src/HOL/Nat.ML
 author paulson Wed Mar 06 12:52:11 1996 +0100 (1996-03-06) changeset 1552 6f71b5d46700 parent 1531 e5eb247ad13c child 1618 372880456b5b permissions -rw-r--r--
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```     1 (*  Title:      HOL/nat
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
```
```     4     Copyright   1991  University of Cambridge
```
```     5
```
```     6 For nat.thy.  Type nat is defined as a set (Nat) over the type ind.
```
```     7 *)
```
```     8
```
```     9 open Nat;
```
```    10
```
```    11 goal Nat.thy "mono(%X. {Zero_Rep} Un (Suc_Rep``X))";
```
```    12 by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1));
```
```    13 qed "Nat_fun_mono";
```
```    14
```
```    15 val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski);
```
```    16
```
```    17 (* Zero is a natural number -- this also justifies the type definition*)
```
```    18 goal Nat.thy "Zero_Rep: Nat";
```
```    19 by (rtac (Nat_unfold RS ssubst) 1);
```
```    20 by (rtac (singletonI RS UnI1) 1);
```
```    21 qed "Zero_RepI";
```
```    22
```
```    23 val prems = goal Nat.thy "i: Nat ==> Suc_Rep(i) : Nat";
```
```    24 by (rtac (Nat_unfold RS ssubst) 1);
```
```    25 by (rtac (imageI RS UnI2) 1);
```
```    26 by (resolve_tac prems 1);
```
```    27 qed "Suc_RepI";
```
```    28
```
```    29 (*** Induction ***)
```
```    30
```
```    31 val major::prems = goal Nat.thy
```
```    32     "[| i: Nat;  P(Zero_Rep);   \
```
```    33 \       !!j. [| j: Nat; P(j) |] ==> P(Suc_Rep(j)) |]  ==> P(i)";
```
```    34 by (rtac ([Nat_def, Nat_fun_mono, major] MRS def_induct) 1);
```
```    35 by (fast_tac (set_cs addIs prems) 1);
```
```    36 qed "Nat_induct";
```
```    37
```
```    38 val prems = goalw Nat.thy [Zero_def,Suc_def]
```
```    39     "[| P(0);   \
```
```    40 \       !!k. P(k) ==> P(Suc(k)) |]  ==> P(n)";
```
```    41 by (rtac (Rep_Nat_inverse RS subst) 1);   (*types force good instantiation*)
```
```    42 by (rtac (Rep_Nat RS Nat_induct) 1);
```
```    43 by (REPEAT (ares_tac prems 1
```
```    44      ORELSE eresolve_tac [Abs_Nat_inverse RS subst] 1));
```
```    45 qed "nat_induct";
```
```    46
```
```    47 (*Perform induction on n. *)
```
```    48 fun nat_ind_tac a i =
```
```    49     EVERY [res_inst_tac [("n",a)] nat_induct i,
```
```    50            rename_last_tac a ["1"] (i+1)];
```
```    51
```
```    52 (*A special form of induction for reasoning about m<n and m-n*)
```
```    53 val prems = goal Nat.thy
```
```    54     "[| !!x. P x 0;  \
```
```    55 \       !!y. P 0 (Suc y);  \
```
```    56 \       !!x y. [| P x y |] ==> P (Suc x) (Suc y)  \
```
```    57 \    |] ==> P m n";
```
```    58 by (res_inst_tac [("x","m")] spec 1);
```
```    59 by (nat_ind_tac "n" 1);
```
```    60 by (rtac allI 2);
```
```    61 by (nat_ind_tac "x" 2);
```
```    62 by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1));
```
```    63 qed "diff_induct";
```
```    64
```
```    65 (*Case analysis on the natural numbers*)
```
```    66 val prems = goal Nat.thy
```
```    67     "[| n=0 ==> P;  !!x. n = Suc(x) ==> P |] ==> P";
```
```    68 by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1);
```
```    69 by (fast_tac (HOL_cs addSEs prems) 1);
```
```    70 by (nat_ind_tac "n" 1);
```
```    71 by (rtac (refl RS disjI1) 1);
```
```    72 by (fast_tac HOL_cs 1);
```
```    73 qed "natE";
```
```    74
```
```    75 (*** Isomorphisms: Abs_Nat and Rep_Nat ***)
```
```    76
```
```    77 (*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat),
```
```    78   since we assume the isomorphism equations will one day be given by Isabelle*)
```
```    79
```
```    80 goal Nat.thy "inj(Rep_Nat)";
```
```    81 by (rtac inj_inverseI 1);
