src/HOL/Arith.ML
author pusch
Tue Feb 25 15:11:56 1997 +0100 (1997-02-25)
changeset 2682 13cdbf95ed92
parent 2498 7914881f47c0
child 2922 580647a879cf
permissions -rw-r--r--
minor changes due to new primrec definitions for +,-,*
     1 (*  Title:      HOL/Arith.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 Proofs about elementary arithmetic: addition, multiplication, etc.
     7 Tests definitions and simplifier.
     8 *)
     9 
    10 open Arith;
    11 
    12 (*** Basic rewrite rules for the arithmetic operators ***)
    13 
    14 goalw Arith.thy [pred_def] "pred 0 = 0";
    15 by(Simp_tac 1);
    16 qed "pred_0";
    17 
    18 goalw Arith.thy [pred_def] "pred(Suc n) = n";
    19 by(Simp_tac 1);
    20 qed "pred_Suc";
    21 
    22 Addsimps [pred_0,pred_Suc];
    23 
    24 (** pred **)
    25 
    26 val prems = goal Arith.thy "n ~= 0 ==> Suc(pred n) = n";
    27 by (res_inst_tac [("n","n")] natE 1);
    28 by (cut_facts_tac prems 1);
    29 by (ALLGOALS Asm_full_simp_tac);
    30 qed "Suc_pred";
    31 Addsimps [Suc_pred];
    32 
    33 (** Difference **)
    34 
    35 qed_goalw "diff_0_eq_0" Arith.thy [pred_def]
    36     "0 - n = 0"
    37  (fn _ => [nat_ind_tac "n" 1,  ALLGOALS Asm_simp_tac]);
    38 
    39 (*Must simplify BEFORE the induction!!  (Else we get a critical pair)
    40   Suc(m) - Suc(n)   rewrites to   pred(Suc(m) - n)  *)
    41 qed_goalw "diff_Suc_Suc" Arith.thy [pred_def]
    42     "Suc(m) - Suc(n) = m - n"
    43  (fn _ =>
    44   [Simp_tac 1, nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
    45 
    46 Addsimps [diff_0_eq_0, diff_Suc_Suc];
    47 
    48 
    49 goal Arith.thy "!!k. 0<k ==> EX j. k = Suc(j)";
    50 by (etac rev_mp 1);
    51 by (nat_ind_tac "k" 1);
    52 by (Simp_tac 1);
    53 by (Fast_tac 1);
    54 val lemma = result();
    55 
    56 (* [| 0 < k; !!j. [| j: nat; k = succ(j) |] ==> Q |] ==> Q *)
    57 bind_thm ("zero_less_natE", lemma RS exE);
    58 
    59 
    60 
    61 (**** Inductive properties of the operators ****)
    62 
    63 (*** Addition ***)
    64 
    65 qed_goal "add_0_right" Arith.thy "m + 0 = m"
    66  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
    67 
    68 qed_goal "add_Suc_right" Arith.thy "m + Suc(n) = Suc(m+n)"
    69  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
    70 
    71 Addsimps [add_0_right,add_Suc_right];
    72 
    73 (*Associative law for addition*)
    74 qed_goal "add_assoc" Arith.thy "(m + n) + k = m + ((n + k)::nat)"
    75  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
    76 
    77 (*Commutative law for addition*)  
    78 qed_goal "add_commute" Arith.thy "m + n = n + (m::nat)"
    79  (fn _ =>  [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
    80 
    81 qed_goal "add_left_commute" Arith.thy "x+(y+z)=y+((x+z)::nat)"
    82  (fn _ => [rtac (add_commute RS trans) 1, rtac (add_assoc RS trans) 1,
    83            rtac (add_commute RS arg_cong) 1]);
    84 
    85 (*Addition is an AC-operator*)
    86 val add_ac = [add_assoc, add_commute, add_left_commute];
    87 
    88 goal Arith.thy "!!k::nat. (k + m = k + n) = (m=n)";
    89 by (nat_ind_tac "k" 1);
    90 by (Simp_tac 1);
    91 by (Asm_simp_tac 1);
    92 qed "add_left_cancel";
    93 
    94 goal Arith.thy "!!k::nat. (m + k = n + k) = (m=n)";
    95 by (nat_ind_tac "k" 1);
