src/HOL/Divides.thy
author haftmann
Sat May 19 11:33:21 2007 +0200 (2007-05-19)
changeset 23017 00c0e4c42396
parent 22993 838c66e760b5
child 23162 b9853c187a1e
permissions -rw-r--r--
uniform module names for code generation
     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 uses "~~/src/Provers/Arith/cancel_div_mod.ML"
    12 begin
    13 
    14 (*We use the same class for div and mod;
    15   moreover, dvd is defined whenever multiplication is*)
    16 class div = type +
    17   fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
    18   fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
    19 begin
    20 
    21 notation
    22   div (infixl "\<^loc>div" 70)
    23 
    24 notation
    25   mod (infixl "\<^loc>mod" 70)
    26 
    27 end
    28 
    29 notation
    30   div (infixl "div" 70)
    31 
    32 notation
    33   mod (infixl "mod" 70)
    34 
    35 instance nat :: Divides.div
    36   div_def: "m div n == wfrec (pred_nat^+)
    37                           (%f j. if j<n | n=0 then 0 else Suc (f (j-n))) m"
    38   mod_def: "m mod n == wfrec (pred_nat^+)
    39                           (%f j. if j<n | n=0 then j else f (j-n)) m" ..
    40 
    41 definition
    42   (*The definition of dvd is polymorphic!*)
    43   dvd  :: "'a::times \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50) where
    44   dvd_def: "m dvd n \<longleftrightarrow> (\<exists>k. n = m*k)"
    45 
    46 definition
    47   quorem :: "(nat*nat) * (nat*nat) => bool" where
    48   (*This definition helps prove the harder properties of div and mod.
    49     It is copied from IntDiv.thy; should it be overloaded?*)
    50   "quorem = (%((a,b), (q,r)).
    51                     a = b*q + r &
    52                     (if 0<b then 0\<le>r & r<b else b<r & r \<le>0))"
    53 
    54 
    55 
    56 subsection{*Initial Lemmas*}
    57 
    58 lemmas wf_less_trans =
    59        def_wfrec [THEN trans, OF eq_reflection wf_pred_nat [THEN wf_trancl],
    60                   standard]
    61 
    62 lemma mod_eq: "(%m. m mod n) =
    63               wfrec (pred_nat^+) (%f j. if j<n | n=0 then j else f (j-n))"
    64 by (simp add: mod_def)
    65 
    66 lemma div_eq: "(%m. m div n) = wfrec (pred_nat^+)
    67                (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"
    68 by (simp add: div_def)
    69 
    70 
    71 (** Aribtrary definitions for division by zero.  Useful to simplify
    72     certain equations **)
    73 
    74 lemma DIVISION_BY_ZERO_DIV [simp]: "a div 0 = (0::nat)"
    75   by (rule div_eq [THEN wf_less_trans], simp)
    76 
    77 lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a::nat)"
    78   by (rule mod_eq [THEN wf_less_trans], simp)
    79 
    80 
    81 subsection{*Remainder*}
    82 
    83 lemma mod_less [simp]: "m<n ==> m mod n = (m::nat)"
    84   by (rule mod_eq [THEN wf_less_trans]) simp
    85 
    86 lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n"
    87   apply (cases "n=0")
    88    apply 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   by (induct m) (simp_all add: mod_geq)
   102 
   103 lemma mod_self [simp]: "n mod n = (0::nat)"
   104   by (cases "n = 0") (simp_all add: mod_geq)
   105 
   106 lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
   107   apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n")
   108    apply (simp add: add_commute)
   109   apply (subst mod_geq [symmetric], simp_all)
   110   done
   111 
   112 lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
   113   by (simp add: add_commute mod_add_self2)
   114 
   115 lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
   116   by (induct k) (simp_all add: add_left_commute [of _ n])
   117 
   118 lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
   119   by (simp add: mult_commute mod_mult_self1)
   120 
   121 lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)"
   122   apply (cases "n = 0", simp)
   123   apply (cases "k = 0", simp)
   124   apply (induct m rule: nat_less_induct)
   125   apply (subst mod_if, simp)
   126   apply (simp add: mod_geq diff_mult_distrib)
   127   done
   128 
   129 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
   130   by (simp add: mult_commute [of k] mod_mult_distrib)
   131 
   132 lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
   133   apply (cases "n = 0", simp)
   134   apply (induct m, simp)
   135   apply (rename_tac k)
   136   apply (cut_tac m = "k * n" and n = n in mod_add_self2)
   137   apply (simp add: add_commute)
   138   done
   139 
   140 lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
   141   by (simp add: mult_commute mod_mult_self_is_0)
   142 
   143 
   144 subsection{*Quotient*}
   145 
   146 lemma div_less [simp]: "m<n ==> m div n = (0::nat)"
   147   by (rule div_eq [THEN wf_less_trans], simp)
   148 
   149 lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"
   150   apply (rule div_eq [THEN wf_less_trans])
   151   apply (simp add: cut_apply less_eq)
   152   done
   153 
   154 (*Avoids the ugly ~m<n above*)
   155 lemma le_div_geq: "[| 0<n;  n\<le>m |] ==> m div n = Suc((m-n) div n)"
   156   by (simp add: div_geq linorder_not_less)
   157 
   158 lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"
   159   by (simp add: div_geq)
   160 
   161 
   162 (*Main Result about quotient and remainder.*)
   163 lemma mod_div_equality: "(m div n)*n + m mod n = (m::nat)"
