src/HOL/Divides.thy
author haftmann
Sun Oct 08 22:28:21 2017 +0200 (20 months ago)
changeset 66808 1907167b6038
parent 66806 a4e82b58d833
child 66810 cc2b490f9dc4
permissions -rw-r--r--
elementary definition of division on natural numbers
paulson@3366
     1
(*  Title:      HOL/Divides.thy
paulson@3366
     2
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@6865
     3
    Copyright   1999  University of Cambridge
huffman@18154
     4
*)
paulson@3366
     5
haftmann@64785
     6
section \<open>More on quotient and remainder\<close>
paulson@3366
     7
nipkow@15131
     8
theory Divides
haftmann@58778
     9
imports Parity
nipkow@15131
    10
begin
paulson@3366
    11
haftmann@64592
    12
subsection \<open>Numeral division with a pragmatic type class\<close>
wenzelm@60758
    13
wenzelm@60758
    14
text \<open>
haftmann@53067
    15
  The following type class contains everything necessary to formulate
haftmann@53067
    16
  a division algorithm in ring structures with numerals, restricted
haftmann@66800
    17
  to its positive segments.  This is its primary motivation, and it
haftmann@53067
    18
  could surely be formulated using a more fine-grained, more algebraic
haftmann@53067
    19
  and less technical class hierarchy.
wenzelm@60758
    20
\<close>
haftmann@53067
    21
haftmann@66806
    22
class unique_euclidean_semiring_numeral = unique_euclidean_semiring + linordered_semidom +
haftmann@59816
    23
  assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
haftmann@53067
    24
    and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
haftmann@53067
    25
    and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
haftmann@53067
    26
    and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
haftmann@53067
    27
    and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
haftmann@53067
    28
    and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
haftmann@53067
    29
    and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@53067
    30
    and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
haftmann@53067
    31
  assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
haftmann@61275
    32
  fixes divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
haftmann@61275
    33
    and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
haftmann@61275
    34
  assumes divmod_def: "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
    35
    and divmod_step_def: "divmod_step l qr = (let (q, r) = qr
haftmann@61275
    36
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
    37
    else (2 * q, r))"
wenzelm@61799
    38
    \<comment> \<open>These are conceptually definitions but force generated code
haftmann@61275
    39
    to be monomorphic wrt. particular instances of this class which
haftmann@61275
    40
    yields a significant speedup.\<close>
haftmann@53067
    41
begin
haftmann@53067
    42
haftmann@66806
    43
subclass unique_euclidean_semiring_parity
haftmann@54226
    44
proof
haftmann@54226
    45
  fix a
haftmann@54226
    46
  show "a mod 2 = 0 \<or> a mod 2 = 1"
haftmann@54226
    47
  proof (rule ccontr)
haftmann@54226
    48
    assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
haftmann@54226
    49
    then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
haftmann@54226
    50
    have "0 < 2" by simp
haftmann@54226
    51
    with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
wenzelm@60758
    52
    with \<open>a mod 2 \<noteq> 0\<close> have "0 < a mod 2" by simp
haftmann@54226
    53
    with discrete have "1 \<le> a mod 2" by simp
wenzelm@60758
    54
    with \<open>a mod 2 \<noteq> 1\<close> have "1 < a mod 2" by simp
haftmann@54226
    55
    with discrete have "2 \<le> a mod 2" by simp
wenzelm@60758
    56
    with \<open>a mod 2 < 2\<close> show False by simp
haftmann@54226
    57
  qed
haftmann@58646
    58
next
haftmann@58646
    59
  show "1 mod 2 = 1"
haftmann@58646
    60
    by (rule mod_less) simp_all
haftmann@58710
    61
next
haftmann@58710
    62
  show "0 \<noteq> 2"
haftmann@58710
    63
    by simp
haftmann@53067
    64
qed
haftmann@53067
    65
haftmann@53067
    66
lemma divmod_digit_1:
haftmann@53067
    67
  assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
haftmann@53067
    68
  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
haftmann@53067
    69
    and "a mod (2 * b) - b = a mod b" (is "?Q")
haftmann@53067
    70
proof -
haftmann@53067
    71
  from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
haftmann@53067
    72
    by (auto intro: trans)
wenzelm@60758
    73
  with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
haftmann@53067
    74
  then have [simp]: "1 \<le> a div b" by (simp add: discrete)
wenzelm@60758
    75
  with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
wenzelm@63040
    76
  define w where "w = a div b mod 2"
wenzelm@63040
    77
  with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
    78
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
    79
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
    80
  from assms w_exhaust have "w = 1"
haftmann@53067
    81
    by (auto simp add: mod_w) (insert mod_less, auto)
haftmann@53067
    82
  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
haftmann@53067
    83
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@64246
    84
    by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
wenzelm@60758
    85
  with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
haftmann@53067
    86
  then show ?P and ?Q
haftmann@60867
    87
    by (simp_all add: div mod add_implies_diff [symmetric])
haftmann@53067
    88
qed
haftmann@53067
    89
haftmann@53067
    90
lemma divmod_digit_0:
haftmann@53067
    91
  assumes "0 < b" and "a mod (2 * b) < b"
haftmann@53067
    92
  shows "2 * (a div (2 * b)) = a div b" (is "?P")
haftmann@53067
    93
    and "a mod (2 * b) = a mod b" (is "?Q")
haftmann@53067
    94
proof -
wenzelm@63040
    95
  define w where "w = a div b mod 2"
wenzelm@63040
    96
  with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
    97
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
    98
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
    99
  moreover have "b \<le> a mod b + b"
haftmann@53067
   100
  proof -
wenzelm@60758
   101
    from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
haftmann@53067
   102
    then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
haftmann@53067
   103
    then show ?thesis by simp
haftmann@53067
   104
  qed
haftmann@53067
   105
  moreover note assms w_exhaust
haftmann@53067
   106
  ultimately have "w = 0" by auto
haftmann@53067
   107
  with mod_w have mod: "a mod (2 * b) = a mod b" by simp
haftmann@53067
   108
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@64246
   109
    by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
wenzelm@60758
   110
  with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
haftmann@53067
   111
  then show ?P and ?Q
haftmann@53067
   112
    by (simp_all add: div mod)
haftmann@53067
   113
qed
haftmann@53067
   114
haftmann@60867
   115
lemma fst_divmod:
haftmann@53067
   116
  "fst (divmod m n) = numeral m div numeral n"
haftmann@53067
   117
  by (simp add: divmod_def)
haftmann@53067
   118
haftmann@60867
   119
lemma snd_divmod:
haftmann@53067
   120
  "snd (divmod m n) = numeral m mod numeral n"
haftmann@53067
   121
  by (simp add: divmod_def)
haftmann@53067
   122
wenzelm@60758
   123
text \<open>
haftmann@53067
   124
  This is a formulation of one step (referring to one digit position)
haftmann@53067
   125
  in school-method division: compare the dividend at the current
haftmann@53070
   126
  digit position with the remainder from previous division steps
haftmann@53067
   127
  and evaluate accordingly.
wenzelm@60758
   128
\<close>
haftmann@53067
   129
haftmann@61275
   130
lemma divmod_step_eq [simp]:
haftmann@53067
   131
  "divmod_step l (q, r) = (if numeral l \<le> r
haftmann@53067
   132
    then (2 * q + 1, r - numeral l) else (2 * q, r))"
haftmann@53067
   133
  by (simp add: divmod_step_def)
haftmann@53067
   134
wenzelm@60758
   135
text \<open>
haftmann@53067
   136
  This is a formulation of school-method division.
haftmann@53067
   137
  If the divisor is smaller than the dividend, terminate.
haftmann@53067
   138
  If not, shift the dividend to the right until termination
haftmann@53067
   139
  occurs and then reiterate single division steps in the
haftmann@53067
   140
  opposite direction.
