |
unfolding real_of_float_mult power_add power_one_right mult_assoc[symmetric] setsum_left_distrib[symmetric]
|
huffman@36770
|
439 |
unfolding mult_commute[where 'a=real] mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
|
hoelzl@29742
|
440 |
by (auto intro!: mult_left_mono)
|
hoelzl@31098
|
441 |
also have "\<dots> \<le> arctan (real x)" using arctan_bounds ..
|
hoelzl@31098
|
442 |
finally have "real (x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan (real x)" . }
|
hoelzl@29742
|
443 |
moreover
|
hoelzl@31098
|
444 |
{ have "arctan (real x) \<le> ?S (Suc n)" using arctan_bounds ..
|
hoelzl@31098
|
445 |
also have "\<dots> \<le> real (x * ub_arctan_horner prec (Suc n) 1 (x*x))"
|
hoelzl@31098
|
446 |
using bounds(2)[of "Suc n"] `0 \<le> real x`
|
huffman@36770
|
447 |
unfolding real_of_float_mult power_add power_one_right mult_assoc[symmetric] setsum_left_distrib[symmetric]
|
huffman@36770
|
448 |
unfolding mult_commute[where 'a=real] mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
|
hoelzl@29742
|
449 |
by (auto intro!: mult_left_mono)
|
hoelzl@31098
|
450 |
finally have "arctan (real x) \<le> real (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . }
|
hoelzl@29742
|
451 |
ultimately show ?thesis by auto
|
hoelzl@29742
|
452 |
qed
|
hoelzl@29742
|
453 |
|
hoelzl@31098
|
454 |
lemma arctan_0_1_bounds: assumes "0 \<le> real x" "real x \<le> 1"
|
hoelzl@31098
|
455 |
shows "arctan (real x) \<in> {real (x * lb_arctan_horner prec (get_even n) 1 (x * x)) .. real (x * ub_arctan_horner prec (get_odd n) 1 (x * x))}"
|
hoelzl@29742
|
456 |
proof (cases "even n")
|
hoelzl@29742
|
457 |
case True
|
hoelzl@29742
|
458 |
obtain n' where "Suc n' = get_odd n" and "odd (Suc n')" using get_odd_ex by auto
|
nipkow@31148
|
459 |
hence "even n'" unfolding even_Suc by auto
|
hoelzl@31098
|
460 |
have "arctan (real x) \<le> real (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
|
hoelzl@31098
|
461 |
unfolding `Suc n' = get_odd n`[symmetric] using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even n'`] by auto
|
hoelzl@29742
|
462 |
moreover
|
hoelzl@31098
|
463 |
have "real (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (real x)"
|
hoelzl@31098
|
464 |
unfolding get_even_def if_P[OF True] using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even n`] by auto
|
hoelzl@29742
|
465 |
ultimately show ?thesis by auto
|
hoelzl@29742
|
466 |
next
|
hoelzl@29742
|
467 |
case False hence "0 < n" by (rule odd_pos)
|
hoelzl@29742
|
468 |
from gr0_implies_Suc[OF this] obtain n' where "n = Suc n'" ..
|
nipkow@31148
|
469 |
from False[unfolded this even_Suc]
|
hoelzl@29742
|
470 |
have "even n'" and "even (Suc (Suc n'))" by auto
|
hoelzl@29742
|
471 |
have "get_odd n = Suc n'" unfolding get_odd_def if_P[OF False] using `n = Suc n'` .
|
hoelzl@29742
|
472 |
|
hoelzl@31098
|
473 |
have "arctan (real x) \<le> real (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
|
hoelzl@31098
|
474 |
unfolding `get_odd n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even n'`] by auto
|
hoelzl@29742
|
475 |
moreover
|
hoelzl@31098
|
476 |
have "real (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (real x)"
|
hoelzl@31098
|
477 |
unfolding get_even_def if_not_P[OF False] unfolding `n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even (Suc (Suc n'))`] by auto
|
hoelzl@29742
|
478 |
ultimately show ?thesis by auto
|
hoelzl@29742
|
479 |
qed
|
hoelzl@29742
|
480 |
|
hoelzl@29742
|
481 |
subsection "Compute \<pi>"
|
hoelzl@29742
|
482 |
|
hoelzl@29742
|
483 |
definition ub_pi :: "nat \<Rightarrow> float" where
|
hoelzl@31809
|
484 |
"ub_pi prec = (let A = rapprox_rat prec 1 5 ;
|
hoelzl@29742
|
485 |
B = lapprox_rat prec 1 239
|
hoelzl@31809
|
486 |
in ((Float 1 2) * ((Float 1 2) * A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (A * A)) -
|
hoelzl@29742
|
487 |
B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 (B * B)))))"
|
hoelzl@29742
|
488 |
|
hoelzl@29742
|
489 |
definition lb_pi :: "nat \<Rightarrow> float" where
|
hoelzl@31809
|
490 |
"lb_pi prec = (let A = lapprox_rat prec 1 5 ;
|
hoelzl@29742
|
491 |
B = rapprox_rat prec 1 239
|
hoelzl@31809
|
492 |
in ((Float 1 2) * ((Float 1 2) * A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (A * A)) -
|
hoelzl@29742
|
493 |
B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (B * B)))))"
|
hoelzl@29742
|
494 |
|
hoelzl@31098
|
495 |
lemma pi_boundaries: "pi \<in> {real (lb_pi n) .. real (ub_pi n)}"
|
hoelzl@29742
|
496 |
proof -
|
hoelzl@29742
|
497 |
have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))" unfolding machin[symmetric] by auto
|
hoelzl@29742
|
498 |
|
hoelzl@29742
|
499 |
{ fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto
|
hoelzl@29742
|
500 |
let ?k = "rapprox_rat prec 1 k"
|
hoelzl@29742
|
501 |
have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
|
hoelzl@31809
|
502 |
|
hoelzl@31098
|
503 |
have "0 \<le> real ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \<le> k`)
|
hoelzl@31098
|
504 |
have "real ?k \<le> 1" unfolding rapprox_rat.simps(2)[OF zero_le_one `0 < k`]
|
hoelzl@29742
|
505 |
by (rule rapprox_posrat_le1, auto simp add: `0 < k` `1 \<le> k`)
|
hoelzl@29742
|
506 |
|
hoelzl@31098
|
507 |
have "1 / real k \<le> real ?k" using rapprox_rat[where x=1 and y=k] by auto
|
hoelzl@31098
|
508 |
hence "arctan (1 / real k) \<le> arctan (real ?k)" by (rule arctan_monotone')
|
hoelzl@31098
|
509 |
also have "\<dots> \<le> real (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))"
|
hoelzl@31098
|
510 |
using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
|
hoelzl@31098
|
511 |
finally have "arctan (1 / (real k)) \<le> real (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" .
|
hoelzl@29742
|
512 |
} note ub_arctan = this
|
hoelzl@29742
|
513 |
|
hoelzl@29742
|
514 |
{ fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
|
hoelzl@29742
|
515 |
let ?k = "lapprox_rat prec 1 k"
|
hoelzl@29742
|
516 |
have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
|
hoelzl@29742
|
517 |
have "1 / real k \<le> 1" using `1 < k` by auto
|
hoelzl@29742
|
518 |
|
hoelzl@31098
|
519 |
have "\<And>n. 0 \<le> real ?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@31098
|
520 |
have "\<And>n. real ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / real k \<le> 1`)
|
hoelzl@29742
|
521 |
|
hoelzl@31098
|
522 |
have "real ?k \<le> 1 / real k" using lapprox_rat[where x=1 and y=k] by auto
|
hoelzl@29742
|
523 |
|
hoelzl@31098
|
524 |
have "real (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (real ?k)"
|
hoelzl@31098
|
525 |
using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
|
hoelzl@31098
|
526 |
also have "\<dots> \<le> arctan (1 / real k)" using `real ?k \<le> 1 / real k` by (rule arctan_monotone')
|
hoelzl@31098
|
527 |
finally have "real (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (1 / (real k))" .
|
hoelzl@29742
|
528 |
} note lb_arctan = this
|
hoelzl@29742
|
529 |
|
hoelzl@31098
|
530 |
have "pi \<le> real (ub_pi n)"
|
hoelzl@31098
|
531 |
unfolding ub_pi_def machin_pi Let_def real_of_float_mult real_of_float_sub unfolding Float_num
|
hoelzl@29742
|
532 |
using lb_arctan[of 239] ub_arctan[of 5]
|
hoelzl@29742
|
533 |
by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
|
hoelzl@29742
|
534 |
moreover
|
hoelzl@31098
|
535 |
have "real (lb_pi n) \<le> pi"
|
hoelzl@31098
|
536 |
unfolding lb_pi_def machin_pi Let_def real_of_float_mult real_of_float_sub Float_num
|
hoelzl@29742
|
537 |
using lb_arctan[of 5] ub_arctan[of 239]
|
hoelzl@29742
|
538 |
by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
|
hoelzl@29742
|
539 |
ultimately show ?thesis by auto
|
hoelzl@29742
|
540 |
qed
|
hoelzl@29742
|
541 |
|
hoelzl@29742
|
542 |
subsection "Compute arcus tangens in the entire domain"
|
hoelzl@29742
|
543 |
|
hoelzl@31467
|
544 |
function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where
|
hoelzl@29742
|
545 |
"lb_arctan prec x = (let ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) ;
|
hoelzl@29742
|
546 |
lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)
|
hoelzl@29742
|
547 |
in (if x < 0 then - ub_arctan prec (-x) else
|
hoelzl@29742
|
548 |
if x \<le> Float 1 -1 then lb_horner x else
|
hoelzl@31467
|
549 |
if x \<le> Float 1 1 then Float 1 1 * lb_horner (float_divl prec x (1 + ub_sqrt prec (1 + x * x)))
|
hoelzl@31467
|
550 |
else (let inv = float_divr prec 1 x
|
hoelzl@31467
|
551 |
in if inv > 1 then 0
|
hoelzl@29742
|
552 |
else lb_pi prec * Float 1 -1 - ub_horner inv)))"
|
hoelzl@29742
|
553 |
|
hoelzl@29742
|
554 |
| "ub_arctan prec x = (let lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) ;
|
hoelzl@29742
|
555 |
ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)
|
hoelzl@29742
|
556 |
in (if x < 0 then - lb_arctan prec (-x) else
|
hoelzl@29742
|
557 |
if x \<le> Float 1 -1 then ub_horner x else
|
hoelzl@31467
|
558 |
if x \<le> Float 1 1 then let y = float_divr prec x (1 + lb_sqrt prec (1 + x * x))
|
hoelzl@31467
|
559 |
in if y > 1 then ub_pi prec * Float 1 -1
|
hoelzl@31467
|
560 |
else Float 1 1 * ub_horner y
|
hoelzl@29742
|
561 |
else ub_pi prec * Float 1 -1 - lb_horner (float_divl prec 1 x)))"
|
hoelzl@29742
|
562 |
by pat_completeness auto
|
hoelzl@29742
|
563 |
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@29742
|
564 |
|
hoelzl@29742
|
565 |
declare ub_arctan_horner.simps[simp del]
|
hoelzl@29742
|
566 |
declare lb_arctan_horner.simps[simp del]
|
hoelzl@29742
|
567 |
|
hoelzl@31098
|
568 |
lemma lb_arctan_bound': assumes "0 \<le> real x"
|
hoelzl@31098
|
569 |
shows "real (lb_arctan prec x) \<le> arctan (real x)"
|
hoelzl@29742
|
570 |
proof -
|
hoelzl@31098
|
571 |
have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> real x` by auto
|
hoelzl@29742
|
572 |
let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
|
hoelzl@29742
|
573 |
and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
|
hoelzl@29742
|
574 |
|
hoelzl@29742
|
575 |
show ?thesis
|
hoelzl@29742
|
576 |
proof (cases "x \<le> Float 1 -1")
|
hoelzl@31098
|
577 |
case True hence "real x \<le> 1" unfolding le_float_def Float_num by auto
|
hoelzl@29742
|
578 |
show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
|
hoelzl@31098
|
579 |
using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
|
hoelzl@29742
|
580 |
next
|
hoelzl@31098
|
581 |
case False hence "0 < real x" unfolding le_float_def Float_num by auto
|
hoelzl@31098
|
582 |
let ?R = "1 + sqrt (1 + real x * real x)"
|
hoelzl@31467
|
583 |
let ?fR = "1 + ub_sqrt prec (1 + x * x)"
|
hoelzl@29742
|
584 |
let ?DIV = "float_divl prec x ?fR"
|
hoelzl@31467
|
585 |
|
hoelzl@31098
|
586 |
have sqr_ge0: "0 \<le> 1 + real x * real x" using sum_power2_ge_zero[of 1 "real x", unfolded numeral_2_eq_2] by auto
|
hoelzl@29742
|
587 |
hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
|
hoelzl@29742
|
588 |
|
hoelzl@31467
|
589 |
have "sqrt (real (1 + x * x)) \<le> real (ub_sqrt prec (1 + x * x))"
|
hoelzl@31467
|
590 |
using bnds_sqrt'[of "1 + x * x"] by auto
|
hoelzl@31467
|
591 |
|
hoelzl@31098
|
592 |
hence "?R \<le> real ?fR" by auto
|
hoelzl@31098
|
593 |
hence "0 < ?fR" and "0 < real ?fR" unfolding less_float_def using `0 < ?R` by auto
|
hoelzl@29742
|
594 |
|
hoelzl@31098
|
595 |
have monotone: "real (float_divl prec x ?fR) \<le> real x / ?R"
|
hoelzl@29742
|
596 |
proof -
|
hoelzl@31098
|
597 |
have "real ?DIV \<le> real x / real ?fR" by (rule float_divl)
|
hoelzl@31098
|
598 |
also have "\<dots> \<le> real x / ?R" by (rule divide_left_mono[OF `?R \<le> real ?fR` `0 \<le> real x` mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 `?R \<le> real ?fR`] divisor_gt0]])
|
hoelzl@29742
|
599 |
finally show ?thesis .
|
hoelzl@29742
|
600 |
qed
|
hoelzl@29742
|
601 |
|
hoelzl@29742
|
602 |
show ?thesis
|
hoelzl@29742
|
603 |
proof (cases "x \<le> Float 1 1")
|
hoelzl@29742
|
604 |
case True
|
hoelzl@31467
|
605 |
|
hoelzl@31098
|
606 |
have "real x \<le> sqrt (real (1 + x * x))" using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
|
hoelzl@31467
|
607 |
also have "\<dots> \<le> real (ub_sqrt prec (1 + x * x))"
|
wenzelm@32962
|
608 |
using bnds_sqrt'[of "1 + x * x"] by auto
|
hoelzl@31098
|
609 |
finally have "real x \<le> real ?fR" by auto
|
hoelzl@31098
|
610 |
moreover have "real ?DIV \<le> real x / real ?fR" by (rule float_divl)
|
hoelzl@31098
|
611 |
ultimately have "real ?DIV \<le> 1" unfolding divide_le_eq_1_pos[OF `0 < real ?fR`, symmetric] by auto
|
hoelzl@29742
|
612 |
|
hoelzl@31098
|
613 |
have "0 \<le> real ?DIV" using float_divl_lower_bound[OF `0 \<le> x` `0 < ?fR`] unfolding le_float_def by auto
|
hoelzl@29742
|
614 |
|
hoelzl@31098
|
615 |
have "real (Float 1 1 * ?lb_horner ?DIV) \<le> 2 * arctan (real (float_divl prec x ?fR))" unfolding real_of_float_mult[of "Float 1 1"] Float_num
|
wenzelm@32962
|
616 |
using arctan_0_1_bounds[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
|
hoelzl@31098
|
617 |
also have "\<dots> \<le> 2 * arctan (real x / ?R)"
|
wenzelm@32962
|
618 |
using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
|
huffman@36770
|
619 |
also have "2 * arctan (real x / ?R) = arctan (real x)" using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
|
hoelzl@29742
|
620 |
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@29742
|
621 |
next
|
hoelzl@29742
|
622 |
case False
|
hoelzl@31098
|
623 |
hence "2 < real x" unfolding le_float_def Float_num by auto
|
hoelzl@31098
|
624 |
hence "1 \<le> real x" by auto
|
hoelzl@29742
|
625 |
|
hoelzl@29742
|
626 |
let "?invx" = "float_divr prec 1 x"
|
hoelzl@31098
|
627 |
have "0 \<le> arctan (real x)" using arctan_monotone'[OF `0 \<le> real x`] using arctan_tan[of 0, unfolded tan_zero] by auto
|
hoelzl@29742
|
628 |
|
hoelzl@29742
|
629 |
show ?thesis
|
hoelzl@29742
|
630 |
proof (cases "1 < ?invx")
|
wenzelm@32962
|
631 |
case True
|
wenzelm@32962
|
632 |
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]
|
wenzelm@32962
|
633 |
using `0 \<le> arctan (real x)` by auto
|
hoelzl@29742
|
634 |
next
|
wenzelm@32962
|
635 |
case False
|
wenzelm@32962
|
636 |
hence "real ?invx \<le> 1" unfolding less_float_def by auto
|
wenzelm@32962
|
637 |
have "0 \<le> real ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> real x`)
|
wenzelm@32962
|
638 |
|
wenzelm@32962
|
639 |
have "1 / real x \<noteq> 0" and "0 < 1 / real x" using `0 < real x` by auto
|
wenzelm@32962
|
640 |
|
wenzelm@32962
|
641 |
have "arctan (1 / real x) \<le> arctan (real ?invx)" unfolding real_of_float_1[symmetric] by (rule arctan_monotone', rule float_divr)
|
wenzelm@32962
|
642 |
also have "\<dots> \<le> real (?ub_horner ?invx)" using arctan_0_1_bounds[OF `0 \<le> real ?invx` `real ?invx \<le> 1`] by auto
|
wenzelm@32962
|
643 |
finally have "pi / 2 - real (?ub_horner ?invx) \<le> arctan (real x)"
|
wenzelm@32962
|
644 |
using `0 \<le> arctan (real x)` arctan_inverse[OF `1 / real x \<noteq> 0`]
|
wenzelm@32962
|
645 |
unfolding real_sgn_pos[OF `0 < 1 / real x`] le_diff_eq by auto
|
wenzelm@32962
|
646 |
moreover
|
huffman@36770
|
647 |
have "real (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding real_of_float_mult Float_num times_divide_eq_right mult_1_left using pi_boundaries by auto
|
wenzelm@32962
|
648 |
ultimately
|
wenzelm@32962
|
649 |
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]
|
wenzelm@32962
|
650 |
by auto
|
hoelzl@29742
|
651 |
qed
|
hoelzl@29742
|
652 |
qed
|
hoelzl@29742
|
653 |
qed
|
hoelzl@29742
|
654 |
qed
|
hoelzl@29742
|
655 |
|
hoelzl@31098
|
656 |
lemma ub_arctan_bound': assumes "0 \<le> real x"
|
hoelzl@31098
|
657 |
shows "arctan (real x) \<le> real (ub_arctan prec x)"
|
hoelzl@29742
|
658 |
proof -
|
hoelzl@31098
|
659 |
have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> real x` by auto
|
hoelzl@29742
|
660 |
|
hoelzl@29742
|
661 |
let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
|
hoelzl@29742
|
662 |
and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
|
hoelzl@29742
|
663 |
|
hoelzl@29742
|
664 |
show ?thesis
|
hoelzl@29742
|
665 |
proof (cases "x \<le> Float 1 -1")
|
hoelzl@31098
|
666 |
case True hence "real x \<le> 1" unfolding le_float_def Float_num by auto
|
hoelzl@29742
|
667 |
show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
|
hoelzl@31098
|
668 |
using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
|
hoelzl@29742
|
669 |
next
|
hoelzl@31098
|
670 |
case False hence "0 < real x" unfolding le_float_def Float_num by auto
|
hoelzl@31098
|
671 |
let ?R = "1 + sqrt (1 + real x * real x)"
|
hoelzl@31467
|
672 |
let ?fR = "1 + lb_sqrt prec (1 + x * x)"
|
hoelzl@29742
|
673 |
let ?DIV = "float_divr prec x ?fR"
|
hoelzl@31467
|
674 |
|
hoelzl@31098
|
675 |
have sqr_ge0: "0 \<le> 1 + real x * real x" using sum_power2_ge_zero[of 1 "real x", unfolded numeral_2_eq_2] by auto
|
hoelzl@31098
|
676 |
hence "0 \<le> real (1 + x*x)" by auto
|
hoelzl@31467
|
677 |
|
hoelzl@29742
|
678 |
hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
|
hoelzl@29742
|
679 |
|
hoelzl@31467
|
680 |
have "real (lb_sqrt prec (1 + x * x)) \<le> sqrt (real (1 + x * x))"
|
hoelzl@31467
|
681 |
using bnds_sqrt'[of "1 + x * x"] by auto
|
hoelzl@31098
|
682 |
hence "real ?fR \<le> ?R" by auto
|
hoelzl@31098
|
683 |
have "0 < real ?fR" unfolding real_of_float_add real_of_float_1 by (rule order_less_le_trans[OF zero_less_one], auto simp add: lb_sqrt_lower_bound[OF `0 \<le> real (1 + x*x)`])
|
hoelzl@29742
|
684 |
|
hoelzl@31098
|
685 |
have monotone: "real x / ?R \<le> real (float_divr prec x ?fR)"
|
hoelzl@29742
|
686 |
proof -
|
hoelzl@31098
|
687 |
from divide_left_mono[OF `real ?fR \<le> ?R` `0 \<le> real x` mult_pos_pos[OF divisor_gt0 `0 < real ?fR`]]
|
hoelzl@31098
|
688 |
have "real x / ?R \<le> real x / real ?fR" .
|
hoelzl@31098
|
689 |
also have "\<dots> \<le> real ?DIV" by (rule float_divr)
|
hoelzl@29742
|
690 |
finally show ?thesis .
|
hoelzl@29742
|
691 |
qed
|
hoelzl@29742
|
692 |
|
hoelzl@29742
|
693 |
show ?thesis
|
hoelzl@29742
|
694 |
proof (cases "x \<le> Float 1 1")
|
hoelzl@29742
|
695 |
case True
|
hoelzl@29742
|
696 |
show ?thesis
|
hoelzl@29742
|
697 |
proof (cases "?DIV > 1")
|
wenzelm@32962
|
698 |
case True
|
huffman@36770
|
699 |
have "pi / 2 \<le> real (ub_pi prec * Float 1 -1)" unfolding real_of_float_mult Float_num times_divide_eq_right mult_1_left using pi_boundaries by auto
|
wenzelm@32962
|
700 |
from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le]
|
wenzelm@32962
|
701 |
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@29742
|
702 |
next
|
wenzelm@32962
|
703 |
case False
|
wenzelm@32962
|
704 |
hence "real ?DIV \<le> 1" unfolding less_float_def by auto
|
wenzelm@32962
|
705 |
|
wenzelm@32962
|
706 |
have "0 \<le> real x / ?R" using `0 \<le> real x` `0 < ?R` unfolding real_0_le_divide_iff by auto
|
wenzelm@32962
|
707 |
hence "0 \<le> real ?DIV" using monotone by (rule order_trans)
|
wenzelm@32962
|
708 |
|
huffman@36770
|
709 |
have "arctan (real x) = 2 * arctan (real x / ?R)" using arctan_half unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
|
wenzelm@32962
|
710 |
also have "\<dots> \<le> 2 * arctan (real ?DIV)"
|
wenzelm@32962
|
711 |
using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
|
wenzelm@32962
|
712 |
also have "\<dots> \<le> real (Float 1 1 * ?ub_horner ?DIV)" unfolding real_of_float_mult[of "Float 1 1"] Float_num
|
wenzelm@32962
|
713 |
using arctan_0_1_bounds[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
|
wenzelm@32962
|
714 |
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@29742
|
715 |
qed
|
hoelzl@29742
|
716 |
next
|
hoelzl@29742
|
717 |
case False
|
hoelzl@31098
|
718 |
hence "2 < real x" unfolding le_float_def Float_num by auto
|
hoelzl@31098
|
719 |
hence "1 \<le> real x" by auto
|
hoelzl@31098
|
720 |
hence "0 < real x" by auto
|
hoelzl@29742
|
721 |
hence "0 < x" unfolding less_float_def by auto
|
hoelzl@29742
|
722 |
|
hoelzl@29742
|
723 |
let "?invx" = "float_divl prec 1 x"
|
hoelzl@31098
|
724 |
have "0 \<le> arctan (real x)" using arctan_monotone'[OF `0 \<le> real x`] using arctan_tan[of 0, unfolded tan_zero] by auto
|
hoelzl@29742
|
725 |
|
hoelzl@31098
|
726 |
have "real ?invx \<le> 1" unfolding less_float_def by (rule order_trans[OF float_divl], auto simp add: `1 \<le> real x` divide_le_eq_1_pos[OF `0 < real x`])
|
hoelzl@31098
|
727 |
have "0 \<le> real ?invx" unfolding real_of_float_0[symmetric] by (rule float_divl_lower_bound[unfolded le_float_def], auto simp add: `0 < x`)
|
hoelzl@31467
|
728 |
|
hoelzl@31098
|
729 |
have "1 / real x \<noteq> 0" and "0 < 1 / real x" using `0 < real x` by auto
|
hoelzl@31467
|
730 |
|
hoelzl@31098
|
731 |
have "real (?lb_horner ?invx) \<le> arctan (real ?invx)" using arctan_0_1_bounds[OF `0 \<le> real ?invx` `real ?invx \<le> 1`] by auto
|
hoelzl@31098
|
732 |
also have "\<dots> \<le> arctan (1 / real x)" unfolding real_of_float_1[symmetric] by (rule arctan_monotone', rule float_divl)
|
hoelzl@31098
|
733 |
finally have "arctan (real x) \<le> pi / 2 - real (?lb_horner ?invx)"
|
wenzelm@32962
|
734 |
using `0 \<le> arctan (real x)` arctan_inverse[OF `1 / real x \<noteq> 0`]
|
wenzelm@32962
|
735 |
unfolding real_sgn_pos[OF `0 < 1 / real x`] le_diff_eq by auto
|
hoelzl@29742
|
736 |
moreover
|
hoelzl@31098
|
737 |
have "pi / 2 \<le> real (ub_pi prec * Float 1 -1)" unfolding real_of_float_mult Float_num times_divide_eq_right mult_1_right using pi_boundaries by auto
|
hoelzl@29742
|
738 |
ultimately
|
hoelzl@29742
|
739 |
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]
|
wenzelm@32962
|
740 |
by auto
|
hoelzl@29742
|
741 |
qed
|
hoelzl@29742
|
742 |
qed
|
hoelzl@29742
|
743 |
qed
|
hoelzl@29742
|
744 |
|
hoelzl@29742
|
745 |
lemma arctan_boundaries:
|
hoelzl@31098
|
746 |
"arctan (real x) \<in> {real (lb_arctan prec x) .. real (ub_arctan prec x)}"
|
hoelzl@29742
|
747 |
proof (cases "0 \<le> x")
|
hoelzl@31098
|
748 |
case True hence "0 \<le> real x" unfolding le_float_def by auto
|
hoelzl@31098
|
749 |
show ?thesis using ub_arctan_bound'[OF `0 \<le> real x`] lb_arctan_bound'[OF `0 \<le> real x`] unfolding atLeastAtMost_iff by auto
|
hoelzl@29742
|
750 |
next
|
hoelzl@29742
|
751 |
let ?mx = "-x"
|
hoelzl@31098
|
752 |
case False hence "x < 0" and "0 \<le> real ?mx" unfolding le_float_def less_float_def by auto
|
hoelzl@31098
|
753 |
hence bounds: "real (lb_arctan prec ?mx) \<le> arctan (real ?mx) \<and> arctan (real ?mx) \<le> real (ub_arctan prec ?mx)"
|
hoelzl@31098
|
754 |
using ub_arctan_bound'[OF `0 \<le> real ?mx`] lb_arctan_bound'[OF `0 \<le> real ?mx`] by auto
|
hoelzl@31098
|
755 |
show ?thesis unfolding real_of_float_minus arctan_minus lb_arctan.simps[where x=x] ub_arctan.simps[where x=x] Let_def if_P[OF `x < 0`]
|
hoelzl@31098
|
756 |
unfolding atLeastAtMost_iff using bounds[unfolded real_of_float_minus arctan_minus] by auto
|
hoelzl@29742
|
757 |
qed
|
hoelzl@29742
|
758 |
|
hoelzl@31098
|
759 |
lemma bnds_arctan: "\<forall> x lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> arctan x \<and> arctan x \<le> real u"
|
hoelzl@29742
|
760 |
proof (rule allI, rule allI, rule allI, rule impI)
|
hoelzl@29742
|
761 |
fix x lx ux
|
hoelzl@31098
|
762 |
assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {real lx .. real ux}"
|
hoelzl@31098
|
763 |
hence l: "lb_arctan prec lx = l " and u: "ub_arctan prec ux = u" and x: "x \<in> {real lx .. real ux}" by auto
|
hoelzl@29742
|
764 |
|
hoelzl@29742
|
765 |
{ from arctan_boundaries[of lx prec, unfolded l]
|
hoelzl@31098
|
766 |
have "real l \<le> arctan (real lx)" by (auto simp del: lb_arctan.simps)
|
hoelzl@29742
|
767 |
also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
|
hoelzl@31098
|
768 |
finally have "real l \<le> arctan x" .
|
hoelzl@29742
|
769 |
} moreover
|
hoelzl@31098
|
770 |
{ have "arctan x \<le> arctan (real ux)" using x by (auto intro: arctan_monotone')
|
hoelzl@31098
|
771 |
also have "\<dots> \<le> real u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
|
hoelzl@31098
|
772 |
finally have "arctan x \<le> real u" .
|
hoelzl@31098
|
773 |
} ultimately show "real l \<le> arctan x \<and> arctan x \<le> real u" ..
|
hoelzl@29742
|
774 |
qed
|
hoelzl@29742
|
775 |
|
hoelzl@29742
|
776 |
section "Sinus and Cosinus"
|
hoelzl@29742
|
777 |
|
hoelzl@29742
|
778 |
subsection "Compute the cosinus and sinus series"
|
hoelzl@29742
|
779 |
|
hoelzl@29742
|
780 |
fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
|
hoelzl@29742
|
781 |
and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
|
hoelzl@29742
|
782 |
"ub_sin_cos_aux prec 0 i k x = 0"
|
hoelzl@31809
|
783 |
| "ub_sin_cos_aux prec (Suc n) i k x =
|
hoelzl@29742
|
784 |
(rapprox_rat prec 1 (int k)) - x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
|
hoelzl@29742
|
785 |
| "lb_sin_cos_aux prec 0 i k x = 0"
|
hoelzl@31809
|
786 |
| "lb_sin_cos_aux prec (Suc n) i k x =
|
hoelzl@29742
|
787 |
(lapprox_rat prec 1 (int k)) - x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
|
hoelzl@29742
|
788 |
|
hoelzl@29742
|
789 |
lemma cos_aux:
|
hoelzl@31098
|
790 |
shows "real (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (real x)^(2 * i))" (is "?lb")
|
hoelzl@31098
|
791 |
and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (real x)^(2 * i)) \<le> real (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
|
hoelzl@29742
|
792 |
proof -
|
hoelzl@31098
|
793 |
have "0 \<le> real (x * x)" unfolding real_of_float_mult by auto
|
hoelzl@29742
|
794 |
let "?f n" = "fact (2 * n)"
|
hoelzl@29742
|
795 |
|
hoelzl@31809
|
796 |
{ fix n
|
haftmann@30971
|
797 |
have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
|
haftmann@30971
|
798 |
have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)"
|
hoelzl@29742
|
799 |
unfolding F by auto } note f_eq = this
|
hoelzl@31809
|
800 |
|
hoelzl@31809
|
801 |
from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
|
hoelzl@31098
|
802 |
OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
|
hoelzl@31098
|
803 |
show "?lb" and "?ub" by (auto simp add: power_mult power2_eq_square[of "real x"])
|
hoelzl@29742
|
804 |
qed
|
hoelzl@29742
|
805 |
|
hoelzl@31098
|
806 |
lemma cos_boundaries: assumes "0 \<le> real x" and "real x \<le> pi / 2"
|
hoelzl@31098
|
807 |
shows "cos (real x) \<in> {real (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. real (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"
|
hoelzl@31098
|
808 |
proof (cases "real x = 0")
|
hoelzl@31098
|
809 |
case False hence "real x \<noteq> 0" by auto
|
hoelzl@31098
|
810 |
hence "0 < x" and "0 < real x" using `0 \<le> real x` unfolding less_float_def by auto
|
hoelzl@31098
|
811 |
have "0 < x * x" using `0 < x` unfolding less_float_def real_of_float_mult real_of_float_0
|
hoelzl@31098
|
812 |
using mult_pos_pos[where a="real x" and b="real x"] by auto
|
hoelzl@29742
|
813 |
|
haftmann@30952
|
814 |
{ fix x n have "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x ^ (2 * i))
|
hoelzl@29742
|
815 |
= (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum")
|
hoelzl@29742
|
816 |
proof -
|
hoelzl@29742
|
817 |
have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
|
hoelzl@31809
|
818 |
also have "\<dots> =
|
hoelzl@29742
|
819 |
(\<Sum> j = 0 ..< n. -1 ^ ((2 * j) div 2) / (real (fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
|
hoelzl@29742
|
820 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then -1 ^ (i div 2) / (real (fact i)) * x ^ i else 0)"
|
hoelzl@29742
|
821 |
unfolding sum_split_even_odd ..
