12531 lines
559 KiB
XML
12531 lines
559 KiB
XML
$( ql.mm - Version of 11-Apr-2012
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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Metamath source file for logic, set theory, numbers, and Hilbert space
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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~~ PUBLIC DOMAIN ~~
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This work is waived of all rights, including copyright, according to the CC0
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Public Domain Dedication. http://creativecommons.org/publicdomain/zero/1.0/
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Norman Megill - email: nm(at)alum(dot)mit(dot)edu - http://metamath.org
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$)
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$( placeholder
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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AUQL - Algebraic Unified Quantum Logic of M. Pavicic
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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$)
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$(
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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Ortholattices
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#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
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$)
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$(
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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Basic syntax and axioms
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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$)
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$( Declare the primitive constant symbols. $)
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$c ( $. $( Left parenthesis $)
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$c ) $. $( Right parenthesis $)
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$c = $. $( Equality (read: 'equals') $)
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$c == $. $( Biconditional (read: 'equivalent') $)
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$c v $. $( Disjunction (read: 'or') $)
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$c ^ $. $( Conjuction (read: 'and') $)
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$c 1 $. $( True constant (upside down ' ) (read: 'true') $)
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$c 0 $. $( False constant ( ' ) (read: 'false') $)
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$c ' $. $( Orthocomplement $)
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$c wff $. $( Well-formed formula symbol (read: 'the following symbol
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sequence is a wff') $)
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$c term $. $( Term $)
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$c |- $. $( Turnstile (read: 'the following symbol sequence is provable' or
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'a proof exists for') $)
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$( Relations as operations $)
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$c C $. $( Commutes relation or commutator operation $)
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$c =< $. $( Less-than-or-equal-to $)
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$c =<2 $. $( Less-than-or-equal-to analogue for terms $)
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$c ->0 $. $( Right arrow (read: 'implies') $)
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$c ->1 $. $( Right arrow (read: 'implies') $)
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$c ->2 $. $( Right arrow (read: 'implies') $)
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$c ->3 $. $( Right arrow (read: 'implies') $)
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$c ->4 $. $( Right arrow (read: 'implies') $)
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$c ->5 $. $( Right arrow (read: 'implies') $)
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$c ==0 $. $( Classical identity $)
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$c ==1 $. $( Asymmetrical identity $)
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$c ==2 $. $( Asymmetrical identity $)
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$c ==3 $. $( Asymmetrical identity $)
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$c ==4 $. $( Asymmetrical identity $)
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$c ==5 $. $( Asymmetrical identity $)
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$c ==OA $. $( Orthoarguesian identity $)
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$c , $. $( Comma $)
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$c <->3 $. $( Biconditional (read: 'equivalent') $)
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$c <->1 $. $( Biconditional (read: 'equivalent') $)
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$c u3 $. $( Disjunction (read: 'or') $)
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$c ^3 $. $( Conjuction (read: 'and') $)
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$( Introduce some variable names we will use to terms. $)
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$v a $.
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$v b $.
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$v c $.
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$v d $.
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$v e $.
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$v f $.
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$v g $.
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$v h $.
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$v j $.
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$v k $.
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$v l $.
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$v i $.
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$v m $.
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$v n $.
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$v p $.
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$v q $.
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$v r $.
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$v t $.
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$v u $.
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$v w $.
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$v x $.
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$v y $.
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$v z $.
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$v a0 a1 a2 b0 b1 b2 c0 c1 c2 p0 p1 p2 $.
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$(
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Specify some variables that we will use to represent terms.
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The fact that a variable represents a wff is relevant only to a theorem
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referring to that variable, so we may use $f hypotheses. The symbol
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` term ` specifies that the variable that follows it represents a term.
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$)
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$( Let variable ` a ` be a term. $)
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wva $f term a $.
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$( Let variable ` b ` be a term. $)
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wvb $f term b $.
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$( Let variable ` c ` be a term. $)
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wvc $f term c $.
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$( Let variable ` d ` be a term. $)
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wvd $f term d $.
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$( Let variable ` e ` be a term. $)
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wve $f term e $.
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$( Let variable ` f ` be a term. $)
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wvf $f term f $.
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$( Let variable ` g ` be a term. $)
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wvg $f term g $.
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$( Let variable ` h ` be a term. $)
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wvh $f term h $.
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$( Let variable ` j ` be a term. $)
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wvj $f term j $.
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$( Let variable ` k ` be a term. $)
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wvk $f term k $.
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$( Let variable ` l ` be a term. $)
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wvl $f term l $.
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$( Let variable ` i ` be a term. $)
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wvi $f term i $.
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$( Let variable ` m ` be a term. $)
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wvm $f term m $.
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$( Let variable ` n ` be a term. $)
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wvn $f term n $.
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$( Let variable ` p ` be a term. $)
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wvp $f term p $.
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$( Let variable ` q ` be a term. $)
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wvq $f term q $.
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$( Let variable ` r ` be a term. $)
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wvr $f term r $.
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$( Let variable ` t ` be a term. $)
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wvt $f term t $.
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$( Let variable ` u ` be a term. $)
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wvu $f term u $.
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$( Let variable ` w ` be a term. $)
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wvw $f term w $.
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$( Let variable ` x ` be a term. $)
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wvx $f term x $.
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$( Let variable ` y ` be a term. $)
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wvy $f term y $.
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$( Let variable ` z ` be a term. $)
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wvz $f term z $.
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$( Let variable ` a0 ` be a term. $)
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wva0 $f term a0 $.
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$( Let variable ` a1 ` be a term. $)
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wva1 $f term a1 $.
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$( Let variable ` a2 ` be a term. $)
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wva2 $f term a2 $.
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$( Let variable ` b0 ` be a term. $)
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wvb0 $f term b0 $.
