src/HOL/Arith.ML
 author paulson Wed Sep 23 10:12:01 1998 +0200 (1998-09-23) changeset 5537 c2bd39a2c0ee parent 5497 497215d66441 child 5598 6b8dee1a6ebb permissions -rw-r--r--
deleted needless parentheses
```     1 (*  Title:      HOL/Arith.ML
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4     Copyright   1998  University of Cambridge
```
```     5
```
```     6 Proofs about elementary arithmetic: addition, multiplication, etc.
```
```     7 Some from the Hoare example from Norbert Galm
```
```     8 *)
```
```     9
```
```    10 (*** Basic rewrite rules for the arithmetic operators ***)
```
```    11
```
```    12
```
```    13 (** Difference **)
```
```    14
```
```    15 qed_goal "diff_0_eq_0" thy
```
```    16     "0 - n = 0"
```
```    17  (fn _ => [induct_tac "n" 1,  ALLGOALS Asm_simp_tac]);
```
```    18
```
```    19 (*Must simplify BEFORE the induction!  (Else we get a critical pair)
```
```    20   Suc(m) - Suc(n)   rewrites to   pred(Suc(m) - n)  *)
```
```    21 qed_goal "diff_Suc_Suc" thy
```
```    22     "Suc(m) - Suc(n) = m - n"
```
```    23  (fn _ =>
```
```    24   [Simp_tac 1, induct_tac "n" 1, ALLGOALS Asm_simp_tac]);
```
```    25
```
```    26 Addsimps [diff_0_eq_0, diff_Suc_Suc];
```
```    27
```
```    28 (* Could be (and is, below) generalized in various ways;
```
```    29    However, none of the generalizations are currently in the simpset,
```
```    30    and I dread to think what happens if I put them in *)
```
```    31 Goal "0 < n ==> Suc(n-1) = n";
```
```    32 by (asm_simp_tac (simpset() addsplits [nat.split]) 1);
```
```    33 qed "Suc_pred";
```
```    34 Addsimps [Suc_pred];
```
```    35
```
```    36 Delsimps [diff_Suc];
```
```    37
```
```    38
```
```    39 (**** Inductive properties of the operators ****)
```
```    40
```
```    41 (*** Addition ***)
```
```    42
```
```    43 qed_goal "add_0_right" thy "m + 0 = m"
```
```    44  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    45
```
```    46 qed_goal "add_Suc_right" thy "m + Suc(n) = Suc(m+n)"
```
```    47  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    48
```
```    49 Addsimps [add_0_right,add_Suc_right];
```
```    50
```
```    51 (*Associative law for addition*)
```
```    52 qed_goal "add_assoc" thy "(m + n) + k = m + ((n + k)::nat)"
```
```    53  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    54
```
```    55 (*Commutative law for addition*)
```
```    56 qed_goal "add_commute" thy "m + n = n + (m::nat)"
```
```    57  (fn _ =>  [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    58
```
```    59 qed_goal "add_left_commute" thy "x+(y+z)=y+((x+z)::nat)"
```
```    60  (fn _ => [rtac (add_commute RS trans) 1, rtac (add_assoc RS trans) 1,
```
```    61            rtac (add_commute RS arg_cong) 1]);
```
```    62
```
```    63 (*Addition is an AC-operator*)
```
```    64 val add_ac = [add_assoc, add_commute, add_left_commute];
```
```    65
```
```    66 Goal "(k + m = k + n) = (m=(n::nat))";
```
```    67 by (induct_tac "k" 1);
```
```    68 by (Simp_tac 1);
```
```    69 by (Asm_simp_tac 1);
```
```    70 qed "add_left_cancel";
```
```    71
```
```    72 Goal "(m + k = n + k) = (m=(n::nat))";
```
```    73 by (induct_tac "k" 1);
```
```    74 by (Simp_tac 1);
```
```    75 by (Asm_simp_tac 1);
```
```    76 qed "add_right_cancel";
```
```    77
```
```    78 Goal "(k + m <= k + n) = (m<=(n::nat))";
```
```    79 by (induct_tac "k" 1);
```
```    80 by (Simp_tac 1);
```
```    81 by (Asm_simp_tac 1);
```
```    82 qed "add_left_cancel_le";
```
```    83
```
```    84 Goal "(k + m < k + n) = (m<(n::nat))";
```
```    85 by (induct_tac "k" 1);
```
```    86 by (Simp_tac 1);
```
```    87 by (Asm_simp_tac 1);
```
```    88 qed "add_left_cancel_less";
```
```    89
```
```    90 Addsimps [add_left_cancel, add_right_cancel,
```
```    91           add_left_cancel_le, add_left_cancel_less];
```
```    92
```
