src/ZF/Ord.ML
author clasohm
Thu Sep 16 12:20:38 1993 +0200 (1993-09-16)
changeset 0 a5a9c433f639
child 6 8ce8c4d13d4d
permissions -rw-r--r--
Initial revision
     1 (*  Title: 	ZF/ordinal.thy
     2     ID:         $Id$
     3     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 For ordinal.thy.  Ordinals in Zermelo-Fraenkel Set Theory 
     7 *)
     8 
     9 open Ord;
    10 
    11 (*** Rules for Transset ***)
    12 
    13 (** Two neat characterisations of Transset **)
    14 
    15 goalw Ord.thy [Transset_def] "Transset(A) <-> A<=Pow(A)";
    16 by (fast_tac ZF_cs 1);
    17 val Transset_iff_Pow = result();
    18 
    19 goalw Ord.thy [Transset_def] "Transset(A) <-> Union(succ(A)) = A";
    20 by (fast_tac (eq_cs addSEs [equalityE]) 1);
    21 val Transset_iff_Union_succ = result();
    22 
    23 (** Consequences of downwards closure **)
    24 
    25 goalw Ord.thy [Transset_def]
    26     "!!C a b. [| Transset(C); {a,b}: C |] ==> a:C & b: C";
    27 by (fast_tac ZF_cs 1);
    28 val Transset_doubleton_D = result();
    29 
    30 val [prem1,prem2] = goalw Ord.thy [Pair_def]
    31     "[| Transset(C); <a,b>: C |] ==> a:C & b: C";
    32 by (cut_facts_tac [prem2] 1);	
    33 by (fast_tac (ZF_cs addSDs [prem1 RS Transset_doubleton_D]) 1);
    34 val Transset_Pair_D = result();
    35 
    36 val prem1::prems = goal Ord.thy
    37     "[| Transset(C); A*B <= C; b: B |] ==> A <= C";
    38 by (cut_facts_tac prems 1);
    39 by (fast_tac (ZF_cs addSDs [prem1 RS Transset_Pair_D]) 1);
    40 val Transset_includes_domain = result();
    41 
    42 val prem1::prems = goal Ord.thy
    43     "[| Transset(C); A*B <= C; a: A |] ==> B <= C";
    44 by (cut_facts_tac prems 1);
    45 by (fast_tac (ZF_cs addSDs [prem1 RS Transset_Pair_D]) 1);
    46 val Transset_includes_range = result();
    47 
    48 val [prem1,prem2] = goalw (merge_theories(Ord.thy,Sum.thy)) [sum_def]
    49     "[| Transset(C); A+B <= C |] ==> A <= C & B <= C";
    50 br (prem2 RS (Un_subset_iff RS iffD1) RS conjE) 1;
    51 by (REPEAT (etac (prem1 RS Transset_includes_range) 1
    52      ORELSE resolve_tac [conjI, singletonI] 1));
    53 val Transset_includes_summands = result();
    54 
    55 val [prem] = goalw (merge_theories(Ord.thy,Sum.thy)) [sum_def]
    56     "Transset(C) ==> (A+B) Int C <= (A Int C) + (B Int C)";
    57 br (Int_Un_distrib RS ssubst) 1;
    58 by (fast_tac (ZF_cs addSDs [prem RS Transset_Pair_D]) 1);
    59 val Transset_sum_Int_subset = result();
    60 
    61 (** Closure properties **)
    62 
    63 goalw Ord.thy [Transset_def] "Transset(0)";
    64 by (fast_tac ZF_cs 1);
    65 val Transset_0 = result();
    66 
    67 goalw Ord.thy [Transset_def]
    68     "!!i j. [| Transset(i);  Transset(j) |] ==> Transset(i Un j)";
    69 by (fast_tac ZF_cs 1);
