
haftmann@58834:  (* Author:     Johannes Hoelzl, TU Muenchen
hoelzl@40881:    Coercions removed by Dmitriy Traytel *)
wenzelm@30122: 
wenzelm@58889: section {* Prove Real Valued Inequalities by Computation *}
wenzelm@30122: 
wenzelm@40892: theory Approximation
wenzelm@41413: imports
wenzelm@41413:   Complex_Main
wenzelm@41413:   "~~/src/HOL/Library/Float"
wenzelm@51544:   Dense_Linear_Order
haftmann@51143:   "~~/src/HOL/Library/Code_Target_Numeral"
haftmann@56923: keywords "approximate" :: diag
hoelzl@29805: begin
hoelzl@29805: 
haftmann@54489: declare powr_one [simp]
haftmann@54489: declare powr_numeral [simp]
haftmann@54489: declare powr_neg_one [simp]
haftmann@54489: declare powr_neg_numeral [simp]
hoelzl@47600: 
hoelzl@29805: section "Horner Scheme"
hoelzl@29805: 
hoelzl@29805: subsection {* Define auxiliary helper @{text horner} function *}
hoelzl@29805: 
hoelzl@31098: primrec horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where
hoelzl@29805: "horner F G 0 i k x       = 0" |
hoelzl@40881: "horner F G (Suc n) i k x = 1 / k - x * horner F G n (F i) (G i k) x"
hoelzl@29805: 
wenzelm@49351: lemma horner_schema':
wenzelm@49351:   fixes x :: real and a :: "nat \<Rightarrow> real"
hoelzl@29805:   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)"
hoelzl@29805: proof -
wenzelm@49351:   have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)"
wenzelm@49351:     by auto
wenzelm@49351:   show ?thesis
haftmann@54230:     unfolding setsum_right_distrib shift_pow uminus_add_conv_diff [symmetric] setsum_negf[symmetric]
wenzelm@49351:     setsum_head_upt_Suc[OF zero_less_Suc]
haftmann@57418:     setsum.reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n  *a n * x^n"] by auto
hoelzl@29805: qed
hoelzl@29805: 
wenzelm@49351: lemma horner_schema:
wenzelm@49351:   fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat"
haftmann@30971:   assumes f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
haftmann@58410:   shows "horner F G n ((F ^^ j') s) (f j') x = (\<Sum> j = 0..< n. (- 1) ^ j * (1 / (f (j' + j))) * x ^ j)"
wenzelm@45129: proof (induct n arbitrary: j')
wenzelm@45129:   case 0
wenzelm@45129:   then show ?case by auto
wenzelm@45129: next
hoelzl@29805:   case (Suc n)
hoelzl@29805:   show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc]
hoelzl@40881:     using horner_schema'[of "\<lambda> j. 1 / (f (j' + j))"] by auto
wenzelm@45129: qed
hoelzl@29805: 
hoelzl@29805: lemma horner_bounds':
hoelzl@40881:   fixes lb :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" and ub :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
hoelzl@31098:   assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
wenzelm@49351:     and lb_0: "\<And> i k x. lb 0 i k x = 0"
wenzelm@49351:     and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 k - x * (ub n (F i) (G i k) x)"
wenzelm@49351:     and ub_0: "\<And> i k x. ub 0 i k x = 0"
wenzelm@49351:     and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 k - x * (lb n (F i) (G i k) x)"
hoelzl@40881:   shows "(lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') x \<and>
hoelzl@40881:          horner F G n ((F ^^ j') s) (f j') x \<le> (ub n ((F ^^ j') s) (f j') x)"
hoelzl@29805:   (is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'")
hoelzl@29805: proof (induct n arbitrary: j')
wenzelm@49351:   case 0
wenzelm@49351:   thus ?case unfolding lb_0 ub_0 horner.simps by auto
hoelzl@29805: next
hoelzl@29805:   case (Suc n)
hoelzl@47599:   thus ?case using lapprox_rat[of prec 1 "f j'"] using rapprox_rat[of 1 "f j'" prec]
hoelzl@47599:     Suc[where j'="Suc j'"] `0 \<le> real x`
hoelzl@47599:     by (auto intro!: add_mono mult_left_mono simp add: lb_Suc ub_Suc field_simps f_Suc)
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: subsection "Theorems for floating point functions implementing the horner scheme"
hoelzl@29805: 
hoelzl@29805: text {*
hoelzl@29805: 
hoelzl@29805: Here @{term_type "f :: nat \<Rightarrow> nat"} is the sequence defining the Taylor series, the coefficients are
hoelzl@29805: all alternating and reciprocs. We use @{term G} and @{term F} to describe the computation of @{term f}.
hoelzl@29805: 
hoelzl@29805: *}
hoelzl@29805: 
wenzelm@49351: lemma horner_bounds:
wenzelm@49351:   fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
hoelzl@31098:   assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
wenzelm@49351:     and lb_0: "\<And> i k x. lb 0 i k x = 0"
wenzelm@49351:     and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 k - x * (ub n (F i) (G i k) x)"
wenzelm@49351:     and ub_0: "\<And> i k x. ub 0 i k x = 0"
wenzelm@49351:     and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 k - x * (lb n (F i) (G i k) x)"
haftmann@58410:   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
haftmann@58410:     "(\<Sum>j=0..<n. (- 1) ^ j * (1 / (f (j' + j))) * (x ^ j)) \<le> (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
hoelzl@29805: proof -
hoelzl@31809:   have "?lb  \<and> ?ub"
hoelzl@31098:     using horner_bounds'[where lb=lb, OF `0 \<le> real x` f_Suc lb_0 lb_Suc ub_0 ub_Suc]
hoelzl@29805:     unfolding horner_schema[where f=f, OF f_Suc] .
hoelzl@29805:   thus "?lb" and "?ub" by auto
hoelzl@29805: qed
hoelzl@29805: 
wenzelm@49351: lemma horner_bounds_nonpos:
wenzelm@49351:   fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
hoelzl@31098:   assumes "real x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
wenzelm@49351:     and lb_0: "\<And> i k x. lb 0 i k x = 0"
wenzelm@49351:     and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 k + x * (ub n (F i) (G i k) x)"
wenzelm@49351:     and ub_0: "\<And> i k x. ub 0 i k x = 0"
wenzelm@49351:     and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 k + x * (lb n (F i) (G i k) x)"
hoelzl@40881:   shows "(lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / (f (j' + j))) * real x ^ j)" (is "?lb") and
hoelzl@40881:     "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real x ^ j) \<le> (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
hoelzl@29805: proof -
hoelzl@47599:   { fix x y z :: float have "x - y * z = x + - y * z" by simp } note diff_mult_minus = this
hoelzl@40881:   have sum_eq: "(\<Sum>j=0..<n. (1 / (f (j' + j))) * real x ^ j) =
haftmann@58410:     (\<Sum>j = 0..<n. (- 1) ^ j * (1 / (f (j' + j))) * real (- x) ^ j)"
hoelzl@47599:     by (auto simp add: field_simps power_mult_distrib[symmetric])
hoelzl@31098:   have "0 \<le> real (-x)" using assms by auto
hoelzl@29805:   from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec
hoelzl@29805:     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,
hoelzl@29805:     OF this f_Suc lb_0 refl ub_0 refl]
hoelzl@29805:   show "?lb" and "?ub" unfolding minus_minus sum_eq
hoelzl@29805:     by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: subsection {* Selectors for next even or odd number *}
hoelzl@29805: 
hoelzl@29805: text {*
hoelzl@29805: 
hoelzl@29805: The horner scheme computes alternating series. To get the upper and lower bounds we need to
hoelzl@29805: guarantee to access a even or odd member. To do this we use @{term get_odd} and @{term get_even}.
hoelzl@29805: 
hoelzl@29805: *}
hoelzl@29805: 
hoelzl@29805: definition get_odd :: "nat \<Rightarrow> nat" where
hoelzl@29805:   "get_odd n = (if odd n then n else (Suc n))"
hoelzl@29805: 
hoelzl@29805: definition get_even :: "nat \<Rightarrow> nat" where
hoelzl@29805:   "get_even n = (if even n then n else (Suc n))"
hoelzl@29805: 
hoelzl@29805: lemma get_odd[simp]: "odd (get_odd n)" unfolding get_odd_def by (cases "odd n", auto)
hoelzl@29805: lemma get_even[simp]: "even (get_even n)" unfolding get_even_def by (cases "even n", auto)
hoelzl@29805: lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)"
hoelzl@54269:   by (auto simp: get_odd_def odd_pos intro!: exI[of _ "n - 1"])
hoelzl@29805: 
haftmann@58709: lemma get_even_double:
haftmann@58709:   "\<exists>i. get_even n = 2 * i" using get_even by (blast elim: evenE)
haftmann@58709: 
haftmann@58709: lemma get_odd_double:
haftmann@58709:   "\<exists>i. get_odd n = 2 * i + 1" using get_odd by (blast elim: oddE)
hoelzl@29805: 
hoelzl@29805: section "Power function"
hoelzl@29805: 
hoelzl@29805: definition float_power_bnds :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
hoelzl@29805: "float_power_bnds n l u = (if odd n \<or> 0 < l then (l ^ n, u ^ n)
hoelzl@29805:                       else if u < 0         then (u ^ n, l ^ n)
hoelzl@29805:                                             else (0, (max (-l) u) ^ n))"
hoelzl@29805: 
hoelzl@54269: lemma float_power_bnds: "(l1, u1) = float_power_bnds n l u \<Longrightarrow> x \<in> {l .. u} \<Longrightarrow> (x::real) ^ n \<in> {l1..u1}"
hoelzl@54269:   by (auto simp: float_power_bnds_def max_def split: split_if_asm
hoelzl@54269:            intro: power_mono_odd power_mono power_mono_even zero_le_even_power)
hoelzl@29805: 
hoelzl@40881: lemma bnds_power: "\<forall> (x::real) l u. (l1, u1) = float_power_bnds n l u \<and> x \<in> {l .. u} \<longrightarrow> l1 \<le> x ^ n \<and> x ^ n \<le> u1"
hoelzl@29805:   using float_power_bnds by auto
hoelzl@29805: 
hoelzl@29805: section "Square root"
hoelzl@29805: 
hoelzl@29805: text {*
hoelzl@29805: 
hoelzl@29805: The square root computation is implemented as newton iteration. As first first step we use the
hoelzl@29805: nearest power of two greater than the square root.
hoelzl@29805: 
hoelzl@29805: *}
hoelzl@29805: 
hoelzl@29805: fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@47599: "sqrt_iteration prec 0 x = Float 1 ((bitlen \<bar>mantissa x\<bar> + exponent x) div 2 + 1)" |
hoelzl@31467: "sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x
haftmann@58410:                                   in Float 1 (- 1) * (y + float_divr prec x y))"
hoelzl@29805: 
hoelzl@47599: lemma compute_sqrt_iteration_base[code]:
hoelzl@47599:   shows "sqrt_iteration prec n (Float m e) =
hoelzl@47599:     (if n = 0 then Float 1 ((if m = 0 then 0 else bitlen \<bar>m\<bar> + e) div 2 + 1)
hoelzl@47599:     else (let y = sqrt_iteration prec (n - 1) (Float m e) in
haftmann@58410:       Float 1 (- 1) * (y + float_divr prec (Float m e) y)))"
hoelzl@47599:   using bitlen_Float by (cases n) simp_all
hoelzl@47599: 
hoelzl@31467: function ub_sqrt lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@31467: "ub_sqrt prec x = (if 0 < x then (sqrt_iteration prec prec x)
hoelzl@31467:               else if x < 0 then - lb_sqrt prec (- x)
hoelzl@31467:                             else 0)" |
hoelzl@31467: "lb_sqrt prec x = (if 0 < x then (float_divl prec x (sqrt_iteration prec prec x))
hoelzl@31467:               else if x < 0 then - ub_sqrt prec (- x)
hoelzl@31467:                             else 0)"
hoelzl@31467: by pat_completeness auto
blanchet@55414: termination by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)
hoelzl@29805: 
hoelzl@31467: declare lb_sqrt.simps[simp del]
hoelzl@31467: declare ub_sqrt.simps[simp del]
hoelzl@29805: 
hoelzl@29805: lemma sqrt_ub_pos_pos_1:
hoelzl@29805:   assumes "sqrt x < b" and "0 < b" and "0 < x"
hoelzl@29805:   shows "sqrt x < (b + x / b)/2"
hoelzl@29805: proof -
wenzelm@53077:   from assms have "0 < (b - sqrt x)\<^sup>2 " by simp
wenzelm@53077:   also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + (sqrt x)\<^sup>2" by algebra
wenzelm@53077:   also have "\<dots> = b\<^sup>2 - 2 * b * sqrt x + x" using assms by simp
wenzelm@53077:   finally have "0 < b\<^sup>2 - 2 * b * sqrt x + x" .
hoelzl@29805:   hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms
hoelzl@29805:     by (simp add: field_simps power2_eq_square)
hoelzl@29805:   thus ?thesis by (simp add: field_simps)
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@31098: lemma sqrt_iteration_bound: assumes "0 < real x"
hoelzl@54269:   shows "sqrt x < sqrt_iteration prec n x"
hoelzl@29805: proof (induct n)
hoelzl@29805:   case 0
hoelzl@29805:   show ?case
hoelzl@29805:   proof (cases x)
hoelzl@29805:     case (Float m e)
hoelzl@47599:     hence "0 < m" using assms powr_gt_zero[of 2 e] by (auto simp: sign_simps)
hoelzl@40881:     hence "0 < sqrt m" by auto
hoelzl@40881: 
hoelzl@47599:     have int_nat_bl: "(nat (bitlen m)) = bitlen m" using bitlen_nonneg by auto
hoelzl@47599: 
hoelzl@47599:     have "x = (m / 2^nat (bitlen m)) * 2 powr (e + (nat (bitlen m)))"
hoelzl@47599:       unfolding Float by (auto simp: powr_realpow[symmetric] field_simps powr_add)
hoelzl@47599:     also have "\<dots> < 1 * 2 powr (e + nat (bitlen m))"
hoelzl@29805:     proof (rule mult_strict_right_mono, auto)
hoelzl@31467:       show "real m < 2^nat (bitlen m)" using bitlen_bounds[OF `0 < m`, THEN conjunct2]
wenzelm@32960:         unfolding real_of_int_less_iff[of m, symmetric] by auto
hoelzl@29805:     qed
hoelzl@47599:     finally have "sqrt x < sqrt (2 powr (e + bitlen m))" unfolding int_nat_bl by auto
hoelzl@47599:     also have "\<dots> \<le> 2 powr ((e + bitlen m) div 2 + 1)"
hoelzl@29805:     proof -
hoelzl@29805:       let ?E = "e + bitlen m"
hoelzl@47599:       have E_mod_pow: "2 powr (?E mod 2) < 4"
hoelzl@29805:       proof (cases "?E mod 2 = 1")
wenzelm@32960:         case True thus ?thesis by auto
hoelzl@29805:       next
wenzelm@32960:         case False
wenzelm@32960:         have "0 \<le> ?E mod 2" by auto
wenzelm@32960:         have "?E mod 2 < 2" by auto
wenzelm@32960:         from this[THEN zless_imp_add1_zle]
wenzelm@32960:         have "?E mod 2 \<le> 0" using False by auto
wenzelm@32960:         from xt1(5)[OF `0 \<le> ?E mod 2` this]
wenzelm@32960:         show ?thesis by auto
hoelzl@29805:       qed
hoelzl@56889:       hence "sqrt (2 powr (?E mod 2)) < sqrt (2 * 2)"
hoelzl@56889:         by (auto simp del: real_sqrt_four)
hoelzl@56889:       hence E_mod_pow: "sqrt (2 powr (?E mod 2)) < 2" by auto
hoelzl@47599: 
hoelzl@47599:       have E_eq: "2 powr ?E = 2 powr (?E div 2 + ?E div 2 + ?E mod 2)" by auto
hoelzl@47599:       have "sqrt (2 powr ?E) = sqrt (2 powr (?E div 2) * 2 powr (?E div 2) * 2 powr (?E mod 2))"
hoelzl@47599:         unfolding E_eq unfolding powr_add[symmetric] by (simp add: int_of_reals del: real_of_ints)
hoelzl@47599:       also have "\<dots> = 2 powr (?E div 2) * sqrt (2 powr (?E mod 2))"
hoelzl@47599:         unfolding real_sqrt_mult[of _ "2 powr (?E mod 2)"] real_sqrt_abs2 by auto
hoelzl@47599:       also have "\<dots> < 2 powr (?E div 2) * 2 powr 1"
wenzelm@32960:         by (rule mult_strict_left_mono, auto intro: E_mod_pow)
haftmann@57512:       also have "\<dots> = 2 powr (?E div 2 + 1)" unfolding add.commute[of _ 1] powr_add[symmetric]
hoelzl@47599:         by simp
hoelzl@29805:       finally show ?thesis by auto
hoelzl@29805:     qed
hoelzl@47599:     finally show ?thesis using `0 < m`
hoelzl@47599:       unfolding Float
hoelzl@47600:       by (subst compute_sqrt_iteration_base) (simp add: ac_simps)
hoelzl@29805:   qed
hoelzl@29805: next
hoelzl@29805:   case (Suc n)
hoelzl@29805:   let ?b = "sqrt_iteration prec n x"
hoelzl@40881:   have "0 < sqrt x" using `0 < real x` by auto
hoelzl@31098:   also have "\<dots> < real ?b" using Suc .
hoelzl@40881:   finally have "sqrt x < (?b + x / ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ `0 < real x`] by auto
hoelzl@40881:   also have "\<dots> \<le> (?b + (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr)
haftmann@58410:   also have "\<dots> = (Float 1 (- 1)) * (?b + (float_divr prec x ?b))" by simp
webertj@49962:   finally show ?case unfolding sqrt_iteration.simps Let_def distrib_left .
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@31098: lemma sqrt_iteration_lower_bound: assumes "0 < real x"
hoelzl@31098:   shows "0 < real (sqrt_iteration prec n x)" (is "0 < ?sqrt")
hoelzl@29805: proof -
hoelzl@40881:   have "0 < sqrt x" using assms by auto
hoelzl@29805:   also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
hoelzl@29805:   finally show ?thesis .
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@31098: lemma lb_sqrt_lower_bound: assumes "0 \<le> real x"
hoelzl@31467:   shows "0 \<le> real (lb_sqrt prec x)"
hoelzl@29805: proof (cases "0 < x")
hoelzl@47600:   case True hence "0 < real x" and "0 \<le> x" using `0 \<le> real x` by auto
hoelzl@47600:   hence "0 < sqrt_iteration prec prec x" using sqrt_iteration_lower_bound by auto
hoelzl@47599:   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
hoelzl@31467:   thus ?thesis unfolding lb_sqrt.simps using True by auto
hoelzl@29805: next
hoelzl@47600:   case False with `0 \<le> real x` have "real x = 0" by auto
hoelzl@47600:   thus ?thesis unfolding lb_sqrt.simps by auto
hoelzl@29805: qed
hoelzl@29805: 
wenzelm@49351: lemma bnds_sqrt': "sqrt x \<in> {(lb_sqrt prec x) .. (ub_sqrt prec x)}"
hoelzl@31467: proof -
hoelzl@31467:   { fix x :: float assume "0 < x"
hoelzl@47600:     hence "0 < real x" and "0 \<le> real x" by auto
hoelzl@40881:     hence sqrt_gt0: "0 < sqrt x" by auto
hoelzl@40881:     hence sqrt_ub: "sqrt x < sqrt_iteration prec prec x" using sqrt_iteration_bound by auto
hoelzl@40881: 
hoelzl@40881:     have "(float_divl prec x (sqrt_iteration prec prec x)) \<le>
hoelzl@40881:           x / (sqrt_iteration prec prec x)" by (rule float_divl)
hoelzl@40881:     also have "\<dots> < x / sqrt x"
hoelzl@31467:       by (rule divide_strict_left_mono[OF sqrt_ub `0 < real x`
hoelzl@31467:                mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
hoelzl@40881:     also have "\<dots> = sqrt x"
hoelzl@40881:       unfolding inverse_eq_iff_eq[of _ "sqrt x", symmetric]
wenzelm@32960:                 sqrt_divide_self_eq[OF `0 \<le> real x`, symmetric] by auto
hoelzl@40881:     finally have "lb_sqrt prec x \<le> sqrt x"
hoelzl@31467:       unfolding lb_sqrt.simps if_P[OF `0 < x`] by auto }
hoelzl@31467:   note lb = this
hoelzl@31467: 
hoelzl@31467:   { fix x :: float assume "0 < x"
hoelzl@47600:     hence "0 < real x" by auto
hoelzl@40881:     hence "0 < sqrt x" by auto
hoelzl@40881:     hence "sqrt x < sqrt_iteration prec prec x"
hoelzl@31467:       using sqrt_iteration_bound by auto
hoelzl@40881:     hence "sqrt x \<le> ub_sqrt prec x"
hoelzl@31467:       unfolding ub_sqrt.simps if_P[OF `0 < x`] by auto }
hoelzl@31467:   note ub = this
hoelzl@31467: 
hoelzl@31467:   show ?thesis
hoelzl@54269:     using lb[of "-x"] ub[of "-x"] lb[of x] ub[of x]
hoelzl@54269:     by (auto simp add: lb_sqrt.simps ub_sqrt.simps real_sqrt_minus)
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@40881: 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"
hoelzl@31467: proof ((rule allI) +, rule impI, erule conjE, rule conjI)
hoelzl@40881:   fix x :: real fix lx ux
hoelzl@31467:   assume "(l, u) = (lb_sqrt prec lx, ub_sqrt prec ux)"
hoelzl@40881:     and x: "x \<in> {lx .. ux}"
hoelzl@31467:   hence l: "l = lb_sqrt prec lx " and u: "u = ub_sqrt prec ux" by auto
hoelzl@29805: 
hoelzl@40881:   have "sqrt lx \<le> sqrt x" using x by auto
hoelzl@31467:   from order_trans[OF _ this]
hoelzl@40881:   show "l \<le> sqrt x" unfolding l using bnds_sqrt'[of lx prec] by auto
hoelzl@40881: 
hoelzl@40881:   have "sqrt x \<le> sqrt ux" using x by auto
hoelzl@31467:   from order_trans[OF this]
hoelzl@40881:   show "sqrt x \<le> u" unfolding u using bnds_sqrt'[of ux prec] by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: section "Arcus tangens and \<pi>"
hoelzl@29805: 
hoelzl@29805: subsection "Compute arcus tangens series"
hoelzl@29805: 
hoelzl@29805: text {*
hoelzl@29805: 
hoelzl@29805: As first step we implement the computation of the arcus tangens series. This is only valid in the range
hoelzl@29805: @{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens.
hoelzl@29805: 
hoelzl@29805: *}
hoelzl@29805: 
hoelzl@29805: fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
hoelzl@29805: and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805:   "ub_arctan_horner prec 0 k x = 0"
hoelzl@31809: | "ub_arctan_horner prec (Suc n) k x =
hoelzl@40881:     (rapprox_rat prec 1 k) - x * (lb_arctan_horner prec n (k + 2) x)"
hoelzl@29805: | "lb_arctan_horner prec 0 k x = 0"
hoelzl@31809: | "lb_arctan_horner prec (Suc n) k x =
hoelzl@40881:     (lapprox_rat prec 1 k) - x * (ub_arctan_horner prec n (k + 2) x)"
hoelzl@29805: 
wenzelm@49351: lemma arctan_0_1_bounds':
wenzelm@49351:   assumes "0 \<le> real x" "real x \<le> 1" and "even n"
hoelzl@40881:   shows "arctan x \<in> {(x * lb_arctan_horner prec n 1 (x * x)) .. (x * ub_arctan_horner prec (Suc n) 1 (x * x))}"
hoelzl@29805: proof -
haftmann@58410:   let ?c = "\<lambda>i. (- 1) ^ i * (1 / (i * 2 + (1::nat)) * real x ^ (i * 2 + 1))"
hoelzl@54269:   let ?S = "\<lambda>n. \<Sum> i=0..<n. ?c i"
hoelzl@29805: 
hoelzl@31098:   have "0 \<le> real (x * x)" by auto
haftmann@58709:   from `even n` obtain m where "2 * m = n" by (blast elim: evenE)
hoelzl@31809: 
hoelzl@40881:   have "arctan x \<in> { ?S n .. ?S (Suc n) }"
hoelzl@31098:   proof (cases "real x = 0")
hoelzl@29805:     case False
hoelzl@31098:     hence "0 < real x" using `0 \<le> real x` by auto
hoelzl@40881:     hence prem: "0 < 1 / (0 * 2 + (1::nat)) * real x ^ (0 * 2 + 1)" by auto
hoelzl@29805: 
hoelzl@31098:     have "\<bar> real x \<bar> \<le> 1"  using `0 \<le> real x` `real x \<le> 1` by auto
hoelzl@29805:     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`]
huffman@56195:     show ?thesis unfolding arctan_series[OF `\<bar> real x \<bar> \<le> 1`] Suc_eq_plus1 atLeast0LessThan .
hoelzl@29805:   qed auto
hoelzl@29805:   note arctan_bounds = this[unfolded atLeastAtMost_iff]
hoelzl@29805: 
hoelzl@29805:   have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
hoelzl@29805: 
hoelzl@31809:   note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0
hoelzl@29805:     and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
hoelzl@31809:     and ub="\<lambda>n i k x. ub_arctan_horner prec n k x",
hoelzl@31098:     OF `0 \<le> real (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps]
hoelzl@29805: 
hoelzl@40881:   { have "(x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n"
hoelzl@31098:       using bounds(1) `0 \<le> real x`
haftmann@57512:       unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
haftmann@57512:       unfolding mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
hoelzl@29805:       by (auto intro!: mult_left_mono)
hoelzl@40881:     also have "\<dots> \<le> arctan x" using arctan_bounds ..
hoelzl@40881:     finally have "(x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan x" . }
hoelzl@29805:   moreover
hoelzl@40881:   { have "arctan x \<le> ?S (Suc n)" using arctan_bounds ..
hoelzl@40881:     also have "\<dots> \<le> (x * ub_arctan_horner prec (Suc n) 1 (x*x))"
hoelzl@31098:       using bounds(2)[of "Suc n"] `0 \<le> real x`
haftmann@57512:       unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
haftmann@57512:       unfolding mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
hoelzl@29805:       by (auto intro!: mult_left_mono)
hoelzl@40881:     finally have "arctan x \<le> (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . }
hoelzl@29805:   ultimately show ?thesis by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@31098: lemma arctan_0_1_bounds: assumes "0 \<le> real x" "real x \<le> 1"
hoelzl@40881:   shows "arctan x \<in> {(x * lb_arctan_horner prec (get_even n) 1 (x * x)) .. (x * ub_arctan_horner prec (get_odd n) 1 (x * x))}"
hoelzl@54269:   using
hoelzl@54269:     arctan_0_1_bounds'[OF assms, of n prec]
hoelzl@54269:     arctan_0_1_bounds'[OF assms, of "n + 1" prec]
hoelzl@54269:     arctan_0_1_bounds'[OF assms, of "n - 1" prec]
hoelzl@54269:   by (auto simp: get_even_def get_odd_def odd_pos simp del: ub_arctan_horner.simps lb_arctan_horner.simps)
hoelzl@29805: 
hoelzl@29805: subsection "Compute \<pi>"
hoelzl@29805: 
hoelzl@29805: definition ub_pi :: "nat \<Rightarrow> float" where
hoelzl@31809:   "ub_pi prec = (let A = rapprox_rat prec 1 5 ;
hoelzl@29805:                      B = lapprox_rat prec 1 239
hoelzl@31809:                  in ((Float 1 2) * ((Float 1 2) * A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (A * A)) -
hoelzl@29805:                                                   B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 (B * B)))))"
hoelzl@29805: 
hoelzl@29805: definition lb_pi :: "nat \<Rightarrow> float" where
hoelzl@31809:   "lb_pi prec = (let A = lapprox_rat prec 1 5 ;
hoelzl@29805:                      B = rapprox_rat prec 1 239
hoelzl@31809:                  in ((Float 1 2) * ((Float 1 2) * A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (A * A)) -
hoelzl@29805:                                                   B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (B * B)))))"
hoelzl@29805: 
hoelzl@40881: lemma pi_boundaries: "pi \<in> {(lb_pi n) .. (ub_pi n)}"
hoelzl@29805: proof -
hoelzl@29805:   have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))" unfolding machin[symmetric] by auto
hoelzl@29805: 
hoelzl@29805:   { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto
hoelzl@29805:     let ?k = "rapprox_rat prec 1 k"
hoelzl@29805:     have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
hoelzl@31809: 
hoelzl@31098:     have "0 \<le> real ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \<le> k`)
hoelzl@47599:     have "real ?k \<le> 1" 
hoelzl@47599:       by (rule rapprox_rat_le1, auto simp add: `0 < k` `1 \<le> k`)
hoelzl@29805: 
hoelzl@40881:     have "1 / k \<le> ?k" using rapprox_rat[where x=1 and y=k] by auto
hoelzl@40881:     hence "arctan (1 / k) \<le> arctan ?k" by (rule arctan_monotone')
hoelzl@40881:     also have "\<dots> \<le> (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))"
hoelzl@31098:       using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
hoelzl@40881:     finally have "arctan (1 / k) \<le> ?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k)" .
