src/HOL/Integ/IntDiv.thy
 author nipkow Mon Aug 16 14:22:27 2004 +0200 (2004-08-16) changeset 15131 c69542757a4d parent 15101 d027515e2aa6 child 15140 322485b816ac permissions -rw-r--r--
```     1 (*  Title:      HOL/IntDiv.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1999  University of Cambridge
```
```     5
```
```     6 The division operators div, mod and the divides relation "dvd"
```
```     7
```
```     8 Here is the division algorithm in ML:
```
```     9
```
```    10     fun posDivAlg (a,b) =
```
```    11       if a<b then (0,a)
```
```    12       else let val (q,r) = posDivAlg(a, 2*b)
```
```    13 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
```
```    14 	   end
```
```    15
```
```    16     fun negDivAlg (a,b) =
```
```    17       if 0\<le>a+b then (~1,a+b)
```
```    18       else let val (q,r) = negDivAlg(a, 2*b)
```
```    19 	       in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
```
```    20 	   end;
```
```    21
```
```    22     fun negateSnd (q,r:int) = (q,~r);
```
```    23
```
```    24     fun divAlg (a,b) = if 0\<le>a then
```
```    25 			  if b>0 then posDivAlg (a,b)
```
```    26 			   else if a=0 then (0,0)
```
```    27 				else negateSnd (negDivAlg (~a,~b))
```
```    28 		       else
```
```    29 			  if 0<b then negDivAlg (a,b)
```
```    30 			  else        negateSnd (posDivAlg (~a,~b));
```
```    31 *)
```
```    32
```
```    33
```
```    34 theory IntDiv
```
```    35 import IntArith Recdef
```
```    36 files ("IntDiv_setup.ML")
```
```    37 begin
```
```    38
```
```    39 declare zless_nat_conj [simp]
```
```    40
```
```    41 constdefs
```
```    42   quorem :: "(int*int) * (int*int) => bool"
```
```    43     "quorem == %((a,b), (q,r)).
```
```    44                       a = b*q + r &
```
```    45                       (if 0 < b then 0\<le>r & r<b else b<r & r \<le> 0)"
```
```    46
```
```    47   adjust :: "[int, int*int] => int*int"
```
```    48     "adjust b == %(q,r). if 0 \<le> r-b then (2*q + 1, r-b)
```
```    49                          else (2*q, r)"
```
```    50
```
```    51 (** the division algorithm **)
```
```    52
```
```    53 (*for the case a>=0, b>0*)
```
```    54 consts posDivAlg :: "int*int => int*int"
```
```    55 recdef posDivAlg "inv_image less_than (%(a,b). nat(a - b + 1))"
```
```    56     "posDivAlg (a,b) =
```
```    57        (if (a<b | b\<le>0) then (0,a)
```
```    58         else adjust b (posDivAlg(a, 2*b)))"
```
```    59
```
```    60 (*for the case a<0, b>0*)
```
```    61 consts negDivAlg :: "int*int => int*int"
```
```    62 recdef negDivAlg "inv_image less_than (%(a,b). nat(- a - b))"
```
```    63     "negDivAlg (a,b) =
```
```    64        (if (0\<le>a+b | b\<le>0) then (-1,a+b)
```
```    65         else adjust b (negDivAlg(a, 2*b)))"
```
```    66
```
```    67 (*for the general case b~=0*)
```
```    68
```
```    69 constdefs
```
```    70   negateSnd :: "int*int => int*int"
```
```    71     "negateSnd == %(q,r). (q,-r)"
```
```    72
```
```    73   (*The full division algorithm considers all possible signs for a, b
```
```    74     including the special case a=0, b<0, because negDivAlg requires a<0*)
```
```    75   divAlg :: "int*int => int*int"
```
```    76     "divAlg ==
```
```    77        %(a,b). if 0\<le>a then
```
```    78                   if 0\<le>b then posDivAlg (a,b)
```
```    79                   else if a=0 then (0,0)
```
```    80                        else negateSnd (negDivAlg (-a,-b))
```
```    81                else
```
```    82                   if 0<b then negDivAlg (a,b)
```
```    83                   else         negateSnd (posDivAlg (-a,-b))"
```
```    84
```
```    85 instance
```
```    86   int :: "Divides.div" ..       (*avoid clash with 'div' token*)
```
```    87
```
```    88 defs
```
```    89   div_def:   "a div b == fst (divAlg (a,b))"
```
```    90   mod_def:   "a mod b == snd (divAlg (a,b))"
```
```    91
```
```    92
```
```    93
```
```    94 subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
```
```    95
```
```    96 lemma unique_quotient_lemma:
```
```    97      "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  0 < b;  r < b |]
```
```    98       ==> q' \<le> (q::int)"
```
```    99 apply (subgoal_tac "r' + b * (q'-q) \<le> r")
```
```   100  prefer 2 apply (simp add: right_diff_distrib)
```
```   101 apply (subgoal_tac "0 < b * (1 + q - q') ")
```
```   102 apply (erule_tac [2] order_le_less_trans)
```
```   103  prefer 2 apply (simp add: right_diff_distrib right_distrib)
```
```   104 apply (subgoal_tac "b * q' < b * (1 + q) ")
```
```   105  prefer 2 apply (simp add: right_diff_distrib right_distrib)
```
```   106 apply (simp add: mult_less_cancel_left)
```
```   107 done
```
```   108
```
```   109 lemma unique_quotient_lemma_neg:
```
```   110      "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < 0;  b < r' |]
```
```   111       ==> q \<le> (q'::int)"
```
```   112 by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma,
```
```   113     auto)
```
```   114
```
```   115 lemma unique_quotient:
```
```   116      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  b ~= 0 |]
```
```   117       ==> q = q'"
```
```   118 apply (simp add: quorem_def linorder_neq_iff split: split_if_asm)
```
```   119 apply (blast intro: order_antisym
```
```   120              dest: order_eq_refl [THEN unique_quotient_lemma]
```
```   121              order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
```
```   122 done
```
```   123
```
```   124
```
```   125 lemma unique_remainder:
```
```   126      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  b ~= 0 |]
```
```   127       ==> r = r'"
```
```   128 apply (subgoal_tac "q = q'")
```
```   129  apply (simp add: quorem_def)
```
```   130 apply (blast intro: unique_quotient)
```
```   131 done
```
```   132
```
```   133
```
```   134 subsection{*Correctness of posDivAlg, the Algorithm for Non-Negative Dividends*}
```
```   135
```
```   136 text{*And positive divisors*}
```
```   137
```
```   138 lemma adjust_eq [simp]:
```
```   139      "adjust b (q,r) =
```
```   140       (let diff = r-b in
```
```   141 	if 0 \<le> diff then (2*q + 1, diff)
```
```   142                      else (2*q, r))"
```
```   143 by (simp add: Let_def adjust_def)
```
```   144
```
```   145 declare posDivAlg.simps [simp del]
```
```   146
```
```   147 (**use with a simproc to avoid repeatedly proving the premise*)
```
```   148 lemma posDivAlg_eqn:
```
```   149      "0 < b ==>
```
```   150       posDivAlg (a,b) = (if a<b then (0,a) else adjust b (posDivAlg(a, 2*b)))"
```
```   151 by (rule posDivAlg.simps [THEN trans], simp)
```
```   152
```
```   153 (*Correctness of posDivAlg: it computes quotients correctly*)
```
```   154 lemma posDivAlg_correct [rule_format]:
```
```   155      "0 \<le> a --> 0 < b --> quorem ((a, b), posDivAlg (a, b))"
```
```   156 apply (induct_tac a b rule: posDivAlg.induct, auto)
```
```   157  apply (simp_all add: quorem_def)
```
```   158  (*base case: a<b*)
```
```   159  apply (simp add: posDivAlg_eqn)
```
```   160 (*main argument*)
```
```   161 apply (subst posDivAlg_eqn, simp_all)
```
```   162 apply (erule splitE)
```
```   163 apply (auto simp add: right_distrib Let_def)
```
```   164 done
```
```   165
```
```   166
```
```   167 subsection{*Correctness of negDivAlg, the Algorithm for Negative Dividends*}
```
```   168
```
```   169 text{*And positive divisors*}
```
```   170
```
```   171 declare negDivAlg.simps [simp del]
```
```   172
```
```   173 (**use with a simproc to avoid repeatedly proving the premise*)
```