   164   apply (cases "n = 0", simp)
   165   apply (induct m rule: nat_less_induct)
   166   apply (subst mod_if)
   167   apply (simp_all add: add_assoc div_geq add_diff_inverse)
   168   done
   169 
   170 lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)"
   171   apply (cut_tac m = m and n = n in mod_div_equality)
   172   apply (simp add: mult_commute)
   173   done
   174 
   175 subsection{*Simproc for Cancelling Div and Mod*}
   176 
   177 lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k"
   178   by (simp add: mod_div_equality)
   179 
   180 lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k"
   181   by (simp add: mod_div_equality2)
   182 
   183 ML
   184 {*
   185 structure CancelDivModData =
   186 struct
   187 
   188 val div_name = @{const_name Divides.div};
   189 val mod_name = @{const_name Divides.mod};
   190 val mk_binop = HOLogic.mk_binop;
   191 val mk_sum = NatArithUtils.mk_sum;
   192 val dest_sum = NatArithUtils.dest_sum;
   193 
   194 (*logic*)
   195 
   196 val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
   197 
   198 val trans = trans
   199 
   200 val prove_eq_sums =
   201   let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
   202   in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;
   203 
   204 end;
   205 
   206 structure CancelDivMod = CancelDivModFun(CancelDivModData);
   207 
   208 val cancel_div_mod_proc = NatArithUtils.prep_simproc
   209       ("cancel_div_mod", ["(m::nat) + n"], K CancelDivMod.proc);
   210 
   211 Addsimprocs[cancel_div_mod_proc];
   212 *}
   213 
   214 
   215 (* a simple rearrangement of mod_div_equality: *)
   216 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
   217   by (cut_tac m = m and n = n in mod_div_equality2, arith)
   218 
   219 lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)"
   220   apply (induct m rule: nat_less_induct)
   221   apply (rename_tac m)
   222   apply (case_tac "m<n", simp)
   223   txt{*case @{term "n \<le> m"}*}
   224   apply (simp add: mod_geq)
   225   done
   226 
   227 lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
   228   apply (drule mod_less_divisor [where m = m])
   229   apply simp
   230   done
   231 
   232 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
   233   by (cut_tac m = "m*n" and n = n in mod_div_equality, auto)
   234 
   235 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
   236   by (simp add: mult_commute div_mult_self_is_m)
   237 
   238 (*mod_mult_distrib2 above is the counterpart for remainder*)
   239 
   240 
   241 subsection{*Proving facts about Quotient and Remainder*}
   242 
   243 lemma unique_quotient_lemma:
   244      "[| b*q' + r'  \<le> b*q + r;  x < b;  r < b |]
   245       ==> q' \<le> (q::nat)"
   246   apply (rule leI)
   247   apply (subst less_iff_Suc_add)
   248   apply (auto simp add: add_mult_distrib2)
   249   done
   250 
   251 lemma unique_quotient:
   252      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
   253       ==> q = q'"
   254   apply (simp add: split_ifs quorem_def)
   255   apply (blast intro: order_antisym
   256     dest: order_eq_refl [THEN unique_quotient_lemma] sym)
   257   done
   258 
   259 lemma unique_remainder:
   260      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
   261       ==> r = r'"
   262   apply (subgoal_tac "q = q'")
   263    prefer 2 apply (blast intro: unique_quotient)
   264   apply (simp add: quorem_def)
   265   done
   266 
   267 lemma quorem_div_mod: "0 < b ==> quorem ((a, b), (a div b, a mod b))"
   268   unfolding quorem_def by simp
   269 
   270 lemma quorem_div: "[| quorem((a,b),(q,r));  0 < b |] ==> a div b = q"
   271   by (simp add: quorem_div_mod [THEN unique_quotient])
   272 
   273 lemma quorem_mod: "[| quorem((a,b),(q,r));  0 < b |] ==> a mod b = r"
   274   by (simp add: quorem_div_mod [THEN unique_remainder])
   275 
   276 (** A dividend of zero **)
   277 
   278 lemma div_0 [simp]: "0 div m = (0::nat)"
   279   by (cases "m = 0") simp_all
   280 
   281 lemma mod_0 [simp]: "0 mod m = (0::nat)"
   282   by (cases "m = 0") simp_all
   283 
   284 (** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
   285 
   286 lemma quorem_mult1_eq:
   287      "[| quorem((b,c),(q,r));  0 < c |]
   288       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
   289   by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
   290 
   291 lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
   292   apply (cases "c = 0", simp)
   293   apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
   294   done
   295 
   296 lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
   297   apply (cases "c = 0", simp)
   298   apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
   299   done
   300 
   301 lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
   302   apply (rule trans)
   303    apply (rule_tac s = "b*a mod c" in trans)
   304     apply (rule_tac [2] mod_mult1_eq)
   305    apply (simp_all add: mult_commute)
   306   done
   307 
   308 lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
   309   apply (rule mod_mult1_eq' [THEN trans])
   310   apply (rule mod_mult1_eq)
   311   done
   312 
   313 (** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
   314 
   315 lemma quorem_add1_eq:
   316      "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  0 < c |]
