wneuper/isa: src/HOL/Word/Bit_Representation.thy@0b562d564d5f

     1 (*

     2   Author: Jeremy Dawson, NICTA

     4   Basic definitions to do with integers, expressed using Pls, Min, BIT.

     5 *)

     7 header {* Basic Definitions for Binary Integers *}

     9 theory Bit_Representation

    10 imports Misc_Numeric "~~/src/HOL/Library/Bit"

    11 begin

    13 subsection {* Further properties of numerals *}

    15 definition bitval :: "bit \<Rightarrow> 'a\<Colon>zero_neq_one" where

    16   "bitval = bit_case 0 1"

    18 lemma bitval_simps [simp]:

    19   "bitval 0 = 0"

    20   "bitval 1 = 1"

    21   by (simp_all add: bitval_def)

    23 definition Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90) where

    24   "k BIT b = bitval b + k + k"

    26 definition bin_last :: "int \<Rightarrow> bit" where

    27   "bin_last w = (if w mod 2 = 0 then (0::bit) else (1::bit))"

    29 definition bin_rest :: "int \<Rightarrow> int" where

    30   "bin_rest w = w div 2"

    32 lemma bin_rl_simp [simp]:

    33   "bin_rest w BIT bin_last w = w"

    34   unfolding bin_rest_def bin_last_def Bit_def

    35   using mod_div_equality [of w 2]

    36   by (cases "w mod 2 = 0", simp_all)

    38 lemma bin_rest_BIT: "bin_rest (x BIT b) = x"

    39   unfolding bin_rest_def Bit_def

    40   by (cases b, simp_all)

    42 lemma bin_last_BIT: "bin_last (x BIT b) = b"

    43   unfolding bin_last_def Bit_def

    44   by (cases b, simp_all add: z1pmod2)

    46 lemma BIT_eq_iff [iff]: "u BIT b = v BIT c \<longleftrightarrow> u = v \<and> b = c"

    47   by (metis bin_rest_BIT bin_last_BIT)

    49 lemma BIT_bin_simps [simp]:

    50   "number_of w BIT 0 = number_of (Int.Bit0 w)"

    51   "number_of w BIT 1 = number_of (Int.Bit1 w)"

    52   unfolding Bit_def number_of_is_id numeral_simps by simp_all

    54 lemma BIT_special_simps [simp]:

    55   shows "0 BIT 0 = 0" and "0 BIT 1 = 1" and "1 BIT 0 = 2" and "1 BIT 1 = 3"

    56   unfolding Bit_def by simp_all

    58 lemma bin_last_numeral_simps [simp]:

    59   "bin_last 0 = 0"

    60   "bin_last 1 = 1"

    61   "bin_last -1 = 1"

    62   "bin_last (number_of (Int.Bit0 w)) = 0"

    63   "bin_last (number_of (Int.Bit1 w)) = 1"

    64   unfolding bin_last_def by simp_all

    66 lemma bin_rest_numeral_simps [simp]:

    67   "bin_rest 0 = 0"

    68   "bin_rest 1 = 0"

    69   "bin_rest -1 = -1"

    70   "bin_rest (number_of (Int.Bit0 w)) = number_of w"

    71   "bin_rest (number_of (Int.Bit1 w)) = number_of w"

    72   unfolding bin_rest_def by simp_all

    74 lemma BIT_B0_eq_Bit0: "w BIT 0 = Int.Bit0 w"

    75   unfolding Bit_def Bit0_def by simp

    77 lemma BIT_B1_eq_Bit1: "w BIT 1 = Int.Bit1 w"

    78   unfolding Bit_def Bit1_def by simp

    80 lemmas BIT_simps = BIT_B0_eq_Bit0 BIT_B1_eq_Bit1

    82 lemma number_of_False_cong:

    83   "False \<Longrightarrow> number_of x = number_of y"

    84   by (rule FalseE)

    86 lemma less_Bits:

    87   "(v BIT b < w BIT c) = (v < w | v <= w & b = (0::bit) & c = (1::bit))"

    88   unfolding Bit_def by (auto simp add: bitval_def split: bit.split)

    90 lemma le_Bits:

    91   "(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= (1::bit) | c ~= (0::bit)))"

    92   unfolding Bit_def by (auto simp add: bitval_def split: bit.split)

    94 lemma Bit_B0:

    95   "k BIT (0::bit) = k + k"

    96    by (unfold Bit_def) simp

    98 lemma Bit_B1:

    99   "k BIT (1::bit) = k + k + 1"

   100    by (unfold Bit_def) simp

   102 lemma Bit_B0_2t: "k BIT (0::bit) = 2 * k"

   103   by (rule trans, rule Bit_B0) simp

   105 lemma Bit_B1_2t: "k BIT (1::bit) = 2 * k + 1"

   106   by (rule trans, rule Bit_B1) simp

   108 lemma B_mod_2':

   109   "X = 2 ==> (w BIT (1::bit)) mod X = 1 & (w BIT (0::bit)) mod X = 0"

   110   apply (simp (no_asm) only: Bit_B0 Bit_B1)

   111   apply (simp add: z1pmod2)

   112   done

   114 lemma B1_mod_2 [simp]: "(Int.Bit1 w) mod 2 = 1"

   115   unfolding numeral_simps number_of_is_id by (simp add: z1pmod2)

