src/HOL/Nat.ML
author nipkow
Thu Apr 25 17:36:29 2002 +0200 (2002-04-25)
changeset 13094 643fce75f6cd
parent 12949 b94843ffc0d1
child 13438 527811f00c56
permissions -rw-r--r--
added "m <= n ==> m-n = 0" [simp]
     1 (*  Title:      HOL/Nat.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson and Tobias Nipkow
     4 
     5 Proofs about natural numbers and elementary arithmetic: addition,
     6 multiplication, etc.  Some from the Hoare example from Norbert Galm.
     7 *)
     8 
     9 (** conversion rules for nat_rec **)
    10 
    11 val [nat_rec_0, nat_rec_Suc] = nat.recs;
    12 bind_thm ("nat_rec_0", nat_rec_0);
    13 bind_thm ("nat_rec_Suc", nat_rec_Suc);
    14 
    15 (*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
    16 val prems = Goal
    17     "[| !!n. f(n) == nat_rec c h n |] ==> f(0) = c";
    18 by (simp_tac (simpset() addsimps prems) 1);
    19 qed "def_nat_rec_0";
    20 
    21 val prems = Goal
    22     "[| !!n. f(n) == nat_rec c h n |] ==> f(Suc(n)) = h n (f n)";
    23 by (simp_tac (simpset() addsimps prems) 1);
    24 qed "def_nat_rec_Suc";
    25 
    26 val [nat_case_0, nat_case_Suc] = nat.cases;
    27 bind_thm ("nat_case_0", nat_case_0);
    28 bind_thm ("nat_case_Suc", nat_case_Suc);
    29 
    30 Goal "n ~= 0 ==> EX m. n = Suc m";
    31 by (case_tac "n" 1);
    32 by (REPEAT (Blast_tac 1));
    33 qed "not0_implies_Suc";
    34 
    35 Goal "!!n::nat. m<n ==> n ~= 0";
    36 by (case_tac "n" 1);
    37 by (ALLGOALS Asm_full_simp_tac);
    38 qed "gr_implies_not0";
    39 
    40 Goal "!!n::nat. (n ~= 0) = (0 < n)";
    41 by (case_tac "n" 1);
    42 by Auto_tac;
    43 qed "neq0_conv";
    44 AddIffs [neq0_conv];
    45 
    46 (*This theorem is useful with blast_tac: (n=0 ==> False) ==> 0<n *)
    47 bind_thm ("gr0I", [neq0_conv, notI] MRS iffD1);
    48 
    49 Goal "(0<n) = (EX m. n = Suc m)";
    50 by (fast_tac (claset() addIs [not0_implies_Suc]) 1);
    51 qed "gr0_conv_Suc";
    52 
    53 Goal "!!n::nat. (~(0 < n)) = (n=0)";
    54 by (rtac iffI 1);
    55  by (rtac ccontr 1);
    56  by (ALLGOALS Asm_full_simp_tac);
    57 qed "not_gr0";
    58 AddIffs [not_gr0];
    59 
    60 Goal "(Suc n <= m') --> (? m. m' = Suc m)";
    61 by (induct_tac "m'" 1);
    62 by  Auto_tac;
    63 qed_spec_mp "Suc_le_D";
    64 
    65 (*Useful in certain inductive arguments*)
    66 Goal "(m < Suc n) = (m=0 | (EX j. m = Suc j & j < n))";
    67 by (case_tac "m" 1);
    68 by Auto_tac;
    69 qed "less_Suc_eq_0_disj";
    70 
    71 val prems = Goal "[| P 0; P(Suc 0); !!k. P k ==> P (Suc (Suc k)) |] ==> P n";
    72 by (rtac nat_less_induct 1);
    73 by (case_tac "n" 1);
    74 by (case_tac "nat" 2);
    75 by (ALLGOALS (blast_tac (claset() addIs prems@[less_trans])));
    76 qed "nat_induct2";
    77 
    78 (** LEAST theorems for type "nat" by specialization **)
    79 
    80 bind_thm("LeastI", wellorder_LeastI);
    81 bind_thm("Least_le", wellorder_Least_le);
    82 bind_thm("not_less_Least", wellorder_not_less_Least);
    83 
    84 Goal "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))";
    85 by (case_tac "n" 1);
