src/HOL/Num.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63654 f90e3926e627 child 63913 6b886cadba7c permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Num.thy
```
```     2     Author:     Florian Haftmann
```
```     3     Author:     Brian Huffman
```
```     4 *)
```
```     5
```
```     6 section \<open>Binary Numerals\<close>
```
```     7
```
```     8 theory Num
```
```     9   imports BNF_Least_Fixpoint
```
```    10 begin
```
```    11
```
```    12 subsection \<open>The \<open>num\<close> type\<close>
```
```    13
```
```    14 datatype num = One | Bit0 num | Bit1 num
```
```    15
```
```    16 text \<open>Increment function for type @{typ num}\<close>
```
```    17
```
```    18 primrec inc :: "num \<Rightarrow> num"
```
```    19   where
```
```    20     "inc One = Bit0 One"
```
```    21   | "inc (Bit0 x) = Bit1 x"
```
```    22   | "inc (Bit1 x) = Bit0 (inc x)"
```
```    23
```
```    24 text \<open>Converting between type @{typ num} and type @{typ nat}\<close>
```
```    25
```
```    26 primrec nat_of_num :: "num \<Rightarrow> nat"
```
```    27   where
```
```    28     "nat_of_num One = Suc 0"
```
```    29   | "nat_of_num (Bit0 x) = nat_of_num x + nat_of_num x"
```
```    30   | "nat_of_num (Bit1 x) = Suc (nat_of_num x + nat_of_num x)"
```
```    31
```
```    32 primrec num_of_nat :: "nat \<Rightarrow> num"
```
```    33   where
```
```    34     "num_of_nat 0 = One"
```
```    35   | "num_of_nat (Suc n) = (if 0 < n then inc (num_of_nat n) else One)"
```
```    36
```
```    37 lemma nat_of_num_pos: "0 < nat_of_num x"
```
```    38   by (induct x) simp_all
```
```    39
```
```    40 lemma nat_of_num_neq_0: " nat_of_num x \<noteq> 0"
```
```    41   by (induct x) simp_all
```
```    42
```
```    43 lemma nat_of_num_inc: "nat_of_num (inc x) = Suc (nat_of_num x)"
```
```    44   by (induct x) simp_all
```
```    45
```
```    46 lemma num_of_nat_double: "0 < n \<Longrightarrow> num_of_nat (n + n) = Bit0 (num_of_nat n)"
```
```    47   by (induct n) simp_all
```
```    48
```
```    49 text \<open>Type @{typ num} is isomorphic to the strictly positive natural numbers.\<close>
```
```    50
```
```    51 lemma nat_of_num_inverse: "num_of_nat (nat_of_num x) = x"
```
```    52   by (induct x) (simp_all add: num_of_nat_double nat_of_num_pos)
```
```    53
```
```    54 lemma num_of_nat_inverse: "0 < n \<Longrightarrow> nat_of_num (num_of_nat n) = n"
```
```    55   by (induct n) (simp_all add: nat_of_num_inc)
```
```    56
```
```    57 lemma num_eq_iff: "x = y \<longleftrightarrow> nat_of_num x = nat_of_num y"
```
```    58   apply safe
```
```    59   apply (drule arg_cong [where f=num_of_nat])
```
```    60   apply (simp add: nat_of_num_inverse)
```
```    61   done
```
```    62
```
```    63 lemma num_induct [case_names One inc]:
```
```    64   fixes P :: "num \<Rightarrow> bool"
```
```    65   assumes One: "P One"
```
```    66     and inc: "\<And>x. P x \<Longrightarrow> P (inc x)"
```
```    67   shows "P x"
```
```    68 proof -
```
```    69   obtain n where n: "Suc n = nat_of_num x"
```
```    70     by (cases "nat_of_num x") (simp_all add: nat_of_num_neq_0)
```
```    71   have "P (num_of_nat (Suc n))"
```
```    72   proof (induct n)
```
```    73     case 0
```
```    74     from One show ?case by simp
```
```    75   next
```
```    76     case (Suc n)
```
```    77     then have "P (inc (num_of_nat (Suc n)))" by (rule inc)
```
```    78     then show "P (num_of_nat (Suc (Suc n)))" by simp
```
```    79   qed
```
```    80   with n show "P x"
```
```    81     by (simp add: nat_of_num_inverse)
```
```    82 qed
```
```    83
```
```    84 text \<open>
```
```    85   From now on, there are two possible models for @{typ num}: as positive
```
```    86   naturals (rule \<open>num_induct\<close>) and as digit representation (rules
```
```    87   \<open>num.induct\<close>, \<open>num.cases\<close>).
```
```    88 \<close>
```
```    89
```
```    90
```
```    91 subsection \<open>Numeral operations\<close>
```
```    92
```
```    93 instantiation num :: "{plus,times,linorder}"
```
```    94 begin
```
```    95
```
```    96 definition [code del]: "m + n = num_of_nat (nat_of_num m + nat_of_num n)"
```
```    97
```
```    98 definition [code del]: "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
```
```    99
```
```   100 definition [code del]: "m \<le> n \<longleftrightarrow> nat_of_num m \<le> nat_of_num n"
```
```   101
```
```   102 definition [code del]: "m < n \<longleftrightarrow> nat_of_num m < nat_of_num n"
```
```   103
```
```   104 instance
```
```   105   by standard (auto simp add: less_num_def less_eq_num_def num_eq_iff)
```
```   106
```
```   107 end
```
```   108
```
```   109 lemma nat_of_num_add: "nat_of_num (x + y) = nat_of_num x + nat_of_num y"
```
```   110   unfolding plus_num_def
```
```   111   by (intro num_of_nat_inverse add_pos_pos nat_of_num_pos)
```
```   112
```
```   113 lemma nat_of_num_mult: "nat_of_num (x * y) = nat_of_num x * nat_of_num y"
```
```   114   unfolding times_num_def
```
```   115   by (intro num_of_nat_inverse mult_pos_pos nat_of_num_pos)
```
```   116
```
```   117 lemma add_num_simps [simp, code]:
```
```   118   "One + One = Bit0 One"
```
```   119   "One + Bit0 n = Bit1 n"
```
```   120   "One + Bit1 n = Bit0 (n + One)"
```
```   121   "Bit0 m + One = Bit1 m"
```
```   122   "Bit0 m + Bit0 n = Bit0 (m + n)"
```
```   123   "Bit0 m + Bit1 n = Bit1 (m + n)"
```
```   124   "Bit1 m + One = Bit0 (m + One)"
```
```   125   "Bit1 m + Bit0 n = Bit1 (m + n)"
```
```   126   "Bit1 m + Bit1 n = Bit0 (m + n + One)"
```
```   127   by (simp_all add: num_eq_iff nat_of_num_add)
```
```   128
```
```   129 lemma mult_num_simps [simp, code]:
```
```   130   "m * One = m"
```
```   131   "One * n = n"
```
```   132   "Bit0 m * Bit0 n = Bit0 (Bit0 (m * n))"
```
```   133   "Bit0 m * Bit1 n = Bit0 (m * Bit1 n)"
```
```   134   "Bit1 m * Bit0 n = Bit0 (Bit1 m * n)"
```
```   135   "Bit1 m * Bit1 n = Bit1 (m + n + Bit0 (m * n))"
```
```   136   by (simp_all add: num_eq_iff nat_of_num_add nat_of_num_mult distrib_right distrib_left)
```
```   137
```
```   138 lemma eq_num_simps:
```
```   139   "One = One \<longleftrightarrow> True"
```
```   140   "One = Bit0 n \<longleftrightarrow> False"
```
```   141   "One = Bit1 n \<longleftrightarrow> False"
```
```   142   "Bit0 m = One \<longleftrightarrow> False"
```
```   143   "Bit1 m = One \<longleftrightarrow> False"
```
```   144   "Bit0 m = Bit0 n \<longleftrightarrow> m = n"
```
```   145   "Bit0 m = Bit1 n \<longleftrightarrow> False"
```
```   146   "Bit1 m = Bit0 n \<longleftrightarrow> False"
```
```   147   "Bit1 m = Bit1 n \<longleftrightarrow> m = n"
```
```   148   by simp_all
```
```   149
```
```   150 lemma le_num_simps [simp, code]:
```
```   151   "One \<le> n \<longleftrightarrow> True"
```
```   152   "Bit0 m \<le> One \<longleftrightarrow> False"
```
```   153   "Bit1 m \<le> One \<longleftrightarrow> False"
```
```   154   "Bit0 m \<le> Bit0 n \<longleftrightarrow> m \<le> n"
```
```   155   "Bit0 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
```
```   156   "Bit1 m \<le> Bit1 n \<longleftrightarrow> m \<le> n"
```
```   157   "Bit1 m \<le> Bit0 n \<longleftrightarrow> m < n"
```
```   158   using nat_of_num_pos [of n] nat_of_num_pos [of m]
```
```   159   by (auto simp add: less_eq_num_def less_num_def)
```
```   160
```
```   161 lemma less_num_simps [simp, code]:
```
```   162   "m < One \<longleftrightarrow> False"
```
```   163   "One < Bit0 n \<longleftrightarrow> True"
```
```   164   "One < Bit1 n \<longleftrightarrow> True"
```
