--- a/src/HOL/Decision_Procs/Approximation.thy Mon May 11 08:29:28 2009 -0700
+++ b/src/HOL/Decision_Procs/Approximation.thy Wed Apr 29 20:19:50 2009 +0200
@@ -10,7 +10,7 @@
subsection {* Define auxiliary helper @{text horner} function *}
-fun horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where
+primrec horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where
"horner F G 0 i k x = 0" |
"horner F G (Suc n) i k x = 1 / real k - x * horner F G n (F i) (G i k) x"
@@ -33,32 +33,32 @@
qed auto
lemma horner_bounds':
- assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
+ assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
and lb_0: "\<And> i k x. lb 0 i k x = 0"
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)"
and ub_0: "\<And> i k x. ub 0 i k x = 0"
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)"
- shows "Ifloat (lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') (Ifloat x) \<and>
- horner F G n ((F ^^ j') s) (f j') (Ifloat x) \<le> Ifloat (ub n ((F ^^ j') s) (f j') x)"
+ shows "real (lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') (real x) \<and>
+ horner F G n ((F ^^ j') s) (f j') (real x) \<le> real (ub n ((F ^^ j') s) (f j') x)"
(is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'")
proof (induct n arbitrary: j')
case 0 thus ?case unfolding lb_0 ub_0 horner.simps by auto
next
case (Suc n)
- have "?lb (Suc n) j' \<le> ?horner (Suc n) j'" unfolding lb_Suc ub_Suc horner.simps Ifloat_sub diff_def
+ have "?lb (Suc n) j' \<le> ?horner (Suc n) j'" unfolding lb_Suc ub_Suc horner.simps real_of_float_sub diff_def
proof (rule add_mono)
- show "Ifloat (lapprox_rat prec 1 (int (f j'))) \<le> 1 / real (f j')" using lapprox_rat[of prec 1 "int (f j')"] by auto
- from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct2] `0 \<le> Ifloat x`
- show "- Ifloat (x * ub n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) x) \<le> - (Ifloat x * horner F G n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) (Ifloat x))"
- unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono)
+ 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
+ from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct2] `0 \<le> real x`
+ 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))"
+ unfolding real_of_float_mult neg_le_iff_le by (rule mult_left_mono)
qed
- moreover have "?horner (Suc n) j' \<le> ?ub (Suc n) j'" unfolding ub_Suc ub_Suc horner.simps Ifloat_sub diff_def
+ moreover have "?horner (Suc n) j' \<le> ?ub (Suc n) j'" unfolding ub_Suc ub_Suc horner.simps real_of_float_sub diff_def
proof (rule add_mono)
- show "1 / real (f j') \<le> Ifloat (rapprox_rat prec 1 (int (f j')))" using rapprox_rat[of 1 "int (f j')" prec] by auto
- from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct1] `0 \<le> Ifloat x`
- show "- (Ifloat x * horner F G n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) (Ifloat x)) \<le>
- - Ifloat (x * lb n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) x)"
- unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono)
+ 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
+ from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct1] `0 \<le> real x`
+ show "- (real x * horner F G n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) (real x)) \<le>
+ - real (x * lb n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) x)"
+ unfolding real_of_float_mult neg_le_iff_le by (rule mult_left_mono)
qed
ultimately show ?case by blast
qed
@@ -73,28 +73,28 @@
*}
lemma horner_bounds: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
- assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
+ assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
and lb_0: "\<And> i k x. lb 0 i k x = 0"
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)"
and ub_0: "\<And> i k x. ub 0 i k x = 0"
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)"
- shows "Ifloat (lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. -1 ^ j * (1 / real (f (j' + j))) * Ifloat x ^ j)" (is "?lb") and
- "(\<Sum>j=0..<n. -1 ^ j * (1 / real (f (j' + j))) * (Ifloat x ^ j)) \<le> Ifloat (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
+ 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
+ "(\<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")
proof -
have "?lb \<and> ?ub"
- using horner_bounds'[where lb=lb, OF `0 \<le> Ifloat x` f_Suc lb_0 lb_Suc ub_0 ub_Suc]
+ using horner_bounds'[where lb=lb, OF `0 \<le> real x` f_Suc lb_0 lb_Suc ub_0 ub_Suc]
unfolding horner_schema[where f=f, OF f_Suc] .
thus "?lb" and "?ub" by auto
qed
lemma horner_bounds_nonpos: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
- assumes "Ifloat x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
+ assumes "real x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
and lb_0: "\<And> i k x. lb 0 i k x = 0"
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)"
and ub_0: "\<And> i k x. ub 0 i k x = 0"
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)"
- shows "Ifloat (lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / real (f (j' + j))) * Ifloat x ^ j)" (is "?lb") and
- "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x ^ j)) \<le> Ifloat (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
+ 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
+ "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (real x ^ j)) \<le> real (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
proof -
{ fix x y z :: float have "x - y * z = x + - y * z"
by (cases x, cases y, cases z, simp add: plus_float.simps minus_float_def uminus_float.simps times_float.simps algebra_simps)
@@ -102,19 +102,19 @@
{ fix x :: float have "- (- x) = x" by (cases x, auto simp add: uminus_float.simps) } note minus_minus = this
- have move_minus: "Ifloat (-x) = -1 * Ifloat x" by auto
+ have move_minus: "real (-x) = -1 * real x" by auto
- have sum_eq: "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * Ifloat x ^ j) =
- (\<Sum>j = 0..<n. -1 ^ j * (1 / real (f (j' + j))) * Ifloat (- x) ^ j)"
+ have sum_eq: "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * real x ^ j) =
+ (\<Sum>j = 0..<n. -1 ^ j * (1 / real (f (j' + j))) * real (- x) ^ j)"
proof (rule setsum_cong, simp)
fix j assume "j \<in> {0 ..< n}"
- show "1 / real (f (j' + j)) * Ifloat x ^ j = -1 ^ j * (1 / real (f (j' + j))) * Ifloat (- x) ^ j"
+ show "1 / real (f (j' + j)) * real x ^ j = -1 ^ j * (1 / real (f (j' + j))) * real (- x) ^ j"
unfolding move_minus power_mult_distrib real_mult_assoc[symmetric]
unfolding real_mult_commute unfolding real_mult_assoc[of "-1 ^ j", symmetric] power_mult_distrib[symmetric]
by auto
qed
- have "0 \<le> Ifloat (-x)" using assms by auto
+ have "0 \<le> real (-x)" using assms by auto
from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec
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,
OF this f_Suc lb_0 refl ub_0 refl]
@@ -159,34 +159,34 @@
else if u < 0 then (u ^ n, l ^ n)
else (0, (max (-l) u) ^ n))"
-lemma float_power_bnds: assumes "(l1, u1) = float_power_bnds n l u" and "x \<in> {Ifloat l .. Ifloat u}"
- shows "x ^ n \<in> {Ifloat l1..Ifloat u1}"
+lemma float_power_bnds: assumes "(l1, u1) = float_power_bnds n l u" and "x \<in> {real l .. real u}"
+ shows "x ^ n \<in> {real l1..real u1}"
proof (cases "even n")
case True
show ?thesis
proof (cases "0 < l")
- case True hence "odd n \<or> 0 < l" and "0 \<le> Ifloat l" unfolding less_float_def by auto
+ case True hence "odd n \<or> 0 < l" and "0 \<le> real l" unfolding less_float_def 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 "Ifloat l ^ n \<le> x ^ n" and "x ^ n \<le> Ifloat u ^ n " using `0 \<le> Ifloat l` and assms unfolding atLeastAtMost_iff using power_mono[of "Ifloat l" x] power_mono[of x "Ifloat u"] by auto
+ 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
thus ?thesis using assms `0 < l` unfolding atLeastAtMost_iff l1 u1 float_power less_float_def 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> - Ifloat u" and "- Ifloat u \<le> - x" and "0 \<le> - x" and "-x \<le> - Ifloat l" using assms unfolding less_float_def by auto
- hence "Ifloat u ^ n \<le> x ^ n" and "x ^ n \<le> Ifloat l ^ n" using power_mono[of "-x" "-Ifloat l" n] power_mono[of "-Ifloat u" "-x" n]
+ 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
+ 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 using float_power by auto
next
case False
- have "\<bar>x\<bar> \<le> Ifloat (max (-l) u)"
+ 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 unfolding le_float_def by auto
next
case False thus ?thesis unfolding max_def if_not_P[OF False] using assms unfolding le_float_def by auto
qed
- hence x_abs: "\<bar>x\<bar> \<le> \<bar>Ifloat (max (-l) u)\<bar>" by auto
+ 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 float_power using zero_le_even_power[OF `even n`] power_mono_even[OF `even n` x_abs] by auto
qed
@@ -194,11 +194,11 @@
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 "Ifloat l ^ n \<le> x ^ n" and "x ^ n \<le> Ifloat u ^ n " using assms unfolding atLeastAtMost_iff using power_mono_odd[OF False] by auto
+ 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 float_power less_float_def by auto
qed
-lemma bnds_power: "\<forall> x l u. (l1, u1) = float_power_bnds n l u \<and> x \<in> {Ifloat l .. Ifloat u} \<longrightarrow> Ifloat l1 \<le> x ^ n \<and> x ^ n \<le> Ifloat u1"
+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"
using float_power_bnds by auto
section "Square root"
@@ -234,8 +234,8 @@
thus ?thesis by (simp add: field_simps)
qed
-lemma sqrt_iteration_bound: assumes "0 < Ifloat x"
- shows "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec n x)"
+lemma sqrt_iteration_bound: assumes "0 < real x"
+ shows "sqrt (real x) < real (sqrt_iteration prec n x)"
proof (induct n)
case 0
show ?case
@@ -246,14 +246,14 @@
have int_nat_bl: "int (nat (bitlen m)) = bitlen m" using bitlen_ge0 by auto
- have "Ifloat x = (real m / 2^nat (bitlen m)) * pow2 (e + int (nat (bitlen m)))"
- unfolding pow2_add pow2_int Float Ifloat.simps by auto
+ have "real x = (real m / 2^nat (bitlen m)) * pow2 (e + int (nat (bitlen m)))"
+ unfolding pow2_add pow2_int Float real_of_float_simp by auto
also have "\<dots> < 1 * pow2 (e + int (nat (bitlen m)))"
proof (rule mult_strict_right_mono, auto)
show "real m < 2^nat (bitlen m)" using bitlen_bounds[OF `0 < m`, THEN conjunct2]
unfolding real_of_int_less_iff[of m, symmetric] by auto
qed
- finally have "sqrt (Ifloat x) < sqrt (pow2 (e + bitlen m))" unfolding int_nat_bl by auto
+ finally have "sqrt (real x) < sqrt (pow2 (e + bitlen m))" unfolding int_nat_bl by auto
also have "\<dots> \<le> pow2 ((e + bitlen m) div 2 + 1)"
proof -
let ?E = "e + bitlen m"
@@ -283,110 +283,110 @@
finally show ?thesis by auto
qed
finally show ?thesis
- unfolding Float sqrt_iteration.simps Ifloat.simps by auto
+ unfolding Float sqrt_iteration.simps real_of_float_simp by auto
qed
next
case (Suc n)
let ?b = "sqrt_iteration prec n x"
- have "0 < sqrt (Ifloat x)" using `0 < Ifloat x` by auto
- also have "\<dots> < Ifloat ?b" using Suc .
- finally have "sqrt (Ifloat x) < (Ifloat ?b + Ifloat x / Ifloat ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ `0 < Ifloat x`] by auto
- also have "\<dots> \<le> (Ifloat ?b + Ifloat (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr)
- also have "\<dots> = Ifloat (Float 1 -1) * (Ifloat ?b + Ifloat (float_divr prec x ?b))" by auto
- finally show ?case unfolding sqrt_iteration.simps Let_def Ifloat_mult Ifloat_add right_distrib .
+ have "0 < sqrt (real x)" using `0 < real x` by auto
+ also have "\<dots> < real ?b" using Suc .
+ 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
+ also have "\<dots> \<le> (real ?b + real (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr)
+ also have "\<dots> = real (Float 1 -1) * (real ?b + real (float_divr prec x ?b))" by auto
+ finally show ?case unfolding sqrt_iteration.simps Let_def real_of_float_mult real_of_float_add right_distrib .
qed
-lemma sqrt_iteration_lower_bound: assumes "0 < Ifloat x"
- shows "0 < Ifloat (sqrt_iteration prec n x)" (is "0 < ?sqrt")
+lemma sqrt_iteration_lower_bound: assumes "0 < real x"
+ shows "0 < real (sqrt_iteration prec n x)" (is "0 < ?sqrt")
proof -
- have "0 < sqrt (Ifloat x)" using assms by auto
+ have "0 < sqrt (real x)" using assms by auto
also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
finally show ?thesis .
qed
-lemma lb_sqrt_lower_bound: assumes "0 \<le> Ifloat x"
- shows "0 \<le> Ifloat (the (lb_sqrt prec x))"
+lemma lb_sqrt_lower_bound: assumes "0 \<le> real x"
+ shows "0 \<le> real (the (lb_sqrt prec x))"
proof (cases "0 < x")
- case True hence "0 < Ifloat x" and "0 \<le> x" using `0 \<le> Ifloat x` unfolding less_float_def le_float_def by auto
+ case True hence "0 < real x" and "0 \<le> x" using `0 \<le> real x` unfolding less_float_def le_float_def by auto
hence "0 < sqrt_iteration prec prec x" unfolding less_float_def using sqrt_iteration_lower_bound by auto
- hence "0 \<le> Ifloat (float_divl prec x (sqrt_iteration prec prec x))" using float_divl_lower_bound[OF `0 \<le> x`] unfolding le_float_def by auto
+ 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
thus ?thesis unfolding lb_sqrt_def using True by auto
next
- case False with `0 \<le> Ifloat x` have "Ifloat x = 0" unfolding less_float_def by auto
+ case False with `0 \<le> real x` have "real x = 0" unfolding less_float_def by auto
thus ?thesis unfolding lb_sqrt_def less_float_def by auto
qed
-lemma lb_sqrt_upper_bound: assumes "0 \<le> Ifloat x"
- shows "Ifloat (the (lb_sqrt prec x)) \<le> sqrt (Ifloat x)"
+lemma lb_sqrt_upper_bound: assumes "0 \<le> real x"
+ shows "real (the (lb_sqrt prec x)) \<le> sqrt (real x)"
proof (cases "0 < x")
- case True hence "0 < Ifloat x" and "0 \<le> Ifloat x" unfolding less_float_def by auto
- hence sqrt_gt0: "0 < sqrt (Ifloat x)" by auto
- hence sqrt_ub: "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
+ case True hence "0 < real x" and "0 \<le> real x" unfolding less_float_def by auto
+ hence sqrt_gt0: "0 < sqrt (real x)" by auto
+ hence sqrt_ub: "sqrt (real x) < real (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
- have "Ifloat (float_divl prec x (sqrt_iteration prec prec x)) \<le> Ifloat x / Ifloat (sqrt_iteration prec prec x)" by (rule float_divl)
- also have "\<dots> < Ifloat x / sqrt (Ifloat x)"
- by (rule divide_strict_left_mono[OF sqrt_ub `0 < Ifloat x` mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
- also have "\<dots> = sqrt (Ifloat x)" unfolding inverse_eq_iff_eq[of _ "sqrt (Ifloat x)", symmetric] sqrt_divide_self_eq[OF `0 \<le> Ifloat x`, symmetric] by auto
+ have "real (float_divl prec x (sqrt_iteration prec prec x)) \<le> real x / real (sqrt_iteration prec prec x)" by (rule float_divl)
+ also have "\<dots> < real x / sqrt (real x)"
+ by (rule divide_strict_left_mono[OF sqrt_ub `0 < real x` mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
+ also have "\<dots> = sqrt (real x)" unfolding inverse_eq_iff_eq[of _ "sqrt (real x)", symmetric] sqrt_divide_self_eq[OF `0 \<le> real x`, symmetric] by auto
finally show ?thesis unfolding lb_sqrt_def if_P[OF `0 < x`] by auto
next
- case False with `0 \<le> Ifloat x`
+ case False with `0 \<le> real x`
have "\<not> x < 0" unfolding less_float_def le_float_def by auto
show ?thesis unfolding lb_sqrt_def if_not_P[OF False] if_not_P[OF `\<not> x < 0`] using assms by auto
qed
lemma lb_sqrt: assumes "Some y = lb_sqrt prec x"
- shows "Ifloat y \<le> sqrt (Ifloat x)" and "0 \<le> Ifloat x"
+ shows "real y \<le> sqrt (real x)" and "0 \<le> real x"
proof -
- show "0 \<le> Ifloat x"
+ show "0 \<le> real x"
proof (rule ccontr)
- assume "\<not> 0 \<le> Ifloat x"
+ assume "\<not> 0 \<le> real x"
hence "lb_sqrt prec x = None" unfolding lb_sqrt_def less_float_def by auto
thus False using assms by auto
qed
from lb_sqrt_upper_bound[OF this, of prec]
- show "Ifloat y \<le> sqrt (Ifloat x)" unfolding assms[symmetric] by auto
+ show "real y \<le> sqrt (real x)" unfolding assms[symmetric] by auto
qed
-lemma ub_sqrt_lower_bound: assumes "0 \<le> Ifloat x"
- shows "sqrt (Ifloat x) \<le> Ifloat (the (ub_sqrt prec x))"
+lemma ub_sqrt_lower_bound: assumes "0 \<le> real x"
+ shows "sqrt (real x) \<le> real (the (ub_sqrt prec x))"
proof (cases "0 < x")
- case True hence "0 < Ifloat x" unfolding less_float_def by auto
- hence "0 < sqrt (Ifloat x)" by auto
- hence "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
+ case True hence "0 < real x" unfolding less_float_def by auto
+ hence "0 < sqrt (real x)" by auto
+ hence "sqrt (real x) < real (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
thus ?thesis unfolding ub_sqrt_def if_P[OF `0 < x`] by auto
next
- case False with `0 \<le> Ifloat x`
- have "Ifloat x = 0" unfolding less_float_def le_float_def by auto
+ case False with `0 \<le> real x`
+ have "real x = 0" unfolding less_float_def le_float_def by auto
thus ?thesis unfolding ub_sqrt_def less_float_def le_float_def by auto
qed
lemma ub_sqrt: assumes "Some y = ub_sqrt prec x"
- shows "sqrt (Ifloat x) \<le> Ifloat y" and "0 \<le> Ifloat x"
+ shows "sqrt (real x) \<le> real y" and "0 \<le> real x"
proof -
- show "0 \<le> Ifloat x"
+ show "0 \<le> real x"
proof (rule ccontr)
- assume "\<not> 0 \<le> Ifloat x"
+ assume "\<not> 0 \<le> real x"
hence "ub_sqrt prec x = None" unfolding ub_sqrt_def less_float_def by auto
thus False using assms by auto
qed
from ub_sqrt_lower_bound[OF this, of prec]
- show "sqrt (Ifloat x) \<le> Ifloat y" unfolding assms[symmetric] by auto
+ show "sqrt (real x) \<le> real y" unfolding assms[symmetric] by auto
qed
-lemma bnds_sqrt: "\<forall> x lx ux. (Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> sqrt x \<and> sqrt x \<le> Ifloat u"
+lemma bnds_sqrt: "\<forall> x lx ux. (Some l, Some 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"
proof (rule allI, rule allI, rule allI, rule impI)
fix x lx ux
- assume "(Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
- hence l: "Some l = lb_sqrt prec lx " and u: "Some u = ub_sqrt prec ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
+ assume "(Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {real lx .. real ux}"
+ hence l: "Some l = lb_sqrt prec lx " and u: "Some u = ub_sqrt prec ux" and x: "x \<in> {real lx .. real ux}" by auto
- have "Ifloat lx \<le> x" and "x \<le> Ifloat ux" using x by auto
+ have "real lx \<le> x" and "x \<le> real ux" using x by auto
- from lb_sqrt(1)[OF l] real_sqrt_le_mono[OF `Ifloat lx \<le> x`]
- have "Ifloat l \<le> sqrt x" by (rule order_trans)
+ from lb_sqrt(1)[OF l] real_sqrt_le_mono[OF `real lx \<le> x`]
+ have "real l \<le> sqrt x" by (rule order_trans)
moreover
- from real_sqrt_le_mono[OF `x \<le> Ifloat ux`] ub_sqrt(1)[OF u]
- have "sqrt x \<le> Ifloat u" by (rule order_trans)
- ultimately show "Ifloat l \<le> sqrt x \<and> sqrt x \<le> Ifloat u" ..
+ from real_sqrt_le_mono[OF `x \<le> real ux`] ub_sqrt(1)[OF u]
+ have "sqrt x \<le> real u" by (rule order_trans)
+ ultimately show "real l \<le> sqrt x \<and> sqrt x \<le> real u" ..
qed
section "Arcus tangens and \<pi>"
@@ -409,24 +409,24 @@
| "lb_arctan_horner prec (Suc n) k x =
(lapprox_rat prec 1 (int k)) - x * (ub_arctan_horner prec n (k + 2) x)"
-lemma arctan_0_1_bounds': assumes "0 \<le> Ifloat x" "Ifloat x \<le> 1" and "even n"
- shows "arctan (Ifloat x) \<in> {Ifloat (x * lb_arctan_horner prec n 1 (x * x)) .. Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x * x))}"
+lemma arctan_0_1_bounds': assumes "0 \<le> real x" "real x \<le> 1" and "even n"
+ 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))}"
proof -
- let "?c i" = "-1^i * (1 / real (i * 2 + 1) * Ifloat x ^ (i * 2 + 1))"
+ let "?c i" = "-1^i * (1 / real (i * 2 + 1) * real x ^ (i * 2 + 1))"
let "?S n" = "\<Sum> i=0..<n. ?c i"
- have "0 \<le> Ifloat (x * x)" by auto
+ have "0 \<le> real (x * x)" by auto
from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
- have "arctan (Ifloat x) \<in> { ?S n .. ?S (Suc n) }"
- proof (cases "Ifloat x = 0")
+ have "arctan (real x) \<in> { ?S n .. ?S (Suc n) }"
+ proof (cases "real x = 0")
case False
- hence "0 < Ifloat x" using `0 \<le> Ifloat x` by auto
- hence prem: "0 < 1 / real (0 * 2 + (1::nat)) * Ifloat x ^ (0 * 2 + 1)" by auto
+ hence "0 < real x" using `0 \<le> real x` by auto
+ hence prem: "0 < 1 / real (0 * 2 + (1::nat)) * real x ^ (0 * 2 + 1)" by auto
- have "\<bar> Ifloat x \<bar> \<le> 1" using `0 \<le> Ifloat x` `Ifloat x \<le> 1` by auto
+ have "\<bar> real x \<bar> \<le> 1" using `0 \<le> real x` `real x \<le> 1` by auto
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`]
- show ?thesis unfolding arctan_series[OF `\<bar> Ifloat x \<bar> \<le> 1`] Suc_plus1 .
+ show ?thesis unfolding arctan_series[OF `\<bar> real x \<bar> \<le> 1`] Suc_plus1 .
qed auto
note arctan_bounds = this[unfolded atLeastAtMost_iff]
@@ -435,37 +435,37 @@
note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0
and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
and ub="\<lambda>n i k x. ub_arctan_horner prec n k x",
- OF `0 \<le> Ifloat (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps]
+ OF `0 \<le> real (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps]
- { have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n"
- using bounds(1) `0 \<le> Ifloat x`
- unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
- unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"]
+ { have "real (x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n"
+ using bounds(1) `0 \<le> real x`
+ unfolding real_of_float_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
+ unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
by (auto intro!: mult_left_mono)
- also have "\<dots> \<le> arctan (Ifloat x)" using arctan_bounds ..
- finally have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan (Ifloat x)" . }
+ also have "\<dots> \<le> arctan (real x)" using arctan_bounds ..
+ finally have "real (x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan (real x)" . }
moreover
- { have "arctan (Ifloat x) \<le> ?S (Suc n)" using arctan_bounds ..
- also have "\<dots> \<le> Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))"
- using bounds(2)[of "Suc n"] `0 \<le> Ifloat x`
- unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
- unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"]
+ { have "arctan (real x) \<le> ?S (Suc n)" using arctan_bounds ..
