src/HOL/Arith.ML
author paulson
Fri Sep 26 10:21:14 1997 +0200 (1997-09-26)
changeset 3718 d78cf498a88c
parent 3484 1e93eb09ebb9
child 3724 f33e301a89f5
permissions -rw-r--r--
Minor tidying to use Clarify_tac, etc.
     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 Some from the Hoare example from Norbert Galm
     8 *)
     9 
    10 (*** Basic rewrite rules for the arithmetic operators ***)
    11 
    12 goalw Arith.thy [pred_def] "pred 0 = 0";
    13 by (Simp_tac 1);
    14 qed "pred_0";
    15 
    16 goalw Arith.thy [pred_def] "pred(Suc n) = n";
    17 by (Simp_tac 1);
    18 qed "pred_Suc";
    19 
    20 Addsimps [pred_0,pred_Suc];
    21 
    22 (** pred **)
    23 
    24 val prems = goal Arith.thy "n ~= 0 ==> Suc(pred n) = n";
    25 by (res_inst_tac [("n","n")] natE 1);
    26 by (cut_facts_tac prems 1);
    27 by (ALLGOALS Asm_full_simp_tac);
    28 qed "Suc_pred";
    29 Addsimps [Suc_pred];
    30 
    31 goal Arith.thy "pred(n) <= (n::nat)";
    32 by (res_inst_tac [("n","n")] natE 1);
    33 by (ALLGOALS Asm_simp_tac);
    34 qed "pred_le";
    35 AddIffs [pred_le];
    36 
    37 goalw Arith.thy [pred_def] "m<=n --> pred(m) <= pred(n)";
    38 by(simp_tac (!simpset setloop (split_tac[expand_nat_case])) 1);
    39 qed_spec_mp "pred_le_mono";
    40 
    41 (** Difference **)
    42 
    43 qed_goalw "diff_0_eq_0" Arith.thy [pred_def]
    44     "0 - n = 0"
    45  (fn _ => [induct_tac "n" 1,  ALLGOALS Asm_simp_tac]);
    46 
    47 (*Must simplify BEFORE the induction!!  (Else we get a critical pair)
    48   Suc(m) - Suc(n)   rewrites to   pred(Suc(m) - n)  *)
    49 qed_goalw "diff_Suc_Suc" Arith.thy [pred_def]
    50     "Suc(m) - Suc(n) = m - n"
    51  (fn _ =>
    52   [Simp_tac 1, induct_tac "n" 1, ALLGOALS Asm_simp_tac]);
    53 
    54 Addsimps [diff_0_eq_0, diff_Suc_Suc];
    55 
    56 
    57 (**** Inductive properties of the operators ****)
    58 
    59 (*** Addition ***)
    60 
    61 qed_goal "add_0_right" Arith.thy "m + 0 = m"
    62  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
    63 
    64 qed_goal "add_Suc_right" Arith.thy "m + Suc(n) = Suc(m+n)"
    65  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
    66 
    67 Addsimps [add_0_right,add_Suc_right];
    68 
    69 (*Associative law for addition*)
    70 qed_goal "add_assoc" Arith.thy "(m + n) + k = m + ((n + k)::nat)"
    71  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
    72 
    73 (*Commutative law for addition*)  
    74 qed_goal "add_commute" Arith.thy "m + n = n + (m::nat)"
    75  (fn _ =>  [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
    76 
    77 qed_goal "add_left_commute" Arith.thy "x+(y+z)=y+((x+z)::nat)"
    78  (fn _ => [rtac (add_commute RS trans) 1, rtac (add_assoc RS trans) 1,
    79            rtac (add_commute RS arg_cong) 1]);
    80 
    81 (*Addition is an AC-operator*)
    82 val add_ac = [add_assoc, add_commute, add_left_commute];
    83 
    84 goal Arith.thy "!!k::nat. (k + m = k + n) = (m=n)";
    85 by (induct_tac "k" 1);
