src/HOL/Divides.thy
author wenzelm
Tue Apr 17 00:30:44 2007 +0200 (2007-04-17)
changeset 22718 936f7580937d
parent 22473 753123c89d72
child 22744 5cbe966d67a2
permissions -rw-r--r--
tuned proofs;
     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 = type +
    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 (pred_nat^+)
    36                           (%f j. if j<n | n=0 then j else f (j-n)) m"
    37   div_def:   "m div n == wfrec (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 definition
    46   quorem :: "(nat*nat) * (nat*nat) => bool" where
    47   (*This definition helps prove the harder properties of div and mod.
    48     It is copied from IntDiv.thy; should it be overloaded?*)
    49   "quorem = (%((a,b), (q,r)).
    50                     a = b*q + r &
    51                     (if 0<b then 0\<le>r & r<b else b<r & r \<le>0))"
    52 
    53 
    54 
    55 subsection{*Initial Lemmas*}
    56 
    57 lemmas wf_less_trans =
    58        def_wfrec [THEN trans, OF eq_reflection wf_pred_nat [THEN wf_trancl],
    59                   standard]
    60 
    61 lemma mod_eq: "(%m. m mod n) =
    62               wfrec (pred_nat^+) (%f j. if j<n | n=0 then j else f (j-n))"
    63 by (simp add: mod_def)
    64 
    65 lemma div_eq: "(%m. m div n) = wfrec (pred_nat^+)
    66                (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"
    67 by (simp add: div_def)
    68 
    69 
    70 (** Aribtrary definitions for division by zero.  Useful to simplify
    71     certain equations **)
    72 
    73 lemma DIVISION_BY_ZERO_DIV [simp]: "a div 0 = (0::nat)"
    74   by (rule div_eq [THEN wf_less_trans], simp)
    75 
    76 lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a::nat)"
    77   by (rule mod_eq [THEN wf_less_trans], simp)
    78 
    79 
    80 subsection{*Remainder*}
    81 
    82 lemma mod_less [simp]: "m<n ==> m mod n = (m::nat)"
    83   by (rule mod_eq [THEN wf_less_trans]) simp
    84 
    85 lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n"
    86   apply (cases "n=0")
    87    apply simp
    88   apply (rule mod_eq [THEN wf_less_trans])
    89   apply (simp add: cut_apply less_eq)
    90   done
    91 
    92 (*Avoids the ugly ~m<n above*)
    93 lemma le_mod_geq: "(n::nat) \<le> m ==> m mod n = (m-n) mod n"
    94   by (simp add: mod_geq linorder_not_less)
    95 
    96 lemma mod_if: "m mod (n::nat) = (if m<n then m else (m-n) mod n)"
    97   by (simp add: mod_geq)
    98 
    99 lemma mod_1 [simp]: "m mod Suc 0 = 0"
   100   by (induct m) (simp_all add: mod_geq)
   101 
   102 lemma mod_self [simp]: "n mod n = (0::nat)"
   103   by (cases "n = 0") (simp_all add: mod_geq)
   104 
   105 lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
   106   apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n")
   107    apply (simp add: add_commute)
   108   apply (subst mod_geq [symmetric], simp_all)
   109   done
   110 
   111 lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
   112   by (simp add: add_commute mod_add_self2)
   113 
   114 lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
   115   by (induct k) (simp_all add: add_left_commute [of _ n])
   116 
   117 lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
   118   by (simp add: mult_commute mod_mult_self1)
   119 
   120 lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)"
   121   apply (cases "n = 0", simp)
   122   apply (cases "k = 0", simp)
   123   apply (induct m rule: nat_less_induct)
   124   apply (subst mod_if, simp)
   125   apply (simp add: mod_geq diff_mult_distrib)
   126   done
   127 
   128 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
   129   by (simp add: mult_commute [of k] mod_mult_distrib)
   130 
   131 lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
   132   apply (cases "n = 0", simp)
   133   apply (induct m, simp)
   134   apply (rename_tac k)
   135   apply (cut_tac m = "k * n" and n = n in mod_add_self2)
   136   apply (simp add: add_commute)
   137   done
   138 
   139 lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
   140   by (simp add: mult_commute mod_mult_self_is_0)
   141 
   142 
   143 subsection{*Quotient*}
   144 
   145 lemma div_less [simp]: "m<n ==> m div n = (0::nat)"
   146   by (rule div_eq [THEN wf_less_trans], simp)
   147 
   148 lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"
   149   apply (rule div_eq [THEN wf_less_trans])
   150   apply (simp add: cut_apply less_eq)
   151   done
   152 
   153 (*Avoids the ugly ~m<n above*)
