src/HOL/Divides.thy
 author paulson Tue Oct 19 18:18:45 2004 +0200 (2004-10-19) changeset 15251 bb6f072c8d10 parent 15140 322485b816ac child 15439 71c0f98e31f1 permissions -rw-r--r--
converted some induct_tac to induct
```     1 (*  Title:      HOL/Divides.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1999  University of Cambridge
```
```     5
```
```     6 The division operators div, mod and the divides relation "dvd"
```
```     7 *)
```
```     8
```
```     9 theory Divides
```
```    10 imports NatArith
```
```    11 begin
```
```    12
```
```    13 (*We use the same class for div and mod;
```
```    14   moreover, dvd is defined whenever multiplication is*)
```
```    15 axclass
```
```    16   div < type
```
```    17
```
```    18 instance  nat :: div ..
```
```    19
```
```    20 consts
```
```    21   div  :: "'a::div \<Rightarrow> 'a \<Rightarrow> 'a"          (infixl 70)
```
```    22   mod  :: "'a::div \<Rightarrow> 'a \<Rightarrow> 'a"          (infixl 70)
```
```    23   dvd  :: "'a::times \<Rightarrow> 'a \<Rightarrow> bool"      (infixl 50)
```
```    24
```
```    25
```
```    26 defs
```
```    27
```
```    28   mod_def:   "m mod n == wfrec (trancl pred_nat)
```
```    29                           (%f j. if j<n | n=0 then j else f (j-n)) m"
```
```    30
```
```    31   div_def:   "m div n == wfrec (trancl pred_nat)
```
```    32                           (%f j. if j<n | n=0 then 0 else Suc (f (j-n))) m"
```
```    33
```
```    34 (*The definition of dvd is polymorphic!*)
```
```    35   dvd_def:   "m dvd n == \<exists>k. n = m*k"
```
```    36
```
```    37 (*This definition helps prove the harder properties of div and mod.
```
```    38   It is copied from IntDiv.thy; should it be overloaded?*)
```
```    39 constdefs
```
```    40   quorem :: "(nat*nat) * (nat*nat) => bool"
```
```    41     "quorem == %((a,b), (q,r)).
```
```    42                       a = b*q + r &
```
```    43                       (if 0<b then 0\<le>r & r<b else b<r & r \<le>0)"
```
```    44
```
```    45
```
```    46
```
```    47 subsection{*Initial Lemmas*}
```
```    48
```
```    49 lemmas wf_less_trans =
```
```    50        def_wfrec [THEN trans, OF eq_reflection wf_pred_nat [THEN wf_trancl],
```
```    51                   standard]
```
```    52
```
```    53 lemma mod_eq: "(%m. m mod n) =
```
```    54               wfrec (trancl pred_nat) (%f j. if j<n | n=0 then j else f (j-n))"
```
```    55 by (simp add: mod_def)
```
```    56
```
```    57 lemma div_eq: "(%m. m div n) = wfrec (trancl pred_nat)
```
```    58                (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"
```
```    59 by (simp add: div_def)
```
```    60
```
```    61
```
```    62 (** Aribtrary definitions for division by zero.  Useful to simplify
```
```    63     certain equations **)
```
```    64
```
```    65 lemma DIVISION_BY_ZERO_DIV [simp]: "a div 0 = (0::nat)"
```
```    66 by (rule div_eq [THEN wf_less_trans], simp)
```
```    67
```
```    68 lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a::nat)"
```
```    69 by (rule mod_eq [THEN wf_less_trans], simp)
```
```    70
```
```    71
```
```    72 subsection{*Remainder*}
```
```    73
```
```    74 lemma mod_less [simp]: "m<n ==> m mod n = (m::nat)"
```
```    75 by (rule mod_eq [THEN wf_less_trans], simp)
```
```    76
```
```    77 lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n"
```
```    78 apply (case_tac "n=0", simp)
```
```    79 apply (rule mod_eq [THEN wf_less_trans])
```
```    80 apply (simp add: diff_less cut_apply less_eq)
```
```    81 done
```
```    82
```
```    83 (*Avoids the ugly ~m<n above*)
```
```    84 lemma le_mod_geq: "(n::nat) \<le> m ==> m mod n = (m-n) mod n"
```
```    85 by (simp add: mod_geq not_less_iff_le)
```
```    86
```
```    87 lemma mod_if: "m mod (n::nat) = (if m<n then m else (m-n) mod n)"
```
```    88 by (simp add: mod_geq)
```
```    89
```
```    90 lemma mod_1 [simp]: "m mod Suc 0 = 0"
```
```    91 apply (induct "m")
```
```    92 apply (simp_all (no_asm_simp) add: mod_geq)
```
```    93 done
```
```    94
```
```    95 lemma mod_self [simp]: "n mod n = (0::nat)"
```
```    96 apply (case_tac "n=0")
```
```    97 apply (simp_all add: mod_geq)
```
```    98 done
```
```    99
```
```   100 lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
```
```   101 apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n")
```
```   102 apply (simp add: add_commute)
```
```   103 apply (subst mod_geq [symmetric], simp_all)
```
```   104 done
```
```   105
```
```   106 lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
```
```   107 by (simp add: add_commute mod_add_self2)
```
```   108
```
```   109 lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
```
```   110 apply (induct "k")
```
```   111 apply (simp_all add: add_left_commute [of _ n])
```
```   112 done
```
```   113
```
```   114 lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
```
```   115 by (simp add: mult_commute mod_mult_self1)
```
```   116
```
```   117 lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)"
```
```   118 apply (case_tac "n=0", simp)
```
```   119 apply (case_tac "k=0", simp)
```
```   120 apply (induct "m" rule: nat_less_induct)
```
```   121 apply (subst mod_if, simp)
```
```   122 apply (simp add: mod_geq diff_less diff_mult_distrib)
```
```   123 done
```
```   124
```
