src/HOL/Divides.thy
 author ballarin Thu Dec 11 18:30:26 2008 +0100 (2008-12-11) changeset 29223 e09c53289830 parent 28823 dcbef866c9e2 child 29252 ea97aa6aeba2 permissions -rw-r--r--
Conversion of HOL-Main and ZF to new locales.
```     1 (*  Title:      HOL/Divides.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1999  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {* The division operators div and mod *}
```
```     8
```
```     9 theory Divides
```
```    10 imports Nat Power Product_Type
```
```    11 uses "~~/src/Provers/Arith/cancel_div_mod.ML"
```
```    12 begin
```
```    13
```
```    14 subsection {* Syntactic division operations *}
```
```    15
```
```    16 class div = dvd +
```
```    17   fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
```
```    18     and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
```
```    19
```
```    20
```
```    21 subsection {* Abstract division in commutative semirings. *}
```
```    22
```
```    23 class semiring_div = comm_semiring_1_cancel + div +
```
```    24   assumes mod_div_equality: "a div b * b + a mod b = a"
```
```    25     and div_by_0 [simp]: "a div 0 = 0"
```
```    26     and div_0 [simp]: "0 div a = 0"
```
```    27     and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
```
```    28 begin
```
```    29
```
```    30 text {* @{const div} and @{const mod} *}
```
```    31
```
```    32 lemma mod_div_equality2: "b * (a div b) + a mod b = a"
```
```    33   unfolding mult_commute [of b]
```
```    34   by (rule mod_div_equality)
```
```    35
```
```    36 lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
```
```    37   by (simp add: mod_div_equality)
```
```    38
```
```    39 lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
```
```    40   by (simp add: mod_div_equality2)
```
```    41
```
```    42 lemma mod_by_0 [simp]: "a mod 0 = a"
```
```    43   using mod_div_equality [of a zero] by simp
```
```    44
```
```    45 lemma mod_0 [simp]: "0 mod a = 0"
```
```    46   using mod_div_equality [of zero a] div_0 by simp
```
```    47
```
```    48 lemma div_mult_self2 [simp]:
```
```    49   assumes "b \<noteq> 0"
```
```    50   shows "(a + b * c) div b = c + a div b"
```
```    51   using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
```
```    52
```
```    53 lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
```
```    54 proof (cases "b = 0")
```
```    55   case True then show ?thesis by simp
```
```    56 next
```
```    57   case False
```
```    58   have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
```
```    59     by (simp add: mod_div_equality)
```
```    60   also from False div_mult_self1 [of b a c] have
```
```    61     "\<dots> = (c + a div b) * b + (a + c * b) mod b"
```
```    62       by (simp add: left_distrib add_ac)
```
```    63   finally have "a = a div b * b + (a + c * b) mod b"
```
```    64     by (simp add: add_commute [of a] add_assoc left_distrib)
```
```    65   then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
```
```    66     by (simp add: mod_div_equality)
```
```    67   then show ?thesis by simp
```
```    68 qed
```
```    69
```
```    70 lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
```
```    71   by (simp add: mult_commute [of b])
```
```    72
```
```    73 lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
```
```    74   using div_mult_self2 [of b 0 a] by simp
```
```    75
```
```    76 lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
```
```    77   using div_mult_self1 [of b 0 a] by simp
```
```    78
```
```    79 lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
```
```    80   using mod_mult_self2 [of 0 b a] by simp
```
```    81
```
```    82 lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
```
```    83   using mod_mult_self1 [of 0 a b] by simp
```
```    84
```
```    85 lemma div_by_1 [simp]: "a div 1 = a"
```
```    86   using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
```
```    87
```
```    88 lemma mod_by_1 [simp]: "a mod 1 = 0"
```
```    89 proof -
```
```    90   from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
```
```    91   then have "a + a mod 1 = a + 0" by simp
```
```    92   then show ?thesis by (rule add_left_imp_eq)
```
```    93 qed
```
```    94
```
```    95 lemma mod_self [simp]: "a mod a = 0"
```
```    96   using mod_mult_self2_is_0 [of 1] by simp
```
```    97
```
```    98 lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
```
```    99   using div_mult_self2_is_id [of _ 1] by simp
```
```   100
```
```   101 lemma div_add_self1 [simp]:
```
```   102   assumes "b \<noteq> 0"
```
```   103   shows "(b + a) div b = a div b + 1"
```
```   104   using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
```
```   105
```
```   106 lemma div_add_self2 [simp]:
```
```   107   assumes "b \<noteq> 0"
```
```   108   shows "(a + b) div b = a div b + 1"
```
```   109   using assms div_add_self1 [of b a] by (simp add: add_commute)
```
```   110
```
```   111 lemma mod_add_self1 [simp]:
```
```   112   "(b + a) mod b = a mod b"
```
```   113   using mod_mult_self1 [of a 1 b] by (simp add: add_commute)
```
```   114
```
```   115 lemma mod_add_self2 [simp]:
```
```   116   "(a + b) mod b = a mod b"
```
```   117   using mod_mult_self1 [of a 1 b] by simp
```
```   118
```
```   119 lemma mod_div_decomp:
```
```   120   fixes a b
```
```   121   obtains q r where "q = a div b" and "r = a mod b"
```
```   122     and "a = q * b + r"
```
```   123 proof -
```
```   124   from mod_div_equality have "a = a div b * b + a mod b" by simp
```
```   125   moreover have "a div b = a div b" ..