    86 by (Simp_tac 1);
    87 by (Asm_simp_tac 1);
    88 qed "add_left_cancel";
    89 
    90 goal Arith.thy "!!k::nat. (m + k = n + k) = (m=n)";
    91 by (induct_tac "k" 1);
    92 by (Simp_tac 1);
    93 by (Asm_simp_tac 1);
    94 qed "add_right_cancel";
    95 
    96 goal Arith.thy "!!k::nat. (k + m <= k + n) = (m<=n)";
    97 by (induct_tac "k" 1);
    98 by (Simp_tac 1);
    99 by (Asm_simp_tac 1);
   100 qed "add_left_cancel_le";
   101 
   102 goal Arith.thy "!!k::nat. (k + m < k + n) = (m<n)";
   103 by (induct_tac "k" 1);
   104 by (Simp_tac 1);
   105 by (Asm_simp_tac 1);
   106 qed "add_left_cancel_less";
   107 
   108 Addsimps [add_left_cancel, add_right_cancel,
   109           add_left_cancel_le, add_left_cancel_less];
   110 
   111 (** Reasoning about m+0=0, etc. **)
   112 
   113 goal Arith.thy "(m+n = 0) = (m=0 & n=0)";
   114 by (induct_tac "m" 1);
   115 by (ALLGOALS Asm_simp_tac);
   116 qed "add_is_0";
   117 Addsimps [add_is_0];
   118 
   119 goal Arith.thy "(pred (m+n) = 0) = (m=0 & pred n = 0 | pred m = 0 & n=0)";
   120 by (induct_tac "m" 1);
   121 by (ALLGOALS (fast_tac (!claset addss (!simpset))));
   122 qed "pred_add_is_0";
   123 Addsimps [pred_add_is_0];
   124 
   125 goal Arith.thy "!!n. n ~= 0 ==> m + pred n = pred(m+n)";
   126 by (induct_tac "m" 1);
   127 by (ALLGOALS Asm_simp_tac);
   128 qed "add_pred";
   129 Addsimps [add_pred];
   130 
   131 
   132 (**** Additional theorems about "less than" ****)
   133 
   134 goal Arith.thy "i<j --> (EX k. j = Suc(i+k))";
   135 by (induct_tac "j" 1);
   136 by (Simp_tac 1);
   137 by (blast_tac (!claset addSEs [less_SucE] 
   138                        addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1);
   139 val lemma = result();
   140 
   141 (* [| i<j;  !!x. j = Suc(i+x) ==> Q |] ==> Q *)
   142 bind_thm ("less_natE", lemma RS mp RS exE);
   143 
   144 goal Arith.thy "!!m. m<n --> (? k. n=Suc(m+k))";
   145 by (induct_tac "n" 1);
   146 by (ALLGOALS (simp_tac (!simpset addsimps [less_Suc_eq])));
   147 by (blast_tac (!claset addSEs [less_SucE] 
   148                        addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1);
   149 qed_spec_mp "less_eq_Suc_add";
   150 
   151 goal Arith.thy "n <= ((m + n)::nat)";
   152 by (induct_tac "m" 1);
   153 by (ALLGOALS Simp_tac);
   154 by (etac le_trans 1);
   155 by (rtac (lessI RS less_imp_le) 1);
   156 qed "le_add2";
   157 
   158 goal Arith.thy "n <= ((n + m)::nat)";
   159 by (simp_tac (!simpset addsimps add_ac) 1);
   160 by (rtac le_add2 1);
   161 qed "le_add1";
   162 
   163 bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
   164 bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
   165 
   166 (*"i <= j ==> i <= j+m"*)
   167 bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
   168 
   169 (*"i <= j ==> i <= m+j"*)
   170 bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
   171 
   172 (*"i < j ==> i < j+m"*)
   173 bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
   174 