hoelzl@29805:   } note ub_arctan = this
hoelzl@29805: 
hoelzl@29805:   { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
hoelzl@29805:     let ?k = "lapprox_rat prec 1 k"
hoelzl@29805:     have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
hoelzl@40881:     have "1 / k \<le> 1" using `1 < k` by auto
hoelzl@47599:     have "\<And>n. 0 \<le> real ?k" using lapprox_rat_nonneg[where x=1 and y=k, OF zero_le_one `0 < k`] by (auto simp add: `1 div k = 0`)
hoelzl@40881:     have "\<And>n. real ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / k \<le> 1`)
hoelzl@40881: 
hoelzl@40881:     have "?k \<le> 1 / k" using lapprox_rat[where x=1 and y=k] by auto
hoelzl@40881: 
hoelzl@40881:     have "?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k) \<le> arctan ?k"
hoelzl@31098:       using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
hoelzl@40881:     also have "\<dots> \<le> arctan (1 / k)" using `?k \<le> 1 / k` by (rule arctan_monotone')
hoelzl@40881:     finally have "?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k) \<le> arctan (1 / k)" .
hoelzl@29805:   } note lb_arctan = this
hoelzl@29805: 
hoelzl@54269:   have "pi \<le> ub_pi n \<and> lb_pi n \<le> pi"
hoelzl@54269:     unfolding lb_pi_def ub_pi_def machin_pi Let_def unfolding Float_num
hoelzl@54269:     using lb_arctan[of 5] ub_arctan[of 239] lb_arctan[of 239] ub_arctan[of 5] powr_realpow[of 2 2]
haftmann@54230:     by (auto intro!: mult_left_mono add_mono simp add: uminus_add_conv_diff [symmetric] simp del: uminus_add_conv_diff)
hoelzl@54269:   then show ?thesis by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: subsection "Compute arcus tangens in the entire domain"
hoelzl@29805: 
hoelzl@31467: function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805:   "lb_arctan prec x = (let ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) ;
hoelzl@29805:                            lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)
hoelzl@29805:     in (if x < 0          then - ub_arctan prec (-x) else
haftmann@58410:         if x \<le> Float 1 (- 1) then lb_horner x else
hoelzl@31467:         if x \<le> Float 1 1  then Float 1 1 * lb_horner (float_divl prec x (1 + ub_sqrt prec (1 + x * x)))
hoelzl@31467:                           else (let inv = float_divr prec 1 x
hoelzl@31467:                                 in if inv > 1 then 0
haftmann@58410:                                               else lb_pi prec * Float 1 (- 1) - ub_horner inv)))"
hoelzl@29805: 
hoelzl@29805: | "ub_arctan prec x = (let lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) ;
hoelzl@29805:                            ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)
hoelzl@29805:     in (if x < 0          then - lb_arctan prec (-x) else
haftmann@58410:         if x \<le> Float 1 (- 1) then ub_horner x else
hoelzl@31467:         if x \<le> Float 1 1  then let y = float_divr prec x (1 + lb_sqrt prec (1 + x * x))
haftmann@58410:                                in if y > 1 then ub_pi prec * Float 1 (- 1)
hoelzl@31467:                                            else Float 1 1 * ub_horner y
haftmann@58410:                           else ub_pi prec * Float 1 (- 1) - lb_horner (float_divl prec 1 x)))"
hoelzl@29805: by pat_completeness auto
blanchet@55414: termination by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 0 then 1 else 0))", auto)
hoelzl@29805: 
hoelzl@29805: declare ub_arctan_horner.simps[simp del]
hoelzl@29805: declare lb_arctan_horner.simps[simp del]
hoelzl@29805: 
hoelzl@31098: lemma lb_arctan_bound': assumes "0 \<le> real x"
hoelzl@40881:   shows "lb_arctan prec x \<le> arctan x"
hoelzl@29805: proof -
hoelzl@47600:   have "\<not> x < 0" and "0 \<le> x" using `0 \<le> real x` by auto
hoelzl@29805:   let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805:     and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805: 
hoelzl@29805:   show ?thesis
haftmann@58410:   proof (cases "x \<le> Float 1 (- 1)")
hoelzl@47600:     case True hence "real x \<le> 1" by auto
hoelzl@29805:     show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
hoelzl@31098:       using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
hoelzl@29805:   next
hoelzl@47600:     case False hence "0 < real x" by auto
hoelzl@31098:     let ?R = "1 + sqrt (1 + real x * real x)"
hoelzl@31467:     let ?fR = "1 + ub_sqrt prec (1 + x * x)"
hoelzl@29805:     let ?DIV = "float_divl prec x ?fR"
hoelzl@31467: 
hoelzl@31098:     have sqr_ge0: "0 \<le> 1 + real x * real x" using sum_power2_ge_zero[of 1 "real x", unfolded numeral_2_eq_2] by auto
hoelzl@29805:     hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
hoelzl@29805: 
hoelzl@40881:     have "sqrt (1 + x * x) \<le> ub_sqrt prec (1 + x * x)"
hoelzl@31467:       using bnds_sqrt'[of "1 + x * x"] by auto
hoelzl@31467: 
hoelzl@40881:     hence "?R \<le> ?fR" by auto
hoelzl@47600:     hence "0 < ?fR" and "0 < real ?fR" using `0 < ?R` by auto
hoelzl@29805: 
hoelzl@40881:     have monotone: "(float_divl prec x ?fR) \<le> x / ?R"
hoelzl@29805:     proof -
hoelzl@40881:       have "?DIV \<le> real x / ?fR" by (rule float_divl)
hoelzl@40881:       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]])
hoelzl@29805:       finally show ?thesis .
hoelzl@29805:     qed
hoelzl@29805: 
hoelzl@29805:     show ?thesis
hoelzl@29805:     proof (cases "x \<le> Float 1 1")
hoelzl@29805:       case True
hoelzl@31467: 
hoelzl@40881:       have "x \<le> sqrt (1 + x * x)" using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
hoelzl@40881:       also have "\<dots> \<le> (ub_sqrt prec (1 + x * x))"
wenzelm@32960:         using bnds_sqrt'[of "1 + x * x"] by auto
hoelzl@40881:       finally have "real x \<le> ?fR" by auto
hoelzl@40881:       moreover have "?DIV \<le> real x / ?fR" by (rule float_divl)
hoelzl@31098:       ultimately have "real ?DIV \<le> 1" unfolding divide_le_eq_1_pos[OF `0 < real ?fR`, symmetric] by auto
hoelzl@29805: 
immler@54782:       have "0 \<le> real ?DIV" using float_divl_lower_bound[OF `0 \<le> x`] `0 < ?fR` unfolding less_eq_float_def by auto
hoelzl@47599: 
hoelzl@47599:       have "(Float 1 1 * ?lb_horner ?DIV) \<le> 2 * arctan (float_divl prec x ?fR)"
wenzelm@32960:         using arctan_0_1_bounds[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
hoelzl@40881:       also have "\<dots> \<le> 2 * arctan (x / ?R)"
wenzelm@32960:         using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
hoelzl@40881:       also have "2 * arctan (x / ?R) = arctan x" using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
haftmann@58410:       finally show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 (- 1)`] if_P[OF True] .
hoelzl@29805:     next
hoelzl@29805:       case False
hoelzl@47600:       hence "2 < real x" by auto
hoelzl@31098:       hence "1 \<le> real x" by auto
hoelzl@29805: 
hoelzl@29805:       let "?invx" = "float_divr prec 1 x"
hoelzl@40881:       have "0 \<le> arctan x" using arctan_monotone'[OF `0 \<le> real x`] using arctan_tan[of 0, unfolded tan_zero] by auto
hoelzl@29805: 
hoelzl@29805:       show ?thesis
hoelzl@29805:       proof (cases "1 < ?invx")
wenzelm@32960:         case True
haftmann@58410:         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]
hoelzl@40881:           using `0 \<le> arctan x` by auto
hoelzl@29805:       next
wenzelm@32960:         case False
hoelzl@47600:         hence "real ?invx \<le> 1" by auto
wenzelm@32960:         have "0 \<le> real ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> real x`)
wenzelm@32960: 
hoelzl@40881:         have "1 / x \<noteq> 0" and "0 < 1 / x" using `0 < real x` by auto
hoelzl@40881: 
hoelzl@47601:         have "arctan (1 / x) \<le> arctan ?invx" unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone', rule float_divr)
hoelzl@40881:         also have "\<dots> \<le> (?ub_horner ?invx)" using arctan_0_1_bounds[OF `0 \<le> real ?invx` `real ?invx \<le> 1`] by auto
hoelzl@40881:         finally have "pi / 2 - (?ub_horner ?invx) \<le> arctan x"
hoelzl@40881:           using `0 \<le> arctan x` arctan_inverse[OF `1 / x \<noteq> 0`]
wenzelm@32960:           unfolding real_sgn_pos[OF `0 < 1 / real x`] le_diff_eq by auto
wenzelm@32960:         moreover
haftmann@58410:         have "lb_pi prec * Float 1 (- 1) \<le> pi / 2"
hoelzl@47599:           unfolding Float_num times_divide_eq_right mult_1_left using pi_boundaries by simp
wenzelm@32960:         ultimately
haftmann@58410:         show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 (- 1)`] if_not_P[OF `\<not> x \<le> Float 1 1`] if_not_P[OF False]
wenzelm@32960:           by auto
hoelzl@29805:       qed
hoelzl@29805:     qed
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@31098: lemma ub_arctan_bound': assumes "0 \<le> real x"
hoelzl@40881:   shows "arctan x \<le> ub_arctan prec x"
hoelzl@29805: proof -
hoelzl@47600:   have "\<not> x < 0" and "0 \<le> x" using `0 \<le> real x` by auto
hoelzl@29805: 
hoelzl@29805:   let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805:     and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
hoelzl@29805: 
hoelzl@29805:   show ?thesis
haftmann@58410:   proof (cases "x \<le> Float 1 (- 1)")
hoelzl@47600:     case True hence "real x \<le> 1" by auto
hoelzl@29805:     show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
hoelzl@31098:       using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
hoelzl@29805:   next
hoelzl@47600:     case False hence "0 < real x" by auto
hoelzl@31098:     let ?R = "1 + sqrt (1 + real x * real x)"
hoelzl@31467:     let ?fR = "1 + lb_sqrt prec (1 + x * x)"
hoelzl@29805:     let ?DIV = "float_divr prec x ?fR"
hoelzl@31467: 
hoelzl@31098:     have sqr_ge0: "0 \<le> 1 + real x * real x" using sum_power2_ge_zero[of 1 "real x", unfolded numeral_2_eq_2] by auto
hoelzl@31098:     hence "0 \<le> real (1 + x*x)" by auto
hoelzl@31467: 
hoelzl@29805:     hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
hoelzl@29805: 
hoelzl@40881:     have "lb_sqrt prec (1 + x * x) \<le> sqrt (1 + x * x)"
hoelzl@31467:       using bnds_sqrt'[of "1 + x * x"] by auto
hoelzl@40881:     hence "?fR \<le> ?R" by auto
hoelzl@47599:     have "0 < real ?fR" by (rule order_less_le_trans[OF zero_less_one], auto simp add: lb_sqrt_lower_bound[OF `0 \<le> real (1 + x*x)`])
hoelzl@29805: 
hoelzl@40881:     have monotone: "x / ?R \<le> (float_divr prec x ?fR)"
hoelzl@29805:     proof -
hoelzl@40881:       from divide_left_mono[OF `?fR \<le> ?R` `0 \<le> real x` mult_pos_pos[OF divisor_gt0 `0 < real ?fR`]]
hoelzl@40881:       have "x / ?R \<le> x / ?fR" .
hoelzl@40881:       also have "\<dots> \<le> ?DIV" by (rule float_divr)
hoelzl@29805:       finally show ?thesis .
hoelzl@29805:     qed
hoelzl@29805: 
hoelzl@29805:     show ?thesis
hoelzl@29805:     proof (cases "x \<le> Float 1 1")
hoelzl@29805:       case True
hoelzl@29805:       show ?thesis
hoelzl@29805:       proof (cases "?DIV > 1")
wenzelm@32960:         case True
haftmann@58410:         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
wenzelm@32960:         from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le]
haftmann@58410:         show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 (- 1)`] if_P[OF `x \<le> Float 1 1`] if_P[OF True] .
hoelzl@29805:       next
wenzelm@32960:         case False
hoelzl@47600:         hence "real ?DIV \<le> 1" by auto
wenzelm@32960: 
huffman@44349:         have "0 \<le> x / ?R" using `0 \<le> real x` `0 < ?R` unfolding zero_le_divide_iff by auto
wenzelm@32960:         hence "0 \<le> real ?DIV" using monotone by (rule order_trans)
wenzelm@32960: 
hoelzl@40881:         have "arctan x = 2 * arctan (x / ?R)" using arctan_half unfolding numeral_2_eq_2 power_Suc2 power_0 mult_1_left .
hoelzl@40881:         also have "\<dots> \<le> 2 * arctan (?DIV)"
wenzelm@32960:           using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
hoelzl@47599:         also have "\<dots> \<le> (Float 1 1 * ?ub_horner ?DIV)" unfolding Float_num
wenzelm@32960:           using arctan_0_1_bounds[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
haftmann@58410:         finally show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 (- 1)`] if_P[OF `x \<le> Float 1 1`] if_not_P[OF False] .
hoelzl@29805:       qed
hoelzl@29805:     next
hoelzl@29805:       case False
hoelzl@47600:       hence "2 < real x" by auto
hoelzl@31098:       hence "1 \<le> real x" by auto
hoelzl@31098:       hence "0 < real x" by auto
hoelzl@47600:       hence "0 < x" by auto
hoelzl@29805: 
hoelzl@29805:       let "?invx" = "float_divl prec 1 x"
hoelzl@40881:       have "0 \<le> arctan x" using arctan_monotone'[OF `0 \<le> real x`] using arctan_tan[of 0, unfolded tan_zero] by auto
hoelzl@29805: 
hoelzl@31098:       have "real ?invx \<le> 1" unfolding less_float_def by (rule order_trans[OF float_divl], auto simp add: `1 \<le> real x` divide_le_eq_1_pos[OF `0 < real x`])
hoelzl@47600:       have "0 \<le> real ?invx" using `0 < x` by (intro float_divl_lower_bound) auto
hoelzl@31467: 
hoelzl@40881:       have "1 / x \<noteq> 0" and "0 < 1 / x" using `0 < real x` by auto
hoelzl@40881: 
hoelzl@40881:       have "(?lb_horner ?invx) \<le> arctan (?invx)" using arctan_0_1_bounds[OF `0 \<le> real ?invx` `real ?invx \<le> 1`] by auto
hoelzl@47601:       also have "\<dots> \<le> arctan (1 / x)" unfolding one_float.rep_eq[symmetric] by (rule arctan_monotone', rule float_divl)
hoelzl@40881:       finally have "arctan x \<le> pi / 2 - (?lb_horner ?invx)"
hoelzl@40881:         using `0 \<le> arctan x` arctan_inverse[OF `1 / x \<noteq> 0`]
hoelzl@40881:         unfolding real_sgn_pos[OF `0 < 1 / x`] le_diff_eq by auto
hoelzl@29805:       moreover
haftmann@58410:       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
hoelzl@29805:       ultimately
haftmann@58410:       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]
wenzelm@32960:         by auto
hoelzl@29805:     qed
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma arctan_boundaries:
hoelzl@40881:   "arctan x \<in> {(lb_arctan prec x) .. (ub_arctan prec x)}"
hoelzl@29805: proof (cases "0 \<le> x")
hoelzl@47600:   case True hence "0 \<le> real x" by auto
hoelzl@31098:   show ?thesis using ub_arctan_bound'[OF `0 \<le> real x`] lb_arctan_bound'[OF `0 \<le> real x`] unfolding atLeastAtMost_iff by auto
hoelzl@29805: next
hoelzl@29805:   let ?mx = "-x"
hoelzl@47600:   case False hence "x < 0" and "0 \<le> real ?mx" by auto
hoelzl@40881:   hence bounds: "lb_arctan prec ?mx \<le> arctan ?mx \<and> arctan ?mx \<le> ub_arctan prec ?mx"
hoelzl@31098:     using ub_arctan_bound'[OF `0 \<le> real ?mx`] lb_arctan_bound'[OF `0 \<le> real ?mx`] by auto
hoelzl@47601:   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`]
hoelzl@47601:     unfolding atLeastAtMost_iff using bounds[unfolded minus_float.rep_eq arctan_minus]
hoelzl@47599:     by (simp add: arctan_minus)
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@40881: 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"
hoelzl@29805: proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@40881:   fix x :: real fix lx ux
hoelzl@40881:   assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {lx .. ux}"
hoelzl@40881:   hence l: "lb_arctan prec lx = l " and u: "ub_arctan prec ux = u" and x: "x \<in> {lx .. ux}" by auto
hoelzl@29805: 
hoelzl@29805:   { from arctan_boundaries[of lx prec, unfolded l]
hoelzl@40881:     have "l \<le> arctan lx" by (auto simp del: lb_arctan.simps)
hoelzl@29805:     also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
hoelzl@40881:     finally have "l \<le> arctan x" .
hoelzl@29805:   } moreover
hoelzl@40881:   { have "arctan x \<le> arctan ux" using x by (auto intro: arctan_monotone')
hoelzl@40881:     also have "\<dots> \<le> u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
hoelzl@40881:     finally have "arctan x \<le> u" .
hoelzl@40881:   } ultimately show "l \<le> arctan x \<and> arctan x \<le> u" ..
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: section "Sinus and Cosinus"
hoelzl@29805: 
hoelzl@29805: subsection "Compute the cosinus and sinus series"
hoelzl@29805: 
hoelzl@29805: fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
hoelzl@29805: and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805:   "ub_sin_cos_aux prec 0 i k x = 0"
hoelzl@31809: | "ub_sin_cos_aux prec (Suc n) i k x =
hoelzl@40881:     (rapprox_rat prec 1 k) - x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
hoelzl@29805: | "lb_sin_cos_aux prec 0 i k x = 0"
hoelzl@31809: | "lb_sin_cos_aux prec (Suc n) i k x =
hoelzl@40881:     (lapprox_rat prec 1 k) - x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
hoelzl@47601: 
hoelzl@29805: lemma cos_aux:
haftmann@58410:   shows "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. (- 1) ^ i * (1/real (fact (2 * i))) * x ^(2 * i))" (is "?lb")
haftmann@58410:   and "(\<Sum> i=0..<n. (- 1) ^ i * (1/real (fact (2 * i))) * x^(2 * i)) \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
hoelzl@29805: proof -
hoelzl@47599:   have "0 \<le> real (x * x)" by auto
hoelzl@29805:   let "?f n" = "fact (2 * n)"
hoelzl@29805: 
hoelzl@31809:   { fix n
wenzelm@45129:     have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
haftmann@30971:     have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)"
hoelzl@29805:       unfolding F by auto } note f_eq = this
hoelzl@31809: 
hoelzl@31809:   from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
hoelzl@31098:     OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
hoelzl@31098:   show "?lb" and "?ub" by (auto simp add: power_mult power2_eq_square[of "real x"])
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@40881: lemma cos_boundaries: assumes "0 \<le> real x" and "x \<le> pi / 2"
hoelzl@40881:   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))}"
hoelzl@31098: proof (cases "real x = 0")
hoelzl@31098:   case False hence "real x \<noteq> 0" by auto
hoelzl@47600:   hence "0 < x" and "0 < real x" using `0 \<le> real x` by auto
nipkow@56544:   have "0 < x * x" using `0 < x` by simp
hoelzl@29805: 
haftmann@58410:   { fix x n have "(\<Sum> i=0..<n. (- 1) ^ i * (1/real (fact (2 * i))) * x ^ (2 * i))
haftmann@58410:     = (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum")
hoelzl@29805:   proof -
hoelzl@29805:     have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
hoelzl@31809:     also have "\<dots> =
haftmann@58410:       (\<Sum> j = 0 ..< n. (- 1) ^ ((2 * j) div 2) / (real (fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
haftmann@58410:     also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then (- 1) ^ (i div 2) / (real (fact i)) * x ^ i else 0)"
huffman@56195:       unfolding sum_split_even_odd atLeast0LessThan ..
haftmann@58410:     also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then (- 1) ^ (i div 2) / (real (fact i)) else 0) * x ^ i)"
haftmann@57418:       by (rule setsum.cong) auto
hoelzl@29805:     finally show ?thesis by assumption
hoelzl@29805:   qed } note morph_to_if_power = this
hoelzl@29805: 
hoelzl@29805: 
hoelzl@29805:   { fix n :: nat assume "0 < n"
hoelzl@29805:     hence "0 < 2 * n" by auto
hoelzl@31098:     obtain t where "0 < t" and "t < real x" and
haftmann@58410:       cos_eq: "cos x = (\<Sum> i = 0 ..< 2 * n. (if even(i) then ((- 1) ^ (i div 2))/(real (fact i)) else 0) * (real x) ^ i)
hoelzl@40881:       + (cos (t + 1/2 * (2 * n) * pi) / real (fact (2*n))) * (real x)^(2*n)"
hoelzl@29805:       (is "_ = ?SUM + ?rest / ?fact * ?pow")
huffman@44306:       using Maclaurin_cos_expansion2[OF `0 < real x` `0 < 2 * n`]
huffman@56195:       unfolding cos_coeff_def atLeast0LessThan by auto
hoelzl@29805: 
haftmann@58410:     have "cos t * (- 1) ^ n = cos t * cos (n * pi) + sin t * sin (n * pi)" by auto
hoelzl@40881:     also have "\<dots> = cos (t + n * pi)"  using cos_add by auto
hoelzl@29805:     also have "\<dots> = ?rest" by auto
haftmann@58410:     finally have "cos t * (- 1) ^ n = ?rest" .
hoelzl@29805:     moreover
hoelzl@40881:     have "t \<le> pi / 2" using `t < real x` and `x \<le> pi / 2` by auto
hoelzl@29805:     hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
hoelzl@29805:     ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
hoelzl@29805: 
hoelzl@29805:     have "0 < ?fact" by auto
hoelzl@31098:     have "0 < ?pow" using `0 < real x` by auto
hoelzl@29805: 
hoelzl@29805:     {
hoelzl@29805:       assume "even n"
hoelzl@40881:       have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
wenzelm@32960:         unfolding morph_to_if_power[symmetric] using cos_aux by auto
hoelzl@40881:       also have "\<dots> \<le> cos x"
hoelzl@29805:       proof -
wenzelm@32960:         from even[OF `even n`] `0 < ?fact` `0 < ?pow`
hoelzl@56571:         have "0 \<le> (?rest / ?fact) * ?pow" by simp
wenzelm@32960:         thus ?thesis unfolding cos_eq by auto
hoelzl@29805:       qed
hoelzl@40881:       finally have "(lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos x" .
hoelzl@29805:     } note lb = this
hoelzl@29805: 
hoelzl@29805:     {
hoelzl@29805:       assume "odd n"
hoelzl@40881:       have "cos x \<le> ?SUM"
hoelzl@29805:       proof -
wenzelm@32960:         from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
wenzelm@32960:         have "0 \<le> (- ?rest) / ?fact * ?pow"
wenzelm@32960:           by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
wenzelm@32960:         thus ?thesis unfolding cos_eq by auto
hoelzl@29805:       qed
hoelzl@40881:       also have "\<dots> \<le> (ub_sin_cos_aux prec n 1 1 (x * x))"
wenzelm@32960:         unfolding morph_to_if_power[symmetric] using cos_aux by auto
hoelzl@40881:       finally have "cos x \<le> (ub_sin_cos_aux prec n 1 1 (x * x))" .
hoelzl@29805:     } note ub = this and lb
hoelzl@29805:   } note ub = this(1) and lb = this(2)
hoelzl@29805: 
hoelzl@40881:   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] .
hoelzl@40881:   moreover have "(lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos x"
hoelzl@29805:   proof (cases "0 < get_even n")
hoelzl@29805:     case True show ?thesis using lb[OF True get_even] .
hoelzl@29805:   next
hoelzl@29805:     case False
hoelzl@29805:     hence "get_even n = 0" by auto
hoelzl@40881:     have "- (pi / 2) \<le> x" by (rule order_trans[OF _ `0 < real x`[THEN less_imp_le]], auto)
hoelzl@40881:     with `x \<le> pi / 2`
hoelzl@47601:     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
hoelzl@29805:   qed
hoelzl@29805:   ultimately show ?thesis by auto
hoelzl@29805: next
hoelzl@29805:   case True
hoelzl@29805:   show ?thesis
hoelzl@29805:   proof (cases "n = 0")
hoelzl@31809:     case True
hoelzl@47599:     thus ?thesis unfolding `n = 0` get_even_def get_odd_def
hoelzl@47599:       using `real x = 0` lapprox_rat[where x="-1" and y=1]
hoelzl@47621:       by (auto simp: Float.compute_lapprox_rat Float.compute_rapprox_rat)
hoelzl@29805:   next
hoelzl@29805:     case False with not0_implies_Suc obtain m where "n = Suc m" by blast
hoelzl@31098:     thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `real x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@31098: lemma sin_aux: assumes "0 \<le> real x"
haftmann@58410:   shows "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> (\<Sum> i=0..<n. (- 1) ^ i * (1/real (fact (2 * i + 1))) * x^(2 * i + 1))" (is "?lb")
haftmann@58410:   and "(\<Sum> i=0..<n. (- 1) ^ i * (1/real (fact (2 * i + 1))) * x^(2 * i + 1)) \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
hoelzl@29805: proof -
hoelzl@47599:   have "0 \<le> real (x * x)" by auto
hoelzl@29805:   let "?f n" = "fact (2 * n + 1)"
hoelzl@29805: 
hoelzl@31809:   { fix n
wenzelm@45129:     have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n) auto
haftmann@30971:     have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)"
hoelzl@29805:       unfolding F by auto } note f_eq = this
hoelzl@31809: 
hoelzl@29805:   from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
hoelzl@31098:     OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
hoelzl@47599:   show "?lb" and "?ub" using `0 \<le> real x`
haftmann@57512:     unfolding power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric]
haftmann@57512:     unfolding mult.commute[where 'a=real]
hoelzl@31098:     by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real x"])
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@40881: lemma sin_boundaries: assumes "0 \<le> real x" and "x \<le> pi / 2"
hoelzl@40881:   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))}"
hoelzl@31098: proof (cases "real x = 0")
hoelzl@31098:   case False hence "real x \<noteq> 0" by auto
hoelzl@47600:   hence "0 < x" and "0 < real x" using `0 \<le> real x` by auto
nipkow@56544:   have "0 < x * x" using `0 < x` by simp
hoelzl@29805: 
haftmann@58410:   { fix x n have "(\<Sum> j = 0 ..< n. (- 1) ^ (((2 * j + 1) - Suc 0) div 2) / (real (fact (2 * j + 1))) * x ^(2 * j + 1))
haftmann@58410:     = (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/(real (fact i))) * x ^ i)" (is "?SUM = _")
hoelzl@29805:     proof -
hoelzl@29805:       have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto
hoelzl@29805:       have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" by auto
haftmann@58410:       also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / (real (fact i)) * x ^ i)"
huffman@56195:         unfolding sum_split_even_odd atLeast0LessThan ..
haftmann@58410:       also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then 0 else (- 1) ^ ((i - Suc 0) div 2) / (real (fact i))) * x ^ i)"
haftmann@57418:         by (rule setsum.cong) auto
hoelzl@29805:       finally show ?thesis by assumption
hoelzl@29805:     qed } note setsum_morph = this
hoelzl@29805: 
hoelzl@29805:   { fix n :: nat assume "0 < n"
hoelzl@29805:     hence "0 < 2 * n + 1" by auto
hoelzl@31098:     obtain t where "0 < t" and "t < real x" and
haftmann@58410:       sin_eq: "sin x = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)
hoelzl@40881:       + (sin (t + 1/2 * (2 * n + 1) * pi) / real (fact (2*n + 1))) * (real x)^(2*n + 1)"
hoelzl@29805:       (is "_ = ?SUM + ?rest / ?fact * ?pow")
huffman@44306:       using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < real x`]
huffman@56195:       unfolding sin_coeff_def atLeast0LessThan by auto
hoelzl@29805: 
haftmann@58410:     have "?rest = cos t * (- 1) ^ n" unfolding sin_add cos_add real_of_nat_add distrib_right distrib_left by auto
hoelzl@29805:     moreover
hoelzl@40881:     have "t \<le> pi / 2" using `t < real x` and `x \<le> pi / 2` by auto
hoelzl@29805:     hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
hoelzl@29805:     ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
hoelzl@29805: 
huffman@44305:     have "0 < ?fact" by (simp del: fact_Suc)
hoelzl@31098:     have "0 < ?pow" using `0 < real x` by (rule zero_less_power)
hoelzl@29805: 
hoelzl@29805:     {
hoelzl@29805:       assume "even n"
hoelzl@40881:       have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
haftmann@58410:             (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)"
wenzelm@32960:         using sin_aux[OF `0 \<le> real x`] unfolding setsum_morph[symmetric] by auto
hoelzl@29805:       also have "\<dots> \<le> ?SUM" by auto
hoelzl@40881:       also have "\<dots> \<le> sin x"
hoelzl@29805:       proof -
wenzelm@32960:         from even[OF `even n`] `0 < ?fact` `0 < ?pow`
hoelzl@56571:         have "0 \<le> (?rest / ?fact) * ?pow" by simp
wenzelm@32960:         thus ?thesis unfolding sin_eq by auto
hoelzl@29805:       qed
hoelzl@40881:       finally have "(x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin x" .
hoelzl@29805:     } note lb = this
hoelzl@29805: 
hoelzl@29805:     {
hoelzl@29805:       assume "odd n"
hoelzl@40881:       have "sin x \<le> ?SUM"
hoelzl@29805:       proof -
wenzelm@32960:         from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
wenzelm@32960:         have "0 \<le> (- ?rest) / ?fact * ?pow"
wenzelm@32960:           by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
wenzelm@32960:         thus ?thesis unfolding sin_eq by auto
hoelzl@29805:       qed
haftmann@58410:       also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else ((- 1) ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)"
wenzelm@32960:          by auto
hoelzl@40881:       also have "\<dots> \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))"
wenzelm@32960:         using sin_aux[OF `0 \<le> real x`] unfolding setsum_morph[symmetric] by auto
hoelzl@40881:       finally have "sin x \<le> (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
hoelzl@29805:     } note ub = this and lb
hoelzl@29805:   } note ub = this(1) and lb = this(2)
hoelzl@29805: 
hoelzl@40881:   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] .
