src/HOL/Divides.thy
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     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 instantiation nat :: semiring_div

   527 begin

   528

   529 definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where

   530   "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"

   531

   532 lemma divmod_nat_rel_divmod_nat:

   533   "divmod_nat_rel m n (divmod_nat m n)"

   534 proof -

   535   from divmod_nat_rel_ex

   536     obtain qr where rel: "divmod_nat_rel m n qr" .

   537   then show ?thesis

   538   by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)

   539 qed

   540

   541 lemma divmod_nat_eq:

   542   assumes "divmod_nat_rel m n qr"

   543   shows "divmod_nat m n = qr"

   544   using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)

   545

   546 definition div_nat where

   547   "m div n = fst (divmod_nat m n)"

   548

   549 definition mod_nat where

   550   "m mod n = snd (divmod_nat m n)"

   551

   552 lemma divmod_nat_div_mod:

   553   "divmod_nat m n = (m div n, m mod n)"

   554   unfolding div_nat_def mod_nat_def by simp

   555

   556 lemma div_eq:

   557   assumes "divmod_nat_rel m n (q, r)"

   558   shows "m div n = q"

   559   using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)

   560

   561 lemma mod_eq:

   562   assumes "divmod_nat_rel m n (q, r)"

   563   shows "m mod n = r"

   564   using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)

   565

   566 lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"

   567   by (simp add: div_nat_def mod_nat_def divmod_nat_rel_divmod_nat)

   568

   569 lemma divmod_nat_zero:

   570   "divmod_nat m 0 = (0, m)"

   571 proof -

   572   from divmod_nat_rel [of m 0] show ?thesis

   573     unfolding divmod_nat_div_mod divmod_nat_rel_def by simp

   574 qed

   575

   576 lemma divmod_nat_base:

   577   assumes "m < n"

   578   shows "divmod_nat m n = (0, m)"

   579 proof -

   580   from divmod_nat_rel [of m n] show ?thesis

   581     unfolding divmod_nat_div_mod divmod_nat_rel_def

   582     using assms by (cases "m div n = 0")

   583       (auto simp add: gr0_conv_Suc [of "m div n"])

   584 qed

   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 -

   590   from divmod_nat_rel have divmod_nat_m_n: "divmod_nat_rel m n (m div n, m mod n)" .

   591   with assms have m_div_n: "m div n \<ge> 1"

   592     by (cases "m div n") (auto simp add: divmod_nat_rel_def)

   593   have "divmod_nat_rel (m - n) n (m div n - Suc 0, m mod n)"

   594   proof -

   595     from assms have

   596       "n \<noteq> 0"

   597       "\<And>k. m = Suc k * n + m mod n ==> m - n = (Suc k - Suc 0) * n + m mod n"

   598       by simp_all

   599     then show ?thesis using assms divmod_nat_m_n

   600       by (cases "m div n")

   601          (simp_all only: divmod_nat_rel_def fst_conv snd_conv, simp_all)

   602   qed

   603   with divmod_nat_eq have "divmod_nat (m - n) n = (m div n - Suc 0, m mod n)" by simp

   604   moreover from divmod_nat_div_mod have "divmod_nat (m - n) n = ((m - n) div n, (m - n) mod n)" .

   605   ultimately have "m div n = Suc ((m - n) div n)"

   606     and "m mod n = (m - n) mod n" using m_div_n by simp_all

   607   then show ?thesis using divmod_nat_div_mod by simp

   608 qed

   609

   610 text {* The ''recursion'' equations for @{const div} and @{const mod} *}

   611

   612 lemma div_less [simp]:

   613   fixes m n :: nat

   614   assumes "m < n"

   615   shows "m div n = 0"

   616   using assms divmod_nat_base divmod_nat_div_mod by simp

   617

   618 lemma le_div_geq:

   619   fixes m n :: nat

   620   assumes "0 < n" and "n \<le> m"

   621   shows "m div n = Suc ((m - n) div n)"

   622   using assms divmod_nat_step divmod_nat_div_mod by simp

   623

   624 lemma mod_less [simp]:

   625   fixes m n :: nat

   626   assumes "m < n"

   627   shows "m mod n = m"

   628   using assms divmod_nat_base divmod_nat_div_mod by simp

   629

   630 lemma le_mod_geq:

   631   fixes m n :: nat

   632   assumes "n \<le> m"

   633   shows "m mod n = (m - n) mod n"

   634   using assms divmod_nat_step divmod_nat_div_mod by (cases "n = 0") simp_all

   635

   636 instance proof -

   637   have [simp]: "\<And>n::nat. n div 0 = 0"

   638     by (simp add: div_nat_def divmod_nat_zero)

   639   have [simp]: "\<And>n::nat. 0 div n = 0"

   640   proof -

   641     fix n :: nat

   642     show "0 div n = 0"

   643       by (cases "n = 0") simp_all

   644   qed

   645   show "OFCLASS(nat, semiring_div_class)" proof

   646     fix m n :: nat

   647     show "m div n * n + m mod n = m"

   648       using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)

   649   next

   650     fix m n q :: nat

   651     assume "n \<noteq> 0"

   652     then show "(q + m * n) div n = m + q div n"

   653       by (induct m) (simp_all add: le_div_geq)

   654   next

   655     fix m n q :: nat

   656     assume "m \<noteq> 0"

   657     then show "(m * n) div (m * q) = n div q"

   658     proof (cases "n \<noteq> 0 \<and> q \<noteq> 0")

   659       case False then show ?thesis by auto

   660     next

   661       case True with m \<noteq> 0

   662         have "m > 0" and "n > 0" and "q > 0" by auto

   663       then have "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"

   664         by (auto simp add: divmod_nat_rel_def) (simp_all add: algebra_simps)

   665       moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .

   666       ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .

   667       then show ?thesis by (simp add: div_eq)

   668     qed

   669   qed simp_all

   670 qed

   671

   672 end

   673

   674 lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else

   675   let (q, r) = divmod_nat (m - n) n in (Suc q, r))"

   676 by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step)

   677     (simp add: divmod_nat_div_mod)

   678

   679 text {* Simproc for cancelling @{const div} and @{const mod} *}

   680

   681 ML {*

   682 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod

   683 (

   684   val div_name = @{const_name div};

   685   val mod_name = @{const_name mod};

   686   val mk_binop = HOLogic.mk_binop;

   687   val mk_sum = Nat_Arith.mk_sum;

   688   val dest_sum = Nat_Arith.dest_sum;

   689

   690   val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];

   691

   692   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac

   693     (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))

   694 )

   695 *}

   696

   697 simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}

   698

   699

   700 subsubsection {* Quotient *}

   701

   702 lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"

   703 by (simp add: le_div_geq linorder_not_less)

   704

   705 lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"

   706 by (simp add: div_geq)

   707

   708 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"

   709 by simp

   710

   711 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"

   712 by simp

   713

   714

   715 subsubsection {* Remainder *}

   716

   717 lemma mod_less_divisor [simp]:

   718   fixes m n :: nat

   719   assumes "n > 0"

   720   shows "m mod n < (n::nat)"

   721   using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto

   722

   723 lemma mod_less_eq_dividend [simp]:

   724   fixes m n :: nat

   725   shows "m mod n \<le> m"

   726 proof (rule add_leD2)

   727   from mod_div_equality have "m div n * n + m mod n = m" .

   728   then show "m div n * n + m mod n \<le> m" by auto

   729 qed

   730

   731 lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"

   732 by (simp add: le_mod_geq linorder_not_less)

   733

   734 lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"

   735 by (simp add: le_mod_geq)

   736

   737 lemma mod_1 [simp]: "m mod Suc 0 = 0"

   738 by (induct m) (simp_all add: mod_geq)

   739

   740 lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"

   741   apply (cases "n = 0", simp)

   742   apply (cases "k = 0", simp)

   743   apply (induct m rule: nat_less_induct)

   744   apply (subst mod_if, simp)

   745   apply (simp add: mod_geq diff_mult_distrib)

   746   done

   747

   748 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"

   749 by (simp add: mult_commute [of k] mod_mult_distrib)

   750

   751 (* a simple rearrangement of mod_div_equality: *)

   752 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"

   753 by (cut_tac a = m and b = n in mod_div_equality2, arith)

   754

   755 lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"

   756   apply (drule mod_less_divisor [where m = m])

   757   apply simp

   758   done

   759

   760 subsubsection {* Quotient and Remainder *}

   761

   762 lemma divmod_nat_rel_mult1_eq:

   763   "divmod_nat_rel b c (q, r) \<Longrightarrow> c > 0

   764    \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"

   765 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)

   766

   767 lemma div_mult1_eq:

   768   "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"

   769 apply (cases "c = 0", simp)

   770 apply (blast intro: divmod_nat_rel [THEN divmod_nat_rel_mult1_eq, THEN div_eq])

