src/HOL/Nat.ML
author clasohm
Fri Mar 03 12:02:25 1995 +0100 (1995-03-03)
changeset 923 ff1574a81019
child 962 136308504cd9
permissions -rw-r--r--
new version of HOL with curried function application
     1 (*  Title: 	HOL/nat
     2     ID:         $Id$
     3     Author: 	Tobias Nipkow, Cambridge University Computer Laboratory
     4     Copyright   1991  University of Cambridge
     5 
     6 For nat.thy.  Type nat is defined as a set (Nat) over the type ind.
     7 *)
     8 
     9 open Nat;
    10 
    11 goal Nat.thy "mono(%X. {Zero_Rep} Un (Suc_Rep``X))";
    12 by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1));
    13 qed "Nat_fun_mono";
    14 
    15 val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski);
    16 
    17 (* Zero is a natural number -- this also justifies the type definition*)
    18 goal Nat.thy "Zero_Rep: Nat";
    19 by (rtac (Nat_unfold RS ssubst) 1);
    20 by (rtac (singletonI RS UnI1) 1);
    21 qed "Zero_RepI";
    22 
    23 val prems = goal Nat.thy "i: Nat ==> Suc_Rep(i) : Nat";
    24 by (rtac (Nat_unfold RS ssubst) 1);
    25 by (rtac (imageI RS UnI2) 1);
    26 by (resolve_tac prems 1);
    27 qed "Suc_RepI";
    28 
    29 (*** Induction ***)
    30 
    31 val major::prems = goal Nat.thy
    32     "[| i: Nat;  P(Zero_Rep);   \
    33 \       !!j. [| j: Nat; P(j) |] ==> P(Suc_Rep(j)) |]  ==> P(i)";
    34 by (rtac ([Nat_def, Nat_fun_mono, major] MRS def_induct) 1);
    35 by (fast_tac (set_cs addIs prems) 1);
    36 qed "Nat_induct";
    37 
    38 val prems = goalw Nat.thy [Zero_def,Suc_def]
    39     "[| P(0);   \
    40 \       !!k. P(k) ==> P(Suc(k)) |]  ==> P(n)";
    41 by (rtac (Rep_Nat_inverse RS subst) 1);   (*types force good instantiation*)
    42 by (rtac (Rep_Nat RS Nat_induct) 1);
    43 by (REPEAT (ares_tac prems 1
    44      ORELSE eresolve_tac [Abs_Nat_inverse RS subst] 1));
    45 qed "nat_induct";
    46 
    47 (*Perform induction on n. *)
    48 fun nat_ind_tac a i = 
    49     EVERY [res_inst_tac [("n",a)] nat_induct i,
    50 	   rename_last_tac a ["1"] (i+1)];
    51 
    52 (*A special form of induction for reasoning about m<n and m-n*)
    53 val prems = goal Nat.thy
    54     "[| !!x. P x 0;  \
    55 \       !!y. P 0 (Suc y);  \
    56 \       !!x y. [| P x y |] ==> P (Suc x) (Suc y)  \
    57 \    |] ==> P m n";
    58 by (res_inst_tac [("x","m")] spec 1);
    59 by (nat_ind_tac "n" 1);
    60 by (rtac allI 2);
    61 by (nat_ind_tac "x" 2);
    62 by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1));
    63 qed "diff_induct";
    64 
    65 (*Case analysis on the natural numbers*)
    66 val prems = goal Nat.thy 
    67     "[| n=0 ==> P;  !!x. n = Suc(x) ==> P |] ==> P";
    68 by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1);
    69 by (fast_tac (HOL_cs addSEs prems) 1);
    70 by (nat_ind_tac "n" 1);
    71 by (rtac (refl RS disjI1) 1);
    72 by (fast_tac HOL_cs 1);
    73 qed "natE";
    74 
    75 (*** Isomorphisms: Abs_Nat and Rep_Nat ***)
    76 
    77 (*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat),
    78   since we assume the isomorphism equations will one day be given by Isabelle*)
    79 
    80 goal Nat.thy "inj(Rep_Nat)";
