(* Author: Johannes Hoelzl, TU Muenchen
Coercions removed by Dmitriy Traytel *)
section {* Prove Real Valued Inequalities by Computation *}
theory Approximation
imports
Complex_Main
"~~/src/HOL/Library/Float"
Dense_Linear_Order
"~~/src/HOL/Library/Code_Target_Numeral"
keywords "approximate" :: diag
begin
declare powr_numeral [simp]
declare powr_neg_one [simp]
declare powr_neg_numeral [simp]
section "Horner Scheme"
subsection {* Define auxiliary helper @{text horner} function *}
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 / k - x * horner F G n (F i) (G i k) x"
lemma horner_schema':
fixes x :: real and a :: "nat \<Rightarrow> real"
shows "a 0 - x * (\<Sum> i=0..<n. (-1)^i * a (Suc i) * x^i) = (\<Sum> i=0..<Suc n. (-1)^i * a i * x^i)"
proof -
have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)"
by auto
show ?thesis
unfolding setsum_right_distrib shift_pow uminus_add_conv_diff [symmetric] setsum_negf[symmetric]
setsum_head_upt_Suc[OF zero_less_Suc]
setsum.reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n *a n * x^n"] by auto
qed
lemma horner_schema:
fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat"
assumes f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
shows "horner F G n ((F ^^ j') s) (f j') x = (\<Sum> j = 0..< n. (- 1) ^ j * (1 / (f (j' + j))) * x ^ j)"
proof (induct n arbitrary: j')
case 0
then show ?case by auto
next
case (Suc n)
show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc]
using horner_schema'[of "\<lambda> j. 1 / (f (j' + j))"] by auto
qed
lemma horner_bounds':
fixes lb :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" and ub :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
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 = float_plus_down prec
(lapprox_rat prec 1 k)
(- float_round_up prec (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 = float_plus_up prec
(rapprox_rat prec 1 k)
(- float_round_down prec (x * (lb n (F i) (G i k) x)))"
shows "(lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') x \<and>
horner F G n ((F ^^ j') s) (f j') x \<le> (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)
thus ?case using lapprox_rat[of prec 1 "f j'"] using rapprox_rat[of 1 "f j'" prec]
Suc[where j'="Suc j'"] `0 \<le> real x`
by (auto intro!: add_mono mult_left_mono float_round_down_le float_round_up_le
order_trans[OF add_mono[OF _ float_plus_down_le]]
order_trans[OF _ add_mono[OF _ float_plus_up_le]]
simp add: lb_Suc ub_Suc field_simps f_Suc)
qed
subsection "Theorems for floating point functions implementing the horner scheme"
text {*
Here @{term_type "f :: nat \<Rightarrow> nat"} is the sequence defining the Taylor series, the coefficients are
all alternating and reciprocs. We use @{term G} and @{term F} to describe the computation of @{term f}.
*}
lemma horner_bounds:
fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
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 = float_plus_down prec
(lapprox_rat prec 1 k)
(- float_round_up prec (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 = float_plus_up prec
(rapprox_rat prec 1 k)
(- float_round_down prec (x * (lb n (F i) (G i k) x)))"
shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j))"
(is "?lb")
and "(\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j)) \<le> (ub n ((F ^^ j') s) (f j') x)"
(is "?ub")
proof -
have "?lb \<and> ?ub"
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 "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 = float_plus_down prec
(lapprox_rat prec 1 k)
(float_round_down prec (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 = float_plus_up prec
(rapprox_rat prec 1 k)
(float_round_up prec (x * (lb n (F i) (G i k) x)))"
shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / (f (j' + j))) * real x ^ j)" (is "?lb")
and "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real x ^ j) \<le> (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
proof -
{ fix x y z :: float have "x - y * z = x + - y * z" by simp } note diff_mult_minus = this
have sum_eq: "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real x ^ j) =
(\<Sum>j = 0..<n. (- 1) ^ j * (1 / (f (j' + j))) * real (- x) ^ j)"
by (auto simp add: field_simps power_mult_distrib[symmetric])
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 _ ub_0 _]
show "?lb" and "?ub" unfolding minus_minus sum_eq
by (auto simp: minus_float_round_up_eq minus_float_round_down_eq)
qed
subsection {* Selectors for next even or odd number *}
text {*
The horner scheme computes alternating series. To get the upper and lower bounds we need to
guarantee to access a even or odd member. To do this we use @{term get_odd} and @{term get_even}.
*}
definition get_odd :: "nat \<Rightarrow> nat" where
"get_odd n = (if odd n then n else (Suc n))"
definition get_even :: "nat \<Rightarrow> nat" where
"get_even n = (if even n then n else (Suc n))"
lemma get_odd[simp]: "odd (get_odd n)" unfolding get_odd_def by (cases "odd n", auto)
lemma get_even[simp]: "even (get_even n)" unfolding get_even_def by (cases "even n", auto)
lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)"
by (auto simp: get_odd_def odd_pos intro!: exI[of _ "n - 1"])
lemma get_even_double:
"\<exists>i. get_even n = 2 * i" using get_even by (blast elim: evenE)
lemma get_odd_double:
"\<exists>i. get_odd n = 2 * i + 1" using get_odd by (blast elim: oddE)
section "Power function"
definition float_power_bnds :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
"float_power_bnds prec n l u =
(if 0 < l then (power_down_fl prec l n, power_up_fl prec u n)
else if odd n then
(- power_up_fl prec (abs l) n,
if u < 0 then - power_down_fl prec (abs u) n else power_up_fl prec u n)
else if u < 0 then (power_down_fl prec (abs u) n, power_up_fl prec (abs l) n)
else (0, power_up_fl prec (max (abs l) (abs u)) n))"
lemma le_minus_power_downI: "0 \<le> x \<Longrightarrow> x ^ n \<le> - a \<Longrightarrow> a \<le> - power_down prec x n"
by (subst le_minus_iff) (auto intro: power_down_le power_mono_odd)
lemma float_power_bnds:
"(l1, u1) = float_power_bnds prec n l u \<Longrightarrow> x \<in> {l .. u} \<Longrightarrow> (x::real) ^ n \<in> {l1..u1}"
by (auto
simp: float_power_bnds_def max_def real_power_up_fl real_power_down_fl minus_le_iff
split: split_if_asm
intro!: power_up_le power_down_le le_minus_power_downI
intro: power_mono_odd power_mono power_mono_even zero_le_even_power)
lemma bnds_power:
"\<forall> (x::real) l u. (l1, u1) = float_power_bnds prec n l u \<and> x \<in> {l .. u} \<longrightarrow>
l1 \<le> x ^ n \<and> x ^ n \<le> u1"
using float_power_bnds by auto
section "Square root"
text {*
The square root computation is implemented as newton iteration. As first first step we use the
nearest power of two greater than the square root.
*}
fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
"sqrt_iteration prec 0 x = Float 1 ((bitlen \<bar>mantissa x\<bar> + exponent x) div 2 + 1)" |
"sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x
in Float 1 (- 1) * float_plus_up prec y (float_divr prec x y))"
lemma compute_sqrt_iteration_base[code]:
shows "sqrt_iteration prec n (Float m e) =
(if n = 0 then Float 1 ((if m = 0 then 0 else bitlen \<bar>m\<bar> + e) div 2 + 1)
else (let y = sqrt_iteration prec (n - 1) (Float m e) in
Float 1 (- 1) * float_plus_up prec y (float_divr prec (Float m e) y)))"
using bitlen_Float by (cases n) simp_all
function ub_sqrt lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float" where
"ub_sqrt prec x = (if 0 < x then (sqrt_iteration prec prec x)
else if x < 0 then - lb_sqrt prec (- x)
else 0)" |
"lb_sqrt prec x = (if 0 < x then (float_divl prec x (sqrt_iteration prec prec x))
else if x < 0 then - ub_sqrt prec (- x)
else 0)"
by pat_completeness auto
termination by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)
declare lb_sqrt.simps[simp del]
declare ub_sqrt.simps[simp del]
lemma sqrt_ub_pos_pos_1:
assumes "sqrt x < b" and "0 < b" and "0 < x"
shows "sqrt x < (b + x / b)/2"
proof -
from assms have "0 < (b - sqrt x)\<^sup>2 " by simp
also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + (sqrt x)\<^sup>2" by algebra
also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + x" using assms by simp
finally have "0 < b\<^sup>2 - 2 * b * sqrt x + x" .
hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms
by (simp add: field_simps power2_eq_square)
thus ?thesis by (simp add: field_simps)
qed
lemma sqrt_iteration_bound: assumes "0 < real x"
shows "sqrt x < sqrt_iteration prec n x"
proof (induct n)
case 0
show ?case
proof (cases x)
case (Float m e)
hence "0 < m" using assms powr_gt_zero[of 2 e] by (auto simp: sign_simps)
hence "0 < sqrt m" by auto
have int_nat_bl: "(nat (bitlen m)) = bitlen m" using bitlen_nonneg by auto
have "x = (m / 2^nat (bitlen m)) * 2 powr (e + (nat (bitlen m)))"
unfolding Float by (auto simp: powr_realpow[symmetric] field_simps powr_add)
also have "\<dots> < 1 * 2 powr (e + nat (bitlen m))"
proof (rule mult_strict_right_mono, auto)
show "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 x < sqrt (2 powr (e + bitlen m))" unfolding int_nat_bl by auto
also have "\<dots> \<le> 2 powr ((e + bitlen m) div 2 + 1)"
proof -
let ?E = "e + bitlen m"
have E_mod_pow: "2 powr (?E mod 2) < 4"
proof (cases "?E mod 2 = 1")
case True thus ?thesis by auto
next
case False
have "0 \<le> ?E mod 2" by auto
have "?E mod 2 < 2" by auto
from this[THEN zless_imp_add1_zle]
have "?E mod 2 \<le> 0" using False by auto
from xt1(5)[OF `0 \<le> ?E mod 2` this]
show ?thesis by auto
qed
hence "sqrt (2 powr (?E mod 2)) < sqrt (2 * 2)"
by (auto simp del: real_sqrt_four)
hence E_mod_pow: "sqrt (2 powr (?E mod 2)) < 2" by auto
have E_eq: "2 powr ?E = 2 powr (?E div 2 + ?E div 2 + ?E mod 2)" by auto
have "sqrt (2 powr ?E) = sqrt (2 powr (?E div 2) * 2 powr (?E div 2) * 2 powr (?E mod 2))"
unfolding E_eq unfolding powr_add[symmetric] by (simp add: int_of_reals del: real_of_ints)
also have "\<dots> = 2 powr (?E div 2) * sqrt (2 powr (?E mod 2))"
unfolding real_sqrt_mult[of _ "2 powr (?E mod 2)"] real_sqrt_abs2 by auto
also have "\<dots> < 2 powr (?E div 2) * 2 powr 1"
by (rule mult_strict_left_mono, auto intro: E_mod_pow)
also have "\<dots> = 2 powr (?E div 2 + 1)" unfolding add.commute[of _ 1] powr_add[symmetric]
by simp
finally show ?thesis by auto
qed
finally show ?thesis using `0 < m`
unfolding Float
by (subst compute_sqrt_iteration_base) (simp add: ac_simps)
qed
next
case (Suc n)
let ?b = "sqrt_iteration prec n x"
have "0 < sqrt x" using `0 < real x` by auto
also have "\<dots> < real ?b" using Suc .
finally have "sqrt x < (?b + x / ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ `0 < real x`] by auto
also have "\<dots> \<le> (?b + (float_divr prec x ?b))/2"
by (rule divide_right_mono, auto simp add: float_divr)
also have "\<dots> = (Float 1 (- 1)) * (?b + (float_divr prec x ?b))" by simp
also have "\<dots> \<le> (Float 1 (- 1)) * (float_plus_up prec ?b (float_divr prec x ?b))"
by (auto simp add: algebra_simps float_plus_up_le)
finally show ?case unfolding sqrt_iteration.simps Let_def distrib_left .
qed
lemma sqrt_iteration_lower_bound: assumes "0 < real x"
shows "0 < real (sqrt_iteration prec n x)" (is "0 < ?sqrt")
proof -
have "0 < sqrt 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> real x"
shows "0 \<le> real (lb_sqrt prec x)"
proof (cases "0 < x")
case True hence "0 < real x" and "0 \<le> x" using `0 \<le> real x` by auto
hence "0 < sqrt_iteration prec prec x" using sqrt_iteration_lower_bound 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 less_eq_float_def by auto
thus ?thesis unfolding lb_sqrt.simps using True by auto
next
case False with `0 \<le> real x` have "real x = 0" by auto
thus ?thesis unfolding lb_sqrt.simps by auto
qed
lemma bnds_sqrt': "sqrt x \<in> {(lb_sqrt prec x) .. (ub_sqrt prec x)}"
proof -
{ fix x :: float assume "0 < x"
hence "0 < real x" and "0 \<le> real x" by auto
hence sqrt_gt0: "0 < sqrt x" by auto
hence sqrt_ub: "sqrt x < sqrt_iteration prec prec x" using sqrt_iteration_bound by auto
have "(float_divl prec x (sqrt_iteration prec prec x)) \<le>
x / (sqrt_iteration prec prec x)" by (rule float_divl)
also have "\<dots> < x / sqrt 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 x"
unfolding inverse_eq_iff_eq[of _ "sqrt x", symmetric]
sqrt_divide_self_eq[OF `0 \<le> real x`, symmetric] by auto
finally have "lb_sqrt prec x \<le> sqrt x"
unfolding lb_sqrt.simps if_P[OF `0 < x`] by auto }
note lb = this
{ fix x :: float assume "0 < x"
hence "0 < real x" by auto
hence "0 < sqrt x" by auto
hence "sqrt x < sqrt_iteration prec prec x"
using sqrt_iteration_bound by auto
hence "sqrt x \<le> ub_sqrt prec x"
unfolding ub_sqrt.simps if_P[OF `0 < x`] by auto }
note ub = this
show ?thesis
using lb[of "-x"] ub[of "-x"] lb[of x] ub[of x]
by (auto simp add: lb_sqrt.simps ub_sqrt.simps real_sqrt_minus)
qed
lemma bnds_sqrt: "\<forall> (x::real) lx ux. (l, u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> sqrt x \<and> sqrt x \<le> u"
proof ((rule allI) +, rule impI, erule conjE, rule conjI)
fix x :: real fix lx ux
assume "(l, u) = (lb_sqrt prec lx, ub_sqrt prec ux)"
and x: "x \<in> {lx .. ux}"
hence l: "l = lb_sqrt prec lx " and u: "u = ub_sqrt prec ux" by auto
have "sqrt lx \<le> sqrt x" using x by auto
from order_trans[OF _ this]
show "l \<le> sqrt x" unfolding l using bnds_sqrt'[of lx prec] by auto
have "sqrt x \<le> sqrt ux" using x by auto
from order_trans[OF this]
show "sqrt x \<le> u" unfolding u using bnds_sqrt'[of ux prec] by auto
qed
section "Arcus tangens and \<pi>"
subsection "Compute arcus tangens series"
text {*
As first step we implement the computation of the arcus tangens series. This is only valid in the range
@{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens.
*}
fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
"ub_arctan_horner prec 0 k x = 0"
| "ub_arctan_horner prec (Suc n) k x = float_plus_up prec
(rapprox_rat prec 1 k) (- float_round_down prec (x * (lb_arctan_horner prec n (k + 2) x)))"
| "lb_arctan_horner prec 0 k x = 0"
| "lb_arctan_horner prec (Suc n) k x = float_plus_down prec
(lapprox_rat prec 1 k) (- float_round_up prec (x * (ub_arctan_horner prec n (k + 2) x)))"
lemma arctan_0_1_bounds':
assumes "0 \<le> real y" "real y \<le> 1" and "even n"
shows "arctan (sqrt y) \<in>
{(sqrt y * lb_arctan_horner prec n 1 y) .. (sqrt y * ub_arctan_horner prec (Suc n) 1 y)}"
proof -
let ?c = "\<lambda>i. (- 1) ^ i * (1 / (i * 2 + (1::nat)) * sqrt y ^ (i * 2 + 1))"
let ?S = "\<lambda>n. \<Sum> i=0..<n. ?c i"
have "0 \<le> sqrt y" using assms by auto
have "sqrt y \<le> 1" using assms by auto
from `even n` obtain m where "2 * m = n" by (blast elim: evenE)
have "arctan (sqrt y) \<in> { ?S n .. ?S (Suc n) }"
proof (cases "sqrt y = 0")
case False
hence "0 < sqrt y" using `0 \<le> sqrt y` by auto
hence prem: "0 < 1 / (0 * 2 + (1::nat)) * sqrt y ^ (0 * 2 + 1)" by auto
have "\<bar> sqrt y \<bar> \<le> 1" using `0 \<le> sqrt y` `sqrt y \<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> sqrt y \<bar> \<le> 1`] Suc_eq_plus1 atLeast0LessThan .
qed auto
note arctan_bounds = this[unfolded atLeastAtMost_iff]
have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
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> real y` F lb_arctan_horner.simps ub_arctan_horner.simps]
{ have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> ?S n"
using bounds(1) `0 \<le> sqrt y`
unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
unfolding mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult
by (auto intro!: mult_left_mono)
also have "\<dots> \<le> arctan (sqrt y)" using arctan_bounds ..
finally have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> arctan (sqrt y)" . }
moreover
{ have "arctan (sqrt y) \<le> ?S (Suc n)" using arctan_bounds ..
also have "\<dots> \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)"
using bounds(2)[of "Suc n"] `0 \<le> sqrt y`
unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
unfolding mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult
by (auto intro!: mult_left_mono)
finally have "arctan (sqrt y) \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)" . }
ultimately show ?thesis by auto
qed
lemma arctan_0_1_bounds: assumes "0 \<le> real y" "real y \<le> 1"
shows "arctan (sqrt y) \<in>
{(sqrt y * lb_arctan_horner prec (get_even n) 1 y) ..
