src/ZF/IntDiv_ZF.thy
 author paulson Tue Mar 06 15:15:49 2012 +0000 (2012-03-06) changeset 46820 c656222c4dc1 parent 45602 2a858377c3d2 child 46821 ff6b0c1087f2 permissions -rw-r--r--
```     1 (*  Title:      ZF/IntDiv_ZF.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   1999  University of Cambridge
```
```     4
```
```     5 Here is the division algorithm in ML:
```
```     6
```
```     7     fun posDivAlg (a,b) =
```
```     8       if a<b then (0,a)
```
```     9       else let val (q,r) = posDivAlg(a, 2*b)
```
```    10                in  if 0<=r-b then (2*q+1, r-b) else (2*q, r)
```
```    11            end
```
```    12
```
```    13     fun negDivAlg (a,b) =
```
```    14       if 0<=a+b then (~1,a+b)
```
```    15       else let val (q,r) = negDivAlg(a, 2*b)
```
```    16                in  if 0<=r-b then (2*q+1, r-b) else (2*q, r)
```
```    17            end;
```
```    18
```
```    19     fun negateSnd (q,r:int) = (q,~r);
```
```    20
```
```    21     fun divAlg (a,b) = if 0<=a then
```
```    22                           if b>0 then posDivAlg (a,b)
```
```    23                            else if a=0 then (0,0)
```
```    24                                 else negateSnd (negDivAlg (~a,~b))
```
```    25                        else
```
```    26                           if 0<b then negDivAlg (a,b)
```
```    27                           else        negateSnd (posDivAlg (~a,~b));
```
```    28 *)
```
```    29
```
```    30 header{*The Division Operators Div and Mod*}
```
```    31
```
```    32 theory IntDiv_ZF imports IntArith OrderArith begin
```
```    33
```
```    34 definition
```
```    35   quorem :: "[i,i] => o"  where
```
```    36     "quorem == %<a,b> <q,r>.
```
```    37                       a = b\$*q \$+ r &
```
```    38                       (#0\$<b & #0\$<=r & r\$<b | ~(#0\$<b) & b\$<r & r \$<= #0)"
```
```    39
```
```    40 definition
```
```    41   adjust :: "[i,i] => i"  where
```
```    42     "adjust(b) == %<q,r>. if #0 \$<= r\$-b then <#2\$*q \$+ #1,r\$-b>
```
```    43                           else <#2\$*q,r>"
```
```    44
```
```    45
```
```    46 (** the division algorithm **)
```
```    47
```
```    48 definition
```
```    49   posDivAlg :: "i => i"  where
```
```    50 (*for the case a>=0, b>0*)
```
```    51 (*recdef posDivAlg "inv_image less_than (%(a,b). nat_of(a \$- b \$+ #1))"*)
```
```    52     "posDivAlg(ab) ==
```
```    53        wfrec(measure(int*int, %<a,b>. nat_of (a \$- b \$+ #1)),
```
```    54              ab,
```
```    55              %<a,b> f. if (a\$<b | b\$<=#0) then <#0,a>
```
```    56                        else adjust(b, f ` <a,#2\$*b>))"
```
```    57
```
```    58
```
```    59 (*for the case a<0, b>0*)
```
```    60 definition
```
```    61   negDivAlg :: "i => i"  where
```
```    62 (*recdef negDivAlg "inv_image less_than (%(a,b). nat_of(- a \$- b))"*)
```
```    63     "negDivAlg(ab) ==
```
```    64        wfrec(measure(int*int, %<a,b>. nat_of (\$- a \$- b)),
```
```    65              ab,
```
```    66              %<a,b> f. if (#0 \$<= a\$+b | b\$<=#0) then <#-1,a\$+b>
```
```    67                        else adjust(b, f ` <a,#2\$*b>))"
```
```    68
```
```    69 (*for the general case @{term"b\<noteq>0"}*)
```
```    70
```
```    71 definition
```
```    72   negateSnd :: "i => i"  where
```
```    73     "negateSnd == %<q,r>. <q, \$-r>"
```
```    74
```
```    75   (*The full division algorithm considers all possible signs for a, b
```
```    76     including the special case a=0, b<0, because negDivAlg requires a<0*)
```
```    77 definition
```
```    78   divAlg :: "i => i"  where
```
```    79     "divAlg ==
```
```    80        %<a,b>. if #0 \$<= a then
```
```    81                   if #0 \$<= b then posDivAlg (<a,b>)
```
```    82                   else if a=#0 then <#0,#0>
```
```    83                        else negateSnd (negDivAlg (<\$-a,\$-b>))
```
```    84                else
```
```    85                   if #0\$<b then negDivAlg (<a,b>)
```
```    86                   else         negateSnd (posDivAlg (<\$-a,\$-b>))"
```
```    87
```
```    88 definition
```
```    89   zdiv  :: "[i,i]=>i"                    (infixl "zdiv" 70)  where
```
```    90     "a zdiv b == fst (divAlg (<intify(a), intify(b)>))"
```
```    91
```
```    92 definition
```
```    93   zmod  :: "[i,i]=>i"                    (infixl "zmod" 70)  where
```
```    94     "a zmod b == snd (divAlg (<intify(a), intify(b)>))"
```
```    95
```
```    96
```
```    97 (** Some basic laws by Sidi Ehmety (need linear arithmetic!) **)
```
```    98
```
```    99 lemma zspos_add_zspos_imp_zspos: "[| #0 \$< x;  #0 \$< y |] ==> #0 \$< x \$+ y"
```
```   100 apply (rule_tac y = "y" in zless_trans)
```
```   101 apply (rule_tac [2] zdiff_zless_iff [THEN iffD1])
```
```   102 apply auto
```
```   103 done
```
```   104
```
```   105 lemma zpos_add_zpos_imp_zpos: "[| #0 \$<= x;  #0 \$<= y |] ==> #0 \$<= x \$+ y"
```
```   106 apply (rule_tac y = "y" in zle_trans)
```
```   107 apply (rule_tac [2] zdiff_zle_iff [THEN iffD1])
```
```   108 apply auto
```
```   109 done
```
```   110
```
```   111 lemma zneg_add_zneg_imp_zneg: "[| x \$< #0;  y \$< #0 |] ==> x \$+ y \$< #0"
```
```   112 apply (rule_tac y = "y" in zless_trans)
```
```   113 apply (rule zless_zdiff_iff [THEN iffD1])
```
```   114 apply auto
```
```   115 done
```
```   116
```
```   117 (* this theorem is used below *)
```
```   118 lemma zneg_or_0_add_zneg_or_0_imp_zneg_or_0:
```
```   119      "[| x \$<= #0;  y \$<= #0 |] ==> x \$+ y \$<= #0"
```
```   120 apply (rule_tac y = "y" in zle_trans)
```
```   121 apply (rule zle_zdiff_iff [THEN iffD1])
```
```   122 apply auto
```
```   123 done
```
```   124
```
```   125 lemma zero_lt_zmagnitude: "[| #0 \$< k; k \<in> int |] ==> 0 < zmagnitude(k)"
```
```   126 apply (drule zero_zless_imp_znegative_zminus)
```
```   127 apply (drule_tac [2] zneg_int_of)
```
```   128 apply (auto simp add: zminus_equation [of k])
```
```   129 apply (subgoal_tac "0 < zmagnitude (\$# succ (n))")
```
```   130  apply simp
```
```   131 apply (simp only: zmagnitude_int_of)
```
```   132 apply simp
```
```   133 done
```
```   134
```
```   135
```
```   136 (*** Inequality lemmas involving \$#succ(m) ***)
```
```   137
```
```   138 lemma zless_add_succ_iff:
```
```   139      "(w \$< z \$+ \$# succ(m)) <-> (w \$< z \$+ \$#m | intify(w) = z \$+ \$#m)"
```
```   140 apply (auto simp add: zless_iff_succ_zadd zadd_assoc int_of_add [symmetric])
```
```   141 apply (rule_tac [3] x = "0" in bexI)
```
```   142 apply (cut_tac m = "m" in int_succ_int_1)
```
```   143 apply (cut_tac m = "n" in int_succ_int_1)
```
```   144 apply simp
```
```   145 apply (erule natE)
```
```   146 apply auto
```
```   147 apply (rule_tac x = "succ (n) " in bexI)
```
```   148 apply auto
```
```   149 done
```
```   150
```
```   151 lemma zadd_succ_lemma:
```
```   152      "z \<in> int ==> (w \$+ \$# succ(m) \$<= z) <-> (w \$+ \$#m \$< z)"
```
```   153 apply (simp only: not_zless_iff_zle [THEN iff_sym] zless_add_succ_iff)
```
```   154 apply (auto intro: zle_anti_sym elim: zless_asym
```
```   155             simp add: zless_imp_zle not_zless_iff_zle)
```
```   156 done
```
```   157
```
```   158 lemma zadd_succ_zle_iff: "(w \$+ \$# succ(m) \$<= z) <-> (w \$+ \$#m \$< z)"
```
```   159 apply (cut_tac z = "intify (z)" in zadd_succ_lemma)
```
```   160 apply auto
```
```   161 done
```
```   162
```
```   163 (** Inequality reasoning **)
```
```   164
```
```   165 lemma zless_add1_iff_zle: "(w \$< z \$+ #1) <-> (w\$<=z)"
```
```   166 apply (subgoal_tac "#1 = \$# 1")
```
```   167 apply (simp only: zless_add_succ_iff zle_def)
```
```   168 apply auto
```
```   169 done
```
```   170
```
```   171 lemma add1_zle_iff: "(w \$+ #1 \$<= z) <-> (w \$< z)"
```
```   172 apply (subgoal_tac "#1 = \$# 1")
```
```   173 apply (simp only: zadd_succ_zle_iff)
```
```   174 apply auto
```
```   175 done
```
```   176
```
```   177 lemma add1_left_zle_iff: "(#1 \$+ w \$<= z) <-> (w \$< z)"
```
```   178 apply (subst zadd_commute)
```
```   179 apply (rule add1_zle_iff)
```
```   180 done
```
```   181
```
```   182
```
```   183 (*** Monotonicity of Multiplication ***)
```
```   184
```
```   185 lemma zmult_mono_lemma: "k \<in> nat ==> i \$<= j ==> i \$* \$#k \$<= j \$* \$#k"
```
```   186 apply (induct_tac "k")
```
```   187  prefer 2 apply (subst int_succ_int_1)
```
```   188 apply (simp_all (no_asm_simp) add: zadd_zmult_distrib2 zadd_zle_mono)
```
```   189 done
```
```   190
```
```   191 lemma zmult_zle_mono1: "[| i \$<= j;  #0 \$<= k |] ==> i\$*k \$<= j\$*k"
```
```   192 apply (subgoal_tac "i \$* intify (k) \$<= j \$* intify (k) ")
```
```   193 apply (simp (no_asm_use))
```
```   194 apply (rule_tac b = "intify (k)" in not_zneg_mag [THEN subst])
```
```   195 apply (rule_tac [3] zmult_mono_lemma)
```
```   196 apply auto
```
```   197 apply (simp add: znegative_iff_zless_0 not_zless_iff_zle [THEN iff_sym])
```
```   198 done
```
```   199
```
```   200 lemma zmult_zle_mono1_neg: "[| i \$<= j;  k \$<= #0 |] ==> j\$*k \$<= i\$*k"
```
```   201 apply (rule zminus_zle_zminus [THEN iffD1])
```
```   202 apply (simp del: zmult_zminus_right
```
```   203             add: zmult_zminus_right [symmetric] zmult_zle_mono1 zle_zminus)
```
```   204 done
```
```   205
```
```   206 lemma zmult_zle_mono2: "[| i \$<= j;  #0 \$<= k |] ==> k\$*i \$<= k\$*j"
```
```   207 apply (drule zmult_zle_mono1)
```
```   208 apply (simp_all add: zmult_commute)
```
```   209 done
```
```   210
```
```   211 lemma zmult_zle_mono2_neg: "[| i \$<= j;  k \$<= #0 |] ==> k\$*j \$<= k\$*i"
```
```   212 apply (drule zmult_zle_mono1_neg)
```
```   213 apply (simp_all add: zmult_commute)
```
```   214 done
```
```   215
```
```   216 (* \$<= monotonicity, BOTH arguments*)
```
```   217 lemma zmult_zle_mono:
```
```   218      "[| i \$<= j;  k \$<= l;  #0 \$<= j;  #0 \$<= k |] ==> i\$*k \$<= j\$*l"
```
```   219 apply (erule zmult_zle_mono1 [THEN zle_trans])
```
```   220 apply assumption
```
```   221 apply (erule zmult_zle_mono2)
```
```   222 apply assumption
```
```   223 done
```
```   224
```
```   225
```
```   226 (** strict, in 1st argument; proof is by induction on k>0 **)
```
```   227
```
```   228 lemma zmult_zless_mono2_lemma [rule_format]:
```
```   229      "[| i\$<j; k \<in> nat |] ==> 0<k \<longrightarrow> \$#k \$* i \$< \$#k \$* j"
```
```   230 apply (induct_tac "k")
```
```   231  prefer 2
```
```   232  apply (subst int_succ_int_1)
```
```   233  apply (erule natE)
```
```   234 apply (simp_all add: zadd_zmult_distrib zadd_zless_mono zle_def)
```
```   235 apply (frule nat_0_le)
```
```   236 apply (subgoal_tac "i \$+ (i \$+ \$# xa \$* i) \$< j \$+ (j \$+ \$# xa \$* j) ")
```
```   237 apply (simp (no_asm_use))
```
```   238 apply (rule zadd_zless_mono)