```
```   126   moreover have "a mod b = a mod b" ..
```
```   127   note that ultimately show thesis by blast
```
```   128 qed
```
```   129
```
```   130 lemma dvd_eq_mod_eq_0 [code]: "a dvd b \<longleftrightarrow> b mod a = 0"
```
```   131 proof
```
```   132   assume "b mod a = 0"
```
```   133   with mod_div_equality [of b a] have "b div a * a = b" by simp
```
```   134   then have "b = a * (b div a)" unfolding mult_commute ..
```
```   135   then have "\<exists>c. b = a * c" ..
```
```   136   then show "a dvd b" unfolding dvd_def .
```
```   137 next
```
```   138   assume "a dvd b"
```
```   139   then have "\<exists>c. b = a * c" unfolding dvd_def .
```
```   140   then obtain c where "b = a * c" ..
```
```   141   then have "b mod a = a * c mod a" by simp
```
```   142   then have "b mod a = c * a mod a" by (simp add: mult_commute)
```
```   143   then show "b mod a = 0" by simp
```
```   144 qed
```
```   145
```
```   146 end
```
```   147
```
```   148
```
```   149 subsection {* Division on @{typ nat} *}
```
```   150
```
```   151 text {*
```
```   152   We define @{const div} and @{const mod} on @{typ nat} by means
```
```   153   of a characteristic relation with two input arguments
```
```   154   @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
```
```   155   @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
```
```   156 *}
```
```   157
```
```   158 definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
```
```   159   "divmod_rel m n q r \<longleftrightarrow> m = q * n + r \<and> (if n > 0 then 0 \<le> r \<and> r < n else q = 0)"
```
```   160
```
```   161 text {* @{const divmod_rel} is total: *}
```
```   162
```
```   163 lemma divmod_rel_ex:
```
```   164   obtains q r where "divmod_rel m n q r"
```
```   165 proof (cases "n = 0")
```
```   166   case True with that show thesis
```
```   167     by (auto simp add: divmod_rel_def)
```
```   168 next
```
```   169   case False
```
```   170   have "\<exists>q r. m = q * n + r \<and> r < n"
```
```   171   proof (induct m)
```
```   172     case 0 with `n \<noteq> 0`
```
```   173     have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
```
```   174     then show ?case by blast
```
```   175   next
```
```   176     case (Suc m) then obtain q' r'
```
```   177       where m: "m = q' * n + r'" and n: "r' < n" by auto
```
```   178     then show ?case proof (cases "Suc r' < n")
```
```   179       case True
```
```   180       from m n have "Suc m = q' * n + Suc r'" by simp
```
```   181       with True show ?thesis by blast
```
```   182     next
```
```   183       case False then have "n \<le> Suc r'" by auto
```
```   184       moreover from n have "Suc r' \<le> n" by auto
```
```   185       ultimately have "n = Suc r'" by auto
```
```   186       with m have "Suc m = Suc q' * n + 0" by simp
```
```   187       with `n \<noteq> 0` show ?thesis by blast
```
```   188     qed
```
```   189   qed
```
```   190   with that show thesis
```
```   191     using `n \<noteq> 0` by (auto simp add: divmod_rel_def)
```
```   192 qed
```
```   193
```
```   194 text {* @{const divmod_rel} is injective: *}
```
```   195
```
```   196 lemma divmod_rel_unique_div:
```
```   197   assumes "divmod_rel m n q r"
```
```   198     and "divmod_rel m n q' r'"
```
```   199   shows "q = q'"
```
```   200 proof (cases "n = 0")
```
```   201   case True with assms show ?thesis
```
```   202     by (simp add: divmod_rel_def)
```
```   203 next
```
```   204   case False
```
```   205   have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
```
```   206   apply (rule leI)
```
```   207   apply (subst less_iff_Suc_add)
```
```   208   apply (auto simp add: add_mult_distrib)
```
```   209   done