hoelzl@40881:   moreover have "(x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin x"
hoelzl@29805:   proof (cases "0 < get_even n")
hoelzl@29805:     case True show ?thesis using lb[OF True get_even] .
hoelzl@29805:   next
hoelzl@29805:     case False
hoelzl@29805:     hence "get_even n = 0" by auto
hoelzl@40881:     with `x \<le> pi / 2` `0 \<le> real x`
hoelzl@47601:     show ?thesis unfolding `get_even n = 0` ub_sin_cos_aux.simps minus_float.rep_eq using sin_ge_zero by auto
hoelzl@29805:   qed
hoelzl@29805:   ultimately show ?thesis by auto
hoelzl@29805: next
hoelzl@29805:   case True
hoelzl@29805:   show ?thesis
hoelzl@29805:   proof (cases "n = 0")
hoelzl@31809:     case True
hoelzl@31098:     thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `real x = 0` lapprox_rat[where x="-1" and y=1] by auto
hoelzl@29805:   next
hoelzl@29805:     case False with not0_implies_Suc obtain m where "n = Suc m" by blast
hoelzl@31098:     thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `real x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: subsection "Compute the cosinus in the entire domain"
hoelzl@29805: 
hoelzl@29805: definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805: "lb_cos prec x = (let
hoelzl@29805:     horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ;
hoelzl@29805:     half = \<lambda> x. if x < 0 then - 1 else Float 1 1 * x * x - 1
haftmann@58410:   in if x < Float 1 (- 1) then horner x
haftmann@58410: else if x < 1          then half (horner (x * Float 1 (- 1)))
haftmann@58410:                        else half (half (horner (x * Float 1 (- 2)))))"
hoelzl@29805: 
hoelzl@29805: definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805: "ub_cos prec x = (let
hoelzl@29805:     horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ;
hoelzl@29805:     half = \<lambda> x. Float 1 1 * x * x - 1
haftmann@58410:   in if x < Float 1 (- 1) then horner x
haftmann@58410: else if x < 1          then half (horner (x * Float 1 (- 1)))
haftmann@58410:                        else half (half (horner (x * Float 1 (- 2)))))"
hoelzl@29805: 
hoelzl@40881: lemma lb_cos: assumes "0 \<le> real x" and "x \<le> pi"
hoelzl@40881:   shows "cos x \<in> {(lb_cos prec x) .. (ub_cos prec x)}" (is "?cos x \<in> {(?lb x) .. (?ub x) }")
hoelzl@29805: proof -
hoelzl@29805:   { fix x :: real
hoelzl@29805:     have "cos x = cos (x / 2 + x / 2)" by auto
hoelzl@29805:     also have "\<dots> = cos (x / 2) * cos (x / 2) + sin (x / 2) * sin (x / 2) - sin (x / 2) * sin (x / 2) + cos (x / 2) * cos (x / 2) - 1"
hoelzl@29805:       unfolding cos_add by auto
hoelzl@29805:     also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1" by algebra
hoelzl@29805:     finally have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" .
hoelzl@29805:   } note x_half = this[symmetric]
hoelzl@29805: 
hoelzl@47600:   have "\<not> x < 0" using `0 \<le> real x` by auto
hoelzl@29805:   let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
hoelzl@29805:   let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
hoelzl@29805:   let "?ub_half x" = "Float 1 1 * x * x - 1"
hoelzl@29805:   let "?lb_half x" = "if x < 0 then - 1 else Float 1 1 * x * x - 1"
hoelzl@29805: 
hoelzl@29805:   show ?thesis
haftmann@58410:   proof (cases "x < Float 1 (- 1)")
hoelzl@47600:     case True hence "x \<le> pi / 2" using pi_ge_two by auto
haftmann@58410:     show ?thesis unfolding lb_cos_def[where x=x] ub_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_P[OF `x < Float 1 (- 1)`] Let_def
hoelzl@40881:       using cos_boundaries[OF `0 \<le> real x` `x \<le> pi / 2`] .
hoelzl@29805:   next
hoelzl@29805:     case False
haftmann@58410:     { fix y x :: float let ?x2 = "(x * Float 1 (- 1))"
hoelzl@40881:       assume "y \<le> cos ?x2" and "-pi \<le> x" and "x \<le> pi"
hoelzl@47599:       hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Float_num by auto
hoelzl@29805:       hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
hoelzl@31467: 
hoelzl@40881:       have "(?lb_half y) \<le> cos x"
hoelzl@29805:       proof (cases "y < 0")
wenzelm@32960:         case True show ?thesis using cos_ge_minus_one unfolding if_P[OF True] by auto
hoelzl@29805:       next
wenzelm@32960:         case False
hoelzl@47600:         hence "0 \<le> real y" by auto
hoelzl@40881:         from mult_mono[OF `y \<le> cos ?x2` `y \<le> cos ?x2` `0 \<le> cos ?x2` this]
wenzelm@32960:         have "real y * real y \<le> cos ?x2 * cos ?x2" .
wenzelm@32960:         hence "2 * real y * real y \<le> 2 * cos ?x2 * cos ?x2" by auto
hoelzl@47599:         hence "2 * real y * real y - 1 \<le> 2 * cos (x / 2) * cos (x / 2) - 1" unfolding Float_num by auto
hoelzl@47599:         thus ?thesis unfolding if_not_P[OF False] x_half Float_num by auto
hoelzl@29805:       qed
hoelzl@29805:     } note lb_half = this
hoelzl@31467: 
haftmann@58410:     { fix y x :: float let ?x2 = "(x * Float 1 (- 1))"
hoelzl@40881:       assume ub: "cos ?x2 \<le> y" and "- pi \<le> x" and "x \<le> pi"
hoelzl@47599:       hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Float_num by auto
hoelzl@29805:       hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
hoelzl@31467: 
hoelzl@40881:       have "cos x \<le> (?ub_half y)"
hoelzl@29805:       proof -
wenzelm@32960:         have "0 \<le> real y" using `0 \<le> cos ?x2` ub by (rule order_trans)
wenzelm@32960:         from mult_mono[OF ub ub this `0 \<le> cos ?x2`]
wenzelm@32960:         have "cos ?x2 * cos ?x2 \<le> real y * real y" .
wenzelm@32960:         hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real y * real y" by auto
hoelzl@47599:         hence "2 * cos (x / 2) * cos (x / 2) - 1 \<le> 2 * real y * real y - 1" unfolding Float_num by auto
hoelzl@47599:         thus ?thesis unfolding x_half Float_num by auto
hoelzl@29805:       qed
hoelzl@29805:     } note ub_half = this
hoelzl@31467: 
haftmann@58410:     let ?x2 = "x * Float 1 (- 1)"
haftmann@58410:     let ?x4 = "x * Float 1 (- 1) * Float 1 (- 1)"
hoelzl@31467: 
hoelzl@40881:     have "-pi \<le> x" using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] `0 \<le> real x` by (rule order_trans)
hoelzl@31467: 
hoelzl@29805:     show ?thesis
hoelzl@29805:     proof (cases "x < 1")
hoelzl@47600:       case True hence "real x \<le> 1" by auto
hoelzl@47599:       have "0 \<le> real ?x2" and "?x2 \<le> pi / 2" using pi_ge_two `0 \<le> real x` using assms by auto
hoelzl@29805:       from cos_boundaries[OF this]
hoelzl@40881:       have lb: "(?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> (?ub_horner ?x2)" by auto
hoelzl@40881: 
hoelzl@40881:       have "(?lb x) \<le> ?cos x"
hoelzl@29805:       proof -
hoelzl@40881:         from lb_half[OF lb `-pi \<le> x` `x \<le> pi`]
haftmann@58410:         show ?thesis unfolding lb_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 (- 1)` `x < 1` by auto
hoelzl@29805:       qed
hoelzl@40881:       moreover have "?cos x \<le> (?ub x)"
hoelzl@29805:       proof -
hoelzl@40881:         from ub_half[OF ub `-pi \<le> x` `x \<le> pi`]
haftmann@58410:         show ?thesis unfolding ub_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 (- 1)` `x < 1` by auto
hoelzl@29805:       qed
hoelzl@29805:       ultimately show ?thesis by auto
hoelzl@29805:     next
hoelzl@29805:       case False
hoelzl@47599:       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
hoelzl@29805:       from cos_boundaries[OF this]
hoelzl@40881:       have lb: "(?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> (?ub_horner ?x4)" by auto
hoelzl@31467: 
haftmann@58410:       have eq_4: "?x2 * Float 1 (- 1) = x * Float 1 (- 2)" by transfer simp
hoelzl@31467: 
hoelzl@40881:       have "(?lb x) \<le> ?cos x"
hoelzl@29805:       proof -
hoelzl@47599:         have "-pi \<le> ?x2" and "?x2 \<le> pi" using pi_ge_two `0 \<le> real x` `x \<le> pi` by auto
hoelzl@40881:         from lb_half[OF lb_half[OF lb this] `-pi \<le> x` `x \<le> pi`, unfolded eq_4]
haftmann@58410:         show ?thesis unfolding lb_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x < Float 1 (- 1)`] if_not_P[OF `\<not> x < 1`] Let_def .
hoelzl@29805:       qed
hoelzl@40881:       moreover have "?cos x \<le> (?ub x)"
hoelzl@29805:       proof -
hoelzl@47599:         have "-pi \<le> ?x2" and "?x2 \<le> pi" using pi_ge_two `0 \<le> real x` ` x \<le> pi` by auto
hoelzl@40881:         from ub_half[OF ub_half[OF ub this] `-pi \<le> x` `x \<le> pi`, unfolded eq_4]
haftmann@58410:         show ?thesis unfolding ub_cos_def[where x=x] if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x < Float 1 (- 1)`] if_not_P[OF `\<not> x < 1`] Let_def .
hoelzl@29805:       qed
hoelzl@29805:       ultimately show ?thesis by auto
hoelzl@29805:     qed
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@40881: lemma lb_cos_minus: assumes "-pi \<le> x" and "real x \<le> 0"
hoelzl@40881:   shows "cos (real(-x)) \<in> {(lb_cos prec (-x)) .. (ub_cos prec (-x))}"
hoelzl@29805: proof -
hoelzl@40881:   have "0 \<le> real (-x)" and "(-x) \<le> pi" using `-pi \<le> x` `real x \<le> 0` by auto
hoelzl@29805:   from lb_cos[OF this] show ?thesis .
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@31467: definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
hoelzl@31467: "bnds_cos prec lx ux = (let
hoelzl@47599:     lpi = float_round_down prec (lb_pi prec) ;
hoelzl@47599:     upi = float_round_up prec (ub_pi prec) ;
hoelzl@31467:     k = floor_fl (float_divr prec (lx + lpi) (2 * lpi)) ;
hoelzl@31467:     lx = lx - k * 2 * (if k < 0 then lpi else upi) ;
hoelzl@31467:     ux = ux - k * 2 * (if k < 0 then upi else lpi)
hoelzl@31467:   in   if - lpi \<le> lx \<and> ux \<le> 0    then (lb_cos prec (-lx), ub_cos prec (-ux))
hoelzl@31467:   else if 0 \<le> lx \<and> ux \<le> lpi      then (lb_cos prec ux, ub_cos prec lx)
hoelzl@31467:   else if - lpi \<le> lx \<and> ux \<le> lpi  then (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0)
haftmann@58410:   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))))
haftmann@58410:   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)))
haftmann@58410:                                  else (Float (- 1) 0, Float 1 0))"
hoelzl@29805: 
hoelzl@31467: lemma floor_int:
hoelzl@40881:   obtains k :: int where "real k = (floor_fl f)"
hoelzl@47599:   by (simp add: floor_fl_def)
hoelzl@29805: 
hoelzl@40881: lemma cos_periodic_nat[simp]: fixes n :: nat shows "cos (x + n * (2 * pi)) = cos x"
hoelzl@31467: proof (induct n arbitrary: x)
hoelzl@31467:   case (Suc n)
hoelzl@40881:   have split_pi_off: "x + (Suc n) * (2 * pi) = (x + n * (2 * pi)) + 2 * pi"
webertj@49962:     unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
hoelzl@31467:   show ?case unfolding split_pi_off using Suc by auto
hoelzl@31467: qed auto
hoelzl@31467: 
hoelzl@40881: lemma cos_periodic_int[simp]: fixes i :: int shows "cos (x + i * (2 * pi)) = cos x"
hoelzl@31467: proof (cases "0 \<le> i")
hoelzl@40881:   case True hence i_nat: "real i = nat i" by auto
hoelzl@31467:   show ?thesis unfolding i_nat by auto
hoelzl@31467: next
hoelzl@40881:   case False hence i_nat: "i = - real (nat (-i))" by auto
hoelzl@40881:   have "cos x = cos (x + i * (2 * pi) - i * (2 * pi))" by auto
hoelzl@40881:   also have "\<dots> = cos (x + i * (2 * pi))"
hoelzl@31467:     unfolding i_nat mult_minus_left diff_minus_eq_add by (rule cos_periodic_nat)
hoelzl@31467:   finally show ?thesis by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@40881: 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"
hoelzl@31467: proof ((rule allI | rule impI | erule conjE) +)
hoelzl@40881:   fix x :: real fix lx ux
hoelzl@40881:   assume bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {lx .. ux}"
hoelzl@31467: 
hoelzl@47599:   let ?lpi = "float_round_down prec (lb_pi prec)"
hoelzl@47599:   let ?upi = "float_round_up prec (ub_pi prec)"
hoelzl@31467:   let ?k = "floor_fl (float_divr prec (lx + ?lpi) (2 * ?lpi))"
hoelzl@31467:   let ?lx = "lx - ?k * 2 * (if ?k < 0 then ?lpi else ?upi)"
hoelzl@31467:   let ?ux = "ux - ?k * 2 * (if ?k < 0 then ?upi else ?lpi)"
hoelzl@31467: 
hoelzl@40881:   obtain k :: int where k: "k = real ?k" using floor_int .
hoelzl@40881: 
hoelzl@40881:   have upi: "pi \<le> ?upi" and lpi: "?lpi \<le> pi"
hoelzl@47599:     using float_round_up[of "ub_pi prec" prec] pi_boundaries[of prec]
hoelzl@47599:           float_round_down[of prec "lb_pi prec"] by auto
hoelzl@40881:   hence "?lx \<le> x - k * (2 * pi) \<and> x - k * (2 * pi) \<le> ?ux"
hoelzl@47599:     using x unfolding k[symmetric]
hoelzl@47599:     by (cases "k = 0")
hoelzl@47599:        (auto intro!: add_mono
haftmann@54230:                 simp add: k [symmetric] uminus_add_conv_diff [symmetric]
haftmann@54230:                 simp del: float_of_numeral uminus_add_conv_diff)
hoelzl@31467:   note lx = this[THEN conjunct1] and ux = this[THEN conjunct2]
hoelzl@40881:   hence lx_less_ux: "?lx \<le> real ?ux" by (rule order_trans)
hoelzl@40881: 
hoelzl@40881:   { assume "- ?lpi \<le> ?lx" and x_le_0: "x - k * (2 * pi) \<le> 0"
hoelzl@31467:     with lpi[THEN le_imp_neg_le] lx
hoelzl@40881:     have pi_lx: "- pi \<le> ?lx" and lx_0: "real ?lx \<le> 0"
hoelzl@47600:       by simp_all
hoelzl@29805: 
hoelzl@40881:     have "(lb_cos prec (- ?lx)) \<le> cos (real (- ?lx))"
hoelzl@31467:       using lb_cos_minus[OF pi_lx lx_0] by simp
hoelzl@40881:     also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
hoelzl@31467:       using cos_monotone_minus_pi_0'[OF pi_lx lx x_le_0]
hoelzl@47601:       by (simp only: uminus_float.rep_eq real_of_int_minus
haftmann@54230:         cos_minus mult_minus_left) simp
hoelzl@40881:     finally have "(lb_cos prec (- ?lx)) \<le> cos x"
hoelzl@31467:       unfolding cos_periodic_int . }
hoelzl@31467:   note negative_lx = this
hoelzl@31467: 
hoelzl@40881:   { assume "0 \<le> ?lx" and pi_x: "x - k * (2 * pi) \<le> pi"
hoelzl@31467:     with lx
hoelzl@40881:     have pi_lx: "?lx \<le> pi" and lx_0: "0 \<le> real ?lx"
hoelzl@47600:       by auto
hoelzl@29805: 
hoelzl@40881:     have "cos (x + (-k) * (2 * pi)) \<le> cos ?lx"
hoelzl@31467:       using cos_monotone_0_pi'[OF lx_0 lx pi_x]
hoelzl@47599:       by (simp only: real_of_int_minus
haftmann@54230:         cos_minus mult_minus_left) simp
hoelzl@40881:     also have "\<dots> \<le> (ub_cos prec ?lx)"
hoelzl@31467:       using lb_cos[OF lx_0 pi_lx] by simp
hoelzl@40881:     finally have "cos x \<le> (ub_cos prec ?lx)"
hoelzl@31467:       unfolding cos_periodic_int . }
hoelzl@31467:   note positive_lx = this
hoelzl@31467: 
hoelzl@40881:   { assume pi_x: "- pi \<le> x - k * (2 * pi)" and "?ux \<le> 0"
hoelzl@31467:     with ux
hoelzl@40881:     have pi_ux: "- pi \<le> ?ux" and ux_0: "real ?ux \<le> 0"
hoelzl@47600:       by simp_all
hoelzl@29805: 
hoelzl@40881:     have "cos (x + (-k) * (2 * pi)) \<le> cos (real (- ?ux))"
hoelzl@31467:       using cos_monotone_minus_pi_0'[OF pi_x ux ux_0]
hoelzl@47601:       by (simp only: uminus_float.rep_eq real_of_int_minus
haftmann@54230:           cos_minus mult_minus_left) simp
hoelzl@40881:     also have "\<dots> \<le> (ub_cos prec (- ?ux))"
hoelzl@31467:       using lb_cos_minus[OF pi_ux ux_0, of prec] by simp
hoelzl@40881:     finally have "cos x \<le> (ub_cos prec (- ?ux))"
hoelzl@31467:       unfolding cos_periodic_int . }
hoelzl@31467:   note negative_ux = this
hoelzl@31467: 
hoelzl@40881:   { assume "?ux \<le> ?lpi" and x_ge_0: "0 \<le> x - k * (2 * pi)"
hoelzl@31467:     with lpi ux
hoelzl@40881:     have pi_ux: "?ux \<le> pi" and ux_0: "0 \<le> real ?ux"
hoelzl@47600:       by simp_all
hoelzl@31467: 
hoelzl@40881:     have "(lb_cos prec ?ux) \<le> cos ?ux"
hoelzl@31467:       using lb_cos[OF ux_0 pi_ux] by simp
hoelzl@40881:     also have "\<dots> \<le> cos (x + (-k) * (2 * pi))"
hoelzl@31467:       using cos_monotone_0_pi'[OF x_ge_0 ux pi_ux]
hoelzl@47599:       by (simp only: real_of_int_minus
haftmann@54230:         cos_minus mult_minus_left) simp
hoelzl@40881:     finally have "(lb_cos prec ?ux) \<le> cos x"
hoelzl@31467:       unfolding cos_periodic_int . }
hoelzl@31467:   note positive_ux = this
hoelzl@31467: 
hoelzl@40881:   show "l \<le> cos x \<and> cos x \<le> u"
hoelzl@31467:   proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> 0")
hoelzl@31467:     case True with bnds
hoelzl@31467:     have l: "l = lb_cos prec (-?lx)"
hoelzl@31467:       and u: "u = ub_cos prec (-?ux)"
hoelzl@31467:       by (auto simp add: bnds_cos_def Let_def)
hoelzl@29805: 
hoelzl@31467:     from True lpi[THEN le_imp_neg_le] lx ux
hoelzl@40881:     have "- pi \<le> x - k * (2 * pi)"
hoelzl@40881:       and "x - k * (2 * pi) \<le> 0"
hoelzl@47600:       by auto
hoelzl@31467:     with True negative_ux negative_lx
hoelzl@31467:     show ?thesis unfolding l u by simp
hoelzl@31467:   next case False note 1 = this show ?thesis
hoelzl@31467:   proof (cases "0 \<le> ?lx \<and> ?ux \<le> ?lpi")
hoelzl@31467:     case True with bnds 1
hoelzl@31467:     have l: "l = lb_cos prec ?ux"
hoelzl@31467:       and u: "u = ub_cos prec ?lx"
hoelzl@31467:       by (auto simp add: bnds_cos_def Let_def)
hoelzl@29805: 
hoelzl@31467:     from True lpi lx ux
hoelzl@40881:     have "0 \<le> x - k * (2 * pi)"
hoelzl@40881:       and "x - k * (2 * pi) \<le> pi"
hoelzl@47600:       by auto
hoelzl@31467:     with True positive_ux positive_lx
hoelzl@31467:     show ?thesis unfolding l u by simp
hoelzl@31467:   next case False note 2 = this show ?thesis
hoelzl@31467:   proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> ?lpi")
hoelzl@31467:     case True note Cond = this with bnds 1 2
hoelzl@31467:     have l: "l = min (lb_cos prec (-?lx)) (lb_cos prec ?ux)"
hoelzl@31467:       and u: "u = Float 1 0"
hoelzl@31467:       by (auto simp add: bnds_cos_def Let_def)
hoelzl@29805: 
hoelzl@31467:     show ?thesis unfolding u l using negative_lx positive_ux Cond
hoelzl@47599:       by (cases "x - k * (2 * pi) < 0") (auto simp add: real_of_float_min)
hoelzl@47599: 
hoelzl@31467:   next case False note 3 = this show ?thesis
hoelzl@31467:   proof (cases "0 \<le> ?lx \<and> ?ux \<le> 2 * ?lpi")
hoelzl@31467:     case True note Cond = this with bnds 1 2 3
haftmann@58410:     have l: "l = Float (- 1) 0"
hoelzl@31467:       and u: "u = max (ub_cos prec ?lx) (ub_cos prec (- (?ux - 2 * ?lpi)))"
hoelzl@31467:       by (auto simp add: bnds_cos_def Let_def)
hoelzl@31467: 
hoelzl@31467:     have "cos x \<le> real u"
hoelzl@40881:     proof (cases "x - k * (2 * pi) < pi")
hoelzl@40881:       case True hence "x - k * (2 * pi) \<le> pi" by simp
hoelzl@31467:       from positive_lx[OF Cond[THEN conjunct1] this]
hoelzl@31467:       show ?thesis unfolding u by (simp add: real_of_float_max)
hoelzl@29805:     next
hoelzl@40881:       case False hence "pi \<le> x - k * (2 * pi)" by simp
hoelzl@40881:       hence pi_x: "- pi \<le> x - k * (2 * pi) - 2 * pi" by simp
hoelzl@40881: 
hoelzl@47600:       have "?ux \<le> 2 * pi" using Cond lpi by auto
hoelzl@40881:       hence "x - k * (2 * pi) - 2 * pi \<le> 0" using ux by simp
hoelzl@31467: 
hoelzl@31467:       have ux_0: "real (?ux - 2 * ?lpi) \<le> 0"
hoelzl@47600:         using Cond by auto
hoelzl@31467: 
hoelzl@31467:       from 2 and Cond have "\<not> ?ux \<le> ?lpi" by auto
hoelzl@47600:       hence "- ?lpi \<le> ?ux - 2 * ?lpi" by auto
hoelzl@40881:       hence pi_ux: "- pi \<le> (?ux - 2 * ?lpi)"
hoelzl@47600:         using lpi[THEN le_imp_neg_le] by auto
hoelzl@31467: 
hoelzl@40881:       have x_le_ux: "x - k * (2 * pi) - 2 * pi \<le> (?ux - 2 * ?lpi)"
wenzelm@32960:         using ux lpi by auto
hoelzl@40881:       have "cos x = cos (x + (-k) * (2 * pi) + (-1::int) * (2 * pi))"
wenzelm@32960:         unfolding cos_periodic_int ..
hoelzl@40881:       also have "\<dots> \<le> cos ((?ux - 2 * ?lpi))"
wenzelm@32960:         using cos_monotone_minus_pi_0'[OF pi_x x_le_ux ux_0]
haftmann@54489:         by (simp only: minus_float.rep_eq real_of_int_minus real_of_one
haftmann@54489:           mult_minus_left mult_1_left) simp
hoelzl@40881:       also have "\<dots> = cos ((- (?ux - 2 * ?lpi)))"
hoelzl@47601:         unfolding uminus_float.rep_eq cos_minus ..
hoelzl@40881:       also have "\<dots> \<le> (ub_cos prec (- (?ux - 2 * ?lpi)))"
wenzelm@32960:         using lb_cos_minus[OF pi_ux ux_0] by simp
hoelzl@31467:       finally show ?thesis unfolding u by (simp add: real_of_float_max)
hoelzl@29805:     qed
hoelzl@31467:     thus ?thesis unfolding l by auto
hoelzl@31508:   next case False note 4 = this show ?thesis
hoelzl@31508:   proof (cases "-2 * ?lpi \<le> ?lx \<and> ?ux \<le> 0")
hoelzl@31508:     case True note Cond = this with bnds 1 2 3 4
haftmann@58410:     have l: "l = Float (- 1) 0"
hoelzl@31508:       and u: "u = max (ub_cos prec (?lx + 2 * ?lpi)) (ub_cos prec (-?ux))"
hoelzl@47600:       by (auto simp add: bnds_cos_def Let_def)
hoelzl@31508: 
hoelzl@40881:     have "cos x \<le> u"
hoelzl@40881:     proof (cases "-pi < x - k * (2 * pi)")
hoelzl@40881:       case True hence "-pi \<le> x - k * (2 * pi)" by simp
hoelzl@31508:       from negative_ux[OF this Cond[THEN conjunct2]]
hoelzl@31508:       show ?thesis unfolding u by (simp add: real_of_float_max)
hoelzl@31508:     next
hoelzl@40881:       case False hence "x - k * (2 * pi) \<le> -pi" by simp
hoelzl@40881:       hence pi_x: "x - k * (2 * pi) + 2 * pi \<le> pi" by simp
hoelzl@40881: 
hoelzl@47600:       have "-2 * pi \<le> ?lx" using Cond lpi by auto
hoelzl@40881: 
hoelzl@40881:       hence "0 \<le> x - k * (2 * pi) + 2 * pi" using lx by simp
hoelzl@31508: 
hoelzl@31508:       have lx_0: "0 \<le> real (?lx + 2 * ?lpi)"
hoelzl@47600:         using Cond lpi by auto
hoelzl@31508: 
hoelzl@31508:       from 1 and Cond have "\<not> -?lpi \<le> ?lx" by auto
hoelzl@47600:       hence "?lx + 2 * ?lpi \<le> ?lpi" by auto
hoelzl@40881:       hence pi_lx: "(?lx + 2 * ?lpi) \<le> pi"
hoelzl@47600:         using lpi[THEN le_imp_neg_le] by auto
hoelzl@31508: 
hoelzl@40881:       have lx_le_x: "(?lx + 2 * ?lpi) \<le> x - k * (2 * pi) + 2 * pi"
wenzelm@32960:         using lx lpi by auto
hoelzl@31508: 
hoelzl@40881:       have "cos x = cos (x + (-k) * (2 * pi) + (1 :: int) * (2 * pi))"
wenzelm@32960:         unfolding cos_periodic_int ..
hoelzl@40881:       also have "\<dots> \<le> cos ((?lx + 2 * ?lpi))"
wenzelm@32960:         using cos_monotone_0_pi'[OF lx_0 lx_le_x pi_x]
hoelzl@47601:         by (simp only: minus_float.rep_eq real_of_int_minus real_of_one
haftmann@54489:           mult_minus_left mult_1_left) simp
hoelzl@40881:       also have "\<dots> \<le> (ub_cos prec (?lx + 2 * ?lpi))"
wenzelm@32960:         using lb_cos[OF lx_0 pi_lx] by simp
hoelzl@31508:       finally show ?thesis unfolding u by (simp add: real_of_float_max)
hoelzl@31508:     qed
hoelzl@31508:     thus ?thesis unfolding l by auto
hoelzl@29805:   next
hoelzl@31508:     case False with bnds 1 2 3 4 show ?thesis by (auto simp add: bnds_cos_def Let_def)
hoelzl@31508:   qed qed qed qed qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: section "Exponential function"
hoelzl@29805: 
hoelzl@29805: subsection "Compute the series of the exponential function"
hoelzl@29805: 
hoelzl@29805: fun ub_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" and lb_exp_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805: "ub_exp_horner prec 0 i k x       = 0" |
hoelzl@29805: "ub_exp_horner prec (Suc n) i k x = rapprox_rat prec 1 (int k) + x * lb_exp_horner prec n (i + 1) (k * i) x" |
hoelzl@29805: "lb_exp_horner prec 0 i k x       = 0" |
hoelzl@29805: "lb_exp_horner prec (Suc n) i k x = lapprox_rat prec 1 (int k) + x * ub_exp_horner prec n (i + 1) (k * i) x"
hoelzl@29805: 
hoelzl@31098: lemma bnds_exp_horner: assumes "real x \<le> 0"
hoelzl@40881:   shows "exp x \<in> { lb_exp_horner prec (get_even n) 1 1 x .. ub_exp_horner prec (get_odd n) 1 1 x }"
hoelzl@29805: proof -
hoelzl@29805:   { fix n
haftmann@30971:     have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m" by (induct n, auto)
haftmann@30971:     have "fact (Suc n) = fact n * ((\<lambda>i. i + 1) ^^ n) 1" unfolding F by auto } note f_eq = this
hoelzl@31467: 
hoelzl@29805:   note bounds = horner_bounds_nonpos[where f="fact" and lb="lb_exp_horner prec" and ub="ub_exp_horner prec" and j'=0 and s=1,
hoelzl@29805:     OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]
hoelzl@29805: 
hoelzl@40881:   { have "lb_exp_horner prec (get_even n) 1 1 x \<le> (\<Sum>j = 0..<get_even n. 1 / real (fact j) * real x ^ j)"
hoelzl@29805:       using bounds(1) by auto
hoelzl@40881:     also have "\<dots> \<le> exp x"
hoelzl@29805:     proof -
hoelzl@40881:       obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp x = (\<Sum>m = 0..<get_even n. real x ^ m / real (fact m)) + exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
huffman@56195:         using Maclaurin_exp_le unfolding atLeast0LessThan by blast
hoelzl@31098:       moreover have "0 \<le> exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
hoelzl@56571:         by (auto simp: zero_le_even_power)
nipkow@56536:       ultimately show ?thesis using get_odd exp_gt_zero by auto
hoelzl@29805:     qed
hoelzl@40881:     finally have "lb_exp_horner prec (get_even n) 1 1 x \<le> exp x" .