   175 (*"i < j ==> i < m+j"*)
   176 bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
   177 
   178 goal Arith.thy "!!i. i+j < (k::nat) ==> i<k";
   179 by (etac rev_mp 1);
   180 by (induct_tac "j" 1);
   181 by (ALLGOALS Asm_simp_tac);
   182 by (blast_tac (!claset addDs [Suc_lessD]) 1);
   183 qed "add_lessD1";
   184 
   185 goal Arith.thy "!!i::nat. ~ (i+j < i)";
   186 by (rtac notI 1);
   187 by (etac (add_lessD1 RS less_irrefl) 1);
   188 qed "not_add_less1";
   189 
   190 goal Arith.thy "!!i::nat. ~ (j+i < i)";
   191 by (simp_tac (!simpset addsimps [add_commute, not_add_less1]) 1);
   192 qed "not_add_less2";
   193 AddIffs [not_add_less1, not_add_less2];
   194 
   195 goal Arith.thy "!!k::nat. m <= n ==> m <= n+k";
   196 by (etac le_trans 1);
   197 by (rtac le_add1 1);
   198 qed "le_imp_add_le";
   199 
   200 goal Arith.thy "!!k::nat. m < n ==> m < n+k";
   201 by (etac less_le_trans 1);
   202 by (rtac le_add1 1);
   203 qed "less_imp_add_less";
   204 
   205 goal Arith.thy "m+k<=n --> m<=(n::nat)";
   206 by (induct_tac "k" 1);
   207 by (ALLGOALS Asm_simp_tac);
   208 by (blast_tac (!claset addDs [Suc_leD]) 1);
   209 qed_spec_mp "add_leD1";
   210 
   211 goal Arith.thy "!!n::nat. m+k<=n ==> k<=n";
   212 by (full_simp_tac (!simpset addsimps [add_commute]) 1);
   213 by (etac add_leD1 1);
   214 qed_spec_mp "add_leD2";
   215 
   216 goal Arith.thy "!!n::nat. m+k<=n ==> m<=n & k<=n";
   217 by (blast_tac (!claset addDs [add_leD1, add_leD2]) 1);
   218 bind_thm ("add_leE", result() RS conjE);
   219 
   220 goal Arith.thy "!!k l::nat. [| k<l; m+l = k+n |] ==> m<n";
   221 by (safe_tac (!claset addSDs [less_eq_Suc_add]));
   222 by (asm_full_simp_tac
   223     (!simpset delsimps [add_Suc_right]
   224                 addsimps ([add_Suc_right RS sym, add_left_cancel] @add_ac)) 1);
   225 by (etac subst 1);
   226 by (simp_tac (!simpset addsimps [less_add_Suc1]) 1);
   227 qed "less_add_eq_less";
   228 
   229 
   230 (*** Monotonicity of Addition ***)
   231 
   232 (*strict, in 1st argument*)
   233 goal Arith.thy "!!i j k::nat. i < j ==> i + k < j + k";
   234 by (induct_tac "k" 1);
   235 by (ALLGOALS Asm_simp_tac);
   236 qed "add_less_mono1";
   237 
   238 (*strict, in both arguments*)
   239 goal Arith.thy "!!i j k::nat. [|i < j; k < l|] ==> i + k < j + l";
   240 by (rtac (add_less_mono1 RS less_trans) 1);
   241 by (REPEAT (assume_tac 1));
   242 by (induct_tac "j" 1);
   243 by (ALLGOALS Asm_simp_tac);
   244 qed "add_less_mono";
   245 
   246 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
   247 val [lt_mono,le] = goal Arith.thy
   248      "[| !!i j::nat. i<j ==> f(i) < f(j);       \
   249 \        i <= j                                 \
   250 \     |] ==> f(i) <= (f(j)::nat)";
   251 by (cut_facts_tac [le] 1);
   252 by (asm_full_simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
   253 by (blast_tac (!claset addSIs [lt_mono]) 1);
   254 qed "less_mono_imp_le_mono";
   255 
   256 (*non-strict, in 1st argument*)
