src/HOL/Nat.ML
 author nipkow Mon Feb 25 10:42:34 2002 +0100 (2002-02-25) changeset 12931 2c0251fada94 parent 12486 0ed8bdd883e0 child 12949 b94843ffc0d1 permissions -rw-r--r--
solved the problem that Larry's simproce cancle_numerals(?) returns
inconsistent inequality w/o rewriting it to False.
```     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 (* a rather special thm needed for arith_tac: m+n = 0 may arise where m or n
```
```   262 contain Suc. This contradiction must be detected. It cannot be detected by
```
```   263 pulling Suc outside because this interferes with simprocs on
```
```   264 numerals. Sigh. *)
```
```   265
```
```   266 Goal "m ~= 0 ==> m+n ~= (0::nat)";
```
```   267 by(Asm_full_simp_tac 1);
```
```   268 qed "add_not_0_if_left_not_0";
```
```   269
```
```   270 Goal "n ~= 0 ==> m+n ~= (0::nat)";
```
```   271 by(Asm_full_simp_tac 1);
```
```   272 qed "add_not_0_if_right_not_0";
```
```   273
```
```   274 (**** Additional theorems about "less than" ****)
```
```   275
```
```   276 (*Deleted less_natE; instead use less_imp_Suc_add RS exE*)
```
```   277 Goal "m<n --> (EX k. n=Suc(m+k))";
```
```   278 by (induct_tac "n" 1);
```
```   279 by (ALLGOALS (simp_tac (simpset() addsimps [order_le_less])));
```
```   280 by (blast_tac (claset() addSEs [less_SucE]
```
```   281                         addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1);
```
```   282 qed_spec_mp "less_imp_Suc_add";
```
```   283
```
```   284 Goal "n <= ((m + n)::nat)";
```
```   285 by (induct_tac "m" 1);
```
```   286 by (ALLGOALS Simp_tac);
```
```   287 by (etac le_SucI 1);
```
```   288 qed "le_add2";
```
```   289
```
```   290 Goal "n <= ((n + m)::nat)";
```
```   291 by (simp_tac (simpset() addsimps add_ac) 1);
```
```   292 by (rtac le_add2 1);
```
```   293 qed "le_add1";
```
```   294
```
```   295 bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
```
```   296 bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
```
```   297
```
```   298 Goal "(m<n) = (EX k. n=Suc(m+k))";
```
```   299 by (blast_tac (claset() addSIs [less_add_Suc1, less_imp_Suc_add]) 1);
```
```   300 qed "less_iff_Suc_add";
```
```   301
```
```   302
```
```   303 (*"i <= j ==> i <= j+m"*)
```
```   304 bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
```
```   305
```
```   306 (*"i <= j ==> i <= m+j"*)
```
```   307 bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
```
```   308
```
```   309 (*"i < j ==> i < j+m"*)
```
```   310 bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
```
```   311
```
```   312 (*"i < j ==> i < m+j"*)
```
```   313 bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
```
```   314
```
```   315 Goal "i+j < (k::nat) --> i<k";
```
```   316 by (induct_tac "j" 1);
```
```   317 by (ALLGOALS Asm_simp_tac);
```
```   318 by (blast_tac (claset() addDs [Suc_lessD]) 1);
```
```   319 qed_spec_mp "add_lessD1";
```
```   320
```
```   321 Goal "~ (i+j < (i::nat))";
```
```   322 by (rtac notI 1);
```
```   323 by (etac (add_lessD1 RS less_irrefl) 1);
```
```   324 qed "not_add_less1";
```
```   325
```
```   326 Goal "~ (j+i < (i::nat))";
```
```   327 by (simp_tac (simpset() addsimps [add_commute, not_add_less1]) 1);
```
```   328 qed "not_add_less2";
```
```   329 AddIffs [not_add_less1, not_add_less2];
```
```   330
```
```   331 Goal "m+k<=n --> m<=(n::nat)";
```
```   332 by (induct_tac "k" 1);
```
```   333 by (ALLGOALS (asm_simp_tac (simpset() addsimps le_simps)));
```
```   334 qed_spec_mp "add_leD1";
```
```   335
```
```   336 Goal "m+k<=n ==> k<=(n::nat)";
```
```   337 by (full_simp_tac (simpset() addsimps [add_commute]) 1);
```
```   338 by (etac add_leD1 1);
