src/HOL/Library/Word.thy
 author nipkow Fri Aug 28 19:35:49 2009 +0200 (2009-08-28) changeset 32438 620a5d8cfa78 parent 30960 fec1a04b7220 child 32456 341c83339aeb permissions -rw-r--r--
tuned proofs
```     1 (*  Title:      HOL/Library/Word.thy
```
```     2     Author:     Sebastian Skalberg (TU Muenchen)
```
```     3 *)
```
```     4
```
```     5 header {* Binary Words *}
```
```     6
```
```     7 theory Word
```
```     8 imports "~~/src/HOL/Main"
```
```     9 begin
```
```    10
```
```    11 subsection {* Auxilary Lemmas *}
```
```    12
```
```    13 lemma max_le [intro!]: "[| x \<le> z; y \<le> z |] ==> max x y \<le> z"
```
```    14   by (simp add: max_def)
```
```    15
```
```    16 lemma max_mono:
```
```    17   fixes x :: "'a::linorder"
```
```    18   assumes mf: "mono f"
```
```    19   shows       "max (f x) (f y) \<le> f (max x y)"
```
```    20 proof -
```
```    21   from mf and le_maxI1 [of x y]
```
```    22   have fx: "f x \<le> f (max x y)" by (rule monoD)
```
```    23   from mf and le_maxI2 [of y x]
```
```    24   have fy: "f y \<le> f (max x y)" by (rule monoD)
```
```    25   from fx and fy
```
```    26   show "max (f x) (f y) \<le> f (max x y)" by auto
```
```    27 qed
```
```    28
```
```    29 declare zero_le_power [intro]
```
```    30   and zero_less_power [intro]
```
```    31
```
```    32 lemma int_nat_two_exp: "2 ^ k = int (2 ^ k)"
```
```    33   by (simp add: zpower_int [symmetric])
```
```    34
```
```    35
```
```    36 subsection {* Bits *}
```
```    37
```
```    38 datatype bit =
```
```    39     Zero ("\<zero>")
```
```    40   | One ("\<one>")
```
```    41
```
```    42 primrec bitval :: "bit => nat" where
```
```    43     "bitval \<zero> = 0"
```
```    44   | "bitval \<one> = 1"
```
```    45
```
```    46 consts
```
```    47   bitnot :: "bit => bit"
```
```    48   bitand :: "bit => bit => bit" (infixr "bitand" 35)
```
```    49   bitor  :: "bit => bit => bit" (infixr "bitor"  30)
```
```    50   bitxor :: "bit => bit => bit" (infixr "bitxor" 30)
```
```    51
```
```    52 notation (xsymbols)
```
```    53   bitnot ("\<not>\<^sub>b _"  40) and
```
```    54   bitand (infixr "\<and>\<^sub>b" 35) and
```
```    55   bitor  (infixr "\<or>\<^sub>b" 30) and
```
```    56   bitxor (infixr "\<oplus>\<^sub>b" 30)
```
```    57
```
```    58 notation (HTML output)
```
```    59   bitnot ("\<not>\<^sub>b _"  40) and
```
```    60   bitand (infixr "\<and>\<^sub>b" 35) and
```
```    61   bitor  (infixr "\<or>\<^sub>b" 30) and
```
```    62   bitxor (infixr "\<oplus>\<^sub>b" 30)
```
```    63
```
```    64 primrec
```
```    65   bitnot_zero: "(bitnot \<zero>) = \<one>"
```
```    66   bitnot_one : "(bitnot \<one>)  = \<zero>"
```
```    67
```
```    68 primrec
```
```    69   bitand_zero: "(\<zero> bitand y) = \<zero>"
```
```    70   bitand_one:  "(\<one> bitand y) = y"
```
```    71
```
```    72 primrec
```
```    73   bitor_zero: "(\<zero> bitor y) = y"
```
```    74   bitor_one:  "(\<one> bitor y) = \<one>"
```
```    75
```
```    76 primrec
```
```    77   bitxor_zero: "(\<zero> bitxor y) = y"
```
```    78   bitxor_one:  "(\<one> bitxor y) = (bitnot y)"
```
```    79
```
```    80 lemma bitnot_bitnot [simp]: "(bitnot (bitnot b)) = b"
```
```    81   by (cases b) simp_all
```
```    82
```
```    83 lemma bitand_cancel [simp]: "(b bitand b) = b"
```
```    84   by (cases b) simp_all
```
```    85
```
```    86 lemma bitor_cancel [simp]: "(b bitor b) = b"
```
```    87   by (cases b) simp_all
```
```    88
```
```    89 lemma bitxor_cancel [simp]: "(b bitxor b) = \<zero>"
```
```    90   by (cases b) simp_all
```
```    91
```
```    92
```
```    93 subsection {* Bit Vectors *}
```
```    94
```
```    95 text {* First, a couple of theorems expressing case analysis and
```
```    96 induction principles for bit vectors. *}
```
```    97
```
```    98 lemma bit_list_cases:
```
```    99   assumes empty: "w = [] ==> P w"
```
```   100   and     zero:  "!!bs. w = \<zero> # bs ==> P w"
```
```   101   and     one:   "!!bs. w = \<one> # bs ==> P w"
```
```   102   shows   "P w"
```
```   103 proof (cases w)
```
```   104   assume "w = []"
```
```   105   thus ?thesis by (rule empty)
```
```   106 next
```
```   107   fix b bs
```
```   108   assume [simp]: "w = b # bs"
```
```   109   show "P w"
```
```   110   proof (cases b)
```
```   111     assume "b = \<zero>"
```
```   112     hence "w = \<zero> # bs" by simp
```
```   113     thus ?thesis by (rule zero)
```
```   114   next
```
```   115     assume "b = \<one>"
```
```   116     hence "w = \<one> # bs" by simp
```
```   117     thus ?thesis by (rule one)
```
```   118   qed
```
```   119 qed
```
```   120
```
```   121 lemma bit_list_induct:
```
```   122   assumes empty: "P []"
```
```   123   and     zero:  "!!bs. P bs ==> P (\<zero>#bs)"
```
```   124   and     one:   "!!bs. P bs ==> P (\<one>#bs)"
```
```   125   shows   "P w"
```
```   126 proof (induct w, simp_all add: empty)
```
```   127   fix b bs
```
```   128   assume "P bs"
```
```   129   then show "P (b#bs)"
```
```   130     by (cases b) (auto intro!: zero one)
```
```   131 qed
```
```   132
```
```   133 definition
```
```   134   bv_msb :: "bit list => bit" where
```
```   135   "bv_msb w = (if w = [] then \<zero> else hd w)"
```
```   136
```
```   137 definition
```
```   138   bv_extend :: "[nat,bit,bit list]=>bit list" where
```
```   139   "bv_extend i b w = (replicate (i - length w) b) @ w"
```
```   140
```
```   141 definition
```
```   142   bv_not :: "bit list => bit list" where
```
```   143   "bv_not w = map bitnot w"
```
```   144
```
```   145 lemma bv_length_extend [simp]: "length w \<le> i ==> length (bv_extend i b w) = i"
```
```   146   by (simp add: bv_extend_def)
```
```   147
```
```   148 lemma bv_not_Nil [simp]: "bv_not [] = []"
```
```   149   by (simp add: bv_not_def)
```
```   150
```
```   151 lemma bv_not_Cons [simp]: "bv_not (b#bs) = (bitnot b) # bv_not bs"
```
```   152   by (simp add: bv_not_def)
```
```   153
```
```   154 lemma bv_not_bv_not [simp]: "bv_not (bv_not w) = w"
```
```   155   by (rule bit_list_induct [of _ w]) simp_all
```
```   156
```
```   157 lemma bv_msb_Nil [simp]: "bv_msb [] = \<zero>"
```
```   158   by (simp add: bv_msb_def)
```
```   159
```
```   160 lemma bv_msb_Cons [simp]: "bv_msb (b#bs) = b"
```
```   161   by (simp add: bv_msb_def)
```
```   162
```
```   163 lemma bv_msb_bv_not [simp]: "0 < length w ==> bv_msb (bv_not w) = (bitnot (bv_msb w))"
```
```   164   by (cases w) simp_all
```
```   165
```
```   166 lemma bv_msb_one_length [simp,intro]: "bv_msb w = \<one> ==> 0 < length w"
```
```   167   by (cases w) simp_all
```
```   168
```
```   169 lemma length_bv_not [simp]: "length (bv_not w) = length w"
```
```   170   by (induct w) simp_all
```
```   171
```
```   172 definition
```
```   173   bv_to_nat :: "bit list => nat" where
```
```   174   "bv_to_nat = foldl (%bn b. 2 * bn + bitval b) 0"
```
```   175
```
```   176 lemma bv_to_nat_Nil [simp]: "bv_to_nat [] = 0"
```
```   177   by (simp add: bv_to_nat_def)
```
```   178
```
```   179 lemma bv_to_nat_helper [simp]: "bv_to_nat (b # bs) = bitval b * 2 ^ length bs + bv_to_nat bs"
```
```   180 proof -
```
```   181   let ?bv_to_nat' = "foldl (\<lambda>bn b. 2 * bn + bitval b)"
```
```   182   have helper: "\<And>base. ?bv_to_nat' base bs = base * 2 ^ length bs + ?bv_to_nat' 0 bs"
```
```   183   proof (induct bs)
```
```   184     case Nil
```
```   185     show ?case by simp
```
```   186   next
```
```   187     case (Cons x xs base)
```
```   188     show ?case
```
```   189       apply (simp only: foldl.simps)
```
```   190       apply (subst Cons [of "2 * base + bitval x"])
```
```   191       apply simp
```
```   192       apply (subst Cons [of "bitval x"])
```
```   193       apply (simp add: add_mult_distrib)
```
```   194       done
```
```   195   qed
```
```   196   show ?thesis by (simp add: bv_to_nat_def) (rule helper)
```
```   197 qed
```
```   198
```
```   199 lemma bv_to_nat0 [simp]: "bv_to_nat (\<zero>#bs) = bv_to_nat bs"
```
```   200   by simp
```
```   201
```
```   202 lemma bv_to_nat1 [simp]: "bv_to_nat (\<one>#bs) = 2 ^ length bs + bv_to_nat bs"
```
```   203   by simp
```
```   204
```
```   205 lemma bv_to_nat_upper_range: "bv_to_nat w < 2 ^ length w"
```
```   206 proof (induct w, simp_all)
```
```   207   fix b bs
```
```   208   assume "bv_to_nat bs < 2 ^ length bs"
```
```   209   show "bitval b * 2 ^ length bs + bv_to_nat bs < 2 * 2 ^ length bs"
```
```   210   proof (cases b, simp_all)
```
```   211     have "bv_to_nat bs < 2 ^ length bs" by fact
```
```   212     also have "... < 2 * 2 ^ length bs" by auto
```
```   213     finally show "bv_to_nat bs < 2 * 2 ^ length bs" by simp
```
```   214   next
```
```   215     have "bv_to_nat bs < 2 ^ length bs" by fact
```
```   216     hence "2 ^ length bs + bv_to_nat bs < 2 ^ length bs + 2 ^ length bs" by arith
```
```   217     also have "... = 2 * (2 ^ length bs)" by simp
```
```   218     finally show "bv_to_nat bs < 2 ^ length bs" by simp
```
```   219   qed
```
```   220 qed
```
```   221
```
```   222 lemma bv_extend_longer [simp]:
```
```   223   assumes wn: "n \<le> length w"
```
```   224   shows       "bv_extend n b w = w"
```
```   225   by (simp add: bv_extend_def wn)
```
```   226
```
```   227 lemma bv_extend_shorter [simp]:
```
```   228   assumes wn: "length w < n"
```
```   229   shows       "bv_extend n b w = bv_extend n b (b#w)"
```
```   230 proof -
```
```   231   from wn
```
```   232   have s: "n - Suc (length w) + 1 = n - length w"
```
```   233     by arith
```
```   234   have "bv_extend n b w = replicate (n - length w) b @ w"
```
```   235     by (simp add: bv_extend_def)
```
```   236   also have "... = replicate (n - Suc (length w) + 1) b @ w"
```
```   237     by (subst s) rule
```
```   238   also have "... = (replicate (n - Suc (length w)) b @ replicate 1 b) @ w"
```
```   239     by (subst replicate_add) rule
```
```   240   also have "... = replicate (n - Suc (length w)) b @ b # w"
```
```   241     by simp
```
```   242   also have "... = bv_extend n b (b#w)"
```
```   243     by (simp add: bv_extend_def)
```
```   244   finally show "bv_extend n b w = bv_extend n b (b#w)" .
```
```   245 qed
```
```   246
```
```   247 consts
```
```   248   rem_initial :: "bit => bit list => bit list"
```
```   249 primrec
```
```   250   "rem_initial b [] = []"
```
```   251   "rem_initial b (x#xs) = (if b = x then rem_initial b xs else x#xs)"
```
```   252
```
```   253 lemma rem_initial_length: "length (rem_initial b w) \<le> length w"
```
```   254   by (rule bit_list_induct [of _ w],simp_all (no_asm),safe,simp_all)
```
```   255
```
```   256 lemma rem_initial_equal:
```
```   257   assumes p: "length (rem_initial b w) = length w"
```
```   258   shows      "rem_initial b w = w"
```
```   259 proof -
```
```   260   have "length (rem_initial b w) = length w --> rem_initial b w = w"
```
```   261   proof (induct w, simp_all, clarify)
```
```   262     fix xs
```
```   263     assume "length (rem_initial b xs) = length xs --> rem_initial b xs = xs"
```
```   264     assume f: "length (rem_initial b xs) = Suc (length xs)"
```
```   265     with rem_initial_length [of b xs]
```
```   266     show "rem_initial b xs = b#xs"
```
```   267       by auto
```
```   268   qed
```
```   269   from this and p show ?thesis ..
```
```   270 qed
```
```   271
```
```   272 lemma bv_extend_rem_initial: "bv_extend (length w) b (rem_initial b w) = w"
```
```   273 proof (induct w, simp_all, safe)
```
```   274   fix xs
```
```   275   assume ind: "bv_extend (length xs) b (rem_initial b xs) = xs"
```
```   276   from rem_initial_length [of b xs]
```
```   277   have [simp]: "Suc (length xs) - length (rem_initial b xs) =
```
```   278       1 + (length xs - length (rem_initial b xs))"
```
```   279     by arith
```
```   280   have "bv_extend (Suc (length xs)) b (rem_initial b xs) =
```
```   281       replicate (Suc (length xs) - length (rem_initial b xs)) b @ (rem_initial b xs)"
```
```   282     by (simp add: bv_extend_def)
```
```   283   also have "... =
```
```   284       replicate (1 + (length xs - length (rem_initial b xs))) b @ rem_initial b xs"
```
```   285     by simp
```
```   286   also have "... =
```
```   287       (replicate 1 b @ replicate (length xs - length (rem_initial b xs)) b) @ rem_initial b xs"
```
```   288     by (subst replicate_add) (rule refl)
```
```   289   also have "... = b # bv_extend (length xs) b (rem_initial b xs)"
```
```   290     by (auto simp add: bv_extend_def [symmetric])
```
```   291   also have "... = b # xs"
```
```   292     by (simp add: ind)
```
```   293   finally show "bv_extend (Suc (length xs)) b (rem_initial b xs) = b # xs" .
