src/HOL/NatDef.ML
 author nipkow Mon Apr 27 16:45:11 1998 +0200 (1998-04-27) changeset 4830 bd73675adbed parent 4821 bfbaea156f43 child 5069 3ea049f7979d permissions -rw-r--r--
Renamed expand_const -> split_const.
```     1 (*  Title:      HOL/NatDef.ML
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
```
```     4     Copyright   1991  University of Cambridge
```
```     5
```
```     6 Blast_tac proofs here can get PROOF FAILED of Ord theorems like order_refl
```
```     7 and order_less_irrefl.  We do not add the "nat" versions to the basic claset
```
```     8 because the type will be promoted to type class "order".
```
```     9 *)
```
```    10
```
```    11 goal 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 thy "Zero_Rep: Nat";
```
```    19 by (stac Nat_unfold 1);
```
```    20 by (rtac (singletonI RS UnI1) 1);
```
```    21 qed "Zero_RepI";
```
```    22
```
```    23 val prems = goal thy "i: Nat ==> Suc_Rep(i) : Nat";
```
```    24 by (stac Nat_unfold 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 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 (blast_tac (claset() addIs prems) 1);
```
```    36 qed "Nat_induct";
```
```    37
```
```    38 val prems = goalw thy [Zero_def,Suc_def]
```
```    39     "[| P(0);   \
```
```    40 \       !!n. P(n) ==> P(Suc(n)) |]  ==> 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 local fun raw_nat_ind_tac a i =
```
```    49     res_inst_tac [("n",a)] nat_induct i  THEN  rename_last_tac a [""] (i+1)
```
```    50 in
```
```    51 val nat_ind_tac = Datatype.occs_in_prems raw_nat_ind_tac
```
```    52 end;
```
```    53
```
```    54 (*A special form of induction for reasoning about m<n and m-n*)
```
```    55 val prems = goal thy
```
```    56     "[| !!x. P x 0;  \
```
```    57 \       !!y. P 0 (Suc y);  \
```
```    58 \       !!x y. [| P x y |] ==> P (Suc x) (Suc y)  \
```
```    59 \    |] ==> P m n";
```
```    60 by (res_inst_tac [("x","m")] spec 1);
```
```    61 by (nat_ind_tac "n" 1);
```
```    62 by (rtac allI 2);
```
```    63 by (nat_ind_tac "x" 2);
```
```    64 by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1));
```
```    65 qed "diff_induct";
```
```    66
```
```    67 (*Case analysis on the natural numbers*)
```
```    68 val prems = goal thy
```
```    69     "[| n=0 ==> P;  !!x. n = Suc(x) ==> P |] ==> P";
```
```    70 by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1);
```
```    71 by (fast_tac (claset() addSEs prems) 1);
```
```    72 by (nat_ind_tac "n" 1);
```
```    73 by (rtac (refl RS disjI1) 1);
```
```    74 by (Blast_tac 1);
```
```    75 qed "natE";
```
```    76
```
```    77
```
```    78 (*** Isomorphisms: Abs_Nat and Rep_Nat ***)
```
```    79
```
```    80 (*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat),
```
```    81   since we assume the isomorphism equations will one day be given by Isabelle*)
```
```    82
```
```    83 goal thy "inj(Rep_Nat)";
```
```    84 by (rtac inj_inverseI 1);
```
```    85 by (rtac Rep_Nat_inverse 1);
```
```    86 qed "inj_Rep_Nat";
```
```    87
```
```    88 goal thy "inj_on Abs_Nat Nat";
```
```    89 by (rtac inj_on_inverseI 1);
```
```    90 by (etac Abs_Nat_inverse 1);
```
```    91 qed "inj_on_Abs_Nat";
```
```    92
```
```    93 (*** Distinctness of constructors ***)
```
```    94
```
```    95 goalw thy [Zero_def,Suc_def] "Suc(m) ~= 0";
```
```    96 by (rtac (inj_on_Abs_Nat RS inj_on_contraD) 1);
```
```    97 by (rtac Suc_Rep_not_Zero_Rep 1);
```
```    98 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1));
```
```    99 qed "Suc_not_Zero";
```
```   100
```
```   101 bind_thm ("Zero_not_Suc", Suc_not_Zero RS not_sym);
```
```   102
```
```   103 AddIffs [Suc_not_Zero,Zero_not_Suc];
```
