src/HOL/Divides.thy
author nipkow
Mon Aug 16 14:22:27 2004 +0200 (2004-08-16)
changeset 15131 c69542757a4d
parent 14640 b31870c50c68
child 15140 322485b816ac
permissions -rw-r--r--
New theory header syntax.
     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 The division operators div, mod and the divides relation "dvd"
     7 *)
     8 
     9 theory Divides
    10 import NatArith
    11 begin
    12 
    13 (*We use the same class for div and mod;
    14   moreover, dvd is defined whenever multiplication is*)
    15 axclass
    16   div < type
    17 
    18 instance  nat :: div ..
    19 
    20 consts
    21   div  :: "'a::div \<Rightarrow> 'a \<Rightarrow> 'a"          (infixl 70)
    22   mod  :: "'a::div \<Rightarrow> 'a \<Rightarrow> 'a"          (infixl 70)
    23   dvd  :: "'a::times \<Rightarrow> 'a \<Rightarrow> bool"      (infixl 50)
    24 
    25 
    26 defs
    27 
    28   mod_def:   "m mod n == wfrec (trancl pred_nat)
    29                           (%f j. if j<n | n=0 then j else f (j-n)) m"
    30 
    31   div_def:   "m div n == wfrec (trancl pred_nat) 
    32                           (%f j. if j<n | n=0 then 0 else Suc (f (j-n))) m"
    33 
    34 (*The definition of dvd is polymorphic!*)
    35   dvd_def:   "m dvd n == \<exists>k. n = m*k"
    36 
    37 (*This definition helps prove the harder properties of div and mod.
    38   It is copied from IntDiv.thy; should it be overloaded?*)
    39 constdefs
    40   quorem :: "(nat*nat) * (nat*nat) => bool"
    41     "quorem == %((a,b), (q,r)).
    42                       a = b*q + r &
    43                       (if 0<b then 0\<le>r & r<b else b<r & r \<le>0)"
    44 
    45 
    46 
    47 subsection{*Initial Lemmas*}
    48 
    49 lemmas wf_less_trans = 
    50        def_wfrec [THEN trans, OF eq_reflection wf_pred_nat [THEN wf_trancl],
    51                   standard]
    52 
    53 lemma mod_eq: "(%m. m mod n) = 
    54               wfrec (trancl pred_nat) (%f j. if j<n | n=0 then j else f (j-n))"
    55 by (simp add: mod_def)
    56 
    57 lemma div_eq: "(%m. m div n) = wfrec (trancl pred_nat)  
    58                (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"
    59 by (simp add: div_def)
    60 
    61 
    62 (** Aribtrary definitions for division by zero.  Useful to simplify 
    63     certain equations **)
    64 
    65 lemma DIVISION_BY_ZERO_DIV [simp]: "a div 0 = (0::nat)"
    66 by (rule div_eq [THEN wf_less_trans], simp)
    67 
    68 lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a::nat)"
    69 by (rule mod_eq [THEN wf_less_trans], simp)
    70 
    71 
    72 subsection{*Remainder*}
    73 
    74 lemma mod_less [simp]: "m<n ==> m mod n = (m::nat)"
    75 by (rule mod_eq [THEN wf_less_trans], simp)
    76 
    77 lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n"
    78 apply (case_tac "n=0", simp) 
    79 apply (rule mod_eq [THEN wf_less_trans])
    80 apply (simp add: diff_less cut_apply less_eq)
    81 done
    82 
    83 (*Avoids the ugly ~m<n above*)
    84 lemma le_mod_geq: "(n::nat) \<le> m ==> m mod n = (m-n) mod n"
    85 by (simp add: mod_geq not_less_iff_le)
    86 
    87 lemma mod_if: "m mod (n::nat) = (if m<n then m else (m-n) mod n)"
    88 by (simp add: mod_geq)
    89 
    90 lemma mod_1 [simp]: "m mod Suc 0 = 0"
    91 apply (induct_tac "m")
    92 apply (simp_all (no_asm_simp) add: mod_geq)
    93 done
    94 
    95 lemma mod_self [simp]: "n mod n = (0::nat)"
    96 apply (case_tac "n=0")
    97 apply (simp_all add: mod_geq)
    98 done
    99 
   100 lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
   101 apply (subgoal_tac " (n + m) mod n = (n+m-n) mod n") 
   102 apply (simp add: add_commute)
   103 apply (subst mod_geq [symmetric], simp_all)
   104 done
   105 
   106 lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
   107 by (simp add: add_commute mod_add_self2)
   108 
   109 lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
   110 apply (induct_tac "k")
   111 apply (simp_all add: add_left_commute [of _ n])
   112 done
   113 
   114 lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
   115 by (simp add: mult_commute mod_mult_self1)
   116 
   117 lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)"
   118 apply (case_tac "n=0", simp)
   119 apply (case_tac "k=0", simp)
   120 apply (induct_tac "m" rule: nat_less_induct)
   121 apply (subst mod_if, simp)
   122 apply (simp add: mod_geq diff_less diff_mult_distrib)
   123 done
   124 
   125 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
