src/ZF/WF.ML
 author lcp Tue Jun 21 17:20:34 1994 +0200 (1994-06-21) changeset 435 ca5356bd315a parent 6 8ce8c4d13d4d child 437 435875e4b21d permissions -rw-r--r--
Addition of cardinals and order types, various tidying
```     1 (*  Title: 	ZF/wf.ML
```
```     2     ID:         \$Id\$
```
```     3     Author: 	Tobias Nipkow and Lawrence C Paulson
```
```     4     Copyright   1992  University of Cambridge
```
```     5
```
```     6 For wf.thy.  Well-founded Recursion
```
```     7
```
```     8 Derived first for transitive relations, and finally for arbitrary WF relations
```
```     9 via wf_trancl and trans_trancl.
```
```    10
```
```    11 It is difficult to derive this general case directly, using r^+ instead of
```
```    12 r.  In is_recfun, the two occurrences of the relation must have the same
```
```    13 form.  Inserting r^+ in the_recfun or wftrec yields a recursion rule with
```
```    14 r^+ -`` {a} instead of r-``{a}.  This recursion rule is stronger in
```
```    15 principle, but harder to use, especially to prove wfrec_eclose_eq in
```
```    16 epsilon.ML.  Expanding out the definition of wftrec in wfrec would yield
```
```    17 a mess.
```
```    18 *)
```
```    19
```
```    20 open WF;
```
```    21
```
```    22
```
```    23 (*** Well-founded relations ***)
```
```    24
```
```    25 (** Equivalences between wf and wf_on **)
```
```    26
```
```    27 goalw WF.thy [wf_def, wf_on_def] "!!A r. wf(r) ==> wf[A](r)";
```
```    28 by (fast_tac ZF_cs 1);
```
```    29 val wf_imp_wf_on = result();
```
```    30
```
```    31 goalw WF.thy [wf_def, wf_on_def] "!!r. wf[field(r)](r) ==> wf(r)";
```
```    32 by (fast_tac ZF_cs 1);
```
```    33 val wf_on_field_imp_wf = result();
```
```    34
```
```    35 goal WF.thy "wf(r) <-> wf[field(r)](r)";
```
```    36 by (fast_tac (ZF_cs addSEs [wf_imp_wf_on, wf_on_field_imp_wf]) 1);
```
```    37 val wf_iff_wf_on_field = result();
```
```    38
```
```    39 goalw WF.thy [wf_on_def, wf_def] "!!A B r. [| wf[A](r);  B<=A |] ==> wf[B](r)";
```
```    40 by (fast_tac ZF_cs 1);
```
```    41 val wf_on_subset_A = result();
```
```    42
```
```    43 goalw WF.thy [wf_on_def, wf_def] "!!A r s. [| wf[A](r);  s<=r |] ==> wf[A](s)";
```
```    44 by (fast_tac ZF_cs 1);
```
```    45 val wf_on_subset_r = result();
```
```    46
```
```    47 (** Introduction rules for wf_on **)
```
```    48
```
```    49 (*If every non-empty subset of A has an r-minimal element then wf[A](r).*)
```
```    50 val [prem] = goalw WF.thy [wf_on_def, wf_def]
```
```    51     "[| !!Z u. [| Z<=A;  u:Z;  ALL x:Z. EX y:Z. <y,x>:r |] ==> False |] \
```
```    52 \    ==>  wf[A](r)";
```
```    53 by (rtac (equals0I RS disjCI RS allI) 1);
```
```    54 by (res_inst_tac [ ("Z", "Z") ] prem 1);
```
```    55 by (ALLGOALS (fast_tac ZF_cs));
```
```    56 val wf_onI = result();
```
```    57
```
```    58 (*If r allows well-founded induction over A then wf[A](r)
```
```    59   Premise is equivalent to
```
```    60   !!B. ALL x:A. (ALL y. <y,x>: r --> y:B) --> x:B ==> A<=B  *)
```
```    61 val [prem] = goal WF.thy
```
```    62     "[| !!y B. [| ALL x:A. (ALL y:A. <y,x>:r --> y:B) --> x:B;   y:A  \
```
```    63 \              |] ==> y:B |] \
```
```    64 \    ==>  wf[A](r)";
```
```    65 br wf_onI 1;
```
```    66 by (res_inst_tac [ ("c", "u") ] (prem RS DiffE) 1);
