src/HOL/Numeral.thy
 author wenzelm Thu Oct 04 20:29:42 2007 +0200 (2007-10-04) changeset 24850 0cfd722ab579 parent 24630 351a308ab58d child 25089 04b8456f7754 permissions -rw-r--r--
Name.uu, Name.aT;
```     1 (*  Title:      HOL/Numeral.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1994  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {* Arithmetic on Binary Integers *}
```
```     8
```
```     9 theory Numeral
```
```    10 imports Datatype IntDef
```
```    11 uses
```
```    12   ("Tools/numeral.ML")
```
```    13   ("Tools/numeral_syntax.ML")
```
```    14 begin
```
```    15
```
```    16 subsection {* Binary representation *}
```
```    17
```
```    18 text {*
```
```    19   This formalization defines binary arithmetic in terms of the integers
```
```    20   rather than using a datatype. This avoids multiple representations (leading
```
```    21   zeroes, etc.)  See @{text "ZF/Tools/twos-compl.ML"}, function @{text
```
```    22   int_of_binary}, for the numerical interpretation.
```
```    23
```
```    24   The representation expects that @{text "(m mod 2)"} is 0 or 1,
```
```    25   even if m is negative;
```
```    26   For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
```
```    27   @{text "-5 = (-3)*2 + 1"}.
```
```    28 *}
```
```    29
```
```    30 datatype bit = B0 | B1
```
```    31
```
```    32 text{*
```
```    33   Type @{typ bit} avoids the use of type @{typ bool}, which would make
```
```    34   all of the rewrite rules higher-order.
```
```    35 *}
```
```    36
```
```    37 definition
```
```    38   Pls :: int where
```
```    39   [code func del]: "Pls = 0"
```
```    40
```
```    41 definition
```
```    42   Min :: int where
```
```    43   [code func del]: "Min = - 1"
```
```    44
```
```    45 definition
```
```    46   Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90) where
```
```    47   [code func del]: "k BIT b = (case b of B0 \<Rightarrow> 0 | B1 \<Rightarrow> 1) + k + k"
```
```    48
```
```    49 class number = type + -- {* for numeric types: nat, int, real, \dots *}
```
```    50   fixes number_of :: "int \<Rightarrow> 'a"
```
```    51
```
```    52 use "Tools/numeral.ML"
```
```    53
```
```    54 syntax
```
```    55   "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
```
```    56
```
```    57 use "Tools/numeral_syntax.ML"
```
```    58 setup NumeralSyntax.setup
```
```    59
```
```    60 abbreviation
```
```    61   "Numeral0 \<equiv> number_of Pls"
```
```    62
```
```    63 abbreviation
```
```    64   "Numeral1 \<equiv> number_of (Pls BIT B1)"
```
```    65
```
```    66 lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"
```
```    67   -- {* Unfold all @{text let}s involving constants *}
```
```    68   unfolding Let_def ..
