src/HOL/Nat.ML
 author paulson Fri Sep 15 12:39:57 2000 +0200 (2000-09-15) changeset 9969 4753185f1dd2 parent 9870 2374ba026fc6 child 10173 1d097572d23b permissions -rw-r--r--
renamed (most of...) the select rules
```     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 Goal "!!n::nat. (0 ~= n) = (0 < n)";
```
```    47 by (case_tac "n" 1);
```
```    48 by Auto_tac;
```
```    49 qed "zero_neq_conv";
```
```    50 AddIffs [zero_neq_conv];
```
```    51
```
```    52 (*This theorem is useful with blast_tac: (n=0 ==> False) ==> 0<n *)
```
```    53 bind_thm ("gr0I", [neq0_conv, notI] MRS iffD1);
```
```    54
```
```    55 Goal "(0<n) = (EX m. n = Suc m)";
```
```    56 by(fast_tac (claset() addIs [not0_implies_Suc]) 1);
```
```    57 qed "gr0_conv_Suc";
```
```    58
```
```    59 Goal "!!n::nat. (~(0 < n)) = (n=0)";
```
```    60 by (rtac iffI 1);
```
```    61  by (etac swap 1);
```
```    62  by (ALLGOALS Asm_full_simp_tac);
```
```    63 qed "not_gr0";
```
```    64 AddIffs [not_gr0];
```
```    65
```
```    66 Goal "(Suc n <= m') --> (? m. m' = Suc m)";
```
```    67 by (induct_tac "m'" 1);
```
```    68 by  Auto_tac;
```
```    69 qed_spec_mp "Suc_le_D";
```
```    70
```
```    71 (*Useful in certain inductive arguments*)
```
```    72 Goal "(m < Suc n) = (m=0 | (EX j. m = Suc j & j < n))";
```
```    73 by (case_tac "m" 1);
```
```    74 by Auto_tac;
```
```    75 qed "less_Suc_eq_0_disj";
```
```    76
```
```    77 Goalw [Least_nat_def]
```
```    78  "[| ? n. P(Suc n); ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))";
```
```    79 by (rtac some_equality 1);
```
```    80 by (fold_goals_tac [Least_nat_def]);
```
```    81 by (safe_tac (claset() addSEs [LeastI]));
```
```    82 by (rename_tac "j" 1);
```
```    83 by (case_tac "j" 1);
```
```    84 by (Blast_tac 1);
```
```    85 by (blast_tac (claset() addDs [Suc_less_SucD, not_less_Least]) 1);
```
```    86 by (rename_tac "k n" 1);
```
```    87 by (case_tac "k" 1);
```
```    88 by (Blast_tac 1);
```
```    89 by (hyp_subst_tac 1);
```
```    90 by (rewtac Least_nat_def);
```
```    91 by (rtac (some_equality RS arg_cong RS sym) 1);
```
```    92 by (blast_tac (claset() addDs [Suc_mono]) 1);
```
```    93 by (cut_inst_tac [("m","m")] less_linear 1);
```
```    94 by (blast_tac (claset() addIs [Suc_mono]) 1);
```
```    95 qed "Least_Suc";
```
```    96
```
```    97 val prems = Goal "[| P 0; P 1; !!k. P k ==> P (Suc (Suc k)) |] ==> P n";
```
```    98 by (rtac nat_less_induct 1);
```
```    99 by (case_tac "n" 1);
```
```   100 by (case_tac "nat" 2);
```
```   101 by (ALLGOALS (blast_tac (claset() addIs prems@[less_trans])));
```
```   102 qed "nat_induct2";
```
```   103
```
```   104 Goal "min 0 n = (0::nat)";
```
```   105 by (rtac min_leastL 1);
```
```   106 by (Simp_tac 1);
```
```   107 qed "min_0L";
```
```   108
```
```   109 Goal "min n 0 = (0::nat)";
```
```   110 by (rtac min_leastR 1);
```
```   111 by (Simp_tac 1);
```
```   112 qed "min_0R";
```
```   113
```
```   114 Goalw [min_def] "min (Suc m) (Suc n) = Suc(min m n)";
```
```   115 by (Simp_tac 1);
```
```   116 qed "min_Suc_Suc";
```
```   117
```
```   118 Addsimps [min_0L,min_0R,min_Suc_Suc];
```
```   119
```
```   120 Goalw [max_def] "max 0 n = (n::nat)";
```
```   121 by (Simp_tac 1);
```
```   122 qed "max_0L";
```
```   123
```
```   124 Goalw [max_def] "max n 0 = (n::nat)";
```
```   125 by (Simp_tac 1);
```
```   126 qed "max_0R";
```
```   127
```
```   128 Goalw [max_def] "max (Suc m) (Suc n) = Suc(max m n)";
