src/HOL/Decision_Procs/Approximation.thy
author wenzelm
Thu Feb 26 20:56:59 2009 +0100 (2009-02-26)
changeset 30122 1c912a9d8200
parent 29823 0ab754d13ccd
child 30273 ecd6f0ca62ea
permissions -rw-r--r--
standard headers;
eliminated non-ASCII chars, which are fragile in the age of unicode;
     1 (*  Title:      HOL/Reflection/Approximation.thy
     2     Author:     Johannes Hoelzl <hoelzl@in.tum.de> 2008 / 2009
     3 *)
     4 
     5 header {* Prove unequations about real numbers by computation *}
     6 
     7 theory Approximation
     8 imports Complex_Main Float Reflection Dense_Linear_Order Efficient_Nat
     9 begin
    10 
    11 section "Horner Scheme"
    12 
    13 subsection {* Define auxiliary helper @{text horner} function *}
    14 
    15 fun horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where
    16 "horner F G 0 i k x       = 0" |
    17 "horner F G (Suc n) i k x = 1 / real k - x * horner F G n (F i) (G i k) x"
    18 
    19 lemma horner_schema': fixes x :: real  and a :: "nat \<Rightarrow> real"
    20   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)"
    21 proof -
    22   have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)" by auto
    23   show ?thesis unfolding setsum_right_distrib shift_pow real_diff_def setsum_negf[symmetric] setsum_head_upt_Suc[OF zero_less_Suc]
    24     setsum_reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n  *a n * x^n"] by auto
    25 qed
    26 
    27 lemma horner_schema: fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat"
    28   assumes f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
    29   shows "horner F G n ((F^j') s) (f j') x = (\<Sum> j = 0..< n. -1^j * (1 / real (f (j' + j))) * x^j)"
    30 proof (induct n arbitrary: i k j')
    31   case (Suc n)
    32 
    33   show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc]
    34     using horner_schema'[of "\<lambda> j. 1 / real (f (j' + j))"] by auto
    35 qed auto
    36 
    37 lemma horner_bounds':
    38   assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
    39   and lb_0: "\<And> i k x. lb 0 i k x = 0"
    40   and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)"
    41   and ub_0: "\<And> i k x. ub 0 i k x = 0"
    42   and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)"
    43   shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> horner F G n ((F^j') s) (f j') (Ifloat x) \<and> 
    44          horner F G n ((F^j') s) (f j') (Ifloat x) \<le> Ifloat (ub n ((F^j') s) (f j') x)"
    45   (is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'")
    46 proof (induct n arbitrary: j')
    47   case 0 thus ?case unfolding lb_0 ub_0 horner.simps by auto
    48 next
    49   case (Suc n)
    50   have "?lb (Suc n) j' \<le> ?horner (Suc n) j'" unfolding lb_Suc ub_Suc horner.simps Ifloat_sub diff_def
    51   proof (rule add_mono)
    52     show "Ifloat (lapprox_rat prec 1 (int (f j'))) \<le> 1 / real (f j')" using lapprox_rat[of prec 1  "int (f j')"] by auto
    53     from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct2] `0 \<le> Ifloat x`
    54     show "- Ifloat (x * ub n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) x) \<le> - (Ifloat x * horner F G n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) (Ifloat x))"
    55       unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono)
    56   qed
    57   moreover have "?horner (Suc n) j' \<le> ?ub (Suc n) j'" unfolding ub_Suc ub_Suc horner.simps Ifloat_sub diff_def
    58   proof (rule add_mono)
    59     show "1 / real (f j') \<le> Ifloat (rapprox_rat prec 1 (int (f j')))" using rapprox_rat[of 1 "int (f j')" prec] by auto
    60     from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct1] `0 \<le> Ifloat x`
    61     show "- (Ifloat x * horner F G n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) (Ifloat x)) \<le> 
    62           - Ifloat (x * lb n (F ((F ^ j') s)) (G ((F ^ j') s) (f j')) x)"
    63       unfolding Ifloat_mult neg_le_iff_le by (rule mult_left_mono)
    64   qed
    65   ultimately show ?case by blast
    66 qed
    67 
    68 subsection "Theorems for floating point functions implementing the horner scheme"
    69 
    70 text {*
    71 
    72 Here @{term_type "f :: nat \<Rightarrow> nat"} is the sequence defining the Taylor series, the coefficients are
    73 all alternating and reciprocs. We use @{term G} and @{term F} to describe the computation of @{term f}.
    74 
    75 *}
    76 
    77 lemma horner_bounds: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
    78   assumes "0 \<le> Ifloat x" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
    79   and lb_0: "\<And> i k x. lb 0 i k x = 0"
    80   and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) - x * (ub n (F i) (G i k) x)"
    81   and ub_0: "\<And> i k x. ub 0 i k x = 0"
    82   and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) - x * (lb n (F i) (G i k) x)"
    83   shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> (\<Sum>j=0..<n. -1^j * (1 / real (f (j' + j))) * (Ifloat x)^j)" (is "?lb") and 
    84         "(\<Sum>j=0..<n. -1^j * (1 / real (f (j' + j))) * (Ifloat x)^j) \<le> Ifloat (ub n ((F^j') s) (f j') x)" (is "?ub")
    85 proof -
    86   have "?lb  \<and> ?ub" 
    87     using horner_bounds'[where lb=lb, OF `0 \<le> Ifloat x` f_Suc lb_0 lb_Suc ub_0 ub_Suc]
    88     unfolding horner_schema[where f=f, OF f_Suc] .
    89   thus "?lb" and "?ub" by auto
    90 qed
    91 
    92 lemma horner_bounds_nonpos: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
    93   assumes "Ifloat x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F^n) s) (f n)"
    94   and lb_0: "\<And> i k x. lb 0 i k x = 0"
    95   and lb_Suc: "\<And> n i k x. lb (Suc n) i k x = lapprox_rat prec 1 (int k) + x * (ub n (F i) (G i k) x)"
    96   and ub_0: "\<And> i k x. ub 0 i k x = 0"
    97   and ub_Suc: "\<And> n i k x. ub (Suc n) i k x = rapprox_rat prec 1 (int k) + x * (lb n (F i) (G i k) x)"
    98   shows "Ifloat (lb n ((F^j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j)" (is "?lb") and 
    99         "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j) \<le> Ifloat (ub n ((F^j') s) (f j') x)" (is "?ub")
   100 proof -
   101   { fix x y z :: float have "x - y * z = x + - y * z"
   102       by (cases x, cases y, cases z, simp add: plus_float.simps minus_float.simps uminus_float.simps times_float.simps algebra_simps)
   103   } note diff_mult_minus = this
   104 
   105   { fix x :: float have "- (- x) = x" by (cases x, auto simp add: uminus_float.simps) } note minus_minus = this
   106 
   107   have move_minus: "Ifloat (-x) = -1 * Ifloat x" by auto
   108 
   109   have sum_eq: "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (Ifloat x)^j) = 
   110     (\<Sum>j = 0..<n. -1 ^ j * (1 / real (f (j' + j))) * Ifloat (- x) ^ j)"
   111   proof (rule setsum_cong, simp)
   112     fix j assume "j \<in> {0 ..< n}"
   113     show "1 / real (f (j' + j)) * Ifloat x ^ j = -1 ^ j * (1 / real (f (j' + j))) * Ifloat (- x) ^ j"
   114       unfolding move_minus power_mult_distrib real_mult_assoc[symmetric]
   115       unfolding real_mult_commute unfolding real_mult_assoc[of "-1^j", symmetric] power_mult_distrib[symmetric]
   116       by auto
   117   qed
   118 
   119   have "0 \<le> Ifloat (-x)" using assms by auto
   120   from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec
   121     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,
   122     OF this f_Suc lb_0 refl ub_0 refl]
   123   show "?lb" and "?ub" unfolding minus_minus sum_eq
   124     by auto
   125 qed
   126 
   127 subsection {* Selectors for next even or odd number *}
   128 
   129 text {*
   130 
   131 The horner scheme computes alternating series. To get the upper and lower bounds we need to
   132 guarantee to access a even or odd member. To do this we use @{term get_odd} and @{term get_even}.
   133 
   134 *}
   135 
   136 definition get_odd :: "nat \<Rightarrow> nat" where
   137   "get_odd n = (if odd n then n else (Suc n))"
   138 
   139 definition get_even :: "nat \<Rightarrow> nat" where
   140   "get_even n = (if even n then n else (Suc n))"
   141 
   142 lemma get_odd[simp]: "odd (get_odd n)" unfolding get_odd_def by (cases "odd n", auto)
   143 lemma get_even[simp]: "even (get_even n)" unfolding get_even_def by (cases "even n", auto)
   144 lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)"
   145 proof (cases "odd n")
   146   case True hence "0 < n" by (rule odd_pos)
   147   from gr0_implies_Suc[OF this] obtain k where "Suc k = n" by auto 
   148   thus ?thesis unfolding get_odd_def if_P[OF True] using True[unfolded `Suc k = n`[symmetric]] by blast
   149 next
   150   case False hence "odd (Suc n)" by auto
   151   thus ?thesis unfolding get_odd_def if_not_P[OF False] by blast
   152 qed
   153 
   154 lemma get_even_double: "\<exists>i. get_even n = 2 * i" using get_even[unfolded even_mult_two_ex] .
   155 lemma get_odd_double: "\<exists>i. get_odd n = 2 * i + 1" using get_odd[unfolded odd_Suc_mult_two_ex] by auto
   156 
   157 section "Power function"
   158 
   159 definition float_power_bnds :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
   160 "float_power_bnds n l u = (if odd n \<or> 0 < l then (l ^ n, u ^ n)
   161                       else if u < 0         then (u ^ n, l ^ n)
   162                                             else (0, (max (-l) u) ^ n))"
   163 
   164 lemma float_power_bnds: assumes "(l1, u1) = float_power_bnds n l u" and "x \<in> {Ifloat l .. Ifloat u}"
   165   shows "x^n \<in> {Ifloat l1..Ifloat u1}"
   166 proof (cases "even n")
   167   case True 
   168   show ?thesis
   169   proof (cases "0 < l")
   170     case True hence "odd n \<or> 0 < l" and "0 \<le> Ifloat l" unfolding less_float_def by auto
   171     have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \<or> 0 < l`] by auto
   172     have "Ifloat l^n \<le> x^n" and "x^n \<le> Ifloat u^n " using `0 \<le> Ifloat l` and assms unfolding atLeastAtMost_iff using power_mono[of "Ifloat l" x] power_mono[of x "Ifloat u"] by auto
   173     thus ?thesis using assms `0 < l` unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto
   174   next
   175     case False hence P: "\<not> (odd n \<or> 0 < l)" using `even n` by auto
   176     show ?thesis
   177     proof (cases "u < 0")
   178       case True hence "0 \<le> - Ifloat u" and "- Ifloat u \<le> - x" and "0 \<le> - x" and "-x \<le> - Ifloat l" using assms unfolding less_float_def by auto
   179       hence "Ifloat u^n \<le> x^n" and "x^n \<le> Ifloat l^n" using power_mono[of  "-x" "-Ifloat l" n] power_mono[of "-Ifloat u" "-x" n] 
   180 	unfolding power_minus_even[OF `even n`] by auto
   181       moreover have u1: "u1 = l ^ n" and l1: "l1 = u ^ n" using assms unfolding float_power_bnds_def if_not_P[OF P] if_P[OF True] by auto
   182       ultimately show ?thesis using float_power by auto
   183     next
   184       case False 
   185       have "\<bar>x\<bar> \<le> Ifloat (max (-l) u)"
   186       proof (cases "-l \<le> u")
   187 	case True thus ?thesis unfolding max_def if_P[OF True] using assms unfolding le_float_def by auto
   188       next
   189 	case False thus ?thesis unfolding max_def if_not_P[OF False] using assms unfolding le_float_def by auto
   190       qed
   191       hence x_abs: "\<bar>x\<bar> \<le> \<bar>Ifloat (max (-l) u)\<bar>" by auto
   192       have u1: "u1 = (max (-l) u) ^ n" and l1: "l1 = 0" using assms unfolding float_power_bnds_def if_not_P[OF P] if_not_P[OF False] by auto
   193       show ?thesis unfolding atLeastAtMost_iff l1 u1 float_power using zero_le_even_power[OF `even n`] power_mono_even[OF `even n` x_abs] by auto
   194     qed
   195   qed
   196 next
   197   case False hence "odd n \<or> 0 < l" by auto
   198   have u1: "u1 = u ^ n" and l1: "l1 = l ^ n" using assms unfolding float_power_bnds_def if_P[OF `odd n \<or> 0 < l`] by auto
   199   have "Ifloat l^n \<le> x^n" and "x^n \<le> Ifloat u^n " using assms unfolding atLeastAtMost_iff using power_mono_odd[OF False] by auto
   200   thus ?thesis unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto
   201 qed
   202 
   203 lemma bnds_power: "\<forall> x l u. (l1, u1) = float_power_bnds n l u \<and> x \<in> {Ifloat l .. Ifloat u} \<longrightarrow> Ifloat l1 \<le> x^n \<and> x^n \<le> Ifloat u1"
   204   using float_power_bnds by auto
   205 
   206 section "Square root"
   207 
   208 text {*
   209 
   210 The square root computation is implemented as newton iteration. As first first step we use the
   211 nearest power of two greater than the square root.
   212 
   213 *}
   214 
   215 fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
   216 "sqrt_iteration prec 0 (Float m e) = Float 1 ((e + bitlen m) div 2 + 1)" |
   217 "sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x 
   218                                   in Float 1 -1 * (y + float_divr prec x y))"
   219 
   220 definition ub_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float option" where 
   221 "ub_sqrt prec x = (if 0 < x then Some (sqrt_iteration prec prec x) else if x < 0 then None else Some 0)"
   222 
   223 definition lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float option" where
   224 "lb_sqrt prec x = (if 0 < x then Some (float_divl prec x (sqrt_iteration prec prec x)) else if x < 0 then None else Some 0)"
   225 
   226 lemma sqrt_ub_pos_pos_1:
   227   assumes "sqrt x < b" and "0 < b" and "0 < x"
   228   shows "sqrt x < (b + x / b)/2"
   229 proof -
   230   from assms have "0 < (b - sqrt x) ^ 2 " by simp
   231   also have "\<dots> = b ^ 2 - 2 * b * sqrt x + (sqrt x) ^ 2" by algebra
   232   also have "\<dots> = b ^ 2 - 2 * b * sqrt x + x" using assms by (simp add: real_sqrt_pow2)
   233   finally have "0 < b ^ 2 - 2 * b * sqrt x + x" by assumption
   234   hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms
   235     by (simp add: field_simps power2_eq_square)
   236   thus ?thesis by (simp add: field_simps)
   237 qed
   238 
   239 lemma sqrt_iteration_bound: assumes "0 < Ifloat x"
   240   shows "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec n x)"
   241 proof (induct n)
   242   case 0
   243   show ?case
   244   proof (cases x)
   245     case (Float m e)
   246     hence "0 < m" using float_pos_m_pos[unfolded less_float_def] assms by auto
   247     hence "0 < sqrt (real m)" by auto
   248 
   249     have int_nat_bl: "int (nat (bitlen m)) = bitlen m" using bitlen_ge0 by auto
   250 
   251     have "Ifloat x = (real m / 2^nat (bitlen m)) * pow2 (e + int (nat (bitlen m)))"
   252       unfolding pow2_add pow2_int Float Ifloat.simps by auto
   253     also have "\<dots> < 1 * pow2 (e + int (nat (bitlen m)))"
   254     proof (rule mult_strict_right_mono, auto)
   255       show "real m < 2^nat (bitlen m)" using bitlen_bounds[OF `0 < m`, THEN conjunct2] 
   256 	unfolding real_of_int_less_iff[of m, symmetric] by auto
   257     qed
   258     finally have "sqrt (Ifloat x) < sqrt (pow2 (e + bitlen m))" unfolding int_nat_bl by auto
   259     also have "\<dots> \<le> pow2 ((e + bitlen m) div 2 + 1)"
   260     proof -
   261       let ?E = "e + bitlen m"
   262       have E_mod_pow: "pow2 (?E mod 2) < 4"
   263       proof (cases "?E mod 2 = 1")
   264 	case True thus ?thesis by auto
   265       next
   266 	case False 
   267 	have "0 \<le> ?E mod 2" by auto 
   268 	have "?E mod 2 < 2" by auto
   269 	from this[THEN zless_imp_add1_zle]
   270 	have "?E mod 2 \<le> 0" using False by auto
   271 	from xt1(5)[OF `0 \<le> ?E mod 2` this]
   272 	show ?thesis by auto
   273       qed
   274       hence "sqrt (pow2 (?E mod 2)) < sqrt (2 * 2)" by auto
   275       hence E_mod_pow: "sqrt (pow2 (?E mod 2)) < 2" unfolding real_sqrt_abs2 by auto
   276 
   277       have E_eq: "pow2 ?E = pow2 (?E div 2 + ?E div 2 + ?E mod 2)" by auto
   278       have "sqrt (pow2 ?E) = sqrt (pow2 (?E div 2) * pow2 (?E div 2) * pow2 (?E mod 2))"
   279 	unfolding E_eq unfolding pow2_add ..
   280       also have "\<dots> = pow2 (?E div 2) * sqrt (pow2 (?E mod 2))"
   281 	unfolding real_sqrt_mult[of _ "pow2 (?E mod 2)"] real_sqrt_abs2 by auto
   282       also have "\<dots> < pow2 (?E div 2) * 2" 
   283 	by (rule mult_strict_left_mono, auto intro: E_mod_pow)
   284       also have "\<dots> = pow2 (?E div 2 + 1)" unfolding zadd_commute[of _ 1] pow2_add1 by auto
   285       finally show ?thesis by auto
   286     qed
   287     finally show ?thesis 
   288       unfolding Float sqrt_iteration.simps Ifloat.simps by auto
   289   qed
   290 next
   291   case (Suc n)
   292   let ?b = "sqrt_iteration prec n x"
   293   have "0 < sqrt (Ifloat x)" using `0 < Ifloat x` by auto
   294   also have "\<dots> < Ifloat ?b" using Suc .
   295   finally have "sqrt (Ifloat x) < (Ifloat ?b + Ifloat x / Ifloat ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ `0 < Ifloat x`] by auto
   296   also have "\<dots> \<le> (Ifloat ?b + Ifloat (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr)
   297   also have "\<dots> = Ifloat (Float 1 -1) * (Ifloat ?b + Ifloat (float_divr prec x ?b))" by auto
   298   finally show ?case unfolding sqrt_iteration.simps Let_def Ifloat_mult Ifloat_add right_distrib .
   299 qed
   300 
   301 lemma sqrt_iteration_lower_bound: assumes "0 < Ifloat x"
   302   shows "0 < Ifloat (sqrt_iteration prec n x)" (is "0 < ?sqrt")
   303 proof -
   304   have "0 < sqrt (Ifloat x)" using assms by auto
   305   also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
   306   finally show ?thesis .
