src/HOL/Divides.thy
author huffman
Tue Mar 27 10:34:12 2012 +0200 (2012-03-27)
changeset 47137 7f5f0531cae6
parent 47136 5b6c5641498a
child 47138 f8cf96545eed
permissions -rw-r--r--
shorten a proof
     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]: "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(fastforce 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_swap:
   313   assumes "c dvd b"
   314   shows "a * (b div c) = (a * b) div c"
   315 proof -
   316   from assms have "b div c * (a div 1) = b * a div (c * 1)"
   317     by (simp only: div_mult_div_if_dvd one_dvd)
   318   then show ?thesis by (simp add: mult_commute)
   319 qed
   320    
   321 lemma div_mult_mult2 [simp]:
   322   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
   323   by (drule div_mult_mult1) (simp add: mult_commute)
   324 
   325 lemma div_mult_mult1_if [simp]:
   326   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
   327   by simp_all
   328 
   329 lemma mod_mult_mult1:
   330   "(c * a) mod (c * b) = c * (a mod b)"
   331 proof (cases "c = 0")
   332   case True then show ?thesis by simp
   333 next
   334   case False
   335   from mod_div_equality
   336   have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
   337   with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
   338     = c * a + c * (a mod b)" by (simp add: algebra_simps)
   339   with mod_div_equality show ?thesis by simp 
   340 qed
   341   
   342 lemma mod_mult_mult2:
   343   "(a * c) mod (b * c) = (a mod b) * c"
   344   using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
   345 
   346 lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
   347   unfolding dvd_def by (auto simp add: mod_mult_mult1)
   348 
   349 lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
   350 by (blast intro: dvd_mod_imp_dvd dvd_mod)
   351 
   352 lemma div_power:
   353   "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
   354 apply (induct n)
   355  apply simp
   356 apply(simp add: div_mult_div_if_dvd dvd_power_same)
   357 done
   358 
   359 lemma dvd_div_eq_mult:
   360   assumes "a \<noteq> 0" and "a dvd b"  
   361   shows "b div a = c \<longleftrightarrow> b = c * a"
   362 proof
   363   assume "b = c * a"
   364   then show "b div a = c" by (simp add: assms)
   365 next
   366   assume "b div a = c"
   367   then have "b div a * a = c * a" by simp
   368   moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self)
   369   ultimately show "b = c * a" by simp
   370 qed
   371    
   372 lemma dvd_div_div_eq_mult:
   373   assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
   374   shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
   375   using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)
   376 
   377 end
   378 
   379 class ring_div = semiring_div + comm_ring_1
   380 begin
   381 
   382 subclass ring_1_no_zero_divisors ..
   383 
   384 text {* Negation respects modular equivalence. *}
   385 
   386 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
   387 proof -
   388   have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
   389     by (simp only: mod_div_equality)
   390   also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
   391     by (simp only: minus_add_distrib minus_mult_left add_ac)
   392   also have "\<dots> = (- (a mod b)) mod b"
   393     by (rule mod_mult_self1)
   394   finally show ?thesis .
   395 qed
   396 
   397 lemma mod_minus_cong:
   398   assumes "a mod b = a' mod b"
   399   shows "(- a) mod b = (- a') mod b"
   400 proof -
   401   have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
   402     unfolding assms ..
   403   thus ?thesis
   404     by (simp only: mod_minus_eq [symmetric])
   405 qed
   406 
   407 text {* Subtraction respects modular equivalence. *}
   408 
   409 lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
   410   unfolding diff_minus
   411   by (intro mod_add_cong mod_minus_cong) simp_all
   412 
   413 lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
   414   unfolding diff_minus
   415   by (intro mod_add_cong mod_minus_cong) simp_all
   416 
   417 lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
   418   unfolding diff_minus
   419   by (intro mod_add_cong mod_minus_cong) simp_all
   420 
   421 lemma mod_diff_cong:
   422   assumes "a mod c = a' mod c"
   423   assumes "b mod c = b' mod c"
   424   shows "(a - b) mod c = (a' - b') mod c"
   425   unfolding diff_minus using assms
   426   by (intro mod_add_cong mod_minus_cong)
   427 
   428 lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
   429 apply (case_tac "y = 0") apply simp
   430 apply (auto simp add: dvd_def)
   431 apply (subgoal_tac "-(y * k) = y * - k")
   432  apply (erule ssubst)
   433  apply (erule div_mult_self1_is_id)
   434 apply simp
   435 done
   436 
   437 lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
   438 apply (case_tac "y = 0") apply simp
   439 apply (auto simp add: dvd_def)
   440 apply (subgoal_tac "y * k = -y * -k")
   441  apply (erule ssubst)
   442  apply (rule div_mult_self1_is_id)
   443  apply simp
   444 apply simp
   445 done
   446 
   447 end
   448 
   449 
   450 subsection {* Division on @{typ nat} *}
   451 
   452 text {*
   453   We define @{const div} and @{const mod} on @{typ nat} by means
   454   of a characteristic relation with two input arguments
   455   @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
   456   @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
   457 *}
   458 
   459 definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
   460   "divmod_nat_rel m n qr \<longleftrightarrow>
   461     m = fst qr * n + snd qr \<and>
   462       (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)"
   463 
   464 text {* @{const divmod_nat_rel} is total: *}
   465 
   466 lemma divmod_nat_rel_ex:
   467   obtains q r where "divmod_nat_rel m n (q, r)"
   468 proof (cases "n = 0")
   469   case True  with that show thesis
   470     by (auto simp add: divmod_nat_rel_def)
   471 next
   472   case False
   473   have "\<exists>q r. m = q * n + r \<and> r < n"
   474   proof (induct m)
   475     case 0 with `n \<noteq> 0`
   476     have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
   477     then show ?case by blast
   478   next
   479     case (Suc m) then obtain q' r'
   480       where m: "m = q' * n + r'" and n: "r' < n" by auto
   481     then show ?case proof (cases "Suc r' < n")
   482       case True
   483       from m n have "Suc m = q' * n + Suc r'" by simp
   484       with True show ?thesis by blast
   485     next
   486       case False then have "n \<le> Suc r'" by auto
   487       moreover from n have "Suc r' \<le> n" by auto
   488       ultimately have "n = Suc r'" by auto
   489       with m have "Suc m = Suc q' * n + 0" by simp
   490       with `n \<noteq> 0` show ?thesis by blast
   491     qed
   492   qed
   493   with that show thesis
   494     using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
   495 qed
   496 
   497 text {* @{const divmod_nat_rel} is injective: *}
   498 
   499 lemma divmod_nat_rel_unique:
   500   assumes "divmod_nat_rel m n qr"
   501     and "divmod_nat_rel m n qr'"
   502   shows "qr = qr'"
   503 proof (cases "n = 0")
   504   case True with assms show ?thesis
   505     by (cases qr, cases qr')
   506       (simp add: divmod_nat_rel_def)
   507 next
   508   case False
   509   have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
   510   apply (rule leI)
   511   apply (subst less_iff_Suc_add)
   512   apply (auto simp add: add_mult_distrib)
   513   done
   514   from `n \<noteq> 0` assms have "fst qr = fst qr'"
   515     by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
   516   moreover from this assms have "snd qr = snd qr'"
   517     by (simp add: divmod_nat_rel_def)
   518   ultimately show ?thesis by (cases qr, cases qr') simp
   519 qed
   520 
   521 text {*
   522   We instantiate divisibility on the natural numbers by
   523   means of @{const divmod_nat_rel}:
   524 *}
   525 
   526 definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
   527   "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
   528 
   529 lemma divmod_nat_rel_divmod_nat:
   530   "divmod_nat_rel m n (divmod_nat m n)"
   531 proof -
   532   from divmod_nat_rel_ex
   533     obtain qr where rel: "divmod_nat_rel m n qr" .