   771 done

   772

   773 lemma divmod_nat_rel_add1_eq:

   774   "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) \<Longrightarrow>  c > 0

   775    \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"

   776 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)

   777

   778 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)

   779 lemma div_add1_eq:

   780   "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"

   781 apply (cases "c = 0", simp)

   782 apply (blast intro: divmod_nat_rel_add1_eq [THEN div_eq] divmod_nat_rel)

   783 done

   784

   785 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"

   786   apply (cut_tac m = q and n = c in mod_less_divisor)

   787   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)

   788   apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)

   789   apply (simp add: add_mult_distrib2)

   790   done

   791

   792 lemma divmod_nat_rel_mult2_eq:

   793   "divmod_nat_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c

   794    \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"

   795 by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)

   796

   797 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"

   798   apply (cases "b = 0", simp)

   799   apply (cases "c = 0", simp)

   800   apply (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_eq])

   801   done

   802

   803 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"

   804   apply (cases "b = 0", simp)

   805   apply (cases "c = 0", simp)

   806   apply (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_eq])

   807   done

   808

   809

   810 subsubsection{*Further Facts about Quotient and Remainder*}

   811

   812 lemma div_1 [simp]: "m div Suc 0 = m"

   813 by (induct m) (simp_all add: div_geq)

   814

   815

   816 (* Monotonicity of div in first argument *)

   817 lemma div_le_mono [rule_format (no_asm)]:

   818     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"

   819 apply (case_tac "k=0", simp)

   820 apply (induct "n" rule: nat_less_induct, clarify)

   821 apply (case_tac "n<k")

   822 (* 1  case n<k *)

   823 apply simp

   824 (* 2  case n >= k *)

   825 apply (case_tac "m<k")

   826 (* 2.1  case m<k *)

   827 apply simp

   828 (* 2.2  case m>=k *)

   829 apply (simp add: div_geq diff_le_mono)

   830 done

   831

   832 (* Antimonotonicity of div in second argument *)

   833 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"

   834 apply (subgoal_tac "0<n")

   835  prefer 2 apply simp

   836 apply (induct_tac k rule: nat_less_induct)

   837 apply (rename_tac "k")

   838 apply (case_tac "k<n", simp)

   839 apply (subgoal_tac "~ (k<m) ")

   840  prefer 2 apply simp

   841 apply (simp add: div_geq)

   842 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")

   843  prefer 2

   844  apply (blast intro: div_le_mono diff_le_mono2)

   845 apply (rule le_trans, simp)

   846 apply (simp)

   847 done

   848

   849 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"

   850 apply (case_tac "n=0", simp)

   851 apply (subgoal_tac "m div n \<le> m div 1", simp)

   852 apply (rule div_le_mono2)

   853 apply (simp_all (no_asm_simp))

   854 done

   855

   856 (* Similar for "less than" *)

   857 lemma div_less_dividend [rule_format]:

   858      "!!n::nat. 1<n ==> 0 < m --> m div n < m"

   859 apply (induct_tac m rule: nat_less_induct)

   860 apply (rename_tac "m")

   861 apply (case_tac "m<n", simp)

   862 apply (subgoal_tac "0<n")

   863  prefer 2 apply simp

   864 apply (simp add: div_geq)

   865 apply (case_tac "n<m")

   866  apply (subgoal_tac "(m-n) div n < (m-n) ")

   867   apply (rule impI less_trans_Suc)+

   868 apply assumption

   869   apply (simp_all)

   870 done

   871

   872 declare div_less_dividend [simp]

   873

   874 text{*A fact for the mutilated chess board*}

   875 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"

   876 apply (case_tac "n=0", simp)

   877 apply (induct "m" rule: nat_less_induct)

   878 apply (case_tac "Suc (na) <n")

   879 (* case Suc(na) < n *)

   880 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)

   881 (* case n \<le> Suc(na) *)

   882 apply (simp add: linorder_not_less le_Suc_eq mod_geq)

   883 apply (auto simp add: Suc_diff_le le_mod_geq)

   884 done

   885

   886 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"

   887 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)

   888

   889 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]

   890

   891 (*Loses information, namely we also have r<d provided d is nonzero*)

   892 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"

   893   apply (cut_tac a = m in mod_div_equality)

   894   apply (simp only: add_ac)

   895   apply (blast intro: sym)

   896   done

   897

   898 lemma split_div:

   899  "P(n div k :: nat) =

   900  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"

   901  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")

   902 proof

   903   assume P: ?P

   904   show ?Q

   905   proof (cases)

   906     assume "k = 0"

   907     with P show ?Q by simp

   908   next

   909     assume not0: "k \<noteq> 0"

   910     thus ?Q

   911     proof (simp, intro allI impI)

   912       fix i j

   913       assume n: "n = k*i + j" and j: "j < k"

   914       show "P i"

   915       proof (cases)

   916         assume "i = 0"

   917         with n j P show "P i" by simp

   918       next

   919         assume "i \<noteq> 0"

   920         with not0 n j P show "P i" by(simp add:add_ac)

   921       qed

   922     qed

   923   qed

   924 next

   925   assume Q: ?Q

   926   show ?P

   927   proof (cases)

   928     assume "k = 0"

   929     with Q show ?P by simp

   930   next

   931     assume not0: "k \<noteq> 0"

   932     with Q have R: ?R by simp

   933     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]

   934     show ?P by simp

   935   qed

   936 qed

   937

   938 lemma split_div_lemma:

   939   assumes "0 < n"

   940   shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")

   941 proof

   942   assume ?rhs

   943   with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp

   944   then have A: "n * q \<le> m" by simp

   945   have "n - (m mod n) > 0" using mod_less_divisor assms by auto

   946   then have "m < m + (n - (m mod n))" by simp

   947   then have "m < n + (m - (m mod n))" by simp

   948   with nq have "m < n + n * q" by simp

   949   then have B: "m < n * Suc q" by simp

   950   from A B show ?lhs ..

   951 next

   952   assume P: ?lhs

   953   then have "divmod_nat_rel m n (q, m - n * q)"

   954     unfolding divmod_nat_rel_def by (auto simp add: mult_ac)

   955   with divmod_nat_rel_unique divmod_nat_rel [of m n]

   956   have "(q, m - n * q) = (m div n, m mod n)" by auto

   957   then show ?rhs by simp

   958 qed

   959

   960 theorem split_div':

   961   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>

   962    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"

   963   apply (case_tac "0 < n")

   964   apply (simp only: add: split_div_lemma)

   965   apply simp_all

   966   done

   967

   968 lemma split_mod:

   969  "P(n mod k :: nat) =

   970  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"

   971  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")

   972 proof

   973   assume P: ?P

   974   show ?Q

   975   proof (cases)

   976     assume "k = 0"

   977     with P show ?Q by simp

   978   next

   979     assume not0: "k \<noteq> 0"

   980     thus ?Q

   981     proof (simp, intro allI impI)

   982       fix i j

   983       assume "n = k*i + j" "j < k"

   984       thus "P j" using not0 P by(simp add:add_ac mult_ac)

   985     qed

   986   qed

   987 next

   988   assume Q: ?Q

   989   show ?P

   990   proof (cases)

   991     assume "k = 0"

   992     with Q show ?P by simp

   993   next

   994     assume not0: "k \<noteq> 0"

   995     with Q have R: ?R by simp

   996     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]

   997     show ?P by simp

   998   qed

   999 qed

  1000

  1001 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"

  1002   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in

  1003     subst [OF mod_div_equality [of _ n]])

  1004   apply arith

  1005   done

  1006

  1007 lemma div_mod_equality':

  1008   fixes m n :: nat

  1009   shows "m div n * n = m - m mod n"

  1010 proof -

  1011   have "m mod n \<le> m mod n" ..