wenzelm@60758
   141
\<close>
haftmann@53067
   142
haftmann@60867
   143
lemma divmod_divmod_step:
haftmann@53067
   144
  "divmod m n = (if m < n then (0, numeral m)
haftmann@53067
   145
    else divmod_step n (divmod m (Num.Bit0 n)))"
haftmann@53067
   146
proof (cases "m < n")
haftmann@53067
   147
  case True then have "numeral m < numeral n" by simp
haftmann@53067
   148
  then show ?thesis
haftmann@60867
   149
    by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)
haftmann@53067
   150
next
haftmann@53067
   151
  case False
haftmann@53067
   152
  have "divmod m n =
haftmann@53067
   153
    divmod_step n (numeral m div (2 * numeral n),
haftmann@53067
   154
      numeral m mod (2 * numeral n))"
haftmann@53067
   155
  proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
haftmann@53067
   156
    case True
haftmann@60867
   157
    with divmod_step_eq
haftmann@53067
   158
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   159
        (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
haftmann@60867
   160
        by simp
haftmann@53067
   161
    moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
haftmann@53067
   162
      have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
haftmann@53067
   163
      and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
haftmann@53067
   164
      by simp_all
haftmann@53067
   165
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   166
  next
haftmann@53067
   167
    case False then have *: "numeral m mod (2 * numeral n) < numeral n"
haftmann@53067
   168
      by (simp add: not_le)
haftmann@60867
   169
    with divmod_step_eq
haftmann@53067
   170
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   171
        (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
haftmann@60867
   172
        by auto
haftmann@53067
   173
    moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
haftmann@53067
   174
      have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
haftmann@53067
   175
      and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
haftmann@53067
   176
      by (simp_all only: zero_less_numeral)
haftmann@53067
   177
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   178
  qed
haftmann@53067
   179
  then have "divmod m n =
haftmann@53067
   180
    divmod_step n (numeral m div numeral (Num.Bit0 n),
haftmann@53067
   181
      numeral m mod numeral (Num.Bit0 n))"
lp15@60562
   182
    by (simp only: numeral.simps distrib mult_1)
haftmann@53067
   183
  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
haftmann@53067
   184
    by (simp add: divmod_def)
haftmann@53067
   185
  with False show ?thesis by simp
haftmann@53067
   186
qed
haftmann@53067
   187
wenzelm@61799
   188
text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
haftmann@60867
   189
haftmann@61275
   190
lemma divmod_trivial [simp]:
haftmann@60867
   191
  "divmod Num.One Num.One = (numeral Num.One, 0)"
haftmann@60867
   192
  "divmod (Num.Bit0 m) Num.One = (numeral (Num.Bit0 m), 0)"
haftmann@60867
   193
  "divmod (Num.Bit1 m) Num.One = (numeral (Num.Bit1 m), 0)"
haftmann@60867
   194
  "divmod num.One (num.Bit0 n) = (0, Numeral1)"
haftmann@60867
   195
  "divmod num.One (num.Bit1 n) = (0, Numeral1)"
haftmann@60867
   196
  using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
haftmann@60867
   197
haftmann@60867
   198
text \<open>Division by an even number is a right-shift\<close>
haftmann@58953
   199
haftmann@61275
   200
lemma divmod_cancel [simp]:
haftmann@53069
   201
  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
haftmann@53069
   202
  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
haftmann@53069
   203
proof -
haftmann@53069
   204
  have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
haftmann@53069
   205
    "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
haftmann@53069
   206
    by (simp_all only: numeral_mult numeral.simps distrib) simp_all
haftmann@53069
   207
  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
haftmann@53069
   208
  then show ?P and ?Q
haftmann@60867
   209
    by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
haftmann@60867
   210
      div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]
haftmann@60867
   211
      add.commute del: numeral_times_numeral)
haftmann@58953
   212
qed
haftmann@58953
   213
haftmann@60867
   214
text \<open>The really hard work\<close>
haftmann@60867
   215
haftmann@61275
   216
lemma divmod_steps [simp]:
haftmann@60867
   217
  "divmod (num.Bit0 m) (num.Bit1 n) =
haftmann@60867
   218
      (if m \<le> n then (0, numeral (num.Bit0 m))
haftmann@60867
   219
       else divmod_step (num.Bit1 n)
haftmann@60867
   220
             (divmod (num.Bit0 m)
haftmann@60867
   221
               (num.Bit0 (num.Bit1 n))))"
haftmann@60867
   222
  "divmod (num.Bit1 m) (num.Bit1 n) =
haftmann@60867
   223
      (if m < n then (0, numeral (num.Bit1 m))
haftmann@60867
   224
       else divmod_step (num.Bit1 n)
haftmann@60867
   225
             (divmod (num.Bit1 m)
haftmann@60867
   226
               (num.Bit0 (num.Bit1 n))))"
haftmann@60867
   227
  by (simp_all add: divmod_divmod_step)
haftmann@60867
   228
haftmann@61275
   229
lemmas divmod_algorithm_code = divmod_step_eq divmod_trivial divmod_cancel divmod_steps  
haftmann@61275
   230
wenzelm@60758
   231
text \<open>Special case: divisibility\<close>
haftmann@58953
   232
haftmann@58953
   233
definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
haftmann@58953
   234
where
haftmann@58953
   235
  "divides_aux qr \<longleftrightarrow> snd qr = 0"
haftmann@58953
   236
haftmann@58953
   237
lemma divides_aux_eq [simp]:
haftmann@58953
   238
  "divides_aux (q, r) \<longleftrightarrow> r = 0"
haftmann@58953
   239
  by (simp add: divides_aux_def)
haftmann@58953
   240
haftmann@58953
   241
lemma dvd_numeral_simp [simp]:
haftmann@58953
   242
  "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
haftmann@58953
   243
  by (simp add: divmod_def mod_eq_0_iff_dvd)
haftmann@53069
   244
haftmann@60867
   245
text \<open>Generic computation of quotient and remainder\<close>  
haftmann@60867
   246
haftmann@60867
   247
lemma numeral_div_numeral [simp]: 
haftmann@60867
   248
  "numeral k div numeral l = fst (divmod k l)"
haftmann@60867
   249
  by (simp add: fst_divmod)
haftmann@60867
   250
haftmann@60867
   251
lemma numeral_mod_numeral [simp]: 
haftmann@60867
   252
  "numeral k mod numeral l = snd (divmod k l)"
haftmann@60867
   253
  by (simp add: snd_divmod)
haftmann@60867
   254
haftmann@60867
   255
lemma one_div_numeral [simp]:
haftmann@60867
   256
  "1 div numeral n = fst (divmod num.One n)"
haftmann@60867
   257
  by (simp add: fst_divmod)
haftmann@60867
   258
haftmann@60867
   259
lemma one_mod_numeral [simp]:
haftmann@60867
   260
  "1 mod numeral n = snd (divmod num.One n)"
haftmann@60867
   261
  by (simp add: snd_divmod)
haftmann@64630
   262
haftmann@64630
   263
text \<open>Computing congruences modulo \<open>2 ^ q\<close>\<close>
haftmann@64630
   264
haftmann@64630
   265
lemma cong_exp_iff_simps:
haftmann@64630
   266
  "numeral n mod numeral Num.One = 0
haftmann@64630
   267
    \<longleftrightarrow> True"
haftmann@64630
   268
  "numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) = 0
haftmann@64630
   269
    \<longleftrightarrow> numeral n mod numeral q = 0"
haftmann@64630
   270
  "numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) = 0
haftmann@64630
   271
    \<longleftrightarrow> False"
haftmann@64630
   272
  "numeral m mod numeral Num.One = (numeral n mod numeral Num.One)
haftmann@64630
   273
    \<longleftrightarrow> True"
haftmann@64630
   274
  "numeral Num.One mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
haftmann@64630
   275
    \<longleftrightarrow> True"
haftmann@64630
   276
  "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
haftmann@64630
   277
    \<longleftrightarrow> False"
haftmann@64630
   278
  "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
haftmann@64630
   279
    \<longleftrightarrow> (numeral n mod numeral q) = 0"
haftmann@64630
   280
  "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
haftmann@64630
   281
    \<longleftrightarrow> False"
haftmann@64630
   282
  "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
haftmann@64630
   283
    \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
haftmann@64630
   284
  "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
haftmann@64630
   285
    \<longleftrightarrow> False"
haftmann@64630
   286
  "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
haftmann@64630
   287
    \<longleftrightarrow> (numeral m mod numeral q) = 0"
haftmann@64630
   288
  "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
haftmann@64630
   289
    \<longleftrightarrow> False"
haftmann@64630
   290
  "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
haftmann@64630
   291
    \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
haftmann@64630
   292
  by (auto simp add: case_prod_beta dest: arg_cong [of _ _ even])
haftmann@64630
   293
haftmann@53067
   294
end
haftmann@53067
   295
haftmann@66808
   296
hide_fact (open) div_less mod_less mod_less_eq_dividend mod_mult2_eq div_mult2_eq
haftmann@66808
   297
lp15@60562
   298
wenzelm@60758
   299
subsection \<open>Division on @{typ nat}\<close>
wenzelm@60758
   300
haftmann@66806
   301
instantiation nat :: unique_euclidean_semiring_numeral
haftmann@61275
   302
begin
haftmann@61275
   303
haftmann@61275
   304
definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
haftmann@61275
   305
where
haftmann@61275
   306
  divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
   307
haftmann@61275
   308
definition divmod_step_nat :: "num \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
haftmann@61275
   309
where
haftmann@61275
   310
  "divmod_step_nat l qr = (let (q, r) = qr
haftmann@61275
   311
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
   312
    else (2 * q, r))"
haftmann@61275
   313
haftmann@66808
   314
instance by standard
haftmann@66808
   315
  (auto simp add: divmod'_nat_def divmod_step_nat_def div_greater_zero_iff div_mult2_eq mod_mult2_eq)
haftmann@61275
   316
haftmann@61275
   317
end
haftmann@61275
   318
haftmann@61275
   319
declare divmod_algorithm_code [where ?'a = nat, code]
paulson@14267
   320
haftmann@66808
   321
lemma odd_Suc_minus_one [simp]: "odd n \<Longrightarrow> Suc (n - Suc 0) = n"
haftmann@66808
   322
  by (auto elim: oddE)
lp15@60562
   323
haftmann@58778
   324
lemma even_Suc_div_two [simp]:
haftmann@58778
   325
  "even n \<Longrightarrow> Suc n div 2 = n div 2"
haftmann@58778
   326
  using even_succ_div_two [of n] by simp
lp15@60562
   327
haftmann@58778
   328
lemma odd_Suc_div_two [simp]:
haftmann@58778
   329
  "odd n \<Longrightarrow> Suc n div 2 = Suc (n div 2)"
haftmann@58778
   330
  using odd_succ_div_two [of n] by simp
haftmann@58778
   331
haftmann@58834
   332
lemma odd_two_times_div_two_nat [simp]:
haftmann@60352
   333
  assumes "odd n"
haftmann@60352
   334