    96 by (Simp_tac 1);
    97 by (Asm_simp_tac 1);
    98 qed "add_right_cancel";
    99 
   100 goal Arith.thy "!!k::nat. (k + m <= k + n) = (m<=n)";
   101 by (nat_ind_tac "k" 1);
   102 by (Simp_tac 1);
   103 by (Asm_simp_tac 1);
   104 qed "add_left_cancel_le";
   105 
   106 goal Arith.thy "!!k::nat. (k + m < k + n) = (m<n)";
   107 by (nat_ind_tac "k" 1);
   108 by (Simp_tac 1);
   109 by (Asm_simp_tac 1);
   110 qed "add_left_cancel_less";
   111 
   112 Addsimps [add_left_cancel, add_right_cancel,
   113           add_left_cancel_le, add_left_cancel_less];
   114 
   115 goal Arith.thy "(m+n = 0) = (m=0 & n=0)";
   116 by (nat_ind_tac "m" 1);
   117 by (ALLGOALS Asm_simp_tac);
   118 qed "add_is_0";
   119 Addsimps [add_is_0];
   120 
   121 goal Arith.thy "!!n. n ~= 0 ==> m + pred n = pred(m+n)";
   122 by (nat_ind_tac "m" 1);
   123 by (ALLGOALS Asm_simp_tac);
   124 qed "add_pred";
   125 Addsimps [add_pred];
   126 
   127 (*** Multiplication ***)
   128 
   129 (*right annihilation in product*)
   130 qed_goal "mult_0_right" Arith.thy "m * 0 = 0"
   131  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
   132 
   133 (*right Sucessor law for multiplication*)
   134 qed_goal "mult_Suc_right" Arith.thy  "m * Suc(n) = m + (m * n)"
   135  (fn _ => [nat_ind_tac "m" 1,
   136            ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
   137 
   138 Addsimps [mult_0_right,mult_Suc_right];
   139 
   140 goal Arith.thy "1 * n = n";
   141 by (Asm_simp_tac 1);
   142 qed "mult_1";
   143 
   144 goal Arith.thy "n * 1 = n";
   145 by (Asm_simp_tac 1);
   146 qed "mult_1_right";
   147 
   148 (*Commutative law for multiplication*)
   149 qed_goal "mult_commute" Arith.thy "m * n = n * (m::nat)"
   150  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
   151 
   152 (*addition distributes over multiplication*)
   153 qed_goal "add_mult_distrib" Arith.thy "(m + n)*k = (m*k) + ((n*k)::nat)"
   154  (fn _ => [nat_ind_tac "m" 1,
   155            ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
   156 
   157 qed_goal "add_mult_distrib2" Arith.thy "k*(m + n) = (k*m) + ((k*n)::nat)"
   158  (fn _ => [nat_ind_tac "m" 1,
   159            ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
   160 
   161 (*Associative law for multiplication*)
   162 qed_goal "mult_assoc" Arith.thy "(m * n) * k = m * ((n * k)::nat)"
   163   (fn _ => [nat_ind_tac "m" 1, 
   164             ALLGOALS (asm_simp_tac (!simpset addsimps [add_mult_distrib]))]);
   165 
   166 qed_goal "mult_left_commute" Arith.thy "x*(y*z) = y*((x*z)::nat)"
   167  (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1,
   168            rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]);
   169 
   170 val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
   171 
   172 (*** Difference ***)
   173 
   174 qed_goal "pred_Suc_diff" Arith.thy "pred(Suc m - n) = m - n"
   175  (fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
   176 Addsimps [pred_Suc_diff];
   177 
   178 qed_goal "diff_self_eq_0" Arith.thy "m - m = 0"
   179  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
   180 Addsimps [diff_self_eq_0];
   181 
   182 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
   183 val [prem] = goal Arith.thy "[| ~ m<n |] ==> n+(m-n) = (m::nat)";
   184 by (rtac (prem RS rev_mp) 1);
