src/ZF/ex/proplog.ML
author clasohm
Thu Sep 16 12:20:38 1993 +0200 (1993-09-16)
changeset 0 a5a9c433f639
child 7 268f93ab3bc4
permissions -rw-r--r--
Initial revision
     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 val prop_congs = mk_typed_congs Prop.thy 
    17 		   [("b", "[i,i,i]=>i"), ("d", "[i,i,i,i]=>i")];
    18 
    19 (** conversion rules **)
    20 
    21 goal PropLog.thy "prop_rec(Fls,b,c,d) = b";
    22 by (rtac (prop_rec_def RS def_Vrec RS trans) 1);
    23 by (rewrite_goals_tac Prop.con_defs);
    24 by (SIMP_TAC rank_ss 1);
    25 val prop_rec_Fls = result();
    26 
    27 goal PropLog.thy "prop_rec(#v,b,c,d) = c(v)";
    28 by (rtac (prop_rec_def RS def_Vrec RS trans) 1);
    29 by (rewrite_goals_tac Prop.con_defs);
    30 by (SIMP_TAC (rank_ss addcongs prop_congs) 1);
    31 val prop_rec_Var = result();
    32 
    33 goal PropLog.thy "prop_rec(p=>q,b,c,d) = \
    34 \      d(p, q, prop_rec(p,b,c,d), prop_rec(q,b,c,d))";
    35 by (rtac (prop_rec_def RS def_Vrec RS trans) 1);
    36 by (rewrite_goals_tac Prop.con_defs);
    37 by (SIMP_TAC (rank_ss addcongs prop_congs) 1);
    38 val prop_rec_Imp = result();
    39 
    40 val prop_rec_ss = 
    41     arith_ss addrews [prop_rec_Fls, prop_rec_Var, prop_rec_Imp];
    42 
    43 (*** Semantics of propositional logic ***)
    44 
    45 (** The function is_true **)
    46 
    47 goalw PropLog.thy [is_true_def] "is_true(Fls,t) <-> False";
    48 by (SIMP_TAC (prop_rec_ss addrews [one_not_0 RS not_sym]) 1);
    49 val is_true_Fls = result();
    50 
    51 goalw PropLog.thy [is_true_def] "is_true(#v,t) <-> v:t";
    52 by (SIMP_TAC (prop_rec_ss addrews [one_not_0 RS not_sym] 
    53 	      addsplits [expand_if]) 1);
    54 val is_true_Var = result();
    55 
    56 goalw PropLog.thy [is_true_def]
    57     "is_true(p=>q,t) <-> (is_true(p,t)-->is_true(q,t))";
    58 by (SIMP_TAC (prop_rec_ss addsplits [expand_if]) 1);
    59 val is_true_Imp = result();
    60 
    61 (** The function hyps **)
    62 
    63 goalw PropLog.thy [hyps_def] "hyps(Fls,t) = 0";
    64 by (SIMP_TAC prop_rec_ss 1);
    65 val hyps_Fls = result();
    66 
    67 goalw PropLog.thy [hyps_def] "hyps(#v,t) = {if(v:t, #v, #v=>Fls)}";
    68 by (SIMP_TAC prop_rec_ss 1);
    69 val hyps_Var = result();
    70 
    71 goalw PropLog.thy [hyps_def] "hyps(p=>q,t) = hyps(p,t) Un hyps(q,t)";
    72 by (SIMP_TAC prop_rec_ss 1);
    73 val hyps_Imp = result();
    74 
    75 val prop_ss = prop_rec_ss 
    76     addcongs Prop.congs
    77     addcongs (mk_congs PropLog.thy ["Fin", "thms", "op |=","is_true","hyps"])
    78     addrews Prop.intrs
    79     addrews [is_true_Fls, is_true_Var, is_true_Imp,
    80 	     hyps_Fls, hyps_Var, hyps_Imp];
    81 
    82 (*** Proof theory of propositional logic ***)
    83 
    84 structure PropThms = Inductive_Fun
    85  (val thy = PropLog.thy;
    86   val rec_doms = [("thms","prop")];
    87   val sintrs = 
    88       ["[| p:H;  p:prop |] ==> H |- p",
    89        "[| p:prop;  q:prop |] ==> H |- p=>q=>p",
    90        "[| p:prop;  q:prop;  r:prop |] ==> H |- (p=>q=>r) => (p=>q) => p=>r",
    91        "p:prop ==> H |- ((p=>Fls) => Fls) => p",
    92        "[| H |- p=>q;  H |- p;  p:prop;  q:prop |] ==> H |- q"];
    93   val monos = [];
    94   val con_defs = [];
    95   val type_intrs = Prop.intrs;
    96   val type_elims = []);
    97 
