src/HOL/Arith.ML
author paulson
Mon Sep 07 10:40:17 1998 +0200 (1998-09-07)
changeset 5429 0833486c23ce
parent 5427 26c9a7c0b36b
child 5485 0cd451e46a20
permissions -rw-r--r--
tidying
     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 [less_Suc_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);
   190 by (blast_tac (claset() addDs [Suc_leD]) 1);
   191 qed_spec_mp "add_leD1";
   192 
   193 Goal "m+k<=n ==> k<=(n::nat)";
   194 by (full_simp_tac (simpset() addsimps [add_commute]) 1);
   195 by (etac add_leD1 1);
   196 qed_spec_mp "add_leD2";
   197 
   198 Goal "m+k<=n ==> m<=n & k<=(n::nat)";
   199 by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1);
   200 bind_thm ("add_leE", result() RS conjE);
   201 
   202 (*needs !!k for add_ac to work*)
   203 Goal "!!k:: nat. [| k<l;  m+l = k+n |] ==> m<n";
   204 by (safe_tac (claset() addSDs [less_eq_Suc_add]));
   205 by (asm_full_simp_tac
   206     (simpset() delsimps [add_Suc_right]
   207                addsimps ([add_Suc_right RS sym, add_left_cancel] @ add_ac)) 1);
   208 by (etac subst 1);
   209 by (simp_tac (simpset() addsimps [less_add_Suc1]) 1);
   210 qed "less_add_eq_less";
   211 
   212 
   213 (*** Monotonicity of Addition ***)
   214 
   215 (*strict, in 1st argument*)
   216 Goal "i < j ==> i + k < j + (k::nat)";
   217 by (induct_tac "k" 1);
   218 by (ALLGOALS Asm_simp_tac);
   219 qed "add_less_mono1";
   220 
   221 (*strict, in both arguments*)
   222 Goal "[|i < j; k < l|] ==> i + k < j + (l::nat)";
   223 by (rtac (add_less_mono1 RS less_trans) 1);
   224 by (REPEAT (assume_tac 1));
   225 by (induct_tac "j" 1);
   226 by (ALLGOALS Asm_simp_tac);
   227 qed "add_less_mono";
   228 
   229 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
   230 val [lt_mono,le] = Goal
   231      "[| !!i j::nat. i<j ==> f(i) < f(j);       \
   232 \        i <= j                                 \
   233 \     |] ==> f(i) <= (f(j)::nat)";
   234 by (cut_facts_tac [le] 1);
   235 by (asm_full_simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
   236 by (blast_tac (claset() addSIs [lt_mono]) 1);
   237 qed "less_mono_imp_le_mono";
   238 
   239 (*non-strict, in 1st argument*)
   240 Goal "i<=j ==> i + k <= j + (k::nat)";
   241 by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1);
   242 by (etac add_less_mono1 1);
   243 by (assume_tac 1);
   244 qed "add_le_mono1";
   245 
   246 (*non-strict, in both arguments*)
   247 Goal "[|i<=j;  k<=l |] ==> i + k <= j + (l::nat)";
   248 by (etac (add_le_mono1 RS le_trans) 1);
   249 by (simp_tac (simpset() addsimps [add_commute]) 1);
   250 qed "add_le_mono";
   251 
   252 
   253 (*** Multiplication ***)
   254 
   255 (*right annihilation in product*)
   256 qed_goal "mult_0_right" thy "m * 0 = 0"
   257  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
   258 
   259 (*right successor law for multiplication*)
   260 qed_goal "mult_Suc_right" thy  "m * Suc(n) = m + (m * n)"
   261  (fn _ => [induct_tac "m" 1,
   262            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
   263 
   264 Addsimps [mult_0_right, mult_Suc_right];
   265 
   266 Goal "1 * n = n";
   267 by (Asm_simp_tac 1);
   268 qed "mult_1";
   269 
   270 Goal "n * 1 = n";
   271 by (Asm_simp_tac 1);
   272 qed "mult_1_right";
   273 
   274 (*Commutative law for multiplication*)
