src/HOL/Real/Float.thy
 author webertj Wed Jul 26 19:23:04 2006 +0200 (2006-07-26) changeset 20217 25b068a99d2b parent 19765 dfe940911617 child 20485 3078fd2eec7b permissions -rw-r--r--
linear arithmetic splits certain operators (e.g. min, max, abs)
 obua@16782 ` 1` ```(* Title: HOL/Real/Float.thy ``` obua@16782 ` 2` ``` ID: \$Id\$ ``` obua@16782 ` 3` ``` Author: Steven Obua ``` obua@16782 ` 4` ```*) ``` obua@16782 ` 5` wenzelm@16890 ` 6` ```theory Float imports Real begin ``` obua@16782 ` 7` wenzelm@19765 ` 8` ```definition ``` obua@16782 ` 9` ``` pow2 :: "int \ real" ``` wenzelm@19765 ` 10` ``` "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))" ``` obua@16782 ` 11` ``` float :: "int * int \ real" ``` wenzelm@19765 ` 12` ``` "float x = real (fst x) * pow2 (snd x)" ``` obua@16782 ` 13` obua@16782 ` 14` ```lemma pow2_0[simp]: "pow2 0 = 1" ``` obua@16782 ` 15` ```by (simp add: pow2_def) ``` obua@16782 ` 16` obua@16782 ` 17` ```lemma pow2_1[simp]: "pow2 1 = 2" ``` obua@16782 ` 18` ```by (simp add: pow2_def) ``` obua@16782 ` 19` obua@16782 ` 20` ```lemma pow2_neg: "pow2 x = inverse (pow2 (-x))" ``` obua@16782 ` 21` ```by (simp add: pow2_def) ``` obua@16782 ` 22` wenzelm@19765 ` 23` ```lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)" ``` obua@16782 ` 24` ```proof - ``` obua@16782 ` 25` ``` have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith ``` obua@16782 ` 26` ``` have g: "! a b. a - -1 = a + (1::int)" by arith ``` obua@16782 ` 27` ``` have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)" ``` obua@16782 ` 28` ``` apply (auto, induct_tac n) ``` obua@16782 ` 29` ``` apply (simp_all add: pow2_def) ``` obua@16782 ` 30` ``` apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if]) ``` webertj@20217 ` 31` ``` by (auto simp add: h) ``` obua@16782 ` 32` ``` show ?thesis ``` obua@16782 ` 33` ``` proof (induct a) ``` obua@16782 ` 34` ``` case (1 n) ``` obua@16782 ` 35` ``` from pos show ?case by (simp add: ring_eq_simps) ``` obua@16782 ` 36` ``` next ``` obua@16782 ` 37` ``` case (2 n) ``` obua@16782 ` 38` ``` show ?case ``` obua@16782 ` 39` ``` apply (auto) ``` obua@16782 ` 40` ``` apply (subst pow2_neg[of "- int n"]) ``` obua@16782 ` 41` ``` apply (subst pow2_neg[of "-1 - int n"]) ``` obua@16782 ` 42` ``` apply (auto simp add: g pos) ``` obua@16782 ` 43` ``` done ``` wenzelm@19765 ` 44` ``` qed ``` obua@16782 ` 45` ```qed ``` wenzelm@19765 ` 46` obua@16782 ` 47` ```lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)" ``` obua@16782 ` 48` ```proof (induct b) ``` wenzelm@19765 ` 49` ``` case (1 n) ``` obua@16782 ` 50` ``` show ?case ``` obua@16782 ` 51` ``` proof (induct n) ``` obua@16782 ` 52` ``` case 0 ``` obua@16782 ` 53` ``` show ?case by simp ``` obua@16782 ` 54` ``` next ``` obua@16782 ` 55` ``` case (Suc m) ``` obua@16782 ` 56` ``` show ?case by (auto simp add: ring_eq_simps pow2_add1 prems) ``` obua@16782 ` 57` ``` qed ``` obua@16782 ` 58` ```next ``` obua@16782 ` 59` ``` case (2 n) ``` wenzelm@19765 ` 60` ``` show ?case ``` obua@16782 ` 61` ``` proof (induct n) ``` obua@16782 ` 62` ``` case 0 ``` wenzelm@19765 ` 63` ``` show ?case ``` obua@16782 ` 64` ``` apply (auto) ``` obua@16782 ` 65` ``` apply (subst pow2_neg[of "a + -1"]) ``` obua@16782 ` 66` ``` apply (subst pow2_neg[of "-1"]) ``` obua@16782 ` 67` ``` apply (simp) ``` obua@16782 ` 68` ``` apply (insert pow2_add1[of "-a"]) ``` obua@16782 ` 69` ``` apply (simp