src/HOL/Divides.thy
 author huffman Tue Mar 27 15:40:11 2012 +0200 (2012-03-27) changeset 47162 9d7d919b9fd8 parent 47160 8ada79014cb2 child 47163 248376f8881d permissions -rw-r--r--
remove redundant lemma
     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 mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"

   347   by (fact mod_mult_mult2 [symmetric])

   348

   349 lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"

   350   by (fact mod_mult_mult1 [symmetric])

   351

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

   353   unfolding dvd_def by (auto simp add: mod_mult_mult1)

   354

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

   356 by (blast intro: dvd_mod_imp_dvd dvd_mod)

   357

   358 lemma div_power:

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

   360 apply (induct n)

   361  apply simp

   362 apply(simp add: div_mult_div_if_dvd dvd_power_same)

   363 done

   364

   365 lemma dvd_div_eq_mult:

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

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

   368 proof

   369   assume "b = c * a"

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

   371 next

   372   assume "b div a = c"

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

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

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

   376 qed

   377

   378 lemma dvd_div_div_eq_mult:

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

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

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

   382

   383 end

   384

   385 class ring_div = semiring_div + comm_ring_1

   386 begin

   387

   388 subclass ring_1_no_zero_divisors ..

   389

   390 text {* Negation respects modular equivalence. *}

   391

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

   393 proof -

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

   395     by (simp only: mod_div_equality)

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

   397     by (simp only: minus_add_distrib minus_mult_left add_ac)

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

   399     by (rule mod_mult_self1)

   400   finally show ?thesis .

   401 qed

   402

   403 lemma mod_minus_cong:

   404   assumes "a mod b = a' mod b"

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

   406 proof -

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

   408     unfolding assms ..

   409   thus ?thesis

   410     by (simp only: mod_minus_eq [symmetric])

   411 qed

   412

   413 text {* Subtraction respects modular equivalence. *}

   414

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

   416   unfolding diff_minus

   417   by (intro mod_add_cong mod_minus_cong) simp_all

   418

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

   420   unfolding diff_minus

   421   by (intro mod_add_cong mod_minus_cong) simp_all

   422

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

   424   unfolding diff_minus

   425   by (intro mod_add_cong mod_minus_cong) simp_all

   426

   427 lemma mod_diff_cong:

   428   assumes "a mod c = a' mod c"

   429   assumes "b mod c = b' mod c"

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

   431   unfolding diff_minus using assms

   432   by (intro mod_add_cong mod_minus_cong)

   433

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

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

   436 apply (auto simp add: dvd_def)

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

   438  apply (erule ssubst)

   439  apply (erule div_mult_self1_is_id)

   440 apply simp

   441 done

   442

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

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

   445 apply (auto simp add: dvd_def)

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

   447  apply (erule ssubst)

   448  apply (rule div_mult_self1_is_id)

   449  apply simp

   450 apply simp

   451 done

   452

   453 lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"

   454   using div_mult_mult1 [of "- 1" a b]

   455   unfolding neg_equal_0_iff_equal by simp

   456

   457 lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"

   458   using mod_mult_mult1 [of "- 1" a b] by simp

   459

   460 lemma div_minus_right: "a div (-b) = (-a) div b"

   461   using div_minus_minus [of "-a" b] by simp

   462

   463 lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"

   464   using mod_minus_minus [of "-a" b] by simp

   465

   466 lemma div_minus1_right [simp]: "a div (-1) = -a"

   467   using div_minus_right [of a 1] by simp

   468

   469 lemma mod_minus1_right [simp]: "a mod (-1) = 0"

   470   using mod_minus_right [of a 1] by simp

   471

   472 end

   473

   474

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

   476

   477 text {*

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

   479   of a characteristic relation with two input arguments

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

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

   482 *}

   483

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

   485   "divmod_nat_rel m n qr \<longleftrightarrow>

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

   487       (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)"

   488

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

   490

   491 lemma divmod_nat_rel_ex:

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

   493 proof (cases "n = 0")

   494   case True  with that show thesis

   495     by (auto simp add: divmod_nat_rel_def)

   496 next

   497   case False

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

   499   proof (induct m)

   500     case 0 with n \<noteq> 0

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

   502     then show ?case by blast

   503   next

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

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

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

   507       case True

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

   509       with True show ?thesis by blast

   510     next

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

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

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

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

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

   516     qed

   517   qed

   518   with that show thesis

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

   520 qed

   521

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

   523

   524 lemma divmod_nat_rel_unique:

   525   assumes "divmod_nat_rel m n qr"

   526     and "divmod_nat_rel m n qr'"

   527   shows "qr = qr'"

   528 proof (cases "n = 0")

   529   case True with assms show ?thesis

   530     by (cases qr, cases qr')

   531       (simp add: divmod_nat_rel_def)

   532 next

   533   case False

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

   535   apply (rule leI)

   536   apply (subst less_iff_Suc_add)

   537   apply (auto simp add: add_mult_distrib)

   538   done

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

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

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

   542     by (simp add: divmod_nat_rel_def)

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

   544 qed

   545

   546 text {*

   547   We instantiate divisibility on the natural numbers by

   548   means of @{const divmod_nat_rel}:

   549 *}

   550

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

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

   553

   554 lemma divmod_nat_rel_divmod_nat:

   555   "divmod_nat_rel m n (divmod_nat m n)"

   556 proof -

   557   from divmod_nat_rel_ex

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

   559   then show ?thesis

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

   561 qed

   562

   563 lemma divmod_nat_unique:

   564   assumes "divmod_nat_rel m n qr"

   565   shows "divmod_nat m n = qr"

   566   using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)

   567

   568 instantiation nat :: semiring_div

   569 begin

   570

   571 definition div_nat where

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

   573

   574 lemma fst_divmod_nat [simp]:

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

   576   by (simp add: div_nat_def)

   577

   578 definition mod_nat where

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

   580

   581 lemma snd_divmod_nat [simp]:

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

   583   by (simp add: mod_nat_def)

   584

   585 lemma divmod_nat_div_mod:

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

   587   by (simp add: prod_eq_iff)

   588

   589 lemma div_nat_unique:

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

   591   shows "m div n = q"

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

   593

   594 lemma mod_nat_unique:

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

   596   shows "m mod n = r"

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

   598

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

   600   using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)

   601

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

   603   by (simp add: divmod_nat_unique divmod_nat_rel_def)

   604

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

   606   by (simp add: divmod_nat_unique divmod_nat_rel_def)

   607

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

   609   by (simp add: divmod_nat_unique divmod_nat_rel_def)

   610

   611 lemma divmod_nat_step:

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

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

   614 proof (rule divmod_nat_unique)

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

   616     by (rule divmod_nat_rel)

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

   618     unfolding divmod_nat_rel_def using assms by auto

   619 qed

   620

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

   622

   623 lemma div_less [simp]:

   624   fixes m n :: nat

   625   assumes "m < n"

   626   shows "m div n = 0"

   627   using assms divmod_nat_base by (simp add: prod_eq_iff)

   628

   629 lemma le_div_geq:

   630   fixes m n :: nat

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

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

   633   using assms divmod_nat_step by (simp add: prod_eq_iff)

   634

   635 lemma mod_less [simp]:

   636   fixes m n :: nat

   637   assumes "m < n"

   638   shows "m mod n = m"

   639   using assms divmod_nat_base by (simp add: prod_eq_iff)

   640

   641 lemma le_mod_geq:

   642   fixes m n :: nat

   643   assumes "n \<le> m"

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

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

   646

   647 instance proof

   648   fix m n :: nat

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

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

   651 next

   652   fix m n q :: nat

   653   assume "n \<noteq> 0"

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

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

   656 next

   657   fix m n q :: nat

   658   assume "m \<noteq> 0"

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

   660     unfolding divmod_nat_rel_def

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

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

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

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

   665 next

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

   667     by (simp add: div_nat_def divmod_nat_zero)

   668 next

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

   670     by (simp add: div_nat_def divmod_nat_zero_left)