   275 qed_goal "mult_commute" thy "m * n = n * (m::nat)"
   276  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
   277 
   278 (*addition distributes over multiplication*)
   279 qed_goal "add_mult_distrib" thy "(m + n)*k = (m*k) + ((n*k)::nat)"
   280  (fn _ => [induct_tac "m" 1,
   281            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
   282 
   283 qed_goal "add_mult_distrib2" thy "k*(m + n) = (k*m) + ((k*n)::nat)"
   284  (fn _ => [induct_tac "m" 1,
   285            ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
   286 
   287 (*Associative law for multiplication*)
   288 qed_goal "mult_assoc" thy "(m * n) * k = m * ((n * k)::nat)"
   289   (fn _ => [induct_tac "m" 1, 
   290             ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib]))]);
   291 
   292 qed_goal "mult_left_commute" thy "x*(y*z) = y*((x*z)::nat)"
   293  (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1,
   294            rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]);
   295 
   296 val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
   297 
   298 Goal "(m*n = 0) = (m=0 | n=0)";
   299 by (induct_tac "m" 1);
   300 by (induct_tac "n" 2);
   301 by (ALLGOALS Asm_simp_tac);
   302 qed "mult_is_0";
   303 Addsimps [mult_is_0];
   304 
   305 Goal "m <= m*(m::nat)";
   306 by (induct_tac "m" 1);
   307 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym])));
   308 by (etac (le_add2 RSN (2,le_trans)) 1);
   309 qed "le_square";
   310 
   311 
   312 (*** Difference ***)
   313 
   314 
   315 qed_goal "diff_self_eq_0" thy "m - m = 0"
   316  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
   317 Addsimps [diff_self_eq_0];
   318 
   319 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
   320 Goal "~ m<n --> n+(m-n) = (m::nat)";
   321 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   322 by (ALLGOALS Asm_simp_tac);
   323 qed_spec_mp "add_diff_inverse";
   324 
   325 Goal "n<=m ==> n+(m-n) = (m::nat)";
   326 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);
   327 qed "le_add_diff_inverse";
   328 
   329 Goal "n<=m ==> (m-n)+n = (m::nat)";
   330 by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);
   331 qed "le_add_diff_inverse2";
   332 
   333 Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];
   334 
   335 
   336 (*** More results about difference ***)
   337 
   338 Goal "n <= m ==> Suc(m)-n = Suc(m-n)";
   339 by (etac rev_mp 1);
   340 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   341 by (ALLGOALS Asm_simp_tac);
   342 qed "Suc_diff_le";
   343 
   344 Goal "n<=(l::nat) --> Suc l - n + m = Suc (l - n + m)";
   345 by (res_inst_tac [("m","n"),("n","l")] diff_induct 1);
   346 by (ALLGOALS Asm_simp_tac);
   347 qed_spec_mp "Suc_diff_add_le";
   348 
   349 Goal "m - n < Suc(m)";
   350 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   351 by (etac less_SucE 3);
   352 by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
   353 qed "diff_less_Suc";
   354 
   355 Goal "m - n <= (m::nat)";
   356 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
   357 by (ALLGOALS Asm_simp_tac);
   358 qed "diff_le_self";
   359 Addsimps [diff_le_self];
   360 
   361 (* j<k ==> j-n < k *)
   362 bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);
   363 
