src/HOL/NatDef.ML
author paulson
Thu Sep 23 13:06:31 1999 +0200 (1999-09-23)
changeset 7584 5be4bb8e4e3f
parent 7064 b053e0ab9f60
child 8555 17325ee838ab
permissions -rw-r--r--
tidied; added lemma restrict_to_left
     1 (*  Title:      HOL/NatDef.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
     4     Copyright   1991  University of Cambridge
     5 *)
     6 
     7 Goal "mono(%X. {Zero_Rep} Un (Suc_Rep``X))";
     8 by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1));
     9 qed "Nat_fun_mono";
    10 
    11 val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski);
    12 
    13 (* Zero is a natural number -- this also justifies the type definition*)
    14 Goal "Zero_Rep: Nat";
    15 by (stac Nat_unfold 1);
    16 by (rtac (singletonI RS UnI1) 1);
    17 qed "Zero_RepI";
    18 
    19 Goal "i: Nat ==> Suc_Rep(i) : Nat";
    20 by (stac Nat_unfold 1);
    21 by (rtac (imageI RS UnI2) 1);
    22 by (assume_tac 1);
    23 qed "Suc_RepI";
    24 
    25 (*** Induction ***)
    26 
    27 val major::prems = Goal
    28     "[| i: Nat;  P(Zero_Rep);   \
    29 \       !!j. [| j: Nat; P(j) |] ==> P(Suc_Rep(j)) |]  ==> P(i)";
    30 by (rtac ([Nat_def, Nat_fun_mono, major] MRS def_induct) 1);
    31 by (blast_tac (claset() addIs prems) 1);
    32 qed "Nat_induct";
    33 
    34 val prems = Goalw [Zero_def,Suc_def]
    35     "[| P(0);   \
    36 \       !!n. P(n) ==> P(Suc(n)) |]  ==> P(n)";
    37 by (rtac (Rep_Nat_inverse RS subst) 1);   (*types force good instantiation*)
    38 by (rtac (Rep_Nat RS Nat_induct) 1);
    39 by (REPEAT (ares_tac prems 1
    40      ORELSE eresolve_tac [Abs_Nat_inverse RS subst] 1));
    41 qed "nat_induct";
    42 
    43 (*Perform induction on n. *)
    44 fun nat_ind_tac a i = 
    45   res_inst_tac [("n",a)] nat_induct i  THEN  rename_last_tac a [""] (i+1);
    46 
    47 (*A special form of induction for reasoning about m<n and m-n*)
    48 val prems = Goal
    49     "[| !!x. P x 0;  \
    50 \       !!y. P 0 (Suc y);  \
    51 \       !!x y. [| P x y |] ==> P (Suc x) (Suc y)  \
    52 \    |] ==> P m n";
    53 by (res_inst_tac [("x","m")] spec 1);
    54 by (nat_ind_tac "n" 1);
    55 by (rtac allI 2);
    56 by (nat_ind_tac "x" 2);
    57 by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1));
    58 qed "diff_induct";
    59 
    60 (*** Isomorphisms: Abs_Nat and Rep_Nat ***)
    61 
    62 (*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat),
    63   since we assume the isomorphism equations will one day be given by Isabelle*)
    64 
    65 Goal "inj(Rep_Nat)";
    66 by (rtac inj_inverseI 1);
    67 by (rtac Rep_Nat_inverse 1);
    68 qed "inj_Rep_Nat";
    69 
    70 Goal "inj_on Abs_Nat Nat";
    71 by (rtac inj_on_inverseI 1);
    72 by (etac Abs_Nat_inverse 1);
    73 qed "inj_on_Abs_Nat";
    74 
    75 (*** Distinctness of constructors ***)
    76 
    77 Goalw [Zero_def,Suc_def] "Suc(m) ~= 0";
    78 by (rtac (inj_on_Abs_Nat RS inj_on_contraD) 1);
    79 by (rtac Suc_Rep_not_Zero_Rep 1);
    80 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1));
    81 qed "Suc_not_Zero";
    82 
    83 bind_thm ("Zero_not_Suc", Suc_not_Zero RS not_sym);
    84 
    85 AddIffs [Suc_not_Zero,Zero_not_Suc];
