src/HOL/Divides.thy
 author paulson Fri Mar 05 15:26:04 2004 +0100 (2004-03-05) changeset 14437 92f6aa05b7bb parent 14430 5cb24165a2e1 child 14640 b31870c50c68 permissions -rw-r--r--
some new results
```     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 = NatArith:
```
```    10
```
```    11 (*We use the same class for div and mod;
```
```    12   moreover, dvd is defined whenever multiplication is*)
```
```    13 axclass
```
```    14   div < type
```
```    15
```
```    16 instance  nat :: div ..
```
```    17
```
```    18 consts
```
```    19   div  :: "'a::div \<Rightarrow> 'a \<Rightarrow> 'a"          (infixl 70)
```
```    20   mod  :: "'a::div \<Rightarrow> 'a \<Rightarrow> 'a"          (infixl 70)
```
```    21   dvd  :: "'a::times \<Rightarrow> 'a \<Rightarrow> bool"      (infixl 50)
```
```    22
```
```    23
```
```    24 defs
```
```    25
```
```    26   mod_def:   "m mod n == wfrec (trancl pred_nat)
```
```    27                           (%f j. if j<n | n=0 then j else f (j-n)) m"
```
```    28
```
```    29   div_def:   "m div n == wfrec (trancl pred_nat)
```
```    30                           (%f j. if j<n | n=0 then 0 else Suc (f (j-n))) m"
```
```    31
```
```    32 (*The definition of dvd is polymorphic!*)
```
```    33   dvd_def:   "m dvd n == \<exists>k. n = m*k"
```
```    34
```
```    35 (*This definition helps prove the harder properties of div and mod.
```
```    36   It is copied from IntDiv.thy; should it be overloaded?*)
```
```    37 constdefs
```
```    38   quorem :: "(nat*nat) * (nat*nat) => bool"
```
```    39     "quorem == %((a,b), (q,r)).
```
```    40                       a = b*q + r &
```
```    41                       (if 0<b then 0\<le>r & r<b else b<r & r \<le>0)"
```
```    42
```
```    43
```
```    44
```
```    45 subsection{*Initial Lemmas*}
```
```    46
```
```    47 lemmas wf_less_trans =
```
```    48        def_wfrec [THEN trans, OF eq_reflection wf_pred_nat [THEN wf_trancl],
```
```    49                   standard]
```
```    50
```
```    51 lemma mod_eq: "(%m. m mod n) =
```
```    52               wfrec (trancl pred_nat) (%f j. if j<n | n=0 then j else f (j-n))"
```
```    53 by (simp add: mod_def)
```
```    54
```
```    55 lemma div_eq: "(%m. m div n) = wfrec (trancl pred_nat)
```
```    56                (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"
```
```    57 by (simp add: div_def)
```
```    58
```
```    59
```
```    60 (** Aribtrary definitions for division by zero.  Useful to simplify
```
```    61     certain equations **)
```
```    62
```
```    63 lemma DIVISION_BY_ZERO_DIV [simp]: "a div 0 = (0::nat)"
```
```    64 by (rule div_eq [THEN wf_less_trans], simp)
```
```    65
```
```    66 lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a::nat)"
```
```    67 by (rule mod_eq [THEN wf_less_trans], simp)
```
```    68
```
```    69
```
```    70 subsection{*Remainder*}
```
```    71
```
```    72 lemma mod_less [simp]: "m<n ==> m mod n = (m::nat)"
```
```    73 by (rule mod_eq [THEN wf_less_trans], simp)
```
```    74
```
```    75 lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n"
```
```    76 apply (case_tac "n=0", simp)
```
```    77 apply (rule mod_eq [THEN wf_less_trans])
```
```    78 apply (simp add: diff_less cut_apply less_eq)
```
```    79 done
```
```    80
```
```    81 (*Avoids the ugly ~m<n above*)
```
```    82 lemma le_mod_geq: "(n::nat) \<le> m ==> m mod n = (m-n) mod n"
```
```    83 by (simp add: mod_geq not_less_iff_le)
```
```    84
```
```    85 lemma mod_if: "m mod (n::nat) = (if m<n then m else (m-n) mod n)"
```
```    86 by (simp add: mod_geq)
```
```    87
```
```    88 lemma mod_1 [simp]: "m mod Suc 0 = 0"
```
```    89 apply (induct_tac "m")
```
```    90 apply (simp_all (no_asm_simp) add: mod_geq)
```
```    91 done
```
```    92
```
```    93 lemma mod_self [simp]: "n mod n = (0::nat)"
```
```    94 apply (case_tac "n=0")
```
```    95 apply (simp_all add: mod_geq)
```
```    96 done
```
```    97
```
```    98 lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
```
```    99 apply (subgoal_tac " (n + m) mod n = (n+m-n) mod n")
```
```   100 apply (simp add: add_commute)
```
```   101 apply (subst mod_geq [symmetric], simp_all)
```
```   102 done
```
```   103
```
```   104 lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
```
```   105 by (simp add: add_commute mod_add_self2)
```
```   106
```
```   107 lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
```
```   108 apply (induct_tac "k")
```
```   109 apply (simp_all add: add_left_commute [of _ n])