```    93 (** Reasoning about m+0=0, etc. **)
```
```    94
```
```    95 Goal "(m+n = 0) = (m=0 & n=0)";
```
```    96 by (induct_tac "m" 1);
```
```    97 by (ALLGOALS Asm_simp_tac);
```
```    98 qed "add_is_0";
```
```    99 AddIffs [add_is_0];
```
```   100
```
```   101 Goal "(0<m+n) = (0<m | 0<n)";
```
```   102 by (simp_tac (simpset() delsimps [neq0_conv] addsimps [neq0_conv RS sym]) 1);
```
```   103 qed "add_gr_0";
```
```   104 AddIffs [add_gr_0];
```
```   105
```
```   106 (* FIXME: really needed?? *)
```
```   107 Goal "((m+n)-1 = 0) = (m=0 & n-1 = 0 | m-1 = 0 & n=0)";
```
```   108 by (exhaust_tac "m" 1);
```
```   109 by (ALLGOALS (fast_tac (claset() addss (simpset()))));
```
```   110 qed "pred_add_is_0";
```
```   111 Addsimps [pred_add_is_0];
```
```   112
```
```   113 (* Could be generalized, eg to "k<n ==> m+(n-(Suc k)) = (m+n)-(Suc k)" *)
```
```   114 Goal "0<n ==> m + (n-1) = (m+n)-1";
```
```   115 by (exhaust_tac "m" 1);
```
```   116 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_Suc]
```
```   117                                       addsplits [nat.split])));
```
```   118 qed "add_pred";
```
```   119 Addsimps [add_pred];
```
```   120
```
```   121 Goal "m + n = m ==> n = 0";
```
```   122 by (dtac (add_0_right RS ssubst) 1);
```
```   123 by (asm_full_simp_tac (simpset() addsimps [add_assoc]
```
```   124                                  delsimps [add_0_right]) 1);
```
```   125 qed "add_eq_self_zero";
```
```   126
```
```   127
```
```   128 (**** Additional theorems about "less than" ****)
```
```   129
```
```   130 (*Deleted less_natE; instead use less_eq_Suc_add RS exE*)
```
```   131 Goal "m<n --> (? k. n=Suc(m+k))";
```
```   132 by (induct_tac "n" 1);
```
```   133 by (ALLGOALS (simp_tac (simpset() addsimps [le_eq_less_or_eq])));
```
```   134 by (blast_tac (claset() addSEs [less_SucE]
```
```   135                         addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1);
```
```   136 qed_spec_mp "less_eq_Suc_add";
```
```   137
```
```   138 Goal "n <= ((m + n)::nat)";
```
```   139 by (induct_tac "m" 1);
```
```   140 by (ALLGOALS Simp_tac);
```
```   141 by (etac le_trans 1);
```
```   142 by (rtac (lessI RS less_imp_le) 1);
```
```   143 qed "le_add2";
```
```   144
```
```   145 Goal "n <= ((n + m)::nat)";
```
```   146 by (simp_tac (simpset() addsimps add_ac) 1);
```
```   147 by (rtac le_add2 1);
```
```   148 qed "le_add1";
```
```   149
```
```   150 bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
```
```   151 bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
```
```   152
```
```   153 Goal "(m<n) = (? k. n=Suc(m+k))";
```
```   154 by (blast_tac (claset() addSIs [less_add_Suc1, less_eq_Suc_add]) 1);
```
```   155 qed "less_iff_Suc_add";
```
```   156
```
```   157
```
```   158 (*"i <= j ==> i <= j+m"*)
```
```   159 bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
```
```   160
```
```   161 (*"i <= j ==> i <= m+j"*)
```
```   162 bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
```
```   163
```
```   164 (*"i < j ==> i < j+m"*)
```
```   165 bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
```
```   166
```
```   167 (*"i < j ==> i < m+j"*)
```
```   168 bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
```
```   169
```
```   170 Goal "i+j < (k::nat) ==> i<k";
```
```   171 by (etac rev_mp 1);
```
```   172 by (induct_tac "j" 1);
```
```   173 by (ALLGOALS Asm_simp_tac);
```
```   174 by (blast_tac (claset() addDs [Suc_lessD]) 1);
```
```   175 qed "add_lessD1";
```
```   176
```
```   177 Goal "~ (i+j < (i::nat))";
```
```   178 by (rtac notI 1);
```
```   179 by (etac (add_lessD1 RS less_irrefl) 1);
```
```   180 qed "not_add_less1";
```
```   181
```
```   182 Goal "~ (j+i < (i::nat))";
```
```   183 by (simp_tac (simpset() addsimps [add_commute, not_add_less1]) 1);
```
```   184 qed "not_add_less2";
```
```   185 AddIffs [not_add_less1, not_add_less2];
```
```   186
```