```
```   294 qed
```
```   295
```
```   296 lemma rem_initial_append1:
```
```   297   assumes "rem_initial b xs ~= []"
```
```   298   shows   "rem_initial b (xs @ ys) = rem_initial b xs @ ys"
```
```   299   using assms by (induct xs) auto
```
```   300
```
```   301 lemma rem_initial_append2:
```
```   302   assumes "rem_initial b xs = []"
```
```   303   shows   "rem_initial b (xs @ ys) = rem_initial b ys"
```
```   304   using assms by (induct xs) auto
```
```   305
```
```   306 definition
```
```   307   norm_unsigned :: "bit list => bit list" where
```
```   308   "norm_unsigned = rem_initial \<zero>"
```
```   309
```
```   310 lemma norm_unsigned_Nil [simp]: "norm_unsigned [] = []"
```
```   311   by (simp add: norm_unsigned_def)
```
```   312
```
```   313 lemma norm_unsigned_Cons0 [simp]: "norm_unsigned (\<zero>#bs) = norm_unsigned bs"
```
```   314   by (simp add: norm_unsigned_def)
```
```   315
```
```   316 lemma norm_unsigned_Cons1 [simp]: "norm_unsigned (\<one>#bs) = \<one>#bs"
```
```   317   by (simp add: norm_unsigned_def)
```
```   318
```
```   319 lemma norm_unsigned_idem [simp]: "norm_unsigned (norm_unsigned w) = norm_unsigned w"
```
```   320   by (rule bit_list_induct [of _ w],simp_all)
```
```   321
```
```   322 consts
```
```   323   nat_to_bv_helper :: "nat => bit list => bit list"
```
```   324 recdef nat_to_bv_helper "measure (\<lambda>n. n)"
```
```   325   "nat_to_bv_helper n = (%bs. (if n = 0 then bs
```
```   326                                else nat_to_bv_helper (n div 2) ((if n mod 2 = 0 then \<zero> else \<one>)#bs)))"
```
```   327
```
```   328 definition
```
```   329   nat_to_bv :: "nat => bit list" where
```
```   330   "nat_to_bv n = nat_to_bv_helper n []"
```
```   331
```
```   332 lemma nat_to_bv0 [simp]: "nat_to_bv 0 = []"
```
```   333   by (simp add: nat_to_bv_def)
```
```   334
```
```   335 lemmas [simp del] = nat_to_bv_helper.simps
```
```   336
```
```   337 lemma n_div_2_cases:
```
```   338   assumes zero: "(n::nat) = 0 ==> R"
```
```   339   and     div : "[| n div 2 < n ; 0 < n |] ==> R"
```
```   340   shows         "R"
```
```   341 proof (cases "n = 0")
```
```   342   assume "n = 0"
```
```   343   thus R by (rule zero)
```
```   344 next
```
```   345   assume "n ~= 0"
```
```   346   hence "0 < n" by simp
```
```   347   hence "n div 2 < n" by arith
```
```   348   from this and `0 < n` show R by (rule div)
```
```   349 qed
```
```   350
```
```   351 lemma int_wf_ge_induct:
```
```   352   assumes ind :  "!!i::int. (!!j. [| k \<le> j ; j < i |] ==> P j) ==> P i"
```
```   353   shows          "P i"
```
```   354 proof (rule wf_induct_rule [OF wf_int_ge_less_than])
```
```   355   fix x
```
```   356   assume ih: "(\<And>y\<Colon>int. (y, x) \<in> int_ge_less_than k \<Longrightarrow> P y)"
```
```   357   thus "P x"
```
```   358     by (rule ind) (simp add: int_ge_less_than_def)
```
```   359 qed
```
```   360
```
```   361 lemma unfold_nat_to_bv_helper:
```
```   362   "nat_to_bv_helper b l = nat_to_bv_helper b [] @ l"
```
```   363 proof -
```
```   364   have "\<forall>l. nat_to_bv_helper b l = nat_to_bv_helper b [] @ l"
```
```   365   proof (induct b rule: less_induct)
```
```   366     fix n
```
```   367     assume ind: "!!j. j < n \<Longrightarrow> \<forall> l. nat_to_bv_helper j l = nat_to_bv_helper j [] @ l"
```
```   368     show "\<forall>l. nat_to_bv_helper n l = nat_to_bv_helper n [] @ l"
```
```   369     proof
```
```   370       fix l
```
```   371       show "nat_to_bv_helper n l = nat_to_bv_helper n [] @ l"
```
```   372       proof (cases "n < 0")
```
```   373         assume "n < 0"
```
```   374         thus ?thesis
```
```   375           by (simp add: nat_to_bv_helper.simps)
```
```   376       next
```
```   377         assume "~n < 0"
```
```   378         show ?thesis
```
```   379         proof (rule n_div_2_cases [of n])
```
```   380           assume [simp]: "n = 0"
```
```   381           show ?thesis
```
```   382             apply (simp only: nat_to_bv_helper.simps [of n])
```
```   383             apply simp
```
```   384             done
```
```   385         next
```
```   386           assume n2n: "n div 2 < n"
```
```   387           assume [simp]: "0 < n"
```
```   388           hence n20: "0 \<le> n div 2"
```
```   389             by arith
```
```   390           from ind [of "n div 2"] and n2n n20
```
```   391           have ind': "\<forall>l. nat_to_bv_helper (n div 2) l = nat_to_bv_helper (n div 2) [] @ l"
```
```   392             by blast
```
```   393           show ?thesis
```
```   394             apply (simp only: nat_to_bv_helper.simps [of n])
```
```   395             apply (cases "n=0")
```
```   396             apply simp
```
```   397             apply (simp only: if_False)
```
```   398             apply simp
```
```   399             apply (subst spec [OF ind',of "\<zero>#l"])
```
```   400             apply (subst spec [OF ind',of "\<one>#l"])
```
```   401             apply (subst spec [OF ind',of "[\<one>]"])
```
```   402             apply (subst spec [OF ind',of "[\<zero>]"])
```
```   403             apply simp
```
```   404             done
```
```   405         qed
```
```   406       qed
```
```   407     qed
```
```   408   qed
```
```   409   thus ?thesis ..
```
```   410 qed
```
```   411
```
```   412 lemma nat_to_bv_non0 [simp]: "n\<noteq>0 ==> nat_to_bv n = nat_to_bv (n div 2) @ [if n mod 2 = 0 then \<zero> else \<one>]"
```
```   413 proof -
```
```   414   assume [simp]: "n\<noteq>0"
```
```   415   show ?thesis
```
```   416     apply (subst nat_to_bv_def [of n])
```
```   417     apply (simp only: nat_to_bv_helper.simps [of n])
```
```   418     apply (subst unfold_nat_to_bv_helper)
```
```   419     using prems
```
```   420     apply (simp)
```
```   421     apply (subst nat_to_bv_def [of "n div 2"])
```
```   422     apply auto
```
```   423     done
```
```   424 qed
```
```   425
```
```   426 lemma bv_to_nat_dist_append:
```
```   427   "bv_to_nat (l1 @ l2) = bv_to_nat l1 * 2 ^ length l2 + bv_to_nat l2"
```
```   428 proof -
```
```   429   have "\<forall>l2. bv_to_nat (l1 @ l2) = bv_to_nat l1 * 2 ^ length l2 + bv_to_nat l2"
```
```   430   proof (induct l1, simp_all)
```
```   431     fix x xs
```
```   432     assume ind: "\<forall>l2. bv_to_nat (xs @ l2) = bv_to_nat xs * 2 ^ length l2 + bv_to_nat l2"
```
```   433     show "\<forall>l2::bit list. bitval x * 2 ^ (length xs + length l2) + bv_to_nat xs * 2 ^ length l2 = (bitval x * 2 ^ length xs + bv_to_nat xs) * 2 ^ length l2"
```
```   434     proof
```
```   435       fix l2
```
```   436       show "bitval x * 2 ^ (length xs + length l2) + bv_to_nat xs * 2 ^ length l2 = (bitval x * 2 ^ length xs + bv_to_nat xs) * 2 ^ length l2"
```
```   437       proof -
```
```   438         have "(2::nat) ^ (length xs + length l2) = 2 ^ length xs * 2 ^ length l2"
```
```   439           by (induct "length xs",simp_all)
```
```   440         hence "bitval x * 2 ^ (length xs + length l2) + bv_to_nat xs * 2 ^ length l2 =
```
```   441           bitval x * 2 ^ length xs * 2 ^ length l2 + bv_to_nat xs * 2 ^ length l2"
```
```   442           by simp
```
```   443         also have "... = (bitval x * 2 ^ length xs + bv_to_nat xs) * 2 ^ length l2"
```
```   444           by (simp add: ring_distribs)
```
```   445         finally show ?thesis by simp
```
```   446       qed
```
```   447     qed
```
```   448   qed
```
```   449   thus ?thesis ..
```
```   450 qed
```
```   451
```
```   452 lemma bv_nat_bv [simp]: "bv_to_nat (nat_to_bv n) = n"
```
```   453 proof (induct n rule: less_induct)
```
```   454   fix n
```
```   455   assume ind: "!!j. j < n \<Longrightarrow> bv_to_nat (nat_to_bv j) = j"
```
```   456   show "bv_to_nat (nat_to_bv n) = n"
```
```   457   proof (rule n_div_2_cases [of n])
```
```   458     assume "n = 0" then show ?thesis by simp
```
```   459   next
```
```   460     assume nn: "n div 2 < n"
```
```   461     assume n0: "0 < n"
```
```   462     from ind and nn
```
```   463     have ind': "bv_to_nat (nat_to_bv (n div 2)) = n div 2" by blast
```
```   464     from n0 have n0': "n \<noteq> 0" by simp
```
```   465     show ?thesis
```
```   466       apply (subst nat_to_bv_def)
```
```   467       apply (simp only: nat_to_bv_helper.simps [of n])
```
```   468       apply (simp only: n0' if_False)
```
```   469       apply (subst unfold_nat_to_bv_helper)
```
```   470       apply (subst bv_to_nat_dist_append)
```
```   471       apply (fold nat_to_bv_def)
```
```   472       apply (simp add: ind' split del: split_if)
```
```   473       apply (cases "n mod 2 = 0")
```
```   474       proof (simp_all)
```
```   475         assume "n mod 2 = 0"
```
```   476         with mod_div_equality [of n 2]
```
```   477         show "n div 2 * 2 = n" by simp
```
```   478       next
```
```   479         assume "n mod 2 = Suc 0"
```
```   480         with mod_div_equality [of n 2]
```
```   481         show "Suc (n div 2 * 2) = n" by arith
```
```   482       qed
```
```   483   qed
```
```   484 qed
```
```   485
```
```   486 lemma bv_to_nat_type [simp]: "bv_to_nat (norm_unsigned w) = bv_to_nat w"
```
```   487   by (rule bit_list_induct) simp_all
```
```   488
```
```   489 lemma length_norm_unsigned_le [simp]: "length (norm_unsigned w) \<le> length w"
```
```   490   by (rule bit_list_induct) simp_all
```
```   491
```
```   492 lemma bv_to_nat_rew_msb: "bv_msb w = \<one> ==> bv_to_nat w = 2 ^ (length w - 1) + bv_to_nat (tl w)"
```
```   493   by (rule bit_list_cases [of w]) simp_all
```
```   494
```
```   495 lemma norm_unsigned_result: "norm_unsigned xs = [] \<or> bv_msb (norm_unsigned xs) = \<one>"
```
```   496 proof (rule length_induct [of _ xs])
```
```   497   fix xs :: "bit list"
```
```   498   assume ind: "\<forall>ys. length ys < length xs --> norm_unsigned ys = [] \<or> bv_msb (norm_unsigned ys) = \<one>"
```
```   499   show "norm_unsigned xs = [] \<or> bv_msb (norm_unsigned xs) = \<one>"
```
```   500   proof (rule bit_list_cases [of xs],simp_all)
```
```   501     fix bs
```
```   502     assume [simp]: "xs = \<zero>#bs"
```
```   503     from ind
```
```   504     have "length bs < length xs --> norm_unsigned bs = [] \<or> bv_msb (norm_unsigned bs) = \<one>" ..
```
```   505     thus "norm_unsigned bs = [] \<or> bv_msb (norm_unsigned bs) = \<one>" by simp
```
```   506   qed
```
```   507 qed
```
```   508
```
```   509 lemma norm_empty_bv_to_nat_zero:
```
```   510   assumes nw: "norm_unsigned w = []"
```
```   511   shows       "bv_to_nat w = 0"
```
```   512 proof -
```
```   513   have "bv_to_nat w = bv_to_nat (norm_unsigned w)" by simp
```
```   514   also have "... = bv_to_nat []" by (subst nw) (rule refl)
```
```   515   also have "... = 0" by simp
```
```   516   finally show ?thesis .
```
```   517 qed
```
```   518
```
```   519 lemma bv_to_nat_lower_limit:
```
```   520   assumes w0: "0 < bv_to_nat w"
```
```   521   shows "2 ^ (length (norm_unsigned w) - 1) \<le> bv_to_nat w"
```
```   522 proof -
```
```   523   from w0 and norm_unsigned_result [of w]
```
```   524   have msbw: "bv_msb (norm_unsigned w) = \<one>"
```
```   525     by (auto simp add: norm_empty_bv_to_nat_zero)
```
```   526   have "2 ^ (length (norm_unsigned w) - 1) \<le> bv_to_nat (norm_unsigned w)"
```
```   527     by (subst bv_to_nat_rew_msb [OF msbw],simp)
```
```   528   thus ?thesis by simp
```
```   529 qed
```
```   530
```
```   531 lemmas [simp del] = nat_to_bv_non0
```
```   532
```
```   533 lemma norm_unsigned_length [intro!]: "length (norm_unsigned w) \<le> length w"
```
```   534 by (subst norm_unsigned_def,rule rem_initial_length)
```
```   535
```
```   536 lemma norm_unsigned_equal:
```
```   537   "length (norm_unsigned w) = length w ==> norm_unsigned w = w"
```
```   538 by (simp add: norm_unsigned_def,rule rem_initial_equal)
```
```   539
```
```   540 lemma bv_extend_norm_unsigned: "bv_extend (length w) \<zero> (norm_unsigned w) = w"
```
```   541 by (simp add: norm_unsigned_def,rule bv_extend_rem_initial)
```
```   542
```
```   543 lemma norm_unsigned_append1 [simp]:
```
```   544   "norm_unsigned xs \<noteq> [] ==> norm_unsigned (xs @ ys) = norm_unsigned xs @ ys"
```
```   545 by (simp add: norm_unsigned_def,rule rem_initial_append1)
```
```   546
```
```   547 lemma norm_unsigned_append2 [simp]:
```
```   548   "norm_unsigned xs = [] ==> norm_unsigned (xs @ ys) = norm_unsigned ys"
```
```   549 by (simp add: norm_unsigned_def,rule rem_initial_append2)
```
```   550
```
```   551 lemma bv_to_nat_zero_imp_empty:
```
```   552   "bv_to_nat w = 0 \<Longrightarrow> norm_unsigned w = []"
```
```   553 by (atomize (full), induct w rule: bit_list_induct) simp_all
```
```   554
```
```   555 lemma bv_to_nat_nzero_imp_nempty:
```
```   556   "bv_to_nat w \<noteq> 0 \<Longrightarrow> norm_unsigned w \<noteq> []"
```
```   557 by (induct w rule: bit_list_induct) simp_all
```
```   558
```
```   559 lemma nat_helper1:
```
```   560   assumes ass: "nat_to_bv (bv_to_nat w) = norm_unsigned w"
```
```   561   shows        "nat_to_bv (2 * bv_to_nat w + bitval x) = norm_unsigned (w @ [x])"
```
```   562 proof (cases x)
```
```   563   assume [simp]: "x = \<one>"
```
```   564   show ?thesis
```
```   565     apply (simp add: nat_to_bv_non0)
```
```   566     apply safe
```
```   567   proof -
```
```   568     fix q
```
```   569     assume "Suc (2 * bv_to_nat w) = 2 * q"
```
```   570     hence orig: "(2 * bv_to_nat w + 1) mod 2 = 2 * q mod 2" (is "?lhs = ?rhs")
```
```   571       by simp
```
```   572     have "?lhs = (1 + 2 * bv_to_nat w) mod 2"
```
```   573       by (simp add: add_commute)
```
```   574     also have "... = 1"
```
```   575       by (subst mod_add_eq) simp
```
```   576     finally have eq1: "?lhs = 1" .
```
```   577     have "?rhs  = 0" by simp
```
```   578     with orig and eq1
```
```   579     show "nat_to_bv (Suc (2 * bv_to_nat w) div 2) @ [\<zero>] = norm_unsigned (w @ [\<one>])"
```
```   580       by simp
```
```   581   next
```
```   582     have "nat_to_bv (Suc (2 * bv_to_nat w) div 2) @ [\<one>] =
```
```   583         nat_to_bv ((1 + 2 * bv_to_nat w) div 2) @ [\<one>]"
```
```   584       by (simp add: add_commute)
```
```   585     also have "... = nat_to_bv (bv_to_nat w) @ [\<one>]"
```
```   586       by (subst div_add1_eq) simp
```
```   587     also have "... = norm_unsigned w @ [\<one>]"
```
```   588       by (subst ass) (rule refl)
```
```   589     also have "... = norm_unsigned (w @ [\<one>])"
```
```   590       by (cases "norm_unsigned w") simp_all
```
```   591     finally show "nat_to_bv (Suc (2 * bv_to_nat w) div 2) @ [\<one>] = norm_unsigned (w @ [\<one>])" .
