src/HOL/NatDef.ML
author paulson
Wed Nov 05 13:23:46 1997 +0100 (1997-11-05)
changeset 4153 e534c4c32d54
parent 4104 84433b1ab826
child 4356 0dfd34f0d33d
permissions -rw-r--r--
Ran expandshort, especially to introduce Safe_tac
     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 thy "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 thy "Zero_Rep: Nat";
    15 by (stac Nat_unfold 1);
    16 by (rtac (singletonI RS UnI1) 1);
    17 qed "Zero_RepI";
    18 
    19 val prems = goal thy "i: Nat ==> Suc_Rep(i) : Nat";
    20 by (stac Nat_unfold 1);
    21 by (rtac (imageI RS UnI2) 1);
    22 by (resolve_tac prems 1);
    23 qed "Suc_RepI";
    24 
    25 (*** Induction ***)
    26 
    27 val major::prems = goal thy
    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 thy [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 local fun raw_nat_ind_tac a i = 
    45     res_inst_tac [("n",a)] nat_induct i  THEN  rename_last_tac a [""] (i+1)
    46 in
    47 val nat_ind_tac = Datatype.occs_in_prems raw_nat_ind_tac
    48 end;
    49 
    50 (*A special form of induction for reasoning about m<n and m-n*)
    51 val prems = goal thy
    52     "[| !!x. P x 0;  \
    53 \       !!y. P 0 (Suc y);  \
    54 \       !!x y. [| P x y |] ==> P (Suc x) (Suc y)  \
    55 \    |] ==> P m n";
    56 by (res_inst_tac [("x","m")] spec 1);
    57 by (nat_ind_tac "n" 1);
    58 by (rtac allI 2);
    59 by (nat_ind_tac "x" 2);
    60 by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1));
    61 qed "diff_induct";
    62 
    63 (*Case analysis on the natural numbers*)
    64 val prems = goal thy 
    65     "[| n=0 ==> P;  !!x. n = Suc(x) ==> P |] ==> P";
    66 by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1);
    67 by (fast_tac (claset() addSEs prems) 1);
    68 by (nat_ind_tac "n" 1);
    69 by (rtac (refl RS disjI1) 1);
    70 by (Blast_tac 1);
    71 qed "natE";
    72 
    73 
    74 (*** Isomorphisms: Abs_Nat and Rep_Nat ***)
    75 
    76 (*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat),
    77   since we assume the isomorphism equations will one day be given by Isabelle*)
    78 
    79 goal thy "inj(Rep_Nat)";
    80 by (rtac inj_inverseI 1);
    81 by (rtac Rep_Nat_inverse 1);
    82 qed "inj_Rep_Nat";
    83 
    84 goal thy "inj_onto Abs_Nat Nat";
    85 by (rtac inj_onto_inverseI 1);
    86 by (etac Abs_Nat_inverse 1);
    87 qed "inj_onto_Abs_Nat";
    88 
    89 (*** Distinctness of constructors ***)
    90 
    91 goalw thy [Zero_def,Suc_def] "Suc(m) ~= 0";
    92 by (rtac (inj_onto_Abs_Nat RS inj_onto_contraD) 1);
    93 by (rtac Suc_Rep_not_Zero_Rep 1);
    94 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1));
    95 qed "Suc_not_Zero";
    96 
    97 bind_thm ("Zero_not_Suc", Suc_not_Zero RS not_sym);
    98 
    99 AddIffs [Suc_not_Zero,Zero_not_Suc];
   100 
   101 bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE));