   117 lemma B0_mod_2 [simp]: "(Int.Bit0 w) mod 2 = 0"

   118   unfolding numeral_simps number_of_is_id by simp

   120 lemma neB1E [elim!]:

   121   assumes ne: "y \<noteq> (1::bit)"

   122   assumes y: "y = (0::bit) \<Longrightarrow> P"

   123   shows "P"

   124   apply (rule y)

   125   apply (cases y rule: bit.exhaust, simp)

   126   apply (simp add: ne)

   127   done

   129 lemma bin_ex_rl: "EX w b. w BIT b = bin"

   130   apply (unfold Bit_def)

   131   apply (cases "even bin")

   132    apply (clarsimp simp: even_equiv_def)

   133    apply (auto simp: odd_equiv_def bitval_def split: bit.split)

   134   done

   136 lemma bin_exhaust:

   137   assumes Q: "\<And>x b. bin = x BIT b \<Longrightarrow> Q"

   138   shows "Q"

   139   apply (insert bin_ex_rl [of bin])

   140   apply (erule exE)+

   141   apply (rule Q)

   142   apply force

   143   done

   146 subsection {* Destructors for binary integers *}

   148 definition bin_rl :: "int \<Rightarrow> int \<times> bit" where

   149   "bin_rl w = (bin_rest w, bin_last w)"

   151 lemma bin_rl_char: "bin_rl w = (r, l) \<longleftrightarrow> r BIT l = w"

   152   unfolding bin_rl_def by (auto simp: bin_rest_BIT bin_last_BIT)

   154 primrec bin_nth where

   155   Z: "bin_nth w 0 = (bin_last w = (1::bit))"

   156   | Suc: "bin_nth w (Suc n) = bin_nth (bin_rest w) n"

   158 lemma bin_rl_simps [simp]:

   159   "bin_rl Int.Pls = (Int.Pls, (0::bit))"

   160   "bin_rl Int.Min = (Int.Min, (1::bit))"

   161   "bin_rl (Int.Bit0 r) = (r, (0::bit))"

   162   "bin_rl (Int.Bit1 r) = (r, (1::bit))"

   163   "bin_rl (r BIT b) = (r, b)"

   164   unfolding bin_rl_char by (simp_all add: BIT_simps)

   166 lemma bin_abs_lem:

   167   "bin = (w BIT b) ==> ~ bin = Int.Min --> ~ bin = Int.Pls -->

   168     nat (abs w) < nat (abs bin)"

   169   apply (clarsimp simp add: bin_rl_char)

   170   apply (unfold Pls_def Min_def Bit_def)

   171   apply (cases b)

   172    apply (clarsimp, arith)

   173   apply (clarsimp, arith)

   174   done

   176 lemma bin_induct:

   177   assumes PPls: "P Int.Pls"

   178     and PMin: "P Int.Min"

   179     and PBit: "!!bin bit. P bin ==> P (bin BIT bit)"

   180   shows "P bin"

   181   apply (rule_tac P=P and a=bin and f1="nat o abs"

   182                   in wf_measure [THEN wf_induct])

   183   apply (simp add: measure_def inv_image_def)

   184   apply (case_tac x rule: bin_exhaust)

   185   apply (frule bin_abs_lem)

   186   apply (auto simp add : PPls PMin PBit)

   187   done

   189 lemma numeral_induct:

   190   assumes Pls: "P Int.Pls"

   191   assumes Min: "P Int.Min"

   192   assumes Bit0: "\<And>w. \<lbrakk>P w; w \<noteq> Int.Pls\<rbrakk> \<Longrightarrow> P (Int.Bit0 w)"

   193   assumes Bit1: "\<And>w. \<lbrakk>P w; w \<noteq> Int.Min\<rbrakk> \<Longrightarrow> P (Int.Bit1 w)"

   194   shows "P x"

   195   apply (induct x rule: bin_induct)

   196     apply (rule Pls)

   197    apply (rule Min)

   198   apply (case_tac bit)

   199    apply (case_tac "bin = Int.Pls")

   200     apply (simp add: BIT_simps)

   201    apply (simp add: Bit0 BIT_simps)

   202   apply (case_tac "bin = Int.Min")

   203    apply (simp add: BIT_simps)

   204   apply (simp add: Bit1 BIT_simps)

   205   done

   207 lemma bin_rest_simps [simp]:

   208   "bin_rest Int.Pls = Int.Pls"

   209   "bin_rest Int.Min = Int.Min"

   210   "bin_rest (Int.Bit0 w) = w"

   211   "bin_rest (Int.Bit1 w) = w"

   212   "bin_rest (w BIT b) = w"

   213   using bin_rl_simps bin_rl_def by auto

   215 lemma bin_last_simps [simp]:

   216   "bin_last Int.Pls = (0::bit)"

   217   "bin_last Int.Min = (1::bit)"

   218   "bin_last (Int.Bit0 w) = (0::bit)"

   219   "bin_last (Int.Bit1 w) = (1::bit)"

   220   "bin_last (w BIT b) = b"

   221   using bin_rl_simps bin_rl_def by auto

   223 lemma Bit_div2 [simp]: "(w BIT b) div 2 = w"

   224   unfolding bin_rest_def [symmetric] by auto

   226 lemma Bit0_div2 [simp]: "(Int.Bit0 w) div 2 = w"