   317       ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
   318   by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
   319 
   320 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
   321 lemma div_add1_eq:
   322      "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
   323   apply (cases "c = 0", simp)
   324   apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod)
   325   done
   326 
   327 lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
   328   apply (cases "c = 0", simp)
   329   apply (blast intro: quorem_div_mod quorem_div_mod quorem_add1_eq [THEN quorem_mod])
   330   done
   331 
   332 
   333 subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*}
   334 
   335 (** first, a lemma to bound the remainder **)
   336 
   337 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
   338   apply (cut_tac m = q and n = c in mod_less_divisor)
   339   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
   340   apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
   341   apply (simp add: add_mult_distrib2)
   342   done
   343 
   344 lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]
   345       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
   346   by (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)
   347 
   348 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
   349   apply (cases "b = 0", simp)
   350   apply (cases "c = 0", simp)
   351   apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
   352   done
   353 
   354 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
   355   apply (cases "b = 0", simp)
   356   apply (cases "c = 0", simp)
   357   apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
   358   done
   359 
   360 
   361 subsection{*Cancellation of Common Factors in Division*}
   362 
   363 lemma div_mult_mult_lemma:
   364     "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
   365   by (auto simp add: div_mult2_eq)
   366 
   367 lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
   368   apply (cases "b = 0")
   369   apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
   370   done
   371 
   372 lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
   373   apply (drule div_mult_mult1)
   374   apply (auto simp add: mult_commute)
   375   done
   376 
   377 
   378 (*Distribution of Factors over Remainders:
   379 
   380 Could prove these as in Integ/IntDiv.ML, but we already have
   381 mod_mult_distrib and mod_mult_distrib2 above!
   382 
   383 Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)"
   384 qed "mod_mult_mult1";
   385 
   386 Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)";
   387 qed "mod_mult_mult2";
   388  ***)
   389 
   390 subsection{*Further Facts about Quotient and Remainder*}
   391 
   392 lemma div_1 [simp]: "m div Suc 0 = m"
   393   by (induct m) (simp_all add: div_geq)
   394 
   395 lemma div_self [simp]: "0<n ==> n div n = (1::nat)"
   396   by (simp add: div_geq)
   397 
   398 lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
   399   apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
   400    apply (simp add: add_commute)
   401   apply (subst div_geq [symmetric], simp_all)
   402   done
   403 
   404 lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
   405   by (simp add: add_commute div_add_self2)
   406 
   407 lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
   408   apply (subst div_add1_eq)
   409   apply (subst div_mult1_eq, simp)
   410   done
   411 
   412 lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
   413   by (simp add: mult_commute div_mult_self1)
   414 
   415 
   416 (* Monotonicity of div in first argument *)
   417 lemma div_le_mono [rule_format (no_asm)]:
   418     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
   419 apply (case_tac "k=0", simp)
   420 apply (induct "n" rule: nat_less_induct, clarify)
   421 apply (case_tac "n<k")
   422 (* 1  case n<k *)
   423 apply simp
   424 (* 2  case n >= k *)
   425 apply (case_tac "m<k")
   426 (* 2.1  case m<k *)
   427 apply simp
   428 (* 2.2  case m>=k *)
   429 apply (simp add: div_geq diff_le_mono)
   430 done
   431 
   432 (* Antimonotonicity of div in second argument *)
   433 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
   434 apply (subgoal_tac "0<n")
   435  prefer 2 apply simp
   436 apply (induct_tac k rule: nat_less_induct)
   437 apply (rename_tac "k")
   438 apply (case_tac "k<n", simp)
   439 apply (subgoal_tac "~ (k<m) ")
   440  prefer 2 apply simp
   441 apply (simp add: div_geq)
   442 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
   443  prefer 2
   444  apply (blast intro: div_le_mono diff_le_mono2)
   445 apply (rule le_trans, simp)
   446 apply (simp)
   447 done
   448 
   449 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
   450 apply (case_tac "n=0", simp)
   451 apply (subgoal_tac "m div n \<le> m div 1", simp)
   452 apply (rule div_le_mono2)
   453 apply (simp_all (no_asm_simp))
   454 done
   455 
   456 (* Similar for "less than" *)
   457 lemma div_less_dividend [rule_format]:
   458      "!!n::nat. 1<n ==> 0 < m --> m div n < m"
   459 apply (induct_tac m rule: nat_less_induct)
   460 apply (rename_tac "m")
   461 apply (case_tac "m<n", simp)
   462 apply (subgoal_tac "0<n")
   463  prefer 2 apply simp
   464 apply (simp add: div_geq)
   465 apply (case_tac "n<m")
