src/HOL/Divides.thy
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```     1 (*  Title:      HOL/Divides.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1999  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 section \<open>More on quotient and remainder\<close>
```
```     7
```
```     8 theory Divides
```
```     9 imports Parity
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Numeral division with a pragmatic type class\<close>
```
```    13
```
```    14 text \<open>
```
```    15   The following type class contains everything necessary to formulate
```
```    16   a division algorithm in ring structures with numerals, restricted
```
```    17   to its positive segments.  This is its primary motivation, and it
```
```    18   could surely be formulated using a more fine-grained, more algebraic
```
```    19   and less technical class hierarchy.
```
```    20 \<close>
```
```    21
```
```    22 class unique_euclidean_semiring_numeral = semiring_parity + linordered_semidom +
```
```    23   assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
```
```    24     and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
```
```    25     and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
```
```    26     and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
```
```    27     and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
```
```    28     and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
```
```    29     and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
```
```    30     and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
```
```    31   assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
```
```    32   fixes divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
```
```    33     and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
```
```    34   assumes divmod_def: "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```    35     and divmod_step_def: "divmod_step l qr = (let (q, r) = qr
```
```    36     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```    37     else (2 * q, r))"
```
```    38     \<comment> \<open>These are conceptually definitions but force generated code
```
```    39     to be monomorphic wrt. particular instances of this class which
```
```    40     yields a significant speedup.\<close>
```
```    41 begin
```
```    42
```
```    43 lemma divmod_digit_1:
```
```    44   assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
```
```    45   shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
```
```    46     and "a mod (2 * b) - b = a mod b" (is "?Q")
```
```    47 proof -
```
```    48   from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
```
```    49     by (auto intro: trans)
```
```    50   with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
```
```    51   then have [simp]: "1 \<le> a div b" by (simp add: discrete)
```
```    52   with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
```
```    53   define w where "w = a div b mod 2"
```
```    54   then have w_exhaust: "w = 0 \<or> w = 1" by auto
```
```    55   have mod_w: "a mod (2 * b) = a mod b + b * w"
```
```    56     by (simp add: w_def mod_mult2_eq ac_simps)
```
```    57   from assms w_exhaust have "w = 1"
```
```    58     by (auto simp add: mod_w) (insert mod_less, auto)
```
```    59   with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
```
```    60   have "2 * (a div (2 * b)) = a div b - w"
```
```    61     by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
```
```    62   with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
```
```    63   then show ?P and ?Q
```
```    64     by (simp_all add: div mod add_implies_diff [symmetric])
```
```    65 qed
```
```    66
```
```    67 lemma divmod_digit_0:
```
```    68   assumes "0 < b" and "a mod (2 * b) < b"
```
```    69   shows "2 * (a div (2 * b)) = a div b" (is "?P")
```
```    70     and "a mod (2 * b) = a mod b" (is "?Q")
```
```    71 proof -
```
```    72   define w where "w = a div b mod 2"
```
```    73   then have w_exhaust: "w = 0 \<or> w = 1" by auto
```
```    74   have mod_w: "a mod (2 * b) = a mod b + b * w"
```
```    75     by (simp add: w_def mod_mult2_eq ac_simps)
```
```    76   moreover have "b \<le> a mod b + b"
```
```    77   proof -
```
```    78     from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
```
```    79     then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
```
```    80     then show ?thesis by simp
```
```    81   qed
```
```    82   moreover note assms w_exhaust
```
```    83   ultimately have "w = 0" by auto
```
```    84   with mod_w have mod: "a mod (2 * b) = a mod b" by simp
```
```    85   have "2 * (a div (2 * b)) = a div b - w"
```
```    86     by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
```
```    87   with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
```
```    88   then show ?P and ?Q
```
```    89     by (simp_all add: div mod)
```
```    90 qed
```
```    91
```
```    92 lemma fst_divmod:
```
```    93   "fst (divmod m n) = numeral m div numeral n"
```
```    94   by (simp add: divmod_def)
```
```    95
```
```    96 lemma snd_divmod:
```
```    97   "snd (divmod m n) = numeral m mod numeral n"
```
```    98   by (simp add: divmod_def)
```
```    99
```
```   100 text \<open>
```
```   101   This is a formulation of one step (referring to one digit position)
```
```   102   in school-method division: compare the dividend at the current
```
```   103   digit position with the remainder from previous division steps
```
```   104   and evaluate accordingly.
```
```   105 \<close>
```
```   106
```
```   107 lemma divmod_step_eq [simp]:
```
```   108   "divmod_step l (q, r) = (if numeral l \<le> r
```
```   109     then (2 * q + 1, r - numeral l) else (2 * q, r))"
```
```   110   by (simp add: divmod_step_def)
```
```   111
```
```   112 text \<open>
```
```   113   This is a formulation of school-method division.
```
```   114   If the divisor is smaller than the dividend, terminate.
```
```   115   If not, shift the dividend to the right until termination
```
```   116   occurs and then reiterate single division steps in the
```
```   117   opposite direction.
```
```   118 \<close>
```
```   119
```
```   120 lemma divmod_divmod_step:
```
```   121   "divmod m n = (if m < n then (0, numeral m)
```
```   122     else divmod_step n (divmod m (Num.Bit0 n)))"
```
```   123 proof (cases "m < n")
```
```   124   case True then have "numeral m < numeral n" by simp
```
```   125   then show ?thesis
```
```   126     by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)
```
```   127 next
```
```   128   case False
```
```   129   have "divmod m n =
```
```   130     divmod_step n (numeral m div (2 * numeral n),
```
```   131       numeral m mod (2 * numeral n))"
```
```   132   proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
```
```   133     case True
```
```   134     with divmod_step_eq
```
```   135       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
```
```   136         (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
```
```   137         by simp
```
```   138     moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
```
```   139       have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
```
```   140       and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
```
```   141       by simp_all
```
```   142     ultimately show ?thesis by (simp only: divmod_def)
```
```   143   next
```
```   144     case False then have *: "numeral m mod (2 * numeral n) < numeral n"
```
```   145       by (simp add: not_le)
```
```   146     with divmod_step_eq
```
```   147       have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
```
```   148         (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
```
```   149         by auto
```
```   150     moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
```
```   151       have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
```
```   152       and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
```