+ also have "\<dots> \<le> real (x * ub_arctan_horner prec (Suc n) 1 (x*x))"
+ using bounds(2)[of "Suc n"] `0 \<le> real x`
+ unfolding real_of_float_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
+ unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
by (auto intro!: mult_left_mono)
- finally have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . }
+ finally have "arctan (real x) \<le> real (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . }
ultimately show ?thesis by auto
qed
-lemma arctan_0_1_bounds: assumes "0 \<le> Ifloat x" "Ifloat x \<le> 1"
- shows "arctan (Ifloat x) \<in> {Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) .. Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))}"
+lemma arctan_0_1_bounds: assumes "0 \<le> real x" "real x \<le> 1"
+ 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))}"
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_nat_Suc by auto
- have "arctan (Ifloat x) \<le> Ifloat (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> Ifloat x` `Ifloat x \<le> 1` `even n'`] by auto
+ have "arctan (real x) \<le> real (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 "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (Ifloat x)"
- unfolding get_even_def if_P[OF True] using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even n`] by auto
+ have "real (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (real 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)
@@ -474,11 +474,11 @@
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 (Ifloat x) \<le> Ifloat (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> Ifloat x` `Ifloat x \<le> 1` `even n'`] by auto
+ have "arctan (real x) \<le> real (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 "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (Ifloat x)"
- unfolding get_even_def if_not_P[OF False] unfolding `n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even (Suc (Suc n'))`] by auto
+ have "real (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (real 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
@@ -496,7 +496,7 @@
in ((Float 1 2) * ((Float 1 2) * A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (A * A)) -
B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (B * B)))))"
-lemma pi_boundaries: "pi \<in> {Ifloat (lb_pi n) .. Ifloat (ub_pi n)}"
+lemma pi_boundaries: "pi \<in> {real (lb_pi n) .. real (ub_pi n)}"
proof -
have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))" unfolding machin[symmetric] by auto
@@ -504,15 +504,15 @@
let ?k = "rapprox_rat prec 1 k"
have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
- have "0 \<le> Ifloat ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \<le> k`)
- have "Ifloat ?k \<le> 1" unfolding rapprox_rat.simps(2)[OF zero_le_one `0 < k`]
+ have "0 \<le> real ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \<le> k`)
+ have "real ?k \<le> 1" unfolding rapprox_rat.simps(2)[OF zero_le_one `0 < k`]
by (rule rapprox_posrat_le1, auto simp add: `0 < k` `1 \<le> k`)
- have "1 / real k \<le> Ifloat ?k" using rapprox_rat[where x=1 and y=k] by auto
- hence "arctan (1 / real k) \<le> arctan (Ifloat ?k)" by (rule arctan_monotone')
- also have "\<dots> \<le> Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))"
- using arctan_0_1_bounds[OF `0 \<le> Ifloat ?k` `Ifloat ?k \<le> 1`] by auto
- finally have "arctan (1 / (real k)) \<le> Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" .
+ have "1 / real k \<le> real ?k" using rapprox_rat[where x=1 and y=k] by auto
+ hence "arctan (1 / real k) \<le> arctan (real ?k)" by (rule arctan_monotone')
+ also have "\<dots> \<le> real (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))"
+ using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
+ finally have "arctan (1 / (real k)) \<le> real (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" .
} note ub_arctan = this
{ fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
@@ -520,24 +520,24 @@
have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
have "1 / real k \<le> 1" using `1 < k` by auto
- have "\<And>n. 0 \<le> Ifloat ?k" using lapprox_rat_bottom[where x=1 and y=k, OF zero_le_one `0 < k`] by (auto simp add: `1 div k = 0`)
- have "\<And>n. Ifloat ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / real k \<le> 1`)
+ 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`)
+ have "\<And>n. real ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / real k \<le> 1`)
- have "Ifloat ?k \<le> 1 / real k" using lapprox_rat[where x=1 and y=k] by auto
+ have "real ?k \<le> 1 / real k" using lapprox_rat[where x=1 and y=k] by auto
- have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (Ifloat ?k)"
- using arctan_0_1_bounds[OF `0 \<le> Ifloat ?k` `Ifloat ?k \<le> 1`] by auto
- also have "\<dots> \<le> arctan (1 / real k)" using `Ifloat ?k \<le> 1 / real k` by (rule arctan_monotone')
- finally have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (1 / (real k))" .
+ have "real (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (real ?k)"
+ using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
+ also have "\<dots> \<le> arctan (1 / real k)" using `real ?k \<le> 1 / real k` by (rule arctan_monotone')
+ finally have "real (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (1 / (real k))" .
} note lb_arctan = this
- have "pi \<le> Ifloat (ub_pi n)"
- unfolding ub_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub unfolding Float_num
+ have "pi \<le> real (ub_pi n)"
+ unfolding ub_pi_def machin_pi Let_def real_of_float_mult real_of_float_sub unfolding Float_num
using lb_arctan[of 239] ub_arctan[of 5]
by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
moreover
- have "Ifloat (lb_pi n) \<le> pi"
- unfolding lb_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub Float_num
+ have "real (lb_pi n) \<le> pi"
+ unfolding lb_pi_def machin_pi Let_def real_of_float_mult real_of_float_sub Float_num
using lb_arctan[of 5] ub_arctan[of 239]
by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
ultimately show ?thesis by auto
@@ -569,35 +569,35 @@
declare ub_arctan_horner.simps[simp del]
declare lb_arctan_horner.simps[simp del]
-lemma lb_arctan_bound': assumes "0 \<le> Ifloat x"
- shows "Ifloat (lb_arctan prec x) \<le> arctan (Ifloat x)"
+lemma lb_arctan_bound': assumes "0 \<le> real x"
+ shows "real (lb_arctan prec x) \<le> arctan (real x)"
proof -
- have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> Ifloat x` by auto
+ have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> real x` by auto
let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
show ?thesis
proof (cases "x \<le> Float 1 -1")
- case True hence "Ifloat x \<le> 1" unfolding le_float_def Float_num by auto
+ case True hence "real x \<le> 1" unfolding le_float_def Float_num by auto
show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
- using arctan_0_1_bounds[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1`] by auto
+ using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
next
- case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto
- let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)"
+ case False hence "0 < real x" unfolding le_float_def Float_num by auto
+ let ?R = "1 + sqrt (1 + real x * real x)"
let ?fR = "1 + the (ub_sqrt prec (1 + x * x))"
let ?DIV = "float_divl prec x ?fR"
- have sqr_ge0: "0 \<le> 1 + Ifloat x * Ifloat x" using sum_power2_ge_zero[of 1 "Ifloat x", unfolded numeral_2_eq_2] by auto
+ 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
hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
- have "sqrt (Ifloat (1 + x * x)) \<le> Ifloat (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0)
- hence "?R \<le> Ifloat ?fR" by auto
- hence "0 < ?fR" and "0 < Ifloat ?fR" unfolding less_float_def using `0 < ?R` by auto
+ have "sqrt (real (1 + x * x)) \<le> real (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0)
+ hence "?R \<le> real ?fR" by auto
+ hence "0 < ?fR" and "0 < real ?fR" unfolding less_float_def using `0 < ?R` by auto
- have monotone: "Ifloat (float_divl prec x ?fR) \<le> Ifloat x / ?R"
+ have monotone: "real (float_divl prec x ?fR) \<le> real x / ?R"
proof -
- have "Ifloat ?DIV \<le> Ifloat x / Ifloat ?fR" by (rule float_divl)
- also have "\<dots> \<le> Ifloat x / ?R" by (rule divide_left_mono[OF `?R \<le> Ifloat ?fR` `0 \<le> Ifloat x` mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 `?R \<le> Ifloat ?fR`] divisor_gt0]])
+ have "real ?DIV \<le> real x / real ?fR" by (rule float_divl)
+ 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]])
finally show ?thesis .
qed
@@ -605,47 +605,47 @@
proof (cases "x \<le> Float 1 1")
case True
- have "Ifloat x \<le> sqrt (Ifloat (1 + x * x))" using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
- also have "\<dots> \<le> Ifloat (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0)
- finally have "Ifloat x \<le> Ifloat ?fR" by auto
- moreover have "Ifloat ?DIV \<le> Ifloat x / Ifloat ?fR" by (rule float_divl)
- ultimately have "Ifloat ?DIV \<le> 1" unfolding divide_le_eq_1_pos[OF `0 < Ifloat ?fR`, symmetric] by auto
+ 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
+ also have "\<dots> \<le> real (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0)
+ finally have "real x \<le> real ?fR" by auto
+ moreover have "real ?DIV \<le> real x / real ?fR" by (rule float_divl)
+ ultimately have "real ?DIV \<le> 1" unfolding divide_le_eq_1_pos[OF `0 < real ?fR`, symmetric] by auto
- have "0 \<le> Ifloat ?DIV" using float_divl_lower_bound[OF `0 \<le> x` `0 < ?fR`] unfolding le_float_def by auto
+ have "0 \<le> real ?DIV" using float_divl_lower_bound[OF `0 \<le> x` `0 < ?fR`] unfolding le_float_def by auto
- have "Ifloat (Float 1 1 * ?lb_horner ?DIV) \<le> 2 * arctan (Ifloat (float_divl prec x ?fR))" unfolding Ifloat_mult[of "Float 1 1"] Float_num
- using arctan_0_1_bounds[OF `0 \<le> Ifloat ?DIV` `Ifloat ?DIV \<le> 1`] by auto
- also have "\<dots> \<le> 2 * arctan (Ifloat x / ?R)"
+ 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
+ using arctan_0_1_bounds[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
+ also have "\<dots> \<le> 2 * arctan (real x / ?R)"
using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
- also have "2 * arctan (Ifloat x / ?R) = arctan (Ifloat x)" using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 real_mult_1 .
+ also have "2 * arctan (real x / ?R) = arctan (real x)" using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 real_mult_1 .
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] .
next
case False
- hence "2 < Ifloat x" unfolding le_float_def Float_num by auto
- hence "1 \<le> Ifloat x" by auto
+ hence "2 < real x" unfolding le_float_def Float_num by auto
+ hence "1 \<le> real x" by auto
let "?invx" = "float_divr prec 1 x"
- have "0 \<le> arctan (Ifloat x)" using arctan_monotone'[OF `0 \<le> Ifloat x`] using arctan_tan[of 0, unfolded tan_zero] by auto
+ have "0 \<le> arctan (real x)" using arctan_monotone'[OF `0 \<le> real x`] using arctan_tan[of 0, unfolded tan_zero] by auto
show ?thesis
proof (cases "1 < ?invx")
case True
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]
- using `0 \<le> arctan (Ifloat x)` by auto
+ using `0 \<le> arctan (real x)` by auto
next
case False
- hence "Ifloat ?invx \<le> 1" unfolding less_float_def by auto
- have "0 \<le> Ifloat ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> Ifloat x`)
+ hence "real ?invx \<le> 1" unfolding less_float_def by auto
+ have "0 \<le> real ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> real x`)
- have "1 / Ifloat x \<noteq> 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto
+ have "1 / real x \<noteq> 0" and "0 < 1 / real x" using `0 < real x` by auto
- have "arctan (1 / Ifloat x) \<le> arctan (Ifloat ?invx)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divr)
- also have "\<dots> \<le> Ifloat (?ub_horner ?invx)" using arctan_0_1_bounds[OF `0 \<le> Ifloat ?invx` `Ifloat ?invx \<le> 1`] by auto
- finally have "pi / 2 - Ifloat (?ub_horner ?invx) \<le> arctan (Ifloat x)"
- using `0 \<le> arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \<noteq> 0`]
- unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto
+ have "arctan (1 / real x) \<le> arctan (real ?invx)" unfolding real_of_float_1[symmetric] by (rule arctan_monotone', rule float_divr)
+ also have "\<dots> \<le> real (?ub_horner ?invx)" using arctan_0_1_bounds[OF `0 \<le> real ?invx` `real ?invx \<le> 1`] by auto
+ finally have "pi / 2 - real (?ub_horner ?invx) \<le> arctan (real x)"
+ using `0 \<le> arctan (real x)` arctan_inverse[OF `1 / real x \<noteq> 0`]
+ unfolding real_sgn_pos[OF `0 < 1 / real x`] le_diff_eq by auto
moreover
- have "Ifloat (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding Ifloat_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
+ have "real (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding real_of_float_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
ultimately
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]
by auto
@@ -654,39 +654,39 @@
qed
qed
-lemma ub_arctan_bound': assumes "0 \<le> Ifloat x"
- shows "arctan (Ifloat x) \<le> Ifloat (ub_arctan prec x)"
+lemma ub_arctan_bound': assumes "0 \<le> real x"
+ shows "arctan (real x) \<le> real (ub_arctan prec x)"
proof -
- have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> Ifloat x` by auto
+ have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> real x` by auto
let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
show ?thesis
proof (cases "x \<le> Float 1 -1")
- case True hence "Ifloat x \<le> 1" unfolding le_float_def Float_num by auto
+ case True hence "real x \<le> 1" unfolding le_float_def Float_num by auto
show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
- using arctan_0_1_bounds[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1`] by auto
+ using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
next
- case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto
- let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)"
+ case False hence "0 < real x" unfolding le_float_def Float_num by auto
+ let ?R = "1 + sqrt (1 + real x * real x)"
let ?fR = "1 + the (lb_sqrt prec (1 + x * x))"
let ?DIV = "float_divr prec x ?fR"
- have sqr_ge0: "0 \<le> 1 + Ifloat x * Ifloat x" using sum_power2_ge_zero[of 1 "Ifloat x", unfolded numeral_2_eq_2] by auto
- hence "0 \<le> Ifloat (1 + x*x)" by auto
+ 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
+ hence "0 \<le> real (1 + x*x)" by auto
hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
- have "Ifloat (the (lb_sqrt prec (1 + x * x))) \<le> sqrt (Ifloat (1 + x * x))" by (rule lb_sqrt_upper_bound, auto simp add: sqr_ge0)
- hence "Ifloat ?fR \<le> ?R" by auto
- have "0 < Ifloat ?fR" unfolding Ifloat_add Ifloat_1 by (rule order_less_le_trans[OF zero_less_one], auto simp add: lb_sqrt_lower_bound[OF `0 \<le> Ifloat (1 + x*x)`])
+ have "real (the (lb_sqrt prec (1 + x * x))) \<le> sqrt (real (1 + x * x))" by (rule lb_sqrt_upper_bound, auto simp add: sqr_ge0)
+ hence "real ?fR \<le> ?R" by auto
+ 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)`])
- have monotone: "Ifloat x / ?R \<le> Ifloat (float_divr prec x ?fR)"
+ have monotone: "real x / ?R \<le> real (float_divr prec x ?fR)"
proof -
- from divide_left_mono[OF `Ifloat ?fR \<le> ?R` `0 \<le> Ifloat x` mult_pos_pos[OF divisor_gt0 `0 < Ifloat ?fR`]]
- have "Ifloat x / ?R \<le> Ifloat x / Ifloat ?fR" .
- also have "\<dots> \<le> Ifloat ?DIV" by (rule float_divr)
+ from divide_left_mono[OF `real ?fR \<le> ?R` `0 \<le> real x` mult_pos_pos[OF divisor_gt0 `0 < real ?fR`]]
+ have "real x / ?R \<le> real x / real ?fR" .
+ also have "\<dots> \<le> real ?DIV" by (rule float_divr)
finally show ?thesis .
qed
@@ -696,45 +696,45 @@
show ?thesis
proof (cases "?DIV > 1")
case True
- have "pi / 2 \<le> Ifloat (ub_pi prec * Float 1 -1)" unfolding Ifloat_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
+ have "pi / 2 \<le> real (ub_pi prec * Float 1 -1)" unfolding real_of_float_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le]
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] .
next
case False
- hence "Ifloat ?DIV \<le> 1" unfolding less_float_def by auto
+ hence "real ?DIV \<le> 1" unfolding less_float_def by auto
- have "0 \<le> Ifloat x / ?R" using `0 \<le> Ifloat x` `0 < ?R` unfolding real_0_le_divide_iff by auto
- hence "0 \<le> Ifloat ?DIV" using monotone by (rule order_trans)
+ have "0 \<le> real x / ?R" using `0 \<le> real x` `0 < ?R` unfolding real_0_le_divide_iff by auto
+ hence "0 \<le> real ?DIV" using monotone by (rule order_trans)
- have "arctan (Ifloat x) = 2 * arctan (Ifloat x / ?R)" using arctan_half unfolding numeral_2_eq_2 power_Suc2 power_0 real_mult_1 .
- also have "\<dots> \<le> 2 * arctan (Ifloat ?DIV)"
+ have "arctan (real x) = 2 * arctan (real x / ?R)" using arctan_half unfolding numeral_2_eq_2 power_Suc2 power_0 real_mult_1 .
+ also have "\<dots> \<le> 2 * arctan (real ?DIV)"
using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
- also have "\<dots> \<le> Ifloat (Float 1 1 * ?ub_horner ?DIV)" unfolding Ifloat_mult[of "Float 1 1"] Float_num
- using arctan_0_1_bounds[OF `0 \<le> Ifloat ?DIV` `Ifloat ?DIV \<le> 1`] by auto
+ also have "\<dots> \<le> real (Float 1 1 * ?ub_horner ?DIV)" unfolding real_of_float_mult[of "Float 1 1"] Float_num
+ using arctan_0_1_bounds[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
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] .
qed
next
case False
- hence "2 < Ifloat x" unfolding le_float_def Float_num by auto
- hence "1 \<le> Ifloat x" by auto
- hence "0 < Ifloat x" by auto
+ hence "2 < real x" unfolding le_float_def Float_num by auto
+ hence "1 \<le> real x" by auto
+ hence "0 < real x" by auto
hence "0 < x" unfolding less_float_def by auto
let "?invx" = "float_divl prec 1 x"
- have "0 \<le> arctan (Ifloat x)" using arctan_monotone'[OF `0 \<le> Ifloat x`] using arctan_tan[of 0, unfolded tan_zero] by auto
+ have "0 \<le> arctan (real x)" using arctan_monotone'[OF `0 \<le> real x`] using arctan_tan[of 0, unfolded tan_zero] by auto
- have "Ifloat ?invx \<le> 1" unfolding less_float_def by (rule order_trans[OF float_divl], auto simp add: `1 \<le> Ifloat x` divide_le_eq_1_pos[OF `0 < Ifloat x`])
- have "0 \<le> Ifloat ?invx" unfolding Ifloat_0[symmetric] by (rule float_divl_lower_bound[unfolded le_float_def], auto simp add: `0 < x`)
+ 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`])
+ 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`)
- have "1 / Ifloat x \<noteq> 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto
+ have "1 / real x \<noteq> 0" and "0 < 1 / real x" using `0 < real x` by auto
- have "Ifloat (?lb_horner ?invx) \<le> arctan (Ifloat ?invx)" using arctan_0_1_bounds[OF `0 \<le> Ifloat ?invx` `Ifloat ?invx \<le> 1`] by auto
- also have "\<dots> \<le> arctan (1 / Ifloat x)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divl)
- finally have "arctan (Ifloat x) \<le> pi / 2 - Ifloat (?lb_horner ?invx)"
- using `0 \<le> arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \<noteq> 0`]
- unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto
+ 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
+ also have "\<dots> \<le> arctan (1 / real x)" unfolding real_of_float_1[symmetric] by (rule arctan_monotone', rule float_divl)
+ finally have "arctan (real x) \<le> pi / 2 - real (?lb_horner ?invx)"
+ using `0 \<le> arctan (real x)` arctan_inverse[OF `1 / real x \<noteq> 0`]
+ unfolding real_sgn_pos[OF `0 < 1 / real x`] le_diff_eq by auto
moreover
- have "pi / 2 \<le> Ifloat (ub_pi prec * Float 1 -1)" unfolding Ifloat_mult Float_num times_divide_eq_right mult_1_right using pi_boundaries by auto
+ 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
ultimately
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]
by auto
@@ -743,34 +743,34 @@
qed
lemma arctan_boundaries:
- "arctan (Ifloat x) \<in> {Ifloat (lb_arctan prec x) .. Ifloat (ub_arctan prec x)}"
+ "arctan (real x) \<in> {real (lb_arctan prec x) .. real (ub_arctan prec x)}"
proof (cases "0 \<le> x")
- case True hence "0 \<le> Ifloat x" unfolding le_float_def by auto
- show ?thesis using ub_arctan_bound'[OF `0 \<le> Ifloat x`] lb_arctan_bound'[OF `0 \<le> Ifloat x`] unfolding atLeastAtMost_iff by auto
+ case True hence "0 \<le> real x" unfolding le_float_def by auto
+ show ?thesis using ub_arctan_bound'[OF `0 \<le> real x`] lb_arctan_bound'[OF `0 \<le> real x`] unfolding atLeastAtMost_iff by auto
next
let ?mx = "-x"
- case False hence "x < 0" and "0 \<le> Ifloat ?mx" unfolding le_float_def less_float_def by auto
- hence bounds: "Ifloat (lb_arctan prec ?mx) \<le> arctan (Ifloat ?mx) \<and> arctan (Ifloat ?mx) \<le> Ifloat (ub_arctan prec ?mx)"
- using ub_arctan_bound'[OF `0 \<le> Ifloat ?mx`] lb_arctan_bound'[OF `0 \<le> Ifloat ?mx`] by auto
- show ?thesis unfolding Ifloat_minus arctan_minus lb_arctan.simps[where x=x] ub_arctan.simps[where x=x] Let_def if_P[OF `x < 0`]
- unfolding atLeastAtMost_iff using bounds[unfolded Ifloat_minus arctan_minus] by auto
+ case False hence "x < 0" and "0 \<le> real ?mx" unfolding le_float_def less_float_def by auto
+ hence bounds: "real (lb_arctan prec ?mx) \<le> arctan (real ?mx) \<and> arctan (real ?mx) \<le> real (ub_arctan prec ?mx)"
+ using ub_arctan_bound'[OF `0 \<le> real ?mx`] lb_arctan_bound'[OF `0 \<le> real ?mx`] by auto
+ 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`]
+ unfolding atLeastAtMost_iff using bounds[unfolded real_of_float_minus arctan_minus] by auto
qed
-lemma bnds_arctan: "\<forall> x lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> arctan x \<and> arctan x \<le> Ifloat u"
+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"
proof (rule allI, rule allI, rule allI, rule impI)
fix x lx ux
- assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
- hence l: "lb_arctan prec lx = l " and u: "ub_arctan prec ux = u" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
+ assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {real lx .. real ux}"
+ hence l: "lb_arctan prec lx = l " and u: "ub_arctan prec ux = u" and x: "x \<in> {real lx .. real ux}" by auto
{ from arctan_boundaries[of lx prec, unfolded l]
- have "Ifloat l \<le> arctan (Ifloat lx)" by (auto simp del: lb_arctan.simps)
+ have "real l \<le> arctan (real lx)" by (auto simp del: lb_arctan.simps)
also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
- finally have "Ifloat l \<le> arctan x" .
+ finally have "real l \<le> arctan x" .
} moreover
- { have "arctan x \<le> arctan (Ifloat ux)" using x by (auto intro: arctan_monotone')
- also have "\<dots> \<le> Ifloat u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
- finally have "arctan x \<le> Ifloat u" .
- } ultimately show "Ifloat l \<le> arctan x \<and> arctan x \<le> Ifloat u" ..
+ { have "arctan x \<le> arctan (real ux)" using x by (auto intro: arctan_monotone')
+ also have "\<dots> \<le> real u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
+ finally have "arctan x \<le> real u" .
+ } ultimately show "real l \<le> arctan x \<and> arctan x \<le> real u" ..
qed
section "Sinus and Cosinus"
@@ -787,10 +787,10 @@
(lapprox_rat prec 1 (int k)) - x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
lemma cos_aux:
- shows "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (Ifloat x)^(2 * i))" (is "?lb")
- and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (Ifloat x)^(2 * i)) \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
+ 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")
+ 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")
proof -
- have "0 \<le> Ifloat (x * x)" unfolding Ifloat_mult by auto
+ have "0 \<le> real (x * x)" unfolding real_of_float_mult by auto
let "?f n" = "fact (2 * n)"
{ fix n
@@ -799,17 +799,17 @@
unfolding F by auto } note f_eq = this
from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
- OF `0 \<le> Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
- show "?lb" and "?ub" by (auto simp add: power_mult power2_eq_square[of "Ifloat x"])
+ OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
+ show "?lb" and "?ub" by (auto simp add: power_mult power2_eq_square[of "real x"])
qed
-lemma cos_boundaries: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
- shows "cos (Ifloat x) \<in> {Ifloat (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. Ifloat (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"
-proof (cases "Ifloat x = 0")
- case False hence "Ifloat x \<noteq> 0" by auto
- hence "0 < x" and "0 < Ifloat x" using `0 \<le> Ifloat x` unfolding less_float_def by auto
- have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0
- using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto
+lemma cos_boundaries: assumes "0 \<le> real x" and "real x \<le> pi / 2"
+ 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))}"
+proof (cases "real x = 0")
+ case False hence "real x \<noteq> 0" by auto
+ hence "0 < x" and "0 < real x" using `0 \<le> real x` unfolding less_float_def by auto
+ have "0 < x * x" using `0 < x` unfolding less_float_def real_of_float_mult real_of_float_0
+ using mult_pos_pos[where a="real x" and b="real x"] by auto
{ fix x n have "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x ^ (2 * i))
= (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum")
@@ -827,62 +827,62 @@
{ fix n :: nat assume "0 < n"
hence "0 < 2 * n" by auto
- obtain t where "0 < t" and "t < Ifloat x" and
- cos_eq: "cos (Ifloat x) = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * (Ifloat x) ^ i)
- + (cos (t + 1/2 * real (2 * n) * pi) / real (fact (2*n))) * (Ifloat x)^(2*n)"
+ obtain t where "0 < t" and "t < real x" and
+ 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)
+ + (cos (t + 1/2 * real (2 * n) * pi) / real (fact (2*n))) * (real x)^(2*n)"
(is "_ = ?SUM + ?rest / ?fact * ?pow")
- using Maclaurin_cos_expansion2[OF `0 < Ifloat x` `0 < 2 * n`] by auto
+ using Maclaurin_cos_expansion2[OF `0 < real x` `0 < 2 * n`] by auto
have "cos t * -1^n = cos t * cos (real n * pi) + sin t * sin (real n * pi)" by auto
also have "\<dots> = cos (t + real n * pi)" using cos_add by auto
also have "\<dots> = ?rest" by auto
finally have "cos t * -1^n = ?rest" .
moreover
- have "t \<le> pi / 2" using `t < Ifloat x` and `Ifloat x \<le> pi / 2` by auto
+ have "t \<le> pi / 2" using `t < real x` and `real x \<le> pi / 2` by auto
hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
have "0 < ?fact" by auto
- have "0 < ?pow" using `0 < Ifloat x` by auto
+ have "0 < ?pow" using `0 < real x` by auto
{
assume "even n"
- have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
+ have "real (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
unfolding morph_to_if_power[symmetric] using cos_aux by auto
- also have "\<dots> \<le> cos (Ifloat x)"
+ also have "\<dots> \<le> cos (real x)"
proof -
from even[OF `even n`] `0 < ?fact` `0 < ?pow`
have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
thus ?thesis unfolding cos_eq by auto
qed
- finally have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos (Ifloat x)" .