    86 
    87 bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE));
    88 val Zero_neq_Suc = sym RS Suc_neq_Zero;
    89 
    90 (** Injectiveness of Suc **)
    91 
    92 Goalw [Suc_def] "inj(Suc)";
    93 by (rtac injI 1);
    94 by (dtac (inj_on_Abs_Nat RS inj_onD) 1);
    95 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1));
    96 by (dtac (inj_Suc_Rep RS injD) 1);
    97 by (etac (inj_Rep_Nat RS injD) 1);
    98 qed "inj_Suc";
    99 
   100 val Suc_inject = inj_Suc RS injD;
   101 
   102 Goal "(Suc(m)=Suc(n)) = (m=n)";
   103 by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]); 
   104 qed "Suc_Suc_eq";
   105 
   106 AddIffs [Suc_Suc_eq];
   107 
   108 Goal "n ~= Suc(n)";
   109 by (nat_ind_tac "n" 1);
   110 by (ALLGOALS Asm_simp_tac);
   111 qed "n_not_Suc_n";
   112 
   113 bind_thm ("Suc_n_not_n", n_not_Suc_n RS not_sym);
   114 
   115 (*** Basic properties of "less than" ***)
   116 
   117 Goalw [wf_def, pred_nat_def] "wf(pred_nat)";
   118 by (Clarify_tac 1);
   119 by (nat_ind_tac "x" 1);
   120 by (ALLGOALS Blast_tac);
   121 qed "wf_pred_nat";
   122 
   123 (*Used in TFL/post.sml*)
   124 Goalw [less_def] "(m,n) : pred_nat^+ = (m<n)";
   125 by (rtac refl 1);
   126 qed "less_eq";
   127 
   128 (** Introduction properties **)
   129 
   130 Goalw [less_def] "[| i<j;  j<k |] ==> i<(k::nat)";
   131 by (rtac (trans_trancl RS transD) 1);
   132 by (assume_tac 1);
   133 by (assume_tac 1);
   134 qed "less_trans";
   135 
   136 Goalw [less_def, pred_nat_def] "n < Suc(n)";
   137 by (simp_tac (simpset() addsimps [r_into_trancl]) 1);
   138 qed "lessI";
   139 AddIffs [lessI];
   140 
   141 (* i<j ==> i<Suc(j) *)
   142 bind_thm("less_SucI", lessI RSN (2, less_trans));
   143 Addsimps [less_SucI];
   144 
   145 Goal "0 < Suc(n)";
   146 by (nat_ind_tac "n" 1);
   147 by (rtac lessI 1);
   148 by (etac less_trans 1);
   149 by (rtac lessI 1);
   150 qed "zero_less_Suc";
   151 AddIffs [zero_less_Suc];
   152 
   153 (** Elimination properties **)
   154 
   155 Goalw [less_def] "n<m ==> ~ m<(n::nat)";
   156 by (blast_tac (claset() addIs [wf_pred_nat, wf_trancl RS wf_asym])1);
   157 qed "less_not_sym";
   158 
   159 (* [| n<m; ~P ==> m<n |] ==> P *)
   160 bind_thm ("less_asym", less_not_sym RS swap);
   161 
   162 Goalw [less_def] "~ n<(n::nat)";
   163 by (rtac notI 1);
   164 by (etac (wf_pred_nat RS wf_trancl RS wf_irrefl) 1);
   165 qed "less_not_refl";
   166 
   167 (* n<n ==> R *)
   168 bind_thm ("less_irrefl", (less_not_refl RS notE));
   169 AddSEs [less_irrefl];
   170 
   171 Goal "n<m ==> m ~= (n::nat)";
   172 by (Blast_tac 1);
   173 qed "less_not_refl2";
   174 
   175 (* s < t ==> s ~= t *)
   176 bind_thm ("less_not_refl3", less_not_refl2 RS not_sym);
   177 
   178 
   179 val major::prems = Goalw [less_def, pred_nat_def]
   180     "[| i<k;  k=Suc(i) ==> P;  !!j. [| i<j;  k=Suc(j) |] ==> P \
   181 \    |] ==> P";
   182 by (rtac (major RS tranclE) 1);
   183 by (ALLGOALS Full_simp_tac); 
   184 by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE'
   185                   eresolve_tac (prems@[asm_rl, Pair_inject])));
   186 qed "lessE";
   187 
   188 Goal "~ n<0";
   189 by (rtac notI 1);