   364 Goal "!!i::nat. i-j-k = i - (j+k)";
   365 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
   366 by (ALLGOALS Asm_simp_tac);
   367 qed "diff_diff_left";
   368 
   369 Goal "(Suc m - n) - Suc k = m - n - k";
   370 by (simp_tac (simpset() addsimps [diff_diff_left]) 1);
   371 qed "Suc_diff_diff";
   372 Addsimps [Suc_diff_diff];
   373 
   374 Goal "0<n ==> n - Suc i < n";
   375 by (exhaust_tac "n" 1);
   376 by Safe_tac;
   377 by (asm_simp_tac (simpset() addsimps [le_eq_less_Suc RS sym]) 1);
   378 qed "diff_Suc_less";
   379 Addsimps [diff_Suc_less];
   380 
   381 Goal "i<n ==> n - Suc i < n - i";
   382 by (exhaust_tac "n" 1);
   383 by Safe_tac;
   384 by (asm_simp_tac (simpset() addsimps [le_eq_less_Suc, Suc_diff_le]) 1);
   385 qed "diff_Suc_less_diff";
   386 
   387 Goal "m - n <= Suc m - n";
   388 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   389 by (ALLGOALS Asm_simp_tac);
   390 qed "diff_le_Suc_diff";
   391 
   392 (*This and the next few suggested by Florian Kammueller*)
   393 Goal "!!i::nat. i-j-k = i-k-j";
   394 by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);
   395 qed "diff_commute";
   396 
   397 Goal "k<=j --> j<=i --> i - (j - k) = i - j + (k::nat)";
   398 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
   399 by (ALLGOALS Asm_simp_tac);
   400 by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
   401 qed_spec_mp "diff_diff_right";
   402 
   403 Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)";
   404 by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
   405 by (ALLGOALS Asm_simp_tac);
   406 qed_spec_mp "diff_add_assoc";
   407 
   408 Goal "k <= (j::nat) --> (j + i) - k = i + (j - k)";
   409 by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);
   410 qed_spec_mp "diff_add_assoc2";
   411 
   412 Goal "(n+m) - n = (m::nat)";
   413 by (induct_tac "n" 1);
   414 by (ALLGOALS Asm_simp_tac);
   415 qed "diff_add_inverse";
   416 Addsimps [diff_add_inverse];
   417 
   418 Goal "(m+n) - n = (m::nat)";
   419 by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);
   420 qed "diff_add_inverse2";
   421 Addsimps [diff_add_inverse2];
   422 
   423 Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)";
   424 by Safe_tac;
   425 by (ALLGOALS Asm_simp_tac);
   426 qed "le_imp_diff_is_add";
   427 
   428 Goal "(m-n = 0) = (m <= n)";
   429 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   430 by (ALLGOALS (asm_simp_tac (simpset() addsimps [le_eq_less_Suc])));
   431 qed "diff_is_0_eq";
   432 Addsimps [diff_is_0_eq RS iffD2];
   433 
   434 Goal "m-n = 0  -->  n-m = 0  -->  m=n";
   435 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   436 by (REPEAT(Simp_tac 1 THEN TRY(atac 1)));
   437 qed_spec_mp "diffs0_imp_equal";
   438 
   439 Goal "(0<n-m) = (m<n)";
   440 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   441 by (ALLGOALS Asm_simp_tac);
   442 qed "zero_less_diff";
   443 Addsimps [zero_less_diff];
   444 
   445 Goal "i < j  ==> ? k. 0<k & i+k = j";
   446 by (res_inst_tac [("x","j - i")] exI 1);
   447 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);
   448 qed "less_imp_add_positive";
   449 
   450 Goal "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
   451 by (simp_tac (simpset() addsimps [leI, Suc_le_eq, Suc_diff_le]) 1);
   452 qed "if_Suc_diff_le";