+ finally have "real (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos (real x)" .
} note lb = this
{
assume "odd n"
- have "cos (Ifloat x) \<le> ?SUM"
+ have "cos (real x) \<le> ?SUM"
proof -
from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
have "0 \<le> (- ?rest) / ?fact * ?pow"
by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
thus ?thesis unfolding cos_eq by auto
qed
- also have "\<dots> \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))"
+ also have "\<dots> \<le> real (ub_sin_cos_aux prec n 1 1 (x * x))"
unfolding morph_to_if_power[symmetric] using cos_aux by auto
- finally have "cos (Ifloat x) \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" .
+ finally have "cos (real x) \<le> real (ub_sin_cos_aux prec n 1 1 (x * x))" .
} note ub = this and lb
} note ub = this(1) and lb = this(2)
- have "cos (Ifloat x) \<le> Ifloat (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
- moreover have "Ifloat (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos (Ifloat x)"
+ 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] .
+ moreover have "real (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos (real x)"
proof (cases "0 < get_even n")
case True show ?thesis using lb[OF True get_even] .
next
case False
hence "get_even n = 0" by auto
- have "- (pi / 2) \<le> Ifloat x" by (rule order_trans[OF _ `0 < Ifloat x`[THEN less_imp_le]], auto)
- with `Ifloat x \<le> pi / 2`
- show ?thesis unfolding `get_even n = 0` lb_sin_cos_aux.simps Ifloat_minus Ifloat_0 using cos_ge_zero by auto
+ have "- (pi / 2) \<le> real x" by (rule order_trans[OF _ `0 < real x`[THEN less_imp_le]], auto)
+ with `real x \<le> pi / 2`
+ 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
qed
ultimately show ?thesis by auto
next
@@ -890,18 +890,18 @@
show ?thesis
proof (cases "n = 0")
case True
- thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `Ifloat x = 0` lapprox_rat[where x="-1" and y=1] by auto
+ 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
next
case False with not0_implies_Suc obtain m where "n = Suc m" by blast
- thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `Ifloat x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
+ 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)
qed
qed
-lemma sin_aux: assumes "0 \<le> Ifloat x"
- shows "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (Ifloat x)^(2 * i + 1))" (is "?lb")
- and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (Ifloat x)^(2 * i + 1)) \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
+lemma sin_aux: assumes "0 \<le> real x"
+ 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")
+ 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")
proof -
- have "0 \<le> Ifloat (x * x)" unfolding Ifloat_mult by auto
+ have "0 \<le> real (x * x)" unfolding real_of_float_mult by auto
let "?f n" = "fact (2 * n + 1)"
{ fix n
@@ -910,20 +910,20 @@
unfolding F by auto } note f_eq = this
from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
- OF `0 \<le> Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
- show "?lb" and "?ub" using `0 \<le> Ifloat x` unfolding Ifloat_mult
+ OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
+ show "?lb" and "?ub" using `0 \<le> real x` unfolding real_of_float_mult
unfolding power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
unfolding real_mult_commute
- by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "Ifloat x"])
+ by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real x"])
qed
-lemma sin_boundaries: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
- shows "sin (Ifloat x) \<in> {Ifloat (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. Ifloat (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}"
-proof (cases "Ifloat x = 0")
- case False hence "Ifloat x \<noteq> 0" by auto
- hence "0 < x" and "0 < Ifloat x" using `0 \<le> Ifloat x` unfolding less_float_def by auto
- have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0
- using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto
+lemma sin_boundaries: assumes "0 \<le> real x" and "real x \<le> pi / 2"
+ 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))}"
+proof (cases "real x = 0")
+ case False hence "real x \<noteq> 0" by auto
+ hence "0 < x" and "0 < real x" using `0 \<le> real x` unfolding less_float_def by auto
+ have "0 < x * x" using `0 < x` unfolding less_float_def real_of_float_mult real_of_float_0
+ using mult_pos_pos[where a="real x" and b="real x"] by auto
{ 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))
= (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * x ^ i)" (is "?SUM = _")
@@ -939,62 +939,62 @@
{ fix n :: nat assume "0 < n"
hence "0 < 2 * n + 1" by auto
- obtain t where "0 < t" and "t < Ifloat x" and
- sin_eq: "sin (Ifloat x) = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)
- + (sin (t + 1/2 * real (2 * n + 1) * pi) / real (fact (2*n + 1))) * (Ifloat x)^(2*n + 1)"
+ obtain t where "0 < t" and "t < real x" and
+ 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)
+ + (sin (t + 1/2 * real (2 * n + 1) * pi) / real (fact (2*n + 1))) * (real x)^(2*n + 1)"
(is "_ = ?SUM + ?rest / ?fact * ?pow")
- using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < Ifloat x`] by auto
+ using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < real x`] by auto
have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add left_distrib right_distrib by auto
moreover
- have "t \<le> pi / 2" using `t < Ifloat x` and `Ifloat x \<le> pi / 2` by auto
+ have "t \<le> pi / 2" using `t < real x` and `real x \<le> pi / 2` by auto
hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
have "0 < ?fact" by (rule real_of_nat_fact_gt_zero)
- have "0 < ?pow" using `0 < Ifloat x` by (rule zero_less_power)
+ have "0 < ?pow" using `0 < real x` by (rule zero_less_power)
{
assume "even n"
- have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
- (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)"
- using sin_aux[OF `0 \<le> Ifloat x`] unfolding setsum_morph[symmetric] by auto
+ have "real (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
+ (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)"
+ using sin_aux[OF `0 \<le> real x`] unfolding setsum_morph[symmetric] by auto
also have "\<dots> \<le> ?SUM" by auto
- also have "\<dots> \<le> sin (Ifloat x)"
+ also have "\<dots> \<le> sin (real x)"
proof -
from even[OF `even n`] `0 < ?fact` `0 < ?pow`
have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
thus ?thesis unfolding sin_eq by auto
qed
- finally have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin (Ifloat x)" .
+ finally have "real (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin (real x)" .
} note lb = this
{
assume "odd n"
- have "sin (Ifloat x) \<le> ?SUM"
+ have "sin (real x) \<le> ?SUM"
proof -
from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
have "0 \<le> (- ?rest) / ?fact * ?pow"
by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
thus ?thesis unfolding sin_eq by auto
qed
- also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)"
+ 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)"
by auto
- also have "\<dots> \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))"
- using sin_aux[OF `0 \<le> Ifloat x`] unfolding setsum_morph[symmetric] by auto
- finally have "sin (Ifloat x) \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
+ also have "\<dots> \<le> real (x * ub_sin_cos_aux prec n 2 1 (x * x))"
+ using sin_aux[OF `0 \<le> real x`] unfolding setsum_morph[symmetric] by auto
+ finally have "sin (real x) \<le> real (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
} note ub = this and lb
} note ub = this(1) and lb = this(2)
- have "sin (Ifloat x) \<le> Ifloat (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
- moreover have "Ifloat (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin (Ifloat x)"
+ 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] .
+ moreover have "real (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin (real x)"
proof (cases "0 < get_even n")
case True show ?thesis using lb[OF True get_even] .
next
case False
hence "get_even n = 0" by auto
- with `Ifloat x \<le> pi / 2` `0 \<le> Ifloat x`
- show ?thesis unfolding `get_even n = 0` ub_sin_cos_aux.simps Ifloat_minus Ifloat_0 using sin_ge_zero by auto
+ with `real x \<le> pi / 2` `0 \<le> real x`
+ 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
qed
ultimately show ?thesis by auto
next
@@ -1002,10 +1002,10 @@
show ?thesis
proof (cases "n = 0")
case True
- thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `Ifloat x = 0` lapprox_rat[where x="-1" and y=1] by auto
+ 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
next
case False with not0_implies_Suc obtain m where "n = Suc m" by blast
- thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `Ifloat x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
+ 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)
qed
qed
@@ -1034,8 +1034,8 @@
else if 0 \<le> lx then (lb_cos prec ux, ub_cos prec lx)
else (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0))"
-lemma lb_cos: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi"
- shows "cos (Ifloat x) \<in> {Ifloat (lb_cos prec x) .. Ifloat (ub_cos prec x)}" (is "?cos x \<in> { Ifloat (?lb x) .. Ifloat (?ub x) }")
+lemma lb_cos: assumes "0 \<le> real x" and "real x \<le> pi"
+ 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) }")
proof -
{ fix x :: real
have "cos x = cos (x / 2 + x / 2)" by auto
@@ -1045,7 +1045,7 @@
finally have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" .
} note x_half = this[symmetric]
- have "\<not> x < 0" using `0 \<le> Ifloat x` unfolding less_float_def by auto
+ have "\<not> x < 0" using `0 \<le> real x` unfolding less_float_def by auto
let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
let "?ub_half x" = "Float 1 1 * x * x - 1"
@@ -1053,88 +1053,88 @@
show ?thesis
proof (cases "x < Float 1 -1")
- case True hence "Ifloat x \<le> pi / 2" unfolding less_float_def using pi_ge_two by auto
+ case True hence "real x \<le> pi / 2" unfolding less_float_def using pi_ge_two by auto
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
- using cos_boundaries[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi / 2`] .
+ using cos_boundaries[OF `0 \<le> real x` `real x \<le> pi / 2`] .
next
case False
- { fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)"
- assume "Ifloat y \<le> cos ?x2" and "-pi \<le> Ifloat x" and "Ifloat x \<le> pi"
- hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto
+ { fix y x :: float let ?x2 = "real (x * Float 1 -1)"
+ assume "real y \<le> cos ?x2" and "-pi \<le> real x" and "real x \<le> pi"
+ hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding real_of_float_mult Float_num by auto
hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
- have "Ifloat (?lb_half y) \<le> cos (Ifloat x)"
+ have "real (?lb_half y) \<le> cos (real x)"
proof (cases "y < 0")
case True show ?thesis using cos_ge_minus_one unfolding if_P[OF True] by auto
next
case False
- hence "0 \<le> Ifloat y" unfolding less_float_def by auto
- from mult_mono[OF `Ifloat y \<le> cos ?x2` `Ifloat y \<le> cos ?x2` `0 \<le> cos ?x2` this]
- have "Ifloat y * Ifloat y \<le> cos ?x2 * cos ?x2" .
- hence "2 * Ifloat y * Ifloat y \<le> 2 * cos ?x2 * cos ?x2" by auto
- hence "2 * Ifloat y * Ifloat y - 1 \<le> 2 * cos (Ifloat x / 2) * cos (Ifloat x / 2) - 1" unfolding Float_num Ifloat_mult by auto
- thus ?thesis unfolding if_not_P[OF False] x_half Float_num Ifloat_mult Ifloat_sub by auto
+ hence "0 \<le> real y" unfolding less_float_def by auto
+ from mult_mono[OF `real y \<le> cos ?x2` `real y \<le> cos ?x2` `0 \<le> cos ?x2` this]
+ have "real y * real y \<le> cos ?x2 * cos ?x2" .
+ hence "2 * real y * real y \<le> 2 * cos ?x2 * cos ?x2" by auto
+ 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
+ thus ?thesis unfolding if_not_P[OF False] x_half Float_num real_of_float_mult real_of_float_sub by auto
qed
} note lb_half = this
- { fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)"
- assume ub: "cos ?x2 \<le> Ifloat y" and "- pi \<le> Ifloat x" and "Ifloat x \<le> pi"
- hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto
+ { fix y x :: float let ?x2 = "real (x * Float 1 -1)"
+ assume ub: "cos ?x2 \<le> real y" and "- pi \<le> real x" and "real x \<le> pi"
+ hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding real_of_float_mult Float_num by auto
hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
- have "cos (Ifloat x) \<le> Ifloat (?ub_half y)"
+ have "cos (real x) \<le> real (?ub_half y)"
proof -
- have "0 \<le> Ifloat y" using `0 \<le> cos ?x2` ub by (rule order_trans)
+ have "0 \<le> real y" using `0 \<le> cos ?x2` ub by (rule order_trans)
from mult_mono[OF ub ub this `0 \<le> cos ?x2`]
- have "cos ?x2 * cos ?x2 \<le> Ifloat y * Ifloat y" .
- hence "2 * cos ?x2 * cos ?x2 \<le> 2 * Ifloat y * Ifloat y" by auto
- hence "2 * cos (Ifloat x / 2) * cos (Ifloat x / 2) - 1 \<le> 2 * Ifloat y * Ifloat y - 1" unfolding Float_num Ifloat_mult by auto
- thus ?thesis unfolding x_half Ifloat_mult Float_num Ifloat_sub by auto
+ have "cos ?x2 * cos ?x2 \<le> real y * real y" .
+ hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real y * real y" by auto
+ 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
+ thus ?thesis unfolding x_half real_of_float_mult Float_num real_of_float_sub by auto
qed
} note ub_half = this
let ?x2 = "x * Float 1 -1"
let ?x4 = "x * Float 1 -1 * Float 1 -1"
- have "-pi \<le> Ifloat x" using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] `0 \<le> Ifloat x` by (rule order_trans)
+ 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)
show ?thesis
proof (cases "x < 1")
- case True hence "Ifloat x \<le> 1" unfolding less_float_def by auto
- have "0 \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi / 2" using pi_ge_two `0 \<le> Ifloat x` unfolding Ifloat_mult Float_num using assms by auto
+ case True hence "real x \<le> 1" unfolding less_float_def by auto
+ 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
from cos_boundaries[OF this]
- have lb: "Ifloat (?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> Ifloat (?ub_horner ?x2)" by auto
+ have lb: "real (?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> real (?ub_horner ?x2)" by auto
- have "Ifloat (?lb x) \<le> ?cos x"
+ have "real (?lb x) \<le> ?cos x"
proof -
- from lb_half[OF lb `-pi \<le> Ifloat x` `Ifloat x \<le> pi`]
+ from lb_half[OF lb `-pi \<le> real x` `real x \<le> pi`]
show ?thesis unfolding lb_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto
qed
- moreover have "?cos x \<le> Ifloat (?ub x)"
+ moreover have "?cos x \<le> real (?ub x)"
proof -
- from ub_half[OF ub `-pi \<le> Ifloat x` `Ifloat x \<le> pi`]
+ from ub_half[OF ub `-pi \<le> real x` `real x \<le> pi`]
show ?thesis unfolding ub_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto
qed
ultimately show ?thesis by auto
next
case False
- have "0 \<le> Ifloat ?x4" and "Ifloat ?x4 \<le> pi / 2" using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` unfolding Ifloat_mult Float_num by auto
+ 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
from cos_boundaries[OF this]
- have lb: "Ifloat (?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> Ifloat (?ub_horner ?x4)" by auto
+ have lb: "real (?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> real (?ub_horner ?x4)" by auto
have eq_4: "?x2 * Float 1 -1 = x * Float 1 -2" by (cases x, auto simp add: times_float.simps)
- have "Ifloat (?lb x) \<le> ?cos x"
+ have "real (?lb x) \<le> ?cos x"
proof -
- have "-pi \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi" unfolding Ifloat_mult Float_num using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto
- from lb_half[OF lb_half[OF lb this] `-pi \<le> Ifloat x` `Ifloat x \<le> pi`, unfolded eq_4]
+ 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
+ from lb_half[OF lb_half[OF lb this] `-pi \<le> real x` `real x \<le> pi`, unfolded eq_4]
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 .
qed
- moreover have "?cos x \<le> Ifloat (?ub x)"
+ moreover have "?cos x \<le> real (?ub x)"
proof -
- have "-pi \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi" unfolding Ifloat_mult Float_num using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto
- from ub_half[OF ub_half[OF ub this] `-pi \<le> Ifloat x` `Ifloat x \<le> pi`, unfolded eq_4]
+ 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
+ from ub_half[OF ub_half[OF ub this] `-pi \<le> real x` `real x \<le> pi`, unfolded eq_4]
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 .
qed
ultimately show ?thesis by auto
@@ -1142,38 +1142,38 @@
qed
qed
-lemma lb_cos_minus: assumes "-pi \<le> Ifloat x" and "Ifloat x \<le> 0"
- shows "cos (Ifloat (-x)) \<in> {Ifloat (lb_cos prec (-x)) .. Ifloat (ub_cos prec (-x))}"
+lemma lb_cos_minus: assumes "-pi \<le> real x" and "real x \<le> 0"
+ shows "cos (real (-x)) \<in> {real (lb_cos prec (-x)) .. real (ub_cos prec (-x))}"
proof -
- have "0 \<le> Ifloat (-x)" and "Ifloat (-x) \<le> pi" using `-pi \<le> Ifloat x` `Ifloat x \<le> 0` by auto
+ have "0 \<le> real (-x)" and "real (-x) \<le> pi" using `-pi \<le> real x` `real x \<le> 0` by auto
from lb_cos[OF this] show ?thesis .
qed
-lemma bnds_cos: "\<forall> x lx ux. (l, u) = bnds_cos prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> cos x \<and> cos x \<le> Ifloat u"
+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"
proof (rule allI, rule allI, rule allI, rule impI)
fix x lx ux
- assume "(l, u) = bnds_cos prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux}"
- hence bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
+ assume "(l, u) = bnds_cos prec lx ux \<and> x \<in> {real lx .. real ux}"
+ hence bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {real lx .. real ux}" by auto
let ?lpi = "lb_pi prec"
- have [intro!]: "Ifloat lx \<le> Ifloat ux" using x by auto
+ have [intro!]: "real lx \<le> real ux" using x by auto
hence "lx \<le> ux" unfolding le_float_def .
- show "Ifloat l \<le> cos x \<and> cos x \<le> Ifloat u"
+ show "real l \<le> cos x \<and> cos x \<le> real u"
proof (cases "lx < -?lpi \<or> ux > ?lpi")
case True
show ?thesis using bnds unfolding bnds_cos_def if_P[OF True] Let_def using cos_le_one cos_ge_minus_one by auto
next
case False note not_out = this
- hence lpi_lx: "- Ifloat ?lpi \<le> Ifloat lx" and lpi_ux: "Ifloat ux \<le> Ifloat ?lpi" unfolding le_float_def less_float_def by auto
+ hence lpi_lx: "- real ?lpi \<le> real lx" and lpi_ux: "real ux \<le> real ?lpi" unfolding le_float_def less_float_def by auto
from pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1, THEN le_imp_neg_le] lpi_lx
- have "- pi \<le> Ifloat lx" by (rule order_trans)
- hence "- pi \<le> x" and "- pi \<le> Ifloat ux" and "x \<le> Ifloat ux" using x by auto
+ have "- pi \<le> real lx" by (rule order_trans)
+ hence "- pi \<le> x" and "- pi \<le> real ux" and "x \<le> real ux" using x by auto
from lpi_ux pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1]
- have "Ifloat ux \<le> pi" by (rule order_trans)
- hence "x \<le> pi" and "Ifloat lx \<le> pi" and "Ifloat lx \<le> x" using x by auto
+ have "real ux \<le> pi" by (rule order_trans)
+ hence "x \<le> pi" and "real lx \<le> pi" and "real lx \<le> x" using x by auto
note lb_cos_minus_bottom = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct1]
note lb_cos_minus_top = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct2]
@@ -1182,50 +1182,50 @@
show ?thesis
proof (cases "ux \<le> 0")
- case True hence "Ifloat ux \<le> 0" unfolding le_float_def by auto
- hence "x \<le> 0" and "Ifloat lx \<le> 0" using x by auto
+ case True hence "real ux \<le> 0" unfolding le_float_def by auto
+ hence "x \<le> 0" and "real lx \<le> 0" using x by auto
- { have "Ifloat (lb_cos prec (-lx)) \<le> cos (Ifloat (-lx))" using lb_cos_minus_bottom[OF `-pi \<le> Ifloat lx` `Ifloat lx \<le> 0`] .
- also have "\<dots> \<le> cos x" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> 0`] .
- finally have "Ifloat (lb_cos prec (-lx)) \<le> cos x" . }
+ { have "real (lb_cos prec (-lx)) \<le> cos (real (-lx))" using lb_cos_minus_bottom[OF `-pi \<le> real lx` `real lx \<le> 0`] .
+ also have "\<dots> \<le> cos x" unfolding real_of_float_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> real lx` `real lx \<le> x` `x \<le> 0`] .
+ finally have "real (lb_cos prec (-lx)) \<le> cos x" . }
moreover
- { have "cos x \<le> cos (Ifloat (-ux))" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> 0`] .
- also have "\<dots> \<le> Ifloat (ub_cos prec (-ux))" using lb_cos_minus_top[OF `-pi \<le> Ifloat ux` `Ifloat ux \<le> 0`] .
- finally have "cos x \<le> Ifloat (ub_cos prec (-ux))" . }
+ { have "cos x \<le> cos (real (-ux))" unfolding real_of_float_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> x` `x \<le> real ux` `real ux \<le> 0`] .
+ also have "\<dots> \<le> real (ub_cos prec (-ux))" using lb_cos_minus_top[OF `-pi \<le> real ux` `real ux \<le> 0`] .
+ finally have "cos x \<le> real (ub_cos prec (-ux))" . }
ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_P[OF True] by auto
next
case False note not_ux = this
show ?thesis
proof (cases "0 \<le> lx")
- case True hence "0 \<le> Ifloat lx" unfolding le_float_def by auto
- hence "0 \<le> x" and "0 \<le> Ifloat ux" using x by auto
+ case True hence "0 \<le> real lx" unfolding le_float_def by auto
+ hence "0 \<le> x" and "0 \<le> real ux" using x by auto
- { have "Ifloat (lb_cos prec ux) \<le> cos (Ifloat ux)" using lb_cos_bottom[OF `0 \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
- also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
- finally have "Ifloat (lb_cos prec ux) \<le> cos x" . }
+ { have "real (lb_cos prec ux) \<le> cos (real ux)" using lb_cos_bottom[OF `0 \<le> real ux` `real ux \<le> pi`] .
+ also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> real ux` `real ux \<le> pi`] .
+ finally have "real (lb_cos prec ux) \<le> cos x" . }
moreover
- { have "cos x \<le> cos (Ifloat lx)" using cos_monotone_0_pi'[OF `0 \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> pi`] .
- also have "\<dots> \<le> Ifloat (ub_cos prec lx)" using lb_cos_top[OF `0 \<le> Ifloat lx` `Ifloat lx \<le> pi`] .
- finally have "cos x \<le> Ifloat (ub_cos prec lx)" . }
+ { have "cos x \<le> cos (real lx)" using cos_monotone_0_pi'[OF `0 \<le> real lx` `real lx \<le> x` `x \<le> pi`] .
+ also have "\<dots> \<le> real (ub_cos prec lx)" using lb_cos_top[OF `0 \<le> real lx` `real lx \<le> pi`] .
+ finally have "cos x \<le> real (ub_cos prec lx)" . }
ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_not_P[OF not_ux] if_P[OF True] by auto
next
case False with not_ux
- have "Ifloat lx \<le> 0" and "0 \<le> Ifloat ux" unfolding le_float_def by auto
+ have "real lx \<le> 0" and "0 \<le> real ux" unfolding le_float_def by auto
- have "Ifloat (min (lb_cos prec (-lx)) (lb_cos prec ux)) \<le> cos x"
+ have "real (min (lb_cos prec (-lx)) (lb_cos prec ux)) \<le> cos x"
proof (cases "x \<le> 0")
case True
- have "Ifloat (lb_cos prec (-lx)) \<le> cos (Ifloat (-lx))" using lb_cos_minus_bottom[OF `-pi \<le> Ifloat lx` `Ifloat lx \<le> 0`] .
- also have "\<dots> \<le> cos x" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> 0`] .
- finally show ?thesis unfolding Ifloat_min by auto
+ have "real (lb_cos prec (-lx)) \<le> cos (real (-lx))" using lb_cos_minus_bottom[OF `-pi \<le> real lx` `real lx \<le> 0`] .
+ also have "\<dots> \<le> cos x" unfolding real_of_float_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> real lx` `real lx \<le> x` `x \<le> 0`] .
+ finally show ?thesis unfolding real_of_float_min by auto
next
case False hence "0 \<le> x" by auto
- have "Ifloat (lb_cos prec ux) \<le> cos (Ifloat ux)" using lb_cos_bottom[OF `0 \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
- also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
- finally show ?thesis unfolding Ifloat_min by auto
+ have "real (lb_cos prec ux) \<le> cos (real ux)" using lb_cos_bottom[OF `0 \<le> real ux` `real ux \<le> pi`] .
+ also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> real ux` `real ux \<le> pi`] .
