src/HOL/Decision_Procs/Approximation.thy
 author nipkow Wed Jun 24 09:41:14 2009 +0200 (2009-06-24) changeset 31790 05c92381363c parent 31508 1ea1c70aba00 child 31809 06372924e86c permissions -rw-r--r--
corrected and unified thm names
```     1 (* Author:     Johannes Hoelzl <hoelzl@in.tum.de> 2008 / 2009 *)
```
```     2
```
```     3 header {* Prove Real Valued Inequalities by Computation *}
```
```     4
```
```     5 theory Approximation
```
```     6 imports Complex_Main Float Reflection Dense_Linear_Order Efficient_Nat
```
```     7 begin
```
```     8
```
```     9 section "Horner Scheme"
```
```    10
```
```    11 subsection {* Define auxiliary helper @{text horner} function *}
```
```    12
```
```    13 primrec horner :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real \<Rightarrow> real" where
```
```    14 "horner F G 0 i k x       = 0" |
```
```    15 "horner F G (Suc n) i k x = 1 / real k - x * horner F G n (F i) (G i k) x"
```
```    16
```
```    17 lemma horner_schema': fixes x :: real  and a :: "nat \<Rightarrow> real"
```
```    18   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)"
```
```    19 proof -
```
```    20   have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)" by auto
```
```    21   show ?thesis unfolding setsum_right_distrib shift_pow real_diff_def setsum_negf[symmetric] setsum_head_upt_Suc[OF zero_less_Suc]
```
```    22     setsum_reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n  *a n * x^n"] by auto
```
```    23 qed
```
```    24
```
```    25 lemma horner_schema: fixes f :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat" and F :: "nat \<Rightarrow> nat"
```
```    26   assumes f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
```
```    27   shows "horner F G n ((F ^^ j') s) (f j') x = (\<Sum> j = 0..< n. -1 ^ j * (1 / real (f (j' + j))) * x ^ j)"
```
```    28 proof (induct n arbitrary: i k j')
```
```    29   case (Suc n)
```
```    30
```
```    31   show ?case unfolding horner.simps Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc]
```
```    32     using horner_schema'[of "\<lambda> j. 1 / real (f (j' + j))"] by auto
```
```    33 qed auto
```
```    34
```
```    35 lemma horner_bounds':
```
```    36   assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
```
```    37   and lb_0: "\<And> i k x. lb 0 i k x = 0"
```
```    38   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)"
```
```    39   and ub_0: "\<And> i k x. ub 0 i k x = 0"
```
```    40   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)"
```
```    41   shows "real (lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') (real x) \<and>
```
```    42          horner F G n ((F ^^ j') s) (f j') (real x) \<le> real (ub n ((F ^^ j') s) (f j') x)"
```
```    43   (is "?lb n j' \<le> ?horner n j' \<and> ?horner n j' \<le> ?ub n j'")
```
```    44 proof (induct n arbitrary: j')
```
```    45   case 0 thus ?case unfolding lb_0 ub_0 horner.simps by auto
```
```    46 next
```
```    47   case (Suc n)
```
```    48   have "?lb (Suc n) j' \<le> ?horner (Suc n) j'" unfolding lb_Suc ub_Suc horner.simps real_of_float_sub diff_def
```
```    49   proof (rule add_mono)
```
```    50     show "real (lapprox_rat prec 1 (int (f j'))) \<le> 1 / real (f j')" using lapprox_rat[of prec 1  "int (f j')"] by auto
```
```    51     from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct2] `0 \<le> real x`
```
```    52     show "- real (x * ub n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) x) \<le> - (real x * horner F G n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) (real x))"
```
```    53       unfolding real_of_float_mult neg_le_iff_le by (rule mult_left_mono)
```
```    54   qed
```
```    55   moreover have "?horner (Suc n) j' \<le> ?ub (Suc n) j'" unfolding ub_Suc ub_Suc horner.simps real_of_float_sub diff_def
```
```    56   proof (rule add_mono)
```
```    57     show "1 / real (f j') \<le> real (rapprox_rat prec 1 (int (f j')))" using rapprox_rat[of 1 "int (f j')" prec] by auto
```
```    58     from Suc[where j'="Suc j'", unfolded funpow.simps comp_def f_Suc, THEN conjunct1] `0 \<le> real x`
```
```    59     show "- (real x * horner F G n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) (real x)) \<le>
```
```    60           - real (x * lb n (F ((F ^^ j') s)) (G ((F ^^ j') s) (f j')) x)"
```
```    61       unfolding real_of_float_mult neg_le_iff_le by (rule mult_left_mono)
```
```    62   qed
```
```    63   ultimately show ?case by blast
```
```    64 qed
```
```    65
```
```    66 subsection "Theorems for floating point functions implementing the horner scheme"
```
```    67
```
```    68 text {*
```
```    69
```
```    70 Here @{term_type "f :: nat \<Rightarrow> nat"} is the sequence defining the Taylor series, the coefficients are
```
```    71 all alternating and reciprocs. We use @{term G} and @{term F} to describe the computation of @{term f}.
```
```    72
```
```    73 *}
```
```    74
```
```    75 lemma horner_bounds: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
```
```    76   assumes "0 \<le> real x" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
```
```    77   and lb_0: "\<And> i k x. lb 0 i k x = 0"
```
```    78   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)"
```
```    79   and ub_0: "\<And> i k x. ub 0 i k x = 0"
```
```    80   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)"
```
```    81   shows "real (lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. -1 ^ j * (1 / real (f (j' + j))) * real x ^ j)" (is "?lb") and
```
```    82     "(\<Sum>j=0..<n. -1 ^ j * (1 / real (f (j' + j))) * (real x ^ j)) \<le> real (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
```
```    83 proof -
```
```    84   have "?lb  \<and> ?ub"
```
```    85     using horner_bounds'[where lb=lb, OF `0 \<le> real x` f_Suc lb_0 lb_Suc ub_0 ub_Suc]
```
```    86     unfolding horner_schema[where f=f, OF f_Suc] .
```
```    87   thus "?lb" and "?ub" by auto
```
```    88 qed
```
```    89
```
```    90 lemma horner_bounds_nonpos: fixes F :: "nat \<Rightarrow> nat" and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"
```
```    91   assumes "real x \<le> 0" and f_Suc: "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"
```
```    92   and lb_0: "\<And> i k x. lb 0 i k x = 0"
```
```    93   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)"
```
```    94   and ub_0: "\<And> i k x. ub 0 i k x = 0"
```
```    95   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)"
```
```    96   shows "real (lb n ((F ^^ j') s) (f j') x) \<le> (\<Sum>j=0..<n. (1 / real (f (j' + j))) * real x ^ j)" (is "?lb") and
```
```    97     "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * (real x ^ j)) \<le> real (ub n ((F ^^ j') s) (f j') x)" (is "?ub")
```
```    98 proof -
```
```    99   { fix x y z :: float have "x - y * z = x + - y * z"
```
```   100       by (cases x, cases y, cases z, simp add: plus_float.simps minus_float_def uminus_float.simps times_float.simps algebra_simps)
```
```   101   } note diff_mult_minus = this
```
```   102
```
```   103   { fix x :: float have "- (- x) = x" by (cases x, auto simp add: uminus_float.simps) } note minus_minus = this
```
```   104
```
```   105   have move_minus: "real (-x) = -1 * real x" by auto
```
```   106
```
```   107   have sum_eq: "(\<Sum>j=0..<n. (1 / real (f (j' + j))) * real x ^ j) =
```
```   108     (\<Sum>j = 0..<n. -1 ^ j * (1 / real (f (j' + j))) * real (- x) ^ j)"
```
```   109   proof (rule setsum_cong, simp)
```
```   110     fix j assume "j \<in> {0 ..< n}"
```
```   111     show "1 / real (f (j' + j)) * real x ^ j = -1 ^ j * (1 / real (f (j' + j))) * real (- x) ^ j"
```
```   112       unfolding move_minus power_mult_distrib real_mult_assoc[symmetric]
```
```   113       unfolding real_mult_commute unfolding real_mult_assoc[of "-1 ^ j", symmetric] power_mult_distrib[symmetric]
```
```   114       by auto
```
```   115   qed
```
```   116
```
```   117   have "0 \<le> real (-x)" using assms by auto
```
```   118   from horner_bounds[where G=G and F=F and f=f and s=s and prec=prec
```
```   119     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,
```
```   120     OF this f_Suc lb_0 refl ub_0 refl]
```
```   121   show "?lb" and "?ub" unfolding minus_minus sum_eq
```
```   122     by auto
```
```   123 qed
```
```   124
```
```   125 subsection {* Selectors for next even or odd number *}
```
```   126
```
```   127 text {*
```
```   128
```
```   129 The horner scheme computes alternating series. To get the upper and lower bounds we need to
```
```   130 guarantee to access a even or odd member. To do this we use @{term get_odd} and @{term get_even}.
```
```   131
```
```   132 *}
```
```   133
```
```   134 definition get_odd :: "nat \<Rightarrow> nat" where
```
```   135   "get_odd n = (if odd n then n else (Suc n))"
```
```   136
```
```   137 definition get_even :: "nat \<Rightarrow> nat" where
```
```   138   "get_even n = (if even n then n else (Suc n))"
```
```   139
```
```   140 lemma get_odd[simp]: "odd (get_odd n)" unfolding get_odd_def by (cases "odd n", auto)
```
```   141 lemma get_even[simp]: "even (get_even n)" unfolding get_even_def by (cases "even n", auto)
```
```   142 lemma get_odd_ex: "\<exists> k. Suc k = get_odd n \<and> odd (Suc k)"
```
```   143 proof (cases "odd n")
```
```   144   case True hence "0 < n" by (rule odd_pos)
```
```   145   from gr0_implies_Suc[OF this] obtain k where "Suc k = n" by auto
```
```   146   thus ?thesis unfolding get_odd_def if_P[OF True] using True[unfolded `Suc k = n`[symmetric]] by blast
```
```   147 next
```
```   148   case False hence "odd (Suc n)" by auto
```
```   149   thus ?thesis unfolding get_odd_def if_not_P[OF False] by blast
```
```   150 qed
```
```   151
```
```   152 lemma get_even_double: "\<exists>i. get_even n = 2 * i" using get_even[unfolded even_mult_two_ex] .
```
```   153 lemma get_odd_double: "\<exists>i. get_odd n = 2 * i + 1" using get_odd[unfolded odd_Suc_mult_two_ex] by auto
```
```   154
```
```   155 section "Power function"
```
```   156
```
```   157 definition float_power_bnds :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
```
```   158 "float_power_bnds n l u = (if odd n \<or> 0 < l then (l ^ n, u ^ n)
```
```   159                       else if u < 0         then (u ^ n, l ^ n)
```
```   160                                             else (0, (max (-l) u) ^ n))"
```
```   161
```
```   162 lemma float_power_bnds: assumes "(l1, u1) = float_power_bnds n l u" and "x \<in> {real l .. real u}"
```
```   163   shows "x ^ n \<in> {real l1..real u1}"
```
```   164 proof (cases "even n")
```
```   165   case True
```
```   166   show ?thesis
```
```   167   proof (cases "0 < l")
```
```   168     case True hence "odd n \<or> 0 < l" and "0 \<le> real l" unfolding less_float_def by auto
```
```   169     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
```
```   170     have "real l ^ n \<le> x ^ n" and "x ^ n \<le> real u ^ n " using `0 \<le> real l` and assms unfolding atLeastAtMost_iff using power_mono[of "real l" x] power_mono[of x "real u"] by auto
```
```   171     thus ?thesis using assms `0 < l` unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto
```
```   172   next
```
```   173     case False hence P: "\<not> (odd n \<or> 0 < l)" using `even n` by auto
```
```   174     show ?thesis
```
```   175     proof (cases "u < 0")
```
```   176       case True hence "0 \<le> - real u" and "- real u \<le> - x" and "0 \<le> - x" and "-x \<le> - real l" using assms unfolding less_float_def by auto
```
```   177       hence "real u ^ n \<le> x ^ n" and "x ^ n \<le> real l ^ n" using power_mono[of  "-x" "-real l" n] power_mono[of "-real u" "-x" n]
```
```   178 	unfolding power_minus_even[OF `even n`] by auto
```
```   179       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
```
```   180       ultimately show ?thesis using float_power by auto
```
```   181     next
```
```   182       case False
```
```   183       have "\<bar>x\<bar> \<le> real (max (-l) u)"
```
```   184       proof (cases "-l \<le> u")
```
```   185 	case True thus ?thesis unfolding max_def if_P[OF True] using assms unfolding le_float_def by auto
```
```   186       next
```
```   187 	case False thus ?thesis unfolding max_def if_not_P[OF False] using assms unfolding le_float_def by auto
```
```   188       qed
```
```   189       hence x_abs: "\<bar>x\<bar> \<le> \<bar>real (max (-l) u)\<bar>" by auto
```
```   190       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
```
```   191       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
```
```   192     qed
```
```   193   qed
```
```   194 next
```
```   195   case False hence "odd n \<or> 0 < l" by auto
```
```   196   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
```
```   197   have "real l ^ n \<le> x ^ n" and "x ^ n \<le> real u ^ n " using assms unfolding atLeastAtMost_iff using power_mono_odd[OF False] by auto
```
```   198   thus ?thesis unfolding atLeastAtMost_iff l1 u1 float_power less_float_def by auto
```
```   199 qed
```
```   200
```
```   201 lemma bnds_power: "\<forall> x l u. (l1, u1) = float_power_bnds n l u \<and> x \<in> {real l .. real u} \<longrightarrow> real l1 \<le> x ^ n \<and> x ^ n \<le> real u1"
```
```   202   using float_power_bnds by auto
```
```   203
```
```   204 section "Square root"
```
```   205
```
```   206 text {*
```
```   207
```
```   208 The square root computation is implemented as newton iteration. As first first step we use the
```
```   209 nearest power of two greater than the square root.
```
```   210
```
```   211 *}
```
```   212
```
```   213 fun sqrt_iteration :: "nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
```
```   214 "sqrt_iteration prec 0 (Float m e) = Float 1 ((e + bitlen m) div 2 + 1)" |
```
```   215 "sqrt_iteration prec (Suc m) x = (let y = sqrt_iteration prec m x
```
```   216                                   in Float 1 -1 * (y + float_divr prec x y))"
```
```   217
```
```   218 function ub_sqrt lb_sqrt :: "nat \<Rightarrow> float \<Rightarrow> float" where
```
```   219 "ub_sqrt prec x = (if 0 < x then (sqrt_iteration prec prec x)
```
```   220               else if x < 0 then - lb_sqrt prec (- x)
```
```   221                             else 0)" |
```
```   222 "lb_sqrt prec x = (if 0 < x then (float_divl prec x (sqrt_iteration prec prec x))
```
```   223               else if x < 0 then - ub_sqrt prec (- x)
```
```   224                             else 0)"
```
```   225 by pat_completeness auto
```
```   226 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)
```
```   227
```
```   228 declare lb_sqrt.simps[simp del]
```
```   229 declare ub_sqrt.simps[simp del]
```
```   230
```
```   231 lemma sqrt_ub_pos_pos_1:
```
```   232   assumes "sqrt x < b" and "0 < b" and "0 < x"
```
```   233   shows "sqrt x < (b + x / b)/2"
```
```   234 proof -
```
```   235   from assms have "0 < (b - sqrt x) ^ 2 " by simp
```
```   236   also have "\<dots> = b ^ 2 - 2 * b * sqrt x + (sqrt x) ^ 2" by algebra
```
```   237   also have "\<dots> = b ^ 2 - 2 * b * sqrt x + x" using assms by (simp add: real_sqrt_pow2)
```
```   238   finally have "0 < b ^ 2 - 2 * b * sqrt x + x" by assumption
```
```   239   hence "0 < b / 2 - sqrt x + x / (2 * b)" using assms
```
```   240     by (simp add: field_simps power2_eq_square)
```
```   241   thus ?thesis by (simp add: field_simps)
```
```   242 qed
```
```   243
```
```   244 lemma sqrt_iteration_bound: assumes "0 < real x"
```
```   245   shows "sqrt (real x) < real (sqrt_iteration prec n x)"
```
```   246 proof (induct n)
```
```   247   case 0
```
```   248   show ?case
```
```   249   proof (cases x)
```
```   250     case (Float m e)
```
```   251     hence "0 < m" using float_pos_m_pos[unfolded less_float_def] assms by auto
```
```   252     hence "0 < sqrt (real m)" by auto
```
```   253
```
```   254     have int_nat_bl: "int (nat (bitlen m)) = bitlen m" using bitlen_ge0 by auto
```
```   255
```
```   256     have "real x = (real m / 2^nat (bitlen m)) * pow2 (e + int (nat (bitlen m)))"
```
```   257       unfolding pow2_add pow2_int Float real_of_float_simp by auto
```
```   258     also have "\<dots> < 1 * pow2 (e + int (nat (bitlen m)))"
```
```   259     proof (rule mult_strict_right_mono, auto)
```
```   260       show "real m < 2^nat (bitlen m)" using bitlen_bounds[OF `0 < m`, THEN conjunct2]
```
```   261 	unfolding real_of_int_less_iff[of m, symmetric] by auto
```
```   262     qed
```
```   263     finally have "sqrt (real x) < sqrt (pow2 (e + bitlen m))" unfolding int_nat_bl by auto
```
```   264     also have "\<dots> \<le> pow2 ((e + bitlen m) div 2 + 1)"
```
```   265     proof -
```
```   266       let ?E = "e + bitlen m"
```
```   267       have E_mod_pow: "pow2 (?E mod 2) < 4"
```
```   268       proof (cases "?E mod 2 = 1")
```
```   269 	case True thus ?thesis by auto
```
```   270       next
```
```   271 	case False
```
```   272 	have "0 \<le> ?E mod 2" by auto
```
```   273 	have "?E mod 2 < 2" by auto
```
```   274 	from this[THEN zless_imp_add1_zle]
```
```   275 	have "?E mod 2 \<le> 0" using False by auto
```
```   276 	from xt1(5)[OF `0 \<le> ?E mod 2` this]
```
```   277 	show ?thesis by auto
```
```   278       qed
```
```   279       hence "sqrt (pow2 (?E mod 2)) < sqrt (2 * 2)" by auto
```
```   280       hence E_mod_pow: "sqrt (pow2 (?E mod 2)) < 2" unfolding real_sqrt_abs2 by auto
```
```   281
```
```   282       have E_eq: "pow2 ?E = pow2 (?E div 2 + ?E div 2 + ?E mod 2)" by auto
```
```   283       have "sqrt (pow2 ?E) = sqrt (pow2 (?E div 2) * pow2 (?E div 2) * pow2 (?E mod 2))"
```
```   284 	unfolding E_eq unfolding pow2_add ..
```
```   285       also have "\<dots> = pow2 (?E div 2) * sqrt (pow2 (?E mod 2))"
```
```   286 	unfolding real_sqrt_mult[of _ "pow2 (?E mod 2)"] real_sqrt_abs2 by auto
```
```   287       also have "\<dots> < pow2 (?E div 2) * 2"
```
```   288 	by (rule mult_strict_left_mono, auto intro: E_mod_pow)
```
```   289       also have "\<dots> = pow2 (?E div 2 + 1)" unfolding zadd_commute[of _ 1] pow2_add1 by auto
```
```   290       finally show ?thesis by auto
```
```   291     qed
```
```   292     finally show ?thesis
```
```   293       unfolding Float sqrt_iteration.simps real_of_float_simp by auto
```
```   294   qed
```
```   295 next
```
```   296   case (Suc n)
```
```   297   let ?b = "sqrt_iteration prec n x"
```
```   298   have "0 < sqrt (real x)" using `0 < real x` by auto
```
```   299   also have "\<dots> < real ?b" using Suc .
```
```   300   finally have "sqrt (real x) < (real ?b + real x / real ?b)/2" using sqrt_ub_pos_pos_1[OF Suc _ `0 < real x`] by auto
```
```   301   also have "\<dots> \<le> (real ?b + real (float_divr prec x ?b))/2" by (rule divide_right_mono, auto simp add: float_divr)
```
```   302   also have "\<dots> = real (Float 1 -1) * (real ?b + real (float_divr prec x ?b))" by auto
```
```   303   finally show ?case unfolding sqrt_iteration.simps Let_def real_of_float_mult real_of_float_add right_distrib .
```
```   304 qed
```
```   305
```
```   306 lemma sqrt_iteration_lower_bound: assumes "0 < real x"
```
```   307   shows "0 < real (sqrt_iteration prec n x)" (is "0 < ?sqrt")
```
```   308 proof -
```
```   309   have "0 < sqrt (real x)" using assms by auto
```
```   310   also have "\<dots> < ?sqrt" using sqrt_iteration_bound[OF assms] .
```
```   311   finally show ?thesis .
```
```   312 qed
```
```   313
```
```   314 lemma lb_sqrt_lower_bound: assumes "0 \<le> real x"
```
```   315   shows "0 \<le> real (lb_sqrt prec x)"
```
```   316 proof (cases "0 < x")
```
```   317   case True hence "0 < real x" and "0 \<le> x" using `0 \<le> real x` unfolding less_float_def le_float_def by auto
```
```   318   hence "0 < sqrt_iteration prec prec x" unfolding less_float_def using sqrt_iteration_lower_bound by auto
```
```   319   hence "0 \<le> real (float_divl prec x (sqrt_iteration prec prec x))" using float_divl_lower_bound[OF `0 \<le> x`] unfolding le_float_def by auto
```
```   320   thus ?thesis unfolding lb_sqrt.simps using True by auto
```
```   321 next
```
```   322   case False with `0 \<le> real x` have "real x = 0" unfolding less_float_def by auto
```
```   323   thus ?thesis unfolding lb_sqrt.simps less_float_def by auto
```
```   324 qed
```
```   325
```
```   326 lemma bnds_sqrt':
```
```   327   shows "sqrt (real x) \<in> { real (lb_sqrt prec x) .. real (ub_sqrt prec x) }"
```
```   328 proof -
```
```   329   { fix x :: float assume "0 < x"
```
```   330     hence "0 < real x" and "0 \<le> real x" unfolding less_float_def by auto
```
```   331     hence sqrt_gt0: "0 < sqrt (real x)" by auto
```
```   332     hence sqrt_ub: "sqrt (real x) < real (sqrt_iteration prec prec x)" using sqrt_iteration_bound by auto
```
```   333
```
```   334     have "real (float_divl prec x (sqrt_iteration prec prec x)) \<le>
```
```   335           real x / real (sqrt_iteration prec prec x)" by (rule float_divl)
```
```   336     also have "\<dots> < real x / sqrt (real x)"
```
```   337       by (rule divide_strict_left_mono[OF sqrt_ub `0 < real x`
```
```   338                mult_pos_pos[OF order_less_trans[OF sqrt_gt0 sqrt_ub] sqrt_gt0]])
```
```   339     also have "\<dots> = sqrt (real x)"
```
```   340       unfolding inverse_eq_iff_eq[of _ "sqrt (real x)", symmetric]
```
```   341 	        sqrt_divide_self_eq[OF `0 \<le> real x`, symmetric] by auto
```
```   342     finally have "real (lb_sqrt prec x) \<le> sqrt (real x)"
```
```   343       unfolding lb_sqrt.simps if_P[OF `0 < x`] by auto }
```
```   344   note lb = this
```
```   345
```
```   346   { fix x :: float assume "0 < x"
```
```   347     hence "0 < real x" unfolding less_float_def by auto
```
```   348     hence "0 < sqrt (real x)" by auto
```
```   349     hence "sqrt (real x) < real (sqrt_iteration prec prec x)"
```
```   350       using sqrt_iteration_bound by auto
```
```   351     hence "sqrt (real x) \<le> real (ub_sqrt prec x)"
```
```   352       unfolding ub_sqrt.simps if_P[OF `0 < x`] by auto }
```
```   353   note ub = this
```
```   354
```
```   355   show ?thesis
```
```   356   proof (cases "0 < x")
```
```   357     case True with lb ub show ?thesis by auto
```
```   358   next case False show ?thesis
```
```   359   proof (cases "real x = 0")
```
```   360     case True thus ?thesis
```
```   361       by (auto simp add: less_float_def lb_sqrt.simps ub_sqrt.simps)
```
```   362   next
```
```   363     case False with `\<not> 0 < x` have "x < 0" and "0 < -x"
```
```   364       by (auto simp add: less_float_def)
```
```   365
```
```   366     with `\<not> 0 < x`
```
```   367     show ?thesis using lb[OF `0 < -x`] ub[OF `0 < -x`]
```
```   368       by (auto simp add: real_sqrt_minus lb_sqrt.simps ub_sqrt.simps)
```
```   369   qed qed
```
```   370 qed
```
```   371
```
```   372 lemma bnds_sqrt: "\<forall> x lx ux. (l, u) = (lb_sqrt prec lx, ub_sqrt prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> sqrt x \<and> sqrt x \<le> real u"
```
```   373 proof ((rule allI) +, rule impI, erule conjE, rule conjI)
```
```   374   fix x lx ux
```
```   375   assume "(l, u) = (lb_sqrt prec lx, ub_sqrt prec ux)"
```
```   376     and x: "x \<in> {real lx .. real ux}"
```
```   377   hence l: "l = lb_sqrt prec lx " and u: "u = ub_sqrt prec ux" by auto
```
```   378
```
```   379   have "sqrt (real lx) \<le> sqrt x" using x by auto
```
```   380   from order_trans[OF _ this]
```
```   381   show "real l \<le> sqrt x" unfolding l using bnds_sqrt'[of lx prec] by auto
```
```   382
```
```   383   have "sqrt x \<le> sqrt (real ux)" using x by auto
```
```   384   from order_trans[OF this]
```
```   385   show "sqrt x \<le> real u" unfolding u using bnds_sqrt'[of ux prec] by auto
```
```   386 qed
```
```   387
```
```   388 section "Arcus tangens and \<pi>"
```
```   389
```
```   390 subsection "Compute arcus tangens series"
```
```   391
```
```   392 text {*
```
```   393
```
```   394 As first step we implement the computation of the arcus tangens series. This is only valid in the range
```
```   395 @{term "{-1 :: real .. 1}"}. This is used to compute \<pi> and then the entire arcus tangens.