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$( Let variable ` b1 ` be a term. $)
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wvb1 $f term b1 $.
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$( Let variable ` b2 ` be a term. $)
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wvb2 $f term b2 $.
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$( Let variable ` c0 ` be a term. $)
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wvc0 $f term c0 $.
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$( Let variable ` c1 ` be a term. $)
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wvc1 $f term c1 $.
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$( Let variable ` c2 ` be a term. $)
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wvc2 $f term c2 $.
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$( Let variable ` p0 ` be a term. $)
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wvp0 $f term p0 $.
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$( Let variable ` p1 ` be a term. $)
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wvp1 $f term p1 $.
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$( Let variable ` p2 ` be a term. $)
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wvp2 $f term p2 $.
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$(
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Recursively define terms and wffs.
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$)
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$( If ` a ` and ` b ` are terms, ` a = b ` is a wff. $)
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wb $a wff a = b $.
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$( If ` a ` and ` b ` are terms, ` a =< b ` is a wff. $)
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wle $a wff a =< b $.
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$( If ` a ` and ` b ` are terms, ` a C b ` is a wff. $)
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wc $a wff a C b $.
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$( If ` a ` is a term, so is ` a ' ` . $)
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wn $a term a ' $.
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$( If ` a ` and ` b ` are terms, so is ` ( a == b ) ` . $)
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tb $a term ( a == b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a v b ) ` . $)
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wo $a term ( a v b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ^ b ) ` . $)
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wa $a term ( a ^ b ) $.
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$(
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@( If ` a ` and ` b ` are terms, so is ` ( a ' b ) ` . @)
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wp @a term ( a ' b ) @.
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$)
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$( The logical true constant is a term. $)
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wt $a term 1 $.
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$( The logical false constant is a term. $)
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wf $a term 0 $.
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$( If ` a ` and ` b ` are terms, so is ` ( a =<2 b ) ` . $)
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wle2 $a term ( a =<2 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ->0 b ) ` . $)
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wi0 $a term ( a ->0 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ->1 b ) ` . $)
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wi1 $a term ( a ->1 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ->2 b ) ` . $)
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wi2 $a term ( a ->2 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ->3 b ) ` . $)
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wi3 $a term ( a ->3 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ->4 b ) ` . $)
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wi4 $a term ( a ->4 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ->5 b ) ` . $)
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wi5 $a term ( a ->5 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ==0 b ) ` . $)
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wid0 $a term ( a ==0 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ==1 b ) ` . $)
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wid1 $a term ( a ==1 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ==2 b ) ` . $)
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wid2 $a term ( a ==2 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ==3 b ) ` . $)
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wid3 $a term ( a ==3 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ==4 b ) ` . $)
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wid4 $a term ( a ==4 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ==5 b ) ` . $)
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wid5 $a term ( a ==5 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a <->3 b ) ` . $)
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wb3 $a term ( a <->3 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a <->3 b ) ` . $)
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wb1 $a term ( a <->1 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a u3 b ) ` . $)
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wo3 $a term ( a u3 b ) $.
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$( If ` a ` and ` b ` are terms, so is ` ( a ^3 b ) ` . $)
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wan3 $a term ( a ^3 b ) $.
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$( If ` a ` , ` b ` , and ` c ` are terms, so is ` ( a == c ==OA b ) ` . $)
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wid3oa $a term ( a == c ==OA b ) $.
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$( If ` a ` , ` b ` , ` c ` , and ` d ` are terms, so is
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` ( a == c , d ==OA b ) ` . $)
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wid4oa $a term ( a == c , d ==OA b ) $.
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$( If ` a ` and ` b ` are terms, so is ` C ( a , b ) ` . $)
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wcmtr $a term C ( a , b ) $.
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$( Axiom for ortholattices. $)
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ax-a1 $a |- a = a ' ' $.
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$( Axiom for ortholattices. $)
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ax-a2 $a |- ( a v b ) = ( b v a ) $.
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$( Axiom for ortholattices. $)
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ax-a3 $a |- ( ( a v b ) v c ) = ( a v ( b v c ) ) $.
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$( Axiom for ortholattices. $)
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ax-a4 $a |- ( a v ( b v b ' ) ) = ( b v b ' ) $.
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$(
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ax-a5 $a |- ( a v ( a ' v b ' ) ' ) = a $.
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$)
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$( Axiom for ortholattices. $)
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ax-a5 $a |- ( a v ( a ' v b ) ' ) = a $.
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$(
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df-b $a |- ( a == b ) =
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( ( a ' ' v b ' ' ) ' v ( a ' v b ' ) ' ) $.
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$)
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${
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r1.1 $e |- a = b $.
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$( Inference rule for ortholattices. $)
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ax-r1 $a |- b = a $.
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$}
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${
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r2.1 $e |- a = b $.
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r2.2 $e |- b = c $.
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$( Inference rule for ortholattices. $)
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ax-r2 $a |- a = c $.
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$}
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$( Axiom ~ax-r3 is the orthomodular axiom and will be introduced
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when we start to use it. $)
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${
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r4.1 $e |- a = b $.
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$( Inference rule for ortholattices. $)
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ax-r4 $a |- a ' = b ' $.
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$}
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${
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r5.1 $e |- a = b $.
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$( Inference rule for ortholattices. $)
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ax-r5 $a |- ( a v c ) = ( b v c ) $.
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$}
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$( Define biconditional. $)
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df-b $a |- ( a == b ) = ( ( a ' v b ' ) ' v ( a v b ) ' ) $.
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$( Define conjunction. $)
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df-a $a |- ( a ^ b ) = ( a ' v b ' ) ' $.
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$( Define true. $)
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df-t $a |- 1 = ( a v a ' ) $.
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$( Define false. $)
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df-f $a |- 0 = 1 ' $.