|
hoelzl@29742
|
822 |
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then -1 ^ (i div 2) / (real (fact i)) else 0) * x ^ i)"
|
hoelzl@29742
|
823 |
by (rule setsum_cong2) auto
|
hoelzl@29742
|
824 |
finally show ?thesis by assumption
|
hoelzl@29742
|
825 |
qed } note morph_to_if_power = this
|
hoelzl@29742
|
826 |
|
hoelzl@29742
|
827 |
|
hoelzl@29742
|
828 |
{ fix n :: nat assume "0 < n"
|
hoelzl@29742
|
829 |
hence "0 < 2 * n" by auto
|
hoelzl@31098
|
830 |
obtain t where "0 < t" and "t < real x" and
|
hoelzl@31809
|
831 |
cos_eq: "cos (real x) = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * (real x) ^ i)
|
hoelzl@31809
|
832 |
+ (cos (t + 1/2 * real (2 * n) * pi) / real (fact (2*n))) * (real x)^(2*n)"
|
hoelzl@29742
|
833 |
(is "_ = ?SUM + ?rest / ?fact * ?pow")
|
hoelzl@31098
|
834 |
using Maclaurin_cos_expansion2[OF `0 < real x` `0 < 2 * n`] by auto
|
hoelzl@29742
|
835 |
|
hoelzl@29742
|
836 |
have "cos t * -1^n = cos t * cos (real n * pi) + sin t * sin (real n * pi)" by auto
|
hoelzl@29742
|
837 |
also have "\<dots> = cos (t + real n * pi)" using cos_add by auto
|
hoelzl@29742
|
838 |
also have "\<dots> = ?rest" by auto
|
hoelzl@29742
|
839 |
finally have "cos t * -1^n = ?rest" .
|
hoelzl@29742
|
840 |
moreover
|
hoelzl@31098
|
841 |
have "t \<le> pi / 2" using `t < real x` and `real x \<le> pi / 2` by auto
|
hoelzl@29742
|
842 |
hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
|
hoelzl@29742
|
843 |
ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
|
hoelzl@29742
|
844 |
|
hoelzl@29742
|
845 |
have "0 < ?fact" by auto
|
hoelzl@31098
|
846 |
have "0 < ?pow" using `0 < real x` by auto
|
hoelzl@29742
|
847 |
|
hoelzl@29742
|
848 |
{
|
hoelzl@29742
|
849 |
assume "even n"
|
hoelzl@31098
|
850 |
have "real (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
|
wenzelm@32962
|
851 |
unfolding morph_to_if_power[symmetric] using cos_aux by auto
|
hoelzl@31098
|
852 |
also have "\<dots> \<le> cos (real x)"
|
hoelzl@29742
|
853 |
proof -
|
wenzelm@32962
|
854 |
from even[OF `even n`] `0 < ?fact` `0 < ?pow`
|
wenzelm@32962
|
855 |
have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
|
wenzelm@32962
|
856 |
thus ?thesis unfolding cos_eq by auto
|
hoelzl@29742
|
857 |
qed
|
hoelzl@31098
|
858 |
finally have "real (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos (real x)" .
|
hoelzl@29742
|
859 |
} note lb = this
|
hoelzl@29742
|
860 |
|
hoelzl@29742
|
861 |
{
|
hoelzl@29742
|
862 |
assume "odd n"
|
hoelzl@31098
|
863 |
have "cos (real x) \<le> ?SUM"
|
hoelzl@29742
|
864 |
proof -
|
wenzelm@32962
|
865 |
from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
|
wenzelm@32962
|
866 |
have "0 \<le> (- ?rest) / ?fact * ?pow"
|
wenzelm@32962
|
867 |
by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
|
wenzelm@32962
|
868 |
thus ?thesis unfolding cos_eq by auto
|
hoelzl@29742
|
869 |
qed
|
hoelzl@31098
|
870 |
also have "\<dots> \<le> real (ub_sin_cos_aux prec n 1 1 (x * x))"
|
wenzelm@32962
|
871 |
unfolding morph_to_if_power[symmetric] using cos_aux by auto
|
hoelzl@31098
|
872 |
finally have "cos (real x) \<le> real (ub_sin_cos_aux prec n 1 1 (x * x))" .
|
hoelzl@29742
|
873 |
} note ub = this and lb
|
hoelzl@29742
|
874 |
} note ub = this(1) and lb = this(2)
|
hoelzl@29742
|
875 |
|
hoelzl@31098
|
876 |
have "cos (real x) \<le> real (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
|
hoelzl@31809
|
877 |
moreover have "real (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos (real x)"
|
hoelzl@29742
|
878 |
proof (cases "0 < get_even n")
|
hoelzl@29742
|
879 |
case True show ?thesis using lb[OF True get_even] .
|
hoelzl@29742
|
880 |
next
|
hoelzl@29742
|
881 |
case False
|
hoelzl@29742
|
882 |
hence "get_even n = 0" by auto
|
hoelzl@31098
|
883 |
have "- (pi / 2) \<le> real x" by (rule order_trans[OF _ `0 < real x`[THEN less_imp_le]], auto)
|
hoelzl@31098
|
884 |
with `real x \<le> pi / 2`
|
hoelzl@31098
|
885 |
show ?thesis unfolding `get_even n = 0` lb_sin_cos_aux.simps real_of_float_minus real_of_float_0 using cos_ge_zero by auto
|
hoelzl@29742
|
886 |
qed
|
hoelzl@29742
|
887 |
ultimately show ?thesis by auto
|
hoelzl@29742
|
888 |
next
|
hoelzl@29742
|
889 |
case True
|
hoelzl@29742
|
890 |
show ?thesis
|
hoelzl@29742
|
891 |
proof (cases "n = 0")
|
hoelzl@31809
|
892 |
case True
|
hoelzl@31098
|
893 |
thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `real x = 0` lapprox_rat[where x="-1" and y=1] by auto
|
hoelzl@29742
|
894 |
next
|
hoelzl@29742
|
895 |
case False with not0_implies_Suc obtain m where "n = Suc m" by blast
|
hoelzl@31098
|
896 |
thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `real 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@29742
|
897 |
qed
|
hoelzl@29742
|
898 |
qed
|
hoelzl@29742
|
899 |
|
hoelzl@31098
|
900 |
lemma sin_aux: assumes "0 \<le> real x"
|
hoelzl@31098
|
901 |
shows "real (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (real x)^(2 * i + 1))" (is "?lb")
|
hoelzl@31098
|
902 |
and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (real x)^(2 * i + 1)) \<le> real (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
|
hoelzl@29742
|
903 |
proof -
|
hoelzl@31098
|
904 |
have "0 \<le> real (x * x)" unfolding real_of_float_mult by auto
|
hoelzl@29742
|
905 |
let "?f n" = "fact (2 * n + 1)"
|
hoelzl@29742
|
906 |
|
hoelzl@31809
|
907 |
{ fix n
|
haftmann@30971
|
908 |
have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
|
haftmann@30971
|
909 |
have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)"
|
hoelzl@29742
|
910 |
unfolding F by auto } note f_eq = this
|
hoelzl@31809
|
911 |
|
hoelzl@29742
|
912 |
from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
|
hoelzl@31098
|
913 |
OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
|
hoelzl@31098
|
914 |
show "?lb" and "?ub" using `0 \<le> real x` unfolding real_of_float_mult
|
huffman@36770
|
915 |
unfolding power_add power_one_right mult_assoc[symmetric] setsum_left_distrib[symmetric]
|
huffman@36770
|
916 |
unfolding mult_commute[where 'a=real]
|
hoelzl@31098
|
917 |
by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real x"])
|
hoelzl@29742
|
918 |
qed
|
hoelzl@29742
|
919 |
|
hoelzl@31098
|
920 |
lemma sin_boundaries: assumes "0 \<le> real x" and "real x \<le> pi / 2"
|
hoelzl@31098
|
921 |
shows "sin (real x) \<in> {real (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. real (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}"
|
hoelzl@31098
|
922 |
proof (cases "real x = 0")
|
hoelzl@31098
|
923 |
case False hence "real x \<noteq> 0" by auto
|
hoelzl@31098
|
924 |
hence "0 < x" and "0 < real x" using `0 \<le> real x` unfolding less_float_def by auto
|
hoelzl@31098
|
925 |
have "0 < x * x" using `0 < x` unfolding less_float_def real_of_float_mult real_of_float_0
|
hoelzl@31098
|
926 |
using mult_pos_pos[where a="real x" and b="real x"] by auto
|
hoelzl@29742
|
927 |
|
hoelzl@29742
|
928 |
{ 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@29742
|
929 |
= (\<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@29742
|
930 |
proof -
|
hoelzl@29742
|
931 |
have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto
|
hoelzl@29742
|
932 |
have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" by auto
|
hoelzl@29742
|
933 |
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)"
|
wenzelm@32962
|
934 |
unfolding sum_split_even_odd ..
|
hoelzl@29742
|
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)"
|
wenzelm@32962
|
936 |
by (rule setsum_cong2) auto
|
hoelzl@29742
|
937 |
finally show ?thesis by assumption
|
hoelzl@29742
|
938 |
qed } note setsum_morph = this
|
hoelzl@29742
|
939 |
|
hoelzl@29742
|
940 |
{ fix n :: nat assume "0 < n"
|
hoelzl@29742
|
941 |
hence "0 < 2 * n + 1" by auto
|
hoelzl@31098
|
942 |
obtain t where "0 < t" and "t < real x" and
|
hoelzl@31809
|
943 |
sin_eq: "sin (real x) = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)
|
hoelzl@31809
|
944 |
+ (sin (t + 1/2 * real (2 * n + 1) * pi) / real (fact (2*n + 1))) * (real x)^(2*n + 1)"
|
hoelzl@29742
|
945 |
(is "_ = ?SUM + ?rest / ?fact * ?pow")
|
hoelzl@31098
|
946 |
using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < real x`] by auto
|
hoelzl@29742
|
947 |
|
hoelzl@29742
|
948 |
have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add left_distrib right_distrib by auto
|
hoelzl@29742
|
949 |
moreover
|
hoelzl@31098
|
950 |
have "t \<le> pi / 2" using `t < real x` and `real x \<le> pi / 2` by auto
|
hoelzl@29742
|
951 |
hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
|
hoelzl@29742
|
952 |
ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
|
hoelzl@29742
|
953 |
|
hoelzl@29742
|
954 |
have "0 < ?fact" by (rule real_of_nat_fact_gt_zero)
|
hoelzl@31098
|
955 |
have "0 < ?pow" using `0 < real x` by (rule zero_less_power)
|
hoelzl@29742
|
956 |
|
hoelzl@29742
|
957 |
{
|
hoelzl@29742
|
958 |
assume "even n"
|
hoelzl@31809
|
959 |
have "real (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
|
hoelzl@31098
|
960 |
(\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)"
|
wenzelm@32962
|
961 |
using sin_aux[OF `0 \<le> real x`] unfolding setsum_morph[symmetric] by auto
|
hoelzl@29742
|
962 |
also have "\<dots> \<le> ?SUM" by auto
|
hoelzl@31098
|
963 |
also have "\<dots> \<le> sin (real x)"
|
hoelzl@29742
|
964 |
proof -
|
wenzelm@32962
|
965 |
from even[OF `even n`] `0 < ?fact` `0 < ?pow`
|
wenzelm@32962
|
966 |
have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
|
wenzelm@32962
|
967 |
thus ?thesis unfolding sin_eq by auto
|
hoelzl@29742
|
968 |
qed
|
hoelzl@31098
|
969 |
finally have "real (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin (real x)" .
|
hoelzl@29742
|
970 |
} note lb = this
|
hoelzl@29742
|
971 |
|
hoelzl@29742
|
972 |
{
|
hoelzl@29742
|
973 |
assume "odd n"
|
hoelzl@31098
|
974 |
have "sin (real x) \<le> ?SUM"
|
hoelzl@29742
|
975 |
proof -
|
wenzelm@32962
|
976 |
from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
|
wenzelm@32962
|
977 |
have "0 \<le> (- ?rest) / ?fact * ?pow"
|
wenzelm@32962
|
978 |
by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
|
wenzelm@32962
|
979 |
thus ?thesis unfolding sin_eq by auto
|
hoelzl@29742
|
980 |
qed
|
hoelzl@31098
|
981 |
also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)"
|
wenzelm@32962
|
982 |
by auto
|
hoelzl@31809
|
983 |
also have "\<dots> \<le> real (x * ub_sin_cos_aux prec n 2 1 (x * x))"
|
wenzelm@32962
|
984 |
using sin_aux[OF `0 \<le> real x`] unfolding setsum_morph[symmetric] by auto
|
hoelzl@31098
|
985 |
finally have "sin (real x) \<le> real (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
|
hoelzl@29742
|
986 |
} note ub = this and lb
|
hoelzl@29742
|
987 |
} note ub = this(1) and lb = this(2)
|
hoelzl@29742
|
988 |
|
hoelzl@31098
|
989 |
have "sin (real x) \<le> real (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
|
hoelzl@31809
|
990 |
moreover have "real (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin (real x)"
|
hoelzl@29742
|
991 |
proof (cases "0 < get_even n")
|
hoelzl@29742
|
992 |
case True show ?thesis using lb[OF True get_even] .
|
hoelzl@29742
|
993 |
next
|
hoelzl@29742
|
994 |
case False
|
hoelzl@29742
|
995 |
hence "get_even n = 0" by auto
|
hoelzl@31098
|
996 |
with `real x \<le> pi / 2` `0 \<le> real x`
|
hoelzl@31098
|
997 |
show ?thesis unfolding `get_even n = 0` ub_sin_cos_aux.simps real_of_float_minus real_of_float_0 using sin_ge_zero by auto
|
hoelzl@29742
|
998 |
qed
|
hoelzl@29742
|
999 |
ultimately show ?thesis by auto
|
hoelzl@29742
|
1000 |
next
|
hoelzl@29742
|
1001 |
case True
|
hoelzl@29742
|
1002 |
show ?thesis
|
hoelzl@29742
|
1003 |
proof (cases "n = 0")
|
hoelzl@31809
|
1004 |
case True
|
hoelzl@31098
|
1005 |
thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `real x = 0` lapprox_rat[where x="-1" and y=1] by auto
|
hoelzl@29742
|
1006 |
next
|
hoelzl@29742
|
1007 |
case False with not0_implies_Suc obtain m where "n = Suc m" by blast
|
hoelzl@31098
|
1008 |
thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `real 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@29742
|
1009 |
qed
|
hoelzl@29742
|
1010 |
qed
|
hoelzl@29742
|
1011 |
|
hoelzl@29742
|
1012 |
subsection "Compute the cosinus in the entire domain"
|
hoelzl@29742
|
1013 |
|
hoelzl@29742
|
1014 |
definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
|
hoelzl@29742
|
1015 |
"lb_cos prec x = (let
|
hoelzl@29742
|
1016 |
horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ;
|
hoelzl@29742
|
1017 |
half = \<lambda> x. if x < 0 then - 1 else Float 1 1 * x * x - 1
|
hoelzl@29742
|
1018 |
in if x < Float 1 -1 then horner x
|
hoelzl@29742
|
1019 |
else if x < 1 then half (horner (x * Float 1 -1))
|
hoelzl@29742
|
1020 |
else half (half (horner (x * Float 1 -2))))"
|
hoelzl@29742
|
1021 |
|
hoelzl@29742
|
1022 |
definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
|
hoelzl@29742
|
1023 |
"ub_cos prec x = (let
|
hoelzl@29742
|
1024 |
horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ;
|
hoelzl@29742
|
1025 |
half = \<lambda> x. Float 1 1 * x * x - 1
|
hoelzl@29742
|
1026 |
in if x < Float 1 -1 then horner x
|
hoelzl@29742
|
1027 |
else if x < 1 then half (horner (x * Float 1 -1))
|
hoelzl@29742
|
1028 |
else half (half (horner (x * Float 1 -2))))"
|
hoelzl@29742
|
1029 |
|
hoelzl@31467
|
1030 |
lemma lb_cos: assumes "0 \<le> real x" and "real x \<le> pi"
|
hoelzl@31098
|
1031 |
shows "cos (real x) \<in> {real (lb_cos prec x) .. real (ub_cos prec x)}" (is "?cos x \<in> { real (?lb x) .. real (?ub x) }")
|
hoelzl@29742
|
1032 |
proof -
|
hoelzl@29742
|
1033 |
{ fix x :: real
|
hoelzl@29742
|
1034 |
have "cos x = cos (x / 2 + x / 2)" by auto
|
hoelzl@29742
|
1035 |
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@29742
|
1036 |
unfolding cos_add by auto
|
hoelzl@29742
|
1037 |
also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1" by algebra
|
hoelzl@29742
|
1038 |
finally have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" .
|
hoelzl@29742
|
1039 |
} note x_half = this[symmetric]
|
hoelzl@29742
|
1040 |
|
hoelzl@31098
|
1041 |
have "\<not> x < 0" using `0 \<le> real x` unfolding less_float_def by auto
|
hoelzl@29742
|
1042 |
let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
|
hoelzl@29742
|
1043 |
let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
|
hoelzl@29742
|
1044 |
let "?ub_half x" = "Float 1 1 * x * x - 1"
|
hoelzl@29742
|
1045 |
let "?lb_half x" = "if x < 0 then - 1 else Float 1 1 * x * x - 1"
|
hoelzl@29742
|
1046 |
|
hoelzl@29742
|
1047 |
show ?thesis
|
hoelzl@29742
|
1048 |
proof (cases "x < Float 1 -1")
|
hoelzl@31098
|
1049 |
case True hence "real x \<le> pi / 2" unfolding less_float_def using pi_ge_two by auto
|
hoelzl@29742
|
1050 |
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@31098
|
1051 |
using cos_boundaries[OF `0 \<le> real x` `real x \<le> pi / 2`] .
|
hoelzl@29742
|
1052 |
next
|
hoelzl@29742
|
1053 |
case False
|
hoelzl@31098
|
1054 |
{ fix y x :: float let ?x2 = "real (x * Float 1 -1)"
|
hoelzl@31098
|
1055 |
assume "real y \<le> cos ?x2" and "-pi \<le> real x" and "real x \<le> pi"
|
hoelzl@31098
|
1056 |
hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding real_of_float_mult Float_num by auto
|
hoelzl@29742
|
1057 |
hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
|
hoelzl@31467
|
1058 |
|
hoelzl@31098
|
1059 |
have "real (?lb_half y) \<le> cos (real x)"
|
hoelzl@29742
|
1060 |
proof (cases "y < 0")
|
wenzelm@32962
|
1061 |
case True show ?thesis using cos_ge_minus_one unfolding if_P[OF True] by auto
|
hoelzl@29742
|
1062 |
next
|
wenzelm@32962
|
1063 |
case False
|
wenzelm@32962
|
1064 |
hence "0 \<le> real y" unfolding less_float_def by auto
|
wenzelm@32962
|
1065 |
from mult_mono[OF `real y \<le> cos ?x2` `real y \<le> cos ?x2` `0 \<le> cos ?x2` this]
|
wenzelm@32962
|
1066 |
have "real y * real y \<le> cos ?x2 * cos ?x2" .
|
wenzelm@32962
|
1067 |
hence "2 * real y * real y \<le> 2 * cos ?x2 * cos ?x2" by auto
|
wenzelm@32962
|
1068 |
hence "2 * real y * real y - 1 \<le> 2 * cos (real x / 2) * cos (real x / 2) - 1" unfolding Float_num real_of_float_mult by auto
|
wenzelm@32962
|
1069 |
thus ?thesis unfolding if_not_P[OF False] x_half Float_num real_of_float_mult real_of_float_sub by auto
|
hoelzl@29742
|
1070 |
qed
|
hoelzl@29742
|
1071 |
} note lb_half = this
|
hoelzl@31467
|
1072 |
|
hoelzl@31098
|
1073 |
{ fix y x :: float let ?x2 = "real (x * Float 1 -1)"
|
hoelzl@31098
|
1074 |
assume ub: "cos ?x2 \<le> real y" and "- pi \<le> real x" and "real x \<le> pi"
|
hoelzl@31098
|
1075 |
hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding real_of_float_mult Float_num by auto
|
hoelzl@29742
|
1076 |
hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
|
hoelzl@31467
|
1077 |
|
hoelzl@31098
|
1078 |
have "cos (real x) \<le> real (?ub_half y)"
|
hoelzl@29742
|
1079 |
proof -
|
wenzelm@32962
|
1080 |
have "0 \<le> real y" using `0 \<le> cos ?x2` ub by (rule order_trans)
|
wenzelm@32962
|
1081 |
from mult_mono[OF ub ub this `0 \<le> cos ?x2`]
|
wenzelm@32962
|
1082 |
have "cos ?x2 * cos ?x2 \<le> real y * real y" .
|
wenzelm@32962
|
1083 |
hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real y * real y" by auto
|
wenzelm@32962
|
1084 |
hence "2 * cos (real x / 2) * cos (real x / 2) - 1 \<le> 2 * real y * real y - 1" unfolding Float_num real_of_float_mult by auto
|
wenzelm@32962
|
1085 |
thus ?thesis unfolding x_half real_of_float_mult Float_num real_of_float_sub by auto
|
hoelzl@29742
|
1086 |
qed
|
hoelzl@29742
|
1087 |
} note ub_half = this
|
hoelzl@31467
|
1088 |
|
hoelzl@29742
|
1089 |
let ?x2 = "x * Float 1 -1"
|
hoelzl@29742
|
1090 |
let ?x4 = "x * Float 1 -1 * Float 1 -1"
|
hoelzl@31467
|
1091 |
|
hoelzl@31098
|
1092 |
have "-pi \<le> real x" using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] `0 \<le> real x` by (rule order_trans)
|
hoelzl@31467
|
1093 |
|
hoelzl@29742
|
1094 |
show ?thesis
|
hoelzl@29742
|
1095 |
proof (cases "x < 1")
|
hoelzl@31098
|
1096 |
case True hence "real x \<le> 1" unfolding less_float_def by auto
|
hoelzl@31098
|
1097 |
have "0 \<le> real ?x2" and "real ?x2 \<le> pi / 2" using pi_ge_two `0 \<le> real x` unfolding real_of_float_mult Float_num using assms by auto
|
hoelzl@29742
|
1098 |
from cos_boundaries[OF this]
|
hoelzl@31098
|
1099 |
have lb: "real (?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> real (?ub_horner ?x2)" by auto
|
hoelzl@31467
|
1100 |
|
hoelzl@31098
|
1101 |
have "real (?lb x) \<le> ?cos x"
|
hoelzl@29742
|
1102 |
proof -
|
wenzelm@32962
|
1103 |
from lb_half[OF lb `-pi \<le> real x` `real x \<le> pi`]
|
wenzelm@32962
|
1104 |
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@29742
|
1105 |
qed
|
hoelzl@31098
|
1106 |
moreover have "?cos x \<le> real (?ub x)"
|
hoelzl@29742
|
1107 |
proof -
|
wenzelm@32962
|
1108 |
from ub_half[OF ub `-pi \<le> real x` `real x \<le> pi`]
|
wenzelm@32962
|
1109 |
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@29742
|
1110 |
qed
|
hoelzl@29742
|
1111 |
ultimately show ?thesis by auto
|
hoelzl@29742
|
1112 |
next
|
hoelzl@29742
|
1113 |
case False
|
hoelzl@31098
|
1114 |
have "0 \<le> real ?x4" and "real ?x4 \<le> pi / 2" using pi_ge_two `0 \<le> real x` `real x \<le> pi` unfolding real_of_float_mult Float_num by auto
|
hoelzl@29742
|
1115 |
from cos_boundaries[OF this]
|
hoelzl@31098
|
1116 |
have lb: "real (?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> real (?ub_horner ?x4)" by auto
|
hoelzl@31467
|
1117 |
|
hoelzl@29742
|
1118 |
have eq_4: "?x2 * Float 1 -1 = x * Float 1 -2" by (cases x, auto simp add: times_float.simps)
|
hoelzl@31467
|
1119 |
|
hoelzl@31098
|
1120 |
have "real (?lb x) \<le> ?cos x"
|
hoelzl@29742
|
1121 |
proof -
|
wenzelm@32962
|
1122 |
have "-pi \<le> real ?x2" and "real ?x2 \<le> pi" unfolding real_of_float_mult Float_num using pi_ge_two `0 \<le> real x` `real x \<le> pi` by auto
|
wenzelm@32962
|
1123 |
from lb_half[OF lb_half[OF lb this] `-pi \<le> real x` `real x \<le> pi`, unfolded eq_4]
|
wenzelm@32962
|
1124 |
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@29742
|
1125 |
qed
|
hoelzl@31098
|
1126 |
moreover have "?cos x \<le> real (?ub x)"
|
hoelzl@29742
|
1127 |
proof -
|
wenzelm@32962
|
1128 |
have "-pi \<le> real ?x2" and "real ?x2 \<le> pi" unfolding real_of_float_mult Float_num using pi_ge_two `0 \<le> real x` `real x \<le> pi` by auto
|
wenzelm@32962
|
1129 |
from ub_half[OF ub_half[OF ub this] `-pi \<le> real x` `real x \<le> pi`, unfolded eq_4]
|
wenzelm@32962
|
1130 |
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@29742
|
1131 |
qed
|
hoelzl@29742
|
1132 |
ultimately show ?thesis by auto
|
hoelzl@29742
|
1133 |
qed
|
hoelzl@29742
|
1134 |
qed
|
hoelzl@29742
|
1135 |
qed
|
hoelzl@29742
|
1136 |
|
hoelzl@31467
|
1137 |
lemma lb_cos_minus: assumes "-pi \<le> real x" and "real x \<le> 0"
|
hoelzl@31098
|
1138 |
shows "cos (real (-x)) \<in> {real (lb_cos prec (-x)) .. real (ub_cos prec (-x))}"
|
hoelzl@29742
|
1139 |
proof -
|
hoelzl@31098
|
1140 |
have "0 \<le> real (-x)" and "real (-x) \<le> pi" using `-pi \<le> real x` `real x \<le> 0` by auto
|
hoelzl@29742
|
1141 |
from lb_cos[OF this] show ?thesis .
|
hoelzl@29742
|
1142 |
qed
|
hoelzl@29742
|
1143 |
|
hoelzl@31467
|
1144 |
definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
|
hoelzl@31467
|
1145 |
"bnds_cos prec lx ux = (let
|
hoelzl@31467
|
1146 |
lpi = round_down prec (lb_pi prec) ;
|
hoelzl@31467
|
1147 |
upi = round_up prec (ub_pi prec) ;
|
hoelzl@31467
|
1148 |
k = floor_fl (float_divr prec (lx + lpi) (2 * lpi)) ;
|
hoelzl@31467
|
1149 |
lx = lx - k * 2 * (if k < 0 then lpi else upi) ;
|
hoelzl@31467
|
1150 |
ux = ux - k * 2 * (if k < 0 then upi else lpi)
|
hoelzl@31467
|
1151 |
in if - lpi \<le> lx \<and> ux \<le> 0 then (lb_cos prec (-lx), ub_cos prec (-ux))
|
hoelzl@31467
|
1152 |
else if 0 \<le> lx \<and> ux \<le> lpi then (lb_cos prec ux, ub_cos prec lx)
|
hoelzl@31467
|
1153 |
else if - lpi \<le> lx \<and> ux \<le> lpi then (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0)
|
hoelzl@31467
|
1154 |
else if 0 \<le> lx \<and> ux \<le> 2 * lpi then (Float -1 0, max (ub_cos prec lx) (ub_cos prec (- (ux - 2 * lpi))))
|
hoelzl@31508
|
1155 |
else if -2 * lpi \<le> lx \<and> ux \<le> 0 then (Float -1 0, max (ub_cos prec (lx + 2 * lpi)) (ub_cos prec (-ux)))
|
hoelzl@31467
|
1156 |
else (Float -1 0, Float 1 0))"
|
hoelzl@31467
|
1157 |
|
hoelzl@31467
|
1158 |
lemma floor_int:
|
hoelzl@31467
|
1159 |
obtains k :: int where "real k = real (floor_fl f)"
|
hoelzl@31467
|
1160 |
proof -
|
hoelzl@31467
|
1161 |
assume *: "\<And> k :: int. real k = real (floor_fl f) \<Longrightarrow> thesis"
|
hoelzl@31467
|
1162 |
obtain m e where fl: "Float m e = floor_fl f" by (cases "floor_fl f", auto)
|
hoelzl@31467
|
1163 |
from floor_pos_exp[OF this]
|
hoelzl@31467
|
1164 |
have "real (m* 2^(nat e)) = real (floor_fl f)"
|
hoelzl@31467
|
1165 |
by (auto simp add: fl[symmetric] real_of_float_def pow2_def)
|
hoelzl@31467
|
1166 |
from *[OF this] show thesis by blast
|
hoelzl@31467
|
1167 |
qed
|
hoelzl@31467
|
1168 |
|
hoelzl@31467
|
1169 |
lemma float_remove_real_numeral[simp]: "real (number_of k :: float) = number_of k"
|
hoelzl@31467
|
1170 |
proof -
|
hoelzl@31467
|
1171 |
have "real (number_of k :: float) = real k"
|
hoelzl@31467
|
1172 |
unfolding number_of_float_def real_of_float_def pow2_def by auto
|
hoelzl@31467
|
1173 |
also have "\<dots> = real (number_of k :: int)"
|
hoelzl@31467
|
1174 |
by (simp add: number_of_is_id)
|
hoelzl@31467
|
1175 |
finally show ?thesis by auto
|
hoelzl@31467
|
1176 |
qed
|
hoelzl@31467
|
1177 |
|
hoelzl@31467
|
1178 |
lemma cos_periodic_nat[simp]: fixes n :: nat shows "cos (x + real n * 2 * pi) = cos x"
|
hoelzl@31467
|
1179 |
proof (induct n arbitrary: x)
|
hoelzl@31467
|
1180 |
case (Suc n)
|
hoelzl@31467
|
1181 |
have split_pi_off: "x + real (Suc n) * 2 * pi = (x + real n * 2 * pi) + 2 * pi"
|
huffman@36770
|
1182 |
unfolding Suc_eq_plus1 real_of_nat_add real_of_one left_distrib by auto
|
hoelzl@31467
|
1183 |
show ?case unfolding split_pi_off using Suc by auto
|
hoelzl@31467
|
1184 |
qed auto
|
hoelzl@31467
|
1185 |
|
hoelzl@31467
|
1186 |
lemma cos_periodic_int[simp]: fixes i :: int shows "cos (x + real i * 2 * pi) = cos x"
|
hoelzl@31467
|
1187 |
proof (cases "0 \<le> i")
|
hoelzl@31467
|
1188 |
case True hence i_nat: "real i = real (nat i)" by auto
|
hoelzl@31467
|
1189 |
show ?thesis unfolding i_nat by auto
|
hoelzl@31467
|
1190 |
next
|
hoelzl@31467
|
1191 |
case False hence i_nat: "real i = - real (nat (-i))" by auto
|
hoelzl@31467
|
1192 |
have "cos x = cos (x + real i * 2 * pi - real i * 2 * pi)" by auto
|
hoelzl@31467
|
1193 |
also have "\<dots> = cos (x + real i * 2 * pi)"
|
hoelzl@31467
|
1194 |
unfolding i_nat mult_minus_left diff_minus_eq_add by (rule cos_periodic_nat)
|
hoelzl@31467
|
1195 |
finally show ?thesis by auto
|
hoelzl@31467
|
1196 |
qed
|
hoelzl@31467
|
1197 |
|
hoelzl@31098
|
1198 |
lemma bnds_cos: "\<forall> x lx ux. (l, u) = bnds_cos prec lx ux \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> cos x \<and> cos x \<le> real u"
|
hoelzl@31467
|
1199 |
proof ((rule allI | rule impI | erule conjE) +)
|
hoelzl@29742
|
1200 |
fix x lx ux
|
hoelzl@31467
|
1201 |
assume bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {real lx .. real ux}"
|
hoelzl@29742
|
1202 |
|
hoelzl@31467
|
1203 |
let ?lpi = "round_down prec (lb_pi prec)"
|
hoelzl@31467
|
1204 |
let ?upi = "round_up prec (ub_pi prec)"
|
hoelzl@31467
|
1205 |
let ?k = "floor_fl (float_divr prec (lx + ?lpi) (2 * ?lpi))"
|
hoelzl@31467
|
1206 |
let ?lx = "lx - ?k * 2 * (if ?k < 0 then ?lpi else ?upi)"
|
hoelzl@31467
|
1207 |
let ?ux = "ux - ?k * 2 * (if ?k < 0 then ?upi else ?lpi)"
|
hoelzl@31467
|
1208 |
|
hoelzl@31467
|
1209 |
obtain k :: int where k: "real k = real ?k" using floor_int .