```   125 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
```
```   126 by (simp add: mult_commute [of k] mod_mult_distrib)
```
```   127
```
```   128 lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
```
```   129 apply (case_tac "n=0", simp)
```
```   130 apply (induct "m", simp)
```
```   131 apply (rename_tac "k")
```
```   132 apply (cut_tac m = "k*n" and n = n in mod_add_self2)
```
```   133 apply (simp add: add_commute)
```
```   134 done
```
```   135
```
```   136 lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
```
```   137 by (simp add: mult_commute mod_mult_self_is_0)
```
```   138
```
```   139
```
```   140 subsection{*Quotient*}
```
```   141
```
```   142 lemma div_less [simp]: "m<n ==> m div n = (0::nat)"
```
```   143 by (rule div_eq [THEN wf_less_trans], simp)
```
```   144
```
```   145 lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"
```
```   146 apply (rule div_eq [THEN wf_less_trans])
```
```   147 apply (simp add: diff_less cut_apply less_eq)
```
```   148 done
```
```   149
```
```   150 (*Avoids the ugly ~m<n above*)
```
```   151 lemma le_div_geq: "[| 0<n;  n\<le>m |] ==> m div n = Suc((m-n) div n)"
```
```   152 by (simp add: div_geq not_less_iff_le)
```
```   153
```
```   154 lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"
```
```   155 by (simp add: div_geq)
```
```   156
```
```   157
```
```   158 (*Main Result about quotient and remainder.*)
```
```   159 lemma mod_div_equality: "(m div n)*n + m mod n = (m::nat)"
```
```   160 apply (case_tac "n=0", simp)
```
```   161 apply (induct "m" rule: nat_less_induct)
```
```   162 apply (subst mod_if)
```
```   163 apply (simp_all (no_asm_simp) add: add_assoc div_geq add_diff_inverse diff_less)
```
```   164 done
```
```   165
```
```   166 lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)"
```
```   167 apply(cut_tac m = m and n = n in mod_div_equality)
```
```   168 apply(simp add: mult_commute)
```
```   169 done
```
```   170
```
```   171 subsection{*Simproc for Cancelling Div and Mod*}
```
```   172
```
```   173 lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k"
```
```   174 apply(simp add: mod_div_equality)
```
```   175 done
```
```   176
```
```   177 lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k"
```
```   178 apply(simp add: mod_div_equality2)
```
```   179 done
```
```   180
```
```   181 ML
```
```   182 {*
```
```   183 val div_mod_equality = thm "div_mod_equality";
```
```   184 val div_mod_equality2 = thm "div_mod_equality2";
```
```   185
```
```   186
```
```   187 structure CancelDivModData =
```
```   188 struct
```
```   189
```
```   190 val div_name = "Divides.op div";
```
```   191 val mod_name = "Divides.op mod";
```
```   192 val mk_binop = HOLogic.mk_binop;
```
```   193 val mk_sum = NatArithUtils.mk_sum;
```
```   194 val dest_sum = NatArithUtils.dest_sum;
```
```   195
```
```   196 (*logic*)
```
```   197
```
```   198 val div_mod_eqs = map mk_meta_eq [div_mod_equality,div_mod_equality2]
```
```   199
```
```   200 val trans = trans
```
```   201
```
```   202 val prove_eq_sums =
```
```   203   let val simps = add_0 :: add_0_right :: add_ac
```
```   204   in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all simps) end
```
```   205
```
```   206 end;
```
```   207
```
```   208 structure CancelDivMod = CancelDivModFun(CancelDivModData);
```
```   209
```
```   210 val cancel_div_mod_proc = NatArithUtils.prep_simproc
```
```   211       ("cancel_div_mod", ["(m::nat) + n"], CancelDivMod.proc);
```
```   212
```
```   213 Addsimprocs[cancel_div_mod_proc];
```
```   214 *}
```
```   215
```
```   216
```
```   217 (* a simple rearrangement of mod_div_equality: *)
```
```   218 lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
```
```   219 by (cut_tac m = m and n = n in mod_div_equality2, arith)
```
```   220
```
```   221 lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)"
```
```   222 apply (induct "m" rule: nat_less_induct)
```
```   223 apply (case_tac "na<n", simp)
```
```   224 txt{*case @{term "n \<le> na"}*}
```
```   225 apply (simp add: mod_geq diff_less)
```
```   226 done
```
```   227
```
```   228 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
```
```   229 by (cut_tac m = "m*n" and n = n in mod_div_equality, auto)
```
```   230
```
```   231 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
```
```   232 by (simp add: mult_commute div_mult_self_is_m)
```
```   233
```
```   234 (*mod_mult_distrib2 above is the counterpart for remainder*)
```
```   235
```
```   236
```
```   237 subsection{*Proving facts about Quotient and Remainder*}
```
```   238
```
```   239 lemma unique_quotient_lemma:
```
```   240      "[| b*q' + r'  \<le> b*q + r;  0 < b;  r < b |]
```
```   241       ==> q' \<le> (q::nat)"
```
```   242 apply (rule leI)
```
```   243 apply (subst less_iff_Suc_add)
```
```   244 apply (auto simp add: add_mult_distrib2)
```
```   245 done
```
```   246
```
```   247 lemma unique_quotient:
```
```   248      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
```
```   249       ==> q = q'"
```
```   250 apply (simp add: split_ifs quorem_def)
```
```   251 apply (blast intro: order_antisym
```
```   252              dest: order_eq_refl [THEN unique_quotient_lemma] sym)+
```
```   253 done
```
```   254
```
```   255 lemma unique_remainder:
```
```   256      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
```