```   104
```
```   105 bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE));
```
```   106 val Zero_neq_Suc = sym RS Suc_neq_Zero;
```
```   107
```
```   108 (** Injectiveness of Suc **)
```
```   109
```
```   110 goalw thy [Suc_def] "inj(Suc)";
```
```   111 by (rtac injI 1);
```
```   112 by (dtac (inj_on_Abs_Nat RS inj_onD) 1);
```
```   113 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1));
```
```   114 by (dtac (inj_Suc_Rep RS injD) 1);
```
```   115 by (etac (inj_Rep_Nat RS injD) 1);
```
```   116 qed "inj_Suc";
```
```   117
```
```   118 val Suc_inject = inj_Suc RS injD;
```
```   119
```
```   120 goal thy "(Suc(m)=Suc(n)) = (m=n)";
```
```   121 by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]);
```
```   122 qed "Suc_Suc_eq";
```
```   123
```
```   124 AddIffs [Suc_Suc_eq];
```
```   125
```
```   126 goal thy "n ~= Suc(n)";
```
```   127 by (nat_ind_tac "n" 1);
```
```   128 by (ALLGOALS Asm_simp_tac);
```
```   129 qed "n_not_Suc_n";
```
```   130
```
```   131 bind_thm ("Suc_n_not_n", n_not_Suc_n RS not_sym);
```
```   132
```
```   133 goal thy "!!n. n ~= 0 ==> EX m. n = Suc m";
```
```   134 by (rtac natE 1);
```
```   135 by (REPEAT (Blast_tac 1));
```
```   136 qed "not0_implies_Suc";
```
```   137
```
```   138
```
```   139 (*** nat_case -- the selection operator for nat ***)
```
```   140
```
```   141 goalw thy [nat_case_def] "nat_case a f 0 = a";
```
```   142 by (Blast_tac 1);
```
```   143 qed "nat_case_0";
```
```   144
```
```   145 goalw thy [nat_case_def] "nat_case a f (Suc k) = f(k)";
```
```   146 by (Blast_tac 1);
```
```   147 qed "nat_case_Suc";
```
```   148
```
```   149 goalw thy [wf_def, pred_nat_def] "wf(pred_nat)";
```
```   150 by (Clarify_tac 1);
```
```   151 by (nat_ind_tac "x" 1);
```
```   152 by (ALLGOALS Blast_tac);
```
```   153 qed "wf_pred_nat";
```
```   154
```
```   155
```
```   156 (*** nat_rec -- by wf recursion on pred_nat ***)
```
```   157
```
```   158 (* The unrolling rule for nat_rec *)
```
```   159 goal thy
```
```   160    "nat_rec c d = wfrec pred_nat (%f. nat_case c (%m. d m (f m)))";
```
```   161   by (simp_tac (HOL_ss addsimps [nat_rec_def]) 1);
```
```   162 bind_thm("nat_rec_unfold", wf_pred_nat RS
```
```   163                             ((result() RS eq_reflection) RS def_wfrec));
```
```   164
```
```   165 (*---------------------------------------------------------------------------
```
```   166  * Old:
```
```   167  * bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec)));
```
```   168  *---------------------------------------------------------------------------*)
```
```   169
```
```   170 (** conversion rules **)
```
```   171
```
```   172 goal thy "nat_rec c h 0 = c";
```
```   173 by (rtac (nat_rec_unfold RS trans) 1);
```
```   174 by (simp_tac (simpset() addsimps [nat_case_0]) 1);
```
```   175 qed "nat_rec_0";
```
```   176
```
```   177 goal thy "nat_rec c h (Suc n) = h n (nat_rec c h n)";
```
```   178 by (rtac (nat_rec_unfold RS trans) 1);
```
```   179 by (simp_tac (simpset() addsimps [nat_case_Suc, pred_nat_def, cut_apply]) 1);
```
```   180 qed "nat_rec_Suc";
```
```   181
```
```   182 (*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
```
```   183 val [rew] = goal thy
```
```   184     "[| !!n. f(n) == nat_rec c h n |] ==> f(0) = c";
```
```   185 by (rewtac rew);
```
```   186 by (rtac nat_rec_0 1);
```
```   187 qed "def_nat_rec_0";
```
```   188
```
```   189 val [rew] = goal thy
```
```   190     "[| !!n. f(n) == nat_rec c h n |] ==> f(Suc(n)) = h n (f n)";
```
```   191 by (rewtac rew);
```
```   192 by (rtac nat_rec_Suc 1);
```
```   193 qed "def_nat_rec_Suc";
```
```   194
```
```   195 fun nat_recs def =
```
