wneuper/isa: src/HOL/Word/Bit_Representation.thy@cce7e6197a46

     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 = Int.Pls"

   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 =

   334     (case bin_last bin of (1::bit) => Int.Min | (0::bit) => Int.Pls)"

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

   337 lemma [code]:

   338   "sbintrunc 0 bin =

   339     (case bin_last bin of (1::bit) => - 1 | (0::bit) => 0)"

   340   "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"

   341   apply simp_all

   342   apply (simp only: Pls_def Min_def)

   343   done

   345 lemma sign_bintr: "bin_sign (bintrunc n w) = Int.Pls"

   346   by (induct n arbitrary: w) auto

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

   349   apply (induct n arbitrary: w)

   350   apply (simp add: Pls_def)

   351   apply (simp add: bin_last_def bin_rest_def Bit_def zmod_zmult2_eq)

   352   done

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

   355   apply (induct n arbitrary: w)

   356    apply clarsimp

   357    apply (subst mod_add_left_eq)

   358    apply (simp add: bin_last_def)

   359    apply (simp add: number_of_eq)

   360   apply (simp add: Pls_def)

   361   apply (simp add: bin_last_def bin_rest_def Bit_def

   362               cong: number_of_False_cong)

   363   apply (clarsimp simp: mod_mult_mult1 [symmetric]

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

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

   366      apply auto

   367   apply (auto simp: even_def)

   368   done

   370 subsection "Simplifications for (s)bintrunc"

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

   373   by (induct n) (auto simp add: Int.Pls_def)

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

   376   by (induct n) (auto simp add: Int.Pls_def)

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

   379   by (induct n) (auto, simp add: number_of_is_id)

   381 lemma bintrunc_Suc_numeral:

   382   "bintrunc (Suc n) 1 = 1"

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

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

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

   386   by simp_all

   388 lemma sbintrunc_0_numeral [simp]:

   389   "sbintrunc 0 1 = -1"

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

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

   392   by (simp_all, unfold Pls_def number_of_is_id, simp_all)

   394 lemma sbintrunc_Suc_numeral:

   395   "sbintrunc (Suc n) 1 = 1"

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

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

   398   by simp_all

   400 lemma bit_bool:

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

   402   by (cases b') auto

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

   406 lemma bin_sign_lem: "(bin_sign (sbintrunc n bin) = Int.Min) = bin_nth bin n"

   407   apply (induct n arbitrary: bin)

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

   409   done

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

   412   apply (induct n arbitrary: w m)

   413    apply (case_tac m, auto)[1]

   414   apply (case_tac m, auto)[1]

   415   done

   417 lemma nth_sbintr:

   418   "bin_nth (sbintrunc m w) n =

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

   420   apply (induct n arbitrary: w m)

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

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

   423   done

   425 lemma bin_nth_Bit:

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

   427   by (cases n) auto

   429 lemma bin_nth_Bit0:

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

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

   433 lemma bin_nth_Bit1:

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

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

   437 lemma bintrunc_bintrunc_l:

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

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

   441 lemma sbintrunc_sbintrunc_l:

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

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

   445 lemma bintrunc_bintrunc_ge:

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

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

   449 lemma bintrunc_bintrunc_min [simp]:

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

   451   apply (rule bin_eqI)

   452   apply (auto simp: nth_bintr)

   453   done

   455 lemma sbintrunc_sbintrunc_min [simp]:

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

   457   apply (rule bin_eqI)

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

   459   done

   461 lemmas bintrunc_Pls =

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

   464 lemmas bintrunc_Min [simp] =

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

   467 lemmas bintrunc_BIT  [simp] =

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

   470 lemma bintrunc_Bit0 [simp]:

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

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

   474 lemma bintrunc_Bit1 [simp]:

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

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

   478 lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT

   479   bintrunc_Bit0 bintrunc_Bit1

   480   bintrunc_Suc_numeral

   482 lemmas sbintrunc_Suc_Pls =

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

   485 lemmas sbintrunc_Suc_Min =

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

   488 lemmas sbintrunc_Suc_BIT [simp] =

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

   491 lemma sbintrunc_Suc_Bit0 [simp]:

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

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

   495 lemma sbintrunc_Suc_Bit1 [simp]:

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

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

   499 lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT

   500   sbintrunc_Suc_Bit0 sbintrunc_Suc_Bit1

   501   sbintrunc_Suc_numeral

   503 lemmas sbintrunc_Pls =

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

   505                simplified bin_last_simps bin_rest_simps bit.simps]

   507 lemmas sbintrunc_Min =

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

   509                simplified bin_last_simps bin_rest_simps bit.simps]

   511 lemmas sbintrunc_0_BIT_B0 [simp] =

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

   513                simplified bin_last_simps bin_rest_simps bit.simps] for w

   515 lemmas sbintrunc_0_BIT_B1 [simp] =

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

   517                simplified bin_last_simps bin_rest_simps bit.simps] for w

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

   520   using sbintrunc_0_BIT_B0 by simp

   522 lemma sbintrunc_0_Bit1 [simp]: "sbintrunc 0 (Int.Bit1 w) = Int.Min"

   523   using sbintrunc_0_BIT_B1 by simp

   525 lemmas sbintrunc_0_simps =

   526   sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1

   527   sbintrunc_0_Bit0 sbintrunc_0_Bit1

   529 lemmas bintrunc_simps = bintrunc.Z bintrunc_Sucs

   530 lemmas sbintrunc_simps = sbintrunc_0_simps sbintrunc_Sucs

   532 lemma bintrunc_minus:

   533   "0 < n ==> bintrunc (Suc (n - 1)) w = bintrunc n w"

   534   by auto

   536 lemma sbintrunc_minus:

   537   "0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w"

   538   by auto

   540 lemmas bintrunc_minus_simps =

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

   542 lemmas sbintrunc_minus_simps =

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

   545 lemma bintrunc_n_Pls [simp]:

   546   "bintrunc n Int.Pls = Int.Pls"

   547   by (induct n) (auto simp: BIT_simps)

   549 lemma sbintrunc_n_PM [simp]:

   550   "sbintrunc n Int.Pls = Int.Pls"

   551   "sbintrunc n Int.Min = Int.Min"

   552   by (induct n) (auto simp: BIT_simps)

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

   556 lemmas bintrunc_BIT_I = trans [OF bintrunc_BIT thobini1]

   557 lemmas bintrunc_Min_I = trans [OF bintrunc_Min thobini1]

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

   560 lemmas bintrunc_Pls_minus_I = bmsts(1)

   561 lemmas bintrunc_Min_minus_I = bmsts(2)

   562 lemmas bintrunc_BIT_minus_I = bmsts(3)

   564 lemma bintrunc_Suc_lem:

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

   566   by auto

   568 lemmas bintrunc_Suc_Ialts =

   569   bintrunc_Min_I [THEN bintrunc_Suc_lem]

   570   bintrunc_BIT_I [THEN bintrunc_Suc_lem]

   572 lemmas sbintrunc_BIT_I = trans [OF sbintrunc_Suc_BIT thobini1]

   574 lemmas sbintrunc_Suc_Is =

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

   577 lemmas sbintrunc_Suc_minus_Is =

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

   580 lemma sbintrunc_Suc_lem:

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

   582   by auto

   584 lemmas sbintrunc_Suc_Ialts =

   585   sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem]

   587 lemma sbintrunc_bintrunc_lt:

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

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

   591 lemma bintrunc_sbintrunc_le:

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

   593   apply (rule bin_eqI)

   594   apply (auto simp: nth_sbintr nth_bintr)

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

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

   597   done

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

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

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

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

   604 lemma bintrunc_sbintrunc' [simp]:

   605   "0 < n \<Longrightarrow> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w"

