src/HOL/Divides.thy
author haftmann
Fri Nov 27 08:41:10 2009 +0100 (2009-11-27)
changeset 33963 977b94b64905
parent 33804 39b494e8c055
child 34126 8a2c5d7aff51
permissions -rw-r--r--
renamed former datatype.ML to datatype_data.ML; datatype.ML provides uniform view on datatype.ML and datatype_rep_proofs.ML
     1 (*  Title:      HOL/Divides.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   1999  University of Cambridge
     4 *)
     5 
     6 header {* The division operators div and mod *}
     7 
     8 theory Divides
     9 imports Nat_Numeral Nat_Transfer
    10 uses "~~/src/Provers/Arith/cancel_div_mod.ML"
    11 begin
    12 
    13 subsection {* Syntactic division operations *}
    14 
    15 class div = dvd +
    16   fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
    17     and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
    18 
    19 
    20 subsection {* Abstract division in commutative semirings. *}
    21 
    22 class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
    23   assumes mod_div_equality: "a div b * b + a mod b = a"
    24     and div_by_0 [simp]: "a div 0 = 0"
    25     and div_0 [simp]: "0 div a = 0"
    26     and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
    27     and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
    28 begin
    29 
    30 text {* @{const div} and @{const mod} *}
    31 
    32 lemma mod_div_equality2: "b * (a div b) + a mod b = a"
    33   unfolding mult_commute [of b]
    34   by (rule mod_div_equality)
    35 
    36 lemma mod_div_equality': "a mod b + a div b * b = a"
    37   using mod_div_equality [of a b]
    38   by (simp only: add_ac)
    39 
    40 lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
    41   by (simp add: mod_div_equality)
    42 
    43 lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
    44   by (simp add: mod_div_equality2)
    45 
    46 lemma mod_by_0 [simp]: "a mod 0 = a"
    47   using mod_div_equality [of a zero] by simp
    48 
    49 lemma mod_0 [simp]: "0 mod a = 0"
    50   using mod_div_equality [of zero a] div_0 by simp
    51 
    52 lemma div_mult_self2 [simp]:
    53   assumes "b \<noteq> 0"
    54   shows "(a + b * c) div b = c + a div b"
    55   using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
    56 
    57 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
    58 proof (cases "b = 0")
    59   case True then show ?thesis by simp
    60 next
    61   case False
    62   have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
    63     by (simp add: mod_div_equality)
    64   also from False div_mult_self1 [of b a c] have
    65     "\<dots> = (c + a div b) * b + (a + c * b) mod b"
    66       by (simp add: algebra_simps)
    67   finally have "a = a div b * b + (a + c * b) mod b"
    68     by (simp add: add_commute [of a] add_assoc left_distrib)
    69   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
    70     by (simp add: mod_div_equality)
    71   then show ?thesis by simp
    72 qed
    73 
    74 lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
    75   by (simp add: mult_commute [of b])
    76 
    77 lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
    78   using div_mult_self2 [of b 0 a] by simp
    79 
    80 lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
    81   using div_mult_self1 [of b 0 a] by simp
    82 
    83 lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
    84   using mod_mult_self2 [of 0 b a] by simp
    85 
    86 lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
    87   using mod_mult_self1 [of 0 a b] by simp
    88 
    89 lemma div_by_1 [simp]: "a div 1 = a"
    90   using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
    91 
    92 lemma mod_by_1 [simp]: "a mod 1 = 0"
    93 proof -
    94   from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
    95   then have "a + a mod 1 = a + 0" by simp
    96   then show ?thesis by (rule add_left_imp_eq)
    97 qed
    98 
    99 lemma mod_self [simp]: "a mod a = 0"
   100   using mod_mult_self2_is_0 [of 1] by simp
   101 
   102 lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
   103   using div_mult_self2_is_id [of _ 1] by simp
   104 
   105 lemma div_add_self1 [simp]:
   106   assumes "b \<noteq> 0"
   107   shows "(b + a) div b = a div b + 1"
   108   using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
   109 
   110 lemma div_add_self2 [simp]:
   111   assumes "b \<noteq> 0"
   112   shows "(a + b) div b = a div b + 1"
   113   using assms div_add_self1 [of b a] by (simp add: add_commute)
   114 
   115 lemma mod_add_self1 [simp]:
   116   "(b + a) mod b = a mod b"
   117   using mod_mult_self1 [of a 1 b] by (simp add: add_commute)
   118 
   119 lemma mod_add_self2 [simp]:
   120   "(a + b) mod b = a mod b"
   121   using mod_mult_self1 [of a 1 b] by simp
   122 
   123 lemma mod_div_decomp:
   124   fixes a b
   125   obtains q r where "q = a div b" and "r = a mod b"
   126     and "a = q * b + r"
   127 proof -
   128   from mod_div_equality have "a = a div b * b + a mod b" by simp
   129   moreover have "a div b = a div b" ..
   130   moreover have "a mod b = a mod b" ..
   131   note that ultimately show thesis by blast
   132 qed
   133 
   134 lemma dvd_eq_mod_eq_0 [code, code_unfold, code_inline del]: "a dvd b \<longleftrightarrow> b mod a = 0"
   135 proof
   136   assume "b mod a = 0"
   137   with mod_div_equality [of b a] have "b div a * a = b" by simp
   138   then have "b = a * (b div a)" unfolding mult_commute ..
   139   then have "\<exists>c. b = a * c" ..
   140   then show "a dvd b" unfolding dvd_def .
   141 next
   142   assume "a dvd b"
   143   then have "\<exists>c. b = a * c" unfolding dvd_def .
   144   then obtain c where "b = a * c" ..
   145   then have "b mod a = a * c mod a" by simp
   146   then have "b mod a = c * a mod a" by (simp add: mult_commute)
   147   then show "b mod a = 0" by simp
   148 qed
   149 
   150 lemma mod_div_trivial [simp]: "a mod b div b = 0"
   151 proof (cases "b = 0")
   152   assume "b = 0"
   153   thus ?thesis by simp
   154 next
   155   assume "b \<noteq> 0"
   156   hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
   157     by (rule div_mult_self1 [symmetric])
   158   also have "\<dots> = a div b"
   159     by (simp only: mod_div_equality')
   160   also have "\<dots> = a div b + 0"
   161     by simp
   162   finally show ?thesis
   163     by (rule add_left_imp_eq)
   164 qed
   165 
   166 lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"
   167 proof -
   168   have "a mod b mod b = (a mod b + a div b * b) mod b"
   169     by (simp only: mod_mult_self1)
   170   also have "\<dots> = a mod b"
   171     by (simp only: mod_div_equality')
   172   finally show ?thesis .
   173 qed
   174 
   175 lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
   176 by (rule dvd_eq_mod_eq_0[THEN iffD1])
   177 
   178 lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
   179 by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
   180 
   181 lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"
   182 by (drule dvd_div_mult_self) (simp add: mult_commute)
   183 
   184 lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
   185 apply (cases "a = 0")
   186  apply simp
   187 apply (auto simp: dvd_def mult_assoc)
   188 done
   189 
   190 lemma div_dvd_div[simp]:
   191   "a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"
   192 apply (cases "a = 0")
   193  apply simp
   194 apply (unfold dvd_def)
   195 apply auto
   196  apply(blast intro:mult_assoc[symmetric])
   197 apply(fastsimp simp add: mult_assoc)
   198 done
   199 
   200 lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"
   201   apply (subgoal_tac "k dvd (m div n) *n + m mod n")
   202    apply (simp add: mod_div_equality)
   203   apply (simp only: dvd_add dvd_mult)
   204   done
   205 
   206 text {* Addition respects modular equivalence. *}
   207 
   208 lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"
   209 proof -
   210   have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
   211     by (simp only: mod_div_equality)
   212   also have "\<dots> = (a mod c + b + a div c * c) mod c"
   213     by (simp only: add_ac)
   214   also have "\<dots> = (a mod c + b) mod c"
   215     by (rule mod_mult_self1)
   216   finally show ?thesis .
   217 qed
   218 
   219 lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"
   220 proof -
   221   have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
   222     by (simp only: mod_div_equality)
   223   also have "\<dots> = (a + b mod c + b div c * c) mod c"
   224     by (simp only: add_ac)
   225   also have "\<dots> = (a + b mod c) mod c"
   226     by (rule mod_mult_self1)
   227   finally show ?thesis .
   228 qed
   229 
   230 lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"
   231 by (rule trans [OF mod_add_left_eq mod_add_right_eq])
   232 
   233 lemma mod_add_cong:
   234   assumes "a mod c = a' mod c"
   235   assumes "b mod c = b' mod c"
   236   shows "(a + b) mod c = (a' + b') mod c"
   237 proof -
   238   have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
   239     unfolding assms ..
   240   thus ?thesis
   241     by (simp only: mod_add_eq [symmetric])
   242 qed
   243 
   244 lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y
   245   \<Longrightarrow> (x + y) div z = x div z + y div z"
   246 by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)
   247 
   248 text {* Multiplication respects modular equivalence. *}
   249 
   250 lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"
   251 proof -
   252   have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
   253     by (simp only: mod_div_equality)
   254   also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
   255     by (simp only: algebra_simps)
   256   also have "\<dots> = (a mod c * b) mod c"
   257     by (rule mod_mult_self1)
   258   finally show ?thesis .
   259 qed
   260 
   261 lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"
   262 proof -
   263   have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
   264     by (simp only: mod_div_equality)
   265   also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
   266     by (simp only: algebra_simps)
   267   also have "\<dots> = (a * (b mod c)) mod c"
   268     by (rule mod_mult_self1)
   269   finally show ?thesis .
   270 qed
   271 
   272 lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
   273 by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])
   274 
   275 lemma mod_mult_cong:
   276   assumes "a mod c = a' mod c"
   277   assumes "b mod c = b' mod c"
   278   shows "(a * b) mod c = (a' * b') mod c"
   279 proof -
   280   have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
   281     unfolding assms ..
   282   thus ?thesis
   283     by (simp only: mod_mult_eq [symmetric])
   284 qed
   285 
   286 lemma mod_mod_cancel:
   287   assumes "c dvd b"
   288   shows "a mod b mod c = a mod c"
   289 proof -
   290   from `c dvd b` obtain k where "b = c * k"
   291     by (rule dvdE)
   292   have "a mod b mod c = a mod (c * k) mod c"
   293     by (simp only: `b = c * k`)
   294   also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
   295     by (simp only: mod_mult_self1)
   296   also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
   297     by (simp only: add_ac mult_ac)
   298   also have "\<dots> = a mod c"
   299     by (simp only: mod_div_equality)
   300   finally show ?thesis .
