src/HOL/Word/BinGeneral.thy
 author nipkow Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) changeset 30313 b2441b0c8d38 parent 30034 60f64f112174 child 30940 663af91c0720 permissions -rw-r--r--
```     1 (*
```
```     2   Author: Jeremy Dawson, NICTA
```
```     3
```
```     4   contains basic definition to do with integers
```
```     5   expressed using Pls, Min, BIT and important resulting theorems,
```
```     6   in particular, bin_rec and related work
```
```     7 *)
```
```     8
```
```     9 header {* Basic Definitions for Binary Integers *}
```
```    10
```
```    11 theory BinGeneral
```
```    12 imports Num_Lemmas
```
```    13 begin
```
```    14
```
```    15 subsection {* Further properties of numerals *}
```
```    16
```
```    17 datatype bit = B0 | B1
```
```    18
```
```    19 definition
```
```    20   Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90) where
```
```    21   "k BIT b = (case b of B0 \<Rightarrow> 0 | B1 \<Rightarrow> 1) + k + k"
```
```    22
```
```    23 lemma BIT_B0_eq_Bit0 [simp]: "w BIT B0 = Int.Bit0 w"
```
```    24   unfolding Bit_def Bit0_def by simp
```
```    25
```
```    26 lemma BIT_B1_eq_Bit1 [simp]: "w BIT B1 = Int.Bit1 w"
```
```    27   unfolding Bit_def Bit1_def by simp
```
```    28
```
```    29 lemmas BIT_simps = BIT_B0_eq_Bit0 BIT_B1_eq_Bit1
```
```    30
```
```    31 hide (open) const B0 B1
```
```    32
```
```    33 lemma Min_ne_Pls [iff]:
```
```    34   "Int.Min ~= Int.Pls"
```
```    35   unfolding Min_def Pls_def by auto
```
```    36
```
```    37 lemmas Pls_ne_Min [iff] = Min_ne_Pls [symmetric]
```
```    38
```
```    39 lemmas PlsMin_defs [intro!] =
```
```    40   Pls_def Min_def Pls_def [symmetric] Min_def [symmetric]
```
```    41
```
```    42 lemmas PlsMin_simps [simp] = PlsMin_defs [THEN Eq_TrueI]
```
```    43
```
```    44 lemma number_of_False_cong:
```
```    45   "False \<Longrightarrow> number_of x = number_of y"
```
```    46   by (rule FalseE)
```
```    47
```
```    48 (** ways in which type Bin resembles a datatype **)
```
```    49
```
```    50 lemma BIT_eq: "u BIT b = v BIT c ==> u = v & b = c"
```
```    51   apply (unfold Bit_def)
```
```    52   apply (simp (no_asm_use) split: bit.split_asm)
```
```    53      apply simp_all
```
```    54    apply (drule_tac f=even in arg_cong, clarsimp)+
```
```    55   done
```
```    56
```
```    57 lemmas BIT_eqE [elim!] = BIT_eq [THEN conjE, standard]
```
```    58
```
```    59 lemma BIT_eq_iff [simp]:
```
```    60   "(u BIT b = v BIT c) = (u = v \<and> b = c)"
```
```    61   by (rule iffI) auto
```
```    62
```
```    63 lemmas BIT_eqI [intro!] = conjI [THEN BIT_eq_iff [THEN iffD2]]
```
```    64
```
```    65 lemma less_Bits:
```
```    66   "(v BIT b < w BIT c) = (v < w | v <= w & b = bit.B0 & c = bit.B1)"
```
```    67   unfolding Bit_def by (auto split: bit.split)
```
```    68
```
```    69 lemma le_Bits:
```
```    70   "(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= bit.B1 | c ~= bit.B0))"
```
```    71   unfolding Bit_def by (auto split: bit.split)
```
```    72
```
```    73 lemma no_no [simp]: "number_of (number_of i) = i"
```
```    74   unfolding number_of_eq by simp
```
```    75
```
```    76 lemma Bit_B0:
```
```    77   "k BIT bit.B0 = k + k"
```
```    78    by (unfold Bit_def) simp
```
```    79
```
```    80 lemma Bit_B1:
```
```    81   "k BIT bit.B1 = k + k + 1"
```
```    82    by (unfold Bit_def) simp
```
```    83
```
```    84 lemma Bit_B0_2t: "k BIT bit.B0 = 2 * k"
```
```    85   by (rule trans, rule Bit_B0) simp
```
```    86
```
```    87 lemma Bit_B1_2t: "k BIT bit.B1 = 2 * k + 1"
```
```    88   by (rule trans, rule Bit_B1) simp
```
```    89
```
```    90 lemma B_mod_2':
```
```    91   "X = 2 ==> (w BIT bit.B1) mod X = 1 & (w BIT bit.B0) mod X = 0"
```
```    92   apply (simp (no_asm) only: Bit_B0 Bit_B1)
```
```    93   apply (simp add: z1pmod2)
```
```    94   done
```
```    95
```
```    96 lemma B1_mod_2 [simp]: "(Int.Bit1 w) mod 2 = 1"
```
```    97   unfolding numeral_simps number_of_is_id by (simp add: z1pmod2)
```
```    98
```
```    99 lemma B0_mod_2 [simp]: "(Int.Bit0 w) mod 2 = 0"
```
```   100   unfolding numeral_simps number_of_is_id by simp
```
```   101
```
```   102 lemma neB1E [elim!]:
```
```   103   assumes ne: "y \<noteq> bit.B1"
```
```   104   assumes y: "y = bit.B0 \<Longrightarrow> P"
```
```   105   shows "P"
```
```   106   apply (rule y)
```
```   107   apply (cases y rule: bit.exhaust, simp)
```
```   108   apply (simp add: ne)
```
```   109   done
```
```   110
```
```   111 lemma bin_ex_rl: "EX w b. w BIT b = bin"
```
```   112   apply (unfold Bit_def)
```
```   113   apply (cases "even bin")
```
```   114    apply (clarsimp simp: even_equiv_def)
```
```   115    apply (auto simp: odd_equiv_def split: bit.split)
```
```   116   done
```
```   117
```
```   118 lemma bin_exhaust:
```
```   119   assumes Q: "\<And>x b. bin = x BIT b \<Longrightarrow> Q"
```
```   120   shows "Q"
```
```   121   apply (insert bin_ex_rl [of bin])
```
```   122   apply (erule exE)+
```
```   123   apply (rule Q)
```
```   124   apply force
```
```   125   done
```
```   126
```
```   127
```
```   128 subsection {* Destructors for binary integers *}
```
```   129
```
```   130 definition bin_rl :: "int \<Rightarrow> int \<times> bit" where
```