    70 val Transset_Un = result();
    71 
    72 goalw Ord.thy [Transset_def]
    73     "!!i j. [| Transset(i);  Transset(j) |] ==> Transset(i Int j)";
    74 by (fast_tac ZF_cs 1);
    75 val Transset_Int = result();
    76 
    77 goalw Ord.thy [Transset_def] "!!i. Transset(i) ==> Transset(succ(i))";
    78 by (fast_tac ZF_cs 1);
    79 val Transset_succ = result();
    80 
    81 goalw Ord.thy [Transset_def] "!!i. Transset(i) ==> Transset(Pow(i))";
    82 by (fast_tac ZF_cs 1);
    83 val Transset_Pow = result();
    84 
    85 goalw Ord.thy [Transset_def] "!!A. Transset(A) ==> Transset(Union(A))";
    86 by (fast_tac ZF_cs 1);
    87 val Transset_Union = result();
    88 
    89 val [Transprem] = goalw Ord.thy [Transset_def]
    90     "[| !!i. i:A ==> Transset(i) |] ==> Transset(Union(A))";
    91 by (fast_tac (ZF_cs addEs [Transprem RS bspec RS subsetD]) 1);
    92 val Transset_Union_family = result();
    93 
    94 val [prem,Transprem] = goalw Ord.thy [Transset_def]
    95     "[| j:A;  !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))";
    96 by (cut_facts_tac [prem] 1);
    97 by (fast_tac (ZF_cs addEs [Transprem RS bspec RS subsetD]) 1);
    98 val Transset_Inter_family = result();
    99 
   100 (*** Natural Deduction rules for Ord ***)
   101 
   102 val prems = goalw Ord.thy [Ord_def]
   103     "[| Transset(i);  !!x. x:i ==> Transset(x) |]  ==>  Ord(i) ";
   104 by (REPEAT (ares_tac (prems@[ballI,conjI]) 1));
   105 val OrdI = result();
   106 
   107 val [major] = goalw Ord.thy [Ord_def]
   108     "Ord(i) ==> Transset(i)";
   109 by (rtac (major RS conjunct1) 1);
   110 val Ord_is_Transset = result();
   111 
   112 val [major,minor] = goalw Ord.thy [Ord_def]
   113     "[| Ord(i);  j:i |] ==> Transset(j) ";
   114 by (rtac (minor RS (major RS conjunct2 RS bspec)) 1);
   115 val Ord_contains_Transset = result();
   116 
   117 (*** Lemmas for ordinals ***)
   118 
   119 goalw Ord.thy [Ord_def,Transset_def] "!!i j. [| Ord(i);  j:i |] ==> Ord(j) ";
   120 by (fast_tac ZF_cs 1);
   121 val Ord_in_Ord = result();
   122 
   123 goal Ord.thy "!!i j. [| Ord(i);  Transset(j);  j<=i |] ==> Ord(j)";
   124 by (REPEAT (ares_tac [OrdI] 1
   125      ORELSE eresolve_tac [Ord_contains_Transset, subsetD] 1));
   126 val Ord_subset_Ord = result();
   127 
   128 goalw Ord.thy [Ord_def,Transset_def] "!!i j. [| j:i;  Ord(i) |] ==> j<=i";
   129 by (fast_tac ZF_cs 1);
   130 val OrdmemD = result();
   131 
   132 goal Ord.thy "!!i j k. [| i:j;  j:k;  Ord(k) |] ==> i:k";
   133 by (REPEAT (ares_tac [OrdmemD RS subsetD] 1));
   134 val Ord_trans = result();
   135 
   136 goal Ord.thy "!!i j. [| i:j;  Ord(j) |] ==> succ(i) <= j";
   137 by (REPEAT (ares_tac [OrdmemD RSN (2,succ_subsetI)] 1));
   138 val Ord_succ_subsetI = result();
   139 
   140 