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

   608 lemma sbintrunc_bintrunc' [simp]:

   609   "0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"

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

   612 lemma bin_sbin_eq_iff:

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

   614    sbintrunc n x = sbintrunc n y"

   615   apply (rule iffI)

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

   617    apply simp

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

   619   apply simp

   620   done

   622 lemma bin_sbin_eq_iff':

   623   "0 < n \<Longrightarrow> bintrunc n x = bintrunc n y <->

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

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

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

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

   630 lemmas bintrunc_bintrunc_l' = le_add1 [THEN bintrunc_bintrunc_l]

   631 lemmas sbintrunc_sbintrunc_l' = le_add1 [THEN sbintrunc_sbintrunc_l]

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

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

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

   637 lemmas nat_non0_gr =

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

   640 lemmas bintrunc_pred_simps [simp] =

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

   643 lemmas sbintrunc_pred_simps [simp] =

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

   646 lemma no_bintr_alt:

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

   648   by (simp add: number_of_eq bintrunc_mod2p)

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

   651   by (rule ext) (rule bintrunc_mod2p)

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

   654   apply (unfold no_bintr_alt1)

   655   apply (auto simp add: image_iff)

   656   apply (rule exI)

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

   658   done

   660 lemma no_bintr:

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

   662   by (simp add : bintrunc_mod2p number_of_eq)

   664 lemma no_sbintr_alt2:

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

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

   668 lemma no_sbintr:

   669   "number_of (sbintrunc n w) =

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

   671   by (simp add : no_sbintr_alt2 number_of_eq)

   673 lemma range_sbintrunc:

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

   675   apply (unfold no_sbintr_alt2)

   676   apply (auto simp add: image_iff eq_diff_eq)

   677   apply (rule exI)

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

   679   done

   681 lemma sb_inc_lem:

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

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

   684   apply (rule TrueI)

   685   done

   687 lemma sb_inc_lem':

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

   689   by (rule sb_inc_lem) simp

   691 lemma sbintrunc_inc:

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

   693   unfolding no_sbintr_alt2 by (drule sb_inc_lem') simp

   695 lemma sb_dec_lem:

   696   "(0::int) <= - (2^k) + a ==> (a + 2^k) mod (2 * 2 ^ k) <= - (2 ^ k) + a"

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

   698     simplified zless2p, OF _ TrueI, simplified])

   700 lemma sb_dec_lem':

   701   "(2::int) ^ k <= a ==> (a + 2 ^ k) mod (2 * 2 ^ k) <= - (2 ^ k) + a"

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

   704 lemma sbintrunc_dec:

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

   706   unfolding no_sbintr_alt2 by (drule sb_dec_lem') simp

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

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

   711 lemmas brdmod1s' [symmetric] =

   712   mod_add_left_eq mod_add_right_eq

   713   zmod_zsub_left_eq zmod_zsub_right_eq

   714   zmod_zmult1_eq zmod_zmult1_eq_rev

   716 lemmas brdmods' [symmetric] =

   717   zpower_zmod' [symmetric]

   718   trans [OF mod_add_left_eq mod_add_right_eq]

   719   trans [OF zmod_zsub_left_eq zmod_zsub_right_eq]

   720   trans [OF zmod_zmult1_eq zmod_zmult1_eq_rev]

   721   zmod_uminus' [symmetric]

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

   723   zmod_zsub_left_eq [where b = "1"]

   725 lemmas bintr_arith1s =

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

   727 lemmas bintr_ariths =

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

   730 lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p]

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

   733   by (simp add : no_bintr m2pths)

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

   736   by (simp add : no_bintr m2pths)

   738 lemma bintr_Min:

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

   740   by (simp add : no_bintr m1mod2k)

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

   743   by (simp add : no_sbintr m2pths)

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

   746   by (simp add : no_sbintr m2pths)

   748 lemma bintrunc_Suc:

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

   750   by (case_tac bin rule: bin_exhaust) auto

   752 lemma sign_Pls_ge_0:

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

   754   by (induct bin rule: numeral_induct) auto

   756 lemma sign_Min_lt_0:

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

   758   by (induct bin rule: numeral_induct) auto

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

   762 lemma bin_rest_trunc:

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

   764   by (induct n arbitrary: bin) auto

   766 lemma bin_rest_power_trunc [rule_format] :

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

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

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

   771 lemma bin_rest_trunc_i:

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

   773   by auto

   775 lemma bin_rest_strunc:

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

   777   by (induct n arbitrary: bin) auto

   779 lemma bintrunc_rest [simp]:

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

   781   apply (induct n arbitrary: bin, simp)

   782   apply (case_tac bin rule: bin_exhaust)

   783   apply (auto simp: bintrunc_bintrunc_l)

   784   done

   786 lemma sbintrunc_rest [simp]:

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

   788   apply (induct n arbitrary: bin, simp)

   789   apply (case_tac bin rule: bin_exhaust)

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

   791   done

   793 lemma bintrunc_rest':

   794   "bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n"

   795   by (rule ext) auto

   797 lemma sbintrunc_rest' :

   798   "sbintrunc n o bin_rest o sbintrunc n = bin_rest o sbintrunc n"

   799   by (rule ext) auto

   801 lemma rco_lem:

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

   803   apply (rule ext)

   804   apply (induct_tac n)

   805    apply (simp_all (no_asm))

   806   apply (drule fun_cong)

   807   apply (unfold o_def)

   808   apply (erule trans)

   809   apply simp

   810   done

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

   813   apply (rule ext)

   814   apply (induct n)

   815    apply (simp_all add: o_def)

   816   done

   818 lemmas rco_bintr = bintrunc_rest'

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

   820 lemmas rco_sbintr = sbintrunc_rest'

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

   823 subsection {* Splitting and concatenation *}

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

   826   Z: "bin_split 0 w = (w, Int.Pls)"

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

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

   830 lemma [code]:

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

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

   833   by (simp_all add: Pls_def)

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

   836   Z: "bin_cat w 0 v = w"

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

   839 subsection {* Miscellaneous lemmas *}

   841 lemma funpow_minus_simp:

   842   "0 < n \<Longrightarrow> f ^^ n = f \<circ> f ^^ (n - 1)"

   843   by (cases n) simp_all

   845 lemmas funpow_pred_simp [simp] =

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

   848 lemmas replicate_minus_simp =

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

   851 lemmas replicate_pred_simp [simp] =

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

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

   856 lemmas power_minus_simp =

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

   859 lemmas power_pred_simp =

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

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

   863 lemma list_exhaust_size_gt0:

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

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

   866   apply (cases y, simp)

   867   apply (rule y)

   868   apply fastforce

   869   done

   871 lemma list_exhaust_size_eq0:

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

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

   874   apply (cases y)

   875    apply (rule y, simp)

   876   apply simp

   877   done

   879 lemma size_Cons_lem_eq:

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

   881   by auto

   883 lemma size_Cons_lem_eq_bin:

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

   885     size list = number_of k"

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

   888 lemmas ls_splits = prod.split prod.split_asm split_if_asm

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

   891   by (cases y) auto

   893 lemma B1_ass_B0:

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

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

   896   apply (rule classical)

   897   apply (drule not_B1_is_B0)

   898   apply (erule y)

   899   done

   901 -- "simplifications for specific word lengths"

   902 lemmas n2s_ths [THEN eq_reflection] = add_2_eq_Suc add_2_eq_Suc'

   904 lemmas s2n_ths = n2s_ths [symmetric]

   906 end

author	huffman
	Tue, 27 Dec 2011 12:49:03 +0100
changeset 46870	cce7e6197a46
parent 46825	f67d3bb5f09c
child 46871	871bdab23f5c
permissions	-rw-r--r--