  shows "2 * (n div 2) = n - (1 :: nat)"
haftmann@60352
   335
proof -
haftmann@60352
   336
  from assms have "2 * (n div 2) + 1 = n"
haftmann@60352
   337
    by (rule odd_two_times_div_two_succ)
haftmann@60352
   338
  then have "Suc (2 * (n div 2)) - 1 = n - 1"
haftmann@60352
   339
    by simp
haftmann@60352
   340
  then show ?thesis
haftmann@60352
   341
    by simp
haftmann@60352
   342
qed
haftmann@58778
   343
haftmann@58778
   344
lemma parity_induct [case_names zero even odd]:
haftmann@58778
   345
  assumes zero: "P 0"
haftmann@58778
   346
  assumes even: "\<And>n. P n \<Longrightarrow> P (2 * n)"
haftmann@58778
   347
  assumes odd: "\<And>n. P n \<Longrightarrow> P (Suc (2 * n))"
haftmann@58778
   348
  shows "P n"
haftmann@58778
   349
proof (induct n rule: less_induct)
haftmann@58778
   350
  case (less n)
haftmann@58778
   351
  show "P n"
haftmann@58778
   352
  proof (cases "n = 0")
haftmann@58778
   353
    case True with zero show ?thesis by simp
haftmann@58778
   354
  next
haftmann@58778
   355
    case False
haftmann@58778
   356
    with less have hyp: "P (n div 2)" by simp
haftmann@58778
   357
    show ?thesis
haftmann@58778
   358
    proof (cases "even n")
haftmann@58778
   359
      case True
haftmann@58778
   360
      with hyp even [of "n div 2"] show ?thesis
haftmann@58834
   361
        by simp
haftmann@58778
   362
    next
haftmann@58778
   363
      case False
lp15@60562
   364
      with hyp odd [of "n div 2"] show ?thesis
haftmann@58834
   365
        by simp
haftmann@58778
   366
    qed
haftmann@58778
   367
  qed
haftmann@58778
   368
qed
haftmann@58778
   369
haftmann@66808
   370
lemma mod_2_not_eq_zero_eq_one_nat:
haftmann@66808
   371
  fixes n :: nat
haftmann@66808
   372
  shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
haftmann@66808
   373
  by (fact not_mod_2_eq_0_eq_1)
haftmann@66808
   374
haftmann@60868
   375
lemma Suc_0_div_numeral [simp]:
haftmann@60868
   376
  fixes k l :: num
haftmann@60868
   377
  shows "Suc 0 div numeral k = fst (divmod Num.One k)"
haftmann@60868
   378
  by (simp_all add: fst_divmod)
haftmann@60868
   379
haftmann@60868
   380
lemma Suc_0_mod_numeral [simp]:
haftmann@60868
   381
  fixes k l :: num
haftmann@60868
   382
  shows "Suc 0 mod numeral k = snd (divmod Num.One k)"
haftmann@60868
   383
  by (simp_all add: snd_divmod)
haftmann@60868
   384
haftmann@66808
   385
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
haftmann@66808
   386
  where "divmod_nat m n = (m div n, m mod n)"
haftmann@66808
   387
haftmann@66808
   388
lemma fst_divmod_nat [simp]:
haftmann@66808
   389
  "fst (divmod_nat m n) = m div n"
haftmann@66808
   390
  by (simp add: divmod_nat_def)
haftmann@66808
   391
haftmann@66808
   392
lemma snd_divmod_nat [simp]:
haftmann@66808
   393
  "snd (divmod_nat m n) = m mod n"
haftmann@66808
   394
  by (simp add: divmod_nat_def)
haftmann@66808
   395
haftmann@66808
   396
lemma divmod_nat_if [code]:
haftmann@66808
   397
  "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@66808
   398
    let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
haftmann@66808
   399
  by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
haftmann@66808
   400
haftmann@66808
   401
lemma [code]:
haftmann@66808
   402
  "m div n = fst (divmod_nat m n)"
haftmann@66808
   403
  "m mod n = snd (divmod_nat m n)"
haftmann@66808
   404
  by simp_all
haftmann@66808
   405
haftmann@33361
   406
wenzelm@60758
   407
subsection \<open>Division on @{typ int}\<close>
haftmann@33361
   408
haftmann@64592
   409
context
haftmann@64592
   410
begin
haftmann@64592
   411
haftmann@64635
   412
inductive eucl_rel_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool"
haftmann@64635
   413
  where eucl_rel_int_by0: "eucl_rel_int k 0 (0, k)"
haftmann@64635
   414
  | eucl_rel_int_dividesI: "l \<noteq> 0 \<Longrightarrow> k = q * l \<Longrightarrow> eucl_rel_int k l (q, 0)"
haftmann@64635
   415
  | eucl_rel_int_remainderI: "sgn r = sgn l \<Longrightarrow> \<bar>r\<bar> < \<bar>l\<bar>
haftmann@64635
   416
      \<Longrightarrow> k = q * l + r \<Longrightarrow> eucl_rel_int k l (q, r)"
haftmann@64635
   417
haftmann@64635
   418
lemma eucl_rel_int_iff:    
haftmann@64635
   419
  "eucl_rel_int k l (q, r) \<longleftrightarrow> 
haftmann@64635
   420
    k = l * q + r \<and>
haftmann@64635
   421
     (if 0 < l then 0 \<le> r \<and> r < l else if l < 0 then l < r \<and> r \<le> 0 else q = 0)"
haftmann@64635
   422
  by (cases "r = 0")
haftmann@64635
   423
    (auto elim!: eucl_rel_int.cases intro: eucl_rel_int_by0 eucl_rel_int_dividesI eucl_rel_int_remainderI
haftmann@64635
   424
    simp add: ac_simps sgn_1_pos sgn_1_neg)
haftmann@33361
   425
haftmann@33361
   426
lemma unique_quotient_lemma:
haftmann@60868
   427
  "b * q' + r' \<le> b * q + r \<Longrightarrow> 0 \<le> r' \<Longrightarrow> r' < b \<Longrightarrow> r < b \<Longrightarrow> q' \<le> (q::int)"
haftmann@33361
   428
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
haftmann@33361
   429
 prefer 2 apply (simp add: right_diff_distrib)
haftmann@33361
   430
apply (subgoal_tac "0 < b * (1 + q - q') ")
haftmann@33361
   431
apply (erule_tac [2] order_le_less_trans)
webertj@49962
   432
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
   433
apply (subgoal_tac "b * q' < b * (1 + q) ")
webertj@49962
   434
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
   435
apply (simp add: mult_less_cancel_left)
haftmann@33361
   436
done
haftmann@33361
   437
haftmann@33361
   438
lemma unique_quotient_lemma_neg:
haftmann@60868
   439
  "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
haftmann@60868
   440
  by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma) auto
haftmann@33361
   441
haftmann@33361
   442
lemma unique_quotient:
haftmann@64635
   443
  "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> q = q'"
haftmann@64635
   444
  apply (simp add: eucl_rel_int_iff linorder_neq_iff split: if_split_asm)
haftmann@64635
   445
  apply (blast intro: order_antisym
haftmann@64635
   446
    dest: order_eq_refl [THEN unique_quotient_lemma]
haftmann@64635
   447
    order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
haftmann@64635
   448
  done
haftmann@33361
   449
haftmann@33361
   450
lemma unique_remainder:
haftmann@64635
   451
  "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> r = r'"
haftmann@33361
   452
apply (subgoal_tac "q = q'")
haftmann@64635
   453
 apply (simp add: eucl_rel_int_iff)
haftmann@33361
   454
apply (blast intro: unique_quotient)
haftmann@33361
   455
done
haftmann@33361
   456
haftmann@64592
   457
end
haftmann@64592
   458
haftmann@64592
   459
instantiation int :: "{idom_modulo, normalization_semidom}"
haftmann@60868
   460
begin
haftmann@60868
   461
haftmann@64592
   462
definition normalize_int :: "int \<Rightarrow> int"
haftmann@64592
   463
  where [simp]: "normalize = (abs :: int \<Rightarrow> int)"
haftmann@64592
   464
haftmann@64592
   465
definition unit_factor_int :: "int \<Rightarrow> int"
haftmann@64592
   466
  where [simp]: "unit_factor = (sgn :: int \<Rightarrow> int)"
haftmann@64592
   467
haftmann@64592
   468
definition divide_int :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@60868
   469
  where "k div l = (if l = 0 \<or> k = 0 then 0
haftmann@60868
   470
    else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
haftmann@60868
   471
      then int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
haftmann@60868
   472
      else
haftmann@60868
   473
        if l dvd k then - int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
haftmann@60868
   474
        else - int (Suc (nat \<bar>k\<bar> div nat \<bar>l\<bar>)))"
haftmann@60868
   475
haftmann@64592
   476
definition modulo_int :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@60868
   477
  where "k mod l = (if l = 0 then k else if l dvd k then 0
haftmann@60868
   478
    else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
haftmann@60868
   479
      then sgn l * int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)
haftmann@60868
   480
      else sgn l * (\<bar>l\<bar> - int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)))"
haftmann@60868
   481
haftmann@64635
   482
lemma eucl_rel_int:
haftmann@64635
   483
  "eucl_rel_int k l (k div l, k mod l)"
haftmann@64592
   484
proof (cases k rule: int_cases3)
haftmann@64592
   485
  case zero
haftmann@64592
   486
  then show ?thesis
haftmann@64635
   487
    by (simp add: eucl_rel_int_iff divide_int_def modulo_int_def)
haftmann@64592
   488
next
haftmann@64592
   489
  case (pos n)
haftmann@64592
   490
  then show ?thesis
haftmann@64592
   491
    using div_mult_mod_eq [of n]
haftmann@64592
   492
    by (cases l rule: int_cases3)
haftmann@64592
   493
      (auto simp del: of_nat_mult of_nat_add
haftmann@64592
   494
        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
haftmann@64635
   495
        eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
haftmann@64592
   496
next
haftmann@64592
   497
  case (neg n)
haftmann@64592
   498
  then show ?thesis
haftmann@64592
   499
    using div_mult_mod_eq [of n]
haftmann@64592
   500
    by (cases l rule: int_cases3)
haftmann@64592
   501
      (auto simp del: of_nat_mult of_nat_add
haftmann@64592
   502
        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
haftmann@64635
   503
        eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
haftmann@64592
   504
qed
haftmann@33361
   505
huffman@47141
   506
lemma divmod_int_unique:
haftmann@64635
   507
  assumes "eucl_rel_int k l (q, r)"
haftmann@60868
   508
  shows div_int_unique: "k div l = q" and mod_int_unique: "k mod l = r"
haftmann@64635
   509
  using assms eucl_rel_int [of k l]
haftmann@60868
   510
  using unique_quotient [of k l] unique_remainder [of k l]
haftmann@60868
   511
  by auto
haftmann@64592
   512
haftmann@64592
   513
instance proof
haftmann@64592
   514
  fix k l :: int
haftmann@64592
   515
  show "k div l * l + k mod l = k"
haftmann@64635
   516
    using eucl_rel_int [of k l]
haftmann@64635
   517
    unfolding eucl_rel_int_iff by (simp add: ac_simps)
huffman@47141
   518
next
haftmann@64592
   519
  fix k :: int show "k div 0 = 0"
haftmann@64635
   520
    by (rule div_int_unique, simp add: eucl_rel_int_iff)
huffman@47141
   521
next
haftmann@64592
   522
  fix k l :: int
haftmann@64592
   523
  assume "l \<noteq> 0"
haftmann@64592
   524
  then show "k * l div l = k"
haftmann@64635
   525
    by (auto simp add: eucl_rel_int_iff ac_simps intro: div_int_unique [of _ _ _ 0])
haftmann@64848
   526
qed (auto simp add: sgn_mult mult_sgn_abs abs_eq_iff')
huffman@47141
   527
haftmann@60429
   528
end
haftmann@60429
   529
haftmann@66806
   530
ML \<open>
haftmann@66806
   531
structure Cancel_Div_Mod_Int = Cancel_Div_Mod
haftmann@66806
   532
(
haftmann@66806
   533
  val div_name = @{const_name divide};
haftmann@66806
   534
  val mod_name = @{const_name modulo};
haftmann@66806
   535
  val mk_binop = HOLogic.mk_binop;
haftmann@66806
   536
  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