   227   using Bit_div2 [where b="(0::bit)"] by (simp add: BIT_simps)

   229 lemma Bit1_div2 [simp]: "(Int.Bit1 w) div 2 = w"

   230   using Bit_div2 [where b="(1::bit)"] by (simp add: BIT_simps)

   232 lemma bin_nth_lem [rule_format]:

   233   "ALL y. bin_nth x = bin_nth y --> x = y"

   234   apply (induct x rule: bin_induct)

   235     apply safe

   236     apply (erule rev_mp)

   237     apply (induct_tac y rule: bin_induct)

   238       apply (safe del: subset_antisym)

   239       apply (drule_tac x=0 in fun_cong, force)

   240      apply (erule notE, rule ext,

   241             drule_tac x="Suc x" in fun_cong, force)

   242     apply (drule_tac x=0 in fun_cong, force simp: BIT_simps)

   243    apply (erule rev_mp)

   244    apply (induct_tac y rule: bin_induct)

   245      apply (safe del: subset_antisym)

   246      apply (drule_tac x=0 in fun_cong, force)

   247     apply (erule notE, rule ext,

   248            drule_tac x="Suc x" in fun_cong, force)

   249    apply (drule_tac x=0 in fun_cong, force simp: BIT_simps)

   250   apply (case_tac y rule: bin_exhaust)

   251   apply clarify

   252   apply (erule allE)

   253   apply (erule impE)

   254    prefer 2

   255    apply (erule conjI)

   256    apply (drule_tac x=0 in fun_cong, force)

   257   apply (rule ext)

   258   apply (drule_tac x="Suc ?x" in fun_cong, force)

   259   done

   261 lemma bin_nth_eq_iff: "(bin_nth x = bin_nth y) = (x = y)"

   262   by (auto elim: bin_nth_lem)

   264 lemmas bin_eqI = ext [THEN bin_nth_eq_iff [THEN iffD1]]

   266 lemma bin_eq_iff: "x = y \<longleftrightarrow> (\<forall>n. bin_nth x n = bin_nth y n)"

   267   by (auto intro!: bin_nth_lem del: equalityI)

   269 lemma bin_nth_zero [simp]: "\<not> bin_nth 0 n"

   270   by (induct n) auto

   272 lemma bin_nth_Pls [simp]: "~ bin_nth Int.Pls n"

   273   by (induct n) auto

   275 lemma bin_nth_minus1 [simp]: "bin_nth -1 n"

   276   by (induct n) auto

   278 lemma bin_nth_Min [simp]: "bin_nth Int.Min n"

   279   by (induct n) auto

   281 lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 = (b = (1::bit))"

   282   by auto

   284 lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n"

   285   by auto

   287 lemma bin_nth_minus [simp]: "0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)"

   288   by (cases n) auto

   290 lemma bin_nth_minus_Bit0 [simp]:

   291   "0 < n ==> bin_nth (Int.Bit0 w) n = bin_nth w (n - 1)"

   292   using bin_nth_minus [where b="(0::bit)"] by (simp add: BIT_simps)

   294 lemma bin_nth_minus_Bit1 [simp]:

   295   "0 < n ==> bin_nth (Int.Bit1 w) n = bin_nth w (n - 1)"

   296   using bin_nth_minus [where b="(1::bit)"] by (simp add: BIT_simps)

   298 lemmas bin_nth_0 = bin_nth.simps(1)

   299 lemmas bin_nth_Suc = bin_nth.simps(2)

   301 lemmas bin_nth_simps =

   302   bin_nth_0 bin_nth_Suc bin_nth_Pls bin_nth_Min bin_nth_minus

   303   bin_nth_minus_Bit0 bin_nth_minus_Bit1

   306 subsection {* Truncating binary integers *}

   308 definition bin_sign :: "int \<Rightarrow> int" where

   309   bin_sign_def: "bin_sign k = (if k \<ge> 0 then 0 else - 1)"

   311 lemma bin_sign_simps [simp]:

   312   "bin_sign 0 = 0"

   313   "bin_sign -1 = -1"

   314   "bin_sign (number_of (Int.Bit0 w)) = bin_sign (number_of w)"

   315   "bin_sign (number_of (Int.Bit1 w)) = bin_sign (number_of w)"

   316   "bin_sign Int.Pls = Int.Pls"

   317   "bin_sign Int.Min = Int.Min"

   318   "bin_sign (Int.Bit0 w) = bin_sign w"

   319   "bin_sign (Int.Bit1 w) = bin_sign w"

   320   "bin_sign (w BIT b) = bin_sign w"

   321   unfolding bin_sign_def numeral_simps Bit_def bitval_def number_of_is_id

   322   by (simp_all split: bit.split)

   324 lemma bin_sign_rest [simp]:

   325   "bin_sign (bin_rest w) = bin_sign w"

   326   by (cases w rule: bin_exhaust) auto

   328 primrec bintrunc :: "nat \<Rightarrow> int \<Rightarrow> int" where

   329   Z : "bintrunc 0 bin = 0"

   330 | Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)"

   332 primrec sbintrunc :: "nat => int => int" where

   333   Z : "sbintrunc 0 bin = (case bin_last bin of (1::bit) \<Rightarrow> -1 | (0::bit) \<Rightarrow> 0)"

   334 | Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"