```
```   129 by (Simp_tac 1);
```
```   130 qed "max_Suc_Suc";
```
```   131
```
```   132 Addsimps [max_0L,max_0R,max_Suc_Suc];
```
```   133
```
```   134
```
```   135 (*** Basic rewrite rules for the arithmetic operators ***)
```
```   136
```
```   137 (** Difference **)
```
```   138
```
```   139 Goal "0 - n = (0::nat)";
```
```   140 by (induct_tac "n" 1);
```
```   141 by (ALLGOALS Asm_simp_tac);
```
```   142 qed "diff_0_eq_0";
```
```   143
```
```   144 (*Must simplify BEFORE the induction!  (Else we get a critical pair)
```
```   145   Suc(m) - Suc(n)   rewrites to   pred(Suc(m) - n)  *)
```
```   146 Goal "Suc(m) - Suc(n) = m - n";
```
```   147 by (Simp_tac 1);
```
```   148 by (induct_tac "n" 1);
```
```   149 by (ALLGOALS Asm_simp_tac);
```
```   150 qed "diff_Suc_Suc";
```
```   151
```
```   152 Addsimps [diff_0_eq_0, diff_Suc_Suc];
```
```   153
```
```   154 (* Could be (and is, below) generalized in various ways;
```
```   155    However, none of the generalizations are currently in the simpset,
```
```   156    and I dread to think what happens if I put them in *)
```
```   157 Goal "0 < n ==> Suc(n-1) = n";
```
```   158 by (asm_simp_tac (simpset() addsplits [nat.split]) 1);
```
```   159 qed "Suc_pred";
```
```   160 Addsimps [Suc_pred];
```
```   161
```
```   162 Delsimps [diff_Suc];
```
```   163
```
```   164
```
```   165 (**** Inductive properties of the operators ****)
```
```   166
```
```   167 (*** Addition ***)
```
```   168
```
```   169 Goal "m + 0 = (m::nat)";
```
```   170 by (induct_tac "m" 1);
```
```   171 by (ALLGOALS Asm_simp_tac);
```
```   172 qed "add_0_right";
```
```   173
```
```   174 Goal "m + Suc(n) = Suc(m+n)";
```
```   175 by (induct_tac "m" 1);
```
```   176 by (ALLGOALS Asm_simp_tac);
```
```   177 qed "add_Suc_right";
```
```   178
```
```   179 Addsimps [add_0_right,add_Suc_right];
```
```   180
```
```   181
```
```   182 (*Associative law for addition*)
```
```   183 Goal "(m + n) + k = m + ((n + k)::nat)";
```
```   184 by (induct_tac "m" 1);
```
```   185 by (ALLGOALS Asm_simp_tac);
```
```   186 qed "add_assoc";
```
```   187
```
```   188 (*Commutative law for addition*)
```
```   189 Goal "m + n = n + (m::nat)";
```
```   190 by (induct_tac "m" 1);
```
```   191 by (ALLGOALS Asm_simp_tac);
```
```   192 qed "add_commute";
```
```   193
```
```   194 Goal "x+(y+z)=y+((x+z)::nat)";
```
```   195 by (rtac (add_commute RS trans) 1);
```
```   196 by (rtac (add_assoc RS trans) 1);
```
```   197 by (rtac (add_commute RS arg_cong) 1);
```
```   198 qed "add_left_commute";
```
```   199
```
```   200 (*Addition is an AC-operator*)
```
```   201 bind_thms ("add_ac", [add_assoc, add_commute, add_left_commute]);
```
```   202
```
```   203 Goal "(k + m = k + n) = (m=(n::nat))";
```
```   204 by (induct_tac "k" 1);
```
```   205 by (Simp_tac 1);
```
```   206 by (Asm_simp_tac 1);
```
```   207 qed "add_left_cancel";
```
```   208
```
```   209 Goal "(m + k = n + k) = (m=(n::nat))";
```
```   210 by (induct_tac "k" 1);
```
```   211 by (Simp_tac 1);
```
```   212 by (Asm_simp_tac 1);
```
```   213 qed "add_right_cancel";
```
```   214
```
```   215 Goal "(k + m <= k + n) = (m<=(n::nat))";
```
```   216 by (induct_tac "k" 1);
```
```   217 by (Simp_tac 1);
```
```   218 by (Asm_simp_tac 1);
```
```   219 qed "add_left_cancel_le";
```
```   220
```
```   221 Goal "(k + m < k + n) = (m<(n::nat))";
```
```   222 by (induct_tac "k" 1);
```
```   223 by (Simp_tac 1);
```