```   131   [code del]: "bin_rl w = (THE (r, l). w = r BIT l)"
```
```   132
```
```   133 lemma bin_rl_char: "(bin_rl w = (r, l)) = (r BIT l = w)"
```
```   134   apply (unfold bin_rl_def)
```
```   135   apply safe
```
```   136    apply (cases w rule: bin_exhaust)
```
```   137    apply auto
```
```   138   done
```
```   139
```
```   140 definition
```
```   141   bin_rest_def [code del]: "bin_rest w = fst (bin_rl w)"
```
```   142
```
```   143 definition
```
```   144   bin_last_def [code del] : "bin_last w = snd (bin_rl w)"
```
```   145
```
```   146 primrec bin_nth where
```
```   147   Z: "bin_nth w 0 = (bin_last w = bit.B1)"
```
```   148   | Suc: "bin_nth w (Suc n) = bin_nth (bin_rest w) n"
```
```   149
```
```   150 lemma bin_rl: "bin_rl w = (bin_rest w, bin_last w)"
```
```   151   unfolding bin_rest_def bin_last_def by auto
```
```   152
```
```   153 lemma bin_rl_simps [simp]:
```
```   154   "bin_rl Int.Pls = (Int.Pls, bit.B0)"
```
```   155   "bin_rl Int.Min = (Int.Min, bit.B1)"
```
```   156   "bin_rl (Int.Bit0 r) = (r, bit.B0)"
```
```   157   "bin_rl (Int.Bit1 r) = (r, bit.B1)"
```
```   158   "bin_rl (r BIT b) = (r, b)"
```
```   159   unfolding bin_rl_char by simp_all
```
```   160
```
```   161 declare bin_rl_simps(1-4) [code]
```
```   162
```
```   163 lemmas bin_rl_simp [simp] = iffD1 [OF bin_rl_char bin_rl]
```
```   164
```
```   165 lemma bin_abs_lem:
```
```   166   "bin = (w BIT b) ==> ~ bin = Int.Min --> ~ bin = Int.Pls -->
```
```   167     nat (abs w) < nat (abs bin)"
```
```   168   apply (clarsimp simp add: bin_rl_char)
```
```   169   apply (unfold Pls_def Min_def Bit_def)
```
```   170   apply (cases b)
```
```   171    apply (clarsimp, arith)
```
```   172   apply (clarsimp, arith)
```
```   173   done
```
```   174
```
```   175 lemma bin_induct:
```
```   176   assumes PPls: "P Int.Pls"
```
```   177     and PMin: "P Int.Min"
```
```   178     and PBit: "!!bin bit. P bin ==> P (bin BIT bit)"
```
```   179   shows "P bin"
```
```   180   apply (rule_tac P=P and a=bin and f1="nat o abs"
```
```   181                   in wf_measure [THEN wf_induct])
```
```   182   apply (simp add: measure_def inv_image_def)
```
```   183   apply (case_tac x rule: bin_exhaust)
```
```   184   apply (frule bin_abs_lem)
```
```   185   apply (auto simp add : PPls PMin PBit)
```
```   186   done
```
```   187
```
```   188 lemma numeral_induct:
```
```   189   assumes Pls: "P Int.Pls"
```
```   190   assumes Min: "P Int.Min"
```
```   191   assumes Bit0: "\<And>w. \<lbrakk>P w; w \<noteq> Int.Pls\<rbrakk> \<Longrightarrow> P (Int.Bit0 w)"
```
```   192   assumes Bit1: "\<And>w. \<lbrakk>P w; w \<noteq> Int.Min\<rbrakk> \<Longrightarrow> P (Int.Bit1 w)"
```
```   193   shows "P x"
```
```   194   apply (induct x rule: bin_induct)
```
```   195     apply (rule Pls)
```
```   196    apply (rule Min)
```
```   197   apply (case_tac bit)
```
```   198    apply (case_tac "bin = Int.Pls")
```
```   199     apply simp
```
```   200    apply (simp add: Bit0)
```
```   201   apply (case_tac "bin = Int.Min")
```
```   202    apply simp
```
```   203   apply (simp add: Bit1)
```
```   204   done
```
```   205
```
```   206 lemma bin_rest_simps [simp]:
```
```   207   "bin_rest Int.Pls = Int.Pls"
```
```   208   "bin_rest Int.Min = Int.Min"
```
```   209   "bin_rest (Int.Bit0 w) = w"
```
```   210   "bin_rest (Int.Bit1 w) = w"
```
```   211   "bin_rest (w BIT b) = w"
```
```   212   unfolding bin_rest_def by auto
```
```   213
```
```   214 declare bin_rest_simps(1-4) [code]
```
```   215
```
```   216 lemma bin_last_simps [simp]:
```
```   217   "bin_last Int.Pls = bit.B0"
```
```   218   "bin_last Int.Min = bit.B1"
```
```   219   "bin_last (Int.Bit0 w) = bit.B0"
```
```   220   "bin_last (Int.Bit1 w) = bit.B1"
```
```   221   "bin_last (w BIT b) = b"
```
```   222   unfolding bin_last_def by auto
```
```   223
```
```   224 declare bin_last_simps(1-4) [code]
```
```   225
```
```   226 lemma bin_r_l_extras [simp]:
```
```   227   "bin_last 0 = bit.B0"
```
```   228   "bin_last (- 1) = bit.B1"
```
```   229   "bin_last -1 = bit.B1"
```
```   230   "bin_last 1 = bit.B1"
```
```   231   "bin_rest 1 = 0"
```
```   232   "bin_rest 0 = 0"
```
```   233   "bin_rest (- 1) = - 1"
```
```   234   "bin_rest -1 = -1"
```
```   235   apply (unfold number_of_Min)
```
```   236   apply (unfold Pls_def [symmetric] Min_def [symmetric])
```
```   237   apply (unfold numeral_1_eq_1 [symmetric])
```
```   238   apply (auto simp: number_of_eq)
```
```   239   done
```
```   240
```
```   241 lemma bin_last_mod:
```
```   242   "bin_last w = (if w mod 2 = 0 then bit.B0 else bit.B1)"
```
```   243   apply (case_tac w rule: bin_exhaust)
```
```   244   apply (case_tac b)
```
```   245    apply auto
```
```   246   done
```
```   247
```
```   248 lemma bin_rest_div:
```
```   249   "bin_rest w = w div 2"
```
```   250   apply (case_tac w rule: bin_exhaust)
```
```   251   apply (rule trans)
```
```   252    apply clarsimp
```
```   253    apply (rule refl)
```
```   254   apply (drule trans)
```
```   255    apply (rule Bit_def)
```
```   256   apply (simp add: z1pdiv2 split: bit.split)
```
```   257   done
```
```   258
```
```   259 lemma Bit_div2 [simp]: "(w BIT b) div 2 = w"
```
```   260   unfolding bin_rest_div [symmetric] by auto
```
```   261
```