   154 lemma le_div_geq: "[| 0<n;  n\<le>m |] ==> m div n = Suc((m-n) div n)"
   155   by (simp add: div_geq linorder_not_less)
   156 
   157 lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"
   158   by (simp add: div_geq)
   159 
   160 
   161 (*Main Result about quotient and remainder.*)
   162 lemma mod_div_equality: "(m div n)*n + m mod n = (m::nat)"
   163   apply (cases "n = 0", simp)
   164   apply (induct m rule: nat_less_induct)
   165   apply (subst mod_if)
   166   apply (simp_all add: add_assoc div_geq add_diff_inverse)
   167   done
   168 
   169 lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)"
   170   apply (cut_tac m = m and n = n in mod_div_equality)
   171   apply (simp add: mult_commute)
   172   done
   173 
   174 subsection{*Simproc for Cancelling Div and Mod*}
   175 
   176 lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k"
   177   by (simp add: mod_div_equality)
   178 
   179 lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k"
   180   by (simp add: mod_div_equality2)
   181 
   182 ML
   183 {*
   184 structure CancelDivModData =
   185 struct
   186 
   187 val div_name = @{const_name Divides.div};
   188 val mod_name = @{const_name Divides.mod};
   189 val mk_binop = HOLogic.mk_binop;
   190 val mk_sum = NatArithUtils.mk_sum;
   191 val dest_sum = NatArithUtils.dest_sum;
   192 
   193 (*logic*)
   194 
   195 val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
   196 
   197 val trans = trans
   198 
   199 val prove_eq_sums =
   200   let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
   201   in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all_tac simps) end;
   202 
   203 end;
   204 
   205 structure CancelDivMod = CancelDivModFun(CancelDivModData);
   206 
   207 val cancel_div_mod_proc = NatArithUtils.prep_simproc
   208       ("cancel_div_mod", ["(m::nat) + n"], K CancelDivMod.proc);
   209 
   210 Addsimprocs[cancel_div_mod_proc];
   211 *}
   212 
   213 
   214 (* a simple rearrangement of mod_div_equality: *)
   215 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
   216   by (cut_tac m = m and n = n in mod_div_equality2, arith)
   217 
   218 lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)"
   219   apply (induct m rule: nat_less_induct)
   220   apply (rename_tac m)
   221   apply (case_tac "m<n", simp)
   222   txt{*case @{term "n \<le> m"}*}
   223   apply (simp add: mod_geq)
   224   done
   225 
   226 lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
   227   apply (drule mod_less_divisor [where m = m])
   228   apply simp
   229   done
   230 
   231 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
   232   by (cut_tac m = "m*n" and n = n in mod_div_equality, auto)
   233 
   234 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
   235   by (simp add: mult_commute div_mult_self_is_m)
   236 
   237 (*mod_mult_distrib2 above is the counterpart for remainder*)
   238 
   239 
   240 subsection{*Proving facts about Quotient and Remainder*}
   241 
   242 lemma unique_quotient_lemma:
   243      "[| b*q' + r'  \<le> b*q + r;  x < b;  r < b |]
   244       ==> q' \<le> (q::nat)"
   245   apply (rule leI)
   246   apply (subst less_iff_Suc_add)
   247   apply (auto simp add: add_mult_distrib2)
   248   done
   249 
   250 lemma unique_quotient:
   251      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
   252       ==> q = q'"
   253   apply (simp add: split_ifs quorem_def)
   254   apply (blast intro: order_antisym
   255     dest: order_eq_refl [THEN unique_quotient_lemma] sym)
   256   done
   257 
   258 lemma unique_remainder:
   259      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
   260       ==> r = r'"
   261   apply (subgoal_tac "q = q'")
   262    prefer 2 apply (blast intro: unique_quotient)
   263   apply (simp add: quorem_def)
   264   done
   265 
   266 lemma quorem_div_mod: "0 < b ==> quorem ((a, b), (a div b, a mod b))"
   267   unfolding quorem_def by simp
   268 
   269 lemma quorem_div: "[| quorem((a,b),(q,r));  0 < b |] ==> a div b = q"
   270   by (simp add: quorem_div_mod [THEN unique_quotient])
   271 
   272 lemma quorem_mod: "[| quorem((a,b),(q,r));  0 < b |] ==> a mod b = r"
   273   by (simp add: quorem_div_mod [THEN unique_remainder])
   274 
   275 (** A dividend of zero **)
   276 
   277 lemma div_0 [simp]: "0 div m = (0::nat)"
   278   by (cases "m = 0") simp_all
   279 
   280 lemma mod_0 [simp]: "0 mod m = (0::nat)"
   281   by (cases "m = 0") simp_all
   282 
   283 (** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