   453 
   454 Goal "Suc(m)-n <= Suc(m-n)";
   455 by (simp_tac (simpset() addsimps [if_Suc_diff_le]) 1);
   456 qed "diff_Suc_le_Suc_diff";
   457 
   458 Goal "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
   459 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
   460 by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
   461 qed "zero_induct_lemma";
   462 
   463 val prems = Goal "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
   464 by (rtac (diff_self_eq_0 RS subst) 1);
   465 by (rtac (zero_induct_lemma RS mp RS mp) 1);
   466 by (REPEAT (ares_tac ([impI,allI]@prems) 1));
   467 qed "zero_induct";
   468 
   469 Goal "(k+m) - (k+n) = m - (n::nat)";
   470 by (induct_tac "k" 1);
   471 by (ALLGOALS Asm_simp_tac);
   472 qed "diff_cancel";
   473 Addsimps [diff_cancel];
   474 
   475 Goal "(m+k) - (n+k) = m - (n::nat)";
   476 val add_commute_k = read_instantiate [("n","k")] add_commute;
   477 by (asm_simp_tac (simpset() addsimps ([add_commute_k])) 1);
   478 qed "diff_cancel2";
   479 Addsimps [diff_cancel2];
   480 
   481 (*From Clemens Ballarin, proof by lcp*)
   482 Goal "[| k<=n; n<=m |] ==> (m-k) - (n-k) = m-(n::nat)";
   483 by (REPEAT (etac rev_mp 1));
   484 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   485 by (ALLGOALS Asm_simp_tac);
   486 (*a confluence problem*)
   487 by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
   488 qed "diff_right_cancel";
   489 
   490 Goal "n - (n+m) = 0";
   491 by (induct_tac "n" 1);
   492 by (ALLGOALS Asm_simp_tac);
   493 qed "diff_add_0";
   494 Addsimps [diff_add_0];
   495 
   496 
   497 (** Difference distributes over multiplication **)
   498 
   499 Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";
   500 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   501 by (ALLGOALS Asm_simp_tac);
   502 qed "diff_mult_distrib" ;
   503 
   504 Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";
   505 val mult_commute_k = read_instantiate [("m","k")] mult_commute;
   506 by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);
   507 qed "diff_mult_distrib2" ;
   508 (*NOT added as rewrites, since sometimes they are used from right-to-left*)
   509 
   510 
   511 (*** Monotonicity of Multiplication ***)
   512 
   513 Goal "i <= (j::nat) ==> i*k<=j*k";
   514 by (induct_tac "k" 1);
   515 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));
   516 qed "mult_le_mono1";
   517 
   518 (*<=monotonicity, BOTH arguments*)
   519 Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l";
   520 by (etac (mult_le_mono1 RS le_trans) 1);
   521 by (rtac le_trans 1);
   522 by (stac mult_commute 2);
   523 by (etac mult_le_mono1 2);
   524 by (simp_tac (simpset() addsimps [mult_commute]) 1);
   525 qed "mult_le_mono";
   526 
   527 (*strict, in 1st argument; proof is by induction on k>0*)
   528 Goal "[| i<j; 0<k |] ==> k*i < k*j";
   529 by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1);
   530 by (Asm_simp_tac 1);
   531 by (induct_tac "x" 1);
   532 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono])));
   533 qed "mult_less_mono2";
   534 
   535 Goal "[| i<j; 0<k |] ==> i*k < j*k";
   536 by (dtac mult_less_mono2 1);
   537 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute])));
   538 qed "mult_less_mono1";
   539 
   540 Goal "(0 < m*n) = (0<m & 0<n)";