   534   then show ?thesis
   535   by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
   536 qed
   537 
   538 lemma divmod_nat_unique:
   539   assumes "divmod_nat_rel m n qr" 
   540   shows "divmod_nat m n = qr"
   541   using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
   542 
   543 instantiation nat :: semiring_div
   544 begin
   545 
   546 definition div_nat where
   547   "m div n = fst (divmod_nat m n)"
   548 
   549 lemma fst_divmod_nat [simp]:
   550   "fst (divmod_nat m n) = m div n"
   551   by (simp add: div_nat_def)
   552 
   553 definition mod_nat where
   554   "m mod n = snd (divmod_nat m n)"
   555 
   556 lemma snd_divmod_nat [simp]:
   557   "snd (divmod_nat m n) = m mod n"
   558   by (simp add: mod_nat_def)
   559 
   560 lemma divmod_nat_div_mod:
   561   "divmod_nat m n = (m div n, m mod n)"
   562   by (simp add: prod_eq_iff)
   563 
   564 lemma div_nat_unique:
   565   assumes "divmod_nat_rel m n (q, r)" 
   566   shows "m div n = q"
   567   using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
   568 
   569 lemma mod_nat_unique:
   570   assumes "divmod_nat_rel m n (q, r)" 
   571   shows "m mod n = r"
   572   using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
   573 
   574 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
   575   using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
   576 
   577 lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
   578   by (simp add: divmod_nat_unique divmod_nat_rel_def)
   579 
   580 lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
   581   by (simp add: divmod_nat_unique divmod_nat_rel_def)
   582 
   583 lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
   584   by (simp add: divmod_nat_unique divmod_nat_rel_def)
   585 
   586 lemma divmod_nat_step:
   587   assumes "0 < n" and "n \<le> m"
   588   shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
   589 proof (rule divmod_nat_unique)
   590   have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"
   591     by (rule divmod_nat_rel)
   592   thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"
   593     unfolding divmod_nat_rel_def using assms by auto
   594 qed
   595 
   596 text {* The ''recursion'' equations for @{const div} and @{const mod} *}
   597 
   598 lemma div_less [simp]:
   599   fixes m n :: nat
   600   assumes "m < n"
   601   shows "m div n = 0"
   602   using assms divmod_nat_base by (simp add: prod_eq_iff)
   603 
   604 lemma le_div_geq:
   605   fixes m n :: nat
   606   assumes "0 < n" and "n \<le> m"
   607   shows "m div n = Suc ((m - n) div n)"
   608   using assms divmod_nat_step by (simp add: prod_eq_iff)
   609 
   610 lemma mod_less [simp]:
   611   fixes m n :: nat
   612   assumes "m < n"
   613   shows "m mod n = m"
   614   using assms divmod_nat_base by (simp add: prod_eq_iff)
   615 
   616 lemma le_mod_geq:
   617   fixes m n :: nat
   618   assumes "n \<le> m"
   619   shows "m mod n = (m - n) mod n"
   620   using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
   621 
   622 instance proof
   623   fix m n :: nat
   624   show "m div n * n + m mod n = m"
   625     using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
   626 next
   627   fix m n q :: nat
   628   assume "n \<noteq> 0"
   629   then show "(q + m * n) div n = m + q div n"
   630     by (induct m) (simp_all add: le_div_geq)
   631 next
   632   fix m n q :: nat
   633   assume "m \<noteq> 0"
   634   hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
   635     unfolding divmod_nat_rel_def
   636     by (auto split: split_if_asm, simp_all add: algebra_simps)
   637   moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
   638   ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
   639   thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
   640 next
   641   fix n :: nat show "n div 0 = 0"
   642     by (simp add: div_nat_def divmod_nat_zero)
   643 next
   644   fix n :: nat show "0 div n = 0"
   645     by (simp add: div_nat_def divmod_nat_zero_left)
   646 qed
   647 
   648 end
   649 
   650 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
   651   let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
   652   by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)
   653 
   654 text {* Simproc for cancelling @{const div} and @{const mod} *}
   655 
   656 ML {*
   657 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
   658 (
   659   val div_name = @{const_name div};
   660   val mod_name = @{const_name mod};
   661   val mk_binop = HOLogic.mk_binop;
   662   val mk_sum = Nat_Arith.mk_sum;
   663   val dest_sum = Nat_Arith.dest_sum;
   664 
   665   val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
   666 
   667   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
   668     (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))
   669 )
   670 *}
   671 
   672 simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}
   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)
   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 by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
   745 
   746 lemma divmod_nat_rel_add1_eq:
   747   "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
   748    \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
   749 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
   750 
   751 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
   752 lemma div_add1_eq:
   753   "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
   754 by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
   755 
   756 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
   757   apply (cut_tac m = q and n = c in mod_less_divisor)
   758   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
   759   apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
   760   apply (simp add: add_mult_distrib2)
   761   done
   762 
   763 lemma divmod_nat_rel_mult2_eq:
   764   "divmod_nat_rel a b (q, r)
   765    \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
   766 by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)
   767 
   768 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
   769 by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
   770 
   771 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
   772 by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
   773 
   774 
   775 subsubsection {* Further Facts about Quotient and Remainder *}
   776 
   777 lemma div_1 [simp]: "m div Suc 0 = m"
   778 by (induct m) (simp_all add: div_geq)
   779 
   780 (* Monotonicity of div in first argument *)
   781 lemma div_le_mono [rule_format (no_asm)]:
   782     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
   783 apply (case_tac "k=0", simp)
   784 apply (induct "n" rule: nat_less_induct, clarify)
   785 apply (case_tac "n<k")
   786 (* 1  case n<k *)
   787 apply simp
   788 (* 2  case n >= k *)
   789 apply (case_tac "m<k")
   790 (* 2.1  case m<k *)
   791 apply simp
   792 (* 2.2  case m>=k *)
   793 apply (simp add: div_geq diff_le_mono)
   794 done
   795 
   796 (* Antimonotonicity of div in second argument *)
   797 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
   798 apply (subgoal_tac "0<n")
   799  prefer 2 apply simp
   800 apply (induct_tac k rule: nat_less_induct)
   801 apply (rename_tac "k")
   802 apply (case_tac "k<n", simp)
   803 apply (subgoal_tac "~ (k<m) ")
   804  prefer 2 apply simp
   805 apply (simp add: div_geq)
   806 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
   807  prefer 2
   808  apply (blast intro: div_le_mono diff_le_mono2)
   809 apply (rule le_trans, simp)
   810 apply (simp)
   811 done
   812 
   813 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
   814 apply (case_tac "n=0", simp)
   815 apply (subgoal_tac "m div n \<le> m div 1", simp)
   816 apply (rule div_le_mono2)
   817 apply (simp_all (no_asm_simp))
   818 done
   819 
   820 (* Similar for "less than" *)
   821 lemma div_less_dividend [rule_format]:
   822      "!!n::nat. 1<n ==> 0 < m --> m div n < m"
   823 apply (induct_tac m rule: nat_less_induct)
   824 apply (rename_tac "m")
   825 apply (case_tac "m<n", simp)
   826 apply (subgoal_tac "0<n")
   827  prefer 2 apply simp
   828 apply (simp add: div_geq)
   829 apply (case_tac "n<m")
   830  apply (subgoal_tac "(m-n) div n < (m-n) ")
   831   apply (rule impI less_trans_Suc)+
   832 apply assumption
   833   apply (simp_all)
   834 done
   835 
   836 declare div_less_dividend [simp]
   837 
   838 text{*A fact for the mutilated chess board*}
   839 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
   840 apply (case_tac "n=0", simp)
   841 apply (induct "m" rule: nat_less_induct)
   842 apply (case_tac "Suc (na) <n")
   843 (* case Suc(na) < n *)
   844 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
   845 (* case n \<le> Suc(na) *)
   846 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
   847 apply (auto simp add: Suc_diff_le le_mod_geq)
   848 done
   849 
   850 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
   851 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
   852 
   853 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
   854 
   855 (*Loses information, namely we also have r<d provided d is nonzero*)
   856 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
   857   apply (cut_tac a = m in mod_div_equality)
   858   apply (simp only: add_ac)
   859   apply (blast intro: sym)
   860   done
   861 
   862 lemma split_div:
   863  "P(n div k :: nat) =
   864  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
   865  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   866 proof
   867   assume P: ?P
   868   show ?Q
   869   proof (cases)
   870     assume "k = 0"
   871     with P show ?Q by simp
   872   next
   873     assume not0: "k \<noteq> 0"
   874     thus ?Q
   875     proof (simp, intro allI impI)
   876       fix i j
   877       assume n: "n = k*i + j" and j: "j < k"
   878       show "P i"
   879       proof (cases)
   880         assume "i = 0"
   881         with n j P show "P i" by simp
   882       next
   883         assume "i \<noteq> 0"
   884         with not0 n j P show "P i" by(simp add:add_ac)
   885       qed
   886     qed
   887   qed
   888 next
   889   assume Q: ?Q
   890   show ?P
   891   proof (cases)
   892     assume "k = 0"
   893     with Q show ?P by simp
   894   next
   895     assume not0: "k \<noteq> 0"
   896     with Q have R: ?R by simp
   897     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   898     show ?P by simp
   899   qed
   900 qed
   901 
   902 lemma split_div_lemma:
   903   assumes "0 < n"
   904   shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
   905 proof
   906   assume ?rhs
   907   with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
   908   then have A: "n * q \<le> m" by simp
   909   have "n - (m mod n) > 0" using mod_less_divisor assms by auto
   910   then have "m < m + (n - (m mod n))" by simp
   911   then have "m < n + (m - (m mod n))" by simp
   912   with nq have "m < n + n * q" by simp
   913   then have B: "m < n * Suc q" by simp
   914   from A B show ?lhs ..
