src/HOL/Num.thy
author haftmann
Fri Nov 01 18:51:14 2013 +0100 (2013-11-01)
changeset 54230 b1d955791529
parent 53064 7e3f39bc41df
child 54249 ce00f2fef556
permissions -rw-r--r--
more simplification rules on unary and binary minus
huffman@47108
     1
(*  Title:      HOL/Num.thy
huffman@47108
     2
    Author:     Florian Haftmann
huffman@47108
     3
    Author:     Brian Huffman
huffman@47108
     4
*)
huffman@47108
     5
huffman@47108
     6
header {* Binary Numerals *}
huffman@47108
     7
huffman@47108
     8
theory Num
huffman@47191
     9
imports Datatype
huffman@47108
    10
begin
huffman@47108
    11
huffman@47108
    12
subsection {* The @{text num} type *}
huffman@47108
    13
huffman@47108
    14
datatype num = One | Bit0 num | Bit1 num
huffman@47108
    15
huffman@47108
    16
text {* Increment function for type @{typ num} *}
huffman@47108
    17
huffman@47108
    18
primrec inc :: "num \<Rightarrow> num" where
huffman@47108
    19
  "inc One = Bit0 One" |
huffman@47108
    20
  "inc (Bit0 x) = Bit1 x" |
huffman@47108
    21
  "inc (Bit1 x) = Bit0 (inc x)"
huffman@47108
    22
huffman@47108
    23
text {* Converting between type @{typ num} and type @{typ nat} *}
huffman@47108
    24
huffman@47108
    25
primrec nat_of_num :: "num \<Rightarrow> nat" where
huffman@47108
    26
  "nat_of_num One = Suc 0" |
huffman@47108
    27
  "nat_of_num (Bit0 x) = nat_of_num x + nat_of_num x" |
huffman@47108
    28
  "nat_of_num (Bit1 x) = Suc (nat_of_num x + nat_of_num x)"
huffman@47108
    29
huffman@47108
    30
primrec num_of_nat :: "nat \<Rightarrow> num" where
huffman@47108
    31
  "num_of_nat 0 = One" |
huffman@47108
    32
  "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
huffman@47108
    33
huffman@47108
    34
lemma nat_of_num_pos: "0 < nat_of_num x"
huffman@47108
    35
  by (induct x) simp_all
huffman@47108
    36
huffman@47108
    37
lemma nat_of_num_neq_0: " nat_of_num x \<noteq> 0"
huffman@47108
    38
  by (induct x) simp_all
huffman@47108
    39
huffman@47108
    40
lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
huffman@47108
    41
  by (induct x) simp_all
huffman@47108
    42
huffman@47108
    43
lemma num_of_nat_double:
huffman@47108
    44
  "0 < n \<Longrightarrow> num_of_nat (n + n) = Bit0 (num_of_nat n)"
huffman@47108
    45
  by (induct n) simp_all
huffman@47108
    46
huffman@47108
    47
text {*
huffman@47108
    48
  Type @{typ num} is isomorphic to the strictly positive
huffman@47108
    49
  natural numbers.
huffman@47108
    50
*}
huffman@47108
    51
huffman@47108
    52
lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
huffman@47108
    53
  by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
huffman@47108
    54
huffman@47108
    55
lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
huffman@47108
    56
  by (induct n) (simp_all add: nat_of_num_inc)
huffman@47108
    57
huffman@47108
    58
lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
huffman@47108
    59
  apply safe
huffman@47108
    60
  apply (drule arg_cong [where f=num_of_nat])
huffman@47108
    61
  apply (simp add: nat_of_num_inverse)
huffman@47108
    62
  done
huffman@47108
    63
huffman@47108
    64
lemma num_induct [case_names One inc]:
huffman@47108
    65
  fixes P :: "num \<Rightarrow> bool"
huffman@47108
    66
  assumes One: "P One"
huffman@47108
    67
    and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
huffman@47108
    68
  shows "P x"
huffman@47108
    69
proof -
huffman@47108
    70
  obtain n where n: "Suc n = nat_of_num x"
huffman@47108
    71
    by (cases "nat_of_num x", simp_all add: nat_of_num_neq_0)
huffman@47108
    72
  have "P (num_of_nat (Suc n))"
huffman@47108
    73
  proof (induct n)
huffman@47108
    74
    case 0 show ?case using One by simp
huffman@47108
    75
  next
huffman@47108
    76
    case (Suc n)
huffman@47108
    77
    then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
huffman@47108
    78
    then show "P (num_of_nat (Suc (Suc n)))" by simp
huffman@47108
    79
  qed
huffman@47108
    80
  with n show "P x"
huffman@47108
    81
    by (simp add: nat_of_num_inverse)
huffman@47108
    82
qed
huffman@47108
    83
huffman@47108
    84
text {*
huffman@47108
    85
  From now on, there are two possible models for @{typ num}:
huffman@47108
    86
  as positive naturals (rule @{text "num_induct"})
huffman@47108
    87
  and as digit representation (rules @{text "num.induct"}, @{text "num.cases"}).
huffman@47108
    88
*}
huffman@47108
    89
huffman@47108
    90
huffman@47108
    91
subsection {* Numeral operations *}
huffman@47108
    92
huffman@47108
    93
instantiation num :: "{plus,times,linorder}"
huffman@47108
    94
begin
huffman@47108
    95
huffman@47108
    96
definition [code del]:
huffman@47108
    97
  "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
huffman@47108
    98
huffman@47108
    99
definition [code del]:
huffman@47108
   100
  "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
huffman@47108
   101
huffman@47108
   102
definition [code del]:
huffman@47108
   103
  "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
huffman@47108
   104
huffman@47108
   105
definition [code del]:
huffman@47108
   106
  "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
huffman@47108
   107
huffman@47108
   108
instance
huffman@47108
   109
  by (default, auto simp add: less_num_def less_eq_num_def num_eq_iff)
huffman@47108
   110
huffman@47108
   111
end
huffman@47108
   112
huffman@47108
   113
lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
huffman@47108
   114
  unfolding plus_num_def
huffman@47108
   115
  by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
huffman@47108
   116
huffman@47108
   117
lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
huffman@47108
   118
  unfolding times_num_def
huffman@47108
   119
  by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
huffman@47108
   120
huffman@47108
   121
lemma add_num_simps [simp, code]:
huffman@47108
   122
  "One + One = Bit0 One"
huffman@47108
   123
  "One + Bit0 n = Bit1 n"
huffman@47108
   124
  "One + Bit1 n = Bit0 (n + One)"
huffman@47108
   125
