src/HOL/Word/Word.thy
author haftmann
Thu Oct 31 11:44:20 2013 +0100 (2013-10-31)
changeset 54221 56587960e444
parent 53080 d815e25ead03
child 54224 9fda41a04c32
permissions -rw-r--r--
more lemmas on division
haftmann@29628
     1
(*  Title:      HOL/Word/Word.thy
wenzelm@46124
     2
    Author:     Jeremy Dawson and Gerwin Klein, NICTA
kleing@24333
     3
*)
kleing@24333
     4
haftmann@37660
     5
header {* A type of finite bit strings *}
huffman@24350
     6
haftmann@29628
     7
theory Word
wenzelm@41413
     8
imports
wenzelm@41413
     9
  Type_Length
wenzelm@41413
    10
  "~~/src/HOL/Library/Boolean_Algebra"
wenzelm@41413
    11
  Bool_List_Representation
haftmann@53062
    12
  Misc_Typedef
haftmann@53062
    13
  Word_Miscellaneous
haftmann@37660
    14
begin
haftmann@37660
    15
haftmann@37660
    16
text {* see @{text "Examples/WordExamples.thy"} for examples *}
haftmann@37660
    17
haftmann@37660
    18
subsection {* Type definition *}
haftmann@37660
    19
wenzelm@49834
    20
typedef 'a word = "{(0::int) ..< 2^len_of TYPE('a::len0)}"
haftmann@37660
    21
  morphisms uint Abs_word by auto
haftmann@37660
    22
huffman@47108
    23
lemma uint_nonnegative:
huffman@47108
    24
  "0 \<le> uint w"
huffman@47108
    25
  using word.uint [of w] by simp
huffman@47108
    26
huffman@47108
    27
lemma uint_bounded:
huffman@47108
    28
  fixes w :: "'a::len0 word"
huffman@47108
    29
  shows "uint w < 2 ^ len_of TYPE('a)"
huffman@47108
    30
  using word.uint [of w] by simp
huffman@47108
    31
huffman@47108
    32
lemma uint_idem:
huffman@47108
    33
  fixes w :: "'a::len0 word"
huffman@47108
    34
  shows "uint w mod 2 ^ len_of TYPE('a) = uint w"
huffman@47108
    35
  using uint_nonnegative uint_bounded by (rule mod_pos_pos_trivial)
huffman@47108
    36
haftmann@37660
    37
definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word" where
haftmann@37660
    38
  -- {* representation of words using unsigned or signed bins, 
haftmann@37660
    39
        only difference in these is the type class *}
huffman@47108
    40
  "word_of_int k = Abs_word (k mod 2 ^ len_of TYPE('a))" 
huffman@47108
    41
huffman@47108
    42
lemma uint_word_of_int:
huffman@47108
    43
  "uint (word_of_int k :: 'a::len0 word) = k mod 2 ^ len_of TYPE('a)"
huffman@47108
    44
  by (auto simp add: word_of_int_def intro: Abs_word_inverse)
huffman@47108
    45
huffman@47108
    46
lemma word_of_int_uint:
huffman@47108
    47
  "word_of_int (uint w) = w"
huffman@47108
    48
  by (simp add: word_of_int_def uint_idem uint_inverse)
huffman@47108
    49
huffman@47108
    50
lemma word_uint_eq_iff:
huffman@47108
    51
  "a = b \<longleftrightarrow> uint a = uint b"
huffman@47108
    52
  by (simp add: uint_inject)
huffman@47108
    53
huffman@47108
    54
lemma word_uint_eqI:
huffman@47108
    55
  "uint a = uint b \<Longrightarrow> a = b"
huffman@47108
    56
  by (simp add: word_uint_eq_iff)
huffman@47108
    57
huffman@47108
    58
huffman@47108
    59
subsection {* Basic code generation setup *}
huffman@47108
    60
huffman@47108
    61
definition Word :: "int \<Rightarrow> 'a::len0 word"
huffman@47108
    62
where
huffman@47108
    63
  [code_post]: "Word = word_of_int"
huffman@47108
    64
huffman@47108
    65
lemma [code abstype]:
huffman@47108
    66
  "Word (uint w) = w"
huffman@47108
    67
  by (simp add: Word_def word_of_int_uint)
huffman@47108
    68
huffman@47108
    69
declare uint_word_of_int [code abstract]
huffman@47108
    70
huffman@47108
    71
instantiation word :: (len0) equal
huffman@47108
    72
begin
huffman@47108
    73
huffman@47108
    74
definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where
huffman@47108
    75
  "equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)"
huffman@47108
    76
huffman@47108
    77
instance proof
huffman@47108
    78
qed (simp add: equal equal_word_def word_uint_eq_iff)
huffman@47108
    79
huffman@47108
    80
end
huffman@45545
    81
haftmann@37751
    82
notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
    83
notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@37660
    84
haftmann@37660
    85
instantiation word :: ("{len0, typerep}") random
haftmann@37660
    86
begin
haftmann@37660
    87
haftmann@37660
    88
definition
huffman@47108
    89
  "random_word i = Random.range i \<circ>\<rightarrow> (\<lambda>k. Pair (
haftmann@51143
    90
     let j = word_of_int (int_of_integer (integer_of_natural k)) :: 'a word
haftmann@37660
    91
     in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))"
haftmann@37660
    92
haftmann@37660
    93
instance ..
haftmann@37660
    94
haftmann@37660
    95
end
haftmann@37660
    96
haftmann@37751
    97
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
    98
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@37660
    99
haftmann@37660
   100
haftmann@37660
   101
subsection {* Type conversions and casting *}
haftmann@37660
   102
haftmann@37660
   103
definition sint :: "'a :: len word => int" where
haftmann@37660
   104
  -- {* treats the most-significant-bit as a sign bit *}
haftmann@37660
   105
  sint_uint: "sint w = sbintrunc (len_of TYPE ('a) - 1) (uint w)"
haftmann@37660
   106
haftmann@37660
   107
definition unat :: "'a :: len0 word => nat" where
haftmann@37660
   108
  "unat w = nat (uint w)"
haftmann@37660
   109
haftmann@37660
   110
definition uints :: "nat => int set" where
haftmann@37660
   111
  -- "the sets of integers representing the words"
haftmann@37660
   112
  "uints n = range (bintrunc n)"
haftmann@37660
   113
haftmann@37660
   114
definition sints :: "nat => int set" where
haftmann@37660
   115
  "sints n = range (sbintrunc (n - 1))"
haftmann@37660
   116
haftmann@37660
   117
definition unats :: "nat => nat set" where
haftmann@37660
   118
  "unats n = {i. i < 2 ^ n}"
haftmann@37660
   119
haftmann@37660
   120
definition norm_sint :: "nat => int => int" where
haftmann@37660
   121
  "norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)"
haftmann@37660
   122
haftmann@37660
   123
definition scast :: "'a :: len word => 'b :: len word" where
haftmann@37660
   124
  -- "cast a word to a different length"
haftmann@37660
   125
  "scast w = word_of_int (sint w)"
haftmann@37660
   126
haftmann@37660
   127
definition ucast :: "'a :: len0 word => 'b :: len0 word" where
haftmann@37660
   128
  "ucast w = word_of_int (uint w)"
haftmann@37660
   129
haftmann@37660
   130
instantiation word :: (len0) size
haftmann@37660
   131
begin
haftmann@37660
   132
haftmann@37660
   133
definition
haftmann@37660
   134
  word_size: "size (w :: 'a word) = len_of TYPE('a)"
haftmann@37660
   135
haftmann@37660
   136
instance ..
haftmann@37660
   137
haftmann@37660
   138
end
haftmann@37660
   139
haftmann@37660
   140
definition source_size :: "('a :: len0 word => 'b) => nat" where
haftmann@37660
   141
  -- "whether a cast (or other) function is to a longer or shorter length"
haftmann@37660
   142
  "source_size c = (let arb = undefined ; x = c arb in size arb)"  
haftmann@37660
   143
haftmann@37660
   144
definition target_size :: "('a => 'b :: len0 word) => nat" where
haftmann@37660
   145
  "target_size c = size (c undefined)"
haftmann@37660
   146
haftmann@37660
   147
definition is_up :: "('a :: len0 word => 'b :: len0 word) => bool" where
haftmann@37660
   148
  "is_up c \<longleftrightarrow> source_size c <= target_size c"
haftmann@37660
   149
haftmann@37660
   150
definition is_down :: "('a :: len0 word => 'b :: len0 word) => bool" where
haftmann@37660
   151
  "is_down c \<longleftrightarrow> target_size c <= source_size c"
haftmann@37660
   152
haftmann@37660
   153
definition of_bl :: "bool list => 'a :: len0 word" where
haftmann@37660
   154
  "of_bl bl = word_of_int (bl_to_bin bl)"
haftmann@37660
   155
haftmann@37660
   156
definition to_bl :: "'a :: len0 word => bool list" where
haftmann@37660
   157
  "to_bl w = bin_to_bl (len_of TYPE ('a)) (uint w)"
haftmann@37660
   158
haftmann@37660
   159
definition word_reverse :: "'a :: len0 word => 'a word" where
haftmann@37660
   160
  "word_reverse w = of_bl (rev (to_bl w))"
haftmann@37660
   161
haftmann@37660
   162
definition word_int_case :: "(int => 'b) => ('a :: len0 word) => 'b" where
haftmann@37660
   163
  "word_int_case f w = f (uint w)"
haftmann@37660
   164
haftmann@37660
   165
translations
wenzelm@46136
   166
  "case x of XCONST of_int y => b" == "CONST word_int_case (%y. b) x"
wenzelm@46136
   167
  "case x of (XCONST of_int :: 'a) y => b" => "CONST word_int_case (%y. b) x"
haftmann@37660
   168
huffman@45545
   169
subsection {* Type-definition locale instantiations *}
huffman@45545
   170
huffman@45805
   171
lemma word_size_gt_0 [iff]: "0 < size (w::'a::len word)"
huffman@45805
   172
  by (fact xtr1 [OF word_size len_gt_0])
huffman@45805
   173
huffman@45545
   174
lemmas lens_gt_0 = word_size_gt_0 len_gt_0
wenzelm@45604
   175
lemmas lens_not_0 [iff] = lens_gt_0 [THEN gr_implies_not0]
huffman@45545
   176
huffman@45545
   177
lemma uints_num: "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}"
huffman@45545
   178
  by (simp add: uints_def range_bintrunc)
huffman@45545
   179
huffman@45545
   180
lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}"
huffman@45545
   181
  by (simp add: sints_def range_sbintrunc)
huffman@45545
   182
huffman@45545
   183
lemma 
huffman@45545
   184
  uint_0:"0 <= uint x" and 
huffman@45545
   185
  uint_lt: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
huffman@45816
   186
  by (auto simp: uint [unfolded atLeastLessThan_iff])
huffman@45545
   187
huffman@45545
   188
lemma uint_mod_same:
huffman@45545
   189
  "uint x mod 2 ^ len_of TYPE('a) = uint (x::'a::len0 word)"
huffman@45545
   190
  by (simp add: int_mod_eq uint_lt uint_0)
huffman@45545
   191
huffman@45545
   192
lemma td_ext_uint: 
huffman@45545
   193
  "td_ext (uint :: 'a word => int) word_of_int (uints (len_of TYPE('a::len0))) 
huffman@45545
   194
    (%w::int. w mod 2 ^ len_of TYPE('a))"
huffman@45545
   195
  apply (unfold td_ext_def')
huffman@45545
   196
  apply (simp add: uints_num word_of_int_def bintrunc_mod2p)
huffman@45545
   197
  apply (simp add: uint_mod_same uint_0 uint_lt
huffman@45545
   198
                   word.uint_inverse word.Abs_word_inverse int_mod_lem)
huffman@45545
   199
  done
huffman@45545
   200
huffman@45545
   201
interpretation word_uint:
huffman@45545
   202
  td_ext "uint::'a::len0 word \<Rightarrow> int" 
huffman@45545
   203
         word_of_int 
huffman@45545
   204
         "uints (len_of TYPE('a::len0))"
huffman@45545
   205
         "\<lambda>w. w mod 2 ^ len_of TYPE('a::len0)"
huffman@45545
   206
  by (rule td_ext_uint)
huffman@46013
   207
huffman@45545
   208
lemmas td_uint = word_uint.td_thm
huffman@45545
   209
huffman@46013
   210
lemmas int_word_uint = word_uint.eq_norm
huffman@46013
   211
huffman@45545
   212
lemmas td_ext_ubin = td_ext_uint 
huffman@45816
   213
  [unfolded len_gt_0 no_bintr_alt1 [symmetric]]
huffman@45545
   214
huffman@45545
   215
interpretation word_ubin:
huffman@45545
   216
  td_ext "uint::'a::len0 word \<Rightarrow> int" 
huffman@45545
   217
         word_of_int 
huffman@45545
   218
         "uints (len_of TYPE('a::len0))"
huffman@45545
   219
         "bintrunc (len_of TYPE('a::len0))"
huffman@45545
   220
  by (rule td_ext_ubin)
huffman@45545
   221
huffman@45545
   222
lemma split_word_all:
huffman@45545
   223
  "(\<And>x::'a::len0 word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))"
huffman@45545
   224
proof
huffman@45545
   225
  fix x :: "'a word"
huffman@45545
   226
  assume "\<And>x. PROP P (word_of_int x)"
huffman@45545
   227
  hence "PROP P (word_of_int (uint x))" .
huffman@45545
   228
  thus "PROP P x" by simp
huffman@45545
   229
qed
haftmann@37660
   230
huffman@47372
   231
subsection {* Correspondence relation for theorem transfer *}
huffman@47372
   232
huffman@47372
   233
definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool"
huffman@47372
   234
  where "cr_word \<equiv> (\<lambda>x y. word_of_int x = y)"
huffman@47372
   235
huffman@47374
   236
lemma Quotient_word:
huffman@47374
   237
  "Quotient (\<lambda>x y. bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y)
huffman@47374
   238
    word_of_int uint (cr_word :: _ \<Rightarrow> 'a::len0 word \<Rightarrow> bool)"
huffman@47374
   239
  unfolding Quotient_alt_def cr_word_def
huffman@47374
   240
  by (simp add: word_ubin.norm_eq_iff)
huffman@47374
   241
huffman@47377
   242
lemma reflp_word:
huffman@47377
   243
  "reflp (\<lambda>x y. bintrunc (len_of TYPE('a::len0)) x = bintrunc (len_of TYPE('a)) y)"
huffman@47377
   244
  by (simp add: reflp_def)
huffman@47374
   245
kuncar@47941
   246
setup_lifting(no_code) Quotient_word reflp_word
huffman@47374
   247
kuncar@47521
   248
text {* TODO: The next lemma could be generated automatically. *}
huffman@47372
   249
huffman@47372
   250
lemma uint_transfer [transfer_rule]:
kuncar@51375
   251
  "(fun_rel pcr_word op =) (bintrunc (len_of TYPE('a)))
huffman@47372
   252
    (uint :: 'a::len0 word \<Rightarrow> int)"
kuncar@51375
   253
  unfolding fun_rel_def word.pcr_cr_eq cr_word_def by (simp add: word_ubin.eq_norm)
huffman@47372
   254
haftmann@37660
   255
subsection  "Arithmetic operations"
haftmann@37660
   256
huffman@47387
   257
lift_definition word_succ :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x + 1"
huffman@47374
   258
  by (metis bintr_ariths(6))
huffman@47374
   259
huffman@47387
   260
lift_definition word_pred :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x - 1"
huffman@47374
   261
  by (metis bintr_ariths(7))
huffman@45545
   262
huffman@47108
   263
instantiation word :: (len0) "{neg_numeral, Divides.div, comm_monoid_mult, comm_ring}"
haftmann@37660
   264
begin
haftmann@37660
   265
huffman@47387
   266
lift_definition zero_word :: "'a word" is "0" .
huffman@47387
   267
huffman@47387
   268
lift_definition one_word :: "'a word" is "1" .
huffman@47387
   269
huffman@47387
   270
lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op +"
huffman@47374
   271
  by (metis bintr_ariths(2))
huffman@47374
   272
huffman@47387
   273
lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op -"
huffman@47374
   274
  by (metis bintr_ariths(3))
huffman@47374
   275
huffman@47387
   276
lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" is uminus
huffman@47374
   277
  by (metis bintr_ariths(5))
huffman@47374
   278
huffman@47387
   279
lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op *"
huffman@47374
   280
  by (metis bintr_ariths(4))
haftmann@37660
   281
haftmann@37660
   282
definition
haftmann@37660
   283
  word_div_def: "a div b = word_of_int (uint a div uint b)"
haftmann@37660
   284
haftmann@37660
   285
definition
haftmann@37660
   286
  word_mod_def: "a mod b = word_of_int (uint a mod uint b)"
haftmann@37660
   287
huffman@47374
   288
instance
huffman@47374
   289
  by default (transfer, simp add: algebra_simps)+
huffman@47374
   290
huffman@47374
   291
end
huffman@47374
   292
huffman@47374
   293
text {* Legacy theorems: *}
huffman@47374
   294
huffman@47611
   295
lemma word_arith_wis [code]: shows
huffman@47374
   296
  word_add_def: "a + b = word_of_int (uint a + uint b)" and
huffman@47374
   297
  word_sub_wi: "a - b = word_of_int (uint a - uint b)" and
huffman@47374
   298
  word_mult_def: "a * b = word_of_int (uint a * uint b)" and
huffman@47374
   299
  word_minus_def: "- a = word_of_int (- uint a)" and
huffman@47374
   300
  word_succ_alt: "word_succ a = word_of_int (uint a + 1)" and
huffman@47374
   301
  word_pred_alt: "word_pred a = word_of_int (uint a - 1)" and
huffman@47374
   302
  word_0_wi: "0 = word_of_int 0" and
huffman@47374
   303
  word_1_wi: "1 = word_of_int 1"
huffman@47374
   304
  unfolding plus_word_def minus_word_def times_word_def uminus_word_def
huffman@47374
   305
  unfolding word_succ_def word_pred_def zero_word_def one_word_def
huffman@47374
   306
  by simp_all
huffman@45545
   307
huffman@45545
   308
lemmas arths = 
wenzelm@45604
   309
  bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm]
huffman@45545
   310
huffman@45545
   311
lemma wi_homs: 
huffman@45545
   312
  shows
huffman@45545
   313
  wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and
huffman@46013
   314
  wi_hom_sub: "word_of_int a - word_of_int b = word_of_int (a - b)" and
huffman@45545
   315
  wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and
huffman@45545
   316
  wi_hom_neg: "- word_of_int a = word_of_int (- a)" and
huffman@46000
   317
  wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" and
huffman@46000
   318
  wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)"
huffman@47374
   319
  by (transfer, simp)+
huffman@45545
   320
huffman@45545
   321
lemmas wi_hom_syms = wi_homs [symmetric]
huffman@45545
   322
huffman@46013
   323
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi
huffman@46009
   324
huffman@46009
   325
lemmas word_of_int_hom_syms = word_of_int_homs [symmetric]
huffman@45545
   326
huffman@45545
   327
instance word :: (len) comm_ring_1
huffman@45810
   328
proof
huffman@45810
   329
  have "0 < len_of TYPE('a)" by (rule len_gt_0)
huffman@45810
   330
  then show "(0::'a word) \<noteq> 1"
huffman@47372
   331
    by - (transfer, auto simp add: gr0_conv_Suc)
huffman@45810
   332
qed
huffman@45545
   333
huffman@45545
   334
lemma word_of_nat: "of_nat n = word_of_int (int n)"
huffman@45545
   335
  by (induct n) (auto simp add : word_of_int_hom_syms)
huffman@45545
   336
huffman@45545
   337
lemma word_of_int: "of_int = word_of_int"
huffman@45545
   338
  apply (rule ext)
huffman@45545
   339
  apply (case_tac x rule: int_diff_cases)
huffman@46013
   340
  apply (simp add: word_of_nat wi_hom_sub)
huffman@45545
   341
  done
huffman@45545
   342
haftmann@37660
   343
definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where
haftmann@40827
   344
  "a udvd b = (EX n>=0. uint b = n * uint a)"
haftmann@37660
   345
huffman@45547
   346
huffman@45547
   347
subsection "Ordering"
huffman@45547
   348
huffman@45547
   349
instantiation word :: (len0) linorder
huffman@45547
   350
begin
huffman@45547
   351
haftmann@37660
   352
definition
haftmann@37660
   353
  word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
haftmann@37660
   354
haftmann@37660
   355
definition
huffman@47108
   356
  word_less_def: "a < b \<longleftrightarrow> uint a < uint b"
haftmann@37660
   357
huffman@45547
   358
instance
huffman@45547
   359
  by default (auto simp: word_less_def word_le_def)
huffman@45547
   360
huffman@45547
   361
end
huffman@45547
   362
haftmann@37660
   363
definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) where
haftmann@40827
   364
  "a <=s b = (sint a <= sint b)"
haftmann@37660
   365
haftmann@37660
   366
definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) where
haftmann@40827
   367
  "(x <s y) = (x <=s y & x ~= y)"
haftmann@37660
   368
haftmann@37660
   369
haftmann@37660
   370
subsection "Bit-wise operations"
haftmann@37660
   371
haftmann@37660
   372
instantiation word :: (len0) bits
haftmann@37660
   373
begin
haftmann@37660
   374
huffman@47387
   375
lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT
huffman@47374
   376
  by (metis bin_trunc_not)
huffman@47374
   377
huffman@47387
   378
lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND
huffman@47374
   379
  by (metis bin_trunc_and)
huffman@47374
   380
huffman@47387
   381
lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR
huffman@47374
   382
  by (metis bin_trunc_or)
huffman@47374
   383
huffman@47387
   384
lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR
huffman@47374
   385
  by (metis bin_trunc_xor)
haftmann@37660
   386
haftmann@37660
   387
definition
haftmann@37660
   388
  word_test_bit_def: "test_bit a = bin_nth (uint a)"
haftmann@37660
   389
haftmann@37660
   390
definition
haftmann@37660
   391
  word_set_bit_def: "set_bit a n x =
haftmann@37660
   392
   word_of_int (bin_sc n (If x 1 0) (uint a))"
haftmann@37660
   393
haftmann@37660
   394
definition
haftmann@37660
   395
  word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)"
haftmann@37660
   396
haftmann@37660
   397
definition
haftmann@37660
   398
  word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = 1"
haftmann@37660
   399
haftmann@37660
   400
definition shiftl1 :: "'a word \<Rightarrow> 'a word" where
haftmann@37660
   401
  "shiftl1 w = word_of_int (uint w BIT 0)"
haftmann@37660
   402
haftmann@37660
   403
definition shiftr1 :: "'a word \<Rightarrow> 'a word" where
haftmann@37660
   404
  -- "shift right as unsigned or as signed, ie logical or arithmetic"
haftmann@37660
   405
  "shiftr1 w = word_of_int (bin_rest (uint w))"
haftmann@37660
   406
haftmann@37660
   407
definition
haftmann@37660
   408
  shiftl_def: "w << n = (shiftl1 ^^ n) w"
haftmann@37660
   409
haftmann@37660
   410
definition
haftmann@37660
   411
  shiftr_def: "w >> n = (shiftr1 ^^ n) w"
haftmann@37660
   412
haftmann@37660
   413
instance ..
haftmann@37660
   414
haftmann@37660
   415
end
haftmann@37660
   416
huffman@47611
   417
lemma [code]: shows
huffman@47374
   418
  word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))" and
huffman@47374
   419
  word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)" and
huffman@47374
   420
  word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)" and
huffman@47374
   421
  word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
huffman@47374
   422
  unfolding bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def
huffman@47374
   423
  by simp_all
huffman@47374
   424
haftmann@37660
   425
instantiation word :: (len) bitss
haftmann@37660
   426
begin
haftmann@37660
   427
haftmann@37660
   428
definition
haftmann@37660
   429
  word_msb_def: 
huffman@46001
   430
  "msb a \<longleftrightarrow> bin_sign (sint a) = -1"
haftmann@37660
   431
haftmann@37660
   432
instance ..
haftmann@37660
   433
haftmann@37660
   434
end
haftmann@37660
   435
haftmann@37660
   436
definition setBit :: "'a :: len0 word => nat => 'a word" where 
haftmann@40827
   437
  "setBit w n = set_bit w n True"
haftmann@37660
   438
haftmann@37660
   439
definition clearBit :: "'a :: len0 word => nat => 'a word" where
haftmann@40827
   440
  "clearBit w n = set_bit w n False"
haftmann@37660
   441
haftmann@37660
   442
haftmann@37660
   443
subsection "Shift operations"
haftmann@37660
   444
haftmann@37660
   445
definition sshiftr1 :: "'a :: len word => 'a word" where 
haftmann@40827
   446
  "sshiftr1 w = word_of_int (bin_rest (sint w))"
haftmann@37660
   447
haftmann@37660
   448
definition bshiftr1 :: "bool => 'a :: len word => 'a word" where
haftmann@40827
   449
  "bshiftr1 b w = of_bl (b # butlast (to_bl w))"
haftmann@37660
   450
haftmann@37660
   451
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where
haftmann@40827
   452
  "w >>> n = (sshiftr1 ^^ n) w"
haftmann@37660
   453
haftmann@37660
   454
definition mask :: "nat => 'a::len word" where
haftmann@40827
   455
  "mask n = (1 << n) - 1"
haftmann@37660
   456
haftmann@37660
   457
definition revcast :: "'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   458
  "revcast w =  of_bl (takefill False (len_of TYPE('b)) (to_bl w))"
haftmann@37660
   459
haftmann@37660
   460
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   461
  "slice1 n w = of_bl (takefill False n (to_bl w))"
haftmann@37660
   462
haftmann@37660
   463
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   464
  "slice n w = slice1 (size w - n) w"
haftmann@37660
   465
haftmann@37660
   466
haftmann@37660
   467
subsection "Rotation"
haftmann@37660
   468
haftmann@37660
   469
definition rotater1 :: "'a list => 'a list" where
haftmann@40827
   470
  "rotater1 ys = 
haftmann@40827
   471
    (case ys of [] => [] | x # xs => last ys # butlast ys)"
haftmann@37660
   472
haftmann@37660
   473
definition rotater :: "nat => 'a list => 'a list" where
haftmann@40827
   474
  "rotater n = rotater1 ^^ n"
haftmann@37660
   475
haftmann@37660
   476
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   477
  "word_rotr n w = of_bl (rotater n (to_bl w))"
haftmann@37660
   478
haftmann@37660
   479
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   480
  "word_rotl n w = of_bl (rotate n (to_bl w))"
haftmann@37660
   481
haftmann@37660
   482
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   483
  "word_roti i w = (if i >= 0 then word_rotr (nat i) w
haftmann@40827
   484
                    else word_rotl (nat (- i)) w)"
haftmann@37660
   485
haftmann@37660
   486
haftmann@37660
   487
subsection "Split and cat operations"
haftmann@37660
   488
haftmann@37660
   489
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where
haftmann@40827
   490
  "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))"
haftmann@37660
   491
haftmann@37660
   492
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where
haftmann@40827
   493
  "word_split a = 
haftmann@40827
   494
   (case bin_split (len_of TYPE ('c)) (uint a) of 
haftmann@40827
   495
     (u, v) => (word_of_int u, word_of_int v))"
haftmann@37660
   496
haftmann@37660
   497
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where
haftmann@40827
   498
  "word_rcat ws = 
haftmann@37660
   499
  word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"
haftmann@37660
   500
haftmann@37660
   501
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where
haftmann@40827
   502
  "word_rsplit w = 
haftmann@37660
   503
  map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"
haftmann@37660
   504
haftmann@37660
   505
definition max_word :: "'a::len word" -- "Largest representable machine integer." where
haftmann@40827
   506
  "max_word = word_of_int (2 ^ len_of TYPE('a) - 1)"
haftmann@37660
   507
haftmann@37660
   508
primrec of_bool :: "bool \<Rightarrow> 'a::len word" where
haftmann@37660
   509
  "of_bool False = 0"
haftmann@37660
   510
| "of_bool True = 1"
haftmann@37660
   511
huffman@45805
   512
(* FIXME: only provide one theorem name *)
haftmann@37660
   513
lemmas of_nth_def = word_set_bits_def
haftmann@37660
   514
huffman@46010
   515
subsection {* Theorems about typedefs *}
huffman@46010
   516
haftmann@37660
   517
lemma sint_sbintrunc': 
haftmann@37660
   518
  "sint (word_of_int bin :: 'a word) = 
haftmann@37660
   519
    (sbintrunc (len_of TYPE ('a :: len) - 1) bin)"
haftmann@37660
   520
  unfolding sint_uint 
haftmann@37660
   521
  by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt)
haftmann@37660
   522
haftmann@37660
   523
lemma uint_sint: 
haftmann@37660
   524
  "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))"
haftmann@37660
   525
  unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le)
haftmann@37660
   526
huffman@46057
   527
lemma bintr_uint:
huffman@46057
   528
  fixes w :: "'a::len0 word"
huffman@46057
   529
  shows "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w"
haftmann@37660
   530
  apply (subst word_ubin.norm_Rep [symmetric]) 
haftmann@37660
   531
  apply (simp only: bintrunc_bintrunc_min word_size)
haftmann@37660
   532
  apply (simp add: min_max.inf_absorb2)
haftmann@37660
   533
  done
haftmann@37660
   534
huffman@46057
   535
lemma wi_bintr:
huffman@46057
   536
  "len_of TYPE('a::len0) \<le> n \<Longrightarrow>
huffman@46057
   537
    word_of_int (bintrunc n w) = (word_of_int w :: 'a word)"
haftmann@37660
   538
  by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1)
haftmann@37660
   539
haftmann@37660
   540
lemma td_ext_sbin: 
haftmann@37660
   541
  "td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) 
haftmann@37660
   542
    (sbintrunc (len_of TYPE('a) - 1))"
haftmann@37660
   543
  apply (unfold td_ext_def' sint_uint)
haftmann@37660
   544
  apply (simp add : word_ubin.eq_norm)
haftmann@37660
   545
  apply (cases "len_of TYPE('a)")
haftmann@37660
   546
   apply (auto simp add : sints_def)
haftmann@37660
   547
  apply (rule sym [THEN trans])
haftmann@37660
   548
  apply (rule word_ubin.Abs_norm)
haftmann@37660