```   165   "Bit0 m < Bit0 n \<longleftrightarrow> m < n"
```
```   166   "Bit0 m < Bit1 n \<longleftrightarrow> m \<le> n"
```
```   167   "Bit1 m < Bit1 n \<longleftrightarrow> m < n"
```
```   168   "Bit1 m < Bit0 n \<longleftrightarrow> m < n"
```
```   169   using nat_of_num_pos [of n] nat_of_num_pos [of m]
```
```   170   by (auto simp add: less_eq_num_def less_num_def)
```
```   171
```
```   172 lemma le_num_One_iff: "x \<le> num.One \<longleftrightarrow> x = num.One"
```
```   173   by (simp add: antisym_conv)
```
```   174
```
```   175 text \<open>Rules using \<open>One\<close> and \<open>inc\<close> as constructors.\<close>
```
```   176
```
```   177 lemma add_One: "x + One = inc x"
```
```   178   by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
```
```   179
```
```   180 lemma add_One_commute: "One + n = n + One"
```
```   181   by (induct n) simp_all
```
```   182
```
```   183 lemma add_inc: "x + inc y = inc (x + y)"
```
```   184   by (simp add: num_eq_iff nat_of_num_add nat_of_num_inc)
```
```   185
```
```   186 lemma mult_inc: "x * inc y = x * y + x"
```
```   187   by (simp add: num_eq_iff nat_of_num_mult nat_of_num_add nat_of_num_inc)
```
```   188
```
```   189 text \<open>The @{const num_of_nat} conversion.\<close>
```
```   190
```
```   191 lemma num_of_nat_One: "n \<le> 1 \<Longrightarrow> num_of_nat n = One"
```
```   192   by (cases n) simp_all
```
```   193
```
```   194 lemma num_of_nat_plus_distrib:
```
```   195   "0 < m \<Longrightarrow> 0 < n \<Longrightarrow> num_of_nat (m + n) = num_of_nat m + num_of_nat n"
```
```   196   by (induct n) (auto simp add: add_One add_One_commute add_inc)
```
```   197
```
```   198 text \<open>A double-and-decrement function.\<close>
```
```   199
```
```   200 primrec BitM :: "num \<Rightarrow> num"
```
```   201   where
```
```   202     "BitM One = One"
```
```   203   | "BitM (Bit0 n) = Bit1 (BitM n)"
```
```   204   | "BitM (Bit1 n) = Bit1 (Bit0 n)"
```
```   205
```
```   206 lemma BitM_plus_one: "BitM n + One = Bit0 n"
```
```   207   by (induct n) simp_all
```
```   208
```
```   209 lemma one_plus_BitM: "One + BitM n = Bit0 n"
```
```   210   unfolding add_One_commute BitM_plus_one ..
```
```   211
```
```   212 text \<open>Squaring and exponentiation.\<close>
```
```   213
```
```   214 primrec sqr :: "num \<Rightarrow> num"
```
```   215   where
```
```   216     "sqr One = One"
```
```   217   | "sqr (Bit0 n) = Bit0 (Bit0 (sqr n))"
```
```   218   | "sqr (Bit1 n) = Bit1 (Bit0 (sqr n + n))"
```
```   219
```
```   220 primrec pow :: "num \<Rightarrow> num \<Rightarrow> num"
```
```   221   where
```
```   222     "pow x One = x"
```
```   223   | "pow x (Bit0 y) = sqr (pow x y)"
```
```   224   | "pow x (Bit1 y) = sqr (pow x y) * x"
```
```   225
```
```   226 lemma nat_of_num_sqr: "nat_of_num (sqr x) = nat_of_num x * nat_of_num x"
```
```   227   by (induct x) (simp_all add: algebra_simps nat_of_num_add)
```
```   228
```
```   229 lemma sqr_conv_mult: "sqr x = x * x"
```
```   230   by (simp add: num_eq_iff nat_of_num_sqr nat_of_num_mult)
```
```   231
```
```   232
```
```   233 subsection \<open>Binary numerals\<close>
```
```   234
```
```   235 text \<open>
```
```   236   We embed binary representations into a generic algebraic
```
```   237   structure using \<open>numeral\<close>.
```
```   238 \<close>
```
```   239
```
```   240 class numeral = one + semigroup_add
```
```   241 begin
```
```   242
```
```   243 primrec numeral :: "num \<Rightarrow> 'a"
```
```   244   where
```
```   245     numeral_One: "numeral One = 1"
```
```   246   | numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n"
```
```   247   | numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"
```
```   248
```
```   249 lemma numeral_code [code]:
```
```   250   "numeral One = 1"
```
```   251   "numeral (Bit0 n) = (let m = numeral n in m + m)"
```
```   252   "numeral (Bit1 n) = (let m = numeral n in m + m + 1)"
```
```   253   by (simp_all add: Let_def)
```
```   254
```
```   255 lemma one_plus_numeral_commute: "1 + numeral x = numeral x + 1"
```
```   256 proof (induct x)
```
```   257   case One
```
```   258   then show ?case by simp
```
```   259 next
```
```   260   case Bit0
```
```   261   then show ?case by (simp add: add.assoc [symmetric]) (simp add: add.assoc)
```
```   262 next
```
```   263   case Bit1
```
```   264   then show ?case by (simp add: add.assoc [symmetric]) (simp add: add.assoc)
```
```   265 qed
```
```   266
```
```   267 lemma numeral_inc: "numeral (inc x) = numeral x + 1"
```
```   268 proof (induct x)
```
```   269   case One
```
```   270   then show ?case by simp
```
```   271 next
```
```   272   case Bit0
```
```   273   then show ?case by simp
```
```   274 next
```
```   275   case (Bit1 x)
```
```   276   have "numeral x + (1 + numeral x) + 1 = numeral x + (numeral x + 1) + 1"
```
```   277     by (simp only: one_plus_numeral_commute)
```
```   278   with Bit1 show ?case
```
```   279     by (simp add: add.assoc)
```
```   280 qed
```
```   281
```
```   282 declare numeral.simps [simp del]
```
```   283
```
```   284 abbreviation "Numeral1 \<equiv> numeral One"
```
```   285
```
```   286 declare numeral_One [code_post]
```
```   287
```
```   288 end
```
```   289
```
```   290 text \<open>Numeral syntax.\<close>
```
```   291
```
```   292 syntax
```
```   293   "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
```
```   294
```
```   295 ML_file "Tools/numeral.ML"
```
```   296
```
```   297 parse_translation \<open>
```
```   298   let
```
```   299     fun numeral_tr [(c as Const (@{syntax_const "_constrain"}, _)) \$ t \$ u] =
```
```   300           c \$ numeral_tr [t] \$ u
```
```   301       | numeral_tr [Const (num, _)] =
```
```   302           (Numeral.mk_number_syntax o #value o Lexicon.read_num) num
```
```   303       | numeral_tr ts = raise TERM ("numeral_tr", ts);
```
```   304   in [(@{syntax_const "_Numeral"}, K numeral_tr)] end
```
```   305 \<close>
```
```   306
```
```   307 typed_print_translation \<open>
```
```   308   let
```
```   309     fun num_tr' ctxt T [n] =
```
```   310       let
```
```   311         val k = Numeral.dest_num_syntax n;
```
```   312         val t' =
```
```   313           Syntax.const @{syntax_const "_Numeral"} \$
```
```   314             Syntax.free (string_of_int k);
```
```   315       in
```
```   316         (case T of
```
```   317           Type (@{type_name fun}, [_, T']) =>
```
```   318             if Printer.type_emphasis ctxt T' then
```
```   319               Syntax.const @{syntax_const "_constrain"} \$ t' \$
```
```   320                 Syntax_Phases.term_of_typ ctxt T'
```
```   321             else t'
```
```   322         | _ => if T = dummyT then t' else raise Match)
```
```   323       end;
```
```   324   in
```
```   325    [(@{const_syntax numeral}, num_tr')]
```
```   326   end
```
```   327 \<close>
```
```   328
```
```   329
```
```   330 subsection \<open>Class-specific numeral rules\<close>
```
```   331
```
```   332 text \<open>@{const numeral} is a morphism.\<close>
```
```   333
```
```   334
```
```   335 subsubsection \<open>Structures with addition: class \<open>numeral\<close>\<close>
```
```   336
```
```   337 context numeral
```
```   338 begin
```
```   339
```
```   340 lemma numeral_add: "numeral (m + n) = numeral m + numeral n"
```
```   341   by (induct n rule: num_induct)
```
```   342     (simp_all only: numeral_One add_One add_inc numeral_inc add.assoc)
```
```   343
```
```   344 lemma numeral_plus_numeral: "numeral m + numeral n = numeral (m + n)"
```
```   345   by (rule numeral_add [symmetric])
```
```   346
```