hoelzl@29805:   } moreover
hoelzl@31809:   {
hoelzl@31098:     have x_less_zero: "real x ^ get_odd n \<le> 0"
hoelzl@31098:     proof (cases "real x = 0")
hoelzl@29805:       case True
hoelzl@29805:       have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
hoelzl@29805:       thus ?thesis unfolding True power_0_left by auto
hoelzl@29805:     next
hoelzl@31098:       case False hence "real x < 0" using `real x \<le> 0` by auto
haftmann@46545:       show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq `real x < 0`)
hoelzl@29805:     qed
hoelzl@29805: 
hoelzl@40881:     obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp x = (\<Sum>m = 0..<get_odd n. (real x) ^ m / real (fact m)) + exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n)"
huffman@56195:       using Maclaurin_exp_le unfolding atLeast0LessThan by blast
hoelzl@31098:     moreover have "exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n) \<le> 0"
haftmann@46545:       by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero)
hoelzl@40881:     ultimately have "exp x \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * real x ^ j)"
nipkow@56536:       using get_odd exp_gt_zero by auto
hoelzl@40881:     also have "\<dots> \<le> ub_exp_horner prec (get_odd n) 1 1 x"
hoelzl@29805:       using bounds(2) by auto
hoelzl@40881:     finally have "exp x \<le> ub_exp_horner prec (get_odd n) 1 1 x" .
hoelzl@29805:   } ultimately show ?thesis by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: subsection "Compute the exponential function on the entire domain"
hoelzl@29805: 
hoelzl@29805: function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805: "lb_exp prec x = (if 0 < x then float_divl prec 1 (ub_exp prec (-x))
hoelzl@31809:              else let
haftmann@58410:                 horner = (\<lambda> x. let  y = lb_exp_horner prec (get_even (prec + 2)) 1 1 x  in if y \<le> 0 then Float 1 (- 2) else y)
hoelzl@47599:              in if x < - 1 then (horner (float_divl prec x (- floor_fl x))) ^ nat (- int_floor_fl x)
hoelzl@29805:                            else horner x)" |
hoelzl@29805: "ub_exp prec x = (if 0 < x    then float_divr prec 1 (lb_exp prec (-x))
hoelzl@47599:              else if x < - 1  then ub_exp_horner prec (get_odd (prec + 2)) 1 1 (float_divr prec x (- floor_fl x)) ^ (nat (- int_floor_fl x))
hoelzl@29805:                               else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
hoelzl@29805: by pat_completeness auto
blanchet@55414: termination by (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if 0 < x then 1 else 0))", auto)
hoelzl@29805: 
hoelzl@29805: lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)"
hoelzl@29805: proof -
hoelzl@29805:   have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
hoelzl@29805: 
haftmann@58410:   have "1 / 4 = (Float 1 (- 2))" unfolding Float_num by auto
hoelzl@40881:   also have "\<dots> \<le> lb_exp_horner 1 (get_even 4) 1 1 (- 1)"
hoelzl@31809:     unfolding get_even_def eq4
hoelzl@47621:     by (auto simp add: Float.compute_lapprox_rat Float.compute_rapprox_rat
hoelzl@47621:                   Float.compute_lapprox_posrat Float.compute_rapprox_posrat rat_precision_def Float.compute_bitlen)
hoelzl@40881:   also have "\<dots> \<le> exp (- 1 :: float)" using bnds_exp_horner[where x="- 1"] by auto
hoelzl@47600:   finally show ?thesis by simp
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma lb_exp_pos: assumes "\<not> 0 < x" shows "0 < lb_exp prec x"
hoelzl@29805: proof -
hoelzl@29805:   let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
haftmann@58410:   let "?horner x" = "let  y = ?lb_horner x  in if y \<le> 0 then Float 1 (- 2) else y"
hoelzl@47600:   have pos_horner: "\<And> x. 0 < ?horner x" unfolding Let_def by (cases "?lb_horner x \<le> 0", auto)
hoelzl@29805:   moreover { fix x :: float fix num :: nat
hoelzl@47600:     have "0 < real (?horner x) ^ num" using `0 < ?horner x` by simp
hoelzl@47599:     also have "\<dots> = (?horner x) ^ num" by auto
hoelzl@31098:     finally have "0 < real ((?horner x) ^ num)" .
hoelzl@29805:   }
hoelzl@29805:   ultimately show ?thesis
haftmann@30968:     unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def
hoelzl@47600:     by (cases "floor_fl x", cases "x < - 1", auto)
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma exp_boundaries': assumes "x \<le> 0"
hoelzl@40881:   shows "exp x \<in> { (lb_exp prec x) .. (ub_exp prec x)}"
hoelzl@29805: proof -
hoelzl@29805:   let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
hoelzl@29805:   let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"
hoelzl@29805: 
hoelzl@47600:   have "real x \<le> 0" and "\<not> x > 0" using `x \<le> 0` by auto
hoelzl@29805:   show ?thesis
hoelzl@29805:   proof (cases "x < - 1")
hoelzl@47600:     case False hence "- 1 \<le> real x" by auto
hoelzl@29805:     show ?thesis
hoelzl@29805:     proof (cases "?lb_exp_horner x \<le> 0")
hoelzl@47600:       from `\<not> x < - 1` have "- 1 \<le> real x" by auto
hoelzl@40881:       hence "exp (- 1) \<le> exp x" unfolding exp_le_cancel_iff .
hoelzl@29805:       from order_trans[OF exp_m1_ge_quarter this]
haftmann@58410:       have "Float 1 (- 2) \<le> exp x" unfolding Float_num .
hoelzl@29805:       moreover case True
hoelzl@31098:       ultimately show ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by auto
hoelzl@29805:     next
hoelzl@31098:       case False thus ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by (auto simp add: Let_def)
hoelzl@29805:     qed
hoelzl@29805:   next
hoelzl@29805:     case True
hoelzl@31809: 
hoelzl@47599:     let ?num = "nat (- int_floor_fl x)"
hoelzl@47599: 
hoelzl@47600:     have "real (int_floor_fl x) < - 1" using int_floor_fl[of x] `x < - 1`
hoelzl@47600:       by simp
hoelzl@47599:     hence "real (int_floor_fl x) < 0" by simp
hoelzl@47599:     hence "int_floor_fl x < 0" by auto
hoelzl@47599:     hence "1 \<le> - int_floor_fl x" by auto
hoelzl@47599:     hence "0 < nat (- int_floor_fl x)" by auto
hoelzl@47599:     hence "0 < ?num"  by auto
hoelzl@29805:     hence "real ?num \<noteq> 0" by auto
hoelzl@47599:     have num_eq: "real ?num = - int_floor_fl x" using `0 < nat (- int_floor_fl x)` by auto
hoelzl@47599:     have "0 < - int_floor_fl x" using `0 < ?num`[unfolded real_of_nat_less_iff[symmetric]] by simp
hoelzl@47599:     hence "real (int_floor_fl x) < 0" unfolding less_float_def by auto
hoelzl@47599:     have fl_eq: "real (- int_floor_fl x) = real (- floor_fl x)"
hoelzl@47599:       by (simp add: floor_fl_def int_floor_fl_def)
hoelzl@47599:     from `0 < - int_floor_fl x` have "0 < real (- floor_fl x)"
hoelzl@47599:       by (simp add: floor_fl_def int_floor_fl_def)
hoelzl@47599:     from `real (int_floor_fl x) < 0` have "real (floor_fl x) < 0"
hoelzl@47599:       by (simp add: floor_fl_def int_floor_fl_def)
hoelzl@40881:     have "exp x \<le> ub_exp prec x"
hoelzl@29805:     proof -
hoelzl@31809:       have div_less_zero: "real (float_divr prec x (- floor_fl x)) \<le> 0"
hoelzl@47599:         using float_divr_nonpos_pos_upper_bound[OF `real x \<le> 0` `0 < real (- floor_fl x)`]
hoelzl@47601:         unfolding less_eq_float_def zero_float.rep_eq .
hoelzl@31809: 
hoelzl@56479:       have "exp x = exp (?num * (x / ?num))" using `real ?num \<noteq> 0` by auto
hoelzl@40881:       also have "\<dots> = exp (x / ?num) ^ ?num" unfolding exp_real_of_nat_mult ..
hoelzl@47599:       also have "\<dots> \<le> exp (float_divr prec x (- floor_fl x)) ^ ?num" unfolding num_eq fl_eq
wenzelm@32960:         by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
hoelzl@47599:       also have "\<dots> \<le> (?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num"
hoelzl@47599:         unfolding real_of_float_power
wenzelm@32960:         by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
hoelzl@47599:       finally show ?thesis unfolding ub_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] floor_fl_def Let_def .
hoelzl@29805:     qed
hoelzl@31809:     moreover
hoelzl@40881:     have "lb_exp prec x \<le> exp x"
hoelzl@29805:     proof -
hoelzl@47599:       let ?divl = "float_divl prec x (- floor_fl x)"
hoelzl@29805:       let ?horner = "?lb_exp_horner ?divl"
hoelzl@31809: 
hoelzl@29805:       show ?thesis
hoelzl@29805:       proof (cases "?horner \<le> 0")
hoelzl@47600:         case False hence "0 \<le> real ?horner" by auto
wenzelm@32960: 
wenzelm@32960:         have div_less_zero: "real (float_divl prec x (- floor_fl x)) \<le> 0"
hoelzl@56479:           using `real (floor_fl x) < 0` `real x \<le> 0` by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
hoelzl@56479: 
hoelzl@40881:         have "(?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num \<le>
hoelzl@47599:           exp (float_divl prec x (- floor_fl x)) ^ ?num"
hoelzl@47599:           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)
hoelzl@47599:         also have "\<dots> \<le> exp (x / ?num) ^ ?num" unfolding num_eq fl_eq
hoelzl@47601:           using float_divl by (auto intro!: power_mono simp del: uminus_float.rep_eq)
hoelzl@40881:         also have "\<dots> = exp (?num * (x / ?num))" unfolding exp_real_of_nat_mult ..
hoelzl@56479:         also have "\<dots> = exp x" using `real ?num \<noteq> 0` by auto
wenzelm@32960:         finally show ?thesis
hoelzl@47599:           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
hoelzl@29805:       next
wenzelm@32960:         case True
wenzelm@32960:         have "real (floor_fl x) \<noteq> 0" and "real (floor_fl x) \<le> 0" using `real (floor_fl x) < 0` by auto
wenzelm@32960:         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`]]
hoelzl@47601:         have "- 1 \<le> x / (- floor_fl x)" unfolding minus_float.rep_eq by auto
wenzelm@32960:         from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
haftmann@58410:         have "Float 1 (- 2) \<le> exp (x / (- floor_fl x))" unfolding Float_num .
haftmann@58410:         hence "real (Float 1 (- 2)) ^ ?num \<le> exp (x / (- floor_fl x)) ^ ?num"
haftmann@46545:           by (auto intro!: power_mono)
hoelzl@56479:         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
wenzelm@32960:         finally show ?thesis
hoelzl@47599:           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 .
hoelzl@29805:       qed
hoelzl@29805:     qed
hoelzl@29805:     ultimately show ?thesis by auto
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@40881: lemma exp_boundaries: "exp x \<in> { lb_exp prec x .. ub_exp prec x }"
hoelzl@29805: proof -
hoelzl@29805:   show ?thesis
hoelzl@29805:   proof (cases "0 < x")
hoelzl@47600:     case False hence "x \<le> 0" by auto
hoelzl@29805:     from exp_boundaries'[OF this] show ?thesis .
hoelzl@29805:   next
hoelzl@47600:     case True hence "-x \<le> 0" by auto
hoelzl@31809: 
hoelzl@40881:     have "lb_exp prec x \<le> exp x"
hoelzl@29805:     proof -
hoelzl@29805:       from exp_boundaries'[OF `-x \<le> 0`]
hoelzl@47601:       have ub_exp: "exp (- real x) \<le> ub_exp prec (-x)" unfolding atLeastAtMost_iff minus_float.rep_eq by auto
hoelzl@40881: 
hoelzl@40881:       have "float_divl prec 1 (ub_exp prec (-x)) \<le> 1 / ub_exp prec (-x)" using float_divl[where x=1] by auto
hoelzl@40881:       also have "\<dots> \<le> exp x"
wenzelm@32960:         using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] exp_gt_zero, symmetric]]
wenzelm@32960:         unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide by auto
hoelzl@29805:       finally show ?thesis unfolding lb_exp.simps if_P[OF True] .
hoelzl@29805:     qed
hoelzl@29805:     moreover
hoelzl@40881:     have "exp x \<le> ub_exp prec x"
hoelzl@29805:     proof -
hoelzl@47600:       have "\<not> 0 < -x" using `0 < x` by auto
hoelzl@31809: 
hoelzl@29805:       from exp_boundaries'[OF `-x \<le> 0`]
hoelzl@47601:       have lb_exp: "lb_exp prec (-x) \<le> exp (- real x)" unfolding atLeastAtMost_iff minus_float.rep_eq by auto
hoelzl@40881: 
hoelzl@40881:       have "exp x \<le> (1 :: float) / lb_exp prec (-x)"
hoelzl@47600:         using lb_exp lb_exp_pos[OF `\<not> 0 < -x`, of prec]
hoelzl@47600:         by (simp del: lb_exp.simps add: exp_minus inverse_eq_divide field_simps)
hoelzl@40881:       also have "\<dots> \<le> float_divr prec 1 (lb_exp prec (-x))" using float_divr .
hoelzl@29805:       finally show ?thesis unfolding ub_exp.simps if_P[OF True] .
hoelzl@29805:     qed
hoelzl@29805:     ultimately show ?thesis by auto
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@40881: 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"
hoelzl@29805: proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@40881:   fix x::real and lx ux
hoelzl@40881:   assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {lx .. ux}"
hoelzl@40881:   hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {lx .. ux}" by auto
hoelzl@29805: 
hoelzl@29805:   { from exp_boundaries[of lx prec, unfolded l]
hoelzl@40881:     have "l \<le> exp lx" by (auto simp del: lb_exp.simps)
hoelzl@29805:     also have "\<dots> \<le> exp x" using x by auto
hoelzl@40881:     finally have "l \<le> exp x" .
hoelzl@29805:   } moreover
hoelzl@40881:   { have "exp x \<le> exp ux" using x by auto
hoelzl@40881:     also have "\<dots> \<le> u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
hoelzl@40881:     finally have "exp x \<le> u" .
hoelzl@40881:   } ultimately show "l \<le> exp x \<and> exp x \<le> u" ..
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: section "Logarithm"
hoelzl@29805: 
hoelzl@29805: subsection "Compute the logarithm series"
hoelzl@29805: 
hoelzl@31809: fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
hoelzl@29805: and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
hoelzl@29805: "ub_ln_horner prec 0 i x       = 0" |
hoelzl@29805: "ub_ln_horner prec (Suc n) i x = rapprox_rat prec 1 (int i) - x * lb_ln_horner prec n (Suc i) x" |
hoelzl@29805: "lb_ln_horner prec 0 i x       = 0" |
hoelzl@29805: "lb_ln_horner prec (Suc n) i x = lapprox_rat prec 1 (int i) - x * ub_ln_horner prec n (Suc i) x"
hoelzl@29805: 
hoelzl@29805: lemma ln_bounds:
hoelzl@29805:   assumes "0 \<le> x" and "x < 1"
haftmann@58410:   shows "(\<Sum>i=0..<2*n. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i)) \<le> ln (x + 1)" (is "?lb")
haftmann@58410:   and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. (- 1) ^ i * (1 / real (i + 1)) * x ^ (Suc i))" (is "?ub")
hoelzl@29805: proof -
haftmann@30952:   let "?a n" = "(1/real (n +1)) * x ^ (Suc n)"
hoelzl@29805: 
haftmann@58410:   have ln_eq: "(\<Sum> i. (- 1) ^ i * ?a i) = ln (x + 1)"
hoelzl@29805:     using ln_series[of "x + 1"] `0 \<le> x` `x < 1` by auto
hoelzl@29805: 
hoelzl@29805:   have "norm x < 1" using assms by auto
hoelzl@31809:   have "?a ----> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric]
huffman@44568:     using tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF `norm x < 1`]]] by auto
nipkow@56536:   { fix n have "0 \<le> ?a n" by (rule mult_nonneg_nonneg, auto simp: `0 \<le> x`) }
hoelzl@29805:   { fix n have "?a (Suc n) \<le> ?a n" unfolding inverse_eq_divide[symmetric]
hoelzl@29805:     proof (rule mult_mono)
nipkow@56536:       show "0 \<le> x ^ Suc (Suc n)" by (auto simp add: `0 \<le> x`)
haftmann@57512:       have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 mult.assoc[symmetric]
nipkow@56536:         by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto simp: `0 \<le> x`)
hoelzl@29805:       thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
hoelzl@29805:     qed auto }
hoelzl@29805:   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]
huffman@56195:   show "?lb" and "?ub" unfolding atLeast0LessThan by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@31809: lemma ln_float_bounds:
hoelzl@31098:   assumes "0 \<le> real x" and "real x < 1"
hoelzl@40881:   shows "x * lb_ln_horner prec (get_even n) 1 x \<le> ln (x + 1)" (is "?lb \<le> ?ln")
hoelzl@40881:   and "ln (x + 1) \<le> x * ub_ln_horner prec (get_odd n) 1 x" (is "?ln \<le> ?ub")
hoelzl@29805: proof -
hoelzl@29805:   obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
hoelzl@29805:   obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..
hoelzl@29805: 
haftmann@58410:   let "?s n" = "(- 1) ^ n * (1 / real (1 + n)) * (real x)^(Suc n)"
hoelzl@29805: 
haftmann@57512:   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
hoelzl@29805:     using horner_bounds(1)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*ev",
hoelzl@31098:       OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
hoelzl@29805:     by (rule mult_right_mono)
hoelzl@31098:   also have "\<dots> \<le> ?ln" using ln_bounds(1)[OF `0 \<le> real x` `real x < 1`] by auto
hoelzl@31809:   finally show "?lb \<le> ?ln" .
hoelzl@29805: 
hoelzl@31098:   have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> real x` `real x < 1`] by auto
haftmann@57512:   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
hoelzl@29805:     using horner_bounds(2)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*od+1",
hoelzl@31098:       OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
hoelzl@29805:     by (rule mult_right_mono)
hoelzl@31809:   finally show "?ln \<le> ?ub" .
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma ln_add: assumes "0 < x" and "0 < y" shows "ln (x + y) = ln x + ln (1 + y / x)"
hoelzl@29805: proof -
hoelzl@29805:   have "x \<noteq> 0" using assms by auto
webertj@49962:   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
hoelzl@31809:   moreover
nipkow@56541:   have "0 < y / x" using assms by auto
hoelzl@29805:   hence "0 < 1 + y / x" by auto
hoelzl@29805:   ultimately show ?thesis using ln_mult assms by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: subsection "Compute the logarithm of 2"
hoelzl@29805: 
hoelzl@31809: definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3
haftmann@58410:                                         in (Float 1 (- 1) * ub_ln_horner prec (get_odd prec) 1 (Float 1 (- 1))) +
hoelzl@29805:                                            (third * ub_ln_horner prec (get_odd prec) 1 third))"
hoelzl@31809: definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3
haftmann@58410:                                         in (Float 1 (- 1) * lb_ln_horner prec (get_even prec) 1 (Float 1 (- 1))) +
hoelzl@29805:                                            (third * lb_ln_horner prec (get_even prec) 1 third))"
hoelzl@29805: 
hoelzl@40881: lemma ub_ln2: "ln 2 \<le> ub_ln2 prec" (is "?ub_ln2")
hoelzl@40881:   and lb_ln2: "lb_ln2 prec \<le> ln 2" (is "?lb_ln2")
hoelzl@29805: proof -
hoelzl@29805:   let ?uthird = "rapprox_rat (max prec 1) 1 3"
hoelzl@29805:   let ?lthird = "lapprox_rat prec 1 3"
hoelzl@29805: 
hoelzl@29805:   have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1)"
hoelzl@29805:     using ln_add[of "3 / 2" "1 / 2"] by auto
hoelzl@40881:   have lb3: "?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
hoelzl@31098:   hence lb3_ub: "real ?lthird < 1" by auto
hoelzl@47599:   have lb3_lb: "0 \<le> real ?lthird" using lapprox_rat_nonneg[of 1 3] by auto
hoelzl@40881:   have ub3: "1 / 3 \<le> ?uthird" using rapprox_rat[of 1 3] by auto
hoelzl@31098:   hence ub3_lb: "0 \<le> real ?uthird" by auto
hoelzl@29805: 
haftmann@58410:   have lb2: "0 \<le> real (Float 1 (- 1))" and ub2: "real (Float 1 (- 1)) < 1" unfolding Float_num by auto
hoelzl@29805: 
hoelzl@29805:   have "0 \<le> (1::int)" and "0 < (3::int)" by auto
hoelzl@47621:   have ub3_ub: "real ?uthird < 1" by (simp add: Float.compute_rapprox_rat rapprox_posrat_less1)
hoelzl@29805: 
hoelzl@29805:   have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
hoelzl@31098:   have uthird_gt0: "0 < real ?uthird + 1" using ub3_lb by auto
hoelzl@31098:   have lthird_gt0: "0 < real ?lthird + 1" using lb3_lb by auto
hoelzl@29805: 
hoelzl@47601:   show ?ub_ln2 unfolding ub_ln2_def Let_def plus_float.rep_eq ln2_sum Float_num(4)[symmetric]
hoelzl@29805:   proof (rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
hoelzl@31098:     have "ln (1 / 3 + 1) \<le> ln (real ?uthird + 1)" unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
hoelzl@40881:     also have "\<dots> \<le> ?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird"
hoelzl@29805:       using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
hoelzl@40881:     finally show "ln (1 / 3 + 1) \<le> ?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird" .
hoelzl@29805:   qed
hoelzl@47601:   show ?lb_ln2 unfolding lb_ln2_def Let_def plus_float.rep_eq ln2_sum Float_num(4)[symmetric]
hoelzl@29805:   proof (rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
hoelzl@40881:     have "?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird \<le> ln (real ?lthird + 1)"
hoelzl@29805:       using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
hoelzl@29805:     also have "\<dots> \<le> ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto
hoelzl@40881:     finally show "?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird \<le> ln (1 / 3 + 1)" .