```
```   239 apply (simp_all (no_asm_simp) add: zle_def)
```
```   240 done
```
```   241
```
```   242 lemma zmult_zless_mono2: "[| i\$<j;  #0 \$< k |] ==> k\$*i \$< k\$*j"
```
```   243 apply (subgoal_tac "intify (k) \$* i \$< intify (k) \$* j")
```
```   244 apply (simp (no_asm_use))
```
```   245 apply (rule_tac b = "intify (k)" in not_zneg_mag [THEN subst])
```
```   246 apply (rule_tac [3] zmult_zless_mono2_lemma)
```
```   247 apply auto
```
```   248 apply (simp add: znegative_iff_zless_0)
```
```   249 apply (drule zless_trans, assumption)
```
```   250 apply (auto simp add: zero_lt_zmagnitude)
```
```   251 done
```
```   252
```
```   253 lemma zmult_zless_mono1: "[| i\$<j;  #0 \$< k |] ==> i\$*k \$< j\$*k"
```
```   254 apply (drule zmult_zless_mono2)
```
```   255 apply (simp_all add: zmult_commute)
```
```   256 done
```
```   257
```
```   258 (* < monotonicity, BOTH arguments*)
```
```   259 lemma zmult_zless_mono:
```
```   260      "[| i \$< j;  k \$< l;  #0 \$< j;  #0 \$< k |] ==> i\$*k \$< j\$*l"
```
```   261 apply (erule zmult_zless_mono1 [THEN zless_trans])
```
```   262 apply assumption
```
```   263 apply (erule zmult_zless_mono2)
```
```   264 apply assumption
```
```   265 done
```
```   266
```
```   267 lemma zmult_zless_mono1_neg: "[| i \$< j;  k \$< #0 |] ==> j\$*k \$< i\$*k"
```
```   268 apply (rule zminus_zless_zminus [THEN iffD1])
```
```   269 apply (simp del: zmult_zminus_right
```
```   270             add: zmult_zminus_right [symmetric] zmult_zless_mono1 zless_zminus)
```
```   271 done
```
```   272
```
```   273 lemma zmult_zless_mono2_neg: "[| i \$< j;  k \$< #0 |] ==> k\$*j \$< k\$*i"
```
```   274 apply (rule zminus_zless_zminus [THEN iffD1])
```
```   275 apply (simp del: zmult_zminus
```
```   276             add: zmult_zminus [symmetric] zmult_zless_mono2 zless_zminus)
```
```   277 done
```
```   278
```
```   279
```
```   280 (** Products of zeroes **)
```
```   281
```
```   282 lemma zmult_eq_lemma:
```
```   283      "[| m \<in> int; n \<in> int |] ==> (m = #0 | n = #0) <-> (m\$*n = #0)"
```
```   284 apply (case_tac "m \$< #0")
```
```   285 apply (auto simp add: not_zless_iff_zle zle_def neq_iff_zless)
```
```   286 apply (force dest: zmult_zless_mono1_neg zmult_zless_mono1)+
```
```   287 done
```
```   288
```
```   289 lemma zmult_eq_0_iff [iff]: "(m\$*n = #0) <-> (intify(m) = #0 | intify(n) = #0)"
```
```   290 apply (simp add: zmult_eq_lemma)
```
```   291 done
```
```   292
```
```   293
```
```   294 (** Cancellation laws for k*m < k*n and m*k < n*k, also for @{text"\<le>"} and =,
```
```   295     but not (yet?) for k*m < n*k. **)
```
```   296
```
```   297 lemma zmult_zless_lemma:
```
```   298      "[| k \<in> int; m \<in> int; n \<in> int |]
```
```   299       ==> (m\$*k \$< n\$*k) <-> ((#0 \$< k & m\$<n) | (k \$< #0 & n\$<m))"
```
```   300 apply (case_tac "k = #0")
```
```   301 apply (auto simp add: neq_iff_zless zmult_zless_mono1 zmult_zless_mono1_neg)
```
```   302 apply (auto simp add: not_zless_iff_zle
```
```   303                       not_zle_iff_zless [THEN iff_sym, of "m\$*k"]
```
```   304                       not_zle_iff_zless [THEN iff_sym, of m])
```
```   305 apply (auto elim: notE
```
```   306             simp add: zless_imp_zle zmult_zle_mono1 zmult_zle_mono1_neg)
```
```   307 done
```
```   308
```
```   309 lemma zmult_zless_cancel2:
```
```   310      "(m\$*k \$< n\$*k) <-> ((#0 \$< k & m\$<n) | (k \$< #0 & n\$<m))"
```
```   311 apply (cut_tac k = "intify (k)" and m = "intify (m)" and n = "intify (n)"
```
```   312        in zmult_zless_lemma)
```
```   313 apply auto
```
```   314 done
```
```   315
```
```   316 lemma zmult_zless_cancel1:
```
```   317      "(k\$*m \$< k\$*n) <-> ((#0 \$< k & m\$<n) | (k \$< #0 & n\$<m))"
```
```   318 by (simp add: zmult_commute [of k] zmult_zless_cancel2)
```
```   319
```
```   320 lemma zmult_zle_cancel2:
```
```   321      "(m\$*k \$<= n\$*k) <-> ((#0 \$< k \<longrightarrow> m\$<=n) & (k \$< #0 \<longrightarrow> n\$<=m))"
```
```   322 by (auto simp add: not_zless_iff_zle [THEN iff_sym] zmult_zless_cancel2)
```
```   323
```
```   324 lemma zmult_zle_cancel1:
```
```   325      "(k\$*m \$<= k\$*n) <-> ((#0 \$< k \<longrightarrow> m\$<=n) & (k \$< #0 \<longrightarrow> n\$<=m))"
```
```   326 by (auto simp add: not_zless_iff_zle [THEN iff_sym] zmult_zless_cancel1)
```
```   327
```
```   328 lemma int_eq_iff_zle: "[| m \<in> int; n \<in> int |] ==> m=n <-> (m \$<= n & n \$<= m)"
```
```   329 apply (blast intro: zle_refl zle_anti_sym)
```
```   330 done
```
```   331
```
```   332 lemma zmult_cancel2_lemma:
```
```   333      "[| k \<in> int; m \<in> int; n \<in> int |] ==> (m\$*k = n\$*k) <-> (k=#0 | m=n)"
```
```   334 apply (simp add: int_eq_iff_zle [of "m\$*k"] int_eq_iff_zle [of m])
```
```   335 apply (auto simp add: zmult_zle_cancel2 neq_iff_zless)
```
```   336 done
```
```   337
```
```   338 lemma zmult_cancel2 [simp]:
```
```   339      "(m\$*k = n\$*k) <-> (intify(k) = #0 | intify(m) = intify(n))"
```
```   340 apply (rule iff_trans)
```
```   341 apply (rule_tac [2] zmult_cancel2_lemma)
```
```   342 apply auto
```
```   343 done
```
```   344
```
```   345 lemma zmult_cancel1 [simp]:
```
```   346      "(k\$*m = k\$*n) <-> (intify(k) = #0 | intify(m) = intify(n))"
```
```   347 by (simp add: zmult_commute [of k] zmult_cancel2)
```
```   348
```
```   349
```
```   350 subsection{* Uniqueness and monotonicity of quotients and remainders *}
```
```   351
```
```   352 lemma unique_quotient_lemma:
```
```   353      "[| b\$*q' \$+ r' \$<= b\$*q \$+ r;  #0 \$<= r';  #0 \$< b;  r \$< b |]
```
```   354       ==> q' \$<= q"
```
```   355 apply (subgoal_tac "r' \$+ b \$* (q'\$-q) \$<= r")
```
```   356  prefer 2 apply (simp add: zdiff_zmult_distrib2 zadd_ac zcompare_rls)
```
```   357 apply (subgoal_tac "#0 \$< b \$* (#1 \$+ q \$- q') ")
```
```   358  prefer 2
```
```   359  apply (erule zle_zless_trans)
```
```   360  apply (simp add: zdiff_zmult_distrib2 zadd_zmult_distrib2 zadd_ac zcompare_rls)
```
```   361  apply (erule zle_zless_trans)
```
```   362  apply (simp add: );
```
```   363 apply (subgoal_tac "b \$* q' \$< b \$* (#1 \$+ q)")
```
```   364  prefer 2
```
```   365  apply (simp add: zdiff_zmult_distrib2 zadd_zmult_distrib2 zadd_ac zcompare_rls)
```
```   366 apply (auto elim: zless_asym
```
```   367         simp add: zmult_zless_cancel1 zless_add1_iff_zle zadd_ac zcompare_rls)
```
```   368 done
```
```   369
```
```   370 lemma unique_quotient_lemma_neg:
```
```   371      "[| b\$*q' \$+ r' \$<= b\$*q \$+ r;  r \$<= #0;  b \$< #0;  b \$< r' |]
```
```   372       ==> q \$<= q'"
```
```   373 apply (rule_tac b = "\$-b" and r = "\$-r'" and r' = "\$-r"
```
```   374        in unique_quotient_lemma)
```
```   375 apply (auto simp del: zminus_zadd_distrib
```
```   376             simp add: zminus_zadd_distrib [symmetric] zle_zminus zless_zminus)
```
```   377 done
```
```   378
```
```   379
```
```   380 lemma unique_quotient:
```
```   381      "[| quorem (<a,b>, <q,r>);  quorem (<a,b>, <q',r'>);  b \<in> int; b \<noteq> #0;
```
```   382          q \<in> int; q' \<in> int |] ==> q = q'"
```
```   383 apply (simp add: split_ifs quorem_def neq_iff_zless)
```
```   384 apply safe
```
```   385 apply simp_all
```
```   386 apply (blast intro: zle_anti_sym
```
```   387              dest: zle_eq_refl [THEN unique_quotient_lemma]
```
```   388                    zle_eq_refl [THEN unique_quotient_lemma_neg] sym)+
```
```   389 done
```
```   390
```
```   391 lemma unique_remainder:
```
```   392      "[| quorem (<a,b>, <q,r>);  quorem (<a,b>, <q',r'>);  b \<in> int; b \<noteq> #0;
```
```   393          q \<in> int; q' \<in> int;
```
```   394          r \<in> int; r' \<in> int |] ==> r = r'"
```
```   395 apply (subgoal_tac "q = q'")
```
```   396  prefer 2 apply (blast intro: unique_quotient)
```
```   397 apply (simp add: quorem_def)
```
```   398 done
```
```   399
```
```   400
```
```   401 subsection{*Correctness of posDivAlg,
```
```   402            the Division Algorithm for @{text "a\<ge>0"} and @{text "b>0"} *}
```
```   403
```
```   404 lemma adjust_eq [simp]:
```
```   405      "adjust(b, <q,r>) = (let diff = r\$-b in
```
```   406                           if #0 \$<= diff then <#2\$*q \$+ #1,diff>
```
```   407                                          else <#2\$*q,r>)"
```
```   408 by (simp add: Let_def adjust_def)
```
```   409
```
```   410
```
```   411 lemma posDivAlg_termination:
```
```   412      "[| #0 \$< b; ~ a \$< b |]
```
```   413       ==> nat_of(a \$- #2 \$\<times> b \$+ #1) < nat_of(a \$- b \$+ #1)"
```
```   414 apply (simp (no_asm) add: zless_nat_conj)
```
```   415 apply (simp add: not_zless_iff_zle zless_add1_iff_zle zcompare_rls)
```
```   416 done
```
```   417
```
```   418 lemmas posDivAlg_unfold = def_wfrec [OF posDivAlg_def wf_measure]
```
```   419
```
```   420 lemma posDivAlg_eqn:
```
```   421      "[| #0 \$< b; a \<in> int; b \<in> int |] ==>
```
```   422       posDivAlg(<a,b>) =
```
```   423        (if a\$<b then <#0,a> else adjust(b, posDivAlg (<a, #2\$*b>)))"
```
```   424 apply (rule posDivAlg_unfold [THEN trans])
```
```   425 apply (simp add: vimage_iff not_zless_iff_zle [THEN iff_sym])
```
```   426 apply (blast intro: posDivAlg_termination)
```
```   427 done
```
```   428
```
```   429 lemma posDivAlg_induct_lemma [rule_format]:
```
```   430   assumes prem:
```
```   431         "!!a b. [| a \<in> int; b \<in> int;
```
```   432                    ~ (a \$< b | b \$<= #0) \<longrightarrow> P(<a, #2 \$* b>) |] ==> P(<a,b>)"
```
```   433   shows "<u,v> \<in> int*int \<longrightarrow> P(<u,v>)"
```
```   434 apply (rule_tac a = "<u,v>" in wf_induct)
```
```   435 apply (rule_tac A = "int*int" and f = "%<a,b>.nat_of (a \$- b \$+ #1)"
```
```   436        in wf_measure)
```
```   437 apply clarify
```
```   438 apply (rule prem)
```
```   439 apply (drule_tac [3] x = "<xa, #2 \$\<times> y>" in spec)
```
```   440 apply auto
```
```   441 apply (simp add: not_zle_iff_zless posDivAlg_termination)
```
```   442 done
```
```   443
```
```   444
```
```   445 lemma posDivAlg_induct [consumes 2]:
```
```   446   assumes u_int: "u \<in> int"
```
```   447       and v_int: "v \<in> int"
```
```   448       and ih: "!!a b. [| a \<in> int; b \<in> int;
```
```   449                      ~ (a \$< b | b \$<= #0) \<longrightarrow> P(a, #2 \$* b) |] ==> P(a,b)"
```
```   450   shows "P(u,v)"
```
```   451 apply (subgoal_tac "(%<x,y>. P (x,y)) (<u,v>)")
```
```   452 apply simp
```
```   453 apply (rule posDivAlg_induct_lemma)
```
```   454 apply (simp (no_asm_use))
```
```   455 apply (rule ih)
```
```   456 apply (auto simp add: u_int v_int)
```
```   457 done
```
```   458
```
```   459 (*FIXME: use intify in integ_of so that we always have @{term"integ_of w \<in> int"}.
