src/HOL/Arith.ML
author oheimb
Fri Oct 23 20:44:34 1998 +0200 (1998-10-23)
changeset 5758 27a2b36efd95
parent 5654 8b872d546b9e
child 5771 7c2c8cf20221
permissions -rw-r--r--
corrected auto_tac (applications of unsafe wrappers)
     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 (exhaust_tac "m" 1);
    97 by (Auto_tac);
    98 qed "add_is_0";
    99 AddIffs [add_is_0];
   100 
   101 Goal "(0 = m+n) = (m=0 & n=0)";
   102 by (exhaust_tac "m" 1);
   103 by (Auto_tac);
   104 qed "zero_is_add";
   105 AddIffs [zero_is_add];
   106 
   107 Goal "(m+n=1) = (m=1 & n=0 | m=0 & n=1)";
   108 by(exhaust_tac "m" 1);
   109 by(Auto_tac);
   110 qed "add_is_1";
   111 
   112 Goal "(1=m+n) = (m=1 & n=0 | m=0 & n=1)";
   113 by(exhaust_tac "m" 1);
   114 by(Auto_tac);
   115 qed "one_is_add";
   116 
   117 Goal "(0<m+n) = (0<m | 0<n)";
   118 by (simp_tac (simpset() delsimps [neq0_conv] addsimps [neq0_conv RS sym]) 1);
   119 qed "add_gr_0";
   120 AddIffs [add_gr_0];
   121 
   122 (* FIXME: really needed?? *)
   123 Goal "((m+n)-1 = 0) = (m=0 & n-1 = 0 | m-1 = 0 & n=0)";
   124 by (exhaust_tac "m" 1);
   125 by (ALLGOALS (fast_tac (claset() addss (simpset()))));
   126 qed "pred_add_is_0";
   127 Addsimps [pred_add_is_0];
   128 
   129 (* Could be generalized, eg to "k<n ==> m+(n-(Suc k)) = (m+n)-(Suc k)" *)
   130 Goal "0<n ==> m + (n-1) = (m+n)-1";
   131 by (exhaust_tac "m" 1);
   132 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_Suc]
   133                                       addsplits [nat.split])));
   134 qed "add_pred";
   135 Addsimps [add_pred];
   136 
   137 Goal "m + n = m ==> n = 0";
   138 by (dtac (add_0_right RS ssubst) 1);
   139 by (asm_full_simp_tac (simpset() addsimps [add_assoc]
   140                                  delsimps [add_0_right]) 1);
   141 qed "add_eq_self_zero";
   142 
   143 
   144 (**** Additional theorems about "less than" ****)
   145 
   146 (*Deleted less_natE; instead use less_eq_Suc_add RS exE*)
   147 Goal "m<n --> (? k. n=Suc(m+k))";
   148 by (induct_tac "n" 1);
   149 by (ALLGOALS (simp_tac (simpset() addsimps [order_le_less])));
   150 by (blast_tac (claset() addSEs [less_SucE] 
   151                         addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1);
   152 qed_spec_mp "less_eq_Suc_add";
   153 
   154 Goal "n <= ((m + n)::nat)";
   155 by (induct_tac "m" 1);
   156 by (ALLGOALS Simp_tac);
   157 qed "le_add2";
   158 
   159 Goal "n <= ((n + m)::nat)";
   160 by (simp_tac (simpset() addsimps add_ac) 1);
   161 by (rtac le_add2 1);
   162 qed "le_add1";
   163 
   164 bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
   165 bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
   166 
   167 Goal "(m<n) = (? k. n=Suc(m+k))";
   168 by (blast_tac (claset() addSIs [less_add_Suc1, less_eq_Suc_add]) 1);
   169 qed "less_iff_Suc_add";
   170 
   171 
   172 (*"i <= j ==> i <= j+m"*)
   173 bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
   174 
   175 (*"i <= j ==> i <= m+j"*)
   176 bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
   177 
   178 (*"i < j ==> i < j+m"*)
   179 bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
   180 
   181 (*"i < j ==> i < m+j"*)
   182 bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
   183 
   184 Goal "i+j < (k::nat) --> i<k";
   185 by (induct_tac "j" 1);
   186 by (ALLGOALS Asm_simp_tac);
