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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/interp simpbool.boot}
+\author{The Axiom Team}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{License}
+<<license>>=
+-- Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
+-- All rights reserved.
+--
+-- Redistribution and use in source and binary forms, with or without
+-- modification, are permitted provided that the following conditions are
+-- met:
+--
+-- - Redistributions of source code must retain the above copyright
+-- notice, this list of conditions and the following disclaimer.
+--
+-- - Redistributions in binary form must reproduce the above copyright
+-- notice, this list of conditions and the following disclaimer in
+-- the documentation and/or other materials provided with the
+-- distribution.
+--
+-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
+-- names of its contributors may be used to endorse or promote products
+-- derived from this software without specific prior written permission.
+--
+-- THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
+-- IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
+-- TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
+-- PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
+-- OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
+-- EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
+-- PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+-- PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
+-- LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+-- NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+-- SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+@
+<<*>>=
+<<license>>
+
+)package "BOOT"
+
+simpBool x == dnf2pf reduceDnf be x
+
+reduceDnf u ==
+-- (OR (AND ..b..) b) ==> (OR b )
+ atom u => u
+ for x in u repeat
+ ok := true
+ for y in u repeat
+ x = y => 'skip
+ dnfContains(x,y) => return (ok := false)
+ ok = true => acc := [x,:acc]
+ nreverse acc
+
+dnfContains([a,b],[c,d]) == fn(a,c) and fn(b,d) where
+ fn(x,y) == and/[member(u,x) for u in y]
+
+prove x ==
+ world := [p for y in listOfUserIds x | (p := getPredicate y)] =>
+ 'false = be mkpf([['NOT,x],:world],'AND) => true
+ 'false = be mkpf([x,:world],'AND) => false
+ x
+ 'false = (y := be x) => 'false
+ y = 'true => true
+ dnf2pf y
+
+simpBoolGiven(x,world) ==
+ world =>
+ 'false = be mkpf([['NOT,x],:world],'AND) => true
+ 'false = (y := be mkpf([x,:world],'AND)) => false
+ (u := andReduce(dnf2pf y,world)) is ['AND,:v] and
+ (w := SETDIFFERENCE(v,world)) ^= v => simpBool ['AND,:w]
+ u
+ 'false = (y := be x) => false
+ 'true = y => true
+ dnf2pf y
+
+andReduce(x,y) ==
+ x is ['AND,:r] =>
+ y is ['AND,:s] => mkpf(S_-(r,s),'AND)
+ mkpf(S_-(r,[s]),'AND)
+ x
+dnf2pf(x) ==
+ x = 'true => 'T
+ x = 'false => nil
+ atom x => x
+ mkpf(
+ [mkpf([:[k for k in b],:[['not,k] for k in a]],'AND) for [a,b] in x],'OR)
+be x == b2dnf x
+b2dnf x ==
+ x = 'T => 'true
+ x = NIL => 'false
+ atom x => bassert x
+ [op,:argl] := x
+ MEMQ(op,'(AND and)) => band argl
+ MEMQ(op,'(OR or)) => bor argl
+ MEMQ(op,'(NOT not)) => bnot first argl
+ bassert x
+band x ==
+ x is [h,:t] => andDnf(b2dnf h,band t)
+ 'true
+bor x ==
+ x is [a,:b] => orDnf(b2dnf a,bor b)
+ 'false
+bnot x == notDnf b2dnf x
+bassert x == [[nil,[x]]]
+bassertNot x == [[[x],nil]]
+------------------------Disjunctive Normal Form Code-----------------------
+-- dnf is true | false | [coaf ... ]
+-- coaf is true | false | [item ... ]
+-- item is anything
+
+orDnf(a,b) == -- or: (dnf, dnf) -> dnf
+ a = 'false => b
+ b = 'false => a
+ a = 'true or b = 'true => 'true
+ null a => b --null list means false
+ a is [c] = coafOrDnf(c,b)
+ coafOrDnf(first a,orDnf(rest a,b))
+
+andDnf(a,b) == -- and: (dnf, dnf) -> dnf
+ a = 'true => b
+ b = 'true => a
+ a = 'false or b = 'false => 'false
+ null a => 'false --null list means false
+ a is [c] => coafAndDnf(c,b)
+ x := coafAndDnf(first a,b)
+ y := andDnf(rest a,b)
+ x = 'false => y
+ y = 'false => x
+ ordUnion(x,y)
+
+notDnf l == -- not: dnf -> dnf
+ l = 'true => 'false
+ l = 'false => 'true
+ null l => 'true --null list means false
+ l is [x] => notCoaf x
+ andDnf(notCoaf first l,notDnf rest l)
+
+coafOrDnf(a,l) == -- or: (coaf, dnf) -> dnf
+ a = 'true or l = 'true => 'true
+ a = 'false => l
+ member(a,l) => l
+ y := notCoaf a
+ x := ordIntersection(y,l)
+ null x => orDel(a,l)
+ x = l => 'true
+ x = y => ordSetDiff(l,x)
+ ordUnion(notDnf ordSetDiff(y,x),l)
+
+coafAndDnf(a,b) == --and: (coaf, dnf) -> dnf
+ a = 'true => b
+ a = 'false => 'false
+ [c,:r] := b
+ null r => coafAndCoaf(a,c)
+ x := coafAndCoaf(a,c) --dnf
+ y := coafAndDnf(a,r) --dnf
+ x = 'false => y
+ y = 'false => x
+ ordUnion(x,y)
+
+coafAndCoaf([a,b],[p,q]) == --and: (coaf,coaf) -> dnf
+ ordIntersection(a,q) or ordIntersection(b,p) => 'false
+ [[ordUnion(a,p),ordUnion(b,q)]]
+
+notCoaf [a,b] == [:[[nil,[x]] for x in a],:[[[x],nil] for x in b]]
+
+list1 l ==
+ l isnt [h,:t] => nil
+ null h => list1 t
+ [[h,nil,nil],:list1 t]
+list2 l ==
+ l isnt [h,:t] => nil
+ null h => list2 t
+ [[nil,h,nil],:list2 t]
+list3 l ==
+ l isnt [h,:t] => nil
+ null h => list3 t
+ [[nil,nil,h],:list3 t]
+orDel(a,l) ==
+ l is [h,:t] =>
+ a = h => t
+ ?ORDER(a,h) => [a,:l]
+ [h,:orDel(a,t)]
+ [a]
+ordList l ==
+ l is [h,:t] and t => orDel(h,ordList t)
+ l
+ordUnion(a,b) ==
+ a isnt [c,:r] => b
+ b isnt [d,:s] => a
+ c=d => [c,:ordUnion(r,s)]
+ ?ORDER(a,b) => [c,:ordUnion(r,b)]
+ [d,:ordUnion(s,a)]
+ordIntersection(a,b) ==
+ a isnt [h,:t] => nil
+ member(h,b) => [h,:ordIntersection(t,b)]
+ ordIntersection(t,b)
+ordSetDiff(a,b) ==
+ b isnt [h,:t] => a
+ member(h,a) => ordSetDiff(delete(h,a),t)
+ ordSetDiff(a,t)
+-------------
+testPredList u ==
+ for x in u repeat
+ y := simpBool x
+ x = y => nil
+ pp x
+ pp '"==========>"
+ pp y
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}