src/HOL/Divides.thy
author haftmann
Wed Dec 27 19:10:00 2006 +0100 (2006-12-27)
changeset 21911 e29bcab0c81c
parent 21408 fff1731da03b
child 22261 9e185f78e7d4
permissions -rw-r--r--
added OCaml code generation (without dictionaries)
     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, mod and the divides relation "dvd" *}
     8 
     9 theory Divides
    10 imports Datatype Power
    11 begin
    12 
    13 (*We use the same class for div and mod;
    14   moreover, dvd is defined whenever multiplication is*)
    15 class div =
    16   fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
    17   fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
    18 begin
    19 
    20 notation
    21   div (infixl "\<^loc>div" 70)
    22 
    23 notation
    24   mod (infixl "\<^loc>mod" 70)
    25 
    26 end
    27 
    28 notation
    29   div (infixl "div" 70)
    30 
    31 notation
    32   mod (infixl "mod" 70)
    33 
    34 instance nat :: "Divides.div"
    35   mod_def: "m mod n == wfrec (trancl pred_nat)
    36                           (%f j. if j<n | n=0 then j else f (j-n)) m"
    37   div_def:   "m div n == wfrec (trancl pred_nat) 
    38                           (%f j. if j<n | n=0 then 0 else Suc (f (j-n))) m" ..
    39 
    40 definition
    41   (*The definition of dvd is polymorphic!*)
    42   dvd  :: "'a::times \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where
    43   dvd_def: "m dvd n \<longleftrightarrow> (\<exists>k. n = m*k)"
    44 
    45 consts
    46   quorem :: "(nat*nat) * (nat*nat) => bool"
    47 
    48 defs
    49   (*This definition helps prove the harder properties of div and mod.
    50     It is copied from IntDiv.thy; should it be overloaded?*)
    51   quorem_def: "quorem \<equiv> (%((a,b), (q,r)).
    52                     a = b*q + r &
    53                     (if 0<b then 0\<le>r & r<b else b<r & r \<le>0))"
    54 
    55 
    56 
    57 subsection{*Initial Lemmas*}
    58 
    59 lemmas wf_less_trans = 
    60        def_wfrec [THEN trans, OF eq_reflection wf_pred_nat [THEN wf_trancl],
    61                   standard]
    62 
    63 lemma mod_eq: "(%m. m mod n) = 
    64               wfrec (trancl pred_nat) (%f j. if j<n | n=0 then j else f (j-n))"
    65 by (simp add: mod_def)
    66 
    67 lemma div_eq: "(%m. m div n) = wfrec (trancl pred_nat)  
    68                (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"
    69 by (simp add: div_def)
    70 
    71 
    72 (** Aribtrary definitions for division by zero.  Useful to simplify 
    73     certain equations **)
    74 
    75 lemma DIVISION_BY_ZERO_DIV [simp]: "a div 0 = (0::nat)"
    76 by (rule div_eq [THEN wf_less_trans], simp)
    77 
    78 lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a::nat)"
    79 by (rule mod_eq [THEN wf_less_trans], simp)
    80 
    81 
    82 subsection{*Remainder*}
    83 
    84 lemma mod_less [simp]: "m<n ==> m mod n = (m::nat)"
    85 by (rule mod_eq [THEN wf_less_trans], simp)
    86 
    87 lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n"
    88 apply (case_tac "n=0", simp) 
    89 apply (rule mod_eq [THEN wf_less_trans])
    90 apply (simp add: cut_apply less_eq)
    91 done
    92 
    93 (*Avoids the ugly ~m<n above*)
    94 lemma le_mod_geq: "(n::nat) \<le> m ==> m mod n = (m-n) mod n"
    95 by (simp add: mod_geq linorder_not_less)
    96 
    97 lemma mod_if: "m mod (n::nat) = (if m<n then m else (m-n) mod n)"
    98 by (simp add: mod_geq)
    99 
   100 lemma mod_1 [simp]: "m mod Suc 0 = 0"
   101 apply (induct "m")
   102 apply (simp_all (no_asm_simp) add: mod_geq)
   103 done
   104 
   105 lemma mod_self [simp]: "n mod n = (0::nat)"
   106 apply (case_tac "n=0")
   107 apply (simp_all add: mod_geq)
   108 done
   109 
   110 lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
   111 apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n") 
   112 apply (simp add: add_commute)
   113 apply (subst mod_geq [symmetric], simp_all)
   114 done
   115 
   116 lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
   117 by (simp add: add_commute mod_add_self2)
   118 
   119 lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
   120 apply (induct "k")
   121 apply (simp_all add: add_left_commute [of _ n])
   122 done
   123 
   124 lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
   125 by (simp add: mult_commute mod_mult_self1)
   126 
   127 lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)"
   128 apply (case_tac "n=0", simp)
   129 apply (case_tac "k=0", simp)
   130 apply (induct "m" rule: nat_less_induct)
   131 apply (subst mod_if, simp)