   307 qed
   308 
   309 lemma lb_sqrt_lower_bound: assumes "0 \<le> Ifloat x"
   310   shows "0 \<le> Ifloat (the (lb_sqrt prec x))"
   311 proof (cases "0 < x")
   312   case True hence "0 < Ifloat x" and "0 \<le> x" using `0 \<le> Ifloat x` unfolding less_float_def le_float_def by auto
   313   hence "0 < sqrt_iteration prec prec x" unfolding less_float_def using sqrt_iteration_lower_bound by auto 
   314   hence "0 \<le> Ifloat (float_divl prec x (sqrt_iteration prec prec x))" using float_divl_lower_bound[OF `0 \<le> x`] unfolding le_float_def by auto
   315   thus ?thesis unfolding lb_sqrt_def using True by auto
   316 next
   317   case False with `0 \<le> Ifloat x` have "Ifloat x = 0" unfolding less_float_def by auto
   318   thus ?thesis unfolding lb_sqrt_def less_float_def by auto
   319 qed
   320 
   321 lemma lb_sqrt_upper_bound: assumes "0 \<le> Ifloat x"
   322   shows "Ifloat (the (lb_sqrt prec x)) \<le> sqrt (Ifloat x)"
   323 proof (cases "0 < x")
   324   case True hence "0 < Ifloat x" and "0 \<le> Ifloat x" unfolding less_float_def by auto
   325   hence sqrt_gt0: "0 < sqrt (Ifloat x)" by auto
   326   hence sqrt_ub: "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
   327   
   328   have "Ifloat (float_divl prec x (sqrt_iteration prec prec x)) \<le> Ifloat x / Ifloat (sqrt_iteration prec prec x)" by (rule float_divl)
   329   also have "\<dots> < Ifloat x / sqrt (Ifloat x)" 
   330     by (rule divide_strict_left_mono[OF sqrt_ub `0 < Ifloat x` mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
   331   also have "\<dots> = sqrt (Ifloat x)" unfolding inverse_eq_iff_eq[of _ "sqrt (Ifloat x)", symmetric] sqrt_divide_self_eq[OF `0 \<le> Ifloat x`, symmetric] by auto
   332   finally show ?thesis unfolding lb_sqrt_def if_P[OF `0 < x`] by auto
   333 next
   334   case False with `0 \<le> Ifloat x`
   335   have "\<not> x < 0" unfolding less_float_def le_float_def by auto
   336   show ?thesis unfolding lb_sqrt_def if_not_P[OF False] if_not_P[OF `\<not> x < 0`] using assms by auto
   337 qed
   338 
   339 lemma lb_sqrt: assumes "Some y = lb_sqrt prec x"
   340   shows "Ifloat y \<le> sqrt (Ifloat x)" and "0 \<le> Ifloat x"
   341 proof -
   342   show "0 \<le> Ifloat x"
   343   proof (rule ccontr)
   344     assume "\<not> 0 \<le> Ifloat x"
   345     hence "lb_sqrt prec x = None" unfolding lb_sqrt_def less_float_def by auto
   346     thus False using assms by auto
   347   qed
   348   from lb_sqrt_upper_bound[OF this, of prec]
   349   show "Ifloat y \<le> sqrt (Ifloat x)" unfolding assms[symmetric] by auto
   350 qed
   351 
   352 lemma ub_sqrt_lower_bound: assumes "0 \<le> Ifloat x"
   353   shows "sqrt (Ifloat x) \<le> Ifloat (the (ub_sqrt prec x))"
   354 proof (cases "0 < x")
   355   case True hence "0 < Ifloat x" unfolding less_float_def by auto
   356   hence "0 < sqrt (Ifloat x)" by auto
   357   hence "sqrt (Ifloat x) < Ifloat (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
   358   thus ?thesis unfolding ub_sqrt_def if_P[OF `0 < x`] by auto
   359 next
   360   case False with `0 \<le> Ifloat x`
   361   have "Ifloat x = 0" unfolding less_float_def le_float_def by auto
   362   thus ?thesis unfolding ub_sqrt_def less_float_def le_float_def by auto
   363 qed
   364 
   365 lemma ub_sqrt: assumes "Some y = ub_sqrt prec x"
   366   shows "sqrt (Ifloat x) \<le> Ifloat y" and "0 \<le> Ifloat x"
   367 proof -
   368   show "0 \<le> Ifloat x"
   369   proof (rule ccontr)
   370     assume "\<not> 0 \<le> Ifloat x"
   371     hence "ub_sqrt prec x = None" unfolding ub_sqrt_def less_float_def by auto
   372     thus False using assms by auto
   373   qed
   374   from ub_sqrt_lower_bound[OF this, of prec]
   375   show "sqrt (Ifloat x) \<le> Ifloat y" unfolding assms[symmetric] by auto
   376 qed
   377 
   378 lemma bnds_sqrt: "\<forall> x lx ux. (Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> sqrt x \<and> sqrt x \<le> Ifloat u"
   379 proof (rule allI, rule allI, rule allI, rule impI)
   380   fix x lx ux
   381   assume "(Some l, Some u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
   382   hence l: "Some l = lb_sqrt prec lx " and u: "Some u = ub_sqrt prec ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
   383   
   384   have "Ifloat lx \<le> x" and "x \<le> Ifloat ux" using x by auto
   385 
   386   from lb_sqrt(1)[OF l] real_sqrt_le_mono[OF `Ifloat lx \<le> x`]
   387   have "Ifloat l \<le> sqrt x" by (rule order_trans)
   388   moreover
   389   from real_sqrt_le_mono[OF `x \<le> Ifloat ux`] ub_sqrt(1)[OF u]
   390   have "sqrt x \<le> Ifloat u" by (rule order_trans)
   391   ultimately show "Ifloat l \<le> sqrt x \<and> sqrt x \<le> Ifloat u" ..
   392 qed
   393 
   394 section "Arcus tangens and \<pi>"
   395 
   396 subsection "Compute arcus tangens series"
   397 
   398 text {*
   399 
   400 As first step we implement the computation of the arcus tangens series. This is only valid in the range
   401 @{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens.
   402 
   403 *}
   404 
   405 fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
   406 and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
   407   "ub_arctan_horner prec 0 k x = 0"
   408 | "ub_arctan_horner prec (Suc n) k x = 
   409     (rapprox_rat prec 1 (int k)) - x * (lb_arctan_horner prec n (k + 2) x)"
   410 | "lb_arctan_horner prec 0 k x = 0"
   411 | "lb_arctan_horner prec (Suc n) k x = 
   412     (lapprox_rat prec 1 (int k)) - x * (ub_arctan_horner prec n (k + 2) x)"
   413 
   414 lemma arctan_0_1_bounds': assumes "0 \<le> Ifloat x" "Ifloat x \<le> 1" and "even n"
   415   shows "arctan (Ifloat x) \<in> {Ifloat (x * lb_arctan_horner prec n 1 (x * x)) .. Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x * x))}"
   416 proof -
   417   let "?c i" = "-1^i * (1 / real (i * 2 + 1) * Ifloat x ^ (i * 2 + 1))"
   418   let "?S n" = "\<Sum> i=0..<n. ?c i"
   419 
   420   have "0 \<le> Ifloat (x * x)" by auto
   421   from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
   422   
   423   have "arctan (Ifloat x) \<in> { ?S n .. ?S (Suc n) }"
   424   proof (cases "Ifloat x = 0")
   425     case False
   426     hence "0 < Ifloat x" using `0 \<le> Ifloat x` by auto
   427     hence prem: "0 < 1 / real (0 * 2 + (1::nat)) * Ifloat x ^ (0 * 2 + 1)" by auto 
   428 
   429     have "\<bar> Ifloat x \<bar> \<le> 1"  using `0 \<le> Ifloat x` `Ifloat x \<le> 1` by auto
   430     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`]
   431     show ?thesis unfolding arctan_series[OF `\<bar> Ifloat x \<bar> \<le> 1`] Suc_plus1  .
   432   qed auto
   433   note arctan_bounds = this[unfolded atLeastAtMost_iff]
   434 
   435   have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
   436 
   437   note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0 
   438     and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
   439     and ub="\<lambda>n i k x. ub_arctan_horner prec n k x", 
   440     OF `0 \<le> Ifloat (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps]
   441 
   442   { have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n"
   443       using bounds(1) `0 \<le> Ifloat x`
   444       unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
   445       unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"]
   446       by (auto intro!: mult_left_mono)
   447     also have "\<dots> \<le> arctan (Ifloat x)" using arctan_bounds ..
   448     finally have "Ifloat (x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan (Ifloat x)" . }
   449   moreover
   450   { have "arctan (Ifloat x) \<le> ?S (Suc n)" using arctan_bounds ..
   451     also have "\<dots> \<le> Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))"
   452       using bounds(2)[of "Suc n"] `0 \<le> Ifloat x`
   453       unfolding Ifloat_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
   454       unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "Ifloat x"]
   455       by (auto intro!: mult_left_mono)
   456     finally have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . }
   457   ultimately show ?thesis by auto
   458 qed
   459 
   460 lemma arctan_0_1_bounds: assumes "0 \<le> Ifloat x" "Ifloat x \<le> 1"
   461   shows "arctan (Ifloat x) \<in> {Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) .. Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))}"
   462 proof (cases "even n")
   463   case True
   464   obtain n' where "Suc n' = get_odd n" and "odd (Suc n')" using get_odd_ex by auto
   465   hence "even n'" unfolding even_nat_Suc by auto
   466   have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
   467     unfolding `Suc n' = get_odd n`[symmetric] using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even n'`] by auto
   468   moreover
   469   have "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (Ifloat x)"
   470     unfolding get_even_def if_P[OF True] using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even n`] by auto
   471   ultimately show ?thesis by auto
   472 next
   473   case False hence "0 < n" by (rule odd_pos)
   474   from gr0_implies_Suc[OF this] obtain n' where "n = Suc n'" ..
   475   from False[unfolded this even_nat_Suc]
   476   have "even n'" and "even (Suc (Suc n'))" by auto
   477   have "get_odd n = Suc n'" unfolding get_odd_def if_P[OF False] using `n = Suc n'` .
   478 
   479   have "arctan (Ifloat x) \<le> Ifloat (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
   480     unfolding `get_odd n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even n'`] by auto
   481   moreover
   482   have "Ifloat (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (Ifloat x)"
   483     unfolding get_even_def if_not_P[OF False] unfolding `n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1` `even (Suc (Suc n'))`] by auto
   484   ultimately show ?thesis by auto
   485 qed
   486 
   487 subsection "Compute \<pi>"
   488 
   489 definition ub_pi :: "nat \<Rightarrow> float" where
   490   "ub_pi prec = (let A = rapprox_rat prec 1 5 ; 
   491                      B = lapprox_rat prec 1 239
   492                  in ((Float 1 2) * ((Float 1 2) * A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (A * A)) - 
   493                                                   B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 (B * B)))))"
   494 
   495 definition lb_pi :: "nat \<Rightarrow> float" where
   496   "lb_pi prec = (let A = lapprox_rat prec 1 5 ; 
   497                      B = rapprox_rat prec 1 239
   498                  in ((Float 1 2) * ((Float 1 2) * A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (A * A)) - 
   499                                                   B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (B * B)))))"
   500 
   501 lemma pi_boundaries: "pi \<in> {Ifloat (lb_pi n) .. Ifloat (ub_pi n)}"
   502 proof -
   503   have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))" unfolding machin[symmetric] by auto
   504 
   505   { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto
   506     let ?k = "rapprox_rat prec 1 k"
   507     have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
   508       
   509     have "0 \<le> Ifloat ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \<le> k`)
   510     have "Ifloat ?k \<le> 1" unfolding rapprox_rat.simps(2)[OF zero_le_one `0 < k`]
   511       by (rule rapprox_posrat_le1, auto simp add: `0 < k` `1 \<le> k`)
   512 
   513     have "1 / real k \<le> Ifloat ?k" using rapprox_rat[where x=1 and y=k] by auto
   514     hence "arctan (1 / real k) \<le> arctan (Ifloat ?k)" by (rule arctan_monotone')
   515     also have "\<dots> \<le> Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))"
   516       using arctan_0_1_bounds[OF `0 \<le> Ifloat ?k` `Ifloat ?k \<le> 1`] by auto
   517     finally have "arctan (1 / (real k)) \<le> Ifloat (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" .
   518   } note ub_arctan = this
   519 
   520   { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
   521     let ?k = "lapprox_rat prec 1 k"
   522     have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
   523     have "1 / real k \<le> 1" using `1 < k` by auto
   524 
   525     have "\<And>n. 0 \<le> Ifloat ?k" using lapprox_rat_bottom[where x=1 and y=k, OF zero_le_one `0 < k`] by (auto simp add: `1 div k = 0`)
   526     have "\<And>n. Ifloat ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / real k \<le> 1`)
   527 
   528     have "Ifloat ?k \<le> 1 / real k" using lapprox_rat[where x=1 and y=k] by auto
   529 
   530     have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (Ifloat ?k)"
   531       using arctan_0_1_bounds[OF `0 \<le> Ifloat ?k` `Ifloat ?k \<le> 1`] by auto
   532     also have "\<dots> \<le> arctan (1 / real k)" using `Ifloat ?k \<le> 1 / real k` by (rule arctan_monotone')
   533     finally have "Ifloat (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (1 / (real k))" .
   534   } note lb_arctan = this
   535 
   536   have "pi \<le> Ifloat (ub_pi n)"
   537     unfolding ub_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub unfolding Float_num
   538     using lb_arctan[of 239] ub_arctan[of 5]
   539     by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
   540   moreover
   541   have "Ifloat (lb_pi n) \<le> pi"
   542     unfolding lb_pi_def machin_pi Let_def Ifloat_mult Ifloat_sub Float_num
   543     using lb_arctan[of 5] ub_arctan[of 239]
   544     by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
   545   ultimately show ?thesis by auto
   546 qed
   547 
   548 subsection "Compute arcus tangens in the entire domain"
   549 
   550 function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where 
   551   "lb_arctan prec x = (let ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) ;
   552                            lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)
   553     in (if x < 0          then - ub_arctan prec (-x) else
   554         if x \<le> Float 1 -1 then lb_horner x else
   555         if x \<le> Float 1 1  then Float 1 1 * lb_horner (float_divl prec x (1 + the (ub_sqrt prec (1 + x * x))))
   556                           else (let inv = float_divr prec 1 x 
   557                                 in if inv > 1 then 0 
   558                                               else lb_pi prec * Float 1 -1 - ub_horner inv)))"
   559 
   560 | "ub_arctan prec x = (let lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) ;
   561                            ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)
   562     in (if x < 0          then - lb_arctan prec (-x) else
   563         if x \<le> Float 1 -1 then ub_horner x else
   564         if x \<le> Float 1 1  then let y = float_divr prec x (1 + the (lb_sqrt prec (1 + x * x)))
   565                                in if y > 1 then ub_pi prec * Float 1 -1 
   566                                            else Float 1 1 * ub_horner y 
   567                           else ub_pi prec * Float 1 -1 - lb_horner (float_divl prec 1 x)))"
   568 by pat_completeness auto
   569 termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 0 then 1 else 0))", auto simp add: less_float_def)
   570 
   571 declare ub_arctan_horner.simps[simp del]
   572 declare lb_arctan_horner.simps[simp del]
   573 
   574 lemma lb_arctan_bound': assumes "0 \<le> Ifloat x"
   575   shows "Ifloat (lb_arctan prec x) \<le> arctan (Ifloat x)"
   576 proof -
   577   have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> Ifloat x` by auto
   578   let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
   579     and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
   580 
   581   show ?thesis
   582   proof (cases "x \<le> Float 1 -1")
   583     case True hence "Ifloat x \<le> 1" unfolding le_float_def Float_num by auto
   584     show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
   585       using arctan_0_1_bounds[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1`] by auto
   586   next
   587     case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto
   588     let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)"
   589     let ?fR = "1 + the (ub_sqrt prec (1 + x * x))"
   590     let ?DIV = "float_divl prec x ?fR"
   591     
   592     have sqr_ge0: "0 \<le> 1 + Ifloat x * Ifloat x" using sum_power2_ge_zero[of 1 "Ifloat x", unfolded numeral_2_eq_2] by auto
   593     hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
   594 
   595     have "sqrt (Ifloat (1 + x * x)) \<le> Ifloat (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0)
   596     hence "?R \<le> Ifloat ?fR" by auto
   597     hence "0 < ?fR" and "0 < Ifloat ?fR" unfolding less_float_def using `0 < ?R` by auto
   598 
   599     have monotone: "Ifloat (float_divl prec x ?fR) \<le> Ifloat x / ?R"
   600     proof -
   601       have "Ifloat ?DIV \<le> Ifloat x / Ifloat ?fR" by (rule float_divl)
   602       also have "\<dots> \<le> Ifloat x / ?R" by (rule divide_left_mono[OF `?R \<le> Ifloat ?fR` `0 \<le> Ifloat x` mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 `?R \<le> Ifloat ?fR`] divisor_gt0]])
   603       finally show ?thesis .
   604     qed
   605 
   606     show ?thesis
   607     proof (cases "x \<le> Float 1 1")
   608       case True
   609       
   610       have "Ifloat x \<le> sqrt (Ifloat (1 + x * x))" using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
   611       also have "\<dots> \<le> Ifloat (the (ub_sqrt prec (1 + x * x)))" by (rule ub_sqrt_lower_bound, auto simp add: sqr_ge0)
   612       finally have "Ifloat x \<le> Ifloat ?fR" by auto
   613       moreover have "Ifloat ?DIV \<le> Ifloat x / Ifloat ?fR" by (rule float_divl)
   614       ultimately have "Ifloat ?DIV \<le> 1" unfolding divide_le_eq_1_pos[OF `0 < Ifloat ?fR`, symmetric] by auto
   615 
   616       have "0 \<le> Ifloat ?DIV" using float_divl_lower_bound[OF `0 \<le> x` `0 < ?fR`] unfolding le_float_def by auto
   617 
   618       have "Ifloat (Float 1 1 * ?lb_horner ?DIV) \<le> 2 * arctan (Ifloat (float_divl prec x ?fR))" unfolding Ifloat_mult[of "Float 1 1"] Float_num
   619 	using arctan_0_1_bounds[OF `0 \<le> Ifloat ?DIV` `Ifloat ?DIV \<le> 1`] by auto
   620       also have "\<dots> \<le> 2 * arctan (Ifloat x / ?R)"
   621 	using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
   622       also have "2 * arctan (Ifloat x / ?R) = arctan (Ifloat x)" using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 realpow_0 real_mult_1 . 
   623       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] .