```
```   339 qed_spec_mp "add_leD2";
```
```   340
```
```   341 Goal "m+k<=n ==> m<=n & k<=(n::nat)";
```
```   342 by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1);
```
```   343 bind_thm ("add_leE", result() RS conjE);
```
```   344
```
```   345 (*needs !!k for add_ac to work*)
```
```   346 Goal "!!k:: nat. [| k<l;  m+l = k+n |] ==> m<n";
```
```   347 by (force_tac (claset(),
```
```   348               simpset() delsimps [add_Suc_right]
```
```   349                         addsimps [less_iff_Suc_add,
```
```   350                                   add_Suc_right RS sym] @ add_ac) 1);
```
```   351 qed "less_add_eq_less";
```
```   352
```
```   353
```
```   354 (*** Monotonicity of Addition ***)
```
```   355
```
```   356 (*strict, in 1st argument*)
```
```   357 Goal "i < j ==> i + k < j + (k::nat)";
```
```   358 by (induct_tac "k" 1);
```
```   359 by (ALLGOALS Asm_simp_tac);
```
```   360 qed "add_less_mono1";
```
```   361
```
```   362 (*strict, in both arguments*)
```
```   363 Goal "[|i < j; k < l|] ==> i + k < j + (l::nat)";
```
```   364 by (rtac (add_less_mono1 RS less_trans) 1);
```
```   365 by (REPEAT (assume_tac 1));
```
```   366 by (induct_tac "j" 1);
```
```   367 by (ALLGOALS Asm_simp_tac);
```
```   368 qed "add_less_mono";
```
```   369
```
```   370 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
```
```   371 val [lt_mono,le] = Goal
```
```   372      "[| !!i j::nat. i<j ==> f(i) < f(j);       \
```
```   373 \        i <= j                                 \
```
```   374 \     |] ==> f(i) <= (f(j)::nat)";
```
```   375 by (cut_facts_tac [le] 1);
```
```   376 by (asm_full_simp_tac (simpset() addsimps [order_le_less]) 1);
```
```   377 by (blast_tac (claset() addSIs [lt_mono]) 1);
```
```   378 qed "less_mono_imp_le_mono";
```
```   379
```
```   380 (*non-strict, in 1st argument*)
```
```   381 Goal "i<=j ==> i + k <= j + (k::nat)";
```
```   382 by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1);
```
```   383 by (etac add_less_mono1 1);
```
```   384 by (assume_tac 1);
```
```   385 qed "add_le_mono1";
```
```   386
```
```   387 (*non-strict, in both arguments*)
```
```   388 Goal "[|i<=j;  k<=l |] ==> i + k <= j + (l::nat)";
```
```   389 by (etac (add_le_mono1 RS le_trans) 1);
```
```   390 by (simp_tac (simpset() addsimps [add_commute]) 1);
```
```   391 qed "add_le_mono";
```
```   392
```
```   393
```
```   394 (*** Multiplication ***)
```
```   395
```
```   396 (*right annihilation in product*)
```
```   397 Goal "!!m::nat. m * 0 = 0";
```
```   398 by (induct_tac "m" 1);
```
```   399 by (ALLGOALS Asm_simp_tac);
```
```   400 qed "mult_0_right";
```
```   401
```
```   402 (*right successor law for multiplication*)
```
```   403 Goal  "m * Suc(n) = m + (m * n)";
```
```   404 by (induct_tac "m" 1);
```
```   405 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
```
```   406 qed "mult_Suc_right";
```
```   407
```
```   408 Addsimps [mult_0_right, mult_Suc_right];
```
```   409
```
```   410 Goal "(1::nat) * n = n";
```
```   411 by (Asm_simp_tac 1);
```
```   412 qed "mult_1";
```
```   413
```
```   414 Goal "n * (1::nat) = n";
```
```   415 by (Asm_simp_tac 1);
```
```   416 qed "mult_1_right";
```
```   417
```
```   418 (*Commutative law for multiplication*)
```
```   419 Goal "m * n = n * (m::nat)";
```
```   420 by (induct_tac "m" 1);
```
```   421 by (ALLGOALS Asm_simp_tac);
```
```   422 qed "mult_commute";
```
```   423
```
```   424 (*addition distributes over multiplication*)
```
```   425 Goal "(m + n)*k = (m*k) + ((n*k)::nat)";
```
```   426 by (induct_tac "m" 1);
```
```   427 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
```