   301 qed
   302 
   303 lemma div_mult_div_if_dvd:
   304   "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
   305   apply (cases "y = 0", simp)
   306   apply (cases "z = 0", simp)
   307   apply (auto elim!: dvdE simp add: algebra_simps)
   308   apply (subst mult_assoc [symmetric])
   309   apply (simp add: no_zero_divisors)
   310   done
   311 
   312 lemma div_mult_mult2 [simp]:
   313   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
   314   by (drule div_mult_mult1) (simp add: mult_commute)
   315 
   316 lemma div_mult_mult1_if [simp]:
   317   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
   318   by simp_all
   319 
   320 lemma mod_mult_mult1:
   321   "(c * a) mod (c * b) = c * (a mod b)"
   322 proof (cases "c = 0")
   323   case True then show ?thesis by simp
   324 next
   325   case False
   326   from mod_div_equality
   327   have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
   328   with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
   329     = c * a + c * (a mod b)" by (simp add: algebra_simps)
   330   with mod_div_equality show ?thesis by simp 
   331 qed
   332   
   333 lemma mod_mult_mult2:
   334   "(a * c) mod (b * c) = (a mod b) * c"
   335   using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
   336 
   337 lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
   338   unfolding dvd_def by (auto simp add: mod_mult_mult1)
   339 
   340 lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
   341 by (blast intro: dvd_mod_imp_dvd dvd_mod)
   342 
   343 lemma div_power:
   344   "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
   345 apply (induct n)
   346  apply simp
   347 apply(simp add: div_mult_div_if_dvd dvd_power_same)
   348 done
   349 
   350 end
   351 
   352 class ring_div = semiring_div + idom
   353 begin
   354 
   355 text {* Negation respects modular equivalence. *}
   356 
   357 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
   358 proof -
   359   have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
   360     by (simp only: mod_div_equality)
   361   also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
   362     by (simp only: minus_add_distrib minus_mult_left add_ac)
   363   also have "\<dots> = (- (a mod b)) mod b"
   364     by (rule mod_mult_self1)
   365   finally show ?thesis .
   366 qed
   367 
   368 lemma mod_minus_cong:
   369   assumes "a mod b = a' mod b"
   370   shows "(- a) mod b = (- a') mod b"
   371 proof -
   372   have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
   373     unfolding assms ..
   374   thus ?thesis
   375     by (simp only: mod_minus_eq [symmetric])
   376 qed
   377 
   378 text {* Subtraction respects modular equivalence. *}
   379 
   380 lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
   381   unfolding diff_minus
   382   by (intro mod_add_cong mod_minus_cong) simp_all
   383 
   384 lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
   385   unfolding diff_minus
   386   by (intro mod_add_cong mod_minus_cong) simp_all
   387 
   388 lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
   389   unfolding diff_minus
   390   by (intro mod_add_cong mod_minus_cong) simp_all
   391 
   392 lemma mod_diff_cong:
   393   assumes "a mod c = a' mod c"
   394   assumes "b mod c = b' mod c"
   395   shows "(a - b) mod c = (a' - b') mod c"
   396   unfolding diff_minus using assms
   397   by (intro mod_add_cong mod_minus_cong)
   398 
   399 lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
   400 apply (case_tac "y = 0") apply simp
   401 apply (auto simp add: dvd_def)
   402 apply (subgoal_tac "-(y * k) = y * - k")
   403  apply (erule ssubst)
   404  apply (erule div_mult_self1_is_id)
   405 apply simp
   406 done
   407 
   408 lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
   409 apply (case_tac "y = 0") apply simp
   410 apply (auto simp add: dvd_def)
   411 apply (subgoal_tac "y * k = -y * -k")
   412  apply (erule ssubst)
   413  apply (rule div_mult_self1_is_id)
   414  apply simp
   415 apply simp
   416 done
   417 
   418 end
   419 
   420 
   421 subsection {* Division on @{typ nat} *}
   422 
   423 text {*
   424   We define @{const div} and @{const mod} on @{typ nat} by means
   425   of a characteristic relation with two input arguments
   426   @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
   427   @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
   428 *}
   429 
   430 definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
   431   "divmod_nat_rel m n qr \<longleftrightarrow>
   432     m = fst qr * n + snd qr \<and>
   433       (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"
   434 
   435 text {* @{const divmod_nat_rel} is total: *}
   436 
   437 lemma divmod_nat_rel_ex:
   438   obtains q r where "divmod_nat_rel m n (q, r)"
   439 proof (cases "n = 0")
   440   case True  with that show thesis
   441     by (auto simp add: divmod_nat_rel_def)
   442 next
   443   case False
   444   have "\<exists>q r. m = q * n + r \<and> r < n"
   445   proof (induct m)
   446     case 0 with `n \<noteq> 0`
   447     have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
   448     then show ?case by blast
   449   next
   450     case (Suc m) then obtain q' r'
   451       where m: "m = q' * n + r'" and n: "r' < n" by auto
   452     then show ?case proof (cases "Suc r' < n")
   453       case True
   454       from m n have "Suc m = q' * n + Suc r'" by simp
   455       with True show ?thesis by blast
   456     next
   457       case False then have "n \<le> Suc r'" by auto
   458       moreover from n have "Suc r' \<le> n" by auto
   459       ultimately have "n = Suc r'" by auto
   460       with m have "Suc m = Suc q' * n + 0" by simp
   461       with `n \<noteq> 0` show ?thesis by blast
   462     qed
   463   qed
   464   with that show thesis
   465     using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
   466 qed
   467 
   468 text {* @{const divmod_nat_rel} is injective: *}
   469 
   470 lemma divmod_nat_rel_unique:
   471   assumes "divmod_nat_rel m n qr"
   472     and "divmod_nat_rel m n qr'"
   473   shows "qr = qr'"
   474 proof (cases "n = 0")
   475   case True with assms show ?thesis
   476     by (cases qr, cases qr')
   477       (simp add: divmod_nat_rel_def)
   478 next
   479   case False
   480   have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
   481   apply (rule leI)
   482   apply (subst less_iff_Suc_add)
   483   apply (auto simp add: add_mult_distrib)
   484   done
   485   from `n \<noteq> 0` assms have "fst qr = fst qr'"
   486     by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
   487   moreover from this assms have "snd qr = snd qr'"
   488     by (simp add: divmod_nat_rel_def)
   489   ultimately show ?thesis by (cases qr, cases qr') simp
   490 qed
   491 
   492 text {*
   493   We instantiate divisibility on the natural numbers by
   494   means of @{const divmod_nat_rel}:
   495 *}
   496 
   497 instantiation nat :: semiring_div
   498 begin
   499 
   500 definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
   501   [code del]: "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
   502 
   503 lemma divmod_nat_rel_divmod_nat:
   504   "divmod_nat_rel m n (divmod_nat m n)"
   505 proof -
   506   from divmod_nat_rel_ex
   507     obtain qr where rel: "divmod_nat_rel m n qr" .
   508   then show ?thesis
   509   by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
   510 qed
   511 
   512 lemma divmod_nat_eq:
   513   assumes "divmod_nat_rel m n qr" 
   514   shows "divmod_nat m n = qr"
   515   using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
   516 
   517 definition div_nat where
   518   "m div n = fst (divmod_nat m n)"
   519 
   520 definition mod_nat where
   521   "m mod n = snd (divmod_nat m n)"
   522 
   523 lemma divmod_nat_div_mod:
   524   "divmod_nat m n = (m div n, m mod n)"
   525   unfolding div_nat_def mod_nat_def by simp
   526 
   527 lemma div_eq:
   528   assumes "divmod_nat_rel m n (q, r)" 
   529   shows "m div n = q"
   530   using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)
   531 
   532 lemma mod_eq:
   533   assumes "divmod_nat_rel m n (q, r)" 
   534   shows "m mod n = r"
   535   using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)
   536 
   537 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
   538   by (simp add: div_nat_def mod_nat_def divmod_nat_rel_divmod_nat)
   539 
   540 lemma divmod_nat_zero:
   541   "divmod_nat m 0 = (0, m)"
   542 proof -
   543   from divmod_nat_rel [of m 0] show ?thesis
   544     unfolding divmod_nat_div_mod divmod_nat_rel_def by simp
   545 qed
   546 
   547 lemma divmod_nat_base:
   548   assumes "m < n"
   549   shows "divmod_nat m n = (0, m)"
   550 proof -
   551   from divmod_nat_rel [of m n] show ?thesis
   552     unfolding divmod_nat_div_mod divmod_nat_rel_def
   553     using assms by (cases "m div n = 0")
   554       (auto simp add: gr0_conv_Suc [of "m div n"])
   555 qed
   556 
   557 lemma divmod_nat_step:
   558   assumes "0 < n" and "n \<le> m"
   559   shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
   560 proof -
   561   from divmod_nat_rel have divmod_nat_m_n: "divmod_nat_rel m n (m div n, m mod n)" .
   562   with assms have m_div_n: "m div n \<ge> 1"
   563     by (cases "m div n") (auto simp add: divmod_nat_rel_def)
   564   from assms divmod_nat_m_n have "divmod_nat_rel (m - n) n (m div n - Suc 0, m mod n)"
   565     by (cases "m div n") (auto simp add: divmod_nat_rel_def)
   566   with divmod_nat_eq have "divmod_nat (m - n) n = (m div n - Suc 0, m mod n)" by simp
   567   moreover from divmod_nat_div_mod have "divmod_nat (m - n) n = ((m - n) div n, (m - n) mod n)" .