   624     next
   625       case False
   626       hence "2 < Ifloat x" unfolding le_float_def Float_num by auto
   627       hence "1 \<le> Ifloat x" by auto
   628 
   629       let "?invx" = "float_divr prec 1 x"
   630       have "0 \<le> arctan (Ifloat x)" using arctan_monotone'[OF `0 \<le> Ifloat x`] using arctan_tan[of 0, unfolded tan_zero] by auto
   631 
   632       show ?thesis
   633       proof (cases "1 < ?invx")
   634 	case True
   635 	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] 
   636 	  using `0 \<le> arctan (Ifloat x)` by auto
   637       next
   638 	case False
   639 	hence "Ifloat ?invx \<le> 1" unfolding less_float_def by auto
   640 	have "0 \<le> Ifloat ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> Ifloat x`)
   641 
   642 	have "1 / Ifloat x \<noteq> 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto
   643 	
   644 	have "arctan (1 / Ifloat x) \<le> arctan (Ifloat ?invx)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divr)
   645 	also have "\<dots> \<le> Ifloat (?ub_horner ?invx)" using arctan_0_1_bounds[OF `0 \<le> Ifloat ?invx` `Ifloat ?invx \<le> 1`] by auto
   646 	finally have "pi / 2 - Ifloat (?ub_horner ?invx) \<le> arctan (Ifloat x)" 
   647 	  using `0 \<le> arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \<noteq> 0`] 
   648 	  unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto
   649 	moreover
   650 	have "Ifloat (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding Ifloat_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
   651 	ultimately
   652 	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]
   653 	  by auto
   654       qed
   655     qed
   656   qed
   657 qed
   658 
   659 lemma ub_arctan_bound': assumes "0 \<le> Ifloat x"
   660   shows "arctan (Ifloat x) \<le> Ifloat (ub_arctan prec x)"
   661 proof -
   662   have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> Ifloat x` by auto
   663 
   664   let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
   665     and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
   666 
   667   show ?thesis
   668   proof (cases "x \<le> Float 1 -1")
   669     case True hence "Ifloat x \<le> 1" unfolding le_float_def Float_num by auto
   670     show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
   671       using arctan_0_1_bounds[OF `0 \<le> Ifloat x` `Ifloat x \<le> 1`] by auto
   672   next
   673     case False hence "0 < Ifloat x" unfolding le_float_def Float_num by auto
   674     let ?R = "1 + sqrt (1 + Ifloat x * Ifloat x)"
   675     let ?fR = "1 + the (lb_sqrt prec (1 + x * x))"
   676     let ?DIV = "float_divr prec x ?fR"
   677     
   678     have sqr_ge0: "0 \<le> 1 + Ifloat x * Ifloat x" using sum_power2_ge_zero[of 1 "Ifloat x", unfolded numeral_2_eq_2] by auto
   679     hence "0 \<le> Ifloat (1 + x*x)" by auto
   680     
   681     hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
   682 
   683     have "Ifloat (the (lb_sqrt prec (1 + x * x))) \<le> sqrt (Ifloat (1 + x * x))" by (rule lb_sqrt_upper_bound, auto simp add: sqr_ge0)
   684     hence "Ifloat ?fR \<le> ?R" by auto
   685     have "0 < Ifloat ?fR" unfolding Ifloat_add Ifloat_1 by (rule order_less_le_trans[OF zero_less_one], auto simp add: lb_sqrt_lower_bound[OF `0 \<le> Ifloat (1 + x*x)`])
   686 
   687     have monotone: "Ifloat x / ?R \<le> Ifloat (float_divr prec x ?fR)"
   688     proof -
   689       from divide_left_mono[OF `Ifloat ?fR \<le> ?R` `0 \<le> Ifloat x` mult_pos_pos[OF divisor_gt0 `0 < Ifloat ?fR`]]
   690       have "Ifloat x / ?R \<le> Ifloat x / Ifloat ?fR" .
   691       also have "\<dots> \<le> Ifloat ?DIV" by (rule float_divr)
   692       finally show ?thesis .
   693     qed
   694 
   695     show ?thesis
   696     proof (cases "x \<le> Float 1 1")
   697       case True
   698       show ?thesis
   699       proof (cases "?DIV > 1")
   700 	case True
   701 	have "pi / 2 \<le> Ifloat (ub_pi prec * Float 1 -1)" unfolding Ifloat_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
   702 	from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le]
   703 	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] .
   704       next
   705 	case False
   706 	hence "Ifloat ?DIV \<le> 1" unfolding less_float_def by auto
   707       
   708 	have "0 \<le> Ifloat x / ?R" using `0 \<le> Ifloat x` `0 < ?R` unfolding real_0_le_divide_iff by auto
   709 	hence "0 \<le> Ifloat ?DIV" using monotone by (rule order_trans)
   710 
   711 	have "arctan (Ifloat x) = 2 * arctan (Ifloat x / ?R)" using arctan_half unfolding numeral_2_eq_2 power_Suc2 realpow_0 real_mult_1 .
   712 	also have "\<dots> \<le> 2 * arctan (Ifloat ?DIV)"
   713 	  using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
   714 	also have "\<dots> \<le> Ifloat (Float 1 1 * ?ub_horner ?DIV)" unfolding Ifloat_mult[of "Float 1 1"] Float_num
   715 	  using arctan_0_1_bounds[OF `0 \<le> Ifloat ?DIV` `Ifloat ?DIV \<le> 1`] by auto
   716 	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] .
   717       qed
   718     next
   719       case False
   720       hence "2 < Ifloat x" unfolding le_float_def Float_num by auto
   721       hence "1 \<le> Ifloat x" by auto
   722       hence "0 < Ifloat x" by auto
   723       hence "0 < x" unfolding less_float_def by auto
   724 
   725       let "?invx" = "float_divl prec 1 x"
   726       have "0 \<le> arctan (Ifloat x)" using arctan_monotone'[OF `0 \<le> Ifloat x`] using arctan_tan[of 0, unfolded tan_zero] by auto
   727 
   728       have "Ifloat ?invx \<le> 1" unfolding less_float_def by (rule order_trans[OF float_divl], auto simp add: `1 \<le> Ifloat x` divide_le_eq_1_pos[OF `0 < Ifloat x`])
   729       have "0 \<le> Ifloat ?invx" unfolding Ifloat_0[symmetric] by (rule float_divl_lower_bound[unfolded le_float_def], auto simp add: `0 < x`)
   730 	
   731       have "1 / Ifloat x \<noteq> 0" and "0 < 1 / Ifloat x" using `0 < Ifloat x` by auto
   732       
   733       have "Ifloat (?lb_horner ?invx) \<le> arctan (Ifloat ?invx)" using arctan_0_1_bounds[OF `0 \<le> Ifloat ?invx` `Ifloat ?invx \<le> 1`] by auto
   734       also have "\<dots> \<le> arctan (1 / Ifloat x)" unfolding Ifloat_1[symmetric] by (rule arctan_monotone', rule float_divl)
   735       finally have "arctan (Ifloat x) \<le> pi / 2 - Ifloat (?lb_horner ?invx)"
   736 	using `0 \<le> arctan (Ifloat x)` arctan_inverse[OF `1 / Ifloat x \<noteq> 0`] 
   737 	unfolding real_sgn_pos[OF `0 < 1 / Ifloat x`] le_diff_eq by auto
   738       moreover
   739       have "pi / 2 \<le> Ifloat (ub_pi prec * Float 1 -1)" unfolding Ifloat_mult Float_num times_divide_eq_right mult_1_right using pi_boundaries by auto
   740       ultimately
   741       show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF `\<not> x \<le> Float 1 1`] if_not_P[OF False]
   742 	by auto
   743     qed
   744   qed
   745 qed
   746 
   747 lemma arctan_boundaries:
   748   "arctan (Ifloat x) \<in> {Ifloat (lb_arctan prec x) .. Ifloat (ub_arctan prec x)}"
   749 proof (cases "0 \<le> x")
   750   case True hence "0 \<le> Ifloat x" unfolding le_float_def by auto
   751   show ?thesis using ub_arctan_bound'[OF `0 \<le> Ifloat x`] lb_arctan_bound'[OF `0 \<le> Ifloat x`] unfolding atLeastAtMost_iff by auto
   752 next
   753   let ?mx = "-x"
   754   case False hence "x < 0" and "0 \<le> Ifloat ?mx" unfolding le_float_def less_float_def by auto
   755   hence bounds: "Ifloat (lb_arctan prec ?mx) \<le> arctan (Ifloat ?mx) \<and> arctan (Ifloat ?mx) \<le> Ifloat (ub_arctan prec ?mx)"
   756     using ub_arctan_bound'[OF `0 \<le> Ifloat ?mx`] lb_arctan_bound'[OF `0 \<le> Ifloat ?mx`] by auto
   757   show ?thesis unfolding Ifloat_minus arctan_minus lb_arctan.simps[where x=x] ub_arctan.simps[where x=x] Let_def if_P[OF `x < 0`]
   758     unfolding atLeastAtMost_iff using bounds[unfolded Ifloat_minus arctan_minus] by auto
   759 qed
   760 
   761 lemma bnds_arctan: "\<forall> x lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> arctan x \<and> arctan x \<le> Ifloat u"
   762 proof (rule allI, rule allI, rule allI, rule impI)
   763   fix x lx ux
   764   assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
   765   hence l: "lb_arctan prec lx = l " and u: "ub_arctan prec ux = u" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
   766 
   767   { from arctan_boundaries[of lx prec, unfolded l]
   768     have "Ifloat l \<le> arctan (Ifloat lx)" by (auto simp del: lb_arctan.simps)
   769     also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
   770     finally have "Ifloat l \<le> arctan x" .
   771   } moreover
   772   { have "arctan x \<le> arctan (Ifloat ux)" using x by (auto intro: arctan_monotone')
   773     also have "\<dots> \<le> Ifloat u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
   774     finally have "arctan x \<le> Ifloat u" .
   775   } ultimately show "Ifloat l \<le> arctan x \<and> arctan x \<le> Ifloat u" ..
   776 qed
   777 
   778 section "Sinus and Cosinus"
   779 
   780 subsection "Compute the cosinus and sinus series"
   781 
   782 fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
   783 and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
   784   "ub_sin_cos_aux prec 0 i k x = 0"
   785 | "ub_sin_cos_aux prec (Suc n) i k x = 
   786     (rapprox_rat prec 1 (int k)) - x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
   787 | "lb_sin_cos_aux prec 0 i k x = 0"
   788 | "lb_sin_cos_aux prec (Suc n) i k x = 
   789     (lapprox_rat prec 1 (int k)) - x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
   790 
   791 lemma cos_aux:
   792   shows "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (Ifloat x)^(2 * i))" (is "?lb")
   793   and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (Ifloat x)^(2 * i)) \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
   794 proof -
   795   have "0 \<le> Ifloat (x * x)" unfolding Ifloat_mult by auto
   796   let "?f n" = "fact (2 * n)"
   797 
   798   { fix n 
   799     have F: "\<And>m. ((\<lambda>i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
   800     have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^ n) 1 * (((\<lambda>i. i + 2) ^ n) 1 + 1)"
   801       unfolding F by auto } note f_eq = this
   802     
   803   from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0, 
   804     OF `0 \<le> Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
   805   show "?lb" and "?ub" by (auto simp add: power_mult power2_eq_square[of "Ifloat x"])
   806 qed
   807 
   808 lemma cos_boundaries: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
   809   shows "cos (Ifloat x) \<in> {Ifloat (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. Ifloat (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"
   810 proof (cases "Ifloat x = 0")
   811   case False hence "Ifloat x \<noteq> 0" by auto
   812   hence "0 < x" and "0 < Ifloat x" using `0 \<le> Ifloat x` unfolding less_float_def by auto
   813   have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0
   814     using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto
   815 
   816   { fix x n have "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x^(2 * i))
   817     = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum")
   818   proof -
   819     have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
   820     also have "\<dots> = 
   821       (\<Sum> j = 0 ..< n. -1 ^ ((2 * j) div 2) / (real (fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
   822     also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then -1 ^ (i div 2) / (real (fact i)) * x ^ i else 0)"
   823       unfolding sum_split_even_odd ..
   824     also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then -1 ^ (i div 2) / (real (fact i)) else 0) * x ^ i)"
   825       by (rule setsum_cong2) auto
   826     finally show ?thesis by assumption
   827   qed } note morph_to_if_power = this
   828 
   829 
   830   { fix n :: nat assume "0 < n"
   831     hence "0 < 2 * n" by auto
   832     obtain t where "0 < t" and "t < Ifloat x" and
   833       cos_eq: "cos (Ifloat x) = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * (Ifloat x) ^ i) 
   834       + (cos (t + 1/2 * real (2 * n) * pi) / real (fact (2*n))) * (Ifloat x)^(2*n)" 
   835       (is "_ = ?SUM + ?rest / ?fact * ?pow")
   836       using Maclaurin_cos_expansion2[OF `0 < Ifloat x` `0 < 2 * n`] by auto
   837 
   838     have "cos t * -1^n = cos t * cos (real n * pi) + sin t * sin (real n * pi)" by auto
   839     also have "\<dots> = cos (t + real n * pi)"  using cos_add by auto
   840     also have "\<dots> = ?rest" by auto
   841     finally have "cos t * -1^n = ?rest" .
   842     moreover
   843     have "t \<le> pi / 2" using `t < Ifloat x` and `Ifloat x \<le> pi / 2` by auto
   844     hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
   845     ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
   846 
   847     have "0 < ?fact" by auto
   848     have "0 < ?pow" using `0 < Ifloat x` by auto
   849 
   850     {
   851       assume "even n"
   852       have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
   853 	unfolding morph_to_if_power[symmetric] using cos_aux by auto 
   854       also have "\<dots> \<le> cos (Ifloat x)"
   855       proof -
   856 	from even[OF `even n`] `0 < ?fact` `0 < ?pow`
   857 	have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
   858 	thus ?thesis unfolding cos_eq by auto
   859       qed
   860       finally have "Ifloat (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos (Ifloat x)" .
   861     } note lb = this
   862 
   863     {
   864       assume "odd n"
   865       have "cos (Ifloat x) \<le> ?SUM"
   866       proof -
   867 	from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
   868 	have "0 \<le> (- ?rest) / ?fact * ?pow"
   869 	  by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
   870 	thus ?thesis unfolding cos_eq by auto
   871       qed
   872       also have "\<dots> \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))"
   873 	unfolding morph_to_if_power[symmetric] using cos_aux by auto
   874       finally have "cos (Ifloat x) \<le> Ifloat (ub_sin_cos_aux prec n 1 1 (x * x))" .
   875     } note ub = this and lb
   876   } note ub = this(1) and lb = this(2)
   877 
   878   have "cos (Ifloat x) \<le> Ifloat (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
   879   moreover have "Ifloat (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos (Ifloat x)" 
   880   proof (cases "0 < get_even n")
   881     case True show ?thesis using lb[OF True get_even] .
   882   next
   883     case False
   884     hence "get_even n = 0" by auto
   885     have "- (pi / 2) \<le> Ifloat x" by (rule order_trans[OF _ `0 < Ifloat x`[THEN less_imp_le]], auto)
   886     with `Ifloat x \<le> pi / 2`
   887     show ?thesis unfolding `get_even n = 0` lb_sin_cos_aux.simps Ifloat_minus Ifloat_0 using cos_ge_zero by auto
   888   qed
   889   ultimately show ?thesis by auto
   890 next
   891   case True
   892   show ?thesis
   893   proof (cases "n = 0")
   894     case True 
   895     thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `Ifloat x = 0` lapprox_rat[where x="-1" and y=1] by auto
   896   next
   897     case False with not0_implies_Suc obtain m where "n = Suc m" by blast
   898     thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `Ifloat x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
   899   qed
   900 qed
   901 
   902 lemma sin_aux: assumes "0 \<le> Ifloat x"
   903   shows "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (Ifloat x)^(2 * i + 1))" (is "?lb")
   904   and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (Ifloat x)^(2 * i + 1)) \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
   905 proof -
   906   have "0 \<le> Ifloat (x * x)" unfolding Ifloat_mult by auto
   907   let "?f n" = "fact (2 * n + 1)"
   908 
   909   { fix n 
   910     have F: "\<And>m. ((\<lambda>i. i + 2) ^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
   911     have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^ n) 2 * (((\<lambda>i. i + 2) ^ n) 2 + 1)"
   912       unfolding F by auto } note f_eq = this
   913     
   914   from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
   915     OF `0 \<le> Ifloat (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
   916   show "?lb" and "?ub" using `0 \<le> Ifloat x` unfolding Ifloat_mult
   917     unfolding power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
   918     unfolding real_mult_commute
   919     by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "Ifloat x"])
   920 qed
   921 
   922 lemma sin_boundaries: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
   923   shows "sin (Ifloat x) \<in> {Ifloat (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. Ifloat (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}"
   924 proof (cases "Ifloat x = 0")
   925   case False hence "Ifloat x \<noteq> 0" by auto
   926   hence "0 < x" and "0 < Ifloat x" using `0 \<le> Ifloat x` unfolding less_float_def by auto
   927   have "0 < x * x" using `0 < x` unfolding less_float_def Ifloat_mult Ifloat_0
   928     using mult_pos_pos[where a="Ifloat x" and b="Ifloat x"] by auto
   929 
   930   { 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))
   931     = (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * x ^ i)" (is "?SUM = _")
   932     proof -
   933       have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto
   934       have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" by auto
   935       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)"
   936 	unfolding sum_split_even_odd ..
   937       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)"
   938 	by (rule setsum_cong2) auto
   939       finally show ?thesis by assumption
   940     qed } note setsum_morph = this
   941 
   942   { fix n :: nat assume "0 < n"
   943     hence "0 < 2 * n + 1" by auto
   944     obtain t where "0 < t" and "t < Ifloat x" and
   945       sin_eq: "sin (Ifloat x) = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i) 
   946       + (sin (t + 1/2 * real (2 * n + 1) * pi) / real (fact (2*n + 1))) * (Ifloat x)^(2*n + 1)" 
   947       (is "_ = ?SUM + ?rest / ?fact * ?pow")
   948       using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < Ifloat x`] by auto
   949 
   950     have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add left_distrib right_distrib by auto
   951     moreover
   952     have "t \<le> pi / 2" using `t < Ifloat x` and `Ifloat x \<le> pi / 2` by auto
   953     hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
   954     ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
   955 
   956     have "0 < ?fact" by (rule real_of_nat_fact_gt_zero)
   957     have "0 < ?pow" using `0 < Ifloat x` by (rule zero_less_power)
   958 
   959     {
   960       assume "even n"
   961       have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> 
   962             (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)"
   963 	using sin_aux[OF `0 \<le> Ifloat x`] unfolding setsum_morph[symmetric] by auto
   964       also have "\<dots> \<le> ?SUM" by auto
   965       also have "\<dots> \<le> sin (Ifloat x)"
   966       proof -
   967 	from even[OF `even n`] `0 < ?fact` `0 < ?pow`
   968 	have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
   969 	thus ?thesis unfolding sin_eq by auto
   970       qed
   971       finally have "Ifloat (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin (Ifloat x)" .
   972     } note lb = this
   973 
   974     {
   975       assume "odd n"
   976       have "sin (Ifloat x) \<le> ?SUM"
   977       proof -
   978 	from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
   979 	have "0 \<le> (- ?rest) / ?fact * ?pow"
   980 	  by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
   981 	thus ?thesis unfolding sin_eq by auto
   982       qed
   983       also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (Ifloat x) ^ i)"
   984 	 by auto
   985       also have "\<dots> \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" 
   986 	using sin_aux[OF `0 \<le> Ifloat x`] unfolding setsum_morph[symmetric] by auto
   987       finally have "sin (Ifloat x) \<le> Ifloat (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
   988     } note ub = this and lb
   989   } note ub = this(1) and lb = this(2)
   990 
   991   have "sin (Ifloat x) \<le> Ifloat (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
   992   moreover have "Ifloat (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin (Ifloat x)" 
   993   proof (cases "0 < get_even n")
   994     case True show ?thesis using lb[OF True get_even] .
   995   next
   996     case False
   997     hence "get_even n = 0" by auto
   998     with `Ifloat x \<le> pi / 2` `0 \<le> Ifloat x`
   999     show ?thesis unfolding `get_even n = 0` ub_sin_cos_aux.simps Ifloat_minus Ifloat_0 using sin_ge_zero by auto
  1000   qed
  1001   ultimately show ?thesis by auto
  1002 next
  1003   case True
  1004   show ?thesis
  1005   proof (cases "n = 0")
  1006     case True 
  1007     thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `Ifloat x = 0` lapprox_rat[where x="-1" and y=1] by auto
  1008   next
  1009     case False with not0_implies_Suc obtain m where "n = Suc m" by blast
  1010     thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `Ifloat x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
  1011   qed
  1012 qed
  1013 
  1014 subsection "Compute the cosinus in the entire domain"
  1015 
  1016 definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
  1017 "lb_cos prec x = (let
  1018     horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ;
  1019     half = \<lambda> x. if x < 0 then - 1 else Float 1 1 * x * x - 1
  1020   in if x < Float 1 -1 then horner x
  1021 else if x < 1          then half (horner (x * Float 1 -1))
  1022                        else half (half (horner (x * Float 1 -2))))"
  1023 
  1024 definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
  1025 "ub_cos prec x = (let
  1026     horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ;
  1027     half = \<lambda> x. Float 1 1 * x * x - 1
  1028   in if x < Float 1 -1 then horner x
  1029 else if x < 1          then half (horner (x * Float 1 -1))
  1030                        else half (half (horner (x * Float 1 -2))))"
  1031 
  1032 definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
  1033 "bnds_cos prec lx ux = (let  lpi = lb_pi prec
  1034   in   if lx < -lpi \<or> ux > lpi   then (Float -1 0, Float 1 0)
  1035   else if ux \<le> 0                 then (lb_cos prec (-lx), ub_cos prec (-ux))
  1036   else if 0 \<le> lx                 then (lb_cos prec ux, ub_cos prec lx)
  1037                                  else (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0))"
  1038 
  1039 lemma lb_cos: assumes "0 \<le> Ifloat x" and "Ifloat x \<le> pi" 
  1040   shows "cos (Ifloat x) \<in> {Ifloat (lb_cos prec x) .. Ifloat (ub_cos prec x)}" (is "?cos x \<in> { Ifloat (?lb x) .. Ifloat (?ub x) }")
  1041 proof -
  1042   { fix x :: real
  1043     have "cos x = cos (x / 2 + x / 2)" by auto
  1044     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"
  1045       unfolding cos_add by auto
  1046     also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1" by algebra
  1047     finally have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" .