```   174 lemma negDivAlg_eqn:
```
```   175      "0 < b ==>
```
```   176       negDivAlg (a,b) =
```
```   177        (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg(a, 2*b)))"
```
```   178 by (rule negDivAlg.simps [THEN trans], simp)
```
```   179
```
```   180 (*Correctness of negDivAlg: it computes quotients correctly
```
```   181   It doesn't work if a=0 because the 0/b equals 0, not -1*)
```
```   182 lemma negDivAlg_correct [rule_format]:
```
```   183      "a < 0 --> 0 < b --> quorem ((a, b), negDivAlg (a, b))"
```
```   184 apply (induct_tac a b rule: negDivAlg.induct, auto)
```
```   185  apply (simp_all add: quorem_def)
```
```   186  (*base case: 0\<le>a+b*)
```
```   187  apply (simp add: negDivAlg_eqn)
```
```   188 (*main argument*)
```
```   189 apply (subst negDivAlg_eqn, assumption)
```
```   190 apply (erule splitE)
```
```   191 apply (auto simp add: right_distrib Let_def)
```
```   192 done
```
```   193
```
```   194
```
```   195 subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
```
```   196
```
```   197 (*the case a=0*)
```
```   198 lemma quorem_0: "b ~= 0 ==> quorem ((0,b), (0,0))"
```
```   199 by (auto simp add: quorem_def linorder_neq_iff)
```
```   200
```
```   201 lemma posDivAlg_0 [simp]: "posDivAlg (0, b) = (0, 0)"
```
```   202 by (subst posDivAlg.simps, auto)
```
```   203
```
```   204 lemma negDivAlg_minus1 [simp]: "negDivAlg (-1, b) = (-1, b - 1)"
```
```   205 by (subst negDivAlg.simps, auto)
```
```   206
```
```   207 lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
```
```   208 by (unfold negateSnd_def, auto)
```
```   209
```
```   210 lemma quorem_neg: "quorem ((-a,-b), qr) ==> quorem ((a,b), negateSnd qr)"
```
```   211 by (auto simp add: split_ifs quorem_def)
```
```   212
```
```   213 lemma divAlg_correct: "b ~= 0 ==> quorem ((a,b), divAlg(a,b))"
```
```   214 by (force simp add: linorder_neq_iff quorem_0 divAlg_def quorem_neg
```
```   215                     posDivAlg_correct negDivAlg_correct)
```
```   216
```
```   217 (** Arbitrary definitions for division by zero.  Useful to simplify
```
```   218     certain equations **)
```
```   219
```
```   220 lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
```
```   221 by (simp add: div_def mod_def divAlg_def posDivAlg.simps)
```
```   222
```
```   223 (** Basic laws about division and remainder **)
```
```   224
```
```   225 lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
```
```   226 apply (case_tac "b = 0", simp)
```
```   227 apply (cut_tac a = a and b = b in divAlg_correct)
```
```   228 apply (auto simp add: quorem_def div_def mod_def)
```
```   229 done
```
```   230
```
```   231 lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
```
```   232 by(simp add: zmod_zdiv_equality[symmetric])
```
```   233
```
```   234 lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
```
```   235 by(simp add: zmult_commute zmod_zdiv_equality[symmetric])
```
```   236
```
```   237 use "IntDiv_setup.ML"
```
```   238
```
```   239 lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
```
```   240 apply (cut_tac a = a and b = b in divAlg_correct)
```
```   241 apply (auto simp add: quorem_def mod_def)
```
```   242 done
```
```   243
```
```   244 lemmas pos_mod_sign[simp]  = pos_mod_conj [THEN conjunct1, standard]
```
```   245    and pos_mod_bound[simp] = pos_mod_conj [THEN conjunct2, standard]
```
```   246
```
```   247 lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
```
```   248 apply (cut_tac a = a and b = b in divAlg_correct)
```
```   249 apply (auto simp add: quorem_def div_def mod_def)
```
```   250 done
```
```   251
```
```   252 lemmas neg_mod_sign[simp]  = neg_mod_conj [THEN conjunct1, standard]
```
```   253    and neg_mod_bound[simp] = neg_mod_conj [THEN conjunct2, standard]
```
```   254
```
```   255
```
```   256
```
```   257 (** proving general properties of div and mod **)
```
```   258
```
```   259 lemma quorem_div_mod: "b ~= 0 ==> quorem ((a, b), (a div b, a mod b))"
```
```   260 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
```
```   261 apply (force simp add: quorem_def linorder_neq_iff)
```
```   262 done
```
```   263
```
```   264 lemma quorem_div: "[| quorem((a,b),(q,r));  b ~= 0 |] ==> a div b = q"
```
```   265 by (simp add: quorem_div_mod [THEN unique_quotient])
```
```   266
```
```   267 lemma quorem_mod: "[| quorem((a,b),(q,r));  b ~= 0 |] ==> a mod b = r"
```
```   268 by (simp add: quorem_div_mod [THEN unique_remainder])
```
```   269
```
```   270 lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
```
```   271 apply (rule quorem_div)
```
```   272 apply (auto simp add: quorem_def)
```
```   273 done
```
```   274
```
```   275 lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
```
```   276 apply (rule quorem_div)
```
```   277 apply (auto simp add: quorem_def)
```
```   278 done
```
```   279
```
```   280 lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
```
```   281 apply (rule quorem_div)
```
```   282 apply (auto simp add: quorem_def)
```
```   283 done
```
```   284
```
```   285 (*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
```
```   286
```
```   287 lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
```
```   288 apply (rule_tac q = 0 in quorem_mod)
```
```   289 apply (auto simp add: quorem_def)
```
```   290 done
```
```   291
```
```   292 lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
```
```   293 apply (rule_tac q = 0 in quorem_mod)
```
```   294 apply (auto simp add: quorem_def)
```
```   295 done
```
```   296
```
```   297 lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
```
```   298 apply (rule_tac q = "-1" in quorem_mod)
```
```   299 apply (auto simp add: quorem_def)
```
```   300 done
```
```   301
```
```   302 (*There is no mod_neg_pos_trivial...*)
```
```   303
```
```   304
```
```   305 (*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
```
```   306 lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
```
```   307 apply (case_tac "b = 0", simp)
```
```   308 apply (simp add: quorem_div_mod [THEN quorem_neg, simplified,
```
```   309                                  THEN quorem_div, THEN sym])
```
```   310
```
```   311 done
```
```   312
```
```   313 (*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
```
```   314 lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
```
```   315 apply (case_tac "b = 0", simp)
```
```   316 apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod],
```
```   317        auto)
```
```   318 done
```
```   319
```
```   320 subsection{*div, mod and unary minus*}
```
```   321
```
```   322 lemma zminus1_lemma:
```
```   323      "quorem((a,b),(q,r))
```
```   324       ==> quorem ((-a,b), (if r=0 then -q else -q - 1),
```
```   325                           (if r=0 then 0 else b-r))"
```
```   326 by (force simp add: split_ifs quorem_def linorder_neq_iff right_diff_distrib)
```
```   327
```
```   328
```
```   329 lemma zdiv_zminus1_eq_if:
```
```   330      "b ~= (0::int)
```
```   331       ==> (-a) div b =
```
```   332           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
```
```   333 by (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_div])
```
```   334
```
```   335 lemma zmod_zminus1_eq_if:
```
```   336      "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
```
```   337 apply (case_tac "b = 0", simp)
```
```   338 apply (blast intro: quorem_div_mod [THEN zminus1_lemma, THEN quorem_mod])
```
```   339 done
```
```   340
```
```   341 lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
```
```   342 by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
```
```   343