   185 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   186 by (ALLGOALS (Asm_simp_tac));
   187 qed "add_diff_inverse";
   188 
   189 
   190 (*** Remainder ***)
   191 
   192 goal Arith.thy "m - n < Suc(m)";
   193 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   194 by (etac less_SucE 3);
   195 by (ALLGOALS (asm_simp_tac (!simpset addsimps [less_Suc_eq])));
   196 qed "diff_less_Suc";
   197 
   198 goal Arith.thy "!!m::nat. m - n <= m";
   199 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
   200 by (ALLGOALS Asm_simp_tac);
   201 qed "diff_le_self";
   202 
   203 goal Arith.thy "!!n::nat. (n+m) - n = m";
   204 by (nat_ind_tac "n" 1);
   205 by (ALLGOALS Asm_simp_tac);
   206 qed "diff_add_inverse";
   207 
   208 goal Arith.thy "!!n::nat.(m+n) - n = m";
   209 by (res_inst_tac [("m1","m")] (add_commute RS ssubst) 1);
   210 by (REPEAT (ares_tac [diff_add_inverse] 1));
   211 qed "diff_add_inverse2";
   212 
   213 goal Arith.thy "!!k::nat. (k+m) - (k+n) = m - n";
   214 by (nat_ind_tac "k" 1);
   215 by (ALLGOALS Asm_simp_tac);
   216 qed "diff_cancel";
   217 Addsimps [diff_cancel];
   218 
   219 goal Arith.thy "!!m::nat. (m+k) - (n+k) = m - n";
   220 val add_commute_k = read_instantiate [("n","k")] add_commute;
   221 by (asm_simp_tac (!simpset addsimps ([add_commute_k])) 1);
   222 qed "diff_cancel2";
   223 Addsimps [diff_cancel2];
   224 
   225 goal Arith.thy "!!n::nat. n - (n+m) = 0";
   226 by (nat_ind_tac "n" 1);
   227 by (ALLGOALS Asm_simp_tac);
   228 qed "diff_add_0";
   229 Addsimps [diff_add_0];
   230 
   231 (** Difference distributes over multiplication **)
   232 
   233 goal Arith.thy "!!m::nat. (m - n) * k = (m * k) - (n * k)";
   234 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   235 by (ALLGOALS Asm_simp_tac);
   236 qed "diff_mult_distrib" ;
   237 
   238 goal Arith.thy "!!m::nat. k * (m - n) = (k * m) - (k * n)";
   239 val mult_commute_k = read_instantiate [("m","k")] mult_commute;
   240 by (simp_tac (!simpset addsimps [diff_mult_distrib, mult_commute_k]) 1);
   241 qed "diff_mult_distrib2" ;
   242 (*NOT added as rewrites, since sometimes they are used from right-to-left*)
   243 
   244 
   245 (** Less-then properties **)
   246 
   247 (*In ordinary notation: if 0<n and n<=m then m-n < m *)
   248 goal Arith.thy "!!m. [| 0<n; ~ m<n |] ==> m - n < m";
   249 by (subgoal_tac "0<n --> ~ m<n --> m - n < m" 1);
   250 by (Fast_tac 1);
   251 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   252 by (ALLGOALS(asm_simp_tac(!simpset addsimps [diff_less_Suc])));
   253 qed "diff_less";
   254 
   255 val wf_less_trans = wf_pred_nat RS wf_trancl RSN (2, def_wfrec RS trans);
   256 
   257 goalw Nat.thy [less_def] "(m,n) : pred_nat^+ = (m<n)";
   258 by (rtac refl 1);
   259 qed "less_eq";
   260 
   261 goal Arith.thy "(%m. m mod n) = wfrec (trancl pred_nat) \
   262              \                      (%f j. if j<n then j else f (j-n))";
   263 by (simp_tac (HOL_ss addsimps [mod_def]) 1);
   264 val mod_def1 = result() RS eq_reflection;
   265 
   266 goal Arith.thy "!!m. m<n ==> m mod n = m";
   267 by (rtac (mod_def1 RS wf_less_trans) 1);
   268 by (Asm_simp_tac 1);
   269 qed "mod_less";
   270 
   271 goal Arith.thy "!!m. [| 0<n;  ~m<n |] ==> m mod n = (m-n) mod n";
   272 by (rtac (mod_def1 RS wf_less_trans) 1);
