src/HOL/Hyperreal/IntFloor.thy
author kleing
Wed Apr 14 14:13:05 2004 +0200 (2004-04-14)
changeset 14565 c6dc17aab88a
parent 14425 0a76d4633bb6
permissions -rw-r--r--
use more symbols in HTML output
     1 (*  Title:       IntFloor.thy
     2     Author:      Jacques D. Fleuriot
     3     Copyright:   2001,2002  University of Edinburgh
     4 Converted to Isar and polished by lcp
     5 *)
     6 
     7 header{*Floor and Ceiling Functions from the Reals to the Integers*}
     8 
     9 theory IntFloor = Integration:
    10 
    11 constdefs
    12 
    13   floor :: "real => int"
    14    "floor r == (LEAST n. r < real (n + (1::int)))"
    15 
    16   ceiling :: "real => int"
    17     "ceiling r == - floor (- r)"
    18 
    19 syntax (xsymbols)
    20   floor :: "real => int"     ("\<lfloor>_\<rfloor>")
    21   ceiling :: "real => int"   ("\<lceil>_\<rceil>")
    22 
    23 syntax (HTML output)
    24   floor :: "real => int"     ("\<lfloor>_\<rfloor>")
    25   ceiling :: "real => int"   ("\<lceil>_\<rceil>")
    26 
    27 
    28 lemma number_of_less_real_of_int_iff [simp]:
    29      "((number_of n) < real (m::int)) = (number_of n < m)"
    30 apply auto
    31 apply (rule real_of_int_less_iff [THEN iffD1])
    32 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
    33 done
    34 
    35 lemma number_of_less_real_of_int_iff2 [simp]:
    36      "(real (m::int) < (number_of n)) = (m < number_of n)"
    37 apply auto
    38 apply (rule real_of_int_less_iff [THEN iffD1])
    39 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)
    40 done
    41 
    42 lemma number_of_le_real_of_int_iff [simp]:
    43      "((number_of n) \<le> real (m::int)) = (number_of n \<le> m)"
    44 by (simp add: linorder_not_less [symmetric])
    45 
    46 lemma number_of_le_real_of_int_iff2 [simp]:
    47      "(real (m::int) \<le> (number_of n)) = (m \<le> number_of n)"
    48 by (simp add: linorder_not_less [symmetric])
    49 
    50 lemma floor_zero [simp]: "floor 0 = 0"
    51 apply (simp add: floor_def)
    52 apply (rule Least_equality, auto)
    53 done
    54 
    55 lemma floor_real_of_nat_zero [simp]: "floor (real (0::nat)) = 0"
    56 by auto
    57 
    58 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"
    59 apply (simp only: floor_def)
    60 apply (rule Least_equality)
    61 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
    62 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
    63 apply (simp_all add: real_of_int_real_of_nat)
    64 done
    65 
    66 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"
    67 apply (simp only: floor_def)
    68 apply (rule Least_equality)
    69 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
    70 apply (drule_tac [2] real_of_int_minus [THEN subst])
    71 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1])
    72 apply (simp_all add: real_of_int_real_of_nat)
    73 done
    74 
    75 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"
    76 apply (simp only: floor_def)
    77 apply (rule Least_equality)
    78 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
    79 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
    80 done
    81 
    82 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"
    83 apply (simp only: floor_def)
    84 apply (rule Least_equality)
    85 apply (drule_tac [2] real_of_int_minus [THEN subst])
    86 apply (drule_tac [2] real_of_int_real_of_nat [THEN ssubst])
    87 apply (drule_tac [2] real_of_int_less_iff [THEN iffD1], auto)
    88 done
    89 
    90 lemma reals_Archimedean6:
    91      "0 \<le> r ==> \<exists>(n::nat). real (n - 1) \<le> r & r < real (n)"
    92 apply (insert reals_Archimedean2 [of r], safe)
    93 apply (frule_tac P = "%k. r < real k" and k = n and m = "%x. x"
    94        in ex_has_least_nat, auto)
    95 apply (rule_tac x = x in exI)
    96 apply (case_tac x, simp)
    97 apply (rename_tac x')
    98 apply (drule_tac x = x' in spec, simp)
    99 done
   100 
   101 lemma reals_Archimedean6a: "0 \<le> r ==> \<exists>n. real (n) \<le> r & r < real (Suc n)"
   102 by (drule reals_Archimedean6, auto)
   103 
   104 lemma reals_Archimedean_6b_int:
   105      "0 \<le> r ==> \<exists>n. real n \<le> r & r < real ((n::int) + 1)"
   106 apply (drule reals_Archimedean6a, auto)
   107 apply (rule_tac x = "int n" in exI)