```
```   592   qed
```
```   593 next
```
```   594   assume [simp]: "x = \<zero>"
```
```   595   show ?thesis
```
```   596   proof (cases "bv_to_nat w = 0")
```
```   597     assume "bv_to_nat w = 0"
```
```   598     thus ?thesis
```
```   599       by (simp add: bv_to_nat_zero_imp_empty)
```
```   600   next
```
```   601     assume "bv_to_nat w \<noteq> 0"
```
```   602     thus ?thesis
```
```   603       apply simp
```
```   604       apply (subst nat_to_bv_non0)
```
```   605       apply simp
```
```   606       apply auto
```
```   607       apply (subst ass)
```
```   608       apply (cases "norm_unsigned w")
```
```   609       apply (simp_all add: norm_empty_bv_to_nat_zero)
```
```   610       done
```
```   611   qed
```
```   612 qed
```
```   613
```
```   614 lemma nat_helper2: "nat_to_bv (2 ^ length xs + bv_to_nat xs) = \<one> # xs"
```
```   615 proof -
```
```   616   have "\<forall>xs. nat_to_bv (2 ^ length (rev xs) + bv_to_nat (rev xs)) = \<one> # (rev xs)" (is "\<forall>xs. ?P xs")
```
```   617   proof
```
```   618     fix xs
```
```   619     show "?P xs"
```
```   620     proof (rule length_induct [of _ xs])
```
```   621       fix xs :: "bit list"
```
```   622       assume ind: "\<forall>ys. length ys < length xs --> ?P ys"
```
```   623       show "?P xs"
```
```   624       proof (cases xs)
```
```   625         assume "xs = []"
```
```   626         then show ?thesis by (simp add: nat_to_bv_non0)
```
```   627       next
```
```   628         fix y ys
```
```   629         assume [simp]: "xs = y # ys"
```
```   630         show ?thesis
```
```   631           apply simp
```
```   632           apply (subst bv_to_nat_dist_append)
```
```   633           apply simp
```
```   634         proof -
```
```   635           have "nat_to_bv (2 * 2 ^ length ys + (bv_to_nat (rev ys) * 2 + bitval y)) =
```
```   636             nat_to_bv (2 * (2 ^ length ys + bv_to_nat (rev ys)) + bitval y)"
```
```   637             by (simp add: add_ac mult_ac)
```
```   638           also have "... = nat_to_bv (2 * (bv_to_nat (\<one>#rev ys)) + bitval y)"
```
```   639             by simp
```
```   640           also have "... = norm_unsigned (\<one>#rev ys) @ [y]"
```
```   641           proof -
```
```   642             from ind
```
```   643             have "nat_to_bv (2 ^ length (rev ys) + bv_to_nat (rev ys)) = \<one> # rev ys"
```
```   644               by auto
```
```   645             hence [simp]: "nat_to_bv (2 ^ length ys + bv_to_nat (rev ys)) = \<one> # rev ys"
```
```   646               by simp
```
```   647             show ?thesis
```
```   648               apply (subst nat_helper1)
```
```   649               apply simp_all
```
```   650               done
```
```   651           qed
```
```   652           also have "... = (\<one>#rev ys) @ [y]"
```
```   653             by simp
```
```   654           also have "... = \<one> # rev ys @ [y]"
```
```   655             by simp
```
```   656           finally show "nat_to_bv (2 * 2 ^ length ys + (bv_to_nat (rev ys) * 2 + bitval y)) =
```
```   657 	      \<one> # rev ys @ [y]" .
```
```   658         qed
```
```   659       qed
```
```   660     qed
```
```   661   qed
```
```   662   hence "nat_to_bv (2 ^ length (rev (rev xs)) + bv_to_nat (rev (rev xs))) =
```
```   663       \<one> # rev (rev xs)" ..
```
```   664   thus ?thesis by simp
```
```   665 qed
```
```   666
```
```   667 lemma nat_bv_nat [simp]: "nat_to_bv (bv_to_nat w) = norm_unsigned w"
```
```   668 proof (rule bit_list_induct [of _ w],simp_all)
```
```   669   fix xs
```
```   670   assume "nat_to_bv (bv_to_nat xs) = norm_unsigned xs"
```
```   671   have "bv_to_nat xs = bv_to_nat (norm_unsigned xs)" by simp
```
```   672   have "bv_to_nat xs < 2 ^ length xs"
```
```   673     by (rule bv_to_nat_upper_range)
```
```   674   show "nat_to_bv (2 ^ length xs + bv_to_nat xs) = \<one> # xs"
```
```   675     by (rule nat_helper2)
```
```   676 qed
```
```   677
```
```   678 lemma bv_to_nat_qinj:
```
```   679   assumes one: "bv_to_nat xs = bv_to_nat ys"
```
```   680   and     len: "length xs = length ys"
```
```   681   shows        "xs = ys"
```
```   682 proof -
```
```   683   from one
```
```   684   have "nat_to_bv (bv_to_nat xs) = nat_to_bv (bv_to_nat ys)"
```
```   685     by simp
```
```   686   hence xsys: "norm_unsigned xs = norm_unsigned ys"
```
```   687     by simp
```
```   688   have "xs = bv_extend (length xs) \<zero> (norm_unsigned xs)"
```
```   689     by (simp add: bv_extend_norm_unsigned)
```
```   690   also have "... = bv_extend (length ys) \<zero> (norm_unsigned ys)"
```
```   691     by (simp add: xsys len)
```
```   692   also have "... = ys"
```
```   693     by (simp add: bv_extend_norm_unsigned)
```
```   694   finally show ?thesis .
```
```   695 qed
```
```   696
```
```   697 lemma norm_unsigned_nat_to_bv [simp]:
```
```   698   "norm_unsigned (nat_to_bv n) = nat_to_bv n"
```
```   699 proof -
```
```   700   have "norm_unsigned (nat_to_bv n) = nat_to_bv (bv_to_nat (norm_unsigned (nat_to_bv n)))"
```
```   701     by (subst nat_bv_nat) simp
```
```   702   also have "... = nat_to_bv n" by simp
```
```   703   finally show ?thesis .
```
```   704 qed
```
```   705
```
```   706 lemma length_nat_to_bv_upper_limit:
```
```   707   assumes nk: "n \<le> 2 ^ k - 1"
```
```   708   shows       "length (nat_to_bv n) \<le> k"
```
```   709 proof (cases "n = 0")
```
```   710   case True
```
```   711   thus ?thesis
```
```   712     by (simp add: nat_to_bv_def nat_to_bv_helper.simps)
```
```   713 next
```
```   714   case False
```
```   715   hence n0: "0 < n" by simp
```
```   716   show ?thesis
```
```   717   proof (rule ccontr)
```
```   718     assume "~ length (nat_to_bv n) \<le> k"
```
```   719     hence "k < length (nat_to_bv n)" by simp
```
```   720     hence "k \<le> length (nat_to_bv n) - 1" by arith
```
```   721     hence "(2::nat) ^ k \<le> 2 ^ (length (nat_to_bv n) - 1)" by simp
```
```   722     also have "... = 2 ^ (length (norm_unsigned (nat_to_bv n)) - 1)" by simp
```
```   723     also have "... \<le> bv_to_nat (nat_to_bv n)"
```
```   724       by (rule bv_to_nat_lower_limit) (simp add: n0)
```
```   725     also have "... = n" by simp
```
```   726     finally have "2 ^ k \<le> n" .
```
```   727     with n0 have "2 ^ k - 1 < n" by arith
```
```   728     with nk show False by simp
```
```   729   qed
```
```   730 qed
```
```   731
```
```   732 lemma length_nat_to_bv_lower_limit:
```
```   733   assumes nk: "2 ^ k \<le> n"
```
```   734   shows       "k < length (nat_to_bv n)"
```
```   735 proof (rule ccontr)
```
```   736   assume "~ k < length (nat_to_bv n)"
```
```   737   hence lnk: "length (nat_to_bv n) \<le> k" by simp
```
```   738   have "n = bv_to_nat (nat_to_bv n)" by simp
```
```   739   also have "... < 2 ^ length (nat_to_bv n)"
```
```   740     by (rule bv_to_nat_upper_range)
```
```   741   also from lnk have "... \<le> 2 ^ k" by simp
```
```   742   finally have "n < 2 ^ k" .
```
```   743   with nk show False by simp
```
```   744 qed
```
```   745
```
```   746
```
```   747 subsection {* Unsigned Arithmetic Operations *}
```
```   748
```
```   749 definition
```
```   750   bv_add :: "[bit list, bit list ] => bit list" where
```
```   751   "bv_add w1 w2 = nat_to_bv (bv_to_nat w1 + bv_to_nat w2)"
```
```   752
```
```   753 lemma bv_add_type1 [simp]: "bv_add (norm_unsigned w1) w2 = bv_add w1 w2"
```
```   754   by (simp add: bv_add_def)
```
```   755
```
```   756 lemma bv_add_type2 [simp]: "bv_add w1 (norm_unsigned w2) = bv_add w1 w2"
```
```   757   by (simp add: bv_add_def)
```
```   758
```
```   759 lemma bv_add_returntype [simp]: "norm_unsigned (bv_add w1 w2) = bv_add w1 w2"
```
```   760   by (simp add: bv_add_def)
```
```   761
```
```   762 lemma bv_add_length: "length (bv_add w1 w2) \<le> Suc (max (length w1) (length w2))"
```
```   763 proof (unfold bv_add_def,rule length_nat_to_bv_upper_limit)
```
```   764   from bv_to_nat_upper_range [of w1] and bv_to_nat_upper_range [of w2]
```
```   765   have "bv_to_nat w1 + bv_to_nat w2 \<le> (2 ^ length w1 - 1) + (2 ^ length w2 - 1)"
```
```   766     by arith
```
```   767   also have "... \<le>
```
```   768       max (2 ^ length w1 - 1) (2 ^ length w2 - 1) + max (2 ^ length w1 - 1) (2 ^ length w2 - 1)"
```
```   769     by (rule add_mono,safe intro!: le_maxI1 le_maxI2)
```
```   770   also have "... = 2 * max (2 ^ length w1 - 1) (2 ^ length w2 - 1)" by simp
```
```   771   also have "... \<le> 2 ^ Suc (max (length w1) (length w2)) - 2"
```
```   772   proof (cases "length w1 \<le> length w2")
```
```   773     assume w1w2: "length w1 \<le> length w2"
```
```   774     hence "(2::nat) ^ length w1 \<le> 2 ^ length w2" by simp
```
```   775     hence "(2::nat) ^ length w1 - 1 \<le> 2 ^ length w2 - 1" by arith
```
```   776     with w1w2 show ?thesis
```
```   777       by (simp add: diff_mult_distrib2 split: split_max)
```
```   778   next
```
```   779     assume [simp]: "~ (length w1 \<le> length w2)"
```
```   780     have "~ ((2::nat) ^ length w1 - 1 \<le> 2 ^ length w2 - 1)"
```
```   781     proof
```
```   782       assume "(2::nat) ^ length w1 - 1 \<le> 2 ^ length w2 - 1"
```
```   783       hence "((2::nat) ^ length w1 - 1) + 1 \<le> (2 ^ length w2 - 1) + 1"
```
```   784         by (rule add_right_mono)
```
```   785       hence "(2::nat) ^ length w1 \<le> 2 ^ length w2" by simp
```
```   786       hence "length w1 \<le> length w2" by simp
```
```   787       thus False by simp
```
```   788     qed
```
```   789     thus ?thesis
```
```   790       by (simp add: diff_mult_distrib2 split: split_max)
```
```   791   qed
```
```   792   finally show "bv_to_nat w1 + bv_to_nat w2 \<le> 2 ^ Suc (max (length w1) (length w2)) - 1"
```
```   793     by arith
```
```   794 qed
```
```   795
```
```   796 definition
```
```   797   bv_mult :: "[bit list, bit list ] => bit list" where
```
```   798   "bv_mult w1 w2 = nat_to_bv (bv_to_nat w1 * bv_to_nat w2)"
```
```   799
```
```   800 lemma bv_mult_type1 [simp]: "bv_mult (norm_unsigned w1) w2 = bv_mult w1 w2"
```
```   801   by (simp add: bv_mult_def)
```
```   802
```
```   803 lemma bv_mult_type2 [simp]: "bv_mult w1 (norm_unsigned w2) = bv_mult w1 w2"
```
```   804   by (simp add: bv_mult_def)
```
```   805
```
```   806 lemma bv_mult_returntype [simp]: "norm_unsigned (bv_mult w1 w2) = bv_mult w1 w2"
```
```   807   by (simp add: bv_mult_def)
```
```   808
```
```   809 lemma bv_mult_length: "length (bv_mult w1 w2) \<le> length w1 + length w2"
```
```   810 proof (unfold bv_mult_def,rule length_nat_to_bv_upper_limit)
```
```   811   from bv_to_nat_upper_range [of w1] and bv_to_nat_upper_range [of w2]
```
```   812   have h: "bv_to_nat w1 \<le> 2 ^ length w1 - 1 \<and> bv_to_nat w2 \<le> 2 ^ length w2 - 1"
```
```   813     by arith
```
```   814   have "bv_to_nat w1 * bv_to_nat w2 \<le> (2 ^ length w1 - 1) * (2 ^ length w2 - 1)"
```
```   815     apply (cut_tac h)
```
```   816     apply (rule mult_mono)
```
```   817     apply auto
```
```   818     done
```
```   819   also have "... < 2 ^ length w1 * 2 ^ length w2"
```
```   820     by (rule mult_strict_mono,auto)
```
```   821   also have "... = 2 ^ (length w1 + length w2)"
```
```   822     by (simp add: power_add)
```
```   823   finally show "bv_to_nat w1 * bv_to_nat w2 \<le> 2 ^ (length w1 + length w2) - 1"
```
```   824     by arith
```
```   825 qed
```
```   826
```
```   827 subsection {* Signed Vectors *}
```
```   828
```
```   829 consts
```
```   830   norm_signed :: "bit list => bit list"
```
```   831 primrec
```
```   832   norm_signed_Nil: "norm_signed [] = []"
```
```   833   norm_signed_Cons: "norm_signed (b#bs) =
```
```   834     (case b of
```
```   835       \<zero> => if norm_unsigned bs = [] then [] else b#norm_unsigned bs
```
```   836     | \<one> => b#rem_initial b bs)"
```
```   837
```
```   838 lemma norm_signed0 [simp]: "norm_signed [\<zero>] = []"
```
```   839   by simp
```
```   840
```
```   841 lemma norm_signed1 [simp]: "norm_signed [\<one>] = [\<one>]"
```
```   842   by simp
```
```   843
```
```   844 lemma norm_signed01 [simp]: "norm_signed (\<zero>#\<one>#xs) = \<zero>#\<one>#xs"
```
```   845   by simp
```
```   846
```
```   847 lemma norm_signed00 [simp]: "norm_signed (\<zero>#\<zero>#xs) = norm_signed (\<zero>#xs)"
```
```   848   by simp
```
```   849
```
```   850 lemma norm_signed10 [simp]: "norm_signed (\<one>#\<zero>#xs) = \<one>#\<zero>#xs"
```
```   851   by simp
```
```   852
```
```   853 lemma norm_signed11 [simp]: "norm_signed (\<one>#\<one>#xs) = norm_signed (\<one>#xs)"
```
```   854   by simp
```
```   855
```
```   856 lemmas [simp del] = norm_signed_Cons
```
```   857
```
```   858 definition
```
```   859   int_to_bv :: "int => bit list" where
```
```   860   "int_to_bv n = (if 0 \<le> n
```
```   861                  then norm_signed (\<zero>#nat_to_bv (nat n))
```
```   862                  else norm_signed (bv_not (\<zero>#nat_to_bv (nat (-n- 1)))))"
```
```   863
```
```   864 lemma int_to_bv_ge0 [simp]: "0 \<le> n ==> int_to_bv n = norm_signed (\<zero> # nat_to_bv (nat n))"
```
```   865   by (simp add: int_to_bv_def)
```
```   866
```
```   867 lemma int_to_bv_lt0 [simp]:
```
```   868     "n < 0 ==> int_to_bv n = norm_signed (bv_not (\<zero>#nat_to_bv (nat (-n- 1))))"
```
```   869   by (simp add: int_to_bv_def)
```
```   870
```
```   871 lemma norm_signed_idem [simp]: "norm_signed (norm_signed w) = norm_signed w"
```
```   872 proof (rule bit_list_induct [of _ w], simp_all)
```
```   873   fix xs
```
```   874   assume eq: "norm_signed (norm_signed xs) = norm_signed xs"
```
```   875   show "norm_signed (norm_signed (\<zero>#xs)) = norm_signed (\<zero>#xs)"
```
```   876   proof (rule bit_list_cases [of xs],simp_all)
```
```   877     fix ys
```
```   878     assume "xs = \<zero>#ys"
```
```   879     from this [symmetric] and eq
```
```   880     show "norm_signed (norm_signed (\<zero>#ys)) = norm_signed (\<zero>#ys)"
```
```   881       by simp
```
```   882   qed
```
```   883 next
```
```   884   fix xs
```
```   885   assume eq: "norm_signed (norm_signed xs) = norm_signed xs"
```
```   886   show "norm_signed (norm_signed (\<one>#xs)) = norm_signed (\<one>#xs)"
```
```   887   proof (rule bit_list_cases [of xs],simp_all)
```
```   888     fix ys
```
```   889     assume "xs = \<one>#ys"
```
```   890     from this [symmetric] and eq
```
```   891     show "norm_signed (norm_signed (\<one>#ys)) = norm_signed (\<one>#ys)"
```
```   892       by simp
```
```   893   qed
```
```   894 qed
```
```   895
```
```   896 definition
```
```   897   bv_to_int :: "bit list => int" where
```
```   898   "bv_to_int w =
```
```   899     (case bv_msb w of \<zero> => int (bv_to_nat w)
```
```   900     | \<one> => - int (bv_to_nat (bv_not w) + 1))"
```
```   901
```
```   902 lemma bv_to_int_Nil [simp]: "bv_to_int [] = 0"
```
```   903   by (simp add: bv_to_int_def)
```
```   904
```
```   905 lemma bv_to_int_Cons0 [simp]: "bv_to_int (\<zero>#bs) = int (bv_to_nat bs)"
```
```   906   by (simp add: bv_to_int_def)
```
```   907
```
```   908 lemma bv_to_int_Cons1 [simp]: "bv_to_int (\<one>#bs) = - int (bv_to_nat (bv_not bs) + 1)"
```
```   909   by (simp add: bv_to_int_def)
```
```   910
```
```   911 lemma bv_to_int_type [simp]: "bv_to_int (norm_signed w) = bv_to_int w"
```
```   912 proof (rule bit_list_induct [of _ w], simp_all)
```
```   913   fix xs
```
```   914   assume ind: "bv_to_int (norm_signed xs) = bv_to_int xs"
```
```   915   show "bv_to_int (norm_signed (\<zero>#xs)) = int (bv_to_nat xs)"
```
```   916   proof (rule bit_list_cases [of xs], simp_all)
```
```   917     fix ys
```
```   918     assume [simp]: "xs = \<zero>#ys"
```
```   919     from ind
```
```   920     show "bv_to_int (norm_signed (\<zero>#ys)) = int (bv_to_nat ys)"
```
```   921       by simp
```
```   922   qed
```
```   923 next
```
```   924   fix xs
```
```   925   assume ind: "bv_to_int (norm_signed xs) = bv_to_int xs"
```
```   926   show "bv_to_int (norm_signed (\<one>#xs)) = -1 - int (bv_to_nat (bv_not xs))"
```
```   927   proof (rule bit_list_cases [of xs], simp_all)
```
```   928     fix ys
```
```   929     assume [simp]: "xs = \<one>#ys"
```
```   930     from ind
```
```   931     show "bv_to_int (norm_signed (\<one>#ys)) = -1 - int (bv_to_nat (bv_not ys))"
```
```   932       by simp
```
```   933   qed
```
```   934 qed
```
```   935
```
```   936 lemma bv_to_int_upper_range: "bv_to_int w < 2 ^ (length w - 1)"
```
```   937 proof (rule bit_list_cases [of w],simp_all)
```
```   938   fix bs
```
```   939   from bv_to_nat_upper_range
```
```   940   show "int (bv_to_nat bs) < 2 ^ length bs"
```
```   941     by (simp add: int_nat_two_exp)
```
```   942 next
```
```   943   fix bs
```
```   944   have "-1 - int (bv_to_nat (bv_not bs)) \<le> 0" by simp
```
```   945   also have "... < 2 ^ length bs" by (induct bs) simp_all
```
```   946   finally show "-1 - int (bv_to_nat (bv_not bs)) < 2 ^ length bs" .