```   262 lemma Bit0_div2 [simp]: "(Int.Bit0 w) div 2 = w"
```
```   263   using Bit_div2 [where b=bit.B0] by simp
```
```   264
```
```   265 lemma Bit1_div2 [simp]: "(Int.Bit1 w) div 2 = w"
```
```   266   using Bit_div2 [where b=bit.B1] by simp
```
```   267
```
```   268 lemma bin_nth_lem [rule_format]:
```
```   269   "ALL y. bin_nth x = bin_nth y --> x = y"
```
```   270   apply (induct x rule: bin_induct)
```
```   271     apply safe
```
```   272     apply (erule rev_mp)
```
```   273     apply (induct_tac y rule: bin_induct)
```
```   274       apply (safe del: subset_antisym)
```
```   275       apply (drule_tac x=0 in fun_cong, force)
```
```   276      apply (erule notE, rule ext,
```
```   277             drule_tac x="Suc x" in fun_cong, force)
```
```   278     apply (drule_tac x=0 in fun_cong, force)
```
```   279    apply (erule rev_mp)
```
```   280    apply (induct_tac y rule: bin_induct)
```
```   281      apply (safe del: subset_antisym)
```
```   282      apply (drule_tac x=0 in fun_cong, force)
```
```   283     apply (erule notE, rule ext,
```
```   284            drule_tac x="Suc x" in fun_cong, force)
```
```   285    apply (drule_tac x=0 in fun_cong, force)
```
```   286   apply (case_tac y rule: bin_exhaust)
```
```   287   apply clarify
```
```   288   apply (erule allE)
```
```   289   apply (erule impE)
```
```   290    prefer 2
```
```   291    apply (erule BIT_eqI)
```
```   292    apply (drule_tac x=0 in fun_cong, force)
```
```   293   apply (rule ext)
```
```   294   apply (drule_tac x="Suc ?x" in fun_cong, force)
```
```   295   done
```
```   296
```
```   297 lemma bin_nth_eq_iff: "(bin_nth x = bin_nth y) = (x = y)"
```
```   298   by (auto elim: bin_nth_lem)
```
```   299
```
```   300 lemmas bin_eqI = ext [THEN bin_nth_eq_iff [THEN iffD1], standard]
```
```   301
```
```   302 lemma bin_nth_Pls [simp]: "~ bin_nth Int.Pls n"
```
```   303   by (induct n) auto
```
```   304
```
```   305 lemma bin_nth_Min [simp]: "bin_nth Int.Min n"
```
```   306   by (induct n) auto
```
```   307
```
```   308 lemma bin_nth_0_BIT: "bin_nth (w BIT b) 0 = (b = bit.B1)"
```
```   309   by auto
```
```   310
```
```   311 lemma bin_nth_Suc_BIT: "bin_nth (w BIT b) (Suc n) = bin_nth w n"
```
```   312   by auto
```
```   313
```
```   314 lemma bin_nth_minus [simp]: "0 < n ==> bin_nth (w BIT b) n = bin_nth w (n - 1)"
```
```   315   by (cases n) auto
```
```   316
```
```   317 lemma bin_nth_minus_Bit0 [simp]:
```
```   318   "0 < n ==> bin_nth (Int.Bit0 w) n = bin_nth w (n - 1)"
```
```   319   using bin_nth_minus [where b=bit.B0] by simp
```
```   320
```
```   321 lemma bin_nth_minus_Bit1 [simp]:
```
```   322   "0 < n ==> bin_nth (Int.Bit1 w) n = bin_nth w (n - 1)"
```
```   323   using bin_nth_minus [where b=bit.B1] by simp
```
```   324
```
```   325 lemmas bin_nth_0 = bin_nth.simps(1)
```
```   326 lemmas bin_nth_Suc = bin_nth.simps(2)
```
```   327
```
```   328 lemmas bin_nth_simps =
```
```   329   bin_nth_0 bin_nth_Suc bin_nth_Pls bin_nth_Min bin_nth_minus
```
```   330   bin_nth_minus_Bit0 bin_nth_minus_Bit1
```
```   331
```
```   332
```
```   333 subsection {* Recursion combinator for binary integers *}
```
```   334
```
```   335 lemma brlem: "(bin = Int.Min) = (- bin + Int.pred 0 = 0)"
```
```   336   unfolding Min_def pred_def by arith
```
```   337
```
```   338 function
```
```   339   bin_rec :: "'a \<Rightarrow> 'a \<Rightarrow> (int \<Rightarrow> bit \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> int \<Rightarrow> 'a"
```
```   340 where
```
```   341   "bin_rec f1 f2 f3 bin = (if bin = Int.Pls then f1
```
```   342     else if bin = Int.Min then f2
```
```   343     else case bin_rl bin of (w, b) => f3 w b (bin_rec f1 f2 f3 w))"
```
```   344   by pat_completeness auto
```
```   345
```
```   346 termination
```
```   347   apply (relation "measure (nat o abs o snd o snd o snd)")
```
```   348    apply simp
```
```   349   apply (simp add: Pls_def brlem)
```
```   350   apply (clarsimp simp: bin_rl_char pred_def)
```
```   351   apply (frule thin_rl [THEN refl [THEN bin_abs_lem [rule_format]]])
```
```   352     apply (unfold Pls_def Min_def number_of_eq)
```
```   353     prefer 2
```
```   354     apply (erule asm_rl)
```
```   355    apply auto
```
```   356   done
```
```   357
```
```   358 declare bin_rec.simps [simp del]
```
```   359
```
```   360 lemma bin_rec_PM:
```
```   361   "f = bin_rec f1 f2 f3 ==> f Int.Pls = f1 & f Int.Min = f2"
```
```   362   by (auto simp add: bin_rec.simps)
```
```   363
```
```   364 lemma bin_rec_Pls: "bin_rec f1 f2 f3 Int.Pls = f1"
```
```   365   by (simp add: bin_rec.simps)
```
```   366
```
```   367 lemma bin_rec_Min: "bin_rec f1 f2 f3 Int.Min = f2"
```
```   368   by (simp add: bin_rec.simps)
```
```   369
```
```   370 lemma bin_rec_Bit0:
```
```   371   "f3 Int.Pls bit.B0 f1 = f1 \<Longrightarrow>
```
```   372     bin_rec f1 f2 f3 (Int.Bit0 w) = f3 w bit.B0 (bin_rec f1 f2 f3 w)"
```
```   373   by (simp add: bin_rec_Pls bin_rec.simps [of _ _ _ "Int.Bit0 w"])
```
```   374
```
```   375 lemma bin_rec_Bit1:
```
```   376   "f3 Int.Min bit.B1 f2 = f2 \<Longrightarrow>
```
```   377     bin_rec f1 f2 f3 (Int.Bit1 w) = f3 w bit.B1 (bin_rec f1 f2 f3 w)"
```
```   378   by (simp add: bin_rec_Min bin_rec.simps [of _ _ _ "Int.Bit1 w"])
```
```   379
```
```   380 lemma bin_rec_Bit:
```
```   381   "f = bin_rec f1 f2 f3  ==> f3 Int.Pls bit.B0 f1 = f1 ==>
```
```   382     f3 Int.Min bit.B1 f2 = f2 ==> f (w BIT b) = f3 w b (f w)"
```
```   383   by (cases b, simp add: bin_rec_Bit0, simp add: bin_rec_Bit1)
```
```   384
```
```   385 lemmas bin_rec_simps = refl [THEN bin_rec_Bit] bin_rec_Pls bin_rec_Min
```
```   386   bin_rec_Bit0 bin_rec_Bit1
```
```   387
```
```   388
```
```   389 subsection {* Truncating binary integers *}
```
```   390
```
```   391 definition
```
```   392   bin_sign_def [code del] : "bin_sign = bin_rec Int.Pls Int.Min (%w b s. s)"
```
```   393
```
```   394 lemma bin_sign_simps [simp]:
```
```   395   "bin_sign Int.Pls = Int.Pls"
```
```   396   "bin_sign Int.Min = Int.Min"
```
```   397   "bin_sign (Int.Bit0 w) = bin_sign w"
```
```   398   "bin_sign (Int.Bit1 w) = bin_sign w"
```
```   399   "bin_sign (w BIT b) = bin_sign w"
```
```   400   unfolding bin_sign_def by (auto simp: bin_rec_simps)
```
```   401
```
```   402 declare bin_sign_simps(1-4) [code]
```
```   403
```
```   404 lemma bin_sign_rest [simp]:
```
```   405   "bin_sign (bin_rest w) = (bin_sign w)"
```
```   406   by (cases w rule: bin_exhaust) auto
```
```   407
```
```   408 consts
```
```   409   bintrunc :: "nat => int => int"
```
```   410 primrec
```
```   411   Z : "bintrunc 0 bin = Int.Pls"
```
```   412   Suc : "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT (bin_last bin)"
```
```   413
```
```   414 consts
```
```   415   sbintrunc :: "nat => int => int"
```
```   416 primrec
```
```   417   Z : "sbintrunc 0 bin =
```
```   418     (case bin_last bin of bit.B1 => Int.Min | bit.B0 => Int.Pls)"
```
```   419   Suc : "sbintrunc (Suc n) bin = sbintrunc n (bin_rest bin) BIT (bin_last bin)"
```
```   420
```
```   421 lemma sign_bintr:
```
```   422   "!!w. bin_sign (bintrunc n w) = Int.Pls"
```
```   423   by (induct n) auto
```
```   424
```
```   425 lemma bintrunc_mod2p:
```
```   426   "!!w. bintrunc n w = (w mod 2 ^ n :: int)"
```
```   427   apply (induct n, clarsimp)
```
```   428   apply (simp add: bin_last_mod bin_rest_div Bit_def zmod_zmult2_eq
```
```   429               cong: number_of_False_cong)
```
```   430   done
```
```   431
```
```   432 lemma sbintrunc_mod2p:
```
```   433   "!!w. sbintrunc n w = ((w + 2 ^ n) mod 2 ^ (Suc n) - 2 ^ n :: int)"
```
```   434   apply (induct n)
```
```   435    apply clarsimp
```
```   436    apply (subst mod_add_left_eq)
```
```   437    apply (simp add: bin_last_mod)
```
```   438    apply (simp add: number_of_eq)
```
```   439   apply clarsimp
```
```   440   apply (simp add: bin_last_mod bin_rest_div Bit_def
```
```   441               cong: number_of_False_cong)
```
```   442   apply (clarsimp simp: zmod_zmult_zmult1 [symmetric]
```
```   443          zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2 [THEN sym]]])
```
```   444   apply (rule trans [symmetric, OF _ emep1])
```
```   445      apply auto
```
```   446   apply (auto simp: even_def)
```
```   447   done
```
```   448
```
```   449 subsection "Simplifications for (s)bintrunc"
```
```   450
```
```   451 lemma bit_bool:
```
```   452   "(b = (b' = bit.B1)) = (b' = (if b then bit.B1 else bit.B0))"
```
```   453   by (cases b') auto
```
```   454
```
```   455 lemmas bit_bool1 [simp] = refl [THEN bit_bool [THEN iffD1], symmetric]
```
```   456
```
```   457 lemma bin_sign_lem:
```
```   458   "!!bin. (bin_sign (sbintrunc n bin) = Int.Min) = bin_nth bin n"
```
```   459   apply (induct n)
```
```   460    apply (case_tac bin rule: bin_exhaust, case_tac b, auto)+
```
```   461   done
```
```   462
```
```   463 lemma nth_bintr:
```
```   464   "!!w m. bin_nth (bintrunc m w) n = (n < m & bin_nth w n)"
```
```   465   apply (induct n)
```
```   466    apply (case_tac m, auto)[1]
```
```   467   apply (case_tac m, auto)[1]
```
```   468   done
```
```   469
```
```   470 lemma nth_sbintr:
```
```   471   "!!w m. bin_nth (sbintrunc m w) n =
```
```   472           (if n < m then bin_nth w n else bin_nth w m)"
```
```   473   apply (induct n)
```
```   474    apply (case_tac m, simp_all split: bit.splits)[1]
```
```   475   apply (case_tac m, simp_all split: bit.splits)[1]
```
```   476   done
```
```   477
```
```   478 lemma bin_nth_Bit:
```
```   479   "bin_nth (w BIT b) n = (n = 0 & b = bit.B1 | (EX m. n = Suc m & bin_nth w m))"
```
```   480   by (cases n) auto
```
```   481
```
```   482 lemma bin_nth_Bit0:
```
```   483   "bin_nth (Int.Bit0 w) n = (EX m. n = Suc m & bin_nth w m)"
```
```   484   using bin_nth_Bit [where b=bit.B0] by simp
```
```   485
```
```   486 lemma bin_nth_Bit1:
```
```   487   "bin_nth (Int.Bit1 w) n = (n = 0 | (EX m. n = Suc m & bin_nth w m))"
```
```   488   using bin_nth_Bit [where b=bit.B1] by simp
```
```   489
```
```   490 lemma bintrunc_bintrunc_l:
```
```   491   "n <= m ==> (bintrunc m (bintrunc n w) = bintrunc n w)"
```
```   492   by (rule bin_eqI) (auto simp add : nth_bintr)
```
```   493
```
```   494 lemma sbintrunc_sbintrunc_l:
```
```   495   "n <= m ==> (sbintrunc m (sbintrunc n w) = sbintrunc n w)"
```
```   496   by (rule bin_eqI) (auto simp: nth_sbintr min_def)
```
```   497
```
```   498 lemma bintrunc_bintrunc_ge:
```
```   499   "n <= m ==> (bintrunc n (bintrunc m w) = bintrunc n w)"
```
```   500   by (rule bin_eqI) (auto simp: nth_bintr)
```
```   501