```
```   110 done
```
```   111
```
```   112 lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
```
```   113 by (simp add: mult_commute mod_mult_self1)
```
```   114
```
```   115 lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)"
```
```   116 apply (case_tac "n=0", simp)
```
```   117 apply (case_tac "k=0", simp)
```
```   118 apply (induct_tac "m" rule: nat_less_induct)
```
```   119 apply (subst mod_if, simp)
```
```   120 apply (simp add: mod_geq diff_less diff_mult_distrib)
```
```   121 done
```
```   122
```
```   123 lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
```
```   124 by (simp add: mult_commute [of k] mod_mult_distrib)
```
```   125
```
```   126 lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
```
```   127 apply (case_tac "n=0", simp)
```
```   128 apply (induct_tac "m", simp)
```
```   129 apply (rename_tac "k")
```
```   130 apply (cut_tac m = "k*n" and n = n in mod_add_self2)
```
```   131 apply (simp add: add_commute)
```
```   132 done
```
```   133
```
```   134 lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
```
```   135 by (simp add: mult_commute mod_mult_self_is_0)
```
```   136
```
```   137
```
```   138 subsection{*Quotient*}
```
```   139
```
```   140 lemma div_less [simp]: "m<n ==> m div n = (0::nat)"
```
```   141 by (rule div_eq [THEN wf_less_trans], simp)
```
```   142
```
```   143 lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"
```
```   144 apply (rule div_eq [THEN wf_less_trans])
```
```   145 apply (simp add: diff_less cut_apply less_eq)
```
```   146 done
```
```   147
```
```   148 (*Avoids the ugly ~m<n above*)
```
```   149 lemma le_div_geq: "[| 0<n;  n\<le>m |] ==> m div n = Suc((m-n) div n)"
```
```   150 by (simp add: div_geq not_less_iff_le)
```
```   151
```
```   152 lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"
```
```   153 by (simp add: div_geq)
```
```   154
```
```   155
```
```   156 (*Main Result about quotient and remainder.*)
```
```   157 lemma mod_div_equality: "(m div n)*n + m mod n = (m::nat)"
```
```   158 apply (case_tac "n=0", simp)
```
```   159 apply (induct_tac "m" rule: nat_less_induct)
```
```   160 apply (subst mod_if)
```
```   161 apply (simp_all (no_asm_simp) add: add_assoc div_geq add_diff_inverse diff_less)
```
```   162 done
```
```   163
```
```   164 lemma mod_div_equality2: "n * (m div n) + m mod n = (m::nat)"
```
```   165 apply(cut_tac m = m and n = n in mod_div_equality)
```
```   166 apply(simp add: mult_commute)
```
```   167 done
```
```   168
```
```   169 subsection{*Simproc for Cancelling Div and Mod*}
```
```   170
```
```   171 lemma div_mod_equality: "((m div n)*n + m mod n) + k = (m::nat) + k"
```
```   172 apply(simp add: mod_div_equality)
```
```   173 done
```
```   174
```
```   175 lemma div_mod_equality2: "(n*(m div n) + m mod n) + k = (m::nat) + k"
```
```   176 apply(simp add: mod_div_equality2)
```
```   177 done
```
```   178
```
```   179 ML
```
```   180 {*
```
```   181 val div_mod_equality = thm "div_mod_equality";
```
```   182 val div_mod_equality2 = thm "div_mod_equality2";
```
```   183
```
```   184
```
```   185 structure CancelDivModData =
```
```   186 struct
```
```   187
```
```   188 val div_name = "Divides.op div";
```
```   189 val mod_name = "Divides.op 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 [div_mod_equality,div_mod_equality2]
```
```   197
```
```   198 val trans = trans
```
```   199
```
```   200 val prove_eq_sums =
```
```   201   let val simps = add_0 :: add_0_right :: add_ac
```
```   202   in NatArithUtils.prove_conv all_tac (NatArithUtils.simp_all 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"], 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_tac "m" rule: nat_less_induct)
```
```   221 apply (case_tac "na<n", simp)
```
```   222 txt{*case @{term "n \<le> na"}*}
```
```   223 apply (simp add: mod_geq diff_less)
```
```   224 done
```
```   225
```
```   226 lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
```
```   227 by (cut_tac m = "m*n" and n = n in mod_div_equality, auto)
```
```   228
```
```   229 lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
```
```   230 by (simp add: mult_commute div_mult_self_is_m)
```
```   231
```
```   232 (*mod_mult_distrib2 above is the counterpart for remainder*)
```
```   233
```
```   234
```
```   235 subsection{*Proving facts about Quotient and Remainder*}
```
```   236
```
```   237 lemma unique_quotient_lemma:
```
```   238      "[| b*q' + r'  \<le> b*q + r;  0 < b;  r < b |]
```
```   239       ==> q' \<le> (q::nat)"
```
```   240 apply (rule leI)