   568   ultimately have "m div n = Suc ((m - n) div n)"
   569     and "m mod n = (m - n) mod n" using m_div_n by simp_all
   570   then show ?thesis using divmod_nat_div_mod by simp
   571 qed
   572 
   573 text {* The ''recursion'' equations for @{const div} and @{const mod} *}
   574 
   575 lemma div_less [simp]:
   576   fixes m n :: nat
   577   assumes "m < n"
   578   shows "m div n = 0"
   579   using assms divmod_nat_base divmod_nat_div_mod by simp
   580 
   581 lemma le_div_geq:
   582   fixes m n :: nat
   583   assumes "0 < n" and "n \<le> m"
   584   shows "m div n = Suc ((m - n) div n)"
   585   using assms divmod_nat_step divmod_nat_div_mod by simp
   586 
   587 lemma mod_less [simp]:
   588   fixes m n :: nat
   589   assumes "m < n"
   590   shows "m mod n = m"
   591   using assms divmod_nat_base divmod_nat_div_mod by simp
   592 
   593 lemma le_mod_geq:
   594   fixes m n :: nat
   595   assumes "n \<le> m"
   596   shows "m mod n = (m - n) mod n"
   597   using assms divmod_nat_step divmod_nat_div_mod by (cases "n = 0") simp_all
   598 
   599 instance proof -
   600   have [simp]: "\<And>n::nat. n div 0 = 0"
   601     by (simp add: div_nat_def divmod_nat_zero)
   602   have [simp]: "\<And>n::nat. 0 div n = 0"
   603   proof -
   604     fix n :: nat
   605     show "0 div n = 0"
   606       by (cases "n = 0") simp_all
   607   qed
   608   show "OFCLASS(nat, semiring_div_class)" proof
   609     fix m n :: nat
   610     show "m div n * n + m mod n = m"
   611       using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
   612   next
   613     fix m n q :: nat
   614     assume "n \<noteq> 0"
   615     then show "(q + m * n) div n = m + q div n"
   616       by (induct m) (simp_all add: le_div_geq)
   617   next
   618     fix m n q :: nat
   619     assume "m \<noteq> 0"
   620     then show "(m * n) div (m * q) = n div q"
   621     proof (cases "n \<noteq> 0 \<and> q \<noteq> 0")
   622       case False then show ?thesis by auto
   623     next
   624       case True with `m \<noteq> 0`
   625         have "m > 0" and "n > 0" and "q > 0" by auto
   626       then have "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
   627         by (auto simp add: divmod_nat_rel_def) (simp_all add: algebra_simps)
   628       moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
   629       ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
   630       then show ?thesis by (simp add: div_eq)
   631     qed
   632   qed simp_all
   633 qed
   634 
   635 end
   636 
   637 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
   638   let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
   639 by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step)
   640     (simp add: divmod_nat_div_mod)
   641 
   642 text {* Simproc for cancelling @{const div} and @{const mod} *}
   643 
   644 ML {*
   645 local
   646 
   647 structure CancelDivMod = CancelDivModFun(struct
   648 
   649   val div_name = @{const_name div};
   650   val mod_name = @{const_name mod};
   651   val mk_binop = HOLogic.mk_binop;
   652   val mk_sum = Nat_Arith.mk_sum;
   653   val dest_sum = Nat_Arith.dest_sum;
   654 
   655   val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
   656 
   657   val trans = trans;
   658 
   659   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
   660     (@{thm monoid_add_class.add_0_left} :: @{thm monoid_add_class.add_0_right} :: @{thms add_ac}))
   661 
   662 end)
   663 
   664 in
   665 
   666 val cancel_div_mod_nat_proc = Simplifier.simproc @{theory}
   667   "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
   668 
   669 val _ = Addsimprocs [cancel_div_mod_nat_proc];
   670 
   671 end
   672 *}
   673 
   674 
   675 subsubsection {* Quotient *}
   676 
   677 lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
   678 by (simp add: le_div_geq linorder_not_less)
   679 
   680 lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
   681 by (simp add: div_geq)
   682 
   683 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
   684 by simp
   685 
   686 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
   687 by simp
   688 
   689 
   690 subsubsection {* Remainder *}
   691 
   692 lemma mod_less_divisor [simp]:
   693   fixes m n :: nat
   694   assumes "n > 0"
   695   shows "m mod n < (n::nat)"
   696   using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
   697 
   698 lemma mod_less_eq_dividend [simp]:
   699   fixes m n :: nat
   700   shows "m mod n \<le> m"
   701 proof (rule add_leD2)
   702   from mod_div_equality have "m div n * n + m mod n = m" .
   703   then show "m div n * n + m mod n \<le> m" by auto
   704 qed
   705 
   706 lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
   707 by (simp add: le_mod_geq linorder_not_less)
   708 
   709 lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
   710 by (simp add: le_mod_geq)
   711 
   712 lemma mod_1 [simp]: "m mod Suc 0 = 0"
   713 by (induct m) (simp_all add: mod_geq)
   714 
   715 lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
   716   apply (cases "n = 0", simp)
   717   apply (cases "k = 0", simp)
   718   apply (induct m rule: nat_less_induct)
   719   apply (subst mod_if, simp)
   720   apply (simp add: mod_geq diff_mult_distrib)
   721   done
   722 
   723 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
   724 by (simp add: mult_commute [of k] mod_mult_distrib)
   725 
   726 (* a simple rearrangement of mod_div_equality: *)
   727 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
   728 by (cut_tac a = m and b = n in mod_div_equality2, arith)
   729 
   730 lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
   731   apply (drule mod_less_divisor [where m = m])
   732   apply simp
   733   done
   734 
   735 subsubsection {* Quotient and Remainder *}
   736 
   737 lemma divmod_nat_rel_mult1_eq:
   738   "divmod_nat_rel b c (q, r) \<Longrightarrow> c > 0
   739    \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
   740 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
   741 
   742 lemma div_mult1_eq:
   743   "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
   744 apply (cases "c = 0", simp)
   745 apply (blast intro: divmod_nat_rel [THEN divmod_nat_rel_mult1_eq, THEN div_eq])
   746 done
   747 
   748 lemma divmod_nat_rel_add1_eq:
   749   "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) \<Longrightarrow>  c > 0
   750    \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
   751 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
   752 
   753 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
   754 lemma div_add1_eq:
   755   "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
   756 apply (cases "c = 0", simp)
   757 apply (blast intro: divmod_nat_rel_add1_eq [THEN div_eq] divmod_nat_rel)
   758 done
   759 
   760 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
   761   apply (cut_tac m = q and n = c in mod_less_divisor)
   762   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
   763   apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
   764   apply (simp add: add_mult_distrib2)
   765   done
   766 
   767 lemma divmod_nat_rel_mult2_eq:
   768   "divmod_nat_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c
   769    \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
   770 by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)
   771 
   772 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
   773   apply (cases "b = 0", simp)
   774   apply (cases "c = 0", simp)
   775   apply (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_eq])
   776   done
   777 
   778 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
   779   apply (cases "b = 0", simp)
   780   apply (cases "c = 0", simp)
   781   apply (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_eq])
   782   done
   783 
   784 
   785 subsubsection{*Further Facts about Quotient and Remainder*}
   786 
   787 lemma div_1 [simp]: "m div Suc 0 = m"
   788 by (induct m) (simp_all add: div_geq)
   789 
   790 
   791 (* Monotonicity of div in first argument *)
   792 lemma div_le_mono [rule_format (no_asm)]:
   793     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
   794 apply (case_tac "k=0", simp)
   795 apply (induct "n" rule: nat_less_induct, clarify)
   796 apply (case_tac "n<k")
   797 (* 1  case n<k *)
   798 apply simp
   799 (* 2  case n >= k *)
   800 apply (case_tac "m<k")
   801 (* 2.1  case m<k *)
   802 apply simp
   803 (* 2.2  case m>=k *)
   804 apply (simp add: div_geq diff_le_mono)
   805 done
   806 
   807 (* Antimonotonicity of div in second argument *)
   808 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
   809 apply (subgoal_tac "0<n")
   810  prefer 2 apply simp
   811 apply (induct_tac k rule: nat_less_induct)
   812 apply (rename_tac "k")
   813 apply (case_tac "k<n", simp)
   814 apply (subgoal_tac "~ (k<m) ")
   815  prefer 2 apply simp
   816 apply (simp add: div_geq)
   817 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
   818  prefer 2
   819  apply (blast intro: div_le_mono diff_le_mono2)
   820 apply (rule le_trans, simp)
   821 apply (simp)
   822 done
   823 
   824 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
   825 apply (case_tac "n=0", simp)
   826 apply (subgoal_tac "m div n \<le> m div 1", simp)
   827 apply (rule div_le_mono2)
   828 apply (simp_all (no_asm_simp))
   829 done
   830 
   831 (* Similar for "less than" *)
   832 lemma div_less_dividend [rule_format]:
   833      "!!n::nat. 1<n ==> 0 < m --> m div n < m"
   834 apply (induct_tac m rule: nat_less_induct)
   835 apply (rename_tac "m")
   836 apply (case_tac "m<n", simp)
   837 apply (subgoal_tac "0<n")
   838  prefer 2 apply simp
   839 apply (simp add: div_geq)
   840 apply (case_tac "n<m")
   841  apply (subgoal_tac "(m-n) div n < (m-n) ")
   842   apply (rule impI less_trans_Suc)+
   843 apply assumption
   844   apply (simp_all)
   845 done
   846 
   847 declare div_less_dividend [simp]
   848 
   849 text{*A fact for the mutilated chess board*}
   850 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
   851 apply (case_tac "n=0", simp)
   852 apply (induct "m" rule: nat_less_induct)
   853 apply (case_tac "Suc (na) <n")
   854 (* case Suc(na) < n *)
   855 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
   856 (* case n \<le> Suc(na) *)
   857 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
   858 apply (auto simp add: Suc_diff_le le_mod_geq)
   859 done
   860 
   861 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
   862 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
   863 
   864 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
   865 
   866 (*Loses information, namely we also have r<d provided d is nonzero*)
   867 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
   868   apply (cut_tac a = m in mod_div_equality)
   869   apply (simp only: add_ac)
   870   apply (blast intro: sym)
   871   done
   872 
   873 lemma split_div:
   874  "P(n div k :: nat) =
   875  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
   876  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   877 proof
   878   assume P: ?P
   879   show ?Q
   880   proof (cases)
   881     assume "k = 0"
   882     with P show ?Q by simp
   883   next
   884     assume not0: "k \<noteq> 0"
   885     thus ?Q
   886     proof (simp, intro allI impI)
   887       fix i j
   888       assume n: "n = k*i + j" and j: "j < k"
   889       show "P i"
   890       proof (cases)
   891         assume "i = 0"
   892         with n j P show "P i" by simp
   893       next
   894         assume "i \<noteq> 0"
   895         with not0 n j P show "P i" by(simp add:add_ac)
   896       qed
   897     qed
   898   qed
   899 next
   900   assume Q: ?Q
   901   show ?P
   902   proof (cases)
   903     assume "k = 0"
   904     with Q show ?P by simp
   905   next
   906     assume not0: "k \<noteq> 0"
   907     with Q have R: ?R by simp
   908     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   909     show ?P by simp
   910   qed
   911 qed
   912 
   913 lemma split_div_lemma:
   914   assumes "0 < n"
   915   shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
   916 proof
   917   assume ?rhs
   918   with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
   919   then have A: "n * q \<le> m" by simp
   920   have "n - (m mod n) > 0" using mod_less_divisor assms by auto
   921   then have "m < m + (n - (m mod n))" by simp
   922   then have "m < n + (m - (m mod n))" by simp
   923   with nq have "m < n + n * q" by simp
   924   then have B: "m < n * Suc q" by simp
   925   from A B show ?lhs ..