  1012   from div_mod_equality have

  1013     "m div n * n + m mod n - m mod n = m - m mod n" by simp

  1014   with diff_add_assoc [OF m mod n \<le> m mod n, of "m div n * n"] have

  1015     "m div n * n + (m mod n - m mod n) = m - m mod n"

  1016     by simp

  1017   then show ?thesis by simp

  1018 qed

  1019

  1020

  1021 subsubsection {*An induction'' law for modulus arithmetic.*}

  1022

  1023 lemma mod_induct_0:

  1024   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"

  1025   and base: "P i" and i: "i<p"

  1026   shows "P 0"

  1027 proof (rule ccontr)

  1028   assume contra: "\<not>(P 0)"

  1029   from i have p: "0<p" by simp

  1030   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")

  1031   proof

  1032     fix k

  1033     show "?A k"

  1034     proof (induct k)

  1035       show "?A 0" by simp  -- "by contradiction"

  1036     next

  1037       fix n

  1038       assume ih: "?A n"

  1039       show "?A (Suc n)"

  1040       proof (clarsimp)

  1041         assume y: "P (p - Suc n)"

  1042         have n: "Suc n < p"

  1043         proof (rule ccontr)

  1044           assume "\<not>(Suc n < p)"

  1045           hence "p - Suc n = 0"

  1046             by simp

  1047           with y contra show "False"

  1048             by simp

  1049         qed

  1050         hence n2: "Suc (p - Suc n) = p-n" by arith

  1051         from p have "p - Suc n < p" by arith

  1052         with y step have z: "P ((Suc (p - Suc n)) mod p)"

  1053           by blast

  1054         show "False"

  1055         proof (cases "n=0")

  1056           case True

  1057           with z n2 contra show ?thesis by simp

  1058         next

  1059           case False

  1060           with p have "p-n < p" by arith

  1061           with z n2 False ih show ?thesis by simp

  1062         qed

  1063       qed

  1064     qed

  1065   qed

  1066   moreover

  1067   from i obtain k where "0<k \<and> i+k=p"

  1068     by (blast dest: less_imp_add_positive)

  1069   hence "0<k \<and> i=p-k" by auto

  1070   moreover

  1071   note base

  1072   ultimately

  1073   show "False" by blast

  1074 qed

  1075

  1076 lemma mod_induct:

  1077   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"

  1078   and base: "P i" and i: "i<p" and j: "j<p"

  1079   shows "P j"

  1080 proof -

  1081   have "\<forall>j<p. P j"

  1082   proof

  1083     fix j

  1084     show "j<p \<longrightarrow> P j" (is "?A j")

  1085     proof (induct j)

  1086       from step base i show "?A 0"

  1087         by (auto elim: mod_induct_0)

  1088     next

  1089       fix k

  1090       assume ih: "?A k"

  1091       show "?A (Suc k)"

  1092       proof

  1093         assume suc: "Suc k < p"

  1094         hence k: "k<p" by simp

  1095         with ih have "P k" ..

  1096         with step k have "P (Suc k mod p)"

  1097           by blast

  1098         moreover

  1099         from suc have "Suc k mod p = Suc k"

  1100           by simp

  1101         ultimately

  1102         show "P (Suc k)" by simp

  1103       qed

  1104     qed

  1105   qed

  1106   with j show ?thesis by blast

  1107 qed

  1108

  1109 lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"

  1110 by (auto simp add: numeral_2_eq_2 le_div_geq)

  1111

  1112 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"

  1113 by (simp add: nat_mult_2 [symmetric])

  1114

  1115 lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"

  1116 apply (subgoal_tac "m mod 2 < 2")

  1117 apply (erule less_2_cases [THEN disjE])

  1118 apply (simp_all (no_asm_simp) add: Let_def mod_Suc)

  1119 done

  1120

  1121 lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"

  1122 proof -

  1123   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }

  1124   moreover have "m mod 2 < 2" by simp

  1125   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .

  1126   then show ?thesis by auto

  1127 qed

  1128

  1129 text{*These lemmas collapse some needless occurrences of Suc:

  1130     at least three Sucs, since two and fewer are rewritten back to Suc again!

  1131     We already have some rules to simplify operands smaller than 3.*}

  1132

  1133 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"

  1134 by (simp add: Suc3_eq_add_3)

  1135

  1136 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"

  1137 by (simp add: Suc3_eq_add_3)

  1138

  1139 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"

  1140 by (simp add: Suc3_eq_add_3)

  1141

  1142 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"

  1143 by (simp add: Suc3_eq_add_3)

  1144

  1145 lemmas Suc_div_eq_add3_div_number_of [simp] = Suc_div_eq_add3_div [of _ "number_of v"] for v

  1146 lemmas Suc_mod_eq_add3_mod_number_of [simp] = Suc_mod_eq_add3_mod [of _ "number_of v"] for v

  1147

  1148

  1149 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"

  1150 apply (induct "m")

  1151 apply (simp_all add: mod_Suc)

  1152 done

  1153

  1154 declare Suc_times_mod_eq [of "number_of w", simp] for w

  1155

  1156 lemma [simp]: "n div k \<le> (Suc n) div k"

  1157 by (simp add: div_le_mono)

  1158

  1159 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"

  1160 by (cases n) simp_all

  1161

  1162 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"

  1163 proof -

  1164   from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all

  1165   from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp

  1166 qed

  1167

  1168   (* Potential use of algebra : Equality modulo n*)

  1169 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"

  1170 by (simp add: mult_ac add_ac)

  1171

  1172 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"

  1173 proof -

  1174   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp

  1175   also have "... = Suc m mod n" by (rule mod_mult_self3)

  1176   finally show ?thesis .

  1177 qed

  1178

  1179 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"

  1180 apply (subst mod_Suc [of m])

  1181 apply (subst mod_Suc [of "m mod n"], simp)

  1182 done

  1183

  1184

  1185 subsection {* Division on @{typ int} *}

  1186

  1187 definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where

  1188     --{*definition of quotient and remainder*}

  1189     [code]: "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>

  1190                (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"

  1191

  1192 definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where

  1193     --{*for the division algorithm*}

  1194     [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)

  1195                          else (2 * q, r))"

  1196

  1197 text{*algorithm for the case @{text "a\<ge>0, b>0"}*}

  1198 function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1199   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)

  1200      else adjust b (posDivAlg a (2 * b)))"

  1201 by auto

  1202 termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")

  1203   (auto simp add: mult_2)

  1204

  1205 text{*algorithm for the case @{text "a<0, b>0"}*}

  1206 function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1207   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)

  1208      else adjust b (negDivAlg a (2 * b)))"

  1209 by auto

  1210 termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")

  1211   (auto simp add: mult_2)

  1212

  1213 text{*algorithm for the general case @{term "b\<noteq>0"}*}

  1214 definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where

  1215   [code_unfold]: "negateSnd = apsnd uminus"

  1216

  1217 definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1218     --{*The full division algorithm considers all possible signs for a, b

  1219        including the special case @{text "a=0, b<0"} because

  1220        @{term negDivAlg} requires @{term "a<0"}.*}

  1221   "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b

  1222                   else if a = 0 then (0, 0)

  1223                        else negateSnd (negDivAlg (-a) (-b))

  1224                else

  1225                   if 0 < b then negDivAlg a b

  1226                   else negateSnd (posDivAlg (-a) (-b)))"

  1227

  1228 instantiation int :: Divides.div

  1229 begin

  1230

  1231 definition

  1232   "a div b = fst (divmod_int a b)"

  1233

  1234 definition

  1235  "a mod b = snd (divmod_int a b)"

  1236

  1237 instance ..

  1238

  1239 end

  1240

  1241 lemma divmod_int_mod_div:

  1242   "divmod_int p q = (p div q, p mod q)"

  1243   by (auto simp add: div_int_def mod_int_def)

  1244

  1245 text{*

  1246 Here is the division algorithm in ML:

  1247

  1248 \begin{verbatim}

  1249     fun posDivAlg (a,b) =

  1250       if a<b then (0,a)

  1251       else let val (q,r) = posDivAlg(a, 2*b)

  1252                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)

  1253            end

  1254

  1255     fun negDivAlg (a,b) =

  1256       if 0\<le>a+b then (~1,a+b)

  1257       else let val (q,r) = negDivAlg(a, 2*b)

  1258                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)

  1259            end;

  1260

  1261     fun negateSnd (q,r:int) = (q,~r);

  1262

  1263     fun divmod (a,b) = if 0\<le>a then

  1264                           if b>0 then posDivAlg (a,b)

  1265                            else if a=0 then (0,0)

  1266                                 else negateSnd (negDivAlg (~a,~b))

  1267                        else

  1268                           if 0<b then negDivAlg (a,b)

  1269                           else        negateSnd (posDivAlg (~a,~b));

  1270 \end{verbatim}

  1271 *}

  1272

  1273

  1274

  1275 subsubsection{*Uniqueness and Monotonicity of Quotients and Remainders*}

  1276

  1277 lemma unique_quotient_lemma:

  1278      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]

  1279       ==> q' \<le> (q::int)"

  1280 apply (subgoal_tac "r' + b * (q'-q) \<le> r")

  1281  prefer 2 apply (simp add: right_diff_distrib)

  1282 apply (subgoal_tac "0 < b * (1 + q - q') ")

  1283 apply (erule_tac [2] order_le_less_trans)

  1284  prefer 2 apply (simp add: right_diff_distrib right_distrib)

  1285 apply (subgoal_tac "b * q' < b * (1 + q) ")

  1286  prefer 2 apply (simp add: right_diff_distrib right_distrib)

  1287 apply (simp add: mult_less_cancel_left)

  1288 done

  1289

  1290 lemma unique_quotient_lemma_neg:

  1291      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]

  1292       ==> q \<le> (q'::int)"

  1293 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,

  1294     auto)

  1295

  1296 lemma unique_quotient:

  1297      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]

  1298       ==> q = q'"