```
```   241 apply (subst less_iff_Suc_add)
```
```   242 apply (auto simp add: add_mult_distrib2)
```
```   243 done
```
```   244
```
```   245 lemma unique_quotient:
```
```   246      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
```
```   247       ==> q = q'"
```
```   248 apply (simp add: split_ifs quorem_def)
```
```   249 apply (blast intro: order_antisym
```
```   250              dest: order_eq_refl [THEN unique_quotient_lemma] sym)+
```
```   251 done
```
```   252
```
```   253 lemma unique_remainder:
```
```   254      "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
```
```   255       ==> r = r'"
```
```   256 apply (subgoal_tac "q = q'")
```
```   257 prefer 2 apply (blast intro: unique_quotient)
```
```   258 apply (simp add: quorem_def)
```
```   259 done
```
```   260
```
```   261 lemma quorem_div_mod: "0 < b ==> quorem ((a, b), (a div b, a mod b))"
```
```   262 by (auto simp add: quorem_def)
```
```   263
```
```   264 lemma quorem_div: "[| quorem((a,b),(q,r));  0 < b |] ==> a div b = q"
```
```   265 by (simp add: quorem_div_mod [THEN unique_quotient])
```
```   266
```
```   267 lemma quorem_mod: "[| quorem((a,b),(q,r));  0 < b |] ==> a mod b = r"
```
```   268 by (simp add: quorem_div_mod [THEN unique_remainder])
```
```   269
```
```   270 (** A dividend of zero **)
```
```   271
```
```   272 lemma div_0 [simp]: "0 div m = (0::nat)"
```
```   273 by (case_tac "m=0", simp_all)
```
```   274
```
```   275 lemma mod_0 [simp]: "0 mod m = (0::nat)"
```
```   276 by (case_tac "m=0", simp_all)
```
```   277
```
```   278 (** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
```
```   279
```
```   280 lemma quorem_mult1_eq:
```
```   281      "[| quorem((b,c),(q,r));  0 < c |]
```
```   282       ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
```
```   283 apply (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
```
```   284 done
```
```   285
```
```   286 lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
```
```   287 apply (case_tac "c = 0", simp)
```
```   288 apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
```
```   289 done
```
```   290
```
```   291 lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
```
```   292 apply (case_tac "c = 0", simp)
```
```   293 apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
```
```   294 done
```
```   295
```
```   296 lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
```
```   297 apply (rule trans)
```
```   298 apply (rule_tac s = "b*a mod c" in trans)
```
```   299 apply (rule_tac [2] mod_mult1_eq)
```
```   300 apply (simp_all (no_asm) add: mult_commute)
```
```   301 done
```
```   302
```
```   303 lemma mod_mult_distrib_mod: "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
```
```   304 apply (rule mod_mult1_eq' [THEN trans])
```
```   305 apply (rule mod_mult1_eq)
```
```   306 done
```
```   307
```
```   308 (** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
```
```   309
```
```   310 lemma quorem_add1_eq:
```
```   311      "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  0 < c |]
```
```   312       ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
```
```   313 by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
```
```   314
```
```   315 (*NOT suitable for rewriting: the RHS has an instance of the LHS*)
```
```   316 lemma div_add1_eq:
```
```   317      "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
```
```   318 apply (case_tac "c = 0", simp)
```
```   319 apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod quorem_div_mod)
```
```   320 done
```
```   321
```
```   322 lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
```
```   323 apply (case_tac "c = 0", simp)
```
```   324 apply (blast intro: quorem_div_mod quorem_div_mod
```
```   325                     quorem_add1_eq [THEN quorem_mod])
```
```   326 done
```
```   327
```
```   328
```
```   329 subsection{*Proving @{term "a div (b*c) = (a div b) div c"}*}
```
```   330
```
```   331 (** first, a lemma to bound the remainder **)
```
```   332
```
```   333 lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
```
```   334 apply (cut_tac m = q and n = c in mod_less_divisor)
```
```   335 apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
```
```   336 apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
```
```   337 apply (simp add: add_mult_distrib2)
```
```   338 done
```
```   339
```
```   340 lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]
```
```   341       ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
```