```   224 by (Asm_simp_tac 1);
```
```   225 qed "add_left_cancel_less";
```
```   226
```
```   227 Addsimps [add_left_cancel, add_right_cancel,
```
```   228           add_left_cancel_le, add_left_cancel_less];
```
```   229
```
```   230 (** Reasoning about m+0=0, etc. **)
```
```   231
```
```   232 Goal "!!m::nat. (m+n = 0) = (m=0 & n=0)";
```
```   233 by (case_tac "m" 1);
```
```   234 by (Auto_tac);
```
```   235 qed "add_is_0";
```
```   236 AddIffs [add_is_0];
```
```   237
```
```   238 Goal "!!m::nat. (0 = m+n) = (m=0 & n=0)";
```
```   239 by (case_tac "m" 1);
```
```   240 by (Auto_tac);
```
```   241 qed "zero_is_add";
```
```   242 AddIffs [zero_is_add];
```
```   243
```
```   244 Goal "!!m::nat. (m+n=1) = (m=1 & n=0 | m=0 & n=1)";
```
```   245 by (case_tac "m" 1);
```
```   246 by (Auto_tac);
```
```   247 qed "add_is_1";
```
```   248
```
```   249 Goal "!!m::nat. (1=m+n) = (m=1 & n=0 | m=0 & n=1)";
```
```   250 by (case_tac "m" 1);
```
```   251 by (Auto_tac);
```
```   252 qed "one_is_add";
```
```   253
```
```   254 Goal "!!m::nat. (0<m+n) = (0<m | 0<n)";
```
```   255 by (simp_tac (simpset() delsimps [neq0_conv] addsimps [neq0_conv RS sym]) 1);
```
```   256 qed "add_gr_0";
```
```   257 AddIffs [add_gr_0];
```
```   258
```
```   259 Goal "!!m::nat. m + n = m ==> n = 0";
```
```   260 by (dtac (add_0_right RS ssubst) 1);
```
```   261 by (asm_full_simp_tac (simpset() addsimps [add_assoc]
```
```   262                                  delsimps [add_0_right]) 1);
```
```   263 qed "add_eq_self_zero";
```
```   264
```
```   265
```
```   266 (**** Additional theorems about "less than" ****)
```
```   267
```
```   268 (*Deleted less_natE; instead use less_eq_Suc_add RS exE*)
```
```   269 Goal "m<n --> (EX k. n=Suc(m+k))";
```
```   270 by (induct_tac "n" 1);
```
```   271 by (ALLGOALS (simp_tac (simpset() addsimps [order_le_less])));
```
```   272 by (blast_tac (claset() addSEs [less_SucE]
```
```   273                         addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1);
```
```   274 qed_spec_mp "less_eq_Suc_add";
```
```   275
```
```   276 Goal "n <= ((m + n)::nat)";
```
```   277 by (induct_tac "m" 1);
```
```   278 by (ALLGOALS Simp_tac);
```
```   279 by (etac le_SucI 1);
```
```   280 qed "le_add2";
```
```   281
```
```   282 Goal "n <= ((n + m)::nat)";
```
```   283 by (simp_tac (simpset() addsimps add_ac) 1);
```
```   284 by (rtac le_add2 1);
```
```   285 qed "le_add1";
```
```   286
```
```   287 bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
```
```   288 bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
```
```   289
```
```   290 Goal "(m<n) = (EX k. n=Suc(m+k))";
```
```   291 by (blast_tac (claset() addSIs [less_add_Suc1, less_eq_Suc_add]) 1);
```
```   292 qed "less_iff_Suc_add";
```
```   293
```
```   294
```
```   295 (*"i <= j ==> i <= j+m"*)
```
```   296 bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
```
```   297
```
```   298 (*"i <= j ==> i <= m+j"*)
```
```   299 bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
```
```   300
```
```   301 (*"i < j ==> i < j+m"*)
```
```   302 bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
```
```   303
```
```   304 (*"i < j ==> i < m+j"*)
```
```   305 bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
```
```   306
```
```   307 Goal "i+j < (k::nat) --> i<k";
```
```   308 by (induct_tac "j" 1);
```
```   309 by (ALLGOALS Asm_simp_tac);
```
```   310 by (blast_tac (claset() addDs [Suc_lessD]) 1);
```
```   311 qed_spec_mp "add_lessD1";