```
```   210   from `n \<noteq> 0` assms show ?thesis
```
```   211     by (auto simp add: divmod_rel_def
```
```   212       intro: order_antisym dest: aux sym)
```
```   213 qed
```
```   214
```
```   215 lemma divmod_rel_unique_mod:
```
```   216   assumes "divmod_rel m n q r"
```
```   217     and "divmod_rel m n q' r'"
```
```   218   shows "r = r'"
```
```   219 proof -
```
```   220   from assms have "q = q'" by (rule divmod_rel_unique_div)
```
```   221   with assms show ?thesis by (simp add: divmod_rel_def)
```
```   222 qed
```
```   223
```
```   224 text {*
```
```   225   We instantiate divisibility on the natural numbers by
```
```   226   means of @{const divmod_rel}:
```
```   227 *}
```
```   228
```
```   229 instantiation nat :: semiring_div
```
```   230 begin
```
```   231
```
```   232 definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
```
```   233   [code del]: "divmod m n = (THE (q, r). divmod_rel m n q r)"
```
```   234
```
```   235 definition div_nat where
```
```   236   "m div n = fst (divmod m n)"
```
```   237
```
```   238 definition mod_nat where
```
```   239   "m mod n = snd (divmod m n)"
```
```   240
```
```   241 lemma divmod_div_mod:
```
```   242   "divmod m n = (m div n, m mod n)"
```
```   243   unfolding div_nat_def mod_nat_def by simp
```
```   244
```
```   245 lemma divmod_eq:
```
```   246   assumes "divmod_rel m n q r"
```
```   247   shows "divmod m n = (q, r)"
```
```   248   using assms by (auto simp add: divmod_def
```
```   249     dest: divmod_rel_unique_div divmod_rel_unique_mod)
```
```   250
```
```   251 lemma div_eq:
```
```   252   assumes "divmod_rel m n q r"
```
```   253   shows "m div n = q"
```
```   254   using assms by (auto dest: divmod_eq simp add: div_nat_def)
```
```   255
```
```   256 lemma mod_eq:
```
```   257   assumes "divmod_rel m n q r"
```
```   258   shows "m mod n = r"
```
```   259   using assms by (auto dest: divmod_eq simp add: mod_nat_def)
```
```   260
```
```   261 lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)"
```
```   262 proof -
```
```   263   from divmod_rel_ex
```
```   264     obtain q r where rel: "divmod_rel m n q r" .
```
```   265   moreover with div_eq mod_eq have "m div n = q" and "m mod n = r"
```
```   266     by simp_all
```
```   267   ultimately show ?thesis by simp
```
```   268 qed
```
```   269
```
```   270 lemma divmod_zero:
```
```   271   "divmod m 0 = (0, m)"
```
```   272 proof -
```
```   273   from divmod_rel [of m 0] show ?thesis
```
```   274     unfolding divmod_div_mod divmod_rel_def by simp
```
```   275 qed
```
```   276
```
```   277 lemma divmod_base:
```
```   278   assumes "m < n"
```
```   279   shows "divmod m n = (0, m)"
```
```   280 proof -
```
```   281   from divmod_rel [of m n] show ?thesis
```
```   282     unfolding divmod_div_mod divmod_rel_def
```
```   283     using assms by (cases "m div n = 0")
```
```   284       (auto simp add: gr0_conv_Suc [of "m div n"])
```
```   285 qed
```
```   286
```
```   287 lemma divmod_step:
```
```   288   assumes "0 < n" and "n \<le> m"
```
```   289   shows "divmod m n = (Suc ((m - n) div n), (m - n) mod n)"
```
```   290 proof -
```
```   291   from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" .
```
```   292   with assms have m_div_n: "m div n \<ge> 1"
```
```   293     by (cases "m div n") (auto simp add: divmod_rel_def)
```
```   294   from assms divmod_m_n have "divmod_rel (m - n) n (m div n - 1) (m mod n)"
```
```   295     by (cases "m div n") (auto simp add: divmod_rel_def)
```
```   296   with divmod_eq have "divmod (m - n) n = (m div n - 1, m mod n)" by simp
```
```   297   moreover from divmod_div_mod have "divmod (m - n) n = ((m - n) div n, (m - n) mod n)" .