(sqrt y * ub_arctan_horner prec (get_odd n) 1 y)}"
using
arctan_0_1_bounds'[OF assms, of n prec]
arctan_0_1_bounds'[OF assms, of "n + 1" prec]
arctan_0_1_bounds'[OF assms, of "n - 1" prec]
by (auto simp: get_even_def get_odd_def odd_pos simp del: ub_arctan_horner.simps
lb_arctan_horner.simps)
lemma arctan_lower_bound:
assumes "0 \<le> x"
shows "x / (1 + x\<^sup>2) \<le> arctan x" (is "?l x \<le> _")
proof -
have "?l x - arctan x \<le> ?l 0 - arctan 0"
using assms
by (intro DERIV_nonpos_imp_nonincreasing[where f="\<lambda>x. ?l x - arctan x"])
(auto intro!: derivative_eq_intros simp: add_nonneg_eq_0_iff field_simps)
thus ?thesis by simp
qed
lemma arctan_divide_mono: "0 < x \<Longrightarrow> x \<le> y \<Longrightarrow> arctan y / y \<le> arctan x / x"
by (rule DERIV_nonpos_imp_nonincreasing[where f="\<lambda>x. arctan x / x"])
(auto intro!: derivative_eq_intros divide_nonpos_nonneg
simp: inverse_eq_divide arctan_lower_bound)
lemma arctan_mult_mono: "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> x * arctan y \<le> y * arctan x"
using arctan_divide_mono[of x y] by (cases "x = 0") (simp_all add: field_simps)
lemma arctan_mult_le:
assumes "0 \<le> x" "x \<le> y" "y * z \<le> arctan y"
shows "x * z \<le> arctan x"
proof cases
assume "x \<noteq> 0"
with assms have "z \<le> arctan y / y" by (simp add: field_simps)
also have "\<dots> \<le> arctan x / x" using assms `x \<noteq> 0` by (auto intro!: arctan_divide_mono)
finally show ?thesis using assms `x \<noteq> 0` by (simp add: field_simps)
qed simp
lemma arctan_le_mult:
assumes "0 < x" "x \<le> y" "arctan x \<le> x * z"
shows "arctan y \<le> y * z"
proof -
from assms have "arctan y / y \<le> arctan x / x" by (auto intro!: arctan_divide_mono)
also have "\<dots> \<le> z" using assms by (auto simp: field_simps)
finally show ?thesis using assms by (simp add: field_simps)
qed
lemma arctan_0_1_bounds_le:
assumes "0 \<le> x" "x \<le> 1" "0 < real xl" "real xl \<le> x * x" "x * x \<le> real xu" "real xu \<le> 1"
shows "arctan x \<in>
{x * lb_arctan_horner p1 (get_even n) 1 xu .. x * ub_arctan_horner p2 (get_odd n) 1 xl}"
proof -
from assms have "real xl \<le> 1" "sqrt (real xl) \<le> x" "x \<le> sqrt (real xu)" "0 \<le> real xu"
"0 \<le> real xl" "0 < sqrt (real xl)"
by (auto intro!: real_le_rsqrt real_le_lsqrt simp: power2_eq_square)
from arctan_0_1_bounds[OF `0 \<le> real xu` `real xu \<le> 1`]
have "sqrt (real xu) * real (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan (sqrt (real xu))"
by simp
from arctan_mult_le[OF `0 \<le> x` `x \<le> sqrt _` this]
have "x * real (lb_arctan_horner p1 (get_even n) 1 xu) \<le> arctan x" .
moreover
from arctan_0_1_bounds[OF `0 \<le> real xl` `real xl \<le> 1`]
have "arctan (sqrt (real xl)) \<le> sqrt (real xl) * real (ub_arctan_horner p2 (get_odd n) 1 xl)"
by simp
from arctan_le_mult[OF `0 < sqrt xl` `sqrt xl \<le> x` this]
have "arctan x \<le> x * real (ub_arctan_horner p2 (get_odd n) 1 xl)" .
ultimately show ?thesis by simp
qed
lemma mult_nonneg_le_one: fixes a::real assumes "0 \<le> a" "0 \<le> b" "a \<le> 1" "b \<le> 1" shows "a * b \<le> 1"
proof -
have "a * b \<le> 1 * 1"
by (intro mult_mono assms) simp_all
thus ?thesis by simp
qed
lemma arctan_0_1_bounds_round:
assumes "0 \<le> real x" "real x \<le> 1"
shows "arctan x \<in>
{real x * lb_arctan_horner p1 (get_even n) 1 (float_round_up (Suc p2) (x * x)) ..
real x * ub_arctan_horner p3 (get_odd n) 1 (float_round_down (Suc p4) (x * x))}"
using assms
apply (cases "x > 0")
apply (intro arctan_0_1_bounds_le)
apply (auto simp: float_round_down.rep_eq float_round_up.rep_eq
intro!: truncate_up_le1 mult_nonneg_le_one truncate_down_le truncate_up_le truncate_down_pos
mult_pos_pos)
done
subsection "Compute \<pi>"
definition ub_pi :: "nat \<Rightarrow> float" where
"ub_pi prec =
(let
A = rapprox_rat prec 1 5 ;
B = lapprox_rat prec 1 239
in ((Float 1 2) * float_plus_up prec
((Float 1 2) * float_round_up prec (A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1
(float_round_down (Suc prec) (A * A)))))
(- float_round_down prec (B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1
(float_round_up (Suc prec) (B * B)))))))"
definition lb_pi :: "nat \<Rightarrow> float" where
"lb_pi prec =
(let
A = lapprox_rat prec 1 5 ;
B = rapprox_rat prec 1 239
in ((Float 1 2) * float_plus_down prec
((Float 1 2) * float_round_down prec (A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1
(float_round_up (Suc prec) (A * A)))))
(- float_round_up prec (B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1
(float_round_down (Suc prec) (B * B)))))))"
lemma pi_boundaries: "pi \<in> {(lb_pi n) .. (ub_pi n)}"
proof -
have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))"
unfolding machin[symmetric] by auto
{ fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto
let ?k = "rapprox_rat prec 1 k"
let ?kl = "float_round_down (Suc prec) (?k * ?k)"
have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
have "0 \<le> real ?k" by (rule order_trans[OF _ rapprox_rat]) (auto simp add: `0 \<le> k`)
have "real ?k \<le> 1"
by (auto simp add: `0 < k` `1 \<le> k` less_imp_le
intro!: mult_nonneg_le_one order_trans[OF _ rapprox_rat] rapprox_rat_le1)
have "1 / k \<le> ?k" using rapprox_rat[where x=1 and y=k] by auto
hence "arctan (1 / k) \<le> arctan ?k" by (rule arctan_monotone')
also have "\<dots> \<le> (?k * ub_arctan_horner prec (get_odd n) 1 ?kl)"
using arctan_0_1_bounds_round[OF `0 \<le> real ?k` `real ?k \<le> 1`]
by auto
finally have "arctan (1 / k) \<le> ?k * ub_arctan_horner prec (get_odd n) 1 ?kl" .
} note ub_arctan = this
{ fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
let ?k = "lapprox_rat prec 1 k"
let ?ku = "float_round_up (Suc prec) (?k * ?k)"
have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
have "1 / k \<le> 1" using `1 < k` by auto
have "0 \<le> real ?k" using lapprox_rat_nonneg[where x=1 and y=k, OF zero_le_one `0 \<le> k`]
by (auto simp add: `1 div k = 0`)
have "0 \<le> real (?k * ?k)" by simp
have "real ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / k \<le> 1`)
hence "real (?k * ?k) \<le> 1" using `0 \<le> real ?k` by (auto intro!: mult_nonneg_le_one)
have "?k \<le> 1 / k" using lapprox_rat[where x=1 and y=k] by auto
have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan ?k"
using arctan_0_1_bounds_round[OF `0 \<le> real ?k` `real ?k \<le> 1`]
by auto
also have "\<dots> \<le> arctan (1 / k)" using `?k \<le> 1 / k` by (rule arctan_monotone')
finally have "?k * lb_arctan_horner prec (get_even n) 1 ?ku \<le> arctan (1 / k)" .
} note lb_arctan = this
have "pi \<le> ub_pi n "
unfolding ub_pi_def machin_pi Let_def times_float.rep_eq Float_num
using lb_arctan[of 239] ub_arctan[of 5] powr_realpow[of 2 2]
by (intro mult_left_mono float_plus_up_le float_plus_down_le)
(auto intro!: mult_left_mono float_round_down_le float_round_up_le diff_mono)
moreover have "lb_pi n \<le> pi"
unfolding lb_pi_def machin_pi Let_def times_float.rep_eq Float_num
using lb_arctan[of 5] ub_arctan[of 239]
by (intro mult_left_mono float_plus_up_le float_plus_down_le)
(auto intro!: mult_left_mono float_round_down_le float_round_up_le diff_mono)
ultimately show ?thesis by auto
qed
subsection "Compute arcus tangens in the entire domain"
function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where
"lb_arctan prec x =
(let
ub_horner = \<lambda> x. float_round_up prec
(x *
ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)));
lb_horner = \<lambda> x. float_round_down prec
(x *
lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x)))
in
if x < 0 then - ub_arctan prec (-x)
else if x \<le> Float 1 (- 1) then lb_horner x
else if x \<le> Float 1 1 then
Float 1 1 *
lb_horner
(float_divl prec x
(float_plus_up prec 1
(ub_sqrt prec (float_plus_up prec 1 (float_round_up prec (x * x))))))
else let inv = float_divr prec 1 x in
if inv > 1 then 0
else float_plus_down prec (lb_pi prec * Float 1 (- 1)) ( - ub_horner inv))"
| "ub_arctan prec x =
(let
lb_horner = \<lambda> x. float_round_down prec
(x *
lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))) ;
ub_horner = \<lambda> x. float_round_up prec
(x *
ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)))
in if x < 0 then - lb_arctan prec (-x)
else if x \<le> Float 1 (- 1) then ub_horner x
else if x \<le> Float 1 1 then
let y = float_divr prec x
(float_plus_down
(Suc prec) 1 (lb_sqrt prec (float_plus_down prec 1 (float_round_down prec (x * x)))))
in if y > 1 then ub_pi prec * Float 1 (- 1) else Float 1 1 * ub_horner y
else float_plus_up prec (ub_pi prec * Float 1 (- 1)) ( - lb_horner (float_divl prec 1 x)))"
by pat_completeness auto
termination
by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)
declare ub_arctan_horner.simps[simp del]
declare lb_arctan_horner.simps[simp del]
lemma lb_arctan_bound': assumes "0 \<le> real x"
shows "lb_arctan prec x \<le> arctan x"
proof -
have "\<not> x < 0" and "0 \<le> x"
using `0 \<le> real x` by (auto intro!: truncate_up_le )
let "?ub_horner x" =
"x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x))"
and "?lb_horner x" =
"x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))"
show ?thesis
proof (cases "x \<le> Float 1 (- 1)")
case True hence "real x \<le> 1" by simp
from arctan_0_1_bounds_round[OF `0 \<le> real x` `real x \<le> 1`]
show ?thesis
unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True] using `0 \<le> x`
by (auto intro!: float_round_down_le)
next
case False hence "0 < real x" by auto
let ?R = "1 + sqrt (1 + real x * real x)"
let ?sxx = "float_plus_up prec 1 (float_round_up prec (x * x))"
let ?fR = "float_plus_up prec 1 (ub_sqrt prec ?sxx)"
let ?DIV = "float_divl prec x ?fR"
have divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
have "sqrt (1 + x*x) \<le> sqrt ?sxx"
by (auto simp: float_plus_up.rep_eq plus_up_def float_round_up.rep_eq intro!: truncate_up_le)
also have "\<dots> \<le> ub_sqrt prec ?sxx"
using bnds_sqrt'[of ?sxx prec] by auto
finally
have "sqrt (1 + x*x) \<le> ub_sqrt prec ?sxx" .
hence "?R \<le> ?fR" by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le)
hence "0 < ?fR" and "0 < real ?fR" using `0 < ?R` by auto
have monotone: "?DIV \<le> x / ?R"
proof -
have "?DIV \<le> real x / ?fR" by (rule float_divl)
also have "\<dots> \<le> x / ?R" by (rule divide_left_mono[OF `?R \<le> ?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
show ?thesis
proof (cases "x \<le> Float 1 1")
case True
have "x \<le> sqrt (1 + x * x)" using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
also note `\<dots> \<le> (ub_sqrt prec ?sxx)`
finally have "real x \<le> ?fR" by (auto simp: float_plus_up.rep_eq plus_up_def intro!: truncate_up_le)
moreover have "?DIV \<le> real x / ?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> real ?DIV" using float_divl_lower_bound[OF `0 \<le> x`] `0 < ?fR` unfolding less_eq_float_def by auto
from arctan_0_1_bounds_round[OF `0 \<le> real (?DIV)` `real (?DIV) \<le> 1`]
have "Float 1 1 * ?lb_horner ?DIV \<le> 2 * arctan ?DIV" by simp
also have "\<dots> \<le> 2 * arctan (x / ?R)"
using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono arctan_monotone')
also have "2 * arctan (x / ?R) = arctan x" using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
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]
by (auto simp: float_round_down.rep_eq intro!: order_trans[OF mult_left_mono[OF truncate_down]])
next
case False
hence "2 < real x" by auto
hence "1 \<le> real x" by auto
let "?invx" = "float_divr prec 1 x"
have "0 \<le> arctan 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 x` by auto
next
case False
hence "real ?invx \<le> 1" by auto
have "0 \<le> real ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> real x`)
have "1 / x \<noteq> 0" and "0 < 1 / x" using `0 < real x` by auto
have "arctan (1 / x) \<le> arctan ?invx" unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone', rule float_divr)
also have "\<dots> \<le> ?ub_horner ?invx" using arctan_0_1_bounds_round[OF `0 \<le> real ?invx` `real ?invx \<le> 1`]
by (auto intro!: float_round_up_le)
also note float_round_up
finally have "pi / 2 - float_round_up prec (?ub_horner ?invx) \<le> arctan x"
using `0 \<le> arctan x` arctan_inverse[OF `1 / x \<noteq> 0`]
unfolding real_sgn_pos[OF `0 < 1 / real x`] le_diff_eq by auto
moreover
have "lb_pi prec * Float 1 (- 1) \<le> pi / 2"
unfolding Float_num times_divide_eq_right mult_1_left using pi_boundaries by simp
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 intro!: float_plus_down_le)
qed
qed
qed
qed
lemma ub_arctan_bound': assumes "0 \<le> real x"
shows "arctan x \<le> ub_arctan prec x"
proof -
have "\<not> x < 0" and "0 \<le> x" using `0 \<le> real x` by auto
let "?ub_horner x" = "float_round_up prec (x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x)))"
and "?lb_horner x" = "float_round_down prec (x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x)))"
show ?thesis
proof (cases "x \<le> Float 1 (- 1)")
case True hence "real x \<le> 1" 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_round[OF `0 \<le> real x` `real x \<le> 1`]
by (auto intro!: float_round_up_le)
next
case False hence "0 < real x" by auto
let ?R = "1 + sqrt (1 + real x * real x)"
let ?sxx = "float_plus_down prec 1 (float_round_down prec (x * x))"
let ?fR = "float_plus_down (Suc prec) 1 (lb_sqrt prec ?sxx)"
let ?DIV = "float_divr prec x ?fR"
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 "lb_sqrt prec ?sxx \<le> sqrt ?sxx"
using bnds_sqrt'[of ?sxx] by auto
also have "\<dots> \<le> sqrt (1 + x*x)"
by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq truncate_down_le)
finally have "lb_sqrt prec ?sxx \<le> sqrt (1 + x*x)" .
hence "?fR \<le> ?R" by (auto simp: float_plus_down.rep_eq plus_down_def truncate_down_le)
have "0 < real ?fR"
by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq
intro!: truncate_down_ge1 lb_sqrt_lower_bound order_less_le_trans[OF zero_less_one]
truncate_down_nonneg add_nonneg_nonneg)
have monotone: "x / ?R \<le> (float_divr prec x ?fR)"
proof -
from divide_left_mono[OF `?fR \<le> ?R` `0 \<le> real x` mult_pos_pos[OF divisor_gt0 `0 < real ?fR`]]
have "x / ?R \<le> x / ?fR" .
also have "\<dots> \<le> ?DIV" by (rule float_divr)
finally show ?thesis .
qed
show ?thesis
proof (cases "x \<le> Float 1 1")
case True
show ?thesis
proof (cases "?DIV > 1")
case True
have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)" unfolding Float_num times_divide_eq_right mult_1_left 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 "real ?DIV \<le> 1" by auto
have "0 \<le> x / ?R" using `0 \<le> real x` `0 < ?R` unfolding zero_le_divide_iff by auto
hence "0 \<le> real ?DIV" using monotone by (rule order_trans)
have "arctan x = 2 * arctan (x / ?R)" using arctan_half unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
also have "\<dots> \<le> 2 * arctan (?DIV)"
using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
also have "\<dots> \<le> (Float 1 1 * ?ub_horner ?DIV)" unfolding Float_num
using arctan_0_1_bounds_round[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`]
by (auto intro!: float_round_up_le)
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 < real x" by auto
hence "1 \<le> real x" by auto
hence "0 < real x" by auto
hence "0 < x" by auto
let "?invx" = "float_divl prec 1 x"
have "0 \<le> arctan x" using arctan_monotone'[OF `0 \<le> real x`] using arctan_tan[of 0, unfolded tan_zero] by auto
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" using `0 < x` by (intro float_divl_lower_bound) auto
have "1 / x \<noteq> 0" and "0 < 1 / x" using `0 < real x` by auto
have "(?lb_horner ?invx) \<le> arctan (?invx)" using arctan_0_1_bounds_round[OF `0 \<le> real ?invx` `real ?invx \<le> 1`]
by (auto intro!: float_round_down_le)
also have "\<dots> \<le> arctan (1 / x)" unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone', rule float_divl)
finally have "arctan x \<le> pi / 2 - (?lb_horner ?invx)"
using `0 \<le> arctan x` arctan_inverse[OF `1 / x \<noteq> 0`]
unfolding real_sgn_pos[OF `0 < 1 / x`] le_diff_eq by auto
moreover
have "pi / 2 \<le> ub_pi prec * Float 1 (- 1)" unfolding 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 False]
by (auto intro!: float_round_up_le float_plus_up_le)
qed
qed
qed
lemma arctan_boundaries:
"arctan x \<in> {(lb_arctan prec x) .. (ub_arctan prec x)}"
proof (cases "0 \<le> x")
case True hence "0 \<le> real x" 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> real ?mx" by auto
hence bounds: "lb_arctan prec ?mx \<le> arctan ?mx \<and> arctan ?mx \<le> 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 minus_float.rep_eq 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 minus_float.rep_eq arctan_minus]
by (simp add: arctan_minus)
qed
lemma bnds_arctan: "\<forall> (x::real) lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> arctan x \<and> arctan x \<le> u"
proof (rule allI, rule allI, rule allI, rule impI)
fix x :: real fix lx ux
assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux}"
hence l: "lb_arctan prec lx = l " and u: "ub_arctan prec ux = u" and x: "x \<in> {lx .. ux}" by auto
{ from arctan_boundaries[of lx prec, unfolded l]
have "l \<le> arctan lx" by (auto simp del: lb_arctan.simps)
also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
finally have "l \<le> arctan x" .
} moreover
{ have "arctan x \<le> arctan ux" using x by (auto intro: arctan_monotone')
also have "\<dots> \<le> u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
finally have "arctan x \<le> u" .
} ultimately show "l \<le> arctan x \<and> arctan x \<le> u" ..
qed
section "Sinus and Cosinus"
subsection "Compute the cosinus and sinus series"
fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
"ub_sin_cos_aux prec 0 i k x = 0"
| "ub_sin_cos_aux prec (Suc n) i k x = float_plus_up prec
(rapprox_rat prec 1 k) (-
float_round_down prec (x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)))"
| "lb_sin_cos_aux prec 0 i k x = 0"
| "lb_sin_cos_aux prec (Suc n) i k x = float_plus_down prec
(lapprox_rat prec 1 k) (-
float_round_up prec (x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)))"
lemma cos_aux:
shows "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x ^(2 * i))" (is "?lb")
and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i))) * x^(2 * i)) \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
proof -
have "0 \<le> real (x * x)" by auto
let "?f n" = "fact (2 * n) :: nat"
{ fix n
have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)"
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> 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 lb_sin_cos_aux_zero_le_one: "lb_sin_cos_aux prec n i j 0 \<le> 1"
by (cases j n rule: nat.exhaust[case_product nat.exhaust])
(auto intro!: float_plus_down_le order_trans[OF lapprox_rat])
lemma one_le_ub_sin_cos_aux: "odd n \<Longrightarrow> 1 \<le> ub_sin_cos_aux prec n i (Suc 0) 0"
by (cases n) (auto intro!: float_plus_up_le order_trans[OF _ rapprox_rat])
lemma cos_boundaries: assumes "0 \<le> real x" and "x \<le> pi / 2"
shows "cos x \<in> {(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. (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` by auto
have "0 < x * x" using `0 < x` by simp
{ fix x n have "(\<Sum> i=0..<n. (-1::real) ^ i * (1/(fact (2 * i))) * x ^ (2 * i))
= (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum")
proof -
have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
also have "\<dots> =
(\<Sum> j = 0 ..< n. (- 1) ^ ((2 * j) div 2) / ((fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then (- 1) ^ (i div 2) / ((fact i)) * x ^ i else 0)"
unfolding sum_split_even_odd atLeast0LessThan ..