   257 goal Arith.thy "!!i j k::nat. i<=j ==> i + k <= j + k";
   258 by (res_inst_tac [("f", "%j.j+k")] less_mono_imp_le_mono 1);
   259 by (etac add_less_mono1 1);
   260 by (assume_tac 1);
   261 qed "add_le_mono1";
   262 
   263 (*non-strict, in both arguments*)
   264 goal Arith.thy "!!k l::nat. [|i<=j;  k<=l |] ==> i + k <= j + l";
   265 by (etac (add_le_mono1 RS le_trans) 1);
   266 by (simp_tac (!simpset addsimps [add_commute]) 1);
   267 (*j moves to the end because it is free while k, l are bound*)
   268 by (etac add_le_mono1 1);
   269 qed "add_le_mono";
   270 
   271 
   272 (*** Multiplication ***)
   273 
   274 (*right annihilation in product*)
   275 qed_goal "mult_0_right" Arith.thy "m * 0 = 0"
   276  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
   277 
   278 (*right successor law for multiplication*)
   279 qed_goal "mult_Suc_right" Arith.thy  "m * Suc(n) = m + (m * n)"
   280  (fn _ => [induct_tac "m" 1,
   281            ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
   282 
   283 Addsimps [mult_0_right, mult_Suc_right];
   284 
   285 goal Arith.thy "1 * n = n";
   286 by (Asm_simp_tac 1);
   287 qed "mult_1";
   288 
   289 goal Arith.thy "n * 1 = n";
   290 by (Asm_simp_tac 1);
   291 qed "mult_1_right";
   292 
   293 (*Commutative law for multiplication*)
   294 qed_goal "mult_commute" Arith.thy "m * n = n * (m::nat)"
   295  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
   296 
   297 (*addition distributes over multiplication*)
   298 qed_goal "add_mult_distrib" Arith.thy "(m + n)*k = (m*k) + ((n*k)::nat)"
   299  (fn _ => [induct_tac "m" 1,
   300            ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
   301 
   302 qed_goal "add_mult_distrib2" Arith.thy "k*(m + n) = (k*m) + ((k*n)::nat)"
   303  (fn _ => [induct_tac "m" 1,
   304            ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
   305 
   306 (*Associative law for multiplication*)
   307 qed_goal "mult_assoc" Arith.thy "(m * n) * k = m * ((n * k)::nat)"
   308   (fn _ => [induct_tac "m" 1, 
   309             ALLGOALS (asm_simp_tac (!simpset addsimps [add_mult_distrib]))]);
   310 
   311 qed_goal "mult_left_commute" Arith.thy "x*(y*z) = y*((x*z)::nat)"
   312  (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1,
   313            rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]);
   314 
   315 val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
   316 
   317 goal Arith.thy "(m*n = 0) = (m=0 | n=0)";
   318 by (induct_tac "m" 1);
   319 by (induct_tac "n" 2);
   320 by (ALLGOALS Asm_simp_tac);
   321 qed "mult_is_0";
   322 Addsimps [mult_is_0];
   323 
   324 
   325 (*** Difference ***)
   326 
   327 qed_goal "pred_Suc_diff" Arith.thy "pred(Suc m - n) = m - n"
   328  (fn _ => [induct_tac "n" 1, ALLGOALS Asm_simp_tac]);
   329 Addsimps [pred_Suc_diff];
   330 
   331 qed_goal "diff_self_eq_0" Arith.thy "m - m = 0"
   332  (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
   333 Addsimps [diff_self_eq_0];
   334 