```
```   298   ultimately have "m div n = Suc ((m - n) div n)"
```
```   299     and "m mod n = (m - n) mod n" using m_div_n by simp_all
```
```   300   then show ?thesis using divmod_div_mod by simp
```
```   301 qed
```
```   302
```
```   303 text {* The ''recursion'' equations for @{const div} and @{const mod} *}
```
```   304
```
```   305 lemma div_less [simp]:
```
```   306   fixes m n :: nat
```
```   307   assumes "m < n"
```
```   308   shows "m div n = 0"
```
```   309   using assms divmod_base divmod_div_mod by simp
```
```   310
```
```   311 lemma le_div_geq:
```
```   312   fixes m n :: nat
```
```   313   assumes "0 < n" and "n \<le> m"
```
```   314   shows "m div n = Suc ((m - n) div n)"
```
```   315   using assms divmod_step divmod_div_mod by simp
```
```   316
```
```   317 lemma mod_less [simp]:
```
```   318   fixes m n :: nat
```
```   319   assumes "m < n"
```
```   320   shows "m mod n = m"
```
```   321   using assms divmod_base divmod_div_mod by simp
```
```   322
```
```   323 lemma le_mod_geq:
```
```   324   fixes m n :: nat
```
```   325   assumes "n \<le> m"
```
```   326   shows "m mod n = (m - n) mod n"
```
```   327   using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all
```
```   328
```
```   329 instance proof
```
```   330   fix m n :: nat show "m div n * n + m mod n = m"
```
```   331     using divmod_rel [of m n] by (simp add: divmod_rel_def)
```
```   332 next
```
```   333   fix n :: nat show "n div 0 = 0"
```
```   334     using divmod_zero divmod_div_mod [of n 0] by simp
```
```   335 next
```
```   336   fix n :: nat show "0 div n = 0"
```
```   337     using divmod_rel [of 0 n] by (cases n) (simp_all add: divmod_rel_def)
```
```   338 next
```
```   339   fix m n q :: nat assume "n \<noteq> 0" then show "(q + m * n) div n = m + q div n"
```
```   340     by (induct m) (simp_all add: le_div_geq)
```
```   341 qed
```
```   342
```
```   343 end
```
```   344
```
```   345 text {* Simproc for cancelling @{const div} and @{const mod} *}
```
```   346
```
```   347 (*lemmas mod_div_equality_nat = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard]
```
```   348 lemmas mod_div_equality2_nat = mod_div_equality2 [of "n\<Colon>nat" m, standard*)
```
```   349
```
```   350 ML {*
```
```   351 structure CancelDivModData =
```
```   352 struct
```
```   353
```
```   354 val div_name = @{const_name div};
```
```   355 val mod_name = @{const_name mod};
```
```   356 val mk_binop = HOLogic.mk_binop;
```
```   357 val mk_sum = ArithData.mk_sum;
```
```   358 val dest_sum = ArithData.dest_sum;
```
```   359
```
```   360 (*logic*)
```
```   361
```
```   362 val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
```
```   363
```
```   364 val trans = trans
```
```   365
```
```   366 val prove_eq_sums =
```
```   367   let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
```
```   368   in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
```
```   369
```
```   370 end;
```
```   371
```
```   372 structure CancelDivMod = CancelDivModFun(CancelDivModData);
```
```   373
```
```   374 val cancel_div_mod_proc = Simplifier.simproc (the_context ())
```
```   375   "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
```
```   376
```
```   377 Addsimprocs[cancel_div_mod_proc];
```
```   378 *}
```
```   379
```
```   380 text {* code generator setup *}
```
```   381
```
```   382 lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else
```
```   383   let (q, r) = divmod (m - n) n in (Suc q, r))"
```
```   384   by (simp add: divmod_zero divmod_base divmod_step)
```
```   385     (simp add: divmod_div_mod)
```
```   386
```
```   387 code_modulename SML
```
```   388   Divides Nat
```
```   389
```
```   390 code_modulename OCaml
```
```   391   Divides Nat
```
```   392
```
```   393 code_modulename Haskell
```
```   394   Divides Nat
```
```   395
```
```   396
```
```   397 subsubsection {* Quotient *}
```
```   398
```
```   399 lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
```
```   400   by (simp add: le_div_geq linorder_not_less)
```
```   401
```
```   402 lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
```
```   403   by (simp add: div_geq)
```
```   404
```
```   405 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
```
```   406   by simp
```
```   407
```
```   408 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
```
```   409   by simp
```
```   410
```
```   411
```
```   412 subsubsection {* Remainder *}
```
```   413
```
```   414 lemma mod_less_divisor [simp]:
```
```   415   fixes m n :: nat
```
```   416   assumes "n > 0"
```
```   417   shows "m mod n < (n::nat)"
```
```   418   using assms divmod_rel unfolding divmod_rel_def by auto
```
```   419
```
```   420 lemma mod_less_eq_dividend [simp]:
```
```   421   fixes m n :: nat
```
```   422   shows "m mod n \<le> m"
```
```   423 proof (rule add_leD2)
```
```   424   from mod_div_equality have "m div n * n + m mod n = m" .