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then (- 1) ^ (i div 2) / ((fact i)) else 0) * x ^ i)"
by (rule setsum.cong) auto
finally show ?thesis by assumption
qed } note morph_to_if_power = this
{ fix n :: nat assume "0 < n"
hence "0 < 2 * n" by auto
obtain t where "0 < t" and "t < real x" and
cos_eq: "cos x = (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * (real x) ^ i)
+ (cos (t + 1/2 * (2 * n) * pi) / (fact (2*n))) * (real x)^(2*n)"
(is "_ = ?SUM + ?rest / ?fact * ?pow")
using Maclaurin_cos_expansion2[OF `0 < real x` `0 < 2 * n`]
unfolding cos_coeff_def atLeast0LessThan by auto
have "cos t * (- 1) ^ n = cos t * cos (n * pi) + sin t * sin (n * pi)" by auto
also have "\<dots> = cos (t + n * pi)" by (simp add: cos_add)
also have "\<dots> = ?rest" by auto
finally have "cos t * (- 1) ^ n = ?rest" .
moreover
have "t \<le> pi / 2" using `t < real x` and `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 < real x` by auto
{
assume "even n"
have "(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 x"
proof -
from even[OF `even n`] `0 < ?fact` `0 < ?pow`
have "0 \<le> (?rest / ?fact) * ?pow" by simp
thus ?thesis unfolding cos_eq by auto
qed
finally have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos x" .
} note lb = this
{
assume "odd n"
have "cos 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> (ub_sin_cos_aux prec n 1 1 (x * x))"
unfolding morph_to_if_power[symmetric] using cos_aux by auto
finally have "cos x \<le> (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 x \<le> (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
moreover have "(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos 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> x" by (rule order_trans[OF _ `0 < real x`[THEN less_imp_le]], auto)
with `x \<le> pi / 2`
show ?thesis unfolding `get_even n = 0` lb_sin_cos_aux.simps minus_float.rep_eq zero_float.rep_eq using cos_ge_zero by auto
qed
ultimately show ?thesis by auto
next
case True
hence "x = 0"
by transfer
thus ?thesis
using lb_sin_cos_aux_zero_le_one one_le_ub_sin_cos_aux
by simp
qed
lemma sin_aux: assumes "0 \<le> real x"
shows "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> (\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1))" (is "?lb")
and "(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1)) \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
proof -
have "0 \<le> real (x * x)" by auto
let "?f n" = "fact (2 * n + 1) :: nat"
{ fix n
have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)"
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> 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 power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
unfolding mult.commute[where 'a=real] real_fact_nat
by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real x"])
qed
lemma sin_boundaries: assumes "0 \<le> real x" and "x \<le> pi / 2"
shows "sin x \<in> {(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. (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` by auto
have "0 < x * x" using `0 < x` by simp
{ fix x::real and n
have "(\<Sum>j = 0 ..< n. (- 1) ^ (((2 * j + 1) - Suc 0) div 2) / ((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))/((fact i))) * x ^ i)" (is "?SUM = _")
proof -
have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto
have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" by auto
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i)) * x ^ i)"
unfolding sum_split_even_odd atLeast0LessThan ..
also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i))) * x ^ i)"
by (rule setsum.cong) auto
finally show ?thesis by assumption
qed } note setsum_morph = this
{ fix n :: nat assume "0 < n"
hence "0 < 2 * n + 1" by auto
obtain t where "0 < t" and "t < real x" and
sin_eq: "sin x = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/((fact i))) * (real x) ^ i)
+ (sin (t + 1/2 * (2 * n + 1) * pi) / (fact (2*n + 1))) * (real x)^(2*n + 1)"
(is "_ = ?SUM + ?rest / ?fact * ?pow")
using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < real x`]
unfolding sin_coeff_def atLeast0LessThan by auto
have "?rest = cos t * (- 1) ^ n" unfolding sin_add cos_add real_of_nat_add distrib_right distrib_left by auto
moreover
have "t \<le> pi / 2" using `t < real x` and `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 (simp del: fact_Suc)
have "0 < ?pow" using `0 < real x` by (rule zero_less_power)
{
assume "even n"
have "(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))/((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 x"
proof -
from even[OF `even n`] `0 < ?fact` `0 < ?pow`
have "0 \<le> (?rest / ?fact) * ?pow" by simp
thus ?thesis unfolding sin_eq by auto
qed
finally have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin x" .
} note lb = this
{
assume "odd n"
have "sin 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))/((fact i))) * (real x) ^ i)"
by auto
also have "\<dots> \<le> (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 x \<le> (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 x \<le> (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 "(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin 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 `x \<le> pi / 2` `0 \<le> real x`
show ?thesis unfolding `get_even n = 0` ub_sin_cos_aux.simps minus_float.rep_eq using sin_ge_zero by auto
qed
ultimately show ?thesis by auto
next
case True
show ?thesis
proof (cases "n = 0")
case True
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 `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
subsection "Compute the cosinus in the entire domain"
definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
"lb_cos prec x = (let
horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ;
half = \<lambda> x. if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1)
in if x < Float 1 (- 1) then horner x
else if x < 1 then half (horner (x * Float 1 (- 1)))
else half (half (horner (x * Float 1 (- 2)))))"
definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
"ub_cos prec x = (let
horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ;
half = \<lambda> x. float_plus_up prec (Float 1 1 * x * x) (- 1)
in if x < Float 1 (- 1) then horner x
else if x < 1 then half (horner (x * Float 1 (- 1)))
else half (half (horner (x * Float 1 (- 2)))))"
lemma lb_cos: assumes "0 \<le> real x" and "x \<le> pi"
shows "cos x \<in> {(lb_cos prec x) .. (ub_cos prec x)}" (is "?cos x \<in> {(?lb x) .. (?ub x) }")
proof -
{ fix x :: real
have "cos x = cos (x / 2 + x / 2)" by auto
also have "\<dots> = cos (x / 2) * cos (x / 2) + sin (x / 2) * sin (x / 2) - sin (x / 2) * sin (x / 2) + cos (x / 2) * cos (x / 2) - 1"
unfolding cos_add by auto
also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1" by algebra
finally have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" .
} note x_half = this[symmetric]
have "\<not> x < 0" using `0 \<le> real x` 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_plus_up prec (Float 1 1 * x * x) (- 1)"
let "?lb_half x" = "if x < 0 then - 1 else float_plus_down prec (Float 1 1 * x * x) (- 1)"
show ?thesis
proof (cases "x < Float 1 (- 1)")
case True hence "x \<le> pi / 2" 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> real x` `x \<le> pi / 2`] .
next
case False
{ fix y x :: float let ?x2 = "(x * Float 1 (- 1))"
assume "y \<le> cos ?x2" and "-pi \<le> x" and "x \<le> pi"
hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Float_num by auto
hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
have "(?lb_half y) \<le> cos 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> real y" by auto
from mult_mono[OF `y \<le> cos ?x2` `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 (x / 2) * cos (x / 2) - 1" unfolding Float_num by auto
thus ?thesis unfolding if_not_P[OF False] x_half Float_num
by (auto intro!: float_plus_down_le)
qed
} note lb_half = this
{ fix y x :: float let ?x2 = "(x * Float 1 (- 1))"
assume ub: "cos ?x2 \<le> y" and "- pi \<le> x" and "x \<le> pi"
hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Float_num by auto
hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
have "cos x \<le> (?ub_half y)"
proof -
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> real y * real y" .
hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real y * real y" by auto
hence "2 * cos (x / 2) * cos (x / 2) - 1 \<le> 2 * real y * real y - 1" unfolding Float_num by auto
thus ?thesis unfolding x_half Float_num
by (auto intro!: float_plus_up_le)
qed
} note ub_half = this
let ?x2 = "x * Float 1 (- 1)"
let ?x4 = "x * Float 1 (- 1) * Float 1 (- 1)"
have "-pi \<le> 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 "real x \<le> 1" by auto
have "0 \<le> real ?x2" and "?x2 \<le> pi / 2" using pi_ge_two `0 \<le> real x` using assms by auto
from cos_boundaries[OF this]
have lb: "(?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> (?ub_horner ?x2)" by auto
have "(?lb x) \<le> ?cos x"
proof -
from lb_half[OF lb `-pi \<le> x` `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> (?ub x)"
proof -
from ub_half[OF ub `-pi \<le> x` `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> real ?x4" and "?x4 \<le> pi / 2" using pi_ge_two `0 \<le> real x` `x \<le> pi` unfolding Float_num by auto
from cos_boundaries[OF this]
have lb: "(?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> (?ub_horner ?x4)" by auto
have eq_4: "?x2 * Float 1 (- 1) = x * Float 1 (- 2)" by transfer simp
have "(?lb x) \<le> ?cos x"
proof -
have "-pi \<le> ?x2" and "?x2 \<le> pi" using pi_ge_two `0 \<le> real x` `x \<le> pi` by auto
from lb_half[OF lb_half[OF lb this] `-pi \<le> x` `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> (?ub x)"
proof -
have "-pi \<le> ?x2" and "?x2 \<le> pi" using pi_ge_two `0 \<le> real x` ` x \<le> pi` by auto
from ub_half[OF ub_half[OF ub this] `-pi \<le> x` `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
qed
qed
qed
lemma lb_cos_minus: assumes "-pi \<le> x" and "real x \<le> 0"
shows "cos (real(-x)) \<in> {(lb_cos prec (-x)) .. (ub_cos prec (-x))}"
proof -
have "0 \<le> real (-x)" and "(-x) \<le> pi" using `-pi \<le> x` `real x \<le> 0` by auto
from lb_cos[OF this] show ?thesis .
qed
definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
"bnds_cos prec lx ux = (let
lpi = float_round_down prec (lb_pi prec) ;
upi = float_round_up prec (ub_pi prec) ;
k = floor_fl (float_divr prec (lx + lpi) (2 * lpi)) ;
lx = float_plus_down prec lx (- k * 2 * (if k < 0 then lpi else upi)) ;
ux = float_plus_up prec ux (- k * 2 * (if k < 0 then upi else lpi))
in if - lpi \<le> lx \<and> ux \<le> 0 then (lb_cos prec (-lx), ub_cos prec (-ux))
else if 0 \<le> lx \<and> ux \<le> lpi then (lb_cos prec ux, ub_cos prec lx)
else if - lpi \<le> lx \<and> ux \<le> lpi then (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0)
else if 0 \<le> lx \<and> ux \<le> 2 * lpi then (Float (- 1) 0, max (ub_cos prec lx) (ub_cos prec (- (ux - 2 * lpi))))
else if -2 * lpi \<le> lx \<and> ux \<le> 0 then (Float (- 1) 0, max (ub_cos prec (lx + 2 * lpi)) (ub_cos prec (-ux)))
else (Float (- 1) 0, Float 1 0))"
lemma floor_int:
obtains k :: int where "real k = (floor_fl f)"
by (simp add: floor_fl_def)
lemma cos_periodic_nat[simp]: fixes n :: nat shows "cos (x + n * (2 * pi)) = cos x"
proof (induct n arbitrary: x)
case (Suc n)
have split_pi_off: "x + (Suc n) * (2 * pi) = (x + n * (2 * pi)) + 2 * pi"
unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
show ?case unfolding split_pi_off using Suc by auto
qed auto
lemma cos_periodic_int[simp]: fixes i :: int shows "cos (x + i * (2 * pi)) = cos x"
proof (cases "0 \<le> i")
case True hence i_nat: "real i = nat i" by auto
show ?thesis unfolding i_nat by auto
next
case False hence i_nat: "i = - real (nat (-i))" by auto
have "cos x = cos (x + i * (2 * pi) - i * (2 * pi))" by auto
also have "\<dots> = cos (x + i * (2 * pi))"
unfolding i_nat mult_minus_left diff_minus_eq_add by (rule cos_periodic_nat)
finally show ?thesis by auto
qed
lemma bnds_cos: "\<forall> (x::real) lx ux. (l, u) = bnds_cos prec lx ux \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> cos x \<and> cos x \<le> u"
proof ((rule allI | rule impI | erule conjE) +)
fix x :: real fix lx ux
assume bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {lx .. ux}"
let ?lpi = "float_round_down prec (lb_pi prec)"
let ?upi = "float_round_up prec (ub_pi prec)"
let ?k = "floor_fl (float_divr prec (lx + ?lpi) (2 * ?lpi))"
let ?lx2 = "(- ?k * 2 * (if ?k < 0 then ?lpi else ?upi))"
let ?ux2 = "(- ?k * 2 * (if ?k < 0 then ?upi else ?lpi))"
let ?lx = "float_plus_down prec lx ?lx2"
let ?ux = "float_plus_up prec ux ?ux2"
obtain k :: int where k: "k = real ?k" using floor_int .
have upi: "pi \<le> ?upi" and lpi: "?lpi \<le> pi"
using float_round_up[of "ub_pi prec" prec] pi_boundaries[of prec]
float_round_down[of prec "lb_pi prec"] by auto
hence "lx + ?lx2 \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ux + ?ux2"
using x
by (cases "k = 0")
(auto intro!: add_mono
simp add: k [symmetric] uminus_add_conv_diff [symmetric]
simp del: float_of_numeral uminus_add_conv_diff)
hence "?lx \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ?ux"
by (auto intro!: float_plus_down_le float_plus_up_le)
note lx = this[THEN conjunct1] and ux = this[THEN conjunct2]
hence lx_less_ux: "?lx \<le> real ?ux" by (rule order_trans)
{ assume "- ?lpi \<le> ?lx" and x_le_0: "x - k * (2 * pi) \<le> 0"
with lpi[THEN le_imp_neg_le] lx
have pi_lx: "- pi \<le> ?lx" and lx_0: "real ?lx \<le> 0"
by simp_all
have "(lb_cos prec (- ?lx)) \<le> cos (real (- ?lx))"
using lb_cos_minus[OF pi_lx lx_0] by simp
also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
using cos_monotone_minus_pi_0'[OF pi_lx lx x_le_0]
by (simp only: uminus_float.rep_eq real_of_int_minus
cos_minus mult_minus_left) simp
finally have "(lb_cos prec (- ?lx)) \<le> cos x"
unfolding cos_periodic_int . }
note negative_lx = this
{ assume "0 \<le> ?lx" and pi_x: "x - k * (2 * pi) \<le> pi"
with lx
have pi_lx: "?lx \<le> pi" and lx_0: "0 \<le> real ?lx"
by auto
have "cos (x + (-k) * (2 * pi)) \<le> cos ?lx"
using cos_monotone_0_pi_le[OF lx_0 lx pi_x]
by (simp only: real_of_int_minus
cos_minus mult_minus_left) simp
also have "\<dots> \<le> (ub_cos prec ?lx)"
using lb_cos[OF lx_0 pi_lx] by simp
finally have "cos x \<le> (ub_cos prec ?lx)"
unfolding cos_periodic_int . }
note positive_lx = this
{ assume pi_x: "- pi \<le> x - k * (2 * pi)" and "?ux \<le> 0"
with ux
have pi_ux: "- pi \<le> ?ux" and ux_0: "real ?ux \<le> 0"
by simp_all
have "cos (x + (-k) * (2 * pi)) \<le> cos (real (- ?ux))"
using cos_monotone_minus_pi_0'[OF pi_x ux ux_0]
by (simp only: uminus_float.rep_eq real_of_int_minus
cos_minus mult_minus_left) simp
also have "\<dots> \<le> (ub_cos prec (- ?ux))"
using lb_cos_minus[OF pi_ux ux_0, of prec] by simp
finally have "cos x \<le> (ub_cos prec (- ?ux))"
unfolding cos_periodic_int . }
note negative_ux = this
{ assume "?ux \<le> ?lpi" and x_ge_0: "0 \<le> x - k * (2 * pi)"
with lpi ux
have pi_ux: "?ux \<le> pi" and ux_0: "0 \<le> real ?ux"
by simp_all
have "(lb_cos prec ?ux) \<le> cos ?ux"
using lb_cos[OF ux_0 pi_ux] by simp
also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
using cos_monotone_0_pi_le[OF x_ge_0 ux pi_ux]
by (simp only: real_of_int_minus
cos_minus mult_minus_left) simp
finally have "(lb_cos prec ?ux) \<le> cos x"
unfolding cos_periodic_int . }
note positive_ux = this
show "l \<le> cos x \<and> cos x \<le> u"
proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> 0")
case True with bnds
have l: "l = lb_cos prec (-?lx)"
and u: "u = ub_cos prec (-?ux)"
by (auto simp add: bnds_cos_def Let_def)
from True lpi[THEN le_imp_neg_le] lx ux
have "- pi \<le> x - k * (2 * pi)"
and "x - k * (2 * pi) \<le> 0"
by auto
with True negative_ux negative_lx
show ?thesis unfolding l u by simp
next case False note 1 = this show ?thesis
proof (cases "0 \<le> ?lx \<and> ?ux \<le> ?lpi")
case True with bnds 1
have l: "l = lb_cos prec ?ux"
and u: "u = ub_cos prec ?lx"
by (auto simp add: bnds_cos_def Let_def)
from True lpi lx ux
have "0 \<le> x - k * (2 * pi)"
and "x - k * (2 * pi) \<le> pi"
by auto
with True positive_ux positive_lx
show ?thesis unfolding l u by simp
next case False note 2 = this show ?thesis
proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> ?lpi")
case True note Cond = this with bnds 1 2
have l: "l = min (lb_cos prec (-?lx)) (lb_cos prec ?ux)"
and u: "u = Float 1 0"
by (auto simp add: bnds_cos_def Let_def)
show ?thesis unfolding u l using negative_lx positive_ux Cond
by (cases "x - k * (2 * pi) < 0") (auto simp add: real_of_float_min)
next case False note 3 = this show ?thesis
proof (cases "0 \<le> ?lx \<and> ?ux \<le> 2 * ?lpi")
case True note Cond = this with bnds 1 2 3
have l: "l = Float (- 1) 0"
and u: "u = max (ub_cos prec ?lx) (ub_cos prec (- (?ux - 2 * ?lpi)))"
by (auto simp add: bnds_cos_def Let_def)
have "cos x \<le> real u"
proof (cases "x - k * (2 * pi) < pi")
case True hence "x - k * (2 * pi) \<le> pi" by simp
from positive_lx[OF Cond[THEN conjunct1] this]
show ?thesis unfolding u by (simp add: real_of_float_max)
next
case False hence "pi \<le> x - k * (2 * pi)" by simp
hence pi_x: "- pi \<le> x - k * (2 * pi) - 2 * pi" by simp
have "?ux \<le> 2 * pi" using Cond lpi by auto
hence "x - k * (2 * pi) - 2 * pi \<le> 0" using ux by simp
have ux_0: "real (?ux - 2 * ?lpi) \<le> 0"
using Cond by auto
from 2 and Cond have "\<not> ?ux \<le> ?lpi" by auto
hence "- ?lpi \<le> ?ux - 2 * ?lpi" by auto
hence pi_ux: "- pi \<le> (?ux - 2 * ?lpi)"
using lpi[THEN le_imp_neg_le] by auto
have x_le_ux: "x - k * (2 * pi) - 2 * pi \<le> (?ux - 2 * ?lpi)"
using ux lpi by auto
have "cos x = cos (x + (-k) * (2 * pi) + (-1::int) * (2 * pi))"
unfolding cos_periodic_int ..
also have "\<dots> \<le> cos ((?ux - 2 * ?lpi))"
using cos_monotone_minus_pi_0'[OF pi_x x_le_ux ux_0]
by (simp only: minus_float.rep_eq real_of_int_minus real_of_one
mult_minus_left mult_1_left) simp
also have "\<dots> = cos ((- (?ux - 2 * ?lpi)))"
unfolding uminus_float.rep_eq cos_minus ..