add: ring_eq_simps) ``` obua@16782 ` 70` ``` apply (subst pow2_neg[of "-a"]) ``` obua@16782 ` 71` ``` apply (simp) ``` obua@16782 ` 72` ``` done ``` obua@16782 ` 73` ``` case (Suc m) ``` wenzelm@19765 ` 74` ``` have a: "int m - (a + -2) = 1 + (int m - a + 1)" by arith ``` obua@16782 ` 75` ``` have b: "int m - -2 = 1 + (int m + 1)" by arith ``` obua@16782 ` 76` ``` show ?case ``` obua@16782 ` 77` ``` apply (auto) ``` obua@16782 ` 78` ``` apply (subst pow2_neg[of "a + (-2 - int m)"]) ``` obua@16782 ` 79` ``` apply (subst pow2_neg[of "-2 - int m"]) ``` obua@16782 ` 80` ``` apply (auto simp add: ring_eq_simps) ``` obua@16782 ` 81` ``` apply (subst a) ``` obua@16782 ` 82` ``` apply (subst b) ``` obua@16782 ` 83` ``` apply (simp only: pow2_add1) ``` obua@16782 ` 84` ``` apply (subst pow2_neg[of "int m - a + 1"]) ``` obua@16782 ` 85` ``` apply (subst pow2_neg[of "int m + 1"]) ``` obua@16782 ` 86` ``` apply auto ``` obua@16782 ` 87` ``` apply (insert prems) ``` obua@16782 ` 88` ``` apply (auto simp add: ring_eq_simps) ``` obua@16782 ` 89` ``` done ``` obua@16782 ` 90` ``` qed ``` obua@16782 ` 91` ```qed ``` obua@16782 ` 92` wenzelm@19765 ` 93` ```lemma "float (a, e) + float (b, e) = float (a + b, e)" ``` obua@16782 ` 94` ```by (simp add: float_def ring_eq_simps) ``` obua@16782 ` 95` wenzelm@19765 ` 96` ```definition ``` obua@16782 ` 97` ``` int_of_real :: "real \ int" ``` wenzelm@19765 ` 98` ``` "int_of_real x = (SOME y. real y = x)" ``` obua@16782 ` 99` ``` real_is_int :: "real \ bool" ``` wenzelm@19765 ` 100` ``` "real_is_int x = (EX (u::int). x = real u)" ``` obua@16782 ` 101` obua@16782 ` 102` ```lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))" ``` obua@16782 ` 103` ```by (auto simp add: real_is_int_def int_of_real_def) ``` obua@16782 ` 104` obua@16782 ` 105` ```lemma float_transfer: "real_is_int ((real a)*(pow2 c)) \ float (a, b) = float (int_of_real ((real a)*(pow2 c)), b - c)" ``` obua@16782 ` 106` ```by (simp add: float_def real_is_int_def2 pow2_add[symmetric]) ``` obua@16782 ` 107` obua@16782 ` 108` ```lemma pow2_int: "pow2 (int c) = (2::real)^c" ``` obua@16782 ` 109` ```by (simp add: pow2_def) ``` obua@16782 ` 110` wenzelm@19765 ` 111` ```lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)" ``` obua@16782 ` 112` ```by (simp add: float_def pow2_int[symmetric] pow2_add[symmetric]) ``` obua@16782 ` 113` obua@16782 ` 114` ```lemma real_is_int_real[simp]: "real_is_int (real (x::int))" ``` obua@16782 ` 115` ```by (auto simp add: real_is_int_def int_of_real_def) ``` obua@16782 ` 116` obua@16782 ` 117` ```lemma int_of_real_real[simp]: "int_of_real (real x) = x" ``` obua@16782 ` 118` ```by (simp add: int_of_real_def) ``` obua@16782 ` 119` obua@16782 ` 120` ```lemma real_int_of_real[simp]: "real_is_int x \ real (int_of_real x) = x" ``` obua@16782 ` 121` ```by (auto simp add: int_of_real_def real_is_int_def) ``` obua@16782 ` 122` obua@16782 ` 123` ```lemma real_is_int_add_int_of_real: "real_is_int a \ real_is_int b \ (int_of_real (a+b)) = (int_of_real a) + (int_of_real b)" ``` obua@16782 ` 124` ```by (auto simp add: int_of_real_def real_is_int_def) ``` obua@16782 ` 125` obua@16782 ` 126` ```lemma real_is_int_add[simp]: "real_is_int a \ real_is_int b \ real_is_int (a+b)" ``` obua@16782 ` 127` ```apply (subst real_is_int_def2) ``` obua@16782 ` 128` ```apply (simp add: real_is_int_add_int_of_real real_int_of_real) ``` obua@16782 ` 129` ```done ``` obua@16782 ` 130` obua@16782 ` 131` ```lemma int_of_real_sub: "real_is_int a \ real_is_int b \ (int_of_real (a-b)) = (int_of_real a) - (int_of_real b)" ``` obua@16782 ` 132` ```by (auto simp add: int_of_real_def real_is_int_def) ``` obua@16782 ` 133` obua@16782 ` 134` ```lemma real_is_int_sub[simp]: "real_is_int a \ real_is_int b \ real_is_int (a-b)" ``` obua@16782 ` 135` ```apply (subst real_is_int_def2) ``` obua@16782 ` 136` ```apply (simp add: int_of_real_sub real_int_of_real) ``` obua@16782 ` 137` ```done ``` obua@16782 ` 138` obua@16782 ` 139` ```lemma real_is_int_rep: "real_is_int x \ ?! (a::int). real a = x" ``` obua@16782 ` 140` ```by (auto simp add: real_is_int_def) ``` obua@16782 ` 141` wenzelm@19765 ` 142` ```lemma int_of_real_mult: ``` obua@16782 ` 143` ``` assumes "real_is_int a" "real_is_int b" ``` obua@16782 ` 144` ``` shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)" ``` obua@16782 ` 145` ```proof - ``` obua@16782 ` 146` ``` from prems have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto) ``` obua@16782 ` 147` ``` from prems have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto) ``` obua@16782 ` 148` ``` from a obtain a'::int where a':"a = real a'" by auto ``` obua@16782 ` 149` ``` from b obtain b'::int where b':"b = real b'" by auto ``` obua@16782 ` 150` ``` have r: "real a' * real b' = real (a' * b')" by auto ``` obua@16782 ` 151` ``` show ?thesis ``` obua@16782 ` 152` ``` apply (simp add: a' b') ``` obua@16782 ` 153` ``` apply (subst r) ``` obua@16782 ` 154` ``` apply (simp only: int_of_real_real) ``` obua@16782 ` 155` ``` done ``` obua@16782 ` 156` ```qed ``` obua@16782 ` 157` obua@16782 ` 158` ```lemma real_is_int_mult[simp]: "real_is_int a \ real_is_int b \ real_is_int (a*b)" ``` obua@16782 ` 159` ```apply (subst real_is_int_def2) ``` obua@16782 ` 160` ```apply (simp add: int_of_real_mult) ``` obua@16782 ` 161` ```done ``` obua@16782 ` 162` obua@16782 ` 163` ```lemma real_is_int_0[simp]: "real_is_int (0::real)" ``` obua@16782 ` 164` ```by (simp add: real_is_int_def int_of_real_def) ``` obua@16782 ` 165` obua@16782 ` 166` ```lemma real_is_int_1[simp]: "real_is_int (1::real)" ``` obua@16782 ` 167` ```proof - ``` obua@16782 ` 168` ``` have "real_is_int (1::real) = real_is_int(real (1::int))" by auto ``` obua@16782 ` 169` ``` also have "\ = True" by (simp only: real_is_int_real) ``` obua@16782 ` 170` ``` ultimately show ?thesis by auto ``` obua@16782 ` 171` ```qed ``` obua@16782 ` 172` obua@16782 ` 173` ```lemma real_is_int_n1: "real_is_int (-1::real)" ``` obua@16782 ` 174` ```proof - ``` obua@16782 ` 175` ``` have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto ``` obua@16782 ` 176` ``` also have "\ = True" by (simp only: real_is_int_real) ``` obua@16782 ` 177` ``` ultimately show ?thesis by auto ``` obua@16782 ` 178` ```qed ``` obua@16782 ` 179` obua@16782 ` 180` ```lemma real_is_int_number_of[simp]: "real_is_int ((number_of::bin\real) x)" ``` obua@16782 ` 181` ```proof - ``` obua@16782 ` 182` ``` have neg1: "real_is_int (-1::real)" ``` obua@16782 ` 183` ``` proof - ``` obua@16782 ` 184` ``` have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto ``` obua@16782 ` 185` ``` also have "\ = True" by (simp only: real_is_int_real) ``` obua@16782 ` 186` ``` ultimately show ?thesis by auto ``` obua@16782 ` 187` ``` qed ``` wenzelm@19765 ` 188` wenzelm@19765 ` 189` ``` { ``` obua@16782 ` 190` ``` fix