```   187 Goal "m+k<=n --> m<=(n::nat)";
```
```   188 by (induct_tac "k" 1);
```
```   189 by (ALLGOALS (asm_simp_tac (simpset() addsimps le_simps)));
```
```   190 qed_spec_mp "add_leD1";
```
```   191
```
```   192 Goal "m+k<=n ==> k<=(n::nat)";
```
```   193 by (full_simp_tac (simpset() addsimps [add_commute]) 1);
```
```   194 by (etac add_leD1 1);
```
```   195 qed_spec_mp "add_leD2";
```
```   196
```
```   197 Goal "m+k<=n ==> m<=n & k<=(n::nat)";
```
```   198 by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1);
```
```   199 bind_thm ("add_leE", result() RS conjE);
```
```   200
```
```   201 (*needs !!k for add_ac to work*)
```
```   202 Goal "!!k:: nat. [| k<l;  m+l = k+n |] ==> m<n";
```
```   203 by (auto_tac (claset(),
```
```   204 	      simpset() delsimps [add_Suc_right]
```
```   205 	                addsimps [less_iff_Suc_add,
```
```   206 				  add_Suc_right RS sym] @ add_ac));
```
```   207 qed "less_add_eq_less";
```
```   208
```
```   209
```
```   210 (*** Monotonicity of Addition ***)
```
```   211
```
```   212 (*strict, in 1st argument*)
```
```   213 Goal "i < j ==> i + k < j + (k::nat)";
```
```   214 by (induct_tac "k" 1);
```
```   215 by (ALLGOALS Asm_simp_tac);
```
```   216 qed "add_less_mono1";
```
```   217
```
```   218 (*strict, in both arguments*)
```
```   219 Goal "[|i < j; k < l|] ==> i + k < j + (l::nat)";
```
```   220 by (rtac (add_less_mono1 RS less_trans) 1);
```
```   221 by (REPEAT (assume_tac 1));
```
```   222 by (induct_tac "j" 1);
```
```   223 by (ALLGOALS Asm_simp_tac);
```
```   224 qed "add_less_mono";
```
```   225
```
```   226 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
```
```   227 val [lt_mono,le] = Goal
```
```   228      "[| !!i j::nat. i<j ==> f(i) < f(j);       \
```
```   229 \        i <= j                                 \
```
```   230 \     |] ==> f(i) <= (f(j)::nat)";
```
```   231 by (cut_facts_tac [le] 1);
```
```   232 by (asm_full_simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
```
```   233 by (blast_tac (claset() addSIs [lt_mono]) 1);
```
```   234 qed "less_mono_imp_le_mono";
```
```   235
```
```   236 (*non-strict, in 1st argument*)
```
```   237 Goal "i<=j ==> i + k <= j + (k::nat)";
```
```   238 by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1);
```
```   239 by (etac add_less_mono1 1);
```
```   240 by (assume_tac 1);
```
```   241 qed "add_le_mono1";
```
```   242
```
```   243 (*non-strict, in both arguments*)
```
```   244 Goal "[|i<=j;  k<=l |] ==> i + k <= j + (l::nat)";
```
```   245 by (etac (add_le_mono1 RS le_trans) 1);
```
```   246 by (simp_tac (simpset() addsimps [add_commute]) 1);
```
```   247 qed "add_le_mono";
```
```   248
```
```   249
```
```   250 (*** Multiplication ***)
```
```   251
```
```   252 (*right annihilation in product*)
```
```   253 qed_goal "mult_0_right" thy "m * 0 = 0"
```
```   254  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```   255
```
```   256 (*right successor law for multiplication*)
```
```   257 qed_goal "mult_Suc_right" thy  "m * Suc(n) = m + (m * n)"
```
```   258  (fn _ => [induct_tac "m" 1,
```
```   259            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
```
```   260
```
```   261 Addsimps [mult_0_right, mult_Suc_right];
```
```   262
```
```   263 Goal "1 * n = n";
```
```   264 by (Asm_simp_tac 1);
```
```   265 qed "mult_1";
```
```   266
```
```   267 Goal "n * 1 = n";
```
```   268 by (Asm_simp_tac 1);
```
```   269 qed "mult_1_right";
```
```   270
```
```   271 (*Commutative law for multiplication*)
```
```   272 qed_goal "mult_commute" thy "m * n = n * (m::nat)"
```
```   273  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```   274
```
```   275 (*addition distributes over multiplication*)
```
```   276 qed_goal "add_mult_distrib" thy "(m + n)*k = (m*k) + ((n*k)::nat)"
```
```   277  (fn _ => [induct_tac "m" 1,
```
```   278            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