```   153       by (simp_all only: zero_less_numeral)
```
```   154     ultimately show ?thesis by (simp only: divmod_def)
```
```   155   qed
```
```   156   then have "divmod m n =
```
```   157     divmod_step n (numeral m div numeral (Num.Bit0 n),
```
```   158       numeral m mod numeral (Num.Bit0 n))"
```
```   159     by (simp only: numeral.simps distrib mult_1)
```
```   160   then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
```
```   161     by (simp add: divmod_def)
```
```   162   with False show ?thesis by simp
```
```   163 qed
```
```   164
```
```   165 text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
```
```   166
```
```   167 lemma divmod_trivial [simp]:
```
```   168   "divmod Num.One Num.One = (numeral Num.One, 0)"
```
```   169   "divmod (Num.Bit0 m) Num.One = (numeral (Num.Bit0 m), 0)"
```
```   170   "divmod (Num.Bit1 m) Num.One = (numeral (Num.Bit1 m), 0)"
```
```   171   "divmod num.One (num.Bit0 n) = (0, Numeral1)"
```
```   172   "divmod num.One (num.Bit1 n) = (0, Numeral1)"
```
```   173   using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
```
```   174
```
```   175 text \<open>Division by an even number is a right-shift\<close>
```
```   176
```
```   177 lemma divmod_cancel [simp]:
```
```   178   "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
```
```   179   "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
```
```   180 proof -
```
```   181   have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
```
```   182     "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
```
```   183     by (simp_all only: numeral_mult numeral.simps distrib) simp_all
```
```   184   have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
```
```   185   then show ?P and ?Q
```
```   186     by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
```
```   187       div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]
```
```   188       add.commute del: numeral_times_numeral)
```
```   189 qed
```
```   190
```
```   191 text \<open>The really hard work\<close>
```
```   192
```
```   193 lemma divmod_steps [simp]:
```
```   194   "divmod (num.Bit0 m) (num.Bit1 n) =
```
```   195       (if m \<le> n then (0, numeral (num.Bit0 m))
```
```   196        else divmod_step (num.Bit1 n)
```
```   197              (divmod (num.Bit0 m)
```
```   198                (num.Bit0 (num.Bit1 n))))"
```
```   199   "divmod (num.Bit1 m) (num.Bit1 n) =
```
```   200       (if m < n then (0, numeral (num.Bit1 m))
```
```   201        else divmod_step (num.Bit1 n)
```
```   202              (divmod (num.Bit1 m)
```
```   203                (num.Bit0 (num.Bit1 n))))"
```
```   204   by (simp_all add: divmod_divmod_step)
```
```   205
```
```   206 lemmas divmod_algorithm_code = divmod_step_eq divmod_trivial divmod_cancel divmod_steps
```
```   207
```
```   208 text \<open>Special case: divisibility\<close>
```
```   209
```
```   210 definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
```
```   211 where
```
```   212   "divides_aux qr \<longleftrightarrow> snd qr = 0"
```
```   213
```
```   214 lemma divides_aux_eq [simp]:
```
```   215   "divides_aux (q, r) \<longleftrightarrow> r = 0"
```
```   216   by (simp add: divides_aux_def)
```
```   217
```
```   218 lemma dvd_numeral_simp [simp]:
```
```   219   "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
```
```   220   by (simp add: divmod_def mod_eq_0_iff_dvd)
```
```   221
```
```   222 text \<open>Generic computation of quotient and remainder\<close>
```
```   223
```
```   224 lemma numeral_div_numeral [simp]:
```
```   225   "numeral k div numeral l = fst (divmod k l)"
```
```   226   by (simp add: fst_divmod)
```
```   227
```
```   228 lemma numeral_mod_numeral [simp]:
```
```   229   "numeral k mod numeral l = snd (divmod k l)"
```
```   230   by (simp add: snd_divmod)
```
```   231
```
```   232 lemma one_div_numeral [simp]:
```
```   233   "1 div numeral n = fst (divmod num.One n)"
```
```   234   by (simp add: fst_divmod)
```
```   235
```
```   236 lemma one_mod_numeral [simp]:
```
```   237   "1 mod numeral n = snd (divmod num.One n)"
```
```   238   by (simp add: snd_divmod)
```
```   239
```
```   240 text \<open>Computing congruences modulo \<open>2 ^ q\<close>\<close>
```
```   241
```
```   242 lemma cong_exp_iff_simps:
```
```   243   "numeral n mod numeral Num.One = 0
```
```   244     \<longleftrightarrow> True"
```
```   245   "numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) = 0
```
```   246     \<longleftrightarrow> numeral n mod numeral q = 0"
```
```   247   "numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) = 0
```
```   248     \<longleftrightarrow> False"
```
```   249   "numeral m mod numeral Num.One = (numeral n mod numeral Num.One)
```
```   250     \<longleftrightarrow> True"
```
```   251   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
```
```   252     \<longleftrightarrow> True"
```
```   253   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
```
```   254     \<longleftrightarrow> False"
```
```   255   "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
```
```   256     \<longleftrightarrow> (numeral n mod numeral q) = 0"
```
```   257   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
```
```   258     \<longleftrightarrow> False"
```
```   259   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
```
```   260     \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
```
```   261   "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
```
```   262     \<longleftrightarrow> False"
```
```   263   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
```
```   264     \<longleftrightarrow> (numeral m mod numeral q) = 0"
```
```   265   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
```
```   266     \<longleftrightarrow> False"
```
```   267   "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
```
```   268     \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
```
```   269   by (auto simp add: case_prod_beta dest: arg_cong [of _ _ even])
```
```   270
```
```   271 end
```
```   272
```
```   273 hide_fact (open) div_less mod_less mod_less_eq_dividend mod_mult2_eq div_mult2_eq
```
```   274
```
```   275
```
```   276 subsection \<open>More on division\<close>
```
```   277
```
```   278 instantiation nat :: unique_euclidean_semiring_numeral
```
```   279 begin
```
```   280
```
```   281 definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
```
```   282 where
```
```   283   divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```   284
```
```   285 definition divmod_step_nat :: "num \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
```
```   286 where
```
```   287   "divmod_step_nat l qr = (let (q, r) = qr
```
```   288     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```   289     else (2 * q, r))"
```
```   290
```
```   291 instance by standard
```
```   292   (auto simp add: divmod'_nat_def divmod_step_nat_def div_greater_zero_iff div_mult2_eq mod_mult2_eq)
```
```   293
```
```   294 end
```
```   295
```
```   296 declare divmod_algorithm_code [where ?'a = nat, code]
```
```   297
```
```   298 lemma Suc_0_div_numeral [simp]:
```
```   299   fixes k l :: num
```
```   300   shows "Suc 0 div numeral k = fst (divmod Num.One k)"
```
```   301   by (simp_all add: fst_divmod)
```
```   302
```
```   303 lemma Suc_0_mod_numeral [simp]:
```
```   304   fixes k l :: num
```
```   305   shows "Suc 0 mod numeral k = snd (divmod Num.One k)"
```
```   306   by (simp_all add: snd_divmod)
```
```   307
```
```   308 definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat"
```
```   309   where "divmod_nat m n = (m div n, m mod n)"
```
```   310
```
```   311 lemma fst_divmod_nat [simp]:
```
```   312   "fst (divmod_nat m n) = m div n"
```
```   313   by (simp add: divmod_nat_def)
```
```   314
```
```   315 lemma snd_divmod_nat [simp]:
```
```   316   "snd (divmod_nat m n) = m mod n"
```
```   317   by (simp add: divmod_nat_def)
```
```   318
```
```   319 lemma divmod_nat_if [code]:
```
```   320   "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