   132 apply (simp add: mod_geq diff_mult_distrib)
   133 done
   134 
   135 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
   136 by (simp add: mult_commute [of k] mod_mult_distrib)
   137 
   138 lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
   139 apply (case_tac "n=0", simp)
   140 apply (induct "m", simp)
   141 apply (rename_tac "k")
   142 apply (cut_tac m = "k*n" and n = n in mod_add_self2)
   143 apply (simp add: add_commute)
   144 done
   145 
   146 lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
   147 by (simp add: mult_commute mod_mult_self_is_0)
   148 
   149 
   150 subsection{*Quotient*}
   151 
   152 lemma div_less [simp]: "m<n ==> m div n = (0::nat)"
   153 by (rule div_eq [THEN wf_less_trans], simp)
   154 
   155 lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"
   156 apply (rule div_eq [THEN wf_less_trans])
   157 apply (simp add: cut_apply less_eq)
   158 done
   159 
   160 (*Avoids the ugly ~m<n above*)
   161 lemma le_div_geq: "[| 0<n;  n\<le>m |] ==> m div n = Suc((m-n) div n)"
   162 by (simp add: div_geq linorder_not_less)
   163 
   164 lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"
   165 by (simp add: div_geq)
   166 
   167 
   168 (*Main Result about quotient and remainder.*)
   169 lemma mod_div_equality: "(m div n)*n + m mod n = (m::nat)"
   170 apply (case_tac "n=0", simp)
   171 apply (induct "m" rule: nat_less_induct)
   172 apply (subst mod_if)
   173 apply (simp_all (no_asm_simp) add: add_assoc div_geq add_diff_inverse)
   174 done
   175 
   176 lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)"
   177 apply(cut_tac m = m and n = n in mod_div_equality)
   178 apply(simp add: mult_commute)
   179 done
   180 
   181 subsection{*Simproc for Cancelling Div and Mod*}
   182 
   183 lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k"
   184 apply(simp add: mod_div_equality)
   185 done
   186 
   187 lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k"
   188 apply(simp add: mod_div_equality2)
   189 done
   190 
   191 ML
   192 {*
   193 val div_mod_equality = thm "div_mod_equality";
   194 val div_mod_equality2 = thm "div_mod_equality2";
   195 
   196 
   197 structure CancelDivModData =
   198 struct
   199 
   200 val div_name = "Divides.div";
   201 val mod_name = "Divides.mod";
   202 val mk_binop = HOLogic.mk_binop;
   203 val mk_sum = NatArithUtils.mk_sum;
   204 val dest_sum = NatArithUtils.dest_sum;
   205 
   206 (*logic*)
   207 
   208 val div_mod_eqs = map mk_meta_eq [div_mod_equality,div_mod_equality2]
   209 
   210 val trans = trans
   211 
   212 val prove_eq_sums =
   213   let val simps = add_0 :: add_0_right :: add_ac
   214   in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;
   215 
   216 end;
   217 
   218 structure CancelDivMod = CancelDivModFun(CancelDivModData);
   219 
   220 val cancel_div_mod_proc = NatArithUtils.prep_simproc
   221       ("cancel_div_mod", ["(m::nat) + n"], K CancelDivMod.proc);
   222 
   223 Addsimprocs[cancel_div_mod_proc];
   224 *}
   225 
   226 
   227 (* a simple rearrangement of mod_div_equality: *)
   228 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
   229 by (cut_tac m = m and n = n in mod_div_equality2, arith)
   230 
   231 lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)"
   232 apply (induct "m" rule: nat_less_induct)
   233 apply (case_tac "na<n", simp) 
   234 txt{*case @{term "n \<le> na"}*}
   235 apply (simp add: mod_geq)
   236 done
   237 
   238 lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
   239 apply(drule mod_less_divisor[where m = m])
   240 apply simp
   241 done
   242 
   243 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
   244 by (cut_tac m = "m*n" and n = n in mod_div_equality, auto)
   245 
   246 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
   247 by (simp add: mult_commute div_mult_self_is_m)
   248 
   249 (*mod_mult_distrib2 above is the counterpart for remainder*)
   250 
   251 
   252 subsection{*Proving facts about Quotient and Remainder*}
   253 
   254 lemma unique_quotient_lemma:
   255      "[| b*q' + r'  \<le> b*q + r;  x < b;  r < b |]  
   256       ==> q' \<le> (q::nat)"
   257 apply (rule leI)
   258 apply (subst less_iff_Suc_add)
   259 apply (auto simp add: add_mult_distrib2)
   260 done
   261 
   262 lemma unique_quotient:
   263      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]  
   264       ==> q = q'"
   265 apply (simp add: split_ifs quorem_def)
   266 apply (blast intro: order_antisym 
   267              dest: order_eq_refl [THEN unique_quotient_lemma] sym)
   268 done
   269 
   270 lemma unique_remainder:
   271      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]  
   272       ==> r = r'"
   273 apply (subgoal_tac "q = q'")
   274 prefer 2 apply (blast intro: unique_quotient)
   275 apply (simp add: quorem_def)
   276 done
   277 
   278 lemma quorem_div_mod: "0 < b ==> quorem ((a, b), (a div b, a mod b))"
   279   unfolding quorem_def by simp 
   280 
   281 lemma quorem_div: "[| quorem((a,b),(q,r));  0 < b |] ==> a div b = q"
   282 by (simp add: quorem_div_mod [THEN unique_quotient])
   283 
   284 lemma quorem_mod: "[| quorem((a,b),(q,r));  0 < b |] ==> a mod b = r"
   285 by (simp add: quorem_div_mod [THEN unique_remainder])
   286 
   287 (** A dividend of zero **)
   288 
   289 lemma div_0 [simp]: "0 div m = (0::nat)"
   290 by (case_tac "m=0", simp_all)
   291 
   292 lemma mod_0 [simp]: "0 mod m = (0::nat)"
   293 by (case_tac "m=0", simp_all)
   294 
   295 (** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
   296 
   297 lemma quorem_mult1_eq:
   298      "[| quorem((b,c),(q,r));  0 < c |]  
   299       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
   300 apply (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
   301 done
   302 
   303 lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
   304 apply (case_tac "c = 0", simp)
   305 apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
   306 done
   307 
   308 lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
   309 apply (case_tac "c = 0", simp)
   310 apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
   311 done
   312 
   313 lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
   314 apply (rule trans)
   315 apply (rule_tac s = "b*a mod c" in trans)
   316 apply (rule_tac [2] mod_mult1_eq)
   317 apply (simp_all (no_asm) add: mult_commute)
   318 done
   319 
   320 lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
   321 apply (rule mod_mult1_eq' [THEN trans])
   322 apply (rule mod_mult1_eq)
   323 done
   324 
   325 (** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
   326 
   327 lemma quorem_add1_eq:
   328      "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  0 < c |]  
   329       ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
   330 by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
   331 
   332 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
   333 lemma div_add1_eq:
   334      "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
   335 apply (case_tac "c = 0", simp)
   336 apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod)
   337 done
   338 
   339 lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
   340 apply (case_tac "c = 0", simp)
   341 apply (blast intro: quorem_div_mod quorem_div_mod
   342                     quorem_add1_eq [THEN quorem_mod])
   343 done
   344 
   345 
   346 subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*}
   347 
   348 (** first, a lemma to bound the remainder **)
   349 
   350 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
   351 apply (cut_tac m = q and n = c in mod_less_divisor)
   352 apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
   353 apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
   354 apply (simp add: add_mult_distrib2)
   355 done
   356 
   357 lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]  
   358       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
   359 apply (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)
   360 done
   361 
   362 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
   363 apply (case_tac "b=0", simp)
   364 apply (case_tac "c=0", simp)
   365 apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
   366 done
   367 
   368 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
   369 apply (case_tac "b=0", simp)
   370 apply (case_tac "c=0", simp)
   371 apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
   372 done
   373 
   374 
   375 subsection{*Cancellation of Common Factors in Division*}
   376 
   377 lemma div_mult_mult_lemma:
   378      "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
   379 by (auto simp add: div_mult2_eq)
   380 
   381 lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
   382 apply (case_tac "b = 0")
   383 apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
   384 done
   385 
   386 lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
   387 apply (drule div_mult_mult1)
   388 apply (auto simp add: mult_commute)
   389 done
   390 
   391 
   392 (*Distribution of Factors over Remainders:
   393 
   394 Could prove these as in Integ/IntDiv.ML, but we already have
   395 mod_mult_distrib and mod_mult_distrib2 above!
