src/HOL/Divides.thy
author wenzelm
Thu May 31 18:16:50 2007 +0200 (2007-05-31)
changeset 23162 b9853c187a1e
parent 23017 00c0e4c42396
child 23684 8c508c4dc53b
permissions -rw-r--r--
removed dead code;
     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 subsection{*Further Facts about Quotient and Remainder*}
   379 
   380 lemma div_1 [simp]: "m div Suc 0 = m"
   381   by (induct m) (simp_all add: div_geq)
   382 
   383 lemma div_self [simp]: "0<n ==> n div n = (1::nat)"
   384   by (simp add: div_geq)
   385 
   386 lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
   387   apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
   388    apply (simp add: add_commute)
   389   apply (subst div_geq [symmetric], simp_all)
   390   done
   391 
   392 lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
   393   by (simp add: add_commute div_add_self2)
   394 
   395 lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
   396   apply (subst div_add1_eq)
   397   apply (subst div_mult1_eq, simp)
   398   done
   399 
   400 lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
   401   by (simp add: mult_commute div_mult_self1)
   402 
   403 
   404 (* Monotonicity of div in first argument *)
   405 lemma div_le_mono [rule_format (no_asm)]:
   406     "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
   407 apply (case_tac "k=0", simp)
   408 apply (induct "n" rule: nat_less_induct, clarify)
   409 apply (case_tac "n<k")
   410 (* 1  case n<k *)
   411 apply simp
   412 (* 2  case n >= k *)
   413 apply (case_tac "m<k")
   414 (* 2.1  case m<k *)
   415 apply simp
   416 (* 2.2  case m>=k *)
   417 apply (simp add: div_geq diff_le_mono)
   418 done
   419 
   420 (* Antimonotonicity of div in second argument *)
   421 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
   422 apply (subgoal_tac "0<n")
   423  prefer 2 apply simp
   424 apply (induct_tac k rule: nat_less_induct)
   425 apply (rename_tac "k")
   426 apply (case_tac "k<n", simp)
   427 apply (subgoal_tac "~ (k<m) ")
   428  prefer 2 apply simp
   429 apply (simp add: div_geq)
   430 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
   431  prefer 2
   432  apply (blast intro: div_le_mono diff_le_mono2)
   433 apply (rule le_trans, simp)
   434 apply (simp)
   435 done
   436 
   437 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
   438 apply (case_tac "n=0", simp)
   439 apply (subgoal_tac "m div n \<le> m div 1", simp)
   440 apply (rule div_le_mono2)
   441 apply (simp_all (no_asm_simp))
   442 done
   443 
   444 (* Similar for "less than" *)
   445 lemma div_less_dividend [rule_format]:
   446      "!!n::nat. 1<n ==> 0 < m --> m div n < m"
   447 apply (induct_tac m rule: nat_less_induct)
   448 apply (rename_tac "m")
   449 apply (case_tac "m<n", simp)
   450 apply (subgoal_tac "0<n")
   451  prefer 2 apply simp
   452 apply (simp add: div_geq)
   453 apply (case_tac "n<m")
   454  apply (subgoal_tac "(m-n) div n < (m-n) ")
   455   apply (rule impI less_trans_Suc)+
   456 apply assumption
   457   apply (simp_all)
   458 done
   459 
   460 declare div_less_dividend [simp]
   461 
   462 text{*A fact for the mutilated chess board*}
   463 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
   464 apply (case_tac "n=0", simp)
   465 apply (induct "m" rule: nat_less_induct)
   466 apply (case_tac "Suc (na) <n")
   467 (* case Suc(na) < n *)
   468 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
   469 (* case n \<le> Suc(na) *)
   470 apply (simp add: linorder_not_less le_Suc_eq mod_geq)
   471 apply (auto simp add: Suc_diff_le le_mod_geq)
   472 done
   473 
   474 lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
   475   by (cases "n = 0") auto
   476 
   477 lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
   478   by (cases "n = 0") auto
   479 
   480 
   481 subsection{*The Divides Relation*}
   482 
   483 lemma dvdI [intro?]: "n = m * k ==> m dvd n"
   484   unfolding dvd_def by blast
   485 
   486 lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
   487   unfolding dvd_def by blast
   488 
   489 lemma dvd_0_right [iff]: "m dvd (0::nat)"
   490   unfolding dvd_def by (blast intro: mult_0_right [symmetric])
   491 
   492 lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
   493   by (force simp add: dvd_def)
   494 
   495 lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
   496   by (blast intro: dvd_0_left)
   497 
   498 lemma dvd_1_left [iff]: "Suc 0 dvd k"
   499   unfolding dvd_def by simp
   500 
   501 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