```
```   344 lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
```
```   345 by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
```
```   346
```
```   347 lemma zdiv_zminus2_eq_if:
```
```   348      "b ~= (0::int)
```
```   349       ==> a div (-b) =
```
```   350           (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
```
```   351 by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
```
```   352
```
```   353 lemma zmod_zminus2_eq_if:
```
```   354      "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
```
```   355 by (simp add: zmod_zminus1_eq_if zmod_zminus2)
```
```   356
```
```   357
```
```   358 subsection{*Division of a Number by Itself*}
```
```   359
```
```   360 lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
```
```   361 apply (subgoal_tac "0 < a*q")
```
```   362  apply (simp add: zero_less_mult_iff, arith)
```
```   363 done
```
```   364
```
```   365 lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
```
```   366 apply (subgoal_tac "0 \<le> a* (1-q) ")
```
```   367  apply (simp add: zero_le_mult_iff)
```
```   368 apply (simp add: right_diff_distrib)
```
```   369 done
```
```   370
```
```   371 lemma self_quotient: "[| quorem((a,a),(q,r));  a ~= (0::int) |] ==> q = 1"
```
```   372 apply (simp add: split_ifs quorem_def linorder_neq_iff)
```
```   373 apply (rule order_antisym, safe, simp_all (no_asm_use))
```
```   374 apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
```
```   375 apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
```
```   376 apply (force intro: self_quotient_aux1 self_quotient_aux2 simp only: zadd_commute zmult_zminus)+
```
```   377 done
```
```   378
```
```   379 lemma self_remainder: "[| quorem((a,a),(q,r));  a ~= (0::int) |] ==> r = 0"
```
```   380 apply (frule self_quotient, assumption)
```
```   381 apply (simp add: quorem_def)
```
```   382 done
```
```   383
```
```   384 lemma zdiv_self [simp]: "a ~= 0 ==> a div a = (1::int)"
```
```   385 by (simp add: quorem_div_mod [THEN self_quotient])
```
```   386
```
```   387 (*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
```
```   388 lemma zmod_self [simp]: "a mod a = (0::int)"
```
```   389 apply (case_tac "a = 0", simp)
```
```   390 apply (simp add: quorem_div_mod [THEN self_remainder])
```
```   391 done
```
```   392
```
```   393
```
```   394 subsection{*Computation of Division and Remainder*}
```
```   395
```
```   396 lemma zdiv_zero [simp]: "(0::int) div b = 0"
```
```   397 by (simp add: div_def divAlg_def)
```
```   398
```
```   399 lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
```
```   400 by (simp add: div_def divAlg_def)
```
```   401
```
```   402 lemma zmod_zero [simp]: "(0::int) mod b = 0"
```
```   403 by (simp add: mod_def divAlg_def)
```
```   404
```
```   405 lemma zdiv_minus1: "(0::int) < b ==> -1 div b = -1"
```
```   406 by (simp add: div_def divAlg_def)
```
```   407
```
```   408 lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
```
```   409 by (simp add: mod_def divAlg_def)
```
```   410
```
```   411 (** a positive, b positive **)
```
```   412
```
```   413 lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg(a,b))"
```
```   414 by (simp add: div_def divAlg_def)
```
```   415
```
```   416 lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg(a,b))"
```
```   417 by (simp add: mod_def divAlg_def)
```
```   418
```
```   419 (** a negative, b positive **)
```
```   420
```
```   421 lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg(a,b))"
```
```   422 by (simp add: div_def divAlg_def)
```
```   423
```
```   424 lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg(a,b))"
```
```   425 by (simp add: mod_def divAlg_def)
```
```   426
```
```   427 (** a positive, b negative **)
```
```   428
```
```   429 lemma div_pos_neg:
```
```   430      "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd(negDivAlg(-a,-b)))"
```
```   431 by (simp add: div_def divAlg_def)
```
```   432
```
```   433 lemma mod_pos_neg:
```
```   434      "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd(negDivAlg(-a,-b)))"
```
```   435 by (simp add: mod_def divAlg_def)
```
```   436
```
```   437 (** a negative, b negative **)
```
```   438
```
```   439 lemma div_neg_neg:
```
```   440      "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd(posDivAlg(-a,-b)))"
```
```   441 by (simp add: div_def divAlg_def)
```
```   442
```
```   443 lemma mod_neg_neg:
```
```   444      "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd(posDivAlg(-a,-b)))"
```
```   445 by (simp add: mod_def divAlg_def)
```
```   446
```
```   447 text {*Simplify expresions in which div and mod combine numerical constants*}
```
```   448
```
```   449 declare div_pos_pos [of "number_of v" "number_of w", standard, simp]
```
```   450 declare div_neg_pos [of "number_of v" "number_of w", standard, simp]
```
```   451 declare div_pos_neg [of "number_of v" "number_of w", standard, simp]
```
```   452 declare div_neg_neg [of "number_of v" "number_of w", standard, simp]
```
```   453
```
```   454 declare mod_pos_pos [of "number_of v" "number_of w", standard, simp]
```
```   455 declare mod_neg_pos [of "number_of v" "number_of w", standard, simp]
```
```   456 declare mod_pos_neg [of "number_of v" "number_of w", standard, simp]
```
```   457 declare mod_neg_neg [of "number_of v" "number_of w", standard, simp]
```
```   458
```
```   459 declare posDivAlg_eqn [of "number_of v" "number_of w", standard, simp]
```
```   460 declare negDivAlg_eqn [of "number_of v" "number_of w", standard, simp]
```
```   461
```
```   462
```
```   463 (** Special-case simplification **)
```
```   464
```
```   465 lemma zmod_1 [simp]: "a mod (1::int) = 0"
```
```   466 apply (cut_tac a = a and b = 1 in pos_mod_sign)
```
```   467 apply (cut_tac [2] a = a and b = 1 in pos_mod_bound)
```
```   468 apply (auto simp del:pos_mod_bound pos_mod_sign)
```
```   469 done
```
```   470
```
```   471 lemma zdiv_1 [simp]: "a div (1::int) = a"
```
```   472 by (cut_tac a = a and b = 1 in zmod_zdiv_equality, auto)
```
```   473
```
```   474 lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
```
```   475 apply (cut_tac a = a and b = "-1" in neg_mod_sign)
```
```   476 apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
```
```   477 apply (auto simp del: neg_mod_sign neg_mod_bound)
```
```   478 done
```
```   479
```
```   480 lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
```
```   481 by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
```
```   482
```
```   483 (** The last remaining special cases for constant arithmetic:
```
```   484     1 div z and 1 mod z **)
```
```   485
```
```   486 declare div_pos_pos [OF int_0_less_1, of "number_of w", standard, simp]
```
```   487 declare div_pos_neg [OF int_0_less_1, of "number_of w", standard, simp]
```
```   488 declare mod_pos_pos [OF int_0_less_1, of "number_of w", standard, simp]
```
```   489 declare mod_pos_neg [OF int_0_less_1, of "number_of w", standard, simp]
```
```   490
```
```   491 declare posDivAlg_eqn [of concl: 1 "number_of w", standard, simp]
```
```   492 declare negDivAlg_eqn [of concl: 1 "number_of w", standard, simp]
```
```   493
```
```   494
```
```   495 subsection{*Monotonicity in the First Argument (Dividend)*}
```
```   496
```
```   497 lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
```
```   498 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
```
```   499 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
```
```   500 apply (rule unique_quotient_lemma)
```
```   501 apply (erule subst)
```
```   502 apply (erule subst)
```
```   503 apply (simp_all)