```
```    82 by (rtac Rep_Nat_inverse 1);
```
```    83 qed "inj_Rep_Nat";
```
```    84
```
```    85 goal Nat.thy "inj_onto Abs_Nat Nat";
```
```    86 by (rtac inj_onto_inverseI 1);
```
```    87 by (etac Abs_Nat_inverse 1);
```
```    88 qed "inj_onto_Abs_Nat";
```
```    89
```
```    90 (*** Distinctness of constructors ***)
```
```    91
```
```    92 goalw Nat.thy [Zero_def,Suc_def] "Suc(m) ~= 0";
```
```    93 by (rtac (inj_onto_Abs_Nat RS inj_onto_contraD) 1);
```
```    94 by (rtac Suc_Rep_not_Zero_Rep 1);
```
```    95 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1));
```
```    96 qed "Suc_not_Zero";
```
```    97
```
```    98 bind_thm ("Zero_not_Suc", (Suc_not_Zero RS not_sym));
```
```    99
```
```   100 Addsimps [Suc_not_Zero,Zero_not_Suc];
```
```   101
```
```   102 bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE));
```
```   103 val Zero_neq_Suc = sym RS Suc_neq_Zero;
```
```   104
```
```   105 (** Injectiveness of Suc **)
```
```   106
```
```   107 goalw Nat.thy [Suc_def] "inj(Suc)";
```
```   108 by (rtac injI 1);
```
```   109 by (dtac (inj_onto_Abs_Nat RS inj_ontoD) 1);
```
```   110 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1));
```
```   111 by (dtac (inj_Suc_Rep RS injD) 1);
```
```   112 by (etac (inj_Rep_Nat RS injD) 1);
```
```   113 qed "inj_Suc";
```
```   114
```
```   115 val Suc_inject = inj_Suc RS injD;
```
```   116
```
```   117 goal Nat.thy "(Suc(m)=Suc(n)) = (m=n)";
```
```   118 by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]);
```
```   119 qed "Suc_Suc_eq";
```
```   120
```
```   121 goal Nat.thy "n ~= Suc(n)";
```
```   122 by (nat_ind_tac "n" 1);
```
```   123 by (ALLGOALS(asm_simp_tac (!simpset addsimps [Suc_Suc_eq])));
```
```   124 qed "n_not_Suc_n";
```
```   125
```
```   126 val Suc_n_not_n = n_not_Suc_n RS not_sym;
```
```   127
```
```   128 (*** nat_case -- the selection operator for nat ***)
```
```   129
```
```   130 goalw Nat.thy [nat_case_def] "nat_case a f 0 = a";
```
```   131 by (fast_tac (set_cs addIs [select_equality] addEs [Zero_neq_Suc]) 1);
```
```   132 qed "nat_case_0";
```
```   133
```
```   134 goalw Nat.thy [nat_case_def] "nat_case a f (Suc k) = f(k)";
```
```   135 by (fast_tac (set_cs addIs [select_equality]
```
```   136                        addEs [make_elim Suc_inject, Suc_neq_Zero]) 1);
```
```   137 qed "nat_case_Suc";
```
```   138
```
```   139 (** Introduction rules for 'pred_nat' **)
```
```   140
```
```   141 goalw Nat.thy [pred_nat_def] "(n, Suc(n)) : pred_nat";
```
```   142 by (fast_tac set_cs 1);
```
```   143 qed "pred_natI";
```
```   144
```
```   145 val major::prems = goalw Nat.thy [pred_nat_def]
```
```   146     "[| p : pred_nat;  !!x n. [| p = (n, Suc(n)) |] ==> R \
```
```   147 \    |] ==> R";
```
```   148 by (rtac (major RS CollectE) 1);
```
```   149 by (REPEAT (eresolve_tac ([asm_rl,exE]@prems) 1));
```
```   150 qed "pred_natE";
```
```   151
```
```   152 goalw Nat.thy [wf_def] "wf(pred_nat)";
```
```   153 by (strip_tac 1);
```
```   154 by (nat_ind_tac "x" 1);
```
```   155 by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject,
```
```   156                              make_elim Suc_inject]) 2);
```
```   157 by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, Zero_neq_Suc]) 1);
```
```   158 qed "wf_pred_nat";
```
```   159
```
```   160
```
```   161 (*** nat_rec -- by wf recursion on pred_nat ***)
```
```   162
```
```   163 (* The unrolling rule for nat_rec *)
```
```   164 goal Nat.thy
```
```   165    "(%n. nat_rec n c d) = wfrec pred_nat (%f. nat_case ?c (%m. ?d m (f m)))";
```
```   166   by (simp_tac (HOL_ss addsimps [nat_rec_def]) 1);
```
```   167 bind_thm("nat_rec_unfold", wf_pred_nat RS
```