   466  apply (subgoal_tac "(m-n) div n < (m-n) ")
   467   apply (rule impI less_trans_Suc)+
   468 apply assumption
   469   apply (simp_all)
   470 done
   471 
   472 declare div_less_dividend [simp]
   473 
   474 text{*A fact for the mutilated chess board*}
   475 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
   476 apply (case_tac "n=0", simp)
   477 apply (induct "m" rule: nat_less_induct)
   478 apply (case_tac "Suc (na) <n")
   479 (* case Suc(na) < n *)
   480 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
   481 (* case n \<le> Suc(na) *)
   482 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
   483 apply (auto simp add: Suc_diff_le le_mod_geq)
   484 done
   485 
   486 lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
   487   by (cases "n = 0") auto
   488 
   489 lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
   490   by (cases "n = 0") auto
   491 
   492 
   493 subsection{*The Divides Relation*}
   494 
   495 lemma dvdI [intro?]: "n = m * k ==> m dvd n"
   496   unfolding dvd_def by blast
   497 
   498 lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
   499   unfolding dvd_def by blast
   500 
   501 lemma dvd_0_right [iff]: "m dvd (0::nat)"
   502   unfolding dvd_def by (blast intro: mult_0_right [symmetric])
   503 
   504 lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
   505   by (force simp add: dvd_def)
   506 
   507 lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
   508   by (blast intro: dvd_0_left)
   509 
   510 lemma dvd_1_left [iff]: "Suc 0 dvd k"
   511   unfolding dvd_def by simp
   512 
   513 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
   514   by (simp add: dvd_def)
   515 
   516 lemma dvd_refl [simp]: "m dvd (m::nat)"
   517   unfolding dvd_def by (blast intro: mult_1_right [symmetric])
   518 
   519 lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"
   520   unfolding dvd_def by (blast intro: mult_assoc)
   521 
   522 lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
   523   unfolding dvd_def
   524   by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
   525 
   526 lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
   527   unfolding dvd_def
   528   by (blast intro: add_mult_distrib2 [symmetric])
   529 
   530 lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
   531   unfolding dvd_def
   532   by (blast intro: diff_mult_distrib2 [symmetric])
   533 
   534 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
   535   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
   536   apply (blast intro: dvd_add)
   537   done
   538 
   539 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
   540   by (drule_tac m = m in dvd_diff, auto)
   541 
   542 lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
   543   unfolding dvd_def by (blast intro: mult_left_commute)
   544 
   545 lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
   546   apply (subst mult_commute)
   547   apply (erule dvd_mult)
   548   done
   549 
   550 lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
   551   by (rule dvd_refl [THEN dvd_mult])
   552 
   553 lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
   554   by (rule dvd_refl [THEN dvd_mult2])
   555 
   556 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
   557   apply (rule iffI)
   558    apply (erule_tac [2] dvd_add)
   559    apply (rule_tac [2] dvd_refl)
   560   apply (subgoal_tac "n = (n+k) -k")
   561    prefer 2 apply simp
   562   apply (erule ssubst)
   563   apply (erule dvd_diff)
   564   apply (rule dvd_refl)
   565   done
   566 
   567 lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
   568   unfolding dvd_def
   569   apply (case_tac "n = 0", auto)
   570   apply (blast intro: mod_mult_distrib2 [symmetric])
   571   done
   572 
   573 lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
   574   apply (subgoal_tac "k dvd (m div n) *n + m mod n")
   575    apply (simp add: mod_div_equality)
   576   apply (simp only: dvd_add dvd_mult)
   577   done
   578 
   579 lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
   580   by (blast intro: dvd_mod_imp_dvd dvd_mod)
   581 
   582 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
   583   unfolding dvd_def
   584   apply (erule exE)
   585   apply (simp add: mult_ac)
   586   done
   587 
   588 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
   589   apply auto
   590    apply (subgoal_tac "m*n dvd m*1")
   591    apply (drule dvd_mult_cancel, auto)
   592   done
   593 
   594 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
   595   apply (subst mult_commute)
   596   apply (erule dvd_mult_cancel1)
   597   done
   598 
   599 lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
   600   apply (unfold dvd_def, clarify)
   601   apply (rule_tac x = "k*ka" in exI)
   602   apply (simp add: mult_ac)
   603   done
   604 
   605 lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
   606   by (simp add: dvd_def mult_assoc, blast)
   607 
   608 lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
   609   apply (unfold dvd_def, clarify)
   610   apply (rule_tac x = "i*k" in exI)
   611   apply (simp add: mult_ac)
   612   done
   613 
   614 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
   615   apply (unfold dvd_def, clarify)
   616   apply (simp_all (no_asm_use) add: zero_less_mult_iff)
   617   apply (erule conjE)