    86 by Auto_tac;  
    87 by (ftac LeastI 1); 
    88 by (dres_inst_tac [("P","%x. P (Suc x)")] LeastI 1);
    89 by (subgoal_tac "(LEAST x. P x) <= Suc (LEAST x. P (Suc x))" 1); 
    90 by (etac Least_le 2); 
    91 by (case_tac "LEAST x. P x" 1);
    92 by Auto_tac;  
    93 by (dres_inst_tac [("P","%x. P (Suc x)")] Least_le 1);
    94 by (blast_tac (claset() addIs [order_antisym]) 1); 
    95 qed "Least_Suc";
    96 
    97 Goal "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)";
    98 by (eatac (Least_Suc RS ssubst) 1 1);
    99 by (Asm_simp_tac 1);
   100 qed "Least_Suc2";
   101 
   102 
   103 (** min and max **)
   104 
   105 Goal "min 0 n = (0::nat)";
   106 by (rtac min_leastL 1);
   107 by (Simp_tac 1);
   108 qed "min_0L";
   109 
   110 Goal "min n 0 = (0::nat)";
   111 by (rtac min_leastR 1);
   112 by (Simp_tac 1);
   113 qed "min_0R";
   114 
   115 Goal "min (Suc m) (Suc n) = Suc (min m n)";
   116 by (simp_tac (simpset() addsimps [min_of_mono]) 1);
   117 qed "min_Suc_Suc";
   118 
   119 Addsimps [min_0L,min_0R,min_Suc_Suc];
   120 
   121 Goal "max 0 n = (n::nat)";
   122 by (rtac max_leastL 1);
   123 by (Simp_tac 1);
   124 qed "max_0L";
   125 
   126 Goal "max n 0 = (n::nat)";
   127 by (rtac max_leastR 1);
   128 by (Simp_tac 1);
   129 qed "max_0R";
   130 
   131 Goal "max (Suc m) (Suc n) = Suc(max m n)";
   132 by (simp_tac (simpset() addsimps [max_of_mono]) 1);
   133 qed "max_Suc_Suc";
   134 
   135 Addsimps [max_0L,max_0R,max_Suc_Suc];
   136 
   137 
   138 (*** Basic rewrite rules for the arithmetic operators ***)
   139 
   140 (** Difference **)
   141 
   142 Goal "0 - n = (0::nat)";
   143 by (induct_tac "n" 1);
   144 by (ALLGOALS Asm_simp_tac);
   145 qed "diff_0_eq_0";
   146 
   147 (*Must simplify BEFORE the induction!  (Else we get a critical pair)
   148   Suc(m) - Suc(n)   rewrites to   pred(Suc(m) - n)  *)
   149 Goal "Suc(m) - Suc(n) = m - n";
   150 by (Simp_tac 1);
   151 by (induct_tac "n" 1);
   152 by (ALLGOALS Asm_simp_tac);
   153 qed "diff_Suc_Suc";
   154 
   155 Addsimps [diff_0_eq_0, diff_Suc_Suc];
   156 
   157 (* Could be (and is, below) generalized in various ways;
   158    However, none of the generalizations are currently in the simpset,
   159    and I dread to think what happens if I put them in *)
   160 Goal "0 < n ==> Suc(n - Suc 0) = n";
   161 by (asm_simp_tac (simpset() addsplits [nat.split]) 1);
   162 qed "Suc_pred";
   163 Addsimps [Suc_pred];
   164 
   165 Delsimps [diff_Suc];
   166 
   167 
   168 (**** Inductive properties of the operators ****)
   169 
   170 (*** Addition ***)
   171 
   172 Goal "m + 0 = (m::nat)";
   173 by (induct_tac "m" 1);
   174 by (ALLGOALS Asm_simp_tac);
   175 qed "add_0_right";
   176 
   177 Goal "m + Suc(n) = Suc(m+n)";
   178 by (induct_tac "m" 1);
   179 by (ALLGOALS Asm_simp_tac);
   180 qed "add_Suc_right";
   181 
   182 Addsimps [add_0_right,add_Suc_right];
   183 
   184 
   185 (*Associative law for addition*)
   186 Goal "(m + n) + k = m + ((n + k)::nat)";
   187 by (induct_tac "m" 1);
   188 by (ALLGOALS Asm_simp_tac);
