src/HOL/Divides.thy
 author huffman Tue Mar 27 12:42:54 2012 +0200 (2012-03-27) changeset 47141 02d6b816e4b3 parent 47140 97c3676c5c94 child 47142 d64fa2ca54b8 permissions -rw-r--r--
move int::ring_div instance upward, simplify several proofs
     1 (*  Title:      HOL/Divides.thy

     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     3     Copyright   1999  University of Cambridge

     4 *)

     5

     6 header {* The division operators div and mod *}

     7

     8 theory Divides

     9 imports Nat_Numeral Nat_Transfer

    10 uses "~~/src/Provers/Arith/cancel_div_mod.ML"

    11 begin

    12

    13 subsection {* Syntactic division operations *}

    14

    15 class div = dvd +

    16   fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)

    17     and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)

    18

    19

    20 subsection {* Abstract division in commutative semirings. *}

    21

    22 class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +

    23   assumes mod_div_equality: "a div b * b + a mod b = a"

    24     and div_by_0 [simp]: "a div 0 = 0"

    25     and div_0 [simp]: "0 div a = 0"

    26     and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"

    27     and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"

    28 begin

    29

    30 text {* @{const div} and @{const mod} *}

    31

    32 lemma mod_div_equality2: "b * (a div b) + a mod b = a"

    33   unfolding mult_commute [of b]

    34   by (rule mod_div_equality)

    35

    36 lemma mod_div_equality': "a mod b + a div b * b = a"

    37   using mod_div_equality [of a b]

    38   by (simp only: add_ac)

    39

    40 lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"

    41   by (simp add: mod_div_equality)

    42

    43 lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"

    44   by (simp add: mod_div_equality2)

    45

    46 lemma mod_by_0 [simp]: "a mod 0 = a"

    47   using mod_div_equality [of a zero] by simp

    48

    49 lemma mod_0 [simp]: "0 mod a = 0"

    50   using mod_div_equality [of zero a] div_0 by simp

    51

    52 lemma div_mult_self2 [simp]:

    53   assumes "b \<noteq> 0"

    54   shows "(a + b * c) div b = c + a div b"

    55   using assms div_mult_self1 [of b a c] by (simp add: mult_commute)

    56

    57 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"

    58 proof (cases "b = 0")

    59   case True then show ?thesis by simp

    60 next

    61   case False

    62   have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"

    63     by (simp add: mod_div_equality)

    64   also from False div_mult_self1 [of b a c] have

    65     "\<dots> = (c + a div b) * b + (a + c * b) mod b"

    66       by (simp add: algebra_simps)

    67   finally have "a = a div b * b + (a + c * b) mod b"

    68     by (simp add: add_commute [of a] add_assoc left_distrib)

    69   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"

    70     by (simp add: mod_div_equality)

    71   then show ?thesis by simp

    72 qed

    73

    74 lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"

    75   by (simp add: mult_commute [of b])

    76

    77 lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"

    78   using div_mult_self2 [of b 0 a] by simp

    79

    80 lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"

    81   using div_mult_self1 [of b 0 a] by simp

    82

    83 lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"

    84   using mod_mult_self2 [of 0 b a] by simp

    85

    86 lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"

    87   using mod_mult_self1 [of 0 a b] by simp

    88

    89 lemma div_by_1 [simp]: "a div 1 = a"

    90   using div_mult_self2_is_id [of 1 a] zero_neq_one by simp

    91

    92 lemma mod_by_1 [simp]: "a mod 1 = 0"

    93 proof -

    94   from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp

    95   then have "a + a mod 1 = a + 0" by simp

    96   then show ?thesis by (rule add_left_imp_eq)

    97 qed

    98

    99 lemma mod_self [simp]: "a mod a = 0"

   100   using mod_mult_self2_is_0 [of 1] by simp

   101

   102 lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"

   103   using div_mult_self2_is_id [of _ 1] by simp

   104

   105 lemma div_add_self1 [simp]:

   106   assumes "b \<noteq> 0"

   107   shows "(b + a) div b = a div b + 1"

   108   using assms div_mult_self1 [of b a 1] by (simp add: add_commute)

   109

   110 lemma div_add_self2 [simp]:

   111   assumes "b \<noteq> 0"

   112   shows "(a + b) div b = a div b + 1"

   113   using assms div_add_self1 [of b a] by (simp add: add_commute)

   114

   115 lemma mod_add_self1 [simp]:

   116   "(b + a) mod b = a mod b"

   117   using mod_mult_self1 [of a 1 b] by (simp add: add_commute)

   118

   119 lemma mod_add_self2 [simp]:

   120   "(a + b) mod b = a mod b"

   121   using mod_mult_self1 [of a 1 b] by simp

   122

   123 lemma mod_div_decomp:

   124   fixes a b

   125   obtains q r where "q = a div b" and "r = a mod b"

   126     and "a = q * b + r"

   127 proof -

   128   from mod_div_equality have "a = a div b * b + a mod b" by simp

   129   moreover have "a div b = a div b" ..

   130   moreover have "a mod b = a mod b" ..

   131   note that ultimately show thesis by blast

   132 qed

   133

   134 lemma dvd_eq_mod_eq_0 [code]: "a dvd b \<longleftrightarrow> b mod a = 0"

   135 proof

   136   assume "b mod a = 0"

   137   with mod_div_equality [of b a] have "b div a * a = b" by simp

   138   then have "b = a * (b div a)" unfolding mult_commute ..

   139   then have "\<exists>c. b = a * c" ..

   140   then show "a dvd b" unfolding dvd_def .

   141 next

   142   assume "a dvd b"

   143   then have "\<exists>c. b = a * c" unfolding dvd_def .

   144   then obtain c where "b = a * c" ..

   145   then have "b mod a = a * c mod a" by simp

   146   then have "b mod a = c * a mod a" by (simp add: mult_commute)

   147   then show "b mod a = 0" by simp

   148 qed

   149

   150 lemma mod_div_trivial [simp]: "a mod b div b = 0"

   151 proof (cases "b = 0")

   152   assume "b = 0"

   153   thus ?thesis by simp

   154 next

   155   assume "b \<noteq> 0"

   156   hence "a div b + a mod b div b = (a mod b + a div b * b) div b"

   157     by (rule div_mult_self1 [symmetric])

   158   also have "\<dots> = a div b"

   159     by (simp only: mod_div_equality')

   160   also have "\<dots> = a div b + 0"

   161     by simp

   162   finally show ?thesis

   163     by (rule add_left_imp_eq)

   164 qed

   165

   166 lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"

   167 proof -

   168   have "a mod b mod b = (a mod b + a div b * b) mod b"

   169     by (simp only: mod_mult_self1)

   170   also have "\<dots> = a mod b"

   171     by (simp only: mod_div_equality')

   172   finally show ?thesis .

   173 qed

   174

   175 lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"

   176 by (rule dvd_eq_mod_eq_0[THEN iffD1])

   177

   178 lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"

   179 by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)

   180

   181 lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"

   182 by (drule dvd_div_mult_self) (simp add: mult_commute)

   183

   184 lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"

   185 apply (cases "a = 0")

   186  apply simp

   187 apply (auto simp: dvd_def mult_assoc)

   188 done

   189

   190 lemma div_dvd_div[simp]:

   191   "a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"

   192 apply (cases "a = 0")

   193  apply simp

   194 apply (unfold dvd_def)

   195 apply auto

   196  apply(blast intro:mult_assoc[symmetric])

   197 apply(fastforce simp add: mult_assoc)

   198 done

   199

   200 lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"

   201   apply (subgoal_tac "k dvd (m div n) *n + m mod n")

   202    apply (simp add: mod_div_equality)

   203   apply (simp only: dvd_add dvd_mult)

   204   done

   205

   206 text {* Addition respects modular equivalence. *}

   207

   208 lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"

   209 proof -

   210   have "(a + b) mod c = (a div c * c + a mod c + b) mod c"

   211     by (simp only: mod_div_equality)

   212   also have "\<dots> = (a mod c + b + a div c * c) mod c"

   213     by (simp only: add_ac)

   214   also have "\<dots> = (a mod c + b) mod c"

   215     by (rule mod_mult_self1)

   216   finally show ?thesis .

   217 qed

   218

   219 lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"

   220 proof -

   221   have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"

   222     by (simp only: mod_div_equality)

   223   also have "\<dots> = (a + b mod c + b div c * c) mod c"

   224     by (simp only: add_ac)

   225   also have "\<dots> = (a + b mod c) mod c"

   226     by (rule mod_mult_self1)

   227   finally show ?thesis .

   228 qed

   229

   230 lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"

   231 by (rule trans [OF mod_add_left_eq mod_add_right_eq])

   232

   233 lemma mod_add_cong:

   234   assumes "a mod c = a' mod c"

   235   assumes "b mod c = b' mod c"

   236   shows "(a + b) mod c = (a' + b') mod c"

   237 proof -

   238   have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"

   239     unfolding assms ..

   240   thus ?thesis

   241     by (simp only: mod_add_eq [symmetric])

   242 qed

   243

   244 lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y

   245   \<Longrightarrow> (x + y) div z = x div z + y div z"

   246 by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)

   247

   248 text {* Multiplication respects modular equivalence. *}

   249

   250 lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"

   251 proof -

   252   have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"

   253     by (simp only: mod_div_equality)

   254   also have "\<dots> = (a mod c * b + a div c * b * c) mod c"

   255     by (simp only: algebra_simps)

   256   also have "\<dots> = (a mod c * b) mod c"

   257     by (rule mod_mult_self1)

   258   finally show ?thesis .

   259 qed

   260

   261 lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"

   262 proof -

   263   have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"

   264     by (simp only: mod_div_equality)

   265   also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"

   266     by (simp only: algebra_simps)

   267   also have "\<dots> = (a * (b mod c)) mod c"

   268     by (rule mod_mult_self1)

   269   finally show ?thesis .