  "Bit0 m + One = Bit1 m"
huffman@47108
   126
  "Bit0 m + Bit0 n = Bit0 (m + n)"
huffman@47108
   127
  "Bit0 m + Bit1 n = Bit1 (m + n)"
huffman@47108
   128
  "Bit1 m + One = Bit0 (m + One)"
huffman@47108
   129
  "Bit1 m + Bit0 n = Bit1 (m + n)"
huffman@47108
   130
  "Bit1 m + Bit1 n = Bit0 (m + n + One)"
huffman@47108
   131
  by (simp_all add: num_eq_iff nat_of_num_add)
huffman@47108
   132
huffman@47108
   133
lemma mult_num_simps [simp, code]:
huffman@47108
   134
  "m * One = m"
huffman@47108
   135
  "One * n = n"
huffman@47108
   136
  "Bit0 m * Bit0 n = Bit0 (Bit0 (m * n))"
huffman@47108
   137
  "Bit0 m * Bit1 n = Bit0 (m * Bit1 n)"
huffman@47108
   138
  "Bit1 m * Bit0 n = Bit0 (Bit1 m * n)"
huffman@47108
   139
  "Bit1 m * Bit1 n = Bit1 (m + n + Bit0 (m * n))"
huffman@47108
   140
  by (simp_all add: num_eq_iff nat_of_num_add
webertj@49962
   141
    nat_of_num_mult distrib_right distrib_left)
huffman@47108
   142
huffman@47108
   143
lemma eq_num_simps:
huffman@47108
   144
  "One = One \<longleftrightarrow> True"
huffman@47108
   145
  "One = Bit0 n \<longleftrightarrow> False"
huffman@47108
   146
  "One = Bit1 n \<longleftrightarrow> False"
huffman@47108
   147
  "Bit0 m = One \<longleftrightarrow> False"
huffman@47108
   148
  "Bit1 m = One \<longleftrightarrow> False"
huffman@47108
   149
  "Bit0 m = Bit0 n \<longleftrightarrow> m = n"
huffman@47108
   150
  "Bit0 m = Bit1 n \<longleftrightarrow> False"
huffman@47108
   151
  "Bit1 m = Bit0 n \<longleftrightarrow> False"
huffman@47108
   152
  "Bit1 m = Bit1 n \<longleftrightarrow> m = n"
huffman@47108
   153
  by simp_all
huffman@47108
   154
huffman@47108
   155
lemma le_num_simps [simp, code]:
huffman@47108
   156
  "One \<le> n \<longleftrightarrow> True"
huffman@47108
   157
  "Bit0 m \<le> One \<longleftrightarrow> False"
huffman@47108
   158
  "Bit1 m \<le> One \<longleftrightarrow> False"
huffman@47108
   159
  "Bit0 m \<le> Bit0 n \<longleftrightarrow> m \<le> n"
huffman@47108
   160
  "Bit0 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
huffman@47108
   161
  "Bit1 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
huffman@47108
   162
  "Bit1 m \<le> Bit0 n \<longleftrightarrow> m < n"
huffman@47108
   163
  using nat_of_num_pos [of n] nat_of_num_pos [of m]
huffman@47108
   164
  by (auto simp add: less_eq_num_def less_num_def)
huffman@47108
   165
huffman@47108
   166
lemma less_num_simps [simp, code]:
huffman@47108
   167
  "m < One \<longleftrightarrow> False"
huffman@47108
   168
  "One < Bit0 n \<longleftrightarrow> True"
huffman@47108
   169
  "One < Bit1 n \<longleftrightarrow> True"
huffman@47108
   170
  "Bit0 m < Bit0 n \<longleftrightarrow> m < n"
huffman@47108
   171
  "Bit0 m < Bit1 n \<longleftrightarrow> m \<le> n"
huffman@47108
   172
  "Bit1 m < Bit1 n \<longleftrightarrow> m < n"
huffman@47108
   173
  "Bit1 m < Bit0 n \<longleftrightarrow> m < n"
huffman@47108
   174
  using nat_of_num_pos [of n] nat_of_num_pos [of m]
huffman@47108
   175
  by (auto simp add: less_eq_num_def less_num_def)
huffman@47108
   176
huffman@47108
   177
text {* Rules using @{text One} and @{text inc} as constructors *}
huffman@47108
   178
huffman@47108
   179
lemma add_One: "x + One = inc x"
huffman@47108
   180
  by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
huffman@47108
   181
huffman@47108
   182
lemma add_One_commute: "One + n = n + One"
huffman@47108
   183
  by (induct n) simp_all
huffman@47108
   184
huffman@47108
   185
lemma add_inc: "x + inc y = inc (x + y)"
huffman@47108
   186
  by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
huffman@47108
   187
huffman@47108
   188
lemma mult_inc: "x * inc y = x * y + x"
huffman@47108
   189
  by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
huffman@47108
   190
huffman@47108
   191
text {* The @{const num_of_nat} conversion *}
huffman@47108
   192
huffman@47108
   193
lemma num_of_nat_One:
huffman@47300
   194
  "n \<le> 1 \<Longrightarrow> num_of_nat n = One"
huffman@47108
   195
  by (cases n) simp_all
huffman@47108
   196
huffman@47108
   197
lemma num_of_nat_plus_distrib:
huffman@47108
   198
  "0 < m \<Longrightarrow> 0 < n \<Longrightarrow> num_of_nat (m + n) = num_of_nat m + num_of_nat n"
huffman@47108
   199
  by (induct n) (auto simp add: add_One add_One_commute add_inc)
huffman@47108
   200
huffman@47108
   201
text {* A double-and-decrement function *}
huffman@47108
   202
huffman@47108
   203
primrec BitM :: "num \<Rightarrow> num" where
huffman@47108
   204
  "BitM One = One" |
huffman@47108
   205
  "BitM (Bit0 n) = Bit1 (BitM n)" |
huffman@47108
   206
  "BitM (Bit1 n) = Bit1 (Bit0 n)"
huffman@47108
   207
huffman@47108
   208
lemma BitM_plus_one: "BitM n + One = Bit0 n"
huffman@47108
   209
  by (induct n) simp_all
huffman@47108
   210
huffman@47108
   211
lemma one_plus_BitM: "One + BitM n = Bit0 n"
huffman@47108
   212
  unfolding add_One_commute BitM_plus_one ..
huffman@47108
   213
huffman@47108
   214
text {* Squaring and exponentiation *}
huffman@47108
   215
huffman@47108
   216
primrec sqr :: "num \<Rightarrow> num" where
huffman@47108
   217
  "sqr One = One" |
huffman@47108
   218
  "sqr (Bit0 n) = Bit0 (Bit0 (sqr n))" |
huffman@47108
   219
  "sqr (Bit1 n) = Bit1 (Bit0 (sqr n + n))"
huffman@47108
   220
huffman@47108
   221
primrec pow :: "num \<Rightarrow> num \<Rightarrow> num" where
huffman@47108
   222
  "pow x One = x" |
huffman@47108
   223
  "pow x (Bit0 y) = sqr (pow x y)" |
huffman@47191
   224
  "pow x (Bit1 y) = sqr (pow x y) * x"
huffman@47108
   225
huffman@47108
   226
lemma nat_of_num_sqr: "nat_of_num (sqr x) = nat_of_num x * nat_of_num x"
huffman@47108
   227
  by (induct x, simp_all add: algebra_simps nat_of_num_add)
huffman@47108
   228
huffman@47108
   229
lemma sqr_conv_mult: "sqr x = x * x"
huffman@47108
   230
  by (simp add: num_eq_iff nat_of_num_sqr nat_of_num_mult)
huffman@47108
   231
huffman@47108
   232
huffman@47211
   233
subsection {* Binary numerals *}
huffman@47108
   234
huffman@47108
   235
text {*
huffman@47211
   236
  We embed binary representations into a generic algebraic
huffman@47108
   237
  structure using @{text numeral}.