   671 qed

   672

   673 end

   674

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

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

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

   678

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

   680

   681 ML {*

   682 structure Cancel_Div_Mod_Nat = Cancel_Div_Mod

   683 (

   684   val div_name = @{const_name div};

   685   val mod_name = @{const_name mod};

   686   val mk_binop = HOLogic.mk_binop;

   687   val mk_sum = Nat_Arith.mk_sum;

   688   val dest_sum = Nat_Arith.dest_sum;

   689

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

   691

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

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

   694 )

   695 *}

   696

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

   698

   699

   700 subsubsection {* Quotient *}

   701

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

   703 by (simp add: le_div_geq linorder_not_less)

   704

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

   706 by (simp add: div_geq)

   707

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

   709 by simp

   710

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

   712 by simp

   713

   714

   715 subsubsection {* Remainder *}

   716

   717 lemma mod_less_divisor [simp]:

   718   fixes m n :: nat

   719   assumes "n > 0"

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

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

   722

   723 lemma mod_less_eq_dividend [simp]:

   724   fixes m n :: nat

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

   726 proof (rule add_leD2)

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

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

   729 qed

   730

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

   732 by (simp add: le_mod_geq linorder_not_less)

   733

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

   735 by (simp add: le_mod_geq)

   736

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

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

   739

   740 (* a simple rearrangement of mod_div_equality: *)

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

   742   using mod_div_equality2 [of n m] by arith

   743

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

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

   746   apply simp

   747   done

   748

   749 subsubsection {* Quotient and Remainder *}

   750

   751 lemma divmod_nat_rel_mult1_eq:

   752   "divmod_nat_rel b c (q, r)

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

   754 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)

   755

   756 lemma div_mult1_eq:

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

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

   759

   760 lemma divmod_nat_rel_add1_eq:

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

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

   763 by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)

   764

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

   766 lemma div_add1_eq:

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

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

   769

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

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

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

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

   774   apply (simp add: add_mult_distrib2)

   775   done

   776

   777 lemma divmod_nat_rel_mult2_eq:

   778   "divmod_nat_rel a b (q, r)

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

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

   781

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

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

   784

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

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

   787

   788

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

   790

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

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

   793

   794 (* Monotonicity of div in first argument *)

   795 lemma div_le_mono [rule_format (no_asm)]:

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

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

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

   799 apply (case_tac "n<k")

   800 (* 1  case n<k *)

   801 apply simp

   802 (* 2  case n >= k *)

   803 apply (case_tac "m<k")

   804 (* 2.1  case m<k *)

   805 apply simp

   806 (* 2.2  case m>=k *)

   807 apply (simp add: div_geq diff_le_mono)

   808 done

   809

   810 (* Antimonotonicity of div in second argument *)

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

   812 apply (subgoal_tac "0<n")

   813  prefer 2 apply simp

   814 apply (induct_tac k rule: nat_less_induct)

   815 apply (rename_tac "k")

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

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

   818  prefer 2 apply simp

   819 apply (simp add: div_geq)

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

   821  prefer 2

   822  apply (blast intro: div_le_mono diff_le_mono2)

   823 apply (rule le_trans, simp)

   824 apply (simp)

   825 done

   826

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

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

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

   830 apply (rule div_le_mono2)

   831 apply (simp_all (no_asm_simp))

   832 done

   833

   834 (* Similar for "less than" *)

   835 lemma div_less_dividend [simp]:

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

   837 apply (induct m rule: nat_less_induct)

   838 apply (rename_tac "m")

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

   840 apply (subgoal_tac "0<n")

   841  prefer 2 apply simp

   842 apply (simp add: div_geq)

   843 apply (case_tac "n<m")

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

   845   apply (rule impI less_trans_Suc)+

   846 apply assumption

   847   apply (simp_all)

   848 done

   849

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

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

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

   853 apply (induct "m" rule: nat_less_induct)

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

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

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

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

   858 apply (simp add: linorder_not_less le_Suc_eq mod_geq)

   859 apply (auto simp add: Suc_diff_le le_mod_geq)

   860 done

   861

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

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

   864

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

   866

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

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

   869   apply (cut_tac a = m in mod_div_equality)

   870   apply (simp only: add_ac)

   871   apply (blast intro: sym)

   872   done

   873

   874 lemma split_div:

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

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

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

   878 proof

   879   assume P: ?P

   880   show ?Q

   881   proof (cases)

   882     assume "k = 0"

   883     with P show ?Q by simp

   884   next

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

   886     thus ?Q

   887     proof (simp, intro allI impI)

   888       fix i j

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

   890       show "P i"

   891       proof (cases)

   892         assume "i = 0"

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

   894       next

   895         assume "i \<noteq> 0"

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

   897       qed

   898     qed

   899   qed

   900 next

   901   assume Q: ?Q

   902   show ?P

   903   proof (cases)

   904     assume "k = 0"

   905     with Q show ?P by simp

   906   next

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

   908     with Q have R: ?R by simp

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

   910     show ?P by simp

   911   qed

   912 qed

   913

   914 lemma split_div_lemma:

   915   assumes "0 < n"

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

   917 proof

   918   assume ?rhs

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

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

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

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

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

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

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

   926   from A B show ?lhs ..

   927 next

   928   assume P: ?lhs

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

   930     unfolding divmod_nat_rel_def by (auto simp add: mult_ac)

   931   with divmod_nat_rel_unique divmod_nat_rel [of m n]

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

   933   then show ?rhs by simp

   934 qed

   935

   936 theorem split_div':

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

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

   939   apply (case_tac "0 < n")

   940   apply (simp only: add: split_div_lemma)

   941   apply simp_all

   942   done

   943

   944 lemma split_mod:

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

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

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

   948 proof

   949   assume P: ?P

   950   show ?Q

   951   proof (cases)

   952     assume "k = 0"

   953     with P show ?Q by simp

   954   next

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

   956     thus ?Q

   957     proof (simp, intro allI impI)

   958       fix i j

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

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

   961     qed

   962   qed

   963 next

   964   assume Q: ?Q

   965   show ?P

   966   proof (cases)

   967     assume "k = 0"

   968     with Q show ?P by simp

   969   next

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

   971     with Q have R: ?R by simp

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

   973     show ?P by simp

   974   qed

   975 qed

   976

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

   978   using mod_div_equality [of m n] by arith

   979

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

   981   using mod_div_equality [of m n] by arith

   982 (* FIXME: very similar to mult_div_cancel *)

   983

   984

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

   986

   987 lemma mod_induct_0:

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

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

   990   shows "P 0"

   991 proof (rule ccontr)

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

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

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

   995   proof

   996     fix k

   997     show "?A k"

   998     proof (induct k)

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

  1000     next

  1001       fix n

  1002       assume ih: "?A n"

  1003       show "?A (Suc n)"

  1004       proof (clarsimp)

  1005         assume y: "P (p - Suc n)"

  1006         have n: "Suc n < p"

  1007         proof (rule ccontr)

  1008           assume "\<not>(Suc n < p)"

  1009           hence "p - Suc n = 0"

  1010             by simp

  1011           with y contra show "False"

  1012             by simp

  1013         qed

  1014         hence n2: "Suc (p - Suc n) = p-n" by arith

  1015         from p have "p - Suc n < p" by arith

  1016         with y step have z: "P ((Suc (p - Suc n)) mod p)"

  1017           by blast

  1018         show "False"

  1019         proof (cases "n=0")

  1020           case True

  1021           with z n2 contra show ?thesis by simp

  1022         next

  1023           case False

  1024           with p have "p-n < p" by arith

  1025           with z n2 False ih show ?thesis by simp

  1026         qed

  1027       qed

  1028     qed

  1029   qed

  1030   moreover

  1031   from i obtain k where "0<k \<and> i+k=p"

  1032     by (blast dest: less_imp_add_positive)

  1033   hence "0<k \<and> i=p-k" by auto

  1034   moreover

  1035   note base

  1036   ultimately

  1037   show "False" by blast

  1038 qed

  1039

  1040 lemma mod_induct:

  1041   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"

  1042   and base: "P i" and i: "i<p" and j: "j<p"

  1043   shows "P j"

  1044 proof -

  1045   have "\<forall>j<p. P j"

  1046   proof

  1047     fix j

  1048     show "j<p \<longrightarrow> P j" (is "?A j")

  1049     proof (induct j)

  1050       from step base i show "?A 0"

  1051         by (auto elim: mod_induct_0)

  1052     next

  1053       fix k

  1054       assume ih: "?A k"

  1055       show "?A (Suc k)"

  1056       proof

  1057         assume suc: "Suc k < p"

  1058         hence k: "k<p" by simp

  1059         with ih have "P k" ..

