isabelle: src/HOL/Nat.ML@6f41a494f3b1

     1 (*  Title:      HOL/nat

     2     ID:         $Id$

     3     Author:     Tobias Nipkow, Cambridge University Computer Laboratory

     4     Copyright   1991  University of Cambridge

     6 For nat.thy.  Type nat is defined as a set (Nat) over the type ind.

     7 *)

     9 open Nat;

    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";

    15 val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski);

    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";

    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";

    29 (*** Induction ***)

    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 (!claset addIs prems) 1);

    36 qed "Nat_induct";

    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";

    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)];

    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";

    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 (!claset addSEs prems) 1);

    70 by (nat_ind_tac "n" 1);

    71 by (rtac (refl RS disjI1) 1);

    72 by (Fast_tac 1);

    73 qed "natE";

    75 (*** Isomorphisms: Abs_Nat and Rep_Nat ***)

    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*)

    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";

    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";

    90 (*** Distinctness of constructors ***)

    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";

    98 bind_thm ("Zero_not_Suc", (Suc_not_Zero RS not_sym));

   100 Addsimps [Suc_not_Zero,Zero_not_Suc];

   102 bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE));

   103 val Zero_neq_Suc = sym RS Suc_neq_Zero;

   105 (** Injectiveness of Suc **)

   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";

   115 val Suc_inject = inj_Suc RS injD;

   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";

   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";

   126 bind_thm ("Suc_n_not_n", n_not_Suc_n RS not_sym);

   128 (*** nat_case -- the selection operator for nat ***)

   130 goalw Nat.thy [nat_case_def] "nat_case a f 0 = a";

   131 by (fast_tac (!claset addIs [select_equality] addEs [Zero_neq_Suc]) 1);

   132 qed "nat_case_0";

   134 goalw Nat.thy [nat_case_def] "nat_case a f (Suc k) = f(k)";

   135 by (fast_tac (!claset addIs [select_equality]

   136                        addEs [make_elim Suc_inject, Suc_neq_Zero]) 1);

   137 qed "nat_case_Suc";

   139 (** Introduction rules for 'pred_nat' **)

   141 goalw Nat.thy [pred_nat_def] "(n, Suc(n)) : pred_nat";

   142 by (Fast_tac 1);

   143 qed "pred_natI";

   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";

   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 (!claset addSEs [mp, pred_natE, Pair_inject,

   156                              make_elim Suc_inject]) 2);

   157 by (fast_tac (!claset addSEs [mp, pred_natE, Pair_inject, Zero_neq_Suc]) 1);

   158 qed "wf_pred_nat";

   161 (*** nat_rec -- by wf recursion on pred_nat ***)

   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));

   170 (*---------------------------------------------------------------------------

   171  * Old:

   172  * bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec)));

   173  *---------------------------------------------------------------------------*)

   175 (** conversion rules **)

   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";

   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";

   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";

   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";

   200 fun nat_recs def =

   201       [standard (def RS def_nat_rec_0),

   202        standard (def RS def_nat_rec_Suc)];

   205 (*** Basic properties of "less than" ***)

   207 (** Introduction properties **)

   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";

   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];

   220 (* i<j ==> i<Suc(j) *)

   221 val less_SucI = lessI RSN (2, less_trans);

   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];

   231 (** Elimination properties **)

   233 val prems = goalw Nat.thy [less_def] "n<m ==> ~ m<(n::nat)";

   234 by (fast_tac (!claset addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1);

   235 qed "less_not_sym";

   237 (* [| n(m; m(n |] ==> R *)

   238 bind_thm ("less_asym", (less_not_sym RS notE));

   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_irrefl) 1);

   243 qed "less_not_refl";

   245 (* n(n ==> R *)

   246 bind_thm ("less_irrefl", (less_not_refl RS notE));

   248 goal Nat.thy "!!m. n<m ==> m ~= (n::nat)";

   249 by (fast_tac (!claset addEs [less_irrefl]) 1);

   250 qed "less_not_refl2";

   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";

   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];

   270 (* n<0 ==> R *)

   271 bind_thm ("less_zeroE", (not_less0 RS notE));

