src/ZF/ex/PropLog.ML
author lcp
Tue Aug 16 18:58:42 1994 +0200 (1994-08-16)
changeset 532 851df239ac8b
parent 515 abcc438e7c27
child 760 f0200e91b272
permissions -rw-r--r--
ZF/Makefile,ROOT.ML, ZF/ex/Integ.thy: updated for EquivClass
     1 (*  Title: 	ZF/ex/prop-log.ML
     2     ID:         $Id$
     3     Author: 	Tobias Nipkow & Lawrence C Paulson
     4     Copyright   1992  University of Cambridge
     5 
     6 For ex/prop-log.thy.  Inductive definition of propositional logic.
     7 Soundness and completeness w.r.t. truth-tables.
     8 
     9 Prove: If H|=p then G|=p where G:Fin(H)
    10 *)
    11 
    12 open PropLog;
    13 
    14 (*** prop_rec -- by Vset recursion ***)
    15 
    16 (** conversion rules **)
    17 
    18 goal PropLog.thy "prop_rec(Fls,b,c,d) = b";
    19 by (rtac (prop_rec_def RS def_Vrec RS trans) 1);
    20 by (rewrite_goals_tac prop.con_defs);
    21 by (simp_tac rank_ss 1);
    22 val prop_rec_Fls = result();
    23 
    24 goal PropLog.thy "prop_rec(#v,b,c,d) = c(v)";
    25 by (rtac (prop_rec_def RS def_Vrec RS trans) 1);
    26 by (rewrite_goals_tac prop.con_defs);
    27 by (simp_tac rank_ss 1);
    28 val prop_rec_Var = result();
    29 
    30 goal PropLog.thy "prop_rec(p=>q,b,c,d) = \
    31 \      d(p, q, prop_rec(p,b,c,d), prop_rec(q,b,c,d))";
    32 by (rtac (prop_rec_def RS def_Vrec RS trans) 1);
    33 by (rewrite_goals_tac prop.con_defs);
    34 by (simp_tac rank_ss 1);
    35 val prop_rec_Imp = result();
    36 
    37 val prop_rec_ss = 
    38     arith_ss addsimps [prop_rec_Fls, prop_rec_Var, prop_rec_Imp];
    39 
    40 (*** Semantics of propositional logic ***)
    41 
    42 (** The function is_true **)
    43 
    44 goalw PropLog.thy [is_true_def] "is_true(Fls,t) <-> False";
    45 by (simp_tac (prop_rec_ss addsimps [one_not_0 RS not_sym]) 1);
    46 val is_true_Fls = result();
    47 
    48 goalw PropLog.thy [is_true_def] "is_true(#v,t) <-> v:t";
    49 by (simp_tac (prop_rec_ss addsimps [one_not_0 RS not_sym] 
    50 	      setloop (split_tac [expand_if])) 1);
    51 val is_true_Var = result();
    52 
    53 goalw PropLog.thy [is_true_def]
    54     "is_true(p=>q,t) <-> (is_true(p,t)-->is_true(q,t))";
    55 by (simp_tac (prop_rec_ss setloop (split_tac [expand_if])) 1);
    56 val is_true_Imp = result();
    57 
    58 (** The function hyps **)
    59 
    60 goalw PropLog.thy [hyps_def] "hyps(Fls,t) = 0";
    61 by (simp_tac prop_rec_ss 1);
    62 val hyps_Fls = result();
    63 
    64 goalw PropLog.thy [hyps_def] "hyps(#v,t) = {if(v:t, #v, #v=>Fls)}";
    65 by (simp_tac prop_rec_ss 1);
    66 val hyps_Var = result();
    67 
    68 goalw PropLog.thy [hyps_def] "hyps(p=>q,t) = hyps(p,t) Un hyps(q,t)";
    69 by (simp_tac prop_rec_ss 1);
    70 val hyps_Imp = result();
    71 
    72 val prop_ss = prop_rec_ss 
    73     addsimps prop.intrs
    74     addsimps [is_true_Fls, is_true_Var, is_true_Imp,
    75 	      hyps_Fls, hyps_Var, hyps_Imp];
    76 
    77 (*** Proof theory of propositional logic ***)
    78 
    79 goalw PropLog.thy thms.defs "!!G H. G<=H ==> thms(G) <= thms(H)";
    80 by (rtac lfp_mono 1);
    81 by (REPEAT (rtac thms.bnd_mono 1));
    82 by (REPEAT (ares_tac (univ_mono::basic_monos) 1));
    83 val thms_mono = result();
    84 
    85 val thms_in_pl = thms.dom_subset RS subsetD;