```
```   396
```
```   397 *}
```
```   398
```
```   399 fun ub_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
```
```   400 and lb_arctan_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
```
```   401   "ub_arctan_horner prec 0 k x = 0"
```
```   402 | "ub_arctan_horner prec (Suc n) k x =
```
```   403     (rapprox_rat prec 1 (int k)) - x * (lb_arctan_horner prec n (k + 2) x)"
```
```   404 | "lb_arctan_horner prec 0 k x = 0"
```
```   405 | "lb_arctan_horner prec (Suc n) k x =
```
```   406     (lapprox_rat prec 1 (int k)) - x * (ub_arctan_horner prec n (k + 2) x)"
```
```   407
```
```   408 lemma arctan_0_1_bounds': assumes "0 \<le> real x" "real x \<le> 1" and "even n"
```
```   409   shows "arctan (real x) \<in> {real (x * lb_arctan_horner prec n 1 (x * x)) .. real (x * ub_arctan_horner prec (Suc n) 1 (x * x))}"
```
```   410 proof -
```
```   411   let "?c i" = "-1^i * (1 / real (i * 2 + 1) * real x ^ (i * 2 + 1))"
```
```   412   let "?S n" = "\<Sum> i=0..<n. ?c i"
```
```   413
```
```   414   have "0 \<le> real (x * x)" by auto
```
```   415   from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
```
```   416
```
```   417   have "arctan (real x) \<in> { ?S n .. ?S (Suc n) }"
```
```   418   proof (cases "real x = 0")
```
```   419     case False
```
```   420     hence "0 < real x" using `0 \<le> real x` by auto
```
```   421     hence prem: "0 < 1 / real (0 * 2 + (1::nat)) * real x ^ (0 * 2 + 1)" by auto
```
```   422
```
```   423     have "\<bar> real x \<bar> \<le> 1"  using `0 \<le> real x` `real x \<le> 1` by auto
```
```   424     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`]
```
```   425     show ?thesis unfolding arctan_series[OF `\<bar> real x \<bar> \<le> 1`] Suc_eq_plus1  .
```
```   426   qed auto
```
```   427   note arctan_bounds = this[unfolded atLeastAtMost_iff]
```
```   428
```
```   429   have F: "\<And>n. 2 * Suc n + 1 = 2 * n + 1 + 2" by auto
```
```   430
```
```   431   note bounds = horner_bounds[where s=1 and f="\<lambda>i. 2 * i + 1" and j'=0
```
```   432     and lb="\<lambda>n i k x. lb_arctan_horner prec n k x"
```
```   433     and ub="\<lambda>n i k x. ub_arctan_horner prec n k x",
```
```   434     OF `0 \<le> real (x*x)` F lb_arctan_horner.simps ub_arctan_horner.simps]
```
```   435
```
```   436   { have "real (x * lb_arctan_horner prec n 1 (x*x)) \<le> ?S n"
```
```   437       using bounds(1) `0 \<le> real x`
```
```   438       unfolding real_of_float_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
```
```   439       unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
```
```   440       by (auto intro!: mult_left_mono)
```
```   441     also have "\<dots> \<le> arctan (real x)" using arctan_bounds ..
```
```   442     finally have "real (x * lb_arctan_horner prec n 1 (x*x)) \<le> arctan (real x)" . }
```
```   443   moreover
```
```   444   { have "arctan (real x) \<le> ?S (Suc n)" using arctan_bounds ..
```
```   445     also have "\<dots> \<le> real (x * ub_arctan_horner prec (Suc n) 1 (x*x))"
```
```   446       using bounds(2)[of "Suc n"] `0 \<le> real x`
```
```   447       unfolding real_of_float_mult power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
```
```   448       unfolding real_mult_commute mult_commute[of _ "2::nat"] power_mult power2_eq_square[of "real x"]
```
```   449       by (auto intro!: mult_left_mono)
```
```   450     finally have "arctan (real x) \<le> real (x * ub_arctan_horner prec (Suc n) 1 (x*x))" . }
```
```   451   ultimately show ?thesis by auto
```
```   452 qed
```
```   453
```
```   454 lemma arctan_0_1_bounds: assumes "0 \<le> real x" "real x \<le> 1"
```
```   455   shows "arctan (real x) \<in> {real (x * lb_arctan_horner prec (get_even n) 1 (x * x)) .. real (x * ub_arctan_horner prec (get_odd n) 1 (x * x))}"
```
```   456 proof (cases "even n")
```
```   457   case True
```
```   458   obtain n' where "Suc n' = get_odd n" and "odd (Suc n')" using get_odd_ex by auto
```
```   459   hence "even n'" unfolding even_Suc by auto
```
```   460   have "arctan (real x) \<le> real (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
```
```   461     unfolding `Suc n' = get_odd n`[symmetric] using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even n'`] by auto
```
```   462   moreover
```
```   463   have "real (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (real x)"
```
```   464     unfolding get_even_def if_P[OF True] using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even n`] by auto
```
```   465   ultimately show ?thesis by auto
```
```   466 next
```
```   467   case False hence "0 < n" by (rule odd_pos)
```
```   468   from gr0_implies_Suc[OF this] obtain n' where "n = Suc n'" ..
```
```   469   from False[unfolded this even_Suc]
```
```   470   have "even n'" and "even (Suc (Suc n'))" by auto
```
```   471   have "get_odd n = Suc n'" unfolding get_odd_def if_P[OF False] using `n = Suc n'` .
```
```   472
```
```   473   have "arctan (real x) \<le> real (x * ub_arctan_horner prec (get_odd n) 1 (x * x))"
```
```   474     unfolding `get_odd n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even n'`] by auto
```
```   475   moreover
```
```   476   have "real (x * lb_arctan_horner prec (get_even n) 1 (x * x)) \<le> arctan (real x)"
```
```   477     unfolding get_even_def if_not_P[OF False] unfolding `n = Suc n'` using arctan_0_1_bounds'[OF `0 \<le> real x` `real x \<le> 1` `even (Suc (Suc n'))`] by auto
```
```   478   ultimately show ?thesis by auto
```
```   479 qed
```
```   480
```
```   481 subsection "Compute \<pi>"
```
```   482
```
```   483 definition ub_pi :: "nat \<Rightarrow> float" where
```
```   484   "ub_pi prec = (let A = rapprox_rat prec 1 5 ;
```
```   485                      B = lapprox_rat prec 1 239
```
```   486                  in ((Float 1 2) * ((Float 1 2) * A * (ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (A * A)) -
```
```   487                                                   B * (lb_arctan_horner prec (get_even (prec div 14 + 1)) 1 (B * B)))))"
```
```   488
```
```   489 definition lb_pi :: "nat \<Rightarrow> float" where
```
```   490   "lb_pi prec = (let A = lapprox_rat prec 1 5 ;
```
```   491                      B = rapprox_rat prec 1 239
```
```   492                  in ((Float 1 2) * ((Float 1 2) * A * (lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (A * A)) -
```
```   493                                                   B * (ub_arctan_horner prec (get_odd (prec div 14 + 1)) 1 (B * B)))))"
```
```   494
```
```   495 lemma pi_boundaries: "pi \<in> {real (lb_pi n) .. real (ub_pi n)}"
```
```   496 proof -
```
```   497   have machin_pi: "pi = 4 * (4 * arctan (1 / 5) - arctan (1 / 239))" unfolding machin[symmetric] by auto
```
```   498
```
```   499   { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" and "1 \<le> k" by auto
```
```   500     let ?k = "rapprox_rat prec 1 k"
```
```   501     have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
```
```   502
```
```   503     have "0 \<le> real ?k" by (rule order_trans[OF _ rapprox_rat], auto simp add: `0 \<le> k`)
```
```   504     have "real ?k \<le> 1" unfolding rapprox_rat.simps(2)[OF zero_le_one `0 < k`]
```
```   505       by (rule rapprox_posrat_le1, auto simp add: `0 < k` `1 \<le> k`)
```
```   506
```
```   507     have "1 / real k \<le> real ?k" using rapprox_rat[where x=1 and y=k] by auto
```
```   508     hence "arctan (1 / real k) \<le> arctan (real ?k)" by (rule arctan_monotone')
```
```   509     also have "\<dots> \<le> real (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))"
```
```   510       using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
```
```   511     finally have "arctan (1 / (real k)) \<le> real (?k * ub_arctan_horner prec (get_odd n) 1 (?k * ?k))" .
```
```   512   } note ub_arctan = this
```
```   513
```
```   514   { fix prec n :: nat fix k :: int assume "1 < k" hence "0 \<le> k" and "0 < k" by auto
```
```   515     let ?k = "lapprox_rat prec 1 k"
```
```   516     have "1 div k = 0" using div_pos_pos_trivial[OF _ `1 < k`] by auto
```
```   517     have "1 / real k \<le> 1" using `1 < k` by auto
```
```   518
```
```   519     have "\<And>n. 0 \<le> real ?k" using lapprox_rat_bottom[where x=1 and y=k, OF zero_le_one `0 < k`] by (auto simp add: `1 div k = 0`)
```
```   520     have "\<And>n. real ?k \<le> 1" using lapprox_rat by (rule order_trans, auto simp add: `1 / real k \<le> 1`)
```
```   521
```
```   522     have "real ?k \<le> 1 / real k" using lapprox_rat[where x=1 and y=k] by auto
```
```   523
```
```   524     have "real (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (real ?k)"
```
```   525       using arctan_0_1_bounds[OF `0 \<le> real ?k` `real ?k \<le> 1`] by auto
```
```   526     also have "\<dots> \<le> arctan (1 / real k)" using `real ?k \<le> 1 / real k` by (rule arctan_monotone')
```
```   527     finally have "real (?k * lb_arctan_horner prec (get_even n) 1 (?k * ?k)) \<le> arctan (1 / (real k))" .
```
```   528   } note lb_arctan = this
```
```   529
```
```   530   have "pi \<le> real (ub_pi n)"
```
```   531     unfolding ub_pi_def machin_pi Let_def real_of_float_mult real_of_float_sub unfolding Float_num
```
```   532     using lb_arctan[of 239] ub_arctan[of 5]
```
```   533     by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
```
```   534   moreover
```
```   535   have "real (lb_pi n) \<le> pi"
```
```   536     unfolding lb_pi_def machin_pi Let_def real_of_float_mult real_of_float_sub Float_num
```
```   537     using lb_arctan[of 5] ub_arctan[of 239]
```
```   538     by (auto intro!: mult_left_mono add_mono simp add: diff_minus simp del: lapprox_rat.simps rapprox_rat.simps)
```
```   539   ultimately show ?thesis by auto
```
```   540 qed
```
```   541
```
```   542 subsection "Compute arcus tangens in the entire domain"
```
```   543
```
```   544 function lb_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" and ub_arctan :: "nat \<Rightarrow> float \<Rightarrow> float" where
```
```   545   "lb_arctan prec x = (let ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x) ;
```
```   546                            lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)
```
```   547     in (if x < 0          then - ub_arctan prec (-x) else
```
```   548         if x \<le> Float 1 -1 then lb_horner x else
```
```   549         if x \<le> Float 1 1  then Float 1 1 * lb_horner (float_divl prec x (1 + ub_sqrt prec (1 + x * x)))
```
```   550                           else (let inv = float_divr prec 1 x
```
```   551                                 in if inv > 1 then 0
```
```   552                                               else lb_pi prec * Float 1 -1 - ub_horner inv)))"
```
```   553
```
```   554 | "ub_arctan prec x = (let lb_horner = \<lambda> x. x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x) ;
```
```   555                            ub_horner = \<lambda> x. x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)
```
```   556     in (if x < 0          then - lb_arctan prec (-x) else
```
```   557         if x \<le> Float 1 -1 then ub_horner x else
```
```   558         if x \<le> Float 1 1  then let y = float_divr prec x (1 + lb_sqrt prec (1 + x * x))
```
```   559                                in if y > 1 then ub_pi prec * Float 1 -1
```
```   560                                            else Float 1 1 * ub_horner y
```
```   561                           else ub_pi prec * Float 1 -1 - lb_horner (float_divl prec 1 x)))"
```
```   562 by pat_completeness auto
```
```   563 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)
```
```   564
```
```   565 declare ub_arctan_horner.simps[simp del]
```
```   566 declare lb_arctan_horner.simps[simp del]
```
```   567
```
```   568 lemma lb_arctan_bound': assumes "0 \<le> real x"
```
```   569   shows "real (lb_arctan prec x) \<le> arctan (real x)"
```
```   570 proof -
```
```   571   have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> real x` by auto
```
```   572   let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
```
```   573     and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
```
```   574
```
```   575   show ?thesis
```
```   576   proof (cases "x \<le> Float 1 -1")
```
```   577     case True hence "real x \<le> 1" unfolding le_float_def Float_num by auto
```
```   578     show ?thesis unfolding lb_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
```
```   579       using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
```
```   580   next
```
```   581     case False hence "0 < real x" unfolding le_float_def Float_num by auto
```
```   582     let ?R = "1 + sqrt (1 + real x * real x)"
```
```   583     let ?fR = "1 + ub_sqrt prec (1 + x * x)"
```
```   584     let ?DIV = "float_divl prec x ?fR"
```
```   585
```
```   586     have sqr_ge0: "0 \<le> 1 + real x * real x" using sum_power2_ge_zero[of 1 "real x", unfolded numeral_2_eq_2] by auto
```
```   587     hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
```
```   588
```
```   589     have "sqrt (real (1 + x * x)) \<le> real (ub_sqrt prec (1 + x * x))"
```
```   590       using bnds_sqrt'[of "1 + x * x"] by auto
```
```   591
```
```   592     hence "?R \<le> real ?fR" by auto
```
```   593     hence "0 < ?fR" and "0 < real ?fR" unfolding less_float_def using `0 < ?R` by auto
```
```   594
```
```   595     have monotone: "real (float_divl prec x ?fR) \<le> real x / ?R"
```
```   596     proof -
```
```   597       have "real ?DIV \<le> real x / real ?fR" by (rule float_divl)
```
```   598       also have "\<dots> \<le> real x / ?R" by (rule divide_left_mono[OF `?R \<le> real ?fR` `0 \<le> real x` mult_pos_pos[OF order_less_le_trans[OF divisor_gt0 `?R \<le> real ?fR`] divisor_gt0]])
```
```   599       finally show ?thesis .
```
```   600     qed
```
```   601
```
```   602     show ?thesis
```
```   603     proof (cases "x \<le> Float 1 1")
```
```   604       case True
```
```   605
```
```   606       have "real x \<le> sqrt (real (1 + x * x))" using real_sqrt_sum_squares_ge2[where x=1, unfolded numeral_2_eq_2] by auto
```
```   607       also have "\<dots> \<le> real (ub_sqrt prec (1 + x * x))"
```
```   608 	using bnds_sqrt'[of "1 + x * x"] by auto
```
```   609       finally have "real x \<le> real ?fR" by auto
```
```   610       moreover have "real ?DIV \<le> real x / real ?fR" by (rule float_divl)
```
```   611       ultimately have "real ?DIV \<le> 1" unfolding divide_le_eq_1_pos[OF `0 < real ?fR`, symmetric] by auto
```
```   612
```
```   613       have "0 \<le> real ?DIV" using float_divl_lower_bound[OF `0 \<le> x` `0 < ?fR`] unfolding le_float_def by auto
```
```   614
```
```   615       have "real (Float 1 1 * ?lb_horner ?DIV) \<le> 2 * arctan (real (float_divl prec x ?fR))" unfolding real_of_float_mult[of "Float 1 1"] Float_num
```
```   616 	using arctan_0_1_bounds[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
```
```   617       also have "\<dots> \<le> 2 * arctan (real x / ?R)"
```
```   618 	using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
```
```   619       also have "2 * arctan (real x / ?R) = arctan (real x)" using arctan_half[symmetric] unfolding numeral_2_eq_2 power_Suc2 power_0 real_mult_1 .
```
```   620       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] .
```
```   621     next
```
```   622       case False
```
```   623       hence "2 < real x" unfolding le_float_def Float_num by auto
```
```   624       hence "1 \<le> real x" by auto
```
```   625
```
```   626       let "?invx" = "float_divr prec 1 x"
```
```   627       have "0 \<le> arctan (real x)" using arctan_monotone'[OF `0 \<le> real x`] using arctan_tan[of 0, unfolded tan_zero] by auto
```
```   628
```
```   629       show ?thesis
```
```   630       proof (cases "1 < ?invx")
```
```   631 	case True
```
```   632 	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]
```
```   633 	  using `0 \<le> arctan (real x)` by auto
```
```   634       next
```
```   635 	case False
```
```   636 	hence "real ?invx \<le> 1" unfolding less_float_def by auto
```
```   637 	have "0 \<le> real ?invx" by (rule order_trans[OF _ float_divr], auto simp add: `0 \<le> real x`)
```
```   638
```
```   639 	have "1 / real x \<noteq> 0" and "0 < 1 / real x" using `0 < real x` by auto
```
```   640
```
```   641 	have "arctan (1 / real x) \<le> arctan (real ?invx)" unfolding real_of_float_1[symmetric] by (rule arctan_monotone', rule float_divr)
```
```   642 	also have "\<dots> \<le> real (?ub_horner ?invx)" using arctan_0_1_bounds[OF `0 \<le> real ?invx` `real ?invx \<le> 1`] by auto
```
```   643 	finally have "pi / 2 - real (?ub_horner ?invx) \<le> arctan (real x)"
```
```   644 	  using `0 \<le> arctan (real x)` arctan_inverse[OF `1 / real x \<noteq> 0`]
```
```   645 	  unfolding real_sgn_pos[OF `0 < 1 / real x`] le_diff_eq by auto
```
```   646 	moreover
```
```   647 	have "real (lb_pi prec * Float 1 -1) \<le> pi / 2" unfolding real_of_float_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
```
```   648 	ultimately
```
```   649 	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]
```
```   650 	  by auto
```
```   651       qed
```
```   652     qed
```
```   653   qed
```
```   654 qed
```
```   655
```
```   656 lemma ub_arctan_bound': assumes "0 \<le> real x"
```
```   657   shows "arctan (real x) \<le> real (ub_arctan prec x)"
```
```   658 proof -
```
```   659   have "\<not> x < 0" and "0 \<le> x" unfolding less_float_def le_float_def using `0 \<le> real x` by auto
```
```   660
```
```   661   let "?ub_horner x" = "x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (x * x)"
```
```   662     and "?lb_horner x" = "x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (x * x)"
```
```   663
```
```   664   show ?thesis
```
```   665   proof (cases "x \<le> Float 1 -1")
```
```   666     case True hence "real x \<le> 1" unfolding le_float_def Float_num by auto
```
```   667     show ?thesis unfolding ub_arctan.simps Let_def if_not_P[OF `\<not> x < 0`] if_P[OF True]
```
```   668       using arctan_0_1_bounds[OF `0 \<le> real x` `real x \<le> 1`] by auto
```
```   669   next
```
```   670     case False hence "0 < real x" unfolding le_float_def Float_num by auto
```
```   671     let ?R = "1 + sqrt (1 + real x * real x)"
```
```   672     let ?fR = "1 + lb_sqrt prec (1 + x * x)"
```
```   673     let ?DIV = "float_divr prec x ?fR"
```
```   674
```
```   675     have sqr_ge0: "0 \<le> 1 + real x * real x" using sum_power2_ge_zero[of 1 "real x", unfolded numeral_2_eq_2] by auto
```
```   676     hence "0 \<le> real (1 + x*x)" by auto
```
```   677
```
```   678     hence divisor_gt0: "0 < ?R" by (auto intro: add_pos_nonneg)
```
```   679
```
```   680     have "real (lb_sqrt prec (1 + x * x)) \<le> sqrt (real (1 + x * x))"
```
```   681       using bnds_sqrt'[of "1 + x * x"] by auto
```
```   682     hence "real ?fR \<le> ?R" by auto
```
```   683     have "0 < real ?fR" unfolding real_of_float_add real_of_float_1 by (rule order_less_le_trans[OF zero_less_one], auto simp add: lb_sqrt_lower_bound[OF `0 \<le> real (1 + x*x)`])
```
```   684
```
```   685     have monotone: "real x / ?R \<le> real (float_divr prec x ?fR)"
```
```   686     proof -
```
```   687       from divide_left_mono[OF `real ?fR \<le> ?R` `0 \<le> real x` mult_pos_pos[OF divisor_gt0 `0 < real ?fR`]]
```
```   688       have "real x / ?R \<le> real x / real ?fR" .
```
```   689       also have "\<dots> \<le> real ?DIV" by (rule float_divr)
```
```   690       finally show ?thesis .
```
```   691     qed
```
```   692
```
```   693     show ?thesis
```
```   694     proof (cases "x \<le> Float 1 1")
```
```   695       case True
```
```   696       show ?thesis
```
```   697       proof (cases "?DIV > 1")
```
```   698 	case True
```
```   699 	have "pi / 2 \<le> real (ub_pi prec * Float 1 -1)" unfolding real_of_float_mult Float_num times_divide_eq_right real_mult_1 using pi_boundaries by auto
```
```   700 	from order_less_le_trans[OF arctan_ubound this, THEN less_imp_le]
```
```   701 	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] .
```
```   702       next
```
```   703 	case False
```
```   704 	hence "real ?DIV \<le> 1" unfolding less_float_def by auto
```
```   705
```
```   706 	have "0 \<le> real x / ?R" using `0 \<le> real x` `0 < ?R` unfolding real_0_le_divide_iff by auto
```
```   707 	hence "0 \<le> real ?DIV" using monotone by (rule order_trans)
```
```   708
```
```   709 	have "arctan (real x) = 2 * arctan (real x / ?R)" using arctan_half unfolding numeral_2_eq_2 power_Suc2 power_0 real_mult_1 .