   187 qed_spec_mp "add_lessD1";
   188 
   189 Goal "~ (i+j < (i::nat))";
   190 by (rtac notI 1);
   191 by (etac (add_lessD1 RS less_irrefl) 1);
   192 qed "not_add_less1";
   193 
   194 Goal "~ (j+i < (i::nat))";
   195 by (simp_tac (simpset() addsimps [add_commute, not_add_less1]) 1);
   196 qed "not_add_less2";
   197 AddIffs [not_add_less1, not_add_less2];
   198 
   199 Goal "m+k<=n --> m<=(n::nat)";
   200 by (induct_tac "k" 1);
   201 by (ALLGOALS (asm_simp_tac (simpset() addsimps le_simps)));
   202 qed_spec_mp "add_leD1";
   203 
   204 Goal "m+k<=n ==> k<=(n::nat)";
   205 by (full_simp_tac (simpset() addsimps [add_commute]) 1);
   206 by (etac add_leD1 1);
   207 qed_spec_mp "add_leD2";
   208 
   209 Goal "m+k<=n ==> m<=n & k<=(n::nat)";
   210 by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1);
   211 bind_thm ("add_leE", result() RS conjE);
   212 
   213 (*needs !!k for add_ac to work*)
   214 Goal "!!k:: nat. [| k<l;  m+l = k+n |] ==> m<n";
   215 by (force_tac (claset(),
   216 	      simpset() delsimps [add_Suc_right]
   217 	                addsimps [less_iff_Suc_add,
   218 				  add_Suc_right RS sym] @ add_ac) 1);
   219 qed "less_add_eq_less";
   220 
   221 
   222 (*** Monotonicity of Addition ***)
   223 
   224 (*strict, in 1st argument*)
   225 Goal "i < j ==> i + k < j + (k::nat)";
   226 by (induct_tac "k" 1);
   227 by (ALLGOALS Asm_simp_tac);
   228 qed "add_less_mono1";
   229 
   230 (*strict, in both arguments*)
   231 Goal "[|i < j; k < l|] ==> i + k < j + (l::nat)";
   232 by (rtac (add_less_mono1 RS less_trans) 1);
   233 by (REPEAT (assume_tac 1));
   234 by (induct_tac "j" 1);
   235 by (ALLGOALS Asm_simp_tac);
   236 qed "add_less_mono";
   237 
   238 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
   239 val [lt_mono,le] = Goal
   240      "[| !!i j::nat. i<j ==> f(i) < f(j);       \
   241 \        i <= j                                 \
   242 \     |] ==> f(i) <= (f(j)::nat)";
   243 by (cut_facts_tac [le] 1);
   244 by (asm_full_simp_tac (simpset() addsimps [order_le_less]) 1);
   245 by (blast_tac (claset() addSIs [lt_mono]) 1);
   246 qed "less_mono_imp_le_mono";
   247 
   248 (*non-strict, in 1st argument*)
   249 Goal "i<=j ==> i + k <= j + (k::nat)";
   250 by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1);
   251 by (etac add_less_mono1 1);
   252 by (assume_tac 1);
   253 qed "add_le_mono1";
   254 
   255 (*non-strict, in both arguments*)
   256 Goal "[|i<=j;  k<=l |] ==> i + k <= j + (l::nat)";
   257 by (etac (add_le_mono1 RS le_trans) 1);
   258 by (simp_tac (simpset() addsimps [add_commute]) 1);
   259 qed "add_le_mono";
   260 
   261 
   262 (*** Multiplication ***)
   263 
   264 (*right annihilation in product*)
   265 qed_goal "mult_0_right" thy "m * 0 = 0"
   266  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
   267 
   268 (*right successor law for multiplication*)
   269 qed_goal "mult_Suc_right" thy  "m * Suc(n) = m + (m * n)"
   270  (fn _ => [induct_tac "m" 1,
   271            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
   272 
   273 Addsimps [mult_0_right, mult_Suc_right];
   274 
   275 Goal "1 * n = n";
   276 by (Asm_simp_tac 1);
   277 qed "mult_1";
   278 
   279 Goal "n * 1 = n";
   280 by (Asm_simp_tac 1);
   281 qed "mult_1_right";
   282 
   283 (*Commutative law for multiplication*)