   502   by (simp add: dvd_def)
   503 
   504 lemma dvd_refl [simp]: "m dvd (m::nat)"
   505   unfolding dvd_def by (blast intro: mult_1_right [symmetric])
   506 
   507 lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"
   508   unfolding dvd_def by (blast intro: mult_assoc)
   509 
   510 lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
   511   unfolding dvd_def
   512   by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
   513 
   514 lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
   515   unfolding dvd_def
   516   by (blast intro: add_mult_distrib2 [symmetric])
   517 
   518 lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
   519   unfolding dvd_def
   520   by (blast intro: diff_mult_distrib2 [symmetric])
   521 
   522 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
   523   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
   524   apply (blast intro: dvd_add)
   525   done
   526 
   527 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
   528   by (drule_tac m = m in dvd_diff, auto)
   529 
   530 lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
   531   unfolding dvd_def by (blast intro: mult_left_commute)
   532 
   533 lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
   534   apply (subst mult_commute)
   535   apply (erule dvd_mult)
   536   done
   537 
   538 lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
   539   by (rule dvd_refl [THEN dvd_mult])
   540 
   541 lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
   542   by (rule dvd_refl [THEN dvd_mult2])
   543 
   544 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
   545   apply (rule iffI)
   546    apply (erule_tac [2] dvd_add)
   547    apply (rule_tac [2] dvd_refl)
   548   apply (subgoal_tac "n = (n+k) -k")
   549    prefer 2 apply simp
   550   apply (erule ssubst)
   551   apply (erule dvd_diff)
   552   apply (rule dvd_refl)
   553   done
   554 
   555 lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
   556   unfolding dvd_def
   557   apply (case_tac "n = 0", auto)
   558   apply (blast intro: mod_mult_distrib2 [symmetric])
   559   done
   560 
   561 lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
   562   apply (subgoal_tac "k dvd (m div n) *n + m mod n")
   563    apply (simp add: mod_div_equality)
   564   apply (simp only: dvd_add dvd_mult)
   565   done
   566 
   567 lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
   568   by (blast intro: dvd_mod_imp_dvd dvd_mod)
   569 
   570 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
   571   unfolding dvd_def
   572   apply (erule exE)
   573   apply (simp add: mult_ac)
   574   done
   575 
   576 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
   577   apply auto
   578    apply (subgoal_tac "m*n dvd m*1")
   579    apply (drule dvd_mult_cancel, auto)
   580   done
   581 
   582 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
   583   apply (subst mult_commute)
   584   apply (erule dvd_mult_cancel1)
   585   done
   586 
   587 lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
   588   apply (unfold dvd_def, clarify)
   589   apply (rule_tac x = "k*ka" in exI)
   590   apply (simp add: mult_ac)
   591   done
   592 
   593 lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
   594   by (simp add: dvd_def mult_assoc, blast)
   595 
   596 lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
   597   apply (unfold dvd_def, clarify)
   598   apply (rule_tac x = "i*k" in exI)
   599   apply (simp add: mult_ac)
   600   done
   601 
   602 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
   603   apply (unfold dvd_def, clarify)
   604   apply (simp_all (no_asm_use) add: zero_less_mult_iff)
   605   apply (erule conjE)
   606   apply (rule le_trans)
   607    apply (rule_tac [2] le_refl [THEN mult_le_mono])
   608    apply (erule_tac [2] Suc_leI, simp)
   609   done
   610 
   611 lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"
   612   apply (unfold dvd_def)
   613   apply (case_tac "k=0", simp, safe)
   614    apply (simp add: mult_commute)
   615   apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])
   616   apply (subst mult_commute, simp)
   617   done
   618 
   619 lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
   620   apply (subgoal_tac "m mod n = 0")
   621    apply (simp add: mult_div_cancel)
   622   apply (simp only: dvd_eq_mod_eq_0)
   623   done
   624 
   625 lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
   626   apply (unfold dvd_def)
   627   apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