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$( Define classical conditional. $)
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df-i0 $a |- ( a ->0 b ) = ( a ' v b ) $.
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$( Define Sasaki (Mittelstaedt) conditional. $)
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df-i1 $a |- ( a ->1 b ) = ( a ' v ( a ^ b ) ) $.
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$( Define Dishkant conditional. $)
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df-i2 $a |- ( a ->2 b ) = ( b v ( a ' ^ b ' ) ) $.
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$( Define Kalmbach conditional. $)
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df-i3 $a |- ( a ->3 b ) = ( ( ( a ' ^ b ) v ( a ' ^ b ' ) ) v
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( a ^ ( a ' v b ) ) ) $.
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$( Define non-tollens conditional. $)
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df-i4 $a |- ( a ->4 b ) = ( ( ( a ^ b ) v ( a ' ^ b ) ) v
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( ( a ' v b ) ^ b ' ) ) $.
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$( Define relevance conditional. $)
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df-i5 $a |- ( a ->5 b ) = ( ( ( a ^ b ) v ( a ' ^ b ) ) v
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( a ' ^ b ' ) ) $.
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$( Define classical identity. $)
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df-id0 $a |- ( a ==0 b ) = ( ( a ' v b ) ^ ( b ' v a ) ) $.
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$( Define asymmetrical identity (for "Non-Orthomodular Models..." paper). $)
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df-id1 $a |- ( a ==1 b ) = ( ( a v b ' ) ^ ( a ' v ( a ^ b ) ) ) $.
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$( Define asymmetrical identity (for "Non-Orthomodular Models..." paper). $)
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df-id2 $a |- ( a ==2 b ) = ( ( a v b ' ) ^ ( b v ( a ' ^ b ' ) ) ) $.
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$( Define asymmetrical identity (for "Non-Orthomodular Models..." paper). $)
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df-id3 $a |- ( a ==3 b ) = ( ( a ' v b ) ^ ( a v ( a ' ^ b ' ) ) ) $.
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$( Define asymmetrical identity (for "Non-Orthomodular Models..." paper). $)
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df-id4 $a |- ( a ==4 b ) = ( ( a ' v b ) ^ ( b ' v ( a ^ b ) ) ) $.
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$( Defined disjunction. $)
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df-o3 $a |- ( a u3 b ) = ( a ' ->3 ( a ' ->3 b ) ) $.
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$( Defined conjunction. $)
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df-a3 $a |- ( a ^3 b ) = ( a ' u3 b ' ) ' $.
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$( Defined biconditional. $)
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df-b3 $a |- ( a <->3 b ) = ( ( a ->3 b ) ^ ( b ->3 a ) ) $.
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$( The 3-variable orthoarguesian identity term. $)
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df-id3oa $a |- ( a == c ==OA b ) = ( ( ( a ->1 c ) ^ ( b ->1 c ) )
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v ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ) $.
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$( The 4-variable orthoarguesian identity term. $)
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df-id4oa $a |- ( a == c , d ==OA b ) = ( ( a == d ==OA b ) v
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( ( a == d ==OA c ) ^ ( b == d ==OA c ) ) ) $.
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$(
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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Basic lemmas
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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$)
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$( Identity law. $)
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id $p |- a = a $=
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( wn ax-a1 ax-r1 ax-r2 ) AABBZAACZAFGDE $.
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$( [9-Aug-97] $)
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$( Justification of definition ~df-t of true ( ` 1 ` ). This shows that the
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definition is independent of the variable used to define it. $)
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tt $p |- ( a v a ' ) = ( b v b ' ) $=
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( wn wo ax-a4 ax-r1 ax-a2 ax-r2 ) AACDZIBBCDZDZJIJIDZKLIJAEFJIGHIBEH $.
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$( [9-Aug-97] $)
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${
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cm.1 $e |- a = b $.
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$( Commutative inference rule for ortholattices. $)
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cm $p |- b = a $=
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( ax-r1 ) ABCD $.
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$( [26-May-2008] $) $( [26-May-2008] $)
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$}
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${
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tr.1 $e |- a = b $.
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tr.2 $e |- b = c $.
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$( Transitive inference rule for ortholattices. $)
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tr $p |- a = c $=
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( ax-r2 ) ABCDEF $.
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$( [26-May-2008] $) $( [26-May-2008] $)
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$}
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${
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3tr1.1 $e |- a = b $.
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3tr1.2 $e |- c = a $.
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3tr1.3 $e |- d = b $.
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$( Transitive inference useful for introducing definitions. $)
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3tr1 $p |- c = d $=
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( ax-r1 ax-r2 ) CADFABDEDBGHII $.
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$( [10-Aug-97] $)
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$}
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${
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3tr2.1 $e |- a = b $.
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3tr2.2 $e |- a = c $.
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3tr2.3 $e |- b = d $.
|
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$( Transitive inference useful for eliminating definitions. $)
|
|
3tr2 $p |- c = d $=
|
|
( ax-r1 3tr1 ) ABCDEACFHBDGHI $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
3tr.1 $e |- a = b $.
|
|
3tr.2 $e |- b = c $.
|
|
3tr.3 $e |- c = d $.
|
|
$( Triple transitive inference. $)
|
|
3tr $p |- a = d $=
|
|
( ax-r2 ) ACDABCEFHGH $.
|
|
$( [20-Sep-98] $)
|
|
$}
|
|
|
|
${
|
|
con1.1 $e |- a ' = b ' $.
|
|
$( Contraposition inference. $)
|
|
con1 $p |- a = b $=
|
|
( wn ax-r4 ax-a1 3tr1 ) ADZDBDZDABHICEAFBFG $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
con2.1 $e |- a = b ' $.
|
|
$( Contraposition inference. $)
|
|
con2 $p |- a ' = b $=
|
|
( wn ax-r4 ax-a1 ax-r1 ax-r2 ) ADBDZDZBAICEBJBFGH $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
con3.1 $e |- a ' = b $.