|
hoelzl@31467
|
1210 |
|
hoelzl@31467
|
1211 |
have upi: "pi \<le> real ?upi" and lpi: "real ?lpi \<le> pi"
|
hoelzl@31467
|
1212 |
using round_up[of "ub_pi prec" prec] pi_boundaries[of prec]
|
hoelzl@31467
|
1213 |
round_down[of prec "lb_pi prec"] by auto
|
hoelzl@31467
|
1214 |
hence "real ?lx \<le> x - real k * 2 * pi \<and> x - real k * 2 * pi \<le> real ?ux"
|
hoelzl@31467
|
1215 |
using x by (cases "k = 0") (auto intro!: add_mono
|
huffman@36770
|
1216 |
simp add: diff_def k[symmetric] less_float_def)
|
hoelzl@31467
|
1217 |
note lx = this[THEN conjunct1] and ux = this[THEN conjunct2]
|
hoelzl@31467
|
1218 |
hence lx_less_ux: "real ?lx \<le> real ?ux" by (rule order_trans)
|
hoelzl@31467
|
1219 |
|
hoelzl@31467
|
1220 |
{ assume "- ?lpi \<le> ?lx" and x_le_0: "x - real k * 2 * pi \<le> 0"
|
hoelzl@31467
|
1221 |
with lpi[THEN le_imp_neg_le] lx
|
hoelzl@31467
|
1222 |
have pi_lx: "- pi \<le> real ?lx" and lx_0: "real ?lx \<le> 0"
|
hoelzl@31467
|
1223 |
by (simp_all add: le_float_def)
|
hoelzl@31467
|
1224 |
|
hoelzl@31467
|
1225 |
have "real (lb_cos prec (- ?lx)) \<le> cos (real (- ?lx))"
|
hoelzl@31467
|
1226 |
using lb_cos_minus[OF pi_lx lx_0] by simp
|
hoelzl@31467
|
1227 |
also have "\<dots> \<le> cos (x + real (-k) * 2 * pi)"
|
hoelzl@31467
|
1228 |
using cos_monotone_minus_pi_0'[OF pi_lx lx x_le_0]
|
hoelzl@31467
|
1229 |
by (simp only: real_of_float_minus real_of_int_minus
|
huffman@36770
|
1230 |
cos_minus diff_def mult_minus_left)
|
hoelzl@31467
|
1231 |
finally have "real (lb_cos prec (- ?lx)) \<le> cos x"
|
hoelzl@31467
|
1232 |
unfolding cos_periodic_int . }
|
hoelzl@31467
|
1233 |
note negative_lx = this
|
hoelzl@31467
|
1234 |
|
hoelzl@31467
|
1235 |
{ assume "0 \<le> ?lx" and pi_x: "x - real k * 2 * pi \<le> pi"
|
hoelzl@31467
|
1236 |
with lx
|
hoelzl@31467
|
1237 |
have pi_lx: "real ?lx \<le> pi" and lx_0: "0 \<le> real ?lx"
|
hoelzl@31467
|
1238 |
by (auto simp add: le_float_def)
|
hoelzl@31467
|
1239 |
|
hoelzl@31467
|
1240 |
have "cos (x + real (-k) * 2 * pi) \<le> cos (real ?lx)"
|
hoelzl@31467
|
1241 |
using cos_monotone_0_pi'[OF lx_0 lx pi_x]
|
hoelzl@31467
|
1242 |
by (simp only: real_of_float_minus real_of_int_minus
|
huffman@36770
|
1243 |
cos_minus diff_def mult_minus_left)
|
hoelzl@31467
|
1244 |
also have "\<dots> \<le> real (ub_cos prec ?lx)"
|
hoelzl@31467
|
1245 |
using lb_cos[OF lx_0 pi_lx] by simp
|
hoelzl@31467
|
1246 |
finally have "cos x \<le> real (ub_cos prec ?lx)"
|
hoelzl@31467
|
1247 |
unfolding cos_periodic_int . }
|
hoelzl@31467
|
1248 |
note positive_lx = this
|
hoelzl@31467
|
1249 |
|
hoelzl@31467
|
1250 |
{ assume pi_x: "- pi \<le> x - real k * 2 * pi" and "?ux \<le> 0"
|
hoelzl@31467
|
1251 |
with ux
|
hoelzl@31467
|
1252 |
have pi_ux: "- pi \<le> real ?ux" and ux_0: "real ?ux \<le> 0"
|
hoelzl@31467
|
1253 |
by (simp_all add: le_float_def)
|
hoelzl@31467
|
1254 |
|
hoelzl@31467
|
1255 |
have "cos (x + real (-k) * 2 * pi) \<le> cos (real (- ?ux))"
|
hoelzl@31467
|
1256 |
using cos_monotone_minus_pi_0'[OF pi_x ux ux_0]
|
hoelzl@31467
|
1257 |
by (simp only: real_of_float_minus real_of_int_minus
|
huffman@36770
|
1258 |
cos_minus diff_def mult_minus_left)
|
hoelzl@31467
|
1259 |
also have "\<dots> \<le> real (ub_cos prec (- ?ux))"
|
hoelzl@31467
|
1260 |
using lb_cos_minus[OF pi_ux ux_0, of prec] by simp
|
hoelzl@31467
|
1261 |
finally have "cos x \<le> real (ub_cos prec (- ?ux))"
|
hoelzl@31467
|
1262 |
unfolding cos_periodic_int . }
|
hoelzl@31467
|
1263 |
note negative_ux = this
|
hoelzl@31467
|
1264 |
|
hoelzl@31467
|
1265 |
{ assume "?ux \<le> ?lpi" and x_ge_0: "0 \<le> x - real k * 2 * pi"
|
hoelzl@31467
|
1266 |
with lpi ux
|
hoelzl@31467
|
1267 |
have pi_ux: "real ?ux \<le> pi" and ux_0: "0 \<le> real ?ux"
|
hoelzl@31467
|
1268 |
by (simp_all add: le_float_def)
|
hoelzl@31467
|
1269 |
|
hoelzl@31467
|
1270 |
have "real (lb_cos prec ?ux) \<le> cos (real ?ux)"
|
hoelzl@31467
|
1271 |
using lb_cos[OF ux_0 pi_ux] by simp
|
hoelzl@31467
|
1272 |
also have "\<dots> \<le> cos (x + real (-k) * 2 * pi)"
|
hoelzl@31467
|
1273 |
using cos_monotone_0_pi'[OF x_ge_0 ux pi_ux]
|
hoelzl@31467
|
1274 |
by (simp only: real_of_float_minus real_of_int_minus
|
huffman@36770
|
1275 |
cos_minus diff_def mult_minus_left)
|
hoelzl@31467
|
1276 |
finally have "real (lb_cos prec ?ux) \<le> cos x"
|
hoelzl@31467
|
1277 |
unfolding cos_periodic_int . }
|
hoelzl@31467
|
1278 |
note positive_ux = this
|
hoelzl@29742
|
1279 |
|
hoelzl@31098
|
1280 |
show "real l \<le> cos x \<and> cos x \<le> real u"
|
hoelzl@31467
|
1281 |
proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> 0")
|
hoelzl@31467
|
1282 |
case True with bnds
|
hoelzl@31467
|
1283 |
have l: "l = lb_cos prec (-?lx)"
|
hoelzl@31467
|
1284 |
and u: "u = ub_cos prec (-?ux)"
|
hoelzl@31467
|
1285 |
by (auto simp add: bnds_cos_def Let_def)
|
hoelzl@31467
|
1286 |
|
hoelzl@31467
|
1287 |
from True lpi[THEN le_imp_neg_le] lx ux
|
hoelzl@31467
|
1288 |
have "- pi \<le> x - real k * 2 * pi"
|
hoelzl@31467
|
1289 |
and "x - real k * 2 * pi \<le> 0"
|
hoelzl@31467
|
1290 |
by (auto simp add: le_float_def)
|
hoelzl@31467
|
1291 |
with True negative_ux negative_lx
|
hoelzl@31467
|
1292 |
show ?thesis unfolding l u by simp
|
hoelzl@31467
|
1293 |
next case False note 1 = this show ?thesis
|
hoelzl@31467
|
1294 |
proof (cases "0 \<le> ?lx \<and> ?ux \<le> ?lpi")
|
hoelzl@31467
|
1295 |
case True with bnds 1
|
hoelzl@31467
|
1296 |
have l: "l = lb_cos prec ?ux"
|
hoelzl@31467
|
1297 |
and u: "u = ub_cos prec ?lx"
|
hoelzl@31467
|
1298 |
by (auto simp add: bnds_cos_def Let_def)
|
hoelzl@31467
|
1299 |
|
hoelzl@31467
|
1300 |
from True lpi lx ux
|
hoelzl@31467
|
1301 |
have "0 \<le> x - real k * 2 * pi"
|
hoelzl@31467
|
1302 |
and "x - real k * 2 * pi \<le> pi"
|
hoelzl@31467
|
1303 |
by (auto simp add: le_float_def)
|
hoelzl@31467
|
1304 |
with True positive_ux positive_lx
|
hoelzl@31467
|
1305 |
show ?thesis unfolding l u by simp
|
hoelzl@31467
|
1306 |
next case False note 2 = this show ?thesis
|
hoelzl@31467
|
1307 |
proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> ?lpi")
|
hoelzl@31467
|
1308 |
case True note Cond = this with bnds 1 2
|
hoelzl@31467
|
1309 |
have l: "l = min (lb_cos prec (-?lx)) (lb_cos prec ?ux)"
|
hoelzl@31467
|
1310 |
and u: "u = Float 1 0"
|
hoelzl@31467
|
1311 |
by (auto simp add: bnds_cos_def Let_def)
|
hoelzl@31467
|
1312 |
|
hoelzl@31467
|
1313 |
show ?thesis unfolding u l using negative_lx positive_ux Cond
|
hoelzl@31467
|
1314 |
by (cases "x - real k * 2 * pi < 0", simp_all add: real_of_float_min)
|
hoelzl@31467
|
1315 |
next case False note 3 = this show ?thesis
|
hoelzl@31467
|
1316 |
proof (cases "0 \<le> ?lx \<and> ?ux \<le> 2 * ?lpi")
|
hoelzl@31467
|
1317 |
case True note Cond = this with bnds 1 2 3
|
hoelzl@31467
|
1318 |
have l: "l = Float -1 0"
|
hoelzl@31467
|
1319 |
and u: "u = max (ub_cos prec ?lx) (ub_cos prec (- (?ux - 2 * ?lpi)))"
|
hoelzl@31467
|
1320 |
by (auto simp add: bnds_cos_def Let_def)
|
hoelzl@31467
|
1321 |
|
hoelzl@31467
|
1322 |
have "cos x \<le> real u"
|
hoelzl@31467
|
1323 |
proof (cases "x - real k * 2 * pi < pi")
|
hoelzl@31467
|
1324 |
case True hence "x - real k * 2 * pi \<le> pi" by simp
|
hoelzl@31467
|
1325 |
from positive_lx[OF Cond[THEN conjunct1] this]
|
hoelzl@31467
|
1326 |
show ?thesis unfolding u by (simp add: real_of_float_max)
|
hoelzl@31467
|
1327 |
next
|
hoelzl@31467
|
1328 |
case False hence "pi \<le> x - real k * 2 * pi" by simp
|
hoelzl@31467
|
1329 |
hence pi_x: "- pi \<le> x - real k * 2 * pi - 2 * pi" by simp
|
hoelzl@31467
|
1330 |
|
hoelzl@31467
|
1331 |
have "real ?ux \<le> 2 * pi" using Cond lpi by (auto simp add: le_float_def)
|
hoelzl@31467
|
1332 |
hence "x - real k * 2 * pi - 2 * pi \<le> 0" using ux by simp
|
hoelzl@31467
|
1333 |
|
hoelzl@31467
|
1334 |
have ux_0: "real (?ux - 2 * ?lpi) \<le> 0"
|
wenzelm@32962
|
1335 |
using Cond by (auto simp add: le_float_def)
|
hoelzl@31467
|
1336 |
|
hoelzl@31467
|
1337 |
from 2 and Cond have "\<not> ?ux \<le> ?lpi" by auto
|
hoelzl@31467
|
1338 |
hence "- ?lpi \<le> ?ux - 2 * ?lpi" by (auto simp add: le_float_def)
|
hoelzl@31467
|
1339 |
hence pi_ux: "- pi \<le> real (?ux - 2 * ?lpi)"
|
wenzelm@32962
|
1340 |
using lpi[THEN le_imp_neg_le] by (auto simp add: le_float_def)
|
hoelzl@31467
|
1341 |
|
hoelzl@31467
|
1342 |
have x_le_ux: "x - real k * 2 * pi - 2 * pi \<le> real (?ux - 2 * ?lpi)"
|
wenzelm@32962
|
1343 |
using ux lpi by auto
|
hoelzl@31467
|
1344 |
|
hoelzl@31467
|
1345 |
have "cos x = cos (x + real (-k) * 2 * pi + real (-1 :: int) * 2 * pi)"
|
wenzelm@32962
|
1346 |
unfolding cos_periodic_int ..
|
hoelzl@31467
|
1347 |
also have "\<dots> \<le> cos (real (?ux - 2 * ?lpi))"
|
wenzelm@32962
|
1348 |
using cos_monotone_minus_pi_0'[OF pi_x x_le_ux ux_0]
|
wenzelm@32962
|
1349 |
by (simp only: real_of_float_minus real_of_int_minus real_of_one
|
huffman@36770
|
1350 |
number_of_Min diff_def mult_minus_left mult_1_left)
|
hoelzl@31467
|
1351 |
also have "\<dots> = cos (real (- (?ux - 2 * ?lpi)))"
|
wenzelm@32962
|
1352 |
unfolding real_of_float_minus cos_minus ..
|
hoelzl@31467
|
1353 |
also have "\<dots> \<le> real (ub_cos prec (- (?ux - 2 * ?lpi)))"
|
wenzelm@32962
|
1354 |
using lb_cos_minus[OF pi_ux ux_0] by simp
|
hoelzl@31467
|
1355 |
finally show ?thesis unfolding u by (simp add: real_of_float_max)
|
hoelzl@31467
|
1356 |
qed
|
hoelzl@31467
|
1357 |
thus ?thesis unfolding l by auto
|
hoelzl@31508
|
1358 |
next case False note 4 = this show ?thesis
|
hoelzl@31508
|
1359 |
proof (cases "-2 * ?lpi \<le> ?lx \<and> ?ux \<le> 0")
|
hoelzl@31508
|
1360 |
case True note Cond = this with bnds 1 2 3 4
|
hoelzl@31508
|
1361 |
have l: "l = Float -1 0"
|
hoelzl@31508
|
1362 |
and u: "u = max (ub_cos prec (?lx + 2 * ?lpi)) (ub_cos prec (-?ux))"
|
hoelzl@31508
|
1363 |
by (auto simp add: bnds_cos_def Let_def)
|
hoelzl@31508
|
1364 |
|
hoelzl@31508
|
1365 |
have "cos x \<le> real u"
|
hoelzl@31508
|
1366 |
proof (cases "-pi < x - real k * 2 * pi")
|
hoelzl@31508
|
1367 |
case True hence "-pi \<le> x - real k * 2 * pi" by simp
|
hoelzl@31508
|
1368 |
from negative_ux[OF this Cond[THEN conjunct2]]
|
hoelzl@31508
|
1369 |
show ?thesis unfolding u by (simp add: real_of_float_max)
|
hoelzl@31508
|
1370 |
next
|
hoelzl@31508
|
1371 |
case False hence "x - real k * 2 * pi \<le> -pi" by simp
|
hoelzl@31508
|
1372 |
hence pi_x: "x - real k * 2 * pi + 2 * pi \<le> pi" by simp
|
hoelzl@31508
|
1373 |
|
hoelzl@31508
|
1374 |
have "-2 * pi \<le> real ?lx" using Cond lpi by (auto simp add: le_float_def)
|
hoelzl@31508
|
1375 |
|
hoelzl@31508
|
1376 |
hence "0 \<le> x - real k * 2 * pi + 2 * pi" using lx by simp
|
hoelzl@31508
|
1377 |
|
hoelzl@31508
|
1378 |
have lx_0: "0 \<le> real (?lx + 2 * ?lpi)"
|
wenzelm@32962
|
1379 |
using Cond lpi by (auto simp add: le_float_def)
|
hoelzl@31508
|
1380 |
|
hoelzl@31508
|
1381 |
from 1 and Cond have "\<not> -?lpi \<le> ?lx" by auto
|
hoelzl@31508
|
1382 |
hence "?lx + 2 * ?lpi \<le> ?lpi" by (auto simp add: le_float_def)
|
hoelzl@31508
|
1383 |
hence pi_lx: "real (?lx + 2 * ?lpi) \<le> pi"
|
wenzelm@32962
|
1384 |
using lpi[THEN le_imp_neg_le] by (auto simp add: le_float_def)
|
hoelzl@31508
|
1385 |
|
hoelzl@31508
|
1386 |
have lx_le_x: "real (?lx + 2 * ?lpi) \<le> x - real k * 2 * pi + 2 * pi"
|
wenzelm@32962
|
1387 |
using lx lpi by auto
|
hoelzl@31508
|
1388 |
|
hoelzl@31508
|
1389 |
have "cos x = cos (x + real (-k) * 2 * pi + real (1 :: int) * 2 * pi)"
|
wenzelm@32962
|
1390 |
unfolding cos_periodic_int ..
|
hoelzl@31508
|
1391 |
also have "\<dots> \<le> cos (real (?lx + 2 * ?lpi))"
|
wenzelm@32962
|
1392 |
using cos_monotone_0_pi'[OF lx_0 lx_le_x pi_x]
|
wenzelm@32962
|
1393 |
by (simp only: real_of_float_minus real_of_int_minus real_of_one
|
huffman@36770
|
1394 |
number_of_Min diff_def mult_minus_left mult_1_left)
|
hoelzl@31508
|
1395 |
also have "\<dots> \<le> real (ub_cos prec (?lx + 2 * ?lpi))"
|
wenzelm@32962
|
1396 |
using lb_cos[OF lx_0 pi_lx] by simp
|
hoelzl@31508
|
1397 |
finally show ?thesis unfolding u by (simp add: real_of_float_max)
|
hoelzl@31508
|
1398 |
qed
|
hoelzl@31508
|
1399 |
thus ?thesis unfolding l by auto
|
hoelzl@29742
|
1400 |
next
|
hoelzl@31508
|
1401 |
case False with bnds 1 2 3 4 show ?thesis by (auto simp add: bnds_cos_def Let_def)
|
hoelzl@31508
|
1402 |
qed qed qed qed qed
|
hoelzl@29742
|
1403 |
qed
|
hoelzl@29742
|
1404 |
|
hoelzl@29742
|
1405 |
section "Exponential function"
|
hoelzl@29742
|
1406 |
|
hoelzl@29742
|
1407 |
subsection "Compute the series of the exponential function"
|
hoelzl@29742
|
1408 |
|
hoelzl@29742
|
1409 |
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@29742
|
1410 |
"ub_exp_horner prec 0 i k x = 0" |
|
hoelzl@29742
|
1411 |
"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@29742
|
1412 |
"lb_exp_horner prec 0 i k x = 0" |
|
hoelzl@29742
|
1413 |
"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@29742
|
1414 |
|
hoelzl@31098
|
1415 |
lemma bnds_exp_horner: assumes "real x \<le> 0"
|
hoelzl@31098
|
1416 |
shows "exp (real x) \<in> { real (lb_exp_horner prec (get_even n) 1 1 x) .. real (ub_exp_horner prec (get_odd n) 1 1 x) }"
|
hoelzl@29742
|
1417 |
proof -
|
hoelzl@29742
|
1418 |
{ fix n
|
haftmann@30971
|
1419 |
have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m" by (induct n, auto)
|
haftmann@30971
|
1420 |
have "fact (Suc n) = fact n * ((\<lambda>i. i + 1) ^^ n) 1" unfolding F by auto } note f_eq = this
|
hoelzl@31467
|
1421 |
|
hoelzl@29742
|
1422 |
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@29742
|
1423 |
OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]
|
hoelzl@29742
|
1424 |
|
hoelzl@31098
|
1425 |
{ have "real (lb_exp_horner prec (get_even n) 1 1 x) \<le> (\<Sum>j = 0..<get_even n. 1 / real (fact j) * real x ^ j)"
|
hoelzl@29742
|
1426 |
using bounds(1) by auto
|
hoelzl@31098
|
1427 |
also have "\<dots> \<le> exp (real x)"
|
hoelzl@29742
|
1428 |
proof -
|
hoelzl@31098
|
1429 |
obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp (real x) = (\<Sum>m = 0..<get_even n. (real x) ^ m / real (fact m)) + exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
|
wenzelm@32962
|
1430 |
using Maclaurin_exp_le by blast
|
hoelzl@31098
|
1431 |
moreover have "0 \<le> exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
|
wenzelm@32962
|
1432 |
by (auto intro!: mult_nonneg_nonneg divide_nonneg_pos simp add: get_even zero_le_even_power exp_gt_zero)
|
hoelzl@29742
|
1433 |
ultimately show ?thesis
|
haftmann@35028
|
1434 |
using get_odd exp_gt_zero by (auto intro!: mult_nonneg_nonneg)
|
hoelzl@29742
|
1435 |
qed
|
hoelzl@31098
|
1436 |
finally have "real (lb_exp_horner prec (get_even n) 1 1 x) \<le> exp (real x)" .
|
hoelzl@29742
|
1437 |
} moreover
|
hoelzl@31809
|
1438 |
{
|
hoelzl@31098
|
1439 |
have x_less_zero: "real x ^ get_odd n \<le> 0"
|
hoelzl@31098
|
1440 |
proof (cases "real x = 0")
|
hoelzl@29742
|
1441 |
case True
|
hoelzl@29742
|
1442 |
have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
|
hoelzl@29742
|
1443 |
thus ?thesis unfolding True power_0_left by auto
|
hoelzl@29742
|
1444 |
next
|
hoelzl@31098
|
1445 |
case False hence "real x < 0" using `real x \<le> 0` by auto
|
hoelzl@31098
|
1446 |
show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq get_odd `real x < 0`)
|
hoelzl@29742
|
1447 |
qed
|
hoelzl@29742
|
1448 |
|
hoelzl@31098
|
1449 |
obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp (real x) = (\<Sum>m = 0..<get_odd n. (real x) ^ m / real (fact m)) + exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n)"
|
hoelzl@29742
|
1450 |
using Maclaurin_exp_le by blast
|
hoelzl@31098
|
1451 |
moreover have "exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n) \<le> 0"
|
hoelzl@29742
|
1452 |
by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero exp_gt_zero)
|
hoelzl@31098
|
1453 |
ultimately have "exp (real x) \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * real x ^ j)"
|
haftmann@35028
|
1454 |
using get_odd exp_gt_zero by (auto intro!: mult_nonneg_nonneg)
|
hoelzl@31098
|
1455 |
also have "\<dots> \<le> real (ub_exp_horner prec (get_odd n) 1 1 x)"
|
hoelzl@29742
|
1456 |
using bounds(2) by auto
|
hoelzl@31098
|
1457 |
finally have "exp (real x) \<le> real (ub_exp_horner prec (get_odd n) 1 1 x)" .
|
hoelzl@29742
|
1458 |
} ultimately show ?thesis by auto
|
hoelzl@29742
|
1459 |
qed
|
hoelzl@29742
|
1460 |
|
hoelzl@29742
|
1461 |
subsection "Compute the exponential function on the entire domain"
|
hoelzl@29742
|
1462 |
|
hoelzl@29742
|
1463 |
function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where
|
hoelzl@29742
|
1464 |
"lb_exp prec x = (if 0 < x then float_divl prec 1 (ub_exp prec (-x))
|
hoelzl@31809
|
1465 |
else let
|
hoelzl@29742
|
1466 |
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@29742
|
1467 |
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@29742
|
1468 |
else horner x)" |
|
hoelzl@29742
|
1469 |
"ub_exp prec x = (if 0 < x then float_divr prec 1 (lb_exp prec (-x))
|
hoelzl@31809
|
1470 |
else if x < - 1 then (case floor_fl x of (Float m e) \<Rightarrow>
|
hoelzl@29742
|
1471 |
(ub_exp_horner prec (get_odd (prec + 2)) 1 1 (float_divr prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e))
|
hoelzl@29742
|
1472 |
else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
|
hoelzl@29742
|
1473 |
by pat_completeness auto
|
hoelzl@29742
|
1474 |
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@29742
|
1475 |
|
hoelzl@29742
|
1476 |
lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)"
|
hoelzl@29742
|
1477 |
proof -
|
hoelzl@29742
|
1478 |
have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
|
hoelzl@29742
|
1479 |
|
hoelzl@31098
|
1480 |
have "1 / 4 = real (Float 1 -2)" unfolding Float_num by auto
|
hoelzl@31098
|
1481 |
also have "\<dots> \<le> real (lb_exp_horner 1 (get_even 4) 1 1 (- 1))"
|
hoelzl@31809
|
1482 |
unfolding get_even_def eq4
|
hoelzl@29742
|
1483 |
by (auto simp add: lapprox_posrat_def rapprox_posrat_def normfloat.simps)
|
hoelzl@31098
|
1484 |
also have "\<dots> \<le> exp (real (- 1 :: float))" using bnds_exp_horner[where x="- 1"] by auto
|
hoelzl@31809
|
1485 |
finally show ?thesis unfolding real_of_float_minus real_of_float_1 .
|
hoelzl@29742
|
1486 |
qed
|
hoelzl@29742
|
1487 |
|
hoelzl@29742
|
1488 |
lemma lb_exp_pos: assumes "\<not> 0 < x" shows "0 < lb_exp prec x"
|
hoelzl@29742
|
1489 |
proof -
|
hoelzl@29742
|
1490 |
let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
|
hoelzl@29742
|
1491 |
let "?horner x" = "let y = ?lb_horner x in if y \<le> 0 then Float 1 -2 else y"
|
hoelzl@29742
|
1492 |
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@29742
|
1493 |
moreover { fix x :: float fix num :: nat
|
hoelzl@31098
|
1494 |
have "0 < real (?horner x) ^ num" using `0 < ?horner x`[unfolded less_float_def real_of_float_0] by (rule zero_less_power)
|
hoelzl@31098
|
1495 |
also have "\<dots> = real ((?horner x) ^ num)" using float_power by auto
|
hoelzl@31098
|
1496 |
finally have "0 < real ((?horner x) ^ num)" .
|
hoelzl@29742
|
1497 |
}
|
hoelzl@29742
|
1498 |
ultimately show ?thesis
|
haftmann@30968
|
1499 |
unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def
|
haftmann@30968
|
1500 |
by (cases "floor_fl x", cases "x < - 1", auto simp add: float_power le_float_def less_float_def)
|
hoelzl@29742
|
1501 |
qed
|
hoelzl@29742
|
1502 |
|
hoelzl@29742
|
1503 |
lemma exp_boundaries': assumes "x \<le> 0"
|
hoelzl@31098
|
1504 |
shows "exp (real x) \<in> { real (lb_exp prec x) .. real (ub_exp prec x)}"
|
hoelzl@29742
|
1505 |
proof -
|
hoelzl@29742
|
1506 |
let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
|
hoelzl@29742
|
1507 |
let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"
|
hoelzl@29742
|
1508 |
|
hoelzl@31098
|
1509 |
have "real x \<le> 0" and "\<not> x > 0" using `x \<le> 0` unfolding le_float_def less_float_def by auto
|
hoelzl@29742
|
1510 |
show ?thesis
|
hoelzl@29742
|
1511 |
proof (cases "x < - 1")
|
hoelzl@31098
|
1512 |
case False hence "- 1 \<le> real x" unfolding less_float_def by auto
|
hoelzl@29742
|
1513 |
show ?thesis
|
hoelzl@29742
|
1514 |
proof (cases "?lb_exp_horner x \<le> 0")
|
hoelzl@31098
|
1515 |
from `\<not> x < - 1` have "- 1 \<le> real x" unfolding less_float_def by auto
|
hoelzl@31098
|
1516 |
hence "exp (- 1) \<le> exp (real x)" unfolding exp_le_cancel_iff .
|
hoelzl@29742
|
1517 |
from order_trans[OF exp_m1_ge_quarter this]
|
hoelzl@31098
|
1518 |
have "real (Float 1 -2) \<le> exp (real x)" unfolding Float_num .
|
hoelzl@29742
|
1519 |
moreover case True
|
hoelzl@31098
|
1520 |
ultimately show ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by auto
|
hoelzl@29742
|
1521 |
next
|
hoelzl@31098
|
1522 |
case False thus ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by (auto simp add: Let_def)
|
hoelzl@29742
|
1523 |
qed
|
hoelzl@29742
|
1524 |
next
|
hoelzl@29742
|
1525 |
case True
|
hoelzl@31809
|
1526 |
|
hoelzl@29742
|
1527 |
obtain m e where Float_floor: "floor_fl x = Float m e" by (cases "floor_fl x", auto)
|
hoelzl@29742
|
1528 |
let ?num = "nat (- m) * 2 ^ nat e"
|
hoelzl@31809
|
1529 |
|
hoelzl@31098
|
1530 |
have "real (floor_fl x) < - 1" using floor_fl `x < - 1` unfolding le_float_def less_float_def real_of_float_minus real_of_float_1 by (rule order_le_less_trans)
|
hoelzl@31098
|
1531 |
hence "real (floor_fl x) < 0" unfolding Float_floor real_of_float_simp using zero_less_pow2[of xe] by auto
|
hoelzl@29742
|
1532 |
hence "m < 0"
|
hoelzl@31098
|
1533 |
unfolding less_float_def real_of_float_0 Float_floor real_of_float_simp
|
huffman@36770
|
1534 |
unfolding pos_prod_lt[OF zero_less_pow2[of e], unfolded mult_commute] by auto
|
hoelzl@29742
|
1535 |
hence "1 \<le> - m" by auto
|
hoelzl@29742
|
1536 |
hence "0 < nat (- m)" by auto
|
hoelzl@29742
|
1537 |
moreover
|
hoelzl@29742
|
1538 |
have "0 \<le> e" using floor_pos_exp Float_floor[symmetric] by auto
|
hoelzl@29742
|
1539 |
hence "(0::nat) < 2 ^ nat e" by auto
|
hoelzl@29742
|
1540 |
ultimately have "0 < ?num" by auto
|
hoelzl@29742
|
1541 |
hence "real ?num \<noteq> 0" by auto
|
hoelzl@29742
|
1542 |
have e_nat: "int (nat e) = e" using `0 \<le> e` by auto
|
hoelzl@31098
|
1543 |
have num_eq: "real ?num = real (- floor_fl x)" using `0 < nat (- m)`
|
huffman@35346
|
1544 |
unfolding Float_floor real_of_float_minus real_of_float_simp real_of_nat_mult pow2_int[of "nat e", unfolded e_nat] real_of_nat_power by auto
|
hoelzl@31098
|
1545 |
have "0 < - floor_fl x" using `0 < ?num`[unfolded real_of_nat_less_iff[symmetric]] unfolding less_float_def num_eq[symmetric] real_of_float_0 real_of_nat_zero .
|
hoelzl@31098
|
1546 |
hence "real (floor_fl x) < 0" unfolding less_float_def by auto
|
hoelzl@31809
|
1547 |
|
hoelzl@31098
|
1548 |
have "exp (real x) \<le> real (ub_exp prec x)"
|
hoelzl@29742
|
1549 |
proof -
|
hoelzl@31809
|
1550 |
have div_less_zero: "real (float_divr prec x (- floor_fl x)) \<le> 0"
|
wenzelm@32962
|
1551 |
using float_divr_nonpos_pos_upper_bound[OF `x \<le> 0` `0 < - floor_fl x`] unfolding le_float_def real_of_float_0 .
|
hoelzl@31809
|
1552 |
|
hoelzl@31098
|
1553 |
have "exp (real x) = exp (real ?num * (real x / real ?num))" using `real ?num \<noteq> 0` by auto
|
hoelzl@31098
|
1554 |
also have "\<dots> = exp (real x / real ?num) ^ ?num" unfolding exp_real_of_nat_mult ..
|
hoelzl@31098
|
1555 |
also have "\<dots> \<le> exp (real (float_divr prec x (- floor_fl x))) ^ ?num" unfolding num_eq
|
wenzelm@32962
|
1556 |
by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
|
hoelzl@31098
|
1557 |
also have "\<dots> \<le> real ((?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num)" unfolding float_power
|
wenzelm@32962
|
1558 |
by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
|
hoelzl@29742
|
1559 |
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@29742
|
1560 |
qed
|
hoelzl@31809
|
1561 |
moreover
|
hoelzl@31098
|
1562 |
have "real (lb_exp prec x) \<le> exp (real x)"
|
hoelzl@29742
|
1563 |
proof -
|
hoelzl@29742
|
1564 |
let ?divl = "float_divl prec x (- Float m e)"
|
hoelzl@29742
|
1565 |
let ?horner = "?lb_exp_horner ?divl"
|
hoelzl@31809
|
1566 |
|
hoelzl@29742
|
1567 |
show ?thesis
|
hoelzl@29742
|
1568 |
proof (cases "?horner \<le> 0")
|
wenzelm@32962
|
1569 |
case False hence "0 \<le> real ?horner" unfolding le_float_def by auto
|
wenzelm@32962
|
1570 |
|
wenzelm@32962
|
1571 |
have div_less_zero: "real (float_divl prec x (- floor_fl x)) \<le> 0"
|
wenzelm@32962
|
1572 |
using `real (floor_fl x) < 0` `real x \<le> 0` by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
|
wenzelm@32962
|
1573 |
|
wenzelm@32962
|
1574 |
have "real ((?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num) \<le>
|
hoelzl@31809
|
1575 |
exp (real (float_divl prec x (- floor_fl x))) ^ ?num" unfolding float_power
|
wenzelm@32962
|
1576 |
using `0 \<le> real ?horner`[unfolded Float_floor[symmetric]] bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1] by (auto intro!: power_mono)
|
wenzelm@32962
|
1577 |
also have "\<dots> \<le> exp (real x / real ?num) ^ ?num" unfolding num_eq
|
wenzelm@32962
|
1578 |
using float_divl by (auto intro!: power_mono simp del: real_of_float_minus)
|
wenzelm@32962
|
1579 |
also have "\<dots> = exp (real ?num * (real x / real ?num))" unfolding exp_real_of_nat_mult ..
|
wenzelm@32962
|
1580 |
also have "\<dots> = exp (real x)" using `real ?num \<noteq> 0` by auto
|
wenzelm@32962
|
1581 |
finally show ?thesis
|
wenzelm@32962
|
1582 |
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@29742
|
1583 |
next
|
wenzelm@32962
|
1584 |
case True
|
wenzelm@32962
|
1585 |
have "real (floor_fl x) \<noteq> 0" and "real (floor_fl x) \<le> 0" using `real (floor_fl x) < 0` by auto
|
wenzelm@32962
|
1586 |
from divide_right_mono_neg[OF floor_fl[of x] `real (floor_fl x) \<le> 0`, unfolded divide_self[OF `real (floor_fl x) \<noteq> 0`]]
|
wenzelm@32962
|
1587 |
have "- 1 \<le> real x / real (- floor_fl x)" unfolding real_of_float_minus by auto
|
wenzelm@32962
|
1588 |
from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
|
wenzelm@32962
|
1589 |
have "real (Float 1 -2) \<le> exp (real x / real (- floor_fl x))" unfolding Float_num .