```   347 lemma numeral_plus_one: "numeral n + 1 = numeral (n + One)"
```
```   348   using numeral_add [of n One] by (simp add: numeral_One)
```
```   349
```
```   350 lemma one_plus_numeral: "1 + numeral n = numeral (One + n)"
```
```   351   using numeral_add [of One n] by (simp add: numeral_One)
```
```   352
```
```   353 lemma one_add_one: "1 + 1 = 2"
```
```   354   using numeral_add [of One One] by (simp add: numeral_One)
```
```   355
```
```   356 lemmas add_numeral_special =
```
```   357   numeral_plus_one one_plus_numeral one_add_one
```
```   358
```
```   359 end
```
```   360
```
```   361
```
```   362 subsubsection \<open>Structures with negation: class \<open>neg_numeral\<close>\<close>
```
```   363
```
```   364 class neg_numeral = numeral + group_add
```
```   365 begin
```
```   366
```
```   367 lemma uminus_numeral_One: "- Numeral1 = - 1"
```
```   368   by (simp add: numeral_One)
```
```   369
```
```   370 text \<open>Numerals form an abelian subgroup.\<close>
```
```   371
```
```   372 inductive is_num :: "'a \<Rightarrow> bool"
```
```   373   where
```
```   374     "is_num 1"
```
```   375   | "is_num x \<Longrightarrow> is_num (- x)"
```
```   376   | "is_num x \<Longrightarrow> is_num y \<Longrightarrow> is_num (x + y)"
```
```   377
```
```   378 lemma is_num_numeral: "is_num (numeral k)"
```
```   379   by (induct k) (simp_all add: numeral.simps is_num.intros)
```
```   380
```
```   381 lemma is_num_add_commute: "is_num x \<Longrightarrow> is_num y \<Longrightarrow> x + y = y + x"
```
```   382   apply (induct x rule: is_num.induct)
```
```   383     apply (induct y rule: is_num.induct)
```
```   384       apply simp
```
```   385      apply (rule_tac a=x in add_left_imp_eq)
```
```   386      apply (rule_tac a=x in add_right_imp_eq)
```
```   387      apply (simp add: add.assoc)
```
```   388     apply (simp add: add.assoc [symmetric])
```
```   389     apply (simp add: add.assoc)
```
```   390    apply (rule_tac a=x in add_left_imp_eq)
```
```   391    apply (rule_tac a=x in add_right_imp_eq)
```
```   392    apply (simp add: add.assoc)
```
```   393   apply (simp add: add.assoc)
```
```   394   apply (simp add: add.assoc [symmetric])
```
```   395   done
```
```   396
```
```   397 lemma is_num_add_left_commute: "is_num x \<Longrightarrow> is_num y \<Longrightarrow> x + (y + z) = y + (x + z)"
```
```   398   by (simp only: add.assoc [symmetric] is_num_add_commute)
```
```   399
```
```   400 lemmas is_num_normalize =
```
```   401   add.assoc is_num_add_commute is_num_add_left_commute
```
```   402   is_num.intros is_num_numeral
```
```   403   minus_add
```
```   404
```
```   405 definition dbl :: "'a \<Rightarrow> 'a"
```
```   406   where "dbl x = x + x"
```
```   407
```
```   408 definition dbl_inc :: "'a \<Rightarrow> 'a"
```
```   409   where "dbl_inc x = x + x + 1"
```
```   410
```
```   411 definition dbl_dec :: "'a \<Rightarrow> 'a"
```
```   412   where "dbl_dec x = x + x - 1"
```
```   413
```
```   414 definition sub :: "num \<Rightarrow> num \<Rightarrow> 'a"
```
```   415   where "sub k l = numeral k - numeral l"
```
```   416
```
```   417 lemma numeral_BitM: "numeral (BitM n) = numeral (Bit0 n) - 1"
```
```   418   by (simp only: BitM_plus_one [symmetric] numeral_add numeral_One eq_diff_eq)
```
```   419
```
```   420 lemma dbl_simps [simp]:
```
```   421   "dbl (- numeral k) = - dbl (numeral k)"
```
```   422   "dbl 0 = 0"
```
```   423   "dbl 1 = 2"
```
```   424   "dbl (- 1) = - 2"
```
```   425   "dbl (numeral k) = numeral (Bit0 k)"
```
```   426   by (simp_all add: dbl_def numeral.simps minus_add)
```
```   427
```
```   428 lemma dbl_inc_simps [simp]:
```
```   429   "dbl_inc (- numeral k) = - dbl_dec (numeral k)"
```
```   430   "dbl_inc 0 = 1"
```
```   431   "dbl_inc 1 = 3"
```
```   432   "dbl_inc (- 1) = - 1"
```
```   433   "dbl_inc (numeral k) = numeral (Bit1 k)"
```
```   434   by (simp_all add: dbl_inc_def dbl_dec_def numeral.simps numeral_BitM is_num_normalize algebra_simps
```
```   435       del: add_uminus_conv_diff)
```
```   436
```
```   437 lemma dbl_dec_simps [simp]:
```
```   438   "dbl_dec (- numeral k) = - dbl_inc (numeral k)"
```
```   439   "dbl_dec 0 = - 1"
```
```   440   "dbl_dec 1 = 1"
```
```   441   "dbl_dec (- 1) = - 3"
```
```   442   "dbl_dec (numeral k) = numeral (BitM k)"
```
```   443   by (simp_all add: dbl_dec_def dbl_inc_def numeral.simps numeral_BitM is_num_normalize)
```
```   444
```
```   445 lemma sub_num_simps [simp]:
```
```   446   "sub One One = 0"
```
```   447   "sub One (Bit0 l) = - numeral (BitM l)"
```
```   448   "sub One (Bit1 l) = - numeral (Bit0 l)"
```
```   449   "sub (Bit0 k) One = numeral (BitM k)"
```
```   450   "sub (Bit1 k) One = numeral (Bit0 k)"
```
```   451   "sub (Bit0 k) (Bit0 l) = dbl (sub k l)"
```
```   452   "sub (Bit0 k) (Bit1 l) = dbl_dec (sub k l)"
```
```   453   "sub (Bit1 k) (Bit0 l) = dbl_inc (sub k l)"
```
```   454   "sub (Bit1 k) (Bit1 l) = dbl (sub k l)"
```
```   455   by (simp_all add: dbl_def dbl_dec_def dbl_inc_def sub_def numeral.simps
```
```   456     numeral_BitM is_num_normalize del: add_uminus_conv_diff add: diff_conv_add_uminus)
```
```   457
```
```   458 lemma add_neg_numeral_simps:
```
```   459   "numeral m + - numeral n = sub m n"
```
```   460   "- numeral m + numeral n = sub n m"
```
```   461   "- numeral m + - numeral n = - (numeral m + numeral n)"
```
```   462   by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
```
```   463       del: add_uminus_conv_diff add: diff_conv_add_uminus)
```
```   464
```
```   465 lemma add_neg_numeral_special:
```
```   466   "1 + - numeral m = sub One m"
```
```   467   "- numeral m + 1 = sub One m"
```
```   468   "numeral m + - 1 = sub m One"
```
```   469   "- 1 + numeral n = sub n One"
```
```   470   "- 1 + - numeral n = - numeral (inc n)"
```
```   471   "- numeral m + - 1 = - numeral (inc m)"
```
```   472   "1 + - 1 = 0"
```
```   473   "- 1 + 1 = 0"
```
```   474   "- 1 + - 1 = - 2"
```
```   475   by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize right_minus numeral_inc
```
```   476       del: add_uminus_conv_diff add: diff_conv_add_uminus)
```
```   477
```
```   478 lemma diff_numeral_simps:
```
```   479   "numeral m - numeral n = sub m n"
```
```   480   "numeral m - - numeral n = numeral (m + n)"
```
```   481   "- numeral m - numeral n = - numeral (m + n)"
```
```   482   "- numeral m - - numeral n = sub n m"
```
```   483   by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize
```
```   484       del: add_uminus_conv_diff add: diff_conv_add_uminus)
```
```   485
```
```   486 lemma diff_numeral_special:
```
```   487   "1 - numeral n = sub One n"
```
```   488   "numeral m - 1 = sub m One"
```
```   489   "1 - - numeral n = numeral (One + n)"
```
```   490   "- numeral m - 1 = - numeral (m + One)"
```
```   491   "- 1 - numeral n = - numeral (inc n)"
```
```   492   "numeral m - - 1 = numeral (inc m)"
```
```   493   "- 1 - - numeral n = sub n One"
```
```   494   "- numeral m - - 1 = sub One m"
```
```   495   "1 - 1 = 0"
```
```   496   "- 1 - 1 = - 2"
```
```   497   "1 - - 1 = 2"
```
```   498   "- 1 - - 1 = 0"
```
```   499   by (simp_all add: sub_def numeral_add numeral.simps is_num_normalize numeral_inc
```
```   500       del: add_uminus_conv_diff add: diff_conv_add_uminus)
```
```   501
```
```   502 end
```
```   503
```
```   504
```
```   505 subsubsection \<open>Structures with multiplication: class \<open>semiring_numeral\<close>\<close>
```
```   506
```
```   507 class semiring_numeral = semiring + monoid_mult
```
```   508 begin
```
```   509
```
```   510 subclass numeral ..