```
```   425   then show "m div n * n + m mod n \<le> m" by auto
```
```   426 qed
```
```   427
```
```   428 lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
```
```   429   by (simp add: le_mod_geq linorder_not_less)
```
```   430
```
```   431 lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
```
```   432   by (simp add: le_mod_geq)
```
```   433
```
```   434 lemma mod_1 [simp]: "m mod Suc 0 = 0"
```
```   435   by (induct m) (simp_all add: mod_geq)
```
```   436
```
```   437 lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
```
```   438   apply (cases "n = 0", simp)
```
```   439   apply (cases "k = 0", simp)
```
```   440   apply (induct m rule: nat_less_induct)
```
```   441   apply (subst mod_if, simp)
```
```   442   apply (simp add: mod_geq diff_mult_distrib)
```
```   443   done
```
```   444
```
```   445 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
```
```   446   by (simp add: mult_commute [of k] mod_mult_distrib)
```
```   447
```
```   448 (* a simple rearrangement of mod_div_equality: *)
```
```   449 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
```
```   450   by (cut_tac a = m and b = n in mod_div_equality2, arith)
```
```   451
```
```   452 lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
```
```   453   apply (drule mod_less_divisor [where m = m])
```
```   454   apply simp
```
```   455   done
```
```   456
```
```   457 subsubsection {* Quotient and Remainder *}
```
```   458
```
```   459 lemma divmod_rel_mult1_eq:
```
```   460   "[| divmod_rel b c q r; c > 0 |]
```
```   461    ==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)"
```
```   462 by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
```
```   463
```
```   464 lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
```
```   465 apply (cases "c = 0", simp)
```
```   466 apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq])
```
```   467 done
```
```   468
```
```   469 lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
```
```   470 apply (cases "c = 0", simp)
```
```   471 apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN mod_eq])
```
```   472 done
```
```   473
```
```   474 lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
```
```   475   apply (rule trans)
```
```   476    apply (rule_tac s = "b*a mod c" in trans)
```
```   477     apply (rule_tac [2] mod_mult1_eq)
```
```   478    apply (simp_all add: mult_commute)
```
```   479   done
```
```   480
```
```   481 lemma mod_mult_distrib_mod:
```
```   482   "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
```
```   483 apply (rule mod_mult1_eq' [THEN trans])
```
```   484 apply (rule mod_mult1_eq)
```
```   485 done
```
```   486
```
```   487 lemma divmod_rel_add1_eq:
```
```   488   "[| divmod_rel a c aq ar; divmod_rel b c bq br;  c > 0 |]
```
```   489    ==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)"
```
```   490 by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
```
```   491
```
```   492 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```   493 lemma div_add1_eq:
```
```   494   "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```   495 apply (cases "c = 0", simp)
```
```   496 apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel)
```
```   497 done
```
```   498
```
```   499 lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
```
```   500 apply (cases "c = 0", simp)
```
```   501 apply (blast intro: divmod_rel_add1_eq [THEN mod_eq] divmod_rel)
```
```   502 done
```
```   503
```
```   504 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
```
```   505   apply (cut_tac m = q and n = c in mod_less_divisor)
```
```   506   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
```
```   507   apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
```
```   508   apply (simp add: add_mult_distrib2)
```
```   509   done
```
```   510
```
```   511 lemma divmod_rel_mult2_eq: "[| divmod_rel a b q r;  0 < b;  0 < c |]
```
```   512       ==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)"
```
```   513   by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
```
```   514
```
```   515 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
```
```   516   apply (cases "b = 0", simp)
```
```   517   apply (cases "c = 0", simp)
```
```   518   apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq])
```
```   519   done
```
```   520
```
```   521 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
```
```   522   apply (cases "b = 0", simp)
```
```   523   apply (cases "c = 0", simp)
```
```   524   apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq])
```
```   525   done
```
```   526
```
```   527
```
```   528 subsubsection{*Cancellation of Common Factors in Division*}
```
```   529
```
```   530 lemma div_mult_mult_lemma:
```
```   531     "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
```
```   532   by (auto simp add: div_mult2_eq)
```
```   533
```
```   534 lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
```
```   535   apply (cases "b = 0")
```
```   536   apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
```
```   537   done
```
```   538
```
```   539 lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
```
```   540   apply (drule div_mult_mult1)
```
```   541   apply (auto simp add: mult_commute)
```
```   542   done
```
```   543
```
```   544
```
```   545 subsubsection{*Further Facts about Quotient and Remainder*}
```
```   546
```
