src/HOL/Arith.ML
 author wenzelm Thu Jan 23 14:19:16 1997 +0100 (1997-01-23) changeset 2545 d10abc8c11fb parent 2498 7914881f47c0 child 2682 13cdbf95ed92 permissions -rw-r--r--
```     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 val [add_0,add_Suc] = nat_recs add_def;
```
```    23 val [mult_0,mult_Suc] = nat_recs mult_def;
```
```    24 store_thm("add_0",add_0);
```
```    25 store_thm("add_Suc",add_Suc);
```
```    26 store_thm("mult_0",mult_0);
```
```    27 store_thm("mult_Suc",mult_Suc);
```
```    28 Addsimps [pred_0,pred_Suc,add_0,add_Suc,mult_0,mult_Suc];
```
```    29
```
```    30 (** pred **)
```
```    31
```
```    32 val prems = goal Arith.thy "n ~= 0 ==> Suc(pred n) = n";
```
```    33 by (res_inst_tac [("n","n")] natE 1);
```
```    34 by (cut_facts_tac prems 1);
```
```    35 by (ALLGOALS Asm_full_simp_tac);
```
```    36 qed "Suc_pred";
```
```    37 Addsimps [Suc_pred];
```
```    38
```
```    39 (** Difference **)
```
```    40
```
```    41 bind_thm("diff_0", diff_def RS def_nat_rec_0);
```
```    42
```
```    43 qed_goalw "diff_0_eq_0" Arith.thy [diff_def, pred_def]
```
```    44     "0 - n = 0"
```
```    45  (fn _ => [nat_ind_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 [diff_def, pred_def]
```
```    50     "Suc(m) - Suc(n) = m - n"
```
```    51  (fn _ =>
```
```    52   [Simp_tac 1, nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
```
```    53
```
```    54 Addsimps [diff_0, diff_0_eq_0, diff_Suc_Suc];
```
```    55
```
```    56
```
```    57 goal Arith.thy "!!k. 0<k ==> EX j. k = Suc(j)";
```
```    58 by (etac rev_mp 1);
```
```    59 by (nat_ind_tac "k" 1);
```
```    60 by (Simp_tac 1);
```
```    61 by (Fast_tac 1);
```
```    62 val lemma = result();
```
```    63
```
```    64 (* [| 0 < k; !!j. [| j: nat; k = succ(j) |] ==> Q |] ==> Q *)
```
```    65 bind_thm ("zero_less_natE", lemma RS exE);
```
```    66
```
```    67
```
```    68
```
```    69 (**** Inductive properties of the operators ****)
```
```    70
```
```    71 (*** Addition ***)
```
```    72
```
```    73 qed_goal "add_0_right" Arith.thy "m + 0 = m"
```
```    74  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    75
```
```    76 qed_goal "add_Suc_right" Arith.thy "m + Suc(n) = Suc(m+n)"
```
```    77  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    78
```
```    79 Addsimps [add_0_right,add_Suc_right];
```
```    80
```
```    81 (*Associative law for addition*)
```
```    82 qed_goal "add_assoc" Arith.thy "(m + n) + k = m + ((n + k)::nat)"
```
```    83  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    84
```
```    85 (*Commutative law for addition*)
```
```    86 qed_goal "add_commute" Arith.thy "m + n = n + (m::nat)"
```
```    87  (fn _ =>  [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```    88
```
```    89 qed_goal "add_left_commute" Arith.thy "x+(y+z)=y+((x+z)::nat)"
```
```    90  (fn _ => [rtac (add_commute RS trans) 1, rtac (add_assoc RS trans) 1,
```
```    91            rtac (add_commute RS arg_cong) 1]);
```
```    92
```
```    93 (*Addition is an AC-operator*)
```
```    94 val add_ac = [add_assoc, add_commute, add_left_commute];
```
```    95
```
```    96 goal Arith.thy "!!k::nat. (k + m = k + n) = (m=n)";
```
```    97 by (nat_ind_tac "k" 1);
```
```    98 by (Simp_tac 1);
```
```    99 by (Asm_simp_tac 1);
```