|
|
$( Contraposition inference. $)
|
|
con3 $p |- a = b ' $=
|
|
( wn ax-a1 ax-r4 ax-r2 ) AADZDBDAEHBCFG $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
con4.1 $e |- a = b $.
|
|
$( Contraposition inference. $)
|
|
con4 $p |- a ' = b ' $=
|
|
( ax-r4 ) ABCD $.
|
|
$( [31-Mar-2011] $) $( [26-May-2008] $)
|
|
$}
|
|
|
|
${
|
|
lor.1 $e |- a = b $.
|
|
$( Inference introducing disjunct to left. $)
|
|
lor $p |- ( c v a ) = ( c v b ) $=
|
|
( wo ax-r5 ax-a2 3tr1 ) ACEBCECAECBEABCDFCAGCBGH $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Inference introducing disjunct to right. $)
|
|
ror $p |- ( a v c ) = ( b v c ) $=
|
|
( ax-r5 ) ABCDE $.
|
|
$( [31-Mar-2011] $) $( [26-May-2008] $)
|
|
$}
|
|
|
|
${
|
|
2or.1 $e |- a = b $.
|
|
2or.2 $e |- c = d $.
|
|
$( Join both sides with disjunction. $)
|
|
2or $p |- ( a v c ) = ( b v d ) $=
|
|
( wo lor ax-r5 ax-r2 ) ACGADGBDGCDAFHABDEIJ $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
$( Commutative law. $)
|
|
orcom $p |- ( a v b ) = ( b v a ) $=
|
|
( ax-a2 ) ABC $.
|
|
$( [31-Mar-2011] $) $( [27-May-2008] $)
|
|
|
|
$( Commutative law. $)
|
|
ancom $p |- ( a ^ b ) = ( b ^ a ) $=
|
|
( wn wo wa ax-a2 ax-r4 df-a 3tr1 ) ACZBCZDZCKJDZCABEBAELMJKFGABHBAHI $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Associative law. $)
|
|
orass $p |- ( ( a v b ) v c ) = ( a v ( b v c ) ) $=
|
|
( ax-a3 ) ABCD $.
|
|
$( [31-Mar-2011] $) $( [27-May-2008] $)
|
|
|
|
$( Associative law. $)
|
|
anass $p |- ( ( a ^ b ) ^ c ) = ( a ^ ( b ^ c ) ) $=
|
|
( wa wn wo ax-a3 df-a con2 ax-r5 lor 3tr1 ax-r4 ) ABDZEZCEZFZEAEZBCDZEZFZEN
|
|
CDASDQUARBEZFZPFRUBPFZFQUARUBPGOUCPNUCABHIJTUDRSUDBCHIKLMNCHASHL $.
|
|
$( [12-Aug-97] $)
|
|
|
|
${
|
|
lan.1 $e |- a = b $.
|
|
$( Introduce conjunct on left. $)
|
|
lan $p |- ( c ^ a ) = ( c ^ b ) $=
|
|
( wn wo wa ax-r4 lor df-a 3tr1 ) CEZAEZFZELBEZFZECAGCBGNPMOLABDHIHCAJCBJK
|
|
$.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
ran.1 $e |- a = b $.
|
|
$( Introduce conjunct on right. $)
|
|
ran $p |- ( a ^ c ) = ( b ^ c ) $=
|
|
( wa lan ancom 3tr1 ) CAECBEACEBCEABCDFACGBCGH $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
2an.1 $e |- a = b $.
|
|
2an.2 $e |- c = d $.
|
|
$( Conjoin both sides of hypotheses. $)
|
|
2an $p |- ( a ^ c ) = ( b ^ d ) $=
|
|
( wa lan ran ax-r2 ) ACGADGBDGCDAFHABDEIJ $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
$( Swap disjuncts. $)
|
|
or12 $p |- ( a v ( b v c ) ) = ( b v ( a v c ) ) $=
|
|
( wo ax-a2 ax-r5 ax-a3 3tr2 ) ABDZCDBADZCDABCDDBACDDIJCABEFABCGBACGH $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Swap conjuncts. $)
|
|
an12 $p |- ( a ^ ( b ^ c ) ) = ( b ^ ( a ^ c ) ) $=
|
|
( wa ancom ran anass 3tr2 ) ABDZCDBADZCDABCDDBACDDIJCABEFABCGBACGH $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Swap disjuncts. $)
|
|
or32 $p |- ( ( a v b ) v c ) = ( ( a v c ) v b ) $=
|
|
( wo ax-a2 lor ax-a3 3tr1 ) ABCDZDACBDZDABDCDACDBDIJABCEFABCGACBGH $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Swap conjuncts. $)
|
|
an32 $p |- ( ( a ^ b ) ^ c ) = ( ( a ^ c ) ^ b ) $=
|
|
( wa ancom lan anass 3tr1 ) ABCDZDACBDZDABDCDACDBDIJABCEFABCGACBGH $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Swap disjuncts. $)
|
|
or4 $p |- ( ( a v b ) v ( c v d ) ) = ( ( a v c ) v ( b v d ) ) $=
|
|
( wo or12 lor ax-a3 3tr1 ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHACLHI $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Rearrange disjuncts. $)
|
|
or42 $p |- ( ( a v b ) v ( c v d ) ) = ( ( a v d ) v ( b v c ) ) $=
|
|
( wo ax-a2 lor or4 ax-r2 ) ABEZCDEZEJDCEZEADEBCEEKLJCDFGABDCHI $.