```   547 lemma div_1 [simp]: "m div Suc 0 = m"
```
```   548   by (induct m) (simp_all add: div_geq)
```
```   549
```
```   550
```
```   551 (* Monotonicity of div in first argument *)
```
```   552 lemma div_le_mono [rule_format (no_asm)]:
```
```   553     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
```
```   554 apply (case_tac "k=0", simp)
```
```   555 apply (induct "n" rule: nat_less_induct, clarify)
```
```   556 apply (case_tac "n<k")
```
```   557 (* 1  case n<k *)
```
```   558 apply simp
```
```   559 (* 2  case n >= k *)
```
```   560 apply (case_tac "m<k")
```
```   561 (* 2.1  case m<k *)
```
```   562 apply simp
```
```   563 (* 2.2  case m>=k *)
```
```   564 apply (simp add: div_geq diff_le_mono)
```
```   565 done
```
```   566
```
```   567 (* Antimonotonicity of div in second argument *)
```
```   568 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
```
```   569 apply (subgoal_tac "0<n")
```
```   570  prefer 2 apply simp
```
```   571 apply (induct_tac k rule: nat_less_induct)
```
```   572 apply (rename_tac "k")
```
```   573 apply (case_tac "k<n", simp)
```
```   574 apply (subgoal_tac "~ (k<m) ")
```
```   575  prefer 2 apply simp
```
```   576 apply (simp add: div_geq)
```
```   577 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
```
```   578  prefer 2
```
```   579  apply (blast intro: div_le_mono diff_le_mono2)
```
```   580 apply (rule le_trans, simp)
```
```   581 apply (simp)
```
```   582 done
```
```   583
```
```   584 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
```
```   585 apply (case_tac "n=0", simp)
```
```   586 apply (subgoal_tac "m div n \<le> m div 1", simp)
```
```   587 apply (rule div_le_mono2)
```
```   588 apply (simp_all (no_asm_simp))
```
```   589 done
```
```   590
```
```   591 (* Similar for "less than" *)
```
```   592 lemma div_less_dividend [rule_format]:
```
```   593      "!!n::nat. 1<n ==> 0 < m --> m div n < m"
```
```   594 apply (induct_tac m rule: nat_less_induct)
```
```   595 apply (rename_tac "m")
```
```   596 apply (case_tac "m<n", simp)
```
```   597 apply (subgoal_tac "0<n")
```
```   598  prefer 2 apply simp
```
```   599 apply (simp add: div_geq)
```
```   600 apply (case_tac "n<m")
```
```   601  apply (subgoal_tac "(m-n) div n < (m-n) ")
```
```   602   apply (rule impI less_trans_Suc)+
```
```   603 apply assumption
```
```   604   apply (simp_all)
```
```   605 done
```
```   606
```
```   607 declare div_less_dividend [simp]
```
```   608
```
```   609 text{*A fact for the mutilated chess board*}
```
```   610 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
```
```   611 apply (case_tac "n=0", simp)
```
```   612 apply (induct "m" rule: nat_less_induct)
```
```   613 apply (case_tac "Suc (na) <n")
```
```   614 (* case Suc(na) < n *)
```
```   615 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
```
```   616 (* case n \<le> Suc(na) *)
```
```   617 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
```
```   618 apply (auto simp add: Suc_diff_le le_mod_geq)
```
```   619 done
```
```   620
```
```   621 lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
```
```   622   by (cases "n = 0") auto
```
```   623
```
```   624 lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
```
```   625   by (cases "n = 0") auto
```
```   626
```
```   627
```
```   628 subsubsection {* The Divides Relation *}
```
```   629
```
```   630 lemma dvd_1_left [iff]: "Suc 0 dvd k"
```
```   631   unfolding dvd_def by simp
```
```   632
```
```   633 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
```
```   634   by (simp add: dvd_def)
```
```   635
```
```   636 lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
```
```   637   unfolding dvd_def
```
```   638   by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
```
```   639
```
```   640 text {* @{term "op dvd"} is a partial order *}
```
```   641
```
```   642 class_interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"]
```
```   643   proof qed (auto intro: dvd_refl dvd_trans dvd_anti_sym)
```
```   644
```
```   645 lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
```
```   646   unfolding dvd_def
```
```   647   by (blast intro: diff_mult_distrib2 [symmetric])
```
```   648
```
```   649 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
```
```   650   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
```
```   651   apply (blast intro: dvd_add)
```
```   652   done
```
```   653
```
```   654 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
```
```   655   by (drule_tac m = m in dvd_diff, auto)
```
```   656
```
```   657 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
```
```   658   apply (rule iffI)
```
```   659    apply (erule_tac [2] dvd_add)
```
```   660    apply (rule_tac [2] dvd_refl)
```
```   661   apply (subgoal_tac "n = (n+k) -k")
```
```   662    prefer 2 apply simp
```
```   663   apply (erule ssubst)
```
```   664   apply (erule dvd_diff)
```
```   665   apply (rule dvd_refl)
```
```   666   done
```
```   667
```
```   668 lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
```
```   669   unfolding dvd_def
```
```   670   apply (case_tac "n = 0", auto)
```
```   671   apply (blast intro: mod_mult_distrib2 [symmetric])
```
```   672   done
```
```   673
```
```   674 lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
```
```   675   apply (subgoal_tac "k dvd (m div n) *n + m mod n")
```
```   676    apply (simp add: mod_div_equality)
```