```
```   710 	also have "\<dots> \<le> 2 * arctan (real ?DIV)"
```
```   711 	  using arctan_monotone'[OF monotone] by (auto intro!: mult_left_mono)
```
```   712 	also have "\<dots> \<le> real (Float 1 1 * ?ub_horner ?DIV)" unfolding real_of_float_mult[of "Float 1 1"] Float_num
```
```   713 	  using arctan_0_1_bounds[OF `0 \<le> real ?DIV` `real ?DIV \<le> 1`] by auto
```
```   714 	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] .
```
```   715       qed
```
```   716     next
```
```   717       case False
```
```   718       hence "2 < real x" unfolding le_float_def Float_num by auto
```
```   719       hence "1 \<le> real x" by auto
```
```   720       hence "0 < real x" by auto
```
```   721       hence "0 < x" unfolding less_float_def by auto
```
```   722
```
```   723       let "?invx" = "float_divl prec 1 x"
```
```   724       have "0 \<le> arctan (real x)" using arctan_monotone'[OF `0 \<le> real x`] using arctan_tan[of 0, unfolded tan_zero] by auto
```
```   725
```
```   726       have "real ?invx \<le> 1" unfolding less_float_def by (rule order_trans[OF float_divl], auto simp add: `1 \<le> real x` divide_le_eq_1_pos[OF `0 < real x`])
```
```   727       have "0 \<le> real ?invx" unfolding real_of_float_0[symmetric] by (rule float_divl_lower_bound[unfolded le_float_def], auto simp add: `0 < x`)
```
```   728
```
```   729       have "1 / real x \<noteq> 0" and "0 < 1 / real x" using `0 < real x` by auto
```
```   730
```
```   731       have "real (?lb_horner ?invx) \<le> arctan (real ?invx)" using arctan_0_1_bounds[OF `0 \<le> real ?invx` `real ?invx \<le> 1`] by auto
```
```   732       also have "\<dots> \<le> arctan (1 / real x)" unfolding real_of_float_1[symmetric] by (rule arctan_monotone', rule float_divl)
```
```   733       finally have "arctan (real x) \<le> pi / 2 - real (?lb_horner ?invx)"
```
```   734 	using `0 \<le> arctan (real x)` arctan_inverse[OF `1 / real x \<noteq> 0`]
```
```   735 	unfolding real_sgn_pos[OF `0 < 1 / real x`] le_diff_eq by auto
```
```   736       moreover
```
```   737       have "pi / 2 \<le> real (ub_pi prec * Float 1 -1)" unfolding real_of_float_mult Float_num times_divide_eq_right mult_1_right using pi_boundaries by auto
```
```   738       ultimately
```
```   739       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]
```
```   740 	by auto
```
```   741     qed
```
```   742   qed
```
```   743 qed
```
```   744
```
```   745 lemma arctan_boundaries:
```
```   746   "arctan (real x) \<in> {real (lb_arctan prec x) .. real (ub_arctan prec x)}"
```
```   747 proof (cases "0 \<le> x")
```
```   748   case True hence "0 \<le> real x" unfolding le_float_def by auto
```
```   749   show ?thesis using ub_arctan_bound'[OF `0 \<le> real x`] lb_arctan_bound'[OF `0 \<le> real x`] unfolding atLeastAtMost_iff by auto
```
```   750 next
```
```   751   let ?mx = "-x"
```
```   752   case False hence "x < 0" and "0 \<le> real ?mx" unfolding le_float_def less_float_def by auto
```
```   753   hence bounds: "real (lb_arctan prec ?mx) \<le> arctan (real ?mx) \<and> arctan (real ?mx) \<le> real (ub_arctan prec ?mx)"
```
```   754     using ub_arctan_bound'[OF `0 \<le> real ?mx`] lb_arctan_bound'[OF `0 \<le> real ?mx`] by auto
```
```   755   show ?thesis unfolding real_of_float_minus arctan_minus lb_arctan.simps[where x=x] ub_arctan.simps[where x=x] Let_def if_P[OF `x < 0`]
```
```   756     unfolding atLeastAtMost_iff using bounds[unfolded real_of_float_minus arctan_minus] by auto
```
```   757 qed
```
```   758
```
```   759 lemma bnds_arctan: "\<forall> x lx ux. (l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> arctan x \<and> arctan x \<le> real u"
```
```   760 proof (rule allI, rule allI, rule allI, rule impI)
```
```   761   fix x lx ux
```
```   762   assume "(l, u) = (lb_arctan prec lx, ub_arctan prec ux) \<and> x \<in> {real lx .. real ux}"
```
```   763   hence l: "lb_arctan prec lx = l " and u: "ub_arctan prec ux = u" and x: "x \<in> {real lx .. real ux}" by auto
```
```   764
```
```   765   { from arctan_boundaries[of lx prec, unfolded l]
```
```   766     have "real l \<le> arctan (real lx)" by (auto simp del: lb_arctan.simps)
```
```   767     also have "\<dots> \<le> arctan x" using x by (auto intro: arctan_monotone')
```
```   768     finally have "real l \<le> arctan x" .
```
```   769   } moreover
```
```   770   { have "arctan x \<le> arctan (real ux)" using x by (auto intro: arctan_monotone')
```
```   771     also have "\<dots> \<le> real u" using arctan_boundaries[of ux prec, unfolded u] by (auto simp del: ub_arctan.simps)
```
```   772     finally have "arctan x \<le> real u" .
```
```   773   } ultimately show "real l \<le> arctan x \<and> arctan x \<le> real u" ..
```
```   774 qed
```
```   775
```
```   776 section "Sinus and Cosinus"
```
```   777
```
```   778 subsection "Compute the cosinus and sinus series"
```
```   779
```
```   780 fun ub_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
```
```   781 and lb_sin_cos_aux :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
```
```   782   "ub_sin_cos_aux prec 0 i k x = 0"
```
```   783 | "ub_sin_cos_aux prec (Suc n) i k x =
```
```   784     (rapprox_rat prec 1 (int k)) - x * (lb_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
```
```   785 | "lb_sin_cos_aux prec 0 i k x = 0"
```
```   786 | "lb_sin_cos_aux prec (Suc n) i k x =
```
```   787     (lapprox_rat prec 1 (int k)) - x * (ub_sin_cos_aux prec n (i + 2) (k * i * (i + 1)) x)"
```
```   788
```
```   789 lemma cos_aux:
```
```   790   shows "real (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (real x)^(2 * i))" (is "?lb")
```
```   791   and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * (real x)^(2 * i)) \<le> real (ub_sin_cos_aux prec n 1 1 (x * x))" (is "?ub")
```
```   792 proof -
```
```   793   have "0 \<le> real (x * x)" unfolding real_of_float_mult by auto
```
```   794   let "?f n" = "fact (2 * n)"
```
```   795
```
```   796   { fix n
```
```   797     have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
```
```   798     have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 1 * (((\<lambda>i. i + 2) ^^ n) 1 + 1)"
```
```   799       unfolding F by auto } note f_eq = this
```
```   800
```
```   801   from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
```
```   802     OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
```
```   803   show "?lb" and "?ub" by (auto simp add: power_mult power2_eq_square[of "real x"])
```
```   804 qed
```
```   805
```
```   806 lemma cos_boundaries: assumes "0 \<le> real x" and "real x \<le> pi / 2"
```
```   807   shows "cos (real x) \<in> {real (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) .. real (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"
```
```   808 proof (cases "real x = 0")
```
```   809   case False hence "real x \<noteq> 0" by auto
```
```   810   hence "0 < x" and "0 < real x" using `0 \<le> real x` unfolding less_float_def by auto
```
```   811   have "0 < x * x" using `0 < x` unfolding less_float_def real_of_float_mult real_of_float_0
```
```   812     using mult_pos_pos[where a="real x" and b="real x"] by auto
```
```   813
```
```   814   { fix x n have "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i))) * x ^ (2 * i))
```
```   815     = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * x ^ i)" (is "?sum = ?ifsum")
```
```   816   proof -
```
```   817     have "?sum = ?sum + (\<Sum> j = 0 ..< n. 0)" by auto
```
```   818     also have "\<dots> =
```
```   819       (\<Sum> j = 0 ..< n. -1 ^ ((2 * j) div 2) / (real (fact (2 * j))) * x ^(2 * j)) + (\<Sum> j = 0 ..< n. 0)" by auto
```
```   820     also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. if even i then -1 ^ (i div 2) / (real (fact i)) * x ^ i else 0)"
```
```   821       unfolding sum_split_even_odd ..
```
```   822     also have "\<dots> = (\<Sum> i = 0 ..< 2 * n. (if even i then -1 ^ (i div 2) / (real (fact i)) else 0) * x ^ i)"
```
```   823       by (rule setsum_cong2) auto
```
```   824     finally show ?thesis by assumption
```
```   825   qed } note morph_to_if_power = this
```
```   826
```
```   827
```
```   828   { fix n :: nat assume "0 < n"
```
```   829     hence "0 < 2 * n" by auto
```
```   830     obtain t where "0 < t" and "t < real x" and
```
```   831       cos_eq: "cos (real x) = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * (real x) ^ i)
```
```   832       + (cos (t + 1/2 * real (2 * n) * pi) / real (fact (2*n))) * (real x)^(2*n)"
```
```   833       (is "_ = ?SUM + ?rest / ?fact * ?pow")
```
```   834       using Maclaurin_cos_expansion2[OF `0 < real x` `0 < 2 * n`] by auto
```
```   835
```
```   836     have "cos t * -1^n = cos t * cos (real n * pi) + sin t * sin (real n * pi)" by auto
```
```   837     also have "\<dots> = cos (t + real n * pi)"  using cos_add by auto
```
```   838     also have "\<dots> = ?rest" by auto
```
```   839     finally have "cos t * -1^n = ?rest" .
```
```   840     moreover
```
```   841     have "t \<le> pi / 2" using `t < real x` and `real x \<le> pi / 2` by auto
```
```   842     hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
```
```   843     ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
```
```   844
```
```   845     have "0 < ?fact" by auto
```
```   846     have "0 < ?pow" using `0 < real x` by auto
```
```   847
```
```   848     {
```
```   849       assume "even n"
```
```   850       have "real (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> ?SUM"
```
```   851 	unfolding morph_to_if_power[symmetric] using cos_aux by auto
```
```   852       also have "\<dots> \<le> cos (real x)"
```
```   853       proof -
```
```   854 	from even[OF `even n`] `0 < ?fact` `0 < ?pow`
```
```   855 	have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
```
```   856 	thus ?thesis unfolding cos_eq by auto
```
```   857       qed
```
```   858       finally have "real (lb_sin_cos_aux prec n 1 1 (x * x)) \<le> cos (real x)" .
```
```   859     } note lb = this
```
```   860
```
```   861     {
```
```   862       assume "odd n"
```
```   863       have "cos (real x) \<le> ?SUM"
```
```   864       proof -
```
```   865 	from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
```
```   866 	have "0 \<le> (- ?rest) / ?fact * ?pow"
```
```   867 	  by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
```
```   868 	thus ?thesis unfolding cos_eq by auto
```
```   869       qed
```
```   870       also have "\<dots> \<le> real (ub_sin_cos_aux prec n 1 1 (x * x))"
```
```   871 	unfolding morph_to_if_power[symmetric] using cos_aux by auto
```
```   872       finally have "cos (real x) \<le> real (ub_sin_cos_aux prec n 1 1 (x * x))" .
```
```   873     } note ub = this and lb
```
```   874   } note ub = this(1) and lb = this(2)
```
```   875
```
```   876   have "cos (real x) \<le> real (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
```
```   877   moreover have "real (lb_sin_cos_aux prec (get_even n) 1 1 (x * x)) \<le> cos (real x)"
```
```   878   proof (cases "0 < get_even n")
```
```   879     case True show ?thesis using lb[OF True get_even] .
```
```   880   next
```
```   881     case False
```
```   882     hence "get_even n = 0" by auto
```
```   883     have "- (pi / 2) \<le> real x" by (rule order_trans[OF _ `0 < real x`[THEN less_imp_le]], auto)
```
```   884     with `real x \<le> pi / 2`
```
```   885     show ?thesis unfolding `get_even n = 0` lb_sin_cos_aux.simps real_of_float_minus real_of_float_0 using cos_ge_zero by auto
```
```   886   qed
```
```   887   ultimately show ?thesis by auto
```
```   888 next
```
```   889   case True
```
```   890   show ?thesis
```
```   891   proof (cases "n = 0")
```
```   892     case True
```
```   893     thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `real x = 0` lapprox_rat[where x="-1" and y=1] by auto
```
```   894   next
```
```   895     case False with not0_implies_Suc obtain m where "n = Suc m" by blast
```
```   896     thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `real x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
```
```   897   qed
```
```   898 qed
```
```   899
```
```   900 lemma sin_aux: assumes "0 \<le> real x"
```
```   901   shows "real (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> (\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (real x)^(2 * i + 1))" (is "?lb")
```
```   902   and "(\<Sum> i=0..<n. -1^i * (1/real (fact (2 * i + 1))) * (real x)^(2 * i + 1)) \<le> real (x * ub_sin_cos_aux prec n 2 1 (x * x))" (is "?ub")
```
```   903 proof -
```
```   904   have "0 \<le> real (x * x)" unfolding real_of_float_mult by auto
```
```   905   let "?f n" = "fact (2 * n + 1)"
```
```   906
```
```   907   { fix n
```
```   908     have F: "\<And>m. ((\<lambda>i. i + 2) ^^ n) m = m + 2 * n" by (induct n arbitrary: m, auto)
```
```   909     have "?f (Suc n) = ?f n * ((\<lambda>i. i + 2) ^^ n) 2 * (((\<lambda>i. i + 2) ^^ n) 2 + 1)"
```
```   910       unfolding F by auto } note f_eq = this
```
```   911
```
```   912   from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
```
```   913     OF `0 \<le> real (x * x)` f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
```
```   914   show "?lb" and "?ub" using `0 \<le> real x` unfolding real_of_float_mult
```
```   915     unfolding power_add power_one_right real_mult_assoc[symmetric] setsum_left_distrib[symmetric]
```
```   916     unfolding real_mult_commute
```
```   917     by (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real x"])
```
```   918 qed
```
```   919
```
```   920 lemma sin_boundaries: assumes "0 \<le> real x" and "real x \<le> pi / 2"
```
```   921   shows "sin (real x) \<in> {real (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) .. real (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))}"
```
```   922 proof (cases "real x = 0")
```
```   923   case False hence "real x \<noteq> 0" by auto
```
```   924   hence "0 < x" and "0 < real x" using `0 \<le> real x` unfolding less_float_def by auto
```
```   925   have "0 < x * x" using `0 < x` unfolding less_float_def real_of_float_mult real_of_float_0
```
```   926     using mult_pos_pos[where a="real x" and b="real x"] by auto
```
```   927
```
```   928   { 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))
```
```   929     = (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * x ^ i)" (is "?SUM = _")
```
```   930     proof -
```
```   931       have pow: "!!i. x ^ (2 * i + 1) = x * x ^ (2 * i)" by auto
```
```   932       have "?SUM = (\<Sum> j = 0 ..< n. 0) + ?SUM" by auto
```
```   933       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)"
```
```   934 	unfolding sum_split_even_odd ..
```
```   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 	by (rule setsum_cong2) auto
```
```   937       finally show ?thesis by assumption
```
```   938     qed } note setsum_morph = this
```
```   939
```
```   940   { fix n :: nat assume "0 < n"
```
```   941     hence "0 < 2 * n + 1" by auto
```
```   942     obtain t where "0 < t" and "t < real x" and
```
```   943       sin_eq: "sin (real x) = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)
```
```   944       + (sin (t + 1/2 * real (2 * n + 1) * pi) / real (fact (2*n + 1))) * (real x)^(2*n + 1)"
```
```   945       (is "_ = ?SUM + ?rest / ?fact * ?pow")
```
```   946       using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < real x`] by auto
```
```   947
```
```   948     have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add left_distrib right_distrib by auto
```
```   949     moreover
```
```   950     have "t \<le> pi / 2" using `t < real x` and `real x \<le> pi / 2` by auto
```
```   951     hence "0 \<le> cos t" using `0 < t` and cos_ge_zero by auto
```
```   952     ultimately have even: "even n \<Longrightarrow> 0 \<le> ?rest" and odd: "odd n \<Longrightarrow> 0 \<le> - ?rest " by auto
```
```   953
```
```   954     have "0 < ?fact" by (rule real_of_nat_fact_gt_zero)
```
```   955     have "0 < ?pow" using `0 < real x` by (rule zero_less_power)
```
```   956
```
```   957     {
```
```   958       assume "even n"
```
```   959       have "real (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le>
```
```   960             (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)"
```
```   961 	using sin_aux[OF `0 \<le> real x`] unfolding setsum_morph[symmetric] by auto
```
```   962       also have "\<dots> \<le> ?SUM" by auto
```
```   963       also have "\<dots> \<le> sin (real x)"
```
```   964       proof -
```
```   965 	from even[OF `even n`] `0 < ?fact` `0 < ?pow`
```
```   966 	have "0 \<le> (?rest / ?fact) * ?pow" by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
```
```   967 	thus ?thesis unfolding sin_eq by auto
```
```   968       qed
```
```   969       finally have "real (x * lb_sin_cos_aux prec n 2 1 (x * x)) \<le> sin (real x)" .
```
```   970     } note lb = this
```
```   971
```
```   972     {
```
```   973       assume "odd n"
```
```   974       have "sin (real x) \<le> ?SUM"
```
```   975       proof -
```
```   976 	from `0 < ?fact` and `0 < ?pow` and odd[OF `odd n`]
```
```   977 	have "0 \<le> (- ?rest) / ?fact * ?pow"
```
```   978 	  by (metis mult_nonneg_nonneg divide_nonneg_pos less_imp_le)
```
```   979 	thus ?thesis unfolding sin_eq by auto
```
```   980       qed
```
```   981       also have "\<dots> \<le> (\<Sum> i = 0 ..< 2 * n. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)"
```
```   982 	 by auto
```
```   983       also have "\<dots> \<le> real (x * ub_sin_cos_aux prec n 2 1 (x * x))"
```
```   984 	using sin_aux[OF `0 \<le> real x`] unfolding setsum_morph[symmetric] by auto
```
```   985       finally have "sin (real x) \<le> real (x * ub_sin_cos_aux prec n 2 1 (x * x))" .
```
```   986     } note ub = this and lb
```
```   987   } note ub = this(1) and lb = this(2)
```
```   988
```
```   989   have "sin (real x) \<le> real (x * ub_sin_cos_aux prec (get_odd n) 2 1 (x * x))" using ub[OF odd_pos[OF get_odd] get_odd] .
```
```   990   moreover have "real (x * lb_sin_cos_aux prec (get_even n) 2 1 (x * x)) \<le> sin (real x)"
```
```   991   proof (cases "0 < get_even n")
```
```   992     case True show ?thesis using lb[OF True get_even] .
```
```   993   next
```
```   994     case False
```
```   995     hence "get_even n = 0" by auto
```
```   996     with `real x \<le> pi / 2` `0 \<le> real x`
```
```   997     show ?thesis unfolding `get_even n = 0` ub_sin_cos_aux.simps real_of_float_minus real_of_float_0 using sin_ge_zero by auto
```
```   998   qed
```
```   999   ultimately show ?thesis by auto
```
```  1000 next
```
```  1001   case True
```
```  1002   show ?thesis
```
```  1003   proof (cases "n = 0")
```
```  1004     case True
```
```  1005     thus ?thesis unfolding `n = 0` get_even_def get_odd_def using `real x = 0` lapprox_rat[where x="-1" and y=1] by auto
```
```  1006   next
```
```  1007     case False with not0_implies_Suc obtain m where "n = Suc m" by blast
```
```  1008     thus ?thesis unfolding `n = Suc m` get_even_def get_odd_def using `real x = 0` rapprox_rat[where x=1 and y=1] lapprox_rat[where x=1 and y=1] by (cases "even (Suc m)", auto)
```
```  1009   qed
```
```  1010 qed
```
```  1011
```
```  1012 subsection "Compute the cosinus in the entire domain"
```
```  1013
```
```  1014 definition lb_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
```
```  1015 "lb_cos prec x = (let
```
```  1016     horner = \<lambda> x. lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x) ;
```
```  1017     half = \<lambda> x. if x < 0 then - 1 else Float 1 1 * x * x - 1
```
```  1018   in if x < Float 1 -1 then horner x
```
```  1019 else if x < 1          then half (horner (x * Float 1 -1))
```
```  1020                        else half (half (horner (x * Float 1 -2))))"
```
```  1021
```
```  1022 definition ub_cos :: "nat \<Rightarrow> float \<Rightarrow> float" where
```
```  1023 "ub_cos prec x = (let
```
```  1024     horner = \<lambda> x. ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x) ;
```
```  1025     half = \<lambda> x. Float 1 1 * x * x - 1
```
```  1026   in if x < Float 1 -1 then horner x
```
```  1027 else if x < 1          then half (horner (x * Float 1 -1))
```
```  1028                        else half (half (horner (x * Float 1 -2))))"
```
```  1029
```
```  1030 lemma lb_cos: assumes "0 \<le> real x" and "real x \<le> pi"
```
```  1031   shows "cos (real x) \<in> {real (lb_cos prec x) .. real (ub_cos prec x)}" (is "?cos x \<in> { real (?lb x) .. real (?ub x) }")
```
```  1032 proof -
```
```  1033   { fix x :: real
```
```  1034     have "cos x = cos (x / 2 + x / 2)" by auto
```
```  1035     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"
```
```  1036       unfolding cos_add by auto
```
```  1037     also have "\<dots> = 2 * cos (x / 2) * cos (x / 2) - 1" by algebra
```
```  1038     finally have "cos x = 2 * cos (x / 2) * cos (x / 2) - 1" .
```
```  1039   } note x_half = this[symmetric]
```
```  1040
```
```  1041   have "\<not> x < 0" using `0 \<le> real x` unfolding less_float_def by auto
```
```  1042   let "?ub_horner x" = "ub_sin_cos_aux prec (get_odd (prec div 4 + 1)) 1 1 (x * x)"
```
```  1043   let "?lb_horner x" = "lb_sin_cos_aux prec (get_even (prec div 4 + 1)) 1 1 (x * x)"
```
```  1044   let "?ub_half x" = "Float 1 1 * x * x - 1"
```
```  1045   let "?lb_half x" = "if x < 0 then - 1 else Float 1 1 * x * x - 1"
```
```  1046
```
```  1047   show ?thesis
```
```  1048   proof (cases "x < Float 1 -1")
```
```  1049     case True hence "real x \<le> pi / 2" unfolding less_float_def using pi_ge_two by auto
```
```  1050     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
```
```  1051       using cos_boundaries[OF `0 \<le> real x` `real x \<le> pi / 2`] .
```
```  1052   next
```
```  1053     case False
```
```  1054     { fix y x :: float let ?x2 = "real (x * Float 1 -1)"
```
```  1055       assume "real y \<le> cos ?x2" and "-pi \<le> real x" and "real x \<le> pi"
```
```  1056       hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding real_of_float_mult Float_num by auto
```
```  1057       hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
```
```  1058
```
```  1059       have "real (?lb_half y) \<le> cos (real x)"
```
```  1060       proof (cases "y < 0")
```
```  1061 	case True show ?thesis using cos_ge_minus_one unfolding if_P[OF True] by auto
```
```  1062       next
```
```  1063 	case False
```
```  1064 	hence "0 \<le> real y" unfolding less_float_def by auto
```
```  1065 	from mult_mono[OF `real y \<le> cos ?x2` `real y \<le> cos ?x2` `0 \<le> cos ?x2` this]
```
```  1066 	have "real y * real y \<le> cos ?x2 * cos ?x2" .