```   257       ==> r = r'"
```
```   258 apply (subgoal_tac "q = q'")
```
```   259 prefer 2 apply (blast intro: unique_quotient)
```
```   260 apply (simp add: quorem_def)
```
```   261 done
```
```   262
```
```   263 lemma quorem_div_mod: "0 < b ==> quorem ((a, b), (a div b, a mod b))"
```
```   264 by (auto simp add: quorem_def)
```
```   265
```
```   266 lemma quorem_div: "[| quorem((a,b),(q,r));  0 < b |] ==> a div b = q"
```
```   267 by (simp add: quorem_div_mod [THEN unique_quotient])
```
```   268
```
```   269 lemma quorem_mod: "[| quorem((a,b),(q,r));  0 < b |] ==> a mod b = r"
```
```   270 by (simp add: quorem_div_mod [THEN unique_remainder])
```
```   271
```
```   272 (** A dividend of zero **)
```
```   273
```
```   274 lemma div_0 [simp]: "0 div m = (0::nat)"
```
```   275 by (case_tac "m=0", simp_all)
```
```   276
```
```   277 lemma mod_0 [simp]: "0 mod m = (0::nat)"
```
```   278 by (case_tac "m=0", simp_all)
```
```   279
```
```   280 (** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
```
```   281
```
```   282 lemma quorem_mult1_eq:
```
```   283      "[| quorem((b,c),(q,r));  0 < c |]
```
```   284       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
```
```   285 apply (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
```
```   286 done
```
```   287
```
```   288 lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
```
```   289 apply (case_tac "c = 0", simp)
```
```   290 apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
```
```   291 done
```
```   292
```
```   293 lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
```
```   294 apply (case_tac "c = 0", simp)
```
```   295 apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
```
```   296 done
```
```   297
```
```   298 lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
```
```   299 apply (rule trans)
```
```   300 apply (rule_tac s = "b*a mod c" in trans)
```
```   301 apply (rule_tac [2] mod_mult1_eq)
```
```   302 apply (simp_all (no_asm) add: mult_commute)
```
```   303 done
```
```   304
```
```   305 lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
```
```   306 apply (rule mod_mult1_eq' [THEN trans])
```
```   307 apply (rule mod_mult1_eq)
```
```   308 done
```
```   309
```
```   310 (** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
```
```   311
```
```   312 lemma quorem_add1_eq:
```
```   313      "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  0 < c |]
```
```   314       ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
```
```   315 by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
```
```   316
```
```   317 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```   318 lemma div_add1_eq:
```
```   319      "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```   320 apply (case_tac "c = 0", simp)
```
```   321 apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod)
```
```   322 done
```
```   323
```
```   324 lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
```
```   325 apply (case_tac "c = 0", simp)
```
```   326 apply (blast intro: quorem_div_mod quorem_div_mod
```
```   327                     quorem_add1_eq [THEN quorem_mod])
```
```   328 done
```
```   329
```
```   330
```
```   331 subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*}
```
```   332
```
```   333 (** first, a lemma to bound the remainder **)
```
```   334
```
```   335 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
```
```   336 apply (cut_tac m = q and n = c in mod_less_divisor)
```
```   337 apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
```
```   338 apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
```
```   339 apply (simp add: add_mult_distrib2)
```
```   340 done
```
```   341
```
```   342 lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]
```
```   343       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
```
```   344 apply (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)
```
```   345 done
```
```   346
```
```   347 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
```
```   348 apply (case_tac "b=0", simp)
```
```   349 apply (case_tac "c=0", simp)
```
```   350 apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
```
```   351 done
```
```   352
```
```   353 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
```
```   354 apply (case_tac "b=0", simp)
```
```   355 apply (case_tac "c=0", simp)
```
```   356 apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
```
```   357 done
```
```   358
```
```   359
```
```   360 subsection{*Cancellation of Common Factors in Division*}
```
```   361
```
```   362 lemma div_mult_mult_lemma:
```
```   363      "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
```
```   364 by (auto simp add: div_mult2_eq)
```
```   365
```
```   366 lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
```
```   367 apply (case_tac "b = 0")
```
```   368 apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
```
```   369 done
```
```   370
```
```   371 lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
```
```   372 apply (drule div_mult_mult1)
```
```   373 apply (auto simp add: mult_commute)
```
```   374 done
```
```   375
```
```   376
```
```   377 (*Distribution of Factors over Remainders:
```
```   378
```
```   379 Could prove these as in Integ/IntDiv.ML, but we already have
```
```   380 mod_mult_distrib and mod_mult_distrib2 above!