   270 qed

   271

   272 lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"

   273 by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])

   274

   275 lemma mod_mult_cong:

   276   assumes "a mod c = a' mod c"

   277   assumes "b mod c = b' mod c"

   278   shows "(a * b) mod c = (a' * b') mod c"

   279 proof -

   280   have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"

   281     unfolding assms ..

   282   thus ?thesis

   283     by (simp only: mod_mult_eq [symmetric])

   284 qed

   285

   286 lemma mod_mod_cancel:

   287   assumes "c dvd b"

   288   shows "a mod b mod c = a mod c"

   289 proof -

   290   from c dvd b obtain k where "b = c * k"

   291     by (rule dvdE)

   292   have "a mod b mod c = a mod (c * k) mod c"

   293     by (simp only: b = c * k)

   294   also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"

   295     by (simp only: mod_mult_self1)

   296   also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"

   297     by (simp only: add_ac mult_ac)

   298   also have "\<dots> = a mod c"

   299     by (simp only: mod_div_equality)

   300   finally show ?thesis .

   301 qed

   302

   303 lemma div_mult_div_if_dvd:

   304   "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"

   305   apply (cases "y = 0", simp)

   306   apply (cases "z = 0", simp)

   307   apply (auto elim!: dvdE simp add: algebra_simps)

   308   apply (subst mult_assoc [symmetric])

   309   apply (simp add: no_zero_divisors)

   310   done

   311

   312 lemma div_mult_swap:

   313   assumes "c dvd b"

   314   shows "a * (b div c) = (a * b) div c"

   315 proof -

   316   from assms have "b div c * (a div 1) = b * a div (c * 1)"

   317     by (simp only: div_mult_div_if_dvd one_dvd)

   318   then show ?thesis by (simp add: mult_commute)

   319 qed

   320

   321 lemma div_mult_mult2 [simp]:

   322   "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"

   323   by (drule div_mult_mult1) (simp add: mult_commute)

   324

   325 lemma div_mult_mult1_if [simp]:

   326   "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"

   327   by simp_all

   328

   329 lemma mod_mult_mult1:

   330   "(c * a) mod (c * b) = c * (a mod b)"

   331 proof (cases "c = 0")

   332   case True then show ?thesis by simp

   333 next

   334   case False

   335   from mod_div_equality

   336   have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .

   337   with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)

   338     = c * a + c * (a mod b)" by (simp add: algebra_simps)

   339   with mod_div_equality show ?thesis by simp

   340 qed

   341

   342 lemma mod_mult_mult2:

   343   "(a * c) mod (b * c) = (a mod b) * c"

   344   using mod_mult_mult1 [of c a b] by (simp add: mult_commute)

   345

   346 lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"

   347   unfolding dvd_def by (auto simp add: mod_mult_mult1)

   348

   349 lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"

   350 by (blast intro: dvd_mod_imp_dvd dvd_mod)

   351

   352 lemma div_power:

   353   "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"

   354 apply (induct n)

   355  apply simp

   356 apply(simp add: div_mult_div_if_dvd dvd_power_same)

   357 done

   358

   359 lemma dvd_div_eq_mult:

   360   assumes "a \<noteq> 0" and "a dvd b"

   361   shows "b div a = c \<longleftrightarrow> b = c * a"

   362 proof

   363   assume "b = c * a"

   364   then show "b div a = c" by (simp add: assms)

   365 next

   366   assume "b div a = c"

   367   then have "b div a * a = c * a" by simp

   368   moreover from a dvd b have "b div a * a = b" by (simp add: dvd_div_mult_self)

   369   ultimately show "b = c * a" by simp

   370 qed

   371

   372 lemma dvd_div_div_eq_mult:

   373   assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"

   374   shows "b div a = d div c \<longleftrightarrow> b * c = a * d"

   375   using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)

   376

   377 end

   378

   379 class ring_div = semiring_div + comm_ring_1

   380 begin

   381

   382 subclass ring_1_no_zero_divisors ..

   383

   384 text {* Negation respects modular equivalence. *}

   385

   386 lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"

   387 proof -

   388   have "(- a) mod b = (- (a div b * b + a mod b)) mod b"

   389     by (simp only: mod_div_equality)

   390   also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"

   391     by (simp only: minus_add_distrib minus_mult_left add_ac)

   392   also have "\<dots> = (- (a mod b)) mod b"

   393     by (rule mod_mult_self1)

   394   finally show ?thesis .

   395 qed

   396

   397 lemma mod_minus_cong:

   398   assumes "a mod b = a' mod b"

   399   shows "(- a) mod b = (- a') mod b"

   400 proof -

   401   have "(- (a mod b)) mod b = (- (a' mod b)) mod b"

   402     unfolding assms ..

   403   thus ?thesis

   404     by (simp only: mod_minus_eq [symmetric])

   405 qed

   406

   407 text {* Subtraction respects modular equivalence. *}

   408

   409 lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"

   410   unfolding diff_minus

   411   by (intro mod_add_cong mod_minus_cong) simp_all

   412

   413 lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"

   414   unfolding diff_minus

   415   by (intro mod_add_cong mod_minus_cong) simp_all

   416

   417 lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"

   418   unfolding diff_minus

   419   by (intro mod_add_cong mod_minus_cong) simp_all

   420

   421 lemma mod_diff_cong:

   422   assumes "a mod c = a' mod c"

   423   assumes "b mod c = b' mod c"

   424   shows "(a - b) mod c = (a' - b') mod c"

   425   unfolding diff_minus using assms

   426   by (intro mod_add_cong mod_minus_cong)

   427

   428 lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"

   429 apply (case_tac "y = 0") apply simp

   430 apply (auto simp add: dvd_def)

   431 apply (subgoal_tac "-(y * k) = y * - k")

   432  apply (erule ssubst)

   433  apply (erule div_mult_self1_is_id)

   434 apply simp

   435 done

   436

   437 lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"

   438 apply (case_tac "y = 0") apply simp

   439 apply (auto simp add: dvd_def)

   440 apply (subgoal_tac "y * k = -y * -k")

   441  apply (erule ssubst)

   442  apply (rule div_mult_self1_is_id)

   443  apply simp

   444 apply simp

   445 done

   446

   447 end

   448

   449

   450 subsection {* Division on @{typ nat} *}

   451

   452 text {*

   453   We define @{const div} and @{const mod} on @{typ nat} by means

   454   of a characteristic relation with two input arguments

   455   @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments

   456   @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).

   457 *}

   458

   459 definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where

   460   "divmod_nat_rel m n qr \<longleftrightarrow>

   461     m = fst qr * n + snd qr \<and>

   462       (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"

   463

   464 text {* @{const divmod_nat_rel} is total: *}

   465

   466 lemma divmod_nat_rel_ex:

   467   obtains q r where "divmod_nat_rel m n (q, r)"

   468 proof (cases "n = 0")

   469   case True  with that show thesis

   470     by (auto simp add: divmod_nat_rel_def)

   471 next

   472   case False

   473   have "\<exists>q r. m = q * n + r \<and> r < n"

   474   proof (induct m)

   475     case 0 with n \<noteq> 0

   476     have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp

   477     then show ?case by blast

   478   next

   479     case (Suc m) then obtain q' r'

   480       where m: "m = q' * n + r'" and n: "r' < n" by auto

   481     then show ?case proof (cases "Suc r' < n")

   482       case True

   483       from m n have "Suc m = q' * n + Suc r'" by simp

   484       with True show ?thesis by blast

   485     next

   486       case False then have "n \<le> Suc r'" by auto

   487       moreover from n have "Suc r' \<le> n" by auto

   488       ultimately have "n = Suc r'" by auto

   489       with m have "Suc m = Suc q' * n + 0" by simp

   490       with n \<noteq> 0 show ?thesis by blast

   491     qed

   492   qed

   493   with that show thesis

   494     using n \<noteq> 0 by (auto simp add: divmod_nat_rel_def)

   495 qed

   496

   497 text {* @{const divmod_nat_rel} is injective: *}

   498

   499 lemma divmod_nat_rel_unique:

   500   assumes "divmod_nat_rel m n qr"

   501     and "divmod_nat_rel m n qr'"

   502   shows "qr = qr'"

   503 proof (cases "n = 0")

   504   case True with assms show ?thesis

   505     by (cases qr, cases qr')

   506       (simp add: divmod_nat_rel_def)

   507 next

   508   case False

   509   have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"

   510   apply (rule leI)

   511   apply (subst less_iff_Suc_add)

   512   apply (auto simp add: add_mult_distrib)

   513   done

   514   from n \<noteq> 0 assms have "fst qr = fst qr'"

   515     by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)

   516   moreover from this assms have "snd qr = snd qr'"

   517     by (simp add: divmod_nat_rel_def)

   518   ultimately show ?thesis by (cases qr, cases qr') simp

   519 qed

   520

   521 text {*

   522   We instantiate divisibility on the natural numbers by

   523   means of @{const divmod_nat_rel}:

   524 *}

   525

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

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

   528

   529 lemma divmod_nat_rel_divmod_nat:

   530   "divmod_nat_rel m n (divmod_nat m n)"

   531 proof -

   532   from divmod_nat_rel_ex

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

   534   then show ?thesis

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

   536 qed

   537

   538 lemma divmod_nat_unique:

   539   assumes "divmod_nat_rel m n qr"

   540   shows "divmod_nat m n = qr"

   541   using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)

   542

   543 instantiation nat :: semiring_div

   544 begin

   545

   546 definition div_nat where

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

   548

   549 lemma fst_divmod_nat [simp]:

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

   551   by (simp add: div_nat_def)

   552

   553 definition mod_nat where

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

   555

   556 lemma snd_divmod_nat [simp]:

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

   558   by (simp add: mod_nat_def)

   559

   560 lemma divmod_nat_div_mod:

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

   562   by (simp add: prod_eq_iff)

   563

   564 lemma div_nat_unique:

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

   566   shows "m div n = q"

   567   using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)

   568

   569 lemma mod_nat_unique:

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

   571   shows "m mod n = r"

   572   using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)

   573

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

   575   using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)

   576

   577 lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"