```   100 qed "add_left_cancel";
```
```   101
```
```   102 goal Arith.thy "!!k::nat. (m + k = n + k) = (m=n)";
```
```   103 by (nat_ind_tac "k" 1);
```
```   104 by (Simp_tac 1);
```
```   105 by (Asm_simp_tac 1);
```
```   106 qed "add_right_cancel";
```
```   107
```
```   108 goal Arith.thy "!!k::nat. (k + m <= k + n) = (m<=n)";
```
```   109 by (nat_ind_tac "k" 1);
```
```   110 by (Simp_tac 1);
```
```   111 by (Asm_simp_tac 1);
```
```   112 qed "add_left_cancel_le";
```
```   113
```
```   114 goal Arith.thy "!!k::nat. (k + m < k + n) = (m<n)";
```
```   115 by (nat_ind_tac "k" 1);
```
```   116 by (Simp_tac 1);
```
```   117 by (Asm_simp_tac 1);
```
```   118 qed "add_left_cancel_less";
```
```   119
```
```   120 Addsimps [add_left_cancel, add_right_cancel,
```
```   121           add_left_cancel_le, add_left_cancel_less];
```
```   122
```
```   123 goal Arith.thy "(m+n = 0) = (m=0 & n=0)";
```
```   124 by (nat_ind_tac "m" 1);
```
```   125 by (ALLGOALS Asm_simp_tac);
```
```   126 qed "add_is_0";
```
```   127 Addsimps [add_is_0];
```
```   128
```
```   129 goal Arith.thy "!!n. n ~= 0 ==> m + pred n = pred(m+n)";
```
```   130 by (nat_ind_tac "m" 1);
```
```   131 by (ALLGOALS Asm_simp_tac);
```
```   132 qed "add_pred";
```
```   133 Addsimps [add_pred];
```
```   134
```
```   135 (*** Multiplication ***)
```
```   136
```
```   137 (*right annihilation in product*)
```
```   138 qed_goal "mult_0_right" Arith.thy "m * 0 = 0"
```
```   139  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```   140
```
```   141 (*right Sucessor law for multiplication*)
```
```   142 qed_goal "mult_Suc_right" Arith.thy  "m * Suc(n) = m + (m * n)"
```
```   143  (fn _ => [nat_ind_tac "m" 1,
```
```   144            ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
```
```   145
```
```   146 Addsimps [mult_0_right,mult_Suc_right];
```
```   147
```
```   148 goal Arith.thy "1 * n = n";
```
```   149 by (Asm_simp_tac 1);
```
```   150 qed "mult_1";
```
```   151
```
```   152 goal Arith.thy "n * 1 = n";
```
```   153 by (Asm_simp_tac 1);
```
```   154 qed "mult_1_right";
```
```   155
```
```   156 (*Commutative law for multiplication*)
```
```   157 qed_goal "mult_commute" Arith.thy "m * n = n * (m::nat)"
```
```   158  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```   159
```
```   160 (*addition distributes over multiplication*)
```
```   161 qed_goal "add_mult_distrib" Arith.thy "(m + n)*k = (m*k) + ((n*k)::nat)"
```
```   162  (fn _ => [nat_ind_tac "m" 1,
```
```   163            ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
```
```   164
```
```   165 qed_goal "add_mult_distrib2" Arith.thy "k*(m + n) = (k*m) + ((k*n)::nat)"
```
```   166  (fn _ => [nat_ind_tac "m" 1,
```
```   167            ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
```
```   168
```
```   169 (*Associative law for multiplication*)
```
```   170 qed_goal "mult_assoc" Arith.thy "(m * n) * k = m * ((n * k)::nat)"
```
```   171   (fn _ => [nat_ind_tac "m" 1,
```
```   172             ALLGOALS (asm_simp_tac (!simpset addsimps [add_mult_distrib]))]);
```
```   173
```
```   174 qed_goal "mult_left_commute" Arith.thy "x*(y*z) = y*((x*z)::nat)"
```
```   175  (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1,
```
```   176            rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]);