  1299 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)

  1300 apply (blast intro: order_antisym

  1301              dest: order_eq_refl [THEN unique_quotient_lemma]

  1302              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+

  1303 done

  1304

  1305

  1306 lemma unique_remainder:

  1307      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]

  1308       ==> r = r'"

  1309 apply (subgoal_tac "q = q'")

  1310  apply (simp add: divmod_int_rel_def)

  1311 apply (blast intro: unique_quotient)

  1312 done

  1313

  1314

  1315 subsubsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}

  1316

  1317 text{*And positive divisors*}

  1318

  1319 lemma adjust_eq [simp]:

  1320      "adjust b (q,r) =

  1321       (let diff = r-b in

  1322         if 0 \<le> diff then (2*q + 1, diff)

  1323                      else (2*q, r))"

  1324 by (simp add: Let_def adjust_def)

  1325

  1326 declare posDivAlg.simps [simp del]

  1327

  1328 text{*use with a simproc to avoid repeatedly proving the premise*}

  1329 lemma posDivAlg_eqn:

  1330      "0 < b ==>

  1331       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"

  1332 by (rule posDivAlg.simps [THEN trans], simp)

  1333

  1334 text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}

  1335 theorem posDivAlg_correct:

  1336   assumes "0 \<le> a" and "0 < b"

  1337   shows "divmod_int_rel a b (posDivAlg a b)"

  1338   using assms

  1339   apply (induct a b rule: posDivAlg.induct)

  1340   apply auto

  1341   apply (simp add: divmod_int_rel_def)

  1342   apply (subst posDivAlg_eqn, simp add: right_distrib)

  1343   apply (case_tac "a < b")

  1344   apply simp_all

  1345   apply (erule splitE)

  1346   apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)

  1347   done

  1348

  1349

  1350 subsubsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}

  1351

  1352 text{*And positive divisors*}

  1353

  1354 declare negDivAlg.simps [simp del]

  1355

  1356 text{*use with a simproc to avoid repeatedly proving the premise*}

  1357 lemma negDivAlg_eqn:

  1358      "0 < b ==>

  1359       negDivAlg a b =

  1360        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"

  1361 by (rule negDivAlg.simps [THEN trans], simp)

  1362

  1363 (*Correctness of negDivAlg: it computes quotients correctly

  1364   It doesn't work if a=0 because the 0/b equals 0, not -1*)

  1365 lemma negDivAlg_correct:

  1366   assumes "a < 0" and "b > 0"

  1367   shows "divmod_int_rel a b (negDivAlg a b)"

  1368   using assms

  1369   apply (induct a b rule: negDivAlg.induct)

  1370   apply (auto simp add: linorder_not_le)

  1371   apply (simp add: divmod_int_rel_def)

  1372   apply (subst negDivAlg_eqn, assumption)

  1373   apply (case_tac "a + b < (0\<Colon>int)")

  1374   apply simp_all

  1375   apply (erule splitE)

  1376   apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)

  1377   done

  1378

  1379

  1380 subsubsection{*Existence Shown by Proving the Division Algorithm to be Correct*}

  1381

  1382 (*the case a=0*)

  1383 lemma divmod_int_rel_0: "b \<noteq> 0 ==> divmod_int_rel 0 b (0, 0)"

  1384 by (auto simp add: divmod_int_rel_def linorder_neq_iff)

  1385

  1386 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"

  1387 by (subst posDivAlg.simps, auto)

  1388

  1389 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"

  1390 by (subst negDivAlg.simps, auto)

  1391

  1392 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"

  1393 by (simp add: negateSnd_def)

  1394

  1395 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (negateSnd qr)"

  1396 by (auto simp add: split_ifs divmod_int_rel_def)

  1397

  1398 lemma divmod_int_correct: "b \<noteq> 0 ==> divmod_int_rel a b (divmod_int a b)"

  1399 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg

  1400                     posDivAlg_correct negDivAlg_correct)

  1401

  1402 text{*Arbitrary definitions for division by zero.  Useful to simplify

  1403     certain equations.*}

  1404

  1405 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"

  1406 by (simp add: div_int_def mod_int_def divmod_int_def posDivAlg.simps)

  1407

  1408

  1409 text{*Basic laws about division and remainder*}

  1410

  1411 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"

  1412 apply (case_tac "b = 0", simp)

  1413 apply (cut_tac a = a and b = b in divmod_int_correct)

  1414 apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)

  1415 done

  1416

  1417 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"

  1418 by(simp add: zmod_zdiv_equality[symmetric])

  1419

  1420 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"

  1421 by(simp add: mult_commute zmod_zdiv_equality[symmetric])

  1422

  1423 text {* Tool setup *}

  1424

  1425 ML {*

  1426 structure Cancel_Div_Mod_Int = Cancel_Div_Mod

  1427 (

  1428   val div_name = @{const_name div};

  1429   val mod_name = @{const_name mod};

  1430   val mk_binop = HOLogic.mk_binop;

  1431   val mk_sum = Arith_Data.mk_sum HOLogic.intT;

  1432   val dest_sum = Arith_Data.dest_sum;

  1433

  1434   val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];

  1435

  1436   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac

  1437     (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))

  1438 )

  1439 *}

  1440

  1441 simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}

  1442

  1443 lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"

  1444 apply (cut_tac a = a and b = b in divmod_int_correct)

  1445 apply (auto simp add: divmod_int_rel_def mod_int_def)

  1446 done

  1447

  1448 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]

  1449    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]

  1450

  1451 lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"

  1452 apply (cut_tac a = a and b = b in divmod_int_correct)

  1453 apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)

  1454 done

  1455

  1456 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]

  1457    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]

  1458

  1459

  1460 subsubsection{*General Properties of div and mod*}

  1461

  1462 lemma divmod_int_rel_div_mod: "b \<noteq> 0 ==> divmod_int_rel a b (a div b, a mod b)"

  1463 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1464 apply (force simp add: divmod_int_rel_def linorder_neq_iff)

  1465 done

  1466

  1467 lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"

  1468 by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])

  1469

  1470 lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"

  1471 by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])

  1472

  1473 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"

  1474 apply (rule divmod_int_rel_div)

  1475 apply (auto simp add: divmod_int_rel_def)

  1476 done

  1477

  1478 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"

  1479 apply (rule divmod_int_rel_div)

  1480 apply (auto simp add: divmod_int_rel_def)

  1481 done

  1482

  1483 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"

  1484 apply (rule divmod_int_rel_div)

  1485 apply (auto simp add: divmod_int_rel_def)

  1486 done

  1487

  1488 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)

  1489

  1490 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"

  1491 apply (rule_tac q = 0 in divmod_int_rel_mod)

  1492 apply (auto simp add: divmod_int_rel_def)

  1493 done

  1494

  1495 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"

  1496 apply (rule_tac q = 0 in divmod_int_rel_mod)

  1497 apply (auto simp add: divmod_int_rel_def)

  1498 done

  1499

  1500 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"

  1501 apply (rule_tac q = "-1" in divmod_int_rel_mod)

  1502 apply (auto simp add: divmod_int_rel_def)

  1503 done

  1504

  1505 text{*There is no @{text mod_neg_pos_trivial}.*}

  1506

  1507

  1508 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)

  1509 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"

  1510 apply (case_tac "b = 0", simp)

  1511 apply (simp add: divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified,

  1512                                  THEN divmod_int_rel_div, THEN sym])

  1513

  1514 done

  1515

  1516 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)

  1517 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"

  1518 apply (case_tac "b = 0", simp)

  1519 apply (subst divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, THEN divmod_int_rel_mod],

  1520        auto)

  1521 done

  1522

  1523

  1524 subsubsection{*Laws for div and mod with Unary Minus*}

  1525

  1526 lemma zminus1_lemma:

  1527      "divmod_int_rel a b (q, r)

  1528       ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,

  1529                           if r=0 then 0 else b-r)"

  1530 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)

  1531

  1532

  1533 lemma zdiv_zminus1_eq_if:

  1534      "b \<noteq> (0::int)

  1535       ==> (-a) div b =

  1536           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"

  1537 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_div])

  1538

  1539 lemma zmod_zminus1_eq_if:

  1540      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"

  1541 apply (case_tac "b = 0", simp)

  1542 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_mod])

  1543 done

  1544

  1545 lemma zmod_zminus1_not_zero:

  1546   fixes k l :: int

  1547   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"

  1548   unfolding zmod_zminus1_eq_if by auto

  1549

  1550 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"

  1551 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)

  1552

  1553 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"

  1554 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)

  1555

  1556 lemma zdiv_zminus2_eq_if:

  1557      "b \<noteq> (0::int)

  1558       ==> a div (-b) =

  1559           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"

  1560 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)

  1561

  1562 lemma zmod_zminus2_eq_if:

  1563      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"