  1060         with step k have "P (Suc k mod p)"

  1061           by blast

  1062         moreover

  1063         from suc have "Suc k mod p = Suc k"

  1064           by simp

  1065         ultimately

  1066         show "P (Suc k)" by simp

  1067       qed

  1068     qed

  1069   qed

  1070   with j show ?thesis by blast

  1071 qed

  1072

  1073 lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"

  1074   by (simp add: numeral_2_eq_2 le_div_geq)

  1075

  1076 lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"

  1077   by (simp add: numeral_2_eq_2 le_mod_geq)

  1078

  1079 lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"

  1080 by (simp add: nat_mult_2 [symmetric])

  1081

  1082 lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"

  1083 proof -

  1084   { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }

  1085   moreover have "m mod 2 < 2" by simp

  1086   ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .

  1087   then show ?thesis by auto

  1088 qed

  1089

  1090 text{*These lemmas collapse some needless occurrences of Suc:

  1091     at least three Sucs, since two and fewer are rewritten back to Suc again!

  1092     We already have some rules to simplify operands smaller than 3.*}

  1093

  1094 lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"

  1095 by (simp add: Suc3_eq_add_3)

  1096

  1097 lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"

  1098 by (simp add: Suc3_eq_add_3)

  1099

  1100 lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"

  1101 by (simp add: Suc3_eq_add_3)

  1102

  1103 lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"

  1104 by (simp add: Suc3_eq_add_3)

  1105

  1106 lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v

  1107 lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v

  1108

  1109

  1110 lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"

  1111 apply (induct "m")

  1112 apply (simp_all add: mod_Suc)

  1113 done

  1114

  1115 declare Suc_times_mod_eq [of "numeral w", simp] for w

  1116

  1117 lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"

  1118 by (simp add: div_le_mono)

  1119

  1120 lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"

  1121 by (cases n) simp_all

  1122

  1123 lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"

  1124 proof -

  1125   from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all

  1126   from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp

  1127 qed

  1128

  1129   (* Potential use of algebra : Equality modulo n*)

  1130 lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"

  1131 by (simp add: mult_ac add_ac)

  1132

  1133 lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"

  1134 proof -

  1135   have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp

  1136   also have "... = Suc m mod n" by (rule mod_mult_self3)

  1137   finally show ?thesis .

  1138 qed

  1139

  1140 lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"

  1141 apply (subst mod_Suc [of m])

  1142 apply (subst mod_Suc [of "m mod n"], simp)

  1143 done

  1144

  1145 lemma mod_2_not_eq_zero_eq_one_nat:

  1146   fixes n :: nat

  1147   shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"

  1148   by simp

  1149

  1150

  1151 subsection {* Division on @{typ int} *}

  1152

  1153 definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where

  1154     --{*definition of quotient and remainder*}

  1155   "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>

  1156     (if 0 < b then 0 \<le> r \<and> r < b else if b < 0 then b < r \<and> r \<le> 0 else q = 0))"

  1157

  1158 definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where

  1159     --{*for the division algorithm*}

  1160     "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)

  1161                          else (2 * q, r))"

  1162

  1163 text{*algorithm for the case @{text "a\<ge>0, b>0"}*}

  1164 function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1165   "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)

  1166      else adjust b (posDivAlg a (2 * b)))"

  1167 by auto

  1168 termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")

  1169   (auto simp add: mult_2)

  1170

  1171 text{*algorithm for the case @{text "a<0, b>0"}*}

  1172 function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1173   "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)

  1174      else adjust b (negDivAlg a (2 * b)))"

  1175 by auto

  1176 termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")

  1177   (auto simp add: mult_2)

  1178

  1179 text{*algorithm for the general case @{term "b\<noteq>0"}*}

  1180

  1181 definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  1182     --{*The full division algorithm considers all possible signs for a, b

  1183        including the special case @{text "a=0, b<0"} because

  1184        @{term negDivAlg} requires @{term "a<0"}.*}

  1185   "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b

  1186                   else if a = 0 then (0, 0)

  1187                        else apsnd uminus (negDivAlg (-a) (-b))

  1188                else

  1189                   if 0 < b then negDivAlg a b

  1190                   else apsnd uminus (posDivAlg (-a) (-b)))"

  1191

  1192 instantiation int :: Divides.div

  1193 begin

  1194

  1195 definition div_int where

  1196   "a div b = fst (divmod_int a b)"

  1197

  1198 lemma fst_divmod_int [simp]:

  1199   "fst (divmod_int a b) = a div b"

  1200   by (simp add: div_int_def)

  1201

  1202 definition mod_int where

  1203   "a mod b = snd (divmod_int a b)"

  1204

  1205 lemma snd_divmod_int [simp]:

  1206   "snd (divmod_int a b) = a mod b"

  1207   by (simp add: mod_int_def)

  1208

  1209 instance ..

  1210

  1211 end

  1212

  1213 lemma divmod_int_mod_div:

  1214   "divmod_int p q = (p div q, p mod q)"

  1215   by (simp add: prod_eq_iff)

  1216

  1217 text{*

  1218 Here is the division algorithm in ML:

  1219

  1220 \begin{verbatim}

  1221     fun posDivAlg (a,b) =

  1222       if a<b then (0,a)

  1223       else let val (q,r) = posDivAlg(a, 2*b)

  1224                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)

  1225            end

  1226

  1227     fun negDivAlg (a,b) =

  1228       if 0\<le>a+b then (~1,a+b)

  1229       else let val (q,r) = negDivAlg(a, 2*b)

  1230                in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)

  1231            end;

  1232

  1233     fun negateSnd (q,r:int) = (q,~r);

  1234

  1235     fun divmod (a,b) = if 0\<le>a then

  1236                           if b>0 then posDivAlg (a,b)

  1237                            else if a=0 then (0,0)

  1238                                 else negateSnd (negDivAlg (~a,~b))

  1239                        else

  1240                           if 0<b then negDivAlg (a,b)

  1241                           else        negateSnd (posDivAlg (~a,~b));

  1242 \end{verbatim}

  1243 *}

  1244

  1245

  1246 subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}

  1247

  1248 lemma unique_quotient_lemma:

  1249      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]

  1250       ==> q' \<le> (q::int)"

  1251 apply (subgoal_tac "r' + b * (q'-q) \<le> r")

  1252  prefer 2 apply (simp add: right_diff_distrib)

  1253 apply (subgoal_tac "0 < b * (1 + q - q') ")

  1254 apply (erule_tac [2] order_le_less_trans)

  1255  prefer 2 apply (simp add: right_diff_distrib right_distrib)

  1256 apply (subgoal_tac "b * q' < b * (1 + q) ")

  1257  prefer 2 apply (simp add: right_diff_distrib right_distrib)

  1258 apply (simp add: mult_less_cancel_left)

  1259 done

  1260

  1261 lemma unique_quotient_lemma_neg:

  1262      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]

  1263       ==> q \<le> (q'::int)"

  1264 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,

  1265     auto)

  1266

  1267 lemma unique_quotient:

  1268      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]

  1269       ==> q = q'"

  1270 apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)

  1271 apply (blast intro: order_antisym

  1272              dest: order_eq_refl [THEN unique_quotient_lemma]

  1273              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+

  1274 done

  1275

  1276

  1277 lemma unique_remainder:

  1278      "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]

  1279       ==> r = r'"

  1280 apply (subgoal_tac "q = q'")

  1281  apply (simp add: divmod_int_rel_def)

  1282 apply (blast intro: unique_quotient)