    81 by (rtac inj_inverseI 1);
    82 by (rtac Rep_Nat_inverse 1);
    83 qed "inj_Rep_Nat";
    84 
    85 goal Nat.thy "inj_onto Abs_Nat Nat";
    86 by (rtac inj_onto_inverseI 1);
    87 by (etac Abs_Nat_inverse 1);
    88 qed "inj_onto_Abs_Nat";
    89 
    90 (*** Distinctness of constructors ***)
    91 
    92 goalw Nat.thy [Zero_def,Suc_def] "Suc(m) ~= 0";
    93 by (rtac (inj_onto_Abs_Nat RS inj_onto_contraD) 1);
    94 by (rtac Suc_Rep_not_Zero_Rep 1);
    95 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1));
    96 qed "Suc_not_Zero";
    97 
    98 bind_thm ("Zero_not_Suc", (Suc_not_Zero RS not_sym));
    99 
   100 bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE));
   101 val Zero_neq_Suc = sym RS Suc_neq_Zero;
   102 
   103 (** Injectiveness of Suc **)
   104 
   105 goalw Nat.thy [Suc_def] "inj(Suc)";
   106 by (rtac injI 1);
   107 by (dtac (inj_onto_Abs_Nat RS inj_ontoD) 1);
   108 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1));
   109 by (dtac (inj_Suc_Rep RS injD) 1);
   110 by (etac (inj_Rep_Nat RS injD) 1);
   111 qed "inj_Suc";
   112 
   113 val Suc_inject = inj_Suc RS injD;;
   114 
   115 goal Nat.thy "(Suc(m)=Suc(n)) = (m=n)";
   116 by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]); 
   117 qed "Suc_Suc_eq";
   118 
   119 goal Nat.thy "n ~= Suc(n)";
   120 by (nat_ind_tac "n" 1);
   121 by (ALLGOALS(asm_simp_tac (HOL_ss addsimps [Zero_not_Suc,Suc_Suc_eq])));
   122 qed "n_not_Suc_n";
   123 
   124 val Suc_n_not_n = n_not_Suc_n RS not_sym;
   125 
   126 (*** nat_case -- the selection operator for nat ***)
   127 
   128 goalw Nat.thy [nat_case_def] "nat_case a f 0 = a";
   129 by (fast_tac (set_cs addIs [select_equality] addEs [Zero_neq_Suc]) 1);
   130 qed "nat_case_0";
   131 
   132 goalw Nat.thy [nat_case_def] "nat_case a f (Suc k) = f(k)";
   133 by (fast_tac (set_cs addIs [select_equality] 
   134 	               addEs [make_elim Suc_inject, Suc_neq_Zero]) 1);
   135 qed "nat_case_Suc";
   136 
   137 (** Introduction rules for 'pred_nat' **)
   138 
   139 goalw Nat.thy [pred_nat_def] "<n, Suc(n)> : pred_nat";
   140 by (fast_tac set_cs 1);
   141 qed "pred_natI";
   142 
   143 val major::prems = goalw Nat.thy [pred_nat_def]
   144     "[| p : pred_nat;  !!x n. [| p = <n, Suc(n)> |] ==> R \
   145 \    |] ==> R";
   146 by (rtac (major RS CollectE) 1);
   147 by (REPEAT (eresolve_tac ([asm_rl,exE]@prems) 1));
   148 qed "pred_natE";
   149 
   150 goalw Nat.thy [wf_def] "wf(pred_nat)";
   151 by (strip_tac 1);
   152 by (nat_ind_tac "x" 1);
   153 by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, 
   154 			     make_elim Suc_inject]) 2);
   155 by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, Zero_neq_Suc]) 1);
   156 qed "wf_pred_nat";
   157 
   158 
   159 (*** nat_rec -- by wf recursion on pred_nat ***)
   160 
   161 bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec)));
   162 
   163 (** conversion rules **)
   164 
   165 goal Nat.thy "nat_rec 0 c h = c";
   166 by (rtac (nat_rec_unfold RS trans) 1);
   167 by (simp_tac (HOL_ss addsimps [nat_case_0]) 1);
   168 qed "nat_rec_0";
   169 
   170 goal Nat.thy "nat_rec (Suc n) c h = h n (nat_rec n c h)";