```
```   502 lemma bintrunc_bintrunc_min [simp]:
```
```   503   "bintrunc m (bintrunc n w) = bintrunc (min m n) w"
```
```   504   apply (unfold min_def)
```
```   505   apply (rule bin_eqI)
```
```   506   apply (auto simp: nth_bintr)
```
```   507   done
```
```   508
```
```   509 lemma sbintrunc_sbintrunc_min [simp]:
```
```   510   "sbintrunc m (sbintrunc n w) = sbintrunc (min m n) w"
```
```   511   apply (unfold min_def)
```
```   512   apply (rule bin_eqI)
```
```   513   apply (auto simp: nth_sbintr)
```
```   514   done
```
```   515
```
```   516 lemmas bintrunc_Pls =
```
```   517   bintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps, standard]
```
```   518
```
```   519 lemmas bintrunc_Min [simp] =
```
```   520   bintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps, standard]
```
```   521
```
```   522 lemmas bintrunc_BIT  [simp] =
```
```   523   bintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps, standard]
```
```   524
```
```   525 lemma bintrunc_Bit0 [simp]:
```
```   526   "bintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (bintrunc n w)"
```
```   527   using bintrunc_BIT [where b=bit.B0] by simp
```
```   528
```
```   529 lemma bintrunc_Bit1 [simp]:
```
```   530   "bintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (bintrunc n w)"
```
```   531   using bintrunc_BIT [where b=bit.B1] by simp
```
```   532
```
```   533 lemmas bintrunc_Sucs = bintrunc_Pls bintrunc_Min bintrunc_BIT
```
```   534   bintrunc_Bit0 bintrunc_Bit1
```
```   535
```
```   536 lemmas sbintrunc_Suc_Pls =
```
```   537   sbintrunc.Suc [where bin="Int.Pls", simplified bin_last_simps bin_rest_simps, standard]
```
```   538
```
```   539 lemmas sbintrunc_Suc_Min =
```
```   540   sbintrunc.Suc [where bin="Int.Min", simplified bin_last_simps bin_rest_simps, standard]
```
```   541
```
```   542 lemmas sbintrunc_Suc_BIT [simp] =
```
```   543   sbintrunc.Suc [where bin="w BIT b", simplified bin_last_simps bin_rest_simps, standard]
```
```   544
```
```   545 lemma sbintrunc_Suc_Bit0 [simp]:
```
```   546   "sbintrunc (Suc n) (Int.Bit0 w) = Int.Bit0 (sbintrunc n w)"
```
```   547   using sbintrunc_Suc_BIT [where b=bit.B0] by simp
```
```   548
```
```   549 lemma sbintrunc_Suc_Bit1 [simp]:
```
```   550   "sbintrunc (Suc n) (Int.Bit1 w) = Int.Bit1 (sbintrunc n w)"
```
```   551   using sbintrunc_Suc_BIT [where b=bit.B1] by simp
```
```   552
```
```   553 lemmas sbintrunc_Sucs = sbintrunc_Suc_Pls sbintrunc_Suc_Min sbintrunc_Suc_BIT
```
```   554   sbintrunc_Suc_Bit0 sbintrunc_Suc_Bit1
```
```   555
```
```   556 lemmas sbintrunc_Pls =
```
```   557   sbintrunc.Z [where bin="Int.Pls",
```
```   558                simplified bin_last_simps bin_rest_simps bit.simps, standard]
```
```   559
```
```   560 lemmas sbintrunc_Min =
```
```   561   sbintrunc.Z [where bin="Int.Min",
```
```   562                simplified bin_last_simps bin_rest_simps bit.simps, standard]
```
```   563
```
```   564 lemmas sbintrunc_0_BIT_B0 [simp] =
```
```   565   sbintrunc.Z [where bin="w BIT bit.B0",
```
```   566                simplified bin_last_simps bin_rest_simps bit.simps, standard]
```
```   567
```
```   568 lemmas sbintrunc_0_BIT_B1 [simp] =
```
```   569   sbintrunc.Z [where bin="w BIT bit.B1",
```
```   570                simplified bin_last_simps bin_rest_simps bit.simps, standard]
```
```   571
```
```   572 lemma sbintrunc_0_Bit0 [simp]: "sbintrunc 0 (Int.Bit0 w) = Int.Pls"
```
```   573   using sbintrunc_0_BIT_B0 by simp
```
```   574
```
```   575 lemma sbintrunc_0_Bit1 [simp]: "sbintrunc 0 (Int.Bit1 w) = Int.Min"
```
```   576   using sbintrunc_0_BIT_B1 by simp
```
```   577
```
```   578 lemmas sbintrunc_0_simps =
```
```   579   sbintrunc_Pls sbintrunc_Min sbintrunc_0_BIT_B0 sbintrunc_0_BIT_B1
```
```   580   sbintrunc_0_Bit0 sbintrunc_0_Bit1
```
```   581
```
```   582 lemmas bintrunc_simps = bintrunc.Z bintrunc_Sucs
```
```   583 lemmas sbintrunc_simps = sbintrunc_0_simps sbintrunc_Sucs
```
```   584
```
```   585 lemma bintrunc_minus:
```
```   586   "0 < n ==> bintrunc (Suc (n - 1)) w = bintrunc n w"
```
```   587   by auto
```
```   588
```
```   589 lemma sbintrunc_minus:
```
```   590   "0 < n ==> sbintrunc (Suc (n - 1)) w = sbintrunc n w"
```
```   591   by auto
```
```   592
```
```   593 lemmas bintrunc_minus_simps =
```
```   594   bintrunc_Sucs [THEN [2] bintrunc_minus [symmetric, THEN trans], standard]
```
```   595 lemmas sbintrunc_minus_simps =
```
```   596   sbintrunc_Sucs [THEN [2] sbintrunc_minus [symmetric, THEN trans], standard]
```
```   597
```
```   598 lemma bintrunc_n_Pls [simp]:
```
```   599   "bintrunc n Int.Pls = Int.Pls"
```
```   600   by (induct n) auto
```
```   601
```
```   602 lemma sbintrunc_n_PM [simp]:
```
```   603   "sbintrunc n Int.Pls = Int.Pls"
```
```   604   "sbintrunc n Int.Min = Int.Min"
```
```   605   by (induct n) auto
```
```   606
```
```   607 lemmas thobini1 = arg_cong [where f = "%w. w BIT b", standard]
```
```   608
```
```   609 lemmas bintrunc_BIT_I = trans [OF bintrunc_BIT thobini1]
```
```   610 lemmas bintrunc_Min_I = trans [OF bintrunc_Min thobini1]
```
```   611
```
```   612 lemmas bmsts = bintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans], standard]
```
```   613 lemmas bintrunc_Pls_minus_I = bmsts(1)
```
```   614 lemmas bintrunc_Min_minus_I = bmsts(2)