  1283 done

  1284

  1285

  1286 subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}

  1287

  1288 text{*And positive divisors*}

  1289

  1290 lemma adjust_eq [simp]:

  1291      "adjust b (q, r) =

  1292       (let diff = r - b in

  1293         if 0 \<le> diff then (2 * q + 1, diff)

  1294                      else (2*q, r))"

  1295   by (simp add: Let_def adjust_def)

  1296

  1297 declare posDivAlg.simps [simp del]

  1298

  1299 text{*use with a simproc to avoid repeatedly proving the premise*}

  1300 lemma posDivAlg_eqn:

  1301      "0 < b ==>

  1302       posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"

  1303 by (rule posDivAlg.simps [THEN trans], simp)

  1304

  1305 text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}

  1306 theorem posDivAlg_correct:

  1307   assumes "0 \<le> a" and "0 < b"

  1308   shows "divmod_int_rel a b (posDivAlg a b)"

  1309   using assms

  1310   apply (induct a b rule: posDivAlg.induct)

  1311   apply auto

  1312   apply (simp add: divmod_int_rel_def)

  1313   apply (subst posDivAlg_eqn, simp add: right_distrib)

  1314   apply (case_tac "a < b")

  1315   apply simp_all

  1316   apply (erule splitE)

  1317   apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)

  1318   done

  1319

  1320

  1321 subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}

  1322

  1323 text{*And positive divisors*}

  1324

  1325 declare negDivAlg.simps [simp del]

  1326

  1327 text{*use with a simproc to avoid repeatedly proving the premise*}

  1328 lemma negDivAlg_eqn:

  1329      "0 < b ==>

  1330       negDivAlg a b =

  1331        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"

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

  1333

  1334 (*Correctness of negDivAlg: it computes quotients correctly

  1335   It doesn't work if a=0 because the 0/b equals 0, not -1*)

  1336 lemma negDivAlg_correct:

  1337   assumes "a < 0" and "b > 0"

  1338   shows "divmod_int_rel a b (negDivAlg a b)"

  1339   using assms

  1340   apply (induct a b rule: negDivAlg.induct)

  1341   apply (auto simp add: linorder_not_le)

  1342   apply (simp add: divmod_int_rel_def)

  1343   apply (subst negDivAlg_eqn, assumption)

  1344   apply (case_tac "a + b < (0\<Colon>int)")

  1345   apply simp_all

  1346   apply (erule splitE)

  1347   apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)

  1348   done

  1349

  1350

  1351 subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}

  1352

  1353 (*the case a=0*)

  1354 lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"

  1355 by (auto simp add: divmod_int_rel_def linorder_neq_iff)

  1356

  1357 lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"

  1358 by (subst posDivAlg.simps, auto)

  1359

  1360 lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"

  1361 by (subst posDivAlg.simps, auto)

  1362

  1363 lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"

  1364 by (subst negDivAlg.simps, auto)

  1365

  1366 lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"

  1367 by (auto simp add: divmod_int_rel_def)

  1368

  1369 lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"

  1370 apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)

  1371 by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg

  1372                     posDivAlg_correct negDivAlg_correct)

  1373

  1374 lemma divmod_int_unique:

  1375   assumes "divmod_int_rel a b qr"

  1376   shows "divmod_int a b = qr"

  1377   using assms divmod_int_correct [of a b]

  1378   using unique_quotient [of a b] unique_remainder [of a b]

  1379   by (metis pair_collapse)

  1380

  1381 lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"

  1382   using divmod_int_correct by (simp add: divmod_int_mod_div)

  1383

  1384 lemma div_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a div b = q"

  1385   by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])

  1386

  1387 lemma mod_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a mod b = r"

  1388   by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])

  1389

  1390 instance int :: ring_div

  1391 proof

  1392   fix a b :: int

  1393   show "a div b * b + a mod b = a"

  1394     using divmod_int_rel_div_mod [of a b]

  1395     unfolding divmod_int_rel_def by (simp add: mult_commute)

  1396 next

  1397   fix a b c :: int

  1398   assume "b \<noteq> 0"

  1399   hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"

  1400     using divmod_int_rel_div_mod [of a b]

  1401     unfolding divmod_int_rel_def by (auto simp: algebra_simps)

  1402   thus "(a + c * b) div b = c + a div b"

  1403     by (rule div_int_unique)

  1404 next

  1405   fix a b c :: int

  1406   assume "c \<noteq> 0"

  1407   hence "\<And>q r. divmod_int_rel a b (q, r)

  1408     \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"

  1409     unfolding divmod_int_rel_def

  1410     by - (rule linorder_cases [of 0 b], auto simp: algebra_simps

  1411       mult_less_0_iff zero_less_mult_iff mult_strict_right_mono

  1412       mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)

  1413   hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"

  1414     using divmod_int_rel_div_mod [of a b] .

  1415   thus "(c * a) div (c * b) = a div b"

  1416     by (rule div_int_unique)

  1417 next

  1418   fix a :: int show "a div 0 = 0"

  1419     by (rule div_int_unique, simp add: divmod_int_rel_def)

  1420 next

  1421   fix a :: int show "0 div a = 0"

  1422     by (rule div_int_unique, auto simp add: divmod_int_rel_def)

  1423 qed

  1424

  1425 text{*Basic laws about division and remainder*}

  1426

  1427 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"

  1428   by (fact mod_div_equality2 [symmetric])

  1429

  1430 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"

  1431   by (fact div_mod_equality2)

  1432

  1433 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"

  1434   by (fact div_mod_equality)

  1435

  1436 text {* Tool setup *}

  1437

  1438 (* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)

  1439 lemmas add_0s = add_0_left add_0_right

  1440

  1441 ML {*

  1442 structure Cancel_Div_Mod_Int = Cancel_Div_Mod

  1443 (

  1444   val div_name = @{const_name div};

  1445   val mod_name = @{const_name mod};

  1446   val mk_binop = HOLogic.mk_binop;

  1447   val mk_sum = Arith_Data.mk_sum HOLogic.intT;

  1448   val dest_sum = Arith_Data.dest_sum;

  1449

  1450   val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];

  1451

  1452   val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac

  1453     (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))

  1454 )

  1455 *}

  1456

  1457 simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}

  1458

  1459 lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"

  1460   using divmod_int_correct [of a b]

  1461   by (auto simp add: divmod_int_rel_def prod_eq_iff)

  1462

  1463 lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]

  1464    and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]

  1465

  1466 lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"

  1467   using divmod_int_correct [of a b]

  1468   by (auto simp add: divmod_int_rel_def prod_eq_iff)

  1469

  1470 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]

  1471    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]

  1472

  1473

  1474 subsubsection {* General Properties of div and mod *}

  1475

  1476 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"

  1477 apply (rule div_int_unique)

  1478 apply (auto simp add: divmod_int_rel_def)

  1479 done

  1480

  1481 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"

  1482 apply (rule div_int_unique)

  1483 apply (auto simp add: divmod_int_rel_def)

  1484 done

  1485

  1486 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"

  1487 apply (rule div_int_unique)

  1488 apply (auto simp add: divmod_int_rel_def)

  1489 done

  1490

  1491 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)

  1492

  1493 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"

  1494 apply (rule_tac q = 0 in mod_int_unique)

  1495 apply (auto simp add: divmod_int_rel_def)

  1496 done

  1497

  1498 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"

  1499 apply (rule_tac q = 0 in mod_int_unique)

  1500 apply (auto simp add: divmod_int_rel_def)

  1501 done

  1502

  1503 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"

  1504 apply (rule_tac q = "-1" in mod_int_unique)

  1505 apply (auto simp add: divmod_int_rel_def)

  1506 done

  1507

  1508 text{*There is no @{text mod_neg_pos_trivial}.*}

  1509

  1510

  1511 subsubsection {* Laws for div and mod with Unary Minus *}

  1512

  1513 lemma zminus1_lemma:

  1514      "divmod_int_rel a b (q, r) ==> b \<noteq> 0

  1515       ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,

  1516                           if r=0 then 0 else b-r)"