|
wenzelm@32962
|
1590 |
hence "real (Float 1 -2) ^ ?num \<le> exp (real x / real (- floor_fl x)) ^ ?num"
|
wenzelm@32962
|
1591 |
by (auto intro!: power_mono simp add: Float_num)
|
wenzelm@32962
|
1592 |
also have "\<dots> = exp (real x)" unfolding num_eq exp_real_of_nat_mult[symmetric] using `real (floor_fl x) \<noteq> 0` by auto
|
wenzelm@32962
|
1593 |
finally show ?thesis
|
wenzelm@32962
|
1594 |
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@29742
|
1595 |
qed
|
hoelzl@29742
|
1596 |
qed
|
hoelzl@29742
|
1597 |
ultimately show ?thesis by auto
|
hoelzl@29742
|
1598 |
qed
|
hoelzl@29742
|
1599 |
qed
|
hoelzl@29742
|
1600 |
|
hoelzl@31098
|
1601 |
lemma exp_boundaries: "exp (real x) \<in> { real (lb_exp prec x) .. real (ub_exp prec x)}"
|
hoelzl@29742
|
1602 |
proof -
|
hoelzl@29742
|
1603 |
show ?thesis
|
hoelzl@29742
|
1604 |
proof (cases "0 < x")
|
hoelzl@31809
|
1605 |
case False hence "x \<le> 0" unfolding less_float_def le_float_def by auto
|
hoelzl@29742
|
1606 |
from exp_boundaries'[OF this] show ?thesis .
|
hoelzl@29742
|
1607 |
next
|
hoelzl@29742
|
1608 |
case True hence "-x \<le> 0" unfolding less_float_def le_float_def by auto
|
hoelzl@31809
|
1609 |
|
hoelzl@31098
|
1610 |
have "real (lb_exp prec x) \<le> exp (real x)"
|
hoelzl@29742
|
1611 |
proof -
|
hoelzl@29742
|
1612 |
from exp_boundaries'[OF `-x \<le> 0`]
|
hoelzl@31098
|
1613 |
have ub_exp: "exp (- real x) \<le> real (ub_exp prec (-x))" unfolding atLeastAtMost_iff real_of_float_minus by auto
|
hoelzl@31809
|
1614 |
|
hoelzl@31098
|
1615 |
have "real (float_divl prec 1 (ub_exp prec (-x))) \<le> 1 / real (ub_exp prec (-x))" using float_divl[where x=1] by auto
|
hoelzl@31098
|
1616 |
also have "\<dots> \<le> exp (real x)"
|
wenzelm@32962
|
1617 |
using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] exp_gt_zero, symmetric]]
|
wenzelm@32962
|
1618 |
unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide by auto
|
hoelzl@29742
|
1619 |
finally show ?thesis unfolding lb_exp.simps if_P[OF True] .
|
hoelzl@29742
|
1620 |
qed
|
hoelzl@29742
|
1621 |
moreover
|
hoelzl@31098
|
1622 |
have "exp (real x) \<le> real (ub_exp prec x)"
|
hoelzl@29742
|
1623 |
proof -
|
hoelzl@29742
|
1624 |
have "\<not> 0 < -x" using `0 < x` unfolding less_float_def by auto
|
hoelzl@31809
|
1625 |
|
hoelzl@29742
|
1626 |
from exp_boundaries'[OF `-x \<le> 0`]
|
hoelzl@31098
|
1627 |
have lb_exp: "real (lb_exp prec (-x)) \<le> exp (- real x)" unfolding atLeastAtMost_iff real_of_float_minus by auto
|
hoelzl@31809
|
1628 |
|
hoelzl@31098
|
1629 |
have "exp (real x) \<le> real (1 :: float) / real (lb_exp prec (-x))"
|
wenzelm@32962
|
1630 |
using lb_exp[unfolded inverse_le_iff_le[OF exp_gt_zero lb_exp_pos[OF `\<not> 0 < -x`, unfolded less_float_def real_of_float_0],
|
wenzelm@32962
|
1631 |
symmetric]]
|
wenzelm@32962
|
1632 |
unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide real_of_float_1 by auto
|
hoelzl@31098
|
1633 |
also have "\<dots> \<le> real (float_divr prec 1 (lb_exp prec (-x)))" using float_divr .
|
hoelzl@29742
|
1634 |
finally show ?thesis unfolding ub_exp.simps if_P[OF True] .
|
hoelzl@29742
|
1635 |
qed
|
hoelzl@29742
|
1636 |
ultimately show ?thesis by auto
|
hoelzl@29742
|
1637 |
qed
|
hoelzl@29742
|
1638 |
qed
|
hoelzl@29742
|
1639 |
|
hoelzl@31098
|
1640 |
lemma bnds_exp: "\<forall> x lx ux. (l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> exp x \<and> exp x \<le> real u"
|
hoelzl@29742
|
1641 |
proof (rule allI, rule allI, rule allI, rule impI)
|
hoelzl@29742
|
1642 |
fix x lx ux
|
hoelzl@31098
|
1643 |
assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {real lx .. real ux}"
|
hoelzl@31098
|
1644 |
hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {real lx .. real ux}" by auto
|
hoelzl@29742
|
1645 |
|
hoelzl@29742
|
1646 |
{ from exp_boundaries[of lx prec, unfolded l]
|
hoelzl@31098
|
1647 |
have "real l \<le> exp (real lx)" by (auto simp del: lb_exp.simps)
|
hoelzl@29742
|
1648 |
also have "\<dots> \<le> exp x" using x by auto
|
hoelzl@31098
|
1649 |
finally have "real l \<le> exp x" .
|
hoelzl@29742
|
1650 |
} moreover
|
hoelzl@31098
|
1651 |
{ have "exp x \<le> exp (real ux)" using x by auto
|
hoelzl@31098
|
1652 |
also have "\<dots> \<le> real u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
|
hoelzl@31098
|
1653 |
finally have "exp x \<le> real u" .
|
hoelzl@31098
|
1654 |
} ultimately show "real l \<le> exp x \<and> exp x \<le> real u" ..
|
hoelzl@29742
|
1655 |
qed
|
hoelzl@29742
|
1656 |
|
hoelzl@29742
|
1657 |
section "Logarithm"
|
hoelzl@29742
|
1658 |
|
hoelzl@29742
|
1659 |
subsection "Compute the logarithm series"
|
hoelzl@29742
|
1660 |
|
hoelzl@31809
|
1661 |
fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
|
hoelzl@29742
|
1662 |
and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
|
hoelzl@29742
|
1663 |
"ub_ln_horner prec 0 i x = 0" |
|
hoelzl@29742
|
1664 |
"ub_ln_horner prec (Suc n) i x = rapprox_rat prec 1 (int i) - x * lb_ln_horner prec n (Suc i) x" |
|
hoelzl@29742
|
1665 |
"lb_ln_horner prec 0 i x = 0" |
|
hoelzl@29742
|
1666 |
"lb_ln_horner prec (Suc n) i x = lapprox_rat prec 1 (int i) - x * ub_ln_horner prec n (Suc i) x"
|
hoelzl@29742
|
1667 |
|
hoelzl@29742
|
1668 |
lemma ln_bounds:
|
hoelzl@29742
|
1669 |
assumes "0 \<le> x" and "x < 1"
|
haftmann@30952
|
1670 |
shows "(\<Sum>i=0..<2*n. -1^i * (1 / real (i + 1)) * x ^ (Suc i)) \<le> ln (x + 1)" (is "?lb")
|
haftmann@30952
|
1671 |
and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. -1^i * (1 / real (i + 1)) * x ^ (Suc i))" (is "?ub")
|
hoelzl@29742
|
1672 |
proof -
|
haftmann@30952
|
1673 |
let "?a n" = "(1/real (n +1)) * x ^ (Suc n)"
|
hoelzl@29742
|
1674 |
|
hoelzl@29742
|
1675 |
have ln_eq: "(\<Sum> i. -1^i * ?a i) = ln (x + 1)"
|
hoelzl@29742
|
1676 |
using ln_series[of "x + 1"] `0 \<le> x` `x < 1` by auto
|
hoelzl@29742
|
1677 |
|
hoelzl@29742
|
1678 |
have "norm x < 1" using assms by auto
|
hoelzl@31809
|
1679 |
have "?a ----> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric]
|
hoelzl@29742
|
1680 |
using LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF `norm x < 1`]]] by auto
|
hoelzl@29742
|
1681 |
{ fix n have "0 \<le> ?a n" by (rule mult_nonneg_nonneg, auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`) }
|
hoelzl@29742
|
1682 |
{ fix n have "?a (Suc n) \<le> ?a n" unfolding inverse_eq_divide[symmetric]
|
hoelzl@29742
|
1683 |
proof (rule mult_mono)
|
hoelzl@29742
|
1684 |
show "0 \<le> x ^ Suc (Suc n)" by (auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
|
huffman@36770
|
1685 |
have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 mult_assoc[symmetric]
|
wenzelm@32962
|
1686 |
by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
|
hoelzl@29742
|
1687 |
thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
|
hoelzl@29742
|
1688 |
qed auto }
|
hoelzl@29742
|
1689 |
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@29742
|
1690 |
show "?lb" and "?ub" by auto
|
hoelzl@29742
|
1691 |
qed
|
hoelzl@29742
|
1692 |
|
hoelzl@31809
|
1693 |
lemma ln_float_bounds:
|
hoelzl@31098
|
1694 |
assumes "0 \<le> real x" and "real x < 1"
|
hoelzl@31098
|
1695 |
shows "real (x * lb_ln_horner prec (get_even n) 1 x) \<le> ln (real x + 1)" (is "?lb \<le> ?ln")
|
hoelzl@31098
|
1696 |
and "ln (real x + 1) \<le> real (x * ub_ln_horner prec (get_odd n) 1 x)" (is "?ln \<le> ?ub")
|
hoelzl@29742
|
1697 |
proof -
|
hoelzl@29742
|
1698 |
obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
|
hoelzl@29742
|
1699 |
obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..
|
hoelzl@29742
|
1700 |
|
hoelzl@31098
|
1701 |
let "?s n" = "-1^n * (1 / real (1 + n)) * (real x)^(Suc n)"
|
hoelzl@29742
|
1702 |
|
huffman@36770
|
1703 |
have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 mult_assoc[symmetric] real_of_float_mult setsum_left_distrib[symmetric] unfolding mult_commute[of "real x"] ev
|
hoelzl@29742
|
1704 |
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@31098
|
1705 |
OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
|
hoelzl@29742
|
1706 |
by (rule mult_right_mono)
|
hoelzl@31098
|
1707 |
also have "\<dots> \<le> ?ln" using ln_bounds(1)[OF `0 \<le> real x` `real x < 1`] by auto
|
hoelzl@31809
|
1708 |
finally show "?lb \<le> ?ln" .
|
hoelzl@29742
|
1709 |
|
hoelzl@31098
|
1710 |
have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> real x` `real x < 1`] by auto
|
huffman@36770
|
1711 |
also have "\<dots> \<le> ?ub" unfolding power_Suc2 mult_assoc[symmetric] real_of_float_mult setsum_left_distrib[symmetric] unfolding mult_commute[of "real x"] od
|
hoelzl@29742
|
1712 |
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@31098
|
1713 |
OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
|
hoelzl@29742
|
1714 |
by (rule mult_right_mono)
|
hoelzl@31809
|
1715 |
finally show "?ln \<le> ?ub" .
|
hoelzl@29742
|
1716 |
qed
|
hoelzl@29742
|
1717 |
|
hoelzl@29742
|
1718 |
lemma ln_add: assumes "0 < x" and "0 < y" shows "ln (x + y) = ln x + ln (1 + y / x)"
|
hoelzl@29742
|
1719 |
proof -
|
hoelzl@29742
|
1720 |
have "x \<noteq> 0" using assms by auto
|
hoelzl@29742
|
1721 |
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@31809
|
1722 |
moreover
|
hoelzl@29742
|
1723 |
have "0 < y / x" using assms divide_pos_pos by auto
|
hoelzl@29742
|
1724 |
hence "0 < 1 + y / x" by auto
|
hoelzl@29742
|
1725 |
ultimately show ?thesis using ln_mult assms by auto
|
hoelzl@29742
|
1726 |
qed
|
hoelzl@29742
|
1727 |
|
hoelzl@29742
|
1728 |
subsection "Compute the logarithm of 2"
|
hoelzl@29742
|
1729 |
|
hoelzl@31809
|
1730 |
definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3
|
hoelzl@31809
|
1731 |
in (Float 1 -1 * ub_ln_horner prec (get_odd prec) 1 (Float 1 -1)) +
|
hoelzl@29742
|
1732 |
(third * ub_ln_horner prec (get_odd prec) 1 third))"
|
hoelzl@31809
|
1733 |
definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3
|
hoelzl@31809
|
1734 |
in (Float 1 -1 * lb_ln_horner prec (get_even prec) 1 (Float 1 -1)) +
|
hoelzl@29742
|
1735 |
(third * lb_ln_horner prec (get_even prec) 1 third))"
|
hoelzl@29742
|
1736 |
|
hoelzl@31098
|
1737 |
lemma ub_ln2: "ln 2 \<le> real (ub_ln2 prec)" (is "?ub_ln2")
|
hoelzl@31098
|
1738 |
and lb_ln2: "real (lb_ln2 prec) \<le> ln 2" (is "?lb_ln2")
|
hoelzl@29742
|
1739 |
proof -
|
hoelzl@29742
|
1740 |
let ?uthird = "rapprox_rat (max prec 1) 1 3"
|
hoelzl@29742
|
1741 |
let ?lthird = "lapprox_rat prec 1 3"
|
hoelzl@29742
|
1742 |
|
hoelzl@29742
|
1743 |
have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1)"
|
hoelzl@29742
|
1744 |
using ln_add[of "3 / 2" "1 / 2"] by auto
|
hoelzl@31098
|
1745 |
have lb3: "real ?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
|
hoelzl@31098
|
1746 |
hence lb3_ub: "real ?lthird < 1" by auto
|
hoelzl@31098
|
1747 |
have lb3_lb: "0 \<le> real ?lthird" using lapprox_rat_bottom[of 1 3] by auto
|
hoelzl@31098
|
1748 |
have ub3: "1 / 3 \<le> real ?uthird" using rapprox_rat[of 1 3] by auto
|
hoelzl@31098
|
1749 |
hence ub3_lb: "0 \<le> real ?uthird" by auto
|
hoelzl@29742
|
1750 |
|
hoelzl@31098
|
1751 |
have lb2: "0 \<le> real (Float 1 -1)" and ub2: "real (Float 1 -1) < 1" unfolding Float_num by auto
|
hoelzl@29742
|
1752 |
|
hoelzl@29742
|
1753 |
have "0 \<le> (1::int)" and "0 < (3::int)" by auto
|
hoelzl@31098
|
1754 |
have ub3_ub: "real ?uthird < 1" unfolding rapprox_rat.simps(2)[OF `0 \<le> 1` `0 < 3`]
|
hoelzl@29742
|
1755 |
by (rule rapprox_posrat_less1, auto)
|
hoelzl@29742
|
1756 |
|
hoelzl@29742
|
1757 |
have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
|
hoelzl@31098
|
1758 |
have uthird_gt0: "0 < real ?uthird + 1" using ub3_lb by auto
|
hoelzl@31098
|
1759 |
have lthird_gt0: "0 < real ?lthird + 1" using lb3_lb by auto
|
hoelzl@29742
|
1760 |
|
hoelzl@31098
|
1761 |
show ?ub_ln2 unfolding ub_ln2_def Let_def real_of_float_add ln2_sum Float_num(4)[symmetric]
|
hoelzl@29742
|
1762 |
proof (rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
|
hoelzl@31098
|
1763 |
have "ln (1 / 3 + 1) \<le> ln (real ?uthird + 1)" unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
|
hoelzl@31098
|
1764 |
also have "\<dots> \<le> real (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)"
|
hoelzl@29742
|
1765 |
using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
|
hoelzl@31098
|
1766 |
finally show "ln (1 / 3 + 1) \<le> real (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
|
hoelzl@29742
|
1767 |
qed
|
hoelzl@31098
|
1768 |
show ?lb_ln2 unfolding lb_ln2_def Let_def real_of_float_add ln2_sum Float_num(4)[symmetric]
|
hoelzl@29742
|
1769 |
proof (rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
|
hoelzl@31098
|
1770 |
have "real (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (real ?lthird + 1)"
|
hoelzl@29742
|
1771 |
using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
|
hoelzl@29742
|
1772 |
also have "\<dots> \<le> ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto
|
hoelzl@31098
|
1773 |
finally show "real (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (1 / 3 + 1)" .
|
hoelzl@29742
|
1774 |
qed
|
hoelzl@29742
|
1775 |
qed
|
hoelzl@29742
|
1776 |
|
hoelzl@29742
|
1777 |
subsection "Compute the logarithm in the entire domain"
|
hoelzl@29742
|
1778 |
|
hoelzl@29742
|
1779 |
function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where
|
hoelzl@31468
|
1780 |
"ub_ln prec x = (if x \<le> 0 then None
|
hoelzl@31468
|
1781 |
else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
|
hoelzl@31468
|
1782 |
else let horner = \<lambda>x. x * ub_ln_horner prec (get_odd prec) 1 x in
|
hoelzl@31468
|
1783 |
if x \<le> Float 3 -1 then Some (horner (x - 1))
|
hoelzl@31468
|
1784 |
else if x < Float 1 1 then Some (horner (Float 1 -1) + horner (x * rapprox_rat prec 2 3 - 1))
|
hoelzl@31468
|
1785 |
else let l = bitlen (mantissa x) - 1 in
|
hoelzl@31468
|
1786 |
Some (ub_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l) - 1)))" |
|
hoelzl@31468
|
1787 |
"lb_ln prec x = (if x \<le> 0 then None
|
hoelzl@31468
|
1788 |
else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x)))
|
hoelzl@31468
|
1789 |
else let horner = \<lambda>x. x * lb_ln_horner prec (get_even prec) 1 x in
|
hoelzl@31468
|
1790 |
if x \<le> Float 3 -1 then Some (horner (x - 1))
|
hoelzl@31468
|
1791 |
else if x < Float 1 1 then Some (horner (Float 1 -1) +
|
hoelzl@31468
|
1792 |
horner (max (x * lapprox_rat prec 2 3 - 1) 0))
|
hoelzl@31468
|
1793 |
else let l = bitlen (mantissa x) - 1 in
|
hoelzl@31468
|
1794 |
Some (lb_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l) - 1)))"
|
hoelzl@29742
|
1795 |
by pat_completeness auto
|
hoelzl@29742
|
1796 |
|
hoelzl@29742
|
1797 |
termination proof (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 1 then 1 else 0))", auto)
|
hoelzl@29742
|
1798 |
fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divl (max prec (Suc 0)) 1 x < 1"
|
hoelzl@29742
|
1799 |
hence "0 < x" and "0 < max prec (Suc 0)" unfolding less_float_def le_float_def by auto
|
hoelzl@29742
|
1800 |
from float_divl_pos_less1_bound[OF `0 < x` `x < 1` `0 < max prec (Suc 0)`]
|
hoelzl@29742
|
1801 |
show False using `float_divl (max prec (Suc 0)) 1 x < 1` unfolding less_float_def le_float_def by auto
|
hoelzl@29742
|
1802 |
next
|
hoelzl@29742
|
1803 |
fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divr prec 1 x < 1"
|
hoelzl@29742
|
1804 |
hence "0 < x" unfolding less_float_def le_float_def by auto
|
hoelzl@29742
|
1805 |
from float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`, of prec]
|
hoelzl@29742
|
1806 |
show False using `float_divr prec 1 x < 1` unfolding less_float_def le_float_def by auto
|
hoelzl@29742
|
1807 |
qed
|
hoelzl@29742
|
1808 |
|
hoelzl@31098
|
1809 |
lemma ln_shifted_float: assumes "0 < m" shows "ln (real (Float m e)) = ln 2 * real (e + (bitlen m - 1)) + ln (real (Float m (- (bitlen m - 1))))"
|
hoelzl@29742
|
1810 |
proof -
|
hoelzl@29742
|
1811 |
let ?B = "2^nat (bitlen m - 1)"
|
hoelzl@29742
|
1812 |
have "0 < real m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0" using assms by auto
|
hoelzl@29742
|
1813 |
hence "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto
|
hoelzl@31468
|
1814 |
show ?thesis
|
hoelzl@29742
|
1815 |
proof (cases "0 \<le> e")
|
hoelzl@29742
|
1816 |
case True
|
hoelzl@29742
|
1817 |
show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
|
hoelzl@31468
|
1818 |
unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`]
|
hoelzl@31468
|
1819 |
unfolding real_of_float_ge0_exp[OF True] ln_mult[OF `0 < real m` `0 < 2^nat e`]
|
hoelzl@29742
|
1820 |
ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` True by auto
|
hoelzl@29742
|
1821 |
next
|
hoelzl@29742
|
1822 |
case False hence "0 < -e" by auto
|
hoelzl@29742
|
1823 |
hence pow_gt0: "(0::real) < 2^nat (-e)" by auto
|
hoelzl@29742
|
1824 |
hence inv_gt0: "(0::real) < inverse (2^nat (-e))" by auto
|
hoelzl@29742
|
1825 |
show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
|
hoelzl@31468
|
1826 |
unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`]
|
hoelzl@31098
|
1827 |
unfolding real_of_float_nge0_exp[OF False] ln_mult[OF `0 < real m` inv_gt0] ln_inverse[OF pow_gt0]
|
hoelzl@29742
|
1828 |
ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` False by auto
|
hoelzl@29742
|
1829 |
qed
|
hoelzl@29742
|
1830 |
qed
|
hoelzl@29742
|
1831 |
|
hoelzl@29742
|
1832 |
lemma ub_ln_lb_ln_bounds': assumes "1 \<le> x"
|
hoelzl@31098
|
1833 |
shows "real (the (lb_ln prec x)) \<le> ln (real x) \<and> ln (real x) \<le> real (the (ub_ln prec x))"
|
hoelzl@29742
|
1834 |
(is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
|
hoelzl@29742
|
1835 |
proof (cases "x < Float 1 1")
|
hoelzl@31468
|
1836 |
case True
|
hoelzl@31468
|
1837 |
hence "real (x - 1) < 1" and "real x < 2" unfolding less_float_def Float_num by auto
|
hoelzl@29742
|
1838 |
have "\<not> x \<le> 0" and "\<not> x < 1" using `1 \<le> x` unfolding less_float_def le_float_def by auto
|
hoelzl@31098
|
1839 |
hence "0 \<le> real (x - 1)" using `1 \<le> x` unfolding less_float_def Float_num by auto
|
hoelzl@31468
|
1840 |
|
hoelzl@31468
|
1841 |
have [simp]: "real (Float 3 -1) = 3 / 2" by (simp add: real_of_float_def pow2_def)
|
hoelzl@31468
|
1842 |
|
hoelzl@31468
|
1843 |
show ?thesis
|
hoelzl@31468
|
1844 |
proof (cases "x \<le> Float 3 -1")
|
hoelzl@31468
|
1845 |
case True
|
hoelzl@31468
|
1846 |
show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
|
hoelzl@31468
|
1847 |
using ln_float_bounds[OF `0 \<le> real (x - 1)` `real (x - 1) < 1`, of prec] `\<not> x \<le> 0` `\<not> x < 1` True
|
hoelzl@31468
|
1848 |
by auto
|
hoelzl@31468
|
1849 |
next
|
hoelzl@31468
|
1850 |
case False hence *: "3 / 2 < real x" by (auto simp add: le_float_def)
|
hoelzl@31468
|
1851 |
|
hoelzl@31468
|
1852 |
with ln_add[of "3 / 2" "real x - 3 / 2"]
|
hoelzl@31468
|
1853 |
have add: "ln (real x) = ln (3 / 2) + ln (real x * 2 / 3)"
|
hoelzl@31468
|
1854 |
by (auto simp add: algebra_simps diff_divide_distrib)
|
hoelzl@31468
|
1855 |
|
hoelzl@31468
|
1856 |
let "?ub_horner x" = "x * ub_ln_horner prec (get_odd prec) 1 x"
|
hoelzl@31468
|
1857 |
let "?lb_horner x" = "x * lb_ln_horner prec (get_even prec) 1 x"
|
hoelzl@31468
|
1858 |
|
hoelzl@31468
|
1859 |
{ have up: "real (rapprox_rat prec 2 3) \<le> 1"
|
wenzelm@32962
|
1860 |
by (rule rapprox_rat_le1) simp_all
|
hoelzl@31468
|
1861 |
have low: "2 / 3 \<le> real (rapprox_rat prec 2 3)"
|
wenzelm@32962
|
1862 |
by (rule order_trans[OF _ rapprox_rat]) simp
|
hoelzl@31468
|
1863 |
from mult_less_le_imp_less[OF * low] *
|
hoelzl@31468
|
1864 |
have pos: "0 < real (x * rapprox_rat prec 2 3 - 1)" by auto
|
hoelzl@31468
|
1865 |
|
hoelzl@31468
|
1866 |
have "ln (real x * 2/3)
|
wenzelm@32962
|
1867 |
\<le> ln (real (x * rapprox_rat prec 2 3 - 1) + 1)"
|
hoelzl@31468
|
1868 |
proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
|
wenzelm@32962
|
1869 |
show "real x * 2 / 3 \<le> real (x * rapprox_rat prec 2 3 - 1) + 1"
|
wenzelm@32962
|
1870 |
using * low by auto
|
wenzelm@32962
|
1871 |
show "0 < real x * 2 / 3" using * by simp
|
wenzelm@32962
|
1872 |
show "0 < real (x * rapprox_rat prec 2 3 - 1) + 1" using pos by auto
|
hoelzl@31468
|
1873 |
qed
|
hoelzl@31468
|
1874 |
also have "\<dots> \<le> real (?ub_horner (x * rapprox_rat prec 2 3 - 1))"
|
hoelzl@31468
|
1875 |
proof (rule ln_float_bounds(2))
|
wenzelm@32962
|
1876 |
from mult_less_le_imp_less[OF `real x < 2` up] low *
|
wenzelm@32962
|
1877 |
show "real (x * rapprox_rat prec 2 3 - 1) < 1" by auto
|
wenzelm@32962
|
1878 |
show "0 \<le> real (x * rapprox_rat prec 2 3 - 1)" using pos by auto
|
hoelzl@31468
|
1879 |
qed
|
hoelzl@31468
|
1880 |
finally have "ln (real x)
|
wenzelm@32962
|
1881 |
\<le> real (?ub_horner (Float 1 -1))
|
wenzelm@32962
|
1882 |
+ real (?ub_horner (x * rapprox_rat prec 2 3 - 1))"
|
wenzelm@32962
|
1883 |
using ln_float_bounds(2)[of "Float 1 -1" prec prec] add by auto }
|
hoelzl@31468
|
1884 |
moreover
|
hoelzl@31468
|
1885 |
{ let ?max = "max (x * lapprox_rat prec 2 3 - 1) 0"
|
hoelzl@31468
|
1886 |
|
hoelzl@31468
|
1887 |
have up: "real (lapprox_rat prec 2 3) \<le> 2/3"
|
wenzelm@32962
|
1888 |
by (rule order_trans[OF lapprox_rat], simp)
|
hoelzl@31468
|
1889 |
|
hoelzl@31468
|
1890 |
have low: "0 \<le> real (lapprox_rat prec 2 3)"
|
wenzelm@32962
|
1891 |
using lapprox_rat_bottom[of 2 3 prec] by simp
|
hoelzl@31468
|
1892 |
|
hoelzl@31468
|
1893 |
have "real (?lb_horner ?max)
|
wenzelm@32962
|
1894 |
\<le> ln (real ?max + 1)"
|
hoelzl@31468
|
1895 |
proof (rule ln_float_bounds(1))
|
wenzelm@32962
|
1896 |
from mult_less_le_imp_less[OF `real x < 2` up] * low
|
wenzelm@32962
|
1897 |
show "real ?max < 1" by (cases "real (lapprox_rat prec 2 3) = 0",
|
wenzelm@32962
|
1898 |
auto simp add: real_of_float_max)
|
wenzelm@32962
|
1899 |
show "0 \<le> real ?max" by (auto simp add: real_of_float_max)
|
hoelzl@31468
|
1900 |
qed
|
hoelzl@31468
|
1901 |
also have "\<dots> \<le> ln (real x * 2/3)"
|
hoelzl@31468
|
1902 |
proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
|
wenzelm@32962
|
1903 |
show "0 < real ?max + 1" by (auto simp add: real_of_float_max)
|
wenzelm@32962
|
1904 |
show "0 < real x * 2/3" using * by auto
|
wenzelm@32962
|
1905 |
show "real ?max + 1 \<le> real x * 2/3" using * up
|
wenzelm@32962
|
1906 |
by (cases "0 < real x * real (lapprox_posrat prec 2 3) - 1",
|
wenzelm@32962
|
1907 |
auto simp add: real_of_float_max min_max.sup_absorb1)
|
hoelzl@31468
|
1908 |
qed
|
hoelzl@31468
|
1909 |
finally have "real (?lb_horner (Float 1 -1)) + real (?lb_horner ?max)
|
wenzelm@32962
|
1910 |
\<le> ln (real x)"
|
wenzelm@32962
|
1911 |
using ln_float_bounds(1)[of "Float 1 -1" prec prec] add by auto }
|
hoelzl@31468
|
1912 |
ultimately
|
hoelzl@31468
|
1913 |
show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
|
hoelzl@31468
|
1914 |
using `\<not> x \<le> 0` `\<not> x < 1` True False by auto
|
hoelzl@31468
|
1915 |
qed
|
hoelzl@29742
|
1916 |
next
|
hoelzl@29742
|
1917 |
case False
|
hoelzl@31468
|
1918 |
hence "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" "\<not> x \<le> Float 3 -1"
|
hoelzl@31468
|
1919 |
using `1 \<le> x` unfolding less_float_def le_float_def real_of_float_simp pow2_def
|
hoelzl@31468
|
1920 |
by auto
|
hoelzl@29742
|
1921 |
show ?thesis
|
hoelzl@29742
|
1922 |
proof (cases x)
|
hoelzl@29742
|
1923 |
case (Float m e)
|
hoelzl@29742
|
1924 |
let ?s = "Float (e + (bitlen m - 1)) 0"
|
hoelzl@29742
|
1925 |
let ?x = "Float m (- (bitlen m - 1))"
|
hoelzl@29742
|
1926 |
|
hoelzl@29742
|
1927 |
have "0 < m" and "m \<noteq> 0" using float_pos_m_pos `0 < x` Float by auto
|
hoelzl@29742
|
1928 |
|
hoelzl@29742
|
1929 |
{
|
hoelzl@31098
|
1930 |
have "real (lb_ln2 prec * ?s) \<le> ln 2 * real (e + (bitlen m - 1))" (is "?lb2 \<le> _")
|
wenzelm@32962
|
1931 |
unfolding real_of_float_mult real_of_float_ge0_exp[OF order_refl] nat_0 power_0 mult_1_right
|
wenzelm@32962
|
1932 |
using lb_ln2[of prec]
|
hoelzl@29742
|
1933 |
proof (rule mult_right_mono)
|
wenzelm@32962
|
1934 |
have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
|
wenzelm@32962
|
1935 |
from float_gt1_scale[OF this]
|
wenzelm@32962
|
1936 |
show "0 \<le> real (e + (bitlen m - 1))" by auto
|
hoelzl@29742
|
1937 |
qed
|
hoelzl@29742
|
1938 |
moreover
|
hoelzl@29742
|
1939 |
from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
|
hoelzl@31098
|
1940 |
have "0 \<le> real (?x - 1)" and "real (?x - 1) < 1" by auto
|
hoelzl@29742
|
1941 |
from ln_float_bounds(1)[OF this]
|
hoelzl@31098
|
1942 |
have "real ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln (real ?x)" (is "?lb_horner \<le> _") by auto
|
hoelzl@31098
|
1943 |
ultimately have "?lb2 + ?lb_horner \<le> ln (real x)"
|
wenzelm@32962
|
1944 |
unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
|
hoelzl@31468
|
1945 |
}
|
hoelzl@29742
|
1946 |
moreover
|
hoelzl@29742
|
1947 |
{
|
hoelzl@29742
|
1948 |
from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
|
hoelzl@31098
|
1949 |
have "0 \<le> real (?x - 1)" and "real (?x - 1) < 1" by auto
|
hoelzl@29742
|
1950 |
from ln_float_bounds(2)[OF this]
|
hoelzl@31098
|
1951 |
have "ln (real ?x) \<le> real ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))" (is "_ \<le> ?ub_horner") by auto
|
hoelzl@29742
|
1952 |
moreover
|
hoelzl@31098
|
1953 |
have "ln 2 * real (e + (bitlen m - 1)) \<le> real (ub_ln2 prec * ?s)" (is "_ \<le> ?ub2")
|
wenzelm@32962
|
1954 |
unfolding real_of_float_mult real_of_float_ge0_exp[OF order_refl] nat_0 power_0 mult_1_right
|
wenzelm@32962
|
1955 |
using ub_ln2[of prec]
|
hoelzl@29742
|
1956 |
proof (rule mult_right_mono)
|
wenzelm@32962
|
1957 |
have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
|
wenzelm@32962
|
1958 |
from float_gt1_scale[OF this]
|
wenzelm@32962
|
1959 |
show "0 \<le> real (e + (bitlen m - 1))" by auto
|
hoelzl@29742
|
1960 |
qed
|
hoelzl@31098
|
1961 |
ultimately have "ln (real x) \<le> ?ub2 + ?ub_horner"
|
wenzelm@32962
|
1962 |
unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
|
hoelzl@29742
|
1963 |
}
|
hoelzl@29742
|
1964 |
ultimately show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps
|
hoelzl@31468
|
1965 |
unfolding if_not_P[OF `\<not> x \<le> 0`] if_not_P[OF `\<not> x < 1`] if_not_P[OF False] if_not_P[OF `\<not> x \<le> Float 3 -1`] Let_def
|
hoelzl@31468
|
1966 |
unfolding scale.simps[of m e, unfolded Float[symmetric]] mantissa.simps[of m e, unfolded Float[symmetric]] real_of_float_add
|
hoelzl@31468
|
1967 |
by auto
|
hoelzl@29742
|
1968 |
qed
|
hoelzl@29742
|
1969 |
qed
|
hoelzl@29742
|
1970 |
|
hoelzl@29742
|
1971 |
lemma ub_ln_lb_ln_bounds: assumes "0 < x"
|
hoelzl@31098
|
1972 |
shows "real (the (lb_ln prec x)) \<le> ln (real x) \<and> ln (real x) \<le> real (the (ub_ln prec x))"
|
hoelzl@29742
|
1973 |
(is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
|
hoelzl@29742
|
1974 |
proof (cases "x < 1")
|
hoelzl@29742
|
1975 |
case False hence "1 \<le> x" unfolding less_float_def le_float_def by auto
|
hoelzl@29742
|
1976 |
show ?thesis using ub_ln_lb_ln_bounds'[OF `1 \<le> x`] .