```
```   460     then this rewrite can work for all constants!!*)
```
```   461 lemma intify_eq_0_iff_zle: "intify(m) = #0 <-> (m \$<= #0 & #0 \$<= m)"
```
```   462 apply (simp (no_asm) add: int_eq_iff_zle)
```
```   463 done
```
```   464
```
```   465
```
```   466 subsection{* Some convenient biconditionals for products of signs *}
```
```   467
```
```   468 lemma zmult_pos: "[| #0 \$< i; #0 \$< j |] ==> #0 \$< i \$* j"
```
```   469 apply (drule zmult_zless_mono1)
```
```   470 apply auto
```
```   471 done
```
```   472
```
```   473 lemma zmult_neg: "[| i \$< #0; j \$< #0 |] ==> #0 \$< i \$* j"
```
```   474 apply (drule zmult_zless_mono1_neg)
```
```   475 apply auto
```
```   476 done
```
```   477
```
```   478 lemma zmult_pos_neg: "[| #0 \$< i; j \$< #0 |] ==> i \$* j \$< #0"
```
```   479 apply (drule zmult_zless_mono1_neg)
```
```   480 apply auto
```
```   481 done
```
```   482
```
```   483 (** Inequality reasoning **)
```
```   484
```
```   485 lemma int_0_less_lemma:
```
```   486      "[| x \<in> int; y \<in> int |]
```
```   487       ==> (#0 \$< x \$* y) <-> (#0 \$< x & #0 \$< y | x \$< #0 & y \$< #0)"
```
```   488 apply (auto simp add: zle_def not_zless_iff_zle zmult_pos zmult_neg)
```
```   489 apply (rule ccontr)
```
```   490 apply (rule_tac [2] ccontr)
```
```   491 apply (auto simp add: zle_def not_zless_iff_zle)
```
```   492 apply (erule_tac P = "#0\$< x\$* y" in rev_mp)
```
```   493 apply (erule_tac [2] P = "#0\$< x\$* y" in rev_mp)
```
```   494 apply (drule zmult_pos_neg, assumption)
```
```   495  prefer 2
```
```   496  apply (drule zmult_pos_neg, assumption)
```
```   497 apply (auto dest: zless_not_sym simp add: zmult_commute)
```
```   498 done
```
```   499
```
```   500 lemma int_0_less_mult_iff:
```
```   501      "(#0 \$< x \$* y) <-> (#0 \$< x & #0 \$< y | x \$< #0 & y \$< #0)"
```
```   502 apply (cut_tac x = "intify (x)" and y = "intify (y)" in int_0_less_lemma)
```
```   503 apply auto
```
```   504 done
```
```   505
```
```   506 lemma int_0_le_lemma:
```
```   507      "[| x \<in> int; y \<in> int |]
```
```   508       ==> (#0 \$<= x \$* y) <-> (#0 \$<= x & #0 \$<= y | x \$<= #0 & y \$<= #0)"
```
```   509 by (auto simp add: zle_def not_zless_iff_zle int_0_less_mult_iff)
```
```   510
```
```   511 lemma int_0_le_mult_iff:
```
```   512      "(#0 \$<= x \$* y) <-> ((#0 \$<= x & #0 \$<= y) | (x \$<= #0 & y \$<= #0))"
```
```   513 apply (cut_tac x = "intify (x)" and y = "intify (y)" in int_0_le_lemma)
```
```   514 apply auto
```
```   515 done
```
```   516
```
```   517 lemma zmult_less_0_iff:
```
```   518      "(x \$* y \$< #0) <-> (#0 \$< x & y \$< #0 | x \$< #0 & #0 \$< y)"
```
```   519 apply (auto simp add: int_0_le_mult_iff not_zle_iff_zless [THEN iff_sym])
```
```   520 apply (auto dest: zless_not_sym simp add: not_zle_iff_zless)
```
```   521 done
```
```   522
```
```   523 lemma zmult_le_0_iff:
```
```   524      "(x \$* y \$<= #0) <-> (#0 \$<= x & y \$<= #0 | x \$<= #0 & #0 \$<= y)"
```
```   525 by (auto dest: zless_not_sym
```
```   526          simp add: int_0_less_mult_iff not_zless_iff_zle [THEN iff_sym])
```
```   527
```
```   528
```
```   529 (*Typechecking for posDivAlg*)
```
```   530 lemma posDivAlg_type [rule_format]:
```
```   531      "[| a \<in> int; b \<in> int |] ==> posDivAlg(<a,b>) \<in> int * int"
```
```   532 apply (rule_tac u = "a" and v = "b" in posDivAlg_induct)
```
```   533 apply assumption+
```
```   534 apply (case_tac "#0 \$< ba")
```
```   535  apply (simp add: posDivAlg_eqn adjust_def integ_of_type
```
```   536              split add: split_if_asm)
```
```   537  apply clarify
```
```   538  apply (simp add: int_0_less_mult_iff not_zle_iff_zless)
```
```   539 apply (simp add: not_zless_iff_zle)
```
```   540 apply (subst posDivAlg_unfold)
```
```   541 apply simp
```
```   542 done
```
```   543
```
```   544 (*Correctness of posDivAlg: it computes quotients correctly*)
```
```   545 lemma posDivAlg_correct [rule_format]:
```
```   546      "[| a \<in> int; b \<in> int |]
```
```   547       ==> #0 \$<= a \<longrightarrow> #0 \$< b \<longrightarrow> quorem (<a,b>, posDivAlg(<a,b>))"
```
```   548 apply (rule_tac u = "a" and v = "b" in posDivAlg_induct)
```
```   549 apply auto
```
```   550    apply (simp_all add: quorem_def)
```
```   551    txt{*base case: a<b*}
```
```   552    apply (simp add: posDivAlg_eqn)
```
```   553   apply (simp add: not_zless_iff_zle [THEN iff_sym])
```
```   554  apply (simp add: int_0_less_mult_iff)
```
```   555 txt{*main argument*}
```
```   556 apply (subst posDivAlg_eqn)
```
```   557 apply (simp_all (no_asm_simp))
```
```   558 apply (erule splitE)
```
```   559 apply (rule posDivAlg_type)
```
```   560 apply (simp_all add: int_0_less_mult_iff)
```
```   561 apply (auto simp add: zadd_zmult_distrib2 Let_def)
```
```   562 txt{*now just linear arithmetic*}
```
```   563 apply (simp add: not_zle_iff_zless zdiff_zless_iff)
```
```   564 done
```
```   565
```
```   566
```
```   567 subsection{*Correctness of negDivAlg, the division algorithm for a<0 and b>0*}
```
```   568
```
```   569 lemma negDivAlg_termination:
```
```   570      "[| #0 \$< b; a \$+ b \$< #0 |]
```
```   571       ==> nat_of(\$- a \$- #2 \$* b) < nat_of(\$- a \$- b)"
```
```   572 apply (simp (no_asm) add: zless_nat_conj)
```
```   573 apply (simp add: zcompare_rls not_zle_iff_zless zless_zdiff_iff [THEN iff_sym]
```
```   574                  zless_zminus)
```
```   575 done
```
```   576
```
```   577 lemmas negDivAlg_unfold = def_wfrec [OF negDivAlg_def wf_measure]
```
```   578
```
```   579 lemma negDivAlg_eqn:
```
```   580      "[| #0 \$< b; a \<in> int; b \<in> int |] ==>
```
```   581       negDivAlg(<a,b>) =
```
```   582        (if #0 \$<= a\$+b then <#-1,a\$+b>
```
```   583                        else adjust(b, negDivAlg (<a, #2\$*b>)))"
```
```   584 apply (rule negDivAlg_unfold [THEN trans])
```
```   585 apply (simp (no_asm_simp) add: vimage_iff not_zless_iff_zle [THEN iff_sym])
```
```   586 apply (blast intro: negDivAlg_termination)
```
```   587 done
```
```   588
```
```   589 lemma negDivAlg_induct_lemma [rule_format]:
```
```   590   assumes prem:
```
```   591         "!!a b. [| a \<in> int; b \<in> int;
```
```   592                    ~ (#0 \$<= a \$+ b | b \$<= #0) \<longrightarrow> P(<a, #2 \$* b>) |]
```
```   593                 ==> P(<a,b>)"
```
```   594   shows "<u,v> \<in> int*int \<longrightarrow> P(<u,v>)"
```
```   595 apply (rule_tac a = "<u,v>" in wf_induct)
```
```   596 apply (rule_tac A = "int*int" and f = "%<a,b>.nat_of (\$- a \$- b)"
```
```   597        in wf_measure)
```
```   598 apply clarify
```
```   599 apply (rule prem)
```
```   600 apply (drule_tac [3] x = "<xa, #2 \$\<times> y>" in spec)
```
```   601 apply auto
```
```   602 apply (simp add: not_zle_iff_zless negDivAlg_termination)
```
```   603 done
```
```   604
```
```   605 lemma negDivAlg_induct [consumes 2]:
```
```   606   assumes u_int: "u \<in> int"
```
```   607       and v_int: "v \<in> int"
```
```   608       and ih: "!!a b. [| a \<in> int; b \<in> int;
```
```   609                          ~ (#0 \$<= a \$+ b | b \$<= #0) \<longrightarrow> P(a, #2 \$* b) |]
```
```   610                       ==> P(a,b)"
```
```   611   shows "P(u,v)"
```
```   612 apply (subgoal_tac " (%<x,y>. P (x,y)) (<u,v>)")
```
```   613 apply simp
```
```   614 apply (rule negDivAlg_induct_lemma)
```
```   615 apply (simp (no_asm_use))
```
```   616 apply (rule ih)
```
```   617 apply (auto simp add: u_int v_int)
```
```   618 done
```
```   619
```
```   620
```
```   621 (*Typechecking for negDivAlg*)
```
```   622 lemma negDivAlg_type:
```
```   623      "[| a \<in> int; b \<in> int |] ==> negDivAlg(<a,b>) \<in> int * int"
```
```   624 apply (rule_tac u = "a" and v = "b" in negDivAlg_induct)
```
```   625 apply assumption+
```
```   626 apply (case_tac "#0 \$< ba")
```
```   627  apply (simp add: negDivAlg_eqn adjust_def integ_of_type
```
```   628              split add: split_if_asm)
```
```   629  apply clarify
```
```   630  apply (simp add: int_0_less_mult_iff not_zle_iff_zless)
```
```   631 apply (simp add: not_zless_iff_zle)
```
```   632 apply (subst negDivAlg_unfold)
```
```   633 apply simp
```
```   634 done
```
```   635
```
```   636
```
```   637 (*Correctness of negDivAlg: it computes quotients correctly
```
```   638   It doesn't work if a=0 because the 0/b=0 rather than -1*)
```
```   639 lemma negDivAlg_correct [rule_format]:
```
```   640      "[| a \<in> int; b \<in> int |]
```
```   641       ==> a \$< #0 \<longrightarrow> #0 \$< b \<longrightarrow> quorem (<a,b>, negDivAlg(<a,b>))"
```
```   642 apply (rule_tac u = "a" and v = "b" in negDivAlg_induct)
```
```   643   apply auto
```
```   644    apply (simp_all add: quorem_def)
```
```   645    txt{*base case: @{term "0\$<=a\$+b"}*}
```
```   646    apply (simp add: negDivAlg_eqn)
```
```   647   apply (simp add: not_zless_iff_zle [THEN iff_sym])
```
```   648  apply (simp add: int_0_less_mult_iff)
```
```   649 txt{*main argument*}
```
```   650 apply (subst negDivAlg_eqn)
```
```   651 apply (simp_all (no_asm_simp))
```
```   652 apply (erule splitE)
```
```   653 apply (rule negDivAlg_type)
```
```   654 apply (simp_all add: int_0_less_mult_iff)
```
```   655 apply (auto simp add: zadd_zmult_distrib2 Let_def)
```
```   656 txt{*now just linear arithmetic*}
```
```   657 apply (simp add: not_zle_iff_zless zdiff_zless_iff)
```
```   658 done
```
```   659
```
```   660
```
```   661 subsection{* Existence shown by proving the division algorithm to be correct *}
```
```   662
```
```   663 (*the case a=0*)
```
```   664 lemma quorem_0: "[|b \<noteq> #0;  b \<in> int|] ==> quorem (<#0,b>, <#0,#0>)"
```
```   665 by (force simp add: quorem_def neq_iff_zless)
```
```   666
```
```   667 lemma posDivAlg_zero_divisor: "posDivAlg(<a,#0>) = <#0,a>"
```
```   668 apply (subst posDivAlg_unfold)
```
```   669 apply simp
```
```   670 done
```
```   671
```
```   672 lemma posDivAlg_0 [simp]: "posDivAlg (<#0,b>) = <#0,#0>"
```
```   673 apply (subst posDivAlg_unfold)
```
```   674 apply (simp add: not_zle_iff_zless)
```
```   675 done
```
```   676
```
```   677
```
```   678 (*Needed below.  Actually it's an equivalence.*)
```
```   679 lemma linear_arith_lemma: "~ (#0 \$<= #-1 \$+ b) ==> (b \$<= #0)"
```
```   680 apply (simp add: not_zle_iff_zless)
```