   102 val Zero_neq_Suc = sym RS Suc_neq_Zero;
   103 
   104 (** Injectiveness of Suc **)
   105 
   106 goalw thy [Suc_def] "inj(Suc)";
   107 by (rtac injI 1);
   108 by (dtac (inj_onto_Abs_Nat RS inj_ontoD) 1);
   109 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1));
   110 by (dtac (inj_Suc_Rep RS injD) 1);
   111 by (etac (inj_Rep_Nat RS injD) 1);
   112 qed "inj_Suc";
   113 
   114 val Suc_inject = inj_Suc RS injD;
   115 
   116 goal thy "(Suc(m)=Suc(n)) = (m=n)";
   117 by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]); 
   118 qed "Suc_Suc_eq";
   119 
   120 AddIffs [Suc_Suc_eq];
   121 
   122 goal thy "n ~= Suc(n)";
   123 by (nat_ind_tac "n" 1);
   124 by (ALLGOALS Asm_simp_tac);
   125 qed "n_not_Suc_n";
   126 
   127 bind_thm ("Suc_n_not_n", n_not_Suc_n RS not_sym);
   128 
   129 goal thy "!!n. n ~= 0 ==> EX m. n = Suc m";
   130 by (rtac natE 1);
   131 by (REPEAT (Blast_tac 1));
   132 qed "not0_implies_Suc";
   133 
   134 
   135 (*** nat_case -- the selection operator for nat ***)
   136 
   137 goalw thy [nat_case_def] "nat_case a f 0 = a";
   138 by (blast_tac (claset() addIs [select_equality]) 1);
   139 qed "nat_case_0";
   140 
   141 goalw thy [nat_case_def] "nat_case a f (Suc k) = f(k)";
   142 by (blast_tac (claset() addIs [select_equality]) 1);
   143 qed "nat_case_Suc";
   144 
   145 goalw thy [wf_def, pred_nat_def] "wf(pred_nat)";
   146 by (Clarify_tac 1);
   147 by (nat_ind_tac "x" 1);
   148 by (ALLGOALS Blast_tac);
   149 qed "wf_pred_nat";
   150 
   151 
   152 (*** nat_rec -- by wf recursion on pred_nat ***)
   153 
   154 (* The unrolling rule for nat_rec *)
   155 goal thy
   156    "(%n. nat_rec c d n) = wfrec pred_nat (%f. nat_case ?c (%m. ?d m (f m)))";
   157   by (simp_tac (HOL_ss addsimps [nat_rec_def]) 1);
   158 bind_thm("nat_rec_unfold", wf_pred_nat RS 
   159                             ((result() RS eq_reflection) RS def_wfrec));
   160 
   161 (*---------------------------------------------------------------------------
   162  * Old:
   163  * bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec))); 
   164  *---------------------------------------------------------------------------*)
   165 
   166 (** conversion rules **)
   167 
   168 goal thy "nat_rec c h 0 = c";
   169 by (rtac (nat_rec_unfold RS trans) 1);
   170 by (simp_tac (simpset() addsimps [nat_case_0]) 1);
   171 qed "nat_rec_0";
   172 
   173 goal thy "nat_rec c h (Suc n) = h n (nat_rec c h n)";
   174 by (rtac (nat_rec_unfold RS trans) 1);
   175 by (simp_tac (simpset() addsimps [nat_case_Suc, pred_nat_def, cut_apply]) 1);
   176 qed "nat_rec_Suc";
   177 
   178 (*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
   179 val [rew] = goal thy
   180     "[| !!n. f(n) == nat_rec c h n |] ==> f(0) = c";
   181 by (rewtac rew);
   182 by (rtac nat_rec_0 1);
   183 qed "def_nat_rec_0";
   184 
   185 val [rew] = goal thy