```
```    67 by (contr_tac 3);
```
```    68 by (fast_tac ZF_cs 2);
```
```    69 by (fast_tac ZF_cs 1);
```
```    70 val wf_onI2 = result();
```
```    71
```
```    72
```
```    73 (** Well-founded Induction **)
```
```    74
```
```    75 (*Consider the least z in domain(r) Un {a} such that P(z) does not hold...*)
```
```    76 val major::prems = goalw WF.thy [wf_def]
```
```    77     "[| wf(r);          \
```
```    78 \       !!x.[| ALL y. <y,x>: r --> P(y) |] ==> P(x) \
```
```    79 \    |]  ==>  P(a)";
```
```    80 by (res_inst_tac [ ("x", "{z:domain(r) Un {a}. ~P(z)}") ]  (major RS allE) 1);
```
```    81 by (etac disjE 1);
```
```    82 by (rtac classical 1);
```
```    83 by (etac equals0D 1);
```
```    84 by (etac (singletonI RS UnI2 RS CollectI) 1);
```
```    85 by (etac bexE 1);
```
```    86 by (etac CollectE 1);
```
```    87 by (etac swap 1);
```
```    88 by (resolve_tac prems 1);
```
```    89 by (fast_tac ZF_cs 1);
```
```    90 val wf_induct = result();
```
```    91
```
```    92 (*Perform induction on i, then prove the wf(r) subgoal using prems. *)
```
```    93 fun wf_ind_tac a prems i =
```
```    94     EVERY [res_inst_tac [("a",a)] wf_induct i,
```
```    95 	   rename_last_tac a ["1"] (i+1),
```
```    96 	   ares_tac prems i];
```
```    97
```
```    98 (*The form of this rule is designed to match wfI2*)
```
```    99 val wfr::amem::prems = goal WF.thy
```
```   100     "[| wf(r);  a:A;  field(r)<=A;  \
```
```   101 \       !!x.[| x: A;  ALL y. <y,x>: r --> P(y) |] ==> P(x) \
```
```   102 \    |]  ==>  P(a)";
```
```   103 by (rtac (amem RS rev_mp) 1);
```
```   104 by (wf_ind_tac "a" [wfr] 1);
```
```   105 by (rtac impI 1);
```
```   106 by (eresolve_tac prems 1);
```
```   107 by (fast_tac (ZF_cs addIs (prems RL [subsetD])) 1);
```
```   108 val wf_induct2 = result();
```
```   109
```
```   110 goal ZF.thy "!!r A. field(r Int A*A) <= A";
```
```   111 by (fast_tac ZF_cs 1);
```
```   112 val field_Int_square = result();
```
```   113
```
```   114 val wfr::amem::prems = goalw WF.thy [wf_on_def]
```
```   115     "[| wf[A](r);  a:A;  					\
```
```   116 \       !!x.[| x: A;  ALL y:A. <y,x>: r --> P(y) |] ==> P(x) 	\
```
```   117 \    |]  ==>  P(a)";
```
```   118 by (rtac ([wfr, amem, field_Int_square] MRS wf_induct2) 1);
```
```   119 by (REPEAT (ares_tac prems 1));
```
```   120 by (fast_tac ZF_cs 1);
```
```   121 val wf_on_induct = result();
```
```   122
```
```   123 fun wf_on_ind_tac a prems i =
```
```   124     EVERY [res_inst_tac [("a",a)] wf_on_induct i,
```
```   125 	   rename_last_tac a ["1"] (i+2),
```
```   126 	   REPEAT (ares_tac prems i)];
```
```   127
```
```   128 (*If r allows well-founded induction then wf(r)*)
```
```   129 val [subs,indhyp] = goal WF.thy
```
```   130     "[| field(r)<=A;  \
```
```   131 \       !!y B. [| ALL x:A. (ALL y:A. <y,x>:r --> y:B) --> x:B;   y:A  \
```
```   132 \              |] ==> y:B |] \
```
```   133 \    ==>  wf(r)";
```
```   134 br ([wf_onI2, subs] MRS (wf_on_subset_A RS wf_on_field_imp_wf)) 1;
```
```   135 by (REPEAT (ares_tac [indhyp] 1));
```
```   136 val wfI2 = result();
```
```   137
```
```   138
```
```   139 (*** Properties of well-founded relations ***)
```
```   140
```
```   141 goal WF.thy "!!r. wf(r) ==> <a,a> ~: r";
```
```   142 by (wf_ind_tac "a" [] 1);
```
```   143 by (fast_tac ZF_cs 1);