   190 by (etac lessE 1);
   191 by (etac Zero_neq_Suc 1);
   192 by (etac Zero_neq_Suc 1);
   193 qed "not_less0";
   194 
   195 AddIffs [not_less0];
   196 
   197 (* n<0 ==> R *)
   198 bind_thm ("less_zeroE", not_less0 RS notE);
   199 
   200 val [major,less,eq] = Goal
   201     "[| m < Suc(n);  m<n ==> P;  m=n ==> P |] ==> P";
   202 by (rtac (major RS lessE) 1);
   203 by (rtac eq 1);
   204 by (Blast_tac 1);
   205 by (rtac less 1);
   206 by (Blast_tac 1);
   207 qed "less_SucE";
   208 
   209 Goal "(m < Suc(n)) = (m < n | m = n)";
   210 by (blast_tac (claset() addSEs [less_SucE] addIs [less_trans]) 1);
   211 qed "less_Suc_eq";
   212 
   213 Goal "(n<1) = (n=0)";
   214 by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
   215 qed "less_one";
   216 AddIffs [less_one];
   217 
   218 Goal "m<n ==> Suc(m) < Suc(n)";
   219 by (etac rev_mp 1);
   220 by (nat_ind_tac "n" 1);
   221 by (ALLGOALS (fast_tac (claset() addEs [less_trans, lessE])));
   222 qed "Suc_mono";
   223 
   224 (*"Less than" is a linear ordering*)
   225 Goal "m<n | m=n | n<(m::nat)";
   226 by (nat_ind_tac "m" 1);
   227 by (nat_ind_tac "n" 1);
   228 by (rtac (refl RS disjI1 RS disjI2) 1);
   229 by (rtac (zero_less_Suc RS disjI1) 1);
   230 by (blast_tac (claset() addIs [Suc_mono, less_SucI] addEs [lessE]) 1);
   231 qed "less_linear";
   232 
   233 Goal "!!m::nat. (m ~= n) = (m<n | n<m)";
   234 by (cut_facts_tac [less_linear] 1);
   235 by (Blast_tac 1);
   236 qed "nat_neq_iff";
   237 
   238 val [major,eqCase,lessCase] = Goal 
   239    "[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m";
   240 by (rtac (less_linear RS disjE) 1);
   241 by (etac disjE 2);
   242 by (etac lessCase 1);
   243 by (etac (sym RS eqCase) 1);
   244 by (etac major 1);
   245 qed "nat_less_cases";
   246 
   247 
   248 (** Inductive (?) properties **)
   249 
   250 Goal "[| m<n; Suc m ~= n |] ==> Suc(m) < n";
   251 by (full_simp_tac (simpset() addsimps [nat_neq_iff]) 1);
   252 by (blast_tac (claset() addSEs [less_irrefl, less_SucE] addEs [less_asym]) 1);
   253 qed "Suc_lessI";
   254 
   255 Goal "Suc(m) < n ==> m<n";
   256 by (etac rev_mp 1);
   257 by (nat_ind_tac "n" 1);
   258 by (ALLGOALS (fast_tac (claset() addSIs [lessI RS less_SucI]
   259                                  addEs  [less_trans, lessE])));
   260 qed "Suc_lessD";
   261 
   262 val [major,minor] = Goal 
   263     "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
   264 \    |] ==> P";
   265 by (rtac (major RS lessE) 1);
   266 by (etac (lessI RS minor) 1);
   267 by (etac (Suc_lessD RS minor) 1);
   268 by (assume_tac 1);
   269 qed "Suc_lessE";
   270 
   271 Goal "Suc(m) < Suc(n) ==> m<n";
   272 by (blast_tac (claset() addEs [lessE, make_elim Suc_lessD]) 1);
   273 qed "Suc_less_SucD";
   274 
   275 
   276 Goal "(Suc(m) < Suc(n)) = (m<n)";
   277 by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
   278 qed "Suc_less_eq";
   279 Addsimps [Suc_less_eq];
   280 
   281 (*Goal "~(Suc(n) < n)";
   282 by (blast_tac (claset() addEs [Suc_lessD RS less_irrefl]) 1);
   283 qed "not_Suc_n_less_n";
   284 Addsimps [not_Suc_n_less_n];*)
   285 
   286 Goal "i<j ==> j<k --> Suc i < k";
   287 by (nat_ind_tac "k" 1);