   284 qed_goal "mult_commute" thy "m * n = n * (m::nat)"
   285  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
   286 
   287 (*addition distributes over multiplication*)
   288 qed_goal "add_mult_distrib" thy "(m + n)*k = (m*k) + ((n*k)::nat)"
   289  (fn _ => [induct_tac "m" 1,
   290            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
   291 
   292 qed_goal "add_mult_distrib2" thy "k*(m + n) = (k*m) + ((k*n)::nat)"
   293  (fn _ => [induct_tac "m" 1,
   294            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
   295 
   296 (*Associative law for multiplication*)
   297 qed_goal "mult_assoc" thy "(m * n) * k = m * ((n * k)::nat)"
   298   (fn _ => [induct_tac "m" 1, 
   299             ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib]))]);
   300 
   301 qed_goal "mult_left_commute" thy "x*(y*z) = y*((x*z)::nat)"
   302  (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1,
   303            rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]);
   304 
   305 val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
   306 
   307 Goal "(m*n = 0) = (m=0 | n=0)";
   308 by (induct_tac "m" 1);
   309 by (induct_tac "n" 2);
   310 by (ALLGOALS Asm_simp_tac);
   311 qed "mult_is_0";
   312 Addsimps [mult_is_0];
   313 
   314 Goal "m <= m*(m::nat)";
   315 by (induct_tac "m" 1);
   316 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym])));
   317 by (etac (le_add2 RSN (2,le_trans)) 1);
   318 qed "le_square";
   319 
   320 
   321 (*** Difference ***)
   322 
   323 
   324 qed_goal "diff_self_eq_0" thy "m - m = 0"
   325  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
   326 Addsimps [diff_self_eq_0];
   327 
   328 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
   329 Goal "~ m<n --> n+(m-n) = (m::nat)";
   330 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   331 by (ALLGOALS Asm_simp_tac);
   332 qed_spec_mp "add_diff_inverse";
   333 
   334 Goal "n<=m ==> n+(m-n) = (m::nat)";
   335 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);
   336 qed "le_add_diff_inverse";
   337 
   338 Goal "n<=m ==> (m-n)+n = (m::nat)";
   339 by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);
   340 qed "le_add_diff_inverse2";
   341 
   342 Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];
   343 
   344 
   345 (*** More results about difference ***)
   346 
   347 Goal "n <= m ==> Suc(m)-n = Suc(m-n)";
   348 by (etac rev_mp 1);
   349 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   350 by (ALLGOALS Asm_simp_tac);
   351 qed "Suc_diff_le";
   352 
   353 Goal "n<=(l::nat) --> Suc l - n + m = Suc (l - n + m)";
   354 by (res_inst_tac [("m","n"),("n","l")] diff_induct 1);
   355 by (ALLGOALS Asm_simp_tac);
   356 qed_spec_mp "Suc_diff_add_le";
   357 
   358 Goal "m - n < Suc(m)";
   359 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   360 by (etac less_SucE 3);
   361 by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
   362 qed "diff_less_Suc";
   363 
   364 Goal "m - n <= (m::nat)";
   365 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
   366 by (ALLGOALS Asm_simp_tac);
   367 qed "diff_le_self";
   368 Addsimps [diff_le_self];
   369 
   370 (* j<k ==> j-n < k *)
   371 bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);
   372 
   373 Goal "!!i::nat. i-j-k = i - (j+k)";