```
```   321     let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
```
```   322   by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
```
```   323
```
```   324 lemma [code]:
```
```   325   "m div n = fst (divmod_nat m n)"
```
```   326   "m mod n = snd (divmod_nat m n)"
```
```   327   by simp_all
```
```   328
```
```   329 inductive eucl_rel_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool"
```
```   330   where eucl_rel_int_by0: "eucl_rel_int k 0 (0, k)"
```
```   331   | eucl_rel_int_dividesI: "l \<noteq> 0 \<Longrightarrow> k = q * l \<Longrightarrow> eucl_rel_int k l (q, 0)"
```
```   332   | eucl_rel_int_remainderI: "sgn r = sgn l \<Longrightarrow> \<bar>r\<bar> < \<bar>l\<bar>
```
```   333       \<Longrightarrow> k = q * l + r \<Longrightarrow> eucl_rel_int k l (q, r)"
```
```   334
```
```   335 lemma eucl_rel_int_iff:
```
```   336   "eucl_rel_int k l (q, r) \<longleftrightarrow>
```
```   337     k = l * q + r \<and>
```
```   338      (if 0 < l then 0 \<le> r \<and> r < l else if l < 0 then l < r \<and> r \<le> 0 else q = 0)"
```
```   339   by (cases "r = 0")
```
```   340     (auto elim!: eucl_rel_int.cases intro: eucl_rel_int_by0 eucl_rel_int_dividesI eucl_rel_int_remainderI
```
```   341     simp add: ac_simps sgn_1_pos sgn_1_neg)
```
```   342
```
```   343 lemma unique_quotient_lemma:
```
```   344   "b * q' + r' \<le> b * q + r \<Longrightarrow> 0 \<le> r' \<Longrightarrow> r' < b \<Longrightarrow> r < b \<Longrightarrow> q' \<le> (q::int)"
```
```   345 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
```
```   346  prefer 2 apply (simp add: right_diff_distrib)
```
```   347 apply (subgoal_tac "0 < b * (1 + q - q') ")
```
```   348 apply (erule_tac [2] order_le_less_trans)
```
```   349  prefer 2 apply (simp add: right_diff_distrib distrib_left)
```
```   350 apply (subgoal_tac "b * q' < b * (1 + q) ")
```
```   351  prefer 2 apply (simp add: right_diff_distrib distrib_left)
```
```   352 apply (simp add: mult_less_cancel_left)
```
```   353 done
```
```   354
```
```   355 lemma unique_quotient_lemma_neg:
```
```   356   "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
```
```   357   by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma) auto
```
```   358
```
```   359 lemma unique_quotient:
```
```   360   "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> q = q'"
```
```   361   apply (simp add: eucl_rel_int_iff linorder_neq_iff split: if_split_asm)
```
```   362   apply (blast intro: order_antisym
```
```   363     dest: order_eq_refl [THEN unique_quotient_lemma]
```
```   364     order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
```
```   365   done
```
```   366
```
```   367 lemma unique_remainder:
```
```   368   "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> r = r'"
```
```   369 apply (subgoal_tac "q = q'")
```
```   370  apply (simp add: eucl_rel_int_iff)
```
```   371 apply (blast intro: unique_quotient)
```
```   372 done
```
```   373
```
```   374 lemma eucl_rel_int:
```
```   375   "eucl_rel_int k l (k div l, k mod l)"
```
```   376 proof (cases k rule: int_cases3)
```
```   377   case zero
```
```   378   then show ?thesis
```
```   379     by (simp add: eucl_rel_int_iff divide_int_def modulo_int_def)
```
```   380 next
```
```   381   case (pos n)
```
```   382   then show ?thesis
```
```   383     using div_mult_mod_eq [of n]
```
```   384     by (cases l rule: int_cases3)
```
```   385       (auto simp del: of_nat_mult of_nat_add
```
```   386         simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
```
```   387         eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
```
```   388 next
```
```   389   case (neg n)
```
```   390   then show ?thesis
```
```   391     using div_mult_mod_eq [of n]
```
```   392     by (cases l rule: int_cases3)
```
```   393       (auto simp del: of_nat_mult of_nat_add
```
```   394         simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
```
```   395         eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
```
```   396 qed
```
```   397
```
```   398 lemma divmod_int_unique:
```
```   399   assumes "eucl_rel_int k l (q, r)"
```
```   400   shows div_int_unique: "k div l = q" and mod_int_unique: "k mod l = r"
```
```   401   using assms eucl_rel_int [of k l]
```
```   402   using unique_quotient [of k l] unique_remainder [of k l]
```
```   403   by auto
```
```   404
```
```   405 lemma div_abs_eq_div_nat:
```
```   406   "\<bar>k\<bar> div \<bar>l\<bar> = int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)"
```
```   407   by (simp add: divide_int_def)
```
```   408
```
```   409 lemma mod_abs_eq_div_nat:
```
```   410   "\<bar>k\<bar> mod \<bar>l\<bar> = int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)"
```
```   411   by (simp add: modulo_int_def)
```
```   412
```
```   413 lemma zdiv_int:
```
```   414   "int (a div b) = int a div int b"
```
```   415   by (simp add: divide_int_def sgn_1_pos)
```
```   416
```
```   417 lemma zmod_int:
```
```   418   "int (a mod b) = int a mod int b"
```
```   419   by (simp add: modulo_int_def sgn_1_pos)
```
```   420
```
```   421 lemma div_sgn_abs_cancel:
```
```   422   fixes k l v :: int
```
```   423   assumes "v \<noteq> 0"
```
```   424   shows "(sgn v * \<bar>k\<bar>) div (sgn v * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
```
```   425 proof -
```
```   426   from assms have "sgn v = - 1 \<or> sgn v = 1"
```
```   427     by (cases "v \<ge> 0") auto
```
```   428   then show ?thesis
```
```   429     using assms unfolding divide_int_def [of "sgn v * \<bar>k\<bar>" "sgn v * \<bar>l\<bar>"]
```
```   430     by (fastforce simp add: not_less div_abs_eq_div_nat)
```
```   431 qed
```
```   432
```
```   433 lemma div_eq_sgn_abs:
```
```   434   fixes k l v :: int
```
```   435   assumes "sgn k = sgn l"
```
```   436   shows "k div l = \<bar>k\<bar> div \<bar>l\<bar>"
```
```   437 proof (cases "l = 0")
```
```   438   case True
```
```   439   then show ?thesis
```
```   440     by simp
```
```   441 next
```
```   442   case False
```
```   443   with assms have "(sgn k * \<bar>k\<bar>) div (sgn l * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
```
```   444     using div_sgn_abs_cancel [of l k l] by simp
```
```   445   then show ?thesis
```
```   446     by (simp add: sgn_mult_abs)
```
```   447 qed
```
```   448
```
```   449 lemma div_dvd_sgn_abs:
```
```   450   fixes k l :: int
```
```   451   assumes "l dvd k"
```
```   452   shows "k div l = (sgn k * sgn l) * (\<bar>k\<bar> div \<bar>l\<bar>)"
```
```   453 proof (cases "k = 0 \<or> l = 0")
```
```   454   case True
```
```   455   then show ?thesis
```
```   456     by auto
```
```   457 next
```
```   458   case False
```
```   459   then have "k \<noteq> 0" and "l \<noteq> 0"
```
```   460     by auto
```
```   461   show ?thesis
```
```   462   proof (cases "sgn l = sgn k")
```
```   463     case True
```
```   464     then show ?thesis
```
```   465       by (simp add: div_eq_sgn_abs)
```
```   466   next
```
```   467     case False
```
```   468     with \<open>k \<noteq> 0\<close> \<open>l \<noteq> 0\<close>
```
```   469     have "sgn l * sgn k = - 1"
```
```   470       by (simp add: sgn_if split: if_splits)
```
```   471     with assms show ?thesis
```
```   472       unfolding divide_int_def [of k l]
```
```   473       by (auto simp add: zdiv_int ac_simps)
```
```   474   qed
```
```   475 qed
```
```   476
```
```   477 lemma div_noneq_sgn_abs:
```
```   478   fixes k l :: int
```
```   479   assumes "l \<noteq> 0"
```
```   480   assumes "sgn k \<noteq> sgn l"
```
```   481   shows "k div l = - (\<bar>k\<bar> div \<bar>l\<bar>) - of_bool (\<not> l dvd k)"
```
```   482   using assms
```
```   483   by (simp only: divide_int_def [of k l], auto simp add: not_less zdiv_int)
```
```   484
```
```   485 text\<open>Basic laws about division and remainder\<close>
```
```   486
```
```   487 lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
```
```   488   using eucl_rel_int [of a b]
```
```   489   by (auto simp add: eucl_rel_int_iff prod_eq_iff)
