src/HOL/Word/Word.thy
author haftmann
Sat, 01 Mar 2014 17:08:39 +0100
changeset 55818 d8b2f50705d0
parent 55817 0bc0217387a5
child 55833 6fe16c8a6474
permissions -rw-r--r--
more precise imports; avoid duplicated simp rules in fact collections; dropped redundancy
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
29628
d9294387ab0e entry point for Word library now named Word
haftmann
parents: 27137
diff changeset
     1
(*  Title:      HOL/Word/Word.thy
46124
3ee75fe01986 misc tuning;
wenzelm
parents: 46064
diff changeset
     2
    Author:     Jeremy Dawson and Gerwin Klein, NICTA
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
     3
*)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
     4
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
     5
header {* A type of finite bit strings *}
24350
4d74f37c6367 headers for document generation
huffman
parents: 24333
diff changeset
     6
29628
d9294387ab0e entry point for Word library now named Word
haftmann
parents: 27137
diff changeset
     7
theory Word
41413
64cd30d6b0b8 explicit file specifications -- avoid secondary load path;
wenzelm
parents: 41060
diff changeset
     8
imports
64cd30d6b0b8 explicit file specifications -- avoid secondary load path;
wenzelm
parents: 41060
diff changeset
     9
  Type_Length
64cd30d6b0b8 explicit file specifications -- avoid secondary load path;
wenzelm
parents: 41060
diff changeset
    10
  "~~/src/HOL/Library/Boolean_Algebra"
54854
3324a0078636 prefer "Bits" as theory name for abstract bit operations, similar to "Orderings", "Lattices", "Groups" etc.
haftmann
parents: 54849
diff changeset
    11
  Bits_Bit
41413
64cd30d6b0b8 explicit file specifications -- avoid secondary load path;
wenzelm
parents: 41060
diff changeset
    12
  Bool_List_Representation
53062
3af1a6020014 some vague grouping of related theorems, with slight tuning of headings and sorting out of dubious lemmas into separate theory
haftmann
parents: 51717
diff changeset
    13
  Misc_Typedef
3af1a6020014 some vague grouping of related theorems, with slight tuning of headings and sorting out of dubious lemmas into separate theory
haftmann
parents: 51717
diff changeset
    14
  Word_Miscellaneous
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
    15
begin
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
    16
54743
b9ae4a2f615b more antiquotations;
wenzelm
parents: 54742
diff changeset
    17
text {* See @{file "Examples/WordExamples.thy"} for examples. *}
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
    18
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
    19
subsection {* Type definition *}
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
    20
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    21
typedef 'a word = "{(0::int) ..< 2 ^ len_of TYPE('a::len0)}"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
    22
  morphisms uint Abs_word by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
    23
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    24
lemma uint_nonnegative:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    25
  "0 \<le> uint w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    26
  using word.uint [of w] by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    27
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    28
lemma uint_bounded:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    29
  fixes w :: "'a::len0 word"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    30
  shows "uint w < 2 ^ len_of TYPE('a)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    31
  using word.uint [of w] by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    32
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    33
lemma uint_idem:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    34
  fixes w :: "'a::len0 word"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    35
  shows "uint w mod 2 ^ len_of TYPE('a) = uint w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    36
  using uint_nonnegative uint_bounded by (rule mod_pos_pos_trivial)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    37
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    38
lemma word_uint_eq_iff:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    39
  "a = b \<longleftrightarrow> uint a = uint b"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    40
  by (simp add: uint_inject)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    41
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    42
lemma word_uint_eqI:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    43
  "uint a = uint b \<Longrightarrow> a = b"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    44
  by (simp add: word_uint_eq_iff)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    45
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
    46
definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
    47
where
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    48
  -- {* representation of words using unsigned or signed bins,
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    49
    only difference in these is the type class *}
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    50
  "word_of_int k = Abs_word (k mod 2 ^ len_of TYPE('a))"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    51
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    52
lemma uint_word_of_int:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    53
  "uint (word_of_int k :: 'a::len0 word) = k mod 2 ^ len_of TYPE('a)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    54
  by (auto simp add: word_of_int_def intro: Abs_word_inverse)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    55
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    56
lemma word_of_int_uint:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    57
  "word_of_int (uint w) = w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    58
  by (simp add: word_of_int_def uint_idem uint_inverse)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
    59
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    60
lemma split_word_all:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    61
  "(\<And>x::'a::len0 word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    62
proof
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    63
  fix x :: "'a word"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    64
  assume "\<And>x. PROP P (word_of_int x)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    65
  then have "PROP P (word_of_int (uint x))" .
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    66
  find_theorems word_of_int uint
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    67
  then show "PROP P x" by (simp add: word_of_int_uint)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    68
qed
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    69
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    70
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    71
subsection {* Type conversions and casting *}
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    72
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    73
definition sint :: "'a::len word \<Rightarrow> int"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    74
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    75
  -- {* treats the most-significant-bit as a sign bit *}
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    76
  sint_uint: "sint w = sbintrunc (len_of TYPE ('a) - 1) (uint w)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    77
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    78
definition unat :: "'a::len0 word \<Rightarrow> nat"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    79
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    80
  "unat w = nat (uint w)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    81
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    82
definition uints :: "nat \<Rightarrow> int set"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    83
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    84
  -- "the sets of integers representing the words"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    85
  "uints n = range (bintrunc n)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    86
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    87
definition sints :: "nat \<Rightarrow> int set"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    88
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    89
  "sints n = range (sbintrunc (n - 1))"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    90
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    91
lemma uints_num:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    92
  "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    93
  by (simp add: uints_def range_bintrunc)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    94
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    95
lemma sints_num:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    96
  "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    97
  by (simp add: sints_def range_sbintrunc)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    98
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
    99
definition unats :: "nat \<Rightarrow> nat set"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   100
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   101
  "unats n = {i. i < 2 ^ n}"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   102
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   103
definition norm_sint :: "nat \<Rightarrow> int \<Rightarrow> int"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   104
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   105
  "norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   106
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   107
definition scast :: "'a::len word \<Rightarrow> 'b::len word"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   108
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   109
  -- "cast a word to a different length"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   110
  "scast w = word_of_int (sint w)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   111
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   112
definition ucast :: "'a::len0 word \<Rightarrow> 'b::len0 word"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   113
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   114
  "ucast w = word_of_int (uint w)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   115
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   116
instantiation word :: (len0) size
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   117
begin
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   118
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   119
definition
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   120
  word_size: "size (w :: 'a word) = len_of TYPE('a)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   121
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   122
instance ..