```
```   177
```
```   178 val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
```
```   179
```
```   180 (*** Difference ***)
```
```   181
```
```   182 qed_goal "diff_self_eq_0" Arith.thy "m - m = 0"
```
```   183  (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
```
```   184 Addsimps [diff_self_eq_0];
```
```   185
```
```   186 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
```
```   187 val [prem] = goal Arith.thy "[| ~ m<n |] ==> n+(m-n) = (m::nat)";
```
```   188 by (rtac (prem RS rev_mp) 1);
```
```   189 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   190 by (ALLGOALS (Asm_simp_tac));
```
```   191 qed "add_diff_inverse";
```
```   192
```
```   193
```
```   194 (*** Remainder ***)
```
```   195
```
```   196 goal Arith.thy "m - n < Suc(m)";
```
```   197 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   198 by (etac less_SucE 3);
```
```   199 by (ALLGOALS (asm_simp_tac (!simpset addsimps [less_Suc_eq])));
```
```   200 qed "diff_less_Suc";
```
```   201
```
```   202 goal Arith.thy "!!m::nat. m - n <= m";
```
```   203 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
```
```   204 by (ALLGOALS Asm_simp_tac);
```
```   205 qed "diff_le_self";
```
```   206
```
```   207 goal Arith.thy "!!n::nat. (n+m) - n = m";
```
```   208 by (nat_ind_tac "n" 1);
```
```   209 by (ALLGOALS Asm_simp_tac);
```
```   210 qed "diff_add_inverse";
```
```   211
```
```   212 goal Arith.thy "!!n::nat.(m+n) - n = m";
```
```   213 by (res_inst_tac [("m1","m")] (add_commute RS ssubst) 1);
```
```   214 by (REPEAT (ares_tac [diff_add_inverse] 1));
```
```   215 qed "diff_add_inverse2";
```
```   216
```
```   217 goal Arith.thy "!!k::nat. (k+m) - (k+n) = m - n";
```
```   218 by (nat_ind_tac "k" 1);
```
```   219 by (ALLGOALS Asm_simp_tac);
```
```   220 qed "diff_cancel";
```
```   221 Addsimps [diff_cancel];
```
```   222
```
```   223 goal Arith.thy "!!m::nat. (m+k) - (n+k) = m - n";
```
```   224 val add_commute_k = read_instantiate [("n","k")] add_commute;
```
```   225 by (asm_simp_tac (!simpset addsimps ([add_commute_k])) 1);
```
```   226 qed "diff_cancel2";
```
```   227 Addsimps [diff_cancel2];
```
```   228
```
```   229 goal Arith.thy "!!n::nat. n - (n+m) = 0";
```
```   230 by (nat_ind_tac "n" 1);
```
```   231 by (ALLGOALS Asm_simp_tac);
```
```   232 qed "diff_add_0";
```
```   233 Addsimps [diff_add_0];
```
```   234
```
```   235 (** Difference distributes over multiplication **)
```
```   236
```
```   237 goal Arith.thy "!!m::nat. (m - n) * k = (m * k) - (n * k)";
```
```   238 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   239 by (ALLGOALS Asm_simp_tac);
```
```   240 qed "diff_mult_distrib" ;
```
```   241
```
```   242 goal Arith.thy "!!m::nat. k * (m - n) = (k * m) - (k * n)";
```
```   243 val mult_commute_k = read_instantiate [("m","k")] mult_commute;
```
```   244 by (simp_tac (!simpset addsimps [diff_mult_distrib, mult_commute_k]) 1);
```
```   245 qed "diff_mult_distrib2" ;
```
```   246 (*NOT added as rewrites, since sometimes they are used from right-to-left*)
```
```   247
```
```   248
```
```   249 (** Less-then properties **)
```
```   250
```
```   251 (*In ordinary notation: if 0<n and n<=m then m-n < m *)
```
```   252 goal Arith.thy "!!m. [| 0<n; ~ m<n |] ==> m - n < m";
```
```   253 by (subgoal_tac "0<n --> ~ m<n --> m - n < m" 1);