```   681 apply (drule zminus_zless_zminus [THEN iffD2])
```
```   682 apply (simp add: zadd_commute zless_add1_iff_zle zle_zminus)
```
```   683 done
```
```   684
```
```   685 lemma negDivAlg_minus1 [simp]: "negDivAlg (<#-1,b>) = <#-1, b\$-#1>"
```
```   686 apply (subst negDivAlg_unfold)
```
```   687 apply (simp add: linear_arith_lemma integ_of_type vimage_iff)
```
```   688 done
```
```   689
```
```   690 lemma negateSnd_eq [simp]: "negateSnd (<q,r>) = <q, \$-r>"
```
```   691 apply (unfold negateSnd_def)
```
```   692 apply auto
```
```   693 done
```
```   694
```
```   695 lemma negateSnd_type: "qr \<in> int * int ==> negateSnd (qr) \<in> int * int"
```
```   696 apply (unfold negateSnd_def)
```
```   697 apply auto
```
```   698 done
```
```   699
```
```   700 lemma quorem_neg:
```
```   701      "[|quorem (<\$-a,\$-b>, qr);  a \<in> int;  b \<in> int;  qr \<in> int * int|]
```
```   702       ==> quorem (<a,b>, negateSnd(qr))"
```
```   703 apply clarify
```
```   704 apply (auto elim: zless_asym simp add: quorem_def zless_zminus)
```
```   705 txt{*linear arithmetic from here on*}
```
```   706 apply (simp_all add: zminus_equation [of a] zminus_zless)
```
```   707 apply (cut_tac [2] z = "b" and w = "#0" in zless_linear)
```
```   708 apply (cut_tac [1] z = "b" and w = "#0" in zless_linear)
```
```   709 apply auto
```
```   710 apply (blast dest: zle_zless_trans)+
```
```   711 done
```
```   712
```
```   713 lemma divAlg_correct:
```
```   714      "[|b \<noteq> #0;  a \<in> int;  b \<in> int|] ==> quorem (<a,b>, divAlg(<a,b>))"
```
```   715 apply (auto simp add: quorem_0 divAlg_def)
```
```   716 apply (safe intro!: quorem_neg posDivAlg_correct negDivAlg_correct
```
```   717                     posDivAlg_type negDivAlg_type)
```
```   718 apply (auto simp add: quorem_def neq_iff_zless)
```
```   719 txt{*linear arithmetic from here on*}
```
```   720 apply (auto simp add: zle_def)
```
```   721 done
```
```   722
```
```   723 lemma divAlg_type: "[|a \<in> int;  b \<in> int|] ==> divAlg(<a,b>) \<in> int * int"
```
```   724 apply (auto simp add: divAlg_def)
```
```   725 apply (auto simp add: posDivAlg_type negDivAlg_type negateSnd_type)
```
```   726 done
```
```   727
```
```   728
```
```   729 (** intify cancellation **)
```
```   730
```
```   731 lemma zdiv_intify1 [simp]: "intify(x) zdiv y = x zdiv y"
```
```   732 apply (simp (no_asm) add: zdiv_def)
```
```   733 done
```
```   734
```
```   735 lemma zdiv_intify2 [simp]: "x zdiv intify(y) = x zdiv y"
```
```   736 apply (simp (no_asm) add: zdiv_def)
```
```   737 done
```
```   738
```
```   739 lemma zdiv_type [iff,TC]: "z zdiv w \<in> int"
```
```   740 apply (unfold zdiv_def)
```
```   741 apply (blast intro: fst_type divAlg_type)
```
```   742 done
```
```   743
```
```   744 lemma zmod_intify1 [simp]: "intify(x) zmod y = x zmod y"
```
```   745 apply (simp (no_asm) add: zmod_def)
```
```   746 done
```
```   747
```
```   748 lemma zmod_intify2 [simp]: "x zmod intify(y) = x zmod y"
```
```   749 apply (simp (no_asm) add: zmod_def)
```
```   750 done
```
```   751
```
```   752 lemma zmod_type [iff,TC]: "z zmod w \<in> int"
```
```   753 apply (unfold zmod_def)
```
```   754 apply (rule snd_type)
```
```   755 apply (blast intro: divAlg_type)
```
```   756 done
```
```   757
```
```   758
```
```   759 (** Arbitrary definitions for division by zero.  Useful to simplify
```
```   760     certain equations **)
```
```   761
```
```   762 lemma DIVISION_BY_ZERO_ZDIV: "a zdiv #0 = #0"
```
```   763 apply (simp (no_asm) add: zdiv_def divAlg_def posDivAlg_zero_divisor)
```
```   764 done  (*NOT for adding to default simpset*)
```
```   765
```
```   766 lemma DIVISION_BY_ZERO_ZMOD: "a zmod #0 = intify(a)"
```
```   767 apply (simp (no_asm) add: zmod_def divAlg_def posDivAlg_zero_divisor)
```
```   768 done  (*NOT for adding to default simpset*)
```
```   769
```
```   770
```
```   771
```
```   772 (** Basic laws about division and remainder **)
```
```   773
```
```   774 lemma raw_zmod_zdiv_equality:
```
```   775      "[| a \<in> int; b \<in> int |] ==> a = b \$* (a zdiv b) \$+ (a zmod b)"
```
```   776 apply (case_tac "b = #0")
```
```   777  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```   778 apply (cut_tac a = "a" and b = "b" in divAlg_correct)
```
```   779 apply (auto simp add: quorem_def zdiv_def zmod_def split_def)
```
```   780 done
```
```   781
```
```   782 lemma zmod_zdiv_equality: "intify(a) = b \$* (a zdiv b) \$+ (a zmod b)"
```
```   783 apply (rule trans)
```
```   784 apply (rule_tac b = "intify (b)" in raw_zmod_zdiv_equality)
```
```   785 apply auto
```
```   786 done
```
```   787
```
```   788 lemma pos_mod: "#0 \$< b ==> #0 \$<= a zmod b & a zmod b \$< b"
```
```   789 apply (cut_tac a = "intify (a)" and b = "intify (b)" in divAlg_correct)
```
```   790 apply (auto simp add: intify_eq_0_iff_zle quorem_def zmod_def split_def)
```
```   791 apply (blast dest: zle_zless_trans)+
```
```   792 done
```
```   793
```
```   794 lemmas pos_mod_sign = pos_mod [THEN conjunct1]
```
```   795   and pos_mod_bound = pos_mod [THEN conjunct2]
```
```   796
```
```   797 lemma neg_mod: "b \$< #0 ==> a zmod b \$<= #0 & b \$< a zmod b"
```
```   798 apply (cut_tac a = "intify (a)" and b = "intify (b)" in divAlg_correct)
```
```   799 apply (auto simp add: intify_eq_0_iff_zle quorem_def zmod_def split_def)
```
```   800 apply (blast dest: zle_zless_trans)
```
```   801 apply (blast dest: zless_trans)+
```
```   802 done
```
```   803
```
```   804 lemmas neg_mod_sign = neg_mod [THEN conjunct1]
```
```   805   and neg_mod_bound = neg_mod [THEN conjunct2]
```
```   806
```
```   807
```
```   808 (** proving general properties of zdiv and zmod **)
```
```   809
```
```   810 lemma quorem_div_mod:
```
```   811      "[|b \<noteq> #0;  a \<in> int;  b \<in> int |]
```
```   812       ==> quorem (<a,b>, <a zdiv b, a zmod b>)"
```
```   813 apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
```
```   814 apply (auto simp add: quorem_def neq_iff_zless pos_mod_sign pos_mod_bound
```
```   815                       neg_mod_sign neg_mod_bound)
```
```   816 done
```
```   817
```
```   818 (*Surely quorem(<a,b>,<q,r>) implies @{term"a \<in> int"}, but it doesn't matter*)
```
```   819 lemma quorem_div:
```
```   820      "[| quorem(<a,b>,<q,r>);  b \<noteq> #0;  a \<in> int;  b \<in> int;  q \<in> int |]
```
```   821       ==> a zdiv b = q"
```
```   822 by (blast intro: quorem_div_mod [THEN unique_quotient])
```
```   823
```
```   824 lemma quorem_mod:
```
```   825      "[| quorem(<a,b>,<q,r>); b \<noteq> #0; a \<in> int; b \<in> int; q \<in> int; r \<in> int |]
```
```   826       ==> a zmod b = r"
```
```   827 by (blast intro: quorem_div_mod [THEN unique_remainder])
```
```   828
```
```   829 lemma zdiv_pos_pos_trivial_raw:
```
```   830      "[| a \<in> int;  b \<in> int;  #0 \$<= a;  a \$< b |] ==> a zdiv b = #0"
```
```   831 apply (rule quorem_div)
```
```   832 apply (auto simp add: quorem_def)
```
```   833 (*linear arithmetic*)
```
```   834 apply (blast dest: zle_zless_trans)+
```
```   835 done
```
```   836
```
```   837 lemma zdiv_pos_pos_trivial: "[| #0 \$<= a;  a \$< b |] ==> a zdiv b = #0"
```
```   838 apply (cut_tac a = "intify (a)" and b = "intify (b)"
```
```   839        in zdiv_pos_pos_trivial_raw)
```
```   840 apply auto
```
```   841 done
```
```   842
```
```   843 lemma zdiv_neg_neg_trivial_raw:
```
```   844      "[| a \<in> int;  b \<in> int;  a \$<= #0;  b \$< a |] ==> a zdiv b = #0"
```
```   845 apply (rule_tac r = "a" in quorem_div)
```
```   846 apply (auto simp add: quorem_def)
```
```   847 (*linear arithmetic*)
```
```   848 apply (blast dest: zle_zless_trans zless_trans)+
```
```   849 done
```
```   850
```
```   851 lemma zdiv_neg_neg_trivial: "[| a \$<= #0;  b \$< a |] ==> a zdiv b = #0"
```
```   852 apply (cut_tac a = "intify (a)" and b = "intify (b)"
```
```   853        in zdiv_neg_neg_trivial_raw)
```
```   854 apply auto
```
```   855 done
```
```   856
```
```   857 lemma zadd_le_0_lemma: "[| a\$+b \$<= #0;  #0 \$< a;  #0 \$< b |] ==> False"
```
```   858 apply (drule_tac z' = "#0" and z = "b" in zadd_zless_mono)
```
```   859 apply (auto simp add: zle_def)
```
```   860 apply (blast dest: zless_trans)
```
```   861 done
```
```   862
```
```   863 lemma zdiv_pos_neg_trivial_raw:
```
```   864      "[| a \<in> int;  b \<in> int;  #0 \$< a;  a\$+b \$<= #0 |] ==> a zdiv b = #-1"
```
```   865 apply (rule_tac r = "a \$+ b" in quorem_div)
```
```   866 apply (auto simp add: quorem_def)
```
```   867 (*linear arithmetic*)
```
```   868 apply (blast dest: zadd_le_0_lemma zle_zless_trans)+
```
```   869 done
```
```   870
```
```   871 lemma zdiv_pos_neg_trivial: "[| #0 \$< a;  a\$+b \$<= #0 |] ==> a zdiv b = #-1"
```
```   872 apply (cut_tac a = "intify (a)" and b = "intify (b)"
```
```   873        in zdiv_pos_neg_trivial_raw)
```
```   874 apply auto
```
```   875 done
```
```   876
```
```   877 (*There is no zdiv_neg_pos_trivial because  #0 zdiv b = #0 would supersede it*)
```
```   878
```
```   879
```
```   880 lemma zmod_pos_pos_trivial_raw:
```
```   881      "[| a \<in> int;  b \<in> int;  #0 \$<= a;  a \$< b |] ==> a zmod b = a"
```
```   882 apply (rule_tac q = "#0" in quorem_mod)
```
```   883 apply (auto simp add: quorem_def)
```
```   884 (*linear arithmetic*)
```
```   885 apply (blast dest: zle_zless_trans)+
```
```   886 done
```
```   887
```
```   888 lemma zmod_pos_pos_trivial: "[| #0 \$<= a;  a \$< b |] ==> a zmod b = intify(a)"
```
```   889 apply (cut_tac a = "intify (a)" and b = "intify (b)"
```
```   890        in zmod_pos_pos_trivial_raw)
```
```   891 apply auto
```
```   892 done
```
```   893
```
```   894 lemma zmod_neg_neg_trivial_raw:
```
```   895      "[| a \<in> int;  b \<in> int;  a \$<= #0;  b \$< a |] ==> a zmod b = a"
```
```   896 apply (rule_tac q = "#0" in quorem_mod)
```
```   897 apply (auto simp add: quorem_def)
```
```   898 (*linear arithmetic*)
```
```   899 apply (blast dest: zle_zless_trans zless_trans)+
```
```   900 done
```
```   901
```
```   902 lemma zmod_neg_neg_trivial: "[| a \$<= #0;  b \$< a |] ==> a zmod b = intify(a)"
```
```   903 apply (cut_tac a = "intify (a)" and b = "intify (b)"
```
```   904        in zmod_neg_neg_trivial_raw)
```
```   905 apply auto
```
```   906 done
```
```   907
```
```   908 lemma zmod_pos_neg_trivial_raw:
```