   186     "[| !!n. f(n) == nat_rec c h n |] ==> f(Suc(n)) = h n (f n)";
   187 by (rewtac rew);
   188 by (rtac nat_rec_Suc 1);
   189 qed "def_nat_rec_Suc";
   190 
   191 fun nat_recs def =
   192       [standard (def RS def_nat_rec_0),
   193        standard (def RS def_nat_rec_Suc)];
   194 
   195 
   196 (*** Basic properties of "less than" ***)
   197 
   198 (*Used in TFL/post.sml*)
   199 goalw thy [less_def] "(m,n) : pred_nat^+ = (m<n)";
   200 by (rtac refl 1);
   201 qed "less_eq";
   202 
   203 (** Introduction properties **)
   204 
   205 val prems = goalw thy [less_def] "[| i<j;  j<k |] ==> i<(k::nat)";
   206 by (rtac (trans_trancl RS transD) 1);
   207 by (resolve_tac prems 1);
   208 by (resolve_tac prems 1);
   209 qed "less_trans";
   210 
   211 goalw thy [less_def, pred_nat_def] "n < Suc(n)";
   212 by (simp_tac (simpset() addsimps [r_into_trancl]) 1);
   213 qed "lessI";
   214 AddIffs [lessI];
   215 
   216 (* i<j ==> i<Suc(j) *)
   217 bind_thm("less_SucI", lessI RSN (2, less_trans));
   218 Addsimps [less_SucI];
   219 
   220 goal thy "0 < Suc(n)";
   221 by (nat_ind_tac "n" 1);
   222 by (rtac lessI 1);
   223 by (etac less_trans 1);
   224 by (rtac lessI 1);
   225 qed "zero_less_Suc";
   226 AddIffs [zero_less_Suc];
   227 
   228 (** Elimination properties **)
   229 
   230 val prems = goalw thy [less_def] "n<m ==> ~ m<(n::nat)";
   231 by (blast_tac (claset() addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1);
   232 qed "less_not_sym";
   233 
   234 (* [| n<m; m<n |] ==> R *)
   235 bind_thm ("less_asym", (less_not_sym RS notE));
   236 
   237 goalw thy [less_def] "~ n<(n::nat)";
   238 by (rtac notI 1);
   239 by (etac (wf_pred_nat RS wf_trancl RS wf_irrefl) 1);
   240 qed "less_not_refl";
   241 
   242 (* n<n ==> R *)
   243 bind_thm ("less_irrefl", (less_not_refl RS notE));
   244 
   245 goal thy "!!m. n<m ==> m ~= (n::nat)";
   246 by (blast_tac (claset() addSEs [less_irrefl]) 1);
   247 qed "less_not_refl2";
   248 
   249 
   250 val major::prems = goalw thy [less_def, pred_nat_def]
   251     "[| i<k;  k=Suc(i) ==> P;  !!j. [| i<j;  k=Suc(j) |] ==> P \
   252 \    |] ==> P";
   253 by (rtac (major RS tranclE) 1);
   254 by (ALLGOALS Full_simp_tac); 
   255 by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE'
   256                   eresolve_tac (prems@[asm_rl, Pair_inject])));
   257 qed "lessE";
   258 
   259 goal thy "~ n<0";
   260 by (rtac notI 1);
   261 by (etac lessE 1);
   262 by (etac Zero_neq_Suc 1);
   263 by (etac Zero_neq_Suc 1);
   264 qed "not_less0";
   265 
   266 AddIffs [not_less0];
   267 
   268 (* n<0 ==> R *)
   269 bind_thm ("less_zeroE", not_less0 RS notE);
   270 
   271 val [major,less,eq] = goal thy
   272     "[| m < Suc(n);  m<n ==> P;  m=n ==> P |] ==> P";
   273 by (rtac (major RS lessE) 1);
   274 by (rtac eq 1);
   275 by (Blast_tac 1);
   276 by (rtac less 1);