|
hoelzl@29742
|
1977 |
next
|
hoelzl@29742
|
1978 |
case True have "\<not> x \<le> 0" using `0 < x` unfolding less_float_def le_float_def by auto
|
hoelzl@29742
|
1979 |
|
hoelzl@31098
|
1980 |
have "0 < real x" and "real x \<noteq> 0" using `0 < x` unfolding less_float_def by auto
|
hoelzl@31098
|
1981 |
hence A: "0 < 1 / real x" by auto
|
hoelzl@29742
|
1982 |
|
hoelzl@29742
|
1983 |
{
|
hoelzl@29742
|
1984 |
let ?divl = "float_divl (max prec 1) 1 x"
|
hoelzl@29742
|
1985 |
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@31098
|
1986 |
hence B: "0 < real ?divl" unfolding le_float_def by auto
|
hoelzl@31468
|
1987 |
|
hoelzl@31098
|
1988 |
have "ln (real ?divl) \<le> ln (1 / real x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
|
hoelzl@31098
|
1989 |
hence "ln (real x) \<le> - ln (real ?divl)" unfolding nonzero_inverse_eq_divide[OF `real x \<noteq> 0`, symmetric] ln_inverse[OF `0 < real x`] by auto
|
hoelzl@31468
|
1990 |
from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le]
|
hoelzl@31098
|
1991 |
have "?ln \<le> real (- the (lb_ln prec ?divl))" unfolding real_of_float_minus by (rule order_trans)
|
hoelzl@29742
|
1992 |
} moreover
|
hoelzl@29742
|
1993 |
{
|
hoelzl@29742
|
1994 |
let ?divr = "float_divr prec 1 x"
|
hoelzl@29742
|
1995 |
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@31098
|
1996 |
hence B: "0 < real ?divr" unfolding le_float_def by auto
|
hoelzl@31468
|
1997 |
|
hoelzl@31098
|
1998 |
have "ln (1 / real x) \<le> ln (real ?divr)" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
|
hoelzl@31098
|
1999 |
hence "- ln (real ?divr) \<le> ln (real x)" unfolding nonzero_inverse_eq_divide[OF `real x \<noteq> 0`, symmetric] ln_inverse[OF `0 < real x`] by auto
|
hoelzl@29742
|
2000 |
from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
|
hoelzl@31098
|
2001 |
have "real (- the (ub_ln prec ?divr)) \<le> ?ln" unfolding real_of_float_minus by (rule order_trans)
|
hoelzl@29742
|
2002 |
}
|
hoelzl@29742
|
2003 |
ultimately show ?thesis unfolding lb_ln.simps[where x=x] ub_ln.simps[where x=x]
|
hoelzl@29742
|
2004 |
unfolding if_not_P[OF `\<not> x \<le> 0`] if_P[OF True] by auto
|
hoelzl@29742
|
2005 |
qed
|
hoelzl@29742
|
2006 |
|
hoelzl@29742
|
2007 |
lemma lb_ln: assumes "Some y = lb_ln prec x"
|
hoelzl@31098
|
2008 |
shows "real y \<le> ln (real x)" and "0 < real x"
|
hoelzl@29742
|
2009 |
proof -
|
hoelzl@29742
|
2010 |
have "0 < x"
|
hoelzl@29742
|
2011 |
proof (rule ccontr)
|
hoelzl@29742
|
2012 |
assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
|
hoelzl@29742
|
2013 |
thus False using assms by auto
|
hoelzl@29742
|
2014 |
qed
|
hoelzl@31098
|
2015 |
thus "0 < real x" unfolding less_float_def by auto
|
hoelzl@31098
|
2016 |
have "real (the (lb_ln prec x)) \<le> ln (real x)" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
|
hoelzl@31098
|
2017 |
thus "real y \<le> ln (real x)" unfolding assms[symmetric] by auto
|
hoelzl@29742
|
2018 |
qed
|
hoelzl@29742
|
2019 |
|
hoelzl@29742
|
2020 |
lemma ub_ln: assumes "Some y = ub_ln prec x"
|
hoelzl@31098
|
2021 |
shows "ln (real x) \<le> real y" and "0 < real x"
|
hoelzl@29742
|
2022 |
proof -
|
hoelzl@29742
|
2023 |
have "0 < x"
|
hoelzl@29742
|
2024 |
proof (rule ccontr)
|
hoelzl@29742
|
2025 |
assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
|
hoelzl@29742
|
2026 |
thus False using assms by auto
|
hoelzl@29742
|
2027 |
qed
|
hoelzl@31098
|
2028 |
thus "0 < real x" unfolding less_float_def by auto
|
hoelzl@31098
|
2029 |
have "ln (real x) \<le> real (the (ub_ln prec x))" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
|
hoelzl@31098
|
2030 |
thus "ln (real x) \<le> real y" unfolding assms[symmetric] by auto
|
hoelzl@29742
|
2031 |
qed
|
hoelzl@29742
|
2032 |
|
hoelzl@31098
|
2033 |
lemma bnds_ln: "\<forall> x lx ux. (Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> ln x \<and> ln x \<le> real u"
|
hoelzl@29742
|
2034 |
proof (rule allI, rule allI, rule allI, rule impI)
|
hoelzl@29742
|
2035 |
fix x lx ux
|
hoelzl@31098
|
2036 |
assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {real lx .. real ux}"
|
hoelzl@31098
|
2037 |
hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {real lx .. real ux}" by auto
|
hoelzl@29742
|
2038 |
|
hoelzl@31098
|
2039 |
have "ln (real ux) \<le> real u" and "0 < real ux" using ub_ln u by auto
|
hoelzl@31098
|
2040 |
have "real l \<le> ln (real lx)" and "0 < real lx" and "0 < x" using lb_ln[OF l] x by auto
|
hoelzl@29742
|
2041 |
|
hoelzl@31467
|
2042 |
from ln_le_cancel_iff[OF `0 < real lx` `0 < x`] `real l \<le> ln (real lx)`
|
hoelzl@31098
|
2043 |
have "real l \<le> ln x" using x unfolding atLeastAtMost_iff by auto
|
hoelzl@29742
|
2044 |
moreover
|
hoelzl@31467
|
2045 |
from ln_le_cancel_iff[OF `0 < x` `0 < real ux`] `ln (real ux) \<le> real u`
|
hoelzl@31098
|
2046 |
have "ln x \<le> real u" using x unfolding atLeastAtMost_iff by auto
|
hoelzl@31098
|
2047 |
ultimately show "real l \<le> ln x \<and> ln x \<le> real u" ..
|
hoelzl@29742
|
2048 |
qed
|
hoelzl@29742
|
2049 |
|
hoelzl@29742
|
2050 |
section "Implement floatarith"
|
hoelzl@29742
|
2051 |
|
hoelzl@29742
|
2052 |
subsection "Define syntax and semantics"
|
hoelzl@29742
|
2053 |
|
hoelzl@29742
|
2054 |
datatype floatarith
|
hoelzl@29742
|
2055 |
= Add floatarith floatarith
|
hoelzl@29742
|
2056 |
| Minus floatarith
|
hoelzl@29742
|
2057 |
| Mult floatarith floatarith
|
hoelzl@29742
|
2058 |
| Inverse floatarith
|
hoelzl@29742
|
2059 |
| Cos floatarith
|
hoelzl@29742
|
2060 |
| Arctan floatarith
|
hoelzl@29742
|
2061 |
| Abs floatarith
|
hoelzl@29742
|
2062 |
| Max floatarith floatarith
|
hoelzl@29742
|
2063 |
| Min floatarith floatarith
|
hoelzl@29742
|
2064 |
| Pi
|
hoelzl@29742
|
2065 |
| Sqrt floatarith
|
hoelzl@29742
|
2066 |
| Exp floatarith
|
hoelzl@29742
|
2067 |
| Ln floatarith
|
hoelzl@29742
|
2068 |
| Power floatarith nat
|
hoelzl@32919
|
2069 |
| Var nat
|
hoelzl@29742
|
2070 |
| Num float
|
hoelzl@29742
|
2071 |
|
hoelzl@31862
|
2072 |
fun interpret_floatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real" where
|
hoelzl@31098
|
2073 |
"interpret_floatarith (Add a b) vs = (interpret_floatarith a vs) + (interpret_floatarith b vs)" |
|
hoelzl@31098
|
2074 |
"interpret_floatarith (Minus a) vs = - (interpret_floatarith a vs)" |
|
hoelzl@31098
|
2075 |
"interpret_floatarith (Mult a b) vs = (interpret_floatarith a vs) * (interpret_floatarith b vs)" |
|
hoelzl@31098
|
2076 |
"interpret_floatarith (Inverse a) vs = inverse (interpret_floatarith a vs)" |
|
hoelzl@31098
|
2077 |
"interpret_floatarith (Cos a) vs = cos (interpret_floatarith a vs)" |
|
hoelzl@31098
|
2078 |
"interpret_floatarith (Arctan a) vs = arctan (interpret_floatarith a vs)" |
|
hoelzl@31098
|
2079 |
"interpret_floatarith (Min a b) vs = min (interpret_floatarith a vs) (interpret_floatarith b vs)" |
|
hoelzl@31098
|
2080 |
"interpret_floatarith (Max a b) vs = max (interpret_floatarith a vs) (interpret_floatarith b vs)" |
|
hoelzl@31098
|
2081 |
"interpret_floatarith (Abs a) vs = abs (interpret_floatarith a vs)" |
|
hoelzl@31098
|
2082 |
"interpret_floatarith Pi vs = pi" |
|
hoelzl@31098
|
2083 |
"interpret_floatarith (Sqrt a) vs = sqrt (interpret_floatarith a vs)" |
|
hoelzl@31098
|
2084 |
"interpret_floatarith (Exp a) vs = exp (interpret_floatarith a vs)" |
|
hoelzl@31098
|
2085 |
"interpret_floatarith (Ln a) vs = ln (interpret_floatarith a vs)" |
|
hoelzl@31098
|
2086 |
"interpret_floatarith (Power a n) vs = (interpret_floatarith a vs)^n" |
|
hoelzl@31098
|
2087 |
"interpret_floatarith (Num f) vs = real f" |
|
hoelzl@32919
|
2088 |
"interpret_floatarith (Var n) vs = vs ! n"
|
hoelzl@29742
|
2089 |
|
hoelzl@31811
|
2090 |
lemma interpret_floatarith_divide: "interpret_floatarith (Mult a (Inverse b)) vs = (interpret_floatarith a vs) / (interpret_floatarith b vs)"
|
huffman@36770
|
2091 |
unfolding divide_inverse interpret_floatarith.simps ..
|
hoelzl@31811
|
2092 |
|
hoelzl@31811
|
2093 |
lemma interpret_floatarith_diff: "interpret_floatarith (Add a (Minus b)) vs = (interpret_floatarith a vs) - (interpret_floatarith b vs)"
|
huffman@36770
|
2094 |
unfolding diff_def interpret_floatarith.simps ..
|
hoelzl@31811
|
2095 |
|
hoelzl@31811
|
2096 |
lemma interpret_floatarith_sin: "interpret_floatarith (Cos (Add (Mult Pi (Num (Float 1 -1))) (Minus a))) vs =
|
hoelzl@31811
|
2097 |
sin (interpret_floatarith a vs)"
|
hoelzl@31811
|
2098 |
unfolding sin_cos_eq interpret_floatarith.simps
|
huffman@36770
|
2099 |
interpret_floatarith_divide interpret_floatarith_diff diff_def
|
hoelzl@31811
|
2100 |
by auto
|
hoelzl@31811
|
2101 |
|
hoelzl@31811
|
2102 |
lemma interpret_floatarith_tan:
|
hoelzl@31811
|
2103 |
"interpret_floatarith (Mult (Cos (Add (Mult Pi (Num (Float 1 -1))) (Minus a))) (Inverse (Cos a))) vs =
|
hoelzl@31811
|
2104 |
tan (interpret_floatarith a vs)"
|
huffman@36770
|
2105 |
unfolding interpret_floatarith.simps(3,4) interpret_floatarith_sin tan_def divide_inverse
|
hoelzl@31811
|
2106 |
by auto
|
hoelzl@31811
|
2107 |
|
hoelzl@31811
|
2108 |
lemma interpret_floatarith_powr: "interpret_floatarith (Exp (Mult b (Ln a))) vs = (interpret_floatarith a vs) powr (interpret_floatarith b vs)"
|
hoelzl@31811
|
2109 |
unfolding powr_def interpret_floatarith.simps ..
|
hoelzl@31811
|
2110 |
|
hoelzl@31811
|
2111 |
lemma interpret_floatarith_log: "interpret_floatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (interpret_floatarith b vs) (interpret_floatarith x vs)"
|
huffman@36770
|
2112 |
unfolding log_def interpret_floatarith.simps divide_inverse ..
|
hoelzl@31811
|
2113 |
|
hoelzl@31811
|
2114 |
lemma interpret_floatarith_num:
|
hoelzl@31811
|
2115 |
shows "interpret_floatarith (Num (Float 0 0)) vs = 0"
|
hoelzl@31811
|
2116 |
and "interpret_floatarith (Num (Float 1 0)) vs = 1"
|
hoelzl@31811
|
2117 |
and "interpret_floatarith (Num (Float (number_of a) 0)) vs = number_of a" by auto
|
hoelzl@31811
|
2118 |
|
hoelzl@29742
|
2119 |
subsection "Implement approximation function"
|
hoelzl@29742
|
2120 |
|
hoelzl@29742
|
2121 |
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@29742
|
2122 |
"lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" |
|
hoelzl@29742
|
2123 |
"lift_bin' a b f = None"
|
hoelzl@29742
|
2124 |
|
hoelzl@29742
|
2125 |
fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where
|
hoelzl@29742
|
2126 |
"lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u)
|
hoelzl@29742
|
2127 |
| t \<Rightarrow> None)" |
|
hoelzl@29742
|
2128 |
"lift_un b f = None"
|
hoelzl@29742
|
2129 |
|
hoelzl@29742
|
2130 |
fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
|
hoelzl@29742
|
2131 |
"lift_un' (Some (l1, u1)) f = Some (f l1 u1)" |
|
hoelzl@29742
|
2132 |
"lift_un' b f = None"
|
hoelzl@29742
|
2133 |
|
hoelzl@31811
|
2134 |
definition
|
hoelzl@31811
|
2135 |
"bounded_by xs vs \<longleftrightarrow>
|
hoelzl@31811
|
2136 |
(\<forall> i < length vs. case vs ! i of None \<Rightarrow> True
|
hoelzl@31811
|
2137 |
| Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u })"
|
hoelzl@31811
|
2138 |
|
hoelzl@31811
|
2139 |
lemma bounded_byE:
|
hoelzl@31811
|
2140 |
assumes "bounded_by xs vs"
|
hoelzl@31811
|
2141 |
shows "\<And> i. i < length vs \<Longrightarrow> case vs ! i of None \<Rightarrow> True
|
hoelzl@31811
|
2142 |
| Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u }"
|
hoelzl@31811
|
2143 |
using assms bounded_by_def by blast
|
hoelzl@31811
|
2144 |
|
hoelzl@31811
|
2145 |
lemma bounded_by_update:
|
hoelzl@31811
|
2146 |
assumes "bounded_by xs vs"
|
hoelzl@31811
|
2147 |
and bnd: "xs ! i \<in> { real l .. real u }"
|
hoelzl@31811
|
2148 |
shows "bounded_by xs (vs[i := Some (l,u)])"
|
hoelzl@31811
|
2149 |
proof -
|
hoelzl@31811
|
2150 |
{ fix j
|
hoelzl@31811
|
2151 |
let ?vs = "vs[i := Some (l,u)]"
|
hoelzl@31811
|
2152 |
assume "j < length ?vs" hence [simp]: "j < length vs" by simp
|
hoelzl@31811
|
2153 |
have "case ?vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> xs ! j \<in> { real l .. real u }"
|
hoelzl@31811
|
2154 |
proof (cases "?vs ! j")
|
hoelzl@31811
|
2155 |
case (Some b)
|
hoelzl@31811
|
2156 |
thus ?thesis
|
hoelzl@31811
|
2157 |
proof (cases "i = j")
|
hoelzl@31811
|
2158 |
case True
|
hoelzl@31811
|
2159 |
thus ?thesis using `?vs ! j = Some b` and bnd by auto
|
hoelzl@31811
|
2160 |
next
|
hoelzl@31811
|
2161 |
case False
|
hoelzl@31811
|
2162 |
thus ?thesis using `bounded_by xs vs` unfolding bounded_by_def by auto
|
hoelzl@31811
|
2163 |
qed
|
hoelzl@31811
|
2164 |
qed auto }
|
hoelzl@31811
|
2165 |
thus ?thesis unfolding bounded_by_def by auto
|
hoelzl@31811
|
2166 |
qed
|
hoelzl@31811
|
2167 |
|
hoelzl@31811
|
2168 |
lemma bounded_by_None:
|
hoelzl@31811
|
2169 |
shows "bounded_by xs (replicate (length xs) None)"
|
hoelzl@31811
|
2170 |
unfolding bounded_by_def by auto
|
hoelzl@31811
|
2171 |
|
hoelzl@31811
|
2172 |
fun approx approx' :: "nat \<Rightarrow> floatarith \<Rightarrow> (float * float) option list \<Rightarrow> (float * float) option" where
|
hoelzl@29742
|
2173 |
"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@31811
|
2174 |
"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@29742
|
2175 |
"approx prec (Minus a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (-u, -l))" |
|
hoelzl@29742
|
2176 |
"approx prec (Mult a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs)
|
hoelzl@31809
|
2177 |
(\<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@29742
|
2178 |
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@29742
|
2179 |
"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@29742
|
2180 |
"approx prec (Cos a) bs = lift_un' (approx' prec a bs) (bnds_cos prec)" |
|
hoelzl@29742
|
2181 |
"approx prec Pi bs = Some (lb_pi prec, ub_pi prec)" |
|
hoelzl@29742
|
2182 |
"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@29742
|
2183 |
"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@29742
|
2184 |
"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@29742
|
2185 |
"approx prec (Arctan a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_arctan prec l, ub_arctan prec u))" |
|
hoelzl@31467
|
2186 |
"approx prec (Sqrt a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" |
|
hoelzl@29742
|
2187 |
"approx prec (Exp a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" |
|
hoelzl@29742
|
2188 |
"approx prec (Ln a) bs = lift_un (approx' prec a bs) (\<lambda> l u. (lb_ln prec l, ub_ln prec u))" |
|
hoelzl@29742
|
2189 |
"approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds n)" |
|
hoelzl@29742
|
2190 |
"approx prec (Num f) bs = Some (f, f)" |
|
hoelzl@32919
|
2191 |
"approx prec (Var i) bs = (if i < length bs then bs ! i else None)"
|
hoelzl@29742
|
2192 |
|
hoelzl@29742
|
2193 |
lemma lift_bin'_ex:
|
hoelzl@29742
|
2194 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f"
|
hoelzl@29742
|
2195 |
shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b"
|
hoelzl@29742
|
2196 |
proof (cases a)
|
hoelzl@29742
|
2197 |
case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
|
hoelzl@29742
|
2198 |
thus ?thesis using lift_bin'_Some by auto
|
hoelzl@29742
|
2199 |
next
|
hoelzl@29742
|
2200 |
case (Some a')
|
hoelzl@29742
|
2201 |
show ?thesis
|
hoelzl@29742
|
2202 |
proof (cases b)
|
hoelzl@29742
|
2203 |
case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
|
hoelzl@29742
|
2204 |
thus ?thesis using lift_bin'_Some by auto
|
hoelzl@29742
|
2205 |
next
|
hoelzl@29742
|
2206 |
case (Some b')
|
hoelzl@29742
|
2207 |
obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
|
hoelzl@29742
|
2208 |
obtain lb ub where b': "b' = (lb, ub)" by (cases b', auto)
|
hoelzl@29742
|
2209 |
thus ?thesis unfolding `a = Some a'` `b = Some b'` a' b' by auto
|
hoelzl@29742
|
2210 |
qed
|
hoelzl@29742
|
2211 |
qed
|
hoelzl@29742
|
2212 |
|
hoelzl@29742
|
2213 |
lemma lift_bin'_f:
|
hoelzl@29742
|
2214 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f"
|
hoelzl@29742
|
2215 |
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@29742
|
2216 |
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@29742
|
2217 |
proof -
|
hoelzl@29742
|
2218 |
obtain l1 u1 l2 u2
|
hoelzl@29742
|
2219 |
where Sa: "Some (l1, u1) = g a" and Sb: "Some (l2, u2) = g b" using lift_bin'_ex[OF assms(1)] by auto
|
hoelzl@31809
|
2220 |
have lu: "(l, u) = f l1 u1 l2 u2" using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto
|
hoelzl@29742
|
2221 |
have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" unfolding lu[symmetric] by auto
|
hoelzl@31809
|
2222 |
thus ?thesis using Pa[OF Sa] Pb[OF Sb] by auto
|
hoelzl@29742
|
2223 |
qed
|
hoelzl@29742
|
2224 |
|
hoelzl@29742
|
2225 |
lemma approx_approx':
|
hoelzl@31098
|
2226 |
assumes Pa: "\<And>l u. Some (l, u) = approx prec a vs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u"
|
hoelzl@29742
|
2227 |
and approx': "Some (l, u) = approx' prec a vs"
|
hoelzl@31098
|
2228 |
shows "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u"
|
hoelzl@29742
|
2229 |
proof -
|
hoelzl@29742
|
2230 |
obtain l' u' where S: "Some (l', u') = approx prec a vs"
|
hoelzl@29742
|
2231 |
using approx' unfolding approx'.simps by (cases "approx prec a vs", auto)
|
hoelzl@29742
|
2232 |
have l': "l = round_down prec l'" and u': "u = round_up prec u'"
|
hoelzl@29742
|
2233 |
using approx' unfolding approx'.simps S[symmetric] by auto
|
hoelzl@31809
|
2234 |
show ?thesis unfolding l' u'
|
hoelzl@29742
|
2235 |
using order_trans[OF Pa[OF S, THEN conjunct2] round_up[of u']]
|
hoelzl@29742
|
2236 |
using order_trans[OF round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto
|
hoelzl@29742
|
2237 |
qed
|
hoelzl@29742
|
2238 |
|
hoelzl@29742
|
2239 |
lemma lift_bin':
|
hoelzl@29742
|
2240 |
assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f"
|
hoelzl@31098
|
2241 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
|
hoelzl@31098
|
2242 |
and Pb: "\<And>l u. Some (l, u) = approx prec b bs \<Longrightarrow> real l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> real u"
|
hoelzl@31809
|
2243 |
shows "\<exists> l1 u1 l2 u2. (real l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u1) \<and>
|
hoelzl@31809
|
2244 |
(real l2 \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> real u2) \<and>
|
hoelzl@29742
|
2245 |
l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
|
hoelzl@29742
|
2246 |
proof -
|
hoelzl@29742
|
2247 |
{ fix l u assume "Some (l, u) = approx' prec a bs"
|
hoelzl@29742
|
2248 |
with approx_approx'[of prec a bs, OF _ this] Pa
|
hoelzl@31098
|
2249 |
have "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" by auto } note Pa = this
|
hoelzl@29742
|
2250 |
{ fix l u assume "Some (l, u) = approx' prec b bs"
|
hoelzl@29742
|
2251 |
with approx_approx'[of prec b bs, OF _ this] Pb
|
hoelzl@31098
|
2252 |
have "real l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> real u" by auto } note Pb = this
|
hoelzl@29742
|
2253 |
|
hoelzl@29742
|
2254 |
from lift_bin'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb]
|
hoelzl@29742
|
2255 |
show ?thesis by auto
|
hoelzl@29742
|
2256 |
qed
|
hoelzl@29742
|
2257 |
|
hoelzl@29742
|
2258 |
lemma lift_un'_ex:
|
hoelzl@29742
|
2259 |
assumes lift_un'_Some: "Some (l, u) = lift_un' a f"
|
hoelzl@29742
|
2260 |
shows "\<exists> l u. Some (l, u) = a"
|
hoelzl@29742
|
2261 |
proof (cases a)
|
hoelzl@29742
|
2262 |
case None hence "None = lift_un' a f" unfolding None lift_un'.simps ..
|
hoelzl@29742
|
2263 |
thus ?thesis using lift_un'_Some by auto
|
hoelzl@29742
|
2264 |
next
|
hoelzl@29742
|
2265 |
case (Some a')
|
hoelzl@29742
|
2266 |
obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
|
hoelzl@29742
|
2267 |
thus ?thesis unfolding `a = Some a'` a' by auto
|
hoelzl@29742
|
2268 |
qed
|
hoelzl@29742
|
2269 |
|
hoelzl@29742
|
2270 |
lemma lift_un'_f:
|
hoelzl@29742
|
2271 |
assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f"
|
hoelzl@29742
|
2272 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
|
hoelzl@29742
|
2273 |
shows "\<exists> l1 u1. P l1 u1 a \<and> l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
|
hoelzl@29742
|
2274 |
proof -
|
hoelzl@29742
|
2275 |
obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un'_ex[OF assms(1)] by auto
|
hoelzl@29742
|
2276 |
have lu: "(l, u) = f l1 u1" using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto
|
hoelzl@29742
|
2277 |
have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" unfolding lu[symmetric] by auto
|
hoelzl@29742
|
2278 |
thus ?thesis using Pa[OF Sa] by auto
|
hoelzl@29742
|
2279 |
qed
|
hoelzl@29742
|
2280 |
|
hoelzl@29742
|
2281 |
lemma lift_un':
|
hoelzl@29742
|
2282 |
assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
|
hoelzl@31098
|
2283 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
|
hoelzl@31809
|
2284 |
shows "\<exists> l1 u1. (real l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u1) \<and>
|
hoelzl@29742
|
2285 |
l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
|
hoelzl@29742
|
2286 |
proof -
|
hoelzl@29742
|
2287 |
{ fix l u assume "Some (l, u) = approx' prec a bs"
|
hoelzl@29742
|
2288 |
with approx_approx'[of prec a bs, OF _ this] Pa
|
hoelzl@31098
|
2289 |
have "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" by auto } note Pa = this
|
hoelzl@29742
|
2290 |
from lift_un'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa]
|
hoelzl@29742
|
2291 |
show ?thesis by auto
|
hoelzl@29742
|
2292 |
qed
|
hoelzl@29742
|
2293 |
|
hoelzl@29742
|
2294 |
lemma lift_un'_bnds:
|
hoelzl@31098
|
2295 |
assumes bnds: "\<forall> x lx ux. (l, u) = f lx ux \<and> x \<in> { real lx .. real ux } \<longrightarrow> real l \<le> f' x \<and> f' x \<le> real u"
|
hoelzl@29742
|
2296 |
and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
|
hoelzl@31098
|
2297 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u"
|
hoelzl@31098
|
2298 |
shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
|
hoelzl@29742
|
2299 |
proof -
|
hoelzl@29742
|
2300 |
from lift_un'[OF lift_un'_Some Pa]
|
hoelzl@31098
|
2301 |
obtain l1 u1 where "real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1" and "l = fst (f l1 u1)" and "u = snd (f l1 u1)" by blast
|
hoelzl@31098
|
2302 |
hence "(l, u) = f l1 u1" and "interpret_floatarith a xs \<in> {real l1 .. real u1}" by auto
|
hoelzl@29742
|
2303 |
thus ?thesis using bnds by auto
|
hoelzl@29742
|
2304 |
qed
|
hoelzl@29742
|
2305 |
|
hoelzl@29742
|
2306 |
lemma lift_un_ex:
|
hoelzl@29742
|
2307 |
assumes lift_un_Some: "Some (l, u) = lift_un a f"
|
hoelzl@29742
|
2308 |
shows "\<exists> l u. Some (l, u) = a"
|
hoelzl@29742
|
2309 |
proof (cases a)
|
hoelzl@29742
|
2310 |
case None hence "None = lift_un a f" unfolding None lift_un.simps ..
|
hoelzl@29742
|
2311 |
thus ?thesis using lift_un_Some by auto
|
hoelzl@29742
|
2312 |
next
|
hoelzl@29742
|
2313 |
case (Some a')
|
hoelzl@29742
|
2314 |
obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
|
hoelzl@29742
|
2315 |
thus ?thesis unfolding `a = Some a'` a' by auto
|
hoelzl@29742
|
2316 |
qed
|
hoelzl@29742
|
2317 |
|
hoelzl@29742
|
2318 |
lemma lift_un_f:
|
hoelzl@29742
|
2319 |
assumes lift_un_Some: "Some (l, u) = lift_un (g a) f"
|
hoelzl@29742
|
2320 |
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
|
hoelzl@29742
|
2321 |
shows "\<exists> l1 u1. P l1 u1 a \<and> Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
|
hoelzl@29742
|
2322 |
proof -
|
hoelzl@29742
|
2323 |
obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un_ex[OF assms(1)] by auto
|
hoelzl@29742
|
2324 |
have "fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None"
|
hoelzl@29742
|
2325 |
proof (rule ccontr)
|
hoelzl@29742
|
2326 |
assume "\<not> (fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None)"
|
hoelzl@29742
|
2327 |
hence or: "fst (f l1 u1) = None \<or> snd (f l1 u1) = None" by auto
|
hoelzl@31809
|
2328 |
hence "lift_un (g a) f = None"
|
hoelzl@29742
|
2329 |
proof (cases "fst (f l1 u1) = None")
|
hoelzl@29742
|
2330 |
case True
|
hoelzl@29742
|
2331 |
then obtain b where b: "f l1 u1 = (None, b)" by (cases "f l1 u1", auto)
|
hoelzl@29742
|
2332 |
thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
|
hoelzl@29742
|
2333 |
next
|
hoelzl@29742
|
2334 |
case False hence "snd (f l1 u1) = None" using or by auto
|
hoelzl@29742
|
2335 |
with False obtain b where b: "f l1 u1 = (Some b, None)" by (cases "f l1 u1", auto)
|
hoelzl@29742
|
2336 |
thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
|
hoelzl@29742
|
2337 |
qed
|
hoelzl@29742
|
2338 |
thus False using lift_un_Some by auto
|
hoelzl@29742
|
2339 |
qed
|
hoelzl@29742
|
2340 |
then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" by (cases "f l1 u1", auto)
|
hoelzl@29742
|
2341 |
from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f]
|
hoelzl@29742
|
2342 |
have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" unfolding f by auto
|
hoelzl@29742
|
2343 |
thus ?thesis unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto
|
hoelzl@29742
|
2344 |
qed
|
hoelzl@29742
|
2345 |
|
hoelzl@29742
|
2346 |
lemma lift_un:
|
hoelzl@29742
|
2347 |
assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
|
hoelzl@31098
|
2348 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
|
hoelzl@31809
|
2349 |
shows "\<exists> l1 u1. (real l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u1) \<and>
|
hoelzl@29742
|
2350 |
Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
|
hoelzl@29742
|
2351 |
proof -
|
hoelzl@29742
|
2352 |
{ fix l u assume "Some (l, u) = approx' prec a bs"
|
hoelzl@29742
|
2353 |
with approx_approx'[of prec a bs, OF _ this] Pa
|
hoelzl@31098
|
2354 |
have "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" by auto } note Pa = this
|
hoelzl@29742
|
2355 |
from lift_un_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa]
|
hoelzl@29742
|
2356 |
show ?thesis by auto
|
hoelzl@29742
|
2357 |
qed
|
hoelzl@29742
|
2358 |
|
hoelzl@29742
|
2359 |
lemma lift_un_bnds:
|
hoelzl@31098
|
2360 |
assumes bnds: "\<forall> x lx ux. (Some l, Some u) = f lx ux \<and> x \<in> { real lx .. real ux } \<longrightarrow> real l \<le> f' x \<and> f' x \<le> real u"
|
hoelzl@29742
|
2361 |
and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
|
hoelzl@31098
|
2362 |
and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u"
|
hoelzl@31098
|
2363 |
shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
|
hoelzl@29742
|
2364 |
proof -
|
hoelzl@29742
|
2365 |
from lift_un[OF lift_un_Some Pa]
|
hoelzl@31098
|
2366 |
obtain l1 u1 where "real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1" and "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" by blast
|
hoelzl@31098
|
2367 |
hence "(Some l, Some u) = f l1 u1" and "interpret_floatarith a xs \<in> {real l1 .. real u1}" by auto
|
hoelzl@29742
|
2368 |
thus ?thesis using bnds by auto
|
hoelzl@29742
|
2369 |
qed
|
hoelzl@29742
|
2370 |
|
hoelzl@29742
|
2371 |
lemma approx:
|
hoelzl@29742
|
2372 |
assumes "bounded_by xs vs"
|
hoelzl@29742
|
2373 |
and "Some (l, u) = approx prec arith vs" (is "_ = ?g arith")
|
hoelzl@31098
|
2374 |
shows "real l \<le> interpret_floatarith arith xs \<and> interpret_floatarith arith xs \<le> real u" (is "?P l u arith")
|
hoelzl@31809
|
2375 |
using `Some (l, u) = approx prec arith vs`
|
hoelzl@29742
|
2376 |
proof (induct arith arbitrary: l u x)
|
hoelzl@29742
|
2377 |
case (Add a b)
|
hoelzl@29742
|
2378 |
from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps
|
hoelzl@29742
|
2379 |
obtain l1 u1 l2 u2 where "l = l1 + l2" and "u = u1 + u2"
|
hoelzl@31098
|
2380 |
"real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1"
|
hoelzl@31098
|
2381 |
"real l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> real u2" unfolding fst_conv snd_conv by blast
|
hoelzl@31098
|
2382 |
thus ?case unfolding interpret_floatarith.simps by auto
|
hoelzl@29742
|
2383 |
next
|
hoelzl@29742
|
2384 |
case (Minus a)
|
hoelzl@29742
|
2385 |
from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps
|
hoelzl@29742
|
2386 |
obtain l1 u1 where "l = -u1" and "u = -l1"
|
hoelzl@31098
|
2387 |
"real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1" unfolding fst_conv snd_conv by blast
|
hoelzl@31098
|
2388 |
thus ?case unfolding interpret_floatarith.simps using real_of_float_minus by auto
|
hoelzl@29742
|
2389 |
next
|
hoelzl@29742
|
2390 |
case (Mult a b)
|
hoelzl@29742
|
2391 |
from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps
|
hoelzl@31809
|
2392 |
obtain l1 u1 l2 u2
|
hoelzl@29742
|
2393 |
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@29742
|
2394 |
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@31098
|
2395 |
and "real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1"
|
hoelzl@31098
|
2396 |
and "real l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> real u2" unfolding fst_conv snd_conv by blast
|
hoelzl@31809
|
2397 |
thus ?case unfolding interpret_floatarith.simps l u real_of_float_add real_of_float_mult real_of_float_nprt real_of_float_pprt
|
hoelzl@29742
|
2398 |
using mult_le_prts mult_ge_prts by auto
|
hoelzl@29742
|
2399 |
next
|
hoelzl@29742
|
2400 |
case (Inverse a)
|
hoelzl@29742
|
2401 |
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@31809
|
2402 |
obtain l1 u1 where l': "Some l = (if 0 < l1 \<or> u1 < 0 then Some (float_divl prec 1 u1) else None)"
|
hoelzl@29742
|
2403 |
and u': "Some u = (if 0 < l1 \<or> u1 < 0 then Some (float_divr prec 1 l1) else None)"
|
hoelzl@31098
|
2404 |
and l1: "real l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> real u1" by blast
|
hoelzl@29742
|
2405 |
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@31098
|
2406 |
moreover have l1_le_u1: "real l1 \<le> real u1" using l1 u1 by auto
|
hoelzl@31098
|
2407 |
ultimately have "real l1 \<noteq> 0" and "real u1 \<noteq> 0" unfolding less_float_def by auto
|
hoelzl@29742
|
2408 |
|
hoelzl@31098
|
2409 |
have inv: "inverse (real u1) \<le> inverse (interpret_floatarith a xs)
|
hoelzl@31098
|
2410 |
\<and> inverse (interpret_floatarith a xs) \<le> inverse (real l1)"
|
hoelzl@29742
|
2411 |
proof (cases "0 < l1")
|
hoelzl@31809
|
2412 |
case True hence "0 < real u1" and "0 < real l1" "0 < interpret_floatarith a xs"
|
hoelzl@29742
|
2413 |
unfolding less_float_def using l1_le_u1 l1 by auto
|
hoelzl@29742
|
2414 |
show ?thesis
|
hoelzl@31098
|
2415 |
unfolding inverse_le_iff_le[OF `0 < real u1` `0 < interpret_floatarith a xs`]
|
wenzelm@32962
|
2416 |
inverse_le_iff_le[OF `0 < interpret_floatarith a xs` `0 < real l1`]
|
hoelzl@29742
|
2417 |
using l1 u1 by auto
|
hoelzl@29742
|
2418 |
next
|
hoelzl@29742
|
2419 |
case False hence "u1 < 0" using either by blast
|
hoelzl@31809
|
2420 |
hence "real u1 < 0" and "real l1 < 0" "interpret_floatarith a xs < 0"
|
hoelzl@29742
|
2421 |
unfolding less_float_def using l1_le_u1 u1 by auto
|
hoelzl@29742
|
2422 |
show ?thesis
|
hoelzl@31098
|
2423 |
unfolding inverse_le_iff_le_neg[OF `real u1 < 0` `interpret_floatarith a xs < 0`]
|
wenzelm@32962
|
2424 |
inverse_le_iff_le_neg[OF `interpret_floatarith a xs < 0` `real l1 < 0`]
|
hoelzl@29742
|
2425 |
using l1 u1 by auto
|
hoelzl@29742
|
2426 |
qed
|
hoelzl@31468
|
2427 |
|
hoelzl@29742
|
2428 |
from l' have "l = float_divl prec 1 u1" by (cases "0 < l1 \<or> u1 < 0", auto)
|
hoelzl@31098
|
2429 |
hence "real l \<le> inverse (real u1)" unfolding nonzero_inverse_eq_divide[OF `real u1 \<noteq> 0`] using float_divl[of prec 1 u1] by auto
|
hoelzl@31098
|
2430 |
also have "\<dots> \<le> inverse (interpret_floatarith a xs)" using inv by auto
|
hoelzl@31098
|
2431 |
finally have "real l \<le> inverse (interpret_floatarith a xs)" .