```
```   279
```
```   280 qed_goal "add_mult_distrib2" thy "k*(m + n) = (k*m) + ((k*n)::nat)"
```
```   281  (fn _ => [induct_tac "m" 1,
```
```   282            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
```
```   283
```
```   284 (*Associative law for multiplication*)
```
```   285 qed_goal "mult_assoc" thy "(m * n) * k = m * ((n * k)::nat)"
```
```   286   (fn _ => [induct_tac "m" 1,
```
```   287             ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib]))]);
```
```   288
```
```   289 qed_goal "mult_left_commute" thy "x*(y*z) = y*((x*z)::nat)"
```
```   290  (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1,
```
```   291            rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]);
```
```   292
```
```   293 val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
```
```   294
```
```   295 Goal "(m*n = 0) = (m=0 | n=0)";
```
```   296 by (induct_tac "m" 1);
```
```   297 by (induct_tac "n" 2);
```
```   298 by (ALLGOALS Asm_simp_tac);
```
```   299 qed "mult_is_0";
```
```   300 Addsimps [mult_is_0];
```
```   301
```
```   302 Goal "m <= m*(m::nat)";
```
```   303 by (induct_tac "m" 1);
```
```   304 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym])));
```
```   305 by (etac (le_add2 RSN (2,le_trans)) 1);
```
```   306 qed "le_square";
```
```   307
```
```   308
```
```   309 (*** Difference ***)
```
```   310
```
```   311
```
```   312 qed_goal "diff_self_eq_0" thy "m - m = 0"
```
```   313  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```   314 Addsimps [diff_self_eq_0];
```
```   315
```
```   316 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
```
```   317 Goal "~ m<n --> n+(m-n) = (m::nat)";
```
```   318 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   319 by (ALLGOALS Asm_simp_tac);
```
```   320 qed_spec_mp "add_diff_inverse";
```
```   321
```
```   322 Goal "n<=m ==> n+(m-n) = (m::nat)";
```
```   323 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);
```
```   324 qed "le_add_diff_inverse";
```
```   325
```
```   326 Goal "n<=m ==> (m-n)+n = (m::nat)";
```
```   327 by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);
```
```   328 qed "le_add_diff_inverse2";
```
```   329
```
```   330 Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];
```
```   331
```
```   332
```
```   333 (*** More results about difference ***)
```
```   334
```
```   335 Goal "n <= m ==> Suc(m)-n = Suc(m-n)";
```
```   336 by (etac rev_mp 1);
```
```   337 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   338 by (ALLGOALS Asm_simp_tac);
```
```   339 qed "Suc_diff_le";
```
```   340
```
```   341 Goal "n<=(l::nat) --> Suc l - n + m = Suc (l - n + m)";
```
```   342 by (res_inst_tac [("m","n"),("n","l")] diff_induct 1);
```
```   343 by (ALLGOALS Asm_simp_tac);
```
```   344 qed_spec_mp "Suc_diff_add_le";
```
```   345
```
```   346 Goal "m - n < Suc(m)";
```
```   347 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   348 by (etac less_SucE 3);
```
```   349 by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
```
```   350 qed "diff_less_Suc";
```
```   351
```
```   352 Goal "m - n <= (m::nat)";
```
```   353 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
```
```   354 by (ALLGOALS Asm_simp_tac);
```
```   355 qed "diff_le_self";
```
```   356 Addsimps [diff_le_self];
```
```   357
```
```   358 (* j<k ==> j-n < k *)
```
```   359 bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);
```
```   360
```
```   361 Goal "!!i::nat. i-j-k = i - (j+k)";
```
```   362 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
```   363 by (ALLGOALS Asm_simp_tac);
```
```   364 qed "diff_diff_left";
```
```   365
```
```   366 Goal "(Suc m - n) - Suc k = m - n - k";
```
```   367 by (simp_tac (simpset() addsimps [diff_diff_left]) 1);
```
```   368 qed "Suc_diff_diff";
```
```   369 Addsimps [Suc_diff_diff];
```
```   370
```
```   371 Goal "0<n ==> n - Suc i < n";
