(* Author: Johannes Hoelzl, TU Muenchen Coercions removed by Dmitriy Traytel
This file contains only general material about computing lower/upper bounds on real functions. Approximation.thy contains the actual approximation algorithm and the approximation oracle. This is in order to make a clear separation between "morally immaculate" material about upper/lower bounds and the trusted oracle/reflection.
*)
theory Approximation_Bounds imports
Complex_Main "HOL-Library.Interval_Float"
Dense_Linear_Order begin
primrec horner :: "(nat \ nat) \ (nat \ nat \ nat) \ nat \ nat \ nat \ real \ 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 \ real" shows"a 0 - x * (\ i=0.. i=0.. proof - have shift_pow: "\i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)" by auto show ?thesis unfolding sum_distrib_left shift_pow uminus_add_conv_diff [symmetric] sum_negf[symmetric]
sum.atLeast_Suc_lessThan[OF zero_less_Suc]
sum.reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\ n. (-1)^n *a n * x^n"] by auto qed
lemma horner_schema: fixes f :: "nat \ nat" and G :: "nat \ nat \ nat" and F :: "nat \ nat" assumes f_Suc: "\n. f (Suc n) = G ((F ^^ n) s) (f n)" shows"horner F G n ((F ^^ j') s) (f j') x = (\ j = 0..< n. (- 1) ^ j * (1 / (f (j' + j))) * x ^ j)" proof (induct n arbitrary: j') case 0 thenshow ?caseby auto next case (Suc n) show ?caseunfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc] using horner_schema'[of "\ j. 1 / (f (j' + j))"] by auto qed
lemma horner_bounds': fixes lb :: "nat \ nat \ nat \ float \ float" and ub :: "nat \ nat \ nat \ float \float" assumes"0 \ real_of_float x" and f_Suc: "\n. f (Suc n) = G ((F ^^ n) s) (f n)" and lb_0: "\ i k x. lb 0 i k x = 0" and lb_Suc: "\ 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: "\ i k x. ub 0 i k x = 0" and ub_Suc: "\ 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) \ horner F G n ((F ^^ j') s) (f j') x \
horner F G n ((F ^^ j') s) (f j') x \<le> (ub n ((F ^^ j') s) (f j') x)"
(is"?lb n j' \ ?horner n j' \ ?horner n j' \ ?ub n j'") proof (induct n arbitrary: j') case 0 thus ?caseunfolding lb_0 ub_0 horner.simps by auto next case (Suc n) thus ?caseusing lapprox_rat[of prec 1 "f j'"] using rapprox_rat[of 1 "f j'" prec]
Suc[where j'="Suc j'"] \0 \ real_of_float 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\<open>
Here \<^term_type>\<open>f :: nat \<Rightarrow> nat\<close> is the sequence defining the Taylor series, the coefficients are
all alternating and reciprocs. We use\<^term>\<open>G\<close> and \<^term>\<open>F\<close> to describe the computation of \<^term>\<open>f\<close>.
\<close>
lemma horner_bounds: fixes F :: "nat \ nat" and G :: "nat \ nat \ nat" assumes"0 \ real_of_float x" and f_Suc: "\n. f (Suc n) = G ((F ^^ n) s) (f n)" and lb_0: "\ i k x. lb 0 i k x = 0" and lb_Suc: "\ 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: "\ i k x. ub 0 i k x = 0" and ub_Suc: "\ 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) \ (\j=0..
(is"?lb") and"(\j=0.. (ub n ((F ^^ j') s) (f j') x)"
(is"?ub") proof - have"?lb \ ?ub" using horner_bounds'[where lb=lb, OF \0 \ real_of_float x\ f_Suc lb_0 lb_Suc ub_0 ub_Suc] unfolding horner_schema[where f=f, OF f_Suc] by simp thus"?lb"and"?ub"by auto qed
lemma horner_bounds_nonpos: fixes F :: "nat \ nat" and G :: "nat \ nat \ nat" assumes"real_of_float x \ 0" and f_Suc: "\n. f (Suc n) = G ((F ^^ n) s) (f n)" and lb_0: "\ i k x. lb 0 i k x = 0" and lb_Suc: "\ 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: "\ i k x. ub 0 i k x = 0" and ub_Suc: "\ 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) \ (\j=0.. and"(\j=0.. (ub n ((F ^^ j') s) (f j') x)" (is "?ub") proof - have diff_mult_minus: "x - y * z = x + - y * z"for x y z :: float by simp have sum_eq: "(\j=0..