   374 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
   375 by (ALLGOALS Asm_simp_tac);
   376 qed "diff_diff_left";
   377 
   378 Goal "(Suc m - n) - Suc k = m - n - k";
   379 by (simp_tac (simpset() addsimps [diff_diff_left]) 1);
   380 qed "Suc_diff_diff";
   381 Addsimps [Suc_diff_diff];
   382 
   383 Goal "0<n ==> n - Suc i < n";
   384 by (exhaust_tac "n" 1);
   385 by Safe_tac;
   386 by (asm_simp_tac (simpset() addsimps le_simps) 1);
   387 qed "diff_Suc_less";
   388 Addsimps [diff_Suc_less];
   389 
   390 Goal "i<n ==> n - Suc i < n - i";
   391 by (exhaust_tac "n" 1);
   392 by (auto_tac (claset(),
   393 	      simpset() addsimps [Suc_diff_le]@le_simps));
   394 qed "diff_Suc_less_diff";
   395 
   396 (*This and the next few suggested by Florian Kammueller*)
   397 Goal "!!i::nat. i-j-k = i-k-j";
   398 by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);
   399 qed "diff_commute";
   400 
   401 Goal "k<=j --> j<=i --> i - (j - k) = i - j + (k::nat)";
   402 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
   403 by (ALLGOALS Asm_simp_tac);
   404 by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
   405 qed_spec_mp "diff_diff_right";
   406 
   407 Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)";
   408 by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
   409 by (ALLGOALS Asm_simp_tac);
   410 qed_spec_mp "diff_add_assoc";
   411 
   412 Goal "k <= (j::nat) --> (j + i) - k = i + (j - k)";
   413 by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);
   414 qed_spec_mp "diff_add_assoc2";
   415 
   416 Goal "(n+m) - n = (m::nat)";
   417 by (induct_tac "n" 1);
   418 by (ALLGOALS Asm_simp_tac);
   419 qed "diff_add_inverse";
   420 Addsimps [diff_add_inverse];
   421 
   422 Goal "(m+n) - n = (m::nat)";
   423 by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);
   424 qed "diff_add_inverse2";
   425 Addsimps [diff_add_inverse2];
   426 
   427 Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)";
   428 by Safe_tac;
   429 by (ALLGOALS Asm_simp_tac);
   430 qed "le_imp_diff_is_add";
   431 
   432 Goal "(m-n = 0) = (m <= n)";
   433 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   434 by (ALLGOALS Asm_simp_tac);
   435 qed "diff_is_0_eq";
   436 Addsimps [diff_is_0_eq RS iffD2];
   437 
   438 Goal "m-n = 0  -->  n-m = 0  -->  m=n";
   439 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   440 by (REPEAT(Simp_tac 1 THEN TRY(atac 1)));
   441 qed_spec_mp "diffs0_imp_equal";
   442 
   443 Goal "(0<n-m) = (m<n)";
   444 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   445 by (ALLGOALS Asm_simp_tac);
   446 qed "zero_less_diff";
   447 Addsimps [zero_less_diff];
   448 
   449 Goal "i < j  ==> ? k. 0<k & i+k = j";
   450 by (res_inst_tac [("x","j - i")] exI 1);
   451 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);
   452 qed "less_imp_add_positive";
   453 
   454 Goal "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
   455 by (simp_tac (simpset() addsimps [leI, Suc_le_eq, Suc_diff_le]) 1);
   456 qed "if_Suc_diff_le";
   457 
   458 Goal "Suc(m)-n <= Suc(m-n)";
   459 by (simp_tac (simpset() addsimps [if_Suc_diff_le]) 1);
   460 qed "diff_Suc_le_Suc_diff";
   461 
   462 Goal "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
   463 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
   464 by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