x::int ``` obua@16782 ` 191` ``` have "!! y. real_is_int ((number_of::bin\real) (Abs_Bin x))" ``` obua@16782 ` 192` ``` apply (simp add: number_of_eq) ``` obua@16782 ` 193` ``` apply (subst Abs_Bin_inverse) ``` obua@16782 ` 194` ``` apply (simp add: Bin_def) ``` obua@16782 ` 195` ``` apply (induct x) ``` obua@16782 ` 196` ``` apply (induct_tac n) ``` obua@16782 ` 197` ``` apply (simp) ``` obua@16782 ` 198` ``` apply (simp) ``` obua@16782 ` 199` ``` apply (induct_tac n) ``` obua@16782 ` 200` ``` apply (simp add: neg1) ``` obua@16782 ` 201` ``` proof - ``` obua@16782 ` 202` ``` fix n :: nat ``` obua@16782 ` 203` ``` assume rn: "(real_is_int (of_int (- (int (Suc n)))))" ``` obua@16782 ` 204` ``` have s: "-(int (Suc (Suc n))) = -1 + - (int (Suc n))" by simp ``` obua@16782 ` 205` ``` show "real_is_int (of_int (- (int (Suc (Suc n)))))" ``` wenzelm@19765 ` 206` ``` apply (simp only: s of_int_add) ``` wenzelm@19765 ` 207` ``` apply (rule real_is_int_add) ``` wenzelm@19765 ` 208` ``` apply (simp add: neg1) ``` wenzelm@19765 ` 209` ``` apply (simp only: rn) ``` wenzelm@19765 ` 210` ``` done ``` obua@16782 ` 211` ``` qed ``` obua@16782 ` 212` ``` } ``` obua@16782 ` 213` ``` note Abs_Bin = this ``` obua@16782 ` 214` ``` { ``` obua@16782 ` 215` ``` fix x :: bin ``` obua@16782 ` 216` ``` have "? u. x = Abs_Bin u" ``` obua@16782 ` 217` ``` apply (rule exI[where x = "Rep_Bin x"]) ``` obua@16782 ` 218` ``` apply (simp add: Rep_Bin_inverse) ``` obua@16782 ` 219` ``` done ``` obua@16782 ` 220` ``` } ``` obua@16782 ` 221` ``` then obtain u::int where "x = Abs_Bin u" by auto ``` obua@16782 ` 222` ``` with Abs_Bin show ?thesis by auto ``` obua@16782 ` 223` ```qed ``` obua@16782 ` 224` obua@16782 ` 225` ```lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)" ``` obua@16782 ` 226` ```by (simp add: int_of_real_def) ``` obua@16782 ` 227` obua@16782 ` 228` ```lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)" ``` wenzelm@19765 ` 229` ```proof - ``` obua@16782 ` 230` ``` have 1: "(1::real) = real (1::int)" by auto ``` obua@16782 ` 231` ``` show ?thesis by (simp only: 1 int_of_real_real) ``` obua@16782 ` 232` ```qed ``` obua@16782 ` 233` obua@16782 ` 234` ```lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b" ``` obua@16782 ` 235` ```proof - ``` obua@16782 ` 236` ``` have "real_is_int (number_of b)" by simp ``` obua@16782 ` 237` ``` then have uu: "?! u::int. number_of b = real u" by (auto simp add: real_is_int_rep) ``` obua@16782 ` 238` ``` then obtain u::int where u:"number_of b = real u" by auto ``` wenzelm@19765 ` 239` ``` have "number_of b = real ((number_of b)::int)" ``` obua@16782 ` 240` ``` by (simp add: number_of_eq real_of_int_def) ``` wenzelm@19765 ` 241` ``` have ub: "number_of b = real ((number_of b)::int)" ``` obua@16782 ` 242` ``` by (simp add: number_of_eq real_of_int_def) ``` obua@16782 ` 243` ``` from uu u ub have unb: "u = number_of b" ``` obua@16782 ` 244` ``` by blast ``` obua@16782 ` 245` ``` have "int_of_real (number_of b) = u" by (simp add: u) ``` obua@16782 ` 246` ``` with unb show ?thesis by simp ``` obua@16782 ` 247` ```qed ``` obua@16782 ` 248` obua@16782 ` 249` ```lemma float_transfer_even: "even a \ float (a, b) = float (a div 2, b+1)" ``` obua@16782 ` 250` ``` apply (subst float_transfer[where a="a" and b="b" and c="-1", simplified]) ``` obua@16782 ` 251` ``` apply (simp_all add: pow2_def even_def real_is_int_def ring_eq_simps) ``` obua@16782 ` 252` ``` apply (auto) ``` obua@16782 ` 253` ```proof - ``` obua@16782 ` 254` ``` fix q::int ``` obua@16782 ` 255` ``` have a:"b - (-1\int) = (1\int) + b" by arith ``` wenzelm@19765 ` 256` ``` show "(float (q, (b - (-1\int)))) = (float (q, ((1\int) + b)))" ``` obua@16782 ` 257` ``` by (simp add: a) ``` obua@16782 ` 258` ```qed ``` wenzelm@19765 ` 259` obua@16782 ` 260` ```consts ``` obua@16782 ` 261` ``` norm_float :: "int*int \ int*int" ``` obua@16782 ` 262` obua@16782 ` 263` ```lemma int_div_zdiv: "int (a div b) = (int a) div (int b)" ``` obua@16782 ` 264` ```apply (subst split_div, auto) ``` obua@16782 ` 265` ```apply (subst split_zdiv, auto) ``` obua@16782 ` 266` ```apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient) ``` obua@16782 ` 267` ```apply (auto simp add: IntDiv.quorem_def int_eq_of_nat) ``` obua@16782 ` 268` ```done ``` obua@16782 ` 269` obua@16782 ` 270` ```lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)" ``` obua@16782 ` 271` ```apply (subst split_mod, auto) ``` obua@16782 ` 272` ```apply (subst split_zmod, auto) ``` obua@16782 ` 273` ```apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia in IntDiv.unique_remainder) ``` obua@16782 ` 274` ```apply (auto simp add: IntDiv.quorem_def int_eq_of_nat) ``` obua@16782 ` 275` ```done ``` obua@16782 ` 276` obua@16782 ` 277` ```lemma abs_div_2_less: "a \ 0 \ a \ -1 \ abs((a::int) div 2) < abs a" ``` obua@16782 ` 278` ```by arith ``` obua@16782 ` 279` obua@16782 ` 280` ```lemma terminating_norm_float: "\a. (a::int) \ 0 \ even a \ a \ 0 \ \a div 2\ < \a\" ``` obua@16782 ` 281` ```apply (auto) ``` obua@16782 ` 282` ```apply (rule abs_div_2_less) ``` obua@16782 ` 283` ```apply (auto) ``` obua@16782 ` 284` ```done ``` obua@16782 ` 285` wenzelm@19765 ` 286` ```ML {* simp_depth_limit := 2 *} ``` obua@16782 ` 287` ```recdef norm_float "measure (% (a,b). nat (abs a))" ``` obua@16782 ` 288` ``` "norm_float (a,b) = (if (a \ 0) & (even a) then norm_float (a div 2, b+1) else (if a=0 then (0,0) else (a,b)))" ``` obua@16782 ` 289` ```(hints simp: terminating_norm_float) ``` obua@16782 ` 290` ```ML {* simp_depth_limit := 1000 *} ``` obua@16782 ` 291` obua@16782 ` 292` ```lemma norm_float: "float x = float (norm_float x)" ``` obua@16782 ` 293` ```proof - ``` obua@16782 ` 294` ``` { ``` wenzelm@19765 ` 295` ``` fix a b :: int ``` wenzelm@19765 ` 296` ``` have norm_float_pair: "float (a,b) = float (norm_float (a,b))" ``` obua@16782 ` 297` ``` proof (induct a b rule: norm_float.induct) ``` obua@16782 ` 298` ``` case (1 u v) ``` wenzelm@19765 ` 299` ``` show ?case ``` obua@16782 ` 300` ``` proof cases ``` wenzelm@19765 ` 301` ``` assume u: "u \ 0 \ even u" ``` wenzelm@19765 ` 302` ``` with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2, v + 1))" by auto ``` wenzelm@19765 ` 303` ``` with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even) ``` wenzelm@19765 ` 304` ``` then show ?thesis ``` wenzelm@19765 ` 305` ``` apply (subst norm_float.simps) ``` wenzelm@19765 ` 306` ``` apply (simp add: ind) ``` wenzelm@19765 ` 307` ``` done ``` obua@16782 ` 308` ``` next ``` wenzelm@19765 ` 309` ``` assume "~(u \ 0 \ even u)" ``` wenzelm@19765 ` 310` ``` then show ?thesis ``` wenzelm@19765 ` 311` ``` by (simp add: prems float_def) ``` obua@16782 ` 312` ``` qed ``` obua@16782 ` 313` ``` qed ``` obua@16782 ` 314` ``` } ``` obua@16782 ` 315` ``` note helper = this ``` obua@16782 ` 