src/HOL/Integ/Numeral.thy
 author haftmann Tue Sep 19 15:22:03 2006 +0200 (2006-09-19) changeset 20596 3950e65f48f8 parent 20500 11da1ce8dbd8 child 20699 0cc77abb185a permissions -rw-r--r--
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```     1 (*  Title:      HOL/Integ/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 IntDef Datatype
```
```    11 uses "../Tools/numeral_syntax.ML"
```
```    12 begin
```
```    13
```
```    14 text {*
```
```    15   This formalization defines binary arithmetic in terms of the integers
```
```    16   rather than using a datatype. This avoids multiple representations (leading
```
```    17   zeroes, etc.)  See @{text "ZF/Integ/twos-compl.ML"}, function @{text
```
```    18   int_of_binary}, for the numerical interpretation.
```
```    19
```
```    20   The representation expects that @{text "(m mod 2)"} is 0 or 1,
```
```    21   even if m is negative;
```
```    22   For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
```
```    23   @{text "-5 = (-3)*2 + 1"}.
```
```    24 *}
```
```    25
```
```    26 text{*
```
```    27   This datatype avoids the use of type @{typ bool}, which would make
```
```    28   all of the rewrite rules higher-order.
```
```    29 *}
```
```    30
```
```    31 datatype bit = B0 | B1
```
```    32
```
```    33 constdefs
```
```    34   Pls :: int
```
```    35   "Pls == 0"
```
```    36   Min :: int
```
```    37   "Min == - 1"
```
```    38   Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90)
```
```    39   "k BIT b == (case b of B0 \<Rightarrow> 0 | B1 \<Rightarrow> 1) + k + k"
```
```    40
```
```    41 axclass
```
```    42   number < type  -- {* for numeric types: nat, int, real, \dots *}
```
```    43
```
```    44 consts
```
```    45   number_of :: "int \<Rightarrow> 'a::number"
```
```    46
```
```    47 syntax
```
```    48   "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
```
```    49
```
```    50 setup NumeralSyntax.setup
```
```    51
```
```    52 abbreviation
```
```    53   "Numeral0 \<equiv> number_of Pls"
```
```    54   "Numeral1 \<equiv> number_of (Pls BIT B1)"
```
```    55
```
```    56 lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"
```
```    57   -- {* Unfold all @{text let}s involving constants *}
```
```    58   unfolding Let_def ..
```
```    59
```
```    60 lemma Let_0 [simp]: "Let 0 f = f 0"
```
```    61   unfolding Let_def ..
```
```    62
```
```    63 lemma Let_1 [simp]: "Let 1 f = f 1"
```
```    64   unfolding Let_def ..
```
```    65
```
```    66 definition
```
```    67   succ :: "int \<Rightarrow> int"
```
```    68   "succ k = k + 1"
```
```    69   pred :: "int \<Rightarrow> int"
```
```    70   "pred k = k - 1"
```
```    71
```
```    72 lemmas numeral_simps =
```
```    73   succ_def pred_def Pls_def Min_def Bit_def
```
```    74
```
```    75 text {* Removal of leading zeroes *}
```
```    76
```
```    77 lemma Pls_0_eq [simp]:
```
```    78   "Pls BIT B0 = Pls"
```
```    79   unfolding numeral_simps by simp
```
```    80
```
```    81 lemma Min_1_eq [simp]:
```
```    82   "Min BIT B1 = Min"
```
```    83   unfolding numeral_simps by simp
```
```    84
```
```    85
```
```    86 subsection {* The Functions @{term succ}, @{term pred} and @{term uminus} *}
```
```    87
```
```    88 lemma succ_Pls [simp]:
```
```    89   "succ Pls = Pls BIT B1"
```
```    90   unfolding numeral_simps by simp
```
```    91
```
```    92 lemma succ_Min [simp]:
```
```    93   "succ Min = Pls"
```
```    94   unfolding numeral_simps by simp
```
```    95
```
```    96 lemma succ_1 [simp]:
```
```    97   "succ (k BIT B1) = succ k BIT B0"
```
```    98   unfolding numeral_simps by simp
```
```    99
```
```   100 lemma succ_0 [simp]:
```
```   101   "succ (k BIT B0) = k BIT B1"
```
```   102   unfolding numeral_simps by simp
```
```   103
```
```   104 lemma pred_Pls [simp]:
```
```   105   "pred Pls = Min"
```
```   106   unfolding numeral_simps by simp
```
```   107
```
```   108 lemma pred_Min [simp]:
```
```   109   "pred Min = Min BIT B0"
```