   335 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
   336 goal Arith.thy "~ m<n --> n+(m-n) = (m::nat)";
   337 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   338 by (ALLGOALS Asm_simp_tac);
   339 qed_spec_mp "add_diff_inverse";
   340 
   341 goal Arith.thy "!!m. n<=m ==> n+(m-n) = (m::nat)";
   342 by (asm_simp_tac (!simpset addsimps [add_diff_inverse, not_less_iff_le]) 1);
   343 qed "le_add_diff_inverse";
   344 
   345 goal Arith.thy "!!m. n<=m ==> (m-n)+n = (m::nat)";
   346 by (asm_simp_tac (!simpset addsimps [le_add_diff_inverse, add_commute]) 1);
   347 qed "le_add_diff_inverse2";
   348 
   349 Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];
   350 Delsimps  [diff_Suc];
   351 
   352 
   353 (*** More results about difference ***)
   354 
   355 val [prem] = goal Arith.thy "n < Suc(m) ==> Suc(m)-n = Suc(m-n)";
   356 by (rtac (prem RS rev_mp) 1);
   357 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   358 by (ALLGOALS Asm_simp_tac);
   359 qed "Suc_diff_n";
   360 
   361 goal Arith.thy "m - n < Suc(m)";
   362 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   363 by (etac less_SucE 3);
   364 by (ALLGOALS (asm_simp_tac (!simpset addsimps [less_Suc_eq])));
   365 qed "diff_less_Suc";
   366 
   367 goal Arith.thy "!!m::nat. m - n <= m";
   368 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
   369 by (ALLGOALS Asm_simp_tac);
   370 qed "diff_le_self";
   371 
   372 goal Arith.thy "!!i::nat. i-j-k = i - (j+k)";
   373 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
   374 by (ALLGOALS Asm_simp_tac);
   375 qed "diff_diff_left";
   376 
   377 (*This and the next few suggested by Florian Kammueller*)
   378 goal Arith.thy "!!i::nat. i-j-k = i-k-j";
   379 by (simp_tac (!simpset addsimps [diff_diff_left, add_commute]) 1);
   380 qed "diff_commute";
   381 
   382 goal Arith.thy "!!i j k:: nat. k<=j --> j<=i --> i - (j - k) = i - j + k";
   383 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
   384 by (ALLGOALS Asm_simp_tac);
   385 by (asm_simp_tac
   386     (!simpset addsimps [Suc_diff_n, le_imp_less_Suc, le_Suc_eq]) 1);
   387 qed_spec_mp "diff_diff_right";
   388 
   389 goal Arith.thy "!!i j k:: nat. k<=j --> (i + j) - k = i + (j - k)";
   390 by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
   391 by (ALLGOALS Asm_simp_tac);
   392 qed_spec_mp "diff_add_assoc";
   393 
   394 goal Arith.thy "!!n::nat. (n+m) - n = m";
   395 by (induct_tac "n" 1);
   396 by (ALLGOALS Asm_simp_tac);
   397 qed "diff_add_inverse";
   398 Addsimps [diff_add_inverse];
   399 
   400 goal Arith.thy "!!n::nat.(m+n) - n = m";
   401 by (simp_tac (!simpset addsimps [diff_add_assoc]) 1);
   402 qed "diff_add_inverse2";
   403 Addsimps [diff_add_inverse2];
   404 
   405 goal Arith.thy "!!i j k::nat. i<=j ==> (j-i=k) = (j=k+i)";
   406 by (Step_tac 1);
   407 by (ALLGOALS Asm_simp_tac);
   408 qed "le_imp_diff_is_add";
   409 
   410 val [prem] = goal Arith.thy "m < Suc(n) ==> m-n = 0";
   411 by (rtac (prem RS rev_mp) 1);