huffman@47108
   238
*}
huffman@47108
   239
huffman@47108
   240
class numeral = one + semigroup_add
huffman@47108
   241
begin
huffman@47108
   242
huffman@47108
   243
primrec numeral :: "num \<Rightarrow> 'a" where
huffman@47108
   244
  numeral_One: "numeral One = 1" |
huffman@47108
   245
  numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n" |
huffman@47108
   246
  numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"
huffman@47108
   247
haftmann@50817
   248
lemma numeral_code [code]:
haftmann@50817
   249
  "numeral One = 1"
haftmann@50817
   250
  "numeral (Bit0 n) = (let m = numeral n in m + m)"
haftmann@50817
   251
  "numeral (Bit1 n) = (let m = numeral n in m + m + 1)"
haftmann@50817
   252
  by (simp_all add: Let_def)
haftmann@50817
   253
  
huffman@47108
   254
lemma one_plus_numeral_commute: "1 + numeral x = numeral x + 1"
huffman@47108
   255
  apply (induct x)
huffman@47108
   256
  apply simp
huffman@47108
   257
  apply (simp add: add_assoc [symmetric], simp add: add_assoc)
huffman@47108
   258
  apply (simp add: add_assoc [symmetric], simp add: add_assoc)
huffman@47108
   259
  done
huffman@47108
   260
huffman@47108
   261
lemma numeral_inc: "numeral (inc x) = numeral x + 1"
huffman@47108
   262
proof (induct x)
huffman@47108
   263
  case (Bit1 x)
huffman@47108
   264
  have "numeral x + (1 + numeral x) + 1 = numeral x + (numeral x + 1) + 1"
huffman@47108
   265
    by (simp only: one_plus_numeral_commute)
huffman@47108
   266
  with Bit1 show ?case
huffman@47108
   267
    by (simp add: add_assoc)
huffman@47108
   268
qed simp_all
huffman@47108
   269
huffman@47108
   270
declare numeral.simps [simp del]
huffman@47108
   271
huffman@47108
   272
abbreviation "Numeral1 \<equiv> numeral One"
huffman@47108
   273
huffman@47108
   274
declare numeral_One [code_post]
huffman@47108
   275
huffman@47108
   276
end
huffman@47108
   277
huffman@47108
   278
text {* Negative numerals. *}
huffman@47108
   279
huffman@47108
   280
class neg_numeral = numeral + group_add
huffman@47108
   281
begin
huffman@47108
   282
huffman@47108
   283
definition neg_numeral :: "num \<Rightarrow> 'a" where
huffman@47108
   284
  "neg_numeral k = - numeral k"
huffman@47108
   285
huffman@47108
   286
end
huffman@47108
   287
huffman@47108
   288
text {* Numeral syntax. *}
huffman@47108
   289
huffman@47108
   290
syntax
huffman@47108
   291
  "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
huffman@47108
   292
huffman@47108
   293
parse_translation {*
wenzelm@52143
   294
  let
wenzelm@52143
   295
    fun num_of_int n =
wenzelm@52143
   296
      if n > 0 then
wenzelm@52143
   297
        (case IntInf.quotRem (n, 2) of
wenzelm@52143
   298
          (0, 1) => Syntax.const @{const_name One}
wenzelm@52143
   299
        | (n, 0) => Syntax.const @{const_name Bit0} $ num_of_int n
wenzelm@52143
   300
        | (n, 1) => Syntax.const @{const_name Bit1} $ num_of_int n)
wenzelm@52143
   301
      else raise Match
wenzelm@52143
   302
    val pos = Syntax.const @{const_name numeral}
wenzelm@52143
   303
    val neg = Syntax.const @{const_name neg_numeral}
wenzelm@52143
   304
    val one = Syntax.const @{const_name Groups.one}
wenzelm@52143
   305
    val zero = Syntax.const @{const_name Groups.zero}
wenzelm@52143
   306
    fun numeral_tr [(c as Const (@{syntax_const "_constrain"}, _)) $ t $ u] =
wenzelm@52143
   307
          c $ numeral_tr [t] $ u
wenzelm@52143
   308
      | numeral_tr [Const (num, _)] =
wenzelm@52143
   309
          let
wenzelm@52143
   310
            val {value, ...} = Lexicon.read_xnum num;
wenzelm@52143
   311
          in
wenzelm@52143
   312
            if value = 0 then zero else
wenzelm@52143
   313
            if value > 0
wenzelm@52143
   314
            then pos $ num_of_int value
wenzelm@52143
   315
            else neg $ num_of_int (~value)
wenzelm@52143
   316
          end
wenzelm@52143
   317
      | numeral_tr ts = raise TERM ("numeral_tr", ts);
wenzelm@52143
   318
  in [("_Numeral", K numeral_tr)] end
huffman@47108
   319
*}
huffman@47108
   320
wenzelm@52143
   321
typed_print_translation {*
wenzelm@52143
   322
  let
wenzelm@52143
   323
    fun dest_num (Const (@{const_syntax Bit0}, _) $ n) = 2 * dest_num n
wenzelm@52143
   324
      | dest_num (Const (@{const_syntax Bit1}, _) $ n) = 2 * dest_num n + 1
wenzelm@52143
   325
      | dest_num (Const (@{const_syntax One}, _)) = 1;
wenzelm@52143
   326
    fun num_tr' sign ctxt T [n] =
wenzelm@52143
   327
      let
wenzelm@52143
   328
        val k = dest_num n;
wenzelm@52187
   329
        val t' =
wenzelm@52187
   330
          Syntax.const @{syntax_const "_Numeral"} $
wenzelm@52187
   331
            Syntax.free (sign ^ string_of_int k);
wenzelm@52143
   332
      in
wenzelm@52143
   333
        (case T of
wenzelm@52143
   334
          Type (@{type_name fun}, [_, T']) =>
wenzelm@52210
   335
            if Printer.type_emphasis ctxt T' then
wenzelm@52210
   336
              Syntax.const @{syntax_const "_constrain"} $ t' $
wenzelm@52210
   337
                Syntax_Phases.term_of_typ ctxt T'
wenzelm@52210
   338
            else t'
wenzelm@52187
   339
        | _ => if T = dummyT then t' else raise Match)
wenzelm@52143
   340
      end;
wenzelm@52143
   341
  in
wenzelm@52143
   342
   [(@{const_syntax numeral}, num_tr' ""),
wenzelm@52143
   343
    (@{const_syntax neg_numeral}, num_tr' "-")]
wenzelm@52143
   344
  end
huffman@47108
   345
*}
huffman@47108
   346
wenzelm@48891
   347
ML_file "Tools/numeral.ML"
huffman@47228
   348
huffman@47228
   349
huffman@47108
   350
subsection {* Class-specific numeral rules *}
huffman@47108
   351
huffman@47108
   352
text {*
huffman@47108
   353
  @{const numeral} is a morphism.
huffman@47108
   354
*}
huffman@47108
   355
huffman@47108
   356
subsubsection {* Structures with addition: class @{text numeral} *}
huffman@47108
   357
huffman@47108
   358
context numeral
huffman@47108
   359
begin
huffman@47108
   360
huffman@47108
   361
lemma numeral_add: "numeral (m + n) = numeral m + numeral n"
huffman@47108
   362
  by (induct n rule: num_induct)
huffman@47108
   363
     (simp_all only: numeral_One add_One add_inc numeral_inc add_assoc)
huffman@47108
   364
huffman@47108
   365
lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"
huffman@47108
   366
  by (rule numeral_add [symmetric])
huffman@47108
   367
huffman@47108
   368
lemma numeral_plus_one: "numeral n + 1 = numeral (n + One)"
huffman@47108
   369
  using numeral_add [of n One] by (simp add: numeral_One)
huffman@47108
   370
huffman@47108
   371
lemma one_plus_numeral: "1 + numeral n = numeral (One + n)"
huffman@47108
   372
  using numeral_add [of One n] by (simp add: numeral_One)
huffman@47108
   373
huffman@47108
   374
lemma one_add_one: "1 + 1 = 2"
huffman@47108
   375
  using numeral_add [of One One] by (simp add: numeral_One)
huffman@47108
   376
huffman@47108
   377
lemmas add_numeral_special =
huffman@47108
   378
  numeral_plus_one one_plus_numeral one_add_one
huffman@47108
   379
huffman@47108
   380
end
huffman@47108
   381
huffman@47108
   382
subsubsection {*
huffman@47108
   383
  Structures with negation: class @{text neg_numeral}
huffman@47108
   384
*}
huffman@47108
   385
huffman@47108
   386
context neg_numeral
huffman@47108
   387
begin
huffman@47108
   388
huffman@47108
   389
text {* Numerals form an abelian subgroup. *}
huffman@47108
   390
huffman@47108
   391
inductive is_num :: "'a \<Rightarrow> bool" where
huffman@47108
   392
  "is_num 1" |
huffman@47108
   393
  "is_num x \<Longrightarrow> is_num (- x)" |
huffman@47108
   394
  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> is_num (x + y)"
huffman@47108
   395
huffman@47108
   396
lemma is_num_numeral: "is_num (numeral k)"
huffman@47108
   397
  by (induct k, simp_all add: numeral.simps is_num.intros)
huffman@47108