   578   by (simp add: divmod_nat_unique divmod_nat_rel_def)

   579

   580 lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"

   581   by (simp add: divmod_nat_unique divmod_nat_rel_def)

   582

   583 lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"

   584   by (simp add: divmod_nat_unique divmod_nat_rel_def)

   585

   586 lemma divmod_nat_step:

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

   588   shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"

   589 proof (rule divmod_nat_unique)

   590   have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"

   591     by (rule divmod_nat_rel)

   592   thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"

   593     unfolding divmod_nat_rel_def using assms by auto

   594 qed

   595

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

   597

   598 lemma div_less [simp]:

   599   fixes m n :: nat

   600   assumes "m < n"

   601   shows "m div n = 0"

   602   using assms divmod_nat_base by (simp add: prod_eq_iff)

   603

   604 lemma le_div_geq:

   605   fixes m n :: nat

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

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

   608   using assms divmod_nat_step by (simp add: prod_eq_iff)

   609

   610 lemma mod_less [simp]:

   611   fixes m n :: nat

   612   assumes "m < n"

   613   shows "m mod n = m"

   614   using assms divmod_nat_base by (simp add: prod_eq_iff)

   615

   616 lemma le_mod_geq:

   617   fixes m n :: nat

   618   assumes "n \<le> m"

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

   620   using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)

   621

   622 instance proof

   623   fix m n :: nat

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

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

   626 next

   627   fix m n q :: nat

   628   assume "n \<noteq> 0"

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

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

   631 next

   632   fix m n q :: nat

   633   assume "m \<noteq> 0"

   634   hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"

   635     unfolding divmod_nat_rel_def

   636     by (auto split: split_if_asm, simp_all add: algebra_simps)

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

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

   639   thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)

   640 next

   641   fix n :: nat show "n div 0 = 0"

   642     by (simp add: div_nat_def divmod_nat_zero)

   643 next

   644   fix n :: nat show "0 div n = 0"

   645     by (simp add: div_nat_def divmod_nat_zero_left)

   646 qed

   647

   648 end

   649

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

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

   652   by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)

   653

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

   655

   656 ML {*

   657 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod

   658 (

   659   val div_name = @{const_name div};

   660   val mod_name = @{const_name mod};

   661   val mk_binop = HOLogic.mk_binop;

   662   val mk_sum = Nat_Arith.mk_sum;

   663   val dest_sum = Nat_Arith.dest_sum;

   664

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

   666

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

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

   669 )

   670 *}

   671

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

   673

   674

   675 subsubsection {* Quotient *}

   676

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

   678 by (simp add: le_div_geq linorder_not_less)

   679

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

   681 by (simp add: div_geq)

   682

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

   684 by simp

   685

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

   687 by simp

   688

   689

   690 subsubsection {* Remainder *}

   691

   692 lemma mod_less_divisor [simp]:

   693   fixes m n :: nat

   694   assumes "n > 0"

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

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

   697

   698 lemma mod_less_eq_dividend [simp]:

   699   fixes m n :: nat

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

   701 proof (rule add_leD2)

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

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

   704 qed

   705

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

   707 by (simp add: le_mod_geq linorder_not_less)

   708

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

   710 by (simp add: le_mod_geq)

   711

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

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

   714

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

   716   by (fact mod_mult_mult2 [symmetric]) (* FIXME: generalize *)

   717

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

   719   by (fact mod_mult_mult1 [symmetric]) (* FIXME: generalize *)

   720

   721 (* a simple rearrangement of mod_div_equality: *)

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

   723   using mod_div_equality2 [of n m] by arith

   724

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

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

   727   apply simp

   728   done

   729

   730 subsubsection {* Quotient and Remainder *}

   731

   732 lemma divmod_nat_rel_mult1_eq:

   733   "divmod_nat_rel b c (q, r)

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

   735 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)

   736

   737 lemma div_mult1_eq:

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

   739 by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)

   740

   741 lemma divmod_nat_rel_add1_eq:

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

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

   744 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)

   745

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

   747 lemma div_add1_eq:

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

   749 by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)

   750

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

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

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

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

   755   apply (simp add: add_mult_distrib2)

   756   done

   757

   758 lemma divmod_nat_rel_mult2_eq:

   759   "divmod_nat_rel a b (q, r)

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

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

   762

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

   764 by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])

   765

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

   767 by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])

   768

   769

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

   771

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

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

   774

   775 (* Monotonicity of div in first argument *)

   776 lemma div_le_mono [rule_format (no_asm)]:

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

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

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

   780 apply (case_tac "n<k")

   781 (* 1  case n<k *)

   782 apply simp

   783 (* 2  case n >= k *)

   784 apply (case_tac "m<k")

   785 (* 2.1  case m<k *)

   786 apply simp

   787 (* 2.2  case m>=k *)

   788 apply (simp add: div_geq diff_le_mono)

   789 done

   790

   791 (* Antimonotonicity of div in second argument *)

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

   793 apply (subgoal_tac "0<n")

   794  prefer 2 apply simp

   795 apply (induct_tac k rule: nat_less_induct)

   796 apply (rename_tac "k")

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

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

   799  prefer 2 apply simp

   800 apply (simp add: div_geq)

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

   802  prefer 2

   803  apply (blast intro: div_le_mono diff_le_mono2)

   804 apply (rule le_trans, simp)

   805 apply (simp)

   806 done

   807

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

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

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

   811 apply (rule div_le_mono2)

   812 apply (simp_all (no_asm_simp))

   813 done

   814

   815 (* Similar for "less than" *)

   816 lemma div_less_dividend [simp]:

   817   "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"

   818 apply (induct m rule: nat_less_induct)

   819 apply (rename_tac "m")

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

   821 apply (subgoal_tac "0<n")

   822  prefer 2 apply simp

   823 apply (simp add: div_geq)

   824 apply (case_tac "n<m")

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

   826   apply (rule impI less_trans_Suc)+

   827 apply assumption

   828   apply (simp_all)

   829 done

   830

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

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

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

   834 apply (induct "m" rule: nat_less_induct)

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

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

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

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

   839 apply (simp add: linorder_not_less le_Suc_eq mod_geq)

   840 apply (auto simp add: Suc_diff_le le_mod_geq)

   841 done

   842

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

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

   845

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

   847

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

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

   850   apply (cut_tac a = m in mod_div_equality)

   851   apply (simp only: add_ac)

   852   apply (blast intro: sym)

   853   done

   854

   855 lemma split_div:

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

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

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

   859 proof

   860   assume P: ?P

   861   show ?Q

   862   proof (cases)

   863     assume "k = 0"

   864     with P show ?Q by simp

   865   next

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

   867     thus ?Q

   868     proof (simp, intro allI impI)

   869       fix i j

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

   871       show "P i"

   872       proof (cases)

   873         assume "i = 0"

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

   875       next

   876         assume "i \<noteq> 0"

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

   878       qed

   879     qed

   880   qed

   881 next

   882   assume Q: ?Q

   883   show ?P

   884   proof (cases)

   885     assume "k = 0"

   886     with Q show ?P by simp

   887   next

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

   889     with Q have R: ?R by simp

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

   891     show ?P by simp

   892   qed

   893 qed

   894

   895 lemma split_div_lemma:

   896   assumes "0 < n"

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

   898 proof

   899   assume ?rhs

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

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

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

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

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

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

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

   907   from A B show ?lhs ..

   908 next

   909   assume P: ?lhs

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

   911     unfolding divmod_nat_rel_def by (auto simp add: mult_ac)

   912   with divmod_nat_rel_unique divmod_nat_rel [of m n]

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

   914   then show ?rhs by simp

   915 qed

   916

   917 theorem split_div':

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

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

   920   apply (case_tac "0 < n")

   921   apply (simp only: add: split_div_lemma)

   922   apply simp_all

   923   done

   924

   925 lemma split_mod:

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

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

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

   929 proof

   930   assume P: ?P

   931   show ?Q

   932   proof (cases)

   933     assume "k = 0"

   934     with P show ?Q by simp

   935   next

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

   937     thus ?Q

   938     proof (simp, intro allI impI)

   939       fix i j

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

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

   942     qed

   943   qed

   944 next

   945   assume Q: ?Q

   946   show ?P

   947   proof (cases)

   948     assume "k = 0"

   949     with Q show ?P by simp

   950   next

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

   952     with Q have R: ?R by simp

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

   954     show ?P by simp

   955   qed

   956 qed

   957

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

   959   using mod_div_equality [of m n] by arith

   960

   961 lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"

   962   using mod_div_equality [of m n] by arith

   963 (* FIXME: very similar to mult_div_cancel *)

   964

   965

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

   967

   968 lemma mod_induct_0:

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

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

   971   shows "P 0"

   972 proof (rule ccontr)

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

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

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

   976   proof

   977     fix k

   978     show "?A k"

   979     proof (induct k)

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

   981     next

   982       fix n

   983       assume ih: "?A n"

   984       show "?A (Suc n)"

   985       proof (clarsimp)

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

   987         have n: "Suc n < p"

   988         proof (rule ccontr)

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

   990           hence "p - Suc n = 0"

   991             by simp

   992           with y contra show "False"

   993             by simp

   994         qed

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

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

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

   998           by blast

   999         show "False"

  1000         proof (cases "n=0")

  1001           case True

  1002           with z n2 contra show ?thesis by simp

  1003         next

  1004           case False

  1005           with p have "p-n < p" by arith

  1006           with z n2 False ih show ?thesis by simp

  1007         qed

  1008       qed

  1009     qed

  1010   qed

  1011   moreover

  1012   from i obtain k where "0<k \<and> i+k=p"

  1013     by (blast dest: less_imp_add_positive)

  1014   hence "0<k \<and> i=p-k" by auto

  1015   moreover

  1016   note base

  1017   ultimately

  1018   show "False" by blast

  1019 qed

  1020

  1021 lemma mod_induct:

  1022   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"

  1023   and base: "P i" and i: "i<p" and j: "j<p"

  1024   shows "P j"

  1025 proof -

  1026   have "\<forall>j<p. P j"

  1027   proof

  1028     fix j

  1029     show "j<p \<longrightarrow> P j" (is "?A j")

  1030     proof (induct j)

  1031       from step base i show "?A 0"

  1032         by (auto elim: mod_induct_0)

  1033     next

  1034       fix k

  1035       assume ih: "?A k"

  1036       show "?A (Suc k)"

  1037       proof

  1038         assume suc: "Suc k < p"

  1039         hence k: "k<p" by simp

  1040         with ih have "P k" ..