+ finally show ?thesis unfolding real_of_float_min by auto
qed
- moreover have "cos x \<le> Ifloat (Float 1 0)" by auto
+ moreover have "cos x \<le> real (Float 1 0)" by auto
ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_not_P[OF not_ux] if_not_P[OF False] by auto
qed
qed
@@ -1254,45 +1254,45 @@
in if lx \<le> - half_pi \<or> half_pi \<le> ux then (Float -1 0, Float 1 0)
else (lb_sin prec lx, ub_sin prec ux))"
-lemma lb_sin: assumes "- (pi / 2) \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
- shows "sin (Ifloat x) \<in> { Ifloat (lb_sin prec x) .. Ifloat (ub_sin prec x) }" (is "?sin x \<in> { ?lb x .. ?ub x}")
+lemma lb_sin: assumes "- (pi / 2) \<le> real x" and "real x \<le> pi / 2"
+ shows "sin (real x) \<in> { real (lb_sin prec x) .. real (ub_sin prec x) }" (is "?sin x \<in> { ?lb x .. ?ub x}")
proof -
- { fix x :: float assume "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
- hence "\<not> (x < 0)" and "- (pi / 2) \<le> Ifloat x" unfolding less_float_def using pi_ge_two by auto
+ { fix x :: float assume "0 \<le> real x" and "real x \<le> pi / 2"
+ hence "\<not> (x < 0)" and "- (pi / 2) \<le> real x" unfolding less_float_def using pi_ge_two by auto
- have "Ifloat x \<le> pi" using `Ifloat x \<le> pi / 2` using pi_ge_two by auto
+ have "real x \<le> pi" using `real x \<le> pi / 2` using pi_ge_two by auto
have "?sin x \<in> { ?lb x .. ?ub x}"
proof (cases "x \<le> Float 1 -1")
- case True from sin_boundaries[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi / 2`]
+ case True from sin_boundaries[OF `0 \<le> real x` `real x \<le> pi / 2`]
show ?thesis unfolding lb_sin.simps[of prec x] ub_sin.simps[of prec x] if_not_P[OF `\<not> (x < 0)`] if_P[OF True] Let_def .
next
case False
- have "0 \<le> cos (Ifloat x)" using cos_ge_zero[OF _ `Ifloat x \<le> pi /2`] `0 \<le> Ifloat x` pi_ge_two by auto
- have "0 \<le> sin (Ifloat x)" using `0 \<le> Ifloat x` and `Ifloat x \<le> pi / 2` using sin_ge_zero by auto
+ have "0 \<le> cos (real x)" using cos_ge_zero[OF _ `real x \<le> pi /2`] `0 \<le> real x` pi_ge_two by auto
+ have "0 \<le> sin (real x)" using `0 \<le> real x` and `real x \<le> pi / 2` using sin_ge_zero by auto
have "?sin x \<le> ?ub x"
proof (cases "lb_cos prec x < 0")
case True
have "?sin x \<le> 1" using sin_le_one .
- also have "\<dots> \<le> Ifloat (the (ub_sqrt prec 1))" using ub_sqrt_lower_bound[where prec=prec and x=1] unfolding Ifloat_1 by auto
+ also have "\<dots> \<le> real (the (ub_sqrt prec 1))" using ub_sqrt_lower_bound[where prec=prec and x=1] unfolding real_of_float_1 by auto
finally show ?thesis unfolding ub_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] Let_def .
next
- case False hence "0 \<le> Ifloat (lb_cos prec x)" unfolding less_float_def by auto
+ case False hence "0 \<le> real (lb_cos prec x)" unfolding less_float_def by auto
- have "sin (Ifloat x) = sqrt (1 - cos (Ifloat x) ^ 2)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \<le> sin (Ifloat x)` by auto
- also have "\<dots> \<le> sqrt (Ifloat (1 - lb_cos prec x * lb_cos prec x))"
+ have "sin (real x) = sqrt (1 - cos (real x) ^ 2)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \<le> sin (real x)` by auto
+ also have "\<dots> \<le> sqrt (real (1 - lb_cos prec x * lb_cos prec x))"
proof (rule real_sqrt_le_mono)
- have "Ifloat (lb_cos prec x * lb_cos prec x) \<le> cos (Ifloat x) ^ 2" unfolding numeral_2_eq_2 power_Suc2 power_0 Ifloat_mult
- using `0 \<le> Ifloat (lb_cos prec x)` lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] `0 \<le> cos (Ifloat x)` by(auto intro!: mult_mono)
- thus "1 - cos (Ifloat x) ^ 2 \<le> Ifloat (1 - lb_cos prec x * lb_cos prec x)" unfolding Ifloat_sub Ifloat_1 by auto
+ have "real (lb_cos prec x * lb_cos prec x) \<le> cos (real x) ^ 2" unfolding numeral_2_eq_2 power_Suc2 power_0 real_of_float_mult
+ using `0 \<le> real (lb_cos prec x)` lb_cos[OF `0 \<le> real x` `real x \<le> pi`] `0 \<le> cos (real x)` by(auto intro!: mult_mono)
+ thus "1 - cos (real x) ^ 2 \<le> real (1 - lb_cos prec x * lb_cos prec x)" unfolding real_of_float_sub real_of_float_1 by auto
qed
- also have "\<dots> \<le> Ifloat (the (ub_sqrt prec (1 - lb_cos prec x * lb_cos prec x)))"
+ also have "\<dots> \<le> real (the (ub_sqrt prec (1 - lb_cos prec x * lb_cos prec x)))"
proof (rule ub_sqrt_lower_bound)
- have "Ifloat (lb_cos prec x) \<le> cos (Ifloat x)" using lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] by auto
+ have "real (lb_cos prec x) \<le> cos (real x)" using lb_cos[OF `0 \<le> real x` `real x \<le> pi`] by auto
from mult_mono[OF order_trans[OF this cos_le_one] order_trans[OF this cos_le_one]]
- have "Ifloat (lb_cos prec x) * Ifloat (lb_cos prec x) \<le> 1" using `0 \<le> Ifloat (lb_cos prec x)` by auto
- thus "0 \<le> Ifloat (1 - lb_cos prec x * lb_cos prec x)" by auto
+ have "real (lb_cos prec x) * real (lb_cos prec x) \<le> 1" using `0 \<le> real (lb_cos prec x)` by auto
+ thus "0 \<le> real (1 - lb_cos prec x * lb_cos prec x)" by auto
qed
finally show ?thesis unfolding ub_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] Let_def .
qed
@@ -1301,25 +1301,25 @@
proof (cases "1 < ub_cos prec x")
case True
show ?thesis unfolding lb_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] Let_def
- by (rule order_trans[OF _ sin_ge_zero[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`]])
- (auto simp add: lb_sqrt_upper_bound[where prec=prec and x=0, unfolded Ifloat_0 real_sqrt_zero])
+ by (rule order_trans[OF _ sin_ge_zero[OF `0 \<le> real x` `real x \<le> pi`]])
+ (auto simp add: lb_sqrt_upper_bound[where prec=prec and x=0, unfolded real_of_float_0 real_sqrt_zero])
next
- case False hence "Ifloat (ub_cos prec x) \<le> 1" unfolding less_float_def by auto
- have "0 \<le> Ifloat (ub_cos prec x)" using order_trans[OF `0 \<le> cos (Ifloat x)`] lb_cos `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto
+ case False hence "real (ub_cos prec x) \<le> 1" unfolding less_float_def by auto
+ have "0 \<le> real (ub_cos prec x)" using order_trans[OF `0 \<le> cos (real x)`] lb_cos `0 \<le> real x` `real x \<le> pi` by auto
- have "Ifloat (the (lb_sqrt prec (1 - ub_cos prec x * ub_cos prec x))) \<le> sqrt (Ifloat (1 - ub_cos prec x * ub_cos prec x))"
+ have "real (the (lb_sqrt prec (1 - ub_cos prec x * ub_cos prec x))) \<le> sqrt (real (1 - ub_cos prec x * ub_cos prec x))"
proof (rule lb_sqrt_upper_bound)
- from mult_mono[OF `Ifloat (ub_cos prec x) \<le> 1` `Ifloat (ub_cos prec x) \<le> 1`] `0 \<le> Ifloat (ub_cos prec x)`
- have "Ifloat (ub_cos prec x) * Ifloat (ub_cos prec x) \<le> 1" by auto
- thus "0 \<le> Ifloat (1 - ub_cos prec x * ub_cos prec x)" by auto
+ from mult_mono[OF `real (ub_cos prec x) \<le> 1` `real (ub_cos prec x) \<le> 1`] `0 \<le> real (ub_cos prec x)`
+ have "real (ub_cos prec x) * real (ub_cos prec x) \<le> 1" by auto
+ thus "0 \<le> real (1 - ub_cos prec x * ub_cos prec x)" by auto
qed
- also have "\<dots> \<le> sqrt (1 - cos (Ifloat x) ^ 2)"
+ also have "\<dots> \<le> sqrt (1 - cos (real x) ^ 2)"
proof (rule real_sqrt_le_mono)
- have "cos (Ifloat x) ^ 2 \<le> Ifloat (ub_cos prec x * ub_cos prec x)" unfolding numeral_2_eq_2 power_Suc2 power_0 Ifloat_mult
- using `0 \<le> Ifloat (ub_cos prec x)` lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] `0 \<le> cos (Ifloat x)` by(auto intro!: mult_mono)
- thus "Ifloat (1 - ub_cos prec x * ub_cos prec x) \<le> 1 - cos (Ifloat x) ^ 2" unfolding Ifloat_sub Ifloat_1 by auto
+ have "cos (real x) ^ 2 \<le> real (ub_cos prec x * ub_cos prec x)" unfolding numeral_2_eq_2 power_Suc2 power_0 real_of_float_mult
+ using `0 \<le> real (ub_cos prec x)` lb_cos[OF `0 \<le> real x` `real x \<le> pi`] `0 \<le> cos (real x)` by(auto intro!: mult_mono)
+ thus "real (1 - ub_cos prec x * ub_cos prec x) \<le> 1 - cos (real x) ^ 2" unfolding real_of_float_sub real_of_float_1 by auto
qed
- also have "\<dots> = sin (Ifloat x)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \<le> sin (Ifloat x)` by auto
+ also have "\<dots> = sin (real x)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \<le> sin (real x)` by auto
finally show ?thesis unfolding lb_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] Let_def .
qed
ultimately show ?thesis by auto
@@ -1329,40 +1329,40 @@
show ?thesis
proof (cases "x < 0")
case True
- hence "0 \<le> Ifloat (-x)" and "Ifloat (- x) \<le> pi / 2" using `-(pi/2) \<le> Ifloat x` unfolding less_float_def by auto
+ hence "0 \<le> real (-x)" and "real (- x) \<le> pi / 2" using `-(pi/2) \<le> real x` unfolding less_float_def by auto
from for_pos[OF this]
- show ?thesis unfolding Ifloat_minus sin_minus lb_sin.simps[of prec x] ub_sin.simps[of prec x] if_P[OF True] Let_def atLeastAtMost_iff by auto
+ show ?thesis unfolding real_of_float_minus sin_minus lb_sin.simps[of prec x] ub_sin.simps[of prec x] if_P[OF True] Let_def atLeastAtMost_iff by auto
next
- case False hence "0 \<le> Ifloat x" unfolding less_float_def by auto
- from for_pos[OF this `Ifloat x \<le> pi /2`]
+ case False hence "0 \<le> real x" unfolding less_float_def by auto
+ from for_pos[OF this `real x \<le> pi /2`]
show ?thesis .
qed
qed
-lemma bnds_sin: "\<forall> x lx ux. (l, u) = bnds_sin prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> sin x \<and> sin x \<le> Ifloat u"
+lemma bnds_sin: "\<forall> x lx ux. (l, u) = bnds_sin prec lx ux \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> sin x \<and> sin x \<le> real u"
proof (rule allI, rule allI, rule allI, rule impI)
fix x lx ux
- assume "(l, u) = bnds_sin prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux}"
- hence bnds: "(l, u) = bnds_sin prec lx ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
- show "Ifloat l \<le> sin x \<and> sin x \<le> Ifloat u"
+ assume "(l, u) = bnds_sin prec lx ux \<and> x \<in> {real lx .. real ux}"
+ hence bnds: "(l, u) = bnds_sin prec lx ux" and x: "x \<in> {real lx .. real ux}" by auto
+ show "real l \<le> sin x \<and> sin x \<le> real u"
proof (cases "lx \<le> - (lb_pi prec * Float 1 -1) \<or> lb_pi prec * Float 1 -1 \<le> ux")
case True show ?thesis using bnds unfolding bnds_sin_def if_P[OF True] Let_def by auto
next
case False
hence "- lb_pi prec * Float 1 -1 \<le> lx" and "ux \<le> lb_pi prec * Float 1 -1" unfolding le_float_def by auto
- moreover have "Ifloat (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding Ifloat_mult using pi_boundaries by auto
- ultimately have "- (pi / 2) \<le> Ifloat lx" and "Ifloat ux \<le> pi / 2" and "Ifloat lx \<le> Ifloat ux" unfolding le_float_def using x by auto
- hence "- (pi / 2) \<le> Ifloat ux" and "Ifloat lx \<le> pi / 2" by auto
+ moreover have "real (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding real_of_float_mult using pi_boundaries by auto
+ ultimately have "- (pi / 2) \<le> real lx" and "real ux \<le> pi / 2" and "real lx \<le> real ux" unfolding le_float_def using x by auto
+ hence "- (pi / 2) \<le> real ux" and "real lx \<le> pi / 2" by auto
- have "- (pi / 2) \<le> x""x \<le> pi / 2" using `Ifloat ux \<le> pi / 2` `- (pi /2) \<le> Ifloat lx` x by auto
+ have "- (pi / 2) \<le> x""x \<le> pi / 2" using `real ux \<le> pi / 2` `- (pi /2) \<le> real lx` x by auto
- { have "Ifloat (lb_sin prec lx) \<le> sin (Ifloat lx)" using lb_sin[OF `- (pi / 2) \<le> Ifloat lx` `Ifloat lx \<le> pi / 2`] unfolding atLeastAtMost_iff by auto
- also have "\<dots> \<le> sin x" using sin_monotone_2pi' `- (pi / 2) \<le> Ifloat lx` x `x \<le> pi / 2` by auto
- finally have "Ifloat (lb_sin prec lx) \<le> sin x" . }
+ { have "real (lb_sin prec lx) \<le> sin (real lx)" using lb_sin[OF `- (pi / 2) \<le> real lx` `real lx \<le> pi / 2`] unfolding atLeastAtMost_iff by auto
+ also have "\<dots> \<le> sin x" using sin_monotone_2pi' `- (pi / 2) \<le> real lx` x `x \<le> pi / 2` by auto
+ finally have "real (lb_sin prec lx) \<le> sin x" . }
moreover
- { have "sin x \<le> sin (Ifloat ux)" using sin_monotone_2pi' `- (pi / 2) \<le> x` x `Ifloat ux \<le> pi / 2` by auto
- also have "\<dots> \<le> Ifloat (ub_sin prec ux)" using lb_sin[OF `- (pi / 2) \<le> Ifloat ux` `Ifloat ux \<le> pi / 2`] unfolding atLeastAtMost_iff by auto
- finally have "sin x \<le> Ifloat (ub_sin prec ux)" . }
+ { have "sin x \<le> sin (real ux)" using sin_monotone_2pi' `- (pi / 2) \<le> x` x `real ux \<le> pi / 2` by auto
+ also have "\<dots> \<le> real (ub_sin prec ux)" using lb_sin[OF `- (pi / 2) \<le> real ux` `real ux \<le> pi / 2`] unfolding atLeastAtMost_iff by auto
+ finally have "sin x \<le> real (ub_sin prec ux)" . }
ultimately
show ?thesis using bnds unfolding bnds_sin_def if_not_P[OF False] Let_def by auto
qed
@@ -1378,8 +1378,8 @@
"lb_exp_horner prec 0 i k x = 0" |
"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"
-lemma bnds_exp_horner: assumes "Ifloat x \<le> 0"
- shows "exp (Ifloat x) \<in> { Ifloat (lb_exp_horner prec (get_even n) 1 1 x) .. Ifloat (ub_exp_horner prec (get_odd n) 1 1 x) }"
+lemma bnds_exp_horner: assumes "real x \<le> 0"
+ 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) }"
proof -
{ fix n
have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m" by (induct n, auto)
@@ -1388,39 +1388,39 @@
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,
OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]
- { have "Ifloat (lb_exp_horner prec (get_even n) 1 1 x) \<le> (\<Sum>j = 0..<get_even n. 1 / real (fact j) * Ifloat x ^ j)"
+ { 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)"
using bounds(1) by auto
- also have "\<dots> \<le> exp (Ifloat x)"
+ also have "\<dots> \<le> exp (real x)"
proof -
- obtain t where "\<bar>t\<bar> \<le> \<bar>Ifloat x\<bar>" and "exp (Ifloat x) = (\<Sum>m = 0..<get_even n. (Ifloat x) ^ m / real (fact m)) + exp t / real (fact (get_even n)) * (Ifloat x) ^ (get_even n)"
+ 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)"
using Maclaurin_exp_le by blast
- moreover have "0 \<le> exp t / real (fact (get_even n)) * (Ifloat x) ^ (get_even n)"
+ moreover have "0 \<le> exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
by (auto intro!: mult_nonneg_nonneg divide_nonneg_pos simp add: get_even zero_le_even_power exp_gt_zero)
ultimately show ?thesis
using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
qed
- finally have "Ifloat (lb_exp_horner prec (get_even n) 1 1 x) \<le> exp (Ifloat x)" .
+ finally have "real (lb_exp_horner prec (get_even n) 1 1 x) \<le> exp (real x)" .
} moreover
{
- have x_less_zero: "Ifloat x ^ get_odd n \<le> 0"
- proof (cases "Ifloat x = 0")
+ have x_less_zero: "real x ^ get_odd n \<le> 0"
+ proof (cases "real x = 0")
case True
have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
thus ?thesis unfolding True power_0_left by auto
next
- case False hence "Ifloat x < 0" using `Ifloat x \<le> 0` by auto
- show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq get_odd `Ifloat x < 0`)
+ case False hence "real x < 0" using `real x \<le> 0` by auto
+ show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq get_odd `real x < 0`)
qed
- obtain t where "\<bar>t\<bar> \<le> \<bar>Ifloat x\<bar>" and "exp (Ifloat x) = (\<Sum>m = 0..<get_odd n. (Ifloat x) ^ m / real (fact m)) + exp t / real (fact (get_odd n)) * (Ifloat x) ^ (get_odd n)"
+ 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)"
using Maclaurin_exp_le by blast
- moreover have "exp t / real (fact (get_odd n)) * (Ifloat x) ^ (get_odd n) \<le> 0"
+ moreover have "exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n) \<le> 0"
by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero exp_gt_zero)
- ultimately have "exp (Ifloat x) \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * Ifloat x ^ j)"
+ ultimately have "exp (real x) \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * real x ^ j)"
using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
- also have "\<dots> \<le> Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)"
+ also have "\<dots> \<le> real (ub_exp_horner prec (get_odd n) 1 1 x)"
using bounds(2) by auto
- finally have "exp (Ifloat x) \<le> Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)" .
+ finally have "exp (real x) \<le> real (ub_exp_horner prec (get_odd n) 1 1 x)" .
} ultimately show ?thesis by auto
qed
@@ -1443,12 +1443,12 @@
proof -
have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
- have "1 / 4 = Ifloat (Float 1 -2)" unfolding Float_num by auto
- also have "\<dots> \<le> Ifloat (lb_exp_horner 1 (get_even 4) 1 1 (- 1))"
+ have "1 / 4 = real (Float 1 -2)" unfolding Float_num by auto
+ also have "\<dots> \<le> real (lb_exp_horner 1 (get_even 4) 1 1 (- 1))"
unfolding get_even_def eq4
by (auto simp add: lapprox_posrat_def rapprox_posrat_def normfloat.simps)
- also have "\<dots> \<le> exp (Ifloat (- 1))" using bnds_exp_horner[where x="- 1"] by auto
- finally show ?thesis unfolding Ifloat_minus Ifloat_1 .
+ also have "\<dots> \<le> exp (real (- 1 :: float))" using bnds_exp_horner[where x="- 1"] by auto
+ finally show ?thesis unfolding real_of_float_minus real_of_float_1 .
qed
lemma lb_exp_pos: assumes "\<not> 0 < x" shows "0 < lb_exp prec x"
@@ -1457,9 +1457,9 @@
let "?horner x" = "let y = ?lb_horner x in if y \<le> 0 then Float 1 -2 else y"
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)
moreover { fix x :: float fix num :: nat
- have "0 < Ifloat (?horner x) ^ num" using `0 < ?horner x`[unfolded less_float_def Ifloat_0] by (rule zero_less_power)
- also have "\<dots> = Ifloat ((?horner x) ^ num)" using float_power by auto
- finally have "0 < Ifloat ((?horner x) ^ num)" .
+ have "0 < real (?horner x) ^ num" using `0 < ?horner x`[unfolded less_float_def real_of_float_0] by (rule zero_less_power)
+ also have "\<dots> = real ((?horner x) ^ num)" using float_power by auto
+ finally have "0 < real ((?horner x) ^ num)" .
}
ultimately show ?thesis
unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def
@@ -1467,25 +1467,25 @@
qed
lemma exp_boundaries': assumes "x \<le> 0"
- shows "exp (Ifloat x) \<in> { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}"
+ shows "exp (real x) \<in> { real (lb_exp prec x) .. real (ub_exp prec x)}"
proof -
let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"
- have "Ifloat x \<le> 0" and "\<not> x > 0" using `x \<le> 0` unfolding le_float_def less_float_def by auto
+ have "real x \<le> 0" and "\<not> x > 0" using `x \<le> 0` unfolding le_float_def less_float_def by auto
show ?thesis
proof (cases "x < - 1")
- case False hence "- 1 \<le> Ifloat x" unfolding less_float_def by auto
+ case False hence "- 1 \<le> real x" unfolding less_float_def by auto
show ?thesis
proof (cases "?lb_exp_horner x \<le> 0")
- from `\<not> x < - 1` have "- 1 \<le> Ifloat x" unfolding less_float_def by auto
- hence "exp (- 1) \<le> exp (Ifloat x)" unfolding exp_le_cancel_iff .
+ from `\<not> x < - 1` have "- 1 \<le> real x" unfolding less_float_def by auto
+ hence "exp (- 1) \<le> exp (real x)" unfolding exp_le_cancel_iff .
from order_trans[OF exp_m1_ge_quarter this]
- have "Ifloat (Float 1 -2) \<le> exp (Ifloat x)" unfolding Float_num .
+ have "real (Float 1 -2) \<le> exp (real x)" unfolding Float_num .
moreover case True
- ultimately show ?thesis using bnds_exp_horner `Ifloat x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by auto
+ ultimately show ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by auto
next
- case False thus ?thesis using bnds_exp_horner `Ifloat x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by (auto simp add: Let_def)
+ case False thus ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by (auto simp add: Let_def)
qed
next
case True
@@ -1493,10 +1493,10 @@
obtain m e where Float_floor: "floor_fl x = Float m e" by (cases "floor_fl x", auto)
let ?num = "nat (- m) * 2 ^ nat e"
- have "Ifloat (floor_fl x) < - 1" using floor_fl `x < - 1` unfolding le_float_def less_float_def Ifloat_minus Ifloat_1 by (rule order_le_less_trans)
- hence "Ifloat (floor_fl x) < 0" unfolding Float_floor Ifloat.simps using zero_less_pow2[of xe] by auto
+ 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)
+ hence "real (floor_fl x) < 0" unfolding Float_floor real_of_float_simp using zero_less_pow2[of xe] by auto
hence "m < 0"
- unfolding less_float_def Ifloat_0 Float_floor Ifloat.simps
+ unfolding less_float_def real_of_float_0 Float_floor real_of_float_simp
unfolding pos_prod_lt[OF zero_less_pow2[of e], unfolded real_mult_commute] by auto
hence "1 \<le> - m" by auto
hence "0 < nat (- m)" by auto
@@ -1506,56 +1506,56 @@
ultimately have "0 < ?num" by auto
hence "real ?num \<noteq> 0" by auto
have e_nat: "int (nat e) = e" using `0 \<le> e` by auto
- have num_eq: "real ?num = Ifloat (- floor_fl x)" using `0 < nat (- m)`
- unfolding Float_floor Ifloat_minus Ifloat.simps real_of_nat_mult pow2_int[of "nat e", unfolded e_nat] realpow_real_of_nat[symmetric] by auto
- have "0 < - floor_fl x" using `0 < ?num`[unfolded real_of_nat_less_iff[symmetric]] unfolding less_float_def num_eq[symmetric] Ifloat_0 real_of_nat_zero .
- hence "Ifloat (floor_fl x) < 0" unfolding less_float_def by auto
+ have num_eq: "real ?num = real (- floor_fl x)" using `0 < nat (- m)`
+ unfolding Float_floor real_of_float_minus real_of_float_simp real_of_nat_mult pow2_int[of "nat e", unfolded e_nat] realpow_real_of_nat[symmetric] by auto
+ 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 .
+ hence "real (floor_fl x) < 0" unfolding less_float_def by auto
- have "exp (Ifloat x) \<le> Ifloat (ub_exp prec x)"
+ have "exp (real x) \<le> real (ub_exp prec x)"
proof -
- have div_less_zero: "Ifloat (float_divr prec x (- floor_fl x)) \<le> 0"
- using float_divr_nonpos_pos_upper_bound[OF `x \<le> 0` `0 < - floor_fl x`] unfolding le_float_def Ifloat_0 .
+ have div_less_zero: "real (float_divr prec x (- floor_fl x)) \<le> 0"
+ using float_divr_nonpos_pos_upper_bound[OF `x \<le> 0` `0 < - floor_fl x`] unfolding le_float_def real_of_float_0 .