```   168                             ((result() RS eq_reflection) RS def_wfrec));
```
```   169
```
```   170 (*---------------------------------------------------------------------------
```
```   171  * Old:
```
```   172  * bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec)));
```
```   173  *---------------------------------------------------------------------------*)
```
```   174
```
```   175 (** conversion rules **)
```
```   176
```
```   177 goal Nat.thy "nat_rec 0 c h = c";
```
```   178 by (rtac (nat_rec_unfold RS trans) 1);
```
```   179 by (simp_tac (!simpset addsimps [nat_case_0]) 1);
```
```   180 qed "nat_rec_0";
```
```   181
```
```   182 goal Nat.thy "nat_rec (Suc n) c h = h n (nat_rec n c h)";
```
```   183 by (rtac (nat_rec_unfold RS trans) 1);
```
```   184 by (simp_tac (!simpset addsimps [nat_case_Suc, pred_natI, cut_apply]) 1);
```
```   185 qed "nat_rec_Suc";
```
```   186
```
```   187 (*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
```
```   188 val [rew] = goal Nat.thy
```
```   189     "[| !!n. f(n) == nat_rec n c h |] ==> f(0) = c";
```
```   190 by (rewtac rew);
```
```   191 by (rtac nat_rec_0 1);
```
```   192 qed "def_nat_rec_0";
```
```   193
```
```   194 val [rew] = goal Nat.thy
```
```   195     "[| !!n. f(n) == nat_rec n c h |] ==> f(Suc(n)) = h n (f n)";
```
```   196 by (rewtac rew);
```
```   197 by (rtac nat_rec_Suc 1);
```
```   198 qed "def_nat_rec_Suc";
```
```   199
```
```   200 fun nat_recs def =
```
```   201       [standard (def RS def_nat_rec_0),
```
```   202        standard (def RS def_nat_rec_Suc)];
```
```   203
```
```   204
```
```   205 (*** Basic properties of "less than" ***)
```
```   206
```
```   207 (** Introduction properties **)
```
```   208
```
```   209 val prems = goalw Nat.thy [less_def] "[| i<j;  j<k |] ==> i<(k::nat)";
```
```   210 by (rtac (trans_trancl RS transD) 1);
```
```   211 by (resolve_tac prems 1);
```
```   212 by (resolve_tac prems 1);
```
```   213 qed "less_trans";
```
```   214
```
```   215 goalw Nat.thy [less_def] "n < Suc(n)";
```
```   216 by (rtac (pred_natI RS r_into_trancl) 1);
```
```   217 qed "lessI";
```
```   218 Addsimps [lessI];
```
```   219
```
```   220 (* i(j ==> i(Suc(j) *)
```
```   221 val less_SucI = lessI RSN (2, less_trans);
```
```   222
```
```   223 goal Nat.thy "0 < Suc(n)";
```
```   224 by (nat_ind_tac "n" 1);
```
```   225 by (rtac lessI 1);
```
```   226 by (etac less_trans 1);
```
```   227 by (rtac lessI 1);
```
```   228 qed "zero_less_Suc";
```
```   229 Addsimps [zero_less_Suc];
```
```   230
```
```   231 (** Elimination properties **)
```
```   232
```
```   233 val prems = goalw Nat.thy [less_def] "n<m ==> ~ m<(n::nat)";
```
```   234 by (fast_tac (HOL_cs addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1);
```
```   235 qed "less_not_sym";
```
```   236
```
```   237 (* [| n(m; m(n |] ==> R *)
```
```   238 bind_thm ("less_asym", (less_not_sym RS notE));
```
```   239
```
```   240 goalw Nat.thy [less_def] "~ n<(n::nat)";
```
```   241 by (rtac notI 1);
```
```   242 by (etac (wf_pred_nat RS wf_trancl RS wf_anti_refl) 1);
```
```   243 qed "less_not_refl";
```
```   244
```
```   245 (* n(n ==> R *)
```
```   246 bind_thm ("less_anti_refl", (less_not_refl RS notE));
```
```   247
```
```   248 goal Nat.thy "!!m. n<m ==> m ~= (n::nat)";
```
```   249 by (fast_tac (HOL_cs addEs [less_anti_refl]) 1);
```
```   250 qed "less_not_refl2";
```
```   251
```
```   252
```
```   253 val major::prems = goalw Nat.thy [less_def]
```
```   254     "[| i<k;  k=Suc(i) ==> P;  !!j. [| i<j;  k=Suc(j) |] ==> P \
```
```   255 \    |] ==> P";
```
```   256 by (rtac (major RS tranclE) 1);
```
```   257 by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE'