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   123
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   124
end
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   125
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   126
lemma word_size_gt_0 [iff]:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   127
  "0 < size (w::'a::len word)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   128
  by (simp add: word_size)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   129
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   130
lemmas lens_gt_0 = word_size_gt_0 len_gt_0
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   131
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   132
lemma lens_not_0 [iff]:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   133
  shows "size (w::'a::len word) \<noteq> 0"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   134
  and "len_of TYPE('a::len) \<noteq> 0"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   135
  by auto
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   136
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   137
definition source_size :: "('a::len0 word \<Rightarrow> 'b) \<Rightarrow> nat"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   138
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   139
  -- "whether a cast (or other) function is to a longer or shorter length"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   140
  "source_size c = (let arb = undefined ; x = c arb in size arb)"  
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   141
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   142
definition target_size :: "('a \<Rightarrow> 'b::len0 word) \<Rightarrow> nat"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   143
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   144
  "target_size c = size (c undefined)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   145
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   146
definition is_up :: "('a::len0 word \<Rightarrow> 'b::len0 word) \<Rightarrow> bool"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   147
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   148
  "is_up c \<longleftrightarrow> source_size c \<le> target_size c"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   149
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   150
definition is_down :: "('a :: len0 word \<Rightarrow> 'b :: len0 word) \<Rightarrow> bool"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   151
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   152
  "is_down c \<longleftrightarrow> target_size c \<le> source_size c"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   153
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   154
definition of_bl :: "bool list \<Rightarrow> 'a::len0 word"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   155
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   156
  "of_bl bl = word_of_int (bl_to_bin bl)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   157
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   158
definition to_bl :: "'a::len0 word \<Rightarrow> bool list"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   159
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   160
  "to_bl w = bin_to_bl (len_of TYPE ('a)) (uint w)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   161
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   162
definition word_reverse :: "'a::len0 word \<Rightarrow> 'a word"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   163
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   164
  "word_reverse w = of_bl (rev (to_bl w))"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   165
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   166
definition word_int_case :: "(int \<Rightarrow> 'b) \<Rightarrow> 'a::len0 word \<Rightarrow> 'b" 
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   167
where
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   168
  "word_int_case f w = f (uint w)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   169
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   170
translations
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   171
  "case x of XCONST of_int y => b" == "CONST word_int_case (%y. b) x"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   172
  "case x of (XCONST of_int :: 'a) y => b" => "CONST word_int_case (%y. b) x"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   173
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   174
55817
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   175
subsection {* Correspondence relation for theorem transfer *}
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   176
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   177
definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool"
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   178
where
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   179
  "cr_word = (\<lambda>x y. word_of_int x = y)"
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   180
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   181
lemma Quotient_word:
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   182
  "Quotient (\<lambda>x y. bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y)
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   183
    word_of_int uint (cr_word :: _ \<Rightarrow> 'a::len0 word \<Rightarrow> bool)"
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   184
  unfolding Quotient_alt_def cr_word_def
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   185
  by (simp add: no_bintr_alt1 word_of_int_uint) (simp add: word_of_int_def Abs_word_inject)
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   186
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   187
lemma reflp_word:
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   188
  "reflp (\<lambda>x y. bintrunc (len_of TYPE('a::len0)) x = bintrunc (len_of TYPE('a)) y)"
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   189
  by (simp add: reflp_def)
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   190
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   191
setup_lifting (no_code) Quotient_word reflp_word
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   192
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   193
text {* TODO: The next lemma could be generated automatically. *}
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   194
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   195
lemma uint_transfer [transfer_rule]:
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   196
  "(fun_rel pcr_word op =) (bintrunc (len_of TYPE('a)))
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   197
    (uint :: 'a::len0 word \<Rightarrow> int)"
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   198
  unfolding fun_rel_def word.pcr_cr_eq cr_word_def
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   199
  by (simp add: no_bintr_alt1 uint_word_of_int)
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   200
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   201
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   202
subsection {* Basic code generation setup *}
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   203
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   204
definition Word :: "int \<Rightarrow> 'a::len0 word"
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   205
where
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   206
  [code_post]: "Word = word_of_int"
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   207
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   208
lemma [code abstype]:
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   209
  "Word (uint w) = w"
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   210
  by (simp add: Word_def word_of_int_uint)
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   211
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   212
declare uint_word_of_int [code abstract]
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   213
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   214
instantiation word :: (len0) equal
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   215
begin
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   216
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   217
definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   218
where
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   219
  "equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)"
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   220
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   221
instance proof
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   222
qed (simp add: equal equal_word_def word_uint_eq_iff)
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   223
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   224
end
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   225
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   226
notation fcomp (infixl "\<circ>>" 60)
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   227
notation scomp (infixl "\<circ>\<rightarrow>" 60)
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   228
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   229
instantiation word :: ("{len0, typerep}") random
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   230
begin
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   231
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   232
definition
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   233
  "random_word i = Random.range i \<circ>\<rightarrow> (\<lambda>k. Pair (
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   234
     let j = word_of_int (int_of_integer (integer_of_natural k)) :: 'a word
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   235
     in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))"
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   236
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   237
instance ..
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   238
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   239
end
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   240
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   241
no_notation fcomp (infixl "\<circ>>" 60)
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   242
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   243
0bc0217387a5 earlier setup of transfer, without dependency on psychodelic interpretations
haftmann
parents: 55816
diff changeset
   244
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   245
subsection {* Type-definition locale instantiations *}
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   246
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   247
lemmas uint_0 = uint_nonnegative (* FIXME duplicate *)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   248
lemmas uint_lt = uint_bounded (* FIXME duplicate *)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   249
lemmas uint_mod_same = uint_idem (* FIXME duplicate *)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   250
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   251
lemma td_ext_uint: 
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   252
  "td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (len_of TYPE('a::len0))) 
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   253
    (\<lambda>w::int. w mod 2 ^ len_of TYPE('a))"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   254
  apply (unfold td_ext_def')
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   255
  apply (simp add: uints_num word_of_int_def bintrunc_mod2p)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   256
  apply (simp add: uint_mod_same uint_0 uint_lt
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   257
                   word.uint_inverse word.Abs_word_inverse int_mod_lem)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   258
  done
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   259
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   260
interpretation word_uint:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   261
  td_ext "uint::'a::len0 word \<Rightarrow> int" 
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   262
         word_of_int 
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   263
         "uints (len_of TYPE('a::len0))"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   264
         "\<lambda>w. w mod 2 ^ len_of TYPE('a::len0)"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   265
  by (fact td_ext_uint)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   266
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   267
lemmas td_uint = word_uint.td_thm
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   268
lemmas int_word_uint = word_uint.eq_norm
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   269
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   270
lemma td_ext_ubin:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   271
  "td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (len_of TYPE('a::len0)))
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   272
    (bintrunc (len_of TYPE('a)))"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   273
  by (unfold no_bintr_alt1) (fact td_ext_uint)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   274
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   275
interpretation word_ubin:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   276
  td_ext "uint::'a::len0 word \<Rightarrow> int" 
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   277
         word_of_int 
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   278
         "uints (len_of TYPE('a::len0))"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   279
         "bintrunc (len_of TYPE('a::len0))"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   280
  by (fact td_ext_ubin)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   281
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   282
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   283
subsection {* Arithmetic operations *}
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   284
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   285
lift_definition word_succ :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x + 1"
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   286
  by (metis bintr_ariths(6))
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   287
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   288
lift_definition word_pred :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x - 1"
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   289
  by (metis bintr_ariths(7))
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   290
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   291
instantiation word :: (len0) "{neg_numeral, Divides.div, comm_monoid_mult, comm_ring}"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   292
begin
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   293
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   294
lift_definition zero_word :: "'a word" is "0" .