- have "exp (Ifloat x) = exp (real ?num * (Ifloat x / real ?num))" using `real ?num \<noteq> 0` by auto
- also have "\<dots> = exp (Ifloat x / real ?num) ^ ?num" unfolding exp_real_of_nat_mult ..
- also have "\<dots> \<le> exp (Ifloat (float_divr prec x (- floor_fl x))) ^ ?num" unfolding num_eq
+ have "exp (real x) = exp (real ?num * (real x / real ?num))" using `real ?num \<noteq> 0` by auto
+ also have "\<dots> = exp (real x / real ?num) ^ ?num" unfolding exp_real_of_nat_mult ..
+ also have "\<dots> \<le> exp (real (float_divr prec x (- floor_fl x))) ^ ?num" unfolding num_eq
by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
- also have "\<dots> \<le> Ifloat ((?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num)" unfolding float_power
+ also have "\<dots> \<le> real ((?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num)" unfolding float_power
by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
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 .
qed
moreover
- have "Ifloat (lb_exp prec x) \<le> exp (Ifloat x)"
+ have "real (lb_exp prec x) \<le> exp (real x)"
proof -
let ?divl = "float_divl prec x (- Float m e)"
let ?horner = "?lb_exp_horner ?divl"
show ?thesis
proof (cases "?horner \<le> 0")
- case False hence "0 \<le> Ifloat ?horner" unfolding le_float_def by auto
+ case False hence "0 \<le> real ?horner" unfolding le_float_def by auto
- have div_less_zero: "Ifloat (float_divl prec x (- floor_fl x)) \<le> 0"
- using `Ifloat (floor_fl x) < 0` `Ifloat x \<le> 0` by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
+ have div_less_zero: "real (float_divl prec x (- floor_fl x)) \<le> 0"
+ using `real (floor_fl x) < 0` `real x \<le> 0` by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
- have "Ifloat ((?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num) \<le>
- exp (Ifloat (float_divl prec x (- floor_fl x))) ^ ?num" unfolding float_power
- using `0 \<le> Ifloat ?horner`[unfolded Float_floor[symmetric]] bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1] by (auto intro!: power_mono)
- also have "\<dots> \<le> exp (Ifloat x / real ?num) ^ ?num" unfolding num_eq
- using float_divl by (auto intro!: power_mono simp del: Ifloat_minus)
- also have "\<dots> = exp (real ?num * (Ifloat x / real ?num))" unfolding exp_real_of_nat_mult ..
- also have "\<dots> = exp (Ifloat x)" using `real ?num \<noteq> 0` by auto
+ have "real ((?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num) \<le>
+ exp (real (float_divl prec x (- floor_fl x))) ^ ?num" unfolding float_power
+ 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)
+ also have "\<dots> \<le> exp (real x / real ?num) ^ ?num" unfolding num_eq
+ using float_divl by (auto intro!: power_mono simp del: real_of_float_minus)
+ also have "\<dots> = exp (real ?num * (real x / real ?num))" unfolding exp_real_of_nat_mult ..
+ also have "\<dots> = exp (real x)" using `real ?num \<noteq> 0` by auto
finally show ?thesis
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
next
case True
- have "Ifloat (floor_fl x) \<noteq> 0" and "Ifloat (floor_fl x) \<le> 0" using `Ifloat (floor_fl x) < 0` by auto
- from divide_right_mono_neg[OF floor_fl[of x] `Ifloat (floor_fl x) \<le> 0`, unfolded divide_self[OF `Ifloat (floor_fl x) \<noteq> 0`]]
- have "- 1 \<le> Ifloat x / Ifloat (- floor_fl x)" unfolding Ifloat_minus by auto
+ have "real (floor_fl x) \<noteq> 0" and "real (floor_fl x) \<le> 0" using `real (floor_fl x) < 0` by auto
+ 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`]]
+ have "- 1 \<le> real x / real (- floor_fl x)" unfolding real_of_float_minus by auto
from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
- have "Ifloat (Float 1 -2) \<le> exp (Ifloat x / Ifloat (- floor_fl x))" unfolding Float_num .
- hence "Ifloat (Float 1 -2) ^ ?num \<le> exp (Ifloat x / Ifloat (- floor_fl x)) ^ ?num"
+ have "real (Float 1 -2) \<le> exp (real x / real (- floor_fl x))" unfolding Float_num .
+ hence "real (Float 1 -2) ^ ?num \<le> exp (real x / real (- floor_fl x)) ^ ?num"
by (auto intro!: power_mono simp add: Float_num)
- also have "\<dots> = exp (Ifloat x)" unfolding num_eq exp_real_of_nat_mult[symmetric] using `Ifloat (floor_fl x) \<noteq> 0` by auto
+ also have "\<dots> = exp (real x)" unfolding num_eq exp_real_of_nat_mult[symmetric] using `real (floor_fl x) \<noteq> 0` by auto
finally show ?thesis
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 .
qed
@@ -1564,7 +1564,7 @@
qed
qed
-lemma exp_boundaries: "exp (Ifloat x) \<in> { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}"
+lemma exp_boundaries: "exp (real x) \<in> { real (lb_exp prec x) .. real (ub_exp prec x)}"
proof -
show ?thesis
proof (cases "0 < x")
@@ -1573,50 +1573,51 @@
next
case True hence "-x \<le> 0" unfolding less_float_def le_float_def by auto
- have "Ifloat (lb_exp prec x) \<le> exp (Ifloat x)"
+ have "real (lb_exp prec x) \<le> exp (real x)"
proof -
from exp_boundaries'[OF `-x \<le> 0`]
- have ub_exp: "exp (- Ifloat x) \<le> Ifloat (ub_exp prec (-x))" unfolding atLeastAtMost_iff Ifloat_minus by auto
+ have ub_exp: "exp (- real x) \<le> real (ub_exp prec (-x))" unfolding atLeastAtMost_iff real_of_float_minus by auto
- have "Ifloat (float_divl prec 1 (ub_exp prec (-x))) \<le> Ifloat 1 / Ifloat (ub_exp prec (-x))" using float_divl .
- also have "Ifloat 1 / Ifloat (ub_exp prec (-x)) \<le> exp (Ifloat x)"
+ 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
+ also have "\<dots> \<le> exp (real x)"
using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] exp_gt_zero, symmetric]]
unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide by auto
finally show ?thesis unfolding lb_exp.simps if_P[OF True] .
qed
moreover
- have "exp (Ifloat x) \<le> Ifloat (ub_exp prec x)"
+ have "exp (real x) \<le> real (ub_exp prec x)"
proof -
have "\<not> 0 < -x" using `0 < x` unfolding less_float_def by auto
from exp_boundaries'[OF `-x \<le> 0`]
- have lb_exp: "Ifloat (lb_exp prec (-x)) \<le> exp (- Ifloat x)" unfolding atLeastAtMost_iff Ifloat_minus by auto
+ have lb_exp: "real (lb_exp prec (-x)) \<le> exp (- real x)" unfolding atLeastAtMost_iff real_of_float_minus by auto
- have "exp (Ifloat x) \<le> Ifloat 1 / Ifloat (lb_exp prec (-x))"
- using lb_exp[unfolded inverse_le_iff_le[OF exp_gt_zero lb_exp_pos[OF `\<not> 0 < -x`, unfolded less_float_def Ifloat_0], symmetric]]
- unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide Ifloat_1 by auto
- also have "\<dots> \<le> Ifloat (float_divr prec 1 (lb_exp prec (-x)))" using float_divr .
+ have "exp (real x) \<le> real (1 :: float) / real (lb_exp prec (-x))"
+ 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],
+ symmetric]]
+ unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide real_of_float_1 by auto
+ also have "\<dots> \<le> real (float_divr prec 1 (lb_exp prec (-x)))" using float_divr .
finally show ?thesis unfolding ub_exp.simps if_P[OF True] .
qed
ultimately show ?thesis by auto
qed
qed
-lemma bnds_exp: "\<forall> x lx ux. (l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> exp x \<and> exp x \<le> Ifloat u"
+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"
proof (rule allI, rule allI, rule allI, rule impI)
fix x lx ux
- assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
- hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
+ assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {real lx .. real ux}"
+ hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {real lx .. real ux}" by auto
{ from exp_boundaries[of lx prec, unfolded l]
- have "Ifloat l \<le> exp (Ifloat lx)" by (auto simp del: lb_exp.simps)
+ have "real l \<le> exp (real lx)" by (auto simp del: lb_exp.simps)
also have "\<dots> \<le> exp x" using x by auto
- finally have "Ifloat l \<le> exp x" .
+ finally have "real l \<le> exp x" .
} moreover
- { have "exp x \<le> exp (Ifloat ux)" using x by auto
- also have "\<dots> \<le> Ifloat u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
- finally have "exp x \<le> Ifloat u" .
- } ultimately show "Ifloat l \<le> exp x \<and> exp x \<le> Ifloat u" ..
+ { have "exp x \<le> exp (real ux)" using x by auto
+ also have "\<dots> \<le> real u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
+ finally have "exp x \<le> real u" .
+ } ultimately show "real l \<le> exp x \<and> exp x \<le> real u" ..
qed
section "Logarithm"
@@ -1656,26 +1657,26 @@
qed
lemma ln_float_bounds:
- assumes "0 \<le> Ifloat x" and "Ifloat x < 1"
- shows "Ifloat (x * lb_ln_horner prec (get_even n) 1 x) \<le> ln (Ifloat x + 1)" (is "?lb \<le> ?ln")
- and "ln (Ifloat x + 1) \<le> Ifloat (x * ub_ln_horner prec (get_odd n) 1 x)" (is "?ln \<le> ?ub")
+ assumes "0 \<le> real x" and "real x < 1"
+ shows "real (x * lb_ln_horner prec (get_even n) 1 x) \<le> ln (real x + 1)" (is "?lb \<le> ?ln")
+ and "ln (real x + 1) \<le> real (x * ub_ln_horner prec (get_odd n) 1 x)" (is "?ln \<le> ?ub")
proof -
obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..
- let "?s n" = "-1^n * (1 / real (1 + n)) * (Ifloat x)^(Suc n)"
+ let "?s n" = "-1^n * (1 / real (1 + n)) * (real x)^(Suc n)"
- have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 real_mult_assoc[symmetric] Ifloat_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "Ifloat x"] ev
+ have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 real_mult_assoc[symmetric] real_of_float_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "real x"] ev
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",
- OF `0 \<le> Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> Ifloat x`
+ OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
by (rule mult_right_mono)
- also have "\<dots> \<le> ?ln" using ln_bounds(1)[OF `0 \<le> Ifloat x` `Ifloat x < 1`] by auto
+ also have "\<dots> \<le> ?ln" using ln_bounds(1)[OF `0 \<le> real x` `real x < 1`] by auto
finally show "?lb \<le> ?ln" .
- have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> Ifloat x` `Ifloat x < 1`] by auto
- also have "\<dots> \<le> ?ub" unfolding power_Suc2 real_mult_assoc[symmetric] Ifloat_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "Ifloat x"] od
+ have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> real x` `real x < 1`] by auto
+ also have "\<dots> \<le> ?ub" unfolding power_Suc2 real_mult_assoc[symmetric] real_of_float_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "real x"] od
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",
- OF `0 \<le> Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> Ifloat x`
+ OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
by (rule mult_right_mono)
finally show "?ln \<le> ?ub" .
qed
@@ -1699,43 +1700,43 @@
in (Float 1 -1 * lb_ln_horner prec (get_even prec) 1 (Float 1 -1)) +
(third * lb_ln_horner prec (get_even prec) 1 third))"
-lemma ub_ln2: "ln 2 \<le> Ifloat (ub_ln2 prec)" (is "?ub_ln2")
- and lb_ln2: "Ifloat (lb_ln2 prec) \<le> ln 2" (is "?lb_ln2")
+lemma ub_ln2: "ln 2 \<le> real (ub_ln2 prec)" (is "?ub_ln2")
+ and lb_ln2: "real (lb_ln2 prec) \<le> ln 2" (is "?lb_ln2")
proof -
let ?uthird = "rapprox_rat (max prec 1) 1 3"
let ?lthird = "lapprox_rat prec 1 3"
have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1)"
using ln_add[of "3 / 2" "1 / 2"] by auto
- have lb3: "Ifloat ?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
- hence lb3_ub: "Ifloat ?lthird < 1" by auto
- have lb3_lb: "0 \<le> Ifloat ?lthird" using lapprox_rat_bottom[of 1 3] by auto
- have ub3: "1 / 3 \<le> Ifloat ?uthird" using rapprox_rat[of 1 3] by auto
- hence ub3_lb: "0 \<le> Ifloat ?uthird" by auto
+ have lb3: "real ?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
+ hence lb3_ub: "real ?lthird < 1" by auto
+ have lb3_lb: "0 \<le> real ?lthird" using lapprox_rat_bottom[of 1 3] by auto
+ have ub3: "1 / 3 \<le> real ?uthird" using rapprox_rat[of 1 3] by auto
+ hence ub3_lb: "0 \<le> real ?uthird" by auto
- have lb2: "0 \<le> Ifloat (Float 1 -1)" and ub2: "Ifloat (Float 1 -1) < 1" unfolding Float_num by auto
+ have lb2: "0 \<le> real (Float 1 -1)" and ub2: "real (Float 1 -1) < 1" unfolding Float_num by auto
have "0 \<le> (1::int)" and "0 < (3::int)" by auto
- have ub3_ub: "Ifloat ?uthird < 1" unfolding rapprox_rat.simps(2)[OF `0 \<le> 1` `0 < 3`]
+ have ub3_ub: "real ?uthird < 1" unfolding rapprox_rat.simps(2)[OF `0 \<le> 1` `0 < 3`]
by (rule rapprox_posrat_less1, auto)
have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
- have uthird_gt0: "0 < Ifloat ?uthird + 1" using ub3_lb by auto
- have lthird_gt0: "0 < Ifloat ?lthird + 1" using lb3_lb by auto
+ have uthird_gt0: "0 < real ?uthird + 1" using ub3_lb by auto
+ have lthird_gt0: "0 < real ?lthird + 1" using lb3_lb by auto
- show ?ub_ln2 unfolding ub_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric]
+ show ?ub_ln2 unfolding ub_ln2_def Let_def real_of_float_add ln2_sum Float_num(4)[symmetric]
proof (rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
- have "ln (1 / 3 + 1) \<le> ln (Ifloat ?uthird + 1)" unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
- also have "\<dots> \<le> Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)"
+ have "ln (1 / 3 + 1) \<le> ln (real ?uthird + 1)" unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
+ also have "\<dots> \<le> real (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)"
using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
- finally show "ln (1 / 3 + 1) \<le> Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
+ finally show "ln (1 / 3 + 1) \<le> real (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
qed
- show ?lb_ln2 unfolding lb_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric]
+ show ?lb_ln2 unfolding lb_ln2_def Let_def real_of_float_add ln2_sum Float_num(4)[symmetric]
proof (rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
- have "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (Ifloat ?lthird + 1)"
+ have "real (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (real ?lthird + 1)"
using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
also have "\<dots> \<le> ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto
- finally show "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (1 / 3 + 1)" .
+ finally show "real (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (1 / 3 + 1)" .
qed
qed
@@ -1768,7 +1769,7 @@
show False using `float_divr prec 1 x < 1` unfolding less_float_def le_float_def by auto
qed
-lemma ln_shifted_float: assumes "0 < m" shows "ln (Ifloat (Float m e)) = ln 2 * real (e + (bitlen m - 1)) + ln (Ifloat (Float m (- (bitlen m - 1))))"
+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))))"
proof -
let ?B = "2^nat (bitlen m - 1)"
have "0 < real m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0" using assms by auto
@@ -1778,7 +1779,7 @@
case True
show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`]
- unfolding Ifloat_ge0_exp[OF True] ln_mult[OF `0 < real m` `0 < 2^nat e`]
+ unfolding real_of_float_ge0_exp[OF True] ln_mult[OF `0 < real m` `0 < 2^nat e`]
ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` True by auto
next
case False hence "0 < -e" by auto
@@ -1786,20 +1787,20 @@
hence inv_gt0: "(0::real) < inverse (2^nat (-e))" by auto
show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`]
- unfolding Ifloat_nge0_exp[OF False] ln_mult[OF `0 < real m` inv_gt0] ln_inverse[OF pow_gt0]
+ unfolding real_of_float_nge0_exp[OF False] ln_mult[OF `0 < real m` inv_gt0] ln_inverse[OF pow_gt0]
ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` False by auto
qed
qed
lemma ub_ln_lb_ln_bounds': assumes "1 \<le> x"
- shows "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x) \<and> ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))"
+ shows "real (the (lb_ln prec x)) \<le> ln (real x) \<and> ln (real x) \<le> real (the (ub_ln prec x))"
(is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
proof (cases "x < Float 1 1")
- case True hence "Ifloat (x - 1) < 1" unfolding less_float_def Float_num by auto
+ case True hence "real (x - 1) < 1" unfolding less_float_def Float_num by auto
have "\<not> x \<le> 0" and "\<not> x < 1" using `1 \<le> x` unfolding less_float_def le_float_def by auto
- hence "0 \<le> Ifloat (x - 1)" using `1 \<le> x` unfolding less_float_def Float_num by auto
+ hence "0 \<le> real (x - 1)" using `1 \<le> x` unfolding less_float_def Float_num by auto
show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
- using ln_float_bounds[OF `0 \<le> Ifloat (x - 1)` `Ifloat (x - 1) < 1`] `\<not> x \<le> 0` `\<not> x < 1` True by auto
+ using ln_float_bounds[OF `0 \<le> real (x - 1)` `real (x - 1) < 1`] `\<not> x \<le> 0` `\<not> x < 1` True by auto
next
case False
have "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" using `1 \<le> x` unfolding less_float_def le_float_def by auto
@@ -1812,8 +1813,8 @@
have "0 < m" and "m \<noteq> 0" using float_pos_m_pos `0 < x` Float by auto
{
- have "Ifloat (lb_ln2 prec * ?s) \<le> ln 2 * real (e + (bitlen m - 1))" (is "?lb2 \<le> _")
- unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 power_0 mult_1_right
+ have "real (lb_ln2 prec * ?s) \<le> ln 2 * real (e + (bitlen m - 1))" (is "?lb2 \<le> _")
+ unfolding real_of_float_mult real_of_float_ge0_exp[OF order_refl] nat_0 power_0 mult_1_right
using lb_ln2[of prec]
proof (rule mult_right_mono)
have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
@@ -1822,38 +1823,38 @@
qed
moreover
from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
- have "0 \<le> Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto
+ have "0 \<le> real (?x - 1)" and "real (?x - 1) < 1" by auto
from ln_float_bounds(1)[OF this]
- have "Ifloat ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln (Ifloat ?x)" (is "?lb_horner \<le> _") by auto
- ultimately have "?lb2 + ?lb_horner \<le> ln (Ifloat x)"
+ have "real ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln (real ?x)" (is "?lb_horner \<le> _") by auto
+ ultimately have "?lb2 + ?lb_horner \<le> ln (real x)"
unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
}
moreover
{
from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
- have "0 \<le> Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto
+ have "0 \<le> real (?x - 1)" and "real (?x - 1) < 1" by auto
from ln_float_bounds(2)[OF this]
- have "ln (Ifloat ?x) \<le> Ifloat ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))" (is "_ \<le> ?ub_horner") by auto
+ have "ln (real ?x) \<le> real ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))" (is "_ \<le> ?ub_horner") by auto
moreover
- have "ln 2 * real (e + (bitlen m - 1)) \<le> Ifloat (ub_ln2 prec * ?s)" (is "_ \<le> ?ub2")
- unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 power_0 mult_1_right
+ have "ln 2 * real (e + (bitlen m - 1)) \<le> real (ub_ln2 prec * ?s)" (is "_ \<le> ?ub2")
+ unfolding real_of_float_mult real_of_float_ge0_exp[OF order_refl] nat_0 power_0 mult_1_right
using ub_ln2[of prec]
proof (rule mult_right_mono)
have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
from float_gt1_scale[OF this]
show "0 \<le> real (e + (bitlen m - 1))" by auto
qed
- ultimately have "ln (Ifloat x) \<le> ?ub2 + ?ub_horner"
+ ultimately have "ln (real x) \<le> ?ub2 + ?ub_horner"
unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
}
ultimately show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps
unfolding if_not_P[OF `\<not> x \<le> 0`] if_not_P[OF `\<not> x < 1`] if_not_P[OF False] Let_def
- unfolding scale.simps[of m e, unfolded Float[symmetric]] mantissa.simps[of m e, unfolded Float[symmetric]] Ifloat_add by auto
+ unfolding scale.simps[of m e, unfolded Float[symmetric]] mantissa.simps[of m e, unfolded Float[symmetric]] real_of_float_add by auto
qed
qed
lemma ub_ln_lb_ln_bounds: assumes "0 < x"
- shows "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x) \<and> ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))"
+ shows "real (the (lb_ln prec x)) \<le> ln (real x) \<and> ln (real x) \<le> real (the (ub_ln prec x))"
(is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
proof (cases "x < 1")
case False hence "1 \<le> x" unfolding less_float_def le_float_def by auto
@@ -1861,74 +1862,74 @@
next
case True have "\<not> x \<le> 0" using `0 < x` unfolding less_float_def le_float_def by auto
- have "0 < Ifloat x" and "Ifloat x \<noteq> 0" using `0 < x` unfolding less_float_def by auto
- hence A: "0 < 1 / Ifloat x" by auto
+ have "0 < real x" and "real x \<noteq> 0" using `0 < x` unfolding less_float_def by auto
+ hence A: "0 < 1 / real x" by auto
{
let ?divl = "float_divl (max prec 1) 1 x"
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
- hence B: "0 < Ifloat ?divl" unfolding le_float_def by auto
+ hence B: "0 < real ?divl" unfolding le_float_def by auto
- have "ln (Ifloat ?divl) \<le> ln (1 / Ifloat x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
- hence "ln (Ifloat x) \<le> - ln (Ifloat ?divl)" unfolding nonzero_inverse_eq_divide[OF `Ifloat x \<noteq> 0`, symmetric] ln_inverse[OF `0 < Ifloat x`] by auto
+ 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
+ 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
from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le]
- have "?ln \<le> Ifloat (- the (lb_ln prec ?divl))" unfolding Ifloat_minus by (rule order_trans)
+ have "?ln \<le> real (- the (lb_ln prec ?divl))" unfolding real_of_float_minus by (rule order_trans)
} moreover
{
let ?divr = "float_divr prec 1 x"
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
- hence B: "0 < Ifloat ?divr" unfolding le_float_def by auto
+ hence B: "0 < real ?divr" unfolding le_float_def by auto
- have "ln (1 / Ifloat x) \<le> ln (Ifloat ?divr)" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
- hence "- ln (Ifloat ?divr) \<le> ln (Ifloat x)" unfolding nonzero_inverse_eq_divide[OF `Ifloat x \<noteq> 0`, symmetric] ln_inverse[OF `0 < Ifloat x`] by auto
+ 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
+ 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
from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
- have "Ifloat (- the (ub_ln prec ?divr)) \<le> ?ln" unfolding Ifloat_minus by (rule order_trans)
+ have "real (- the (ub_ln prec ?divr)) \<le> ?ln" unfolding real_of_float_minus by (rule order_trans)
}
ultimately show ?thesis unfolding lb_ln.simps[where x=x] ub_ln.simps[where x=x]
unfolding if_not_P[OF `\<not> x \<le> 0`] if_P[OF True] by auto
qed
lemma lb_ln: assumes "Some y = lb_ln prec x"
- shows "Ifloat y \<le> ln (Ifloat x)" and "0 < Ifloat x"
+ shows "real y \<le> ln (real x)" and "0 < real x"
proof -
have "0 < x"
proof (rule ccontr)
assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
thus False using assms by auto
qed
- thus "0 < Ifloat x" unfolding less_float_def by auto
- have "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x)" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
- thus "Ifloat y \<le> ln (Ifloat x)" unfolding assms[symmetric] by auto
+ thus "0 < real x" unfolding less_float_def by auto
+ have "real (the (lb_ln prec x)) \<le> ln (real x)" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
+ thus "real y \<le> ln (real x)" unfolding assms[symmetric] by auto
qed
lemma ub_ln: assumes "Some y = ub_ln prec x"
- shows "ln (Ifloat x) \<le> Ifloat y" and "0 < Ifloat x"
+ shows "ln (real x) \<le> real y" and "0 < real x"
proof -
have "0 < x"
proof (rule ccontr)
assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
thus False using assms by auto
qed
- thus "0 < Ifloat x" unfolding less_float_def by auto
- have "ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
- thus "ln (Ifloat x) \<le> Ifloat y" unfolding assms[symmetric] by auto
+ thus "0 < real x" unfolding less_float_def by auto
+ have "ln (real x) \<le> real (the (ub_ln prec x))" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
+ thus "ln (real x) \<le> real y" unfolding assms[symmetric] by auto
qed
-lemma bnds_ln: "\<forall> x lx ux. (Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> ln x \<and> ln x \<le> Ifloat u"
+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"
proof (rule allI, rule allI, rule allI, rule impI)
fix x lx ux
- assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
- hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
+ assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {real lx .. real ux}"
+ 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
- have "ln (Ifloat ux) \<le> Ifloat u" and "0 < Ifloat ux" using ub_ln u by auto
- have "Ifloat l \<le> ln (Ifloat lx)" and "0 < Ifloat lx" and "0 < x" using lb_ln[OF l] x by auto
+ have "ln (real ux) \<le> real u" and "0 < real ux" using ub_ln u by auto
+ have "real l \<le> ln (real lx)" and "0 < real lx" and "0 < x" using lb_ln[OF l] x by auto
- from ln_le_cancel_iff[OF `0 < Ifloat lx` `0 < x`] `Ifloat l \<le> ln (Ifloat lx)`
- have "Ifloat l \<le> ln x" using x unfolding atLeastAtMost_iff by auto
+ from ln_le_cancel_iff[OF `0 < real lx` `0 < x`] `real l \<le> ln (real lx)`
+ have "real l \<le> ln x" using x unfolding atLeastAtMost_iff by auto
moreover
- from ln_le_cancel_iff[OF `0 < x` `0 < Ifloat ux`] `ln (Ifloat ux) \<le> Ifloat u`
- have "ln x \<le> Ifloat u" using x unfolding atLeastAtMost_iff by auto
- ultimately show "Ifloat l \<le> ln x \<and> ln x \<le> Ifloat u" ..
