ex/PL.ML

-(*  Title: 	HOL/ex/prop-log.ML
+(*  Title: 	HOL/ex/pl.ML
     ID:         $Id$
     Author: 	Tobias Nipkow & Lawrence C Paulson
-    Copyright   1993  TU Muenchen & University of Cambridge
-
-For ex/prop-log.thy.  Inductive definition of propositional logic.
-Soundness and completeness w.r.t. truth-tables.
+    Copyright   1994  TU Muenchen & University of Cambridge
+
+Soundness and completeness of propositional logic w.r.t. truth-tables.
 
 Prove: If H|=p then G|=p where G:Fin(H)
 *)
 
 open PL;

 (* [| H |- p->false;  H |- p;  q: pl |] ==> H |- q *)
 val conseq_notE = standard (conseq_MP RS conseq_falseE);
 
 (** The function eval **)
 
-val pl_ss = set_ss addsimps [pl_rec_var,pl_rec_false,pl_rec_imp];
+val pl_ss = set_ss addsimps [pl_rec_var,pl_rec_false,pl_rec_imp]
+                            @ PL0.inject @ PL0.ineq;
 
 goalw PL.thy [eval_def] "tt[false] = False";
 by (simp_tac pl_ss 1);
 val eval_false = result();
 

 val imp_false = result();
 
 (*This formulation is required for strong induction hypotheses*)
 goal PL.thy "hyps(p,tt) |- if(tt[p], p, p->false)";
 by (rtac (expand_if RS iffD2) 1);
-by(res_inst_tac[("x","p")]spec 1);
-by (rtac pl_ind 1);
+by(PL0.induct_tac "p" 1);
 by (ALLGOALS (simp_tac (pl_ss addsimps [conseq_I, conseq_H])));
 by (fast_tac (set_cs addIs [weaken_left_Un1, weaken_left_Un2, 
 			   weaken_right, imp_false]
                     addSEs [false_imp]) 1);
 val hyps_conseq_if = result();

 
 (*** Completeness -- lemmas for reducing the set of assumptions ***)
 
 (*For the case hyps(p,t)-insert(#v,Y) |- p;
   we also have hyps(p,t)-{#v} <= hyps(p, t-{v}) *)
-goal PL.thy "!p.hyps(p, t-{v}) <= insert((#v)->false, hyps(p,t)-{#v})";
-by (rtac pl_ind 1);
+goal PL.thy "hyps(p, t-{v}) <= insert((#v)->false, hyps(p,t)-{#v})";
+by (PL0.induct_tac "p" 1);
 by (simp_tac pl_ss 1);
 by (simp_tac (pl_ss setloop (split_tac [expand_if])) 1);
-by (fast_tac (set_cs addSEs [sym RS var_neq_imp] addSDs [var_inject]) 1);
-by (simp_tac pl_ss 1);
-by (fast_tac set_cs 1);
-val hyps_Diff = result() RS spec;
+by (simp_tac pl_ss 1);
+by (fast_tac set_cs 1);
+val hyps_Diff = result();
 
 (*For the case hyps(p,t)-insert(#v -> false,Y) |- p;
   we also have hyps(p,t)-{(#v)->false} <= hyps(p, insert(v,t)) *)
-goal PL.thy "!p.hyps(p, insert(v,t)) <= insert(#v, hyps(p,t)-{(#v)->false})";
-by (rtac pl_ind 1);
-by (simp_tac pl_ss 1);
-by (simp_tac (pl_ss setloop (split_tac [expand_if])) 1);
-by (fast_tac (set_cs addSEs [var_neq_imp, imp_inject] addSDs [var_inject]) 1);
-by (simp_tac pl_ss 1);
-by (fast_tac set_cs 1);
-val hyps_insert = result() RS spec;
+goal PL.thy "hyps(p, insert(v,t)) <= insert(#v, hyps(p,t)-{(#v)->false})";
+by (PL0.induct_tac "p" 1);
+by (simp_tac pl_ss 1);
+by (simp_tac (pl_ss addsimps [insert_subset]
+                    setloop (split_tac [expand_if])) 1);
+by (simp_tac pl_ss 1);
+by (fast_tac set_cs 1);
+val hyps_insert = result();
 
 (** Two lemmas for use with weaken_left **)
 
 goal Set.thy "B-C <= insert(a, B-insert(a,C))";
 by (fast_tac set_cs 1);

 by (fast_tac set_cs 1);
 val insert_Diff_subset2 = result();
 
 (*The set hyps(p,t) is finite, and elements have the form #v or #v->false;
  could probably prove the stronger hyps(p,t) : Fin(hyps(p,{}) Un hyps(p,nat))*)
-goal PL.thy "!p.hyps(p,t) : Fin(UN v:{x.True}. {#v, (#v)->false})";
-by (rtac pl_ind 1);
+goal PL.thy "hyps(p,t) : Fin(UN v:{x.True}. {#v, (#v)->false})";
+by (PL0.induct_tac "p" 1);
 by (ALLGOALS (simp_tac (pl_ss setloop (split_tac [expand_if])) THEN'
               fast_tac (set_cs addSIs [Fin_0I, Fin_insertI, Fin_UnI])));
-val hyps_finite = result() RS spec;
+val hyps_finite = result();
 
 val Diff_weaken_left = subset_refl RSN (2, Diff_mono) RS weaken_left;
 
 (*Induction on the finite set of assumptions hyps(p,t0).
   We may repeatedly subtract assumptions until none are left!*)

 by (fast_tac (set_cs addSEs [soundness, finite RS completeness]) 1);
 val conseq_iff = result();
 
 writeln"Reached end of file.";
 
-