  1041         with step k have "P (Suc k mod p)"

  1042           by blast

  1043         moreover

  1044         from suc have "Suc k mod p = Suc k"

  1045           by simp

  1046         ultimately

  1047         show "P (Suc k)" by simp

  1048       qed

  1049     qed

  1050   qed

  1051   with j show ?thesis by blast

  1052 qed

  1053

  1054 lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"

  1055   by (simp add: numeral_2_eq_2 le_div_geq)

  1056

  1057 lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"

  1058   by (simp add: numeral_2_eq_2 le_mod_geq)

  1059

  1060 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"

  1061 by (simp add: nat_mult_2 [symmetric])

  1062

  1063 lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"

  1064 proof -

  1065   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }

  1066   moreover have "m mod 2 < 2" by simp

  1067   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .

  1068   then show ?thesis by auto

  1069 qed

  1070

  1071 text{*These lemmas collapse some needless occurrences of Suc:

  1072     at least three Sucs, since two and fewer are rewritten back to Suc again!

  1073     We already have some rules to simplify operands smaller than 3.*}

  1074

  1075 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"

  1076 by (simp add: Suc3_eq_add_3)

  1077

  1078 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"

  1079 by (simp add: Suc3_eq_add_3)

  1080

  1081 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"

  1082 by (simp add: Suc3_eq_add_3)

  1083

  1084 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"

  1085 by (simp add: Suc3_eq_add_3)

  1086

  1087 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v

  1088 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v

  1089

  1090

  1091 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"

  1092 apply (induct "m")

  1093 apply (simp_all add: mod_Suc)

  1094 done

  1095

  1096 declare Suc_times_mod_eq [of "numeral w", simp] for w

  1097

  1098 lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"

  1099 by (simp add: div_le_mono)

  1100

  1101 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"

  1102 by (cases n) simp_all

  1103

  1104 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"

  1105 proof -

  1106   from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all

  1107   from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp

  1108 qed

  1109

  1110   (* Potential use of algebra : Equality modulo n*)

  1111 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"

  1112 by (simp add: mult_ac add_ac)

  1113

  1114 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"

  1115 proof -

  1116   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp

  1117   also have "... = Suc m mod n" by (rule mod_mult_self3)

  1118   finally show ?thesis .

  1119 qed

  1120

  1121 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"

  1122 apply (subst mod_Suc [of m])

  1123 apply (subst mod_Suc [of "m mod n"], simp)

  1124 done

  1125

  1126 lemma mod_2_not_eq_zero_eq_one_nat:

  1127   fixes n :: nat

  1128   shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"

  1129   by simp

  1130

  1131

  1132 subsection {* Division on @{typ int} *}

  1133

  1134 definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where

  1135     --{*definition of quotient and remainder*}

  1136   "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>

  1137     (if 0 < b then 0 \<le> r \<and> r < b else if b < 0 then b < r \<and> r \<le> 0 else q = 0))"

  1138

  1139 definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where

  1140     --{*for the division algorithm*}

  1141     "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)

  1142                          else (2 * q, r))"

  1143

  1144 text{*algorithm for the case @{text "a\<ge>0, b>0"}*}

  1145 function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1146   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)

  1147      else adjust b (posDivAlg a (2 * b)))"

  1148 by auto

  1149 termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")

  1150   (auto simp add: mult_2)

  1151

  1152 text{*algorithm for the case @{text "a<0, b>0"}*}

  1153 function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1154   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)

  1155      else adjust b (negDivAlg a (2 * b)))"

  1156 by auto

  1157 termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")

  1158   (auto simp add: mult_2)

  1159

  1160 text{*algorithm for the general case @{term "b\<noteq>0"}*}

  1161

  1162 definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1163     --{*The full division algorithm considers all possible signs for a, b

  1164        including the special case @{text "a=0, b<0"} because

  1165        @{term negDivAlg} requires @{term "a<0"}.*}

  1166   "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b

  1167                   else if a = 0 then (0, 0)

  1168                        else apsnd uminus (negDivAlg (-a) (-b))

  1169                else

  1170                   if 0 < b then negDivAlg a b

  1171                   else apsnd uminus (posDivAlg (-a) (-b)))"

  1172

  1173 instantiation int :: Divides.div

  1174 begin

  1175

  1176 definition div_int where

  1177   "a div b = fst (divmod_int a b)"

  1178

  1179 lemma fst_divmod_int [simp]:

  1180   "fst (divmod_int a b) = a div b"

  1181   by (simp add: div_int_def)

  1182

  1183 definition mod_int where

  1184   "a mod b = snd (divmod_int a b)"

  1185

  1186 lemma snd_divmod_int [simp]:

  1187   "snd (divmod_int a b) = a mod b"

  1188   by (simp add: mod_int_def)

  1189

  1190 instance ..

  1191

  1192 end

  1193

  1194 lemma divmod_int_mod_div:

  1195   "divmod_int p q = (p div q, p mod q)"

  1196   by (simp add: prod_eq_iff)

  1197

  1198 text{*

  1199 Here is the division algorithm in ML:

  1200

  1201 \begin{verbatim}

  1202     fun posDivAlg (a,b) =

  1203       if a<b then (0,a)

  1204       else let val (q,r) = posDivAlg(a, 2*b)

  1205                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)

  1206            end

  1207

  1208     fun negDivAlg (a,b) =

  1209       if 0\<le>a+b then (~1,a+b)

  1210       else let val (q,r) = negDivAlg(a, 2*b)

  1211                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)

  1212            end;

  1213

  1214     fun negateSnd (q,r:int) = (q,~r);

  1215

  1216     fun divmod (a,b) = if 0\<le>a then

  1217                           if b>0 then posDivAlg (a,b)

  1218                            else if a=0 then (0,0)

  1219                                 else negateSnd (negDivAlg (~a,~b))

  1220                        else

  1221                           if 0<b then negDivAlg (a,b)

  1222                           else        negateSnd (posDivAlg (~a,~b));

  1223 \end{verbatim}

  1224 *}

  1225

  1226

  1227 subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}

  1228

  1229 lemma unique_quotient_lemma:

  1230      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]

  1231       ==> q' \<le> (q::int)"

  1232 apply (subgoal_tac "r' + b * (q'-q) \<le> r")

  1233  prefer 2 apply (simp add: right_diff_distrib)

  1234 apply (subgoal_tac "0 < b * (1 + q - q') ")

  1235 apply (erule_tac [2] order_le_less_trans)

  1236  prefer 2 apply (simp add: right_diff_distrib right_distrib)

  1237 apply (subgoal_tac "b * q' < b * (1 + q) ")

  1238  prefer 2 apply (simp add: right_diff_distrib right_distrib)

  1239 apply (simp add: mult_less_cancel_left)

  1240 done

  1241

  1242 lemma unique_quotient_lemma_neg:

  1243      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]

  1244       ==> q \<le> (q'::int)"

  1245 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,

  1246     auto)

  1247

  1248 lemma unique_quotient:

  1249      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]

  1250       ==> q = q'"

  1251 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)

  1252 apply (blast intro: order_antisym

  1253              dest: order_eq_refl [THEN unique_quotient_lemma]

  1254              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+

  1255 done

  1256

  1257

  1258 lemma unique_remainder:

  1259      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]

  1260       ==> r = r'"

  1261 apply (subgoal_tac "q = q'")

  1262  apply (simp add: divmod_int_rel_def)

  1263 apply (blast intro: unique_quotient)

  1264 done

  1265

  1266

  1267 subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}

  1268

  1269 text{*And positive divisors*}

  1270

  1271 lemma adjust_eq [simp]:

  1272      "adjust b (q, r) =

  1273       (let diff = r - b in

  1274         if 0 \<le> diff then (2 * q + 1, diff)

  1275                      else (2*q, r))"

  1276   by (simp add: Let_def adjust_def)

  1277

  1278 declare posDivAlg.simps [simp del]

  1279

  1280 text{*use with a simproc to avoid repeatedly proving the premise*}

  1281 lemma posDivAlg_eqn:

  1282      "0 < b ==>

  1283       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"

  1284 by (rule posDivAlg.simps [THEN trans], simp)

  1285

  1286 text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}

  1287 theorem posDivAlg_correct:

  1288   assumes "0 \<le> a" and "0 < b"

  1289   shows "divmod_int_rel a b (posDivAlg a b)"

  1290   using assms

  1291   apply (induct a b rule: posDivAlg.induct)

  1292   apply auto

  1293   apply (simp add: divmod_int_rel_def)

  1294   apply (subst posDivAlg_eqn, simp add: right_distrib)

  1295   apply (case_tac "a < b")

  1296   apply simp_all

  1297   apply (erule splitE)

  1298   apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)

  1299   done

  1300

  1301

  1302 subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}

  1303

  1304 text{*And positive divisors*}

  1305

  1306 declare negDivAlg.simps [simp del]

  1307

  1308 text{*use with a simproc to avoid repeatedly proving the premise*}

  1309 lemma negDivAlg_eqn:

  1310      "0 < b ==>

  1311       negDivAlg a b =

  1312        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"

  1313 by (rule negDivAlg.simps [THEN trans], simp)

  1314

  1315 (*Correctness of negDivAlg: it computes quotients correctly

  1316   It doesn't work if a=0 because the 0/b equals 0, not -1*)

  1317 lemma negDivAlg_correct:

  1318   assumes "a < 0" and "b > 0"

  1319   shows "divmod_int_rel a b (negDivAlg a b)"