   549
  apply (simp only: bintrunc_sbintrunc)
haftmann@37660
   550
  apply (drule sym)
haftmann@37660
   551
  apply simp
haftmann@37660
   552
  done
haftmann@37660
   553
haftmann@37660
   554
lemmas td_ext_sint = td_ext_sbin 
haftmann@37660
   555
  [simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]]
haftmann@37660
   556
haftmann@37660
   557
(* We do sint before sbin, before sint is the user version
haftmann@37660
   558
   and interpretations do not produce thm duplicates. I.e. 
haftmann@37660
   559
   we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD,
haftmann@37660
   560
   because the latter is the same thm as the former *)
haftmann@37660
   561
interpretation word_sint:
haftmann@37660
   562
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   563
          word_of_int 
haftmann@37660
   564
          "sints (len_of TYPE('a::len))"
haftmann@37660
   565
          "%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) -
haftmann@37660
   566
               2 ^ (len_of TYPE('a::len) - 1)"
haftmann@37660
   567
  by (rule td_ext_sint)
haftmann@37660
   568
haftmann@37660
   569
interpretation word_sbin:
haftmann@37660
   570
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   571
          word_of_int 
haftmann@37660
   572
          "sints (len_of TYPE('a::len))"
haftmann@37660
   573
          "sbintrunc (len_of TYPE('a::len) - 1)"
haftmann@37660
   574
  by (rule td_ext_sbin)
haftmann@37660
   575
wenzelm@45604
   576
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
haftmann@37660
   577
haftmann@37660
   578
lemmas td_sint = word_sint.td
haftmann@37660
   579
haftmann@37660
   580
lemma to_bl_def': 
haftmann@37660
   581
  "(to_bl :: 'a :: len0 word => bool list) =
haftmann@37660
   582
    bin_to_bl (len_of TYPE('a)) o uint"
wenzelm@44762
   583
  by (auto simp: to_bl_def)
haftmann@37660
   584
huffman@47108
   585
lemmas word_reverse_no_def [simp] = word_reverse_def [of "numeral w"] for w
haftmann@37660
   586
huffman@45805
   587
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
huffman@45805
   588
  by (fact uints_def [unfolded no_bintr_alt1])
huffman@45805
   589
huffman@47108
   590
lemma word_numeral_alt:
huffman@47108
   591
  "numeral b = word_of_int (numeral b)"
huffman@47108
   592
  by (induct b, simp_all only: numeral.simps word_of_int_homs)
huffman@47108
   593
huffman@47108
   594
declare word_numeral_alt [symmetric, code_abbrev]
huffman@47108
   595
huffman@47108
   596
lemma word_neg_numeral_alt:
huffman@47108
   597
  "neg_numeral b = word_of_int (neg_numeral b)"
huffman@47108
   598
  by (simp only: neg_numeral_def word_numeral_alt wi_hom_neg)
huffman@47108
   599
huffman@47108
   600
declare word_neg_numeral_alt [symmetric, code_abbrev]
huffman@47108
   601
huffman@47372
   602
lemma word_numeral_transfer [transfer_rule]:
kuncar@51375
   603
  "(fun_rel op = pcr_word) numeral numeral"
kuncar@51375
   604
  "(fun_rel op = pcr_word) neg_numeral neg_numeral"
kuncar@51375
   605
  unfolding fun_rel_def word.pcr_cr_eq cr_word_def word_numeral_alt word_neg_numeral_alt
huffman@47372
   606
  by simp_all
huffman@47372
   607
huffman@45805
   608
lemma uint_bintrunc [simp]:
huffman@47108
   609
  "uint (numeral bin :: 'a word) = 
huffman@47108
   610
    bintrunc (len_of TYPE ('a :: len0)) (numeral bin)"
huffman@47108
   611
  unfolding word_numeral_alt by (rule word_ubin.eq_norm)
huffman@47108
   612
huffman@47108
   613
lemma uint_bintrunc_neg [simp]: "uint (neg_numeral bin :: 'a word) = 
huffman@47108
   614
    bintrunc (len_of TYPE ('a :: len0)) (neg_numeral bin)"
huffman@47108
   615
  by (simp only: word_neg_numeral_alt word_ubin.eq_norm)
haftmann@37660
   616
huffman@45805
   617
lemma sint_sbintrunc [simp]:
huffman@47108
   618
  "sint (numeral bin :: 'a word) = 
huffman@47108
   619
    sbintrunc (len_of TYPE ('a :: len) - 1) (numeral bin)"
huffman@47108
   620
  by (simp only: word_numeral_alt word_sbin.eq_norm)
huffman@47108
   621
huffman@47108
   622
lemma sint_sbintrunc_neg [simp]: "sint (neg_numeral bin :: 'a word) = 
huffman@47108
   623
    sbintrunc (len_of TYPE ('a :: len) - 1) (neg_numeral bin)"
huffman@47108
   624
  by (simp only: word_neg_numeral_alt word_sbin.eq_norm)
haftmann@37660
   625
huffman@45805
   626
lemma unat_bintrunc [simp]:
huffman@47108
   627
  "unat (numeral bin :: 'a :: len0 word) =
huffman@47108
   628
    nat (bintrunc (len_of TYPE('a)) (numeral bin))"
huffman@47108
   629
  by (simp only: unat_def uint_bintrunc)
huffman@47108
   630
huffman@47108
   631
lemma unat_bintrunc_neg [simp]:
huffman@47108
   632
  "unat (neg_numeral bin :: 'a :: len0 word) =
huffman@47108
   633
    nat (bintrunc (len_of TYPE('a)) (neg_numeral bin))"
huffman@47108
   634
  by (simp only: unat_def uint_bintrunc_neg)
haftmann@37660
   635
haftmann@40827
   636
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w"
haftmann@37660
   637
  apply (unfold word_size)
haftmann@37660
   638
  apply (rule word_uint.Rep_eqD)
haftmann@37660
   639
  apply (rule box_equals)
haftmann@37660
   640
    defer
haftmann@37660
   641
    apply (rule word_ubin.norm_Rep)+
haftmann@37660
   642
  apply simp
haftmann@37660
   643
  done
haftmann@37660
   644
huffman@45805
   645
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)"
huffman@45805
   646
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   647
huffman@45805
   648
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
huffman@45805
   649
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   650
huffman@45805
   651
lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint (x::'a::len word)"
huffman@45805
   652
  using word_sint.Rep [of x] by (simp add: sints_num)
huffman@45805
   653
huffman@45805
   654
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a) - 1)"
huffman@45805
   655
  using word_sint.Rep [of x] by (simp add: sints_num)
haftmann@37660
   656
haftmann@37660
   657
lemma sign_uint_Pls [simp]: 
huffman@46604
   658
  "bin_sign (uint x) = 0"
huffman@47108
   659
  by (simp add: sign_Pls_ge_0)
haftmann@37660
   660
huffman@45805
   661
lemma uint_m2p_neg: "uint (x::'a::len0 word) - 2 ^ len_of TYPE('a) < 0"
huffman@45805
   662
  by (simp only: diff_less_0_iff_less uint_lt2p)
huffman@45805
   663
huffman@45805
   664
lemma uint_m2p_not_non_neg:
huffman@45805
   665
  "\<not> 0 \<le> uint (x::'a::len0 word) - 2 ^ len_of TYPE('a)"
huffman@45805
   666
  by (simp only: not_le uint_m2p_neg)
haftmann@37660
   667
haftmann@37660
   668
lemma lt2p_lem:
haftmann@40827
   669
  "len_of TYPE('a) <= n \<Longrightarrow> uint (w :: 'a :: len0 word) < 2 ^ n"
haftmann@37660
   670
  by (rule xtr8 [OF _ uint_lt2p]) simp
haftmann@37660
   671
huffman@45805
   672
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
huffman@45805
   673
  by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1])
haftmann@37660
   674
haftmann@40827
   675
lemma uint_nat: "uint w = int (unat w)"
haftmann@37660
   676
  unfolding unat_def by auto
haftmann@37660
   677
huffman@47108
   678
lemma uint_numeral:
huffman@47108
   679
  "uint (numeral b :: 'a :: len0 word) = numeral b mod 2 ^ len_of TYPE('a)"
huffman@47108
   680
  unfolding word_numeral_alt
haftmann@37660
   681
  by (simp only: int_word_uint)
haftmann@37660
   682
huffman@47108
   683
lemma uint_neg_numeral:
huffman@47108
   684
  "uint (neg_numeral b :: 'a :: len0 word) = neg_numeral b mod 2 ^ len_of TYPE('a)"
huffman@47108
   685
  unfolding word_neg_numeral_alt
huffman@47108
   686
  by (simp only: int_word_uint)
huffman@47108
   687
huffman@47108
   688
lemma unat_numeral: 
huffman@47108
   689
  "unat (numeral b::'a::len0 word) = numeral b mod 2 ^ len_of TYPE ('a)"
haftmann@37660
   690
  apply (unfold unat_def)
huffman@47108
   691
  apply (clarsimp simp only: uint_numeral)
haftmann@37660
   692
  apply (rule nat_mod_distrib [THEN trans])
huffman@47108
   693
    apply (rule zero_le_numeral)
haftmann@37660
   694
   apply (simp_all add: nat_power_eq)
haftmann@37660
   695
  done
haftmann@37660
   696
huffman@47108
   697
lemma sint_numeral: "sint (numeral b :: 'a :: len word) = (numeral b + 
haftmann@37660
   698
    2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
haftmann@37660
   699
    2 ^ (len_of TYPE('a) - 1)"
huffman@47108
   700
  unfolding word_numeral_alt by (rule int_word_sint)
huffman@47108
   701
huffman@47108
   702
lemma word_of_int_0 [simp, code_post]: "word_of_int 0 = 0"
huffman@45958
   703
  unfolding word_0_wi ..
huffman@45958
   704
huffman@47108
   705
lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1"
huffman@45958
   706
  unfolding word_1_wi ..
huffman@45958
   707
huffman@47108
   708
lemma word_of_int_numeral [simp] : 
huffman@47108
   709
  "(word_of_int (numeral bin) :: 'a :: len0 word) = (numeral bin)"
huffman@47108
   710
  unfolding word_numeral_alt ..
huffman@47108
   711
huffman@47108
   712
lemma word_of_int_neg_numeral [simp]:
huffman@47108
   713
  "(word_of_int (neg_numeral bin) :: 'a :: len0 word) = (neg_numeral bin)"
huffman@47108
   714
  unfolding neg_numeral_def word_numeral_alt wi_hom_syms ..
haftmann@37660
   715
haftmann@37660
   716
lemma word_int_case_wi: 
haftmann@37660
   717
  "word_int_case f (word_of_int i :: 'b word) = 
haftmann@37660
   718
    f (i mod 2 ^ len_of TYPE('b::len0))"
haftmann@37660
   719
  unfolding word_int_case_def by (simp add: word_uint.eq_norm)
haftmann@37660
   720
haftmann@37660
   721
lemma word_int_split: 
haftmann@37660
   722
  "P (word_int_case f x) = 
haftmann@37660
   723
    (ALL i. x = (word_of_int i :: 'b :: len0 word) & 
haftmann@37660
   724
      0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))"
haftmann@37660
   725
  unfolding word_int_case_def
haftmann@37660
   726
  by (auto simp: word_uint.eq_norm int_mod_eq')
haftmann@37660
   727
haftmann@37660
   728
lemma word_int_split_asm: 
haftmann@37660
   729
  "P (word_int_case f x) = 
haftmann@37660
   730
    (~ (EX n. x = (word_of_int n :: 'b::len0 word) &
haftmann@37660
   731
      0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))"
haftmann@37660
   732
  unfolding word_int_case_def
haftmann@37660
   733
  by (auto simp: word_uint.eq_norm int_mod_eq')
huffman@45805
   734
wenzelm@45604
   735
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
wenzelm@45604
   736
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]
haftmann@37660
   737
haftmann@37660
   738
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w"
haftmann@37660
   739
  unfolding word_size by (rule uint_range')
haftmann@37660
   740
haftmann@37660
   741
lemma sint_range_size:
haftmann@37660
   742
  "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)"
haftmann@37660
   743
  unfolding word_size by (rule sint_range')
haftmann@37660
   744
huffman@45805
   745
lemma sint_above_size: "2 ^ (size (w::'a::len word) - 1) \<le> x \<Longrightarrow> sint w < x"
huffman@45805
   746
  unfolding word_size by (rule less_le_trans [OF sint_lt])
huffman@45805
   747
huffman@45805
   748
lemma sint_below_size:
huffman@45805
   749
  "x \<le> - (2 ^ (size (w::'a::len word) - 1)) \<Longrightarrow> x \<le> sint w"
huffman@45805
   750
  unfolding word_size by (rule order_trans [OF _ sint_ge])
haftmann@37660
   751
huffman@46010
   752
subsection {* Testing bits *}
huffman@46010
   753
haftmann@37660
   754
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)"
haftmann@37660
   755
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)
haftmann@37660
   756
haftmann@37660
   757
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w"
haftmann@37660
   758
  apply (unfold word_test_bit_def)
haftmann@37660
   759
  apply (subst word_ubin.norm_Rep [symmetric])
haftmann@37660
   760
  apply (simp only: nth_bintr word_size)
haftmann@37660
   761
  apply fast
haftmann@37660
   762
  done
haftmann@37660
   763
huffman@46021
   764
lemma word_eq_iff:
huffman@46021
   765
  fixes x y :: "'a::len0 word"
huffman@46021
   766
  shows "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)"
huffman@46021
   767
  unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric]
huffman@46021
   768
  by (metis test_bit_size [unfolded word_size])
huffman@46021
   769
huffman@46023
   770
lemma word_eqI [rule_format]:
haftmann@37660
   771
  fixes u :: "'a::len0 word"
haftmann@40827
   772
  shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v"
huffman@46021
   773
  by (simp add: word_size word_eq_iff)
haftmann@37660
   774
huffman@45805
   775
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x"
huffman@45805
   776
  by simp
haftmann@37660
   777
haftmann@37660
   778
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)"
haftmann@37660
   779
  unfolding word_test_bit_def word_size
haftmann@37660
   780
  by (simp add: nth_bintr [symmetric])
haftmann@37660
   781
haftmann@37660
   782
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
haftmann@37660
   783
huffman@46057
   784
lemma bin_nth_uint_imp:
huffman@46057
   785
  "bin_nth (uint (w::'a::len0 word)) n \<Longrightarrow> n < len_of TYPE('a)"
haftmann@37660
   786
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
haftmann@37660
   787
  apply (subst word_ubin.norm_Rep)
haftmann@37660
   788
  apply assumption
haftmann@37660
   789
  done
haftmann@37660
   790
huffman@46057
   791
lemma bin_nth_sint:
huffman@46057
   792
  fixes w :: "'a::len word"
huffman@46057
   793
  shows "len_of TYPE('a) \<le> n \<Longrightarrow>
huffman@46057
   794
    bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a) - 1)"
haftmann@37660
   795
  apply (subst word_sbin.norm_Rep [symmetric])
huffman@46057
   796
  apply (auto simp add: nth_sbintr)
haftmann@37660
   797
  done
haftmann@37660
   798
haftmann@37660
   799
(* type definitions theorem for in terms of equivalent bool list *)
haftmann@37660
   800
lemma td_bl: 
haftmann@37660
   801
  "type_definition (to_bl :: 'a::len0 word => bool list) 
haftmann@37660
   802
                   of_bl  
haftmann@37660
   803
                   {bl. length bl = len_of TYPE('a)}"
haftmann@37660
   804
  apply (unfold type_definition_def of_bl_def to_bl_def)
haftmann@37660
   805
  apply (simp add: word_ubin.eq_norm)
haftmann@37660
   806
  apply safe
haftmann@37660
   807
  apply (drule sym)
haftmann@37660
   808
  apply simp
haftmann@37660
   809
  done
haftmann@37660
   810
haftmann@37660
   811
interpretation word_bl:
haftmann@37660
   812
  type_definition "to_bl :: 'a::len0 word => bool list"
haftmann@37660
   813
                  of_bl  
haftmann@37660
   814
                  "{bl. length bl = len_of TYPE('a::len0)}"
haftmann@37660
   815
  by (rule td_bl)
haftmann@37660
   816
huffman@45816
   817
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]
wenzelm@45538
   818
haftmann@40827
   819
lemma word_size_bl: "size w = size (to_bl w)"
haftmann@37660
   820
  unfolding word_size by auto
haftmann@37660
   821
haftmann@37660
   822
lemma to_bl_use_of_bl:
haftmann@37660
   823
  "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
huffman@45816
   824
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])
haftmann@37660
   825
haftmann@37660
   826
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
haftmann@37660
   827
  unfolding word_reverse_def by (simp add: word_bl.Abs_inverse)
haftmann@37660
   828
haftmann@37660
   829
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
haftmann@37660
   830
  unfolding word_reverse_def by (simp add : word_bl.Abs_inverse)
haftmann@37660
   831
haftmann@40827
   832
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
huffman@47108
   833
  by (metis word_rev_rev)
haftmann@37660
   834
huffman@45805
   835
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u"
huffman@45805
   836
  by simp
huffman@45805
   837
huffman@45805
   838
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))"
huffman@45805
   839
  unfolding word_bl_Rep' by (rule len_gt_0)
huffman@45805
   840
huffman@45805
   841
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []"
huffman@45805
   842
  by (fact length_bl_gt_0 [unfolded length_greater_0_conv])
huffman@45805
   843
huffman@45805
   844
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0"
huffman@45805
   845
  by (fact length_bl_gt_0 [THEN gr_implies_not0])
haftmann@37660
   846
huffman@46001
   847
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
haftmann@37660
   848
  apply (unfold to_bl_def sint_uint)
haftmann@37660
   849
  apply (rule trans [OF _ bl_sbin_sign])
haftmann@37660
   850
  apply simp
haftmann@37660
   851
  done
haftmann@37660
   852
haftmann@37660
   853
lemma of_bl_drop': 
haftmann@40827
   854
  "lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> 
haftmann@37660
   855
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
haftmann@37660
   856
  apply (unfold of_bl_def)
haftmann@37660
   857
  apply (clarsimp simp add : trunc_bl2bin [symmetric])
haftmann@37660
   858
  done
haftmann@37660
   859
haftmann@37660
   860
lemma test_bit_of_bl:  
haftmann@37660
   861
  "(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)"
haftmann@37660
   862
  apply (unfold of_bl_def word_test_bit_def)
haftmann@37660
   863
  apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl)
haftmann@37660
   864
  done
haftmann@37660
   865
haftmann@37660
   866
lemma no_of_bl: 
huffman@47108
   867
  "(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) (numeral bin))"
huffman@47108
   868
  unfolding of_bl_def by simp
haftmann@37660
   869
haftmann@40827
   870
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
haftmann@37660
   871
  unfolding word_size to_bl_def by auto
haftmann@37660
   872
haftmann@37660
   873
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   874
  unfolding uint_bl by (simp add : word_size)
haftmann@37660
   875
haftmann@37660
   876
lemma to_bl_of_bin: 
haftmann@37660
   877
  "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin"
haftmann@37660
   878
  unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size)
haftmann@37660
   879
huffman@47108
   880
lemma to_bl_numeral [simp]:
huffman@47108
   881
  "to_bl (numeral bin::'a::len0 word) =
huffman@47108
   882
    bin_to_bl (len_of TYPE('a)) (numeral bin)"
huffman@47108
   883
  unfolding word_numeral_alt by (rule to_bl_of_bin)
huffman@47108
   884
huffman@47108
   885
lemma to_bl_neg_numeral [simp]:
huffman@47108
   886
  "to_bl (neg_numeral bin::'a::len0 word) =
huffman@47108
   887
    bin_to_bl (len_of TYPE('a)) (neg_numeral bin)"
huffman@47108
   888
  unfolding word_neg_numeral_alt by (rule to_bl_of_bin)
haftmann@37660
   889
haftmann@37660
   890
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   891
  unfolding uint_bl by (simp add : word_size)
huffman@46011
   892
huffman@46011
   893
lemma uint_bl_bin:
huffman@46011
   894
  fixes x :: "'a::len0 word"
huffman@46011
   895
  shows "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x"
huffman@46011
   896
  by (rule trans [OF bin_bl_bin word_ubin.norm_Rep])
wenzelm@45604
   897
haftmann@37660
   898
(* naturals *)
haftmann@37660
   899
lemma uints_unats: "uints n = int ` unats n"
haftmann@37660
   900
  apply (unfold unats_def uints_num)
haftmann@37660
   901
  apply safe
haftmann@37660
   902
  apply (rule_tac image_eqI)
haftmann@37660
   903
  apply (erule_tac nat_0_le [symmetric])
haftmann@37660
   904
  apply auto
haftmann@37660
   905
  apply (erule_tac nat_less_iff [THEN iffD2])
haftmann@37660
   906
  apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1])
haftmann@37660
   907
  apply (auto simp add : nat_power_eq int_power)
haftmann@37660
   908
  done
haftmann@37660
   909
haftmann@37660
   910
lemma unats_uints: "unats n = nat ` uints n"
haftmann@37660
   911
  by (auto simp add : uints_unats image_iff)
haftmann@37660
   912
huffman@46962
   913
lemmas bintr_num = word_ubin.norm_eq_iff
huffman@47108
   914
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
huffman@46962
   915
lemmas sbintr_num = word_sbin.norm_eq_iff
huffman@47108
   916
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
haftmann@37660
   917
haftmann@37660
   918
lemma num_of_bintr':
huffman@47108
   919
  "bintrunc (len_of TYPE('a :: len0)) (numeral a) = (numeral b) \<Longrightarrow> 
huffman@47108
   920
    numeral a = (numeral b :: 'a word)"
huffman@46962
   921
  unfolding bintr_num by (erule subst, simp)
haftmann@37660
   922
haftmann@37660
   923
lemma num_of_sbintr':
huffman@47108
   924
  "sbintrunc (len_of TYPE('a :: len) - 1) (numeral a) = (numeral b) \<Longrightarrow> 
huffman@47108
   925
    numeral a = (numeral b :: 'a word)"
huffman@46962
   926
  unfolding sbintr_num by (erule subst, simp)
huffman@46962
   927
huffman@46962
   928
lemma num_abs_bintr:
huffman@47108
   929
  "(numeral x :: 'a word) =
huffman@47108
   930
    word_of_int (bintrunc (len_of TYPE('a::len0)) (numeral x))"
huffman@47108
   931
  by (simp only: word_ubin.Abs_norm word_numeral_alt)
huffman@46962
   932
huffman@46962
   933
lemma num_abs_sbintr:
huffman@47108
   934
  "(numeral x :: 'a word) =
huffman@47108
   935
    word_of_int (sbintrunc (len_of TYPE('a::len) - 1) (numeral x))"
huffman@47108
   936
  by (simp only: word_sbin.Abs_norm word_numeral_alt)
huffman@46962
   937
haftmann@37660
   938
(** cast - note, no arg for new length, as it's determined by type of result,
haftmann@37660
   939
  thus in "cast w = w, the type means cast to length of w! **)
haftmann@37660
   940
haftmann@37660
   941
lemma ucast_id: "ucast w = w"
haftmann@37660
   942
  unfolding ucast_def by auto
haftmann@37660
   943
haftmann@37660
   944
lemma scast_id: "scast w = w"
haftmann@37660
   945
  unfolding scast_def by auto
haftmann@37660
   946
haftmann@40827
   947
lemma ucast_bl: "ucast w = of_bl (to_bl w)"
haftmann@37660
   948
  unfolding ucast_def of_bl_def uint_bl
haftmann@37660
   949
  by (auto simp add : word_size)
haftmann@37660
   950
haftmann@37660
   951
lemma nth_ucast: 
haftmann@37660
   952
  "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))"
haftmann@37660
   953
  apply (unfold ucast_def test_bit_bin)
haftmann@37660
   954
  apply (simp add: word_ubin.eq_norm nth_bintr word_size) 
haftmann@37660
   955
  apply (fast elim!: bin_nth_uint_imp)
haftmann@37660
   956
  done
haftmann@37660
   957
haftmann@37660
   958
(* for literal u(s)cast *)
haftmann@37660
   959
huffman@46001
   960
lemma ucast_bintr [simp]:
huffman@47108
   961
  "ucast (numeral w ::'a::len0 word) = 
huffman@47108
   962
   word_of_int (bintrunc (len_of TYPE('a)) (numeral w))"
haftmann@37660
   963
  unfolding ucast_def by simp
huffman@47108
   964
(* TODO: neg_numeral *)
haftmann@37660
   965
huffman@46001
   966
lemma scast_sbintr [simp]:
huffman@47108
   967
  "scast (numeral w ::'a::len word) = 
huffman@47108
   968
   word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (numeral w))"
haftmann@37660
   969
  unfolding scast_def by simp
haftmann@37660
   970
huffman@46011
   971
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)"
huffman@46011
   972
  unfolding source_size_def word_size Let_def ..
huffman@46011
   973
huffman@46011
   974
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)"
huffman@46011
   975
  unfolding target_size_def word_size Let_def ..
huffman@46011
   976
huffman@46011
   977
lemma is_down:
huffman@46011
   978
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
huffman@46011
   979
  shows "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)"
huffman@46011
   980
  unfolding is_down_def source_size target_size ..
huffman@46011
   981
huffman@46011
   982
lemma is_up:
huffman@46011
   983
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
huffman@46011
   984
  shows "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)"
huffman@46011
   985
  unfolding is_up_def source_size target_size ..
haftmann@37660
   986
wenzelm@45604
   987
lemmas is_up_down = trans [OF is_up is_down [symmetric]]
haftmann@37660
   988
huffman@45811
   989
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast"
haftmann@37660
   990
  apply (unfold is_down)
haftmann@37660
   991
  apply safe
haftmann@37660
   992
  apply (rule ext)
haftmann@37660
   993
  apply (unfold ucast_def scast_def uint_sint)
haftmann@37660
   994
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
   995
  apply simp
haftmann@37660
   996
  done
haftmann@37660
   997
huffman@45811
   998
lemma word_rev_tf:
huffman@45811
   999
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
  1000
    rev (takefill False (len_of TYPE('a)) (rev bl))"
haftmann@37660
  1001
  unfolding of_bl_def uint_bl
haftmann@37660
  1002
  by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size)
haftmann@37660
  1003
huffman@45811
  1004
lemma word_rep_drop:
huffman@45811
  1005
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
  1006
    replicate (len_of TYPE('a) - length bl) False @
huffman@45811
  1007
    drop (length bl - len_of TYPE('a)) bl"
huffman@45811
  1008
  by (simp add: word_rev_tf takefill_alt rev_take)
haftmann@37660
  1009
haftmann@37660
  1010
lemma to_bl_ucast: 
haftmann@37660
  1011
  "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 
haftmann@37660
  1012
   replicate (len_of TYPE('a) - len_of TYPE('b)) False @
haftmann@37660
  1013
   drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)"
haftmann@37660
  1014
  apply (unfold ucast_bl)
haftmann@37660
  1015
  apply (rule trans)
haftmann@37660
  1016
   apply (rule word_rep_drop)
haftmann@37660
  1017
  apply simp
haftmann@37660
  1018
  done
haftmann@37660
  1019
huffman@45811
  1020
lemma ucast_up_app [OF refl]:
haftmann@40827
  1021
  "uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
haftmann@37660
  1022
    to_bl (uc w) = replicate n False @ (to_bl w)"
haftmann@37660
  1023
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
  1024
huffman@45811
  1025
lemma ucast_down_drop [OF refl]:
haftmann@40827
  1026
  "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
haftmann@37660
  1027
    to_bl (uc w) = drop n (to_bl w)"
haftmann@37660
  1028
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
  1029
huffman@45811
  1030
lemma scast_down_drop [OF refl]:
haftmann@40827
  1031
  "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
haftmann@37660
  1032
    to_bl (sc w) = drop n (to_bl w)"
haftmann@37660
  1033
  apply (subgoal_tac "sc = ucast")
haftmann@37660
  1034
   apply safe
haftmann@37660
  1035
   apply simp
huffman@45811
  1036
   apply (erule ucast_down_drop)
huffman@45811
  1037
  apply (rule down_cast_same [symmetric])
haftmann@37660
  1038
  apply (simp add : source_size target_size is_down)
haftmann@37660
  1039
  done
haftmann@37660
  1040
huffman@45811
  1041
lemma sint_up_scast [OF refl]:
haftmann@40827
  1042
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w"
haftmann@37660
  1043
  apply (unfold is_up)
haftmann@37660
  1044
  apply safe
haftmann@37660
  1045
  apply (simp add: scast_def word_sbin.eq_norm)
haftmann@37660
  1046
  apply (rule box_equals)
haftmann@37660
  1047
    prefer 3
haftmann@37660
  1048
    apply (rule word_sbin.norm_Rep)
haftmann@37660
  1049
   apply (rule sbintrunc_sbintrunc_l)
haftmann@37660
  1050
   defer
haftmann@37660
  1051
   apply (subst word_sbin.norm_Rep)
haftmann@37660
  1052
   apply (rule refl)
haftmann@37660
  1053
  apply simp
haftmann@37660
  1054
  done
haftmann@37660
  1055
huffman@45811
  1056
lemma uint_up_ucast [OF refl]:
haftmann@40827
  1057
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w"
haftmann@37660
  1058
  apply (unfold is_up)
haftmann@37660
  1059
  apply safe
haftmann@37660
  1060
  apply (rule bin_eqI)
haftmann@37660
  1061
  apply (fold word_test_bit_def)
haftmann@37660
  1062
  apply (auto simp add: nth_ucast)
haftmann@37660
  1063
  apply (auto simp add: test_bit_bin)
haftmann@37660
  1064
  done
huffman@45811
  1065
huffman@45811
  1066
lemma ucast_up_ucast [OF refl]:
huffman@45811
  1067
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w"
haftmann@37660
  1068
  apply (simp (no_asm) add: ucast_def)
haftmann@37660
  1069
  apply (clarsimp simp add: uint_up_ucast)
haftmann@37660
  1070
  done
haftmann@37660
  1071
    