   108 apply (simp add: real_of_int_real_of_nat real_of_nat_Suc)
   109 done
   110 
   111 lemma reals_Archimedean_6c_int:
   112      "r < 0 ==> \<exists>n. real n \<le> r & r < real ((n::int) + 1)"
   113 apply (rule reals_Archimedean_6b_int [of "-r", THEN exE], simp, auto)
   114 apply (rename_tac n)
   115 apply (drule real_le_imp_less_or_eq, auto)
   116 apply (rule_tac x = "- n - 1" in exI)
   117 apply (rule_tac [2] x = "- n" in exI, auto)
   118 done
   119 
   120 lemma real_lb_ub_int: " \<exists>(n::int). real n \<le> r & r < real ((n::int) + 1)"
   121 apply (case_tac "r < 0")
   122 apply (blast intro: reals_Archimedean_6c_int)
   123 apply (simp only: linorder_not_less)
   124 apply (blast intro: reals_Archimedean_6b_int reals_Archimedean_6c_int)
   125 done
   126 
   127 lemma lemma_floor:
   128   assumes a1: "real m \<le> r" and a2: "r < real n + 1"
   129   shows "m \<le> (n::int)"
   130 proof -
   131   have "real m < real n + 1" by (rule order_le_less_trans)
   132   also have "... = real(n+1)" by simp
   133   finally have "m < n+1" by (simp only: real_of_int_less_iff)
   134   thus ?thesis by arith
   135 qed
   136 
   137 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"
   138 apply (simp add: floor_def Least_def)
   139 apply (insert real_lb_ub_int [of r], safe)
   140 apply (rule theI2, auto)
   141 done
   142 
   143 lemma floor_le: "x < y ==> floor x \<le> floor y"
   144 apply (simp add: floor_def Least_def)
   145 apply (insert real_lb_ub_int [of x])
   146 apply (insert real_lb_ub_int [of y], safe)
   147 apply (rule theI2)
   148 apply (rule_tac [3] theI2, auto)
   149 done
   150 
   151 lemma floor_le2: "x \<le> y ==> floor x \<le> floor y"
   152 by (auto dest: real_le_imp_less_or_eq simp add: floor_le)
   153 
   154 lemma lemma_floor2: "real na < real (x::int) + 1 ==> na \<le> x"
   155 by (auto intro: lemma_floor)
   156 
   157 lemma real_of_int_floor_cancel [simp]:
   158     "(real (floor x) = x) = (\<exists>n::int. x = real n)"
   159 apply (simp add: floor_def Least_def)
   160 apply (insert real_lb_ub_int [of x], erule exE)
   161 apply (rule theI2)
   162 apply (auto intro: lemma_floor)
   163 done
   164 
   165 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"
   166 apply (simp add: floor_def)
   167 apply (rule Least_equality)
   168 apply (auto intro: lemma_floor)
   169 done
   170 
   171 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"
   172 apply (simp add: floor_def)
   173 apply (rule Least_equality)
   174 apply (auto intro: lemma_floor)
   175 done
   176 
   177 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
   178 apply (rule inj_int [THEN injD])
   179 apply (simp add: real_of_nat_Suc)
   180 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "of_nat n"])
   181 done
   182 
   183 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
   184 apply (drule order_le_imp_less_or_eq)
   185 apply (auto intro: floor_eq3)
   186 done
   187 
   188 lemma floor_number_of_eq [simp]:
   189      "floor(number_of n :: real) = (number_of n :: int)"
   190 apply (subst real_number_of [symmetric])
   191 apply (rule floor_real_of_int)
   192 done
   193 
   194 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"
   195 apply (simp add: floor_def Least_def)
   196 apply (insert real_lb_ub_int [of r], safe)
   197 apply (rule theI2)
   198 apply (auto intro: lemma_floor)
   199 done
   200 
   201 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"
   202 apply (insert real_of_int_floor_ge_diff_one [of r])
   203 apply (auto simp del: real_of_int_floor_ge_diff_one)
   204 done
   205 
   206 
   207 subsection{*Ceiling Function for Positive Reals*}
   208 
   209 lemma ceiling_zero [simp]: "ceiling 0 = 0"
   210 by (simp add: ceiling_def)
   211 
   212 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"
   213 by (simp add: ceiling_def)
   214 
   215 lemma ceiling_real_of_nat_zero [simp]: "ceiling (real (0::nat)) = 0"
   216 by auto
   217 
   218 lemma ceiling_floor [simp]: "ceiling (real (floor r)) = floor r"
   219 by (simp add: ceiling_def)
   220 
   221 lemma floor_ceiling [simp]: "floor (real (ceiling r)) = ceiling r"
   222 by (simp add: ceiling_def)
   223 
   224 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"
   225 apply (simp add: ceiling_def)
   226 apply (subst le_minus_iff, simp)