   284 
   285 lemma quorem_mult1_eq:
   286      "[| quorem((b,c),(q,r));  0 < c |]
   287       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
   288   by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
   289 
   290 lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
   291   apply (cases "c = 0", simp)
   292   apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
   293   done
   294 
   295 lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
   296   apply (cases "c = 0", simp)
   297   apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
   298   done
   299 
   300 lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
   301   apply (rule trans)
   302    apply (rule_tac s = "b*a mod c" in trans)
   303     apply (rule_tac [2] mod_mult1_eq)
   304    apply (simp_all add: mult_commute)
   305   done
   306 
   307 lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
   308   apply (rule mod_mult1_eq' [THEN trans])
   309   apply (rule mod_mult1_eq)
   310   done
   311 
   312 (** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
   313 
   314 lemma quorem_add1_eq:
   315      "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  0 < c |]
   316       ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
   317   by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
   318 
   319 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
   320 lemma div_add1_eq:
   321      "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
   322   apply (cases "c = 0", simp)
   323   apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod)
   324   done
   325 
   326 lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
   327   apply (cases "c = 0", simp)
   328   apply (blast intro: quorem_div_mod quorem_div_mod quorem_add1_eq [THEN quorem_mod])
   329   done
   330 
   331 
   332 subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*}
   333 
   334 (** first, a lemma to bound the remainder **)
   335 
   336 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
   337   apply (cut_tac m = q and n = c in mod_less_divisor)
   338   apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
   339   apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
   340   apply (simp add: add_mult_distrib2)
   341   done
   342 
   343 lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]
   344       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
   345   by (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)
   346 
   347 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
   348   apply (cases "b = 0", simp)
   349   apply (cases "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 (cases "b = 0", simp)
   355   apply (cases "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 (cases "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   by (induct m) (simp_all add: div_geq)
   393 
   394 lemma div_self [simp]: "0<n ==> n div n = (1::nat)"
   395   by (simp add: div_geq)
   396 
   397 lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
   398   apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
   399    apply (simp add: add_commute)
   400   apply (subst div_geq [symmetric], simp_all)
   401   done
   402 
   403 lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
   404   by (simp add: add_commute div_add_self2)
   405 
   406 lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
   407   apply (subst div_add1_eq)
   408   apply (subst div_mult1_eq, simp)
   409   done
   410 
   411 lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
   412   by (simp add: mult_commute div_mult_self1)
   413 
   414 
   415 (* Monotonicity of div in first argument *)
   416 lemma div_le_mono [rule_format (no_asm)]:
   417     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
   418 apply (case_tac "k=0", simp)
   419 apply (induct "n" rule: nat_less_induct, clarify)
   420 apply (case_tac "n<k")
   421 (* 1  case n<k *)
   422 apply simp
   423 (* 2  case n >= k *)
   424 apply (case_tac "m<k")
   425 (* 2.1  case m<k *)
   426 apply simp
   427 (* 2.2  case m>=k *)
   428 apply (simp add: div_geq diff_le_mono)
   429 done
   430 
   431 (* Antimonotonicity of div in second argument *)
   432 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
   433 apply (subgoal_tac "0<n")
   434  prefer 2 apply simp
   435 apply (induct_tac k rule: nat_less_induct)
   436 apply (rename_tac "k")
   437 apply (case_tac "k<n", simp)
   438 apply (subgoal_tac "~ (k<m) ")
   439  prefer 2 apply simp
   440 apply (simp add: div_geq)