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   295
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   296
lift_definition one_word :: "'a word" is "1" .
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   297
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   298
lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op +"
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   299
  by (metis bintr_ariths(2))
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   300
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   301
lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op -"
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   302
  by (metis bintr_ariths(3))
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   303
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   304
lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" is uminus
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   305
  by (metis bintr_ariths(5))
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   306
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   307
lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op *"
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   308
  by (metis bintr_ariths(4))
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   309
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   310
definition
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   311
  word_div_def: "a div b = word_of_int (uint a div uint b)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   312
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   313
definition
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   314
  word_mod_def: "a mod b = word_of_int (uint a mod uint b)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   315
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   316
instance
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   317
  by default (transfer, simp add: algebra_simps)+
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   318
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   319
end
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   320
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   321
text {* Legacy theorems: *}
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   322
47611
e3c699a1fae6 add code lemmas for word operations
huffman
parents: 47567
diff changeset
   323
lemma word_arith_wis [code]: shows
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   324
  word_add_def: "a + b = word_of_int (uint a + uint b)" and
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   325
  word_sub_wi: "a - b = word_of_int (uint a - uint b)" and
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   326
  word_mult_def: "a * b = word_of_int (uint a * uint b)" and
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   327
  word_minus_def: "- a = word_of_int (- uint a)" and
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   328
  word_succ_alt: "word_succ a = word_of_int (uint a + 1)" and
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   329
  word_pred_alt: "word_pred a = word_of_int (uint a - 1)" and
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   330
  word_0_wi: "0 = word_of_int 0" and
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   331
  word_1_wi: "1 = word_of_int 1"
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   332
  unfolding plus_word_def minus_word_def times_word_def uminus_word_def
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   333
  unfolding word_succ_def word_pred_def zero_word_def one_word_def
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   334
  by simp_all
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   335
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   336
lemmas arths = 
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   337
  bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm]
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   338
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   339
lemma wi_homs: 
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   340
  shows
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   341
  wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and
46013
d2f179d26133 remove some duplicate lemmas
huffman
parents: 46012
diff changeset
   342
  wi_hom_sub: "word_of_int a - word_of_int b = word_of_int (a - b)" and
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   343
  wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   344
  wi_hom_neg: "- word_of_int a = word_of_int (- a)" and
46000
871bdab23f5c remove some uses of Int.succ and Int.pred
huffman
parents: 45998
diff changeset
   345
  wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" and
871bdab23f5c remove some uses of Int.succ and Int.pred
huffman
parents: 45998
diff changeset
   346
  wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)"
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   347
  by (transfer, simp)+
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   348
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   349
lemmas wi_hom_syms = wi_homs [symmetric]
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   350
46013
d2f179d26133 remove some duplicate lemmas
huffman
parents: 46012
diff changeset
   351
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi
46009
5cb7ef5bfef2 remove duplicate lemma lists
huffman
parents: 46001
diff changeset
   352
5cb7ef5bfef2 remove duplicate lemma lists
huffman
parents: 46001
diff changeset
   353
lemmas word_of_int_hom_syms = word_of_int_homs [symmetric]
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   354
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   355
instance word :: (len) comm_ring_1
45810
024947a0e492 prove class instances without extra lemmas
huffman
parents: 45809
diff changeset
   356
proof
024947a0e492 prove class instances without extra lemmas
huffman
parents: 45809
diff changeset
   357
  have "0 < len_of TYPE('a)" by (rule len_gt_0)
024947a0e492 prove class instances without extra lemmas
huffman
parents: 45809
diff changeset
   358
  then show "(0::'a word) \<noteq> 1"
47372
9ab4e22dac7b configure transfer method for 'a word -> int
huffman
parents: 47168
diff changeset
   359
    by - (transfer, auto simp add: gr0_conv_Suc)
45810
024947a0e492 prove class instances without extra lemmas
huffman
parents: 45809
diff changeset
   360
qed
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   361
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   362
lemma word_of_nat: "of_nat n = word_of_int (int n)"
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   363
  by (induct n) (auto simp add : word_of_int_hom_syms)
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   364
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   365
lemma word_of_int: "of_int = word_of_int"
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   366
  apply (rule ext)
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   367
  apply (case_tac x rule: int_diff_cases)
46013
d2f179d26133 remove some duplicate lemmas
huffman
parents: 46012
diff changeset
   368
  apply (simp add: word_of_nat wi_hom_sub)
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   369
  done
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   370
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   371
definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50)
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   372
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   373
  "a udvd b = (EX n>=0. uint b = n * uint a)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   374
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   375
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   376
subsection {* Ordering *}
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   377
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   378
instantiation word :: (len0) linorder
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   379
begin
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   380
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   381
definition
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   382
  word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   383
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   384
definition
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   385
  word_less_def: "a < b \<longleftrightarrow> uint a < uint b"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   386
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   387
instance
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   388
  by default (auto simp: word_less_def word_le_def)
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   389
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   390
end
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   391
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   392
definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50)
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   393
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   394
  "a <=s b = (sint a <= sint b)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   395
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   396
definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50)
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   397
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   398
  "(x <s y) = (x <=s y & x ~= y)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   399
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   400
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   401
subsection {* Bit-wise operations *}
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   402
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   403
instantiation word :: (len0) bits
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   404
begin
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   405
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   406
lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   407
  by (metis bin_trunc_not)
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   408
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   409
lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   410
  by (metis bin_trunc_and)
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   411
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   412
lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   413
  by (metis bin_trunc_or)
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   414
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   415
lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   416
  by (metis bin_trunc_xor)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   417
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   418
definition
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   419
  word_test_bit_def: "test_bit a = bin_nth (uint a)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   420
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   421
definition
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   422
  word_set_bit_def: "set_bit a n x =
54847
d6cf9a5b9be9 prefer plain bool over dedicated type for binary digits
haftmann
parents: 54743
diff changeset
   423
   word_of_int (bin_sc n x (uint a))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   424
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   425
definition
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   426
  word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   427
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   428
definition
54847
d6cf9a5b9be9 prefer plain bool over dedicated type for binary digits
haftmann
parents: 54743
diff changeset
   429
  word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   430
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   431
definition shiftl1 :: "'a word \<Rightarrow> 'a word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   432
where
54847
d6cf9a5b9be9 prefer plain bool over dedicated type for binary digits
haftmann
parents: 54743
diff changeset
   433
  "shiftl1 w = word_of_int (uint w BIT False)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   434
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   435
definition shiftr1 :: "'a word \<Rightarrow> 'a word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   436
where
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   437
  -- "shift right as unsigned or as signed, ie logical or arithmetic"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   438
  "shiftr1 w = word_of_int (bin_rest (uint w))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   439
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   440
definition
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   441
  shiftl_def: "w << n = (shiftl1 ^^ n) w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   442
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   443
definition
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   444
  shiftr_def: "w >> n = (shiftr1 ^^ n) w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   445
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   446
instance ..