```   196       [standard (def RS def_nat_rec_0),
```
```   197        standard (def RS def_nat_rec_Suc)];
```
```   198
```
```   199
```
```   200 (*** Basic properties of "less than" ***)
```
```   201
```
```   202 (*Used in TFL/post.sml*)
```
```   203 goalw thy [less_def] "(m,n) : pred_nat^+ = (m<n)";
```
```   204 by (rtac refl 1);
```
```   205 qed "less_eq";
```
```   206
```
```   207 (** Introduction properties **)
```
```   208
```
```   209 val prems = goalw 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 thy [less_def, pred_nat_def] "n < Suc(n)";
```
```   216 by (simp_tac (simpset() addsimps [r_into_trancl]) 1);
```
```   217 qed "lessI";
```
```   218 AddIffs [lessI];
```
```   219
```
```   220 (* i<j ==> i<Suc(j) *)
```
```   221 bind_thm("less_SucI", lessI RSN (2, less_trans));
```
```   222 Addsimps [less_SucI];
```
```   223
```
```   224 goal thy "0 < Suc(n)";
```
```   225 by (nat_ind_tac "n" 1);
```
```   226 by (rtac lessI 1);
```
```   227 by (etac less_trans 1);
```
```   228 by (rtac lessI 1);
```
```   229 qed "zero_less_Suc";
```
```   230 AddIffs [zero_less_Suc];
```
```   231
```
```   232 (** Elimination properties **)
```
```   233
```
```   234 val prems = goalw thy [less_def] "n<m ==> ~ m<(n::nat)";
```
```   235 by (blast_tac (claset() addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1);
```
```   236 qed "less_not_sym";
```
```   237
```
```   238 (* [| n<m; m<n |] ==> R *)
```
```   239 bind_thm ("less_asym", (less_not_sym RS notE));
```
```   240
```
```   241 goalw thy [less_def] "~ n<(n::nat)";
```
```   242 by (rtac notI 1);
```
```   243 by (etac (wf_pred_nat RS wf_trancl RS wf_irrefl) 1);
```
```   244 qed "less_not_refl";
```
```   245
```
```   246 (* n<n ==> R *)
```
```   247 bind_thm ("less_irrefl", (less_not_refl RS notE));
```
```   248
```
```   249 goal thy "!!m. n<m ==> m ~= (n::nat)";
```
```   250 by (blast_tac (claset() addSEs [less_irrefl]) 1);
```
```   251 qed "less_not_refl2";
```
```   252
```
```   253
```
```   254 val major::prems = goalw thy [less_def, pred_nat_def]
```
```   255     "[| i<k;  k=Suc(i) ==> P;  !!j. [| i<j;  k=Suc(j) |] ==> P \
```
```   256 \    |] ==> P";
```
```   257 by (rtac (major RS tranclE) 1);
```
```   258 by (ALLGOALS Full_simp_tac);
```
```   259 by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE'
```
```   260                   eresolve_tac (prems@[asm_rl, Pair_inject])));
```
```   261 qed "lessE";
```
```   262
```
```   263 goal thy "~ n<0";
```
```   264 by (rtac notI 1);
```
```   265 by (etac lessE 1);
```
```   266 by (etac Zero_neq_Suc 1);
```
```   267 by (etac Zero_neq_Suc 1);
```
```   268 qed "not_less0";
```
```   269
```
```   270 AddIffs [not_less0];
```
```   271
```
```   272 (* n<0 ==> R *)
```
```   273 bind_thm ("less_zeroE", not_less0 RS notE);
```
```   274
```
```   275 val [major,less,eq] = goal thy
```
```   276     "[| m < Suc(n);  m<n ==> P;  m=n ==> P |] ==> P";
```
```   277 by (rtac (major RS lessE) 1);
```
```   278 by (rtac eq 1);
```
```   279 by (Blast_tac 1);
```
```   280 by (rtac less 1);
```
```   281 by (Blast_tac 1);
```
```   282 qed "less_SucE";
```
```   283
```
```   284 goal thy "(m < Suc(n)) = (m < n | m = n)";
```
```   285 by (blast_tac (claset() addSEs [less_SucE] addIs [less_trans]) 1);
```
```   286 qed "less_Suc_eq";
```
```   287
```
```   288 goal thy "(n<1) = (n=0)";
```
```   289 by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
```
```   290 qed "less_one";
```
```   291 AddIffs [less_one];
```
```   292
```
```   293 val prems = goal thy "m<n ==> n ~= 0";
```
```   294 by (res_inst_tac [("n","n")] natE 1);
```
```   295 by (cut_facts_tac prems 1);
```
```   296 by (ALLGOALS Asm_full_simp_tac);
```