   126 by (simp add: mult_commute [of k] mod_mult_distrib)
   127 
   128 lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
   129 apply (case_tac "n=0", simp)
   130 apply (induct_tac "m", simp)
   131 apply (rename_tac "k")
   132 apply (cut_tac m = "k*n" and n = n in mod_add_self2)
   133 apply (simp add: add_commute)
   134 done
   135 
   136 lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
   137 by (simp add: mult_commute mod_mult_self_is_0)
   138 
   139 
   140 subsection{*Quotient*}
   141 
   142 lemma div_less [simp]: "m<n ==> m div n = (0::nat)"
   143 by (rule div_eq [THEN wf_less_trans], simp)
   144 
   145 lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"
   146 apply (rule div_eq [THEN wf_less_trans])
   147 apply (simp add: diff_less cut_apply less_eq)
   148 done
   149 
   150 (*Avoids the ugly ~m<n above*)
   151 lemma le_div_geq: "[| 0<n;  n\<le>m |] ==> m div n = Suc((m-n) div n)"
   152 by (simp add: div_geq not_less_iff_le)
   153 
   154 lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"
   155 by (simp add: div_geq)
   156 
   157 
   158 (*Main Result about quotient and remainder.*)
   159 lemma mod_div_equality: "(m div n)*n + m mod n = (m::nat)"
   160 apply (case_tac "n=0", simp)
   161 apply (induct_tac "m" rule: nat_less_induct)
   162 apply (subst mod_if)
   163 apply (simp_all (no_asm_simp) add: add_assoc div_geq add_diff_inverse diff_less)
   164 done
   165 
   166 lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)"
   167 apply(cut_tac m = m and n = n in mod_div_equality)
   168 apply(simp add: mult_commute)
   169 done
   170 
   171 subsection{*Simproc for Cancelling Div and Mod*}
   172 
   173 lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k"
   174 apply(simp add: mod_div_equality)
   175 done
   176 
   177 lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k"
   178 apply(simp add: mod_div_equality2)
   179 done
   180 
   181 ML
   182 {*
   183 val div_mod_equality = thm "div_mod_equality";
   184 val div_mod_equality2 = thm "div_mod_equality2";
   185 
   186 
   187 structure CancelDivModData =
   188 struct
   189 
   190 val div_name = "Divides.op div";
   191 val mod_name = "Divides.op mod";
   192 val mk_binop = HOLogic.mk_binop;
   193 val mk_sum = NatArithUtils.mk_sum;
   194 val dest_sum = NatArithUtils.dest_sum;
   195 
   196 (*logic*)
   197 
   198 val div_mod_eqs = map mk_meta_eq [div_mod_equality,div_mod_equality2]
   199 
   200 val trans = trans
   201 
   202 val prove_eq_sums =
   203   let val simps = add_0 :: add_0_right :: add_ac
   204   in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all simps) end
   205 
   206 end;
   207 
   208 structure CancelDivMod = CancelDivModFun(CancelDivModData);
   209 
   210 val cancel_div_mod_proc = NatArithUtils.prep_simproc
   211       ("cancel_div_mod", ["(m::nat) + n"], CancelDivMod.proc);
   212 
   213 Addsimprocs[cancel_div_mod_proc];
   214 *}
   215 
   216 
   217 (* a simple rearrangement of mod_div_equality: *)
   218 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
   219 by (cut_tac m = m and n = n in mod_div_equality2, arith)
   220 
   221 lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)"
   222 apply (induct_tac "m" rule: nat_less_induct)
   223 apply (case_tac "na<n", simp) 
   224 txt{*case @{term "n \<le> na"}*}
   225 apply (simp add: mod_geq diff_less)
   226 done
   227 
   228 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
   229 by (cut_tac m = "m*n" and n = n in mod_div_equality, auto)
   230 
   231 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
   232 by (simp add: mult_commute div_mult_self_is_m)
   233 
   234 (*mod_mult_distrib2 above is the counterpart for remainder*)
   235 
   236 
   237 subsection{*Proving facts about Quotient and Remainder*}
   238 
   239 lemma unique_quotient_lemma:
   240      "[| b*q' + r'  \<le> b*q + r;  0 < b;  r < b |]  
   241       ==> q' \<le> (q::nat)"
   242 apply (rule leI)
   243 apply (subst less_iff_Suc_add)
   244 apply (auto simp add: add_mult_distrib2)
   245 done
   246 
   247 lemma unique_quotient:
   248      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]  
   249       ==> q = q'"
   250 apply (simp add: split_ifs quorem_def)
   251 apply (blast intro: order_antisym 
   252              dest: order_eq_refl [THEN unique_quotient_lemma] sym)+
   253 done
   254 
   255 lemma unique_remainder:
   256      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]  
   257       ==> r = r'"