```
```   254 by (Fast_tac 1);
```
```   255 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   256 by (ALLGOALS(asm_simp_tac(!simpset addsimps [diff_less_Suc])));
```
```   257 qed "diff_less";
```
```   258
```
```   259 val wf_less_trans = wf_pred_nat RS wf_trancl RSN (2, def_wfrec RS trans);
```
```   260
```
```   261 goalw Nat.thy [less_def] "(m,n) : pred_nat^+ = (m<n)";
```
```   262 by (rtac refl 1);
```
```   263 qed "less_eq";
```
```   264
```
```   265 goal Arith.thy "(%m. m mod n) = wfrec (trancl pred_nat) \
```
```   266              \                      (%f j. if j<n then j else f (j-n))";
```
```   267 by (simp_tac (HOL_ss addsimps [mod_def]) 1);
```
```   268 val mod_def1 = result() RS eq_reflection;
```
```   269
```
```   270 goal Arith.thy "!!m. m<n ==> m mod n = m";
```
```   271 by (rtac (mod_def1 RS wf_less_trans) 1);
```
```   272 by (Asm_simp_tac 1);
```
```   273 qed "mod_less";
```
```   274
```
```   275 goal Arith.thy "!!m. [| 0<n;  ~m<n |] ==> m mod n = (m-n) mod n";
```
```   276 by (rtac (mod_def1 RS wf_less_trans) 1);
```
```   277 by (asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1);
```
```   278 qed "mod_geq";
```
```   279
```
```   280
```
```   281 (*** Quotient ***)
```
```   282
```
```   283 goal Arith.thy "(%m. m div n) = wfrec (trancl pred_nat) \
```
```   284                         \            (%f j. if j<n then 0 else Suc (f (j-n)))";
```
```   285 by (simp_tac (HOL_ss addsimps [div_def]) 1);
```
```   286 val div_def1 = result() RS eq_reflection;
```
```   287
```
```   288 goal Arith.thy "!!m. m<n ==> m div n = 0";
```
```   289 by (rtac (div_def1 RS wf_less_trans) 1);
```
```   290 by (Asm_simp_tac 1);
```
```   291 qed "div_less";
```
```   292
```
```   293 goal Arith.thy "!!M. [| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)";
```
```   294 by (rtac (div_def1 RS wf_less_trans) 1);
```
```   295 by (asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1);
```
```   296 qed "div_geq";
```
```   297
```
```   298 (*Main Result about quotient and remainder.*)
```
```   299 goal Arith.thy "!!m. 0<n ==> (m div n)*n + m mod n = m";
```
```   300 by (res_inst_tac [("n","m")] less_induct 1);
```
```   301 by (rename_tac "k" 1);    (*Variable name used in line below*)
```
```   302 by (case_tac "k<n" 1);
```
```   303 by (ALLGOALS (asm_simp_tac(!simpset addsimps ([add_assoc] @
```
```   304                        [mod_less, mod_geq, div_less, div_geq,
```
```   305                         add_diff_inverse, diff_less]))));
```
```   306 qed "mod_div_equality";
```
```   307
```
```   308
```
```   309 (*** More results about difference ***)
```
```   310
```
```   311 val [prem] = goal Arith.thy "m < Suc(n) ==> m-n = 0";
```
```   312 by (rtac (prem RS rev_mp) 1);
```
```   313 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   314 by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
```
```   315 by (ALLGOALS (Asm_simp_tac));
```
```   316 qed "less_imp_diff_is_0";
```
```   317
```
```   318 val prems = goal Arith.thy "m-n = 0  -->  n-m = 0  -->  m=n";
```
```   319 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   320 by (REPEAT(Simp_tac 1 THEN TRY(atac 1)));
```
```   321 qed_spec_mp "diffs0_imp_equal";
```
```   322
```
```   323 val [prem] = goal Arith.thy "m<n ==> 0<n-m";
```