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   447
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   448
end
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   449
47611
e3c699a1fae6 add code lemmas for word operations
huffman
parents: 47567
diff changeset
   450
lemma [code]: shows
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   451
  word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))" and
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   452
  word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)" and
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   453
  word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)" and
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   454
  word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   455
  unfolding bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   456
  by simp_all
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   457
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   458
instantiation word :: (len) bitss
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   459
begin
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   460
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   461
definition
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   462
  word_msb_def: 
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   463
  "msb a \<longleftrightarrow> bin_sign (sint a) = -1"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   464
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   465
instance ..
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   466
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   467
end
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   468
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   469
definition setBit :: "'a :: len0 word => nat => 'a word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   470
where 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   471
  "setBit w n = set_bit w n True"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   472
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   473
definition clearBit :: "'a :: len0 word => nat => 'a word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   474
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   475
  "clearBit w n = set_bit w n False"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   476
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   477
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   478
subsection {* Shift operations *}
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   479
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   480
definition sshiftr1 :: "'a :: len word => 'a word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   481
where 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   482
  "sshiftr1 w = word_of_int (bin_rest (sint w))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   483
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   484
definition bshiftr1 :: "bool => 'a :: len word => 'a word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   485
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   486
  "bshiftr1 b w = of_bl (b # butlast (to_bl w))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   487
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   488
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55)
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   489
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   490
  "w >>> n = (sshiftr1 ^^ n) w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   491
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   492
definition mask :: "nat => 'a::len word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   493
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   494
  "mask n = (1 << n) - 1"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   495
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   496
definition revcast :: "'a :: len0 word => 'b :: len0 word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   497
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   498
  "revcast w =  of_bl (takefill False (len_of TYPE('b)) (to_bl w))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   499
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   500
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   501
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   502
  "slice1 n w = of_bl (takefill False n (to_bl w))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   503
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   504
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   505
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   506
  "slice n w = slice1 (size w - n) w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   507
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   508
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   509
subsection {* Rotation *}
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   510
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   511
definition rotater1 :: "'a list => 'a list"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   512
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   513
  "rotater1 ys = 
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   514
    (case ys of [] => [] | x # xs => last ys # butlast ys)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   515
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   516
definition rotater :: "nat => 'a list => 'a list"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   517
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   518
  "rotater n = rotater1 ^^ n"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   519
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   520
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   521
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   522
  "word_rotr n w = of_bl (rotater n (to_bl w))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   523
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   524
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   525
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   526
  "word_rotl n w = of_bl (rotate n (to_bl w))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   527
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   528
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   529
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   530
  "word_roti i w = (if i >= 0 then word_rotr (nat i) w
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   531
                    else word_rotl (nat (- i)) w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   532
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   533
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   534
subsection {* Split and cat operations *}
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   535
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   536
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   537
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   538
  "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   539
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   540
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   541
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   542
  "word_split a = 
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   543
   (case bin_split (len_of TYPE ('c)) (uint a) of 
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   544
     (u, v) => (word_of_int u, word_of_int v))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   545
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   546
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   547
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   548
  "word_rcat ws = 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   549
  word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   550
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   551
definition word_rsplit :: "'a :: len0 word => 'b :: len word list"
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   552
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   553
  "word_rsplit w = 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   554
  map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   555
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   556
definition max_word :: "'a::len word" -- "Largest representable machine integer."
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   557
where
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   558
  "max_word = word_of_int (2 ^ len_of TYPE('a) - 1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   559
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   560
lemmas of_nth_def = word_set_bits_def (* FIXME duplicate *)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   561
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   562
46010
ebbc2d5cd720 add section headings
huffman
parents: 46009
diff changeset
   563
subsection {* Theorems about typedefs *}
ebbc2d5cd720 add section headings
huffman
parents: 46009
diff changeset
   564
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   565
lemma sint_sbintrunc': 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   566
  "sint (word_of_int bin :: 'a word) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   567
    (sbintrunc (len_of TYPE ('a :: len) - 1) bin)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   568
  unfolding sint_uint 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   569
  by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   570
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   571
lemma uint_sint: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   572
  "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   573
  unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   574
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   575
lemma bintr_uint:
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   576
  fixes w :: "'a::len0 word"
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   577
  shows "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   578
  apply (subst word_ubin.norm_Rep [symmetric]) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   579
  apply (simp only: bintrunc_bintrunc_min word_size)
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54854
diff changeset
   580
  apply (simp add: min.absorb2)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   581
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   582
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   583
lemma wi_bintr:
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   584
  "len_of TYPE('a::len0) \<le> n \<Longrightarrow>
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   585
    word_of_int (bintrunc n w) = (word_of_int w :: 'a word)"
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54854
diff changeset
   586
  by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min.absorb1)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   587
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   588
lemma td_ext_sbin: 
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   589
  "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (len_of TYPE('a::len))) 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   590
    (sbintrunc (len_of TYPE('a) - 1))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   591
  apply (unfold td_ext_def' sint_uint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   592
  apply (simp add : word_ubin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   593
  apply (cases "len_of TYPE('a)")
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   594
   apply (auto simp add : sints_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   595
  apply (rule sym [THEN trans])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   596
  apply (rule word_ubin.Abs_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   597
  apply (simp only: bintrunc_sbintrunc)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   598
  apply (drule sym)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   599
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   600
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   601
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   602
lemma td_ext_sint:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   603
  "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (len_of TYPE('a::len)))
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   604
     (\<lambda>w. (w + 2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   605
         2 ^ (len_of TYPE('a) - 1))"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   606
  using td_ext_sbin [where ?'a = 'a] by (simp add: no_sbintr_alt2)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   607
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   608
(* We do sint before sbin, before sint is the user version
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   609
   and interpretations do not produce thm duplicates. I.e. 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   610
   we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD,
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   611
   because the latter is the same thm as the former *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   612
interpretation word_sint:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   613
  td_ext "sint ::'a::len word => int" 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   614