|
|
$( [4-Mar-06] $)
|
|
|
|
$( Swap conjuncts. $)
|
|
an4 $p |- ( ( a ^ b ) ^ ( c ^ d ) ) = ( ( a ^ c ) ^ ( b ^ d ) ) $=
|
|
( wa an12 lan anass 3tr1 ) ABCDEZEZEACBDEZEZEABEJEACELEKMABCDFGABJHACLHI $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Disjunction expressed with conjunction. $)
|
|
oran $p |- ( a v b ) = ( a ' ^ b ' ) ' $=
|
|
( wn wo wa ax-a1 2or df-a ax-r4 3tr1 ) ACZCZBCZCZDZOCZCABDKMEZCOFALBNAFBFGQ
|
|
PKMHIJ $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Conjunction expressed with disjunction. $)
|
|
anor1 $p |- ( a ^ b ' ) = ( a ' v b ) ' $=
|
|
( wn wa wo df-a ax-a1 ax-r1 lor ax-r4 ax-r2 ) ABCZDACZLCZEZCMBEZCALFOPNBMBN
|
|
BGHIJK $.
|
|
$( [12-Aug-97] $)
|
|
|
|
$( Conjunction expressed with disjunction. $)
|
|
anor2 $p |- ( a ' ^ b ) = ( a v b ' ) ' $=
|
|
( wn wa wo df-a ax-a1 ax-r1 ax-r5 ax-r4 ax-r2 ) ACZBDLCZBCZEZCANEZCLBFOPMAN
|
|
AMAGHIJK $.
|
|
$( [12-Aug-97] $)
|
|
|
|
$( Conjunction expressed with disjunction. $)
|
|
anor3 $p |- ( a ' ^ b ' ) = ( a v b ) ' $=
|
|
( wn wa wo oran ax-r1 con3 ) ACBCDZABEZJICABFGH $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Disjunction expressed with conjunction. $)
|
|
oran1 $p |- ( a v b ' ) = ( a ' ^ b ) ' $=
|
|
( wn wo wa anor2 ax-r1 con3 ) ABCDZACBEZJICABFGH $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Disjunction expressed with conjunction. $)
|
|
oran2 $p |- ( a ' v b ) = ( a ^ b ' ) ' $=
|
|
( wn wo wa anor1 ax-r1 con3 ) ACBDZABCEZJICABFGH $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Disjunction expressed with conjunction. $)
|
|
oran3 $p |- ( a ' v b ' ) = ( a ^ b ) ' $=
|
|
( wn wo wa df-a ax-r1 con3 ) ACBCDZABEZJICABFGH $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Biconditional expressed with others. $)
|
|
dfb $p |- ( a == b ) = ( ( a ^ b ) v ( a ' ^ b ' ) ) $=
|
|
( tb wn wo wa df-b df-a ax-r1 oran con2 2or ax-r2 ) ABCADZBDZEDZABEZDZEABFZ
|
|
NOFZEABGPSRTSPABHIQTABJKLM $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Negated biconditional. $)
|
|
dfnb $p |- ( a == b ) ' = ( ( a v b ) ^ ( a ' v b ' ) ) $=
|
|
( wa wn wo tb oran con2 ancom ax-r2 dfb ax-r4 df-a ax-r1 2an 3tr1 ) ABCZADZ
|
|
BDZCZEZDZTDZQDZCZABFZDABEZRSEZCUBUDUCCZUEUAUIQTGHUDUCIJUFUAABKLUGUCUHUDABGU
|
|
DUHQUHABMHNOP $.
|
|
$( [30-Aug-97] $)
|
|
|
|
$( Commutative law. $)
|
|
bicom $p |- ( a == b ) = ( b == a ) $=
|
|
( wa wn wo tb ancom 2or dfb 3tr1 ) ABCZADZBDZCZEBACZMLCZEABFBAFKONPABGLMGHA
|
|
BIBAIJ $.
|
|
$( [10-Aug-97] $)
|
|
|
|
${
|
|
lbi.1 $e |- a = b $.
|
|
$( Introduce biconditional to the left. $)
|
|
lbi $p |- ( c == a ) = ( c == b ) $=
|
|
( wa wn wo tb lan ax-r4 2or dfb 3tr1 ) CAEZCFZAFZEZGCBEZOBFZEZGCAHCBHNRQT
|
|
ABCDIPSOABDJIKCALCBLM $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
rbi.1 $e |- a = b $.
|
|
$( Introduce biconditional to the right. $)
|
|
rbi $p |- ( a == c ) = ( b == c ) $=
|
|
( tb lbi bicom 3tr1 ) CAECBEACEBCEABCDFACGBCGH $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
2bi.1 $e |- a = b $.
|
|
2bi.2 $e |- c = d $.
|
|
$( Join both sides with biconditional. $)
|
|
2bi $p |- ( a == c ) = ( b == d ) $=
|
|
( tb lbi rbi ax-r2 ) ACGADGBDGCDAFHABDEIJ $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
$( Alternate defintion of "false". $)
|
|
dff2 $p |- 0 = ( a v a ' ) ' $=
|
|
( wf wt wn wo df-f df-t ax-r4 ax-r2 ) BCDAADEZDFCJAGHI $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Alternate defintion of "false". $)
|
|
dff $p |- 0 = ( a ^ a ' ) $=
|
|
( wf wn wo wa dff2 ancom anor2 ax-r2 ax-r1 ) BAACZDCZAKEZAFMLMKAELAKGAAHIJI
|
|
$.
|
|
$( [29-Aug-97] $)
|
|
|
|
$( Disjunction with 0. $)
|
|
or0 $p |- ( a v 0 ) = a $=
|
|
( wf wo wn dff2 ax-a2 ax-r4 ax-r2 lor ax-a5 ) ABCAADZACZDZCABMABAKCZDMAENLA
|
|
KFGHIAAJH $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Disjunction with 0. $)
|
|
or0r $p |- ( 0 v a ) = a $=
|
|
( wf wo ax-a2 or0 ax-r2 ) BACABCABADAEF $.