also have "\<dots> \<le> (ub_cos prec (- (?ux - 2 * ?lpi)))"
using lb_cos_minus[OF pi_ux ux_0] by simp
finally show ?thesis unfolding u by (simp add: real_of_float_max)
qed
thus ?thesis unfolding l by auto
next case False note 4 = this show ?thesis
proof (cases "-2 * ?lpi \<le> ?lx \<and> ?ux \<le> 0")
case True note Cond = this with bnds 1 2 3 4
have l: "l = Float (- 1) 0"
and u: "u = max (ub_cos prec (?lx + 2 * ?lpi)) (ub_cos prec (-?ux))"
by (auto simp add: bnds_cos_def Let_def)
have "cos x \<le> u"
proof (cases "-pi < x - k * (2 * pi)")
case True hence "-pi \<le> x - k * (2 * pi)" by simp
from negative_ux[OF this Cond[THEN conjunct2]]
show ?thesis unfolding u by (simp add: real_of_float_max)
next
case False hence "x - k * (2 * pi) \<le> -pi" by simp
hence pi_x: "x - k * (2 * pi) + 2 * pi \<le> pi" by simp
have "-2 * pi \<le> ?lx" using Cond lpi by auto
hence "0 \<le> x - k * (2 * pi) + 2 * pi" using lx by simp
have lx_0: "0 \<le> real (?lx + 2 * ?lpi)"
using Cond lpi by auto
from 1 and Cond have "\<not> -?lpi \<le> ?lx" by auto
hence "?lx + 2 * ?lpi \<le> ?lpi" by auto
hence pi_lx: "(?lx + 2 * ?lpi) \<le> pi"
using lpi[THEN le_imp_neg_le] by auto
have lx_le_x: "(?lx + 2 * ?lpi) \<le> x - k * (2 * pi) + 2 * pi"
using lx lpi by auto
have "cos x = cos (x + (-k) * (2 * pi) + (1 :: int) * (2 * pi))"
unfolding cos_periodic_int ..
also have "\<dots> \<le> cos ((?lx + 2 * ?lpi))"
using cos_monotone_0_pi_le[OF lx_0 lx_le_x pi_x]
by (simp only: minus_float.rep_eq real_of_int_minus real_of_one
mult_minus_left mult_1_left) simp
also have "\<dots> \<le> (ub_cos prec (?lx + 2 * ?lpi))"
using lb_cos[OF lx_0 pi_lx] by simp
finally show ?thesis unfolding u by (simp add: real_of_float_max)
qed
thus ?thesis unfolding l by auto
next
case False with bnds 1 2 3 4 show ?thesis by (auto simp add: bnds_cos_def Let_def)
qed qed qed qed qed
qed
section "Exponential function"
subsection "Compute the series of the exponential function"
fun ub_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" and lb_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
"ub_exp_horner prec 0 i k x = 0" |
"ub_exp_horner prec (Suc n) i k x = float_plus_up prec
(rapprox_rat prec 1 (int k)) (float_round_up prec (x * lb_exp_horner prec n (i + 1) (k * i) x))" |
"lb_exp_horner prec 0 i k x = 0" |
"lb_exp_horner prec (Suc n) i k x = float_plus_down prec
(lapprox_rat prec 1 (int k)) (float_round_down prec (x * ub_exp_horner prec n (i + 1) (k * i) x))"
lemma bnds_exp_horner: assumes "real x \<le> 0"
shows "exp x \<in> { lb_exp_horner prec (get_even n) 1 1 x .. 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)
have "fact (Suc n) = fact n * ((\<lambda>i::nat. i + 1) ^^ n) 1" unfolding F by auto
} note f_eq = this
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 "lb_exp_horner prec (get_even n) 1 1 x \<le> (\<Sum>j = 0..<get_even n. 1 / (fact j) * real x ^ j)"
using bounds(1) by auto
also have "\<dots> \<le> exp x"
proof -
obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp x = (\<Sum>m = 0..<get_even n. real x ^ m / (fact m)) + exp t / (fact (get_even n)) * (real x) ^ (get_even n)"
using Maclaurin_exp_le unfolding atLeast0LessThan by blast
moreover have "0 \<le> exp t / (fact (get_even n)) * (real x) ^ (get_even n)"
by (auto simp: zero_le_even_power)
ultimately show ?thesis using get_odd exp_gt_zero by auto
qed
finally have "lb_exp_horner prec (get_even n) 1 1 x \<le> exp x" .
} moreover
{
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 "real x < 0" using `real x \<le> 0` by auto
show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq `real x < 0`)
qed
obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp x = (\<Sum>m = 0..<get_odd n. (real x) ^ m / (fact m)) + exp t / (fact (get_odd n)) * (real x) ^ (get_odd n)"
using Maclaurin_exp_le unfolding atLeast0LessThan by blast
moreover have "exp t / (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)
ultimately have "exp x \<le> (\<Sum>j = 0..<get_odd n. 1 / (fact j) * real x ^ j)"
using get_odd exp_gt_zero by auto
also have "\<dots> \<le> ub_exp_horner prec (get_odd n) 1 1 x"
using bounds(2) by auto
finally have "exp x \<le> ub_exp_horner prec (get_odd n) 1 1 x" .
} ultimately show ?thesis by auto
qed
lemma ub_exp_horner_nonneg: "real x \<le> 0 \<Longrightarrow> 0 \<le> real (ub_exp_horner prec (get_odd n) (Suc 0) (Suc 0) x)"
using bnds_exp_horner[of x prec n]
by (intro order_trans[OF exp_ge_zero]) auto
subsection "Compute the exponential function on the entire domain"
function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where
"lb_exp prec x =
(if 0 < x then float_divl prec 1 (ub_exp prec (-x))
else
let
horner = (\<lambda> x. let y = lb_exp_horner prec (get_even (prec + 2)) 1 1 x in
if y \<le> 0 then Float 1 (- 2) else y)
in
if x < - 1 then
power_down_fl prec (horner (float_divl prec x (- floor_fl x))) (nat (- int_floor_fl x))
else horner x)" |
"ub_exp prec x =
(if 0 < x then float_divr prec 1 (lb_exp prec (-x))
else if x < - 1 then
power_up_fl prec
(ub_exp_horner prec (get_odd (prec + 2)) 1 1
(float_divr prec x (- floor_fl x))) (nat (- int_floor_fl x))
else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
by pat_completeness auto
termination
by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if 0 < x then 1 else 0))", auto)
lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)"
proof -
have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
have "1 / 4 = (Float 1 (- 2))" unfolding Float_num by auto
also have "\<dots> \<le> lb_exp_horner 3 (get_even 3) 1 1 (- 1)"
by code_simp
also have "\<dots> \<le> exp (- 1 :: float)" using bnds_exp_horner[where x="- 1"] by auto
finally show ?thesis by simp
qed
lemma lb_exp_pos: assumes "\<not> 0 < x" shows "0 < lb_exp prec x"
proof -
let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
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)
moreover { fix x :: float fix num :: nat
have "0 < real (?horner x) ^ num" using `0 < ?horner x` by simp
also have "\<dots> = (?horner x) ^ num" 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
by (cases "floor_fl x", cases "x < - 1", auto simp: real_power_up_fl real_power_down_fl
intro!: power_up_less power_down_pos)
qed
lemma exp_boundaries': assumes "x \<le> 0"
shows "exp x \<in> { (lb_exp prec x) .. (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 "real x \<le> 0" and "\<not> x > 0" using `x \<le> 0` by auto
show ?thesis
proof (cases "x < - 1")
case False hence "- 1 \<le> real x" by auto
show ?thesis
proof (cases "?lb_exp_horner x \<le> 0")
from `\<not> x < - 1` have "- 1 \<le> real x" by auto
hence "exp (- 1) \<le> exp x" unfolding exp_le_cancel_iff .
from order_trans[OF exp_m1_ge_quarter this]
have "Float 1 (- 2) \<le> exp x" unfolding Float_num .
moreover case True
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 `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by (auto simp add: Let_def)
qed
next
case True
let ?num = "nat (- int_floor_fl x)"
have "real (int_floor_fl x) < - 1" using int_floor_fl[of x] `x < - 1`
by simp
hence "real (int_floor_fl x) < 0" by simp
hence "int_floor_fl x < 0" by auto
hence "1 \<le> - int_floor_fl x" by auto
hence "0 < nat (- int_floor_fl x)" by auto
hence "0 < ?num" by auto
hence "real ?num \<noteq> 0" by auto
have num_eq: "real ?num = - int_floor_fl x" using `0 < nat (- int_floor_fl x)` by auto
have "0 < - int_floor_fl x" using `0 < ?num`[unfolded real_of_nat_less_iff[symmetric]] by simp
hence "real (int_floor_fl x) < 0" unfolding less_float_def by auto
have fl_eq: "real (- int_floor_fl x) = real (- floor_fl x)"
by (simp add: floor_fl_def int_floor_fl_def)
from `0 < - int_floor_fl x` have "0 \<le> real (- floor_fl x)"
by (simp add: floor_fl_def int_floor_fl_def)
from `real (int_floor_fl x) < 0` have "real (floor_fl x) < 0"
by (simp add: floor_fl_def int_floor_fl_def)
have "exp x \<le> ub_exp prec x"
proof -
have div_less_zero: "real (float_divr prec x (- floor_fl x)) \<le> 0"
using float_divr_nonpos_pos_upper_bound[OF `real x \<le> 0` `0 \<le> real (- floor_fl x)`]
unfolding less_eq_float_def zero_float.rep_eq .
have "exp x = exp (?num * (x / ?num))" using `real ?num \<noteq> 0` by auto
also have "\<dots> = exp (x / ?num) ^ ?num" unfolding exp_real_of_nat_mult ..
also have "\<dots> \<le> exp (float_divr prec x (- floor_fl x)) ^ ?num" unfolding num_eq fl_eq
by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
also have "\<dots> \<le> (?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num"
unfolding real_of_float_power
by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
also have "\<dots> \<le> real (power_up_fl prec (?ub_exp_horner (float_divr prec x (- floor_fl x))) ?num)"
by (auto simp add: real_power_up_fl intro!: power_up ub_exp_horner_nonneg div_less_zero)
finally show ?thesis unfolding ub_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] floor_fl_def Let_def
.
qed
moreover
have "lb_exp prec x \<le> exp x"
proof -
let ?divl = "float_divl prec x (- floor_fl x)"
let ?horner = "?lb_exp_horner ?divl"
show ?thesis
proof (cases "?horner \<le> 0")
case False hence "0 \<le> real ?horner" by auto
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 "(?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num \<le>
exp (float_divl prec x (- floor_fl x)) ^ ?num"
using `0 \<le> real ?horner`[unfolded floor_fl_def[symmetric]] bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1] by (auto intro!: power_mono)
also have "\<dots> \<le> exp (x / ?num) ^ ?num" unfolding num_eq fl_eq
using float_divl by (auto intro!: power_mono simp del: uminus_float.rep_eq)
also have "\<dots> = exp (?num * (x / ?num))" unfolding exp_real_of_nat_mult ..
also have "\<dots> = exp x" using `real ?num \<noteq> 0` by auto
finally show ?thesis using False
unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] int_floor_fl_def Let_def if_not_P[OF False]
by (auto simp: real_power_down_fl intro!: power_down_le)
next
case True
have "power_down_fl prec (Float 1 (- 2)) ?num \<le> (Float 1 (- 2)) ^ ?num"
by (metis Float_le_zero_iff less_imp_le linorder_not_less not_numeral_le_zero numeral_One power_down_fl)
then have "power_down_fl prec (Float 1 (- 2)) ?num \<le> real (Float 1 (- 2)) ^ ?num"
by simp
also
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> x / (- floor_fl x)" unfolding minus_float.rep_eq by auto
from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
have "Float 1 (- 2) \<le> exp (x / (- floor_fl x))" unfolding Float_num .
hence "real (Float 1 (- 2)) ^ ?num \<le> exp (x / (- floor_fl x)) ^ ?num"
by (metis Float_num(5) power_mono zero_le_divide_1_iff zero_le_numeral)
also have "\<dots> = exp x" unfolding num_eq fl_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`] int_floor_fl_def Let_def if_P[OF True] real_of_float_power
.
qed
qed
ultimately show ?thesis by auto
qed
qed
lemma exp_boundaries: "exp x \<in> { lb_exp prec x .. ub_exp prec x }"
proof -
show ?thesis
proof (cases "0 < x")
case False hence "x \<le> 0" by auto
from exp_boundaries'[OF this] show ?thesis .
next
case True hence "-x \<le> 0" by auto
have "lb_exp prec x \<le> exp x"
proof -
from exp_boundaries'[OF `-x \<le> 0`]
have ub_exp: "exp (- real x) \<le> ub_exp prec (-x)" unfolding atLeastAtMost_iff minus_float.rep_eq by auto
have "float_divl prec 1 (ub_exp prec (-x)) \<le> 1 / ub_exp prec (-x)" using float_divl[where x=1] by auto
also have "\<dots> \<le> exp 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 x \<le> ub_exp prec x"
proof -
have "\<not> 0 < -x" using `0 < x` by auto
from exp_boundaries'[OF `-x \<le> 0`]
have lb_exp: "lb_exp prec (-x) \<le> exp (- real x)" unfolding atLeastAtMost_iff minus_float.rep_eq by auto
have "exp x \<le> (1 :: float) / lb_exp prec (-x)"
using lb_exp lb_exp_pos[OF `\<not> 0 < -x`, of prec]
by (simp del: lb_exp.simps add: exp_minus inverse_eq_divide field_simps)
also have "\<dots> \<le> 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::real) lx ux. (l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> exp x \<and> exp x \<le> u"
proof (rule allI, rule allI, rule allI, rule impI)
fix x::real and lx ux
assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux}"
hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {lx .. ux}" by auto
{ from exp_boundaries[of lx prec, unfolded l]
have "l \<le> exp lx" by (auto simp del: lb_exp.simps)
also have "\<dots> \<le> exp x" using x by auto
finally have "l \<le> exp x" .
} moreover
{ have "exp x \<le> exp ux" using x by auto
also have "\<dots> \<le> u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
finally have "exp x \<le> u" .
} ultimately show "l \<le> exp x \<and> exp x \<le> u" ..
qed
section "Logarithm"
subsection "Compute the logarithm series"
fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
"ub_ln_horner prec 0 i x = 0" |
"ub_ln_horner prec (Suc n) i x = float_plus_up prec
(rapprox_rat prec 1 (int i)) (- float_round_down prec (x * lb_ln_horner prec n (Suc i) x))" |
"lb_ln_horner prec 0 i x = 0" |
"lb_ln_horner prec (Suc n) i x = float_plus_down prec
(lapprox_rat prec 1 (int i)) (- float_round_up prec (x * ub_ln_horner prec n (Suc i) x))"
lemma ln_bounds:
assumes "0 \<le> x" and "x < 1"
shows "(\<Sum>i=0..<2*n. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i)) \<le> ln (x + 1)" (is "?lb")
and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i))" (is "?ub")
proof -
let "?a n" = "(1/real (n +1)) * x ^ (Suc n)"
have ln_eq: "(\<Sum> i. (- 1) ^ i * ?a i) = ln (x + 1)"
using ln_series[of "x + 1"] `0 \<le> x` `x < 1` by auto
have "norm x < 1" using assms by auto
have "?a ----> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric]
using tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF `norm x < 1`]]] by auto
{ fix n have "0 \<le> ?a n" by (rule mult_nonneg_nonneg, auto simp: `0 \<le> x`) }
{ fix n have "?a (Suc n) \<le> ?a n" unfolding inverse_eq_divide[symmetric]
proof (rule mult_mono)
show "0 \<le> x ^ Suc (Suc n)" by (auto simp add: `0 \<le> x`)
have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 mult.assoc[symmetric]
by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto simp: `0 \<le> x`)
thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
qed auto }
from summable_Leibniz'(2,4)[OF `?a ----> 0` `\<And>n. 0 \<le> ?a n`, OF `\<And>n. ?a (Suc n) \<le> ?a n`, unfolded ln_eq]
show "?lb" and "?ub" unfolding atLeast0LessThan by auto
qed
lemma ln_float_bounds:
assumes "0 \<le> real x" and "real x < 1"
shows "x * lb_ln_horner prec (get_even n) 1 x \<le> ln (x + 1)" (is "?lb \<le> ?ln")
and "ln (x + 1) \<le> 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)) * (real x)^(Suc n)"
have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric] unfolding 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> 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> 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> real x` `real x < 1`] by auto
also have "\<dots> \<le> ?ub" unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric] unfolding 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> 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
lemma ln_add: assumes "0 < x" and "0 < y" shows "ln (x + y) = ln x + ln (1 + y / x)"
proof -
have "x \<noteq> 0" using assms by auto
have "x + y = x * (1 + y / x)" unfolding distrib_left times_divide_eq_right nonzero_mult_divide_cancel_left[OF `x \<noteq> 0`] by auto
moreover
have "0 < y / x" using assms by auto
hence "0 < 1 + y / x" by auto
ultimately show ?thesis using ln_mult assms by auto
qed
subsection "Compute the logarithm of 2"
definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3
in float_plus_up prec
((Float 1 (- 1) * ub_ln_horner prec (get_odd prec) 1 (Float 1 (- 1))))
(float_round_up prec (third * ub_ln_horner prec (get_odd prec) 1 third)))"
definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3
in float_plus_down prec
((Float 1 (- 1) * lb_ln_horner prec (get_even prec) 1 (Float 1 (- 1))))
(float_round_down prec (third * lb_ln_horner prec (get_even prec) 1 third)))"
lemma ub_ln2: "ln 2 \<le> ub_ln2 prec" (is "?ub_ln2")
and lb_ln2: "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: "?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_nonneg[of 1 3] by auto
have ub3: "1 / 3 \<le> ?uthird" using rapprox_rat[of 1 3] by auto
hence ub3_lb: "0 \<le> real ?uthird" 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: "real ?uthird < 1"
by (simp add: Float.compute_rapprox_rat Float.compute_lapprox_rat rapprox_posrat_less1)
have third_gt0: "(0 :: real) < 1 / 3 + 1" 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 ln2_sum Float_num(4)[symmetric]
proof (rule float_plus_up_le, rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
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> ?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird"
using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
also note float_round_up
finally show "ln (1 / 3 + 1) \<le> float_round_up prec (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
qed
show ?lb_ln2 unfolding lb_ln2_def Let_def ln2_sum Float_num(4)[symmetric]
proof (rule float_plus_down_le, rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
have "?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] .
note float_round_down_le[OF this]
also have "\<dots> \<le> ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto
finally show "float_round_down prec (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (1 / 3 + 1)" .