   273 by (asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1);
   274 qed "mod_geq";
   275 
   276 
   277 (*** Quotient ***)
   278 
   279 goal Arith.thy "(%m. m div n) = wfrec (trancl pred_nat) \
   280                         \            (%f j. if j<n then 0 else Suc (f (j-n)))";
   281 by (simp_tac (HOL_ss addsimps [div_def]) 1);
   282 val div_def1 = result() RS eq_reflection;
   283 
   284 goal Arith.thy "!!m. m<n ==> m div n = 0";
   285 by (rtac (div_def1 RS wf_less_trans) 1);
   286 by (Asm_simp_tac 1);
   287 qed "div_less";
   288 
   289 goal Arith.thy "!!M. [| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)";
   290 by (rtac (div_def1 RS wf_less_trans) 1);
   291 by (asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1);
   292 qed "div_geq";
   293 
   294 (*Main Result about quotient and remainder.*)
   295 goal Arith.thy "!!m. 0<n ==> (m div n)*n + m mod n = m";
   296 by (res_inst_tac [("n","m")] less_induct 1);
   297 by (rename_tac "k" 1);    (*Variable name used in line below*)
   298 by (case_tac "k<n" 1);
   299 by (ALLGOALS (asm_simp_tac(!simpset addsimps ([add_assoc] @
   300                        [mod_less, mod_geq, div_less, div_geq,
   301                         add_diff_inverse, diff_less]))));
   302 qed "mod_div_equality";
   303 
   304 
   305 (*** More results about difference ***)
   306 
   307 val [prem] = goal Arith.thy "m < Suc(n) ==> m-n = 0";
   308 by (rtac (prem RS rev_mp) 1);
   309 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   310 by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
   311 by (ALLGOALS (Asm_simp_tac));
   312 qed "less_imp_diff_is_0";
   313 
   314 val prems = goal Arith.thy "m-n = 0  -->  n-m = 0  -->  m=n";
   315 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   316 by (REPEAT(Simp_tac 1 THEN TRY(atac 1)));
   317 qed_spec_mp "diffs0_imp_equal";
   318 
   319 val [prem] = goal Arith.thy "m<n ==> 0<n-m";
   320 by (rtac (prem RS rev_mp) 1);
   321 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   322 by (ALLGOALS (Asm_simp_tac));
   323 qed "less_imp_diff_positive";
   324 
   325 val [prem] = goal Arith.thy "n < Suc(m) ==> Suc(m)-n = Suc(m-n)";
   326 by (rtac (prem RS rev_mp) 1);
   327 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   328 by (ALLGOALS (Asm_simp_tac));
   329 qed "Suc_diff_n";
   330 
   331 goal Arith.thy "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
   332 by (simp_tac (!simpset addsimps [less_imp_diff_is_0, not_less_eq, Suc_diff_n]
   333                     setloop (split_tac [expand_if])) 1);
   334 qed "if_Suc_diff_n";
   335 
   336 goal Arith.thy "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
   337 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
   338 by (ALLGOALS (strip_tac THEN' Simp_tac THEN' TRY o Fast_tac));
   339 qed "zero_induct_lemma";
   340 
   341 val prems = goal Arith.thy "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
   342 by (rtac (diff_self_eq_0 RS subst) 1);
   343 by (rtac (zero_induct_lemma RS mp RS mp) 1);
   344 by (REPEAT (ares_tac ([impI,allI]@prems) 1));
   345 qed "zero_induct";
   346 
   347 (*13 July 1992: loaded in 105.7s*)
   348 
   349 
   350 (*** Further facts about mod (mainly for mutilated checkerboard ***)
   351 
   352 goal Arith.thy
   353     "!!m n. 0<n ==> \