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: subsection "Compute the logarithm in the entire domain"
hoelzl@29805: 
hoelzl@29805: function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where
hoelzl@31468: "ub_ln prec x = (if x \<le> 0          then None
hoelzl@31468:             else if x < 1          then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
hoelzl@31468:             else let horner = \<lambda>x. x * ub_ln_horner prec (get_odd prec) 1 x in
haftmann@58410:                  if x \<le> Float 3 (- 1) then Some (horner (x - 1))
haftmann@58410:             else if x < Float 1 1  then Some (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1))
hoelzl@31468:                                    else let l = bitlen (mantissa x) - 1 in
hoelzl@47599:                                         Some (ub_ln2 prec * (Float (exponent x + l) 0) + horner (Float (mantissa x) (- l) - 1)))" |
hoelzl@31468: "lb_ln prec x = (if x \<le> 0          then None
hoelzl@31468:             else if x < 1          then Some (- the (ub_ln prec (float_divr prec 1 x)))
hoelzl@31468:             else let horner = \<lambda>x. x * lb_ln_horner prec (get_even prec) 1 x in
haftmann@58410:                  if x \<le> Float 3 (- 1) then Some (horner (x - 1))
haftmann@58410:             else if x < Float 1 1  then Some (horner (Float 1 (- 1)) +
hoelzl@31468:                                               horner (max (x * lapprox_rat prec 2 3 - 1) 0))
hoelzl@31468:                                    else let l = bitlen (mantissa x) - 1 in
hoelzl@47599:                                         Some (lb_ln2 prec * (Float (exponent x + l) 0) + horner (Float (mantissa x) (- l) - 1)))"
hoelzl@29805: by pat_completeness auto
hoelzl@29805: 
blanchet@55414: termination proof (relation "measure (\<lambda> v. let (prec, x) = case_sum id id v in (if x < 1 then 1 else 0))", auto)
hoelzl@47600:   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"
hoelzl@47600:   hence "0 < real x" "1 \<le> max prec (Suc 0)" "real x < 1" by auto
hoelzl@47599:   from float_divl_pos_less1_bound[OF `0 < real x` `real x < 1` `1 \<le> max prec (Suc 0)`]
hoelzl@47600:   show False using `real (float_divl (max prec (Suc 0)) 1 x) < 1` by auto
hoelzl@29805: next
hoelzl@47600:   fix prec x assume "\<not> real x \<le> 0" and "real x < 1" and "real (float_divr prec 1 x) < 1"
hoelzl@47600:   hence "0 < x" by auto
hoelzl@47600:   from float_divr_pos_less1_lower_bound[OF `0 < x`, of prec] `real x < 1`
hoelzl@47600:   show False using `real (float_divr prec 1 x) < 1` by auto
hoelzl@47599: qed
hoelzl@47599: 
hoelzl@47599: lemma float_pos_eq_mantissa_pos:  "x > 0 \<longleftrightarrow> mantissa x > 0"
hoelzl@47599:   apply (subst Float_mantissa_exponent[of x, symmetric])
hoelzl@47599:   apply (auto simp add: zero_less_mult_iff zero_float_def powr_gt_zero[of 2 "exponent x"] dest: less_zeroE)
hoelzl@47599:   using powr_gt_zero[of 2 "exponent x"]
hoelzl@47599:   apply simp
hoelzl@47599:   done
hoelzl@47599: 
hoelzl@47599: lemma Float_pos_eq_mantissa_pos:  "Float m e > 0 \<longleftrightarrow> m > 0"
hoelzl@47599:   using powr_gt_zero[of 2 "e"]
hoelzl@54269:   by (auto simp add: zero_less_mult_iff zero_float_def simp del: powr_gt_zero dest: less_zeroE)
hoelzl@47599: 
hoelzl@47599: lemma Float_representation_aux:
hoelzl@47599:   fixes m e
hoelzl@47599:   defines "x \<equiv> Float m e"
hoelzl@47599:   assumes "x > 0"
hoelzl@47599:   shows "Float (exponent x + (bitlen (mantissa x) - 1)) 0 = Float (e + (bitlen m - 1)) 0" (is ?th1)
hoelzl@47599:     and "Float (mantissa x) (- (bitlen (mantissa x) - 1)) = Float m ( - (bitlen m - 1))"  (is ?th2)
hoelzl@47599: proof -
hoelzl@47599:   from assms have mantissa_pos: "m > 0" "mantissa x > 0"
hoelzl@47600:     using Float_pos_eq_mantissa_pos[of m e] float_pos_eq_mantissa_pos[of x] by simp_all
hoelzl@47600:   thus ?th1 using bitlen_Float[of m e] assms by (auto simp add: zero_less_mult_iff intro!: arg_cong2[where f=Float])
hoelzl@47600:   have "x \<noteq> float_of 0"
hoelzl@47600:     unfolding zero_float_def[symmetric] using `0 < x` by auto
hoelzl@47599:   from denormalize_shift[OF assms(1) this] guess i . note i = this
hoelzl@47600: 
hoelzl@47599:   have "2 powr (1 - (real (bitlen (mantissa x)) + real i)) =
hoelzl@47599:     2 powr (1 - (real (bitlen (mantissa x)))) * inverse (2 powr (real i))"
hoelzl@47599:     by (simp add: powr_minus[symmetric] powr_add[symmetric] field_simps)
hoelzl@47599:   hence "real (mantissa x) * 2 powr (1 - real (bitlen (mantissa x))) =
hoelzl@47599:     (real (mantissa x) * 2 ^ i) * 2 powr (1 - real (bitlen (mantissa x * 2 ^ i)))"
hoelzl@47599:     using `mantissa x > 0` by (simp add: powr_realpow)
hoelzl@47600:   then show ?th2
hoelzl@47600:     unfolding i by transfer auto
hoelzl@47599: qed
hoelzl@47599: 
hoelzl@47599: lemma compute_ln[code]:
hoelzl@47599:   fixes m e
hoelzl@47599:   defines "x \<equiv> Float m e"
hoelzl@47599:   shows "ub_ln prec x = (if x \<le> 0          then None
hoelzl@47599:               else if x < 1          then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
hoelzl@47599:             else let horner = \<lambda>x. x * ub_ln_horner prec (get_odd prec) 1 x in
haftmann@58410:                  if x \<le> Float 3 (- 1) then Some (horner (x - 1))
haftmann@58410:             else if x < Float 1 1  then Some (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1))
hoelzl@47599:                                    else let l = bitlen m - 1 in
hoelzl@47599:                                         Some (ub_ln2 prec * (Float (e + l) 0) + horner (Float m (- l) - 1)))"
hoelzl@47599:     (is ?th1)
hoelzl@47599:   and "lb_ln prec x = (if x \<le> 0          then None
hoelzl@47599:             else if x < 1          then Some (- the (ub_ln prec (float_divr prec 1 x)))
hoelzl@47599:             else let horner = \<lambda>x. x * lb_ln_horner prec (get_even prec) 1 x in
haftmann@58410:                  if x \<le> Float 3 (- 1) then Some (horner (x - 1))
haftmann@58410:             else if x < Float 1 1  then Some (horner (Float 1 (- 1)) +
hoelzl@47599:                                               horner (max (x * lapprox_rat prec 2 3 - 1) 0))
hoelzl@47599:                                    else let l = bitlen m - 1 in
hoelzl@47599:                                         Some (lb_ln2 prec * (Float (e + l) 0) + horner (Float m (- l) - 1)))"
hoelzl@47599:     (is ?th2)
hoelzl@47599: proof -
hoelzl@47599:   from assms Float_pos_eq_mantissa_pos have "x > 0 \<Longrightarrow> m > 0" by simp
hoelzl@47599:   thus ?th1 ?th2 using Float_representation_aux[of m e] unfolding x_def[symmetric]
hoelzl@47600:     by (auto dest: not_leE)
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@40881: lemma ln_shifted_float: assumes "0 < m" shows "ln (Float m e) = ln 2 * (e + (bitlen m - 1)) + ln (Float m (- (bitlen m - 1)))"
hoelzl@29805: proof -
hoelzl@29805:   let ?B = "2^nat (bitlen m - 1)"
hoelzl@47599:   def bl \<equiv> "bitlen m - 1"
hoelzl@29805:   have "0 < real m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0" using assms by auto
hoelzl@47599:   hence "0 \<le> bl" by (simp add: bitlen_def bl_def)
hoelzl@31468:   show ?thesis
hoelzl@29805:   proof (cases "0 \<le> e")
hoelzl@47599:     case True 
hoelzl@47599:     thus ?thesis
hoelzl@47599:       unfolding bl_def[symmetric] using `0 < real m` `0 \<le> bl`
hoelzl@47599:       apply (simp add: ln_mult)
hoelzl@47599:       apply (cases "e=0")
hoelzl@47599:         apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr)
hoelzl@47599:         apply (cases "bl = 0", simp_all add: powr_minus ln_inverse ln_powr field_simps)
hoelzl@47599:       done
hoelzl@29805:   next
hoelzl@29805:     case False hence "0 < -e" by auto
hoelzl@47599:     have lne: "ln (2 powr real e) = ln (inverse (2 powr - e))" by (simp add: powr_minus)
hoelzl@29805:     hence pow_gt0: "(0::real) < 2^nat (-e)" by auto
hoelzl@29805:     hence inv_gt0: "(0::real) < inverse (2^nat (-e))" by auto
hoelzl@47599:     show ?thesis using False unfolding bl_def[symmetric] using `0 < real m` `0 \<le> bl`
hoelzl@56483:       by (auto simp add: lne ln_mult ln_powr ln_div field_simps)
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma ub_ln_lb_ln_bounds': assumes "1 \<le> x"
hoelzl@40881:   shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)"
hoelzl@29805:   (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
hoelzl@29805: proof (cases "x < Float 1 1")
hoelzl@31468:   case True
hoelzl@47600:   hence "real (x - 1) < 1" and "real x < 2" by auto
hoelzl@47600:   have "\<not> x \<le> 0" and "\<not> x < 1" using `1 \<le> x` by auto
hoelzl@47600:   hence "0 \<le> real (x - 1)" using `1 \<le> x` by auto
hoelzl@31468: 
haftmann@58410:   have [simp]: "(Float 3 (- 1)) = 3 / 2" by simp
hoelzl@31468: 
hoelzl@31468:   show ?thesis
haftmann@58410:   proof (cases "x \<le> Float 3 (- 1)")
hoelzl@31468:     case True
hoelzl@31468:     show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
hoelzl@31468:       using ln_float_bounds[OF `0 \<le> real (x - 1)` `real (x - 1) < 1`, of prec] `\<not> x \<le> 0` `\<not> x < 1` True
hoelzl@31468:       by auto
hoelzl@31468:   next
hoelzl@47600:     case False hence *: "3 / 2 < x" by auto
hoelzl@40881: 
hoelzl@40881:     with ln_add[of "3 / 2" "x - 3 / 2"]
hoelzl@40881:     have add: "ln x = ln (3 / 2) + ln (real x * 2 / 3)"
hoelzl@31468:       by (auto simp add: algebra_simps diff_divide_distrib)
hoelzl@31468: 
hoelzl@31468:     let "?ub_horner x" = "x * ub_ln_horner prec (get_odd prec) 1 x"
hoelzl@31468:     let "?lb_horner x" = "x * lb_ln_horner prec (get_even prec) 1 x"
hoelzl@31468: 
hoelzl@31468:     { have up: "real (rapprox_rat prec 2 3) \<le> 1"
wenzelm@32960:         by (rule rapprox_rat_le1) simp_all
hoelzl@40881:       have low: "2 / 3 \<le> rapprox_rat prec 2 3"
wenzelm@32960:         by (rule order_trans[OF _ rapprox_rat]) simp
hoelzl@31468:       from mult_less_le_imp_less[OF * low] *
hoelzl@31468:       have pos: "0 < real (x * rapprox_rat prec 2 3 - 1)" by auto
hoelzl@31468: 
hoelzl@31468:       have "ln (real x * 2/3)
wenzelm@32960:         \<le> ln (real (x * rapprox_rat prec 2 3 - 1) + 1)"
hoelzl@31468:       proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
wenzelm@32960:         show "real x * 2 / 3 \<le> real (x * rapprox_rat prec 2 3 - 1) + 1"
wenzelm@32960:           using * low by auto
wenzelm@32960:         show "0 < real x * 2 / 3" using * by simp
wenzelm@32960:         show "0 < real (x * rapprox_rat prec 2 3 - 1) + 1" using pos by auto
hoelzl@31468:       qed
hoelzl@40881:       also have "\<dots> \<le> ?ub_horner (x * rapprox_rat prec 2 3 - 1)"
hoelzl@31468:       proof (rule ln_float_bounds(2))
wenzelm@32960:         from mult_less_le_imp_less[OF `real x < 2` up] low *
wenzelm@32960:         show "real (x * rapprox_rat prec 2 3 - 1) < 1" by auto
wenzelm@32960:         show "0 \<le> real (x * rapprox_rat prec 2 3 - 1)" using pos by auto
hoelzl@31468:       qed
hoelzl@40881:       finally have "ln x
haftmann@58410:         \<le> ?ub_horner (Float 1 (- 1))
hoelzl@40881:           + ?ub_horner (x * rapprox_rat prec 2 3 - 1)"
haftmann@58410:         using ln_float_bounds(2)[of "Float 1 (- 1)" prec prec] add by auto }
hoelzl@31468:     moreover
hoelzl@31468:     { let ?max = "max (x * lapprox_rat prec 2 3 - 1) 0"
hoelzl@31468: 
hoelzl@40881:       have up: "lapprox_rat prec 2 3 \<le> 2/3"
wenzelm@32960:         by (rule order_trans[OF lapprox_rat], simp)
hoelzl@31468: 
hoelzl@31468:       have low: "0 \<le> real (lapprox_rat prec 2 3)"
hoelzl@47599:         using lapprox_rat_nonneg[of 2 3 prec] by simp
hoelzl@31468: 
hoelzl@40881:       have "?lb_horner ?max
wenzelm@32960:         \<le> ln (real ?max + 1)"
hoelzl@31468:       proof (rule ln_float_bounds(1))
wenzelm@32960:         from mult_less_le_imp_less[OF `real x < 2` up] * low
wenzelm@32960:         show "real ?max < 1" by (cases "real (lapprox_rat prec 2 3) = 0",
wenzelm@32960:           auto simp add: real_of_float_max)
wenzelm@32960:         show "0 \<le> real ?max" by (auto simp add: real_of_float_max)
hoelzl@31468:       qed
hoelzl@31468:       also have "\<dots> \<le> ln (real x * 2/3)"
hoelzl@31468:       proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
wenzelm@32960:         show "0 < real ?max + 1" by (auto simp add: real_of_float_max)
wenzelm@32960:         show "0 < real x * 2/3" using * by auto
wenzelm@32960:         show "real ?max + 1 \<le> real x * 2/3" using * up
wenzelm@32960:           by (cases "0 < real x * real (lapprox_posrat prec 2 3) - 1",
hoelzl@47599:               auto simp add: max_def)
hoelzl@31468:       qed
haftmann@58410:       finally have "?lb_horner (Float 1 (- 1)) + ?lb_horner ?max
hoelzl@40881:         \<le> ln x"
haftmann@58410:         using ln_float_bounds(1)[of "Float 1 (- 1)" prec prec] add by auto }
hoelzl@31468:     ultimately
hoelzl@31468:     show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
hoelzl@31468:       using `\<not> x \<le> 0` `\<not> x < 1` True False by auto
hoelzl@31468:   qed
hoelzl@29805: next
hoelzl@29805:   case False
haftmann@58410:   hence "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" "\<not> x \<le> Float 3 (- 1)"
hoelzl@47600:     using `1 \<le> x` by auto
hoelzl@29805:   show ?thesis
hoelzl@47599:   proof -
hoelzl@47599:     def m \<equiv> "mantissa x"
hoelzl@47599:     def e \<equiv> "exponent x"
hoelzl@47599:     from Float_mantissa_exponent[of x] have Float: "x = Float m e" by (simp add: m_def e_def)
hoelzl@29805:     let ?s = "Float (e + (bitlen m - 1)) 0"
hoelzl@29805:     let ?x = "Float m (- (bitlen m - 1))"
hoelzl@29805: 
hoelzl@47599:     have "0 < m" and "m \<noteq> 0" using `0 < x` Float powr_gt_zero[of 2 e]
hoelzl@47600:       by (auto simp: zero_less_mult_iff)
hoelzl@47599:     def bl \<equiv> "bitlen m - 1" hence "bl \<ge> 0" using `m > 0` by (simp add: bitlen_def)
hoelzl@47599:     have "1 \<le> Float m e" using `1 \<le> x` Float unfolding less_eq_float_def by auto
hoelzl@47599:     from bitlen_div[OF `0 < m`] float_gt1_scale[OF `1 \<le> Float m e`] `bl \<ge> 0`
hoelzl@47599:     have x_bnds: "0 \<le> real (?x - 1)" "real (?x - 1) < 1"
hoelzl@47599:       unfolding bl_def[symmetric]
hoelzl@47599:       by (auto simp: powr_realpow[symmetric] field_simps inverse_eq_divide)
hoelzl@47599:          (auto simp : powr_minus field_simps inverse_eq_divide)
hoelzl@29805: 
hoelzl@29805:     {
hoelzl@40881:       have "lb_ln2 prec * ?s \<le> ln 2 * (e + (bitlen m - 1))" (is "?lb2 \<le> _")
hoelzl@47601:         unfolding nat_0 power_0 mult_1_right times_float.rep_eq
wenzelm@32960:         using lb_ln2[of prec]
hoelzl@47599:       proof (rule mult_mono)
hoelzl@47599:         from float_gt1_scale[OF `1 \<le> Float m e`]
hoelzl@47599:         show "0 \<le> real (Float (e + (bitlen m - 1)) 0)" by simp
hoelzl@47599:       qed auto
hoelzl@29805:       moreover
hoelzl@47599:       from ln_float_bounds(1)[OF x_bnds]
hoelzl@40881:       have "(?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1) \<le> ln ?x" (is "?lb_horner \<le> _") by auto
hoelzl@40881:       ultimately have "?lb2 + ?lb_horner \<le> ln x"
wenzelm@32960:         unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
hoelzl@31468:     }
hoelzl@29805:     moreover
hoelzl@29805:     {
hoelzl@47599:       from ln_float_bounds(2)[OF x_bnds]
hoelzl@40881:       have "ln ?x \<le> (?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1)" (is "_ \<le> ?ub_horner") by auto
hoelzl@29805:       moreover
hoelzl@40881:       have "ln 2 * (e + (bitlen m - 1)) \<le> ub_ln2 prec * ?s" (is "_ \<le> ?ub2")
hoelzl@47601:         unfolding nat_0 power_0 mult_1_right times_float.rep_eq
wenzelm@32960:         using ub_ln2[of prec]
hoelzl@47599:       proof (rule mult_mono)
hoelzl@47599:         from float_gt1_scale[OF `1 \<le> Float m e`]
wenzelm@32960:         show "0 \<le> real (e + (bitlen m - 1))" by auto
hoelzl@47599:       next
hoelzl@47599:         have "0 \<le> ln 2" by simp
hoelzl@47599:         thus "0 \<le> real (ub_ln2 prec)" using ub_ln2[of prec] by arith
hoelzl@47599:       qed auto
hoelzl@40881:       ultimately have "ln x \<le> ?ub2 + ?ub_horner"
wenzelm@32960:         unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
hoelzl@29805:     }
hoelzl@29805:     ultimately show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps
haftmann@58410:       unfolding if_not_P[OF `\<not> x \<le> 0`] if_not_P[OF `\<not> x < 1`] if_not_P[OF False] if_not_P[OF `\<not> x \<le> Float 3 (- 1)`] Let_def
hoelzl@47601:       unfolding plus_float.rep_eq e_def[symmetric] m_def[symmetric] by simp
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
wenzelm@49351: lemma ub_ln_lb_ln_bounds:
wenzelm@49351:   assumes "0 < x"
hoelzl@40881:   shows "the (lb_ln prec x) \<le> ln x \<and> ln x \<le> the (ub_ln prec x)"
hoelzl@29805:   (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
hoelzl@29805: proof (cases "x < 1")
hoelzl@47599:   case False hence "1 \<le> x" unfolding less_float_def less_eq_float_def by auto
hoelzl@29805:   show ?thesis using ub_ln_lb_ln_bounds'[OF `1 \<le> x`] .
hoelzl@29805: next
hoelzl@47600:   case True have "\<not> x \<le> 0" using `0 < x` by auto
hoelzl@47600:   from True have "real x < 1" by simp
hoelzl@47600:   have "0 < real x" and "real x \<noteq> 0" using `0 < x` by auto
hoelzl@31098:   hence A: "0 < 1 / real x" by auto
hoelzl@29805: 
hoelzl@29805:   {
hoelzl@29805:     let ?divl = "float_divl (max prec 1) 1 x"
hoelzl@47600:     have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF `0 < real x` `real x < 1`] by auto
hoelzl@47600:     hence B: "0 < real ?divl" by auto
hoelzl@31468: 
hoelzl@40881:     have "ln ?divl \<le> ln (1 / x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
hoelzl@40881:     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
hoelzl@31468:     from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le]
hoelzl@47601:     have "?ln \<le> - the (lb_ln prec ?divl)" unfolding uminus_float.rep_eq by (rule order_trans)
hoelzl@29805:   } moreover
hoelzl@29805:   {
hoelzl@29805:     let ?divr = "float_divr prec 1 x"
hoelzl@47599:     have A': "1 \<le> ?divr" using float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`] unfolding less_eq_float_def less_float_def by auto
hoelzl@47600:     hence B: "0 < real ?divr" by auto
hoelzl@31468: 
hoelzl@40881:     have "ln (1 / x) \<le> ln ?divr" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
hoelzl@40881:     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
hoelzl@29805:     from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
hoelzl@47601:     have "- the (ub_ln prec ?divr) \<le> ?ln" unfolding uminus_float.rep_eq by (rule order_trans)
hoelzl@29805:   }
hoelzl@29805:   ultimately show ?thesis unfolding lb_ln.simps[where x=x]  ub_ln.simps[where x=x]
hoelzl@29805:     unfolding if_not_P[OF `\<not> x \<le> 0`] if_P[OF True] by auto
hoelzl@29805: qed
hoelzl@29805: 
wenzelm@49351: lemma lb_ln:
wenzelm@49351:   assumes "Some y = lb_ln prec x"
hoelzl@40881:   shows "y \<le> ln x" and "0 < real x"
hoelzl@29805: proof -
hoelzl@29805:   have "0 < x"
hoelzl@29805:   proof (rule ccontr)
hoelzl@47599:     assume "\<not> 0 < x" hence "x \<le> 0" unfolding less_eq_float_def less_float_def by auto
hoelzl@29805:     thus False using assms by auto
hoelzl@29805:   qed
hoelzl@47600:   thus "0 < real x" by auto
hoelzl@40881:   have "the (lb_ln prec x) \<le> ln x" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
hoelzl@40881:   thus "y \<le> ln x" unfolding assms[symmetric] by auto
hoelzl@29805: qed
hoelzl@29805: 
wenzelm@49351: lemma ub_ln:
wenzelm@49351:   assumes "Some y = ub_ln prec x"
hoelzl@40881:   shows "ln x \<le> y" and "0 < real x"
hoelzl@29805: proof -
hoelzl@29805:   have "0 < x"
hoelzl@29805:   proof (rule ccontr)
hoelzl@47600:     assume "\<not> 0 < x" hence "x \<le> 0" by auto
hoelzl@29805:     thus False using assms by auto
hoelzl@29805:   qed
hoelzl@47600:   thus "0 < real x" by auto
hoelzl@40881:   have "ln x \<le> the (ub_ln prec x)" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
hoelzl@40881:   thus "ln x \<le> y" unfolding assms[symmetric] by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@40881: 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"
hoelzl@29805: proof (rule allI, rule allI, rule allI, rule impI)
hoelzl@40881:   fix x::real and lx ux
hoelzl@40881:   assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {lx .. ux}"
hoelzl@40881:   hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {lx .. ux}" by auto
hoelzl@40881: 
hoelzl@40881:   have "ln ux \<le> u" and "0 < real ux" using ub_ln u by auto
hoelzl@40881:   have "l \<le> ln lx" and "0 < real lx" and "0 < x" using lb_ln[OF l] x by auto
hoelzl@40881: 
hoelzl@40881:   from ln_le_cancel_iff[OF `0 < real lx` `0 < x`] `l \<le> ln lx`
hoelzl@40881:   have "l \<le> ln x" using x unfolding atLeastAtMost_iff by auto
hoelzl@29805:   moreover
hoelzl@40881:   from ln_le_cancel_iff[OF `0 < x` `0 < real ux`] `ln ux \<le> real u`
hoelzl@40881:   have "ln x \<le> u" using x unfolding atLeastAtMost_iff by auto
hoelzl@40881:   ultimately show "l \<le> ln x \<and> ln x \<le> u" ..
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: section "Implement floatarith"
hoelzl@29805: 
hoelzl@29805: subsection "Define syntax and semantics"
hoelzl@29805: 
blanchet@58310: datatype floatarith
hoelzl@29805:   = Add floatarith floatarith
hoelzl@29805:   | Minus floatarith
hoelzl@29805:   | Mult floatarith floatarith
hoelzl@29805:   | Inverse floatarith
hoelzl@29805:   | Cos floatarith
hoelzl@29805:   | Arctan floatarith
hoelzl@29805:   | Abs floatarith
hoelzl@29805:   | Max floatarith floatarith
hoelzl@29805:   | Min floatarith floatarith
hoelzl@29805:   | Pi
hoelzl@29805:   | Sqrt floatarith
hoelzl@29805:   | Exp floatarith
hoelzl@29805:   | Ln floatarith
hoelzl@29805:   | Power floatarith nat
hoelzl@32919:   | Var nat
hoelzl@29805:   | Num float
hoelzl@29805: 
hoelzl@31863: fun interpret_floatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real" where
hoelzl@31098: "interpret_floatarith (Add a b) vs   = (interpret_floatarith a vs) + (interpret_floatarith b vs)" |
hoelzl@31098: "interpret_floatarith (Minus a) vs    = - (interpret_floatarith a vs)" |
hoelzl@31098: "interpret_floatarith (Mult a b) vs   = (interpret_floatarith a vs) * (interpret_floatarith b vs)" |
hoelzl@31098: "interpret_floatarith (Inverse a) vs  = inverse (interpret_floatarith a vs)" |
hoelzl@31098: "interpret_floatarith (Cos a) vs      = cos (interpret_floatarith a vs)" |
hoelzl@31098: "interpret_floatarith (Arctan a) vs   = arctan (interpret_floatarith a vs)" |
hoelzl@31098: "interpret_floatarith (Min a b) vs    = min (interpret_floatarith a vs) (interpret_floatarith b vs)" |
hoelzl@31098: "interpret_floatarith (Max a b) vs    = max (interpret_floatarith a vs) (interpret_floatarith b vs)" |
hoelzl@31098: "interpret_floatarith (Abs a) vs      = abs (interpret_floatarith a vs)" |
hoelzl@31098: "interpret_floatarith Pi vs           = pi" |
hoelzl@31098: "interpret_floatarith (Sqrt a) vs     = sqrt (interpret_floatarith a vs)" |
hoelzl@31098: "interpret_floatarith (Exp a) vs      = exp (interpret_floatarith a vs)" |
hoelzl@31098: "interpret_floatarith (Ln a) vs       = ln (interpret_floatarith a vs)" |
hoelzl@31098: "interpret_floatarith (Power a n) vs  = (interpret_floatarith a vs)^n" |
hoelzl@40881: "interpret_floatarith (Num f) vs      = f" |
hoelzl@32919: "interpret_floatarith (Var n) vs     = vs ! n"
hoelzl@29805: 
hoelzl@31811: lemma interpret_floatarith_divide: "interpret_floatarith (Mult a (Inverse b)) vs = (interpret_floatarith a vs) / (interpret_floatarith b vs)"
huffman@36778:   unfolding divide_inverse interpret_floatarith.simps ..
hoelzl@31811: 
hoelzl@31811: lemma interpret_floatarith_diff: "interpret_floatarith (Add a (Minus b)) vs = (interpret_floatarith a vs) - (interpret_floatarith b vs)"
haftmann@54230:   unfolding interpret_floatarith.simps by simp
hoelzl@31811: 
haftmann@58410: lemma interpret_floatarith_sin: "interpret_floatarith (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) vs =
hoelzl@31811:   sin (interpret_floatarith a vs)"
hoelzl@31811:   unfolding sin_cos_eq interpret_floatarith.simps
haftmann@54230:             interpret_floatarith_divide interpret_floatarith_diff
hoelzl@31811:   by auto
hoelzl@31811: 
hoelzl@31811: lemma interpret_floatarith_tan:
haftmann@58410:   "interpret_floatarith (Mult (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) (Inverse (Cos a))) vs =
hoelzl@31811:    tan (interpret_floatarith a vs)"
huffman@36778:   unfolding interpret_floatarith.simps(3,4) interpret_floatarith_sin tan_def divide_inverse
hoelzl@31811:   by auto
hoelzl@31811: 
hoelzl@31811: lemma interpret_floatarith_powr: "interpret_floatarith (Exp (Mult b (Ln a))) vs = (interpret_floatarith a vs) powr (interpret_floatarith b vs)"
hoelzl@31811:   unfolding powr_def interpret_floatarith.simps ..
hoelzl@31811: 
hoelzl@31811: lemma interpret_floatarith_log: "interpret_floatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (interpret_floatarith b vs) (interpret_floatarith x vs)"
huffman@36778:   unfolding log_def interpret_floatarith.simps divide_inverse ..
hoelzl@31811: 
hoelzl@31811: lemma interpret_floatarith_num:
hoelzl@31811:   shows "interpret_floatarith (Num (Float 0 0)) vs = 0"
hoelzl@31811:   and "interpret_floatarith (Num (Float 1 0)) vs = 1"
haftmann@54489:   and "interpret_floatarith (Num (Float (- 1) 0)) vs = - 1"
huffman@47108:   and "interpret_floatarith (Num (Float (numeral a) 0)) vs = numeral a"
haftmann@54489:   and "interpret_floatarith (Num (Float (- numeral a) 0)) vs = - numeral a" by auto
hoelzl@31811: 
hoelzl@29805: subsection "Implement approximation function"
hoelzl@29805: 
hoelzl@29805: fun lift_bin' :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
hoelzl@29805: "lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" |
hoelzl@29805: "lift_bin' a b f = None"
hoelzl@29805: 
hoelzl@29805: fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where
hoelzl@29805: "lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u)
hoelzl@29805:                                              | t \<Rightarrow> None)" |
hoelzl@29805: "lift_un b f = None"
hoelzl@29805: 
hoelzl@29805: fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
hoelzl@29805: "lift_un' (Some (l1, u1)) f = Some (f l1 u1)" |
hoelzl@29805: "lift_un' b f = None"
hoelzl@29805: 
hoelzl@31811: definition
hoelzl@31811: "bounded_by xs vs \<longleftrightarrow>
hoelzl@31811:   (\<forall> i < length vs. case vs ! i of None \<Rightarrow> True
hoelzl@31811:          | Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u })"
hoelzl@31811: 
hoelzl@31811: lemma bounded_byE:
hoelzl@31811:   assumes "bounded_by xs vs"
hoelzl@31811:   shows "\<And> i. i < length vs \<Longrightarrow> case vs ! i of None \<Rightarrow> True
hoelzl@31811:          | Some (l, u) \<Rightarrow> xs ! i \<in> { real l .. real u }"
hoelzl@31811:   using assms bounded_by_def by blast
hoelzl@31811: 
hoelzl@31811: lemma bounded_by_update:
hoelzl@31811:   assumes "bounded_by xs vs"
hoelzl@31811:   and bnd: "xs ! i \<in> { real l .. real u }"
hoelzl@31811:   shows "bounded_by xs (vs[i := Some (l,u)])"
hoelzl@31811: proof -
hoelzl@31811: { fix j
hoelzl@31811:   let ?vs = "vs[i := Some (l,u)]"
hoelzl@31811:   assume "j < length ?vs" hence [simp]: "j < length vs" by simp
hoelzl@31811:   have "case ?vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> xs ! j \<in> { real l .. real u }"
hoelzl@31811:   proof (cases "?vs ! j")
hoelzl@31811:     case (Some b)
hoelzl@31811:     thus ?thesis
hoelzl@31811:     proof (cases "i = j")
hoelzl@31811:       case True
hoelzl@31811:       thus ?thesis using `?vs ! j = Some b` and bnd by auto
hoelzl@31811:     next
hoelzl@31811:       case False
hoelzl@31811:       thus ?thesis using `bounded_by xs vs` unfolding bounded_by_def by auto
hoelzl@31811:     qed
hoelzl@31811:   qed auto }
hoelzl@31811:   thus ?thesis unfolding bounded_by_def by auto
hoelzl@31811: qed
hoelzl@31811: 
hoelzl@31811: lemma bounded_by_None:
hoelzl@31811:   shows "bounded_by xs (replicate (length xs) None)"
hoelzl@31811:   unfolding bounded_by_def by auto
hoelzl@31811: 
hoelzl@31811: fun approx approx' :: "nat \<Rightarrow> floatarith \<Rightarrow> (float * float) option list \<Rightarrow> (float * float) option" where
hoelzl@47599: "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)" |
hoelzl@31811: "approx prec (Add a b) bs   = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (l1 + l2, u1 + u2))" |
hoelzl@29805: "approx prec (Minus a) bs   = lift_un' (approx' prec a bs) (\<lambda> l u. (-u, -l))" |
hoelzl@29805: "approx prec (Mult a b) bs  = lift_bin' (approx' prec a bs) (approx' prec b bs)
hoelzl@47599:                                     (\<lambda> a1 a2 b1 b2. (nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1,
hoelzl@47599:                                                      pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1))" |
hoelzl@29805: "approx prec (Inverse a) bs = lift_un (approx' prec a bs) (\<lambda> l u. if (0 < l \<or> u < 0) then (Some (float_divl prec 1 u), Some (float_divr prec 1 l)) else (None, None))" |
hoelzl@29805: "approx prec (Cos a) bs     = lift_un' (approx' prec a bs) (bnds_cos prec)" |
hoelzl@29805: "approx prec Pi bs          = Some (lb_pi prec, ub_pi prec)" |
hoelzl@29805: "approx prec (Min a b) bs   = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (min l1 l2, min u1 u2))" |
hoelzl@29805: "approx prec (Max a b) bs   = lift_bin' (approx' prec a bs) (approx' prec b bs) (\<lambda> l1 u1 l2 u2. (max l1 l2, max u1 u2))" |
hoelzl@29805: "approx prec (Abs a) bs     = lift_un' (approx' prec a bs) (\<lambda>l u. (if l < 0 \<and> 0 < u then 0 else min \<bar>l\<bar> \<bar>u\<bar>, max \<bar>l\<bar> \<bar>u\<bar>))" |
hoelzl@29805: "approx prec (Arctan a) bs  = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_arctan prec l, ub_arctan prec u))" |
hoelzl@31467: "approx prec (Sqrt a) bs    = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" |
hoelzl@29805: "approx prec (Exp a) bs     = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" |
hoelzl@29805: "approx prec (Ln a) bs      = lift_un (approx' prec a bs) (\<lambda> l u. (lb_ln prec l, ub_ln prec u))" |
hoelzl@29805: "approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds n)" |
hoelzl@29805: "approx prec (Num f) bs     = Some (f, f)" |
hoelzl@32919: "approx prec (Var i) bs    = (if i < length bs then bs ! i else None)"
hoelzl@29805: 
hoelzl@29805: lemma lift_bin'_ex:
hoelzl@29805:   assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f"
hoelzl@29805:   shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b"
hoelzl@29805: proof (cases a)
hoelzl@29805:   case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
hoelzl@29805:   thus ?thesis using lift_bin'_Some by auto
hoelzl@29805: next
hoelzl@29805:   case (Some a')
hoelzl@29805:   show ?thesis
hoelzl@29805:   proof (cases b)
hoelzl@29805:     case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
hoelzl@29805:     thus ?thesis using lift_bin'_Some by auto
hoelzl@29805:   next
hoelzl@29805:     case (Some b')
hoelzl@29805:     obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
hoelzl@29805:     obtain lb ub where b': "b' = (lb, ub)" by (cases b', auto)
hoelzl@29805:     thus ?thesis unfolding `a = Some a'` `b = Some b'` a' b' by auto
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma lift_bin'_f:
hoelzl@29805:   assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f"
hoelzl@29805:   and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a" and Pb: "\<And>l u. Some (l, u) = g b \<Longrightarrow> P l u b"
hoelzl@29805:   shows "\<exists> l1 u1 l2 u2. P l1 u1 a \<and> P l2 u2 b \<and> l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
hoelzl@29805: proof -
hoelzl@29805:   obtain l1 u1 l2 u2
hoelzl@29805:     where Sa: "Some (l1, u1) = g a" and Sb: "Some (l2, u2) = g b" using lift_bin'_ex[OF assms(1)] by auto
hoelzl@31809:   have lu: "(l, u) = f l1 u1 l2 u2" using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto
hoelzl@29805:   have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" unfolding lu[symmetric] by auto
hoelzl@31809:   thus ?thesis using Pa[OF Sa] Pb[OF Sb] by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma approx_approx':
hoelzl@40881:   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"
hoelzl@29805:   and approx': "Some (l, u) = approx' prec a vs"
hoelzl@40881:   shows "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u"
hoelzl@29805: proof -
hoelzl@29805:   obtain l' u' where S: "Some (l', u') = approx prec a vs"
hoelzl@29805:     using approx' unfolding approx'.simps by (cases "approx prec a vs", auto)
hoelzl@47599:   have l': "l = float_round_down prec l'" and u': "u = float_round_up prec u'"
hoelzl@29805:     using approx' unfolding approx'.simps S[symmetric] by auto
hoelzl@31809:   show ?thesis unfolding l' u'
hoelzl@47599:     using order_trans[OF Pa[OF S, THEN conjunct2] float_round_up[of u']]
hoelzl@47599:     using order_trans[OF float_round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma lift_bin':
hoelzl@29805:   assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f"
hoelzl@40881:   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")
hoelzl@40881:   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"
hoelzl@40881:   shows "\<exists> l1 u1 l2 u2. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and>
hoelzl@40881:                         (l2 \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u2) \<and>
hoelzl@29805:                         l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
hoelzl@29805: proof -
hoelzl@29805:   { fix l u assume "Some (l, u) = approx' prec a bs"
hoelzl@29805:     with approx_approx'[of prec a bs, OF _ this] Pa
hoelzl@40881:     have "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" by auto } note Pa = this
hoelzl@29805:   { fix l u assume "Some (l, u) = approx' prec b bs"
hoelzl@29805:     with approx_approx'[of prec b bs, OF _ this] Pb
hoelzl@40881:     have "l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> u" by auto } note Pb = this
hoelzl@29805: 
hoelzl@29805:   from lift_bin'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb]
hoelzl@29805:   show ?thesis by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma lift_un'_ex:
hoelzl@29805:   assumes lift_un'_Some: "Some (l, u) = lift_un' a f"
hoelzl@29805:   shows "\<exists> l u. Some (l, u) = a"
hoelzl@29805: proof (cases a)
hoelzl@29805:   case None hence "None = lift_un' a f" unfolding None lift_un'.simps ..