```   909      "[| a \<in> int;  b \<in> int;  #0 \$< a;  a\$+b \$<= #0 |] ==> a zmod b = a\$+b"
```
```   910 apply (rule_tac q = "#-1" in quorem_mod)
```
```   911 apply (auto simp add: quorem_def)
```
```   912 (*linear arithmetic*)
```
```   913 apply (blast dest: zadd_le_0_lemma zle_zless_trans)+
```
```   914 done
```
```   915
```
```   916 lemma zmod_pos_neg_trivial: "[| #0 \$< a;  a\$+b \$<= #0 |] ==> a zmod b = a\$+b"
```
```   917 apply (cut_tac a = "intify (a)" and b = "intify (b)"
```
```   918        in zmod_pos_neg_trivial_raw)
```
```   919 apply auto
```
```   920 done
```
```   921
```
```   922 (*There is no zmod_neg_pos_trivial...*)
```
```   923
```
```   924
```
```   925 (*Simpler laws such as -a zdiv b = -(a zdiv b) FAIL*)
```
```   926
```
```   927 lemma zdiv_zminus_zminus_raw:
```
```   928      "[|a \<in> int;  b \<in> int|] ==> (\$-a) zdiv (\$-b) = a zdiv b"
```
```   929 apply (case_tac "b = #0")
```
```   930  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```   931 apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_div])
```
```   932 apply auto
```
```   933 done
```
```   934
```
```   935 lemma zdiv_zminus_zminus [simp]: "(\$-a) zdiv (\$-b) = a zdiv b"
```
```   936 apply (cut_tac a = "intify (a)" and b = "intify (b)" in zdiv_zminus_zminus_raw)
```
```   937 apply auto
```
```   938 done
```
```   939
```
```   940 (*Simpler laws such as -a zmod b = -(a zmod b) FAIL*)
```
```   941 lemma zmod_zminus_zminus_raw:
```
```   942      "[|a \<in> int;  b \<in> int|] ==> (\$-a) zmod (\$-b) = \$- (a zmod b)"
```
```   943 apply (case_tac "b = #0")
```
```   944  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```   945 apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod])
```
```   946 apply auto
```
```   947 done
```
```   948
```
```   949 lemma zmod_zminus_zminus [simp]: "(\$-a) zmod (\$-b) = \$- (a zmod b)"
```
```   950 apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_zminus_zminus_raw)
```
```   951 apply auto
```
```   952 done
```
```   953
```
```   954
```
```   955 subsection{* division of a number by itself *}
```
```   956
```
```   957 lemma self_quotient_aux1: "[| #0 \$< a; a = r \$+ a\$*q; r \$< a |] ==> #1 \$<= q"
```
```   958 apply (subgoal_tac "#0 \$< a\$*q")
```
```   959 apply (cut_tac w = "#0" and z = "q" in add1_zle_iff)
```
```   960 apply (simp add: int_0_less_mult_iff)
```
```   961 apply (blast dest: zless_trans)
```
```   962 (*linear arithmetic...*)
```
```   963 apply (drule_tac t = "%x. x \$- r" in subst_context)
```
```   964 apply (drule sym)
```
```   965 apply (simp add: zcompare_rls)
```
```   966 done
```
```   967
```
```   968 lemma self_quotient_aux2: "[| #0 \$< a; a = r \$+ a\$*q; #0 \$<= r |] ==> q \$<= #1"
```
```   969 apply (subgoal_tac "#0 \$<= a\$* (#1\$-q)")
```
```   970  apply (simp add: int_0_le_mult_iff zcompare_rls)
```
```   971  apply (blast dest: zle_zless_trans)
```
```   972 apply (simp add: zdiff_zmult_distrib2)
```
```   973 apply (drule_tac t = "%x. x \$- a \$* q" in subst_context)
```
```   974 apply (simp add: zcompare_rls)
```
```   975 done
```
```   976
```
```   977 lemma self_quotient:
```
```   978      "[| quorem(<a,a>,<q,r>);  a \<in> int;  q \<in> int;  a \<noteq> #0|] ==> q = #1"
```
```   979 apply (simp add: split_ifs quorem_def neq_iff_zless)
```
```   980 apply (rule zle_anti_sym)
```
```   981 apply safe
```
```   982 apply auto
```
```   983 prefer 4 apply (blast dest: zless_trans)
```
```   984 apply (blast dest: zless_trans)
```
```   985 apply (rule_tac [3] a = "\$-a" and r = "\$-r" in self_quotient_aux1)
```
```   986 apply (rule_tac a = "\$-a" and r = "\$-r" in self_quotient_aux2)
```
```   987 apply (rule_tac [6] zminus_equation [THEN iffD1])
```
```   988 apply (rule_tac [2] zminus_equation [THEN iffD1])
```
```   989 apply (force intro: self_quotient_aux1 self_quotient_aux2
```
```   990   simp add: zadd_commute zmult_zminus)+
```
```   991 done
```
```   992
```
```   993 lemma self_remainder:
```
```   994      "[|quorem(<a,a>,<q,r>); a \<in> int; q \<in> int; r \<in> int; a \<noteq> #0|] ==> r = #0"
```
```   995 apply (frule self_quotient)
```
```   996 apply (auto simp add: quorem_def)
```
```   997 done
```
```   998
```
```   999 lemma zdiv_self_raw: "[|a \<noteq> #0; a \<in> int|] ==> a zdiv a = #1"
```
```  1000 apply (blast intro: quorem_div_mod [THEN self_quotient])
```
```  1001 done
```
```  1002
```
```  1003 lemma zdiv_self [simp]: "intify(a) \<noteq> #0 ==> a zdiv a = #1"
```
```  1004 apply (drule zdiv_self_raw)
```
```  1005 apply auto
```
```  1006 done
```
```  1007
```
```  1008 (*Here we have 0 zmod 0 = 0, also assumed by Knuth (who puts m zmod 0 = 0) *)
```
```  1009 lemma zmod_self_raw: "a \<in> int ==> a zmod a = #0"
```
```  1010 apply (case_tac "a = #0")
```
```  1011  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1012 apply (blast intro: quorem_div_mod [THEN self_remainder])
```
```  1013 done
```
```  1014
```
```  1015 lemma zmod_self [simp]: "a zmod a = #0"
```
```  1016 apply (cut_tac a = "intify (a)" in zmod_self_raw)
```
```  1017 apply auto
```
```  1018 done
```
```  1019
```
```  1020
```
```  1021 subsection{* Computation of division and remainder *}
```
```  1022
```
```  1023 lemma zdiv_zero [simp]: "#0 zdiv b = #0"
```
```  1024 apply (simp (no_asm) add: zdiv_def divAlg_def)
```
```  1025 done
```
```  1026
```
```  1027 lemma zdiv_eq_minus1: "#0 \$< b ==> #-1 zdiv b = #-1"
```
```  1028 apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
```
```  1029 done
```
```  1030
```
```  1031 lemma zmod_zero [simp]: "#0 zmod b = #0"
```
```  1032 apply (simp (no_asm) add: zmod_def divAlg_def)
```
```  1033 done
```
```  1034
```
```  1035 lemma zdiv_minus1: "#0 \$< b ==> #-1 zdiv b = #-1"
```
```  1036 apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
```
```  1037 done
```
```  1038
```
```  1039 lemma zmod_minus1: "#0 \$< b ==> #-1 zmod b = b \$- #1"
```
```  1040 apply (simp (no_asm_simp) add: zmod_def divAlg_def)
```
```  1041 done
```
```  1042
```
```  1043 (** a positive, b positive **)
```
```  1044
```
```  1045 lemma zdiv_pos_pos: "[| #0 \$< a;  #0 \$<= b |]
```
```  1046       ==> a zdiv b = fst (posDivAlg(<intify(a), intify(b)>))"
```
```  1047 apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
```
```  1048 apply (auto simp add: zle_def)
```
```  1049 done
```
```  1050
```
```  1051 lemma zmod_pos_pos:
```
```  1052      "[| #0 \$< a;  #0 \$<= b |]
```
```  1053       ==> a zmod b = snd (posDivAlg(<intify(a), intify(b)>))"
```
```  1054 apply (simp (no_asm_simp) add: zmod_def divAlg_def)
```
```  1055 apply (auto simp add: zle_def)
```
```  1056 done
```
```  1057
```
```  1058 (** a negative, b positive **)
```
```  1059
```
```  1060 lemma zdiv_neg_pos:
```
```  1061      "[| a \$< #0;  #0 \$< b |]
```
```  1062       ==> a zdiv b = fst (negDivAlg(<intify(a), intify(b)>))"
```
```  1063 apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
```
```  1064 apply (blast dest: zle_zless_trans)
```
```  1065 done
```
```  1066
```
```  1067 lemma zmod_neg_pos:
```
```  1068      "[| a \$< #0;  #0 \$< b |]
```
```  1069       ==> a zmod b = snd (negDivAlg(<intify(a), intify(b)>))"
```
```  1070 apply (simp (no_asm_simp) add: zmod_def divAlg_def)
```
```  1071 apply (blast dest: zle_zless_trans)
```
```  1072 done
```
```  1073
```
```  1074 (** a positive, b negative **)
```
```  1075
```
```  1076 lemma zdiv_pos_neg:
```
```  1077      "[| #0 \$< a;  b \$< #0 |]
```
```  1078       ==> a zdiv b = fst (negateSnd(negDivAlg (<\$-a, \$-b>)))"
```
```  1079 apply (simp (no_asm_simp) add: zdiv_def divAlg_def intify_eq_0_iff_zle)
```
```  1080 apply auto
```
```  1081 apply (blast dest: zle_zless_trans)+
```
```  1082 apply (blast dest: zless_trans)
```
```  1083 apply (blast intro: zless_imp_zle)
```
```  1084 done
```
```  1085
```
```  1086 lemma zmod_pos_neg:
```
```  1087      "[| #0 \$< a;  b \$< #0 |]
```
```  1088       ==> a zmod b = snd (negateSnd(negDivAlg (<\$-a, \$-b>)))"
```
```  1089 apply (simp (no_asm_simp) add: zmod_def divAlg_def intify_eq_0_iff_zle)
```
```  1090 apply auto
```
```  1091 apply (blast dest: zle_zless_trans)+
```
```  1092 apply (blast dest: zless_trans)
```
```  1093 apply (blast intro: zless_imp_zle)
```
```  1094 done
```
```  1095
```
```  1096 (** a negative, b negative **)
```
```  1097
```
```  1098 lemma zdiv_neg_neg:
```
```  1099      "[| a \$< #0;  b \$<= #0 |]
```
```  1100       ==> a zdiv b = fst (negateSnd(posDivAlg(<\$-a, \$-b>)))"
```
```  1101 apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
```
```  1102 apply auto
```
```  1103 apply (blast dest!: zle_zless_trans)+
```
```  1104 done
```
```  1105
```
```  1106 lemma zmod_neg_neg:
```
```  1107      "[| a \$< #0;  b \$<= #0 |]
```
```  1108       ==> a zmod b = snd (negateSnd(posDivAlg(<\$-a, \$-b>)))"
```
```  1109 apply (simp (no_asm_simp) add: zmod_def divAlg_def)
```
```  1110 apply auto
```
```  1111 apply (blast dest!: zle_zless_trans)+
```
```  1112 done
```
```  1113
```
```  1114 declare zdiv_pos_pos [of "integ_of (v)" "integ_of (w)", simp] for v w
```
```  1115 declare zdiv_neg_pos [of "integ_of (v)" "integ_of (w)", simp] for v w
```
```  1116 declare zdiv_pos_neg [of "integ_of (v)" "integ_of (w)", simp] for v w
```
```  1117 declare zdiv_neg_neg [of "integ_of (v)" "integ_of (w)", simp] for v w
```
```  1118 declare zmod_pos_pos [of "integ_of (v)" "integ_of (w)", simp] for v w
```
```  1119 declare zmod_neg_pos [of "integ_of (v)" "integ_of (w)", simp] for v w
```
```  1120 declare zmod_pos_neg [of "integ_of (v)" "integ_of (w)", simp] for v w
```
```  1121 declare zmod_neg_neg [of "integ_of (v)" "integ_of (w)", simp] for v w
```
```  1122 declare posDivAlg_eqn [of concl: "integ_of (v)" "integ_of (w)", simp] for v w
```
```  1123 declare negDivAlg_eqn [of concl: "integ_of (v)" "integ_of (w)", simp] for v w
```
```  1124
```
```  1125
```
```  1126 (** Special-case simplification **)
```
```  1127
```
```  1128 lemma zmod_1 [simp]: "a zmod #1 = #0"
```
```  1129 apply (cut_tac a = "a" and b = "#1" in pos_mod_sign)
```
```  1130 apply (cut_tac [2] a = "a" and b = "#1" in pos_mod_bound)
```
```  1131 apply auto
```
```  1132 (*arithmetic*)
```
```  1133 apply (drule add1_zle_iff [THEN iffD2])
```
```  1134 apply (rule zle_anti_sym)
```
```  1135 apply auto
```
```  1136 done
```
```  1137
```
```  1138 lemma zdiv_1 [simp]: "a zdiv #1 = intify(a)"