   412 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   413 by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
   414 by (ALLGOALS Asm_simp_tac);
   415 qed "less_imp_diff_is_0";
   416 
   417 val prems = goal Arith.thy "m-n = 0  -->  n-m = 0  -->  m=n";
   418 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   419 by (REPEAT(Simp_tac 1 THEN TRY(atac 1)));
   420 qed_spec_mp "diffs0_imp_equal";
   421 
   422 val [prem] = goal Arith.thy "m<n ==> 0<n-m";
   423 by (rtac (prem RS rev_mp) 1);
   424 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   425 by (ALLGOALS Asm_simp_tac);
   426 qed "less_imp_diff_positive";
   427 
   428 goal Arith.thy "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
   429 by (simp_tac (!simpset addsimps [less_imp_diff_is_0, not_less_eq, Suc_diff_n]
   430                     setloop (split_tac [expand_if])) 1);
   431 qed "if_Suc_diff_n";
   432 
   433 goal Arith.thy "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
   434 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
   435 by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
   436 qed "zero_induct_lemma";
   437 
   438 val prems = goal Arith.thy "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
   439 by (rtac (diff_self_eq_0 RS subst) 1);
   440 by (rtac (zero_induct_lemma RS mp RS mp) 1);
   441 by (REPEAT (ares_tac ([impI,allI]@prems) 1));
   442 qed "zero_induct";
   443 
   444 goal Arith.thy "!!k::nat. (k+m) - (k+n) = m - n";
   445 by (induct_tac "k" 1);
   446 by (ALLGOALS Asm_simp_tac);
   447 qed "diff_cancel";
   448 Addsimps [diff_cancel];
   449 
   450 goal Arith.thy "!!m::nat. (m+k) - (n+k) = m - n";
   451 val add_commute_k = read_instantiate [("n","k")] add_commute;
   452 by (asm_simp_tac (!simpset addsimps ([add_commute_k])) 1);
   453 qed "diff_cancel2";
   454 Addsimps [diff_cancel2];
   455 
   456 (*From Clemens Ballarin*)
   457 goal Arith.thy "!!n::nat. [| k<=n; n<=m |] ==> (m-k) - (n-k) = m-n";
   458 by (subgoal_tac "k<=n --> n<=m --> (m-k) - (n-k) = m-n" 1);
   459 by (Asm_full_simp_tac 1);
   460 by (induct_tac "k" 1);
   461 by (Simp_tac 1);
   462 (* Induction step *)
   463 by (subgoal_tac "Suc na <= m --> n <= m --> Suc na <= n --> \
   464 \                Suc (m - Suc na) - Suc (n - Suc na) = m-n" 1);
   465 by (Asm_full_simp_tac 1);
   466 by (blast_tac (!claset addIs [le_trans]) 1);
   467 by (auto_tac (!claset addIs [Suc_leD], !simpset delsimps [diff_Suc_Suc]));
   468 by (asm_full_simp_tac (!simpset delsimps [Suc_less_eq] 
   469 		       addsimps [Suc_diff_n RS sym, le_eq_less_Suc]) 1);
   470 qed "diff_right_cancel";
   471 
   472 goal Arith.thy "!!n::nat. n - (n+m) = 0";
   473 by (induct_tac "n" 1);
   474 by (ALLGOALS Asm_simp_tac);
   475 qed "diff_add_0";
   476 Addsimps [diff_add_0];
   477 
   478 (** Difference distributes over multiplication **)
   479 
   480 goal Arith.thy "!!m::nat. (m - n) * k = (m * k) - (n * k)";
   481 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   482 by (ALLGOALS Asm_simp_tac);
