--- a/src/HOL/Decision_Procs/Approximation.thy Tue Nov 05 13:23:27 2013 +0100
+++ b/src/HOL/Decision_Procs/Approximation.thy Tue Nov 05 15:10:59 2013 +0100
@@ -132,14 +132,7 @@
lemma get_odd[simp]: "odd (get_odd n)" unfolding get_odd_def by (cases "odd n", auto)
lemma get_even[simp]: "even (get_even n)" unfolding get_even_def by (cases "even n", auto)
lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)"
-proof (cases "odd n")
- case True hence "0 < n" by (rule odd_pos)
- from gr0_implies_Suc[OF this] obtain k where "Suc k = n" by auto
- thus ?thesis unfolding get_odd_def if_P[OF True] using True[unfolded `Suc k = n`[symmetric]] by blast
-next
- case False hence "odd (Suc n)" by auto
- thus ?thesis unfolding get_odd_def if_not_P[OF False] by blast
-qed
+ by (auto simp: get_odd_def odd_pos intro!: exI[of _ "n - 1"])
lemma get_even_double: "\<exists>i. get_even n = 2 * i" using get_even[unfolded even_mult_two_ex] .
lemma get_odd_double: "\<exists>i. get_odd n = 2 * i + 1" using get_odd[unfolded odd_Suc_mult_two_ex] by auto
@@ -151,47 +144,9 @@
else if u < 0 then (u ^ n, l ^ n)
else (0, (max (-l) u) ^ n))"
-lemma float_power_bnds: fixes x :: real
- assumes "(l1, u1) = float_power_bnds n l u" and "x \<in> {l .. u}"
- shows "x ^ n \<in> {l1..u1}"
-proof (cases "even n")
- case True
- show ?thesis
- proof (cases "0 < l")
- case True hence "odd n \<or> 0 < l" and "0 \<le> real l" by auto
- 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
- have "real l ^ n \<le> x ^ n" and "x ^ n \<le> real u ^ n " using `0 \<le> real l` assms
- by (auto simp: power_mono)
- thus ?thesis using assms `0 < l` unfolding l1 u1 by auto
- next
- case False hence P: "\<not> (odd n \<or> 0 < l)" using `even n` by auto
- show ?thesis
- proof (cases "u < 0")
- case True hence "0 \<le> - real u" and "- real u \<le> - x" and "0 \<le> - x" and "-x \<le> - real l" using assms by auto
- 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]
- unfolding power_minus_even[OF `even n`] by auto
- 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
- ultimately show ?thesis by auto
- next
- case False
- have "\<bar>x\<bar> \<le> real (max (-l) u)"
- proof (cases "-l \<le> u")
- case True thus ?thesis unfolding max_def if_P[OF True] using assms by auto
- next
- case False thus ?thesis unfolding max_def if_not_P[OF False] using assms by auto
- qed
- hence x_abs: "\<bar>x\<bar> \<le> \<bar>real (max (-l) u)\<bar>" by auto
- 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
- show ?thesis unfolding atLeastAtMost_iff l1 u1 using zero_le_even_power[OF `even n`] power_mono_even[OF `even n` x_abs] by auto
- qed
- qed
-next
- case False hence "odd n \<or> 0 < l" by auto
- 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
- 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
- thus ?thesis unfolding atLeastAtMost_iff l1 u1 less_float_def by auto
-qed
+lemma float_power_bnds: "(l1, u1) = float_power_bnds n l u \<Longrightarrow> x \<in> {l .. u} \<Longrightarrow> (x::real) ^ n \<in> {l1..u1}"
+ by (auto simp: float_power_bnds_def max_def split: split_if_asm
+ intro: power_mono_odd power_mono power_mono_even zero_le_even_power)
lemma bnds_power: "\<forall> (x::real) l u. (l1, u1) = float_power_bnds n l u \<and> x \<in> {l .. u} \<longrightarrow> l1 \<le> x ^ n \<and> x ^ n \<le> u1"
using float_power_bnds by auto
@@ -244,7 +199,7 @@
qed
lemma sqrt_iteration_bound: assumes "0 < real x"
- shows "sqrt x < (sqrt_iteration prec n x)"
+ shows "sqrt x < sqrt_iteration prec n x"
proof (induct n)
case 0
show ?case
@@ -356,20 +311,8 @@
note ub = this
show ?thesis
- proof (cases "0 < x")
- case True with lb ub show ?thesis by auto
- next case False show ?thesis
- proof (cases "real x = 0")
- case True thus ?thesis
- by (auto simp add: lb_sqrt.simps ub_sqrt.simps)
- next
- case False with `\<not> 0 < x` have "x < 0" and "0 < -x"
- by auto
-
- with `\<not> 0 < x`