   277 by (Blast_tac 1);
   278 qed "less_SucE";
   279 
   280 goal thy "(m < Suc(n)) = (m < n | m = n)";
   281 by (blast_tac (claset() addSEs [less_SucE] addIs [less_trans]) 1);
   282 qed "less_Suc_eq";
   283 
   284 goal thy "(n<1) = (n=0)";
   285 by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
   286 qed "less_one";
   287 AddIffs [less_one];
   288 
   289 val prems = goal thy "m<n ==> n ~= 0";
   290 by (res_inst_tac [("n","n")] natE 1);
   291 by (cut_facts_tac prems 1);
   292 by (ALLGOALS Asm_full_simp_tac);
   293 qed "gr_implies_not0";
   294 Addsimps [gr_implies_not0];
   295 
   296 qed_goal "zero_less_eq" thy "0 < n = (n ~= 0)" (fn _ => [
   297         rtac iffI 1,
   298         etac gr_implies_not0 1,
   299         rtac natE 1,
   300         contr_tac 1,
   301         etac ssubst 1,
   302         rtac zero_less_Suc 1]);
   303 
   304 (** Inductive (?) properties **)
   305 
   306 val [prem] = goal thy "Suc(m) < n ==> m<n";
   307 by (rtac (prem RS rev_mp) 1);
   308 by (nat_ind_tac "n" 1);
   309 by (ALLGOALS (fast_tac (claset() addSIs [lessI RS less_SucI]
   310                                 addEs  [less_trans, lessE])));
   311 qed "Suc_lessD";
   312 
   313 val [major,minor] = goal thy 
   314     "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
   315 \    |] ==> P";
   316 by (rtac (major RS lessE) 1);
   317 by (etac (lessI RS minor) 1);
   318 by (etac (Suc_lessD RS minor) 1);
   319 by (assume_tac 1);
   320 qed "Suc_lessE";
   321 
   322 goal thy "!!m n. Suc(m) < Suc(n) ==> m<n";
   323 by (blast_tac (claset() addEs [lessE, make_elim Suc_lessD]) 1);
   324 qed "Suc_less_SucD";
   325 
   326 goal thy "!!m n. m<n ==> Suc(m) < Suc(n)";
   327 by (etac rev_mp 1);
   328 by (nat_ind_tac "n" 1);
   329 by (ALLGOALS (fast_tac (claset() addEs  [less_trans, lessE])));
   330 qed "Suc_mono";
   331 
   332 
   333 goal thy "(Suc(m) < Suc(n)) = (m<n)";
   334 by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
   335 qed "Suc_less_eq";
   336 Addsimps [Suc_less_eq];
   337 
   338 goal thy "~(Suc(n) < n)";
   339 by (blast_tac (claset() addEs [Suc_lessD RS less_irrefl]) 1);
   340 qed "not_Suc_n_less_n";
   341 Addsimps [not_Suc_n_less_n];
   342 
   343 goal thy "!!i. i<j ==> j<k --> Suc i < k";
   344 by (nat_ind_tac "k" 1);
   345 by (ALLGOALS (asm_simp_tac (simpset())));
   346 by (asm_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
   347 by (blast_tac (claset() addDs [Suc_lessD]) 1);
   348 qed_spec_mp "less_trans_Suc";
   349 
   350 (*"Less than" is a linear ordering*)
   351 goal thy "m<n | m=n | n<(m::nat)";
   352 by (nat_ind_tac "m" 1);
   353 by (nat_ind_tac "n" 1);
   354 by (rtac (refl RS disjI1 RS disjI2) 1);
   355 by (rtac (zero_less_Suc RS disjI1) 1);
   356 by (blast_tac (claset() addIs [Suc_mono, less_SucI] addEs [lessE]) 1);
   357 qed "less_linear";
   358 
   359 qed_goal "nat_less_cases" thy 