  1320   using assms

  1321   apply (induct a b rule: negDivAlg.induct)

  1322   apply (auto simp add: linorder_not_le)

  1323   apply (simp add: divmod_int_rel_def)

  1324   apply (subst negDivAlg_eqn, assumption)

  1325   apply (case_tac "a + b < (0\<Colon>int)")

  1326   apply simp_all

  1327   apply (erule splitE)

  1328   apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)

  1329   done

  1330

  1331

  1332 subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}

  1333

  1334 (*the case a=0*)

  1335 lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"

  1336 by (auto simp add: divmod_int_rel_def linorder_neq_iff)

  1337

  1338 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"

  1339 by (subst posDivAlg.simps, auto)

  1340

  1341 lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"

  1342 by (subst posDivAlg.simps, auto)

  1343

  1344 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"

  1345 by (subst negDivAlg.simps, auto)

  1346

  1347 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"

  1348 by (auto simp add: divmod_int_rel_def)

  1349

  1350 lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"

  1351 apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)

  1352 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg

  1353                     posDivAlg_correct negDivAlg_correct)

  1354

  1355 lemma divmod_int_unique:

  1356   assumes "divmod_int_rel a b qr"

  1357   shows "divmod_int a b = qr"

  1358   using assms divmod_int_correct [of a b]

  1359   using unique_quotient [of a b] unique_remainder [of a b]

  1360   by (metis pair_collapse)

  1361

  1362 lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"

  1363   using divmod_int_correct by (simp add: divmod_int_mod_div)

  1364

  1365 lemma div_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a div b = q"

  1366   by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])

  1367

  1368 lemma mod_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a mod b = r"

  1369   by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])

  1370

  1371 instance int :: ring_div

  1372 proof

  1373   fix a b :: int

  1374   show "a div b * b + a mod b = a"

  1375     using divmod_int_rel_div_mod [of a b]

  1376     unfolding divmod_int_rel_def by (simp add: mult_commute)

  1377 next

  1378   fix a b c :: int

  1379   assume "b \<noteq> 0"

  1380   hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"

  1381     using divmod_int_rel_div_mod [of a b]

  1382     unfolding divmod_int_rel_def by (auto simp: algebra_simps)

  1383   thus "(a + c * b) div b = c + a div b"

  1384     by (rule div_int_unique)

  1385 next

  1386   fix a b c :: int

  1387   assume "c \<noteq> 0"

  1388   hence "\<And>q r. divmod_int_rel a b (q, r)

  1389     \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"

  1390     unfolding divmod_int_rel_def

  1391     by - (rule linorder_cases [of 0 b], auto simp: algebra_simps

  1392       mult_less_0_iff zero_less_mult_iff mult_strict_right_mono

  1393       mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)

  1394   hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"

  1395     using divmod_int_rel_div_mod [of a b] .

  1396   thus "(c * a) div (c * b) = a div b"

  1397     by (rule div_int_unique)

  1398 next

  1399   fix a :: int show "a div 0 = 0"

  1400     by (rule div_int_unique, simp add: divmod_int_rel_def)

  1401 next

  1402   fix a :: int show "0 div a = 0"

  1403     by (rule div_int_unique, auto simp add: divmod_int_rel_def)

  1404 qed

  1405

  1406 text{*Arbitrary definitions for division by zero.  Useful to simplify

  1407     certain equations.*}

  1408

  1409 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"

  1410   by simp (* FIXME: delete *)

  1411

  1412 text{*Basic laws about division and remainder*}

  1413

  1414 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"

  1415   by (fact mod_div_equality2 [symmetric])

  1416

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

  1418   by (fact div_mod_equality2)

  1419

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

  1421   by (fact div_mod_equality)

  1422

  1423 text {* Tool setup *}

  1424

  1425 (* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)

  1426 lemmas add_0s = add_0_left add_0_right

  1427

  1428 ML {*

  1429 structure Cancel_Div_Mod_Int = Cancel_Div_Mod

  1430 (

  1431   val div_name = @{const_name div};

  1432   val mod_name = @{const_name mod};

  1433   val mk_binop = HOLogic.mk_binop;

  1434   val mk_sum = Arith_Data.mk_sum HOLogic.intT;

  1435   val dest_sum = Arith_Data.dest_sum;

  1436

  1437   val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];

  1438

  1439   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac

  1440     (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))

  1441 )

  1442 *}

  1443

  1444 simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}

  1445

  1446 lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"

  1447   using divmod_int_correct [of a b]

  1448   by (auto simp add: divmod_int_rel_def prod_eq_iff)

  1449

  1450 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]

  1451    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]

  1452

  1453 lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"

  1454   using divmod_int_correct [of a b]

  1455   by (auto simp add: divmod_int_rel_def prod_eq_iff)

  1456

  1457 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]

  1458    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]

  1459

  1460

  1461 subsubsection {* General Properties of div and mod *}

  1462

  1463 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"

  1464 apply (rule div_int_unique)

  1465 apply (auto simp add: divmod_int_rel_def)

  1466 done

  1467

  1468 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"

  1469 apply (rule div_int_unique)

  1470 apply (auto simp add: divmod_int_rel_def)

  1471 done

  1472

  1473 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"

  1474 apply (rule div_int_unique)

  1475 apply (auto simp add: divmod_int_rel_def)

  1476 done

  1477

  1478 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)

  1479

  1480 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"

  1481 apply (rule_tac q = 0 in mod_int_unique)

  1482 apply (auto simp add: divmod_int_rel_def)

  1483 done

  1484

  1485 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"

  1486 apply (rule_tac q = 0 in mod_int_unique)

  1487 apply (auto simp add: divmod_int_rel_def)

  1488 done

  1489

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

  1491 apply (rule_tac q = "-1" in mod_int_unique)

  1492 apply (auto simp add: divmod_int_rel_def)

  1493 done

  1494

  1495 text{*There is no @{text mod_neg_pos_trivial}.*}

  1496

  1497

  1498 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)

  1499 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"

  1500   using div_mult_mult1 [of "-1" a b] by simp (* FIXME: generalize *)

  1501

  1502 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)

  1503 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"

  1504   using mod_mult_mult1 [of "-1" a b] by simp (* FIXME: generalize *)

  1505

  1506

  1507 subsubsection {* Laws for div and mod with Unary Minus *}

  1508

  1509 lemma zminus1_lemma:

  1510      "divmod_int_rel a b (q, r) ==> b \<noteq> 0

  1511       ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,

  1512                           if r=0 then 0 else b-r)"

  1513 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)

  1514

  1515

  1516 lemma zdiv_zminus1_eq_if:

  1517      "b \<noteq> (0::int)

  1518       ==> (-a) div b =

  1519           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"

  1520 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])

  1521

  1522 lemma zmod_zminus1_eq_if:

  1523      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"

  1524 apply (case_tac "b = 0", simp)

  1525 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])

  1526 done

  1527

  1528 lemma zmod_zminus1_not_zero:

  1529   fixes k l :: int

  1530   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"

  1531   unfolding zmod_zminus1_eq_if by auto

  1532

  1533 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"

  1534   using zdiv_zminus_zminus [of "-a" b] by simp (* FIXME: generalize *)

  1535

  1536 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"

  1537   using zmod_zminus_zminus [of "-a" b] by simp (* FIXME: generalize*)

  1538

  1539 lemma zdiv_zminus2_eq_if:

  1540      "b \<noteq> (0::int)

  1541       ==> a div (-b) =

  1542           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"

  1543 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)

  1544

  1545 lemma zmod_zminus2_eq_if:

  1546      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"

  1547 by (simp add: zmod_zminus1_eq_if zmod_zminus2)

  1548

  1549 lemma zmod_zminus2_not_zero:

  1550   fixes k l :: int

  1551   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"

  1552   unfolding zmod_zminus2_eq_if by auto

  1553

  1554

  1555 subsubsection {* Division of a Number by Itself *}

  1556

  1557 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"

  1558 apply (subgoal_tac "0 < a*q")

  1559  apply (simp add: zero_less_mult_iff, arith)

  1560 done

  1561

  1562 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"

  1563 apply (subgoal_tac "0 \<le> a* (1-q) ")

  1564  apply (simp add: zero_le_mult_iff)

  1565 apply (simp add: right_diff_distrib)

  1566 done

  1567

  1568 lemma self_quotient: "[| divmod_int_rel a a (q, r); a \<noteq> 0 |] ==> q = 1"

  1569 apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)

  1570 apply (rule order_antisym, safe, simp_all)

  1571 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)

  1572 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)

  1573 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+

  1574 done

  1575

  1576 lemma self_remainder: "[| divmod_int_rel a a (q, r); a \<noteq> 0 |] ==> r = 0"

  1577 apply (frule (1) self_quotient)

  1578 apply (simp add: divmod_int_rel_def)

  1579 done

  1580

  1581 lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"

  1582   by (fact div_self) (* FIXME: delete *)

  1583

  1584 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)

  1585 lemma zmod_self [simp]: "a mod a = (0::int)"

  1586   by (fact mod_self) (* FIXME: delete *)

  1587

  1588

  1589 subsubsection {* Computation of Division and Remainder *}

  1590

  1591 lemma zdiv_zero [simp]: "(0::int) div b = 0"

  1592   by (fact div_0) (* FIXME: delete *)

  1593

  1594 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"

  1595 by (simp add: div_int_def divmod_int_def)

  1596

  1597 lemma zmod_zero [simp]: "(0::int) mod b = 0"

  1598   by (fact mod_0) (* FIXME: delete *)

  1599

  1600 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"

  1601 by (simp add: mod_int_def divmod_int_def)

  1602

  1603 text{*a positive, b positive *}

  1604

  1605 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"

  1606 by (simp add: div_int_def divmod_int_def)

  1607

  1608 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"

  1609 by (simp add: mod_int_def divmod_int_def)

  1610

  1611 text{*a negative, b positive *}

  1612

  1613 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"

  1614 by (simp add: div_int_def divmod_int_def)

  1615

  1616 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"

  1617 by (simp add: mod_int_def divmod_int_def)