```
```   947 qed
```
```   948
```
```   949 lemma bv_to_int_lower_range: "- (2 ^ (length w - 1)) \<le> bv_to_int w"
```
```   950 proof (rule bit_list_cases [of w],simp_all)
```
```   951   fix bs :: "bit list"
```
```   952   have "- (2 ^ length bs) \<le> (0::int)" by (induct bs) simp_all
```
```   953   also have "... \<le> int (bv_to_nat bs)" by simp
```
```   954   finally show "- (2 ^ length bs) \<le> int (bv_to_nat bs)" .
```
```   955 next
```
```   956   fix bs
```
```   957   from bv_to_nat_upper_range [of "bv_not bs"]
```
```   958   show "- (2 ^ length bs) \<le> -1 - int (bv_to_nat (bv_not bs))"
```
```   959     by (simp add: int_nat_two_exp)
```
```   960 qed
```
```   961
```
```   962 lemma int_bv_int [simp]: "int_to_bv (bv_to_int w) = norm_signed w"
```
```   963 proof (rule bit_list_cases [of w],simp)
```
```   964   fix xs
```
```   965   assume [simp]: "w = \<zero>#xs"
```
```   966   show ?thesis
```
```   967     apply simp
```
```   968     apply (subst norm_signed_Cons [of "\<zero>" "xs"])
```
```   969     apply simp
```
```   970     using norm_unsigned_result [of xs]
```
```   971     apply safe
```
```   972     apply (rule bit_list_cases [of "norm_unsigned xs"])
```
```   973     apply simp_all
```
```   974     done
```
```   975 next
```
```   976   fix xs
```
```   977   assume [simp]: "w = \<one>#xs"
```
```   978   show ?thesis
```
```   979     apply (simp del: int_to_bv_lt0)
```
```   980     apply (rule bit_list_induct [of _ xs])
```
```   981     apply simp
```
```   982     apply (subst int_to_bv_lt0)
```
```   983     apply (subgoal_tac "- int (bv_to_nat (bv_not (\<zero> # bs))) + -1 < 0 + 0")
```
```   984     apply simp
```
```   985     apply (rule add_le_less_mono)
```
```   986     apply simp
```
```   987     apply simp
```
```   988     apply (simp del: bv_to_nat1 bv_to_nat_helper)
```
```   989     apply simp
```
```   990     done
```
```   991 qed
```
```   992
```
```   993 lemma bv_int_bv [simp]: "bv_to_int (int_to_bv i) = i"
```
```   994   by (cases "0 \<le> i") simp_all
```
```   995
```
```   996 lemma bv_msb_norm [simp]: "bv_msb (norm_signed w) = bv_msb w"
```
```   997   by (rule bit_list_cases [of w]) (simp_all add: norm_signed_Cons)
```
```   998
```
```   999 lemma norm_signed_length: "length (norm_signed w) \<le> length w"
```
```  1000   apply (cases w, simp_all)
```
```  1001   apply (subst norm_signed_Cons)
```
```  1002   apply (case_tac a, simp_all)
```
```  1003   apply (rule rem_initial_length)
```
```  1004   done
```
```  1005
```
```  1006 lemma norm_signed_equal: "length (norm_signed w) = length w ==> norm_signed w = w"
```
```  1007 proof (rule bit_list_cases [of w], simp_all)
```
```  1008   fix xs
```
```  1009   assume "length (norm_signed (\<zero>#xs)) = Suc (length xs)"
```
```  1010   thus "norm_signed (\<zero>#xs) = \<zero>#xs"
```
```  1011     apply (simp add: norm_signed_Cons)
```
```  1012     apply safe
```
```  1013     apply simp_all
```
```  1014     apply (rule norm_unsigned_equal)
```
```  1015     apply assumption
```
```  1016     done
```
```  1017 next
```
```  1018   fix xs
```
```  1019   assume "length (norm_signed (\<one>#xs)) = Suc (length xs)"
```
```  1020   thus "norm_signed (\<one>#xs) = \<one>#xs"
```
```  1021     apply (simp add: norm_signed_Cons)
```
```  1022     apply (rule rem_initial_equal)
```
```  1023     apply assumption
```
```  1024     done
```
```  1025 qed
```
```  1026
```
```  1027 lemma bv_extend_norm_signed: "bv_msb w = b ==> bv_extend (length w) b (norm_signed w) = w"
```
```  1028 proof (rule bit_list_cases [of w],simp_all)
```
```  1029   fix xs
```
```  1030   show "bv_extend (Suc (length xs)) \<zero> (norm_signed (\<zero>#xs)) = \<zero>#xs"
```
```  1031   proof (simp add: norm_signed_list_def,auto)
```
```  1032     assume "norm_unsigned xs = []"
```
```  1033     hence xx: "rem_initial \<zero> xs = []"
```
```  1034       by (simp add: norm_unsigned_def)
```
```  1035     have "bv_extend (Suc (length xs)) \<zero> (\<zero>#rem_initial \<zero> xs) = \<zero>#xs"
```
```  1036       apply (simp add: bv_extend_def replicate_app_Cons_same)
```
```  1037       apply (fold bv_extend_def)
```
```  1038       apply (rule bv_extend_rem_initial)
```
```  1039       done
```
```  1040     thus "bv_extend (Suc (length xs)) \<zero> [\<zero>] = \<zero>#xs"
```
```  1041       by (simp add: xx)
```
```  1042   next
```
```  1043     show "bv_extend (Suc (length xs)) \<zero> (\<zero>#norm_unsigned xs) = \<zero>#xs"
```
```  1044       apply (simp add: norm_unsigned_def)
```
```  1045       apply (simp add: bv_extend_def replicate_app_Cons_same)
```
```  1046       apply (fold bv_extend_def)
```
```  1047       apply (rule bv_extend_rem_initial)
```
```  1048       done
```
```  1049   qed
```
```  1050 next
```
```  1051   fix xs
```
```  1052   show "bv_extend (Suc (length xs)) \<one> (norm_signed (\<one>#xs)) = \<one>#xs"
```
```  1053     apply (simp add: norm_signed_Cons)
```
```  1054     apply (simp add: bv_extend_def replicate_app_Cons_same)
```
```  1055     apply (fold bv_extend_def)
```
```  1056     apply (rule bv_extend_rem_initial)
```
```  1057     done
```
```  1058 qed
```
```  1059
```
```  1060 lemma bv_to_int_qinj:
```
```  1061   assumes one: "bv_to_int xs = bv_to_int ys"
```
```  1062   and     len: "length xs = length ys"
```
```  1063   shows        "xs = ys"
```
```  1064 proof -
```
```  1065   from one
```
```  1066   have "int_to_bv (bv_to_int xs) = int_to_bv (bv_to_int ys)" by simp
```
```  1067   hence xsys: "norm_signed xs = norm_signed ys" by simp
```
```  1068   hence xsys': "bv_msb xs = bv_msb ys"
```
```  1069   proof -
```
```  1070     have "bv_msb xs = bv_msb (norm_signed xs)" by simp
```
```  1071     also have "... = bv_msb (norm_signed ys)" by (simp add: xsys)
```
```  1072     also have "... = bv_msb ys" by simp
```
```  1073     finally show ?thesis .
```
```  1074   qed
```
```  1075   have "xs = bv_extend (length xs) (bv_msb xs) (norm_signed xs)"
```
```  1076     by (simp add: bv_extend_norm_signed)
```
```  1077   also have "... = bv_extend (length ys) (bv_msb ys) (norm_signed ys)"
```
```  1078     by (simp add: xsys xsys' len)
```
```  1079   also have "... = ys"
```
```  1080     by (simp add: bv_extend_norm_signed)
```
```  1081   finally show ?thesis .
```
```  1082 qed
```
```  1083
```
```  1084 lemma int_to_bv_returntype [simp]: "norm_signed (int_to_bv w) = int_to_bv w"
```
```  1085   by (simp add: int_to_bv_def)
```
```  1086
```
```  1087 lemma bv_to_int_msb0: "0 \<le> bv_to_int w1 ==> bv_msb w1 = \<zero>"
```
```  1088   by (rule bit_list_cases,simp_all)
```
```  1089
```
```  1090 lemma bv_to_int_msb1: "bv_to_int w1 < 0 ==> bv_msb w1 = \<one>"
```
```  1091   by (rule bit_list_cases,simp_all)
```
```  1092
```
```  1093 lemma bv_to_int_lower_limit_gt0:
```
```  1094   assumes w0: "0 < bv_to_int w"
```
```  1095   shows       "2 ^ (length (norm_signed w) - 2) \<le> bv_to_int w"
```
```  1096 proof -
```
```  1097   from w0
```
```  1098   have "0 \<le> bv_to_int w" by simp
```
```  1099   hence [simp]: "bv_msb w = \<zero>" by (rule bv_to_int_msb0)
```
```  1100   have "2 ^ (length (norm_signed w) - 2) \<le> bv_to_int (norm_signed w)"
```
```  1101   proof (rule bit_list_cases [of w])
```
```  1102     assume "w = []"
```
```  1103     with w0 show ?thesis by simp
```
```  1104   next
```
```  1105     fix w'
```
```  1106     assume weq: "w = \<zero> # w'"
```
```  1107     thus ?thesis
```
```  1108     proof (simp add: norm_signed_Cons,safe)
```
```  1109       assume "norm_unsigned w' = []"
```
```  1110       with weq and w0 show False
```
```  1111 	by (simp add: norm_empty_bv_to_nat_zero)
```
```  1112     next
```
```  1113       assume w'0: "norm_unsigned w' \<noteq> []"
```
```  1114       have "0 < bv_to_nat w'"
```
```  1115       proof (rule ccontr)
```
```  1116         assume "~ (0 < bv_to_nat w')"
```
```  1117         hence "bv_to_nat w' = 0"
```
```  1118           by arith
```
```  1119         hence "norm_unsigned w' = []"
```
```  1120           by (simp add: bv_to_nat_zero_imp_empty)
```
```  1121         with w'0
```
```  1122         show False by simp
```
```  1123       qed
```
```  1124       with bv_to_nat_lower_limit [of w']
```
```  1125       show "2 ^ (length (norm_unsigned w') - Suc 0) \<le> int (bv_to_nat w')"
```
```  1126         by (simp add: int_nat_two_exp)
```
```  1127     qed
```
```  1128   next
```
```  1129     fix w'
```
```  1130     assume "w = \<one> # w'"
```
```  1131     from w0 have "bv_msb w = \<zero>" by simp
```
```  1132     with prems show ?thesis by simp
```
```  1133   qed
```
```  1134   also have "...  = bv_to_int w" by simp
```
```  1135   finally show ?thesis .
```
```  1136 qed
```
```  1137
```
```  1138 lemma norm_signed_result: "norm_signed w = [] \<or> norm_signed w = [\<one>] \<or> bv_msb (norm_signed w) \<noteq> bv_msb (tl (norm_signed w))"
```
```  1139   apply (rule bit_list_cases [of w],simp_all)
```
```  1140   apply (case_tac "bs",simp_all)
```
```  1141   apply (case_tac "a",simp_all)
```
```  1142   apply (simp add: norm_signed_Cons)
```
```  1143   apply safe
```
```  1144   apply simp
```
```  1145 proof -
```
```  1146   fix l
```
```  1147   assume msb: "\<zero> = bv_msb (norm_unsigned l)"
```
```  1148   assume "norm_unsigned l \<noteq> []"
```
```  1149   with norm_unsigned_result [of l]
```
```  1150   have "bv_msb (norm_unsigned l) = \<one>" by simp
```
```  1151   with msb show False by simp
```
```  1152 next
```
```  1153   fix xs
```
```  1154   assume p: "\<one> = bv_msb (tl (norm_signed (\<one> # xs)))"
```
```  1155   have "\<one> \<noteq> bv_msb (tl (norm_signed (\<one> # xs)))"
```
```  1156     by (rule bit_list_induct [of _ xs],simp_all)
```
```  1157   with p show False by simp
```
```  1158 qed
```
```  1159
```
```  1160 lemma bv_to_int_upper_limit_lem1:
```
```  1161   assumes w0: "bv_to_int w < -1"
```
```  1162   shows       "bv_to_int w < - (2 ^ (length (norm_signed w) - 2))"
```
```  1163 proof -
```
```  1164   from w0
```
```  1165   have "bv_to_int w < 0" by simp
```
```  1166   hence msbw [simp]: "bv_msb w = \<one>"
```
```  1167     by (rule bv_to_int_msb1)
```
```  1168   have "bv_to_int w = bv_to_int (norm_signed w)" by simp
```
```  1169   also from norm_signed_result [of w]
```
```  1170   have "... < - (2 ^ (length (norm_signed w) - 2))"
```
```  1171   proof safe
```
```  1172     assume "norm_signed w = []"
```
```  1173     hence "bv_to_int (norm_signed w) = 0" by simp
```
```  1174     with w0 show ?thesis by simp
```
```  1175   next
```
```  1176     assume "norm_signed w = [\<one>]"
```
```  1177     hence "bv_to_int (norm_signed w) = -1" by simp
```
```  1178     with w0 show ?thesis by simp
```
```  1179   next
```
```  1180     assume "bv_msb (norm_signed w) \<noteq> bv_msb (tl (norm_signed w))"
```
```  1181     hence msb_tl: "\<one> \<noteq> bv_msb (tl (norm_signed w))" by simp
```
```  1182     show "bv_to_int (norm_signed w) < - (2 ^ (length (norm_signed w) - 2))"
```
```  1183     proof (rule bit_list_cases [of "norm_signed w"])
```
```  1184       assume "norm_signed w = []"
```
```  1185       hence "bv_to_int (norm_signed w) = 0" by simp
```
```  1186       with w0 show ?thesis by simp
```
```  1187     next
```
```  1188       fix w'
```
```  1189       assume nw: "norm_signed w = \<zero> # w'"
```
```  1190       from msbw have "bv_msb (norm_signed w) = \<one>" by simp
```
```  1191       with nw show ?thesis by simp
```
```  1192     next
```
```  1193       fix w'
```
```  1194       assume weq: "norm_signed w = \<one> # w'"
```
```  1195       show ?thesis
```
```  1196       proof (rule bit_list_cases [of w'])
```
```  1197         assume w'eq: "w' = []"
```
```  1198         from w0 have "bv_to_int (norm_signed w) < -1" by simp
```
```  1199         with w'eq and weq show ?thesis by simp
```
```  1200       next
```
```  1201         fix w''
```
```  1202         assume w'eq: "w' = \<zero> # w''"
```
```  1203         show ?thesis
```
```  1204           apply (simp add: weq w'eq)
```
```  1205           apply (subgoal_tac "- int (bv_to_nat (bv_not w'')) + -1 < 0 + 0")
```
```  1206           apply (simp add: int_nat_two_exp)
```
```  1207           apply (rule add_le_less_mono)
```
```  1208           apply simp_all
```
```  1209           done
```
```  1210       next
```
```  1211         fix w''
```
```  1212         assume w'eq: "w' = \<one> # w''"
```
```  1213         with weq and msb_tl show ?thesis by simp
```
```  1214       qed
```
```  1215     qed
```
```  1216   qed
```
```  1217   finally show ?thesis .