   227 done
   228 
   229 lemma ceiling_le: "x < y ==> ceiling x \<le> ceiling y"
   230 by (simp add: floor_le ceiling_def)
   231 
   232 lemma ceiling_le2: "x \<le> y ==> ceiling x \<le> ceiling y"
   233 by (simp add: floor_le2 ceiling_def)
   234 
   235 lemma real_of_int_ceiling_cancel [simp]:
   236      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"
   237 apply (auto simp add: ceiling_def)
   238 apply (drule arg_cong [where f = uminus], auto)
   239 apply (rule_tac x = "-n" in exI, auto)
   240 done
   241 
   242 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"
   243 apply (simp add: ceiling_def)
   244 apply (rule minus_equation_iff [THEN iffD1])
   245 apply (simp add: floor_eq [where n = "-(n+1)"])
   246 done
   247 
   248 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"
   249 by (simp add: ceiling_def floor_eq2 [where n = "-(n+1)"])
   250 
   251 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"
   252 by (simp add: ceiling_def floor_eq2 [where n = "-n"])
   253 
   254 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"
   255 by (simp add: ceiling_def)
   256 
   257 lemma ceiling_number_of_eq [simp]:
   258      "ceiling (number_of n :: real) = (number_of n)"
   259 apply (subst real_number_of [symmetric])
   260 apply (rule ceiling_real_of_int)
   261 done
   262 
   263 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"
   264 apply (rule neg_le_iff_le [THEN iffD1])
   265 apply (simp add: ceiling_def diff_minus)
   266 done
   267 
   268 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"
   269 apply (insert real_of_int_ceiling_diff_one_le [of r])
   270 apply (simp del: real_of_int_ceiling_diff_one_le)
   271 done
   272 
   273 ML
   274 {*
   275 val number_of_less_real_of_int_iff = thm "number_of_less_real_of_int_iff";
   276 val number_of_less_real_of_int_iff2 = thm "number_of_less_real_of_int_iff2";
   277 val number_of_le_real_of_int_iff = thm "number_of_le_real_of_int_iff";
   278 val number_of_le_real_of_int_iff2 = thm "number_of_le_real_of_int_iff2";
   279 val floor_zero = thm "floor_zero";
   280 val floor_real_of_nat_zero = thm "floor_real_of_nat_zero";
   281 val floor_real_of_nat = thm "floor_real_of_nat";
   282 val floor_minus_real_of_nat = thm "floor_minus_real_of_nat";
   283 val floor_real_of_int = thm "floor_real_of_int";
   284 val floor_minus_real_of_int = thm "floor_minus_real_of_int";
   285 val reals_Archimedean6 = thm "reals_Archimedean6";
   286 val reals_Archimedean6a = thm "reals_Archimedean6a";
   287 val reals_Archimedean_6b_int = thm "reals_Archimedean_6b_int";
   288 val reals_Archimedean_6c_int = thm "reals_Archimedean_6c_int";
   289 val real_lb_ub_int = thm "real_lb_ub_int";
   290 val lemma_floor = thm "lemma_floor";
   291 val real_of_int_floor_le = thm "real_of_int_floor_le";
   292 val floor_le = thm "floor_le";
   293 val floor_le2 = thm "floor_le2";
   294 val lemma_floor2 = thm "lemma_floor2";
   295 val real_of_int_floor_cancel = thm "real_of_int_floor_cancel";
   296 val floor_eq = thm "floor_eq";
   297 val floor_eq2 = thm "floor_eq2";
   298 val floor_eq3 = thm "floor_eq3";
   299 val floor_eq4 = thm "floor_eq4";
   300 val floor_number_of_eq = thm "floor_number_of_eq";
   301 val real_of_int_floor_ge_diff_one = thm "real_of_int_floor_ge_diff_one";
   302 val real_of_int_floor_add_one_ge = thm "real_of_int_floor_add_one_ge";
   303 val ceiling_zero = thm "ceiling_zero";
   304 val ceiling_real_of_nat = thm "ceiling_real_of_nat";
   305 val ceiling_real_of_nat_zero = thm "ceiling_real_of_nat_zero";
   306 val ceiling_floor = thm "ceiling_floor";
   307 val floor_ceiling = thm "floor_ceiling";
   308 val real_of_int_ceiling_ge = thm "real_of_int_ceiling_ge";
   309 val ceiling_le = thm "ceiling_le";
   310 val ceiling_le2 = thm "ceiling_le2";
   311 val real_of_int_ceiling_cancel = thm "real_of_int_ceiling_cancel";
   312 val ceiling_eq = thm "ceiling_eq";
   313 val ceiling_eq2 = thm "ceiling_eq2";
   314 val ceiling_eq3 = thm "ceiling_eq3";
   315 val ceiling_real_of_int = thm "ceiling_real_of_int";
   316 val ceiling_number_of_eq = thm "ceiling_number_of_eq";
   317 val real_of_int_ceiling_diff_one_le = thm "real_of_int_ceiling_diff_one_le";
   318 val real_of_int_ceiling_le_add_one = thm "real_of_int_ceiling_le_add_one";
   319 *}
   320 
   321 
   322 end