```
```   258                   eresolve_tac (prems@[pred_natE, Pair_inject])));
```
```   259 by (rtac refl 1);
```
```   260 qed "lessE";
```
```   261
```
```   262 goal Nat.thy "~ n<0";
```
```   263 by (rtac notI 1);
```
```   264 by (etac lessE 1);
```
```   265 by (etac Zero_neq_Suc 1);
```
```   266 by (etac Zero_neq_Suc 1);
```
```   267 qed "not_less0";
```
```   268 Addsimps [not_less0];
```
```   269
```
```   270 (* n<0 ==> R *)
```
```   271 bind_thm ("less_zeroE", (not_less0 RS notE));
```
```   272
```
```   273 val [major,less,eq] = goal Nat.thy
```
```   274     "[| m < Suc(n);  m<n ==> P;  m=n ==> P |] ==> P";
```
```   275 by (rtac (major RS lessE) 1);
```
```   276 by (rtac eq 1);
```
```   277 by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
```
```   278 by (rtac less 1);
```
```   279 by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
```
```   280 qed "less_SucE";
```
```   281
```
```   282 goal Nat.thy "(m < Suc(n)) = (m < n | m = n)";
```
```   283 by (fast_tac (HOL_cs addSIs [lessI]
```
```   284                      addEs  [less_trans, less_SucE]) 1);
```
```   285 qed "less_Suc_eq";
```
```   286
```
```   287 val prems = goal Nat.thy "m<n ==> n ~= 0";
```
```   288 by (res_inst_tac [("n","n")] natE 1);
```
```   289 by (cut_facts_tac prems 1);
```
```   290 by (ALLGOALS Asm_full_simp_tac);
```
```   291 qed "gr_implies_not0";
```
```   292 Addsimps [gr_implies_not0];
```
```   293
```
```   294 (** Inductive (?) properties **)
```
```   295
```
```   296 val [prem] = goal Nat.thy "Suc(m) < n ==> m<n";
```
```   297 by (rtac (prem RS rev_mp) 1);
```
```   298 by (nat_ind_tac "n" 1);
```
```   299 by (rtac impI 1);
```
```   300 by (etac less_zeroE 1);
```
```   301 by (fast_tac (HOL_cs addSIs [lessI RS less_SucI]
```
```   302                      addSDs [Suc_inject]
```
```   303                      addEs  [less_trans, lessE]) 1);
```
```   304 qed "Suc_lessD";
```
```   305
```
```   306 val [major,minor] = goal Nat.thy
```
```   307     "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
```
```   308 \    |] ==> P";
```
```   309 by (rtac (major RS lessE) 1);
```
```   310 by (etac (lessI RS minor) 1);
```
```   311 by (etac (Suc_lessD RS minor) 1);
```
```   312 by (assume_tac 1);
```
```   313 qed "Suc_lessE";
```
```   314
```
```   315 val [major] = goal Nat.thy "Suc(m) < Suc(n) ==> m<n";
```
```   316 by (rtac (major RS lessE) 1);
```
```   317 by (REPEAT (rtac lessI 1
```
```   318      ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1));
```
```   319 qed "Suc_less_SucD";
```
```   320
```
```   321 val prems = goal Nat.thy "m<n ==> Suc(m) < Suc(n)";
```
```   322 by (subgoal_tac "m<n --> Suc(m) < Suc(n)" 1);
```
```   323 by (fast_tac (HOL_cs addIs prems) 1);
```
```   324 by (nat_ind_tac "n" 1);
```
```   325 by (rtac impI 1);
```
```   326 by (etac less_zeroE 1);
```
```   327 by (fast_tac (HOL_cs addSIs [lessI]
```
```   328                      addSDs [Suc_inject]
```
```   329                      addEs  [less_trans, lessE]) 1);
```
```   330 qed "Suc_mono";
```
```   331
```
```   332 goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)";
```
```   333 by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
```
```   334 qed "Suc_less_eq";
```
```   335 Addsimps [Suc_less_eq];
```
```   336
```
```   337 goal Nat.thy "~(Suc(n) < n)";
```
```   338 by (fast_tac (HOL_cs addEs [Suc_lessD RS less_anti_refl]) 1);
```
```   339 qed "not_Suc_n_less_n";
```
```   340 Addsimps [not_Suc_n_less_n];
```
```   341
```
```   342 goal Nat.thy "!!i. i<j ==> j<k --> Suc i < k";
```
```   343 by (nat_ind_tac "k" 1);
```
```   344 by (ALLGOALS (asm_simp_tac (!simpset addsimps [less_Suc_eq])));
```