   398
huffman@47108
   399
lemma is_num_add_commute:
huffman@47108
   400
  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + y = y + x"
huffman@47108
   401
  apply (induct x rule: is_num.induct)
huffman@47108
   402
  apply (induct y rule: is_num.induct)
huffman@47108
   403
  apply simp
huffman@47108
   404
  apply (rule_tac a=x in add_left_imp_eq)
huffman@47108
   405
  apply (rule_tac a=x in add_right_imp_eq)
huffman@47108
   406
  apply (simp add: add_assoc minus_add_cancel)
huffman@47108
   407
  apply (simp add: add_assoc [symmetric], simp add: add_assoc)
huffman@47108
   408
  apply (rule_tac a=x in add_left_imp_eq)
huffman@47108
   409
  apply (rule_tac a=x in add_right_imp_eq)
haftmann@54230
   410
  apply (simp add: add_assoc)
huffman@47108
   411
  apply (simp add: add_assoc, simp add: add_assoc [symmetric])
huffman@47108
   412
  done
huffman@47108
   413
huffman@47108
   414
lemma is_num_add_left_commute:
huffman@47108
   415
  "\<lbrakk>is_num x; is_num y\<rbrakk> \<Longrightarrow> x + (y + z) = y + (x + z)"
huffman@47108
   416
  by (simp only: add_assoc [symmetric] is_num_add_commute)
huffman@47108
   417
huffman@47108
   418
lemmas is_num_normalize =
huffman@47108
   419
  add_assoc is_num_add_commute is_num_add_left_commute
huffman@47108
   420
  is_num.intros is_num_numeral
haftmann@54230
   421
  minus_add
huffman@47108
   422
huffman@47108
   423
definition dbl :: "'a \<Rightarrow> 'a" where "dbl x = x + x"
huffman@47108
   424
definition dbl_inc :: "'a \<Rightarrow> 'a" where "dbl_inc x = x + x + 1"
huffman@47108
   425
definition dbl_dec :: "'a \<Rightarrow> 'a" where "dbl_dec x = x + x - 1"
huffman@47108
   426
huffman@47108
   427
definition sub :: "num \<Rightarrow> num \<Rightarrow> 'a" where
huffman@47108
   428
  "sub k l = numeral k - numeral l"
huffman@47108
   429
huffman@47108
   430
lemma numeral_BitM: "numeral (BitM n) = numeral (Bit0 n) - 1"
huffman@47108
   431
  by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)
huffman@47108
   432
huffman@47108
   433
lemma dbl_simps [simp]:
huffman@47108
   434
  "dbl (neg_numeral k) = neg_numeral (Bit0 k)"
huffman@47108
   435
  "dbl 0 = 0"
huffman@47108
   436
  "dbl 1 = 2"
huffman@47108
   437
  "dbl (numeral k) = numeral (Bit0 k)"
haftmann@54230
   438
  by (simp_all add: dbl_def neg_numeral_def numeral.simps minus_add)
huffman@47108
   439
huffman@47108
   440
lemma dbl_inc_simps [simp]:
huffman@47108
   441
  "dbl_inc (neg_numeral k) = neg_numeral (BitM k)"
huffman@47108
   442
  "dbl_inc 0 = 1"
huffman@47108
   443
  "dbl_inc 1 = 3"
huffman@47108
   444
  "dbl_inc (numeral k) = numeral (Bit1 k)"
haftmann@54230
   445
  by (simp_all add: dbl_inc_def neg_numeral_def numeral.simps numeral_BitM is_num_normalize algebra_simps del: add_uminus_conv_diff)
huffman@47108
   446
huffman@47108
   447
lemma dbl_dec_simps [simp]:
huffman@47108
   448
  "dbl_dec (neg_numeral k) = neg_numeral (Bit1 k)"
huffman@47108
   449
  "dbl_dec 0 = -1"
huffman@47108
   450
  "dbl_dec 1 = 1"
huffman@47108
   451
  "dbl_dec (numeral k) = numeral (BitM k)"
haftmann@54230
   452
  by (simp_all add: dbl_dec_def neg_numeral_def numeral.simps numeral_BitM is_num_normalize)
huffman@47108
   453
huffman@47108
   454
lemma sub_num_simps [simp]:
huffman@47108
   455
  "sub One One = 0"
huffman@47108
   456
  "sub One (Bit0 l) = neg_numeral (BitM l)"
huffman@47108
   457
  "sub One (Bit1 l) = neg_numeral (Bit0 l)"
huffman@47108
   458
  "sub (Bit0 k) One = numeral (BitM k)"
huffman@47108
   459
  "sub (Bit1 k) One = numeral (Bit0 k)"
huffman@47108
   460
  "sub (Bit0 k) (Bit0 l) = dbl (sub k l)"
huffman@47108
   461
  "sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"
huffman@47108
   462
  "sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"
huffman@47108
   463
  "sub (Bit1 k) (Bit1 l) = dbl (sub k l)"
haftmann@54230
   464
  by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def neg_numeral_def numeral.simps
haftmann@54230
   465
    numeral_BitM is_num_normalize del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   466
huffman@47108
   467
lemma add_neg_numeral_simps:
huffman@47108
   468
  "numeral m + neg_numeral n = sub m n"
huffman@47108
   469
  "neg_numeral m + numeral n = sub n m"
huffman@47108
   470
  "neg_numeral m + neg_numeral n = neg_numeral (m + n)"
haftmann@54230
   471
  by (simp_all add: sub_def neg_numeral_def numeral_add numeral.simps is_num_normalize
haftmann@54230
   472
    del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   473
huffman@47108
   474
lemma add_neg_numeral_special:
huffman@47108
   475
  "1 + neg_numeral m = sub One m"
huffman@47108
   476
  "neg_numeral m + 1 = sub One m"
haftmann@54230
   477
  by (simp_all add: sub_def neg_numeral_def numeral_add numeral.simps is_num_normalize)
huffman@47108
   478
huffman@47108
   479
lemma diff_numeral_simps:
huffman@47108
   480
  "numeral m - numeral n = sub m n"
huffman@47108
   481
  "numeral m - neg_numeral n = numeral (m + n)"
huffman@47108
   482
  "neg_numeral m - numeral n = neg_numeral (m + n)"
huffman@47108
   483
  "neg_numeral m - neg_numeral n = sub n m"
haftmann@54230
   484
  by (simp_all add: neg_numeral_def sub_def numeral_add numeral.simps is_num_normalize
haftmann@54230
   485
    del: add_uminus_conv_diff add: diff_conv_add_uminus)
huffman@47108
   486
huffman@47108
   487
lemma diff_numeral_special:
huffman@47108
   488
  "1 - numeral n = sub One n"
huffman@47108
   489
  "1 - neg_numeral n = numeral (One + n)"
huffman@47108
   490
  "numeral m - 1 = sub m One"
huffman@47108
   491
  "neg_numeral m - 1 = neg_numeral (m + One)"
haftmann@54230
   492
  by (simp_all add: neg_numeral_def sub_def numeral_add numeral.simps add: is_num_normalize)
huffman@47108
   493
huffman@47108
   494
lemma minus_one: "- 1 = -1"
huffman@47108
   495
  unfolding neg_numeral_def numeral.simps ..
huffman@47108
   496
huffman@47108
   497
lemma minus_numeral: "- numeral n = neg_numeral n"
huffman@47108
   498
  unfolding neg_numeral_def ..
huffman@47108
   499
huffman@47108
   500
lemma minus_neg_numeral: "- neg_numeral n = numeral n"
huffman@47108
   501
  unfolding neg_numeral_def by simp
huffman@47108
   502
huffman@47108
   503
lemmas minus_numeral_simps [simp] =
huffman@47108
   504
  minus_one minus_numeral minus_neg_numeral
huffman@47108
   505
huffman@47108
   506
end
huffman@47108
   507
huffman@47108
   508
subsubsection {*
huffman@47108
   509
  Structures with multiplication: class @{text semiring_numeral}
huffman@47108
   510
*}
huffman@47108
   511
huffman@47108
   512
class semiring_numeral = semiring + monoid_mult
huffman@47108
   513
begin
huffman@47108
   514
huffman@47108
   515
subclass numeral ..
huffman@47108
   516
huffman@47108
   517
lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
huffman@47108
   518
  apply (induct n rule: num_induct)
huffman@47108
   519
  apply (simp add: numeral_One)
webertj@49962
   520
  apply (simp add: mult_inc numeral_inc numeral_add distrib_left)
huffman@47108
   521
  done
huffman@47108
   522
huffman@47108
   523
lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"
huffman@47108
   524
  by (rule numeral_mult [symmetric])
huffman@47108
   525
haftmann@53064
   526
lemma mult_2: "2 * z = z + z"
haftmann@53064
   527
  unfolding one_add_one [symmetric] distrib_right by simp
haftmann@53064
   528
haftmann@53064
   529
lemma mult_2_right: "z * 2 = z + z"
haftmann@53064
   530
  unfolding one_add_one [symmetric] distrib_left by simp
haftmann@53064
   531
huffman@47108
   532
end
huffman@47108
   533
huffman@47108
   534
subsubsection {*
huffman@47108
   535
  Structures with a zero: class @{text semiring_1}
huffman@47108
   536
*}
huffman@47108
   537
huffman@47108
   538
context semiring_1
huffman@47108
   539
begin
huffman@47108
   540
huffman@47108
   541
subclass semiring_numeral ..