```   110   unfolding numeral_simps by simp
```
```   111
```
```   112 lemma pred_1 [simp]:
```
```   113   "pred (k BIT B1) = k BIT B0"
```
```   114   unfolding numeral_simps by simp
```
```   115
```
```   116 lemma pred_0 [simp]:
```
```   117   "pred (k BIT B0) = pred k BIT B1"
```
```   118   unfolding numeral_simps by simp
```
```   119
```
```   120 lemma minus_Pls [simp]:
```
```   121   "- Pls = Pls"
```
```   122   unfolding numeral_simps by simp
```
```   123
```
```   124 lemma minus_Min [simp]:
```
```   125   "- Min = Pls BIT B1"
```
```   126   unfolding numeral_simps by simp
```
```   127
```
```   128 lemma minus_1 [simp]:
```
```   129   "- (k BIT B1) = pred (- k) BIT B1"
```
```   130   unfolding numeral_simps by simp
```
```   131
```
```   132 lemma minus_0 [simp]:
```
```   133   "- (k BIT B0) = (- k) BIT B0"
```
```   134   unfolding numeral_simps by simp
```
```   135
```
```   136
```
```   137 subsection {*
```
```   138   Binary Addition and Multiplication: @{term "op + \<Colon> int \<Rightarrow> int \<Rightarrow> int"}
```
```   139     and @{term "op * \<Colon> int \<Rightarrow> int \<Rightarrow> int"}
```
```   140 *}
```
```   141
```
```   142 lemma add_Pls [simp]:
```
```   143   "Pls + k = k"
```
```   144   unfolding numeral_simps by simp
```
```   145
```
```   146 lemma add_Min [simp]:
```
```   147   "Min + k = pred k"
```
```   148   unfolding numeral_simps by simp
```
```   149
```
```   150 lemma add_BIT_11 [simp]:
```
```   151   "(k BIT B1) + (l BIT B1) = (k + succ l) BIT B0"
```
```   152   unfolding numeral_simps by simp
```
```   153
```
```   154 lemma add_BIT_10 [simp]:
```
```   155   "(k BIT B1) + (l BIT B0) = (k + l) BIT B1"
```
```   156   unfolding numeral_simps by simp
```
```   157
```
```   158 lemma add_BIT_0 [simp]:
```
```   159   "(k BIT B0) + (l BIT b) = (k + l) BIT b"
```
```   160   unfolding numeral_simps by simp
```
```   161
```
```   162 lemma add_Pls_right [simp]:
```
```   163   "k + Pls = k"
```
```   164   unfolding numeral_simps by simp
```
```   165
```
```   166 lemma add_Min_right [simp]:
```
```   167   "k + Min = pred k"
```
```   168   unfolding numeral_simps by simp
```
```   169
```
```   170 lemma mult_Pls [simp]:
```
```   171   "Pls * w = Pls"
```
```   172   unfolding numeral_simps by simp
```
```   173
```
```   174 lemma mult_Min [simp]:
```
```   175   "Min * k = - k"
```
```   176   unfolding numeral_simps by simp
```
```   177
```
```   178 lemma mult_num1 [simp]:
```
```   179   "(k BIT B1) * l = ((k * l) BIT B0) + l"
```
```   180   unfolding numeral_simps int_distrib by simp
```
```   181
```
```   182 lemma mult_num0 [simp]:
```
```   183   "(k BIT B0) * l = (k * l) BIT B0"
```
```   184   unfolding numeral_simps int_distrib by simp
```
```   185
```
```   186
```
```   187
```
```   188 subsection {* Converting Numerals to Rings: @{term number_of} *}
```
```   189
```
```   190 axclass number_ring \<subseteq> number, comm_ring_1
```
```   191   number_of_eq: "number_of k = of_int k"
```
```   192
```
```   193 lemma number_of_succ:
```
```   194   "number_of (succ k) = (1 + number_of k ::'a::number_ring)"
```
```   195   unfolding number_of_eq numeral_simps by simp
```
```   196
```
```   197 lemma number_of_pred:
```
```   198   "number_of (pred w) = (- 1 + number_of w ::'a::number_ring)"
```
```   199   unfolding number_of_eq numeral_simps by simp
```
```   200
```
```   201 lemma number_of_minus:
```
```   202   "number_of (uminus w) = (- (number_of w)::'a::number_ring)"
```
```   203   unfolding number_of_eq numeral_simps by simp
```
```   204
```
```   205 lemma number_of_add:
```
```   206   "number_of (v + w) = (number_of v + number_of w::'a::number_ring)"
```
```   207   unfolding number_of_eq numeral_simps by simp
```
```   208
```
```   209 lemma number_of_mult:
```
```   210   "number_of (v * w) = (number_of v * number_of w::'a::number_ring)"
```
```   211   unfolding number_of_eq numeral_simps by simp
```
```   212
```
```   213 text {*
```
```   214   The correctness of shifting.