```
```   490
```
```   491 lemmas pos_mod_sign = pos_mod_conj [THEN conjunct1]
```
```   492    and pos_mod_bound = pos_mod_conj [THEN conjunct2]
```
```   493
```
```   494 lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
```
```   495   using eucl_rel_int [of a b]
```
```   496   by (auto simp add: eucl_rel_int_iff prod_eq_iff)
```
```   497
```
```   498 lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
```
```   499    and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
```
```   500
```
```   501
```
```   502 subsubsection \<open>General Properties of div and mod\<close>
```
```   503
```
```   504 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
```
```   505 apply (rule div_int_unique)
```
```   506 apply (auto simp add: eucl_rel_int_iff)
```
```   507 done
```
```   508
```
```   509 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
```
```   510 apply (rule div_int_unique)
```
```   511 apply (auto simp add: eucl_rel_int_iff)
```
```   512 done
```
```   513
```
```   514 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
```
```   515 apply (rule div_int_unique)
```
```   516 apply (auto simp add: eucl_rel_int_iff)
```
```   517 done
```
```   518
```
```   519 lemma div_positive_int:
```
```   520   "k div l > 0" if "k \<ge> l" and "l > 0" for k l :: int
```
```   521   using that by (simp add: divide_int_def div_positive)
```
```   522
```
```   523 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
```
```   524
```
```   525 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
```
```   526 apply (rule_tac q = 0 in mod_int_unique)
```
```   527 apply (auto simp add: eucl_rel_int_iff)
```
```   528 done
```
```   529
```
```   530 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
```
```   531 apply (rule_tac q = 0 in mod_int_unique)
```
```   532 apply (auto simp add: eucl_rel_int_iff)
```
```   533 done
```
```   534
```
```   535 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
```
```   536 apply (rule_tac q = "-1" in mod_int_unique)
```
```   537 apply (auto simp add: eucl_rel_int_iff)
```
```   538 done
```
```   539
```
```   540 text\<open>There is no \<open>mod_neg_pos_trivial\<close>.\<close>
```
```   541
```
```   542
```
```   543 subsubsection \<open>Laws for div and mod with Unary Minus\<close>
```
```   544
```
```   545 lemma zminus1_lemma:
```
```   546      "eucl_rel_int a b (q, r) ==> b \<noteq> 0
```
```   547       ==> eucl_rel_int (-a) b (if r=0 then -q else -q - 1,
```
```   548                           if r=0 then 0 else b-r)"
```
```   549 by (force simp add: eucl_rel_int_iff right_diff_distrib)
```
```   550
```
```   551
```
```   552 lemma zdiv_zminus1_eq_if:
```
```   553      "b \<noteq> (0::int)
```
```   554       ==> (-a) div b =
```
```   555           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
```
```   556 by (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN div_int_unique])
```
```   557
```
```   558 lemma zmod_zminus1_eq_if:
```
```   559      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
```
```   560 apply (case_tac "b = 0", simp)
```
```   561 apply (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN mod_int_unique])
```
```   562 done
```
```   563
```
```   564 lemma zmod_zminus1_not_zero:
```
```   565   fixes k l :: int
```
```   566   shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
```
```   567   by (simp add: mod_eq_0_iff_dvd)
```
```   568
```
```   569 lemma zmod_zminus2_not_zero:
```
```   570   fixes k l :: int
```
```   571   shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
```
```   572   by (simp add: mod_eq_0_iff_dvd)
```
```   573
```
```   574 lemma zdiv_zminus2_eq_if:
```
```   575   "b \<noteq> (0::int)
```
```   576       ==> a div (-b) =
```
```   577           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
```
```   578   by (auto simp add: zdiv_zminus1_eq_if div_minus_right)
```
```   579
```
```   580 lemma zmod_zminus2_eq_if:
```
```   581   "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
```
```   582   by (auto simp add: zmod_zminus1_eq_if mod_minus_right)
```
```   583
```
```   584
```
```   585 subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
```
```   586
```
```   587 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
```
```   588 using mult_div_mod_eq [symmetric, of a b]
```
```   589 using mult_div_mod_eq [symmetric, of a' b]
```
```   590 apply -
```
```   591 apply (rule unique_quotient_lemma)
```
```   592 apply (erule subst)
```
```   593 apply (erule subst, simp_all)
```
```   594 done
```
```   595
```
```   596 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
```
```   597 using mult_div_mod_eq [symmetric, of a b]
```
```   598 using mult_div_mod_eq [symmetric, of a' b]
```
```   599 apply -
```
```   600 apply (rule unique_quotient_lemma_neg)
```
```   601 apply (erule subst)
```
```   602 apply (erule subst, simp_all)
```
```   603 done
```
```   604
```
```   605
```
```   606 subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
```
```   607
```
```   608 lemma q_pos_lemma:
```
```   609      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
```
```   610 apply (subgoal_tac "0 < b'* (q' + 1) ")
```
```   611  apply (simp add: zero_less_mult_iff)
```
```   612 apply (simp add: distrib_left)
```
```   613 done
```
```   614
```
```   615 lemma zdiv_mono2_lemma:
```
```   616      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
```
```   617          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
```
```   618       ==> q \<le> (q'::int)"
```
```   619 apply (frule q_pos_lemma, assumption+)
```
```   620 apply (subgoal_tac "b*q < b* (q' + 1) ")
```
```   621  apply (simp add: mult_less_cancel_left)
```
```   622 apply (subgoal_tac "b*q = r' - r + b'*q'")
```
```   623  prefer 2 apply simp
```
```   624 apply (simp (no_asm_simp) add: distrib_left)
```
```   625 apply (subst add.commute, rule add_less_le_mono, arith)
```
```   626 apply (rule mult_right_mono, auto)
```
```   627 done
```
```   628
```
```   629 lemma zdiv_mono2:
```
```   630      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
```
```   631 apply (subgoal_tac "b \<noteq> 0")
```
```   632   prefer 2 apply arith
```
```   633 using mult_div_mod_eq [symmetric, of a b]
```
```   634 using mult_div_mod_eq [symmetric, of a b']
```
```   635 apply -
```
```   636 apply (rule zdiv_mono2_lemma)
```
```   637 apply (erule subst)
```
```   638 apply (erule subst, simp_all)
```
```   639 done
```
```   640
```
```   641 lemma q_neg_lemma:
```
```   642      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
```
```   643 apply (subgoal_tac "b'*q' < 0")
```
```   644  apply (simp add: mult_less_0_iff, arith)
```
```   645 done
```
```   646
```
```   647 lemma zdiv_mono2_neg_lemma:
```
```   648      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
```
```   649          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
```
```   650       ==> q' \<le> (q::int)"
```
```   651 apply (frule q_neg_lemma, assumption+)
```
```   652 apply (subgoal_tac "b*q' < b* (q + 1) ")
```
```   653  apply (simp add: mult_less_cancel_left)
```
```   654 apply (simp add: distrib_left)
```
```   655 apply (subgoal_tac "b*q' \<le> b'*q'")
```
```   656  prefer 2 apply (simp add: mult_right_mono_neg, arith)
```
```   657 done
```
```   658
```
```   659 lemma zdiv_mono2_neg:
```
```   660      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
```
```   661 using mult_div_mod_eq [symmetric, of a b]
```
```   662 using mult_div_mod_eq [symmetric, of a b']
```
```   663 apply -
```
```   664 apply (rule zdiv_mono2_neg_lemma)
```
```   665 apply (erule subst)
```
```   666 apply (erule subst, simp_all)
```
```   667 done
```
```   668
```
```   669
```
```   670 subsubsection \<open>More Algebraic Laws for div and mod\<close>
```
```   671
```
```   672 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
```
```   673   by (fact div_mult1_eq)
```
```   674
```
```   675 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```   676 lemma zdiv_zadd1_eq:
```