   618   apply (rule le_trans)
   619    apply (rule_tac [2] le_refl [THEN mult_le_mono])
   620    apply (erule_tac [2] Suc_leI, simp)
   621   done
   622 
   623 lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"
   624   apply (unfold dvd_def)
   625   apply (case_tac "k=0", simp, safe)
   626    apply (simp add: mult_commute)
   627   apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])
   628   apply (subst mult_commute, simp)
   629   done
   630 
   631 lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
   632   apply (subgoal_tac "m mod n = 0")
   633    apply (simp add: mult_div_cancel)
   634   apply (simp only: dvd_eq_mod_eq_0)
   635   done
   636 
   637 lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
   638   apply (unfold dvd_def)
   639   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
   640   apply (simp add: power_add)
   641   done
   642 
   643 lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
   644   by (induct n) auto
   645 
   646 lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
   647   apply (induct j)
   648    apply (simp_all add: le_Suc_eq)
   649   apply (blast dest!: dvd_mult_right)
   650   done
   651 
   652 lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
   653   apply (rule power_le_imp_le_exp, assumption)
   654   apply (erule dvd_imp_le, simp)
   655   done
   656 
   657 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
   658   by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
   659 
   660 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
   661 
   662 (*Loses information, namely we also have r<d provided d is nonzero*)
   663 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
   664   apply (cut_tac m = m in mod_div_equality)
   665   apply (simp only: add_ac)
   666   apply (blast intro: sym)
   667   done
   668 
   669 
   670 lemma split_div:
   671  "P(n div k :: nat) =
   672  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
   673  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   674 proof
   675   assume P: ?P
   676   show ?Q
   677   proof (cases)
   678     assume "k = 0"
   679     with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
   680   next
   681     assume not0: "k \<noteq> 0"
   682     thus ?Q
   683     proof (simp, intro allI impI)
   684       fix i j
   685       assume n: "n = k*i + j" and j: "j < k"
   686       show "P i"
   687       proof (cases)
   688         assume "i = 0"
   689         with n j P show "P i" by simp
   690       next
   691         assume "i \<noteq> 0"
   692         with not0 n j P show "P i" by(simp add:add_ac)
   693       qed
   694     qed
   695   qed
   696 next
   697   assume Q: ?Q
   698   show ?P
   699   proof (cases)
   700     assume "k = 0"
   701     with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
   702   next
   703     assume not0: "k \<noteq> 0"
   704     with Q have R: ?R by simp
   705     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   706     show ?P by simp
   707   qed
   708 qed
   709 
   710 lemma split_div_lemma:
   711   "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
   712   apply (rule iffI)
   713   apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
   714 prefer 3; apply assumption
   715   apply (simp_all add: quorem_def) apply arith
   716   apply (rule conjI)
   717   apply (rule_tac P="%x. n * (m div n) \<le> x" in
   718     subst [OF mod_div_equality [of _ n]])
   719   apply (simp only: add: mult_ac)
   720   apply (rule_tac P="%x. x < n + n * (m div n)" in
   721     subst [OF mod_div_equality [of _ n]])
   722   apply (simp only: add: mult_ac add_ac)
   723   apply (rule add_less_mono1, simp)
   724   done
   725 
   726 theorem split_div':
   727   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
   728    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
   729   apply (case_tac "0 < n")
   730   apply (simp only: add: split_div_lemma)
   731   apply (simp_all add: DIVISION_BY_ZERO_DIV)
   732   done
   733 
   734 lemma split_mod:
   735  "P(n mod k :: nat) =
   736  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
   737  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   738 proof
   739   assume P: ?P
   740   show ?Q
   741   proof (cases)
   742     assume "k = 0"
   743     with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
   744   next
   745     assume not0: "k \<noteq> 0"
   746     thus ?Q
   747     proof (simp, intro allI impI)
   748       fix i j
   749       assume "n = k*i + j" "j < k"
   750       thus "P j" using not0 P by(simp add:add_ac mult_ac)
   751     qed
   752   qed
   753 next
   754   assume Q: ?Q
   755   show ?P
   756   proof (cases)
   757     assume "k = 0"
   758     with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
   759   next
   760     assume not0: "k \<noteq> 0"
   761     with Q have R: ?R by simp
   762     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   763     show ?P by simp
   764   qed
   765 qed
   766 
   767 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
   768   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
   769     subst [OF mod_div_equality [of _ n]])
   770   apply arith
   771   done
   772 
   773 lemma div_mod_equality':
   774   fixes m n :: nat
   775   shows "m div n * n = m - m mod n"
   776 proof -
   777   have "m mod n \<le> m mod n" ..