   915 next
   916   assume P: ?lhs
   917   then have "divmod_nat_rel m n (q, m - n * q)"
   918     unfolding divmod_nat_rel_def by (auto simp add: mult_ac)
   919   with divmod_nat_rel_unique divmod_nat_rel [of m n]
   920   have "(q, m - n * q) = (m div n, m mod n)" by auto
   921   then show ?rhs by simp
   922 qed
   923 
   924 theorem split_div':
   925   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
   926    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
   927   apply (case_tac "0 < n")
   928   apply (simp only: add: split_div_lemma)
   929   apply simp_all
   930   done
   931 
   932 lemma split_mod:
   933  "P(n mod k :: nat) =
   934  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
   935  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   936 proof
   937   assume P: ?P
   938   show ?Q
   939   proof (cases)
   940     assume "k = 0"
   941     with P show ?Q by simp
   942   next
   943     assume not0: "k \<noteq> 0"
   944     thus ?Q
   945     proof (simp, intro allI impI)
   946       fix i j
   947       assume "n = k*i + j" "j < k"
   948       thus "P j" using not0 P by(simp add:add_ac mult_ac)
   949     qed
   950   qed
   951 next
   952   assume Q: ?Q
   953   show ?P
   954   proof (cases)
   955     assume "k = 0"
   956     with Q show ?P by simp
   957   next
   958     assume not0: "k \<noteq> 0"
   959     with Q have R: ?R by simp
   960     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   961     show ?P by simp
   962   qed
   963 qed
   964 
   965 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
   966   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
   967     subst [OF mod_div_equality [of _ n]])
   968   apply arith
   969   done
   970 
   971 lemma div_mod_equality':
   972   fixes m n :: nat
   973   shows "m div n * n = m - m mod n"
   974 proof -
   975   have "m mod n \<le> m mod n" ..
   976   from div_mod_equality have 
   977     "m div n * n + m mod n - m mod n = m - m mod n" by simp
   978   with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
   979     "m div n * n + (m mod n - m mod n) = m - m mod n"
   980     by simp
   981   then show ?thesis by simp
   982 qed
   983 
   984 
   985 subsubsection {* An ``induction'' law for modulus arithmetic. *}
   986 
   987 lemma mod_induct_0:
   988   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
   989   and base: "P i" and i: "i<p"
   990   shows "P 0"
   991 proof (rule ccontr)
   992   assume contra: "\<not>(P 0)"
   993   from i have p: "0<p" by simp
   994   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
   995   proof
   996     fix k
   997     show "?A k"
   998     proof (induct k)
   999       show "?A 0" by simp  -- "by contradiction"
  1000     next
  1001       fix n
  1002       assume ih: "?A n"
  1003       show "?A (Suc n)"
  1004       proof (clarsimp)
  1005         assume y: "P (p - Suc n)"
  1006         have n: "Suc n < p"
  1007         proof (rule ccontr)
  1008           assume "\<not>(Suc n < p)"
  1009           hence "p - Suc n = 0"
  1010             by simp
  1011           with y contra show "False"
  1012             by simp
  1013         qed
  1014         hence n2: "Suc (p - Suc n) = p-n" by arith
  1015         from p have "p - Suc n < p" by arith
  1016         with y step have z: "P ((Suc (p - Suc n)) mod p)"
  1017           by blast
  1018         show "False"
  1019         proof (cases "n=0")
  1020           case True
  1021           with z n2 contra show ?thesis by simp
  1022         next
  1023           case False
  1024           with p have "p-n < p" by arith
  1025           with z n2 False ih show ?thesis by simp
  1026         qed
  1027       qed
  1028     qed
  1029   qed
  1030   moreover
  1031   from i obtain k where "0<k \<and> i+k=p"
  1032     by (blast dest: less_imp_add_positive)
  1033   hence "0<k \<and> i=p-k" by auto
  1034   moreover
  1035   note base
  1036   ultimately
  1037   show "False" by blast
  1038 qed
  1039 
  1040 lemma mod_induct:
  1041   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
  1042   and base: "P i" and i: "i<p" and j: "j<p"
  1043   shows "P j"
  1044 proof -
  1045   have "\<forall>j<p. P j"
  1046   proof
  1047     fix j
  1048     show "j<p \<longrightarrow> P j" (is "?A j")
  1049     proof (induct j)
  1050       from step base i show "?A 0"
  1051         by (auto elim: mod_induct_0)
  1052     next
  1053       fix k
  1054       assume ih: "?A k"
  1055       show "?A (Suc k)"
  1056       proof
  1057         assume suc: "Suc k < p"
  1058         hence k: "k<p" by simp
  1059         with ih have "P k" ..
  1060         with step k have "P (Suc k mod p)"
  1061           by blast
  1062         moreover
  1063         from suc have "Suc k mod p = Suc k"
  1064           by simp
  1065         ultimately
  1066         show "P (Suc k)" by simp
  1067       qed
  1068     qed
  1069   qed
  1070   with j show ?thesis by blast
  1071 qed
  1072 
  1073 lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
  1074 by (auto simp add: numeral_2_eq_2 le_div_geq)
  1075 
  1076 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
  1077 by (simp add: nat_mult_2 [symmetric])
  1078 
  1079 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
  1080 apply (subgoal_tac "m mod 2 < 2")
  1081 apply (erule less_2_cases [THEN disjE])
  1082 apply (simp_all (no_asm_simp) add: Let_def mod_Suc)
  1083 done
  1084 
  1085 lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
  1086 proof -
  1087   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
  1088   moreover have "m mod 2 < 2" by simp
  1089   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
  1090   then show ?thesis by auto
  1091 qed
  1092 
  1093 text{*These lemmas collapse some needless occurrences of Suc:
  1094     at least three Sucs, since two and fewer are rewritten back to Suc again!
  1095     We already have some rules to simplify operands smaller than 3.*}
  1096 
  1097 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
  1098 by (simp add: Suc3_eq_add_3)
  1099 
  1100 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
  1101 by (simp add: Suc3_eq_add_3)
  1102 
  1103 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
  1104 by (simp add: Suc3_eq_add_3)
  1105 
  1106 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
  1107 by (simp add: Suc3_eq_add_3)
  1108 
  1109 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
  1110 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
  1111 
  1112 
  1113 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
  1114 apply (induct "m")
  1115 apply (simp_all add: mod_Suc)
  1116 done
  1117 
  1118 declare Suc_times_mod_eq [of "numeral w", simp] for w
  1119 
  1120 lemma [simp]: "n div k \<le> (Suc n) div k"
  1121 by (simp add: div_le_mono) 
  1122 
  1123 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
  1124 by (cases n) simp_all
  1125 
  1126 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
  1127 proof -
  1128   from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
  1129   from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp 
  1130 qed
  1131 
  1132   (* Potential use of algebra : Equality modulo n*)
  1133 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
  1134 by (simp add: mult_ac add_ac)
  1135 
  1136 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
  1137 proof -
  1138   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
  1139   also have "... = Suc m mod n" by (rule mod_mult_self3) 
  1140   finally show ?thesis .
  1141 qed
  1142 
  1143 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
  1144 apply (subst mod_Suc [of m]) 
  1145 apply (subst mod_Suc [of "m mod n"], simp) 
  1146 done
  1147 
  1148 lemma mod_2_not_eq_zero_eq_one_nat:
  1149   fixes n :: nat
  1150   shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
  1151   by simp
  1152 
  1153 
  1154 subsection {* Division on @{typ int} *}
  1155 
  1156 definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
  1157     --{*definition of quotient and remainder*}
  1158     "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
  1159                (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
  1160 
  1161 definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
  1162     --{*for the division algorithm*}
  1163     "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
  1164                          else (2 * q, r))"
  1165 
  1166 text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
  1167 function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  1168   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
  1169      else adjust b (posDivAlg a (2 * b)))"
  1170 by auto
  1171 termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
  1172   (auto simp add: mult_2)
  1173 
  1174 text{*algorithm for the case @{text "a<0, b>0"}*}
  1175 function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  1176   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
  1177      else adjust b (negDivAlg a (2 * b)))"
  1178 by auto
  1179 termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
  1180   (auto simp add: mult_2)
  1181 
  1182 text{*algorithm for the general case @{term "b\<noteq>0"}*}
  1183 
  1184 definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  1185     --{*The full division algorithm considers all possible signs for a, b
  1186        including the special case @{text "a=0, b<0"} because 
  1187        @{term negDivAlg} requires @{term "a<0"}.*}
  1188   "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
  1189                   else if a = 0 then (0, 0)
  1190                        else apsnd uminus (negDivAlg (-a) (-b))
  1191                else 
  1192                   if 0 < b then negDivAlg a b
  1193                   else apsnd uminus (posDivAlg (-a) (-b)))"
  1194 
  1195 instantiation int :: Divides.div
  1196 begin
  1197 
  1198 definition div_int where
  1199   "a div b = fst (divmod_int a b)"
  1200 
  1201 lemma fst_divmod_int [simp]:
  1202   "fst (divmod_int a b) = a div b"
  1203   by (simp add: div_int_def)
  1204 
  1205 definition mod_int where
  1206   "a mod b = snd (divmod_int a b)"
  1207 
  1208 lemma snd_divmod_int [simp]:
  1209   "snd (divmod_int a b) = a mod b"
  1210   by (simp add: mod_int_def)
  1211 
  1212 instance ..