```   342 apply (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)
```
```   343 done
```
```   344
```
```   345 lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
```
```   346 apply (case_tac "b=0", simp)
```
```   347 apply (case_tac "c=0", simp)
```
```   348 apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
```
```   349 done
```
```   350
```
```   351 lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
```
```   352 apply (case_tac "b=0", simp)
```
```   353 apply (case_tac "c=0", simp)
```
```   354 apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
```
```   355 done
```
```   356
```
```   357
```
```   358 subsection{*Cancellation of Common Factors in Division*}
```
```   359
```
```   360 lemma div_mult_mult_lemma:
```
```   361      "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
```
```   362 by (auto simp add: div_mult2_eq)
```
```   363
```
```   364 lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
```
```   365 apply (case_tac "b = 0")
```
```   366 apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
```
```   367 done
```
```   368
```
```   369 lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
```
```   370 apply (drule div_mult_mult1)
```
```   371 apply (auto simp add: mult_commute)
```
```   372 done
```
```   373
```
```   374
```
```   375 (*Distribution of Factors over Remainders:
```
```   376
```
```   377 Could prove these as in Integ/IntDiv.ML, but we already have
```
```   378 mod_mult_distrib and mod_mult_distrib2 above!
```
```   379
```
```   380 Goal "(c*a) mod (c*b) = (c::nat) * (a mod b)"
```
```   381 qed "mod_mult_mult1";
```
```   382
```
```   383 Goal "(a*c) mod (b*c) = (a mod b) * (c::nat)";
```
```   384 qed "mod_mult_mult2";
```
```   385  ***)
```
```   386
```
```   387 subsection{*Further Facts about Quotient and Remainder*}
```
```   388
```
```   389 lemma div_1 [simp]: "m div Suc 0 = m"
```
```   390 apply (induct_tac "m")
```
```   391 apply (simp_all (no_asm_simp) add: div_geq)
```
```   392 done
```
```   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_tac "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_less 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 add: diff_less)
```
```   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, simp]:
```
```   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 add: diff_less)
```
```   469 done
```
```   470
```
```   471 text{*A fact for the mutilated chess board*}
```
```   472 lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
```
```   473 apply (case_tac "n=0", simp)
```
```   474 apply (induct_tac "m" rule: nat_less_induct)
```
```   475 apply (case_tac "Suc (na) <n")
```
```   476 (* case Suc(na) < n *)
```
```   477 apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
```
```   478 (* case n \<le> Suc(na) *)
```
```   479 apply (simp add: not_less_iff_le le_Suc_eq mod_geq)
```
```   480 apply (auto simp add: Suc_diff_le diff_less le_mod_geq)
```
```   481 done
```
```   482
```
```   483 lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
```
```   484 by (case_tac "n=0", auto)
```
```   485
```
```   486 lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
```
```   487 by (case_tac "n=0", auto)
```
```   488
```
```   489
```
```   490 subsection{*The Divides Relation*}
```
```   491
```
```   492 lemma dvdI [intro?]: "n = m * k ==> m dvd n"
```
```   493 by (unfold dvd_def, blast)
```
```   494
```
```   495 lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
```
```   496 by (unfold dvd_def, blast)
```
```   497
```
```   498 lemma dvd_0_right [iff]: "m dvd (0::nat)"
```
```   499 apply (unfold dvd_def)
```
```   500 apply (blast intro: mult_0_right [symmetric])
```
```   501 done
```
```   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 by (unfold dvd_def, 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 apply (unfold dvd_def)
```
```   517 apply (blast intro: mult_1_right [symmetric])
```
```   518 done
```
```   519
```
```   520 lemma dvd_trans [trans]: "[| m dvd n; n dvd p |] ==> m dvd (p::nat)"
```
```   521 apply (unfold dvd_def)
```
```   522 apply (blast intro: mult_assoc)
```
```   523 done
```
```   524
```
```   525 lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
```
```   526 apply (unfold dvd_def)
```
```   527 apply (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
```
```   528 done
```
```   529
```
```   530 lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
```
```   531 apply (unfold dvd_def)
```
```   532 apply (blast intro: add_mult_distrib2 [symmetric])
```
```   533 done
```
```   534