   396 
   397 Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)"
   398 qed "mod_mult_mult1";
   399 
   400 Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)";
   401 qed "mod_mult_mult2";
   402  ***)
   403 
   404 subsection{*Further Facts about Quotient and Remainder*}
   405 
   406 lemma div_1 [simp]: "m div Suc 0 = m"
   407 apply (induct "m")
   408 apply (simp_all (no_asm_simp) add: div_geq)
   409 done
   410 
   411 lemma div_self [simp]: "0<n ==> n div n = (1::nat)"
   412 by (simp add: div_geq)
   413 
   414 lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
   415 apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
   416 apply (simp add: add_commute)
   417 apply (subst div_geq [symmetric], simp_all)
   418 done
   419 
   420 lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
   421 by (simp add: add_commute div_add_self2)
   422 
   423 lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
   424 apply (subst div_add1_eq)
   425 apply (subst div_mult1_eq, simp)
   426 done
   427 
   428 lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
   429 by (simp add: mult_commute div_mult_self1)
   430 
   431 
   432 (* Monotonicity of div in first argument *)
   433 lemma div_le_mono [rule_format (no_asm)]:
   434      "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
   435 apply (case_tac "k=0", simp)
   436 apply (induct "n" rule: nat_less_induct, clarify)
   437 apply (case_tac "n<k")
   438 (* 1  case n<k *)
   439 apply simp
   440 (* 2  case n >= k *)
   441 apply (case_tac "m<k")
   442 (* 2.1  case m<k *)
   443 apply simp
   444 (* 2.2  case m>=k *)
   445 apply (simp add: div_geq diff_le_mono)
   446 done
   447 
   448 (* Antimonotonicity of div in second argument *)
   449 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
   450 apply (subgoal_tac "0<n")
   451  prefer 2 apply simp 
   452 apply (induct_tac k rule: nat_less_induct)
   453 apply (rename_tac "k")
   454 apply (case_tac "k<n", simp)
   455 apply (subgoal_tac "~ (k<m) ")
   456  prefer 2 apply simp 
   457 apply (simp add: div_geq)
   458 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
   459  prefer 2
   460  apply (blast intro: div_le_mono diff_le_mono2)
   461 apply (rule le_trans, simp)
   462 apply (simp)
   463 done
   464 
   465 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
   466 apply (case_tac "n=0", simp)
   467 apply (subgoal_tac "m div n \<le> m div 1", simp)
   468 apply (rule div_le_mono2)
   469 apply (simp_all (no_asm_simp))
   470 done
   471 
   472 (* Similar for "less than" *) 
   473 lemma div_less_dividend [rule_format]:
   474      "!!n::nat. 1<n ==> 0 < m --> m div n < m"
   475 apply (induct_tac m rule: nat_less_induct)
   476 apply (rename_tac "m")
   477 apply (case_tac "m<n", simp)
   478 apply (subgoal_tac "0<n")
   479  prefer 2 apply simp 
   480 apply (simp add: div_geq)
   481 apply (case_tac "n<m")
   482  apply (subgoal_tac "(m-n) div n < (m-n) ")
   483   apply (rule impI less_trans_Suc)+
   484 apply assumption
   485   apply (simp_all)
   486 done
   487 
   488 declare div_less_dividend [simp]
   489 
   490 text{*A fact for the mutilated chess board*}
   491 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
   492 apply (case_tac "n=0", simp)
   493 apply (induct "m" rule: nat_less_induct)
   494 apply (case_tac "Suc (na) <n")
   495 (* case Suc(na) < n *)
   496 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
   497 (* case n \<le> Suc(na) *)
   498 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
   499 apply (auto simp add: Suc_diff_le le_mod_geq)
   500 done
   501 
   502 lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
   503 by (case_tac "n=0", auto)
   504 
   505 lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
   506 by (case_tac "n=0", auto)
   507 
   508 
   509 subsection{*The Divides Relation*}
   510 
   511 lemma dvdI [intro?]: "n = m * k ==> m dvd n"
   512 by (unfold dvd_def, blast)
   513 
   514 lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
   515 by (unfold dvd_def, blast)
   516 
   517 lemma dvd_0_right [iff]: "m dvd (0::nat)"
   518 apply (unfold dvd_def)
   519 apply (blast intro: mult_0_right [symmetric])
   520 done
   521 
   522 lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
   523 by (force simp add: dvd_def)
   524 
   525 lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
   526 by (blast intro: dvd_0_left)
   527 
   528 lemma dvd_1_left [iff]: "Suc 0 dvd k"
   529 by (unfold dvd_def, simp)
   530 
   531 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
   532 by (simp add: dvd_def)
   533 
   534 lemma dvd_refl [simp]: "m dvd (m::nat)"
   535 apply (unfold dvd_def)
   536 apply (blast intro: mult_1_right [symmetric])
   537 done
   538 
   539 lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"
   540 apply (unfold dvd_def)
   541 apply (blast intro: mult_assoc)
   542 done
   543 