|
|
$( [26-Nov-97] $)
|
|
|
|
$( Disjunction with 1. $)
|
|
or1 $p |- ( a v 1 ) = 1 $=
|
|
( wt wo wn df-t lor ax-a4 ax-r2 ax-r1 ) ABCZAADCZBJAKCKBKAAEZFAAGHBKLIH $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Disjunction with 1. $)
|
|
or1r $p |- ( 1 v a ) = 1 $=
|
|
( wt wo ax-a2 or1 ax-r2 ) BACABCBBADAEF $.
|
|
$( [26-Nov-97] $)
|
|
|
|
$( Conjunction with 1. $)
|
|
an1 $p |- ( a ^ 1 ) = a $=
|
|
( wt wa wn wo df-a wf df-f ax-r1 lor or0 ax-r2 con2 ) ABCADZBDZEZDAABFPAPNG
|
|
ENOGNGOHIJNKLML $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Conjunction with 1. $)
|
|
an1r $p |- ( 1 ^ a ) = a $=
|
|
( wt wa ancom an1 ax-r2 ) BACABCABADAEF $.
|
|
$( [26-Nov-97] $)
|
|
|
|
$( Conjunction with 0. $)
|
|
an0 $p |- ( a ^ 0 ) = 0 $=
|
|
( wf wa wn wo df-a wt or1 df-f con2 lor 3tr1 ax-r2 ) ABCADZBDZEZDBABFPBNGEG
|
|
PONHOGNBGIJZKQLJM $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Conjunction with 0. $)
|
|
an0r $p |- ( 0 ^ a ) = 0 $=
|
|
( wf wa ancom an0 ax-r2 ) BACABCBBADAEF $.
|
|
$( [26-Nov-97] $)
|
|
|
|
$( Idempotent law. $)
|
|
oridm $p |- ( a v a ) = a $=
|
|
( wo wn wf ax-a1 or0 ax-r1 ax-r4 ax-r2 lor ax-a5 ) AABAACZDBZCZBAANAALCNAEL
|
|
MMLLFGHIJADKI $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Idempotent law. $)
|
|
anidm $p |- ( a ^ a ) = a $=
|
|
( wa wn wo df-a oridm con2 ax-r2 ) AABACZIDZCAAAEJAIFGH $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Distribution of disjunction over disjunction. $)
|
|
orordi $p |- ( a v ( b v c ) ) =
|
|
( ( a v b ) v ( a v c ) ) $=
|
|
( wo oridm ax-r1 ax-r5 or4 ax-r2 ) ABCDZDAADZJDABDACDDAKJKAAEFGAABCHI $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Distribution of disjunction over disjunction. $)
|
|
orordir $p |- ( ( a v b ) v c ) =
|
|
( ( a v c ) v ( b v c ) ) $=
|
|
( wo oridm ax-r1 lor or4 ax-r2 ) ABDZCDJCCDZDACDBCDDCKJKCCEFGABCCHI $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Distribution of conjunction over conjunction. $)
|
|
anandi $p |- ( a ^ ( b ^ c ) ) =
|
|
( ( a ^ b ) ^ ( a ^ c ) ) $=
|
|
( wa anidm ax-r1 ran an4 ax-r2 ) ABCDZDAADZJDABDACDDAKJKAAEFGAABCHI $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Distribution of conjunction over conjunction. $)
|
|
anandir $p |- ( ( a ^ b ) ^ c ) =
|
|
( ( a ^ c ) ^ ( b ^ c ) ) $=
|
|
( wa anidm ax-r1 lan an4 ax-r2 ) ABDZCDJCCDZDACDBCDDCKJKCCEFGABCCHI $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Identity law. $)
|
|
biid $p |- ( a == a ) = 1 $=
|
|
( wa wn wo tb wt anidm 2or dfb df-t 3tr1 ) AABZACZMBZDAMDAAEFLANMAGMGHAAIAJ
|
|
K $.
|
|
$( [10-Aug-97] $)
|
|
|
|
$( Identity law. $)
|
|
1b $p |- ( 1 == a ) = a $=
|
|
( wt tb wa wn wo dfb wf ancom df-f ax-r1 lan ax-r2 2or an1 an0 or0 ) BACBAD
|
|
ZBEZAEZDZFZABAGUBAHFZAUBABDZTHDZFUCRUDUAUEBAIUATSDUESTISHTHSJKLMNUDAUEHAOTP
|
|
NMAQMM $.
|
|
$( [10-Aug-97] $)
|
|
|
|
${
|
|
bi1.1 $e |- a = b $.
|
|
$( Identity inference. $)
|
|
bi1 $p |- ( a == b ) = 1 $=
|
|
( tb wt rbi biid ax-r2 ) ABDBBDEABBCFBGH $.
|
|
$( [30-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
1bi.1 $e |- a = b $.
|
|
$( Identity inference. $)
|
|
1bi $p |- 1 = ( a == b ) $=
|
|
( tb wt bi1 ax-r1 ) ABDEABCFG $.
|
|
$( [30-Aug-97] $)
|
|
$}
|
|
|
|
$( Absorption law. $)
|
|
orabs $p |- ( a v ( a ^ b ) ) = a $=
|
|
( wa wo wn df-a lor ax-a5 ax-r2 ) AABCZDAAEBEZDEZDAJLAABFGAKHI $.
|
|
$( [11-Aug-97] $)
|
|
|
|
$( Absorption law. $)
|
|
anabs $p |- ( a ^ ( a v b ) ) = a $=
|
|
( wo wa wn ax-a1 ax-r5 lan df-a ax-r2 ax-a5 con2 ) AABCZDZAEZOEZBCZECZEZANA
|
|
QDSMQAAPBAFGHAQIJRAOBKLJ $.