```
```  1067 	hence "2 * real y * real y \<le> 2 * cos ?x2 * cos ?x2" by auto
```
```  1068 	hence "2 * real y * real y - 1 \<le> 2 * cos (real x / 2) * cos (real x / 2) - 1" unfolding Float_num real_of_float_mult by auto
```
```  1069 	thus ?thesis unfolding if_not_P[OF False] x_half Float_num real_of_float_mult real_of_float_sub by auto
```
```  1070       qed
```
```  1071     } note lb_half = this
```
```  1072
```
```  1073     { fix y x :: float let ?x2 = "real (x * Float 1 -1)"
```
```  1074       assume ub: "cos ?x2 \<le> real y" and "- pi \<le> real x" and "real x \<le> pi"
```
```  1075       hence "- (pi / 2) \<le> ?x2" and "?x2 \<le> pi / 2" using pi_ge_two unfolding real_of_float_mult Float_num by auto
```
```  1076       hence "0 \<le> cos ?x2" by (rule cos_ge_zero)
```
```  1077
```
```  1078       have "cos (real x) \<le> real (?ub_half y)"
```
```  1079       proof -
```
```  1080 	have "0 \<le> real y" using `0 \<le> cos ?x2` ub by (rule order_trans)
```
```  1081 	from mult_mono[OF ub ub this `0 \<le> cos ?x2`]
```
```  1082 	have "cos ?x2 * cos ?x2 \<le> real y * real y" .
```
```  1083 	hence "2 * cos ?x2 * cos ?x2 \<le> 2 * real y * real y" by auto
```
```  1084 	hence "2 * cos (real x / 2) * cos (real x / 2) - 1 \<le> 2 * real y * real y - 1" unfolding Float_num real_of_float_mult by auto
```
```  1085 	thus ?thesis unfolding x_half real_of_float_mult Float_num real_of_float_sub by auto
```
```  1086       qed
```
```  1087     } note ub_half = this
```
```  1088
```
```  1089     let ?x2 = "x * Float 1 -1"
```
```  1090     let ?x4 = "x * Float 1 -1 * Float 1 -1"
```
```  1091
```
```  1092     have "-pi \<le> real x" using pi_ge_zero[THEN le_imp_neg_le, unfolded minus_zero] `0 \<le> real x` by (rule order_trans)
```
```  1093
```
```  1094     show ?thesis
```
```  1095     proof (cases "x < 1")
```
```  1096       case True hence "real x \<le> 1" unfolding less_float_def by auto
```
```  1097       have "0 \<le> real ?x2" and "real ?x2 \<le> pi / 2" using pi_ge_two `0 \<le> real x` unfolding real_of_float_mult Float_num using assms by auto
```
```  1098       from cos_boundaries[OF this]
```
```  1099       have lb: "real (?lb_horner ?x2) \<le> ?cos ?x2" and ub: "?cos ?x2 \<le> real (?ub_horner ?x2)" by auto
```
```  1100
```
```  1101       have "real (?lb x) \<le> ?cos x"
```
```  1102       proof -
```
```  1103 	from lb_half[OF lb `-pi \<le> real x` `real x \<le> pi`]
```
```  1104 	show ?thesis unfolding lb_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto
```
```  1105       qed
```
```  1106       moreover have "?cos x \<le> real (?ub x)"
```
```  1107       proof -
```
```  1108 	from ub_half[OF ub `-pi \<le> real x` `real x \<le> pi`]
```
```  1109 	show ?thesis unfolding ub_cos_def[where x=x] Let_def using `\<not> x < 0` `\<not> x < Float 1 -1` `x < 1` by auto
```
```  1110       qed
```
```  1111       ultimately show ?thesis by auto
```
```  1112     next
```
```  1113       case False
```
```  1114       have "0 \<le> real ?x4" and "real ?x4 \<le> pi / 2" using pi_ge_two `0 \<le> real x` `real x \<le> pi` unfolding real_of_float_mult Float_num by auto
```
```  1115       from cos_boundaries[OF this]
```
```  1116       have lb: "real (?lb_horner ?x4) \<le> ?cos ?x4" and ub: "?cos ?x4 \<le> real (?ub_horner ?x4)" by auto
```
```  1117
```
```  1118       have eq_4: "?x2 * Float 1 -1 = x * Float 1 -2" by (cases x, auto simp add: times_float.simps)
```
```  1119
```
```  1120       have "real (?lb x) \<le> ?cos x"
```
```  1121       proof -
```
```  1122 	have "-pi \<le> real ?x2" and "real ?x2 \<le> pi" unfolding real_of_float_mult Float_num using pi_ge_two `0 \<le> real x` `real x \<le> pi` by auto
```
```  1123 	from lb_half[OF lb_half[OF lb this] `-pi \<le> real x` `real x \<le> pi`, unfolded eq_4]
```
```  1124 	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 .
```
```  1125       qed
```
```  1126       moreover have "?cos x \<le> real (?ub x)"
```
```  1127       proof -
```
```  1128 	have "-pi \<le> real ?x2" and "real ?x2 \<le> pi" unfolding real_of_float_mult Float_num using pi_ge_two `0 \<le> real x` `real x \<le> pi` by auto
```
```  1129 	from ub_half[OF ub_half[OF ub this] `-pi \<le> real x` `real x \<le> pi`, unfolded eq_4]
```
```  1130 	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 .
```
```  1131       qed
```
```  1132       ultimately show ?thesis by auto
```
```  1133     qed
```
```  1134   qed
```
```  1135 qed
```
```  1136
```
```  1137 lemma lb_cos_minus: assumes "-pi \<le> real x" and "real x \<le> 0"
```
```  1138   shows "cos (real (-x)) \<in> {real (lb_cos prec (-x)) .. real (ub_cos prec (-x))}"
```
```  1139 proof -
```
```  1140   have "0 \<le> real (-x)" and "real (-x) \<le> pi" using `-pi \<le> real x` `real x \<le> 0` by auto
```
```  1141   from lb_cos[OF this] show ?thesis .
```
```  1142 qed
```
```  1143
```
```  1144 definition bnds_cos :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float * float" where
```
```  1145 "bnds_cos prec lx ux = (let
```
```  1146     lpi = round_down prec (lb_pi prec) ;
```
```  1147     upi = round_up prec (ub_pi prec) ;
```
```  1148     k = floor_fl (float_divr prec (lx + lpi) (2 * lpi)) ;
```
```  1149     lx = lx - k * 2 * (if k < 0 then lpi else upi) ;
```
```  1150     ux = ux - k * 2 * (if k < 0 then upi else lpi)
```
```  1151   in   if - lpi \<le> lx \<and> ux \<le> 0    then (lb_cos prec (-lx), ub_cos prec (-ux))
```
```  1152   else if 0 \<le> lx \<and> ux \<le> lpi      then (lb_cos prec ux, ub_cos prec lx)
```
```  1153   else if - lpi \<le> lx \<and> ux \<le> lpi  then (min (lb_cos prec (-lx)) (lb_cos prec ux), Float 1 0)
```
```  1154   else if 0 \<le> lx \<and> ux \<le> 2 * lpi  then (Float -1 0, max (ub_cos prec lx) (ub_cos prec (- (ux - 2 * lpi))))
```
```  1155   else if -2 * lpi \<le> lx \<and> ux \<le> 0 then (Float -1 0, max (ub_cos prec (lx + 2 * lpi)) (ub_cos prec (-ux)))
```
```  1156                                  else (Float -1 0, Float 1 0))"
```
```  1157
```
```  1158 lemma floor_int:
```
```  1159   obtains k :: int where "real k = real (floor_fl f)"
```
```  1160 proof -
```
```  1161   assume *: "\<And> k :: int. real k = real (floor_fl f) \<Longrightarrow> thesis"
```
```  1162   obtain m e where fl: "Float m e = floor_fl f" by (cases "floor_fl f", auto)
```
```  1163   from floor_pos_exp[OF this]
```
```  1164   have "real (m* 2^(nat e)) = real (floor_fl f)"
```
```  1165     by (auto simp add: fl[symmetric] real_of_float_def pow2_def)
```
```  1166   from *[OF this] show thesis by blast
```
```  1167 qed
```
```  1168
```
```  1169 lemma float_remove_real_numeral[simp]: "real (number_of k :: float) = number_of k"
```
```  1170 proof -
```
```  1171   have "real (number_of k :: float) = real k"
```
```  1172     unfolding number_of_float_def real_of_float_def pow2_def by auto
```
```  1173   also have "\<dots> = real (number_of k :: int)"
```
```  1174     by (simp add: number_of_is_id)
```
```  1175   finally show ?thesis by auto
```
```  1176 qed
```
```  1177
```
```  1178 lemma cos_periodic_nat[simp]: fixes n :: nat shows "cos (x + real n * 2 * pi) = cos x"
```
```  1179 proof (induct n arbitrary: x)
```
```  1180   case (Suc n)
```
```  1181   have split_pi_off: "x + real (Suc n) * 2 * pi = (x + real n * 2 * pi) + 2 * pi"
```
```  1182     unfolding Suc_eq_plus1 real_of_nat_add real_of_one real_add_mult_distrib by auto
```
```  1183   show ?case unfolding split_pi_off using Suc by auto
```
```  1184 qed auto
```
```  1185
```
```  1186 lemma cos_periodic_int[simp]: fixes i :: int shows "cos (x + real i * 2 * pi) = cos x"
```
```  1187 proof (cases "0 \<le> i")
```
```  1188   case True hence i_nat: "real i = real (nat i)" by auto
```
```  1189   show ?thesis unfolding i_nat by auto
```
```  1190 next
```
```  1191   case False hence i_nat: "real i = - real (nat (-i))" by auto
```
```  1192   have "cos x = cos (x + real i * 2 * pi - real i * 2 * pi)" by auto
```
```  1193   also have "\<dots> = cos (x + real i * 2 * pi)"
```
```  1194     unfolding i_nat mult_minus_left diff_minus_eq_add by (rule cos_periodic_nat)
```
```  1195   finally show ?thesis by auto
```
```  1196 qed
```
```  1197
```
```  1198 lemma bnds_cos: "\<forall> x lx ux. (l, u) = bnds_cos prec lx ux \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> cos x \<and> cos x \<le> real u"
```
```  1199 proof ((rule allI | rule impI | erule conjE) +)
```
```  1200   fix x lx ux
```
```  1201   assume bnds: "(l, u) = bnds_cos prec lx ux" and x: "x \<in> {real lx .. real ux}"
```
```  1202
```
```  1203   let ?lpi = "round_down prec (lb_pi prec)"
```
```  1204   let ?upi = "round_up prec (ub_pi prec)"
```
```  1205   let ?k = "floor_fl (float_divr prec (lx + ?lpi) (2 * ?lpi))"
```
```  1206   let ?lx = "lx - ?k * 2 * (if ?k < 0 then ?lpi else ?upi)"
```
```  1207   let ?ux = "ux - ?k * 2 * (if ?k < 0 then ?upi else ?lpi)"
```
```  1208
```
```  1209   obtain k :: int where k: "real k = real ?k" using floor_int .
```
```  1210
```
```  1211   have upi: "pi \<le> real ?upi" and lpi: "real ?lpi \<le> pi"
```
```  1212     using round_up[of "ub_pi prec" prec] pi_boundaries[of prec]
```
```  1213           round_down[of prec "lb_pi prec"] by auto
```
```  1214   hence "real ?lx \<le> x - real k * 2 * pi \<and> x - real k * 2 * pi \<le> real ?ux"
```
```  1215     using x by (cases "k = 0") (auto intro!: add_mono
```
```  1216                 simp add: real_diff_def k[symmetric] less_float_def)
```
```  1217   note lx = this[THEN conjunct1] and ux = this[THEN conjunct2]
```
```  1218   hence lx_less_ux: "real ?lx \<le> real ?ux" by (rule order_trans)
```
```  1219
```
```  1220   { assume "- ?lpi \<le> ?lx" and x_le_0: "x - real k * 2 * pi \<le> 0"
```
```  1221     with lpi[THEN le_imp_neg_le] lx
```
```  1222     have pi_lx: "- pi \<le> real ?lx" and lx_0: "real ?lx \<le> 0"
```
```  1223       by (simp_all add: le_float_def)
```
```  1224
```
```  1225     have "real (lb_cos prec (- ?lx)) \<le> cos (real (- ?lx))"
```
```  1226       using lb_cos_minus[OF pi_lx lx_0] by simp
```
```  1227     also have "\<dots> \<le> cos (x + real (-k) * 2 * pi)"
```
```  1228       using cos_monotone_minus_pi_0'[OF pi_lx lx x_le_0]
```
```  1229       by (simp only: real_of_float_minus real_of_int_minus
```
```  1230 	cos_minus real_diff_def mult_minus_left)
```
```  1231     finally have "real (lb_cos prec (- ?lx)) \<le> cos x"
```
```  1232       unfolding cos_periodic_int . }
```
```  1233   note negative_lx = this
```
```  1234
```
```  1235   { assume "0 \<le> ?lx" and pi_x: "x - real k * 2 * pi \<le> pi"
```
```  1236     with lx
```
```  1237     have pi_lx: "real ?lx \<le> pi" and lx_0: "0 \<le> real ?lx"
```
```  1238       by (auto simp add: le_float_def)
```
```  1239
```
```  1240     have "cos (x + real (-k) * 2 * pi) \<le> cos (real ?lx)"
```
```  1241       using cos_monotone_0_pi'[OF lx_0 lx pi_x]
```
```  1242       by (simp only: real_of_float_minus real_of_int_minus
```
```  1243 	cos_minus real_diff_def mult_minus_left)
```
```  1244     also have "\<dots> \<le> real (ub_cos prec ?lx)"
```
```  1245       using lb_cos[OF lx_0 pi_lx] by simp
```
```  1246     finally have "cos x \<le> real (ub_cos prec ?lx)"
```
```  1247       unfolding cos_periodic_int . }
```
```  1248   note positive_lx = this
```
```  1249
```
```  1250   { assume pi_x: "- pi \<le> x - real k * 2 * pi" and "?ux \<le> 0"
```
```  1251     with ux
```
```  1252     have pi_ux: "- pi \<le> real ?ux" and ux_0: "real ?ux \<le> 0"
```
```  1253       by (simp_all add: le_float_def)
```
```  1254
```
```  1255     have "cos (x + real (-k) * 2 * pi) \<le> cos (real (- ?ux))"
```
```  1256       using cos_monotone_minus_pi_0'[OF pi_x ux ux_0]
```
```  1257       by (simp only: real_of_float_minus real_of_int_minus
```
```  1258 	  cos_minus real_diff_def mult_minus_left)
```
```  1259     also have "\<dots> \<le> real (ub_cos prec (- ?ux))"
```
```  1260       using lb_cos_minus[OF pi_ux ux_0, of prec] by simp
```
```  1261     finally have "cos x \<le> real (ub_cos prec (- ?ux))"
```
```  1262       unfolding cos_periodic_int . }
```
```  1263   note negative_ux = this
```
```  1264
```
```  1265   { assume "?ux \<le> ?lpi" and x_ge_0: "0 \<le> x - real k * 2 * pi"
```
```  1266     with lpi ux
```
```  1267     have pi_ux: "real ?ux \<le> pi" and ux_0: "0 \<le> real ?ux"
```
```  1268       by (simp_all add: le_float_def)
```
```  1269
```
```  1270     have "real (lb_cos prec ?ux) \<le> cos (real ?ux)"
```
```  1271       using lb_cos[OF ux_0 pi_ux] by simp
```
```  1272     also have "\<dots> \<le> cos (x + real (-k) * 2 * pi)"
```
```  1273       using cos_monotone_0_pi'[OF x_ge_0 ux pi_ux]
```
```  1274       by (simp only: real_of_float_minus real_of_int_minus
```
```  1275 	cos_minus real_diff_def mult_minus_left)
```
```  1276     finally have "real (lb_cos prec ?ux) \<le> cos x"
```
```  1277       unfolding cos_periodic_int . }
```
```  1278   note positive_ux = this
```
```  1279
```
```  1280   show "real l \<le> cos x \<and> cos x \<le> real u"
```
```  1281   proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> 0")
```
```  1282     case True with bnds
```
```  1283     have l: "l = lb_cos prec (-?lx)"
```
```  1284       and u: "u = ub_cos prec (-?ux)"
```
```  1285       by (auto simp add: bnds_cos_def Let_def)
```
```  1286
```
```  1287     from True lpi[THEN le_imp_neg_le] lx ux
```
```  1288     have "- pi \<le> x - real k * 2 * pi"
```
```  1289       and "x - real k * 2 * pi \<le> 0"
```
```  1290       by (auto simp add: le_float_def)
```
```  1291     with True negative_ux negative_lx
```
```  1292     show ?thesis unfolding l u by simp
```
```  1293   next case False note 1 = this show ?thesis
```
```  1294   proof (cases "0 \<le> ?lx \<and> ?ux \<le> ?lpi")
```
```  1295     case True with bnds 1
```
```  1296     have l: "l = lb_cos prec ?ux"
```
```  1297       and u: "u = ub_cos prec ?lx"
```
```  1298       by (auto simp add: bnds_cos_def Let_def)
```
```  1299
```
```  1300     from True lpi lx ux
```
```  1301     have "0 \<le> x - real k * 2 * pi"
```
```  1302       and "x - real k * 2 * pi \<le> pi"
```
```  1303       by (auto simp add: le_float_def)
```
```  1304     with True positive_ux positive_lx
```
```  1305     show ?thesis unfolding l u by simp
```
```  1306   next case False note 2 = this show ?thesis
```
```  1307   proof (cases "- ?lpi \<le> ?lx \<and> ?ux \<le> ?lpi")
```
```  1308     case True note Cond = this with bnds 1 2
```
```  1309     have l: "l = min (lb_cos prec (-?lx)) (lb_cos prec ?ux)"
```
```  1310       and u: "u = Float 1 0"
```
```  1311       by (auto simp add: bnds_cos_def Let_def)
```
```  1312
```
```  1313     show ?thesis unfolding u l using negative_lx positive_ux Cond
```
```  1314       by (cases "x - real k * 2 * pi < 0", simp_all add: real_of_float_min)
```
```  1315   next case False note 3 = this show ?thesis
```
```  1316   proof (cases "0 \<le> ?lx \<and> ?ux \<le> 2 * ?lpi")
```
```  1317     case True note Cond = this with bnds 1 2 3
```
```  1318     have l: "l = Float -1 0"
```
```  1319       and u: "u = max (ub_cos prec ?lx) (ub_cos prec (- (?ux - 2 * ?lpi)))"
```
```  1320       by (auto simp add: bnds_cos_def Let_def)
```
```  1321
```
```  1322     have "cos x \<le> real u"
```
```  1323     proof (cases "x - real k * 2 * pi < pi")
```
```  1324       case True hence "x - real k * 2 * pi \<le> pi" by simp
```
```  1325       from positive_lx[OF Cond[THEN conjunct1] this]
```
```  1326       show ?thesis unfolding u by (simp add: real_of_float_max)
```
```  1327     next
```
```  1328       case False hence "pi \<le> x - real k * 2 * pi" by simp
```
```  1329       hence pi_x: "- pi \<le> x - real k * 2 * pi - 2 * pi" by simp
```
```  1330
```
```  1331       have "real ?ux \<le> 2 * pi" using Cond lpi by (auto simp add: le_float_def)
```
```  1332       hence "x - real k * 2 * pi - 2 * pi \<le> 0" using ux by simp
```
```  1333
```
```  1334       have ux_0: "real (?ux - 2 * ?lpi) \<le> 0"
```
```  1335 	using Cond by (auto simp add: le_float_def)
```
```  1336
```
```  1337       from 2 and Cond have "\<not> ?ux \<le> ?lpi" by auto
```
```  1338       hence "- ?lpi \<le> ?ux - 2 * ?lpi" by (auto simp add: le_float_def)
```
```  1339       hence pi_ux: "- pi \<le> real (?ux - 2 * ?lpi)"
```
```  1340 	using lpi[THEN le_imp_neg_le] by (auto simp add: le_float_def)
```
```  1341
```
```  1342       have x_le_ux: "x - real k * 2 * pi - 2 * pi \<le> real (?ux - 2 * ?lpi)"
```
```  1343 	using ux lpi by auto
```
```  1344
```
```  1345       have "cos x = cos (x + real (-k) * 2 * pi + real (-1 :: int) * 2 * pi)"
```
```  1346 	unfolding cos_periodic_int ..
```
```  1347       also have "\<dots> \<le> cos (real (?ux - 2 * ?lpi))"
```
```  1348 	using cos_monotone_minus_pi_0'[OF pi_x x_le_ux ux_0]
```
```  1349 	by (simp only: real_of_float_minus real_of_int_minus real_of_one
```
```  1350 	    number_of_Min real_diff_def mult_minus_left real_mult_1)
```
```  1351       also have "\<dots> = cos (real (- (?ux - 2 * ?lpi)))"
```
```  1352 	unfolding real_of_float_minus cos_minus ..
```
```  1353       also have "\<dots> \<le> real (ub_cos prec (- (?ux - 2 * ?lpi)))"
```
```  1354 	using lb_cos_minus[OF pi_ux ux_0] by simp
```
```  1355       finally show ?thesis unfolding u by (simp add: real_of_float_max)
```
```  1356     qed
```
```  1357     thus ?thesis unfolding l by auto
```
```  1358   next case False note 4 = this show ?thesis
```
```  1359   proof (cases "-2 * ?lpi \<le> ?lx \<and> ?ux \<le> 0")
```
```  1360     case True note Cond = this with bnds 1 2 3 4
```
```  1361     have l: "l = Float -1 0"
```
```  1362       and u: "u = max (ub_cos prec (?lx + 2 * ?lpi)) (ub_cos prec (-?ux))"
```
```  1363       by (auto simp add: bnds_cos_def Let_def)
```
```  1364
```
```  1365     have "cos x \<le> real u"
```
```  1366     proof (cases "-pi < x - real k * 2 * pi")
```
```  1367       case True hence "-pi \<le> x - real k * 2 * pi" by simp
```
```  1368       from negative_ux[OF this Cond[THEN conjunct2]]
```
```  1369       show ?thesis unfolding u by (simp add: real_of_float_max)
```
```  1370     next
```
```  1371       case False hence "x - real k * 2 * pi \<le> -pi" by simp
```
```  1372       hence pi_x: "x - real k * 2 * pi + 2 * pi \<le> pi" by simp
```
```  1373
```
```  1374       have "-2 * pi \<le> real ?lx" using Cond lpi by (auto simp add: le_float_def)
```
```  1375
```
```  1376       hence "0 \<le> x - real k * 2 * pi + 2 * pi" using lx by simp
```
```  1377
```
```  1378       have lx_0: "0 \<le> real (?lx + 2 * ?lpi)"
```
```  1379 	using Cond lpi by (auto simp add: le_float_def)
```
```  1380
```
```  1381       from 1 and Cond have "\<not> -?lpi \<le> ?lx" by auto
```
```  1382       hence "?lx + 2 * ?lpi \<le> ?lpi" by (auto simp add: le_float_def)
```
```  1383       hence pi_lx: "real (?lx + 2 * ?lpi) \<le> pi"
```
```  1384 	using lpi[THEN le_imp_neg_le] by (auto simp add: le_float_def)
```
```  1385
```
```  1386       have lx_le_x: "real (?lx + 2 * ?lpi) \<le> x - real k * 2 * pi + 2 * pi"
```
```  1387 	using lx lpi by auto
```
```  1388
```
```  1389       have "cos x = cos (x + real (-k) * 2 * pi + real (1 :: int) * 2 * pi)"
```
```  1390 	unfolding cos_periodic_int ..