   258 apply (subgoal_tac "q = q'")
   259 prefer 2 apply (blast intro: unique_quotient)
   260 apply (simp add: quorem_def)
   261 done
   262 
   263 lemma quorem_div_mod: "0 < b ==> quorem ((a, b), (a div b, a mod b))"
   264 by (auto simp add: quorem_def)
   265 
   266 lemma quorem_div: "[| quorem((a,b),(q,r));  0 < b |] ==> a div b = q"
   267 by (simp add: quorem_div_mod [THEN unique_quotient])
   268 
   269 lemma quorem_mod: "[| quorem((a,b),(q,r));  0 < b |] ==> a mod b = r"
   270 by (simp add: quorem_div_mod [THEN unique_remainder])
   271 
   272 (** A dividend of zero **)
   273 
   274 lemma div_0 [simp]: "0 div m = (0::nat)"
   275 by (case_tac "m=0", simp_all)
   276 
   277 lemma mod_0 [simp]: "0 mod m = (0::nat)"
   278 by (case_tac "m=0", simp_all)
   279 
   280 (** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
   281 
   282 lemma quorem_mult1_eq:
   283      "[| quorem((b,c),(q,r));  0 < c |]  
   284       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
   285 apply (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
   286 done
   287 
   288 lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
   289 apply (case_tac "c = 0", simp)
   290 apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
   291 done
   292 
   293 lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
   294 apply (case_tac "c = 0", simp)
   295 apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
   296 done
   297 
   298 lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
   299 apply (rule trans)
   300 apply (rule_tac s = "b*a mod c" in trans)
   301 apply (rule_tac [2] mod_mult1_eq)
   302 apply (simp_all (no_asm) add: mult_commute)
   303 done
   304 
   305 lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
   306 apply (rule mod_mult1_eq' [THEN trans])
   307 apply (rule mod_mult1_eq)
   308 done
   309 
   310 (** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
   311 
   312 lemma quorem_add1_eq:
   313      "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  0 < c |]  
   314       ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
   315 by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
   316 
   317 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
   318 lemma div_add1_eq:
   319      "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
   320 apply (case_tac "c = 0", simp)
   321 apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod)
   322 done
   323 
   324 lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
   325 apply (case_tac "c = 0", simp)
   326 apply (blast intro: quorem_div_mod quorem_div_mod
   327                     quorem_add1_eq [THEN quorem_mod])
   328 done
   329 
   330 
   331 subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*}
   332 
   333 (** first, a lemma to bound the remainder **)
   334 
   335 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
   336 apply (cut_tac m = q and n = c in mod_less_divisor)
   337 apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
   338 apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
   339 apply (simp add: add_mult_distrib2)
   340 done
   341 
   342 lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]  
   343       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
   344 apply (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)
   345 done
   346 
   347 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
   348 apply (case_tac "b=0", simp)
   349 apply (case_tac "c=0", simp)
   350 apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
   351 done
   352 
   353 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
   354 apply (case_tac "b=0", simp)
   355 apply (case_tac "c=0", simp)
   356 apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
   357 done
   358 
   359 
   360 subsection{*Cancellation of Common Factors in Division*}
   361 
   362 lemma div_mult_mult_lemma:
   363      "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
   364 by (auto simp add: div_mult2_eq)
   365 
   366 lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
   367 apply (case_tac "b = 0")
   368 apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
   369 done
   370 
   371 lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
   372 apply (drule div_mult_mult1)
   373 apply (auto simp add: mult_commute)
   374 done
   375 
   376 
   377 (*Distribution of Factors over Remainders:
   378 
   379 Could prove these as in Integ/IntDiv.ML, but we already have
   380 mod_mult_distrib and mod_mult_distrib2 above!