```
```   504 done
```
```   505
```
```   506 lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
```
```   507 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
```
```   508 apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
```
```   509 apply (rule unique_quotient_lemma_neg)
```
```   510 apply (erule subst)
```
```   511 apply (erule subst)
```
```   512 apply (simp_all)
```
```   513 done
```
```   514
```
```   515
```
```   516 subsection{*Monotonicity in the Second Argument (Divisor)*}
```
```   517
```
```   518 lemma q_pos_lemma:
```
```   519      "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
```
```   520 apply (subgoal_tac "0 < b'* (q' + 1) ")
```
```   521  apply (simp add: zero_less_mult_iff)
```
```   522 apply (simp add: right_distrib)
```
```   523 done
```
```   524
```
```   525 lemma zdiv_mono2_lemma:
```
```   526      "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
```
```   527          r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
```
```   528       ==> q \<le> (q'::int)"
```
```   529 apply (frule q_pos_lemma, assumption+)
```
```   530 apply (subgoal_tac "b*q < b* (q' + 1) ")
```
```   531  apply (simp add: mult_less_cancel_left)
```
```   532 apply (subgoal_tac "b*q = r' - r + b'*q'")
```
```   533  prefer 2 apply simp
```
```   534 apply (simp (no_asm_simp) add: right_distrib)
```
```   535 apply (subst zadd_commute, rule zadd_zless_mono, arith)
```
```   536 apply (rule mult_right_mono, auto)
```
```   537 done
```
```   538
```
```   539 lemma zdiv_mono2:
```
```   540      "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
```
```   541 apply (subgoal_tac "b ~= 0")
```
```   542  prefer 2 apply arith
```
```   543 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
```
```   544 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
```
```   545 apply (rule zdiv_mono2_lemma)
```
```   546 apply (erule subst)
```
```   547 apply (erule subst)
```
```   548 apply (simp_all)
```
```   549 done
```
```   550
```
```   551 lemma q_neg_lemma:
```
```   552      "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
```
```   553 apply (subgoal_tac "b'*q' < 0")
```
```   554  apply (simp add: mult_less_0_iff, arith)
```
```   555 done
```
```   556
```
```   557 lemma zdiv_mono2_neg_lemma:
```
```   558      "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
```
```   559          r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
```
```   560       ==> q' \<le> (q::int)"
```
```   561 apply (frule q_neg_lemma, assumption+)
```
```   562 apply (subgoal_tac "b*q' < b* (q + 1) ")
```
```   563  apply (simp add: mult_less_cancel_left)
```
```   564 apply (simp add: right_distrib)
```
```   565 apply (subgoal_tac "b*q' \<le> b'*q'")
```
```   566  prefer 2 apply (simp add: mult_right_mono_neg)
```
```   567 apply (subgoal_tac "b'*q' < b + b*q")
```
```   568  apply arith
```
```   569 apply simp
```
```   570 done
```
```   571
```
```   572 lemma zdiv_mono2_neg:
```
```   573      "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
```
```   574 apply (cut_tac a = a and b = b in zmod_zdiv_equality)
```
```   575 apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
```
```   576 apply (rule zdiv_mono2_neg_lemma)
```
```   577 apply (erule subst)
```
```   578 apply (erule subst)
```
```   579 apply (simp_all)
```
```   580 done
```
```   581
```
```   582
```
```   583 subsection{*More Algebraic Laws for div and mod*}
```
```   584
```
```   585 (** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
```
```   586
```
```   587 lemma zmult1_lemma:
```
```   588      "[| quorem((b,c),(q,r));  c ~= 0 |]
```
```   589       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
```
```   590 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
```
```   591
```
```   592 lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
```
```   593 apply (case_tac "c = 0", simp)
```
```   594 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])
```
```   595 done
```
```   596
```
```   597 lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
```
```   598 apply (case_tac "c = 0", simp)
```
```   599 apply (blast intro: quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])
```
```   600 done
```
```   601
```
```   602 lemma zmod_zmult1_eq': "(a*b) mod (c::int) = ((a mod c) * b) mod c"
```
```   603 apply (rule trans)
```
```   604 apply (rule_tac s = "b*a mod c" in trans)
```
```   605 apply (rule_tac [2] zmod_zmult1_eq)
```
```   606 apply (simp_all add: zmult_commute)
```
```   607 done
```
```   608
```
```   609 lemma zmod_zmult_distrib: "(a*b) mod (c::int) = ((a mod c) * (b mod c)) mod c"
```
```   610 apply (rule zmod_zmult1_eq' [THEN trans])
```
```   611 apply (rule zmod_zmult1_eq)
```
```   612 done
```
```   613
```
```   614 lemma zdiv_zmult_self1 [simp]: "b ~= (0::int) ==> (a*b) div b = a"
```
```   615 by (simp add: zdiv_zmult1_eq)
```
```   616
```
```   617 lemma zdiv_zmult_self2 [simp]: "b ~= (0::int) ==> (b*a) div b = a"
```
```   618 by (subst zmult_commute, erule zdiv_zmult_self1)
```
```   619
```
```   620 lemma zmod_zmult_self1 [simp]: "(a*b) mod b = (0::int)"
```
```   621 by (simp add: zmod_zmult1_eq)
```
```   622
```
```   623 lemma zmod_zmult_self2 [simp]: "(b*a) mod b = (0::int)"
```
```   624 by (simp add: zmult_commute zmod_zmult1_eq)
```
```   625
```
```   626 lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
```
```   627 proof
```
```   628   assume "m mod d = 0"
```
```   629   with zmod_zdiv_equality[of m d] show "EX q::int. m = d*q" by auto
```
```   630 next
```
```   631   assume "EX q::int. m = d*q"
```
```   632   thus "m mod d = 0" by auto
```
```   633 qed
```
```   634
```
```   635 declare zmod_eq_0_iff [THEN iffD1, dest!]
```
```   636
```
```   637 (** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
```
```   638
```
```   639 lemma zadd1_lemma:
```
```   640      "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c ~= 0 |]
```
```   641       ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
```
```   642 by (force simp add: split_ifs quorem_def linorder_neq_iff right_distrib)
```
```   643
```
```   644 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```   645 lemma zdiv_zadd1_eq:
```
```   646      "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```   647 apply (case_tac "c = 0", simp)
```
```   648 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_div)
```
```   649 done
```
```   650
```
```   651 lemma zmod_zadd1_eq: "(a+b) mod (c::int) = (a mod c + b mod c) mod c"
```
```   652 apply (case_tac "c = 0", simp)
```
```   653 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod] quorem_mod)
```
```   654 done
```
```   655
```
```   656 lemma mod_div_trivial [simp]: "(a mod b) div b = (0::int)"
```
```   657 apply (case_tac "b = 0", simp)
```
```   658 apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
```
```   659 done
```
```   660
```
```   661 lemma mod_mod_trivial [simp]: "(a mod b) mod b = a mod (b::int)"
```
```   662 apply (case_tac "b = 0", simp)
```
```   663 apply (force simp add: linorder_neq_iff mod_pos_pos_trivial mod_neg_neg_trivial)
```
```   664 done
```
```   665
```
```   666 lemma zmod_zadd_left_eq: "(a+b) mod (c::int) = ((a mod c) + b) mod c"
```
```   667 apply (rule trans [symmetric])
```
```   668 apply (rule zmod_zadd1_eq, simp)
```
```   669 apply (rule zmod_zadd1_eq [symmetric])
```
```   670 done
```
```   671
```
```   672 lemma zmod_zadd_right_eq: "(a+b) mod (c::int) = (a + (b mod c)) mod c"
```
```   673 apply (rule trans [symmetric])
```
```   674 apply (rule zmod_zadd1_eq, simp)
```
```   675 apply (rule zmod_zadd1_eq [symmetric])