```
```   215   But it doesn't seem to give a measurable speed-up.
```
```   216 *}
```
```   217
```
```   218 lemma double_number_of_BIT:
```
```   219   "(1 + 1) * number_of w = (number_of (w BIT B0) ::'a::number_ring)"
```
```   220   unfolding number_of_eq numeral_simps left_distrib by simp
```
```   221
```
```   222 text {*
```
```   223   Converting numerals 0 and 1 to their abstract versions.
```
```   224 *}
```
```   225
```
```   226 lemma numeral_0_eq_0 [simp]:
```
```   227   "Numeral0 = (0::'a::number_ring)"
```
```   228   unfolding number_of_eq numeral_simps by simp
```
```   229
```
```   230 lemma numeral_1_eq_1 [simp]:
```
```   231   "Numeral1 = (1::'a::number_ring)"
```
```   232   unfolding number_of_eq numeral_simps by simp
```
```   233
```
```   234 text {*
```
```   235   Special-case simplification for small constants.
```
```   236 *}
```
```   237
```
```   238 text{*
```
```   239   Unary minus for the abstract constant 1. Cannot be inserted
```
```   240   as a simprule until later: it is @{text number_of_Min} re-oriented!
```
```   241 *}
```
```   242
```
```   243 lemma numeral_m1_eq_minus_1:
```
```   244   "(-1::'a::number_ring) = - 1"
```
```   245   unfolding number_of_eq numeral_simps by simp
```
```   246
```
```   247 lemma mult_minus1 [simp]:
```
```   248   "-1 * z = -(z::'a::number_ring)"
```
```   249   unfolding number_of_eq numeral_simps by simp
```
```   250
```
```   251 lemma mult_minus1_right [simp]:
```
```   252   "z * -1 = -(z::'a::number_ring)"
```
```   253   unfolding number_of_eq numeral_simps by simp
```
```   254
```
```   255 (*Negation of a coefficient*)
```
```   256 lemma minus_number_of_mult [simp]:
```
```   257    "- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)"
```
```   258    unfolding number_of_eq by simp
```
```   259
```
```   260 text {* Subtraction *}
```
```   261
```
```   262 lemma diff_number_of_eq:
```
```   263   "number_of v - number_of w =
```
```   264     (number_of (v + uminus w)::'a::number_ring)"
```
```   265   unfolding number_of_eq by simp
```
```   266
```
```   267 lemma number_of_Pls:
```
```   268   "number_of Pls = (0::'a::number_ring)"
```
```   269   unfolding number_of_eq numeral_simps by simp
```
```   270
```
```   271 lemma number_of_Min:
```
```   272   "number_of Min = (- 1::'a::number_ring)"
```
```   273   unfolding number_of_eq numeral_simps by simp
```
```   274
```
```   275 lemma number_of_BIT:
```
```   276   "number_of(w BIT x) = (case x of B0 => 0 | B1 => (1::'a::number_ring))
```
```   277     + (number_of w) + (number_of w)"
```
```   278   unfolding number_of_eq numeral_simps by (simp split: bit.split)
```
```   279
```
```   280
```
```   281 subsection {* Equality of Binary Numbers *}
```
```   282
```
```   283 text {* First version by Norbert Voelker *}
```
```   284
```
```   285 lemma eq_number_of_eq:
```
```   286   "((number_of x::'a::number_ring) = number_of y) =
```
```   287    iszero (number_of (x + uminus y) :: 'a)"
```
```   288   unfolding iszero_def number_of_add number_of_minus
```
```   289   by (simp add: compare_rls)
```
```   290
```
```   291 lemma iszero_number_of_Pls:
```
```   292   "iszero ((number_of Pls)::'a::number_ring)"
```
```   293   unfolding iszero_def numeral_0_eq_0 ..