          word_of_int 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   615
          "sints (len_of TYPE('a::len))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   616
          "%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) -
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   617
               2 ^ (len_of TYPE('a::len) - 1)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   618
  by (rule td_ext_sint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   619
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   620
interpretation word_sbin:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   621
  td_ext "sint ::'a::len word => int" 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   622
          word_of_int 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   623
          "sints (len_of TYPE('a::len))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   624
          "sbintrunc (len_of TYPE('a::len) - 1)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   625
  by (rule td_ext_sbin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   626
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   627
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   628
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   629
lemmas td_sint = word_sint.td
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   630
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   631
lemma to_bl_def': 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   632
  "(to_bl :: 'a :: len0 word => bool list) =
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   633
    bin_to_bl (len_of TYPE('a)) o uint"
44762
8f9d09241a68 tuned proofs;
wenzelm
parents: 42793
diff changeset
   634
  by (auto simp: to_bl_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   635
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   636
lemmas word_reverse_no_def [simp] = word_reverse_def [of "numeral w"] for w
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   637
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   638
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   639
  by (fact uints_def [unfolded no_bintr_alt1])
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   640
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   641
lemma word_numeral_alt:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   642
  "numeral b = word_of_int (numeral b)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   643
  by (induct b, simp_all only: numeral.simps word_of_int_homs)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   644
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   645
declare word_numeral_alt [symmetric, code_abbrev]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   646
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   647
lemma word_neg_numeral_alt:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   648
  "- numeral b = word_of_int (- numeral b)"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   649
  by (simp only: word_numeral_alt wi_hom_neg)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   650
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   651
declare word_neg_numeral_alt [symmetric, code_abbrev]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   652
47372
9ab4e22dac7b configure transfer method for 'a word -> int
huffman
parents: 47168
diff changeset
   653
lemma word_numeral_transfer [transfer_rule]:
51375
d9e62d9c98de patch Isabelle ditribution to conform to changes regarding the parametricity
kuncar
parents: 51286
diff changeset
   654
  "(fun_rel op = pcr_word) numeral numeral"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   655
  "(fun_rel op = pcr_word) (- numeral) (- numeral)"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   656
  apply (simp_all add: fun_rel_def word.pcr_cr_eq cr_word_def)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   657
  using word_numeral_alt [symmetric] word_neg_numeral_alt [symmetric] by blast+
47372
9ab4e22dac7b configure transfer method for 'a word -> int
huffman
parents: 47168
diff changeset
   658
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   659
lemma uint_bintrunc [simp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   660
  "uint (numeral bin :: 'a word) = 
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   661
    bintrunc (len_of TYPE ('a :: len0)) (numeral bin)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   662
  unfolding word_numeral_alt by (rule word_ubin.eq_norm)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   663
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   664
lemma uint_bintrunc_neg [simp]: "uint (- numeral bin :: 'a word) = 
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   665
    bintrunc (len_of TYPE ('a :: len0)) (- numeral bin)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   666
  by (simp only: word_neg_numeral_alt word_ubin.eq_norm)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   667
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   668
lemma sint_sbintrunc [simp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   669
  "sint (numeral bin :: 'a word) = 
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   670
    sbintrunc (len_of TYPE ('a :: len) - 1) (numeral bin)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   671
  by (simp only: word_numeral_alt word_sbin.eq_norm)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   672
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   673
lemma sint_sbintrunc_neg [simp]: "sint (- numeral bin :: 'a word) = 
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   674
    sbintrunc (len_of TYPE ('a :: len) - 1) (- numeral bin)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   675
  by (simp only: word_neg_numeral_alt word_sbin.eq_norm)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   676
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   677
lemma unat_bintrunc [simp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   678
  "unat (numeral bin :: 'a :: len0 word) =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   679
    nat (bintrunc (len_of TYPE('a)) (numeral bin))"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   680
  by (simp only: unat_def uint_bintrunc)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   681
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   682
lemma unat_bintrunc_neg [simp]:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   683
  "unat (- numeral bin :: 'a :: len0 word) =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   684
    nat (bintrunc (len_of TYPE('a)) (- numeral bin))"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   685
  by (simp only: unat_def uint_bintrunc_neg)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   686
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   687
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   688
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   689
  apply (rule word_uint.Rep_eqD)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   690
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   691
    defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   692
    apply (rule word_ubin.norm_Rep)+
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   693
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   694
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   695
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   696
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   697
  using word_uint.Rep [of x] by (simp add: uints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   698
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   699
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   700
  using word_uint.Rep [of x] by (simp add: uints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   701
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   702
lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint (x::'a::len word)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   703
  using word_sint.Rep [of x] by (simp add: sints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   704
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   705
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a) - 1)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   706
  using word_sint.Rep [of x] by (simp add: sints_num)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   707
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   708
lemma sign_uint_Pls [simp]: 
46604
9f9e85264e4d make uses of bin_sign respect int/bin distinction
huffman
parents: 46603
diff changeset
   709
  "bin_sign (uint x) = 0"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   710
  by (simp add: sign_Pls_ge_0)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   711
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   712
lemma uint_m2p_neg: "uint (x::'a::len0 word) - 2 ^ len_of TYPE('a) < 0"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   713
  by (simp only: diff_less_0_iff_less uint_lt2p)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   714
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   715
lemma uint_m2p_not_non_neg:
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   716
  "\<not> 0 \<le> uint (x::'a::len0 word) - 2 ^ len_of TYPE('a)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   717
  by (simp only: not_le uint_m2p_neg)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   718
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   719
lemma lt2p_lem:
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   720
  "len_of TYPE('a) \<le> n \<Longrightarrow> uint (w :: 'a::len0 word) < 2 ^ n"
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   721
  by (metis bintr_uint bintrunc_mod2p int_mod_lem zless2p)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   722
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   723
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   724
  by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   725
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   726
lemma uint_nat: "uint w = int (unat w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   727
  unfolding unat_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   728
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   729
lemma uint_numeral:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   730
  "uint (numeral b :: 'a :: len0 word) = numeral b mod 2 ^ len_of TYPE('a)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   731
  unfolding word_numeral_alt
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   732
  by (simp only: int_word_uint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   733
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   734
lemma uint_neg_numeral:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   735
  "uint (- numeral b :: 'a :: len0 word) = - numeral b mod 2 ^ len_of TYPE('a)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   736
  unfolding word_neg_numeral_alt
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   737
  by (simp only: int_word_uint)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   738
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   739
lemma unat_numeral: 
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   740
  "unat (numeral b::'a::len0 word) = numeral b mod 2 ^ len_of TYPE ('a)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   741
  apply (unfold unat_def)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   742
  apply (clarsimp simp only: uint_numeral)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   743
  apply (rule nat_mod_distrib [THEN trans])
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   744
    apply (rule zero_le_numeral)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   745
   apply (simp_all add: nat_power_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   746
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   747
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   748
lemma sint_numeral: "sint (numeral b :: 'a :: len word) = (numeral b + 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   749
    2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   750
    2 ^ (len_of TYPE('a) - 1)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   751
  unfolding word_numeral_alt by (rule int_word_sint)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   752
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   753
lemma word_of_int_0 [simp, code_post]:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   754
  "word_of_int 0 = 0"
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   755
  unfolding word_0_wi ..
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   756
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   757
lemma word_of_int_1 [simp, code_post]:
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   758
  "word_of_int 1 = 1"
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   759
  unfolding word_1_wi ..
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   760
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   761
lemma word_of_int_neg_1 [simp]: "word_of_int (- 1) = - 1"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   762
  by (simp add: wi_hom_syms)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   763
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   764
lemma word_of_int_numeral [simp] : 
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   765
  "(word_of_int (numeral bin) :: 'a :: len0 word) = (numeral bin)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   766
  unfolding word_numeral_alt ..