huffman@47108
   542
huffman@47108
   543
lemma of_nat_numeral [simp]: "of_nat (numeral n) = numeral n"
huffman@47108
   544
  by (induct n,
huffman@47108
   545
    simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)
huffman@47108
   546
huffman@47108
   547
end
huffman@47108
   548
haftmann@51143
   549
lemma nat_of_num_numeral [code_abbrev]:
haftmann@51143
   550
  "nat_of_num = numeral"
huffman@47108
   551
proof
huffman@47108
   552
  fix n
huffman@47108
   553
  have "numeral n = nat_of_num n"
huffman@47108
   554
    by (induct n) (simp_all add: numeral.simps)
huffman@47108
   555
  then show "nat_of_num n = numeral n" by simp
huffman@47108
   556
qed
huffman@47108
   557
haftmann@51143
   558
lemma nat_of_num_code [code]:
haftmann@51143
   559
  "nat_of_num One = 1"
haftmann@51143
   560
  "nat_of_num (Bit0 n) = (let m = nat_of_num n in m + m)"
haftmann@51143
   561
  "nat_of_num (Bit1 n) = (let m = nat_of_num n in Suc (m + m))"
haftmann@51143
   562
  by (simp_all add: Let_def)
haftmann@51143
   563
huffman@47108
   564
subsubsection {*
huffman@47108
   565
  Equality: class @{text semiring_char_0}
huffman@47108
   566
*}
huffman@47108
   567
huffman@47108
   568
context semiring_char_0
huffman@47108
   569
begin
huffman@47108
   570
huffman@47108
   571
lemma numeral_eq_iff: "numeral m = numeral n \<longleftrightarrow> m = n"
huffman@47108
   572
  unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
huffman@47108
   573
    of_nat_eq_iff num_eq_iff ..
huffman@47108
   574
huffman@47108
   575
lemma numeral_eq_one_iff: "numeral n = 1 \<longleftrightarrow> n = One"
huffman@47108
   576
  by (rule numeral_eq_iff [of n One, unfolded numeral_One])
huffman@47108
   577
huffman@47108
   578
lemma one_eq_numeral_iff: "1 = numeral n \<longleftrightarrow> One = n"
huffman@47108
   579
  by (rule numeral_eq_iff [of One n, unfolded numeral_One])
huffman@47108
   580
huffman@47108
   581
lemma numeral_neq_zero: "numeral n \<noteq> 0"
huffman@47108
   582
  unfolding of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
huffman@47108
   583
  by (simp add: nat_of_num_pos)
huffman@47108
   584
huffman@47108
   585
lemma zero_neq_numeral: "0 \<noteq> numeral n"
huffman@47108
   586
  unfolding eq_commute [of 0] by (rule numeral_neq_zero)
huffman@47108
   587
huffman@47108
   588
lemmas eq_numeral_simps [simp] =
huffman@47108
   589
  numeral_eq_iff
huffman@47108
   590
  numeral_eq_one_iff
huffman@47108
   591
  one_eq_numeral_iff
huffman@47108
   592
  numeral_neq_zero
huffman@47108
   593
  zero_neq_numeral
huffman@47108
   594
huffman@47108
   595
end
huffman@47108
   596
huffman@47108
   597
subsubsection {*
huffman@47108
   598
  Comparisons: class @{text linordered_semidom}
huffman@47108
   599
*}
huffman@47108
   600
huffman@47108
   601
text {*  Could be perhaps more general than here. *}
huffman@47108
   602
huffman@47108
   603
context linordered_semidom
huffman@47108
   604
begin
huffman@47108
   605
huffman@47108
   606
lemma numeral_le_iff: "numeral m \<le> numeral n \<longleftrightarrow> m \<le> n"
huffman@47108
   607
proof -
huffman@47108
   608
  have "of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n"
huffman@47108
   609
    unfolding less_eq_num_def nat_of_num_numeral of_nat_le_iff ..
huffman@47108
   610
  then show ?thesis by simp
huffman@47108
   611
qed
huffman@47108
   612
huffman@47108
   613
lemma one_le_numeral: "1 \<le> numeral n"
huffman@47108
   614
using numeral_le_iff [of One n] by (simp add: numeral_One)
huffman@47108
   615
huffman@47108
   616
lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> One"
huffman@47108
   617
using numeral_le_iff [of n One] by (simp add: numeral_One)
huffman@47108
   618
huffman@47108
   619
lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"
huffman@47108
   620
proof -
huffman@47108
   621
  have "of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n"
huffman@47108
   622
    unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..
huffman@47108
   623
  then show ?thesis by simp
huffman@47108
   624
qed
huffman@47108
   625
huffman@47108
   626
lemma not_numeral_less_one: "\<not> numeral n < 1"
huffman@47108
   627
  using numeral_less_iff [of n One] by (simp add: numeral_One)
huffman@47108
   628
huffman@47108
   629
lemma one_less_numeral_iff: "1 < numeral n \<longleftrightarrow> One < n"
huffman@47108
   630
  using numeral_less_iff [of One n] by (simp add: numeral_One)
huffman@47108
   631
huffman@47108
   632
lemma zero_le_numeral: "0 \<le> numeral n"
huffman@47108
   633
  by (induct n) (simp_all add: numeral.simps)
huffman@47108
   634
huffman@47108
   635
lemma zero_less_numeral: "0 < numeral n"
huffman@47108
   636
  by (induct n) (simp_all add: numeral.simps add_pos_pos)
huffman@47108
   637
huffman@47108
   638
lemma not_numeral_le_zero: "\<not> numeral n \<le> 0"
huffman@47108
   639
  by (simp add: not_le zero_less_numeral)
huffman@47108
   640
huffman@47108
   641
lemma not_numeral_less_zero: "\<not> numeral n < 0"
huffman@47108
   642
  by (simp add: not_less zero_le_numeral)
huffman@47108
   643
huffman@47108
   644
lemmas le_numeral_extra =
huffman@47108
   645
  zero_le_one not_one_le_zero
huffman@47108
   646
  order_refl [of 0] order_refl [of 1]
huffman@47108
   647
huffman@47108
   648
lemmas less_numeral_extra =
huffman@47108
   649
  zero_less_one not_one_less_zero
huffman@47108
   650
  less_irrefl [of 0] less_irrefl [of 1]
huffman@47108
   651
huffman@47108
   652
lemmas le_numeral_simps [simp] =
huffman@47108
   653
  numeral_le_iff
huffman@47108
   654
  one_le_numeral
huffman@47108
   655
  numeral_le_one_iff
huffman@47108
   656
  zero_le_numeral
huffman@47108
   657
  not_numeral_le_zero
huffman@47108
   658
huffman@47108
   659
lemmas less_numeral_simps [simp] =
huffman@47108
   660
  numeral_less_iff
huffman@47108
   661
  one_less_numeral_iff
huffman@47108
   662
  not_numeral_less_one
huffman@47108
   663
  zero_less_numeral
huffman@47108
   664
  not_numeral_less_zero
huffman@47108
   665
huffman@47108
   666
end
huffman@47108
   667
huffman@47108
   668
subsubsection {*
huffman@47108
   669
  Multiplication and negation: class @{text ring_1}
huffman@47108
   670
*}
huffman@47108
   671
huffman@47108
   672
context ring_1
huffman@47108
   673
begin
huffman@47108
   674
huffman@47108
   675
subclass neg_numeral ..
huffman@47108
   676
huffman@47108
   677
lemma mult_neg_numeral_simps:
huffman@47108
   678
  "neg_numeral m * neg_numeral n = numeral (m * n)"
huffman@47108
   679
  "neg_numeral m * numeral n = neg_numeral (m * n)"
huffman@47108
   680
  "numeral m * neg_numeral n = neg_numeral (m * n)"
huffman@47108
   681
  unfolding neg_numeral_def mult_minus_left mult_minus_right
huffman@47108
   682
  by (simp_all only: minus_minus numeral_mult)
huffman@47108
   683
huffman@47108
   684
lemma mult_minus1 [simp]: "-1 * z = - z"
huffman@47108
   685
  unfolding neg_numeral_def numeral.simps mult_minus_left by simp
huffman@47108
   686
huffman@47108
   687
lemma mult_minus1_right [simp]: "z * -1 = - z"
huffman@47108
   688
  unfolding neg_numeral_def numeral.simps mult_minus_right by simp
huffman@47108
   689
huffman@47108
   690
end
huffman@47108
   691
huffman@47108
   692
subsubsection {*
huffman@47108
   693
  Equality using @{text iszero} for rings with non-zero characteristic
huffman@47108
   694
*}
huffman@47108
   695
huffman@47108
   696
context ring_1
huffman@47108
   697
begin
huffman@47108
   698
huffman@47108
   699
definition iszero :: "'a \<Rightarrow> bool"
huffman@47108
   700
  where "iszero z \<longleftrightarrow> z = 0"
huffman@47108
   701
huffman@47108
   702
lemma iszero_0 [simp]: "iszero 0"
huffman@47108
   703
  by (simp add: iszero_def)
huffman@47108
   704
huffman@47108
   705
lemma not_iszero_1 [simp]: "\<not> iszero 1"
huffman@47108
   706
  by (simp add: iszero_def)
huffman@47108
   707
huffman@47108
   708
lemma not_iszero_Numeral1: "\<not> iszero Numeral1"
huffman@47108
   709
  by (simp add: numeral_One)
huffman@47108
   710
huffman@47108
   711
lemma iszero_neg_numeral [simp]:
huffman@47108
   712
  "iszero (neg_numeral w) \<longleftrightarrow> iszero (numeral w)"
huffman@47108
   713
  unfolding iszero_def neg_numeral_def
huffman@47108
   714
  by (rule neg_equal_0_iff_equal)
huffman@47108
   715
huffman@47108
   716
lemma eq_iff_iszero_diff: "x = y \<longleftrightarrow> iszero (x - y)"
huffman@47108
   717
  unfolding iszero_def by (rule eq_iff_diff_eq_0)
huffman@47108
   718
huffman@47108
   719
text {* The @{text "eq_numeral_iff_iszero"} lemmas are not declared
huffman@47108
   720
@{text "[simp]"} by default, because for rings of characteristic zero,
huffman@47108
   721
better simp rules are possible. For a type like integers mod @{text
huffman@47108
   722
"n"}, type-instantiated versions of these rules should be added to the
huffman@47108
   723
simplifier, along with a type-specific rule for deciding propositions
huffman@47108
   724
of the form @{text "iszero (numeral w)"}.