```
```    69
```
```    70 definition
```
```    71   succ :: "int \<Rightarrow> int" where
```
```    72   [code func del]: "succ k = k + 1"
```
```    73
```
```    74 definition
```
```    75   pred :: "int \<Rightarrow> int" where
```
```    76   [code func del]: "pred k = k - 1"
```
```    77
```
```    78 lemmas
```
```    79   max_number_of [simp] = max_def
```
```    80     [of "number_of u" "number_of v", standard, simp]
```
```    81 and
```
```    82   min_number_of [simp] = min_def
```
```    83     [of "number_of u" "number_of v", standard, simp]
```
```    84   -- {* unfolding @{text minx} and @{text max} on numerals *}
```
```    85
```
```    86 lemmas numeral_simps =
```
```    87   succ_def pred_def Pls_def Min_def Bit_def
```
```    88
```
```    89 text {* Removal of leading zeroes *}
```
```    90
```
```    91 lemma Pls_0_eq [simp, code post]:
```
```    92   "Pls BIT B0 = Pls"
```
```    93   unfolding numeral_simps by simp
```
```    94
```
```    95 lemma Min_1_eq [simp, code post]:
```
```    96   "Min BIT B1 = Min"
```
```    97   unfolding numeral_simps by simp
```
```    98
```
```    99
```
```   100 subsection {* The Functions @{term succ}, @{term pred} and @{term uminus} *}
```
```   101
```
```   102 lemma succ_Pls [simp]:
```
```   103   "succ Pls = Pls BIT B1"
```
```   104   unfolding numeral_simps by simp
```
```   105
```
```   106 lemma succ_Min [simp]:
```
```   107   "succ Min = Pls"
```
```   108   unfolding numeral_simps by simp
```
```   109
```
```   110 lemma succ_1 [simp]:
```
```   111   "succ (k BIT B1) = succ k BIT B0"
```
```   112   unfolding numeral_simps by simp
```
```   113
```
```   114 lemma succ_0 [simp]:
```
```   115   "succ (k BIT B0) = k BIT B1"
```
```   116   unfolding numeral_simps by simp
```
```   117
```
```   118 lemma pred_Pls [simp]:
```
```   119   "pred Pls = Min"
```
```   120   unfolding numeral_simps by simp
```
```   121
```
```   122 lemma pred_Min [simp]:
```
```   123   "pred Min = Min BIT B0"
```
```   124   unfolding numeral_simps by simp
```
```   125
```
```   126 lemma pred_1 [simp]:
```
```   127   "pred (k BIT B1) = k BIT B0"
```
```   128   unfolding numeral_simps by simp
```
```   129
```
```   130 lemma pred_0 [simp]:
```
```   131   "pred (k BIT B0) = pred k BIT B1"
```
```   132   unfolding numeral_simps by simp
```
```   133
```
```   134 lemma minus_Pls [simp]:
```
```   135   "- Pls = Pls"
```
```   136   unfolding numeral_simps by simp
```
```   137
```
```   138 lemma minus_Min [simp]:
```
```   139   "- Min = Pls BIT B1"
```
```   140   unfolding numeral_simps by simp
```
```   141
```
```   142 lemma minus_1 [simp]:
```
```   143   "- (k BIT B1) = pred (- k) BIT B1"
```
```   144   unfolding numeral_simps by simp
```
```   145
```
```   146 lemma minus_0 [simp]:
```
```   147   "- (k BIT B0) = (- k) BIT B0"
```
```   148   unfolding numeral_simps by simp
```
```   149
```
```   150
```
```   151 subsection {*
```
```   152   Binary Addition and Multiplication: @{term "op + \<Colon> int \<Rightarrow> int \<Rightarrow> int"}
```
```   153     and @{term "op * \<Colon> int \<Rightarrow> int \<Rightarrow> int"}
```
```   154 *}
```
```   155
```
```   156 lemma add_Pls [simp]:
```
```   157   "Pls + k = k"
```
```   158   unfolding numeral_simps by simp
```
```   159
```
```   160 lemma add_Min [simp]:
```
```   161   "Min + k = pred k"
```
```   162   unfolding numeral_simps by simp
```
```   163
```
```   164 lemma add_BIT_11 [simp]:
```
```   165   "(k BIT B1) + (l BIT B1) = (k + succ l) BIT B0"
```
```   166   unfolding numeral_simps by simp
```
```   167
```
```   168 lemma add_BIT_10 [simp]:
```
```   169   "(k BIT B1) + (l BIT B0) = (k + l) BIT B1"