```
```  1391       also have "\<dots> \<le> cos (real (?lx + 2 * ?lpi))"
```
```  1392 	using cos_monotone_0_pi'[OF lx_0 lx_le_x pi_x]
```
```  1393 	by (simp only: real_of_float_minus real_of_int_minus real_of_one
```
```  1394 	  number_of_Min real_diff_def mult_minus_left real_mult_1)
```
```  1395       also have "\<dots> \<le> real (ub_cos prec (?lx + 2 * ?lpi))"
```
```  1396 	using lb_cos[OF lx_0 pi_lx] by simp
```
```  1397       finally show ?thesis unfolding u by (simp add: real_of_float_max)
```
```  1398     qed
```
```  1399     thus ?thesis unfolding l by auto
```
```  1400   next
```
```  1401     case False with bnds 1 2 3 4 show ?thesis by (auto simp add: bnds_cos_def Let_def)
```
```  1402   qed qed qed qed qed
```
```  1403 qed
```
```  1404
```
```  1405 section "Exponential function"
```
```  1406
```
```  1407 subsection "Compute the series of the exponential function"
```
```  1408
```
```  1409 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
```
```  1410 "ub_exp_horner prec 0 i k x       = 0" |
```
```  1411 "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" |
```
```  1412 "lb_exp_horner prec 0 i k x       = 0" |
```
```  1413 "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"
```
```  1414
```
```  1415 lemma bnds_exp_horner: assumes "real x \<le> 0"
```
```  1416   shows "exp (real x) \<in> { real (lb_exp_horner prec (get_even n) 1 1 x) .. real (ub_exp_horner prec (get_odd n) 1 1 x) }"
```
```  1417 proof -
```
```  1418   { fix n
```
```  1419     have F: "\<And> m. ((\<lambda>i. i + 1) ^^ n) m = n + m" by (induct n, auto)
```
```  1420     have "fact (Suc n) = fact n * ((\<lambda>i. i + 1) ^^ n) 1" unfolding F by auto } note f_eq = this
```
```  1421
```
```  1422   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,
```
```  1423     OF assms f_eq lb_exp_horner.simps ub_exp_horner.simps]
```
```  1424
```
```  1425   { have "real (lb_exp_horner prec (get_even n) 1 1 x) \<le> (\<Sum>j = 0..<get_even n. 1 / real (fact j) * real x ^ j)"
```
```  1426       using bounds(1) by auto
```
```  1427     also have "\<dots> \<le> exp (real x)"
```
```  1428     proof -
```
```  1429       obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp (real x) = (\<Sum>m = 0..<get_even n. (real x) ^ m / real (fact m)) + exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
```
```  1430 	using Maclaurin_exp_le by blast
```
```  1431       moreover have "0 \<le> exp t / real (fact (get_even n)) * (real x) ^ (get_even n)"
```
```  1432 	by (auto intro!: mult_nonneg_nonneg divide_nonneg_pos simp add: get_even zero_le_even_power exp_gt_zero)
```
```  1433       ultimately show ?thesis
```
```  1434 	using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
```
```  1435     qed
```
```  1436     finally have "real (lb_exp_horner prec (get_even n) 1 1 x) \<le> exp (real x)" .
```
```  1437   } moreover
```
```  1438   {
```
```  1439     have x_less_zero: "real x ^ get_odd n \<le> 0"
```
```  1440     proof (cases "real x = 0")
```
```  1441       case True
```
```  1442       have "(get_odd n) \<noteq> 0" using get_odd[THEN odd_pos] by auto
```
```  1443       thus ?thesis unfolding True power_0_left by auto
```
```  1444     next
```
```  1445       case False hence "real x < 0" using `real x \<le> 0` by auto
```
```  1446       show ?thesis by (rule less_imp_le, auto simp add: power_less_zero_eq get_odd `real x < 0`)
```
```  1447     qed
```
```  1448
```
```  1449     obtain t where "\<bar>t\<bar> \<le> \<bar>real x\<bar>" and "exp (real x) = (\<Sum>m = 0..<get_odd n. (real x) ^ m / real (fact m)) + exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n)"
```
```  1450       using Maclaurin_exp_le by blast
```
```  1451     moreover have "exp t / real (fact (get_odd n)) * (real x) ^ (get_odd n) \<le> 0"
```
```  1452       by (auto intro!: mult_nonneg_nonpos divide_nonpos_pos simp add: x_less_zero exp_gt_zero)
```
```  1453     ultimately have "exp (real x) \<le> (\<Sum>j = 0..<get_odd n. 1 / real (fact j) * real x ^ j)"
```
```  1454       using get_odd exp_gt_zero by (auto intro!: pordered_cancel_semiring_class.mult_nonneg_nonneg)
```
```  1455     also have "\<dots> \<le> real (ub_exp_horner prec (get_odd n) 1 1 x)"
```
```  1456       using bounds(2) by auto
```
```  1457     finally have "exp (real x) \<le> real (ub_exp_horner prec (get_odd n) 1 1 x)" .
```
```  1458   } ultimately show ?thesis by auto
```
```  1459 qed
```
```  1460
```
```  1461 subsection "Compute the exponential function on the entire domain"
```
```  1462
```
```  1463 function ub_exp :: "nat \<Rightarrow> float \<Rightarrow> float" and lb_exp :: "nat \<Rightarrow> float \<Rightarrow> float" where
```
```  1464 "lb_exp prec x = (if 0 < x then float_divl prec 1 (ub_exp prec (-x))
```
```  1465              else let
```
```  1466                 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)
```
```  1467              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))
```
```  1468                            else horner x)" |
```
```  1469 "ub_exp prec x = (if 0 < x    then float_divr prec 1 (lb_exp prec (-x))
```
```  1470              else if x < - 1  then (case floor_fl x of (Float m e) \<Rightarrow>
```
```  1471                                     (ub_exp_horner prec (get_odd (prec + 2)) 1 1 (float_divr prec x (- Float m e))) ^ (nat (-m) * 2 ^ nat e))
```
```  1472                               else ub_exp_horner prec (get_odd (prec + 2)) 1 1 x)"
```
```  1473 by pat_completeness auto
```
```  1474 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)
```
```  1475
```
```  1476 lemma exp_m1_ge_quarter: "(1 / 4 :: real) \<le> exp (- 1)"
```
```  1477 proof -
```
```  1478   have eq4: "4 = Suc (Suc (Suc (Suc 0)))" by auto
```
```  1479
```
```  1480   have "1 / 4 = real (Float 1 -2)" unfolding Float_num by auto
```
```  1481   also have "\<dots> \<le> real (lb_exp_horner 1 (get_even 4) 1 1 (- 1))"
```
```  1482     unfolding get_even_def eq4
```
```  1483     by (auto simp add: lapprox_posrat_def rapprox_posrat_def normfloat.simps)
```
```  1484   also have "\<dots> \<le> exp (real (- 1 :: float))" using bnds_exp_horner[where x="- 1"] by auto
```
```  1485   finally show ?thesis unfolding real_of_float_minus real_of_float_1 .
```
```  1486 qed
```
```  1487
```
```  1488 lemma lb_exp_pos: assumes "\<not> 0 < x" shows "0 < lb_exp prec x"
```
```  1489 proof -
```
```  1490   let "?lb_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
```
```  1491   let "?horner x" = "let  y = ?lb_horner x  in if y \<le> 0 then Float 1 -2 else y"
```
```  1492   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)
```
```  1493   moreover { fix x :: float fix num :: nat
```
```  1494     have "0 < real (?horner x) ^ num" using `0 < ?horner x`[unfolded less_float_def real_of_float_0] by (rule zero_less_power)
```
```  1495     also have "\<dots> = real ((?horner x) ^ num)" using float_power by auto
```
```  1496     finally have "0 < real ((?horner x) ^ num)" .
```
```  1497   }
```
```  1498   ultimately show ?thesis
```
```  1499     unfolding lb_exp.simps if_not_P[OF `\<not> 0 < x`] Let_def
```
```  1500     by (cases "floor_fl x", cases "x < - 1", auto simp add: float_power le_float_def less_float_def)
```
```  1501 qed
```
```  1502
```
```  1503 lemma exp_boundaries': assumes "x \<le> 0"
```
```  1504   shows "exp (real x) \<in> { real (lb_exp prec x) .. real (ub_exp prec x)}"
```
```  1505 proof -
```
```  1506   let "?lb_exp_horner x" = "lb_exp_horner prec (get_even (prec + 2)) 1 1 x"
```
```  1507   let "?ub_exp_horner x" = "ub_exp_horner prec (get_odd (prec + 2)) 1 1 x"
```
```  1508
```
```  1509   have "real x \<le> 0" and "\<not> x > 0" using `x \<le> 0` unfolding le_float_def less_float_def by auto
```
```  1510   show ?thesis
```
```  1511   proof (cases "x < - 1")
```
```  1512     case False hence "- 1 \<le> real x" unfolding less_float_def by auto
```
```  1513     show ?thesis
```
```  1514     proof (cases "?lb_exp_horner x \<le> 0")
```
```  1515       from `\<not> x < - 1` have "- 1 \<le> real x" unfolding less_float_def by auto
```
```  1516       hence "exp (- 1) \<le> exp (real x)" unfolding exp_le_cancel_iff .
```
```  1517       from order_trans[OF exp_m1_ge_quarter this]
```
```  1518       have "real (Float 1 -2) \<le> exp (real x)" unfolding Float_num .
```
```  1519       moreover case True
```
```  1520       ultimately show ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by auto
```
```  1521     next
```
```  1522       case False thus ?thesis using bnds_exp_horner `real x \<le> 0` `\<not> x > 0` `\<not> x < - 1` by (auto simp add: Let_def)
```
```  1523     qed
```
```  1524   next
```
```  1525     case True
```
```  1526
```
```  1527     obtain m e where Float_floor: "floor_fl x = Float m e" by (cases "floor_fl x", auto)
```
```  1528     let ?num = "nat (- m) * 2 ^ nat e"
```
```  1529
```
```  1530     have "real (floor_fl x) < - 1" using floor_fl `x < - 1` unfolding le_float_def less_float_def real_of_float_minus real_of_float_1 by (rule order_le_less_trans)
```
```  1531     hence "real (floor_fl x) < 0" unfolding Float_floor real_of_float_simp using zero_less_pow2[of xe] by auto
```
```  1532     hence "m < 0"
```
```  1533       unfolding less_float_def real_of_float_0 Float_floor real_of_float_simp
```
```  1534       unfolding pos_prod_lt[OF zero_less_pow2[of e], unfolded real_mult_commute] by auto
```
```  1535     hence "1 \<le> - m" by auto
```
```  1536     hence "0 < nat (- m)" by auto
```
```  1537     moreover
```
```  1538     have "0 \<le> e" using floor_pos_exp Float_floor[symmetric] by auto
```
```  1539     hence "(0::nat) < 2 ^ nat e" by auto
```
```  1540     ultimately have "0 < ?num"  by auto
```
```  1541     hence "real ?num \<noteq> 0" by auto
```
```  1542     have e_nat: "int (nat e) = e" using `0 \<le> e` by auto
```
```  1543     have num_eq: "real ?num = real (- floor_fl x)" using `0 < nat (- m)`
```
```  1544       unfolding Float_floor real_of_float_minus real_of_float_simp real_of_nat_mult pow2_int[of "nat e", unfolded e_nat] realpow_real_of_nat[symmetric] by auto
```
```  1545     have "0 < - floor_fl x" using `0 < ?num`[unfolded real_of_nat_less_iff[symmetric]] unfolding less_float_def num_eq[symmetric] real_of_float_0 real_of_nat_zero .
```
```  1546     hence "real (floor_fl x) < 0" unfolding less_float_def by auto
```
```  1547
```
```  1548     have "exp (real x) \<le> real (ub_exp prec x)"
```
```  1549     proof -
```
```  1550       have div_less_zero: "real (float_divr prec x (- floor_fl x)) \<le> 0"
```
```  1551 	using float_divr_nonpos_pos_upper_bound[OF `x \<le> 0` `0 < - floor_fl x`] unfolding le_float_def real_of_float_0 .
```
```  1552
```
```  1553       have "exp (real x) = exp (real ?num * (real x / real ?num))" using `real ?num \<noteq> 0` by auto
```
```  1554       also have "\<dots> = exp (real x / real ?num) ^ ?num" unfolding exp_real_of_nat_mult ..
```
```  1555       also have "\<dots> \<le> exp (real (float_divr prec x (- floor_fl x))) ^ ?num" unfolding num_eq
```
```  1556 	by (rule power_mono, rule exp_le_cancel_iff[THEN iffD2], rule float_divr) auto
```
```  1557       also have "\<dots> \<le> real ((?ub_exp_horner (float_divr prec x (- floor_fl x))) ^ ?num)" unfolding float_power
```
```  1558 	by (rule power_mono, rule bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct2], auto)
```
```  1559       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 .
```
```  1560     qed
```
```  1561     moreover
```
```  1562     have "real (lb_exp prec x) \<le> exp (real x)"
```
```  1563     proof -
```
```  1564       let ?divl = "float_divl prec x (- Float m e)"
```
```  1565       let ?horner = "?lb_exp_horner ?divl"
```
```  1566
```
```  1567       show ?thesis
```
```  1568       proof (cases "?horner \<le> 0")
```
```  1569 	case False hence "0 \<le> real ?horner" unfolding le_float_def by auto
```
```  1570
```
```  1571 	have div_less_zero: "real (float_divl prec x (- floor_fl x)) \<le> 0"
```
```  1572 	  using `real (floor_fl x) < 0` `real x \<le> 0` by (auto intro!: order_trans[OF float_divl] divide_nonpos_neg)
```
```  1573
```
```  1574 	have "real ((?lb_exp_horner (float_divl prec x (- floor_fl x))) ^ ?num) \<le>
```
```  1575           exp (real (float_divl prec x (- floor_fl x))) ^ ?num" unfolding float_power
```
```  1576 	  using `0 \<le> real ?horner`[unfolded Float_floor[symmetric]] bnds_exp_horner[OF div_less_zero, unfolded atLeastAtMost_iff, THEN conjunct1] by (auto intro!: power_mono)
```
```  1577 	also have "\<dots> \<le> exp (real x / real ?num) ^ ?num" unfolding num_eq
```
```  1578 	  using float_divl by (auto intro!: power_mono simp del: real_of_float_minus)
```
```  1579 	also have "\<dots> = exp (real ?num * (real x / real ?num))" unfolding exp_real_of_nat_mult ..
```
```  1580 	also have "\<dots> = exp (real x)" using `real ?num \<noteq> 0` by auto
```
```  1581 	finally show ?thesis
```
```  1582 	  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
```
```  1583       next
```
```  1584 	case True
```
```  1585 	have "real (floor_fl x) \<noteq> 0" and "real (floor_fl x) \<le> 0" using `real (floor_fl x) < 0` by auto
```
```  1586 	from divide_right_mono_neg[OF floor_fl[of x] `real (floor_fl x) \<le> 0`, unfolded divide_self[OF `real (floor_fl x) \<noteq> 0`]]
```
```  1587 	have "- 1 \<le> real x / real (- floor_fl x)" unfolding real_of_float_minus by auto
```
```  1588 	from order_trans[OF exp_m1_ge_quarter this[unfolded exp_le_cancel_iff[where x="- 1", symmetric]]]
```
```  1589 	have "real (Float 1 -2) \<le> exp (real x / real (- floor_fl x))" unfolding Float_num .
```
```  1590 	hence "real (Float 1 -2) ^ ?num \<le> exp (real x / real (- floor_fl x)) ^ ?num"
```
```  1591 	  by (auto intro!: power_mono simp add: Float_num)
```
```  1592 	also have "\<dots> = exp (real x)" unfolding num_eq exp_real_of_nat_mult[symmetric] using `real (floor_fl x) \<noteq> 0` by auto
```
```  1593 	finally show ?thesis
```
```  1594 	  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 .
```
```  1595       qed
```
```  1596     qed
```
```  1597     ultimately show ?thesis by auto
```
```  1598   qed
```
```  1599 qed
```
```  1600
```
```  1601 lemma exp_boundaries: "exp (real x) \<in> { real (lb_exp prec x) .. real (ub_exp prec x)}"
```
```  1602 proof -
```
```  1603   show ?thesis
```
```  1604   proof (cases "0 < x")
```
```  1605     case False hence "x \<le> 0" unfolding less_float_def le_float_def by auto
```
```  1606     from exp_boundaries'[OF this] show ?thesis .
```
```  1607   next
```
```  1608     case True hence "-x \<le> 0" unfolding less_float_def le_float_def by auto
```
```  1609
```
```  1610     have "real (lb_exp prec x) \<le> exp (real x)"
```
```  1611     proof -
```
```  1612       from exp_boundaries'[OF `-x \<le> 0`]
```
```  1613       have ub_exp: "exp (- real x) \<le> real (ub_exp prec (-x))" unfolding atLeastAtMost_iff real_of_float_minus by auto
```
```  1614
```
```  1615       have "real (float_divl prec 1 (ub_exp prec (-x))) \<le> 1 / real (ub_exp prec (-x))" using float_divl[where x=1] by auto
```
```  1616       also have "\<dots> \<le> exp (real x)"
```
```  1617 	using ub_exp[unfolded inverse_le_iff_le[OF order_less_le_trans[OF exp_gt_zero ub_exp] exp_gt_zero, symmetric]]
```
```  1618 	unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide by auto
```
```  1619       finally show ?thesis unfolding lb_exp.simps if_P[OF True] .
```
```  1620     qed
```
```  1621     moreover
```
```  1622     have "exp (real x) \<le> real (ub_exp prec x)"
```
```  1623     proof -
```
```  1624       have "\<not> 0 < -x" using `0 < x` unfolding less_float_def by auto
```
```  1625
```
```  1626       from exp_boundaries'[OF `-x \<le> 0`]
```
```  1627       have lb_exp: "real (lb_exp prec (-x)) \<le> exp (- real x)" unfolding atLeastAtMost_iff real_of_float_minus by auto
```
```  1628
```
```  1629       have "exp (real x) \<le> real (1 :: float) / real (lb_exp prec (-x))"
```
```  1630 	using lb_exp[unfolded inverse_le_iff_le[OF exp_gt_zero lb_exp_pos[OF `\<not> 0 < -x`, unfolded less_float_def real_of_float_0],
```
```  1631 	                                        symmetric]]
```
```  1632 	unfolding exp_minus nonzero_inverse_inverse_eq[OF exp_not_eq_zero] inverse_eq_divide real_of_float_1 by auto
```
```  1633       also have "\<dots> \<le> real (float_divr prec 1 (lb_exp prec (-x)))" using float_divr .
```
```  1634       finally show ?thesis unfolding ub_exp.simps if_P[OF True] .
```
```  1635     qed
```
```  1636     ultimately show ?thesis by auto
```
```  1637   qed
```
```  1638 qed
```
```  1639
```
```  1640 lemma bnds_exp: "\<forall> x lx ux. (l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> exp x \<and> exp x \<le> real u"
```
```  1641 proof (rule allI, rule allI, rule allI, rule impI)
```
```  1642   fix x lx ux
```
```  1643   assume "(l, u) = (lb_exp prec lx, ub_exp prec ux) \<and> x \<in> {real lx .. real ux}"
```
```  1644   hence l: "lb_exp prec lx = l " and u: "ub_exp prec ux = u" and x: "x \<in> {real lx .. real ux}" by auto
```
```  1645
```
```  1646   { from exp_boundaries[of lx prec, unfolded l]
```
```  1647     have "real l \<le> exp (real lx)" by (auto simp del: lb_exp.simps)
```
```  1648     also have "\<dots> \<le> exp x" using x by auto
```
```  1649     finally have "real l \<le> exp x" .
```
```  1650   } moreover
```
```  1651   { have "exp x \<le> exp (real ux)" using x by auto
```
```  1652     also have "\<dots> \<le> real u" using exp_boundaries[of ux prec, unfolded u] by (auto simp del: ub_exp.simps)
```
```  1653     finally have "exp x \<le> real u" .
```
```  1654   } ultimately show "real l \<le> exp x \<and> exp x \<le> real u" ..
```
```  1655 qed
```
```  1656
```
```  1657 section "Logarithm"
```
```  1658
```
```  1659 subsection "Compute the logarithm series"
```
```  1660
```
```  1661 fun ub_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"
```
```  1662 and lb_ln_horner :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float" where
```
```  1663 "ub_ln_horner prec 0 i x       = 0" |
```
```  1664 "ub_ln_horner prec (Suc n) i x = rapprox_rat prec 1 (int i) - x * lb_ln_horner prec n (Suc i) x" |
```
```  1665 "lb_ln_horner prec 0 i x       = 0" |
```
```  1666 "lb_ln_horner prec (Suc n) i x = lapprox_rat prec 1 (int i) - x * ub_ln_horner prec n (Suc i) x"
```
```  1667
```
```  1668 lemma ln_bounds:
```
```  1669   assumes "0 \<le> x" and "x < 1"
```
```  1670   shows "(\<Sum>i=0..<2*n. -1^i * (1 / real (i + 1)) * x ^ (Suc i)) \<le> ln (x + 1)" (is "?lb")
```
```  1671   and "ln (x + 1) \<le> (\<Sum>i=0..<2*n + 1. -1^i * (1 / real (i + 1)) * x ^ (Suc i))" (is "?ub")
```
```  1672 proof -
```
```  1673   let "?a n" = "(1/real (n +1)) * x ^ (Suc n)"
```
```  1674
```
```  1675   have ln_eq: "(\<Sum> i. -1^i * ?a i) = ln (x + 1)"
```
```  1676     using ln_series[of "x + 1"] `0 \<le> x` `x < 1` by auto
```
```  1677
```
```  1678   have "norm x < 1" using assms by auto
```
```  1679   have "?a ----> 0" unfolding Suc_eq_plus1[symmetric] inverse_eq_divide[symmetric]
```
```  1680     using LIMSEQ_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_Suc[OF LIMSEQ_power_zero[OF `norm x < 1`]]] by auto
```
```  1681   { fix n have "0 \<le> ?a n" by (rule mult_nonneg_nonneg, auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`) }
```
```  1682   { fix n have "?a (Suc n) \<le> ?a n" unfolding inverse_eq_divide[symmetric]
```
```  1683     proof (rule mult_mono)
```
```  1684       show "0 \<le> x ^ Suc (Suc n)" by (auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
```
```  1685       have "x ^ Suc (Suc n) \<le> x ^ Suc n * 1" unfolding power_Suc2 real_mult_assoc[symmetric]
```
```  1686 	by (rule mult_left_mono, fact less_imp_le[OF `x < 1`], auto intro!: mult_nonneg_nonneg simp add: `0 \<le> x`)
```
```  1687       thus "x ^ Suc (Suc n) \<le> x ^ Suc n" by auto
```
```  1688     qed auto }
```
```  1689   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]
```
```  1690   show "?lb" and "?ub" by auto
```
```  1691 qed
```
```  1692
```
```  1693 lemma ln_float_bounds:
```
```  1694   assumes "0 \<le> real x" and "real x < 1"
```
```  1695   shows "real (x * lb_ln_horner prec (get_even n) 1 x) \<le> ln (real x + 1)" (is "?lb \<le> ?ln")
```
```  1696   and "ln (real x + 1) \<le> real (x * ub_ln_horner prec (get_odd n) 1 x)" (is "?ln \<le> ?ub")
```
```  1697 proof -
```
```  1698   obtain ev where ev: "get_even n = 2 * ev" using get_even_double ..
```
```  1699   obtain od where od: "get_odd n = 2 * od + 1" using get_odd_double ..
```
```  1700
```
```  1701   let "?s n" = "-1^n * (1 / real (1 + n)) * (real x)^(Suc n)"
```
```  1702
```
```  1703   have "?lb \<le> setsum ?s {0 ..< 2 * ev}" unfolding power_Suc2 real_mult_assoc[symmetric] real_of_float_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "real x"] ev
```
```  1704     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",
```
```  1705       OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
```
```  1706     by (rule mult_right_mono)
```
```  1707   also have "\<dots> \<le> ?ln" using ln_bounds(1)[OF `0 \<le> real x` `real x < 1`] by auto
```
```  1708   finally show "?lb \<le> ?ln" .