```
```   312
```
```   313 Goal "~ (i+j < (i::nat))";
```
```   314 by (rtac notI 1);
```
```   315 by (etac (add_lessD1 RS less_irrefl) 1);
```
```   316 qed "not_add_less1";
```
```   317
```
```   318 Goal "~ (j+i < (i::nat))";
```
```   319 by (simp_tac (simpset() addsimps [add_commute, not_add_less1]) 1);
```
```   320 qed "not_add_less2";
```
```   321 AddIffs [not_add_less1, not_add_less2];
```
```   322
```
```   323 Goal "m+k<=n --> m<=(n::nat)";
```
```   324 by (induct_tac "k" 1);
```
```   325 by (ALLGOALS (asm_simp_tac (simpset() addsimps le_simps)));
```
```   326 qed_spec_mp "add_leD1";
```
```   327
```
```   328 Goal "m+k<=n ==> k<=(n::nat)";
```
```   329 by (full_simp_tac (simpset() addsimps [add_commute]) 1);
```
```   330 by (etac add_leD1 1);
```
```   331 qed_spec_mp "add_leD2";
```
```   332
```
```   333 Goal "m+k<=n ==> m<=n & k<=(n::nat)";
```
```   334 by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1);
```
```   335 bind_thm ("add_leE", result() RS conjE);
```
```   336
```
```   337 (*needs !!k for add_ac to work*)
```
```   338 Goal "!!k:: nat. [| k<l;  m+l = k+n |] ==> m<n";
```
```   339 by (force_tac (claset(),
```
```   340               simpset() delsimps [add_Suc_right]
```
```   341                         addsimps [less_iff_Suc_add,
```
```   342                                   add_Suc_right RS sym] @ add_ac) 1);
```
```   343 qed "less_add_eq_less";
```
```   344
```
```   345
```
```   346 (*** Monotonicity of Addition ***)
```
```   347
```
```   348 (*strict, in 1st argument*)
```
```   349 Goal "i < j ==> i + k < j + (k::nat)";
```
```   350 by (induct_tac "k" 1);
```
```   351 by (ALLGOALS Asm_simp_tac);
```
```   352 qed "add_less_mono1";
```
```   353
```
```   354 (*strict, in both arguments*)
```
```   355 Goal "[|i < j; k < l|] ==> i + k < j + (l::nat)";
```
```   356 by (rtac (add_less_mono1 RS less_trans) 1);
```
```   357 by (REPEAT (assume_tac 1));
```
```   358 by (induct_tac "j" 1);
```
```   359 by (ALLGOALS Asm_simp_tac);
```
```   360 qed "add_less_mono";
```
```   361
```
```   362 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
```
```   363 val [lt_mono,le] = Goal
```
```   364      "[| !!i j::nat. i<j ==> f(i) < f(j);       \
```
```   365 \        i <= j                                 \
```
```   366 \     |] ==> f(i) <= (f(j)::nat)";
```
```   367 by (cut_facts_tac [le] 1);
```
```   368 by (asm_full_simp_tac (simpset() addsimps [order_le_less]) 1);
```
```   369 by (blast_tac (claset() addSIs [lt_mono]) 1);
```
```   370 qed "less_mono_imp_le_mono";
```
```   371
```
```   372 (*non-strict, in 1st argument*)
```
```   373 Goal "i<=j ==> i + k <= j + (k::nat)";
```
```   374 by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1);
```
```   375 by (etac add_less_mono1 1);
```
```   376 by (assume_tac 1);
```
```   377 qed "add_le_mono1";
```
```   378
```
```   379 (*non-strict, in both arguments*)
```
```   380 Goal "[|i<=j;  k<=l |] ==> i + k <= j + (l::nat)";
```
```   381 by (etac (add_le_mono1 RS le_trans) 1);
```
```   382 by (simp_tac (simpset() addsimps [add_commute]) 1);
```
```   383 qed "add_le_mono";
```
```   384
```
```   385
```
```   386 (*** Multiplication ***)
```
```   387
```
```   388 (*right annihilation in product*)
```
```   389 Goal "!!m::nat. m * 0 = 0";
```
```   390 by (induct_tac "m" 1);
```
```   391 by (ALLGOALS Asm_simp_tac);
```
```   392 qed "mult_0_right";
```
```   393
```
```   394 (*right successor law for multiplication*)