   778   from div_mod_equality have 
   779     "m div n * n + m mod n - m mod n = m - m mod n" by simp
   780   with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
   781     "m div n * n + (m mod n - m mod n) = m - m mod n"
   782     by simp
   783   then show ?thesis by simp
   784 qed
   785 
   786 
   787 subsection {*An ``induction'' law for modulus arithmetic.*}
   788 
   789 lemma mod_induct_0:
   790   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
   791   and base: "P i" and i: "i<p"
   792   shows "P 0"
   793 proof (rule ccontr)
   794   assume contra: "\<not>(P 0)"
   795   from i have p: "0<p" by simp
   796   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
   797   proof
   798     fix k
   799     show "?A k"
   800     proof (induct k)
   801       show "?A 0" by simp  -- "by contradiction"
   802     next
   803       fix n
   804       assume ih: "?A n"
   805       show "?A (Suc n)"
   806       proof (clarsimp)
   807         assume y: "P (p - Suc n)"
   808         have n: "Suc n < p"
   809         proof (rule ccontr)
   810           assume "\<not>(Suc n < p)"
   811           hence "p - Suc n = 0"
   812             by simp
   813           with y contra show "False"
   814             by simp
   815         qed
   816         hence n2: "Suc (p - Suc n) = p-n" by arith
   817         from p have "p - Suc n < p" by arith
   818         with y step have z: "P ((Suc (p - Suc n)) mod p)"
   819           by blast
   820         show "False"
   821         proof (cases "n=0")
   822           case True
   823           with z n2 contra show ?thesis by simp
   824         next
   825           case False
   826           with p have "p-n < p" by arith
   827           with z n2 False ih show ?thesis by simp
   828         qed
   829       qed
   830     qed
   831   qed
   832   moreover
   833   from i obtain k where "0<k \<and> i+k=p"
   834     by (blast dest: less_imp_add_positive)
   835   hence "0<k \<and> i=p-k" by auto
   836   moreover
   837   note base
   838   ultimately
   839   show "False" by blast
   840 qed
   841 
   842 lemma mod_induct:
   843   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
   844   and base: "P i" and i: "i<p" and j: "j<p"
   845   shows "P j"
   846 proof -
   847   have "\<forall>j<p. P j"
   848   proof
   849     fix j
   850     show "j<p \<longrightarrow> P j" (is "?A j")
   851     proof (induct j)
   852       from step base i show "?A 0"
   853         by (auto elim: mod_induct_0)
   854     next
   855       fix k
   856       assume ih: "?A k"
   857       show "?A (Suc k)"
   858       proof
   859         assume suc: "Suc k < p"
   860         hence k: "k<p" by simp
   861         with ih have "P k" ..
   862         with step k have "P (Suc k mod p)"
   863           by blast
   864         moreover
   865         from suc have "Suc k mod p = Suc k"
   866           by simp
   867         ultimately
   868         show "P (Suc k)" by simp
   869       qed
   870     qed
   871   qed
   872   with j show ?thesis by blast
   873 qed
   874 
   875 
   876 lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
   877   apply (rule trans [symmetric])
   878    apply (rule mod_add1_eq, simp)
   879   apply (rule mod_add1_eq [symmetric])
   880   done
   881 
   882 lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
   883   apply (rule trans [symmetric])
   884    apply (rule mod_add1_eq, simp)
   885   apply (rule mod_add1_eq [symmetric])
   886   done
   887 
   888 lemma mod_div_decomp:
   889   fixes n k :: nat
   890   obtains m q where "m = n div k" and "q = n mod k"
   891     and "n = m * k + q"
   892 proof -
   893   from mod_div_equality have "n = n div k * k + n mod k" by auto
   894   moreover have "n div k = n div k" ..