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   767
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   768
lemma word_of_int_neg_numeral [simp]:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   769
  "(word_of_int (- numeral bin) :: 'a :: len0 word) = (- numeral bin)"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   770
  unfolding word_numeral_alt wi_hom_syms ..
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   771
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   772
lemma word_int_case_wi: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   773
  "word_int_case f (word_of_int i :: 'b word) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   774
    f (i mod 2 ^ len_of TYPE('b::len0))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   775
  unfolding word_int_case_def by (simp add: word_uint.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   776
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   777
lemma word_int_split: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   778
  "P (word_int_case f x) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   779
    (ALL i. x = (word_of_int i :: 'b :: len0 word) & 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   780
      0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   781
  unfolding word_int_case_def
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   782
  by (auto simp: word_uint.eq_norm mod_pos_pos_trivial)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   783
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   784
lemma word_int_split_asm: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   785
  "P (word_int_case f x) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   786
    (~ (EX n. x = (word_of_int n :: 'b::len0 word) &
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   787
      0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   788
  unfolding word_int_case_def
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   789
  by (auto simp: word_uint.eq_norm mod_pos_pos_trivial)
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   790
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   791
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   792
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   793
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   794
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   795
  unfolding word_size by (rule uint_range')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   796
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   797
lemma sint_range_size:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   798
  "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   799
  unfolding word_size by (rule sint_range')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   800
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   801
lemma sint_above_size: "2 ^ (size (w::'a::len word) - 1) \<le> x \<Longrightarrow> sint w < x"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   802
  unfolding word_size by (rule less_le_trans [OF sint_lt])
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   803
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   804
lemma sint_below_size:
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   805
  "x \<le> - (2 ^ (size (w::'a::len word) - 1)) \<Longrightarrow> x \<le> sint w"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   806
  unfolding word_size by (rule order_trans [OF _ sint_ge])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   807
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   808
46010
ebbc2d5cd720 add section headings
huffman
parents: 46009
diff changeset
   809
subsection {* Testing bits *}
ebbc2d5cd720 add section headings
huffman
parents: 46009
diff changeset
   810
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   811
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   812
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   813
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   814
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   815
  apply (unfold word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   816
  apply (subst word_ubin.norm_Rep [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   817
  apply (simp only: nth_bintr word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   818
  apply fast
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   819
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   820
46021
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   821
lemma word_eq_iff:
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   822
  fixes x y :: "'a::len0 word"
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   823
  shows "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)"
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   824
  unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric]
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   825
  by (metis test_bit_size [unfolded word_size])
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   826
46023
fad87bb608fc restate some lemmas to respect int/bin distinction
huffman
parents: 46022
diff changeset
   827
lemma word_eqI [rule_format]:
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   828
  fixes u :: "'a::len0 word"
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   829
  shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v"
46021
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   830
  by (simp add: word_size word_eq_iff)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   831
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   832
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   833
  by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   834
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   835
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   836
  unfolding word_test_bit_def word_size
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   837
  by (simp add: nth_bintr [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   838
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   839
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   840
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   841
lemma bin_nth_uint_imp:
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   842
  "bin_nth (uint (w::'a::len0 word)) n \<Longrightarrow> n < len_of TYPE('a)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   843
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   844
  apply (subst word_ubin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   845
  apply assumption
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   846
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   847
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   848
lemma bin_nth_sint:
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   849
  fixes w :: "'a::len word"
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   850
  shows "len_of TYPE('a) \<le> n \<Longrightarrow>
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   851
    bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a) - 1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   852
  apply (subst word_sbin.norm_Rep [symmetric])
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   853
  apply (auto simp add: nth_sbintr)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   854
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   855
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   856
(* type definitions theorem for in terms of equivalent bool list *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   857
lemma td_bl: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   858
  "type_definition (to_bl :: 'a::len0 word => bool list) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   859
                   of_bl  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   860
                   {bl. length bl = len_of TYPE('a)}"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   861
  apply (unfold type_definition_def of_bl_def to_bl_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   862
  apply (simp add: word_ubin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   863
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   864
  apply (drule sym)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   865
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   866
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   867
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   868
interpretation word_bl:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   869
  type_definition "to_bl :: 'a::len0 word => bool list"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   870
                  of_bl  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   871
                  "{bl. length bl = len_of TYPE('a::len0)}"
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   872
  by (fact td_bl)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   873
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
   874
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]
45538
1fffa81b9b83 eliminated slightly odd Rep' with dynamically-scoped [simplified];
wenzelm
parents: 45529
diff changeset
   875
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   876
lemma word_size_bl: "size w = size (to_bl w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   877
  unfolding word_size by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   878
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   879
lemma to_bl_use_of_bl:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   880
  "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
   881
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   882
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   883
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   884
  unfolding word_reverse_def by (simp add: word_bl.Abs_inverse)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   885
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   886
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   887
  unfolding word_reverse_def by (simp add : word_bl.Abs_inverse)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   888
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   889
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   890
  by (metis word_rev_rev)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   891
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   892
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   893
  by simp
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   894
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   895
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   896
  unfolding word_bl_Rep' by (rule len_gt_0)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   897
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   898
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   899
  by (fact length_bl_gt_0 [unfolded length_greater_0_conv])
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   900
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   901
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   902
  by (fact length_bl_gt_0 [THEN gr_implies_not0])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   903
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   904
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   905
  apply (unfold to_bl_def sint_uint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   906
  apply (rule trans [OF _ bl_sbin_sign])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   907
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   908
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   909
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   910
lemma of_bl_drop': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   911
  "lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   912
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   913
  apply (unfold of_bl_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   914
  apply (clarsimp simp add : trunc_bl2bin [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   915
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   916
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   917
lemma test_bit_of_bl:  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   918
  "(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   919
  apply (unfold of_bl_def word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   920
  apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   921
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   922
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   923
lemma no_of_bl: 
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   924
  "(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) (numeral bin))"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   925
  unfolding of_bl_def by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   926
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   927
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   928
  unfolding word_size to_bl_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   929
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   930
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   931
  unfolding uint_bl by (simp add : word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   932
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   933
lemma to_bl_of_bin: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   934
  "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   935
  unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   936
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   937
lemma to_bl_numeral [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   938
  "to_bl (numeral bin::'a::len0 word) =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   939
    bin_to_bl (len_of TYPE('a)) (numeral bin)"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   940
  unfolding word_numeral_alt by (rule to_bl_of_bin)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   941
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   942
lemma to_bl_neg_numeral [simp]:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   943
  "to_bl (- numeral bin::'a::len0 word) =
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   944
    bin_to_bl (len_of TYPE('a)) (- numeral bin)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   945
  unfolding word_neg_numeral_alt by (rule to_bl_of_bin)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   946