qed
qed
subsection "Compute the logarithm in the entire domain"
function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where
"ub_ln prec x = (if x \<le> 0 then None
else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
else let horner = \<lambda>x. float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x) in
if x \<le> Float 3 (- 1) then Some (horner (x - 1))
else if x < Float 1 1 then Some (float_round_up prec (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1)))
else let l = bitlen (mantissa x) - 1 in
Some (float_plus_up prec (float_round_up prec (ub_ln2 prec * (Float (exponent x + l) 0))) (horner (Float (mantissa x) (- l) - 1))))" |
"lb_ln prec x = (if x \<le> 0 then None
else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x)))
else let horner = \<lambda>x. float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x) in
if x \<le> Float 3 (- 1) then Some (horner (x - 1))
else if x < Float 1 1 then Some (float_round_down prec (horner (Float 1 (- 1)) +
horner (max (x * lapprox_rat prec 2 3 - 1) 0)))
else let l = bitlen (mantissa x) - 1 in
Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * (Float (exponent x + l) 0))) (horner (Float (mantissa x) (- l) - 1))))"
by pat_completeness auto
termination proof (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 1 then 1 else 0))", auto)
fix prec and x :: float assume "\<not> real x \<le> 0" and "real x < 1" and "real (float_divl (max prec (Suc 0)) 1 x) < 1"
hence "0 < real x" "1 \<le> max prec (Suc 0)" "real x < 1" by auto
from float_divl_pos_less1_bound[OF `0 < real x` `real x < 1`[THEN less_imp_le] `1 \<le> max prec (Suc 0)`]
show False using `real (float_divl (max prec (Suc 0)) 1 x) < 1` by auto
next
fix prec x assume "\<not> real x \<le> 0" and "real x < 1" and "real (float_divr prec 1 x) < 1"
hence "0 < x" by auto
from float_divr_pos_less1_lower_bound[OF `0 < x`, of prec] `real x < 1`
show False using `real (float_divr prec 1 x) < 1` by auto
qed
lemma float_pos_eq_mantissa_pos: "x > 0 \<longleftrightarrow> mantissa x > 0"
apply (subst Float_mantissa_exponent[of x, symmetric])
apply (auto simp add: zero_less_mult_iff zero_float_def powr_gt_zero[of 2 "exponent x"] dest: less_zeroE)
using powr_gt_zero[of 2 "exponent x"]
apply simp
done
lemma Float_pos_eq_mantissa_pos: "Float m e > 0 \<longleftrightarrow> m > 0"
using powr_gt_zero[of 2 "e"]
by (auto simp add: zero_less_mult_iff zero_float_def simp del: powr_gt_zero dest: less_zeroE)
lemma Float_representation_aux:
fixes m e
defines "x \<equiv> Float m e"
assumes "x > 0"
shows "Float (exponent x + (bitlen (mantissa x) - 1)) 0 = Float (e + (bitlen m - 1)) 0" (is ?th1)
and "Float (mantissa x) (- (bitlen (mantissa x) - 1)) = Float m ( - (bitlen m - 1))" (is ?th2)
proof -
from assms have mantissa_pos: "m > 0" "mantissa x > 0"
using Float_pos_eq_mantissa_pos[of m e] float_pos_eq_mantissa_pos[of x] by simp_all
thus ?th1 using bitlen_Float[of m e] assms by (auto simp add: zero_less_mult_iff intro!: arg_cong2[where f=Float])
have "x \<noteq> float_of 0"
unfolding zero_float_def[symmetric] using `0 < x` by auto
from denormalize_shift[OF assms(1) this] guess i . note i = this
have "2 powr (1 - (real (bitlen (mantissa x)) + real i)) =
2 powr (1 - (real (bitlen (mantissa x)))) * inverse (2 powr (real i))"
by (simp add: powr_minus[symmetric] powr_add[symmetric] field_simps)
hence "real (mantissa x) * 2 powr (1 - real (bitlen (mantissa x))) =
(real (mantissa x) * 2 ^ i) * 2 powr (1 - real (bitlen (mantissa x * 2 ^ i)))"
using `mantissa x > 0` by (simp add: powr_realpow)
then show ?th2
unfolding i by transfer auto
qed
lemma compute_ln[code]:
fixes m e
defines "x \<equiv> Float m e"
shows "ub_ln prec x = (if x \<le> 0 then None
else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
else let horner = \<lambda>x. float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x) in
if x \<le> Float 3 (- 1) then Some (horner (x - 1))
else if x < Float 1 1 then Some (float_round_up prec (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1)))
else let l = bitlen m - 1 in
Some (float_plus_up prec (float_round_up prec (ub_ln2 prec * (Float (e + l) 0))) (horner (Float m (- l) - 1))))"
(is ?th1)
and "lb_ln prec x = (if x \<le> 0 then None
else if x < 1 then Some (- the (ub_ln prec (float_divr prec 1 x)))
else let horner = \<lambda>x. float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x) in
if x \<le> Float 3 (- 1) then Some (horner (x - 1))
else if x < Float 1 1 then Some (float_round_down prec (horner (Float 1 (- 1)) +
horner (max (x * lapprox_rat prec 2 3 - 1) 0)))
else let l = bitlen m - 1 in
Some (float_plus_down prec (float_round_down prec (lb_ln2 prec * (Float (e + l) 0))) (horner (Float m (- l) - 1))))"
(is ?th2)
proof -
from assms Float_pos_eq_mantissa_pos have "x > 0 \<Longrightarrow> m > 0" by simp
thus ?th1 ?th2 using Float_representation_aux[of m e] unfolding x_def[symmetric]
by (auto dest: not_leE)
qed
lemma ln_shifted_float: assumes "0 < m" shows "ln (Float m e) = ln 2 * (e + (bitlen m - 1)) + ln (Float m (- (bitlen m - 1)))"
proof -
let ?B = "2^nat (bitlen m - 1)"
def bl \<equiv> "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
hence "0 \<le> bl" by (simp add: bitlen_def bl_def)
show ?thesis
proof (cases "0 \<le> e")
case True
thus ?thesis
unfolding bl_def[symmetric] using `0 < real m` `0 \<le> bl`
apply (simp add: ln_mult)
apply (cases "e=0")
apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr)
apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr field_simps)
done
next
case False hence "0 < -e" by auto
have lne: "ln (2 powr real e) = ln (inverse (2 powr - e))" by (simp add: powr_minus)
hence pow_gt0: "(0::real) < 2^nat (-e)" by auto
hence inv_gt0: "(0::real) < inverse (2^nat (-e))" by auto
show ?thesis using False unfolding bl_def[symmetric] using `0 < real m` `0 \<le> bl`
by (auto simp add: lne ln_mult ln_powr ln_div field_simps)
qed
qed
lemma ub_ln_lb_ln_bounds': assumes "1 \<le> x"
shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)"
(is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
proof (cases "x < Float 1 1")
case True
hence "real (x - 1) < 1" and "real x < 2" by auto
have "\<not> x \<le> 0" and "\<not> x < 1" using `1 \<le> x` by auto
hence "0 \<le> real (x - 1)" using `1 \<le> x` by auto
have [simp]: "(Float 3 (- 1)) = 3 / 2" by simp
show ?thesis
proof (cases "x \<le> Float 3 (- 1)")
case True
show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
using ln_float_bounds[OF `0 \<le> real (x - 1)` `real (x - 1) < 1`, of prec] `\<not> x \<le> 0` `\<not> x < 1` True
by (auto intro!: float_round_down_le float_round_up_le)
next
case False hence *: "3 / 2 < x" by auto
with ln_add[of "3 / 2" "x - 3 / 2"]
have add: "ln x = ln (3 / 2) + ln (real x * 2 / 3)"
by (auto simp add: algebra_simps diff_divide_distrib)
let "?ub_horner x" = "float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x)"
let "?lb_horner x" = "float_round_down prec (x * lb_ln_horner prec (get_even prec) 1 x)"
{ have up: "real (rapprox_rat prec 2 3) \<le> 1"
by (rule rapprox_rat_le1) simp_all
have low: "2 / 3 \<le> rapprox_rat prec 2 3"
by (rule order_trans[OF _ rapprox_rat]) simp
from mult_less_le_imp_less[OF * low] *
have pos: "0 < real (x * rapprox_rat prec 2 3 - 1)" by auto
have "ln (real x * 2/3)
\<le> ln (real (x * rapprox_rat prec 2 3 - 1) + 1)"
proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
show "real x * 2 / 3 \<le> real (x * rapprox_rat prec 2 3 - 1) + 1"
using * low by auto
show "0 < real x * 2 / 3" using * by simp
show "0 < real (x * rapprox_rat prec 2 3 - 1) + 1" using pos by auto
qed
also have "\<dots> \<le> ?ub_horner (x * rapprox_rat prec 2 3 - 1)"
proof (rule float_round_up_le, rule ln_float_bounds(2))
from mult_less_le_imp_less[OF `real x < 2` up] low *
show "real (x * rapprox_rat prec 2 3 - 1) < 1" by auto
show "0 \<le> real (x * rapprox_rat prec 2 3 - 1)" using pos by auto
qed
finally have "ln x \<le> ?ub_horner (Float 1 (-1))
+ ?ub_horner ((x * rapprox_rat prec 2 3 - 1))"
using ln_float_bounds(2)[of "Float 1 (- 1)" prec prec] add
by (auto intro!: add_mono float_round_up_le)
note float_round_up_le[OF this, of prec]
}
moreover
{ let ?max = "max (x * lapprox_rat prec 2 3 - 1) 0"
have up: "lapprox_rat prec 2 3 \<le> 2/3"
by (rule order_trans[OF lapprox_rat], simp)
have low: "0 \<le> real (lapprox_rat prec 2 3)"
using lapprox_rat_nonneg[of 2 3 prec] by simp
have "?lb_horner ?max
\<le> ln (real ?max + 1)"
proof (rule float_round_down_le, rule ln_float_bounds(1))
from mult_less_le_imp_less[OF `real x < 2` up] * low
show "real ?max < 1" by (cases "real (lapprox_rat prec 2 3) = 0",
auto simp add: real_of_float_max)
show "0 \<le> real ?max" by (auto simp add: real_of_float_max)
qed
also have "\<dots> \<le> ln (real x * 2/3)"
proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
show "0 < real ?max + 1" by (auto simp add: real_of_float_max)
show "0 < real x * 2/3" using * by auto
show "real ?max + 1 \<le> real x * 2/3" using * up
by (cases "0 < real x * real (lapprox_posrat prec 2 3) - 1",
auto simp add: max_def)
qed
finally have "?lb_horner (Float 1 (- 1)) + ?lb_horner ?max \<le> ln x"
using ln_float_bounds(1)[of "Float 1 (- 1)" prec prec] add
by (auto intro!: add_mono float_round_down_le)
note float_round_down_le[OF this, of prec]
}
ultimately
show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
using `\<not> x \<le> 0` `\<not> x < 1` True False by auto
qed
next
case False
hence "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" "\<not> x \<le> Float 3 (- 1)"
using `1 \<le> x` by auto
show ?thesis
proof -
def m \<equiv> "mantissa x"
def e \<equiv> "exponent x"
from Float_mantissa_exponent[of x] have Float: "x = Float m e" by (simp add: m_def e_def)
let ?s = "Float (e + (bitlen m - 1)) 0"
let ?x = "Float m (- (bitlen m - 1))"
have "0 < m" and "m \<noteq> 0" using `0 < x` Float powr_gt_zero[of 2 e]
by (auto simp: zero_less_mult_iff)
def bl \<equiv> "bitlen m - 1" hence "bl \<ge> 0" using `m > 0` by (simp add: bitlen_def)
have "1 \<le> Float m e" using `1 \<le> x` Float unfolding less_eq_float_def by auto
from bitlen_div[OF `0 < m`] float_gt1_scale[OF `1 \<le> Float m e`] `bl \<ge> 0`
have x_bnds: "0 \<le> real (?x - 1)" "real (?x - 1) < 1"
unfolding bl_def[symmetric]
by (auto simp: powr_realpow[symmetric] field_simps inverse_eq_divide)
(auto simp : powr_minus field_simps inverse_eq_divide)
{
have "float_round_down prec (lb_ln2 prec * ?s) \<le> ln 2 * (e + (bitlen m - 1))" (is "real ?lb2 \<le> _")
apply (rule float_round_down_le)
unfolding nat_0 power_0 mult_1_right times_float.rep_eq
using lb_ln2[of prec]
proof (rule mult_mono)
from float_gt1_scale[OF `1 \<le> Float m e`]
show "0 \<le> real (Float (e + (bitlen m - 1)) 0)" by simp
qed auto
moreover
from ln_float_bounds(1)[OF x_bnds]
have "float_round_down prec ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln ?x" (is "real ?lb_horner \<le> _")
by (auto intro!: float_round_down_le)
ultimately have "float_plus_down prec ?lb2 ?lb_horner \<le> ln x"
unfolding Float ln_shifted_float[OF `0 < m`, of e] by (auto intro!: float_plus_down_le)
}
moreover
{
from ln_float_bounds(2)[OF x_bnds]
have "ln ?x \<le> float_round_up prec ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))" (is "_ \<le> real ?ub_horner")
by (auto intro!: float_round_up_le)
moreover
have "ln 2 * (e + (bitlen m - 1)) \<le> float_round_up prec (ub_ln2 prec * ?s)" (is "_ \<le> real ?ub2")
apply (rule float_round_up_le)
unfolding nat_0 power_0 mult_1_right times_float.rep_eq
using ub_ln2[of prec]
proof (rule mult_mono)
from float_gt1_scale[OF `1 \<le> Float m e`]
show "0 \<le> real (e + (bitlen m - 1))" by auto
next
have "0 \<le> ln 2" by simp
thus "0 \<le> real (ub_ln2 prec)" using ub_ln2[of prec] by arith
qed auto
ultimately have "ln x \<le> float_plus_up prec ?ub2 ?ub_horner"
unfolding Float ln_shifted_float[OF `0 < m`, of e]
by (auto intro!: float_plus_up_le)
}
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] if_not_P[OF `\<not> x \<le> Float 3 (- 1)`] Let_def
unfolding plus_float.rep_eq e_def[symmetric] m_def[symmetric] by simp
qed
qed
lemma ub_ln_lb_ln_bounds:
assumes "0 < x"
shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> 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 less_eq_float_def by auto
show ?thesis using ub_ln_lb_ln_bounds'[OF `1 \<le> x`] .
next
case True have "\<not> x \<le> 0" using `0 < x` by auto
from True have "real x \<le> 1" "x \<le> 1" by simp_all
have "0 < real x" and "real x \<noteq> 0" using `0 < x` 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 < real x` `real x \<le> 1`] by auto
hence B: "0 < real ?divl" by auto
have "ln ?divl \<le> ln (1 / x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
hence "ln x \<le> - ln ?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> - the (lb_ln prec ?divl)" unfolding uminus_float.rep_eq 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 \<le> 1`] unfolding less_eq_float_def less_float_def by auto
hence B: "0 < real ?divr" by auto
have "ln (1 / x) \<le> ln ?divr" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
hence "- ln ?divr \<le> ln 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 "- the (ub_ln prec ?divr) \<le> ?ln" unfolding uminus_float.rep_eq 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 "y \<le> ln x" and "0 < real x"
proof -
have "0 < x"
proof (rule ccontr)
assume "\<not> 0 < x" hence "x \<le> 0" unfolding less_eq_float_def less_float_def by auto
thus False using assms by auto
qed
thus "0 < real x" by auto
have "the (lb_ln prec x) \<le> ln x" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
thus "y \<le> ln x" unfolding assms[symmetric] by auto
qed
lemma ub_ln:
assumes "Some y = ub_ln prec x"
shows "ln x \<le> y" and "0 < real x"
proof -
have "0 < x"
proof (rule ccontr)
assume "\<not> 0 < x" hence "x \<le> 0" by auto
thus False using assms by auto
qed
thus "0 < real x" by auto
have "ln x \<le> the (ub_ln prec x)" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
thus "ln x \<le> y" unfolding assms[symmetric] by auto
qed
lemma bnds_ln: "\<forall> (x::real) lx ux. (Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux} \<longrightarrow> l \<le> ln x \<and> ln x \<le> u"
proof (rule allI, rule allI, rule allI, rule impI)
fix x::real and lx ux
assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux}"
hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {lx .. ux}" by auto
have "ln ux \<le> u" and "0 < real ux" using ub_ln u by auto
have "l \<le> ln lx" and "0 < real lx" and "0 < x" using lb_ln[OF l] x by auto
from ln_le_cancel_iff[OF `0 < real lx` `0 < x`] `l \<le> ln lx`
have "l \<le> ln x" using x unfolding atLeastAtMost_iff by auto
moreover
from ln_le_cancel_iff[OF `0 < x` `0 < real ux`] `ln ux \<le> real u`
have "ln x \<le> u" using x unfolding atLeastAtMost_iff by auto
ultimately show "l \<le> ln x \<and> ln x \<le> u" ..
qed
section "Implement floatarith"
subsection "Define syntax and semantics"
datatype floatarith
= Add floatarith floatarith
| Minus floatarith
| Mult floatarith floatarith
| Inverse floatarith
| Cos floatarith
| Arctan floatarith
| Abs floatarith
| Max floatarith floatarith
| Min floatarith floatarith
| Pi
| Sqrt floatarith
| Exp floatarith
| Ln floatarith
| Power floatarith nat
| Var nat
| Num float
fun interpret_floatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real" where
"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 (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 = f" |
"interpret_floatarith (Var n) vs = vs ! n"
lemma interpret_floatarith_divide: "interpret_floatarith (Mult a (Inverse b)) vs = (interpret_floatarith a vs) / (interpret_floatarith b vs)"
unfolding divide_inverse interpret_floatarith.simps ..
lemma interpret_floatarith_diff: "interpret_floatarith (Add a (Minus b)) vs = (interpret_floatarith a vs) - (interpret_floatarith b vs)"
unfolding interpret_floatarith.simps by simp
lemma interpret_floatarith_sin: "interpret_floatarith (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) vs =
sin (interpret_floatarith a vs)"
unfolding sin_cos_eq interpret_floatarith.simps
interpret_floatarith_divide interpret_floatarith_diff
by auto
lemma interpret_floatarith_tan:
"interpret_floatarith (Mult (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) (Inverse (Cos a))) vs =
tan (interpret_floatarith a vs)"
unfolding interpret_floatarith.simps(3,4) interpret_floatarith_sin tan_def divide_inverse
by auto
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 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 divide_inverse ..
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 (- 1) 0)) vs = - 1"
and "interpret_floatarith (Num (Float (numeral a) 0)) vs = numeral a"
and "interpret_floatarith (Num (Float (- numeral a) 0)) vs = - numeral a" by auto
subsection "Implement approximation function"
fun lift_bin' :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
"lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" |
"lift_bin' a b f = None"
fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where
"lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u)
| t \<Rightarrow> None)" |
"lift_un b f = None"
fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
"lift_un' (Some (l1, u1)) f = Some (f l1 u1)" |
"lift_un' b f = None"
definition
"bounded_by xs vs \<longleftrightarrow>
(\<forall> i < length vs. case vs ! i of None \<Rightarrow> True
| Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u })"
lemma bounded_byE:
assumes "bounded_by xs vs"
shows "\<And> i. i < length vs \<Longrightarrow> case vs ! i of None \<Rightarrow> True
| Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u }"
using assms bounded_by_def by blast
lemma bounded_by_update:
assumes "bounded_by xs vs"
and bnd: "xs ! i \<in> { real l .. real u }"
shows "bounded_by xs (vs[i := Some (l,u)])"
proof -
{ fix j
let ?vs = "vs[i := Some (l,u)]"
assume "j < length ?vs" hence [simp]: "j < length vs" by simp
have "case ?vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> xs ! j \<in> { real l .. real u }"
proof (cases "?vs ! j")
case (Some b)
thus ?thesis
proof (cases "i = j")
case True
thus ?thesis using `?vs ! j = Some b` and bnd by auto
next
case False
thus ?thesis using `bounded_by xs vs` unfolding bounded_by_def by auto
qed
qed auto }
thus ?thesis unfolding bounded_by_def by auto
qed
lemma bounded_by_None:
shows "bounded_by xs (replicate (length xs) None)"
unfolding bounded_by_def by auto
fun approx approx' :: "nat \<Rightarrow> floatarith \<Rightarrow> (float * float) option list \<Rightarrow> (float * float) option" where
"approx' prec a bs = (case (approx prec a bs) of Some (l, u) \<Rightarrow> Some (float_round_down prec l, float_round_up prec u) | None \<Rightarrow> None)" |
"approx prec (Add a b) bs =
lift_bin' (approx' prec a bs) (approx' prec b bs)
(\<lambda> l1 u1 l2 u2. (float_plus_down prec l1 l2, float_plus_up prec u1 u2))" |
"approx prec (Minus a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (-u, -l))" |
"approx prec (Mult a b) bs =
lift_bin' (approx' prec a bs) (approx' prec b bs)
(\<lambda> a1 a2 b1 b2.