(\<Sum>j = 0..<n. (- 1) ^ j * (1 / (f (j' + j))) * real_of_float (- x) ^ j)" by (auto simp add: field_simps power_mult_distrib[symmetric]) have"0 \ real_of_float (-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="\ n i k x. lb n i k (-x)" and ub="\ 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 \<open>Selectors for next even or odd number\<close>
text\<open>
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>\<open>get_odd\<close> and \<^term>\<open>get_even\<close>. \<close>
definition get_odd :: "nat \ nat" where "get_odd n = (if odd n then n else (Suc n))"
definition get_even :: "nat \ 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: "\ k. Suc k = get_odd n \ odd (Suc k)" by (auto simp: get_odd_def odd_pos intro!: exI[of _ "n - 1"])
lemma get_even_double: "\i. get_even n = 2 * i" using get_even by (blast elim: evenE)
lemma get_odd_double: "\i. get_odd n = 2 * i + 1" using get_odd by (blast elim: oddE)
section "Power function"
definition float_power_bnds :: "nat \ nat \ float \ float \ 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 \<bar>l\<bar> n, if u < 0 then - power_down_fl prec \<bar>u\<bar> n else power_up_fl prec u n)
else if u < 0 then (power_down_fl prec \<bar>u\<bar> n, power_up_fl prec \<bar>l\<bar> n)
else (0, power_up_fl prec (max \<bar>l\<bar> \<bar>u\<bar>) n))"
lemma le_minus_power_downI: "0 \ x \ x ^ n \ - a \ a \ - 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 \ x \ {l .. u} \ (x::real) ^ n \ {l1..u1}" by (auto
simp: float_power_bnds_def max_def real_power_up_fl real_power_down_fl minus_le_iff
split: if_split_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: "\(x::real) l u. (l1, u1) = float_power_bnds prec n l u \ x \ {l .. u} \
l1 \<le> x ^ n \<and> x ^ n \<le> u1" using float_power_bnds by auto
lift_definition power_float_interval :: "nat \ nat \ float interval \ float interval" is"\p n (l, u). float_power_bnds p n l u" using float_power_bnds by (auto simp: bnds_power dest!: float_power_bnds[OF sym])
lemma lower_power_float_interval: "lower (power_float_interval p n x) = fst (float_power_bnds p n (lower x) (upper x))" by transfer auto lemma upper_power_float_interval: "upper (power_float_interval p n x) = snd (float_power_bnds p n (lower x) (upper x))" by transfer auto
lemma power_float_intervalI: "x \\<^sub>r X \ x ^ n \\<^sub>r power_float_interval p n X" using float_power_bnds[OF prod.collapse] by (auto simp: set_of_eq lower_power_float_interval upper_power_float_interval)
lemma power_float_interval_mono: "set_of (power_float_interval prec n A) \<subseteq> set_of (power_float_interval prec n B)" if"set_of A \ set_of B" proof -
define la where"la = real_of_float (lower A)"
define ua where"ua = real_of_float (upper A)"
define lb where"lb = real_of_float (lower B)"
define ub where"ub = real_of_float (upper B)" have ineqs: "lb \ la" "la \ ua" "ua \ ub" "lb \ ub" using that lower_le_upper[of A] lower_le_upper[of B] by (auto simp: la_def ua_def lb_def ub_def set_of_eq) show ?thesis using ineqs by (simp add: set_of_subset_iff float_power_bnds_def max_def
power_down_fl.rep_eq power_up_fl.rep_eq
lower_power_float_interval upper_power_float_interval
la_def[symmetric] ua_def[symmetric] lb_def[symmetric] ub_def[symmetric])
(auto intro!: power_down_mono power_up_mono intro: order_trans[where y=0]) qed
lemma lower_plus_float_interval: "lower (plus_float_interval prec ivl ivl') = float_plus_down prec (lower ivl) (lower ivl')" by transfer auto lemma upper_plus_float_interval: "upper (plus_float_interval prec ivl ivl') = float_plus_up prec (upper ivl) (upper ivl')" by transfer auto
lemma mult_float_interval_ge: "real_interval A + real_interval B \ real_interval (plus_float_interval prec A B)" unfolding less_eq_interval_def by transfer
(auto simp: lower_plus_float_interval upper_plus_float_interval
intro!: order.trans[OF float_plus_down] order.trans[OF _ float_plus_up])
lemma plus_float_interval: "set_of (real_interval A) + set_of (real_interval B) \
set_of (real_interval (plus_float_interval prec A B))" proof - have"set_of (real_interval A) + set_of (real_interval B) \
set_of (real_interval A + real_interval B)" by (simp add: set_of_plus) alsohave"\ \ set_of (real_interval (plus_float_interval prec A B))" using mult_float_interval_ge[of A B prec] by (simp add: set_of_subset_iff') finallyshow ?thesis . qed
lemma plus_float_intervalI: "x + y \\<^sub>r plus_float_interval prec A B" if"x \\<^sub>i real_interval A" "y \\<^sub>i real_interval B" using plus_float_interval[of A B] that by auto