   189 qed "add_assoc";
   190 
   191 (*Commutative law for addition*)
   192 Goal "m + n = n + (m::nat)";
   193 by (induct_tac "m" 1);
   194 by (ALLGOALS Asm_simp_tac);
   195 qed "add_commute";
   196 
   197 Goal "x+(y+z)=y+((x+z)::nat)";
   198 by (rtac (add_commute RS trans) 1);
   199 by (rtac (add_assoc RS trans) 1);
   200 by (rtac (add_commute RS arg_cong) 1);
   201 qed "add_left_commute";
   202 
   203 (*Addition is an AC-operator*)
   204 bind_thms ("add_ac", [add_assoc, add_commute, add_left_commute]);
   205 
   206 Goal "(k + m = k + n) = (m=(n::nat))";
   207 by (induct_tac "k" 1);
   208 by (Simp_tac 1);
   209 by (Asm_simp_tac 1);
   210 qed "add_left_cancel";
   211 
   212 Goal "(m + k = n + k) = (m=(n::nat))";
   213 by (induct_tac "k" 1);
   214 by (Simp_tac 1);
   215 by (Asm_simp_tac 1);
   216 qed "add_right_cancel";
   217 
   218 Goal "(k + m <= k + n) = (m<=(n::nat))";
   219 by (induct_tac "k" 1);
   220 by (Simp_tac 1);
   221 by (Asm_simp_tac 1);
   222 qed "add_left_cancel_le";
   223 
   224 Goal "(k + m < k + n) = (m<(n::nat))";
   225 by (induct_tac "k" 1);
   226 by (Simp_tac 1);
   227 by (Asm_simp_tac 1);
   228 qed "add_left_cancel_less";
   229 
   230 Addsimps [add_left_cancel, add_right_cancel,
   231           add_left_cancel_le, add_left_cancel_less];
   232 
   233 (** Reasoning about m+0=0, etc. **)
   234 
   235 Goal "!!m::nat. (m+n = 0) = (m=0 & n=0)";
   236 by (case_tac "m" 1);
   237 by (Auto_tac);
   238 qed "add_is_0";
   239 AddIffs [add_is_0];
   240 
   241 Goal "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)";
   242 by (case_tac "m" 1);
   243 by (Auto_tac);
   244 qed "add_is_1";
   245 
   246 Goal "(Suc 0 = m+n) = (m = Suc 0 & n=0 | m=0 & n = Suc 0)";
   247 by (rtac ([eq_commute, add_is_1] MRS trans) 1);
   248 qed "one_is_add";
   249 
   250 Goal "!!m::nat. (0<m+n) = (0<m | 0<n)";
   251 by (simp_tac (simpset() delsimps [neq0_conv] addsimps [neq0_conv RS sym]) 1);
   252 qed "add_gr_0";
   253 AddIffs [add_gr_0];
   254 
   255 Goal "!!m::nat. m + n = m ==> n = 0";
   256 by (dtac (add_0_right RS ssubst) 1);
   257 by (asm_full_simp_tac (simpset() addsimps [add_assoc]
   258                                  delsimps [add_0_right]) 1);
   259 qed "add_eq_self_zero";
   260 
   261 (**** Additional theorems about "less than" ****)
   262 
   263 (*Deleted less_natE; instead use less_imp_Suc_add RS exE*)
   264 Goal "m<n --> (EX k. n=Suc(m+k))";
   265 by (induct_tac "n" 1);
   266 by (ALLGOALS (simp_tac (simpset() addsimps [order_le_less])));
   267 by (blast_tac (claset() addSEs [less_SucE]
   268                         addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1);
   269 qed_spec_mp "less_imp_Suc_add";
   270 
   271 Goal "n <= ((m + n)::nat)";
   272 by (induct_tac "m" 1);
   273 by (ALLGOALS Simp_tac);
   274 by (etac le_SucI 1);
   275 qed "le_add2";
   276 
   277 Goal "n <= ((n + m)::nat)";
   278 by (simp_tac (simpset() addsimps add_ac) 1);
   279 by (rtac le_add2 1);
   280 qed "le_add1";
   281 
   282 bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