```
```  1218 qed
```
```  1219
```
```  1220 lemma length_int_to_bv_upper_limit_gt0:
```
```  1221   assumes w0: "0 < i"
```
```  1222   and     wk: "i \<le> 2 ^ (k - 1) - 1"
```
```  1223   shows       "length (int_to_bv i) \<le> k"
```
```  1224 proof (rule ccontr)
```
```  1225   from w0 wk
```
```  1226   have k1: "1 < k"
```
```  1227     by (cases "k - 1",simp_all)
```
```  1228   assume "~ length (int_to_bv i) \<le> k"
```
```  1229   hence "k < length (int_to_bv i)" by simp
```
```  1230   hence "k \<le> length (int_to_bv i) - 1" by arith
```
```  1231   hence a: "k - 1 \<le> length (int_to_bv i) - 2" by arith
```
```  1232   hence "(2::int) ^ (k - 1) \<le> 2 ^ (length (int_to_bv i) - 2)" by simp
```
```  1233   also have "... \<le> i"
```
```  1234   proof -
```
```  1235     have "2 ^ (length (norm_signed (int_to_bv i)) - 2) \<le> bv_to_int (int_to_bv i)"
```
```  1236     proof (rule bv_to_int_lower_limit_gt0)
```
```  1237       from w0 show "0 < bv_to_int (int_to_bv i)" by simp
```
```  1238     qed
```
```  1239     thus ?thesis by simp
```
```  1240   qed
```
```  1241   finally have "2 ^ (k - 1) \<le> i" .
```
```  1242   with wk show False by simp
```
```  1243 qed
```
```  1244
```
```  1245 lemma pos_length_pos:
```
```  1246   assumes i0: "0 < bv_to_int w"
```
```  1247   shows       "0 < length w"
```
```  1248 proof -
```
```  1249   from norm_signed_result [of w]
```
```  1250   have "0 < length (norm_signed w)"
```
```  1251   proof (auto)
```
```  1252     assume ii: "norm_signed w = []"
```
```  1253     have "bv_to_int (norm_signed w) = 0" by (subst ii) simp
```
```  1254     hence "bv_to_int w = 0" by simp
```
```  1255     with i0 show False by simp
```
```  1256   next
```
```  1257     assume ii: "norm_signed w = []"
```
```  1258     assume jj: "bv_msb w \<noteq> \<zero>"
```
```  1259     have "\<zero> = bv_msb (norm_signed w)"
```
```  1260       by (subst ii) simp
```
```  1261     also have "... \<noteq> \<zero>"
```
```  1262       by (simp add: jj)
```
```  1263     finally show False by simp
```
```  1264   qed
```
```  1265   also have "... \<le> length w"
```
```  1266     by (rule norm_signed_length)
```
```  1267   finally show ?thesis .
```
```  1268 qed
```
```  1269
```
```  1270 lemma neg_length_pos:
```
```  1271   assumes i0: "bv_to_int w < -1"
```
```  1272   shows       "0 < length w"
```
```  1273 proof -
```
```  1274   from norm_signed_result [of w]
```
```  1275   have "0 < length (norm_signed w)"
```
```  1276   proof (auto)
```
```  1277     assume ii: "norm_signed w = []"
```
```  1278     have "bv_to_int (norm_signed w) = 0"
```
```  1279       by (subst ii) simp
```
```  1280     hence "bv_to_int w = 0" by simp
```
```  1281     with i0 show False by simp
```
```  1282   next
```
```  1283     assume ii: "norm_signed w = []"
```
```  1284     assume jj: "bv_msb w \<noteq> \<zero>"
```
```  1285     have "\<zero> = bv_msb (norm_signed w)" by (subst ii) simp
```
```  1286     also have "... \<noteq> \<zero>" by (simp add: jj)
```
```  1287     finally show False by simp
```
```  1288   qed
```
```  1289   also have "... \<le> length w"
```
```  1290     by (rule norm_signed_length)
```
```  1291   finally show ?thesis .
```
```  1292 qed
```
```  1293
```
```  1294 lemma length_int_to_bv_lower_limit_gt0:
```
```  1295   assumes wk: "2 ^ (k - 1) \<le> i"
```
```  1296   shows       "k < length (int_to_bv i)"
```
```  1297 proof (rule ccontr)
```
```  1298   have "0 < (2::int) ^ (k - 1)"
```
```  1299     by (rule zero_less_power) simp
```
```  1300   also have "... \<le> i" by (rule wk)
```
```  1301   finally have i0: "0 < i" .
```
```  1302   have lii0: "0 < length (int_to_bv i)"
```
```  1303     apply (rule pos_length_pos)
```
```  1304     apply (simp,rule i0)
```
```  1305     done
```
```  1306   assume "~ k < length (int_to_bv i)"
```
```  1307   hence "length (int_to_bv i) \<le> k" by simp
```
```  1308   with lii0
```
```  1309   have a: "length (int_to_bv i) - 1 \<le> k - 1"
```
```  1310     by arith
```
```  1311   have "i < 2 ^ (length (int_to_bv i) - 1)"
```
```  1312   proof -
```
```  1313     have "i = bv_to_int (int_to_bv i)"
```
```  1314       by simp
```
```  1315     also have "... < 2 ^ (length (int_to_bv i) - 1)"
```
```  1316       by (rule bv_to_int_upper_range)
```
```  1317     finally show ?thesis .
```
```  1318   qed
```
```  1319   also have "(2::int) ^ (length (int_to_bv i) - 1) \<le> 2 ^ (k - 1)" using a
```
```  1320     by simp
```
```  1321   finally have "i < 2 ^ (k - 1)" .
```
```  1322   with wk show False by simp
```
```  1323 qed
```
```  1324
```
```  1325 lemma length_int_to_bv_upper_limit_lem1:
```
```  1326   assumes w1: "i < -1"
```
```  1327   and     wk: "- (2 ^ (k - 1)) \<le> i"
```
```  1328   shows       "length (int_to_bv i) \<le> k"
```
```  1329 proof (rule ccontr)
```
```  1330   from w1 wk
```
```  1331   have k1: "1 < k" by (cases "k - 1") simp_all
```
```  1332   assume "~ length (int_to_bv i) \<le> k"
```
```  1333   hence "k < length (int_to_bv i)" by simp
```
```  1334   hence "k \<le> length (int_to_bv i) - 1" by arith
```
```  1335   hence a: "k - 1 \<le> length (int_to_bv i) - 2" by arith
```
```  1336   have "i < - (2 ^ (length (int_to_bv i) - 2))"
```
```  1337   proof -
```
```  1338     have "i = bv_to_int (int_to_bv i)"
```
```  1339       by simp
```
```  1340     also have "... < - (2 ^ (length (norm_signed (int_to_bv i)) - 2))"
```
```  1341       by (rule bv_to_int_upper_limit_lem1,simp,rule w1)
```
```  1342     finally show ?thesis by simp
```
```  1343   qed
```
```  1344   also have "... \<le> -(2 ^ (k - 1))"
```
```  1345   proof -
```
```  1346     have "(2::int) ^ (k - 1) \<le> 2 ^ (length (int_to_bv i) - 2)" using a by simp
```
```  1347     thus ?thesis by simp
```
```  1348   qed
```
```  1349   finally have "i < -(2 ^ (k - 1))" .
```
```  1350   with wk show False by simp
```
```  1351 qed
```
```  1352
```
```  1353 lemma length_int_to_bv_lower_limit_lem1:
```
```  1354   assumes wk: "i < -(2 ^ (k - 1))"
```
```  1355   shows       "k < length (int_to_bv i)"
```
```  1356 proof (rule ccontr)
```
```  1357   from wk have "i \<le> -(2 ^ (k - 1)) - 1" by simp
```
```  1358   also have "... < -1"
```
```  1359   proof -
```
```  1360     have "0 < (2::int) ^ (k - 1)"
```
```  1361       by (rule zero_less_power) simp
```
```  1362     hence "-((2::int) ^ (k - 1)) < 0" by simp
```
```  1363     thus ?thesis by simp
```
```  1364   qed
```
```  1365   finally have i1: "i < -1" .
```
```  1366   have lii0: "0 < length (int_to_bv i)"
```
```  1367     apply (rule neg_length_pos)
```
```  1368     apply (simp, rule i1)
```
```  1369     done
```
```  1370   assume "~ k < length (int_to_bv i)"
```
```  1371   hence "length (int_to_bv i) \<le> k"
```
```  1372     by simp
```
```  1373   with lii0 have a: "length (int_to_bv i) - 1 \<le> k - 1" by arith
```
```  1374   hence "(2::int) ^ (length (int_to_bv i) - 1) \<le> 2 ^ (k - 1)" by simp
```
```  1375   hence "-((2::int) ^ (k - 1)) \<le> - (2 ^ (length (int_to_bv i) - 1))" by simp
```
```  1376   also have "... \<le> i"
```
```  1377   proof -
```
```  1378     have "- (2 ^ (length (int_to_bv i) - 1)) \<le> bv_to_int (int_to_bv i)"
```
```  1379       by (rule bv_to_int_lower_range)
```
```  1380     also have "... = i"
```
```  1381       by simp
```
```  1382     finally show ?thesis .
```
```  1383   qed
```
```  1384   finally have "-(2 ^ (k - 1)) \<le> i" .
```
```  1385   with wk show False by simp
```
```  1386 qed
```
```  1387
```
```  1388
```
```  1389 subsection {* Signed Arithmetic Operations *}
```
```  1390
```
```  1391 subsubsection {* Conversion from unsigned to signed *}
```
```  1392
```
```  1393 definition
```
```  1394   utos :: "bit list => bit list" where
```
```  1395   "utos w = norm_signed (\<zero> # w)"
```
```  1396
```
```  1397 lemma utos_type [simp]: "utos (norm_unsigned w) = utos w"
```
```  1398   by (simp add: utos_def norm_signed_Cons)
```
```  1399
```
```  1400 lemma utos_returntype [simp]: "norm_signed (utos w) = utos w"
```
```  1401   by (simp add: utos_def)
```
```  1402
```
```  1403 lemma utos_length: "length (utos w) \<le> Suc (length w)"
```
```  1404   by (simp add: utos_def norm_signed_Cons)
```
```  1405
```
```  1406 lemma bv_to_int_utos: "bv_to_int (utos w) = int (bv_to_nat w)"
```
```  1407 proof (simp add: utos_def norm_signed_Cons, safe)
```
```  1408   assume "norm_unsigned w = []"
```
```  1409   hence "bv_to_nat (norm_unsigned w) = 0" by simp
```
```  1410   thus "bv_to_nat w = 0" by simp
```
```  1411 qed
```
```  1412
```
```  1413
```
```  1414 subsubsection {* Unary minus *}
```
```  1415
```
```  1416 definition
```
```  1417   bv_uminus :: "bit list => bit list" where
```
```  1418   "bv_uminus w = int_to_bv (- bv_to_int w)"
```
```  1419
```
```  1420 lemma bv_uminus_type [simp]: "bv_uminus (norm_signed w) = bv_uminus w"
```
```  1421   by (simp add: bv_uminus_def)
```
```  1422
```
```  1423 lemma bv_uminus_returntype [simp]: "norm_signed (bv_uminus w) = bv_uminus w"
```
```  1424   by (simp add: bv_uminus_def)
```
```  1425
```
```  1426 lemma bv_uminus_length: "length (bv_uminus w) \<le> Suc (length w)"
```
```  1427 proof -
```
```  1428   have "1 < -bv_to_int w \<or> -bv_to_int w = 1 \<or> -bv_to_int w = 0 \<or> -bv_to_int w = -1 \<or> -bv_to_int w < -1"
```
```  1429     by arith
```
```  1430   thus ?thesis
```
```  1431   proof safe
```
```  1432     assume p: "1 < - bv_to_int w"
```
```  1433     have lw: "0 < length w"
```
```  1434       apply (rule neg_length_pos)
```
```  1435       using p
```
```  1436       apply simp
```
```  1437       done
```
```  1438     show ?thesis
```
```  1439     proof (simp add: bv_uminus_def,rule length_int_to_bv_upper_limit_gt0,simp_all)
```
```  1440       from prems show "bv_to_int w < 0" by simp
```
```  1441     next
```
```  1442       have "-(2^(length w - 1)) \<le> bv_to_int w"
```
```  1443         by (rule bv_to_int_lower_range)
```
```  1444       hence "- bv_to_int w \<le> 2^(length w - 1)" by simp
```
```  1445       also from lw have "... < 2 ^ length w" by simp
```
```  1446       finally show "- bv_to_int w < 2 ^ length w" by simp
```
```  1447     qed
```
```  1448   next
```
```  1449     assume p: "- bv_to_int w = 1"
```
```  1450     hence lw: "0 < length w" by (cases w) simp_all
```
```  1451     from p
```
```  1452     show ?thesis
```
```  1453       apply (simp add: bv_uminus_def)
```
```  1454       using lw
```
```  1455       apply (simp (no_asm) add: nat_to_bv_non0)
```
```  1456       done
```
```  1457   next
```
```  1458     assume "- bv_to_int w = 0"
```
```  1459     thus ?thesis by (simp add: bv_uminus_def)
```
```  1460   next
```
```  1461     assume p: "- bv_to_int w = -1"
```
```  1462     thus ?thesis by (simp add: bv_uminus_def)
```
```  1463   next
```
```  1464     assume p: "- bv_to_int w < -1"
```
```  1465     show ?thesis
```
```  1466       apply (simp add: bv_uminus_def)
```
```  1467       apply (rule length_int_to_bv_upper_limit_lem1)
```
```  1468       apply (rule p)
```
```  1469       apply simp
```
```  1470     proof -
```
```  1471       have "bv_to_int w < 2 ^ (length w - 1)"
```
```  1472         by (rule bv_to_int_upper_range)
```
```  1473       also have "... \<le> 2 ^ length w" by simp
```
```  1474       finally show "bv_to_int w \<le> 2 ^ length w" by simp
```
```  1475     qed
```
```  1476   qed
```
```  1477 qed
```
```  1478
```
```  1479 lemma bv_uminus_length_utos: "length (bv_uminus (utos w)) \<le> Suc (length w)"
```
```  1480 proof -
```
```  1481   have "-bv_to_int (utos w) = 0 \<or> -bv_to_int (utos w) = -1 \<or> -bv_to_int (utos w) < -1"
```
```  1482     by (simp add: bv_to_int_utos, arith)
```
```  1483   thus ?thesis
```
```  1484   proof safe
```
```  1485     assume "-bv_to_int (utos w) = 0"
```
```  1486     thus ?thesis by (simp add: bv_uminus_def)
```
```  1487   next
```
```  1488     assume "-bv_to_int (utos w) = -1"
```
```  1489     thus ?thesis by (simp add: bv_uminus_def)
```
```  1490   next
```
```  1491     assume p: "-bv_to_int (utos w) < -1"
```
```  1492     show ?thesis
```
```  1493       apply (simp add: bv_uminus_def)
```
```  1494       apply (rule length_int_to_bv_upper_limit_lem1)
```
```  1495       apply (rule p)
```
```  1496       apply (simp add: bv_to_int_utos)
```
```  1497       using bv_to_nat_upper_range [of w]
```
```  1498       apply (simp add: int_nat_two_exp)
```
```  1499       done
```
```  1500   qed
```
```  1501 qed
```
```  1502
```
```  1503 definition
```
```  1504   bv_sadd :: "[bit list, bit list ] => bit list" where
```
```  1505   "bv_sadd w1 w2 = int_to_bv (bv_to_int w1 + bv_to_int w2)"
```
```  1506
```
```  1507 lemma bv_sadd_type1 [simp]: "bv_sadd (norm_signed w1) w2 = bv_sadd w1 w2"
```
```  1508   by (simp add: bv_sadd_def)
```
```  1509
```
```  1510 lemma bv_sadd_type2 [simp]: "bv_sadd w1 (norm_signed w2) = bv_sadd w1 w2"
```
```  1511   by (simp add: bv_sadd_def)
```
```  1512
```
```  1513 lemma bv_sadd_returntype [simp]: "norm_signed (bv_sadd w1 w2) = bv_sadd w1 w2"
```
```  1514   by (simp add: bv_sadd_def)
```
```  1515
```
```  1516 lemma adder_helper:
```
```  1517   assumes lw: "0 < max (length w1) (length w2)"
```
```  1518   shows   "((2::int) ^ (length w1 - 1)) + (2 ^ (length w2 - 1)) \<le> 2 ^ max (length w1) (length w2)"
```
```  1519 proof -
```
```  1520   have "((2::int) ^ (length w1 - 1)) + (2 ^ (length w2 - 1)) \<le>
```
```  1521       2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1)"
```
```  1522     by (auto simp:max_def)
```
```  1523   also have "... = 2 ^ max (length w1) (length w2)"
```
```  1524   proof -
```
```  1525     from lw
```
```  1526     show ?thesis
```
```  1527       apply simp
```
```  1528       apply (subst power_Suc [symmetric])
```
```  1529       apply simp
```
```  1530       done
```
```  1531   qed
```
```  1532   finally show ?thesis .