   381 
   382 Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)"
   383 qed "mod_mult_mult1";
   384 
   385 Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)";
   386 qed "mod_mult_mult2";
   387  ***)
   388 
   389 subsection{*Further Facts about Quotient and Remainder*}
   390 
   391 lemma div_1 [simp]: "m div Suc 0 = m"
   392 apply (induct_tac "m")
   393 apply (simp_all (no_asm_simp) add: div_geq)
   394 done
   395 
   396 lemma div_self [simp]: "0<n ==> n div n = (1::nat)"
   397 by (simp add: div_geq)
   398 
   399 lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
   400 apply (subgoal_tac " (n + m) div n = Suc ((n+m-n) div n) ")
   401 apply (simp add: add_commute)
   402 apply (subst div_geq [symmetric], simp_all)
   403 done
   404 
   405 lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
   406 by (simp add: add_commute div_add_self2)
   407 
   408 lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
   409 apply (subst div_add1_eq)
   410 apply (subst div_mult1_eq, simp)
   411 done
   412 
   413 lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
   414 by (simp add: mult_commute div_mult_self1)
   415 
   416 
   417 (* Monotonicity of div in first argument *)
   418 lemma div_le_mono [rule_format (no_asm)]:
   419      "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
   420 apply (case_tac "k=0", simp)
   421 apply (induct_tac "n" rule: nat_less_induct, clarify)
   422 apply (case_tac "n<k")
   423 (* 1  case n<k *)
   424 apply simp
   425 (* 2  case n >= k *)
   426 apply (case_tac "m<k")
   427 (* 2.1  case m<k *)
   428 apply simp
   429 (* 2.2  case m>=k *)
   430 apply (simp add: div_geq diff_less diff_le_mono)
   431 done
   432 
   433 (* Antimonotonicity of div in second argument *)
   434 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
   435 apply (subgoal_tac "0<n")
   436  prefer 2 apply simp 
   437 apply (induct_tac "k" rule: nat_less_induct)
   438 apply (rename_tac "k")
   439 apply (case_tac "k<n", simp)
   440 apply (subgoal_tac "~ (k<m) ")
   441  prefer 2 apply simp 
   442 apply (simp add: div_geq)
   443 apply (subgoal_tac " (k-n) div n \<le> (k-m) div n")
   444  prefer 2
   445  apply (blast intro: div_le_mono diff_le_mono2)
   446 apply (rule le_trans, simp)
   447 apply (simp add: diff_less)
   448 done
   449 
   450 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
   451 apply (case_tac "n=0", simp)
   452 apply (subgoal_tac "m div n \<le> m div 1", simp)
   453 apply (rule div_le_mono2)
   454 apply (simp_all (no_asm_simp))
   455 done
   456 
   457 (* Similar for "less than" *) 
   458 lemma div_less_dividend [rule_format, simp]:
   459      "!!n::nat. 1<n ==> 0 < m --> m div n < m"
   460 apply (induct_tac "m" rule: nat_less_induct)
   461 apply (rename_tac "m")
   462 apply (case_tac "m<n", simp)
   463 apply (subgoal_tac "0<n")
   464  prefer 2 apply simp 
   465 apply (simp add: div_geq)
   466 apply (case_tac "n<m")
   467  apply (subgoal_tac " (m-n) div n < (m-n) ")
   468   apply (rule impI less_trans_Suc)+
   469 apply assumption
   470   apply (simp_all add: diff_less)
   471 done
   472 
   473 text{*A fact for the mutilated chess board*}
   474 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
   475 apply (case_tac "n=0", simp)
   476 apply (induct_tac "m" rule: nat_less_induct)
   477 apply (case_tac "Suc (na) <n")
   478 (* case Suc(na) < n *)
   479 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
   480 (* case n \<le> Suc(na) *)
   481 apply (simp add: not_less_iff_le le_Suc_eq mod_geq)
   482 apply (auto simp add: Suc_diff_le diff_less le_mod_geq)
   483 done
   484 
   485 lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
   486 by (case_tac "n=0", auto)
   487 
   488 lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
   489 by (case_tac "n=0", auto)
   490 
   491 
   492 subsection{*The Divides Relation*}
   493 
   494 lemma dvdI [intro?]: "n = m * k ==> m dvd n"
   495 by (unfold dvd_def, blast)
   496 
   497 lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
   498 by (unfold dvd_def, blast)
   499 
   500 lemma dvd_0_right [iff]: "m dvd (0::nat)"
   501 apply (unfold dvd_def)
   502 apply (blast intro: mult_0_right [symmetric])
   503 done
   504 
   505 lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
   506 by (force simp add: dvd_def)
   507 
   508 lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
   509 by (blast intro: dvd_0_left)
   510 
   511 lemma dvd_1_left [iff]: "Suc 0 dvd k"
   512 by (unfold dvd_def, simp)
   513 