```
```   381
```
```   382 Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)"
```
```   383 qed "mod_mult_mult1";
```
```   384
```
```   385 Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)";
```
```   386 qed "mod_mult_mult2";
```
```   387  ***)
```
```   388
```
```   389 subsection{*Further Facts about Quotient and Remainder*}
```
```   390
```
```   391 lemma div_1 [simp]: "m div Suc 0 = m"
```
```   392 apply (induct "m")
```
```   393 apply (simp_all (no_asm_simp) add: div_geq)
```
```   394 done
```
```   395
```
```   396 lemma div_self [simp]: "0<n ==> n div n = (1::nat)"
```
```   397 by (simp add: div_geq)
```
```   398
```
```   399 lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
```
```   400 apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
```
```   401 apply (simp add: add_commute)
```
```   402 apply (subst div_geq [symmetric], simp_all)
```
```   403 done
```
```   404
```
```   405 lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
```
```   406 by (simp add: add_commute div_add_self2)
```
```   407
```
```   408 lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
```
```   409 apply (subst div_add1_eq)
```
```   410 apply (subst div_mult1_eq, simp)
```
```   411 done
```
```   412
```
```   413 lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
```
```   414 by (simp add: mult_commute div_mult_self1)
```
```   415
```
```   416
```
```   417 (* Monotonicity of div in first argument *)
```
```   418 lemma div_le_mono [rule_format (no_asm)]:
```
```   419      "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
```
```   420 apply (case_tac "k=0", simp)
```
```   421 apply (induct "n" rule: nat_less_induct, clarify)
```
```   422 apply (case_tac "n<k")
```
```   423 (* 1  case n<k *)
```
```   424 apply simp
```
```   425 (* 2  case n >= k *)
```
```   426 apply (case_tac "m<k")
```
```   427 (* 2.1  case m<k *)
```
```   428 apply simp
```
```   429 (* 2.2  case m>=k *)
```
```   430 apply (simp add: div_geq diff_less diff_le_mono)
```
```   431 done
```
```   432
```
```   433 (* Antimonotonicity of div in second argument *)
```
```   434 lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
```
```   435 apply (subgoal_tac "0<n")
```
```   436  prefer 2 apply simp
```
```   437 apply (induct_tac k rule: nat_less_induct)
```
```   438 apply (rename_tac "k")
```
```   439 apply (case_tac "k<n", simp)
```
```   440 apply (subgoal_tac "~ (k<m) ")
```
```   441  prefer 2 apply simp
```
```   442 apply (simp add: div_geq)
```
```   443 apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
```
```   444  prefer 2
```
```   445  apply (blast intro: div_le_mono diff_le_mono2)
```
```   446 apply (rule le_trans, simp)
```
```   447 apply (simp add: diff_less)
```
```   448 done
```
```   449
```
```   450 lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
```
```   451 apply (case_tac "n=0", simp)
```
```   452 apply (subgoal_tac "m div n \<le> m div 1", simp)
```
```   453 apply (rule div_le_mono2)
```
```   454 apply (simp_all (no_asm_simp))
```
```   455 done
```
```   456
```
```   457 (* Similar for "less than" *)
```
```   458 lemma div_less_dividend [rule_format, simp]:
```
```   459      "!!n::nat. 1<n ==> 0 < m --> m div n < m"
```
```   460 apply (induct_tac m rule: nat_less_induct)
```
```   461 apply (rename_tac "m")
```
```   462 apply (case_tac "m<n", simp)
```
```   463 apply (subgoal_tac "0<n")
```
```   464  prefer 2 apply simp
```
```   465 apply (simp add: div_geq)
```
```   466 apply (case_tac "n<m")
```
```   467  apply (subgoal_tac "(m-n) div n < (m-n) ")
```
```   468   apply (rule impI less_trans_Suc)+
```
```   469 apply assumption
```
```   470   apply (simp_all add: diff_less)
```
```   471 done
```
```   472
```
```   473 text{*A fact for the mutilated chess board*}
```
```   474 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
```
```   475 apply (case_tac "n=0", simp)
```
```   476 apply (induct "m" rule: nat_less_induct)
```
```   477 apply (case_tac "Suc (na) <n")
```
```   478 (* case Suc(na) < n *)
```
```   479 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
```
```   480 (* case n \<le> Suc(na) *)
```
```   481 apply (simp add: not_less_iff_le le_Suc_eq mod_geq)
```
```   482 apply (auto simp add: Suc_diff_le diff_less le_mod_geq)
```
```   483 done
```
```   484
```
```   485 lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
```
```   486 by (case_tac "n=0", auto)
```
```   487
```
```   488 lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
```
```   489 by (case_tac "n=0", auto)
```
```   490
```
```   491
```
```   492 subsection{*The Divides Relation*}
```
```   493
```
```   494 lemma dvdI [intro?]: "n = m * k ==> m dvd n"
```
```   495 by (unfold dvd_def, blast)
```
```   496
```
```   497 lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
```
```   498 by (unfold dvd_def, blast)
```
```   499
```
```   500 lemma dvd_0_right [iff]: "m dvd (0::nat)"
```
```   501 apply (unfold dvd_def)
```
```   502 apply (blast intro: mult_0_right [symmetric])
```
```   503 done
```
```   504
```
```   505 lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
```
```   506 by (force simp add: dvd_def)
```
```   507
```
```   508 lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
```
```   509 by (blast intro: dvd_0_left)
```
```   510
```
```   511 lemma dvd_1_left [iff]: "Suc 0 dvd k"
```
```   512 by (unfold dvd_def, simp)
```
```   513
```