   141 (*** The construction of ordinals: 0, succ, Union ***)
   142 
   143 goal Ord.thy "Ord(0)";
   144 by (REPEAT (ares_tac [OrdI,Transset_0] 1 ORELSE etac emptyE 1));
   145 val Ord_0 = result();
   146 
   147 goal Ord.thy "!!i. Ord(i) ==> Ord(succ(i))";
   148 by (REPEAT (ares_tac [OrdI,Transset_succ] 1
   149      ORELSE eresolve_tac [succE,ssubst,Ord_is_Transset,
   150 			  Ord_contains_Transset] 1));
   151 val Ord_succ = result();
   152 
   153 val nonempty::prems = goal Ord.thy
   154     "[| j:A;  !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))";
   155 by (rtac (nonempty RS Transset_Inter_family RS OrdI) 1);
   156 by (rtac Ord_is_Transset 1);
   157 by (REPEAT (ares_tac ([Ord_contains_Transset,nonempty]@prems) 1
   158      ORELSE etac InterD 1));
   159 val Ord_Inter = result();
   160 
   161 val jmemA::prems = goal Ord.thy
   162     "[| j:A;  !!x. x:A ==> Ord(B(x)) |] ==> Ord(INT x:A. B(x))";
   163 by (rtac (jmemA RS RepFunI RS Ord_Inter) 1);
   164 by (etac RepFunE 1);
   165 by (etac ssubst 1);
   166 by (eresolve_tac prems 1);
   167 val Ord_INT = result();
   168 
   169 
   170 (*** Natural Deduction rules for Memrel ***)
   171 
   172 goalw Ord.thy [Memrel_def] "<a,b> : Memrel(A) <-> a:b & a:A & b:A";
   173 by (fast_tac ZF_cs 1);
   174 val Memrel_iff = result();
   175 
   176 val prems = goal Ord.thy "[| a: b;  a: A;  b: A |]  ==>  <a,b> : Memrel(A)";
   177 by (REPEAT (resolve_tac (prems@[conjI, Memrel_iff RS iffD2]) 1));
   178 val MemrelI = result();
   179 
   180 val [major,minor] = goal Ord.thy
   181     "[| <a,b> : Memrel(A);  \
   182 \       [| a: A;  b: A;  a:b |]  ==> P \
   183 \    |]  ==> P";
   184 by (rtac (major RS (Memrel_iff RS iffD1) RS conjE) 1);
   185 by (etac conjE 1);
   186 by (rtac minor 1);
   187 by (REPEAT (assume_tac 1));
   188 val MemrelE = result();
   189 
   190 (*The membership relation (as a set) is well-founded.
   191   Proof idea: show A<=B by applying the foundation axiom to A-B *)
   192 goalw Ord.thy [wf_def] "wf(Memrel(A))";
   193 by (EVERY1 [rtac (foundation RS disjE RS allI),
   194 	    etac disjI1,
   195 	    etac bexE, 
   196 	    rtac (impI RS allI RS bexI RS disjI2),
   197 	    etac MemrelE,
   198 	    etac bspec,
   199 	    REPEAT o assume_tac]);
   200 val wf_Memrel = result();
   201 
   202 (*** Transfinite induction ***)
   203 
   204 (*Epsilon induction over a transitive set*)
   205 val major::prems = goalw Ord.thy [Transset_def]
   206     "[| i: k;  Transset(k);                          \
   207 \       !!x.[| x: k;  ALL y:x. P(y) |] ==> P(x) \
   208 \    |]  ==>  P(i)";
   209 by (rtac (major RS (wf_Memrel RS wf_induct2)) 1);
   210 by (fast_tac (ZF_cs addEs [MemrelE]) 1);
   211 by (resolve_tac prems 1);
   212 by (assume_tac 1);
   213 by (cut_facts_tac prems 1);