```
```   170   unfolding numeral_simps by simp
```
```   171
```
```   172 lemma add_BIT_0 [simp]:
```
```   173   "(k BIT B0) + (l BIT b) = (k + l) BIT b"
```
```   174   unfolding numeral_simps by simp
```
```   175
```
```   176 lemma add_Pls_right [simp]:
```
```   177   "k + Pls = k"
```
```   178   unfolding numeral_simps by simp
```
```   179
```
```   180 lemma add_Min_right [simp]:
```
```   181   "k + Min = pred k"
```
```   182   unfolding numeral_simps by simp
```
```   183
```
```   184 lemma mult_Pls [simp]:
```
```   185   "Pls * w = Pls"
```
```   186   unfolding numeral_simps by simp
```
```   187
```
```   188 lemma mult_Min [simp]:
```
```   189   "Min * k = - k"
```
```   190   unfolding numeral_simps by simp
```
```   191
```
```   192 lemma mult_num1 [simp]:
```
```   193   "(k BIT B1) * l = ((k * l) BIT B0) + l"
```
```   194   unfolding numeral_simps int_distrib by simp
```
```   195
```
```   196 lemma mult_num0 [simp]:
```
```   197   "(k BIT B0) * l = (k * l) BIT B0"
```
```   198   unfolding numeral_simps int_distrib by simp
```
```   199
```
```   200
```
```   201
```
```   202 subsection {* Converting Numerals to Rings: @{term number_of} *}
```
```   203
```
```   204 axclass number_ring \<subseteq> number, comm_ring_1
```
```   205   number_of_eq: "number_of k = of_int k"
```
```   206
```
```   207 text {* self-embedding of the integers *}
```
```   208
```
```   209 instance int :: number_ring
```
```   210   int_number_of_def: "number_of w \<equiv> of_int w"
```
```   211   by intro_classes (simp only: int_number_of_def)
```
```   212
```
```   213 lemmas [code func del] = int_number_of_def
```
```   214
```
```   215 lemma number_of_is_id:
```
```   216   "number_of (k::int) = k"
```
```   217   unfolding int_number_of_def by simp
```
```   218
```
```   219 lemma number_of_succ:
```
```   220   "number_of (succ k) = (1 + number_of k ::'a::number_ring)"
```
```   221   unfolding number_of_eq numeral_simps by simp
```
```   222
```
```   223 lemma number_of_pred:
```
```   224   "number_of (pred w) = (- 1 + number_of w ::'a::number_ring)"
```
```   225   unfolding number_of_eq numeral_simps by simp
```
```   226
```
```   227 lemma number_of_minus:
```
```   228   "number_of (uminus w) = (- (number_of w)::'a::number_ring)"
```
```   229   unfolding number_of_eq numeral_simps by simp
```
```   230
```
```   231 lemma number_of_add:
```
```   232   "number_of (v + w) = (number_of v + number_of w::'a::number_ring)"
```
```   233   unfolding number_of_eq numeral_simps by simp
```
```   234
```
```   235 lemma number_of_mult:
```
```   236   "number_of (v * w) = (number_of v * number_of w::'a::number_ring)"
```
```   237   unfolding number_of_eq numeral_simps by simp
```
```   238
```
```   239 text {*
```
```   240   The correctness of shifting.
```
```   241   But it doesn't seem to give a measurable speed-up.
```
```   242 *}
```
```   243
```
```   244 lemma double_number_of_BIT:
```
```   245   "(1 + 1) * number_of w = (number_of (w BIT B0) ::'a::number_ring)"
```
```   246   unfolding number_of_eq numeral_simps left_distrib by simp
```
```   247
```
```   248 text {*
```
```   249   Converting numerals 0 and 1 to their abstract versions.
```
```   250 *}
```
```   251
```
```   252 lemma numeral_0_eq_0 [simp]:
```
```   253   "Numeral0 = (0::'a::number_ring)"
```
```   254   unfolding number_of_eq numeral_simps by simp
```
```   255
```
```   256 lemma numeral_1_eq_1 [simp]:
```
```   257   "Numeral1 = (1::'a::number_ring)"
```
```   258   unfolding number_of_eq numeral_simps by simp
```
```   259
```
```   260 text {*
```
```   261   Special-case simplification for small constants.