huffman@45811
  1072
lemma scast_up_scast [OF refl]:
huffman@45811
  1073
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w"
haftmann@37660
  1074
  apply (simp (no_asm) add: scast_def)
haftmann@37660
  1075
  apply (clarsimp simp add: sint_up_scast)
haftmann@37660
  1076
  done
haftmann@37660
  1077
    
huffman@45811
  1078
lemma ucast_of_bl_up [OF refl]:
haftmann@40827
  1079
  "w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl"
haftmann@37660
  1080
  by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI)
haftmann@37660
  1081
haftmann@37660
  1082
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id]
haftmann@37660
  1083
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id]
haftmann@37660
  1084
haftmann@37660
  1085
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2]
haftmann@37660
  1086
lemmas isdus = is_up_down [where c = "scast", THEN iffD2]
haftmann@37660
  1087
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
haftmann@37660
  1088
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]
haftmann@37660
  1089
haftmann@37660
  1090
lemma up_ucast_surj:
haftmann@40827
  1091
  "is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1092
   surj (ucast :: 'a word => 'b word)"
haftmann@37660
  1093
  by (rule surjI, erule ucast_up_ucast_id)
haftmann@37660
  1094
haftmann@37660
  1095
lemma up_scast_surj:
haftmann@40827
  1096
  "is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1097
   surj (scast :: 'a word => 'b word)"
haftmann@37660
  1098
  by (rule surjI, erule scast_up_scast_id)
haftmann@37660
  1099
haftmann@37660
  1100
lemma down_scast_inj:
haftmann@40827
  1101
  "is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1102
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1103
  by (rule inj_on_inverseI, erule scast_down_scast_id)
haftmann@37660
  1104
haftmann@37660
  1105
lemma down_ucast_inj:
haftmann@40827
  1106
  "is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1107
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1108
  by (rule inj_on_inverseI, erule ucast_down_ucast_id)
haftmann@37660
  1109
haftmann@37660
  1110
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
haftmann@37660
  1111
  by (rule word_bl.Rep_eqD) (simp add: word_rep_drop)
huffman@45811
  1112
huffman@46646
  1113
lemma ucast_down_wi [OF refl]:
huffman@46646
  1114
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x"
huffman@46646
  1115
  apply (unfold is_down)
haftmann@37660
  1116
  apply (clarsimp simp add: ucast_def word_ubin.eq_norm)
haftmann@37660
  1117
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1118
  apply (erule bintrunc_bintrunc_ge)
haftmann@37660
  1119
  done
huffman@45811
  1120
huffman@46646
  1121
lemma ucast_down_no [OF refl]:
huffman@47108
  1122
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin"
huffman@47108
  1123
  unfolding word_numeral_alt by clarify (rule ucast_down_wi)
huffman@46646
  1124
huffman@45811
  1125
lemma ucast_down_bl [OF refl]:
huffman@45811
  1126
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl"
huffman@46646
  1127
  unfolding of_bl_def by clarify (erule ucast_down_wi)
haftmann@37660
  1128
haftmann@37660
  1129
lemmas slice_def' = slice_def [unfolded word_size]
haftmann@37660
  1130
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
haftmann@37660
  1131
haftmann@37660
  1132
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def
haftmann@37660
  1133
haftmann@37660
  1134
haftmann@37660
  1135
subsection {* Word Arithmetic *}
haftmann@37660
  1136
haftmann@37660
  1137
lemma word_less_alt: "(a < b) = (uint a < uint b)"
huffman@46012
  1138
  unfolding word_less_def word_le_def by (simp add: less_le)
haftmann@37660
  1139
haftmann@37660
  1140
lemma signed_linorder: "class.linorder word_sle word_sless"
wenzelm@46124
  1141
  by default (unfold word_sle_def word_sless_def, auto)
haftmann@37660
  1142
haftmann@37660
  1143
interpretation signed: linorder "word_sle" "word_sless"
haftmann@37660
  1144
  by (rule signed_linorder)
haftmann@37660
  1145
haftmann@37660
  1146
lemma udvdI: 
haftmann@40827
  1147
  "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b"
haftmann@37660
  1148
  by (auto simp: udvd_def)
haftmann@37660
  1149
huffman@47108
  1150
lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1151
huffman@47108
  1152
lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1153
huffman@47108
  1154
lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1155
huffman@47108
  1156
lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1157
huffman@47108
  1158
lemmas word_sless_no [simp] = word_sless_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1159
huffman@47108
  1160
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b
haftmann@37660
  1161
huffman@46020
  1162
lemma word_1_no: "(1::'a::len0 word) = Numeral1"
huffman@47108
  1163
  by (simp add: word_numeral_alt)
haftmann@37660
  1164
haftmann@40827
  1165
lemma word_m1_wi: "-1 = word_of_int -1" 
huffman@47108
  1166
  by (rule word_neg_numeral_alt)
haftmann@37660
  1167
huffman@46648
  1168
lemma word_0_bl [simp]: "of_bl [] = 0"
huffman@46648
  1169
  unfolding of_bl_def by simp
haftmann@37660
  1170
haftmann@37660
  1171
lemma word_1_bl: "of_bl [True] = 1" 
huffman@46648
  1172
  unfolding of_bl_def by (simp add: bl_to_bin_def)
huffman@46648
  1173
huffman@46648
  1174
lemma uint_eq_0 [simp]: "uint 0 = 0"
huffman@46648
  1175
  unfolding word_0_wi word_ubin.eq_norm by simp
haftmann@37660
  1176
huffman@45995
  1177
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
huffman@46648
  1178
  by (simp add: of_bl_def bl_to_bin_rep_False)
haftmann@37660
  1179
huffman@45805
  1180
lemma to_bl_0 [simp]:
haftmann@37660
  1181
  "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False"
haftmann@37660
  1182
  unfolding uint_bl
huffman@46617
  1183
  by (simp add: word_size bin_to_bl_zero)
haftmann@37660
  1184
haftmann@37660
  1185
lemma uint_0_iff: "(uint x = 0) = (x = 0)"
haftmann@37660
  1186
  by (auto intro!: word_uint.Rep_eqD)
haftmann@37660
  1187
haftmann@37660
  1188
lemma unat_0_iff: "(unat x = 0) = (x = 0)"
haftmann@37660
  1189
  unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff)
haftmann@37660
  1190
haftmann@37660
  1191
lemma unat_0 [simp]: "unat 0 = 0"
haftmann@37660
  1192
  unfolding unat_def by auto
haftmann@37660
  1193
haftmann@40827
  1194
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)"
haftmann@37660
  1195
  apply (unfold word_size)
haftmann@37660
  1196
  apply (rule box_equals)
haftmann@37660
  1197
    defer
haftmann@37660
  1198
    apply (rule word_uint.Rep_inverse)+
haftmann@37660
  1199
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1200
  apply simp
haftmann@37660
  1201
  done
haftmann@37660
  1202
huffman@45816
  1203
lemmas size_0_same = size_0_same' [unfolded word_size]
haftmann@37660
  1204
haftmann@37660
  1205
lemmas unat_eq_0 = unat_0_iff
haftmann@37660
  1206
lemmas unat_eq_zero = unat_0_iff
haftmann@37660
  1207
haftmann@37660
  1208
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)"
haftmann@37660
  1209
by (auto simp: unat_0_iff [symmetric])
haftmann@37660
  1210
huffman@45958
  1211
lemma ucast_0 [simp]: "ucast 0 = 0"
huffman@45995
  1212
  unfolding ucast_def by simp
huffman@45958
  1213
huffman@45958
  1214
lemma sint_0 [simp]: "sint 0 = 0"
huffman@45958
  1215
  unfolding sint_uint by simp
huffman@45958
  1216
huffman@45958
  1217
lemma scast_0 [simp]: "scast 0 = 0"
huffman@45995
  1218
  unfolding scast_def by simp
haftmann@37660
  1219
haftmann@37660
  1220
lemma sint_n1 [simp] : "sint -1 = -1"
huffman@45958
  1221
  unfolding word_m1_wi by (simp add: word_sbin.eq_norm)
huffman@45958
  1222
huffman@45958
  1223
lemma scast_n1 [simp]: "scast -1 = -1"
huffman@45958
  1224
  unfolding scast_def by simp
huffman@45958
  1225
huffman@45958
  1226
lemma uint_1 [simp]: "uint (1::'a::len word) = 1"
haftmann@37660
  1227
  unfolding word_1_wi
huffman@45995
  1228
  by (simp add: word_ubin.eq_norm bintrunc_minus_simps del: word_of_int_1)
huffman@45958
  1229
huffman@45958
  1230
lemma unat_1 [simp]: "unat (1::'a::len word) = 1"
huffman@45958
  1231
  unfolding unat_def by simp
huffman@45958
  1232
huffman@45958
  1233
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1"
huffman@45995
  1234
  unfolding ucast_def by simp
haftmann@37660
  1235
haftmann@37660
  1236
(* now, to get the weaker results analogous to word_div/mod_def *)
haftmann@37660
  1237
haftmann@37660
  1238
lemmas word_arith_alts = 
huffman@46000
  1239
  word_sub_wi
huffman@46000
  1240
  word_arith_wis (* FIXME: duplicate *)
huffman@46000
  1241
haftmann@37660
  1242
subsection  "Transferring goals from words to ints"
haftmann@37660
  1243
haftmann@37660
  1244
lemma word_ths:  
haftmann@37660
  1245
  shows
haftmann@37660
  1246
  word_succ_p1:   "word_succ a = a + 1" and
haftmann@37660
  1247
  word_pred_m1:   "word_pred a = a - 1" and
haftmann@37660
  1248
  word_pred_succ: "word_pred (word_succ a) = a" and
haftmann@37660
  1249
  word_succ_pred: "word_succ (word_pred a) = a" and
haftmann@37660
  1250
  word_mult_succ: "word_succ a * b = b + a * b"
huffman@47374
  1251
  by (transfer, simp add: algebra_simps)+
haftmann@37660
  1252
huffman@45816
  1253
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y"
huffman@45816
  1254
  by simp
haftmann@37660
  1255
haftmann@37660
  1256
lemmas uint_word_ariths = 
wenzelm@45604
  1257
  word_arith_alts [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1258
haftmann@37660
  1259
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p]
haftmann@37660
  1260
haftmann@37660
  1261
(* similar expressions for sint (arith operations) *)
haftmann@37660
  1262
lemmas sint_word_ariths = uint_word_arith_bintrs
haftmann@37660
  1263
  [THEN uint_sint [symmetric, THEN trans],
haftmann@37660
  1264
  unfolded uint_sint bintr_arith1s bintr_ariths 
wenzelm@45604
  1265
    len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep]
wenzelm@45604
  1266
wenzelm@45604
  1267
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
wenzelm@45604
  1268
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1269
haftmann@37660
  1270
lemma word_pred_0_n1: "word_pred 0 = word_of_int -1"
huffman@47374
  1271
  unfolding word_pred_m1 by simp
haftmann@37660
  1272
haftmann@37660
  1273
lemma succ_pred_no [simp]:
huffman@47108
  1274
  "word_succ (numeral w) = numeral w + 1"
huffman@47108
  1275
  "word_pred (numeral w) = numeral w - 1"
huffman@47108
  1276
  "word_succ (neg_numeral w) = neg_numeral w + 1"
huffman@47108
  1277
  "word_pred (neg_numeral w) = neg_numeral w - 1"
huffman@47108
  1278
  unfolding word_succ_p1 word_pred_m1 by simp_all
haftmann@37660
  1279
haftmann@37660
  1280
lemma word_sp_01 [simp] : 
haftmann@37660
  1281
  "word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0"
huffman@47108
  1282
  unfolding word_succ_p1 word_pred_m1 by simp_all
haftmann@37660
  1283
haftmann@37660
  1284
(* alternative approach to lifting arithmetic equalities *)
haftmann@37660
  1285
lemma word_of_int_Ex:
haftmann@37660
  1286
  "\<exists>y. x = word_of_int y"
haftmann@37660
  1287
  by (rule_tac x="uint x" in exI) simp
haftmann@37660
  1288
haftmann@37660
  1289
haftmann@37660
  1290
subsection "Order on fixed-length words"
haftmann@37660
  1291
haftmann@37660
  1292
lemma word_zero_le [simp] :
haftmann@37660
  1293
  "0 <= (y :: 'a :: len0 word)"
haftmann@37660
  1294
  unfolding word_le_def by auto
haftmann@37660
  1295
  