- show ?thesis using lb[OF `0 < -x`] ub[OF `0 < -x`]
- by (auto simp add: real_sqrt_minus lb_sqrt.simps ub_sqrt.simps)
- qed qed
+ using lb[of "-x"] ub[of "-x"] lb[of x] ub[of x]
+ by (auto simp add: lb_sqrt.simps ub_sqrt.simps real_sqrt_minus)
qed
lemma bnds_sqrt: "\<forall> (x::real) lx ux. (l, u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> sqrt x \<and> sqrt x \<le> u"
@@ -412,8 +355,8 @@
assumes "0 \<le> real x" "real x \<le> 1" and "even n"
shows "arctan x \<in> {(x * lb_arctan_horner prec n 1 (x * x)) .. (x * ub_arctan_horner prec (Suc n) 1 (x * x))}"
proof -
- let "?c i" = "-1^i * (1 / (i * 2 + (1::nat)) * real x ^ (i * 2 + 1))"
- let "?S n" = "\<Sum> i=0..<n. ?c i"
+ let ?c = "\<lambda>i. -1^i * (1 / (i * 2 + (1::nat)) * real x ^ (i * 2 + 1))"
+ let ?S = "\<lambda>n. \<Sum> i=0..<n. ?c i"
have "0 \<le> real (x * x)" by auto
from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
@@ -457,30 +400,11 @@
lemma arctan_0_1_bounds: assumes "0 \<le> real x" "real x \<le> 1"
shows "arctan x \<in> {(x * lb_arctan_horner prec (get_even n) 1 (x * x)) .. (x * ub_arctan_horner prec (get_odd n) 1 (x * x))}"
-proof (cases "even n")
- case True
- obtain n' where "Suc n' = get_odd n" and "odd (Suc n')" using get_odd_ex by auto
- hence "even n'" unfolding even_Suc by auto
- have "arctan x \<le> x * ub_arctan_horner prec (get_odd n) 1 (x * x)"
- 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
- moreover
- have "x * lb_arctan_horner prec (get_even n) 1 (x * x) \<le> arctan x"
- 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
- ultimately show ?thesis by auto
-next
- case False hence "0 < n" by (rule odd_pos)
- from gr0_implies_Suc[OF this] obtain n' where "n = Suc n'" ..
- from False[unfolded this even_Suc]
- have "even n'" and "even (Suc (Suc n'))" by auto
- have "get_odd n = Suc n'" unfolding get_odd_def if_P[OF False] using `n = Suc n'` .
-
- have "arctan x \<le> x * ub_arctan_horner prec (get_odd n) 1 (x * x)"
- unfolding `get_odd n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even n'`] by auto
- moreover
- have "(x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan x"
- 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
- ultimately show ?thesis by auto
-qed
+ using
+ arctan_0_1_bounds'[OF assms, of n prec]
+ arctan_0_1_bounds'[OF assms, of "n + 1" prec]
+ arctan_0_1_bounds'[OF assms, of "n - 1" prec]
+ by (auto simp: get_even_def get_odd_def odd_pos simp del: ub_arctan_horner.simps lb_arctan_horner.simps)
subsection "Compute \<pi>"
@@ -530,16 +454,11 @@
finally have "?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k) \<le> arctan (1 / k)" .
} note lb_arctan = this
- have "pi \<le> ub_pi n"
- unfolding ub_pi_def machin_pi Let_def unfolding Float_num
- using lb_arctan[of 239] ub_arctan[of 5] powr_realpow[of 2 2]
+ have "pi \<le> ub_pi n \<and> lb_pi n \<le> pi"
+ unfolding lb_pi_def ub_pi_def machin_pi Let_def unfolding Float_num
+ using lb_arctan[of 5] ub_arctan[of 239] lb_arctan[of 239] ub_arctan[of 5] powr_realpow[of 2 2]
by (auto intro!: mult_left_mono add_mono simp add: uminus_add_conv_diff [symmetric] simp del: uminus_add_conv_diff)
- moreover
- have "lb_pi n \<le> pi"
- unfolding lb_pi_def machin_pi Let_def Float_num
- using lb_arctan[of 5] ub_arctan[of 239] powr_realpow[of 2 2]
- by (auto intro!: mult_left_mono add_mono simp add: uminus_add_conv_diff [symmetric] simp del: uminus_add_conv_diff)
- ultimately show ?thesis by auto
+ then show ?thesis by auto
qed
subsection "Compute arcus tangens in the entire domain"
@@ -1808,10 +1727,8 @@
done
lemma Float_pos_eq_mantissa_pos: "Float m e > 0 \<longleftrightarrow> m > 0"
- apply (auto simp add: zero_less_mult_iff zero_float_def powr_gt_zero[of 2 "exponent x"] dest: less_zeroE)
using powr_gt_zero[of 2 "e"]
- apply simp
- done
+ by (auto simp add: zero_less_mult_iff zero_float_def simp del: powr_gt_zero dest: less_zeroE)
lemma Float_representation_aux:
fixes m e