hoelzl@29805:   thus ?thesis using lift_un'_Some by auto
hoelzl@29805: next
hoelzl@29805:   case (Some a')
hoelzl@29805:   obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
hoelzl@29805:   thus ?thesis unfolding `a = Some a'` a' by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma lift_un'_f:
hoelzl@29805:   assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f"
hoelzl@29805:   and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
hoelzl@29805:   shows "\<exists> l1 u1. P l1 u1 a \<and> l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
hoelzl@29805: proof -
hoelzl@29805:   obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un'_ex[OF assms(1)] by auto
hoelzl@29805:   have lu: "(l, u) = f l1 u1" using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto
hoelzl@29805:   have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" unfolding lu[symmetric] by auto
hoelzl@29805:   thus ?thesis using Pa[OF Sa] by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma lift_un':
hoelzl@29805:   assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
hoelzl@40881:   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")
hoelzl@40881:   shows "\<exists> l1 u1. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and>
hoelzl@29805:                         l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
hoelzl@29805: proof -
hoelzl@29805:   { fix l u assume "Some (l, u) = approx' prec a bs"
hoelzl@29805:     with approx_approx'[of prec a bs, OF _ this] Pa
hoelzl@40881:     have "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" by auto } note Pa = this
hoelzl@29805:   from lift_un'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa]
hoelzl@29805:   show ?thesis by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma lift_un'_bnds:
hoelzl@40881:   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"
hoelzl@29805:   and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
hoelzl@40881:   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"
hoelzl@31098:   shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
hoelzl@29805: proof -
hoelzl@29805:   from lift_un'[OF lift_un'_Some Pa]
hoelzl@40881:   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
hoelzl@40881:   hence "(l, u) = f l1 u1" and "interpret_floatarith a xs \<in> {l1 .. u1}" by auto
hoelzl@29805:   thus ?thesis using bnds by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma lift_un_ex:
hoelzl@29805:   assumes lift_un_Some: "Some (l, u) = lift_un a f"
hoelzl@29805:   shows "\<exists> l u. Some (l, u) = a"
hoelzl@29805: proof (cases a)
hoelzl@29805:   case None hence "None = lift_un a f" unfolding None lift_un.simps ..
hoelzl@29805:   thus ?thesis using lift_un_Some by auto
hoelzl@29805: next
hoelzl@29805:   case (Some a')
hoelzl@29805:   obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
hoelzl@29805:   thus ?thesis unfolding `a = Some a'` a' by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma lift_un_f:
hoelzl@29805:   assumes lift_un_Some: "Some (l, u) = lift_un (g a) f"
hoelzl@29805:   and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
hoelzl@29805:   shows "\<exists> l1 u1. P l1 u1 a \<and> Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
hoelzl@29805: proof -
hoelzl@29805:   obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un_ex[OF assms(1)] by auto
hoelzl@29805:   have "fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None"
hoelzl@29805:   proof (rule ccontr)
hoelzl@29805:     assume "\<not> (fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None)"
hoelzl@29805:     hence or: "fst (f l1 u1) = None \<or> snd (f l1 u1) = None" by auto
hoelzl@31809:     hence "lift_un (g a) f = None"
hoelzl@29805:     proof (cases "fst (f l1 u1) = None")
hoelzl@29805:       case True
hoelzl@29805:       then obtain b where b: "f l1 u1 = (None, b)" by (cases "f l1 u1", auto)
hoelzl@29805:       thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
hoelzl@29805:     next
hoelzl@29805:       case False hence "snd (f l1 u1) = None" using or by auto
hoelzl@29805:       with False obtain b where b: "f l1 u1 = (Some b, None)" by (cases "f l1 u1", auto)
hoelzl@29805:       thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
hoelzl@29805:     qed
hoelzl@29805:     thus False using lift_un_Some by auto
hoelzl@29805:   qed
hoelzl@29805:   then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" by (cases "f l1 u1", auto)
hoelzl@29805:   from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f]
hoelzl@29805:   have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" unfolding f by auto
hoelzl@29805:   thus ?thesis unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma lift_un:
hoelzl@29805:   assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
hoelzl@40881:   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")
hoelzl@40881:   shows "\<exists> l1 u1. (l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u1) \<and>
hoelzl@29805:                   Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
hoelzl@29805: proof -
hoelzl@29805:   { fix l u assume "Some (l, u) = approx' prec a bs"
hoelzl@29805:     with approx_approx'[of prec a bs, OF _ this] Pa
hoelzl@40881:     have "l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> u" by auto } note Pa = this
hoelzl@29805:   from lift_un_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa]
hoelzl@29805:   show ?thesis by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma lift_un_bnds:
hoelzl@40881:   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"
hoelzl@29805:   and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
hoelzl@40881:   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"
hoelzl@31098:   shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
hoelzl@29805: proof -
hoelzl@29805:   from lift_un[OF lift_un_Some Pa]
hoelzl@40881:   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
hoelzl@40881:   hence "(Some l, Some u) = f l1 u1" and "interpret_floatarith a xs \<in> {l1 .. u1}" by auto
hoelzl@29805:   thus ?thesis using bnds by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@29805: lemma approx:
hoelzl@29805:   assumes "bounded_by xs vs"
hoelzl@29805:   and "Some (l, u) = approx prec arith vs" (is "_ = ?g arith")
hoelzl@40881:   shows "l \<le> interpret_floatarith arith xs \<and> interpret_floatarith arith xs \<le> u" (is "?P l u arith")
hoelzl@31809:   using `Some (l, u) = approx prec arith vs`
wenzelm@45129: proof (induct arith arbitrary: l u)
hoelzl@29805:   case (Add a b)
hoelzl@29805:   from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps
hoelzl@29805:   obtain l1 u1 l2 u2 where "l = l1 + l2" and "u = u1 + u2"
hoelzl@40881:     "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1"
hoelzl@40881:     "l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> u2" unfolding fst_conv snd_conv by blast
hoelzl@31098:   thus ?case unfolding interpret_floatarith.simps by auto
hoelzl@29805: next
hoelzl@29805:   case (Minus a)
hoelzl@29805:   from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps
hoelzl@29805:   obtain l1 u1 where "l = -u1" and "u = -l1"
hoelzl@40881:     "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1" unfolding fst_conv snd_conv by blast
hoelzl@47601:   thus ?case unfolding interpret_floatarith.simps using minus_float.rep_eq by auto
hoelzl@29805: next
hoelzl@29805:   case (Mult a b)
hoelzl@29805:   from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps
hoelzl@31809:   obtain l1 u1 l2 u2
hoelzl@47599:     where l: "l = nprt l1 * pprt u2 + nprt u1 * nprt u2 + pprt l1 * pprt l2 + pprt u1 * nprt l2"
hoelzl@47599:     and u: "u = pprt u1 * pprt u2 + pprt l1 * nprt u2 + nprt u1 * pprt l2 + nprt l1 * nprt l2"
hoelzl@40881:     and "l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> u1"
hoelzl@40881:     and "l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> u2" unfolding fst_conv snd_conv by blast
hoelzl@47599:   thus ?case unfolding interpret_floatarith.simps l u
hoelzl@29805:     using mult_le_prts mult_ge_prts by auto
hoelzl@29805: next
hoelzl@29805:   case (Inverse a)
hoelzl@29805:   from lift_un[OF Inverse.prems[unfolded approx.simps], unfolded if_distrib[of fst] if_distrib[of snd] fst_conv snd_conv] Inverse.hyps
hoelzl@31809:   obtain l1 u1 where l': "Some l = (if 0 < l1 \<or> u1 < 0 then Some (float_divl prec 1 u1) else None)"
hoelzl@29805:     and u': "Some u = (if 0 < l1 \<or> u1 < 0 then Some (float_divr prec 1 l1) else None)"
hoelzl@40881:     and l1: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1" by blast
hoelzl@29805:   have either: "0 < l1 \<or> u1 < 0" proof (rule ccontr) assume P: "\<not> (0 < l1 \<or> u1 < 0)" show False using l' unfolding if_not_P[OF P] by auto qed
hoelzl@31098:   moreover have l1_le_u1: "real l1 \<le> real u1" using l1 u1 by auto
hoelzl@47600:   ultimately have "real l1 \<noteq> 0" and "real u1 \<noteq> 0" by auto
hoelzl@29805: 
hoelzl@40881:   have inv: "inverse u1 \<le> inverse (interpret_floatarith a xs)
hoelzl@40881:            \<and> inverse (interpret_floatarith a xs) \<le> inverse l1"
hoelzl@29805:   proof (cases "0 < l1")
hoelzl@31809:     case True hence "0 < real u1" and "0 < real l1" "0 < interpret_floatarith a xs"
hoelzl@47600:       using l1_le_u1 l1 by auto
hoelzl@29805:     show ?thesis
hoelzl@31098:       unfolding inverse_le_iff_le[OF `0 < real u1` `0 < interpret_floatarith a xs`]
wenzelm@32960:         inverse_le_iff_le[OF `0 < interpret_floatarith a xs` `0 < real l1`]
hoelzl@29805:       using l1 u1 by auto
hoelzl@29805:   next
hoelzl@29805:     case False hence "u1 < 0" using either by blast
hoelzl@31809:     hence "real u1 < 0" and "real l1 < 0" "interpret_floatarith a xs < 0"
hoelzl@47600:       using l1_le_u1 u1 by auto
hoelzl@29805:     show ?thesis
hoelzl@31098:       unfolding inverse_le_iff_le_neg[OF `real u1 < 0` `interpret_floatarith a xs < 0`]
wenzelm@32960:         inverse_le_iff_le_neg[OF `interpret_floatarith a xs < 0` `real l1 < 0`]
hoelzl@29805:       using l1 u1 by auto
hoelzl@29805:   qed
hoelzl@31468: 
hoelzl@29805:   from l' have "l = float_divl prec 1 u1" by (cases "0 < l1 \<or> u1 < 0", auto)
hoelzl@40881:   hence "l \<le> inverse u1" unfolding nonzero_inverse_eq_divide[OF `real u1 \<noteq> 0`] using float_divl[of prec 1 u1] by auto
hoelzl@31098:   also have "\<dots> \<le> inverse (interpret_floatarith a xs)" using inv by auto
hoelzl@40881:   finally have "l \<le> inverse (interpret_floatarith a xs)" .
hoelzl@29805:   moreover
hoelzl@29805:   from u' have "u = float_divr prec 1 l1" by (cases "0 < l1 \<or> u1 < 0", auto)
hoelzl@40881:   hence "inverse l1 \<le> u" unfolding nonzero_inverse_eq_divide[OF `real l1 \<noteq> 0`] using float_divr[of 1 l1 prec] by auto
hoelzl@40881:   hence "inverse (interpret_floatarith a xs) \<le> u" by (rule order_trans[OF inv[THEN conjunct2]])
hoelzl@31098:   ultimately show ?case unfolding interpret_floatarith.simps using l1 u1 by auto
hoelzl@29805: next
hoelzl@29805:   case (Abs x)
hoelzl@29805:   from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps
hoelzl@29805:   obtain l1 u1 where l': "l = (if l1 < 0 \<and> 0 < u1 then 0 else min \<bar>l1\<bar> \<bar>u1\<bar>)" and u': "u = max \<bar>l1\<bar> \<bar>u1\<bar>"
hoelzl@40881:     and l1: "l1 \<le> interpret_floatarith x xs" and u1: "interpret_floatarith x xs \<le> u1" by blast
hoelzl@47600:   thus ?case unfolding l' u' by (cases "l1 < 0 \<and> 0 < u1", auto simp add: real_of_float_min real_of_float_max)
hoelzl@29805: next
hoelzl@29805:   case (Min a b)
hoelzl@29805:   from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps
hoelzl@29805:   obtain l1 u1 l2 u2 where l': "l = min l1 l2" and u': "u = min u1 u2"
hoelzl@40881:     and l1: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1"
hoelzl@40881:     and l1: "l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> u2" by blast
hoelzl@31098:   thus ?case unfolding l' u' by (auto simp add: real_of_float_min)
hoelzl@29805: next
hoelzl@29805:   case (Max a b)
hoelzl@29805:   from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps
hoelzl@29805:   obtain l1 u1 l2 u2 where l': "l = max l1 l2" and u': "u = max u1 u2"
hoelzl@40881:     and l1: "l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> u1"
hoelzl@40881:     and l1: "l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> u2" by blast
hoelzl@31098:   thus ?case unfolding l' u' by (auto simp add: real_of_float_max)
hoelzl@29805: next case (Cos a) with lift_un'_bnds[OF bnds_cos] show ?case by auto
hoelzl@29805: next case (Arctan a) with lift_un'_bnds[OF bnds_arctan] show ?case by auto
hoelzl@29805: next case Pi with pi_boundaries show ?case by auto
hoelzl@31467: next case (Sqrt a) with lift_un'_bnds[OF bnds_sqrt] show ?case by auto
hoelzl@29805: next case (Exp a) with lift_un'_bnds[OF bnds_exp] show ?case by auto
hoelzl@29805: next case (Ln a) with lift_un_bnds[OF bnds_ln] show ?case by auto
hoelzl@29805: next case (Power a n) with lift_un'_bnds[OF bnds_power] show ?case by auto
hoelzl@29805: next case (Num f) thus ?case by auto
hoelzl@29805: next
hoelzl@32919:   case (Var n)
hoelzl@31811:   from this[symmetric] `bounded_by xs vs`[THEN bounded_byE, of n]
hoelzl@31811:   show ?case by (cases "n < length vs", auto)
hoelzl@31811: qed
hoelzl@31811: 
blanchet@58310: datatype form = Bound floatarith floatarith floatarith form
hoelzl@31811:               | Assign floatarith floatarith form
hoelzl@31811:               | Less floatarith floatarith
hoelzl@31811:               | LessEqual floatarith floatarith
hoelzl@31811:               | AtLeastAtMost floatarith floatarith floatarith
hoelzl@31811: 
hoelzl@31811: fun interpret_form :: "form \<Rightarrow> real list \<Rightarrow> bool" where
hoelzl@31811: "interpret_form (Bound x a b f) vs = (interpret_floatarith x vs \<in> { interpret_floatarith a vs .. interpret_floatarith b vs } \<longrightarrow> interpret_form f vs)" |
hoelzl@31811: "interpret_form (Assign x a f) vs  = (interpret_floatarith x vs = interpret_floatarith a vs \<longrightarrow> interpret_form f vs)" |
hoelzl@31811: "interpret_form (Less a b) vs      = (interpret_floatarith a vs < interpret_floatarith b vs)" |
hoelzl@31811: "interpret_form (LessEqual a b) vs = (interpret_floatarith a vs \<le> interpret_floatarith b vs)" |
hoelzl@31811: "interpret_form (AtLeastAtMost x a b) vs = (interpret_floatarith x vs \<in> { interpret_floatarith a vs .. interpret_floatarith b vs })"
hoelzl@31811: 
hoelzl@31811: fun approx_form' and approx_form :: "nat \<Rightarrow> form \<Rightarrow> (float * float) option list \<Rightarrow> nat list \<Rightarrow> bool" where
hoelzl@31811: "approx_form' prec f 0 n l u bs ss = approx_form prec f (bs[n := Some (l, u)]) ss" |
hoelzl@31811: "approx_form' prec f (Suc s) n l u bs ss =
haftmann@58410:   (let m = (l + u) * Float 1 (- 1)
hoelzl@32919:    in (if approx_form' prec f s n l m bs ss then approx_form' prec f s n m u bs ss else False))" |
hoelzl@32919: "approx_form prec (Bound (Var n) a b f) bs ss =
hoelzl@31811:    (case (approx prec a bs, approx prec b bs)
hoelzl@31811:    of (Some (l, _), Some (_, u)) \<Rightarrow> approx_form' prec f (ss ! n) n l u bs ss
hoelzl@31811:     | _ \<Rightarrow> False)" |
hoelzl@32919: "approx_form prec (Assign (Var n) a f) bs ss =
hoelzl@31811:    (case (approx prec a bs)
hoelzl@31811:    of (Some (l, u)) \<Rightarrow> approx_form' prec f (ss ! n) n l u bs ss
hoelzl@31811:     | _ \<Rightarrow> False)" |
hoelzl@31811: "approx_form prec (Less a b) bs ss =
hoelzl@31811:    (case (approx prec a bs, approx prec b bs)
hoelzl@31811:    of (Some (l, u), Some (l', u')) \<Rightarrow> u < l'
hoelzl@31811:     | _ \<Rightarrow> False)" |
hoelzl@31811: "approx_form prec (LessEqual a b) bs ss =
hoelzl@31811:    (case (approx prec a bs, approx prec b bs)
hoelzl@31811:    of (Some (l, u), Some (l', u')) \<Rightarrow> u \<le> l'
hoelzl@31811:     | _ \<Rightarrow> False)" |
hoelzl@31811: "approx_form prec (AtLeastAtMost x a b) bs ss =
hoelzl@31811:    (case (approx prec x bs, approx prec a bs, approx prec b bs)
hoelzl@31811:    of (Some (lx, ux), Some (l, u), Some (l', u')) \<Rightarrow> u \<le> lx \<and> ux \<le> l'
hoelzl@31811:     | _ \<Rightarrow> False)" |
hoelzl@31811: "approx_form _ _ _ _ = False"
hoelzl@31811: 
hoelzl@32919: lemma lazy_conj: "(if A then B else False) = (A \<and> B)" by simp
hoelzl@32919: 
hoelzl@31811: lemma approx_form_approx_form':
hoelzl@40881:   assumes "approx_form' prec f s n l u bs ss" and "(x::real) \<in> { l .. u }"
hoelzl@40881:   obtains l' u' where "x \<in> { l' .. u' }"
wenzelm@49351:     and "approx_form prec f (bs[n := Some (l', u')]) ss"
hoelzl@31811: using assms proof (induct s arbitrary: l u)
hoelzl@31811:   case 0
hoelzl@31811:   from this(1)[of l u] this(2,3)
hoelzl@31811:   show thesis by auto
hoelzl@31811: next
hoelzl@31811:   case (Suc s)
hoelzl@31811: 
haftmann@58410:   let ?m = "(l + u) * Float 1 (- 1)"
hoelzl@40881:   have "real l \<le> ?m" and "?m \<le> real u"
hoelzl@47599:     unfolding less_eq_float_def using Suc.prems by auto
hoelzl@31811: 
hoelzl@40881:   with `x \<in> { l .. u }`
hoelzl@40881:   have "x \<in> { l .. ?m} \<or> x \<in> { ?m .. u }" by auto
hoelzl@31811:   thus thesis
hoelzl@31811:   proof (rule disjE)
hoelzl@40881:     assume *: "x \<in> { l .. ?m }"
hoelzl@31811:     with Suc.hyps[OF _ _ *] Suc.prems
hoelzl@32919:     show thesis by (simp add: Let_def lazy_conj)
hoelzl@29805:   next
hoelzl@40881:     assume *: "x \<in> { ?m .. u }"
hoelzl@31811:     with Suc.hyps[OF _ _ *] Suc.prems
hoelzl@32919:     show thesis by (simp add: Let_def lazy_conj)
hoelzl@29805:   qed
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@31811: lemma approx_form_aux:
hoelzl@31811:   assumes "approx_form prec f vs ss"
wenzelm@49351:     and "bounded_by xs vs"
hoelzl@31811:   shows "interpret_form f xs"
hoelzl@31811: using assms proof (induct f arbitrary: vs)
hoelzl@31811:   case (Bound x a b f)
hoelzl@31811:   then obtain n
hoelzl@32919:     where x_eq: "x = Var n" by (cases x) auto
hoelzl@31811: 
hoelzl@31811:   with Bound.prems obtain l u' l' u
hoelzl@31811:     where l_eq: "Some (l, u') = approx prec a vs"
hoelzl@31811:     and u_eq: "Some (l', u) = approx prec b vs"
hoelzl@31811:     and approx_form': "approx_form' prec f (ss ! n) n l u vs ss"
haftmann@37411:     by (cases "approx prec a vs", simp) (cases "approx prec b vs", auto)
hoelzl@40881: 
hoelzl@31811:   { assume "xs ! n \<in> { interpret_floatarith a xs .. interpret_floatarith b xs }"
hoelzl@31811:     with approx[OF Bound.prems(2) l_eq] and approx[OF Bound.prems(2) u_eq]
hoelzl@40881:     have "xs ! n \<in> { l .. u}" by auto
hoelzl@31811: 
hoelzl@31811:     from approx_form_approx_form'[OF approx_form' this]
hoelzl@40881:     obtain lx ux where bnds: "xs ! n \<in> { lx .. ux }"
hoelzl@31811:       and approx_form: "approx_form prec f (vs[n := Some (lx, ux)]) ss" .
hoelzl@31811: 
hoelzl@31811:     from `bounded_by xs vs` bnds
hoelzl@31811:     have "bounded_by xs (vs[n := Some (lx, ux)])" by (rule bounded_by_update)
hoelzl@31811:     with Bound.hyps[OF approx_form]
hoelzl@31811:     have "interpret_form f xs" by blast }
hoelzl@31811:   thus ?case using interpret_form.simps x_eq and interpret_floatarith.simps by simp
hoelzl@31811: next
hoelzl@31811:   case (Assign x a f)
hoelzl@31811:   then obtain n
hoelzl@32919:     where x_eq: "x = Var n" by (cases x) auto
hoelzl@31811: 
hoelzl@47599:   with Assign.prems obtain l u
hoelzl@31811:     where bnd_eq: "Some (l, u) = approx prec a vs"
hoelzl@32919:     and x_eq: "x = Var n"
hoelzl@31811:     and approx_form': "approx_form' prec f (ss ! n) n l u vs ss"
hoelzl@31811:     by (cases "approx prec a vs") auto
hoelzl@31811: 
hoelzl@31811:   { assume bnds: "xs ! n = interpret_floatarith a xs"
hoelzl@31811:     with approx[OF Assign.prems(2) bnd_eq]
hoelzl@40881:     have "xs ! n \<in> { l .. u}" by auto
hoelzl@31811:     from approx_form_approx_form'[OF approx_form' this]
hoelzl@40881:     obtain lx ux where bnds: "xs ! n \<in> { lx .. ux }"
hoelzl@31811:       and approx_form: "approx_form prec f (vs[n := Some (lx, ux)]) ss" .