    86 
    87 val ImpE = prop.mk_cases prop.con_defs "p=>q : prop";
    88 
    89 (*Stronger Modus Ponens rule: no typechecking!*)
    90 goal PropLog.thy "!!p q H. [| H |- p=>q;  H |- p |] ==> H |- q";
    91 by (rtac thms.MP 1);
    92 by (REPEAT (eresolve_tac [asm_rl, thms_in_pl, thms_in_pl RS ImpE] 1));
    93 val thms_MP = result();
    94 
    95 (*Rule is called I for Identity Combinator, not for Introduction*)
    96 goal PropLog.thy "!!p H. p:prop ==> H |- p=>p";
    97 by (rtac (thms.S RS thms_MP RS thms_MP) 1);
    98 by (rtac thms.K 5);
    99 by (rtac thms.K 4);
   100 by (REPEAT (ares_tac prop.intrs 1));
   101 val thms_I = result();
   102 
   103 (** Weakening, left and right **)
   104 
   105 (* [| G<=H;  G|-p |] ==> H|-p   Order of premises is convenient with RS*)
   106 val weaken_left = standard (thms_mono RS subsetD);
   107 
   108 (* H |- p ==> cons(a,H) |- p *)
   109 val weaken_left_cons = subset_consI RS weaken_left;
   110 
   111 val weaken_left_Un1  = Un_upper1 RS weaken_left;
   112 val weaken_left_Un2  = Un_upper2 RS weaken_left;
   113 
   114 goal PropLog.thy "!!H p q. [| H |- q;  p:prop |] ==> H |- p=>q";
   115 by (rtac (thms.K RS thms_MP) 1);
   116 by (REPEAT (ares_tac [thms_in_pl] 1));
   117 val weaken_right = result();
   118 
   119 (*The deduction theorem*)
   120 goal PropLog.thy "!!p q H. [| cons(p,H) |- q;  p:prop |] ==>  H |- p=>q";
   121 by (etac thms.induct 1);
   122 by (fast_tac (ZF_cs addIs [thms_I, thms.H RS weaken_right]) 1);
   123 by (fast_tac (ZF_cs addIs [thms.K RS weaken_right]) 1);
   124 by (fast_tac (ZF_cs addIs [thms.S RS weaken_right]) 1);
   125 by (fast_tac (ZF_cs addIs [thms.DN RS weaken_right]) 1);
   126 by (fast_tac (ZF_cs addIs [thms.S RS thms_MP RS thms_MP]) 1);
   127 val deduction = result();
   128 
   129 
   130 (*The cut rule*)
   131 goal PropLog.thy "!!H p q. [| H|-p;  cons(p,H) |- q |] ==>  H |- q";
   132 by (rtac (deduction RS thms_MP) 1);
   133 by (REPEAT (ares_tac [thms_in_pl] 1));
   134 val cut = result();
   135 
   136 goal PropLog.thy "!!H p. [| H |- Fls; p:prop |] ==> H |- p";
   137 by (rtac (thms.DN RS thms_MP) 1);
   138 by (rtac weaken_right 2);
   139 by (REPEAT (ares_tac (prop.intrs@[consI1]) 1));
   140 val thms_FlsE = result();
   141 
   142 (* [| H |- p=>Fls;  H |- p;  q: prop |] ==> H |- q *)
   143 val thms_notE = standard (thms_MP RS thms_FlsE);
   144 
   145 (*Soundness of the rules wrt truth-table semantics*)
   146 goalw PropLog.thy [logcon_def] "!!H. H |- p ==> H |= p";
   147 by (etac thms.induct 1);
   148 by (fast_tac (ZF_cs addSDs [is_true_Imp RS iffD1 RS mp]) 5);
   149 by (ALLGOALS (asm_simp_tac prop_ss));
   150 val soundness = result();
   151 
   152 (*** Towards the completeness proof ***)
   153 
   154 val [premf,premq] = goal PropLog.thy
   155     "[| H |- p=>Fls; q: prop |] ==> H |- p=>q";
   156 by (rtac (premf RS thms_in_pl RS ImpE) 1);
   157 by (rtac deduction 1);
   158 by (rtac (premf RS weaken_left_cons RS thms_notE) 1);
   159 by (REPEAT (ares_tac [premq, consI1, thms.H] 1));
   160 val Fls_Imp = result();
   161 
   162 val [premp,premq] = goal PropLog.thy
   163     "[| H |- p;  H |- q=>Fls |] ==> H |- (p=>q)=>Fls";
   164 by (cut_facts_tac ([premp,premq] RL [thms_in_pl]) 1);
   165 by (etac ImpE 1);
   166 by (rtac deduction 1);