```
```   262 *}
```
```   263
```
```   264 text{*
```
```   265   Unary minus for the abstract constant 1. Cannot be inserted
```
```   266   as a simprule until later: it is @{text number_of_Min} re-oriented!
```
```   267 *}
```
```   268
```
```   269 lemma numeral_m1_eq_minus_1:
```
```   270   "(-1::'a::number_ring) = - 1"
```
```   271   unfolding number_of_eq numeral_simps by simp
```
```   272
```
```   273 lemma mult_minus1 [simp]:
```
```   274   "-1 * z = -(z::'a::number_ring)"
```
```   275   unfolding number_of_eq numeral_simps by simp
```
```   276
```
```   277 lemma mult_minus1_right [simp]:
```
```   278   "z * -1 = -(z::'a::number_ring)"
```
```   279   unfolding number_of_eq numeral_simps by simp
```
```   280
```
```   281 (*Negation of a coefficient*)
```
```   282 lemma minus_number_of_mult [simp]:
```
```   283    "- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)"
```
```   284    unfolding number_of_eq by simp
```
```   285
```
```   286 text {* Subtraction *}
```
```   287
```
```   288 lemma diff_number_of_eq:
```
```   289   "number_of v - number_of w =
```
```   290     (number_of (v + uminus w)::'a::number_ring)"
```
```   291   unfolding number_of_eq by simp
```
```   292
```
```   293 lemma number_of_Pls:
```
```   294   "number_of Pls = (0::'a::number_ring)"
```
```   295   unfolding number_of_eq numeral_simps by simp
```
```   296
```
```   297 lemma number_of_Min:
```
```   298   "number_of Min = (- 1::'a::number_ring)"
```
```   299   unfolding number_of_eq numeral_simps by simp
```
```   300
```
```   301 lemma number_of_BIT:
```
```   302   "number_of(w BIT x) = (case x of B0 => 0 | B1 => (1::'a::number_ring))
```
```   303     + (number_of w) + (number_of w)"
```
```   304   unfolding number_of_eq numeral_simps by (simp split: bit.split)
```
```   305
```
```   306
```
```   307 subsection {* Equality of Binary Numbers *}
```
```   308
```
```   309 text {* First version by Norbert Voelker *}
```
```   310
```
```   311 lemma eq_number_of_eq:
```
```   312   "((number_of x::'a::number_ring) = number_of y) =
```
```   313    iszero (number_of (x + uminus y) :: 'a)"
```
```   314   unfolding iszero_def number_of_add number_of_minus
```
```   315   by (simp add: compare_rls)
```
```   316
```
```   317 lemma iszero_number_of_Pls:
```
```   318   "iszero ((number_of Pls)::'a::number_ring)"
```
```   319   unfolding iszero_def numeral_0_eq_0 ..
```
```   320
```
```   321 lemma nonzero_number_of_Min:
```
```   322   "~ iszero ((number_of Min)::'a::number_ring)"
```
```   323   unfolding iszero_def numeral_m1_eq_minus_1 by simp
```
```   324
```
```   325
```
```   326 subsection {* Comparisons, for Ordered Rings *}
```
```   327
```
```   328 lemma double_eq_0_iff:
```
```   329   "(a + a = 0) = (a = (0::'a::ordered_idom))"
```
```   330 proof -
```
```   331   have "a + a = (1 + 1) * a" unfolding left_distrib by simp
```
```   332   with zero_less_two [where 'a = 'a]
```
```   333   show ?thesis by force
```
```   334 qed
```
```   335
```
```   336 lemma le_imp_0_less:
```
```   337   assumes le: "0 \<le> z"
```
```   338   shows "(0::int) < 1 + z"
```
```   339 proof -
```
```   340   have "0 \<le> z" by fact
```
```   341   also have "... < z + 1" by (rule less_add_one)
```
```   342   also have "... = 1 + z" by (simp add: add_ac)
```
```   343   finally show "0 < 1 + z" .