   288 by (ALLGOALS (asm_simp_tac (simpset())));
   289 by (asm_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
   290 by (blast_tac (claset() addDs [Suc_lessD]) 1);
   291 qed_spec_mp "less_trans_Suc";
   292 
   293 (*Can be used with less_Suc_eq to get n=m | n<m *)
   294 Goal "(~ m < n) = (n < Suc(m))";
   295 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   296 by (ALLGOALS Asm_simp_tac);
   297 qed "not_less_eq";
   298 
   299 (*Complete induction, aka course-of-values induction*)
   300 val prems = Goalw [less_def]
   301     "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
   302 by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
   303 by (eresolve_tac prems 1);
   304 qed "less_induct";
   305 
   306 (*** Properties of <= ***)
   307 
   308 (*Was le_eq_less_Suc, but this orientation is more useful*)
   309 Goalw [le_def] "(m < Suc n) = (m <= n)";
   310 by (rtac (not_less_eq RS sym) 1);
   311 qed "less_Suc_eq_le";
   312 
   313 (*  m<=n ==> m < Suc n  *)
   314 bind_thm ("le_imp_less_Suc", less_Suc_eq_le RS iffD2);
   315 
   316 Goalw [le_def] "0 <= n";
   317 by (rtac not_less0 1);
   318 qed "le0";
   319 AddIffs [le0];
   320 
   321 Goalw [le_def] "~ Suc n <= n";
   322 by (Simp_tac 1);
   323 qed "Suc_n_not_le_n";
   324 
   325 Goalw [le_def] "(i <= 0) = (i = 0)";
   326 by (nat_ind_tac "i" 1);
   327 by (ALLGOALS Asm_simp_tac);
   328 qed "le_0_eq";
   329 AddIffs [le_0_eq];
   330 
   331 Goal "(m <= Suc(n)) = (m<=n | m = Suc n)";
   332 by (simp_tac (simpset() delsimps [less_Suc_eq_le]
   333 			addsimps [less_Suc_eq_le RS sym, less_Suc_eq]) 1);
   334 qed "le_Suc_eq";
   335 
   336 (* [| m <= Suc n;  m <= n ==> R;  m = Suc n ==> R |] ==> R *)
   337 bind_thm ("le_SucE", le_Suc_eq RS iffD1 RS disjE);
   338 
   339 Goalw [le_def] "~n<m ==> m<=(n::nat)";
   340 by (assume_tac 1);
   341 qed "leI";
   342 
   343 Goalw [le_def] "m<=n ==> ~ n < (m::nat)";
   344 by (assume_tac 1);
   345 qed "leD";
   346 
   347 val leE = make_elim leD;
   348 
   349 Goal "(~n<m) = (m<=(n::nat))";
   350 by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
   351 qed "not_less_iff_le";
   352 
   353 Goalw [le_def] "~ m <= n ==> n<(m::nat)";
   354 by (Blast_tac 1);
   355 qed "not_leE";
   356 
   357 Goalw [le_def] "(~n<=m) = (m<(n::nat))";
   358 by (Simp_tac 1);
   359 qed "not_le_iff_less";
   360 
   361 Goalw [le_def] "m < n ==> Suc(m) <= n";
   362 by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
   363 by (blast_tac (claset() addSEs [less_irrefl,less_asym]) 1);
   364 qed "Suc_leI";  (*formerly called lessD*)
   365 
   366 Goalw [le_def] "Suc(m) <= n ==> m <= n";
   367 by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
   368 qed "Suc_leD";
   369 
   370 (* stronger version of Suc_leD *)
   371 Goalw [le_def] "Suc m <= n ==> m < n";
   372 by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
   373 by (cut_facts_tac [less_linear] 1);
   374 by (Blast_tac 1);
   375 qed "Suc_le_lessD";
   376 
   377 Goal "(Suc m <= n) = (m < n)";
   378 by (blast_tac (claset() addIs [Suc_leI, Suc_le_lessD]) 1);
   379 qed "Suc_le_eq";
   380 
   381 Goalw [le_def] "m <= n ==> m <= Suc n";