   541 by (induct_tac "m" 1);
   542 by (induct_tac "n" 2);
   543 by (ALLGOALS Asm_simp_tac);
   544 qed "zero_less_mult_iff";
   545 Addsimps [zero_less_mult_iff];
   546 
   547 Goal "(m*n = 1) = (m=1 & n=1)";
   548 by (induct_tac "m" 1);
   549 by (Simp_tac 1);
   550 by (induct_tac "n" 1);
   551 by (Simp_tac 1);
   552 by (fast_tac (claset() addss simpset()) 1);
   553 qed "mult_eq_1_iff";
   554 Addsimps [mult_eq_1_iff];
   555 
   556 Goal "0<k ==> (m*k < n*k) = (m<n)";
   557 by (safe_tac (claset() addSIs [mult_less_mono1]));
   558 by (cut_facts_tac [less_linear] 1);
   559 by (blast_tac (claset() addIs [mult_less_mono1] addEs [less_asym]) 1);
   560 qed "mult_less_cancel2";
   561 
   562 Goal "0<k ==> (k*m < k*n) = (m<n)";
   563 by (dtac mult_less_cancel2 1);
   564 by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
   565 qed "mult_less_cancel1";
   566 Addsimps [mult_less_cancel1, mult_less_cancel2];
   567 
   568 Goal "(Suc k * m < Suc k * n) = (m < n)";
   569 by (rtac mult_less_cancel1 1);
   570 by (Simp_tac 1);
   571 qed "Suc_mult_less_cancel1";
   572 
   573 Goalw [le_def] "(Suc k * m <= Suc k * n) = (m <= n)";
   574 by (simp_tac (simpset_of HOL.thy) 1);
   575 by (rtac Suc_mult_less_cancel1 1);
   576 qed "Suc_mult_le_cancel1";
   577 
   578 Goal "0<k ==> (m*k = n*k) = (m=n)";
   579 by (cut_facts_tac [less_linear] 1);
   580 by Safe_tac;
   581 by (assume_tac 2);
   582 by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
   583 by (ALLGOALS Asm_full_simp_tac);
   584 qed "mult_cancel2";
   585 
   586 Goal "0<k ==> (k*m = k*n) = (m=n)";
   587 by (dtac mult_cancel2 1);
   588 by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
   589 qed "mult_cancel1";
   590 Addsimps [mult_cancel1, mult_cancel2];
   591 
   592 Goal "(Suc k * m = Suc k * n) = (m = n)";
   593 by (rtac mult_cancel1 1);
   594 by (Simp_tac 1);
   595 qed "Suc_mult_cancel1";
   596 
   597 
   598 (** Lemma for gcd **)
   599 
   600 Goal "m = m*n ==> n=1 | m=0";
   601 by (dtac sym 1);
   602 by (rtac disjCI 1);
   603 by (rtac nat_less_cases 1 THEN assume_tac 2);
   604 by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);
   605 by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);
   606 qed "mult_eq_self_implies_10";
   607 
   608 
   609 (*** Subtraction laws -- mostly from Clemens Ballarin ***)
   610 
   611 Goal "[| a < (b::nat); c <= a |] ==> a-c < b-c";
   612 by (subgoal_tac "c+(a-c) < c+(b-c)" 1);
   613 by (Full_simp_tac 1);
   614 by (subgoal_tac "c <= b" 1);
   615 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 2);
   616 by (Asm_simp_tac 1);
   617 qed "diff_less_mono";
   618 
   619 Goal "a+b < (c::nat) ==> a < c-b";
   620 by (dtac diff_less_mono 1);
   621 by (rtac le_add2 1);
   622 by (Asm_full_simp_tac 1);
   623 qed "add_less_imp_less_diff";
   624 
   625 Goal "(i < j-k) = (i+k < (j::nat))";
   626 br iffI 1;
   627  by(case_tac "k <= j" 1);
   628   bd le_add_diff_inverse2 1;
   629   by(dres_inst_tac [("k","k")] add_less_mono1 1);
   630   by(Asm_full_simp_tac 1);
   631  by(rotate_tac 1 1);
   632  by(asm_full_simp_tac (simpset() addSolver cut_trans_tac) 1);
   633 be add_less_imp_less_diff 1;
   634 qed "less_diff_conv";
   635 
   636 Goal "Suc i <= n ==> Suc (n - Suc i) = n - i";
   637 by (asm_simp_tac (simpset() addsimps [Suc_diff_le RS sym]) 1);