   336 lemma sign_bintr: "bin_sign (bintrunc n w) = 0"

   337   by (induct n arbitrary: w) auto

   339 lemma bintrunc_mod2p: "bintrunc n w = (w mod 2 ^ n)"

   340   apply (induct n arbitrary: w)

   341   apply simp

   342   apply (simp add: bin_last_def bin_rest_def Bit_def zmod_zmult2_eq)

   343   done

   345 lemma sbintrunc_mod2p: "sbintrunc n w = (w + 2 ^ n) mod 2 ^ (Suc n) - 2 ^ n"

   346   apply (induct n arbitrary: w)

   347    apply clarsimp

   348    apply (subst mod_add_left_eq)

   349    apply (simp add: bin_last_def)

   350   apply simp

   351   apply (simp add: bin_last_def bin_rest_def Bit_def)

   352   apply (clarsimp simp: mod_mult_mult1 [symmetric]

   353          zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2 [THEN sym]]])

   354   apply (rule trans [symmetric, OF _ emep1])

   355      apply auto

   356   apply (auto simp: even_def)

   357   done

   359 subsection "Simplifications for (s)bintrunc"

   361 lemma bintrunc_n_0 [simp]: "bintrunc n 0 = 0"

   362   by (induct n) auto

   364 lemma sbintrunc_n_0 [simp]: "sbintrunc n 0 = 0"

   365   by (induct n) auto

   367 lemma sbintrunc_n_minus1 [simp]: "sbintrunc n -1 = -1"

   368   by (induct n) auto

   370 lemma bintrunc_Suc_numeral:

   371   "bintrunc (Suc n) 1 = 1"

   372   "bintrunc (Suc n) -1 = bintrunc n -1 BIT 1"

   373   "bintrunc (Suc n) (number_of (Int.Bit0 w)) = bintrunc n (number_of w) BIT 0"

   374   "bintrunc (Suc n) (number_of (Int.Bit1 w)) = bintrunc n (number_of w) BIT 1"

   375   by simp_all

   377 lemma sbintrunc_0_numeral [simp]:

   378   "sbintrunc 0 1 = -1"

   379   "sbintrunc 0 (number_of (Int.Bit0 w)) = 0"

   380   "sbintrunc 0 (number_of (Int.Bit1 w)) = -1"

   381   by simp_all

   383 lemma sbintrunc_Suc_numeral:

   384   "sbintrunc (Suc n) 1 = 1"

   385   "sbintrunc (Suc n) (number_of (Int.Bit0 w)) = sbintrunc n (number_of w) BIT 0"

   386   "sbintrunc (Suc n) (number_of (Int.Bit1 w)) = sbintrunc n (number_of w) BIT 1"

   387   by simp_all

   389 lemma bit_bool:

   390   "(b = (b' = (1::bit))) = (b' = (if b then (1::bit) else (0::bit)))"

   391   by (cases b') auto

   393 lemmas bit_bool1 [simp] = refl [THEN bit_bool [THEN iffD1], symmetric]

   395 lemma bin_sign_lem: "(bin_sign (sbintrunc n bin) = -1) = bin_nth bin n"

   396   apply (induct n arbitrary: bin)

   397    apply (case_tac bin rule: bin_exhaust, case_tac b, auto)+

   398   done

   400 lemma nth_bintr: "bin_nth (bintrunc m w) n = (n < m & bin_nth w n)"

   401   apply (induct n arbitrary: w m)

   402    apply (case_tac m, auto)[1]

   403   apply (case_tac m, auto)[1]

   404   done

   406 lemma nth_sbintr:

   407   "bin_nth (sbintrunc m w) n =

   408           (if n < m then bin_nth w n else bin_nth w m)"

   409   apply (induct n arbitrary: w m)

   410    apply (case_tac m, simp_all split: bit.splits)[1]

   411   apply (case_tac m, simp_all split: bit.splits)[1]

   412   done

   414 lemma bin_nth_Bit:

   415   "bin_nth (w BIT b) n = (n = 0 & b = (1::bit) | (EX m. n = Suc m & bin_nth w m))"

   416   by (cases n) auto

   418 lemma bin_nth_Bit0:

   419   "bin_nth (Int.Bit0 w) n = (EX m. n = Suc m & bin_nth w m)"

   420   using bin_nth_Bit [where b="(0::bit)"] by (simp add: BIT_simps)

   422 lemma bin_nth_Bit1:

   423   "bin_nth (Int.Bit1 w) n = (n = 0 | (EX m. n = Suc m & bin_nth w m))"

   424   using bin_nth_Bit [where b="(1::bit)"] by (simp add: BIT_simps)

   426 lemma bintrunc_bintrunc_l:

   427   "n <= m ==> (bintrunc m (bintrunc n w) = bintrunc n w)"

   428   by (rule bin_eqI) (auto simp add : nth_bintr)

   430 lemma sbintrunc_sbintrunc_l:

   431   "n <= m ==> (sbintrunc m (sbintrunc n w) = sbintrunc n w)"

   432   by (rule bin_eqI) (auto simp: nth_sbintr)

   434 lemma bintrunc_bintrunc_ge:

   435   "n <= m ==> (bintrunc n (bintrunc m w) = bintrunc n w)"

   436   by (rule bin_eqI) (auto simp: nth_bintr)