   382 by (blast_tac (claset() addDs [Suc_lessD]) 1);
   383 qed "le_SucI";
   384 Addsimps[le_SucI];
   385 
   386 (*bind_thm ("le_Suc", not_Suc_n_less_n RS leI);*)
   387 
   388 Goalw [le_def] "m < n ==> m <= (n::nat)";
   389 by (blast_tac (claset() addEs [less_asym]) 1);
   390 qed "less_imp_le";
   391 
   392 (*For instance, (Suc m < Suc n)  =   (Suc m <= n)  =  (m<n) *)
   393 val le_simps = [less_imp_le, less_Suc_eq_le, Suc_le_eq];
   394 
   395 
   396 (** Equivalence of m<=n and  m<n | m=n **)
   397 
   398 Goalw [le_def] "m <= n ==> m < n | m=(n::nat)";
   399 by (cut_facts_tac [less_linear] 1);
   400 by (blast_tac (claset() addEs [less_irrefl,less_asym]) 1);
   401 qed "le_imp_less_or_eq";
   402 
   403 Goalw [le_def] "m<n | m=n ==> m <=(n::nat)";
   404 by (cut_facts_tac [less_linear] 1);
   405 by (blast_tac (claset() addSEs [less_irrefl] addEs [less_asym]) 1);
   406 qed "less_or_eq_imp_le";
   407 
   408 Goal "(m <= (n::nat)) = (m < n | m=n)";
   409 by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
   410 qed "le_eq_less_or_eq";
   411 
   412 (*Useful with Blast_tac.   m=n ==> m<=n *)
   413 bind_thm ("eq_imp_le", disjI2 RS less_or_eq_imp_le);
   414 
   415 Goal "n <= (n::nat)";
   416 by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
   417 qed "le_refl";
   418 
   419 
   420 Goal "[| i <= j; j < k |] ==> i < (k::nat)";
   421 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
   422 	                addIs [less_trans]) 1);
   423 qed "le_less_trans";
   424 
   425 Goal "[| i < j; j <= k |] ==> i < (k::nat)";
   426 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
   427 	                addIs [less_trans]) 1);
   428 qed "less_le_trans";
   429 
   430 Goal "[| i <= j; j <= k |] ==> i <= (k::nat)";
   431 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
   432 	                addIs [less_or_eq_imp_le, less_trans]) 1);
   433 qed "le_trans";
   434 
   435 Goal "[| m <= n; n <= m |] ==> m = (n::nat)";
   436 (*order_less_irrefl could make this proof fail*)
   437 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
   438 	                addSEs [less_irrefl] addEs [less_asym]) 1);
   439 qed "le_anti_sym";
   440 
   441 Goal "(Suc(n) <= Suc(m)) = (n <= m)";
   442 by (simp_tac (simpset() addsimps le_simps) 1);
   443 qed "Suc_le_mono";
   444 
   445 AddIffs [Suc_le_mono];
   446 
   447 (* Axiom 'order_less_le' of class 'order': *)
   448 Goal "(m::nat) < n = (m <= n & m ~= n)";
   449 by (simp_tac (simpset() addsimps [le_def, nat_neq_iff]) 1);
   450 by (blast_tac (claset() addSEs [less_asym]) 1);
   451 qed "nat_less_le";
   452 
   453 (* [| m <= n; m ~= n |] ==> m < n *)
   454 bind_thm ("le_neq_implies_less", [nat_less_le, conjI] MRS iffD2);
   455 
   456 (* Axiom 'linorder_linear' of class 'linorder': *)
   457 Goal "(m::nat) <= n | n <= m";
   458 by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
   459 by (cut_facts_tac [less_linear] 1);
   460 by (Blast_tac 1);
   461 qed "nat_le_linear";
   462 
   463 Goal "~ n < m ==> (n < Suc m) = (n = m)";
   464 by (blast_tac (claset() addSEs [less_SucE]) 1);
   465 qed "not_less_less_Suc_eq";
   466 
   467 
   468 (*Rewrite (n < Suc m) to (n=m) if  ~ n<m or m<=n hold.