```
```   344 qed
```
```   345
```
```   346 lemma odd_nonzero:
```
```   347   "1 + z + z \<noteq> (0::int)";
```
```   348 proof (cases z rule: int_cases)
```
```   349   case (nonneg n)
```
```   350   have le: "0 \<le> z+z" by (simp add: nonneg add_increasing)
```
```   351   thus ?thesis using  le_imp_0_less [OF le]
```
```   352     by (auto simp add: add_assoc)
```
```   353 next
```
```   354   case (neg n)
```
```   355   show ?thesis
```
```   356   proof
```
```   357     assume eq: "1 + z + z = 0"
```
```   358     have "0 < 1 + (int n + int n)"
```
```   359       by (simp add: le_imp_0_less add_increasing)
```
```   360     also have "... = - (1 + z + z)"
```
```   361       by (simp add: neg add_assoc [symmetric])
```
```   362     also have "... = 0" by (simp add: eq)
```
```   363     finally have "0<0" ..
```
```   364     thus False by blast
```
```   365   qed
```
```   366 qed
```
```   367
```
```   368 text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
```
```   369
```
```   370 lemma Ints_double_eq_0_iff:
```
```   371   assumes in_Ints: "a \<in> Ints"
```
```   372   shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
```
```   373 proof -
```
```   374   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```   375   then obtain z where a: "a = of_int z" ..
```
```   376   show ?thesis
```
```   377   proof
```
```   378     assume "a = 0"
```
```   379     thus "a + a = 0" by simp
```
```   380   next
```
```   381     assume eq: "a + a = 0"
```
```   382     hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```   383     hence "z + z = 0" by (simp only: of_int_eq_iff)
```
```   384     hence "z = 0" by (simp only: double_eq_0_iff)
```
```   385     thus "a = 0" by (simp add: a)
```
```   386   qed
```
```   387 qed
```
```   388
```
```   389 lemma Ints_odd_nonzero:
```
```   390   assumes in_Ints: "a \<in> Ints"
```
```   391   shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
```
```   392 proof -
```
```   393   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```   394   then obtain z where a: "a = of_int z" ..
```
```   395   show ?thesis
```
```   396   proof
```
```   397     assume eq: "1 + a + a = 0"
```
```   398     hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```   399     hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
```
```   400     with odd_nonzero show False by blast
```
```   401   qed
```
```   402 qed
```
```   403
```
```   404 lemma Ints_number_of:
```
```   405   "(number_of w :: 'a::number_ring) \<in> Ints"
```
```   406   unfolding number_of_eq Ints_def by simp
```
```   407
```
```   408 lemma iszero_number_of_BIT:
```
```   409   "iszero (number_of (w BIT x)::'a) =
```
```   410    (x = B0 \<and> iszero (number_of w::'a::{ring_char_0,number_ring}))"
```
```   411   by (simp add: iszero_def number_of_eq numeral_simps Ints_double_eq_0_iff
```
```   412     Ints_odd_nonzero Ints_def split: bit.split)
```
```   413
```
```   414 lemma iszero_number_of_0:
```
```   415   "iszero (number_of (w BIT B0) :: 'a::{ring_char_0,number_ring}) =
```
```   416   iszero (number_of w :: 'a)"
```
```   417   by (simp only: iszero_number_of_BIT simp_thms)
```
```   418
```
```   419 lemma iszero_number_of_1:
```
```   420   "~ iszero (number_of (w BIT B1)::'a::{ring_char_0,number_ring})"
```
```   421   by (simp add: iszero_number_of_BIT)
```
```   422
```
```   423
```
```   424 subsection {* The Less-Than Relation *}
```
```   425
```
```   426 lemma less_number_of_eq_neg:
```
```   427   "((number_of x::'a::{ordered_idom,number_ring}) < number_of y)
```
```   428   = neg (number_of (x + uminus y) :: 'a)"
```
```   429 apply (subst less_iff_diff_less_0)
```
```   430 apply (simp add: neg_def diff_minus number_of_add number_of_minus)
```
```   431 done
```
```   432
```
```   433 text {*
```
```   434   If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
```
```   435   @{term Numeral0} IS @{term "number_of Pls"}
```
```   436 *}
```
```   437
```