```   514 lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
```
```   515 by (simp add: dvd_def)
```
```   516
```
```   517 lemma dvd_refl [simp]: "m dvd (m::nat)"
```
```   518 apply (unfold dvd_def)
```
```   519 apply (blast intro: mult_1_right [symmetric])
```
```   520 done
```
```   521
```
```   522 lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"
```
```   523 apply (unfold dvd_def)
```
```   524 apply (blast intro: mult_assoc)
```
```   525 done
```
```   526
```
```   527 lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
```
```   528 apply (unfold dvd_def)
```
```   529 apply (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
```
```   530 done
```
```   531
```
```   532 lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
```
```   533 apply (unfold dvd_def)
```
```   534 apply (blast intro: add_mult_distrib2 [symmetric])
```
```   535 done
```
```   536
```
```   537 lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
```
```   538 apply (unfold dvd_def)
```
```   539 apply (blast intro: diff_mult_distrib2 [symmetric])
```
```   540 done
```
```   541
```
```   542 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
```
```   543 apply (erule not_less_iff_le [THEN iffD2, THEN add_diff_inverse, THEN subst])
```
```   544 apply (blast intro: dvd_add)
```
```   545 done
```
```   546
```
```   547 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
```
```   548 by (drule_tac m = m in dvd_diff, auto)
```
```   549
```
```   550 lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
```
```   551 apply (unfold dvd_def)
```
```   552 apply (blast intro: mult_left_commute)
```
```   553 done
```
```   554
```
```   555 lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
```
```   556 apply (subst mult_commute)
```
```   557 apply (erule dvd_mult)
```
```   558 done
```
```   559
```
```   560 (* k dvd (m*k) *)
```
```   561 declare dvd_refl [THEN dvd_mult, iff] dvd_refl [THEN dvd_mult2, iff]
```
```   562
```
```   563 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
```
```   564 apply (rule iffI)
```
```   565 apply (erule_tac [2] dvd_add)
```
```   566 apply (rule_tac [2] dvd_refl)
```
```   567 apply (subgoal_tac "n = (n+k) -k")
```
```   568  prefer 2 apply simp
```
```   569 apply (erule ssubst)
```
```   570 apply (erule dvd_diff)
```
```   571 apply (rule dvd_refl)
```
```   572 done
```
```   573
```
```   574 lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
```
```   575 apply (unfold dvd_def)
```
```   576 apply (case_tac "n=0", auto)
```
```   577 apply (blast intro: mod_mult_distrib2 [symmetric])
```
```   578 done
```
```   579
```
```   580 lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
```
```   581 apply (subgoal_tac "k dvd (m div n) *n + m mod n")
```
```   582  apply (simp add: mod_div_equality)
```
```   583 apply (simp only: dvd_add dvd_mult)
```
```   584 done
```
```   585
```
```   586 lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
```
```   587 by (blast intro: dvd_mod_imp_dvd dvd_mod)
```
```   588
```
```   589 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
```
```   590 apply (unfold dvd_def)
```
```   591 apply (erule exE)
```
```   592 apply (simp add: mult_ac)
```
```   593 done
```
```   594
```
```   595 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
```
```   596 apply auto
```
```   597 apply (subgoal_tac "m*n dvd m*1")
```
```   598 apply (drule dvd_mult_cancel, auto)
```
```   599 done
```
```   600
```
```   601 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
```
```   602 apply (subst mult_commute)
```
```   603 apply (erule dvd_mult_cancel1)
```
```   604 done
```
```   605
```
```   606 lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
```
```   607 apply (unfold dvd_def, clarify)
```
```   608 apply (rule_tac x = "k*ka" in exI)
```
```   609 apply (simp add: mult_ac)
```
```   610 done
```
```   611
```
```   612 lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
```
```   613 by (simp add: dvd_def mult_assoc, blast)
```
```   614
```
```   615 lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
```
```   616 apply (unfold dvd_def, clarify)
```
```   617 apply (rule_tac x = "i*k" in exI)
```
```   618 apply (simp add: mult_ac)
```
```   619 done
```
```   620
```
```   621 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
```
```   622 apply (unfold dvd_def, clarify)
```
```   623 apply (simp_all (no_asm_use) add: zero_less_mult_iff)
```
```   624 apply (erule conjE)
```
```   625 apply (rule le_trans)
```
```   626 apply (rule_tac [2] le_refl [THEN mult_le_mono])
```
```   627 apply (erule_tac [2] Suc_leI, simp)
```
```   628 done
```
```   629
```
```   630 lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"
```
```   631 apply (unfold dvd_def)
```
```   632 apply (case_tac "k=0", simp, safe)
```
```   633 apply (simp add: mult_commute)
```
```   634 apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])
```
```   635 apply (subst mult_commute, simp)
```
```   636 done
```
```   637
```
```   638 lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
```
```   639 apply (subgoal_tac "m mod n = 0")
```
```   640  apply (simp add: mult_div_cancel)
```
```   641 apply (simp only: dvd_eq_mod_eq_0)
```
```   642 done
```
```   643
```
```   644 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
```
```   645 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```   646 declare mod_eq_0_iff [THEN iffD1, dest!]