   167 by (rtac (premq RS weaken_left_cons RS thms_MP) 1);
   168 by (rtac (consI1 RS thms.H RS thms_MP) 1);
   169 by (rtac (premp RS weaken_left_cons) 2);
   170 by (REPEAT (ares_tac prop.intrs 1));
   171 val Imp_Fls = result();
   172 
   173 (*Typical example of strengthening the induction formula*)
   174 val [major] = goal PropLog.thy 
   175     "p: prop ==> hyps(p,t) |- if(is_true(p,t), p, p=>Fls)";
   176 by (rtac (expand_if RS iffD2) 1);
   177 by (rtac (major RS prop.induct) 1);
   178 by (ALLGOALS (asm_simp_tac (prop_ss addsimps [thms_I, thms.H])));
   179 by (safe_tac (ZF_cs addSEs [Fls_Imp RS weaken_left_Un1, 
   180 			    Fls_Imp RS weaken_left_Un2]));
   181 by (ALLGOALS (fast_tac (ZF_cs addIs [weaken_left_Un1, weaken_left_Un2, 
   182 				     weaken_right, Imp_Fls])));
   183 val hyps_thms_if = result();
   184 
   185 (*Key lemma for completeness; yields a set of assumptions satisfying p*)
   186 val [premp,sat] = goalw PropLog.thy [logcon_def]
   187     "[| p: prop;  0 |= p |] ==> hyps(p,t) |- p";
   188 by (rtac (sat RS spec RS mp RS if_P RS subst) 1 THEN
   189     rtac (premp RS hyps_thms_if) 2);
   190 by (fast_tac ZF_cs 1);
   191 val logcon_thms_p = result();
   192 
   193 (*For proving certain theorems in our new propositional logic*)
   194 val thms_cs = 
   195     ZF_cs addSIs (prop.intrs @ [deduction])
   196           addIs [thms_in_pl, thms.H, thms.H RS thms_MP];
   197 
   198 (*The excluded middle in the form of an elimination rule*)
   199 val prems = goal PropLog.thy
   200     "[| p: prop;  q: prop |] ==> H |- (p=>q) => ((p=>Fls)=>q) => q";
   201 by (rtac (deduction RS deduction) 1);
   202 by (rtac (thms.DN RS thms_MP) 1);
   203 by (ALLGOALS (best_tac (thms_cs addSIs prems)));
   204 val thms_excluded_middle = result();
   205 
   206 (*Hard to prove directly because it requires cuts*)
   207 val prems = goal PropLog.thy
   208     "[| cons(p,H) |- q;  cons(p=>Fls,H) |- q;  p: prop |] ==> H |- q";
   209 by (rtac (thms_excluded_middle RS thms_MP RS thms_MP) 1);
   210 by (REPEAT (resolve_tac (prems@prop.intrs@[deduction,thms_in_pl]) 1));
   211 val thms_excluded_middle_rule = result();
   212 
   213 (*** Completeness -- lemmas for reducing the set of assumptions ***)
   214 
   215 (*For the case hyps(p,t)-cons(#v,Y) |- p;
   216   we also have hyps(p,t)-{#v} <= hyps(p, t-{v}) *)
   217 val [major] = goal PropLog.thy
   218     "p: prop ==> hyps(p, t-{v}) <= cons(#v=>Fls, hyps(p,t)-{#v})";
   219 by (rtac (major RS prop.induct) 1);
   220 by (simp_tac prop_ss 1);
   221 by (asm_simp_tac (prop_ss setloop (split_tac [expand_if])) 1);
   222 by (fast_tac (ZF_cs addSEs prop.free_SEs) 1);
   223 by (asm_simp_tac prop_ss 1);
   224 by (fast_tac ZF_cs 1);
   225 val hyps_Diff = result();
   226 
   227 (*For the case hyps(p,t)-cons(#v => Fls,Y) |- p;
   228   we also have hyps(p,t)-{#v=>Fls} <= hyps(p, cons(v,t)) *)
   229 val [major] = goal PropLog.thy
   230     "p: prop ==> hyps(p, cons(v,t)) <= cons(#v, hyps(p,t)-{#v=>Fls})";
   231 by (rtac (major RS prop.induct) 1);
   232 by (simp_tac prop_ss 1);
   233 by (asm_simp_tac (prop_ss setloop (split_tac [expand_if])) 1);
   234 by (fast_tac (ZF_cs addSEs prop.free_SEs) 1);
   235 by (asm_simp_tac prop_ss 1);
   236 by (fast_tac ZF_cs 1);
   237 val hyps_cons = result();
   238 