```
```  1139 apply (cut_tac a = "a" and b = "#1" in zmod_zdiv_equality)
```
```  1140 apply auto
```
```  1141 done
```
```  1142
```
```  1143 lemma zmod_minus1_right [simp]: "a zmod #-1 = #0"
```
```  1144 apply (cut_tac a = "a" and b = "#-1" in neg_mod_sign)
```
```  1145 apply (cut_tac [2] a = "a" and b = "#-1" in neg_mod_bound)
```
```  1146 apply auto
```
```  1147 (*arithmetic*)
```
```  1148 apply (drule add1_zle_iff [THEN iffD2])
```
```  1149 apply (rule zle_anti_sym)
```
```  1150 apply auto
```
```  1151 done
```
```  1152
```
```  1153 lemma zdiv_minus1_right_raw: "a \<in> int ==> a zdiv #-1 = \$-a"
```
```  1154 apply (cut_tac a = "a" and b = "#-1" in zmod_zdiv_equality)
```
```  1155 apply auto
```
```  1156 apply (rule equation_zminus [THEN iffD2])
```
```  1157 apply auto
```
```  1158 done
```
```  1159
```
```  1160 lemma zdiv_minus1_right: "a zdiv #-1 = \$-a"
```
```  1161 apply (cut_tac a = "intify (a)" in zdiv_minus1_right_raw)
```
```  1162 apply auto
```
```  1163 done
```
```  1164 declare zdiv_minus1_right [simp]
```
```  1165
```
```  1166
```
```  1167 subsection{* Monotonicity in the first argument (divisor) *}
```
```  1168
```
```  1169 lemma zdiv_mono1: "[| a \$<= a';  #0 \$< b |] ==> a zdiv b \$<= a' zdiv b"
```
```  1170 apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
```
```  1171 apply (cut_tac a = "a'" and b = "b" in zmod_zdiv_equality)
```
```  1172 apply (rule unique_quotient_lemma)
```
```  1173 apply (erule subst)
```
```  1174 apply (erule subst)
```
```  1175 apply (simp_all (no_asm_simp) add: pos_mod_sign pos_mod_bound)
```
```  1176 done
```
```  1177
```
```  1178 lemma zdiv_mono1_neg: "[| a \$<= a';  b \$< #0 |] ==> a' zdiv b \$<= a zdiv b"
```
```  1179 apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
```
```  1180 apply (cut_tac a = "a'" and b = "b" in zmod_zdiv_equality)
```
```  1181 apply (rule unique_quotient_lemma_neg)
```
```  1182 apply (erule subst)
```
```  1183 apply (erule subst)
```
```  1184 apply (simp_all (no_asm_simp) add: neg_mod_sign neg_mod_bound)
```
```  1185 done
```
```  1186
```
```  1187
```
```  1188 subsection{* Monotonicity in the second argument (dividend) *}
```
```  1189
```
```  1190 lemma q_pos_lemma:
```
```  1191      "[| #0 \$<= b'\$*q' \$+ r'; r' \$< b';  #0 \$< b' |] ==> #0 \$<= q'"
```
```  1192 apply (subgoal_tac "#0 \$< b'\$* (q' \$+ #1)")
```
```  1193  apply (simp add: int_0_less_mult_iff)
```
```  1194  apply (blast dest: zless_trans intro: zless_add1_iff_zle [THEN iffD1])
```
```  1195 apply (simp add: zadd_zmult_distrib2)
```
```  1196 apply (erule zle_zless_trans)
```
```  1197 apply (erule zadd_zless_mono2)
```
```  1198 done
```
```  1199
```
```  1200 lemma zdiv_mono2_lemma:
```
```  1201      "[| b\$*q \$+ r = b'\$*q' \$+ r';  #0 \$<= b'\$*q' \$+ r';
```
```  1202          r' \$< b';  #0 \$<= r;  #0 \$< b';  b' \$<= b |]
```
```  1203       ==> q \$<= q'"
```
```  1204 apply (frule q_pos_lemma, assumption+)
```
```  1205 apply (subgoal_tac "b\$*q \$< b\$* (q' \$+ #1)")
```
```  1206  apply (simp add: zmult_zless_cancel1)
```
```  1207  apply (force dest: zless_add1_iff_zle [THEN iffD1] zless_trans zless_zle_trans)
```
```  1208 apply (subgoal_tac "b\$*q = r' \$- r \$+ b'\$*q'")
```
```  1209  prefer 2 apply (simp add: zcompare_rls)
```
```  1210 apply (simp (no_asm_simp) add: zadd_zmult_distrib2)
```
```  1211 apply (subst zadd_commute [of "b \$\<times> q'"], rule zadd_zless_mono)
```
```  1212  prefer 2 apply (blast intro: zmult_zle_mono1)
```
```  1213 apply (subgoal_tac "r' \$+ #0 \$< b \$+ r")
```
```  1214  apply (simp add: zcompare_rls)
```
```  1215 apply (rule zadd_zless_mono)
```
```  1216  apply auto
```
```  1217 apply (blast dest: zless_zle_trans)
```
```  1218 done
```
```  1219
```
```  1220
```
```  1221 lemma zdiv_mono2_raw:
```
```  1222      "[| #0 \$<= a;  #0 \$< b';  b' \$<= b;  a \<in> int |]
```
```  1223       ==> a zdiv b \$<= a zdiv b'"
```
```  1224 apply (subgoal_tac "#0 \$< b")
```
```  1225  prefer 2 apply (blast dest: zless_zle_trans)
```
```  1226 apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
```
```  1227 apply (cut_tac a = "a" and b = "b'" in zmod_zdiv_equality)
```
```  1228 apply (rule zdiv_mono2_lemma)
```
```  1229 apply (erule subst)
```
```  1230 apply (erule subst)
```
```  1231 apply (simp_all add: pos_mod_sign pos_mod_bound)
```
```  1232 done
```
```  1233
```
```  1234 lemma zdiv_mono2:
```
```  1235      "[| #0 \$<= a;  #0 \$< b';  b' \$<= b |]
```
```  1236       ==> a zdiv b \$<= a zdiv b'"
```
```  1237 apply (cut_tac a = "intify (a)" in zdiv_mono2_raw)
```
```  1238 apply auto
```
```  1239 done
```
```  1240
```
```  1241 lemma q_neg_lemma:
```
```  1242      "[| b'\$*q' \$+ r' \$< #0;  #0 \$<= r';  #0 \$< b' |] ==> q' \$< #0"
```
```  1243 apply (subgoal_tac "b'\$*q' \$< #0")
```
```  1244  prefer 2 apply (force intro: zle_zless_trans)
```
```  1245 apply (simp add: zmult_less_0_iff)
```
```  1246 apply (blast dest: zless_trans)
```
```  1247 done
```
```  1248
```
```  1249
```
```  1250
```
```  1251 lemma zdiv_mono2_neg_lemma:
```
```  1252      "[| b\$*q \$+ r = b'\$*q' \$+ r';  b'\$*q' \$+ r' \$< #0;
```
```  1253          r \$< b;  #0 \$<= r';  #0 \$< b';  b' \$<= b |]
```
```  1254       ==> q' \$<= q"
```
```  1255 apply (subgoal_tac "#0 \$< b")
```
```  1256  prefer 2 apply (blast dest: zless_zle_trans)
```
```  1257 apply (frule q_neg_lemma, assumption+)
```
```  1258 apply (subgoal_tac "b\$*q' \$< b\$* (q \$+ #1)")
```
```  1259  apply (simp add: zmult_zless_cancel1)
```
```  1260  apply (blast dest: zless_trans zless_add1_iff_zle [THEN iffD1])
```
```  1261 apply (simp (no_asm_simp) add: zadd_zmult_distrib2)
```
```  1262 apply (subgoal_tac "b\$*q' \$<= b'\$*q'")
```
```  1263  prefer 2
```
```  1264  apply (simp add: zmult_zle_cancel2)
```
```  1265  apply (blast dest: zless_trans)
```
```  1266 apply (subgoal_tac "b'\$*q' \$+ r \$< b \$+ (b\$*q \$+ r)")
```
```  1267  prefer 2
```
```  1268  apply (erule ssubst)
```
```  1269  apply simp
```
```  1270  apply (drule_tac w' = "r" and z' = "#0" in zadd_zless_mono)
```
```  1271   apply (assumption)
```
```  1272  apply simp
```
```  1273 apply (simp (no_asm_use) add: zadd_commute)
```
```  1274 apply (rule zle_zless_trans)
```
```  1275  prefer 2 apply (assumption)
```
```  1276 apply (simp (no_asm_simp) add: zmult_zle_cancel2)
```
```  1277 apply (blast dest: zless_trans)
```
```  1278 done
```
```  1279
```
```  1280 lemma zdiv_mono2_neg_raw:
```
```  1281      "[| a \$< #0;  #0 \$< b';  b' \$<= b;  a \<in> int |]
```
```  1282       ==> a zdiv b' \$<= a zdiv b"
```
```  1283 apply (subgoal_tac "#0 \$< b")
```
```  1284  prefer 2 apply (blast dest: zless_zle_trans)
```
```  1285 apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
```
```  1286 apply (cut_tac a = "a" and b = "b'" in zmod_zdiv_equality)
```
```  1287 apply (rule zdiv_mono2_neg_lemma)
```
```  1288 apply (erule subst)
```
```  1289 apply (erule subst)
```
```  1290 apply (simp_all add: pos_mod_sign pos_mod_bound)
```
```  1291 done
```
```  1292
```
```  1293 lemma zdiv_mono2_neg: "[| a \$< #0;  #0 \$< b';  b' \$<= b |]
```
```  1294       ==> a zdiv b' \$<= a zdiv b"
```
```  1295 apply (cut_tac a = "intify (a)" in zdiv_mono2_neg_raw)
```
```  1296 apply auto
```
```  1297 done
```
```  1298
```
```  1299
```
```  1300
```
```  1301 subsection{* More algebraic laws for zdiv and zmod *}
```
```  1302
```
```  1303 (** proving (a*b) zdiv c = a \$* (b zdiv c) \$+ a * (b zmod c) **)
```
```  1304
```
```  1305 lemma zmult1_lemma:
```
```  1306      "[| quorem(<b,c>, <q,r>);  c \<in> int;  c \<noteq> #0 |]
```
```  1307       ==> quorem (<a\$*b, c>, <a\$*q \$+ (a\$*r) zdiv c, (a\$*r) zmod c>)"
```
```  1308 apply (auto simp add: split_ifs quorem_def neq_iff_zless zadd_zmult_distrib2
```
```  1309                       pos_mod_sign pos_mod_bound neg_mod_sign neg_mod_bound)
```
```  1310 apply (auto intro: raw_zmod_zdiv_equality)
```
```  1311 done
```
```  1312
```
```  1313 lemma zdiv_zmult1_eq_raw:
```
```  1314      "[|b \<in> int;  c \<in> int|]
```
```  1315       ==> (a\$*b) zdiv c = a\$*(b zdiv c) \$+ a\$*(b zmod c) zdiv c"
```
```  1316 apply (case_tac "c = #0")
```
```  1317  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1318 apply (rule quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])
```
```  1319 apply auto
```
```  1320 done
```
```  1321
```
```  1322 lemma zdiv_zmult1_eq: "(a\$*b) zdiv c = a\$*(b zdiv c) \$+ a\$*(b zmod c) zdiv c"
```
```  1323 apply (cut_tac b = "intify (b)" and c = "intify (c)" in zdiv_zmult1_eq_raw)
```
```  1324 apply auto
```
```  1325 done
```
```  1326
```
```  1327 lemma zmod_zmult1_eq_raw:
```
```  1328      "[|b \<in> int;  c \<in> int|] ==> (a\$*b) zmod c = a\$*(b zmod c) zmod c"
```
```  1329 apply (case_tac "c = #0")
```
```  1330  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1331 apply (rule quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])
```
```  1332 apply auto
```
```  1333 done
```
```  1334
```
```  1335 lemma zmod_zmult1_eq: "(a\$*b) zmod c = a\$*(b zmod c) zmod c"
```
```  1336 apply (cut_tac b = "intify (b)" and c = "intify (c)" in zmod_zmult1_eq_raw)
```
```  1337 apply auto
```
```  1338 done
```
```  1339
```
```  1340 lemma zmod_zmult1_eq': "(a\$*b) zmod c = ((a zmod c) \$* b) zmod c"
```
```  1341 apply (rule trans)
```
```  1342 apply (rule_tac b = " (b \$* a) zmod c" in trans)
```
```  1343 apply (rule_tac [2] zmod_zmult1_eq)
```
```  1344 apply (simp_all (no_asm) add: zmult_commute)
```
```  1345 done
```
```  1346
```
```  1347 lemma zmod_zmult_distrib: "(a\$*b) zmod c = ((a zmod c) \$* (b zmod c)) zmod c"
```
```  1348 apply (rule zmod_zmult1_eq' [THEN trans])
```
```  1349 apply (rule zmod_zmult1_eq)
```
```  1350 done
```
```  1351
```
```  1352 lemma zdiv_zmult_self1 [simp]: "intify(b) \<noteq> #0 ==> (a\$*b) zdiv b = intify(a)"
```
```  1353 apply (simp (no_asm_simp) add: zdiv_zmult1_eq)
```
```  1354 done
```
```  1355
```
```  1356 lemma zdiv_zmult_self2 [simp]: "intify(b) \<noteq> #0 ==> (b\$*a) zdiv b = intify(a)"
```
```  1357 apply (subst zmult_commute , erule zdiv_zmult_self1)
```
```  1358 done
```
```  1359
```
```  1360 lemma zmod_zmult_self1 [simp]: "(a\$*b) zmod b = #0"
```
```  1361 apply (simp (no_asm) add: zmod_zmult1_eq)
```
```  1362 done
```
```  1363
```
```  1364 lemma zmod_zmult_self2 [simp]: "(b\$*a) zmod b = #0"
```