  1618

  1619 text{*a positive, b negative *}

  1620

  1621 lemma div_pos_neg:

  1622      "[| 0 < a;  b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"

  1623 by (simp add: div_int_def divmod_int_def)

  1624

  1625 lemma mod_pos_neg:

  1626      "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"

  1627 by (simp add: mod_int_def divmod_int_def)

  1628

  1629 text{*a negative, b negative *}

  1630

  1631 lemma div_neg_neg:

  1632      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"

  1633 by (simp add: div_int_def divmod_int_def)

  1634

  1635 lemma mod_neg_neg:

  1636      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"

  1637 by (simp add: mod_int_def divmod_int_def)

  1638

  1639 text {*Simplify expresions in which div and mod combine numerical constants*}

  1640

  1641 lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"

  1642   by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)

  1643

  1644 lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"

  1645   by (rule div_int_unique [of a b q r],

  1646     simp add: divmod_int_rel_def)

  1647

  1648 lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"

  1649   by (rule mod_int_unique [of a b q r],

  1650     simp add: divmod_int_rel_def)

  1651

  1652 lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"

  1653   by (rule mod_int_unique [of a b q r],

  1654     simp add: divmod_int_rel_def)

  1655

  1656 (* simprocs adapted from HOL/ex/Binary.thy *)

  1657 ML {*

  1658 local

  1659   val mk_number = HOLogic.mk_number HOLogic.intT

  1660   val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}

  1661   val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}

  1662   val zero = @{term "0 :: int"}

  1663   val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}

  1664   val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}

  1665   val simps = @{thms arith_simps} @ @{thms rel_simps} @

  1666     map (fn th => th RS sym) [@{thm numeral_1_eq_1}]

  1667   fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)

  1668     (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps simps))));

  1669   fun binary_proc proc ss ct =

  1670     (case Thm.term_of ct of

  1671       _ $t$ u =>

  1672       (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of

  1673         SOME args => proc (Simplifier.the_context ss) args

  1674       | NONE => NONE)

  1675     | _ => NONE);

  1676 in

  1677   fun divmod_proc posrule negrule =

  1678     binary_proc (fn ctxt => fn ((a, t), (b, u)) =>

  1679       if b = 0 then NONE else let

  1680         val (q, r) = pairself mk_number (Integer.div_mod a b)

  1681         val goal1 = HOLogic.mk_eq (t, plus $(times$ u $q)$ r)

  1682         val (goal2, goal3, rule) = if b > 0

  1683           then (le $zero$ r, less $r$ u, posrule RS eq_reflection)

  1684           else (le $r$ zero, less $u$ r, negrule RS eq_reflection)

  1685       in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)

  1686 end

  1687 *}

  1688

  1689 simproc_setup binary_int_div

  1690   ("numeral m div numeral n :: int" |

  1691    "numeral m div neg_numeral n :: int" |

  1692    "neg_numeral m div numeral n :: int" |

  1693    "neg_numeral m div neg_numeral n :: int") =

  1694   {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}

  1695

  1696 simproc_setup binary_int_mod

  1697   ("numeral m mod numeral n :: int" |

  1698    "numeral m mod neg_numeral n :: int" |

  1699    "neg_numeral m mod numeral n :: int" |

  1700    "neg_numeral m mod neg_numeral n :: int") =

  1701   {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}

  1702

  1703 lemmas posDivAlg_eqn_numeral [simp] =

  1704     posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w

  1705

  1706 lemmas negDivAlg_eqn_numeral [simp] =

  1707     negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w

  1708

  1709

  1710 text{*Special-case simplification *}

  1711

  1712 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"

  1713 apply (cut_tac a = a and b = "-1" in neg_mod_sign)

  1714 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)

  1715 apply (auto simp del: neg_mod_sign neg_mod_bound)

  1716 done (* FIXME: generalize *)

  1717

  1718 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"

  1719 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)

  1720 (* FIXME: generalize *)

  1721

  1722 (** The last remaining special cases for constant arithmetic:

  1723     1 div z and 1 mod z **)

  1724

  1725 lemmas div_pos_pos_1_numeral [simp] =

  1726   div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w

  1727

  1728 lemmas div_pos_neg_1_numeral [simp] =

  1729   div_pos_neg [OF zero_less_one, of "neg_numeral w",

  1730   OF neg_numeral_less_zero] for w

  1731

  1732 lemmas mod_pos_pos_1_numeral [simp] =

  1733   mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w

  1734

  1735 lemmas mod_pos_neg_1_numeral [simp] =

  1736   mod_pos_neg [OF zero_less_one, of "neg_numeral w",

  1737   OF neg_numeral_less_zero] for w

  1738

  1739 lemmas posDivAlg_eqn_1_numeral [simp] =

  1740     posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w

  1741

  1742 lemmas negDivAlg_eqn_1_numeral [simp] =

  1743     negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w

  1744

  1745

  1746 subsubsection {* Monotonicity in the First Argument (Dividend) *}

  1747

  1748 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"

  1749 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1750 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)

  1751 apply (rule unique_quotient_lemma)

  1752 apply (erule subst)

  1753 apply (erule subst, simp_all)

  1754 done

  1755

  1756 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"

  1757 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1758 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)

  1759 apply (rule unique_quotient_lemma_neg)

  1760 apply (erule subst)

  1761 apply (erule subst, simp_all)

  1762 done

  1763

  1764

  1765 subsubsection {* Monotonicity in the Second Argument (Divisor) *}

  1766

  1767 lemma q_pos_lemma:

  1768      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"

  1769 apply (subgoal_tac "0 < b'* (q' + 1) ")

  1770  apply (simp add: zero_less_mult_iff)

  1771 apply (simp add: right_distrib)

  1772 done

  1773

  1774 lemma zdiv_mono2_lemma:

  1775      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';

  1776          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]

  1777       ==> q \<le> (q'::int)"

  1778 apply (frule q_pos_lemma, assumption+)

  1779 apply (subgoal_tac "b*q < b* (q' + 1) ")

  1780  apply (simp add: mult_less_cancel_left)

  1781 apply (subgoal_tac "b*q = r' - r + b'*q'")

  1782  prefer 2 apply simp

  1783 apply (simp (no_asm_simp) add: right_distrib)

  1784 apply (subst add_commute, rule add_less_le_mono, arith)

  1785 apply (rule mult_right_mono, auto)

  1786 done

  1787

  1788 lemma zdiv_mono2:

  1789      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"

  1790 apply (subgoal_tac "b \<noteq> 0")

  1791  prefer 2 apply arith

  1792 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1793 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)

  1794 apply (rule zdiv_mono2_lemma)

  1795 apply (erule subst)

  1796 apply (erule subst, simp_all)

  1797 done

  1798

  1799 lemma q_neg_lemma:

  1800      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"

  1801 apply (subgoal_tac "b'*q' < 0")

  1802  apply (simp add: mult_less_0_iff, arith)

  1803 done

  1804

  1805 lemma zdiv_mono2_neg_lemma:

  1806      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;

  1807          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]

  1808       ==> q' \<le> (q::int)"

  1809 apply (frule q_neg_lemma, assumption+)

  1810 apply (subgoal_tac "b*q' < b* (q + 1) ")

  1811  apply (simp add: mult_less_cancel_left)

  1812 apply (simp add: right_distrib)

  1813 apply (subgoal_tac "b*q' \<le> b'*q'")

  1814  prefer 2 apply (simp add: mult_right_mono_neg, arith)

  1815 done

  1816

  1817 lemma zdiv_mono2_neg:

  1818      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"

  1819 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1820 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)

  1821 apply (rule zdiv_mono2_neg_lemma)

  1822 apply (erule subst)

  1823 apply (erule subst, simp_all)

  1824 done

  1825

  1826

  1827 subsubsection {* More Algebraic Laws for div and mod *}

  1828

  1829 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}

  1830

  1831 lemma zmult1_lemma:

  1832      "[| divmod_int_rel b c (q, r) |]

  1833       ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"

  1834 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)

  1835

  1836 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"

  1837 apply (case_tac "c = 0", simp)

  1838 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])

  1839 done

  1840

  1841 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"

  1842   by (fact mod_mult_right_eq) (* FIXME: delete *)

  1843

  1844 lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"

  1845   by (fact mod_div_trivial) (* FIXME: delete *)

  1846

  1847 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}

  1848

  1849 lemma zadd1_lemma:

  1850      "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]

  1851       ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"

  1852 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)

  1853

  1854 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)

  1855 lemma zdiv_zadd1_eq:

  1856      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"

  1857 apply (case_tac "c = 0", simp)

  1858 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)

  1859 done

  1860

  1861 lemma posDivAlg_div_mod:

  1862   assumes "k \<ge> 0"

  1863   and "l \<ge> 0"

  1864   shows "posDivAlg k l = (k div l, k mod l)"

  1865 proof (cases "l = 0")

  1866   case True then show ?thesis by (simp add: posDivAlg.simps)

  1867 next

  1868   case False with assms posDivAlg_correct

  1869     have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"

  1870     by simp

  1871   from div_int_unique [OF this] mod_int_unique [OF this]

  1872   show ?thesis by simp

  1873 qed

  1874

  1875 lemma negDivAlg_div_mod:

  1876   assumes "k < 0"

  1877   and "l > 0"

  1878   shows "negDivAlg k l = (k div l, k mod l)"

  1879 proof -

  1880   from assms have "l \<noteq> 0" by simp

  1881   from assms negDivAlg_correct

  1882     have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"

  1883     by simp

  1884   from div_int_unique [OF this] mod_int_unique [OF this]

  1885   show ?thesis by simp

  1886 qed

  1887

  1888 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"

  1889 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)

  1890

  1891 (* REVISIT: should this be generalized to all semiring_div types? *)

  1892 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]

  1893

  1894 lemma zmod_zdiv_equality':

  1895   "(m\<Colon>int) mod n = m - (m div n) * n"

