src/HOL/Archimedean_Field.thy
 author haftmann Fri Jun 11 17:14:02 2010 +0200 (2010-06-11) changeset 37407 61dd8c145da7 parent 35028 108662d50512 child 37765 26bdfb7b680b permissions -rw-r--r--
declare lex_prod_def [code del]
 huffman@30096 ` 1` ```(* Title: Archimedean_Field.thy ``` huffman@30096 ` 2` ``` Author: Brian Huffman ``` huffman@30096 ` 3` ```*) ``` huffman@30096 ` 4` huffman@30096 ` 5` ```header {* Archimedean Fields, Floor and Ceiling Functions *} ``` huffman@30096 ` 6` huffman@30096 ` 7` ```theory Archimedean_Field ``` huffman@30096 ` 8` ```imports Main ``` huffman@30096 ` 9` ```begin ``` huffman@30096 ` 10` huffman@30096 ` 11` ```subsection {* Class of Archimedean fields *} ``` huffman@30096 ` 12` huffman@30096 ` 13` ```text {* Archimedean fields have no infinite elements. *} ``` huffman@30096 ` 14` haftmann@35028 ` 15` ```class archimedean_field = linordered_field + number_ring + ``` huffman@30096 ` 16` ``` assumes ex_le_of_int: "\z. x \ of_int z" ``` huffman@30096 ` 17` huffman@30096 ` 18` ```lemma ex_less_of_int: ``` huffman@30096 ` 19` ``` fixes x :: "'a::archimedean_field" shows "\z. x < of_int z" ``` huffman@30096 ` 20` ```proof - ``` huffman@30096 ` 21` ``` from ex_le_of_int obtain z where "x \ of_int z" .. ``` huffman@30096 ` 22` ``` then have "x < of_int (z + 1)" by simp ``` huffman@30096 ` 23` ``` then show ?thesis .. ``` huffman@30096 ` 24` ```qed ``` huffman@30096 ` 25` huffman@30096 ` 26` ```lemma ex_of_int_less: ``` huffman@30096 ` 27` ``` fixes x :: "'a::archimedean_field" shows "\z. of_int z < x" ``` huffman@30096 ` 28` ```proof - ``` huffman@30096 ` 29` ``` from ex_less_of_int obtain z where "- x < of_int z" .. ``` huffman@30096 ` 30` ``` then have "of_int (- z) < x" by simp ``` huffman@30096 ` 31` ``` then show ?thesis .. ``` huffman@30096 ` 32` ```qed ``` huffman@30096 ` 33` huffman@30096 ` 34` ```lemma ex_less_of_nat: ``` huffman@30096 ` 35` ``` fixes x :: "'a::archimedean_field" shows "\n. x < of_nat n" ``` huffman@30096 ` 36` ```proof - ``` huffman@30096 ` 37` ``` obtain z where "x < of_int z" using ex_less_of_int .. ``` huffman@30096 ` 38` ``` also have "\ \ of_int (int (nat z))" by simp ``` huffman@30096 ` 39` ``` also have "\ = of_nat (nat z)" by (simp only: of_int_of_nat_eq) ``` huffman@30096 ` 40` ``` finally show ?thesis .. ``` huffman@30096 ` 41` ```qed ``` huffman@30096 ` 42` huffman@30096 ` 43` ```lemma ex_le_of_nat: ``` huffman@30096 ` 44` ``` fixes x :: "'a::archimedean_field" shows "\n. x \ of_nat n" ``` huffman@30096 ` 45` ```proof - ``` huffman@30096 ` 46` ``` obtain n where "x < of_nat n" using ex_less_of_nat .. ``` huffman@30096 ` 47` ``` then have "x \ of_nat n" by simp ``` huffman@30096 ` 48` ``` then show ?thesis .. ``` huffman@30096 ` 49` ```qed ``` huffman@30096 ` 50` huffman@30096 ` 51` ```text {* Archimedean