```
```   535 lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
```
```   536 apply (unfold dvd_def)
```
```   537 apply (blast intro: diff_mult_distrib2 [symmetric])
```
```   538 done
```
```   539
```
```   540 lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
```
```   541 apply (erule not_less_iff_le [THEN iffD2, THEN add_diff_inverse, THEN subst])
```
```   542 apply (blast intro: dvd_add)
```
```   543 done
```
```   544
```
```   545 lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
```
```   546 by (drule_tac m = m in dvd_diff, auto)
```
```   547
```
```   548 lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
```
```   549 apply (unfold dvd_def)
```
```   550 apply (blast intro: mult_left_commute)
```
```   551 done
```
```   552
```
```   553 lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
```
```   554 apply (subst mult_commute)
```
```   555 apply (erule dvd_mult)
```
```   556 done
```
```   557
```
```   558 (* k dvd (m*k) *)
```
```   559 declare dvd_refl [THEN dvd_mult, iff] dvd_refl [THEN dvd_mult2, iff]
```
```   560
```
```   561 lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
```
```   562 apply (rule iffI)
```
```   563 apply (erule_tac [2] dvd_add)
```
```   564 apply (rule_tac [2] dvd_refl)
```
```   565 apply (subgoal_tac "n = (n+k) -k")
```
```   566  prefer 2 apply simp
```
```   567 apply (erule ssubst)
```
```   568 apply (erule dvd_diff)
```
```   569 apply (rule dvd_refl)
```
```   570 done
```
```   571
```
```   572 lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
```
```   573 apply (unfold dvd_def)
```
```   574 apply (case_tac "n=0", auto)
```
```   575 apply (blast intro: mod_mult_distrib2 [symmetric])
```
```   576 done
```
```   577
```
```   578 lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
```
```   579 apply (subgoal_tac "k dvd (m div n) *n + m mod n")
```
```   580  apply (simp add: mod_div_equality)
```
```   581 apply (simp only: dvd_add dvd_mult)
```
```   582 done
```
```   583
```
```   584 lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
```
```   585 by (blast intro: dvd_mod_imp_dvd dvd_mod)
```
```   586
```
```   587 lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
```
```   588 apply (unfold dvd_def)
```
```   589 apply (erule exE)
```
```   590 apply (simp add: mult_ac)
```
```   591 done
```
```   592
```
```   593 lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
```
```   594 apply auto
```
```   595 apply (subgoal_tac "m*n dvd m*1")
```
```   596 apply (drule dvd_mult_cancel, auto)
```
```   597 done
```
```   598
```
```   599 lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
```
```   600 apply (subst mult_commute)
```
```   601 apply (erule dvd_mult_cancel1)
```
```   602 done
```
```   603
```
```   604 lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
```
```   605 apply (unfold dvd_def, clarify)
```
```   606 apply (rule_tac x = "k*ka" in exI)
```
```   607 apply (simp add: mult_ac)
```
```   608 done
```
```   609
```
```   610 lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
```
```   611 by (simp add: dvd_def mult_assoc, blast)
```
```   612
```
```   613 lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
```
```   614 apply (unfold dvd_def, clarify)
```
```   615 apply (rule_tac x = "i*k" in exI)
```
```   616 apply (simp add: mult_ac)
```
```   617 done
```
```   618
```
```   619 lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
```
```   620 apply (unfold dvd_def, clarify)
```
```   621 apply (simp_all (no_asm_use) add: zero_less_mult_iff)
```
```   622 apply (erule conjE)
```
```   623 apply (rule le_trans)
```
```   624 apply (rule_tac [2] le_refl [THEN mult_le_mono])
```
```   625 apply (erule_tac [2] Suc_leI, simp)
```
```   626 done
```
```   627
```
```   628 lemma dvd_eq_mod_eq_0: "!!k::nat. (k dvd n) = (n mod k = 0)"
```
```   629 apply (unfold dvd_def)
```
```   630 apply (case_tac "k=0", simp, safe)
```
```   631 apply (simp add: mult_commute)
```
```   632 apply (rule_tac t = n and n1 = k in mod_div_equality [THEN subst])
```
```   633 apply (subst mult_commute, simp)
```
```   634 done
```
```   635
```
```   636 lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
```
```   637 apply (subgoal_tac "m mod n = 0")
```
```   638  apply (simp add: mult_div_cancel)
```
```   639 apply (simp only: dvd_eq_mod_eq_0)
```
```   640 done
```
```   641
```
```   642 lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
```
```   643 by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
```
```   644 declare mod_eq_0_iff [THEN iffD1, dest!]
