src/ZF/Fixedpt.ML
author clasohm
Thu Sep 16 12:20:38 1993 +0200 (1993-09-16)
changeset 0 a5a9c433f639
child 14 1c0926788772
permissions -rw-r--r--
Initial revision
     1 (*  Title: 	ZF/fixedpt.ML
     2     ID:         $Id$
     3     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1992  University of Cambridge
     5 
     6 For fixedpt.thy.  Least and greatest fixed points; the Knaster-Tarski Theorem
     7 
     8 Proved in the lattice of subsets of D, namely Pow(D), with Inter as glb
     9 *)
    10 
    11 open Fixedpt;
    12 
    13 (*** Monotone operators ***)
    14 
    15 val prems = goalw Fixedpt.thy [bnd_mono_def]
    16     "[| h(D)<=D;  \
    17 \       !!W X. [| W<=D;  X<=D;  W<=X |] ==> h(W) <= h(X)  \
    18 \    |] ==> bnd_mono(D,h)";  
    19 by (REPEAT (ares_tac (prems@[conjI,allI,impI]) 1
    20      ORELSE etac subset_trans 1));
    21 val bnd_monoI = result();
    22 
    23 val [major] = goalw Fixedpt.thy [bnd_mono_def] "bnd_mono(D,h) ==> h(D) <= D";
    24 by (rtac (major RS conjunct1) 1);
    25 val bnd_monoD1 = result();
    26 
    27 val major::prems = goalw Fixedpt.thy [bnd_mono_def]
    28     "[| bnd_mono(D,h);  W<=X;  X<=D |] ==> h(W) <= h(X)";
    29 by (rtac (major RS conjunct2 RS spec RS spec RS mp RS mp) 1);
    30 by (REPEAT (resolve_tac prems 1));
    31 val bnd_monoD2 = result();
    32 
    33 val [major,minor] = goal Fixedpt.thy
    34     "[| bnd_mono(D,h);  X<=D |] ==> h(X) <= D";
    35 by (rtac (major RS bnd_monoD2 RS subset_trans) 1);
    36 by (rtac (major RS bnd_monoD1) 3);
    37 by (rtac minor 1);
    38 by (rtac subset_refl 1);
    39 val bnd_mono_subset = result();
    40 
    41 goal Fixedpt.thy "!!A B. [| bnd_mono(D,h);  A <= D;  B <= D |] ==> \
    42 \                         h(A) Un h(B) <= h(A Un B)";
    43 by (REPEAT (ares_tac [Un_upper1, Un_upper2, Un_least] 1
    44      ORELSE etac bnd_monoD2 1));
    45 val bnd_mono_Un = result();
    46 
    47 (*Useful??*)
    48 goal Fixedpt.thy "!!A B. [| bnd_mono(D,h);  A <= D;  B <= D |] ==> \
    49 \                        h(A Int B) <= h(A) Int h(B)";
    50 by (REPEAT (ares_tac [Int_lower1, Int_lower2, Int_greatest] 1
    51      ORELSE etac bnd_monoD2 1));
    52 val bnd_mono_Int = result();
    53 
    54 (**** Proof of Knaster-Tarski Theorem for the lfp ****)
    55 
    56 (*lfp is contained in each pre-fixedpoint*)
    57 val prems = goalw Fixedpt.thy [lfp_def]
    58     "[| h(A) <= A;  A<=D |] ==> lfp(D,h) <= A";
    59 by (rtac (PowI RS CollectI RS Inter_lower) 1);
    60 by (REPEAT (resolve_tac prems 1));
    61 val lfp_lowerbound = result();
    62 
    63 (*Unfolding the defn of Inter dispenses with the premise bnd_mono(D,h)!*)
    64 goalw Fixedpt.thy [lfp_def,Inter_def] "lfp(D,h) <= D";
    65 by (fast_tac ZF_cs 1);
    66 val lfp_subset = result();
    67 
    68 (*Used in datatype package*)
    69 val [rew] = goal Fixedpt.thy "A==lfp(D,h) ==> A <= D";
    70 by (rewtac rew);
    71 by (rtac lfp_subset 1);
    72 val def_lfp_subset = result();
    73 
    74 val subset0_cs = FOL_cs
    75   addSIs [ballI, InterI, CollectI, PowI, empty_subsetI]
    76   addIs [bexI, UnionI, ReplaceI, RepFunI]
    77   addSEs [bexE, make_elim PowD, UnionE, ReplaceE, RepFunE,
    78 	  CollectE, emptyE]
    79   addEs [rev_ballE, InterD, make_elim InterD, subsetD];
    80 