```
```   372 by (exhaust_tac "n" 1);
```
```   373 by Safe_tac;
```
```   374 by (asm_simp_tac (simpset() addsimps le_simps) 1);
```
```   375 qed "diff_Suc_less";
```
```   376 Addsimps [diff_Suc_less];
```
```   377
```
```   378 Goal "i<n ==> n - Suc i < n - i";
```
```   379 by (exhaust_tac "n" 1);
```
```   380 by (auto_tac (claset(),
```
```   381 	      simpset() addsimps [Suc_diff_le]@le_simps));
```
```   382 qed "diff_Suc_less_diff";
```
```   383
```
```   384 Goal "m - n <= Suc m - n";
```
```   385 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   386 by (ALLGOALS Asm_simp_tac);
```
```   387 qed "diff_le_Suc_diff";
```
```   388
```
```   389 (*This and the next few suggested by Florian Kammueller*)
```
```   390 Goal "!!i::nat. i-j-k = i-k-j";
```
```   391 by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);
```
```   392 qed "diff_commute";
```
```   393
```
```   394 Goal "k<=j --> j<=i --> i - (j - k) = i - j + (k::nat)";
```
```   395 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
```   396 by (ALLGOALS Asm_simp_tac);
```
```   397 by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
```
```   398 qed_spec_mp "diff_diff_right";
```
```   399
```
```   400 Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)";
```
```   401 by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
```
```   402 by (ALLGOALS Asm_simp_tac);
```
```   403 qed_spec_mp "diff_add_assoc";
```
```   404
```
```   405 Goal "k <= (j::nat) --> (j + i) - k = i + (j - k)";
```
```   406 by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);
```
```   407 qed_spec_mp "diff_add_assoc2";
```
```   408
```
```   409 Goal "(n+m) - n = (m::nat)";
```
```   410 by (induct_tac "n" 1);
```
```   411 by (ALLGOALS Asm_simp_tac);
```
```   412 qed "diff_add_inverse";
```
```   413 Addsimps [diff_add_inverse];
```
```   414
```
```   415 Goal "(m+n) - n = (m::nat)";
```
```   416 by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);
```
```   417 qed "diff_add_inverse2";
```
```   418 Addsimps [diff_add_inverse2];
```
```   419
```
```   420 Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)";
```
```   421 by Safe_tac;
```
```   422 by (ALLGOALS Asm_simp_tac);
```
```   423 qed "le_imp_diff_is_add";
```
```   424
```
```   425 Goal "(m-n = 0) = (m <= n)";
```
```   426 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   427 by (ALLGOALS Asm_simp_tac);
```
```   428 qed "diff_is_0_eq";
```
```   429 Addsimps [diff_is_0_eq RS iffD2];
```
```   430
```
```   431 Goal "m-n = 0  -->  n-m = 0  -->  m=n";
```
```   432 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   433 by (REPEAT(Simp_tac 1 THEN TRY(atac 1)));
```
```   434 qed_spec_mp "diffs0_imp_equal";
```
```   435
```
```   436 Goal "(0<n-m) = (m<n)";
```
```   437 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   438 by (ALLGOALS Asm_simp_tac);
```
```   439 qed "zero_less_diff";
```
```   440 Addsimps [zero_less_diff];
```
```   441
```
```   442 Goal "i < j  ==> ? k. 0<k & i+k = j";
```
```   443 by (res_inst_tac [("x","j - i")] exI 1);
```
```   444 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);
```
```   445 qed "less_imp_add_positive";
```
```   446
```
```   447 Goal "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
```
```   448 by (simp_tac (simpset() addsimps [leI, Suc_le_eq, Suc_diff_le]) 1);
```
```   449 qed "if_Suc_diff_le";
```
```   450
```
```   451 Goal "Suc(m)-n <= Suc(m-n)";
```
```   452 by (simp_tac (simpset() addsimps [if_Suc_diff_le]) 1);
```
```   453 qed "diff_Suc_le_Suc_diff";
```
```   454
```
```   455 Goal "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
```
```   456 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
```
```   457 by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
```
```   458 qed "zero_induct_lemma";
```
```   459
```
```   460 val prems = Goal "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
```