   360    "[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m"
   361 ( fn [major,eqCase,lessCase] =>
   362         [
   363         (rtac (less_linear RS disjE) 1),
   364         (etac disjE 2),
   365         (etac lessCase 1),
   366         (etac (sym RS eqCase) 1),
   367         (etac major 1)
   368         ]);
   369 
   370 (*Can be used with less_Suc_eq to get n=m | n<m *)
   371 goal thy "(~ m < n) = (n < Suc(m))";
   372 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   373 by (ALLGOALS Asm_simp_tac);
   374 qed "not_less_eq";
   375 
   376 (*Complete induction, aka course-of-values induction*)
   377 val prems = goalw thy [less_def]
   378     "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
   379 by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
   380 by (eresolve_tac prems 1);
   381 qed "less_induct";
   382 
   383 qed_goal "nat_induct2" thy 
   384 "[| P 0; P 1; !!k. P k ==> P (Suc (Suc k)) |] ==> P n" (fn prems => [
   385         cut_facts_tac prems 1,
   386         rtac less_induct 1,
   387         res_inst_tac [("n","n")] natE 1,
   388          hyp_subst_tac 1,
   389          atac 1,
   390         hyp_subst_tac 1,
   391         res_inst_tac [("n","x")] natE 1,
   392          hyp_subst_tac 1,
   393          atac 1,
   394         hyp_subst_tac 1,
   395         resolve_tac prems 1,
   396         dtac spec 1,
   397         etac mp 1,
   398         rtac (lessI RS less_trans) 1,
   399         rtac (lessI RS Suc_mono) 1]);
   400 
   401 (*** Properties of <= ***)
   402 
   403 goalw thy [le_def] "(m <= n) = (m < Suc n)";
   404 by (rtac not_less_eq 1);
   405 qed "le_eq_less_Suc";
   406 
   407 (*  m<=n ==> m < Suc n  *)
   408 bind_thm ("le_imp_less_Suc", le_eq_less_Suc RS iffD1);
   409 
   410 goalw thy [le_def] "0 <= n";
   411 by (rtac not_less0 1);
   412 qed "le0";
   413 
   414 goalw thy [le_def] "~ Suc n <= n";
   415 by (Simp_tac 1);
   416 qed "Suc_n_not_le_n";
   417 
   418 goalw thy [le_def] "(i <= 0) = (i = 0)";
   419 by (nat_ind_tac "i" 1);
   420 by (ALLGOALS Asm_simp_tac);
   421 qed "le_0_eq";
   422 
   423 Addsimps [(*less_Suc_eq, makes simpset non-confluent*) le0, le_0_eq,
   424           Suc_n_not_le_n,
   425           n_not_Suc_n, Suc_n_not_n,
   426           nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];
   427 
   428 goal thy "!!m. (m <= Suc(n)) = (m<=n | m = Suc n)";
   429 by (simp_tac (simpset() addsimps [le_eq_less_Suc]) 1);
   430 by (blast_tac (claset() addSEs [less_SucE] addIs [less_SucI]) 1);
   431 qed "le_Suc_eq";
   432 
   433 (*
   434 goal thy "(Suc m < n | Suc m = n) = (m < n)";
   435 by (stac (less_Suc_eq RS sym) 1);
   436 by (rtac Suc_less_eq 1);
   437 qed "Suc_le_eq";
   438 
   439 this could make the simpset (with less_Suc_eq added again) more confluent,
   440 but less_Suc_eq makes additional problems with terms of the form 0 < Suc (...)