   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 (!claset addSDs [Suc_inject]) 1);

   278 by (rtac less 1);

   279 by (fast_tac (!claset addSDs [Suc_inject]) 1);

   280 qed "less_SucE";

   282 goal Nat.thy "(m < Suc(n)) = (m < n | m = n)";

   283 by (fast_tac (!claset addSIs [lessI]

   284                      addEs  [less_trans, less_SucE]) 1);

   285 qed "less_Suc_eq";

   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];

   294 qed_goal "zero_less_eq" Nat.thy "0 < n = (n ~= 0)" (fn _ => [

   295 	rtac iffI 1,

   296 	etac gr_implies_not0 1,

   297 	rtac natE 1,

   298 	contr_tac 1,

   299 	etac ssubst 1,

   300 	rtac zero_less_Suc 1]);

   302 (** Inductive (?) properties **)

   304 val [prem] = goal Nat.thy "Suc(m) < n ==> m<n";

   305 by (rtac (prem RS rev_mp) 1);

   306 by (nat_ind_tac "n" 1);

   307 by (rtac impI 1);

   308 by (etac less_zeroE 1);

   309 by (fast_tac (!claset addSIs [lessI RS less_SucI]

   310                      addSDs [Suc_inject]

   311                      addEs  [less_trans, lessE]) 1);

   312 qed "Suc_lessD";

   314 val [major,minor] = goal Nat.thy

   315     "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \

   316 \    |] ==> P";

   317 by (rtac (major RS lessE) 1);

   318 by (etac (lessI RS minor) 1);

   319 by (etac (Suc_lessD RS minor) 1);

   320 by (assume_tac 1);

   321 qed "Suc_lessE";

   323 val [major] = goal Nat.thy "Suc(m) < Suc(n) ==> m<n";

   324 by (rtac (major RS lessE) 1);

   325 by (REPEAT (rtac lessI 1

   326      ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1));

   327 qed "Suc_less_SucD";

   329 val prems = goal Nat.thy "m<n ==> Suc(m) < Suc(n)";

   330 by (subgoal_tac "m<n --> Suc(m) < Suc(n)" 1);

   331 by (fast_tac (!claset addIs prems) 1);

   332 by (nat_ind_tac "n" 1);

   333 by (rtac impI 1);

   334 by (etac less_zeroE 1);

   335 by (fast_tac (!claset addSIs [lessI]

   336                      addSDs [Suc_inject]

   337                      addEs  [less_trans, lessE]) 1);

   338 qed "Suc_mono";

   341 (*

   342 goal Nat.thy "(Suc m < n | Suc m = n) = (m < n)";

   345 goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)";

   346 by(stac less_Suc_eq 1);

   347 by(rtac

   348 *)

   350 goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)";

   351 by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);

   352 qed "Suc_less_eq";

   353 Addsimps [Suc_less_eq];

   355 goal Nat.thy "~(Suc(n) < n)";

   356 by (fast_tac (!claset addEs [Suc_lessD RS less_irrefl]) 1);

   357 qed "not_Suc_n_less_n";

   358 Addsimps [not_Suc_n_less_n];

   360 goal Nat.thy "!!i. i<j ==> j<k --> Suc i < k";

   361 by (nat_ind_tac "k" 1);

   362 by (ALLGOALS (asm_simp_tac (!simpset)));

   363 by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1);

   364 by (fast_tac (!claset addDs [Suc_lessD]) 1);

   365 qed_spec_mp "less_trans_Suc";

   367 (*"Less than" is a linear ordering*)

   368 goal Nat.thy "m<n | m=n | n<(m::nat)";

   369 by (nat_ind_tac "m" 1);

   370 by (nat_ind_tac "n" 1);

   371 by (rtac (refl RS disjI1 RS disjI2) 1);

   372 by (rtac (zero_less_Suc RS disjI1) 1);

   373 by (fast_tac (!claset addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1);

   374 qed "less_linear";

   376 qed_goal "nat_less_cases" Nat.thy

   377    "[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m"