  1564 by (simp add: zmod_zminus1_eq_if zmod_zminus2)

  1565

  1566 lemma zmod_zminus2_not_zero:

  1567   fixes k l :: int

  1568   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"

  1569   unfolding zmod_zminus2_eq_if by auto

  1570

  1571

  1572 subsubsection{*Division of a Number by Itself*}

  1573

  1574 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"

  1575 apply (subgoal_tac "0 < a*q")

  1576  apply (simp add: zero_less_mult_iff, arith)

  1577 done

  1578

  1579 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"

  1580 apply (subgoal_tac "0 \<le> a* (1-q) ")

  1581  apply (simp add: zero_le_mult_iff)

  1582 apply (simp add: right_diff_distrib)

  1583 done

  1584

  1585 lemma self_quotient: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"

  1586 apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)

  1587 apply (rule order_antisym, safe, simp_all)

  1588 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)

  1589 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)

  1590 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+

  1591 done

  1592

  1593 lemma self_remainder: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"

  1594 apply (frule self_quotient, assumption)

  1595 apply (simp add: divmod_int_rel_def)

  1596 done

  1597

  1598 lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"

  1599 by (simp add: divmod_int_rel_div_mod [THEN self_quotient])

  1600

  1601 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)

  1602 lemma zmod_self [simp]: "a mod a = (0::int)"

  1603 apply (case_tac "a = 0", simp)

  1604 apply (simp add: divmod_int_rel_div_mod [THEN self_remainder])

  1605 done

  1606

  1607

  1608 subsubsection{*Computation of Division and Remainder*}

  1609

  1610 lemma zdiv_zero [simp]: "(0::int) div b = 0"

  1611 by (simp add: div_int_def divmod_int_def)

  1612

  1613 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"

  1614 by (simp add: div_int_def divmod_int_def)

  1615

  1616 lemma zmod_zero [simp]: "(0::int) mod b = 0"

  1617 by (simp add: mod_int_def divmod_int_def)

  1618

  1619 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"

  1620 by (simp add: mod_int_def divmod_int_def)

  1621

  1622 text{*a positive, b positive *}

  1623

  1624 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"

  1625 by (simp add: div_int_def divmod_int_def)

  1626

  1627 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"

  1628 by (simp add: mod_int_def divmod_int_def)

  1629

  1630 text{*a negative, b positive *}

  1631

  1632 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"

  1633 by (simp add: div_int_def divmod_int_def)

  1634

  1635 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"

  1636 by (simp add: mod_int_def divmod_int_def)

  1637

  1638 text{*a positive, b negative *}

  1639

  1640 lemma div_pos_neg:

  1641      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"

  1642 by (simp add: div_int_def divmod_int_def)

  1643

  1644 lemma mod_pos_neg:

  1645      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"

  1646 by (simp add: mod_int_def divmod_int_def)

  1647

  1648 text{*a negative, b negative *}

  1649

  1650 lemma div_neg_neg:

  1651      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"

  1652 by (simp add: div_int_def divmod_int_def)

  1653

  1654 lemma mod_neg_neg:

  1655      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"

  1656 by (simp add: mod_int_def divmod_int_def)

  1657

  1658 text {*Simplify expresions in which div and mod combine numerical constants*}

  1659

  1660 lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"

  1661   by (rule divmod_int_rel_div [of a b q r],

  1662     simp add: divmod_int_rel_def, simp)

  1663

  1664 lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"

  1665   by (rule divmod_int_rel_div [of a b q r],

  1666     simp add: divmod_int_rel_def, simp)

  1667

  1668 lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"

  1669   by (rule divmod_int_rel_mod [of a b q r],

  1670     simp add: divmod_int_rel_def, simp)

  1671

  1672 lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"

  1673   by (rule divmod_int_rel_mod [of a b q r],

  1674     simp add: divmod_int_rel_def, simp)

  1675

  1676 lemmas arithmetic_simps =

  1677   arith_simps

  1678   add_special

  1679   add_0_left

  1680   add_0_right

  1681   mult_zero_left

  1682   mult_zero_right

  1683   mult_1_left

  1684   mult_1_right

  1685

  1686 (* simprocs adapted from HOL/ex/Binary.thy *)

  1687 ML {*

  1688 local

  1689   val mk_number = HOLogic.mk_number HOLogic.intT

  1690   val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}

  1691   val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}

  1692   val zero = @{term "0 :: int"}

  1693   val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}

  1694   val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}

  1695   val simps = @{thms arith_simps} @ @{thms rel_simps} @

  1696     map (fn th => th RS sym) [@{thm numeral_0_eq_0}, @{thm numeral_1_eq_1}]

  1697   fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)

  1698     (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps simps))));

  1699   fun binary_proc proc ss ct =

  1700     (case Thm.term_of ct of

  1701       _ $t$ u =>

  1702       (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of

  1703         SOME args => proc (Simplifier.the_context ss) args

  1704       | NONE => NONE)

  1705     | _ => NONE);

  1706 in

  1707   fun divmod_proc posrule negrule =

  1708     binary_proc (fn ctxt => fn ((a, t), (b, u)) =>

  1709       if b = 0 then NONE else let

  1710         val (q, r) = pairself mk_number (Integer.div_mod a b)

  1711         val goal1 = HOLogic.mk_eq (t, plus $(times$ u $q)$ r)

  1712         val (goal2, goal3, rule) = if b > 0

  1713           then (le $zero$ r, less $r$ u, posrule RS eq_reflection)

  1714           else (le $r$ zero, less $u$ r, negrule RS eq_reflection)

  1715       in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)

  1716 end

  1717 *}

  1718

  1719 simproc_setup binary_int_div ("number_of m div number_of n :: int") =

  1720   {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}

  1721

  1722 simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =

  1723   {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}

  1724

  1725 lemmas posDivAlg_eqn_number_of [simp] = posDivAlg_eqn [of "number_of v" "number_of w"] for v w

  1726 lemmas negDivAlg_eqn_number_of [simp] = negDivAlg_eqn [of "number_of v" "number_of w"] for v w

  1727

  1728

  1729 text{*Special-case simplification *}

  1730

  1731 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"

  1732 apply (cut_tac a = a and b = "-1" in neg_mod_sign)

  1733 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)

  1734 apply (auto simp del: neg_mod_sign neg_mod_bound)

  1735 done

  1736

  1737 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"

  1738 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)

  1739

  1740 (** The last remaining special cases for constant arithmetic:

  1741     1 div z and 1 mod z **)

  1742

  1743 lemmas div_pos_pos_1_number_of [simp] = div_pos_pos [OF zero_less_one, of "number_of w"] for w

  1744 lemmas div_pos_neg_1_number_of [simp] = div_pos_neg [OF zero_less_one, of "number_of w"] for w

  1745 lemmas mod_pos_pos_1_number_of [simp] = mod_pos_pos [OF zero_less_one, of "number_of w"] for w

  1746 lemmas mod_pos_neg_1_number_of [simp] = mod_pos_neg [OF zero_less_one, of "number_of w"] for w

  1747 lemmas posDivAlg_eqn_1_number_of [simp] = posDivAlg_eqn [of concl: 1 "number_of w"] for w

  1748 lemmas negDivAlg_eqn_1_number_of [simp] = negDivAlg_eqn [of concl: 1 "number_of w"] for w

  1749

  1750

  1751 subsubsection{*Monotonicity in the First Argument (Dividend)*}

  1752

  1753 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"

  1754 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1755 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)

  1756 apply (rule unique_quotient_lemma)

  1757 apply (erule subst)

  1758 apply (erule subst, simp_all)

  1759 done

  1760

  1761 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"

  1762 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1763 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)

  1764 apply (rule unique_quotient_lemma_neg)

  1765 apply (erule subst)

  1766 apply (erule subst, simp_all)

  1767 done

  1768

  1769

  1770 subsubsection{*Monotonicity in the Second Argument (Divisor)*}

  1771

  1772 lemma q_pos_lemma:

  1773      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"

  1774 apply (subgoal_tac "0 < b'* (q' + 1) ")

  1775  apply (simp add: zero_less_mult_iff)

  1776 apply (simp add: right_distrib)

  1777 done

  1778

  1779 lemma zdiv_mono2_lemma:

  1780      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';

  1781          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]

  1782       ==> q \<le> (q'::int)"

  1783 apply (frule q_pos_lemma, assumption+)

  1784 apply (subgoal_tac "b*q < b* (q' + 1) ")

  1785  apply (simp add: mult_less_cancel_left)

  1786 apply (subgoal_tac "b*q = r' - r + b'*q'")

  1787  prefer 2 apply simp

  1788 apply (simp (no_asm_simp) add: right_distrib)

  1789 apply (subst add_commute, rule add_less_le_mono, arith)

  1790 apply (rule mult_right_mono, auto)

  1791 done

  1792

  1793 lemma zdiv_mono2:

  1794      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"

  1795 apply (subgoal_tac "b \<noteq> 0")

  1796  prefer 2 apply arith

  1797 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1798 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)

  1799 apply (rule zdiv_mono2_lemma)

  1800 apply (erule subst)

  1801 apply (erule subst, simp_all)

  1802 done

  1803

  1804 lemma q_neg_lemma:

  1805      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"

  1806 apply (subgoal_tac "b'*q' < 0")

  1807  apply (simp add: mult_less_0_iff, arith)

  1808 done

  1809

  1810 lemma zdiv_mono2_neg_lemma:

  1811      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;

  1812          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]

  1813       ==> q' \<le> (q::int)"

  1814 apply (frule q_neg_lemma, assumption+)

  1815 apply (subgoal_tac "b*q' < b* (q + 1) ")

  1816  apply (simp add: mult_less_cancel_left)

  1817 apply (simp add: right_distrib)

  1818 apply (subgoal_tac "b*q' \<le> b'*q'")

  1819  prefer 2 apply (simp add: mult_right_mono_neg, arith)

  1820 done

  1821

  1822 lemma zdiv_mono2_neg:

  1823      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"

  1824 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1825 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)

  1826 apply (rule zdiv_mono2_neg_lemma)

  1827 apply (erule subst)

  1828 apply (erule subst, simp_all)

  1829 done

  1830

  1831

  1832 subsubsection{*More Algebraic Laws for div and mod*}

  1833

  1834 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}

  1835

  1836 lemma zmult1_lemma:

  1837      "[| divmod_int_rel b c (q, r);  c \<noteq> 0 |]

  1838       ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"

  1839 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)

  1840

  1841 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"

  1842 apply (case_tac "c = 0", simp)

  1843 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_div])

  1844 done

  1845

  1846 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"

  1847 apply (case_tac "c = 0", simp)

  1848 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_mod])

  1849 done

  1850

  1851 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"

  1852 apply (case_tac "b = 0", simp)

  1853 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)

  1854 done

  1855

  1856 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}

  1857

  1858 lemma zadd1_lemma:

  1859      "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br);  c \<noteq> 0 |]

  1860       ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"

  1861 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)

  1862

  1863 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)

  1864 lemma zdiv_zadd1_eq:

  1865      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"

  1866 apply (case_tac "c = 0", simp)

  1867 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] divmod_int_rel_div)

  1868 done

  1869

  1870 instance int :: ring_div

  1871 proof

  1872   fix a b c :: int

  1873   assume not0: "b \<noteq> 0"

  1874   show "(a + c * b) div b = c + a div b"

  1875     unfolding zdiv_zadd1_eq [of a "c * b"] using not0

  1876       by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)

  1877 next

  1878   fix a b c :: int

  1879   assume "a \<noteq> 0"

  1880   then show "(a * b) div (a * c) = b div c"

  1881   proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")

  1882     case False then show ?thesis by auto

  1883   next

  1884     case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto

  1885     with a \<noteq> 0

  1886     have "\<And>q r. divmod_int_rel b c (q, r) \<Longrightarrow> divmod_int_rel (a * b) (a * c) (q, a * r)"

  1887       apply (auto simp add: divmod_int_rel_def)

  1888       apply (auto simp add: algebra_simps)

  1889       apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff mult_commute [of a] mult_less_cancel_right)

  1890       done

  1891     moreover with c \<noteq> 0 divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto

  1892     ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .

  1893     moreover from  a \<noteq> 0 c \<noteq> 0 have "a * c \<noteq> 0" by simp

  1894     ultimately show ?thesis by (rule divmod_int_rel_div)

  1895   qed

  1896 qed auto

  1897

  1898 lemma posDivAlg_div_mod:

  1899   assumes "k \<ge> 0"

  1900   and "l \<ge> 0"

  1901   shows "posDivAlg k l = (k div l, k mod l)"

  1902 proof (cases "l = 0")

  1903   case True then show ?thesis by (simp add: posDivAlg.simps)

  1904 next

  1905   case False with assms posDivAlg_correct

  1906     have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"

  1907     by simp

  1908   from divmod_int_rel_div [OF this l \<noteq> 0] divmod_int_rel_mod [OF this l \<noteq> 0]

  1909   show ?thesis by simp

  1910 qed

  1911

  1912 lemma negDivAlg_div_mod:

  1913   assumes "k < 0"

  1914   and "l > 0"

  1915   shows "negDivAlg k l = (k div l, k mod l)"

  1916 proof -

  1917   from assms have "l \<noteq> 0" by simp

  1918   from assms negDivAlg_correct

  1919     have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"

  1920     by simp

  1921   from divmod_int_rel_div [OF this l \<noteq> 0] divmod_int_rel_mod [OF this l \<noteq> 0]

  1922   show ?thesis by simp

  1923 qed

  1924

  1925 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"

  1926 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)

  1927

  1928 (* REVISIT: should this be generalized to all semiring_div types? *)

  1929 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]

  1930

  1931

  1932 subsubsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}

  1933

  1934 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but

  1935   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems

  1936   to cause particular problems.*)

  1937

  1938 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}

  1939

  1940 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"

  1941 apply (subgoal_tac "b * (c - q mod c) < r * 1")

  1942  apply (simp add: algebra_simps)

  1943 apply (rule order_le_less_trans)

  1944  apply (erule_tac [2] mult_strict_right_mono)

  1945  apply (rule mult_left_mono_neg)

  1946   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)

  1947  apply (simp)

  1948 apply (simp)

  1949 done

  1950

  1951 lemma zmult2_lemma_aux2:

  1952      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"

  1953 apply (subgoal_tac "b * (q mod c) \<le> 0")

  1954  apply arith

  1955 apply (simp add: mult_le_0_iff)

  1956 done

  1957

  1958 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"

  1959 apply (subgoal_tac "0 \<le> b * (q mod c) ")

  1960 apply arith

  1961 apply (simp add: zero_le_mult_iff)

  1962 done

  1963

  1964 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"

  1965 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")

  1966  apply (simp add: right_diff_distrib)

  1967 apply (rule order_less_le_trans)

  1968  apply (erule mult_strict_right_mono)

  1969  apply (rule_tac [2] mult_left_mono)

  1970   apply simp

  1971  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)

  1972 apply simp

  1973 done

  1974

  1975 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r);  b \<noteq> 0;  0 < c |]

  1976       ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"

  1977 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff

  1978                    zero_less_mult_iff right_distrib [symmetric]

  1979                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)

  1980

  1981 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"

  1982 apply (case_tac "b = 0", simp)

  1983 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_div])

  1984 done

  1985

  1986 lemma zmod_zmult2_eq:

  1987      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"

  1988 apply (case_tac "b = 0", simp)

  1989 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_mod])

  1990 done

  1991

  1992

  1993 subsubsection {*Splitting Rules for div and mod*}

  1994

  1995 text{*The proofs of the two lemmas below are essentially identical*}

  1996

  1997 lemma split_pos_lemma:

  1998  "0<k ==>

  1999     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"

  2000 apply (rule iffI, clarify)

  2001  apply (erule_tac P="P ?x ?y" in rev_mp)

  2002  apply (subst mod_add_eq)

  2003  apply (subst zdiv_zadd1_eq)

  2004  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)

  2005 txt{*converse direction*}

  2006 apply (drule_tac x = "n div k" in spec)

  2007 apply (drule_tac x = "n mod k" in spec, simp)

  2008 done

  2009

  2010 lemma split_neg_lemma:

  2011  "k<0 ==>

  2012     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"

  2013 apply (rule iffI, clarify)

  2014  apply (erule_tac P="P ?x ?y" in rev_mp)

  2015  apply (subst mod_add_eq)

  2016  apply (subst zdiv_zadd1_eq)

  2017  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)

  2018 txt{*converse direction*}

  2019 apply (drule_tac x = "n div k" in spec)

  2020 apply (drule_tac x = "n mod k" in spec, simp)

  2021 done

  2022

  2023 lemma split_zdiv:

  2024  "P(n div k :: int) =

  2025   ((k = 0 --> P 0) &

  2026    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &

  2027    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"

  2028 apply (case_tac "k=0", simp)

  2029 apply (simp only: linorder_neq_iff)

  2030 apply (erule disjE)

  2031  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]

  2032                       split_neg_lemma [of concl: "%x y. P x"])

  2033 done

  2034

  2035 lemma split_zmod:

  2036  "P(n mod k :: int) =

  2037   ((k = 0 --> P n) &

  2038    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &

  2039    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"

  2040 apply (case_tac "k=0", simp)

  2041 apply (simp only: linorder_neq_iff)

  2042 apply (erule disjE)

  2043  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]

  2044                       split_neg_lemma [of concl: "%x y. P y"])