```
```   511
```
```   512 lemma numeral_mult: "numeral (m * n) = numeral m * numeral n"
```
```   513   by (induct n rule: num_induct)
```
```   514     (simp_all add: numeral_One mult_inc numeral_inc numeral_add distrib_left)
```
```   515
```
```   516 lemma numeral_times_numeral: "numeral m * numeral n = numeral (m * n)"
```
```   517   by (rule numeral_mult [symmetric])
```
```   518
```
```   519 lemma mult_2: "2 * z = z + z"
```
```   520   by (simp add: one_add_one [symmetric] distrib_right)
```
```   521
```
```   522 lemma mult_2_right: "z * 2 = z + z"
```
```   523   by (simp add: one_add_one [symmetric] distrib_left)
```
```   524
```
```   525 end
```
```   526
```
```   527
```
```   528 subsubsection \<open>Structures with a zero: class \<open>semiring_1\<close>\<close>
```
```   529
```
```   530 context semiring_1
```
```   531 begin
```
```   532
```
```   533 subclass semiring_numeral ..
```
```   534
```
```   535 lemma of_nat_numeral [simp]: "of_nat (numeral n) = numeral n"
```
```   536   by (induct n) (simp_all only: numeral.simps numeral_class.numeral.simps of_nat_add of_nat_1)
```
```   537
```
```   538 end
```
```   539
```
```   540 lemma nat_of_num_numeral [code_abbrev]: "nat_of_num = numeral"
```
```   541 proof
```
```   542   fix n
```
```   543   have "numeral n = nat_of_num n"
```
```   544     by (induct n) (simp_all add: numeral.simps)
```
```   545   then show "nat_of_num n = numeral n"
```
```   546     by simp
```
```   547 qed
```
```   548
```
```   549 lemma nat_of_num_code [code]:
```
```   550   "nat_of_num One = 1"
```
```   551   "nat_of_num (Bit0 n) = (let m = nat_of_num n in m + m)"
```
```   552   "nat_of_num (Bit1 n) = (let m = nat_of_num n in Suc (m + m))"
```
```   553   by (simp_all add: Let_def)
```
```   554
```
```   555
```
```   556 subsubsection \<open>Equality: class \<open>semiring_char_0\<close>\<close>
```
```   557
```
```   558 context semiring_char_0
```
```   559 begin
```
```   560
```
```   561 lemma numeral_eq_iff: "numeral m = numeral n \<longleftrightarrow> m = n"
```
```   562   by (simp only: of_nat_numeral [symmetric] nat_of_num_numeral [symmetric]
```
```   563     of_nat_eq_iff num_eq_iff)
```
```   564
```
```   565 lemma numeral_eq_one_iff: "numeral n = 1 \<longleftrightarrow> n = One"
```
```   566   by (rule numeral_eq_iff [of n One, unfolded numeral_One])
```
```   567
```
```   568 lemma one_eq_numeral_iff: "1 = numeral n \<longleftrightarrow> One = n"
```
```   569   by (rule numeral_eq_iff [of One n, unfolded numeral_One])
```
```   570
```
```   571 lemma numeral_neq_zero: "numeral n \<noteq> 0"
```
```   572   by (simp add: of_nat_numeral [symmetric] nat_of_num_numeral [symmetric] nat_of_num_pos)
```
```   573
```
```   574 lemma zero_neq_numeral: "0 \<noteq> numeral n"
```
```   575   unfolding eq_commute [of 0] by (rule numeral_neq_zero)
```
```   576
```
```   577 lemmas eq_numeral_simps [simp] =
```
```   578   numeral_eq_iff
```
```   579   numeral_eq_one_iff
```
```   580   one_eq_numeral_iff
```
```   581   numeral_neq_zero
```
```   582   zero_neq_numeral
```
```   583
```
```   584 end
```
```   585
```
```   586
```
```   587 subsubsection \<open>Comparisons: class \<open>linordered_semidom\<close>\<close>
```
```   588
```
```   589 text \<open>Could be perhaps more general than here.\<close>
```
```   590
```
```   591 context linordered_semidom
```
```   592 begin
```
```   593
```
```   594 lemma numeral_le_iff: "numeral m \<le> numeral n \<longleftrightarrow> m \<le> n"
```
```   595 proof -
```
```   596   have "of_nat (numeral m) \<le> of_nat (numeral n) \<longleftrightarrow> m \<le> n"
```
```   597     by (simp only: less_eq_num_def nat_of_num_numeral of_nat_le_iff)
```
```   598   then show ?thesis by simp
```
```   599 qed
```
```   600
```
```   601 lemma one_le_numeral: "1 \<le> numeral n"
```
```   602   using numeral_le_iff [of One n] by (simp add: numeral_One)
```
```   603
```
```   604 lemma numeral_le_one_iff: "numeral n \<le> 1 \<longleftrightarrow> n \<le> One"
```
```   605   using numeral_le_iff [of n One] by (simp add: numeral_One)
```
```   606
```
```   607 lemma numeral_less_iff: "numeral m < numeral n \<longleftrightarrow> m < n"
```
```   608 proof -
```
```   609   have "of_nat (numeral m) < of_nat (numeral n) \<longleftrightarrow> m < n"
```
```   610     unfolding less_num_def nat_of_num_numeral of_nat_less_iff ..