   441 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
   442  prefer 2
   443  apply (blast intro: div_le_mono diff_le_mono2)
   444 apply (rule le_trans, simp)
   445 apply (simp)
   446 done
   447 
   448 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
   449 apply (case_tac "n=0", simp)
   450 apply (subgoal_tac "m div n \<le> m div 1", simp)
   451 apply (rule div_le_mono2)
   452 apply (simp_all (no_asm_simp))
   453 done
   454 
   455 (* Similar for "less than" *)
   456 lemma div_less_dividend [rule_format]:
   457      "!!n::nat. 1<n ==> 0 < m --> m div n < m"
   458 apply (induct_tac m rule: nat_less_induct)
   459 apply (rename_tac "m")
   460 apply (case_tac "m<n", simp)
   461 apply (subgoal_tac "0<n")
   462  prefer 2 apply simp
   463 apply (simp add: div_geq)
   464 apply (case_tac "n<m")
   465  apply (subgoal_tac "(m-n) div n < (m-n) ")
   466   apply (rule impI less_trans_Suc)+
   467 apply assumption
   468   apply (simp_all)
   469 done
   470 
   471 declare div_less_dividend [simp]
   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 "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: linorder_not_less le_Suc_eq mod_geq)
   482 apply (auto simp add: Suc_diff_le le_mod_geq)
   483 done
   484 
   485 lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
   486   by (cases "n = 0") auto
   487 
   488 lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
   489   by (cases "n = 0") auto
   490 
   491 
   492 subsection{*The Divides Relation*}
   493 
   494 lemma dvdI [intro?]: "n = m * k ==> m dvd n"
   495   unfolding dvd_def by blast
   496 
   497 lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
   498   unfolding dvd_def by blast
   499 
   500 lemma dvd_0_right [iff]: "m dvd (0::nat)"
   501   unfolding dvd_def by (blast intro: mult_0_right [symmetric])
   502 
   503 lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
   504   by (force simp add: dvd_def)
   505 
   506 lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
   507   by (blast intro: dvd_0_left)
   508 
   509 lemma dvd_1_left [iff]: "Suc 0 dvd k"
   510   unfolding dvd_def by simp
   511 
   512 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
   513   by (simp add: dvd_def)
   514 
   515 lemma dvd_refl [simp]: "m dvd (m::nat)"
   516   unfolding dvd_def by (blast intro: mult_1_right [symmetric])
   517 
   518 lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"
   519   unfolding dvd_def by (blast intro: mult_assoc)
   520 
   521 lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
   522   unfolding dvd_def
   523   by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
   524 
   525 lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
   526   unfolding dvd_def
   527   by (blast intro: add_mult_distrib2 [symmetric])
   528 
   529 lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
   530   unfolding dvd_def
   531   by (blast intro: diff_mult_distrib2 [symmetric])
   532 
   533 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
   534   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
   535   apply (blast intro: dvd_add)
   536   done
   537 
   538 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
   539   by (drule_tac m = m in dvd_diff, auto)
   540 
   541 lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
   542   unfolding dvd_def by (blast intro: mult_left_commute)
   543 
   544 lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
   545   apply (subst mult_commute)
   546   apply (erule dvd_mult)
   547   done
   548 
   549 lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
   550   by (rule dvd_refl [THEN dvd_mult])
   551 
   552 lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
   553   by (rule dvd_refl [THEN dvd_mult2])
   554 
   555 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
   556   apply (rule iffI)
   557    apply (erule_tac [2] dvd_add)
   558    apply (rule_tac [2] dvd_refl)
   559   apply (subgoal_tac "n = (n+k) -k")
   560    prefer 2 apply simp
   561   apply (erule ssubst)
   562   apply (erule dvd_diff)
   563   apply (rule dvd_refl)
   564   done
   565 
   566 lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
   567   unfolding dvd_def
   568   apply (case_tac "n = 0", auto)
   569   apply (blast intro: mod_mult_distrib2 [symmetric])
   570   done
   571 
   572 lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
   573   apply (subgoal_tac "k dvd (m div n) *n + m mod n")
   574    apply (simp add: mod_div_equality)