```
```   615 lemmas bintrunc_BIT_minus_I = bmsts(3)
```
```   616
```
```   617 lemma bintrunc_0_Min: "bintrunc 0 Int.Min = Int.Pls"
```
```   618   by auto
```
```   619 lemma bintrunc_0_BIT: "bintrunc 0 (w BIT b) = Int.Pls"
```
```   620   by auto
```
```   621
```
```   622 lemma bintrunc_Suc_lem:
```
```   623   "bintrunc (Suc n) x = y ==> m = Suc n ==> bintrunc m x = y"
```
```   624   by auto
```
```   625
```
```   626 lemmas bintrunc_Suc_Ialts =
```
```   627   bintrunc_Min_I [THEN bintrunc_Suc_lem, standard]
```
```   628   bintrunc_BIT_I [THEN bintrunc_Suc_lem, standard]
```
```   629
```
```   630 lemmas sbintrunc_BIT_I = trans [OF sbintrunc_Suc_BIT thobini1]
```
```   631
```
```   632 lemmas sbintrunc_Suc_Is =
```
```   633   sbintrunc_Sucs(1-3) [THEN thobini1 [THEN [2] trans], standard]
```
```   634
```
```   635 lemmas sbintrunc_Suc_minus_Is =
```
```   636   sbintrunc_minus_simps(1-3) [THEN thobini1 [THEN [2] trans], standard]
```
```   637
```
```   638 lemma sbintrunc_Suc_lem:
```
```   639   "sbintrunc (Suc n) x = y ==> m = Suc n ==> sbintrunc m x = y"
```
```   640   by auto
```
```   641
```
```   642 lemmas sbintrunc_Suc_Ialts =
```
```   643   sbintrunc_Suc_Is [THEN sbintrunc_Suc_lem, standard]
```
```   644
```
```   645 lemma sbintrunc_bintrunc_lt:
```
```   646   "m > n ==> sbintrunc n (bintrunc m w) = sbintrunc n w"
```
```   647   by (rule bin_eqI) (auto simp: nth_sbintr nth_bintr)
```
```   648
```
```   649 lemma bintrunc_sbintrunc_le:
```
```   650   "m <= Suc n ==> bintrunc m (sbintrunc n w) = bintrunc m w"
```
```   651   apply (rule bin_eqI)
```
```   652   apply (auto simp: nth_sbintr nth_bintr)
```
```   653    apply (subgoal_tac "x=n", safe, arith+)[1]
```
```   654   apply (subgoal_tac "x=n", safe, arith+)[1]
```
```   655   done
```
```   656
```
```   657 lemmas bintrunc_sbintrunc [simp] = order_refl [THEN bintrunc_sbintrunc_le]
```
```   658 lemmas sbintrunc_bintrunc [simp] = lessI [THEN sbintrunc_bintrunc_lt]
```
```   659 lemmas bintrunc_bintrunc [simp] = order_refl [THEN bintrunc_bintrunc_l]
```
```   660 lemmas sbintrunc_sbintrunc [simp] = order_refl [THEN sbintrunc_sbintrunc_l]
```
```   661
```
```   662 lemma bintrunc_sbintrunc' [simp]:
```
```   663   "0 < n \<Longrightarrow> bintrunc n (sbintrunc (n - 1) w) = bintrunc n w"
```
```   664   by (cases n) (auto simp del: bintrunc.Suc)
```
```   665
```
```   666 lemma sbintrunc_bintrunc' [simp]:
```
```   667   "0 < n \<Longrightarrow> sbintrunc (n - 1) (bintrunc n w) = sbintrunc (n - 1) w"
```
```   668   by (cases n) (auto simp del: bintrunc.Suc)
```
```   669
```
```   670 lemma bin_sbin_eq_iff:
```
```   671   "bintrunc (Suc n) x = bintrunc (Suc n) y <->
```
```   672    sbintrunc n x = sbintrunc n y"
```
```   673   apply (rule iffI)
```
```   674    apply (rule box_equals [OF _ sbintrunc_bintrunc sbintrunc_bintrunc])
```
```   675    apply simp
```
```   676   apply (rule box_equals [OF _ bintrunc_sbintrunc bintrunc_sbintrunc])
```
```   677   apply simp
```
```   678   done
```
```   679
```
```   680 lemma bin_sbin_eq_iff':
```
```   681   "0 < n \<Longrightarrow> bintrunc n x = bintrunc n y <->
```
```   682             sbintrunc (n - 1) x = sbintrunc (n - 1) y"
```
```   683   by (cases n) (simp_all add: bin_sbin_eq_iff del: bintrunc.Suc)
```
```   684
```
```   685 lemmas bintrunc_sbintruncS0 [simp] = bintrunc_sbintrunc' [unfolded One_nat_def]
```
```   686 lemmas sbintrunc_bintruncS0 [simp] = sbintrunc_bintrunc' [unfolded One_nat_def]
```
```   687
```
```   688 lemmas bintrunc_bintrunc_l' = le_add1 [THEN bintrunc_bintrunc_l]
```
```   689 lemmas sbintrunc_sbintrunc_l' = le_add1 [THEN sbintrunc_sbintrunc_l]
```
```   690
```
```   691 (* although bintrunc_minus_simps, if added to default simpset,
```
```   692   tends to get applied where it's not wanted in developing the theories,
```
```   693   we get a version for when the word length is given literally *)
```
```   694
```
```   695 lemmas nat_non0_gr =
```
```   696   trans [OF iszero_def [THEN Not_eq_iff [THEN iffD2]] refl, standard]
```
```   697
```
```   698 lemmas bintrunc_pred_simps [simp] =
```
```   699   bintrunc_minus_simps [of "number_of bin", simplified nobm1, standard]
```
```   700
```
```   701 lemmas sbintrunc_pred_simps [simp] =
```
```   702   sbintrunc_minus_simps [of "number_of bin", simplified nobm1, standard]
```
```   703
```
```   704 lemma no_bintr_alt:
```
```   705   "number_of (bintrunc n w) = w mod 2 ^ n"
```
```   706   by (simp add: number_of_eq bintrunc_mod2p)
```
```   707
```
```   708 lemma no_bintr_alt1: "bintrunc n = (%w. w mod 2 ^ n :: int)"
```
```   709   by (rule ext) (rule bintrunc_mod2p)
```
```   710
```
```   711 lemma range_bintrunc: "range (bintrunc n) = {i. 0 <= i & i < 2 ^ n}"
```
```   712   apply (unfold no_bintr_alt1)
```
```   713   apply (auto simp add: image_iff)
```
```   714   apply (rule exI)
```
```   715   apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
```
```   716   done
```
```   717
```
```   718 lemma no_bintr:
```
```   719   "number_of (bintrunc n w) = (number_of w mod 2 ^ n :: int)"
```
```   720   by (simp add : bintrunc_mod2p number_of_eq)
```
```   721
```
```   722 lemma no_sbintr_alt2:
```
```   723   "sbintrunc n = (%w. (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
```