    98 goalw PropThms.thy PropThms.defs "!!G H. G<=H ==> thms(G) <= thms(H)";
    99 by (rtac lfp_mono 1);
   100 by (REPEAT (rtac PropThms.bnd_mono 1));
   101 by (REPEAT (ares_tac (univ_mono::basic_monos) 1));
   102 val thms_mono = result();
   103 
   104 val thms_in_pl = PropThms.dom_subset RS subsetD;
   105 
   106 val [thms_H, thms_K, thms_S, thms_DN, weak_thms_MP] = PropThms.intrs;
   107 
   108 (*Modus Ponens rule -- this stronger version avoids typecheck*)
   109 goal PropThms.thy "!!p q H. [| H |- p=>q;  H |- p |] ==> H |- q";
   110 by (rtac weak_thms_MP 1);
   111 by (REPEAT (eresolve_tac [asm_rl, thms_in_pl, thms_in_pl RS ImpE] 1));
   112 val thms_MP = result();
   113 
   114 (*Rule is called I for Identity Combinator, not for Introduction*)
   115 goal PropThms.thy "!!p H. p:prop ==> H |- p=>p";
   116 by (rtac (thms_S RS thms_MP RS thms_MP) 1);
   117 by (rtac thms_K 5);
   118 by (rtac thms_K 4);
   119 by (REPEAT (ares_tac [ImpI] 1));
   120 val thms_I = result();
   121 
   122 (** Weakening, left and right **)
   123 
   124 (* [| G<=H;  G|-p |] ==> H|-p   Order of premises is convenient with RS*)
   125 val weaken_left = standard (thms_mono RS subsetD);
   126 
   127 (* H |- p ==> cons(a,H) |- p *)
   128 val weaken_left_cons = subset_consI RS weaken_left;
   129 
   130 val weaken_left_Un1  = Un_upper1 RS weaken_left;
   131 val weaken_left_Un2  = Un_upper2 RS weaken_left;
   132 
   133 goal PropThms.thy "!!H p q. [| H |- q;  p:prop |] ==> H |- p=>q";
   134 by (rtac (thms_K RS thms_MP) 1);
   135 by (REPEAT (ares_tac [thms_in_pl] 1));
   136 val weaken_right = result();
   137 
   138 (*The deduction theorem*)
   139 goal PropThms.thy "!!p q H. [| cons(p,H) |- q;  p:prop |] ==>  H |- p=>q";
   140 by (etac PropThms.induct 1);
   141 by (fast_tac (ZF_cs addIs [thms_I, thms_H RS weaken_right]) 1);
   142 by (fast_tac (ZF_cs addIs [thms_K RS weaken_right]) 1);
   143 by (fast_tac (ZF_cs addIs [thms_S RS weaken_right]) 1);
   144 by (fast_tac (ZF_cs addIs [thms_DN RS weaken_right]) 1);
   145 by (fast_tac (ZF_cs addIs [thms_S RS thms_MP RS thms_MP]) 1);
   146 val deduction = result();
   147 
   148 
   149 (*The cut rule*)
   150 goal PropThms.thy "!!H p q. [| H|-p;  cons(p,H) |- q |] ==>  H |- q";
   151 by (rtac (deduction RS thms_MP) 1);
   152 by (REPEAT (ares_tac [thms_in_pl] 1));
   153 val cut = result();
   154 
   155 goal PropThms.thy "!!H p. [| H |- Fls; p:prop |] ==> H |- p";
   156 by (rtac (thms_DN RS thms_MP) 1);
   157 by (rtac weaken_right 2);
   158 by (REPEAT (ares_tac (Prop.intrs@[consI1]) 1));
   159 val thms_FlsE = result();
   160 
   161 (* [| H |- p=>Fls;  H |- p;  q: prop |] ==> H |- q *)
   162 val thms_notE = standard (thms_MP RS thms_FlsE);
   163 
   164 (*Soundness of the rules wrt truth-table semantics*)
   165 val [major] = goalw PropThms.thy [sat_def] "H |- p ==> H |= p";
   166 by (rtac (major RS PropThms.induct) 1);
   167 by (fast_tac (ZF_cs addSDs [is_true_Imp RS iffD1 RS mp]) 5);
   168 by (ALLGOALS (SIMP_TAC prop_ss));
   169 val soundness = result();
   170 
   171 (*** Towards the completeness proof ***)
   172 
   173 val [premf,premq] = goal PropThms.thy
   174     "[| H |- p=>Fls; q: prop |] ==> H |- p=>q";
   175 by (rtac (premf RS thms_in_pl RS ImpE) 1);
   176 by (rtac deduction 1);
   177 by (rtac (premf RS weaken_left_cons RS thms_notE) 1);
   178 by (REPEAT (ares_tac [premq, consI1, thms_H] 1));