  1213 
  1214 end
  1215 
  1216 lemma divmod_int_mod_div:
  1217   "divmod_int p q = (p div q, p mod q)"
  1218   by (simp add: prod_eq_iff)
  1219 
  1220 text{*
  1221 Here is the division algorithm in ML:
  1222 
  1223 \begin{verbatim}
  1224     fun posDivAlg (a,b) =
  1225       if a<b then (0,a)
  1226       else let val (q,r) = posDivAlg(a, 2*b)
  1227                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
  1228            end
  1229 
  1230     fun negDivAlg (a,b) =
  1231       if 0\<le>a+b then (~1,a+b)
  1232       else let val (q,r) = negDivAlg(a, 2*b)
  1233                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
  1234            end;
  1235 
  1236     fun negateSnd (q,r:int) = (q,~r);
  1237 
  1238     fun divmod (a,b) = if 0\<le>a then 
  1239                           if b>0 then posDivAlg (a,b) 
  1240                            else if a=0 then (0,0)
  1241                                 else negateSnd (negDivAlg (~a,~b))
  1242                        else 
  1243                           if 0<b then negDivAlg (a,b)
  1244                           else        negateSnd (posDivAlg (~a,~b));
  1245 \end{verbatim}
  1246 *}
  1247 
  1248 
  1249 subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}
  1250 
  1251 lemma unique_quotient_lemma:
  1252      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
  1253       ==> q' \<le> (q::int)"
  1254 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
  1255  prefer 2 apply (simp add: right_diff_distrib)
  1256 apply (subgoal_tac "0 < b * (1 + q - q') ")
  1257 apply (erule_tac [2] order_le_less_trans)
  1258  prefer 2 apply (simp add: right_diff_distrib right_distrib)
  1259 apply (subgoal_tac "b * q' < b * (1 + q) ")
  1260  prefer 2 apply (simp add: right_diff_distrib right_distrib)
  1261 apply (simp add: mult_less_cancel_left)
  1262 done
  1263 
  1264 lemma unique_quotient_lemma_neg:
  1265      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
  1266       ==> q \<le> (q'::int)"
  1267 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
  1268     auto)
  1269 
  1270 lemma unique_quotient:
  1271      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
  1272       ==> q = q'"
  1273 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
  1274 apply (blast intro: order_antisym
  1275              dest: order_eq_refl [THEN unique_quotient_lemma] 
  1276              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
  1277 done
  1278 
  1279 
  1280 lemma unique_remainder:
  1281      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
  1282       ==> r = r'"
  1283 apply (subgoal_tac "q = q'")
  1284  apply (simp add: divmod_int_rel_def)
  1285 apply (blast intro: unique_quotient)
  1286 done
  1287 
  1288 
  1289 subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}
  1290 
  1291 text{*And positive divisors*}
  1292 
  1293 lemma adjust_eq [simp]:
  1294      "adjust b (q, r) = 
  1295       (let diff = r - b in  
  1296         if 0 \<le> diff then (2 * q + 1, diff)   
  1297                      else (2*q, r))"
  1298   by (simp add: Let_def adjust_def)
  1299 
  1300 declare posDivAlg.simps [simp del]
  1301 
  1302 text{*use with a simproc to avoid repeatedly proving the premise*}
  1303 lemma posDivAlg_eqn:
  1304      "0 < b ==>  
  1305       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
  1306 by (rule posDivAlg.simps [THEN trans], simp)
  1307 
  1308 text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
  1309 theorem posDivAlg_correct:
  1310   assumes "0 \<le> a" and "0 < b"
  1311   shows "divmod_int_rel a b (posDivAlg a b)"
  1312   using assms
  1313   apply (induct a b rule: posDivAlg.induct)
  1314   apply auto
  1315   apply (simp add: divmod_int_rel_def)
  1316   apply (subst posDivAlg_eqn, simp add: right_distrib)
  1317   apply (case_tac "a < b")
  1318   apply simp_all
  1319   apply (erule splitE)
  1320   apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
  1321   done
  1322 
  1323 
  1324 subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}
  1325 
  1326 text{*And positive divisors*}
  1327 
  1328 declare negDivAlg.simps [simp del]
  1329 
  1330 text{*use with a simproc to avoid repeatedly proving the premise*}
  1331 lemma negDivAlg_eqn:
  1332      "0 < b ==>  
  1333       negDivAlg a b =       
  1334        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
  1335 by (rule negDivAlg.simps [THEN trans], simp)
  1336 
  1337 (*Correctness of negDivAlg: it computes quotients correctly
  1338   It doesn't work if a=0 because the 0/b equals 0, not -1*)
  1339 lemma negDivAlg_correct:
  1340   assumes "a < 0" and "b > 0"
  1341   shows "divmod_int_rel a b (negDivAlg a b)"
  1342   using assms
  1343   apply (induct a b rule: negDivAlg.induct)
  1344   apply (auto simp add: linorder_not_le)
  1345   apply (simp add: divmod_int_rel_def)
  1346   apply (subst negDivAlg_eqn, assumption)
  1347   apply (case_tac "a + b < (0\<Colon>int)")
  1348   apply simp_all
  1349   apply (erule splitE)
  1350   apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
  1351   done
  1352 
  1353 
  1354 subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}
  1355 
  1356 (*the case a=0*)
  1357 lemma divmod_int_rel_0: "b \<noteq> 0 ==> divmod_int_rel 0 b (0, 0)"
  1358 by (auto simp add: divmod_int_rel_def linorder_neq_iff)
  1359 
  1360 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
  1361 by (subst posDivAlg.simps, auto)
  1362 
  1363 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
  1364 by (subst negDivAlg.simps, auto)
  1365 
  1366 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"
  1367 by (auto simp add: split_ifs divmod_int_rel_def)
  1368 
  1369 lemma divmod_int_correct: "b \<noteq> 0 ==> divmod_int_rel a b (divmod_int a b)"
  1370 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
  1371                     posDivAlg_correct negDivAlg_correct)
  1372 
  1373 text{*Arbitrary definitions for division by zero.  Useful to simplify 
  1374     certain equations.*}
  1375 
  1376 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
  1377 by (simp add: div_int_def mod_int_def divmod_int_def posDivAlg.simps)  
  1378 
  1379 
  1380 text{*Basic laws about division and remainder*}
  1381 
  1382 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
  1383 apply (case_tac "b = 0", simp)
  1384 apply (cut_tac a = a and b = b in divmod_int_correct)
  1385 apply (auto simp add: divmod_int_rel_def prod_eq_iff)
  1386 done
  1387 
  1388 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
  1389 by(simp add: zmod_zdiv_equality[symmetric])
  1390 
  1391 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
  1392 by(simp add: mult_commute zmod_zdiv_equality[symmetric])
  1393 
  1394 text {* Tool setup *}
  1395 
  1396 (* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)
  1397 lemmas add_0s = add_0_left add_0_right
  1398 
  1399 ML {*
  1400 structure Cancel_Div_Mod_Int = Cancel_Div_Mod
  1401 (
  1402   val div_name = @{const_name div};
  1403   val mod_name = @{const_name mod};
  1404   val mk_binop = HOLogic.mk_binop;
  1405   val mk_sum = Arith_Data.mk_sum HOLogic.intT;
  1406   val dest_sum = Arith_Data.dest_sum;
  1407 
  1408   val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];
  1409 
  1410   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
  1411     (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
  1412 )
  1413 *}
  1414 
  1415 simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}
  1416 
  1417 lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
  1418 apply (cut_tac a = a and b = b in divmod_int_correct)
  1419 apply (auto simp add: divmod_int_rel_def prod_eq_iff)
  1420 done
  1421 
  1422 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
  1423    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
  1424 
  1425 lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
  1426 apply (cut_tac a = a and b = b in divmod_int_correct)
  1427 apply (auto simp add: divmod_int_rel_def prod_eq_iff)
  1428 done
  1429 
  1430 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
  1431    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
  1432 
  1433 
  1434 subsubsection {* General Properties of div and mod *}
  1435 
  1436 lemma divmod_int_rel_div_mod: "b \<noteq> 0 ==> divmod_int_rel a b (a div b, a mod b)"
  1437 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
  1438 apply (force simp add: divmod_int_rel_def linorder_neq_iff)
  1439 done
  1440 
  1441 lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r) |] ==> a div b = q"
  1442 apply (cases "b = 0")
  1443 apply (simp add: divmod_int_rel_def)
  1444 by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
  1445 
  1446 lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r) |] ==> a mod b = r"
  1447 apply (cases "b = 0")
  1448 apply (simp add: divmod_int_rel_def)
  1449 by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
  1450 
  1451 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
  1452 apply (rule divmod_int_rel_div)
  1453 apply (auto simp add: divmod_int_rel_def)
  1454 done
  1455 
  1456 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
  1457 apply (rule divmod_int_rel_div)
  1458 apply (auto simp add: divmod_int_rel_def)
  1459 done
  1460 
  1461 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
  1462 apply (rule divmod_int_rel_div)
  1463 apply (auto simp add: divmod_int_rel_def)
  1464 done
  1465 
  1466 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
  1467 
  1468 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
  1469 apply (rule_tac q = 0 in divmod_int_rel_mod)
  1470 apply (auto simp add: divmod_int_rel_def)
  1471 done
  1472 
  1473 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
  1474 apply (rule_tac q = 0 in divmod_int_rel_mod)
  1475 apply (auto simp add: divmod_int_rel_def)
  1476 done
  1477 
  1478 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
  1479 apply (rule_tac q = "-1" in divmod_int_rel_mod)