```   724   by (rule ext) (simp add : sbintrunc_mod2p)
```
```   725
```
```   726 lemma no_sbintr:
```
```   727   "number_of (sbintrunc n w) =
```
```   728    ((number_of w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n :: int)"
```
```   729   by (simp add : no_sbintr_alt2 number_of_eq)
```
```   730
```
```   731 lemma range_sbintrunc:
```
```   732   "range (sbintrunc n) = {i. - (2 ^ n) <= i & i < 2 ^ n}"
```
```   733   apply (unfold no_sbintr_alt2)
```
```   734   apply (auto simp add: image_iff eq_diff_eq)
```
```   735   apply (rule exI)
```
```   736   apply (auto intro: int_mod_lem [THEN iffD1, symmetric])
```
```   737   done
```
```   738
```
```   739 lemma sb_inc_lem:
```
```   740   "(a::int) + 2^k < 0 \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"
```
```   741   apply (erule int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k", simplified zless2p])
```
```   742   apply (rule TrueI)
```
```   743   done
```
```   744
```
```   745 lemma sb_inc_lem':
```
```   746   "(a::int) < - (2^k) \<Longrightarrow> a + 2^k + 2^(Suc k) <= (a + 2^k) mod 2^(Suc k)"
```
```   747   by (rule iffD1 [OF less_diff_eq, THEN sb_inc_lem, simplified OrderedGroup.diff_0])
```
```   748
```
```   749 lemma sbintrunc_inc:
```
```   750   "x < - (2^n) ==> x + 2^(Suc n) <= sbintrunc n x"
```
```   751   unfolding no_sbintr_alt2 by (drule sb_inc_lem') simp
```
```   752
```
```   753 lemma sb_dec_lem:
```
```   754   "(0::int) <= - (2^k) + a ==> (a + 2^k) mod (2 * 2 ^ k) <= - (2 ^ k) + a"
```
```   755   by (rule int_mod_le' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k",
```
```   756     simplified zless2p, OF _ TrueI, simplified])
```
```   757
```
```   758 lemma sb_dec_lem':
```
```   759   "(2::int) ^ k <= a ==> (a + 2 ^ k) mod (2 * 2 ^ k) <= - (2 ^ k) + a"
```
```   760   by (rule iffD1 [OF diff_le_eq', THEN sb_dec_lem, simplified])
```
```   761
```
```   762 lemma sbintrunc_dec:
```
```   763   "x >= (2 ^ n) ==> x - 2 ^ (Suc n) >= sbintrunc n x"
```
```   764   unfolding no_sbintr_alt2 by (drule sb_dec_lem') simp
```
```   765
```
```   766 lemmas zmod_uminus' = zmod_uminus [where b="c", standard]
```
```   767 lemmas zpower_zmod' = zpower_zmod [where m="c" and y="k", standard]
```
```   768
```
```   769 lemmas brdmod1s' [symmetric] =
```
```   770   mod_add_left_eq mod_add_right_eq
```
```   771   zmod_zsub_left_eq zmod_zsub_right_eq
```
```   772   zmod_zmult1_eq zmod_zmult1_eq_rev
```
```   773
```
```   774 lemmas brdmods' [symmetric] =
```
```   775   zpower_zmod' [symmetric]
```
```   776   trans [OF mod_add_left_eq mod_add_right_eq]
```
```   777   trans [OF zmod_zsub_left_eq zmod_zsub_right_eq]
```
```   778   trans [OF zmod_zmult1_eq zmod_zmult1_eq_rev]
```
```   779   zmod_uminus' [symmetric]
```
```   780   mod_add_left_eq [where b = "1::int"]
```
```   781   zmod_zsub_left_eq [where b = "1"]
```
```   782
```
```   783 lemmas bintr_arith1s =
```
```   784   brdmod1s' [where c="2^n::int", folded pred_def succ_def bintrunc_mod2p, standard]
```
```   785 lemmas bintr_ariths =
```
```   786   brdmods' [where c="2^n::int", folded pred_def succ_def bintrunc_mod2p, standard]
```
```   787
```
```   788 lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p, standard]
```
```   789
```
```   790 lemma bintr_ge0: "(0 :: int) <= number_of (bintrunc n w)"
```
```   791   by (simp add : no_bintr m2pths)
```
```   792
```
```   793 lemma bintr_lt2p: "number_of (bintrunc n w) < (2 ^ n :: int)"
```
```   794   by (simp add : no_bintr m2pths)
```
```   795
```
```   796 lemma bintr_Min:
```
```   797   "number_of (bintrunc n Int.Min) = (2 ^ n :: int) - 1"
```
```   798   by (simp add : no_bintr m1mod2k)
```
```   799
```
```   800 lemma sbintr_ge: "(- (2 ^ n) :: int) <= number_of (sbintrunc n w)"
```
```   801   by (simp add : no_sbintr m2pths)
```
```   802
```
```   803 lemma sbintr_lt: "number_of (sbintrunc n w) < (2 ^ n :: int)"
```
```   804   by (simp add : no_sbintr m2pths)
```
```   805
```
```   806 lemma bintrunc_Suc:
```
```   807   "bintrunc (Suc n) bin = bintrunc n (bin_rest bin) BIT bin_last bin"
```
```   808   by (case_tac bin rule: bin_exhaust) auto
```
```   809
```
```   810 lemma sign_Pls_ge_0:
```
```   811   "(bin_sign bin = Int.Pls) = (number_of bin >= (0 :: int))"
```
```   812   by (induct bin rule: numeral_induct) auto
```
```   813
```
```   814 lemma sign_Min_lt_0:
```
```   815   "(bin_sign bin = Int.Min) = (number_of bin < (0 :: int))"
```
```   816   by (induct bin rule: numeral_induct) auto
```
```   817
```
```   818 lemmas sign_Min_neg = trans [OF sign_Min_lt_0 neg_def [symmetric]]
```
```   819
```
```   820 lemma bin_rest_trunc:
```
```   821   "!!bin. (bin_rest (bintrunc n bin)) = bintrunc (n - 1) (bin_rest bin)"
```
```   822   by (induct n) auto
```
```   823
```
```   824 lemma bin_rest_power_trunc [rule_format] :
```
```   825   "(bin_rest ^ k) (bintrunc n bin) =
```
```   826     bintrunc (n - k) ((bin_rest ^ k) bin)"
```
```   827   by (induct k) (auto simp: bin_rest_trunc)
```
```   828
```
```   829 lemma bin_rest_trunc_i:
```
```   830   "bintrunc n (bin_rest bin) = bin_rest (bintrunc (Suc n) bin)"
```
```   831   by auto
```
```   832
```
```   833 lemma bin_rest_strunc:
```
```   834   "!!bin. bin_rest (sbintrunc (Suc n) bin) = sbintrunc n (bin_rest bin)"
```
```   835   by (induct n) auto
```
```   836