```
```   676 done
```
```   677
```
```   678 lemma zdiv_zadd_self1[simp]: "a ~= (0::int) ==> (a+b) div a = b div a + 1"
```
```   679 by (simp add: zdiv_zadd1_eq)
```
```   680
```
```   681 lemma zdiv_zadd_self2[simp]: "a ~= (0::int) ==> (b+a) div a = b div a + 1"
```
```   682 by (simp add: zdiv_zadd1_eq)
```
```   683
```
```   684 lemma zmod_zadd_self1[simp]: "(a+b) mod a = b mod (a::int)"
```
```   685 apply (case_tac "a = 0", simp)
```
```   686 apply (simp add: zmod_zadd1_eq)
```
```   687 done
```
```   688
```
```   689 lemma zmod_zadd_self2[simp]: "(b+a) mod a = b mod (a::int)"
```
```   690 apply (case_tac "a = 0", simp)
```
```   691 apply (simp add: zmod_zadd1_eq)
```
```   692 done
```
```   693
```
```   694
```
```   695 subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
```
```   696
```
```   697 (*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
```
```   698   7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
```
```   699   to cause particular problems.*)
```
```   700
```
```   701 (** first, four lemmas to bound the remainder for the cases b<0 and b>0 **)
```
```   702
```
```   703 lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
```
```   704 apply (subgoal_tac "b * (c - q mod c) < r * 1")
```
```   705 apply (simp add: right_diff_distrib)
```
```   706 apply (rule order_le_less_trans)
```
```   707 apply (erule_tac [2] mult_strict_right_mono)
```
```   708 apply (rule mult_left_mono_neg)
```
```   709 apply (auto simp add: compare_rls zadd_commute [of 1]
```
```   710                       add1_zle_eq pos_mod_bound)
```
```   711 done
```
```   712
```
```   713 lemma zmult2_lemma_aux2: "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
```
```   714 apply (subgoal_tac "b * (q mod c) \<le> 0")
```
```   715  apply arith
```
```   716 apply (simp add: mult_le_0_iff)
```
```   717 done
```
```   718
```
```   719 lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
```
```   720 apply (subgoal_tac "0 \<le> b * (q mod c) ")
```
```   721 apply arith
```
```   722 apply (simp add: zero_le_mult_iff)
```
```   723 done
```
```   724
```
```   725 lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
```
```   726 apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
```
```   727 apply (simp add: right_diff_distrib)
```
```   728 apply (rule order_less_le_trans)
```
```   729 apply (erule mult_strict_right_mono)
```
```   730 apply (rule_tac [2] mult_left_mono)
```
```   731 apply (auto simp add: compare_rls zadd_commute [of 1]
```
```   732                       add1_zle_eq pos_mod_bound)
```
```   733 done
```
```   734
```
```   735 lemma zmult2_lemma: "[| quorem ((a,b), (q,r));  b ~= 0;  0 < c |]
```
```   736       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
```
```   737 by (auto simp add: mult_ac quorem_def linorder_neq_iff
```
```   738                    zero_less_mult_iff right_distrib [symmetric]
```
```   739                    zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
```
```   740
```
```   741 lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
```
```   742 apply (case_tac "b = 0", simp)
```
```   743 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_div])
```
```   744 done
```
```   745
```
```   746 lemma zmod_zmult2_eq:
```
```   747      "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
```
```   748 apply (case_tac "b = 0", simp)
```
```   749 apply (force simp add: quorem_div_mod [THEN zmult2_lemma, THEN quorem_mod])
```
```   750 done
```
```   751
```
```   752
```
```   753 subsection{*Cancellation of Common Factors in div*}
```
```   754
```
```   755 lemma zdiv_zmult_zmult1_aux1: "[| (0::int) < b;  c ~= 0 |] ==> (c*a) div (c*b) = a div b"
```
```   756 by (subst zdiv_zmult2_eq, auto)
```
```   757
```
```   758 lemma zdiv_zmult_zmult1_aux2: "[| b < (0::int);  c ~= 0 |] ==> (c*a) div (c*b) = a div b"
```
```   759 apply (subgoal_tac " (c * (-a)) div (c * (-b)) = (-a) div (-b) ")
```
```   760 apply (rule_tac [2] zdiv_zmult_zmult1_aux1, auto)
```
```   761 done
```
```   762
```
```   763 lemma zdiv_zmult_zmult1: "c ~= (0::int) ==> (c*a) div (c*b) = a div b"
```
```   764 apply (case_tac "b = 0", simp)
```
```   765 apply (auto simp add: linorder_neq_iff zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
```
```   766 done
```
```   767
```
```   768 lemma zdiv_zmult_zmult2: "c ~= (0::int) ==> (a*c) div (b*c) = a div b"
```
```   769 apply (drule zdiv_zmult_zmult1)
```
```   770 apply (auto simp add: zmult_commute)
```
```   771 done
```
```   772
```
```   773
```
```   774
```
```   775 subsection{*Distribution of Factors over mod*}
```
```   776
```
```   777 lemma zmod_zmult_zmult1_aux1: "[| (0::int) < b;  c ~= 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
```
```   778 by (subst zmod_zmult2_eq, auto)
```
```   779
```
```   780 lemma zmod_zmult_zmult1_aux2: "[| b < (0::int);  c ~= 0 |] ==> (c*a) mod (c*b) = c * (a mod b)"
```
```   781 apply (subgoal_tac " (c * (-a)) mod (c * (-b)) = c * ((-a) mod (-b))")
```
```   782 apply (rule_tac [2] zmod_zmult_zmult1_aux1, auto)
```
```   783 done
```
```   784
```
```   785 lemma zmod_zmult_zmult1: "(c*a) mod (c*b) = (c::int) * (a mod b)"
```
```   786 apply (case_tac "b = 0", simp)
```
```   787 apply (case_tac "c = 0", simp)
```
```   788 apply (auto simp add: linorder_neq_iff zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
```
```   789 done
```
```   790
```
```   791 lemma zmod_zmult_zmult2: "(a*c) mod (b*c) = (a mod b) * (c::int)"
```
```   792 apply (cut_tac c = c in zmod_zmult_zmult1)
```
```   793 apply (auto simp add: zmult_commute)
```
```   794 done
```
```   795
```
```   796
```
```   797 subsection {*Splitting Rules for div and mod*}
```
```   798
```
```   799 text{*The proofs of the two lemmas below are essentially identical*}
```
```   800
```
```   801 lemma split_pos_lemma:
```
```   802  "0<k ==>
```
```   803     P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
```
```   804 apply (rule iffI)
```
```   805  apply clarify
```
```   806  apply (erule_tac P="P ?x ?y" in rev_mp)
```
```   807  apply (subst zmod_zadd1_eq)
```
```   808  apply (subst zdiv_zadd1_eq)
```
```   809  apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
```
```   810 txt{*converse direction*}
```
```   811 apply (drule_tac x = "n div k" in spec)
```
```   812 apply (drule_tac x = "n mod k" in spec)
```
```   813 apply (simp)
```
```   814 done
```
```   815
```
```   816 lemma split_neg_lemma:
```
```   817  "k<0 ==>
```
```   818     P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
```
```   819 apply (rule iffI)
```
```   820  apply clarify
```
```   821  apply (erule_tac P="P ?x ?y" in rev_mp)
```
```   822  apply (subst zmod_zadd1_eq)
```
```   823  apply (subst zdiv_zadd1_eq)
```
```   824  apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
```
```   825 txt{*converse direction*}
```
```   826 apply (drule_tac x = "n div k" in spec)
```
```   827 apply (drule_tac x = "n mod k" in spec)
```
```   828 apply (simp)
```
```   829 done
```
```   830
```
```   831 lemma split_zdiv:
```
```   832  "P(n div k :: int) =
```
```   833   ((k = 0 --> P 0) &
```
```   834    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
```
```   835    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
```
```   836 apply (case_tac "k=0")
```
```   837  apply (simp)
```
```   838 apply (simp only: linorder_neq_iff)
```
```   839 apply (erule disjE)
```
```   840  apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
```
```   841                       split_neg_lemma [of concl: "%x y. P x"])
```
```   842 done