```
```   294
```
```   295 lemma nonzero_number_of_Min:
```
```   296   "~ iszero ((number_of Min)::'a::number_ring)"
```
```   297   unfolding iszero_def numeral_m1_eq_minus_1 by simp
```
```   298
```
```   299
```
```   300 subsection {* Comparisons, for Ordered Rings *}
```
```   301
```
```   302 lemma double_eq_0_iff:
```
```   303   "(a + a = 0) = (a = (0::'a::ordered_idom))"
```
```   304 proof -
```
```   305   have "a + a = (1 + 1) * a" unfolding left_distrib by simp
```
```   306   with zero_less_two [where 'a = 'a]
```
```   307   show ?thesis by force
```
```   308 qed
```
```   309
```
```   310 lemma le_imp_0_less:
```
```   311   assumes le: "0 \<le> z"
```
```   312   shows "(0::int) < 1 + z"
```
```   313 proof -
```
```   314   have "0 \<le> z" .
```
```   315   also have "... < z + 1" by (rule less_add_one)
```
```   316   also have "... = 1 + z" by (simp add: add_ac)
```
```   317   finally show "0 < 1 + z" .
```
```   318 qed
```
```   319
```
```   320 lemma odd_nonzero:
```
```   321   "1 + z + z \<noteq> (0::int)";
```
```   322 proof (cases z rule: int_cases)
```
```   323   case (nonneg n)
```
```   324   have le: "0 \<le> z+z" by (simp add: nonneg add_increasing)
```
```   325   thus ?thesis using  le_imp_0_less [OF le]
```
```   326     by (auto simp add: add_assoc)
```
```   327 next
```
```   328   case (neg n)
```
```   329   show ?thesis
```
```   330   proof
```
```   331     assume eq: "1 + z + z = 0"
```
```   332     have "0 < 1 + (int n + int n)"
```
```   333       by (simp add: le_imp_0_less add_increasing)
```
```   334     also have "... = - (1 + z + z)"
```
```   335       by (simp add: neg add_assoc [symmetric])
```
```   336     also have "... = 0" by (simp add: eq)
```
```   337     finally have "0<0" ..
```
```   338     thus False by blast
```
```   339   qed
```
```   340 qed
```
```   341
```
```   342 text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
```
```   343
```
```   344 lemma Ints_odd_nonzero:
```
```   345   assumes in_Ints: "a \<in> Ints"
```
```   346   shows "1 + a + a \<noteq> (0::'a::ordered_idom)"
```
```   347 proof -
```
```   348   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```   349   then obtain z where a: "a = of_int z" ..