lemma plus_float_interval_mono: "plus_float_interval prec A B \ plus_float_interval prec X Y" if"A \ X" "B \ Y" using that by (auto simp: less_eq_interval_def lower_plus_float_interval upper_plus_float_interval
float_plus_down.rep_eq float_plus_up.rep_eq plus_down_mono plus_up_mono)
lemma lower_mult_float_interval: "lower (mult_float_interval p x y) = fst (bnds_mult p (lower x) (upper x) (lower y) (upper y))" by transfer auto lemma upper_mult_float_interval: "upper (mult_float_interval p x y) = snd (bnds_mult p (lower x) (upper x) (lower y) (upper y))" by transfer auto
lemma mult_float_interval: "set_of (real_interval A) * set_of (real_interval B) \
set_of (real_interval (mult_float_interval prec A B))" proof - let ?bm = "bnds_mult prec (lower A) (upper A) (lower B) (upper B)" show ?thesis using bnds_mult[of "fst ?bm""snd ?bm", simplified, OF refl] by (auto simp: set_of_eq set_times_def upper_mult_float_interval lower_mult_float_interval) qed
lemma mult_float_intervalI: "x * y \\<^sub>r mult_float_interval prec A B" if"x \\<^sub>i real_interval A" "y \\<^sub>i real_interval B" using mult_float_interval[of A B] that by auto
lemma mult_float_interval_mono': "set_of (mult_float_interval prec A B) \ set_of (mult_float_interval prec X Y)" if"set_of A \ set_of X" "set_of B \ set_of Y" using that apply transfer unfolding bnds_mult_def atLeastatMost_subset_iff float_plus_down.rep_eq float_plus_up.rep_eq by (auto simp: float_plus_down.rep_eq float_plus_up.rep_eq mult_float_mono1 mult_float_mono2)
lemma mult_float_interval_mono: "mult_float_interval prec A B \ mult_float_interval prec X Y" if"A \ X" "B \ Y" using mult_float_interval_mono'[of A X B Y prec] that by (simp add: set_of_subset_iff')
lemma map_bnds: assumes"(lf, uf) = map_bnds lb ub prec (l, u)" assumes"mono f" assumes"x \ {real_of_float l..real_of_float u}" assumes"real_of_float (lb prec l) \ f (real_of_float l)" assumes"real_of_float (ub prec u) \ f (real_of_float u)" shows"f x \ {real_of_float lf..real_of_float uf}" proof - from assms have"real_of_float lf = real_of_float (lb prec l)" by (simp add: map_bnds_def) alsohave"real_of_float (lb prec l) \ f (real_of_float l)" by fact alsofrom assms have"\ \ f x" by (intro monoD[OF \<open>mono f\<close>]) auto finallyhave lf: "real_of_float lf \ f x" .
from assms have"f x \ f (real_of_float u)" by (intro monoD[OF \<open>mono f\<close>]) auto alsohave"\ \ real_of_float (ub prec u)" by fact alsofrom assms have"\ = real_of_float uf" by (simp add: map_bnds_def) finallyhave uf: "f x \ real_of_float uf" .
from lf uf show ?thesis by simp qed
section "Square root"
text\<open>
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. \<close>
fun sqrt_iteration :: "nat \ nat \ float \ float" where "sqrt_iteration prec 0 x = Float 1 ((bitlen \mantissa x\ + 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 \ float \ 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 terminationby (relation "measure (\ v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)
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 alsohave"\ = b\<^sup>2 - 2 * b * sqrt x + (sqrt x)\<^sup>2" by algebra alsohave"\ = b\<^sup>2 - 2 * b * sqrt x + x" using assms by simp finallyhave"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_of_float 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 by (auto simp: algebra_split_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) alsohave"\ < 1 * 2 powr (e + nat (bitlen m))" proof (rule mult_strict_right_mono, auto) show"m < 2^nat (bitlen m)" using bitlen_bounds[OF \<open>0 < m\<close>, THEN conjunct2] unfolding of_int_less_iff[of m, symmetric] by auto qed finallyhave"sqrt x < sqrt (2 powr (e + bitlen m))" unfolding int_nat_bl by auto alsohave"\ \ 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 \ ?E mod 2" by auto have"?E mod 2 < 2"by auto from this[THEN zless_imp_add1_zle] have"?E mod 2 \ 0" using False by auto from xt1(5)[OF \<open>0 \<le> ?E mod 2\<close> this] show ?thesis by auto qed hence"sqrt (2 powr (?E mod 2)) < sqrt (2 * 2)" by (intro real_sqrt_less_mono) auto 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 (metis of_int_add) alsohave"\ = 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 alsohave"\ < 2 powr (?E div 2) * 2 powr 1" by (rule mult_strict_left_mono) (auto intro: E_mod_pow) alsohave"\ = 2 powr (?E div 2 + 1)" unfolding