```
```   645
```
```   646 (*Loses information, namely we also have r<d provided d is nonzero*)
```
```   647 lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
```
```   648 apply (cut_tac m = m in mod_div_equality)
```
```   649 apply (simp only: add_ac)
```
```   650 apply (blast intro: sym)
```
```   651 done
```
```   652
```
```   653
```
```   654 lemma split_div:
```
```   655  "P(n div k :: nat) =
```
```   656  ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
```
```   657  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```   658 proof
```
```   659   assume P: ?P
```
```   660   show ?Q
```
```   661   proof (cases)
```
```   662     assume "k = 0"
```
```   663     with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
```
```   664   next
```
```   665     assume not0: "k \<noteq> 0"
```
```   666     thus ?Q
```
```   667     proof (simp, intro allI impI)
```
```   668       fix i j
```
```   669       assume n: "n = k*i + j" and j: "j < k"
```
```   670       show "P i"
```
```   671       proof (cases)
```
```   672 	assume "i = 0"
```
```   673 	with n j P show "P i" by simp
```
```   674       next
```
```   675 	assume "i \<noteq> 0"
```
```   676 	with not0 n j P show "P i" by(simp add:add_ac)
```
```   677       qed
```
```   678     qed
```
```   679   qed
```
```   680 next
```
```   681   assume Q: ?Q
```
```   682   show ?P
```
```   683   proof (cases)
```
```   684     assume "k = 0"
```
```   685     with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
```
```   686   next
```
```   687     assume not0: "k \<noteq> 0"
```
```   688     with Q have R: ?R by simp
```
```   689     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```   690     show ?P by simp
```
```   691   qed
```
```   692 qed
```
```   693
```
```   694 lemma split_div_lemma:
```
```   695   "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
```
```   696   apply (rule iffI)
```
```   697   apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
```
```   698   apply (simp_all add: quorem_def, arith)
```
```   699   apply (rule conjI)
```
```   700   apply (rule_tac P="%x. n * (m div n) \<le> x" in
```
```   701     subst [OF mod_div_equality [of _ n]])
```
```   702   apply (simp only: add: mult_ac)
```
```   703   apply (rule_tac P="%x. x < n + n * (m div n)" in
```
```   704     subst [OF mod_div_equality [of _ n]])
```
```   705   apply (simp only: add: mult_ac add_ac)
```
```   706   apply (rule add_less_mono1, simp)
```
```   707   done
```
```   708
```
```   709 theorem split_div':
```
```   710   "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
```
```   711    (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
```
```   712   apply (case_tac "0 < n")
```
```   713   apply (simp only: add: split_div_lemma)
```
```   714   apply (simp_all add: DIVISION_BY_ZERO_DIV)
```
```   715   done
```
```   716
```
```   717 lemma split_mod:
```
```   718  "P(n mod k :: nat) =
```
```   719  ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
```
```   720  (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
```
```   721 proof
```
```   722   assume P: ?P
```
```   723   show ?Q
```
```   724   proof (cases)
```
```   725     assume "k = 0"
```
```   726     with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
```
```   727   next
```
```   728     assume not0: "k \<noteq> 0"
```
```   729     thus ?Q
```
```   730     proof (simp, intro allI impI)
```
```   731       fix i j
```
```   732       assume "n = k*i + j" "j < k"
```
```   733       thus "P j" using not0 P by(simp add:add_ac mult_ac)
```
```   734     qed
```
```   735   qed
```
```   736 next
```