   575   apply (simp only: dvd_add dvd_mult)
   576   done
   577 
   578 lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
   579   by (blast intro: dvd_mod_imp_dvd dvd_mod)
   580 
   581 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
   582   unfolding dvd_def
   583   apply (erule exE)
   584   apply (simp add: mult_ac)
   585   done
   586 
   587 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
   588   apply auto
   589    apply (subgoal_tac "m*n dvd m*1")
   590    apply (drule dvd_mult_cancel, auto)
   591   done
   592 
   593 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
   594   apply (subst mult_commute)
   595   apply (erule dvd_mult_cancel1)
   596   done
   597 
   598 lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
   599   apply (unfold dvd_def, clarify)
   600   apply (rule_tac x = "k*ka" in exI)
   601   apply (simp add: mult_ac)
   602   done
   603 
   604 lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
   605   by (simp add: dvd_def mult_assoc, blast)
   606 
   607 lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
   608   apply (unfold dvd_def, clarify)
   609   apply (rule_tac x = "i*k" in exI)
   610   apply (simp add: mult_ac)
   611   done
   612 
   613 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
   614   apply (unfold dvd_def, clarify)
   615   apply (simp_all (no_asm_use) add: zero_less_mult_iff)
   616   apply (erule conjE)
   617   apply (rule le_trans)
   618    apply (rule_tac [2] le_refl [THEN mult_le_mono])
   619    apply (erule_tac [2] Suc_leI, simp)
   620   done
   621 
   622 lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"
   623   apply (unfold dvd_def)
   624   apply (case_tac "k=0", simp, safe)
   625    apply (simp add: mult_commute)
   626   apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])
   627   apply (subst mult_commute, simp)
   628   done
   629 
   630 lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
   631   apply (subgoal_tac "m mod n = 0")
   632    apply (simp add: mult_div_cancel)
   633   apply (simp only: dvd_eq_mod_eq_0)
   634   done
   635 
   636 lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
   637   apply (unfold dvd_def)
   638   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
   639   apply (simp add: power_add)
   640   done
   641 
   642 lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
   643   by (induct n) auto
   644 
   645 lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
   646   apply (induct j)
   647    apply (simp_all add: le_Suc_eq)
   648   apply (blast dest!: dvd_mult_right)
   649   done
   650 
   651 lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
   652   apply (rule power_le_imp_le_exp, assumption)
   653   apply (erule dvd_imp_le, simp)
   654   done
   655 
   656 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
   657   by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
   658 
   659 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
   660 
   661 (*Loses information, namely we also have r<d provided d is nonzero*)
   662 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
   663   apply (cut_tac m = m in mod_div_equality)
   664   apply (simp only: add_ac)
   665   apply (blast intro: sym)
   666   done
   667 
   668 
   669 lemma split_div:
   670  "P(n div k :: nat) =
   671  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
   672  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   673 proof
   674   assume P: ?P
   675   show ?Q
   676   proof (cases)
   677     assume "k = 0"
   678     with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
   679   next
   680     assume not0: "k \<noteq> 0"
   681     thus ?Q
   682     proof (simp, intro allI impI)
   683       fix i j
   684       assume n: "n = k*i + j" and j: "j < k"
   685       show "P i"
   686       proof (cases)
   687         assume "i = 0"
   688         with n j P show "P i" by simp
   689       next
   690         assume "i \<noteq> 0"
   691         with not0 n j P show "P i" by(simp add:add_ac)
   692       qed
   693     qed
   694   qed
   695 next
   696   assume Q: ?Q
   697   show ?P
   698   proof (cases)
   699     assume "k = 0"
   700     with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
   701   next
   702     assume not0: "k \<noteq> 0"
   703     with Q have R: ?R by simp
   704     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   705     show ?P by simp
   706   qed
   707 qed
   708 
   709 lemma split_div_lemma:
   710   "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