   628   apply (simp add: power_add)
   629   done
   630 
   631 lemma nat_zero_less_power_iff [simp]: "(0 < x^n) = (x \<noteq> (0::nat) | n=0)"
   632   by (induct n) auto
   633 
   634 lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
   635   apply (induct j)
   636    apply (simp_all add: le_Suc_eq)
   637   apply (blast dest!: dvd_mult_right)
   638   done
   639 
   640 lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
   641   apply (rule power_le_imp_le_exp, assumption)
   642   apply (erule dvd_imp_le, simp)
   643   done
   644 
   645 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
   646   by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
   647 
   648 lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
   649 
   650 (*Loses information, namely we also have r<d provided d is nonzero*)
   651 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
   652   apply (cut_tac m = m in mod_div_equality)
   653   apply (simp only: add_ac)
   654   apply (blast intro: sym)
   655   done
   656 
   657 
   658 lemma split_div:
   659  "P(n div k :: nat) =
   660  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
   661  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   662 proof
   663   assume P: ?P
   664   show ?Q
   665   proof (cases)
   666     assume "k = 0"
   667     with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
   668   next
   669     assume not0: "k \<noteq> 0"
   670     thus ?Q
   671     proof (simp, intro allI impI)
   672       fix i j
   673       assume n: "n = k*i + j" and j: "j < k"
   674       show "P i"
   675       proof (cases)
   676         assume "i = 0"
   677         with n j P show "P i" by simp
   678       next
   679         assume "i \<noteq> 0"
   680         with not0 n j P show "P i" by(simp add:add_ac)
   681       qed
   682     qed
   683   qed
   684 next
   685   assume Q: ?Q
   686   show ?P
   687   proof (cases)
   688     assume "k = 0"
   689     with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
   690   next
   691     assume not0: "k \<noteq> 0"
   692     with Q have R: ?R by simp
   693     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   694     show ?P by simp
   695   qed
   696 qed
   697 
   698 lemma split_div_lemma:
   699   "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
   700   apply (rule iffI)
   701   apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
   702 prefer 3; apply assumption
   703   apply (simp_all add: quorem_def) apply arith
   704   apply (rule conjI)
   705   apply (rule_tac P="%x. n * (m div n) \<le> x" in
   706     subst [OF mod_div_equality [of _ n]])
   707   apply (simp only: add: mult_ac)
   708   apply (rule_tac P="%x. x < n + n * (m div n)" in
   709     subst [OF mod_div_equality [of _ n]])
   710   apply (simp only: add: mult_ac add_ac)
   711   apply (rule add_less_mono1, simp)
   712   done
   713 
   714 theorem split_div':
   715   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
   716    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
   717   apply (case_tac "0 < n")
   718   apply (simp only: add: split_div_lemma)
   719   apply (simp_all add: DIVISION_BY_ZERO_DIV)
   720   done
   721 
   722 lemma split_mod:
   723  "P(n mod k :: nat) =
   724  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
   725  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
   726 proof
   727   assume P: ?P
   728   show ?Q
   729   proof (cases)
   730     assume "k = 0"
   731     with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
   732   next
   733     assume not0: "k \<noteq> 0"
   734     thus ?Q
   735     proof (simp, intro allI impI)
   736       fix i j
   737       assume "n = k*i + j" "j < k"
   738       thus "P j" using not0 P by(simp add:add_ac mult_ac)
   739     qed
   740   qed
   741 next
   742   assume Q: ?Q
   743   show ?P
   744   proof (cases)
   745     assume "k = 0"
   746     with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
   747   next
   748     assume not0: "k \<noteq> 0"
   749     with Q have R: ?R by simp
   750     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
   751     show ?P by simp
   752   qed
   753 qed
   754 
   755 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
   756   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
   757     subst [OF mod_div_equality [of _ n]])
   758   apply arith
   759   done
   760 
   761 lemma div_mod_equality':
   762   fixes m n :: nat
   763   shows "m div n * n = m - m mod n"
   764 proof -
   765   have "m mod n \<le> m mod n" ..