|
hoelzl@29742
|
2432 |
moreover
|
hoelzl@29742
|
2433 |
from u' have "u = float_divr prec 1 l1" by (cases "0 < l1 \<or> u1 < 0", auto)
|
hoelzl@31098
|
2434 |
hence "inverse (real l1) \<le> real u" unfolding nonzero_inverse_eq_divide[OF `real l1 \<noteq> 0`] using float_divr[of 1 l1 prec] by auto
|
hoelzl@31098
|
2435 |
hence "inverse (interpret_floatarith a xs) \<le> real u" by (rule order_trans[OF inv[THEN conjunct2]])
|
hoelzl@31098
|
2436 |
ultimately show ?case unfolding interpret_floatarith.simps using l1 u1 by auto
|
hoelzl@29742
|
2437 |
next
|
hoelzl@29742
|
2438 |
case (Abs x)
|
hoelzl@29742
|
2439 |
from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps
|
hoelzl@29742
|
2440 |
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@31098
|
2441 |
and l1: "real l1 \<le> interpret_floatarith x xs" and u1: "interpret_floatarith x xs \<le> real u1" by blast
|
hoelzl@31098
|
2442 |
thus ?case unfolding l' u' by (cases "l1 < 0 \<and> 0 < u1", auto simp add: real_of_float_min real_of_float_max real_of_float_abs less_float_def)
|
hoelzl@29742
|
2443 |
next
|
hoelzl@29742
|
2444 |
case (Min a b)
|
hoelzl@29742
|
2445 |
from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps
|
hoelzl@29742
|
2446 |
obtain l1 u1 l2 u2 where l': "l = min l1 l2" and u': "u = min u1 u2"
|
hoelzl@31098
|
2447 |
and l1: "real l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> real u1"
|
hoelzl@31098
|
2448 |
and l1: "real l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> real u2" by blast
|
hoelzl@31098
|
2449 |
thus ?case unfolding l' u' by (auto simp add: real_of_float_min)
|
hoelzl@29742
|
2450 |
next
|
hoelzl@29742
|
2451 |
case (Max a b)
|
hoelzl@29742
|
2452 |
from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps
|
hoelzl@29742
|
2453 |
obtain l1 u1 l2 u2 where l': "l = max l1 l2" and u': "u = max u1 u2"
|
hoelzl@31098
|
2454 |
and l1: "real l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> real u1"
|
hoelzl@31098
|
2455 |
and l1: "real l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> real u2" by blast
|
hoelzl@31098
|
2456 |
thus ?case unfolding l' u' by (auto simp add: real_of_float_max)
|
hoelzl@29742
|
2457 |
next case (Cos a) with lift_un'_bnds[OF bnds_cos] show ?case by auto
|
hoelzl@29742
|
2458 |
next case (Arctan a) with lift_un'_bnds[OF bnds_arctan] show ?case by auto
|
hoelzl@29742
|
2459 |
next case Pi with pi_boundaries show ?case by auto
|
hoelzl@31467
|
2460 |
next case (Sqrt a) with lift_un'_bnds[OF bnds_sqrt] show ?case by auto
|
hoelzl@29742
|
2461 |
next case (Exp a) with lift_un'_bnds[OF bnds_exp] show ?case by auto
|
hoelzl@29742
|
2462 |
next case (Ln a) with lift_un_bnds[OF bnds_ln] show ?case by auto
|
hoelzl@29742
|
2463 |
next case (Power a n) with lift_un'_bnds[OF bnds_power] show ?case by auto
|
hoelzl@29742
|
2464 |
next case (Num f) thus ?case by auto
|
hoelzl@29742
|
2465 |
next
|
hoelzl@32919
|
2466 |
case (Var n)
|
hoelzl@31811
|
2467 |
from this[symmetric] `bounded_by xs vs`[THEN bounded_byE, of n]
|
hoelzl@31811
|
2468 |
show ?case by (cases "n < length vs", auto)
|
hoelzl@31811
|
2469 |
qed
|
hoelzl@31811
|
2470 |
|
hoelzl@31811
|
2471 |
datatype form = Bound floatarith floatarith floatarith form
|
hoelzl@31811
|
2472 |
| Assign floatarith floatarith form
|
hoelzl@31811
|
2473 |
| Less floatarith floatarith
|
hoelzl@31811
|
2474 |
| LessEqual floatarith floatarith
|
hoelzl@31811
|
2475 |
| AtLeastAtMost floatarith floatarith floatarith
|
hoelzl@31811
|
2476 |
|
hoelzl@31811
|
2477 |
fun interpret_form :: "form \<Rightarrow> real list \<Rightarrow> bool" where
|
hoelzl@31811
|
2478 |
"interpret_form (Bound x a b f) vs = (interpret_floatarith x vs \<in> { interpret_floatarith a vs .. interpret_floatarith b vs } \<longrightarrow> interpret_form f vs)" |
|
hoelzl@31811
|
2479 |
"interpret_form (Assign x a f) vs = (interpret_floatarith x vs = interpret_floatarith a vs \<longrightarrow> interpret_form f vs)" |
|
hoelzl@31811
|
2480 |
"interpret_form (Less a b) vs = (interpret_floatarith a vs < interpret_floatarith b vs)" |
|
hoelzl@31811
|
2481 |
"interpret_form (LessEqual a b) vs = (interpret_floatarith a vs \<le> interpret_floatarith b vs)" |
|
hoelzl@31811
|
2482 |
"interpret_form (AtLeastAtMost x a b) vs = (interpret_floatarith x vs \<in> { interpret_floatarith a vs .. interpret_floatarith b vs })"
|
hoelzl@31811
|
2483 |
|
hoelzl@31811
|
2484 |
fun approx_form' and approx_form :: "nat \<Rightarrow> form \<Rightarrow> (float * float) option list \<Rightarrow> nat list \<Rightarrow> bool" where
|
hoelzl@31811
|
2485 |
"approx_form' prec f 0 n l u bs ss = approx_form prec f (bs[n := Some (l, u)]) ss" |
|
hoelzl@31811
|
2486 |
"approx_form' prec f (Suc s) n l u bs ss =
|
hoelzl@31811
|
2487 |
(let m = (l + u) * Float 1 -1
|
hoelzl@32919
|
2488 |
in (if approx_form' prec f s n l m bs ss then approx_form' prec f s n m u bs ss else False))" |
|
hoelzl@32919
|
2489 |
"approx_form prec (Bound (Var n) a b f) bs ss =
|
hoelzl@31811
|
2490 |
(case (approx prec a bs, approx prec b bs)
|
hoelzl@31811
|
2491 |
of (Some (l, _), Some (_, u)) \<Rightarrow> approx_form' prec f (ss ! n) n l u bs ss
|
hoelzl@31811
|
2492 |
| _ \<Rightarrow> False)" |
|
hoelzl@32919
|
2493 |
"approx_form prec (Assign (Var n) a f) bs ss =
|
hoelzl@31811
|
2494 |
(case (approx prec a bs)
|
hoelzl@31811
|
2495 |
of (Some (l, u)) \<Rightarrow> approx_form' prec f (ss ! n) n l u bs ss
|
hoelzl@31811
|
2496 |
| _ \<Rightarrow> False)" |
|
hoelzl@31811
|
2497 |
"approx_form prec (Less a b) bs ss =
|
hoelzl@31811
|
2498 |
(case (approx prec a bs, approx prec b bs)
|
hoelzl@31811
|
2499 |
of (Some (l, u), Some (l', u')) \<Rightarrow> u < l'
|
hoelzl@31811
|
2500 |
| _ \<Rightarrow> False)" |
|
hoelzl@31811
|
2501 |
"approx_form prec (LessEqual a b) bs ss =
|
hoelzl@31811
|
2502 |
(case (approx prec a bs, approx prec b bs)
|
hoelzl@31811
|
2503 |
of (Some (l, u), Some (l', u')) \<Rightarrow> u \<le> l'
|
hoelzl@31811
|
2504 |
| _ \<Rightarrow> False)" |
|
hoelzl@31811
|
2505 |
"approx_form prec (AtLeastAtMost x a b) bs ss =
|
hoelzl@31811
|
2506 |
(case (approx prec x bs, approx prec a bs, approx prec b bs)
|
hoelzl@31811
|
2507 |
of (Some (lx, ux), Some (l, u), Some (l', u')) \<Rightarrow> u \<le> lx \<and> ux \<le> l'
|
hoelzl@31811
|
2508 |
| _ \<Rightarrow> False)" |
|
hoelzl@31811
|
2509 |
"approx_form _ _ _ _ = False"
|
hoelzl@31811
|
2510 |
|
hoelzl@32919
|
2511 |
lemma lazy_conj: "(if A then B else False) = (A \<and> B)" by simp
|
hoelzl@32919
|
2512 |
|
hoelzl@31811
|
2513 |
lemma approx_form_approx_form':
|
hoelzl@31811
|
2514 |
assumes "approx_form' prec f s n l u bs ss" and "x \<in> { real l .. real u }"
|
hoelzl@31811
|
2515 |
obtains l' u' where "x \<in> { real l' .. real u' }"
|
hoelzl@31811
|
2516 |
and "approx_form prec f (bs[n := Some (l', u')]) ss"
|
hoelzl@31811
|
2517 |
using assms proof (induct s arbitrary: l u)
|
hoelzl@31811
|
2518 |
case 0
|
hoelzl@31811
|
2519 |
from this(1)[of l u] this(2,3)
|
hoelzl@31811
|
2520 |
show thesis by auto
|
hoelzl@31811
|
2521 |
next
|
hoelzl@31811
|
2522 |
case (Suc s)
|
hoelzl@31811
|
2523 |
|
hoelzl@31811
|
2524 |
let ?m = "(l + u) * Float 1 -1"
|
hoelzl@31811
|
2525 |
have "real l \<le> real ?m" and "real ?m \<le> real u"
|
hoelzl@31811
|
2526 |
unfolding le_float_def using Suc.prems by auto
|
hoelzl@31811
|
2527 |
|
hoelzl@31811
|
2528 |
with `x \<in> { real l .. real u }`
|
hoelzl@31811
|
2529 |
have "x \<in> { real l .. real ?m} \<or> x \<in> { real ?m .. real u }" by auto
|
hoelzl@31811
|
2530 |
thus thesis
|
hoelzl@31811
|
2531 |
proof (rule disjE)
|
hoelzl@31811
|
2532 |
assume *: "x \<in> { real l .. real ?m }"
|
hoelzl@31811
|
2533 |
with Suc.hyps[OF _ _ *] Suc.prems
|
hoelzl@32919
|
2534 |
show thesis by (simp add: Let_def lazy_conj)
|
hoelzl@29742
|
2535 |
next
|
hoelzl@31811
|
2536 |
assume *: "x \<in> { real ?m .. real u }"
|
hoelzl@31811
|
2537 |
with Suc.hyps[OF _ _ *] Suc.prems
|
hoelzl@32919
|
2538 |
show thesis by (simp add: Let_def lazy_conj)
|
hoelzl@29742
|
2539 |
qed
|
hoelzl@29742
|
2540 |
qed
|
hoelzl@29742
|
2541 |
|
hoelzl@31811
|
2542 |
lemma approx_form_aux:
|
hoelzl@31811
|
2543 |
assumes "approx_form prec f vs ss"
|
hoelzl@31811
|
2544 |
and "bounded_by xs vs"
|
hoelzl@31811
|
2545 |
shows "interpret_form f xs"
|
hoelzl@31811
|
2546 |
using assms proof (induct f arbitrary: vs)
|
hoelzl@31811
|
2547 |
case (Bound x a b f)
|
hoelzl@31811
|
2548 |
then obtain n
|
hoelzl@32919
|
2549 |
where x_eq: "x = Var n" by (cases x) auto
|
hoelzl@31811
|
2550 |
|
hoelzl@31811
|
2551 |
with Bound.prems obtain l u' l' u
|
hoelzl@31811
|
2552 |
where l_eq: "Some (l, u') = approx prec a vs"
|
hoelzl@31811
|
2553 |
and u_eq: "Some (l', u) = approx prec b vs"
|
hoelzl@31811
|
2554 |
and approx_form': "approx_form' prec f (ss ! n) n l u vs ss"
|
hoelzl@31811
|
2555 |
by (cases "approx prec a vs", simp,
|
hoelzl@31811
|
2556 |
cases "approx prec b vs", auto) blast
|
hoelzl@31811
|
2557 |
|
hoelzl@31811
|
2558 |
{ assume "xs ! n \<in> { interpret_floatarith a xs .. interpret_floatarith b xs }"
|
hoelzl@31811
|
2559 |
with approx[OF Bound.prems(2) l_eq] and approx[OF Bound.prems(2) u_eq]
|
hoelzl@31811
|
2560 |
have "xs ! n \<in> { real l .. real u}" by auto
|
hoelzl@31811
|
2561 |
|
hoelzl@31811
|
2562 |
from approx_form_approx_form'[OF approx_form' this]
|
hoelzl@31811
|
2563 |
obtain lx ux where bnds: "xs ! n \<in> { real lx .. real ux }"
|
hoelzl@31811
|
2564 |
and approx_form: "approx_form prec f (vs[n := Some (lx, ux)]) ss" .
|
hoelzl@31811
|
2565 |
|
hoelzl@31811
|
2566 |
from `bounded_by xs vs` bnds
|
hoelzl@31811
|
2567 |
have "bounded_by xs (vs[n := Some (lx, ux)])" by (rule bounded_by_update)
|
hoelzl@31811
|
2568 |
with Bound.hyps[OF approx_form]
|
hoelzl@31811
|
2569 |
have "interpret_form f xs" by blast }
|
hoelzl@31811
|
2570 |
thus ?case using interpret_form.simps x_eq and interpret_floatarith.simps by simp
|
hoelzl@31811
|
2571 |
next
|
hoelzl@31811
|
2572 |
case (Assign x a f)
|
hoelzl@31811
|
2573 |
then obtain n
|
hoelzl@32919
|
2574 |
where x_eq: "x = Var n" by (cases x) auto
|
hoelzl@31811
|
2575 |
|
hoelzl@31811
|
2576 |
with Assign.prems obtain l u' l' u
|
hoelzl@31811
|
2577 |
where bnd_eq: "Some (l, u) = approx prec a vs"
|
hoelzl@32919
|
2578 |
and x_eq: "x = Var n"
|
hoelzl@31811
|
2579 |
and approx_form': "approx_form' prec f (ss ! n) n l u vs ss"
|
hoelzl@31811
|
2580 |
by (cases "approx prec a vs") auto
|
hoelzl@31811
|
2581 |
|
hoelzl@31811
|
2582 |
{ assume bnds: "xs ! n = interpret_floatarith a xs"
|
hoelzl@31811
|
2583 |
with approx[OF Assign.prems(2) bnd_eq]
|
hoelzl@31811
|
2584 |
have "xs ! n \<in> { real l .. real u}" by auto
|
hoelzl@31811
|
2585 |
from approx_form_approx_form'[OF approx_form' this]
|
hoelzl@31811
|
2586 |
obtain lx ux where bnds: "xs ! n \<in> { real lx .. real ux }"
|
hoelzl@31811
|
2587 |
and approx_form: "approx_form prec f (vs[n := Some (lx, ux)]) ss" .
|
hoelzl@31811
|
2588 |
|
hoelzl@31811
|
2589 |
from `bounded_by xs vs` bnds
|
hoelzl@31811
|
2590 |
have "bounded_by xs (vs[n := Some (lx, ux)])" by (rule bounded_by_update)
|
hoelzl@31811
|
2591 |
with Assign.hyps[OF approx_form]
|
hoelzl@31811
|
2592 |
have "interpret_form f xs" by blast }
|
hoelzl@31811
|
2593 |
thus ?case using interpret_form.simps x_eq and interpret_floatarith.simps by simp
|
hoelzl@31811
|
2594 |
next
|
hoelzl@29742
|
2595 |
case (Less a b)
|
hoelzl@31811
|
2596 |
then obtain l u l' u'
|
hoelzl@31811
|
2597 |
where l_eq: "Some (l, u) = approx prec a vs"
|
hoelzl@31811
|
2598 |
and u_eq: "Some (l', u') = approx prec b vs"
|
hoelzl@31811
|
2599 |
and inequality: "u < l'"
|
hoelzl@31811
|
2600 |
by (cases "approx prec a vs", auto,
|
hoelzl@31811
|
2601 |
cases "approx prec b vs", auto)
|
hoelzl@31811
|
2602 |
from inequality[unfolded less_float_def] approx[OF Less.prems(2) l_eq] approx[OF Less.prems(2) u_eq]
|
hoelzl@31811
|
2603 |
show ?case by auto
|
hoelzl@29742
|
2604 |
next
|
hoelzl@29742
|
2605 |
case (LessEqual a b)
|
hoelzl@31811
|
2606 |
then obtain l u l' u'
|
hoelzl@31811
|
2607 |
where l_eq: "Some (l, u) = approx prec a vs"
|
hoelzl@31811
|
2608 |
and u_eq: "Some (l', u') = approx prec b vs"
|
hoelzl@31811
|
2609 |
and inequality: "u \<le> l'"
|
hoelzl@31811
|
2610 |
by (cases "approx prec a vs", auto,
|
hoelzl@31811
|
2611 |
cases "approx prec b vs", auto)
|
hoelzl@31811
|
2612 |
from inequality[unfolded le_float_def] approx[OF LessEqual.prems(2) l_eq] approx[OF LessEqual.prems(2) u_eq]
|
hoelzl@31811
|
2613 |
show ?case by auto
|
hoelzl@31811
|
2614 |
next
|
hoelzl@31811
|
2615 |
case (AtLeastAtMost x a b)
|
hoelzl@31811
|
2616 |
then obtain lx ux l u l' u'
|
hoelzl@31811
|
2617 |
where x_eq: "Some (lx, ux) = approx prec x vs"
|
hoelzl@31811
|
2618 |
and l_eq: "Some (l, u) = approx prec a vs"
|
hoelzl@31811
|
2619 |
and u_eq: "Some (l', u') = approx prec b vs"
|
hoelzl@31811
|
2620 |
and inequality: "u \<le> lx \<and> ux \<le> l'"
|
hoelzl@31811
|
2621 |
by (cases "approx prec x vs", auto,
|
hoelzl@31811
|
2622 |
cases "approx prec a vs", auto,
|
hoelzl@31811
|
2623 |
cases "approx prec b vs", auto, blast)
|
hoelzl@31811
|
2624 |
from inequality[unfolded le_float_def] approx[OF AtLeastAtMost.prems(2) l_eq] approx[OF AtLeastAtMost.prems(2) u_eq] approx[OF AtLeastAtMost.prems(2) x_eq]
|
hoelzl@31811
|
2625 |
show ?case by auto
|
hoelzl@29742
|
2626 |
qed
|
hoelzl@29742
|
2627 |
|
hoelzl@31811
|
2628 |
lemma approx_form:
|
hoelzl@31811
|
2629 |
assumes "n = length xs"
|
hoelzl@31811
|
2630 |
assumes "approx_form prec f (replicate n None) ss"
|
hoelzl@31811
|
2631 |
shows "interpret_form f xs"
|
hoelzl@31811
|
2632 |
using approx_form_aux[OF _ bounded_by_None] assms by auto
|
hoelzl@29742
|
2633 |
|
hoelzl@31862
|
2634 |
subsection {* Implementing Taylor series expansion *}
|
hoelzl@31862
|
2635 |
|
hoelzl@31862
|
2636 |
fun isDERIV :: "nat \<Rightarrow> floatarith \<Rightarrow> real list \<Rightarrow> bool" where
|
hoelzl@31862
|
2637 |
"isDERIV x (Add a b) vs = (isDERIV x a vs \<and> isDERIV x b vs)" |
|
hoelzl@31862
|
2638 |
"isDERIV x (Mult a b) vs = (isDERIV x a vs \<and> isDERIV x b vs)" |
|
hoelzl@31862
|
2639 |
"isDERIV x (Minus a) vs = isDERIV x a vs" |
|
hoelzl@31862
|
2640 |
"isDERIV x (Inverse a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs \<noteq> 0)" |
|
hoelzl@31862
|
2641 |
"isDERIV x (Cos a) vs = isDERIV x a vs" |
|
hoelzl@31862
|
2642 |
"isDERIV x (Arctan a) vs = isDERIV x a vs" |
|
hoelzl@31862
|
2643 |
"isDERIV x (Min a b) vs = False" |
|
hoelzl@31862
|
2644 |
"isDERIV x (Max a b) vs = False" |
|
hoelzl@31862
|
2645 |
"isDERIV x (Abs a) vs = False" |
|
hoelzl@31862
|
2646 |
"isDERIV x Pi vs = True" |
|
hoelzl@31862
|
2647 |
"isDERIV x (Sqrt a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs > 0)" |
|
hoelzl@31862
|
2648 |
"isDERIV x (Exp a) vs = isDERIV x a vs" |
|
hoelzl@31862
|
2649 |
"isDERIV x (Ln a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs > 0)" |
|
hoelzl@31862
|
2650 |
"isDERIV x (Power a 0) vs = True" |
|
hoelzl@31862
|
2651 |
"isDERIV x (Power a (Suc n)) vs = isDERIV x a vs" |
|
hoelzl@31862
|
2652 |
"isDERIV x (Num f) vs = True" |
|
hoelzl@32919
|
2653 |
"isDERIV x (Var n) vs = True"
|
hoelzl@31862
|
2654 |
|
hoelzl@31862
|
2655 |
fun DERIV_floatarith :: "nat \<Rightarrow> floatarith \<Rightarrow> floatarith" where
|
hoelzl@31862
|
2656 |
"DERIV_floatarith x (Add a b) = Add (DERIV_floatarith x a) (DERIV_floatarith x b)" |
|
hoelzl@31862
|
2657 |
"DERIV_floatarith x (Mult a b) = Add (Mult a (DERIV_floatarith x b)) (Mult (DERIV_floatarith x a) b)" |
|
hoelzl@31862
|
2658 |
"DERIV_floatarith x (Minus a) = Minus (DERIV_floatarith x a)" |
|
hoelzl@31862
|
2659 |
"DERIV_floatarith x (Inverse a) = Minus (Mult (DERIV_floatarith x a) (Inverse (Power a 2)))" |
|
hoelzl@31862
|
2660 |
"DERIV_floatarith x (Cos a) = Minus (Mult (Cos (Add (Mult Pi (Num (Float 1 -1))) (Minus a))) (DERIV_floatarith x a))" |
|
hoelzl@31862
|
2661 |
"DERIV_floatarith x (Arctan a) = Mult (Inverse (Add (Num 1) (Power a 2))) (DERIV_floatarith x a)" |
|
hoelzl@31862
|
2662 |
"DERIV_floatarith x (Min a b) = Num 0" |
|
hoelzl@31862
|
2663 |
"DERIV_floatarith x (Max a b) = Num 0" |
|
hoelzl@31862
|
2664 |
"DERIV_floatarith x (Abs a) = Num 0" |
|
hoelzl@31862
|
2665 |
"DERIV_floatarith x Pi = Num 0" |
|
hoelzl@31862
|
2666 |
"DERIV_floatarith x (Sqrt a) = (Mult (Inverse (Mult (Sqrt a) (Num 2))) (DERIV_floatarith x a))" |
|
hoelzl@31862
|
2667 |
"DERIV_floatarith x (Exp a) = Mult (Exp a) (DERIV_floatarith x a)" |
|
hoelzl@31862
|
2668 |
"DERIV_floatarith x (Ln a) = Mult (Inverse a) (DERIV_floatarith x a)" |
|
hoelzl@31862
|
2669 |
"DERIV_floatarith x (Power a 0) = Num 0" |
|
hoelzl@31862
|
2670 |
"DERIV_floatarith x (Power a (Suc n)) = Mult (Num (Float (int (Suc n)) 0)) (Mult (Power a n) (DERIV_floatarith x a))" |
|
hoelzl@31862
|
2671 |
"DERIV_floatarith x (Num f) = Num 0" |
|
hoelzl@32919
|
2672 |
"DERIV_floatarith x (Var n) = (if x = n then Num 1 else Num 0)"
|
hoelzl@31862
|
2673 |
|
hoelzl@31862
|
2674 |
lemma DERIV_floatarith:
|
hoelzl@31862
|
2675 |
assumes "n < length vs"
|
hoelzl@31862
|
2676 |
assumes isDERIV: "isDERIV n f (vs[n := x])"
|
hoelzl@31862
|
2677 |
shows "DERIV (\<lambda> x'. interpret_floatarith f (vs[n := x'])) x :>
|
hoelzl@31862
|
2678 |
interpret_floatarith (DERIV_floatarith n f) (vs[n := x])"
|
hoelzl@31862
|
2679 |
(is "DERIV (?i f) x :> _")
|
hoelzl@31862
|
2680 |
using isDERIV proof (induct f arbitrary: x)
|
hoelzl@31880
|
2681 |
case (Inverse a) thus ?case
|
hoelzl@31880
|
2682 |
by (auto intro!: DERIV_intros
|
hoelzl@31862
|
2683 |
simp add: algebra_simps power2_eq_square)
|
hoelzl@31862
|
2684 |
next case (Cos a) thus ?case
|
hoelzl@31880
|
2685 |
by (auto intro!: DERIV_intros
|
hoelzl@31862
|
2686 |
simp del: interpret_floatarith.simps(5)
|
hoelzl@31862
|
2687 |
simp add: interpret_floatarith_sin interpret_floatarith.simps(5)[of a])
|
hoelzl@31862
|
2688 |
next case (Power a n) thus ?case
|
hoelzl@31880
|
2689 |
by (cases n, auto intro!: DERIV_intros
|
hoelzl@31862
|
2690 |
simp del: power_Suc simp add: real_eq_of_nat)
|
hoelzl@31862
|
2691 |
next case (Ln a) thus ?case
|
hoelzl@31880
|
2692 |
by (auto intro!: DERIV_intros simp add: divide_inverse)
|
hoelzl@32919
|
2693 |
next case (Var i) thus ?case using `n < length vs` by auto
|
hoelzl@31880
|
2694 |
qed (auto intro!: DERIV_intros)
|
hoelzl@31862
|
2695 |
|
hoelzl@31862
|
2696 |
declare approx.simps[simp del]
|
hoelzl@31862
|
2697 |
|
hoelzl@31862
|
2698 |
fun isDERIV_approx :: "nat \<Rightarrow> nat \<Rightarrow> floatarith \<Rightarrow> (float * float) option list \<Rightarrow> bool" where
|
hoelzl@31862
|
2699 |
"isDERIV_approx prec x (Add a b) vs = (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs)" |
|
hoelzl@31862
|
2700 |
"isDERIV_approx prec x (Mult a b) vs = (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs)" |
|
hoelzl@31862
|
2701 |
"isDERIV_approx prec x (Minus a) vs = isDERIV_approx prec x a vs" |
|
hoelzl@31862
|
2702 |
"isDERIV_approx prec x (Inverse a) vs =
|
hoelzl@31862
|
2703 |
(isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l \<or> u < 0 | None \<Rightarrow> False))" |
|
hoelzl@31862
|
2704 |
"isDERIV_approx prec x (Cos a) vs = isDERIV_approx prec x a vs" |
|
hoelzl@31862
|
2705 |
"isDERIV_approx prec x (Arctan a) vs = isDERIV_approx prec x a vs" |
|
hoelzl@31862
|
2706 |
"isDERIV_approx prec x (Min a b) vs = False" |
|
hoelzl@31862
|
2707 |
"isDERIV_approx prec x (Max a b) vs = False" |
|
hoelzl@31862
|
2708 |
"isDERIV_approx prec x (Abs a) vs = False" |
|
hoelzl@31862
|
2709 |
"isDERIV_approx prec x Pi vs = True" |
|
hoelzl@31862
|
2710 |
"isDERIV_approx prec x (Sqrt a) vs =
|
hoelzl@31862
|
2711 |
(isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l | None \<Rightarrow> False))" |
|
hoelzl@31862
|
2712 |
"isDERIV_approx prec x (Exp a) vs = isDERIV_approx prec x a vs" |
|
hoelzl@31862
|
2713 |
"isDERIV_approx prec x (Ln a) vs =
|
hoelzl@31862
|
2714 |
(isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l | None \<Rightarrow> False))" |
|
hoelzl@31862
|
2715 |
"isDERIV_approx prec x (Power a 0) vs = True" |
|
hoelzl@31862
|
2716 |
"isDERIV_approx prec x (Power a (Suc n)) vs = isDERIV_approx prec x a vs" |
|
hoelzl@31862
|
2717 |
"isDERIV_approx prec x (Num f) vs = True" |
|
hoelzl@32919
|
2718 |
"isDERIV_approx prec x (Var n) vs = True"
|
hoelzl@31862
|
2719 |
|
hoelzl@31862
|
2720 |
lemma isDERIV_approx:
|
hoelzl@31862
|
2721 |
assumes "bounded_by xs vs"
|
hoelzl@31862
|
2722 |
and isDERIV_approx: "isDERIV_approx prec x f vs"
|
hoelzl@31862
|
2723 |
shows "isDERIV x f xs"
|
hoelzl@31862
|
2724 |
using isDERIV_approx proof (induct f)
|
hoelzl@31862
|
2725 |
case (Inverse a)
|
hoelzl@31862
|
2726 |
then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
|
hoelzl@31862
|
2727 |
and *: "0 < l \<or> u < 0"
|
hoelzl@31862
|
2728 |
by (cases "approx prec a vs", auto)
|
hoelzl@31862
|
2729 |
with approx[OF `bounded_by xs vs` approx_Some]
|
hoelzl@31862
|
2730 |
have "interpret_floatarith a xs \<noteq> 0" unfolding less_float_def by auto
|
hoelzl@31862
|
2731 |
thus ?case using Inverse by auto
|
hoelzl@31862
|
2732 |
next
|
hoelzl@31862
|
2733 |
case (Ln a)
|
hoelzl@31862
|
2734 |
then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
|
hoelzl@31862
|
2735 |
and *: "0 < l"
|
hoelzl@31862
|
2736 |
by (cases "approx prec a vs", auto)
|
hoelzl@31862
|
2737 |
with approx[OF `bounded_by xs vs` approx_Some]
|
hoelzl@31862
|
2738 |
have "0 < interpret_floatarith a xs" unfolding less_float_def by auto
|
hoelzl@31862
|
2739 |
thus ?case using Ln by auto
|
hoelzl@31862
|
2740 |
next
|
hoelzl@31862
|
2741 |
case (Sqrt a)
|
hoelzl@31862
|
2742 |
then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
|
hoelzl@31862
|
2743 |
and *: "0 < l"
|
hoelzl@31862
|
2744 |
by (cases "approx prec a vs", auto)
|
hoelzl@31862
|
2745 |
with approx[OF `bounded_by xs vs` approx_Some]
|
hoelzl@31862
|
2746 |
have "0 < interpret_floatarith a xs" unfolding less_float_def by auto
|
hoelzl@31862
|
2747 |
thus ?case using Sqrt by auto
|
hoelzl@31862
|
2748 |
next
|
hoelzl@31862
|
2749 |
case (Power a n) thus ?case by (cases n, auto)
|
hoelzl@31862
|
2750 |
qed auto
|
hoelzl@31862
|
2751 |
|
hoelzl@31862
|
2752 |
lemma bounded_by_update_var:
|
hoelzl@31862
|
2753 |
assumes "bounded_by xs vs" and "vs ! i = Some (l, u)"
|
hoelzl@31862
|
2754 |
and bnd: "x \<in> { real l .. real u }"
|
hoelzl@31862
|
2755 |