```  1365 apply (simp (no_asm) add: zmult_commute zmod_zmult1_eq)
```
```  1366 done
```
```  1367
```
```  1368
```
```  1369 (** proving (a\$+b) zdiv c =
```
```  1370             a zdiv c \$+ b zdiv c \$+ ((a zmod c \$+ b zmod c) zdiv c) **)
```
```  1371
```
```  1372 lemma zadd1_lemma:
```
```  1373      "[| quorem(<a,c>, <aq,ar>);  quorem(<b,c>, <bq,br>);
```
```  1374          c \<in> int;  c \<noteq> #0 |]
```
```  1375       ==> quorem (<a\$+b, c>, <aq \$+ bq \$+ (ar\$+br) zdiv c, (ar\$+br) zmod c>)"
```
```  1376 apply (auto simp add: split_ifs quorem_def neq_iff_zless zadd_zmult_distrib2
```
```  1377                       pos_mod_sign pos_mod_bound neg_mod_sign neg_mod_bound)
```
```  1378 apply (auto intro: raw_zmod_zdiv_equality)
```
```  1379 done
```
```  1380
```
```  1381 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```  1382 lemma zdiv_zadd1_eq_raw:
```
```  1383      "[|a \<in> int; b \<in> int; c \<in> int|] ==>
```
```  1384       (a\$+b) zdiv c = a zdiv c \$+ b zdiv c \$+ ((a zmod c \$+ b zmod c) zdiv c)"
```
```  1385 apply (case_tac "c = #0")
```
```  1386  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1387 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod,
```
```  1388                                  THEN quorem_div])
```
```  1389 done
```
```  1390
```
```  1391 lemma zdiv_zadd1_eq:
```
```  1392      "(a\$+b) zdiv c = a zdiv c \$+ b zdiv c \$+ ((a zmod c \$+ b zmod c) zdiv c)"
```
```  1393 apply (cut_tac a = "intify (a)" and b = "intify (b)" and c = "intify (c)"
```
```  1394        in zdiv_zadd1_eq_raw)
```
```  1395 apply auto
```
```  1396 done
```
```  1397
```
```  1398 lemma zmod_zadd1_eq_raw:
```
```  1399      "[|a \<in> int; b \<in> int; c \<in> int|]
```
```  1400       ==> (a\$+b) zmod c = (a zmod c \$+ b zmod c) zmod c"
```
```  1401 apply (case_tac "c = #0")
```
```  1402  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1403 apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod,
```
```  1404                                  THEN quorem_mod])
```
```  1405 done
```
```  1406
```
```  1407 lemma zmod_zadd1_eq: "(a\$+b) zmod c = (a zmod c \$+ b zmod c) zmod c"
```
```  1408 apply (cut_tac a = "intify (a)" and b = "intify (b)" and c = "intify (c)"
```
```  1409        in zmod_zadd1_eq_raw)
```
```  1410 apply auto
```
```  1411 done
```
```  1412
```
```  1413 lemma zmod_div_trivial_raw:
```
```  1414      "[|a \<in> int; b \<in> int|] ==> (a zmod b) zdiv b = #0"
```
```  1415 apply (case_tac "b = #0")
```
```  1416  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1417 apply (auto simp add: neq_iff_zless pos_mod_sign pos_mod_bound
```
```  1418          zdiv_pos_pos_trivial neg_mod_sign neg_mod_bound zdiv_neg_neg_trivial)
```
```  1419 done
```
```  1420
```
```  1421 lemma zmod_div_trivial [simp]: "(a zmod b) zdiv b = #0"
```
```  1422 apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_div_trivial_raw)
```
```  1423 apply auto
```
```  1424 done
```
```  1425
```
```  1426 lemma zmod_mod_trivial_raw:
```
```  1427      "[|a \<in> int; b \<in> int|] ==> (a zmod b) zmod b = a zmod b"
```
```  1428 apply (case_tac "b = #0")
```
```  1429  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1430 apply (auto simp add: neq_iff_zless pos_mod_sign pos_mod_bound
```
```  1431        zmod_pos_pos_trivial neg_mod_sign neg_mod_bound zmod_neg_neg_trivial)
```
```  1432 done
```
```  1433
```
```  1434 lemma zmod_mod_trivial [simp]: "(a zmod b) zmod b = a zmod b"
```
```  1435 apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_mod_trivial_raw)
```
```  1436 apply auto
```
```  1437 done
```
```  1438
```
```  1439 lemma zmod_zadd_left_eq: "(a\$+b) zmod c = ((a zmod c) \$+ b) zmod c"
```
```  1440 apply (rule trans [symmetric])
```
```  1441 apply (rule zmod_zadd1_eq)
```
```  1442 apply (simp (no_asm))
```
```  1443 apply (rule zmod_zadd1_eq [symmetric])
```
```  1444 done
```
```  1445
```
```  1446 lemma zmod_zadd_right_eq: "(a\$+b) zmod c = (a \$+ (b zmod c)) zmod c"
```
```  1447 apply (rule trans [symmetric])
```
```  1448 apply (rule zmod_zadd1_eq)
```
```  1449 apply (simp (no_asm))
```
```  1450 apply (rule zmod_zadd1_eq [symmetric])
```
```  1451 done
```
```  1452
```
```  1453
```
```  1454 lemma zdiv_zadd_self1 [simp]:
```
```  1455      "intify(a) \<noteq> #0 ==> (a\$+b) zdiv a = b zdiv a \$+ #1"
```
```  1456 by (simp (no_asm_simp) add: zdiv_zadd1_eq)
```
```  1457
```
```  1458 lemma zdiv_zadd_self2 [simp]:
```
```  1459      "intify(a) \<noteq> #0 ==> (b\$+a) zdiv a = b zdiv a \$+ #1"
```
```  1460 by (simp (no_asm_simp) add: zdiv_zadd1_eq)
```
```  1461
```
```  1462 lemma zmod_zadd_self1 [simp]: "(a\$+b) zmod a = b zmod a"
```
```  1463 apply (case_tac "a = #0")
```
```  1464  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1465 apply (simp (no_asm_simp) add: zmod_zadd1_eq)
```
```  1466 done
```
```  1467
```
```  1468 lemma zmod_zadd_self2 [simp]: "(b\$+a) zmod a = b zmod a"
```
```  1469 apply (case_tac "a = #0")
```
```  1470  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1471 apply (simp (no_asm_simp) add: zmod_zadd1_eq)
```
```  1472 done
```
```  1473
```
```  1474
```
```  1475 subsection{* proving  a zdiv (b*c) = (a zdiv b) zdiv c *}
```
```  1476
```
```  1477 (*The condition c>0 seems necessary.  Consider that 7 zdiv ~6 = ~2 but
```
```  1478   7 zdiv 2 zdiv ~3 = 3 zdiv ~3 = ~1.  The subcase (a zdiv b) zmod c = 0 seems
```
```  1479   to cause particular problems.*)
```
```  1480
```
```  1481 (** first, four lemmas to bound the remainder for the cases b<0 and b>0 **)
```
```  1482
```
```  1483 lemma zdiv_zmult2_aux1:
```
```  1484      "[| #0 \$< c;  b \$< r;  r \$<= #0 |] ==> b\$*c \$< b\$*(q zmod c) \$+ r"
```
```  1485 apply (subgoal_tac "b \$* (c \$- q zmod c) \$< r \$* #1")
```
```  1486 apply (simp add: zdiff_zmult_distrib2 zadd_commute zcompare_rls)
```
```  1487 apply (rule zle_zless_trans)
```
```  1488 apply (erule_tac [2] zmult_zless_mono1)
```
```  1489 apply (rule zmult_zle_mono2_neg)
```
```  1490 apply (auto simp add: zcompare_rls zadd_commute add1_zle_iff pos_mod_bound)
```
```  1491 apply (blast intro: zless_imp_zle dest: zless_zle_trans)
```
```  1492 done
```
```  1493
```
```  1494 lemma zdiv_zmult2_aux2:
```
```  1495      "[| #0 \$< c;   b \$< r;  r \$<= #0 |] ==> b \$* (q zmod c) \$+ r \$<= #0"
```
```  1496 apply (subgoal_tac "b \$* (q zmod c) \$<= #0")
```
```  1497  prefer 2
```
```  1498  apply (simp add: zmult_le_0_iff pos_mod_sign)
```
```  1499  apply (blast intro: zless_imp_zle dest: zless_zle_trans)
```
```  1500 (*arithmetic*)
```
```  1501 apply (drule zadd_zle_mono)
```
```  1502 apply assumption
```
```  1503 apply (simp add: zadd_commute)
```
```  1504 done
```
```  1505
```
```  1506 lemma zdiv_zmult2_aux3:
```
```  1507      "[| #0 \$< c;  #0 \$<= r;  r \$< b |] ==> #0 \$<= b \$* (q zmod c) \$+ r"
```
```  1508 apply (subgoal_tac "#0 \$<= b \$* (q zmod c)")
```
```  1509  prefer 2
```
```  1510  apply (simp add: int_0_le_mult_iff pos_mod_sign)
```
```  1511  apply (blast intro: zless_imp_zle dest: zle_zless_trans)
```
```  1512 (*arithmetic*)
```
```  1513 apply (drule zadd_zle_mono)
```
```  1514 apply assumption
```
```  1515 apply (simp add: zadd_commute)
```
```  1516 done
```
```  1517
```
```  1518 lemma zdiv_zmult2_aux4:
```
```  1519      "[| #0 \$< c; #0 \$<= r; r \$< b |] ==> b \$* (q zmod c) \$+ r \$< b \$* c"
```
```  1520 apply (subgoal_tac "r \$* #1 \$< b \$* (c \$- q zmod c)")
```
```  1521 apply (simp add: zdiff_zmult_distrib2 zadd_commute zcompare_rls)
```
```  1522 apply (rule zless_zle_trans)
```
```  1523 apply (erule zmult_zless_mono1)
```
```  1524 apply (rule_tac [2] zmult_zle_mono2)
```
```  1525 apply (auto simp add: zcompare_rls zadd_commute add1_zle_iff pos_mod_bound)
```
```  1526 apply (blast intro: zless_imp_zle dest: zle_zless_trans)
```
```  1527 done
```
```  1528
```
```  1529 lemma zdiv_zmult2_lemma:
```
```  1530      "[| quorem (<a,b>, <q,r>);  a \<in> int;  b \<in> int;  b \<noteq> #0;  #0 \$< c |]
```
```  1531       ==> quorem (<a,b\$*c>, <q zdiv c, b\$*(q zmod c) \$+ r>)"
```
```  1532 apply (auto simp add: zmult_ac zmod_zdiv_equality [symmetric] quorem_def
```
```  1533                neq_iff_zless int_0_less_mult_iff
```
```  1534                zadd_zmult_distrib2 [symmetric] zdiv_zmult2_aux1 zdiv_zmult2_aux2
```
```  1535                zdiv_zmult2_aux3 zdiv_zmult2_aux4)
```
```  1536 apply (blast dest: zless_trans)+
```
```  1537 done
```
```  1538
```
```  1539 lemma zdiv_zmult2_eq_raw:
```
```  1540      "[|#0 \$< c;  a \<in> int;  b \<in> int|] ==> a zdiv (b\$*c) = (a zdiv b) zdiv c"
```
```  1541 apply (case_tac "b = #0")
```
```  1542  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1543 apply (rule quorem_div_mod [THEN zdiv_zmult2_lemma, THEN quorem_div])
```
```  1544 apply (auto simp add: intify_eq_0_iff_zle)
```
```  1545 apply (blast dest: zle_zless_trans)
```
```  1546 done
```
```  1547
```
```  1548 lemma zdiv_zmult2_eq: "#0 \$< c ==> a zdiv (b\$*c) = (a zdiv b) zdiv c"
```
```  1549 apply (cut_tac a = "intify (a)" and b = "intify (b)" in zdiv_zmult2_eq_raw)
```
```  1550 apply auto
```
```  1551 done
```
```  1552
```
```  1553 lemma zmod_zmult2_eq_raw:
```
```  1554      "[|#0 \$< c;  a \<in> int;  b \<in> int|]
```
```  1555       ==> a zmod (b\$*c) = b\$*(a zdiv b zmod c) \$+ a zmod b"
```
```  1556 apply (case_tac "b = #0")
```
```  1557  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1558 apply (rule quorem_div_mod [THEN zdiv_zmult2_lemma, THEN quorem_mod])
```
```  1559 apply (auto simp add: intify_eq_0_iff_zle)
```
```  1560 apply (blast dest: zle_zless_trans)
```
```  1561 done
```
```  1562
```
```  1563 lemma zmod_zmult2_eq:
```
```  1564      "#0 \$< c ==> a zmod (b\$*c) = b\$*(a zdiv b zmod c) \$+ a zmod b"
```
```  1565 apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_zmult2_eq_raw)
```
```  1566 apply auto
```
```  1567 done
```
```  1568
```
```  1569 subsection{* Cancellation of common factors in "zdiv" *}
```
```  1570
```
```  1571 lemma zdiv_zmult_zmult1_aux1:
```
```  1572      "[| #0 \$< b;  intify(c) \<noteq> #0 |] ==> (c\$*a) zdiv (c\$*b) = a zdiv b"
```
```  1573 apply (subst zdiv_zmult2_eq)
```
```  1574 apply auto
```
```  1575 done
```
```  1576
```
```  1577 lemma zdiv_zmult_zmult1_aux2:
```
```  1578      "[| b \$< #0;  intify(c) \<noteq> #0 |] ==> (c\$*a) zdiv (c\$*b) = a zdiv b"