   469   Not suitable as default simprules because they often lead to looping*)
   470 val not_less_simps = [not_less_less_Suc_eq, leD RS not_less_less_Suc_eq];
   471 
   472 (** LEAST -- the least number operator **)
   473 
   474 Goal "(! m::nat. P m --> n <= m) = (! m. m < n --> ~ P m)";
   475 by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
   476 val lemma = result();
   477 
   478 (* This is an old def of Least for nat, which is derived for compatibility *)
   479 Goalw [Least_def]
   480   "(LEAST n::nat. P n) == (@n. P(n) & (ALL m. m < n --> ~P(m)))";
   481 by (simp_tac (simpset() addsimps [lemma]) 1);
   482 qed "Least_nat_def";
   483 
   484 val [prem1,prem2] = Goalw [Least_nat_def]
   485     "[| P(k::nat);  !!x. x<k ==> ~P(x) |] ==> (LEAST x. P(x)) = k";
   486 by (rtac select_equality 1);
   487 by (blast_tac (claset() addSIs [prem1,prem2]) 1);
   488 by (cut_facts_tac [less_linear] 1);
   489 by (blast_tac (claset() addSIs [prem1] addSDs [prem2]) 1);
   490 qed "Least_equality";
   491 
   492 Goal "P(k::nat) ==> P(LEAST x. P(x))";
   493 by (etac rev_mp 1);
   494 by (res_inst_tac [("n","k")] less_induct 1);
   495 by (rtac impI 1);
   496 by (rtac classical 1);
   497 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
   498 by (assume_tac 1);
   499 by (assume_tac 2);
   500 by (Blast_tac 1);
   501 qed "LeastI";
   502 
   503 (*Proof is almost identical to the one above!*)
   504 Goal "P(k::nat) ==> (LEAST x. P(x)) <= k";
   505 by (etac rev_mp 1);
   506 by (res_inst_tac [("n","k")] less_induct 1);
   507 by (rtac impI 1);
   508 by (rtac classical 1);
   509 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
   510 by (assume_tac 1);
   511 by (rtac le_refl 2);
   512 by (blast_tac (claset() addIs [less_imp_le,le_trans]) 1);
   513 qed "Least_le";
   514 
   515 Goal "k < (LEAST x. P(x)) ==> ~P(k::nat)";
   516 by (rtac notI 1);
   517 by (etac (rewrite_rule [le_def] Least_le RS notE) 1 THEN assume_tac 1);
   518 qed "not_less_Least";
   519 
   520 (* [| m ~= n; m < n ==> P; n < m ==> P |] ==> P *)
   521 bind_thm("nat_neqE", nat_neq_iff RS iffD1 RS disjE);
   522 
   523 Goal "(S::nat set) ~= {} ==> ? x:S. ! y:S. x <= y";
   524 by (cut_facts_tac [wf_pred_nat RS wf_trancl RS (wf_eq_minimal RS iffD1)] 1);
   525 by (dres_inst_tac [("x","S")] spec 1);
   526 by (Asm_full_simp_tac 1);
   527 by (etac impE 1);
   528 by (Force_tac 1);
   529 by (force_tac (claset(), simpset() addsimps [less_eq,not_le_iff_less]) 1);
   530 qed "nonempty_has_least";