shows "bounded_by (xs[i := x]) vs"
|
hoelzl@31862
|
2756 |
proof (cases "i < length xs")
|
hoelzl@31862
|
2757 |
case False thus ?thesis using `bounded_by xs vs` by auto
|
hoelzl@31862
|
2758 |
next
|
hoelzl@31862
|
2759 |
let ?xs = "xs[i := x]"
|
hoelzl@31862
|
2760 |
case True hence "i < length ?xs" by auto
|
hoelzl@31862
|
2761 |
{ fix j
|
hoelzl@31862
|
2762 |
assume "j < length vs"
|
hoelzl@31862
|
2763 |
have "case vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> ?xs ! j \<in> { real l .. real u }"
|
hoelzl@31862
|
2764 |
proof (cases "vs ! j")
|
hoelzl@31862
|
2765 |
case (Some b)
|
hoelzl@31862
|
2766 |
thus ?thesis
|
hoelzl@31862
|
2767 |
proof (cases "i = j")
|
hoelzl@31862
|
2768 |
case True
|
hoelzl@31862
|
2769 |
thus ?thesis using `vs ! i = Some (l, u)` Some and bnd `i < length ?xs`
|
wenzelm@32962
|
2770 |
by auto
|
hoelzl@31862
|
2771 |
next
|
hoelzl@31862
|
2772 |
case False
|
hoelzl@31862
|
2773 |
thus ?thesis using `bounded_by xs vs`[THEN bounded_byE, OF `j < length vs`] Some
|
wenzelm@32962
|
2774 |
by auto
|
hoelzl@31862
|
2775 |
qed
|
hoelzl@31862
|
2776 |
qed auto }
|
hoelzl@31862
|
2777 |
thus ?thesis unfolding bounded_by_def by auto
|
hoelzl@31862
|
2778 |
qed
|
hoelzl@31862
|
2779 |
|
hoelzl@31862
|
2780 |
lemma isDERIV_approx':
|
hoelzl@31862
|
2781 |
assumes "bounded_by xs vs"
|
hoelzl@31862
|
2782 |
and vs_x: "vs ! x = Some (l, u)" and X_in: "X \<in> { real l .. real u }"
|
hoelzl@31862
|
2783 |
and approx: "isDERIV_approx prec x f vs"
|
hoelzl@31862
|
2784 |
shows "isDERIV x f (xs[x := X])"
|
hoelzl@31862
|
2785 |
proof -
|
hoelzl@31862
|
2786 |
note bounded_by_update_var[OF `bounded_by xs vs` vs_x X_in] approx
|
hoelzl@31862
|
2787 |
thus ?thesis by (rule isDERIV_approx)
|
hoelzl@31862
|
2788 |
qed
|
hoelzl@31862
|
2789 |
|
hoelzl@31862
|
2790 |
lemma DERIV_approx:
|
hoelzl@31862
|
2791 |
assumes "n < length xs" and bnd: "bounded_by xs vs"
|
hoelzl@31862
|
2792 |
and isD: "isDERIV_approx prec n f vs"
|
hoelzl@31862
|
2793 |
and app: "Some (l, u) = approx prec (DERIV_floatarith n f) vs" (is "_ = approx _ ?D _")
|
hoelzl@31862
|
2794 |
shows "\<exists>x. real l \<le> x \<and> x \<le> real u \<and>
|
hoelzl@31862
|
2795 |
DERIV (\<lambda> x. interpret_floatarith f (xs[n := x])) (xs!n) :> x"
|
hoelzl@31862
|
2796 |
(is "\<exists> x. _ \<and> _ \<and> DERIV (?i f) _ :> _")
|
hoelzl@31862
|
2797 |
proof (rule exI[of _ "?i ?D (xs!n)"], rule conjI[OF _ conjI])
|
hoelzl@31862
|
2798 |
let "?i f x" = "interpret_floatarith f (xs[n := x])"
|
hoelzl@31862
|
2799 |
from approx[OF bnd app]
|
hoelzl@31862
|
2800 |
show "real l \<le> ?i ?D (xs!n)" and "?i ?D (xs!n) \<le> real u"
|
hoelzl@31862
|
2801 |
using `n < length xs` by auto
|
hoelzl@31862
|
2802 |
from DERIV_floatarith[OF `n < length xs`, of f "xs!n"] isDERIV_approx[OF bnd isD]
|
hoelzl@31862
|
2803 |
show "DERIV (?i f) (xs!n) :> (?i ?D (xs!n))" by simp
|
hoelzl@31862
|
2804 |
qed
|
hoelzl@31862
|
2805 |
|
hoelzl@31862
|
2806 |
fun lift_bin :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float) option) \<Rightarrow> (float * float) option" where
|
hoelzl@31862
|
2807 |
"lift_bin (Some (l1, u1)) (Some (l2, u2)) f = f l1 u1 l2 u2" |
|
hoelzl@31862
|
2808 |
"lift_bin a b f = None"
|
hoelzl@31862
|
2809 |
|
hoelzl@31862
|
2810 |
lemma lift_bin:
|
hoelzl@31862
|
2811 |
assumes lift_bin_Some: "Some (l, u) = lift_bin a b f"
|
hoelzl@31862
|
2812 |
obtains l1 u1 l2 u2
|
hoelzl@31862
|
2813 |
where "a = Some (l1, u1)"
|
hoelzl@31862
|
2814 |
and "b = Some (l2, u2)"
|
hoelzl@31862
|
2815 |
and "f l1 u1 l2 u2 = Some (l, u)"
|
hoelzl@31862
|
2816 |
using assms by (cases a, simp, cases b, simp, auto)
|
hoelzl@31862
|
2817 |
|
hoelzl@31862
|
2818 |
fun approx_tse where
|
hoelzl@31862
|
2819 |
"approx_tse prec n 0 c k f bs = approx prec f bs" |
|
hoelzl@31862
|
2820 |
"approx_tse prec n (Suc s) c k f bs =
|
hoelzl@31862
|
2821 |
(if isDERIV_approx prec n f bs then
|
hoelzl@31862
|
2822 |
lift_bin (approx prec f (bs[n := Some (c,c)]))
|
hoelzl@31862
|
2823 |
(approx_tse prec n s c (Suc k) (DERIV_floatarith n f) bs)
|
hoelzl@31862
|
2824 |
(\<lambda> l1 u1 l2 u2. approx prec
|
hoelzl@32919
|
2825 |
(Add (Var 0)
|
hoelzl@31862
|
2826 |
(Mult (Inverse (Num (Float (int k) 0)))
|
hoelzl@32919
|
2827 |
(Mult (Add (Var (Suc (Suc 0))) (Minus (Num c)))
|
hoelzl@32919
|
2828 |
(Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), bs!n])
|
hoelzl@31862
|
2829 |
else approx prec f bs)"
|
hoelzl@31862
|
2830 |
|
hoelzl@31862
|
2831 |
lemma bounded_by_Cons:
|
hoelzl@31862
|
2832 |
assumes bnd: "bounded_by xs vs"
|
hoelzl@31862
|
2833 |
and x: "x \<in> { real l .. real u }"
|
hoelzl@31862
|
2834 |
shows "bounded_by (x#xs) ((Some (l, u))#vs)"
|
hoelzl@31862
|
2835 |
proof -
|
hoelzl@31862
|
2836 |
{ fix i assume *: "i < length ((Some (l, u))#vs)"
|
hoelzl@31862
|
2837 |
have "case ((Some (l,u))#vs) ! i of Some (l, u) \<Rightarrow> (x#xs)!i \<in> { real l .. real u } | None \<Rightarrow> True"
|
hoelzl@31862
|
2838 |
proof (cases i)
|
hoelzl@31862
|
2839 |
case 0 with x show ?thesis by auto
|
hoelzl@31862
|
2840 |
next
|
hoelzl@31862
|
2841 |
case (Suc i) with * have "i < length vs" by auto
|
hoelzl@31862
|
2842 |
from bnd[THEN bounded_byE, OF this]
|
hoelzl@31862
|
2843 |
show ?thesis unfolding Suc nth_Cons_Suc .
|
hoelzl@31862
|
2844 |
qed }
|
hoelzl@31862
|
2845 |
thus ?thesis by (auto simp add: bounded_by_def)
|
hoelzl@31862
|
2846 |
qed
|
hoelzl@31862
|
2847 |
|
hoelzl@31862
|
2848 |
lemma approx_tse_generic:
|
hoelzl@31862
|
2849 |
assumes "bounded_by xs vs"
|
hoelzl@31862
|
2850 |
and bnd_c: "bounded_by (xs[x := real c]) vs" and "x < length vs" and "x < length xs"
|
hoelzl@31862
|
2851 |
and bnd_x: "vs ! x = Some (lx, ux)"
|
hoelzl@31862
|
2852 |
and ate: "Some (l, u) = approx_tse prec x s c k f vs"
|
hoelzl@31862
|
2853 |
shows "\<exists> n. (\<forall> m < n. \<forall> z \<in> {real lx .. real ux}.
|
hoelzl@31862
|
2854 |
DERIV (\<lambda> y. interpret_floatarith ((DERIV_floatarith x ^^ m) f) (xs[x := y])) z :>
|
hoelzl@31862
|
2855 |
(interpret_floatarith ((DERIV_floatarith x ^^ (Suc m)) f) (xs[x := z])))
|
hoelzl@31862
|
2856 |
\<and> (\<forall> t \<in> {real lx .. real ux}. (\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) *
|
hoelzl@31862
|
2857 |
interpret_floatarith ((DERIV_floatarith x ^^ i) f) (xs[x := real c]) *
|
hoelzl@31862
|
2858 |
(xs!x - real c)^i) +
|
hoelzl@31862
|
2859 |
inverse (real (\<Prod> j \<in> {k..<k+n}. j)) *
|
hoelzl@31862
|
2860 |
interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := t]) *
|
hoelzl@31862
|
2861 |
(xs!x - real c)^n \<in> {real l .. real u})" (is "\<exists> n. ?taylor f k l u n")
|
hoelzl@31862
|
2862 |
using ate proof (induct s arbitrary: k f l u)
|
hoelzl@31862
|
2863 |
case 0
|
hoelzl@31862
|
2864 |
{ fix t assume "t \<in> {real lx .. real ux}"
|
hoelzl@31862
|
2865 |
note bounded_by_update_var[OF `bounded_by xs vs` bnd_x this]
|
hoelzl@31862
|
2866 |
from approx[OF this 0[unfolded approx_tse.simps]]
|
hoelzl@31862
|
2867 |
have "(interpret_floatarith f (xs[x := t])) \<in> {real l .. real u}"
|
hoelzl@31862
|
2868 |
by (auto simp add: algebra_simps)
|
hoelzl@31862
|
2869 |
} thus ?case by (auto intro!: exI[of _ 0])
|
hoelzl@31862
|
2870 |
next
|
hoelzl@31862
|
2871 |
case (Suc s)
|
hoelzl@31862
|
2872 |
show ?case
|
hoelzl@31862
|
2873 |
proof (cases "isDERIV_approx prec x f vs")
|
hoelzl@31862
|
2874 |
case False
|
hoelzl@31862
|
2875 |
note ap = Suc.prems[unfolded approx_tse.simps if_not_P[OF False]]
|
hoelzl@31862
|
2876 |
|
hoelzl@31862
|
2877 |
{ fix t assume "t \<in> {real lx .. real ux}"
|
hoelzl@31862
|
2878 |
note bounded_by_update_var[OF `bounded_by xs vs` bnd_x this]
|
hoelzl@31862
|
2879 |
from approx[OF this ap]
|
hoelzl@31862
|
2880 |
have "(interpret_floatarith f (xs[x := t])) \<in> {real l .. real u}"
|
wenzelm@32962
|
2881 |
by (auto simp add: algebra_simps)
|
hoelzl@31862
|
2882 |
} thus ?thesis by (auto intro!: exI[of _ 0])
|
hoelzl@31862
|
2883 |
next
|
hoelzl@31862
|
2884 |
case True
|
hoelzl@31862
|
2885 |
with Suc.prems
|
hoelzl@31862
|
2886 |
obtain l1 u1 l2 u2
|
hoelzl@31862
|
2887 |
where a: "Some (l1, u1) = approx prec f (vs[x := Some (c,c)])"
|
hoelzl@31862
|
2888 |
and ate: "Some (l2, u2) = approx_tse prec x s c (Suc k) (DERIV_floatarith x f) vs"
|
hoelzl@31862
|
2889 |
and final: "Some (l, u) = approx prec
|
hoelzl@32919
|
2890 |
(Add (Var 0)
|
hoelzl@31862
|
2891 |
(Mult (Inverse (Num (Float (int k) 0)))
|
hoelzl@32919
|
2892 |
(Mult (Add (Var (Suc (Suc 0))) (Minus (Num c)))
|
hoelzl@32919
|
2893 |
(Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), vs!x]"
|
hoelzl@31862
|
2894 |
by (auto elim!: lift_bin) blast
|
hoelzl@31862
|
2895 |
|
hoelzl@31862
|
2896 |
from bnd_c `x < length xs`
|
hoelzl@31862
|
2897 |
have bnd: "bounded_by (xs[x:=real c]) (vs[x:= Some (c,c)])"
|
hoelzl@31862
|
2898 |
by (auto intro!: bounded_by_update)
|
hoelzl@31862
|
2899 |
|
hoelzl@31862
|
2900 |
from approx[OF this a]
|
hoelzl@31862
|
2901 |
have f_c: "interpret_floatarith ((DERIV_floatarith x ^^ 0) f) (xs[x := real c]) \<in> { real l1 .. real u1 }"
|
hoelzl@31862
|
2902 |
(is "?f 0 (real c) \<in> _")
|
hoelzl@31862
|
2903 |
by auto
|
hoelzl@31862
|
2904 |
|
hoelzl@31862
|
2905 |
{ fix f :: "'a \<Rightarrow> 'a" fix n :: nat fix x :: 'a
|
hoelzl@31862
|
2906 |
have "(f ^^ Suc n) x = (f ^^ n) (f x)"
|
wenzelm@32962
|
2907 |
by (induct n, auto) }
|
hoelzl@31862
|
2908 |
note funpow_Suc = this[symmetric]
|
hoelzl@31862
|
2909 |
from Suc.hyps[OF ate, unfolded this]
|
hoelzl@31862
|
2910 |
obtain n
|
hoelzl@31862
|
2911 |
where DERIV_hyp: "\<And> m z. \<lbrakk> m < n ; z \<in> { real lx .. real ux } \<rbrakk> \<Longrightarrow> DERIV (?f (Suc m)) z :> ?f (Suc (Suc m)) z"
|
hoelzl@31862
|
2912 |
and hyp: "\<forall> t \<in> {real lx .. real ux}. (\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {Suc k..<Suc k + i}. j)) * ?f (Suc i) (real c) * (xs!x - real c)^i) +
|
hoelzl@31862
|
2913 |
inverse (real (\<Prod> j \<in> {Suc k..<Suc k + n}. j)) * ?f (Suc n) t * (xs!x - real c)^n \<in> {real l2 .. real u2}"
|
hoelzl@31862
|
2914 |
(is "\<forall> t \<in> _. ?X (Suc k) f n t \<in> _")
|
hoelzl@31862
|
2915 |
by blast
|
hoelzl@31862
|
2916 |
|
hoelzl@31862
|
2917 |
{ fix m z
|
hoelzl@31862
|
2918 |
assume "m < Suc n" and bnd_z: "z \<in> { real lx .. real ux }"
|
hoelzl@31862
|
2919 |
have "DERIV (?f m) z :> ?f (Suc m) z"
|
hoelzl@31862
|
2920 |
proof (cases m)
|
wenzelm@32962
|
2921 |
case 0
|
wenzelm@32962
|
2922 |
with DERIV_floatarith[OF `x < length xs` isDERIV_approx'[OF `bounded_by xs vs` bnd_x bnd_z True]]
|
wenzelm@32962
|
2923 |
show ?thesis by simp
|
hoelzl@31862
|
2924 |
next
|
wenzelm@32962
|
2925 |
case (Suc m')
|
wenzelm@32962
|
2926 |
hence "m' < n" using `m < Suc n` by auto
|
wenzelm@32962
|
2927 |
from DERIV_hyp[OF this bnd_z]
|
wenzelm@32962
|
2928 |
show ?thesis using Suc by simp
|
hoelzl@31862
|
2929 |
qed } note DERIV = this
|
hoelzl@31862
|
2930 |
|
hoelzl@31862
|
2931 |
have "\<And> k i. k < i \<Longrightarrow> {k ..< i} = insert k {Suc k ..< i}" by auto
|
hoelzl@31862
|
2932 |
hence setprod_head_Suc: "\<And> k i. \<Prod> {k ..< k + Suc i} = k * \<Prod> {Suc k ..< Suc k + i}" by auto
|
hoelzl@31862
|
2933 |
have setsum_move0: "\<And> k F. setsum F {0..<Suc k} = F 0 + setsum (\<lambda> k. F (Suc k)) {0..<k}"
|
hoelzl@31862
|
2934 |
unfolding setsum_shift_bounds_Suc_ivl[symmetric]
|
hoelzl@31862
|
2935 |
unfolding setsum_head_upt_Suc[OF zero_less_Suc] ..
|
hoelzl@31862
|
2936 |
def C \<equiv> "xs!x - real c"
|
hoelzl@31862
|
2937 |
|
hoelzl@31862
|
2938 |
{ fix t assume t: "t \<in> {real lx .. real ux}"
|
hoelzl@31862
|
2939 |
hence "bounded_by [xs!x] [vs!x]"
|
wenzelm@32962
|
2940 |
using `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`]
|
wenzelm@32962
|
2941 |
by (cases "vs!x", auto simp add: bounded_by_def)
|
hoelzl@31862
|
2942 |
|
hoelzl@31862
|
2943 |
with hyp[THEN bspec, OF t] f_c
|
hoelzl@31862
|
2944 |
have "bounded_by [?f 0 (real c), ?X (Suc k) f n t, xs!x] [Some (l1, u1), Some (l2, u2), vs!x]"
|
wenzelm@32962
|
2945 |
by (auto intro!: bounded_by_Cons)
|
hoelzl@31862
|
2946 |
from approx[OF this final, unfolded atLeastAtMost_iff[symmetric]]
|
hoelzl@31862
|
2947 |
have "?X (Suc k) f n t * (xs!x - real c) * inverse (real k) + ?f 0 (real c) \<in> {real l .. real u}"
|
wenzelm@32962
|
2948 |
by (auto simp add: algebra_simps)
|
hoelzl@31862
|
2949 |
also have "?X (Suc k) f n t * (xs!x - real c) * inverse (real k) + ?f 0 (real c) =
|
hoelzl@31862
|
2950 |
(\<Sum> i = 0..<Suc n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) * ?f i (real c) * (xs!x - real c)^i) +
|
hoelzl@31862
|
2951 |
inverse (real (\<Prod> j \<in> {k..<k+Suc n}. j)) * ?f (Suc n) t * (xs!x - real c)^Suc n" (is "_ = ?T")
|
wenzelm@32962
|
2952 |
unfolding funpow_Suc C_def[symmetric] setsum_move0 setprod_head_Suc
|
haftmann@35082
|
2953 |
by (auto simp add: algebra_simps)
|
haftmann@35082
|
2954 |
(simp only: mult_left_commute [of _ "inverse (real k)"] setsum_right_distrib [symmetric])
|
hoelzl@31862
|
2955 |
finally have "?T \<in> {real l .. real u}" . }
|
hoelzl@31862
|
2956 |
thus ?thesis using DERIV by blast
|
hoelzl@31862
|
2957 |
qed
|
hoelzl@31862
|
2958 |
qed
|
hoelzl@31862
|
2959 |
|
hoelzl@31862
|
2960 |
lemma setprod_fact: "\<Prod> {1..<1 + k} = fact (k :: nat)"
|
hoelzl@31862
|
2961 |
proof (induct k)
|
hoelzl@31862
|
2962 |
case (Suc k)
|
hoelzl@31862
|
2963 |
have "{ 1 ..< Suc (Suc k) } = insert (Suc k) { 1 ..< Suc k }" by auto
|
hoelzl@31862
|
2964 |
hence "\<Prod> { 1 ..< Suc (Suc k) } = (Suc k) * \<Prod> { 1 ..< Suc k }" by auto
|
hoelzl@31862
|
2965 |
thus ?case using Suc by auto
|
hoelzl@31862
|
2966 |
qed simp
|
hoelzl@31862
|
2967 |
|
hoelzl@31862
|
2968 |
lemma approx_tse:
|
hoelzl@31862
|
2969 |
assumes "bounded_by xs vs"
|
hoelzl@31862
|
2970 |
and bnd_x: "vs ! x = Some (lx, ux)" and bnd_c: "real c \<in> {real lx .. real ux}"
|
hoelzl@31862
|
2971 |
and "x < length vs" and "x < length xs"
|
hoelzl@31862
|
2972 |
and ate: "Some (l, u) = approx_tse prec x s c 1 f vs"
|
hoelzl@31862
|
2973 |
shows "interpret_floatarith f xs \<in> { real l .. real u }"
|
hoelzl@31862
|
2974 |
proof -
|
hoelzl@31862
|
2975 |
def F \<equiv> "\<lambda> n z. interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := z])"
|
hoelzl@31862
|
2976 |
hence F0: "F 0 = (\<lambda> z. interpret_floatarith f (xs[x := z]))" by auto
|
hoelzl@31862
|
2977 |
|
hoelzl@31862
|
2978 |
hence "bounded_by (xs[x := real c]) vs" and "x < length vs" "x < length xs"
|
hoelzl@31862
|
2979 |
using `bounded_by xs vs` bnd_x bnd_c `x < length vs` `x < length xs`
|
hoelzl@31862
|
2980 |
by (auto intro!: bounded_by_update_var)
|
hoelzl@31862
|
2981 |
|
hoelzl@31862
|
2982 |
from approx_tse_generic[OF `bounded_by xs vs` this bnd_x ate]
|
hoelzl@31862
|
2983 |
obtain n
|
hoelzl@31862
|
2984 |
where DERIV: "\<forall> m z. m < n \<and> real lx \<le> z \<and> z \<le> real ux \<longrightarrow> DERIV (F m) z :> F (Suc m) z"
|
hoelzl@31862
|
2985 |
and hyp: "\<And> t. t \<in> {real lx .. real ux} \<Longrightarrow>
|
hoelzl@31862
|
2986 |
(\<Sum> j = 0..<n. inverse (real (fact j)) * F j (real c) * (xs!x - real c)^j) +
|
hoelzl@31862
|
2987 |
inverse (real (fact n)) * F n t * (xs!x - real c)^n
|
hoelzl@31862
|
2988 |
\<in> {real l .. real u}" (is "\<And> t. _ \<Longrightarrow> ?taylor t \<in> _")
|
hoelzl@31862
|
2989 |
unfolding F_def atLeastAtMost_iff[symmetric] setprod_fact by blast
|
hoelzl@31862
|
2990 |
|
hoelzl@31862
|
2991 |
have bnd_xs: "xs ! x \<in> { real lx .. real ux }"
|
hoelzl@31862
|
2992 |
using `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`] bnd_x by auto
|
hoelzl@31862
|
2993 |
|
hoelzl@31862
|
2994 |
show ?thesis
|
hoelzl@31862
|
2995 |
proof (cases n)
|
hoelzl@31862
|
2996 |
case 0 thus ?thesis using hyp[OF bnd_xs] unfolding F_def by auto
|
hoelzl@31862
|
2997 |
next
|
hoelzl@31862
|
2998 |
case (Suc n')
|
hoelzl@31862
|
2999 |
show ?thesis
|
hoelzl@31862
|
3000 |
proof (cases "xs ! x = real c")
|
hoelzl@31862
|
3001 |
case True
|
hoelzl@31862
|
3002 |
from True[symmetric] hyp[OF bnd_xs] Suc show ?thesis
|
wenzelm@32962
|
3003 |
unfolding F_def Suc setsum_head_upt_Suc[OF zero_less_Suc] setsum_shift_bounds_Suc_ivl by auto
|
hoelzl@31862
|
3004 |
next
|
hoelzl@31862
|
3005 |
case False
|
hoelzl@31862
|
3006 |
|
hoelzl@31862
|
3007 |
have "real lx \<le> real c" "real c \<le> real ux" "real lx \<le> xs!x" "xs!x \<le> real ux"
|
wenzelm@32962
|
3008 |
using Suc bnd_c `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`] bnd_x by auto
|
hoelzl@31862
|
3009 |
from Taylor.taylor[OF zero_less_Suc, of F, OF F0 DERIV[unfolded Suc] this False]
|
hoelzl@31862
|
3010 |
obtain t where t_bnd: "if xs ! x < real c then xs ! x < t \<and> t < real c else real c < t \<and> t < xs ! x"
|
wenzelm@32962
|
3011 |
and fl_eq: "interpret_floatarith f (xs[x := xs ! x]) =
|
wenzelm@32962
|
3012 |
(\<Sum>m = 0..<Suc n'. F m (real c) / real (fact m) * (xs ! x - real c) ^ m) +
|
hoelzl@31862
|
3013 |
F (Suc n') t / real (fact (Suc n')) * (xs ! x - real c) ^ Suc n'"
|
wenzelm@32962
|
3014 |
by blast
|
hoelzl@31862
|
3015 |
|
hoelzl@31862
|
3016 |
from t_bnd bnd_xs bnd_c have *: "t \<in> {real lx .. real ux}"
|
wenzelm@32962
|
3017 |
by (cases "xs ! x < real c", auto)
|
hoelzl@31862
|
3018 |
|
hoelzl@31862
|
3019 |
have "interpret_floatarith f (xs[x := xs ! x]) = ?taylor t"
|
wenzelm@32962
|
3020 |
unfolding fl_eq Suc by (auto simp add: algebra_simps divide_inverse)
|
hoelzl@31862
|
3021 |
also have "\<dots> \<in> {real l .. real u}" using * by (rule hyp)
|
hoelzl@31862
|
3022 |
finally show ?thesis by simp
|
hoelzl@31862
|
3023 |
qed
|
hoelzl@31862
|
3024 |
qed
|
hoelzl@31862
|
3025 |
qed
|
hoelzl@31862
|
3026 |
|
hoelzl@31862
|
3027 |
fun approx_tse_form' where
|
hoelzl@31862
|
3028 |
"approx_tse_form' prec t f 0 l u cmp =
|
hoelzl@31862
|
3029 |
(case approx_tse prec 0 t ((l + u) * Float 1 -1) 1 f [Some (l, u)]
|
hoelzl@31862
|
3030 |
of Some (l, u) \<Rightarrow> cmp l u | None \<Rightarrow> False)" |
|
hoelzl@31862
|
3031 |
"approx_tse_form' prec t f (Suc s) l u cmp =
|
hoelzl@31862
|
3032 |
(let m = (l + u) * Float 1 -1
|
hoelzl@32919
|
3033 |
in (if approx_tse_form' prec t f s l m cmp then
|
hoelzl@32919
|
3034 |
approx_tse_form' prec t f s m u cmp else False))"
|
hoelzl@31862
|
3035 |
|
hoelzl@31862
|
3036 |
lemma approx_tse_form':
|
hoelzl@31862
|
3037 |
assumes "approx_tse_form' prec t f s l u cmp" and "x \<in> {real l .. real u}"
|
hoelzl@31862
|
3038 |
shows "\<exists> l' u' ly uy. x \<in> { real l' .. real u' } \<and> real l \<le> real l' \<and> real u' \<le> real u \<and> cmp ly uy \<and>
|
hoelzl@31862
|
3039 |
approx_tse prec 0 t ((l' + u') * Float 1 -1) 1 f [Some (l', u')] = Some (ly, uy)"
|
hoelzl@31862
|
3040 |
using assms proof (induct s arbitrary: l u)
|
hoelzl@31862
|
3041 |
case 0
|
hoelzl@31862
|
3042 |
then obtain ly uy
|
hoelzl@31862
|
3043 |
where *: "approx_tse prec 0 t ((l + u) * Float 1 -1) 1 f [Some (l, u)] = Some (ly, uy)"
|
hoelzl@31862
|
3044 |
and **: "cmp ly uy" by (auto elim!: option_caseE)
|
hoelzl@31862
|
3045 |
with 0 show ?case by (auto intro!: exI)
|
hoelzl@31862
|
3046 |
next
|
hoelzl@31862
|
3047 |
case (Suc s)
|
hoelzl@31862
|
3048 |
let ?m = "(l + u) * Float 1 -1"
|
hoelzl@31862
|
3049 |
from Suc.prems
|
hoelzl@31862
|
3050 |
have l: "approx_tse_form' prec t f s l ?m cmp"
|
hoelzl@31862
|
3051 |
and u: "approx_tse_form' prec t f s ?m u cmp"
|
hoelzl@32919
|
3052 |
by (auto simp add: Let_def lazy_conj)
|
hoelzl@31862
|
3053 |
|
hoelzl@31862
|
3054 |
have m_l: "real l \<le> real ?m" and m_u: "real ?m \<le> real u"
|
hoelzl@31862
|
3055 |
unfolding le_float_def using Suc.prems by auto
|
hoelzl@31862
|
3056 |
|
hoelzl@31862
|
3057 |
with `x \<in> { real l .. real u }`
|
hoelzl@31862
|
3058 |
have "x \<in> { real l .. real ?m} \<or> x \<in> { real ?m .. real u }" by auto
|
hoelzl@31862
|
3059 |
thus ?case
|
hoelzl@31862
|
3060 |
proof (rule disjE)
|
hoelzl@31862
|
3061 |
assume "x \<in> { real l .. real ?m}"
|
hoelzl@31862
|
3062 |
from Suc.hyps[OF l this]
|
hoelzl@31862
|
3063 |
obtain l' u' ly uy
|
hoelzl@31862
|
3064 |
where "x \<in> { real l' .. real u' } \<and> real l \<le> real l' \<and> real u' \<le> real ?m \<and> cmp ly uy \<and>
|
hoelzl@31862
|
3065 |
approx_tse prec 0 t ((l' + u') * Float 1 -1) 1 f [Some (l', u')] = Some (ly, uy)" by blast
|
hoelzl@31862
|
3066 |
with m_u show ?thesis by (auto intro!: exI)
|
hoelzl@31862
|
3067 |
next
|
hoelzl@31862
|
3068 |
assume "x \<in> { real ?m .. real u }"
|
hoelzl@31862
|
3069 |
from Suc.hyps[OF u this]
|
hoelzl@31862
|
3070 |
obtain l' u' ly uy
|
hoelzl@31862
|
3071 |
where "x \<in> { real l' .. real u' } \<and> real ?m \<le> real l' \<and> real u' \<le> real u \<and> cmp ly uy \<and>
|
hoelzl@31862
|
3072 |
approx_tse prec 0 t ((l' + u') * Float 1 -1) 1 f [Some (l', u')] = Some (ly, uy)" by blast
|
hoelzl@31862
|
3073 |
with m_u show ?thesis by (auto intro!: exI)
|
hoelzl@31862
|
3074 |
qed
|
hoelzl@31862
|
3075 |
qed
|
hoelzl@31862
|
3076 |
|
hoelzl@31862
|
3077 |
lemma approx_tse_form'_less:
|
hoelzl@31862
|
3078 |
assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 < l)"
|
hoelzl@31862
|
3079 |
and x: "x \<in> {real l .. real u}"
|
hoelzl@31862
|
3080 |
shows "interpret_floatarith b [x] < interpret_floatarith a [x]"
|
hoelzl@31862
|
3081 |
proof -
|
hoelzl@31862
|
3082 |
from approx_tse_form'[OF tse x]
|
hoelzl@31862
|
3083 |
obtain l' u' ly uy
|
hoelzl@31862
|
3084 |
where x': "x \<in> { real l' .. real u' }" and "real l \<le> real l'"
|
hoelzl@31862
|
3085 |
and "real u' \<le> real u" and "0 < ly"
|
hoelzl@31862
|
3086 |
and tse: "approx_tse prec 0 t ((l' + u') * Float 1 -1) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)"
|
hoelzl@31862
|
3087 |
by blast
|
hoelzl@31862
|
3088 |
|
hoelzl@31862
|
3089 |
hence "bounded_by [x] [Some (l', u')]" by (auto simp add: bounded_by_def)
|
hoelzl@31862
|
3090 |
|
hoelzl@31862
|
3091 |
from approx_tse[OF this _ _ _ _ tse[symmetric], of l' u'] x'
|
hoelzl@31862
|
3092 |
have "real ly \<le> interpret_floatarith a [x] - interpret_floatarith b [x]"
|
hoelzl@31862
|
3093 |
by (auto simp add: diff_minus)
|
hoelzl@31862
|
3094 |
from order_less_le_trans[OF `0 < ly`[unfolded less_float_def] this]
|
hoelzl@31862
|
3095 |
show ?thesis by auto
|
hoelzl@31862
|
3096 |
qed
|
hoelzl@31862
|
3097 |
|
hoelzl@31862
|
3098 |
lemma approx_tse_form'_le:
|
hoelzl@31862
|
3099 |
assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 \<le> l)"