```
```  1579 apply (subgoal_tac " (c \$* (\$-a)) zdiv (c \$* (\$-b)) = (\$-a) zdiv (\$-b)")
```
```  1580 apply (rule_tac [2] zdiv_zmult_zmult1_aux1)
```
```  1581 apply auto
```
```  1582 done
```
```  1583
```
```  1584 lemma zdiv_zmult_zmult1_raw:
```
```  1585      "[|intify(c) \<noteq> #0; b \<in> int|] ==> (c\$*a) zdiv (c\$*b) = a zdiv b"
```
```  1586 apply (case_tac "b = #0")
```
```  1587  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1588 apply (auto simp add: neq_iff_zless [of b]
```
```  1589   zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
```
```  1590 done
```
```  1591
```
```  1592 lemma zdiv_zmult_zmult1: "intify(c) \<noteq> #0 ==> (c\$*a) zdiv (c\$*b) = a zdiv b"
```
```  1593 apply (cut_tac b = "intify (b)" in zdiv_zmult_zmult1_raw)
```
```  1594 apply auto
```
```  1595 done
```
```  1596
```
```  1597 lemma zdiv_zmult_zmult2: "intify(c) \<noteq> #0 ==> (a\$*c) zdiv (b\$*c) = a zdiv b"
```
```  1598 apply (drule zdiv_zmult_zmult1)
```
```  1599 apply (auto simp add: zmult_commute)
```
```  1600 done
```
```  1601
```
```  1602
```
```  1603 subsection{* Distribution of factors over "zmod" *}
```
```  1604
```
```  1605 lemma zmod_zmult_zmult1_aux1:
```
```  1606      "[| #0 \$< b;  intify(c) \<noteq> #0 |]
```
```  1607       ==> (c\$*a) zmod (c\$*b) = c \$* (a zmod b)"
```
```  1608 apply (subst zmod_zmult2_eq)
```
```  1609 apply auto
```
```  1610 done
```
```  1611
```
```  1612 lemma zmod_zmult_zmult1_aux2:
```
```  1613      "[| b \$< #0;  intify(c) \<noteq> #0 |]
```
```  1614       ==> (c\$*a) zmod (c\$*b) = c \$* (a zmod b)"
```
```  1615 apply (subgoal_tac " (c \$* (\$-a)) zmod (c \$* (\$-b)) = c \$* ((\$-a) zmod (\$-b))")
```
```  1616 apply (rule_tac [2] zmod_zmult_zmult1_aux1)
```
```  1617 apply auto
```
```  1618 done
```
```  1619
```
```  1620 lemma zmod_zmult_zmult1_raw:
```
```  1621      "[|b \<in> int; c \<in> int|] ==> (c\$*a) zmod (c\$*b) = c \$* (a zmod b)"
```
```  1622 apply (case_tac "b = #0")
```
```  1623  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1624 apply (case_tac "c = #0")
```
```  1625  apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
```
```  1626 apply (auto simp add: neq_iff_zless [of b]
```
```  1627   zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
```
```  1628 done
```
```  1629
```
```  1630 lemma zmod_zmult_zmult1: "(c\$*a) zmod (c\$*b) = c \$* (a zmod b)"
```
```  1631 apply (cut_tac b = "intify (b)" and c = "intify (c)" in zmod_zmult_zmult1_raw)
```
```  1632 apply auto
```
```  1633 done
```
```  1634
```
```  1635 lemma zmod_zmult_zmult2: "(a\$*c) zmod (b\$*c) = (a zmod b) \$* c"
```
```  1636 apply (cut_tac c = "c" in zmod_zmult_zmult1)
```
```  1637 apply (auto simp add: zmult_commute)
```
```  1638 done
```
```  1639
```
```  1640
```
```  1641 (** Quotients of signs **)
```
```  1642
```
```  1643 lemma zdiv_neg_pos_less0: "[| a \$< #0;  #0 \$< b |] ==> a zdiv b \$< #0"
```
```  1644 apply (subgoal_tac "a zdiv b \$<= #-1")
```
```  1645 apply (erule zle_zless_trans)
```
```  1646 apply (simp (no_asm))
```
```  1647 apply (rule zle_trans)
```
```  1648 apply (rule_tac a' = "#-1" in zdiv_mono1)
```
```  1649 apply (rule zless_add1_iff_zle [THEN iffD1])
```
```  1650 apply (simp (no_asm))
```
```  1651 apply (auto simp add: zdiv_minus1)
```
```  1652 done
```
```  1653
```
```  1654 lemma zdiv_nonneg_neg_le0: "[| #0 \$<= a;  b \$< #0 |] ==> a zdiv b \$<= #0"
```
```  1655 apply (drule zdiv_mono1_neg)
```
```  1656 apply auto
```
```  1657 done
```
```  1658
```
```  1659 lemma pos_imp_zdiv_nonneg_iff: "#0 \$< b ==> (#0 \$<= a zdiv b) <-> (#0 \$<= a)"
```
```  1660 apply auto
```
```  1661 apply (drule_tac [2] zdiv_mono1)
```
```  1662 apply (auto simp add: neq_iff_zless)
```
```  1663 apply (simp (no_asm_use) add: not_zless_iff_zle [THEN iff_sym])
```
```  1664 apply (blast intro: zdiv_neg_pos_less0)
```
```  1665 done
```
```  1666
```
```  1667 lemma neg_imp_zdiv_nonneg_iff: "b \$< #0 ==> (#0 \$<= a zdiv b) <-> (a \$<= #0)"
```
```  1668 apply (subst zdiv_zminus_zminus [symmetric])
```
```  1669 apply (rule iff_trans)
```
```  1670 apply (rule pos_imp_zdiv_nonneg_iff)
```
```  1671 apply auto
```
```  1672 done
```
```  1673
```
```  1674 (*But not (a zdiv b \$<= 0 iff a\$<=0); consider a=1, b=2 when a zdiv b = 0.*)
```
```  1675 lemma pos_imp_zdiv_neg_iff: "#0 \$< b ==> (a zdiv b \$< #0) <-> (a \$< #0)"
```
```  1676 apply (simp (no_asm_simp) add: not_zle_iff_zless [THEN iff_sym])
```
```  1677 apply (erule pos_imp_zdiv_nonneg_iff)
```
```  1678 done
```
```  1679
```
```  1680 (*Again the law fails for \$<=: consider a = -1, b = -2 when a zdiv b = 0*)
```
```  1681 lemma neg_imp_zdiv_neg_iff: "b \$< #0 ==> (a zdiv b \$< #0) <-> (#0 \$< a)"
```
```  1682 apply (simp (no_asm_simp) add: not_zle_iff_zless [THEN iff_sym])
```
```  1683 apply (erule neg_imp_zdiv_nonneg_iff)
```
```  1684 done
```
```  1685
```
```  1686 (*
```
```  1687  THESE REMAIN TO BE CONVERTED -- but aren't that useful!
```
```  1688
```
```  1689  subsection{* Speeding up the division algorithm with shifting *}
```
```  1690
```
```  1691  (** computing "zdiv" by shifting **)
```
```  1692
```
```  1693  lemma pos_zdiv_mult_2: "#0 \$<= a ==> (#1 \$+ #2\$*b) zdiv (#2\$*a) = b zdiv a"
```
```  1694  apply (case_tac "a = #0")
```
```  1695  apply (subgoal_tac "#1 \$<= a")
```
```  1696   apply (arith_tac 2)
```
```  1697  apply (subgoal_tac "#1 \$< a \$* #2")
```
```  1698   apply (arith_tac 2)
```
```  1699  apply (subgoal_tac "#2\$* (#1 \$+ b zmod a) \$<= #2\$*a")
```
```  1700   apply (rule_tac [2] zmult_zle_mono2)
```
```  1701  apply (auto simp add: zadd_commute zmult_commute add1_zle_iff pos_mod_bound)
```
```  1702  apply (subst zdiv_zadd1_eq)
```
```  1703  apply (simp (no_asm_simp) add: zdiv_zmult_zmult2 zmod_zmult_zmult2 zdiv_pos_pos_trivial)
```
```  1704  apply (subst zdiv_pos_pos_trivial)
```
```  1705  apply (simp (no_asm_simp) add: [zmod_pos_pos_trivial pos_mod_sign [THEN zadd_zle_mono1] RSN (2,zle_trans) ])
```
```  1706  apply (auto simp add: zmod_pos_pos_trivial)
```
```  1707  apply (subgoal_tac "#0 \$<= b zmod a")
```
```  1708   apply (asm_simp_tac (simpset () add: pos_mod_sign) 2)
```
```  1709  apply arith
```
```  1710  done
```
```  1711
```
```  1712
```
```  1713  lemma neg_zdiv_mult_2: "a \$<= #0 ==> (#1 \$+ #2\$*b) zdiv (#2\$*a) <-> (b\$+#1) zdiv a"
```
```  1714  apply (subgoal_tac " (#1 \$+ #2\$* (\$-b-#1)) zdiv (#2 \$* (\$-a)) <-> (\$-b-#1) zdiv (\$-a)")
```
```  1715  apply (rule_tac [2] pos_zdiv_mult_2)
```
```  1716  apply (auto simp add: zmult_zminus_right)
```
```  1717  apply (subgoal_tac " (#-1 - (#2 \$* b)) = - (#1 \$+ (#2 \$* b))")
```
```  1718  apply (Simp_tac 2)
```
```  1719  apply (asm_full_simp_tac (HOL_ss add: zdiv_zminus_zminus zdiff_def zminus_zadd_distrib [symmetric])
```
```  1720  done
```
```  1721
```
```  1722
```
```  1723  (*Not clear why this must be proved separately; probably integ_of causes
```
```  1724    simplification problems*)
```
```  1725  lemma lemma: "~ #0 \$<= x ==> x \$<= #0"
```
```  1726  apply auto
```
```  1727  done
```
```  1728
```
```  1729  lemma zdiv_integ_of_BIT: "integ_of (v BIT b) zdiv integ_of (w BIT False) =
```
```  1730            (if ~b | #0 \$<= integ_of w
```
```  1731             then integ_of v zdiv (integ_of w)
```
```  1732             else (integ_of v \$+ #1) zdiv (integ_of w))"
```
```  1733  apply (simp_tac (global_simpset_of Int.thy add: zadd_assoc integ_of_BIT)
```
```  1734  apply (simp (no_asm_simp) del: bin_arith_extra_simps@bin_rel_simps add: zdiv_zmult_zmult1 pos_zdiv_mult_2 lemma neg_zdiv_mult_2)
```
```  1735  done
```
```  1736
```
```  1737  declare zdiv_integ_of_BIT [simp]
```
```  1738
```
```  1739
```
```  1740  (** computing "zmod" by shifting (proofs resemble those for "zdiv") **)
```
```  1741
```
```  1742  lemma pos_zmod_mult_2: "#0 \$<= a ==> (#1 \$+ #2\$*b) zmod (#2\$*a) = #1 \$+ #2 \$* (b zmod a)"
```
```  1743  apply (case_tac "a = #0")
```
```  1744  apply (subgoal_tac "#1 \$<= a")
```
```  1745   apply (arith_tac 2)
```
```  1746  apply (subgoal_tac "#1 \$< a \$* #2")
```
```  1747   apply (arith_tac 2)
```
```  1748  apply (subgoal_tac "#2\$* (#1 \$+ b zmod a) \$<= #2\$*a")
```
```  1749   apply (rule_tac [2] zmult_zle_mono2)
```
```  1750  apply (auto simp add: zadd_commute zmult_commute add1_zle_iff pos_mod_bound)
```
```  1751  apply (subst zmod_zadd1_eq)
```
```  1752  apply (simp (no_asm_simp) add: zmod_zmult_zmult2 zmod_pos_pos_trivial)
```
```  1753  apply (rule zmod_pos_pos_trivial)
```
```  1754  apply (simp (no_asm_simp) # add: [zmod_pos_pos_trivial pos_mod_sign [THEN zadd_zle_mono1] RSN (2,zle_trans) ])
```
```  1755  apply (auto simp add: zmod_pos_pos_trivial)
```
```  1756  apply (subgoal_tac "#0 \$<= b zmod a")
```
```  1757   apply (asm_simp_tac (simpset () add: pos_mod_sign) 2)
```
```  1758  apply arith
```
```  1759  done
```
```  1760
```
```  1761
```
```  1762  lemma neg_zmod_mult_2: "a \$<= #0 ==> (#1 \$+ #2\$*b) zmod (#2\$*a) = #2 \$* ((b\$+#1) zmod a) - #1"
```
```  1763  apply (subgoal_tac " (#1 \$+ #2\$* (\$-b-#1)) zmod (#2\$* (\$-a)) = #1 \$+ #2\$* ((\$-b-#1) zmod (\$-a))")
```
```  1764  apply (rule_tac [2] pos_zmod_mult_2)
```
```  1765  apply (auto simp add: zmult_zminus_right)
```
```  1766  apply (subgoal_tac " (#-1 - (#2 \$* b)) = - (#1 \$+ (#2 \$* b))")
```
```  1767  apply (Simp_tac 2)
```
```  1768  apply (asm_full_simp_tac (HOL_ss add: zmod_zminus_zminus zdiff_def zminus_zadd_distrib [symmetric])
```
```  1769  apply (dtac (zminus_equation [THEN iffD1, symmetric])
```
```  1770  apply auto
```
```  1771  done
```
```  1772
```
```  1773  lemma zmod_integ_of_BIT: "integ_of (v BIT b) zmod integ_of (w BIT False) =
```
```  1774            (if b then
```
```  1775                  if #0 \$<= integ_of w
```
```  1776                  then #2 \$* (integ_of v zmod integ_of w) \$+ #1
```
```  1777                  else #2 \$* ((integ_of v \$+ #1) zmod integ_of w) - #1
```
```  1778             else #2 \$* (integ_of v zmod integ_of w))"
```
```  1779  apply (simp_tac (global_simpset_of Int.thy add: zadd_assoc integ_of_BIT)
```
```  1780  apply (simp (no_asm_simp) del: bin_arith_extra_simps@bin_rel_simps add: zmod_zmult_zmult1 pos_zmod_mult_2 lemma neg_zmod_mult_2)
```
```  1781  done
```
```  1782
```
```  1783  declare zmod_integ_of_BIT [simp]
```
```  1784 *)
```
```  1785
```
```  1786 end
```
```  1787
```