```   428 qed "add_mult_distrib";
```
```   429
```
```   430 Goal "k*(m + n) = (k*m) + ((k*n)::nat)";
```
```   431 by (induct_tac "m" 1);
```
```   432 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
```
```   433 qed "add_mult_distrib2";
```
```   434
```
```   435 (*Associative law for multiplication*)
```
```   436 Goal "(m * n) * k = m * ((n * k)::nat)";
```
```   437 by (induct_tac "m" 1);
```
```   438 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib])));
```
```   439 qed "mult_assoc";
```
```   440
```
```   441 Goal "x*(y*z) = y*((x*z)::nat)";
```
```   442 by (rtac trans 1);
```
```   443 by (rtac mult_commute 1);
```
```   444 by (rtac trans 1);
```
```   445 by (rtac mult_assoc 1);
```
```   446 by (rtac (mult_commute RS arg_cong) 1);
```
```   447 qed "mult_left_commute";
```
```   448
```
```   449 bind_thms ("mult_ac", [mult_assoc,mult_commute,mult_left_commute]);
```
```   450
```
```   451 Goal "!!m::nat. (m*n = 0) = (m=0 | n=0)";
```
```   452 by (induct_tac "m" 1);
```
```   453 by (induct_tac "n" 2);
```
```   454 by (ALLGOALS Asm_simp_tac);
```
```   455 qed "mult_is_0";
```
```   456 Addsimps [mult_is_0];
```
```   457
```
```   458
```
```   459 (*** Difference ***)
```
```   460
```
```   461 Goal "!!m::nat. m - m = 0";
```
```   462 by (induct_tac "m" 1);
```
```   463 by (ALLGOALS Asm_simp_tac);
```
```   464 qed "diff_self_eq_0";
```
```   465
```
```   466 Addsimps [diff_self_eq_0];
```
```   467
```
```   468 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
```
```   469 Goal "~ m<n --> n+(m-n) = (m::nat)";
```
```   470 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   471 by (ALLGOALS Asm_simp_tac);
```
```   472 qed_spec_mp "add_diff_inverse";
```
```   473
```
```   474 Goal "n<=m ==> n+(m-n) = (m::nat)";
```
```   475 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);
```
```   476 qed "le_add_diff_inverse";
```
```   477
```
```   478 Goal "n<=m ==> (m-n)+n = (m::nat)";
```
```   479 by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);
```
```   480 qed "le_add_diff_inverse2";
```
```   481
```
```   482 Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];
```
```   483
```
```   484
```
```   485 (*** More results about difference ***)
```
```   486
```
```   487 Goal "n <= m ==> Suc(m)-n = Suc(m-n)";
```
```   488 by (etac rev_mp 1);
```
```   489 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   490 by (ALLGOALS Asm_simp_tac);
```
```   491 qed "Suc_diff_le";
```
```   492
```
```   493 Goal "m - n < Suc(m)";
```
```   494 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   495 by (etac less_SucE 3);
```
```   496 by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
```
```   497 qed "diff_less_Suc";
```
```   498
```
```   499 Goal "m - n <= (m::nat)";
```
```   500 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
```
```   501 by (ALLGOALS (asm_simp_tac (simpset() addsimps [le_SucI])));
```
```   502 qed "diff_le_self";
```
```   503 Addsimps [diff_le_self];
```
```   504
```
```   505 (* j<k ==> j-n < k *)
```
```   506 bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);
```
```   507
```
```   508 Goal "!!i::nat. i-j-k = i - (j+k)";
```
```   509 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
```   510 by (ALLGOALS Asm_simp_tac);
```
```   511 qed "diff_diff_left";
```
```   512
```
```   513 Goal "(Suc m - n) - Suc k = m - n - k";
```
```   514 by (simp_tac (simpset() addsimps [diff_diff_left]) 1);
```
```   515 qed "Suc_diff_diff";
```
```   516 Addsimps [Suc_diff_diff];
```
```   517
```
```   518 Goal "0<n ==> n - Suc i < n";
```
```   519 by (case_tac "n" 1);
```
```   520 by Safe_tac;
```