   926 next
   927   assume P: ?lhs
   928   then have "divmod_nat_rel m n (q, m - n * q)"
   929     unfolding divmod_nat_rel_def by (auto simp add: mult_ac)
   930   with divmod_nat_rel_unique divmod_nat_rel [of m n]
   931   have "(q, m - n * q) = (m div n, m mod n)" by auto
   932   then show ?rhs by simp
   933 qed
   934 
   935 theorem split_div':
   936   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
   937    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
   938   apply (case_tac "0 < n")
   939   apply (simp only: add: split_div_lemma)
   940   apply simp_all
   941   done
   942 
   943 lemma split_mod:
   944  "P(n mod k :: nat) =
   945  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
   946  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   947 proof
   948   assume P: ?P
   949   show ?Q
   950   proof (cases)
   951     assume "k = 0"
   952     with P show ?Q by simp
   953   next
   954     assume not0: "k \<noteq> 0"
   955     thus ?Q
   956     proof (simp, intro allI impI)
   957       fix i j
   958       assume "n = k*i + j" "j < k"
   959       thus "P j" using not0 P by(simp add:add_ac mult_ac)
   960     qed
   961   qed
   962 next
   963   assume Q: ?Q
   964   show ?P
   965   proof (cases)
   966     assume "k = 0"
   967     with Q show ?P by simp
   968   next
   969     assume not0: "k \<noteq> 0"
   970     with Q have R: ?R by simp
   971     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   972     show ?P by simp
   973   qed
   974 qed
   975 
   976 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
   977   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
   978     subst [OF mod_div_equality [of _ n]])
   979   apply arith
   980   done
   981 
   982 lemma div_mod_equality':
   983   fixes m n :: nat
   984   shows "m div n * n = m - m mod n"
   985 proof -
   986   have "m mod n \<le> m mod n" ..
   987   from div_mod_equality have 
   988     "m div n * n + m mod n - m mod n = m - m mod n" by simp
   989   with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
   990     "m div n * n + (m mod n - m mod n) = m - m mod n"
   991     by simp
   992   then show ?thesis by simp
   993 qed
   994 
   995 
   996 subsubsection {*An ``induction'' law for modulus arithmetic.*}
   997 
   998 lemma mod_induct_0:
   999   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
  1000   and base: "P i" and i: "i<p"
  1001   shows "P 0"
  1002 proof (rule ccontr)
  1003   assume contra: "\<not>(P 0)"
  1004   from i have p: "0<p" by simp
  1005   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
  1006   proof
  1007     fix k
  1008     show "?A k"
  1009     proof (induct k)
  1010       show "?A 0" by simp  -- "by contradiction"
  1011     next
  1012       fix n
  1013       assume ih: "?A n"
  1014       show "?A (Suc n)"
  1015       proof (clarsimp)
  1016         assume y: "P (p - Suc n)"
  1017         have n: "Suc n < p"
  1018         proof (rule ccontr)
  1019           assume "\<not>(Suc n < p)"
  1020           hence "p - Suc n = 0"
  1021             by simp
  1022           with y contra show "False"
  1023             by simp
  1024         qed
  1025         hence n2: "Suc (p - Suc n) = p-n" by arith
  1026         from p have "p - Suc n < p" by arith
  1027         with y step have z: "P ((Suc (p - Suc n)) mod p)"
  1028           by blast
  1029         show "False"
  1030         proof (cases "n=0")
  1031           case True
  1032           with z n2 contra show ?thesis by simp
  1033         next
  1034           case False
  1035           with p have "p-n < p" by arith
  1036           with z n2 False ih show ?thesis by simp
  1037         qed
  1038       qed
  1039     qed
  1040   qed
  1041   moreover
  1042   from i obtain k where "0<k \<and> i+k=p"
  1043     by (blast dest: less_imp_add_positive)
  1044   hence "0<k \<and> i=p-k" by auto
  1045   moreover
  1046   note base
  1047   ultimately
  1048   show "False" by blast
  1049 qed
  1050 
  1051 lemma mod_induct:
  1052   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
  1053   and base: "P i" and i: "i<p" and j: "j<p"
  1054   shows "P j"
  1055 proof -
  1056   have "\<forall>j<p. P j"
  1057   proof
  1058     fix j
  1059     show "j<p \<longrightarrow> P j" (is "?A j")
  1060     proof (induct j)
  1061       from step base i show "?A 0"
  1062         by (auto elim: mod_induct_0)
  1063     next
  1064       fix k
  1065       assume ih: "?A k"
  1066       show "?A (Suc k)"
  1067       proof
  1068         assume suc: "Suc k < p"
  1069         hence k: "k<p" by simp
  1070         with ih have "P k" ..
  1071         with step k have "P (Suc k mod p)"
  1072           by blast
  1073         moreover
  1074         from suc have "Suc k mod p = Suc k"
  1075           by simp
  1076         ultimately
  1077         show "P (Suc k)" by simp
  1078       qed
  1079     qed
  1080   qed
  1081   with j show ?thesis by blast
  1082 qed
  1083 
  1084 lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
  1085 by (auto simp add: numeral_2_eq_2 le_div_geq)
  1086 
  1087 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
  1088 by (simp add: nat_mult_2 [symmetric])
  1089 
  1090 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
  1091 apply (subgoal_tac "m mod 2 < 2")
  1092 apply (erule less_2_cases [THEN disjE])
  1093 apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1)
  1094 done
  1095 
  1096 lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
  1097 proof -
  1098   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (induct n) simp_all }
  1099   moreover have "m mod 2 < 2" by simp
  1100   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
  1101   then show ?thesis by auto
  1102 qed
  1103 
  1104 text{*These lemmas collapse some needless occurrences of Suc:
  1105     at least three Sucs, since two and fewer are rewritten back to Suc again!
  1106     We already have some rules to simplify operands smaller than 3.*}
  1107 
  1108 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
  1109 by (simp add: Suc3_eq_add_3)
  1110 
  1111 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
  1112 by (simp add: Suc3_eq_add_3)
  1113 
  1114 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
  1115 by (simp add: Suc3_eq_add_3)
  1116 
  1117 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
  1118 by (simp add: Suc3_eq_add_3)
  1119 
  1120 lemmas Suc_div_eq_add3_div_number_of =
  1121     Suc_div_eq_add3_div [of _ "number_of v", standard]
  1122 declare Suc_div_eq_add3_div_number_of [simp]
  1123 
  1124 lemmas Suc_mod_eq_add3_mod_number_of =
  1125     Suc_mod_eq_add3_mod [of _ "number_of v", standard]
  1126 declare Suc_mod_eq_add3_mod_number_of [simp]
  1127 
  1128 
  1129 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
  1130 apply (induct "m")
  1131 apply (simp_all add: mod_Suc)
  1132 done
  1133 
  1134 declare Suc_times_mod_eq [of "number_of w", standard, simp]
  1135 
  1136 lemma [simp]: "n div k \<le> (Suc n) div k"
  1137 by (simp add: div_le_mono) 
  1138 
  1139 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
  1140 by (cases n) simp_all
  1141 
  1142 lemma div_2_gt_zero [simp]: "(1::nat) < n ==> 0 < n div 2" 
  1143 using Suc_n_div_2_gt_zero [of "n - 1"] by simp
  1144 
  1145   (* Potential use of algebra : Equality modulo n*)
  1146 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
  1147 by (simp add: mult_ac add_ac)
  1148 
  1149 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
  1150 proof -
  1151   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
  1152   also have "... = Suc m mod n" by (rule mod_mult_self3) 
  1153   finally show ?thesis .
  1154 qed
  1155 
  1156 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
  1157 apply (subst mod_Suc [of m]) 
  1158 apply (subst mod_Suc [of "m mod n"], simp) 
  1159 done
  1160 
  1161 
  1162 subsection {* Division on @{typ int} *}
  1163 
  1164 definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
  1165     --{*definition of quotient and remainder*}
  1166     [code]: "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
  1167                (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
  1168 
  1169 definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
  1170     --{*for the division algorithm*}
  1171     [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
  1172                          else (2 * q, r))"
  1173 
  1174 text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
  1175 function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  1176   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
  1177      else adjust b (posDivAlg a (2 * b)))"
  1178 by auto
  1179 termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
  1180   (auto simp add: mult_2)
  1181 
  1182 text{*algorithm for the case @{text "a<0, b>0"}*}
  1183 function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  1184   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
  1185      else adjust b (negDivAlg a (2 * b)))"
  1186 by auto
  1187 termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
  1188   (auto simp add: mult_2)
  1189 
  1190 text{*algorithm for the general case @{term "b\<noteq>0"}*}
  1191 definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
  1192   [code_unfold]: "negateSnd = apsnd uminus"
  1193 
  1194 definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  1195     --{*The full division algorithm considers all possible signs for a, b
  1196        including the special case @{text "a=0, b<0"} because 
  1197        @{term negDivAlg} requires @{term "a<0"}.*}
  1198   "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
  1199                   else if a = 0 then (0, 0)
  1200                        else negateSnd (negDivAlg (-a) (-b))
  1201                else 
  1202                   if 0 < b then negDivAlg a b
  1203                   else negateSnd (posDivAlg (-a) (-b)))"
  1204 
  1205 instantiation int :: Divides.div
  1206 begin
  1207 
  1208 definition
  1209   "a div b = fst (divmod_int a b)"
  1210 
  1211 definition
  1212  "a mod b = snd (divmod_int a b)"
  1213 
  1214 instance ..