   354 \           Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))";
   355 by (res_inst_tac [("n","m")] less_induct 1);
   356 by (excluded_middle_tac "Suc(na)<n" 1);
   357 (* case Suc(na) < n *)
   358 by (forward_tac [lessI RS less_trans] 2);
   359 by (asm_simp_tac (!simpset addsimps [mod_less, less_not_refl2 RS not_sym]) 2);
   360 (* case n <= Suc(na) *)
   361 by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, mod_geq]) 1);
   362 by (etac (le_imp_less_or_eq RS disjE) 1);
   363 by (asm_simp_tac (!simpset addsimps [Suc_diff_n]) 1);
   364 by (asm_full_simp_tac (!simpset addsimps [not_less_eq RS sym, 
   365                                           diff_less, mod_geq]) 1);
   366 by (asm_simp_tac (!simpset addsimps [mod_less]) 1);
   367 qed "mod_Suc";
   368 
   369 goal Arith.thy "!!m n. 0<n ==> m mod n < n";
   370 by (res_inst_tac [("n","m")] less_induct 1);
   371 by (excluded_middle_tac "na<n" 1);
   372 (*case na<n*)
   373 by (asm_simp_tac (!simpset addsimps [mod_less]) 2);
   374 (*case n le na*)
   375 by (asm_full_simp_tac (!simpset addsimps [mod_geq, diff_less]) 1);
   376 qed "mod_less_divisor";
   377 
   378 
   379 (** Evens and Odds **)
   380 
   381 (*With less_zeroE, causes case analysis on b<2*)
   382 AddSEs [less_SucE];
   383 
   384 goal thy "!!k b. b<2 ==> k mod 2 = b | k mod 2 = (if b=1 then 0 else 1)";
   385 by (subgoal_tac "k mod 2 < 2" 1);
   386 by (asm_simp_tac (!simpset addsimps [mod_less_divisor]) 2);
   387 by (asm_simp_tac (!simpset setloop split_tac [expand_if]) 1);
   388 by (Fast_tac 1);
   389 qed "mod2_cases";
   390 
   391 goal thy "Suc(Suc(m)) mod 2 = m mod 2";
   392 by (subgoal_tac "m mod 2 < 2" 1);
   393 by (asm_simp_tac (!simpset addsimps [mod_less_divisor]) 2);
   394 by (Step_tac 1);
   395 by (ALLGOALS (asm_simp_tac (!simpset addsimps [mod_Suc])));
   396 qed "mod2_Suc_Suc";
   397 Addsimps [mod2_Suc_Suc];
   398 
   399 goal thy "(m+m) mod 2 = 0";
   400 by (nat_ind_tac "m" 1);
   401 by (simp_tac (!simpset addsimps [mod_less]) 1);
   402 by (asm_simp_tac (!simpset addsimps [mod2_Suc_Suc, add_Suc_right]) 1);
   403 qed "mod2_add_self";
   404 Addsimps [mod2_add_self];
   405 
   406 Delrules [less_SucE];
   407 
   408 
   409 (**** Additional theorems about "less than" ****)
   410 
   411 goal Arith.thy "? k::nat. n = n+k";
   412 by (res_inst_tac [("x","0")] exI 1);
   413 by (Simp_tac 1);
   414 val lemma = result();
   415 
   416 goal Arith.thy "!!m. m<n --> (? k. n=Suc(m+k))";
   417 by (nat_ind_tac "n" 1);
   418 by (ALLGOALS (simp_tac (!simpset addsimps [less_Suc_eq])));
   419 by (step_tac (!claset addSIs [lemma]) 1);
   420 by (res_inst_tac [("x","Suc(k)")] exI 1);
   421 by (Simp_tac 1);
   422 qed_spec_mp "less_eq_Suc_add";
   423 
   424 goal Arith.thy "n <= ((m + n)::nat)";
   425 by (nat_ind_tac "m" 1);
   426 by (ALLGOALS Simp_tac);
   427 by (etac le_trans 1);
   428 by (rtac (lessI RS less_imp_le) 1);
   429 qed "le_add2";
   430 
   431 goal Arith.thy "n <= ((n + m)::nat)";
   432 by (simp_tac (!simpset addsimps add_ac) 1);
   433 by (rtac le_add2 1);
   434 qed "le_add1";
   435 
   436 bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
   437 bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
   438 
   439 (*"i <= j ==> i <= j+m"*)
   440 bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