huffman@47108
   725
huffman@47108
   726
bh: Maybe it would not be so bad to just declare these as simp
huffman@47108
   727
rules anyway? I should test whether these rules take precedence over
huffman@47108
   728
the @{text "ring_char_0"} rules in the simplifier.
huffman@47108
   729
*}
huffman@47108
   730
huffman@47108
   731
lemma eq_numeral_iff_iszero:
huffman@47108
   732
  "numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"
huffman@47108
   733
  "numeral x = neg_numeral y \<longleftrightarrow> iszero (numeral (x + y))"
huffman@47108
   734
  "neg_numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
huffman@47108
   735
  "neg_numeral x = neg_numeral y \<longleftrightarrow> iszero (sub y x)"
huffman@47108
   736
  "numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
huffman@47108
   737
  "1 = numeral y \<longleftrightarrow> iszero (sub One y)"
huffman@47108
   738
  "neg_numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
huffman@47108
   739
  "1 = neg_numeral y \<longleftrightarrow> iszero (numeral (One + y))"
huffman@47108
   740
  "numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
huffman@47108
   741
  "0 = numeral y \<longleftrightarrow> iszero (numeral y)"
huffman@47108
   742
  "neg_numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
huffman@47108
   743
  "0 = neg_numeral y \<longleftrightarrow> iszero (numeral y)"
huffman@47108
   744
  unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
huffman@47108
   745
  by simp_all
huffman@47108
   746
huffman@47108
   747
end
huffman@47108
   748
huffman@47108
   749
subsubsection {*
huffman@47108
   750
  Equality and negation: class @{text ring_char_0}
huffman@47108
   751
*}
huffman@47108
   752
huffman@47108
   753
class ring_char_0 = ring_1 + semiring_char_0
huffman@47108
   754
begin
huffman@47108
   755
huffman@47108
   756
lemma not_iszero_numeral [simp]: "\<not> iszero (numeral w)"
huffman@47108
   757
  by (simp add: iszero_def)
huffman@47108
   758
huffman@47108
   759
lemma neg_numeral_eq_iff: "neg_numeral m = neg_numeral n \<longleftrightarrow> m = n"
huffman@47108
   760
  by (simp only: neg_numeral_def neg_equal_iff_equal numeral_eq_iff)
huffman@47108
   761
huffman@47108
   762
lemma numeral_neq_neg_numeral: "numeral m \<noteq> neg_numeral n"
huffman@47108
   763
  unfolding neg_numeral_def eq_neg_iff_add_eq_0
huffman@47108
   764
  by (simp add: numeral_plus_numeral)
huffman@47108
   765
huffman@47108
   766
lemma neg_numeral_neq_numeral: "neg_numeral m \<noteq> numeral n"
huffman@47108
   767
  by (rule numeral_neq_neg_numeral [symmetric])
huffman@47108
   768
huffman@47108
   769
lemma zero_neq_neg_numeral: "0 \<noteq> neg_numeral n"
huffman@47108
   770
  unfolding neg_numeral_def neg_0_equal_iff_equal by simp
huffman@47108
   771
huffman@47108
   772
lemma neg_numeral_neq_zero: "neg_numeral n \<noteq> 0"
huffman@47108
   773
  unfolding neg_numeral_def neg_equal_0_iff_equal by simp
huffman@47108
   774
huffman@47108
   775
lemma one_neq_neg_numeral: "1 \<noteq> neg_numeral n"
huffman@47108
   776
  using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
huffman@47108
   777
huffman@47108
   778
lemma neg_numeral_neq_one: "neg_numeral n \<noteq> 1"
huffman@47108
   779
  using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)
huffman@47108
   780
huffman@47108
   781
lemmas eq_neg_numeral_simps [simp] =
huffman@47108
   782
  neg_numeral_eq_iff
huffman@47108
   783
  numeral_neq_neg_numeral neg_numeral_neq_numeral
huffman@47108
   784
  one_neq_neg_numeral neg_numeral_neq_one
huffman@47108
   785
  zero_neq_neg_numeral neg_numeral_neq_zero
huffman@47108
   786
huffman@47108
   787
end
huffman@47108
   788
huffman@47108
   789
subsubsection {*
huffman@47108
   790
  Structures with negation and order: class @{text linordered_idom}
huffman@47108
   791
*}
huffman@47108
   792
huffman@47108
   793
context linordered_idom
huffman@47108
   794
begin
huffman@47108
   795
huffman@47108
   796
subclass ring_char_0 ..
huffman@47108
   797
huffman@47108
   798
lemma neg_numeral_le_iff: "neg_numeral m \<le> neg_numeral n \<longleftrightarrow> n \<le> m"
huffman@47108
   799
  by (simp only: neg_numeral_def neg_le_iff_le numeral_le_iff)
huffman@47108
   800
huffman@47108
   801
lemma neg_numeral_less_iff: "neg_numeral m < neg_numeral n \<longleftrightarrow> n < m"
huffman@47108
   802
  by (simp only: neg_numeral_def neg_less_iff_less numeral_less_iff)
huffman@47108
   803
huffman@47108
   804
lemma neg_numeral_less_zero: "neg_numeral n < 0"
huffman@47108
   805
  by (simp only: neg_numeral_def neg_less_0_iff_less zero_less_numeral)
huffman@47108
   806
huffman@47108
   807
lemma neg_numeral_le_zero: "neg_numeral n \<le> 0"
huffman@47108
   808
  by (simp only: neg_numeral_def neg_le_0_iff_le zero_le_numeral)
huffman@47108
   809
huffman@47108
   810
lemma not_zero_less_neg_numeral: "\<not> 0 < neg_numeral n"
huffman@47108
   811
  by (simp only: not_less neg_numeral_le_zero)
huffman@47108
   812
huffman@47108
   813
lemma not_zero_le_neg_numeral: "\<not> 0 \<le> neg_numeral n"
huffman@47108
   814
  by (simp only: not_le neg_numeral_less_zero)
huffman@47108
   815
huffman@47108
   816
lemma neg_numeral_less_numeral: "neg_numeral m < numeral n"
huffman@47108
   817
  using neg_numeral_less_zero zero_less_numeral by (rule less_trans)
huffman@47108
   818
huffman@47108
   819
lemma neg_numeral_le_numeral: "neg_numeral m \<le> numeral n"
huffman@47108
   820
  by (simp only: less_imp_le neg_numeral_less_numeral)
huffman@47108
   821
huffman@47108
   822
lemma not_numeral_less_neg_numeral: "\<not> numeral m < neg_numeral n"
huffman@47108
   823
  by (simp only: not_less neg_numeral_le_numeral)
huffman@47108
   824
huffman@47108
   825
lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> neg_numeral n"
huffman@47108
   826
  by (simp only: not_le neg_numeral_less_numeral)
huffman@47108
   827
  
huffman@47108
   828
lemma neg_numeral_less_one: "neg_numeral m < 1"
huffman@47108
   829
  by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])
huffman@47108
   830
huffman@47108
   831
lemma neg_numeral_le_one: "neg_numeral m \<le> 1"
huffman@47108
   832
  by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])
huffman@47108
   833
huffman@47108
   834
lemma not_one_less_neg_numeral: "\<not> 1 < neg_numeral m"
huffman@47108
   835
  by (simp only: not_less neg_numeral_le_one)
huffman@47108
   836
huffman@47108
   837
lemma not_one_le_neg_numeral: "\<not> 1 \<le> neg_numeral m"
huffman@47108
   838
  by (simp only: not_le neg_numeral_less_one)
huffman@47108
   839
huffman@47108
   840
lemma sub_non_negative:
huffman@47108
   841
  "sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"
huffman@47108
   842
  by (simp only: sub_def le_diff_eq) simp
huffman@47108
   843
huffman@47108
   844
lemma sub_positive:
huffman@47108
   845
  "sub n m > 0 \<longleftrightarrow> n > m"
huffman@47108
   846
  by (simp only: sub_def less_diff_eq) simp
huffman@47108