```   461 by (rtac (diff_self_eq_0 RS subst) 1);
```
```   462 by (rtac (zero_induct_lemma RS mp RS mp) 1);
```
```   463 by (REPEAT (ares_tac ([impI,allI]@prems) 1));
```
```   464 qed "zero_induct";
```
```   465
```
```   466 Goal "(k+m) - (k+n) = m - (n::nat)";
```
```   467 by (induct_tac "k" 1);
```
```   468 by (ALLGOALS Asm_simp_tac);
```
```   469 qed "diff_cancel";
```
```   470 Addsimps [diff_cancel];
```
```   471
```
```   472 Goal "(m+k) - (n+k) = m - (n::nat)";
```
```   473 val add_commute_k = read_instantiate [("n","k")] add_commute;
```
```   474 by (asm_simp_tac (simpset() addsimps [add_commute_k]) 1);
```
```   475 qed "diff_cancel2";
```
```   476 Addsimps [diff_cancel2];
```
```   477
```
```   478 (*From Clemens Ballarin, proof by lcp*)
```
```   479 Goal "[| k<=n; n<=m |] ==> (m-k) - (n-k) = m-(n::nat)";
```
```   480 by (REPEAT (etac rev_mp 1));
```
```   481 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   482 by (ALLGOALS Asm_simp_tac);
```
```   483 (*a confluence problem*)
```
```   484 by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
```
```   485 qed "diff_right_cancel";
```
```   486
```
```   487 Goal "n - (n+m) = 0";
```
```   488 by (induct_tac "n" 1);
```
```   489 by (ALLGOALS Asm_simp_tac);
```
```   490 qed "diff_add_0";
```
```   491 Addsimps [diff_add_0];
```
```   492
```
```   493
```
```   494 (** Difference distributes over multiplication **)
```
```   495
```
```   496 Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";
```
```   497 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   498 by (ALLGOALS Asm_simp_tac);
```
```   499 qed "diff_mult_distrib" ;
```
```   500
```
```   501 Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";
```
```   502 val mult_commute_k = read_instantiate [("m","k")] mult_commute;
```
```   503 by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);
```
```   504 qed "diff_mult_distrib2" ;
```
```   505 (*NOT added as rewrites, since sometimes they are used from right-to-left*)
```
```   506
```
```   507
```
```   508 (*** Monotonicity of Multiplication ***)
```
```   509
```
```   510 Goal "i <= (j::nat) ==> i*k<=j*k";
```
```   511 by (induct_tac "k" 1);
```
```   512 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));
```
```   513 qed "mult_le_mono1";
```
```   514
```
```   515 (*<=monotonicity, BOTH arguments*)
```
```   516 Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l";
```
```   517 by (etac (mult_le_mono1 RS le_trans) 1);
```
```   518 by (rtac le_trans 1);
```
```   519 by (stac mult_commute 2);
```
```   520 by (etac mult_le_mono1 2);
```
```   521 by (simp_tac (simpset() addsimps [mult_commute]) 1);
```
```   522 qed "mult_le_mono";
```
```   523
```
```   524 (*strict, in 1st argument; proof is by induction on k>0*)
```
```   525 Goal "[| i<j; 0<k |] ==> k*i < k*j";
```
```   526 by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1);
```
```   527 by (Asm_simp_tac 1);
```
```   528 by (induct_tac "x" 1);
```
```   529 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono])));
```
```   530 qed "mult_less_mono2";
```
```   531
```
```   532 Goal "[| i<j; 0<k |] ==> i*k < j*k";
```
```   533 by (dtac mult_less_mono2 1);
```
```   534 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute])));
```
```   535 qed "mult_less_mono1";
```
```   536
```
```   537 Goal "(0 < m*n) = (0<m & 0<n)";
```
```   538 by (induct_tac "m" 1);
```
```   539 by (induct_tac "n" 2);
```
```   540 by (ALLGOALS Asm_simp_tac);
```
```   541 qed "zero_less_mult_iff";
```
```   542 Addsimps [zero_less_mult_iff];
```
```   543
```
```   544 Goal "(m*n = 1) = (m=1 & n=1)";
```
```   545 by (induct_tac "m" 1);
```
```   546 by (Simp_tac 1);
```
```   547 by (induct_tac "n" 1);
```
```   548 by (Simp_tac 1);
```
```   549 by (fast_tac (claset() addss simpset()) 1);
```
```   550 qed "mult_eq_1_iff";
```
```   551 Addsimps [mult_eq_1_iff];
```
```   552
```
```   553 Goal "0<k ==> (m*k < n*k) = (m<n)";