   441 *)
   442 
   443 (*Prevents simplification of f and g: much faster*)
   444 qed_goal "nat_case_weak_cong" thy
   445   "m=n ==> nat_case a f m = nat_case a f n"
   446   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   447 
   448 qed_goal "nat_rec_weak_cong" thy
   449   "m=n ==> nat_rec a f m = nat_rec a f n"
   450   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   451 
   452 qed_goal "expand_nat_case" thy
   453   "P(nat_case z s n) = ((n=0 --> P z) & (!m. n = Suc m --> P(s m)))"
   454   (fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
   455 
   456 val prems = goalw thy [le_def] "~n<m ==> m<=(n::nat)";
   457 by (resolve_tac prems 1);
   458 qed "leI";
   459 
   460 val prems = goalw thy [le_def] "m<=n ==> ~ n < (m::nat)";
   461 by (resolve_tac prems 1);
   462 qed "leD";
   463 
   464 val leE = make_elim leD;
   465 
   466 goal thy "(~n<m) = (m<=(n::nat))";
   467 by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
   468 qed "not_less_iff_le";
   469 
   470 goalw thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
   471 by (Blast_tac 1);
   472 qed "not_leE";
   473 
   474 goalw thy [le_def] "!!m. m < n ==> Suc(m) <= n";
   475 by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
   476 by (blast_tac (claset() addSEs [less_irrefl,less_asym]) 1);
   477 qed "Suc_leI";  (*formerly called lessD*)
   478 
   479 goalw thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
   480 by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
   481 qed "Suc_leD";
   482 
   483 (* stronger version of Suc_leD *)
   484 goalw thy [le_def] 
   485         "!!m. Suc m <= n ==> m < n";
   486 by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
   487 by (cut_facts_tac [less_linear] 1);
   488 by (Blast_tac 1);
   489 qed "Suc_le_lessD";
   490 
   491 goal thy "(Suc m <= n) = (m < n)";
   492 by (blast_tac (claset() addIs [Suc_leI, Suc_le_lessD]) 1);
   493 qed "Suc_le_eq";
   494 
   495 goalw thy [le_def] "!!m. m <= n ==> m <= Suc n";
   496 by (blast_tac (claset() addDs [Suc_lessD]) 1);
   497 qed "le_SucI";
   498 Addsimps[le_SucI];
   499 
   500 bind_thm ("le_Suc", not_Suc_n_less_n RS leI);
   501 
   502 goalw thy [le_def] "!!m. m < n ==> m <= (n::nat)";
   503 by (blast_tac (claset() addEs [less_asym]) 1);
   504 qed "less_imp_le";
   505 
   506 (** Equivalence of m<=n and  m<n | m=n **)
   507 
   508 goalw thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
   509 by (cut_facts_tac [less_linear] 1);
   510 by (blast_tac (claset() addEs [less_irrefl,less_asym]) 1);
   511 qed "le_imp_less_or_eq";
   512 
   513 goalw thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
   514 by (cut_facts_tac [less_linear] 1);
   515 by (blast_tac (claset() addSEs [less_irrefl] addEs [less_asym]) 1);
   516 qed "less_or_eq_imp_le";
   517 
   518 goal thy "(m <= (n::nat)) = (m < n | m=n)";
   519 by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
   520 qed "le_eq_less_or_eq";
   521 
   522 goal thy "n <= (n::nat)";
   523 by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
   524 qed "le_refl";
   525 
   526 val prems = goal thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
   527 by (dtac le_imp_less_or_eq 1);
   528 by (blast_tac (claset() addIs [less_trans]) 1);
   529 qed "le_less_trans";
   530 
   531 goal thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
   532 by (dtac le_imp_less_or_eq 1);
   533 by (blast_tac (claset() addIs [less_trans]) 1);
   534 qed "less_le_trans";
   535 
   536 goal thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";