```   677      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```   678   by (fact div_add1_eq)
```
```   679
```
```   680 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
```
```   681 by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```   682
```
```   683 (* REVISIT: should this be generalized to all semiring_div types? *)
```
```   684 lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
```
```   685
```
```   686
```
```   687 subsubsection \<open>Proving  @{term "a div (b * c) = (a div b) div c"}\<close>
```
```   688
```
```   689 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
```
```   690   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
```
```   691   to cause particular problems.*)
```
```   692
```
```   693 text\<open>first, four lemmas to bound the remainder for the cases b<0 and b>0\<close>
```
```   694
```
```   695 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b * c < b * (q mod c) + r"
```
```   696 apply (subgoal_tac "b * (c - q mod c) < r * 1")
```
```   697  apply (simp add: algebra_simps)
```
```   698 apply (rule order_le_less_trans)
```
```   699  apply (erule_tac [2] mult_strict_right_mono)
```
```   700  apply (rule mult_left_mono_neg)
```
```   701   using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
```
```   702  apply (simp)
```
```   703 apply (simp)
```
```   704 done
```
```   705
```
```   706 lemma zmult2_lemma_aux2:
```
```   707      "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
```
```   708 apply (subgoal_tac "b * (q mod c) \<le> 0")
```
```   709  apply arith
```
```   710 apply (simp add: mult_le_0_iff)
```
```   711 done
```
```   712
```
```   713 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
```
```   714 apply (subgoal_tac "0 \<le> b * (q mod c) ")
```
```   715 apply arith
```
```   716 apply (simp add: zero_le_mult_iff)
```
```   717 done
```
```   718
```
```   719 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
```
```   720 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
```
```   721  apply (simp add: right_diff_distrib)
```
```   722 apply (rule order_less_le_trans)
```
```   723  apply (erule mult_strict_right_mono)
```
```   724  apply (rule_tac [2] mult_left_mono)
```
```   725   apply simp
```
```   726  using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
```
```   727 apply simp
```
```   728 done
```
```   729
```
```   730 lemma zmult2_lemma: "[| eucl_rel_int a b (q, r); 0 < c |]
```
```   731       ==> eucl_rel_int a (b * c) (q div c, b*(q mod c) + r)"
```
```   732 by (auto simp add: mult.assoc eucl_rel_int_iff linorder_neq_iff
```
```   733                    zero_less_mult_iff distrib_left [symmetric]
```
```   734                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: if_split_asm)
```
```   735
```
```   736 lemma zdiv_zmult2_eq:
```
```   737   fixes a b c :: int
```
```   738   shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
```
```   739 apply (case_tac "b = 0", simp)
```
```   740 apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN div_int_unique])
```
```   741 done
```
```   742
```
```   743 lemma zmod_zmult2_eq:
```
```   744   fixes a b c :: int
```
```   745   shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
```
```   746 apply (case_tac "b = 0", simp)
```
```   747 apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN mod_int_unique])
```
```   748 done
```
```   749
```
```   750 lemma div_pos_geq:
```
```   751   fixes k l :: int
```
```   752   assumes "0 < l" and "l \<le> k"
```
```   753   shows "k div l = (k - l) div l + 1"
```
```   754 proof -
```
```   755   have "k = (k - l) + l" by simp
```
```   756   then obtain j where k: "k = j + l" ..
```
```   757   with assms show ?thesis by (simp add: div_add_self2)
```
```   758 qed
```
```   759
```
```   760 lemma mod_pos_geq:
```
```   761   fixes k l :: int
```
```   762   assumes "0 < l" and "l \<le> k"
```
```   763   shows "k mod l = (k - l) mod l"
```
```   764 proof -
```
```   765   have "k = (k - l) + l" by simp
```
```   766   then obtain j where k: "k = j + l" ..
```
```   767   with assms show ?thesis by simp
```
```   768 qed
```
```   769
```
```   770
```
```   771 subsubsection \<open>Splitting Rules for div and mod\<close>
```
```   772
```
```   773 text\<open>The proofs of the two lemmas below are essentially identical\<close>
```
```   774
```
```   775 lemma split_pos_lemma:
```
```   776  "0<k ==>
```
```   777     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
```
```   778 apply (rule iffI, clarify)
```
```   779  apply (erule_tac P="P x y" for x y in rev_mp)
```
```   780  apply (subst mod_add_eq [symmetric])
```
```   781  apply (subst zdiv_zadd1_eq)
```
```   782  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
```
```   783 txt\<open>converse direction\<close>
```
```   784 apply (drule_tac x = "n div k" in spec)
```
```   785 apply (drule_tac x = "n mod k" in spec, simp)
```
```   786 done
```
```   787
```
```   788 lemma split_neg_lemma:
```
```   789  "k<0 ==>
```
```   790     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
```
```   791 apply (rule iffI, clarify)
```
```   792  apply (erule_tac P="P x y" for x y in rev_mp)
```
```   793  apply (subst mod_add_eq [symmetric])
```
```   794  apply (subst zdiv_zadd1_eq)
```
```   795  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
```
```   796 txt\<open>converse direction\<close>
```
```   797 apply (drule_tac x = "n div k" in spec)
```
```   798 apply (drule_tac x = "n mod k" in spec, simp)
```
```   799 done
```
```   800
```
```   801 lemma split_zdiv:
```
```   802  "P(n div k :: int) =
```
```   803   ((k = 0 --> P 0) &
```
```   804    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
```
```   805    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
```
```   806 apply (case_tac "k=0", simp)
```
```   807 apply (simp only: linorder_neq_iff)
```
```   808 apply (erule disjE)
```
```   809  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
```
```   810                       split_neg_lemma [of concl: "%x y. P x"])
```
```   811 done
```
```   812
```
```   813 lemma split_zmod:
```
```   814  "P(n mod k :: int) =
```
```   815   ((k = 0 --> P n) &
```
```   816    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
```
```   817    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
```
```   818 apply (case_tac "k=0", simp)
```
```   819 apply (simp only: linorder_neq_iff)
```
```   820 apply (erule disjE)
```
```   821  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
```
```   822                       split_neg_lemma [of concl: "%x y. P y"])
```
```   823 done
```
```   824
```
```   825 text \<open>Enable (lin)arith to deal with @{const divide} and @{const modulo}
```
```   826   when these are applied to some constant that is of the form
```
```   827   @{term "numeral k"}:\<close>
```
```   828 declare split_zdiv [of _ _ "numeral k", arith_split] for k
```
```   829 declare split_zmod [of _ _ "numeral k", arith_split] for k
```
```   830
```
```   831
```
```   832 subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
```
```   833
```
```   834 lemma pos_eucl_rel_int_mult_2:
```
```   835   assumes "0 \<le> b"
```
```   836   assumes "eucl_rel_int a b (q, r)"
```
```   837   shows "eucl_rel_int (1 + 2*a) (2*b) (q, 1 + 2*r)"
```
```   838   using assms unfolding eucl_rel_int_iff by auto
```
```   839
```
```   840 lemma neg_eucl_rel_int_mult_2:
```
```   841   assumes "b \<le> 0"
```
```   842   assumes "eucl_rel_int (a + 1) b (q, r)"
```
```   843   shows "eucl_rel_int (1 + 2*a) (2*b) (q, 2*r - 1)"
```
```   844   using assms unfolding eucl_rel_int_iff by auto
```
```   845
```
```   846 text\<open>computing div by shifting\<close>
```
```   847
```
```   848 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
```
```   849   using pos_eucl_rel_int_mult_2 [OF _ eucl_rel_int]
```
```   850   by (rule div_int_unique)
```
```   851
```
```   852 lemma neg_zdiv_mult_2:
```
```   853   assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
```
```   854   using neg_eucl_rel_int_mult_2 [OF A eucl_rel_int]