```
```  1533 qed
```
```  1534
```
```  1535 lemma bv_sadd_length: "length (bv_sadd w1 w2) \<le> Suc (max (length w1) (length w2))"
```
```  1536 proof -
```
```  1537   let ?Q = "bv_to_int w1 + bv_to_int w2"
```
```  1538
```
```  1539   have helper: "?Q \<noteq> 0 ==> 0 < max (length w1) (length w2)"
```
```  1540   proof -
```
```  1541     assume p: "?Q \<noteq> 0"
```
```  1542     show "0 < max (length w1) (length w2)"
```
```  1543     proof (simp add: less_max_iff_disj,rule)
```
```  1544       assume [simp]: "w1 = []"
```
```  1545       show "w2 \<noteq> []"
```
```  1546       proof (rule ccontr,simp)
```
```  1547         assume [simp]: "w2 = []"
```
```  1548         from p show False by simp
```
```  1549       qed
```
```  1550     qed
```
```  1551   qed
```
```  1552
```
```  1553   have "0 < ?Q \<or> ?Q = 0 \<or> ?Q = -1 \<or> ?Q < -1" by arith
```
```  1554   thus ?thesis
```
```  1555   proof safe
```
```  1556     assume "?Q = 0"
```
```  1557     thus ?thesis
```
```  1558       by (simp add: bv_sadd_def)
```
```  1559   next
```
```  1560     assume "?Q = -1"
```
```  1561     thus ?thesis
```
```  1562       by (simp add: bv_sadd_def)
```
```  1563   next
```
```  1564     assume p: "0 < ?Q"
```
```  1565     show ?thesis
```
```  1566       apply (simp add: bv_sadd_def)
```
```  1567       apply (rule length_int_to_bv_upper_limit_gt0)
```
```  1568       apply (rule p)
```
```  1569     proof simp
```
```  1570       from bv_to_int_upper_range [of w2]
```
```  1571       have "bv_to_int w2 \<le> 2 ^ (length w2 - 1)"
```
```  1572         by simp
```
```  1573       with bv_to_int_upper_range [of w1]
```
```  1574       have "bv_to_int w1 + bv_to_int w2 < (2 ^ (length w1 - 1)) + (2 ^ (length w2 - 1))"
```
```  1575         by (rule zadd_zless_mono)
```
```  1576       also have "... \<le> 2 ^ max (length w1) (length w2)"
```
```  1577         apply (rule adder_helper)
```
```  1578         apply (rule helper)
```
```  1579         using p
```
```  1580         apply simp
```
```  1581         done
```
```  1582       finally show "?Q < 2 ^ max (length w1) (length w2)" .
```
```  1583     qed
```
```  1584   next
```
```  1585     assume p: "?Q < -1"
```
```  1586     show ?thesis
```
```  1587       apply (simp add: bv_sadd_def)
```
```  1588       apply (rule length_int_to_bv_upper_limit_lem1,simp_all)
```
```  1589       apply (rule p)
```
```  1590     proof -
```
```  1591       have "(2 ^ (length w1 - 1)) + 2 ^ (length w2 - 1) \<le> (2::int) ^ max (length w1) (length w2)"
```
```  1592         apply (rule adder_helper)
```
```  1593         apply (rule helper)
```
```  1594         using p
```
```  1595         apply simp
```
```  1596         done
```
```  1597       hence "-((2::int) ^ max (length w1) (length w2)) \<le> - (2 ^ (length w1 - 1)) + -(2 ^ (length w2 - 1))"
```
```  1598         by simp
```
```  1599       also have "- (2 ^ (length w1 - 1)) + -(2 ^ (length w2 - 1)) \<le> ?Q"
```
```  1600         apply (rule add_mono)
```
```  1601         apply (rule bv_to_int_lower_range [of w1])
```
```  1602         apply (rule bv_to_int_lower_range [of w2])
```
```  1603         done
```
```  1604       finally show "- (2^max (length w1) (length w2)) \<le> ?Q" .
```
```  1605     qed
```
```  1606   qed
```
```  1607 qed
```
```  1608
```
```  1609 definition
```
```  1610   bv_sub :: "[bit list, bit list] => bit list" where
```
```  1611   "bv_sub w1 w2 = bv_sadd w1 (bv_uminus w2)"
```
```  1612
```
```  1613 lemma bv_sub_type1 [simp]: "bv_sub (norm_signed w1) w2 = bv_sub w1 w2"
```
```  1614   by (simp add: bv_sub_def)
```
```  1615
```
```  1616 lemma bv_sub_type2 [simp]: "bv_sub w1 (norm_signed w2) = bv_sub w1 w2"
```
```  1617   by (simp add: bv_sub_def)
```
```  1618
```
```  1619 lemma bv_sub_returntype [simp]: "norm_signed (bv_sub w1 w2) = bv_sub w1 w2"
```
```  1620   by (simp add: bv_sub_def)
```
```  1621
```
```  1622 lemma bv_sub_length: "length (bv_sub w1 w2) \<le> Suc (max (length w1) (length w2))"
```
```  1623 proof (cases "bv_to_int w2 = 0")
```
```  1624   assume p: "bv_to_int w2 = 0"
```
```  1625   show ?thesis
```
```  1626   proof (simp add: bv_sub_def bv_sadd_def bv_uminus_def p)
```
```  1627     have "length (norm_signed w1) \<le> length w1"
```
```  1628       by (rule norm_signed_length)
```
```  1629     also have "... \<le> max (length w1) (length w2)"
```
```  1630       by (rule le_maxI1)
```
```  1631     also have "... \<le> Suc (max (length w1) (length w2))"
```
```  1632       by arith
```
```  1633     finally show "length (norm_signed w1) \<le> Suc (max (length w1) (length w2))" .
```
```  1634   qed
```
```  1635 next
```
```  1636   assume "bv_to_int w2 \<noteq> 0"
```
```  1637   hence "0 < length w2" by (cases w2,simp_all)
```
```  1638   hence lmw: "0 < max (length w1) (length w2)" by arith
```
```  1639
```
```  1640   let ?Q = "bv_to_int w1 - bv_to_int w2"
```
```  1641
```
```  1642   have "0 < ?Q \<or> ?Q = 0 \<or> ?Q = -1 \<or> ?Q < -1" by arith
```
```  1643   thus ?thesis
```
```  1644   proof safe
```
```  1645     assume "?Q = 0"
```
```  1646     thus ?thesis
```
```  1647       by (simp add: bv_sub_def bv_sadd_def bv_uminus_def)
```
```  1648   next
```
```  1649     assume "?Q = -1"
```
```  1650     thus ?thesis
```
```  1651       by (simp add: bv_sub_def bv_sadd_def bv_uminus_def)
```
```  1652   next
```
```  1653     assume p: "0 < ?Q"
```
```  1654     show ?thesis
```
```  1655       apply (simp add: bv_sub_def bv_sadd_def bv_uminus_def)
```
```  1656       apply (rule length_int_to_bv_upper_limit_gt0)
```
```  1657       apply (rule p)
```
```  1658     proof simp
```
```  1659       from bv_to_int_lower_range [of w2]
```
```  1660       have v2: "- bv_to_int w2 \<le> 2 ^ (length w2 - 1)" by simp
```
```  1661       have "bv_to_int w1 + - bv_to_int w2 < (2 ^ (length w1 - 1)) + (2 ^ (length w2 - 1))"
```
```  1662         apply (rule zadd_zless_mono)
```
```  1663         apply (rule bv_to_int_upper_range [of w1])
```
```  1664         apply (rule v2)
```
```  1665         done
```
```  1666       also have "... \<le> 2 ^ max (length w1) (length w2)"
```
```  1667         apply (rule adder_helper)
```
```  1668         apply (rule lmw)
```
```  1669         done
```
```  1670       finally show "?Q < 2 ^ max (length w1) (length w2)" by simp
```
```  1671     qed
```
```  1672   next
```
```  1673     assume p: "?Q < -1"
```
```  1674     show ?thesis
```
```  1675       apply (simp add: bv_sub_def bv_sadd_def bv_uminus_def)
```
```  1676       apply (rule length_int_to_bv_upper_limit_lem1)
```
```  1677       apply (rule p)
```
```  1678     proof simp
```
```  1679       have "(2 ^ (length w1 - 1)) + 2 ^ (length w2 - 1) \<le> (2::int) ^ max (length w1) (length w2)"
```
```  1680         apply (rule adder_helper)
```
```  1681         apply (rule lmw)
```
```  1682         done
```
```  1683       hence "-((2::int) ^ max (length w1) (length w2)) \<le> - (2 ^ (length w1 - 1)) + -(2 ^ (length w2 - 1))"
```
```  1684         by simp
```
```  1685       also have "- (2 ^ (length w1 - 1)) + -(2 ^ (length w2 - 1)) \<le> bv_to_int w1 + -bv_to_int w2"
```
```  1686         apply (rule add_mono)
```
```  1687         apply (rule bv_to_int_lower_range [of w1])
```
```  1688         using bv_to_int_upper_range [of w2]
```
```  1689         apply simp
```
```  1690         done
```
```  1691       finally show "- (2^max (length w1) (length w2)) \<le> ?Q" by simp
```
```  1692     qed
```
```  1693   qed
```
```  1694 qed
```
```  1695
```
```  1696 definition
```
```  1697   bv_smult :: "[bit list, bit list] => bit list" where
```
```  1698   "bv_smult w1 w2 = int_to_bv (bv_to_int w1 * bv_to_int w2)"
```
```  1699
```
```  1700 lemma bv_smult_type1 [simp]: "bv_smult (norm_signed w1) w2 = bv_smult w1 w2"
```
```  1701   by (simp add: bv_smult_def)
```
```  1702
```
```  1703 lemma bv_smult_type2 [simp]: "bv_smult w1 (norm_signed w2) = bv_smult w1 w2"
```
```  1704   by (simp add: bv_smult_def)
```
```  1705
```
```  1706 lemma bv_smult_returntype [simp]: "norm_signed (bv_smult w1 w2) = bv_smult w1 w2"
```
```  1707   by (simp add: bv_smult_def)
```
```  1708
```
```  1709 lemma bv_smult_length: "length (bv_smult w1 w2) \<le> length w1 + length w2"
```
```  1710 proof -
```
```  1711   let ?Q = "bv_to_int w1 * bv_to_int w2"
```
```  1712
```
```  1713   have lmw: "?Q \<noteq> 0 ==> 0 < length w1 \<and> 0 < length w2" by auto
```
```  1714
```
```  1715   have "0 < ?Q \<or> ?Q = 0 \<or> ?Q = -1 \<or> ?Q < -1" by arith
```
```  1716   thus ?thesis
```
```  1717   proof (safe dest!: iffD1 [OF mult_eq_0_iff])
```
```  1718     assume "bv_to_int w1 = 0"
```
```  1719     thus ?thesis by (simp add: bv_smult_def)
```
```  1720   next
```
```  1721     assume "bv_to_int w2 = 0"
```
```  1722     thus ?thesis by (simp add: bv_smult_def)
```
```  1723   next
```
```  1724     assume p: "?Q = -1"
```
```  1725     show ?thesis
```
```  1726       apply (simp add: bv_smult_def p)
```
```  1727       apply (cut_tac lmw)
```
```  1728       apply arith
```
```  1729       using p
```
```  1730       apply simp
```
```  1731       done
```
```  1732   next
```
```  1733     assume p: "0 < ?Q"
```
```  1734     thus ?thesis
```
```  1735     proof (simp add: zero_less_mult_iff,safe)
```
```  1736       assume bi1: "0 < bv_to_int w1"
```
```  1737       assume bi2: "0 < bv_to_int w2"
```
```  1738       show ?thesis
```
```  1739         apply (simp add: bv_smult_def)
```
```  1740         apply (rule length_int_to_bv_upper_limit_gt0)
```
```  1741         apply (rule p)
```
```  1742       proof simp
```
```  1743         have "?Q < 2 ^ (length w1 - 1) * 2 ^ (length w2 - 1)"
```
```  1744           apply (rule mult_strict_mono)
```
```  1745           apply (rule bv_to_int_upper_range)
```
```  1746           apply (rule bv_to_int_upper_range)
```
```  1747           apply (rule zero_less_power)
```
```  1748           apply simp
```
```  1749           using bi2
```
```  1750           apply simp
```
```  1751           done
```
```  1752         also have "... \<le> 2 ^ (length w1 + length w2 - Suc 0)"
```
```  1753           apply simp
```
```  1754           apply (subst zpower_zadd_distrib [symmetric])
```
```  1755           apply simp
```
```  1756           done
```
```  1757         finally show "?Q < 2 ^ (length w1 + length w2 - Suc 0)" .
```
```  1758       qed
```
```  1759     next
```
```  1760       assume bi1: "bv_to_int w1 < 0"
```
```  1761       assume bi2: "bv_to_int w2 < 0"
```
```  1762       show ?thesis
```
```  1763         apply (simp add: bv_smult_def)
```
```  1764         apply (rule length_int_to_bv_upper_limit_gt0)
```
```  1765         apply (rule p)
```
```  1766       proof simp
```
```  1767         have "-bv_to_int w1 * -bv_to_int w2 \<le> 2 ^ (length w1 - 1) * 2 ^(length w2 - 1)"
```
```  1768           apply (rule mult_mono)
```
```  1769           using bv_to_int_lower_range [of w1]
```
```  1770           apply simp
```
```  1771           using bv_to_int_lower_range [of w2]
```
```  1772           apply simp
```
```  1773           apply (rule zero_le_power,simp)
```
```  1774           using bi2
```
```  1775           apply simp
```
```  1776           done
```
```  1777         hence "?Q \<le> 2 ^ (length w1 - 1) * 2 ^(length w2 - 1)"
```
```  1778           by simp
```
```  1779         also have "... < 2 ^ (length w1 + length w2 - Suc 0)"
```
```  1780           apply simp
```
```  1781           apply (subst zpower_zadd_distrib [symmetric])
```
```  1782           apply simp
```
```  1783           apply (cut_tac lmw)
```
```  1784           apply arith
```
```  1785           apply (cut_tac p)
```
```  1786           apply arith
```
```  1787           done
```
```  1788         finally show "?Q < 2 ^ (length w1 + length w2 - Suc 0)" .