  1215 
  1216 end
  1217 
  1218 lemma divmod_int_mod_div:
  1219   "divmod_int p q = (p div q, p mod q)"
  1220   by (auto simp add: div_int_def mod_int_def)
  1221 
  1222 text{*
  1223 Here is the division algorithm in ML:
  1224 
  1225 \begin{verbatim}
  1226     fun posDivAlg (a,b) =
  1227       if a<b then (0,a)
  1228       else let val (q,r) = posDivAlg(a, 2*b)
  1229                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
  1230            end
  1231 
  1232     fun negDivAlg (a,b) =
  1233       if 0\<le>a+b then (~1,a+b)
  1234       else let val (q,r) = negDivAlg(a, 2*b)
  1235                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
  1236            end;
  1237 
  1238     fun negateSnd (q,r:int) = (q,~r);
  1239 
  1240     fun divmod (a,b) = if 0\<le>a then 
  1241                           if b>0 then posDivAlg (a,b) 
  1242                            else if a=0 then (0,0)
  1243                                 else negateSnd (negDivAlg (~a,~b))
  1244                        else 
  1245                           if 0<b then negDivAlg (a,b)
  1246                           else        negateSnd (posDivAlg (~a,~b));
  1247 \end{verbatim}
  1248 *}
  1249 
  1250 
  1251 
  1252 subsubsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
  1253 
  1254 lemma unique_quotient_lemma:
  1255      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
  1256       ==> q' \<le> (q::int)"
  1257 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
  1258  prefer 2 apply (simp add: right_diff_distrib)
  1259 apply (subgoal_tac "0 < b * (1 + q - q') ")
  1260 apply (erule_tac [2] order_le_less_trans)
  1261  prefer 2 apply (simp add: right_diff_distrib right_distrib)
  1262 apply (subgoal_tac "b * q' < b * (1 + q) ")
  1263  prefer 2 apply (simp add: right_diff_distrib right_distrib)
  1264 apply (simp add: mult_less_cancel_left)
  1265 done
  1266 
  1267 lemma unique_quotient_lemma_neg:
  1268      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
  1269       ==> q \<le> (q'::int)"
  1270 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
  1271     auto)
  1272 
  1273 lemma unique_quotient:
  1274      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
  1275       ==> q = q'"
  1276 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
  1277 apply (blast intro: order_antisym
  1278              dest: order_eq_refl [THEN unique_quotient_lemma] 
  1279              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
  1280 done
  1281 
  1282 
  1283 lemma unique_remainder:
  1284      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
  1285       ==> r = r'"
  1286 apply (subgoal_tac "q = q'")
  1287  apply (simp add: divmod_int_rel_def)
  1288 apply (blast intro: unique_quotient)
  1289 done
  1290 
  1291 
  1292 subsubsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
  1293 
  1294 text{*And positive divisors*}
  1295 
  1296 lemma adjust_eq [simp]:
  1297      "adjust b (q,r) = 
  1298       (let diff = r-b in  
  1299         if 0 \<le> diff then (2*q + 1, diff)   
  1300                      else (2*q, r))"
  1301 by (simp add: Let_def adjust_def)
  1302 
  1303 declare posDivAlg.simps [simp del]
  1304 
  1305 text{*use with a simproc to avoid repeatedly proving the premise*}
  1306 lemma posDivAlg_eqn:
  1307      "0 < b ==>  
  1308       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
  1309 by (rule posDivAlg.simps [THEN trans], simp)
  1310 
  1311 text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
  1312 theorem posDivAlg_correct:
  1313   assumes "0 \<le> a" and "0 < b"
  1314   shows "divmod_int_rel a b (posDivAlg a b)"
  1315 using prems apply (induct a b rule: posDivAlg.induct)
  1316 apply auto
  1317 apply (simp add: divmod_int_rel_def)
  1318 apply (subst posDivAlg_eqn, simp add: right_distrib)
  1319 apply (case_tac "a < b")
  1320 apply simp_all
  1321 apply (erule splitE)
  1322 apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
  1323 done
  1324 
  1325 
  1326 subsubsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
  1327 
  1328 text{*And positive divisors*}
  1329 
  1330 declare negDivAlg.simps [simp del]
  1331 
  1332 text{*use with a simproc to avoid repeatedly proving the premise*}
  1333 lemma negDivAlg_eqn:
  1334      "0 < b ==>  
  1335       negDivAlg a b =       
  1336        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
  1337 by (rule negDivAlg.simps [THEN trans], simp)
  1338 
  1339 (*Correctness of negDivAlg: it computes quotients correctly
  1340   It doesn't work if a=0 because the 0/b equals 0, not -1*)
  1341 lemma negDivAlg_correct:
  1342   assumes "a < 0" and "b > 0"
  1343   shows "divmod_int_rel a b (negDivAlg a b)"
  1344 using prems apply (induct a b rule: negDivAlg.induct)
  1345 apply (auto simp add: linorder_not_le)
  1346 apply (simp add: divmod_int_rel_def)
  1347 apply (subst negDivAlg_eqn, assumption)
  1348 apply (case_tac "a + b < (0\<Colon>int)")
  1349 apply simp_all
  1350 apply (erule splitE)
  1351 apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
  1352 done
  1353 
  1354 
  1355 subsubsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
  1356 
  1357 (*the case a=0*)
  1358 lemma divmod_int_rel_0: "b \<noteq> 0 ==> divmod_int_rel 0 b (0, 0)"
  1359 by (auto simp add: divmod_int_rel_def linorder_neq_iff)
  1360 
  1361 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
  1362 by (subst posDivAlg.simps, auto)
  1363 
  1364 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
  1365 by (subst negDivAlg.simps, auto)
  1366 
  1367 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
  1368 by (simp add: negateSnd_def)
  1369 
  1370 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (negateSnd qr)"
  1371 by (auto simp add: split_ifs divmod_int_rel_def)
  1372 
  1373 lemma divmod_int_correct: "b \<noteq> 0 ==> divmod_int_rel a b (divmod_int a b)"
  1374 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
  1375                     posDivAlg_correct negDivAlg_correct)
  1376 
  1377 text{*Arbitrary definitions for division by zero.  Useful to simplify 
  1378     certain equations.*}
  1379 
  1380 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
  1381 by (simp add: div_int_def mod_int_def divmod_int_def posDivAlg.simps)  
  1382 
  1383 
  1384 text{*Basic laws about division and remainder*}
  1385 
  1386 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
  1387 apply (case_tac "b = 0", simp)
  1388 apply (cut_tac a = a and b = b in divmod_int_correct)
  1389 apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
  1390 done
  1391 
  1392 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
  1393 by(simp add: zmod_zdiv_equality[symmetric])
  1394 
  1395 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
  1396 by(simp add: mult_commute zmod_zdiv_equality[symmetric])
  1397 
  1398 text {* Tool setup *}
  1399 
  1400 ML {*
  1401 local
  1402 
  1403 structure CancelDivMod = CancelDivModFun(struct
  1404 
  1405   val div_name = @{const_name div};
  1406   val mod_name = @{const_name mod};
  1407   val mk_binop = HOLogic.mk_binop;
  1408   val mk_sum = Arith_Data.mk_sum HOLogic.intT;
  1409   val dest_sum = Arith_Data.dest_sum;
  1410 
  1411   val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];
  1412 
  1413   val trans = trans;
  1414 
  1415   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
  1416     (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
  1417 
  1418 end)
  1419 
  1420 in
  1421 
  1422 val cancel_div_mod_int_proc = Simplifier.simproc @{theory}
  1423   "cancel_zdiv_zmod" ["(k::int) + l"] (K CancelDivMod.proc);
  1424 
  1425 val _ = Addsimprocs [cancel_div_mod_int_proc];
  1426 
  1427 end
  1428 *}
  1429 
  1430 lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
  1431 apply (cut_tac a = a and b = b in divmod_int_correct)
  1432 apply (auto simp add: divmod_int_rel_def mod_int_def)
  1433 done
  1434 
  1435 lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
  1436    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
  1437 
  1438 lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
  1439 apply (cut_tac a = a and b = b in divmod_int_correct)
  1440 apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
  1441 done
  1442 
  1443 lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
  1444    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
  1445 
  1446 
  1447 
  1448 subsubsection{*General Properties of div and mod*}
  1449 
  1450 lemma divmod_int_rel_div_mod: "b \<noteq> 0 ==> divmod_int_rel a b (a div b, a mod b)"
  1451 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
  1452 apply (force simp add: divmod_int_rel_def linorder_neq_iff)
  1453 done
  1454 
  1455 lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
  1456 by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
  1457 
  1458 lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
  1459 by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
  1460 
  1461 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
  1462 apply (rule divmod_int_rel_div)
  1463 apply (auto simp add: divmod_int_rel_def)
  1464 done
  1465 
  1466 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
  1467 apply (rule divmod_int_rel_div)
  1468 apply (auto simp add: divmod_int_rel_def)
  1469 done
  1470 
  1471 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
  1472 apply (rule divmod_int_rel_div)
  1473 apply (auto simp add: divmod_int_rel_def)
  1474 done
  1475 
  1476 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
  1477 
  1478 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
  1479 apply (rule_tac q = 0 in divmod_int_rel_mod)
  1480 apply (auto simp add: divmod_int_rel_def)
  1481 done
  1482 
  1483 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
  1484 apply (rule_tac q = 0 in divmod_int_rel_mod)
  1485 apply (auto simp add: divmod_int_rel_def)
  1486 done
  1487 
  1488 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
  1489 apply (rule_tac q = "-1" in divmod_int_rel_mod)
  1490 apply (auto simp add: divmod_int_rel_def)
  1491 done
  1492 
  1493 text{*There is no @{text mod_neg_pos_trivial}.*}
  1494 
  1495 
  1496 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
  1497 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
  1498 apply (case_tac "b = 0", simp)
  1499 apply (simp add: divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, 
  1500                                  THEN divmod_int_rel_div, THEN sym])
  1501 
  1502 done
  1503 
  1504 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
  1505 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
  1506 apply (case_tac "b = 0", simp)
  1507 apply (subst divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, THEN divmod_int_rel_mod],
  1508        auto)
  1509 done
  1510 
  1511 
  1512 subsubsection{*Laws for div and mod with Unary Minus*}
  1513 
  1514 lemma zminus1_lemma:
  1515      "divmod_int_rel a b (q, r)
  1516       ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
  1517                           if r=0 then 0 else b-r)"
  1518 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
  1519 
  1520 
  1521 lemma zdiv_zminus1_eq_if:
  1522      "b \<noteq> (0::int)  
  1523       ==> (-a) div b =  
  1524           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
  1525 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_div])
  1526 
  1527 lemma zmod_zminus1_eq_if:
  1528      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
  1529 apply (case_tac "b = 0", simp)
  1530 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_mod])
  1531 done
  1532 
  1533 lemma zmod_zminus1_not_zero:
  1534   fixes k l :: int
  1535   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
  1536   unfolding zmod_zminus1_eq_if by auto
  1537 
  1538 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
  1539 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
  1540 
  1541 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
  1542 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
  1543 
  1544 lemma zdiv_zminus2_eq_if:
  1545      "b \<noteq> (0::int)  
  1546       ==> a div (-b) =  
  1547           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
  1548 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
  1549 
  1550 lemma zmod_zminus2_eq_if:
  1551      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
  1552 by (simp add: zmod_zminus1_eq_if zmod_zminus2)
  1553 
  1554 lemma zmod_zminus2_not_zero:
  1555   fixes k l :: int
  1556   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