   214 by (fast_tac (ZF_cs addIs [MemrelI]) 1);
   215 val Transset_induct = result();
   216 
   217 (*Induction over an ordinal*)
   218 val Ord_induct = Ord_is_Transset RSN (2, Transset_induct);
   219 
   220 (*Induction over the class of ordinals -- a useful corollary of Ord_induct*)
   221 val [major,indhyp] = goal Ord.thy
   222     "[| Ord(i); \
   223 \       !!x.[| Ord(x);  ALL y:x. P(y) |] ==> P(x) \
   224 \    |]  ==>  P(i)";
   225 by (rtac (major RS Ord_succ RS (succI1 RS Ord_induct)) 1);
   226 by (rtac indhyp 1);
   227 by (rtac (major RS Ord_succ RS Ord_in_Ord) 1);
   228 by (REPEAT (assume_tac 1));
   229 val trans_induct = result();
   230 
   231 (*Perform induction on i, then prove the Ord(i) subgoal using prems. *)
   232 fun trans_ind_tac a prems i = 
   233     EVERY [res_inst_tac [("i",a)] trans_induct i,
   234 	   rename_last_tac a ["1"] (i+1),
   235 	   ares_tac prems i];
   236 
   237 
   238 (*** Fundamental properties of the epsilon ordering (< on ordinals) ***)
   239 
   240 (*Finds contradictions for the following proof*)
   241 val Ord_trans_tac = EVERY' [etac notE, etac Ord_trans, REPEAT o atac];
   242 
   243 (** Proving that "member" is a linear ordering on the ordinals **)
   244 
   245 val prems = goal Ord.thy
   246     "Ord(i) ==> (ALL j. Ord(j) --> i:j | i=j | j:i)";
   247 by (trans_ind_tac "i" prems 1);
   248 by (rtac (impI RS allI) 1);
   249 by (trans_ind_tac "j" [] 1);
   250 by (DEPTH_SOLVE (swap_res_tac [disjCI,equalityI,subsetI] 1
   251      ORELSE ball_tac 1
   252      ORELSE eresolve_tac [impE,disjE,allE] 1 
   253      ORELSE hyp_subst_tac 1
   254      ORELSE Ord_trans_tac 1));
   255 val Ord_linear_lemma = result();
   256 
   257 val ordi::ordj::prems = goal Ord.thy
   258     "[| Ord(i);  Ord(j);  i:j ==> P;  i=j ==> P;  j:i ==> P |] \
   259 \    ==> P";
   260 by (rtac (ordi RS Ord_linear_lemma RS spec RS impE) 1);
   261 by (rtac ordj 1);
   262 by (REPEAT (eresolve_tac (prems@[asm_rl,disjE]) 1)); 
   263 val Ord_linear = result();
   264 
   265 val prems = goal Ord.thy
   266     "[| Ord(i);  Ord(j);  i<=j ==> P;  j<=i ==> P |] \
   267 \    ==> P";
   268 by (res_inst_tac [("i","i"),("j","j")] Ord_linear 1);
   269 by (DEPTH_SOLVE (ares_tac (prems@[subset_refl]) 1
   270           ORELSE eresolve_tac [asm_rl,OrdmemD,ssubst] 1));
   271 val Ord_subset = result();
   272 
   273 goal Ord.thy "!!i j. [| j<=i;  ~ i<=j;  Ord(i);  Ord(j) |] ==> j:i";
   274 by (etac Ord_linear 1);
   275 by (REPEAT (ares_tac [subset_refl] 1
   276      ORELSE eresolve_tac [notE,OrdmemD,ssubst] 1));
   277 val Ord_member = result();
   278 
   279 val [prem] = goal Ord.thy "Ord(i) ==> 0: succ(i)";
   280 by (rtac (empty_subsetI RS Ord_member) 1);
   281 by (fast_tac ZF_cs 1);
   282 by (rtac (prem RS Ord_succ) 1);