   283 bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
   284 
   285 Goal "(m<n) = (EX k. n=Suc(m+k))";
   286 by (blast_tac (claset() addSIs [less_add_Suc1, less_imp_Suc_add]) 1);
   287 qed "less_iff_Suc_add";
   288 
   289 
   290 (*"i <= j ==> i <= j+m"*)
   291 bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
   292 
   293 (*"i <= j ==> i <= m+j"*)
   294 bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
   295 
   296 (*"i < j ==> i < j+m"*)
   297 bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
   298 
   299 (*"i < j ==> i < m+j"*)
   300 bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
   301 
   302 Goal "i+j < (k::nat) --> i<k";
   303 by (induct_tac "j" 1);
   304 by (ALLGOALS Asm_simp_tac);
   305 by (blast_tac (claset() addDs [Suc_lessD]) 1);
   306 qed_spec_mp "add_lessD1";
   307 
   308 Goal "~ (i+j < (i::nat))";
   309 by (rtac notI 1);
   310 by (etac (add_lessD1 RS less_irrefl) 1);
   311 qed "not_add_less1";
   312 
   313 Goal "~ (j+i < (i::nat))";
   314 by (simp_tac (simpset() addsimps [add_commute, not_add_less1]) 1);
   315 qed "not_add_less2";
   316 AddIffs [not_add_less1, not_add_less2];
   317 
   318 Goal "m+k<=n --> m<=(n::nat)";
   319 by (induct_tac "k" 1);
   320 by (ALLGOALS (asm_simp_tac (simpset() addsimps le_simps)));
   321 qed_spec_mp "add_leD1";
   322 
   323 Goal "m+k<=n ==> k<=(n::nat)";
   324 by (full_simp_tac (simpset() addsimps [add_commute]) 1);
   325 by (etac add_leD1 1);
   326 qed_spec_mp "add_leD2";
   327 
   328 Goal "m+k<=n ==> m<=n & k<=(n::nat)";
   329 by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1);
   330 bind_thm ("add_leE", result() RS conjE);
   331 
   332 (*needs !!k for add_ac to work*)
   333 Goal "!!k:: nat. [| k<l;  m+l = k+n |] ==> m<n";
   334 by (force_tac (claset(),
   335               simpset() delsimps [add_Suc_right]
   336                         addsimps [less_iff_Suc_add,
   337                                   add_Suc_right RS sym] @ add_ac) 1);
   338 qed "less_add_eq_less";
   339 
   340 
   341 (*** Monotonicity of Addition ***)
   342 
   343 (*strict, in 1st argument*)
   344 Goal "i < j ==> i + k < j + (k::nat)";
   345 by (induct_tac "k" 1);
   346 by (ALLGOALS Asm_simp_tac);
   347 qed "add_less_mono1";
   348 
   349 (*strict, in both arguments*)
   350 Goal "[|i < j; k < l|] ==> i + k < j + (l::nat)";
   351 by (rtac (add_less_mono1 RS less_trans) 1);
   352 by (REPEAT (assume_tac 1));
   353 by (induct_tac "j" 1);
   354 by (ALLGOALS Asm_simp_tac);
   355 qed "add_less_mono";
   356 
   357 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
   358 val [lt_mono,le] = Goal
   359      "[| !!i j::nat. i<j ==> f(i) < f(j);       \
   360 \        i <= j                                 \
   361 \     |] ==> f(i) <= (f(j)::nat)";
   362 by (cut_facts_tac [le] 1);
   363 by (asm_full_simp_tac (simpset() addsimps [order_le_less]) 1);
   364 by (blast_tac (claset() addSIs [lt_mono]) 1);
   365 qed "less_mono_imp_le_mono";
   366 
   367 (*non-strict, in 1st argument*)
   368 Goal "i<=j ==> i + k <= j + (k::nat)";
   369 by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1);