```
```   611   then show ?thesis by simp
```
```   612 qed
```
```   613
```
```   614 lemma not_numeral_less_one: "\<not> numeral n < 1"
```
```   615   using numeral_less_iff [of n One] by (simp add: numeral_One)
```
```   616
```
```   617 lemma one_less_numeral_iff: "1 < numeral n \<longleftrightarrow> One < n"
```
```   618   using numeral_less_iff [of One n] by (simp add: numeral_One)
```
```   619
```
```   620 lemma zero_le_numeral: "0 \<le> numeral n"
```
```   621   by (induct n) (simp_all add: numeral.simps)
```
```   622
```
```   623 lemma zero_less_numeral: "0 < numeral n"
```
```   624   by (induct n) (simp_all add: numeral.simps add_pos_pos)
```
```   625
```
```   626 lemma not_numeral_le_zero: "\<not> numeral n \<le> 0"
```
```   627   by (simp add: not_le zero_less_numeral)
```
```   628
```
```   629 lemma not_numeral_less_zero: "\<not> numeral n < 0"
```
```   630   by (simp add: not_less zero_le_numeral)
```
```   631
```
```   632 lemmas le_numeral_extra =
```
```   633   zero_le_one not_one_le_zero
```
```   634   order_refl [of 0] order_refl [of 1]
```
```   635
```
```   636 lemmas less_numeral_extra =
```
```   637   zero_less_one not_one_less_zero
```
```   638   less_irrefl [of 0] less_irrefl [of 1]
```
```   639
```
```   640 lemmas le_numeral_simps [simp] =
```
```   641   numeral_le_iff
```
```   642   one_le_numeral
```
```   643   numeral_le_one_iff
```
```   644   zero_le_numeral
```
```   645   not_numeral_le_zero
```
```   646
```
```   647 lemmas less_numeral_simps [simp] =
```
```   648   numeral_less_iff
```
```   649   one_less_numeral_iff
```
```   650   not_numeral_less_one
```
```   651   zero_less_numeral
```
```   652   not_numeral_less_zero
```
```   653
```
```   654 lemma min_0_1 [simp]:
```
```   655   fixes min' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   656   defines "min' \<equiv> min"
```
```   657   shows
```
```   658     "min' 0 1 = 0"
```
```   659     "min' 1 0 = 0"
```
```   660     "min' 0 (numeral x) = 0"
```
```   661     "min' (numeral x) 0 = 0"
```
```   662     "min' 1 (numeral x) = 1"
```
```   663     "min' (numeral x) 1 = 1"
```
```   664   by (simp_all add: min'_def min_def le_num_One_iff)
```
```   665
```
```   666 lemma max_0_1 [simp]:
```
```   667   fixes max' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
```
```   668   defines "max' \<equiv> max"
```
```   669   shows
```
```   670     "max' 0 1 = 1"
```
```   671     "max' 1 0 = 1"
```
```   672     "max' 0 (numeral x) = numeral x"
```
```   673     "max' (numeral x) 0 = numeral x"
```
```   674     "max' 1 (numeral x) = numeral x"
```
```   675     "max' (numeral x) 1 = numeral x"
```
```   676   by (simp_all add: max'_def max_def le_num_One_iff)
```
```   677
```
```   678 end
```
```   679
```
```   680
```
```   681 subsubsection \<open>Multiplication and negation: class \<open>ring_1\<close>\<close>
```
```   682
```
```   683 context ring_1
```
```   684 begin
```
```   685
```
```   686 subclass neg_numeral ..
```
```   687
```
```   688 lemma mult_neg_numeral_simps:
```
```   689   "- numeral m * - numeral n = numeral (m * n)"
```
```   690   "- numeral m * numeral n = - numeral (m * n)"
```
```   691   "numeral m * - numeral n = - numeral (m * n)"
```
```   692   by (simp_all only: mult_minus_left mult_minus_right minus_minus numeral_mult)
```
```   693
```
```   694 lemma mult_minus1 [simp]: "- 1 * z = - z"
```
```   695   by (simp add: numeral.simps)
```
```   696
```
```   697 lemma mult_minus1_right [simp]: "z * - 1 = - z"
```
```   698   by (simp add: numeral.simps)
```
```   699
```
```   700 end
```
```   701
```
```   702
```
```   703 subsubsection \<open>Equality using \<open>iszero\<close> for rings with non-zero characteristic\<close>
```
```   704
```
```   705 context ring_1
```
```   706 begin
```
```   707
```
```   708 definition iszero :: "'a \<Rightarrow> bool"
```
```   709   where "iszero z \<longleftrightarrow> z = 0"
```
```   710
```
```   711 lemma iszero_0 [simp]: "iszero 0"
```
```   712   by (simp add: iszero_def)
```
```   713
```
```   714 lemma not_iszero_1 [simp]: "\<not> iszero 1"
```
```   715   by (simp add: iszero_def)
```
```   716
```
```   717 lemma not_iszero_Numeral1: "\<not> iszero Numeral1"
```
```   718   by (simp add: numeral_One)
```
```   719
```
```   720 lemma not_iszero_neg_1 [simp]: "\<not> iszero (- 1)"
```
```   721   by (simp add: iszero_def)
```
```   722
```
```   723 lemma not_iszero_neg_Numeral1: "\<not> iszero (- Numeral1)"
```
```   724   by (simp add: numeral_One)
```
```   725
```
```   726 lemma iszero_neg_numeral [simp]: "iszero (- numeral w) \<longleftrightarrow> iszero (numeral w)"
```
```   727   unfolding iszero_def by (rule neg_equal_0_iff_equal)
```
```   728
```
```   729 lemma eq_iff_iszero_diff: "x = y \<longleftrightarrow> iszero (x - y)"
```
```   730   unfolding iszero_def by (rule eq_iff_diff_eq_0)
```
```   731
```
```   732 text \<open>
```
```   733   The \<open>eq_numeral_iff_iszero\<close> lemmas are not declared \<open>[simp]\<close> by default,
```
```   734   because for rings of characteristic zero, better simp rules are possible.
```
```   735   For a type like integers mod \<open>n\<close>, type-instantiated versions of these rules
```
```   736   should be added to the simplifier, along with a type-specific rule for
```
```   737   deciding propositions of the form \<open>iszero (numeral w)\<close>.
```
```   738
```
```   739   bh: Maybe it would not be so bad to just declare these as simp rules anyway?
```
```   740   I should test whether these rules take precedence over the \<open>ring_char_0\<close>
```
```   741   rules in the simplifier.
```
```   742 \<close>
```
```   743
```
```   744 lemma eq_numeral_iff_iszero:
```
```   745   "numeral x = numeral y \<longleftrightarrow> iszero (sub x y)"
```
```   746   "numeral x = - numeral y \<longleftrightarrow> iszero (numeral (x + y))"
```
```   747   "- numeral x = numeral y \<longleftrightarrow> iszero (numeral (x + y))"
```
```   748   "- numeral x = - numeral y \<longleftrightarrow> iszero (sub y x)"
```
```   749   "numeral x = 1 \<longleftrightarrow> iszero (sub x One)"
```
```   750   "1 = numeral y \<longleftrightarrow> iszero (sub One y)"
```
```   751   "- numeral x = 1 \<longleftrightarrow> iszero (numeral (x + One))"
```
```   752   "1 = - numeral y \<longleftrightarrow> iszero (numeral (One + y))"
```
```   753   "numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
```
```   754   "0 = numeral y \<longleftrightarrow> iszero (numeral y)"
```
```   755   "- numeral x = 0 \<longleftrightarrow> iszero (numeral x)"
```
```   756   "0 = - numeral y \<longleftrightarrow> iszero (numeral y)"
```
```   757   unfolding eq_iff_iszero_diff diff_numeral_simps diff_numeral_special
```
```   758   by simp_all
```
```   759
```
```   760 end
```
```   761
```
```   762
```
```   763 subsubsection \<open>Equality and negation: class \<open>ring_char_0\<close>\<close>
```
```   764
```
```   765 context ring_char_0
```
```   766 begin
```
```   767
```
```   768 lemma not_iszero_numeral [simp]: "\<not> iszero (numeral w)"
```
```   769   by (simp add: iszero_def)
```
```   770
```
```   771 lemma neg_numeral_eq_iff: "- numeral m = - numeral n \<longleftrightarrow> m = n"
```
```   772   by simp
```
```   773
```
```   774 lemma numeral_neq_neg_numeral: "numeral m \<noteq> - numeral n"
```
```   775   by (simp add: eq_neg_iff_add_eq_0 numeral_plus_numeral)
```
```   776
```
```   777 lemma neg_numeral_neq_numeral: "- numeral m \<noteq> numeral n"
```
```   778   by (rule numeral_neq_neg_numeral [symmetric])
```
```   779
```
```   780 lemma zero_neq_neg_numeral: "0 \<noteq> - numeral n"
```
```   781   by simp
```
```   782
```
```   783 lemma neg_numeral_neq_zero: "- numeral n \<noteq> 0"
```
```   784   by simp
```
```   785
```
```   786 lemma one_neq_neg_numeral: "1 \<noteq> - numeral n"
```
```   787   using numeral_neq_neg_numeral [of One n] by (simp add: numeral_One)
```
```   788
```
```   789 lemma neg_numeral_neq_one: "- numeral n \<noteq> 1"
```
```   790   using neg_numeral_neq_numeral [of n One] by (simp add: numeral_One)
```
```   791
```
```   792 lemma neg_one_neq_numeral: "- 1 \<noteq> numeral n"
```
```   793   using neg_numeral_neq_numeral [of One n] by (simp add: numeral_One)
```
```   794
```
```   795 lemma numeral_neq_neg_one: "numeral n \<noteq> - 1"
```
```   796   using numeral_neq_neg_numeral [of n One] by (simp add: numeral_One)
```
```   797
```
```   798 lemma neg_one_eq_numeral_iff: "- 1 = - numeral n \<longleftrightarrow> n = One"
```
```   799   using neg_numeral_eq_iff [of One n] by (auto simp add: numeral_One)
```
```   800
```
```   801 lemma numeral_eq_neg_one_iff: "- numeral n = - 1 \<longleftrightarrow> n = One"
```
```   802   using neg_numeral_eq_iff [of n One] by (auto simp add: numeral_One)
```
```   803
```
```   804 lemma neg_one_neq_zero: "- 1 \<noteq> 0"
```
```   805   by simp
```
```   806
```
```   807 lemma zero_neq_neg_one: "0 \<noteq> - 1"
```
```   808   by simp
```
```   809
```
```   810 lemma neg_one_neq_one: "- 1 \<noteq> 1"
```
```   811   using neg_numeral_neq_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
```
```   812
```
```   813 lemma one_neq_neg_one: "1 \<noteq> - 1"
```
```   814   using numeral_neq_neg_numeral [of One One] by (simp only: numeral_One not_False_eq_True)
```
```   815
```
```   816 lemmas eq_neg_numeral_simps [simp] =
```
```   817   neg_numeral_eq_iff
```
```   818   numeral_neq_neg_numeral neg_numeral_neq_numeral
```
```   819   one_neq_neg_numeral neg_numeral_neq_one
```
```   820   zero_neq_neg_numeral neg_numeral_neq_zero
```
```   821   neg_one_neq_numeral numeral_neq_neg_one
```
```   822   neg_one_eq_numeral_iff numeral_eq_neg_one_iff
```
```   823   neg_one_neq_zero zero_neq_neg_one
```
```   824   neg_one_neq_one one_neq_neg_one
```
```   825
```
```   826 end
```
```   827
```
```   828
```
```   829 subsubsection \<open>Structures with negation and order: class \<open>linordered_idom\<close>\<close>
```
```   830
```
```   831 context linordered_idom
```
```   832 begin
```
```   833
```
```   834 subclass ring_char_0 ..