  1517 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)

  1518

  1519

  1520 lemma zdiv_zminus1_eq_if:

  1521      "b \<noteq> (0::int)

  1522       ==> (-a) div b =

  1523           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"

  1524 by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])

  1525

  1526 lemma zmod_zminus1_eq_if:

  1527      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"

  1528 apply (case_tac "b = 0", simp)

  1529 apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])

  1530 done

  1531

  1532 lemma zmod_zminus1_not_zero:

  1533   fixes k l :: int

  1534   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"

  1535   unfolding zmod_zminus1_eq_if by auto

  1536

  1537 lemma zdiv_zminus2_eq_if:

  1538      "b \<noteq> (0::int)

  1539       ==> a div (-b) =

  1540           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"

  1541 by (simp add: zdiv_zminus1_eq_if div_minus_right)

  1542

  1543 lemma zmod_zminus2_eq_if:

  1544      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"

  1545 by (simp add: zmod_zminus1_eq_if mod_minus_right)

  1546

  1547 lemma zmod_zminus2_not_zero:

  1548   fixes k l :: int

  1549   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"

  1550   unfolding zmod_zminus2_eq_if by auto

  1551

  1552

  1553 subsubsection {* Computation of Division and Remainder *}

  1554

  1555 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"

  1556 by (simp add: div_int_def divmod_int_def)

  1557

  1558 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"

  1559 by (simp add: mod_int_def divmod_int_def)

  1560

  1561 text{*a positive, b positive *}

  1562

  1563 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"

  1564 by (simp add: div_int_def divmod_int_def)

  1565

  1566 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"

  1567 by (simp add: mod_int_def divmod_int_def)

  1568

  1569 text{*a negative, b positive *}

  1570

  1571 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"

  1572 by (simp add: div_int_def divmod_int_def)

  1573

  1574 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"

  1575 by (simp add: mod_int_def divmod_int_def)

  1576

  1577 text{*a positive, b negative *}

  1578

  1579 lemma div_pos_neg:

  1580      "[| 0 < a;  b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"

  1581 by (simp add: div_int_def divmod_int_def)

  1582

  1583 lemma mod_pos_neg:

  1584      "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"

  1585 by (simp add: mod_int_def divmod_int_def)

  1586

  1587 text{*a negative, b negative *}

  1588

  1589 lemma div_neg_neg:

  1590      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"

  1591 by (simp add: div_int_def divmod_int_def)

  1592

  1593 lemma mod_neg_neg:

  1594      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"

  1595 by (simp add: mod_int_def divmod_int_def)

  1596

  1597 text {*Simplify expresions in which div and mod combine numerical constants*}

  1598

  1599 lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"

  1600   by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)

  1601

  1602 lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"

  1603   by (rule div_int_unique [of a b q r],

  1604     simp add: divmod_int_rel_def)

  1605

  1606 lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"

  1607   by (rule mod_int_unique [of a b q r],

  1608     simp add: divmod_int_rel_def)

  1609

  1610 lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"

  1611   by (rule mod_int_unique [of a b q r],

  1612     simp add: divmod_int_rel_def)

  1613

  1614 (* simprocs adapted from HOL/ex/Binary.thy *)

  1615 ML {*

  1616 local

  1617   val mk_number = HOLogic.mk_number HOLogic.intT

  1618   val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}

  1619   val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}

  1620   val zero = @{term "0 :: int"}

  1621   val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}

  1622   val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}

  1623   val simps = @{thms arith_simps} @ @{thms rel_simps} @

  1624     map (fn th => th RS sym) [@{thm numeral_1_eq_1}]

  1625   fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)

  1626     (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps simps))));

  1627   fun binary_proc proc ss ct =

  1628     (case Thm.term_of ct of

  1629       _ $t$ u =>

  1630       (case try (pairself ((snd o HOLogic.dest_number))) (t, u) of

  1631         SOME args => proc (Simplifier.the_context ss) args

  1632       | NONE => NONE)

  1633     | _ => NONE);

  1634 in

  1635   fun divmod_proc posrule negrule =

  1636     binary_proc (fn ctxt => fn ((a, t), (b, u)) =>

  1637       if b = 0 then NONE else let

  1638         val (q, r) = pairself mk_number (Integer.div_mod a b)

  1639         val goal1 = HOLogic.mk_eq (t, plus $(times$ u $q)$ r)

  1640         val (goal2, goal3, rule) = if b > 0

  1641           then (le $zero$ r, less $r$ u, posrule RS eq_reflection)

  1642           else (le $r$ zero, less $u$ r, negrule RS eq_reflection)

  1643       in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)

  1644 end

  1645 *}

  1646

  1647 simproc_setup binary_int_div

  1648   ("numeral m div numeral n :: int" |

  1649    "numeral m div neg_numeral n :: int" |

  1650    "neg_numeral m div numeral n :: int" |

  1651    "neg_numeral m div neg_numeral n :: int") =

  1652   {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}

  1653

  1654 simproc_setup binary_int_mod

  1655   ("numeral m mod numeral n :: int" |

  1656    "numeral m mod neg_numeral n :: int" |

  1657    "neg_numeral m mod numeral n :: int" |

  1658    "neg_numeral m mod neg_numeral n :: int") =

  1659   {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}

  1660

  1661 lemmas posDivAlg_eqn_numeral [simp] =

  1662     posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w

  1663

  1664 lemmas negDivAlg_eqn_numeral [simp] =

  1665     negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w

  1666

  1667

  1668 text{*Special-case simplification *}

  1669

  1670 (** The last remaining special cases for constant arithmetic:

  1671     1 div z and 1 mod z **)

  1672

  1673 lemmas div_pos_pos_1_numeral [simp] =

  1674   div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w

  1675

  1676 lemmas div_pos_neg_1_numeral [simp] =

  1677   div_pos_neg [OF zero_less_one, of "neg_numeral w",

  1678   OF neg_numeral_less_zero] for w

  1679

  1680 lemmas mod_pos_pos_1_numeral [simp] =

  1681   mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w

  1682

  1683 lemmas mod_pos_neg_1_numeral [simp] =

  1684   mod_pos_neg [OF zero_less_one, of "neg_numeral w",

  1685   OF neg_numeral_less_zero] for w

  1686

  1687 lemmas posDivAlg_eqn_1_numeral [simp] =

  1688     posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w

  1689

  1690 lemmas negDivAlg_eqn_1_numeral [simp] =

  1691     negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w

  1692

  1693

  1694 subsubsection {* Monotonicity in the First Argument (Dividend) *}

  1695

  1696 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"

  1697 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1698 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)

  1699 apply (rule unique_quotient_lemma)

  1700 apply (erule subst)

  1701 apply (erule subst, simp_all)

  1702 done

  1703

  1704 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"

  1705 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1706 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)

  1707 apply (rule unique_quotient_lemma_neg)

  1708 apply (erule subst)

  1709 apply (erule subst, simp_all)

  1710 done

  1711

  1712

  1713 subsubsection {* Monotonicity in the Second Argument (Divisor) *}

  1714

  1715 lemma q_pos_lemma:

  1716      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"

  1717 apply (subgoal_tac "0 < b'* (q' + 1) ")

  1718  apply (simp add: zero_less_mult_iff)

  1719 apply (simp add: right_distrib)

  1720 done

  1721

  1722 lemma zdiv_mono2_lemma:

  1723      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';

  1724          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]

  1725       ==> q \<le> (q'::int)"

  1726 apply (frule q_pos_lemma, assumption+)

  1727 apply (subgoal_tac "b*q < b* (q' + 1) ")

  1728  apply (simp add: mult_less_cancel_left)

  1729 apply (subgoal_tac "b*q = r' - r + b'*q'")

  1730  prefer 2 apply simp

  1731 apply (simp (no_asm_simp) add: right_distrib)

  1732 apply (subst add_commute, rule add_less_le_mono, arith)

  1733 apply (rule mult_right_mono, auto)

  1734 done

  1735

  1736 lemma zdiv_mono2:

  1737      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"

  1738 apply (subgoal_tac "b \<noteq> 0")