   465 qed "zero_induct_lemma";
   466 
   467 val prems = Goal "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
   468 by (rtac (diff_self_eq_0 RS subst) 1);
   469 by (rtac (zero_induct_lemma RS mp RS mp) 1);
   470 by (REPEAT (ares_tac ([impI,allI]@prems) 1));
   471 qed "zero_induct";
   472 
   473 Goal "(k+m) - (k+n) = m - (n::nat)";
   474 by (induct_tac "k" 1);
   475 by (ALLGOALS Asm_simp_tac);
   476 qed "diff_cancel";
   477 Addsimps [diff_cancel];
   478 
   479 Goal "(m+k) - (n+k) = m - (n::nat)";
   480 val add_commute_k = read_instantiate [("n","k")] add_commute;
   481 by (asm_simp_tac (simpset() addsimps [add_commute_k]) 1);
   482 qed "diff_cancel2";
   483 Addsimps [diff_cancel2];
   484 
   485 (*From Clemens Ballarin, proof by lcp*)
   486 Goal "[| k<=n; n<=m |] ==> (m-k) - (n-k) = m-(n::nat)";
   487 by (REPEAT (etac rev_mp 1));
   488 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   489 by (ALLGOALS Asm_simp_tac);
   490 (*a confluence problem*)
   491 by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
   492 qed "diff_right_cancel";
   493 
   494 Goal "n - (n+m) = 0";
   495 by (induct_tac "n" 1);
   496 by (ALLGOALS Asm_simp_tac);
   497 qed "diff_add_0";
   498 Addsimps [diff_add_0];
   499 
   500 
   501 (** Difference distributes over multiplication **)
   502 
   503 Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";
   504 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   505 by (ALLGOALS Asm_simp_tac);
   506 qed "diff_mult_distrib" ;
   507 
   508 Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";
   509 val mult_commute_k = read_instantiate [("m","k")] mult_commute;
   510 by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);
   511 qed "diff_mult_distrib2" ;
   512 (*NOT added as rewrites, since sometimes they are used from right-to-left*)
   513 
   514 
   515 (*** Monotonicity of Multiplication ***)
   516 
   517 Goal "i <= (j::nat) ==> i*k<=j*k";
   518 by (induct_tac "k" 1);
   519 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));
   520 qed "mult_le_mono1";
   521 
   522 (*<=monotonicity, BOTH arguments*)
   523 Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l";
   524 by (etac (mult_le_mono1 RS le_trans) 1);
   525 by (rtac le_trans 1);
   526 by (stac mult_commute 2);
   527 by (etac mult_le_mono1 2);
   528 by (simp_tac (simpset() addsimps [mult_commute]) 1);
   529 qed "mult_le_mono";
   530 
   531 (*strict, in 1st argument; proof is by induction on k>0*)
   532 Goal "[| i<j; 0<k |] ==> k*i < k*j";
   533 by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1);
   534 by (Asm_simp_tac 1);
   535 by (induct_tac "x" 1);
   536 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono])));
   537 qed "mult_less_mono2";
   538 
   539 Goal "[| i<j; 0<k |] ==> i*k < j*k";
   540 by (dtac mult_less_mono2 1);
   541 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute])));
   542 qed "mult_less_mono1";
   543 
   544 Goal "(0 < m*n) = (0<m & 0<n)";
   545 by (induct_tac "m" 1);
   546 by (induct_tac "n" 2);
   547 by (ALLGOALS Asm_simp_tac);
   548 qed "zero_less_mult_iff";
   549 Addsimps [zero_less_mult_iff];
   550 
   551 Goal "(m*n = 1) = (m=1 & n=1)";
   552 by (induct_tac "m" 1);
   553 by (Simp_tac 1);
   554 by (induct_tac "n" 1);
   555 by (Simp_tac 1);
   556 by (fast_tac (claset() addss simpset()) 1);