add.commute[of _ 1] powr_add[symmetric] by simp finallyshow ?thesis by auto qed finallyshow ?thesis using\<open>0 < m\<close> 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\<open>0 < real_of_float x\<close> by auto alsohave"\ < real_of_float ?b" using Suc . finallyhave"sqrt x < (?b + x / ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ \<open>0 < real_of_float x\<close>] by auto alsohave"\ \ (?b + (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr) alsohave"\ = (Float 1 (- 1)) * (?b + (float_divr prec x ?b))" by simp alsohave"\ \ (Float 1 (- 1)) * (float_plus_up prec ?b (float_divr prec x ?b))" by (auto simp add: algebra_simps float_plus_up_le) finallyshow ?case unfolding sqrt_iteration.simps Let_def distrib_left . qed
lemma sqrt_iteration_lower_bound: assumes"0 < real_of_float x" shows"0 < real_of_float (sqrt_iteration prec n x)" (is"0 < ?sqrt") proof - have"0 < sqrt x"using assms by auto alsohave"\ < ?sqrt" using sqrt_iteration_bound[OF assms] . finallyshow ?thesis . qed
lemma lb_sqrt_lower_bound: assumes"0 \ real_of_float x" shows"0 \ real_of_float (lb_sqrt prec x)" proof (cases "0 < x") case True hence"0 < real_of_float x"and"0 \ x" using\<open>0 \<le> real_of_float x\<close> by auto hence"0 < sqrt_iteration prec prec x" using sqrt_iteration_lower_bound by auto hence"0 \ real_of_float (float_divl prec x (sqrt_iteration prec prec x))" using float_divl_lower_bound[OF \<open>0 \<le> x\<close>] unfolding less_eq_float_def by auto thus ?thesis unfolding lb_sqrt.simps using True by auto next case False with\<open>0 \<le> real_of_float x\<close> have "real_of_float x = 0" by auto thus ?thesis unfolding lb_sqrt.simps by auto qed
lemma bnds_sqrt': "sqrt x \ {(lb_sqrt prec x) .. (ub_sqrt prec x)}" proof - have lb: "lb_sqrt prec x \ sqrt x" if "0 < x" for x :: float proof - from that have"0 < real_of_float x"and"0 \ real_of_float 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)) \
x / (sqrt_iteration prec prec x)" by (rule float_divl) alsohave"\ < x / sqrt x" by (rule divide_strict_left_mono[OF sqrt_ub \<open>0 < real_of_float x\<close>
mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]]) alsohave"\ = sqrt x" unfolding inverse_eq_iff_eq[of _ "sqrt x", symmetric]
sqrt_divide_self_eq[OF \<open>0 \<le> real_of_float x\<close>, symmetric] by auto finallyshow ?thesis unfolding lb_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto qed have ub: "sqrt x \ ub_sqrt prec x" if "0 < x" for x :: float proof - from that have"0 < real_of_float x"by auto hence"0 < sqrt x"by auto hence"sqrt x < sqrt_iteration prec prec x" using sqrt_iteration_bound by auto thenshow ?thesis unfolding ub_sqrt.simps if_P[OF \<open>0 < x\<close>] by auto qed 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: "\(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 \ {lx .. ux}" hence l: "l = lb_sqrt prec lx "and u: "u = ub_sqrt prec ux"by auto
have"sqrt lx \ sqrt x" using x by auto from order_trans[OF _ this] show"l \ sqrt x" unfolding l using bnds_sqrt'[of lx prec] by auto
have"sqrt x \ sqrt ux" using x by auto from order_trans[OF this] show"sqrt x \ u" unfolding u using bnds_sqrt'[of ux prec] by auto qed
lift_definition sqrt_float_interval::"nat \ float interval \ float interval" is"\prec. \(lx, ux). (lb_sqrt prec lx, ub_sqrt prec ux)" using bnds_sqrt' by auto (meson order_trans real_sqrt_le_iff)
lemma lower_float_interval: "lower (sqrt_float_interval prec X) = lb_sqrt prec (lower X)" by transfer auto
lemma upper_float_interval: "upper (sqrt_float_interval prec X) = ub_sqrt prec (upper X)" by transfer auto
lemma sqrt_float_intervalI: "sqrt x \\<^sub>r sqrt_float_interval p X" if "x \\<^sub>r X" using sqrt_float_interval[of X p] that by auto
section "Arcus tangens and \"
subsection "Compute arcus tangens series"
text\<open>
As first step we implement the computation of the arcus tangens series. This is only valid in the range \<^term>\<open>{-1 :: real .. 1}\<close>. This is used to compute \<pi> and then the entire arcus tangens. \<close>
fun ub_arctan_horner :: "nat \ nat \ nat \ float \ float" and lb_arctan_horner :: "nat \ nat \ nat \ float \ 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 \ real_of_float y" "real_of_float y \ 1" and"even n" shows"arctan (sqrt y) \
{(sqrt y * lb_arctan_horner prec n 1 y) .. (sqrt y * ub_arctan_horner prec (Suc n) 1 y)}" proof - let ?c = "\i. (- 1) ^ i * (1 / (i * 2 + (1::nat)) * sqrt y ^ (i * 2 + 1))" let ?S = "\n. \ i=0..