   514 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
   515 by (simp add: dvd_def)
   516 
   517 lemma dvd_refl [simp]: "m dvd (m::nat)"
   518 apply (unfold dvd_def)
   519 apply (blast intro: mult_1_right [symmetric])
   520 done
   521 
   522 lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"
   523 apply (unfold dvd_def)
   524 apply (blast intro: mult_assoc)
   525 done
   526 
   527 lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
   528 apply (unfold dvd_def)
   529 apply (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
   530 done
   531 
   532 lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
   533 apply (unfold dvd_def)
   534 apply (blast intro: add_mult_distrib2 [symmetric])
   535 done
   536 
   537 lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
   538 apply (unfold dvd_def)
   539 apply (blast intro: diff_mult_distrib2 [symmetric])
   540 done
   541 
   542 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
   543 apply (erule not_less_iff_le [THEN iffD2, THEN add_diff_inverse, THEN subst])
   544 apply (blast intro: dvd_add)
   545 done
   546 
   547 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
   548 by (drule_tac m = m in dvd_diff, auto)
   549 
   550 lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
   551 apply (unfold dvd_def)
   552 apply (blast intro: mult_left_commute)
   553 done
   554 
   555 lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
   556 apply (subst mult_commute)
   557 apply (erule dvd_mult)
   558 done
   559 
   560 (* k dvd (m*k) *)
   561 declare dvd_refl [THEN dvd_mult, iff] dvd_refl [THEN dvd_mult2, iff]
   562 
   563 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
   564 apply (rule iffI)
   565 apply (erule_tac [2] dvd_add)
   566 apply (rule_tac [2] dvd_refl)
   567 apply (subgoal_tac "n = (n+k) -k")
   568  prefer 2 apply simp 
   569 apply (erule ssubst)
   570 apply (erule dvd_diff)
   571 apply (rule dvd_refl)
   572 done
   573 
   574 lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
   575 apply (unfold dvd_def)
   576 apply (case_tac "n=0", auto)
   577 apply (blast intro: mod_mult_distrib2 [symmetric])
   578 done
   579 
   580 lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
   581 apply (subgoal_tac "k dvd (m div n) *n + m mod n")
   582  apply (simp add: mod_div_equality)
   583 apply (simp only: dvd_add dvd_mult)
   584 done
   585 
   586 lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
   587 by (blast intro: dvd_mod_imp_dvd dvd_mod)
   588 
   589 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
   590 apply (unfold dvd_def)
   591 apply (erule exE)
   592 apply (simp add: mult_ac)
   593 done
   594 
   595 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
   596 apply auto
   597 apply (subgoal_tac "m*n dvd m*1")
   598 apply (drule dvd_mult_cancel, auto)
   599 done
   600 
   601 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
   602 apply (subst mult_commute)
   603 apply (erule dvd_mult_cancel1)
   604 done
   605 
   606 lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
   607 apply (unfold dvd_def, clarify)
   608 apply (rule_tac x = "k*ka" in exI)
   609 apply (simp add: mult_ac)
   610 done
   611 
   612 lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
   613 by (simp add: dvd_def mult_assoc, blast)
   614 
   615 lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
   616 apply (unfold dvd_def, clarify)
   617 apply (rule_tac x = "i*k" in exI)
   618 apply (simp add: mult_ac)
   619 done
   620 
   621 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
   622 apply (unfold dvd_def, clarify)
   623 apply (simp_all (no_asm_use) add: zero_less_mult_iff)
   624 apply (erule conjE)
   625 apply (rule le_trans)
   626 apply (rule_tac [2] le_refl [THEN mult_le_mono])
   627 apply (erule_tac [2] Suc_leI, simp)
   628 done
   629 
   630 lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"
   631 apply (unfold dvd_def)
   632 apply (case_tac "k=0", simp, safe)
   633 apply (simp add: mult_commute)
   634 apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])
   635 apply (subst mult_commute, simp)
   636 done
   637 
   638 lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
   639 apply (subgoal_tac "m mod n = 0")
   640  apply (simp add: mult_div_cancel)
   641 apply (simp only: dvd_eq_mod_eq_0)
   642 done
   643 
   644 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
   645 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
   646 declare mod_eq_0_iff [THEN iffD1, dest!]