```
```   855   by (rule div_int_unique)
```
```   856
```
```   857 (* FIXME: add rules for negative numerals *)
```
```   858 lemma zdiv_numeral_Bit0 [simp]:
```
```   859   "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
```
```   860     numeral v div (numeral w :: int)"
```
```   861   unfolding numeral.simps unfolding mult_2 [symmetric]
```
```   862   by (rule div_mult_mult1, simp)
```
```   863
```
```   864 lemma zdiv_numeral_Bit1 [simp]:
```
```   865   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
```
```   866     (numeral v div (numeral w :: int))"
```
```   867   unfolding numeral.simps
```
```   868   unfolding mult_2 [symmetric] add.commute [of _ 1]
```
```   869   by (rule pos_zdiv_mult_2, simp)
```
```   870
```
```   871 lemma pos_zmod_mult_2:
```
```   872   fixes a b :: int
```
```   873   assumes "0 \<le> a"
```
```   874   shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
```
```   875   using pos_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
```
```   876   by (rule mod_int_unique)
```
```   877
```
```   878 lemma neg_zmod_mult_2:
```
```   879   fixes a b :: int
```
```   880   assumes "a \<le> 0"
```
```   881   shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
```
```   882   using neg_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
```
```   883   by (rule mod_int_unique)
```
```   884
```
```   885 (* FIXME: add rules for negative numerals *)
```
```   886 lemma zmod_numeral_Bit0 [simp]:
```
```   887   "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
```
```   888     (2::int) * (numeral v mod numeral w)"
```
```   889   unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
```
```   890   unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
```
```   891
```
```   892 lemma zmod_numeral_Bit1 [simp]:
```
```   893   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
```
```   894     2 * (numeral v mod numeral w) + (1::int)"
```
```   895   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
```
```   896   unfolding mult_2 [symmetric] add.commute [of _ 1]
```
```   897   by (rule pos_zmod_mult_2, simp)
```
```   898
```
```   899 lemma zdiv_eq_0_iff:
```
```   900  "(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")
```
```   901 proof
```
```   902   assume ?L
```
```   903   have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
```
```   904   with \<open>?L\<close> show ?R by blast
```
```   905 next
```
```   906   assume ?R thus ?L
```
```   907     by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
```
```   908 qed
```
```   909
```
```   910 lemma zmod_trival_iff:
```
```   911   fixes i k :: int
```
```   912   shows "i mod k = i \<longleftrightarrow> k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i"
```
```   913 proof -
```
```   914   have "i mod k = i \<longleftrightarrow> i div k = 0"
```
```   915     by safe (insert div_mult_mod_eq [of i k], auto)
```
```   916   with zdiv_eq_0_iff
```
```   917   show ?thesis
```
```   918     by simp
```
```   919 qed
```
```   920
```
```   921
```
```   922 subsubsection \<open>Quotients of Signs\<close>
```
```   923
```
```   924 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
```
```   925 by (simp add: divide_int_def)
```
```   926
```
```   927 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
```
```   928 by (simp add: modulo_int_def)
```
```   929
```
```   930 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
```
```   931 apply (subgoal_tac "a div b \<le> -1", force)
```
```   932 apply (rule order_trans)
```
```   933 apply (rule_tac a' = "-1" in zdiv_mono1)
```
```   934 apply (auto simp add: div_eq_minus1)
```
```   935 done
```
```   936
```
```   937 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
```
```   938 by (drule zdiv_mono1_neg, auto)
```
```   939
```
```   940 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
```
```   941 by (drule zdiv_mono1, auto)
```
```   942
```
```   943 text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
```
```   944 conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
```
```   945 They should all be simp rules unless that causes too much search.\<close>
```
```   946
```
```   947 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
```
```   948 apply auto
```
```   949 apply (drule_tac [2] zdiv_mono1)
```
```   950 apply (auto simp add: linorder_neq_iff)
```
```   951 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
```
```   952 apply (blast intro: div_neg_pos_less0)
```
```   953 done
```
```   954
```
```   955 lemma pos_imp_zdiv_pos_iff:
```
```   956   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
```
```   957 using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
```
```   958 by arith
```
```   959
```
```   960 lemma neg_imp_zdiv_nonneg_iff:
```
```   961   "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
```
```   962 apply (subst div_minus_minus [symmetric])
```
```   963 apply (subst pos_imp_zdiv_nonneg_iff, auto)
```
```   964 done
```
```   965
```
```   966 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
```
```   967 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
```
```   968 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
```
```   969
```
```   970 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
```
```   971 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
```
```   972 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
```
```   973
```
```   974 lemma nonneg1_imp_zdiv_pos_iff:
```
```   975   "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
```
```   976 apply rule
```
```   977  apply rule
```
```   978   using div_pos_pos_trivial[of a b]apply arith
```
```   979  apply(cases "b=0")apply simp
```
```   980  using div_nonneg_neg_le0[of a b]apply arith
```
```   981 using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
```
```   982 done
```
```   983
```
```   984 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
```
```   985 apply (rule split_zmod[THEN iffD2])
```
```   986 apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
```
```   987 done
```
```   988
```
```   989
```
```   990 subsubsection \<open>Computation of Division and Remainder\<close>
```
```   991
```
```   992 instantiation int :: unique_euclidean_semiring_numeral
```
```   993 begin
```
```   994
```
```   995 definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
```
```   996 where
```
```   997   "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```   998
```
```   999 definition divmod_step_int :: "num \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
```
```  1000 where
```
```  1001   "divmod_step_int l qr = (let (q, r) = qr
```
```  1002     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```  1003     else (2 * q, r))"
```
```  1004
```
```  1005 instance
```
```  1006   by standard (auto intro: zmod_le_nonneg_dividend simp add: divmod_int_def divmod_step_int_def
```
```  1007     pos_imp_zdiv_pos_iff div_pos_pos_trivial mod_pos_pos_trivial zmod_zmult2_eq zdiv_zmult2_eq)
```
```  1008
```
```  1009 end
```
```  1010
```
```  1011 declare divmod_algorithm_code [where ?'a = int, code]
```
```  1012
```
```  1013 context
```
```  1014 begin
```
```  1015
```
```  1016 qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
```
```  1017 where
```
```  1018   "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
```
```  1019
```
```  1020 qualified lemma adjust_div_eq [simp, code]:
```
```  1021   "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
```
```  1022   by (simp add: adjust_div_def)
```
```  1023
```
```  1024 qualified definition adjust_mod :: "int \<Rightarrow> int \<Rightarrow> int"
```
```  1025 where
```
```  1026   [simp]: "adjust_mod l r = (if r = 0 then 0 else l - r)"
```
```  1027
```
```  1028 lemma minus_numeral_div_numeral [simp]:
```
```  1029   "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
```
```  1030 proof -
```
```  1031   have "int (fst (divmod m n)) = fst (divmod m n)"
```
```  1032     by (simp only: fst_divmod divide_int_def) auto
```
```  1033   then show ?thesis