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   947
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   948
  unfolding uint_bl by (simp add : word_size)
46011
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   949
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   950
lemma uint_bl_bin:
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   951
  fixes x :: "'a::len0 word"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   952
  shows "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   953
  by (rule trans [OF bin_bl_bin word_ubin.norm_Rep])
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   954
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   955
(* naturals *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   956
lemma uints_unats: "uints n = int ` unats n"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   957
  apply (unfold unats_def uints_num)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   958
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   959
  apply (rule_tac image_eqI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   960
  apply (erule_tac nat_0_le [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   961
  apply auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   962
  apply (erule_tac nat_less_iff [THEN iffD2])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   963
  apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   964
  apply (auto simp add : nat_power_eq int_power)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   965
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   966
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   967
lemma unats_uints: "unats n = nat ` uints n"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   968
  by (auto simp add : uints_unats image_iff)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   969
46962
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   970
lemmas bintr_num = word_ubin.norm_eq_iff
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   971
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
46962
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   972
lemmas sbintr_num = word_sbin.norm_eq_iff
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   973
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   974
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   975
lemma num_of_bintr':
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   976
  "bintrunc (len_of TYPE('a :: len0)) (numeral a) = (numeral b) \<Longrightarrow> 
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   977
    numeral a = (numeral b :: 'a word)"
46962
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   978
  unfolding bintr_num by (erule subst, simp)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   979
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   980
lemma num_of_sbintr':
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   981
  "sbintrunc (len_of TYPE('a :: len) - 1) (numeral a) = (numeral b) \<Longrightarrow> 
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   982
    numeral a = (numeral b :: 'a word)"
46962
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   983
  unfolding sbintr_num by (erule subst, simp)
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   984
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   985
lemma num_abs_bintr:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   986
  "(numeral x :: 'a word) =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   987
    word_of_int (bintrunc (len_of TYPE('a::len0)) (numeral x))"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   988
  by (simp only: word_ubin.Abs_norm word_numeral_alt)
46962
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   989
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   990
lemma num_abs_sbintr:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   991
  "(numeral x :: 'a word) =
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   992
    word_of_int (sbintrunc (len_of TYPE('a::len) - 1) (numeral x))"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   993
  by (simp only: word_sbin.Abs_norm word_numeral_alt)
46962
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   994
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   995
(** cast - note, no arg for new length, as it's determined by type of result,
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   996
  thus in "cast w = w, the type means cast to length of w! **)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   997
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   998
lemma ucast_id: "ucast w = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   999
  unfolding ucast_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1000
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1001
lemma scast_id: "scast w = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1002
  unfolding scast_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1003
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1004
lemma ucast_bl: "ucast w = of_bl (to_bl w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1005
  unfolding ucast_def of_bl_def uint_bl
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1006
  by (auto simp add : word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1007
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1008
lemma nth_ucast: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1009
  "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1010
  apply (unfold ucast_def test_bit_bin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1011
  apply (simp add: word_ubin.eq_norm nth_bintr word_size) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1012
  apply (fast elim!: bin_nth_uint_imp)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1013
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1014
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1015
(* for literal u(s)cast *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1016
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
  1017
lemma ucast_bintr [simp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1018
  "ucast (numeral w ::'a::len0 word) = 
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1019
   word_of_int (bintrunc (len_of TYPE('a)) (numeral w))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1020
  unfolding ucast_def by simp
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1021
(* TODO: neg_numeral *)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1022
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
  1023
lemma scast_sbintr [simp]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1024
  "scast (numeral w ::'a::len word) = 
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1025
   word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (numeral w))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1026
  unfolding scast_def by simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1027
46011
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1028
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1029
  unfolding source_size_def word_size Let_def ..
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1030
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1031
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1032
  unfolding target_size_def word_size Let_def ..
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1033
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1034
lemma is_down:
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1035
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1036
  shows "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1037
  unfolding is_down_def source_size target_size ..
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1038
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1039
lemma is_up:
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1040
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1041
  shows "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
  1042
  unfolding is_up_def source_size target_size ..
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1043
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1044
lemmas is_up_down = trans [OF is_up is_down [symmetric]]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1045
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1046
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1047
  apply (unfold is_down)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1048
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1049
  apply (rule ext)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1050
  apply (unfold ucast_def scast_def uint_sint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1051
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1052
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1053
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1054
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1055
lemma word_rev_tf:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1056
  "to_bl (of_bl bl::'a::len0 word) =
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1057
    rev (takefill False (len_of TYPE('a)) (rev bl))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1058
  unfolding of_bl_def uint_bl
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1059
  by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1060
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1061
lemma word_rep_drop:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1062
  "to_bl (of_bl bl::'a::len0 word) =
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1063
    replicate (len_of TYPE('a) - length bl) False @
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1064
    drop (length bl - len_of TYPE('a)) bl"
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1065
  by (simp add: word_rev_tf takefill_alt rev_take)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1066
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1067
lemma to_bl_ucast: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1068
  "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1069
   replicate (len_of TYPE('a) - len_of TYPE('b)) False @
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1070
   drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1071
  apply (unfold ucast_bl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1072
  apply (rule trans)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1073
   apply (rule word_rep_drop)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1074
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1075
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1076
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1077
lemma ucast_up_app [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1078
  "uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1079
    to_bl (uc w) = replicate n False @ (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1080
  by (auto simp add : source_size target_size to_bl_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1081
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1082
lemma ucast_down_drop [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1083
  "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1084
    to_bl (uc w) = drop n (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1085
  by (auto simp add : source_size target_size to_bl_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1086
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1087
lemma scast_down_drop [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1088
  "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1089
    to_bl (sc w) = drop n (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1090
  apply (subgoal_tac "sc = ucast")
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1091
   apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1092
   apply simp
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1093
   apply (erule ucast_down_drop)
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1094
  apply (rule down_cast_same [symmetric])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1095
  apply (simp add : source_size target_size is_down)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1096
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1097
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1098
lemma sint_up_scast [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1099
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1100
  apply (unfold is_up)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1101
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1102
  apply (simp add: scast_def word_sbin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1103
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1104
    prefer 3
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1105
    apply (rule word_sbin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1106
   apply (rule sbintrunc_sbintrunc_l)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1107
   defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1108
   apply (subst word_sbin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1109
   apply (rule refl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1110
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1111
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1112
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1113
lemma uint_up_ucast [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1114
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1115
  apply (unfold is_up)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1116
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1117
  apply (rule bin_eqI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1118
  apply (fold word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1119
  apply (auto simp add: nth_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1120
  apply (auto simp add: test_bit_bin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1121
  done
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1122
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1123
lemma ucast_up_ucast [OF refl]:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1124
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1125
  apply (simp (no_asm) add: ucast_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1126
  apply (clarsimp simp add: uint_up_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1127
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1128
    