```
```   554 by (safe_tac (claset() addSIs [mult_less_mono1]));
```
```   555 by (cut_facts_tac [less_linear] 1);
```
```   556 by (blast_tac (claset() addIs [mult_less_mono1] addEs [less_asym]) 1);
```
```   557 qed "mult_less_cancel2";
```
```   558
```
```   559 Goal "0<k ==> (k*m < k*n) = (m<n)";
```
```   560 by (dtac mult_less_cancel2 1);
```
```   561 by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
```
```   562 qed "mult_less_cancel1";
```
```   563 Addsimps [mult_less_cancel1, mult_less_cancel2];
```
```   564
```
```   565 Goal "(Suc k * m < Suc k * n) = (m < n)";
```
```   566 by (rtac mult_less_cancel1 1);
```
```   567 by (Simp_tac 1);
```
```   568 qed "Suc_mult_less_cancel1";
```
```   569
```
```   570 Goalw [le_def] "(Suc k * m <= Suc k * n) = (m <= n)";
```
```   571 by (simp_tac (simpset_of HOL.thy) 1);
```
```   572 by (rtac Suc_mult_less_cancel1 1);
```
```   573 qed "Suc_mult_le_cancel1";
```
```   574
```
```   575 Goal "0<k ==> (m*k = n*k) = (m=n)";
```
```   576 by (cut_facts_tac [less_linear] 1);
```
```   577 by Safe_tac;
```
```   578 by (assume_tac 2);
```
```   579 by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
```
```   580 by (ALLGOALS Asm_full_simp_tac);
```
```   581 qed "mult_cancel2";
```
```   582
```
```   583 Goal "0<k ==> (k*m = k*n) = (m=n)";
```
```   584 by (dtac mult_cancel2 1);
```
```   585 by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
```
```   586 qed "mult_cancel1";
```
```   587 Addsimps [mult_cancel1, mult_cancel2];
```
```   588
```
```   589 Goal "(Suc k * m = Suc k * n) = (m = n)";
```
```   590 by (rtac mult_cancel1 1);
```
```   591 by (Simp_tac 1);
```
```   592 qed "Suc_mult_cancel1";
```
```   593
```
```   594
```
```   595 (** Lemma for gcd **)
```
```   596
```
```   597 Goal "m = m*n ==> n=1 | m=0";
```
```   598 by (dtac sym 1);
```
```   599 by (rtac disjCI 1);
```
```   600 by (rtac nat_less_cases 1 THEN assume_tac 2);
```
```   601 by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);
```
```   602 by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);
```
```   603 qed "mult_eq_self_implies_10";
```
```   604
```
```   605
```
```   606 (*** Subtraction laws -- mostly from Clemens Ballarin ***)
```
```   607
```
```   608 Goal "[| a < (b::nat); c <= a |] ==> a-c < b-c";
```
```   609 by (subgoal_tac "c+(a-c) < c+(b-c)" 1);
```
```   610 by (Full_simp_tac 1);
```
```   611 by (subgoal_tac "c <= b" 1);
```
```   612 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 2);
```
```   613 by (Asm_simp_tac 1);
```
```   614 qed "diff_less_mono";
```
```   615
```
```   616 Goal "a+b < (c::nat) ==> a < c-b";
```
```   617 by (dtac diff_less_mono 1);
```
```   618 by (rtac le_add2 1);
```
```   619 by (Asm_full_simp_tac 1);
```
```   620 qed "add_less_imp_less_diff";
```
```   621
```
```   622 Goal "(i < j-k) = (i+k < (j::nat))";
```
```   623 by (rtac iffI 1);
```
```   624  by (case_tac "k <= j" 1);
```
```   625   by (dtac le_add_diff_inverse2 1);
```
```   626   by (dres_inst_tac [("k","k")] add_less_mono1 1);
```
```   627   by (Asm_full_simp_tac 1);
```
```   628  by (rotate_tac 1 1);
```
```   629  by (asm_full_simp_tac (simpset() addSolver cut_trans_tac) 1);
```
```   630 by (etac add_less_imp_less_diff 1);
```
```   631 qed "less_diff_conv";
```
```   632
```
```   633 Goal "(j-k <= (i::nat)) = (j <= i+k)";
```
```   634 by (simp_tac (simpset() addsimps [less_diff_conv, le_def]) 1);
```
```   635 qed "le_diff_conv";
```
```   636
```
```   637 Goal "k <= j ==> (i <= j-k) = (i+k <= (j::nat))";
```
```   638 by (asm_full_simp_tac
```
```   639     (simpset() delsimps [less_Suc_eq_le]
```
```   640                addsimps [less_Suc_eq_le RS sym, less_diff_conv,
```
```   641 			 Suc_diff_le RS sym]) 1);
```
```   642 qed "le_diff_conv2";
```
```   643
```
```   644 Goal "Suc i <= n ==> Suc (n - Suc i) = n - i";
```