  1048   } note x_half = this[symmetric]
  1049 
  1050   have "\<not> x < 0" using `0 \<le> Ifloat x` unfolding less_float_def by auto
  1051   let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
  1052   let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
  1053   let "?ub_half x" = "Float 1 1 * x * x - 1"
  1054   let "?lb_half x" = "if x < 0 then - 1 else Float 1 1 * x * x - 1"
  1055 
  1056   show ?thesis
  1057   proof (cases "x < Float 1 -1")
  1058     case True hence "Ifloat x \<le> pi / 2" unfolding less_float_def using pi_ge_two by auto
  1059     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
  1060       using cos_boundaries[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi / 2`] .
  1061   next
  1062     case False
  1063     
  1064     { fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)"
  1065       assume "Ifloat y \<le> cos ?x2" and "-pi \<le> Ifloat x" and "Ifloat x \<le> pi"
  1066       hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto
  1067       hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
  1068       
  1069       have "Ifloat (?lb_half y) \<le> cos (Ifloat x)"
  1070       proof (cases "y < 0")
  1071 	case True show ?thesis using cos_ge_minus_one unfolding if_P[OF True] by auto
  1072       next
  1073 	case False
  1074 	hence "0 \<le> Ifloat y" unfolding less_float_def by auto
  1075 	from mult_mono[OF `Ifloat y \<le> cos ?x2` `Ifloat y \<le> cos ?x2` `0 \<le> cos ?x2` this]
  1076 	have "Ifloat y * Ifloat y \<le> cos ?x2 * cos ?x2" .
  1077 	hence "2 * Ifloat y * Ifloat y \<le> 2 * cos ?x2 * cos ?x2" by auto
  1078 	hence "2 * Ifloat y * Ifloat y - 1 \<le> 2 * cos (Ifloat x / 2) * cos (Ifloat x / 2) - 1" unfolding Float_num Ifloat_mult by auto
  1079 	thus ?thesis unfolding if_not_P[OF False] x_half Float_num Ifloat_mult Ifloat_sub by auto
  1080       qed
  1081     } note lb_half = this
  1082     
  1083     { fix y x :: float let ?x2 = "Ifloat (x * Float 1 -1)"
  1084       assume ub: "cos ?x2 \<le> Ifloat y" and "- pi \<le> Ifloat x" and "Ifloat x \<le> pi"
  1085       hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding Ifloat_mult Float_num by auto
  1086       hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
  1087       
  1088       have "cos (Ifloat x) \<le> Ifloat (?ub_half y)"
  1089       proof -
  1090 	have "0 \<le> Ifloat y" using `0 \<le> cos ?x2` ub by (rule order_trans)
  1091 	from mult_mono[OF ub ub this `0 \<le> cos ?x2`]
  1092 	have "cos ?x2 * cos ?x2 \<le> Ifloat y * Ifloat y" .
  1093 	hence "2 * cos ?x2 * cos ?x2 \<le> 2 * Ifloat y * Ifloat y" by auto
  1094 	hence "2 * cos (Ifloat x / 2) * cos (Ifloat x / 2) - 1 \<le> 2 * Ifloat y * Ifloat y - 1" unfolding Float_num Ifloat_mult by auto
  1095 	thus ?thesis unfolding x_half Ifloat_mult Float_num Ifloat_sub by auto
  1096       qed
  1097     } note ub_half = this
  1098     
  1099     let ?x2 = "x * Float 1 -1"
  1100     let ?x4 = "x * Float 1 -1 * Float 1 -1"
  1101     
  1102     have "-pi \<le> Ifloat x" using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] `0 \<le> Ifloat x` by (rule order_trans)
  1103     
  1104     show ?thesis
  1105     proof (cases "x < 1")
  1106       case True hence "Ifloat x \<le> 1" unfolding less_float_def by auto
  1107       have "0 \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi / 2" using pi_ge_two `0 \<le> Ifloat x` unfolding Ifloat_mult Float_num using assms by auto
  1108       from cos_boundaries[OF this]
  1109       have lb: "Ifloat (?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> Ifloat (?ub_horner ?x2)" by auto
  1110       
  1111       have "Ifloat (?lb x) \<le> ?cos x"
  1112       proof -
  1113 	from lb_half[OF lb `-pi \<le> Ifloat x` `Ifloat x \<le> pi`]
  1114 	show ?thesis unfolding lb_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto
  1115       qed
  1116       moreover have "?cos x \<le> Ifloat (?ub x)"
  1117       proof -
  1118 	from ub_half[OF ub `-pi \<le> Ifloat x` `Ifloat x \<le> pi`]
  1119 	show ?thesis unfolding ub_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto 
  1120       qed
  1121       ultimately show ?thesis by auto
  1122     next
  1123       case False
  1124       have "0 \<le> Ifloat ?x4" and "Ifloat ?x4 \<le> pi / 2" using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` unfolding Ifloat_mult Float_num by auto
  1125       from cos_boundaries[OF this]
  1126       have lb: "Ifloat (?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> Ifloat (?ub_horner ?x4)" by auto
  1127       
  1128       have eq_4: "?x2 * Float 1 -1 = x * Float 1 -2" by (cases x, auto simp add: times_float.simps)
  1129       
  1130       have "Ifloat (?lb x) \<le> ?cos x"
  1131       proof -
  1132 	have "-pi \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi" unfolding Ifloat_mult Float_num using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto
  1133 	from lb_half[OF lb_half[OF lb this] `-pi \<le> Ifloat x` `Ifloat x \<le> pi`, unfolded eq_4]
  1134 	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 .
  1135       qed
  1136       moreover have "?cos x \<le> Ifloat (?ub x)"
  1137       proof -
  1138 	have "-pi \<le> Ifloat ?x2" and "Ifloat ?x2 \<le> pi" unfolding Ifloat_mult Float_num using pi_ge_two `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto
  1139 	from ub_half[OF ub_half[OF ub this] `-pi \<le> Ifloat x` `Ifloat x \<le> pi`, unfolded eq_4]
  1140 	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 .
  1141       qed
  1142       ultimately show ?thesis by auto
  1143     qed
  1144   qed
  1145 qed
  1146 
  1147 lemma lb_cos_minus: assumes "-pi \<le> Ifloat x" and "Ifloat x \<le> 0" 
  1148   shows "cos (Ifloat (-x)) \<in> {Ifloat (lb_cos prec (-x)) .. Ifloat (ub_cos prec (-x))}"
  1149 proof -
  1150   have "0 \<le> Ifloat (-x)" and "Ifloat (-x) \<le> pi" using `-pi \<le> Ifloat x` `Ifloat x \<le> 0` by auto
  1151   from lb_cos[OF this] show ?thesis .
  1152 qed
  1153 
  1154 lemma bnds_cos: "\<forall> x lx ux. (l, u) = bnds_cos prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> cos x \<and> cos x \<le> Ifloat u"
  1155 proof (rule allI, rule allI, rule allI, rule impI)
  1156   fix x lx ux
  1157   assume "(l, u) = bnds_cos prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux}"
  1158   hence bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
  1159 
  1160   let ?lpi = "lb_pi prec"  
  1161   have [intro!]: "Ifloat lx \<le> Ifloat ux" using x by auto
  1162   hence "lx \<le> ux" unfolding le_float_def .
  1163 
  1164   show "Ifloat l \<le> cos x \<and> cos x \<le> Ifloat u"
  1165   proof (cases "lx < -?lpi \<or> ux > ?lpi")
  1166     case True
  1167     show ?thesis using bnds unfolding bnds_cos_def if_P[OF True] Let_def using cos_le_one cos_ge_minus_one by auto
  1168   next
  1169     case False note not_out = this
  1170     hence lpi_lx: "- Ifloat ?lpi \<le> Ifloat lx" and lpi_ux: "Ifloat ux \<le> Ifloat ?lpi" unfolding le_float_def less_float_def by auto
  1171 
  1172     from pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1, THEN le_imp_neg_le] lpi_lx
  1173     have "- pi \<le> Ifloat lx" by (rule order_trans)
  1174     hence "- pi \<le> x" and "- pi \<le> Ifloat ux" and "x \<le> Ifloat ux" using x by auto
  1175     
  1176     from lpi_ux pi_boundaries[unfolded atLeastAtMost_iff, THEN conjunct1]
  1177     have "Ifloat ux \<le> pi" by (rule order_trans)
  1178     hence "x \<le> pi" and "Ifloat lx \<le> pi" and "Ifloat lx \<le> x" using x by auto
  1179 
  1180     note lb_cos_minus_bottom = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct1]
  1181     note lb_cos_minus_top = lb_cos_minus[unfolded atLeastAtMost_iff, THEN conjunct2]
  1182     note lb_cos_bottom = lb_cos[unfolded atLeastAtMost_iff, THEN conjunct1]
  1183     note lb_cos_top = lb_cos[unfolded atLeastAtMost_iff, THEN conjunct2]
  1184 
  1185     show ?thesis
  1186     proof (cases "ux \<le> 0")
  1187       case True hence "Ifloat ux \<le> 0" unfolding le_float_def by auto
  1188       hence "x \<le> 0" and "Ifloat lx \<le> 0" using x by auto
  1189       
  1190       { have "Ifloat (lb_cos prec (-lx)) \<le> cos (Ifloat (-lx))" using lb_cos_minus_bottom[OF `-pi \<le> Ifloat lx` `Ifloat lx \<le> 0`] .
  1191 	also have "\<dots> \<le> cos x" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> 0`] .
  1192 	finally have "Ifloat (lb_cos prec (-lx)) \<le> cos x" . }
  1193       moreover
  1194       { have "cos x \<le> cos (Ifloat (-ux))" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> 0`] .
  1195 	also have "\<dots> \<le> Ifloat (ub_cos prec (-ux))" using lb_cos_minus_top[OF `-pi \<le> Ifloat ux` `Ifloat ux \<le> 0`] .
  1196 	finally have "cos x \<le> Ifloat (ub_cos prec (-ux))" . }
  1197       ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_P[OF True] by auto
  1198     next
  1199       case False note not_ux = this
  1200       
  1201       show ?thesis
  1202       proof (cases "0 \<le> lx")
  1203 	case True hence "0 \<le> Ifloat lx" unfolding le_float_def by auto
  1204 	hence "0 \<le> x" and "0 \<le> Ifloat ux" using x by auto
  1205       
  1206 	{ have "Ifloat (lb_cos prec ux) \<le> cos (Ifloat ux)" using lb_cos_bottom[OF `0 \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
  1207 	  also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
  1208 	  finally have "Ifloat (lb_cos prec ux) \<le> cos x" . }
  1209 	moreover
  1210 	{ have "cos x \<le> cos (Ifloat lx)" using cos_monotone_0_pi'[OF `0 \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> pi`] .
  1211 	  also have "\<dots> \<le> Ifloat (ub_cos prec lx)" using lb_cos_top[OF `0 \<le> Ifloat lx` `Ifloat lx \<le> pi`] .
  1212 	  finally have "cos x \<le> Ifloat (ub_cos prec lx)" . }
  1213 	ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_not_P[OF not_ux] if_P[OF True] by auto
  1214       next
  1215 	case False with not_ux
  1216 	have "Ifloat lx \<le> 0" and "0 \<le> Ifloat ux" unfolding le_float_def by auto
  1217 
  1218 	have "Ifloat (min (lb_cos prec (-lx)) (lb_cos prec ux)) \<le> cos x"
  1219 	proof (cases "x \<le> 0")
  1220 	  case True
  1221 	  have "Ifloat (lb_cos prec (-lx)) \<le> cos (Ifloat (-lx))" using lb_cos_minus_bottom[OF `-pi \<le> Ifloat lx` `Ifloat lx \<le> 0`] .
  1222 	  also have "\<dots> \<le> cos x" unfolding Ifloat_minus cos_minus using cos_monotone_minus_pi_0'[OF `- pi \<le> Ifloat lx` `Ifloat lx \<le> x` `x \<le> 0`] .
  1223 	  finally show ?thesis unfolding Ifloat_min by auto
  1224 	next
  1225 	  case False hence "0 \<le> x" by auto
  1226 	  have "Ifloat (lb_cos prec ux) \<le> cos (Ifloat ux)" using lb_cos_bottom[OF `0 \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
  1227 	  also have "\<dots> \<le> cos x" using cos_monotone_0_pi'[OF `0 \<le> x` `x \<le> Ifloat ux` `Ifloat ux \<le> pi`] .
  1228 	  finally show ?thesis unfolding Ifloat_min by auto
  1229 	qed
  1230 	moreover have "cos x \<le> Ifloat (Float 1 0)" by auto
  1231 	ultimately show ?thesis using bnds unfolding bnds_cos_def Let_def if_not_P[OF not_out] if_not_P[OF not_ux] if_not_P[OF False] by auto
  1232       qed
  1233     qed
  1234   qed
  1235 qed
  1236 
  1237 subsection "Compute the sinus in the entire domain"
  1238 
  1239 function lb_sin :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_sin :: "nat \<Rightarrow> float \<Rightarrow> float" where
  1240 "lb_sin prec x = (let sqr_diff = \<lambda> x. if x > 1 then 0 else 1 - x * x 
  1241   in if x < 0           then - ub_sin prec (- x)
  1242 else if x \<le> Float 1 -1  then x * lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 2 1 (x * x)
  1243                         else the (lb_sqrt prec (sqr_diff (ub_cos prec x))))" |
  1244 
  1245 "ub_sin prec x = (let sqr_diff = \<lambda> x. if x < 0 then 1 else 1 - x * x
  1246   in if x < 0           then - lb_sin prec (- x)
  1247 else if x \<le> Float 1 -1  then x * ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 2 1 (x * x)
  1248                         else the (ub_sqrt prec (sqr_diff (lb_cos prec x))))"
  1249 by pat_completeness auto
  1250 termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 0 then 1 else 0))", auto simp add: less_float_def)
  1251 
  1252 definition bnds_sin :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
  1253 "bnds_sin prec lx ux = (let 
  1254     lpi = lb_pi prec ;
  1255     half_pi = lpi * Float 1 -1
  1256   in if lx \<le> - half_pi \<or> half_pi \<le> ux then (Float -1 0, Float 1 0)
  1257                                        else (lb_sin prec lx, ub_sin prec ux))"
  1258 
  1259 lemma lb_sin: assumes "- (pi / 2) \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
  1260   shows "sin (Ifloat x) \<in> { Ifloat (lb_sin prec x) .. Ifloat (ub_sin prec x) }" (is "?sin x \<in> { ?lb x .. ?ub x}")
  1261 proof -
  1262   { fix x :: float assume "0 \<le> Ifloat x" and "Ifloat x \<le> pi / 2"
  1263     hence "\<not> (x < 0)" and "- (pi / 2) \<le> Ifloat x" unfolding less_float_def using pi_ge_two by auto
  1264 
  1265     have "Ifloat x \<le> pi" using `Ifloat x \<le> pi / 2` using pi_ge_two by auto
  1266 
  1267     have "?sin x \<in> { ?lb x .. ?ub x}"
  1268     proof (cases "x \<le> Float 1 -1")
  1269       case True from sin_boundaries[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi / 2`]
  1270       show ?thesis unfolding lb_sin.simps[of prec x] ub_sin.simps[of prec x] if_not_P[OF `\<not> (x < 0)`] if_P[OF True] Let_def .
  1271     next
  1272       case False
  1273       have "0 \<le> cos (Ifloat x)" using cos_ge_zero[OF _ `Ifloat x \<le> pi /2`] `0 \<le> Ifloat x` pi_ge_two by auto
  1274       have "0 \<le> sin (Ifloat x)" using `0 \<le> Ifloat x` and `Ifloat x \<le> pi / 2` using sin_ge_zero by auto
  1275       
  1276       have "?sin x \<le> ?ub x"
  1277       proof (cases "lb_cos prec x < 0")
  1278 	case True
  1279 	have "?sin x \<le> 1" using sin_le_one .
  1280 	also have "\<dots> \<le> Ifloat (the (ub_sqrt prec 1))" using ub_sqrt_lower_bound[where prec=prec and x=1] unfolding Ifloat_1 by auto
  1281 	finally show ?thesis unfolding ub_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] Let_def .
  1282       next
  1283 	case False hence "0 \<le> Ifloat (lb_cos prec x)" unfolding less_float_def by auto
  1284 	
  1285 	have "sin (Ifloat x) = sqrt (1 - cos (Ifloat x) ^ 2)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \<le> sin (Ifloat x)` by auto
  1286 	also have "\<dots> \<le> sqrt (Ifloat (1 - lb_cos prec x * lb_cos prec x))" 
  1287 	proof (rule real_sqrt_le_mono)
  1288 	  have "Ifloat (lb_cos prec x * lb_cos prec x) \<le> cos (Ifloat x) ^ 2" unfolding numeral_2_eq_2 power_Suc2 realpow_0 Ifloat_mult
  1289 	    using `0 \<le> Ifloat (lb_cos prec x)` lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] `0 \<le> cos (Ifloat x)` by(auto intro!: mult_mono)
  1290 	  thus "1 - cos (Ifloat x) ^ 2 \<le> Ifloat (1 - lb_cos prec x * lb_cos prec x)" unfolding Ifloat_sub Ifloat_1 by auto
  1291 	qed
  1292 	also have "\<dots> \<le> Ifloat (the (ub_sqrt prec (1 - lb_cos prec x * lb_cos prec x)))"
  1293 	proof (rule ub_sqrt_lower_bound)
  1294 	  have "Ifloat (lb_cos prec x) \<le> cos (Ifloat x)" using lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] by auto
  1295 	  from mult_mono[OF order_trans[OF this cos_le_one] order_trans[OF this cos_le_one]]
  1296 	  have "Ifloat (lb_cos prec x) * Ifloat (lb_cos prec x) \<le> 1" using `0 \<le> Ifloat (lb_cos prec x)` by auto
  1297 	  thus "0 \<le> Ifloat (1 - lb_cos prec x * lb_cos prec x)" by auto
  1298 	qed
  1299 	finally show ?thesis unfolding ub_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] Let_def .