   370 by (etac add_less_mono1 1);
   371 by (assume_tac 1);
   372 qed "add_le_mono1";
   373 
   374 (*non-strict, in both arguments*)
   375 Goal "[|i<=j;  k<=l |] ==> i + k <= j + (l::nat)";
   376 by (etac (add_le_mono1 RS le_trans) 1);
   377 by (simp_tac (simpset() addsimps [add_commute]) 1);
   378 qed "add_le_mono";
   379 
   380 
   381 (*** Multiplication ***)
   382 
   383 (*right annihilation in product*)
   384 Goal "!!m::nat. m * 0 = 0";
   385 by (induct_tac "m" 1);
   386 by (ALLGOALS Asm_simp_tac);
   387 qed "mult_0_right";
   388 
   389 (*right successor law for multiplication*)
   390 Goal  "m * Suc(n) = m + (m * n)";
   391 by (induct_tac "m" 1);
   392 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
   393 qed "mult_Suc_right";
   394 
   395 Addsimps [mult_0_right, mult_Suc_right];
   396 
   397 Goal "(1::nat) * n = n";
   398 by (Asm_simp_tac 1);
   399 qed "mult_1";
   400 
   401 Goal "n * (1::nat) = n";
   402 by (Asm_simp_tac 1);
   403 qed "mult_1_right";
   404 
   405 (*Commutative law for multiplication*)
   406 Goal "m * n = n * (m::nat)";
   407 by (induct_tac "m" 1);
   408 by (ALLGOALS Asm_simp_tac);
   409 qed "mult_commute";
   410 
   411 (*addition distributes over multiplication*)
   412 Goal "(m + n)*k = (m*k) + ((n*k)::nat)";
   413 by (induct_tac "m" 1);
   414 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
   415 qed "add_mult_distrib";
   416 
   417 Goal "k*(m + n) = (k*m) + ((k*n)::nat)";
   418 by (induct_tac "m" 1);
   419 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
   420 qed "add_mult_distrib2";
   421 
   422 (*Associative law for multiplication*)
   423 Goal "(m * n) * k = m * ((n * k)::nat)";
   424 by (induct_tac "m" 1);
   425 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib])));
   426 qed "mult_assoc";
   427 
   428 Goal "x*(y*z) = y*((x*z)::nat)";
   429 by (rtac trans 1);
   430 by (rtac mult_commute 1);
   431 by (rtac trans 1);
   432 by (rtac mult_assoc 1);
   433 by (rtac (mult_commute RS arg_cong) 1);
   434 qed "mult_left_commute";
   435 
   436 bind_thms ("mult_ac", [mult_assoc,mult_commute,mult_left_commute]);
   437 
   438 Goal "!!m::nat. (m*n = 0) = (m=0 | n=0)";
   439 by (induct_tac "m" 1);
   440 by (induct_tac "n" 2);
   441 by (ALLGOALS Asm_simp_tac);
   442 qed "mult_is_0";
   443 Addsimps [mult_is_0];
   444 
   445 
   446 (*** Difference ***)
   447 
   448 Goal "!!m::nat. m - m = 0";
   449 by (induct_tac "m" 1);
   450 by (ALLGOALS Asm_simp_tac);
   451 qed "diff_self_eq_0";
   452 
   453 Addsimps [diff_self_eq_0];
   454 
   455 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
   456 Goal "~ m<n --> n+(m-n) = (m::nat)";
   457 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   458 by (ALLGOALS Asm_simp_tac);
   459 qed_spec_mp "add_diff_inverse";
   460 
   461 Goal "n<=m ==> n+(m-n) = (m::nat)";
   462 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);
   463 qed "le_add_diff_inverse";
   464 
   465 Goal "n<=m ==> (m-n)+n = (m::nat)";
   466 by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);