```   521 by (asm_simp_tac (simpset() addsimps le_simps) 1);
```
```   522 qed "diff_Suc_less";
```
```   523 Addsimps [diff_Suc_less];
```
```   524
```
```   525 (*This and the next few suggested by Florian Kammueller*)
```
```   526 Goal "!!i::nat. i-j-k = i-k-j";
```
```   527 by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);
```
```   528 qed "diff_commute";
```
```   529
```
```   530 Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)";
```
```   531 by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
```
```   532 by (ALLGOALS Asm_simp_tac);
```
```   533 qed_spec_mp "diff_add_assoc";
```
```   534
```
```   535 Goal "k <= (j::nat) --> (j + i) - k = (j - k) + i";
```
```   536 by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);
```
```   537 qed_spec_mp "diff_add_assoc2";
```
```   538
```
```   539 Goal "(n+m) - n = (m::nat)";
```
```   540 by (induct_tac "n" 1);
```
```   541 by (ALLGOALS Asm_simp_tac);
```
```   542 qed "diff_add_inverse";
```
```   543
```
```   544 Goal "(m+n) - n = (m::nat)";
```
```   545 by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);
```
```   546 qed "diff_add_inverse2";
```
```   547
```
```   548 Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)";
```
```   549 by Safe_tac;
```
```   550 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_add_inverse2])));
```
```   551 qed "le_imp_diff_is_add";
```
```   552
```
```   553 Goal "!!m::nat. (m-n = 0) = (m <= n)";
```
```   554 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   555 by (ALLGOALS Asm_simp_tac);
```
```   556 qed "diff_is_0_eq";
```
```   557 Addsimps [diff_is_0_eq];
```
```   558
```
```   559 Goal "!!m::nat. (0<n-m) = (m<n)";
```
```   560 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   561 by (ALLGOALS Asm_simp_tac);
```
```   562 qed "zero_less_diff";
```
```   563 Addsimps [zero_less_diff];
```
```   564
```
```   565 Goal "i < j  ==> EX k::nat. 0<k & i+k = j";
```
```   566 by (res_inst_tac [("x","j - i")] exI 1);
```
```   567 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);
```
```   568 qed "less_imp_add_positive";
```
```   569
```
```   570 Goal "P(k) --> (ALL n. P(Suc(n))--> P(n)) --> P(k-i)";
```
```   571 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
```
```   572 by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
```
```   573 qed "zero_induct_lemma";
```
```   574
```
```   575 val prems = Goal "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
```
```   576 by (rtac (diff_self_eq_0 RS subst) 1);
```
```   577 by (rtac (zero_induct_lemma RS mp RS mp) 1);
```
```   578 by (REPEAT (ares_tac ([impI,allI]@prems) 1));
```
```   579 qed "zero_induct";
```
```   580
```
```   581 Goal "(k+m) - (k+n) = m - (n::nat)";
```
```   582 by (induct_tac "k" 1);
```
```   583 by (ALLGOALS Asm_simp_tac);
```
```   584 qed "diff_cancel";
```
```   585
```
```   586 Goal "(m+k) - (n+k) = m - (n::nat)";
```
```   587 by (asm_simp_tac
```
```   588     (simpset() addsimps [diff_cancel, inst "n" "k" add_commute]) 1);
```
```   589 qed "diff_cancel2";
```
```   590
```
```   591 Goal "n - (n+m) = (0::nat)";
```
```   592 by (induct_tac "n" 1);
```
```   593 by (ALLGOALS Asm_simp_tac);
```
```   594 qed "diff_add_0";
```
```   595
```
```   596
```
```   597 (** Difference distributes over multiplication **)
```
```   598
```
```   599 Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";
```
```   600 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   601 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_cancel])));
```
```   602 qed "diff_mult_distrib" ;
```
```   603
```
```   604 Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";
```