  1300       qed
  1301       moreover
  1302       have "?lb x \<le> ?sin x"
  1303       proof (cases "1 < ub_cos prec x")
  1304 	case True
  1305 	show ?thesis unfolding lb_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_P[OF True] Let_def 
  1306 	  by (rule order_trans[OF _ sin_ge_zero[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`]]) 
  1307         (auto simp add: lb_sqrt_upper_bound[where prec=prec and x=0, unfolded Ifloat_0 real_sqrt_zero])
  1308       next
  1309 	case False hence "Ifloat (ub_cos prec x) \<le> 1" unfolding less_float_def by auto
  1310 	have "0 \<le> Ifloat (ub_cos prec x)" using order_trans[OF `0 \<le> cos (Ifloat x)`] lb_cos `0 \<le> Ifloat x` `Ifloat x \<le> pi` by auto
  1311 	
  1312 	have "Ifloat (the (lb_sqrt prec (1 - ub_cos prec x * ub_cos prec x))) \<le> sqrt (Ifloat (1 - ub_cos prec x * ub_cos prec x))"
  1313 	proof (rule lb_sqrt_upper_bound)
  1314 	  from mult_mono[OF `Ifloat (ub_cos prec x) \<le> 1` `Ifloat (ub_cos prec x) \<le> 1`] `0 \<le> Ifloat (ub_cos prec x)`
  1315 	  have "Ifloat (ub_cos prec x) * Ifloat (ub_cos prec x) \<le> 1" by auto
  1316 	  thus "0 \<le> Ifloat (1 - ub_cos prec x * ub_cos prec x)" by auto
  1317 	qed
  1318 	also have "\<dots> \<le> sqrt (1 - cos (Ifloat x) ^ 2)"
  1319 	proof (rule real_sqrt_le_mono)
  1320 	  have "cos (Ifloat x) ^ 2 \<le> Ifloat (ub_cos prec x * ub_cos prec x)" unfolding numeral_2_eq_2 power_Suc2 realpow_0 Ifloat_mult
  1321 	    using `0 \<le> Ifloat (ub_cos prec x)` lb_cos[OF `0 \<le> Ifloat x` `Ifloat x \<le> pi`] `0 \<le> cos (Ifloat x)` by(auto intro!: mult_mono)
  1322 	  thus "Ifloat (1 - ub_cos prec x * ub_cos prec x) \<le> 1 - cos (Ifloat x) ^ 2" unfolding Ifloat_sub Ifloat_1 by auto
  1323 	qed
  1324 	also have "\<dots> = sin (Ifloat x)" unfolding sin_squared_eq[symmetric] real_sqrt_abs using `0 \<le> sin (Ifloat x)` by auto
  1325 	finally show ?thesis unfolding lb_sin.simps if_not_P[OF `\<not> (x < 0)`] if_not_P[OF `\<not> x \<le> Float 1 -1`] if_not_P[OF False] Let_def .
  1326       qed
  1327       ultimately show ?thesis by auto
  1328     qed
  1329   } note for_pos = this
  1330 
  1331   show ?thesis
  1332   proof (cases "x < 0")
  1333     case True 
  1334     hence "0 \<le> Ifloat (-x)" and "Ifloat (- x) \<le> pi / 2" using `-(pi/2) \<le> Ifloat x` unfolding less_float_def by auto
  1335     from for_pos[OF this]
  1336     show ?thesis unfolding Ifloat_minus sin_minus lb_sin.simps[of prec x] ub_sin.simps[of prec x] if_P[OF True] Let_def atLeastAtMost_iff by auto
  1337   next
  1338     case False hence "0 \<le> Ifloat x" unfolding less_float_def by auto
  1339     from for_pos[OF this `Ifloat x \<le> pi /2`]
  1340     show ?thesis .
  1341   qed
  1342 qed
  1343 
  1344 lemma bnds_sin: "\<forall> x lx ux. (l, u) = bnds_sin prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> sin x \<and> sin x \<le> Ifloat u"
  1345 proof (rule allI, rule allI, rule allI, rule impI)
  1346   fix x lx ux
  1347   assume "(l, u) = bnds_sin prec lx ux \<and> x \<in> {Ifloat lx .. Ifloat ux}"
  1348   hence bnds: "(l, u) = bnds_sin prec lx ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
  1349   show "Ifloat l \<le> sin x \<and> sin x \<le> Ifloat u"
  1350   proof (cases "lx \<le> - (lb_pi prec * Float 1 -1) \<or> lb_pi prec * Float 1 -1 \<le> ux")
  1351     case True show ?thesis using bnds unfolding bnds_sin_def if_P[OF True] Let_def by auto
  1352   next
  1353     case False
  1354     hence "- lb_pi prec * Float 1 -1 \<le> lx" and "ux \<le> lb_pi prec * Float 1 -1" unfolding le_float_def by auto
  1355     moreover have "Ifloat (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding Ifloat_mult using pi_boundaries by auto
  1356     ultimately have "- (pi / 2) \<le> Ifloat lx" and "Ifloat ux \<le> pi / 2" and "Ifloat lx \<le> Ifloat ux" unfolding le_float_def using x by auto
  1357     hence "- (pi / 2) \<le> Ifloat ux" and "Ifloat lx \<le> pi / 2" by auto
  1358     
  1359     have "- (pi / 2) \<le> x""x \<le> pi / 2" using `Ifloat ux \<le> pi / 2` `- (pi /2) \<le> Ifloat lx` x by auto
  1360     
  1361     { have "Ifloat (lb_sin prec lx) \<le> sin (Ifloat lx)" using lb_sin[OF `- (pi / 2) \<le> Ifloat lx` `Ifloat lx \<le> pi / 2`] unfolding atLeastAtMost_iff by auto
  1362       also have "\<dots> \<le> sin x" using sin_monotone_2pi' `- (pi / 2) \<le> Ifloat lx` x `x \<le> pi / 2` by auto
  1363       finally have "Ifloat (lb_sin prec lx) \<le> sin x" . }
  1364     moreover
  1365     { have "sin x \<le> sin (Ifloat ux)" using sin_monotone_2pi' `- (pi / 2) \<le> x` x `Ifloat ux \<le> pi / 2` by auto
  1366       also have "\<dots> \<le> Ifloat (ub_sin prec ux)" using lb_sin[OF `- (pi / 2) \<le> Ifloat ux` `Ifloat ux \<le> pi / 2`] unfolding atLeastAtMost_iff by auto
  1367       finally have "sin x \<le> Ifloat (ub_sin prec ux)" . }
  1368     ultimately
  1369     show ?thesis using bnds unfolding bnds_sin_def if_not_P[OF False] Let_def by auto
  1370   qed
  1371 qed
  1372 
  1373 section "Exponential function"
  1374 
  1375 subsection "Compute the series of the exponential function"
  1376 
  1377 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
  1378 "ub_exp_horner prec 0 i k x       = 0" |
  1379 "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" |
  1380 "lb_exp_horner prec 0 i k x       = 0" |
  1381 "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"
  1382 
  1383 lemma bnds_exp_horner: assumes "Ifloat x \<le> 0"
  1384   shows "exp (Ifloat x) \<in> { Ifloat (lb_exp_horner prec (get_even n) 1 1 x) .. Ifloat (ub_exp_horner prec (get_odd n) 1 1 x) }"
  1385 proof -
  1386   { fix n
  1387     have F: "\<And> m. ((\<lambda>i. i + 1) ^ n) m = n + m" by (induct n, auto)
  1388     have "fact (Suc n) = fact n * ((\<lambda>i. i + 1) ^ n) 1" unfolding F by auto } note f_eq = this
  1389     
  1390   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,
  1391     OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]
  1392 
  1393   { have "Ifloat (lb_exp_horner prec (get_even n) 1 1 x) \<le> (\<Sum>j = 0..<get_even n. 1 / real (fact j) * Ifloat x ^ j)"
  1394       using bounds(1) by auto
  1395     also have "\<dots> \<le> exp (Ifloat x)"
  1396     proof -
  1397       obtain t where "\<bar>t\<bar> \<le> \<bar>Ifloat x\<bar>" and "exp (Ifloat x) = (\<Sum>m = 0..<get_even n. (Ifloat x) ^ m / real (fact m)) + exp t / real (fact (get_even n)) * (Ifloat x) ^ (get_even n)"
  1398 	using Maclaurin_exp_le by blast
  1399       moreover have "0 \<le> exp t / real (fact (get_even n)) * (Ifloat x) ^ (get_even n)"
  1400 	by (auto intro!: mult_nonneg_nonneg divide_nonneg_pos simp add: get_even zero_le_even_power exp_gt_zero)
  1401       ultimately show ?thesis
  1402 	using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
  1403     qed
  1404     finally have "Ifloat (lb_exp_horner prec (get_even n) 1 1 x) \<le> exp (Ifloat x)" .
  1405   } moreover
  1406   { 
  1407     have x_less_zero: "Ifloat x ^ get_odd n \<le> 0"
  1408     proof (cases "Ifloat x = 0")
  1409       case True
  1410       have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
  1411       thus ?thesis unfolding True power_0_left by auto
  1412     next
  1413       case False hence "Ifloat x < 0" using `Ifloat x \<le> 0` by auto
  1414       show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq get_odd `Ifloat x < 0`)
  1415     qed
  1416 
  1417     obtain t where "\<bar>t\<bar> \<le> \<bar>Ifloat x\<bar>" and "exp (Ifloat x) = (\<Sum>m = 0..<get_odd n. (Ifloat x) ^ m / real (fact m)) + exp t / real (fact (get_odd n)) * (Ifloat x) ^ (get_odd n)"
  1418       using Maclaurin_exp_le by blast
  1419     moreover have "exp t / real (fact (get_odd n)) * (Ifloat x) ^ (get_odd n) \<le> 0"
  1420       by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero exp_gt_zero)
  1421     ultimately have "exp (Ifloat x) \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * Ifloat x ^ j)"
  1422       using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
  1423     also have "\<dots> \<le> Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)"
  1424       using bounds(2) by auto
  1425     finally have "exp (Ifloat x) \<le> Ifloat (ub_exp_horner prec (get_odd n) 1 1 x)" .
  1426   } ultimately show ?thesis by auto
  1427 qed
  1428 
  1429 subsection "Compute the exponential function on the entire domain"
  1430 
  1431 function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where
  1432 "lb_exp prec x = (if 0 < x then float_divl prec 1 (ub_exp prec (-x))
  1433              else let 
  1434                 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)
  1435              in if x < - 1 then (case floor_fl x of (Float m e) \<Rightarrow> (horner (float_divl prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e))
  1436                            else horner x)" |
  1437 "ub_exp prec x = (if 0 < x    then float_divr prec 1 (lb_exp prec (-x))
  1438              else if x < - 1  then (case floor_fl x of (Float m e) \<Rightarrow> 
  1439                                     (ub_exp_horner prec (get_odd (prec + 2)) 1 1 (float_divr prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e))
  1440                               else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
  1441 by pat_completeness auto
  1442 termination by (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if 0 < x then 1 else 0))", auto simp add: less_float_def)
  1443 
  1444 lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)"
  1445 proof -
  1446   have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
  1447 
  1448   have "1 / 4 = Ifloat (Float 1 -2)" unfolding Float_num by auto
  1449   also have "\<dots> \<le> Ifloat (lb_exp_horner 1 (get_even 4) 1 1 (- 1))"
  1450     unfolding get_even_def eq4 
  1451     by (auto simp add: lapprox_posrat_def rapprox_posrat_def normfloat.simps)
  1452   also have "\<dots> \<le> exp (Ifloat (- 1))" using bnds_exp_horner[where x="- 1"] by auto
  1453   finally show ?thesis unfolding Ifloat_minus Ifloat_1 . 
  1454 qed
  1455 
  1456 lemma lb_exp_pos: assumes "\<not> 0 < x" shows "0 < lb_exp prec x"
  1457 proof -
  1458   let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
  1459   let "?horner x" = "let  y = ?lb_horner x  in if y \<le> 0 then Float 1 -2 else y"
  1460   have pos_horner: "\<And> x. 0 < ?horner x" unfolding Let_def by (cases "?lb_horner x \<le> 0", auto simp add: le_float_def less_float_def)
  1461   moreover { fix x :: float fix num :: nat
  1462     have "0 < Ifloat (?horner x) ^ num" using `0 < ?horner x`[unfolded less_float_def Ifloat_0] by (rule zero_less_power)
  1463     also have "\<dots> = Ifloat ((?horner x) ^ num)" using float_power by auto
  1464     finally have "0 < Ifloat ((?horner x) ^ num)" .
  1465   }
  1466   ultimately show ?thesis
  1467     unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def by (cases "floor_fl x", cases "x < - 1", auto simp add: le_float_def less_float_def normfloat) 
  1468 qed
  1469 
  1470 lemma exp_boundaries': assumes "x \<le> 0"
  1471   shows "exp (Ifloat x) \<in> { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}"
  1472 proof -
  1473   let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
  1474   let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"
  1475 
  1476   have "Ifloat x \<le> 0" and "\<not> x > 0" using `x \<le> 0` unfolding le_float_def less_float_def by auto
  1477   show ?thesis
  1478   proof (cases "x < - 1")
  1479     case False hence "- 1 \<le> Ifloat x" unfolding less_float_def by auto
  1480     show ?thesis
  1481     proof (cases "?lb_exp_horner x \<le> 0")
  1482       from `\<not> x < - 1` have "- 1 \<le> Ifloat x" unfolding less_float_def by auto
  1483       hence "exp (- 1) \<le> exp (Ifloat x)" unfolding exp_le_cancel_iff .
  1484       from order_trans[OF exp_m1_ge_quarter this]
  1485       have "Ifloat (Float 1 -2) \<le> exp (Ifloat x)" unfolding Float_num .
  1486       moreover case True
  1487       ultimately show ?thesis using bnds_exp_horner `Ifloat x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by auto
  1488     next
  1489       case False thus ?thesis using bnds_exp_horner `Ifloat x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by (auto simp add: Let_def)
  1490     qed
  1491   next
  1492     case True
  1493     
  1494     obtain m e where Float_floor: "floor_fl x = Float m e" by (cases "floor_fl x", auto)
  1495     let ?num = "nat (- m) * 2 ^ nat e"
  1496     
  1497     have "Ifloat (floor_fl x) < - 1" using floor_fl `x < - 1` unfolding le_float_def less_float_def Ifloat_minus Ifloat_1 by (rule order_le_less_trans)
  1498     hence "Ifloat (floor_fl x) < 0" unfolding Float_floor Ifloat.simps using zero_less_pow2[of xe] by auto
  1499     hence "m < 0"
  1500       unfolding less_float_def Ifloat_0 Float_floor Ifloat.simps
  1501       unfolding pos_prod_lt[OF zero_less_pow2[of e], unfolded real_mult_commute] by auto
  1502     hence "1 \<le> - m" by auto
  1503     hence "0 < nat (- m)" by auto
  1504     moreover
  1505     have "0 \<le> e" using floor_pos_exp Float_floor[symmetric] by auto
  1506     hence "(0::nat) < 2 ^ nat e" by auto
  1507     ultimately have "0 < ?num"  by auto
  1508     hence "real ?num \<noteq> 0" by auto
  1509     have e_nat: "int (nat e) = e" using `0 \<le> e` by auto
  1510     have num_eq: "real ?num = Ifloat (- floor_fl x)" using `0 < nat (- m)`
  1511       unfolding Float_floor Ifloat_minus Ifloat.simps real_of_nat_mult pow2_int[of "nat e", unfolded e_nat] realpow_real_of_nat[symmetric] by auto
  1512     have "0 < - floor_fl x" using `0 < ?num`[unfolded real_of_nat_less_iff[symmetric]] unfolding less_float_def num_eq[symmetric] Ifloat_0 real_of_nat_zero .
  1513     hence "Ifloat (floor_fl x) < 0" unfolding less_float_def by auto
  1514     
  1515     have "exp (Ifloat x) \<le> Ifloat (ub_exp prec x)"
  1516     proof -
  1517       have div_less_zero: "Ifloat (float_divr prec x (- floor_fl x)) \<le> 0" 
  1518 	using float_divr_nonpos_pos_upper_bound[OF `x \<le> 0` `0 < - floor_fl x`] unfolding le_float_def Ifloat_0 .
  1519       
  1520       have "exp (Ifloat x) = exp (real ?num * (Ifloat x / real ?num))" using `real ?num \<noteq> 0` by auto
  1521       also have "\<dots> = exp (Ifloat x / real ?num) ^ ?num" unfolding exp_real_of_nat_mult ..
  1522       also have "\<dots> \<le> exp (Ifloat (float_divr prec x (- floor_fl x))) ^ ?num" unfolding num_eq
  1523 	by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
  1524       also have "\<dots> \<le> Ifloat ((?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num)" unfolding float_power
  1525 	by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
  1526       finally show ?thesis unfolding ub_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def .
  1527     qed
  1528     moreover 
  1529     have "Ifloat (lb_exp prec x) \<le> exp (Ifloat x)"
  1530     proof -
  1531       let ?divl = "float_divl prec x (- Float m e)"
  1532       let ?horner = "?lb_exp_horner ?divl"
  1533       
  1534       show ?thesis
  1535       proof (cases "?horner \<le> 0")
  1536 	case False hence "0 \<le> Ifloat ?horner" unfolding le_float_def by auto
  1537 	
  1538 	have div_less_zero: "Ifloat (float_divl prec x (- floor_fl x)) \<le> 0"
  1539 	  using `Ifloat (floor_fl x) < 0` `Ifloat x \<le> 0` by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
  1540 	
  1541 	have "Ifloat ((?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num) \<le>  
  1542           exp (Ifloat (float_divl prec x (- floor_fl x))) ^ ?num" unfolding float_power 
  1543 	  using `0 \<le> Ifloat ?horner`[unfolded Float_floor[symmetric]] bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1] by (auto intro!: power_mono)
  1544 	also have "\<dots> \<le> exp (Ifloat x / real ?num) ^ ?num" unfolding num_eq
  1545 	  using float_divl by (auto intro!: power_mono simp del: Ifloat_minus)
  1546 	also have "\<dots> = exp (real ?num * (Ifloat x / real ?num))" unfolding exp_real_of_nat_mult ..
  1547 	also have "\<dots> = exp (Ifloat x)" using `real ?num \<noteq> 0` by auto
  1548 	finally show ?thesis
  1549 	  unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def if_not_P[OF False] by auto
  1550       next
  1551 	case True
  1552 	have "Ifloat (floor_fl x) \<noteq> 0" and "Ifloat (floor_fl x) \<le> 0" using `Ifloat (floor_fl x) < 0` by auto
  1553 	from divide_right_mono_neg[OF floor_fl[of x] `Ifloat (floor_fl x) \<le> 0`, unfolded divide_self[OF `Ifloat (floor_fl x) \<noteq> 0`]]
  1554 	have "- 1 \<le> Ifloat x / Ifloat (- floor_fl x)" unfolding Ifloat_minus by auto
  1555 	from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
  1556 	have "Ifloat (Float 1 -2) \<le> exp (Ifloat x / Ifloat (- floor_fl x))" unfolding Float_num .
  1557 	hence "Ifloat (Float 1 -2) ^ ?num \<le> exp (Ifloat x / Ifloat (- floor_fl x)) ^ ?num"
  1558 	  by (auto intro!: power_mono simp add: Float_num)
  1559 	also have "\<dots> = exp (Ifloat x)" unfolding num_eq exp_real_of_nat_mult[symmetric] using `Ifloat (floor_fl x) \<noteq> 0` by auto
  1560 	finally show ?thesis
  1561 	  unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] if_P[OF `x < - 1`] float.cases Float_floor Let_def if_P[OF True] float_power .
  1562       qed
  1563     qed
  1564     ultimately show ?thesis by auto
  1565   qed
  1566 qed
  1567 
  1568 lemma exp_boundaries: "exp (Ifloat x) \<in> { Ifloat (lb_exp prec x) .. Ifloat (ub_exp prec x)}"
  1569 proof -
  1570   show ?thesis
  1571   proof (cases "0 < x")
  1572     case False hence "x \<le> 0" unfolding less_float_def le_float_def by auto 
  1573     from exp_boundaries'[OF this] show ?thesis .