   283 by (rtac Ord_0 1);
   284 val Ord_0_mem_succ = result();
   285 
   286 goal Ord.thy "!!i j. [| Ord(i);  Ord(j) |] ==> j:i <-> j<=i & ~(i<=j)";
   287 by (rtac iffI 1);
   288 by (rtac conjI 1);
   289 by (etac OrdmemD 1);
   290 by (rtac (mem_anti_refl RS notI) 2);
   291 by (etac subsetD 2);
   292 by (REPEAT (eresolve_tac [asm_rl, conjE, Ord_member] 1));
   293 val Ord_member_iff = result();
   294 
   295 goal Ord.thy "!!i. Ord(i) ==> 0:i  <-> ~ i=0";
   296 be (Ord_0 RSN (2,Ord_member_iff) RS iff_trans) 1;
   297 by (fast_tac eq_cs 1);
   298 val Ord_0_member_iff = result();
   299 
   300 (** For ordinals, i: succ(j) means 'less-than or equals' **)
   301 
   302 goal Ord.thy "!!i j. [| j<=i;  Ord(i);  Ord(j) |] ==> j : succ(i)";
   303 by (rtac Ord_member 1);
   304 by (REPEAT (ares_tac [Ord_succ] 3));
   305 by (rtac (mem_anti_refl RS notI) 2);
   306 by (etac subsetD 2);
   307 by (ALLGOALS (fast_tac ZF_cs));
   308 val member_succI = result();
   309 
   310 goalw Ord.thy [Transset_def,Ord_def]
   311     "!!i j. [| i : succ(j);  Ord(j) |] ==> i<=j";
   312 by (fast_tac ZF_cs 1);
   313 val member_succD = result();
   314 
   315 goal Ord.thy "!!i j. [| Ord(i);  Ord(j) |] ==> j:succ(i) <-> j<=i";
   316 by (fast_tac (subset_cs addSEs [member_succI, member_succD]) 1);
   317 val member_succ_iff = result();
   318 
   319 goal Ord.thy
   320     "!!i j. [| Ord(i);  Ord(j) |] ==> i<=succ(j) <-> i<=j | i=succ(j)";
   321 by (ASM_SIMP_TAC (ZF_ss addrews [member_succ_iff RS iff_sym, Ord_succ]) 1);
   322 by (fast_tac ZF_cs 1);
   323 val subset_succ_iff = result();
   324 
   325 goal Ord.thy "!!i j. [| i:succ(j);  j:k;  Ord(k) |] ==> i:k";
   326 by (fast_tac (ZF_cs addEs [Ord_trans]) 1);
   327 val Ord_trans1 = result();
   328 
   329 goal Ord.thy "!!i j. [| i:j;  j:succ(k);  Ord(k) |] ==> i:k";
   330 by (fast_tac (ZF_cs addEs [Ord_trans]) 1);
   331 val Ord_trans2 = result();
   332 
   333 goal Ord.thy "!!i jk. [| i:j;  j<=k;  Ord(k) |] ==> i:k";
   334 by (fast_tac (ZF_cs addEs [Ord_trans]) 1);
   335 val Ord_transsub2 = result();
   336 
   337 goal Ord.thy "!!i j. [| i:j;  Ord(j) |] ==> succ(i) : succ(j)";
   338 by (rtac member_succI 1);
   339 by (REPEAT (ares_tac [subsetI,Ord_succ,Ord_in_Ord] 1   
   340      ORELSE eresolve_tac [succE,Ord_trans,ssubst] 1));
   341 val succ_mem_succI = result();
   342 
   343 goal Ord.thy "!!i j. [| succ(i) : succ(j);  Ord(j) |] ==> i:j";
   344 by (REPEAT (eresolve_tac [asm_rl, make_elim member_succD, succ_subsetE] 1));
   345 val succ_mem_succE = result();
   346 
   347 goal Ord.thy "!!i j. Ord(j) ==> succ(i) : succ(j) <-> i:j";
   348 by (REPEAT (ares_tac [iffI,succ_mem_succI,succ_mem_succE] 1));
   349 val succ_mem_succ_iff = result();
   350 
   351 goal Ord.thy "!!i j. [| i<=j;  Ord(i);  Ord(j) |] ==> succ(i) <= succ(j)";