   438 lemma bintrunc_bintrunc_min [simp]:

   439   "bintrunc m (bintrunc n w) = bintrunc (min m n) w"

   440   apply (rule bin_eqI)

   441   apply (auto simp: nth_bintr)

   442   done

   444 lemma sbintrunc_sbintrunc_min [simp]:

   445   "sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w"

   446   apply (rule bin_eqI)

   447   apply (auto simp: nth_sbintr min_max.inf_absorb1 min_max.inf_absorb2)

   448   done

   450 lemmas bintrunc_Pls =

   451   bintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps]

   453 lemmas bintrunc_Min [simp] =

   454   bintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps]

   456 lemmas bintrunc_BIT  [simp] =

   457   bintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps] for w b

   459 lemma bintrunc_Bit0 [simp]:

   460   "bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)"

   461   using bintrunc_BIT [where b="(0::bit)"] by (simp add: BIT_simps)

   463 lemma bintrunc_Bit1 [simp]:

   464   "bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)"

   465   using bintrunc_BIT [where b="(1::bit)"] by (simp add: BIT_simps)

   467 lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT

   468   bintrunc_Bit0 bintrunc_Bit1

   469   bintrunc_Suc_numeral

   471 lemmas sbintrunc_Suc_Pls =

   472   sbintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps]

   474 lemmas sbintrunc_Suc_Min =

   475   sbintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps]

   477 lemmas sbintrunc_Suc_BIT [simp] =

   478   sbintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps] for w b

   480 lemma sbintrunc_Suc_Bit0 [simp]:

   481   "sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)"

   482   using sbintrunc_Suc_BIT [where b="(0::bit)"] by (simp add: BIT_simps)

   484 lemma sbintrunc_Suc_Bit1 [simp]:

   485   "sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)"

   486   using sbintrunc_Suc_BIT [where b="(1::bit)"] by (simp add: BIT_simps)

   488 lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT

   489   sbintrunc_Suc_Bit0 sbintrunc_Suc_Bit1

   490   sbintrunc_Suc_numeral

   492 lemmas sbintrunc_Pls =

   493   sbintrunc.Z [where bin="Int.Pls",

   494                simplified bin_last_simps bin_rest_simps bit.simps]

   496 lemmas sbintrunc_Min =

   497   sbintrunc.Z [where bin="Int.Min",

   498                simplified bin_last_simps bin_rest_simps bit.simps]

   500 lemmas sbintrunc_0_BIT_B0 [simp] =

   501   sbintrunc.Z [where bin="w BIT (0::bit)",

   502                simplified bin_last_simps bin_rest_simps bit.simps] for w

   504 lemmas sbintrunc_0_BIT_B1 [simp] =

   505   sbintrunc.Z [where bin="w BIT (1::bit)",

   506                simplified bin_last_simps bin_rest_simps bit.simps] for w

   508 lemma sbintrunc_0_Bit0 [simp]: "sbintrunc 0 (Int.Bit0 w) = 0"

   509   using sbintrunc_0_BIT_B0 by simp

   511 lemma sbintrunc_0_Bit1 [simp]: "sbintrunc 0 (Int.Bit1 w) = -1"

   512   using sbintrunc_0_BIT_B1 by simp

   514 lemmas sbintrunc_0_simps =

   515   sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1

   516   sbintrunc_0_Bit0 sbintrunc_0_Bit1

   518 lemmas bintrunc_simps = bintrunc.Z bintrunc_Sucs

   519 lemmas sbintrunc_simps = sbintrunc_0_simps sbintrunc_Sucs

   521 lemma bintrunc_minus:

   522   "0 < n ==> bintrunc (Suc (n - 1)) w = bintrunc n w"

   523   by auto

   525 lemma sbintrunc_minus:

   526   "0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w"

   527   by auto

   529 lemmas bintrunc_minus_simps =

   530   bintrunc_Sucs [THEN [2] bintrunc_minus [symmetric, THEN trans]]

   531 lemmas sbintrunc_minus_simps =

   532   sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans]]

   534 lemma bintrunc_n_Pls [simp]:

   535   "bintrunc n Int.Pls = Int.Pls"

   536   unfolding Pls_def by simp

   538 lemma sbintrunc_n_PM [simp]:

   539   "sbintrunc n Int.Pls = Int.Pls"

   540   "sbintrunc n Int.Min = Int.Min"

   541   unfolding Pls_def Min_def by simp_all

   543 lemmas thobini1 = arg_cong [where f = "%w. w BIT b"] for b

   545 lemmas bintrunc_BIT_I = trans [OF bintrunc_BIT thobini1]

   546 lemmas bintrunc_Min_I = trans [OF bintrunc_Min thobini1]

   548 lemmas bmsts = bintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans]]

   549 lemmas bintrunc_Pls_minus_I = bmsts(1)

   550 lemmas bintrunc_Min_minus_I = bmsts(2)

   551 lemmas bintrunc_BIT_minus_I = bmsts(3)

   553 lemma bintrunc_Suc_lem:

   554   "bintrunc (Suc n) x = y ==> m = Suc n ==> bintrunc m x = y"

   555   by auto

   557 lemmas bintrunc_Suc_Ialts =

   558   bintrunc_Min_I [THEN bintrunc_Suc_lem]