   441 
   442 (*"i <= j ==> i <= m+j"*)
   443 bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
   444 
   445 (*"i < j ==> i < j+m"*)
   446 bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
   447 
   448 (*"i < j ==> i < m+j"*)
   449 bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
   450 
   451 goal Arith.thy "!!i. i+j < (k::nat) ==> i<k";
   452 by (etac rev_mp 1);
   453 by (nat_ind_tac "j" 1);
   454 by (ALLGOALS Asm_simp_tac);
   455 by (fast_tac (!claset addDs [Suc_lessD]) 1);
   456 qed "add_lessD1";
   457 
   458 goal Arith.thy "!!k::nat. m <= n ==> m <= n+k";
   459 by (etac le_trans 1);
   460 by (rtac le_add1 1);
   461 qed "le_imp_add_le";
   462 
   463 goal Arith.thy "!!k::nat. m < n ==> m < n+k";
   464 by (etac less_le_trans 1);
   465 by (rtac le_add1 1);
   466 qed "less_imp_add_less";
   467 
   468 goal Arith.thy "m+k<=n --> m<=(n::nat)";
   469 by (nat_ind_tac "k" 1);
   470 by (ALLGOALS Asm_simp_tac);
   471 by (fast_tac (!claset addDs [Suc_leD]) 1);
   472 qed_spec_mp "add_leD1";
   473 
   474 goal Arith.thy "!!n::nat. m+k<=n ==> k<=n";
   475 by (full_simp_tac (!simpset addsimps [add_commute]) 1);
   476 by (etac add_leD1 1);
   477 qed_spec_mp "add_leD2";
   478 
   479 goal Arith.thy "!!n::nat. m+k<=n ==> m<=n & k<=n";
   480 by (fast_tac (!claset addDs [add_leD1, add_leD2]) 1);
   481 bind_thm ("add_leE", result() RS conjE);
   482 
   483 goal Arith.thy "!!k l::nat. [| k<l; m+l = k+n |] ==> m<n";
   484 by (safe_tac (!claset addSDs [less_eq_Suc_add]));
   485 by (asm_full_simp_tac
   486     (!simpset delsimps [add_Suc_right]
   487                 addsimps ([add_Suc_right RS sym, add_left_cancel] @add_ac)) 1);
   488 by (etac subst 1);
   489 by (simp_tac (!simpset addsimps [less_add_Suc1]) 1);
   490 qed "less_add_eq_less";
   491 
   492 
   493 (*** Monotonicity of Addition ***)
   494 
   495 (*strict, in 1st argument*)
   496 goal Arith.thy "!!i j k::nat. i < j ==> i + k < j + k";
   497 by (nat_ind_tac "k" 1);
   498 by (ALLGOALS Asm_simp_tac);
   499 qed "add_less_mono1";
   500 
   501 (*strict, in both arguments*)
   502 goal Arith.thy "!!i j k::nat. [|i < j; k < l|] ==> i + k < j + l";
   503 by (rtac (add_less_mono1 RS less_trans) 1);
   504 by (REPEAT (assume_tac 1));
   505 by (nat_ind_tac "j" 1);
   506 by (ALLGOALS Asm_simp_tac);
   507 qed "add_less_mono";
   508 
   509 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
   510 val [lt_mono,le] = goal Arith.thy
   511      "[| !!i j::nat. i<j ==> f(i) < f(j);       \
   512 \        i <= j                                 \
   513 \     |] ==> f(i) <= (f(j)::nat)";
   514 by (cut_facts_tac [le] 1);
   515 by (asm_full_simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
   516 by (fast_tac (!claset addSIs [lt_mono]) 1);
   517 qed "less_mono_imp_le_mono";
   518 
   519 (*non-strict, in 1st argument*)
   520 goal Arith.thy "!!i j k::nat. i<=j ==> i + k <= j + k";
   521 by (res_inst_tac [("f", "%j.j+k")] less_mono_imp_le_mono 1);
   522 by (etac add_less_mono1 1);
   523 by (assume_tac 1);
   524 qed "add_le_mono1";
   525 
   526 (*non-strict, in both arguments*)
   527 goal Arith.thy "!!k l::nat. [|i<=j;  k<=l |] ==> i + k <= j + l";
   528 by (etac (add_le_mono1 RS le_trans) 1);
   529 by (simp_tac (!simpset addsimps [add_commute]) 1);