```
```   837 lemma bintrunc_rest [simp]:
```
```   838   "!!bin. bintrunc n (bin_rest (bintrunc n bin)) = bin_rest (bintrunc n bin)"
```
```   839   apply (induct n, simp)
```
```   840   apply (case_tac bin rule: bin_exhaust)
```
```   841   apply (auto simp: bintrunc_bintrunc_l)
```
```   842   done
```
```   843
```
```   844 lemma sbintrunc_rest [simp]:
```
```   845   "!!bin. sbintrunc n (bin_rest (sbintrunc n bin)) = bin_rest (sbintrunc n bin)"
```
```   846   apply (induct n, simp)
```
```   847   apply (case_tac bin rule: bin_exhaust)
```
```   848   apply (auto simp: bintrunc_bintrunc_l split: bit.splits)
```
```   849   done
```
```   850
```
```   851 lemma bintrunc_rest':
```
```   852   "bintrunc n o bin_rest o bintrunc n = bin_rest o bintrunc n"
```
```   853   by (rule ext) auto
```
```   854
```
```   855 lemma sbintrunc_rest' :
```
```   856   "sbintrunc n o bin_rest o sbintrunc n = bin_rest o sbintrunc n"
```
```   857   by (rule ext) auto
```
```   858
```
```   859 lemma rco_lem:
```
```   860   "f o g o f = g o f ==> f o (g o f) ^ n = g ^ n o f"
```
```   861   apply (rule ext)
```
```   862   apply (induct_tac n)
```
```   863    apply (simp_all (no_asm))
```
```   864   apply (drule fun_cong)
```
```   865   apply (unfold o_def)
```
```   866   apply (erule trans)
```
```   867   apply simp
```
```   868   done
```
```   869
```
```   870 lemma rco_alt: "(f o g) ^ n o f = f o (g o f) ^ n"
```
```   871   apply (rule ext)
```
```   872   apply (induct n)
```
```   873    apply (simp_all add: o_def)
```
```   874   done
```
```   875
```
```   876 lemmas rco_bintr = bintrunc_rest'
```
```   877   [THEN rco_lem [THEN fun_cong], unfolded o_def]
```
```   878 lemmas rco_sbintr = sbintrunc_rest'
```
```   879   [THEN rco_lem [THEN fun_cong], unfolded o_def]
```
```   880
```
```   881 subsection {* Splitting and concatenation *}
```
```   882
```
```   883 primrec bin_split :: "nat \<Rightarrow> int \<Rightarrow> int \<times> int" where
```
```   884   Z: "bin_split 0 w = (w, Int.Pls)"
```
```   885   | Suc: "bin_split (Suc n) w = (let (w1, w2) = bin_split n (bin_rest w)
```
```   886         in (w1, w2 BIT bin_last w))"
```
```   887
```
```   888 primrec bin_cat :: "int \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int" where
```
```   889   Z: "bin_cat w 0 v = w"
```
```   890   | Suc: "bin_cat w (Suc n) v = bin_cat w n (bin_rest v) BIT bin_last v"
```
```   891
```
```   892 subsection {* Miscellaneous lemmas *}
```
```   893
```
```   894 lemmas funpow_minus_simp =
```
```   895   trans [OF gen_minus [where f = "power f"] funpow_Suc, standard]
```
```   896
```
```   897 lemmas funpow_pred_simp [simp] =
```
```   898   funpow_minus_simp [of "number_of bin", simplified nobm1, standard]
```
```   899
```
```   900 lemmas replicate_minus_simp =
```
```   901   trans [OF gen_minus [where f = "%n. replicate n x"] replicate.replicate_Suc,
```
```   902          standard]
```
```   903
```
```   904 lemmas replicate_pred_simp [simp] =
```
```   905   replicate_minus_simp [of "number_of bin", simplified nobm1, standard]
```
```   906
```
```   907 lemmas power_Suc_no [simp] = power_Suc [of "number_of a", standard]
```
```   908
```
```   909 lemmas power_minus_simp =
```
```   910   trans [OF gen_minus [where f = "power f"] power_Suc, standard]
```
```   911
```
```   912 lemmas power_pred_simp =
```
```   913   power_minus_simp [of "number_of bin", simplified nobm1, standard]
```
```   914 lemmas power_pred_simp_no [simp] = power_pred_simp [where f= "number_of f", standard]
```
```   915
```
```   916 lemma list_exhaust_size_gt0:
```
```   917   assumes y: "\<And>a list. y = a # list \<Longrightarrow> P"
```
```   918   shows "0 < length y \<Longrightarrow> P"
```
```   919   apply (cases y, simp)
```
```   920   apply (rule y)
```
```   921   apply fastsimp
```
```   922   done
```
```   923
```
```   924 lemma list_exhaust_size_eq0:
```
```   925   assumes y: "y = [] \<Longrightarrow> P"
```
```   926   shows "length y = 0 \<Longrightarrow> P"
```
```   927   apply (cases y)
```
```   928    apply (rule y, simp)
```
```   929   apply simp
```
```   930   done
```
```   931
```
```   932 lemma size_Cons_lem_eq:
```
```   933   "y = xa # list ==> size y = Suc k ==> size list = k"
```
```   934   by auto
```
```   935
```
```   936 lemma size_Cons_lem_eq_bin:
```
```   937   "y = xa # list ==> size y = number_of (Int.succ k) ==>
```
```   938     size list = number_of k"
```
```   939   by (auto simp: pred_def succ_def split add : split_if_asm)
```
```   940
```
```   941 lemmas ls_splits =
```
```   942   prod.split split_split prod.split_asm split_split_asm split_if_asm
```
```   943
```
```   944 lemma not_B1_is_B0: "y \<noteq> bit.B1 \<Longrightarrow> y = bit.B0"
```
```   945   by (cases y) auto
```
```   946
```
```   947 lemma B1_ass_B0:
```
```   948   assumes y: "y = bit.B0 \<Longrightarrow> y = bit.B1"
```
```   949   shows "y = bit.B1"
```
```   950   apply (rule classical)
```
```   951   apply (drule not_B1_is_B0)
```
```   952   apply (erule y)
```
```   953   done
```
```   954
```
```   955 -- "simplifications for specific word lengths"
```
```   956 lemmas n2s_ths [THEN eq_reflection] = add_2_eq_Suc add_2_eq_Suc'
```
```   957
```
```   958 lemmas s2n_ths = n2s_ths [symmetric]
```
```   959
```
```   960 end
```