  1480 apply (auto simp add: divmod_int_rel_def)
  1481 done
  1482 
  1483 text{*There is no @{text mod_neg_pos_trivial}.*}
  1484 
  1485 
  1486 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
  1487 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
  1488 apply (case_tac "b = 0", simp)
  1489 apply (simp add: divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, 
  1490                                  THEN divmod_int_rel_div, THEN sym])
  1491 
  1492 done
  1493 
  1494 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
  1495 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
  1496 apply (case_tac "b = 0", simp)
  1497 apply (subst divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, THEN divmod_int_rel_mod],
  1498        auto)
  1499 done
  1500 
  1501 
  1502 subsubsection {* Laws for div and mod with Unary Minus *}
  1503 
  1504 lemma zminus1_lemma:
  1505      "divmod_int_rel a b (q, r)
  1506       ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
  1507                           if r=0 then 0 else b-r)"
  1508 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
  1509 
  1510 
  1511 lemma zdiv_zminus1_eq_if:
  1512      "b \<noteq> (0::int)  
  1513       ==> (-a) div b =  
  1514           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
  1515 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_div])
  1516 
  1517 lemma zmod_zminus1_eq_if:
  1518      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
  1519 apply (case_tac "b = 0", simp)
  1520 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_mod])
  1521 done
  1522 
  1523 lemma zmod_zminus1_not_zero:
  1524   fixes k l :: int
  1525   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
  1526   unfolding zmod_zminus1_eq_if by auto
  1527 
  1528 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
  1529 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
  1530 
  1531 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
  1532 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
  1533 
  1534 lemma zdiv_zminus2_eq_if:
  1535      "b \<noteq> (0::int)  
  1536       ==> a div (-b) =  
  1537           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
  1538 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
  1539 
  1540 lemma zmod_zminus2_eq_if:
  1541      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
  1542 by (simp add: zmod_zminus1_eq_if zmod_zminus2)
  1543 
  1544 lemma zmod_zminus2_not_zero:
  1545   fixes k l :: int
  1546   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
  1547   unfolding zmod_zminus2_eq_if by auto 
  1548 
  1549 
  1550 subsubsection {* Division of a Number by Itself *}
  1551 
  1552 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
  1553 apply (subgoal_tac "0 < a*q")
  1554  apply (simp add: zero_less_mult_iff, arith)
  1555 done
  1556 
  1557 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
  1558 apply (subgoal_tac "0 \<le> a* (1-q) ")
  1559  apply (simp add: zero_le_mult_iff)
  1560 apply (simp add: right_diff_distrib)
  1561 done
  1562 
  1563 lemma self_quotient: "[| divmod_int_rel a a (q, r) |] ==> q = 1"
  1564 apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)
  1565 apply (rule order_antisym, safe, simp_all)
  1566 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
  1567 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
  1568 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
  1569 done
  1570 
  1571 lemma self_remainder: "[| divmod_int_rel a a (q, r) |] ==> r = 0"
  1572 apply (frule self_quotient)
  1573 apply (simp add: divmod_int_rel_def)
  1574 done
  1575 
  1576 lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
  1577 by (simp add: divmod_int_rel_div_mod [THEN self_quotient])
  1578 
  1579 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
  1580 lemma zmod_self [simp]: "a mod a = (0::int)"
  1581 apply (case_tac "a = 0", simp)
  1582 apply (simp add: divmod_int_rel_div_mod [THEN self_remainder])
  1583 done
  1584 
  1585 
  1586 subsubsection {* Computation of Division and Remainder *}
  1587 
  1588 lemma zdiv_zero [simp]: "(0::int) div b = 0"
  1589 by (simp add: div_int_def divmod_int_def)
  1590 
  1591 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
  1592 by (simp add: div_int_def divmod_int_def)
  1593 
  1594 lemma zmod_zero [simp]: "(0::int) mod b = 0"
  1595 by (simp add: mod_int_def divmod_int_def)
  1596 
  1597 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
  1598 by (simp add: mod_int_def divmod_int_def)
  1599 
  1600 text{*a positive, b positive *}
  1601 
  1602 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
  1603 by (simp add: div_int_def divmod_int_def)
  1604 
  1605 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
  1606 by (simp add: mod_int_def divmod_int_def)
  1607 
  1608 text{*a negative, b positive *}
  1609 
  1610 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
  1611 by (simp add: div_int_def divmod_int_def)
  1612 
  1613 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
  1614 by (simp add: mod_int_def divmod_int_def)
  1615 
  1616 text{*a positive, b negative *}
  1617 
  1618 lemma div_pos_neg:
  1619      "[| 0 < a;  b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"
  1620 by (simp add: div_int_def divmod_int_def)
  1621 
  1622 lemma mod_pos_neg:
  1623      "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"
  1624 by (simp add: mod_int_def divmod_int_def)
  1625 
  1626 text{*a negative, b negative *}
  1627 
  1628 lemma div_neg_neg:
  1629      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"
  1630 by (simp add: div_int_def divmod_int_def)
  1631 
  1632 lemma mod_neg_neg:
  1633      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"
  1634 by (simp add: mod_int_def divmod_int_def)
  1635 
  1636 text {*Simplify expresions in which div and mod combine numerical constants*}
  1637 
  1638 lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
  1639   by (rule divmod_int_rel_div [of a b q r]) (simp add: divmod_int_rel_def)
  1640 
  1641 lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
  1642   by (rule divmod_int_rel_div [of a b q r],
  1643     simp add: divmod_int_rel_def)
  1644 
  1645 lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
  1646   by (rule divmod_int_rel_mod [of a b q r],
  1647     simp add: divmod_int_rel_def)
  1648 
  1649 lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
  1650   by (rule divmod_int_rel_mod [of a b q r],
  1651     simp add: divmod_int_rel_def)
  1652 
  1653 (* simprocs adapted from HOL/ex/Binary.thy *)
  1654 ML {*
  1655 local
  1656   val mk_number = HOLogic.mk_number HOLogic.intT
  1657   val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}
  1658   val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}
  1659   val zero = @{term "0 :: int"}
  1660   val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}
  1661   val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}
  1662   val simps = @{thms arith_simps} @ @{thms rel_simps} @
  1663     map (fn th => th RS sym) [@{thm numeral_1_eq_1}]
  1664   fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)
  1665     (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps simps))));
  1666   fun binary_proc proc ss ct =
  1667     (case Thm.term_of ct of
  1668       _ $ t $ u =>
  1669       (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
  1670         SOME args => proc (Simplifier.the_context ss) args
  1671       | NONE => NONE)
  1672     | _ => NONE);
  1673 in
  1674   fun divmod_proc posrule negrule =
  1675     binary_proc (fn ctxt => fn ((a, t), (b, u)) =>
  1676       if b = 0 then NONE else let
  1677         val (q, r) = pairself mk_number (Integer.div_mod a b)
  1678         val goal1 = HOLogic.mk_eq (t, plus $ (times $ u $ q) $ r)
  1679         val (goal2, goal3, rule) = if b > 0
  1680           then (le $ zero $ r, less $ r $ u, posrule RS eq_reflection)
  1681           else (le $ r $ zero, less $ u $ r, negrule RS eq_reflection)
  1682       in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)
  1683 end
  1684 *}
  1685 
  1686 simproc_setup binary_int_div
  1687   ("numeral m div numeral n :: int" |
  1688    "numeral m div neg_numeral n :: int" |
  1689    "neg_numeral m div numeral n :: int" |
  1690    "neg_numeral m div neg_numeral n :: int") =
  1691   {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}
  1692 
  1693 simproc_setup binary_int_mod
  1694   ("numeral m mod numeral n :: int" |
  1695    "numeral m mod neg_numeral n :: int" |
  1696    "neg_numeral m mod numeral n :: int" |
  1697    "neg_numeral m mod neg_numeral n :: int") =
  1698   {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}
  1699 
  1700 lemmas posDivAlg_eqn_numeral [simp] =
  1701     posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w
  1702 
  1703 lemmas negDivAlg_eqn_numeral [simp] =
  1704     negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w
  1705 
  1706 
  1707 text{*Special-case simplification *}
  1708 
  1709 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
  1710 apply (cut_tac a = a and b = "-1" in neg_mod_sign)
  1711 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
  1712 apply (auto simp del: neg_mod_sign neg_mod_bound)
  1713 done
  1714 
  1715 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
  1716 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
  1717 
  1718 (** The last remaining special cases for constant arithmetic:
  1719     1 div z and 1 mod z **)
  1720 
  1721 lemmas div_pos_pos_1_numeral [simp] =
  1722   div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
  1723 
  1724 lemmas div_pos_neg_1_numeral [simp] =
  1725   div_pos_neg [OF zero_less_one, of "neg_numeral w",
  1726   OF neg_numeral_less_zero] for w
  1727 
  1728 lemmas mod_pos_pos_1_numeral [simp] =
  1729   mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
  1730 
  1731 lemmas mod_pos_neg_1_numeral [simp] =
  1732   mod_pos_neg [OF zero_less_one, of "neg_numeral w",
  1733   OF neg_numeral_less_zero] for w