```
```   144 val wf_not_refl = result();
```
```   145
```
```   146 goal WF.thy "!!r. [| wf(r);  <a,x>:r;  <x,a>:r |] ==> P";
```
```   147 by (subgoal_tac "ALL x. <a,x>:r --> <x,a>:r --> P" 1);
```
```   148 by (wf_ind_tac "a" [] 2);
```
```   149 by (fast_tac ZF_cs 2);
```
```   150 by (fast_tac FOL_cs 1);
```
```   151 val wf_anti_sym = result();
```
```   152
```
```   153 goal WF.thy "!!r. [| wf[A](r); a: A |] ==> <a,a> ~: r";
```
```   154 by (wf_on_ind_tac "a" [] 1);
```
```   155 by (fast_tac ZF_cs 1);
```
```   156 val wf_on_not_refl = result();
```
```   157
```
```   158 goal WF.thy "!!r. [| wf[A](r);  <a,b>:r;  <b,a>:r;  a:A;  b:A |] ==> P";
```
```   159 by (subgoal_tac "ALL y:A. <a,y>:r --> <y,a>:r --> P" 1);
```
```   160 by (wf_on_ind_tac "a" [] 2);
```
```   161 by (fast_tac ZF_cs 2);
```
```   162 by (fast_tac ZF_cs 1);
```
```   163 val wf_on_anti_sym = result();
```
```   164
```
```   165 (*Needed to prove well_ordI.  Could also reason that wf[A](r) means
```
```   166   wf(r Int A*A);  thus wf( (r Int A*A)^+ ) and use wf_not_refl *)
```
```   167 goal WF.thy
```
```   168     "!!r. [| wf[A](r); <a,b>:r; <b,c>:r; <c,a>:r; a:A; b:A; c:A |] ==> P";
```
```   169 by (subgoal_tac
```
```   170     "ALL y:A. ALL z:A. <a,y>:r --> <y,z>:r --> <z,a>:r --> P" 1);
```
```   171 by (wf_on_ind_tac "a" [] 2);
```
```   172 by (fast_tac ZF_cs 2);
```
```   173 by (fast_tac ZF_cs 1);
```
```   174 val wf_on_chain3 = result();
```
```   175
```
```   176
```
```   177 (*retains the universal formula for later use!*)
```
```   178 val bchain_tac = EVERY' [rtac (bspec RS mp), assume_tac, assume_tac ];
```
```   179
```
```   180 (*transitive closure of a WF relation is WF provided A is downwards closed*)
```
```   181 val [wfr,subs] = goal WF.thy
```
```   182     "[| wf[A](r);  r-``A <= A |] ==> wf[A](r^+)";
```
```   183 br wf_onI2 1;
```
```   184 by (bchain_tac 1);
```
```   185 by (eres_inst_tac [("a","y")] (wfr RS wf_on_induct) 1);
```
```   186 by (rtac (impI RS ballI) 1);
```
```   187 by (etac tranclE 1);
```
```   188 by (etac (bspec RS mp) 1 THEN assume_tac 1);
```
```   189 by (fast_tac ZF_cs 1);
```
```   190 by (cut_facts_tac [subs] 1);
```
```   191 (*astar_tac is slightly faster*)
```
```   192 by (best_tac ZF_cs 1);
```
```   193 val wf_on_trancl = result();
```
```   194
```
```   195 goal WF.thy "!!r. wf(r) ==> wf(r^+)";
```
```   196 by (asm_full_simp_tac (ZF_ss addsimps [wf_iff_wf_on_field]) 1);
```
```   197 br (trancl_type RS field_rel_subset RSN (2, wf_on_subset_A)) 1;
```
```   198 be wf_on_trancl 1;
```
```   199 by (fast_tac ZF_cs 1);
```
```   200 val wf_trancl = result();
```
```   201
```
```   202
```
```   203
```
```   204 (** r-``{a} is the set of everything under a in r **)
```
```   205
```
```   206 val underI = standard (vimage_singleton_iff RS iffD2);
```
```   207 val underD = standard (vimage_singleton_iff RS iffD1);
```
```   208
```
```   209 (** is_recfun **)
```
```   210
```
```   211 val [major] = goalw WF.thy [is_recfun_def]
```
```   212     "is_recfun(r,a,H,f) ==> f: r-``{a} -> range(f)";
```
```   213 by (rtac (major RS ssubst) 1);
```
```   214 by (rtac (lamI RS rangeI RS lam_type) 1);
```
```   215 by (assume_tac 1);
```
```   216 val is_recfun_type = result();
```
```   217
```
```   218 val [isrec,rel] = goalw WF.thy [is_recfun_def]