```   345 by (fast_tac (HOL_cs addDs [Suc_lessD]) 1);
```
```   346 qed_spec_mp "less_trans_Suc";
```
```   347
```
```   348 (*"Less than" is a linear ordering*)
```
```   349 goal Nat.thy "m<n | m=n | n<(m::nat)";
```
```   350 by (nat_ind_tac "m" 1);
```
```   351 by (nat_ind_tac "n" 1);
```
```   352 by (rtac (refl RS disjI1 RS disjI2) 1);
```
```   353 by (rtac (zero_less_Suc RS disjI1) 1);
```
```   354 by (fast_tac (HOL_cs addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1);
```
```   355 qed "less_linear";
```
```   356
```
```   357 (*Can be used with less_Suc_eq to get n=m | n<m *)
```
```   358 goal Nat.thy "(~ m < n) = (n < Suc(m))";
```
```   359 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   360 by (ALLGOALS Asm_simp_tac);
```
```   361 qed "not_less_eq";
```
```   362
```
```   363 (*Complete induction, aka course-of-values induction*)
```
```   364 val prems = goalw Nat.thy [less_def]
```
```   365     "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
```
```   366 by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
```
```   367 by (eresolve_tac prems 1);
```
```   368 qed "less_induct";
```
```   369
```
```   370
```
```   371 (*** Properties of <= ***)
```
```   372
```
```   373 goalw Nat.thy [le_def] "0 <= n";
```
```   374 by (rtac not_less0 1);
```
```   375 qed "le0";
```
```   376
```
```   377 goalw Nat.thy [le_def] "~ Suc n <= n";
```
```   378 by (Simp_tac 1);
```
```   379 qed "Suc_n_not_le_n";
```
```   380
```
```   381 goalw Nat.thy [le_def] "(i <= 0) = (i = 0)";
```
```   382 by (nat_ind_tac "i" 1);
```
```   383 by (ALLGOALS Asm_simp_tac);
```
```   384 qed "le_0";
```
```   385
```
```   386 Addsimps [less_not_refl,
```
```   387           less_Suc_eq, le0, le_0,
```
```   388           Suc_Suc_eq, Suc_n_not_le_n,
```
```   389           n_not_Suc_n, Suc_n_not_n,
```
```   390           nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];
```
```   391
```
```   392 (*Prevents simplification of f and g: much faster*)
```
```   393 qed_goal "nat_case_weak_cong" Nat.thy
```
```   394   "m=n ==> nat_case a f m = nat_case a f n"
```
```   395   (fn [prem] => [rtac (prem RS arg_cong) 1]);
```
```   396
```
```   397 qed_goal "nat_rec_weak_cong" Nat.thy
```
```   398   "m=n ==> nat_rec m a f = nat_rec n a f"
```
```   399   (fn [prem] => [rtac (prem RS arg_cong) 1]);
```
```   400
```
```   401 val prems = goalw Nat.thy [le_def] "~(n<m) ==> m<=(n::nat)";
```
```   402 by (resolve_tac prems 1);
```
```   403 qed "leI";
```
```   404
```
```   405 val prems = goalw Nat.thy [le_def] "m<=n ==> ~(n<(m::nat))";
```
```   406 by (resolve_tac prems 1);
```
```   407 qed "leD";
```
```   408
```
```   409 val leE = make_elim leD;
```
```   410
```
```   411 goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
```
```   412 by (fast_tac HOL_cs 1);
```
```   413 qed "not_leE";
```
```   414
```
```   415 goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n";
```
```   416 by (Simp_tac 1);
```
```   417 by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
```
```   418 qed "lessD";
```
```   419
```
```   420 goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
```
```   421 by (Asm_full_simp_tac 1);
```
```   422 by (fast_tac HOL_cs 1);
```
```   423 qed "Suc_leD";
```
```   424
```
```   425 goalw Nat.thy [le_def] "!!m. m <= n ==> m <= Suc n";
```
```   426 by (fast_tac (HOL_cs addDs [Suc_lessD]) 1);
```
```   427 qed "le_SucI";
```
```   428 Addsimps[le_SucI];
```
```   429
```
```   430 goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)";
```
```   431 by (fast_tac (HOL_cs addEs [less_asym]) 1);
```
```   432 qed "less_imp_le";
```
```   433
```
```   434 goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
```