```
```   835
```
```   836 lemma neg_numeral_le_iff: "- numeral m \<le> - numeral n \<longleftrightarrow> n \<le> m"
```
```   837   by (simp only: neg_le_iff_le numeral_le_iff)
```
```   838
```
```   839 lemma neg_numeral_less_iff: "- numeral m < - numeral n \<longleftrightarrow> n < m"
```
```   840   by (simp only: neg_less_iff_less numeral_less_iff)
```
```   841
```
```   842 lemma neg_numeral_less_zero: "- numeral n < 0"
```
```   843   by (simp only: neg_less_0_iff_less zero_less_numeral)
```
```   844
```
```   845 lemma neg_numeral_le_zero: "- numeral n \<le> 0"
```
```   846   by (simp only: neg_le_0_iff_le zero_le_numeral)
```
```   847
```
```   848 lemma not_zero_less_neg_numeral: "\<not> 0 < - numeral n"
```
```   849   by (simp only: not_less neg_numeral_le_zero)
```
```   850
```
```   851 lemma not_zero_le_neg_numeral: "\<not> 0 \<le> - numeral n"
```
```   852   by (simp only: not_le neg_numeral_less_zero)
```
```   853
```
```   854 lemma neg_numeral_less_numeral: "- numeral m < numeral n"
```
```   855   using neg_numeral_less_zero zero_less_numeral by (rule less_trans)
```
```   856
```
```   857 lemma neg_numeral_le_numeral: "- numeral m \<le> numeral n"
```
```   858   by (simp only: less_imp_le neg_numeral_less_numeral)
```
```   859
```
```   860 lemma not_numeral_less_neg_numeral: "\<not> numeral m < - numeral n"
```
```   861   by (simp only: not_less neg_numeral_le_numeral)
```
```   862
```
```   863 lemma not_numeral_le_neg_numeral: "\<not> numeral m \<le> - numeral n"
```
```   864   by (simp only: not_le neg_numeral_less_numeral)
```
```   865
```
```   866 lemma neg_numeral_less_one: "- numeral m < 1"
```
```   867   by (rule neg_numeral_less_numeral [of m One, unfolded numeral_One])
```
```   868
```
```   869 lemma neg_numeral_le_one: "- numeral m \<le> 1"
```
```   870   by (rule neg_numeral_le_numeral [of m One, unfolded numeral_One])
```
```   871
```
```   872 lemma not_one_less_neg_numeral: "\<not> 1 < - numeral m"
```
```   873   by (simp only: not_less neg_numeral_le_one)
```
```   874
```
```   875 lemma not_one_le_neg_numeral: "\<not> 1 \<le> - numeral m"
```
```   876   by (simp only: not_le neg_numeral_less_one)
```
```   877
```
```   878 lemma not_numeral_less_neg_one: "\<not> numeral m < - 1"
```
```   879   using not_numeral_less_neg_numeral [of m One] by (simp add: numeral_One)
```
```   880
```
```   881 lemma not_numeral_le_neg_one: "\<not> numeral m \<le> - 1"
```
```   882   using not_numeral_le_neg_numeral [of m One] by (simp add: numeral_One)
```
```   883
```
```   884 lemma neg_one_less_numeral: "- 1 < numeral m"
```
```   885   using neg_numeral_less_numeral [of One m] by (simp add: numeral_One)
```
```   886
```
```   887 lemma neg_one_le_numeral: "- 1 \<le> numeral m"
```
```   888   using neg_numeral_le_numeral [of One m] by (simp add: numeral_One)
```
```   889
```
```   890 lemma neg_numeral_less_neg_one_iff: "- numeral m < - 1 \<longleftrightarrow> m \<noteq> One"
```
```   891   by (cases m) simp_all
```
```   892
```
```   893 lemma neg_numeral_le_neg_one: "- numeral m \<le> - 1"
```
```   894   by simp
```
```   895
```
```   896 lemma not_neg_one_less_neg_numeral: "\<not> - 1 < - numeral m"
```
```   897   by simp
```
```   898
```
```   899 lemma not_neg_one_le_neg_numeral_iff: "\<not> - 1 \<le> - numeral m \<longleftrightarrow> m \<noteq> One"
```
```   900   by (cases m) simp_all
```
```   901
```
```   902 lemma sub_non_negative: "sub n m \<ge> 0 \<longleftrightarrow> n \<ge> m"
```
```   903   by (simp only: sub_def le_diff_eq) simp
```
```   904
```
```   905 lemma sub_positive: "sub n m > 0 \<longleftrightarrow> n > m"
```
```   906   by (simp only: sub_def less_diff_eq) simp
```
```   907
```
```   908 lemma sub_non_positive: "sub n m \<le> 0 \<longleftrightarrow> n \<le> m"
```
```   909   by (simp only: sub_def diff_le_eq) simp
```
```   910
```
```   911 lemma sub_negative: "sub n m < 0 \<longleftrightarrow> n < m"
```
```   912   by (simp only: sub_def diff_less_eq) simp
```
```   913
```
```   914 lemmas le_neg_numeral_simps [simp] =
```
```   915   neg_numeral_le_iff
```
```   916   neg_numeral_le_numeral not_numeral_le_neg_numeral
```
```   917   neg_numeral_le_zero not_zero_le_neg_numeral
```
```   918   neg_numeral_le_one not_one_le_neg_numeral
```
```   919   neg_one_le_numeral not_numeral_le_neg_one
```
```   920   neg_numeral_le_neg_one not_neg_one_le_neg_numeral_iff
```
```   921
```
```   922 lemma le_minus_one_simps [simp]:
```
```   923   "- 1 \<le> 0"
```
```   924   "- 1 \<le> 1"
```
```   925   "\<not> 0 \<le> - 1"
```
```   926   "\<not> 1 \<le> - 1"
```
```   927   by simp_all
```
```   928
```
```   929 lemmas less_neg_numeral_simps [simp] =
```
```   930   neg_numeral_less_iff
```
```   931   neg_numeral_less_numeral not_numeral_less_neg_numeral
```
```   932   neg_numeral_less_zero not_zero_less_neg_numeral
```
```   933   neg_numeral_less_one not_one_less_neg_numeral
```
```   934   neg_one_less_numeral not_numeral_less_neg_one
```
```   935   neg_numeral_less_neg_one_iff not_neg_one_less_neg_numeral
```
```   936
```
```   937 lemma less_minus_one_simps [simp]:
```
```   938   "- 1 < 0"
```
```   939   "- 1 < 1"
```
```   940   "\<not> 0 < - 1"
```
```   941   "\<not> 1 < - 1"
```
```   942   by (simp_all add: less_le)
```
```   943
```
```   944 lemma abs_numeral [simp]: "\<bar>numeral n\<bar> = numeral n"
```
```   945   by simp
```
```   946
```
```   947 lemma abs_neg_numeral [simp]: "\<bar>- numeral n\<bar> = numeral n"
```
```   948   by (simp only: abs_minus_cancel abs_numeral)
```
```   949
```
```   950 lemma abs_neg_one [simp]: "\<bar>- 1\<bar> = 1"
```
```   951   by simp
```
```   952
```
```   953 end
```
```   954
```
```   955
```
```   956 subsubsection \<open>Natural numbers\<close>
```
```   957