hoelzl@31811: 
hoelzl@31811:     from `bounded_by xs vs` bnds
hoelzl@31811:     have "bounded_by xs (vs[n := Some (lx, ux)])" by (rule bounded_by_update)
hoelzl@31811:     with Assign.hyps[OF approx_form]
hoelzl@31811:     have "interpret_form f xs" by blast }
hoelzl@31811:   thus ?case using interpret_form.simps x_eq and interpret_floatarith.simps by simp
hoelzl@31811: next
hoelzl@29805:   case (Less a b)
hoelzl@31811:   then obtain l u l' u'
hoelzl@31811:     where l_eq: "Some (l, u) = approx prec a vs"
wenzelm@49351:       and u_eq: "Some (l', u') = approx prec b vs"
wenzelm@49351:       and inequality: "u < l'"
hoelzl@31811:     by (cases "approx prec a vs", auto,
hoelzl@31811:       cases "approx prec b vs", auto)
hoelzl@47600:   from inequality approx[OF Less.prems(2) l_eq] approx[OF Less.prems(2) u_eq]
hoelzl@31811:   show ?case by auto
hoelzl@29805: next
hoelzl@29805:   case (LessEqual a b)
hoelzl@31811:   then obtain l u l' u'
hoelzl@31811:     where l_eq: "Some (l, u) = approx prec a vs"
wenzelm@49351:       and u_eq: "Some (l', u') = approx prec b vs"
wenzelm@49351:       and inequality: "u \<le> l'"
hoelzl@31811:     by (cases "approx prec a vs", auto,
hoelzl@31811:       cases "approx prec b vs", auto)
hoelzl@47600:   from inequality approx[OF LessEqual.prems(2) l_eq] approx[OF LessEqual.prems(2) u_eq]
hoelzl@31811:   show ?case by auto
hoelzl@31811: next
hoelzl@31811:   case (AtLeastAtMost x a b)
hoelzl@31811:   then obtain lx ux l u l' u'
hoelzl@31811:     where x_eq: "Some (lx, ux) = approx prec x vs"
hoelzl@31811:     and l_eq: "Some (l, u) = approx prec a vs"
hoelzl@31811:     and u_eq: "Some (l', u') = approx prec b vs"
hoelzl@31811:     and inequality: "u \<le> lx \<and> ux \<le> l'"
hoelzl@31811:     by (cases "approx prec x vs", auto,
hoelzl@31811:       cases "approx prec a vs", auto,
nipkow@56073:       cases "approx prec b vs", auto)
hoelzl@47600:   from inequality approx[OF AtLeastAtMost.prems(2) l_eq] approx[OF AtLeastAtMost.prems(2) u_eq] approx[OF AtLeastAtMost.prems(2) x_eq]
hoelzl@31811:   show ?case by auto
hoelzl@29805: qed
hoelzl@29805: 
hoelzl@31811: lemma approx_form:
hoelzl@31811:   assumes "n = length xs"
hoelzl@31811:   assumes "approx_form prec f (replicate n None) ss"
hoelzl@31811:   shows "interpret_form f xs"
hoelzl@31811:   using approx_form_aux[OF _ bounded_by_None] assms by auto
hoelzl@29805: 
hoelzl@31863: subsection {* Implementing Taylor series expansion *}
hoelzl@31863: 
hoelzl@31863: fun isDERIV :: "nat \<Rightarrow> floatarith \<Rightarrow> real list \<Rightarrow> bool" where
hoelzl@31863: "isDERIV x (Add a b) vs         = (isDERIV x a vs \<and> isDERIV x b vs)" |
hoelzl@31863: "isDERIV x (Mult a b) vs        = (isDERIV x a vs \<and> isDERIV x b vs)" |
hoelzl@31863: "isDERIV x (Minus a) vs         = isDERIV x a vs" |
hoelzl@31863: "isDERIV x (Inverse a) vs       = (isDERIV x a vs \<and> interpret_floatarith a vs \<noteq> 0)" |
hoelzl@31863: "isDERIV x (Cos a) vs           = isDERIV x a vs" |
hoelzl@31863: "isDERIV x (Arctan a) vs        = isDERIV x a vs" |
hoelzl@31863: "isDERIV x (Min a b) vs         = False" |
hoelzl@31863: "isDERIV x (Max a b) vs         = False" |
hoelzl@31863: "isDERIV x (Abs a) vs           = False" |
hoelzl@31863: "isDERIV x Pi vs                = True" |
hoelzl@31863: "isDERIV x (Sqrt a) vs          = (isDERIV x a vs \<and> interpret_floatarith a vs > 0)" |
hoelzl@31863: "isDERIV x (Exp a) vs           = isDERIV x a vs" |
hoelzl@31863: "isDERIV x (Ln a) vs            = (isDERIV x a vs \<and> interpret_floatarith a vs > 0)" |
hoelzl@31863: "isDERIV x (Power a 0) vs       = True" |
hoelzl@31863: "isDERIV x (Power a (Suc n)) vs = isDERIV x a vs" |
hoelzl@31863: "isDERIV x (Num f) vs           = True" |
hoelzl@32919: "isDERIV x (Var n) vs          = True"
hoelzl@31863: 
hoelzl@31863: fun DERIV_floatarith :: "nat \<Rightarrow> floatarith \<Rightarrow> floatarith" where
hoelzl@31863: "DERIV_floatarith x (Add a b)         = Add (DERIV_floatarith x a) (DERIV_floatarith x b)" |
hoelzl@31863: "DERIV_floatarith x (Mult a b)        = Add (Mult a (DERIV_floatarith x b)) (Mult (DERIV_floatarith x a) b)" |
hoelzl@31863: "DERIV_floatarith x (Minus a)         = Minus (DERIV_floatarith x a)" |
hoelzl@31863: "DERIV_floatarith x (Inverse a)       = Minus (Mult (DERIV_floatarith x a) (Inverse (Power a 2)))" |
haftmann@58410: "DERIV_floatarith x (Cos a)           = Minus (Mult (Cos (Add (Mult Pi (Num (Float 1 (- 1)))) (Minus a))) (DERIV_floatarith x a))" |
hoelzl@31863: "DERIV_floatarith x (Arctan a)        = Mult (Inverse (Add (Num 1) (Power a 2))) (DERIV_floatarith x a)" |
hoelzl@31863: "DERIV_floatarith x (Min a b)         = Num 0" |
hoelzl@31863: "DERIV_floatarith x (Max a b)         = Num 0" |
hoelzl@31863: "DERIV_floatarith x (Abs a)           = Num 0" |
hoelzl@31863: "DERIV_floatarith x Pi                = Num 0" |
hoelzl@31863: "DERIV_floatarith x (Sqrt a)          = (Mult (Inverse (Mult (Sqrt a) (Num 2))) (DERIV_floatarith x a))" |
hoelzl@31863: "DERIV_floatarith x (Exp a)           = Mult (Exp a) (DERIV_floatarith x a)" |
hoelzl@31863: "DERIV_floatarith x (Ln a)            = Mult (Inverse a) (DERIV_floatarith x a)" |
hoelzl@31863: "DERIV_floatarith x (Power a 0)       = Num 0" |
hoelzl@31863: "DERIV_floatarith x (Power a (Suc n)) = Mult (Num (Float (int (Suc n)) 0)) (Mult (Power a n) (DERIV_floatarith x a))" |
hoelzl@31863: "DERIV_floatarith x (Num f)           = Num 0" |
hoelzl@32919: "DERIV_floatarith x (Var n)          = (if x = n then Num 1 else Num 0)"
hoelzl@31863: 
hoelzl@31863: lemma DERIV_floatarith:
hoelzl@31863:   assumes "n < length vs"
hoelzl@31863:   assumes isDERIV: "isDERIV n f (vs[n := x])"
hoelzl@31863:   shows "DERIV (\<lambda> x'. interpret_floatarith f (vs[n := x'])) x :>
hoelzl@31863:                interpret_floatarith (DERIV_floatarith n f) (vs[n := x])"
hoelzl@31863:    (is "DERIV (?i f) x :> _")
wenzelm@49351: using isDERIV
wenzelm@49351: proof (induct f arbitrary: x)
wenzelm@49351:   case (Inverse a)
wenzelm@49351:   thus ?case
hoelzl@56381:     by (auto intro!: derivative_eq_intros simp add: algebra_simps power2_eq_square)
wenzelm@49351: next
wenzelm@49351:   case (Cos a)
wenzelm@49351:   thus ?case
hoelzl@56382:     by (auto intro!: derivative_eq_intros
hoelzl@31863:            simp del: interpret_floatarith.simps(5)
hoelzl@31863:            simp add: interpret_floatarith_sin interpret_floatarith.simps(5)[of a])
wenzelm@49351: next
wenzelm@49351:   case (Power a n)
wenzelm@49351:   thus ?case
hoelzl@56382:     by (cases n) (auto intro!: derivative_eq_intros simp del: power_Suc simp add: real_of_nat_def)
wenzelm@49351: next
wenzelm@49351:   case (Ln a)
hoelzl@56382:   thus ?case by (auto intro!: derivative_eq_intros simp add: divide_inverse)
wenzelm@49351: next
wenzelm@49351:   case (Var i)
wenzelm@49351:   thus ?case using `n < length vs` by auto
hoelzl@56381: qed (auto intro!: derivative_eq_intros)
hoelzl@31863: 
hoelzl@31863: declare approx.simps[simp del]
hoelzl@31863: 
hoelzl@31863: fun isDERIV_approx :: "nat \<Rightarrow> nat \<Rightarrow> floatarith \<Rightarrow> (float * float) option list \<Rightarrow> bool" where
hoelzl@31863: "isDERIV_approx prec x (Add a b) vs         = (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs)" |
hoelzl@31863: "isDERIV_approx prec x (Mult a b) vs        = (isDERIV_approx prec x a vs \<and> isDERIV_approx prec x b vs)" |
hoelzl@31863: "isDERIV_approx prec x (Minus a) vs         = isDERIV_approx prec x a vs" |
hoelzl@31863: "isDERIV_approx prec x (Inverse a) vs       =
hoelzl@31863:   (isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l \<or> u < 0 | None \<Rightarrow> False))" |
hoelzl@31863: "isDERIV_approx prec x (Cos a) vs           = isDERIV_approx prec x a vs" |
hoelzl@31863: "isDERIV_approx prec x (Arctan a) vs        = isDERIV_approx prec x a vs" |
hoelzl@31863: "isDERIV_approx prec x (Min a b) vs         = False" |
hoelzl@31863: "isDERIV_approx prec x (Max a b) vs         = False" |
hoelzl@31863: "isDERIV_approx prec x (Abs a) vs           = False" |
hoelzl@31863: "isDERIV_approx prec x Pi vs                = True" |
hoelzl@31863: "isDERIV_approx prec x (Sqrt a) vs          =
hoelzl@31863:   (isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l | None \<Rightarrow> False))" |
hoelzl@31863: "isDERIV_approx prec x (Exp a) vs           = isDERIV_approx prec x a vs" |
hoelzl@31863: "isDERIV_approx prec x (Ln a) vs            =
hoelzl@31863:   (isDERIV_approx prec x a vs \<and> (case approx prec a vs of Some (l, u) \<Rightarrow> 0 < l | None \<Rightarrow> False))" |
hoelzl@31863: "isDERIV_approx prec x (Power a 0) vs       = True" |
hoelzl@31863: "isDERIV_approx prec x (Power a (Suc n)) vs = isDERIV_approx prec x a vs" |
hoelzl@31863: "isDERIV_approx prec x (Num f) vs           = True" |
hoelzl@32919: "isDERIV_approx prec x (Var n) vs          = True"
hoelzl@31863: 
hoelzl@31863: lemma isDERIV_approx:
hoelzl@31863:   assumes "bounded_by xs vs"
wenzelm@49351:     and isDERIV_approx: "isDERIV_approx prec x f vs"
hoelzl@31863:   shows "isDERIV x f xs"
wenzelm@49351:   using isDERIV_approx
wenzelm@49351: proof (induct f)
hoelzl@31863:   case (Inverse a)
hoelzl@31863:   then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
hoelzl@31863:     and *: "0 < l \<or> u < 0"
wenzelm@49351:     by (cases "approx prec a vs") auto
hoelzl@31863:   with approx[OF `bounded_by xs vs` approx_Some]
hoelzl@47600:   have "interpret_floatarith a xs \<noteq> 0" by auto
hoelzl@31863:   thus ?case using Inverse by auto
hoelzl@31863: next
hoelzl@31863:   case (Ln a)
hoelzl@31863:   then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
hoelzl@31863:     and *: "0 < l"
wenzelm@49351:     by (cases "approx prec a vs") auto
hoelzl@31863:   with approx[OF `bounded_by xs vs` approx_Some]
hoelzl@47600:   have "0 < interpret_floatarith a xs" by auto
hoelzl@31863:   thus ?case using Ln by auto
hoelzl@31863: next
hoelzl@31863:   case (Sqrt a)
hoelzl@31863:   then obtain l u where approx_Some: "Some (l, u) = approx prec a vs"
hoelzl@31863:     and *: "0 < l"
wenzelm@49351:     by (cases "approx prec a vs") auto
hoelzl@31863:   with approx[OF `bounded_by xs vs` approx_Some]
hoelzl@47600:   have "0 < interpret_floatarith a xs" by auto
hoelzl@31863:   thus ?case using Sqrt by auto
hoelzl@31863: next
wenzelm@49351:   case (Power a n) thus ?case by (cases n) auto
hoelzl@31863: qed auto
hoelzl@31863: 
hoelzl@31863: lemma bounded_by_update_var:
hoelzl@31863:   assumes "bounded_by xs vs" and "vs ! i = Some (l, u)"
wenzelm@49351:     and bnd: "x \<in> { real l .. real u }"
hoelzl@31863:   shows "bounded_by (xs[i := x]) vs"
hoelzl@31863: proof (cases "i < length xs")
wenzelm@49351:   case False
wenzelm@49351:   thus ?thesis using `bounded_by xs vs` by auto
hoelzl@31863: next
hoelzl@31863:   let ?xs = "xs[i := x]"
hoelzl@31863:   case True hence "i < length ?xs" by auto
wenzelm@49351:   {
wenzelm@49351:     fix j
wenzelm@49351:     assume "j < length vs"
wenzelm@49351:     have "case vs ! j of None \<Rightarrow> True | Some (l, u) \<Rightarrow> ?xs ! j \<in> { real l .. real u }"
wenzelm@49351:     proof (cases "vs ! j")
wenzelm@49351:       case (Some b)
wenzelm@49351:       thus ?thesis
wenzelm@49351:       proof (cases "i = j")
wenzelm@49351:         case True
wenzelm@49351:         thus ?thesis using `vs ! i = Some (l, u)` Some and bnd `i < length ?xs`
wenzelm@49351:           by auto
wenzelm@49351:       next
wenzelm@49351:         case False
wenzelm@49351:         thus ?thesis
wenzelm@49351:           using `bounded_by xs vs`[THEN bounded_byE, OF `j < length vs`] Some by auto
wenzelm@49351:       qed
wenzelm@49351:     qed auto
wenzelm@49351:   }
hoelzl@31863:   thus ?thesis unfolding bounded_by_def by auto
hoelzl@31863: qed
hoelzl@31863: 
hoelzl@31863: lemma isDERIV_approx':
hoelzl@31863:   assumes "bounded_by xs vs"
wenzelm@49351:     and vs_x: "vs ! x = Some (l, u)" and X_in: "X \<in> { real l .. real u }"
wenzelm@49351:     and approx: "isDERIV_approx prec x f vs"
hoelzl@31863:   shows "isDERIV x f (xs[x := X])"
hoelzl@31863: proof -
hoelzl@31863:   note bounded_by_update_var[OF `bounded_by xs vs` vs_x X_in] approx
hoelzl@31863:   thus ?thesis by (rule isDERIV_approx)
hoelzl@31863: qed
hoelzl@31863: 
hoelzl@31863: lemma DERIV_approx:
hoelzl@31863:   assumes "n < length xs" and bnd: "bounded_by xs vs"
wenzelm@49351:     and isD: "isDERIV_approx prec n f vs"
wenzelm@49351:     and app: "Some (l, u) = approx prec (DERIV_floatarith n f) vs" (is "_ = approx _ ?D _")
hoelzl@40881:   shows "\<exists>(x::real). l \<le> x \<and> x \<le> u \<and>
hoelzl@31863:              DERIV (\<lambda> x. interpret_floatarith f (xs[n := x])) (xs!n) :> x"
hoelzl@31863:          (is "\<exists> x. _ \<and> _ \<and> DERIV (?i f) _ :> _")
hoelzl@31863: proof (rule exI[of _ "?i ?D (xs!n)"], rule conjI[OF _ conjI])
hoelzl@31863:   let "?i f x" = "interpret_floatarith f (xs[n := x])"
hoelzl@31863:   from approx[OF bnd app]
hoelzl@40881:   show "l \<le> ?i ?D (xs!n)" and "?i ?D (xs!n) \<le> u"
hoelzl@31863:     using `n < length xs` by auto
hoelzl@31863:   from DERIV_floatarith[OF `n < length xs`, of f "xs!n"] isDERIV_approx[OF bnd isD]
hoelzl@31863:   show "DERIV (?i f) (xs!n) :> (?i ?D (xs!n))" by simp
hoelzl@31863: qed
hoelzl@31863: 
wenzelm@49351: fun lift_bin :: "(float * float) option \<Rightarrow>
wenzelm@49351:     (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float * float) option) \<Rightarrow>
wenzelm@49351:     (float * float) option" where
wenzelm@49351:   "lift_bin (Some (l1, u1)) (Some (l2, u2)) f = f l1 u1 l2 u2"
wenzelm@49351: | "lift_bin a b f = None"
hoelzl@31863: 
hoelzl@31863: lemma lift_bin:
hoelzl@31863:   assumes lift_bin_Some: "Some (l, u) = lift_bin a b f"
hoelzl@31863:   obtains l1 u1 l2 u2
hoelzl@31863:   where "a = Some (l1, u1)"
wenzelm@49351:     and "b = Some (l2, u2)"
wenzelm@49351:     and "f l1 u1 l2 u2 = Some (l, u)"
wenzelm@49351:   using assms by (cases a, simp, cases b, simp, auto)
hoelzl@31863: 
hoelzl@31863: fun approx_tse where
hoelzl@31863: "approx_tse prec n 0 c k f bs = approx prec f bs" |
hoelzl@31863: "approx_tse prec n (Suc s) c k f bs =
hoelzl@31863:   (if isDERIV_approx prec n f bs then
hoelzl@31863:     lift_bin (approx prec f (bs[n := Some (c,c)]))
hoelzl@31863:              (approx_tse prec n s c (Suc k) (DERIV_floatarith n f) bs)
hoelzl@31863:              (\<lambda> l1 u1 l2 u2. approx prec
hoelzl@32919:                  (Add (Var 0)
hoelzl@31863:                       (Mult (Inverse (Num (Float (int k) 0)))
hoelzl@32919:                                  (Mult (Add (Var (Suc (Suc 0))) (Minus (Num c)))
hoelzl@32919:                                        (Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), bs!n])
hoelzl@31863:   else approx prec f bs)"
hoelzl@31863: 
hoelzl@31863: lemma bounded_by_Cons:
hoelzl@31863:   assumes bnd: "bounded_by xs vs"
wenzelm@49351:     and x: "x \<in> { real l .. real u }"
hoelzl@31863:   shows "bounded_by (x#xs) ((Some (l, u))#vs)"
hoelzl@31863: proof -
wenzelm@49351:   {
wenzelm@49351:     fix i assume *: "i < length ((Some (l, u))#vs)"
hoelzl@31863:     have "case ((Some (l,u))#vs) ! i of Some (l, u) \<Rightarrow> (x#xs)!i \<in> { real l .. real u } | None \<Rightarrow> True"
hoelzl@31863:     proof (cases i)
hoelzl@31863:       case 0 with x show ?thesis by auto
hoelzl@31863:     next
hoelzl@31863:       case (Suc i) with * have "i < length vs" by auto
hoelzl@31863:       from bnd[THEN bounded_byE, OF this]
hoelzl@31863:       show ?thesis unfolding Suc nth_Cons_Suc .
wenzelm@49351:     qed
wenzelm@49351:   }
hoelzl@31863:   thus ?thesis by (auto simp add: bounded_by_def)
hoelzl@31863: qed
hoelzl@31863: 
hoelzl@31863: lemma approx_tse_generic:
hoelzl@31863:   assumes "bounded_by xs vs"
wenzelm@49351:     and bnd_c: "bounded_by (xs[x := c]) vs" and "x < length vs" and "x < length xs"
wenzelm@49351:     and bnd_x: "vs ! x = Some (lx, ux)"
wenzelm@49351:     and ate: "Some (l, u) = approx_tse prec x s c k f vs"
hoelzl@40881:   shows "\<exists> n. (\<forall> m < n. \<forall> (z::real) \<in> {lx .. ux}.
hoelzl@31863:       DERIV (\<lambda> y. interpret_floatarith ((DERIV_floatarith x ^^ m) f) (xs[x := y])) z :>
hoelzl@31863:             (interpret_floatarith ((DERIV_floatarith x ^^ (Suc m)) f) (xs[x := z])))
hoelzl@40881:    \<and> (\<forall> (t::real) \<in> {lx .. ux}.  (\<Sum> i = 0..<n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) *
hoelzl@40881:                   interpret_floatarith ((DERIV_floatarith x ^^ i) f) (xs[x := c]) *
hoelzl@40881:                   (xs!x - c)^i) +
hoelzl@31863:       inverse (real (\<Prod> j \<in> {k..<k+n}. j)) *
hoelzl@31863:       interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := t]) *
hoelzl@40881:       (xs!x - c)^n \<in> {l .. u})" (is "\<exists> n. ?taylor f k l u n")
hoelzl@31863: using ate proof (induct s arbitrary: k f l u)
hoelzl@31863:   case 0
wenzelm@49351:   {
wenzelm@49351:     fix t::real assume "t \<in> {lx .. ux}"
hoelzl@31863:     note bounded_by_update_var[OF `bounded_by xs vs` bnd_x this]
hoelzl@31863:     from approx[OF this 0[unfolded approx_tse.simps]]
hoelzl@40881:     have "(interpret_floatarith f (xs[x := t])) \<in> {l .. u}"
hoelzl@31863:       by (auto simp add: algebra_simps)
wenzelm@49351:   }
wenzelm@49351:   thus ?case by (auto intro!: exI[of _ 0])
hoelzl@31863: next
hoelzl@31863:   case (Suc s)
hoelzl@31863:   show ?case
hoelzl@31863:   proof (cases "isDERIV_approx prec x f vs")
hoelzl@31863:     case False
hoelzl@31863:     note ap = Suc.prems[unfolded approx_tse.simps if_not_P[OF False]]
wenzelm@49351:     {
wenzelm@49351:       fix t::real assume "t \<in> {lx .. ux}"
hoelzl@31863:       note bounded_by_update_var[OF `bounded_by xs vs` bnd_x this]
hoelzl@31863:       from approx[OF this ap]
hoelzl@40881:       have "(interpret_floatarith f (xs[x := t])) \<in> {l .. u}"
wenzelm@32960:         by (auto simp add: algebra_simps)
wenzelm@49351:     }
wenzelm@49351:     thus ?thesis by (auto intro!: exI[of _ 0])
hoelzl@31863:   next
hoelzl@31863:     case True
hoelzl@31863:     with Suc.prems
hoelzl@31863:     obtain l1 u1 l2 u2
hoelzl@31863:       where a: "Some (l1, u1) = approx prec f (vs[x := Some (c,c)])"
wenzelm@49351:         and ate: "Some (l2, u2) = approx_tse prec x s c (Suc k) (DERIV_floatarith x f) vs"
wenzelm@49351:         and final: "Some (l, u) = approx prec
wenzelm@49351:           (Add (Var 0)
wenzelm@49351:                (Mult (Inverse (Num (Float (int k) 0)))
wenzelm@49351:                      (Mult (Add (Var (Suc (Suc 0))) (Minus (Num c)))
wenzelm@49351:                            (Var (Suc 0))))) [Some (l1, u1), Some (l2, u2), vs!x]"
nipkow@56073:       by (auto elim!: lift_bin)
hoelzl@31863: 
hoelzl@31863:     from bnd_c `x < length xs`
hoelzl@40881:     have bnd: "bounded_by (xs[x:=c]) (vs[x:= Some (c,c)])"
hoelzl@31863:       by (auto intro!: bounded_by_update)
hoelzl@31863: 
hoelzl@31863:     from approx[OF this a]
hoelzl@40881:     have f_c: "interpret_floatarith ((DERIV_floatarith x ^^ 0) f) (xs[x := c]) \<in> { l1 .. u1 }"
hoelzl@31863:               (is "?f 0 (real c) \<in> _")
hoelzl@31863:       by auto
hoelzl@31863: 
wenzelm@49351:     {
wenzelm@49351:       fix f :: "'a \<Rightarrow> 'a" fix n :: nat fix x :: 'a
hoelzl@31863:       have "(f ^^ Suc n) x = (f ^^ n) (f x)"
wenzelm@49351:         by (induct n) auto
wenzelm@49351:     }
hoelzl@31863:     note funpow_Suc = this[symmetric]
hoelzl@31863:     from Suc.hyps[OF ate, unfolded this]
hoelzl@31863:     obtain n
hoelzl@40881:       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"
hoelzl@40881:       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) +
hoelzl@40881:            inverse (real (\<Prod> j \<in> {Suc k..<Suc k + n}. j)) * ?f (Suc n) t * (xs!x - c)^n \<in> {l2 .. u2}"
hoelzl@31863:           (is "\<forall> t \<in> _. ?X (Suc k) f n t \<in> _")
hoelzl@31863:       by blast
hoelzl@31863: 
wenzelm@49351:     {
wenzelm@49351:       fix m and z::real
hoelzl@40881:       assume "m < Suc n" and bnd_z: "z \<in> { lx .. ux }"
hoelzl@31863:       have "DERIV (?f m) z :> ?f (Suc m) z"
hoelzl@31863:       proof (cases m)
wenzelm@32960:         case 0
wenzelm@32960:         with DERIV_floatarith[OF `x < length xs` isDERIV_approx'[OF `bounded_by xs vs` bnd_x bnd_z True]]
wenzelm@32960:         show ?thesis by simp
hoelzl@31863:       next
wenzelm@32960:         case (Suc m')
wenzelm@32960:         hence "m' < n" using `m < Suc n` by auto
wenzelm@32960:         from DERIV_hyp[OF this bnd_z]
wenzelm@32960:         show ?thesis using Suc by simp
wenzelm@49351:       qed
wenzelm@49351:     } note DERIV = this
hoelzl@31863: 
hoelzl@31863:     have "\<And> k i. k < i \<Longrightarrow> {k ..< i} = insert k {Suc k ..< i}" by auto
hoelzl@31863:     hence setprod_head_Suc: "\<And> k i. \<Prod> {k ..< k + Suc i} = k * \<Prod> {Suc k ..< Suc k + i}" by auto
hoelzl@31863:     have setsum_move0: "\<And> k F. setsum F {0..<Suc k} = F 0 + setsum (\<lambda> k. F (Suc k)) {0..<k}"
hoelzl@31863:       unfolding setsum_shift_bounds_Suc_ivl[symmetric]
hoelzl@31863:       unfolding setsum_head_upt_Suc[OF zero_less_Suc] ..
hoelzl@40881:     def C \<equiv> "xs!x - c"
hoelzl@40881: 
wenzelm@49351:     {
wenzelm@49351:       fix t::real assume t: "t \<in> {lx .. ux}"
hoelzl@31863:       hence "bounded_by [xs!x] [vs!x]"
wenzelm@32960:         using `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`]
wenzelm@32960:         by (cases "vs!x", auto simp add: bounded_by_def)
hoelzl@31863: 
hoelzl@31863:       with hyp[THEN bspec, OF t] f_c
hoelzl@40881:       have "bounded_by [?f 0 c, ?X (Suc k) f n t, xs!x] [Some (l1, u1), Some (l2, u2), vs!x]"
wenzelm@32960:         by (auto intro!: bounded_by_Cons)
hoelzl@31863:       from approx[OF this final, unfolded atLeastAtMost_iff[symmetric]]
hoelzl@40881:       have "?X (Suc k) f n t * (xs!x - real c) * inverse k + ?f 0 c \<in> {l .. u}"
wenzelm@32960:         by (auto simp add: algebra_simps)
hoelzl@40881:       also have "?X (Suc k) f n t * (xs!x - real c) * inverse (real k) + ?f 0 c =
hoelzl@40881:                (\<Sum> i = 0..<Suc n. inverse (real (\<Prod> j \<in> {k..<k+i}. j)) * ?f i c * (xs!x - c)^i) +
hoelzl@40881:                inverse (real (\<Prod> j \<in> {k..<k+Suc n}. j)) * ?f (Suc n) t * (xs!x - c)^Suc n" (is "_ = ?T")
wenzelm@32960:         unfolding funpow_Suc C_def[symmetric] setsum_move0 setprod_head_Suc
haftmann@35082:         by (auto simp add: algebra_simps)
haftmann@57512:           (simp only: mult.left_commute [of _ "inverse (real k)"] setsum_right_distrib [symmetric])
wenzelm@49351:       finally have "?T \<in> {l .. u}" .