have"0 \ sqrt y" using assms by auto have"sqrt y \ 1" using assms by auto from\<open>even n\<close> obtain m where "2 * m = n" by (blast elim: evenE)
have"arctan (sqrt y) \ { ?S n .. ?S (Suc n) }" proof (cases "sqrt y = 0") case True thenshow ?thesis by simp next case False hence"0 < sqrt y"using\<open>0 \<le> sqrt y\<close> by auto hence prem: "0 < 1 / (0 * 2 + (1::nat)) * sqrt y ^ (0 * 2 + 1)"by auto
have"\ sqrt y \ \ 1" using \0 \ sqrt y\ \sqrt y \ 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 \<open>2 * m = n\<close>] show ?thesis unfolding arctan_series[OF \<open>\<bar> sqrt y \<bar> \<le> 1\<close>] Suc_eq_plus1 atLeast0LessThan . qed note arctan_bounds = this[unfolded atLeastAtMost_iff]
have F: "\n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
note bounds = horner_bounds[where s=1 and f="\i. 2 * i + 1" and j'=0 and lb="\n i k x. lb_arctan_horner prec n k x" and ub="\n i k x. ub_arctan_horner prec n k x",
OF \<open>0 \<le> real_of_float y\<close> F lb_arctan_horner.simps ub_arctan_horner.simps]
have"(sqrt y * lb_arctan_horner prec n 1 y) \ arctan (sqrt y)" proof - have"(sqrt y * lb_arctan_horner prec n 1 y) \ ?S n" using bounds(1) \<open>0 \<le> sqrt y\<close> apply (simp only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric]) apply (simp only: mult.commute[where'a=real] mult.commute[of _ "2::nat"] power_mult) apply (auto intro!: mult_left_mono) done alsohave"\ \ arctan (sqrt y)" using arctan_bounds .. finallyshow ?thesis . qed moreover have"arctan (sqrt y) \ (sqrt y * ub_arctan_horner prec (Suc n) 1 y)" proof - have"arctan (sqrt y) \ ?S (Suc n)" using arctan_bounds .. alsohave"\ \ (sqrt y * ub_arctan_horner prec (Suc n) 1 y)" using bounds(2)[of "Suc n"] \<open>0 \<le> sqrt y\<close> apply (simp only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric]) apply (simp only: mult.commute[where'a=real] mult.commute[of _ "2::nat"] power_mult) apply (auto intro!: mult_left_mono) done finallyshow ?thesis . qed ultimatelyshow ?thesis by auto qed
lemma arctan_0_1_bounds: assumes"0 \ real_of_float y" "real_of_float y \ 1" shows"arctan (sqrt y) \
{(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 \ x" shows"x / (1 + x\<^sup>2) \ arctan x" (is "?l x \ _") proof - have"?l x - arctan x \ ?l 0 - arctan 0" using assms by (intro DERIV_nonpos_imp_nonincreasing[where f="\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 \ x \ y \ arctan y / y \ arctan x / x" by (rule DERIV_nonpos_imp_nonincreasing[where f="\x. arctan x / x"])
(auto intro!: derivative_eq_intros divide_nonpos_nonneg
simp: inverse_eq_divide arctan_lower_bound)
lemma arctan_mult_mono: "0 \ x \ x \ y \ x * arctan y \ 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 \ x" "x \ y" "y * z \ arctan y" shows"x * z \ arctan x" proof (cases "x = 0") case True thenshow ?thesis by simp next case False with assms have"z \ arctan y / y" by (simp add: field_simps) alsohave"\ \ arctan x / x" using assms \x \ 0\ by (auto intro!: arctan_divide_mono) finallyshow ?thesis using assms \<open>x \<noteq> 0\<close> by (simp add: field_simps) qed
lemma arctan_le_mult: assumes"0 < x""x \ y" "arctan x \ x * z" shows"arctan y \ y * z" proof - from assms have"arctan y / y \ arctan x / x" by (auto intro!: arctan_divide_mono) alsohave"\ \ z" using assms by (auto simp: field_simps) finallyshow ?thesis using assms by (simp add: field_simps) qed
{ fix prec n :: nat fix k :: int assume"1 < k"hence"0 \ k" and "0 < k" and "1 \ 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 _ \<open>1 < k\<close>] by auto