   378 ( fn prems =>

   379         [

   380         (res_inst_tac [("m1","n"),("n1","m")] (less_linear RS disjE) 1),

   381         (etac disjE 2),

   382         (etac (hd (tl (tl prems))) 1),

   383         (etac (sym RS hd (tl prems)) 1),

   384         (etac (hd prems) 1)

   385         ]);

   387 (*Can be used with less_Suc_eq to get n=m | n<m *)

   388 goal Nat.thy "(~ m < n) = (n < Suc(m))";

   389 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);

   390 by (ALLGOALS Asm_simp_tac);

   391 qed "not_less_eq";

   393 (*Complete induction, aka course-of-values induction*)

   394 val prems = goalw Nat.thy [less_def]

   395     "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";

   396 by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);

   397 by (eresolve_tac prems 1);

   398 qed "less_induct";

   401 (*** Properties of <= ***)

   403 goalw Nat.thy [le_def] "0 <= n";

   404 by (rtac not_less0 1);

   405 qed "le0";

   407 goalw Nat.thy [le_def] "~ Suc n <= n";

   408 by (Simp_tac 1);

   409 qed "Suc_n_not_le_n";

   411 goalw Nat.thy [le_def] "(i <= 0) = (i = 0)";

   412 by (nat_ind_tac "i" 1);

   413 by (ALLGOALS Asm_simp_tac);

   414 qed "le_0";

   416 Addsimps [less_not_refl,

   417           (*less_Suc_eq,*) le0, le_0,

   418           Suc_Suc_eq, Suc_n_not_le_n,

   419           n_not_Suc_n, Suc_n_not_n,

   420           nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];

   422 (*Prevents simplification of f and g: much faster*)

   423 qed_goal "nat_case_weak_cong" Nat.thy

   424   "m=n ==> nat_case a f m = nat_case a f n"

   425   (fn [prem] => [rtac (prem RS arg_cong) 1]);

   427 qed_goal "nat_rec_weak_cong" Nat.thy

   428   "m=n ==> nat_rec m a f = nat_rec n a f"

   429   (fn [prem] => [rtac (prem RS arg_cong) 1]);

   431 val prems = goalw Nat.thy [le_def] "~n<m ==> m<=(n::nat)";

   432 by (resolve_tac prems 1);

   433 qed "leI";

   435 val prems = goalw Nat.thy [le_def] "m<=n ==> ~ n < (m::nat)";

   436 by (resolve_tac prems 1);

   437 qed "leD";

   439 val leE = make_elim leD;

   441 goal Nat.thy "(~n<m) = (m<=(n::nat))";

   442 by (fast_tac (!claset addIs [leI] addEs [leE]) 1);

   443 qed "not_less_iff_le";

   445 goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";

   446 by (Fast_tac 1);

   447 qed "not_leE";

   449 goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n";

   450 by (simp_tac (!simpset addsimps [less_Suc_eq]) 1);

   451 by (fast_tac (!claset addEs [less_irrefl,less_asym]) 1);

   452 qed "lessD";

   454 goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n";

   455 by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1);

   456 by (Fast_tac 1);

   457 qed "Suc_leD";

   459 goalw Nat.thy [le_def] "!!m. m <= n ==> m <= Suc n";

   460 by (fast_tac (!claset addDs [Suc_lessD]) 1);

   461 qed "le_SucI";

   462 Addsimps[le_SucI];

   464 goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)";

   465 by (fast_tac (!claset addEs [less_asym]) 1);

   466 qed "less_imp_le";

   468 goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";

   469 by (cut_facts_tac [less_linear] 1);

   470 by (fast_tac (!claset addEs [less_irrefl,less_asym]) 1);

   471 qed "le_imp_less_or_eq";

   473 goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";

   474 by (cut_facts_tac [less_linear] 1);

   475 by (fast_tac (!claset addEs [less_irrefl,less_asym]) 1);

   476 by (flexflex_tac);

   477 qed "less_or_eq_imp_le";

   479 goal Nat.thy "(x <= (y::nat)) = (x < y | x=y)";

   480 by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));

   481 qed "le_eq_less_or_eq";

   483 goal Nat.thy "n <= (n::nat)";

   484 by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);

   485 qed "le_refl";

   487 val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";