```
```  1034     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
```
```  1035 qed
```
```  1036
```
```  1037 lemma minus_numeral_mod_numeral [simp]:
```
```  1038   "- numeral m mod numeral n = adjust_mod (numeral n) (snd (divmod m n) :: int)"
```
```  1039 proof (cases "snd (divmod m n) = (0::int)")
```
```  1040   case True
```
```  1041   then show ?thesis
```
```  1042     by (simp add: mod_eq_0_iff_dvd divides_aux_def)
```
```  1043 next
```
```  1044   case False
```
```  1045   then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
```
```  1046     by (simp only: snd_divmod modulo_int_def) auto
```
```  1047   then show ?thesis
```
```  1048     by (simp add: divides_aux_def adjust_div_def)
```
```  1049       (simp add: divides_aux_def modulo_int_def)
```
```  1050 qed
```
```  1051
```
```  1052 lemma numeral_div_minus_numeral [simp]:
```
```  1053   "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
```
```  1054 proof -
```
```  1055   have "int (fst (divmod m n)) = fst (divmod m n)"
```
```  1056     by (simp only: fst_divmod divide_int_def) auto
```
```  1057   then show ?thesis
```
```  1058     by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
```
```  1059 qed
```
```  1060
```
```  1061 lemma numeral_mod_minus_numeral [simp]:
```
```  1062   "numeral m mod - numeral n = - adjust_mod (numeral n) (snd (divmod m n) :: int)"
```
```  1063 proof (cases "snd (divmod m n) = (0::int)")
```
```  1064   case True
```
```  1065   then show ?thesis
```
```  1066     by (simp add: mod_eq_0_iff_dvd divides_aux_def)
```
```  1067 next
```
```  1068   case False
```
```  1069   then have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
```
```  1070     by (simp only: snd_divmod modulo_int_def) auto
```
```  1071   then show ?thesis
```
```  1072     by (simp add: divides_aux_def adjust_div_def)
```
```  1073       (simp add: divides_aux_def modulo_int_def)
```
```  1074 qed
```
```  1075
```
```  1076 lemma minus_one_div_numeral [simp]:
```
```  1077   "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
```
```  1078   using minus_numeral_div_numeral [of Num.One n] by simp
```
```  1079
```
```  1080 lemma minus_one_mod_numeral [simp]:
```
```  1081   "- 1 mod numeral n = adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
```
```  1082   using minus_numeral_mod_numeral [of Num.One n] by simp
```
```  1083
```
```  1084 lemma one_div_minus_numeral [simp]:
```
```  1085   "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
```
```  1086   using numeral_div_minus_numeral [of Num.One n] by simp
```
```  1087
```
```  1088 lemma one_mod_minus_numeral [simp]:
```
```  1089   "1 mod - numeral n = - adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
```
```  1090   using numeral_mod_minus_numeral [of Num.One n] by simp
```
```  1091
```
```  1092 end
```
```  1093
```
```  1094
```
```  1095 subsubsection \<open>Further properties\<close>
```
```  1096
```
```  1097 lemma div_int_pos_iff:
```
```  1098   "k div l \<ge> 0 \<longleftrightarrow> k = 0 \<or> l = 0 \<or> k \<ge> 0 \<and> l \<ge> 0
```
```  1099     \<or> k < 0 \<and> l < 0"
```
```  1100   for k l :: int
```
```  1101   apply (cases "k = 0 \<or> l = 0")
```
```  1102    apply (auto simp add: pos_imp_zdiv_nonneg_iff neg_imp_zdiv_nonneg_iff)
```
```  1103   apply (rule ccontr)
```
```  1104   apply (simp add: neg_imp_zdiv_nonneg_iff)
```
```  1105   done
```
```  1106
```
```  1107 lemma mod_int_pos_iff:
```
```  1108   "k mod l \<ge> 0 \<longleftrightarrow> l dvd k \<or> l = 0 \<and> k \<ge> 0 \<or> l > 0"
```
```  1109   for k l :: int
```
```  1110   apply (cases "l > 0")
```
```  1111    apply (simp_all add: dvd_eq_mod_eq_0)
```
```  1112   apply (use neg_mod_conj [of l k] in \<open>auto simp add: le_less not_less\<close>)
```
```  1113   done
```
```  1114
```
```  1115 text \<open>Simplify expresions in which div and mod combine numerical constants\<close>
```
```  1116
```
```  1117 lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
```
```  1118   by (rule div_int_unique [of a b q r]) (simp add: eucl_rel_int_iff)
```
```  1119
```
```  1120 lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
```
```  1121   by (rule div_int_unique [of a b q r],
```
```  1122     simp add: eucl_rel_int_iff)
```
```  1123
```
```  1124 lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
```
```  1125   by (rule mod_int_unique [of a b q r],
```
```  1126     simp add: eucl_rel_int_iff)
```
```  1127
```
```  1128 lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
```
```  1129   by (rule mod_int_unique [of a b q r],
```
```  1130     simp add: eucl_rel_int_iff)
```
```  1131
```
```  1132 lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
```
```  1133 by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
```
```  1134
```
```  1135 text\<open>Suggested by Matthias Daum\<close>
```
```  1136 lemma int_power_div_base:
```
```  1137      "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
```
```  1138 apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
```
```  1139  apply (erule ssubst)
```
```  1140  apply (simp only: power_add)
```
```  1141  apply simp_all
```
```  1142 done
```
```  1143
```
```  1144 text \<open>Distributive laws for function \<open>nat\<close>.\<close>
```
```  1145
```
```  1146 lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
```
```  1147 apply (rule linorder_cases [of y 0])
```
```  1148 apply (simp add: div_nonneg_neg_le0)
```
```  1149 apply simp
```
```  1150 apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
```
```  1151 done
```
```  1152
```
```  1153 (*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
```
```  1154 lemma nat_mod_distrib:
```
```  1155   "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
```
```  1156 apply (case_tac "y = 0", simp)
```
```  1157 apply (simp add: nat_eq_iff zmod_int)
```
```  1158 done
```
```  1159
```
```  1160 text\<open>Suggested by Matthias Daum\<close>
```
```  1161 lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
```
```  1162 apply (subgoal_tac "nat x div nat k < nat x")
```
```  1163  apply (simp add: nat_div_distrib [symmetric])
```
```  1164 apply (rule div_less_dividend, simp_all)
```
```  1165 done
```
```  1166
```
```  1167 lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \<le> x"
```
```  1168   shows "\<exists>q. x = y + n * q"
```
```  1169 proof-
```
```  1170   from xy have th: "int x - int y = int (x - y)" by simp
```
```  1171   from xyn have "int x mod int n = int y mod int n"
```
```  1172     by (simp add: zmod_int [symmetric])
```
```  1173   hence "int n dvd int x - int y" by (simp only: mod_eq_dvd_iff [symmetric])
```
```  1174   hence "n dvd x - y" by (simp add: th zdvd_int)
```
```  1175   then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
```
```  1176 qed
```
```  1177
```
```  1178 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
```
```  1179   (is "?lhs = ?rhs")
```
```  1180 proof
```
```  1181   assume H: "x mod n = y mod n"
```
```  1182   {assume xy: "x \<le> y"
```
```  1183     from H have th: "y mod n = x mod n" by simp
```
```  1184     from nat_mod_eq_lemma[OF th xy] have ?rhs
```
```  1185       apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
```
```  1186   moreover
```
```  1187   {assume xy: "y \<le> x"
```
```  1188     from nat_mod_eq_lemma[OF H xy] have ?rhs
```
```  1189       apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
```
```  1190   ultimately  show ?rhs using linear[of x y] by blast
```
```  1191 next
```
```  1192   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
```
```  1193   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
```
```  1194   thus  ?lhs by simp
```
```  1195 qed
```
```  1196
```
```  1197
```
```  1198 subsubsection \<open>Dedicated simproc for calculation\<close>
```
```  1199
```
```  1200 text \<open>
```
```  1201   There is space for improvement here: the calculation itself
```
```  1202   could be carried out outside the logic, and a generic simproc
```
```  1203   (simplifier setup) for generic calculation would be helpful.