haftmann@66806
   537
  val dest_sum = Arith_Data.dest_sum;
haftmann@66806
   538
haftmann@66806
   539
  val div_mod_eqs = map mk_meta_eq @{thms cancel_div_mod_rules};
haftmann@66806
   540
haftmann@66806
   541
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@66806
   542
    @{thms diff_conv_add_uminus add_0_left add_0_right ac_simps})
haftmann@66806
   543
)
haftmann@66806
   544
\<close>
haftmann@66806
   545
haftmann@66806
   546
simproc_setup cancel_div_mod_int ("(k::int) + l") =
haftmann@66806
   547
  \<open>K Cancel_Div_Mod_Int.proc\<close>
haftmann@66806
   548
haftmann@60517
   549
lemma is_unit_int:
haftmann@60517
   550
  "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
haftmann@60517
   551
  by auto
haftmann@60517
   552
haftmann@64715
   553
lemma zdiv_int:
haftmann@64715
   554
  "int (a div b) = int a div int b"
haftmann@64715
   555
  by (simp add: divide_int_def)
haftmann@64715
   556
haftmann@64715
   557
lemma zmod_int:
haftmann@64715
   558
  "int (a mod b) = int a mod int b"
haftmann@64715
   559
  by (simp add: modulo_int_def int_dvd_iff)
haftmann@64715
   560
haftmann@64715
   561
lemma div_abs_eq_div_nat:
haftmann@64715
   562
  "\<bar>k\<bar> div \<bar>l\<bar> = int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)"
haftmann@64715
   563
  by (simp add: divide_int_def)
haftmann@64715
   564
haftmann@64715
   565
lemma mod_abs_eq_div_nat:
haftmann@64715
   566
  "\<bar>k\<bar> mod \<bar>l\<bar> = int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)"
haftmann@64715
   567
  by (simp add: modulo_int_def dvd_int_unfold_dvd_nat)
haftmann@64715
   568
haftmann@64715
   569
lemma div_sgn_abs_cancel:
haftmann@64715
   570
  fixes k l v :: int
haftmann@64715
   571
  assumes "v \<noteq> 0"
haftmann@64715
   572
  shows "(sgn v * \<bar>k\<bar>) div (sgn v * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@64715
   573
proof -
haftmann@64715
   574
  from assms have "sgn v = - 1 \<or> sgn v = 1"
haftmann@64715
   575
    by (cases "v \<ge> 0") auto
haftmann@64715
   576
  then show ?thesis
blanchet@66630
   577
    using assms unfolding divide_int_def [of "sgn v * \<bar>k\<bar>" "sgn v * \<bar>l\<bar>"]
blanchet@66630
   578
    by (fastforce simp add: not_less div_abs_eq_div_nat)
haftmann@64715
   579
qed
haftmann@64715
   580
haftmann@64715
   581
lemma div_eq_sgn_abs:
haftmann@64715
   582
  fixes k l v :: int
haftmann@64715
   583
  assumes "sgn k = sgn l"
haftmann@64715
   584
  shows "k div l = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@64715
   585
proof (cases "l = 0")
haftmann@64715
   586
  case True
haftmann@64715
   587
  then show ?thesis
haftmann@64715
   588
    by simp
haftmann@64715
   589
next
haftmann@64715
   590
  case False
haftmann@64715
   591
  with assms have "(sgn k * \<bar>k\<bar>) div (sgn l * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@64715
   592
    by (simp add: div_sgn_abs_cancel)
haftmann@64715
   593
  then show ?thesis
haftmann@64715
   594
    by (simp add: sgn_mult_abs)
haftmann@64715
   595
qed
haftmann@64715
   596
haftmann@64715
   597
lemma div_dvd_sgn_abs:
haftmann@64715
   598
  fixes k l :: int
haftmann@64715
   599
  assumes "l dvd k"
haftmann@64715
   600
  shows "k div l = (sgn k * sgn l) * (\<bar>k\<bar> div \<bar>l\<bar>)"
haftmann@64715
   601
proof (cases "k = 0")
haftmann@64715
   602
  case True
haftmann@64715
   603
  then show ?thesis
haftmann@64715
   604
    by simp
haftmann@64715
   605
next
haftmann@64715
   606
  case False
haftmann@64715
   607
  show ?thesis
haftmann@64715
   608
  proof (cases "sgn l = sgn k")
haftmann@64715
   609
    case True
haftmann@64715
   610
    then show ?thesis
haftmann@64715
   611
      by (simp add: div_eq_sgn_abs)
haftmann@64715
   612
  next
haftmann@64715
   613
    case False
haftmann@64715
   614
    with \<open>k \<noteq> 0\<close> assms show ?thesis
haftmann@64715
   615
      unfolding divide_int_def [of k l]
haftmann@64715
   616
        by (auto simp add: zdiv_int)
haftmann@64715
   617
  qed
haftmann@64715
   618
qed
haftmann@64715
   619
haftmann@64715
   620
lemma div_noneq_sgn_abs:
haftmann@64715
   621
  fixes k l :: int
haftmann@64715
   622
  assumes "l \<noteq> 0"
haftmann@64715
   623
  assumes "sgn k \<noteq> sgn l"
haftmann@64715
   624
  shows "k div l = - (\<bar>k\<bar> div \<bar>l\<bar>) - of_bool (\<not> l dvd k)"
haftmann@64715
   625
  using assms
haftmann@64715
   626
  by (simp only: divide_int_def [of k l], auto simp add: not_less zdiv_int)
haftmann@64715
   627
  
haftmann@64715
   628
lemma sgn_mod:
haftmann@64715
   629
  fixes k l :: int
haftmann@64715
   630
  assumes "l \<noteq> 0" "\<not> l dvd k"
haftmann@64715
   631
  shows "sgn (k mod l) = sgn l"
haftmann@64715
   632
proof -
haftmann@64715
   633
  from \<open>\<not> l dvd k\<close>
haftmann@64715
   634
  have "k mod l \<noteq> 0"
haftmann@64715
   635
    by (simp add: dvd_eq_mod_eq_0)
haftmann@64715
   636
  show ?thesis
haftmann@64715
   637
    using \<open>l \<noteq> 0\<close> \<open>\<not> l dvd k\<close>
haftmann@64715
   638
    unfolding modulo_int_def [of k l]
haftmann@64715
   639
    by (auto simp add: sgn_1_pos sgn_1_neg mod_greater_zero_iff_not_dvd nat_dvd_iff not_less
haftmann@64715
   640
      zless_nat_eq_int_zless [symmetric] elim: nonpos_int_cases)
haftmann@64715
   641
qed
haftmann@64715
   642
haftmann@64715
   643
lemma abs_mod_less:
haftmann@64715
   644
  fixes k l :: int
haftmann@64715
   645
  assumes "l \<noteq> 0"
haftmann@64715
   646
  shows "\<bar>k mod l\<bar> < \<bar>l\<bar>"
haftmann@64715
   647
  using assms unfolding modulo_int_def [of k l]
haftmann@64715
   648
  by (auto simp add: not_less int_dvd_iff mod_greater_zero_iff_not_dvd elim: pos_int_cases neg_int_cases nonneg_int_cases nonpos_int_cases)
haftmann@64715
   649
haftmann@66806
   650
instantiation int :: unique_euclidean_ring
haftmann@66806
   651
begin
haftmann@66806
   652
haftmann@66806
   653
definition [simp]:
haftmann@66806
   654
  "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
haftmann@66806
   655
haftmann@66806
   656
definition [simp]:
haftmann@66806
   657
  "uniqueness_constraint_int (k :: int) l \<longleftrightarrow> unit_factor k = unit_factor l"
haftmann@66806
   658
  
haftmann@66806
   659
instance proof
haftmann@66806
   660
  fix l q r:: int
haftmann@66806
   661
  assume "uniqueness_constraint r l"
haftmann@66806
   662
    and "euclidean_size r < euclidean_size l"
haftmann@66806
   663
  then have "sgn r = sgn l" and "\<bar>r\<bar> < \<bar>l\<bar>"
haftmann@66806
   664
    by simp_all
haftmann@66806
   665
  then have "eucl_rel_int (q * l + r) l (q, r)"
haftmann@66806
   666
    by (rule eucl_rel_int_remainderI) simp
haftmann@66806
   667
  then show "(q * l + r) div l = q"
haftmann@64592
   668
    by (rule div_int_unique)
haftmann@66806
   669
qed (use mult_le_mono2 [of 1] in \<open>auto simp add: abs_mult sgn_mult abs_mod_less sgn_mod nat_mult_distrib\<close>)
wenzelm@60758
   670
haftmann@66806
   671
end
haftmann@64592
   672
haftmann@64592
   673
text\<open>Basic laws about division and remainder\<close>
haftmann@64592
   674
huffman@47141
   675
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
haftmann@64635
   676
  using eucl_rel_int [of a b]
haftmann@64635
   677
  by (auto simp add: eucl_rel_int_iff prod_eq_iff)
haftmann@33361
   678
wenzelm@45607
   679
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
wenzelm@45607
   680
   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
haftmann@33361
   681
huffman@47141
   682
lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
haftmann@64635
   683
  using eucl_rel_int [of a b]
haftmann@64635
   684
  by (auto simp add: eucl_rel_int_iff prod_eq_iff)
haftmann@33361
   685
wenzelm@45607
   686
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
wenzelm@45607
   687
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
haftmann@33361
   688
haftmann@33361
   689
wenzelm@60758
   690
subsubsection \<open>General Properties of div and mod\<close>
haftmann@33361
   691
haftmann@33361
   692
lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
huffman@47140
   693
apply (rule div_int_unique)
haftmann@64635
   694
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
   695
done
haftmann@33361
   696
haftmann@33361
   697
lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
huffman@47140
   698
apply (rule div_int_unique)
haftmann@64635
   699
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
   700
done
haftmann@33361
   701
haftmann@33361
   702
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
huffman@47140
   703
apply (rule div_int_unique)
haftmann@64635
   704
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
   705
done
haftmann@33361
   706
haftmann@66801
   707
lemma div_positive_int:
haftmann@66801
   708
  "k div l > 0" if "k \<ge> l" and "l > 0" for k l :: int
haftmann@66801
   709
  using that by (simp add: divide_int_def div_positive)
haftmann@66801
   710
haftmann@33361
   711
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
haftmann@33361
   712
haftmann@33361
   713
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
huffman@47140
   714
apply (rule_tac q = 0 in mod_int_unique)
haftmann@64635
   715
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
   716
done
haftmann@33361
   717
haftmann@33361
   718
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
huffman@47140
   719
apply (rule_tac q = 0 in mod_int_unique)
haftmann@64635
   720
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
   721
done
haftmann@33361
   722
haftmann@33361
   723
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
huffman@47140
   724
apply (rule_tac q = "-1" in mod_int_unique)
haftmann@64635
   725
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
   726
done
haftmann@33361
   727
wenzelm@61799
   728
text\<open>There is no \<open>mod_neg_pos_trivial\<close>.\<close>
wenzelm@60758
   729
wenzelm@60758
   730
wenzelm@60758
   731
subsubsection \<open>Laws for div and mod with Unary Minus\<close>
haftmann@33361
   732
haftmann@33361
   733
lemma zminus1_lemma:
haftmann@64635
   734
     "eucl_rel_int a b (q, r) ==> b \<noteq> 0
haftmann@64635
   735
      ==> eucl_rel_int (-a) b (if r=0 then -q else -q - 1,
haftmann@33361
   736
                          if r=0 then 0 else b-r)"
blanchet@66630
   737
by (force simp add: eucl_rel_int_iff right_diff_distrib)
haftmann@33361
   738
haftmann@33361
   739
haftmann@33361
   740
lemma zdiv_zminus1_eq_if:
lp15@60562
   741
     "b \<noteq> (0::int)
lp15@60562
   742
      ==> (-a) div b =
haftmann@33361
   743
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
haftmann@64635
   744
by (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN div_int_unique])
haftmann@33361
   745
haftmann@33361
   746
lemma zmod_zminus1_eq_if:
haftmann@33361
   747
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
haftmann@33361
   748
apply (case_tac "b = 0", simp)
haftmann@64635
   749
apply (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN mod_int_unique])