+ from ln_le_cancel_iff[OF `0 < x` `0 < real ux`] `ln (real ux) \<le> real u`
+ have "ln x \<le> real u" using x unfolding atLeastAtMost_iff by auto
+ ultimately show "real l \<le> ln x \<and> ln x \<le> real u" ..
qed
@@ -1955,25 +1956,25 @@
| Atom nat
| Num float
-fun Ifloatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real"
+fun interpret_floatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real"
where
-"Ifloatarith (Add a b) vs = (Ifloatarith a vs) + (Ifloatarith b vs)" |
-"Ifloatarith (Minus a) vs = - (Ifloatarith a vs)" |
-"Ifloatarith (Mult a b) vs = (Ifloatarith a vs) * (Ifloatarith b vs)" |
-"Ifloatarith (Inverse a) vs = inverse (Ifloatarith a vs)" |
-"Ifloatarith (Sin a) vs = sin (Ifloatarith a vs)" |
-"Ifloatarith (Cos a) vs = cos (Ifloatarith a vs)" |
-"Ifloatarith (Arctan a) vs = arctan (Ifloatarith a vs)" |
-"Ifloatarith (Min a b) vs = min (Ifloatarith a vs) (Ifloatarith b vs)" |
-"Ifloatarith (Max a b) vs = max (Ifloatarith a vs) (Ifloatarith b vs)" |
-"Ifloatarith (Abs a) vs = abs (Ifloatarith a vs)" |
-"Ifloatarith Pi vs = pi" |
-"Ifloatarith (Sqrt a) vs = sqrt (Ifloatarith a vs)" |
-"Ifloatarith (Exp a) vs = exp (Ifloatarith a vs)" |
-"Ifloatarith (Ln a) vs = ln (Ifloatarith a vs)" |
-"Ifloatarith (Power a n) vs = (Ifloatarith a vs)^n" |
-"Ifloatarith (Num f) vs = Ifloat f" |
-"Ifloatarith (Atom n) vs = vs ! n"
+"interpret_floatarith (Add a b) vs = (interpret_floatarith a vs) + (interpret_floatarith b vs)" |
+"interpret_floatarith (Minus a) vs = - (interpret_floatarith a vs)" |
+"interpret_floatarith (Mult a b) vs = (interpret_floatarith a vs) * (interpret_floatarith b vs)" |
+"interpret_floatarith (Inverse a) vs = inverse (interpret_floatarith a vs)" |
+"interpret_floatarith (Sin a) vs = sin (interpret_floatarith a vs)" |
+"interpret_floatarith (Cos a) vs = cos (interpret_floatarith a vs)" |
+"interpret_floatarith (Arctan a) vs = arctan (interpret_floatarith a vs)" |
+"interpret_floatarith (Min a b) vs = min (interpret_floatarith a vs) (interpret_floatarith b vs)" |
+"interpret_floatarith (Max a b) vs = max (interpret_floatarith a vs) (interpret_floatarith b vs)" |
+"interpret_floatarith (Abs a) vs = abs (interpret_floatarith a vs)" |
+"interpret_floatarith Pi vs = pi" |
+"interpret_floatarith (Sqrt a) vs = sqrt (interpret_floatarith a vs)" |
+"interpret_floatarith (Exp a) vs = exp (interpret_floatarith a vs)" |
+"interpret_floatarith (Ln a) vs = ln (interpret_floatarith a vs)" |
+"interpret_floatarith (Power a n) vs = (interpret_floatarith a vs)^n" |
+"interpret_floatarith (Num f) vs = real f" |
+"interpret_floatarith (Atom n) vs = vs ! n"
subsection "Implement approximation function"
@@ -1996,18 +1997,18 @@
"lift_un' b f = None"
fun bounded_by :: "real list \<Rightarrow> (float * float) list \<Rightarrow> bool " where
-bounded_by_Cons: "bounded_by (v#vs) ((l, u)#bs) = ((Ifloat l \<le> v \<and> v \<le> Ifloat u) \<and> bounded_by vs bs)" |
+bounded_by_Cons: "bounded_by (v#vs) ((l, u)#bs) = ((real l \<le> v \<and> v \<le> real u) \<and> bounded_by vs bs)" |
bounded_by_Nil: "bounded_by [] [] = True" |
"bounded_by _ _ = False"
lemma bounded_by: assumes "bounded_by vs bs" and "i < length bs"
- shows "Ifloat (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> Ifloat (snd (bs ! i))"
+ shows "real (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> real (snd (bs ! i))"
using `bounded_by vs bs` and `i < length bs`
proof (induct arbitrary: i rule: bounded_by.induct)
fix v :: real and vs :: "real list" and l u :: float and bs :: "(float * float) list" and i :: nat
- assume hyp: "\<And>i. \<lbrakk>bounded_by vs bs; i < length bs\<rbrakk> \<Longrightarrow> Ifloat (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> Ifloat (snd (bs ! i))"
+ assume hyp: "\<And>i. \<lbrakk>bounded_by vs bs; i < length bs\<rbrakk> \<Longrightarrow> real (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> real (snd (bs ! i))"
assume bounded: "bounded_by (v # vs) ((l, u) # bs)" and length: "i < length ((l, u) # bs)"
- show "Ifloat (fst (((l, u) # bs) ! i)) \<le> (v # vs) ! i \<and> (v # vs) ! i \<le> Ifloat (snd (((l, u) # bs) ! i))"
+ show "real (fst (((l, u) # bs) ! i)) \<le> (v # vs) ! i \<and> (v # vs) ! i \<le> real (snd (((l, u) # bs) ! i))"
proof (cases i)
case 0
show ?thesis using bounded unfolding 0 nth_Cons_0 fst_conv snd_conv bounded_by.simps ..
@@ -2073,9 +2074,9 @@
qed
lemma approx_approx':
- assumes Pa: "\<And>l u. Some (l, u) = approx prec a vs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
+ 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"
and approx': "Some (l, u) = approx' prec a vs"
- shows "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
+ shows "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u"
proof -
obtain l' u' where S: "Some (l', u') = approx prec a vs"
using approx' unfolding approx'.simps by (cases "approx prec a vs", auto)
@@ -2088,18 +2089,18 @@
lemma lift_bin':
assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f"
- and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
- and Pb: "\<And>l u. Some (l, u) = approx prec b bs \<Longrightarrow> Ifloat l \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u"
- shows "\<exists> l1 u1 l2 u2. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and>
- (Ifloat l2 \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u2) \<and>
+ 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")
+ 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"
+ shows "\<exists> l1 u1 l2 u2. (real l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u1) \<and>
+ (real l2 \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> real u2) \<and>
l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
proof -
{ fix l u assume "Some (l, u) = approx' prec a bs"
with approx_approx'[of prec a bs, OF _ this] Pa
- have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
+ have "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" by auto } note Pa = this
{ fix l u assume "Some (l, u) = approx' prec b bs"
with approx_approx'[of prec b bs, OF _ this] Pb
- have "Ifloat l \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u" by auto } note Pb = this
+ have "real l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> real u" by auto } note Pb = this
from lift_bin'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb]
show ?thesis by auto
@@ -2130,26 +2131,26 @@
lemma lift_un':
assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
- and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
- shows "\<exists> l1 u1. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and>
+ 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")
+ shows "\<exists> l1 u1. (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)"
proof -
{ fix l u assume "Some (l, u) = approx' prec a bs"
with approx_approx'[of prec a bs, OF _ this] Pa
- have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
+ have "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" by auto } note Pa = this
from lift_un'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa]
show ?thesis by auto
qed
lemma lift_un'_bnds:
- assumes bnds: "\<forall> x lx ux. (l, u) = f lx ux \<and> x \<in> { Ifloat lx .. Ifloat ux } \<longrightarrow> Ifloat l \<le> f' x \<and> f' x \<le> Ifloat u"
+ 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"
and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
- and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
- shows "Ifloat l \<le> f' (Ifloatarith a xs) \<and> f' (Ifloatarith a xs) \<le> Ifloat u"
+ 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"
+ shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
proof -
from lift_un'[OF lift_un'_Some Pa]
- obtain l1 u1 where "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" and "l = fst (f l1 u1)" and "u = snd (f l1 u1)" by blast
- hence "(l, u) = f l1 u1" and "Ifloatarith a xs \<in> {Ifloat l1 .. Ifloat u1}" by auto
+ 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
+ hence "(l, u) = f l1 u1" and "interpret_floatarith a xs \<in> {real l1 .. real u1}" by auto
thus ?thesis using bnds by auto
qed
@@ -2195,115 +2196,115 @@
lemma lift_un:
assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
- and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
- shows "\<exists> l1 u1. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and>
+ 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")
+ shows "\<exists> l1 u1. (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)"
proof -
{ fix l u assume "Some (l, u) = approx' prec a bs"
with approx_approx'[of prec a bs, OF _ this] Pa
- have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
+ have "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" by auto } note Pa = this
from lift_un_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa]
show ?thesis by auto
qed
lemma lift_un_bnds:
- assumes bnds: "\<forall> x lx ux. (Some l, Some u) = f lx ux \<and> x \<in> { Ifloat lx .. Ifloat ux } \<longrightarrow> Ifloat l \<le> f' x \<and> f' x \<le> Ifloat u"
+ 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"
and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
- and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
- shows "Ifloat l \<le> f' (Ifloatarith a xs) \<and> f' (Ifloatarith a xs) \<le> Ifloat u"
+ 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"
+ shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
proof -
from lift_un[OF lift_un_Some Pa]
- obtain l1 u1 where "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" and "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" by blast
- hence "(Some l, Some u) = f l1 u1" and "Ifloatarith a xs \<in> {Ifloat l1 .. Ifloat u1}" by auto
+ 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
+ hence "(Some l, Some u) = f l1 u1" and "interpret_floatarith a xs \<in> {real l1 .. real u1}" by auto
thus ?thesis using bnds by auto
qed
lemma approx:
assumes "bounded_by xs vs"
and "Some (l, u) = approx prec arith vs" (is "_ = ?g arith")
- shows "Ifloat l \<le> Ifloatarith arith xs \<and> Ifloatarith arith xs \<le> Ifloat u" (is "?P l u arith")
+ shows "real l \<le> interpret_floatarith arith xs \<and> interpret_floatarith arith xs \<le> real u" (is "?P l u arith")
using `Some (l, u) = approx prec arith vs`
proof (induct arith arbitrary: l u x)
case (Add a b)
from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps
obtain l1 u1 l2 u2 where "l = l1 + l2" and "u = u1 + u2"
- "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1"
- "Ifloat l2 \<le> Ifloatarith b xs" and "Ifloatarith b xs \<le> Ifloat u2" unfolding fst_conv snd_conv by blast
- thus ?case unfolding Ifloatarith.simps by auto
+ "real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1"
+ "real l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> real u2" unfolding fst_conv snd_conv by blast
+ thus ?case unfolding interpret_floatarith.simps by auto
next
case (Minus a)
from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps
obtain l1 u1 where "l = -u1" and "u = -l1"
- "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" unfolding fst_conv snd_conv by blast
- thus ?case unfolding Ifloatarith.simps using Ifloat_minus by auto
+ "real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1" unfolding fst_conv snd_conv by blast
+ thus ?case unfolding interpret_floatarith.simps using real_of_float_minus by auto
next
case (Mult a b)
from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps
obtain l1 u1 l2 u2
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"
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"
- and "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1"
- and "Ifloat l2 \<le> Ifloatarith b xs" and "Ifloatarith b xs \<le> Ifloat u2" unfolding fst_conv snd_conv by blast
- thus ?case unfolding Ifloatarith.simps l u Ifloat_add Ifloat_mult Ifloat_nprt Ifloat_pprt
+ and "real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1"
+ and "real l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> real u2" unfolding fst_conv snd_conv by blast
+ thus ?case unfolding interpret_floatarith.simps l u real_of_float_add real_of_float_mult real_of_float_nprt real_of_float_pprt
using mult_le_prts mult_ge_prts by auto
next
case (Inverse a)
from lift_un[OF Inverse.prems[unfolded approx.simps], unfolded if_distrib[of fst] if_distrib[of snd] fst_conv snd_conv] Inverse.hyps
obtain l1 u1 where l': "Some l = (if 0 < l1 \<or> u1 < 0 then Some (float_divl prec 1 u1) else None)"
and u': "Some u = (if 0 < l1 \<or> u1 < 0 then Some (float_divr prec 1 l1) else None)"
- and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1" by blast
+ and l1: "real l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> real u1" by blast
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
- moreover have l1_le_u1: "Ifloat l1 \<le> Ifloat u1" using l1 u1 by auto
- ultimately have "Ifloat l1 \<noteq> 0" and "Ifloat u1 \<noteq> 0" unfolding less_float_def by auto
+ moreover have l1_le_u1: "real l1 \<le> real u1" using l1 u1 by auto
+ ultimately have "real l1 \<noteq> 0" and "real u1 \<noteq> 0" unfolding less_float_def by auto
- have inv: "inverse (Ifloat u1) \<le> inverse (Ifloatarith a xs)
- \<and> inverse (Ifloatarith a xs) \<le> inverse (Ifloat l1)"
+ have inv: "inverse (real u1) \<le> inverse (interpret_floatarith a xs)
+ \<and> inverse (interpret_floatarith a xs) \<le> inverse (real l1)"
proof (cases "0 < l1")
- case True hence "0 < Ifloat u1" and "0 < Ifloat l1" "0 < Ifloatarith a xs"
+ case True hence "0 < real u1" and "0 < real l1" "0 < interpret_floatarith a xs"
unfolding less_float_def using l1_le_u1 l1 by auto
show ?thesis
- unfolding inverse_le_iff_le[OF `0 < Ifloat u1` `0 < Ifloatarith a xs`]
- inverse_le_iff_le[OF `0 < Ifloatarith a xs` `0 < Ifloat l1`]
+ unfolding inverse_le_iff_le[OF `0 < real u1` `0 < interpret_floatarith a xs`]
+ inverse_le_iff_le[OF `0 < interpret_floatarith a xs` `0 < real l1`]
using l1 u1 by auto
next
case False hence "u1 < 0" using either by blast
- hence "Ifloat u1 < 0" and "Ifloat l1 < 0" "Ifloatarith a xs < 0"
+ hence "real u1 < 0" and "real l1 < 0" "interpret_floatarith a xs < 0"
unfolding less_float_def using l1_le_u1 u1 by auto
show ?thesis
- unfolding inverse_le_iff_le_neg[OF `Ifloat u1 < 0` `Ifloatarith a xs < 0`]
- inverse_le_iff_le_neg[OF `Ifloatarith a xs < 0` `Ifloat l1 < 0`]
+ unfolding inverse_le_iff_le_neg[OF `real u1 < 0` `interpret_floatarith a xs < 0`]
+ inverse_le_iff_le_neg[OF `interpret_floatarith a xs < 0` `real l1 < 0`]
using l1 u1 by auto
qed
from l' have "l = float_divl prec 1 u1" by (cases "0 < l1 \<or> u1 < 0", auto)
- hence "Ifloat l \<le> inverse (Ifloat u1)" unfolding nonzero_inverse_eq_divide[OF `Ifloat u1 \<noteq> 0`] using float_divl[of prec 1 u1] by auto
- also have "\<dots> \<le> inverse (Ifloatarith a xs)" using inv by auto
- finally have "Ifloat l \<le> inverse (Ifloatarith a xs)" .
+ 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
+ also have "\<dots> \<le> inverse (interpret_floatarith a xs)" using inv by auto
+ finally have "real l \<le> inverse (interpret_floatarith a xs)" .
moreover
from u' have "u = float_divr prec 1 l1" by (cases "0 < l1 \<or> u1 < 0", auto)
- hence "inverse (Ifloat l1) \<le> Ifloat u" unfolding nonzero_inverse_eq_divide[OF `Ifloat l1 \<noteq> 0`] using float_divr[of 1 l1 prec] by auto
- hence "inverse (Ifloatarith a xs) \<le> Ifloat u" by (rule order_trans[OF inv[THEN conjunct2]])
- ultimately show ?case unfolding Ifloatarith.simps using l1 u1 by auto
+ 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
+ hence "inverse (interpret_floatarith a xs) \<le> real u" by (rule order_trans[OF inv[THEN conjunct2]])
+ ultimately show ?case unfolding interpret_floatarith.simps using l1 u1 by auto
next
case (Abs x)
from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps
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>"
- and l1: "Ifloat l1 \<le> Ifloatarith x xs" and u1: "Ifloatarith x xs \<le> Ifloat u1" by blast
- thus ?case unfolding l' u' by (cases "l1 < 0 \<and> 0 < u1", auto simp add: Ifloat_min Ifloat_max Ifloat_abs less_float_def)
+ and l1: "real l1 \<le> interpret_floatarith x xs" and u1: "interpret_floatarith x xs \<le> real u1" by blast
+ 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)
next
case (Min a b)
from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps
obtain l1 u1 l2 u2 where l': "l = min l1 l2" and u': "u = min u1 u2"
- and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1"
- and l1: "Ifloat l2 \<le> Ifloatarith b xs" and u1: "Ifloatarith b xs \<le> Ifloat u2" by blast
- thus ?case unfolding l' u' by (auto simp add: Ifloat_min)
+ and l1: "real l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> real u1"
+ and l1: "real l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> real u2" by blast
+ thus ?case unfolding l' u' by (auto simp add: real_of_float_min)
next
case (Max a b)
from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps
obtain l1 u1 l2 u2 where l': "l = max l1 l2" and u': "u = max u1 u2"
- and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1"
- and l1: "Ifloat l2 \<le> Ifloatarith b xs" and u1: "Ifloatarith b xs \<le> Ifloat u2" by blast
- thus ?case unfolding l' u' by (auto simp add: Ifloat_max)
+ and l1: "real l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> real u1"
+ and l1: "real l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> real u2" by blast
+ thus ?case unfolding l' u' by (auto simp add: real_of_float_max)
next case (Sin a) with lift_un'_bnds[OF bnds_sin] show ?case by auto
next case (Cos a) with lift_un'_bnds[OF bnds_cos] show ?case by auto
next case (Arctan a) with lift_un'_bnds[OF bnds_arctan] show ?case by auto
@@ -2325,19 +2326,19 @@
qed
qed
-datatype ApproxEq = Less floatarith floatarith
- | LessEqual floatarith floatarith
+datatype inequality = Less floatarith floatarith
+ | LessEqual floatarith floatarith
-fun uneq :: "ApproxEq \<Rightarrow> real list \<Rightarrow> bool" where
-"uneq (Less a b) vs = (Ifloatarith a vs < Ifloatarith b vs)" |
-"uneq (LessEqual a b) vs = (Ifloatarith a vs \<le> Ifloatarith b vs)"
+fun interpret_inequality :: "inequality \<Rightarrow> real list \<Rightarrow> bool" where
+"interpret_inequality (Less a b) vs = (interpret_floatarith a vs < interpret_floatarith b vs)" |
+"interpret_inequality (LessEqual a b) vs = (interpret_floatarith a vs \<le> interpret_floatarith b vs)"
-fun uneq' :: "nat \<Rightarrow> ApproxEq \<Rightarrow> (float * float) list \<Rightarrow> bool" where
-"uneq' prec (Less a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u < l' | _ \<Rightarrow> False)" |
-"uneq' prec (LessEqual a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u \<le> l' | _ \<Rightarrow> False)"
+fun approx_inequality :: "nat \<Rightarrow> inequality \<Rightarrow> (float * float) list \<Rightarrow> bool" where
+"approx_inequality prec (Less a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u < l' | _ \<Rightarrow> False)" |
+"approx_inequality prec (LessEqual a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u \<le> l' | _ \<Rightarrow> False)"
-lemma uneq_approx: fixes m :: nat assumes "bounded_by vs bs" and "uneq' prec eq bs"
- shows "uneq eq vs"
+lemma approx_inequality: fixes m :: nat assumes "bounded_by vs bs" and "approx_inequality prec eq bs"
+ shows "interpret_inequality eq vs"
proof (cases eq)
case (Less a b)
show ?thesis
@@ -2346,17 +2347,17 @@
case True
then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)"
and b_approx: "approx prec b bs = Some (l', u') " by auto
- with `uneq' prec eq bs` have "Ifloat u < Ifloat l'"
- unfolding Less uneq'.simps less_float_def by auto
+ with `approx_inequality prec eq bs` have "real u < real l'"
+ unfolding Less approx_inequality.simps less_float_def by auto
moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs`
- have "Ifloatarith a vs \<le> Ifloat u" and "Ifloat l' \<le> Ifloatarith b vs"
+ have "interpret_floatarith a vs \<le> real u" and "real l' \<le> interpret_floatarith b vs"
using approx by auto
- ultimately show ?thesis unfolding uneq.simps Less by auto
+ ultimately show ?thesis unfolding interpret_inequality.simps Less by auto
next
case False
hence "approx prec a bs = None \<or> approx prec b bs = None"
unfolding not_Some_eq[symmetric] by auto
- hence "\<not> uneq' prec eq bs" unfolding Less uneq'.simps
+ hence "\<not> approx_inequality prec eq bs" unfolding Less approx_inequality.simps
by (cases "approx prec a bs = None", auto)
thus ?thesis using assms by auto
qed
@@ -2368,66 +2369,70 @@
case True
then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)"
and b_approx: "approx prec b bs = Some (l', u') " by auto
- with `uneq' prec eq bs` have "Ifloat u \<le> Ifloat l'"
- unfolding LessEqual uneq'.simps le_float_def by auto
+ with `approx_inequality prec eq bs` have "real u \<le> real l'"
+ unfolding LessEqual approx_inequality.simps le_float_def by auto
moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs`
- have "Ifloatarith a vs \<le> Ifloat u" and "Ifloat l' \<le> Ifloatarith b vs"
+ have "interpret_floatarith a vs \<le> real u" and "real l' \<le> interpret_floatarith b vs"
using approx by auto
- ultimately show ?thesis unfolding uneq.simps LessEqual by auto
+ ultimately show ?thesis unfolding interpret_inequality.simps LessEqual by auto
next
case False
hence "approx prec a bs = None \<or> approx prec b bs = None"
unfolding not_Some_eq[symmetric] by auto
- hence "\<not> uneq' prec eq bs" unfolding LessEqual uneq'.simps
+ hence "\<not> approx_inequality prec eq bs" unfolding LessEqual approx_inequality.simps
by (cases "approx prec a bs = None", auto)
thus ?thesis using assms by auto
qed
qed
-lemma Ifloatarith_divide: "Ifloatarith (Mult a (Inverse b)) vs = (Ifloatarith a vs) / (Ifloatarith b vs)"
- unfolding real_divide_def Ifloatarith.simps ..
+lemma interpret_floatarith_divide: "interpret_floatarith (Mult a (Inverse b)) vs = (interpret_floatarith a vs) / (interpret_floatarith b vs)"
+ unfolding real_divide_def interpret_floatarith.simps ..
-lemma Ifloatarith_diff: "Ifloatarith (Add a (Minus b)) vs = (Ifloatarith a vs) - (Ifloatarith b vs)"
- unfolding real_diff_def Ifloatarith.simps ..
+lemma interpret_floatarith_diff: "interpret_floatarith (Add a (Minus b)) vs = (interpret_floatarith a vs) - (interpret_floatarith b vs)"
+ unfolding real_diff_def interpret_floatarith.simps ..
+
+lemma interpret_floatarith_tan: "interpret_floatarith (Mult (Sin a) (Inverse (Cos a))) vs = tan (interpret_floatarith a vs)"
+ unfolding tan_def interpret_floatarith.simps real_divide_def ..
-lemma Ifloatarith_tan: "Ifloatarith (Mult (Sin a) (Inverse (Cos a))) vs = tan (Ifloatarith a vs)"
- unfolding tan_def Ifloatarith.simps real_divide_def ..
+lemma interpret_floatarith_powr: "interpret_floatarith (Exp (Mult b (Ln a))) vs = (interpret_floatarith a vs) powr (interpret_floatarith b vs)"
+ unfolding powr_def interpret_floatarith.simps ..
-lemma Ifloatarith_powr: "Ifloatarith (Exp (Mult b (Ln a))) vs = (Ifloatarith a vs) powr (Ifloatarith b vs)"
- unfolding powr_def Ifloatarith.simps ..
+lemma interpret_floatarith_log: "interpret_floatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (interpret_floatarith b vs) (interpret_floatarith x vs)"
+ unfolding log_def interpret_floatarith.simps real_divide_def ..