```
```   395 Goal  "m * Suc(n) = m + (m * n)";
```
```   396 by (induct_tac "m" 1);
```
```   397 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
```
```   398 qed "mult_Suc_right";
```
```   399
```
```   400 Addsimps [mult_0_right, mult_Suc_right];
```
```   401
```
```   402 Goal "1 * n = n";
```
```   403 by (Asm_simp_tac 1);
```
```   404 qed "mult_1";
```
```   405
```
```   406 Goal "n * 1 = n";
```
```   407 by (Asm_simp_tac 1);
```
```   408 qed "mult_1_right";
```
```   409
```
```   410 (*Commutative law for multiplication*)
```
```   411 Goal "m * n = n * (m::nat)";
```
```   412 by (induct_tac "m" 1);
```
```   413 by (ALLGOALS Asm_simp_tac);
```
```   414 qed "mult_commute";
```
```   415
```
```   416 (*addition distributes over multiplication*)
```
```   417 Goal "(m + n)*k = (m*k) + ((n*k)::nat)";
```
```   418 by (induct_tac "m" 1);
```
```   419 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
```
```   420 qed "add_mult_distrib";
```
```   421
```
```   422 Goal "k*(m + n) = (k*m) + ((k*n)::nat)";
```
```   423 by (induct_tac "m" 1);
```
```   424 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
```
```   425 qed "add_mult_distrib2";
```
```   426
```
```   427 (*Associative law for multiplication*)
```
```   428 Goal "(m * n) * k = m * ((n * k)::nat)";
```
```   429 by (induct_tac "m" 1);
```
```   430 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib])));
```
```   431 qed "mult_assoc";
```
```   432
```
```   433 Goal "x*(y*z) = y*((x*z)::nat)";
```
```   434 by (rtac trans 1);
```
```   435 by (rtac mult_commute 1);
```
```   436 by (rtac trans 1);
```
```   437 by (rtac mult_assoc 1);
```
```   438 by (rtac (mult_commute RS arg_cong) 1);
```
```   439 qed "mult_left_commute";
```
```   440
```
```   441 bind_thms ("mult_ac", [mult_assoc,mult_commute,mult_left_commute]);
```
```   442
```
```   443 Goal "!!m::nat. (m*n = 0) = (m=0 | n=0)";
```
```   444 by (induct_tac "m" 1);
```
```   445 by (induct_tac "n" 2);
```
```   446 by (ALLGOALS Asm_simp_tac);
```
```   447 qed "mult_is_0";
```
```   448
```
```   449 Goal "!!m::nat. (0 = m*n) = (0=m | 0=n)";
```
```   450 by (stac eq_commute 1 THEN stac mult_is_0 1);
```
```   451 by Auto_tac;
```
```   452 qed "zero_is_mult";
```
```   453
```
```   454 Addsimps [mult_is_0, zero_is_mult];
```
```   455
```
```   456
```
```   457 (*** Difference ***)
```
```   458
```
```   459 Goal "!!m::nat. m - m = 0";
```
```   460 by (induct_tac "m" 1);
```
```   461 by (ALLGOALS Asm_simp_tac);
```
```   462 qed "diff_self_eq_0";
```
```   463
```
```   464 Addsimps [diff_self_eq_0];
```
```   465
```
```   466 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
```
```   467 Goal "~ m<n --> n+(m-n) = (m::nat)";
```
```   468 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   469 by (ALLGOALS Asm_simp_tac);
```
```   470 qed_spec_mp "add_diff_inverse";
```
```   471
```
```   472 Goal "n<=m ==> n+(m-n) = (m::nat)";
```
```   473 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);
```
```   474 qed "le_add_diff_inverse";
```
```   475
```
```   476 Goal "n<=m ==> (m-n)+n = (m::nat)";
```
```   477 by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);
```
```   478 qed "le_add_diff_inverse2";
```
```   479
```
```   480 Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];
```
```   481
```
```   482
```
```   483 (*** More results about difference ***)
```
```   484
```
```   485 Goal "n <= m ==> Suc(m)-n = Suc(m-n)";
```
```   486 by (etac rev_mp 1);