```   297 qed "gr_implies_not0";
```
```   298
```
```   299 goal thy "(n ~= 0) = (0 < n)";
```
```   300 by (rtac natE 1);
```
```   301 by (Blast_tac 1);
```
```   302 by (Blast_tac 1);
```
```   303 qed "neq0_conv";
```
```   304 AddIffs [neq0_conv];
```
```   305
```
```   306 (*This theorem is useful with blast_tac: (n=0 ==> False) ==> 0<n *)
```
```   307 bind_thm ("gr0I", [neq0_conv, notI] MRS iffD1);
```
```   308
```
```   309 goal thy "(~(0 < n)) = (n=0)";
```
```   310 by (rtac iffI 1);
```
```   311  by (etac swap 1);
```
```   312  by (ALLGOALS Asm_full_simp_tac);
```
```   313 qed "not_gr0";
```
```   314 Addsimps [not_gr0];
```
```   315
```
```   316 goal thy "!!m. m<n ==> 0 < n";
```
```   317 by (dtac gr_implies_not0 1);
```
```   318 by (Asm_full_simp_tac 1);
```
```   319 qed "gr_implies_gr0";
```
```   320 Addsimps [gr_implies_gr0];
```
```   321
```
```   322
```
```   323 goal thy "!!m n. m<n ==> Suc(m) < Suc(n)";
```
```   324 by (etac rev_mp 1);
```
```   325 by (nat_ind_tac "n" 1);
```
```   326 by (ALLGOALS (fast_tac (claset() addEs  [less_trans, lessE])));
```
```   327 qed "Suc_mono";
```
```   328
```
```   329 (*"Less than" is a linear ordering*)
```
```   330 goal thy "m<n | m=n | n<(m::nat)";
```
```   331 by (nat_ind_tac "m" 1);
```
```   332 by (nat_ind_tac "n" 1);
```
```   333 by (rtac (refl RS disjI1 RS disjI2) 1);
```
```   334 by (rtac (zero_less_Suc RS disjI1) 1);
```
```   335 by (blast_tac (claset() addIs [Suc_mono, less_SucI] addEs [lessE]) 1);
```
```   336 qed "less_linear";
```
```   337
```
```   338 goal thy "!!m::nat. (m ~= n) = (m<n | n<m)";
```
```   339 by (cut_facts_tac [less_linear] 1);
```
```   340 by (blast_tac (claset() addSEs [less_irrefl]) 1);
```
```   341 qed "nat_neq_iff";
```
```   342
```
```   343 qed_goal "nat_less_cases" thy
```
```   344    "[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m"
```
```   345 ( fn [major,eqCase,lessCase] =>
```
```   346         [
```
```   347         (rtac (less_linear RS disjE) 1),
```
```   348         (etac disjE 2),
```
```   349         (etac lessCase 1),
```
```   350         (etac (sym RS eqCase) 1),
```
```   351         (etac major 1)
```
```   352         ]);
```
```   353
```
```   354
```
```   355 (** Inductive (?) properties **)
```
```   356
```
```   357 goal thy "!!m. [| m<n; Suc m ~= n |] ==> Suc(m) < n";
```
```   358 by (full_simp_tac (simpset() addsimps [nat_neq_iff]) 1);
```
```   359 by (blast_tac (claset() addSEs [less_irrefl, less_SucE] addEs [less_asym]) 1);
```
```   360 qed "Suc_lessI";
```
```   361
```
```   362 val [prem] = goal thy "Suc(m) < n ==> m<n";
```
```   363 by (rtac (prem RS rev_mp) 1);
```
```   364 by (nat_ind_tac "n" 1);
```
```   365 by (ALLGOALS (fast_tac (claset() addSIs [lessI RS less_SucI]
```
```   366                                  addEs  [less_trans, lessE])));
```
```   367 qed "Suc_lessD";
```
```   368
```
```   369 val [major,minor] = goal thy
```
```   370     "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
```
```   371 \    |] ==> P";
```
```   372 by (rtac (major RS lessE) 1);
```
```   373 by (etac (lessI RS minor) 1);
```
```   374 by (etac (Suc_lessD RS minor) 1);
```
```   375 by (assume_tac 1);
```
```   376 qed "Suc_lessE";
```
```   377
```
```   378 goal thy "!!m n. Suc(m) < Suc(n) ==> m<n";
```
```   379 by (blast_tac (claset() addEs [lessE, make_elim Suc_lessD]) 1);
```
```   380 qed "Suc_less_SucD";
```
```   381
```
```   382
```
```   383 goal thy "(Suc(m) < Suc(n)) = (m<n)";
```
```   384 by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
```
```   385 qed "Suc_less_eq";
```
```   386 Addsimps [Suc_less_eq];
```
```   387
```
```   388 goal thy "~(Suc(n) < n)";
```