   766   from div_mod_equality have 
   767     "m div n * n + m mod n - m mod n = m - m mod n" by simp
   768   with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
   769     "m div n * n + (m mod n - m mod n) = m - m mod n"
   770     by simp
   771   then show ?thesis by simp
   772 qed
   773 
   774 
   775 subsection {*An ``induction'' law for modulus arithmetic.*}
   776 
   777 lemma mod_induct_0:
   778   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
   779   and base: "P i" and i: "i<p"
   780   shows "P 0"
   781 proof (rule ccontr)
   782   assume contra: "\<not>(P 0)"
   783   from i have p: "0<p" by simp
   784   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
   785   proof
   786     fix k
   787     show "?A k"
   788     proof (induct k)
   789       show "?A 0" by simp  -- "by contradiction"
   790     next
   791       fix n
   792       assume ih: "?A n"
   793       show "?A (Suc n)"
   794       proof (clarsimp)
   795         assume y: "P (p - Suc n)"
   796         have n: "Suc n < p"
   797         proof (rule ccontr)
   798           assume "\<not>(Suc n < p)"
   799           hence "p - Suc n = 0"
   800             by simp
   801           with y contra show "False"
   802             by simp
   803         qed
   804         hence n2: "Suc (p - Suc n) = p-n" by arith
   805         from p have "p - Suc n < p" by arith
   806         with y step have z: "P ((Suc (p - Suc n)) mod p)"
   807           by blast
   808         show "False"
   809         proof (cases "n=0")
   810           case True
   811           with z n2 contra show ?thesis by simp
   812         next
   813           case False
   814           with p have "p-n < p" by arith
   815           with z n2 False ih show ?thesis by simp
   816         qed
   817       qed
   818     qed
   819   qed
   820   moreover
   821   from i obtain k where "0<k \<and> i+k=p"
   822     by (blast dest: less_imp_add_positive)
   823   hence "0<k \<and> i=p-k" by auto
   824   moreover
   825   note base
   826   ultimately
   827   show "False" by blast
   828 qed
   829 
   830 lemma mod_induct:
   831   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
   832   and base: "P i" and i: "i<p" and j: "j<p"
   833   shows "P j"
   834 proof -
   835   have "\<forall>j<p. P j"
   836   proof
   837     fix j
   838     show "j<p \<longrightarrow> P j" (is "?A j")
   839     proof (induct j)
   840       from step base i show "?A 0"
   841         by (auto elim: mod_induct_0)
   842     next
   843       fix k
   844       assume ih: "?A k"
   845       show "?A (Suc k)"
   846       proof
   847         assume suc: "Suc k < p"
   848         hence k: "k<p" by simp
   849         with ih have "P k" ..
   850         with step k have "P (Suc k mod p)"
   851           by blast
   852         moreover
   853         from suc have "Suc k mod p = Suc k"
   854           by simp
   855         ultimately
   856         show "P (Suc k)" by simp
   857       qed
   858     qed
   859   qed
   860   with j show ?thesis by blast
   861 qed
   862 
   863 
   864 lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
   865   apply (rule trans [symmetric])
   866    apply (rule mod_add1_eq, simp)
   867   apply (rule mod_add1_eq [symmetric])
   868   done
   869 
   870 lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
   871   apply (rule trans [symmetric])
   872    apply (rule mod_add1_eq, simp)
   873   apply (rule mod_add1_eq [symmetric])
   874   done
   875 
   876 lemma mod_div_decomp:
   877   fixes n k :: nat
   878   obtains m q where "m = n div k" and "q = n mod k"
   879     and "n = m * k + q"
   880 proof -
   881   from mod_div_equality have "n = n div k * k + n mod k" by auto
   882   moreover have "n div k = n div k" ..
   883   moreover have "n mod k = n mod k" ..