```
```  1204 \<close>
```
```  1205
```
```  1206 simproc_setup numeral_divmod
```
```  1207   ("0 div 0 :: 'a :: unique_euclidean_semiring_numeral" | "0 mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
```
```  1208    "0 div 1 :: 'a :: unique_euclidean_semiring_numeral" | "0 mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
```
```  1209    "0 div - 1 :: int" | "0 mod - 1 :: int" |
```
```  1210    "0 div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "0 mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
```
```  1211    "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
```
```  1212    "1 div 0 :: 'a :: unique_euclidean_semiring_numeral" | "1 mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
```
```  1213    "1 div 1 :: 'a :: unique_euclidean_semiring_numeral" | "1 mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
```
```  1214    "1 div - 1 :: int" | "1 mod - 1 :: int" |
```
```  1215    "1 div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "1 mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
```
```  1216    "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
```
```  1217    "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
```
```  1218    "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
```
```  1219    "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
```
```  1220    "numeral a div 0 :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
```
```  1221    "numeral a div 1 :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
```
```  1222    "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
```
```  1223    "numeral a div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
```
```  1224    "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
```
```  1225    "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
```
```  1226    "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
```
```  1227    "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
```
```  1228    "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
```
```  1229    "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") =
```
```  1230 \<open> let
```
```  1231     val if_cong = the (Code.get_case_cong @{theory} @{const_name If});
```
```  1232     fun successful_rewrite ctxt ct =
```
```  1233       let
```
```  1234         val thm = Simplifier.rewrite ctxt ct
```
```  1235       in if Thm.is_reflexive thm then NONE else SOME thm end;
```
```  1236   in fn phi =>
```
```  1237     let
```
```  1238       val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
```
```  1239         one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
```
```  1240         one_div_minus_numeral one_mod_minus_numeral
```
```  1241         numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
```
```  1242         numeral_div_minus_numeral numeral_mod_minus_numeral
```
```  1243         div_minus_minus mod_minus_minus Divides.adjust_div_eq of_bool_eq one_neq_zero
```
```  1244         numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
```
```  1245         divmod_cancel divmod_steps divmod_step_eq fst_conv snd_conv numeral_One
```
```  1246         case_prod_beta rel_simps Divides.adjust_mod_def div_minus1_right mod_minus1_right
```
```  1247         minus_minus numeral_times_numeral mult_zero_right mult_1_right}
```
```  1248         @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
```
```  1249       fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
```
```  1250         (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
```
```  1251     in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
```
```  1252   end;
```
```  1253 \<close>
```
```  1254
```
```  1255
```
```  1256 subsubsection \<open>Code generation\<close>
```
```  1257
```
```  1258 lemma [code]:
```
```  1259   fixes k :: int
```
```  1260   shows
```
```  1261     "k div 0 = 0"
```
```  1262     "k mod 0 = k"
```
```  1263     "0 div k = 0"
```
```  1264     "0 mod k = 0"
```
```  1265     "k div Int.Pos Num.One = k"
```
```  1266     "k mod Int.Pos Num.One = 0"
```
```  1267     "k div Int.Neg Num.One = - k"
```
```  1268     "k mod Int.Neg Num.One = 0"
```
```  1269     "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
```
```  1270     "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
```
```  1271     "Int.Neg m div Int.Pos n = - (Divides.adjust_div (divmod m n) :: int)"
```
```  1272     "Int.Neg m mod Int.Pos n = Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
```
```  1273     "Int.Pos m div Int.Neg n = - (Divides.adjust_div (divmod m n) :: int)"
```
```  1274     "Int.Pos m mod Int.Neg n = - Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
```
```  1275     "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
```
```  1276     "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
```
```  1277   by simp_all
```
```  1278
```
```  1279 code_identifier
```
```  1280   code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```  1281
```
```  1282 lemma dvd_eq_mod_eq_0_numeral:
```
```  1283   "numeral x dvd (numeral y :: 'a) \<longleftrightarrow> numeral y mod numeral x = (0 :: 'a::semidom_modulo)"
```
```  1284   by (fact dvd_eq_mod_eq_0)
```
```  1285
```
```  1286 declare minus_div_mult_eq_mod [symmetric, nitpick_unfold]
```
```  1287
```
```  1288
```
```  1289 subsubsection \<open>Lemmas of doubtful value\<close>
```
```  1290
```
```  1291 lemma mod_mult_self3':
```
```  1292   "Suc (k * n + m) mod n = Suc m mod n"
```
```  1293   by (fact Suc_mod_mult_self3)
```
```  1294
```
```  1295 lemma mod_Suc_eq_Suc_mod:
```
```  1296   "Suc m mod n = Suc (m mod n) mod n"
```
```  1297   by (simp add: mod_simps)
```
```  1298
```
```  1299 lemma div_geq:
```
```  1300   "m div n = Suc ((m - n) div n)" if "0 < n" and " \<not> m < n" for m n :: nat
```
```  1301   by (rule le_div_geq) (use that in \<open>simp_all add: not_less\<close>)
```
```  1302
```
```  1303 lemma mod_geq:
```
```  1304   "m mod n = (m - n) mod n" if "\<not> m < n" for m n :: nat
```
```  1305   by (rule le_mod_geq) (use that in \<open>simp add: not_less\<close>)
```
```  1306
```
```  1307 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
```
```  1308   by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```  1309
```
```  1310 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
```
```  1311
```
```  1312 (*Loses information, namely we also have r<d provided d is nonzero*)
```
```  1313 lemma mod_eqD:
```
```  1314   fixes m d r q :: nat
```
```  1315   assumes "m mod d = r"
```
```  1316   shows "\<exists>q. m = r + q * d"
```
```  1317 proof -
```
```  1318   from div_mult_mod_eq obtain q where "q * d + m mod d = m" by blast
```
```  1319   with assms have "m = r + q * d" by simp
```
```  1320   then show ?thesis ..
```
```  1321 qed
```
```  1322
```
```  1323 lemmas even_times_iff = even_mult_iff -- \<open>FIXME duplicate\<close>
```
```  1324
```
```  1325 lemma mod_2_not_eq_zero_eq_one_nat:
```
```  1326   fixes n :: nat
```
```  1327   shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
```
```  1328   by (fact not_mod_2_eq_0_eq_1)
```
```  1329
```
```  1330 lemma even_int_iff [simp]: "even (int n) \<longleftrightarrow> even n"
```
```  1331   by (fact even_of_nat)
```
```  1332
```
```  1333 lemma is_unit_int:
```
```  1334   "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
```
```  1335   by auto
```
```  1336
```
```  1337 end
```