```
```   843
```
```   844 lemma split_zmod:
```
```   845  "P(n mod k :: int) =
```
```   846   ((k = 0 --> P n) &
```
```   847    (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
```
```   848    (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
```
```   849 apply (case_tac "k=0")
```
```   850  apply (simp)
```
```   851 apply (simp only: linorder_neq_iff)
```
```   852 apply (erule disjE)
```
```   853  apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
```
```   854                       split_neg_lemma [of concl: "%x y. P y"])
```
```   855 done
```
```   856
```
```   857 (* Enable arith to deal with div 2 and mod 2: *)
```
```   858 declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
```
```   859 declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
```
```   860
```
```   861
```
```   862 subsection{*Speeding up the Division Algorithm with Shifting*}
```
```   863
```
```   864 (** computing div by shifting **)
```
```   865
```
```   866 lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
```
```   867 proof cases
```
```   868   assume "a=0"
```
```   869     thus ?thesis by simp
```
```   870 next
```
```   871   assume "a\<noteq>0" and le_a: "0\<le>a"
```
```   872   hence a_pos: "1 \<le> a" by arith
```
```   873   hence one_less_a2: "1 < 2*a" by arith
```
```   874   hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
```
```   875     by (simp add: mult_le_cancel_left zadd_commute [of 1] add1_zle_eq)
```
```   876   with a_pos have "0 \<le> b mod a" by simp
```
```   877   hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
```
```   878     by (simp add: mod_pos_pos_trivial one_less_a2)
```
```   879   with  le_2a
```
```   880   have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
```
```   881     by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
```
```   882                   right_distrib)
```
```   883   thus ?thesis
```
```   884     by (subst zdiv_zadd1_eq,
```
```   885         simp add: zdiv_zmult_zmult1 zmod_zmult_zmult1 one_less_a2
```
```   886                   div_pos_pos_trivial)
```
```   887 qed
```
```   888
```
```   889 lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
```
```   890 apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
```
```   891 apply (rule_tac [2] pos_zdiv_mult_2)
```
```   892 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
```
```   893 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
```
```   894 apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric],
```
```   895        simp)
```
```   896 done
```
```   897
```
```   898
```
```   899 (*Not clear why this must be proved separately; probably number_of causes
```
```   900   simplification problems*)
```
```   901 lemma not_0_le_lemma: "~ 0 \<le> x ==> x \<le> (0::int)"
```
```   902 by auto
```
```   903
```
```   904 lemma zdiv_number_of_BIT[simp]:
```
```   905      "number_of (v BIT b) div number_of (w BIT False) =
```
```   906           (if ~b | (0::int) \<le> number_of w
```
```   907            then number_of v div (number_of w)
```
```   908            else (number_of v + (1::int)) div (number_of w))"
```
```   909 apply (simp only: number_of_eq Bin_simps UNIV_I split: split_if)
```
```   910 apply (simp add: zdiv_zmult_zmult1 pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac)
```
```   911 done
```
```   912
```
```   913
```
```   914 subsection{*Computing mod by Shifting (proofs resemble those for div)*}
```
```   915
```
```   916 lemma pos_zmod_mult_2:
```
```   917      "(0::int) \<le> a ==> (1 + 2*b) mod (2*a) = 1 + 2 * (b mod a)"
```
```   918 apply (case_tac "a = 0", simp)
```
```   919 apply (subgoal_tac "1 \<le> a")
```
```   920  prefer 2 apply arith
```
```   921 apply (subgoal_tac "1 < a * 2")
```
```   922  prefer 2 apply arith
```
```   923 apply (subgoal_tac "2* (1 + b mod a) \<le> 2*a")
```
```   924  apply (rule_tac [2] mult_left_mono)
```
```   925 apply (auto simp add: zadd_commute [of 1] zmult_commute add1_zle_eq
```
```   926                       pos_mod_bound)
```
```   927 apply (subst zmod_zadd1_eq)
```
```   928 apply (simp add: zmod_zmult_zmult2 mod_pos_pos_trivial)
```
```   929 apply (rule mod_pos_pos_trivial)
```
```   930 apply (auto simp add: mod_pos_pos_trivial left_distrib)
```
```   931 apply (subgoal_tac "0 \<le> b mod a", arith)
```
```   932 apply (simp)
```
```   933 done
```
```   934
```
```   935 lemma neg_zmod_mult_2:
```
```   936      "a \<le> (0::int) ==> (1 + 2*b) mod (2*a) = 2 * ((b+1) mod a) - 1"
```
```   937 apply (subgoal_tac "(1 + 2* (-b - 1)) mod (2* (-a)) =
```
```   938                     1 + 2* ((-b - 1) mod (-a))")
```
```   939 apply (rule_tac [2] pos_zmod_mult_2)
```
```   940 apply (auto simp add: minus_mult_right [symmetric] right_diff_distrib)
```
```   941 apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
```
```   942  prefer 2 apply simp
```
```   943 apply (simp only: zmod_zminus_zminus diff_minus minus_add_distrib [symmetric])
```
```   944 done
```
```   945
```
```   946 lemma zmod_number_of_BIT [simp]:
```
```   947      "number_of (v BIT b) mod number_of (w BIT False) =
```
```   948           (if b then
```
```   949                 if (0::int) \<le> number_of w
```
```   950                 then 2 * (number_of v mod number_of w) + 1
```
```   951                 else 2 * ((number_of v + (1::int)) mod number_of w) - 1
```
```   952            else 2 * (number_of v mod number_of w))"
```
```   953 apply (simp only: number_of_eq Bin_simps UNIV_I split: split_if)
```
```   954 apply (simp add: zmod_zmult_zmult1 pos_zmod_mult_2
```
```   955                  not_0_le_lemma neg_zmod_mult_2 add_ac)
```
```   956 done
```
```   957
```
```   958
```
```   959
```
```   960 subsection{*Quotients of Signs*}
```
```   961
```
```   962 lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
```
```   963 apply (subgoal_tac "a div b \<le> -1", force)
```
```   964 apply (rule order_trans)
```
```   965 apply (rule_tac a' = "-1" in zdiv_mono1)
```
```   966 apply (auto simp add: zdiv_minus1)
```
```   967 done
```
```   968
```
```   969 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a;  b < 0 |] ==> a div b \<le> 0"
```
```   970 by (drule zdiv_mono1_neg, auto)
```
```   971
```
```   972 lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
```
```   973 apply auto
```
```   974 apply (drule_tac [2] zdiv_mono1)
```
```   975 apply (auto simp add: linorder_neq_iff)
```
```   976 apply (simp (no_asm_use) add: linorder_not_less [symmetric])
```
```   977 apply (blast intro: div_neg_pos_less0)
```
```   978 done
```
```   979
```
```   980 lemma neg_imp_zdiv_nonneg_iff:
```
```   981      "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
```
```   982 apply (subst zdiv_zminus_zminus [symmetric])
```
```   983 apply (subst pos_imp_zdiv_nonneg_iff, auto)
```
```   984 done
```
```   985
```
```   986 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
```
```   987 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
```
```   988 by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
```
```   989
```
```   990 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
```
```   991 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
```
```   992 by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
```
```   993
```
```   994
```
```   995 subsection {* The Divides Relation *}
```
```   996
```
```   997 lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))"
```
```   998 by(simp add:dvd_def zmod_eq_0_iff)
```
```   999
```
```  1000 lemma zdvd_0_right [iff]: "(m::int) dvd 0"
```
```  1001   apply (unfold dvd_def)
```