   352 by (rtac (member_succI RS succ_mem_succI RS member_succD) 1);
   353 by (REPEAT (ares_tac [Ord_succ] 1));
   354 val Ord_succ_mono = result();
   355 
   356 goal Ord.thy "!!i j k. [| i:k;  j:k;  Ord(k) |] ==> i Un j : k";
   357 by (res_inst_tac [("i","i"),("j","j")] Ord_subset 1);
   358 by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1));
   359 by (ASM_SIMP_TAC (ZF_ss addrews [subset_Un_iff RS iffD1]) 1);
   360 by (rtac (Un_commute RS ssubst) 1);
   361 by (ASM_SIMP_TAC (ZF_ss addrews [subset_Un_iff RS iffD1]) 1);
   362 val Ord_member_UnI = result();
   363 
   364 goal Ord.thy "!!i j k. [| i:k;  j:k;  Ord(k) |] ==> i Int j : k";
   365 by (res_inst_tac [("i","i"),("j","j")] Ord_subset 1);
   366 by (REPEAT (eresolve_tac [asm_rl, Ord_in_Ord] 1));
   367 by (ASM_SIMP_TAC (ZF_ss addrews [subset_Int_iff RS iffD1]) 1);
   368 by (rtac (Int_commute RS ssubst) 1);
   369 by (ASM_SIMP_TAC (ZF_ss addrews [subset_Int_iff RS iffD1]) 1);
   370 val Ord_member_IntI = result();
   371 
   372 
   373 (*** Results about limits ***)
   374 
   375 val prems = goal Ord.thy "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))";
   376 by (rtac (Ord_is_Transset RS Transset_Union_family RS OrdI) 1);
   377 by (REPEAT (etac UnionE 1 ORELSE ares_tac ([Ord_contains_Transset]@prems) 1));
   378 val Ord_Union = result();
   379 
   380 val prems = goal Ord.thy "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(UN x:A. B(x))";
   381 by (rtac Ord_Union 1);
   382 by (etac RepFunE 1);
   383 by (etac ssubst 1);
   384 by (eresolve_tac prems 1);
   385 val Ord_UN = result();
   386 
   387 (*The upper bound must be a successor ordinal --
   388   consider that (UN i:nat.i)=nat although nat is an upper bound of each i*)
   389 val [ordi,limit] = goal Ord.thy
   390     "[| Ord(i);  !!y. y:A ==> f(y): succ(i) |] ==> (UN y:A. f(y)) : succ(i)";
   391 by (rtac (member_succD RS UN_least RS member_succI) 1);
   392 by (REPEAT (ares_tac [ordi, Ord_UN, ordi RS Ord_succ RS Ord_in_Ord,
   393 		      limit] 1));
   394 val sup_least = result();
   395 
   396 val [jmemi,ordi,limit] = goal Ord.thy
   397     "[| j: i;  Ord(i);  !!y. y:A ==> f(y): j |] ==> (UN y:A. succ(f(y))) : i";
   398 by (cut_facts_tac [jmemi RS (ordi RS Ord_in_Ord)] 1);
   399 by (rtac (sup_least RS Ord_trans2) 1);
   400 by (REPEAT (ares_tac [jmemi, ordi, succ_mem_succI, limit] 1));
   401 val sup_least2 = result();
   402 
   403 goal Ord.thy "!!i. Ord(i) ==> (UN y:i. succ(y)) = i";
   404 by (fast_tac (eq_cs addSEs [Ord_trans1]) 1);
   405 val Ord_equality = result();
   406 
   407 (*Holds for all transitive sets, not just ordinals*)
   408 goal Ord.thy "!!i. Ord(i) ==> Union(i) <= i";
   409 by (fast_tac (ZF_cs addSEs [Ord_trans]) 1);
   410 val Ord_Union_subset = result();