|
hoelzl@31862
|
3100 |
and x: "x \<in> {real l .. real u}"
|
hoelzl@31862
|
3101 |
shows "interpret_floatarith b [x] \<le> interpret_floatarith a [x]"
|
hoelzl@31862
|
3102 |
proof -
|
hoelzl@31862
|
3103 |
from approx_tse_form'[OF tse x]
|
hoelzl@31862
|
3104 |
obtain l' u' ly uy
|
hoelzl@31862
|
3105 |
where x': "x \<in> { real l' .. real u' }" and "real l \<le> real l'"
|
hoelzl@31862
|
3106 |
and "real u' \<le> real u" and "0 \<le> ly"
|
hoelzl@31862
|
3107 |
and tse: "approx_tse prec 0 t ((l' + u') * Float 1 -1) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)"
|
hoelzl@31862
|
3108 |
by blast
|
hoelzl@31862
|
3109 |
|
hoelzl@31862
|
3110 |
hence "bounded_by [x] [Some (l', u')]" by (auto simp add: bounded_by_def)
|
hoelzl@31862
|
3111 |
|
hoelzl@31862
|
3112 |
from approx_tse[OF this _ _ _ _ tse[symmetric], of l' u'] x'
|
hoelzl@31862
|
3113 |
have "real ly \<le> interpret_floatarith a [x] - interpret_floatarith b [x]"
|
hoelzl@31862
|
3114 |
by (auto simp add: diff_minus)
|
hoelzl@31862
|
3115 |
from order_trans[OF `0 \<le> ly`[unfolded le_float_def] this]
|
hoelzl@31862
|
3116 |
show ?thesis by auto
|
hoelzl@31862
|
3117 |
qed
|
hoelzl@31862
|
3118 |
|
hoelzl@31862
|
3119 |
definition
|
hoelzl@31862
|
3120 |
"approx_tse_form prec t s f =
|
hoelzl@31862
|
3121 |
(case f
|
hoelzl@32919
|
3122 |
of (Bound x a b f) \<Rightarrow> x = Var 0 \<and>
|
hoelzl@31862
|
3123 |
(case (approx prec a [None], approx prec b [None])
|
hoelzl@31862
|
3124 |
of (Some (l, u), Some (l', u')) \<Rightarrow>
|
hoelzl@31862
|
3125 |
(case f
|
hoelzl@31862
|
3126 |
of Less lf rt \<Rightarrow> approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 < l)
|
hoelzl@31862
|
3127 |
| LessEqual lf rt \<Rightarrow> approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)
|
hoelzl@31862
|
3128 |
| AtLeastAtMost x lf rt \<Rightarrow>
|
hoelzl@32919
|
3129 |
(if approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l) then
|
hoelzl@32919
|
3130 |
approx_tse_form' prec t (Add rt (Minus x)) s l u' (\<lambda> l u. 0 \<le> l) else False)
|
hoelzl@31862
|
3131 |
| _ \<Rightarrow> False)
|
hoelzl@31862
|
3132 |
| _ \<Rightarrow> False)
|
hoelzl@31862
|
3133 |
| _ \<Rightarrow> False)"
|
hoelzl@31862
|
3134 |
|
hoelzl@31862
|
3135 |
lemma approx_tse_form:
|
hoelzl@31862
|
3136 |
assumes "approx_tse_form prec t s f"
|
hoelzl@31862
|
3137 |
shows "interpret_form f [x]"
|
hoelzl@31862
|
3138 |
proof (cases f)
|
hoelzl@31862
|
3139 |
case (Bound i a b f') note f_def = this
|
hoelzl@31862
|
3140 |
with assms obtain l u l' u'
|
hoelzl@31862
|
3141 |
where a: "approx prec a [None] = Some (l, u)"
|
hoelzl@31862
|
3142 |
and b: "approx prec b [None] = Some (l', u')"
|
hoelzl@31862
|
3143 |
unfolding approx_tse_form_def by (auto elim!: option_caseE)
|
hoelzl@31862
|
3144 |
|
hoelzl@32919
|
3145 |
from Bound assms have "i = Var 0" unfolding approx_tse_form_def by auto
|
hoelzl@31862
|
3146 |
hence i: "interpret_floatarith i [x] = x" by auto
|
hoelzl@31862
|
3147 |
|
hoelzl@31862
|
3148 |
{ let "?f z" = "interpret_floatarith z [x]"
|
hoelzl@31862
|
3149 |
assume "?f i \<in> { ?f a .. ?f b }"
|
hoelzl@31862
|
3150 |
with approx[OF _ a[symmetric], of "[x]"] approx[OF _ b[symmetric], of "[x]"]
|
hoelzl@31862
|
3151 |
have bnd: "x \<in> { real l .. real u'}" unfolding bounded_by_def i by auto
|
hoelzl@31862
|
3152 |
|
hoelzl@31862
|
3153 |
have "interpret_form f' [x]"
|
hoelzl@31862
|
3154 |
proof (cases f')
|
hoelzl@31862
|
3155 |
case (Less lf rt)
|
hoelzl@31862
|
3156 |
with Bound a b assms
|
hoelzl@31862
|
3157 |
have "approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 < l)"
|
wenzelm@32962
|
3158 |
unfolding approx_tse_form_def by auto
|
hoelzl@31862
|
3159 |
from approx_tse_form'_less[OF this bnd]
|
hoelzl@31862
|
3160 |
show ?thesis using Less by auto
|
hoelzl@31862
|
3161 |
next
|
hoelzl@31862
|
3162 |
case (LessEqual lf rt)
|
hoelzl@31862
|
3163 |
with Bound a b assms
|
hoelzl@31862
|
3164 |
have "approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)"
|
wenzelm@32962
|
3165 |
unfolding approx_tse_form_def by auto
|
hoelzl@31862
|
3166 |
from approx_tse_form'_le[OF this bnd]
|
hoelzl@31862
|
3167 |
show ?thesis using LessEqual by auto
|
hoelzl@31862
|
3168 |
next
|
hoelzl@31862
|
3169 |
case (AtLeastAtMost x lf rt)
|
hoelzl@31862
|
3170 |
with Bound a b assms
|
hoelzl@31862
|
3171 |
have "approx_tse_form' prec t (Add rt (Minus x)) s l u' (\<lambda> l u. 0 \<le> l)"
|
wenzelm@32962
|
3172 |
and "approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)"
|
wenzelm@32962
|
3173 |
unfolding approx_tse_form_def lazy_conj by auto
|
hoelzl@31862
|
3174 |
from approx_tse_form'_le[OF this(1) bnd] approx_tse_form'_le[OF this(2) bnd]
|
hoelzl@31862
|
3175 |
show ?thesis using AtLeastAtMost by auto
|
hoelzl@31862
|
3176 |
next
|
hoelzl@31862
|
3177 |
case (Bound x a b f') with assms
|
hoelzl@31862
|
3178 |
show ?thesis by (auto elim!: option_caseE simp add: f_def approx_tse_form_def)
|
hoelzl@31862
|
3179 |
next
|
hoelzl@31862
|
3180 |
case (Assign x a f') with assms
|
hoelzl@31862
|
3181 |
show ?thesis by (auto elim!: option_caseE simp add: f_def approx_tse_form_def)
|
hoelzl@31862
|
3182 |
qed } thus ?thesis unfolding f_def by auto
|
hoelzl@31862
|
3183 |
next case Assign with assms show ?thesis by (auto simp add: approx_tse_form_def)
|
hoelzl@31862
|
3184 |
next case LessEqual with assms show ?thesis by (auto simp add: approx_tse_form_def)
|
hoelzl@31862
|
3185 |
next case Less with assms show ?thesis by (auto simp add: approx_tse_form_def)
|
hoelzl@31862
|
3186 |
next case AtLeastAtMost with assms show ?thesis by (auto simp add: approx_tse_form_def)
|
hoelzl@31862
|
3187 |
qed
|
hoelzl@31862
|
3188 |
|
hoelzl@32919
|
3189 |
text {* @{term approx_form_eval} is only used for the {\tt value}-command. *}
|
hoelzl@32919
|
3190 |
|
hoelzl@32919
|
3191 |
fun approx_form_eval :: "nat \<Rightarrow> form \<Rightarrow> (float * float) option list \<Rightarrow> (float * float) option list" where
|
hoelzl@32919
|
3192 |
"approx_form_eval prec (Bound (Var n) a b f) bs =
|
hoelzl@32919
|
3193 |
(case (approx prec a bs, approx prec b bs)
|
hoelzl@32919
|
3194 |
of (Some (l, _), Some (_, u)) \<Rightarrow> approx_form_eval prec f (bs[n := Some (l, u)])
|
hoelzl@32919
|
3195 |
| _ \<Rightarrow> bs)" |
|
hoelzl@32919
|
3196 |
"approx_form_eval prec (Assign (Var n) a f) bs =
|
hoelzl@32919
|
3197 |
(case (approx prec a bs)
|
hoelzl@32919
|
3198 |
of (Some (l, u)) \<Rightarrow> approx_form_eval prec f (bs[n := Some (l, u)])
|
hoelzl@32919
|
3199 |
| _ \<Rightarrow> bs)" |
|
hoelzl@32919
|
3200 |
"approx_form_eval prec (Less a b) bs = bs @ [approx prec a bs, approx prec b bs]" |
|
hoelzl@32919
|
3201 |
"approx_form_eval prec (LessEqual a b) bs = bs @ [approx prec a bs, approx prec b bs]" |
|
hoelzl@32919
|
3202 |
"approx_form_eval prec (AtLeastAtMost x a b) bs =
|
hoelzl@32919
|
3203 |
bs @ [approx prec x bs, approx prec a bs, approx prec b bs]" |
|
hoelzl@32919
|
3204 |
"approx_form_eval _ _ bs = bs"
|
hoelzl@32919
|
3205 |
|
hoelzl@29742
|
3206 |
subsection {* Implement proof method \texttt{approximation} *}
|
hoelzl@29742
|
3207 |
|
hoelzl@31811
|
3208 |
lemmas interpret_form_equations = interpret_form.simps interpret_floatarith.simps interpret_floatarith_num
|
hoelzl@31098
|
3209 |
interpret_floatarith_divide interpret_floatarith_diff interpret_floatarith_tan interpret_floatarith_powr interpret_floatarith_log
|
hoelzl@31467
|
3210 |
interpret_floatarith_sin
|
hoelzl@29742
|
3211 |
|
haftmann@36531
|
3212 |
code_reflect Float_Arith
|
haftmann@36526
|
3213 |
datatypes float = Float
|
haftmann@36526
|
3214 |
and floatarith = Add | Minus | Mult | Inverse | Cos | Arctan
|
haftmann@36526
|
3215 |
| Abs | Max | Min | Pi | Sqrt | Exp | Ln | Power | Var | Num
|
haftmann@36526
|
3216 |
and form = Bound | Assign | Less | LessEqual | AtLeastAtMost
|
haftmann@36526
|
3217 |
functions approx_form approx_tse_form approx' approx_form_eval
|
hoelzl@31099
|
3218 |
|
hoelzl@31099
|
3219 |
ML {*
|
wenzelm@32212
|
3220 |
fun reorder_bounds_tac prems i =
|
hoelzl@29742
|
3221 |
let
|
hoelzl@31811
|
3222 |
fun variable_of_bound (Const ("Trueprop", _) $
|
hoelzl@31811
|
3223 |
(Const (@{const_name "op :"}, _) $
|
hoelzl@31811
|
3224 |
Free (name, _) $ _)) = name
|
hoelzl@31811
|
3225 |
| variable_of_bound (Const ("Trueprop", _) $
|
hoelzl@31811
|
3226 |
(Const ("op =", _) $
|
hoelzl@31811
|
3227 |
Free (name, _) $ _)) = name
|
hoelzl@31811
|
3228 |
| variable_of_bound t = raise TERM ("variable_of_bound", [t])
|
hoelzl@31811
|
3229 |
|
hoelzl@31811
|
3230 |
val variable_bounds
|
hoelzl@31811
|
3231 |
= map (` (variable_of_bound o prop_of)) prems
|
hoelzl@31811
|
3232 |
|
hoelzl@31811
|
3233 |
fun add_deps (name, bnds)
|
hoelzl@32650
|
3234 |
= Graph.add_deps_acyclic (name,
|
hoelzl@32650
|
3235 |
remove (op =) name (Term.add_free_names (prop_of bnds) []))
|
hoelzl@32650
|
3236 |
|
hoelzl@31811
|
3237 |
val order = Graph.empty
|
hoelzl@31811
|
3238 |
|> fold Graph.new_node variable_bounds
|
hoelzl@31811
|
3239 |
|> fold add_deps variable_bounds
|
hoelzl@32650
|
3240 |
|> Graph.strong_conn |> map the_single |> rev
|
hoelzl@31811
|
3241 |
|> map_filter (AList.lookup (op =) variable_bounds)
|
hoelzl@31811
|
3242 |
|
hoelzl@31811
|
3243 |
fun prepend_prem th tac
|
hoelzl@31811
|
3244 |
= tac THEN rtac (th RSN (2, @{thm mp})) i
|
hoelzl@31811
|
3245 |
in
|
hoelzl@31811
|
3246 |
fold prepend_prem order all_tac
|
hoelzl@31811
|
3247 |
end
|
hoelzl@31811
|
3248 |
|
hoelzl@31811
|
3249 |
(* Should be in HOL.thy ? *)
|
hoelzl@29742
|
3250 |
fun gen_eval_tac conv ctxt = CONVERSION (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt)
|
hoelzl@29742
|
3251 |
THEN' rtac TrueI
|
hoelzl@29742
|
3252 |
|
hoelzl@31811
|
3253 |
val form_equations = PureThy.get_thms @{theory} "interpret_form_equations";
|
hoelzl@31811
|
3254 |
|
hoelzl@31862
|
3255 |
fun rewrite_interpret_form_tac ctxt prec splitting taylor i st = let
|
hoelzl@31862
|
3256 |
fun lookup_splitting (Free (name, typ))
|
hoelzl@31862
|
3257 |
= case AList.lookup (op =) splitting name
|
hoelzl@31862
|
3258 |
of SOME s => HOLogic.mk_number @{typ nat} s
|
hoelzl@31862
|
3259 |
| NONE => @{term "0 :: nat"}
|
hoelzl@31811
|
3260 |
val vs = nth (prems_of st) (i - 1)
|
hoelzl@31811
|
3261 |
|> Logic.strip_imp_concl
|
hoelzl@31811
|
3262 |
|> HOLogic.dest_Trueprop
|
hoelzl@31811
|
3263 |
|> Term.strip_comb |> snd |> List.last
|
hoelzl@31811
|
3264 |
|> HOLogic.dest_list
|
hoelzl@31811
|
3265 |
val p = prec
|
hoelzl@31811
|
3266 |
|> HOLogic.mk_number @{typ nat}
|
hoelzl@31811
|
3267 |
|> Thm.cterm_of (ProofContext.theory_of ctxt)
|
hoelzl@31862
|
3268 |
in case taylor
|
hoelzl@31862
|
3269 |
of NONE => let
|
hoelzl@31862
|
3270 |
val n = vs |> length
|
hoelzl@31862
|
3271 |
|> HOLogic.mk_number @{typ nat}
|
hoelzl@31862
|
3272 |
|> Thm.cterm_of (ProofContext.theory_of ctxt)
|
hoelzl@31862
|
3273 |
val s = vs
|
hoelzl@31862
|
3274 |
|> map lookup_splitting
|
hoelzl@31862
|
3275 |
|> HOLogic.mk_list @{typ nat}
|
hoelzl@31862
|
3276 |
|> Thm.cterm_of (ProofContext.theory_of ctxt)
|
hoelzl@31862
|
3277 |
in
|
hoelzl@31862
|
3278 |
(rtac (Thm.instantiate ([], [(@{cpat "?n::nat"}, n),
|
hoelzl@31862
|
3279 |
(@{cpat "?prec::nat"}, p),
|
hoelzl@31862
|
3280 |
(@{cpat "?ss::nat list"}, s)])
|
hoelzl@31862
|
3281 |
@{thm "approx_form"}) i
|
hoelzl@31862
|
3282 |
THEN simp_tac @{simpset} i) st
|
hoelzl@31862
|
3283 |
end
|
hoelzl@31862
|
3284 |
|
hoelzl@31862
|
3285 |
| SOME t => if length vs <> 1 then raise (TERM ("More than one variable used for taylor series expansion", [prop_of st]))
|
hoelzl@31862
|
3286 |
else let
|
hoelzl@31862
|
3287 |
val t = t
|
hoelzl@31862
|
3288 |
|> HOLogic.mk_number @{typ nat}
|
hoelzl@31811
|
3289 |
|> Thm.cterm_of (ProofContext.theory_of ctxt)
|
hoelzl@31862
|
3290 |
val s = vs |> map lookup_splitting |> hd
|
hoelzl@31862
|
3291 |
|> Thm.cterm_of (ProofContext.theory_of ctxt)
|
hoelzl@31862
|
3292 |
in
|
hoelzl@31862
|
3293 |
rtac (Thm.instantiate ([], [(@{cpat "?s::nat"}, s),
|
hoelzl@31862
|
3294 |
(@{cpat "?t::nat"}, t),
|
hoelzl@31862
|
3295 |
(@{cpat "?prec::nat"}, p)])
|
hoelzl@31862
|
3296 |
@{thm "approx_tse_form"}) i st
|
hoelzl@31862
|
3297 |
end
|
hoelzl@31811
|
3298 |
end
|
hoelzl@31811
|
3299 |
|
hoelzl@31811
|
3300 |
(* copied from Tools/induct.ML should probably in args.ML *)
|
hoelzl@31811
|
3301 |
val free = Args.context -- Args.term >> (fn (_, Free (n, t)) => n | (ctxt, t) =>
|
hoelzl@31811
|
3302 |
error ("Bad free variable: " ^ Syntax.string_of_term ctxt t));
|
hoelzl@31811
|
3303 |
|
hoelzl@29742
|
3304 |
*}
|
hoelzl@29742
|
3305 |
|
hoelzl@31811
|
3306 |
lemma intervalE: "a \<le> x \<and> x \<le> b \<Longrightarrow> \<lbrakk> x \<in> { a .. b } \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
|
hoelzl@31811
|
3307 |
by auto
|
hoelzl@31811
|
3308 |
|
hoelzl@31811
|
3309 |
lemma meta_eqE: "x \<equiv> a \<Longrightarrow> \<lbrakk> x = a \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
|
hoelzl@31811
|
3310 |
by auto
|
hoelzl@31811
|
3311 |
|
wenzelm@30549
|
3312 |
method_setup approximation = {*
|
wenzelm@36970
|
3313 |
Scan.lift Parse.nat
|
hoelzl@31862
|
3314 |
--
|
hoelzl@31811
|
3315 |
Scan.optional (Scan.lift (Args.$$$ "splitting" |-- Args.colon)
|
wenzelm@36970
|
3316 |
|-- Parse.and_list' (free --| Scan.lift (Args.$$$ "=") -- Scan.lift Parse.nat)) []
|
hoelzl@31862
|
3317 |
--
|
hoelzl@31862
|
3318 |
Scan.option (Scan.lift (Args.$$$ "taylor" |-- Args.colon)
|
wenzelm@36970
|
3319 |
|-- (free |-- Scan.lift (Args.$$$ "=") |-- Scan.lift Parse.nat))
|
hoelzl@31811
|
3320 |
>>
|
hoelzl@31862
|
3321 |
(fn ((prec, splitting), taylor) => fn ctxt =>
|
wenzelm@30549
|
3322 |
SIMPLE_METHOD' (fn i =>
|
hoelzl@31811
|
3323 |
REPEAT (FIRST' [etac @{thm intervalE},
|
hoelzl@31811
|
3324 |
etac @{thm meta_eqE},
|
hoelzl@31811
|
3325 |
rtac @{thm impI}] i)
|
wenzelm@32286
|
3326 |
THEN Subgoal.FOCUS (fn {prems, ...} => reorder_bounds_tac prems i) @{context} i
|
hoelzl@32650
|
3327 |
THEN DETERM (TRY (filter_prems_tac (K false) i))
|
hoelzl@31811
|
3328 |
THEN DETERM (Reflection.genreify_tac ctxt form_equations NONE i)
|
hoelzl@31862
|
3329 |
THEN rewrite_interpret_form_tac ctxt prec splitting taylor i
|
hoelzl@31811
|
3330 |
THEN gen_eval_tac eval_oracle ctxt i))
|
hoelzl@31811
|
3331 |
*} "real number approximation"
|
hoelzl@31811
|
3332 |
|
hoelzl@31810
|
3333 |
ML {*
|
hoelzl@32919
|
3334 |
fun calculated_subterms (@{const Trueprop} $ t) = calculated_subterms t
|
hoelzl@32919
|
3335 |
| calculated_subterms (@{const "op -->"} $ _ $ t) = calculated_subterms t
|
hoelzl@32919
|
3336 |
| calculated_subterms (@{term "op <= :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = [t1, t2]
|
hoelzl@32919
|
3337 |
| calculated_subterms (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = [t1, t2]
|
hoelzl@32919
|
3338 |
| calculated_subterms (@{term "op : :: real \<Rightarrow> real set \<Rightarrow> bool"} $ t1 $
|
hoelzl@32919
|
3339 |
(@{term "atLeastAtMost :: real \<Rightarrow> real \<Rightarrow> real set"} $ t2 $ t3)) = [t1, t2, t3]
|
hoelzl@32919
|
3340 |
| calculated_subterms t = raise TERM ("calculated_subterms", [t])
|
hoelzl@32919
|
3341 |
|
hoelzl@32919
|
3342 |
fun dest_interpret_form (@{const "interpret_form"} $ b $ xs) = (b, xs)
|
hoelzl@32919
|
3343 |
| dest_interpret_form t = raise TERM ("dest_interpret_form", [t])
|
hoelzl@32919
|
3344 |
|
hoelzl@31810
|
3345 |
fun dest_interpret (@{const "interpret_floatarith"} $ b $ xs) = (b, xs)
|
hoelzl@32919
|
3346 |
| dest_interpret t = raise TERM ("dest_interpret", [t])
|
hoelzl@32919
|
3347 |
|
hoelzl@32919
|
3348 |
|
hoelzl@32919
|
3349 |
fun dest_float (@{const "Float"} $ m $ e) = (snd (HOLogic.dest_number m), snd (HOLogic.dest_number e))
|
hoelzl@32919
|
3350 |
fun dest_ivl (Const (@{const_name "Some"}, _) $
|
hoelzl@32919
|
3351 |
(Const (@{const_name "Pair"}, _) $ u $ l)) = SOME (dest_float u, dest_float l)
|
hoelzl@32919
|
3352 |
| dest_ivl (Const (@{const_name "None"}, _)) = NONE
|
hoelzl@32919
|
3353 |
| dest_ivl t = raise TERM ("dest_result", [t])
|
hoelzl@31810
|
3354 |
|
hoelzl@31810
|
3355 |
fun mk_approx' prec t = (@{const "approx'"}
|
hoelzl@31810
|
3356 |
$ HOLogic.mk_number @{typ nat} prec
|
hoelzl@32650
|
3357 |
$ t $ @{term "[] :: (float * float) option list"})
|
hoelzl@31810
|
3358 |
|
hoelzl@32919
|
3359 |
fun mk_approx_form_eval prec t xs = (@{const "approx_form_eval"}
|
hoelzl@32919
|
3360 |
$ HOLogic.mk_number @{typ nat} prec
|
hoelzl@32919
|
3361 |
$ t $ xs)
|
hoelzl@31810
|
3362 |
|
hoelzl@31810
|
3363 |
fun float2_float10 prec round_down (m, e) = (
|
hoelzl@31810
|
3364 |
let
|
hoelzl@31810
|
3365 |
val (m, e) = (if e < 0 then (m,e) else (m * Integer.pow e 2, 0))
|
hoelzl@31810
|
3366 |
|
hoelzl@31810
|
3367 |
fun frac c p 0 digits cnt = (digits, cnt, 0)
|
hoelzl@31810
|
3368 |
| frac c 0 r digits cnt = (digits, cnt, r)
|
hoelzl@31810
|
3369 |
| frac c p r digits cnt = (let
|
hoelzl@31810
|
3370 |
val (d, r) = Integer.div_mod (r * 10) (Integer.pow (~e) 2)
|
hoelzl@31810
|
3371 |
in frac (c orelse d <> 0) (if d <> 0 orelse c then p - 1 else p) r
|
hoelzl@31810
|
3372 |
(digits * 10 + d) (cnt + 1)
|
hoelzl@31810
|
3373 |
end)
|
hoelzl@31810
|
3374 |
|
hoelzl@31810
|
3375 |
val sgn = Int.sign m
|
hoelzl@31810
|
3376 |
val m = abs m
|
hoelzl@31810
|
3377 |
|
hoelzl@31810
|
3378 |
val round_down = (sgn = 1 andalso round_down) orelse
|
hoelzl@31810
|
3379 |
(sgn = ~1 andalso not round_down)
|
hoelzl@31810
|
3380 |
|
hoelzl@31810
|
3381 |
val (x, r) = Integer.div_mod m (Integer.pow (~e) 2)
|
hoelzl@31810
|
3382 |
|
hoelzl@31810
|
3383 |
val p = ((if x = 0 then prec else prec - (IntInf.log2 x + 1)) * 3) div 10 + 1
|
hoelzl@31810
|
3384 |
|
hoelzl@31810
|
3385 |
val (digits, e10, r) = if p > 0 then frac (x <> 0) p r 0 0 else (0,0,0)
|
hoelzl@31810
|
3386 |
|
hoelzl@31810
|
3387 |
val digits = if round_down orelse r = 0 then digits else digits + 1
|
hoelzl@31810
|
3388 |
|
hoelzl@31810
|
3389 |
in (sgn * (digits + x * (Integer.pow e10 10)), ~e10)
|
hoelzl@31810
|
3390 |
end)
|
hoelzl@31810
|
3391 |
|
hoelzl@31810
|
3392 |
fun mk_result prec (SOME (l, u)) = (let
|
hoelzl@31810
|
3393 |
fun mk_float10 rnd x = (let val (m, e) = float2_float10 prec rnd x
|
hoelzl@31810
|
3394 |
in if e = 0 then HOLogic.mk_number @{typ real} m
|
hoelzl@31810
|
3395 |
else if e = 1 then @{term "divide :: real \<Rightarrow> real \<Rightarrow> real"} $
|
hoelzl@31810
|
3396 |
HOLogic.mk_number @{typ real} m $
|
hoelzl@31810
|
3397 |
@{term "10"}
|
hoelzl@31810
|
3398 |
else @{term "divide :: real \<Rightarrow> real \<Rightarrow> real"} $
|
hoelzl@31810
|
3399 |
HOLogic.mk_number @{typ real} m $
|
hoelzl@31810
|
3400 |
(@{term "power 10 :: nat \<Rightarrow> real"} $
|
hoelzl@31810
|
3401 |
HOLogic.mk_number @{typ nat} (~e)) end)
|
hoelzl@32919
|
3402 |
in @{term "atLeastAtMost :: real \<Rightarrow> real \<Rightarrow> real set"} $ mk_float10 true l $ mk_float10 false u end)
|
hoelzl@31810
|
3403 |
| mk_result prec NONE = @{term "UNIV :: real set"}
|
hoelzl@31810
|
3404 |
|
hoelzl@31810
|
3405 |
fun realify t = let
|
wenzelm@35845
|
3406 |
val t = Logic.varify_global t
|
hoelzl@31810
|
3407 |
val m = map (fn (name, sort) => (name, @{typ real})) (Term.add_tvars t [])
|
hoelzl@31810
|
3408 |
val t = Term.subst_TVars m t
|
hoelzl@31810
|
3409 |
in t end
|
hoelzl@31810
|
3410 |
|
hoelzl@32919
|
3411 |
fun converted_result t =
|
hoelzl@32919
|
3412 |
prop_of t
|
hoelzl@32919
|
3413 |
|> HOLogic.dest_Trueprop
|
hoelzl@32919
|
3414 |
|> HOLogic.dest_eq |> snd
|
hoelzl@32919
|
3415 |
|
hoelzl@32919
|
3416 |
fun apply_tactic context term tactic = cterm_of context term
|
hoelzl@32919
|
3417 |
|> Goal.init
|
hoelzl@32919
|
3418 |
|> SINGLE tactic
|
hoelzl@32919
|
3419 |
|> the |> prems_of |> hd
|
hoelzl@32919
|
3420 |
|
hoelzl@32919
|
3421 |
fun prepare_form context term = apply_tactic context term (
|
hoelzl@32919
|
3422 |
REPEAT (FIRST' [etac @{thm intervalE}, etac @{thm meta_eqE}, rtac @{thm impI}] 1)
|
hoelzl@32919
|
3423 |
THEN Subgoal.FOCUS (fn {prems, ...} => reorder_bounds_tac prems 1) @{context} 1
|
hoelzl@32919
|
3424 |
THEN DETERM (TRY (filter_prems_tac (K false) 1)))
|
hoelzl@32919
|
3425 |
|
hoelzl@32919
|
3426 |
fun reify_form context term = apply_tactic context term
|
hoelzl@32919
|
3427 |
(Reflection.genreify_tac @{context} form_equations NONE 1)
|
hoelzl@32919
|
3428 |
|
hoelzl@32919
|
3429 |
fun approx_form prec ctxt t =
|
hoelzl@32919
|
3430 |
realify t
|
wenzelm@33030
|
3431 |
|> prepare_form (ProofContext.theory_of ctxt)
|
hoelzl@32919
|
3432 |
|> (fn arith_term =>
|
wenzelm@33030
|
3433 |
reify_form (ProofContext.theory_of ctxt) arith_term
|
hoelzl@32919
|
3434 |
|> HOLogic.dest_Trueprop |> dest_interpret_form
|
hoelzl@32919
|
3435 |
|> (fn (data, xs) =>
|
hoelzl@32919
|
3436 |
mk_approx_form_eval prec data (HOLogic.mk_list @{typ "(float * float) option"}
|
hoelzl@32919
|
3437 |
(map (fn _ => @{term "None :: (float * float) option"}) (HOLogic.dest_list xs)))
|
hoelzl@32919
|
3438 |
|> Codegen.eval_term @{theory}
|
hoelzl@32919
|
3439 |
|> HOLogic.dest_list
|
hoelzl@32919
|
3440 |
|> curry ListPair.zip (HOLogic.dest_list xs @ calculated_subterms arith_term)
|
hoelzl@32919
|
3441 |
|> map (fn (elem, s) => @{term "op : :: real \<Rightarrow> real set \<Rightarrow> bool"} $ elem $ mk_result prec (dest_ivl s))
|
hoelzl@32920
|
3442 |
|> foldr1 HOLogic.mk_conj))
|
hoelzl@32919
|
3443 |
|
hoelzl@32919
|
3444 |
fun approx_arith prec ctxt t = realify t
|
hoelzl@31811
|
3445 |
|> Reflection.genreif ctxt form_equations
|
hoelzl@31810
|
3446 |
|> prop_of
|
hoelzl@31810
|
3447 |
|> HOLogic.dest_Trueprop
|
hoelzl@31810
|
3448 |
|> HOLogic.dest_eq |> snd
|
hoelzl@31810
|
3449 |
|> dest_interpret |> fst
|
hoelzl@31810
|
3450 |
|> mk_approx' prec
|
hoelzl@31810
|
3451 |
|> Codegen.eval_term @{theory}
|
hoelzl@32919
|
3452 |
|> dest_ivl
|
hoelzl@31810
|
3453 |
|> mk_result prec
|
hoelzl@32919
|
3454 |
|
hoelzl@32919
|
3455 |
fun approx prec ctxt t = if type_of t = @{typ prop} then approx_form prec ctxt t
|
hoelzl@32919
|
3456 |
else if type_of t = @{typ bool} then approx_form prec ctxt (@{const Trueprop} $ t)
|
hoelzl@32919
|
3457 |
else approx_arith prec ctxt t
|
hoelzl@31810
|
3458 |
*}
|
hoelzl@31810
|
3459 |
|
hoelzl@31810
|
3460 |
setup {*
|
hoelzl@31810
|
3461 |
Value.add_evaluator ("approximate", approx 30)
|
hoelzl@31810
|
3462 |
*}
|
hoelzl@31810
|
3463 |
|
hoelzl@29742
|
3464 |
end
|