fields have no infinitesimal elements. *} ``` huffman@30096 ` 52` huffman@30096 ` 53` ```lemma ex_inverse_of_nat_Suc_less: ``` huffman@30096 ` 54` ``` fixes x :: "'a::archimedean_field" ``` huffman@30096 ` 55` ``` assumes "0 < x" shows "\n. inverse (of_nat (Suc n)) < x" ``` huffman@30096 ` 56` ```proof - ``` huffman@30096 ` 57` ``` from `0 < x` have "0 < inverse x" ``` huffman@30096 ` 58` ``` by (rule positive_imp_inverse_positive) ``` huffman@30096 ` 59` ``` obtain n where "inverse x < of_nat n" ``` huffman@30096 ` 60` ``` using ex_less_of_nat .. ``` huffman@30096 ` 61` ``` then obtain m where "inverse x < of_nat (Suc m)" ``` huffman@30096 ` 62` ``` using `0 < inverse x` by (cases n) (simp_all del: of_nat_Suc) ``` huffman@30096 ` 63` ``` then have "inverse (of_nat (Suc m)) < inverse (inverse x)" ``` huffman@30096 ` 64` ``` using `0 < inverse x` by (rule less_imp_inverse_less) ``` huffman@30096 ` 65` ``` then have "inverse (of_nat (Suc m)) < x" ``` huffman@30096 ` 66` ``` using `0 < x` by (simp add: nonzero_inverse_inverse_eq) ``` huffman@30096 ` 67` ``` then show ?thesis .. ``` huffman@30096 ` 68` ```qed ``` huffman@30096 ` 69` huffman@30096 ` 70` ```lemma ex_inverse_of_nat_less: ``` huffman@30096 ` 71` ``` fixes x :: "'a::archimedean_field" ``` huffman@30096 ` 72` ``` assumes "0 < x" shows "\n>0. inverse (of_nat n) < x" ``` huffman@30096 ` 73` ``` using ex_inverse_of_nat_Suc_less [OF `0 < x`] by auto ``` huffman@30096 ` 74` huffman@30096 ` 75` ```lemma ex_less_of_nat_mult: ``` huffman@30096 ` 76` ``` fixes x :: "'a::archimedean_field" ``` huffman@30096 ` 77` ``` assumes "0 < x" shows "\n. y < of_nat n * x" ``` huffman@30096 ` 78` ```proof - ``` huffman@30096 ` 79` ``` obtain n where "y / x < of_nat n" using ex_less_of_nat .. ``` huffman@30096 ` 80` ``` with `0 < x` have "y < of_nat n * x" by (simp add: pos_divide_less_eq) ``` huffman@30096 ` 81` ``` then show ?thesis .. ``` huffman@30096 ` 82` ```qed ``` huffman@30096 ` 83` huffman@30096 ` 84` huffman@30096 ` 85` ```subsection {* Existence and uniqueness of floor function *} ``` huffman@30096 ` 86` huffman@30096 ` 87` ```lemma exists_least_lemma: ``` huffman@30096 ` 88` ``` assumes "\ P 0" and "\n. P n" ``` huffman@30096 ` 89` ``` shows "\n. \ P n \ P (Suc n)" ``` huffman@30096 ` 90` ```proof - ``` huffman@30096 ` 91` ``` from `\n. P n` have "P (Least P)" by (rule LeastI_ex) ``` huffman@30096 ` 92` ``` with `\ P 0` obtain n where "Least P = Suc n" ``` huffman@30096 ` 93` ``` by (cases "Least P") auto ``` huffman@30096 ` 94` ``` then have "n < Least P" by simp ``` huffman@30096 ` 95` ``` then have "\ P n" by (rule not_less_Least) ``` huffman@30096 ` 96` ``` then have "\ P n \ P (Suc n)" ``` huffman@30096 ` 97` ``` using `P (Least P)` `Least P = Suc n` by simp ``` huffman@30096 ` 98` ``` then show ?thesis .. ``` huffman@30096 ` 99` ```qed ``` huffman@30096 ` 100` huffman@30096 ` 101` ```lemma floor_exists: ``` huffman@30096 ` 102` ``` fixes x :: "'a::archimedean_field" ``` huffman@30096 ` 103` ``` shows "\z. of_int z \ x \ x < of_int (z + 1)" ``` huffman@30096 ` 104` ```proof (cases) ``` huffman@30096 ` 105` ``` assume "0 \ x" ``` huffman@30096 ` 106` ``` then have "\ x < of_nat 0" by simp ``` huffman@30096 ` 107` ``` then have "\n. \ x < of_nat n \ x < of_nat (Suc n)" ``` huffman@30096 ` 108` ``` using ex_less_of_nat by (rule exists_least_lemma) ``` huffman@30096 ` 109` ``` then obtain n where "\ x < of_nat n \ x < of_nat (Suc n)" .. ``` huffman@30096 ` 110` ``` then have "of_int (int n) \ x \ x < of_int (int n + 1)" by simp ``` huffman@30096 ` 111` ``` then show ?thesis .. ``` huffman@30096 ` 112` ```next ``` huffman@30096 ` 113` ``` assume "\ 0 \ x" ``` huffman@30096 ` 114` ``` then have "\ - x \ of_nat 0" by simp ``` huffman@30096 ` 115` ``` then have "\n. \ - x \ of_nat n \ - x \ of_nat (Suc n)" ``` huffman@30096 ` 116` ``` using ex_le_of_nat by (rule exists_least_lemma) ``` huffman@30096 ` 117` ``` then obtain n where "\ - x \ of_nat n \ - x \ of_nat (Suc n)" .. ``` huffman@30096 ` 118` ``` then have "of_int (- int n - 1) \ x \ x < of_int (- int n - 1 + 1)" by simp ``` huffman@30096 ` 119` ``` then show ?thesis .. ``` huffman@30096 ` 120` ```qed ``` huffman@30096 ` 121` huffman@30096 ` 122` ```lemma floor_exists1: ``` huffman@30096 ` 123` ``` fixes x :: "'a::archimedean_field" ``` huffman@30096 ` 124` ``` shows "\!z. of_int z \ x \ x < of_int (z + 1)" ``` huffman@30096 ` 125` ```proof (rule ex_ex1I) ``` huffman@30096 ` 126` ``` show "\z. of_int z \ x \ x < of_int (z + 1)" ``` huffman@30096 ` 127` ``` by (rule floor_exists) ``` huffman@30096 ` 128` ```next ``` huffman@30096 ` 129` ``` fix y z assume ``` huffman@30096 ` 130` ``` "of_int y \ x \ x < of_int (y + 1)" ``` huffman@30096 ` 131` ``` "of_int z \ x \ x < of_int (z + 1)" ``` huffman@30096 ` 132` ``` then have ``` huffman@30096 ` 133` ``` "of_int y \ x" "x < of_int (y + 1)" ``` huffman@30096 ` 134` ``` "of_int z \ x" "x < of_int (z + 1)" ``` huffman@30096 ` 135` ``` by simp_all ``` huffman@30096 ` 136` ``` from le_less_trans [OF `of_int y \ x` `x < of_int (z + 1)`] ``` huffman@30096 ` 137` ``` le_less_trans [OF `of_int z \ x` `x < of_int (y + 1)`] ``` huffman@30096 ` 138` ``` show "y = z" by (simp del: of_int_add) ``` huffman@30096 ` 139` ```qed ``` huffman@30096 ` 140` huffman@30096 ` 141` huffman@30096 ` 142` ```subsection {* Floor function *} ``` huffman@30096 ` 143` huffman@30096 ` 144` ```definition ``` huffman@30096 ` 145` ``` floor :: "'a::archimedean_field \ int" where ``` huffman@30096 ` 146` ``` [code del]: "floor x = (THE z. of_int z \ x \ x < of_int (z + 1))" ``` huffman@30096 ` 147` huffman@30096 ` 148` ```notation (xsymbols) ``` huffman@30096 ` 149` ``` floor ("\_\") ``` huffman@30096 ` 150` huffman@30096 ` 151` ```notation (HTML output) ``` huffman@30096 ` 152` ``` floor ("\_\") ``` huffman@30096 ` 153` huffman@30096 ` 154` ```lemma floor_correct: "of_int (floor x) \ x \ x < of_int (floor x + 1)" ``` huffman@30096 ` 155` ``` unfolding floor_def using floor_exists1 by (rule theI') ``` huffman@30096 ` 156` huffman@30096 ` 157` ```lemma floor_unique: "\of_int z \ x; x < of_int z + 1\ \ floor x = z" ``` huffman@30096 ` 158` ``` using floor_correct [of x] floor_exists1 [of x] by auto ``` huffman@30096 ` 159` huffman@30096 ` 160` ```lemma of_int_floor_le: "of_int (floor x) \ x" ``` huffman@30096 ` 161` ``` using floor_correct .. ``` huffman@30096 ` 162` huffman@30096 ` 163` ```lemma le_floor_iff: "z \ floor x \ of_int z \ x" ``` huffman@30096 ` 164` ```proof ``` huffman@30096 ` 165` ``` assume "z \ floor x" ``` huffman@30096 ` 166` ``` then have "(of_int z :: 'a) \ of_int (floor x)" by simp ``` huffman@30096 ` 167` ``` also have "of_int (floor x) \ x" by (rule of_int_floor_le) ``` huffman@30096 ` 168` ``` finally show "of_int z \ x" . ``` huffman@30096 ` 169` ```next ``` huffman@30096 ` 170` ``` assume "of_int z \ x" ``` huffman@30096 ` 171` ``` also have "x < of_int (floor x + 1)" using floor_correct .. ``` huffman@30096 ` 172` ``` finally show "z \ floor x" by (simp del: of_int_add) ``` huffman@30096 ` 173` ```qed ``` huffman@30096 ` 174` huffman@30096 ` 175` ```lemma floor_less_iff: "floor x < z \ x < of_int z" ``` huffman@30096 ` 176` ``` by (simp add: not_le [symmetric] le_floor_iff) ``` huffman@30096 ` 177` huffman@30096 ` 178` ```lemma less_floor_iff: "z < floor x \ of_int z + 1 \ x" ``` huffman@30096 ` 179` ``` using le_floor_iff [of "z + 1" x] by auto ``` huffman@30096 ` 180` huffman@30096 ` 181` ```lemma floor_le_iff: "floor x \ z \ x < of_int z + 1" ``` huffman@30096 ` 182` ``` by (simp add: not_less [symmetric] less_floor_iff) ``` huffman@30096 ` 183` huffman@30096 ` 184` ```lemma floor_mono: assumes "x \ y" shows "floor x \ floor y" ``` huffman@30096 ` 185` ```proof - ``` huffman@30096 ` 186` ``` have "of_int (floor x) \ x" by (rule of_int_floor_le) ``` huffman@30096 ` 187` ``` also note `x \ y` ``` huffman@30096 ` 188` ``` finally show ?thesis by (simp add: le_floor_iff) ``` huffman@30096 ` 189` ```qed ``` huffman@30096 ` 190` huffman@30096 ` 191` ```lemma floor_less_cancel: "floor x < floor y \ x < y" ``` huffman@30096 ` 192` ``` by (auto simp add: not_le [symmetric] floor_mono) ``` huffman@30096 ` 193` huffman@30096 ` 194` ```lemma floor_of_int [simp]: "floor (of_int z) = z" ``` huffman@30096 ` 195` ``` by (rule floor_unique) simp_all ``` huffman@30096 ` 196` huffman@30096 ` 197` ```lemma floor_of_nat [simp]: "floor (of_nat n) = int n" ``` huffman@30096 ` 