huffman@45816
  1296
lemma word_m1_ge [simp] : "word_pred 0 >= y" (* FIXME: delete *)
haftmann@37660
  1297
  unfolding word_le_def
haftmann@37660
  1298
  by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1299
huffman@45816
  1300
lemma word_n1_ge [simp]: "y \<le> (-1::'a::len0 word)"
huffman@45816
  1301
  unfolding word_le_def
huffman@45816
  1302
  by (simp only: word_m1_wi word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1303
haftmann@37660
  1304
lemmas word_not_simps [simp] = 
haftmann@37660
  1305
  word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD]
haftmann@37660
  1306
huffman@47108
  1307
lemma word_gt_0: "0 < y \<longleftrightarrow> 0 \<noteq> (y :: 'a :: len0 word)"
huffman@47108
  1308
  by (simp add: less_le)
huffman@47108
  1309
huffman@47108
  1310
lemmas word_gt_0_no [simp] = word_gt_0 [of "numeral y"] for y
haftmann@37660
  1311
haftmann@40827
  1312
lemma word_sless_alt: "(a <s b) = (sint a < sint b)"
haftmann@37660
  1313
  unfolding word_sle_def word_sless_def
haftmann@37660
  1314
  by (auto simp add: less_le)
haftmann@37660
  1315
haftmann@37660
  1316
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)"
haftmann@37660
  1317
  unfolding unat_def word_le_def
haftmann@37660
  1318
  by (rule nat_le_eq_zle [symmetric]) simp
haftmann@37660
  1319
haftmann@37660
  1320
lemma word_less_nat_alt: "(a < b) = (unat a < unat b)"
haftmann@37660
  1321
  unfolding unat_def word_less_alt
haftmann@37660
  1322
  by (rule nat_less_eq_zless [symmetric]) simp
haftmann@37660
  1323
  
haftmann@37660
  1324
lemma wi_less: 
haftmann@37660
  1325
  "(word_of_int n < (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1326
    (n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1327
  unfolding word_less_alt by (simp add: word_uint.eq_norm)
haftmann@37660
  1328
haftmann@37660
  1329
lemma wi_le: 
haftmann@37660
  1330
  "(word_of_int n <= (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1331
    (n mod 2 ^ len_of TYPE('a) <= m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1332
  unfolding word_le_def by (simp add: word_uint.eq_norm)
haftmann@37660
  1333
haftmann@37660
  1334
lemma udvd_nat_alt: "a udvd b = (EX n>=0. unat b = n * unat a)"
haftmann@37660
  1335
  apply (unfold udvd_def)
haftmann@37660
  1336
  apply safe
haftmann@37660
  1337
   apply (simp add: unat_def nat_mult_distrib)
haftmann@37660
  1338
  apply (simp add: uint_nat int_mult)
haftmann@37660
  1339
  apply (rule exI)
haftmann@37660
  1340
  apply safe
haftmann@37660
  1341
   prefer 2
haftmann@37660
  1342
   apply (erule notE)
haftmann@37660
  1343
   apply (rule refl)
haftmann@37660
  1344
  apply force
haftmann@37660
  1345
  done
haftmann@37660
  1346
haftmann@37660
  1347
lemma udvd_iff_dvd: "x udvd y <-> unat x dvd unat y"
haftmann@37660
  1348
  unfolding dvd_def udvd_nat_alt by force
haftmann@37660
  1349
wenzelm@45604
  1350
lemmas unat_mono = word_less_nat_alt [THEN iffD1]
haftmann@37660
  1351
haftmann@40827
  1352
lemma unat_minus_one: "x ~= 0 \<Longrightarrow> unat (x - 1) = unat x - 1"
haftmann@37660
  1353
  apply (unfold unat_def)
haftmann@37660
  1354
  apply (simp only: int_word_uint word_arith_alts rdmods)
haftmann@37660
  1355
  apply (subgoal_tac "uint x >= 1")
haftmann@37660
  1356
   prefer 2
haftmann@37660
  1357
   apply (drule contrapos_nn)
haftmann@37660
  1358
    apply (erule word_uint.Rep_inverse' [symmetric])
haftmann@37660
  1359
   apply (insert uint_ge_0 [of x])[1]
haftmann@37660
  1360
   apply arith
haftmann@37660
  1361
  apply (rule box_equals)
haftmann@37660
  1362
    apply (rule nat_diff_distrib)
haftmann@37660
  1363
     prefer 2
haftmann@37660
  1364
     apply assumption
haftmann@37660
  1365
    apply simp
haftmann@37660
  1366
   apply (subst mod_pos_pos_trivial)
haftmann@37660
  1367
     apply arith
haftmann@37660
  1368
    apply (insert uint_lt2p [of x])[1]
haftmann@37660
  1369
    apply arith
haftmann@37660
  1370
   apply (rule refl)
haftmann@37660
  1371
  apply simp
haftmann@37660
  1372
  done
haftmann@37660
  1373
    
haftmann@40827
  1374
lemma measure_unat: "p ~= 0 \<Longrightarrow> unat (p - 1) < unat p"
haftmann@37660
  1375
  by (simp add: unat_minus_one) (simp add: unat_0_iff [symmetric])
haftmann@37660
  1376
  