```  1002   apply (blast intro: mult_zero_right [symmetric])
```
```  1003   done
```
```  1004
```
```  1005 lemma zdvd_0_left [iff]: "(0 dvd (m::int)) = (m = 0)"
```
```  1006   by (unfold dvd_def, auto)
```
```  1007
```
```  1008 lemma zdvd_1_left [iff]: "1 dvd (m::int)"
```
```  1009   by (unfold dvd_def, simp)
```
```  1010
```
```  1011 lemma zdvd_refl [simp]: "m dvd (m::int)"
```
```  1012   apply (unfold dvd_def)
```
```  1013   apply (blast intro: zmult_1_right [symmetric])
```
```  1014   done
```
```  1015
```
```  1016 lemma zdvd_trans: "m dvd n ==> n dvd k ==> m dvd (k::int)"
```
```  1017   apply (unfold dvd_def)
```
```  1018   apply (blast intro: zmult_assoc)
```
```  1019   done
```
```  1020
```
```  1021 lemma zdvd_zminus_iff: "(m dvd -n) = (m dvd (n::int))"
```
```  1022   apply (unfold dvd_def, auto)
```
```  1023    apply (rule_tac [!] x = "-k" in exI, auto)
```
```  1024   done
```
```  1025
```
```  1026 lemma zdvd_zminus2_iff: "(-m dvd n) = (m dvd (n::int))"
```
```  1027   apply (unfold dvd_def, auto)
```
```  1028    apply (rule_tac [!] x = "-k" in exI, auto)
```
```  1029   done
```
```  1030
```
```  1031 lemma zdvd_anti_sym:
```
```  1032     "0 < m ==> 0 < n ==> m dvd n ==> n dvd m ==> m = (n::int)"
```
```  1033   apply (unfold dvd_def, auto)
```
```  1034   apply (simp add: zmult_assoc zmult_eq_self_iff zero_less_mult_iff zmult_eq_1_iff)
```
```  1035   done
```
```  1036
```
```  1037 lemma zdvd_zadd: "k dvd m ==> k dvd n ==> k dvd (m + n :: int)"
```
```  1038   apply (unfold dvd_def)
```
```  1039   apply (blast intro: right_distrib [symmetric])
```
```  1040   done
```
```  1041
```
```  1042 lemma zdvd_zdiff: "k dvd m ==> k dvd n ==> k dvd (m - n :: int)"
```
```  1043   apply (unfold dvd_def)
```
```  1044   apply (blast intro: right_diff_distrib [symmetric])
```
```  1045   done
```
```  1046
```
```  1047 lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
```
```  1048   apply (subgoal_tac "m = n + (m - n)")
```
```  1049    apply (erule ssubst)
```
```  1050    apply (blast intro: zdvd_zadd, simp)
```
```  1051   done
```
```  1052
```
```  1053 lemma zdvd_zmult: "k dvd (n::int) ==> k dvd m * n"
```
```  1054   apply (unfold dvd_def)
```
```  1055   apply (blast intro: mult_left_commute)
```
```  1056   done
```
```  1057
```
```  1058 lemma zdvd_zmult2: "k dvd (m::int) ==> k dvd m * n"
```
```  1059   apply (subst zmult_commute)
```
```  1060   apply (erule zdvd_zmult)
```
```  1061   done
```
```  1062
```
```  1063 lemma [iff]: "(k::int) dvd m * k"
```
```  1064   apply (rule zdvd_zmult)
```
```  1065   apply (rule zdvd_refl)
```
```  1066   done
```
```  1067
```
```  1068 lemma [iff]: "(k::int) dvd k * m"
```
```  1069   apply (rule zdvd_zmult2)
```
```  1070   apply (rule zdvd_refl)
```
```  1071   done
```
```  1072
```
```  1073 lemma zdvd_zmultD2: "j * k dvd n ==> j dvd (n::int)"
```
```  1074   apply (unfold dvd_def)
```
```  1075   apply (simp add: zmult_assoc, blast)
```
```  1076   done
```
```  1077
```
```  1078 lemma zdvd_zmultD: "j * k dvd n ==> k dvd (n::int)"
```
```  1079   apply (rule zdvd_zmultD2)
```
```  1080   apply (subst zmult_commute, assumption)
```
```  1081   done
```
```  1082
```
```  1083 lemma zdvd_zmult_mono: "i dvd m ==> j dvd (n::int) ==> i * j dvd m * n"
```
```  1084   apply (unfold dvd_def, clarify)
```
```  1085   apply (rule_tac x = "k * ka" in exI)
```
```  1086   apply (simp add: mult_ac)
```
```  1087   done
```
```  1088
```
```  1089 lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
```
```  1090   apply (rule iffI)
```
```  1091    apply (erule_tac [2] zdvd_zadd)
```
```  1092    apply (subgoal_tac "n = (n + k * m) - k * m")
```
```  1093     apply (erule ssubst)
```
```  1094     apply (erule zdvd_zdiff, simp_all)
```
```  1095   done
```
```  1096
```
```  1097 lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
```
```  1098   apply (unfold dvd_def)
```
```  1099   apply (auto simp add: zmod_zmult_zmult1)
```
```  1100   done
```
```  1101
```
```  1102 lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
```
```  1103   apply (subgoal_tac "k dvd n * (m div n) + m mod n")
```
```  1104    apply (simp add: zmod_zdiv_equality [symmetric])
```
```  1105   apply (simp only: zdvd_zadd zdvd_zmult2)
```
```  1106   done
```
```  1107
```
```  1108 lemma zdvd_not_zless: "0 < m ==> m < n ==> \<not> n dvd (m::int)"
```
```  1109   apply (unfold dvd_def, auto)
```
```  1110   apply (subgoal_tac "0 < n")
```
```  1111    prefer 2
```
```  1112    apply (blast intro: order_less_trans)
```
```  1113   apply (simp add: zero_less_mult_iff)
```
```  1114   apply (subgoal_tac "n * k < n * 1")
```
```  1115    apply (drule mult_less_cancel_left [THEN iffD1], auto)
```
```  1116   done
```
```  1117
```
```  1118 lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
```
```  1119   apply (auto simp add: dvd_def nat_abs_mult_distrib)
```
```  1120   apply (auto simp add: nat_eq_iff abs_if split add: split_if_asm)
```
```  1121    apply (rule_tac x = "-(int k)" in exI)
```
```  1122   apply (auto simp add: zmult_int [symmetric])
```
```  1123   done
```
```  1124
```
```  1125 lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
```
```  1126   apply (auto simp add: dvd_def abs_if zmult_int [symmetric])
```
```  1127     apply (rule_tac [3] x = "nat k" in exI)
```
```  1128     apply (rule_tac [2] x = "-(int k)" in exI)
```
```  1129     apply (rule_tac x = "nat (-k)" in exI)
```
```  1130     apply (cut_tac [3] k = m in int_less_0_conv)
```
```  1131     apply (cut_tac k = m in int_less_0_conv)
```
```  1132     apply (auto simp add: zero_le_mult_iff mult_less_0_iff
```
```  1133       nat_mult_distrib [symmetric] nat_eq_iff2)
```
```  1134   done
```
```  1135
```
```  1136 lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
```
```  1137   apply (auto simp add: dvd_def zmult_int [symmetric])
```
```  1138   apply (rule_tac x = "nat k" in exI)
```
```  1139   apply (cut_tac k = m in int_less_0_conv)
```
```  1140   apply (auto simp add: zero_le_mult_iff mult_less_0_iff
```
```  1141     nat_mult_distrib [symmetric] nat_eq_iff2)
```
```  1142   done
```
```  1143
```
```  1144 lemma zminus_dvd_iff [iff]: "(-z dvd w) = (z dvd (w::int))"
```
```  1145   apply (auto simp add: dvd_def)
```
```  1146    apply (rule_tac [!] x = "-k" in exI, auto)
```
```  1147   done
```
```  1148
```
```  1149 lemma dvd_zminus_iff [iff]: "(z dvd -w) = (z dvd (w::int))"
```
```  1150   apply (auto simp add: dvd_def)
```
```  1151    apply (drule minus_equation_iff [THEN iffD1])
```
```  1152    apply (rule_tac [!] x = "-k" in exI, auto)
```
```  1153   done
```
```  1154
```
```  1155 lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
```
```  1156   apply (rule_tac z=n in int_cases)
```
```  1157   apply (auto simp add: dvd_int_iff)
```
```  1158   apply (rule_tac z=z in int_cases)
```
```  1159   apply (auto simp add: dvd_imp_le)
```
```  1160   done
```
```  1161
```
```  1162
```
```  1163 subsection{*Integer Powers*}
```
```  1164
```
```  1165 instance int :: power ..
```
```  1166
```
```  1167 primrec
```
```  1168   "p ^ 0 = 1"
```
```  1169   "p ^ (Suc n) = (p::int) * (p ^ n)"
```
```  1170
```
```  1171
```
```  1172 instance int :: recpower
```
```  1173 proof
```
```  1174   fix z :: int
```
```  1175   fix n :: nat
```
```  1176   show "z^0 = 1" by simp
```
```  1177   show "z^(Suc n) = z * (z^n)" by simp
```
```  1178 qed
```
```  1179
```
```  1180
```
```  1181 lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
```
```  1182 apply (induct_tac "y", auto)
```
```  1183 apply (rule zmod_zmult1_eq [THEN trans])
```