   483 qed "diff_mult_distrib" ;
   484 
   485 goal Arith.thy "!!m::nat. k * (m - n) = (k * m) - (k * n)";
   486 val mult_commute_k = read_instantiate [("m","k")] mult_commute;
   487 by (simp_tac (!simpset addsimps [diff_mult_distrib, mult_commute_k]) 1);
   488 qed "diff_mult_distrib2" ;
   489 (*NOT added as rewrites, since sometimes they are used from right-to-left*)
   490 
   491 
   492 (*** Monotonicity of Multiplication ***)
   493 
   494 goal Arith.thy "!!i::nat. i<=j ==> i*k<=j*k";
   495 by (induct_tac "k" 1);
   496 by (ALLGOALS (asm_simp_tac (!simpset addsimps [add_le_mono])));
   497 qed "mult_le_mono1";
   498 
   499 (*<=monotonicity, BOTH arguments*)
   500 goal Arith.thy "!!i::nat. [| i<=j; k<=l |] ==> i*k<=j*l";
   501 by (etac (mult_le_mono1 RS le_trans) 1);
   502 by (rtac le_trans 1);
   503 by (stac mult_commute 2);
   504 by (etac mult_le_mono1 2);
   505 by (simp_tac (!simpset addsimps [mult_commute]) 1);
   506 qed "mult_le_mono";
   507 
   508 (*strict, in 1st argument; proof is by induction on k>0*)
   509 goal Arith.thy "!!i::nat. [| i<j; 0<k |] ==> k*i < k*j";
   510 by (eres_inst_tac [("i","0")] less_natE 1);
   511 by (Asm_simp_tac 1);
   512 by (induct_tac "x" 1);
   513 by (ALLGOALS (asm_simp_tac (!simpset addsimps [add_less_mono])));
   514 qed "mult_less_mono2";
   515 
   516 goal Arith.thy "!!i::nat. [| i<j; 0<k |] ==> i*k < j*k";
   517 by (dtac mult_less_mono2 1);
   518 by (ALLGOALS (asm_full_simp_tac (!simpset addsimps [mult_commute])));
   519 qed "mult_less_mono1";
   520 
   521 goal Arith.thy "(0 < m*n) = (0<m & 0<n)";
   522 by (induct_tac "m" 1);
   523 by (induct_tac "n" 2);
   524 by (ALLGOALS Asm_simp_tac);
   525 qed "zero_less_mult_iff";
   526 
   527 goal Arith.thy "(m*n = 1) = (m=1 & n=1)";
   528 by (induct_tac "m" 1);
   529 by (Simp_tac 1);
   530 by (induct_tac "n" 1);
   531 by (Simp_tac 1);
   532 by (fast_tac (!claset addss !simpset) 1);
   533 qed "mult_eq_1_iff";
   534 
   535 goal Arith.thy "!!k. 0<k ==> (m*k < n*k) = (m<n)";
   536 by (safe_tac (!claset addSIs [mult_less_mono1]));
   537 by (cut_facts_tac [less_linear] 1);
   538 by (blast_tac (!claset addDs [mult_less_mono1] addEs [less_asym]) 1);
   539 qed "mult_less_cancel2";
   540 
   541 goal Arith.thy "!!k. 0<k ==> (k*m < k*n) = (m<n)";
   542 by (dtac mult_less_cancel2 1);
   543 by (asm_full_simp_tac (!simpset addsimps [mult_commute]) 1);
   544 qed "mult_less_cancel1";
   545 Addsimps [mult_less_cancel1, mult_less_cancel2];
   546 
   547 goal Arith.thy "!!k. 0<k ==> (m*k = n*k) = (m=n)";
   548 by (cut_facts_tac [less_linear] 1);
   549 by (Step_tac 1);
   550 by (assume_tac 2);
   551 by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
   552 by (ALLGOALS Asm_full_simp_tac);
   553 qed "mult_cancel2";
   554 
   555 goal Arith.thy "!!k. 0<k ==> (k*m = k*n) = (m=n)";
   556 by (dtac mult_cancel2 1);
   557 by (asm_full_simp_tac (!simpset addsimps [mult_commute]) 1);
   558 qed "mult_cancel1";
   559 Addsimps [mult_cancel1, mult_cancel2];
   560 