   711   apply (rule iffI)
   712   apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
   713 prefer 3; apply assumption
   714   apply (simp_all add: quorem_def) apply arith
   715   apply (rule conjI)
   716   apply (rule_tac P="%x. n * (m div n) \<le> x" in
   717     subst [OF mod_div_equality [of _ n]])
   718   apply (simp only: add: mult_ac)
   719   apply (rule_tac P="%x. x < n + n * (m div n)" in
   720     subst [OF mod_div_equality [of _ n]])
   721   apply (simp only: add: mult_ac add_ac)
   722   apply (rule add_less_mono1, simp)
   723   done
   724 
   725 theorem split_div':
   726   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
   727    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
   728   apply (case_tac "0 < n")
   729   apply (simp only: add: split_div_lemma)
   730   apply (simp_all add: DIVISION_BY_ZERO_DIV)
   731   done
   732 
   733 lemma split_mod:
   734  "P(n mod k :: nat) =
   735  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
   736  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   737 proof
   738   assume P: ?P
   739   show ?Q
   740   proof (cases)
   741     assume "k = 0"
   742     with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
   743   next
   744     assume not0: "k \<noteq> 0"
   745     thus ?Q
   746     proof (simp, intro allI impI)
   747       fix i j
   748       assume "n = k*i + j" "j < k"
   749       thus "P j" using not0 P by(simp add:add_ac mult_ac)
   750     qed
   751   qed
   752 next
   753   assume Q: ?Q
   754   show ?P
   755   proof (cases)
   756     assume "k = 0"
   757     with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
   758   next
   759     assume not0: "k \<noteq> 0"
   760     with Q have R: ?R by simp
   761     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   762     show ?P by simp
   763   qed
   764 qed
   765 
   766 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
   767   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
   768     subst [OF mod_div_equality [of _ n]])
   769   apply arith
   770   done
   771 
   772 subsection {*An ``induction'' law for modulus arithmetic.*}
   773 
   774 lemma mod_induct_0:
   775   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
   776   and base: "P i" and i: "i<p"
   777   shows "P 0"
   778 proof (rule ccontr)
   779   assume contra: "\<not>(P 0)"
   780   from i have p: "0<p" by simp
   781   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
   782   proof
   783     fix k
   784     show "?A k"
   785     proof (induct k)
   786       show "?A 0" by simp  -- "by contradiction"
   787     next
   788       fix n
   789       assume ih: "?A n"
   790       show "?A (Suc n)"
   791       proof (clarsimp)
   792         assume y: "P (p - Suc n)"
   793         have n: "Suc n < p"
   794         proof (rule ccontr)
   795           assume "\<not>(Suc n < p)"
   796           hence "p - Suc n = 0"
   797             by simp
   798           with y contra show "False"
   799             by simp
   800         qed
   801         hence n2: "Suc (p - Suc n) = p-n" by arith
   802         from p have "p - Suc n < p" by arith
   803         with y step have z: "P ((Suc (p - Suc n)) mod p)"
   804           by blast
   805         show "False"
   806         proof (cases "n=0")
   807           case True
   808           with z n2 contra show ?thesis by simp
   809         next
   810           case False
   811           with p have "p-n < p" by arith
   812           with z n2 False ih show ?thesis by simp
   813         qed
   814       qed
   815     qed
   816   qed
   817   moreover
   818   from i obtain k where "0<k \<and> i+k=p"
   819     by (blast dest: less_imp_add_positive)
   820   hence "0<k \<and> i=p-k" by auto
   821   moreover
   822   note base
   823   ultimately
   824   show "False" by blast
   825 qed
   826 
   827 lemma mod_induct:
   828   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
   829   and base: "P i" and i: "i<p" and j: "j<p"
   830   shows "P j"
   831 proof -
   832   have "\<forall>j<p. P j"
   833   proof
   834     fix j
   835     show "j<p \<longrightarrow> P j" (is "?A j")
   836     proof (induct j)
   837       from step base i show "?A 0"
   838         by (auto elim: mod_induct_0)
   839     next
   840       fix k
   841       assume ih: "?A k"
   842       show "?A (Suc k)"
   843       proof
   844         assume suc: "Suc k < p"
   845         hence k: "k<p" by simp
   846         with ih have "P k" ..