wenzelm@45604
  1377
lemmas uint_add_ge0 [simp] = add_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
wenzelm@45604
  1378
lemmas uint_mult_ge0 [simp] = mult_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
haftmann@37660
  1379
haftmann@37660
  1380
lemma uint_sub_lt2p [simp]: 
haftmann@37660
  1381
  "uint (x :: 'a :: len0 word) - uint (y :: 'b :: len0 word) < 
haftmann@37660
  1382
    2 ^ len_of TYPE('a)"
haftmann@37660
  1383
  using uint_ge_0 [of y] uint_lt2p [of x] by arith
haftmann@37660
  1384
haftmann@37660
  1385
haftmann@37660
  1386
subsection "Conditions for the addition (etc) of two words to overflow"
haftmann@37660
  1387
haftmann@37660
  1388
lemma uint_add_lem: 
haftmann@37660
  1389
  "(uint x + uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1390
    (uint (x + y :: 'a :: len0 word) = uint x + uint y)"
haftmann@37660
  1391
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1392
haftmann@37660
  1393
lemma uint_mult_lem: 
haftmann@37660
  1394
  "(uint x * uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1395
    (uint (x * y :: 'a :: len0 word) = uint x * uint y)"
haftmann@37660
  1396
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1397
haftmann@37660
  1398
lemma uint_sub_lem: 
haftmann@37660
  1399
  "(uint x >= uint y) = (uint (x - y) = uint x - uint y)"
haftmann@37660
  1400
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1401
haftmann@37660
  1402
lemma uint_add_le: "uint (x + y) <= uint x + uint y"
haftmann@37660
  1403
  unfolding uint_word_ariths by (auto simp: mod_add_if_z)
haftmann@37660
  1404
haftmann@37660
  1405
lemma uint_sub_ge: "uint (x - y) >= uint x - uint y"
haftmann@37660
  1406
  unfolding uint_word_ariths by (auto simp: mod_sub_if_z)
haftmann@37660
  1407
wenzelm@45604
  1408
lemmas uint_sub_if' = trans [OF uint_word_ariths(1) mod_sub_if_z, simplified]
wenzelm@45604
  1409
lemmas uint_plus_if' = trans [OF uint_word_ariths(2) mod_add_if_z, simplified]
haftmann@37660
  1410
haftmann@37660
  1411
haftmann@37660
  1412
subsection {* Definition of uint\_arith *}
haftmann@37660
  1413
haftmann@37660
  1414
lemma word_of_int_inverse:
haftmann@40827
  1415
  "word_of_int r = a \<Longrightarrow> 0 <= r \<Longrightarrow> r < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  1416
   uint (a::'a::len0 word) = r"
haftmann@37660
  1417
  apply (erule word_uint.Abs_inverse' [rotated])
haftmann@37660
  1418
  apply (simp add: uints_num)
haftmann@37660
  1419
  done
haftmann@37660
  1420
haftmann@37660
  1421
lemma uint_split:
haftmann@37660
  1422
  fixes x::"'a::len0 word"
haftmann@37660
  1423
  shows "P (uint x) = 
haftmann@37660
  1424
         (ALL i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) --> P i)"
haftmann@37660
  1425
  apply (fold word_int_case_def)
haftmann@37660
  1426
  apply (auto dest!: word_of_int_inverse simp: int_word_uint int_mod_eq'
haftmann@37660
  1427
              split: word_int_split)
haftmann@37660
  1428
  done
haftmann@37660
  1429
haftmann@37660
  1430
lemma uint_split_asm:
haftmann@37660
  1431
  fixes x::"'a::len0 word"
haftmann@37660
  1432
  shows "P (uint x) = 
haftmann@37660
  1433
         (~(EX i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) & ~ P i))"
haftmann@37660
  1434
  by (auto dest!: word_of_int_inverse 
haftmann@37660
  1435
           simp: int_word_uint int_mod_eq'
haftmann@37660
  1436
           split: uint_split)
haftmann@37660
  1437
haftmann@37660
  1438
lemmas uint_splits = uint_split uint_split_asm
haftmann@37660
  1439
haftmann@37660
  1440
lemmas uint_arith_simps = 
haftmann@37660
  1441
  word_le_def word_less_alt
haftmann@37660
  1442
  word_uint.Rep_inject [symmetric] 
haftmann@37660
  1443
  uint_sub_if' uint_plus_if'
haftmann@37660
  1444
haftmann@37660
  1445
(* use this to stop, eg, 2 ^ len_of TYPE (32) being simplified *)
haftmann@40827
  1446
lemma power_False_cong: "False \<Longrightarrow> a ^ b = c ^ d" 
haftmann@37660
  1447
  by auto
haftmann@37660
  1448
haftmann@37660
  1449
(* uint_arith_tac: reduce to arithmetic on int, try to solve by arith *)
haftmann@37660
  1450
ML {*
wenzelm@51717
  1451
fun uint_arith_simpset ctxt = 
wenzelm@51717
  1452
  ctxt addsimps @{thms uint_arith_simps}
haftmann@37660
  1453
     delsimps @{thms word_uint.Rep_inject}
wenzelm@45620
  1454
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  1455
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  1456
haftmann@37660
  1457
fun uint_arith_tacs ctxt = 
haftmann@37660
  1458
  let
haftmann@37660
  1459
    fun arith_tac' n t =
haftmann@37660
  1460
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  1461
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  1462
  in 
wenzelm@42793
  1463
    [ clarify_tac ctxt 1,
wenzelm@51717
  1464
      full_simp_tac (uint_arith_simpset ctxt) 1,
wenzelm@51717
  1465
      ALLGOALS (full_simp_tac
wenzelm@51717
  1466
        (put_simpset HOL_ss ctxt
wenzelm@51717
  1467
          |> fold Splitter.add_split @{thms uint_splits}
wenzelm@51717
  1468
          |> fold Simplifier.add_cong @{thms power_False_cong})),
haftmann@37660
  1469
      rewrite_goals_tac @{thms word_size}, 
haftmann@37660
  1470
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  1471
                         REPEAT (etac conjE n) THEN
haftmann@37660
  1472
                         REPEAT (dtac @{thm word_of_int_inverse} n 
haftmann@37660
  1473
                                 THEN atac n 
haftmann@37660
  1474
                                 THEN atac n)),
haftmann@37660
  1475
      TRYALL arith_tac' ]
haftmann@37660
  1476
  end
haftmann@37660
  1477
haftmann@37660
  1478
fun uint_arith_tac ctxt = SELECT_GOAL (EVERY (uint_arith_tacs ctxt))
haftmann@37660
  1479
*}
haftmann@37660
  1480
haftmann@37660
  1481
method_setup uint_arith = 
haftmann@37660
  1482
  {* Scan.succeed (SIMPLE_METHOD' o uint_arith_tac) *}
haftmann@37660
  1483
  "solving word arithmetic via integers and arith"
haftmann@37660
  1484
haftmann@37660
  1485
haftmann@37660
  1486
subsection "More on overflows and monotonicity"
haftmann@37660
  1487
haftmann@37660
  1488
lemma no_plus_overflow_uint_size: 
haftmann@37660
  1489
  "((x :: 'a :: len0 word) <= x + y) = (uint x + uint y < 2 ^ size x)"
haftmann@37660
  1490
  unfolding word_size by uint_arith
haftmann@37660
  1491
haftmann@37660
  1492
lemmas no_olen_add = no_plus_overflow_uint_size [unfolded word_size]
haftmann@37660
  1493
haftmann@37660
  1494
lemma no_ulen_sub: "((x :: 'a :: len0 word) >= x - y) = (uint y <= uint x)"
haftmann@37660
  1495
  by uint_arith
haftmann@37660
  1496
haftmann@37660
  1497
lemma no_olen_add':
haftmann@37660
  1498
  fixes x :: "'a::len0 word"
haftmann@37660
  1499
  shows "(x \<le> y + x) = (uint y + uint x < 2 ^ len_of TYPE('a))"
huffman@45546
  1500
  by (simp add: add_ac no_olen_add)
haftmann@37660
  1501
wenzelm@45604
  1502
lemmas olen_add_eqv = trans [OF no_olen_add no_olen_add' [symmetric]]
wenzelm@45604
  1503
wenzelm@45604
  1504
lemmas uint_plus_simple_iff = trans [OF no_olen_add uint_add_lem]
wenzelm@45604
  1505
lemmas uint_plus_simple = uint_plus_simple_iff [THEN iffD1]
wenzelm@45604
  1506
lemmas uint_minus_simple_iff = trans [OF no_ulen_sub uint_sub_lem]
haftmann@37660
  1507
lemmas uint_minus_simple_alt = uint_sub_lem [folded word_le_def]
haftmann@37660
  1508
lemmas word_sub_le_iff = no_ulen_sub [folded word_le_def]
wenzelm@45604
  1509
lemmas word_sub_le = word_sub_le_iff [THEN iffD2]
haftmann@37660
  1510
haftmann@37660
  1511
lemma word_less_sub1: 
haftmann@40827
  1512
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 < x) = (0 < x - 1)"
haftmann@37660
  1513
  by uint_arith
haftmann@37660
  1514
haftmann@37660
  1515
lemma word_le_sub1: 
haftmann@40827
  1516
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 <= x) = (0 <= x - 1)"
haftmann@37660
  1517
  by uint_arith
haftmann@37660
  1518
haftmann@37660
  1519
lemma sub_wrap_lt: 
haftmann@37660
  1520
  "((x :: 'a :: len0 word) < x - z) = (x < z)"
haftmann@37660
  1521
  by uint_arith
haftmann@37660
  1522
haftmann@37660
  1523
lemma sub_wrap: 
haftmann@37660
  1524
  "((x :: 'a :: len0 word) <= x - z) = (z = 0 | x < z)"
haftmann@37660
  1525
  by uint_arith
haftmann@37660
  1526
haftmann@37660
  1527
lemma plus_minus_not_NULL_ab: 
haftmann@40827
  1528
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> c ~= 0 \<Longrightarrow> x + c ~= 0"
haftmann@37660
  1529
  by uint_arith
haftmann@37660
  1530
haftmann@37660
  1531
lemma plus_minus_no_overflow_ab: 
haftmann@40827
  1532
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> x <= x + c" 
haftmann@37660
  1533
  by uint_arith
haftmann@37660
  1534
haftmann@37660
  1535
lemma le_minus': 
haftmann@40827
  1536
  "(a :: 'a :: len0 word) + c <= b \<Longrightarrow> a <= a + c \<Longrightarrow> c <= b - a"
haftmann@37660
  1537
  by uint_arith
haftmann@37660
  1538
haftmann@37660
  1539
lemma le_plus': 
haftmann@40827
  1540
  "(a :: 'a :: len0 word) <= b \<Longrightarrow> c <= b - a \<Longrightarrow> a + c <= b"
haftmann@37660
  1541
  by uint_arith
haftmann@37660
  1542
haftmann@37660
  1543
lemmas le_plus = le_plus' [rotated]
haftmann@37660
  1544
huffman@46011
  1545
lemmas le_minus = leD [THEN thin_rl, THEN le_minus'] (* FIXME *)
haftmann@37660
  1546
haftmann@37660
  1547
lemma word_plus_mono_right: 
haftmann@40827
  1548
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= x + z \<Longrightarrow> x + y <= x + z"
haftmann@37660
  1549
  by uint_arith
haftmann@37660
  1550
haftmann@37660
  1551
lemma word_less_minus_cancel: 
haftmann@40827
  1552
  "y - x < z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) < z"
haftmann@37660
  1553
  by uint_arith
haftmann@37660
  1554
haftmann@37660
  1555
lemma word_less_minus_mono_left: 
haftmann@40827
  1556
  "(y :: 'a :: len0 word) < z \<Longrightarrow> x <= y \<Longrightarrow> y - x < z - x"
haftmann@37660
  1557
  by uint_arith
haftmann@37660
  1558
haftmann@37660
  1559
lemma word_less_minus_mono:  
haftmann@40827
  1560
  "a < c \<Longrightarrow> d < b \<Longrightarrow> a - b < a \<Longrightarrow> c - d < c 
haftmann@40827
  1561
  \<Longrightarrow> a - b < c - (d::'a::len word)"
haftmann@37660
  1562
  by uint_arith
haftmann@37660
  1563
haftmann@37660
  1564
lemma word_le_minus_cancel: 
haftmann@40827
  1565
  "y - x <= z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) <= z"
haftmann@37660
  1566
  by uint_arith
haftmann@37660
  1567
haftmann@37660
  1568
lemma word_le_minus_mono_left: 
haftmann@40827
  1569
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= y \<Longrightarrow> y - x <= z - x"
haftmann@37660
  1570
  by uint_arith
haftmann@37660
  1571
haftmann@37660
  1572
lemma word_le_minus_mono:  
haftmann@40827
  1573
  "a <= c \<Longrightarrow> d <= b \<Longrightarrow> a - b <= a \<Longrightarrow> c - d <= c 
haftmann@40827
  1574
  \<Longrightarrow> a - b <= c - (d::'a::len word)"
haftmann@37660
  1575
  by uint_arith
haftmann@37660
  1576
haftmann@37660
  1577
lemma plus_le_left_cancel_wrap: 
haftmann@40827
  1578
  "(x :: 'a :: len0 word) + y' < x \<Longrightarrow> x + y < x \<Longrightarrow> (x + y' < x + y) = (y' < y)"
haftmann@37660
  1579
  by uint_arith
haftmann@37660
  1580
haftmann@37660
  1581
lemma plus_le_left_cancel_nowrap: 
haftmann@40827
  1582
  "(x :: 'a :: len0 word) <= x + y' \<Longrightarrow> x <= x + y \<Longrightarrow> 
haftmann@37660
  1583
    (x + y' < x + y) = (y' < y)" 
haftmann@37660
  1584
  by uint_arith
haftmann@37660
  1585
haftmann@37660
  1586
lemma word_plus_mono_right2: 
haftmann@40827
  1587
  "(a :: 'a :: len0 word) <= a + b \<Longrightarrow> c <= b \<Longrightarrow> a <= a + c"
haftmann@37660
  1588
  by uint_arith
haftmann@37660
  1589
haftmann@37660
  1590
lemma word_less_add_right: 
haftmann@40827
  1591
  "(x :: 'a :: len0 word) < y - z \<Longrightarrow> z <= y \<Longrightarrow> x + z < y"
haftmann@37660
  1592
  by uint_arith
haftmann@37660
  1593
haftmann@37660
  1594
lemma word_less_sub_right: 
haftmann@40827
  1595
  "(x :: 'a :: len0 word) < y + z \<Longrightarrow> y <= x \<Longrightarrow> x - y < z"
haftmann@37660
  1596
  by uint_arith
haftmann@37660
  1597
haftmann@37660
  1598
lemma word_le_plus_either: 
haftmann@40827
  1599
  "(x :: 'a :: len0 word) <= y | x <= z \<Longrightarrow> y <= y + z \<Longrightarrow> x <= y + z"
haftmann@37660
  1600
  by uint_arith
haftmann@37660
  1601
haftmann@37660
  1602
lemma word_less_nowrapI: 
haftmann@40827
  1603
  "(x :: 'a :: len0 word) < z - k \<Longrightarrow> k <= z \<Longrightarrow> 0 < k \<Longrightarrow> x < x + k"
haftmann@37660
  1604
  by uint_arith
haftmann@37660
  1605
haftmann@40827
  1606
lemma inc_le: "(i :: 'a :: len word) < m \<Longrightarrow> i + 1 <= m"
haftmann@37660
  1607
  by uint_arith
haftmann@37660
  1608
haftmann@37660
  1609
lemma inc_i: 
haftmann@40827
  1610
  "(1 :: 'a :: len word) <= i \<Longrightarrow> i < m \<Longrightarrow> 1 <= (i + 1) & i + 1 <= m"
haftmann@37660
  1611
  by uint_arith
haftmann@37660
  1612
haftmann@37660
  1613
lemma udvd_incr_lem:
haftmann@40827
  1614
  "up < uq \<Longrightarrow> up = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1615
    uq = ua + n' * uint K \<Longrightarrow> up + uint K <= uq"
haftmann@37660
  1616
  apply clarsimp
haftmann@37660
  1617
  apply (drule less_le_mult)
haftmann@37660
  1618
  apply safe
haftmann@37660
  1619
  done
haftmann@37660
  1620
haftmann@37660
  1621
lemma udvd_incr': 
haftmann@40827
  1622
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1623
    uint q = ua + n' * uint K \<Longrightarrow> p + K <= q" 
haftmann@37660
  1624
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1625
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1626
  apply (erule uint_add_le [THEN order_trans])
haftmann@37660
  1627
  done
haftmann@37660
  1628
haftmann@37660
  1629
lemma udvd_decr': 
haftmann@40827
  1630
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1631
    uint q = ua + n' * uint K \<Longrightarrow> p <= q - K"
haftmann@37660
  1632
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1633
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1634
  apply (drule le_diff_eq [THEN iffD2])
haftmann@37660
  1635
  apply (erule order_trans)
haftmann@37660
  1636
  apply (rule uint_sub_ge)
haftmann@37660
  1637
  done
haftmann@37660
  1638
huffman@45816
  1639
lemmas udvd_incr_lem0 = udvd_incr_lem [where ua=0, unfolded add_0_left]
huffman@45816
  1640
lemmas udvd_incr0 = udvd_incr' [where ua=0, unfolded add_0_left]
huffman@45816
  1641
lemmas udvd_decr0 = udvd_decr' [where ua=0, unfolded add_0_left]
haftmann@37660
  1642
haftmann@37660
  1643
lemma udvd_minus_le': 
haftmann@40827
  1644
  "xy < k \<Longrightarrow> z udvd xy \<Longrightarrow> z udvd k \<Longrightarrow> xy <= k - z"
haftmann@37660
  1645
  apply (unfold udvd_def)
haftmann@37660
  1646
  apply clarify
haftmann@37660
  1647
  apply (erule (2) udvd_decr0)
haftmann@37660
  1648
  done
haftmann@37660
  1649
haftmann@37660
  1650
lemma udvd_incr2_K: 
haftmann@40827
  1651
  "p < a + s \<Longrightarrow> a <= a + s \<Longrightarrow> K udvd s \<Longrightarrow> K udvd p - a \<Longrightarrow> a <= p \<Longrightarrow> 
haftmann@40827
  1652
    0 < K \<Longrightarrow> p <= p + K & p + K <= a + s"
wenzelm@51286
  1653
  using [[simproc del: linordered_ring_less_cancel_factor]]
haftmann@37660
  1654
  apply (unfold udvd_def)
haftmann@37660
  1655
  apply clarify
haftmann@37660
  1656
  apply (simp add: uint_arith_simps split: split_if_asm)
haftmann@37660
  1657
   prefer 2 
haftmann@37660
  1658
   apply (insert uint_range' [of s])[1]
haftmann@37660
  1659
   apply arith
haftmann@37660
  1660
  apply (drule add_commute [THEN xtr1])
haftmann@37660
  1661
  apply (simp add: diff_less_eq [symmetric])
haftmann@37660
  1662
  apply (drule less_le_mult)
haftmann@37660
  1663
   apply arith
haftmann@37660
  1664
  apply simp
haftmann@37660
  1665
  done
haftmann@37660
  1666
haftmann@37660
  1667
(* links with rbl operations *)
haftmann@37660
  1668
lemma word_succ_rbl:
haftmann@40827
  1669
  "to_bl w = bl \<Longrightarrow> to_bl (word_succ w) = (rev (rbl_succ (rev bl)))"
haftmann@37660
  1670
  apply (unfold word_succ_def)
haftmann@37660
  1671
  apply clarify
haftmann@37660
  1672
  apply (simp add: to_bl_of_bin)
huffman@46654
  1673
  apply (simp add: to_bl_def rbl_succ)
haftmann@37660
  1674
  done
haftmann@37660
  1675
haftmann@37660
  1676
lemma word_pred_rbl:
haftmann@40827
  1677
  "to_bl w = bl \<Longrightarrow> to_bl (word_pred w) = (rev (rbl_pred (rev bl)))"
haftmann@37660
  1678
  apply (unfold word_pred_def)
haftmann@37660
  1679
  apply clarify
haftmann@37660
  1680
  apply (simp add: to_bl_of_bin)
huffman@46654
  1681
  apply (simp add: to_bl_def rbl_pred)
haftmann@37660
  1682
  done
haftmann@37660
  1683
haftmann@37660
  1684
lemma word_add_rbl:
haftmann@40827
  1685
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1686
    to_bl (v + w) = (rev (rbl_add (rev vbl) (rev wbl)))"
haftmann@37660
  1687
  apply (unfold word_add_def)
haftmann@37660
  1688
  apply clarify
haftmann@37660
  1689
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1690
  apply (simp add: to_bl_def rbl_add)
haftmann@37660
  1691
  done
haftmann@37660
  1692
haftmann@37660
  1693
lemma word_mult_rbl:
haftmann@40827
  1694
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1695
    to_bl (v * w) = (rev (rbl_mult (rev vbl) (rev wbl)))"
haftmann@37660
  1696
  apply (unfold word_mult_def)
haftmann@37660
  1697
  apply clarify
haftmann@37660
  1698
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1699
  apply (simp add: to_bl_def rbl_mult)
haftmann@37660
  1700
  done
haftmann@37660
  1701
haftmann@37660
  1702
lemma rtb_rbl_ariths:
haftmann@37660
  1703
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys"
haftmann@37660
  1704
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys"
haftmann@40827
  1705
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v * w)) = rbl_mult ys xs"
haftmann@40827
  1706
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v + w)) = rbl_add ys xs"
haftmann@37660
  1707
  by (auto simp: rev_swap [symmetric] word_succ_rbl 
haftmann@37660
  1708
                 word_pred_rbl word_mult_rbl word_add_rbl)
haftmann@37660
  1709
haftmann@37660
  1710
haftmann@37660
  1711
subsection "Arithmetic type class instantiations"
haftmann@37660
  1712
haftmann@37660
  1713
lemmas word_le_0_iff [simp] =
haftmann@37660
  1714
  word_zero_le [THEN leD, THEN linorder_antisym_conv1]
haftmann@37660
  1715
haftmann@37660
  1716
lemma word_of_int_nat: 
haftmann@40827
  1717
  "0 <= x \<Longrightarrow> word_of_int x = of_nat (nat x)"
haftmann@37660
  1718
  by (simp add: of_nat_nat word_of_int)
haftmann@37660
  1719
huffman@46603
  1720
(* note that iszero_def is only for class comm_semiring_1_cancel,
huffman@46603
  1721
   which requires word length >= 1, ie 'a :: len word *) 
huffman@46603
  1722
lemma iszero_word_no [simp]:
huffman@47108
  1723
  "iszero (numeral bin :: 'a :: len word) = 
huffman@47108
  1724
    iszero (bintrunc (len_of TYPE('a)) (numeral bin))"
huffman@47108
  1725
  using word_ubin.norm_eq_iff [where 'a='a, of "numeral bin" 0]
huffman@46603
  1726
  by (simp add: iszero_def [symmetric])
huffman@47108
  1727
    