|
|
$( [11-Aug-97] $)
|
|
|
|
$( Contraposition law. $)
|
|
conb $p |- ( a == b ) = ( a ' == b ' ) $=
|
|
( wa wn wo tb ax-a2 ax-a1 2an lor ax-r2 dfb 3tr1 ) ABCZADZBDZCZEZQODZPDZCZE
|
|
ZABFOPFRQNEUBNQGNUAQASBTAHBHIJKABLOPLM $.
|
|
$( [10-Aug-97] $)
|
|
|
|
${
|
|
leoa.1 $e |- ( a v c ) = b $.
|
|
$( Relation between two methods of expressing "less than or equal to". $)
|
|
leoa $p |- ( a ^ b ) = a $=
|
|
( wa wo ax-r1 lan anabs ax-r2 ) ABEAACFZEABKAKBDGHACIJ $.
|
|
$( [11-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
leao.1 $e |- ( c ^ b ) = a $.
|
|
$( Relation between two methods of expressing "less than or equal to". $)
|
|
leao $p |- ( a v b ) = b $=
|
|
( wo wa ax-a2 ax-r1 ancom ax-r2 lor orabs ) ABEZBBCFZEZBMBAEOABGANBACBFZN
|
|
PADHNPBCIHJKJBCLJ $.
|
|
$( [11-Aug-97] $)
|
|
$}
|
|
|
|
$( Mittelstaedt implication. $)
|
|
mi $p |- ( ( a v b ) == b ) = ( b v ( a ' ^ b ' ) ) $=
|
|
( wo tb wa wn dfb ancom ax-a2 lan anabs ax-r2 oran con2 ran anass anidm 2or
|
|
) ABCZBDSBEZSFZBFZEZCBAFZUBEZCSBGTBUCUETBSEZBSBHUFBBACZEBSUGBABIJBAKLLUCUDU
|
|
BUBEZEZUEUCUEUBEUIUAUEUBSUEABMNOUDUBUBPLUHUBUDUBQJLRL $.
|
|
$( [12-Aug-97] $)
|
|
|
|
$( Dishkant implication. $)
|
|
di $p |- ( ( a ^ b ) == a ) = ( a ' v ( a ^ b ) ) $=
|
|
( wn wo tb wa conb ax-a1 ax-r1 rbi mi ax-r2 ancom df-a 2an lor 3tr1 ) BCZAC
|
|
ZDZCZAEZSRCZSCZFZDZABFZAESUGDUBUACZSEZUFUAAGUITSEUFUHTSTUHTHIJRSKLLUGUAAUGB
|
|
AFZUAABMZBANLJUGUESUGUJUEUKBUCAUDBHAHOLPQ $.
|
|
$( [12-Aug-97] $)
|
|
|
|
$( Lemma in proof of Th. 1 of Pavicic 1987. $)
|
|
omlem1 $p |- ( ( a v ( a ' ^ ( a v b ) ) ) v ( a v b ) ) =
|
|
( a v b ) $=
|
|
( wn wo wa ax-a2 ax-a3 3tr1 ax-r2 ax-r1 oridm ax-r5 ancom 2or orabs 3tr2 )
|
|
AACZABDZEZDZADBDZRADZSDZTRDZRUDRTDUAUCTRFTABGZRASGHUEUCRRQEZDRUBRSUFUBAADZB
|
|
DZRUHUBUHARDUBAABGARFIJUGABAKLIQRMNRQOIP $.
|
|
$( [12-Aug-97] $)
|
|
|
|
$( Lemma in proof of Th. 1 of Pavicic 1987. $)
|
|
omlem2 $p |- ( ( a v b ) ' v ( a v ( a ' ^ ( a v b ) ) ) ) = 1 $=
|
|
( wo wn wa wt ax-a2 anor2 2or ax-a3 ax-r1 df-t 3tr1 ) ABCZDZACZADNEZCZAOCZS
|
|
DZCOAQCCZFPSQTOAGANHIRUAOAQJKSLM $.
|
|
$( [12-Aug-97] $)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Relationship analogues (ordering; commutation)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Define 'less than or equal to' analogue. $)
|
|
df-le $a |- ( a =<2 b ) = ( ( a v b ) == b ) $.
|
|
|
|
$( Since we don't have strong BMP in AUQL, we must add extra definitions
|
|
to eliminate the middle = . $)
|
|
${
|
|
df-le1.1 $e |- ( a v b ) = b $.
|
|
$( Define 'less than or equal to'. See ~ df-le2 for the other
|
|
direction. $)
|
|
df-le1 $a |- a =< b $.
|
|
$}
|
|
|
|
${
|
|
df-le2.1 $e |- a =< b $.
|
|
$( Define 'less than or equal to'. See ~ df-le1 for the other
|
|
direction. $)
|
|
df-le2 $a |- ( a v b ) = b $.
|
|
$}
|
|
|
|
${
|
|
df-c1.1 $e |- a = ( ( a ^ b ) v ( a ^ b ' ) ) $.
|
|
$( Define 'commutes'. See ~ df-c2 for the other direction. $)
|
|
df-c1 $a |- a C b $.
|
|
$}
|
|
|
|
${
|
|
df-c2.1 $e |- a C b $.
|
|
$( Define 'commutes'. See ~ df-c1 for the other direction. $)
|
|
df-c2 $a |- a = ( ( a ^ b ) v ( a ^ b ' ) ) $.
|
|
$}
|
|
|
|
$( Define 'commutator'. $)
|
|
df-cmtr $a |- C ( a , b ) = ( ( ( a ^ b ) v ( a ^ b ' ) ) v
|
|
( ( a ' ^ b ) v ( a ' ^ b ' ) ) ) $.
|
|
|
|
${
|
|
df2le1.1 $e |- ( a ^ b ) = a $.
