ex/PL.ML

 (* [| 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] @ PL0.pl.simps;
+val pl_ss = set_ss addsimps [pl_rec_var,pl_rec_false,pl_rec_imp] @ PL.pl.simps;
 
 goalw PL.thy [eval_def] "tt[false] = False";
 by (simp_tac pl_ss 1);
 val eval_false = result();
 

 
 goalw PL.thy [hyps_def] "hyps(false,tt) = {}";
 by (simp_tac pl_ss 1);
 val hyps_false = result();
 
-goalw PL.thy [hyps_def] "hyps(#v,tt) = {if(v:tt, #v, (#v)->false)}";
+goalw PL.thy [hyps_def] "hyps(#v,tt) = {if(v:tt, #v, #v->false)}";
 by (simp_tac pl_ss 1);
 val hyps_var = result();
 
 goalw PL.thy [hyps_def] "hyps(p->q,tt) = hyps(p,tt) Un hyps(q,tt)";
 by (simp_tac pl_ss 1);

 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(PL0.pl.induct_tac "p" 1);
+by(PL.pl.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 "hyps(p, t-{v}) <= insert((#v)->false, hyps(p,t)-{#v})";
-by (PL0.pl.induct_tac "p" 1);
+goal PL.thy "hyps(p, t-{v}) <= insert(#v->false, hyps(p,t)-{#v})";
+by (PL.pl.induct_tac "p" 1);
 by (simp_tac pl_ss 1);
 by (simp_tac (pl_ss setloop (split_tac [expand_if])) 1);
 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 "hyps(p, insert(v,t)) <= insert(#v, hyps(p,t)-{(#v)->false})";
-by (PL0.pl.induct_tac "p" 1);
+  we also have hyps(p,t)-{#v->false} <= hyps(p, insert(v,t)) *)
+goal PL.thy "hyps(p, insert(v,t)) <= insert(#v, hyps(p,t)-{#v->false})";
+by (PL.pl.induct_tac "p" 1);
 by (simp_tac pl_ss 1);
 by (simp_tac (pl_ss setloop (split_tac [expand_if])) 1);
 by (simp_tac pl_ss 1);
 by (fast_tac set_cs 1);
 val hyps_insert = result();

 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 "hyps(p,t) : Fin(UN v:{x.True}. {#v, (#v)->false})";
-by (PL0.pl.induct_tac "p" 1);
+goal PL.thy "hyps(p,t) : Fin(UN v:{x.True}. {#v, #v->false})";
+by (PL.pl.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();
 
 val Diff_weaken_left = subset_refl RSN (2, Diff_mono) RS weaken_left;