huffman@47108
  1728
text {* Use @{text iszero} to simplify equalities between word numerals. *}
huffman@47108
  1729
huffman@47108
  1730
lemmas word_eq_numeral_iff_iszero [simp] =
huffman@47108
  1731
  eq_numeral_iff_iszero [where 'a="'a::len word"]
huffman@46603
  1732
haftmann@37660
  1733
haftmann@37660
  1734
subsection "Word and nat"
haftmann@37660
  1735
huffman@45811
  1736
lemma td_ext_unat [OF refl]:
haftmann@40827
  1737
  "n = len_of TYPE ('a :: len) \<Longrightarrow> 
haftmann@37660
  1738
    td_ext (unat :: 'a word => nat) of_nat 
haftmann@37660
  1739
    (unats n) (%i. i mod 2 ^ n)"
haftmann@37660
  1740
  apply (unfold td_ext_def' unat_def word_of_nat unats_uints)
haftmann@37660
  1741
  apply (auto intro!: imageI simp add : word_of_int_hom_syms)
haftmann@37660
  1742
  apply (erule word_uint.Abs_inverse [THEN arg_cong])
haftmann@37660
  1743
  apply (simp add: int_word_uint nat_mod_distrib nat_power_eq)
haftmann@37660
  1744
  done
haftmann@37660
  1745
wenzelm@45604
  1746
lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm]
haftmann@37660
  1747
haftmann@37660
  1748
interpretation word_unat:
haftmann@37660
  1749
  td_ext "unat::'a::len word => nat" 
haftmann@37660
  1750
         of_nat 
haftmann@37660
  1751
         "unats (len_of TYPE('a::len))"
haftmann@37660
  1752
         "%i. i mod 2 ^ len_of TYPE('a::len)"
haftmann@37660
  1753
  by (rule td_ext_unat)
haftmann@37660
  1754
haftmann@37660
  1755
lemmas td_unat = word_unat.td_thm
haftmann@37660
  1756
haftmann@37660
  1757
lemmas unat_lt2p [iff] = word_unat.Rep [unfolded unats_def mem_Collect_eq]
haftmann@37660
  1758
haftmann@40827
  1759
lemma unat_le: "y <= unat (z :: 'a :: len word) \<Longrightarrow> y : unats (len_of TYPE ('a))"
haftmann@37660
  1760
  apply (unfold unats_def)
haftmann@37660
  1761
  apply clarsimp
haftmann@37660
  1762
  apply (rule xtrans, rule unat_lt2p, assumption) 
haftmann@37660
  1763
  done
haftmann@37660
  1764
haftmann@37660
  1765
lemma word_nchotomy:
haftmann@37660
  1766
  "ALL w. EX n. (w :: 'a :: len word) = of_nat n & n < 2 ^ len_of TYPE ('a)"
haftmann@37660
  1767
  apply (rule allI)
haftmann@37660
  1768
  apply (rule word_unat.Abs_cases)
haftmann@37660
  1769
  apply (unfold unats_def)
haftmann@37660
  1770
  apply auto
haftmann@37660
  1771
  done
haftmann@37660
  1772
haftmann@37660
  1773
lemma of_nat_eq:
haftmann@37660
  1774
  fixes w :: "'a::len word"
haftmann@37660
  1775
  shows "(of_nat n = w) = (\<exists>q. n = unat w + q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1776
  apply (rule trans)
haftmann@37660
  1777
   apply (rule word_unat.inverse_norm)
haftmann@37660
  1778
  apply (rule iffI)
haftmann@37660
  1779
   apply (rule mod_eqD)
haftmann@37660
  1780
   apply simp
haftmann@37660
  1781
  apply clarsimp
haftmann@37660
  1782
  done
haftmann@37660
  1783
haftmann@37660
  1784
lemma of_nat_eq_size: 
haftmann@37660
  1785
  "(of_nat n = w) = (EX q. n = unat w + q * 2 ^ size w)"
haftmann@37660
  1786
  unfolding word_size by (rule of_nat_eq)
haftmann@37660
  1787
haftmann@37660
  1788
lemma of_nat_0:
haftmann@37660
  1789
  "(of_nat m = (0::'a::len word)) = (\<exists>q. m = q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1790
  by (simp add: of_nat_eq)
haftmann@37660
  1791
huffman@45805
  1792
lemma of_nat_2p [simp]:
huffman@45805
  1793
  "of_nat (2 ^ len_of TYPE('a)) = (0::'a::len word)"
huffman@45805
  1794
  by (fact mult_1 [symmetric, THEN iffD2 [OF of_nat_0 exI]])
haftmann@37660
  1795
haftmann@40827
  1796
lemma of_nat_gt_0: "of_nat k ~= 0 \<Longrightarrow> 0 < k"
haftmann@37660
  1797
  by (cases k) auto
haftmann@37660
  1798
haftmann@37660
  1799
lemma of_nat_neq_0: 
haftmann@40827
  1800
  "0 < k \<Longrightarrow> k < 2 ^ len_of TYPE ('a :: len) \<Longrightarrow> of_nat k ~= (0 :: 'a word)"
haftmann@37660
  1801
  by (clarsimp simp add : of_nat_0)
haftmann@37660
  1802
haftmann@37660
  1803
lemma Abs_fnat_hom_add:
haftmann@37660
  1804
  "of_nat a + of_nat b = of_nat (a + b)"
haftmann@37660
  1805
  by simp
haftmann@37660
  1806
haftmann@37660
  1807
lemma Abs_fnat_hom_mult:
haftmann@37660
  1808
  "of_nat a * of_nat b = (of_nat (a * b) :: 'a :: len word)"
huffman@46013
  1809
  by (simp add: word_of_nat wi_hom_mult zmult_int)
haftmann@37660
  1810
haftmann@37660
  1811
lemma Abs_fnat_hom_Suc:
haftmann@37660
  1812
  "word_succ (of_nat a) = of_nat (Suc a)"
huffman@46013
  1813
  by (simp add: word_of_nat wi_hom_succ add_ac)
haftmann@37660
  1814
haftmann@37660
  1815
lemma Abs_fnat_hom_0: "(0::'a::len word) = of_nat 0"
huffman@45995
  1816
  by simp
haftmann@37660
  1817
haftmann@37660
  1818
lemma Abs_fnat_hom_1: "(1::'a::len word) = of_nat (Suc 0)"
huffman@45995
  1819
  by simp
haftmann@37660
  1820
haftmann@37660
  1821
lemmas Abs_fnat_homs = 
haftmann@37660
  1822
  Abs_fnat_hom_add Abs_fnat_hom_mult Abs_fnat_hom_Suc 
haftmann@37660
  1823
  Abs_fnat_hom_0 Abs_fnat_hom_1
haftmann@37660
  1824
haftmann@37660
  1825
lemma word_arith_nat_add:
haftmann@37660
  1826
  "a + b = of_nat (unat a + unat b)" 
haftmann@37660
  1827
  by simp
haftmann@37660
  1828
haftmann@37660
  1829
lemma word_arith_nat_mult:
haftmann@37660
  1830
  "a * b = of_nat (unat a * unat b)"
huffman@45995
  1831
  by (simp add: of_nat_mult)
haftmann@37660
  1832
    
haftmann@37660
  1833
lemma word_arith_nat_Suc:
haftmann@37660
  1834
  "word_succ a = of_nat (Suc (unat a))"
haftmann@37660
  1835
  by (subst Abs_fnat_hom_Suc [symmetric]) simp
haftmann@37660
  1836
haftmann@37660
  1837
lemma word_arith_nat_div:
haftmann@37660
  1838
  "a div b = of_nat (unat a div unat b)"
haftmann@37660
  1839
  by (simp add: word_div_def word_of_nat zdiv_int uint_nat)
haftmann@37660
  1840
haftmann@37660
  1841
lemma word_arith_nat_mod:
haftmann@37660
  1842
  "a mod b = of_nat (unat a mod unat b)"
haftmann@37660
  1843
  by (simp add: word_mod_def word_of_nat zmod_int uint_nat)
haftmann@37660
  1844
haftmann@37660
  1845
lemmas word_arith_nat_defs =
haftmann@37660
  1846
  word_arith_nat_add word_arith_nat_mult
haftmann@37660
  1847
  word_arith_nat_Suc Abs_fnat_hom_0
haftmann@37660
  1848
  Abs_fnat_hom_1 word_arith_nat_div
haftmann@37660
  1849
  word_arith_nat_mod 
haftmann@37660
  1850
huffman@45816
  1851
lemma unat_cong: "x = y \<Longrightarrow> unat x = unat y"
huffman@45816
  1852
  by simp
haftmann@37660
  1853
  
haftmann@37660
  1854
lemmas unat_word_ariths = word_arith_nat_defs
wenzelm@45604
  1855
  [THEN trans [OF unat_cong unat_of_nat]]
haftmann@37660
  1856
haftmann@37660
  1857
lemmas word_sub_less_iff = word_sub_le_iff
huffman@45816
  1858
  [unfolded linorder_not_less [symmetric] Not_eq_iff]
haftmann@37660
  1859
haftmann@37660
  1860
lemma unat_add_lem: 
haftmann@37660
  1861
  "(unat x + unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1862
    (unat (x + y :: 'a :: len word) = unat x + unat y)"
haftmann@37660
  1863
  unfolding unat_word_ariths
haftmann@37660
  1864
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1865
haftmann@37660
  1866
lemma unat_mult_lem: 
haftmann@37660
  1867
  "(unat x * unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1868
    (unat (x * y :: 'a :: len word) = unat x * unat y)"
haftmann@37660
  1869
  unfolding unat_word_ariths
haftmann@37660
  1870
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1871
wenzelm@45604
  1872
lemmas unat_plus_if' = trans [OF unat_word_ariths(1) mod_nat_add, simplified]
haftmann@37660
  1873
haftmann@37660
  1874
lemma le_no_overflow: 
haftmann@40827
  1875
  "x <= b \<Longrightarrow> a <= a + b \<Longrightarrow> x <= a + (b :: 'a :: len0 word)"
haftmann@37660
  1876
  apply (erule order_trans)
haftmann@37660
  1877
  apply (erule olen_add_eqv [THEN iffD1])
haftmann@37660
  1878
  done
haftmann@37660
  1879
wenzelm@45604
  1880
lemmas un_ui_le = trans [OF word_le_nat_alt [symmetric] word_le_def]
haftmann@37660
  1881
haftmann@37660
  1882
lemma unat_sub_if_size:
haftmann@37660
  1883
  "unat (x - y) = (if unat y <= unat x 
haftmann@37660
  1884
   then unat x - unat y 
haftmann@37660
  1885
   else unat x + 2 ^ size x - unat y)"
haftmann@37660
  1886
  apply (unfold word_size)
haftmann@37660
  1887
  apply (simp add: un_ui_le)
haftmann@37660
  1888
  apply (auto simp add: unat_def uint_sub_if')
haftmann@37660
  1889
   apply (rule nat_diff_distrib)
haftmann@37660
  1890
    prefer 3
haftmann@37660
  1891
    apply (simp add: algebra_simps)
haftmann@37660
  1892
    apply (rule nat_diff_distrib [THEN trans])
haftmann@37660
  1893
      prefer 3
haftmann@37660
  1894
      apply (subst nat_add_distrib)
haftmann@37660
  1895
        prefer 3
haftmann@37660
  1896
        apply (simp add: nat_power_eq)
haftmann@37660
  1897
       apply auto
haftmann@37660
  1898
  apply uint_arith
haftmann@37660
  1899
  done
haftmann@37660
  1900
haftmann@37660
  1901
lemmas unat_sub_if' = unat_sub_if_size [unfolded word_size]
haftmann@37660
  1902
haftmann@37660
  1903
lemma unat_div: "unat ((x :: 'a :: len word) div y) = unat x div unat y"
haftmann@37660
  1904
  apply (simp add : unat_word_ariths)
haftmann@37660
  1905
  apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1906
  apply (rule div_le_dividend)
haftmann@37660
  1907
  done
haftmann@37660
  1908
haftmann@37660
  1909
lemma unat_mod: "unat ((x :: 'a :: len word) mod y) = unat x mod unat y"
haftmann@37660
  1910
  apply (clarsimp simp add : unat_word_ariths)
haftmann@37660
  1911
  apply (cases "unat y")
haftmann@37660
  1912
   prefer 2
haftmann@37660
  1913
   apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1914
   apply (rule mod_le_divisor)
haftmann@37660
  1915
   apply auto
haftmann@37660
  1916
  done
haftmann@37660
  1917
haftmann@37660
  1918
lemma uint_div: "uint ((x :: 'a :: len word) div y) = uint x div uint y"
haftmann@37660
  1919
  unfolding uint_nat by (simp add : unat_div zdiv_int)
haftmann@37660
  1920
haftmann@37660
  1921
lemma uint_mod: "uint ((x :: 'a :: len word) mod y) = uint x mod uint y"
haftmann@37660
  1922
  unfolding uint_nat by (simp add : unat_mod zmod_int)
haftmann@37660
  1923
haftmann@37660
  1924
haftmann@37660
  1925
subsection {* Definition of unat\_arith tactic *}
haftmann@37660
  1926
haftmann@37660
  1927
lemma unat_split:
haftmann@37660
  1928
  fixes x::"'a::len word"
haftmann@37660
  1929
  shows "P (unat x) = 
haftmann@37660
  1930
         (ALL n. of_nat n = x & n < 2^len_of TYPE('a) --> P n)"
haftmann@37660
  1931
  by (auto simp: unat_of_nat)
haftmann@37660
  1932
haftmann@37660
  1933
lemma unat_split_asm:
haftmann@37660
  1934
  fixes x::"'a::len word"
haftmann@37660
  1935
  shows "P (unat x) = 
haftmann@37660
  1936
         (~(EX n. of_nat n = x & n < 2^len_of TYPE('a) & ~ P n))"
haftmann@37660
  1937
  by (auto simp: unat_of_nat)
haftmann@37660
  1938
haftmann@37660
  1939
lemmas of_nat_inverse = 
haftmann@37660
  1940
  word_unat.Abs_inverse' [rotated, unfolded unats_def, simplified]
haftmann@37660
  1941
haftmann@37660
  1942
lemmas unat_splits = unat_split unat_split_asm
haftmann@37660
  1943
haftmann@37660
  1944
lemmas unat_arith_simps =
haftmann@37660
  1945
  word_le_nat_alt word_less_nat_alt
haftmann@37660
  1946
  word_unat.Rep_inject [symmetric]
haftmann@37660
  1947
  unat_sub_if' unat_plus_if' unat_div unat_mod
haftmann@37660
  1948
haftmann@37660
  1949
(* unat_arith_tac: tactic to reduce word arithmetic to nat, 
haftmann@37660
  1950
   try to solve via arith *)
haftmann@37660
  1951
ML {*
wenzelm@51717
  1952
fun unat_arith_simpset ctxt = 
wenzelm@51717
  1953
  ctxt addsimps @{thms unat_arith_simps}
haftmann@37660
  1954
     delsimps @{thms word_unat.Rep_inject}
wenzelm@45620
  1955
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  1956
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  1957
haftmann@37660
  1958
fun unat_arith_tacs ctxt =   
haftmann@37660
  1959
  let
haftmann@37660
  1960
    fun arith_tac' n t =
haftmann@37660
  1961
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  1962
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  1963
  in 
wenzelm@42793
  1964
    [ clarify_tac ctxt 1,
wenzelm@51717
  1965
      full_simp_tac (unat_arith_simpset ctxt) 1,
wenzelm@51717
  1966
      ALLGOALS (full_simp_tac
wenzelm@51717
  1967
        (put_simpset HOL_ss ctxt
wenzelm@51717
  1968
          |> fold Splitter.add_split @{thms unat_splits}
wenzelm@51717
  1969
          |> fold Simplifier.add_cong @{thms power_False_cong})),
haftmann@37660
  1970
      rewrite_goals_tac @{thms word_size}, 
haftmann@37660
  1971
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  1972
                         REPEAT (etac conjE n) THEN
haftmann@37660
  1973
                         REPEAT (dtac @{thm of_nat_inverse} n THEN atac n)),
haftmann@37660
  1974
      TRYALL arith_tac' ] 
haftmann@37660
  1975
  end
haftmann@37660
  1976
haftmann@37660
  1977
fun unat_arith_tac ctxt = SELECT_GOAL (EVERY (unat_arith_tacs ctxt))
haftmann@37660
  1978
*}
haftmann@37660
  1979
haftmann@37660
  1980
method_setup unat_arith = 
haftmann@37660
  1981
  {* Scan.succeed (SIMPLE_METHOD' o unat_arith_tac) *}
haftmann@37660
  1982
  "solving word arithmetic via natural numbers and arith"
haftmann@37660
  1983
haftmann@37660
  1984
lemma no_plus_overflow_unat_size: 
haftmann@37660
  1985
  "((x :: 'a :: len word) <= x + y) = (unat x + unat y < 2 ^ size x)" 
haftmann@37660
  1986
  unfolding word_size by unat_arith
haftmann@37660
  1987
haftmann@37660
  1988
lemmas no_olen_add_nat = no_plus_overflow_unat_size [unfolded word_size]
haftmann@37660
  1989
wenzelm@45604
  1990
lemmas unat_plus_simple = trans [OF no_olen_add_nat unat_add_lem]
haftmann@37660
  1991
haftmann@37660
  1992
lemma word_div_mult: 
haftmann@40827
  1993
  "(0 :: 'a :: len word) < y \<Longrightarrow> unat x * unat y < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  1994
    x * y div y = x"
haftmann@37660
  1995
  apply unat_arith
haftmann@37660
  1996
  apply clarsimp
haftmann@37660
  1997
  apply (subst unat_mult_lem [THEN iffD1])
haftmann@37660
  1998
  apply auto
haftmann@37660
  1999
  done
haftmann@37660
  2000
haftmann@40827
  2001
lemma div_lt': "(i :: 'a :: len word) <= k div x \<Longrightarrow> 
haftmann@37660
  2002
    unat i * unat x < 2 ^ len_of TYPE('a)"
haftmann@37660
  2003
  apply unat_arith
haftmann@37660
  2004
  apply clarsimp
haftmann@37660
  2005
  apply (drule mult_le_mono1)
haftmann@37660
  2006
  apply (erule order_le_less_trans)
haftmann@37660
  2007
  apply (rule xtr7 [OF unat_lt2p div_mult_le])
haftmann@37660
  2008
  done
haftmann@37660
  2009
haftmann@37660
  2010
lemmas div_lt'' = order_less_imp_le [THEN div_lt']
haftmann@37660
  2011
haftmann@40827
  2012
lemma div_lt_mult: "(i :: 'a :: len word) < k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x < k"
haftmann@37660
  2013
  apply (frule div_lt'' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  2014
  apply (simp add: unat_arith_simps)
haftmann@37660
  2015
  apply (drule (1) mult_less_mono1)
haftmann@37660
  2016
  apply (erule order_less_le_trans)
haftmann@37660
  2017
  apply (rule div_mult_le)
haftmann@37660
  2018
  done
haftmann@37660
  2019
haftmann@37660
  2020
lemma div_le_mult: 
haftmann@40827
  2021
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x <= k"
haftmann@37660
  2022
  apply (frule div_lt' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  2023
  apply (simp add: unat_arith_simps)
haftmann@37660
  2024
  apply (drule mult_le_mono1)
haftmann@37660
  2025
  apply (erule order_trans)
haftmann@37660
  2026
  apply (rule div_mult_le)
haftmann@37660
  2027
  done
haftmann@37660
  2028
haftmann@37660
  2029
lemma div_lt_uint': 
haftmann@40827
  2030
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> uint i * uint x < 2 ^ len_of TYPE('a)"
haftmann@37660
  2031
  apply (unfold uint_nat)
haftmann@37660
  2032
  apply (drule div_lt')
haftmann@37660
  2033
  apply (simp add: zmult_int zless_nat_eq_int_zless [symmetric] 
haftmann@37660
  2034
                   nat_power_eq)
haftmann@37660
  2035
  done
haftmann@37660
  2036
haftmann@37660
  2037
lemmas div_lt_uint'' = order_less_imp_le [THEN div_lt_uint']
haftmann@37660
  2038
haftmann@37660
  2039
lemma word_le_exists': 
haftmann@40827
  2040
  "(x :: 'a :: len0 word) <= y \<Longrightarrow> 
haftmann@37660
  2041
    (EX z. y = x + z & uint x + uint z < 2 ^ len_of TYPE('a))"
haftmann@37660
  2042
  apply (rule exI)
haftmann@37660
  2043
  apply (rule conjI)
haftmann@37660
  2044
  apply (rule zadd_diff_inverse)
haftmann@37660
  2045
  apply uint_arith
haftmann@37660
  2046
  done
haftmann@37660
  2047
haftmann@37660
  2048
lemmas plus_minus_not_NULL = order_less_imp_le [THEN plus_minus_not_NULL_ab]
haftmann@37660
  2049
haftmann@37660
  2050
lemmas plus_minus_no_overflow =
haftmann@37660
  2051
  order_less_imp_le [THEN plus_minus_no_overflow_ab]
haftmann@37660
  2052
  