```   389 by (blast_tac (claset() addEs [Suc_lessD RS less_irrefl]) 1);
```
```   390 qed "not_Suc_n_less_n";
```
```   391 Addsimps [not_Suc_n_less_n];
```
```   392
```
```   393 goal thy "!!i. i<j ==> j<k --> Suc i < k";
```
```   394 by (nat_ind_tac "k" 1);
```
```   395 by (ALLGOALS (asm_simp_tac (simpset())));
```
```   396 by (asm_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
```
```   397 by (blast_tac (claset() addDs [Suc_lessD]) 1);
```
```   398 qed_spec_mp "less_trans_Suc";
```
```   399
```
```   400 (*Can be used with less_Suc_eq to get n=m | n<m *)
```
```   401 goal thy "(~ m < n) = (n < Suc(m))";
```
```   402 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
```
```   403 by (ALLGOALS Asm_simp_tac);
```
```   404 qed "not_less_eq";
```
```   405
```
```   406 (*Complete induction, aka course-of-values induction*)
```
```   407 val prems = goalw thy [less_def]
```
```   408     "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
```
```   409 by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
```
```   410 by (eresolve_tac prems 1);
```
```   411 qed "less_induct";
```
```   412
```
```   413 qed_goal "nat_induct2" thy
```
```   414 "[| P 0; P 1; !!k. P k ==> P (Suc (Suc k)) |] ==> P n" (fn prems => [
```
```   415         cut_facts_tac prems 1,
```
```   416         rtac less_induct 1,
```
```   417         res_inst_tac [("n","n")] natE 1,
```
```   418          hyp_subst_tac 1,
```
```   419          atac 1,
```
```   420         hyp_subst_tac 1,
```
```   421         res_inst_tac [("n","x")] natE 1,
```
```   422          hyp_subst_tac 1,
```
```   423          atac 1,
```
```   424         hyp_subst_tac 1,
```
```   425         resolve_tac prems 1,
```
```   426         dtac spec 1,
```
```   427         etac mp 1,
```
```   428         rtac (lessI RS less_trans) 1,
```
```   429         rtac (lessI RS Suc_mono) 1]);
```
```   430
```
```   431 (*** Properties of <= ***)
```
```   432
```
```   433 goalw thy [le_def] "(m <= n) = (m < Suc n)";
```
```   434 by (rtac not_less_eq 1);
```
```   435 qed "le_eq_less_Suc";
```
```   436
```
```   437 (*  m<=n ==> m < Suc n  *)
```
```   438 bind_thm ("le_imp_less_Suc", le_eq_less_Suc RS iffD1);
```
```   439
```
```   440 goalw thy [le_def] "0 <= n";
```
```   441 by (rtac not_less0 1);
```
```   442 qed "le0";
```
```   443
```
```   444 goalw thy [le_def] "~ Suc n <= n";
```
```   445 by (Simp_tac 1);
```
```   446 qed "Suc_n_not_le_n";
```
```   447
```
```   448 goalw thy [le_def] "(i <= 0) = (i = 0)";
```
```   449 by (nat_ind_tac "i" 1);
```
```   450 by (ALLGOALS Asm_simp_tac);
```
```   451 qed "le_0_eq";
```
```   452 AddIffs [le_0_eq];
```
```   453
```
```   454 Addsimps [(*less_Suc_eq, makes simpset non-confluent*) le0, le_0_eq,
```
```   455           Suc_n_not_le_n,
```
```   456           n_not_Suc_n, Suc_n_not_n,
```
```   457           nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];
```
```   458
```
```   459 goal thy "!!m. (m <= Suc(n)) = (m<=n | m = Suc n)";
```
```   460 by (simp_tac (simpset() addsimps [le_eq_less_Suc]) 1);
```
```   461 by (blast_tac (claset() addSEs [less_SucE] addIs [less_SucI]) 1);
```
```   462 qed "le_Suc_eq";
```
```   463
```
```   464 (* [| m <= Suc n;  m <= n ==> R;  m = Suc n ==> R |] ==> R *)
```
```   465 bind_thm ("le_SucE", le_Suc_eq RS iffD1 RS disjE);
```
```   466
```
```   467 (*
```
```   468 goal thy "(Suc m < n | Suc m = n) = (m < n)";
```
```   469 by (stac (less_Suc_eq RS sym) 1);
```
```   470 by (rtac Suc_less_eq 1);
```
```   471 qed "Suc_le_eq";
```
```   472
```
```   473 this could make the simpset (with less_Suc_eq added again) more confluent,
```
```   474 but less_Suc_eq makes additional problems with terms of the form 0 < Suc (...)
