(*  Title:      FOL/ex/mini
    ID:         $Id: mini.ML,v 1.6 1997/10/10 13:55:08 wenzelm Exp $
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge

Classical First-Order Logic

Conversion to nnf/miniscope format: pushing quantifiers in

Demonstration of formula rewriting by proof
*)

val ccontr = FalseE RS classical;

(**** Negation Normal Form ****)

(*** de Morgan laws ***)

fun prove_fun s = 
 (writeln s;  
  prove_goal FOL.thy s
   (fn prems => [ (cut_facts_tac prems 1), 
                  (Cla.fast_tac FOL_cs 1) ]));

val demorgans = map prove_fun
                  ["~(P&Q) <-> ~P | ~Q",
                   "~(P|Q) <-> ~P & ~Q",
                   "~~P <-> P",
                   "~(ALL x. P(x)) <-> (EX x. ~P(x))",
                   "~(EX x. P(x)) <-> (ALL x. ~P(x))"];

(*** Removal of --> and <-> (positive and negative occurrences) ***)
(*Last one is important for computing a compact CNF*)
val nnf_simps = map prove_fun
                ["(P-->Q) <-> (~P | Q)",
                 "~(P-->Q) <-> (P & ~Q)",
                 "(P<->Q) <-> (~P | Q) & (~Q | P)",
                 "~(P<->Q) <-> (P | Q) & (~P | ~Q)"];

(* BEWARE: rewrite rules for <-> can confuse the simplifier!! *)

(*** Pushing in the existential quantifiers ***)

val ex_simps = map prove_fun 
                ["(EX x. P) <-> P",
                 "(EX x. P(x) & Q) <-> (EX x. P(x)) & Q",
                 "(EX x. P & Q(x)) <-> P & (EX x. Q(x))",
                 "(EX x. P(x) | Q(x)) <-> (EX x. P(x)) | (EX x. Q(x))",
                 "(EX x. P(x) | Q) <-> (EX x. P(x)) | Q",
                 "(EX x. P | Q(x)) <-> P | (EX x. Q(x))"];

(*** Pushing in the universal quantifiers ***)

val all_simps = map prove_fun
                ["(ALL x. P) <-> P",
                 "(ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))",
                 "(ALL x. P(x) & Q) <-> (ALL x. P(x)) & Q",
                 "(ALL x. P & Q(x)) <-> P & (ALL x. Q(x))",
                 "(ALL x. P(x) | Q) <-> (ALL x. P(x)) | Q",
                 "(ALL x. P | Q(x)) <-> P | (ALL x. Q(x))"];


val mini_ss=FOL_basic_ss addsimps (demorgans @ nnf_simps @ ex_simps @ all_simps);

val mini_tac = rtac ccontr THEN' asm_full_simp_tac mini_ss;