haftmann@33361
   750
done
haftmann@33361
   751
haftmann@64593
   752
lemma zmod_zminus1_not_zero:
haftmann@33361
   753
  fixes k l :: int
haftmann@33361
   754
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@64592
   755
  by (simp add: mod_eq_0_iff_dvd)
haftmann@64592
   756
haftmann@64593
   757
lemma zmod_zminus2_not_zero:
haftmann@64592
   758
  fixes k l :: int
haftmann@64592
   759
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@64592
   760
  by (simp add: mod_eq_0_iff_dvd)
haftmann@33361
   761
haftmann@33361
   762
lemma zdiv_zminus2_eq_if:
lp15@60562
   763
     "b \<noteq> (0::int)
lp15@60562
   764
      ==> a div (-b) =
haftmann@33361
   765
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47159
   766
by (simp add: zdiv_zminus1_eq_if div_minus_right)
haftmann@33361
   767
haftmann@33361
   768
lemma zmod_zminus2_eq_if:
haftmann@33361
   769
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
huffman@47159
   770
by (simp add: zmod_zminus1_eq_if mod_minus_right)
haftmann@33361
   771
haftmann@33361
   772
wenzelm@60758
   773
subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
haftmann@33361
   774
haftmann@33361
   775
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
haftmann@64246
   776
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
   777
using mult_div_mod_eq [symmetric, of a' b]
haftmann@64246
   778
apply -
haftmann@33361
   779
apply (rule unique_quotient_lemma)
haftmann@33361
   780
apply (erule subst)
haftmann@33361
   781
apply (erule subst, simp_all)
haftmann@33361
   782
done
haftmann@33361
   783
haftmann@33361
   784
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
haftmann@64246
   785
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
   786
using mult_div_mod_eq [symmetric, of a' b]
haftmann@64246
   787
apply -
haftmann@33361
   788
apply (rule unique_quotient_lemma_neg)
haftmann@33361
   789
apply (erule subst)
haftmann@33361
   790
apply (erule subst, simp_all)
haftmann@33361
   791
done
haftmann@33361
   792
haftmann@33361
   793
wenzelm@60758
   794
subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
haftmann@33361
   795
haftmann@33361
   796
lemma q_pos_lemma:
haftmann@33361
   797
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
haftmann@33361
   798
apply (subgoal_tac "0 < b'* (q' + 1) ")
haftmann@33361
   799
 apply (simp add: zero_less_mult_iff)
webertj@49962
   800
apply (simp add: distrib_left)
haftmann@33361
   801
done
haftmann@33361
   802
haftmann@33361
   803
lemma zdiv_mono2_lemma:
lp15@60562
   804
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
lp15@60562
   805
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
haftmann@33361
   806
      ==> q \<le> (q'::int)"
lp15@60562
   807
apply (frule q_pos_lemma, assumption+)
haftmann@33361
   808
apply (subgoal_tac "b*q < b* (q' + 1) ")
haftmann@33361
   809
 apply (simp add: mult_less_cancel_left)
haftmann@33361
   810
apply (subgoal_tac "b*q = r' - r + b'*q'")
haftmann@33361
   811
 prefer 2 apply simp
webertj@49962
   812
apply (simp (no_asm_simp) add: distrib_left)
haftmann@57512
   813
apply (subst add.commute, rule add_less_le_mono, arith)
haftmann@33361
   814
apply (rule mult_right_mono, auto)
haftmann@33361
   815
done
haftmann@33361
   816
haftmann@33361
   817
lemma zdiv_mono2:
haftmann@33361
   818
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
haftmann@33361
   819
apply (subgoal_tac "b \<noteq> 0")
haftmann@64246
   820
  prefer 2 apply arith
haftmann@64246
   821
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
   822
using mult_div_mod_eq [symmetric, of a b']
haftmann@64246
   823
apply -
haftmann@33361
   824
apply (rule zdiv_mono2_lemma)
haftmann@33361
   825
apply (erule subst)
haftmann@33361
   826
apply (erule subst, simp_all)
haftmann@33361
   827
done
haftmann@33361
   828
haftmann@33361
   829
lemma q_neg_lemma:
haftmann@33361
   830
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
haftmann@33361
   831
apply (subgoal_tac "b'*q' < 0")
haftmann@33361
   832
 apply (simp add: mult_less_0_iff, arith)
haftmann@33361
   833
done
haftmann@33361
   834
haftmann@33361
   835
lemma zdiv_mono2_neg_lemma:
lp15@60562
   836
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
lp15@60562
   837
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
haftmann@33361
   838
      ==> q' \<le> (q::int)"
lp15@60562
   839
apply (frule q_neg_lemma, assumption+)
haftmann@33361
   840
apply (subgoal_tac "b*q' < b* (q + 1) ")
haftmann@33361
   841
 apply (simp add: mult_less_cancel_left)
webertj@49962
   842
apply (simp add: distrib_left)
haftmann@33361
   843
apply (subgoal_tac "b*q' \<le> b'*q'")
haftmann@33361
   844
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
haftmann@33361
   845
done
haftmann@33361
   846
haftmann@33361
   847
lemma zdiv_mono2_neg:
haftmann@33361
   848
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
haftmann@64246
   849
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
   850
using mult_div_mod_eq [symmetric, of a b']
haftmann@64246
   851
apply -
haftmann@33361
   852
apply (rule zdiv_mono2_neg_lemma)
haftmann@33361
   853
apply (erule subst)
haftmann@33361
   854
apply (erule subst, simp_all)
haftmann@33361
   855
done
haftmann@33361
   856
haftmann@33361
   857
wenzelm@60758
   858
subsubsection \<open>More Algebraic Laws for div and mod\<close>
wenzelm@60758
   859
wenzelm@60758
   860
text\<open>proving (a*b) div c = a * (b div c) + a * (b mod c)\<close>
haftmann@33361
   861
haftmann@33361
   862
lemma zmult1_lemma:
haftmann@64635
   863
     "[| eucl_rel_int b c (q, r) |]
haftmann@64635
   864
      ==> eucl_rel_int (a * b) c (a*q + a*r div c, a*r mod c)"
haftmann@64635
   865
by (auto simp add: split_ifs eucl_rel_int_iff linorder_neq_iff distrib_left ac_simps)
haftmann@33361
   866
haftmann@33361
   867
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
haftmann@33361
   868
apply (case_tac "c = 0", simp)
haftmann@64635
   869
apply (blast intro: eucl_rel_int [THEN zmult1_lemma, THEN div_int_unique])
haftmann@33361
   870
done
haftmann@33361
   871
wenzelm@60758
   872
text\<open>proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c)\<close>
haftmann@33361
   873
haftmann@33361
   874
lemma zadd1_lemma:
haftmann@64635
   875
     "[| eucl_rel_int a c (aq, ar);  eucl_rel_int b c (bq, br) |]
haftmann@64635
   876
      ==> eucl_rel_int (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
haftmann@64635
   877
by (force simp add: split_ifs eucl_rel_int_iff linorder_neq_iff distrib_left)
haftmann@33361
   878
haftmann@33361
   879
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@33361
   880
lemma zdiv_zadd1_eq:
haftmann@33361
   881
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@33361
   882
apply (case_tac "c = 0", simp)
haftmann@64635
   883
apply (blast intro: zadd1_lemma [OF eucl_rel_int eucl_rel_int] div_int_unique)
haftmann@33361
   884
done
haftmann@33361
   885
haftmann@33361
   886
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
haftmann@33361
   887
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@33361
   888
haftmann@33361
   889
(* REVISIT: should this be generalized to all semiring_div types? *)
haftmann@33361
   890
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
haftmann@33361
   891
haftmann@33361
   892
wenzelm@60758
   893
subsubsection \<open>Proving  @{term "a div (b * c) = (a div b) div c"}\<close>
haftmann@33361
   894
haftmann@33361
   895
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
haftmann@33361
   896
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
haftmann@33361
   897
  to cause particular problems.*)
haftmann@33361
   898
wenzelm@60758
   899
text\<open>first, four lemmas to bound the remainder for the cases b<0 and b>0\<close>
haftmann@33361
   900
blanchet@55085
   901
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b * c < b * (q mod c) + r"
haftmann@33361
   902
apply (subgoal_tac "b * (c - q mod c) < r * 1")
haftmann@33361
   903
 apply (simp add: algebra_simps)
haftmann@33361
   904
apply (rule order_le_less_trans)
haftmann@33361
   905
 apply (erule_tac [2] mult_strict_right_mono)
haftmann@33361
   906
 apply (rule mult_left_mono_neg)
huffman@35216
   907
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
haftmann@33361
   908
 apply (simp)
haftmann@33361
   909
apply (simp)
haftmann@33361
   910
done
haftmann@33361
   911
haftmann@33361
   912
lemma zmult2_lemma_aux2:
haftmann@33361
   913
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
haftmann@33361
   914
apply (subgoal_tac "b * (q mod c) \<le> 0")
haftmann@33361
   915
 apply arith
haftmann@33361
   916
apply (simp add: mult_le_0_iff)
haftmann@33361
   917
done
haftmann@33361
   918
haftmann@33361
   919
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
haftmann@33361
   920
apply (subgoal_tac "0 \<le> b * (q mod c) ")
haftmann@33361
   921
apply arith
haftmann@33361
   922
apply (simp add: zero_le_mult_iff)
haftmann@33361
   923
done
haftmann@33361
   924
haftmann@33361
   925
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
haftmann@33361
   926
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
haftmann@33361
   927
 apply (simp add: right_diff_distrib)
haftmann@33361
   928
apply (rule order_less_le_trans)
haftmann@33361
   929
 apply (erule mult_strict_right_mono)
haftmann@33361
   930
 apply (rule_tac [2] mult_left_mono)
haftmann@33361
   931
  apply simp
huffman@35216
   932
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
haftmann@33361
   933
apply simp
haftmann@33361
   934
done
haftmann@33361
   935
haftmann@64635
   936
lemma zmult2_lemma: "[| eucl_rel_int a b (q, r); 0 < c |]
haftmann@64635
   937
      ==> eucl_rel_int a (b * c) (q div c, b*(q mod c) + r)"
haftmann@64635
   938
by (auto simp add: mult.assoc eucl_rel_int_iff linorder_neq_iff
lp15@60562
   939
                   zero_less_mult_iff distrib_left [symmetric]
nipkow@62390
   940
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: if_split_asm)
haftmann@33361
   941
haftmann@53068
   942
lemma zdiv_zmult2_eq:
haftmann@53068
   943
  fixes a b c :: int
haftmann@53068
   944
  shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
haftmann@33361
   945
apply (case_tac "b = 0", simp)
haftmann@64635
   946
apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN div_int_unique])
haftmann@33361
   947
done
haftmann@33361
   948
haftmann@33361
   949
lemma zmod_zmult2_eq:
haftmann@53068
   950
  fixes a b c :: int
haftmann@53068
   951
  shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@33361
   952
apply (case_tac "b = 0", simp)
haftmann@64635
   953
apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN mod_int_unique])
haftmann@33361
   954
done
haftmann@33361
   955
huffman@47108
   956
lemma div_pos_geq:
huffman@47108
   957
  fixes k l :: int
huffman@47108
   958
  assumes "0 < l" and "l \<le> k"
huffman@47108
   959
  shows "k div l = (k - l) div l + 1"
huffman@47108
   960
proof -
huffman@47108
   961
  have "k = (k - l) + l" by simp
huffman@47108
   962
  then obtain j where k: "k = j + l" ..