  2045 done

  2046

  2047 text {* Enable (lin)arith to deal with @{const div} and @{const mod}

  2048   when these are applied to some constant that is of the form

  2049   @{term "number_of k"}: *}

  2050 declare split_zdiv [of _ _ "number_of k", arith_split] for k

  2051 declare split_zmod [of _ _ "number_of k", arith_split] for k

  2052

  2053

  2054 subsubsection{*Speeding up the Division Algorithm with Shifting*}

  2055

  2056 text{*computing div by shifting *}

  2057

  2058 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"

  2059 proof cases

  2060   assume "a=0"

  2061     thus ?thesis by simp

  2062 next

  2063   assume "a\<noteq>0" and le_a: "0\<le>a"

  2064   hence a_pos: "1 \<le> a" by arith

  2065   hence one_less_a2: "1 < 2 * a" by arith

  2066   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"

  2067     unfolding mult_le_cancel_left

  2068     by (simp add: add1_zle_eq add_commute [of 1])

  2069   with a_pos have "0 \<le> b mod a" by simp

  2070   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"

  2071     by (simp add: mod_pos_pos_trivial one_less_a2)

  2072   with  le_2a

  2073   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"

  2074     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2

  2075                   right_distrib)

  2076   thus ?thesis

  2077     by (subst zdiv_zadd1_eq,

  2078         simp add: mod_mult_mult1 one_less_a2

  2079                   div_pos_pos_trivial)

  2080 qed

  2081

  2082 lemma neg_zdiv_mult_2:

  2083   assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"

  2084 proof -

  2085   have R: "1 + - (2 * (b + 1)) = - (1 + 2 * b)" by simp

  2086   have "(1 + 2 * (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a)"

  2087     by (rule pos_zdiv_mult_2, simp add: A)

  2088   thus ?thesis

  2089     by (simp only: R zdiv_zminus_zminus diff_minus

  2090       minus_add_distrib [symmetric] mult_minus_right)

  2091 qed

  2092

  2093 lemma zdiv_number_of_Bit0 [simp]:

  2094      "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =

  2095           number_of v div (number_of w :: int)"

  2096 by (simp only: number_of_eq numeral_simps) (simp add: mult_2 [symmetric])

  2097

  2098 lemma zdiv_number_of_Bit1 [simp]:

  2099      "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =

  2100           (if (0::int) \<le> number_of w

  2101            then number_of v div (number_of w)

  2102            else (number_of v + (1::int)) div (number_of w))"

  2103 apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if)

  2104 apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac mult_2 [symmetric])

  2105 done

  2106

  2107

  2108 subsubsection{*Computing mod by Shifting (proofs resemble those for div)*}

  2109

  2110 lemma pos_zmod_mult_2:

  2111   fixes a b :: int

  2112   assumes "0 \<le> a"

  2113   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"

  2114 proof (cases "0 < a")

  2115   case False with assms show ?thesis by simp

  2116 next

  2117   case True

  2118   then have "b mod a < a" by (rule pos_mod_bound)

  2119   then have "1 + b mod a \<le> a" by simp

  2120   then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp

  2121   from 0 < a have "0 \<le> b mod a" by (rule pos_mod_sign)

  2122   then have B: "0 \<le> 1 + 2 * (b mod a)" by simp

  2123   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)"

  2124     using 0 < a and A

  2125     by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)

  2126   then show ?thesis by (subst mod_add_eq)

  2127 qed

  2128

  2129 lemma neg_zmod_mult_2:

  2130   fixes a b :: int

  2131   assumes "a \<le> 0"

  2132   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"

  2133 proof -

  2134   from assms have "0 \<le> - a" by auto

  2135   then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"

  2136     by (rule pos_zmod_mult_2)

  2137   then show ?thesis by (simp add: zmod_zminus2 algebra_simps)

  2138      (simp add: diff_minus add_ac)

  2139 qed

  2140

  2141 lemma zmod_number_of_Bit0 [simp]:

  2142      "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =

  2143       (2::int) * (number_of v mod number_of w)"

  2144 apply (simp only: number_of_eq numeral_simps)

  2145 apply (simp add: mod_mult_mult1 pos_zmod_mult_2

  2146                  neg_zmod_mult_2 add_ac mult_2 [symmetric])

  2147 done

  2148

  2149 lemma zmod_number_of_Bit1 [simp]:

  2150      "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =

  2151       (if (0::int) \<le> number_of w

  2152                 then 2 * (number_of v mod number_of w) + 1

  2153                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"

  2154 apply (simp only: number_of_eq numeral_simps)

  2155 apply (simp add: mod_mult_mult1 pos_zmod_mult_2

  2156                  neg_zmod_mult_2 add_ac mult_2 [symmetric])

  2157 done

  2158

  2159

  2160 lemma zdiv_eq_0_iff:

  2161  "(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")

  2162 proof

  2163   assume ?L

  2164   have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp

  2165   with ?L show ?R by blast

  2166 next

  2167   assume ?R thus ?L

  2168     by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)

  2169 qed

  2170

  2171

  2172 subsubsection{*Quotients of Signs*}

  2173

  2174 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"

  2175 apply (subgoal_tac "a div b \<le> -1", force)

  2176 apply (rule order_trans)

  2177 apply (rule_tac a' = "-1" in zdiv_mono1)

  2178 apply (auto simp add: div_eq_minus1)

  2179 done

  2180

  2181 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"

  2182 by (drule zdiv_mono1_neg, auto)

  2183

  2184 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"

  2185 by (drule zdiv_mono1, auto)

  2186

  2187 text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}

  2188 conditional upon the sign of @{text a} or @{text b}. There are many more.

  2189 They should all be simp rules unless that causes too much search. *}

  2190

  2191 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"

  2192 apply auto

  2193 apply (drule_tac [2] zdiv_mono1)

  2194 apply (auto simp add: linorder_neq_iff)

  2195 apply (simp (no_asm_use) add: linorder_not_less [symmetric])

  2196 apply (blast intro: div_neg_pos_less0)

  2197 done

  2198

  2199 lemma neg_imp_zdiv_nonneg_iff:

  2200   "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"

  2201 apply (subst zdiv_zminus_zminus [symmetric])

  2202 apply (subst pos_imp_zdiv_nonneg_iff, auto)

  2203 done

  2204

  2205 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)

  2206 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"

  2207 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)

  2208

  2209 lemma pos_imp_zdiv_pos_iff:

  2210   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"

  2211 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]

  2212 by arith

  2213

  2214 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)

  2215 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"

  2216 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)

  2217

  2218 lemma nonneg1_imp_zdiv_pos_iff:

  2219   "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"

  2220 apply rule

  2221  apply rule

  2222   using div_pos_pos_trivial[of a b]apply arith

  2223  apply(cases "b=0")apply simp

  2224  using div_nonneg_neg_le0[of a b]apply arith

  2225 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp

  2226 done

  2227

  2228

  2229 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"

  2230 apply (rule split_zmod[THEN iffD2])

  2231 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)

  2232 done

  2233

  2234

  2235 subsubsection {* The Divides Relation *}

  2236

  2237 lemmas zdvd_iff_zmod_eq_0_number_of [simp] =

  2238   dvd_eq_mod_eq_0 [of "number_of x" "number_of y"] for x y :: int

  2239

  2240 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"

  2241   by (rule dvd_mod) (* TODO: remove *)

  2242

  2243 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"

  2244   by (rule dvd_mod_imp_dvd) (* TODO: remove *)

  2245

  2246 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"

  2247   using zmod_zdiv_equality[where a="m" and b="n"]

  2248   by (simp add: algebra_simps)

  2249

  2250 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"

  2251 apply (induct "y", auto)

  2252 apply (rule zmod_zmult1_eq [THEN trans])

  2253 apply (simp (no_asm_simp))

  2254 apply (rule mod_mult_eq [symmetric])

  2255 done

  2256

  2257 lemma zdiv_int: "int (a div b) = (int a) div (int b)"

  2258 apply (subst split_div, auto)

  2259 apply (subst split_zdiv, auto)

  2260 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)

  2261 apply (auto simp add: divmod_int_rel_def of_nat_mult)

  2262 done

  2263

  2264 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"

  2265 apply (subst split_mod, auto)

  2266 apply (subst split_zmod, auto)

  2267 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia

  2268        in unique_remainder)

  2269 apply (auto simp add: divmod_int_rel_def of_nat_mult)

  2270 done

  2271

  2272 lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"

  2273 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)

  2274

  2275 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"

  2276 apply (subgoal_tac "m mod n = 0")

  2277  apply (simp add: zmult_div_cancel)

  2278 apply (simp only: dvd_eq_mod_eq_0)

  2279 done

  2280

  2281 text{*Suggested by Matthias Daum*}

  2282 lemma int_power_div_base:

  2283      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"

  2284 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")

  2285  apply (erule ssubst)

  2286  apply (simp only: power_add)