have"0 \ real_of_float ?k" by (rule order_trans[OF _ rapprox_rat]) (auto simp add: \0 \ k\) have"real_of_float ?k \ 1" by (auto simp add: \<open>0 < k\<close> \<open>1 \<le> k\<close> less_imp_le
intro!: mult_le_one order_trans[OF _ rapprox_rat] rapprox_rat_le1) have"1 / k \ ?k" using rapprox_rat[where x=1 and y=k] by auto hence"arctan (1 / k) \ arctan ?k" by (rule arctan_monotone') alsohave"\ \ (?k * ub_arctan_horner prec (get_odd n) 1 ?kl)" using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>] by auto finallyhave"arctan (1 / k) \ ?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 \ 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 _ \<open>1 < k\<close>] by auto have"1 / k \ 1" using \1 < k\ by auto have"0 \ real_of_float ?k" using lapprox_rat_nonneg[where x=1 and y=k, OF zero_le_one \0 \ k\] by (auto simp add: \<open>1 div k = 0\<close>) have"0 \ real_of_float (?k * ?k)" by simp have"real_of_float ?k \ 1" using lapprox_rat by (rule order_trans, auto simp add: \1 / k \ 1\) hence"real_of_float (?k * ?k) \ 1" using \0 \ real_of_float ?k\ by (auto intro!: mult_le_one)
have"?k \ 1 / k" using lapprox_rat[where x=1 and y=k] by auto
have"?k * lb_arctan_horner prec (get_even n) 1 ?ku \ arctan ?k" using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?k\<close> \<open>real_of_float ?k \<le> 1\<close>] by auto alsohave"\ \ arctan (1 / k)" using \?k \ 1 / k\ by (rule arctan_monotone') finallyhave"?k * lb_arctan_horner prec (get_even n) 1 ?ku \ arctan (1 / k)" .
} note lb_arctan = this
have"pi \ 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) moreoverhave"lb_pi n \ 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) ultimatelyshow ?thesis by auto qed
lift_definition pi_float_interval::"nat \ float interval" is "\prec. (lb_pi prec, ub_pi prec)" using pi_boundaries by (auto intro: order_trans)
lemma lower_pi_float_interval: "lower (pi_float_interval prec) = lb_pi prec" by transfer auto lemma upper_pi_float_interval: "upper (pi_float_interval prec) = ub_pi prec" by transfer auto lemma pi_float_interval: "pi \ set_of (real_interval (pi_float_interval prec))" using pi_boundaries by (auto simp: set_of_eq lower_pi_float_interval upper_pi_float_interval)
subsection "Compute arcus tangens in the entire domain"
function lb_arctan :: "nat \ float \ float" and ub_arctan :: "nat \ float \ 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))) inif 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))))) inif 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 (\ v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)
show ?thesis proof (cases "x \ Float 1 (- 1)") case True hence"real_of_float x \ 1" by auto show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>] if_P[OF True] using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x \<le> 1\<close>] by (auto intro!: float_round_up_le) next case False hence"0 < real_of_float x"by auto let ?R = "1 + sqrt (1 + real_of_float x * real_of_float 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 \ 1 + real_of_float x * real_of_float x" using sum_power2_ge_zero[of 1 "real_of_float x", unfolded numeral_2_eq_2] by auto hence"0 \ real_of_float (1 + x*x)" by auto
have"lb_sqrt prec ?sxx \ sqrt ?sxx" using bnds_sqrt'[of ?sxx] by auto alsohave"\ \ sqrt (1 + x*x)" by (auto simp: float_plus_down.rep_eq plus_down_def float_round_down.rep_eq truncate_down_le) finallyhave"lb_sqrt prec ?sxx \ sqrt (1 + x*x)" . hence"?fR \ ?R" by (auto simp: float_plus_down.rep_eq plus_down_def truncate_down_le) have"0 < real_of_float ?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 \ (float_divr prec x ?fR)" proof - from divide_left_mono[OF \<open>?fR \<le> ?R\<close> \<open>0 \<le> real_of_float x\<close> mult_pos_pos[OF divisor_gt0 \<open>0 < real_of_float ?fR\<close>]] have"x / ?R \ x / ?fR" . alsohave"\ \ ?DIV" by (rule float_divr) finallyshow ?thesis . qed