```   438 lemma not_neg_number_of_Pls:
```
```   439   "~ neg (number_of Pls ::'a::{ordered_idom,number_ring})"
```
```   440   by (simp add: neg_def numeral_0_eq_0)
```
```   441
```
```   442 lemma neg_number_of_Min:
```
```   443   "neg (number_of Min ::'a::{ordered_idom,number_ring})"
```
```   444   by (simp add: neg_def zero_less_one numeral_m1_eq_minus_1)
```
```   445
```
```   446 lemma double_less_0_iff:
```
```   447   "(a + a < 0) = (a < (0::'a::ordered_idom))"
```
```   448 proof -
```
```   449   have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
```
```   450   also have "... = (a < 0)"
```
```   451     by (simp add: mult_less_0_iff zero_less_two
```
```   452                   order_less_not_sym [OF zero_less_two])
```
```   453   finally show ?thesis .
```
```   454 qed
```
```   455
```
```   456 lemma odd_less_0:
```
```   457   "(1 + z + z < 0) = (z < (0::int))";
```
```   458 proof (cases z rule: int_cases)
```
```   459   case (nonneg n)
```
```   460   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
```
```   461                              le_imp_0_less [THEN order_less_imp_le])
```
```   462 next
```
```   463   case (neg n)
```
```   464   thus ?thesis by (simp del: of_nat_Suc of_nat_add
```
```   465     add: compare_rls of_nat_1 [symmetric] of_nat_add [symmetric])
```
```   466 qed
```
```   467
```
```   468 text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
```
```   469
```
```   470 lemma Ints_odd_less_0:
```
```   471   assumes in_Ints: "a \<in> Ints"
```
```   472   shows "(1 + a + a < 0) = (a < (0::'a::ordered_idom))";
```
```   473 proof -
```
```   474   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```   475   then obtain z where a: "a = of_int z" ..
```
```   476   hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
```
```   477     by (simp add: a)
```
```   478   also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
```
```   479   also have "... = (a < 0)" by (simp add: a)
```
```   480   finally show ?thesis .
```
```   481 qed
```
```   482
```
```   483 lemma neg_number_of_BIT:
```
```   484   "neg (number_of (w BIT x)::'a) =
```
```   485   neg (number_of w :: 'a::{ordered_idom,number_ring})"
```
```   486   by (simp add: neg_def number_of_eq numeral_simps double_less_0_iff
```
```   487     Ints_odd_less_0 Ints_def split: bit.split)
```
```   488
```
```   489
```
```   490 text {* Less-Than or Equals *}
```
```   491
```
```   492 text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *}
```
```   493
```
```   494 lemmas le_number_of_eq_not_less =
```
```   495   linorder_not_less [of "number_of w" "number_of v", symmetric,
```
```   496   standard]
```
```   497
```
```   498 lemma le_number_of_eq:
```
```   499     "((number_of x::'a::{ordered_idom,number_ring}) \<le> number_of y)
```
```   500      = (~ (neg (number_of (y + uminus x) :: 'a)))"
```
```   501 by (simp add: le_number_of_eq_not_less less_number_of_eq_neg)
```
```   502
```
```   503
```
```   504 text {* Absolute value (@{term abs}) *}
```
```   505
```
```   506 lemma abs_number_of:
```
```   507   "abs(number_of x::'a::{ordered_idom,number_ring}) =
```
```   508    (if number_of x < (0::'a) then -number_of x else number_of x)"
```
```   509   by (simp add: abs_if)
```
```   510
```
```   511
```
```   512 text {* Re-orientation of the equation nnn=x *}
```
```   513
```
```   514 lemma number_of_reorient:
```
```   515   "(number_of w = x) = (x = number_of w)"
```
```   516   by auto
```
```   517
```
```   518
```
```   519 subsection {* Simplification of arithmetic operations on integer constants. *}
```
```   520
```
```   521 lemmas arith_extra_simps [standard, simp] =
```
```   522   number_of_add [symmetric]
```
```   523   number_of_minus [symmetric] numeral_m1_eq_minus_1 [symmetric]
```
```   524   number_of_mult [symmetric]
```
```   525   diff_number_of_eq abs_number_of
```
```   526
```
```   527 text {*
```
```   528   For making a minimal simpset, one must include these default simprules.