   179 val Fls_Imp = result();
   180 
   181 val [premp,premq] = goal PropThms.thy
   182     "[| H |- p;  H |- q=>Fls |] ==> H |- (p=>q)=>Fls";
   183 by (cut_facts_tac ([premp,premq] RL [thms_in_pl]) 1);
   184 by (etac ImpE 1);
   185 by (rtac deduction 1);
   186 by (rtac (premq RS weaken_left_cons RS thms_MP) 1);
   187 by (rtac (consI1 RS thms_H RS thms_MP) 1);
   188 by (rtac (premp RS weaken_left_cons) 2);
   189 by (REPEAT (ares_tac Prop.intrs 1));
   190 val Imp_Fls = result();
   191 
   192 (*Typical example of strengthening the induction formula*)
   193 val [major] = goal PropThms.thy 
   194     "p: prop ==> hyps(p,t) |- if(is_true(p,t), p, p=>Fls)";
   195 by (rtac (expand_if RS iffD2) 1);
   196 by (rtac (major RS Prop.induct) 1);
   197 by (ALLGOALS (ASM_SIMP_TAC (prop_ss addrews [thms_I, thms_H])));
   198 by (fast_tac (ZF_cs addIs [weaken_left_Un1, weaken_left_Un2, 
   199 			   weaken_right, Imp_Fls]
   200                     addSEs [Fls_Imp]) 1);
   201 val hyps_thms_if = result();
   202 
   203 (*Key lemma for completeness; yields a set of assumptions satisfying p*)
   204 val [premp,sat] = goalw PropThms.thy [sat_def]
   205     "[| p: prop;  0 |= p |] ==> hyps(p,t) |- p";
   206 by (rtac (sat RS spec RS mp RS if_P RS subst) 1 THEN
   207     rtac (premp RS hyps_thms_if) 2);
   208 by (fast_tac ZF_cs 1);
   209 val sat_thms_p = result();
   210 
   211 (*For proving certain theorems in our new propositional logic*)
   212 val thms_cs = 
   213     ZF_cs addSIs [FlsI, VarI, ImpI, deduction]
   214           addIs [thms_in_pl, thms_H, thms_H RS thms_MP];
   215 
   216 (*The excluded middle in the form of an elimination rule*)
   217 val prems = goal PropThms.thy
   218     "[| p: prop;  q: prop |] ==> H |- (p=>q) => ((p=>Fls)=>q) => q";
   219 by (rtac (deduction RS deduction) 1);
   220 by (rtac (thms_DN RS thms_MP) 1);
   221 by (ALLGOALS (best_tac (thms_cs addSIs prems)));
   222 val thms_excluded_middle = result();
   223 
   224 (*Hard to prove directly because it requires cuts*)
   225 val prems = goal PropThms.thy
   226     "[| cons(p,H) |- q;  cons(p=>Fls,H) |- q;  p: prop |] ==> H |- q";
   227 by (rtac (thms_excluded_middle RS thms_MP RS thms_MP) 1);
   228 by (REPEAT (resolve_tac (prems@Prop.intrs@[deduction,thms_in_pl]) 1));
   229 val thms_excluded_middle_rule = result();
   230 
   231 (*** Completeness -- lemmas for reducing the set of assumptions ***)
   232 
   233 (*For the case hyps(p,t)-cons(#v,Y) |- p;
   234   we also have hyps(p,t)-{#v} <= hyps(p, t-{v}) *)
   235 val [major] = goal PropThms.thy
   236     "p: prop ==> hyps(p, t-{v}) <= cons(#v=>Fls, hyps(p,t)-{#v})";
   237 by (rtac (major RS Prop.induct) 1);
   238 by (SIMP_TAC prop_ss 1);
   239 by (ASM_SIMP_TAC (prop_ss addsplits [expand_if]) 1);
   240 by (fast_tac (ZF_cs addSEs Prop.free_SEs) 1);
   241 by (ASM_SIMP_TAC prop_ss 1);
   242 by (fast_tac ZF_cs 1);
   243 val hyps_Diff = result();
   244 
   245 (*For the case hyps(p,t)-cons(#v => Fls,Y) |- p;
   246   we also have hyps(p,t)-{#v=>Fls} <= hyps(p, cons(v,t)) *)
   247 val [major] = goal PropThms.thy
   248     "p: prop ==> hyps(p, cons(v,t)) <= cons(#v, hyps(p,t)-{#v=>Fls})";
   249 by (rtac (major RS Prop.induct) 1);
   250 by (SIMP_TAC prop_ss 1);
   251 by (ASM_SIMP_TAC (prop_ss addsplits [expand_if]) 1);
   252 by (fast_tac (ZF_cs addSEs Prop.free_SEs) 1);