```   737   assume Q: ?Q
```
```   738   show ?P
```
```   739   proof (cases)
```
```   740     assume "k = 0"
```
```   741     with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
```
```   742   next
```
```   743     assume not0: "k \<noteq> 0"
```
```   744     with Q have R: ?R by simp
```
```   745     from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
```
```   746     show ?P by simp
```
```   747   qed
```
```   748 qed
```
```   749
```
```   750 theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
```
```   751   apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
```
```   752     subst [OF mod_div_equality [of _ n]])
```
```   753   apply arith
```
```   754   done
```
```   755
```
```   756 ML
```
```   757 {*
```
```   758 val div_def = thm "div_def"
```
```   759 val mod_def = thm "mod_def"
```
```   760 val dvd_def = thm "dvd_def"
```
```   761 val quorem_def = thm "quorem_def"
```
```   762
```
```   763 val wf_less_trans = thm "wf_less_trans";
```
```   764 val mod_eq = thm "mod_eq";
```
```   765 val div_eq = thm "div_eq";
```
```   766 val DIVISION_BY_ZERO_DIV = thm "DIVISION_BY_ZERO_DIV";
```
```   767 val DIVISION_BY_ZERO_MOD = thm "DIVISION_BY_ZERO_MOD";
```
```   768 val mod_less = thm "mod_less";
```
```   769 val mod_geq = thm "mod_geq";
```
```   770 val le_mod_geq = thm "le_mod_geq";
```
```   771 val mod_if = thm "mod_if";
```
```   772 val mod_1 = thm "mod_1";
```
```   773 val mod_self = thm "mod_self";
```
```   774 val mod_add_self2 = thm "mod_add_self2";
```
```   775 val mod_add_self1 = thm "mod_add_self1";
```
```   776 val mod_mult_self1 = thm "mod_mult_self1";
```
```   777 val mod_mult_self2 = thm "mod_mult_self2";
```
```   778 val mod_mult_distrib = thm "mod_mult_distrib";
```
```   779 val mod_mult_distrib2 = thm "mod_mult_distrib2";
```
```   780 val mod_mult_self_is_0 = thm "mod_mult_self_is_0";
```
```   781 val mod_mult_self1_is_0 = thm "mod_mult_self1_is_0";
```
```   782 val div_less = thm "div_less";
```
```   783 val div_geq = thm "div_geq";
```
```   784 val le_div_geq = thm "le_div_geq";
```
```   785 val div_if = thm "div_if";
```
```   786 val mod_div_equality = thm "mod_div_equality";
```
```   787 val mod_div_equality2 = thm "mod_div_equality2";
```
```   788 val div_mod_equality = thm "div_mod_equality";
```
```   789 val div_mod_equality2 = thm "div_mod_equality2";
```
```   790 val mult_div_cancel = thm "mult_div_cancel";
```
```   791 val mod_less_divisor = thm "mod_less_divisor";
```
```   792 val div_mult_self_is_m = thm "div_mult_self_is_m";
```
```   793 val div_mult_self1_is_m = thm "div_mult_self1_is_m";
```
```   794 val unique_quotient_lemma = thm "unique_quotient_lemma";
```
```   795 val unique_quotient = thm "unique_quotient";
```
```   796 val unique_remainder = thm "unique_remainder";
```
```   797 val div_0 = thm "div_0";
```
```   798 val mod_0 = thm "mod_0";
```
```   799 val div_mult1_eq = thm "div_mult1_eq";
```
```   800 val mod_mult1_eq = thm "mod_mult1_eq";
```
```   801 val mod_mult1_eq' = thm "mod_mult1_eq'";
```
```   802 val mod_mult_distrib_mod = thm "mod_mult_distrib_mod";
```
```   803 val div_add1_eq = thm "div_add1_eq";
```
```   804 val mod_add1_eq = thm "mod_add1_eq";
```
```   805 val mod_lemma = thm "mod_lemma";
```
```   806 val div_mult2_eq = thm "div_mult2_eq";
```
```   807 val mod_mult2_eq = thm "mod_mult2_eq";
```
```   808 val div_mult_mult_lemma = thm "div_mult_mult_lemma";
```
```   809 val div_mult_mult1 = thm "div_mult_mult1";
```
```   810 val div_mult_mult2 = thm "div_mult_mult2";
```
```   811 val div_1 = thm "div_1";
```
```   812 val div_self = thm "div_self";
```
```   813 val div_add_self2 = thm "div_add_self2";