    81 val subset_cs = subset0_cs 
    82   addSIs [subset_refl,cons_subsetI,subset_consI,Union_least,UN_least,Un_least,
    83 	  Inter_greatest,Int_greatest,RepFun_subset]
    84   addSIs [Un_upper1,Un_upper2,Int_lower1,Int_lower2]
    85   addIs  [Union_upper,Inter_lower]
    86   addSEs [cons_subsetE];
    87 
    88 val prems = goalw Fixedpt.thy [lfp_def]
    89     "[| h(D) <= D;  !!X. [| h(X) <= X;  X<=D |] ==> A<=X |] ==> \
    90 \    A <= lfp(D,h)";
    91 br (Pow_top RS CollectI RS Inter_greatest) 1;
    92 by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [CollectE,PowD] 1));
    93 val lfp_greatest = result();
    94 
    95 val hmono::prems = goal Fixedpt.thy
    96     "[| bnd_mono(D,h);  h(A)<=A;  A<=D |] ==> h(lfp(D,h)) <= A";
    97 by (rtac (hmono RS bnd_monoD2 RS subset_trans) 1);
    98 by (rtac lfp_lowerbound 1);
    99 by (REPEAT (resolve_tac prems 1));
   100 val lfp_lemma1 = result();
   101 
   102 val [hmono] = goal Fixedpt.thy
   103     "bnd_mono(D,h) ==> h(lfp(D,h)) <= lfp(D,h)";
   104 by (rtac (bnd_monoD1 RS lfp_greatest) 1);
   105 by (rtac lfp_lemma1 2);
   106 by (REPEAT (ares_tac [hmono] 1));
   107 val lfp_lemma2 = result();
   108 
   109 val [hmono] = goal Fixedpt.thy
   110     "bnd_mono(D,h) ==> lfp(D,h) <= h(lfp(D,h))";
   111 by (rtac lfp_lowerbound 1);
   112 by (rtac (hmono RS bnd_monoD2) 1);
   113 by (rtac (hmono RS lfp_lemma2) 1);
   114 by (rtac (hmono RS bnd_mono_subset) 2);
   115 by (REPEAT (rtac lfp_subset 1));
   116 val lfp_lemma3 = result();
   117 
   118 val prems = goal Fixedpt.thy
   119     "bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h))";
   120 by (REPEAT (resolve_tac (prems@[equalityI,lfp_lemma2,lfp_lemma3]) 1));
   121 val lfp_Tarski = result();
   122 
   123 (*Definition form, to control unfolding*)
   124 val [rew,mono] = goal Fixedpt.thy
   125     "[| A==lfp(D,h);  bnd_mono(D,h) |] ==> A = h(A)";
   126 by (rewtac rew);
   127 by (rtac (mono RS lfp_Tarski) 1);
   128 val def_lfp_Tarski = result();
   129 
   130 (*** General induction rule for least fixedpoints ***)
   131 
   132 val [hmono,indstep] = goal Fixedpt.thy
   133     "[| bnd_mono(D,h);  !!x. x : h(Collect(lfp(D,h),P)) ==> P(x) \
   134 \    |] ==> h(Collect(lfp(D,h),P)) <= Collect(lfp(D,h),P)";
   135 by (rtac subsetI 1);
   136 by (rtac CollectI 1);
   137 by (etac indstep 2);
   138 by (rtac (hmono RS lfp_lemma2 RS subsetD) 1);
   139 by (rtac (hmono RS bnd_monoD2 RS subsetD) 1);
   140 by (REPEAT (ares_tac [Collect_subset, lfp_subset] 1));
   141 val Collect_is_pre_fixedpt = result();
   142 
   143 (*This rule yields an induction hypothesis in which the components of a
   144   data structure may be assumed to be elements of lfp(D,h)*)
   145 val prems = goal Fixedpt.thy
   146     "[| bnd_mono(D,h);  a : lfp(D,h);   		\
   147 \       !!x. x : h(Collect(lfp(D,h),P)) ==> P(x) 	\
   148 \    |] ==> P(a)";
   149 by (rtac (Collect_is_pre_fixedpt RS lfp_lowerbound RS subsetD RS CollectD2) 1);
   150 by (rtac (lfp_subset RS (Collect_subset RS subset_trans)) 3);
   151 by (REPEAT (ares_tac prems 1));
   152 val induct = result();
   153 
   154 (*Definition form, to control unfolding*)
   155 val rew::prems = goal Fixedpt.thy
   156     "[| A == lfp(D,h);  bnd_mono(D,h);  a:A;   \
   157 \       !!x. x : h(Collect(A,P)) ==> P(x) \
   158 \    |] ==> P(a)";