```
```  1789       qed
```
```  1790     qed
```
```  1791   next
```
```  1792     assume p: "?Q < -1"
```
```  1793     show ?thesis
```
```  1794       apply (subst bv_smult_def)
```
```  1795       apply (rule length_int_to_bv_upper_limit_lem1)
```
```  1796       apply (rule p)
```
```  1797     proof simp
```
```  1798       have "(2::int) ^ (length w1 - 1) * 2 ^(length w2 - 1) \<le> 2 ^ (length w1 + length w2 - Suc 0)"
```
```  1799         apply simp
```
```  1800         apply (subst zpower_zadd_distrib [symmetric])
```
```  1801         apply simp
```
```  1802         done
```
```  1803       hence "-((2::int) ^ (length w1 + length w2 - Suc 0)) \<le> -(2^(length w1 - 1) * 2 ^ (length w2 - 1))"
```
```  1804         by simp
```
```  1805       also have "... \<le> ?Q"
```
```  1806       proof -
```
```  1807         from p
```
```  1808         have q: "bv_to_int w1 * bv_to_int w2 < 0"
```
```  1809           by simp
```
```  1810         thus ?thesis
```
```  1811         proof (simp add: mult_less_0_iff,safe)
```
```  1812           assume bi1: "0 < bv_to_int w1"
```
```  1813           assume bi2: "bv_to_int w2 < 0"
```
```  1814           have "-bv_to_int w2 * bv_to_int w1 \<le> ((2::int)^(length w2 - 1)) * (2 ^ (length w1 - 1))"
```
```  1815             apply (rule mult_mono)
```
```  1816             using bv_to_int_lower_range [of w2]
```
```  1817             apply simp
```
```  1818             using bv_to_int_upper_range [of w1]
```
```  1819             apply simp
```
```  1820             apply (rule zero_le_power,simp)
```
```  1821             using bi1
```
```  1822             apply simp
```
```  1823             done
```
```  1824           hence "-?Q \<le> ((2::int)^(length w1 - 1)) * (2 ^ (length w2 - 1))"
```
```  1825             by (simp add: zmult_ac)
```
```  1826           thus "-(((2::int)^(length w1 - Suc 0)) * (2 ^ (length w2 - Suc 0))) \<le> ?Q"
```
```  1827             by simp
```
```  1828         next
```
```  1829           assume bi1: "bv_to_int w1 < 0"
```
```  1830           assume bi2: "0 < bv_to_int w2"
```
```  1831           have "-bv_to_int w1 * bv_to_int w2 \<le> ((2::int)^(length w1 - 1)) * (2 ^ (length w2 - 1))"
```
```  1832             apply (rule mult_mono)
```
```  1833             using bv_to_int_lower_range [of w1]
```
```  1834             apply simp
```
```  1835             using bv_to_int_upper_range [of w2]
```
```  1836             apply simp
```
```  1837             apply (rule zero_le_power,simp)
```
```  1838             using bi2
```
```  1839             apply simp
```
```  1840             done
```
```  1841           hence "-?Q \<le> ((2::int)^(length w1 - 1)) * (2 ^ (length w2 - 1))"
```
```  1842             by (simp add: zmult_ac)
```
```  1843           thus "-(((2::int)^(length w1 - Suc 0)) * (2 ^ (length w2 - Suc 0))) \<le> ?Q"
```
```  1844             by simp
```
```  1845         qed
```
```  1846       qed
```
```  1847       finally show "-(2 ^ (length w1 + length w2 - Suc 0)) \<le> ?Q" .
```
```  1848     qed
```
```  1849   qed
```
```  1850 qed
```
```  1851
```
```  1852 lemma bv_msb_one: "bv_msb w = \<one> ==> bv_to_nat w \<noteq> 0"
```
```  1853 by (cases w) simp_all
```
```  1854
```
```  1855 lemma bv_smult_length_utos: "length (bv_smult (utos w1) w2) \<le> length w1 + length w2"
```
```  1856 proof -
```
```  1857   let ?Q = "bv_to_int (utos w1) * bv_to_int w2"
```
```  1858
```
```  1859   have lmw: "?Q \<noteq> 0 ==> 0 < length (utos w1) \<and> 0 < length w2" by auto
```
```  1860
```
```  1861   have "0 < ?Q \<or> ?Q = 0 \<or> ?Q = -1 \<or> ?Q < -1" by arith
```
```  1862   thus ?thesis
```
```  1863   proof (safe dest!: iffD1 [OF mult_eq_0_iff])
```
```  1864     assume "bv_to_int (utos w1) = 0"
```
```  1865     thus ?thesis by (simp add: bv_smult_def)
```
```  1866   next
```
```  1867     assume "bv_to_int w2 = 0"
```
```  1868     thus ?thesis by (simp add: bv_smult_def)
```
```  1869   next
```
```  1870     assume p: "0 < ?Q"
```
```  1871     thus ?thesis
```
```  1872     proof (simp add: zero_less_mult_iff,safe)
```
```  1873       assume biw2: "0 < bv_to_int w2"
```
```  1874       show ?thesis
```
```  1875         apply (simp add: bv_smult_def)
```
```  1876         apply (rule length_int_to_bv_upper_limit_gt0)
```
```  1877         apply (rule p)
```
```  1878       proof simp
```
```  1879         have "?Q < 2 ^ length w1 * 2 ^ (length w2 - 1)"
```
```  1880           apply (rule mult_strict_mono)
```
```  1881           apply (simp add: bv_to_int_utos int_nat_two_exp)
```
```  1882           apply (rule bv_to_nat_upper_range)
```
```  1883           apply (rule bv_to_int_upper_range)
```
```  1884           apply (rule zero_less_power,simp)
```
```  1885           using biw2
```
```  1886           apply simp
```
```  1887           done
```
```  1888         also have "... \<le> 2 ^ (length w1 + length w2 - Suc 0)"
```
```  1889           apply simp
```
```  1890           apply (subst zpower_zadd_distrib [symmetric])
```
```  1891           apply simp
```
```  1892           apply (cut_tac lmw)
```
```  1893           apply arith
```
```  1894           using p
```
```  1895           apply auto
```
```  1896           done
```
```  1897         finally show "?Q < 2 ^ (length w1 + length w2 - Suc 0)" .
```
```  1898       qed
```
```  1899     next
```
```  1900       assume "bv_to_int (utos w1) < 0"
```
```  1901       thus ?thesis by (simp add: bv_to_int_utos)
```
```  1902     qed
```
```  1903   next
```
```  1904     assume p: "?Q = -1"
```
```  1905     thus ?thesis
```
```  1906       apply (simp add: bv_smult_def)
```
```  1907       apply (cut_tac lmw)
```
```  1908       apply arith
```
```  1909       apply simp
```
```  1910       done
```
```  1911   next
```
```  1912     assume p: "?Q < -1"
```
```  1913     show ?thesis
```
```  1914       apply (subst bv_smult_def)
```
```  1915       apply (rule length_int_to_bv_upper_limit_lem1)
```
```  1916       apply (rule p)
```
```  1917     proof simp
```
```  1918       have "(2::int) ^ length w1 * 2 ^(length w2 - 1) \<le> 2 ^ (length w1 + length w2 - Suc 0)"
```
```  1919         apply simp
```
```  1920         apply (subst zpower_zadd_distrib [symmetric])
```
```  1921         apply simp
```
```  1922         apply (cut_tac lmw)
```
```  1923         apply arith
```
```  1924         apply (cut_tac p)
```
```  1925         apply arith
```
```  1926         done
```
```  1927       hence "-((2::int) ^ (length w1 + length w2 - Suc 0)) \<le> -(2^ length w1 * 2 ^ (length w2 - 1))"
```
```  1928         by simp
```
```  1929       also have "... \<le> ?Q"
```
```  1930       proof -
```
```  1931         from p
```
```  1932         have q: "bv_to_int (utos w1) * bv_to_int w2 < 0"
```
```  1933           by simp
```
```  1934         thus ?thesis
```
```  1935         proof (simp add: mult_less_0_iff,safe)
```
```  1936           assume bi1: "0 < bv_to_int (utos w1)"
```
```  1937           assume bi2: "bv_to_int w2 < 0"
```
```  1938           have "-bv_to_int w2 * bv_to_int (utos w1) \<le> ((2::int)^(length w2 - 1)) * (2 ^ length w1)"
```
```  1939             apply (rule mult_mono)
```
```  1940             using bv_to_int_lower_range [of w2]
```
```  1941             apply simp
```
```  1942             apply (simp add: bv_to_int_utos)
```
```  1943             using bv_to_nat_upper_range [of w1]
```
```  1944             apply (simp add: int_nat_two_exp)
```
```  1945             apply (rule zero_le_power,simp)
```
```  1946             using bi1
```
```  1947             apply simp
```
```  1948             done
```
```  1949           hence "-?Q \<le> ((2::int)^length w1) * (2 ^ (length w2 - 1))"
```
```  1950             by (simp add: zmult_ac)
```
```  1951           thus "-(((2::int)^length w1) * (2 ^ (length w2 - Suc 0))) \<le> ?Q"
```
```  1952             by simp
```
```  1953         next
```
```  1954           assume bi1: "bv_to_int (utos w1) < 0"
```
```  1955           thus "-(((2::int)^length w1) * (2 ^ (length w2 - Suc 0))) \<le> ?Q"
```
```  1956             by (simp add: bv_to_int_utos)
```
```  1957         qed
```
```  1958       qed
```
```  1959       finally show "-(2 ^ (length w1 + length w2 - Suc 0)) \<le> ?Q" .
```
```  1960     qed
```
```  1961   qed
```
```  1962 qed
```
```  1963
```
```  1964 lemma bv_smult_sym: "bv_smult w1 w2 = bv_smult w2 w1"
```
```  1965   by (simp add: bv_smult_def zmult_ac)
```
```  1966
```
```  1967 subsection {* Structural operations *}
```
```  1968
```
```  1969 definition
```
```  1970   bv_select :: "[bit list,nat] => bit" where
```
```  1971   "bv_select w i = w ! (length w - 1 - i)"
```
```  1972
```
```  1973 definition
```
```  1974   bv_chop :: "[bit list,nat] => bit list * bit list" where
```
```  1975   "bv_chop w i = (let len = length w in (take (len - i) w,drop (len - i) w))"
```
```  1976
```
```  1977 definition
```
```  1978   bv_slice :: "[bit list,nat*nat] => bit list" where
```
```  1979   "bv_slice w = (\<lambda>(b,e). fst (bv_chop (snd (bv_chop w (b+1))) e))"
```
```  1980
```
```  1981 lemma bv_select_rev:
```
```  1982   assumes notnull: "n < length w"
```
```  1983   shows            "bv_select w n = rev w ! n"
```
```  1984 proof -
```
```  1985   have "\<forall>n. n < length w --> bv_select w n = rev w ! n"
```
```  1986   proof (rule length_induct [of _ w],auto simp add: bv_select_def)
```
```  1987     fix xs :: "bit list"
```
```  1988     fix n
```
```  1989     assume ind: "\<forall>ys::bit list. length ys < length xs --> (\<forall>n. n < length ys --> ys ! (length ys - Suc n) = rev ys ! n)"
```
```  1990     assume notx: "n < length xs"
```
```  1991     show "xs ! (length xs - Suc n) = rev xs ! n"
```
```  1992     proof (cases xs)
```
```  1993       assume "xs = []"
```
```  1994       with notx show ?thesis by simp
```
```  1995     next
```
```  1996       fix y ys
```
```  1997       assume [simp]: "xs = y # ys"
```
```  1998       show ?thesis
```
```  1999       proof (auto simp add: nth_append)
```
```  2000         assume noty: "n < length ys"
```
```  2001         from spec [OF ind,of ys]
```
```  2002         have "\<forall>n. n < length ys --> ys ! (length ys - Suc n) = rev ys ! n"
```
```  2003           by simp
```
```  2004         hence "n < length ys --> ys ! (length ys - Suc n) = rev ys ! n" ..
```
```  2005 	from this and noty
```
```  2006         have "ys ! (length ys - Suc n) = rev ys ! n" ..
```
```  2007         thus "(y # ys) ! (length ys - n) = rev ys ! n"
```
```  2008           by (simp add: nth_Cons' noty linorder_not_less [symmetric])
```
```  2009       next
```
```  2010         assume "~ n < length ys"
```
```  2011         hence x: "length ys \<le> n" by simp
```
```  2012         from notx have "n < Suc (length ys)" by simp
```
```  2013         hence "n \<le> length ys" by simp
```
```  2014         with x have "length ys = n" by simp
```
```  2015         thus "y = [y] ! (n - length ys)" by simp
```
```  2016       qed
```
```  2017     qed
```
```  2018   qed
```
```  2019   then have "n < length w --> bv_select w n = rev w ! n" ..
```
```  2020   from this and notnull show ?thesis ..