   647 
   648 (*Loses information, namely we also have r<d provided d is nonzero*)
   649 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
   650 apply (cut_tac m = m in mod_div_equality)
   651 apply (simp only: add_ac)
   652 apply (blast intro: sym)
   653 done
   654 
   655 
   656 lemma split_div:
   657  "P(n div k :: nat) =
   658  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
   659  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   660 proof
   661   assume P: ?P
   662   show ?Q
   663   proof (cases)
   664     assume "k = 0"
   665     with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
   666   next
   667     assume not0: "k \<noteq> 0"
   668     thus ?Q
   669     proof (simp, intro allI impI)
   670       fix i j
   671       assume n: "n = k*i + j" and j: "j < k"
   672       show "P i"
   673       proof (cases)
   674 	assume "i = 0"
   675 	with n j P show "P i" by simp
   676       next
   677 	assume "i \<noteq> 0"
   678 	with not0 n j P show "P i" by(simp add:add_ac)
   679       qed
   680     qed
   681   qed
   682 next
   683   assume Q: ?Q
   684   show ?P
   685   proof (cases)
   686     assume "k = 0"
   687     with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
   688   next
   689     assume not0: "k \<noteq> 0"
   690     with Q have R: ?R by simp
   691     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   692     show ?P by simp
   693   qed
   694 qed
   695 
   696 lemma split_div_lemma:
   697   "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
   698   apply (rule iffI)
   699   apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
   700   apply (simp_all add: quorem_def, arith)
   701   apply (rule conjI)
   702   apply (rule_tac P="%x. n * (m div n) \<le> x" in
   703     subst [OF mod_div_equality [of _ n]])
   704   apply (simp only: add: mult_ac)
   705   apply (rule_tac P="%x. x < n + n * (m div n)" in
   706     subst [OF mod_div_equality [of _ n]])
   707   apply (simp only: add: mult_ac add_ac)
   708   apply (rule add_less_mono1, simp)
   709   done
   710 
   711 theorem split_div':
   712   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
   713    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
   714   apply (case_tac "0 < n")
   715   apply (simp only: add: split_div_lemma)
   716   apply (simp_all add: DIVISION_BY_ZERO_DIV)
   717   done
   718 
   719 lemma split_mod:
   720  "P(n mod k :: nat) =
   721  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
   722  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   723 proof
   724   assume P: ?P
   725   show ?Q
   726   proof (cases)
   727     assume "k = 0"
   728     with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
   729   next
   730     assume not0: "k \<noteq> 0"
   731     thus ?Q
   732     proof (simp, intro allI impI)
   733       fix i j
   734       assume "n = k*i + j" "j < k"
   735       thus "P j" using not0 P by(simp add:add_ac mult_ac)
   736     qed
   737   qed
   738 next
   739   assume Q: ?Q
   740   show ?P
   741   proof (cases)
   742     assume "k = 0"
   743     with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
   744   next
   745     assume not0: "k \<noteq> 0"
   746     with Q have R: ?R by simp
   747     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   748     show ?P by simp
   749   qed
   750 qed
   751 
   752 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
   753   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
   754     subst [OF mod_div_equality [of _ n]])
   755   apply arith
   756   done
   757 
   758 subsection {*An ``induction'' law for modulus arithmetic.*}
   759 
   760 lemma mod_induct_0:
   761   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
   762   and base: "P i" and i: "i<p"
   763   shows "P 0"
   764 proof (rule ccontr)
   765   assume contra: "\<not>(P 0)"
   766   from i have p: "0<p" by simp