```   324 by (rtac (prem RS rev_mp) 1);
```
```   325 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   326 by (ALLGOALS (Asm_simp_tac));
```
```   327 qed "less_imp_diff_positive";
```
```   328
```
```   329 val [prem] = goal Arith.thy "n < Suc(m) ==> Suc(m)-n = Suc(m-n)";
```
```   330 by (rtac (prem RS rev_mp) 1);
```
```   331 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   332 by (ALLGOALS (Asm_simp_tac));
```
```   333 qed "Suc_diff_n";
```
```   334
```
```   335 goal Arith.thy "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
```
```   336 by (simp_tac (!simpset addsimps [less_imp_diff_is_0, not_less_eq, Suc_diff_n]
```
```   337                     setloop (split_tac [expand_if])) 1);
```
```   338 qed "if_Suc_diff_n";
```
```   339
```
```   340 goal Arith.thy "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
```
```   341 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
```
```   342 by (ALLGOALS (strip_tac THEN' Simp_tac THEN' TRY o Fast_tac));
```
```   343 qed "zero_induct_lemma";
```
```   344
```
```   345 val prems = goal Arith.thy "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
```
```   346 by (rtac (diff_self_eq_0 RS subst) 1);
```
```   347 by (rtac (zero_induct_lemma RS mp RS mp) 1);
```
```   348 by (REPEAT (ares_tac ([impI,allI]@prems) 1));
```
```   349 qed "zero_induct";
```
```   350
```
```   351 (*13 July 1992: loaded in 105.7s*)
```
```   352
```
```   353
```
```   354 (*** Further facts about mod (mainly for mutilated checkerboard ***)
```
```   355
```
```   356 goal Arith.thy
```
```   357     "!!m n. 0<n ==> \
```
```   358 \           Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))";
```
```   359 by (res_inst_tac [("n","m")] less_induct 1);
```
```   360 by (excluded_middle_tac "Suc(na)<n" 1);
```
```   361 (* case Suc(na) < n *)
```
```   362 by (forward_tac [lessI RS less_trans] 2);
```
```   363 by (asm_simp_tac (!simpset addsimps [mod_less, less_not_refl2 RS not_sym]) 2);
```
```   364 (* case n <= Suc(na) *)
```
```   365 by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, mod_geq]) 1);
```
```   366 by (etac (le_imp_less_or_eq RS disjE) 1);
```
```   367 by (asm_simp_tac (!simpset addsimps [Suc_diff_n]) 1);
```
```   368 by (asm_full_simp_tac (!simpset addsimps [not_less_eq RS sym,
```
```   369                                           diff_less, mod_geq]) 1);
```
```   370 by (asm_simp_tac (!simpset addsimps [mod_less]) 1);
```
```   371 qed "mod_Suc";
```
```   372
```
```   373 goal Arith.thy "!!m n. 0<n ==> m mod n < n";
```
```   374 by (res_inst_tac [("n","m")] less_induct 1);
```
```   375 by (excluded_middle_tac "na<n" 1);
```
```   376 (*case na<n*)
```
```   377 by (asm_simp_tac (!simpset addsimps [mod_less]) 2);
```
```   378 (*case n le na*)
```
```   379 by (asm_full_simp_tac (!simpset addsimps [mod_geq, diff_less]) 1);
```
```   380 qed "mod_less_divisor";
```
```   381
```
```   382
```
```   383 (** Evens and Odds **)
```
```   384
```
```   385 (*With less_zeroE, causes case analysis on b<2*)
```
```   386 AddSEs [less_SucE];
```
```   387
```
```   388 goal thy "!!k b. b<2 ==> k mod 2 = b | k mod 2 = (if b=1 then 0 else 1)";
```
```   389 by (subgoal_tac "k mod 2 < 2" 1);
```
```   390 by (asm_simp_tac (!simpset addsimps [mod_less_divisor]) 2);
```
```   391 by (asm_simp_tac (!simpset setloop split_tac [expand_if]) 1);