   895   moreover have "n mod k = n mod k" ..
   896   note that ultimately show thesis by blast
   897 qed
   898 
   899 
   900 subsection {* Code generation for div, mod and dvd on nat *}
   901 
   902 definition [code func del]:
   903   "divmod (m\<Colon>nat) n = (m div n, m mod n)"
   904 
   905 lemma divmod_zero [code]: "divmod m 0 = (0, m)"
   906   unfolding divmod_def by simp
   907 
   908 lemma divmod_succ [code]:
   909   "divmod m (Suc k) = (if m < Suc k then (0, m) else
   910     let
   911       (p, q) = divmod (m - Suc k) (Suc k)
   912     in (Suc p, q))"
   913   unfolding divmod_def Let_def split_def
   914   by (auto intro: div_geq mod_geq)
   915 
   916 lemma div_divmod [code]: "m div n = fst (divmod m n)"
   917   unfolding divmod_def by simp
   918 
   919 lemma mod_divmod [code]: "m mod n = snd (divmod m n)"
   920   unfolding divmod_def by simp
   921 
   922 definition
   923   dvd_nat :: "nat \<Rightarrow> nat \<Rightarrow> bool"
   924 where
   925   "dvd_nat m n \<longleftrightarrow> n mod m = (0 \<Colon> nat)"
   926 
   927 lemma [code inline]:
   928   "op dvd = dvd_nat"
   929   by (auto simp add: dvd_nat_def dvd_eq_mod_eq_0 expand_fun_eq)
   930 
   931 code_modulename SML
   932   Divides Nat
   933 
   934 code_modulename OCaml
   935   Divides Nat
   936 
   937 code_modulename Haskell
   938   Divides Nat
   939 
   940 hide (open) const divmod dvd_nat
   941 
   942 subsection {* Legacy bindings *}
   943 
   944 ML
   945 {*
   946 val div_def = thm "div_def"
   947 val mod_def = thm "mod_def"
   948 val dvd_def = thm "dvd_def"
   949 val quorem_def = thm "quorem_def"
   950 
   951 val wf_less_trans = thm "wf_less_trans";
   952 val mod_eq = thm "mod_eq";
   953 val div_eq = thm "div_eq";
   954 val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";
   955 val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";
   956 val mod_less = thm "mod_less";
   957 val mod_geq = thm "mod_geq";
   958 val le_mod_geq = thm "le_mod_geq";
   959 val mod_if = thm "mod_if";
   960 val mod_1 = thm "mod_1";
   961 val mod_self = thm "mod_self";
   962 val mod_add_self2 = thm "mod_add_self2";
   963 val mod_add_self1 = thm "mod_add_self1";
   964 val mod_mult_self1 = thm "mod_mult_self1";
   965 val mod_mult_self2 = thm "mod_mult_self2";
   966 val mod_mult_distrib = thm "mod_mult_distrib";
   967 val mod_mult_distrib2 = thm "mod_mult_distrib2";
   968 val mod_mult_self_is_0 = thm "mod_mult_self_is_0";
   969 val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0";
   970 val div_less = thm "div_less";
   971 val div_geq = thm "div_geq";
   972 val le_div_geq = thm "le_div_geq";
   973 val div_if = thm "div_if";
   974 val mod_div_equality = thm "mod_div_equality";
   975 val mod_div_equality2 = thm "mod_div_equality2";
   976 val div_mod_equality = thm "div_mod_equality";
   977 val div_mod_equality2 = thm "div_mod_equality2";
   978 val mult_div_cancel = thm "mult_div_cancel";
   979 val mod_less_divisor = thm "mod_less_divisor";
   980 val div_mult_self_is_m = thm "div_mult_self_is_m";
   981 val div_mult_self1_is_m = thm "div_mult_self1_is_m";
   982 val unique_quotient_lemma = thm "unique_quotient_lemma";
   983 val unique_quotient = thm "unique_quotient";
   984 val unique_remainder = thm "unique_remainder";
   985 val div_0 = thm "div_0";
   986 val mod_0 = thm "mod_0";
   987 val div_mult1_eq = thm "div_mult1_eq";
   988 val mod_mult1_eq = thm "mod_mult1_eq";
   989 val mod_mult1_eq' = thm "mod_mult1_eq'";
   990 val mod_mult_distrib_mod = thm "mod_mult_distrib_mod";
   991 val div_add1_eq = thm "div_add1_eq";