```
```   487 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   488 by (ALLGOALS Asm_simp_tac);
```
```   489 qed "Suc_diff_le";
```
```   490
```
```   491 Goal "m - n < Suc(m)";
```
```   492 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   493 by (etac less_SucE 3);
```
```   494 by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
```
```   495 qed "diff_less_Suc";
```
```   496
```
```   497 Goal "m - n <= (m::nat)";
```
```   498 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
```
```   499 by (ALLGOALS (asm_simp_tac (simpset() addsimps [le_SucI])));
```
```   500 qed "diff_le_self";
```
```   501 Addsimps [diff_le_self];
```
```   502
```
```   503 (* j<k ==> j-n < k *)
```
```   504 bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);
```
```   505
```
```   506 Goal "!!i::nat. i-j-k = i - (j+k)";
```
```   507 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
```   508 by (ALLGOALS Asm_simp_tac);
```
```   509 qed "diff_diff_left";
```
```   510
```
```   511 Goal "(Suc m - n) - Suc k = m - n - k";
```
```   512 by (simp_tac (simpset() addsimps [diff_diff_left]) 1);
```
```   513 qed "Suc_diff_diff";
```
```   514 Addsimps [Suc_diff_diff];
```
```   515
```
```   516 Goal "0<n ==> n - Suc i < n";
```
```   517 by (case_tac "n" 1);
```
```   518 by Safe_tac;
```
```   519 by (asm_simp_tac (simpset() addsimps le_simps) 1);
```
```   520 qed "diff_Suc_less";
```
```   521 Addsimps [diff_Suc_less];
```
```   522
```
```   523 (*This and the next few suggested by Florian Kammueller*)
```
```   524 Goal "!!i::nat. i-j-k = i-k-j";
```
```   525 by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);
```
```   526 qed "diff_commute";
```
```   527
```
```   528 Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)";
```
```   529 by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
```
```   530 by (ALLGOALS Asm_simp_tac);
```
```   531 qed_spec_mp "diff_add_assoc";
```
```   532
```
```   533 Goal "k <= (j::nat) --> (j + i) - k = (j - k) + i";
```
```   534 by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);
```
```   535 qed_spec_mp "diff_add_assoc2";
```
```   536
```
```   537 Goal "(n+m) - n = (m::nat)";
```
```   538 by (induct_tac "n" 1);
```
```   539 by (ALLGOALS Asm_simp_tac);
```
```   540 qed "diff_add_inverse";
```
```   541
```
```   542 Goal "(m+n) - n = (m::nat)";
```
```   543 by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);
```
```   544 qed "diff_add_inverse2";
```
```   545
```
```   546 Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)";
```
```   547 by Safe_tac;
```
```   548 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_add_inverse2])));
```
```   549 qed "le_imp_diff_is_add";
```
```   550
```
```   551 Goal "!!m::nat. (m-n = 0) = (m <= n)";
```
```   552 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   553 by (ALLGOALS Asm_simp_tac);
```
```   554 qed "diff_is_0_eq";
```
```   555
```
```   556 Goal "!!m::nat. (0 = m-n) = (m <= n)";
```
```   557 by (stac (diff_is_0_eq RS sym) 1);
```
```   558 by (rtac eq_sym_conv 1);
```
```   559 qed "zero_is_diff_eq";
```
```   560 Addsimps [diff_is_0_eq, zero_is_diff_eq];
```
```   561
```
```   562 Goal "!!m::nat. (0<n-m) = (m<n)";
```
```   563 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   564 by (ALLGOALS Asm_simp_tac);
```
```   565 qed "zero_less_diff";
```
```   566 Addsimps [zero_less_diff];
```
```   567
```
```   568 Goal "i < j  ==> EX k::nat. 0<k & i+k = j";
```
```   569 by (res_inst_tac [("x","j - i")] exI 1);