```
```   392 by (Fast_tac 1);
```
```   393 qed "mod2_cases";
```
```   394
```
```   395 goal thy "Suc(Suc(m)) mod 2 = m mod 2";
```
```   396 by (subgoal_tac "m mod 2 < 2" 1);
```
```   397 by (asm_simp_tac (!simpset addsimps [mod_less_divisor]) 2);
```
```   398 by (Step_tac 1);
```
```   399 by (ALLGOALS (asm_simp_tac (!simpset addsimps [mod_Suc])));
```
```   400 qed "mod2_Suc_Suc";
```
```   401 Addsimps [mod2_Suc_Suc];
```
```   402
```
```   403 goal thy "(m+m) mod 2 = 0";
```
```   404 by (nat_ind_tac "m" 1);
```
```   405 by (simp_tac (!simpset addsimps [mod_less]) 1);
```
```   406 by (asm_simp_tac (!simpset addsimps [mod2_Suc_Suc, add_Suc_right]) 1);
```
```   407 qed "mod2_add_self";
```
```   408 Addsimps [mod2_add_self];
```
```   409
```
```   410 Delrules [less_SucE];
```
```   411
```
```   412
```
```   413 (**** Additional theorems about "less than" ****)
```
```   414
```
```   415 goal Arith.thy "? k::nat. n = n+k";
```
```   416 by (res_inst_tac [("x","0")] exI 1);
```
```   417 by (Simp_tac 1);
```
```   418 val lemma = result();
```
```   419
```
```   420 goal Arith.thy "!!m. m<n --> (? k. n=Suc(m+k))";
```
```   421 by (nat_ind_tac "n" 1);
```
```   422 by (ALLGOALS (simp_tac (!simpset addsimps [less_Suc_eq])));
```
```   423 by (step_tac (!claset addSIs [lemma]) 1);
```
```   424 by (res_inst_tac [("x","Suc(k)")] exI 1);
```
```   425 by (Simp_tac 1);
```
```   426 qed_spec_mp "less_eq_Suc_add";
```
```   427
```
```   428 goal Arith.thy "n <= ((m + n)::nat)";
```
```   429 by (nat_ind_tac "m" 1);
```
```   430 by (ALLGOALS Simp_tac);
```
```   431 by (etac le_trans 1);
```
```   432 by (rtac (lessI RS less_imp_le) 1);
```
```   433 qed "le_add2";
```
```   434
```
```   435 goal Arith.thy "n <= ((n + m)::nat)";
```
```   436 by (simp_tac (!simpset addsimps add_ac) 1);
```
```   437 by (rtac le_add2 1);
```
```   438 qed "le_add1";
```
```   439
```
```   440 bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
```
```   441 bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
```
```   442
```
```   443 (*"i <= j ==> i <= j+m"*)
```
```   444 bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
```
```   445
```
```   446 (*"i <= j ==> i <= m+j"*)
```
```   447 bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
```
```   448
```
```   449 (*"i < j ==> i < j+m"*)
```
```   450 bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
```
```   451
```
```   452 (*"i < j ==> i < m+j"*)
```
```   453 bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
```
```   454
```
```   455 goal Arith.thy "!!i. i+j < (k::nat) ==> i<k";
```
```   456 by (etac rev_mp 1);
```
```   457 by (nat_ind_tac "j" 1);
```
```   458 by (ALLGOALS Asm_simp_tac);
```
```   459 by (fast_tac (!claset addDs [Suc_lessD]) 1);
```
```   460 qed "add_lessD1";
```
```   461
```
```   462 goal Arith.thy "!!k::nat. m <= n ==> m <= n+k";
```
```   463 by (etac le_trans 1);
```
```   464 by (rtac le_add1 1);
```
```   465 qed "le_imp_add_le";
```
```   466
```
```   467 goal Arith.thy "!!k::nat. m < n ==> m < n+k";
```
```   468 by (etac less_le_trans 1);
```
```   469 by (rtac le_add1 1);
```
```   470 qed "less_imp_add_less";
```
```   471
```
```   472 goal Arith.thy "m+k<=n --> m<=(n::nat)";
```
```   473 by (nat_ind_tac "k" 1);
```
```   474 by (ALLGOALS Asm_simp_tac);