   557 qed "mult_eq_1_iff";
   558 Addsimps [mult_eq_1_iff];
   559 
   560 Goal "0<k ==> (m*k < n*k) = (m<n)";
   561 by (safe_tac (claset() addSIs [mult_less_mono1]));
   562 by (cut_facts_tac [less_linear] 1);
   563 by (blast_tac (claset() addIs [mult_less_mono1] addEs [less_asym]) 1);
   564 qed "mult_less_cancel2";
   565 
   566 Goal "0<k ==> (k*m < k*n) = (m<n)";
   567 by (dtac mult_less_cancel2 1);
   568 by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
   569 qed "mult_less_cancel1";
   570 Addsimps [mult_less_cancel1, mult_less_cancel2];
   571 
   572 Goal "(Suc k * m < Suc k * n) = (m < n)";
   573 by (rtac mult_less_cancel1 1);
   574 by (Simp_tac 1);
   575 qed "Suc_mult_less_cancel1";
   576 
   577 Goalw [le_def] "(Suc k * m <= Suc k * n) = (m <= n)";
   578 by (simp_tac (simpset_of HOL.thy) 1);
   579 by (rtac Suc_mult_less_cancel1 1);
   580 qed "Suc_mult_le_cancel1";
   581 
   582 Goal "0<k ==> (m*k = n*k) = (m=n)";
   583 by (cut_facts_tac [less_linear] 1);
   584 by Safe_tac;
   585 by (assume_tac 2);
   586 by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
   587 by (ALLGOALS Asm_full_simp_tac);
   588 qed "mult_cancel2";
   589 
   590 Goal "0<k ==> (k*m = k*n) = (m=n)";
   591 by (dtac mult_cancel2 1);
   592 by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
   593 qed "mult_cancel1";
   594 Addsimps [mult_cancel1, mult_cancel2];
   595 
   596 Goal "(Suc k * m = Suc k * n) = (m = n)";
   597 by (rtac mult_cancel1 1);
   598 by (Simp_tac 1);
   599 qed "Suc_mult_cancel1";
   600 
   601 
   602 (** Lemma for gcd **)
   603 
   604 Goal "m = m*n ==> n=1 | m=0";
   605 by (dtac sym 1);
   606 by (rtac disjCI 1);
   607 by (rtac nat_less_cases 1 THEN assume_tac 2);
   608 by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);
   609 by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);
   610 qed "mult_eq_self_implies_10";
   611 
   612 
   613 (*** Subtraction laws -- mostly from Clemens Ballarin ***)
   614 
   615 Goal "[| a < (b::nat); c <= a |] ==> a-c < b-c";
   616 by (subgoal_tac "c+(a-c) < c+(b-c)" 1);
   617 by (Full_simp_tac 1);
   618 by (subgoal_tac "c <= b" 1);
   619 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 2);
   620 by (Asm_simp_tac 1);
   621 qed "diff_less_mono";
   622 
   623 Goal "a+b < (c::nat) ==> a < c-b";
   624 by (dtac diff_less_mono 1);
   625 by (rtac le_add2 1);
   626 by (Asm_full_simp_tac 1);
   627 qed "add_less_imp_less_diff";
   628 
   629 Goal "(i < j-k) = (i+k < (j::nat))";
   630 by (rtac iffI 1);
   631  by (case_tac "k <= j" 1);
   632   by (dtac le_add_diff_inverse2 1);
   633   by (dres_inst_tac [("k","k")] add_less_mono1 1);
   634   by (Asm_full_simp_tac 1);
   635  by (rotate_tac 1 1);
   636  by (asm_full_simp_tac (simpset() addSolver cut_trans_tac) 1);
   637 by (etac add_less_imp_less_diff 1);
   638 qed "less_diff_conv";
   639 
   640 Goal "(j-k <= (i::nat)) = (j <= i+k)";
   641 by (simp_tac (simpset() addsimps [less_diff_conv, le_def]) 1);
   642 qed "le_diff_conv";
   643 
   644 Goal "k <= j ==> (i <= j-k) = (i+k <= (j::nat))";
   645 by (asm_full_simp_tac
   646     (simpset() delsimps [less_Suc_eq_le]
   647                addsimps [less_Suc_eq_le RS sym, less_diff_conv,
   648 			 Suc_diff_le RS sym]) 1);
   649 qed "le_diff_conv2";
   650 
   651 Goal "Suc i <= n ==> Suc (n - Suc i) = n - i";