   467 qed "le_add_diff_inverse2";
   468 
   469 Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];
   470 
   471 
   472 (*** More results about difference ***)
   473 
   474 Goal "n <= m ==> Suc(m)-n = Suc(m-n)";
   475 by (etac rev_mp 1);
   476 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   477 by (ALLGOALS Asm_simp_tac);
   478 qed "Suc_diff_le";
   479 
   480 Goal "m - n < Suc(m)";
   481 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   482 by (etac less_SucE 3);
   483 by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
   484 qed "diff_less_Suc";
   485 
   486 Goal "m - n <= (m::nat)";
   487 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
   488 by (ALLGOALS (asm_simp_tac (simpset() addsimps [le_SucI])));
   489 qed "diff_le_self";
   490 Addsimps [diff_le_self];
   491 
   492 (* j<k ==> j-n < k *)
   493 bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);
   494 
   495 Goal "!!i::nat. i-j-k = i - (j+k)";
   496 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
   497 by (ALLGOALS Asm_simp_tac);
   498 qed "diff_diff_left";
   499 
   500 Goal "(Suc m - n) - Suc k = m - n - k";
   501 by (simp_tac (simpset() addsimps [diff_diff_left]) 1);
   502 qed "Suc_diff_diff";
   503 Addsimps [Suc_diff_diff];
   504 
   505 Goal "0<n ==> n - Suc i < n";
   506 by (case_tac "n" 1);
   507 by Safe_tac;
   508 by (asm_simp_tac (simpset() addsimps le_simps) 1);
   509 qed "diff_Suc_less";
   510 Addsimps [diff_Suc_less];
   511 
   512 (*This and the next few suggested by Florian Kammueller*)
   513 Goal "!!i::nat. i-j-k = i-k-j";
   514 by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);
   515 qed "diff_commute";
   516 
   517 Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)";
   518 by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
   519 by (ALLGOALS Asm_simp_tac);
   520 qed_spec_mp "diff_add_assoc";
   521 
   522 Goal "k <= (j::nat) --> (j + i) - k = (j - k) + i";
   523 by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);
   524 qed_spec_mp "diff_add_assoc2";
   525 
   526 Goal "(n+m) - n = (m::nat)";
   527 by (induct_tac "n" 1);
   528 by (ALLGOALS Asm_simp_tac);
   529 qed "diff_add_inverse";
   530 
   531 Goal "(m+n) - n = (m::nat)";
   532 by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);
   533 qed "diff_add_inverse2";
   534 
   535 Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)";
   536 by Safe_tac;
   537 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_add_inverse2])));
   538 qed "le_imp_diff_is_add";
   539 
   540 Goal "!!m::nat. (m-n = 0) = (m <= n)";
   541 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   542 by (ALLGOALS Asm_simp_tac);
   543 qed "diff_is_0_eq";
   544 Addsimps [diff_is_0_eq, diff_is_0_eq RS iffD2];
   545 
   546 Goal "!!m::nat. (0<n-m) = (m<n)";
   547 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   548 by (ALLGOALS Asm_simp_tac);
   549 qed "zero_less_diff";
   550 Addsimps [zero_less_diff];
   551 
   552 Goal "i < j  ==> EX k::nat. 0<k & i+k = j";
   553 by (res_inst_tac [("x","j - i")] exI 1);
   554 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);