```   435 by (cut_facts_tac [less_linear] 1);
```
```   436 by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
```
```   437 qed "le_imp_less_or_eq";
```
```   438
```
```   439 goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
```
```   440 by (cut_facts_tac [less_linear] 1);
```
```   441 by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
```
```   442 by (flexflex_tac);
```
```   443 qed "less_or_eq_imp_le";
```
```   444
```
```   445 goal Nat.thy "(x <= (y::nat)) = (x < y | x=y)";
```
```   446 by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
```
```   447 qed "le_eq_less_or_eq";
```
```   448
```
```   449 goal Nat.thy "n <= (n::nat)";
```
```   450 by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
```
```   451 qed "le_refl";
```
```   452
```
```   453 val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
```
```   454 by (dtac le_imp_less_or_eq 1);
```
```   455 by (fast_tac (HOL_cs addIs [less_trans]) 1);
```
```   456 qed "le_less_trans";
```
```   457
```
```   458 goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
```
```   459 by (dtac le_imp_less_or_eq 1);
```
```   460 by (fast_tac (HOL_cs addIs [less_trans]) 1);
```
```   461 qed "less_le_trans";
```
```   462
```
```   463 goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";
```
```   464 by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
```
```   465           rtac less_or_eq_imp_le, fast_tac (HOL_cs addIs [less_trans])]);
```
```   466 qed "le_trans";
```
```   467
```
```   468 val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";
```
```   469 by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
```
```   470           fast_tac (HOL_cs addEs [less_anti_refl,less_asym])]);
```
```   471 qed "le_anti_sym";
```
```   472
```
```   473 goal Nat.thy "(Suc(n) <= Suc(m)) = (n <= m)";
```
```   474 by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
```
```   475 qed "Suc_le_mono";
```
```   476
```
```   477 Addsimps [le_refl,Suc_le_mono];
```
```   478
```
```   479
```
```   480 (** LEAST -- the least number operator **)
```
```   481
```
```   482 val [prem1,prem2] = goalw Nat.thy [Least_def]
```
```   483     "[| P(k);  !!x. x<k ==> ~P(x) |] ==> (LEAST x.P(x)) = k";
```
```   484 by (rtac select_equality 1);
```
```   485 by (fast_tac (HOL_cs addSIs [prem1,prem2]) 1);
```
```   486 by (cut_facts_tac [less_linear] 1);
```
```   487 by (fast_tac (HOL_cs addSIs [prem1] addSDs [prem2]) 1);
```
```   488 qed "Least_equality";
```
```   489
```
```   490 val [prem] = goal Nat.thy "P(k) ==> P(LEAST x.P(x))";
```
```   491 by (rtac (prem RS rev_mp) 1);
```
```   492 by (res_inst_tac [("n","k")] less_induct 1);
```
```   493 by (rtac impI 1);
```
```   494 by (rtac classical 1);
```
```   495 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
```
```   496 by (assume_tac 1);
```
```   497 by (assume_tac 2);
```
```   498 by (fast_tac HOL_cs 1);
```
```   499 qed "LeastI";
```
```   500
```
```   501 (*Proof is almost identical to the one above!*)
```
```   502 val [prem] = goal Nat.thy "P(k) ==> (LEAST x.P(x)) <= k";
```
```   503 by (rtac (prem RS rev_mp) 1);
```
```   504 by (res_inst_tac [("n","k")] less_induct 1);
```
```   505 by (rtac impI 1);
```
```   506 by (rtac classical 1);
```
```   507 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
```
```   508 by (assume_tac 1);
```
```   509 by (rtac le_refl 2);
```
```   510 by (fast_tac (HOL_cs addIs [less_imp_le,le_trans]) 1);
```
```   511 qed "Least_le";
```
```   512
```
```   513 val [prem] = goal Nat.thy "k < (LEAST x.P(x)) ==> ~P(k)";
```
```   514 by (rtac notI 1);
```
```   515 by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
```
```   516 by (rtac prem 1);
```
```   517 qed "not_less_Least";
```