(float_plus_down prec (nprt a1 * pprt b2)
(float_plus_down prec (nprt a2 * nprt b2)
(float_plus_down prec (pprt a1 * pprt b1) (pprt a2 * nprt b1))),
float_plus_up prec (pprt a2 * pprt b2)
(float_plus_up prec (pprt a1 * nprt b2)
(float_plus_up prec (nprt a2 * pprt b1) (nprt a1 * nprt b1)))))" |
"approx prec (Inverse a) bs = lift_un (approx' prec a bs) (\<lambda> l u. if (0 < l \<or> u < 0) then (Some (float_divl prec 1 u), Some (float_divr prec 1 l)) else (None, None))" |
"approx prec (Cos a) bs = lift_un' (approx' prec a bs) (bnds_cos prec)" |
"approx prec Pi bs = Some (lb_pi prec, ub_pi prec)" |
"approx prec (Min a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (min l1 l2, min u1 u2))" |
"approx prec (Max a b) bs = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (max l1 l2, max u1 u2))" |
"approx prec (Abs a) bs = lift_un' (approx' prec a bs) (\<lambda>l u. (if l < 0 \<and> 0 < u then 0 else min \<bar>l\<bar> \<bar>u\<bar>, max \<bar>l\<bar> \<bar>u\<bar>))" |
"approx prec (Arctan a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_arctan prec l, ub_arctan prec u))" |
"approx prec (Sqrt a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" |
"approx prec (Exp a) bs = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" |
"approx prec (Ln a) bs = lift_un (approx' prec a bs) (\<lambda> l u. (lb_ln prec l, ub_ln prec u))" |
"approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds prec n)" |
"approx prec (Num f) bs = Some (f, f)" |
"approx prec (Var i) bs = (if i < length bs then bs ! i else None)"
lemma lift_bin'_ex:
assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f"
shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b"
proof (cases a)
case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
thus ?thesis using lift_bin'_Some by auto
next
case (Some a')
show ?thesis
proof (cases b)
case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
thus ?thesis using lift_bin'_Some by auto
next
case (Some b')
obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
obtain lb ub where b': "b' = (lb, ub)" by (cases b', auto)
thus ?thesis unfolding `a = Some a'` `b = Some b'` a' b' by auto
qed
qed
lemma lift_bin'_f:
assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f"
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" and Pb: "\<And>l u. Some (l, u) = g b \<Longrightarrow> P l u b"
shows "\<exists> l1 u1 l2 u2. P l1 u1 a \<and> P l2 u2 b \<and> l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
proof -
obtain l1 u1 l2 u2
where Sa: "Some (l1, u1) = g a" and Sb: "Some (l2, u2) = g b" using lift_bin'_ex[OF assms(1)] by auto
have lu: "(l, u) = f l1 u1 l2 u2" using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto
have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" unfolding lu[symmetric] by auto
thus ?thesis using Pa[OF Sa] Pb[OF Sb] by auto
qed
lemma approx_approx':
assumes Pa: "\<And>l u. Some (l, u) = approx prec a vs \<Longrightarrow> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u"
and approx': "Some (l, u) = approx' prec a vs"
shows "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> 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)
have l': "l = float_round_down prec l'" and u': "u = float_round_up prec u'"
using approx' unfolding approx'.simps S[symmetric] by auto
show ?thesis unfolding l' u'
using order_trans[OF Pa[OF S, THEN conjunct2] float_round_up[of u']]
using order_trans[OF float_round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto
qed
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> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> 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> l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u"
shows "\<exists> l1 u1 l2 u2. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and>
(l2 \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> 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 "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> 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 "l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> 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
qed
lemma lift_un'_ex:
assumes lift_un'_Some: "Some (l, u) = lift_un' a f"
shows "\<exists> l u. Some (l, u) = a"
proof (cases a)
case None hence "None = lift_un' a f" unfolding None lift_un'.simps ..
thus ?thesis using lift_un'_Some by auto
next
case (Some a')
obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
thus ?thesis unfolding `a = Some a'` a' by auto
qed
lemma lift_un'_f:
assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f"
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
shows "\<exists> l1 u1. P l1 u1 a \<and> l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
proof -
obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un'_ex[OF assms(1)] by auto
have lu: "(l, u) = f l1 u1" using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto
have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" unfolding lu[symmetric] by auto
thus ?thesis using Pa[OF Sa] by auto
qed
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> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
shows "\<exists> l1 u1. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> 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 "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> 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::real) lx ux. (l, u) = f lx ux \<and> x \<in> { lx .. ux } \<longrightarrow> l \<le> f' x \<and> f' x \<le> 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> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> 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 "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> 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> {l1 .. u1}" by auto
thus ?thesis using bnds by auto
qed
lemma lift_un_ex:
assumes lift_un_Some: "Some (l, u) = lift_un a f"
shows "\<exists> l u. Some (l, u) = a"
proof (cases a)
case None hence "None = lift_un a f" unfolding None lift_un.simps ..
thus ?thesis using lift_un_Some by auto
next
case (Some a')
obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
thus ?thesis unfolding `a = Some a'` a' by auto
qed
lemma lift_un_f:
assumes lift_un_Some: "Some (l, u) = lift_un (g a) f"
and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
shows "\<exists> l1 u1. P l1 u1 a \<and> Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
proof -
obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un_ex[OF assms(1)] by auto
have "fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None"
proof (rule ccontr)
assume "\<not> (fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None)"
hence or: "fst (f l1 u1) = None \<or> snd (f l1 u1) = None" by auto
hence "lift_un (g a) f = None"
proof (cases "fst (f l1 u1) = None")
case True
then obtain b where b: "f l1 u1 = (None, b)" by (cases "f l1 u1", auto)
thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
next
case False hence "snd (f l1 u1) = None" using or by auto
with False obtain b where b: "f l1 u1 = (Some b, None)" by (cases "f l1 u1", auto)
thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
qed
thus False using lift_un_Some by auto
qed
then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" by (cases "f l1 u1", auto)
from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f]
have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" unfolding f by auto
thus ?thesis unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto
qed
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> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
shows "\<exists> l1 u1. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> 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 "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> 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::real) lx ux. (Some l, Some u) = f lx ux \<and> x \<in> { lx .. ux } \<longrightarrow> l \<le> f' x \<and> f' x \<le> 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> l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> 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 "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> 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> {l1 .. 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 "l \<le> interpret_floatarith arith xs \<and> interpret_floatarith arith xs \<le> u" (is "?P l u arith")
using `Some (l, u) = approx prec arith vs`
proof (induct arith arbitrary: l u)
case (Add a b)
from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps
obtain l1 u1 l2 u2 where "l = float_plus_down prec l1 l2" and "u = float_plus_up prec u1 u2"
"l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1"
"l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> u2" unfolding fst_conv snd_conv by blast
thus ?case unfolding interpret_floatarith.simps by (auto intro!: float_plus_up_le float_plus_down_le)
next
case (Minus a)
from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps
obtain l1 u1 where "l = -u1" and "u = -l1"
"l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1" unfolding fst_conv snd_conv by blast
thus ?case unfolding interpret_floatarith.simps using minus_float.rep_eq 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_plus_down prec (nprt l1 * pprt u2) (float_plus_down prec (nprt u1 * nprt u2) (float_plus_down prec (pprt l1 * pprt l2) (pprt u1 * nprt l2)))"
and u: "u = float_plus_up prec (pprt u1 * pprt u2) (float_plus_up prec (pprt l1 * nprt u2) (float_plus_up prec (nprt u1 * pprt l2) (nprt l1 * nprt l2)))"
and "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1"
and "l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> u2" unfolding fst_conv snd_conv by blast
hence bnds:
"nprt l1 * pprt u2 + nprt u1 * nprt u2 + pprt l1 * pprt l2 + pprt u1 * nprt l2 \<le> interpret_floatarith (Mult a b) xs" (is "?l \<le> _")
"interpret_floatarith (Mult a b) xs \<le> pprt u1 * pprt u2 + pprt l1 * nprt u2 + nprt u1 * pprt l2 + nprt l1 * nprt l2" (is "_ \<le> ?u")
unfolding interpret_floatarith.simps l u
using mult_le_prts mult_ge_prts by auto
from l u have "l \<le> ?l" "?u \<le> u"
by (auto intro!: float_plus_up_le float_plus_down_le)
thus ?case using bnds by simp
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: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> 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: "real l1 \<le> real u1" using l1 u1 by auto
ultimately have "real l1 \<noteq> 0" and "real u1 \<noteq> 0" by auto
have inv: "inverse u1 \<le> inverse (interpret_floatarith a xs)
\<and> inverse (interpret_floatarith a xs) \<le> inverse l1"
proof (cases "0 < l1")
case True hence "0 < real u1" and "0 < real l1" "0 < interpret_floatarith a xs"
using l1_le_u1 l1 by auto
show ?thesis
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 "real u1 < 0" and "real l1 < 0" "interpret_floatarith a xs < 0"
using l1_le_u1 u1 by auto
show ?thesis
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 "l \<le> inverse 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 "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 l1 \<le> 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> 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: "l1 \<le> interpret_floatarith x xs" and u1: "interpret_floatarith x xs \<le> 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)
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: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1"
and l1: "l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> 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: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1"
and l1: "l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> u2" by blast
thus ?case unfolding l' u' by (auto simp add: real_of_float_max)
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
next case Pi with pi_boundaries show ?case by auto
next case (Sqrt a) with lift_un'_bnds[OF bnds_sqrt] show ?case by auto
next case (Exp a) with lift_un'_bnds[OF bnds_exp] show ?case by auto
next case (Ln a) with lift_un_bnds[OF bnds_ln] show ?case by auto
next case (Power a n) with lift_un'_bnds[OF bnds_power] show ?case by auto
next case (Num f) thus ?case by auto
next
case (Var n)
from this[symmetric] `bounded_by xs vs`[THEN bounded_byE, of n]
show ?case by (cases "n < length vs", auto)
qed
datatype form = Bound floatarith floatarith floatarith form
| Assign floatarith floatarith form
| Less floatarith floatarith
| LessEqual floatarith floatarith
| AtLeastAtMost floatarith floatarith floatarith
| Conj form form
| Disj form form
fun interpret_form :: "form \<Rightarrow> real list \<Rightarrow> bool" where
"interpret_form (Bound x a b f) vs = (interpret_floatarith x vs \<in> { interpret_floatarith a vs .. interpret_floatarith b vs } \<longrightarrow> interpret_form f vs)" |
"interpret_form (Assign x a f) vs = (interpret_floatarith x vs = interpret_floatarith a vs \<longrightarrow> interpret_form f vs)" |
"interpret_form (Less a b) vs = (interpret_floatarith a vs < interpret_floatarith b vs)" |
"interpret_form (LessEqual a b) vs = (interpret_floatarith a vs \<le> interpret_floatarith b vs)" |
"interpret_form (AtLeastAtMost x a b) vs = (interpret_floatarith x vs \<in> { interpret_floatarith a vs .. interpret_floatarith b vs })" |
"interpret_form (Conj f g) vs \<longleftrightarrow> interpret_form f vs \<and> interpret_form g vs" |
"interpret_form (Disj f g) vs \<longleftrightarrow> interpret_form f vs \<or> interpret_form g vs"
fun approx_form' and approx_form :: "nat \<Rightarrow> form \<Rightarrow> (float * float) option list \<Rightarrow> nat list \<Rightarrow> bool" where
"approx_form' prec f 0 n l u bs ss = approx_form prec f (bs[n := Some (l, u)]) ss" |
"approx_form' prec f (Suc s) n l u bs ss =
(let m = (l + u) * Float 1 (- 1)
in (if approx_form' prec f s n l m bs ss then approx_form' prec f s n m u bs ss else False))" |
"approx_form prec (Bound (Var n) a b f) bs ss =
(case (approx prec a bs, approx prec b bs)
of (Some (l, _), Some (_, u)) \<Rightarrow> approx_form' prec f (ss ! n) n l u bs ss
| _ \<Rightarrow> False)" |
"approx_form prec (Assign (Var n) a f) bs ss =
(case (approx prec a bs)
of (Some (l, u)) \<Rightarrow> approx_form' prec f (ss ! n) n l u bs ss
| _ \<Rightarrow> False)" |
"approx_form prec (Less a b) bs ss =
(case (approx prec a bs, approx prec b bs)
of (Some (l, u), Some (l', u')) \<Rightarrow> float_plus_up prec u (-l') < 0
| _ \<Rightarrow> False)" |
"approx_form prec (LessEqual a b) bs ss =
(case (approx prec a bs, approx prec b bs)
of (Some (l, u), Some (l', u')) \<Rightarrow> float_plus_up prec u (-l') \<le> 0
| _ \<Rightarrow> False)" |
"approx_form prec (AtLeastAtMost x a b) bs ss =
(case (approx prec x bs, approx prec a bs, approx prec b bs)
of (Some (lx, ux), Some (l, u), Some (l', u')) \<Rightarrow> float_plus_up prec u (-lx) \<le> 0 \<and> float_plus_up prec ux (-l') \<le> 0
| _ \<Rightarrow> False)" |
"approx_form prec (Conj a b) bs ss \<longleftrightarrow> approx_form prec a bs ss \<and> approx_form prec b bs ss" |
"approx_form prec (Disj a b) bs ss \<longleftrightarrow> approx_form prec a bs ss \<or> approx_form prec b bs ss" |
"approx_form _ _ _ _ = False"
lemma lazy_conj: "(if A then B else False) = (A \<and> B)" by simp
lemma approx_form_approx_form':
assumes "approx_form' prec f s n l u bs ss" and "(x::real) \<in> { l .. u }"
obtains l' u' where "x \<in> { l' .. u' }"
and "approx_form prec f (bs[n := Some (l', u')]) ss"
using assms proof (induct s arbitrary: l u)
case 0
from this(1)[of l u] this(2,3)
show thesis by auto
next
case (Suc s)
let ?m = "(l + u) * Float 1 (- 1)"
have "real l \<le> ?m" and "?m \<le> real u"
unfolding less_eq_float_def using Suc.prems by auto
with `x \<in> { l .. u }`
have "x \<in> { l .. ?m} \<or> x \<in> { ?m .. u }" by auto
thus thesis
proof (rule disjE)
assume *: "x \<in> { l .. ?m }"
with Suc.hyps[OF _ _ *] Suc.prems
show thesis by (simp add: Let_def lazy_conj)
next
assume *: "x \<in> { ?m .. u }"
with Suc.hyps[OF _ _ *] Suc.prems
show thesis by (simp add: Let_def lazy_conj)
qed
qed
lemma approx_form_aux:
assumes "approx_form prec f vs ss"
and "bounded_by xs vs"
shows "interpret_form f xs"
using assms proof (induct f arbitrary: vs)
case (Bound x a b f)
then obtain n
where x_eq: "x = Var n" by (cases x) auto
with Bound.prems obtain l u' l' u
where l_eq: "Some (l, u') = approx prec a vs"
and u_eq: "Some (l', u) = approx prec b vs"
and approx_form': "approx_form' prec f (ss ! n) n l u vs ss"
by (cases "approx prec a vs", simp) (cases "approx prec b vs", auto)
{ assume "xs ! n \<in> { interpret_floatarith a xs .. interpret_floatarith b xs }"
with approx[OF Bound.prems(2) l_eq] and approx[OF Bound.prems(2) u_eq]
have "xs ! n \<in> { l .. u}" by auto
from approx_form_approx_form'[OF approx_form' this]
obtain lx ux where bnds: "xs ! n \<in> { lx .. ux }"
and approx_form: "approx_form prec f (vs[n := Some (lx, ux)]) ss" .
from `bounded_by xs vs` bnds
have "bounded_by xs (vs[n := Some (lx, ux)])" by (rule bounded_by_update)
with Bound.hyps[OF approx_form]
have "interpret_form f xs" by blast }
thus ?case using interpret_form.simps x_eq and interpret_floatarith.simps by simp
next
case (Assign x a f)
then obtain n
where x_eq: "x = Var n" by (cases x) auto
with Assign.prems obtain l u
where bnd_eq: "Some (l, u) = approx prec a vs"
and x_eq: "x = Var n"
and approx_form': "approx_form' prec f (ss ! n) n l u vs ss"
by (cases "approx prec a vs") auto
{ assume bnds: "xs ! n = interpret_floatarith a xs"
with approx[OF Assign.prems(2) bnd_eq]
have "xs ! n \<in> { l .. u}" by auto
from approx_form_approx_form'[OF approx_form' this]
obtain lx ux where bnds: "xs ! n \<in> { lx .. ux }"
and approx_form: "approx_form prec f (vs[n := Some (lx, ux)]) ss" .