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1129
lemma scast_up_scast [OF refl]:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1130
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1131
  apply (simp (no_asm) add: scast_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1132
  apply (clarsimp simp add: sint_up_scast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1133
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1134
    
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1135
lemma ucast_of_bl_up [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1136
  "w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1137
  by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1138
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1139
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1140
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1141
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1142
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1143
lemmas isdus = is_up_down [where c = "scast", THEN iffD2]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1144
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1145
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1146
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1147
lemma up_ucast_surj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1148
  "is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1149
   surj (ucast :: 'a word => 'b word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1150
  by (rule surjI, erule ucast_up_ucast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1151
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1152
lemma up_scast_surj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1153
  "is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1154
   surj (scast :: 'a word => 'b word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1155
  by (rule surjI, erule scast_up_scast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1156
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1157
lemma down_scast_inj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1158
  "is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1159
   inj_on (ucast :: 'a word => 'b word) A"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1160
  by (rule inj_on_inverseI, erule scast_down_scast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1161
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1162
lemma down_ucast_inj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1163
  "is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1164
   inj_on (ucast :: 'a word => 'b word) A"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1165
  by (rule inj_on_inverseI, erule ucast_down_ucast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1166
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1167
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1168
  by (rule word_bl.Rep_eqD) (simp add: word_rep_drop)
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1169
46646
0abbf6dd09ee remove ill-formed lemma of_bl_no; adapt proofs
huffman
parents: 46645
diff changeset
  1170
lemma ucast_down_wi [OF refl]:
0abbf6dd09ee remove ill-formed lemma of_bl_no; adapt proofs
huffman
parents: 46645
diff changeset
  1171
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x"
0abbf6dd09ee remove ill-formed lemma of_bl_no; adapt proofs
huffman
parents: 46645
diff changeset
  1172
  apply (unfold is_down)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1173
  apply (clarsimp simp add: ucast_def word_ubin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1174
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1175
  apply (erule bintrunc_bintrunc_ge)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1176
  done
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1177
46646
0abbf6dd09ee remove ill-formed lemma of_bl_no; adapt proofs
huffman
parents: 46645
diff changeset
  1178
lemma ucast_down_no [OF refl]:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1179
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1180
  unfolding word_numeral_alt by clarify (rule ucast_down_wi)
46646
0abbf6dd09ee remove ill-formed lemma of_bl_no; adapt proofs
huffman
parents: 46645
diff changeset
  1181
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1182
lemma ucast_down_bl [OF refl]:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1183
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl"
46646
0abbf6dd09ee remove ill-formed lemma of_bl_no; adapt proofs
huffman
parents: 46645
diff changeset
  1184
  unfolding of_bl_def by clarify (erule ucast_down_wi)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1185
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1186
lemmas slice_def' = slice_def [unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1187
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1188
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1189
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1190
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1191
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1192
subsection {* Word Arithmetic *}
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1193
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1194
lemma word_less_alt: "(a < b) = (uint a < uint b)"
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1195
  by (fact word_less_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1196
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1197
lemma signed_linorder: "class.linorder word_sle word_sless"
46124
3ee75fe01986 misc tuning;
wenzelm
parents: 46064
diff changeset
  1198
  by default (unfold word_sle_def word_sless_def, auto)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1199
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1200
interpretation signed: linorder "word_sle" "word_sless"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1201
  by (rule signed_linorder)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1202
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1203
lemma udvdI: 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1204
  "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1205
  by (auto simp: udvd_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1206
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1207
lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1208
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1209
lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1210
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1211
lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1212
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1213
lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1214
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1215
lemmas word_sless_no [simp] = word_sless_def [of "numeral a" "numeral b"] for a b
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1216
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1217
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1218
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
  1219
lemma word_m1_wi: "- 1 = word_of_int (- 1)" 
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
  1220
  using word_neg_numeral_alt [of Num.One] by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1221
46648
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1222
lemma word_0_bl [simp]: "of_bl [] = 0"
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1223
  unfolding of_bl_def by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1224
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1225
lemma word_1_bl: "of_bl [True] = 1" 
46648
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1226
  unfolding of_bl_def by (simp add: bl_to_bin_def)
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1227
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1228
lemma uint_eq_0 [simp]: "uint 0 = 0"
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1229
  unfolding word_0_wi word_ubin.eq_norm by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1230
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1231
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
46648
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1232
  by (simp add: of_bl_def bl_to_bin_rep_False)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1233
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1234
lemma to_bl_0 [simp]:
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1235
  "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1236
  unfolding uint_bl
46617
8c5d10d41391 make bool list functions respect int/bin distinction
huffman
parents: 46604
diff changeset
  1237
  by (simp add: word_size bin_to_bl_zero)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1238
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1239
lemma uint_0_iff:
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1240
  "uint x = 0 \<longleftrightarrow> x = 0"
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1241
  by (simp add: word_uint_eq_iff)
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1242
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1243
lemma unat_0_iff:
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1244
  "unat x = 0 \<longleftrightarrow> x = 0"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1245
  unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1246
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1247
lemma unat_0 [simp]:
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1248
  "unat 0 = 0"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1249
  unfolding unat_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1250
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1251
lemma size_0_same':
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1252
  "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1253
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1254
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1255
    defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1256
    apply (rule word_uint.Rep_inverse)+
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1257
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1258
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1259
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1260
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1261
lemmas size_0_same = size_0_same' [unfolded word_size]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1262
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1263
lemmas unat_eq_0 = unat_0_iff
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1264
lemmas unat_eq_zero = unat_0_iff
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1265
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1266
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1267
by (auto simp: unat_0_iff [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1268
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1269
lemma ucast_0 [simp]: "ucast 0 = 0"
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1270
  unfolding ucast_def by simp
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1271
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1272
lemma sint_0 [simp]: "sint 0 = 0"
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1273
  unfolding sint_uint by simp
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1274
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1275
lemma scast_0 [simp]: "scast 0 = 0"
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1276
  unfolding scast_def by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1277
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1278
lemma sint_n1 [simp] : "sint -1 = -1"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
  1279
  unfolding word_m1_wi word_sbin.eq_norm by simp
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
  1280
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
  1281
lemma scast_n1 [simp]: "scast (- 1) = - 1"
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1282
  unfolding scast_def by simp
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1283
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1284
lemma uint_1 [simp]: "uint (1::'a::len word) = 1"
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1285
  by (simp only: word_1_wi word_ubin.eq_norm) (simp add: bintrunc_minus_simps(4))
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1286
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1287
lemma unat_1 [simp]: "unat (1::'a::len word) = 1"
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1288
  unfolding unat_def by simp
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1289
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1290
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1"
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1291
  unfolding ucast_def by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1292
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1293
(* now, to get the weaker results analogous to word_div/mod_def *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1294
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
  1295
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
  1296
subsection {* Transferring goals from words to ints *}
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1297
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1298
lemma word_ths:  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1299
  shows
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1300
  word_succ_p1:   "word_succ a = a + 1" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1301
  word_pred_m1:   "word_pred a = a - 1" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1302
  word_pred_succ: "word_pred (word_succ a) = a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1303
  word_succ_pred: "word_succ (word_pred a) = a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1304
  word_mult_succ: "word_succ a * b = b + a * b"
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
  1305
  by (transfer, simp add: algebra_simps)+
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1306
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1307
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y"
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1308
  by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1309
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1310
lemma uint_word_ariths:
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1311
  fixes a b :: "'a::len0 word"
d8b2f50705d0 more precise imports;
haftmann
parents: 55817