```
```   814 val div_add_self1 = thm "div_add_self1";
```
```   815 val div_mult_self1 = thm "div_mult_self1";
```
```   816 val div_mult_self2 = thm "div_mult_self2";
```
```   817 val div_le_mono = thm "div_le_mono";
```
```   818 val div_le_mono2 = thm "div_le_mono2";
```
```   819 val div_le_dividend = thm "div_le_dividend";
```
```   820 val div_less_dividend = thm "div_less_dividend";
```
```   821 val mod_Suc = thm "mod_Suc";
```
```   822 val dvdI = thm "dvdI";
```
```   823 val dvdE = thm "dvdE";
```
```   824 val dvd_0_right = thm "dvd_0_right";
```
```   825 val dvd_0_left = thm "dvd_0_left";
```
```   826 val dvd_0_left_iff = thm "dvd_0_left_iff";
```
```   827 val dvd_1_left = thm "dvd_1_left";
```
```   828 val dvd_1_iff_1 = thm "dvd_1_iff_1";
```
```   829 val dvd_refl = thm "dvd_refl";
```
```   830 val dvd_trans = thm "dvd_trans";
```
```   831 val dvd_anti_sym = thm "dvd_anti_sym";
```
```   832 val dvd_add = thm "dvd_add";
```
```   833 val dvd_diff = thm "dvd_diff";
```
```   834 val dvd_diffD = thm "dvd_diffD";
```
```   835 val dvd_diffD1 = thm "dvd_diffD1";
```
```   836 val dvd_mult = thm "dvd_mult";
```
```   837 val dvd_mult2 = thm "dvd_mult2";
```
```   838 val dvd_reduce = thm "dvd_reduce";
```
```   839 val dvd_mod = thm "dvd_mod";
```
```   840 val dvd_mod_imp_dvd = thm "dvd_mod_imp_dvd";
```
```   841 val dvd_mod_iff = thm "dvd_mod_iff";
```
```   842 val dvd_mult_cancel = thm "dvd_mult_cancel";
```
```   843 val dvd_mult_cancel1 = thm "dvd_mult_cancel1";
```
```   844 val dvd_mult_cancel2 = thm "dvd_mult_cancel2";
```
```   845 val mult_dvd_mono = thm "mult_dvd_mono";
```
```   846 val dvd_mult_left = thm "dvd_mult_left";
```
```   847 val dvd_mult_right = thm "dvd_mult_right";
```
```   848 val dvd_imp_le = thm "dvd_imp_le";
```
```   849 val dvd_eq_mod_eq_0 = thm "dvd_eq_mod_eq_0";
```
```   850 val dvd_mult_div_cancel = thm "dvd_mult_div_cancel";
```
```   851 val mod_eq_0_iff = thm "mod_eq_0_iff";
```
```   852 val mod_eqD = thm "mod_eqD";
```
```   853 *}
```
```   854
```
```   855
```
```   856 (*
```
```   857 lemma split_div:
```
```   858 assumes m: "m \<noteq> 0"
```
```   859 shows "P(n div m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P i)"
```
```   860        (is "?P = ?Q")
```
```   861 proof
```
```   862   assume P: ?P
```
```   863   show ?Q
```
```   864   proof (intro allI impI)
```
```   865     fix i j
```
```   866     assume n: "n = m*i + j" and j: "j < m"
```
```   867     show "P i"
```
```   868     proof (cases)
```
```   869       assume "i = 0"
```
```   870       with n j P show "P i" by simp
```
```   871     next
```
```   872       assume "i \<noteq> 0"
```
```   873       with n j P show "P i" by (simp add:add_ac div_mult_self1)
```
```   874     qed
```
```   875   qed
```
```   876 next
```
```   877   assume Q: ?Q
```
```   878   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
```
```   879   show ?P by simp
```
```   880 qed
```
```   881
```
```   882 lemma split_mod:
```
```   883 assumes m: "m \<noteq> 0"
```
```   884 shows "P(n mod m :: nat) = (!i. !j<m. n = m*i + j \<longrightarrow> P j)"
```
```   885        (is "?P = ?Q")
```
```   886 proof
```
```   887   assume P: ?P
```
```   888   show ?Q
```
```   889   proof (intro allI impI)
```
```   890     fix i j
```
```   891     assume "n = m*i + j" "j < m"
```
```   892     thus "P j" using m P by(simp add:add_ac mult_ac)
```
```   893   qed
```
```   894 next
```
```   895   assume Q: ?Q
```
```   896   from m Q[THEN spec,of "n div m",THEN spec, of "n mod m"]
```
```   897   show ?P by simp
```
```   898 qed
```
```   899 *)
```
```   900 end
```