from `bounded_by xs vs` bnds
have "bounded_by xs (vs[n := Some (lx, ux)])" by (rule bounded_by_update)
with Assign.hyps[OF approx_form]
have "interpret_form f xs" by blast }
thus ?case using interpret_form.simps x_eq and interpret_floatarith.simps by simp
next
case (Less a b)
then obtain l u l' u'
where l_eq: "Some (l, u) = approx prec a vs"
and u_eq: "Some (l', u') = approx prec b vs"
and inequality: "real (float_plus_up prec u (-l')) < 0"
by (cases "approx prec a vs", auto,
cases "approx prec b vs", auto)
from le_less_trans[OF float_plus_up inequality]
approx[OF Less.prems(2) l_eq] approx[OF Less.prems(2) u_eq]
show ?case by auto
next
case (LessEqual a b)
then obtain l u l' u'
where l_eq: "Some (l, u) = approx prec a vs"
and u_eq: "Some (l', u') = approx prec b vs"
and inequality: "real (float_plus_up prec u (-l')) \<le> 0"
by (cases "approx prec a vs", auto,
cases "approx prec b vs", auto)
from order_trans[OF float_plus_up inequality]
approx[OF LessEqual.prems(2) l_eq] approx[OF LessEqual.prems(2) u_eq]
show ?case by auto
next
case (AtLeastAtMost x a b)
then obtain lx ux l u l' u'
where x_eq: "Some (lx, ux) = approx prec x vs"
and l_eq: "Some (l, u) = approx prec a vs"
and u_eq: "Some (l', u') = approx prec b vs"
and inequality: "real (float_plus_up prec u (-lx)) \<le> 0" "real (float_plus_up prec ux (-l')) \<le> 0"
by (cases "approx prec x vs", auto,
cases "approx prec a vs", auto,
cases "approx prec b vs", auto)
from order_trans[OF float_plus_up inequality(1)] order_trans[OF float_plus_up inequality(2)]
approx[OF AtLeastAtMost.prems(2) l_eq] approx[OF AtLeastAtMost.prems(2) u_eq] approx[OF AtLeastAtMost.prems(2) x_eq]
show ?case by auto
qed auto
lemma approx_form:
assumes "n = length xs"
assumes "approx_form prec f (replicate n None) ss"
shows "interpret_form f xs"
using approx_form_aux[OF _ bounded_by_None] assms by auto
subsection {* Implementing Taylor series expansion *}
fun isDERIV :: "nat \<Rightarrow> floatarith \<Rightarrow> real list \<Rightarrow> bool" where
"isDERIV x (Add a b) vs = (isDERIV x a vs \<and> isDERIV x b vs)" |
"isDERIV x (Mult a b) vs = (isDERIV x a vs \<and> isDERIV x b vs)" |
"isDERIV x (Minus a) vs = isDERIV x a vs" |
"isDERIV x (Inverse a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs \<noteq> 0)" |
"isDERIV x (Cos a) vs = isDERIV x a vs" |
"isDERIV x (Arctan a) vs = isDERIV x a vs" |
"isDERIV x (Min a b) vs = False" |
"isDERIV x (Max a b) vs = False" |
"isDERIV x (Abs a) vs = False" |
"isDERIV x Pi vs = True" |
"isDERIV x (Sqrt a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs > 0)" |
"isDERIV x (Exp a) vs = isDERIV x a vs" |
"isDERIV x (Ln a) vs = (isDERIV x a vs \<and> interpret_floatarith a vs > 0)" |
"isDERIV x (Power a 0) vs = True" |
"isDERIV x (Power a (Suc n)) vs = isDERIV x a vs" |
"isDERIV x (Num f) vs = True" |
"isDERIV x (Var n) vs = True"
fun DERIV_floatarith :: "nat \<Rightarrow> floatarith \<Rightarrow> floatarith" where
"DERIV_floatarith x (Add a b) = Add (DERIV_floatarith x a) (DERIV_floatarith x b)" |
"DERIV_floatarith x (Mult a b) = Add (Mult a (DERIV_floatarith x b)) (Mult (DERIV_floatarith x a) b)" |
"DERIV_floatarith x (Minus a) = Minus (DERIV_floatarith x a)" |
"DERIV_floatarith x (Inverse a) = Minus (Mult (DERIV_floatarith x a) (Inverse (Power a 2)))" |
"DERIV_floatarith x (Cos a) = Minus (Mult (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) (DERIV_floatarith x a))" |
"DERIV_floatarith x (Arctan a) = Mult (Inverse (Add (Num 1) (Power a 2))) (DERIV_floatarith x a)" |
"DERIV_floatarith x (Min a b) = Num 0" |
"DERIV_floatarith x (Max a b) = Num 0" |
"DERIV_floatarith x (Abs a) = Num 0" |
"DERIV_floatarith x Pi = Num 0" |
"DERIV_floatarith x (Sqrt a) = (Mult (Inverse (Mult (Sqrt a) (Num 2))) (DERIV_floatarith x a))" |
"DERIV_floatarith x (Exp a) = Mult (Exp a) (DERIV_floatarith x a)" |
"DERIV_floatarith x (Ln a) = Mult (Inverse a) (DERIV_floatarith x a)" |
"DERIV_floatarith x (Power a 0) = Num 0" |
"DERIV_floatarith x (Power a (Suc n)) = Mult (Num (Float (int (Suc n)) 0)) (Mult (Power a n) (DERIV_floatarith x a))" |
"DERIV_floatarith x (Num f) = Num 0" |
"DERIV_floatarith x (Var n) = (if x = n then Num 1 else Num 0)"
lemma DERIV_floatarith:
assumes "n < length vs"
assumes isDERIV: "isDERIV n f (vs[n := x])"
shows "DERIV (\<lambda> x'. interpret_floatarith f (vs[n := x'])) x :>
interpret_floatarith (DERIV_floatarith n f) (vs[n := x])"
(is "DERIV (?i f) x :> _")
using isDERIV
proof (induct f arbitrary: x)
case (Inverse a)
thus ?case
by (auto intro!: derivative_eq_intros simp add: algebra_simps power2_eq_square)
next
case (Cos a)
thus ?case
by (auto intro!: derivative_eq_intros
simp del: interpret_floatarith.simps(5)
simp add: interpret_floatarith_sin interpret_floatarith.simps(5)[of a])
next
case (Power a n)
thus ?case
by (cases n) (auto intro!: derivative_eq_intros simp del: power_Suc simp add: real_of_nat_def)
next
case (Ln a)
thus ?case by (auto intro!: derivative_eq_intros simp add: divide_inverse)
next
case (Var i)
thus ?case using `n < length vs` by auto
qed (auto intro!: derivative_eq_intros)
declare approx.simps[simp del]
fun isDERIV_approx :: "nat \<Rightarrow> nat \<Rightarrow> floatarith \<Rightarrow> (float * float) option list \<Rightarrow> bool" where
"isDERIV_approx prec x (Add a b) vs = (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs)" |
"isDERIV_approx prec x (Mult a b) vs = (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs)" |
"isDERIV_approx prec x (Minus a) vs = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Inverse a) vs =
(isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l \<or> u < 0 | None \<Rightarrow> False))" |
"isDERIV_approx prec x (Cos a) vs = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Arctan a) vs = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Min a b) vs = False" |
"isDERIV_approx prec x (Max a b) vs = False" |
"isDERIV_approx prec x (Abs a) vs = False" |
"isDERIV_approx prec x Pi vs = True" |
"isDERIV_approx prec x (Sqrt a) vs =
(isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l | None \<Rightarrow> False))" |
"isDERIV_approx prec x (Exp a) vs = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Ln a) vs =
(isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l | None \<Rightarrow> False))" |
"isDERIV_approx prec x (Power a 0) vs = True" |
"isDERIV_approx prec x (Power a (Suc n)) vs = isDERIV_approx prec x a vs" |
"isDERIV_approx prec x (Num f) vs = True" |
"isDERIV_approx prec x (Var n) vs = True"
lemma isDERIV_approx:
assumes "bounded_by xs vs"
and isDERIV_approx: "isDERIV_approx prec x f vs"
shows "isDERIV x f xs"
using isDERIV_approx
proof (induct f)
case (Inverse a)
then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
and *: "0 < l \<or> u < 0"
by (cases "approx prec a vs") auto
with approx[OF `bounded_by xs vs` approx_Some]
have "interpret_floatarith a xs \<noteq> 0" by auto
thus ?case using Inverse by auto
next
case (Ln a)
then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
and *: "0 < l"
by (cases "approx prec a vs") auto
with approx[OF `bounded_by xs vs` approx_Some]
have "0 < interpret_floatarith a xs" by auto
thus ?case using Ln by auto
next
case (Sqrt a)
then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
and *: "0 < l"
by (cases "approx prec a vs") auto
with approx[OF `bounded_by xs vs` approx_Some]
have "0 < interpret_floatarith a xs" by auto
thus ?case using Sqrt by auto
next
case (Power a n) thus ?case by (cases n) auto
qed auto
lemma bounded_by_update_var:
assumes "bounded_by xs vs" and "vs ! i = Some (l, u)"
and bnd: "x \<in> { real l .. real u }"
shows "bounded_by (xs[i := x]) vs"
proof (cases "i < length xs")
case False
thus ?thesis using `bounded_by xs vs` by auto
next
let ?xs = "xs[i := x]"
case True hence "i < length ?xs" by auto
{
fix j
assume "j < length vs"
have "case vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> ?xs ! j \<in> { real l .. real u }"
proof (cases "vs ! j")
case (Some b)
thus ?thesis
proof (cases "i = j")
case True
thus ?thesis using `vs ! i = Some (l, u)` Some and bnd `i < length ?xs`
by auto
next
case False
thus ?thesis
using `bounded_by xs vs`[THEN bounded_byE, OF `j < length vs`] Some by auto
qed
qed auto
}
thus ?thesis unfolding bounded_by_def by auto
qed
lemma isDERIV_approx':
assumes "bounded_by xs vs"
and vs_x: "vs ! x = Some (l, u)" and X_in: "X \<in> { real l .. real u }"
and approx: "isDERIV_approx prec x f vs"
shows "isDERIV x f (xs[x := X])"
proof -
note bounded_by_update_var[OF `bounded_by xs vs` vs_x X_in] approx
thus ?thesis by (rule isDERIV_approx)
qed
lemma DERIV_approx:
assumes "n < length xs" and bnd: "bounded_by xs vs"
and isD: "isDERIV_approx prec n f vs"
and app: "Some (l, u) = approx prec (DERIV_floatarith n f) vs" (is "_ = approx _ ?D _")
shows "\<exists>(x::real). l \<le> x \<and> x \<le> u \<and>
DERIV (\<lambda> x. interpret_floatarith f (xs[n := x])) (xs!n) :> x"
(is "\<exists> x. _ \<and> _ \<and> DERIV (?i f) _ :> _")
proof (rule exI[of _ "?i ?D (xs!n)"], rule conjI[OF _ conjI])
let "?i f x" = "interpret_floatarith f (xs[n := x])"
from approx[OF bnd app]
show "l \<le> ?i ?D (xs!n)" and "?i ?D (xs!n) \<le> u"
using `n < length xs` by auto
from DERIV_floatarith[OF `n < length xs`, of f "xs!n"] isDERIV_approx[OF bnd isD]
show "DERIV (?i f) (xs!n) :> (?i ?D (xs!n))" by simp
qed
fun lift_bin :: "(float * float) option \<Rightarrow>
(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float) option) \<Rightarrow>
(float * float) option" where
"lift_bin (Some (l1, u1)) (Some (l2, u2)) f = f l1 u1 l2 u2"
| "lift_bin a b f = None"
lemma lift_bin:
assumes lift_bin_Some: "Some (l, u) = lift_bin a b f"
obtains l1 u1 l2 u2
where "a = Some (l1, u1)"
and "b = Some (l2, u2)"
and "f l1 u1 l2 u2 = Some (l, u)"
using assms by (cases a, simp, cases b, simp, auto)
fun approx_tse where
"approx_tse prec n 0 c k f bs = approx prec f bs" |
"approx_tse prec n (Suc s) c k f bs =
(if isDERIV_approx prec n f bs then
lift_bin (approx prec f (bs[n := Some (c,c)]))
(approx_tse prec n s c (Suc k) (DERIV_floatarith n f) bs)
(\<lambda> l1 u1 l2 u2. approx prec
(Add (Var 0)
(Mult (Inverse (Num (Float (int k) 0)))
(Mult (Add (Var (Suc (Suc 0))) (Minus (Num c)))
(Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), bs!n])
else approx prec f bs)"
lemma bounded_by_Cons:
assumes bnd: "bounded_by xs vs"
and x: "x \<in> { real l .. real u }"
shows "bounded_by (x#xs) ((Some (l, u))#vs)"
proof -
{
fix i assume *: "i < length ((Some (l, u))#vs)"
have "case ((Some (l,u))#vs) ! i of Some (l, u) \<Rightarrow> (x#xs)!i \<in> { real l .. real u } | None \<Rightarrow> True"
proof (cases i)
case 0 with x show ?thesis by auto
next
case (Suc i) with * have "i < length vs" by auto
from bnd[THEN bounded_byE, OF this]
show ?thesis unfolding Suc nth_Cons_Suc .
qed
}
thus ?thesis by (auto simp add: bounded_by_def)
qed
lemma approx_tse_generic:
assumes "bounded_by xs vs"
and bnd_c: "bounded_by (xs[x := c]) vs" and "x < length vs" and "x < length xs"
and bnd_x: "vs ! x = Some (lx, ux)"
and ate: "Some (l, u) = approx_tse prec x s c k f vs"
shows "\<exists> n. (\<forall> m < n. \<forall> (z::real) \<in> {lx .. ux}.
DERIV (\<lambda> y. interpret_floatarith ((DERIV_floatarith x ^^ m) f) (xs[x := y])) z :>
(interpret_floatarith ((DERIV_floatarith x ^^ (Suc m)) f) (xs[x := z])))
\<and> (\<forall> (t::real) \<in> {lx .. ux}. (\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) *
interpret_floatarith ((DERIV_floatarith x ^^ i) f) (xs[x := c]) *
(xs!x - c)^i) +
inverse (real (\<Prod> j \<in> {k..<k+n}. j)) *
interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := t]) *
(xs!x - c)^n \<in> {l .. u})" (is "\<exists> n. ?taylor f k l u n")
using ate proof (induct s arbitrary: k f l u)
case 0
{
fix t::real assume "t \<in> {lx .. ux}"
note bounded_by_update_var[OF `bounded_by xs vs` bnd_x this]
from approx[OF this 0[unfolded approx_tse.simps]]
have "(interpret_floatarith f (xs[x := t])) \<in> {l .. u}"
by (auto simp add: algebra_simps)
}
thus ?case by (auto intro!: exI[of _ 0])
next
case (Suc s)
show ?case
proof (cases "isDERIV_approx prec x f vs")
case False
note ap = Suc.prems[unfolded approx_tse.simps if_not_P[OF False]]
{
fix t::real assume "t \<in> {lx .. ux}"
note bounded_by_update_var[OF `bounded_by xs vs` bnd_x this]
from approx[OF this ap]
have "(interpret_floatarith f (xs[x := t])) \<in> {l .. u}"
by (auto simp add: algebra_simps)
}
thus ?thesis by (auto intro!: exI[of _ 0])
next
case True
with Suc.prems
obtain l1 u1 l2 u2
where a: "Some (l1, u1) = approx prec f (vs[x := Some (c,c)])"
and ate: "Some (l2, u2) = approx_tse prec x s c (Suc k) (DERIV_floatarith x f) vs"
and final: "Some (l, u) = approx prec
(Add (Var 0)
(Mult (Inverse (Num (Float (int k) 0)))
(Mult (Add (Var (Suc (Suc 0))) (Minus (Num c)))
(Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), vs!x]"
by (auto elim!: lift_bin)
from bnd_c `x < length xs`
have bnd: "bounded_by (xs[x:=c]) (vs[x:= Some (c,c)])"
by (auto intro!: bounded_by_update)
from approx[OF this a]
have f_c: "interpret_floatarith ((DERIV_floatarith x ^^ 0) f) (xs[x := c]) \<in> { l1 .. u1 }"
(is "?f 0 (real c) \<in> _")
by auto
{
fix f :: "'a \<Rightarrow> 'a" fix n :: nat fix x :: 'a
have "(f ^^ Suc n) x = (f ^^ n) (f x)"
by (induct n) auto
}
note funpow_Suc = this[symmetric]
from Suc.hyps[OF ate, unfolded this]
obtain n
where DERIV_hyp: "\<And> m z. \<lbrakk> m < n ; (z::real) \<in> { lx .. ux } \<rbrakk> \<Longrightarrow> DERIV (?f (Suc m)) z :> ?f (Suc (Suc m)) z"
and hyp: "\<forall> t \<in> {real lx .. real ux}. (\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {Suc k..<Suc k + i}. j)) * ?f (Suc i) c * (xs!x - c)^i) +
inverse (real (\<Prod> j \<in> {Suc k..<Suc k + n}. j)) * ?f (Suc n) t * (xs!x - c)^n \<in> {l2 .. u2}"
(is "\<forall> t \<in> _. ?X (Suc k) f n t \<in> _")
by blast
{
fix m and z::real
assume "m < Suc n" and bnd_z: "z \<in> { lx .. ux }"
have "DERIV (?f m) z :> ?f (Suc m) z"
proof (cases m)
case 0
with DERIV_floatarith[OF `x < length xs` isDERIV_approx'[OF `bounded_by xs vs` bnd_x bnd_z True]]
show ?thesis by simp
next
case (Suc m')
hence "m' < n" using `m < Suc n` by auto
from DERIV_hyp[OF this bnd_z]
show ?thesis using Suc by simp
qed
} note DERIV = this
have "\<And> k i. k < i \<Longrightarrow> {k ..< i} = insert k {Suc k ..< i}" by auto
hence setprod_head_Suc: "\<And> k i. \<Prod> {k ..< k + Suc i} = k * \<Prod> {Suc k ..< Suc k + i}" by auto
have setsum_move0: "\<And> k F. setsum F {0..<Suc k} = F 0 + setsum (\<lambda> k. F (Suc k)) {0..<k}"
unfolding setsum_shift_bounds_Suc_ivl[symmetric]
unfolding setsum_head_upt_Suc[OF zero_less_Suc] ..
def C \<equiv> "xs!x - c"
{
fix t::real assume t: "t \<in> {lx .. ux}"
hence "bounded_by [xs!x] [vs!x]"
using `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`]
by (cases "vs!x", auto simp add: bounded_by_def)
with hyp[THEN bspec, OF t] f_c
have "bounded_by [?f 0 c, ?X (Suc k) f n t, xs!x] [Some (l1, u1), Some (l2, u2), vs!x]"
by (auto intro!: bounded_by_Cons)
from approx[OF this final, unfolded atLeastAtMost_iff[symmetric]]
have "?X (Suc k) f n t * (xs!x - real c) * inverse k + ?f 0 c \<in> {l .. u}"
by (auto simp add: algebra_simps)
also have "?X (Suc k) f n t * (xs!x - real c) * inverse (real k) + ?f 0 c =
(\<Sum> i = 0..<Suc n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) * ?f i c * (xs!x - c)^i) +
inverse (real (\<Prod> j \<in> {k..<k+Suc n}. j)) * ?f (Suc n) t * (xs!x - c)^Suc n" (is "_ = ?T")
unfolding funpow_Suc C_def[symmetric] setsum_move0 setprod_head_Suc
by (auto simp add: algebra_simps)
(simp only: mult.left_commute [of _ "inverse (real k)"] setsum_right_distrib [symmetric])
finally have "?T \<in> {l .. u}" .