```
```  2021 qed
```
```  2022
```
```  2023 lemma bv_chop_append: "bv_chop (w1 @ w2) (length w2) = (w1,w2)"
```
```  2024   by (simp add: bv_chop_def Let_def)
```
```  2025
```
```  2026 lemma append_bv_chop_id: "fst (bv_chop w l) @ snd (bv_chop w l) = w"
```
```  2027   by (simp add: bv_chop_def Let_def)
```
```  2028
```
```  2029 lemma bv_chop_length_fst [simp]: "length (fst (bv_chop w i)) = length w - i"
```
```  2030   by (simp add: bv_chop_def Let_def)
```
```  2031
```
```  2032 lemma bv_chop_length_snd [simp]: "length (snd (bv_chop w i)) = min i (length w)"
```
```  2033   by (simp add: bv_chop_def Let_def)
```
```  2034
```
```  2035 lemma bv_slice_length [simp]: "[| j \<le> i; i < length w |] ==> length (bv_slice w (i,j)) = i - j + 1"
```
```  2036   by (auto simp add: bv_slice_def)
```
```  2037
```
```  2038 definition
```
```  2039   length_nat :: "nat => nat" where
```
```  2040   [code del]: "length_nat x = (LEAST n. x < 2 ^ n)"
```
```  2041
```
```  2042 lemma length_nat: "length (nat_to_bv n) = length_nat n"
```
```  2043   apply (simp add: length_nat_def)
```
```  2044   apply (rule Least_equality [symmetric])
```
```  2045   prefer 2
```
```  2046   apply (rule length_nat_to_bv_upper_limit)
```
```  2047   apply arith
```
```  2048   apply (rule ccontr)
```
```  2049 proof -
```
```  2050   assume "~ n < 2 ^ length (nat_to_bv n)"
```
```  2051   hence "2 ^ length (nat_to_bv n) \<le> n" by simp
```
```  2052   hence "length (nat_to_bv n) < length (nat_to_bv n)"
```
```  2053     by (rule length_nat_to_bv_lower_limit)
```
```  2054   thus False by simp
```
```  2055 qed
```
```  2056
```
```  2057 lemma length_nat_0 [simp]: "length_nat 0 = 0"
```
```  2058   by (simp add: length_nat_def Least_equality)
```
```  2059
```
```  2060 lemma length_nat_non0:
```
```  2061   assumes n0: "n \<noteq> 0"
```
```  2062   shows       "length_nat n = Suc (length_nat (n div 2))"
```
```  2063   apply (simp add: length_nat [symmetric])
```
```  2064   apply (subst nat_to_bv_non0 [of n])
```
```  2065   apply (simp_all add: n0)
```
```  2066   done
```
```  2067
```
```  2068 definition
```
```  2069   length_int :: "int => nat" where
```
```  2070   "length_int x =
```
```  2071     (if 0 < x then Suc (length_nat (nat x))
```
```  2072     else if x = 0 then 0
```
```  2073     else Suc (length_nat (nat (-x - 1))))"
```
```  2074
```
```  2075 lemma length_int: "length (int_to_bv i) = length_int i"
```
```  2076 proof (cases "0 < i")
```
```  2077   assume i0: "0 < i"
```
```  2078   hence "length (int_to_bv i) =
```
```  2079       length (norm_signed (\<zero> # norm_unsigned (nat_to_bv (nat i))))" by simp
```
```  2080   also from norm_unsigned_result [of "nat_to_bv (nat i)"]
```
```  2081   have "... = Suc (length_nat (nat i))"
```
```  2082     apply safe
```
```  2083     apply (simp del: norm_unsigned_nat_to_bv)
```
```  2084     apply (drule norm_empty_bv_to_nat_zero)
```
```  2085     using prems
```
```  2086     apply simp
```
```  2087     apply (cases "norm_unsigned (nat_to_bv (nat i))")
```
```  2088     apply (drule norm_empty_bv_to_nat_zero [of "nat_to_bv (nat i)"])
```
```  2089     using prems
```
```  2090     apply simp
```
```  2091     apply simp
```
```  2092     using prems
```
```  2093     apply (simp add: length_nat [symmetric])
```
```  2094     done
```
```  2095   finally show ?thesis
```
```  2096     using i0
```
```  2097     by (simp add: length_int_def)
```
```  2098 next
```
```  2099   assume "~ 0 < i"
```
```  2100   hence i0: "i \<le> 0" by simp
```
```  2101   show ?thesis
```
```  2102   proof (cases "i = 0")
```
```  2103     assume "i = 0"
```
```  2104     thus ?thesis by (simp add: length_int_def)
```
```  2105   next
```
```  2106     assume "i \<noteq> 0"
```
```  2107     with i0 have i0: "i < 0" by simp
```
```  2108     hence "length (int_to_bv i) =
```
```  2109         length (norm_signed (\<one> # bv_not (norm_unsigned (nat_to_bv (nat (- i) - 1)))))"
```
```  2110       by (simp add: int_to_bv_def nat_diff_distrib)
```
```  2111     also from norm_unsigned_result [of "nat_to_bv (nat (- i) - 1)"]
```
```  2112     have "... = Suc (length_nat (nat (- i) - 1))"
```
```  2113       apply safe
```
```  2114       apply (simp del: norm_unsigned_nat_to_bv)
```
```  2115       apply (drule norm_empty_bv_to_nat_zero [of "nat_to_bv (nat (-i) - Suc 0)"])
```
```  2116       using prems
```
```  2117       apply simp
```
```  2118       apply (cases "- i - 1 = 0")
```
```  2119       apply simp
```
```  2120       apply (simp add: length_nat [symmetric])
```
```  2121       apply (cases "norm_unsigned (nat_to_bv (nat (- i) - 1))")
```
```  2122       apply simp
```
```  2123       apply simp
```
```  2124       done
```
```  2125     finally
```
```  2126     show ?thesis
```
```  2127       using i0 by (simp add: length_int_def nat_diff_distrib del: int_to_bv_lt0)
```
```  2128   qed
```
```  2129 qed
```
```  2130
```
```  2131 lemma length_int_0 [simp]: "length_int 0 = 0"
```
```  2132   by (simp add: length_int_def)
```
```  2133
```
```  2134 lemma length_int_gt0: "0 < i ==> length_int i = Suc (length_nat (nat i))"
```
```  2135   by (simp add: length_int_def)
```
```  2136
```
```  2137 lemma length_int_lt0: "i < 0 ==> length_int i = Suc (length_nat (nat (- i) - 1))"
```
```  2138   by (simp add: length_int_def nat_diff_distrib)
```
```  2139
```
```  2140 lemma bv_chopI: "[| w = w1 @ w2 ; i = length w2 |] ==> bv_chop w i = (w1,w2)"
```
```  2141   by (simp add: bv_chop_def Let_def)
```
```  2142
```
```  2143 lemma bv_sliceI: "[| j \<le> i ; i < length w ; w = w1 @ w2 @ w3 ; Suc i = length w2 + j ; j = length w3  |] ==> bv_slice w (i,j) = w2"
```
```  2144   apply (simp add: bv_slice_def)
```
```  2145   apply (subst bv_chopI [of "w1 @ w2 @ w3" w1 "w2 @ w3"])
```
```  2146   apply simp
```
```  2147   apply simp
```
```  2148   apply simp
```
```  2149   apply (subst bv_chopI [of "w2 @ w3" w2 w3],simp_all)
```
```  2150   done
```
```  2151
```
```  2152 lemma bv_slice_bv_slice:
```
```  2153   assumes ki: "k \<le> i"
```
```  2154   and     ij: "i \<le> j"
```
```  2155   and     jl: "j \<le> l"
```
```  2156   and     lw: "l < length w"
```
```  2157   shows       "bv_slice w (j,i) = bv_slice (bv_slice w (l,k)) (j-k,i-k)"
```
```  2158 proof -
```
```  2159   def w1  == "fst (bv_chop w (Suc l))"
```
```  2160   and w2  == "fst (bv_chop (snd (bv_chop w (Suc l))) (Suc j))"
```
```  2161   and w3  == "fst (bv_chop (snd (bv_chop (snd (bv_chop w (Suc l))) (Suc j))) i)"
```
```  2162   and w4  == "fst (bv_chop (snd (bv_chop (snd (bv_chop (snd (bv_chop w (Suc l))) (Suc j))) i)) k)"
```
```  2163   and w5  == "snd (bv_chop (snd (bv_chop (snd (bv_chop (snd (bv_chop w (Suc l))) (Suc j))) i)) k)"
```
```  2164   note w_defs = this
```
```  2165
```
```  2166   have w_def: "w = w1 @ w2 @ w3 @ w4 @ w5"
```
```  2167     by (simp add: w_defs append_bv_chop_id)
```
```  2168
```
```  2169   from ki ij jl lw
```
```  2170   show ?thesis
```
```  2171     apply (subst bv_sliceI [where ?j = i and ?i = j and ?w = w and ?w1.0 = "w1 @ w2" and ?w2.0 = w3 and ?w3.0 = "w4 @ w5"])
```
```  2172     apply simp_all
```
```  2173     apply (rule w_def)
```
```  2174     apply (simp add: w_defs)
```
```  2175     apply (simp add: w_defs)
```
```  2176     apply (subst bv_sliceI [where ?j = k and ?i = l and ?w = w and ?w1.0 = w1 and ?w2.0 = "w2 @ w3 @ w4" and ?w3.0 = w5])
```
```  2177     apply simp_all
```
```  2178     apply (rule w_def)
```
```  2179     apply (simp add: w_defs)
```
```  2180     apply (simp add: w_defs)
```
```  2181     apply (subst bv_sliceI [where ?j = "i-k" and ?i = "j-k" and ?w = "w2 @ w3 @ w4" and ?w1.0 = w2 and ?w2.0 = w3 and ?w3.0 = w4])
```
```  2182     apply simp_all
```
```  2183     apply (simp_all add: w_defs)
```
```  2184     done
```
```  2185 qed
```
```  2186
```
```  2187 lemma bv_to_nat_extend [simp]: "bv_to_nat (bv_extend n \<zero> w) = bv_to_nat w"
```
```  2188   apply (simp add: bv_extend_def)
```
```  2189   apply (subst bv_to_nat_dist_append)
```
```  2190   apply simp
```
```  2191   apply (induct "n - length w")
```
```  2192    apply simp_all
```
```  2193   done
```
```  2194
```
```  2195 lemma bv_msb_extend_same [simp]: "bv_msb w = b ==> bv_msb (bv_extend n b w) = b"
```
```  2196   apply (simp add: bv_extend_def)
```
```  2197   apply (induct "n - length w")
```
```  2198    apply simp_all
```
```  2199   done
```
```  2200
```
```  2201 lemma bv_to_int_extend [simp]:
```
```  2202   assumes a: "bv_msb w = b"
```
```  2203   shows      "bv_to_int (bv_extend n b w) = bv_to_int w"
```
```  2204 proof (cases "bv_msb w")
```
```  2205   assume [simp]: "bv_msb w = \<zero>"
```
```  2206   with a have [simp]: "b = \<zero>" by simp
```
```  2207   show ?thesis by (simp add: bv_to_int_def)
```
```  2208 next
```
```  2209   assume [simp]: "bv_msb w = \<one>"
```
```  2210   with a have [simp]: "b = \<one>" by simp
```
```  2211   show ?thesis
```
```  2212     apply (simp add: bv_to_int_def)
```
```  2213     apply (simp add: bv_extend_def)
```
```  2214     apply (induct "n - length w",simp_all)
```
```  2215     done
```
```  2216 qed
```
```  2217
```
```  2218 lemma length_nat_mono [simp]: "x \<le> y ==> length_nat x \<le> length_nat y"
```
```  2219 proof (rule ccontr)
```
```  2220   assume xy: "x \<le> y"
```
```  2221   assume "~ length_nat x \<le> length_nat y"
```
```  2222   hence lxly: "length_nat y < length_nat x"
```
```  2223     by simp
```
```  2224   hence "length_nat y < (LEAST n. x < 2 ^ n)"
```
```  2225     by (simp add: length_nat_def)
```
```  2226   hence "~ x < 2 ^ length_nat y"
```
```  2227     by (rule not_less_Least)
```
```  2228   hence xx: "2 ^ length_nat y \<le> x"
```
```  2229     by simp
```
```  2230   have yy: "y < 2 ^ length_nat y"
```
```  2231     apply (simp add: length_nat_def)
```
```  2232     apply (rule LeastI)
```
```  2233     apply (subgoal_tac "y < 2 ^ y",assumption)
```
```  2234     apply (cases "0 \<le> y")
```
```  2235     apply (induct y,simp_all)
```
```  2236     done
```
```  2237   with xx have "y < x" by simp
```
```  2238   with xy show False by simp
```
```  2239 qed
```
```  2240
```
```  2241 lemma length_nat_mono_int [simp]: "x \<le> y ==> length_nat x \<le> length_nat y"
```
```  2242   by (rule length_nat_mono) arith
```
```  2243
```
```  2244 lemma length_nat_pos [simp,intro!]: "0 < x ==> 0 < length_nat x"
```
```  2245   by (simp add: length_nat_non0)
```
```  2246
```
```  2247 lemma length_int_mono_gt0: "[| 0 \<le> x ; x \<le> y |] ==> length_int x \<le> length_int y"
```
```  2248   by (cases "x = 0") (simp_all add: length_int_gt0 nat_le_eq_zle)
```
```  2249
```
```  2250 lemma length_int_mono_lt0: "[| x \<le> y ; y \<le> 0 |] ==> length_int y \<le> length_int x"
```
```  2251   by (cases "y = 0") (simp_all add: length_int_lt0)
```
```  2252
```
```  2253 lemmas [simp] = length_nat_non0
```
```  2254
```
```  2255 lemma "nat_to_bv (number_of Int.Pls) = []"
```
```  2256   by simp
```
```  2257
```
```  2258 consts
```
```  2259   fast_bv_to_nat_helper :: "[bit list, int] => int"
```
```  2260 primrec
```
```  2261   fast_bv_to_nat_Nil: "fast_bv_to_nat_helper [] k = k"
```
```  2262   fast_bv_to_nat_Cons: "fast_bv_to_nat_helper (b#bs) k =
```
```  2263     fast_bv_to_nat_helper bs ((bit_case Int.Bit0 Int.Bit1 b) k)"
```
```  2264
```
```  2265 declare fast_bv_to_nat_helper.simps [code del]
```
```  2266
```
```  2267 lemma fast_bv_to_nat_Cons0: "fast_bv_to_nat_helper (\<zero>#bs) bin =
```
```  2268     fast_bv_to_nat_helper bs (Int.Bit0 bin)"
```
```  2269   by simp
```
```  2270
```
```  2271 lemma fast_bv_to_nat_Cons1: "fast_bv_to_nat_helper (\<one>#bs) bin =
```
```  2272     fast_bv_to_nat_helper bs (Int.Bit1 bin)"
```
```  2273   by simp
```
```  2274
```
```  2275 lemma fast_bv_to_nat_def:
```
```  2276   "bv_to_nat bs == number_of (fast_bv_to_nat_helper bs Int.Pls)"
```
```  2277 proof (simp add: bv_to_nat_def)
```
```  2278   have "\<forall> bin. \<not> (neg (number_of bin :: int)) \<longrightarrow> (foldl (%bn b. 2 * bn + bitval b) (number_of bin) bs) = number_of (fast_bv_to_nat_helper bs bin)"
```
```  2279     apply (induct bs,simp)
```
```  2280     apply (case_tac a,simp_all)
```
```  2281     done
```
```  2282   thus "foldl (\<lambda>bn b. 2 * bn + bitval b) 0 bs \<equiv> number_of (fast_bv_to_nat_helper bs Int.Pls)"
```
```  2283     by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
```
```  2284 qed
```
```  2285
```
```  2286 declare fast_bv_to_nat_Cons [simp del]
```
```  2287 declare fast_bv_to_nat_Cons0 [simp]
```
```  2288 declare fast_bv_to_nat_Cons1 [simp]
```
```  2289
```
```  2290 setup {*
```
```  2291 (*comments containing lcp are the removal of fast_bv_of_nat*)
```
```  2292 let
```
```  2293   fun is_const_bool (Const("True",_)) = true
```
```  2294     | is_const_bool (Const("False",_)) = true
```
```  2295     | is_const_bool _ = false
```
```  2296   fun is_const_bit (Const("Word.bit.Zero",_)) = true
```
```  2297     | is_const_bit (Const("Word.bit.One",_)) = true
```
```  2298     | is_const_bit _ = false
```
```  2299   fun num_is_usable (Const(@{const_name Int.Pls},_)) = true
```
```  2300     | num_is_usable (Const(@{const_name Int.Min},_)) = false
```
```  2301     | num_is_usable (Const(@{const_name Int.Bit0},_) \$ w) =
```
```  2302         num_is_usable w
```
```  2303     | num_is_usable (Const(@{const_name Int.Bit1},_) \$ w) =
```
```  2304         num_is_usable w
```
```  2305     | num_is_usable _ = false
```
```  2306   fun vec_is_usable (Const("List.list.Nil",_)) = true
```
```  2307     | vec_is_usable (Const("List.list.Cons",_) \$ b \$ bs) =
```
```  2308         vec_is_usable bs andalso is_const_bit b
```
```  2309     | vec_is_usable _ = false
```
```  2310   (*lcp** val fast1_th = PureThy.get_thm thy "Word.fast_nat_to_bv_def"*)
```
```  2311   val fast2_th = @{thm "Word.fast_bv_to_nat_def"};
```
```  2312   (*lcp** fun f sg thms (Const("Word.nat_to_bv",_) \$ (Const(@{const_name Int.number_of},_) \$ t)) =
```
```  2313     if num_is_usable t
```
```  2314       then SOME (Drule.cterm_instantiate [(cterm_of sg (Var (("w", 0), @{typ int})), cterm_of sg t)] fast1_th)
```
```  2315       else NONE
```
```  2316     | f _ _ _ = NONE *)
```
```  2317   fun g sg thms (Const("Word.bv_to_nat",_) \$ (t as (Const("List.list.Cons",_) \$ _ \$ _))) =
```
```  2318         if vec_is_usable t then
```
```  2319           SOME (Drule.cterm_instantiate [(cterm_of sg (Var(("bs",0),Type("List.list",[Type("Word.bit",[])]))),cterm_of sg t)] fast2_th)
```
```  2320         else NONE
```
```  2321     | g _ _ _ = NONE
```
```  2322   (*lcp** val simproc1 = Simplifier.simproc thy "nat_to_bv" ["Word.nat_to_bv (number_of w)"] f *)
```
```  2323   val simproc2 = Simplifier.simproc @{theory} "bv_to_nat" ["Word.bv_to_nat (x # xs)"] g
```
```  2324 in
```
```  2325   Simplifier.map_simpset (fn ss => ss addsimprocs [(*lcp*simproc1,*)simproc2])
```
```  2326 end*}
```
```  2327
```
```  2328 declare bv_to_nat1 [simp del]
```
```  2329 declare bv_to_nat_helper [simp del]
```
```  2330
```
```  2331 definition
```
```  2332   bv_mapzip :: "[bit => bit => bit,bit list, bit list] => bit list" where
```
```  2333   "bv_mapzip f w1 w2 =
```
```  2334     (let g = bv_extend (max (length w1) (length w2)) \<zero>
```
```  2335      in map (split f) (zip (g w1) (g w2)))"
```
```  2336
```
```  2337 lemma bv_length_bv_mapzip [simp]:
```
```  2338     "length (bv_mapzip f w1 w2) = max (length w1) (length w2)"
```
```  2339   by (simp add: bv_mapzip_def Let_def split: split_max)
```
```  2340
```
```  2341 lemma bv_mapzip_Nil [simp]: "bv_mapzip f [] [] = []"
```
```  2342   by (simp add: bv_mapzip_def Let_def)
```
```  2343
```
```  2344 lemma bv_mapzip_Cons [simp]: "length w1 = length w2 ==>
```
```  2345     bv_mapzip f (x#w1) (y#w2) = f x y # bv_mapzip f w1 w2"
```
```  2346   by (simp add: bv_mapzip_def Let_def)
```
```  2347
```
```  2348 end
```