   544 lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
   545 apply (unfold dvd_def)
   546 apply (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
   547 done
   548 
   549 lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
   550 apply (unfold dvd_def)
   551 apply (blast intro: add_mult_distrib2 [symmetric])
   552 done
   553 
   554 lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
   555 apply (unfold dvd_def)
   556 apply (blast intro: diff_mult_distrib2 [symmetric])
   557 done
   558 
   559 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
   560 apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
   561 apply (blast intro: dvd_add)
   562 done
   563 
   564 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
   565 by (drule_tac m = m in dvd_diff, auto)
   566 
   567 lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
   568 apply (unfold dvd_def)
   569 apply (blast intro: mult_left_commute)
   570 done
   571 
   572 lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
   573 apply (subst mult_commute)
   574 apply (erule dvd_mult)
   575 done
   576 
   577 lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
   578 by (rule dvd_refl [THEN dvd_mult])
   579 
   580 lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
   581 by (rule dvd_refl [THEN dvd_mult2])
   582 
   583 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
   584 apply (rule iffI)
   585 apply (erule_tac [2] dvd_add)
   586 apply (rule_tac [2] dvd_refl)
   587 apply (subgoal_tac "n = (n+k) -k")
   588  prefer 2 apply simp 
   589 apply (erule ssubst)
   590 apply (erule dvd_diff)
   591 apply (rule dvd_refl)
   592 done
   593 
   594 lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
   595 apply (unfold dvd_def)
   596 apply (case_tac "n=0", auto)
   597 apply (blast intro: mod_mult_distrib2 [symmetric])
   598 done
   599 
   600 lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
   601 apply (subgoal_tac "k dvd (m div n) *n + m mod n")
   602  apply (simp add: mod_div_equality)
   603 apply (simp only: dvd_add dvd_mult)
   604 done
   605 
   606 lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
   607 by (blast intro: dvd_mod_imp_dvd dvd_mod)
   608 
   609 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
   610 apply (unfold dvd_def)
   611 apply (erule exE)
   612 apply (simp add: mult_ac)
   613 done
   614 
   615 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
   616 apply auto
   617 apply (subgoal_tac "m*n dvd m*1")
   618 apply (drule dvd_mult_cancel, auto)
   619 done
   620 
   621 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
   622 apply (subst mult_commute)
   623 apply (erule dvd_mult_cancel1)
   624 done
   625 
   626 lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
   627 apply (unfold dvd_def, clarify)
   628 apply (rule_tac x = "k*ka" in exI)
   629 apply (simp add: mult_ac)
   630 done
   631 
   632 lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
   633 by (simp add: dvd_def mult_assoc, blast)
   634 
   635 lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
   636 apply (unfold dvd_def, clarify)
   637 apply (rule_tac x = "i*k" in exI)
   638 apply (simp add: mult_ac)
   639 done
   640 
   641 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
   642 apply (unfold dvd_def, clarify)
   643 apply (simp_all (no_asm_use) add: zero_less_mult_iff)
   644 apply (erule conjE)
   645 apply (rule le_trans)
   646 apply (rule_tac [2] le_refl [THEN mult_le_mono])
   647 apply (erule_tac [2] Suc_leI, simp)
   648 done
   649 
   650 lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"
   651 apply (unfold dvd_def)
   652 apply (case_tac "k=0", simp, safe)
   653 apply (simp add: mult_commute)
   654 apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])
   655 apply (subst mult_commute, simp)
   656 done
   657 
   658 lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
   659 apply (subgoal_tac "m mod n = 0")
   660  apply (simp add: mult_div_cancel)
   661 apply (simp only: dvd_eq_mod_eq_0)
   662 done
   663 
   664 lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
   665 apply (unfold dvd_def)
   666 apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
   667 apply (simp add: power_add)
   668 done
   669 
   670 lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
   671 by (induct "n", auto)
   672 
   673 lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
   674 apply (induct "j")
   675 apply (simp_all add: le_Suc_eq)
   676 apply (blast dest!: dvd_mult_right)
   677 done
   678 
   679 lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
   680 apply (rule power_le_imp_le_exp, assumption)
   681 apply (erule dvd_imp_le, simp)
   682 done
   683 
   684 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
   685 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
   686 
   687 lemmas mod_eq_0D = mod_eq_0_iff [THEN iffD1]
   688 declare mod_eq_0D [dest!]