```
```   529   Also include @{text simp_thms}.
```
```   530 *}
```
```   531
```
```   532 lemmas arith_simps =
```
```   533   bit.distinct
```
```   534   Pls_0_eq Min_1_eq
```
```   535   pred_Pls pred_Min pred_1 pred_0
```
```   536   succ_Pls succ_Min succ_1 succ_0
```
```   537   add_Pls add_Min add_BIT_0 add_BIT_10 add_BIT_11
```
```   538   minus_Pls minus_Min minus_1 minus_0
```
```   539   mult_Pls mult_Min mult_num1 mult_num0
```
```   540   add_Pls_right add_Min_right
```
```   541   abs_zero abs_one arith_extra_simps
```
```   542
```
```   543 text {* Simplification of relational operations *}
```
```   544
```
```   545 lemmas rel_simps [simp] =
```
```   546   eq_number_of_eq iszero_0 nonzero_number_of_Min
```
```   547   iszero_number_of_0 iszero_number_of_1
```
```   548   less_number_of_eq_neg
```
```   549   not_neg_number_of_Pls not_neg_0 not_neg_1 not_iszero_1
```
```   550   neg_number_of_Min neg_number_of_BIT
```
```   551   le_number_of_eq
```
```   552 (* iszero_number_of_Pls would never be used
```
```   553    because its lhs simplifies to "iszero 0" *)
```
```   554
```
```   555
```
```   556 subsection {* Simplification of arithmetic when nested to the right. *}
```
```   557
```
```   558 lemma add_number_of_left [simp]:
```
```   559   "number_of v + (number_of w + z) =
```
```   560    (number_of(v + w) + z::'a::number_ring)"
```
```   561   by (simp add: add_assoc [symmetric])
```
```   562
```
```   563 lemma mult_number_of_left [simp]:
```
```   564   "number_of v * (number_of w * z) =
```
```   565    (number_of(v * w) * z::'a::number_ring)"
```
```   566   by (simp add: mult_assoc [symmetric])
```
```   567
```
```   568 lemma add_number_of_diff1:
```
```   569   "number_of v + (number_of w - c) =
```
```   570   number_of(v + w) - (c::'a::number_ring)"
```
```   571   by (simp add: diff_minus add_number_of_left)
```
```   572
```
```   573 lemma add_number_of_diff2 [simp]:
```
```   574   "number_of v + (c - number_of w) =
```
```   575    number_of (v + uminus w) + (c::'a::number_ring)"
```
```   576 apply (subst diff_number_of_eq [symmetric])
```
```   577 apply (simp only: compare_rls)
```
```   578 done
```
```   579
```
```   580
```
```   581 subsection {* Configuration of the code generator *}
```
```   582
```
```   583 instance int :: eq ..