   239 (** Two lemmas for use with weaken_left **)
   240 
   241 goal ZF.thy "B-C <= cons(a, B-cons(a,C))";
   242 by (fast_tac ZF_cs 1);
   243 val cons_Diff_same = result();
   244 
   245 goal ZF.thy "cons(a, B-{c}) - D <= cons(a, B-cons(c,D))";
   246 by (fast_tac ZF_cs 1);
   247 val cons_Diff_subset2 = result();
   248 
   249 (*The set hyps(p,t) is finite, and elements have the form #v or #v=>Fls;
   250  could probably prove the stronger hyps(p,t) : Fin(hyps(p,0) Un hyps(p,nat))*)
   251 val [major] = goal PropLog.thy
   252     "p: prop ==> hyps(p,t) : Fin(UN v:nat. {#v, #v=>Fls})";
   253 by (rtac (major RS prop.induct) 1);
   254 by (asm_simp_tac (prop_ss addsimps (Fin.intrs @ [UN_I, cons_iff])
   255 		  setloop (split_tac [expand_if])) 2);
   256 by (ALLGOALS (asm_simp_tac (prop_ss addsimps [Un_0, Fin.emptyI, Fin_UnI])));
   257 val hyps_finite = result();
   258 
   259 val Diff_weaken_left = subset_refl RSN (2, Diff_mono) RS weaken_left;
   260 
   261 (*Induction on the finite set of assumptions hyps(p,t0).
   262   We may repeatedly subtract assumptions until none are left!*)
   263 val [premp,sat] = goal PropLog.thy
   264     "[| p: prop;  0 |= p |] ==> ALL t. hyps(p,t) - hyps(p,t0) |- p";
   265 by (rtac (premp RS hyps_finite RS Fin_induct) 1);
   266 by (simp_tac (prop_ss addsimps [premp, sat, logcon_thms_p, Diff_0]) 1);
   267 by (safe_tac ZF_cs);
   268 (*Case hyps(p,t)-cons(#v,Y) |- p *)
   269 by (rtac thms_excluded_middle_rule 1);
   270 by (etac prop.Var_I 3);
   271 by (rtac (cons_Diff_same RS weaken_left) 1);
   272 by (etac spec 1);
   273 by (rtac (cons_Diff_subset2 RS weaken_left) 1);
   274 by (rtac (premp RS hyps_Diff RS Diff_weaken_left) 1);
   275 by (etac spec 1);
   276 (*Case hyps(p,t)-cons(#v => Fls,Y) |- p *)
   277 by (rtac thms_excluded_middle_rule 1);
   278 by (etac prop.Var_I 3);
   279 by (rtac (cons_Diff_same RS weaken_left) 2);
   280 by (etac spec 2);
   281 by (rtac (cons_Diff_subset2 RS weaken_left) 1);
   282 by (rtac (premp RS hyps_cons RS Diff_weaken_left) 1);
   283 by (etac spec 1);
   284 val completeness_0_lemma = result();
   285 
   286 (*The base case for completeness*)
   287 val [premp,sat] = goal PropLog.thy "[| p: prop;  0 |= p |] ==> 0 |- p";
   288 by (rtac (Diff_cancel RS subst) 1);
   289 by (rtac (sat RS (premp RS completeness_0_lemma RS spec)) 1);
   290 val completeness_0 = result();
   291 
   292 (*A semantic analogue of the Deduction Theorem*)
   293 goalw PropLog.thy [logcon_def] "!!H p q. [| cons(p,H) |= q |] ==> H |= p=>q";
   294 by (simp_tac prop_ss 1);
   295 by (fast_tac ZF_cs 1);
   296 val logcon_Imp = result();
   297 
   298 goal PropLog.thy "!!H. H: Fin(prop) ==> ALL p:prop. H |= p --> H |- p";
   299 by (etac Fin_induct 1);
   300 by (safe_tac (ZF_cs addSIs [completeness_0]));
   301 by (rtac (weaken_left_cons RS thms_MP) 1);
   302 by (fast_tac (ZF_cs addSIs (logcon_Imp::prop.intrs)) 1);
   303 by (fast_tac thms_cs 1);
   304 val completeness_lemma = result();
   305 
   306 val completeness = completeness_lemma RS bspec RS mp;
   307 
   308 val [finite] = goal PropLog.thy "H: Fin(prop) ==> H |- p <-> H |= p & p:prop";
   309 by (fast_tac (ZF_cs addSEs [soundness, finite RS completeness, 
   310 			    thms_in_pl]) 1);
   311 val thms_iff = result();
   312 
   313 writeln"Reached end of file.";