  1896   using mod_div_equality [of m n] by arith

  1897

  1898

  1899 subsubsection {* Proving  @{term "a div (b*c) = (a div b) div c"} *}

  1900

  1901 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but

  1902   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems

  1903   to cause particular problems.*)

  1904

  1905 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}

  1906

  1907 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"

  1908 apply (subgoal_tac "b * (c - q mod c) < r * 1")

  1909  apply (simp add: algebra_simps)

  1910 apply (rule order_le_less_trans)

  1911  apply (erule_tac [2] mult_strict_right_mono)

  1912  apply (rule mult_left_mono_neg)

  1913   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)

  1914  apply (simp)

  1915 apply (simp)

  1916 done

  1917

  1918 lemma zmult2_lemma_aux2:

  1919      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"

  1920 apply (subgoal_tac "b * (q mod c) \<le> 0")

  1921  apply arith

  1922 apply (simp add: mult_le_0_iff)

  1923 done

  1924

  1925 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"

  1926 apply (subgoal_tac "0 \<le> b * (q mod c) ")

  1927 apply arith

  1928 apply (simp add: zero_le_mult_iff)

  1929 done

  1930

  1931 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"

  1932 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")

  1933  apply (simp add: right_diff_distrib)

  1934 apply (rule order_less_le_trans)

  1935  apply (erule mult_strict_right_mono)

  1936  apply (rule_tac [2] mult_left_mono)

  1937   apply simp

  1938  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)

  1939 apply simp

  1940 done

  1941

  1942 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]

  1943       ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"

  1944 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff

  1945                    zero_less_mult_iff right_distrib [symmetric]

  1946                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)

  1947

  1948 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"

  1949 apply (case_tac "b = 0", simp)

  1950 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])

  1951 done

  1952

  1953 lemma zmod_zmult2_eq:

  1954      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"

  1955 apply (case_tac "b = 0", simp)

  1956 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])

  1957 done

  1958

  1959 lemma div_pos_geq:

  1960   fixes k l :: int

  1961   assumes "0 < l" and "l \<le> k"

  1962   shows "k div l = (k - l) div l + 1"

  1963 proof -

  1964   have "k = (k - l) + l" by simp

  1965   then obtain j where k: "k = j + l" ..

  1966   with assms show ?thesis by simp

  1967 qed

  1968

  1969 lemma mod_pos_geq:

  1970   fixes k l :: int

  1971   assumes "0 < l" and "l \<le> k"

  1972   shows "k mod l = (k - l) mod l"

  1973 proof -

  1974   have "k = (k - l) + l" by simp

  1975   then obtain j where k: "k = j + l" ..

  1976   with assms show ?thesis by simp

  1977 qed

  1978

  1979

  1980 subsubsection {* Splitting Rules for div and mod *}

  1981

  1982 text{*The proofs of the two lemmas below are essentially identical*}

  1983

  1984 lemma split_pos_lemma:

  1985  "0<k ==>

  1986     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"

  1987 apply (rule iffI, clarify)

  1988  apply (erule_tac P="P ?x ?y" in rev_mp)

  1989  apply (subst mod_add_eq)

  1990  apply (subst zdiv_zadd1_eq)

  1991  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)

  1992 txt{*converse direction*}

  1993 apply (drule_tac x = "n div k" in spec)

  1994 apply (drule_tac x = "n mod k" in spec, simp)

  1995 done

  1996

  1997 lemma split_neg_lemma:

  1998  "k<0 ==>

  1999     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & 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_neg_neg_trivial mod_neg_neg_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_zdiv:

  2011  "P(n div k :: int) =

  2012   ((k = 0 --> P 0) &

  2013    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &

  2014    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"

  2015 apply (case_tac "k=0", simp)

  2016 apply (simp only: linorder_neq_iff)

  2017 apply (erule disjE)

  2018  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]

  2019                       split_neg_lemma [of concl: "%x y. P x"])

  2020 done

  2021

  2022 lemma split_zmod:

  2023  "P(n mod k :: int) =

  2024   ((k = 0 --> P n) &

  2025    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &

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

  2027 apply (case_tac "k=0", simp)

  2028 apply (simp only: linorder_neq_iff)

  2029 apply (erule disjE)

  2030  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]

  2031                       split_neg_lemma [of concl: "%x y. P y"])

  2032 done

  2033

  2034 text {* Enable (lin)arith to deal with @{const div} and @{const mod}

  2035   when these are applied to some constant that is of the form

  2036   @{term "numeral k"}: *}

  2037 declare split_zdiv [of _ _ "numeral k", arith_split] for k

  2038 declare split_zmod [of _ _ "numeral k", arith_split] for k

  2039

  2040

  2041 subsubsection {* Speeding up the Division Algorithm with Shifting *}

  2042

  2043 text{*computing div by shifting *}

  2044

  2045 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"

  2046 proof cases

  2047   assume "a=0"

  2048     thus ?thesis by simp

  2049 next

  2050   assume "a\<noteq>0" and le_a: "0\<le>a"

  2051   hence a_pos: "1 \<le> a" by arith

  2052   hence one_less_a2: "1 < 2 * a" by arith

  2053   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"

  2054     unfolding mult_le_cancel_left

  2055     by (simp add: add1_zle_eq add_commute [of 1])

  2056   with a_pos have "0 \<le> b mod a" by simp

  2057   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"

  2058     by (simp add: mod_pos_pos_trivial one_less_a2)

  2059   with  le_2a

  2060   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"

  2061     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2

  2062                   right_distrib)

  2063   thus ?thesis

  2064     by (subst zdiv_zadd1_eq,

  2065         simp add: mod_mult_mult1 one_less_a2

  2066                   div_pos_pos_trivial)

  2067 qed

  2068

  2069 lemma neg_zdiv_mult_2:

  2070   assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"

  2071 proof -

  2072   have R: "1 + - (2 * (b + 1)) = - (1 + 2 * b)" by simp

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

  2074     by (rule pos_zdiv_mult_2, simp add: A)

  2075   thus ?thesis

  2076     by (simp only: R zdiv_zminus_zminus diff_minus

  2077       minus_add_distrib [symmetric] mult_minus_right)

  2078 qed

  2079

  2080 (* FIXME: add rules for negative numerals *)

  2081 lemma zdiv_numeral_Bit0 [simp]:

  2082   "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =

  2083     numeral v div (numeral w :: int)"

  2084   unfolding numeral.simps unfolding mult_2 [symmetric]

  2085   by (rule div_mult_mult1, simp)

  2086

  2087 lemma zdiv_numeral_Bit1 [simp]:

  2088   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =

  2089     (numeral v div (numeral w :: int))"

  2090   unfolding numeral.simps

  2091   unfolding mult_2 [symmetric] add_commute [of _ 1]

  2092   by (rule pos_zdiv_mult_2, simp)

  2093

  2094

  2095 subsubsection {* Computing mod by Shifting (proofs resemble those for div) *}

  2096

  2097 lemma pos_zmod_mult_2:

  2098   fixes a b :: int

  2099   assumes "0 \<le> a"

  2100   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"

  2101 proof (cases "0 < a")

  2102   case False with assms show ?thesis by simp

  2103 next

  2104   case True

  2105   then have "b mod a < a" by (rule pos_mod_bound)

  2106   then have "1 + b mod a \<le> a" by simp

  2107   then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp

  2108   from 0 < a have "0 \<le> b mod a" by (rule pos_mod_sign)

  2109   then have B: "0 \<le> 1 + 2 * (b mod a)" by simp

  2110   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)"

  2111     using 0 < a and A

  2112     by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)

  2113   then show ?thesis by (subst mod_add_eq)

  2114 qed

  2115

  2116 lemma neg_zmod_mult_2:

  2117   fixes a b :: int

  2118   assumes "a \<le> 0"

  2119   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"

  2120 proof -

  2121   from assms have "0 \<le> - a" by auto

  2122   then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"

  2123     by (rule pos_zmod_mult_2)

  2124   then show ?thesis by (simp add: zmod_zminus2 algebra_simps)

  2125      (simp add: diff_minus add_ac)

  2126 qed

  2127

  2128 (* FIXME: add rules for negative numerals *)

  2129 lemma zmod_numeral_Bit0 [simp]:

  2130   "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =

  2131     (2::int) * (numeral v mod numeral w)"

  2132   unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]

  2133   unfolding mult_2 [symmetric] by (rule mod_mult_mult1)

  2134

  2135 lemma zmod_numeral_Bit1 [simp]:

  2136   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =

  2137     2 * (numeral v mod numeral w) + (1::int)"

  2138   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]

  2139   unfolding mult_2 [symmetric] add_commute [of _ 1]

  2140   by (rule pos_zmod_mult_2, simp)

  2141

  2142 lemma zdiv_eq_0_iff:

  2143  "(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")

  2144 proof

  2145   assume ?L

  2146   have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp

  2147   with ?L show ?R by blast

  2148 next

  2149   assume ?R thus ?L

  2150     by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)

  2151 qed

  2152

  2153

  2154 subsubsection {* Quotients of Signs *}

  2155

  2156 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"

  2157 apply (subgoal_tac "a div b \<le> -1", force)

  2158 apply (rule order_trans)

  2159 apply (rule_tac a' = "-1" in zdiv_mono1)

  2160 apply (auto simp add: div_eq_minus1)

  2161 done

  2162

  2163 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"

  2164 by (drule zdiv_mono1_neg, auto)

  2165

  2166 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"

  2167 by (drule zdiv_mono1, auto)

  2168

  2169 text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}

  2170 conditional upon the sign of @{text a} or @{text b}. There are many more.