haftmann@37660
  2053
lemmas mcs = word_less_minus_cancel word_less_minus_mono_left
haftmann@37660
  2054
  word_le_minus_cancel word_le_minus_mono_left
haftmann@37660
  2055
wenzelm@45604
  2056
lemmas word_l_diffs = mcs [where y = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2057
lemmas word_diff_ls = mcs [where z = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2058
lemmas word_plus_mcs = word_diff_ls [where y = "v + x", unfolded add_diff_cancel] for v x
haftmann@37660
  2059
haftmann@37660
  2060
lemmas le_unat_uoi = unat_le [THEN word_unat.Abs_inverse]
haftmann@37660
  2061
haftmann@37660
  2062
lemmas thd = refl [THEN [2] split_div_lemma [THEN iffD2], THEN conjunct1]
haftmann@37660
  2063
haftmann@37660
  2064
lemma thd1:
haftmann@37660
  2065
  "a div b * b \<le> (a::nat)"
haftmann@37660
  2066
  using gt_or_eq_0 [of b]
haftmann@37660
  2067
  apply (rule disjE)
haftmann@37660
  2068
   apply (erule xtr4 [OF thd mult_commute])
haftmann@37660
  2069
  apply clarsimp
haftmann@37660
  2070
  done
haftmann@37660
  2071
wenzelm@45604
  2072
lemmas uno_simps [THEN le_unat_uoi] = mod_le_divisor div_le_dividend thd1 
haftmann@37660
  2073
haftmann@37660
  2074
lemma word_mod_div_equality:
haftmann@37660
  2075
  "(n div b) * b + (n mod b) = (n :: 'a :: len word)"
haftmann@37660
  2076
  apply (unfold word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2077
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2078
  apply (erule disjE)
haftmann@37660
  2079
   apply (simp add: mod_div_equality uno_simps)
haftmann@37660
  2080
  apply simp
haftmann@37660
  2081
  done
haftmann@37660
  2082
haftmann@37660
  2083
lemma word_div_mult_le: "a div b * b <= (a::'a::len word)"
haftmann@37660
  2084
  apply (unfold word_le_nat_alt word_arith_nat_defs)
haftmann@37660
  2085
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2086
  apply (erule disjE)
haftmann@37660
  2087
   apply (simp add: div_mult_le uno_simps)
haftmann@37660
  2088
  apply simp
haftmann@37660
  2089
  done
haftmann@37660
  2090
haftmann@40827
  2091
lemma word_mod_less_divisor: "0 < n \<Longrightarrow> m mod n < (n :: 'a :: len word)"
haftmann@37660
  2092
  apply (simp only: word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2093
  apply (clarsimp simp add : uno_simps)
haftmann@37660
  2094
  done
haftmann@37660
  2095
haftmann@37660
  2096
lemma word_of_int_power_hom: 
haftmann@37660
  2097
  "word_of_int a ^ n = (word_of_int (a ^ n) :: 'a :: len word)"
huffman@45995
  2098
  by (induct n) (simp_all add: wi_hom_mult [symmetric])
haftmann@37660
  2099
haftmann@37660
  2100
lemma word_arith_power_alt: 
haftmann@37660
  2101
  "a ^ n = (word_of_int (uint a ^ n) :: 'a :: len word)"
haftmann@37660
  2102
  by (simp add : word_of_int_power_hom [symmetric])
haftmann@37660
  2103
haftmann@37660
  2104
lemma of_bl_length_less: 
haftmann@40827
  2105
  "length x = k \<Longrightarrow> k < len_of TYPE('a) \<Longrightarrow> (of_bl x :: 'a :: len word) < 2 ^ k"
huffman@47108
  2106
  apply (unfold of_bl_def word_less_alt word_numeral_alt)
haftmann@37660
  2107
  apply safe
haftmann@37660
  2108
  apply (simp (no_asm) add: word_of_int_power_hom word_uint.eq_norm 
huffman@47108
  2109
                       del: word_of_int_numeral)
haftmann@37660
  2110
  apply (simp add: mod_pos_pos_trivial)
haftmann@37660
  2111
  apply (subst mod_pos_pos_trivial)
haftmann@37660
  2112
    apply (rule bl_to_bin_ge0)
haftmann@37660
  2113
   apply (rule order_less_trans)
haftmann@37660
  2114
    apply (rule bl_to_bin_lt2p)
haftmann@37660
  2115
   apply simp
huffman@46646
  2116
  apply (rule bl_to_bin_lt2p)
haftmann@37660
  2117
  done
haftmann@37660
  2118
haftmann@37660
  2119
haftmann@37660
  2120
subsection "Cardinality, finiteness of set of words"
haftmann@37660
  2121
huffman@45809
  2122
instance word :: (len0) finite
huffman@45809
  2123
  by (default, simp add: type_definition.univ [OF type_definition_word])
huffman@45809
  2124
huffman@45809
  2125
lemma card_word: "CARD('a::len0 word) = 2 ^ len_of TYPE('a)"
huffman@45809
  2126
  by (simp add: type_definition.card [OF type_definition_word] nat_power_eq)
haftmann@37660
  2127
haftmann@37660
  2128
lemma card_word_size: 
huffman@45809
  2129
  "card (UNIV :: 'a :: len0 word set) = (2 ^ size (x :: 'a word))"
haftmann@37660
  2130
unfolding word_size by (rule card_word)
haftmann@37660
  2131
haftmann@37660
  2132
haftmann@37660
  2133
subsection {* Bitwise Operations on Words *}
haftmann@37660
  2134
haftmann@37660
  2135
lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or
haftmann@37660
  2136
  
haftmann@37660
  2137
(* following definitions require both arithmetic and bit-wise word operations *)
haftmann@37660
  2138
haftmann@37660
  2139
(* to get word_no_log_defs from word_log_defs, using bin_log_bintrs *)
haftmann@37660
  2140
lemmas wils1 = bin_log_bintrs [THEN word_ubin.norm_eq_iff [THEN iffD1],
wenzelm@45604
  2141
  folded word_ubin.eq_norm, THEN eq_reflection]
haftmann@37660
  2142
haftmann@37660
  2143
(* the binary operations only *)
huffman@46013
  2144
(* BH: why is this needed? *)
haftmann@37660
  2145
lemmas word_log_binary_defs = 
haftmann@37660
  2146
  word_and_def word_or_def word_xor_def
haftmann@37660
  2147
huffman@46011
  2148
lemma word_wi_log_defs:
huffman@46011
  2149
  "NOT word_of_int a = word_of_int (NOT a)"
huffman@46011
  2150
  "word_of_int a AND word_of_int b = word_of_int (a AND b)"
huffman@46011
  2151
  "word_of_int a OR word_of_int b = word_of_int (a OR b)"
huffman@46011
  2152
  "word_of_int a XOR word_of_int b = word_of_int (a XOR b)"
huffman@47374
  2153
  by (transfer, rule refl)+
huffman@47372
  2154
huffman@46011
  2155
lemma word_no_log_defs [simp]:
huffman@47108
  2156
  "NOT (numeral a) = word_of_int (NOT (numeral a))"
huffman@47108
  2157
  "NOT (neg_numeral a) = word_of_int (NOT (neg_numeral a))"
huffman@47108
  2158
  "numeral a AND numeral b = word_of_int (numeral a AND numeral b)"
huffman@47108
  2159
  "numeral a AND neg_numeral b = word_of_int (numeral a AND neg_numeral b)"
huffman@47108
  2160
  "neg_numeral a AND numeral b = word_of_int (neg_numeral a AND numeral b)"
huffman@47108
  2161
  "neg_numeral a AND neg_numeral b = word_of_int (neg_numeral a AND neg_numeral b)"
huffman@47108
  2162
  "numeral a OR numeral b = word_of_int (numeral a OR numeral b)"
huffman@47108
  2163
  "numeral a OR neg_numeral b = word_of_int (numeral a OR neg_numeral b)"
huffman@47108
  2164
  "neg_numeral a OR numeral b = word_of_int (neg_numeral a OR numeral b)"
huffman@47108
  2165
  "neg_numeral a OR neg_numeral b = word_of_int (neg_numeral a OR neg_numeral b)"
huffman@47108
  2166
  "numeral a XOR numeral b = word_of_int (numeral a XOR numeral b)"
huffman@47108
  2167
  "numeral a XOR neg_numeral b = word_of_int (numeral a XOR neg_numeral b)"
huffman@47108
  2168
  "neg_numeral a XOR numeral b = word_of_int (neg_numeral a XOR numeral b)"
huffman@47108
  2169
  "neg_numeral a XOR neg_numeral b = word_of_int (neg_numeral a XOR neg_numeral b)"
huffman@47372
  2170
  by (transfer, rule refl)+
haftmann@37660
  2171
huffman@46064
  2172
text {* Special cases for when one of the arguments equals 1. *}
huffman@46064
  2173
huffman@46064
  2174
lemma word_bitwise_1_simps [simp]:
huffman@46064
  2175
  "NOT (1::'a::len0 word) = -2"
huffman@47108
  2176
  "1 AND numeral b = word_of_int (1 AND numeral b)"
huffman@47108
  2177
  "1 AND neg_numeral b = word_of_int (1 AND neg_numeral b)"
huffman@47108
  2178
  "numeral a AND 1 = word_of_int (numeral a AND 1)"
huffman@47108
  2179
  "neg_numeral a AND 1 = word_of_int (neg_numeral a AND 1)"
huffman@47108
  2180
  "1 OR numeral b = word_of_int (1 OR numeral b)"
huffman@47108
  2181
  "1 OR neg_numeral b = word_of_int (1 OR neg_numeral b)"
huffman@47108
  2182
  "numeral a OR 1 = word_of_int (numeral a OR 1)"
huffman@47108
  2183
  "neg_numeral a OR 1 = word_of_int (neg_numeral a OR 1)"
huffman@47108
  2184
  "1 XOR numeral b = word_of_int (1 XOR numeral b)"
huffman@47108
  2185
  "1 XOR neg_numeral b = word_of_int (1 XOR neg_numeral b)"
huffman@47108
  2186
  "numeral a XOR 1 = word_of_int (numeral a XOR 1)"
huffman@47108
  2187
  "neg_numeral a XOR 1 = word_of_int (neg_numeral a XOR 1)"
huffman@47372
  2188
  by (transfer, simp)+
huffman@46064
  2189
haftmann@37660
  2190
lemma uint_or: "uint (x OR y) = (uint x) OR (uint y)"
huffman@47372
  2191
  by (transfer, simp add: bin_trunc_ao)
haftmann@37660
  2192
haftmann@37660
  2193
lemma uint_and: "uint (x AND y) = (uint x) AND (uint y)"
huffman@47372
  2194
  by (transfer, simp add: bin_trunc_ao)
huffman@47372
  2195
huffman@47372
  2196
lemma test_bit_wi [simp]:
huffman@47372
  2197
  "(word_of_int x::'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a) \<and> bin_nth x n"
huffman@47372
  2198
  unfolding word_test_bit_def
huffman@47372
  2199
  by (simp add: word_ubin.eq_norm nth_bintr)
huffman@47372
  2200
huffman@47372
  2201
lemma word_test_bit_transfer [transfer_rule]:
kuncar@51375
  2202
  "(fun_rel pcr_word (fun_rel op = op =))
huffman@47372
  2203
    (\<lambda>x n. n < len_of TYPE('a) \<and> bin_nth x n) (test_bit :: 'a::len0 word \<Rightarrow> _)"
kuncar@51375
  2204
  unfolding fun_rel_def word.pcr_cr_eq cr_word_def by simp
haftmann@37660
  2205
haftmann@37660
  2206
lemma word_ops_nth_size:
haftmann@40827
  2207
  "n < size (x::'a::len0 word) \<Longrightarrow> 
haftmann@37660
  2208
    (x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2209
    (x AND y) !! n = (x !! n & y !! n) & 
haftmann@37660
  2210
    (x XOR y) !! n = (x !! n ~= y !! n) & 
haftmann@37660
  2211
    (NOT x) !! n = (~ x !! n)"
huffman@47372
  2212
  unfolding word_size by transfer (simp add: bin_nth_ops)
haftmann@37660
  2213
haftmann@37660
  2214
lemma word_ao_nth:
haftmann@37660
  2215
  fixes x :: "'a::len0 word"
haftmann@37660
  2216
  shows "(x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2217
         (x AND y) !! n = (x !! n & y !! n)"
huffman@47372
  2218
  by transfer (auto simp add: bin_nth_ops)
huffman@46023
  2219
huffman@47108
  2220
lemma test_bit_numeral [simp]:
huffman@47108
  2221
  "(numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
huffman@47108
  2222
    n < len_of TYPE('a) \<and> bin_nth (numeral w) n"
huffman@47372
  2223
  by transfer (rule refl)
huffman@47108
  2224
huffman@47108
  2225
lemma test_bit_neg_numeral [simp]:
huffman@47108
  2226
  "(neg_numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
huffman@47108
  2227
    n < len_of TYPE('a) \<and> bin_nth (neg_numeral w) n"
huffman@47372
  2228
  by transfer (rule refl)
huffman@46023
  2229
huffman@46172
  2230
lemma test_bit_1 [simp]: "(1::'a::len word) !! n \<longleftrightarrow> n = 0"
huffman@47372
  2231
  by transfer auto
huffman@46172
  2232
  
huffman@46023
  2233
lemma nth_0 [simp]: "~ (0::'a::len0 word) !! n"
huffman@47372
  2234
  by transfer simp
huffman@46023
  2235
huffman@47108
  2236
lemma nth_minus1 [simp]: "(-1::'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a)"
huffman@47372
  2237
  by transfer simp
huffman@47108
  2238
haftmann@37660
  2239
(* get from commutativity, associativity etc of int_and etc
haftmann@37660
  2240
  to same for word_and etc *)
haftmann@37660
  2241
haftmann@37660
  2242
lemmas bwsimps = 
huffman@46013
  2243
  wi_hom_add
haftmann@37660
  2244
  word_wi_log_defs
haftmann@37660
  2245
haftmann@37660
  2246
lemma word_bw_assocs:
haftmann@37660
  2247
  fixes x :: "'a::len0 word"
haftmann@37660
  2248
  shows
haftmann@37660
  2249
  "(x AND y) AND z = x AND y AND z"
haftmann@37660
  2250
  "(x OR y) OR z = x OR y OR z"
haftmann@37660
  2251
  "(x XOR y) XOR z = x XOR y XOR z"
huffman@46022
  2252
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2253
  
haftmann@37660
  2254
lemma word_bw_comms:
haftmann@37660
  2255
  fixes x :: "'a::len0 word"
haftmann@37660
  2256
  shows
haftmann@37660
  2257
  "x AND y = y AND x"
haftmann@37660
  2258
  "x OR y = y OR x"
haftmann@37660
  2259
  "x XOR y = y XOR x"
huffman@46022
  2260
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2261
  