   537 by (EVERY1[dtac le_imp_less_or_eq, 
   538            dtac le_imp_less_or_eq,
   539            rtac less_or_eq_imp_le, 
   540            blast_tac (claset() addIs [less_trans])]);
   541 qed "le_trans";
   542 
   543 goal thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";
   544 by (EVERY1[dtac le_imp_less_or_eq, 
   545            dtac le_imp_less_or_eq,
   546            blast_tac (claset() addEs [less_irrefl,less_asym])]);
   547 qed "le_anti_sym";
   548 
   549 goal thy "(Suc(n) <= Suc(m)) = (n <= m)";
   550 by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
   551 qed "Suc_le_mono";
   552 
   553 AddIffs [Suc_le_mono];
   554 
   555 (* Axiom 'order_le_less' of class 'order': *)
   556 goal thy "(m::nat) < n = (m <= n & m ~= n)";
   557 by (rtac iffI 1);
   558  by (rtac conjI 1);
   559   by (etac less_imp_le 1);
   560  by (etac (less_not_refl2 RS not_sym) 1);
   561 by (blast_tac (claset() addSDs [le_imp_less_or_eq]) 1);
   562 qed "nat_less_le";
   563 
   564 (** LEAST -- the least number operator **)
   565 
   566 goal thy "(! m::nat. P m --> n <= m) = (! m. m < n --> ~ P m)";
   567 by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
   568 val lemma = result();
   569 
   570 (* This is an old def of Least for nat, which is derived for compatibility *)
   571 goalw thy [Least_def]
   572   "(LEAST n::nat. P n) == (@n. P(n) & (ALL m. m < n --> ~P(m)))";
   573 by (simp_tac (simpset() addsimps [lemma]) 1);
   574 by (rtac eq_reflection 1);
   575 by (rtac refl 1);
   576 qed "Least_nat_def";
   577 
   578 val [prem1,prem2] = goalw thy [Least_nat_def]
   579     "[| P(k::nat);  !!x. x<k ==> ~P(x) |] ==> (LEAST x. P(x)) = k";
   580 by (rtac select_equality 1);
   581 by (blast_tac (claset() addSIs [prem1,prem2]) 1);
   582 by (cut_facts_tac [less_linear] 1);
   583 by (blast_tac (claset() addSIs [prem1] addSDs [prem2]) 1);
   584 qed "Least_equality";
   585 
   586 val [prem] = goal thy "P(k::nat) ==> P(LEAST x. P(x))";
   587 by (rtac (prem RS rev_mp) 1);
   588 by (res_inst_tac [("n","k")] less_induct 1);
   589 by (rtac impI 1);
   590 by (rtac classical 1);
   591 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
   592 by (assume_tac 1);
   593 by (assume_tac 2);
   594 by (Blast_tac 1);
   595 qed "LeastI";
   596 
   597 (*Proof is almost identical to the one above!*)
   598 val [prem] = goal thy "P(k::nat) ==> (LEAST x. P(x)) <= k";
   599 by (rtac (prem RS rev_mp) 1);
   600 by (res_inst_tac [("n","k")] less_induct 1);
   601 by (rtac impI 1);
   602 by (rtac classical 1);
   603 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
   604 by (assume_tac 1);
   605 by (rtac le_refl 2);
   606 by (blast_tac (claset() addIs [less_imp_le,le_trans]) 1);
   607 qed "Least_le";
   608 
   609 val [prem] = goal thy "k < (LEAST x. P(x)) ==> ~P(k::nat)";
   610 by (rtac notI 1);
   611 by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
   612 by (rtac prem 1);
   613 qed "not_less_Least";
   614 
   615 qed_goalw "Least_Suc" thy [Least_nat_def]
   616  "!!P. [| ? n. P(Suc n); ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
   617  (fn _ => [
   618         rtac select_equality 1,
   619         fold_goals_tac [Least_nat_def],
   620         safe_tac (claset() addSEs [LeastI]),
   621         rename_tac "j" 1,
   622         res_inst_tac [("n","j")] natE 1,
   623         Blast_tac 1,
   624         blast_tac (claset() addDs [Suc_less_SucD, not_less_Least]) 1,
   625         rename_tac "k n" 1,