   847
huffman@47108
   848
lemma sub_non_positive:
huffman@47108
   849
  "sub n m \<le> 0 \<longleftrightarrow> n \<le> m"
huffman@47108
   850
  by (simp only: sub_def diff_le_eq) simp
huffman@47108
   851
huffman@47108
   852
lemma sub_negative:
huffman@47108
   853
  "sub n m < 0 \<longleftrightarrow> n < m"
huffman@47108
   854
  by (simp only: sub_def diff_less_eq) simp
huffman@47108
   855
huffman@47108
   856
lemmas le_neg_numeral_simps [simp] =
huffman@47108
   857
  neg_numeral_le_iff
huffman@47108
   858
  neg_numeral_le_numeral not_numeral_le_neg_numeral
huffman@47108
   859
  neg_numeral_le_zero not_zero_le_neg_numeral
huffman@47108
   860
  neg_numeral_le_one not_one_le_neg_numeral
huffman@47108
   861
huffman@47108
   862
lemmas less_neg_numeral_simps [simp] =
huffman@47108
   863
  neg_numeral_less_iff
huffman@47108
   864
  neg_numeral_less_numeral not_numeral_less_neg_numeral
huffman@47108
   865
  neg_numeral_less_zero not_zero_less_neg_numeral
huffman@47108
   866
  neg_numeral_less_one not_one_less_neg_numeral
huffman@47108
   867
huffman@47108
   868
lemma abs_numeral [simp]: "abs (numeral n) = numeral n"
huffman@47108
   869
  by simp
huffman@47108
   870
huffman@47108
   871
lemma abs_neg_numeral [simp]: "abs (neg_numeral n) = numeral n"
huffman@47108
   872
  by (simp only: neg_numeral_def abs_minus_cancel abs_numeral)
huffman@47108
   873
huffman@47108
   874
end
huffman@47108
   875
huffman@47108
   876
subsubsection {*
huffman@47108
   877
  Natural numbers
huffman@47108
   878
*}
huffman@47108
   879
huffman@47299
   880
lemma Suc_1 [simp]: "Suc 1 = 2"
huffman@47299
   881
  unfolding Suc_eq_plus1 by (rule one_add_one)
huffman@47299
   882
huffman@47108
   883
lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"
huffman@47299
   884
  unfolding Suc_eq_plus1 by (rule numeral_plus_one)
huffman@47108
   885
huffman@47209
   886
definition pred_numeral :: "num \<Rightarrow> nat"
huffman@47209
   887
  where [code del]: "pred_numeral k = numeral k - 1"
huffman@47209
   888
huffman@47209
   889
lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
huffman@47209
   890
  unfolding pred_numeral_def by simp
huffman@47209
   891
huffman@47220
   892
lemma eval_nat_numeral:
huffman@47108
   893
  "numeral One = Suc 0"
huffman@47108
   894
  "numeral (Bit0 n) = Suc (numeral (BitM n))"
huffman@47108
   895
  "numeral (Bit1 n) = Suc (numeral (Bit0 n))"
huffman@47108
   896
  by (simp_all add: numeral.simps BitM_plus_one)
huffman@47108
   897
huffman@47209
   898
lemma pred_numeral_simps [simp]:
huffman@47300
   899
  "pred_numeral One = 0"
huffman@47300
   900
  "pred_numeral (Bit0 k) = numeral (BitM k)"
huffman@47300
   901
  "pred_numeral (Bit1 k) = numeral (Bit0 k)"
huffman@47220
   902
  unfolding pred_numeral_def eval_nat_numeral
huffman@47209
   903
  by (simp_all only: diff_Suc_Suc diff_0)
huffman@47209
   904
huffman@47192
   905
lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
huffman@47220
   906
  by (simp add: eval_nat_numeral)
huffman@47192
   907
huffman@47192
   908
lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
huffman@47220
   909
  by (simp add: eval_nat_numeral)
huffman@47192
   910
huffman@47207
   911
lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
huffman@47207
   912
  by (simp only: numeral_One One_nat_def)
huffman@47207
   913
huffman@47207
   914
lemma Suc_nat_number_of_add:
huffman@47300
   915
  "Suc (numeral v + n) = numeral (v + One) + n"
huffman@47207
   916
  by simp
huffman@47207
   917
huffman@47207
   918
(*Maps #n to n for n = 1, 2*)
huffman@47207
   919
lemmas numerals = numeral_One [where 'a=nat] numeral_2_eq_2
huffman@47207
   920
huffman@47209
   921
text {* Comparisons involving @{term Suc}. *}
huffman@47209
   922
huffman@47209
   923
lemma eq_numeral_Suc [simp]: "numeral k = Suc n \<longleftrightarrow> pred_numeral k = n"
huffman@47209
   924
  by (simp add: numeral_eq_Suc)
huffman@47209
   925
huffman@47209
   926
lemma Suc_eq_numeral [simp]: "Suc n = numeral k \<longleftrightarrow> n = pred_numeral k"
huffman@47209
   927
  by (simp add: numeral_eq_Suc)
huffman@47209
   928
huffman@47209
   929
lemma less_numeral_Suc [simp]: "numeral k < Suc n \<longleftrightarrow> pred_numeral k < n"
huffman@47209
   930
  by (simp add: numeral_eq_Suc)
huffman@47209
   931
huffman@47209
   932
lemma less_Suc_numeral [simp]: "Suc n < numeral k \<longleftrightarrow> n < pred_numeral k"
huffman@47209
   933
  by (simp add: numeral_eq_Suc)
huffman@47209
   934
huffman@47209
   935
lemma le_numeral_Suc [simp]: "numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n"
huffman@47209
   936
  by (simp add: numeral_eq_Suc)
huffman@47209
   937
huffman@47209
   938
lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"
huffman@47209
   939
  by (simp add: numeral_eq_Suc)
huffman@47209
   940
huffman@47218
   941
lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"
huffman@47218
   942
  by (simp add: numeral_eq_Suc)
huffman@47218
   943
huffman@47218
   944
lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"
huffman@47218
   945
  by (simp add: numeral_eq_Suc)
huffman@47218
   946
huffman@47209
   947
lemma max_Suc_numeral [simp]:
huffman@47209
   948
  "max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
huffman@47209
   949
  by (simp add: numeral_eq_Suc)
huffman@47209
   950
huffman@47209
   951
lemma max_numeral_Suc [simp]:
huffman@47209
   952
  "max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"
huffman@47209
   953
  by (simp add: numeral_eq_Suc)
huffman@47209
   954
huffman@47209
   955
lemma min_Suc_numeral [simp]:
huffman@47209
   956
  "min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"
huffman@47209
   957
  by (simp add: numeral_eq_Suc)
huffman@47209
   958
huffman@47209
   959
lemma min_numeral_Suc [simp]:
huffman@47209
   960
  "min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
huffman@47209
   961
  by (simp add: numeral_eq_Suc)
huffman@47209
   962
huffman@47216
   963
text {* For @{term nat_case} and @{term nat_rec}. *}
huffman@47216
   964
huffman@47216
   965
lemma nat_case_numeral [simp]:
huffman@47216
   966
  "nat_case a f (numeral v) = (let pv = pred_numeral v in f pv)"
huffman@47216
   967
  by (simp add: numeral_eq_Suc)
huffman@47216
   968
huffman@47216
   969
lemma nat_case_add_eq_if [simp]:
huffman@47216
   970
  "nat_case a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"
huffman@47216
   971
  by (simp add: numeral_eq_Suc)
huffman@47216
   972
huffman@47216
   973
lemma nat_rec_numeral [simp]:
huffman@47216
   974
  "nat_rec a f (numeral v) =
huffman@47216
   975
    (let pv = pred_numeral v in f pv (nat_rec a f pv))"
huffman@47216
   976
  by (simp add: numeral_eq_Suc Let_def)
huffman@47216
   977
huffman@47216
   978
lemma nat_rec_add_eq_if [simp]:
huffman@47216
   979
  "nat_rec a f (numeral v + n) =
huffman@47216
   980
    (let pv = pred_numeral v in f (pv + n) (nat_rec a f (pv + n)))"
huffman@47216
   981