show ?thesis proof (cases "x \ Float 1 1") case True show ?thesis proof (cases "?DIV > 1") case True have"pi / 2 \ 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 \<open>\<not> x < 0\<close>]
if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF \<open>x \<le> Float 1 1\<close>] if_P[OF True] . next case False hence"real_of_float ?DIV \ 1" by auto
have"0 \ x / ?R" using\<open>0 \<le> real_of_float x\<close> \<open>0 < ?R\<close> unfolding zero_le_divide_iff by auto hence"0 \ real_of_float ?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 . alsohave"\ \ 2 * arctan (?DIV)" using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono) alsohave"\ \ (Float 1 1 * ?ub_horner ?DIV)" unfolding Float_num using arctan_0_1_bounds_round[OF \<open>0 \<le> real_of_float ?DIV\<close> \<open>real_of_float ?DIV \<le> 1\<close>] by (auto intro!: float_round_up_le) finallyshow ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF \<open>\<not> x < 0\<close>]
if_not_P[OF \<open>\<not> x \<le> Float 1 (- 1)\<close>] if_P[OF \<open>x \<le> Float 1 1\<close>] if_not_P[OF False] . qed next case False hence"2 < real_of_float x"by auto hence"1 \ real_of_float x" by auto hence"0 < real_of_float x"by auto hence"0 < x"by auto
let"?invx" = "float_divl prec 1 x" have"0 \ arctan x" using arctan_monotone'[OF \0 \ real_of_float x\] and arctan_tan[of 0, unfolded tan_zero] by auto
lemma lb_arctan: "arctan (real_of_float x) \ y \ real_of_float (lb_arctan prec x) \ y" and ub_arctan: "y \ arctan x \ y \ ub_arctan prec x" for x::float and y::real using arctan_boundaries[of x prec] by auto
lemma lower_arctan_float_interval: "lower (arctan_float_interval p x) = lb_arctan p (lower x)" by transfer auto lemma upper_arctan_float_interval: "upper (arctan_float_interval p x) = ub_arctan p (upper x)" by transfer auto
lemma arctan_float_intervalI: "arctan x \\<^sub>r arctan_float_interval p X" if "x \\<^sub>r X" using arctan_float_interval[of X p] that by auto
section "Sinus and Cosinus"
subsection "Compute the cosinus and sinus series"
fun ub_sin_cos_aux :: "nat \ nat \ nat \ nat \ float \ float" and lb_sin_cos_aux :: "nat \ nat \ nat \ nat \ float \ 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)) \ (\ i=0.. and"(\ i=0.. (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub") proof - have"0 \ real_of_float (x * x)" by auto let"?f n" = "fact (2 * n) :: nat" have f_eq: "?f (Suc n) = ?f n * ((\i. i + 2) ^^ n) 1 * (((\i. i + 2) ^^ n) 1 + 1)" for n proof - have"\m. ((\i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto thenshow ?thesis by auto qed from horner_bounds[where lb="lb_sin_cos_aux prec"and ub="ub_sin_cos_aux prec"and j'=0,
OF \<open>0 \<le> real_of_float (x * x)\<close> 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_of_float x"]) qed
lemma lb_sin_cos_aux_zero_le_one: "lb_sin_cos_aux prec n i j 0 \ 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 \ 1 \ 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 \ real_of_float x" and "x \ pi / 2" shows"cos x \ {(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_of_float x = 0") case False hence"real_of_float x \ 0" by auto hence"0 < x"and"0 < real_of_float x" using\<open>0 \<le> real_of_float x\<close> by auto have"0 < x * x" using\<open>0 < x\<close> by simp
have morph_to_if_power: "(\ i=0..