  1574   next
  1575     case True hence "-x \<le> 0" unfolding less_float_def le_float_def by auto
  1576     
  1577     have "Ifloat (lb_exp prec x) \<le> exp (Ifloat x)"
  1578     proof -
  1579       from exp_boundaries'[OF `-x \<le> 0`]
  1580       have ub_exp: "exp (- Ifloat x) \<le> Ifloat (ub_exp prec (-x))" unfolding atLeastAtMost_iff Ifloat_minus by auto
  1581       
  1582       have "Ifloat (float_divl prec 1 (ub_exp prec (-x))) \<le> Ifloat 1 / Ifloat (ub_exp prec (-x))" using float_divl .
  1583       also have "Ifloat 1 / Ifloat (ub_exp prec (-x)) \<le> exp (Ifloat x)"
  1584 	using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] exp_gt_zero, symmetric]]
  1585 	unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide by auto
  1586       finally show ?thesis unfolding lb_exp.simps if_P[OF True] .
  1587     qed
  1588     moreover
  1589     have "exp (Ifloat x) \<le> Ifloat (ub_exp prec x)"
  1590     proof -
  1591       have "\<not> 0 < -x" using `0 < x` unfolding less_float_def by auto
  1592       
  1593       from exp_boundaries'[OF `-x \<le> 0`]
  1594       have lb_exp: "Ifloat (lb_exp prec (-x)) \<le> exp (- Ifloat x)" unfolding atLeastAtMost_iff Ifloat_minus by auto
  1595       
  1596       have "exp (Ifloat x) \<le> Ifloat 1 / Ifloat (lb_exp prec (-x))"
  1597 	using lb_exp[unfolded inverse_le_iff_le[OF exp_gt_zero lb_exp_pos[OF `\<not> 0 < -x`, unfolded less_float_def Ifloat_0], symmetric]]
  1598 	unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide Ifloat_1 by auto
  1599       also have "\<dots> \<le> Ifloat (float_divr prec 1 (lb_exp prec (-x)))" using float_divr .
  1600       finally show ?thesis unfolding ub_exp.simps if_P[OF True] .
  1601     qed
  1602     ultimately show ?thesis by auto
  1603   qed
  1604 qed
  1605 
  1606 lemma bnds_exp: "\<forall> x lx ux. (l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> exp x \<and> exp x \<le> Ifloat u"
  1607 proof (rule allI, rule allI, rule allI, rule impI)
  1608   fix x lx ux
  1609   assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
  1610   hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
  1611 
  1612   { from exp_boundaries[of lx prec, unfolded l]
  1613     have "Ifloat l \<le> exp (Ifloat lx)" by (auto simp del: lb_exp.simps)
  1614     also have "\<dots> \<le> exp x" using x by auto
  1615     finally have "Ifloat l \<le> exp x" .
  1616   } moreover
  1617   { have "exp x \<le> exp (Ifloat ux)" using x by auto
  1618     also have "\<dots> \<le> Ifloat u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
  1619     finally have "exp x \<le> Ifloat u" .
  1620   } ultimately show "Ifloat l \<le> exp x \<and> exp x \<le> Ifloat u" ..
  1621 qed
  1622 
  1623 section "Logarithm"
  1624 
  1625 subsection "Compute the logarithm series"
  1626 
  1627 fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" 
  1628 and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
  1629 "ub_ln_horner prec 0 i x       = 0" |
  1630 "ub_ln_horner prec (Suc n) i x = rapprox_rat prec 1 (int i) - x * lb_ln_horner prec n (Suc i) x" |
  1631 "lb_ln_horner prec 0 i x       = 0" |
  1632 "lb_ln_horner prec (Suc n) i x = lapprox_rat prec 1 (int i) - x * ub_ln_horner prec n (Suc i) x"
  1633 
  1634 lemma ln_bounds:
  1635   assumes "0 \<le> x" and "x < 1"
  1636   shows "(\<Sum>i=0..<2*n. -1^i * (1 / real (i + 1)) * x^(Suc i)) \<le> ln (x + 1)" (is "?lb")
  1637   and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. -1^i * (1 / real (i + 1)) * x^(Suc i))" (is "?ub")
  1638 proof -
  1639   let "?a n" = "(1/real (n +1)) * x^(Suc n)"
  1640 
  1641   have ln_eq: "(\<Sum> i. -1^i * ?a i) = ln (x + 1)"
  1642     using ln_series[of "x + 1"] `0 \<le> x` `x < 1` by auto
  1643 
  1644   have "norm x < 1" using assms by auto
  1645   have "?a ----> 0" unfolding Suc_plus1[symmetric] inverse_eq_divide[symmetric] 
  1646     using LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF `norm x < 1`]]] by auto
  1647   { fix n have "0 \<le> ?a n" by (rule mult_nonneg_nonneg, auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`) }
  1648   { fix n have "?a (Suc n) \<le> ?a n" unfolding inverse_eq_divide[symmetric]
  1649     proof (rule mult_mono)
  1650       show "0 \<le> x ^ Suc (Suc n)" by (auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
  1651       have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 real_mult_assoc[symmetric] 
  1652 	by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
  1653       thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
  1654     qed auto }
  1655   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]
  1656   show "?lb" and "?ub" by auto
  1657 qed
  1658 
  1659 lemma ln_float_bounds: 
  1660   assumes "0 \<le> Ifloat x" and "Ifloat x < 1"
  1661   shows "Ifloat (x * lb_ln_horner prec (get_even n) 1 x) \<le> ln (Ifloat x + 1)" (is "?lb \<le> ?ln")
  1662   and "ln (Ifloat x + 1) \<le> Ifloat (x * ub_ln_horner prec (get_odd n) 1 x)" (is "?ln \<le> ?ub")
  1663 proof -
  1664   obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
  1665   obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..
  1666 
  1667   let "?s n" = "-1^n * (1 / real (1 + n)) * (Ifloat x)^(Suc n)"
  1668 
  1669   have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 real_mult_assoc[symmetric] Ifloat_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "Ifloat x"] ev
  1670     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",
  1671       OF `0 \<le> Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> Ifloat x`
  1672     by (rule mult_right_mono)
  1673   also have "\<dots> \<le> ?ln" using ln_bounds(1)[OF `0 \<le> Ifloat x` `Ifloat x < 1`] by auto
  1674   finally show "?lb \<le> ?ln" . 
  1675 
  1676   have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> Ifloat x` `Ifloat x < 1`] by auto
  1677   also have "\<dots> \<le> ?ub" unfolding power_Suc2 real_mult_assoc[symmetric] Ifloat_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "Ifloat x"] od
  1678     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",
  1679       OF `0 \<le> Ifloat x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> Ifloat x`
  1680     by (rule mult_right_mono)
  1681   finally show "?ln \<le> ?ub" . 
  1682 qed
  1683 
  1684 lemma ln_add: assumes "0 < x" and "0 < y" shows "ln (x + y) = ln x + ln (1 + y / x)"
  1685 proof -
  1686   have "x \<noteq> 0" using assms by auto
  1687   have "x + y = x * (1 + y / x)" unfolding right_distrib times_divide_eq_right nonzero_mult_divide_cancel_left[OF `x \<noteq> 0`] by auto
  1688   moreover 
  1689   have "0 < y / x" using assms divide_pos_pos by auto
  1690   hence "0 < 1 + y / x" by auto
  1691   ultimately show ?thesis using ln_mult assms by auto
  1692 qed
  1693 
  1694 subsection "Compute the logarithm of 2"
  1695 
  1696 definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3 
  1697                                         in (Float 1 -1 * ub_ln_horner prec (get_odd prec) 1 (Float 1 -1)) + 
  1698                                            (third * ub_ln_horner prec (get_odd prec) 1 third))"
  1699 definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3 
  1700                                         in (Float 1 -1 * lb_ln_horner prec (get_even prec) 1 (Float 1 -1)) + 
  1701                                            (third * lb_ln_horner prec (get_even prec) 1 third))"
  1702 
  1703 lemma ub_ln2: "ln 2 \<le> Ifloat (ub_ln2 prec)" (is "?ub_ln2")
  1704   and lb_ln2: "Ifloat (lb_ln2 prec) \<le> ln 2" (is "?lb_ln2")
  1705 proof -
  1706   let ?uthird = "rapprox_rat (max prec 1) 1 3"
  1707   let ?lthird = "lapprox_rat prec 1 3"
  1708 
  1709   have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1)"
  1710     using ln_add[of "3 / 2" "1 / 2"] by auto
  1711   have lb3: "Ifloat ?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
  1712   hence lb3_ub: "Ifloat ?lthird < 1" by auto
  1713   have lb3_lb: "0 \<le> Ifloat ?lthird" using lapprox_rat_bottom[of 1 3] by auto
  1714   have ub3: "1 / 3 \<le> Ifloat ?uthird" using rapprox_rat[of 1 3] by auto
  1715   hence ub3_lb: "0 \<le> Ifloat ?uthird" by auto
  1716 
  1717   have lb2: "0 \<le> Ifloat (Float 1 -1)" and ub2: "Ifloat (Float 1 -1) < 1" unfolding Float_num by auto
  1718 
  1719   have "0 \<le> (1::int)" and "0 < (3::int)" by auto
  1720   have ub3_ub: "Ifloat ?uthird < 1" unfolding rapprox_rat.simps(2)[OF `0 \<le> 1` `0 < 3`]
  1721     by (rule rapprox_posrat_less1, auto)
  1722 
  1723   have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
  1724   have uthird_gt0: "0 < Ifloat ?uthird + 1" using ub3_lb by auto
  1725   have lthird_gt0: "0 < Ifloat ?lthird + 1" using lb3_lb by auto
  1726 
  1727   show ?ub_ln2 unfolding ub_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric]
  1728   proof (rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
  1729     have "ln (1 / 3 + 1) \<le> ln (Ifloat ?uthird + 1)" unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
  1730     also have "\<dots> \<le> Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)"
  1731       using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
  1732     finally show "ln (1 / 3 + 1) \<le> Ifloat (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
  1733   qed
  1734   show ?lb_ln2 unfolding lb_ln2_def Let_def Ifloat_add ln2_sum Float_num(4)[symmetric]
  1735   proof (rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
  1736     have "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (Ifloat ?lthird + 1)"
  1737       using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
  1738     also have "\<dots> \<le> ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto
  1739     finally show "Ifloat (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (1 / 3 + 1)" .
  1740   qed
  1741 qed
  1742 
  1743 subsection "Compute the logarithm in the entire domain"
  1744 
  1745 function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where
  1746 "ub_ln prec x = (if x \<le> 0         then None
  1747             else if x < 1         then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
  1748             else let horner = \<lambda>x. (x - 1) * ub_ln_horner prec (get_odd prec) 1 (x - 1) in
  1749                  if x < Float 1 1 then Some (horner x)
  1750                                   else let l = bitlen (mantissa x) - 1 in 
  1751                                        Some (ub_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l))))" |
  1752 "lb_ln prec x = (if x \<le> 0         then None
  1753             else if x < 1         then Some (- the (ub_ln prec (float_divr prec 1 x)))
  1754             else let horner = \<lambda>x. (x - 1) * lb_ln_horner prec (get_even prec) 1 (x - 1) in
  1755                  if x < Float 1 1 then Some (horner x)
  1756                                   else let l = bitlen (mantissa x) - 1 in 
  1757                                        Some (lb_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l))))"
  1758 by pat_completeness auto
  1759 
  1760 termination proof (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 1 then 1 else 0))", auto)
  1761   fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divl (max prec (Suc 0)) 1 x < 1"
  1762   hence "0 < x" and "0 < max prec (Suc 0)" unfolding less_float_def le_float_def by auto
  1763   from float_divl_pos_less1_bound[OF `0 < x` `x < 1` `0 < max prec (Suc 0)`]
  1764   show False using `float_divl (max prec (Suc 0)) 1 x < 1` unfolding less_float_def le_float_def by auto
  1765 next
  1766   fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divr prec 1 x < 1"
  1767   hence "0 < x" unfolding less_float_def le_float_def by auto
  1768   from float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`, of prec]
  1769   show False using `float_divr prec 1 x < 1` unfolding less_float_def le_float_def by auto
  1770 qed
  1771 
  1772 lemma ln_shifted_float: assumes "0 < m" shows "ln (Ifloat (Float m e)) = ln 2 * real (e + (bitlen m - 1)) + ln (Ifloat (Float m (- (bitlen m - 1))))"
  1773 proof -
  1774   let ?B = "2^nat (bitlen m - 1)"
  1775   have "0 < real m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0" using assms by auto
  1776   hence "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto
  1777   show ?thesis 
  1778   proof (cases "0 \<le> e")
  1779     case True
  1780     show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
  1781       unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`] 
  1782       unfolding Ifloat_ge0_exp[OF True] ln_mult[OF `0 < real m` `0 < 2^nat e`] 
  1783       ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` True by auto
  1784   next
  1785     case False hence "0 < -e" by auto
  1786     hence pow_gt0: "(0::real) < 2^nat (-e)" by auto
  1787     hence inv_gt0: "(0::real) < inverse (2^nat (-e))" by auto
  1788     show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
  1789       unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`] 
  1790       unfolding Ifloat_nge0_exp[OF False] ln_mult[OF `0 < real m` inv_gt0] ln_inverse[OF pow_gt0]
  1791       ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` False by auto
  1792   qed
  1793 qed
  1794 
  1795 lemma ub_ln_lb_ln_bounds': assumes "1 \<le> x"
  1796   shows "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x) \<and> ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))"
  1797   (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
  1798 proof (cases "x < Float 1 1")
  1799   case True hence "Ifloat (x - 1) < 1" unfolding less_float_def Float_num by auto
  1800   have "\<not> x \<le> 0" and "\<not> x < 1" using `1 \<le> x` unfolding less_float_def le_float_def by auto
  1801   hence "0 \<le> Ifloat (x - 1)" using `1 \<le> x` unfolding less_float_def Float_num by auto
  1802   show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
  1803     using ln_float_bounds[OF `0 \<le> Ifloat (x - 1)` `Ifloat (x - 1) < 1`] `\<not> x \<le> 0` `\<not> x < 1` True by auto
  1804 next
  1805   case False
  1806   have "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" using `1 \<le> x` unfolding less_float_def le_float_def by auto
  1807   show ?thesis
  1808   proof (cases x)
  1809     case (Float m e)
  1810     let ?s = "Float (e + (bitlen m - 1)) 0"
  1811     let ?x = "Float m (- (bitlen m - 1))"
  1812 
  1813     have "0 < m" and "m \<noteq> 0" using float_pos_m_pos `0 < x` Float by auto
  1814 
  1815     {
  1816       have "Ifloat (lb_ln2 prec * ?s) \<le> ln 2 * real (e + (bitlen m - 1))" (is "?lb2 \<le> _")
  1817 	unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 realpow_0 mult_1_right
  1818 	using lb_ln2[of prec]
  1819       proof (rule mult_right_mono)
  1820 	have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
  1821 	from float_gt1_scale[OF this]
  1822 	show "0 \<le> real (e + (bitlen m - 1))" by auto
  1823       qed
  1824       moreover
  1825       from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
  1826       have "0 \<le> Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto
  1827       from ln_float_bounds(1)[OF this]
  1828       have "Ifloat ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln (Ifloat ?x)" (is "?lb_horner \<le> _") by auto
  1829       ultimately have "?lb2 + ?lb_horner \<le> ln (Ifloat x)"
  1830 	unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
  1831     } 
  1832     moreover
  1833     {
  1834       from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
  1835       have "0 \<le> Ifloat (?x - 1)" and "Ifloat (?x - 1) < 1" by auto
  1836       from ln_float_bounds(2)[OF this]
  1837       have "ln (Ifloat ?x) \<le> Ifloat ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))" (is "_ \<le> ?ub_horner") by auto
  1838       moreover
  1839       have "ln 2 * real (e + (bitlen m - 1)) \<le> Ifloat (ub_ln2 prec * ?s)" (is "_ \<le> ?ub2")
  1840 	unfolding Ifloat_mult Ifloat_ge0_exp[OF order_refl] nat_0 realpow_0 mult_1_right
  1841 	using ub_ln2[of prec] 
  1842       proof (rule mult_right_mono)
  1843 	have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
  1844 	from float_gt1_scale[OF this]
  1845 	show "0 \<le> real (e + (bitlen m - 1))" by auto
  1846       qed
  1847       ultimately have "ln (Ifloat x) \<le> ?ub2 + ?ub_horner"
  1848 	unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
  1849     }
  1850     ultimately show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps
  1851       unfolding if_not_P[OF `\<not> x \<le> 0`] if_not_P[OF `\<not> x < 1`] if_not_P[OF False] Let_def
  1852       unfolding scale.simps[of m e, unfolded Float[symmetric]] mantissa.simps[of m e, unfolded Float[symmetric]] Ifloat_add by auto
  1853   qed
  1854 qed
  1855 
  1856 lemma ub_ln_lb_ln_bounds: assumes "0 < x"
  1857   shows "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x) \<and> ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))"
  1858   (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
  1859 proof (cases "x < 1")
  1860   case False hence "1 \<le> x" unfolding less_float_def le_float_def by auto
  1861   show ?thesis using ub_ln_lb_ln_bounds'[OF `1 \<le> x`] .
