(* Title: FOL/ex/Miniscope.thy
ID: $Id$
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.
*)
theory Miniscope
imports FOL
begin
lemmas ccontr = FalseE [THEN classical]
subsection {* Negation Normal Form *}
subsubsection {* de Morgan laws *}
lemma demorgans:
"~(P&Q) <-> ~P | ~Q"
"~(P|Q) <-> ~P & ~Q"
"~~P <-> P"
"!!P. ~(ALL x. P(x)) <-> (EX x. ~P(x))"
"!!P. ~(EX x. P(x)) <-> (ALL x. ~P(x))"
by blast+
(*** Removal of --> and <-> (positive and negative occurrences) ***)
(*Last one is important for computing a compact CNF*)
lemma nnf_simps:
"(P-->Q) <-> (~P | Q)"
"~(P-->Q) <-> (P & ~Q)"
"(P<->Q) <-> (~P | Q) & (~Q | P)"
"~(P<->Q) <-> (P | Q) & (~P | ~Q)"
by blast+
(* BEWARE: rewrite rules for <-> can confuse the simplifier!! *)
subsubsection {* Pushing in the existential quantifiers *}
lemma ex_simps:
"(EX x. P) <-> P"
"!!P Q. (EX x. P(x) & Q) <-> (EX x. P(x)) & Q"
"!!P Q. (EX x. P & Q(x)) <-> P & (EX x. Q(x))"
"!!P Q. (EX x. P(x) | Q(x)) <-> (EX x. P(x)) | (EX x. Q(x))"
"!!P Q. (EX x. P(x) | Q) <-> (EX x. P(x)) | Q"
"!!P Q. (EX x. P | Q(x)) <-> P | (EX x. Q(x))"
by blast+
subsubsection {* Pushing in the universal quantifiers *}
lemma all_simps:
"(ALL x. P) <-> P"
"!!P Q. (ALL x. P(x) & Q(x)) <-> (ALL x. P(x)) & (ALL x. Q(x))"
"!!P Q. (ALL x. P(x) & Q) <-> (ALL x. P(x)) & Q"
"!!P Q. (ALL x. P & Q(x)) <-> P & (ALL x. Q(x))"
"!!P Q. (ALL x. P(x) | Q) <-> (ALL x. P(x)) | Q"
"!!P Q. (ALL x. P | Q(x)) <-> P | (ALL x. Q(x))"
by blast+
lemmas mini_simps = demorgans nnf_simps ex_simps all_simps
ML {*
val mini_ss = simpset() addsimps (thms "mini_simps");
val mini_tac = rtac (thm "ccontr") THEN' asm_full_simp_tac mini_ss;
*}
end