```
```   475 *)
```
```   476
```
```   477 (*Prevents simplification of f and g: much faster*)
```
```   478 qed_goal "nat_case_weak_cong" thy
```
```   479   "m=n ==> nat_case a f m = nat_case a f n"
```
```   480   (fn [prem] => [rtac (prem RS arg_cong) 1]);
```
```   481
```
```   482 qed_goal "nat_rec_weak_cong" thy
```
```   483   "m=n ==> nat_rec a f m = nat_rec a f n"
```
```   484   (fn [prem] => [rtac (prem RS arg_cong) 1]);
```
```   485
```
```   486 qed_goal "split_nat_case" thy
```
```   487   "P(nat_case z s n) = ((n=0 --> P z) & (!m. n = Suc m --> P(s m)))"
```
```   488   (fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
```
```   489
```
```   490 val prems = goalw thy [le_def] "~n<m ==> m<=(n::nat)";
```
```   491 by (resolve_tac prems 1);
```
```   492 qed "leI";
```
```   493
```
```   494 val prems = goalw thy [le_def] "m<=n ==> ~ n < (m::nat)";
```
```   495 by (resolve_tac prems 1);
```
```   496 qed "leD";
```
```   497
```
```   498 val leE = make_elim leD;
```
```   499
```
```   500 goal thy "(~n<m) = (m<=(n::nat))";
```
```   501 by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
```
```   502 qed "not_less_iff_le";
```
```   503
```
```   504 goalw thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
```
```   505 by (Blast_tac 1);
```
```   506 qed "not_leE";
```
```   507
```
```   508 goalw thy [le_def] "(~n<=m) = (m<(n::nat))";
```
```   509 by (Simp_tac 1);
```
```   510 qed "not_le_iff_less";
```
```   511
```
```   512 goalw thy [le_def] "!!m. m < n ==> Suc(m) <= n";
```
```   513 by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
```
```   514 by (blast_tac (claset() addSEs [less_irrefl,less_asym]) 1);
```
```   515 qed "Suc_leI";  (*formerly called lessD*)
```
```   516
```
```   517 goalw thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
```
```   518 by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
```
```   519 qed "Suc_leD";
```
```   520
```
```   521 (* stronger version of Suc_leD *)
```
```   522 goalw thy [le_def]
```
```   523         "!!m. Suc m <= n ==> m < n";
```
```   524 by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
```
```   525 by (cut_facts_tac [less_linear] 1);
```
```   526 by (Blast_tac 1);
```
```   527 qed "Suc_le_lessD";
```
```   528
```
```   529 goal thy "(Suc m <= n) = (m < n)";
```
```   530 by (blast_tac (claset() addIs [Suc_leI, Suc_le_lessD]) 1);
```
```   531 qed "Suc_le_eq";
```
```   532
```
```   533 goalw thy [le_def] "!!m. m <= n ==> m <= Suc n";
```
```   534 by (blast_tac (claset() addDs [Suc_lessD]) 1);
```
```   535 qed "le_SucI";
```
```   536 Addsimps[le_SucI];
```
```   537
```
```   538 bind_thm ("le_Suc", not_Suc_n_less_n RS leI);
```
```   539
```
```   540 goalw thy [le_def] "!!m. m < n ==> m <= (n::nat)";
```
```   541 by (blast_tac (claset() addEs [less_asym]) 1);
```
```   542 qed "less_imp_le";
```
```   543
```
```   544 (** Equivalence of m<=n and  m<n | m=n **)
```
```   545
```
```   546 goalw thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
```
```   547 by (cut_facts_tac [less_linear] 1);
```
```   548 by (blast_tac (claset() addEs [less_irrefl,less_asym]) 1);
```
```   549 qed "le_imp_less_or_eq";
```
```   550
```
```   551 goalw thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
```
```   552 by (cut_facts_tac [less_linear] 1);
```
```   553 by (blast_tac (claset() addSEs [less_irrefl] addEs [less_asym]) 1);
```
```   554 qed "less_or_eq_imp_le";
```
```   555
```
```   556 goal thy "(m <= (n::nat)) = (m < n | m=n)";
```
```   557 by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
```
```   558 qed "le_eq_less_or_eq";
```
```   559
```
```   560 (*Useful with Blast_tac.   m=n ==> m<=n *)
```
```   561 bind_thm ("eq_imp_le", disjI2 RS less_or_eq_imp_le);
```
```   562
```
```   563 goal thy "n <= (n::nat)";
```
```   564 by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
```
```   565 qed "le_refl";
```
```   566
```
```   567 goal thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
```
```   568 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
```
```   569 	                addIs [less_trans]) 1);
```
```   570 qed "le_less_trans";
```
```   571
```
```   572 goal thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
```
```   573 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
```
```   574 	                addIs [less_trans]) 1);
```
```   575 qed "less_le_trans";
```
```   576
```
```   577 goal thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";
```
```   578 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
```
```   579 	                addIs [less_or_eq_imp_le, less_trans]) 1);
```