```
```   570 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);
```
```   571 qed "less_imp_add_positive";
```
```   572
```
```   573 Goal "P(k) --> (ALL n. P(Suc(n))--> P(n)) --> P(k-i)";
```
```   574 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
```
```   575 by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
```
```   576 qed "zero_induct_lemma";
```
```   577
```
```   578 val prems = Goal "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
```
```   579 by (rtac (diff_self_eq_0 RS subst) 1);
```
```   580 by (rtac (zero_induct_lemma RS mp RS mp) 1);
```
```   581 by (REPEAT (ares_tac ([impI,allI]@prems) 1));
```
```   582 qed "zero_induct";
```
```   583
```
```   584 Goal "(k+m) - (k+n) = m - (n::nat)";
```
```   585 by (induct_tac "k" 1);
```
```   586 by (ALLGOALS Asm_simp_tac);
```
```   587 qed "diff_cancel";
```
```   588
```
```   589 Goal "(m+k) - (n+k) = m - (n::nat)";
```
```   590 by (asm_simp_tac
```
```   591     (simpset() addsimps [diff_cancel, inst "n" "k" add_commute]) 1);
```
```   592 qed "diff_cancel2";
```
```   593
```
```   594 Goal "n - (n+m) = (0::nat)";
```
```   595 by (induct_tac "n" 1);
```
```   596 by (ALLGOALS Asm_simp_tac);
```
```   597 qed "diff_add_0";
```
```   598
```
```   599
```
```   600 (** Difference distributes over multiplication **)
```
```   601
```
```   602 Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";
```
```   603 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   604 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_cancel])));
```
```   605 qed "diff_mult_distrib" ;
```
```   606
```
```   607 Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";
```
```   608 val mult_commute_k = read_instantiate [("m","k")] mult_commute;
```
```   609 by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);
```
```   610 qed "diff_mult_distrib2" ;
```
```   611 (*NOT added as rewrites, since sometimes they are used from right-to-left*)
```
```   612
```
```   613
```
```   614 (*** Monotonicity of Multiplication ***)
```
```   615
```
```   616 Goal "i <= (j::nat) ==> i*k<=j*k";
```
```   617 by (induct_tac "k" 1);
```
```   618 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));
```
```   619 qed "mult_le_mono1";
```
```   620
```
```   621 Goal "i <= (j::nat) ==> k*i <= k*j";
```
```   622 by (dtac mult_le_mono1 1);
```
```   623 by (asm_simp_tac (simpset() addsimps [mult_commute]) 1);
```
```   624 qed "mult_le_mono2";
```
```   625
```
```   626 (* <= monotonicity, BOTH arguments*)
```
```   627 Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l";
```
```   628 by (etac (mult_le_mono1 RS le_trans) 1);
```
```   629 by (etac mult_le_mono2 1);
```
```   630 qed "mult_le_mono";
```
```   631
```
```   632 (*strict, in 1st argument; proof is by induction on k>0*)
```
```   633 Goal "!!i::nat. [| i<j; 0<k |] ==> k*i < k*j";
```
```   634 by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1);
```
```   635 by (Asm_simp_tac 1);
```
```   636 by (induct_tac "x" 1);
```
```   637 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono])));
```
```   638 qed "mult_less_mono2";
```
```   639
```
```   640 Goal "!!i::nat. [| i<j; 0<k |] ==> i*k < j*k";
```
```   641 by (dtac mult_less_mono2 1);
```
```   642 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute])));
```
```   643 qed "mult_less_mono1";
```
```   644
```
```   645 Goal "!!m::nat. (0 < m*n) = (0<m & 0<n)";
```
```   646 by (induct_tac "m" 1);
```
```   647 by (case_tac "n" 2);