   555 qed "less_imp_add_positive";
   556 
   557 Goal "P(k) --> (ALL n. P(Suc(n))--> P(n)) --> P(k-i)";
   558 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
   559 by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
   560 qed "zero_induct_lemma";
   561 
   562 val prems = Goal "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
   563 by (rtac (diff_self_eq_0 RS subst) 1);
   564 by (rtac (zero_induct_lemma RS mp RS mp) 1);
   565 by (REPEAT (ares_tac ([impI,allI]@prems) 1));
   566 qed "zero_induct";
   567 
   568 Goal "(k+m) - (k+n) = m - (n::nat)";
   569 by (induct_tac "k" 1);
   570 by (ALLGOALS Asm_simp_tac);
   571 qed "diff_cancel";
   572 
   573 Goal "(m+k) - (n+k) = m - (n::nat)";
   574 by (asm_simp_tac
   575     (simpset() addsimps [diff_cancel, inst "n" "k" add_commute]) 1);
   576 qed "diff_cancel2";
   577 
   578 Goal "n - (n+m) = (0::nat)";
   579 by (induct_tac "n" 1);
   580 by (ALLGOALS Asm_simp_tac);
   581 qed "diff_add_0";
   582 
   583 
   584 (** Difference distributes over multiplication **)
   585 
   586 Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";
   587 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   588 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_cancel])));
   589 qed "diff_mult_distrib" ;
   590 
   591 Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";
   592 val mult_commute_k = read_instantiate [("m","k")] mult_commute;
   593 by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);
   594 qed "diff_mult_distrib2" ;
   595 (*NOT added as rewrites, since sometimes they are used from right-to-left*)
   596 
   597 bind_thms ("nat_distrib",
   598   [add_mult_distrib, add_mult_distrib2, diff_mult_distrib, diff_mult_distrib2]);
   599 
   600 
   601 (*** Monotonicity of Multiplication ***)
   602 
   603 Goal "i <= (j::nat) ==> i*k<=j*k";
   604 by (induct_tac "k" 1);
   605 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));
   606 qed "mult_le_mono1";
   607 
   608 Goal "i <= (j::nat) ==> k*i <= k*j";
   609 by (dtac mult_le_mono1 1);
   610 by (asm_simp_tac (simpset() addsimps [mult_commute]) 1);
   611 qed "mult_le_mono2";
   612 
   613 (* <= monotonicity, BOTH arguments*)
   614 Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l";
   615 by (etac (mult_le_mono1 RS le_trans) 1);
   616 by (etac mult_le_mono2 1);
   617 qed "mult_le_mono";
   618 
   619 (*strict, in 1st argument; proof is by induction on k>0*)
   620 Goal "!!i::nat. [| i<j; 0<k |] ==> k*i < k*j";
   621 by (eres_inst_tac [("m1","0")] (less_imp_Suc_add RS exE) 1);
   622 by (Asm_simp_tac 1);
   623 by (induct_tac "x" 1);
   624 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono])));
   625 qed "mult_less_mono2";
   626 
   627 Goal "!!i::nat. [| i<j; 0<k |] ==> i*k < j*k";
   628 by (dtac mult_less_mono2 1);
   629 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute])));
   630 qed "mult_less_mono1";
   631 
   632 Goal "!!m::nat. (0 < m*n) = (0<m & 0<n)";
   633 by (induct_tac "m" 1);
   634 by (case_tac "n" 2);
   635 by (ALLGOALS Asm_simp_tac);