```
```   475 by (fast_tac (!claset addDs [Suc_leD]) 1);
```
```   476 qed_spec_mp "add_leD1";
```
```   477
```
```   478 goal Arith.thy "!!n::nat. m+k<=n ==> k<=n";
```
```   479 by (full_simp_tac (!simpset addsimps [add_commute]) 1);
```
```   480 by (etac add_leD1 1);
```
```   481 qed_spec_mp "add_leD2";
```
```   482
```
```   483 goal Arith.thy "!!n::nat. m+k<=n ==> m<=n & k<=n";
```
```   484 by (fast_tac (!claset addDs [add_leD1, add_leD2]) 1);
```
```   485 bind_thm ("add_leE", result() RS conjE);
```
```   486
```
```   487 goal Arith.thy "!!k l::nat. [| k<l; m+l = k+n |] ==> m<n";
```
```   488 by (safe_tac (!claset addSDs [less_eq_Suc_add]));
```
```   489 by (asm_full_simp_tac
```
```   490     (!simpset delsimps [add_Suc_right]
```
```   491                 addsimps ([add_Suc_right RS sym, add_left_cancel] @add_ac)) 1);
```
```   492 by (etac subst 1);
```
```   493 by (simp_tac (!simpset addsimps [less_add_Suc1]) 1);
```
```   494 qed "less_add_eq_less";
```
```   495
```
```   496
```
```   497 (*** Monotonicity of Addition ***)
```
```   498
```
```   499 (*strict, in 1st argument*)
```
```   500 goal Arith.thy "!!i j k::nat. i < j ==> i + k < j + k";
```
```   501 by (nat_ind_tac "k" 1);
```
```   502 by (ALLGOALS Asm_simp_tac);
```
```   503 qed "add_less_mono1";
```
```   504
```
```   505 (*strict, in both arguments*)
```
```   506 goal Arith.thy "!!i j k::nat. [|i < j; k < l|] ==> i + k < j + l";
```
```   507 by (rtac (add_less_mono1 RS less_trans) 1);
```
```   508 by (REPEAT (assume_tac 1));
```
```   509 by (nat_ind_tac "j" 1);
```
```   510 by (ALLGOALS Asm_simp_tac);
```
```   511 qed "add_less_mono";
```
```   512
```
```   513 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
```
```   514 val [lt_mono,le] = goal Arith.thy
```
```   515      "[| !!i j::nat. i<j ==> f(i) < f(j);       \
```
```   516 \        i <= j                                 \
```
```   517 \     |] ==> f(i) <= (f(j)::nat)";
```
```   518 by (cut_facts_tac [le] 1);
```
```   519 by (asm_full_simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
```
```   520 by (fast_tac (!claset addSIs [lt_mono]) 1);
```
```   521 qed "less_mono_imp_le_mono";
```
```   522
```
```   523 (*non-strict, in 1st argument*)
```
```   524 goal Arith.thy "!!i j k::nat. i<=j ==> i + k <= j + k";
```
```   525 by (res_inst_tac [("f", "%j.j+k")] less_mono_imp_le_mono 1);
```
```   526 by (etac add_less_mono1 1);
```
```   527 by (assume_tac 1);
```
```   528 qed "add_le_mono1";
```
```   529
```
```   530 (*non-strict, in both arguments*)
```
```   531 goal Arith.thy "!!k l::nat. [|i<=j;  k<=l |] ==> i + k <= j + l";
```
```   532 by (etac (add_le_mono1 RS le_trans) 1);
```
```   533 by (simp_tac (!simpset addsimps [add_commute]) 1);
```
```   534 (*j moves to the end because it is free while k, l are bound*)
```
```   535 by (etac add_le_mono1 1);
```
```   536 qed "add_le_mono";
```
```   537
```
```   538 (*** Monotonicity of Multiplication ***)
```
```   539
```
```   540 goal Arith.thy "!!i::nat. i<=j ==> i*k<=j*k";
```
```   541 by (nat_ind_tac "k" 1);
```
```   542 by (ALLGOALS (asm_simp_tac (!simpset addsimps [add_le_mono])));
```
```   543 qed "mult_le_mono1";
```
```   544
```
```   545 (*<=monotonicity, BOTH arguments*)
```
```   546 goal Arith.thy "!!i::nat. [| i<=j; k<=l |] ==> i*k<=j*l";