-lemma Ifloatarith_log: "Ifloatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (Ifloatarith b vs) (Ifloatarith x vs)"
- unfolding log_def Ifloatarith.simps real_divide_def ..
-
-lemma Ifloatarith_num: shows "Ifloatarith (Num (Float 0 0)) vs = 0" and "Ifloatarith (Num (Float 1 0)) vs = 1" and "Ifloatarith (Num (Float (number_of a) 0)) vs = number_of a" by auto
+lemma interpret_floatarith_num:
+ shows "interpret_floatarith (Num (Float 0 0)) vs = 0"
+ and "interpret_floatarith (Num (Float 1 0)) vs = 1"
+ and "interpret_floatarith (Num (Float (number_of a) 0)) vs = number_of a" by auto
subsection {* Implement proof method \texttt{approximation} *}
-lemma bounded_divl: assumes "Ifloat a / Ifloat b \<le> x" shows "Ifloat (float_divl p a b) \<le> x" by (rule order_trans[OF _ assms], rule float_divl)
-lemma bounded_divr: assumes "x \<le> Ifloat a / Ifloat b" shows "x \<le> Ifloat (float_divr p a b)" by (rule order_trans[OF assms _], rule float_divr)
-lemma bounded_num: shows "Ifloat (Float 5 1) = 10" and "Ifloat (Float 0 0) = 0" and "Ifloat (Float 1 0) = 1" and "Ifloat (Float (number_of n) 0) = (number_of n)"
- and "0 * pow2 e = Ifloat (Float 0 e)" and "1 * pow2 e = Ifloat (Float 1 e)" and "number_of m * pow2 e = Ifloat (Float (number_of m) e)"
- and "Ifloat (Float (number_of A) (int B)) = (number_of A) * 2^B"
- and "Ifloat (Float 1 (int B)) = 2^B"
- and "Ifloat (Float (number_of A) (- int B)) = (number_of A) / 2^B"
- and "Ifloat (Float 1 (- int B)) = 1 / 2^B"
- by (auto simp add: Ifloat.simps pow2_def real_divide_def)
+lemma bounded_divl: assumes "real a / real b \<le> x" shows "real (float_divl p a b) \<le> x" by (rule order_trans[OF _ assms], rule float_divl)
+lemma bounded_divr: assumes "x \<le> real a / real b" shows "x \<le> real (float_divr p a b)" by (rule order_trans[OF assms _], rule float_divr)
+lemma bounded_num: shows "real (Float 5 1) = 10" and "real (Float 0 0) = 0" and "real (Float 1 0) = 1" and "real (Float (number_of n) 0) = (number_of n)"
+ and "0 * pow2 e = real (Float 0 e)" and "1 * pow2 e = real (Float 1 e)" and "number_of m * pow2 e = real (Float (number_of m) e)"
+ and "real (Float (number_of A) (int B)) = (number_of A) * 2^B"
+ and "real (Float 1 (int B)) = 2^B"
+ and "real (Float (number_of A) (- int B)) = (number_of A) / 2^B"
+ and "real (Float 1 (- int B)) = 1 / 2^B"
+ by (auto simp add: real_of_float_simp pow2_def real_divide_def)
lemmas bounded_by_equations = bounded_by_Cons bounded_by_Nil float_power bounded_divl bounded_divr bounded_num HOL.simp_thms
-lemmas uneq_equations = uneq.simps Ifloatarith.simps Ifloatarith_num Ifloatarith_divide Ifloatarith_diff Ifloatarith_tan Ifloatarith_powr Ifloatarith_log
+lemmas interpret_inequality_equations = interpret_inequality.simps interpret_floatarith.simps interpret_floatarith_num
+ interpret_floatarith_divide interpret_floatarith_diff interpret_floatarith_tan interpret_floatarith_powr interpret_floatarith_log
ML {*
- val uneq_equations = PureThy.get_thms @{theory} "uneq_equations";
+ val ineq_equations = PureThy.get_thms @{theory} "interpret_inequality_equations";
val bounded_by_equations = PureThy.get_thms @{theory} "bounded_by_equations";
val bounded_by_simpset = (HOL_basic_ss addsimps bounded_by_equations)
- fun reify_uneq ctxt i = (fn st =>
+ fun reify_ineq ctxt i = (fn st =>
let
val to = HOLogic.dest_Trueprop (Logic.strip_imp_concl (List.nth (prems_of st, i - 1)))
- in (Reflection.genreify_tac ctxt uneq_equations (SOME to) i) st
+ in (Reflection.genreify_tac ctxt ineq_equations (SOME to) i) st
end)
- fun rule_uneq ctxt prec i thm = let
+ fun rule_ineq ctxt prec i thm = let
fun conv_num typ = HOLogic.dest_number #> snd #> HOLogic.mk_number typ
val to_natc = conv_num @{typ "nat"} #> Thm.cterm_of (ProofContext.theory_of ctxt)
val to_nat = conv_num @{typ "nat"}
@@ -2488,7 +2493,7 @@
val bs = (HOLogic.dest_list #> map lift_var #> HOLogic.mk_list @{typ "float * float"}) vs
val map = [(@{cpat "?prec::nat"}, to_natc prec),
(@{cpat "?bs::(float * float) list"}, Thm.cterm_of (ProofContext.theory_of ctxt) bs)]
- in rtac (Thm.instantiate ([], map) @{thm "uneq_approx"}) i thm end
+ in rtac (Thm.instantiate ([], map) @{thm "approx_inequality"}) i thm end
val eval_tac = CSUBGOAL (fn (ct, i) => rtac (eval_oracle ct) i)
@@ -2501,8 +2506,8 @@
Args.term >>
(fn prec => fn ctxt =>
SIMPLE_METHOD' (fn i =>
- (DETERM (reify_uneq ctxt i)
- THEN rule_uneq ctxt prec i
+ (DETERM (reify_ineq ctxt i)
+ THEN rule_ineq ctxt prec i
THEN Simplifier.asm_full_simp_tac bounded_by_simpset i
THEN (TRY (filter_prems_tac (fn t => false) i))
THEN (gen_eval_tac eval_oracle ctxt) i)))
--- a/src/HOL/Library/Float.thy Mon May 11 08:29:28 2009 -0700
+++ b/src/HOL/Library/Float.thy Wed Apr 29 20:19:50 2009 +0200
@@ -15,8 +15,17 @@
datatype float = Float int int
-primrec Ifloat :: "float \<Rightarrow> real" where
- "Ifloat (Float a b) = real a * pow2 b"
+primrec of_float :: "float \<Rightarrow> real" where
+ "of_float (Float a b) = real a * pow2 b"
+
+defs (overloaded)
+ real_of_float_def [code unfold]: "real == of_float"
+
+primrec mantissa :: "float \<Rightarrow> int" where
+ "mantissa (Float a b) = a"
+
+primrec scale :: "float \<Rightarrow> int" where
+ "scale (Float a b) = b"
instantiation float :: zero begin
definition zero_float where "0 = Float 0 0"
@@ -33,20 +42,17 @@
instance ..
end
-primrec mantissa :: "float \<Rightarrow> int" where
- "mantissa (Float a b) = a"
+lemma real_of_float_simp[simp]: "real (Float a b) = real a * pow2 b"
+ unfolding real_of_float_def using of_float.simps .
-primrec scale :: "float \<Rightarrow> int" where
- "scale (Float a b) = b"
-
-lemma Ifloat_neg_exp: "e < 0 \<Longrightarrow> Ifloat (Float m e) = real m * inverse (2^nat (-e))" by auto
-lemma Ifloat_nge0_exp: "\<not> 0 \<le> e \<Longrightarrow> Ifloat (Float m e) = real m * inverse (2^nat (-e))" by auto
-lemma Ifloat_ge0_exp: "0 \<le> e \<Longrightarrow> Ifloat (Float m e) = real m * (2^nat e)" by auto
+lemma real_of_float_neg_exp: "e < 0 \<Longrightarrow> real (Float m e) = real m * inverse (2^nat (-e))" by auto
+lemma real_of_float_nge0_exp: "\<not> 0 \<le> e \<Longrightarrow> real (Float m e) = real m * inverse (2^nat (-e))" by auto
+lemma real_of_float_ge0_exp: "0 \<le> e \<Longrightarrow> real (Float m e) = real m * (2^nat e)" by auto
lemma Float_num[simp]: shows
- "Ifloat (Float 1 0) = 1" and "Ifloat (Float 1 1) = 2" and "Ifloat (Float 1 2) = 4" and
- "Ifloat (Float 1 -1) = 1/2" and "Ifloat (Float 1 -2) = 1/4" and "Ifloat (Float 1 -3) = 1/8" and
- "Ifloat (Float -1 0) = -1" and "Ifloat (Float (number_of n) 0) = number_of n"
+ "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
+ "real (Float 1 -1) = 1/2" and "real (Float 1 -2) = 1/4" and "real (Float 1 -3) = 1/8" and
+ "real (Float -1 0) = -1" and "real (Float (number_of n) 0) = number_of n"
by auto
lemma pow2_0[simp]: "pow2 0 = 1" by simp
@@ -131,7 +137,7 @@
lemma float_split2: "(\<forall> a b. x \<noteq> Float a b) = False" by (auto simp add: float_split)
-lemma float_zero[simp]: "Ifloat (Float 0 e) = 0" by simp
+lemma float_zero[simp]: "real (Float 0 e) = 0" by simp
lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
by arith
@@ -142,7 +148,7 @@
termination by (relation "measure (nat o abs o mantissa)") (auto intro: abs_div_2_less)
declare normfloat.simps[simp del]
-theorem normfloat[symmetric, simp]: "Ifloat f = Ifloat (normfloat f)"
+theorem normfloat[symmetric, simp]: "real f = real (normfloat f)"
proof (induct f rule: normfloat.induct)
case (1 a b)
have real2: "2 = real (2::int)"
@@ -217,7 +223,7 @@
lemma float_eq_odd_helper:
assumes odd: "odd a'"
- and floateq: "Ifloat (Float a b) = Ifloat (Float a' b')"
+ and floateq: "real (Float a b) = real (Float a' b')"
shows "b \<le> b'"
proof -
{
@@ -267,7 +273,7 @@
lemma float_eq_odd:
assumes odd1: "odd a"
and odd2: "odd a'"
- and floateq: "Ifloat (Float a b) = Ifloat (Float a' b')"
+ and floateq: "real (Float a b) = real (Float a' b')"
shows "a = a' \<and> b = b'"
proof -
from
@@ -278,14 +284,14 @@
qed
theorem normfloat_unique:
- assumes Ifloat_eq: "Ifloat f = Ifloat g"
+ assumes real_of_float_eq: "real f = real g"
shows "normfloat f = normfloat g"
proof -
from float_split[of "normfloat f"] obtain a b where normf:"normfloat f = Float a b" by auto
from float_split[of "normfloat g"] obtain a' b' where normg:"normfloat g = Float a' b'" by auto
- have "Ifloat (normfloat f) = Ifloat (normfloat g)"
- by (simp add: Ifloat_eq)
- then have float_eq: "Ifloat (Float a b) = Ifloat (Float a' b')"
+ have "real (normfloat f) = real (normfloat g)"
+ by (simp add: real_of_float_eq)
+ then have float_eq: "real (Float a b) = real (Float a' b')"
by (simp add: normf normg)
have ab: "odd a \<or> (a = 0 \<and> b = 0)" by (rule normfloat_imp_odd_or_zero[OF normf])
have ab': "odd a' \<or> (a' = 0 \<and> b' = 0)" by (rule normfloat_imp_odd_or_zero[OF normg])
@@ -341,32 +347,32 @@
"float_nprt (Float a e) = (if 0 <= a then 0 else (Float a e))"
instantiation float :: ord begin
-definition le_float_def: "z \<le> w \<equiv> Ifloat z \<le> Ifloat w"
-definition less_float_def: "z < w \<equiv> Ifloat z < Ifloat w"
+definition le_float_def: "z \<le> (w :: float) \<equiv> real z \<le> real w"
+definition less_float_def: "z < (w :: float) \<equiv> real z < real w"
instance ..
end
-lemma Ifloat_add[simp]: "Ifloat (a + b) = Ifloat a + Ifloat b"
+lemma real_of_float_add[simp]: "real (a + b) = real a + real (b :: float)"
by (cases a, cases b, simp add: algebra_simps plus_float.simps,
auto simp add: pow2_int[symmetric] pow2_add[symmetric])
-lemma Ifloat_minus[simp]: "Ifloat (- a) = - Ifloat a"
+lemma real_of_float_minus[simp]: "real (- a) = - real (a :: float)"
by (cases a, simp add: uminus_float.simps)
-lemma Ifloat_sub[simp]: "Ifloat (a - b) = Ifloat a - Ifloat b"
+lemma real_of_float_sub[simp]: "real (a - b) = real a - real (b :: float)"
by (cases a, cases b, simp add: minus_float_def)
-lemma Ifloat_mult[simp]: "Ifloat (a*b) = Ifloat a * Ifloat b"
+lemma real_of_float_mult[simp]: "real (a*b) = real a * real (b :: float)"
by (cases a, cases b, simp add: times_float.simps pow2_add)
-lemma Ifloat_0[simp]: "Ifloat 0 = 0"
+lemma real_of_float_0[simp]: "real (0 :: float) = 0"
by (auto simp add: zero_float_def float_zero)
-lemma Ifloat_1[simp]: "Ifloat 1 = 1"
+lemma real_of_float_1[simp]: "real (1 :: float) = 1"
by (auto simp add: one_float_def)
lemma zero_le_float:
- "(0 <= Ifloat (Float a b)) = (0 <= a)"
+ "(0 <= real (Float a b)) = (0 <= a)"
apply auto
apply (auto simp add: zero_le_mult_iff)
apply (insert zero_less_pow2[of b])
@@ -374,19 +380,19 @@
done
lemma float_le_zero:
- "(Ifloat (Float a b) <= 0) = (a <= 0)"
+ "(real (Float a b) <= 0) = (a <= 0)"
apply auto
apply (auto simp add: mult_le_0_iff)
apply (insert zero_less_pow2[of b])
apply auto
done
-declare Ifloat.simps[simp del]
+declare real_of_float_simp[simp del]
-lemma Ifloat_pprt[simp]: "Ifloat (float_pprt a) = pprt (Ifloat a)"
+lemma real_of_float_pprt[simp]: "real (float_pprt a) = pprt (real a)"
by (cases a, auto simp add: float_pprt.simps zero_le_float float_le_zero float_zero)
-lemma Ifloat_nprt[simp]: "Ifloat (float_nprt a) = nprt (Ifloat a)"
+lemma real_of_float_nprt[simp]: "real (float_nprt a) = nprt (real a)"
by (cases a, auto simp add: float_nprt.simps zero_le_float float_le_zero float_zero)
instance float :: ab_semigroup_add
@@ -440,10 +446,10 @@
lemma float_less_simp: "((x::float) < y) = (0 < y - x)"
by (auto simp add: less_float_def)
-lemma Ifloat_min: "Ifloat (min x y) = min (Ifloat x) (Ifloat y)" unfolding min_def le_float_def by auto
-lemma Ifloat_max: "Ifloat (max a b) = max (Ifloat a) (Ifloat b)" unfolding max_def le_float_def by auto
+lemma real_of_float_min: "real (min x y :: float) = min (real x) (real y)" unfolding min_def le_float_def by auto
+lemma real_of_float_max: "real (max a b :: float) = max (real a) (real b)" unfolding max_def le_float_def by auto
-lemma float_power: "Ifloat (x ^ n) = Ifloat x ^ n"
+lemma float_power: "real (x ^ n :: float) = real x ^ n"
by (induct n) simp_all
lemma zero_le_pow2[simp]: "0 \<le> pow2 s"
@@ -467,7 +473,7 @@
done
lemma float_pos_m_pos: "0 < Float m e \<Longrightarrow> 0 < m"
- unfolding less_float_def Ifloat.simps Ifloat_0 zero_less_mult_iff
+ unfolding less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff
by auto
lemma float_pos_less1_e_neg: assumes "0 < Float m e" and "Float m e < 1" shows "e < 0"
@@ -480,7 +486,7 @@
assume "\<not> e < 0" hence "0 \<le> e" by auto
hence "1 \<le> pow2 e" unfolding pow2_def by auto
from mult_mono[OF `1 \<le> real m` this `0 \<le> real m`]
- have "1 \<le> Float m e" by (simp add: le_float_def Ifloat.simps)
+ have "1 \<le> Float m e" by (simp add: le_float_def real_of_float_simp)
thus False using `Float m e < 1` unfolding less_float_def le_float_def by auto
qed
qed
@@ -490,7 +496,7 @@
have "e < 0" using float_pos_less1_e_neg assms by auto
have "\<And>x. (0::real) < 2^x" by auto
have "real m < 2^(nat (-e))" using `Float m e < 1`
- unfolding less_float_def Ifloat_neg_exp[OF `e < 0`] Ifloat_1
+ unfolding less_float_def real_of_float_neg_exp[OF `e < 0`] real_of_float_1
real_mult_less_iff1[of _ _ 1, OF `0 < 2^(nat (-e))`, symmetric]
real_mult_assoc by auto
thus ?thesis unfolding real_of_int_less_iff[symmetric] by auto
@@ -588,7 +594,7 @@
hence "0 < m" using float_pos_m_pos by auto
hence "m \<noteq> 0" and "1 < (2::int)" by auto
case False let ?S = "2^(nat (-e))"
- have "1 \<le> real m * inverse ?S" using assms unfolding le_float_def Ifloat_nge0_exp[OF False] by auto
+ have "1 \<le> real m * inverse ?S" using assms unfolding le_float_def real_of_float_nge0_exp[OF False] by auto
hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
hence "?S \<le> real m" unfolding mult_assoc by auto
hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
@@ -598,12 +604,12 @@
thus ?thesis by auto
qed
-lemma normalized_float: assumes "m \<noteq> 0" shows "Ifloat (Float m (- (bitlen m - 1))) = real m / 2^nat (bitlen m - 1)"
+lemma normalized_float: assumes "m \<noteq> 0" shows "real (Float m (- (bitlen m - 1))) = real m / 2^nat (bitlen m - 1)"
proof (cases "- (bitlen m - 1) = 0")
- case True show ?thesis unfolding Ifloat.simps pow2_def using True by auto
+ case True show ?thesis unfolding real_of_float_simp pow2_def using True by auto
next
case False hence P: "\<not> 0 \<le> - (bitlen m - 1)" using bitlen_ge1[OF `m \<noteq> 0`] by auto
- show ?thesis unfolding Ifloat_nge0_exp[OF P] real_divide_def by auto
+ show ?thesis unfolding real_of_float_nge0_exp[OF P] real_divide_def by auto
qed
lemma bitlen_Pls: "bitlen (Int.Pls) = Int.Pls" by (subst Pls_def, subst Pls_def, simp)
@@ -660,7 +666,7 @@
lemma lapprox_posrat:
assumes x: "0 \<le> x"
and y: "0 < y"
- shows "Ifloat (lapprox_posrat prec x y) \<le> real x / real y"
+ shows "real (lapprox_posrat prec x y) \<le> real x / real y"
proof -
let ?l = "nat (int prec + bitlen y - bitlen x)"
@@ -668,7 +674,7 @@
by (rule mult_right_mono, fact real_of_int_div4, simp)
also have "\<dots> \<le> (real x / real y) * 2^?l * inverse (2^?l)" by auto
finally have "real (x * 2^?l div y) * inverse (2^?l) \<le> real x / real y" unfolding real_mult_assoc by auto
- thus ?thesis unfolding lapprox_posrat_def Let_def normfloat Ifloat.simps
+ thus ?thesis unfolding lapprox_posrat_def Let_def normfloat real_of_float_simp
unfolding pow2_minus pow2_int minus_minus .
qed
@@ -688,19 +694,19 @@
qed
lemma lapprox_posrat_bottom: assumes "0 < y"
- shows "real (x div y) \<le> Ifloat (lapprox_posrat n x y)"
+ shows "real (x div y) \<le> real (lapprox_posrat n x y)"
proof -
have pow: "\<And>x. (0::int) < 2^x" by auto
show ?thesis
- unfolding lapprox_posrat_def Let_def Ifloat_add normfloat Ifloat.simps pow2_minus pow2_int
+ unfolding lapprox_posrat_def Let_def real_of_float_add normfloat real_of_float_simp pow2_minus pow2_int
using real_of_int_div_mult[OF `0 < y` pow] by auto
qed
lemma lapprox_posrat_nonneg: assumes "0 \<le> x" and "0 < y"
- shows "0 \<le> Ifloat (lapprox_posrat n x y)"
+ shows "0 \<le> real (lapprox_posrat n x y)"
proof -
show ?thesis
- unfolding lapprox_posrat_def Let_def Ifloat_add normfloat Ifloat.simps pow2_minus pow2_int
+ unfolding lapprox_posrat_def Let_def real_of_float_add normfloat real_of_float_simp pow2_minus pow2_int
using pos_imp_zdiv_nonneg_iff[OF `0 < y`] assms by (auto intro!: mult_nonneg_nonneg)
qed
@@ -716,7 +722,7 @@
lemma rapprox_posrat:
assumes x: "0 \<le> x"
and y: "0 < y"
- shows "real x / real y \<le> Ifloat (rapprox_posrat prec x y)"
+ shows "real x / real y \<le> real (rapprox_posrat prec x y)"
proof -
let ?l = "nat (int prec + bitlen y - bitlen x)" let ?X = "x * 2^?l"
show ?thesis
@@ -727,7 +733,7 @@
also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
thus ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True]
- unfolding Ifloat.simps pow2_minus pow2_int minus_minus by auto
+ unfolding real_of_float_simp pow2_minus pow2_int minus_minus by auto
next
case False
have "0 \<le> real y" and "real y \<noteq> 0" using `0 < y` by auto
@@ -742,13 +748,13 @@
also have "\<dots> = real y * real (?X div y + 1) / real y / 2^?l" by auto
also have "\<dots> = real (?X div y + 1) * inverse (2^?l)" unfolding nonzero_mult_divide_cancel_left[OF `real y \<noteq> 0`]
unfolding real_divide_def ..
- finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat Ifloat.simps if_not_P[OF False]
+ finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
unfolding pow2_minus pow2_int minus_minus by auto
qed
qed
lemma rapprox_posrat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"
- shows "Ifloat (rapprox_posrat n x y) \<le> 1"
+ shows "real (rapprox_posrat n x y) \<le> 1"
proof -
let ?l = "nat (int n + bitlen y - bitlen x)" let ?X = "x * 2^?l"
show ?thesis
@@ -760,7 +766,7 @@
finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
also have "real x / real y \<le> 1" using `0 \<le> x` and `0 < y` and `x \<le> y` by auto
finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True]
- unfolding Ifloat.simps pow2_minus pow2_int minus_minus by auto
+ unfolding real_of_float_simp pow2_minus pow2_int minus_minus by auto
next
case False
have "x \<noteq> y"
@@ -781,7 +787,7 @@
unfolding real_of_int_le_iff[of _ "2^?l", symmetric] real_of_int_power[symmetric] real_number_of
by (rule mult_right_mono, auto)
hence "real (?X div y + 1) * inverse (2^?l) \<le> 1" by auto
- thus ?thesis unfolding rapprox_posrat_def Let_def normfloat Ifloat.simps if_not_P[OF False]
+ thus ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
unfolding pow2_minus pow2_int minus_minus by auto
qed
qed
@@ -798,9 +804,9 @@
qed
lemma rapprox_posrat_less1: assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
- shows "Ifloat (rapprox_posrat n x y) < 1"
+ shows "real (rapprox_posrat n x y) < 1"
proof (cases "x = 0")
- case True thus ?thesis unfolding rapprox_posrat_def True Let_def normfloat Ifloat.simps by auto
+ case True thus ?thesis unfolding rapprox_posrat_def True Let_def normfloat real_of_float_simp by auto
next
case False hence "0 < x" using `0 \<le> x` by auto
hence "x < y" using assms by auto
@@ -814,7 +820,7 @@
also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
also have "real x / real y < 1" using `0 \<le> x` and `0 < y` and `x < y` by auto
- finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat Ifloat.simps if_P[OF True]
+ finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_P[OF True]
unfolding pow2_minus pow2_int minus_minus by auto
next
case False
@@ -855,7 +861,7 @@
unfolding real_of_int_less_iff[of _ "2^?l", symmetric] real_of_int_power[symmetric] real_number_of
by (rule mult_strict_right_mono, auto)
hence "real (?X div y + 1) * inverse (2^?l) < 1" by auto
- thus ?thesis unfolding rapprox_posrat_def Let_def normfloat Ifloat.simps if_not_P[OF False]
+ thus ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]
unfolding pow2_minus pow2_int minus_minus by auto
qed
qed
@@ -890,7 +896,7 @@
else (if 0 < y then - (rapprox_posrat prec (-x) y) else lapprox_posrat prec (-x) (-y)))"
by auto
-lemma lapprox_rat: "Ifloat (lapprox_rat prec x y) \<le> real x / real y"
+lemma lapprox_rat: "real (lapprox_rat prec x y) \<le> real x / real y"
proof -
have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto
show ?thesis
@@ -917,7 +923,7 @@
qed
lemma lapprox_rat_bottom: assumes "0 \<le> x" and "0 < y"
- shows "real (x div y) \<le> Ifloat (lapprox_rat n x y)"
+ shows "real (x div y) \<le> real (lapprox_rat n x y)"
unfolding lapprox_rat.simps(2)[OF assms] using lapprox_posrat_bottom[OF `0<y`] .
function rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
@@ -935,7 +941,7 @@
(if 0 < y then - (lapprox_posrat prec (-x) y) else rapprox_posrat prec (-x) (-y)))"
by auto
-lemma rapprox_rat: "real x / real y \<le> Ifloat (rapprox_rat prec x y)"
+lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"
proof -
have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto
show ?thesis
@@ -962,19 +968,19 @@
qed
lemma rapprox_rat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"
- shows "Ifloat (rapprox_rat n x y) \<le> 1"
+ shows "real (rapprox_rat n x y) \<le> 1"
unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`] using rapprox_posrat_le1[OF assms] .
lemma rapprox_rat_neg: assumes "x < 0" and "0 < y"
- shows "Ifloat (rapprox_rat n x y) \<le> 0"
+ shows "real (rapprox_rat n x y) \<le> 0"
unfolding rapprox_rat.simps(3)[OF assms] using lapprox_posrat_nonneg[of "-x" y n] assms by auto
lemma rapprox_rat_nonneg_neg: assumes "0 \<le> x" and "y < 0"
- shows "Ifloat (rapprox_rat n x y) \<le> 0"
+ shows "real (rapprox_rat n x y) \<le> 0"
unfolding rapprox_rat.simps(5)[OF assms] using lapprox_posrat_nonneg[of x "-y" n] assms by auto
lemma rapprox_rat_nonpos_pos: assumes "x \<le> 0" and "0 < y"
- shows "Ifloat (rapprox_rat n x y) \<le> 0"
+ shows "real (rapprox_rat n x y) \<le> 0"
proof (cases "x = 0")
case True hence "0 \<le> x" by auto show ?thesis unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`]
unfolding True rapprox_posrat_def Let_def by auto
@@ -992,7 +998,7 @@
in
f * l)"
-lemma float_divl: "Ifloat (float_divl prec x y) \<le> Ifloat x / Ifloat y"
+lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"
proof -
from float_split[of x] obtain mx sx where x: "x = Float mx sx" by auto
from float_split[of y] obtain my sy where y: "y = Float my sy" by auto
@@ -1013,29 +1019,29 @@
apply (subst pow2_add[symmetric])
apply (simp add: field_simps)
done
- then have "Ifloat (lapprox_rat prec mx my) \<le> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
+ then have "real (lapprox_rat prec mx my) \<le> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
by (rule order_trans[OF lapprox_rat])
- then have "Ifloat (lapprox_rat prec mx my) * pow2 (sx - sy) \<le> real mx * pow2 sx / (real my * pow2 sy)"
+ then have "real (lapprox_rat prec mx my) * pow2 (sx - sy) \<le> real mx * pow2 sx / (real my * pow2 sy)"
apply (subst pos_le_divide_eq[symmetric])
apply simp_all
done
- then have "pow2 (sx - sy) * Ifloat (lapprox_rat prec mx my) \<le> real mx * pow2 sx / (real my * pow2 sy)"
+ then have "pow2 (sx - sy) * real (lapprox_rat prec mx my) \<le> real mx * pow2 sx / (real my * pow2 sy)"
by (simp add: algebra_simps)
then show ?thesis
- by (simp add: x y Let_def Ifloat.simps)
+ by (simp add: x y Let_def real_of_float_simp)
qed
lemma float_divl_lower_bound: assumes "0 \<le> x" and "0 < y" shows "0 \<le> float_divl prec x y"
proof (cases x, cases y)
fix xm xe ym ye :: int
assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
- have "0 \<le> xm" using `0 \<le> x`[unfolded x_eq le_float_def Ifloat.simps Ifloat_0 zero_le_mult_iff] by auto
- have "0 < ym" using `0 < y`[unfolded y_eq less_float_def Ifloat.simps Ifloat_0 zero_less_mult_iff] by auto
+ have "0 \<le> xm" using `0 \<le> x`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 zero_le_mult_iff] by auto
+ have "0 < ym" using `0 < y`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff] by auto
- have "\<And>n. 0 \<le> Ifloat (Float 1 n)" unfolding Ifloat.simps using zero_le_pow2 by auto
- moreover have "0 \<le> Ifloat (lapprox_rat prec xm ym)" by (rule order_trans[OF _ lapprox_rat_bottom[OF `0 \<le> xm` `0 < ym`]], auto simp add: `0 \<le> xm` pos_imp_zdiv_nonneg_iff[OF `0 < ym`])
+ have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto
+ moreover have "0 \<le> real (lapprox_rat prec xm ym)" by (rule order_trans[OF _ lapprox_rat_bottom[OF `0 \<le> xm` `0 < ym`]], auto simp add: `0 \<le> xm` pos_imp_zdiv_nonneg_iff[OF `0 < ym`])
ultimately show "0 \<le> float_divl prec x y"
- unfolding x_eq y_eq float_divl.simps Let_def le_float_def Ifloat_0 by (auto intro!: mult_nonneg_nonneg)
+ unfolding x_eq y_eq float_divl.simps Let_def le_float_def real_of_float_0 by (auto intro!: mult_nonneg_nonneg)
qed
lemma float_divl_pos_less1_bound: assumes "0 < x" and "x < 1" and "0 < prec" shows "1 \<le> float_divl prec 1 x"
@@ -1076,7 +1082,7 @@
show ?thesis
unfolding one_float_def Float float_divl.simps Let_def lapprox_rat.simps(2)[OF zero_le_one `0 < m`] lapprox_posrat_def `bitlen 1 = 1`
- unfolding le_float_def Ifloat_mult normfloat Ifloat.simps pow2_minus pow2_int e_nat
+ unfolding le_float_def real_of_float_mult normfloat real_of_float_simp pow2_minus pow2_int e_nat
using `1 \<le> 2^?e * ?d` by (auto simp add: pow2_def)
qed
@@ -1089,7 +1095,7 @@
in
f * r)"
-lemma float_divr: "Ifloat x / Ifloat y \<le> Ifloat (float_divr prec x y)"
+lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"
proof -
from float_split[of x] obtain mx sx where x: "x = Float mx sx" by auto
from float_split[of y] obtain my sy where y: "y = Float my sy" by auto
@@ -1109,20 +1115,20 @@
apply (subst pow2_add[symmetric])
apply (simp add: field_simps)
done
- then have "Ifloat (rapprox_rat prec mx my) \<ge> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
+ then have "real (rapprox_rat prec mx my) \<ge> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
by (rule order_trans[OF _ rapprox_rat])
- then have "Ifloat (rapprox_rat prec mx my) * pow2 (sx - sy) \<ge> real mx * pow2 sx / (real my * pow2 sy)"
+ then have "real (rapprox_rat prec mx my) * pow2 (sx - sy) \<ge> real mx * pow2 sx / (real my * pow2 sy)"
apply (subst pos_divide_le_eq[symmetric])
apply simp_all
done
then show ?thesis
- by (simp add: x y Let_def algebra_simps Ifloat.simps)
+ by (simp add: x y Let_def algebra_simps real_of_float_simp)
qed
lemma float_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> float_divr prec 1 x"
proof -
- have "1 \<le> 1 / Ifloat x" using `0 < x` and `x < 1` unfolding less_float_def by auto
- also have "\<dots> \<le> Ifloat (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto
+ have "1 \<le> 1 / real x" using `0 < x` and `x < 1` unfolding less_float_def by auto
+ also have "\<dots> \<le> real (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto
finally show ?thesis unfolding le_float_def by auto
qed
@@ -1130,27 +1136,27 @@
proof (cases x, cases y)
fix xm xe ym ye :: int
assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
- have "xm \<le> 0" using `x \<le> 0`[unfolded x_eq le_float_def Ifloat.simps Ifloat_0 mult_le_0_iff] by auto
- have "0 < ym" using `0 < y`[unfolded y_eq less_float_def Ifloat.simps Ifloat_0 zero_less_mult_iff] by auto
+ have "xm \<le> 0" using `x \<le> 0`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 mult_le_0_iff] by auto
+ have "0 < ym" using `0 < y`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff] by auto
- have "\<And>n. 0 \<le> Ifloat (Float 1 n)" unfolding Ifloat.simps using zero_le_pow2 by auto
- moreover have "Ifloat (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonpos_pos[OF `xm \<le> 0` `0 < ym`] .
+ have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto
+ moreover have "real (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonpos_pos[OF `xm \<le> 0` `0 < ym`] .
ultimately show "float_divr prec x y \<le> 0"
- unfolding x_eq y_eq float_divr.simps Let_def le_float_def Ifloat_0 Ifloat_mult by (auto intro!: mult_nonneg_nonpos)
+ unfolding x_eq y_eq float_divr.simps Let_def le_float_def real_of_float_0 real_of_float_mult by (auto intro!: mult_nonneg_nonpos)
qed
lemma float_divr_nonneg_neg_upper_bound: assumes "0 \<le> x" and "y < 0" shows "float_divr prec x y \<le> 0"
proof (cases x, cases y)
fix xm xe ym ye :: int
assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
- have "0 \<le> xm" using `0 \<le> x`[unfolded x_eq le_float_def Ifloat.simps Ifloat_0 zero_le_mult_iff] by auto
- have "ym < 0" using `y < 0`[unfolded y_eq less_float_def Ifloat.simps Ifloat_0 mult_less_0_iff] by auto
+ have "0 \<le> xm" using `0 \<le> x`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 zero_le_mult_iff] by auto
+ have "ym < 0" using `y < 0`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 mult_less_0_iff] by auto
hence "0 < - ym" by auto
- have "\<And>n. 0 \<le> Ifloat (Float 1 n)" unfolding Ifloat.simps using zero_le_pow2 by auto
- moreover have "Ifloat (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonneg_neg[OF `0 \<le> xm` `ym < 0`] .
+ have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto
+ moreover have "real (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonneg_neg[OF `0 \<le> xm` `ym < 0`] .
ultimately show "float_divr prec x y \<le> 0"
- unfolding x_eq y_eq float_divr.simps Let_def le_float_def Ifloat_0 Ifloat_mult by (auto intro!: mult_nonneg_nonpos)
+ unfolding x_eq y_eq float_divr.simps Let_def le_float_def real_of_float_0 real_of_float_mult by (auto intro!: mult_nonneg_nonpos)
qed
primrec round_down :: "nat \<Rightarrow> float \<Rightarrow> float" where
@@ -1163,7 +1169,7 @@
if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P in Float (n + (if r = 0 then 0 else 1)) (e + d)
else Float m e)"
-lemma round_up: "Ifloat x \<le> Ifloat (round_up prec x)"
+lemma round_up: "real x \<le> real (round_up prec x)"
proof (cases x)
case (Float m e)
let ?d = "bitlen m - int prec"
@@ -1177,7 +1183,7 @@
proof (cases "m mod ?p = 0")
case True
have m: "m = m div ?p * ?p + 0" unfolding True[symmetric] using zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right, symmetric] .
- have "Ifloat (Float m e) = Ifloat (Float (m div ?p) (e + ?d))" unfolding Ifloat.simps arg_cong[OF m, of real]
+ have "real (Float m e) = real (Float (m div ?p) (e + ?d))" unfolding real_of_float_simp arg_cong[OF m, of real]
by (auto simp add: pow2_add `0 < ?d` pow_d)
thus ?thesis
unfolding Float round_up.simps Let_def if_P[OF `m mod ?p = 0`] if_P[OF `0 < ?d`]
@@ -1186,7 +1192,7 @@
case False
have "m = m div ?p * ?p + m mod ?p" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
also have "\<dots> \<le> (m div ?p + 1) * ?p" unfolding left_distrib zmult_1 by (rule add_left_mono, rule pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
- finally have "Ifloat (Float m e) \<le> Ifloat (Float (m div ?p + 1) (e + ?d))" unfolding Ifloat.simps add_commute[of e]
+ finally have "real (Float m e) \<le> real (Float (m div ?p + 1) (e + ?d))" unfolding real_of_float_simp add_commute[of e]
unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of m, symmetric]
by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
thus ?thesis
@@ -1199,7 +1205,7 @@
qed
qed
-lemma round_down: "Ifloat (round_down prec x) \<le> Ifloat x"
+lemma round_down: "real (round_down prec x) \<le> real x"
proof (cases x)
case (Float m e)
let ?d = "bitlen m - int prec"
@@ -1211,7 +1217,7 @@
hence pow_d: "pow2 ?d = real ?p" unfolding pow2_int[symmetric] power_real_number_of[symmetric] by auto
have "m div ?p * ?p \<le> m div ?p * ?p + m mod ?p" by (auto simp add: pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
also have "\<dots> \<le> m" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
- finally have "Ifloat (Float (m div ?p) (e + ?d)) \<le> Ifloat (Float m e)" unfolding Ifloat.simps add_commute[of e]
+ finally have "real (Float (m div ?p) (e + ?d)) \<le> real (Float m e)" unfolding real_of_float_simp add_commute[of e]
unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of _ m, symmetric]
by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
thus ?thesis
@@ -1235,12 +1241,12 @@
in if l > 0 then Float (m div (2^nat l) + 1) (e + l)
else Float m e)"
-lemma lb_mult: "Ifloat (lb_mult prec x y) \<le> Ifloat (x * y)"
+lemma lb_mult: "real (lb_mult prec x y) \<le> real (x * y)"
proof (cases "normfloat (x * y)")
case (Float m e)
hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)
let ?l = "bitlen m - int prec"
- have "Ifloat (lb_mult prec x y) \<le> Ifloat (normfloat (x * y))"
+ have "real (lb_mult prec x y) \<le> real (normfloat (x * y))"
proof (cases "?l > 0")
case False thus ?thesis unfolding lb_mult_def Float Let_def float.cases by auto
next
@@ -1253,19 +1259,19 @@
also have "\<dots> = real m" unfolding zmod_zdiv_equality[symmetric] ..
finally show ?thesis by auto
qed
- thus ?thesis unfolding lb_mult_def Float Let_def float.cases if_P[OF True] Ifloat.simps pow2_add real_mult_commute real_mult_assoc by auto
+ thus ?thesis unfolding lb_mult_def Float Let_def float.cases if_P[OF True] real_of_float_simp pow2_add real_mult_commute real_mult_assoc by auto
qed
- also have "\<dots> = Ifloat (x * y)" unfolding normfloat ..
+ also have "\<dots> = real (x * y)" unfolding normfloat ..
finally show ?thesis .
qed
-lemma ub_mult: "Ifloat (x * y) \<le> Ifloat (ub_mult prec x y)"
+lemma ub_mult: "real (x * y) \<le> real (ub_mult prec x y)"
proof (cases "normfloat (x * y)")
case (Float m e)
hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)
let ?l = "bitlen m - int prec"
- have "Ifloat (x * y) = Ifloat (normfloat (x * y))" unfolding normfloat ..
- also have "\<dots> \<le> Ifloat (ub_mult prec x y)"
+ have "real (x * y) = real (normfloat (x * y))" unfolding normfloat ..
+ also have "\<dots> \<le> real (ub_mult prec x y)"
proof (cases "?l > 0")
case False thus ?thesis unfolding ub_mult_def Float Let_def float.cases by auto
next
@@ -1280,7 +1286,7 @@
also have "\<dots> \<le> (real (m div 2^(nat ?l)) + 1) * 2^(nat ?l)" unfolding real_add_mult_distrib using mod_uneq by auto
finally show ?thesis unfolding pow2_int[symmetric] using True by auto
qed
- thus ?thesis unfolding ub_mult_def Float Let_def float.cases if_P[OF True] Ifloat.simps pow2_add real_mult_commute real_mult_assoc by auto
+ thus ?thesis unfolding ub_mult_def Float Let_def float.cases if_P[OF True] real_of_float_simp pow2_add real_mult_commute real_mult_assoc by auto
qed
finally show ?thesis .
qed
@@ -1293,28 +1299,28 @@
instance ..
end
-lemma Ifloat_abs: "Ifloat \<bar>x\<bar> = \<bar>Ifloat x\<bar>"
+lemma real_of_float_abs: "real \<bar>x :: float\<bar> = \<bar>real x\<bar>"
proof (cases x)
case (Float m e)
have "\<bar>real m\<bar> * pow2 e = \<bar>real m * pow2 e\<bar>" unfolding abs_mult by auto
- thus ?thesis unfolding Float abs_float_def float_abs.simps Ifloat.simps by auto
+ thus ?thesis unfolding Float abs_float_def float_abs.simps real_of_float_simp by auto
qed
primrec floor_fl :: "float \<Rightarrow> float" where
"floor_fl (Float m e) = (if 0 \<le> e then Float m e
else Float (m div (2 ^ (nat (-e)))) 0)"
-lemma floor_fl: "Ifloat (floor_fl x) \<le> Ifloat x"
+lemma floor_fl: "real (floor_fl x) \<le> real x"
proof (cases x)
case (Float m e)
show ?thesis
proof (cases "0 \<le> e")
case False
hence me_eq: "pow2 (-e) = pow2 (int (nat (-e)))" by auto
- have "Ifloat (Float (m div (2 ^ (nat (-e)))) 0) = real (m div 2 ^ (nat (-e)))" unfolding Ifloat.simps by auto
+ have "real (Float (m div (2 ^ (nat (-e)))) 0) = real (m div 2 ^ (nat (-e)))" unfolding real_of_float_simp by auto
also have "\<dots> \<le> real m / real ((2::int) ^ (nat (-e)))" using real_of_int_div4 .
also have "\<dots> = real m * inverse (2 ^ (nat (-e)))" unfolding power_real_number_of[symmetric] real_divide_def ..
- also have "\<dots> = Ifloat (Float m e)" unfolding Ifloat.simps me_eq pow2_int pow2_neg[of e] ..
+ also have "\<dots> = real (Float m e)" unfolding real_of_float_simp me_eq pow2_int pow2_neg[of e] ..
finally show ?thesis unfolding Float floor_fl.simps if_not_P[OF `\<not> 0 \<le> e`] .
next
case True thus ?thesis unfolding Float by auto
@@ -1333,17 +1339,17 @@
"ceiling_fl (Float m e) = (if 0 \<le> e then Float m e
else Float (m div (2 ^ (nat (-e))) + 1) 0)"
-lemma ceiling_fl: "Ifloat x \<le> Ifloat (ceiling_fl x)"
+lemma ceiling_fl: "real x \<le> real (ceiling_fl x)"
proof (cases x)
case (Float m e)
show ?thesis
proof (cases "0 \<le> e")
case False
hence me_eq: "pow2 (-e) = pow2 (int (nat (-e)))" by auto
- have "Ifloat (Float m e) = real m * inverse (2 ^ (nat (-e)))" unfolding Ifloat.simps me_eq pow2_int pow2_neg[of e] ..
+ have "real (Float m e) = real m * inverse (2 ^ (nat (-e)))" unfolding real_of_float_simp me_eq pow2_int pow2_neg[of e] ..
also have "\<dots> = real m / real ((2::int) ^ (nat (-e)))" unfolding power_real_number_of[symmetric] real_divide_def ..
also have "\<dots> \<le> 1 + real (m div 2 ^ (nat (-e)))" using real_of_int_div3[unfolded diff_le_eq] .
- also have "\<dots> = Ifloat (Float (m div (2 ^ (nat (-e))) + 1) 0)" unfolding Ifloat.simps by auto
+ also have "\<dots> = real (Float (m div (2 ^ (nat (-e))) + 1) 0)" unfolding real_of_float_simp by auto
finally show ?thesis unfolding Float ceiling_fl.simps if_not_P[OF `\<not> 0 \<le> e`] .
next
case True thus ?thesis unfolding Float by auto
@@ -1358,48 +1364,48 @@
definition ub_mod :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
"ub_mod prec x ub lb = x - floor_fl (float_divl prec x ub) * lb"
-lemma lb_mod: fixes k :: int assumes "0 \<le> Ifloat x" and "real k * y \<le> Ifloat x" (is "?k * y \<le> ?x")
- assumes "0 < Ifloat lb" "Ifloat lb \<le> y" (is "?lb \<le> y") "y \<le> Ifloat ub" (is "y \<le> ?ub")
- shows "Ifloat (lb_mod prec x ub lb) \<le> ?x - ?k * y"
+lemma lb_mod: fixes k :: int assumes "0 \<le> real x" and "real k * y \<le> real x" (is "?k * y \<le> ?x")
+ assumes "0 < real lb" "real lb \<le> y" (is "?lb \<le> y") "y \<le> real ub" (is "y \<le> ?ub")
+ shows "real (lb_mod prec x ub lb) \<le> ?x - ?k * y"
proof -
have "?lb \<le> ?ub" by (auto!)
have "0 \<le> ?lb" and "?lb \<noteq> 0" by (auto!)
have "?k * y \<le> ?x" using assms by auto
also have "\<dots> \<le> ?x / ?lb * ?ub" by (metis mult_left_mono[OF `?lb \<le> ?ub` `0 \<le> ?x`] divide_right_mono[OF _ `0 \<le> ?lb` ] times_divide_eq_left nonzero_mult_divide_cancel_right[OF `?lb \<noteq> 0`])
- also have "\<dots> \<le> Ifloat (ceiling_fl (float_divr prec x lb)) * ?ub" by (metis mult_right_mono order_trans `0 \<le> ?lb` `?lb \<le> ?ub` float_divr ceiling_fl)
- finally show ?thesis unfolding lb_mod_def Ifloat_sub Ifloat_mult by auto
+ also have "\<dots> \<le> real (ceiling_fl (float_divr prec x lb)) * ?ub" by (metis mult_right_mono order_trans `0 \<le> ?lb` `?lb \<le> ?ub` float_divr ceiling_fl)
+ finally show ?thesis unfolding lb_mod_def real_of_float_sub real_of_float_mult by auto
qed
-lemma ub_mod: fixes k :: int assumes "0 \<le> Ifloat x" and "Ifloat x \<le> real k * y" (is "?x \<le> ?k * y")
- assumes "0 < Ifloat lb" "Ifloat lb \<le> y" (is "?lb \<le> y") "y \<le> Ifloat ub" (is "y \<le> ?ub")
- shows "?x - ?k * y \<le> Ifloat (ub_mod prec x ub lb)"
+lemma ub_mod: fixes k :: int and x :: float assumes "0 \<le> real x" and "real x \<le> real k * y" (is "?x \<le> ?k * y")
+ assumes "0 < real lb" "real lb \<le> y" (is "?lb \<le> y") "y \<le> real ub" (is "y \<le> ?ub")
+ shows "?x - ?k * y \<le> real (ub_mod prec x ub lb)"
proof -
have "?lb \<le> ?ub" by (auto!)
hence "0 \<le> ?lb" and "0 \<le> ?ub" and "?ub \<noteq> 0" by (auto!)
- have "Ifloat (floor_fl (float_divl prec x ub)) * ?lb \<le> ?x / ?ub * ?lb" by (metis mult_right_mono order_trans `0 \<le> ?lb` `?lb \<le> ?ub` float_divl floor_fl)
+ have "real (floor_fl (float_divl prec x ub)) * ?lb \<le> ?x / ?ub * ?lb" by (metis mult_right_mono order_trans `0 \<le> ?lb` `?lb \<le> ?ub` float_divl floor_fl)
also have "\<dots> \<le> ?x" by (metis mult_left_mono[OF `?lb \<le> ?ub` `0 \<le> ?x`] divide_right_mono[OF _ `0 \<le> ?ub` ] times_divide_eq_left nonzero_mult_divide_cancel_right[OF `?ub \<noteq> 0`])
also have "\<dots> \<le> ?k * y" using assms by auto
- finally show ?thesis unfolding ub_mod_def Ifloat_sub Ifloat_mult by auto
+ finally show ?thesis unfolding ub_mod_def real_of_float_sub real_of_float_mult by auto
qed
lemma le_float_def': "f \<le> g = (case f - g of Float a b \<Rightarrow> a \<le> 0)"
proof -
- have le_transfer: "(f \<le> g) = (Ifloat (f - g) \<le> 0)" by (auto simp add: le_float_def)
+ have le_transfer: "(f \<le> g) = (real (f - g) \<le> 0)" by (auto simp add: le_float_def)
from float_split[of "f - g"] obtain a b where f_diff_g: "f - g = Float a b" by auto
- with le_transfer have le_transfer': "f \<le> g = (Ifloat (Float a b) \<le> 0)" by simp
+ with le_transfer have le_transfer': "f \<le> g = (real (Float a b) \<le> 0)" by simp
show ?thesis by (simp add: le_transfer' f_diff_g float_le_zero)
qed
lemma float_less_zero:
- "(Ifloat (Float a b) < 0) = (a < 0)"
- apply (auto simp add: mult_less_0_iff Ifloat.simps)
+ "(real (Float a b) < 0) = (a < 0)"
+ apply (auto simp add: mult_less_0_iff real_of_float_simp)
done
lemma less_float_def': "f < g = (case f - g of Float a b \<Rightarrow> a < 0)"
proof -
- have less_transfer: "(f < g) = (Ifloat (f - g) < 0)" by (auto simp add: less_float_def)
+ have less_transfer: "(f < g) = (real (f - g) < 0)" by (auto simp add: less_float_def)
from float_split[of "f - g"] obtain a b where f_diff_g: "f - g = Float a b" by auto
- with less_transfer have less_transfer': "f < g = (Ifloat (Float a b) < 0)" by simp
+ with less_transfer have less_transfer': "f < g = (real (Float a b) < 0)" by simp
show ?thesis by (simp add: less_transfer' f_diff_g float_less_zero)
qed