```
```   958 lemma Suc_1 [simp]: "Suc 1 = 2"
```
```   959   unfolding Suc_eq_plus1 by (rule one_add_one)
```
```   960
```
```   961 lemma Suc_numeral [simp]: "Suc (numeral n) = numeral (n + One)"
```
```   962   unfolding Suc_eq_plus1 by (rule numeral_plus_one)
```
```   963
```
```   964 definition pred_numeral :: "num \<Rightarrow> nat"
```
```   965   where [code del]: "pred_numeral k = numeral k - 1"
```
```   966
```
```   967 lemma numeral_eq_Suc: "numeral k = Suc (pred_numeral k)"
```
```   968   by (simp add: pred_numeral_def)
```
```   969
```
```   970 lemma eval_nat_numeral:
```
```   971   "numeral One = Suc 0"
```
```   972   "numeral (Bit0 n) = Suc (numeral (BitM n))"
```
```   973   "numeral (Bit1 n) = Suc (numeral (Bit0 n))"
```
```   974   by (simp_all add: numeral.simps BitM_plus_one)
```
```   975
```
```   976 lemma pred_numeral_simps [simp]:
```
```   977   "pred_numeral One = 0"
```
```   978   "pred_numeral (Bit0 k) = numeral (BitM k)"
```
```   979   "pred_numeral (Bit1 k) = numeral (Bit0 k)"
```
```   980   by (simp_all only: pred_numeral_def eval_nat_numeral diff_Suc_Suc diff_0)
```
```   981
```
```   982 lemma numeral_2_eq_2: "2 = Suc (Suc 0)"
```
```   983   by (simp add: eval_nat_numeral)
```
```   984
```
```   985 lemma numeral_3_eq_3: "3 = Suc (Suc (Suc 0))"
```
```   986   by (simp add: eval_nat_numeral)
```
```   987
```
```   988 lemma numeral_1_eq_Suc_0: "Numeral1 = Suc 0"
```
```   989   by (simp only: numeral_One One_nat_def)
```
```   990
```
```   991 lemma Suc_nat_number_of_add: "Suc (numeral v + n) = numeral (v + One) + n"
```
```   992   by simp
```
```   993
```
```   994 lemma numerals: "Numeral1 = (1::nat)" "2 = Suc (Suc 0)"
```
```   995   by (rule numeral_One) (rule numeral_2_eq_2)
```
```   996
```
```   997 text \<open>Comparisons involving @{term Suc}.\<close>
```
```   998
```
```   999 lemma eq_numeral_Suc [simp]: "numeral k = Suc n \<longleftrightarrow> pred_numeral k = n"
```
```  1000   by (simp add: numeral_eq_Suc)
```
```  1001
```
```  1002 lemma Suc_eq_numeral [simp]: "Suc n = numeral k \<longleftrightarrow> n = pred_numeral k"
```
```  1003   by (simp add: numeral_eq_Suc)
```
```  1004
```
```  1005 lemma less_numeral_Suc [simp]: "numeral k < Suc n \<longleftrightarrow> pred_numeral k < n"
```
```  1006   by (simp add: numeral_eq_Suc)
```
```  1007
```
```  1008 lemma less_Suc_numeral [simp]: "Suc n < numeral k \<longleftrightarrow> n < pred_numeral k"
```
```  1009   by (simp add: numeral_eq_Suc)
```
```  1010
```
```  1011 lemma le_numeral_Suc [simp]: "numeral k \<le> Suc n \<longleftrightarrow> pred_numeral k \<le> n"
```
```  1012   by (simp add: numeral_eq_Suc)
```
```  1013
```
```  1014 lemma le_Suc_numeral [simp]: "Suc n \<le> numeral k \<longleftrightarrow> n \<le> pred_numeral k"
```
```  1015   by (simp add: numeral_eq_Suc)
```
```  1016
```
```  1017 lemma diff_Suc_numeral [simp]: "Suc n - numeral k = n - pred_numeral k"
```
```  1018   by (simp add: numeral_eq_Suc)
```
```  1019
```
```  1020 lemma diff_numeral_Suc [simp]: "numeral k - Suc n = pred_numeral k - n"
```
```  1021   by (simp add: numeral_eq_Suc)
```
```  1022
```
```  1023 lemma max_Suc_numeral [simp]: "max (Suc n) (numeral k) = Suc (max n (pred_numeral k))"
```
```  1024   by (simp add: numeral_eq_Suc)
```
```  1025
```
```  1026 lemma max_numeral_Suc [simp]: "max (numeral k) (Suc n) = Suc (max (pred_numeral k) n)"
```
```  1027   by (simp add: numeral_eq_Suc)
```
```  1028
```
```  1029 lemma min_Suc_numeral [simp]: "min (Suc n) (numeral k) = Suc (min n (pred_numeral k))"
```
```  1030   by (simp add: numeral_eq_Suc)
```
```  1031
```
```  1032 lemma min_numeral_Suc [simp]: "min (numeral k) (Suc n) = Suc (min (pred_numeral k) n)"
```
```  1033   by (simp add: numeral_eq_Suc)
```
```  1034
```
```  1035 text \<open>For @{term case_nat} and @{term rec_nat}.\<close>
```
```  1036
```
```  1037 lemma case_nat_numeral [simp]: "case_nat a f (numeral v) = (let pv = pred_numeral v in f pv)"
```
```  1038   by (simp add: numeral_eq_Suc)
```
```  1039
```
```  1040 lemma case_nat_add_eq_if [simp]:
```
```  1041   "case_nat a f ((numeral v) + n) = (let pv = pred_numeral v in f (pv + n))"
```
```  1042   by (simp add: numeral_eq_Suc)
```
```  1043
```
```  1044 lemma rec_nat_numeral [simp]:
```
```  1045   "rec_nat a f (numeral v) = (let pv = pred_numeral v in f pv (rec_nat a f pv))"
```
```  1046   by (simp add: numeral_eq_Suc Let_def)
```
```  1047
```
```  1048 lemma rec_nat_add_eq_if [simp]:
```
```  1049   "rec_nat a f (numeral v + n) = (let pv = pred_numeral v in f (pv + n) (rec_nat a f (pv + n)))"
```
```  1050   by (simp add: numeral_eq_Suc Let_def)
```
```  1051
```
```  1052 text \<open>Case analysis on @{term "n < 2"}.\<close>
```
```  1053 lemma less_2_cases: "n < 2 \<Longrightarrow> n = 0 \<or> n = Suc 0"
```
```  1054   by (auto simp add: numeral_2_eq_2)
```
```  1055
```
```  1056 text \<open>Removal of Small Numerals: 0, 1 and (in additive positions) 2.\<close>
```
```  1057 text \<open>bh: Are these rules really a good idea?\<close>
```
```  1058
```
```  1059 lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)"
```
```  1060   by simp
```
```  1061
```
```  1062 lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)"
```
```  1063   by simp
```
```  1064
```
```  1065 text \<open>Can be used to eliminate long strings of Sucs, but not by default.\<close>
```
```  1066 lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n"
```
```  1067   by simp
```
```  1068
```
```  1069 lemmas nat_1_add_1 = one_add_one [where 'a=nat] (* legacy *)
```
```  1070
```
```  1071
```
```  1072 subsection \<open>Particular lemmas concerning @{term 2}\<close>
```
```  1073
```
```  1074 context linordered_field
```
```  1075 begin
```
```  1076
```
```  1077 subclass field_char_0 ..