}
thus ?thesis using DERIV by blast
qed
qed
lemma setprod_fact: "real (\<Prod> {1..<1 + k}) = fact (k :: nat)"
using fact_altdef_nat Suc_eq_plus1_left atLeastLessThanSuc_atLeastAtMost real_fact_nat
by presburger
lemma approx_tse:
assumes "bounded_by xs vs"
and bnd_x: "vs ! x = Some (lx, ux)" and bnd_c: "real c \<in> {lx .. ux}"
and "x < length vs" and "x < length xs"
and ate: "Some (l, u) = approx_tse prec x s c 1 f vs"
shows "interpret_floatarith f xs \<in> { l .. u }"
proof -
def F \<equiv> "\<lambda> n z. interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := z])"
hence F0: "F 0 = (\<lambda> z. interpret_floatarith f (xs[x := z]))" by auto
hence "bounded_by (xs[x := c]) vs" and "x < length vs" "x < length xs"
using `bounded_by xs vs` bnd_x bnd_c `x < length vs` `x < length xs`
by (auto intro!: bounded_by_update_var)
from approx_tse_generic[OF `bounded_by xs vs` this bnd_x ate]
obtain n
where DERIV: "\<forall> m z. m < n \<and> real lx \<le> z \<and> z \<le> real ux \<longrightarrow> DERIV (F m) z :> F (Suc m) z"
and hyp: "\<And> (t::real). t \<in> {lx .. ux} \<Longrightarrow>
(\<Sum> j = 0..<n. inverse(fact j) * F j c * (xs!x - c)^j) +
inverse ((fact n)) * F n t * (xs!x - c)^n
\<in> {l .. u}" (is "\<And> t. _ \<Longrightarrow> ?taylor t \<in> _")
unfolding F_def atLeastAtMost_iff[symmetric] setprod_fact
by blast
have bnd_xs: "xs ! x \<in> { lx .. ux }"
using `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`] bnd_x by auto
show ?thesis
proof (cases n)
case 0 thus ?thesis using hyp[OF bnd_xs] unfolding F_def by auto
next
case (Suc n')
show ?thesis
proof (cases "xs ! x = c")
case True
from True[symmetric] hyp[OF bnd_xs] Suc show ?thesis
unfolding F_def Suc setsum_head_upt_Suc[OF zero_less_Suc] setsum_shift_bounds_Suc_ivl by auto
next
case False
have "lx \<le> real c" "real c \<le> ux" "lx \<le> xs!x" "xs!x \<le> ux"
using Suc bnd_c `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`] bnd_x by auto
from Taylor.taylor[OF zero_less_Suc, of F, OF F0 DERIV[unfolded Suc] this False]
obtain t::real where t_bnd: "if xs ! x < c then xs ! x < t \<and> t < c else c < t \<and> t < xs ! x"
and fl_eq: "interpret_floatarith f (xs[x := xs ! x]) =
(\<Sum>m = 0..<Suc n'. F m c / (fact m) * (xs ! x - c) ^ m) +
F (Suc n') t / (fact (Suc n')) * (xs ! x - c) ^ Suc n'"
unfolding atLeast0LessThan by blast
from t_bnd bnd_xs bnd_c have *: "t \<in> {lx .. ux}"
by (cases "xs ! x < c", auto)
have "interpret_floatarith f (xs[x := xs ! x]) = ?taylor t"
unfolding fl_eq Suc by (auto simp add: algebra_simps divide_inverse)
also have "\<dots> \<in> {l .. u}" using * by (rule hyp)
finally show ?thesis by simp
qed
qed
qed
fun approx_tse_form' where
"approx_tse_form' prec t f 0 l u cmp =
(case approx_tse prec 0 t ((l + u) * Float 1 (- 1)) 1 f [Some (l, u)]
of Some (l, u) \<Rightarrow> cmp l u | None \<Rightarrow> False)" |
"approx_tse_form' prec t f (Suc s) l u cmp =
(let m = (l + u) * Float 1 (- 1)
in (if approx_tse_form' prec t f s l m cmp then
approx_tse_form' prec t f s m u cmp else False))"
lemma approx_tse_form':
fixes x :: real
assumes "approx_tse_form' prec t f s l u cmp" and "x \<in> {l .. u}"
shows "\<exists> l' u' ly uy. x \<in> { l' .. u' } \<and> real l \<le> l' \<and> u' \<le> real u \<and> cmp ly uy \<and>
approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)"
using assms proof (induct s arbitrary: l u)
case 0
then obtain ly uy
where *: "approx_tse prec 0 t ((l + u) * Float 1 (- 1)) 1 f [Some (l, u)] = Some (ly, uy)"
and **: "cmp ly uy" by (auto elim!: case_optionE)
with 0 show ?case by auto
next
case (Suc s)
let ?m = "(l + u) * Float 1 (- 1)"
from Suc.prems
have l: "approx_tse_form' prec t f s l ?m cmp"
and u: "approx_tse_form' prec t f s ?m u cmp"
by (auto simp add: Let_def lazy_conj)
have m_l: "real l \<le> ?m" and m_u: "?m \<le> real u"
unfolding less_eq_float_def using Suc.prems by auto
with `x \<in> { l .. u }`
have "x \<in> { l .. ?m} \<or> x \<in> { ?m .. u }" by auto
thus ?case
proof (rule disjE)
assume "x \<in> { l .. ?m}"
from Suc.hyps[OF l this]
obtain l' u' ly uy
where "x \<in> { l' .. u' } \<and> real l \<le> l' \<and> real u' \<le> ?m \<and> cmp ly uy \<and>
approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)" by blast
with m_u show ?thesis by (auto intro!: exI)
next
assume "x \<in> { ?m .. u }"
from Suc.hyps[OF u this]
obtain l' u' ly uy
where "x \<in> { l' .. u' } \<and> ?m \<le> real l' \<and> u' \<le> real u \<and> cmp ly uy \<and>
approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)" by blast
with m_u show ?thesis by (auto intro!: exI)
qed
qed
lemma approx_tse_form'_less:
fixes x :: real
assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 < l)"
and x: "x \<in> {l .. u}"
shows "interpret_floatarith b [x] < interpret_floatarith a [x]"
proof -
from approx_tse_form'[OF tse x]
obtain l' u' ly uy
where x': "x \<in> { l' .. u' }" and "l \<le> real l'"
and "real u' \<le> u" and "0 < ly"
and tse: "approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)"
by blast
hence "bounded_by [x] [Some (l', u')]" by (auto simp add: bounded_by_def)
from approx_tse[OF this _ _ _ _ tse[symmetric], of l' u'] x'
have "ly \<le> interpret_floatarith a [x] - interpret_floatarith b [x]"
by auto
from order_less_le_trans[OF _ this, of 0] `0 < ly`
show ?thesis by auto
qed
lemma approx_tse_form'_le:
fixes x :: real
assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 \<le> l)"
and x: "x \<in> {l .. u}"
shows "interpret_floatarith b [x] \<le> interpret_floatarith a [x]"
proof -
from approx_tse_form'[OF tse x]
obtain l' u' ly uy
where x': "x \<in> { l' .. u' }" and "l \<le> real l'"
and "real u' \<le> u" and "0 \<le> ly"
and tse: "approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)"
by blast
hence "bounded_by [x] [Some (l', u')]" by (auto simp add: bounded_by_def)
from approx_tse[OF this _ _ _ _ tse[symmetric], of l' u'] x'
have "ly \<le> interpret_floatarith a [x] - interpret_floatarith b [x]"
by auto
from order_trans[OF _ this, of 0] `0 \<le> ly`
show ?thesis by auto
qed
fun approx_tse_concl where
"approx_tse_concl prec t (Less lf rt) s l u l' u' \<longleftrightarrow>
approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 < l)" |
"approx_tse_concl prec t (LessEqual lf rt) s l u l' u' \<longleftrightarrow>
approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)" |
"approx_tse_concl prec t (AtLeastAtMost x lf rt) s l u l' u' \<longleftrightarrow>
(if approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l) then
approx_tse_form' prec t (Add rt (Minus x)) s l u' (\<lambda> l u. 0 \<le> l) else False)" |
"approx_tse_concl prec t (Conj f g) s l u l' u' \<longleftrightarrow>
approx_tse_concl prec t f s l u l' u' \<and> approx_tse_concl prec t g s l u l' u'" |
"approx_tse_concl prec t (Disj f g) s l u l' u' \<longleftrightarrow>
approx_tse_concl prec t f s l u l' u' \<or> approx_tse_concl prec t g s l u l' u'" |
"approx_tse_concl _ _ _ _ _ _ _ _ \<longleftrightarrow> False"
definition
"approx_tse_form prec t s f =
(case f
of (Bound x a b f) \<Rightarrow> x = Var 0 \<and>
(case (approx prec a [None], approx prec b [None])
of (Some (l, u), Some (l', u')) \<Rightarrow> approx_tse_concl prec t f s l u l' u'
| _ \<Rightarrow> False)
| _ \<Rightarrow> False)"
lemma approx_tse_form:
assumes "approx_tse_form prec t s f"
shows "interpret_form f [x]"
proof (cases f)
case (Bound i a b f') note f_def = this
with assms obtain l u l' u'
where a: "approx prec a [None] = Some (l, u)"
and b: "approx prec b [None] = Some (l', u')"
unfolding approx_tse_form_def by (auto elim!: case_optionE)
from Bound assms have "i = Var 0" unfolding approx_tse_form_def by auto
hence i: "interpret_floatarith i [x] = x" by auto
{ let "?f z" = "interpret_floatarith z [x]"
assume "?f i \<in> { ?f a .. ?f b }"
with approx[OF _ a[symmetric], of "[x]"] approx[OF _ b[symmetric], of "[x]"]
have bnd: "x \<in> { l .. u'}" unfolding bounded_by_def i by auto
have "interpret_form f' [x]"
using assms[unfolded Bound]
proof (induct f')
case (Less lf rt)
with a b
have "approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 < l)"
unfolding approx_tse_form_def by auto
from approx_tse_form'_less[OF this bnd]
show ?case using Less by auto
next
case (LessEqual lf rt)
with Bound a b assms
have "approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)"
unfolding approx_tse_form_def by auto
from approx_tse_form'_le[OF this bnd]
show ?case using LessEqual by auto
next
case (AtLeastAtMost x lf rt)
with Bound a b assms
have "approx_tse_form' prec t (Add rt (Minus x)) s l u' (\<lambda> l u. 0 \<le> l)"
and "approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)"
unfolding approx_tse_form_def lazy_conj by (auto split: split_if_asm)
from approx_tse_form'_le[OF this(1) bnd] approx_tse_form'_le[OF this(2) bnd]
show ?case using AtLeastAtMost by auto
qed (auto simp: f_def approx_tse_form_def elim!: case_optionE)
} thus ?thesis unfolding f_def by auto
qed (insert assms, auto simp add: approx_tse_form_def)
text {* @{term approx_form_eval} is only used for the {\tt value}-command. *}
fun approx_form_eval :: "nat \<Rightarrow> form \<Rightarrow> (float * float) option list \<Rightarrow> (float * float) option list" where
"approx_form_eval prec (Bound (Var n) a b f) bs =
(case (approx prec a bs, approx prec b bs)
of (Some (l, _), Some (_, u)) \<Rightarrow> approx_form_eval prec f (bs[n := Some (l, u)])
| _ \<Rightarrow> bs)" |
"approx_form_eval prec (Assign (Var n) a f) bs =
(case (approx prec a bs)
of (Some (l, u)) \<Rightarrow> approx_form_eval prec f (bs[n := Some (l, u)])
| _ \<Rightarrow> bs)" |
"approx_form_eval prec (Less a b) bs = bs @ [approx prec a bs, approx prec b bs]" |
"approx_form_eval prec (LessEqual a b) bs = bs @ [approx prec a bs, approx prec b bs]" |
"approx_form_eval prec (AtLeastAtMost x a b) bs =
bs @ [approx prec x bs, approx prec a bs, approx prec b bs]" |
"approx_form_eval _ _ bs = bs"
subsection {* Implement proof method \texttt{approximation} *}
lemmas interpret_form_equations = interpret_form.simps interpret_floatarith.simps interpret_floatarith_num
interpret_floatarith_divide interpret_floatarith_diff interpret_floatarith_tan interpret_floatarith_powr interpret_floatarith_log
interpret_floatarith_sin
oracle approximation_oracle = {* fn (thy, t) =>
let
fun bad t = error ("Bad term: " ^ Syntax.string_of_term_global thy t);
fun term_of_bool true = @{term True}
| term_of_bool false = @{term False};
val mk_int = HOLogic.mk_number @{typ int} o @{code integer_of_int};
fun dest_int (@{term int_of_integer} $ j) = @{code int_of_integer} (snd (HOLogic.dest_number j))
| dest_int i = @{code int_of_integer} (snd (HOLogic.dest_number i));
fun term_of_float (@{code Float} (k, l)) =
@{term Float} $ mk_int k $ mk_int l;
fun term_of_float_float_option NONE = @{term "None :: (float \<times> float) option"}
| term_of_float_float_option (SOME ff) = @{term "Some :: float \<times> float \<Rightarrow> _"}
$ HOLogic.mk_prod (apply2 term_of_float ff);
val term_of_float_float_option_list =
HOLogic.mk_list @{typ "(float \<times> float) option"} o map term_of_float_float_option;
fun nat_of_term t = @{code nat_of_integer}
(HOLogic.dest_nat t handle TERM _ => snd (HOLogic.dest_number t));
fun float_of_term (@{term Float} $ k $ l) =
@{code Float} (dest_int k, dest_int l)
| float_of_term t = bad t;
fun floatarith_of_term (@{term Add} $ a $ b) = @{code Add} (floatarith_of_term a, floatarith_of_term b)
| floatarith_of_term (@{term Minus} $ a) = @{code Minus} (floatarith_of_term a)
| floatarith_of_term (@{term Mult} $ a $ b) = @{code Mult} (floatarith_of_term a, floatarith_of_term b)
| floatarith_of_term (@{term Inverse} $ a) = @{code Inverse} (floatarith_of_term a)
| floatarith_of_term (@{term Cos} $ a) = @{code Cos} (floatarith_of_term a)
| floatarith_of_term (@{term Arctan} $ a) = @{code Arctan} (floatarith_of_term a)
| floatarith_of_term (@{term Abs} $ a) = @{code Abs} (floatarith_of_term a)
| floatarith_of_term (@{term Max} $ a $ b) = @{code Max} (floatarith_of_term a, floatarith_of_term b)
| floatarith_of_term (@{term Min} $ a $ b) = @{code Min} (floatarith_of_term a, floatarith_of_term b)
| floatarith_of_term @{term Pi} = @{code Pi}
| floatarith_of_term (@{term Sqrt} $ a) = @{code Sqrt} (floatarith_of_term a)
| floatarith_of_term (@{term Exp} $ a) = @{code Exp} (floatarith_of_term a)
| floatarith_of_term (@{term Ln} $ a) = @{code Ln} (floatarith_of_term a)
| floatarith_of_term (@{term Power} $ a $ n) =
@{code Power} (floatarith_of_term a, nat_of_term n)
| floatarith_of_term (@{term Var} $ n) = @{code Var} (nat_of_term n)
| floatarith_of_term (@{term Num} $ m) = @{code Num} (float_of_term m)
| floatarith_of_term t = bad t;
fun form_of_term (@{term Bound} $ a $ b $ c $ p) = @{code Bound}
(floatarith_of_term a, floatarith_of_term b, floatarith_of_term c, form_of_term p)
| form_of_term (@{term Assign} $ a $ b $ p) = @{code Assign}
(floatarith_of_term a, floatarith_of_term b, form_of_term p)
| form_of_term (@{term Less} $ a $ b) = @{code Less}
(floatarith_of_term a, floatarith_of_term b)
| form_of_term (@{term LessEqual} $ a $ b) = @{code LessEqual}
(floatarith_of_term a, floatarith_of_term b)
| form_of_term (@{term Conj} $ a $ b) = @{code Conj}
(form_of_term a, form_of_term b)
| form_of_term (@{term Disj} $ a $ b) = @{code Disj}
(form_of_term a, form_of_term b)
| form_of_term (@{term AtLeastAtMost} $ a $ b $ c) = @{code AtLeastAtMost}
(floatarith_of_term a, floatarith_of_term b, floatarith_of_term c)
| form_of_term t = bad t;
fun float_float_option_of_term @{term "None :: (float \<times> float) option"} = NONE
| float_float_option_of_term (@{term "Some :: float \<times> float \<Rightarrow> _"} $ ff) =
SOME (apply2 float_of_term (HOLogic.dest_prod ff))
| float_float_option_of_term (@{term approx'} $ n $ a $ ffs) = @{code approx'}
(nat_of_term n) (floatarith_of_term a) (float_float_option_list_of_term ffs)
| float_float_option_of_term t = bad t
and float_float_option_list_of_term
(@{term "replicate :: _ \<Rightarrow> (float \<times> float) option \<Rightarrow> _"} $ n $ @{term "None :: (float \<times> float) option"}) =
@{code replicate} (nat_of_term n) NONE
| float_float_option_list_of_term (@{term approx_form_eval} $ n $ p $ ffs) =
@{code approx_form_eval} (nat_of_term n) (form_of_term p) (float_float_option_list_of_term ffs)
| float_float_option_list_of_term t = map float_float_option_of_term
(HOLogic.dest_list t);
val nat_list_of_term = map nat_of_term o HOLogic.dest_list ;
fun bool_of_term (@{term approx_form} $ n $ p $ ffs $ ms) = @{code approx_form}
(nat_of_term n) (form_of_term p) (float_float_option_list_of_term ffs) (nat_list_of_term ms)
| bool_of_term (@{term approx_tse_form} $ m $ n $ q $ p) =
@{code approx_tse_form} (nat_of_term m) (nat_of_term n) (nat_of_term q) (form_of_term p)
| bool_of_term t = bad t;
fun eval t = case fastype_of t
of @{typ bool} =>
(term_of_bool o bool_of_term) t
| @{typ "(float \<times> float) option"} =>
(term_of_float_float_option o float_float_option_of_term) t
| @{typ "(float \<times> float) option list"} =>
(term_of_float_float_option_list o float_float_option_list_of_term) t
| _ => bad t;
val normalize = eval o Envir.beta_norm o Envir.eta_long [];
in Thm.global_cterm_of thy (Logic.mk_equals (t, normalize t)) end
*}
ML {*
fun reorder_bounds_tac prems i =
let
fun variable_of_bound (Const (@{const_name Trueprop}, _) $
(Const (@{const_name Set.member}, _) $
Free (name, _) $ _)) = name
| variable_of_bound (Const (@{const_name Trueprop}, _) $
(Const (@{const_name HOL.eq}, _) $
Free (name, _) $ _)) = name
| variable_of_bound t = raise TERM ("variable_of_bound", [t])
val variable_bounds
= map (`(variable_of_bound o Thm.prop_of)) prems
fun add_deps (name, bnds)
= Graph.add_deps_acyclic (name,
remove (op =) name (Term.add_free_names (Thm.prop_of bnds) []))
val order = Graph.empty
|> fold Graph.new_node variable_bounds
|> fold add_deps variable_bounds
|> Graph.strong_conn |> map the_single |> rev
|> map_filter (AList.lookup (op =) variable_bounds)
fun prepend_prem th tac
= tac THEN rtac (th RSN (2, @{thm mp})) i
in
fold prepend_prem order all_tac
end
fun approximation_conv ctxt ct =
approximation_oracle (Proof_Context.theory_of ctxt, Thm.term_of ct |> tap (tracing o Syntax.string_of_term ctxt));
fun approximate ctxt t =
approximation_oracle (Proof_Context.theory_of ctxt, t)
|> Thm.prop_of |> Logic.dest_equals |> snd;
(* Should be in HOL.thy ? *)
fun gen_eval_tac conv ctxt = CONVERSION
(Object_Logic.judgment_conv (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt))
THEN' rtac TrueI
val form_equations = @{thms interpret_form_equations};
fun rewrite_interpret_form_tac ctxt prec splitting taylor i st = let
fun lookup_splitting (Free (name, _))
= case AList.lookup (op =) splitting name
of SOME s => HOLogic.mk_number @{typ nat} s
| NONE => @{term "0 :: nat"}
val vs = nth (Thm.prems_of st) (i - 1)
|> Logic.strip_imp_concl
|> HOLogic.dest_Trueprop
|> Term.strip_comb |> snd |> List.last
|> HOLogic.dest_list
val p = prec
|> HOLogic.mk_number @{typ nat}
|> Thm.cterm_of ctxt
in case taylor
of NONE => let
val n = vs |> length
|> HOLogic.mk_number @{typ nat}
|> Thm.cterm_of ctxt
val s = vs
|> map lookup_splitting
|> HOLogic.mk_list @{typ nat}
|> Thm.cterm_of ctxt
in
(rtac (Thm.instantiate ([], [(@{cpat "?n::nat"}, n),
(@{cpat "?prec::nat"}, p),
(@{cpat "?ss::nat list"}, s)])
@{thm "approx_form"}) i
THEN simp_tac (put_simpset (simpset_of @{context}) ctxt) i) st
end
| SOME t =>
if length vs <> 1
then raise (TERM ("More than one variable used for taylor series expansion", [Thm.prop_of st]))
else let
val t = t
|> HOLogic.mk_number @{typ nat}
|> Thm.cterm_of ctxt
val s = vs |> map lookup_splitting |> hd
|> Thm.cterm_of ctxt
in
rtac (Thm.instantiate ([], [(@{cpat "?s::nat"}, s),
(@{cpat "?t::nat"}, t),
(@{cpat "?prec::nat"}, p)])
@{thm "approx_tse_form"}) i st
end
end
val free = Args.context -- Args.term >> (fn (_, Free (n, _)) => n | (ctxt, t) =>
error ("Bad free variable: " ^ Syntax.string_of_term ctxt t));
*}
lemma intervalE: "a \<le> x \<and> x \<le> b \<Longrightarrow> \<lbrakk> x \<in> { a .. b } \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by auto
lemma meta_eqE: "x \<equiv> a \<Longrightarrow> \<lbrakk> x = a \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by auto
method_setup approximation = {*
Scan.lift Parse.nat
--
Scan.optional (Scan.lift (Args.$$$ "splitting" |-- Args.colon)
|-- Parse.and_list' (free --| Scan.lift (Args.$$$ "=") -- Scan.lift Parse.nat)) []
--
Scan.option (Scan.lift (Args.$$$ "taylor" |-- Args.colon)
|-- (free |-- Scan.lift (Args.$$$ "=") |-- Scan.lift Parse.nat))
>>
(fn ((prec, splitting), taylor) => fn ctxt =>
SIMPLE_METHOD' (fn i =>
REPEAT (FIRST' [etac @{thm intervalE},
etac @{thm meta_eqE},
rtac @{thm impI}] i)
THEN Subgoal.FOCUS (fn {prems, ...} => reorder_bounds_tac prems i) ctxt i
THEN DETERM (TRY (filter_prems_tac ctxt (K false) i))
THEN DETERM (Reification.tac ctxt form_equations NONE i)
THEN rewrite_interpret_form_tac ctxt prec splitting taylor i
THEN gen_eval_tac (approximation_conv ctxt) ctxt i))
*} "real number approximation"
ML_file "approximation.ML"
section "Quickcheck Generator"
ML_file "approximation_generator.ML"
setup "Approximation_Generator.setup"
end