   767   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
   768   proof
   769     fix k
   770     show "?A k"
   771     proof (induct k)
   772       show "?A 0" by simp  -- "by contradiction"
   773     next
   774       fix n
   775       assume ih: "?A n"
   776       show "?A (Suc n)"
   777       proof (clarsimp)
   778 	assume y: "P (p - Suc n)"
   779 	have n: "Suc n < p"
   780 	proof (rule ccontr)
   781 	  assume "\<not>(Suc n < p)"
   782 	  hence "p - Suc n = 0"
   783 	    by simp
   784 	  with y contra show "False"
   785 	    by simp
   786 	qed
   787 	hence n2: "Suc (p - Suc n) = p-n" by arith
   788 	from p have "p - Suc n < p" by arith
   789 	with y step have z: "P ((Suc (p - Suc n)) mod p)"
   790 	  by blast
   791 	show "False"
   792 	proof (cases "n=0")
   793 	  case True
   794 	  with z n2 contra show ?thesis by simp
   795 	next
   796 	  case False
   797 	  with p have "p-n < p" by arith
   798 	  with z n2 False ih show ?thesis by simp
   799 	qed
   800       qed
   801     qed
   802   qed
   803   moreover
   804   from i obtain k where "0<k \<and> i+k=p"
   805     by (blast dest: less_imp_add_positive)
   806   hence "0<k \<and> i=p-k" by auto
   807   moreover
   808   note base
   809   ultimately
   810   show "False" by blast
   811 qed
   812 
   813 lemma mod_induct:
   814   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
   815   and base: "P i" and i: "i<p" and j: "j<p"
   816   shows "P j"
   817 proof -
   818   have "\<forall>j<p. P j"
   819   proof
   820     fix j
   821     show "j<p \<longrightarrow> P j" (is "?A j")
   822     proof (induct j)
   823       from step base i show "?A 0"
   824 	by (auto elim: mod_induct_0)
   825     next
   826       fix k
   827       assume ih: "?A k"
   828       show "?A (Suc k)"
   829       proof
   830 	assume suc: "Suc k < p"
   831 	hence k: "k<p" by simp
   832 	with ih have "P k" ..
   833 	with step k have "P (Suc k mod p)"
   834 	  by blast
   835 	moreover
   836 	from suc have "Suc k mod p = Suc k"
   837 	  by simp
   838 	ultimately
   839 	show "P (Suc k)" by simp
   840       qed
   841     qed
   842   qed
   843   with j show ?thesis by blast
   844 qed
   845 
   846 
   847 ML
   848 {*
   849 val div_def = thm "div_def"
   850 val mod_def = thm "mod_def"
   851 val dvd_def = thm "dvd_def"
   852 val quorem_def = thm "quorem_def"
   853 
   854 val wf_less_trans = thm "wf_less_trans";
   855 val mod_eq = thm "mod_eq";
   856 val div_eq = thm "div_eq";
   857 val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";
   858 val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";
   859 val mod_less = thm "mod_less";
   860 val mod_geq = thm "mod_geq";
   861 val le_mod_geq = thm "le_mod_geq";
   862 val mod_if = thm "mod_if";
   863 val mod_1 = thm "mod_1";
   864 val mod_self = thm "mod_self";
   865 val mod_add_self2 = thm "mod_add_self2";
   866 val mod_add_self1 = thm "mod_add_self1";
   867 val mod_mult_self1 = thm "mod_mult_self1";
   868 val mod_mult_self2 = thm "mod_mult_self2";
   869 val mod_mult_distrib = thm "mod_mult_distrib";
   870 val mod_mult_distrib2 = thm "mod_mult_distrib2";
   871 val mod_mult_self_is_0 = thm "mod_mult_self_is_0";
   872 val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0";
   873 val div_less = thm "div_less";
   874 val div_geq = thm "div_geq";
   875 val le_div_geq = thm "le_div_geq";
   876 val div_if = thm "div_if";
   877 val mod_div_equality = thm "mod_div_equality";
   878 val mod_div_equality2 = thm "mod_div_equality2";
   879 val div_mod_equality = thm "div_mod_equality";
   880 val div_mod_equality2 = thm "div_mod_equality2";
   881 val mult_div_cancel = thm "mult_div_cancel";
   882 val mod_less_divisor = thm "mod_less_divisor";
   883 val div_mult_self_is_m = thm "div_mult_self_is_m";
   884 val div_mult_self1_is_m = thm "div_mult_self1_is_m";
   885 val unique_quotient_lemma = thm "unique_quotient_lemma";