   884   note that ultimately show thesis by blast
   885 qed
   886 
   887 
   888 subsection {* Code generation for div, mod and dvd on nat *}
   889 
   890 definition [code func del]:
   891   "divmod (m\<Colon>nat) n = (m div n, m mod n)"
   892 
   893 lemma divmod_zero [code]: "divmod m 0 = (0, m)"
   894   unfolding divmod_def by simp
   895 
   896 lemma divmod_succ [code]:
   897   "divmod m (Suc k) = (if m < Suc k then (0, m) else
   898     let
   899       (p, q) = divmod (m - Suc k) (Suc k)
   900     in (Suc p, q))"
   901   unfolding divmod_def Let_def split_def
   902   by (auto intro: div_geq mod_geq)
   903 
   904 lemma div_divmod [code]: "m div n = fst (divmod m n)"
   905   unfolding divmod_def by simp
   906 
   907 lemma mod_divmod [code]: "m mod n = snd (divmod m n)"
   908   unfolding divmod_def by simp
   909 
   910 definition
   911   dvd_nat :: "nat \<Rightarrow> nat \<Rightarrow> bool"
   912 where
   913   "dvd_nat m n \<longleftrightarrow> n mod m = (0 \<Colon> nat)"
   914 
   915 lemma [code inline]:
   916   "op dvd = dvd_nat"
   917   by (auto simp add: dvd_nat_def dvd_eq_mod_eq_0 expand_fun_eq)
   918 
   919 code_modulename SML
   920   Divides Nat
   921 
   922 code_modulename OCaml
   923   Divides Nat
   924 
   925 code_modulename Haskell
   926   Divides Nat
   927 
   928 hide (open) const divmod dvd_nat
   929 
   930 subsection {* Legacy bindings *}
   931 
   932 ML
   933 {*
   934 val div_def = thm "div_def"
   935 val mod_def = thm "mod_def"
   936 val dvd_def = thm "dvd_def"
   937 val quorem_def = thm "quorem_def"
   938 
   939 val wf_less_trans = thm "wf_less_trans";
   940 val mod_eq = thm "mod_eq";
   941 val div_eq = thm "div_eq";
   942 val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";
   943 val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";
   944 val mod_less = thm "mod_less";
   945 val mod_geq = thm "mod_geq";
   946 val le_mod_geq = thm "le_mod_geq";
   947 val mod_if = thm "mod_if";
   948 val mod_1 = thm "mod_1";
   949 val mod_self = thm "mod_self";
   950 val mod_add_self2 = thm "mod_add_self2";
   951 val mod_add_self1 = thm "mod_add_self1";
   952 val mod_mult_self1 = thm "mod_mult_self1";
   953 val mod_mult_self2 = thm "mod_mult_self2";
   954 val mod_mult_distrib = thm "mod_mult_distrib";
   955 val mod_mult_distrib2 = thm "mod_mult_distrib2";
   956 val mod_mult_self_is_0 = thm "mod_mult_self_is_0";
   957 val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0";
   958 val div_less = thm "div_less";
   959 val div_geq = thm "div_geq";
   960 val le_div_geq = thm "le_div_geq";
   961 val div_if = thm "div_if";
   962 val mod_div_equality = thm "mod_div_equality";
   963 val mod_div_equality2 = thm "mod_div_equality2";
   964 val div_mod_equality = thm "div_mod_equality";
   965 val div_mod_equality2 = thm "div_mod_equality2";
   966 val mult_div_cancel = thm "mult_div_cancel";
   967 val mod_less_divisor = thm "mod_less_divisor";
   968 val div_mult_self_is_m = thm "div_mult_self_is_m";
   969 val div_mult_self1_is_m = thm "div_mult_self1_is_m";
   970 val unique_quotient_lemma = thm "unique_quotient_lemma";
   971 val unique_quotient = thm "unique_quotient";
   972 val unique_remainder = thm "unique_remainder";
   973 val div_0 = thm "div_0";
   974 val mod_0 = thm "mod_0";
   975 val div_mult1_eq = thm "div_mult1_eq";
   976 val mod_mult1_eq = thm "mod_mult1_eq";
   977 val mod_mult1_eq' = thm "mod_mult1_eq'";
   978 val mod_mult_distrib_mod = thm "mod_mult_distrib_mod";
   979 val div_add1_eq = thm "div_add1_eq";