  1739  prefer 2 apply arith

  1740 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1741 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)

  1742 apply (rule zdiv_mono2_lemma)

  1743 apply (erule subst)

  1744 apply (erule subst, simp_all)

  1745 done

  1746

  1747 lemma q_neg_lemma:

  1748      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"

  1749 apply (subgoal_tac "b'*q' < 0")

  1750  apply (simp add: mult_less_0_iff, arith)

  1751 done

  1752

  1753 lemma zdiv_mono2_neg_lemma:

  1754      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;

  1755          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]

  1756       ==> q' \<le> (q::int)"

  1757 apply (frule q_neg_lemma, assumption+)

  1758 apply (subgoal_tac "b*q' < b* (q + 1) ")

  1759  apply (simp add: mult_less_cancel_left)

  1760 apply (simp add: right_distrib)

  1761 apply (subgoal_tac "b*q' \<le> b'*q'")

  1762  prefer 2 apply (simp add: mult_right_mono_neg, arith)

  1763 done

  1764

  1765 lemma zdiv_mono2_neg:

  1766      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"

  1767 apply (cut_tac a = a and b = b in zmod_zdiv_equality)

  1768 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)

  1769 apply (rule zdiv_mono2_neg_lemma)

  1770 apply (erule subst)

  1771 apply (erule subst, simp_all)

  1772 done

  1773

  1774

  1775 subsubsection {* More Algebraic Laws for div and mod *}

  1776

  1777 text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}

  1778

  1779 lemma zmult1_lemma:

  1780      "[| divmod_int_rel b c (q, r) |]

  1781       ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"

  1782 by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)

  1783

  1784 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"

  1785 apply (case_tac "c = 0", simp)

  1786 apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])

  1787 done

  1788

  1789 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"

  1790   by (fact mod_mult_right_eq) (* FIXME: delete *)

  1791

  1792 text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}

  1793

  1794 lemma zadd1_lemma:

  1795      "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]

  1796       ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"

  1797 by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)

  1798

  1799 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)

  1800 lemma zdiv_zadd1_eq:

  1801      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"

  1802 apply (case_tac "c = 0", simp)

  1803 apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)

  1804 done

  1805

  1806 lemma posDivAlg_div_mod:

  1807   assumes "k \<ge> 0"

  1808   and "l \<ge> 0"

  1809   shows "posDivAlg k l = (k div l, k mod l)"

  1810 proof (cases "l = 0")

  1811   case True then show ?thesis by (simp add: posDivAlg.simps)

  1812 next

  1813   case False with assms posDivAlg_correct

  1814     have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"

  1815     by simp

  1816   from div_int_unique [OF this] mod_int_unique [OF this]

  1817   show ?thesis by simp

  1818 qed

  1819

  1820 lemma negDivAlg_div_mod:

  1821   assumes "k < 0"

  1822   and "l > 0"

  1823   shows "negDivAlg k l = (k div l, k mod l)"

  1824 proof -

  1825   from assms have "l \<noteq> 0" by simp

  1826   from assms negDivAlg_correct

  1827     have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"

  1828     by simp

  1829   from div_int_unique [OF this] mod_int_unique [OF this]

  1830   show ?thesis by simp

  1831 qed

  1832

  1833 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"

  1834 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)

  1835

  1836 (* REVISIT: should this be generalized to all semiring_div types? *)

  1837 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]

  1838

  1839 lemma zmod_zdiv_equality':

  1840   "(m\<Colon>int) mod n = m - (m div n) * n"

  1841   using mod_div_equality [of m n] by arith

  1842

  1843

  1844 subsubsection {* Proving  @{term "a div (b*c) = (a div b) div c"} *}

  1845

  1846 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but

  1847   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems

  1848   to cause particular problems.*)

  1849

  1850 text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}

  1851

  1852 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"

  1853 apply (subgoal_tac "b * (c - q mod c) < r * 1")

  1854  apply (simp add: algebra_simps)

  1855 apply (rule order_le_less_trans)

  1856  apply (erule_tac [2] mult_strict_right_mono)

  1857  apply (rule mult_left_mono_neg)

  1858   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)

  1859  apply (simp)

  1860 apply (simp)

  1861 done

  1862

  1863 lemma zmult2_lemma_aux2:

  1864      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"

  1865 apply (subgoal_tac "b * (q mod c) \<le> 0")

  1866  apply arith

  1867 apply (simp add: mult_le_0_iff)

  1868 done

  1869

  1870 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"

  1871 apply (subgoal_tac "0 \<le> b * (q mod c) ")

  1872 apply arith

  1873 apply (simp add: zero_le_mult_iff)

  1874 done

  1875

  1876 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"

  1877 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")

  1878  apply (simp add: right_diff_distrib)

  1879 apply (rule order_less_le_trans)

  1880  apply (erule mult_strict_right_mono)

  1881  apply (rule_tac [2] mult_left_mono)

  1882   apply simp

  1883  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)

  1884 apply simp

  1885 done

  1886

  1887 lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]

  1888       ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"

  1889 by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff

  1890                    zero_less_mult_iff right_distrib [symmetric]

  1891                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)

  1892

  1893 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"

  1894 apply (case_tac "b = 0", simp)

  1895 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])

  1896 done

  1897

  1898 lemma zmod_zmult2_eq:

  1899      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"

  1900 apply (case_tac "b = 0", simp)

  1901 apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])

  1902 done

  1903

  1904 lemma div_pos_geq:

  1905   fixes k l :: int

  1906   assumes "0 < l" and "l \<le> k"

  1907   shows "k div l = (k - l) div l + 1"

  1908 proof -

  1909   have "k = (k - l) + l" by simp

  1910   then obtain j where k: "k = j + l" ..

  1911   with assms show ?thesis by simp

  1912 qed

  1913

  1914 lemma mod_pos_geq:

  1915   fixes k l :: int

  1916   assumes "0 < l" and "l \<le> k"

  1917   shows "k mod l = (k - l) mod l"

  1918 proof -

  1919   have "k = (k - l) + l" by simp

  1920   then obtain j where k: "k = j + l" ..

  1921   with assms show ?thesis by simp

  1922 qed

  1923

  1924

  1925 subsubsection {* Splitting Rules for div and mod *}

  1926

  1927 text{*The proofs of the two lemmas below are essentially identical*}

  1928

  1929 lemma split_pos_lemma:

  1930  "0<k ==>

  1931     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"

  1932 apply (rule iffI, clarify)

  1933  apply (erule_tac P="P ?x ?y" in rev_mp)

  1934  apply (subst mod_add_eq)

  1935  apply (subst zdiv_zadd1_eq)

  1936  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)

  1937 txt{*converse direction*}

  1938 apply (drule_tac x = "n div k" in spec)

  1939 apply (drule_tac x = "n mod k" in spec, simp)

  1940 done

  1941

  1942 lemma split_neg_lemma:

  1943  "k<0 ==>

  1944     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"

  1945 apply (rule iffI, clarify)

  1946  apply (erule_tac P="P ?x ?y" in rev_mp)

  1947  apply (subst mod_add_eq)

  1948  apply (subst zdiv_zadd1_eq)

  1949  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)

  1950 txt{*converse direction*}

  1951 apply (drule_tac x = "n div k" in spec)

  1952 apply (drule_tac x = "n mod k" in spec, simp)

  1953 done

  1954

  1955 lemma split_zdiv:

  1956  "P(n div k :: int) =

  1957   ((k = 0 --> P 0) &

  1958    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &

  1959    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"

  1960 apply (case_tac "k=0", simp)

  1961 apply (simp only: linorder_neq_iff)

  1962 apply (erule disjE)

  1963  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]

  1964                       split_neg_lemma [of concl: "%x y. P x"])

  1965 done

  1966

  1967 lemma split_zmod:

  1968  "P(n mod k :: int) =

  1969   ((k = 0 --> P n) &

  1970    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &

  1971    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"

  1972 apply (case_tac "k=0", simp)

  1973 apply (simp only: linorder_neq_iff)

  1974 apply (erule disjE)

  1975  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]