   992 val mod_add1_eq = thm "mod_add1_eq";
   993 val mod_add_left_eq = thm "mod_add_left_eq";
   994  val mod_add_right_eq = thm "mod_add_right_eq";
   995 val mod_lemma = thm "mod_lemma";
   996 val div_mult2_eq = thm "div_mult2_eq";
   997 val mod_mult2_eq = thm "mod_mult2_eq";
   998 val div_mult_mult_lemma = thm "div_mult_mult_lemma";
   999 val div_mult_mult1 = thm "div_mult_mult1";
  1000 val div_mult_mult2 = thm "div_mult_mult2";
  1001 val div_1 = thm "div_1";
  1002 val div_self = thm "div_self";
  1003 val div_add_self2 = thm "div_add_self2";
  1004 val div_add_self1 = thm "div_add_self1";
  1005 val div_mult_self1 = thm "div_mult_self1";
  1006 val div_mult_self2 = thm "div_mult_self2";
  1007 val div_le_mono = thm "div_le_mono";
  1008 val div_le_mono2 = thm "div_le_mono2";
  1009 val div_le_dividend = thm "div_le_dividend";
  1010 val div_less_dividend = thm "div_less_dividend";
  1011 val mod_Suc = thm "mod_Suc";
  1012 val dvdI = thm "dvdI";
  1013 val dvdE = thm "dvdE";
  1014 val dvd_0_right = thm "dvd_0_right";
  1015 val dvd_0_left = thm "dvd_0_left";
  1016 val dvd_0_left_iff = thm "dvd_0_left_iff";
  1017 val dvd_1_left = thm "dvd_1_left";
  1018 val dvd_1_iff_1 = thm "dvd_1_iff_1";
  1019 val dvd_refl = thm "dvd_refl";
  1020 val dvd_trans = thm "dvd_trans";
  1021 val dvd_anti_sym = thm "dvd_anti_sym";
  1022 val dvd_add = thm "dvd_add";
  1023 val dvd_diff = thm "dvd_diff";
  1024 val dvd_diffD = thm "dvd_diffD";
  1025 val dvd_diffD1 = thm "dvd_diffD1";
  1026 val dvd_mult = thm "dvd_mult";
  1027 val dvd_mult2 = thm "dvd_mult2";
  1028 val dvd_reduce = thm "dvd_reduce";
  1029 val dvd_mod = thm "dvd_mod";
  1030 val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd";
  1031 val dvd_mod_iff = thm "dvd_mod_iff";
  1032 val dvd_mult_cancel = thm "dvd_mult_cancel";
  1033 val dvd_mult_cancel1 = thm "dvd_mult_cancel1";
  1034 val dvd_mult_cancel2 = thm "dvd_mult_cancel2";
  1035 val mult_dvd_mono = thm "mult_dvd_mono";
  1036 val dvd_mult_left = thm "dvd_mult_left";
  1037 val dvd_mult_right = thm "dvd_mult_right";
  1038 val dvd_imp_le = thm "dvd_imp_le";
  1039 val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0";
  1040 val dvd_mult_div_cancel = thm "dvd_mult_div_cancel";
  1041 val mod_eq_0_iff = thm "mod_eq_0_iff";
  1042 val mod_eqD = thm "mod_eqD";
  1043 *}
  1044 
  1045 (*
  1046 lemma split_div:
  1047 assumes m: "m \<noteq> 0"
  1048 shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)"
  1049        (is "?P = ?Q")
  1050 proof
  1051   assume P: ?P
  1052   show ?Q
  1053   proof (intro allI impI)
  1054     fix i j
  1055     assume n: "n = m*i + j" and j: "j < m"
  1056     show "P i"
  1057     proof (cases)
  1058       assume "i = 0"
  1059       with n j P show "P i" by simp
  1060     next
  1061       assume "i \<noteq> 0"
  1062       with n j P show "P i" by (simp add:add_ac div_mult_self1)
  1063     qed
  1064   qed
  1065 next
  1066   assume Q: ?Q
  1067   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
  1068   show ?P by simp
  1069 qed
  1070 
  1071 lemma split_mod:
  1072 assumes m: "m \<noteq> 0"
  1073 shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)"
  1074        (is "?P = ?Q")
  1075 proof
  1076   assume P: ?P
  1077   show ?Q
  1078   proof (intro allI impI)
  1079     fix i j
  1080     assume "n = m*i + j" "j < m"
  1081     thus "P j" using m P by(simp add:add_ac mult_ac)
  1082   qed
  1083 next
  1084   assume Q: ?Q
  1085   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
  1086   show ?P by simp
  1087 qed
  1088 *)
  1089 end