wenzelm@49351:     }
hoelzl@31863:     thus ?thesis using DERIV by blast
hoelzl@31863:   qed
hoelzl@31863: qed
hoelzl@31863: 
hoelzl@31863: lemma setprod_fact: "\<Prod> {1..<1 + k} = fact (k :: nat)"
hoelzl@31863: proof (induct k)
wenzelm@49351:   case 0
wenzelm@49351:   show ?case by simp
wenzelm@49351: next
hoelzl@31863:   case (Suc k)
hoelzl@31863:   have "{ 1 ..< Suc (Suc k) } = insert (Suc k) { 1 ..< Suc k }" by auto
hoelzl@31863:   hence "\<Prod> { 1 ..< Suc (Suc k) } = (Suc k) * \<Prod> { 1 ..< Suc k }" by auto
hoelzl@31863:   thus ?case using Suc by auto
wenzelm@49351: qed
hoelzl@31863: 
hoelzl@31863: lemma approx_tse:
hoelzl@31863:   assumes "bounded_by xs vs"
wenzelm@49351:     and bnd_x: "vs ! x = Some (lx, ux)" and bnd_c: "real c \<in> {lx .. ux}"
wenzelm@49351:     and "x < length vs" and "x < length xs"
wenzelm@49351:     and ate: "Some (l, u) = approx_tse prec x s c 1 f vs"
hoelzl@40881:   shows "interpret_floatarith f xs \<in> { l .. u }"
hoelzl@31863: proof -
hoelzl@31863:   def F \<equiv> "\<lambda> n z. interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := z])"
hoelzl@31863:   hence F0: "F 0 = (\<lambda> z. interpret_floatarith f (xs[x := z]))" by auto
hoelzl@31863: 
hoelzl@40881:   hence "bounded_by (xs[x := c]) vs" and "x < length vs" "x < length xs"
hoelzl@31863:     using `bounded_by xs vs` bnd_x bnd_c `x < length vs` `x < length xs`
hoelzl@31863:     by (auto intro!: bounded_by_update_var)
hoelzl@31863: 
hoelzl@31863:   from approx_tse_generic[OF `bounded_by xs vs` this bnd_x ate]
hoelzl@31863:   obtain n
hoelzl@31863:     where DERIV: "\<forall> m z. m < n \<and> real lx \<le> z \<and> z \<le> real ux \<longrightarrow> DERIV (F m) z :> F (Suc m) z"
hoelzl@40881:     and hyp: "\<And> (t::real). t \<in> {lx .. ux} \<Longrightarrow>
hoelzl@40881:            (\<Sum> j = 0..<n. inverse (real (fact j)) * F j c * (xs!x - c)^j) +
hoelzl@40881:              inverse (real (fact n)) * F n t * (xs!x - c)^n
hoelzl@40881:              \<in> {l .. u}" (is "\<And> t. _ \<Longrightarrow> ?taylor t \<in> _")
hoelzl@31863:     unfolding F_def atLeastAtMost_iff[symmetric] setprod_fact by blast
hoelzl@31863: 
hoelzl@40881:   have bnd_xs: "xs ! x \<in> { lx .. ux }"
hoelzl@31863:     using `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`] bnd_x by auto
hoelzl@31863: 
hoelzl@31863:   show ?thesis
hoelzl@31863:   proof (cases n)
hoelzl@31863:     case 0 thus ?thesis using hyp[OF bnd_xs] unfolding F_def by auto
hoelzl@31863:   next
hoelzl@31863:     case (Suc n')
hoelzl@31863:     show ?thesis
hoelzl@40881:     proof (cases "xs ! x = c")
hoelzl@31863:       case True
hoelzl@31863:       from True[symmetric] hyp[OF bnd_xs] Suc show ?thesis
wenzelm@32960:         unfolding F_def Suc setsum_head_upt_Suc[OF zero_less_Suc] setsum_shift_bounds_Suc_ivl by auto
hoelzl@31863:     next
hoelzl@31863:       case False
hoelzl@31863: 
hoelzl@40881:       have "lx \<le> real c" "real c \<le> ux" "lx \<le> xs!x" "xs!x \<le> ux"
wenzelm@32960:         using Suc bnd_c `bounded_by xs vs`[THEN bounded_byE, OF `x < length vs`] bnd_x by auto
hoelzl@31863:       from Taylor.taylor[OF zero_less_Suc, of F, OF F0 DERIV[unfolded Suc] this False]
hoelzl@40881:       obtain t::real where t_bnd: "if xs ! x < c then xs ! x < t \<and> t < c else c < t \<and> t < xs ! x"
wenzelm@32960:         and fl_eq: "interpret_floatarith f (xs[x := xs ! x]) =
hoelzl@40881:            (\<Sum>m = 0..<Suc n'. F m c / real (fact m) * (xs ! x - c) ^ m) +
hoelzl@40881:            F (Suc n') t / real (fact (Suc n')) * (xs ! x - c) ^ Suc n'"
huffman@56195:         unfolding atLeast0LessThan by blast
hoelzl@31863: 
hoelzl@40881:       from t_bnd bnd_xs bnd_c have *: "t \<in> {lx .. ux}"
hoelzl@40881:         by (cases "xs ! x < c", auto)
hoelzl@31863: 
hoelzl@31863:       have "interpret_floatarith f (xs[x := xs ! x]) = ?taylor t"
wenzelm@32960:         unfolding fl_eq Suc by (auto simp add: algebra_simps divide_inverse)
hoelzl@40881:       also have "\<dots> \<in> {l .. u}" using * by (rule hyp)
hoelzl@31863:       finally show ?thesis by simp
hoelzl@31863:     qed
hoelzl@31863:   qed
hoelzl@31863: qed
hoelzl@31863: 
hoelzl@31863: fun approx_tse_form' where
hoelzl@31863: "approx_tse_form' prec t f 0 l u cmp =
haftmann@58410:   (case approx_tse prec 0 t ((l + u) * Float 1 (- 1)) 1 f [Some (l, u)]
hoelzl@31863:      of Some (l, u) \<Rightarrow> cmp l u | None \<Rightarrow> False)" |
hoelzl@31863: "approx_tse_form' prec t f (Suc s) l u cmp =
haftmann@58410:   (let m = (l + u) * Float 1 (- 1)
hoelzl@32919:    in (if approx_tse_form' prec t f s l m cmp then
hoelzl@32919:       approx_tse_form' prec t f s m u cmp else False))"
hoelzl@31863: 
hoelzl@31863: lemma approx_tse_form':
hoelzl@40881:   fixes x :: real
hoelzl@40881:   assumes "approx_tse_form' prec t f s l u cmp" and "x \<in> {l .. u}"
hoelzl@40881:   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>
haftmann@58410:                   approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)"
hoelzl@31863: using assms proof (induct s arbitrary: l u)
hoelzl@31863:   case 0
hoelzl@31863:   then obtain ly uy
haftmann@58410:     where *: "approx_tse prec 0 t ((l + u) * Float 1 (- 1)) 1 f [Some (l, u)] = Some (ly, uy)"
blanchet@55413:     and **: "cmp ly uy" by (auto elim!: case_optionE)
haftmann@46545:   with 0 show ?case by auto
hoelzl@31863: next
hoelzl@31863:   case (Suc s)
haftmann@58410:   let ?m = "(l + u) * Float 1 (- 1)"
hoelzl@31863:   from Suc.prems
hoelzl@31863:   have l: "approx_tse_form' prec t f s l ?m cmp"
hoelzl@31863:     and u: "approx_tse_form' prec t f s ?m u cmp"
hoelzl@32919:     by (auto simp add: Let_def lazy_conj)
hoelzl@31863: 
hoelzl@40881:   have m_l: "real l \<le> ?m" and m_u: "?m \<le> real u"
hoelzl@47599:     unfolding less_eq_float_def using Suc.prems by auto
hoelzl@31863: 
hoelzl@40881:   with `x \<in> { l .. u }`
hoelzl@40881:   have "x \<in> { l .. ?m} \<or> x \<in> { ?m .. u }" by auto
hoelzl@31863:   thus ?case
hoelzl@31863:   proof (rule disjE)
hoelzl@40881:     assume "x \<in> { l .. ?m}"
hoelzl@31863:     from Suc.hyps[OF l this]
hoelzl@31863:     obtain l' u' ly uy
hoelzl@40881:       where "x \<in> { l' .. u' } \<and> real l \<le> l' \<and> real u' \<le> ?m \<and> cmp ly uy \<and>
haftmann@58410:                   approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)" by blast
hoelzl@31863:     with m_u show ?thesis by (auto intro!: exI)
hoelzl@31863:   next
hoelzl@40881:     assume "x \<in> { ?m .. u }"
hoelzl@31863:     from Suc.hyps[OF u this]
hoelzl@31863:     obtain l' u' ly uy
hoelzl@40881:       where "x \<in> { l' .. u' } \<and> ?m \<le> real l' \<and> u' \<le> real u \<and> cmp ly uy \<and>
haftmann@58410:                   approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 f [Some (l', u')] = Some (ly, uy)" by blast
hoelzl@31863:     with m_u show ?thesis by (auto intro!: exI)
hoelzl@31863:   qed
hoelzl@31863: qed
hoelzl@31863: 
hoelzl@31863: lemma approx_tse_form'_less:
hoelzl@40881:   fixes x :: real
hoelzl@31863:   assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 < l)"
hoelzl@40881:   and x: "x \<in> {l .. u}"
hoelzl@31863:   shows "interpret_floatarith b [x] < interpret_floatarith a [x]"
hoelzl@31863: proof -
hoelzl@31863:   from approx_tse_form'[OF tse x]
hoelzl@31863:   obtain l' u' ly uy
hoelzl@40881:     where x': "x \<in> { l' .. u' }" and "l \<le> real l'"
hoelzl@40881:     and "real u' \<le> u" and "0 < ly"
haftmann@58410:     and tse: "approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)"
hoelzl@31863:     by blast
hoelzl@31863: 
hoelzl@31863:   hence "bounded_by [x] [Some (l', u')]" by (auto simp add: bounded_by_def)
hoelzl@31863: 
hoelzl@31863:   from approx_tse[OF this _ _ _ _ tse[symmetric], of l' u'] x'
hoelzl@40881:   have "ly \<le> interpret_floatarith a [x] - interpret_floatarith b [x]"
haftmann@54230:     by auto
hoelzl@47600:   from order_less_le_trans[OF _ this, of 0] `0 < ly`
hoelzl@31863:   show ?thesis by auto
hoelzl@31863: qed
hoelzl@31863: 
hoelzl@31863: lemma approx_tse_form'_le:
hoelzl@40881:   fixes x :: real
hoelzl@31863:   assumes tse: "approx_tse_form' prec t (Add a (Minus b)) s l u (\<lambda> l u. 0 \<le> l)"
hoelzl@40881:   and x: "x \<in> {l .. u}"
hoelzl@31863:   shows "interpret_floatarith b [x] \<le> interpret_floatarith a [x]"
hoelzl@31863: proof -
hoelzl@31863:   from approx_tse_form'[OF tse x]
hoelzl@31863:   obtain l' u' ly uy
hoelzl@40881:     where x': "x \<in> { l' .. u' }" and "l \<le> real l'"
hoelzl@40881:     and "real u' \<le> u" and "0 \<le> ly"
haftmann@58410:     and tse: "approx_tse prec 0 t ((l' + u') * Float 1 (- 1)) 1 (Add a (Minus b)) [Some (l', u')] = Some (ly, uy)"
hoelzl@31863:     by blast
hoelzl@31863: 
hoelzl@31863:   hence "bounded_by [x] [Some (l', u')]" by (auto simp add: bounded_by_def)
hoelzl@31863: 
hoelzl@31863:   from approx_tse[OF this _ _ _ _ tse[symmetric], of l' u'] x'
hoelzl@40881:   have "ly \<le> interpret_floatarith a [x] - interpret_floatarith b [x]"
haftmann@54230:     by auto
hoelzl@47600:   from order_trans[OF _ this, of 0] `0 \<le> ly`
hoelzl@31863:   show ?thesis by auto
hoelzl@31863: qed
hoelzl@31863: 
hoelzl@31863: definition
hoelzl@31863: "approx_tse_form prec t s f =
hoelzl@31863:   (case f
hoelzl@32919:    of (Bound x a b f) \<Rightarrow> x = Var 0 \<and>
hoelzl@31863:      (case (approx prec a [None], approx prec b [None])
hoelzl@31863:       of (Some (l, u), Some (l', u')) \<Rightarrow>
hoelzl@31863:         (case f
hoelzl@31863:          of Less lf rt \<Rightarrow> approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 < l)
hoelzl@31863:           | LessEqual lf rt \<Rightarrow> approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)
hoelzl@31863:           | AtLeastAtMost x lf rt \<Rightarrow>
hoelzl@32919:             (if approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l) then
hoelzl@32919:             approx_tse_form' prec t (Add rt (Minus x)) s l u' (\<lambda> l u. 0 \<le> l) else False)
hoelzl@31863:           | _ \<Rightarrow> False)
hoelzl@31863:        | _ \<Rightarrow> False)
hoelzl@31863:    | _ \<Rightarrow> False)"
hoelzl@31863: 
hoelzl@31863: lemma approx_tse_form:
hoelzl@31863:   assumes "approx_tse_form prec t s f"
hoelzl@31863:   shows "interpret_form f [x]"
hoelzl@31863: proof (cases f)
hoelzl@31863:   case (Bound i a b f') note f_def = this
hoelzl@31863:   with assms obtain l u l' u'
hoelzl@31863:     where a: "approx prec a [None] = Some (l, u)"
hoelzl@31863:     and b: "approx prec b [None] = Some (l', u')"
blanchet@55413:     unfolding approx_tse_form_def by (auto elim!: case_optionE)
hoelzl@31863: 
hoelzl@32919:   from Bound assms have "i = Var 0" unfolding approx_tse_form_def by auto
hoelzl@31863:   hence i: "interpret_floatarith i [x] = x" by auto
hoelzl@31863: 
hoelzl@31863:   { let "?f z" = "interpret_floatarith z [x]"
hoelzl@31863:     assume "?f i \<in> { ?f a .. ?f b }"
hoelzl@31863:     with approx[OF _ a[symmetric], of "[x]"] approx[OF _ b[symmetric], of "[x]"]
hoelzl@40881:     have bnd: "x \<in> { l .. u'}" unfolding bounded_by_def i by auto
hoelzl@31863: 
hoelzl@31863:     have "interpret_form f' [x]"
hoelzl@31863:     proof (cases f')
hoelzl@31863:       case (Less lf rt)
hoelzl@31863:       with Bound a b assms
hoelzl@31863:       have "approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 < l)"
wenzelm@32960:         unfolding approx_tse_form_def by auto
hoelzl@31863:       from approx_tse_form'_less[OF this bnd]
hoelzl@31863:       show ?thesis using Less by auto
hoelzl@31863:     next
hoelzl@31863:       case (LessEqual lf rt)
hoelzl@31863:       with Bound a b assms
hoelzl@31863:       have "approx_tse_form' prec t (Add rt (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)"
wenzelm@32960:         unfolding approx_tse_form_def by auto
hoelzl@31863:       from approx_tse_form'_le[OF this bnd]
hoelzl@31863:       show ?thesis using LessEqual by auto
hoelzl@31863:     next
hoelzl@31863:       case (AtLeastAtMost x lf rt)
hoelzl@31863:       with Bound a b assms
hoelzl@31863:       have "approx_tse_form' prec t (Add rt (Minus x)) s l u' (\<lambda> l u. 0 \<le> l)"
wenzelm@32960:         and "approx_tse_form' prec t (Add x (Minus lf)) s l u' (\<lambda> l u. 0 \<le> l)"
wenzelm@32960:         unfolding approx_tse_form_def lazy_conj by auto
hoelzl@31863:       from approx_tse_form'_le[OF this(1) bnd] approx_tse_form'_le[OF this(2) bnd]
hoelzl@31863:       show ?thesis using AtLeastAtMost by auto
hoelzl@31863:     next
hoelzl@31863:       case (Bound x a b f') with assms
blanchet@55413:       show ?thesis by (auto elim!: case_optionE simp add: f_def approx_tse_form_def)
hoelzl@31863:     next
hoelzl@31863:       case (Assign x a f') with assms
blanchet@55413:       show ?thesis by (auto elim!: case_optionE simp add: f_def approx_tse_form_def)
hoelzl@31863:     qed } thus ?thesis unfolding f_def by auto
wenzelm@49351: next
wenzelm@49351:   case Assign
wenzelm@49351:   with assms show ?thesis by (auto simp add: approx_tse_form_def)
wenzelm@49351: next
wenzelm@49351:   case LessEqual
wenzelm@49351:   with assms show ?thesis by (auto simp add: approx_tse_form_def)
wenzelm@49351: next
wenzelm@49351:   case Less
wenzelm@49351:   with assms show ?thesis by (auto simp add: approx_tse_form_def)
wenzelm@49351: next
wenzelm@49351:   case AtLeastAtMost
wenzelm@49351:   with assms show ?thesis by (auto simp add: approx_tse_form_def)
hoelzl@31863: qed
hoelzl@31863: 
hoelzl@32919: text {* @{term approx_form_eval} is only used for the {\tt value}-command. *}
hoelzl@32919: 
hoelzl@32919: fun approx_form_eval :: "nat \<Rightarrow> form \<Rightarrow> (float * float) option list \<Rightarrow> (float * float) option list" where
hoelzl@32919: "approx_form_eval prec (Bound (Var n) a b f) bs =
hoelzl@32919:    (case (approx prec a bs, approx prec b bs)
hoelzl@32919:    of (Some (l, _), Some (_, u)) \<Rightarrow> approx_form_eval prec f (bs[n := Some (l, u)])
hoelzl@32919:     | _ \<Rightarrow> bs)" |
hoelzl@32919: "approx_form_eval prec (Assign (Var n) a f) bs =
hoelzl@32919:    (case (approx prec a bs)
hoelzl@32919:    of (Some (l, u)) \<Rightarrow> approx_form_eval prec f (bs[n := Some (l, u)])
hoelzl@32919:     | _ \<Rightarrow> bs)" |
hoelzl@32919: "approx_form_eval prec (Less a b) bs = bs @ [approx prec a bs, approx prec b bs]" |
hoelzl@32919: "approx_form_eval prec (LessEqual a b) bs = bs @ [approx prec a bs, approx prec b bs]" |
hoelzl@32919: "approx_form_eval prec (AtLeastAtMost x a b) bs =
hoelzl@32919:    bs @ [approx prec x bs, approx prec a bs, approx prec b bs]" |
hoelzl@32919: "approx_form_eval _ _ bs = bs"
hoelzl@32919: 
hoelzl@29805: subsection {* Implement proof method \texttt{approximation} *}
hoelzl@29805: 
hoelzl@31811: lemmas interpret_form_equations = interpret_form.simps interpret_floatarith.simps interpret_floatarith_num
hoelzl@31098:   interpret_floatarith_divide interpret_floatarith_diff interpret_floatarith_tan interpret_floatarith_powr interpret_floatarith_log
hoelzl@31467:   interpret_floatarith_sin
hoelzl@29805: 
haftmann@36985: oracle approximation_oracle = {* fn (thy, t) =>
haftmann@36985: let
haftmann@36985:   fun bad t = error ("Bad term: " ^ Syntax.string_of_term_global thy t);
haftmann@36985: 
wenzelm@38716:   fun term_of_bool true = @{term True}
wenzelm@38716:     | term_of_bool false = @{term False};
haftmann@36985: 
haftmann@51143:   val mk_int = HOLogic.mk_number @{typ int} o @{code integer_of_int};
haftmann@51143:   val dest_int = @{code int_of_integer} o snd o HOLogic.dest_number;
haftmann@51143: 
haftmann@36985:   fun term_of_float (@{code Float} (k, l)) =
haftmann@51143:     @{term Float} $ mk_int k $ mk_int l;
haftmann@36985: 
haftmann@36985:   fun term_of_float_float_option NONE = @{term "None :: (float \<times> float) option"}
haftmann@36985:     | term_of_float_float_option (SOME ff) = @{term "Some :: float \<times> float \<Rightarrow> _"}
haftmann@36985:         $ HOLogic.mk_prod (pairself term_of_float ff);
haftmann@36985: 
haftmann@36985:   val term_of_float_float_option_list =
haftmann@36985:     HOLogic.mk_list @{typ "(float \<times> float) option"} o map term_of_float_float_option;
haftmann@36985: 
haftmann@51143:   fun nat_of_term t = @{code nat_of_integer}
haftmann@51143:     (HOLogic.dest_nat t handle TERM _ => snd (HOLogic.dest_number t));
haftmann@36985: 
haftmann@36985:   fun float_of_term (@{term Float} $ k $ l) =
haftmann@51143:         @{code Float} (dest_int k, dest_int l)
haftmann@36985:     | float_of_term t = bad t;
haftmann@36985: 
haftmann@36985:   fun floatarith_of_term (@{term Add} $ a $ b) = @{code Add} (floatarith_of_term a, floatarith_of_term b)
haftmann@36985:     | floatarith_of_term (@{term Minus} $ a) = @{code Minus} (floatarith_of_term a)
haftmann@36985:     | floatarith_of_term (@{term Mult} $ a $ b) = @{code Mult} (floatarith_of_term a, floatarith_of_term b)
haftmann@36985:     | floatarith_of_term (@{term Inverse} $ a) = @{code Inverse} (floatarith_of_term a)
haftmann@36985:     | floatarith_of_term (@{term Cos} $ a) = @{code Cos} (floatarith_of_term a)
haftmann@36985:     | floatarith_of_term (@{term Arctan} $ a) = @{code Arctan} (floatarith_of_term a)
haftmann@36985:     | floatarith_of_term (@{term Abs} $ a) = @{code Abs} (floatarith_of_term a)
haftmann@36985:     | floatarith_of_term (@{term Max} $ a $ b) = @{code Max} (floatarith_of_term a, floatarith_of_term b)
haftmann@36985:     | floatarith_of_term (@{term Min} $ a $ b) = @{code Min} (floatarith_of_term a, floatarith_of_term b)
haftmann@36985:     | floatarith_of_term @{term Pi} = @{code Pi}
haftmann@36985:     | floatarith_of_term (@{term Sqrt} $ a) = @{code Sqrt} (floatarith_of_term a)
haftmann@36985:     | floatarith_of_term (@{term Exp} $ a) = @{code Exp} (floatarith_of_term a)
haftmann@36985:     | floatarith_of_term (@{term Ln} $ a) = @{code Ln} (floatarith_of_term a)
haftmann@36985:     | floatarith_of_term (@{term Power} $ a $ n) =
haftmann@36985:         @{code Power} (floatarith_of_term a, nat_of_term n)
haftmann@36985:     | floatarith_of_term (@{term Var} $ n) = @{code Var} (nat_of_term n)
haftmann@36985:     | floatarith_of_term (@{term Num} $ m) = @{code Num} (float_of_term m)
haftmann@36985:     | floatarith_of_term t = bad t;
haftmann@36985: 
haftmann@36985:   fun form_of_term (@{term Bound} $ a $ b $ c $ p) = @{code Bound}
haftmann@36985:         (floatarith_of_term a, floatarith_of_term b, floatarith_of_term c, form_of_term p)
haftmann@36985:     | form_of_term (@{term Assign} $ a $ b $ p) = @{code Assign}
haftmann@36985:         (floatarith_of_term a, floatarith_of_term b, form_of_term p)
haftmann@36985:     | form_of_term (@{term Less} $ a $ b) = @{code Less}
haftmann@36985:         (floatarith_of_term a, floatarith_of_term b)
haftmann@36985:     | form_of_term (@{term LessEqual} $ a $ b) = @{code LessEqual}
haftmann@36985:         (floatarith_of_term a, floatarith_of_term b)
haftmann@36985:     | form_of_term (@{term AtLeastAtMost} $ a $ b $ c) = @{code AtLeastAtMost}
haftmann@36985:         (floatarith_of_term a, floatarith_of_term b, floatarith_of_term c)
haftmann@36985:     | form_of_term t = bad t;
haftmann@36985: 
haftmann@36985:   fun float_float_option_of_term @{term "None :: (float \<times> float) option"} = NONE
haftmann@36985:     | float_float_option_of_term (@{term "Some :: float \<times> float \<Rightarrow> _"} $ ff) =
haftmann@36985:         SOME (pairself float_of_term (HOLogic.dest_prod ff))
haftmann@36985:     | float_float_option_of_term (@{term approx'} $ n $ a $ ffs) = @{code approx'}
haftmann@36985:         (nat_of_term n) (floatarith_of_term a) (float_float_option_list_of_term ffs)
haftmann@36985:     | float_float_option_of_term t = bad t
haftmann@36985:   and float_float_option_list_of_term
haftmann@36985:         (@{term "replicate :: _ \<Rightarrow> (float \<times> float) option \<Rightarrow> _"} $ n $ @{term "None :: (float \<times> float) option"}) =
haftmann@36985:           @{code replicate} (nat_of_term n) NONE
haftmann@36985:     | float_float_option_list_of_term (@{term approx_form_eval} $ n $ p $ ffs) =
haftmann@36985:         @{code approx_form_eval} (nat_of_term n) (form_of_term p) (float_float_option_list_of_term ffs)
haftmann@36985:     | float_float_option_list_of_term t = map float_float_option_of_term
haftmann@36985:         (HOLogic.dest_list t);
haftmann@36985: 
haftmann@36985:   val nat_list_of_term = map nat_of_term o HOLogic.dest_list ;
haftmann@36985: 
haftmann@36985:   fun bool_of_term (@{term approx_form} $ n $ p $ ffs $ ms) = @{code approx_form}
haftmann@36985:         (nat_of_term n) (form_of_term p) (float_float_option_list_of_term ffs) (nat_list_of_term ms)
haftmann@36985:     | bool_of_term (@{term approx_tse_form} $ m $ n $ q $ p) =
haftmann@36985:         @{code approx_tse_form} (nat_of_term m) (nat_of_term n) (nat_of_term q) (form_of_term p)
haftmann@36985:     | bool_of_term t = bad t;
haftmann@36985: 
haftmann@36985:   fun eval t = case fastype_of t
haftmann@36985:    of @{typ bool} =>
haftmann@36985:         (term_of_bool o bool_of_term) t
haftmann@36985:     | @{typ "(float \<times> float) option"} =>
haftmann@36985:         (term_of_float_float_option o float_float_option_of_term) t
haftmann@36985:     | @{typ "(float \<times> float) option list"} =>
haftmann@36985:         (term_of_float_float_option_list o float_float_option_list_of_term) t
haftmann@36985:     | _ => bad t;
haftmann@36985: 
wenzelm@52131:   val normalize = eval o Envir.beta_norm o Envir.eta_long [];
haftmann@36985: 
haftmann@36985: in Thm.cterm_of thy (Logic.mk_equals (t, normalize t)) end
haftmann@36985: *}
hoelzl@31099: 
hoelzl@31099: ML {*
wenzelm@32212:   fun reorder_bounds_tac prems i =
hoelzl@29805:     let
haftmann@38558:       fun variable_of_bound (Const (@{const_name Trueprop}, _) $
haftmann@37677:                              (Const (@{const_name Set.member}, _) $
hoelzl@31811:                               Free (name, _) $ _)) = name
haftmann@38558:         | variable_of_bound (Const (@{const_name Trueprop}, _) $
haftmann@38864:                              (Const (@{const_name HOL.eq}, _) $
hoelzl@31811:                               Free (name, _) $ _)) = name
hoelzl@31811:         | variable_of_bound t = raise TERM ("variable_of_bound", [t])
hoelzl@31811: 
hoelzl@31811:       val variable_bounds
hoelzl@31811:         = map (` (variable_of_bound o prop_of)) prems
hoelzl@31811: 
hoelzl@31811:       fun add_deps (name, bnds)
hoelzl@32650:         = Graph.add_deps_acyclic (name,
hoelzl@32650:             remove (op =) name (Term.add_free_names (prop_of bnds) []))
hoelzl@32650: 
hoelzl@31811:       val order = Graph.empty
hoelzl@31811:                   |> fold Graph.new_node variable_bounds
hoelzl@31811:                   |> fold add_deps variable_bounds
hoelzl@32650:                   |> Graph.strong_conn |> map the_single |> rev
hoelzl@31811:                   |> map_filter (AList.lookup (op =) variable_bounds)
hoelzl@31811: 
hoelzl@31811:       fun prepend_prem th tac
hoelzl@31811:         = tac THEN rtac (th RSN (2, @{thm mp})) i
hoelzl@31811:     in
hoelzl@31811:       fold prepend_prem order all_tac
hoelzl@31811:     end
hoelzl@31811: 
haftmann@36985:   fun approximation_conv ctxt ct =
wenzelm@42361:     approximation_oracle (Proof_Context.theory_of ctxt, Thm.term_of ct |> tap (tracing o Syntax.string_of_term ctxt));
haftmann@36985: 
haftmann@36985:   fun approximate ctxt t =
wenzelm@42361:     approximation_oracle (Proof_Context.theory_of ctxt, t)
haftmann@36985:     |> Thm.prop_of |> Logic.dest_equals |> snd;
haftmann@36985: 
hoelzl@31811:   (* Should be in HOL.thy ? *)
haftmann@36985:   fun gen_eval_tac conv ctxt = CONVERSION
haftmann@36985:     (Object_Logic.judgment_conv (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt))
haftmann@36985:     THEN' rtac TrueI
hoelzl@29805: 
wenzelm@39556:   val form_equations = @{thms interpret_form_equations};
hoelzl@31811: 
hoelzl@31863:   fun rewrite_interpret_form_tac ctxt prec splitting taylor i st = let
haftmann@46545:       fun lookup_splitting (Free (name, _))
hoelzl@31863:         = case AList.lookup (op =) splitting name
hoelzl@31863:           of SOME s => HOLogic.mk_number @{typ nat} s
hoelzl@31863:            | NONE => @{term "0 :: nat"}
hoelzl@31811:       val vs = nth (prems_of st) (i - 1)
hoelzl@31811:                |> Logic.strip_imp_concl
hoelzl@31811:                |> HOLogic.dest_Trueprop
hoelzl@31811:                |> Term.strip_comb |> snd |> List.last
hoelzl@31811:                |> HOLogic.dest_list
hoelzl@31811:       val p = prec
hoelzl@31811:               |> HOLogic.mk_number @{typ nat}
wenzelm@42361:               |> Thm.cterm_of (Proof_Context.theory_of ctxt)
hoelzl@31863:     in case taylor
hoelzl@31863:     of NONE => let
hoelzl@31863:          val n = vs |> length
hoelzl@31863:                  |> HOLogic.mk_number @{typ nat}
wenzelm@42361:                  |> Thm.cterm_of (Proof_Context.theory_of ctxt)
hoelzl@31863:          val s = vs
hoelzl@31863:                  |> map lookup_splitting
hoelzl@31863:                  |> HOLogic.mk_list @{typ nat}
wenzelm@42361:                  |> Thm.cterm_of (Proof_Context.theory_of ctxt)
hoelzl@31863:        in
hoelzl@31863:          (rtac (Thm.instantiate ([], [(@{cpat "?n::nat"}, n),
hoelzl@31863:                                      (@{cpat "?prec::nat"}, p),
hoelzl@31863:                                      (@{cpat "?ss::nat list"}, s)])
hoelzl@31863:               @{thm "approx_form"}) i
wenzelm@52090:           THEN simp_tac (put_simpset (simpset_of @{context}) ctxt) i) st
hoelzl@31863:        end
hoelzl@31863: 
hoelzl@31863:      | SOME t => if length vs <> 1 then raise (TERM ("More than one variable used for taylor series expansion", [prop_of st]))
hoelzl@31863:        else let
hoelzl@31863:          val t = t
hoelzl@31863:               |> HOLogic.mk_number @{typ nat}
wenzelm@42361:               |> Thm.cterm_of (Proof_Context.theory_of ctxt)
hoelzl@31863:          val s = vs |> map lookup_splitting |> hd
wenzelm@42361:               |> Thm.cterm_of (Proof_Context.theory_of ctxt)
hoelzl@31863:        in
hoelzl@31863:          rtac (Thm.instantiate ([], [(@{cpat "?s::nat"}, s),
hoelzl@31863:                                      (@{cpat "?t::nat"}, t),
hoelzl@31863:                                      (@{cpat "?prec::nat"}, p)])
hoelzl@31863:               @{thm "approx_tse_form"}) i st
hoelzl@31863:        end
hoelzl@31811:     end
hoelzl@31811: 
haftmann@46545:   val free = Args.context -- Args.term >> (fn (_, Free (n, _)) => n | (ctxt, t) =>
hoelzl@31811:     error ("Bad free variable: " ^ Syntax.string_of_term ctxt t));
hoelzl@29805: *}
hoelzl@29805: 
hoelzl@31811: lemma intervalE: "a \<le> x \<and> x \<le> b \<Longrightarrow> \<lbrakk> x \<in> { a .. b } \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
hoelzl@31811:   by auto
hoelzl@31811: 
hoelzl@31811: lemma meta_eqE: "x \<equiv> a \<Longrightarrow> \<lbrakk> x = a \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
hoelzl@31811:   by auto
hoelzl@31811: 
wenzelm@30549: method_setup approximation = {*
wenzelm@36960:   Scan.lift Parse.nat
hoelzl@31863:   --
hoelzl@31811:   Scan.optional (Scan.lift (Args.$$$ "splitting" |-- Args.colon)
wenzelm@36960:     |-- Parse.and_list' (free --| Scan.lift (Args.$$$ "=") -- Scan.lift Parse.nat)) []
hoelzl@31863:   --
hoelzl@31863:   Scan.option (Scan.lift (Args.$$$ "taylor" |-- Args.colon)
wenzelm@36960:     |-- (free |-- Scan.lift (Args.$$$ "=") |-- Scan.lift Parse.nat))
hoelzl@31811:   >>
hoelzl@31863:   (fn ((prec, splitting), taylor) => fn ctxt =>
wenzelm@30549:     SIMPLE_METHOD' (fn i =>
hoelzl@31811:       REPEAT (FIRST' [etac @{thm intervalE},
hoelzl@31811:                       etac @{thm meta_eqE},
hoelzl@31811:                       rtac @{thm impI}] i)
wenzelm@52090:       THEN Subgoal.FOCUS (fn {prems, ...} => reorder_bounds_tac prems i) ctxt i
hoelzl@32650:       THEN DETERM (TRY (filter_prems_tac (K false) i))
haftmann@52286:       THEN DETERM (Reification.tac ctxt form_equations NONE i)
hoelzl@31863:       THEN rewrite_interpret_form_tac ctxt prec splitting taylor i
haftmann@36985:       THEN gen_eval_tac (approximation_conv ctxt) ctxt i))
wenzelm@55506: *} "real number approximation"
hoelzl@31811: 
haftmann@56813: ML_file "approximation.ML"
haftmann@56813: 
hoelzl@29805: end