   171 by (rtac (nat_rec_unfold RS trans) 1);
   172 by (simp_tac (HOL_ss addsimps [nat_case_Suc, pred_natI, cut_apply]) 1);
   173 qed "nat_rec_Suc";
   174 
   175 (*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
   176 val [rew] = goal Nat.thy
   177     "[| !!n. f(n) == nat_rec n c h |] ==> f(0) = c";
   178 by (rewtac rew);
   179 by (rtac nat_rec_0 1);
   180 qed "def_nat_rec_0";
   181 
   182 val [rew] = goal Nat.thy
   183     "[| !!n. f(n) == nat_rec n c h |] ==> f(Suc(n)) = h n (f n)";
   184 by (rewtac rew);
   185 by (rtac nat_rec_Suc 1);
   186 qed "def_nat_rec_Suc";
   187 
   188 fun nat_recs def =
   189       [standard (def RS def_nat_rec_0),
   190        standard (def RS def_nat_rec_Suc)];
   191 
   192 
   193 (*** Basic properties of "less than" ***)
   194 
   195 (** Introduction properties **)
   196 
   197 val prems = goalw Nat.thy [less_def] "[| i<j;  j<k |] ==> i<(k::nat)";
   198 by (rtac (trans_trancl RS transD) 1);
   199 by (resolve_tac prems 1);
   200 by (resolve_tac prems 1);
   201 qed "less_trans";
   202 
   203 goalw Nat.thy [less_def] "n < Suc(n)";
   204 by (rtac (pred_natI RS r_into_trancl) 1);
   205 qed "lessI";
   206 
   207 (* i<j ==> i<Suc(j) *)
   208 val less_SucI = lessI RSN (2, less_trans);
   209 
   210 goal Nat.thy "0 < Suc(n)";
   211 by (nat_ind_tac "n" 1);
   212 by (rtac lessI 1);
   213 by (etac less_trans 1);
   214 by (rtac lessI 1);
   215 qed "zero_less_Suc";
   216 
   217 (** Elimination properties **)
   218 
   219 val prems = goalw Nat.thy [less_def] "n<m ==> ~ m<(n::nat)";
   220 by(fast_tac (HOL_cs addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1);
   221 qed "less_not_sym";
   222 
   223 (* [| n<m; m<n |] ==> R *)
   224 bind_thm ("less_asym", (less_not_sym RS notE));
   225 
   226 goalw Nat.thy [less_def] "~ n<(n::nat)";
   227 by (rtac notI 1);
   228 by (etac (wf_pred_nat RS wf_trancl RS wf_anti_refl) 1);
   229 qed "less_not_refl";
   230 
   231 (* n<n ==> R *)
   232 bind_thm ("less_anti_refl", (less_not_refl RS notE));
   233 
   234 goal Nat.thy "!!m. n<m ==> m ~= (n::nat)";
   235 by(fast_tac (HOL_cs addEs [less_anti_refl]) 1);
   236 qed "less_not_refl2";
   237 
   238 
   239 val major::prems = goalw Nat.thy [less_def]
   240     "[| i<k;  k=Suc(i) ==> P;  !!j. [| i<j;  k=Suc(j) |] ==> P \
   241 \    |] ==> P";
   242 by (rtac (major RS tranclE) 1);
   243 by (fast_tac (HOL_cs addSEs (prems@[pred_natE, Pair_inject])) 1);
   244 by (fast_tac (HOL_cs addSEs (prems@[pred_natE, Pair_inject])) 1);
   245 qed "lessE";
   246 
   247 goal Nat.thy "~ n<0";
   248 by (rtac notI 1);
   249 by (etac lessE 1);
   250 by (etac Zero_neq_Suc 1);
   251 by (etac Zero_neq_Suc 1);
   252 qed "not_less0";
   253 
   254 (* n<0 ==> R *)
   255 bind_thm ("less_zeroE", (not_less0 RS notE));
   256 
   257 val [major,less,eq] = goal Nat.thy
   258     "[| m < Suc(n);  m<n ==> P;  m=n ==> P |] ==> P";
   259 by (rtac (major RS lessE) 1);
   260 by (rtac eq 1);
   261 by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
   262 by (rtac less 1);
   263 by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
   264 qed "less_SucE";
   265 