   530 (*j moves to the end because it is free while k, l are bound*)
   531 by (etac add_le_mono1 1);
   532 qed "add_le_mono";
   533 
   534 (*** Monotonicity of Multiplication ***)
   535 
   536 goal Arith.thy "!!i::nat. i<=j ==> i*k<=j*k";
   537 by (nat_ind_tac "k" 1);
   538 by (ALLGOALS (asm_simp_tac (!simpset addsimps [add_le_mono])));
   539 qed "mult_le_mono1";
   540 
   541 (*<=monotonicity, BOTH arguments*)
   542 goal Arith.thy "!!i::nat. [| i<=j; k<=l |] ==> i*k<=j*l";
   543 by (etac (mult_le_mono1 RS le_trans) 1);
   544 by (rtac le_trans 1);
   545 by (stac mult_commute 2);
   546 by (etac mult_le_mono1 2);
   547 by (simp_tac (!simpset addsimps [mult_commute]) 1);
   548 qed "mult_le_mono";
   549 
   550 (*strict, in 1st argument; proof is by induction on k>0*)
   551 goal Arith.thy "!!i::nat. [| i<j; 0<k |] ==> k*i < k*j";
   552 by (etac zero_less_natE 1);
   553 by (Asm_simp_tac 1);
   554 by (nat_ind_tac "x" 1);
   555 by (ALLGOALS (asm_simp_tac (!simpset addsimps [add_less_mono])));
   556 qed "mult_less_mono2";
   557 
   558 goal Arith.thy "(0 < m*n) = (0<m & 0<n)";
   559 by (nat_ind_tac "m" 1);
   560 by (nat_ind_tac "n" 2);
   561 by (ALLGOALS Asm_simp_tac);
   562 qed "zero_less_mult_iff";
   563 
   564 goal Arith.thy "(m*n = 1) = (m=1 & n=1)";
   565 by (nat_ind_tac "m" 1);
   566 by (Simp_tac 1);
   567 by (nat_ind_tac "n" 1);
   568 by (Simp_tac 1);
   569 by (fast_tac (!claset addss !simpset) 1);
   570 qed "mult_eq_1_iff";
   571 
   572 (*Cancellation law for division*)
   573 goal Arith.thy "!!k. [| 0<n; 0<k |] ==> (k*m) div (k*n) = m div n";
   574 by (res_inst_tac [("n","m")] less_induct 1);
   575 by (case_tac "na<n" 1);
   576 by (asm_simp_tac (!simpset addsimps [div_less, zero_less_mult_iff, 
   577                                      mult_less_mono2]) 1);
   578 by (subgoal_tac "~ k*na < k*n" 1);
   579 by (asm_simp_tac
   580      (!simpset addsimps [zero_less_mult_iff, div_geq,
   581                          diff_mult_distrib2 RS sym, diff_less]) 1);
   582 by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, 
   583                                           le_refl RS mult_le_mono]) 1);
   584 qed "div_cancel";
   585 
   586 goal Arith.thy "!!k. [| 0<n; 0<k |] ==> (k*m) mod (k*n) = k * (m mod n)";
   587 by (res_inst_tac [("n","m")] less_induct 1);
   588 by (case_tac "na<n" 1);
   589 by (asm_simp_tac (!simpset addsimps [mod_less, zero_less_mult_iff, 
   590                                      mult_less_mono2]) 1);
   591 by (subgoal_tac "~ k*na < k*n" 1);
   592 by (asm_simp_tac
   593      (!simpset addsimps [zero_less_mult_iff, mod_geq,
   594                          diff_mult_distrib2 RS sym, diff_less]) 1);
   595 by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, 
   596                                           le_refl RS mult_le_mono]) 1);
   597 qed "mult_mod_distrib";
   598 
   599 
   600 (** Lemma for gcd **)
   601 
   602 goal Arith.thy "!!m n. m = m*n ==> n=1 | m=0";
   603 by (dtac sym 1);
   604 by (rtac disjCI 1);
   605 by (rtac nat_less_cases 1 THEN assume_tac 2);
   606 by (fast_tac (!claset addSEs [less_SucE] addss !simpset) 1);
   607 by (best_tac (!claset addDs [mult_less_mono2] 
   608                       addss (!simpset addsimps [zero_less_eq RS sym])) 1);
   609 qed "mult_eq_self_implies_10";
   610 
   611