```
```  1078
```
```  1079 lemma half_gt_zero_iff: "0 < a / 2 \<longleftrightarrow> 0 < a"
```
```  1080   by (auto simp add: field_simps)
```
```  1081
```
```  1082 lemma half_gt_zero [simp]: "0 < a \<Longrightarrow> 0 < a / 2"
```
```  1083   by (simp add: half_gt_zero_iff)
```
```  1084
```
```  1085 end
```
```  1086
```
```  1087
```
```  1088 subsection \<open>Numeral equations as default simplification rules\<close>
```
```  1089
```
```  1090 declare (in numeral) numeral_One [simp]
```
```  1091 declare (in numeral) numeral_plus_numeral [simp]
```
```  1092 declare (in numeral) add_numeral_special [simp]
```
```  1093 declare (in neg_numeral) add_neg_numeral_simps [simp]
```
```  1094 declare (in neg_numeral) add_neg_numeral_special [simp]
```
```  1095 declare (in neg_numeral) diff_numeral_simps [simp]
```
```  1096 declare (in neg_numeral) diff_numeral_special [simp]
```
```  1097 declare (in semiring_numeral) numeral_times_numeral [simp]
```
```  1098 declare (in ring_1) mult_neg_numeral_simps [simp]
```
```  1099
```
```  1100 subsection \<open>Setting up simprocs\<close>
```
```  1101
```
```  1102 lemma mult_numeral_1: "Numeral1 * a = a"
```
```  1103   for a :: "'a::semiring_numeral"
```
```  1104   by simp
```
```  1105
```
```  1106 lemma mult_numeral_1_right: "a * Numeral1 = a"
```
```  1107   for a :: "'a::semiring_numeral"
```
```  1108   by simp
```
```  1109
```
```  1110 lemma divide_numeral_1: "a / Numeral1 = a"
```
```  1111   for a :: "'a::field"
```
```  1112   by simp
```
```  1113
```
```  1114 lemma inverse_numeral_1: "inverse Numeral1 = (Numeral1::'a::division_ring)"
```
```  1115   by simp
```
```  1116
```
```  1117 text \<open>
```
```  1118   Theorem lists for the cancellation simprocs. The use of a binary
```
```  1119   numeral for 1 reduces the number of special cases.
```
```  1120 \<close>
```
```  1121
```
```  1122 lemma mult_1s:
```
```  1123   "Numeral1 * a = a"
```
```  1124   "a * Numeral1 = a"
```
```  1125   "- Numeral1 * b = - b"
```
```  1126   "b * - Numeral1 = - b"
```
```  1127   for a :: "'a::semiring_numeral" and b :: "'b::ring_1"
```
```  1128   by simp_all
```
```  1129
```
```  1130 setup \<open>
```
```  1131   Reorient_Proc.add
```
```  1132     (fn Const (@{const_name numeral}, _) \$ _ => true
```
```  1133       | Const (@{const_name uminus}, _) \$ (Const (@{const_name numeral}, _) \$ _) => true
```
```  1134       | _ => false)
```
```  1135 \<close>
```
```  1136
```
```  1137 simproc_setup reorient_numeral ("numeral w = x" | "- numeral w = y") =
```
```  1138   Reorient_Proc.proc
```
```  1139
```
```  1140
```
```  1141 subsubsection \<open>Simplification of arithmetic operations on integer constants\<close>
```
```  1142
```
```  1143 lemmas arith_special = (* already declared simp above *)
```
```  1144   add_numeral_special add_neg_numeral_special
```
```  1145   diff_numeral_special
```
```  1146
```
```  1147 lemmas arith_extra_simps = (* rules already in simpset *)
```
```  1148   numeral_plus_numeral add_neg_numeral_simps add_0_left add_0_right
```
```  1149   minus_zero
```
```  1150   diff_numeral_simps diff_0 diff_0_right
```
```  1151   numeral_times_numeral mult_neg_numeral_simps
```
```  1152   mult_zero_left mult_zero_right
```
```  1153   abs_numeral abs_neg_numeral
```
```  1154
```
```  1155 text \<open>
```
```  1156   For making a minimal simpset, one must include these default simprules.
```
```  1157   Also include \<open>simp_thms\<close>.
```
```  1158 \<close>
```
```  1159
```
```  1160 lemmas arith_simps =
```
```  1161   add_num_simps mult_num_simps sub_num_simps
```
```  1162   BitM.simps dbl_simps dbl_inc_simps dbl_dec_simps
```
```  1163   abs_zero abs_one arith_extra_simps
```
```  1164
```
```  1165 lemmas more_arith_simps =
```
```  1166   neg_le_iff_le
```
```  1167   minus_zero left_minus right_minus
```
```  1168   mult_1_left mult_1_right
```
```  1169   mult_minus_left mult_minus_right
```
```  1170   minus_add_distrib minus_minus mult.assoc
```
```  1171
```
```  1172 lemmas of_nat_simps =
```
```  1173   of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
```
```  1174
```
```  1175 text \<open>Simplification of relational operations.\<close>
```
```  1176
```
```  1177 lemmas eq_numeral_extra =
```
```  1178   zero_neq_one one_neq_zero
```
```  1179
```
```  1180 lemmas rel_simps =
```
```  1181   le_num_simps less_num_simps eq_num_simps
```
```  1182   le_numeral_simps le_neg_numeral_simps le_minus_one_simps le_numeral_extra
```
```  1183   less_numeral_simps less_neg_numeral_simps less_minus_one_simps less_numeral_extra
```
```  1184   eq_numeral_simps eq_neg_numeral_simps eq_numeral_extra
```
```  1185
```
```  1186 lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
```
```  1187   \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
```
```  1188   unfolding Let_def ..
```
```  1189
```
```  1190 lemma Let_neg_numeral [simp]: "Let (- numeral v) f = f (- numeral v)"
```
```  1191   \<comment> \<open>Unfold all \<open>let\<close>s involving constants\<close>
```
```  1192   unfolding Let_def ..
```
```  1193
```
```  1194 declaration \<open>
```
```  1195 let
```
```  1196   fun number_of ctxt T n =
```
```  1197     if not (Sign.of_sort (Proof_Context.theory_of ctxt) (T, @{sort numeral}))
```
```  1198     then raise CTERM ("number_of", [])
```
```  1199     else Numeral.mk_cnumber (Thm.ctyp_of ctxt T) n;
```
```  1200 in
```
```  1201   K (
```
```  1202     Lin_Arith.add_simps
```
```  1203       @{thms arith_simps more_arith_simps rel_simps pred_numeral_simps
```
```  1204         arith_special numeral_One of_nat_simps}
```
```  1205     #> Lin_Arith.add_simps
```
```  1206       @{thms Suc_numeral Let_numeral Let_neg_numeral Let_0 Let_1
```
```  1207         le_Suc_numeral le_numeral_Suc less_Suc_numeral less_numeral_Suc
```
```  1208         Suc_eq_numeral eq_numeral_Suc mult_Suc mult_Suc_right of_nat_numeral}
```
```  1209     #> Lin_Arith.set_number_of number_of)
```
```  1210 end
```
```  1211 \<close>
```
```  1212
```
```  1213
```
```  1214 subsubsection \<open>Simplification of arithmetic when nested to the right\<close>
```
```  1215
```
```  1216 lemma add_numeral_left [simp]: "numeral v + (numeral w + z) = (numeral(v + w) + z)"
```
```  1217   by (simp_all add: add.assoc [symmetric])
```
```  1218
```
```  1219 lemma add_neg_numeral_left [simp]:
```
```  1220   "numeral v + (- numeral w + y) = (sub v w + y)"
```
```  1221   "- numeral v + (numeral w + y) = (sub w v + y)"
```
```  1222   "- numeral v + (- numeral w + y) = (- numeral(v + w) + y)"
```
```  1223   by (simp_all add: add.assoc [symmetric])
```
```  1224
```
```  1225 lemma mult_numeral_left [simp]:
```
```  1226   "numeral v * (numeral w * z) = (numeral(v * w) * z :: 'a::semiring_numeral)"
```
```  1227   "- numeral v * (numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
```
```  1228   "numeral v * (- numeral w * y) = (- numeral(v * w) * y :: 'b::ring_1)"
```
```  1229   "- numeral v * (- numeral w * y) = (numeral(v * w) * y :: 'b::ring_1)"
```
```  1230   by (simp_all add: mult.assoc [symmetric])
```
```  1231
```
```  1232 hide_const (open) One Bit0 Bit1 BitM inc pow sqr sub dbl dbl_inc dbl_dec
```
```  1233
```
```  1234
```
```  1235 subsection \<open>Code module namespace\<close>
```
```  1236
```
```  1237 code_identifier
```
```  1238   code_module Num \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  1239
```
```  1240 end
```