```
```   350   show ?thesis
```
```   351   proof
```
```   352     assume eq: "1 + a + a = 0"
```
```   353     hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
```
```   354     hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
```
```   355     with odd_nonzero show False by blast
```
```   356   qed
```
```   357 qed
```
```   358
```
```   359 lemma Ints_number_of:
```
```   360   "(number_of w :: 'a::number_ring) \<in> Ints"
```
```   361   unfolding number_of_eq Ints_def by simp
```
```   362
```
```   363 lemma iszero_number_of_BIT:
```
```   364   "iszero (number_of (w BIT x)::'a) =
```
```   365    (x = B0 \<and> iszero (number_of w::'a::{ordered_idom,number_ring}))"
```
```   366   by (simp add: iszero_def number_of_eq numeral_simps double_eq_0_iff
```
```   367     Ints_odd_nonzero Ints_def split: bit.split)
```
```   368
```
```   369 lemma iszero_number_of_0:
```
```   370   "iszero (number_of (w BIT B0) :: 'a::{ordered_idom,number_ring}) =
```
```   371   iszero (number_of w :: 'a)"
```
```   372   by (simp only: iszero_number_of_BIT simp_thms)
```
```   373
```
```   374 lemma iszero_number_of_1:
```
```   375   "~ iszero (number_of (w BIT B1)::'a::{ordered_idom,number_ring})"
```
```   376   by (simp add: iszero_number_of_BIT)
```
```   377
```
```   378
```
```   379 subsection {* The Less-Than Relation *}
```
```   380
```
```   381 lemma less_number_of_eq_neg:
```
```   382   "((number_of x::'a::{ordered_idom,number_ring}) < number_of y)
```
```   383   = neg (number_of (x + uminus y) :: 'a)"
```
```   384 apply (subst less_iff_diff_less_0)
```
```   385 apply (simp add: neg_def diff_minus number_of_add number_of_minus)
```
```   386 done
```
```   387
```
```   388 text {*
```
```   389   If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
```
```   390   @{term Numeral0} IS @{term "number_of Pls"}
```
```   391 *}
```
```   392
```
```   393 lemma not_neg_number_of_Pls:
```
```   394   "~ neg (number_of Pls ::'a::{ordered_idom,number_ring})"
```
```   395   by (simp add: neg_def numeral_0_eq_0)
```
```   396
```
```   397 lemma neg_number_of_Min:
```
```   398   "neg (number_of Min ::'a::{ordered_idom,number_ring})"
```
```   399   by (simp add: neg_def zero_less_one numeral_m1_eq_minus_1)
```
```   400
```
```   401 lemma double_less_0_iff:
```
```   402   "(a + a < 0) = (a < (0::'a::ordered_idom))"
```
```   403 proof -
```
```   404   have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
```
```   405   also have "... = (a < 0)"
```
```   406     by (simp add: mult_less_0_iff zero_less_two
```
```   407                   order_less_not_sym [OF zero_less_two])
```
```   408   finally show ?thesis .
```
```   409 qed
```
```   410
```
```   411 lemma odd_less_0:
```
```   412   "(1 + z + z < 0) = (z < (0::int))";
```
```   413 proof (cases z rule: int_cases)
```
```   414   case (nonneg n)
```
```   415   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
```
```   416                              le_imp_0_less [THEN order_less_imp_le])
```
```   417 next
```
```   418   case (neg n)
```
```   419   thus ?thesis by (simp del: int_Suc
```
```   420     add: int_Suc0_eq_1 [symmetric] zadd_int compare_rls)
```
```   421 qed
```
```   422
```
```   423 text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
```
```   424
```
```   425 lemma Ints_odd_less_0:
```
```   426   assumes in_Ints: "a \<in> Ints"
```
```   427   shows "(1 + a + a < 0) = (a < (0::'a::ordered_idom))";
```
```   428 proof -
```
```   429   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
```
```   430   then obtain z where a: "a = of_int z" ..
```
```   431   hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
```
```   432     by (simp add: a)
```
```   433   also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
```
```   434   also have "... = (a < 0)" by (simp add: a)
```
```   435   finally show ?thesis .