haftmann@37660
  2262
lemma word_bw_lcs:
haftmann@37660
  2263
  fixes x :: "'a::len0 word"
haftmann@37660
  2264
  shows
haftmann@37660
  2265
  "y AND x AND z = x AND y AND z"
haftmann@37660
  2266
  "y OR x OR z = x OR y OR z"
haftmann@37660
  2267
  "y XOR x XOR z = x XOR y XOR z"
huffman@46022
  2268
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2269
haftmann@37660
  2270
lemma word_log_esimps [simp]:
haftmann@37660
  2271
  fixes x :: "'a::len0 word"
haftmann@37660
  2272
  shows
haftmann@37660
  2273
  "x AND 0 = 0"
haftmann@37660
  2274
  "x AND -1 = x"
haftmann@37660
  2275
  "x OR 0 = x"
haftmann@37660
  2276
  "x OR -1 = -1"
haftmann@37660
  2277
  "x XOR 0 = x"
haftmann@37660
  2278
  "x XOR -1 = NOT x"
haftmann@37660
  2279
  "0 AND x = 0"
haftmann@37660
  2280
  "-1 AND x = x"
haftmann@37660
  2281
  "0 OR x = x"
haftmann@37660
  2282
  "-1 OR x = -1"
haftmann@37660
  2283
  "0 XOR x = x"
haftmann@37660
  2284
  "-1 XOR x = NOT x"
huffman@46023
  2285
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2286
haftmann@37660
  2287
lemma word_not_dist:
haftmann@37660
  2288
  fixes x :: "'a::len0 word"
haftmann@37660
  2289
  shows
haftmann@37660
  2290
  "NOT (x OR y) = NOT x AND NOT y"
haftmann@37660
  2291
  "NOT (x AND y) = NOT x OR NOT y"
huffman@46022
  2292
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2293
haftmann@37660
  2294
lemma word_bw_same:
haftmann@37660
  2295
  fixes x :: "'a::len0 word"
haftmann@37660
  2296
  shows
haftmann@37660
  2297
  "x AND x = x"
haftmann@37660
  2298
  "x OR x = x"
haftmann@37660
  2299
  "x XOR x = 0"
huffman@46023
  2300
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2301
haftmann@37660
  2302
lemma word_ao_absorbs [simp]:
haftmann@37660
  2303
  fixes x :: "'a::len0 word"
haftmann@37660
  2304
  shows
haftmann@37660
  2305
  "x AND (y OR x) = x"
haftmann@37660
  2306
  "x OR y AND x = x"
haftmann@37660
  2307
  "x AND (x OR y) = x"
haftmann@37660
  2308
  "y AND x OR x = x"
haftmann@37660
  2309
  "(y OR x) AND x = x"
haftmann@37660
  2310
  "x OR x AND y = x"
haftmann@37660
  2311
  "(x OR y) AND x = x"
haftmann@37660
  2312
  "x AND y OR x = x"
huffman@46022
  2313
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2314
haftmann@37660
  2315
lemma word_not_not [simp]:
haftmann@37660
  2316
  "NOT NOT (x::'a::len0 word) = x"
huffman@46022
  2317
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2318
haftmann@37660
  2319
lemma word_ao_dist:
haftmann@37660
  2320
  fixes x :: "'a::len0 word"
haftmann@37660
  2321
  shows "(x OR y) AND z = x AND z OR y AND z"
huffman@46022
  2322
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2323
haftmann@37660
  2324
lemma word_oa_dist:
haftmann@37660
  2325
  fixes x :: "'a::len0 word"
haftmann@37660
  2326
  shows "x AND y OR z = (x OR z) AND (y OR z)"
huffman@46022
  2327
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2328
haftmann@37660
  2329
lemma word_add_not [simp]: 
haftmann@37660
  2330
  fixes x :: "'a::len0 word"
haftmann@37660
  2331
  shows "x + NOT x = -1"
huffman@47372
  2332
  by transfer (simp add: bin_add_not)
haftmann@37660
  2333
haftmann@37660
  2334
lemma word_plus_and_or [simp]:
haftmann@37660
  2335
  fixes x :: "'a::len0 word"
haftmann@37660
  2336
  shows "(x AND y) + (x OR y) = x + y"
huffman@47372
  2337
  by transfer (simp add: plus_and_or)
haftmann@37660
  2338
haftmann@37660
  2339
lemma leoa:   
haftmann@37660
  2340
  fixes x :: "'a::len0 word"
haftmann@40827
  2341
  shows "(w = (x OR y)) \<Longrightarrow> (y = (w AND y))" by auto
haftmann@37660
  2342
lemma leao: 
haftmann@37660
  2343
  fixes x' :: "'a::len0 word"
haftmann@40827
  2344
  shows "(w' = (x' AND y')) \<Longrightarrow> (x' = (x' OR w'))" by auto 
haftmann@37660
  2345
wenzelm@48196
  2346
lemma word_ao_equiv:
wenzelm@48196
  2347
  fixes w w' :: "'a::len0 word"
wenzelm@48196
  2348
  shows "(w = w OR w') = (w' = w AND w')"
wenzelm@48196
  2349
  by (auto intro: leoa leao)
haftmann@37660
  2350
haftmann@37660
  2351
lemma le_word_or2: "x <= x OR (y::'a::len0 word)"
haftmann@37660
  2352
  unfolding word_le_def uint_or
haftmann@37660
  2353
  by (auto intro: le_int_or) 
haftmann@37660
  2354
wenzelm@45604
  2355
lemmas le_word_or1 = xtr3 [OF word_bw_comms (2) le_word_or2]
wenzelm@45604
  2356
lemmas word_and_le1 = xtr3 [OF word_ao_absorbs (4) [symmetric] le_word_or2]
wenzelm@45604
  2357
lemmas word_and_le2 = xtr3 [OF word_ao_absorbs (8) [symmetric] le_word_or2]
haftmann@37660
  2358
haftmann@37660
  2359
lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)" 
huffman@45550
  2360
  unfolding to_bl_def word_log_defs bl_not_bin
huffman@45550
  2361
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2362
haftmann@37660
  2363
lemma bl_word_xor: "to_bl (v XOR w) = map2 op ~= (to_bl v) (to_bl w)" 
haftmann@37660
  2364
  unfolding to_bl_def word_log_defs bl_xor_bin
huffman@45550
  2365
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2366
haftmann@37660
  2367
lemma bl_word_or: "to_bl (v OR w) = map2 op | (to_bl v) (to_bl w)" 
huffman@45550
  2368
  unfolding to_bl_def word_log_defs bl_or_bin
huffman@45550
  2369
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2370
haftmann@37660
  2371
lemma bl_word_and: "to_bl (v AND w) = map2 op & (to_bl v) (to_bl w)" 
huffman@45550
  2372
  unfolding to_bl_def word_log_defs bl_and_bin
huffman@45550
  2373
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2374
haftmann@37660
  2375
lemma word_lsb_alt: "lsb (w::'a::len0 word) = test_bit w 0"
haftmann@37660
  2376
  by (auto simp: word_test_bit_def word_lsb_def)
haftmann@37660
  2377
huffman@45805
  2378
lemma word_lsb_1_0 [simp]: "lsb (1::'a::len word) & ~ lsb (0::'b::len0 word)"
huffman@45550
  2379
  unfolding word_lsb_def uint_eq_0 uint_1 by simp
haftmann@37660
  2380
haftmann@37660
  2381
lemma word_lsb_last: "lsb (w::'a::len word) = last (to_bl w)"
haftmann@37660
  2382
  apply (unfold word_lsb_def uint_bl bin_to_bl_def) 
haftmann@37660
  2383
  apply (rule_tac bin="uint w" in bin_exhaust)
haftmann@37660
  2384
  apply (cases "size w")
haftmann@37660
  2385
   apply auto
haftmann@37660
  2386
   apply (auto simp add: bin_to_bl_aux_alt)
haftmann@37660
  2387
  done
haftmann@37660
  2388
haftmann@37660
  2389
lemma word_lsb_int: "lsb w = (uint w mod 2 = 1)"
huffman@45529
  2390
  unfolding word_lsb_def bin_last_def by auto
haftmann@37660
  2391
haftmann@37660
  2392
lemma word_msb_sint: "msb w = (sint w < 0)" 
huffman@46604
  2393
  unfolding word_msb_def sign_Min_lt_0 ..
haftmann@37660
  2394
huffman@46173
  2395
lemma msb_word_of_int:
huffman@46173
  2396
  "msb (word_of_int x::'a::len word) = bin_nth x (len_of TYPE('a) - 1)"
huffman@46173
  2397
  unfolding word_msb_def by (simp add: word_sbin.eq_norm bin_sign_lem)
huffman@46173
  2398
huffman@47108
  2399
lemma word_msb_numeral [simp]:
huffman@47108
  2400
  "msb (numeral w::'a::len word) = bin_nth (numeral w) (len_of TYPE('a) - 1)"
huffman@47108
  2401
  unfolding word_numeral_alt by (rule msb_word_of_int)
huffman@47108
  2402
huffman@47108
  2403
lemma word_msb_neg_numeral [simp]:
huffman@47108
  2404
  "msb (neg_numeral w::'a::len word) = bin_nth (neg_numeral w) (len_of TYPE('a) - 1)"
huffman@47108
  2405
  unfolding word_neg_numeral_alt by (rule msb_word_of_int)
huffman@46173
  2406
huffman@46173
  2407
lemma word_msb_0 [simp]: "\<not> msb (0::'a::len word)"
huffman@46173
  2408
  unfolding word_msb_def by simp
huffman@46173
  2409
huffman@46173
  2410
lemma word_msb_1 [simp]: "msb (1::'a::len word) \<longleftrightarrow> len_of TYPE('a) = 1"
huffman@46173
  2411
  unfolding word_1_wi msb_word_of_int eq_iff [where 'a=nat]
huffman@46173
  2412
  by (simp add: Suc_le_eq)
huffman@45811
  2413
huffman@45811
  2414
lemma word_msb_nth:
huffman@45811
  2415
  "msb (w::'a::len word) = bin_nth (uint w) (len_of TYPE('a) - 1)"
huffman@46023
  2416
  unfolding word_msb_def sint_uint by (simp add: bin_sign_lem)
haftmann@37660
  2417
haftmann@37660
  2418
lemma word_msb_alt: "msb (w::'a::len word) = hd (to_bl w)"
haftmann@37660
  2419
  apply (unfold word_msb_nth uint_bl)
haftmann@37660
  2420
  apply (subst hd_conv_nth)
haftmann@37660
  2421
  apply (rule length_greater_0_conv [THEN iffD1])
haftmann@37660
  2422
   apply simp
haftmann@37660
  2423
  apply (simp add : nth_bin_to_bl word_size)
haftmann@37660
  2424
  done
haftmann@37660
  2425
huffman@45805
  2426
lemma word_set_nth [simp]:
haftmann@37660
  2427
  "set_bit w n (test_bit w n) = (w::'a::len0 word)"
haftmann@37660
  2428
  unfolding word_test_bit_def word_set_bit_def by auto
haftmann@37660
  2429
haftmann@37660
  2430
lemma bin_nth_uint':
haftmann@37660
  2431
  "bin_nth (uint w) n = (rev (bin_to_bl (size w) (uint w)) ! n & n < size w)"
haftmann@37660
  2432
  apply (unfold word_size)
haftmann@37660
  2433
  apply (safe elim!: bin_nth_uint_imp)
haftmann@37660
  2434
   apply (frule bin_nth_uint_imp)
haftmann@37660
  2435
   apply (fast dest!: bin_nth_bl)+
haftmann@37660
  2436
  done
haftmann@37660
  2437
haftmann@37660
  2438
lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size]
haftmann@37660
  2439
haftmann@37660
  2440
lemma test_bit_bl: "w !! n = (rev (to_bl w) ! n & n < size w)"
haftmann@37660
  2441
  unfolding to_bl_def word_test_bit_def word_size
haftmann@37660
  2442
  by (rule bin_nth_uint)
haftmann@37660
  2443
haftmann@40827
  2444
lemma to_bl_nth: "n < size w \<Longrightarrow> to_bl w ! n = w !! (size w - Suc n)"
haftmann@37660
  2445
  apply (unfold test_bit_bl)
haftmann@37660
  2446
  apply clarsimp
haftmann@37660
  2447
  apply (rule trans)
haftmann@37660
  2448
   apply (rule nth_rev_alt)
haftmann@37660
  2449
   apply (auto simp add: word_size)
haftmann@37660
  2450
  done
haftmann@37660
  2451
haftmann@37660
  2452
lemma test_bit_set: 
haftmann@37660
  2453
  fixes w :: "'a::len0 word"
haftmann@37660
  2454
  shows "(set_bit w n x) !! n = (n < size w & x)"
haftmann@37660
  2455
  unfolding word_size word_test_bit_def word_set_bit_def
haftmann@37660
  2456
  by (clarsimp simp add : word_ubin.eq_norm nth_bintr)
haftmann@37660
  2457
haftmann@37660
  2458
lemma test_bit_set_gen: 
haftmann@37660
  2459
  fixes w :: "'a::len0 word"
haftmann@37660
  2460
  shows "test_bit (set_bit w n x) m = 
haftmann@37660
  2461
         (if m = n then n < size w & x else test_bit w m)"
haftmann@37660
  2462
  apply (unfold word_size word_test_bit_def word_set_bit_def)
haftmann@37660
  2463
  apply (clarsimp simp add: word_ubin.eq_norm nth_bintr bin_nth_sc_gen)
haftmann@37660
  2464
  apply (auto elim!: test_bit_size [unfolded word_size]
haftmann@37660
  2465
              simp add: word_test_bit_def [symmetric])
haftmann@37660
  2466
  done
haftmann@37660
  2467
haftmann@37660
  2468
lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs"
haftmann@37660
  2469
  unfolding of_bl_def bl_to_bin_rep_F by auto
haftmann@37660
  2470
  
huffman@45811
  2471
lemma msb_nth:
haftmann@37660
  2472
  fixes w :: "'a::len word"
huffman@45811
  2473
  shows "msb w = w !! (len_of TYPE('a) - 1)"
huffman@45811
  2474
  unfolding word_msb_nth word_test_bit_def by simp
haftmann@37660
  2475
wenzelm@45604
  2476
lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2477
lemmas msb1 = msb0 [where i = 0]
haftmann@37660
  2478
lemmas word_ops_msb = msb1 [unfolded msb_nth [symmetric, unfolded One_nat_def]]
haftmann@37660
  2479
wenzelm@45604
  2480
lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2481
lemmas word_ops_lsb = lsb0 [unfolded word_lsb_alt]
haftmann@37660
  2482
huffman@45811
  2483
lemma td_ext_nth [OF refl refl refl, unfolded word_size]:
haftmann@40827
  2484
  "n = size (w::'a::len0 word) \<Longrightarrow> ofn = set_bits \<Longrightarrow> [w, ofn g] = l \<Longrightarrow> 
haftmann@37660
  2485
    td_ext test_bit ofn {f. ALL i. f i --> i < n} (%h i. h i & i < n)"
haftmann@37660
  2486
  apply (unfold word_size td_ext_def')
wenzelm@46008
  2487
  apply safe
haftmann@37660
  2488
     apply (rule_tac [3] ext)
haftmann@37660
  2489
     apply (rule_tac [4] ext)
haftmann@37660
  2490
     apply (unfold word_size of_nth_def test_bit_bl)
haftmann@37660
  2491
     apply safe
haftmann@37660
  2492
       defer
haftmann@37660
  2493
       apply (clarsimp simp: word_bl.Abs_inverse)+
haftmann@37660
  2494
  apply (rule word_bl.Rep_inverse')
haftmann@37660
  2495
  apply (rule sym [THEN trans])
haftmann@37660
  2496
  apply (rule bl_of_nth_nth)
haftmann@37660
  2497
  apply simp
haftmann@37660
  2498
  apply (rule bl_of_nth_inj)
haftmann@37660
  2499
  apply (clarsimp simp add : test_bit_bl word_size)
haftmann@37660
  2500
  done
haftmann@37660
  2501
haftmann@37660
  2502
interpretation test_bit:
haftmann@37660
  2503
  td_ext "op !! :: 'a::len0 word => nat => bool"
haftmann@37660
  2504
         set_bits
haftmann@37660
  2505
         "{f. \<forall>i. f i \<longrightarrow> i < len_of TYPE('a::len0)}"
haftmann@37660
  2506
         "(\<lambda>h i. h i \<and> i < len_of TYPE('a::len0))"
haftmann@37660
  2507
  by (rule td_ext_nth)
haftmann@37660
  2508
haftmann@37660
  2509
lemmas td_nth = test_bit.td_thm
haftmann@37660
  2510
huffman@45805
  2511
lemma word_set_set_same [simp]:
haftmann@37660
  2512
  fixes w :: "'a::len0 word"
haftmann@37660
  2513
  shows "set_bit (set_bit w n x) n y = set_bit w n y" 
haftmann@37660
  2514
  by (rule word_eqI) (simp add : test_bit_set_gen word_size)
haftmann@37660
  2515
    
haftmann@37660
  2516
lemma word_set_set_diff: 
haftmann@37660
  2517
  fixes w :: "'a::len0 word"
haftmann@37660
  2518
  assumes "m ~= n"
haftmann@37660
  2519
  shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x" 
wenzelm@41550
  2520
  by (rule word_eqI) (clarsimp simp add: test_bit_set_gen word_size assms)
huffman@46001
  2521
haftmann@37660
  2522
lemma nth_sint: 
haftmann@37660
  2523
  fixes w :: "'a::len word"
haftmann@37660
  2524
  defines "l \<equiv> len_of TYPE ('a)"
haftmann@37660
  2525
  shows "bin_nth (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"
haftmann@37660
  2526
  unfolding sint_uint l_def
haftmann@37660
  2527
  by (clarsimp simp add: nth_sbintr word_test_bit_def [symmetric])
haftmann@37660
  2528
huffman@47108
  2529
lemma word_lsb_numeral [simp]:
huffman@47108
  2530
  "lsb (numeral bin :: 'a :: len word) = (bin_last (numeral bin) = 1)"
huffman@47108
  2531
  unfolding word_lsb_alt test_bit_numeral by simp
huffman@47108
  2532
huffman@47108
  2533
lemma word_lsb_neg_numeral [simp]:
huffman@47108
  2534
  "lsb (neg_numeral bin :: 'a :: len word) = (bin_last (neg_numeral bin) = 1)"
huffman@47108
  2535
  unfolding word_lsb_alt test_bit_neg_numeral by simp
haftmann@37660
  2536
huffman@46173
  2537
lemma set_bit_word_of_int:
huffman@46173
  2538
  "set_bit (word_of_int x) n b = word_of_int (bin_sc n (if b then 1 else 0) x)"
huffman@46173
  2539
  unfolding word_set_bit_def
huffman@46173
  2540
  apply (rule word_eqI)
huffman@46173
  2541
  apply (simp add: word_size bin_nth_sc_gen word_ubin.eq_norm nth_bintr)
huffman@46173
  2542
  done
huffman@46173
  2543
huffman@47108
  2544
lemma word_set_numeral [simp]:
huffman@47108
  2545
  "set_bit (numeral bin::'a::len0 word) n b = 
huffman@47108
  2546
    word_of_int (bin_sc n (if b then 1 else 0) (numeral bin))"
huffman@47108
  2547
  unfolding word_numeral_alt by (rule set_bit_word_of_int)
huffman@47108
  2548
huffman@47108
  2549
lemma word_set_neg_numeral [simp]:
huffman@47108
  2550
  "set_bit (neg_numeral bin::'a::len0 word) n b = 
huffman@47108
  2551
    word_of_int (bin_sc n (if b then 1 else 0) (neg_numeral bin))"
huffman@47108
  2552
  unfolding word_neg_numeral_alt by (rule set_bit_word_of_int)
huffman@46173
  2553
huffman@46173
  2554
lemma word_set_bit_0 [simp]:
huffman@46173
  2555
  "set_bit 0 n b = word_of_int (bin_sc n (if b then 1 else 0) 0)"
huffman@46173
  2556
  unfolding word_0_wi by (rule set_bit_word_of_int)
huffman@46173
  2557
huffman@46173
  2558
lemma word_set_bit_1 [simp]:
huffman@46173
  2559
  "set_bit 1 n b = word_of_int (bin_sc n (if b then 1 else 0) 1)"
huffman@46173
  2560
  unfolding word_1_wi by (rule set_bit_word_of_int)
haftmann@37660
  2561
huffman@45805
  2562
lemma setBit_no [simp]:
huffman@47108
  2563
  "setBit (numeral bin) n = word_of_int (bin_sc n 1 (numeral bin))"
huffman@45805
  2564
  by (simp add: setBit_def)
huffman@45805
  2565
huffman@45805
  2566
lemma clearBit_no [simp]:
huffman@47108
  2567
  "clearBit (numeral bin) n = word_of_int (bin_sc n 0 (numeral bin))"
huffman@45805
  2568
  by (simp add: clearBit_def)
haftmann@37660
  2569
haftmann@37660
  2570
lemma to_bl_n1: 
haftmann@37660
  2571
  "to_bl (-1::'a::len0 word) = replicate (len_of TYPE ('a)) True"
haftmann@37660
  2572
  apply (rule word_bl.Abs_inverse')
haftmann@37660
  2573
   apply simp
haftmann@37660
  2574
  apply (rule word_eqI)
huffman@45805
  2575
  apply (clarsimp simp add: word_size)
haftmann@37660
  2576
  apply (auto simp add: word_bl.Abs_inverse test_bit_bl word_size)
haftmann@37660
  2577
  done
haftmann@37660
  2578
huffman@45805
  2579
lemma word_msb_n1 [simp]: "msb (-1::'a::len word)"
wenzelm@41550
  2580
  unfolding word_msb_alt to_bl_n1 by simp
haftmann@37660
  2581
haftmann@37660
  2582
lemma word_set_nth_iff: 
haftmann@37660
  2583
  "(set_bit w n b = w) = (w !! n = b | n >= size (w::'a::len0 word))"
haftmann@37660
  2584
  apply (rule iffI)
haftmann@37660
  2585
   apply (rule disjCI)
haftmann@37660
  2586
   apply (drule word_eqD)
haftmann@37660
  2587
   apply (erule sym [THEN trans])
haftmann@37660
  2588
   apply (simp add: test_bit_set)
haftmann@37660
  2589
  apply (erule disjE)
haftmann@37660
  2590
   apply clarsimp
haftmann@37660
  2591
  apply (rule word_eqI)
haftmann@37660
  2592
  apply (clarsimp simp add : test_bit_set_gen)
haftmann@37660
  2593
  apply (drule test_bit_size)
haftmann@37660
  2594
  apply force
haftmann@37660
  2595
  done
haftmann@37660
  2596
huffman@45811
  2597
lemma test_bit_2p:
huffman@45811
  2598
  "(word_of_int (2 ^ n)::'a::len word) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a)"
huffman@45811
  2599
  unfolding word_test_bit_def
haftmann@37660
  2600
  by (auto simp add: word_ubin.eq_norm nth_bintr nth_2p_bin)
haftmann@37660
  2601
haftmann@37660
  2602
lemma nth_w2p:
haftmann@37660
  2603
  "((2\<Colon>'a\<Colon>len word) ^ n) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a\<Colon>len)"
haftmann@37660
  2604
  unfolding test_bit_2p [symmetric] word_of_int [symmetric]
haftmann@37660
  2605
  by (simp add:  of_int_power)
haftmann@37660
  2606
haftmann@37660
  2607
lemma uint_2p: 
haftmann@40827
  2608
  "(0::'a::len word) < 2 ^ n \<Longrightarrow> uint (2 ^ n::'a::len word) = 2 ^ n"
haftmann@37660
  2609
  apply (unfold word_arith_power_alt)
haftmann@37660
  2610
  apply (case_tac "len_of TYPE ('a)")
haftmann@37660
  2611
   apply clarsimp
haftmann@37660
  2612
  apply (case_tac "nat")
haftmann@37660
  2613
   apply clarsimp
haftmann@37660
  2614
   apply (case_tac "n")
huffman@46001
  2615
    apply clarsimp
huffman@46001
  2616
   apply clarsimp
haftmann@37660
  2617
  apply (drule word_gt_0 [THEN iffD1])
wenzelm@46124
  2618
  apply (safe intro!: word_eqI bin_nth_lem)
huffman@46001
  2619
     apply (auto simp add: test_bit_2p nth_2p_bin word_test_bit_def [symmetric])
haftmann@37660
  2620
  done
haftmann@37660
  2621
haftmann@37660
  2622
lemma word_of_int_2p: "(word_of_int (2 ^ n) :: 'a :: len word) = 2 ^ n" 
haftmann@37660