eberlm@63499
   963
  with assms show ?thesis by (simp add: div_add_self2)
huffman@47108
   964
qed
huffman@47108
   965
huffman@47108
   966
lemma mod_pos_geq:
huffman@47108
   967
  fixes k l :: int
huffman@47108
   968
  assumes "0 < l" and "l \<le> k"
huffman@47108
   969
  shows "k mod l = (k - l) mod l"
huffman@47108
   970
proof -
huffman@47108
   971
  have "k = (k - l) + l" by simp
huffman@47108
   972
  then obtain j where k: "k = j + l" ..
huffman@47108
   973
  with assms show ?thesis by simp
huffman@47108
   974
qed
huffman@47108
   975
haftmann@33361
   976
wenzelm@60758
   977
subsubsection \<open>Splitting Rules for div and mod\<close>
wenzelm@60758
   978
wenzelm@60758
   979
text\<open>The proofs of the two lemmas below are essentially identical\<close>
haftmann@33361
   980
haftmann@33361
   981
lemma split_pos_lemma:
lp15@60562
   982
 "0<k ==>
haftmann@33361
   983
    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
haftmann@33361
   984
apply (rule iffI, clarify)
lp15@60562
   985
 apply (erule_tac P="P x y" for x y in rev_mp)
haftmann@64593
   986
 apply (subst mod_add_eq [symmetric])
lp15@60562
   987
 apply (subst zdiv_zadd1_eq)
lp15@60562
   988
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
wenzelm@60758
   989
txt\<open>converse direction\<close>
lp15@60562
   990
apply (drule_tac x = "n div k" in spec)
haftmann@33361
   991
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
   992
done
haftmann@33361
   993
haftmann@33361
   994
lemma split_neg_lemma:
haftmann@33361
   995
 "k<0 ==>
haftmann@33361
   996
    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
haftmann@33361
   997
apply (rule iffI, clarify)
lp15@60562
   998
 apply (erule_tac P="P x y" for x y in rev_mp)
haftmann@64593
   999
 apply (subst mod_add_eq [symmetric])
lp15@60562
  1000
 apply (subst zdiv_zadd1_eq)
lp15@60562
  1001
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
wenzelm@60758
  1002
txt\<open>converse direction\<close>
lp15@60562
  1003
apply (drule_tac x = "n div k" in spec)
haftmann@33361
  1004
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  1005
done
haftmann@33361
  1006
haftmann@33361
  1007
lemma split_zdiv:
haftmann@33361
  1008
 "P(n div k :: int) =
lp15@60562
  1009
  ((k = 0 --> P 0) &
lp15@60562
  1010
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
haftmann@33361
  1011
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
haftmann@33361
  1012
apply (case_tac "k=0", simp)
haftmann@33361
  1013
apply (simp only: linorder_neq_iff)
lp15@60562
  1014
apply (erule disjE)
lp15@60562
  1015
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
haftmann@33361
  1016
                      split_neg_lemma [of concl: "%x y. P x"])
haftmann@33361
  1017
done
haftmann@33361
  1018
haftmann@33361
  1019
lemma split_zmod:
haftmann@33361
  1020
 "P(n mod k :: int) =
lp15@60562
  1021
  ((k = 0 --> P n) &
lp15@60562
  1022
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
haftmann@33361
  1023
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
haftmann@33361
  1024
apply (case_tac "k=0", simp)
haftmann@33361
  1025
apply (simp only: linorder_neq_iff)
lp15@60562
  1026
apply (erule disjE)
lp15@60562
  1027
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
haftmann@33361
  1028
                      split_neg_lemma [of concl: "%x y. P y"])
haftmann@33361
  1029
done
haftmann@33361
  1030
haftmann@63950
  1031
text \<open>Enable (lin)arith to deal with @{const divide} and @{const modulo}
webertj@33730
  1032
  when these are applied to some constant that is of the form
wenzelm@60758
  1033
  @{term "numeral k"}:\<close>
huffman@47108
  1034
declare split_zdiv [of _ _ "numeral k", arith_split] for k
huffman@47108
  1035
declare split_zmod [of _ _ "numeral k", arith_split] for k
haftmann@33361
  1036
haftmann@33361
  1037
wenzelm@61799
  1038
subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
huffman@47166
  1039
haftmann@64635
  1040
lemma pos_eucl_rel_int_mult_2:
huffman@47166
  1041
  assumes "0 \<le> b"
haftmann@64635
  1042
  assumes "eucl_rel_int a b (q, r)"
haftmann@64635
  1043
  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 1 + 2*r)"
haftmann@64635
  1044
  using assms unfolding eucl_rel_int_iff by auto
haftmann@64635
  1045
haftmann@64635
  1046
lemma neg_eucl_rel_int_mult_2:
huffman@47166
  1047
  assumes "b \<le> 0"
haftmann@64635
  1048
  assumes "eucl_rel_int (a + 1) b (q, r)"
haftmann@64635
  1049
  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 2*r - 1)"
haftmann@64635
  1050
  using assms unfolding eucl_rel_int_iff by auto
haftmann@33361
  1051
wenzelm@60758
  1052
text\<open>computing div by shifting\<close>
haftmann@33361
  1053
haftmann@33361
  1054
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
haftmann@64635
  1055
  using pos_eucl_rel_int_mult_2 [OF _ eucl_rel_int]
huffman@47166
  1056
  by (rule div_int_unique)
haftmann@33361
  1057
lp15@60562
  1058
lemma neg_zdiv_mult_2:
boehmes@35815
  1059
  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
haftmann@64635
  1060
  using neg_eucl_rel_int_mult_2 [OF A eucl_rel_int]
huffman@47166
  1061
  by (rule div_int_unique)
haftmann@33361
  1062
huffman@47108
  1063
(* FIXME: add rules for negative numerals *)
huffman@47108
  1064
lemma zdiv_numeral_Bit0 [simp]:
huffman@47108
  1065
  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
huffman@47108
  1066
    numeral v div (numeral w :: int)"
huffman@47108
  1067
  unfolding numeral.simps unfolding mult_2 [symmetric]
huffman@47108
  1068
  by (rule div_mult_mult1, simp)
huffman@47108
  1069
huffman@47108
  1070
lemma zdiv_numeral_Bit1 [simp]:
lp15@60562
  1071
  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
huffman@47108
  1072
    (numeral v div (numeral w :: int))"
huffman@47108
  1073
  unfolding numeral.simps
haftmann@57512
  1074
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
  1075
  by (rule pos_zdiv_mult_2, simp)
haftmann@33361
  1076
haftmann@33361
  1077
lemma pos_zmod_mult_2:
haftmann@33361
  1078
  fixes a b :: int
haftmann@33361
  1079
  assumes "0 \<le> a"
haftmann@33361
  1080
  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
haftmann@64635
  1081
  using pos_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
huffman@47166
  1082
  by (rule mod_int_unique)
haftmann@33361
  1083
haftmann@33361
  1084
lemma neg_zmod_mult_2:
haftmann@33361
  1085
  fixes a b :: int
haftmann@33361
  1086
  assumes "a \<le> 0"
haftmann@33361
  1087
  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
haftmann@64635
  1088
  using neg_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
huffman@47166
  1089
  by (rule mod_int_unique)
haftmann@33361
  1090
huffman@47108
  1091
(* FIXME: add rules for negative numerals *)
huffman@47108
  1092
lemma zmod_numeral_Bit0 [simp]:
lp15@60562
  1093
  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
huffman@47108
  1094
    (2::int) * (numeral v mod numeral w)"
huffman@47108
  1095
  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
huffman@47108
  1096
  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
huffman@47108
  1097
huffman@47108
  1098
lemma zmod_numeral_Bit1 [simp]:
huffman@47108
  1099
  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
huffman@47108
  1100
    2 * (numeral v mod numeral w) + (1::int)"
huffman@47108
  1101
  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
haftmann@57512
  1102
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
  1103
  by (rule pos_zmod_mult_2, simp)
haftmann@33361
  1104
nipkow@39489
  1105
lemma zdiv_eq_0_iff:
nipkow@39489
  1106
 "(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")
nipkow@39489
  1107
proof
nipkow@39489
  1108
  assume ?L
nipkow@39489
  1109
  have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
wenzelm@60758
  1110
  with \<open>?L\<close> show ?R by blast
nipkow@39489
  1111
next
nipkow@39489
  1112
  assume ?R thus ?L
nipkow@39489
  1113
    by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
nipkow@39489
  1114
qed
nipkow@39489
  1115
haftmann@63947
  1116
lemma zmod_trival_iff:
haftmann@63947
  1117
  fixes i k :: int
haftmann@63947
  1118
  shows "i mod k = i \<longleftrightarrow> k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i"
haftmann@63947
  1119
proof -
haftmann@63947
  1120
  have "i mod k = i \<longleftrightarrow> i div k = 0"
haftmann@64242
  1121
    by safe (insert div_mult_mod_eq [of i k], auto)
haftmann@63947
  1122
  with zdiv_eq_0_iff
haftmann@63947
  1123
  show ?thesis
haftmann@63947
  1124
    by simp
haftmann@63947
  1125
qed
nipkow@39489
  1126
haftmann@64785
  1127
  
wenzelm@60758
  1128
subsubsection \<open>Quotients of Signs\<close>
haftmann@33361
  1129
haftmann@60868
  1130
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@60868
  1131
by (simp add: divide_int_def)
haftmann@60868
  1132
haftmann@60868
  1133
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@63950
  1134
by (simp add: modulo_int_def)
haftmann@60868
  1135
haftmann@33361
  1136
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
haftmann@33361
  1137
apply (subgoal_tac "a div b \<le> -1", force)
haftmann@33361
  1138
apply (rule order_trans)
haftmann@33361
  1139
apply (rule_tac a' = "-1" in zdiv_mono1)
haftmann@33361
  1140
apply (auto simp add: div_eq_minus1)
haftmann@33361
  1141
done
haftmann@33361
  1142
haftmann@33361
  1143
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
haftmann@33361
  1144
by (drule zdiv_mono1_neg, auto)
haftmann@33361
  1145
haftmann@33361
  1146
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
haftmann@33361
  1147
by (drule zdiv_mono1, auto)
haftmann@33361
  1148
wenzelm@61799
  1149
text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
wenzelm@61799
  1150
conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
wenzelm@60758
  1151
They should all be simp rules unless that causes too much search.\<close>
nipkow@33804
  1152
haftmann@33361
  1153
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
haftmann@33361
  1154
apply auto
haftmann@33361
  1155
apply (drule_tac [2] zdiv_mono1)
haftmann@33361
  1156
apply (auto simp add: linorder_neq_iff)
haftmann@33361
  1157
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
haftmann@33361
  1158
apply (blast intro: div_neg_pos_less0)
haftmann@33361
  1159
done
haftmann@33361
  1160
haftmann@60868
  1161
lemma pos_imp_zdiv_pos_iff:
haftmann@60868
  1162
  "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
haftmann@60868
  1163
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
haftmann@60868
  1164
by arith
haftmann@60868
  1165
haftmann@33361