  1976                       split_neg_lemma [of concl: "%x y. P y"])

  1977 done

  1978

  1979 text {* Enable (lin)arith to deal with @{const div} and @{const mod}

  1980   when these are applied to some constant that is of the form

  1981   @{term "numeral k"}: *}

  1982 declare split_zdiv [of _ _ "numeral k", arith_split] for k

  1983 declare split_zmod [of _ _ "numeral k", arith_split] for k

  1984

  1985

  1986 subsubsection {* Speeding up the Division Algorithm with Shifting *}

  1987

  1988 text{*computing div by shifting *}

  1989

  1990 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"

  1991 proof cases

  1992   assume "a=0"

  1993     thus ?thesis by simp

  1994 next

  1995   assume "a\<noteq>0" and le_a: "0\<le>a"

  1996   hence a_pos: "1 \<le> a" by arith

  1997   hence one_less_a2: "1 < 2 * a" by arith

  1998   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"

  1999     unfolding mult_le_cancel_left

  2000     by (simp add: add1_zle_eq add_commute [of 1])

  2001   with a_pos have "0 \<le> b mod a" by simp

  2002   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"

  2003     by (simp add: mod_pos_pos_trivial one_less_a2)

  2004   with  le_2a

  2005   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"

  2006     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2

  2007                   right_distrib)

  2008   thus ?thesis

  2009     by (subst zdiv_zadd1_eq,

  2010         simp add: mod_mult_mult1 one_less_a2

  2011                   div_pos_pos_trivial)

  2012 qed

  2013

  2014 lemma neg_zdiv_mult_2:

  2015   assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"

  2016 proof -

  2017   have R: "1 + - (2 * (b + 1)) = - (1 + 2 * b)" by simp

  2018   have "(1 + 2 * (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a)"

  2019     by (rule pos_zdiv_mult_2, simp add: A)

  2020   thus ?thesis

  2021     by (simp only: R div_minus_minus diff_minus

  2022       minus_add_distrib [symmetric] mult_minus_right)

  2023 qed

  2024

  2025 (* FIXME: add rules for negative numerals *)

  2026 lemma zdiv_numeral_Bit0 [simp]:

  2027   "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =

  2028     numeral v div (numeral w :: int)"

  2029   unfolding numeral.simps unfolding mult_2 [symmetric]

  2030   by (rule div_mult_mult1, simp)

  2031

  2032 lemma zdiv_numeral_Bit1 [simp]:

  2033   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =

  2034     (numeral v div (numeral w :: int))"

  2035   unfolding numeral.simps

  2036   unfolding mult_2 [symmetric] add_commute [of _ 1]

  2037   by (rule pos_zdiv_mult_2, simp)

  2038

  2039

  2040 subsubsection {* Computing mod by Shifting (proofs resemble those for div) *}

  2041

  2042 lemma pos_zmod_mult_2:

  2043   fixes a b :: int

  2044   assumes "0 \<le> a"

  2045   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"

  2046 proof (cases "0 < a")

  2047   case False with assms show ?thesis by simp

  2048 next

  2049   case True

  2050   then have "b mod a < a" by (rule pos_mod_bound)

  2051   then have "1 + b mod a \<le> a" by simp

  2052   then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp

  2053   from 0 < a have "0 \<le> b mod a" by (rule pos_mod_sign)

  2054   then have B: "0 \<le> 1 + 2 * (b mod a)" by simp

  2055   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)"

  2056     using 0 < a and A

  2057     by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)

  2058   then show ?thesis by (subst mod_add_eq)

  2059 qed

  2060

  2061 lemma neg_zmod_mult_2:

  2062   fixes a b :: int

  2063   assumes "a \<le> 0"

  2064   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"

  2065 proof -

  2066   from assms have "0 \<le> - a" by auto

  2067   then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"

  2068     by (rule pos_zmod_mult_2)

  2069   then show ?thesis by (simp add: mod_minus_right algebra_simps)

  2070      (simp add: diff_minus add_ac)

  2071 qed

  2072

  2073 (* FIXME: add rules for negative numerals *)

  2074 lemma zmod_numeral_Bit0 [simp]:

  2075   "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =

  2076     (2::int) * (numeral v mod numeral w)"

  2077   unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]

  2078   unfolding mult_2 [symmetric] by (rule mod_mult_mult1)

  2079

  2080 lemma zmod_numeral_Bit1 [simp]:

  2081   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =

  2082     2 * (numeral v mod numeral w) + (1::int)"

  2083   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]

  2084   unfolding mult_2 [symmetric] add_commute [of _ 1]

  2085   by (rule pos_zmod_mult_2, simp)

  2086

  2087 lemma zdiv_eq_0_iff:

  2088  "(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")

  2089 proof

  2090   assume ?L

  2091   have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp

  2092   with ?L show ?R by blast

  2093 next

  2094   assume ?R thus ?L

  2095     by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)

  2096 qed

  2097

  2098

  2099 subsubsection {* Quotients of Signs *}

  2100

  2101 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"

  2102 apply (subgoal_tac "a div b \<le> -1", force)

  2103 apply (rule order_trans)

  2104 apply (rule_tac a' = "-1" in zdiv_mono1)

  2105 apply (auto simp add: div_eq_minus1)

  2106 done

  2107

  2108 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"

  2109 by (drule zdiv_mono1_neg, auto)

  2110

  2111 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"

  2112 by (drule zdiv_mono1, auto)

  2113

  2114 text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}

  2115 conditional upon the sign of @{text a} or @{text b}. There are many more.

  2116 They should all be simp rules unless that causes too much search. *}

  2117

  2118 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"

  2119 apply auto

  2120 apply (drule_tac [2] zdiv_mono1)

  2121 apply (auto simp add: linorder_neq_iff)

  2122 apply (simp (no_asm_use) add: linorder_not_less [symmetric])

  2123 apply (blast intro: div_neg_pos_less0)

  2124 done

  2125

  2126 lemma neg_imp_zdiv_nonneg_iff:

  2127   "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"

  2128 apply (subst div_minus_minus [symmetric])

  2129 apply (subst pos_imp_zdiv_nonneg_iff, auto)

  2130 done

  2131

  2132 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)

  2133 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"

  2134 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)

  2135

  2136 lemma pos_imp_zdiv_pos_iff:

  2137   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"

  2138 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]

  2139 by arith

  2140

  2141 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)

  2142 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"

  2143 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)

  2144

  2145 lemma nonneg1_imp_zdiv_pos_iff:

  2146   "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"

  2147 apply rule

  2148  apply rule

  2149   using div_pos_pos_trivial[of a b]apply arith

  2150  apply(cases "b=0")apply simp

  2151  using div_nonneg_neg_le0[of a b]apply arith

  2152 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp

  2153 done

  2154

  2155 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"

  2156 apply (rule split_zmod[THEN iffD2])

  2157 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)

  2158 done

  2159

  2160

  2161 subsubsection {* The Divides Relation *}

  2162

  2163 lemmas zdvd_iff_zmod_eq_0_numeral [simp] =

  2164   dvd_eq_mod_eq_0 [of "numeral x::int" "numeral y::int"]

  2165   dvd_eq_mod_eq_0 [of "numeral x::int" "neg_numeral y::int"]

  2166   dvd_eq_mod_eq_0 [of "neg_numeral x::int" "numeral y::int"]

  2167   dvd_eq_mod_eq_0 [of "neg_numeral x::int" "neg_numeral y::int"] for x y

  2168

  2169 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"

  2170   by (rule dvd_mod) (* TODO: remove *)

  2171

  2172 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"

  2173   by (rule dvd_mod_imp_dvd) (* TODO: remove *)

  2174

  2175 lemmas dvd_eq_mod_eq_0_numeral [simp] =

  2176   dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y

  2177

  2178

  2179 subsubsection {* Further properties *}

  2180

  2181 lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"

  2182   using zmod_zdiv_equality[where a="m" and b="n"]

  2183   by (simp add: algebra_simps) (* FIXME: generalize *)

  2184

  2185 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"

  2186 apply (induct "y", auto)