```
```   648 by (ALLGOALS Asm_simp_tac);
```
```   649 qed "zero_less_mult_iff";
```
```   650 Addsimps [zero_less_mult_iff];
```
```   651
```
```   652 Goal "(1 <= m*n) = (1<=m & 1<=n)";
```
```   653 by (induct_tac "m" 1);
```
```   654 by (case_tac "n" 2);
```
```   655 by (ALLGOALS Asm_simp_tac);
```
```   656 qed "one_le_mult_iff";
```
```   657 Addsimps [one_le_mult_iff];
```
```   658
```
```   659 Goal "(m*n = 1) = (m=1 & n=1)";
```
```   660 by (induct_tac "m" 1);
```
```   661 by (Simp_tac 1);
```
```   662 by (induct_tac "n" 1);
```
```   663 by (Simp_tac 1);
```
```   664 by (fast_tac (claset() addss simpset()) 1);
```
```   665 qed "mult_eq_1_iff";
```
```   666 Addsimps [mult_eq_1_iff];
```
```   667
```
```   668 Goal "!!m::nat. (m*k < n*k) = (0<k & m<n)";
```
```   669 by (safe_tac (claset() addSIs [mult_less_mono1]));
```
```   670 by (case_tac "k" 1);
```
```   671 by Auto_tac;
```
```   672 by (full_simp_tac (simpset() delsimps [le_0_eq]
```
```   673 			     addsimps [linorder_not_le RS sym]) 1);
```
```   674 by (blast_tac (claset() addIs [mult_le_mono1]) 1);
```
```   675 qed "mult_less_cancel2";
```
```   676
```
```   677 Goal "!!m::nat. (k*m < k*n) = (0<k & m<n)";
```
```   678 by (simp_tac (simpset() addsimps [mult_less_cancel2,
```
```   679                                   inst "m" "k" mult_commute]) 1);
```
```   680 qed "mult_less_cancel1";
```
```   681 Addsimps [mult_less_cancel1, mult_less_cancel2];
```
```   682
```
```   683 Goal "!!m::nat. (m*k <= n*k) = (0<k --> m<=n)";
```
```   684 by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
```
```   685 by Auto_tac;
```
```   686 qed "mult_le_cancel2";
```
```   687
```
```   688 Goal "!!m::nat. (k*m <= k*n) = (0<k --> m<=n)";
```
```   689 by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
```
```   690 by Auto_tac;
```
```   691 qed "mult_le_cancel1";
```
```   692 Addsimps [mult_le_cancel1, mult_le_cancel2];
```
```   693
```
```   694 Goal "(m*k = n*k) = (m=n | (k = (0::nat)))";
```
```   695 by (cut_facts_tac [less_linear] 1);
```
```   696 by Safe_tac;
```
```   697 by Auto_tac;
```
```   698 by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
```
```   699 by (ALLGOALS Asm_full_simp_tac);
```
```   700 qed "mult_cancel2";
```
```   701
```
```   702 Goal "(k*m = k*n) = (m=n | (k = (0::nat)))";
```
```   703 by (simp_tac (simpset() addsimps [mult_cancel2, inst "m" "k" mult_commute]) 1);
```
```   704 qed "mult_cancel1";
```
```   705 Addsimps [mult_cancel1, mult_cancel2];
```
```   706
```
```   707 Goal "(Suc k * m < Suc k * n) = (m < n)";
```
```   708 by (stac mult_less_cancel1 1);
```
```   709 by (Simp_tac 1);
```
```   710 qed "Suc_mult_less_cancel1";
```
```   711
```
```   712 Goal "(Suc k * m <= Suc k * n) = (m <= n)";
```
```   713 by (stac mult_le_cancel1 1);
```
```   714 by (Simp_tac 1);
```
```   715 qed "Suc_mult_le_cancel1";
```
```   716
```
```   717 Goal "(Suc k * m = Suc k * n) = (m = n)";
```
```   718 by (stac mult_cancel1 1);
```
```   719 by (Simp_tac 1);
```
```   720 qed "Suc_mult_cancel1";
```
```   721
```
```   722
```
```   723 (*Lemma for gcd*)
```
```   724 Goal "!!m::nat. m = m*n ==> n=1 | m=0";
```
```   725 by (dtac sym 1);
```
```   726 by (rtac disjCI 1);
```
```   727 by (rtac nat_less_cases 1 THEN assume_tac 2);
```
```   728 by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);
```
```   729 by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);
```
```   730 qed "mult_eq_self_implies_10";
```