  1734 
  1735 lemmas posDivAlg_eqn_1_numeral [simp] =
  1736     posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
  1737 
  1738 lemmas negDivAlg_eqn_1_numeral [simp] =
  1739     negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
  1740 
  1741 
  1742 subsubsection {* Monotonicity in the First Argument (Dividend) *}
  1743 
  1744 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
  1745 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
  1746 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
  1747 apply (rule unique_quotient_lemma)
  1748 apply (erule subst)
  1749 apply (erule subst, simp_all)
  1750 done
  1751 
  1752 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
  1753 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
  1754 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
  1755 apply (rule unique_quotient_lemma_neg)
  1756 apply (erule subst)
  1757 apply (erule subst, simp_all)
  1758 done
  1759 
  1760 
  1761 subsubsection {* Monotonicity in the Second Argument (Divisor) *}
  1762 
  1763 lemma q_pos_lemma:
  1764      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
  1765 apply (subgoal_tac "0 < b'* (q' + 1) ")
  1766  apply (simp add: zero_less_mult_iff)
  1767 apply (simp add: right_distrib)
  1768 done
  1769 
  1770 lemma zdiv_mono2_lemma:
  1771      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
  1772          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
  1773       ==> q \<le> (q'::int)"
  1774 apply (frule q_pos_lemma, assumption+) 
  1775 apply (subgoal_tac "b*q < b* (q' + 1) ")
  1776  apply (simp add: mult_less_cancel_left)
  1777 apply (subgoal_tac "b*q = r' - r + b'*q'")
  1778  prefer 2 apply simp
  1779 apply (simp (no_asm_simp) add: right_distrib)
  1780 apply (subst add_commute, rule add_less_le_mono, arith)
  1781 apply (rule mult_right_mono, auto)
  1782 done
  1783 
  1784 lemma zdiv_mono2:
  1785      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
  1786 apply (subgoal_tac "b \<noteq> 0")
  1787  prefer 2 apply arith
  1788 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
  1789 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
  1790 apply (rule zdiv_mono2_lemma)
  1791 apply (erule subst)
  1792 apply (erule subst, simp_all)
  1793 done
  1794 
  1795 lemma q_neg_lemma:
  1796      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
  1797 apply (subgoal_tac "b'*q' < 0")
  1798  apply (simp add: mult_less_0_iff, arith)
  1799 done
  1800 
  1801 lemma zdiv_mono2_neg_lemma:
  1802      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
  1803          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
  1804       ==> q' \<le> (q::int)"
  1805 apply (frule q_neg_lemma, assumption+) 
  1806 apply (subgoal_tac "b*q' < b* (q + 1) ")
  1807  apply (simp add: mult_less_cancel_left)
  1808 apply (simp add: right_distrib)
  1809 apply (subgoal_tac "b*q' \<le> b'*q'")
  1810  prefer 2 apply (simp add: mult_right_mono_neg, arith)
  1811 done
  1812 
  1813 lemma zdiv_mono2_neg:
  1814      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
  1815 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
  1816 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
  1817 apply (rule zdiv_mono2_neg_lemma)
  1818 apply (erule subst)
  1819 apply (erule subst, simp_all)
  1820 done
  1821 
  1822 
  1823 subsubsection {* More Algebraic Laws for div and mod *}
  1824 
  1825 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
  1826 
  1827 lemma zmult1_lemma:
  1828      "[| divmod_int_rel b c (q, r) |]  
  1829       ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
  1830 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)
  1831 
  1832 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (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_div])
  1835 done
  1836 
  1837 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
  1838 apply (case_tac "c = 0", simp)
  1839 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_mod])
  1840 done
  1841 
  1842 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
  1843 apply (case_tac "b = 0", simp)
  1844 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
  1845 done
  1846 
  1847 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
  1848 
  1849 lemma zadd1_lemma:
  1850      "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]  
  1851       ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
  1852 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)
  1853 
  1854 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
  1855 lemma zdiv_zadd1_eq:
  1856      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
  1857 apply (case_tac "c = 0", simp)
  1858 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] divmod_int_rel_div)
  1859 done
  1860 
  1861 instance int :: ring_div
  1862 proof
  1863   fix a b c :: int
  1864   assume not0: "b \<noteq> 0"
  1865   show "(a + c * b) div b = c + a div b"
  1866     unfolding zdiv_zadd1_eq [of a "c * b"] using not0 
  1867       by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
  1868 next
  1869   fix a b c :: int
  1870   assume "a \<noteq> 0"
  1871   then show "(a * b) div (a * c) = b div c"
  1872   proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")
  1873     case False then show ?thesis by auto
  1874   next
  1875     case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto
  1876     with `a \<noteq> 0`
  1877     have "\<And>q r. divmod_int_rel b c (q, r) \<Longrightarrow> divmod_int_rel (a * b) (a * c) (q, a * r)"
  1878       apply (auto simp add: divmod_int_rel_def) 
  1879       apply (auto simp add: algebra_simps)
  1880       apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff mult_commute [of a] mult_less_cancel_right)
  1881       done
  1882     moreover with `c \<noteq> 0` divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto
  1883     ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .
  1884     from this show ?thesis by (rule divmod_int_rel_div)
  1885   qed
  1886 qed auto
  1887 
  1888 lemma posDivAlg_div_mod:
  1889   assumes "k \<ge> 0"
  1890   and "l \<ge> 0"
  1891   shows "posDivAlg k l = (k div l, k mod l)"
  1892 proof (cases "l = 0")
  1893   case True then show ?thesis by (simp add: posDivAlg.simps)
  1894 next
  1895   case False with assms posDivAlg_correct
  1896     have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
  1897     by simp
  1898   from divmod_int_rel_div [OF this] divmod_int_rel_mod [OF this]
  1899   show ?thesis by simp
  1900 qed
  1901 
  1902 lemma negDivAlg_div_mod:
  1903   assumes "k < 0"
  1904   and "l > 0"
  1905   shows "negDivAlg k l = (k div l, k mod l)"
  1906 proof -
  1907   from assms have "l \<noteq> 0" by simp
  1908   from assms negDivAlg_correct
  1909     have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
  1910     by simp
  1911   from divmod_int_rel_div [OF this] divmod_int_rel_mod [OF this]
  1912   show ?thesis by simp
  1913 qed
  1914 
  1915 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
  1916 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
  1917 
  1918 (* REVISIT: should this be generalized to all semiring_div types? *)
  1919 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
  1920 
  1921 lemma zmod_zdiv_equality':
  1922   "(m\<Colon>int) mod n = m - (m div n) * n"
  1923   by (rule_tac P="%x. m mod n = x - (m div n) * n" in subst [OF mod_div_equality [of _ n]])
  1924     arith
  1925 
  1926 
  1927 subsubsection {* Proving  @{term "a div (b*c) = (a div b) div c"} *}
  1928 
  1929 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
  1930   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
  1931   to cause particular problems.*)
  1932 
  1933 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
  1934 
  1935 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
  1936 apply (subgoal_tac "b * (c - q mod c) < r * 1")
  1937  apply (simp add: algebra_simps)
  1938 apply (rule order_le_less_trans)
  1939  apply (erule_tac [2] mult_strict_right_mono)
  1940  apply (rule mult_left_mono_neg)
  1941   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
  1942  apply (simp)
  1943 apply (simp)
  1944 done
  1945 
  1946 lemma zmult2_lemma_aux2:
  1947      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
  1948 apply (subgoal_tac "b * (q mod c) \<le> 0")
  1949  apply arith
  1950 apply (simp add: mult_le_0_iff)
  1951 done
  1952 
  1953 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
  1954 apply (subgoal_tac "0 \<le> b * (q mod c) ")
  1955 apply arith
  1956 apply (simp add: zero_le_mult_iff)
  1957 done
  1958 
  1959 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
  1960 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
  1961  apply (simp add: right_diff_distrib)
  1962 apply (rule order_less_le_trans)
  1963  apply (erule mult_strict_right_mono)
  1964  apply (rule_tac [2] mult_left_mono)
  1965   apply simp
  1966  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
  1967 apply simp
  1968 done
  1969 
  1970 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  
  1971       ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
  1972 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
  1973                    zero_less_mult_iff right_distrib [symmetric] 
  1974                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
  1975 
  1976 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
  1977 apply (case_tac "b = 0", simp)
  1978 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_div])
  1979 done
  1980 
  1981 lemma zmod_zmult2_eq:
  1982      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
  1983 apply (case_tac "b = 0", simp)
  1984 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_mod])
  1985 done
  1986 
  1987 lemma div_pos_geq:
  1988   fixes k l :: int
  1989   assumes "0 < l" and "l \<le> k"
  1990   shows "k div l = (k - l) div l + 1"
  1991 proof -
  1992   have "k = (k - l) + l" by simp
  1993   then obtain j where k: "k = j + l" ..