   159 by (rtac induct 1);
   160 by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems) 1));
   161 val def_induct = result();
   162 
   163 (*This version is useful when "A" is not a subset of D;
   164   second premise could simply be h(D Int A) <= D or !!X. X<=D ==> h(X)<=D *)
   165 val [hsub,hmono] = goal Fixedpt.thy
   166     "[| h(D Int A) <= A;  bnd_mono(D,h) |] ==> lfp(D,h) <= A";
   167 by (rtac (lfp_lowerbound RS subset_trans) 1);
   168 by (rtac (hmono RS bnd_mono_subset RS Int_greatest) 1);
   169 by (REPEAT (resolve_tac [hsub,Int_lower1,Int_lower2] 1));
   170 val lfp_Int_lowerbound = result();
   171 
   172 (*Monotonicity of lfp, where h precedes i under a domain-like partial order
   173   monotonicity of h is not strictly necessary; h must be bounded by D*)
   174 val [hmono,imono,subhi] = goal Fixedpt.thy
   175     "[| bnd_mono(D,h);  bnd_mono(E,i); 		\
   176 \       !!X. X<=D ==> h(X) <= i(X)  |] ==> lfp(D,h) <= lfp(E,i)";
   177 br (bnd_monoD1 RS lfp_greatest) 1;
   178 br imono 1;
   179 by (rtac (hmono RSN (2, lfp_Int_lowerbound)) 1);
   180 by (rtac (Int_lower1 RS subhi RS subset_trans) 1);
   181 by (rtac (imono RS bnd_monoD2 RS subset_trans) 1);
   182 by (REPEAT (ares_tac [Int_lower2] 1));
   183 val lfp_mono = result();
   184 
   185 (*This (unused) version illustrates that monotonicity is not really needed,
   186   but both lfp's must be over the SAME set D;  Inter is anti-monotonic!*)
   187 val [isubD,subhi] = goal Fixedpt.thy
   188     "[| i(D) <= D;  !!X. X<=D ==> h(X) <= i(X)  |] ==> lfp(D,h) <= lfp(D,i)";
   189 br lfp_greatest 1;
   190 br isubD 1;
   191 by (rtac lfp_lowerbound 1);
   192 be (subhi RS subset_trans) 1;
   193 by (REPEAT (assume_tac 1));
   194 val lfp_mono2 = result();
   195 
   196 
   197 (**** Proof of Knaster-Tarski Theorem for the gfp ****)
   198 
   199 (*gfp contains each post-fixedpoint that is contained in D*)
   200 val prems = goalw Fixedpt.thy [gfp_def]
   201     "[| A <= h(A);  A<=D |] ==> A <= gfp(D,h)";
   202 by (rtac (PowI RS CollectI RS Union_upper) 1);
   203 by (REPEAT (resolve_tac prems 1));
   204 val gfp_upperbound = result();
   205 
   206 goalw Fixedpt.thy [gfp_def] "gfp(D,h) <= D";
   207 by (fast_tac ZF_cs 1);
   208 val gfp_subset = result();
   209 
   210 (*Used in datatype package*)
   211 val [rew] = goal Fixedpt.thy "A==gfp(D,h) ==> A <= D";
   212 by (rewtac rew);
   213 by (rtac gfp_subset 1);
   214 val def_gfp_subset = result();
   215 
   216 val hmono::prems = goalw Fixedpt.thy [gfp_def]
   217     "[| bnd_mono(D,h);  !!X. [| X <= h(X);  X<=D |] ==> X<=A |] ==> \
   218 \    gfp(D,h) <= A";
   219 by (fast_tac (subset_cs addIs ((hmono RS bnd_monoD1)::prems)) 1);
   220 val gfp_least = result();
   221 
   222 val hmono::prems = goal Fixedpt.thy
   223     "[| bnd_mono(D,h);  A<=h(A);  A<=D |] ==> A <= h(gfp(D,h))";
   224 by (rtac (hmono RS bnd_monoD2 RSN (2,subset_trans)) 1);
   225 by (rtac gfp_subset 3);
   226 by (rtac gfp_upperbound 2);
   227 by (REPEAT (resolve_tac prems 1));
   228 val gfp_lemma1 = result();
   229 
   230 val [hmono] = goal Fixedpt.thy
   231     "bnd_mono(D,h) ==> gfp(D,h) <= h(gfp(D,h))";
   232 by (rtac gfp_least 1);
   233 by (rtac gfp_lemma1 2);
   234 by (REPEAT (ares_tac [hmono] 1));
   235 val gfp_lemma2 = result();
   236 
   237 val [hmono] = goal Fixedpt.thy