   689 
   690 (*Loses information, namely we also have r<d provided d is nonzero*)
   691 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
   692 apply (cut_tac m = m in mod_div_equality)
   693 apply (simp only: add_ac)
   694 apply (blast intro: sym)
   695 done
   696 
   697 
   698 lemma split_div:
   699  "P(n div k :: nat) =
   700  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
   701  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   702 proof
   703   assume P: ?P
   704   show ?Q
   705   proof (cases)
   706     assume "k = 0"
   707     with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
   708   next
   709     assume not0: "k \<noteq> 0"
   710     thus ?Q
   711     proof (simp, intro allI impI)
   712       fix i j
   713       assume n: "n = k*i + j" and j: "j < k"
   714       show "P i"
   715       proof (cases)
   716 	assume "i = 0"
   717 	with n j P show "P i" by simp
   718       next
   719 	assume "i \<noteq> 0"
   720 	with not0 n j P show "P i" by(simp add:add_ac)
   721       qed
   722     qed
   723   qed
   724 next
   725   assume Q: ?Q
   726   show ?P
   727   proof (cases)
   728     assume "k = 0"
   729     with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
   730   next
   731     assume not0: "k \<noteq> 0"
   732     with Q have R: ?R by simp
   733     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   734     show ?P by simp
   735   qed
   736 qed
   737 
   738 lemma split_div_lemma:
   739   "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
   740   apply (rule iffI)
   741   apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
   742 prefer 3; apply assumption
   743   apply (simp_all add: quorem_def) apply arith
   744   apply (rule conjI)
   745   apply (rule_tac P="%x. n * (m div n) \<le> x" in
   746     subst [OF mod_div_equality [of _ n]])
   747   apply (simp only: add: mult_ac)
   748   apply (rule_tac P="%x. x < n + n * (m div n)" in
   749     subst [OF mod_div_equality [of _ n]])
   750   apply (simp only: add: mult_ac add_ac)
   751   apply (rule add_less_mono1, simp)
   752   done
   753 
   754 theorem split_div':
   755   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
   756    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
   757   apply (case_tac "0 < n")
   758   apply (simp only: add: split_div_lemma)
   759   apply (simp_all add: DIVISION_BY_ZERO_DIV)
   760   done
   761 
   762 lemma split_mod:
   763  "P(n mod k :: nat) =
   764  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
   765  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   766 proof
   767   assume P: ?P
   768   show ?Q
   769   proof (cases)
   770     assume "k = 0"
   771     with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
   772   next
   773     assume not0: "k \<noteq> 0"
   774     thus ?Q
   775     proof (simp, intro allI impI)
   776       fix i j
   777       assume "n = k*i + j" "j < k"
   778       thus "P j" using not0 P by(simp add:add_ac mult_ac)
   779     qed
   780   qed
   781 next
   782   assume Q: ?Q
   783   show ?P
   784   proof (cases)
   785     assume "k = 0"
   786     with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
   787   next
   788     assume not0: "k \<noteq> 0"
   789     with Q have R: ?R by simp
   790     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   791     show ?P by simp
   792   qed
   793 qed
   794 
   795 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
   796   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
   797     subst [OF mod_div_equality [of _ n]])
   798   apply arith
   799   done
   800 
   801 subsection {*An ``induction'' law for modulus arithmetic.*}
   802 
   803 lemma mod_induct_0:
   804   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
   805   and base: "P i" and i: "i<p"
   806   shows "P 0"
   807 proof (rule ccontr)
   808   assume contra: "\<not>(P 0)"
   809   from i have p: "0<p" by simp
   810   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
   811   proof
   812     fix k
   813     show "?A k"
   814     proof (induct k)
   815       show "?A 0" by simp  -- "by contradiction"
   816     next
   817       fix n
   818       assume ih: "?A n"
   819       show "?A (Suc n)"
   820       proof (clarsimp)
   821 	assume y: "P (p - Suc n)"
   822 	have n: "Suc n < p"
   823 	proof (rule ccontr)
   824 	  assume "\<not>(Suc n < p)"
   825 	  hence "p - Suc n = 0"
   826 	    by simp
   827 	  with y contra show "False"
   828 	    by simp
   829 	qed
   830 	hence n2: "Suc (p - Suc n) = p-n" by arith
   831 	from p have "p - Suc n < p" by arith
   832 	with y step have z: "P ((Suc (p - Suc n)) mod p)"
   833 	  by blast
   834 	show "False"
   835 	proof (cases "n=0")
   836 	  case True
   837 	  with z n2 contra show ?thesis by simp
   838 	next
   839 	  case False
   840 	  with p have "p-n < p" by arith
   841 	  with z n2 False ih show ?thesis by simp
   842 	qed
   843       qed
   844     qed
   845   qed
   846   moreover
   847   from i obtain k where "0<k \<and> i+k=p"
   848     by (blast dest: less_imp_add_positive)
   849   hence "0<k \<and> i=p-k" by auto
   850   moreover
   851   note base
   852   ultimately
   853   show "False" by blast
   854 qed
   855 
   856 lemma mod_induct:
   857   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
   858   and base: "P i" and i: "i<p" and j: "j<p"
   859   shows "P j"
   860 proof -
   861   have "\<forall>j<p. P j"
   862   proof
   863     fix j
   864     show "j<p \<longrightarrow> P j" (is "?A j")
   865     proof (induct j)
   866       from step base i show "?A 0"
   867 	by (auto elim: mod_induct_0)
   868     next
   869       fix k
   870       assume ih: "?A k"
   871       show "?A (Suc k)"
   872       proof
   873 	assume suc: "Suc k < p"
   874 	hence k: "k<p" by simp
   875 	with ih have "P k" ..