   559   bintrunc_BIT_I [THEN bintrunc_Suc_lem]

   561 lemmas sbintrunc_BIT_I = trans [OF sbintrunc_Suc_BIT thobini1]

   563 lemmas sbintrunc_Suc_Is =

   564   sbintrunc_Sucs(1-3) [THEN thobini1 [THEN [2] trans]]

   566 lemmas sbintrunc_Suc_minus_Is =

   567   sbintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans]]

   569 lemma sbintrunc_Suc_lem:

   570   "sbintrunc (Suc n) x = y ==> m = Suc n ==> sbintrunc m x = y"

   571   by auto

   573 lemmas sbintrunc_Suc_Ialts =

   574   sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem]

   576 lemma sbintrunc_bintrunc_lt:

   577   "m > n ==> sbintrunc n (bintrunc m w) = sbintrunc n w"

   578   by (rule bin_eqI) (auto simp: nth_sbintr nth_bintr)

   580 lemma bintrunc_sbintrunc_le:

   581   "m <= Suc n ==> bintrunc m (sbintrunc n w) = bintrunc m w"

   582   apply (rule bin_eqI)

   583   apply (auto simp: nth_sbintr nth_bintr)

   584    apply (subgoal_tac "x=n", safe, arith+)[1]

   585   apply (subgoal_tac "x=n", safe, arith+)[1]

   586   done

   588 lemmas bintrunc_sbintrunc [simp] = order_refl [THEN bintrunc_sbintrunc_le]

   589 lemmas sbintrunc_bintrunc [simp] = lessI [THEN sbintrunc_bintrunc_lt]

   590 lemmas bintrunc_bintrunc [simp] = order_refl [THEN bintrunc_bintrunc_l]

   591 lemmas sbintrunc_sbintrunc [simp] = order_refl [THEN sbintrunc_sbintrunc_l]

   593 lemma bintrunc_sbintrunc' [simp]:

   594   "0 < n \<Longrightarrow> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w"

   595   by (cases n) (auto simp del: bintrunc.Suc)

   597 lemma sbintrunc_bintrunc' [simp]:

   598   "0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"

   599   by (cases n) (auto simp del: bintrunc.Suc)

   601 lemma bin_sbin_eq_iff:

   602   "bintrunc (Suc n) x = bintrunc (Suc n) y <->

   603    sbintrunc n x = sbintrunc n y"

   604   apply (rule iffI)

   605    apply (rule box_equals [OF _ sbintrunc_bintrunc sbintrunc_bintrunc])

   606    apply simp

   607   apply (rule box_equals [OF _ bintrunc_sbintrunc bintrunc_sbintrunc])

   608   apply simp

   609   done

   611 lemma bin_sbin_eq_iff':

   612   "0 < n \<Longrightarrow> bintrunc n x = bintrunc n y <->

   613             sbintrunc (n - 1) x = sbintrunc (n - 1) y"

   614   by (cases n) (simp_all add: bin_sbin_eq_iff del: bintrunc.Suc)

   616 lemmas bintrunc_sbintruncS0 [simp] = bintrunc_sbintrunc' [unfolded One_nat_def]

   617 lemmas sbintrunc_bintruncS0 [simp] = sbintrunc_bintrunc' [unfolded One_nat_def]

   619 lemmas bintrunc_bintrunc_l' = le_add1 [THEN bintrunc_bintrunc_l]

   620 lemmas sbintrunc_sbintrunc_l' = le_add1 [THEN sbintrunc_sbintrunc_l]

   622 (* although bintrunc_minus_simps, if added to default simpset,

   623   tends to get applied where it's not wanted in developing the theories,

   624   we get a version for when the word length is given literally *)

   626 lemmas nat_non0_gr =

   627   trans [OF iszero_def [THEN Not_eq_iff [THEN iffD2]] refl]

   629 lemmas bintrunc_pred_simps [simp] =

   630   bintrunc_minus_simps [of "number_of bin", simplified nobm1] for bin

   632 lemmas sbintrunc_pred_simps [simp] =

   633   sbintrunc_minus_simps [of "number_of bin", simplified nobm1] for bin

   635 lemma no_bintr_alt:

   636   "number_of (bintrunc n w) = w mod 2 ^ n"

   637   by (simp add: number_of_eq bintrunc_mod2p)

   639 lemma no_bintr_alt1: "bintrunc n = (%w. w mod 2 ^ n :: int)"

   640   by (rule ext) (rule bintrunc_mod2p)

   642 lemma range_bintrunc: "range (bintrunc n) = {i. 0 <= i & i < 2 ^ n}"

   643   apply (unfold no_bintr_alt1)

   644   apply (auto simp add: image_iff)

   645   apply (rule exI)

   646   apply (auto intro: int_mod_lem [THEN iffD1, symmetric])

   647   done

   649 lemma no_bintr:

   650   "number_of (bintrunc n w) = (number_of w mod 2 ^ n :: int)"

   651   by (simp add : bintrunc_mod2p number_of_eq)

   653 lemma no_sbintr_alt2:

   654   "sbintrunc n = (%w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"

   655   by (rule ext) (simp add : sbintrunc_mod2p)

   657 lemma no_sbintr:

   658   "number_of (sbintrunc n w) =

   659    ((number_of w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"

   660   by (simp add : no_sbintr_alt2 number_of_eq)