  2187 apply (rule mod_mult_right_eq [THEN trans])

  2188 apply (simp (no_asm_simp))

  2189 apply (rule mod_mult_eq [symmetric])

  2190 done (* FIXME: generalize *)

  2191

  2192 lemma zdiv_int: "int (a div b) = (int a) div (int b)"

  2193 apply (subst split_div, auto)

  2194 apply (subst split_zdiv, auto)

  2195 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)

  2196 apply (auto simp add: divmod_int_rel_def of_nat_mult)

  2197 done

  2198

  2199 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"

  2200 apply (subst split_mod, auto)

  2201 apply (subst split_zmod, auto)

  2202 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia

  2203        in unique_remainder)

  2204 apply (auto simp add: divmod_int_rel_def of_nat_mult)

  2205 done

  2206

  2207 lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"

  2208 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)

  2209

  2210 text{*Suggested by Matthias Daum*}

  2211 lemma int_power_div_base:

  2212      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"

  2213 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")

  2214  apply (erule ssubst)

  2215  apply (simp only: power_add)

  2216  apply simp_all

  2217 done

  2218

  2219 text {* by Brian Huffman *}

  2220 lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"

  2221 by (rule mod_minus_eq [symmetric])

  2222

  2223 lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"

  2224 by (rule mod_diff_left_eq [symmetric])

  2225

  2226 lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"

  2227 by (rule mod_diff_right_eq [symmetric])

  2228

  2229 lemmas zmod_simps =

  2230   mod_add_left_eq  [symmetric]

  2231   mod_add_right_eq [symmetric]

  2232   mod_mult_right_eq[symmetric]

  2233   mod_mult_left_eq [symmetric]

  2234   zpower_zmod

  2235   zminus_zmod zdiff_zmod_left zdiff_zmod_right

  2236

  2237 text {* Distributive laws for function @{text nat}. *}

  2238

  2239 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"

  2240 apply (rule linorder_cases [of y 0])

  2241 apply (simp add: div_nonneg_neg_le0)

  2242 apply simp

  2243 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)

  2244 done

  2245

  2246 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)

  2247 lemma nat_mod_distrib:

  2248   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"

  2249 apply (case_tac "y = 0", simp)

  2250 apply (simp add: nat_eq_iff zmod_int)

  2251 done

  2252

  2253 text  {* transfer setup *}

  2254

  2255 lemma transfer_nat_int_functions:

  2256     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"

  2257     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"

  2258   by (auto simp add: nat_div_distrib nat_mod_distrib)

  2259

  2260 lemma transfer_nat_int_function_closures:

  2261     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"

  2262     "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"

  2263   apply (cases "y = 0")

  2264   apply (auto simp add: pos_imp_zdiv_nonneg_iff)

  2265   apply (cases "y = 0")

  2266   apply auto

  2267 done

  2268

  2269 declare transfer_morphism_nat_int [transfer add return:

  2270   transfer_nat_int_functions

  2271   transfer_nat_int_function_closures

  2272 ]

  2273

  2274 lemma transfer_int_nat_functions:

  2275     "(int x) div (int y) = int (x div y)"

  2276     "(int x) mod (int y) = int (x mod y)"

  2277   by (auto simp add: zdiv_int zmod_int)

  2278

  2279 lemma transfer_int_nat_function_closures:

  2280     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"

  2281     "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"

  2282   by (simp_all only: is_nat_def transfer_nat_int_function_closures)

  2283

  2284 declare transfer_morphism_int_nat [transfer add return:

  2285   transfer_int_nat_functions

  2286   transfer_int_nat_function_closures

  2287 ]

  2288

  2289 text{*Suggested by Matthias Daum*}

  2290 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"

  2291 apply (subgoal_tac "nat x div nat k < nat x")

  2292  apply (simp add: nat_div_distrib [symmetric])

  2293 apply (rule Divides.div_less_dividend, simp_all)

  2294 done

  2295

  2296 lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"

  2297 proof

  2298   assume H: "x mod n = y mod n"

  2299   hence "x mod n - y mod n = 0" by simp

  2300   hence "(x mod n - y mod n) mod n = 0" by simp

  2301   hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])

  2302   thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)

  2303 next

  2304   assume H: "n dvd x - y"

  2305   then obtain k where k: "x-y = n*k" unfolding dvd_def by blast

  2306   hence "x = n*k + y" by simp

  2307   hence "x mod n = (n*k + y) mod n" by simp

  2308   thus "x mod n = y mod n" by (simp add: mod_add_left_eq)

  2309 qed

  2310

  2311 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"

  2312   shows "\<exists>q. x = y + n * q"

  2313 proof-

  2314   from xy have th: "int x - int y = int (x - y)" by simp

  2315   from xyn have "int x mod int n = int y mod int n"

  2316     by (simp add: zmod_int [symmetric])

  2317   hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric])

  2318   hence "n dvd x - y" by (simp add: th zdvd_int)

  2319   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith

  2320 qed

  2321

  2322 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"

  2323   (is "?lhs = ?rhs")

  2324 proof

  2325   assume H: "x mod n = y mod n"

  2326   {assume xy: "x \<le> y"

  2327     from H have th: "y mod n = x mod n" by simp

  2328     from nat_mod_eq_lemma[OF th xy] have ?rhs

  2329       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}

  2330   moreover

  2331   {assume xy: "y \<le> x"

  2332     from nat_mod_eq_lemma[OF H xy] have ?rhs

  2333       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}

  2334   ultimately  show ?rhs using linear[of x y] by blast

  2335 next

  2336   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast

  2337   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp

  2338   thus  ?lhs by simp

  2339 qed

  2340

  2341 lemma div_nat_numeral [simp]:

  2342   "(numeral v :: nat) div numeral v' = nat (numeral v div numeral v')"

  2343   by (simp add: nat_div_distrib)

  2344

  2345 lemma one_div_nat_numeral [simp]:

  2346   "Suc 0 div numeral v' = nat (1 div numeral v')"

  2347   by (subst nat_div_distrib, simp_all)

  2348

  2349 lemma mod_nat_numeral [simp]:

  2350   "(numeral v :: nat) mod numeral v' = nat (numeral v mod numeral v')"

  2351   by (simp add: nat_mod_distrib)

  2352

  2353 lemma one_mod_nat_numeral [simp]:

  2354   "Suc 0 mod numeral v' = nat (1 mod numeral v')"

  2355   by (subst nat_mod_distrib) simp_all

  2356

  2357 lemma mod_2_not_eq_zero_eq_one_int:

  2358   fixes k :: int

  2359   shows "k mod 2 \<noteq> 0 \<longleftrightarrow> k mod 2 = 1"

  2360   by auto

  2361

  2362

  2363 subsubsection {* Tools setup *}

  2364

  2365 text {* Nitpick *}

  2366

  2367 lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'

  2368

  2369

  2370 subsubsection {* Code generation *}

  2371

  2372 definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where

  2373   "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"

  2374

  2375 lemma pdivmod_posDivAlg [code]:

  2376   "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"

  2377 by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)

  2378

  2379 lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  2380   apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0

  2381     then pdivmod k l

  2382     else (let (r, s) = pdivmod k l in

  2383        if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  2384 proof -

  2385   have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto

  2386   show ?thesis

  2387     by (simp add: divmod_int_mod_div pdivmod_def)

  2388       (auto simp add: aux not_less not_le zdiv_zminus1_eq_if

  2389       zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)

  2390 qed

  2391

  2392 lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else

  2393   apsnd ((op *) (sgn l)) (if sgn k = sgn l

  2394     then pdivmod k l

  2395     else (let (r, s) = pdivmod k l in

  2396       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"

  2397 proof -

  2398   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"

  2399     by (auto simp add: not_less sgn_if)

  2400   then show ?thesis by (simp add: divmod_int_pdivmod)

  2401 qed

  2402

  2403 code_modulename SML

  2404   Divides Arith

  2405

  2406 code_modulename OCaml

  2407   Divides Arith

  2408

  2409 code_modulename Haskell

  2410   Divides Arith

  2411

  2412 end
`