   980 val mod_add1_eq = thm "mod_add1_eq";
   981 val mod_add_left_eq = thm "mod_add_left_eq";
   982  val mod_add_right_eq = thm "mod_add_right_eq";
   983 val mod_lemma = thm "mod_lemma";
   984 val div_mult2_eq = thm "div_mult2_eq";
   985 val mod_mult2_eq = thm "mod_mult2_eq";
   986 val div_mult_mult_lemma = thm "div_mult_mult_lemma";
   987 val div_mult_mult1 = thm "div_mult_mult1";
   988 val div_mult_mult2 = thm "div_mult_mult2";
   989 val div_1 = thm "div_1";
   990 val div_self = thm "div_self";
   991 val div_add_self2 = thm "div_add_self2";
   992 val div_add_self1 = thm "div_add_self1";
   993 val div_mult_self1 = thm "div_mult_self1";
   994 val div_mult_self2 = thm "div_mult_self2";
   995 val div_le_mono = thm "div_le_mono";
   996 val div_le_mono2 = thm "div_le_mono2";
   997 val div_le_dividend = thm "div_le_dividend";
   998 val div_less_dividend = thm "div_less_dividend";
   999 val mod_Suc = thm "mod_Suc";
  1000 val dvdI = thm "dvdI";
  1001 val dvdE = thm "dvdE";
  1002 val dvd_0_right = thm "dvd_0_right";
  1003 val dvd_0_left = thm "dvd_0_left";
  1004 val dvd_0_left_iff = thm "dvd_0_left_iff";
  1005 val dvd_1_left = thm "dvd_1_left";
  1006 val dvd_1_iff_1 = thm "dvd_1_iff_1";
  1007 val dvd_refl = thm "dvd_refl";
  1008 val dvd_trans = thm "dvd_trans";
  1009 val dvd_anti_sym = thm "dvd_anti_sym";
  1010 val dvd_add = thm "dvd_add";
  1011 val dvd_diff = thm "dvd_diff";
  1012 val dvd_diffD = thm "dvd_diffD";
  1013 val dvd_diffD1 = thm "dvd_diffD1";
  1014 val dvd_mult = thm "dvd_mult";
  1015 val dvd_mult2 = thm "dvd_mult2";
  1016 val dvd_reduce = thm "dvd_reduce";
  1017 val dvd_mod = thm "dvd_mod";
  1018 val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd";
  1019 val dvd_mod_iff = thm "dvd_mod_iff";
  1020 val dvd_mult_cancel = thm "dvd_mult_cancel";
  1021 val dvd_mult_cancel1 = thm "dvd_mult_cancel1";
  1022 val dvd_mult_cancel2 = thm "dvd_mult_cancel2";
  1023 val mult_dvd_mono = thm "mult_dvd_mono";
  1024 val dvd_mult_left = thm "dvd_mult_left";
  1025 val dvd_mult_right = thm "dvd_mult_right";
  1026 val dvd_imp_le = thm "dvd_imp_le";
  1027 val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0";
  1028 val dvd_mult_div_cancel = thm "dvd_mult_div_cancel";
  1029 val mod_eq_0_iff = thm "mod_eq_0_iff";
  1030 val mod_eqD = thm "mod_eqD";
  1031 *}
  1032 
  1033 (*
  1034 lemma split_div:
  1035 assumes m: "m \<noteq> 0"
  1036 shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)"
  1037        (is "?P = ?Q")
  1038 proof
  1039   assume P: ?P
  1040   show ?Q
  1041   proof (intro allI impI)
  1042     fix i j
  1043     assume n: "n = m*i + j" and j: "j < m"
  1044     show "P i"
  1045     proof (cases)
  1046       assume "i = 0"
  1047       with n j P show "P i" by simp
  1048     next
  1049       assume "i \<noteq> 0"
  1050       with n j P show "P i" by (simp add:add_ac div_mult_self1)
  1051     qed
  1052   qed
  1053 next
  1054   assume Q: ?Q
  1055   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
  1056   show ?P by simp
  1057 qed
  1058 
  1059 lemma split_mod:
  1060 assumes m: "m \<noteq> 0"
  1061 shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)"
  1062        (is "?P = ?Q")
  1063 proof
  1064   assume P: ?P
  1065   show ?Q
  1066   proof (intro allI impI)
  1067     fix i j
  1068     assume "n = m*i + j" "j < m"
  1069     thus "P j" using m P by(simp add:add_ac mult_ac)
  1070   qed
  1071 next
  1072   assume Q: ?Q
  1073   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
  1074   show ?P by simp
  1075 qed
  1076 *)
  1077 end