  1166
lemma neg_imp_zdiv_nonneg_iff:
nipkow@33804
  1167
  "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
huffman@47159
  1168
apply (subst div_minus_minus [symmetric])
haftmann@33361
  1169
apply (subst pos_imp_zdiv_nonneg_iff, auto)
haftmann@33361
  1170
done
haftmann@33361
  1171
haftmann@33361
  1172
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
haftmann@33361
  1173
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
haftmann@33361
  1174
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
haftmann@33361
  1175
haftmann@33361
  1176
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
haftmann@33361
  1177
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
haftmann@33361
  1178
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
haftmann@33361
  1179
nipkow@33804
  1180
lemma nonneg1_imp_zdiv_pos_iff:
nipkow@33804
  1181
  "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
nipkow@33804
  1182
apply rule
nipkow@33804
  1183
 apply rule
nipkow@33804
  1184
  using div_pos_pos_trivial[of a b]apply arith
nipkow@33804
  1185
 apply(cases "b=0")apply simp
nipkow@33804
  1186
 using div_nonneg_neg_le0[of a b]apply arith
nipkow@33804
  1187
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
nipkow@33804
  1188
done
nipkow@33804
  1189
nipkow@39489
  1190
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
nipkow@39489
  1191
apply (rule split_zmod[THEN iffD2])
nipkow@44890
  1192
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
nipkow@39489
  1193
done
nipkow@39489
  1194
haftmann@60868
  1195
haftmann@60868
  1196
subsubsection \<open>Computation of Division and Remainder\<close>
haftmann@60868
  1197
haftmann@66806
  1198
instantiation int :: unique_euclidean_semiring_numeral
haftmann@61275
  1199
begin
haftmann@61275
  1200
haftmann@61275
  1201
definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
haftmann@61275
  1202
where
haftmann@61275
  1203
  "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
  1204
haftmann@61275
  1205
definition divmod_step_int :: "num \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
haftmann@61275
  1206
where
haftmann@61275
  1207
  "divmod_step_int l qr = (let (q, r) = qr
haftmann@61275
  1208
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
  1209
    else (2 * q, r))"
haftmann@61275
  1210
haftmann@61275
  1211
instance
haftmann@61275
  1212
  by standard (auto intro: zmod_le_nonneg_dividend simp add: divmod_int_def divmod_step_int_def
haftmann@61275
  1213
    pos_imp_zdiv_pos_iff div_pos_pos_trivial mod_pos_pos_trivial zmod_zmult2_eq zdiv_zmult2_eq)
haftmann@61275
  1214
haftmann@61275
  1215
end
haftmann@61275
  1216
haftmann@61275
  1217
declare divmod_algorithm_code [where ?'a = int, code]
lp15@60562
  1218
haftmann@60930
  1219
context
haftmann@60930
  1220
begin
haftmann@60930
  1221
  
haftmann@60930
  1222
qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
haftmann@60868
  1223
where
haftmann@60868
  1224
  "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
haftmann@60868
  1225
haftmann@60930
  1226
qualified lemma adjust_div_eq [simp, code]:
haftmann@60868
  1227
  "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
haftmann@60868
  1228
  by (simp add: adjust_div_def)
haftmann@60868
  1229
haftmann@60930
  1230
qualified definition adjust_mod :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@60868
  1231
where
haftmann@60868
  1232
  [simp]: "adjust_mod l r = (if r = 0 then 0 else l - r)"
haftmann@60868
  1233
haftmann@60868
  1234
lemma minus_numeral_div_numeral [simp]:
haftmann@60868
  1235
  "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
haftmann@60868
  1236
proof -
haftmann@60868
  1237
  have "int (fst (divmod m n)) = fst (divmod m n)"
haftmann@60868
  1238
    by (simp only: fst_divmod divide_int_def) auto
haftmann@60868
  1239
  then show ?thesis
haftmann@60868
  1240
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
haftmann@60868
  1241
qed
haftmann@60868
  1242
haftmann@60868
  1243
lemma minus_numeral_mod_numeral [simp]:
haftmann@60868
  1244
  "- numeral m mod numeral n = adjust_mod (numeral n) (snd (divmod m n) :: int)"
haftmann@60868
  1245
proof -
haftmann@60868
  1246
  have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
haftmann@63950
  1247
    using that by (simp only: snd_divmod modulo_int_def) auto
haftmann@60868
  1248
  then show ?thesis
haftmann@63950
  1249
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def modulo_int_def)
haftmann@60868
  1250
qed
haftmann@60868
  1251
haftmann@60868
  1252
lemma numeral_div_minus_numeral [simp]:
haftmann@60868
  1253
  "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
haftmann@60868
  1254
proof -
haftmann@60868
  1255
  have "int (fst (divmod m n)) = fst (divmod m n)"
haftmann@60868
  1256
    by (simp only: fst_divmod divide_int_def) auto
haftmann@60868
  1257
  then show ?thesis
haftmann@60868
  1258
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
haftmann@60868
  1259
qed
haftmann@60868
  1260
  
haftmann@60868
  1261
lemma numeral_mod_minus_numeral [simp]:
haftmann@60868
  1262
  "numeral m mod - numeral n = - adjust_mod (numeral n) (snd (divmod m n) :: int)"
haftmann@60868
  1263
proof -
haftmann@60868
  1264
  have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
haftmann@63950
  1265
    using that by (simp only: snd_divmod modulo_int_def) auto
haftmann@60868
  1266
  then show ?thesis
haftmann@63950
  1267
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def modulo_int_def)
haftmann@60868
  1268
qed
haftmann@60868
  1269
haftmann@60868
  1270
lemma minus_one_div_numeral [simp]:
haftmann@60868
  1271
  "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
haftmann@60868
  1272
  using minus_numeral_div_numeral [of Num.One n] by simp  
haftmann@60868
  1273
haftmann@60868
  1274
lemma minus_one_mod_numeral [simp]:
haftmann@60868
  1275
  "- 1 mod numeral n = adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
haftmann@60868
  1276
  using minus_numeral_mod_numeral [of Num.One n] by simp
haftmann@60868
  1277
haftmann@60868
  1278
lemma one_div_minus_numeral [simp]:
haftmann@60868
  1279
  "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
haftmann@60868
  1280
  using numeral_div_minus_numeral [of Num.One n] by simp
haftmann@60868
  1281
  
haftmann@60868
  1282
lemma one_mod_minus_numeral [simp]:
haftmann@60868
  1283
  "1 mod - numeral n = - adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
haftmann@60868
  1284
  using numeral_mod_minus_numeral [of Num.One n] by simp
haftmann@60868
  1285
haftmann@60930
  1286
end
haftmann@60930
  1287
haftmann@60868
  1288
haftmann@60868
  1289
subsubsection \<open>Further properties\<close>
haftmann@60868
  1290
haftmann@60868
  1291
text \<open>Simplify expresions in which div and mod combine numerical constants\<close>
haftmann@60868
  1292
haftmann@60868
  1293
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
haftmann@64635
  1294
  by (rule div_int_unique [of a b q r]) (simp add: eucl_rel_int_iff)
haftmann@60868
  1295
haftmann@60868
  1296
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
haftmann@60868
  1297
  by (rule div_int_unique [of a b q r],
haftmann@64635
  1298
    simp add: eucl_rel_int_iff)
haftmann@60868
  1299
haftmann@60868
  1300
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
haftmann@60868
  1301
  by (rule mod_int_unique [of a b q r],
haftmann@64635
  1302
    simp add: eucl_rel_int_iff)
haftmann@60868
  1303
haftmann@60868
  1304
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
haftmann@60868
  1305
  by (rule mod_int_unique [of a b q r],
haftmann@64635
  1306
    simp add: eucl_rel_int_iff)
haftmann@33361
  1307
wenzelm@61944
  1308
lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
haftmann@33361
  1309
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
haftmann@33361
  1310
wenzelm@60758
  1311
text\<open>Suggested by Matthias Daum\<close>
haftmann@33361
  1312
lemma int_power_div_base:
haftmann@33361
  1313
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
haftmann@33361
  1314
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
haftmann@33361
  1315
 apply (erule ssubst)
haftmann@33361
  1316
 apply (simp only: power_add)
haftmann@33361
  1317
 apply simp_all
haftmann@33361
  1318
done
haftmann@33361
  1319
wenzelm@61799
  1320
text \<open>Distributive laws for function \<open>nat\<close>.\<close>
haftmann@33361
  1321
haftmann@33361
  1322
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
haftmann@33361
  1323
apply (rule linorder_cases [of y 0])
haftmann@33361
  1324
apply (simp add: div_nonneg_neg_le0)
haftmann@33361
  1325
apply simp
haftmann@33361
  1326
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
haftmann@33361
  1327
done
haftmann@33361
  1328
haftmann@33361
  1329
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
haftmann@33361
  1330
lemma nat_mod_distrib:
haftmann@33361
  1331
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
haftmann@33361
  1332
apply (case_tac "y = 0", simp)
haftmann@33361
  1333
apply (simp add: nat_eq_iff zmod_int)
haftmann@33361
  1334
done
haftmann@33361
  1335
wenzelm@60758
  1336
text  \<open>transfer setup\<close>
haftmann@33361
  1337
haftmann@33361
  1338
lemma transfer_nat_int_functions:
haftmann@33361
  1339
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
haftmann@33361
  1340
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
haftmann@33361
  1341
  by (auto simp add: nat_div_distrib nat_mod_distrib)
haftmann@33361
  1342
haftmann@33361
  1343
lemma transfer_nat_int_function_closures:
haftmann@33361
  1344
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
haftmann@33361
  1345
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
haftmann@33361
  1346
  apply (cases "y = 0")
haftmann@33361
  1347
  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
haftmann@33361
  1348
  apply (cases "y = 0")
haftmann@33361
  1349
  apply auto
haftmann@33361
  1350
done
haftmann@33361
  1351
haftmann@35644
  1352
declare transfer_morphism_nat_int [transfer add return:
haftmann@33361
  1353
  transfer_nat_int_functions
haftmann@33361
  1354
  transfer_nat_int_function_closures
haftmann@33361
  1355
]
haftmann@33361
  1356
haftmann@33361
  1357
lemma transfer_int_nat_functions:
haftmann@33361
  1358
    "(int x) div (int y) = int (x div y)"
haftmann@33361
  1359
    "(int x) mod (int y) = int (x mod y)"
haftmann@33361
  1360
  by (auto simp add: zdiv_int zmod_int)
haftmann@33361
  1361
haftmann@33361
  1362
lemma transfer_int_nat_function_closures:
haftmann@33361
  1363
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