```
```   547 by (etac (mult_le_mono1 RS le_trans) 1);
```
```   548 by (rtac le_trans 1);
```
```   549 by (stac mult_commute 2);
```
```   550 by (etac mult_le_mono1 2);
```
```   551 by (simp_tac (!simpset addsimps [mult_commute]) 1);
```
```   552 qed "mult_le_mono";
```
```   553
```
```   554 (*strict, in 1st argument; proof is by induction on k>0*)
```
```   555 goal Arith.thy "!!i::nat. [| i<j; 0<k |] ==> k*i < k*j";
```
```   556 by (etac zero_less_natE 1);
```
```   557 by (Asm_simp_tac 1);
```
```   558 by (nat_ind_tac "x" 1);
```
```   559 by (ALLGOALS (asm_simp_tac (!simpset addsimps [add_less_mono])));
```
```   560 qed "mult_less_mono2";
```
```   561
```
```   562 goal Arith.thy "(0 < m*n) = (0<m & 0<n)";
```
```   563 by (nat_ind_tac "m" 1);
```
```   564 by (nat_ind_tac "n" 2);
```
```   565 by (ALLGOALS Asm_simp_tac);
```
```   566 qed "zero_less_mult_iff";
```
```   567
```
```   568 goal Arith.thy "(m*n = 1) = (m=1 & n=1)";
```
```   569 by (nat_ind_tac "m" 1);
```
```   570 by (Simp_tac 1);
```
```   571 by (nat_ind_tac "n" 1);
```
```   572 by (Simp_tac 1);
```
```   573 by (fast_tac (!claset addss !simpset) 1);
```
```   574 qed "mult_eq_1_iff";
```
```   575
```
```   576 (*Cancellation law for division*)
```
```   577 goal Arith.thy "!!k. [| 0<n; 0<k |] ==> (k*m) div (k*n) = m div n";
```
```   578 by (res_inst_tac [("n","m")] less_induct 1);
```
```   579 by (case_tac "na<n" 1);
```
```   580 by (asm_simp_tac (!simpset addsimps [div_less, zero_less_mult_iff,
```
```   581                                      mult_less_mono2]) 1);
```
```   582 by (subgoal_tac "~ k*na < k*n" 1);
```
```   583 by (asm_simp_tac
```
```   584      (!simpset addsimps [zero_less_mult_iff, div_geq,
```
```   585                          diff_mult_distrib2 RS sym, diff_less]) 1);
```
```   586 by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le,
```
```   587                                           le_refl RS mult_le_mono]) 1);
```
```   588 qed "div_cancel";
```
```   589
```
```   590 goal Arith.thy "!!k. [| 0<n; 0<k |] ==> (k*m) mod (k*n) = k * (m mod n)";
```
```   591 by (res_inst_tac [("n","m")] less_induct 1);
```
```   592 by (case_tac "na<n" 1);
```
```   593 by (asm_simp_tac (!simpset addsimps [mod_less, zero_less_mult_iff,
```
```   594                                      mult_less_mono2]) 1);
```
```   595 by (subgoal_tac "~ k*na < k*n" 1);
```
```   596 by (asm_simp_tac
```
```   597      (!simpset addsimps [zero_less_mult_iff, mod_geq,
```
```   598                          diff_mult_distrib2 RS sym, diff_less]) 1);
```
```   599 by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le,
```
```   600                                           le_refl RS mult_le_mono]) 1);
```
```   601 qed "mult_mod_distrib";
```
```   602
```
```   603
```
```   604 (** Lemma for gcd **)
```
```   605
```
```   606 goal Arith.thy "!!m n. m = m*n ==> n=1 | m=0";
```
```   607 by (dtac sym 1);
```
```   608 by (rtac disjCI 1);
```
```   609 by (rtac nat_less_cases 1 THEN assume_tac 2);
```
```   610 by (fast_tac (!claset addSEs [less_SucE] addss !simpset) 1);
```
```   611 by (best_tac (!claset addDs [mult_less_mono2]
```
```   612                       addss (!simpset addsimps [zero_less_eq RS sym])) 1);
```
```   613 qed "mult_eq_self_implies_10";
```
```   614
```
```   615
```