  2287  apply simp_all

  2288 done

  2289

  2290 text {* by Brian Huffman *}

  2291 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"

  2292 by (rule mod_minus_eq [symmetric])

  2293

  2294 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"

  2295 by (rule mod_diff_left_eq [symmetric])

  2296

  2297 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"

  2298 by (rule mod_diff_right_eq [symmetric])

  2299

  2300 lemmas zmod_simps =

  2301   mod_add_left_eq  [symmetric]

  2302   mod_add_right_eq [symmetric]

  2303   zmod_zmult1_eq   [symmetric]

  2304   mod_mult_left_eq [symmetric]

  2305   zpower_zmod

  2306   zminus_zmod zdiff_zmod_left zdiff_zmod_right

  2307

  2308 text {* Distributive laws for function @{text nat}. *}

  2309

  2310 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"

  2311 apply (rule linorder_cases [of y 0])

  2312 apply (simp add: div_nonneg_neg_le0)

  2313 apply simp

  2314 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)

  2315 done

  2316

  2317 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)

  2318 lemma nat_mod_distrib:

  2319   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"

  2320 apply (case_tac "y = 0", simp)

  2321 apply (simp add: nat_eq_iff zmod_int)

  2322 done

  2323

  2324 text  {* transfer setup *}

  2325

  2326 lemma transfer_nat_int_functions:

  2327     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"

  2328     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"

  2329   by (auto simp add: nat_div_distrib nat_mod_distrib)

  2330

  2331 lemma transfer_nat_int_function_closures:

  2332     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"

  2333     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"

  2334   apply (cases "y = 0")

  2335   apply (auto simp add: pos_imp_zdiv_nonneg_iff)

  2336   apply (cases "y = 0")

  2337   apply auto

  2338 done

  2339

  2340 declare transfer_morphism_nat_int [transfer add return:

  2341   transfer_nat_int_functions

  2342   transfer_nat_int_function_closures

  2343 ]

  2344

  2345 lemma transfer_int_nat_functions:

  2346     "(int x) div (int y) = int (x div y)"

  2347     "(int x) mod (int y) = int (x mod y)"

  2348   by (auto simp add: zdiv_int zmod_int)

  2349

  2350 lemma transfer_int_nat_function_closures:

  2351     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"

  2352     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"

  2353   by (simp_all only: is_nat_def transfer_nat_int_function_closures)

  2354

  2355 declare transfer_morphism_int_nat [transfer add return:

  2356   transfer_int_nat_functions

  2357   transfer_int_nat_function_closures

  2358 ]

  2359

  2360 text{*Suggested by Matthias Daum*}

  2361 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"

  2362 apply (subgoal_tac "nat x div nat k < nat x")

  2363  apply (simp add: nat_div_distrib [symmetric])

  2364 apply (rule Divides.div_less_dividend, simp_all)

  2365 done

  2366

  2367 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"

  2368 proof

  2369   assume H: "x mod n = y mod n"

  2370   hence "x mod n - y mod n = 0" by simp

  2371   hence "(x mod n - y mod n) mod n = 0" by simp

  2372   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])

  2373   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)

  2374 next

  2375   assume H: "n dvd x - y"

  2376   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast

  2377   hence "x = n*k + y" by simp

  2378   hence "x mod n = (n*k + y) mod n" by simp

  2379   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)

  2380 qed

  2381

  2382 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"

  2383   shows "\<exists>q. x = y + n * q"

  2384 proof-

  2385   from xy have th: "int x - int y = int (x - y)" by simp

  2386   from xyn have "int x mod int n = int y mod int n"

  2387     by (simp add: zmod_int[symmetric])

  2388   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])

  2389   hence "n dvd x - y" by (simp add: th zdvd_int)

  2390   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith

  2391 qed

  2392

  2393 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"

  2394   (is "?lhs = ?rhs")

  2395 proof

  2396   assume H: "x mod n = y mod n"

  2397   {assume xy: "x \<le> y"

  2398     from H have th: "y mod n = x mod n" by simp

  2399     from nat_mod_eq_lemma[OF th xy] have ?rhs

  2400       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}

  2401   moreover

  2402   {assume xy: "y \<le> x"

  2403     from nat_mod_eq_lemma[OF H xy] have ?rhs

  2404       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}

  2405   ultimately  show ?rhs using linear[of x y] by blast

  2406 next

  2407   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast

  2408   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp

  2409   thus  ?lhs by simp

  2410 qed

  2411

  2412 lemma div_nat_number_of [simp]:

  2413      "(number_of v :: nat)  div  number_of v' =

  2414           (if neg (number_of v :: int) then 0

  2415            else nat (number_of v div number_of v'))"

  2416   unfolding nat_number_of_def number_of_is_id neg_def

  2417   by (simp add: nat_div_distrib)

  2418

  2419 lemma one_div_nat_number_of [simp]:

  2420      "Suc 0 div number_of v' = nat (1 div number_of v')"

  2421 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])

  2422

  2423 lemma mod_nat_number_of [simp]:

  2424      "(number_of v :: nat)  mod  number_of v' =

  2425         (if neg (number_of v :: int) then 0

  2426          else if neg (number_of v' :: int) then number_of v

  2427          else nat (number_of v mod number_of v'))"

  2428   unfolding nat_number_of_def number_of_is_id neg_def

  2429   by (simp add: nat_mod_distrib)

  2430

  2431 lemma one_mod_nat_number_of [simp]:

  2432      "Suc 0 mod number_of v' =

  2433         (if neg (number_of v' :: int) then Suc 0

  2434          else nat (1 mod number_of v'))"

  2435 by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric])

  2436

  2437 lemmas dvd_eq_mod_eq_0_number_of [simp] =

  2438   dvd_eq_mod_eq_0 [of "number_of x" "number_of y"] for x y

  2439

  2440

  2441 subsubsection {* Nitpick *}

  2442

  2443 lemma zmod_zdiv_equality':

  2444 "(m\<Colon>int) mod n = m - (m div n) * n"

  2445 by (rule_tac P="%x. m mod n = x - (m div n) * n"

  2446     in subst [OF mod_div_equality [of _ n]])

  2447    arith

  2448

  2449 lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'

  2450

  2451

  2452 subsubsection {* Code generation *}

  2453

  2454 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  2455   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"

  2456

  2457 lemma pdivmod_posDivAlg [code]:

  2458   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"

  2459 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)

  2460

  2461 lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  2462   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0

  2463     then pdivmod k l

  2464     else (let (r, s) = pdivmod k l in

  2465       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  2466 proof -

  2467   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto

  2468   show ?thesis

  2469     by (simp add: divmod_int_mod_div pdivmod_def)

  2470       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if

  2471       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)

  2472 qed

  2473

  2474 lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  2475   apsnd ((op *) (sgn l)) (if sgn k = sgn l

  2476     then pdivmod k l

  2477     else (let (r, s) = pdivmod k l in

  2478       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  2479 proof -

  2480   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"

  2481     by (auto simp add: not_less sgn_if)

  2482   then show ?thesis by (simp add: divmod_int_pdivmod)

  2483 qed

  2484

  2485 context ring_1

  2486 begin

  2487

  2488 lemma of_int_num [code]:

  2489   "of_int k = (if k = 0 then 0 else if k < 0 then

  2490      - of_int (- k) else let

  2491        (l, m) = divmod_int k 2;

  2492        l' = of_int l

  2493      in if m = 0 then l' + l' else l' + l' + 1)"

  2494 proof -

  2495   have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow>

  2496     of_int k = of_int (k div 2 * 2 + 1)"

  2497   proof -

  2498     have "k mod 2 < 2" by (auto intro: pos_mod_bound)

  2499     moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)

  2500     moreover assume "k mod 2 \<noteq> 0"

  2501     ultimately have "k mod 2 = 1" by arith

  2502     moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp

  2503     ultimately show ?thesis by auto

  2504   qed

  2505   have aux2: "\<And>x. of_int 2 * x = x + x"

  2506   proof -

  2507     fix x

  2508     have int2: "(2::int) = 1 + 1" by arith

  2509     show "of_int 2 * x = x + x"

  2510     unfolding int2 of_int_add left_distrib by simp

  2511   qed

  2512   have aux3: "\<And>x. x * of_int 2 = x + x"

  2513   proof -

  2514     fix x

  2515     have int2: "(2::int) = 1 + 1" by arith

  2516     show "x * of_int 2 = x + x"

  2517     unfolding int2 of_int_add right_distrib by simp

  2518   qed

  2519   from aux1 show ?thesis by (auto simp add: divmod_int_mod_div Let_def aux2 aux3)

  2520 qed

  2521

  2522 end

  2523

  2524 code_modulename SML

  2525   Divides Arith

  2526

  2527 code_modulename OCaml

  2528   Divides Arith

  2529

  2530 code_modulename Haskell

  2531   Divides Arith

  2532

  2533 end
`