   238     "bnd_mono(D,h) ==> h(gfp(D,h)) <= gfp(D,h)";
   239 by (rtac gfp_upperbound 1);
   240 by (rtac (hmono RS bnd_monoD2) 1);
   241 by (rtac (hmono RS gfp_lemma2) 1);
   242 by (REPEAT (rtac ([hmono, gfp_subset] MRS bnd_mono_subset) 1));
   243 val gfp_lemma3 = result();
   244 
   245 val prems = goal Fixedpt.thy
   246     "bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h))";
   247 by (REPEAT (resolve_tac (prems@[equalityI,gfp_lemma2,gfp_lemma3]) 1));
   248 val gfp_Tarski = result();
   249 
   250 (*Definition form, to control unfolding*)
   251 val [rew,mono] = goal Fixedpt.thy
   252     "[| A==gfp(D,h);  bnd_mono(D,h) |] ==> A = h(A)";
   253 by (rewtac rew);
   254 by (rtac (mono RS gfp_Tarski) 1);
   255 val def_gfp_Tarski = result();
   256 
   257 
   258 (*** Coinduction rules for greatest fixed points ***)
   259 
   260 (*weak version*)
   261 goal Fixedpt.thy "!!X h. [| a: X;  X <= h(X);  X <= D |] ==> a : gfp(D,h)";
   262 by (REPEAT (ares_tac [gfp_upperbound RS subsetD] 1));
   263 val weak_coinduct = result();
   264 
   265 val [subs_h,subs_D,mono] = goal Fixedpt.thy
   266     "[| X <= h(X Un gfp(D,h));  X <= D;  bnd_mono(D,h) |] ==>  \
   267 \    X Un gfp(D,h) <= h(X Un gfp(D,h))";
   268 by (rtac (subs_h RS Un_least) 1);
   269 by (rtac (mono RS gfp_lemma2 RS subset_trans) 1);
   270 by (rtac (Un_upper2 RS subset_trans) 1);
   271 by (rtac ([mono, subs_D, gfp_subset] MRS bnd_mono_Un) 1);
   272 val coinduct_lemma = result();
   273 
   274 (*strong version*)
   275 goal Fixedpt.thy
   276     "!!X D. [| bnd_mono(D,h);  a: X;  X <= h(X Un gfp(D,h));  X <= D |] ==> \
   277 \           a : gfp(D,h)";
   278 by (rtac (coinduct_lemma RSN (2, weak_coinduct)) 1);
   279 by (REPEAT (ares_tac [gfp_subset, UnI1, Un_least] 1));
   280 val coinduct = result();
   281 
   282 (*Definition form, to control unfolding*)
   283 val rew::prems = goal Fixedpt.thy
   284     "[| A == gfp(D,h);  bnd_mono(D,h);  a: X;  X <= h(X Un A);  X <= D |] ==> \
   285 \    a : A";
   286 by (rewtac rew);
   287 by (rtac coinduct 1);
   288 by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems) 1));
   289 val def_coinduct = result();
   290 
   291 (*Lemma used immediately below!*)
   292 val [subsA,XimpP] = goal ZF.thy
   293     "[| X <= A;  !!z. z:X ==> P(z) |] ==> X <= Collect(A,P)";
   294 by (rtac (subsA RS subsetD RS CollectI RS subsetI) 1);
   295 by (assume_tac 1);
   296 by (etac XimpP 1);
   297 val subset_Collect = result();
   298 
   299 (*The version used in the induction/coinduction package*)
   300 val prems = goal Fixedpt.thy
   301     "[| A == gfp(D, %w. Collect(D,P(w)));  bnd_mono(D, %w. Collect(D,P(w)));  \
   302 \       a: X;  X <= D;  !!z. z: X ==> P(X Un A, z) |] ==> \
   303 \    a : A";
   304 by (rtac def_coinduct 1);
   305 by (REPEAT (ares_tac (subset_Collect::prems) 1));
   306 val def_Collect_coinduct = result();
   307 
   308 (*Monotonicity of gfp!*)
   309 val [hmono,subde,subhi] = goal Fixedpt.thy
   310     "[| bnd_mono(D,h);  D <= E; 		\
   311 \       !!X. X<=D ==> h(X) <= i(X)  |] ==> gfp(D,h) <= gfp(E,i)";
   312 by (rtac gfp_upperbound 1);
   313 by (rtac (hmono RS gfp_lemma2 RS subset_trans) 1);
   314 by (rtac (gfp_subset RS subhi) 1);
   315 by (rtac ([gfp_subset, subde] MRS subset_trans) 1);
   316 val gfp_mono = result();
   317