   662 lemma range_sbintrunc:

   663   "range (sbintrunc n) = {i. - (2 ^ n) <= i & i < 2 ^ n}"

   664   apply (unfold no_sbintr_alt2)

   665   apply (auto simp add: image_iff eq_diff_eq)

   666   apply (rule exI)

   667   apply (auto intro: int_mod_lem [THEN iffD1, symmetric])

   668   done

   670 lemma sb_inc_lem:

   671   "(a::int) + 2^k < 0 \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"

   672   apply (erule int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k", simplified zless2p])

   673   apply (rule TrueI)

   674   done

   676 lemma sb_inc_lem':

   677   "(a::int) < - (2^k) \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"

   678   by (rule sb_inc_lem) simp

   680 lemma sbintrunc_inc:

   681   "x < - (2^n) ==> x + 2^(Suc n) <= sbintrunc n x"

   682   unfolding no_sbintr_alt2 by (drule sb_inc_lem') simp

   684 lemma sb_dec_lem:

   685   "(0::int) <= - (2^k) + a ==> (a + 2^k) mod (2 * 2 ^ k) <= - (2 ^ k) + a"

   686   by (rule int_mod_le' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k",

   687     simplified zless2p, OF _ TrueI, simplified])

   689 lemma sb_dec_lem':

   690   "(2::int) ^ k <= a ==> (a + 2 ^ k) mod (2 * 2 ^ k) <= - (2 ^ k) + a"

   691   by (rule iffD1 [OF diff_le_eq', THEN sb_dec_lem, simplified])

   693 lemma sbintrunc_dec:

   694   "x >= (2 ^ n) ==> x - 2 ^ (Suc n) >= sbintrunc n x"

   695   unfolding no_sbintr_alt2 by (drule sb_dec_lem') simp

   697 lemmas zmod_uminus' = zmod_uminus [where b=c] for c

   698 lemmas zpower_zmod' = zpower_zmod [where m=c and y=k] for c k

   700 lemmas brdmod1s' [symmetric] =

   701   mod_add_left_eq mod_add_right_eq

   702   zmod_zsub_left_eq zmod_zsub_right_eq

   703   zmod_zmult1_eq zmod_zmult1_eq_rev

   705 lemmas brdmods' [symmetric] =

   706   zpower_zmod' [symmetric]

   707   trans [OF mod_add_left_eq mod_add_right_eq]

   708   trans [OF zmod_zsub_left_eq zmod_zsub_right_eq]

   709   trans [OF zmod_zmult1_eq zmod_zmult1_eq_rev]

   710   zmod_uminus' [symmetric]

   711   mod_add_left_eq [where b = "1::int"]

   712   zmod_zsub_left_eq [where b = "1"]

   714 lemmas bintr_arith1s =

   715   brdmod1s' [where c="2^n::int", folded bintrunc_mod2p] for n

   716 lemmas bintr_ariths =

   717   brdmods' [where c="2^n::int", folded bintrunc_mod2p] for n

   719 lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p]

   721 lemma bintr_ge0: "(0 :: int) <= number_of (bintrunc n w)"

   722   by (simp add : no_bintr m2pths)

   724 lemma bintr_lt2p: "number_of (bintrunc n w) < (2 ^ n :: int)"

   725   by (simp add : no_bintr m2pths)

   727 lemma bintr_Min:

   728   "number_of (bintrunc n Int.Min) = (2 ^ n :: int) - 1"

   729   by (simp add : no_bintr m1mod2k)

   731 lemma sbintr_ge: "(- (2 ^ n) :: int) <= number_of (sbintrunc n w)"

   732   by (simp add : no_sbintr m2pths)

   734 lemma sbintr_lt: "number_of (sbintrunc n w) < (2 ^ n :: int)"

   735   by (simp add : no_sbintr m2pths)

   737 lemma bintrunc_Suc:

   738   "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT bin_last bin"

   739   by (case_tac bin rule: bin_exhaust) auto

   741 lemma sign_Pls_ge_0:

   742   "(bin_sign bin = Int.Pls) = (number_of bin >= (0 :: int))"

   743   by (induct bin rule: numeral_induct) auto

   745 lemma sign_Min_lt_0:

   746   "(bin_sign bin = Int.Min) = (number_of bin < (0 :: int))"

   747   by (induct bin rule: numeral_induct) auto

   749 lemmas sign_Min_neg = trans [OF sign_Min_lt_0 neg_def [symmetric]]

   751 lemma bin_rest_trunc:

   752   "(bin_rest (bintrunc n bin)) = bintrunc (n - 1) (bin_rest bin)"

   753   by (induct n arbitrary: bin) auto

   755 lemma bin_rest_power_trunc [rule_format] :

   756   "(bin_rest ^^ k) (bintrunc n bin) =

   757     bintrunc (n - k) ((bin_rest ^^ k) bin)"

   758   by (induct k) (auto simp: bin_rest_trunc)

   760 lemma bin_rest_trunc_i:

   761   "bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)"

   762   by auto

   764 lemma bin_rest_strunc:

   765   "bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"

   766   by (induct n arbitrary: bin) auto

   768 lemma bintrunc_rest [simp]:

   769   "bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"

   770   apply (induct n arbitrary: bin, simp)

   771   apply (case_tac bin rule: bin_exhaust)