```
```   436 qed
```
```   437
```
```   438 lemma neg_number_of_BIT:
```
```   439   "neg (number_of (w BIT x)::'a) =
```
```   440   neg (number_of w :: 'a::{ordered_idom,number_ring})"
```
```   441   by (simp add: neg_def number_of_eq numeral_simps double_less_0_iff
```
```   442     Ints_odd_less_0 Ints_def split: bit.split)
```
```   443
```
```   444
```
```   445 text {* Less-Than or Equals *}
```
```   446
```
```   447 text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *}
```
```   448
```
```   449 lemmas le_number_of_eq_not_less =
```
```   450   linorder_not_less [of "number_of w" "number_of v", symmetric,
```
```   451   standard]
```
```   452
```
```   453 lemma le_number_of_eq:
```
```   454     "((number_of x::'a::{ordered_idom,number_ring}) \<le> number_of y)
```
```   455      = (~ (neg (number_of (y + uminus x) :: 'a)))"
```
```   456 by (simp add: le_number_of_eq_not_less less_number_of_eq_neg)
```
```   457
```
```   458
```
```   459 text {* Absolute value (@{term abs}) *}
```
```   460
```
```   461 lemma abs_number_of:
```
```   462   "abs(number_of x::'a::{ordered_idom,number_ring}) =
```
```   463    (if number_of x < (0::'a) then -number_of x else number_of x)"
```
```   464   by (simp add: abs_if)
```
```   465
```
```   466
```
```   467 text {* Re-orientation of the equation nnn=x *}
```
```   468
```
```   469 lemma number_of_reorient:
```
```   470   "(number_of w = x) = (x = number_of w)"
```
```   471   by auto
```
```   472
```
```   473
```
```   474 subsection {* Simplification of arithmetic operations on integer constants. *}
```
```   475
```
```   476 lemmas arith_extra_simps =
```
```   477   number_of_add [symmetric]
```
```   478   number_of_minus [symmetric] numeral_m1_eq_minus_1 [symmetric]
```
```   479   number_of_mult [symmetric]
```
```   480   diff_number_of_eq abs_number_of
```
```   481
```
```   482 text {*
```
```   483   For making a minimal simpset, one must include these default simprules.
```
```   484   Also include @{text simp_thms}.
```
```   485 *}
```
```   486
```
```   487 lemmas arith_simps =
```
```   488   bit.distinct
```
```   489   Pls_0_eq Min_1_eq
```
```   490   pred_Pls pred_Min pred_1 pred_0
```
```   491   succ_Pls succ_Min succ_1 succ_0
```
```   492   add_Pls add_Min add_BIT_0 add_BIT_10 add_BIT_11
```
```   493   minus_Pls minus_Min minus_1 minus_0
```
```   494   mult_Pls mult_Min mult_num1 mult_num0
```
```   495   add_Pls_right add_Min_right
```
```   496   abs_zero abs_one arith_extra_simps
```
```   497
```
```   498 text {* Simplification of relational operations *}
```
```   499
```
```   500 lemmas rel_simps =
```
```   501   eq_number_of_eq iszero_number_of_Pls nonzero_number_of_Min
```
```   502   iszero_number_of_0 iszero_number_of_1
```
```   503   less_number_of_eq_neg
```
```   504   not_neg_number_of_Pls not_neg_0 not_neg_1 not_iszero_1
```
```   505   neg_number_of_Min neg_number_of_BIT
```
```   506   le_number_of_eq
```
```   507
```
```   508 declare arith_extra_simps [simp]
```
```   509 declare rel_simps [simp]
```
```   510
```
```   511
```
```   512 subsection {* Simplification of arithmetic when nested to the right. *}
```
```   513
```
```   514 lemma add_number_of_left [simp]:
```
```   515   "number_of v + (number_of w + z) =
```
```   516    (number_of(v + w) + z::'a::number_ring)"
```
```   517   by (simp add: add_assoc [symmetric])
```
```   518
```
```   519 lemma mult_number_of_left [simp]:
```
```   520   "number_of v * (number_of w * z) =
```
```   521    (number_of(v * w) * z::'a::number_ring)"
```
```   522   by (simp add: mult_assoc [symmetric])
```
```   523
```
```   524 lemma add_number_of_diff1:
```
```   525   "number_of v + (number_of w - c) =
```
```   526   number_of(v + w) - (c::'a::number_ring)"
```
```   527   by (simp add: diff_minus add_number_of_left)
```
```   528
```
```   529 lemma add_number_of_diff2 [simp]:
```
```   530   "number_of v + (c - number_of w) =
```
```   531    number_of (v + uminus w) + (c::'a::number_ring)"
```
```   532 apply (subst diff_number_of_eq [symmetric])
```
```   533 apply (simp only: compare_rls)
```
```   534 done
```
```   535
```
```   536
```
```   537 hide (open) const Pls Min B0 B1 succ pred
```
```   538
```
```   539 end
```