   488 by (dtac le_imp_less_or_eq 1);

   489 by (fast_tac (!claset addIs [less_trans]) 1);

   490 qed "le_less_trans";

   492 goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";

   493 by (dtac le_imp_less_or_eq 1);

   494 by (fast_tac (!claset addIs [less_trans]) 1);

   495 qed "less_le_trans";

   497 goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";

   498 by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,

   499           rtac less_or_eq_imp_le, fast_tac (!claset addIs [less_trans])]);

   500 qed "le_trans";

   502 val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";

   503 by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,

   504           fast_tac (!claset addEs [less_irrefl,less_asym])]);

   505 qed "le_anti_sym";

   507 goal Nat.thy "(Suc(n) <= Suc(m)) = (n <= m)";

   508 by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);

   509 qed "Suc_le_mono";

   511 Addsimps [le_refl,Suc_le_mono];

   514 (** LEAST -- the least number operator **)

   516 val [prem1,prem2] = goalw Nat.thy [Least_def]

   517     "[| P(k);  !!x. x<k ==> ~P(x) |] ==> (LEAST x.P(x)) = k";

   518 by (rtac select_equality 1);

   519 by (fast_tac (!claset addSIs [prem1,prem2]) 1);

   520 by (cut_facts_tac [less_linear] 1);

   521 by (fast_tac (!claset addSIs [prem1] addSDs [prem2]) 1);

   522 qed "Least_equality";

   524 val [prem] = goal Nat.thy "P(k) ==> P(LEAST x.P(x))";

   525 by (rtac (prem RS rev_mp) 1);

   526 by (res_inst_tac [("n","k")] less_induct 1);

   527 by (rtac impI 1);

   528 by (rtac classical 1);

   529 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);

   530 by (assume_tac 1);

   531 by (assume_tac 2);

   532 by (Fast_tac 1);

   533 qed "LeastI";

   535 (*Proof is almost identical to the one above!*)

   536 val [prem] = goal Nat.thy "P(k) ==> (LEAST x.P(x)) <= k";

   537 by (rtac (prem RS rev_mp) 1);

   538 by (res_inst_tac [("n","k")] less_induct 1);

   539 by (rtac impI 1);

   540 by (rtac classical 1);

   541 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);

   542 by (assume_tac 1);

   543 by (rtac le_refl 2);

   544 by (fast_tac (!claset addIs [less_imp_le,le_trans]) 1);

   545 qed "Least_le";

   547 val [prem] = goal Nat.thy "k < (LEAST x.P(x)) ==> ~P(k)";

   548 by (rtac notI 1);

   549 by (etac (rewrite_rule [le_def] Least_le RS notE) 1);

   550 by (rtac prem 1);

   551 qed "not_less_Least";

   553 qed_goalw "Least_Suc" Nat.thy [Least_def]

   554  "[| ? n. P (Suc n); ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P (Suc m))"

   555  (fn prems => [

   556 	cut_facts_tac prems 1,

   557 	rtac select_equality 1,

   558 	fold_goals_tac [Least_def],

   559 	safe_tac (HOL_cs addSEs [LeastI]),

   560 	res_inst_tac [("n","j")] natE 1,

   561 	Fast_tac 1,

   562 	fast_tac (!claset addDs [Suc_less_SucD] addDs [not_less_Least]) 1,

   563 	res_inst_tac [("n","k")] natE 1,

   564 	Fast_tac 1,

   565 	hyp_subst_tac 1,

   566 	rewtac Least_def,

   567 	rtac (select_equality RS arg_cong RS sym) 1,

   568 	safe_tac HOL_cs,

   569 	dtac Suc_mono 1,

   570 	Fast_tac 1,

   571 	cut_facts_tac [less_linear] 1,

   572 	safe_tac HOL_cs,

   573 	atac 2,

   574 	Fast_tac 2,

   575 	dtac Suc_mono 1,

   576 	Fast_tac 1]);

author	berghofe
	Thu May 23 14:37:06 1996 +0200 (1996-05-23)
changeset 1760	6f41a494f3b1
parent 1679	6a82e122b337
child 1777	47c47b7d7aa8
permissions	-rw-r--r--