   636 qed "zero_less_mult_iff";
   637 Addsimps [zero_less_mult_iff];
   638 
   639 Goal "(Suc 0 <= m*n) = (1<=m & 1<=n)";
   640 by (induct_tac "m" 1);
   641 by (case_tac "n" 2);
   642 by (ALLGOALS Asm_simp_tac);
   643 qed "one_le_mult_iff";
   644 Addsimps [one_le_mult_iff];
   645 
   646 Goal "(m*n = Suc 0) = (m=1 & n=1)";
   647 by (induct_tac "m" 1);
   648 by (Simp_tac 1);
   649 by (induct_tac "n" 1);
   650 by (Simp_tac 1);
   651 by (fast_tac (claset() addss simpset()) 1);
   652 qed "mult_eq_1_iff";
   653 Addsimps [mult_eq_1_iff];
   654 
   655 Goal "(Suc 0 = m*n) = (m=1 & n=1)";
   656 by (rtac (mult_eq_1_iff RSN (2,trans)) 1);
   657 by (fast_tac (claset() addss simpset()) 1);
   658 qed "one_eq_mult_iff";
   659 Addsimps [one_eq_mult_iff];
   660 
   661 Goal "!!m::nat. (m*k < n*k) = (0<k & m<n)";
   662 by (safe_tac (claset() addSIs [mult_less_mono1]));
   663 by (case_tac "k" 1);
   664 by Auto_tac;  
   665 by (full_simp_tac (simpset() delsimps [le_0_eq]
   666 			     addsimps [linorder_not_le RS sym]) 1);
   667 by (blast_tac (claset() addIs [mult_le_mono1]) 1); 
   668 qed "mult_less_cancel2";
   669 
   670 Goal "!!m::nat. (k*m < k*n) = (0<k & m<n)";
   671 by (simp_tac (simpset() addsimps [mult_less_cancel2, 
   672                                   inst "m" "k" mult_commute]) 1);
   673 qed "mult_less_cancel1";
   674 Addsimps [mult_less_cancel1, mult_less_cancel2];
   675 
   676 Goal "!!m::nat. (m*k <= n*k) = (0<k --> m<=n)";
   677 by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
   678 by Auto_tac;  
   679 qed "mult_le_cancel2";
   680 
   681 Goal "!!m::nat. (k*m <= k*n) = (0<k --> m<=n)";
   682 by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
   683 by Auto_tac;  
   684 qed "mult_le_cancel1";
   685 Addsimps [mult_le_cancel1, mult_le_cancel2];
   686 
   687 Goal "(m*k = n*k) = (m=n | (k = (0::nat)))";
   688 by (cut_facts_tac [less_linear] 1);
   689 by Safe_tac;
   690 by Auto_tac; 	
   691 by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
   692 by (ALLGOALS Asm_full_simp_tac);
   693 qed "mult_cancel2";
   694 
   695 Goal "(k*m = k*n) = (m=n | (k = (0::nat)))";
   696 by (simp_tac (simpset() addsimps [mult_cancel2, inst "m" "k" mult_commute]) 1);
   697 qed "mult_cancel1";
   698 Addsimps [mult_cancel1, mult_cancel2];
   699 
   700 Goal "(Suc k * m < Suc k * n) = (m < n)";
   701 by (stac mult_less_cancel1 1);
   702 by (Simp_tac 1);
   703 qed "Suc_mult_less_cancel1";
   704 
   705 Goal "(Suc k * m <= Suc k * n) = (m <= n)";
   706 by (stac mult_le_cancel1 1);
   707 by (Simp_tac 1);
   708 qed "Suc_mult_le_cancel1";
   709 
   710 Goal "(Suc k * m = Suc k * n) = (m = n)";
   711 by (stac mult_cancel1 1);
   712 by (Simp_tac 1);
   713 qed "Suc_mult_cancel1";
   714 
   715 
   716 (*Lemma for gcd*)
   717 Goal "!!m::nat. m = m*n ==> n=1 | m=0";
   718 by (dtac sym 1);
   719 by (rtac disjCI 1);
   720 by (rtac nat_less_cases 1 THEN assume_tac 2);
   721 by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);
   722 by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);
   723 qed "mult_eq_self_implies_10";