   886 val unique_quotient = thm "unique_quotient";
   887 val unique_remainder = thm "unique_remainder";
   888 val div_0 = thm "div_0";
   889 val mod_0 = thm "mod_0";
   890 val div_mult1_eq = thm "div_mult1_eq";
   891 val mod_mult1_eq = thm "mod_mult1_eq";
   892 val mod_mult1_eq' = thm "mod_mult1_eq'";
   893 val mod_mult_distrib_mod = thm "mod_mult_distrib_mod";
   894 val div_add1_eq = thm "div_add1_eq";
   895 val mod_add1_eq = thm "mod_add1_eq";
   896 val mod_lemma = thm "mod_lemma";
   897 val div_mult2_eq = thm "div_mult2_eq";
   898 val mod_mult2_eq = thm "mod_mult2_eq";
   899 val div_mult_mult_lemma = thm "div_mult_mult_lemma";
   900 val div_mult_mult1 = thm "div_mult_mult1";
   901 val div_mult_mult2 = thm "div_mult_mult2";
   902 val div_1 = thm "div_1";
   903 val div_self = thm "div_self";
   904 val div_add_self2 = thm "div_add_self2";
   905 val div_add_self1 = thm "div_add_self1";
   906 val div_mult_self1 = thm "div_mult_self1";
   907 val div_mult_self2 = thm "div_mult_self2";
   908 val div_le_mono = thm "div_le_mono";
   909 val div_le_mono2 = thm "div_le_mono2";
   910 val div_le_dividend = thm "div_le_dividend";
   911 val div_less_dividend = thm "div_less_dividend";
   912 val mod_Suc = thm "mod_Suc";
   913 val dvdI = thm "dvdI";
   914 val dvdE = thm "dvdE";
   915 val dvd_0_right = thm "dvd_0_right";
   916 val dvd_0_left = thm "dvd_0_left";
   917 val dvd_0_left_iff = thm "dvd_0_left_iff";
   918 val dvd_1_left = thm "dvd_1_left";
   919 val dvd_1_iff_1 = thm "dvd_1_iff_1";
   920 val dvd_refl = thm "dvd_refl";
   921 val dvd_trans = thm "dvd_trans";
   922 val dvd_anti_sym = thm "dvd_anti_sym";
   923 val dvd_add = thm "dvd_add";
   924 val dvd_diff = thm "dvd_diff";
   925 val dvd_diffD = thm "dvd_diffD";
   926 val dvd_diffD1 = thm "dvd_diffD1";
   927 val dvd_mult = thm "dvd_mult";
   928 val dvd_mult2 = thm "dvd_mult2";
   929 val dvd_reduce = thm "dvd_reduce";
   930 val dvd_mod = thm "dvd_mod";
   931 val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd";
   932 val dvd_mod_iff = thm "dvd_mod_iff";
   933 val dvd_mult_cancel = thm "dvd_mult_cancel";
   934 val dvd_mult_cancel1 = thm "dvd_mult_cancel1";
   935 val dvd_mult_cancel2 = thm "dvd_mult_cancel2";
   936 val mult_dvd_mono = thm "mult_dvd_mono";
   937 val dvd_mult_left = thm "dvd_mult_left";
   938 val dvd_mult_right = thm "dvd_mult_right";
   939 val dvd_imp_le = thm "dvd_imp_le";
   940 val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0";
   941 val dvd_mult_div_cancel = thm "dvd_mult_div_cancel";
   942 val mod_eq_0_iff = thm "mod_eq_0_iff";
   943 val mod_eqD = thm "mod_eqD";
   944 *}
   945 
   946 
   947 (*
   948 lemma split_div:
   949 assumes m: "m \<noteq> 0"
   950 shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)"
   951        (is "?P = ?Q")
   952 proof
   953   assume P: ?P
   954   show ?Q
   955   proof (intro allI impI)
   956     fix i j
   957     assume n: "n = m*i + j" and j: "j < m"
   958     show "P i"
   959     proof (cases)
   960       assume "i = 0"
   961       with n j P show "P i" by simp
   962     next
   963       assume "i \<noteq> 0"
   964       with n j P show "P i" by (simp add:add_ac div_mult_self1)
   965     qed
   966   qed
   967 next
   968   assume Q: ?Q
   969   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
   970   show ?P by simp
   971 qed
   972 
   973 lemma split_mod:
   974 assumes m: "m \<noteq> 0"
   975 shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)"
   976        (is "?P = ?Q")
   977 proof
   978   assume P: ?P
   979   show ?Q
   980   proof (intro allI impI)
   981     fix i j
   982     assume "n = m*i + j" "j < m"
   983     thus "P j" using m P by(simp add:add_ac mult_ac)
   984   qed
   985 next
   986   assume Q: ?Q
   987   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
   988   show ?P by simp
   989 qed
   990 *)
   991 end