```
```  1709
```
```  1710   have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}" using ln_bounds(2)[OF `0 \<le> real x` `real x < 1`] by auto
```
```  1711   also have "\<dots> \<le> ?ub" unfolding power_Suc2 real_mult_assoc[symmetric] real_of_float_mult setsum_left_distrib[symmetric] unfolding real_mult_commute[of "real x"] od
```
```  1712     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",
```
```  1713       OF `0 \<le> real x` refl lb_ln_horner.simps ub_ln_horner.simps] `0 \<le> real x`
```
```  1714     by (rule mult_right_mono)
```
```  1715   finally show "?ln \<le> ?ub" .
```
```  1716 qed
```
```  1717
```
```  1718 lemma ln_add: assumes "0 < x" and "0 < y" shows "ln (x + y) = ln x + ln (1 + y / x)"
```
```  1719 proof -
```
```  1720   have "x \<noteq> 0" using assms by auto
```
```  1721   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
```
```  1722   moreover
```
```  1723   have "0 < y / x" using assms divide_pos_pos by auto
```
```  1724   hence "0 < 1 + y / x" by auto
```
```  1725   ultimately show ?thesis using ln_mult assms by auto
```
```  1726 qed
```
```  1727
```
```  1728 subsection "Compute the logarithm of 2"
```
```  1729
```
```  1730 definition ub_ln2 where "ub_ln2 prec = (let third = rapprox_rat (max prec 1) 1 3
```
```  1731                                         in (Float 1 -1 * ub_ln_horner prec (get_odd prec) 1 (Float 1 -1)) +
```
```  1732                                            (third * ub_ln_horner prec (get_odd prec) 1 third))"
```
```  1733 definition lb_ln2 where "lb_ln2 prec = (let third = lapprox_rat prec 1 3
```
```  1734                                         in (Float 1 -1 * lb_ln_horner prec (get_even prec) 1 (Float 1 -1)) +
```
```  1735                                            (third * lb_ln_horner prec (get_even prec) 1 third))"
```
```  1736
```
```  1737 lemma ub_ln2: "ln 2 \<le> real (ub_ln2 prec)" (is "?ub_ln2")
```
```  1738   and lb_ln2: "real (lb_ln2 prec) \<le> ln 2" (is "?lb_ln2")
```
```  1739 proof -
```
```  1740   let ?uthird = "rapprox_rat (max prec 1) 1 3"
```
```  1741   let ?lthird = "lapprox_rat prec 1 3"
```
```  1742
```
```  1743   have ln2_sum: "ln 2 = ln (1/2 + 1) + ln (1 / 3 + 1)"
```
```  1744     using ln_add[of "3 / 2" "1 / 2"] by auto
```
```  1745   have lb3: "real ?lthird \<le> 1 / 3" using lapprox_rat[of prec 1 3] by auto
```
```  1746   hence lb3_ub: "real ?lthird < 1" by auto
```
```  1747   have lb3_lb: "0 \<le> real ?lthird" using lapprox_rat_bottom[of 1 3] by auto
```
```  1748   have ub3: "1 / 3 \<le> real ?uthird" using rapprox_rat[of 1 3] by auto
```
```  1749   hence ub3_lb: "0 \<le> real ?uthird" by auto
```
```  1750
```
```  1751   have lb2: "0 \<le> real (Float 1 -1)" and ub2: "real (Float 1 -1) < 1" unfolding Float_num by auto
```
```  1752
```
```  1753   have "0 \<le> (1::int)" and "0 < (3::int)" by auto
```
```  1754   have ub3_ub: "real ?uthird < 1" unfolding rapprox_rat.simps(2)[OF `0 \<le> 1` `0 < 3`]
```
```  1755     by (rule rapprox_posrat_less1, auto)
```
```  1756
```
```  1757   have third_gt0: "(0 :: real) < 1 / 3 + 1" by auto
```
```  1758   have uthird_gt0: "0 < real ?uthird + 1" using ub3_lb by auto
```
```  1759   have lthird_gt0: "0 < real ?lthird + 1" using lb3_lb by auto
```
```  1760
```
```  1761   show ?ub_ln2 unfolding ub_ln2_def Let_def real_of_float_add ln2_sum Float_num(4)[symmetric]
```
```  1762   proof (rule add_mono, fact ln_float_bounds(2)[OF lb2 ub2])
```
```  1763     have "ln (1 / 3 + 1) \<le> ln (real ?uthird + 1)" unfolding ln_le_cancel_iff[OF third_gt0 uthird_gt0] using ub3 by auto
```
```  1764     also have "\<dots> \<le> real (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)"
```
```  1765       using ln_float_bounds(2)[OF ub3_lb ub3_ub] .
```
```  1766     finally show "ln (1 / 3 + 1) \<le> real (?uthird * ub_ln_horner prec (get_odd prec) 1 ?uthird)" .
```
```  1767   qed
```
```  1768   show ?lb_ln2 unfolding lb_ln2_def Let_def real_of_float_add ln2_sum Float_num(4)[symmetric]
```
```  1769   proof (rule add_mono, fact ln_float_bounds(1)[OF lb2 ub2])
```
```  1770     have "real (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (real ?lthird + 1)"
```
```  1771       using ln_float_bounds(1)[OF lb3_lb lb3_ub] .
```
```  1772     also have "\<dots> \<le> ln (1 / 3 + 1)" unfolding ln_le_cancel_iff[OF lthird_gt0 third_gt0] using lb3 by auto
```
```  1773     finally show "real (?lthird * lb_ln_horner prec (get_even prec) 1 ?lthird) \<le> ln (1 / 3 + 1)" .
```
```  1774   qed
```
```  1775 qed
```
```  1776
```
```  1777 subsection "Compute the logarithm in the entire domain"
```
```  1778
```
```  1779 function ub_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" and lb_ln :: "nat \<Rightarrow> float \<Rightarrow> float option" where
```
```  1780 "ub_ln prec x = (if x \<le> 0          then None
```
```  1781             else if x < 1          then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x)))
```
```  1782             else let horner = \<lambda>x. x * ub_ln_horner prec (get_odd prec) 1 x in
```
```  1783                  if x \<le> Float 3 -1 then Some (horner (x - 1))
```
```  1784             else if x < Float 1 1  then Some (horner (Float 1 -1) + horner (x * rapprox_rat prec 2 3 - 1))
```
```  1785                                    else let l = bitlen (mantissa x) - 1 in
```
```  1786                                         Some (ub_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l) - 1)))" |
```
```  1787 "lb_ln prec x = (if x \<le> 0          then None
```
```  1788             else if x < 1          then Some (- the (ub_ln prec (float_divr prec 1 x)))
```
```  1789             else let horner = \<lambda>x. x * lb_ln_horner prec (get_even prec) 1 x in
```
```  1790                  if x \<le> Float 3 -1 then Some (horner (x - 1))
```
```  1791             else if x < Float 1 1  then Some (horner (Float 1 -1) +
```
```  1792                                               horner (max (x * lapprox_rat prec 2 3 - 1) 0))
```
```  1793                                    else let l = bitlen (mantissa x) - 1 in
```
```  1794                                         Some (lb_ln2 prec * (Float (scale x + l) 0) + horner (Float (mantissa x) (- l) - 1)))"
```
```  1795 by pat_completeness auto
```
```  1796
```
```  1797 termination proof (relation "measure (\<lambda> v. let (prec, x) = sum_case id id v in (if x < 1 then 1 else 0))", auto)
```
```  1798   fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divl (max prec (Suc 0)) 1 x < 1"
```
```  1799   hence "0 < x" and "0 < max prec (Suc 0)" unfolding less_float_def le_float_def by auto
```
```  1800   from float_divl_pos_less1_bound[OF `0 < x` `x < 1` `0 < max prec (Suc 0)`]
```
```  1801   show False using `float_divl (max prec (Suc 0)) 1 x < 1` unfolding less_float_def le_float_def by auto
```
```  1802 next
```
```  1803   fix prec x assume "\<not> x \<le> 0" and "x < 1" and "float_divr prec 1 x < 1"
```
```  1804   hence "0 < x" unfolding less_float_def le_float_def by auto
```
```  1805   from float_divr_pos_less1_lower_bound[OF `0 < x` `x < 1`, of prec]
```
```  1806   show False using `float_divr prec 1 x < 1` unfolding less_float_def le_float_def by auto
```
```  1807 qed
```
```  1808
```
```  1809 lemma ln_shifted_float: assumes "0 < m" shows "ln (real (Float m e)) = ln 2 * real (e + (bitlen m - 1)) + ln (real (Float m (- (bitlen m - 1))))"
```
```  1810 proof -
```
```  1811   let ?B = "2^nat (bitlen m - 1)"
```
```  1812   have "0 < real m" and "\<And>X. (0 :: real) < 2^X" and "0 < (2 :: real)" and "m \<noteq> 0" using assms by auto
```
```  1813   hence "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto
```
```  1814   show ?thesis
```
```  1815   proof (cases "0 \<le> e")
```
```  1816     case True
```
```  1817     show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
```
```  1818       unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`]
```
```  1819       unfolding real_of_float_ge0_exp[OF True] ln_mult[OF `0 < real m` `0 < 2^nat e`]
```
```  1820       ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` True by auto
```
```  1821   next
```
```  1822     case False hence "0 < -e" by auto
```
```  1823     hence pow_gt0: "(0::real) < 2^nat (-e)" by auto
```
```  1824     hence inv_gt0: "(0::real) < inverse (2^nat (-e))" by auto
```
```  1825     show ?thesis unfolding normalized_float[OF `m \<noteq> 0`]
```
```  1826       unfolding ln_div[OF `0 < real m` `0 < ?B`] real_of_int_add ln_realpow[OF `0 < 2`]
```
```  1827       unfolding real_of_float_nge0_exp[OF False] ln_mult[OF `0 < real m` inv_gt0] ln_inverse[OF pow_gt0]
```
```  1828       ln_realpow[OF `0 < 2`] algebra_simps using `0 \<le> bitlen m - 1` False by auto
```
```  1829   qed
```
```  1830 qed
```
```  1831
```
```  1832 lemma ub_ln_lb_ln_bounds': assumes "1 \<le> x"
```
```  1833   shows "real (the (lb_ln prec x)) \<le> ln (real x) \<and> ln (real x) \<le> real (the (ub_ln prec x))"
```
```  1834   (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
```
```  1835 proof (cases "x < Float 1 1")
```
```  1836   case True
```
```  1837   hence "real (x - 1) < 1" and "real x < 2" unfolding less_float_def Float_num by auto
```
```  1838   have "\<not> x \<le> 0" and "\<not> x < 1" using `1 \<le> x` unfolding less_float_def le_float_def by auto
```
```  1839   hence "0 \<le> real (x - 1)" using `1 \<le> x` unfolding less_float_def Float_num by auto
```
```  1840
```
```  1841   have [simp]: "real (Float 3 -1) = 3 / 2" by (simp add: real_of_float_def pow2_def)
```
```  1842
```
```  1843   show ?thesis
```
```  1844   proof (cases "x \<le> Float 3 -1")
```
```  1845     case True
```
```  1846     show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
```
```  1847       using ln_float_bounds[OF `0 \<le> real (x - 1)` `real (x - 1) < 1`, of prec] `\<not> x \<le> 0` `\<not> x < 1` True
```
```  1848       by auto
```
```  1849   next
```
```  1850     case False hence *: "3 / 2 < real x" by (auto simp add: le_float_def)
```
```  1851
```
```  1852     with ln_add[of "3 / 2" "real x - 3 / 2"]
```
```  1853     have add: "ln (real x) = ln (3 / 2) + ln (real x * 2 / 3)"
```
```  1854       by (auto simp add: algebra_simps diff_divide_distrib)
```
```  1855
```
```  1856     let "?ub_horner x" = "x * ub_ln_horner prec (get_odd prec) 1 x"
```
```  1857     let "?lb_horner x" = "x * lb_ln_horner prec (get_even prec) 1 x"
```
```  1858
```
```  1859     { have up: "real (rapprox_rat prec 2 3) \<le> 1"
```
```  1860 	by (rule rapprox_rat_le1) simp_all
```
```  1861       have low: "2 / 3 \<le> real (rapprox_rat prec 2 3)"
```
```  1862 	by (rule order_trans[OF _ rapprox_rat]) simp
```
```  1863       from mult_less_le_imp_less[OF * low] *
```
```  1864       have pos: "0 < real (x * rapprox_rat prec 2 3 - 1)" by auto
```
```  1865
```
```  1866       have "ln (real x * 2/3)
```
```  1867 	\<le> ln (real (x * rapprox_rat prec 2 3 - 1) + 1)"
```
```  1868       proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
```
```  1869 	show "real x * 2 / 3 \<le> real (x * rapprox_rat prec 2 3 - 1) + 1"
```
```  1870 	  using * low by auto
```
```  1871 	show "0 < real x * 2 / 3" using * by simp
```
```  1872 	show "0 < real (x * rapprox_rat prec 2 3 - 1) + 1" using pos by auto
```
```  1873       qed
```
```  1874       also have "\<dots> \<le> real (?ub_horner (x * rapprox_rat prec 2 3 - 1))"
```
```  1875       proof (rule ln_float_bounds(2))
```
```  1876 	from mult_less_le_imp_less[OF `real x < 2` up] low *
```
```  1877 	show "real (x * rapprox_rat prec 2 3 - 1) < 1" by auto
```
```  1878 	show "0 \<le> real (x * rapprox_rat prec 2 3 - 1)" using pos by auto
```
```  1879       qed
```
```  1880       finally have "ln (real x)
```
```  1881 	\<le> real (?ub_horner (Float 1 -1))
```
```  1882 	  + real (?ub_horner (x * rapprox_rat prec 2 3 - 1))"
```
```  1883 	using ln_float_bounds(2)[of "Float 1 -1" prec prec] add by auto }
```
```  1884     moreover
```
```  1885     { let ?max = "max (x * lapprox_rat prec 2 3 - 1) 0"
```
```  1886
```
```  1887       have up: "real (lapprox_rat prec 2 3) \<le> 2/3"
```
```  1888 	by (rule order_trans[OF lapprox_rat], simp)
```
```  1889
```
```  1890       have low: "0 \<le> real (lapprox_rat prec 2 3)"
```
```  1891 	using lapprox_rat_bottom[of 2 3 prec] by simp
```
```  1892
```
```  1893       have "real (?lb_horner ?max)
```
```  1894 	\<le> ln (real ?max + 1)"
```
```  1895       proof (rule ln_float_bounds(1))
```
```  1896 	from mult_less_le_imp_less[OF `real x < 2` up] * low
```
```  1897 	show "real ?max < 1" by (cases "real (lapprox_rat prec 2 3) = 0",
```
```  1898 	  auto simp add: real_of_float_max)
```
```  1899 	show "0 \<le> real ?max" by (auto simp add: real_of_float_max)
```
```  1900       qed
```
```  1901       also have "\<dots> \<le> ln (real x * 2/3)"
```
```  1902       proof (rule ln_le_cancel_iff[symmetric, THEN iffD1])
```
```  1903 	show "0 < real ?max + 1" by (auto simp add: real_of_float_max)
```
```  1904 	show "0 < real x * 2/3" using * by auto
```
```  1905 	show "real ?max + 1 \<le> real x * 2/3" using * up
```
```  1906 	  by (cases "0 < real x * real (lapprox_posrat prec 2 3) - 1",
```
```  1907 	      auto simp add: real_of_float_max max_def)
```
```  1908       qed
```
```  1909       finally have "real (?lb_horner (Float 1 -1)) + real (?lb_horner ?max)
```
```  1910 	\<le> ln (real x)"
```
```  1911 	using ln_float_bounds(1)[of "Float 1 -1" prec prec] add by auto }
```
```  1912     ultimately
```
```  1913     show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps Let_def
```
```  1914       using `\<not> x \<le> 0` `\<not> x < 1` True False by auto
```
```  1915   qed
```
```  1916 next
```
```  1917   case False
```
```  1918   hence "\<not> x \<le> 0" and "\<not> x < 1" "0 < x" "\<not> x \<le> Float 3 -1"
```
```  1919     using `1 \<le> x` unfolding less_float_def le_float_def real_of_float_simp pow2_def
```
```  1920     by auto
```
```  1921   show ?thesis
```
```  1922   proof (cases x)
```
```  1923     case (Float m e)
```
```  1924     let ?s = "Float (e + (bitlen m - 1)) 0"
```
```  1925     let ?x = "Float m (- (bitlen m - 1))"
```
```  1926
```
```  1927     have "0 < m" and "m \<noteq> 0" using float_pos_m_pos `0 < x` Float by auto
```
```  1928
```
```  1929     {
```
```  1930       have "real (lb_ln2 prec * ?s) \<le> ln 2 * real (e + (bitlen m - 1))" (is "?lb2 \<le> _")
```
```  1931 	unfolding real_of_float_mult real_of_float_ge0_exp[OF order_refl] nat_0 power_0 mult_1_right
```
```  1932 	using lb_ln2[of prec]
```
```  1933       proof (rule mult_right_mono)
```
```  1934 	have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
```
```  1935 	from float_gt1_scale[OF this]
```
```  1936 	show "0 \<le> real (e + (bitlen m - 1))" by auto
```
```  1937       qed
```
```  1938       moreover
```
```  1939       from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
```
```  1940       have "0 \<le> real (?x - 1)" and "real (?x - 1) < 1" by auto
```
```  1941       from ln_float_bounds(1)[OF this]
```
```  1942       have "real ((?x - 1) * lb_ln_horner prec (get_even prec) 1 (?x - 1)) \<le> ln (real ?x)" (is "?lb_horner \<le> _") by auto
```
```  1943       ultimately have "?lb2 + ?lb_horner \<le> ln (real x)"
```
```  1944 	unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
```
```  1945     }
```
```  1946     moreover
```
```  1947     {
```
```  1948       from bitlen_div[OF `0 < m`, unfolded normalized_float[OF `m \<noteq> 0`, symmetric]]
```
```  1949       have "0 \<le> real (?x - 1)" and "real (?x - 1) < 1" by auto
```
```  1950       from ln_float_bounds(2)[OF this]
```
```  1951       have "ln (real ?x) \<le> real ((?x - 1) * ub_ln_horner prec (get_odd prec) 1 (?x - 1))" (is "_ \<le> ?ub_horner") by auto
```
```  1952       moreover
```
```  1953       have "ln 2 * real (e + (bitlen m - 1)) \<le> real (ub_ln2 prec * ?s)" (is "_ \<le> ?ub2")
```
```  1954 	unfolding real_of_float_mult real_of_float_ge0_exp[OF order_refl] nat_0 power_0 mult_1_right
```
```  1955 	using ub_ln2[of prec]
```
```  1956       proof (rule mult_right_mono)
```
```  1957 	have "1 \<le> Float m e" using `1 \<le> x` Float unfolding le_float_def by auto
```
```  1958 	from float_gt1_scale[OF this]
```
```  1959 	show "0 \<le> real (e + (bitlen m - 1))" by auto
```
```  1960       qed
```
```  1961       ultimately have "ln (real x) \<le> ?ub2 + ?ub_horner"
```
```  1962 	unfolding Float ln_shifted_float[OF `0 < m`, of e] by auto
```
```  1963     }
```
```  1964     ultimately show ?thesis unfolding lb_ln.simps unfolding ub_ln.simps
```
```  1965       unfolding if_not_P[OF `\<not> x \<le> 0`] if_not_P[OF `\<not> x < 1`] if_not_P[OF False] if_not_P[OF `\<not> x \<le> Float 3 -1`] Let_def
```
```  1966       unfolding scale.simps[of m e, unfolded Float[symmetric]] mantissa.simps[of m e, unfolded Float[symmetric]] real_of_float_add
```
```  1967       by auto
```
```  1968   qed
```
```  1969 qed
```
```  1970
```
```  1971 lemma ub_ln_lb_ln_bounds: assumes "0 < x"
```
```  1972   shows "real (the (lb_ln prec x)) \<le> ln (real x) \<and> ln (real x) \<le> real (the (ub_ln prec x))"
```
```  1973   (is "?lb \<le> ?ln \<and> ?ln \<le> ?ub")
```
```  1974 proof (cases "x < 1")
```
```  1975   case False hence "1 \<le> x" unfolding less_float_def le_float_def by auto
```
```  1976   show ?thesis using ub_ln_lb_ln_bounds'[OF `1 \<le> x`] .
```
```  1977 next
```
```  1978   case True have "\<not> x \<le> 0" using `0 < x` unfolding less_float_def le_float_def by auto
```
```  1979
```
```  1980   have "0 < real x" and "real x \<noteq> 0" using `0 < x` unfolding less_float_def by auto
```
```  1981   hence A: "0 < 1 / real x" by auto
```
```  1982
```
```  1983   {
```
```  1984     let ?divl = "float_divl (max prec 1) 1 x"
```
```  1985     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
```
```  1986     hence B: "0 < real ?divl" unfolding le_float_def by auto
```
```  1987
```
```  1988     have "ln (real ?divl) \<le> ln (1 / real x)" unfolding ln_le_cancel_iff[OF B A] using float_divl[of _ 1 x] by auto
```
```  1989     hence "ln (real x) \<le> - ln (real ?divl)" unfolding nonzero_inverse_eq_divide[OF `real x \<noteq> 0`, symmetric] ln_inverse[OF `0 < real x`] by auto
```
```  1990     from this ub_ln_lb_ln_bounds'[OF A', THEN conjunct1, THEN le_imp_neg_le]
```
```  1991     have "?ln \<le> real (- the (lb_ln prec ?divl))" unfolding real_of_float_minus by (rule order_trans)
```
```  1992   } moreover
```
```  1993   {
```
```  1994     let ?divr = "float_divr prec 1 x"
```
```  1995     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
```
```  1996     hence B: "0 < real ?divr" unfolding le_float_def by auto
```
```  1997
```
```  1998     have "ln (1 / real x) \<le> ln (real ?divr)" unfolding ln_le_cancel_iff[OF A B] using float_divr[of 1 x] by auto
```
```  1999     hence "- ln (real ?divr) \<le> ln (real x)" unfolding nonzero_inverse_eq_divide[OF `real x \<noteq> 0`, symmetric] ln_inverse[OF `0 < real x`] by auto
```
```  2000     from ub_ln_lb_ln_bounds'[OF A', THEN conjunct2, THEN le_imp_neg_le] this
```
```  2001     have "real (- the (ub_ln prec ?divr)) \<le> ?ln" unfolding real_of_float_minus by (rule order_trans)
```
```  2002   }
```
```  2003   ultimately show ?thesis unfolding lb_ln.simps[where x=x]  ub_ln.simps[where x=x]
```
```  2004     unfolding if_not_P[OF `\<not> x \<le> 0`] if_P[OF True] by auto
```
```  2005 qed
```
```  2006
```
```  2007 lemma lb_ln: assumes "Some y = lb_ln prec x"
```
```  2008   shows "real y \<le> ln (real x)" and "0 < real x"
```
```  2009 proof -
```
```  2010   have "0 < x"
```
```  2011   proof (rule ccontr)
```
```  2012     assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
```
```  2013     thus False using assms by auto
```
```  2014   qed
```
```  2015   thus "0 < real x" unfolding less_float_def by auto
```
```  2016   have "real (the (lb_ln prec x)) \<le> ln (real x)" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
```
```  2017   thus "real y \<le> ln (real x)" unfolding assms[symmetric] by auto
```
```  2018 qed
```
```  2019
```
```  2020 lemma ub_ln: assumes "Some y = ub_ln prec x"
```
```  2021   shows "ln (real x) \<le> real y" and "0 < real x"
```
```  2022 proof -
```
```  2023   have "0 < x"
```
```  2024   proof (rule ccontr)
```
```  2025     assume "\<not> 0 < x" hence "x \<le> 0" unfolding le_float_def less_float_def by auto
```
```  2026     thus False using assms by auto
```
```  2027   qed
```
```  2028   thus "0 < real x" unfolding less_float_def by auto
```
```  2029   have "ln (real x) \<le> real (the (ub_ln prec x))" using ub_ln_lb_ln_bounds[OF `0 < x`] ..