```
```   584
```
```   585 code_datatype Pls Min Bit "number_of \<Colon> int \<Rightarrow> int"
```
```   586
```
```   587 definition
```
```   588   int_aux :: "nat \<Rightarrow> int \<Rightarrow> int" where
```
```   589   "int_aux n i = int n + i"
```
```   590
```
```   591 lemma [code]:
```
```   592   "int_aux 0 i  = i"
```
```   593   "int_aux (Suc n) i = int_aux n (i + 1)" -- {* tail recursive *}
```
```   594   by (simp add: int_aux_def)+
```
```   595
```
```   596 lemma [code unfold]:
```
```   597   "int n = int_aux n 0"
```
```   598   by (simp add: int_aux_def)
```
```   599
```
```   600 definition
```
```   601   nat_aux :: "int \<Rightarrow> nat \<Rightarrow> nat" where
```
```   602   "nat_aux i n = nat i + n"
```
```   603
```
```   604 lemma [code]:
```
```   605   "nat_aux i n = (if i \<le> 0 then n else nat_aux (i - 1) (Suc n))"  -- {* tail recursive *}
```
```   606   by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le
```
```   607     dest: zless_imp_add1_zle)
```
```   608
```
```   609 lemma [code]: "nat i = nat_aux i 0"
```
```   610   by (simp add: nat_aux_def)
```
```   611
```
```   612 lemma zero_is_num_zero [code func, code inline, symmetric, code post]:
```
```   613   "(0\<Colon>int) = Numeral0"
```
```   614   by simp
```
```   615
```
```   616 lemma one_is_num_one [code func, code inline, symmetric, code post]:
```
```   617   "(1\<Colon>int) = Numeral1"
```
```   618   by simp
```
```   619
```
```   620 code_modulename SML
```
```   621   IntDef Integer
```
```   622
```
```   623 code_modulename OCaml
```
```   624   IntDef Integer
```
```   625
```
```   626 code_modulename Haskell
```
```   627   IntDef Integer
```
```   628
```
```   629 code_modulename SML
```
```   630   Numeral Integer
```
```   631
```
```   632 code_modulename OCaml
```
```   633   Numeral Integer
```
```   634
```
```   635 code_modulename Haskell
```
```   636   Numeral Integer
```
```   637
```
```   638 types_code
```
```   639   "int" ("int")
```
```   640 attach (term_of) {*
```
```   641 val term_of_int = HOLogic.mk_number HOLogic.intT;
```
```   642 *}
```
```   643 attach (test) {*
```
```   644 fun gen_int i = one_of [~1, 1] * random_range 0 i;
```
```   645 *}
```
```   646
```
```   647 setup {*
```
```   648 let
```
```   649
```
```   650 fun strip_number_of (@{term "Numeral.number_of :: int => int"} \$ t) = t
```
```   651   | strip_number_of t = t;
```
```   652
```
```   653 fun numeral_codegen thy defs gr dep module b t =
```
```   654   let val i = HOLogic.dest_numeral (strip_number_of t)
```
```   655   in
```
```   656     SOME (fst (Codegen.invoke_tycodegen thy defs dep module false (gr, HOLogic.intT)),
```
```   657       Pretty.str (string_of_int i))
```
```   658   end handle TERM _ => NONE;
```
```   659
```
```   660 in
```
```   661
```
```   662 Codegen.add_codegen "numeral_codegen" numeral_codegen
```
```   663
```
```   664 end
```
```   665 *}
```
```   666
```
```   667 consts_code
```
```   668   "number_of :: int \<Rightarrow> int"    ("(_)")
```
```   669   "0 :: int"                   ("0")
```
```   670   "1 :: int"                   ("1")
```
```   671   "uminus :: int => int"       ("~")
```
```   672   "op + :: int => int => int"  ("(_ +/ _)")
```
```   673   "op * :: int => int => int"  ("(_ */ _)")
```
```   674   "op \<le> :: int => int => bool" ("(_ <=/ _)")
```
```   675   "op < :: int => int => bool" ("(_ </ _)")
```
```   676
```
```   677 quickcheck_params [default_type = int]
```
```   678
```
```   679 (*setup continues in theory Presburger*)
```
```   680
```
```   681 hide (open) const Pls Min B0 B1 succ pred
```
```   682
```
```   683 end
```