   561 
   562 (** Lemma for gcd **)
   563 
   564 goal Arith.thy "!!m n. m = m*n ==> n=1 | m=0";
   565 by (dtac sym 1);
   566 by (rtac disjCI 1);
   567 by (rtac nat_less_cases 1 THEN assume_tac 2);
   568 by (fast_tac (!claset addSEs [less_SucE] addss !simpset) 1);
   569 by (best_tac (!claset addDs [mult_less_mono2] 
   570                       addss (!simpset addsimps [zero_less_eq RS sym])) 1);
   571 qed "mult_eq_self_implies_10";
   572 
   573 
   574 (*** Subtraction laws -- from Clemens Ballarin ***)
   575 
   576 goal Arith.thy "!! a b c::nat. [| a < b; c <= a |] ==> a-c < b-c";
   577 by (subgoal_tac "c+(a-c) < c+(b-c)" 1);
   578 by (Full_simp_tac 1);
   579 by (subgoal_tac "c <= b" 1);
   580 by (blast_tac (!claset addIs [less_imp_le, le_trans]) 2);
   581 by (Asm_simp_tac 1);
   582 qed "diff_less_mono";
   583 
   584 goal Arith.thy "!! a b c::nat. a+b < c ==> a < c-b";
   585 by (dtac diff_less_mono 1);
   586 by (rtac le_add2 1);
   587 by (Asm_full_simp_tac 1);
   588 qed "add_less_imp_less_diff";
   589 
   590 goal Arith.thy "!! n. n <= m ==> Suc m - n = Suc (m - n)";
   591 by (rtac Suc_diff_n 1);
   592 by (asm_full_simp_tac (!simpset addsimps [le_eq_less_Suc]) 1);
   593 qed "Suc_diff_le";
   594 
   595 goal Arith.thy "!! n. Suc i <= n ==> Suc (n - Suc i) = n - i";
   596 by (asm_full_simp_tac
   597     (!simpset addsimps [Suc_diff_n RS sym, le_eq_less_Suc]) 1);
   598 qed "Suc_diff_Suc";
   599 
   600 goal Arith.thy "!! i::nat. i <= n ==> n - (n - i) = i";
   601 by (subgoal_tac "(n-i) + (n - (n-i)) = (n-i) + i" 1);
   602 by (Full_simp_tac 1);
   603 by (asm_simp_tac (!simpset addsimps [diff_le_self, add_commute]) 1);
   604 qed "diff_diff_cancel";
   605 Addsimps [diff_diff_cancel];
   606 
   607 goal Arith.thy "!!k::nat. k <= n ==> m <= n + m - k";
   608 by (etac rev_mp 1);
   609 by (res_inst_tac [("m", "k"), ("n", "n")] diff_induct 1);
   610 by (Simp_tac 1);
   611 by (simp_tac (!simpset addsimps [less_add_Suc2, less_imp_le]) 1);
   612 by (Simp_tac 1);
   613 qed "le_add_diff";
   614 
   615 
   616 (** (Anti)Monotonicity of subtraction -- by Stefan Merz **)
   617 
   618 (* Monotonicity of subtraction in first argument *)
   619 goal Arith.thy "!!n::nat. m<=n --> (m-l) <= (n-l)";
   620 by (induct_tac "n" 1);
   621 by (Simp_tac 1);
   622 by (simp_tac (!simpset addsimps [le_Suc_eq]) 1);
   623 by (rtac impI 1);
   624 by (etac impE 1);
   625 by (atac 1);
   626 by (etac le_trans 1);
   627 by (res_inst_tac [("m1","n")] (pred_Suc_diff RS subst) 1);
   628 by (rtac pred_le 1);
   629 qed_spec_mp "diff_le_mono";
   630 
   631 goal Arith.thy "!!n::nat. m<=n ==> (l-n) <= (l-m)";
   632 by (induct_tac "l" 1);
   633 by (Simp_tac 1);
   634 by (case_tac "n <= l" 1);
   635 by (subgoal_tac "m <= l" 1);
   636 by (asm_simp_tac (!simpset addsimps [Suc_diff_le]) 1);
   637 by (fast_tac (!claset addEs [le_trans]) 1);
   638 by (dtac not_leE 1);
   639 by (asm_simp_tac (!simpset addsimps [if_Suc_diff_n]) 1);
   640 qed_spec_mp "diff_le_mono2";