```
```  2030   thus "ln (real x) \<le> real y" unfolding assms[symmetric] by auto
```
```  2031 qed
```
```  2032
```
```  2033 lemma bnds_ln: "\<forall> x lx ux. (Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {real lx .. real ux} \<longrightarrow> real l \<le> ln x \<and> ln x \<le> real u"
```
```  2034 proof (rule allI, rule allI, rule allI, rule impI)
```
```  2035   fix x lx ux
```
```  2036   assume "(Some l, Some u) = (lb_ln prec lx, ub_ln prec ux) \<and> x \<in> {real lx .. real ux}"
```
```  2037   hence l: "Some l = lb_ln prec lx " and u: "Some u = ub_ln prec ux" and x: "x \<in> {real lx .. real ux}" by auto
```
```  2038
```
```  2039   have "ln (real ux) \<le> real u" and "0 < real ux" using ub_ln u by auto
```
```  2040   have "real l \<le> ln (real lx)" and "0 < real lx" and "0 < x" using lb_ln[OF l] x by auto
```
```  2041
```
```  2042   from ln_le_cancel_iff[OF `0 < real lx` `0 < x`] `real l \<le> ln (real lx)`
```
```  2043   have "real l \<le> ln x" using x unfolding atLeastAtMost_iff by auto
```
```  2044   moreover
```
```  2045   from ln_le_cancel_iff[OF `0 < x` `0 < real ux`] `ln (real ux) \<le> real u`
```
```  2046   have "ln x \<le> real u" using x unfolding atLeastAtMost_iff by auto
```
```  2047   ultimately show "real l \<le> ln x \<and> ln x \<le> real u" ..
```
```  2048 qed
```
```  2049
```
```  2050 section "Implement floatarith"
```
```  2051
```
```  2052 subsection "Define syntax and semantics"
```
```  2053
```
```  2054 datatype floatarith
```
```  2055   = Add floatarith floatarith
```
```  2056   | Minus floatarith
```
```  2057   | Mult floatarith floatarith
```
```  2058   | Inverse floatarith
```
```  2059   | Cos floatarith
```
```  2060   | Arctan floatarith
```
```  2061   | Abs floatarith
```
```  2062   | Max floatarith floatarith
```
```  2063   | Min floatarith floatarith
```
```  2064   | Pi
```
```  2065   | Sqrt floatarith
```
```  2066   | Exp floatarith
```
```  2067   | Ln floatarith
```
```  2068   | Power floatarith nat
```
```  2069   | Atom nat
```
```  2070   | Num float
```
```  2071
```
```  2072 fun interpret_floatarith :: "floatarith \<Rightarrow> real list \<Rightarrow> real"
```
```  2073 where
```
```  2074 "interpret_floatarith (Add a b) vs   = (interpret_floatarith a vs) + (interpret_floatarith b vs)" |
```
```  2075 "interpret_floatarith (Minus a) vs    = - (interpret_floatarith a vs)" |
```
```  2076 "interpret_floatarith (Mult a b) vs   = (interpret_floatarith a vs) * (interpret_floatarith b vs)" |
```
```  2077 "interpret_floatarith (Inverse a) vs  = inverse (interpret_floatarith a vs)" |
```
```  2078 "interpret_floatarith (Cos a) vs      = cos (interpret_floatarith a vs)" |
```
```  2079 "interpret_floatarith (Arctan a) vs   = arctan (interpret_floatarith a vs)" |
```
```  2080 "interpret_floatarith (Min a b) vs    = min (interpret_floatarith a vs) (interpret_floatarith b vs)" |
```
```  2081 "interpret_floatarith (Max a b) vs    = max (interpret_floatarith a vs) (interpret_floatarith b vs)" |
```
```  2082 "interpret_floatarith (Abs a) vs      = abs (interpret_floatarith a vs)" |
```
```  2083 "interpret_floatarith Pi vs           = pi" |
```
```  2084 "interpret_floatarith (Sqrt a) vs     = sqrt (interpret_floatarith a vs)" |
```
```  2085 "interpret_floatarith (Exp a) vs      = exp (interpret_floatarith a vs)" |
```
```  2086 "interpret_floatarith (Ln a) vs       = ln (interpret_floatarith a vs)" |
```
```  2087 "interpret_floatarith (Power a n) vs  = (interpret_floatarith a vs)^n" |
```
```  2088 "interpret_floatarith (Num f) vs      = real f" |
```
```  2089 "interpret_floatarith (Atom n) vs     = vs ! n"
```
```  2090
```
```  2091 subsection "Implement approximation function"
```
```  2092
```
```  2093 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
```
```  2094 "lift_bin' (Some (l1, u1)) (Some (l2, u2)) f = Some (f l1 u1 l2 u2)" |
```
```  2095 "lift_bin' a b f = None"
```
```  2096
```
```  2097 fun lift_un :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> ((float option) * (float option))) \<Rightarrow> (float * float) option" where
```
```  2098 "lift_un (Some (l1, u1)) f = (case (f l1 u1) of (Some l, Some u) \<Rightarrow> Some (l, u)
```
```  2099                                              | t \<Rightarrow> None)" |
```
```  2100 "lift_un b f = None"
```
```  2101
```
```  2102 fun lift_un' :: "(float * float) option \<Rightarrow> (float \<Rightarrow> float \<Rightarrow> (float * float)) \<Rightarrow> (float * float) option" where
```
```  2103 "lift_un' (Some (l1, u1)) f = Some (f l1 u1)" |
```
```  2104 "lift_un' b f = None"
```
```  2105
```
```  2106 fun bounded_by :: "real list \<Rightarrow> (float * float) list \<Rightarrow> bool " where
```
```  2107 bounded_by_Cons: "bounded_by (v#vs) ((l, u)#bs) = ((real l \<le> v \<and> v \<le> real u) \<and> bounded_by vs bs)" |
```
```  2108 bounded_by_Nil: "bounded_by [] [] = True" |
```
```  2109 "bounded_by _ _ = False"
```
```  2110
```
```  2111 lemma bounded_by: assumes "bounded_by vs bs" and "i < length bs"
```
```  2112   shows "real (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> real (snd (bs ! i))"
```
```  2113   using `bounded_by vs bs` and `i < length bs`
```
```  2114 proof (induct arbitrary: i rule: bounded_by.induct)
```
```  2115   fix v :: real and vs :: "real list" and l u :: float and bs :: "(float * float) list" and i :: nat
```
```  2116   assume hyp: "\<And>i. \<lbrakk>bounded_by vs bs; i < length bs\<rbrakk> \<Longrightarrow> real (fst (bs ! i)) \<le> vs ! i \<and> vs ! i \<le> real (snd (bs ! i))"
```
```  2117   assume bounded: "bounded_by (v # vs) ((l, u) # bs)" and length: "i < length ((l, u) # bs)"
```
```  2118   show "real (fst (((l, u) # bs) ! i)) \<le> (v # vs) ! i \<and> (v # vs) ! i \<le> real (snd (((l, u) # bs) ! i))"
```
```  2119   proof (cases i)
```
```  2120     case 0
```
```  2121     show ?thesis using bounded unfolding 0 nth_Cons_0 fst_conv snd_conv bounded_by.simps ..
```
```  2122   next
```
```  2123     case (Suc i) with length have "i < length bs" by auto
```
```  2124     show ?thesis unfolding Suc nth_Cons_Suc bounded_by.simps
```
```  2125       using hyp[OF bounded[unfolded bounded_by.simps, THEN conjunct2] `i < length bs`] .
```
```  2126   qed
```
```  2127 qed auto
```
```  2128
```
```  2129 fun approx approx' :: "nat \<Rightarrow> floatarith \<Rightarrow> (float * float) list \<Rightarrow> (float * float) option" where
```
```  2130 "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)" |
```
```  2131 "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))" |
```
```  2132 "approx prec (Minus a) bs   = lift_un' (approx' prec a bs) (\<lambda> l u. (-u, -l))" |
```
```  2133 "approx prec (Mult a b) bs  = lift_bin' (approx' prec a bs) (approx' prec b bs)
```
```  2134                                     (\<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,
```
```  2135                                                      float_pprt a2 * float_pprt b2 + float_pprt a1 * float_nprt b2 + float_nprt a2 * float_pprt b1 + float_nprt a1 * float_nprt b1))" |
```
```  2136 "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))" |
```
```  2137 "approx prec (Cos a) bs     = lift_un' (approx' prec a bs) (bnds_cos prec)" |
```
```  2138 "approx prec Pi bs          = Some (lb_pi prec, ub_pi prec)" |
```
```  2139 "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))" |
```
```  2140 "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))" |
```
```  2141 "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>))" |
```
```  2142 "approx prec (Arctan a) bs  = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_arctan prec l, ub_arctan prec u))" |
```
```  2143 "approx prec (Sqrt a) bs    = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_sqrt prec l, ub_sqrt prec u))" |
```
```  2144 "approx prec (Exp a) bs     = lift_un' (approx' prec a bs) (\<lambda> l u. (lb_exp prec l, ub_exp prec u))" |
```
```  2145 "approx prec (Ln a) bs      = lift_un (approx' prec a bs) (\<lambda> l u. (lb_ln prec l, ub_ln prec u))" |
```
```  2146 "approx prec (Power a n) bs = lift_un' (approx' prec a bs) (float_power_bnds n)" |
```
```  2147 "approx prec (Num f) bs     = Some (f, f)" |
```
```  2148 "approx prec (Atom i) bs    = (if i < length bs then Some (bs ! i) else None)"
```
```  2149
```
```  2150 lemma lift_bin'_ex:
```
```  2151   assumes lift_bin'_Some: "Some (l, u) = lift_bin' a b f"
```
```  2152   shows "\<exists> l1 u1 l2 u2. Some (l1, u1) = a \<and> Some (l2, u2) = b"
```
```  2153 proof (cases a)
```
```  2154   case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
```
```  2155   thus ?thesis using lift_bin'_Some by auto
```
```  2156 next
```
```  2157   case (Some a')
```
```  2158   show ?thesis
```
```  2159   proof (cases b)
```
```  2160     case None hence "None = lift_bin' a b f" unfolding None lift_bin'.simps ..
```
```  2161     thus ?thesis using lift_bin'_Some by auto
```
```  2162   next
```
```  2163     case (Some b')
```
```  2164     obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
```
```  2165     obtain lb ub where b': "b' = (lb, ub)" by (cases b', auto)
```
```  2166     thus ?thesis unfolding `a = Some a'` `b = Some b'` a' b' by auto
```
```  2167   qed
```
```  2168 qed
```
```  2169
```
```  2170 lemma lift_bin'_f:
```
```  2171   assumes lift_bin'_Some: "Some (l, u) = lift_bin' (g a) (g b) f"
```
```  2172   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"
```
```  2173   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)"
```
```  2174 proof -
```
```  2175   obtain l1 u1 l2 u2
```
```  2176     where Sa: "Some (l1, u1) = g a" and Sb: "Some (l2, u2) = g b" using lift_bin'_ex[OF assms(1)] by auto
```
```  2177   have lu: "(l, u) = f l1 u1 l2 u2" using lift_bin'_Some[unfolded Sa[symmetric] Sb[symmetric] lift_bin'.simps] by auto
```
```  2178   have "l = fst (f l1 u1 l2 u2)" and "u = snd (f l1 u1 l2 u2)" unfolding lu[symmetric] by auto
```
```  2179   thus ?thesis using Pa[OF Sa] Pb[OF Sb] by auto
```
```  2180 qed
```
```  2181
```
```  2182 lemma approx_approx':
```
```  2183   assumes Pa: "\<And>l u. Some (l, u) = approx prec a vs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u"
```
```  2184   and approx': "Some (l, u) = approx' prec a vs"
```
```  2185   shows "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u"
```
```  2186 proof -
```
```  2187   obtain l' u' where S: "Some (l', u') = approx prec a vs"
```
```  2188     using approx' unfolding approx'.simps by (cases "approx prec a vs", auto)
```
```  2189   have l': "l = round_down prec l'" and u': "u = round_up prec u'"
```
```  2190     using approx' unfolding approx'.simps S[symmetric] by auto
```
```  2191   show ?thesis unfolding l' u'
```
```  2192     using order_trans[OF Pa[OF S, THEN conjunct2] round_up[of u']]
```
```  2193     using order_trans[OF round_down[of _ l'] Pa[OF S, THEN conjunct1]] by auto
```
```  2194 qed
```
```  2195
```
```  2196 lemma lift_bin':
```
```  2197   assumes lift_bin'_Some: "Some (l, u) = lift_bin' (approx' prec a bs) (approx' prec b bs) f"
```
```  2198   and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
```
```  2199   and Pb: "\<And>l u. Some (l, u) = approx prec b bs \<Longrightarrow> real l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> real u"
```
```  2200   shows "\<exists> l1 u1 l2 u2. (real l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u1) \<and>
```
```  2201                         (real l2 \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> real u2) \<and>
```
```  2202                         l = fst (f l1 u1 l2 u2) \<and> u = snd (f l1 u1 l2 u2)"
```
```  2203 proof -
```
```  2204   { fix l u assume "Some (l, u) = approx' prec a bs"
```
```  2205     with approx_approx'[of prec a bs, OF _ this] Pa
```
```  2206     have "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" by auto } note Pa = this
```
```  2207   { fix l u assume "Some (l, u) = approx' prec b bs"
```
```  2208     with approx_approx'[of prec b bs, OF _ this] Pb
```
```  2209     have "real l \<le> interpret_floatarith b xs \<and> interpret_floatarith b xs \<le> real u" by auto } note Pb = this
```
```  2210
```
```  2211   from lift_bin'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_bin'_Some, OF Pa Pb]
```
```  2212   show ?thesis by auto
```
```  2213 qed
```
```  2214
```
```  2215 lemma lift_un'_ex:
```
```  2216   assumes lift_un'_Some: "Some (l, u) = lift_un' a f"
```
```  2217   shows "\<exists> l u. Some (l, u) = a"
```
```  2218 proof (cases a)
```
```  2219   case None hence "None = lift_un' a f" unfolding None lift_un'.simps ..
```
```  2220   thus ?thesis using lift_un'_Some by auto
```
```  2221 next
```
```  2222   case (Some a')
```
```  2223   obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
```
```  2224   thus ?thesis unfolding `a = Some a'` a' by auto
```
```  2225 qed
```
```  2226
```
```  2227 lemma lift_un'_f:
```
```  2228   assumes lift_un'_Some: "Some (l, u) = lift_un' (g a) f"
```
```  2229   and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
```
```  2230   shows "\<exists> l1 u1. P l1 u1 a \<and> l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
```
```  2231 proof -
```
```  2232   obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un'_ex[OF assms(1)] by auto
```
```  2233   have lu: "(l, u) = f l1 u1" using lift_un'_Some[unfolded Sa[symmetric] lift_un'.simps] by auto
```
```  2234   have "l = fst (f l1 u1)" and "u = snd (f l1 u1)" unfolding lu[symmetric] by auto
```
```  2235   thus ?thesis using Pa[OF Sa] by auto
```
```  2236 qed
```
```  2237
```
```  2238 lemma lift_un':
```
```  2239   assumes lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
```
```  2240   and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
```
```  2241   shows "\<exists> l1 u1. (real l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u1) \<and>
```
```  2242                         l = fst (f l1 u1) \<and> u = snd (f l1 u1)"
```
```  2243 proof -
```
```  2244   { fix l u assume "Some (l, u) = approx' prec a bs"
```
```  2245     with approx_approx'[of prec a bs, OF _ this] Pa
```
```  2246     have "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" by auto } note Pa = this
```
```  2247   from lift_un'_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un'_Some, OF Pa]
```
```  2248   show ?thesis by auto
```
```  2249 qed
```
```  2250
```
```  2251 lemma lift_un'_bnds:
```
```  2252   assumes bnds: "\<forall> x lx ux. (l, u) = f lx ux \<and> x \<in> { real lx .. real ux } \<longrightarrow> real l \<le> f' x \<and> f' x \<le> real u"
```
```  2253   and lift_un'_Some: "Some (l, u) = lift_un' (approx' prec a bs) f"
```
```  2254   and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u"
```
```  2255   shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
```
```  2256 proof -
```
```  2257   from lift_un'[OF lift_un'_Some Pa]
```
```  2258   obtain l1 u1 where "real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1" and "l = fst (f l1 u1)" and "u = snd (f l1 u1)" by blast
```
```  2259   hence "(l, u) = f l1 u1" and "interpret_floatarith a xs \<in> {real l1 .. real u1}" by auto
```
```  2260   thus ?thesis using bnds by auto
```
```  2261 qed
```
```  2262
```
```  2263 lemma lift_un_ex:
```
```  2264   assumes lift_un_Some: "Some (l, u) = lift_un a f"
```
```  2265   shows "\<exists> l u. Some (l, u) = a"
```
```  2266 proof (cases a)
```
```  2267   case None hence "None = lift_un a f" unfolding None lift_un.simps ..
```
```  2268   thus ?thesis using lift_un_Some by auto
```
```  2269 next
```
```  2270   case (Some a')
```
```  2271   obtain la ua where a': "a' = (la, ua)" by (cases a', auto)
```
```  2272   thus ?thesis unfolding `a = Some a'` a' by auto
```
```  2273 qed
```
```  2274
```
```  2275 lemma lift_un_f:
```
```  2276   assumes lift_un_Some: "Some (l, u) = lift_un (g a) f"
```
```  2277   and Pa: "\<And>l u. Some (l, u) = g a \<Longrightarrow> P l u a"
```
```  2278   shows "\<exists> l1 u1. P l1 u1 a \<and> Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
```
```  2279 proof -
```
```  2280   obtain l1 u1 where Sa: "Some (l1, u1) = g a" using lift_un_ex[OF assms(1)] by auto
```
```  2281   have "fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None"
```
```  2282   proof (rule ccontr)
```
```  2283     assume "\<not> (fst (f l1 u1) \<noteq> None \<and> snd (f l1 u1) \<noteq> None)"
```
```  2284     hence or: "fst (f l1 u1) = None \<or> snd (f l1 u1) = None" by auto
```
```  2285     hence "lift_un (g a) f = None"
```
```  2286     proof (cases "fst (f l1 u1) = None")
```
```  2287       case True
```
```  2288       then obtain b where b: "f l1 u1 = (None, b)" by (cases "f l1 u1", auto)
```
```  2289       thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
```
```  2290     next
```
```  2291       case False hence "snd (f l1 u1) = None" using or by auto
```
```  2292       with False obtain b where b: "f l1 u1 = (Some b, None)" by (cases "f l1 u1", auto)
```
```  2293       thus ?thesis unfolding Sa[symmetric] lift_un.simps b by auto
```
```  2294     qed
```
```  2295     thus False using lift_un_Some by auto
```
```  2296   qed
```
```  2297   then obtain a' b' where f: "f l1 u1 = (Some a', Some b')" by (cases "f l1 u1", auto)
```
```  2298   from lift_un_Some[unfolded Sa[symmetric] lift_un.simps f]
```
```  2299   have "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" unfolding f by auto
```
```  2300   thus ?thesis unfolding Sa[symmetric] lift_un.simps using Pa[OF Sa] by auto
```
```  2301 qed
```
```  2302
```
```  2303 lemma lift_un:
```
```  2304   assumes lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
```
```  2305   and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" (is "\<And>l u. _ = ?g a \<Longrightarrow> ?P l u a")
```
```  2306   shows "\<exists> l1 u1. (real l1 \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u1) \<and>
```
```  2307                   Some l = fst (f l1 u1) \<and> Some u = snd (f l1 u1)"
```
```  2308 proof -
```
```  2309   { fix l u assume "Some (l, u) = approx' prec a bs"
```
```  2310     with approx_approx'[of prec a bs, OF _ this] Pa
```
```  2311     have "real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u" by auto } note Pa = this
```
```  2312   from lift_un_f[where g="\<lambda>a. approx' prec a bs" and P = ?P, OF lift_un_Some, OF Pa]
```
```  2313   show ?thesis by auto
```
```  2314 qed
```
```  2315
```
```  2316 lemma lift_un_bnds:
```
```  2317   assumes bnds: "\<forall> x lx ux. (Some l, Some u) = f lx ux \<and> x \<in> { real lx .. real ux } \<longrightarrow> real l \<le> f' x \<and> f' x \<le> real u"
```
```  2318   and lift_un_Some: "Some (l, u) = lift_un (approx' prec a bs) f"
```
```  2319   and Pa: "\<And>l u. Some (l, u) = approx prec a bs \<Longrightarrow> real l \<le> interpret_floatarith a xs \<and> interpret_floatarith a xs \<le> real u"
```
```  2320   shows "real l \<le> f' (interpret_floatarith a xs) \<and> f' (interpret_floatarith a xs) \<le> real u"
```
```  2321 proof -
```
```  2322   from lift_un[OF lift_un_Some Pa]
```
```  2323   obtain l1 u1 where "real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1" and "Some l = fst (f l1 u1)" and "Some u = snd (f l1 u1)" by blast
```
```  2324   hence "(Some l, Some u) = f l1 u1" and "interpret_floatarith a xs \<in> {real l1 .. real u1}" by auto
```
```  2325   thus ?thesis using bnds by auto
```
```  2326 qed
```
```  2327
```
```  2328 lemma approx:
```
```  2329   assumes "bounded_by xs vs"
```
```  2330   and "Some (l, u) = approx prec arith vs" (is "_ = ?g arith")
```
```  2331   shows "real l \<le> interpret_floatarith arith xs \<and> interpret_floatarith arith xs \<le> real u" (is "?P l u arith")
```
```  2332   using `Some (l, u) = approx prec arith vs`
```
```  2333 proof (induct arith arbitrary: l u x)
```
```  2334   case (Add a b)
```
```  2335   from lift_bin'[OF Add.prems[unfolded approx.simps]] Add.hyps
```
```  2336   obtain l1 u1 l2 u2 where "l = l1 + l2" and "u = u1 + u2"
```
```  2337     "real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1"
```
```  2338     "real l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> real u2" unfolding fst_conv snd_conv by blast
```
```  2339   thus ?case unfolding interpret_floatarith.simps by auto
```
```  2340 next
```
```  2341   case (Minus a)
```
```  2342   from lift_un'[OF Minus.prems[unfolded approx.simps]] Minus.hyps
```
```  2343   obtain l1 u1 where "l = -u1" and "u = -l1"
```
```  2344     "real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1" unfolding fst_conv snd_conv by blast
```
```  2345   thus ?case unfolding interpret_floatarith.simps using real_of_float_minus by auto
```
```  2346 next
```
```  2347   case (Mult a b)
```
```  2348   from lift_bin'[OF Mult.prems[unfolded approx.simps]] Mult.hyps
```
```  2349   obtain l1 u1 l2 u2
```
```  2350     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"
```
```  2351     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"
```
```  2352     and "real l1 \<le> interpret_floatarith a xs" and "interpret_floatarith a xs \<le> real u1"
```
```  2353     and "real l2 \<le> interpret_floatarith b xs" and "interpret_floatarith b xs \<le> real u2" unfolding fst_conv snd_conv by blast
```
```  2354   thus ?case unfolding interpret_floatarith.simps l u real_of_float_add real_of_float_mult real_of_float_nprt real_of_float_pprt
```
```  2355     using mult_le_prts mult_ge_prts by auto
```
```  2356 next
```
```  2357   case (Inverse a)
```
```  2358   from lift_un[OF Inverse.prems[unfolded approx.simps], unfolded if_distrib[of fst] if_distrib[of snd] fst_conv snd_conv] Inverse.hyps
```
```  2359   obtain l1 u1 where l': "Some l = (if 0 < l1 \<or> u1 < 0 then Some (float_divl prec 1 u1) else None)"
```
```  2360     and u': "Some u = (if 0 < l1 \<or> u1 < 0 then Some (float_divr prec 1 l1) else None)"
```
```  2361     and l1: "real l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> real u1" by blast
```
```  2362   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
```
```  2363   moreover have l1_le_u1: "real l1 \<le> real u1" using l1 u1 by auto
```
```  2364   ultimately have "real l1 \<noteq> 0" and "real u1 \<noteq> 0" unfolding less_float_def by auto
```
```  2365
```
```  2366   have inv: "inverse (real u1) \<le> inverse (interpret_floatarith a xs)
```
```  2367            \<and> inverse (interpret_floatarith a xs) \<le> inverse (real l1)"
```
```  2368   proof (cases "0 < l1")
```
```  2369     case True hence "0 < real u1" and "0 < real l1" "0 < interpret_floatarith a xs"
```
```  2370       unfolding less_float_def using l1_le_u1 l1 by auto
```
```  2371     show ?thesis
```
```  2372       unfolding inverse_le_iff_le[OF `0 < real u1` `0 < interpret_floatarith a xs`]
```
```  2373 	inverse_le_iff_le[OF `0 < interpret_floatarith a xs` `0 < real l1`]
```
```  2374       using l1 u1 by auto
```
```  2375   next
```
```  2376     case False hence "u1 < 0" using either by blast
```
```  2377     hence "real u1 < 0" and "real l1 < 0" "interpret_floatarith a xs < 0"
```
```  2378       unfolding less_float_def using l1_le_u1 u1 by auto
```
```  2379     show ?thesis
```
```  2380       unfolding inverse_le_iff_le_neg[OF `real u1 < 0` `interpret_floatarith a xs < 0`]
```
```  2381 	inverse_le_iff_le_neg[OF `interpret_floatarith a xs < 0` `real l1 < 0`]
```
```  2382       using l1 u1 by auto
```
```  2383   qed
```
```  2384
```
```  2385   from l' have "l = float_divl prec 1 u1" by (cases "0 < l1 \<or> u1 < 0", auto)
```
```  2386   hence "real l \<le> inverse (real u1)" unfolding nonzero_inverse_eq_divide[OF `real u1 \<noteq> 0`] using float_divl[of prec 1 u1] by auto
```
```  2387   also have "\<dots> \<le> inverse (interpret_floatarith a xs)" using inv by auto
```
```  2388   finally have "real l \<le> inverse (interpret_floatarith a xs)" .