```
```   219     "[| is_recfun(r,a,H,f); <x,a>:r |] ==> f`x = H(x, restrict(f,r-``{x}))";
```
```   220 by (res_inst_tac [("P", "%x.?t(x) = ?u::i")] (isrec RS ssubst) 1);
```
```   221 by (rtac (rel RS underI RS beta) 1);
```
```   222 val apply_recfun = result();
```
```   223
```
```   224 (*eresolve_tac transD solves <a,b>:r using transitivity AT MOST ONCE
```
```   225   spec RS mp  instantiates induction hypotheses*)
```
```   226 fun indhyp_tac hyps =
```
```   227     resolve_tac (TrueI::refl::hyps) ORELSE'
```
```   228     (cut_facts_tac hyps THEN'
```
```   229        DEPTH_SOLVE_1 o (ares_tac [TrueI, ballI] ORELSE'
```
```   230 		        eresolve_tac [underD, transD, spec RS mp]));
```
```   231
```
```   232 (*** NOTE! some simplifications need a different solver!! ***)
```
```   233 val wf_super_ss = ZF_ss setsolver indhyp_tac;
```
```   234
```
```   235 val prems = goalw WF.thy [is_recfun_def]
```
```   236     "[| wf(r);  trans(r);  is_recfun(r,a,H,f);  is_recfun(r,b,H,g) |] ==> \
```
```   237 \    <x,a>:r --> <x,b>:r --> f`x=g`x";
```
```   238 by (cut_facts_tac prems 1);
```
```   239 by (wf_ind_tac "x" prems 1);
```
```   240 by (REPEAT (rtac impI 1 ORELSE etac ssubst 1));
```
```   241 by (rewtac restrict_def);
```
```   242 by (asm_simp_tac (wf_super_ss addsimps [vimage_singleton_iff]) 1);
```
```   243 val is_recfun_equal_lemma = result();
```
```   244 val is_recfun_equal = standard (is_recfun_equal_lemma RS mp RS mp);
```
```   245
```
```   246 val prems as [wfr,transr,recf,recg,_] = goal WF.thy
```
```   247     "[| wf(r);  trans(r);       \
```
```   248 \       is_recfun(r,a,H,f);  is_recfun(r,b,H,g);  <b,a>:r |] ==> \
```
```   249 \    restrict(f, r-``{b}) = g";
```
```   250 by (cut_facts_tac prems 1);
```
```   251 by (rtac (consI1 RS restrict_type RS fun_extension) 1);
```
```   252 by (etac is_recfun_type 1);
```
```   253 by (ALLGOALS
```
```   254     (asm_simp_tac (wf_super_ss addsimps
```
```   255 		   [ [wfr,transr,recf,recg] MRS is_recfun_equal ])));
```
```   256 val is_recfun_cut = result();
```
```   257
```
```   258 (*** Main Existence Lemma ***)
```
```   259
```
```   260 val prems = goal WF.thy
```
```   261     "[| wf(r); trans(r); is_recfun(r,a,H,f); is_recfun(r,a,H,g) |]  ==>  f=g";
```
```   262 by (cut_facts_tac prems 1);
```
```   263 by (rtac fun_extension 1);
```
```   264 by (REPEAT (ares_tac [is_recfun_equal] 1
```
```   265      ORELSE eresolve_tac [is_recfun_type,underD] 1));
```
```   266 val is_recfun_functional = result();
```
```   267
```
```   268 (*If some f satisfies is_recfun(r,a,H,-) then so does the_recfun(r,a,H) *)
```
```   269 val prems = goalw WF.thy [the_recfun_def]
```
```   270     "[| is_recfun(r,a,H,f);  wf(r);  trans(r) |]  \
```
```   271 \    ==> is_recfun(r, a, H, the_recfun(r,a,H))";
```
```   272 by (rtac (ex1I RS theI) 1);
```
```   273 by (REPEAT (ares_tac (prems@[is_recfun_functional]) 1));
```
```   274 val is_the_recfun = result();
```
```   275
```
```   276 val prems = goal WF.thy
```
```   277     "[| wf(r);  trans(r) |] ==> is_recfun(r, a, H, the_recfun(r,a,H))";
```
```   278 by (cut_facts_tac prems 1);
```
```   279 by (wf_ind_tac "a" prems 1);
```
```   280 by (res_inst_tac [("f", "lam y: r-``{a1}. wftrec(r,y,H)")] is_the_recfun 1);
```
```   281 by (REPEAT (assume_tac 2));
```