  by (simp add: numeral_eq_Suc Let_def)
huffman@47216
   982
huffman@47255
   983
text {* Case analysis on @{term "n < 2"} *}
huffman@47255
   984
huffman@47255
   985
lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"
huffman@47255
   986
  by (auto simp add: numeral_2_eq_2)
huffman@47255
   987
huffman@47255
   988
text {* Removal of Small Numerals: 0, 1 and (in additive positions) 2 *}
huffman@47255
   989
text {* bh: Are these rules really a good idea? *}
huffman@47255
   990
huffman@47255
   991
lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
huffman@47255
   992
  by simp
huffman@47255
   993
huffman@47255
   994
lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
huffman@47255
   995
  by simp
huffman@47255
   996
huffman@47255
   997
text {* Can be used to eliminate long strings of Sucs, but not by default. *}
huffman@47255
   998
huffman@47255
   999
lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
huffman@47255
  1000
  by simp
huffman@47255
  1001
huffman@47255
  1002
lemmas nat_1_add_1 = one_add_one [where 'a=nat] (* legacy *)
huffman@47255
  1003
huffman@47108
  1004
huffman@47108
  1005
subsection {* Numeral equations as default simplification rules *}
huffman@47108
  1006
huffman@47108
  1007
declare (in numeral) numeral_One [simp]
huffman@47108
  1008
declare (in numeral) numeral_plus_numeral [simp]
huffman@47108
  1009
declare (in numeral) add_numeral_special [simp]
huffman@47108
  1010
declare (in neg_numeral) add_neg_numeral_simps [simp]
huffman@47108
  1011
declare (in neg_numeral) add_neg_numeral_special [simp]
huffman@47108
  1012
declare (in neg_numeral) diff_numeral_simps [simp]
huffman@47108
  1013
declare (in neg_numeral) diff_numeral_special [simp]
huffman@47108
  1014
declare (in semiring_numeral) numeral_times_numeral [simp]
huffman@47108
  1015
declare (in ring_1) mult_neg_numeral_simps [simp]
huffman@47108
  1016
huffman@47108
  1017
subsection {* Setting up simprocs *}
huffman@47108
  1018
huffman@47108
  1019
lemma mult_numeral_1: "Numeral1 * a = (a::'a::semiring_numeral)"
huffman@47108
  1020
  by simp
huffman@47108
  1021
huffman@47108
  1022
lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::semiring_numeral)"
huffman@47108
  1023
  by simp
huffman@47108
  1024
huffman@47108
  1025
lemma divide_numeral_1: "a / Numeral1 = (a::'a::field)"
huffman@47108
  1026
  by simp
huffman@47108
  1027
huffman@47108
  1028
lemma inverse_numeral_1:
huffman@47108
  1029
  "inverse Numeral1 = (Numeral1::'a::division_ring)"
huffman@47108
  1030
  by simp
huffman@47108
  1031
huffman@47211
  1032
text{*Theorem lists for the cancellation simprocs. The use of a binary
huffman@47108
  1033
numeral for 1 reduces the number of special cases.*}
huffman@47108
  1034
huffman@47108
  1035
lemmas mult_1s =
huffman@47108
  1036
  mult_numeral_1 mult_numeral_1_right 
huffman@47108
  1037
  mult_minus1 mult_minus1_right
huffman@47108
  1038
huffman@47226
  1039
setup {*
huffman@47226
  1040
  Reorient_Proc.add
huffman@47226
  1041
    (fn Const (@{const_name numeral}, _) $ _ => true
huffman@47226
  1042
    | Const (@{const_name neg_numeral}, _) $ _ => true
huffman@47226
  1043
    | _ => false)
huffman@47226
  1044
*}
huffman@47226
  1045
huffman@47226
  1046
simproc_setup reorient_numeral
huffman@47226
  1047
  ("numeral w = x" | "neg_numeral w = y") = Reorient_Proc.proc
huffman@47226
  1048
huffman@47108
  1049
huffman@47108
  1050
subsubsection {* Simplification of arithmetic operations on integer constants. *}
huffman@47108
  1051
huffman@47108
  1052
lemmas arith_special = (* already declared simp above *)
huffman@47108
  1053
  add_numeral_special add_neg_numeral_special
huffman@47108
  1054
  diff_numeral_special minus_one
huffman@47108
  1055
huffman@47108
  1056
(* rules already in simpset *)
huffman@47108
  1057
lemmas arith_extra_simps =
huffman@47108
  1058
  numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right
huffman@47108
  1059
  minus_numeral minus_neg_numeral minus_zero minus_one
huffman@47108
  1060
  diff_numeral_simps diff_0 diff_0_right
huffman@47108
  1061
  numeral_times_numeral mult_neg_numeral_simps
huffman@47108
  1062
  mult_zero_left mult_zero_right
huffman@47108
  1063
  abs_numeral abs_neg_numeral
huffman@47108
  1064
huffman@47108
  1065
text {*
huffman@47108
  1066
  For making a minimal simpset, one must include these default simprules.
huffman@47108
  1067
  Also include @{text simp_thms}.
huffman@47108
  1068
*}
huffman@47108
  1069
huffman@47108
  1070
lemmas arith_simps =
huffman@47108
  1071
  add_num_simps mult_num_simps sub_num_simps
huffman@47108
  1072
  BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps
huffman@47108
  1073
  abs_zero abs_one arith_extra_simps
huffman@47108
  1074
huffman@47108
  1075
text {* Simplification of relational operations *}
huffman@47108
  1076
huffman@47108
  1077
lemmas eq_numeral_extra =
huffman@47108
  1078
  zero_neq_one one_neq_zero
huffman@47108
  1079
huffman@47108
  1080
lemmas rel_simps =
huffman@47108
  1081
  le_num_simps less_num_simps eq_num_simps
huffman@47108
  1082
  le_numeral_simps le_neg_numeral_simps le_numeral_extra
huffman@47108
  1083
  less_numeral_simps less_neg_numeral_simps less_numeral_extra
huffman@47108
  1084
  eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra
huffman@47108
  1085
huffman@47108
  1086
huffman@47108
  1087
subsubsection {* Simplification of arithmetic when nested to the right. *}
huffman@47108
  1088
huffman@47108
  1089
lemma add_numeral_left [simp]:
huffman@47108
  1090
  "numeral v + (numeral w + z) = (numeral(v + w) + z)"
huffman@47108
  1091
  by (simp_all add: add_assoc [symmetric])
huffman@47108
  1092
huffman@47108
  1093
lemma add_neg_numeral_left [simp]:
huffman@47108
  1094
  "numeral v + (neg_numeral w + y) = (sub v w + y)"
huffman@47108
  1095
  "neg_numeral v + (numeral w + y) = (sub w v + y)"
huffman@47108
  1096
  "neg_numeral v + (neg_numeral w + y) = (neg_numeral(v + w) + y)"
huffman@47108
  1097
  by (simp_all add: add_assoc [symmetric])
huffman@47108
  1098
huffman@47108
  1099
lemma mult_numeral_left [simp]:
huffman@47108
  1100
  "numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
huffman@47108
  1101
  "neg_numeral v * (numeral w * y) = (neg_numeral(v * w) * y :: 'b::ring_1)"
huffman@47108
  1102
  "numeral v * (neg_numeral w * y) = (neg_numeral(v * w) * y :: 'b::ring_1)"
huffman@47108
  1103
  "neg_numeral v * (neg_numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
huffman@47108
  1104
  by (simp_all add: mult_assoc [symmetric])
huffman@47108
  1105
huffman@47108
  1106
hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
huffman@47108
  1107
haftmann@51143
  1108
huffman@47108
  1109
subsection {* code module namespace *}
huffman@47108
  1110
haftmann@52435
  1111
code_identifier
haftmann@52435
  1112
  code_module Num \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
huffman@47108
  1113
huffman@47108
  1114
end
haftmann@50817
  1115
haftmann@51143
  1116