```
```  2389   moreover
```
```  2390   from u' have "u = float_divr prec 1 l1" by (cases "0 < l1 \<or> u1 < 0", auto)
```
```  2391   hence "inverse (real l1) \<le> real u" unfolding nonzero_inverse_eq_divide[OF `real l1 \<noteq> 0`] using float_divr[of 1 l1 prec] by auto
```
```  2392   hence "inverse (interpret_floatarith a xs) \<le> real u" by (rule order_trans[OF inv[THEN conjunct2]])
```
```  2393   ultimately show ?case unfolding interpret_floatarith.simps using l1 u1 by auto
```
```  2394 next
```
```  2395   case (Abs x)
```
```  2396   from lift_un'[OF Abs.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Abs.hyps
```
```  2397   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>"
```
```  2398     and l1: "real l1 \<le> interpret_floatarith x xs" and u1: "interpret_floatarith x xs \<le> real u1" by blast
```
```  2399   thus ?case unfolding l' u' by (cases "l1 < 0 \<and> 0 < u1", auto simp add: real_of_float_min real_of_float_max real_of_float_abs less_float_def)
```
```  2400 next
```
```  2401   case (Min a b)
```
```  2402   from lift_bin'[OF Min.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Min.hyps
```
```  2403   obtain l1 u1 l2 u2 where l': "l = min l1 l2" and u': "u = min u1 u2"
```
```  2404     and l1: "real l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> real u1"
```
```  2405     and l1: "real l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> real u2" by blast
```
```  2406   thus ?case unfolding l' u' by (auto simp add: real_of_float_min)
```
```  2407 next
```
```  2408   case (Max a b)
```
```  2409   from lift_bin'[OF Max.prems[unfolded approx.simps], unfolded fst_conv snd_conv] Max.hyps
```
```  2410   obtain l1 u1 l2 u2 where l': "l = max l1 l2" and u': "u = max u1 u2"
```
```  2411     and l1: "real l1 \<le> interpret_floatarith a xs" and u1: "interpret_floatarith a xs \<le> real u1"
```
```  2412     and l1: "real l2 \<le> interpret_floatarith b xs" and u1: "interpret_floatarith b xs \<le> real u2" by blast
```
```  2413   thus ?case unfolding l' u' by (auto simp add: real_of_float_max)
```
```  2414 next case (Cos a) with lift_un'_bnds[OF bnds_cos] show ?case by auto
```
```  2415 next case (Arctan a) with lift_un'_bnds[OF bnds_arctan] show ?case by auto
```
```  2416 next case Pi with pi_boundaries show ?case by auto
```
```  2417 next case (Sqrt a) with lift_un'_bnds[OF bnds_sqrt] show ?case by auto
```
```  2418 next case (Exp a) with lift_un'_bnds[OF bnds_exp] show ?case by auto
```
```  2419 next case (Ln a) with lift_un_bnds[OF bnds_ln] show ?case by auto
```
```  2420 next case (Power a n) with lift_un'_bnds[OF bnds_power] show ?case by auto
```
```  2421 next case (Num f) thus ?case by auto
```
```  2422 next
```
```  2423   case (Atom n)
```
```  2424   show ?case
```
```  2425   proof (cases "n < length vs")
```
```  2426     case True
```
```  2427     with Atom have "vs ! n = (l, u)" by auto
```
```  2428     thus ?thesis using bounded_by[OF assms(1) True] by auto
```
```  2429   next
```
```  2430     case False thus ?thesis using Atom by auto
```
```  2431   qed
```
```  2432 qed
```
```  2433
```
```  2434 datatype inequality = Less floatarith floatarith
```
```  2435                     | LessEqual floatarith floatarith
```
```  2436
```
```  2437 fun interpret_inequality :: "inequality \<Rightarrow> real list \<Rightarrow> bool" where
```
```  2438 "interpret_inequality (Less a b) vs                   = (interpret_floatarith a vs < interpret_floatarith b vs)" |
```
```  2439 "interpret_inequality (LessEqual a b) vs              = (interpret_floatarith a vs \<le> interpret_floatarith b vs)"
```
```  2440
```
```  2441 fun approx_inequality :: "nat \<Rightarrow> inequality \<Rightarrow> (float * float) list \<Rightarrow> bool" where
```
```  2442 "approx_inequality prec (Less a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u < l' | _ \<Rightarrow> False)" |
```
```  2443 "approx_inequality prec (LessEqual a b) bs = (case (approx prec a bs, approx prec b bs) of (Some (l, u), Some (l', u')) \<Rightarrow> u \<le> l' | _ \<Rightarrow> False)"
```
```  2444
```
```  2445 lemma approx_inequality: fixes m :: nat assumes "bounded_by vs bs" and "approx_inequality prec eq bs"
```
```  2446   shows "interpret_inequality eq vs"
```
```  2447 proof (cases eq)
```
```  2448   case (Less a b)
```
```  2449   show ?thesis
```
```  2450   proof (cases "\<exists> u l u' l'. approx prec a bs = Some (l, u) \<and>
```
```  2451                              approx prec b bs = Some (l', u')")
```
```  2452     case True
```
```  2453     then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)"
```
```  2454       and b_approx: "approx prec b bs = Some (l', u') " by auto
```
```  2455     with `approx_inequality prec eq bs` have "real u < real l'"
```
```  2456       unfolding Less approx_inequality.simps less_float_def by auto
```
```  2457     moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs`
```
```  2458     have "interpret_floatarith a vs \<le> real u" and "real l' \<le> interpret_floatarith b vs"
```
```  2459       using approx by auto
```
```  2460     ultimately show ?thesis unfolding interpret_inequality.simps Less by auto
```
```  2461   next
```
```  2462     case False
```
```  2463     hence "approx prec a bs = None \<or> approx prec b bs = None"
```
```  2464       unfolding not_Some_eq[symmetric] by auto
```
```  2465     hence "\<not> approx_inequality prec eq bs" unfolding Less approx_inequality.simps
```
```  2466       by (cases "approx prec a bs = None", auto)
```
```  2467     thus ?thesis using assms by auto
```
```  2468   qed
```
```  2469 next
```
```  2470   case (LessEqual a b)
```
```  2471   show ?thesis
```
```  2472   proof (cases "\<exists> u l u' l'. approx prec a bs = Some (l, u) \<and>
```
```  2473                              approx prec b bs = Some (l', u')")
```
```  2474     case True
```
```  2475     then obtain l u l' u' where a_approx: "approx prec a bs = Some (l, u)"
```
```  2476       and b_approx: "approx prec b bs = Some (l', u') " by auto
```
```  2477     with `approx_inequality prec eq bs` have "real u \<le> real l'"
```
```  2478       unfolding LessEqual approx_inequality.simps le_float_def by auto
```
```  2479     moreover from a_approx[symmetric] and b_approx[symmetric] and `bounded_by vs bs`
```
```  2480     have "interpret_floatarith a vs \<le> real u" and "real l' \<le> interpret_floatarith b vs"
```
```  2481       using approx by auto
```
```  2482     ultimately show ?thesis unfolding interpret_inequality.simps LessEqual by auto
```
```  2483   next
```
```  2484     case False
```
```  2485     hence "approx prec a bs = None \<or> approx prec b bs = None"
```
```  2486       unfolding not_Some_eq[symmetric] by auto
```
```  2487     hence "\<not> approx_inequality prec eq bs" unfolding LessEqual approx_inequality.simps
```
```  2488       by (cases "approx prec a bs = None", auto)
```
```  2489     thus ?thesis using assms by auto
```
```  2490   qed
```
```  2491 qed
```
```  2492
```
```  2493 lemma interpret_floatarith_divide: "interpret_floatarith (Mult a (Inverse b)) vs = (interpret_floatarith a vs) / (interpret_floatarith b vs)"
```
```  2494   unfolding real_divide_def interpret_floatarith.simps ..
```
```  2495
```
```  2496 lemma interpret_floatarith_diff: "interpret_floatarith (Add a (Minus b)) vs = (interpret_floatarith a vs) - (interpret_floatarith b vs)"
```
```  2497   unfolding real_diff_def interpret_floatarith.simps ..
```
```  2498
```
```  2499 lemma interpret_floatarith_sin: "interpret_floatarith (Cos (Add (Mult Pi (Num (Float 1 -1))) (Minus a))) vs =
```
```  2500   sin (interpret_floatarith a vs)"
```
```  2501   unfolding sin_cos_eq interpret_floatarith.simps
```
```  2502             interpret_floatarith_divide interpret_floatarith_diff real_diff_def
```
```  2503   by auto
```
```  2504
```
```  2505 lemma interpret_floatarith_tan:
```
```  2506   "interpret_floatarith (Mult (Cos (Add (Mult Pi (Num (Float 1 -1))) (Minus a))) (Inverse (Cos a))) vs =
```
```  2507    tan (interpret_floatarith a vs)"
```
```  2508   unfolding interpret_floatarith.simps(3,4) interpret_floatarith_sin tan_def real_divide_def
```
```  2509   by auto
```
```  2510
```
```  2511 lemma interpret_floatarith_powr: "interpret_floatarith (Exp (Mult b (Ln a))) vs = (interpret_floatarith a vs) powr (interpret_floatarith b vs)"
```
```  2512   unfolding powr_def interpret_floatarith.simps ..
```
```  2513
```
```  2514 lemma interpret_floatarith_log: "interpret_floatarith ((Mult (Ln x) (Inverse (Ln b)))) vs = log (interpret_floatarith b vs) (interpret_floatarith x vs)"
```
```  2515   unfolding log_def interpret_floatarith.simps real_divide_def ..
```
```  2516
```
```  2517 lemma interpret_floatarith_num:
```
```  2518   shows "interpret_floatarith (Num (Float 0 0)) vs = 0"
```
```  2519   and "interpret_floatarith (Num (Float 1 0)) vs = 1"
```
```  2520   and "interpret_floatarith (Num (Float (number_of a) 0)) vs = number_of a" by auto
```
```  2521
```
```  2522 subsection {* Implement proof method \texttt{approximation} *}
```
```  2523
```
```  2524 lemma bounded_divl: assumes "real a / real b \<le> x" shows "real (float_divl p a b) \<le> x" by (rule order_trans[OF _ assms], rule float_divl)
```
```  2525 lemma bounded_divr: assumes "x \<le> real a / real b" shows "x \<le> real (float_divr p a b)" by (rule order_trans[OF assms _], rule float_divr)
```
```  2526 lemma bounded_num: shows "real (Float 5 1) = 10" and "real (Float 0 0) = 0" and "real (Float 1 0) = 1" and "real (Float (number_of n) 0) = (number_of n)"
```
```  2527                      and "0 * pow2 e = real (Float 0 e)" and "1 * pow2 e = real (Float 1 e)" and "number_of m * pow2 e = real (Float (number_of m) e)"
```
```  2528                      and "real (Float (number_of A) (int B)) = (number_of A) * 2^B"
```
```  2529                      and "real (Float 1 (int B)) = 2^B"
```
```  2530                      and "real (Float (number_of A) (- int B)) = (number_of A) / 2^B"
```
```  2531                      and "real (Float 1 (- int B)) = 1 / 2^B"
```
```  2532   by (auto simp add: real_of_float_simp pow2_def real_divide_def)
```
```  2533
```
```  2534 lemmas bounded_by_equations = bounded_by_Cons bounded_by_Nil float_power bounded_divl bounded_divr bounded_num HOL.simp_thms
```
```  2535 lemmas interpret_inequality_equations = interpret_inequality.simps interpret_floatarith.simps interpret_floatarith_num
```
```  2536   interpret_floatarith_divide interpret_floatarith_diff interpret_floatarith_tan interpret_floatarith_powr interpret_floatarith_log
```
```  2537   interpret_floatarith_sin
```
```  2538
```
```  2539 ML {*
```
```  2540 structure Float_Arith =
```
```  2541 struct
```
```  2542
```
```  2543 @{code_datatype float = Float}
```
```  2544 @{code_datatype floatarith = Add | Minus | Mult | Inverse | Cos | Arctan
```
```  2545                            | Abs | Max | Min | Pi | Sqrt | Exp | Ln | Power | Atom | Num }
```
```  2546 @{code_datatype inequality = Less | LessEqual }
```
```  2547
```
```  2548 val approx_inequality = @{code approx_inequality}
```
```  2549
```
```  2550 end
```
```  2551 *}
```
```  2552
```
```  2553 code_reserved Eval Float_Arith
```
```  2554
```
```  2555 code_type float (Eval "Float'_Arith.float")
```
```  2556 code_const Float (Eval "Float'_Arith.Float/ (_,/ _)")
```
```  2557
```
```  2558 code_type floatarith (Eval "Float'_Arith.floatarith")
```
```  2559 code_const Add and Minus and Mult and Inverse and Cos and Arctan and Abs and Max and Min and
```
```  2560            Pi and Sqrt  and Exp and Ln and Power and Atom and Num
```
```  2561   (Eval "Float'_Arith.Add/ (_,/ _)" and "Float'_Arith.Minus" and "Float'_Arith.Mult/ (_,/ _)" and
```
```  2562         "Float'_Arith.Inverse" and "Float'_Arith.Cos" and
```
```  2563         "Float'_Arith.Arctan" and "Float'_Arith.Abs" and "Float'_Arith.Max/ (_,/ _)" and
```
```  2564         "Float'_Arith.Min/ (_,/ _)" and "Float'_Arith.Pi" and "Float'_Arith.Sqrt" and
```
```  2565         "Float'_Arith.Exp" and "Float'_Arith.Ln" and "Float'_Arith.Power/ (_,/ _)" and
```
```  2566         "Float'_Arith.Atom" and "Float'_Arith.Num")
```
```  2567
```
```  2568 code_type inequality (Eval "Float'_Arith.inequality")
```
```  2569 code_const Less and LessEqual (Eval "Float'_Arith.Less/ (_,/ _)" and "Float'_Arith.LessEqual/ (_,/ _)")
```
```  2570
```
```  2571 code_const approx_inequality (Eval "Float'_Arith.approx'_inequality")
```
```  2572
```
```  2573 ML {*
```
```  2574   val ineq_equations = PureThy.get_thms @{theory} "interpret_inequality_equations";
```
```  2575   val bounded_by_equations = PureThy.get_thms @{theory} "bounded_by_equations";
```
```  2576   val bounded_by_simpset = (HOL_basic_ss addsimps bounded_by_equations)
```
```  2577
```
```  2578   fun reify_ineq ctxt i = (fn st =>
```
```  2579     let
```
```  2580       val to = HOLogic.dest_Trueprop (Logic.strip_imp_concl (List.nth (prems_of st, i - 1)))
```
```  2581     in (Reflection.genreify_tac ctxt ineq_equations (SOME to) i) st
```
```  2582     end)
```
```  2583
```
```  2584   fun rule_ineq ctxt prec i thm = let
```
```  2585     fun conv_num typ = HOLogic.dest_number #> snd #> HOLogic.mk_number typ
```
```  2586     val to_natc = conv_num @{typ "nat"} #> Thm.cterm_of (ProofContext.theory_of ctxt)
```
```  2587     val to_nat = conv_num @{typ "nat"}
```
```  2588     val to_int = conv_num @{typ "int"}
```
```  2589     fun int_to_float A = @{term "Float"} \$ to_int A \$ @{term "0 :: int"}
```
```  2590
```
```  2591     val prec' = to_nat prec
```
```  2592
```
```  2593     fun bot_float (Const (@{const_name "times"}, _) \$ mantisse \$ (Const (@{const_name "pow2"}, _) \$ exp))
```
```  2594                    = @{term "Float"} \$ to_int mantisse \$ to_int exp
```
```  2595       | bot_float (Const (@{const_name "divide"}, _) \$ mantisse \$ (@{term "power 2 :: nat \<Rightarrow> real"} \$ exp))
```
```  2596                    = @{term "Float"} \$ to_int mantisse \$ (@{term "uminus :: int \<Rightarrow> int"} \$ (@{term "int :: nat \<Rightarrow> int"} \$ to_nat exp))
```
```  2597       | bot_float (Const (@{const_name "times"}, _) \$ mantisse \$ (@{term "power 2 :: nat \<Rightarrow> real"} \$ exp))
```
```  2598                    = @{term "Float"} \$ to_int mantisse \$ (@{term "int :: nat \<Rightarrow> int"} \$ to_nat exp)
```
```  2599       | bot_float (Const (@{const_name "divide"}, _) \$ A \$ (@{term "power 10 :: nat \<Rightarrow> real"} \$ exp))
```
```  2600                    = @{term "float_divl"} \$ prec' \$ (int_to_float A) \$ (@{term "power (Float 5 1)"} \$ to_nat exp)
```
```  2601       | bot_float (Const (@{const_name "divide"}, _) \$ A \$ B)
```
```  2602                    = @{term "float_divl"} \$ prec' \$ int_to_float A \$ int_to_float B
```
```  2603       | bot_float (@{term "power 2 :: nat \<Rightarrow> real"} \$ exp)
```
```  2604                    = @{term "Float 1"} \$ (@{term "int :: nat \<Rightarrow> int"} \$ to_nat exp)
```
```  2605       | bot_float A = int_to_float A
```
```  2606
```
```  2607     fun top_float (Const (@{const_name "times"}, _) \$ mantisse \$ (Const (@{const_name "pow2"}, _) \$ exp))
```
```  2608                    = @{term "Float"} \$ to_int mantisse \$ to_int exp
```
```  2609       | top_float (Const (@{const_name "divide"}, _) \$ mantisse \$ (@{term "power 2 :: nat \<Rightarrow> real"} \$ exp))
```
```  2610                    = @{term "Float"} \$ to_int mantisse \$ (@{term "uminus :: int \<Rightarrow> int"} \$ (@{term "int :: nat \<Rightarrow> int"} \$ to_nat exp))
```
```  2611       | top_float (Const (@{const_name "times"}, _) \$ mantisse \$ (@{term "power 2 :: nat \<Rightarrow> real"} \$ exp))
```
```  2612                    = @{term "Float"} \$ to_int mantisse \$ (@{term "int :: nat \<Rightarrow> int"} \$ to_nat exp)
```
```  2613       | top_float (Const (@{const_name "divide"}, _) \$ A \$ (@{term "power 10 :: nat \<Rightarrow> real"} \$ exp))
```
```  2614                    = @{term "float_divr"} \$ prec' \$ (int_to_float A) \$ (@{term "power (Float 5 1)"} \$ to_nat exp)
```
```  2615       | top_float (Const (@{const_name "divide"}, _) \$ A \$ B)
```
```  2616                    = @{term "float_divr"} \$ prec' \$ int_to_float A \$ int_to_float B
```
```  2617       | top_float (@{term "power 2 :: nat \<Rightarrow> real"} \$ exp)
```
```  2618                    = @{term "Float 1"} \$ (@{term "int :: nat \<Rightarrow> int"} \$ to_nat exp)
```
```  2619       | top_float A = int_to_float A
```
```  2620
```
```  2621     val goal' : term = List.nth (prems_of thm, i - 1)
```
```  2622
```
```  2623     fun lift_bnd (t as (Const (@{const_name "op &"}, _) \$
```
```  2624                         (Const (@{const_name "less_eq"}, _) \$
```
```  2625                          bottom \$ (Free (name, _))) \$
```
```  2626                         (Const (@{const_name "less_eq"}, _) \$ _ \$ top)))
```
```  2627          = ((name, HOLogic.mk_prod (bot_float bottom, top_float top))
```
```  2628             handle TERM (txt, ts) => raise TERM ("Invalid bound number format: " ^
```
```  2629                                   (Syntax.string_of_term ctxt t), [t]))
```
```  2630       | lift_bnd t = raise TERM ("Premisse needs format '<num> <= <var> & <var> <= <num>', but found " ^
```
```  2631                                  (Syntax.string_of_term ctxt t), [t])
```
```  2632     val bound_eqs = map (HOLogic.dest_Trueprop #> lift_bnd)  (Logic.strip_imp_prems goal')
```
```  2633
```
```  2634     fun lift_var (Free (varname, _)) = (case AList.lookup (op =) bound_eqs varname of
```
```  2635                                           SOME bound => bound
```
```  2636                                         | NONE => raise TERM ("No bound equations found for " ^ varname, []))
```
```  2637       | lift_var t = raise TERM ("Can not convert expression " ^
```
```  2638                                  (Syntax.string_of_term ctxt t), [t])
```
```  2639
```
```  2640     val _ \$ vs = HOLogic.dest_Trueprop (Logic.strip_imp_concl goal')
```
```  2641
```
```  2642     val bs = (HOLogic.dest_list #> map lift_var #> HOLogic.mk_list @{typ "float * float"}) vs
```
```  2643     val map = [(@{cpat "?prec::nat"}, to_natc prec),
```
```  2644                (@{cpat "?bs::(float * float) list"}, Thm.cterm_of (ProofContext.theory_of ctxt) bs)]
```
```  2645   in rtac (Thm.instantiate ([], map) @{thm "approx_inequality"}) i thm end
```
```  2646
```
```  2647   val eval_tac = CSUBGOAL (fn (ct, i) => rtac (eval_oracle ct) i)
```
```  2648
```
```  2649   fun gen_eval_tac conv ctxt = CONVERSION (Conv.params_conv (~1) (K (Conv.concl_conv (~1) conv)) ctxt)
```
```  2650                                THEN' rtac TrueI
```
```  2651
```
```  2652 *}
```
```  2653
```
```  2654 method_setup approximation = {*
```
```  2655   Args.term >>
```
```  2656   (fn prec => fn ctxt =>
```
```  2657     SIMPLE_METHOD' (fn i =>
```
```  2658      (DETERM (reify_ineq ctxt i)
```
```  2659       THEN rule_ineq ctxt prec i
```
```  2660       THEN Simplifier.asm_full_simp_tac bounded_by_simpset i
```
```  2661       THEN (TRY (filter_prems_tac (fn t => false) i))
```
```  2662       THEN (gen_eval_tac eval_oracle ctxt) i)))
```
```  2663 *} "real number approximation"
```
```  2664
```
```  2665 end
```