   266 goal Nat.thy "(m < Suc(n)) = (m < n | m = n)";
   267 by (fast_tac (HOL_cs addSIs [lessI]
   268 		     addEs  [less_trans, less_SucE]) 1);
   269 qed "less_Suc_eq";
   270 
   271 
   272 (** Inductive (?) properties **)
   273 
   274 val [prem] = goal Nat.thy "Suc(m) < n ==> m<n";
   275 by (rtac (prem RS rev_mp) 1);
   276 by (nat_ind_tac "n" 1);
   277 by (rtac impI 1);
   278 by (etac less_zeroE 1);
   279 by (fast_tac (HOL_cs addSIs [lessI RS less_SucI]
   280 	 	     addSDs [Suc_inject]
   281 		     addEs  [less_trans, lessE]) 1);
   282 qed "Suc_lessD";
   283 
   284 val [major,minor] = goal Nat.thy 
   285     "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
   286 \    |] ==> P";
   287 by (rtac (major RS lessE) 1);
   288 by (etac (lessI RS minor) 1);
   289 by (etac (Suc_lessD RS minor) 1);
   290 by (assume_tac 1);
   291 qed "Suc_lessE";
   292 
   293 val [major] = goal Nat.thy "Suc(m) < Suc(n) ==> m<n";
   294 by (rtac (major RS lessE) 1);
   295 by (REPEAT (rtac lessI 1
   296      ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1));
   297 qed "Suc_less_SucD";
   298 
   299 val prems = goal Nat.thy "m<n ==> Suc(m) < Suc(n)";
   300 by (subgoal_tac "m<n --> Suc(m) < Suc(n)" 1);
   301 by (fast_tac (HOL_cs addIs prems) 1);
   302 by (nat_ind_tac "n" 1);
   303 by (rtac impI 1);
   304 by (etac less_zeroE 1);
   305 by (fast_tac (HOL_cs addSIs [lessI]
   306 	 	     addSDs [Suc_inject]
   307 		     addEs  [less_trans, lessE]) 1);
   308 qed "Suc_mono";
   309 
   310 goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)";
   311 by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
   312 qed "Suc_less_eq";
   313 
   314 goal Nat.thy "~(Suc(n) < n)";
   315 by(fast_tac (HOL_cs addEs [Suc_lessD RS less_anti_refl]) 1);
   316 qed "not_Suc_n_less_n";
   317 
   318 (*"Less than" is a linear ordering*)
   319 goal Nat.thy "m<n | m=n | n<(m::nat)";
   320 by (nat_ind_tac "m" 1);
   321 by (nat_ind_tac "n" 1);
   322 by (rtac (refl RS disjI1 RS disjI2) 1);
   323 by (rtac (zero_less_Suc RS disjI1) 1);
   324 by (fast_tac (HOL_cs addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1);
   325 qed "less_linear";
   326 
   327 (*Can be used with less_Suc_eq to get n=m | n<m *)
   328 goal Nat.thy "(~ m < n) = (n < Suc(m))";
   329 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
   330 by(ALLGOALS(asm_simp_tac (HOL_ss addsimps
   331                           [not_less0,zero_less_Suc,Suc_less_eq])));
   332 qed "not_less_eq";
   333 
   334 (*Complete induction, aka course-of-values induction*)
   335 val prems = goalw Nat.thy [less_def]
   336     "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
   337 by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
   338 by (eresolve_tac prems 1);
   339 qed "less_induct";
   340 
   341 
   342 (*** Properties of <= ***)
   343 
   344 goalw Nat.thy [le_def] "0 <= n";
   345 by (rtac not_less0 1);
   346 qed "le0";
   347 
   348 val nat_simps = [not_less0, less_not_refl, zero_less_Suc, lessI, 
   349 		 Suc_less_eq, less_Suc_eq, le0, not_Suc_n_less_n,
   350 		 Suc_not_Zero, Zero_not_Suc, Suc_Suc_eq,
   351 		 n_not_Suc_n, Suc_n_not_n,