  1862 next
  1863   case True have "\<not> x \<le> 0" using `0 < x` unfolding less_float_def le_float_def by auto
  1864 
  1865   have "0 < Ifloat x" and "Ifloat x \<noteq> 0" using `0 < x` unfolding less_float_def by auto
  1866   hence A: "0 < 1 / Ifloat x" by auto
  1867 
  1868   {
  1869     let ?divl = "float_divl (max prec 1) 1 x"
  1870     have A': "1 \<le> ?divl" using float_divl_pos_less1_bound[OF `0 < x` `x < 1`] unfolding le_float_def less_float_def by auto
  1871     hence B: "0 < Ifloat ?divl" unfolding le_float_def by auto
  1872     
  1873     have "ln (Ifloat ?divl) \<le> ln (1 / Ifloat x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
  1874     hence "ln (Ifloat x) \<le> - ln (Ifloat ?divl)" unfolding nonzero_inverse_eq_divide[OF `Ifloat x \<noteq> 0`, symmetric] ln_inverse[OF `0 < Ifloat x`] by auto
  1875     from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le] 
  1876     have "?ln \<le> Ifloat (- the (lb_ln prec ?divl))" unfolding Ifloat_minus by (rule order_trans)
  1877   } moreover
  1878   {
  1879     let ?divr = "float_divr prec 1 x"
  1880     have A': "1 \<le> ?divr" using float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`] unfolding le_float_def less_float_def by auto
  1881     hence B: "0 < Ifloat ?divr" unfolding le_float_def by auto
  1882     
  1883     have "ln (1 / Ifloat x) \<le> ln (Ifloat ?divr)" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
  1884     hence "- ln (Ifloat ?divr) \<le> ln (Ifloat x)" unfolding nonzero_inverse_eq_divide[OF `Ifloat x \<noteq> 0`, symmetric] ln_inverse[OF `0 < Ifloat x`] by auto
  1885     from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
  1886     have "Ifloat (- the (ub_ln prec ?divr)) \<le> ?ln" unfolding Ifloat_minus by (rule order_trans)
  1887   }
  1888   ultimately show ?thesis unfolding lb_ln.simps[where x=x]  ub_ln.simps[where x=x]
  1889     unfolding if_not_P[OF `\<not> x \<le> 0`] if_P[OF True] by auto
  1890 qed
  1891 
  1892 lemma lb_ln: assumes "Some y = lb_ln prec x"
  1893   shows "Ifloat y \<le> ln (Ifloat x)" and "0 < Ifloat x"
  1894 proof -
  1895   have "0 < x"
  1896   proof (rule ccontr)
  1897     assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
  1898     thus False using assms by auto
  1899   qed
  1900   thus "0 < Ifloat x" unfolding less_float_def by auto
  1901   have "Ifloat (the (lb_ln prec x)) \<le> ln (Ifloat x)" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
  1902   thus "Ifloat y \<le> ln (Ifloat x)" unfolding assms[symmetric] by auto
  1903 qed
  1904 
  1905 lemma ub_ln: assumes "Some y = ub_ln prec x"
  1906   shows "ln (Ifloat x) \<le> Ifloat y" and "0 < Ifloat x"
  1907 proof -
  1908   have "0 < x"
  1909   proof (rule ccontr)
  1910     assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
  1911     thus False using assms by auto
  1912   qed
  1913   thus "0 < Ifloat x" unfolding less_float_def by auto
  1914   have "ln (Ifloat x) \<le> Ifloat (the (ub_ln prec x))" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
  1915   thus "ln (Ifloat x) \<le> Ifloat y" unfolding assms[symmetric] by auto
  1916 qed
  1917 
  1918 lemma bnds_ln: "\<forall> x lx ux. (Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux} \<longrightarrow> Ifloat l \<le> ln x \<and> ln x \<le> Ifloat u"
  1919 proof (rule allI, rule allI, rule allI, rule impI)
  1920   fix x lx ux
  1921   assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {Ifloat lx .. Ifloat ux}"
  1922   hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {Ifloat lx .. Ifloat ux}" by auto
  1923 
  1924   have "ln (Ifloat ux) \<le> Ifloat u" and "0 < Ifloat ux" using ub_ln u by auto
  1925   have "Ifloat l \<le> ln (Ifloat lx)" and "0 < Ifloat lx" and "0 < x" using lb_ln[OF l] x by auto
  1926 
  1927   from ln_le_cancel_iff[OF `0 < Ifloat lx` `0 < x`] `Ifloat l \<le> ln (Ifloat lx)` 
  1928   have "Ifloat l \<le> ln x" using x unfolding atLeastAtMost_iff by auto
  1929   moreover
  1930   from ln_le_cancel_iff[OF `0 < x` `0 < Ifloat ux`] `ln (Ifloat ux) \<le> Ifloat u` 
  1931   have "ln x \<le> Ifloat u" using x unfolding atLeastAtMost_iff by auto
  1932   ultimately show "Ifloat l \<le> ln x \<and> ln x \<le> Ifloat u" ..
  1933 qed
  1934 
  1935 
  1936 section "Implement floatarith"
  1937 
  1938 subsection "Define syntax and semantics"
  1939 
  1940 datatype floatarith
  1941   = Add floatarith floatarith
  1942   | Minus floatarith
  1943   | Mult floatarith floatarith
  1944   | Inverse floatarith
  1945   | Sin floatarith
  1946   | Cos floatarith
  1947   | Arctan floatarith
  1948   | Abs floatarith
  1949   | Max floatarith floatarith
  1950   | Min floatarith floatarith
  1951   | Pi
  1952   | Sqrt floatarith
  1953   | Exp floatarith
  1954   | Ln floatarith
  1955   | Power floatarith nat
  1956   | Atom nat
  1957   | Num float
  1958 
  1959 fun Ifloatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real"
  1960 where
  1961 "Ifloatarith (Add a b) vs   = (Ifloatarith a vs) + (Ifloatarith b vs)" |
  1962 "Ifloatarith (Minus a) vs    = - (Ifloatarith a vs)" |
  1963 "Ifloatarith (Mult a b) vs   = (Ifloatarith a vs) * (Ifloatarith b vs)" |
  1964 "Ifloatarith (Inverse a) vs  = inverse (Ifloatarith a vs)" |
  1965 "Ifloatarith (Sin a) vs      = sin (Ifloatarith a vs)" |
  1966 "Ifloatarith (Cos a) vs      = cos (Ifloatarith a vs)" |
  1967 "Ifloatarith (Arctan a) vs   = arctan (Ifloatarith a vs)" |
  1968 "Ifloatarith (Min a b) vs    = min (Ifloatarith a vs) (Ifloatarith b vs)" |
  1969 "Ifloatarith (Max a b) vs    = max (Ifloatarith a vs) (Ifloatarith b vs)" |
  1970 "Ifloatarith (Abs a) vs      = abs (Ifloatarith a vs)" |
  1971 "Ifloatarith Pi vs           = pi" |
  1972 "Ifloatarith (Sqrt a) vs     = sqrt (Ifloatarith a vs)" |
  1973 "Ifloatarith (Exp a) vs      = exp (Ifloatarith a vs)" |
  1974 "Ifloatarith (Ln a) vs       = ln (Ifloatarith a vs)" |
  1975 "Ifloatarith (Power a n) vs  = (Ifloatarith a vs)^n" |
  1976 "Ifloatarith (Num f) vs      = Ifloat f" |
  1977 "Ifloatarith (Atom n) vs     = vs ! n"
  1978 
  1979 subsection "Implement approximation function"
  1980 
  1981 fun lift_bin :: "(float * float) option \<Rightarrow> (float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> (float option * float option)) \<Rightarrow> (float * float) option" where
  1982 "lift_bin (Some (l1, u1)) (Some (l2, u2)) f = (case (f l1 u1 l2 u2) of (Some l, Some u) \<Rightarrow> Some (l, u)
  1983                                                                      | t \<Rightarrow> None)" |
  1984 "lift_bin a b f = None"
  1985 
  1986 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
  1987 "lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" |
  1988 "lift_bin' a b f = None"
  1989 
  1990 fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where
  1991 "lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u)
  1992                                              | t \<Rightarrow> None)" |
  1993 "lift_un b f = None"
  1994 
  1995 fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
  1996 "lift_un' (Some (l1, u1)) f = Some (f l1 u1)" |
  1997 "lift_un' b f = None"
  1998 
  1999 fun bounded_by :: "real list \<Rightarrow> (float * float) list \<Rightarrow> bool " where
  2000 bounded_by_Cons: "bounded_by (v#vs) ((l, u)#bs) = ((Ifloat l \<le> v \<and> v \<le> Ifloat u) \<and> bounded_by vs bs)" |
  2001 bounded_by_Nil: "bounded_by [] [] = True" |
  2002 "bounded_by _ _ = False"
  2003 
  2004 lemma bounded_by: assumes "bounded_by vs bs" and "i < length bs"
  2005   shows "Ifloat (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> Ifloat (snd (bs ! i))"
  2006   using `bounded_by vs bs` and `i < length bs`
  2007 proof (induct arbitrary: i rule: bounded_by.induct)
  2008   fix v :: real and vs :: "real list" and l u :: float and bs :: "(float * float) list" and i :: nat
  2009   assume hyp: "\<And>i. \<lbrakk>bounded_by vs bs; i < length bs\<rbrakk> \<Longrightarrow> Ifloat (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> Ifloat (snd (bs ! i))"
  2010   assume bounded: "bounded_by (v # vs) ((l, u) # bs)" and length: "i < length ((l, u) # bs)"
  2011   show "Ifloat (fst (((l, u) # bs) ! i)) \<le> (v # vs) ! i \<and> (v # vs) ! i \<le> Ifloat (snd (((l, u) # bs) ! i))"
  2012   proof (cases i)
  2013     case 0
  2014     show ?thesis using bounded unfolding 0 nth_Cons_0 fst_conv snd_conv bounded_by.simps ..
  2015   next
  2016     case (Suc i) with length have "i < length bs" by auto
  2017     show ?thesis unfolding Suc nth_Cons_Suc bounded_by.simps
  2018       using hyp[OF bounded[unfolded bounded_by.simps, THEN conjunct2] `i < length bs`] .
  2019   qed
  2020 qed auto
  2021 
  2022 fun approx approx' :: "nat \<Rightarrow> floatarith \<Rightarrow> (float * float) list \<Rightarrow> (float * float) option" where
  2023 "approx' prec a bs          = (case (approx prec a bs) of Some (l, u) \<Rightarrow> Some (round_down prec l, round_up prec u) | None \<Rightarrow> None)" |
  2024 "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))" | 
  2025 "approx prec (Minus a) bs   = lift_un' (approx' prec a bs) (\<lambda> l u. (-u, -l))" |
  2026 "approx prec (Mult a b) bs  = lift_bin' (approx' prec a bs) (approx' prec b bs)
  2027                                     (\<lambda> a1 a2 b1 b2. (float_nprt a1 * float_pprt b2 + float_nprt a2 * float_nprt b2 + float_pprt a1 * float_pprt b1 + float_pprt a2 * float_nprt b1, 
  2028                                                      float_pprt a2 * float_pprt b2 + float_pprt a1 * float_nprt b2 + float_nprt a2 * float_pprt b1 + float_nprt a1 * float_nprt b1))" |
  2029 "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))" |
  2030 "approx prec (Sin a) bs     = lift_un' (approx' prec a bs) (bnds_sin prec)" |
  2031 "approx prec (Cos a) bs     = lift_un' (approx' prec a bs) (bnds_cos prec)" |
  2032 "approx prec Pi bs          = Some (lb_pi prec, ub_pi prec)" |
  2033 "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))" |
  2034 "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))" |
  2035 "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>))" |
  2036 "approx prec (Arctan a) bs  = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_arctan prec l, ub_arctan prec u))" |
  2037 "approx prec (Sqrt a) bs    = lift_un (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" |
  2038 "approx prec (Exp a) bs     = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" |
  2039 "approx prec (Ln a) bs      = lift_un (approx' prec a bs) (\<lambda> l u. (lb_ln prec l, ub_ln prec u))" |
  2040 "approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds n)" |
  2041 "approx prec (Num f) bs     = Some (f, f)" |
  2042 "approx prec (Atom i) bs    = (if i < length bs then Some (bs ! i) else None)"
  2043 
  2044 lemma lift_bin'_ex:
  2045   assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f"
  2046   shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b"
  2047 proof (cases a)
  2048   case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
  2049   thus ?thesis using lift_bin'_Some by auto
  2050 next
  2051   case (Some a')
  2052   show ?thesis
  2053   proof (cases b)
  2054     case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
  2055     thus ?thesis using lift_bin'_Some by auto
  2056   next
  2057     case (Some b')
  2058     obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
  2059     obtain lb ub where b': "b' = (lb, ub)" by (cases b', auto)
  2060     thus ?thesis unfolding `a = Some a'` `b = Some b'` a' b' by auto
  2061   qed
  2062 qed
  2063 
  2064 lemma lift_bin'_f:
  2065   assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f"
  2066   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"
  2067   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)"
  2068 proof -
  2069   obtain l1 u1 l2 u2
  2070     where Sa: "Some (l1, u1) = g a" and Sb: "Some (l2, u2) = g b" using lift_bin'_ex[OF assms(1)] by auto
  2071   have lu: "(l, u) = f l1 u1 l2 u2" using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto 
  2072   have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" unfolding lu[symmetric] by auto
  2073   thus ?thesis using Pa[OF Sa] Pb[OF Sb] by auto 
  2074 qed
  2075 
  2076 lemma approx_approx':
  2077   assumes Pa: "\<And>l u. Some (l, u) = approx prec a vs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
  2078   and approx': "Some (l, u) = approx' prec a vs"
  2079   shows "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
  2080 proof -
  2081   obtain l' u' where S: "Some (l', u') = approx prec a vs"
  2082     using approx' unfolding approx'.simps by (cases "approx prec a vs", auto)
  2083   have l': "l = round_down prec l'" and u': "u = round_up prec u'"
  2084     using approx' unfolding approx'.simps S[symmetric] by auto
  2085   show ?thesis unfolding l' u' 
  2086     using order_trans[OF Pa[OF S, THEN conjunct2] round_up[of u']]
  2087     using order_trans[OF round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto
  2088 qed
  2089 
  2090 lemma lift_bin':
  2091   assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f"
  2092   and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
  2093   and Pb: "\<And>l u. Some (l, u) = approx prec b bs \<Longrightarrow> Ifloat l \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u"
  2094   shows "\<exists> l1 u1 l2 u2. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> 
  2095                         (Ifloat l2 \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u2) \<and> 
  2096                         l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
  2097 proof -
  2098   { fix l u assume "Some (l, u) = approx' prec a bs"
  2099     with approx_approx'[of prec a bs, OF _ this] Pa
  2100     have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
  2101   { fix l u assume "Some (l, u) = approx' prec b bs"
  2102     with approx_approx'[of prec b bs, OF _ this] Pb
  2103     have "Ifloat l \<le> Ifloatarith b xs \<and> Ifloatarith b xs \<le> Ifloat u" by auto } note Pb = this
  2104 
  2105   from lift_bin'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb]
  2106   show ?thesis by auto
  2107 qed
  2108 
  2109 lemma lift_un'_ex:
  2110   assumes lift_un'_Some: "Some (l, u) = lift_un' a f"
  2111   shows "\<exists> l u. Some (l, u) = a"
  2112 proof (cases a)
  2113   case None hence "None = lift_un' a f" unfolding None lift_un'.simps ..
  2114   thus ?thesis using lift_un'_Some by auto
  2115 next
  2116   case (Some a')
  2117   obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
  2118   thus ?thesis unfolding `a = Some a'` a' by auto
  2119 qed
  2120 
  2121 lemma lift_un'_f:
  2122   assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f"
  2123   and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
  2124   shows "\<exists> l1 u1. P l1 u1 a \<and> l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
  2125 proof -
  2126   obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un'_ex[OF assms(1)] by auto
  2127   have lu: "(l, u) = f l1 u1" using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto
  2128   have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" unfolding lu[symmetric] by auto
  2129   thus ?thesis using Pa[OF Sa] by auto
  2130 qed
  2131 
  2132 lemma lift_un':
  2133   assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
  2134   and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
  2135   shows "\<exists> l1 u1. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> 
  2136                         l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
  2137 proof -
  2138   { fix l u assume "Some (l, u) = approx' prec a bs"
  2139     with approx_approx'[of prec a bs, OF _ this] Pa
  2140     have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
  2141   from lift_un'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa]
  2142   show ?thesis by auto
  2143 qed
  2144 
  2145 lemma lift_un'_bnds:
  2146   assumes bnds: "\<forall> x lx ux. (l, u) = f lx ux \<and> x \<in> { Ifloat lx .. Ifloat ux } \<longrightarrow> Ifloat l \<le> f' x \<and> f' x \<le> Ifloat u"
  2147   and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
  2148   and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
  2149   shows "Ifloat l \<le> f' (Ifloatarith a xs) \<and> f' (Ifloatarith a xs) \<le> Ifloat u"
  2150 proof -
  2151   from lift_un'[OF lift_un'_Some Pa]
  2152   obtain l1 u1 where "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" and "l = fst (f l1 u1)" and "u = snd (f l1 u1)" by blast
  2153   hence "(l, u) = f l1 u1" and "Ifloatarith a xs \<in> {Ifloat l1 .. Ifloat u1}" by auto
  2154   thus ?thesis using bnds by auto
  2155 qed
  2156 
  2157 lemma lift_un_ex:
  2158   assumes lift_un_Some: "Some (l, u) = lift_un a f"
  2159   shows "\<exists> l u. Some (l, u) = a"
  2160 proof (cases a)
  2161   case None hence "None = lift_un a f" unfolding None lift_un.simps ..
  2162   thus ?thesis using lift_un_Some by auto
  2163 next
  2164   case (Some a')
  2165   obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
  2166   thus ?thesis unfolding `a = Some a'` a' by auto
  2167 qed
  2168 
  2169 lemma lift_un_f:
  2170   assumes lift_un_Some: "Some (l, u) = lift_un (g a) f"
  2171   and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
  2172   shows "\<exists> l1 u1. P l1 u1 a \<and> Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
  2173 proof -
  2174   obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un_ex[OF assms(1)] by auto
  2175   have "fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None"
  2176   proof (rule ccontr)
  2177     assume "\<not> (fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None)"
  2178     hence or: "fst (f l1 u1) = None \<or> snd (f l1 u1) = None" by auto
  2179     hence "lift_un (g a) f = None" 
  2180     proof (cases "fst (f l1 u1) = None")
  2181       case True
  2182       then obtain b where b: "f l1 u1 = (None, b)" by (cases "f l1 u1", auto)
  2183       thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
  2184     next
  2185       case False hence "snd (f l1 u1) = None" using or by auto
  2186       with False obtain b where b: "f l1 u1 = (Some b, None)" by (cases "f l1 u1", auto)
  2187       thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
  2188     qed
  2189     thus False using lift_un_Some by auto
  2190   qed
  2191   then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" by (cases "f l1 u1", auto)
  2192   from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f]
  2193   have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" unfolding f by auto
  2194   thus ?thesis unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto
  2195 qed
  2196 
  2197 lemma lift_un:
  2198   assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
  2199   and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
  2200   shows "\<exists> l1 u1. (Ifloat l1 \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u1) \<and> 
  2201                   Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
  2202 proof -
  2203   { fix l u assume "Some (l, u) = approx' prec a bs"
  2204     with approx_approx'[of prec a bs, OF _ this] Pa
  2205     have "Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u" by auto } note Pa = this
  2206   from lift_un_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa]
  2207   show ?thesis by auto
  2208 qed
  2209 
  2210 lemma lift_un_bnds:
  2211   assumes bnds: "\<forall> x lx ux. (Some l, Some u) = f lx ux \<and> x \<in> { Ifloat lx .. Ifloat ux } \<longrightarrow> Ifloat l \<le> f' x \<and> f' x \<le> Ifloat u"
  2212   and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
  2213   and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> Ifloat l \<le> Ifloatarith a xs \<and> Ifloatarith a xs \<le> Ifloat u"
  2214   shows "Ifloat l \<le> f' (Ifloatarith a xs) \<and> f' (Ifloatarith a xs) \<le> Ifloat u"
  2215 proof -
  2216   from lift_un[OF lift_un_Some Pa]
  2217   obtain l1 u1 where "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" and "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" by blast
  2218   hence "(Some l, Some u) = f l1 u1" and "Ifloatarith a xs \<in> {Ifloat l1 .. Ifloat u1}" by auto
  2219   thus ?thesis using bnds by auto
  2220 qed
  2221 
  2222 lemma approx:
  2223   assumes "bounded_by xs vs"
  2224   and "Some (l, u) = approx prec arith vs" (is "_ = ?g arith")
  2225   shows "Ifloat l \<le> Ifloatarith arith xs \<and> Ifloatarith arith xs \<le> Ifloat u" (is "?P l u arith")
  2226   using `Some (l, u) = approx prec arith vs` 
  2227 proof (induct arith arbitrary: l u x)
  2228   case (Add a b)
  2229   from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps
  2230   obtain l1 u1 l2 u2 where "l = l1 + l2" and "u = u1 + u2"
  2231     "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1"
  2232     "Ifloat l2 \<le> Ifloatarith b xs" and "Ifloatarith b xs \<le> Ifloat u2" unfolding fst_conv snd_conv by blast
  2233   thus ?case unfolding Ifloatarith.simps by auto
  2234 next
  2235   case (Minus a)
  2236   from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps
  2237   obtain l1 u1 where "l = -u1" and "u = -l1"
  2238     "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1" unfolding fst_conv snd_conv by blast
  2239   thus ?case unfolding Ifloatarith.simps using Ifloat_minus by auto
  2240 next
  2241   case (Mult a b)
  2242   from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps
  2243   obtain l1 u1 l2 u2 
  2244     where l: "l = float_nprt l1 * float_pprt u2 + float_nprt u1 * float_nprt u2 + float_pprt l1 * float_pprt l2 + float_pprt u1 * float_nprt l2"
  2245     and u: "u = float_pprt u1 * float_pprt u2 + float_pprt l1 * float_nprt u2 + float_nprt u1 * float_pprt l2 + float_nprt l1 * float_nprt l2"
  2246     and "Ifloat l1 \<le> Ifloatarith a xs" and "Ifloatarith a xs \<le> Ifloat u1"
  2247     and "Ifloat l2 \<le> Ifloatarith b xs" and "Ifloatarith b xs \<le> Ifloat u2" unfolding fst_conv snd_conv by blast
  2248   thus ?case unfolding Ifloatarith.simps l u Ifloat_add Ifloat_mult Ifloat_nprt Ifloat_pprt 
  2249     using mult_le_prts mult_ge_prts by auto
  2250 next
  2251   case (Inverse a)
  2252   from lift_un[OF Inverse.prems[unfolded approx.simps], unfolded if_distrib[of fst] if_distrib[of snd] fst_conv snd_conv] Inverse.hyps
  2253   obtain l1 u1 where l': "Some l = (if 0 < l1 \<or> u1 < 0 then Some (float_divl prec 1 u1) else None)" 
  2254     and u': "Some u = (if 0 < l1 \<or> u1 < 0 then Some (float_divr prec 1 l1) else None)"
  2255     and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1" by blast
  2256   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
  2257   moreover have l1_le_u1: "Ifloat l1 \<le> Ifloat u1" using l1 u1 by auto
  2258   ultimately have "Ifloat l1 \<noteq> 0" and "Ifloat u1 \<noteq> 0" unfolding less_float_def by auto
  2259 
  2260   have inv: "inverse (Ifloat u1) \<le> inverse (Ifloatarith a xs)
  2261            \<and> inverse (Ifloatarith a xs) \<le> inverse (Ifloat l1)"
  2262   proof (cases "0 < l1")
  2263     case True hence "0 < Ifloat u1" and "0 < Ifloat l1" "0 < Ifloatarith a xs" 
  2264       unfolding less_float_def using l1_le_u1 l1 by auto
  2265     show ?thesis
  2266       unfolding inverse_le_iff_le[OF `0 < Ifloat u1` `0 < Ifloatarith a xs`]
  2267 	inverse_le_iff_le[OF `0 < Ifloatarith a xs` `0 < Ifloat l1`]
  2268       using l1 u1 by auto
  2269   next
  2270     case False hence "u1 < 0" using either by blast
  2271     hence "Ifloat u1 < 0" and "Ifloat l1 < 0" "Ifloatarith a xs < 0" 
  2272       unfolding less_float_def using l1_le_u1 u1 by auto
  2273     show ?thesis
  2274       unfolding inverse_le_iff_le_neg[OF `Ifloat u1 < 0` `Ifloatarith a xs < 0`]
  2275 	inverse_le_iff_le_neg[OF `Ifloatarith a xs < 0` `Ifloat l1 < 0`]
  2276       using l1 u1 by auto
  2277   qed
  2278     
  2279   from l' have "l = float_divl prec 1 u1" by (cases "0 < l1 \<or> u1 < 0", auto)
  2280   hence "Ifloat l \<le> inverse (Ifloat u1)" unfolding nonzero_inverse_eq_divide[OF `Ifloat u1 \<noteq> 0`] using float_divl[of prec 1 u1] by auto
  2281   also have "\<dots> \<le> inverse (Ifloatarith a xs)" using inv by auto
  2282   finally have "Ifloat l \<le> inverse (Ifloatarith a xs)" .