  2171 They should all be simp rules unless that causes too much search. *}

  2172

  2173 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"

  2174 apply auto

  2175 apply (drule_tac [2] zdiv_mono1)

  2176 apply (auto simp add: linorder_neq_iff)

  2177 apply (simp (no_asm_use) add: linorder_not_less [symmetric])

  2178 apply (blast intro: div_neg_pos_less0)

  2179 done

  2180

  2181 lemma neg_imp_zdiv_nonneg_iff:

  2182   "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"

  2183 apply (subst zdiv_zminus_zminus [symmetric])

  2184 apply (subst pos_imp_zdiv_nonneg_iff, auto)

  2185 done

  2186

  2187 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)

  2188 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"

  2189 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)

  2190

  2191 lemma pos_imp_zdiv_pos_iff:

  2192   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"

  2193 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]

  2194 by arith

  2195

  2196 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)

  2197 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"

  2198 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)

  2199

  2200 lemma nonneg1_imp_zdiv_pos_iff:

  2201   "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"

  2202 apply rule

  2203  apply rule

  2204   using div_pos_pos_trivial[of a b]apply arith

  2205  apply(cases "b=0")apply simp

  2206  using div_nonneg_neg_le0[of a b]apply arith

  2207 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp

  2208 done

  2209

  2210 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"

  2211 apply (rule split_zmod[THEN iffD2])

  2212 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)

  2213 done

  2214

  2215

  2216 subsubsection {* The Divides Relation *}

  2217

  2218 lemmas zdvd_iff_zmod_eq_0_numeral [simp] =

  2219   dvd_eq_mod_eq_0 [of "numeral x::int" "numeral y::int"]

  2220   dvd_eq_mod_eq_0 [of "numeral x::int" "neg_numeral y::int"]

  2221   dvd_eq_mod_eq_0 [of "neg_numeral x::int" "numeral y::int"]

  2222   dvd_eq_mod_eq_0 [of "neg_numeral x::int" "neg_numeral y::int"] for x y

  2223

  2224 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"

  2225   by (rule dvd_mod) (* TODO: remove *)

  2226

  2227 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"

  2228   by (rule dvd_mod_imp_dvd) (* TODO: remove *)

  2229

  2230 lemmas dvd_eq_mod_eq_0_numeral [simp] =

  2231   dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y

  2232

  2233

  2234 subsubsection {* Further properties *}

  2235

  2236 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"

  2237   using zmod_zdiv_equality[where a="m" and b="n"]

  2238   by (simp add: algebra_simps)

  2239

  2240 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"

  2241 apply (induct "y", auto)

  2242 apply (rule zmod_zmult1_eq [THEN trans])

  2243 apply (simp (no_asm_simp))

  2244 apply (rule mod_mult_eq [symmetric])

  2245 done

  2246

  2247 lemma zdiv_int: "int (a div b) = (int a) div (int b)"

  2248 apply (subst split_div, auto)

  2249 apply (subst split_zdiv, auto)

  2250 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)

  2251 apply (auto simp add: divmod_int_rel_def of_nat_mult)

  2252 done

  2253

  2254 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"

  2255 apply (subst split_mod, auto)

  2256 apply (subst split_zmod, auto)

  2257 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia

  2258        in unique_remainder)

  2259 apply (auto simp add: divmod_int_rel_def of_nat_mult)

  2260 done

  2261

  2262 lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"

  2263 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)

  2264

  2265 lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"

  2266 apply (subgoal_tac "m mod n = 0")

  2267  apply (simp add: zmult_div_cancel)

  2268 apply (simp only: dvd_eq_mod_eq_0)

  2269 done

  2270

  2271 text{*Suggested by Matthias Daum*}

  2272 lemma int_power_div_base:

  2273      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"

  2274 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")

  2275  apply (erule ssubst)

  2276  apply (simp only: power_add)

  2277  apply simp_all

  2278 done

  2279

  2280 text {* by Brian Huffman *}

  2281 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"

  2282 by (rule mod_minus_eq [symmetric])

  2283

  2284 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"

  2285 by (rule mod_diff_left_eq [symmetric])

  2286

  2287 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"

  2288 by (rule mod_diff_right_eq [symmetric])

  2289

  2290 lemmas zmod_simps =

  2291   mod_add_left_eq  [symmetric]

  2292   mod_add_right_eq [symmetric]

  2293   zmod_zmult1_eq   [symmetric]

  2294   mod_mult_left_eq [symmetric]

  2295   zpower_zmod

  2296   zminus_zmod zdiff_zmod_left zdiff_zmod_right

  2297

  2298 text {* Distributive laws for function @{text nat}. *}

  2299

  2300 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"

  2301 apply (rule linorder_cases [of y 0])

  2302 apply (simp add: div_nonneg_neg_le0)

  2303 apply simp

  2304 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)

  2305 done

  2306

  2307 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)

  2308 lemma nat_mod_distrib:

  2309   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"

  2310 apply (case_tac "y = 0", simp)

  2311 apply (simp add: nat_eq_iff zmod_int)

  2312 done

  2313

  2314 text  {* transfer setup *}

  2315

  2316 lemma transfer_nat_int_functions:

  2317     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"

  2318     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"

  2319   by (auto simp add: nat_div_distrib nat_mod_distrib)

  2320

  2321 lemma transfer_nat_int_function_closures:

  2322     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"

  2323     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"

  2324   apply (cases "y = 0")

  2325   apply (auto simp add: pos_imp_zdiv_nonneg_iff)

  2326   apply (cases "y = 0")

  2327   apply auto

  2328 done

  2329

  2330 declare transfer_morphism_nat_int [transfer add return:

  2331   transfer_nat_int_functions

  2332   transfer_nat_int_function_closures

  2333 ]

  2334

  2335 lemma transfer_int_nat_functions:

  2336     "(int x) div (int y) = int (x div y)"

  2337     "(int x) mod (int y) = int (x mod y)"

  2338   by (auto simp add: zdiv_int zmod_int)

  2339

  2340 lemma transfer_int_nat_function_closures:

  2341     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"

  2342     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"

  2343   by (simp_all only: is_nat_def transfer_nat_int_function_closures)

  2344

  2345 declare transfer_morphism_int_nat [transfer add return:

  2346   transfer_int_nat_functions

  2347   transfer_int_nat_function_closures

  2348 ]

  2349

  2350 text{*Suggested by Matthias Daum*}

  2351 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"

  2352 apply (subgoal_tac "nat x div nat k < nat x")

  2353  apply (simp add: nat_div_distrib [symmetric])

  2354 apply (rule Divides.div_less_dividend, simp_all)

  2355 done

  2356

  2357 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"

  2358 proof

  2359   assume H: "x mod n = y mod n"

  2360   hence "x mod n - y mod n = 0" by simp

  2361   hence "(x mod n - y mod n) mod n = 0" by simp

  2362   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])

  2363   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)

  2364 next

  2365   assume H: "n dvd x - y"

  2366   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast

  2367   hence "x = n*k + y" by simp

  2368   hence "x mod n = (n*k + y) mod n" by simp

  2369   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)

  2370 qed

  2371

  2372 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"

  2373   shows "\<exists>q. x = y + n * q"

  2374 proof-

  2375   from xy have th: "int x - int y = int (x - y)" by simp

  2376   from xyn have "int x mod int n = int y mod int n"

  2377     by (simp add: zmod_int [symmetric])

  2378   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])

  2379   hence "n dvd x - y" by (simp add: th zdvd_int)

  2380   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith

  2381 qed

  2382

  2383 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"

  2384   (is "?lhs = ?rhs")

  2385 proof

  2386   assume H: "x mod n = y mod n"

  2387   {assume xy: "x \<le> y"

  2388     from H have th: "y mod n = x mod n" by simp

  2389     from nat_mod_eq_lemma[OF th xy] have ?rhs

  2390       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}

  2391   moreover

  2392   {assume xy: "y \<le> x"

  2393     from nat_mod_eq_lemma[OF H xy] have ?rhs

  2394       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}

  2395   ultimately  show ?rhs using linear[of x y] by blast

  2396 next

  2397   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast

  2398   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp

  2399   thus  ?lhs by simp

  2400 qed

  2401

  2402 lemma div_nat_numeral [simp]:

  2403   "(numeral v :: nat) div numeral v' = nat (numeral v div numeral v')"

  2404   by (simp add: nat_div_distrib)

  2405

  2406 lemma one_div_nat_numeral [simp]:

  2407   "Suc 0 div numeral v' = nat (1 div numeral v')"

  2408   by (subst nat_div_distrib, simp_all)

  2409

  2410 lemma mod_nat_numeral [simp]:

  2411   "(numeral v :: nat) mod numeral v' = nat (numeral v mod numeral v')"

  2412   by (simp add: nat_mod_distrib)

  2413

  2414 lemma one_mod_nat_numeral [simp]:

  2415   "Suc 0 mod numeral v' = nat (1 mod numeral v')"

  2416   by (subst nat_mod_distrib) simp_all

  2417

  2418 lemma mod_2_not_eq_zero_eq_one_int:

  2419   fixes k :: int

  2420   shows "k mod 2 \<noteq> 0 \<longleftrightarrow> k mod 2 = 1"

  2421   by auto

  2422

  2423

  2424 subsubsection {* Tools setup *}

  2425

  2426 text {* Nitpick *}

  2427

  2428 lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'

  2429

  2430

  2431 subsubsection {* Code generation *}

  2432

  2433 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  2434   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"

  2435

  2436 lemma pdivmod_posDivAlg [code]:

  2437   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"

  2438 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)

  2439

  2440 lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  2441   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0

  2442     then pdivmod k l

  2443     else (let (r, s) = pdivmod k l in

  2444        if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  2445 proof -

  2446   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto

  2447   show ?thesis

  2448     by (simp add: divmod_int_mod_div pdivmod_def)

  2449       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if

  2450       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)

  2451 qed

  2452

  2453 lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  2454   apsnd ((op *) (sgn l)) (if sgn k = sgn l

  2455     then pdivmod k l

  2456     else (let (r, s) = pdivmod k l in

  2457       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  2458 proof -

  2459   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"

  2460     by (auto simp add: not_less sgn_if)

  2461   then show ?thesis by (simp add: divmod_int_pdivmod)

  2462 qed

  2463

  2464 code_modulename SML

  2465   Divides Arith

  2466

  2467 code_modulename OCaml

  2468   Divides Arith

  2469

  2470 code_modulename Haskell

  2471   Divides Arith

  2472

  2473 end
`