   626         res_inst_tac [("n","k")] natE 1,
   627         Blast_tac 1,
   628         hyp_subst_tac 1,
   629         rewtac Least_nat_def,
   630         rtac (select_equality RS arg_cong RS sym) 1,
   631         Safe_tac,
   632         dtac Suc_mono 1,
   633         Blast_tac 1,
   634         cut_facts_tac [less_linear] 1,
   635         Safe_tac,
   636         atac 2,
   637         Blast_tac 2,
   638         dtac Suc_mono 1,
   639         Blast_tac 1]);
   640 
   641 
   642 (*** Instantiation of transitivity prover ***)
   643 
   644 structure Less_Arith =
   645 struct
   646 val nat_leI = leI;
   647 val nat_leD = leD;
   648 val lessI = lessI;
   649 val zero_less_Suc = zero_less_Suc;
   650 val less_reflE = less_irrefl;
   651 val less_zeroE = less_zeroE;
   652 val less_incr = Suc_mono;
   653 val less_decr = Suc_less_SucD;
   654 val less_incr_rhs = Suc_mono RS Suc_lessD;
   655 val less_decr_lhs = Suc_lessD;
   656 val less_trans_Suc = less_trans_Suc;
   657 val leI = Suc_leI RS (Suc_le_mono RS iffD1);
   658 val not_lessI = leI RS leD
   659 val not_leI = prove_goal thy "!!m::nat. n < m ==> ~ m <= n"
   660   (fn _ => [etac swap2 1, etac leD 1]);
   661 val eqI = prove_goal thy "!!m. [| m < Suc n; n < Suc m |] ==> m=n"
   662   (fn _ => [etac less_SucE 1,
   663             blast_tac (claset() addSDs [Suc_less_SucD] addSEs [less_irrefl]
   664                               addDs [less_trans_Suc]) 1,
   665             assume_tac 1]);
   666 val leD = le_eq_less_Suc RS iffD1;
   667 val not_lessD = nat_leI RS leD;
   668 val not_leD = not_leE
   669 val eqD1 = prove_goal thy  "!!n. m = n ==> m < Suc n"
   670  (fn _ => [etac subst 1, rtac lessI 1]);
   671 val eqD2 = sym RS eqD1;
   672 
   673 fun is_zero(t) =  t = Const("0",Type("nat",[]));
   674 
   675 fun nnb T = T = Type("fun",[Type("nat",[]),
   676                             Type("fun",[Type("nat",[]),
   677                                         Type("bool",[])])])
   678 
   679 fun decomp_Suc(Const("Suc",_)$t) = let val (a,i) = decomp_Suc t in (a,i+1) end
   680   | decomp_Suc t = (t,0);
   681 
   682 fun decomp2(rel,T,lhs,rhs) =
   683   if not(nnb T) then None else
   684   let val (x,i) = decomp_Suc lhs
   685       val (y,j) = decomp_Suc rhs
   686   in case rel of
   687        "op <"  => Some(x,i,"<",y,j)
   688      | "op <=" => Some(x,i,"<=",y,j)
   689      | "op ="  => Some(x,i,"=",y,j)
   690      | _       => None
   691   end;
   692 
   693 fun negate(Some(x,i,rel,y,j)) = Some(x,i,"~"^rel,y,j)
   694   | negate None = None;
   695 
   696 fun decomp(_$(Const(rel,T)$lhs$rhs)) = decomp2(rel,T,lhs,rhs)
   697   | decomp(_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) =
   698       negate(decomp2(rel,T,lhs,rhs))
   699   | decomp _ = None
   700 
   701 end;
   702 
   703 structure Trans_Tac = Trans_Tac_Fun(Less_Arith);
   704 
   705 open Trans_Tac;
   706 
   707 (*** eliminates ~= in premises, which trans_tac cannot deal with ***)
   708 qed_goal "nat_neqE" thy
   709   "[| (m::nat) ~= n; m < n ==> P; n < m ==> P |] ==> P"
   710   (fn major::prems => [cut_facts_tac [less_linear] 1,
   711                        REPEAT(eresolve_tac ([disjE,major RS notE]@prems) 1)]);
   712 
   713 
   714 
   715 (* add function nat_add_primrec *) 
   716 val (_, nat_add_primrec, _, _) = Datatype.add_datatype
   717 ([], "nat", [("0", [], Mixfix ("0", [], max_pri)), ("Suc", [dtTyp ([],
   718 "nat")], NoSyn)]) (Theory.add_name "Arith" HOL.thy);
   719 (*pretend Arith is part of the basic theory to fool package*)