haftmann@33361
  1364
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
haftmann@33361
  1365
  by (simp_all only: is_nat_def transfer_nat_int_function_closures)
haftmann@33361
  1366
haftmann@35644
  1367
declare transfer_morphism_int_nat [transfer add return:
haftmann@33361
  1368
  transfer_int_nat_functions
haftmann@33361
  1369
  transfer_int_nat_function_closures
haftmann@33361
  1370
]
haftmann@33361
  1371
wenzelm@60758
  1372
text\<open>Suggested by Matthias Daum\<close>
haftmann@33361
  1373
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
haftmann@33361
  1374
apply (subgoal_tac "nat x div nat k < nat x")
nipkow@34225
  1375
 apply (simp add: nat_div_distrib [symmetric])
haftmann@66808
  1376
apply (rule div_less_dividend, simp_all)
haftmann@33361
  1377
done
haftmann@33361
  1378
haftmann@64593
  1379
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \<le> x"
haftmann@33361
  1380
  shows "\<exists>q. x = y + n * q"
haftmann@33361
  1381
proof-
lp15@60562
  1382
  from xy have th: "int x - int y = int (x - y)" by simp
lp15@60562
  1383
  from xyn have "int x mod int n = int y mod int n"
huffman@46551
  1384
    by (simp add: zmod_int [symmetric])
haftmann@64593
  1385
  hence "int n dvd int x - int y" by (simp only: mod_eq_dvd_iff [symmetric])
haftmann@33361
  1386
  hence "n dvd x - y" by (simp add: th zdvd_int)
haftmann@33361
  1387
  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
haftmann@33361
  1388
qed
haftmann@33361
  1389
lp15@60562
  1390
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
haftmann@33361
  1391
  (is "?lhs = ?rhs")
haftmann@33361
  1392
proof
haftmann@33361
  1393
  assume H: "x mod n = y mod n"
haftmann@33361
  1394
  {assume xy: "x \<le> y"
haftmann@33361
  1395
    from H have th: "y mod n = x mod n" by simp
lp15@60562
  1396
    from nat_mod_eq_lemma[OF th xy] have ?rhs
haftmann@33361
  1397
      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
haftmann@33361
  1398
  moreover
haftmann@33361
  1399
  {assume xy: "y \<le> x"
lp15@60562
  1400
    from nat_mod_eq_lemma[OF H xy] have ?rhs
haftmann@33361
  1401
      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
lp15@60562
  1402
  ultimately  show ?rhs using linear[of x y] by blast
haftmann@33361
  1403
next
haftmann@33361
  1404
  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
haftmann@33361
  1405
  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
haftmann@33361
  1406
  thus  ?lhs by simp
haftmann@33361
  1407
qed
haftmann@33361
  1408
haftmann@66808
  1409
haftmann@60868
  1410
subsubsection \<open>Dedicated simproc for calculation\<close>
haftmann@60868
  1411
wenzelm@60758
  1412
text \<open>
haftmann@60868
  1413
  There is space for improvement here: the calculation itself
haftmann@66808
  1414
  could be carried out outside the logic, and a generic simproc
haftmann@60868
  1415
  (simplifier setup) for generic calculation would be helpful. 
wenzelm@60758
  1416
\<close>
haftmann@53067
  1417
haftmann@60868
  1418
simproc_setup numeral_divmod
haftmann@66806
  1419
  ("0 div 0 :: 'a :: unique_euclidean_semiring_numeral" | "0 mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@66806
  1420
   "0 div 1 :: 'a :: unique_euclidean_semiring_numeral" | "0 mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1421
   "0 div - 1 :: int" | "0 mod - 1 :: int" |
haftmann@66806
  1422
   "0 div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "0 mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1423
   "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
haftmann@66806
  1424
   "1 div 0 :: 'a :: unique_euclidean_semiring_numeral" | "1 mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@66806
  1425
   "1 div 1 :: 'a :: unique_euclidean_semiring_numeral" | "1 mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1426
   "1 div - 1 :: int" | "1 mod - 1 :: int" |
haftmann@66806
  1427
   "1 div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "1 mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1428
   "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
haftmann@60868
  1429
   "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
haftmann@60868
  1430
   "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
haftmann@60868
  1431
   "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
haftmann@66806
  1432
   "numeral a div 0 :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@66806
  1433
   "numeral a div 1 :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1434
   "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
haftmann@66806
  1435
   "numeral a div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  1436
   "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
haftmann@60868
  1437
   "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
haftmann@60868
  1438
   "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
haftmann@60868
  1439
   "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
haftmann@60868
  1440
   "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
haftmann@60868
  1441
   "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") =
haftmann@60868
  1442
\<open> let
haftmann@60868
  1443
    val if_cong = the (Code.get_case_cong @{theory} @{const_name If});
haftmann@60868
  1444
    fun successful_rewrite ctxt ct =
haftmann@60868
  1445
      let
haftmann@60868
  1446
        val thm = Simplifier.rewrite ctxt ct
haftmann@60868
  1447
      in if Thm.is_reflexive thm then NONE else SOME thm end;
haftmann@60868
  1448
  in fn phi =>
haftmann@60868
  1449
    let
haftmann@60868
  1450
      val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
haftmann@60868
  1451
        one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
haftmann@60868
  1452
        one_div_minus_numeral one_mod_minus_numeral
haftmann@60868
  1453
        numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
haftmann@60868
  1454
        numeral_div_minus_numeral numeral_mod_minus_numeral
haftmann@60930
  1455
        div_minus_minus mod_minus_minus Divides.adjust_div_eq of_bool_eq one_neq_zero
haftmann@60868
  1456
        numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
haftmann@60868
  1457
        divmod_cancel divmod_steps divmod_step_eq fst_conv snd_conv numeral_One
haftmann@60930
  1458
        case_prod_beta rel_simps Divides.adjust_mod_def div_minus1_right mod_minus1_right
haftmann@60868
  1459
        minus_minus numeral_times_numeral mult_zero_right mult_1_right}
haftmann@60868
  1460
        @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
haftmann@60868
  1461
      fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
haftmann@60868
  1462
        (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
haftmann@60868
  1463
    in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
haftmann@60868
  1464
  end;
haftmann@60868
  1465
\<close>
blanchet@34126
  1466
haftmann@35673
  1467
wenzelm@60758
  1468
subsubsection \<open>Code generation\<close>
haftmann@33361
  1469
haftmann@60868
  1470
lemma [code]:
haftmann@60868
  1471
  fixes k :: int
haftmann@60868
  1472
  shows 
haftmann@60868
  1473
    "k div 0 = 0"
haftmann@60868
  1474
    "k mod 0 = k"
haftmann@60868
  1475
    "0 div k = 0"
haftmann@60868
  1476
    "0 mod k = 0"
haftmann@60868
  1477
    "k div Int.Pos Num.One = k"
haftmann@60868
  1478
    "k mod Int.Pos Num.One = 0"
haftmann@60868
  1479
    "k div Int.Neg Num.One = - k"
haftmann@60868
  1480
    "k mod Int.Neg Num.One = 0"
haftmann@60868
  1481
    "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
haftmann@60868
  1482
    "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
haftmann@60930
  1483
    "Int.Neg m div Int.Pos n = - (Divides.adjust_div (divmod m n) :: int)"
haftmann@60930
  1484
    "Int.Neg m mod Int.Pos n = Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
haftmann@60930
  1485
    "Int.Pos m div Int.Neg n = - (Divides.adjust_div (divmod m n) :: int)"
haftmann@60930
  1486
    "Int.Pos m mod Int.Neg n = - Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
haftmann@60868
  1487
    "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
haftmann@60868
  1488
    "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
haftmann@60868
  1489
  by simp_all
haftmann@53069
  1490
haftmann@52435
  1491
code_identifier
haftmann@52435
  1492
  code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1493
haftmann@60868
  1494
lemma dvd_eq_mod_eq_0_numeral:
haftmann@66806
  1495
  "numeral x dvd (numeral y :: 'a) \<longleftrightarrow> numeral y mod numeral x = (0 :: 'a::semidom_modulo)"
haftmann@60868
  1496
  by (fact dvd_eq_mod_eq_0)
haftmann@60868
  1497
haftmann@64246
  1498
declare minus_div_mult_eq_mod [symmetric, nitpick_unfold]
haftmann@64246
  1499
haftmann@66808
  1500
haftmann@66808
  1501
subsubsection \<open>Lemmas of doubtful value\<close>
haftmann@66808
  1502
haftmann@66808
  1503
lemma mod_mult_self3':
haftmann@66808
  1504
  "Suc (k * n + m) mod n = Suc m mod n"
haftmann@66808
  1505
  by (fact Suc_mod_mult_self3)
haftmann@66808
  1506
haftmann@66808
  1507
lemma mod_Suc_eq_Suc_mod:
haftmann@66808
  1508
  "Suc m mod n = Suc (m mod n) mod n"
haftmann@66808
  1509
  by (simp add: mod_simps)
haftmann@66808
  1510
haftmann@66808
  1511
lemma div_geq:
haftmann@66808
  1512
  "m div n = Suc ((m - n) div n)" if "0 < n" and " \<not> m < n" for m n :: nat
haftmann@66808
  1513
  by (rule le_div_geq) (use that in \<open>simp_all add: not_less\<close>)
haftmann@66808
  1514
haftmann@66808
  1515
lemma mod_geq:
haftmann@66808
  1516
  "m mod n = (m - n) mod n" if "\<not> m < n" for m n :: nat
haftmann@66808
  1517
  by (rule le_mod_geq) (use that in \<open>simp add: not_less\<close>)
haftmann@66808
  1518
haftmann@66808
  1519
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
haftmann@66808
  1520
  by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@66808
  1521
haftmann@66808
  1522
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
haftmann@66808
  1523
haftmann@66808
  1524
(*Loses information, namely we also have r<d provided d is nonzero*)
haftmann@66808
  1525
lemma mod_eqD:
haftmann@66808
  1526
  fixes m d r q :: nat
haftmann@66808
  1527
  assumes "m mod d = r"
haftmann@66808
  1528
  shows "\<exists>q. m = r + q * d"
haftmann@66808
  1529
proof -
haftmann@66808
  1530
  from div_mult_mod_eq obtain q where "q * d + m mod d = m" by blast
haftmann@66808
  1531
  with assms have "m = r + q * d" by simp
haftmann@66808
  1532
  then show ?thesis ..
haftmann@66808
  1533
qed
haftmann@66808
  1534
haftmann@33361
  1535
end