   638 qed "Suc_diff_Suc";
   639 
   640 Goal "i <= (n::nat) ==> n - (n - i) = i";
   641 by (etac rev_mp 1);
   642 by (res_inst_tac [("m","n"),("n","i")] diff_induct 1);
   643 by (ALLGOALS (asm_simp_tac  (simpset() addsimps [Suc_diff_le])));
   644 qed "diff_diff_cancel";
   645 Addsimps [diff_diff_cancel];
   646 
   647 Goal "k <= (n::nat) ==> m <= n + m - k";
   648 by (etac rev_mp 1);
   649 by (res_inst_tac [("m", "k"), ("n", "n")] diff_induct 1);
   650 by (Simp_tac 1);
   651 by (simp_tac (simpset() addsimps [less_add_Suc2, less_imp_le]) 1);
   652 by (Simp_tac 1);
   653 qed "le_add_diff";
   654 
   655 Goal "0<k ==> j<i --> j+k-i < k";
   656 by (res_inst_tac [("m","j"),("n","i")] diff_induct 1);
   657 by (ALLGOALS Asm_simp_tac);
   658 qed_spec_mp "add_diff_less";
   659 
   660 
   661 Goal "m-1 < n ==> m <= n";
   662 by (exhaust_tac "m" 1);
   663 by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
   664 qed "pred_less_imp_le";
   665 
   666 Goal "j<=i ==> i - j < Suc i - j";
   667 by (REPEAT (etac rev_mp 1));
   668 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
   669 by Auto_tac;
   670 qed "diff_less_Suc_diff";
   671 
   672 Goal "i - j <= Suc i - j";
   673 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
   674 by Auto_tac;
   675 qed "diff_le_Suc_diff";
   676 AddIffs [diff_le_Suc_diff];
   677 
   678 Goal "n - Suc i <= n - i";
   679 by (case_tac "i<n" 1);
   680 bd diff_Suc_less_diff 1;
   681 by (auto_tac (claset(), simpset() addsimps [leI RS le_imp_less_Suc]));
   682 qed "diff_Suc_le_diff";
   683 AddIffs [diff_Suc_le_diff];
   684 
   685 Goal "0 < n ==> (m <= n-1) = (m<n)";
   686 by (exhaust_tac "n" 1);
   687 by Auto_tac;
   688 by (ALLGOALS trans_tac);
   689 qed "le_pred_eq";
   690 
   691 Goal "0 < n ==> (m-1 < n) = (m<=n)";
   692 by (exhaust_tac "m" 1);
   693 by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
   694 qed "less_pred_eq";
   695 
   696 (*In ordinary notation: if 0<n and n<=m then m-n < m *)
   697 Goal "[| 0<n; ~ m<n |] ==> m - n < m";
   698 by (subgoal_tac "0<n --> ~ m<n --> m - n < m" 1);
   699 by (Blast_tac 1);
   700 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   701 by (ALLGOALS(asm_simp_tac(simpset() addsimps [diff_less_Suc])));
   702 qed "diff_less";
   703 
   704 Goal "[| 0<n; n<=m |] ==> m - n < m";
   705 by (asm_simp_tac (simpset() addsimps [diff_less, not_less_iff_le]) 1);
   706 qed "le_diff_less";
   707 
   708 
   709 
   710 (** (Anti)Monotonicity of subtraction -- by Stefan Merz **)
   711 
   712 (* Monotonicity of subtraction in first argument *)
   713 Goal "m <= (n::nat) --> (m-l) <= (n-l)";
   714 by (induct_tac "n" 1);
   715 by (Simp_tac 1);
   716 by (simp_tac (simpset() addsimps [le_Suc_eq]) 1);
   717 by (blast_tac (claset() addIs [diff_le_Suc_diff, le_trans]) 1);
   718 qed_spec_mp "diff_le_mono";
   719 
   720 Goal "m <= (n::nat) ==> (l-n) <= (l-m)";
   721 by (induct_tac "l" 1);
   722 by (Simp_tac 1);
   723 by (case_tac "n <= na" 1);
   724 by (subgoal_tac "m <= na" 1);
   725 by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1);
   726 by (fast_tac (claset() addEs [le_trans]) 1);
   727 by (dtac not_leE 1);
   728 by (asm_simp_tac (simpset() addsimps [if_Suc_diff_le]) 1);
   729 qed_spec_mp "diff_le_mono2";