```
```   647
```
```   648 (*Loses information, namely we also have r<d provided d is nonzero*)
```
```   649 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
```
```   650 apply (cut_tac m = m in mod_div_equality)
```
```   651 apply (simp only: add_ac)
```
```   652 apply (blast intro: sym)
```
```   653 done
```
```   654
```
```   655
```
```   656 lemma split_div:
```
```   657  "P(n div k :: nat) =
```
```   658  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
```
```   659  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```   660 proof
```
```   661   assume P: ?P
```
```   662   show ?Q
```
```   663   proof (cases)
```
```   664     assume "k = 0"
```
```   665     with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
```
```   666   next
```
```   667     assume not0: "k \<noteq> 0"
```
```   668     thus ?Q
```
```   669     proof (simp, intro allI impI)
```
```   670       fix i j
```
```   671       assume n: "n = k*i + j" and j: "j < k"
```
```   672       show "P i"
```
```   673       proof (cases)
```
```   674 	assume "i = 0"
```
```   675 	with n j P show "P i" by simp
```
```   676       next
```
```   677 	assume "i \<noteq> 0"
```
```   678 	with not0 n j P show "P i" by(simp add:add_ac)
```
```   679       qed
```
```   680     qed
```
```   681   qed
```
```   682 next
```
```   683   assume Q: ?Q
```
```   684   show ?P
```
```   685   proof (cases)
```
```   686     assume "k = 0"
```
```   687     with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
```
```   688   next
```
```   689     assume not0: "k \<noteq> 0"
```
```   690     with Q have R: ?R by simp
```
```   691     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```   692     show ?P by simp
```
```   693   qed
```
```   694 qed
```
```   695
```
```   696 lemma split_div_lemma:
```
```   697   "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
```
```   698   apply (rule iffI)
```
```   699   apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
```
```   700   apply (simp_all add: quorem_def, arith)
```
```   701   apply (rule conjI)
```
```   702   apply (rule_tac P="%x. n * (m div n) \<le> x" in
```
```   703     subst [OF mod_div_equality [of _ n]])
```
```   704   apply (simp only: add: mult_ac)
```
```   705   apply (rule_tac P="%x. x < n + n * (m div n)" in
```
```   706     subst [OF mod_div_equality [of _ n]])
```
```   707   apply (simp only: add: mult_ac add_ac)
```
```   708   apply (rule add_less_mono1, simp)
```
```   709   done
```
```   710
```
```   711 theorem split_div':
```
```   712   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
```
```   713    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
```
```   714   apply (case_tac "0 < n")
```
```   715   apply (simp only: add: split_div_lemma)
```
```   716   apply (simp_all add: DIVISION_BY_ZERO_DIV)
```
```   717   done
```
```   718
```
```   719 lemma split_mod:
```
```   720  "P(n mod k :: nat) =
```
```   721  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
```
```   722  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```   723 proof
```
```   724   assume P: ?P
```
```   725   show ?Q
```
```   726   proof (cases)
```
```   727     assume "k = 0"
```
```   728     with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
```
```   729   next
```
```   730     assume not0: "k \<noteq> 0"
```
```   731     thus ?Q
```
```   732     proof (simp, intro allI impI)
```
```   733       fix i j
```
```   734       assume "n = k*i + j" "j < k"
```
```   735       thus "P j" using not0 P by(simp add:add_ac mult_ac)
```
```   736     qed
```
```   737   qed
```
```   738 next
```
```   739   assume Q: ?Q
```
```   740   show ?P
```
```   741   proof (cases)
```
```   742     assume "k = 0"
```
```   743     with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
```
```   744   next
```
```   745     assume not0: "k \<noteq> 0"
```
```   746     with Q have R: ?R by simp
```
```   747     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```   748     show ?P by simp
```
```   749   qed
```
```   750 qed
```
```   751
```
```   752 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
```
```   753   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
```
```   754     subst [OF mod_div_equality [of _ n]])
```
```   755   apply arith
```
```   756   done
```
```   757
```
```   758 subsection {*An ``induction'' law for modulus arithmetic.*}
```
```   759
```
```   760 lemma mod_induct_0:
```
```   761   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```   762   and base: "P i" and i: "i<p"
```
```   763   shows "P 0"
```
```   764 proof (rule ccontr)
```
```   765   assume contra: "\<not>(P 0)"
```
```   766   from i have p: "0<p" by simp