```   605 val mult_commute_k = read_instantiate [("m","k")] mult_commute;
```
```   606 by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);
```
```   607 qed "diff_mult_distrib2" ;
```
```   608 (*NOT added as rewrites, since sometimes they are used from right-to-left*)
```
```   609
```
```   610 bind_thms ("nat_distrib",
```
```   611   [add_mult_distrib, add_mult_distrib2, diff_mult_distrib, diff_mult_distrib2]);
```
```   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_imp_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 "(Suc 0 <= 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 = Suc 0) = (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 "(Suc 0 = m*n) = (m=1 & n=1)";
```
```   669 by (rtac (mult_eq_1_iff RSN (2,trans)) 1);
```
```   670 by (fast_tac (claset() addss simpset()) 1);
```
```   671 qed "one_eq_mult_iff";
```
```   672 Addsimps [one_eq_mult_iff];
```
```   673
```
```   674 Goal "!!m::nat. (m*k < n*k) = (0<k & m<n)";
```
```   675 by (safe_tac (claset() addSIs [mult_less_mono1]));
```
```   676 by (case_tac "k" 1);
```
```   677 by Auto_tac;
```
```   678 by (full_simp_tac (simpset() delsimps [le_0_eq]
```
```   679 			     addsimps [linorder_not_le RS sym]) 1);
```
```   680 by (blast_tac (claset() addIs [mult_le_mono1]) 1);
```
```   681 qed "mult_less_cancel2";
```
```   682
```
```   683 Goal "!!m::nat. (k*m < k*n) = (0<k & m<n)";
```
```   684 by (simp_tac (simpset() addsimps [mult_less_cancel2,
```
```   685                                   inst "m" "k" mult_commute]) 1);
```
```   686 qed "mult_less_cancel1";
```
```   687 Addsimps [mult_less_cancel1, mult_less_cancel2];
```
```   688
```
```   689 Goal "!!m::nat. (m*k <= n*k) = (0<k --> m<=n)";
```
```   690 by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
```
```   691 by Auto_tac;
```
```   692 qed "mult_le_cancel2";
```
```   693
```
```   694 Goal "!!m::nat. (k*m <= k*n) = (0<k --> m<=n)";
```
```   695 by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
```
```   696 by Auto_tac;
```
```   697 qed "mult_le_cancel1";
```
```   698 Addsimps [mult_le_cancel1, mult_le_cancel2];
```
```   699
```
```   700 Goal "(m*k = n*k) = (m=n | (k = (0::nat)))";
```
```   701 by (cut_facts_tac [less_linear] 1);
```
```   702 by Safe_tac;
```
```   703 by Auto_tac;
```
```   704 by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
```
```   705 by (ALLGOALS Asm_full_simp_tac);
```
```   706 qed "mult_cancel2";
```
```   707
```
```   708 Goal "(k*m = k*n) = (m=n | (k = (0::nat)))";
```
```   709 by (simp_tac (simpset() addsimps [mult_cancel2, inst "m" "k" mult_commute]) 1);
```
```   710 qed "mult_cancel1";
```
```   711 Addsimps [mult_cancel1, mult_cancel2];
```
```   712
```
```   713 Goal "(Suc k * m < Suc k * n) = (m < n)";
```
```   714 by (stac mult_less_cancel1 1);
```
```   715 by (Simp_tac 1);
```
```   716 qed "Suc_mult_less_cancel1";
```
```   717
```
```   718 Goal "(Suc k * m <= Suc k * n) = (m <= n)";
```
```   719 by (stac mult_le_cancel1 1);
```
```   720 by (Simp_tac 1);
```
```   721 qed "Suc_mult_le_cancel1";
```
```   722
```
```   723 Goal "(Suc k * m = Suc k * n) = (m = n)";
```
```   724 by (stac mult_cancel1 1);
```
```   725 by (Simp_tac 1);
```
```   726 qed "Suc_mult_cancel1";
```
```   727
```
```   728
```
```   729 (*Lemma for gcd*)
```
```   730 Goal "!!m::nat. m = m*n ==> n=1 | m=0";
```
```   731 by (dtac sym 1);
```
```   732 by (rtac disjCI 1);
```
```   733 by (rtac nat_less_cases 1 THEN assume_tac 2);
```
```   734 by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);
```
```   735 by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);
```
```   736 qed "mult_eq_self_implies_10";
```