  2283   moreover
  2284   from u' have "u = float_divr prec 1 l1" by (cases "0 < l1 \<or> u1 < 0", auto)
  2285   hence "inverse (Ifloat l1) \<le> Ifloat u" unfolding nonzero_inverse_eq_divide[OF `Ifloat l1 \<noteq> 0`] using float_divr[of 1 l1 prec] by auto
  2286   hence "inverse (Ifloatarith a xs) \<le> Ifloat u" by (rule order_trans[OF inv[THEN conjunct2]])
  2287   ultimately show ?case unfolding Ifloatarith.simps using l1 u1 by auto
  2288 next
  2289   case (Abs x)
  2290   from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps
  2291   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>"
  2292     and l1: "Ifloat l1 \<le> Ifloatarith x xs" and u1: "Ifloatarith x xs \<le> Ifloat u1" by blast
  2293   thus ?case unfolding l' u' by (cases "l1 < 0 \<and> 0 < u1", auto simp add: Ifloat_min Ifloat_max Ifloat_abs less_float_def)
  2294 next
  2295   case (Min a b)
  2296   from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps
  2297   obtain l1 u1 l2 u2 where l': "l = min l1 l2" and u': "u = min u1 u2"
  2298     and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1"
  2299     and l1: "Ifloat l2 \<le> Ifloatarith b xs" and u1: "Ifloatarith b xs \<le> Ifloat u2" by blast
  2300   thus ?case unfolding l' u' by (auto simp add: Ifloat_min)
  2301 next
  2302   case (Max a b)
  2303   from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps
  2304   obtain l1 u1 l2 u2 where l': "l = max l1 l2" and u': "u = max u1 u2"
  2305     and l1: "Ifloat l1 \<le> Ifloatarith a xs" and u1: "Ifloatarith a xs \<le> Ifloat u1"
  2306     and l1: "Ifloat l2 \<le> Ifloatarith b xs" and u1: "Ifloatarith b xs \<le> Ifloat u2" by blast
  2307   thus ?case unfolding l' u' by (auto simp add: Ifloat_max)
  2308 next case (Sin a) with lift_un'_bnds[OF bnds_sin] show ?case by auto
  2309 next case (Cos a) with lift_un'_bnds[OF bnds_cos] show ?case by auto
  2310 next case (Arctan a) with lift_un'_bnds[OF bnds_arctan] show ?case by auto
  2311 next case Pi with pi_boundaries show ?case by auto
  2312 next case (Sqrt a) with lift_un_bnds[OF bnds_sqrt] show ?case by auto
  2313 next case (Exp a) with lift_un'_bnds[OF bnds_exp] show ?case by auto
  2314 next case (Ln a) with lift_un_bnds[OF bnds_ln] show ?case by auto
  2315 next case (Power a n) with lift_un'_bnds[OF bnds_power] show ?case by auto
  2316 next case (Num f) thus ?case by auto
  2317 next
  2318   case (Atom n) 
  2319   show ?case
  2320   proof (cases "n < length vs")
  2321     case True
  2322     with Atom have "vs ! n = (l, u)" by auto
  2323     thus ?thesis using bounded_by[OF assms(1) True] by auto
  2324   next
  2325     case False thus ?thesis using Atom by auto
  2326   qed
  2327 qed
  2328 
  2329 datatype ApproxEq = Less floatarith floatarith 
  2330                   | LessEqual floatarith floatarith 
  2331 
  2332 fun uneq :: "ApproxEq \<Rightarrow> real list \<Rightarrow> bool" where 
  2333 "uneq (Less a b) vs                   = (Ifloatarith a vs < Ifloatarith b vs)" |
  2334 "uneq (LessEqual a b) vs              = (Ifloatarith a vs \<le> Ifloatarith b vs)"
  2335 
  2336 fun uneq' :: "nat \<Rightarrow> ApproxEq \<Rightarrow> (float * float) list \<Rightarrow> bool" where 
  2337 "uneq' prec (Less a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u < l' | _ \<Rightarrow> False)" |
  2338 "uneq' prec (LessEqual a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u \<le> l' | _ \<Rightarrow> False)"
  2339 
  2340 lemma uneq_approx: fixes m :: nat assumes "bounded_by vs bs" and "uneq' prec eq bs"
  2341   shows "uneq eq vs"
  2342 proof (cases eq)
  2343   case (Less a b)
  2344   show ?thesis
  2345   proof (cases "\<exists> u l u' l'. approx prec a bs = Some (l, u) \<and> 
  2346                              approx prec b bs = Some (l', u')")
  2347     case True
  2348     then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)"
  2349       and b_approx: "approx prec b bs = Some (l', u') " by auto
  2350     with `uneq' prec eq bs` have "Ifloat u < Ifloat l'"
  2351       unfolding Less uneq'.simps less_float_def by auto
  2352     moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs`
  2353     have "Ifloatarith a vs \<le> Ifloat u" and "Ifloat l' \<le> Ifloatarith b vs"
  2354       using approx by auto
  2355     ultimately show ?thesis unfolding uneq.simps Less by auto
  2356   next
  2357     case False
  2358     hence "approx prec a bs = None \<or> approx prec b bs = None"
  2359       unfolding not_Some_eq[symmetric] by auto
  2360     hence "\<not> uneq' prec eq bs" unfolding Less uneq'.simps 
  2361       by (cases "approx prec a bs = None", auto)
  2362     thus ?thesis using assms by auto
  2363   qed
  2364 next
  2365   case (LessEqual a b)
  2366   show ?thesis
  2367   proof (cases "\<exists> u l u' l'. approx prec a bs = Some (l, u) \<and> 
  2368                              approx prec b bs = Some (l', u')")
  2369     case True
  2370     then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)"
  2371       and b_approx: "approx prec b bs = Some (l', u') " by auto
  2372     with `uneq' prec eq bs` have "Ifloat u \<le> Ifloat l'"
  2373       unfolding LessEqual uneq'.simps le_float_def by auto
  2374     moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs`
  2375     have "Ifloatarith a vs \<le> Ifloat u" and "Ifloat l' \<le> Ifloatarith b vs"
  2376       using approx by auto
  2377     ultimately show ?thesis unfolding uneq.simps LessEqual by auto
  2378   next
  2379     case False
  2380     hence "approx prec a bs = None \<or> approx prec b bs = None"
  2381       unfolding not_Some_eq[symmetric] by auto
  2382     hence "\<not> uneq' prec eq bs" unfolding LessEqual uneq'.simps 
  2383       by (cases "approx prec a bs = None", auto)
  2384     thus ?thesis using assms by auto
  2385   qed
  2386 qed
  2387 
  2388 lemma Ifloatarith_divide: "Ifloatarith (Mult a (Inverse b)) vs = (Ifloatarith a vs) / (Ifloatarith b vs)"
  2389   unfolding real_divide_def Ifloatarith.simps ..
  2390 
  2391 lemma Ifloatarith_diff: "Ifloatarith (Add a (Minus b)) vs = (Ifloatarith a vs) - (Ifloatarith b vs)"
  2392   unfolding real_diff_def Ifloatarith.simps ..
  2393 
  2394 lemma Ifloatarith_tan: "Ifloatarith (Mult (Sin a) (Inverse (Cos a))) vs = tan (Ifloatarith a vs)"
  2395   unfolding tan_def Ifloatarith.simps real_divide_def ..
  2396 
  2397 lemma Ifloatarith_powr: "Ifloatarith (Exp (Mult b (Ln a))) vs = (Ifloatarith a vs) powr (Ifloatarith b vs)"
  2398   unfolding powr_def Ifloatarith.simps ..
  2399 
  2400 lemma Ifloatarith_log: "Ifloatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (Ifloatarith b vs) (Ifloatarith x vs)"
  2401   unfolding log_def Ifloatarith.simps real_divide_def ..
  2402 
  2403 lemma Ifloatarith_num: shows "Ifloatarith (Num (Float 0 0)) vs = 0" and "Ifloatarith (Num (Float 1 0)) vs = 1" and "Ifloatarith (Num (Float (number_of a) 0)) vs = number_of a" by auto
  2404 
  2405 subsection {* Implement proof method \texttt{approximation} *}
  2406 
  2407 lemma bounded_divl: assumes "Ifloat a / Ifloat b \<le> x" shows "Ifloat (float_divl p a b) \<le> x" by (rule order_trans[OF _ assms], rule float_divl)
  2408 lemma bounded_divr: assumes "x \<le> Ifloat a / Ifloat b" shows "x \<le> Ifloat (float_divr p a b)" by (rule order_trans[OF assms _], rule float_divr)
  2409 lemma bounded_num: shows "Ifloat (Float 5 1) = 10" and "Ifloat (Float 0 0) = 0" and "Ifloat (Float 1 0) = 1" and "Ifloat (Float (number_of n) 0) = (number_of n)"
  2410                      and "0 * pow2 e = Ifloat (Float 0 e)" and "1 * pow2 e = Ifloat (Float 1 e)" and "number_of m * pow2 e = Ifloat (Float (number_of m) e)"
  2411   by (auto simp add: Ifloat.simps pow2_def)
  2412 
  2413 lemmas bounded_by_equations = bounded_by_Cons bounded_by_Nil float_power bounded_divl bounded_divr bounded_num HOL.simp_thms
  2414 lemmas uneq_equations = uneq.simps Ifloatarith.simps Ifloatarith_num Ifloatarith_divide Ifloatarith_diff Ifloatarith_tan Ifloatarith_powr Ifloatarith_log
  2415 
  2416 lemma "x div (0::int) = 0" by auto -- "What happens in the zero case for div"
  2417 lemma "x mod (0::int) = x" by auto -- "What happens in the zero case for mod"
  2418 
  2419 text {* The following equations must hold for div & mod 
  2420         -- see "The Definition of Standard ML" by R. Milner, M. Tofte and R. Harper (pg. 79) *}
  2421 lemma "d * (i div d) + i mod d = (i::int)" by auto
  2422 lemma "0 < (d :: int) \<Longrightarrow> 0 \<le> i mod d \<and> i mod d < d" by auto
  2423 lemma "(d :: int) < 0 \<Longrightarrow> d < i mod d \<and> i mod d \<le> 0" by auto
  2424 
  2425 code_const "op div :: int \<Rightarrow> int \<Rightarrow> int" (SML "(fn i => fn d => if d = 0 then 0 else i div d)")
  2426 code_const "op mod :: int \<Rightarrow> int \<Rightarrow> int" (SML "(fn i => fn d => if d = 0 then i else i mod d)")
  2427 code_const "divmod :: int \<Rightarrow> int \<Rightarrow> (int * int)" (SML "(fn i => fn d => if d = 0 then (0, i) else IntInf.divMod (i, d))")
  2428 
  2429 ML {*
  2430   val uneq_equations = PureThy.get_thms @{theory} "uneq_equations";
  2431   val bounded_by_equations = PureThy.get_thms @{theory} "bounded_by_equations";
  2432   val bounded_by_simpset = (HOL_basic_ss addsimps bounded_by_equations)
  2433 
  2434   fun reify_uneq ctxt i = (fn st =>
  2435     let
  2436       val to = HOLogic.dest_Trueprop (Logic.strip_imp_concl (List.nth (prems_of st, i - 1)))
  2437     in (Reflection.genreify_tac ctxt uneq_equations (SOME to) i) st
  2438     end)
  2439 
  2440   fun rule_uneq ctxt prec i thm = let
  2441     fun conv_num typ = HOLogic.dest_number #> snd #> HOLogic.mk_number typ
  2442     val to_natc = conv_num @{typ "nat"} #> Thm.cterm_of (ProofContext.theory_of ctxt)
  2443     val to_nat = conv_num @{typ "nat"}
  2444     val to_int = conv_num @{typ "int"}
  2445 
  2446     val prec' = to_nat prec
  2447 
  2448     fun bot_float (Const (@{const_name "times"}, _) $ mantisse $ (Const (@{const_name "pow2"}, _) $ exp))
  2449                    = @{term "Float"} $ to_int mantisse $ to_int exp
  2450       | bot_float (Const (@{const_name "divide"}, _) $ mantisse $ (Const (@{const_name "power"}, _) $ ten $ exp))
  2451                    = @{term "float_divl"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ (@{term "power_float (Float 5 1)"} $ to_nat exp)
  2452       | bot_float (Const (@{const_name "divide"}, _) $ mantisse $ ten)
  2453                    = @{term "float_divl"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ @{term "Float 5 1"}
  2454       | bot_float mantisse = @{term "Float"} $ to_int mantisse $ @{term "0 :: int"}
  2455 
  2456     fun top_float (Const (@{const_name "times"}, _) $ mantisse $ (Const (@{const_name "pow2"}, _) $ exp))
  2457                    = @{term "Float"} $ to_int mantisse $ to_int exp
  2458       | top_float (Const (@{const_name "divide"}, _) $ mantisse $ (Const (@{const_name "power"}, _) $ ten $ exp))
  2459                    = @{term "float_divr"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ (@{term "power_float (Float 5 1)"} $ to_nat exp)
  2460       | top_float (Const (@{const_name "divide"}, _) $ mantisse $ ten)
  2461                    = @{term "float_divr"} $ prec' $ (@{term "Float"} $ to_int mantisse $ @{term "0 :: int"}) $ @{term "Float 5 1"}
  2462       | top_float mantisse = @{term "Float"} $ to_int mantisse $ @{term "0 :: int"}
  2463 
  2464     val goal' : term = List.nth (prems_of thm, i - 1)
  2465 
  2466     fun lift_bnd (t as (Const (@{const_name "op &"}, _) $ 
  2467                         (Const (@{const_name "less_eq"}, _) $ 
  2468                          bottom $ (Free (name, _))) $ 
  2469                         (Const (@{const_name "less_eq"}, _) $ _ $ top)))
  2470          = ((name, HOLogic.mk_prod (bot_float bottom, top_float top))
  2471             handle TERM (txt, ts) => raise TERM ("Premisse needs format '<num> <= <var> & <var> <= <num>', but found " ^
  2472                                   (Syntax.string_of_term ctxt t), [t]))
  2473       | lift_bnd t = raise TERM ("Premisse needs format '<num> <= <var> & <var> <= <num>', but found " ^
  2474                                  (Syntax.string_of_term ctxt t), [t])
  2475     val bound_eqs = map (HOLogic.dest_Trueprop #> lift_bnd)  (Logic.strip_imp_prems goal')
  2476 
  2477     fun lift_var (Free (varname, _)) = (case AList.lookup (op =) bound_eqs varname of
  2478                                           SOME bound => bound
  2479                                         | NONE => raise TERM ("No bound equations found for " ^ varname, []))
  2480       | lift_var t = raise TERM ("Can not convert expression " ^ 
  2481                                  (Syntax.string_of_term ctxt t), [t])
  2482 
  2483     val _ $ vs = HOLogic.dest_Trueprop (Logic.strip_imp_concl goal')
  2484 
  2485     val bs = (HOLogic.dest_list #> map lift_var #> HOLogic.mk_list @{typ "float * float"}) vs
  2486     val map = [(@{cpat "?prec::nat"}, to_natc prec),
  2487                (@{cpat "?bs::(float * float) list"}, Thm.cterm_of (ProofContext.theory_of ctxt) bs)]
  2488   in rtac (Thm.instantiate ([], map) @{thm "uneq_approx"}) i thm end
  2489 
  2490   val eval_tac = CSUBGOAL (fn (ct, i) => rtac (eval_oracle ct) i)
  2491 
  2492   fun gen_eval_tac conv ctxt = CONVERSION (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt)
  2493                                THEN' rtac TrueI
  2494 
  2495 *}
  2496 
  2497 method_setup approximation = {* fn src => 
  2498   Method.syntax Args.term src #>
  2499   (fn (prec, ctxt) => let
  2500    in Method.SIMPLE_METHOD' (fn i =>
  2501      (DETERM (reify_uneq ctxt i)
  2502       THEN rule_uneq ctxt prec i
  2503       THEN Simplifier.asm_full_simp_tac bounded_by_simpset i 
  2504       THEN (TRY (filter_prems_tac (fn t => false) i))
  2505       THEN (gen_eval_tac eval_oracle ctxt) i))
  2506    end)
  2507 *} "real number approximation"
  2508 
  2509 end