```   677   apply (simp only: dvd_add dvd_mult)
```
```   678   done
```
```   679
```
```   680 lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
```
```   681   by (blast intro: dvd_mod_imp_dvd dvd_mod)
```
```   682
```
```   683 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
```
```   684   unfolding dvd_def
```
```   685   apply (erule exE)
```
```   686   apply (simp add: mult_ac)
```
```   687   done
```
```   688
```
```   689 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
```
```   690   apply auto
```
```   691    apply (subgoal_tac "m*n dvd m*1")
```
```   692    apply (drule dvd_mult_cancel, auto)
```
```   693   done
```
```   694
```
```   695 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
```
```   696   apply (subst mult_commute)
```
```   697   apply (erule dvd_mult_cancel1)
```
```   698   done
```
```   699
```
```   700 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
```
```   701   apply (unfold dvd_def, clarify)
```
```   702   apply (simp_all (no_asm_use) add: zero_less_mult_iff)
```
```   703   apply (erule conjE)
```
```   704   apply (rule le_trans)
```
```   705    apply (rule_tac [2] le_refl [THEN mult_le_mono])
```
```   706    apply (erule_tac [2] Suc_leI, simp)
```
```   707   done
```
```   708
```
```   709 lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
```
```   710   apply (subgoal_tac "m mod n = 0")
```
```   711    apply (simp add: mult_div_cancel)
```
```   712   apply (simp only: dvd_eq_mod_eq_0)
```
```   713   done
```
```   714
```
```   715 lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
```
```   716   apply (unfold dvd_def)
```
```   717   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
```
```   718   apply (simp add: power_add)
```
```   719   done
```
```   720
```
```   721 lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
```
```   722   apply (rule trans [symmetric])
```
```   723    apply (rule mod_add1_eq, simp)
```
```   724   apply (rule mod_add1_eq [symmetric])
```
```   725   done
```
```   726
```
```   727 lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
```
```   728   apply (rule trans [symmetric])
```
```   729    apply (rule mod_add1_eq, simp)
```
```   730   apply (rule mod_add1_eq [symmetric])
```
```   731   done
```
```   732
```
```   733 lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
```
```   734   by (induct n) auto
```
```   735
```
```   736 lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
```
```   737   apply (induct j)
```
```   738    apply (simp_all add: le_Suc_eq)
```
```   739   apply (blast dest!: dvd_mult_right)
```
```   740   done
```
```   741
```
```   742 lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
```
```   743   apply (rule power_le_imp_le_exp, assumption)
```
```   744   apply (erule dvd_imp_le, simp)
```
```   745   done
```
```   746
```
```   747 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
```
```   748   by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```   749
```
```   750 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
```
```   751
```
```   752 (*Loses information, namely we also have r<d provided d is nonzero*)
```
```   753 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
```
```   754   apply (cut_tac a = m in mod_div_equality)
```
```   755   apply (simp only: add_ac)
```
```   756   apply (blast intro: sym)
```
```   757   done
```
```   758
```
```   759 lemma split_div:
```
```   760  "P(n div k :: nat) =
```
```   761  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
```
```   762  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```   763 proof
```
```   764   assume P: ?P
```
```   765   show ?Q
```
```   766   proof (cases)
```
```   767     assume "k = 0"
```
```   768     with P show ?Q by simp
```
```   769   next
```
```   770     assume not0: "k \<noteq> 0"
```
```   771     thus ?Q
```
```   772     proof (simp, intro allI impI)
```
```   773       fix i j
```
```   774       assume n: "n = k*i + j" and j: "j < k"
```
```   775       show "P i"
```
```   776       proof (cases)
```
```   777         assume "i = 0"
```
```   778         with n j P show "P i" by simp
```
```   779       next
```
```   780         assume "i \<noteq> 0"
```
```   781         with not0 n j P show "P i" by(simp add:add_ac)
```
```   782       qed
```
```   783     qed
```
```   784   qed
```
```   785 next
```
```   786   assume Q: ?Q
```
```   787   show ?P
```
```   788   proof (cases)
```
```   789     assume "k = 0"
```
```   790     with Q show ?P by simp
```
```   791   next
```
```   792     assume not0: "k \<noteq> 0"
```
```   793     with Q have R: ?R by simp
```
```   794     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```   795     show ?P by simp
```
```   796   qed
```
```   797 qed
```
```   798
```
```   799 lemma split_div_lemma:
```
```   800   assumes "0 < n"
```
```   801   shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
```
```   802 proof
```
```   803   assume ?rhs
```
```   804   with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
```
```   805   then have A: "n * q \<le> m" by simp
```
```   806   have "n - (m mod n) > 0" using mod_less_divisor assms by auto
```
```   807   then have "m < m + (n - (m mod n))" by simp
```
```   808   then have "m < n + (m - (m mod n))" by simp
```
```   809   with nq have "m < n + n * q" by simp
```
```   810   then have B: "m < n * Suc q" by simp
```
```   811   from A B show ?lhs ..