  1994   with assms show ?thesis by simp
  1995 qed
  1996 
  1997 lemma mod_pos_geq:
  1998   fixes k l :: int
  1999   assumes "0 < l" and "l \<le> k"
  2000   shows "k mod l = (k - l) mod l"
  2001 proof -
  2002   have "k = (k - l) + l" by simp
  2003   then obtain j where k: "k = j + l" ..
  2004   with assms show ?thesis by simp
  2005 qed
  2006 
  2007 
  2008 subsubsection {* Splitting Rules for div and mod *}
  2009 
  2010 text{*The proofs of the two lemmas below are essentially identical*}
  2011 
  2012 lemma split_pos_lemma:
  2013  "0<k ==> 
  2014     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
  2015 apply (rule iffI, clarify)
  2016  apply (erule_tac P="P ?x ?y" in rev_mp)  
  2017  apply (subst mod_add_eq) 
  2018  apply (subst zdiv_zadd1_eq) 
  2019  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
  2020 txt{*converse direction*}
  2021 apply (drule_tac x = "n div k" in spec) 
  2022 apply (drule_tac x = "n mod k" in spec, simp)
  2023 done
  2024 
  2025 lemma split_neg_lemma:
  2026  "k<0 ==>
  2027     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
  2028 apply (rule iffI, clarify)
  2029  apply (erule_tac P="P ?x ?y" in rev_mp)  
  2030  apply (subst mod_add_eq) 
  2031  apply (subst zdiv_zadd1_eq) 
  2032  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
  2033 txt{*converse direction*}
  2034 apply (drule_tac x = "n div k" in spec) 
  2035 apply (drule_tac x = "n mod k" in spec, simp)
  2036 done
  2037 
  2038 lemma split_zdiv:
  2039  "P(n div k :: int) =
  2040   ((k = 0 --> P 0) & 
  2041    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
  2042    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
  2043 apply (case_tac "k=0", simp)
  2044 apply (simp only: linorder_neq_iff)
  2045 apply (erule disjE) 
  2046  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
  2047                       split_neg_lemma [of concl: "%x y. P x"])
  2048 done
  2049 
  2050 lemma split_zmod:
  2051  "P(n mod k :: int) =
  2052   ((k = 0 --> P n) & 
  2053    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
  2054    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
  2055 apply (case_tac "k=0", simp)
  2056 apply (simp only: linorder_neq_iff)
  2057 apply (erule disjE) 
  2058  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
  2059                       split_neg_lemma [of concl: "%x y. P y"])
  2060 done
  2061 
  2062 text {* Enable (lin)arith to deal with @{const div} and @{const mod}
  2063   when these are applied to some constant that is of the form
  2064   @{term "numeral k"}: *}
  2065 declare split_zdiv [of _ _ "numeral k", arith_split] for k
  2066 declare split_zmod [of _ _ "numeral k", arith_split] for k
  2067 
  2068 
  2069 subsubsection {* Speeding up the Division Algorithm with Shifting *}
  2070 
  2071 text{*computing div by shifting *}
  2072 
  2073 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
  2074 proof cases
  2075   assume "a=0"
  2076     thus ?thesis by simp
  2077 next
  2078   assume "a\<noteq>0" and le_a: "0\<le>a"   
  2079   hence a_pos: "1 \<le> a" by arith
  2080   hence one_less_a2: "1 < 2 * a" by arith
  2081   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
  2082     unfolding mult_le_cancel_left
  2083     by (simp add: add1_zle_eq add_commute [of 1])
  2084   with a_pos have "0 \<le> b mod a" by simp
  2085   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
  2086     by (simp add: mod_pos_pos_trivial one_less_a2)
  2087   with  le_2a
  2088   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
  2089     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
  2090                   right_distrib) 
  2091   thus ?thesis
  2092     by (subst zdiv_zadd1_eq,
  2093         simp add: mod_mult_mult1 one_less_a2
  2094                   div_pos_pos_trivial)
  2095 qed
  2096 
  2097 lemma neg_zdiv_mult_2: 
  2098   assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
  2099 proof -
  2100   have R: "1 + - (2 * (b + 1)) = - (1 + 2 * b)" by simp
  2101   have "(1 + 2 * (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a)"
  2102     by (rule pos_zdiv_mult_2, simp add: A)
  2103   thus ?thesis
  2104     by (simp only: R zdiv_zminus_zminus diff_minus
  2105       minus_add_distrib [symmetric] mult_minus_right)
  2106 qed
  2107 
  2108 (* FIXME: add rules for negative numerals *)
  2109 lemma zdiv_numeral_Bit0 [simp]:
  2110   "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
  2111     numeral v div (numeral w :: int)"
  2112   unfolding numeral.simps unfolding mult_2 [symmetric]
  2113   by (rule div_mult_mult1, simp)
  2114 
  2115 lemma zdiv_numeral_Bit1 [simp]:
  2116   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  
  2117     (numeral v div (numeral w :: int))"
  2118   unfolding numeral.simps
  2119   unfolding mult_2 [symmetric] add_commute [of _ 1]
  2120   by (rule pos_zdiv_mult_2, simp)
  2121 
  2122 
  2123 subsubsection {* Computing mod by Shifting (proofs resemble those for div) *}
  2124 
  2125 lemma pos_zmod_mult_2:
  2126   fixes a b :: int
  2127   assumes "0 \<le> a"
  2128   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
  2129 proof (cases "0 < a")
  2130   case False with assms show ?thesis by simp
  2131 next
  2132   case True
  2133   then have "b mod a < a" by (rule pos_mod_bound)
  2134   then have "1 + b mod a \<le> a" by simp
  2135   then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp
  2136   from `0 < a` have "0 \<le> b mod a" by (rule pos_mod_sign)
  2137   then have B: "0 \<le> 1 + 2 * (b mod a)" by simp
  2138   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)"
  2139     using `0 < a` and A
  2140     by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)
  2141   then show ?thesis by (subst mod_add_eq)
  2142 qed
  2143 
  2144 lemma neg_zmod_mult_2:
  2145   fixes a b :: int
  2146   assumes "a \<le> 0"
  2147   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
  2148 proof -
  2149   from assms have "0 \<le> - a" by auto
  2150   then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"
  2151     by (rule pos_zmod_mult_2)
  2152   then show ?thesis by (simp add: zmod_zminus2 algebra_simps)
  2153      (simp add: diff_minus add_ac)
  2154 qed
  2155 
  2156 (* FIXME: add rules for negative numerals *)
  2157 lemma zmod_numeral_Bit0 [simp]:
  2158   "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  
  2159     (2::int) * (numeral v mod numeral w)"
  2160   unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
  2161   unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
  2162 
  2163 lemma zmod_numeral_Bit1 [simp]:
  2164   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
  2165     2 * (numeral v mod numeral w) + (1::int)"
  2166   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
  2167   unfolding mult_2 [symmetric] add_commute [of _ 1]
  2168   by (rule pos_zmod_mult_2, simp)
  2169 
  2170 lemma zdiv_eq_0_iff:
  2171  "(i::int) div k = 0 \<longleftrightarrow> k=0 \<or> 0\<le>i \<and> i<k \<or> i\<le>0 \<and> k<i" (is "?L = ?R")
  2172 proof
  2173   assume ?L
  2174   have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
  2175   with `?L` show ?R by blast
  2176 next
  2177   assume ?R thus ?L
  2178     by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
  2179 qed
  2180 
  2181 
  2182 subsubsection {* Quotients of Signs *}
  2183 
  2184 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
  2185 apply (subgoal_tac "a div b \<le> -1", force)
  2186 apply (rule order_trans)
  2187 apply (rule_tac a' = "-1" in zdiv_mono1)
  2188 apply (auto simp add: div_eq_minus1)
  2189 done
  2190 
  2191 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
  2192 by (drule zdiv_mono1_neg, auto)
  2193 
  2194 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
  2195 by (drule zdiv_mono1, auto)
  2196 
  2197 text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}
  2198 conditional upon the sign of @{text a} or @{text b}. There are many more.