   876 	with step k have "P (Suc k mod p)"
   877 	  by blast
   878 	moreover
   879 	from suc have "Suc k mod p = Suc k"
   880 	  by simp
   881 	ultimately
   882 	show "P (Suc k)" by simp
   883       qed
   884     qed
   885   qed
   886   with j show ?thesis by blast
   887 qed
   888 
   889 
   890 lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
   891   apply (rule trans [symmetric])
   892   apply (rule mod_add1_eq, simp)
   893   apply (rule mod_add1_eq [symmetric])
   894   done
   895 
   896 lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
   897 apply (rule trans [symmetric])
   898 apply (rule mod_add1_eq, simp)
   899 apply (rule mod_add1_eq [symmetric])
   900 done
   901 
   902 
   903 subsection {* Code generation for div and mod *}
   904 
   905 definition
   906   "divmod (m\<Colon>nat) n = (m div n, m mod n)"
   907 
   908 lemma divmod_zero [code]:
   909   "divmod m 0 = (0, m)"
   910   unfolding divmod_def by simp
   911 
   912 lemma divmod_succ [code]:
   913   "divmod m (Suc k) = (if m < Suc k then (0, m) else
   914     let
   915       (p, q) = divmod (m - Suc k) (Suc k)
   916     in (Suc p, q)
   917   )"
   918   unfolding divmod_def Let_def split_def
   919   by (auto intro: div_geq mod_geq)
   920 
   921 lemma div_divmod [code]:
   922   "m div n = fst (divmod m n)"
   923   unfolding divmod_def by simp
   924 
   925 lemma mod_divmod [code]:
   926   "m mod n = snd (divmod m n)"
   927   unfolding divmod_def by simp
   928 
   929 code_modulename SML
   930   Divides Integer
   931 
   932 code_modulename OCaml
   933   Divides Integer
   934 
   935 hide (open) const divmod
   936 
   937 
   938 subsection {* Legacy bindings *}
   939 
   940 ML
   941 {*
   942 val div_def = thm "div_def"
   943 val mod_def = thm "mod_def"
   944 val dvd_def = thm "dvd_def"
   945 val quorem_def = thm "quorem_def"
   946 
   947 val wf_less_trans = thm "wf_less_trans";
   948 val mod_eq = thm "mod_eq";
   949 val div_eq = thm "div_eq";
   950 val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";
   951 val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";
   952 val mod_less = thm "mod_less";
   953 val mod_geq = thm "mod_geq";
   954 val le_mod_geq = thm "le_mod_geq";
   955 val mod_if = thm "mod_if";
   956 val mod_1 = thm "mod_1";
   957 val mod_self = thm "mod_self";
   958 val mod_add_self2 = thm "mod_add_self2";
   959 val mod_add_self1 = thm "mod_add_self1";
   960 val mod_mult_self1 = thm "mod_mult_self1";
   961 val mod_mult_self2 = thm "mod_mult_self2";
   962 val mod_mult_distrib = thm "mod_mult_distrib";
   963 val mod_mult_distrib2 = thm "mod_mult_distrib2";
   964 val mod_mult_self_is_0 = thm "mod_mult_self_is_0";
   965 val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0";
   966 val div_less = thm "div_less";
   967 val div_geq = thm "div_geq";
   968 val le_div_geq = thm "le_div_geq";
   969 val div_if = thm "div_if";
   970 val mod_div_equality = thm "mod_div_equality";
   971 val mod_div_equality2 = thm "mod_div_equality2";
   972 val div_mod_equality = thm "div_mod_equality";
   973 val div_mod_equality2 = thm "div_mod_equality2";
   974 val mult_div_cancel = thm "mult_div_cancel";
   975 val mod_less_divisor = thm "mod_less_divisor";
   976 val div_mult_self_is_m = thm "div_mult_self_is_m";
   977 val div_mult_self1_is_m = thm "div_mult_self1_is_m";
   978 val unique_quotient_lemma = thm "unique_quotient_lemma";
   979 val unique_quotient = thm "unique_quotient";
   980 val unique_remainder = thm "unique_remainder";
   981 val div_0 = thm "div_0";
   982 val mod_0 = thm "mod_0";
   983 val div_mult1_eq = thm "div_mult1_eq";
   984 val mod_mult1_eq = thm "mod_mult1_eq";