198` ``` using floor_of_int [of "of_nat n"] by simp ``` huffman@30096 ` 199` huffman@30096 ` 200` ```text {* Floor with numerals *} ``` huffman@30096 ` 201` huffman@30096 ` 202` ```lemma floor_zero [simp]: "floor 0 = 0" ``` huffman@30096 ` 203` ``` using floor_of_int [of 0] by simp ``` huffman@30096 ` 204` huffman@30096 ` 205` ```lemma floor_one [simp]: "floor 1 = 1" ``` huffman@30096 ` 206` ``` using floor_of_int [of 1] by simp ``` huffman@30096 ` 207` huffman@30096 ` 208` ```lemma floor_number_of [simp]: "floor (number_of v) = number_of v" ``` huffman@30096 ` 209` ``` using floor_of_int [of "number_of v"] by simp ``` huffman@30096 ` 210` huffman@30096 ` 211` ```lemma zero_le_floor [simp]: "0 \ floor x \ 0 \ x" ``` huffman@30096 ` 212` ``` by (simp add: le_floor_iff) ``` huffman@30096 ` 213` huffman@30096 ` 214` ```lemma one_le_floor [simp]: "1 \ floor x \ 1 \ x" ``` huffman@30096 ` 215` ``` by (simp add: le_floor_iff) ``` huffman@30096 ` 216` huffman@30096 ` 217` ```lemma number_of_le_floor [simp]: "number_of v \ floor x \ number_of v \ x" ``` huffman@30096 ` 218` ``` by (simp add: le_floor_iff) ``` huffman@30096 ` 219` huffman@30096 ` 220` ```lemma zero_less_floor [simp]: "0 < floor x \ 1 \ x" ``` huffman@30096 ` 221` ``` by (simp add: less_floor_iff) ``` huffman@30096 ` 222` huffman@30096 ` 223` ```lemma one_less_floor [simp]: "1 < floor x \ 2 \ x" ``` huffman@30096 ` 224` ``` by (simp add: less_floor_iff) ``` huffman@30096 ` 225` huffman@30096 ` 226` ```lemma number_of_less_floor [simp]: ``` huffman@30096 ` 227` ``` "number_of v < floor x \ number_of v + 1 \ x" ``` huffman@30096 ` 228` ``` by (simp add: less_floor_iff) ``` huffman@30096 ` 229` huffman@30096 ` 230` ```lemma floor_le_zero [simp]: "floor x \ 0 \ x < 1" ``` huffman@30096 ` 231` ``` by (simp add: floor_le_iff) ``` huffman@30096 ` 232` huffman@30096 ` 233` ```lemma floor_le_one [simp]: "floor x \ 1 \ x < 2" ``` huffman@30096 ` 234` ``` by (simp add: floor_le_iff) ``` huffman@30096 ` 235` huffman@30096 ` 236` ```lemma floor_le_number_of [simp]: ``` huffman@30096 ` 237` ``` "floor x \ number_of v \ x < number_of v + 1" ``` huffman@30096 ` 238` ``` by (simp add: floor_le_iff) ``` huffman@30096 ` 239` huffman@30096 ` 240` ```lemma floor_less_zero [simp]: "floor x < 0 \ x < 0" ``` huffman@30096 ` 241` ``` by (simp add: floor_less_iff) ``` huffman@30096 ` 242` huffman@30096 ` 243` ```lemma floor_less_one [simp]: "floor x < 1 \ x < 1" ``` huffman@30096 ` 244` ``` by (simp add: floor_less_iff) ``` huffman@30096 ` 245` huffman@30096 ` 246` ```lemma floor_less_number_of [simp]: ``` huffman@30096 ` 247` ``` "floor x < number_of v \ x < number_of v" ``` huffman@30096 ` 248` ``` by (simp add: floor_less_iff) ``` huffman@30096 ` 249` huffman@30096 ` 250` ```text {* Addition and subtraction of integers *} ``` huffman@30096 ` 251` huffman@30096 ` 252` ```lemma floor_add_of_int [simp]: "floor (x + of_int z) = floor x + z" ``` huffman@30096 ` 253` ``` using floor_correct [of x] by (simp add: floor_unique) ``` huffman@30096 ` 254` huffman@30096 ` 255` ```lemma floor_add_number_of [simp]: ``` huffman@30096 ` 256` ``` "floor (x + number_of v) = floor x + number_of v" ``` huffman@30096 ` 257` ``` using floor_add_of_int [of x "number_of v"] by simp ``` huffman@30096 ` 258` huffman@30096 ` 259` ```lemma floor_add_one [simp]: "floor (x + 1) = floor x + 1" ``` huffman@30096 ` 260` ``` using floor_add_of_int [of x 1] by simp ``` huffman@30096 ` 261` huffman@30096 ` 262` ```lemma floor_diff_of_int [simp]: "floor (x - of_int z) = floor x - z" ``` huffman@30096 ` 263` ``` using floor_add_of_int [of x "- z"] by (simp add: algebra_simps) ``` huffman@30096 ` 264` huffman@30096 ` 265` ```lemma floor_diff_number_of [simp]: ``` huffman@30096 ` 266` ``` "floor (x - number_of v) = floor x - number_of v" ``` huffman@30096 ` 267` ``` using floor_diff_of_int [of x "number_of v"] by simp ``` huffman@30096 ` 268` huffman@30096 ` 269` ```lemma floor_diff_one [simp]: "floor (x - 1) = floor x - 1" ``` huffman@30096 ` 270` ``` using floor_diff_of_int [of x 1] by simp ``` huffman@30096 ` 271` huffman@30096 ` 272` huffman@30096 ` 273` ```subsection {* Ceiling function *} ``` huffman@30096 ` 274` huffman@30096 ` 275` ```definition ``` huffman@30096 ` 276` ``` ceiling :: "'a::archimedean_field \ int" where ``` huffman@30096 ` 277` ``` [code del]: "ceiling x = - floor (- x)" ``` huffman@30096 ` 278` huffman@30096 ` 279` ```notation (xsymbols) ``` huffman@30096 ` 280` ``` ceiling ("\_\") ``` huffman@30096 ` 281` huffman@30096 ` 282` ```notation (HTML output) ``` huffman@30096 ` 283` ``` ceiling ("\_\") ``` huffman@30096 ` 284` huffman@30096 ` 285` ```lemma ceiling_correct: "of_int (ceiling x) - 1 < x \ x \ of_int (ceiling x)" ``` huffman@30096 ` 286` ``` unfolding ceiling_def using floor_correct [of "- x"] by simp ``` huffman@30096 ` 287` huffman@30096 ` 288` ```lemma ceiling_unique: "\of_int z - 1 < x; x \ of_int z\ \ ceiling x = z" ``` huffman@30096 ` 289` ``` unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp ``` huffman@30096 ` 290` huffman@30096 ` 291` ```lemma le_of_int_ceiling: "x \ of_int (ceiling x)" ``` huffman@30096 ` 292` ``` using ceiling_correct .. ``` huffman@30096 ` 293` huffman@30096 ` 294` ```lemma ceiling_le_iff: "ceiling x \ z \ x \ of_int z" ``` huffman@30096 ` 295` ``` unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto ``` huffman@30096 ` 296` huffman@30096 ` 297` ```lemma less_ceiling_iff: "z < ceiling x \ of_int z < x" ``` huffman@30096 ` 298` ``` by (simp add: not_le [symmetric] ceiling_le_iff) ``` huffman@30096 ` 299` huffman@30096 ` 300` ```lemma ceiling_less_iff: "ceiling x < z \ x \ of_int z - 1" ``` huffman@30096 ` 301` ``` using ceiling_le_iff [of x "z - 1"] by simp ``` huffman@30096 ` 302` huffman@30096 ` 303` ```lemma le_ceiling_iff: "z \ ceiling x \ of_int z - 1 < x" ``` huffman@30096 ` 304` ``` by (simp add: not_less [symmetric] ceiling_less_iff) ``` huffman@30096 ` 305` huffman@30096 ` 306` ```lemma ceiling_mono: "x \ y \ ceiling x \ ceiling y" ``` huffman@30096 ` 307` ``` unfolding ceiling_def by (simp add: floor_mono) ``` huffman@30096 ` 308` huffman@30096 ` 309` ```lemma ceiling_less_cancel: "ceiling x < ceiling y \ x < y" ``` huffman@30096 ` 310` ``` by (auto simp add: not_le [symmetric] ceiling_mono) ``` huffman@30096 ` 311` huffman@30096 ` 312` ```lemma ceiling_of_int [simp]: "ceiling (of_int z) = z" ``` huffman@30096 ` 313` ``` by (rule ceiling_unique) simp_all ``` huffman@30096 ` 314` huffman@30096 ` 315` ```lemma ceiling_of_nat [simp]: "ceiling (of_nat n) = int n" ``` huffman@30096 ` 316` ``` using ceiling_of_int [of "of_nat n"] by simp ``` huffman@30096 ` 317` huffman@30096 ` 318` ```text {* Ceiling with numerals *} ``` huffman@30096 ` 319` huffman@30096 ` 320` ```lemma ceiling_zero [simp]: "ceiling 0 = 0" ``` huffman@30096 ` 321` ``` using ceiling_of_int [of 0] by simp ``` huffman@30096 ` 322` huffman@30096 ` 323` ```lemma ceiling_one [simp]: "ceiling 1 = 1" ``` huffman@30096 ` 324` ``` using ceiling_of_int [of 1] by simp ``` huffman@30096 ` 325` huffman@30096 ` 326` ```lemma ceiling_number_of [simp]: "ceiling (number_of v) = number_of v" ``` huffman@30096 ` 327` ``` using ceiling_of_int [of "number_of v"] by simp ``` huffman@30096 ` 328` huffman@30096 ` 329` ```lemma ceiling_le_zero [simp]: "ceiling x \ 0 \ x \ 0" ``` huffman@30096 ` 330` ``` by (simp add: ceiling_le_iff) ``` huffman@30096 ` 331` huffman@30096 ` 332` ```lemma ceiling_le_one [simp]: "ceiling x \ 1 \ x \ 1" ``` huffman@30096 ` 333` ``` by (simp add: ceiling_le_iff) ``` huffman@30096 ` 334` huffman@30096 ` 335` ```lemma ceiling_le_number_of [simp]: ``` huffman@30096 ` 336` ``` "ceiling x \ number_of v \ x \ number_of v" ``` huffman@30096 ` 337` ``` by (simp add: ceiling_le_iff) ``` huffman@30096 ` 338` huffman@30096 ` 339` ```lemma ceiling_less_zero [simp]: "ceiling x < 0 \ x \ -1" ``` huffman@30096 ` 340` ``` by (simp add: ceiling_less_iff) ``` huffman@30096 ` 341` huffman@30096 ` 342` ```lemma ceiling_less_one [simp]: "ceiling x < 1 \ x \ 0" ``` huffman@30096 ` 343` ``` by (simp add: ceiling_less_iff) ``` huffman@30096 ` 344` huffman@30096 ` 345` ```lemma ceiling_less_number_of [simp]: ``` huffman@30096 ` 346` ``` "ceiling x < number_of v \ x \ number_of v - 1" ``` huffman@30096 ` 347` ``` by (simp add: ceiling_less_iff) ``` huffman@30096 ` 348` huffman@30096 ` 349` ```lemma zero_le_ceiling [simp]: "0 \ ceiling x \ -1 < x" ``` huffman@30096 ` 350` ``` by (simp add: le_ceiling_iff) ``` huffman@30096 ` 351` huffman@30096 ` 352` ```lemma one_le_ceiling [simp]: "1 \ ceiling x \ 0 < x" ``` huffman@30096 ` 353` ``` by (simp add: le_ceiling_iff) ``` huffman@30096 ` 354` huffman@30096 ` 355` ```lemma number_of_le_ceiling [simp]: ``` huffman@30096 ` 356` ``` "number_of v \ ceiling x\ number_of v - 1 < x" ``` huffman@30096 ` 357` ``` by (simp add: le_ceiling_iff) ``` huffman@30096 ` 358` huffman@30096 ` 359` ```lemma zero_less_ceiling [simp]: "0 < ceiling x \ 0 < x" ``` huffman@30096 ` 360` ``` by (simp add: less_ceiling_iff) ``` huffman@30096 ` 361` huffman@30096 ` 362` ```lemma one_less_ceiling [simp]: "1 < ceiling x \ 1 < x" ``` huffman@30096 ` 363` ``` by (simp add: less_ceiling_iff) ``` huffman@30096 ` 364` huffman@30096 ` 365` ```lemma number_of_less_ceiling [simp]: ``` huffman@30096 ` 366` ``` "number_of v < ceiling x \ number_of v < x" ``` huffman@30096 ` 367` ``` by (simp add: less_ceiling_iff) ``` huffman@30096 ` 368` huffman@30096 ` 369` ```text {* Addition and subtraction of integers *} ``` huffman@30096 ` 370` huffman@30096 ` 371` ```lemma ceiling_add_of_int [simp]: "ceiling (x + of_int z) = ceiling x + z" ``` huffman@30096 ` 372` ``` using ceiling_correct [of x] by (simp add: ceiling_unique) ``` huffman@30096 ` 373` huffman@30096 ` 374` ```lemma ceiling_add_number_of [simp]: ``` huffman@30096 ` 375` ``` "ceiling (x + number_of v) = ceiling x + number_of v" ``` huffman@30096 ` 376` ``` using ceiling_add_of_int [of x "number_of v"] by simp ``` huffman@30096 ` 377` huffman@30096 ` 378` ```lemma ceiling_add_one [simp]: "ceiling (x + 1) = ceiling x + 1" ``` huffman@30096 ` 379` ``` using ceiling_add_of_int [of x 1] by simp ``` huffman@30096 ` 380` huffman@30096 ` 381` ```lemma ceiling_diff_of_int [simp]: "ceiling (x - of_int z) = ceiling x - z" ``` huffman@30096 ` 382` ``` using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps) ``` huffman@30096 ` 383` huffman@30096 ` 384` ```lemma ceiling_diff_number_of [simp]: ``` huffman@30096 ` 385` ``` "ceiling (x - number_of v) = ceiling x - number_of v" ``` huffman@30096 ` 386` ``` using ceiling_diff_of_int [of x "number_of v"] by simp ``` huffman@30096 ` 387` huffman@30096 ` 388` ```lemma ceiling_diff_one [simp]: "ceiling (x - 1) = ceiling x - 1" ``` huffman@30096 ` 389` ``` using ceiling_diff_of_int [of x 1] by simp ``` huffman@30096 ` 390` huffman@30096 ` 391` huffman@30096 ` 392` ```subsection {* Negation *} ``` huffman@30096 ` 393` huffman@30102 ` 394` ```lemma floor_minus: "floor (- x) = - ceiling x" ``` huffman@30096 ` 395` ``` unfolding ceiling_def by simp ``` huffman@30096 ` 396` huffman@30102 ` 397` ```lemma ceiling_minus: "ceiling (- x) = - floor x" ``` huffman@30096 ` 398` ``` unfolding ceiling_def by simp ``` huffman@30096 ` 399` huffman@30096 ` 400` ```end ```