  2199 They should all be simp rules unless that causes too much search. *}
  2200 
  2201 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
  2202 apply auto
  2203 apply (drule_tac [2] zdiv_mono1)
  2204 apply (auto simp add: linorder_neq_iff)
  2205 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
  2206 apply (blast intro: div_neg_pos_less0)
  2207 done
  2208 
  2209 lemma neg_imp_zdiv_nonneg_iff:
  2210   "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
  2211 apply (subst zdiv_zminus_zminus [symmetric])
  2212 apply (subst pos_imp_zdiv_nonneg_iff, auto)
  2213 done
  2214 
  2215 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
  2216 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
  2217 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
  2218 
  2219 lemma pos_imp_zdiv_pos_iff:
  2220   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
  2221 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
  2222 by arith
  2223 
  2224 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
  2225 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
  2226 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
  2227 
  2228 lemma nonneg1_imp_zdiv_pos_iff:
  2229   "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
  2230 apply rule
  2231  apply rule
  2232   using div_pos_pos_trivial[of a b]apply arith
  2233  apply(cases "b=0")apply simp
  2234  using div_nonneg_neg_le0[of a b]apply arith
  2235 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
  2236 done
  2237 
  2238 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
  2239 apply (rule split_zmod[THEN iffD2])
  2240 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
  2241 done
  2242 
  2243 
  2244 subsubsection {* The Divides Relation *}
  2245 
  2246 lemmas zdvd_iff_zmod_eq_0_numeral [simp] =
  2247   dvd_eq_mod_eq_0 [of "numeral x::int" "numeral y::int"]
  2248   dvd_eq_mod_eq_0 [of "numeral x::int" "neg_numeral y::int"]
  2249   dvd_eq_mod_eq_0 [of "neg_numeral x::int" "numeral y::int"]
  2250   dvd_eq_mod_eq_0 [of "neg_numeral x::int" "neg_numeral y::int"] for x y
  2251 
  2252 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
  2253   by (rule dvd_mod) (* TODO: remove *)
  2254 
  2255 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
  2256   by (rule dvd_mod_imp_dvd) (* TODO: remove *)
  2257 
  2258 lemmas dvd_eq_mod_eq_0_numeral [simp] =
  2259   dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y
  2260 
  2261 
  2262 subsubsection {* Further properties *}
  2263 
  2264 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
  2265   using zmod_zdiv_equality[where a="m" and b="n"]
  2266   by (simp add: algebra_simps)
  2267 
  2268 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
  2269 apply (induct "y", auto)
  2270 apply (rule zmod_zmult1_eq [THEN trans])
  2271 apply (simp (no_asm_simp))
  2272 apply (rule mod_mult_eq [symmetric])
  2273 done
  2274 
  2275 lemma zdiv_int: "int (a div b) = (int a) div (int b)"
  2276 apply (subst split_div, auto)
  2277 apply (subst split_zdiv, auto)
  2278 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)
  2279 apply (auto simp add: divmod_int_rel_def of_nat_mult)
  2280 done
  2281 
  2282 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
  2283 apply (subst split_mod, auto)
  2284 apply (subst split_zmod, auto)
  2285 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
  2286        in unique_remainder)
  2287 apply (auto simp add: divmod_int_rel_def of_nat_mult)
  2288 done
  2289 
  2290 lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
  2291 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
  2292 
  2293 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
  2294 apply (subgoal_tac "m mod n = 0")
  2295  apply (simp add: zmult_div_cancel)
  2296 apply (simp only: dvd_eq_mod_eq_0)
  2297 done
  2298 
  2299 text{*Suggested by Matthias Daum*}
  2300 lemma int_power_div_base:
  2301      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
  2302 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
  2303  apply (erule ssubst)
  2304  apply (simp only: power_add)
  2305  apply simp_all
  2306 done
  2307 
  2308 text {* by Brian Huffman *}
  2309 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
  2310 by (rule mod_minus_eq [symmetric])
  2311 
  2312 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
  2313 by (rule mod_diff_left_eq [symmetric])
  2314 
  2315 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
  2316 by (rule mod_diff_right_eq [symmetric])
  2317 
  2318 lemmas zmod_simps =
  2319   mod_add_left_eq  [symmetric]
  2320   mod_add_right_eq [symmetric]
  2321   zmod_zmult1_eq   [symmetric]
  2322   mod_mult_left_eq [symmetric]
  2323   zpower_zmod
  2324   zminus_zmod zdiff_zmod_left zdiff_zmod_right
  2325 
  2326 text {* Distributive laws for function @{text nat}. *}
  2327 
  2328 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
  2329 apply (rule linorder_cases [of y 0])
  2330 apply (simp add: div_nonneg_neg_le0)
  2331 apply simp
  2332 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
  2333 done
  2334 
  2335 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
  2336 lemma nat_mod_distrib:
  2337   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
  2338 apply (case_tac "y = 0", simp)
  2339 apply (simp add: nat_eq_iff zmod_int)
  2340 done
  2341 
  2342 text  {* transfer setup *}
  2343 
  2344 lemma transfer_nat_int_functions:
  2345     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
  2346     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
  2347   by (auto simp add: nat_div_distrib nat_mod_distrib)
  2348 
  2349 lemma transfer_nat_int_function_closures:
  2350     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
  2351     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
  2352   apply (cases "y = 0")
  2353   apply (auto simp add: pos_imp_zdiv_nonneg_iff)
  2354   apply (cases "y = 0")
  2355   apply auto
  2356 done
  2357 
  2358 declare transfer_morphism_nat_int [transfer add return:
  2359   transfer_nat_int_functions
  2360   transfer_nat_int_function_closures
  2361 ]
  2362 
  2363 lemma transfer_int_nat_functions:
  2364     "(int x) div (int y) = int (x div y)"
  2365     "(int x) mod (int y) = int (x mod y)"
  2366   by (auto simp add: zdiv_int zmod_int)
  2367 
  2368 lemma transfer_int_nat_function_closures:
  2369     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
  2370     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
  2371   by (simp_all only: is_nat_def transfer_nat_int_function_closures)
  2372 
  2373 declare transfer_morphism_int_nat [transfer add return:
  2374   transfer_int_nat_functions
  2375   transfer_int_nat_function_closures
  2376 ]
  2377 
  2378 text{*Suggested by Matthias Daum*}
  2379 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
  2380 apply (subgoal_tac "nat x div nat k < nat x")
  2381  apply (simp add: nat_div_distrib [symmetric])
  2382 apply (rule Divides.div_less_dividend, simp_all)
  2383 done
  2384 
  2385 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
  2386 proof
  2387   assume H: "x mod n = y mod n"
  2388   hence "x mod n - y mod n = 0" by simp
  2389   hence "(x mod n - y mod n) mod n = 0" by simp 
  2390   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
  2391   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
  2392 next
  2393   assume H: "n dvd x - y"
  2394   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
  2395   hence "x = n*k + y" by simp
  2396   hence "x mod n = (n*k + y) mod n" by simp
  2397   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
  2398 qed
  2399 
  2400 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
  2401   shows "\<exists>q. x = y + n * q"
  2402 proof-
  2403   from xy have th: "int x - int y = int (x - y)" by simp 
  2404   from xyn have "int x mod int n = int y mod int n" 
  2405     by (simp add: zmod_int [symmetric])
  2406   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
  2407   hence "n dvd x - y" by (simp add: th zdvd_int)
  2408   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
  2409 qed
  2410 
  2411 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
  2412   (is "?lhs = ?rhs")
  2413 proof
  2414   assume H: "x mod n = y mod n"
  2415   {assume xy: "x \<le> y"
  2416     from H have th: "y mod n = x mod n" by simp
  2417     from nat_mod_eq_lemma[OF th xy] have ?rhs 
  2418       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
  2419   moreover
  2420   {assume xy: "y \<le> x"
  2421     from nat_mod_eq_lemma[OF H xy] have ?rhs 
  2422       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
  2423   ultimately  show ?rhs using linear[of x y] by blast  
  2424 next
  2425   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
  2426   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
  2427   thus  ?lhs by simp
  2428 qed
  2429 
  2430 lemma div_nat_numeral [simp]:
  2431   "(numeral v :: nat) div numeral v' = nat (numeral v div numeral v')"
  2432   by (simp add: nat_div_distrib)
  2433 
  2434 lemma one_div_nat_numeral [simp]:
  2435   "Suc 0 div numeral v' = nat (1 div numeral v')"
  2436   by (subst nat_div_distrib, simp_all)
  2437 
  2438 lemma mod_nat_numeral [simp]:
  2439   "(numeral v :: nat) mod numeral v' = nat (numeral v mod numeral v')"
  2440   by (simp add: nat_mod_distrib)
  2441 
  2442 lemma one_mod_nat_numeral [simp]:
  2443   "Suc 0 mod numeral v' = nat (1 mod numeral v')"
  2444   by (subst nat_mod_distrib) simp_all
  2445 
  2446 lemma mod_2_not_eq_zero_eq_one_int:
  2447   fixes k :: int
  2448   shows "k mod 2 \<noteq> 0 \<longleftrightarrow> k mod 2 = 1"
  2449   by auto
  2450 
  2451 
  2452 subsubsection {* Tools setup *}
  2453 
  2454 text {* Nitpick *}
  2455 
  2456 lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'
  2457 
  2458 
  2459 subsubsection {* Code generation *}
  2460 
  2461 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
  2462   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
  2463 
  2464 lemma pdivmod_posDivAlg [code]:
  2465   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
  2466 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
  2467 
  2468 lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
  2469   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
  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 aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
  2475   show ?thesis
  2476     by (simp add: divmod_int_mod_div pdivmod_def)
  2477       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
  2478       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
  2479 qed
  2480 
  2481 lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
  2482   apsnd ((op *) (sgn l)) (if sgn k = sgn l
  2483     then pdivmod k l
  2484     else (let (r, s) = pdivmod k l in
  2485       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
  2486 proof -
  2487   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"
  2488     by (auto simp add: not_less sgn_if)
  2489   then show ?thesis by (simp add: divmod_int_pdivmod)
  2490 qed
  2491 
  2492 code_modulename SML
  2493   Divides Arith
  2494 
  2495 code_modulename OCaml
  2496   Divides Arith
  2497 
  2498 code_modulename Haskell
  2499   Divides Arith
  2500 
  2501 end