   772   apply (auto simp: bintrunc_bintrunc_l)

   773   done

   775 lemma sbintrunc_rest [simp]:

   776   "sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"

   777   apply (induct n arbitrary: bin, simp)

   778   apply (case_tac bin rule: bin_exhaust)

   779   apply (auto simp: bintrunc_bintrunc_l split: bit.splits)

   780   done

   782 lemma bintrunc_rest':

   783   "bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n"

   784   by (rule ext) auto

   786 lemma sbintrunc_rest' :

   787   "sbintrunc n o bin_rest o sbintrunc n = bin_rest o sbintrunc n"

   788   by (rule ext) auto

   790 lemma rco_lem:

   791   "f o g o f = g o f ==> f o (g o f) ^^ n = g ^^ n o f"

   792   apply (rule ext)

   793   apply (induct_tac n)

   794    apply (simp_all (no_asm))

   795   apply (drule fun_cong)

   796   apply (unfold o_def)

   797   apply (erule trans)

   798   apply simp

   799   done

   801 lemma rco_alt: "(f o g) ^^ n o f = f o (g o f) ^^ n"

   802   apply (rule ext)

   803   apply (induct n)

   804    apply (simp_all add: o_def)

   805   done

   807 lemmas rco_bintr = bintrunc_rest'

   808   [THEN rco_lem [THEN fun_cong], unfolded o_def]

   809 lemmas rco_sbintr = sbintrunc_rest'

   810   [THEN rco_lem [THEN fun_cong], unfolded o_def]

   812 subsection {* Splitting and concatenation *}

   814 primrec bin_split :: "nat \<Rightarrow> int \<Rightarrow> int \<times> int" where

   815   Z: "bin_split 0 w = (w, 0)"

   816   | Suc: "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w)

   817         in (w1, w2 BIT bin_last w))"

   819 lemma [code]:

   820   "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w) in (w1, w2 BIT bin_last w))"

   821   "bin_split 0 w = (w, 0)"

   822   by (simp_all add: Pls_def)

   824 primrec bin_cat :: "int \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int" where

   825   Z: "bin_cat w 0 v = w"

   826   | Suc: "bin_cat w (Suc n) v = bin_cat w n (bin_rest v) BIT bin_last v"

   828 subsection {* Miscellaneous lemmas *}

   830 lemma funpow_minus_simp:

   831   "0 < n \<Longrightarrow> f ^^ n = f \<circ> f ^^ (n - 1)"

   832   by (cases n) simp_all

   834 lemmas funpow_pred_simp [simp] =

   835   funpow_minus_simp [of "number_of bin", simplified nobm1] for bin

   837 lemmas replicate_minus_simp =

   838   trans [OF gen_minus [where f = "%n. replicate n x"] replicate.replicate_Suc] for x

   840 lemmas replicate_pred_simp [simp] =

   841   replicate_minus_simp [of "number_of bin", simplified nobm1] for bin

   843 lemmas power_Suc_no [simp] = power_Suc [of "number_of a"] for a

   845 lemmas power_minus_simp =

   846   trans [OF gen_minus [where f = "power f"] power_Suc] for f

   848 lemmas power_pred_simp =

   849   power_minus_simp [of "number_of bin", simplified nobm1] for bin

   850 lemmas power_pred_simp_no [simp] = power_pred_simp [where f= "number_of f"] for f

   852 lemma list_exhaust_size_gt0:

   853   assumes y: "\<And>a list. y = a # list \<Longrightarrow> P"

   854   shows "0 < length y \<Longrightarrow> P"

   855   apply (cases y, simp)

   856   apply (rule y)

   857   apply fastforce

   858   done

   860 lemma list_exhaust_size_eq0:

   861   assumes y: "y = [] \<Longrightarrow> P"

   862   shows "length y = 0 \<Longrightarrow> P"

   863   apply (cases y)

   864    apply (rule y, simp)

   865   apply simp

   866   done

   868 lemma size_Cons_lem_eq:

   869   "y = xa # list ==> size y = Suc k ==> size list = k"

   870   by auto

   872 lemma size_Cons_lem_eq_bin:

   873   "y = xa # list ==> size y = number_of (Int.succ k) ==>

   874     size list = number_of k"

   875   by (auto simp: pred_def succ_def split add : split_if_asm)

   877 lemmas ls_splits = prod.split prod.split_asm split_if_asm

   879 lemma not_B1_is_B0: "y \<noteq> (1::bit) \<Longrightarrow> y = (0::bit)"

   880   by (cases y) auto

   882 lemma B1_ass_B0:

   883   assumes y: "y = (0::bit) \<Longrightarrow> y = (1::bit)"

   884   shows "y = (1::bit)"

   885   apply (rule classical)

   886   apply (drule not_B1_is_B0)

   887   apply (erule y)

   888   done

   890 -- "simplifications for specific word lengths"

   891 lemmas n2s_ths [THEN eq_reflection] = add_2_eq_Suc add_2_eq_Suc'

   893 lemmas s2n_ths = n2s_ths [symmetric]

   895 end

author	huffman
	Tue, 27 Dec 2011 15:37:33 +0100
changeset 46872	0b562d564d5f
parent 46871	871bdab23f5c
child 46893	fad87bb608fc
child 46895	c296c75f4cf4
permissions	-rw-r--r--