```   645 by (asm_full_simp_tac (simpset() addsimps [Suc_diff_le RS sym]) 1);
```
```   646 qed "Suc_diff_Suc";
```
```   647
```
```   648 Goal "i <= (n::nat) ==> n - (n - i) = i";
```
```   649 by (etac rev_mp 1);
```
```   650 by (res_inst_tac [("m","n"),("n","i")] diff_induct 1);
```
```   651 by (ALLGOALS (asm_simp_tac  (simpset() addsimps [Suc_diff_le])));
```
```   652 qed "diff_diff_cancel";
```
```   653 Addsimps [diff_diff_cancel];
```
```   654
```
```   655 Goal "k <= (n::nat) ==> m <= n + m - k";
```
```   656 by (etac rev_mp 1);
```
```   657 by (res_inst_tac [("m", "k"), ("n", "n")] diff_induct 1);
```
```   658 by (Simp_tac 1);
```
```   659 by (simp_tac (simpset() addsimps [le_add2, less_imp_le]) 1);
```
```   660 by (Simp_tac 1);
```
```   661 qed "le_add_diff";
```
```   662
```
```   663 Goal "0<k ==> j<i --> j+k-i < k";
```
```   664 by (res_inst_tac [("m","j"),("n","i")] diff_induct 1);
```
```   665 by (ALLGOALS Asm_simp_tac);
```
```   666 qed_spec_mp "add_diff_less";
```
```   667
```
```   668
```
```   669 Goal "m-1 < n ==> m <= n";
```
```   670 by (exhaust_tac "m" 1);
```
```   671 by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
```
```   672 qed "pred_less_imp_le";
```
```   673
```
```   674 Goal "j<=i ==> i - j < Suc i - j";
```
```   675 by (REPEAT (etac rev_mp 1));
```
```   676 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
```   677 by Auto_tac;
```
```   678 qed "diff_less_Suc_diff";
```
```   679
```
```   680 Goal "i - j <= Suc i - j";
```
```   681 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
```
```   682 by Auto_tac;
```
```   683 qed "diff_le_Suc_diff";
```
```   684 AddIffs [diff_le_Suc_diff];
```
```   685
```
```   686 Goal "n - Suc i <= n - i";
```
```   687 by (case_tac "i<n" 1);
```
```   688 by (dtac diff_Suc_less_diff 1);
```
```   689 by (auto_tac (claset(), simpset() addsimps [leI]));
```
```   690 qed "diff_Suc_le_diff";
```
```   691 AddIffs [diff_Suc_le_diff];
```
```   692
```
```   693 Goal "0 < n ==> (m <= n-1) = (m<n)";
```
```   694 by (exhaust_tac "n" 1);
```
```   695 by (auto_tac (claset(), simpset() addsimps le_simps));
```
```   696 qed "le_pred_eq";
```
```   697
```
```   698 Goal "0 < n ==> (m-1 < n) = (m<=n)";
```
```   699 by (exhaust_tac "m" 1);
```
```   700 by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
```
```   701 qed "less_pred_eq";
```
```   702
```
```   703 (*In ordinary notation: if 0<n and n<=m then m-n < m *)
```
```   704 Goal "[| 0<n; ~ m<n |] ==> m - n < m";
```
```   705 by (subgoal_tac "0<n --> ~ m<n --> m - n < m" 1);
```
```   706 by (Blast_tac 1);
```
```   707 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   708 by (ALLGOALS(asm_simp_tac(simpset() addsimps [diff_less_Suc])));
```
```   709 qed "diff_less";
```
```   710
```
```   711 Goal "[| 0<n; n<=m |] ==> m - n < m";
```
```   712 by (asm_simp_tac (simpset() addsimps [diff_less, not_less_iff_le]) 1);
```
```   713 qed "le_diff_less";
```
```   714
```
```   715
```
```   716
```
```   717 (** (Anti)Monotonicity of subtraction -- by Stefan Merz **)
```
```   718
```
```   719 (* Monotonicity of subtraction in first argument *)
```
```   720 Goal "m <= (n::nat) --> (m-l) <= (n-l)";
```
```   721 by (induct_tac "n" 1);
```
```   722 by (Simp_tac 1);
```
```   723 by (simp_tac (simpset() addsimps [le_Suc_eq]) 1);
```
```   724 by (blast_tac (claset() addIs [diff_le_Suc_diff, le_trans]) 1);
```
```   725 qed_spec_mp "diff_le_mono";
```
```   726
```
```   727 Goal "m <= (n::nat) ==> (l-n) <= (l-m)";
```
```   728 by (induct_tac "l" 1);
```
```   729 by (Simp_tac 1);
```
```   730 by (case_tac "n <= na" 1);
```
```   731 by (subgoal_tac "m <= na" 1);
```
```   732 by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1);
```
```   733 by (fast_tac (claset() addEs [le_trans]) 1);
```
```   734 by (dtac not_leE 1);
```
```   735 by (asm_simp_tac (simpset() addsimps [if_Suc_diff_le]) 1);
```
```   736 qed_spec_mp "diff_le_mono2";
```