   352 		 nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];
   353 
   354 val nat_ss0 = sum_ss  addsimps  nat_simps;
   355 
   356 (*Prevents simplification of f and g: much faster*)
   357 qed_goal "nat_case_weak_cong" Nat.thy
   358   "m=n ==> nat_case a f m = nat_case a f n"
   359   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   360 
   361 qed_goal "nat_rec_weak_cong" Nat.thy
   362   "m=n ==> nat_rec m a f = nat_rec n a f"
   363   (fn [prem] => [rtac (prem RS arg_cong) 1]);
   364 
   365 val prems = goalw Nat.thy [le_def] "~(n<m) ==> m<=(n::nat)";
   366 by (resolve_tac prems 1);
   367 qed "leI";
   368 
   369 val prems = goalw Nat.thy [le_def] "m<=n ==> ~(n<(m::nat))";
   370 by (resolve_tac prems 1);
   371 qed "leD";
   372 
   373 val leE = make_elim leD;
   374 
   375 goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
   376 by (fast_tac HOL_cs 1);
   377 qed "not_leE";
   378 
   379 goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n";
   380 by(simp_tac nat_ss0 1);
   381 by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
   382 qed "lessD";
   383 
   384 goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
   385 by(asm_full_simp_tac nat_ss0 1);
   386 by(fast_tac HOL_cs 1);
   387 qed "Suc_leD";
   388 
   389 goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)";
   390 by (fast_tac (HOL_cs addEs [less_asym]) 1);
   391 qed "less_imp_le";
   392 
   393 goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
   394 by (cut_facts_tac [less_linear] 1);
   395 by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
   396 qed "le_imp_less_or_eq";
   397 
   398 goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
   399 by (cut_facts_tac [less_linear] 1);
   400 by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
   401 by (flexflex_tac);
   402 qed "less_or_eq_imp_le";
   403 
   404 goal Nat.thy "(x <= (y::nat)) = (x < y | x=y)";
   405 by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
   406 qed "le_eq_less_or_eq";
   407 
   408 goal Nat.thy "n <= (n::nat)";
   409 by(simp_tac (HOL_ss addsimps [le_eq_less_or_eq]) 1);
   410 qed "le_refl";
   411 
   412 val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
   413 by (dtac le_imp_less_or_eq 1);
   414 by (fast_tac (HOL_cs addIs [less_trans]) 1);
   415 qed "le_less_trans";
   416 
   417 goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
   418 by (dtac le_imp_less_or_eq 1);
   419 by (fast_tac (HOL_cs addIs [less_trans]) 1);
   420 qed "less_le_trans";
   421 
   422 goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";
   423 by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
   424           rtac less_or_eq_imp_le, fast_tac (HOL_cs addIs [less_trans])]);
   425 qed "le_trans";
   426 
   427 val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";
   428 by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
   429           fast_tac (HOL_cs addEs [less_anti_refl,less_asym])]);
   430 qed "le_anti_sym";
   431 
   432 goal Nat.thy "(Suc(n) <= Suc(m)) = (n <= m)";
   433 by (simp_tac (nat_ss0 addsimps [le_eq_less_or_eq]) 1);
   434 qed "Suc_le_mono";
   435 
   436 val nat_ss = nat_ss0 addsimps [le_refl];