```
```   767   have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
```
```   768   proof
```
```   769     fix k
```
```   770     show "?A k"
```
```   771     proof (induct k)
```
```   772       show "?A 0" by simp  -- "by contradiction"
```
```   773     next
```
```   774       fix n
```
```   775       assume ih: "?A n"
```
```   776       show "?A (Suc n)"
```
```   777       proof (clarsimp)
```
```   778 	assume y: "P (p - Suc n)"
```
```   779 	have n: "Suc n < p"
```
```   780 	proof (rule ccontr)
```
```   781 	  assume "\<not>(Suc n < p)"
```
```   782 	  hence "p - Suc n = 0"
```
```   783 	    by simp
```
```   784 	  with y contra show "False"
```
```   785 	    by simp
```
```   786 	qed
```
```   787 	hence n2: "Suc (p - Suc n) = p-n" by arith
```
```   788 	from p have "p - Suc n < p" by arith
```
```   789 	with y step have z: "P ((Suc (p - Suc n)) mod p)"
```
```   790 	  by blast
```
```   791 	show "False"
```
```   792 	proof (cases "n=0")
```
```   793 	  case True
```
```   794 	  with z n2 contra show ?thesis by simp
```
```   795 	next
```
```   796 	  case False
```
```   797 	  with p have "p-n < p" by arith
```
```   798 	  with z n2 False ih show ?thesis by simp
```
```   799 	qed
```
```   800       qed
```
```   801     qed
```
```   802   qed
```
```   803   moreover
```
```   804   from i obtain k where "0<k \<and> i+k=p"
```
```   805     by (blast dest: less_imp_add_positive)
```
```   806   hence "0<k \<and> i=p-k" by auto
```
```   807   moreover
```
```   808   note base
```
```   809   ultimately
```
```   810   show "False" by blast
```
```   811 qed
```
```   812
```
```   813 lemma mod_induct:
```
```   814   assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
```
```   815   and base: "P i" and i: "i<p" and j: "j<p"
```
```   816   shows "P j"
```
```   817 proof -
```
```   818   have "\<forall>j<p. P j"
```
```   819   proof
```
```   820     fix j
```
```   821     show "j<p \<longrightarrow> P j" (is "?A j")
```
```   822     proof (induct j)
```
```   823       from step base i show "?A 0"
```
```   824 	by (auto elim: mod_induct_0)
```
```   825     next
```
```   826       fix k
```
```   827       assume ih: "?A k"
```
```   828       show "?A (Suc k)"
```
```   829       proof
```
```   830 	assume suc: "Suc k < p"
```
```   831 	hence k: "k<p" by simp
```
```   832 	with ih have "P k" ..
```
```   833 	with step k have "P (Suc k mod p)"
```
```   834 	  by blast
```
```   835 	moreover
```
```   836 	from suc have "Suc k mod p = Suc k"
```
```   837 	  by simp
```
```   838 	ultimately
```
```   839 	show "P (Suc k)" by simp
```
```   840       qed
```
```   841     qed
```
```   842   qed
```
```   843   with j show ?thesis by blast
```
```   844 qed
```
```   845
```
```   846
```
```   847 ML
```
```   848 {*
```
```   849 val div_def = thm "div_def"
```
```   850 val mod_def = thm "mod_def"
```
```   851 val dvd_def = thm "dvd_def"
```
```   852 val quorem_def = thm "quorem_def"
```
```   853
```
```   854 val wf_less_trans = thm "wf_less_trans";
```
```   855 val mod_eq = thm "mod_eq";
```
```   856 val div_eq = thm "div_eq";
```
```   857 val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";
```
```   858 val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";
```
```   859 val mod_less = thm "mod_less";
```
```   860 val mod_geq = thm "mod_geq";
```
```   861 val le_mod_geq = thm "le_mod_geq";
```
```   862 val mod_if = thm "mod_if";
```
```   863 val mod_1 = thm "mod_1";
```
```   864 val mod_self = thm "mod_self";
```
```   865 val mod_add_self2 = thm "mod_add_self2";
```
```   866 val mod_add_self1 = thm "mod_add_self1";
```
```   867 val mod_mult_self1 = thm "mod_mult_self1";
```
```   868 val mod_mult_self2 = thm "mod_mult_self2";
```
```   869 val mod_mult_distrib = thm "mod_mult_distrib";
```
```   870 val mod_mult_distrib2 = thm "mod_mult_distrib2";
```
```   871 val mod_mult_self_is_0 = thm "mod_mult_self_is_0";
```
```   872 val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0";
```
```   873 val div_less = thm "div_less";
```
```   874 val div_geq = thm "div_geq";
```
```   875 val le_div_geq = thm "le_div_geq";
```
```   876 val div_if = thm "div_if";
```
```   877 val mod_div_equality = thm "mod_div_equality";
```
```   878 val mod_div_equality2 = thm "mod_div_equality2";
```
```   879 val div_mod_equality = thm "div_mod_equality";
```
```   880 val div_mod_equality2 = thm "div_mod_equality2";
```
```   881 val mult_div_cancel = thm "mult_div_cancel";
```
```   882 val mod_less_divisor = thm "mod_less_divisor";
```
```   883 val div_mult_self_is_m = thm "div_mult_self_is_m";
```
```   884 val div_mult_self1_is_m = thm "div_mult_self1_is_m";