(\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * x ^ i)"
(is"?sum = ?ifsum") for x n proof - have"?sum = ?sum + (\ j = 0 ..< n. 0)" by auto alsohave"\ =
(\<Sum> j = 0 ..< n. (- 1) ^ ((2 * j) div 2) / ((fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto alsohave"\ = (\ i = 0 ..< 2 * n. if even i then (- 1) ^ (i div 2) / ((fact i)) * x ^ i else 0)" unfolding sum_split_even_odd atLeast0LessThan .. alsohave"\ = (\ i = 0 ..< 2 * n. (if even i then (- 1) ^ (i div 2) / ((fact i)) else 0) * x ^ i)" by (rule sum.cong) auto finallyshow ?thesis . qed
{ fix n :: nat assume"0 < n" hence"0 < 2 * n"by auto obtain t where"0 < t"and"t < real_of_float x"and
cos_eq: "cos x = (\ i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/((fact i)) else 0) * (real_of_float x) ^ i)
+ (cos (t + 1/2 * (2 * n) * pi) / (fact (2*n))) * (real_of_float x)^(2*n)"
(is"_ = ?SUM + ?rest / ?fact * ?pow") using Maclaurin_cos_expansion2[OF \<open>0 < real_of_float x\<close> \<open>0 < 2 * n\<close>] unfolding cos_coeff_def atLeast0LessThan by auto
have"cos t * (- 1) ^ n = cos t * cos (n * pi) + sin t * sin (n * pi)"by auto alsohave"\ = cos (t + n * pi)" by (simp add: cos_add) alsohave"\ = ?rest" by auto finallyhave"cos t * (- 1) ^ n = ?rest" . moreover have"t \ pi / 2" using \t < real_of_float x\ and \x \ pi / 2\ by auto hence"0 \ cos t" using \0 < t\ and cos_ge_zero by auto ultimatelyhave even: "even n \ 0 \ ?rest" and odd: "odd n \ 0 \ - ?rest " by auto
have"0 < ?fact"by auto have"0 < ?pow"using\<open>0 < real_of_float x\<close> by auto
{ assume"even n" have"(lb_sin_cos_aux prec n 1 1 (x * x)) \ ?SUM" unfolding morph_to_if_power[symmetric] using cos_aux by auto alsohave"\ \ cos x" proof - from even[OF \<open>even n\<close>] \<open>0 < ?fact\<close> \<open>0 < ?pow\<close> have"0 \ (?rest / ?fact) * ?pow" by simp thus ?thesis unfolding cos_eq by auto qed finallyhave"(lb_sin_cos_aux prec n 1 1 (x * x)) \ cos x" .
} note lb = this
{ assume"odd n" have"cos x \ ?SUM" proof - from\<open>0 < ?fact\<close> and \<open>0 < ?pow\<close> and odd[OF \<open>odd n\<close>] have"0 \ (- ?rest) / ?fact * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le) thus ?thesis unfolding cos_eq by auto qed alsohave"\ \ (ub_sin_cos_aux prec n 1 1 (x * x))" unfolding morph_to_if_power[symmetric] using cos_aux by auto finallyhave"cos x \ (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 \ (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] . moreoverhave"(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \ 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) \ x" by (rule order_trans[OF _ \<open>0 < real_of_float x\<close>[THEN less_imp_le]]) auto with\<open>x \<le> pi / 2\<close> show ?thesis unfolding\<open>get_even n = 0\<close> lb_sin_cos_aux.simps minus_float.rep_eq zero_float.rep_eq using cos_ge_zero by auto qed ultimatelyshow ?thesis by auto next case True hence"x = 0" by (simp add: real_of_float_eq) thus ?thesis using lb_sin_cos_aux_zero_le_one one_le_ub_sin_cos_aux by simp qed
lemma sin_aux: assumes"0 \ real_of_float x" shows"(x * lb_sin_cos_aux prec n 2 1 (x * x)) \
(\<Sum> i=0..<n. (- 1) ^ i * (1/(fact (2 * i + 1))) * x^(2 * i + 1))" (is "?lb") and"(\ i=0..
(x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub") proof - have"0 \ real_of_float (x * x)" by auto let"?f n" = "fact (2 * n + 1) :: nat" have f_eq: "?f (Suc n) = ?f n * ((\i. i + 2) ^^ n) 2 * (((\i. i + 2) ^^ n) 2 + 1)" for n proof - have F: "\m. ((\i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto show ?thesis unfolding F by auto qed from horner_bounds[where lb="lb_sin_cos_aux prec"and ub="ub_sin_cos_aux prec"and j'=0,
OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps] show"?lb"and"?ub"using\<open>0 \<le> real_of_float x\<close> apply (simp_all only: power_add power_one_right mult.assoc[symmetric] sum_distrib_right[symmetric]) apply (simp_all only: mult.commute[where'a=real] of_nat_fact) apply (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real_of_float x"]) done qed
lemma sin_boundaries: assumes"0 \ real_of_float x" and"x \ pi / 2" shows"sin x \ {(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_of_float x = 0") case False hence"real_of_float x \ 0" by auto hence"0 < x"and"0 < real_of_float x" using\<open>0 \<le> real_of_float x\<close> by auto have"0 < x * x" using\<open>0 < x\<close> by simp
have sum_morph: "(\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 = _") for x :: real and n proof - have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto have"?SUM = (\ j = 0 ..< n. 0) + ?SUM" by auto alsohave"\ = (\ 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 .. alsohave"\ = (\ i = 0 ..< 2 * n. (if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / ((fact i))) * x ^ i)" by (rule sum.cong) auto
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