```
```   812 next
```
```   813   assume P: ?lhs
```
```   814   then have "divmod_rel m n q (m - n * q)"
```
```   815     unfolding divmod_rel_def by (auto simp add: mult_ac)
```
```   816   then show ?rhs using divmod_rel by (rule divmod_rel_unique_div)
```
```   817 qed
```
```   818
```
```   819 theorem split_div':
```
```   820   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
```
```   821    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
```
```   822   apply (case_tac "0 < n")
```
```   823   apply (simp only: add: split_div_lemma)
```
```   824   apply simp_all
```
```   825   done
```
```   826
```
```   827 lemma split_mod:
```
```   828  "P(n mod k :: nat) =
```
```   829  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
```
```   830  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```   831 proof
```
```   832   assume P: ?P
```
```   833   show ?Q
```
```   834   proof (cases)
```
```   835     assume "k = 0"
```
```   836     with P show ?Q by simp
```
```   837   next
```
```   838     assume not0: "k \<noteq> 0"
```
```   839     thus ?Q
```
```   840     proof (simp, intro allI impI)
```
```   841       fix i j
```
```   842       assume "n = k*i + j" "j < k"
```
```   843       thus "P j" using not0 P by(simp add:add_ac mult_ac)
```
```   844     qed
```
```   845   qed
```
```   846 next
```
```   847   assume Q: ?Q
```
```   848   show ?P
```
```   849   proof (cases)
```
```   850     assume "k = 0"
```
```   851     with Q show ?P by simp
```
```   852   next
```
```   853     assume not0: "k \<noteq> 0"
```
```   854     with Q have R: ?R by simp
```
```   855     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```   856     show ?P by simp
```
```   857   qed
```
```   858 qed
```
```   859
```
```   860 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
```
```   861   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
```
```   862     subst [OF mod_div_equality [of _ n]])
```
```   863   apply arith
```
```   864   done
```
```   865
```
```   866 lemma div_mod_equality':
```
```   867   fixes m n :: nat
```
```   868   shows "m div n * n = m - m mod n"
```
```   869 proof -
```
```   870   have "m mod n \<le> m mod n" ..
```
```   871   from div_mod_equality have
```
```   872     "m div n * n + m mod n - m mod n = m - m mod n" by simp
```
```   873   with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
```
```   874     "m div n * n + (m mod n - m mod n) = m - m mod n"
```
```   875     by simp
```
```   876   then show ?thesis by simp
```
```   877 qed
```
```   878
```
```   879
```
```   880 subsubsection {*An ``induction'' law for modulus arithmetic.*}
```
```   881
```
```   882 lemma mod_induct_0:
```
```   883   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```   884   and base: "P i" and i: "i<p"
```
```   885   shows "P 0"
```
```   886 proof (rule ccontr)
```
```   887   assume contra: "\<not>(P 0)"
```
```   888   from i have p: "0<p" by simp
```
```   889   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
```
```   890   proof
```
```   891     fix k
```
```   892     show "?A k"
```
```   893     proof (induct k)
```
```   894       show "?A 0" by simp  -- "by contradiction"
```
```   895     next
```
```   896       fix n
```
```   897       assume ih: "?A n"
```
```   898       show "?A (Suc n)"
```
```   899       proof (clarsimp)
```
```   900         assume y: "P (p - Suc n)"
```
```   901         have n: "Suc n < p"
```
```   902         proof (rule ccontr)
```
```   903           assume "\<not>(Suc n < p)"
```
```   904           hence "p - Suc n = 0"
```
```   905             by simp
```
```   906           with y contra show "False"
```
```   907             by simp
```
```   908         qed
```
```   909         hence n2: "Suc (p - Suc n) = p-n" by arith
```
```   910         from p have "p - Suc n < p" by arith
```
```   911         with y step have z: "P ((Suc (p - Suc n)) mod p)"
```
```   912           by blast
```
```   913         show "False"
```
```   914         proof (cases "n=0")
```
```   915           case True
```
```   916           with z n2 contra show ?thesis by simp
```
```   917         next
```
```   918           case False
```
```   919           with p have "p-n < p" by arith
```
```   920           with z n2 False ih show ?thesis by simp
```
```   921         qed
```
```   922       qed
```
```   923     qed
```
```   924   qed
```
```   925   moreover
```
```   926   from i obtain k where "0<k \<and> i+k=p"
```
```   927     by (blast dest: less_imp_add_positive)
```
```   928   hence "0<k \<and> i=p-k" by auto
```
```   929   moreover
```
```   930   note base
```
```   931   ultimately
```
```   932   show "False" by blast
```
```   933 qed
```
```   934
```
```   935 lemma mod_induct:
```
```   936   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```   937   and base: "P i" and i: "i<p" and j: "j<p"
```
```   938   shows "P j"
```
```   939 proof -
```
```   940   have "\<forall>j<p. P j"
```
```   941   proof
```
```   942     fix j
```
```   943     show "j<p \<longrightarrow> P j" (is "?A j")
```
```   944     proof (induct j)
```
```   945       from step base i show "?A 0"
```
```   946         by (auto elim: mod_induct_0)
```
```   947     next
```
```   948       fix k
```
```   949       assume ih: "?A k"
```
```   950       show "?A (Suc k)"
```
```   951       proof
```
```   952         assume suc: "Suc k < p"
```
```   953         hence k: "k<p" by simp
```
```   954         with ih have "P k" ..
```
```   955         with step k have "P (Suc k mod p)"
```
```   956           by blast
```
```   957         moreover
```
```   958         from suc have "Suc k mod p = Suc k"
```
```   959           by simp
```
```   960         ultimately
```
```   961         show "P (Suc k)" by simp
```
```   962       qed
```
```   963     qed
```
```   964   qed
```
```   965   with j show ?thesis by blast
```
```   966 qed
```
```   967
```
```   968 end
```