```
```   885 val unique_quotient_lemma = thm "unique_quotient_lemma";
```
```   886 val unique_quotient = thm "unique_quotient";
```
```   887 val unique_remainder = thm "unique_remainder";
```
```   888 val div_0 = thm "div_0";
```
```   889 val mod_0 = thm "mod_0";
```
```   890 val div_mult1_eq = thm "div_mult1_eq";
```
```   891 val mod_mult1_eq = thm "mod_mult1_eq";
```
```   892 val mod_mult1_eq' = thm "mod_mult1_eq'";
```
```   893 val mod_mult_distrib_mod = thm "mod_mult_distrib_mod";
```
```   894 val div_add1_eq = thm "div_add1_eq";
```
```   895 val mod_add1_eq = thm "mod_add1_eq";
```
```   896 val mod_lemma = thm "mod_lemma";
```
```   897 val div_mult2_eq = thm "div_mult2_eq";
```
```   898 val mod_mult2_eq = thm "mod_mult2_eq";
```
```   899 val div_mult_mult_lemma = thm "div_mult_mult_lemma";
```
```   900 val div_mult_mult1 = thm "div_mult_mult1";
```
```   901 val div_mult_mult2 = thm "div_mult_mult2";
```
```   902 val div_1 = thm "div_1";
```
```   903 val div_self = thm "div_self";
```
```   904 val div_add_self2 = thm "div_add_self2";
```
```   905 val div_add_self1 = thm "div_add_self1";
```
```   906 val div_mult_self1 = thm "div_mult_self1";
```
```   907 val div_mult_self2 = thm "div_mult_self2";
```
```   908 val div_le_mono = thm "div_le_mono";
```
```   909 val div_le_mono2 = thm "div_le_mono2";
```
```   910 val div_le_dividend = thm "div_le_dividend";
```
```   911 val div_less_dividend = thm "div_less_dividend";
```
```   912 val mod_Suc = thm "mod_Suc";
```
```   913 val dvdI = thm "dvdI";
```
```   914 val dvdE = thm "dvdE";
```
```   915 val dvd_0_right = thm "dvd_0_right";
```
```   916 val dvd_0_left = thm "dvd_0_left";
```
```   917 val dvd_0_left_iff = thm "dvd_0_left_iff";
```
```   918 val dvd_1_left = thm "dvd_1_left";
```
```   919 val dvd_1_iff_1 = thm "dvd_1_iff_1";
```
```   920 val dvd_refl = thm "dvd_refl";
```
```   921 val dvd_trans = thm "dvd_trans";
```
```   922 val dvd_anti_sym = thm "dvd_anti_sym";
```
```   923 val dvd_add = thm "dvd_add";
```
```   924 val dvd_diff = thm "dvd_diff";
```
```   925 val dvd_diffD = thm "dvd_diffD";
```
```   926 val dvd_diffD1 = thm "dvd_diffD1";
```
```   927 val dvd_mult = thm "dvd_mult";
```
```   928 val dvd_mult2 = thm "dvd_mult2";
```
```   929 val dvd_reduce = thm "dvd_reduce";
```
```   930 val dvd_mod = thm "dvd_mod";
```
```   931 val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd";
```
```   932 val dvd_mod_iff = thm "dvd_mod_iff";
```
```   933 val dvd_mult_cancel = thm "dvd_mult_cancel";
```
```   934 val dvd_mult_cancel1 = thm "dvd_mult_cancel1";
```
```   935 val dvd_mult_cancel2 = thm "dvd_mult_cancel2";
```
```   936 val mult_dvd_mono = thm "mult_dvd_mono";
```
```   937 val dvd_mult_left = thm "dvd_mult_left";
```
```   938 val dvd_mult_right = thm "dvd_mult_right";
```
```   939 val dvd_imp_le = thm "dvd_imp_le";
```
```   940 val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0";
```
```   941 val dvd_mult_div_cancel = thm "dvd_mult_div_cancel";
```
```   942 val mod_eq_0_iff = thm "mod_eq_0_iff";
```
```   943 val mod_eqD = thm "mod_eqD";
```
```   944 *}
```
```   945
```
```   946
```
```   947 (*
```
```   948 lemma split_div:
```
```   949 assumes m: "m \<noteq> 0"
```
```   950 shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)"
```
```   951        (is "?P = ?Q")
```
```   952 proof
```
```   953   assume P: ?P
```
```   954   show ?Q
```
```   955   proof (intro allI impI)
```
```   956     fix i j
```
```   957     assume n: "n = m*i + j" and j: "j < m"
```
```   958     show "P i"
```
```   959     proof (cases)
```
```   960       assume "i = 0"
```
```   961       with n j P show "P i" by simp
```
```   962     next
```
```   963       assume "i \<noteq> 0"
```
```   964       with n j P show "P i" by (simp add:add_ac div_mult_self1)
```
```   965     qed
```
```   966   qed
```
```   967 next
```
```   968   assume Q: ?Q
```
```   969   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
```
```   970   show ?P by simp
```
```   971 qed
```
```   972
```
```   973 lemma split_mod:
```
```   974 assumes m: "m \<noteq> 0"
```
```   975 shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)"
```
```   976        (is "?P = ?Q")
```
```   977 proof
```
```   978   assume P: ?P
```
```   979   show ?Q
```
```   980   proof (intro allI impI)
```
```   981     fix i j
```
```   982     assume "n = m*i + j" "j < m"
```
```   983     thus "P j" using m P by(simp add:add_ac mult_ac)
```
```   984   qed
```
```   985 next
```
```   986   assume Q: ?Q
```
```   987   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
```
```   988   show ?P by simp
```
```   989 qed
```
```   990 *)
```
```   991 end
```