316` ``` have "? a b. x = (a,b)" by auto ``` obua@16782 ` 317` ``` then obtain a b where "x = (a, b)" by blast ``` obua@16782 ` 318` ``` then show ?thesis by (simp only: helper) ``` obua@16782 ` 319` ```qed ``` obua@16782 ` 320` obua@16782 ` 321` ```lemma pow2_int: "pow2 (int n) = 2^n" ``` obua@16782 ` 322` ``` by (simp add: pow2_def) ``` obua@16782 ` 323` wenzelm@19765 ` 324` ```lemma float_add: ``` wenzelm@19765 ` 325` ``` "float (a1, e1) + float (a2, e2) = ``` wenzelm@19765 ` 326` ``` (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1) ``` obua@16782 ` 327` ``` else float (a1*2^(nat (e1-e2))+a2, e2))" ``` obua@16782 ` 328` ``` apply (simp add: float_def ring_eq_simps) ``` obua@16782 ` 329` ``` apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric]) ``` obua@16782 ` 330` ``` done ``` obua@16782 ` 331` obua@16782 ` 332` ```lemma float_mult: ``` wenzelm@19765 ` 333` ``` "float (a1, e1) * float (a2, e2) = ``` obua@16782 ` 334` ``` (float (a1 * a2, e1 + e2))" ``` obua@16782 ` 335` ``` by (simp add: float_def pow2_add) ``` obua@16782 ` 336` obua@16782 ` 337` ```lemma float_minus: ``` obua@16782 ` 338` ``` "- (float (a,b)) = float (-a, b)" ``` obua@16782 ` 339` ``` by (simp add: float_def) ``` obua@16782 ` 340` obua@16782 ` 341` ```lemma zero_less_pow2: ``` obua@16782 ` 342` ``` "0 < pow2 x" ``` obua@16782 ` 343` ```proof - ``` obua@16782 ` 344` ``` { ``` obua@16782 ` 345` ``` fix y ``` wenzelm@19765 ` 346` ``` have "0 <= y \ 0 < pow2 y" ``` obua@16782 ` 347` ``` by (induct y, induct_tac n, simp_all add: pow2_add) ``` obua@16782 ` 348` ``` } ``` obua@16782 ` 349` ``` note helper=this ``` obua@16782 ` 350` ``` show ?thesis ``` obua@16782 ` 351` ``` apply (case_tac "0 <= x") ``` obua@16782 ` 352` ``` apply (simp add: helper) ``` obua@16782 ` 353` ``` apply (subst pow2_neg) ``` obua@16782 ` 354` ``` apply (simp add: helper) ``` obua@16782 ` 355` ``` done ``` obua@16782 ` 356` ```qed ``` obua@16782 ` 357` obua@16782 ` 358` ```lemma zero_le_float: ``` obua@16782 ` 359` ``` "(0 <= float (a,b)) = (0 <= a)" ``` obua@16782 ` 360` ``` apply (auto simp add: float_def) ``` wenzelm@19765 ` 361` ``` apply (auto simp add: zero_le_mult_iff zero_less_pow2) ``` obua@16782 ` 362` ``` apply (insert zero_less_pow2[of b]) ``` obua@16782 ` 363` ``` apply (simp_all) ``` obua@16782 ` 364` ``` done ``` obua@16782 ` 365` obua@16782 ` 366` ```lemma float_le_zero: ``` obua@16782 ` 367` ``` "(float (a,b) <= 0) = (a <= 0)" ``` obua@16782 ` 368` ``` apply (auto simp add: float_def) ``` obua@16782 ` 369` ``` apply (auto simp add: mult_le_0_iff) ``` obua@16782 ` 370` ``` apply (insert zero_less_pow2[of b]) ``` obua@16782 ` 371` ``` apply auto ``` obua@16782 ` 372` ``` done ``` obua@16782 ` 373` obua@16782 ` 374` ```lemma float_abs: ``` obua@16782 ` 375` ``` "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))" ``` obua@16782 ` 376` ``` apply (auto simp add: abs_if) ``` obua@16782 ` 377` ``` apply (simp_all add: zero_le_float[symmetric, of a b] float_minus) ``` obua@16782 ` 378` ``` done ``` obua@16782 ` 379` obua@16782 ` 380` ```lemma float_zero: ``` obua@16782 ` 381` ``` "float (0, b) = 0" ``` obua@16782 ` 382` ``` by (simp add: float_def) ``` obua@16782 ` 383` obua@16782 ` 384` ```lemma float_pprt: ``` obua@16782 ` 385` ``` "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))" ``` obua@16782 ` 386` ``` by (auto simp add: zero_le_float float_le_zero float_zero) ``` obua@16782 ` 387` obua@16782 ` 388` ```lemma float_nprt: ``` obua@16782 ` 389` ``` "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))" ``` obua@16782 ` 390` ``` by (auto simp add: zero_le_float float_le_zero float_zero) ``` obua@16782 ` 391` obua@16782 ` 392` ```lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1" ``` obua@16782 ` 393` ``` by auto ``` wenzelm@19765 ` 394` obua@16782 ` 395` ```lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)" ``` obua@16782 ` 396` ``` by simp ``` obua@16782 ` 397` obua@16782 ` 398` ```lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)" ``` obua@16782 ` 399` ``` by simp ``` obua@16782 ` 400` obua@16782 ` 401` ```lemma mult_left_one: "1 * a = (a::'a::semiring_1)" ``` obua@16782 ` 402` ``` by simp ``` obua@16782 ` 403` obua@16782 ` 404` ```lemma mult_right_one: "a * 1 = (a::'a::semiring_1)" ``` obua@16782 ` 405` ``` by simp ``` obua@16782 ` 406` obua@16782 ` 407` ```lemma int_pow_0: "(a::int)^(Numeral0) = 1" ``` obua@16782 ` 408` ``` by simp ``` obua@16782 ` 409` obua@16782 ` 410` ```lemma int_pow_1: "(a::int)^(Numeral1) = a" ``` obua@16782 ` 411` ``` by simp ``` obua@16782 ` 412` obua@16782 ` 413` ```lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0" ``` obua@16782 ` 414` ``` by simp ``` obua@16782 ` 415` obua@16782 ` 416` ```lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1" ``` obua@16782 ` 417` ``` by simp ``` obua@16782 ` 418` obua@16782 ` 419` ```lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0" ``` obua@16782 ` 420` ``` by simp ``` obua@16782 ` 421` obua@16782 ` 422` ```lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1" ``` obua@16782 ` 423` ``` by simp ``` obua@16782 ` 424` obua@16782 ` 425` ```lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1" ``` obua@16782 ` 426` ``` by simp ``` obua@16782 ` 427` obua@16782 ` 428` ```lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1" ``` obua@16782 ` 429` ```proof - ``` obua@16782 ` 430` ``` have 1:"((-1)::nat) = 0" ``` obua@16782 ` 431` ``` by simp ``` obua@16782 ` 432` ``` show ?thesis by (simp add: 1) ``` obua@16782 ` 433` ```qed ``` obua@16782 ` 434` obua@16782 ` 435` ```lemma fst_cong: "a=a' \ fst (a,b) = fst (a',b)" ``` obua@16782 ` 436` ``` by simp ``` obua@16782 ` 437` obua@16782 ` 438` ```lemma snd_cong: "b=b' \ snd (a,b) = snd (a,b')" ``` obua@16782 ` 439` ``` by simp ``` obua@16782 ` 440` obua@16782 ` 441` ```lemma lift_bool: "x \ x=True" ``` obua@16782 ` 442` ``` by simp ``` obua@16782 ` 443` obua@16782 ` 444` ```lemma nlift_bool: "~x \ x=False" ``` obua@16782 ` 445` ``` by simp ``` obua@16782 ` 446` obua@16782 ` 447` ```lemma not_false_eq_true: "(~ False) = True" by simp ``` obua@16782 ` 448` obua@16782 ` 449` ```lemma not_true_eq_false: "(~ True) = False" by simp ``` obua@16782 ` 450` obua@16782 ` 451` wenzelm@19765 ` 452` ```lemmas binarith = ``` obua@16782 ` 453` ``` Pls_0_eq Min_1_eq ``` wenzelm@19765 ` 454` ``` bin_pred_Pls bin_pred_Min bin_pred_1 bin_pred_0 ``` obua@16782 ` 455` ``` bin_succ_Pls bin_succ_Min bin_succ_1 bin_succ_0 ``` obua@16782 ` 456` ``` bin_add_Pls bin_add_Min bin_add_BIT_0 bin_add_BIT_10 ``` wenzelm@19765 ` 457` ``` bin_add_BIT_11 bin_minus_Pls bin_minus_Min bin_minus_1 ``` wenzelm@19765 ` 458` ``` bin_minus_0 bin_mult_Pls bin_mult_Min bin_mult_1 bin_mult_0 ``` obua@16782 ` 459` ``` bin_add_Pls_right bin_add_Min_right ``` obua@16782 ` 460` obua@16782 ` 461` ```lemma int_eq_number_of_eq: "(((number_of v)::int)=(number_of w)) = iszero ((number_of (bin_add v (bin_minus w)))::int)" ``` obua@16782 ` 462` ``` by simp ``` obua@16782 ` 463` wenzelm@19765 ` 464` ```lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)" ``` obua@16782 ` 465` ``` by (simp only: iszero_number_of_Pls) ``` obua@16782 ` 466` obua@16782 ` 467` ```lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))" ``` obua@16782 ` 468` ``` by simp ``` obua@16782 ` 469` obua@16782 ` 470` ```lemma int_iszero_number_of_0: "iszero ((number_of (w BIT bit.B0))::int) = iszero ((number_of w)::int)" ``` obua@16782 ` 471` ``` by simp ``` obua@16782 ` 472` wenzelm@19765 ` 473` ```lemma int_iszero_number_of_1: "\ iszero ((number_of (w BIT bit.B1))::int)" ``` obua@16782 ` 474` ``` by simp ``` obua@16782 ` 475` obua@16782 ` 476` ```lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (bin_add x (bin_minus y)))::int)" ``` obua@16782 ` 477` ``` by simp ``` obua@16782 ` 478` wenzelm@19765 ` 479` ```lemma int_not_neg_number_of_Pls: "\ (neg (Numeral0::int))" ``` obua@16782 ` 480` ``` by simp ``` obua@16782 ` 481` obua@16782 ` 482` ```lemma int_neg_number_of_Min: "neg (-1::int)" ``` obua@16782 ` 483` ``` by simp ``` obua@16782 ` 484` obua@16782 ` 485` ```lemma int_neg_number_of_BIT: "neg ((number_of (w BIT x))::int) = neg ((number_of w)::int)" ``` obua@16782 ` 486` ``` by simp ``` obua@16782 ` 487` obua@16782 ` 488` ```lemma int_le_number_of_eq: "(((number_of x)::int) \ number_of y) = (\ neg ((number_of (bin_add y (bin_minus x)))::int))" ``` obua@16782 ` 489` ``` by simp ``` obua@16782 ` 490` wenzelm@19765 ` 491` ```lemmas intarithrel = ``` wenzelm@19765 ` 492` ``` int_eq_number_of_eq ``` wenzelm@19765 ` 493` ``` lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_0 ``` obua@16782 ` 494` ``` lift_bool[OF int_iszero_number_of_1] int_less_number_of_eq_neg nlift_bool[OF int_not_neg_number_of_Pls] lift_bool[OF int_neg_number_of_Min] ``` obua@16782 ` 495` ``` int_neg_number_of_BIT int_le_number_of_eq ``` obua@16782 ` 496` obua@16782 ` 497` ```lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (bin_add v w)" ``` obua@16782 ` 498` ``` by simp ``` obua@16782 ` 499` obua@16782 ` 500` ```lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (bin_add v (bin_minus w))" ``` obua@16782 ` 501` ``` by simp ``` obua@16782 ` 502` obua@16782 ` 503` ```lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (bin_mult v w)" ``` obua@16782 ` 504` ``` by simp ``` obua@16782 ` 505` obua@16782 ` 506` ```lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (bin_minus v)" ``` obua@16782 ` 507` ``` by simp ``` obua@16782 ` 508` obua@16782 ` 509` ```lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym ``` obua@16782 ` 510` obua@16782 ` 511` ```lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of ``` obua@16782 ` 512` wenzelm@19765 ` 513` ```lemmas powerarith = nat_number_of zpower_number_of_even ``` wenzelm@19765 ` 514` ``` zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring] ``` obua@16782 ` 515` ``` zpower_Pls zpower_Min ``` obua@16782 ` 516` obua@16782 ` 517` ```lemmas floatarith[simplified norm_0_1] = float_add float_mult float_minus float_abs zero_le_float float_pprt float_nprt ``` obua@16782 ` 518` obua@16782 ` 519` ```(* for use with the compute oracle *) ``` obua@16782 ` 520` ```lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false ``` obua@16782 ` 521` obua@16782 ` 522` ```end ```