   847         with step k have "P (Suc k mod p)"
   848           by blast
   849         moreover
   850         from suc have "Suc k mod p = Suc k"
   851           by simp
   852         ultimately
   853         show "P (Suc k)" by simp
   854       qed
   855     qed
   856   qed
   857   with j show ?thesis by blast
   858 qed
   859 
   860 
   861 lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
   862   apply (rule trans [symmetric])
   863    apply (rule mod_add1_eq, simp)
   864   apply (rule mod_add1_eq [symmetric])
   865   done
   866 
   867 lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
   868   apply (rule trans [symmetric])
   869    apply (rule mod_add1_eq, simp)
   870   apply (rule mod_add1_eq [symmetric])
   871   done
   872 
   873 
   874 subsection {* Code generation for div and mod *}
   875 
   876 definition
   877   "divmod (m\<Colon>nat) n = (m div n, m mod n)"
   878 
   879 lemma divmod_zero [code]: "divmod m 0 = (0, m)"
   880   unfolding divmod_def by simp
   881 
   882 lemma divmod_succ [code]:
   883   "divmod m (Suc k) = (if m < Suc k then (0, m) else
   884     let
   885       (p, q) = divmod (m - Suc k) (Suc k)
   886     in (Suc p, q))"
   887   unfolding divmod_def Let_def split_def
   888   by (auto intro: div_geq mod_geq)
   889 
   890 lemma div_divmod [code]: "m div n = fst (divmod m n)"
   891   unfolding divmod_def by simp
   892 
   893 lemma mod_divmod [code]: "m mod n = snd (divmod m n)"
   894   unfolding divmod_def by simp
   895 
   896 code_modulename SML
   897   Divides Integer
   898 
   899 code_modulename OCaml
   900   Divides Integer
   901 
   902 hide (open) const divmod
   903 
   904 subsection {* Legacy bindings *}
   905 
   906 ML
   907 {*
   908 val div_def = thm "div_def"
   909 val mod_def = thm "mod_def"
   910 val dvd_def = thm "dvd_def"
   911 val quorem_def = thm "quorem_def"
   912 
   913 val wf_less_trans = thm "wf_less_trans";
   914 val mod_eq = thm "mod_eq";
   915 val div_eq = thm "div_eq";
   916 val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";
   917 val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";
   918 val mod_less = thm "mod_less";
   919 val mod_geq = thm "mod_geq";
   920 val le_mod_geq = thm "le_mod_geq";
   921 val mod_if = thm "mod_if";
   922 val mod_1 = thm "mod_1";
   923 val mod_self = thm "mod_self";
   924 val mod_add_self2 = thm "mod_add_self2";
   925 val mod_add_self1 = thm "mod_add_self1";
   926 val mod_mult_self1 = thm "mod_mult_self1";
   927 val mod_mult_self2 = thm "mod_mult_self2";
   928 val mod_mult_distrib = thm "mod_mult_distrib";
   929 val mod_mult_distrib2 = thm "mod_mult_distrib2";
   930 val mod_mult_self_is_0 = thm "mod_mult_self_is_0";
   931 val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0";
   932 val div_less = thm "div_less";
   933 val div_geq = thm "div_geq";
   934 val le_div_geq = thm "le_div_geq";
   935 val div_if = thm "div_if";
   936 val mod_div_equality = thm "mod_div_equality";
   937 val mod_div_equality2 = thm "mod_div_equality2";
   938 val div_mod_equality = thm "div_mod_equality";
   939 val div_mod_equality2 = thm "div_mod_equality2";
   940 val mult_div_cancel = thm "mult_div_cancel";
   941 val mod_less_divisor = thm "mod_less_divisor";
   942 val div_mult_self_is_m = thm "div_mult_self_is_m";
   943 val div_mult_self1_is_m = thm "div_mult_self1_is_m";
   944 val unique_quotient_lemma = thm "unique_quotient_lemma";
   945 val unique_quotient = thm "unique_quotient";
   946 val unique_remainder = thm "unique_remainder";
   947 val div_0 = thm "div_0";
   948 val mod_0 = thm "mod_0";
   949 val div_mult1_eq = thm "div_mult1_eq";
   950 val mod_mult1_eq = thm "mod_mult1_eq";