  1557   unfolding zmod_zminus2_eq_if by auto 
  1558 
  1559 
  1560 subsubsection{*Division of a Number by Itself*}
  1561 
  1562 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
  1563 apply (subgoal_tac "0 < a*q")
  1564  apply (simp add: zero_less_mult_iff, arith)
  1565 done
  1566 
  1567 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
  1568 apply (subgoal_tac "0 \<le> a* (1-q) ")
  1569  apply (simp add: zero_le_mult_iff)
  1570 apply (simp add: right_diff_distrib)
  1571 done
  1572 
  1573 lemma self_quotient: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
  1574 apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)
  1575 apply (rule order_antisym, safe, simp_all)
  1576 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
  1577 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
  1578 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
  1579 done
  1580 
  1581 lemma self_remainder: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
  1582 apply (frule self_quotient, assumption)
  1583 apply (simp add: divmod_int_rel_def)
  1584 done
  1585 
  1586 lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
  1587 by (simp add: divmod_int_rel_div_mod [THEN self_quotient])
  1588 
  1589 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
  1590 lemma zmod_self [simp]: "a mod a = (0::int)"
  1591 apply (case_tac "a = 0", simp)
  1592 apply (simp add: divmod_int_rel_div_mod [THEN self_remainder])
  1593 done
  1594 
  1595 
  1596 subsubsection{*Computation of Division and Remainder*}
  1597 
  1598 lemma zdiv_zero [simp]: "(0::int) div b = 0"
  1599 by (simp add: div_int_def divmod_int_def)
  1600 
  1601 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
  1602 by (simp add: div_int_def divmod_int_def)
  1603 
  1604 lemma zmod_zero [simp]: "(0::int) mod b = 0"
  1605 by (simp add: mod_int_def divmod_int_def)
  1606 
  1607 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
  1608 by (simp add: mod_int_def divmod_int_def)
  1609 
  1610 text{*a positive, b positive *}
  1611 
  1612 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
  1613 by (simp add: div_int_def divmod_int_def)
  1614 
  1615 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
  1616 by (simp add: mod_int_def divmod_int_def)
  1617 
  1618 text{*a negative, b positive *}
  1619 
  1620 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
  1621 by (simp add: div_int_def divmod_int_def)
  1622 
  1623 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
  1624 by (simp add: mod_int_def divmod_int_def)
  1625 
  1626 text{*a positive, b negative *}
  1627 
  1628 lemma div_pos_neg:
  1629      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
  1630 by (simp add: div_int_def divmod_int_def)
  1631 
  1632 lemma mod_pos_neg:
  1633      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
  1634 by (simp add: mod_int_def divmod_int_def)
  1635 
  1636 text{*a negative, b negative *}
  1637 
  1638 lemma div_neg_neg:
  1639      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
  1640 by (simp add: div_int_def divmod_int_def)
  1641 
  1642 lemma mod_neg_neg:
  1643      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
  1644 by (simp add: mod_int_def divmod_int_def)
  1645 
  1646 text {*Simplify expresions in which div and mod combine numerical constants*}
  1647 
  1648 lemma divmod_int_relI:
  1649   "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
  1650     \<Longrightarrow> divmod_int_rel a b (q, r)"
  1651   unfolding divmod_int_rel_def by simp
  1652 
  1653 lemmas divmod_int_rel_div_eq = divmod_int_relI [THEN divmod_int_rel_div, THEN eq_reflection]
  1654 lemmas divmod_int_rel_mod_eq = divmod_int_relI [THEN divmod_int_rel_mod, THEN eq_reflection]
  1655 lemmas arithmetic_simps =
  1656   arith_simps
  1657   add_special
  1658   OrderedGroup.add_0_left
  1659   OrderedGroup.add_0_right
  1660   mult_zero_left
  1661   mult_zero_right
  1662   mult_1_left
  1663   mult_1_right
  1664 
  1665 (* simprocs adapted from HOL/ex/Binary.thy *)
  1666 ML {*
  1667 local
  1668   val mk_number = HOLogic.mk_number HOLogic.intT;
  1669   fun mk_cert u k l = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $
  1670     (@{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ u $ mk_number k) $
  1671       mk_number l;
  1672   fun prove ctxt prop = Goal.prove ctxt [] [] prop
  1673     (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps @{thms arithmetic_simps}))));
  1674   fun binary_proc proc ss ct =
  1675     (case Thm.term_of ct of
  1676       _ $ t $ u =>
  1677       (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
  1678         SOME args => proc (Simplifier.the_context ss) args
  1679       | NONE => NONE)
  1680     | _ => NONE);
  1681 in
  1682   fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
  1683     if n = 0 then NONE
  1684     else let val (k, l) = Integer.div_mod m n;
  1685     in SOME (rule OF [prove ctxt (Logic.mk_equals (t, mk_cert u k l))]) end);
  1686 end
  1687 *}
  1688 
  1689 simproc_setup binary_int_div ("number_of m div number_of n :: int") =
  1690   {* K (divmod_proc (@{thm divmod_int_rel_div_eq})) *}
  1691 
  1692 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
  1693   {* K (divmod_proc (@{thm divmod_int_rel_mod_eq})) *}
  1694 
  1695 lemmas posDivAlg_eqn_number_of [simp] =
  1696     posDivAlg_eqn [of "number_of v" "number_of w", standard]
  1697 
  1698 lemmas negDivAlg_eqn_number_of [simp] =
  1699     negDivAlg_eqn [of "number_of v" "number_of w", standard]
  1700 
  1701 
  1702 text{*Special-case simplification *}
  1703 
  1704 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
  1705 apply (cut_tac a = a and b = "-1" in neg_mod_sign)
  1706 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
  1707 apply (auto simp del: neg_mod_sign neg_mod_bound)
  1708 done
  1709 
  1710 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
  1711 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
  1712 
  1713 (** The last remaining special cases for constant arithmetic:
  1714     1 div z and 1 mod z **)
  1715 
  1716 lemmas div_pos_pos_1_number_of [simp] =
  1717     div_pos_pos [OF int_0_less_1, of "number_of w", standard]
  1718 
  1719 lemmas div_pos_neg_1_number_of [simp] =
  1720     div_pos_neg [OF int_0_less_1, of "number_of w", standard]
  1721 
  1722 lemmas mod_pos_pos_1_number_of [simp] =
  1723     mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
  1724 
  1725 lemmas mod_pos_neg_1_number_of [simp] =
  1726     mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
  1727 
  1728 
  1729 lemmas posDivAlg_eqn_1_number_of [simp] =
  1730     posDivAlg_eqn [of concl: 1 "number_of w", standard]
  1731 
  1732 lemmas negDivAlg_eqn_1_number_of [simp] =
  1733     negDivAlg_eqn [of concl: 1 "number_of w", standard]
  1734 
  1735 
  1736 
  1737 subsubsection{*Monotonicity in the First Argument (Dividend)*}
  1738 
  1739 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
  1740 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
  1741 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
  1742 apply (rule unique_quotient_lemma)
  1743 apply (erule subst)
  1744 apply (erule subst, simp_all)
  1745 done
  1746 
  1747 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
  1748 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
  1749 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
  1750 apply (rule unique_quotient_lemma_neg)
  1751 apply (erule subst)
  1752 apply (erule subst, simp_all)
  1753 done
  1754 
  1755 
  1756 subsubsection{*Monotonicity in the Second Argument (Divisor)*}
  1757 
  1758 lemma q_pos_lemma:
  1759      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
  1760 apply (subgoal_tac "0 < b'* (q' + 1) ")
  1761  apply (simp add: zero_less_mult_iff)
  1762 apply (simp add: right_distrib)
  1763 done
  1764 
  1765 lemma zdiv_mono2_lemma:
  1766      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
  1767          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
  1768       ==> q \<le> (q'::int)"
  1769 apply (frule q_pos_lemma, assumption+) 
  1770 apply (subgoal_tac "b*q < b* (q' + 1) ")
  1771  apply (simp add: mult_less_cancel_left)
  1772 apply (subgoal_tac "b*q = r' - r + b'*q'")
  1773  prefer 2 apply simp
  1774 apply (simp (no_asm_simp) add: right_distrib)
  1775 apply (subst add_commute, rule zadd_zless_mono, arith)
  1776 apply (rule mult_right_mono, auto)
  1777 done
  1778 
  1779 lemma zdiv_mono2:
  1780      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
  1781 apply (subgoal_tac "b \<noteq> 0")
  1782  prefer 2 apply arith
  1783 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
  1784 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
  1785 apply (rule zdiv_mono2_lemma)
  1786 apply (erule subst)
  1787 apply (erule subst, simp_all)
  1788 done
  1789 
  1790 lemma q_neg_lemma:
  1791      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
  1792 apply (subgoal_tac "b'*q' < 0")
  1793  apply (simp add: mult_less_0_iff, arith)
  1794 done
  1795 
  1796 lemma zdiv_mono2_neg_lemma:
  1797      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
  1798          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
  1799       ==> q' \<le> (q::int)"
  1800 apply (frule q_neg_lemma, assumption+) 
  1801 apply (subgoal_tac "b*q' < b* (q + 1) ")
  1802  apply (simp add: mult_less_cancel_left)
  1803 apply (simp add: right_distrib)
  1804 apply (subgoal_tac "b*q' \<le> b'*q'")
  1805  prefer 2 apply (simp add: mult_right_mono_neg, arith)
  1806 done
  1807 
  1808 lemma zdiv_mono2_neg:
  1809      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
  1810 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
  1811 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
  1812 apply (rule zdiv_mono2_neg_lemma)
  1813 apply (erule subst)
  1814 apply (erule subst, simp_all)
  1815 done
  1816 
  1817 
  1818 subsubsection{*More Algebraic Laws for div and mod*}
  1819 
  1820 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
  1821 
  1822 lemma zmult1_lemma:
  1823      "[| divmod_int_rel b c (q, r);  c \<noteq> 0 |]  
  1824       ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
  1825 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)
  1826 
  1827 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
  1828 apply (case_tac "c = 0", simp)
  1829 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_div])
  1830 done
  1831 
  1832 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
  1833 apply (case_tac "c = 0", simp)
  1834 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_mod])
  1835 done
  1836 
  1837 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
  1838 apply (case_tac "b = 0", simp)
  1839 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
  1840 done
  1841 
  1842 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
  1843 
  1844 lemma zadd1_lemma:
  1845      "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br);  c \<noteq> 0 |]  
  1846       ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
  1847 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)
  1848 
  1849 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
  1850 lemma zdiv_zadd1_eq:
  1851      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
  1852 apply (case_tac "c = 0", simp)
  1853 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] divmod_int_rel_div)
  1854 done
  1855 
  1856 instance int :: ring_div
  1857 proof
  1858   fix a b c :: int
  1859   assume not0: "b \<noteq> 0"
  1860   show "(a + c * b) div b = c + a div b"
  1861     unfolding zdiv_zadd1_eq [of a "c * b"] using not0 
  1862       by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
  1863 next
  1864   fix a b c :: int
  1865   assume "a \<noteq> 0"
  1866   then show "(a * b) div (a * c) = b div c"
  1867   proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")
  1868     case False then show ?thesis by auto
  1869   next
  1870     case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto
  1871     with `a \<noteq> 0`
  1872     have "\<And>q r. divmod_int_rel b c (q, r) \<Longrightarrow> divmod_int_rel (a * b) (a * c) (q, a * r)"
  1873       apply (auto simp add: divmod_int_rel_def) 
  1874       apply (auto simp add: algebra_simps)
  1875       apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff mult_commute [of a] mult_less_cancel_right)
  1876       done
  1877     moreover with `c \<noteq> 0` divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto
  1878     ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .
  1879     moreover from  `a \<noteq> 0` `c \<noteq> 0` have "a * c \<noteq> 0" by simp
  1880     ultimately show ?thesis by (rule divmod_int_rel_div)
  1881   qed
  1882 qed auto
  1883 
  1884 lemma posDivAlg_div_mod:
  1885   assumes "k \<ge> 0"
  1886   and "l \<ge> 0"
  1887   shows "posDivAlg k l = (k div l, k mod l)"
  1888 proof (cases "l = 0")
  1889   case True then show ?thesis by (simp add: posDivAlg.simps)
  1890 next
  1891   case False with assms posDivAlg_correct
  1892     have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
  1893     by simp
  1894   from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
  1895   show ?thesis by simp
  1896 qed
  1897 
  1898 lemma negDivAlg_div_mod:
  1899   assumes "k < 0"
  1900   and "l > 0"
  1901   shows "negDivAlg k l = (k div l, k mod l)"
  1902 proof -
  1903   from assms have "l \<noteq> 0" by simp
  1904   from assms negDivAlg_correct
  1905     have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
  1906     by simp
  1907   from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
  1908   show ?thesis by simp
  1909 qed
  1910 
  1911 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
  1912 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
  1913 
  1914 (* REVISIT: should this be generalized to all semiring_div types? *)
  1915 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
  1916 
  1917 
  1918 subsubsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
  1919 
  1920 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
  1921   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
  1922   to cause particular problems.*)
  1923 
  1924 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
  1925 
  1926 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
  1927 apply (subgoal_tac "b * (c - q mod c) < r * 1")
  1928  apply (simp add: algebra_simps)
  1929 apply (rule order_le_less_trans)
  1930  apply (erule_tac [2] mult_strict_right_mono)
  1931  apply (rule mult_left_mono_neg)
  1932   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)
  1933  apply (simp)
  1934 apply (simp)
  1935 done
  1936 
  1937 lemma zmult2_lemma_aux2:
  1938      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
  1939 apply (subgoal_tac "b * (q mod c) \<le> 0")
  1940  apply arith
  1941 apply (simp add: mult_le_0_iff)
  1942 done
  1943 
  1944 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
  1945 apply (subgoal_tac "0 \<le> b * (q mod c) ")
  1946 apply arith
  1947 apply (simp add: zero_le_mult_iff)
  1948 done
  1949 
  1950 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
  1951 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
  1952  apply (simp add: right_diff_distrib)
  1953 apply (rule order_less_le_trans)
  1954  apply (erule mult_strict_right_mono)
  1955  apply (rule_tac [2] mult_left_mono)
  1956   apply simp
  1957  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)
  1958 apply simp
  1959 done
  1960 
  1961 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r);  b \<noteq> 0;  0 < c |]  
  1962       ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
  1963 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
  1964                    zero_less_mult_iff right_distrib [symmetric] 
  1965                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
  1966 
  1967 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
  1968 apply (case_tac "b = 0", simp)
  1969 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_div])
  1970 done
  1971 
  1972 lemma zmod_zmult2_eq:
  1973      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
  1974 apply (case_tac "b = 0", simp)
  1975 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_mod])
  1976 done
  1977 
  1978 
  1979 subsubsection {*Splitting Rules for div and mod*}
  1980 
  1981 text{*The proofs of the two lemmas below are essentially identical*}
  1982 
  1983 lemma split_pos_lemma:
  1984  "0<k ==> 
  1985     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
  1986 apply (rule iffI, clarify)
  1987  apply (erule_tac P="P ?x ?y" in rev_mp)  
  1988  apply (subst mod_add_eq) 
  1989  apply (subst zdiv_zadd1_eq) 
  1990  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
  1991 txt{*converse direction*}
  1992 apply (drule_tac x = "n div k" in spec) 
  1993 apply (drule_tac x = "n mod k" in spec, simp)
  1994 done
  1995 
  1996 lemma split_neg_lemma:
  1997  "k<0 ==>
  1998     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
  1999 apply (rule iffI, clarify)
  2000  apply (erule_tac P="P ?x ?y" in rev_mp)  
  2001  apply (subst mod_add_eq) 
  2002  apply (subst zdiv_zadd1_eq) 
  2003  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
  2004 txt{*converse direction*}
  2005 apply (drule_tac x = "n div k" in spec) 
  2006 apply (drule_tac x = "n mod k" in spec, simp)
  2007 done
  2008 
  2009 lemma split_zdiv:
  2010  "P(n div k :: int) =
  2011   ((k = 0 --> P 0) & 
  2012    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
  2013    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
  2014 apply (case_tac "k=0", simp)
  2015 apply (simp only: linorder_neq_iff)
  2016 apply (erule disjE) 
  2017  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
  2018                       split_neg_lemma [of concl: "%x y. P x"])
  2019 done
  2020 
  2021 lemma split_zmod:
  2022  "P(n mod k :: int) =
  2023   ((k = 0 --> P n) & 
  2024    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
  2025    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
  2026 apply (case_tac "k=0", simp)
  2027 apply (simp only: linorder_neq_iff)
  2028 apply (erule disjE) 
  2029  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
  2030                       split_neg_lemma [of concl: "%x y. P y"])
  2031 done
  2032 
  2033 text {* Enable (lin)arith to deal with @{const div} and @{const mod}
  2034   when these are applied to some constant that is of the form
  2035   @{term "number_of k"}: *}
  2036 declare split_zdiv [of _ _ "number_of k", standard, arith_split]
  2037 declare split_zmod [of _ _ "number_of k", standard, arith_split]
  2038 
  2039 
  2040 subsubsection{*Speeding up the Division Algorithm with Shifting*}
  2041 
  2042 text{*computing div by shifting *}
  2043 
  2044 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
  2045 proof cases
  2046   assume "a=0"
  2047     thus ?thesis by simp
  2048 next
  2049   assume "a\<noteq>0" and le_a: "0\<le>a"   
  2050   hence a_pos: "1 \<le> a" by arith
  2051   hence one_less_a2: "1 < 2 * a" by arith
  2052   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
  2053     unfolding mult_le_cancel_left
  2054     by (simp add: add1_zle_eq add_commute [of 1])
  2055   with a_pos have "0 \<le> b mod a" by simp
  2056   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
  2057     by (simp add: mod_pos_pos_trivial one_less_a2)
  2058   with  le_2a
  2059   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
  2060     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
  2061                   right_distrib) 
  2062   thus ?thesis
  2063     by (subst zdiv_zadd1_eq,
  2064         simp add: mod_mult_mult1 one_less_a2
  2065                   div_pos_pos_trivial)
  2066 qed
  2067 
  2068 lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
  2069 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
  2070 apply (rule_tac [2] pos_zdiv_mult_2)
  2071 apply (auto simp add: right_diff_distrib)
  2072 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
  2073 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric])
  2074 apply (simp_all add: algebra_simps)
  2075 apply (simp only: ab_diff_minus minus_add_distrib [symmetric] number_of_Min zdiv_zminus_zminus)
  2076 done
  2077 
  2078 lemma zdiv_number_of_Bit0 [simp]:
  2079      "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  
  2080           number_of v div (number_of w :: int)"
  2081 by (simp only: number_of_eq numeral_simps) (simp add: mult_2 [symmetric])
  2082 
  2083 lemma zdiv_number_of_Bit1 [simp]:
  2084      "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  
  2085           (if (0::int) \<le> number_of w                    
  2086            then number_of v div (number_of w)     
  2087            else (number_of v + (1::int)) div (number_of w))"
  2088 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
  2089 apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac mult_2 [symmetric])
  2090 done
  2091 
  2092 
  2093 subsubsection{*Computing mod by Shifting (proofs resemble those for div)*}
  2094 
  2095 lemma pos_zmod_mult_2:
  2096   fixes a b :: int
  2097   assumes "0 \<le> a"
  2098   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
  2099 proof (cases "0 < a")
  2100   case False with assms show ?thesis by simp
  2101 next
  2102   case True
  2103   then have "b mod a < a" by (rule pos_mod_bound)
  2104   then have "1 + b mod a \<le> a" by simp
  2105   then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp
  2106   from `0 < a` have "0 \<le> b mod a" by (rule pos_mod_sign)
  2107   then have B: "0 \<le> 1 + 2 * (b mod a)" by simp
  2108   have "((1\<Colon>int) mod ((2\<Colon>int) * a) + (2\<Colon>int) * b mod ((2\<Colon>int) * a)) mod ((2\<Colon>int) * a) = (1\<Colon>int) + (2\<Colon>int) * (b mod a)"
  2109     using `0 < a` and A
  2110     by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)
  2111   then show ?thesis by (subst mod_add_eq)
  2112 qed
  2113 
  2114 lemma neg_zmod_mult_2:
  2115   fixes a b :: int
  2116   assumes "a \<le> 0"
  2117   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
  2118 proof -
  2119   from assms have "0 \<le> - a" by auto
  2120   then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"
  2121     by (rule pos_zmod_mult_2)
  2122   then show ?thesis by (simp add: zmod_zminus2 algebra_simps)
  2123      (simp add: diff_minus add_ac)
  2124 qed
  2125 
  2126 lemma zmod_number_of_Bit0 [simp]:
  2127      "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  
  2128       (2::int) * (number_of v mod number_of w)"
  2129 apply (simp only: number_of_eq numeral_simps) 
  2130 apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
  2131                  neg_zmod_mult_2 add_ac mult_2 [symmetric])
  2132 done
  2133 
  2134 lemma zmod_number_of_Bit1 [simp]:
  2135      "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  
  2136       (if (0::int) \<le> number_of w  
  2137                 then 2 * (number_of v mod number_of w) + 1     
  2138                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
  2139 apply (simp only: number_of_eq numeral_simps) 
  2140 apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
  2141                  neg_zmod_mult_2 add_ac mult_2 [symmetric])
  2142 done
  2143 
  2144 
  2145 subsubsection{*Quotients of Signs*}
  2146 
  2147 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
  2148 apply (subgoal_tac "a div b \<le> -1", force)
  2149 apply (rule order_trans)
  2150 apply (rule_tac a' = "-1" in zdiv_mono1)
  2151 apply (auto simp add: div_eq_minus1)
  2152 done
  2153 
  2154 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
  2155 by (drule zdiv_mono1_neg, auto)
  2156 
  2157 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
  2158 by (drule zdiv_mono1, auto)
  2159 
  2160 text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}
  2161 conditional upon the sign of @{text a} or @{text b}. There are many more.