```   580 qed "le_trans";
```
```   581
```
```   582 goal thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";
```
```   583 (*order_less_irrefl could make this proof fail*)
```
```   584 by (blast_tac (claset() addSDs [le_imp_less_or_eq]
```
```   585 	                addSEs [less_irrefl] addEs [less_asym]) 1);
```
```   586 qed "le_anti_sym";
```
```   587
```
```   588 goal thy "(Suc(n) <= Suc(m)) = (n <= m)";
```
```   589 by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
```
```   590 qed "Suc_le_mono";
```
```   591
```
```   592 AddIffs [Suc_le_mono];
```
```   593
```
```   594 (* Axiom 'order_le_less' of class 'order': *)
```
```   595 goal thy "(m::nat) < n = (m <= n & m ~= n)";
```
```   596 by (simp_tac (simpset() addsimps [le_def, nat_neq_iff]) 1);
```
```   597 by (blast_tac (claset() addSEs [less_asym]) 1);
```
```   598 qed "nat_less_le";
```
```   599
```
```   600 (* Axiom 'linorder_linear' of class 'linorder': *)
```
```   601 goal thy "(m::nat) <= n | n <= m";
```
```   602 by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
```
```   603 by (cut_facts_tac [less_linear] 1);
```
```   604 by(Blast_tac 1);
```
```   605 qed "nat_le_linear";
```
```   606
```
```   607
```
```   608 (** max
```
```   609
```
```   610 goalw thy [max_def] "!!z::nat. (z <= max x y) = (z <= x | z <= y)";
```
```   611 by (simp_tac (simpset() addsimps [not_le_iff_less]) 1);
```
```   612 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
```
```   613 qed "le_max_iff_disj";
```
```   614
```
```   615 goalw thy [max_def] "!!z::nat. (max x y <= z) = (x <= z & y <= z)";
```
```   616 by (simp_tac (simpset() addsimps [not_le_iff_less]) 1);
```
```   617 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
```
```   618 qed "max_le_iff_conj";
```
```   619
```
```   620
```
```   621 (** min **)
```
```   622
```
```   623 goalw thy [min_def] "!!z::nat. (z <= min x y) = (z <= x & z <= y)";
```
```   624 by (simp_tac (simpset() addsimps [not_le_iff_less]) 1);
```
```   625 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
```
```   626 qed "le_min_iff_conj";
```
```   627
```
```   628 goalw thy [min_def] "!!z::nat. (min x y <= z) = (x <= z | y <= z)";
```
```   629 by (simp_tac (simpset() addsimps [not_le_iff_less] addsplits) 1);
```
```   630 by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
```
```   631 qed "min_le_iff_disj";
```
```   632  **)
```
```   633
```
```   634 (** LEAST -- the least number operator **)
```
```   635
```
```   636 goal thy "(! m::nat. P m --> n <= m) = (! m. m < n --> ~ P m)";
```
```   637 by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
```
```   638 val lemma = result();
```
```   639
```
```   640 (* This is an old def of Least for nat, which is derived for compatibility *)
```
```   641 goalw thy [Least_def]
```
```   642   "(LEAST n::nat. P n) == (@n. P(n) & (ALL m. m < n --> ~P(m)))";
```
```   643 by (simp_tac (simpset() addsimps [lemma]) 1);
```
```   644 qed "Least_nat_def";
```
```   645
```
```   646 val [prem1,prem2] = goalw thy [Least_nat_def]
```
```   647     "[| P(k::nat);  !!x. x<k ==> ~P(x) |] ==> (LEAST x. P(x)) = k";
```
```   648 by (rtac select_equality 1);
```
```   649 by (blast_tac (claset() addSIs [prem1,prem2]) 1);
```
```   650 by (cut_facts_tac [less_linear] 1);
```
```   651 by (blast_tac (claset() addSIs [prem1] addSDs [prem2]) 1);
```
```   652 qed "Least_equality";
```
```   653
```
```   654 val [prem] = goal thy "P(k::nat) ==> P(LEAST x. P(x))";
```
```   655 by (rtac (prem RS rev_mp) 1);
```
```   656 by (res_inst_tac [("n","k")] less_induct 1);
```
```   657 by (rtac impI 1);
```
```   658 by (rtac classical 1);
```
```   659 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
```
```   660 by (assume_tac 1);
```
```   661 by (assume_tac 2);
```
```   662 by (Blast_tac 1);
```
```   663 qed "LeastI";
```
```   664
```
```   665 (*Proof is almost identical to the one above!*)
```
```   666 val [prem] = goal thy "P(k::nat) ==> (LEAST x. P(x)) <= k";
```
```   667 by (rtac (prem RS rev_mp) 1);
```
```   668 by (res_inst_tac [("n","k")] less_induct 1);
```
```   669 by (rtac impI 1);
```
```   670 by (rtac classical 1);
```
```   671 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
```
```   672 by (assume_tac 1);
```
```   673 by (rtac le_refl 2);
```
```   674 by (blast_tac (claset() addIs [less_imp_le,le_trans]) 1);
```
```   675 qed "Least_le";
```
```   676
```
```   677 val [prem] = goal thy "k < (LEAST x. P(x)) ==> ~P(k::nat)";
```
```   678 by (rtac notI 1);
```
```   679 by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
```
```   680 by (rtac prem 1);
```
```   681 qed "not_less_Least";