   951 val mod_mult1_eq' = thm "mod_mult1_eq'";
   952 val mod_mult_distrib_mod = thm "mod_mult_distrib_mod";
   953 val div_add1_eq = thm "div_add1_eq";
   954 val mod_add1_eq = thm "mod_add1_eq";
   955 val mod_add_left_eq = thm "mod_add_left_eq";
   956  val mod_add_right_eq = thm "mod_add_right_eq";
   957 val mod_lemma = thm "mod_lemma";
   958 val div_mult2_eq = thm "div_mult2_eq";
   959 val mod_mult2_eq = thm "mod_mult2_eq";
   960 val div_mult_mult_lemma = thm "div_mult_mult_lemma";
   961 val div_mult_mult1 = thm "div_mult_mult1";
   962 val div_mult_mult2 = thm "div_mult_mult2";
   963 val div_1 = thm "div_1";
   964 val div_self = thm "div_self";
   965 val div_add_self2 = thm "div_add_self2";
   966 val div_add_self1 = thm "div_add_self1";
   967 val div_mult_self1 = thm "div_mult_self1";
   968 val div_mult_self2 = thm "div_mult_self2";
   969 val div_le_mono = thm "div_le_mono";
   970 val div_le_mono2 = thm "div_le_mono2";
   971 val div_le_dividend = thm "div_le_dividend";
   972 val div_less_dividend = thm "div_less_dividend";
   973 val mod_Suc = thm "mod_Suc";
   974 val dvdI = thm "dvdI";
   975 val dvdE = thm "dvdE";
   976 val dvd_0_right = thm "dvd_0_right";
   977 val dvd_0_left = thm "dvd_0_left";
   978 val dvd_0_left_iff = thm "dvd_0_left_iff";
   979 val dvd_1_left = thm "dvd_1_left";
   980 val dvd_1_iff_1 = thm "dvd_1_iff_1";
   981 val dvd_refl = thm "dvd_refl";
   982 val dvd_trans = thm "dvd_trans";
   983 val dvd_anti_sym = thm "dvd_anti_sym";
   984 val dvd_add = thm "dvd_add";
   985 val dvd_diff = thm "dvd_diff";
   986 val dvd_diffD = thm "dvd_diffD";
   987 val dvd_diffD1 = thm "dvd_diffD1";
   988 val dvd_mult = thm "dvd_mult";
   989 val dvd_mult2 = thm "dvd_mult2";
   990 val dvd_reduce = thm "dvd_reduce";
   991 val dvd_mod = thm "dvd_mod";
   992 val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd";
   993 val dvd_mod_iff = thm "dvd_mod_iff";
   994 val dvd_mult_cancel = thm "dvd_mult_cancel";
   995 val dvd_mult_cancel1 = thm "dvd_mult_cancel1";
   996 val dvd_mult_cancel2 = thm "dvd_mult_cancel2";
   997 val mult_dvd_mono = thm "mult_dvd_mono";
   998 val dvd_mult_left = thm "dvd_mult_left";
   999 val dvd_mult_right = thm "dvd_mult_right";
  1000 val dvd_imp_le = thm "dvd_imp_le";
  1001 val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0";
  1002 val dvd_mult_div_cancel = thm "dvd_mult_div_cancel";
  1003 val mod_eq_0_iff = thm "mod_eq_0_iff";
  1004 val mod_eqD = thm "mod_eqD";
  1005 *}
  1006 
  1007 (*
  1008 lemma split_div:
  1009 assumes m: "m \<noteq> 0"
  1010 shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)"
  1011        (is "?P = ?Q")
  1012 proof
  1013   assume P: ?P
  1014   show ?Q
  1015   proof (intro allI impI)
  1016     fix i j
  1017     assume n: "n = m*i + j" and j: "j < m"
  1018     show "P i"
  1019     proof (cases)
  1020       assume "i = 0"
  1021       with n j P show "P i" by simp
  1022     next
  1023       assume "i \<noteq> 0"
  1024       with n j P show "P i" by (simp add:add_ac div_mult_self1)
  1025     qed
  1026   qed
  1027 next
  1028   assume Q: ?Q
  1029   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
  1030   show ?P by simp
  1031 qed
  1032 
  1033 lemma split_mod:
  1034 assumes m: "m \<noteq> 0"
  1035 shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)"
  1036        (is "?P = ?Q")
  1037 proof
  1038   assume P: ?P
  1039   show ?Q
  1040   proof (intro allI impI)
  1041     fix i j
  1042     assume "n = m*i + j" "j < m"
  1043     thus "P j" using m P by(simp add:add_ac mult_ac)
  1044   qed
  1045 next
  1046   assume Q: ?Q
  1047   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
  1048   show ?P by simp
  1049 qed
  1050 *)
  1051 end