```   282 by (rewrite_goals_tac [is_recfun_def, wftrec_def]);
```
```   283 (*Applying the substitution: must keep the quantified assumption!!*)
```
```   284 by (REPEAT (dtac underD 1 ORELSE resolve_tac [refl, lam_cong] 1));
```
```   285 by (fold_tac [is_recfun_def]);
```
```   286 by (rtac (consI1 RS restrict_type RSN (2,fun_extension) RS subst_context) 1);
```
```   287 by (rtac is_recfun_type 1);
```
```   288 by (ALLGOALS
```
```   289     (asm_simp_tac
```
```   290      (wf_super_ss addsimps [underI RS beta, apply_recfun, is_recfun_cut])));
```
```   291 val unfold_the_recfun = result();
```
```   292
```
```   293
```
```   294 (*** Unfolding wftrec ***)
```
```   295
```
```   296 val prems = goal WF.thy
```
```   297     "[| wf(r);  trans(r);  <b,a>:r |] ==> \
```
```   298 \    restrict(the_recfun(r,a,H), r-``{b}) = the_recfun(r,b,H)";
```
```   299 by (REPEAT (ares_tac (prems @ [is_recfun_cut, unfold_the_recfun]) 1));
```
```   300 val the_recfun_cut = result();
```
```   301
```
```   302 (*NOT SUITABLE FOR REWRITING since it is recursive!*)
```
```   303 goalw WF.thy [wftrec_def]
```
```   304     "!!r. [| wf(r);  trans(r) |] ==> \
```
```   305 \         wftrec(r,a,H) = H(a, lam x: r-``{a}. wftrec(r,x,H))";
```
```   306 by (rtac (rewrite_rule [is_recfun_def] unfold_the_recfun RS ssubst) 1);
```
```   307 by (ALLGOALS (asm_simp_tac
```
```   308 	(ZF_ss addsimps [vimage_singleton_iff RS iff_sym, the_recfun_cut])));
```
```   309 val wftrec = result();
```
```   310
```
```   311 (** Removal of the premise trans(r) **)
```
```   312
```
```   313 (*NOT SUITABLE FOR REWRITING since it is recursive!*)
```
```   314 val [wfr] = goalw WF.thy [wfrec_def]
```
```   315     "wf(r) ==> wfrec(r,a,H) = H(a, lam x:r-``{a}. wfrec(r,x,H))";
```
```   316 by (rtac (wfr RS wf_trancl RS wftrec RS ssubst) 1);
```
```   317 by (rtac trans_trancl 1);
```
```   318 by (rtac (vimage_pair_mono RS restrict_lam_eq RS subst_context) 1);
```
```   319 by (etac r_into_trancl 1);
```
```   320 by (rtac subset_refl 1);
```
```   321 val wfrec = result();
```
```   322
```
```   323 (*This form avoids giant explosions in proofs.  NOTE USE OF == *)
```
```   324 val rew::prems = goal WF.thy
```
```   325     "[| !!x. h(x)==wfrec(r,x,H);  wf(r) |] ==> \
```
```   326 \    h(a) = H(a, lam x: r-``{a}. h(x))";
```
```   327 by (rewtac rew);
```
```   328 by (REPEAT (resolve_tac (prems@[wfrec]) 1));
```
```   329 val def_wfrec = result();
```
```   330
```
```   331 val prems = goal WF.thy
```
```   332     "[| wf(r);  a:A;  field(r)<=A;  \
```
```   333 \       !!x u. [| x: A;  u: Pi(r-``{x}, B) |] ==> H(x,u) : B(x)   \
```
```   334 \    |] ==> wfrec(r,a,H) : B(a)";
```
```   335 by (res_inst_tac [("a","a")] wf_induct2 1);
```
```   336 by (rtac (wfrec RS ssubst) 4);
```
```   337 by (REPEAT (ares_tac (prems@[lam_type]) 1
```
```   338      ORELSE eresolve_tac [spec RS mp, underD] 1));
```
```   339 val wfrec_type = result();
```
```   340
```
```   341
```
```   342 goalw WF.thy [wf_on_def, wfrec_on_def]
```
```   343  "!!A r. [| wf[A](r);  a: A |] ==> \
```
```   344 \        wfrec[A](r,a,H) = H(a, lam x: (r-``{a}) Int A. wfrec[A](r,x,H))";
```
```   345 be (wfrec RS trans) 1;
```
```   346 by (asm_simp_tac (ZF_ss addsimps [vimage_Int_square, cons_subset_iff]) 1);
```
```   347 val wfrec_on = result();
```
```   348
```