   652 by (asm_full_simp_tac (simpset() addsimps [Suc_diff_le RS sym]) 1);
   653 qed "Suc_diff_Suc";
   654 
   655 Goal "i <= (n::nat) ==> n - (n - i) = i";
   656 by (etac rev_mp 1);
   657 by (res_inst_tac [("m","n"),("n","i")] diff_induct 1);
   658 by (ALLGOALS (asm_simp_tac  (simpset() addsimps [Suc_diff_le])));
   659 qed "diff_diff_cancel";
   660 Addsimps [diff_diff_cancel];
   661 
   662 Goal "k <= (n::nat) ==> m <= n + m - k";
   663 by (etac rev_mp 1);
   664 by (res_inst_tac [("m", "k"), ("n", "n")] diff_induct 1);
   665 by (Simp_tac 1);
   666 by (simp_tac (simpset() addsimps [le_add2, less_imp_le]) 1);
   667 by (Simp_tac 1);
   668 qed "le_add_diff";
   669 
   670 Goal "0<k ==> j<i --> j+k-i < k";
   671 by (res_inst_tac [("m","j"),("n","i")] diff_induct 1);
   672 by (ALLGOALS Asm_simp_tac);
   673 qed_spec_mp "add_diff_less";
   674 
   675 
   676 Goal "m-1 < n ==> m <= n";
   677 by (exhaust_tac "m" 1);
   678 by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
   679 qed "pred_less_imp_le";
   680 
   681 Goal "j<=i ==> i - j < Suc i - j";
   682 by (REPEAT (etac rev_mp 1));
   683 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
   684 by Auto_tac;
   685 qed "diff_less_Suc_diff";
   686 
   687 Goal "i - j <= Suc i - j";
   688 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
   689 by Auto_tac;
   690 qed "diff_le_Suc_diff";
   691 AddIffs [diff_le_Suc_diff];
   692 
   693 Goal "n - Suc i <= n - i";
   694 by (case_tac "i<n" 1);
   695 by (dtac diff_Suc_less_diff 1);
   696 by (auto_tac (claset(), simpset() addsimps [less_imp_le, leI]));
   697 qed "diff_Suc_le_diff";
   698 AddIffs [diff_Suc_le_diff];
   699 
   700 Goal "0 < n ==> (m <= n-1) = (m<n)";
   701 by (exhaust_tac "n" 1);
   702 by (auto_tac (claset(), simpset() addsimps le_simps));
   703 qed "le_pred_eq";
   704 
   705 Goal "0 < n ==> (m-1 < n) = (m<=n)";
   706 by (exhaust_tac "m" 1);
   707 by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
   708 qed "less_pred_eq";
   709 
   710 (*In ordinary notation: if 0<n and n<=m then m-n < m *)
   711 Goal "[| 0<n; ~ m<n |] ==> m - n < m";
   712 by (subgoal_tac "0<n --> ~ m<n --> m - n < m" 1);
   713 by (Blast_tac 1);
   714 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   715 by (ALLGOALS(asm_simp_tac(simpset() addsimps [diff_less_Suc])));
   716 qed "diff_less";
   717 
   718 Goal "[| 0<n; n<=m |] ==> m - n < m";
   719 by (asm_simp_tac (simpset() addsimps [diff_less, not_less_iff_le]) 1);
   720 qed "le_diff_less";
   721 
   722 
   723 
   724 (** (Anti)Monotonicity of subtraction -- by Stefan Merz **)
   725 
   726 (* Monotonicity of subtraction in first argument *)
   727 Goal "m <= (n::nat) --> (m-l) <= (n-l)";
   728 by (induct_tac "n" 1);
   729 by (Simp_tac 1);
   730 by (simp_tac (simpset() addsimps [le_Suc_eq]) 1);
   731 by (blast_tac (claset() addIs [diff_le_Suc_diff, le_trans]) 1);
   732 qed_spec_mp "diff_le_mono";
   733 
   734 Goal "m <= (n::nat) ==> (l-n) <= (l-m)";
   735 by (induct_tac "l" 1);
   736 by (Simp_tac 1);
   737 by (case_tac "n <= na" 1);
   738 by (subgoal_tac "m <= na" 1);
   739 by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1);
   740 by (fast_tac (claset() addEs [le_trans]) 1);
   741 by (dtac not_leE 1);
   742 by (asm_simp_tac (simpset() addsimps [if_Suc_diff_le]) 1);
   743 qed_spec_mp "diff_le_mono2";