```
```   682
```
```   683 qed_goalw "Least_Suc" thy [Least_nat_def]
```
```   684  "!!P. [| ? n. P(Suc n); ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
```
```   685  (fn _ => [
```
```   686         rtac select_equality 1,
```
```   687         fold_goals_tac [Least_nat_def],
```
```   688         safe_tac (claset() addSEs [LeastI]),
```
```   689         rename_tac "j" 1,
```
```   690         res_inst_tac [("n","j")] natE 1,
```
```   691         Blast_tac 1,
```
```   692         blast_tac (claset() addDs [Suc_less_SucD, not_less_Least]) 1,
```
```   693         rename_tac "k n" 1,
```
```   694         res_inst_tac [("n","k")] natE 1,
```
```   695         Blast_tac 1,
```
```   696         hyp_subst_tac 1,
```
```   697         rewtac Least_nat_def,
```
```   698         rtac (select_equality RS arg_cong RS sym) 1,
```
```   699         Safe_tac,
```
```   700         dtac Suc_mono 1,
```
```   701         Blast_tac 1,
```
```   702         cut_facts_tac [less_linear] 1,
```
```   703         Safe_tac,
```
```   704         atac 2,
```
```   705         Blast_tac 2,
```
```   706         dtac Suc_mono 1,
```
```   707         Blast_tac 1]);
```
```   708
```
```   709
```
```   710 (*** Instantiation of transitivity prover ***)
```
```   711
```
```   712 structure Less_Arith =
```
```   713 struct
```
```   714 val nat_leI = leI;
```
```   715 val nat_leD = leD;
```
```   716 val lessI = lessI;
```
```   717 val zero_less_Suc = zero_less_Suc;
```
```   718 val less_reflE = less_irrefl;
```
```   719 val less_zeroE = less_zeroE;
```
```   720 val less_incr = Suc_mono;
```
```   721 val less_decr = Suc_less_SucD;
```
```   722 val less_incr_rhs = Suc_mono RS Suc_lessD;
```
```   723 val less_decr_lhs = Suc_lessD;
```
```   724 val less_trans_Suc = less_trans_Suc;
```
```   725 val leI = Suc_leI RS (Suc_le_mono RS iffD1);
```
```   726 val not_lessI = leI RS leD
```
```   727 val not_leI = prove_goal thy "!!m::nat. n < m ==> ~ m <= n"
```
```   728   (fn _ => [etac swap2 1, etac leD 1]);
```
```   729 val eqI = prove_goal thy "!!m. [| m < Suc n; n < Suc m |] ==> m=n"
```
```   730   (fn _ => [etac less_SucE 1,
```
```   731             blast_tac (claset() addSDs [Suc_less_SucD] addSEs [less_irrefl]
```
```   732                               addDs [less_trans_Suc]) 1,
```
```   733             assume_tac 1]);
```
```   734 val leD = le_eq_less_Suc RS iffD1;
```
```   735 val not_lessD = nat_leI RS leD;
```
```   736 val not_leD = not_leE
```
```   737 val eqD1 = prove_goal thy  "!!n. m = n ==> m < Suc n"
```
```   738  (fn _ => [etac subst 1, rtac lessI 1]);
```
```   739 val eqD2 = sym RS eqD1;
```
```   740
```
```   741 fun is_zero(t) =  t = Const("0",Type("nat",[]));
```
```   742
```
```   743 fun nnb T = T = Type("fun",[Type("nat",[]),
```
```   744                             Type("fun",[Type("nat",[]),
```
```   745                                         Type("bool",[])])])
```
```   746
```
```   747 fun decomp_Suc(Const("Suc",_)\$t) = let val (a,i) = decomp_Suc t in (a,i+1) end
```
```   748   | decomp_Suc t = (t,0);
```
```   749
```
```   750 fun decomp2(rel,T,lhs,rhs) =
```
```   751   if not(nnb T) then None else
```
```   752   let val (x,i) = decomp_Suc lhs
```
```   753       val (y,j) = decomp_Suc rhs
```
```   754   in case rel of
```
```   755        "op <"  => Some(x,i,"<",y,j)
```
```   756      | "op <=" => Some(x,i,"<=",y,j)
```
```   757      | "op ="  => Some(x,i,"=",y,j)
```
```   758      | _       => None
```
```   759   end;
```
```   760
```
```   761 fun negate(Some(x,i,rel,y,j)) = Some(x,i,"~"^rel,y,j)
```
```   762   | negate None = None;
```
```   763
```
```   764 fun decomp(_\$(Const(rel,T)\$lhs\$rhs)) = decomp2(rel,T,lhs,rhs)
```
```   765   | decomp(_\$(Const("Not",_)\$(Const(rel,T)\$lhs\$rhs))) =
```
```   766       negate(decomp2(rel,T,lhs,rhs))
```
```   767   | decomp _ = None
```
```   768
```
```   769 end;
```
```   770
```
```   771 structure Trans_Tac = Trans_Tac_Fun(Less_Arith);
```
```   772
```
```   773 open Trans_Tac;
```
```   774
```
```   775 (*** eliminates ~= in premises, which trans_tac cannot deal with ***)
```
```   776 bind_thm("nat_neqE", nat_neq_iff RS iffD1 RS disjE);
```
```   777
```
```   778
```
```   779 (* add function nat_add_primrec *)
```
```   780 val (_, nat_add_primrec, _, _) =
```
```   781     Datatype.add_datatype ([], "nat",
```
```   782 			   [("0", [], Mixfix ("0", [], max_pri)),
```
```   783 			    ("Suc", [dtTyp ([], "nat")], NoSyn)])
```
```   784     (Theory.add_name "Arith" HOL.thy);
```
```   785
```
```   786 (*pretend Arith is part of the basic theory to fool package*)
```