   985 val mod_mult1_eq' = thm "mod_mult1_eq'";
   986 val mod_mult_distrib_mod = thm "mod_mult_distrib_mod";
   987 val div_add1_eq = thm "div_add1_eq";
   988 val mod_add1_eq = thm "mod_add1_eq";
   989 val mod_add_left_eq = thm "mod_add_left_eq";
   990  val mod_add_right_eq = thm "mod_add_right_eq";
   991 val mod_lemma = thm "mod_lemma";
   992 val div_mult2_eq = thm "div_mult2_eq";
   993 val mod_mult2_eq = thm "mod_mult2_eq";
   994 val div_mult_mult_lemma = thm "div_mult_mult_lemma";
   995 val div_mult_mult1 = thm "div_mult_mult1";
   996 val div_mult_mult2 = thm "div_mult_mult2";
   997 val div_1 = thm "div_1";
   998 val div_self = thm "div_self";
   999 val div_add_self2 = thm "div_add_self2";
  1000 val div_add_self1 = thm "div_add_self1";
  1001 val div_mult_self1 = thm "div_mult_self1";
  1002 val div_mult_self2 = thm "div_mult_self2";
  1003 val div_le_mono = thm "div_le_mono";
  1004 val div_le_mono2 = thm "div_le_mono2";
  1005 val div_le_dividend = thm "div_le_dividend";
  1006 val div_less_dividend = thm "div_less_dividend";
  1007 val mod_Suc = thm "mod_Suc";
  1008 val dvdI = thm "dvdI";
  1009 val dvdE = thm "dvdE";
  1010 val dvd_0_right = thm "dvd_0_right";
  1011 val dvd_0_left = thm "dvd_0_left";
  1012 val dvd_0_left_iff = thm "dvd_0_left_iff";
  1013 val dvd_1_left = thm "dvd_1_left";
  1014 val dvd_1_iff_1 = thm "dvd_1_iff_1";
  1015 val dvd_refl = thm "dvd_refl";
  1016 val dvd_trans = thm "dvd_trans";
  1017 val dvd_anti_sym = thm "dvd_anti_sym";
  1018 val dvd_add = thm "dvd_add";
  1019 val dvd_diff = thm "dvd_diff";
  1020 val dvd_diffD = thm "dvd_diffD";
  1021 val dvd_diffD1 = thm "dvd_diffD1";
  1022 val dvd_mult = thm "dvd_mult";
  1023 val dvd_mult2 = thm "dvd_mult2";
  1024 val dvd_reduce = thm "dvd_reduce";
  1025 val dvd_mod = thm "dvd_mod";
  1026 val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd";
  1027 val dvd_mod_iff = thm "dvd_mod_iff";
  1028 val dvd_mult_cancel = thm "dvd_mult_cancel";
  1029 val dvd_mult_cancel1 = thm "dvd_mult_cancel1";
  1030 val dvd_mult_cancel2 = thm "dvd_mult_cancel2";
  1031 val mult_dvd_mono = thm "mult_dvd_mono";
  1032 val dvd_mult_left = thm "dvd_mult_left";
  1033 val dvd_mult_right = thm "dvd_mult_right";
  1034 val dvd_imp_le = thm "dvd_imp_le";
  1035 val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0";
  1036 val dvd_mult_div_cancel = thm "dvd_mult_div_cancel";
  1037 val mod_eq_0_iff = thm "mod_eq_0_iff";
  1038 val mod_eqD = thm "mod_eqD";
  1039 *}
  1040 
  1041 
  1042 (*
  1043 lemma split_div:
  1044 assumes m: "m \<noteq> 0"
  1045 shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)"
  1046        (is "?P = ?Q")
  1047 proof
  1048   assume P: ?P
  1049   show ?Q
  1050   proof (intro allI impI)
  1051     fix i j
  1052     assume n: "n = m*i + j" and j: "j < m"
  1053     show "P i"
  1054     proof (cases)
  1055       assume "i = 0"
  1056       with n j P show "P i" by simp
  1057     next
  1058       assume "i \<noteq> 0"
  1059       with n j P show "P i" by (simp add:add_ac div_mult_self1)
  1060     qed
  1061   qed
  1062 next
  1063   assume Q: ?Q
  1064   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
  1065   show ?P by simp
  1066 qed
  1067 
  1068 lemma split_mod:
  1069 assumes m: "m \<noteq> 0"
  1070 shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)"
  1071        (is "?P = ?Q")
  1072 proof
  1073   assume P: ?P
  1074   show ?Q
  1075   proof (intro allI impI)
  1076     fix i j
  1077     assume "n = m*i + j" "j < m"
  1078     thus "P j" using m P by(simp add:add_ac mult_ac)
  1079   qed
  1080 next
  1081   assume Q: ?Q
  1082   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
  1083   show ?P by simp
  1084 qed
  1085 *)
  1086 end