```  1184 apply (simp (no_asm_simp))
```
```  1185 apply (rule zmod_zmult_distrib [symmetric])
```
```  1186 done
```
```  1187
```
```  1188 lemma zpower_zadd_distrib: "x^(y+z) = ((x^y)*(x^z)::int)"
```
```  1189   by (rule Power.power_add)
```
```  1190
```
```  1191 lemma zpower_zpower: "(x^y)^z = (x^(y*z)::int)"
```
```  1192   by (rule Power.power_mult [symmetric])
```
```  1193
```
```  1194 lemma zero_less_zpower_abs_iff [simp]:
```
```  1195      "(0 < (abs x)^n) = (x \<noteq> (0::int) | n=0)"
```
```  1196 apply (induct_tac "n")
```
```  1197 apply (auto simp add: zero_less_mult_iff)
```
```  1198 done
```
```  1199
```
```  1200 lemma zero_le_zpower_abs [simp]: "(0::int) <= (abs x)^n"
```
```  1201 apply (induct_tac "n")
```
```  1202 apply (auto simp add: zero_le_mult_iff)
```
```  1203 done
```
```  1204
```
```  1205 lemma zdiv_int: "int (a div b) = (int a) div (int b)"
```
```  1206 apply (subst split_div, auto)
```
```  1207 apply (subst split_zdiv, auto)
```
```  1208 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
```
```  1209 apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
```
```  1210 done
```
```  1211
```
```  1212 lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
```
```  1213 apply (subst split_mod, auto)
```
```  1214 apply (subst split_zmod, auto)
```
```  1215 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia in IntDiv.unique_remainder)
```
```  1216 apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
```
```  1217 done
```
```  1218
```
```  1219 ML
```
```  1220 {*
```
```  1221 val quorem_def = thm "quorem_def";
```
```  1222
```
```  1223 val unique_quotient = thm "unique_quotient";
```
```  1224 val unique_remainder = thm "unique_remainder";
```
```  1225 val adjust_eq = thm "adjust_eq";
```
```  1226 val posDivAlg_eqn = thm "posDivAlg_eqn";
```
```  1227 val posDivAlg_correct = thm "posDivAlg_correct";
```
```  1228 val negDivAlg_eqn = thm "negDivAlg_eqn";
```
```  1229 val negDivAlg_correct = thm "negDivAlg_correct";
```
```  1230 val quorem_0 = thm "quorem_0";
```
```  1231 val posDivAlg_0 = thm "posDivAlg_0";
```
```  1232 val negDivAlg_minus1 = thm "negDivAlg_minus1";
```
```  1233 val negateSnd_eq = thm "negateSnd_eq";
```
```  1234 val quorem_neg = thm "quorem_neg";
```
```  1235 val divAlg_correct = thm "divAlg_correct";
```
```  1236 val DIVISION_BY_ZERO = thm "DIVISION_BY_ZERO";
```
```  1237 val zmod_zdiv_equality = thm "zmod_zdiv_equality";
```
```  1238 val pos_mod_conj = thm "pos_mod_conj";
```
```  1239 val pos_mod_sign = thm "pos_mod_sign";
```
```  1240 val neg_mod_conj = thm "neg_mod_conj";
```
```  1241 val neg_mod_sign = thm "neg_mod_sign";
```
```  1242 val quorem_div_mod = thm "quorem_div_mod";
```
```  1243 val quorem_div = thm "quorem_div";
```
```  1244 val quorem_mod = thm "quorem_mod";
```
```  1245 val div_pos_pos_trivial = thm "div_pos_pos_trivial";
```
```  1246 val div_neg_neg_trivial = thm "div_neg_neg_trivial";
```
```  1247 val div_pos_neg_trivial = thm "div_pos_neg_trivial";
```
```  1248 val mod_pos_pos_trivial = thm "mod_pos_pos_trivial";
```
```  1249 val mod_neg_neg_trivial = thm "mod_neg_neg_trivial";
```
```  1250 val mod_pos_neg_trivial = thm "mod_pos_neg_trivial";
```
```  1251 val zdiv_zminus_zminus = thm "zdiv_zminus_zminus";
```
```  1252 val zmod_zminus_zminus = thm "zmod_zminus_zminus";
```
```  1253 val zdiv_zminus1_eq_if = thm "zdiv_zminus1_eq_if";
```
```  1254 val zmod_zminus1_eq_if = thm "zmod_zminus1_eq_if";
```
```  1255 val zdiv_zminus2 = thm "zdiv_zminus2";
```
```  1256 val zmod_zminus2 = thm "zmod_zminus2";
```
```  1257 val zdiv_zminus2_eq_if = thm "zdiv_zminus2_eq_if";
```
```  1258 val zmod_zminus2_eq_if = thm "zmod_zminus2_eq_if";
```
```  1259 val self_quotient = thm "self_quotient";
```
```  1260 val self_remainder = thm "self_remainder";
```
```  1261 val zdiv_self = thm "zdiv_self";
```
```  1262 val zmod_self = thm "zmod_self";
```
```  1263 val zdiv_zero = thm "zdiv_zero";
```
```  1264 val div_eq_minus1 = thm "div_eq_minus1";
```
```  1265 val zmod_zero = thm "zmod_zero";
```
```  1266 val zdiv_minus1 = thm "zdiv_minus1";
```
```  1267 val zmod_minus1 = thm "zmod_minus1";
```
```  1268 val div_pos_pos = thm "div_pos_pos";
```
```  1269 val mod_pos_pos = thm "mod_pos_pos";
```
```  1270 val div_neg_pos = thm "div_neg_pos";
```
```  1271 val mod_neg_pos = thm "mod_neg_pos";
```
```  1272 val div_pos_neg = thm "div_pos_neg";
```
```  1273 val mod_pos_neg = thm "mod_pos_neg";
```
```  1274 val div_neg_neg = thm "div_neg_neg";
```
```  1275 val mod_neg_neg = thm "mod_neg_neg";
```
```  1276 val zmod_1 = thm "zmod_1";
```
```  1277 val zdiv_1 = thm "zdiv_1";
```
```  1278 val zmod_minus1_right = thm "zmod_minus1_right";
```
```  1279 val zdiv_minus1_right = thm "zdiv_minus1_right";
```
```  1280 val zdiv_mono1 = thm "zdiv_mono1";
```
```  1281 val zdiv_mono1_neg = thm "zdiv_mono1_neg";
```
```  1282 val zdiv_mono2 = thm "zdiv_mono2";
```
```  1283 val zdiv_mono2_neg = thm "zdiv_mono2_neg";
```
```  1284 val zdiv_zmult1_eq = thm "zdiv_zmult1_eq";
```
```  1285 val zmod_zmult1_eq = thm "zmod_zmult1_eq";
```
```  1286 val zmod_zmult1_eq' = thm "zmod_zmult1_eq'";
```
```  1287 val zmod_zmult_distrib = thm "zmod_zmult_distrib";
```
```  1288 val zdiv_zmult_self1 = thm "zdiv_zmult_self1";
```
```  1289 val zdiv_zmult_self2 = thm "zdiv_zmult_self2";
```
```  1290 val zmod_zmult_self1 = thm "zmod_zmult_self1";
```
```  1291 val zmod_zmult_self2 = thm "zmod_zmult_self2";
```
```  1292 val zmod_eq_0_iff = thm "zmod_eq_0_iff";
```
```  1293 val zdiv_zadd1_eq = thm "zdiv_zadd1_eq";
```
```  1294 val zmod_zadd1_eq = thm "zmod_zadd1_eq";
```
```  1295 val mod_div_trivial = thm "mod_div_trivial";
```
```  1296 val mod_mod_trivial = thm "mod_mod_trivial";
```
```  1297 val zmod_zadd_left_eq = thm "zmod_zadd_left_eq";
```
```  1298 val zmod_zadd_right_eq = thm "zmod_zadd_right_eq";
```
```  1299 val zdiv_zadd_self1 = thm "zdiv_zadd_self1";
```
```  1300 val zdiv_zadd_self2 = thm "zdiv_zadd_self2";
```
```  1301 val zmod_zadd_self1 = thm "zmod_zadd_self1";
```
```  1302 val zmod_zadd_self2 = thm "zmod_zadd_self2";
```
```  1303 val zdiv_zmult2_eq = thm "zdiv_zmult2_eq";
```
```  1304 val zmod_zmult2_eq = thm "zmod_zmult2_eq";
```
```  1305 val zdiv_zmult_zmult1 = thm "zdiv_zmult_zmult1";
```
```  1306 val zdiv_zmult_zmult2 = thm "zdiv_zmult_zmult2";
```
```  1307 val zmod_zmult_zmult1 = thm "zmod_zmult_zmult1";
```
```  1308 val zmod_zmult_zmult2 = thm "zmod_zmult_zmult2";
```
```  1309 val pos_zdiv_mult_2 = thm "pos_zdiv_mult_2";
```
```  1310 val neg_zdiv_mult_2 = thm "neg_zdiv_mult_2";
```
```  1311 val zdiv_number_of_BIT = thm "zdiv_number_of_BIT";
```
```  1312 val pos_zmod_mult_2 = thm "pos_zmod_mult_2";
```
```  1313 val neg_zmod_mult_2 = thm "neg_zmod_mult_2";
```
```  1314 val zmod_number_of_BIT = thm "zmod_number_of_BIT";
```
```  1315 val div_neg_pos_less0 = thm "div_neg_pos_less0";
```
```  1316 val div_nonneg_neg_le0 = thm "div_nonneg_neg_le0";
```
```  1317 val pos_imp_zdiv_nonneg_iff = thm "pos_imp_zdiv_nonneg_iff";
```
```  1318 val neg_imp_zdiv_nonneg_iff = thm "neg_imp_zdiv_nonneg_iff";
```
```  1319 val pos_imp_zdiv_neg_iff = thm "pos_imp_zdiv_neg_iff";
```
```  1320 val neg_imp_zdiv_neg_iff = thm "neg_imp_zdiv_neg_iff";
```
```  1321
```
```  1322 val zpower_zmod = thm "zpower_zmod";
```
```  1323 val zpower_zadd_distrib = thm "zpower_zadd_distrib";
```
```  1324 val zpower_zpower = thm "zpower_zpower";
```
```  1325 val zero_less_zpower_abs_iff = thm "zero_less_zpower_abs_iff";
```
```  1326 val zero_le_zpower_abs = thm "zero_le_zpower_abs";
```
```  1327 *}
```
```  1328
```
```  1329 end
```