   253 by (ASM_SIMP_TAC prop_ss 1);
   254 by (fast_tac ZF_cs 1);
   255 val hyps_cons = result();
   256 
   257 (** Two lemmas for use with weaken_left **)
   258 
   259 goal ZF.thy "B-C <= cons(a, B-cons(a,C))";
   260 by (fast_tac ZF_cs 1);
   261 val cons_Diff_same = result();
   262 
   263 goal ZF.thy "cons(a, B-{c}) - D <= cons(a, B-cons(c,D))";
   264 by (fast_tac ZF_cs 1);
   265 val cons_Diff_subset2 = result();
   266 
   267 (*The set hyps(p,t) is finite, and elements have the form #v or #v=>Fls;
   268  could probably prove the stronger hyps(p,t) : Fin(hyps(p,0) Un hyps(p,nat))*)
   269 val [major] = goal PropThms.thy
   270     "p: prop ==> hyps(p,t) : Fin(UN v:nat. {#v, #v=>Fls})";
   271 by (rtac (major RS Prop.induct) 1);
   272 by (ASM_SIMP_TAC (prop_ss addrews [Fin_0I, Fin_consI, UN_I] 
   273 		  addsplits [expand_if]) 2);
   274 by (ALLGOALS (ASM_SIMP_TAC (prop_ss addrews [Un_0, Fin_0I, Fin_UnI])));
   275 val hyps_finite = result();
   276 
   277 val Diff_weaken_left = subset_refl RSN (2, Diff_mono) RS weaken_left;
   278 
   279 (*Induction on the finite set of assumptions hyps(p,t0).
   280   We may repeatedly subtract assumptions until none are left!*)
   281 val [premp,sat] = goal PropThms.thy
   282     "[| p: prop;  0 |= p |] ==> ALL t. hyps(p,t) - hyps(p,t0) |- p";
   283 by (rtac (premp RS hyps_finite RS Fin_induct) 1);
   284 by (SIMP_TAC (prop_ss addrews [premp, sat, sat_thms_p, Diff_0]) 1);
   285 by (safe_tac ZF_cs);
   286 (*Case hyps(p,t)-cons(#v,Y) |- p *)
   287 by (rtac thms_excluded_middle_rule 1);
   288 by (etac VarI 3);
   289 by (rtac (cons_Diff_same RS weaken_left) 1);
   290 by (etac spec 1);
   291 by (rtac (cons_Diff_subset2 RS weaken_left) 1);
   292 by (rtac (premp RS hyps_Diff RS Diff_weaken_left) 1);
   293 by (etac spec 1);
   294 (*Case hyps(p,t)-cons(#v => Fls,Y) |- p *)
   295 by (rtac thms_excluded_middle_rule 1);
   296 by (etac VarI 3);
   297 by (rtac (cons_Diff_same RS weaken_left) 2);
   298 by (etac spec 2);
   299 by (rtac (cons_Diff_subset2 RS weaken_left) 1);
   300 by (rtac (premp RS hyps_cons RS Diff_weaken_left) 1);
   301 by (etac spec 1);
   302 val completeness_0_lemma = result();
   303 
   304 (*The base case for completeness*)
   305 val [premp,sat] = goal PropThms.thy "[| p: prop;  0 |= p |] ==> 0 |- p";
   306 by (rtac (Diff_cancel RS subst) 1);
   307 by (rtac (sat RS (premp RS completeness_0_lemma RS spec)) 1);
   308 val completeness_0 = result();
   309 
   310 (*A semantic analogue of the Deduction Theorem*)
   311 goalw PropThms.thy [sat_def] "!!H p q. [| cons(p,H) |= q |] ==> H |= p=>q";
   312 by (SIMP_TAC prop_ss 1);
   313 by (fast_tac ZF_cs 1);
   314 val sat_Imp = result();
   315 
   316 goal PropThms.thy "!!H. H: Fin(prop) ==> ALL p:prop. H |= p --> H |- p";
   317 by (etac Fin_induct 1);
   318 by (safe_tac (ZF_cs addSIs [completeness_0]));
   319 by (rtac (weaken_left_cons RS thms_MP) 1);
   320 by (fast_tac (ZF_cs addSIs [sat_Imp,ImpI]) 1);
   321 by (fast_tac thms_cs 1);
   322 val completeness_lemma = result();
   323 
   324 val completeness = completeness_lemma RS bspec RS mp;
   325 
   326 val [finite] = goal PropThms.thy "H: Fin(prop) ==> H |- p <-> H |= p & p:prop";
   327 by (fast_tac (ZF_cs addSEs [soundness, finite RS completeness, 
   328 			    thms_in_pl]) 1);
   329 val thms_iff = result();
   330 
   331 writeln"Reached end of file.";
   332 
   333