  2162 They should all be simp rules unless that causes too much search. *}
  2163 
  2164 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
  2165 apply auto
  2166 apply (drule_tac [2] zdiv_mono1)
  2167 apply (auto simp add: linorder_neq_iff)
  2168 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
  2169 apply (blast intro: div_neg_pos_less0)
  2170 done
  2171 
  2172 lemma neg_imp_zdiv_nonneg_iff:
  2173   "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
  2174 apply (subst zdiv_zminus_zminus [symmetric])
  2175 apply (subst pos_imp_zdiv_nonneg_iff, auto)
  2176 done
  2177 
  2178 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
  2179 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
  2180 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
  2181 
  2182 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
  2183 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
  2184 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
  2185 
  2186 lemma nonneg1_imp_zdiv_pos_iff:
  2187   "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
  2188 apply rule
  2189  apply rule
  2190   using div_pos_pos_trivial[of a b]apply arith
  2191  apply(cases "b=0")apply simp
  2192  using div_nonneg_neg_le0[of a b]apply arith
  2193 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
  2194 done
  2195 
  2196 
  2197 subsubsection {* The Divides Relation *}
  2198 
  2199 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
  2200   dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]
  2201 
  2202 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
  2203   by (rule dvd_mod) (* TODO: remove *)
  2204 
  2205 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
  2206   by (rule dvd_mod_imp_dvd) (* TODO: remove *)
  2207 
  2208 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
  2209   using zmod_zdiv_equality[where a="m" and b="n"]
  2210   by (simp add: algebra_simps)
  2211 
  2212 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
  2213 apply (induct "y", auto)
  2214 apply (rule zmod_zmult1_eq [THEN trans])
  2215 apply (simp (no_asm_simp))
  2216 apply (rule mod_mult_eq [symmetric])
  2217 done
  2218 
  2219 lemma zdiv_int: "int (a div b) = (int a) div (int b)"
  2220 apply (subst split_div, auto)
  2221 apply (subst split_zdiv, auto)
  2222 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)
  2223 apply (auto simp add: divmod_int_rel_def of_nat_mult)
  2224 done
  2225 
  2226 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
  2227 apply (subst split_mod, auto)
  2228 apply (subst split_zmod, auto)
  2229 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
  2230        in unique_remainder)
  2231 apply (auto simp add: divmod_int_rel_def of_nat_mult)
  2232 done
  2233 
  2234 lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
  2235 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
  2236 
  2237 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
  2238 apply (subgoal_tac "m mod n = 0")
  2239  apply (simp add: zmult_div_cancel)
  2240 apply (simp only: dvd_eq_mod_eq_0)
  2241 done
  2242 
  2243 text{*Suggested by Matthias Daum*}
  2244 lemma int_power_div_base:
  2245      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
  2246 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
  2247  apply (erule ssubst)
  2248  apply (simp only: power_add)
  2249  apply simp_all
  2250 done
  2251 
  2252 text {* by Brian Huffman *}
  2253 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
  2254 by (rule mod_minus_eq [symmetric])
  2255 
  2256 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
  2257 by (rule mod_diff_left_eq [symmetric])
  2258 
  2259 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
  2260 by (rule mod_diff_right_eq [symmetric])
  2261 
  2262 lemmas zmod_simps =
  2263   mod_add_left_eq  [symmetric]
  2264   mod_add_right_eq [symmetric]
  2265   zmod_zmult1_eq   [symmetric]
  2266   mod_mult_left_eq [symmetric]
  2267   zpower_zmod
  2268   zminus_zmod zdiff_zmod_left zdiff_zmod_right
  2269 
  2270 text {* Distributive laws for function @{text nat}. *}
  2271 
  2272 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
  2273 apply (rule linorder_cases [of y 0])
  2274 apply (simp add: div_nonneg_neg_le0)
  2275 apply simp
  2276 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
  2277 done
  2278 
  2279 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
  2280 lemma nat_mod_distrib:
  2281   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
  2282 apply (case_tac "y = 0", simp)
  2283 apply (simp add: nat_eq_iff zmod_int)
  2284 done
  2285 
  2286 text  {* transfer setup *}
  2287 
  2288 lemma transfer_nat_int_functions:
  2289     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
  2290     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
  2291   by (auto simp add: nat_div_distrib nat_mod_distrib)
  2292 
  2293 lemma transfer_nat_int_function_closures:
  2294     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
  2295     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
  2296   apply (cases "y = 0")
  2297   apply (auto simp add: pos_imp_zdiv_nonneg_iff)
  2298   apply (cases "y = 0")
  2299   apply auto
  2300 done
  2301 
  2302 declare TransferMorphism_nat_int [transfer add return:
  2303   transfer_nat_int_functions
  2304   transfer_nat_int_function_closures
  2305 ]
  2306 
  2307 lemma transfer_int_nat_functions:
  2308     "(int x) div (int y) = int (x div y)"
  2309     "(int x) mod (int y) = int (x mod y)"
  2310   by (auto simp add: zdiv_int zmod_int)
  2311 
  2312 lemma transfer_int_nat_function_closures:
  2313     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
  2314     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
  2315   by (simp_all only: is_nat_def transfer_nat_int_function_closures)
  2316 
  2317 declare TransferMorphism_int_nat [transfer add return:
  2318   transfer_int_nat_functions
  2319   transfer_int_nat_function_closures
  2320 ]
  2321 
  2322 text{*Suggested by Matthias Daum*}
  2323 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
  2324 apply (subgoal_tac "nat x div nat k < nat x")
  2325  apply (simp (asm_lr) add: nat_div_distrib [symmetric])
  2326 apply (rule Divides.div_less_dividend, simp_all)
  2327 done
  2328 
  2329 text {* code generator setup *}
  2330 
  2331 context ring_1
  2332 begin
  2333 
  2334 lemma of_int_num [code]:
  2335   "of_int k = (if k = 0 then 0 else if k < 0 then
  2336      - of_int (- k) else let
  2337        (l, m) = divmod_int k 2;
  2338        l' = of_int l
  2339      in if m = 0 then l' + l' else l' + l' + 1)"
  2340 proof -
  2341   have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> 
  2342     of_int k = of_int (k div 2 * 2 + 1)"
  2343   proof -
  2344     have "k mod 2 < 2" by (auto intro: pos_mod_bound)
  2345     moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)
  2346     moreover assume "k mod 2 \<noteq> 0"
  2347     ultimately have "k mod 2 = 1" by arith
  2348     moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
  2349     ultimately show ?thesis by auto
  2350   qed
  2351   have aux2: "\<And>x. of_int 2 * x = x + x"
  2352   proof -
  2353     fix x
  2354     have int2: "(2::int) = 1 + 1" by arith
  2355     show "of_int 2 * x = x + x"
  2356     unfolding int2 of_int_add left_distrib by simp
  2357   qed
  2358   have aux3: "\<And>x. x * of_int 2 = x + x"
  2359   proof -
  2360     fix x
  2361     have int2: "(2::int) = 1 + 1" by arith
  2362     show "x * of_int 2 = x + x" 
  2363     unfolding int2 of_int_add right_distrib by simp
  2364   qed
  2365   from aux1 show ?thesis by (auto simp add: divmod_int_mod_div Let_def aux2 aux3)
  2366 qed
  2367 
  2368 end
  2369 
  2370 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
  2371 proof
  2372   assume H: "x mod n = y mod n"
  2373   hence "x mod n - y mod n = 0" by simp
  2374   hence "(x mod n - y mod n) mod n = 0" by simp 
  2375   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
  2376   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
  2377 next
  2378   assume H: "n dvd x - y"
  2379   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
  2380   hence "x = n*k + y" by simp
  2381   hence "x mod n = (n*k + y) mod n" by simp
  2382   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
  2383 qed
  2384 
  2385 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
  2386   shows "\<exists>q. x = y + n * q"
  2387 proof-
  2388   from xy have th: "int x - int y = int (x - y)" by simp 
  2389   from xyn have "int x mod int n = int y mod int n" 
  2390     by (simp add: zmod_int[symmetric])
  2391   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
  2392   hence "n dvd x - y" by (simp add: th zdvd_int)
  2393   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
  2394 qed
  2395 
  2396 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
  2397   (is "?lhs = ?rhs")
  2398 proof
  2399   assume H: "x mod n = y mod n"
  2400   {assume xy: "x \<le> y"
  2401     from H have th: "y mod n = x mod n" by simp
  2402     from nat_mod_eq_lemma[OF th xy] have ?rhs 
  2403       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
  2404   moreover
  2405   {assume xy: "y \<le> x"
  2406     from nat_mod_eq_lemma[OF H xy] have ?rhs 
  2407       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
  2408   ultimately  show ?rhs using linear[of x y] by blast  
  2409 next
  2410   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
  2411   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
  2412   thus  ?lhs by simp
  2413 qed
  2414 
  2415 lemma div_nat_number_of [simp]:
  2416      "(number_of v :: nat)  div  number_of v' =  
  2417           (if neg (number_of v :: int) then 0  
  2418            else nat (number_of v div number_of v'))"
  2419   unfolding nat_number_of_def number_of_is_id neg_def
  2420   by (simp add: nat_div_distrib)
  2421 
  2422 lemma one_div_nat_number_of [simp]:
  2423      "Suc 0 div number_of v' = nat (1 div number_of v')" 
  2424 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
  2425 
  2426 lemma mod_nat_number_of [simp]:
  2427      "(number_of v :: nat)  mod  number_of v' =  
  2428         (if neg (number_of v :: int) then 0  
  2429          else if neg (number_of v' :: int) then number_of v  
  2430          else nat (number_of v mod number_of v'))"
  2431   unfolding nat_number_of_def number_of_is_id neg_def
  2432   by (simp add: nat_mod_distrib)
  2433 
  2434 lemma one_mod_nat_number_of [simp]:
  2435      "Suc 0 mod number_of v' =  
  2436         (if neg (number_of v' :: int) then Suc 0
  2437          else nat (1 mod number_of v'))"
  2438 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
  2439 
  2440 lemmas dvd_eq_mod_eq_0_number_of =
  2441   dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
  2442 
  2443 declare dvd_eq_mod_eq_0_number_of [simp]
  2444 
  2445 
  2446 subsubsection {* Code generation *}
  2447 
  2448 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  2449   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
  2450 
  2451 lemma pdivmod_posDivAlg [code]:
  2452   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
  2453 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
  2454 
  2455 lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
  2456   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
  2457     then pdivmod k l
  2458     else (let (r, s) = pdivmod k l in
  2459       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
  2460 proof -
  2461   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
  2462   show ?thesis
  2463     by (simp add: divmod_int_mod_div pdivmod_def)
  2464       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
  2465       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
  2466 qed
  2467 
  2468 lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
  2469   apsnd ((op *) (sgn l)) (if sgn k = sgn l
  2470     then pdivmod k l
  2471     else (let (r, s) = pdivmod k l in
  2472       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
  2473 proof -
  2474   have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
  2475     by (auto simp add: not_less sgn_if)
  2476   then show ?thesis by (simp add: divmod_int_pdivmod)
  2477 qed
  2478 
  2479 code_modulename SML
  2480   Divides Arith
  2481 
  2482 code_modulename OCaml
  2483   Divides Arith
  2484 
  2485 code_modulename Haskell
  2486   Divides Arith
  2487 
  2488 end