|
|
$( Alternate definition of 'less than or equal to'. $)
|
|
df2le1 $p |- a =< b $=
|
|
( leao df-le1 ) ABABACDE $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
df2le2.1 $e |- a =< b $.
|
|
$( Alternate definition of 'less than or equal to'. $)
|
|
df2le2 $p |- ( a ^ b ) = a $=
|
|
( df-le2 leoa ) ABBABCDE $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
letr.1 $e |- a =< b $.
|
|
letr.2 $e |- b =< c $.
|
|
$( Transitive law for l.e. $)
|
|
letr $p |- a =< c $=
|
|
( wa wo df-le2 ax-r5 ax-r1 ax-a3 3tr2 lan anabs ax-r2 df2le1 ) ACACFAABCG
|
|
ZGZFACRAQABGZCGZCRTQSBCABDHIJBCEHABCKLMAQNOP $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
bltr.1 $e |- a = b $.
|
|
bltr.2 $e |- b =< c $.
|
|
$( Transitive inference. $)
|
|
bltr $p |- a =< c $=
|
|
( wo ax-r5 df-le2 ax-r2 df-le1 ) ACACFBCFCABCDGBCEHIJ $.
|
|
$( [28-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
lbtr.1 $e |- a =< b $.
|
|
lbtr.2 $e |- b = c $.
|
|
$( Transitive inference. $)
|
|
lbtr $p |- a =< c $=
|
|
( wa ax-r1 lan df2le2 ax-r2 df2le1 ) ACACFABFACBABCEGHABDIJK $.
|
|
$( [28-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
le3tr1.1 $e |- a =< b $.
|
|
le3tr1.2 $e |- c = a $.
|
|
le3tr1.3 $e |- d = b $.
|
|
$( Transitive inference useful for introducing definitions. $)
|
|
le3tr1 $p |- c =< d $=
|
|
( bltr ax-r1 lbtr ) CBDCABFEHDBGIJ $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
le3tr2.1 $e |- a =< b $.
|
|
le3tr2.2 $e |- a = c $.
|
|
le3tr2.3 $e |- b = d $.
|
|
$( Transitive inference useful for eliminating definitions. $)
|
|
le3tr2 $p |- c =< d $=
|
|
( ax-r1 le3tr1 ) ABCDEACFHBDGHI $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
bile.1 $e |- a = b $.
|
|
$( Biconditional to l.e. $)
|
|
bile $p |- a =< b $=
|
|
( wo ax-r5 oridm ax-r2 df-le1 ) ABABDBBDBABBCEBFGH $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
$( An ortholattice inequality, corresponding to a theorem provable in Hilbert
|
|
space. Part of Definition 2.1 p. 2092, in M. Pavicic and N. Megill,
|
|
"Quantum and Classical Implicational Algebras with Primitive Implication,"
|
|
_Int. J. of Theor. Phys._ 37, 2091-2098 (1998). $)
|
|
qlhoml1a $p |- a =< a ' ' $=
|
|
( wn ax-a1 bile ) AABBACD $.
|
|
$( [3-Feb-02] $)
|
|
|
|
$( An ortholattice inequality, corresponding to a theorem provable in Hilbert
|
|
space. $)
|
|
qlhoml1b $p |- a ' ' =< a $=
|
|
( wn ax-a1 ax-r1 bile ) ABBZAAFACDE $.
|
|
$( [3-Feb-02] $)
|
|
|
|
${
|
|
lebi.1 $e |- a =< b $.
|
|
lebi.2 $e |- b =< a $.
|
|
$( L.e. to biconditional. $)
|
|
lebi $p |- a = b $=
|
|
( wo df-le2 ax-r1 ax-a2 ax-r2 ) AABEZBABAEZJKABADFGBAHIABCFI $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
$( Anything is l.e. 1. $)
|
|
le1 $p |- a =< 1 $=
|
|
( wt or1 df-le1 ) ABACD $.
|
|
$( [30-Aug-97] $)
|
|
|
|
$( 0 is l.e. anything. $)
|
|
le0 $p |- 0 =< a $=
|
|
( wf wo ax-a2 or0 ax-r2 df-le1 ) BABACABCABADAEFG $.
|
|
$( [30-Aug-97] $)
|
|
|
|
$( Identity law for less-than-or-equal. $)
|
|
leid $p |- a =< a $=
|
|
( id bile ) AAABC $.
|
|
$( [24-Dec-98] $)
|
|
|
|
${
|
|
le.1 $e |- a =< b $.
|
|
$( Add disjunct to right of l.e. $)
|
|
ler $p |- a =< ( b v c ) $=
|
|
( wo ax-a3 ax-r1 df-le2 ax-r5 ax-r2 df-le1 ) ABCEZALEZABEZCEZLOMABCFGNBCA
|
|
BDHIJK $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Add disjunct to right of l.e. $)
|
|
lerr $p |- a =< ( c v b ) $=
|
|
( wo ler ax-a2 lbtr ) ABCECBEABCDFBCGH $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( Add conjunct to left of l.e. $)
|
|
lel $p |- ( a ^ c ) =< b $=
|
|
( wa an32 df2le2 ran ax-r2 df2le1 ) ACEZBKBEABEZCEKACBFLACABDGHIJ $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Add disjunct to right of both sides. $)
|
|
leror $p |- ( a v c ) =< ( b v c ) $=
|
|
( wo orordir ax-r1 df-le2 ax-r5 ax-r2 df-le1 ) ACEZBCEZLMEZABEZCEZMPNABCF
|
|
GOBCABDHIJK $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Add conjunct to right of both sides. $)
|
|
leran $p |- ( a ^ c ) =< ( b ^ c ) $=
|
|
( wa anandir ax-r1 df2le2 ran ax-r2 df2le1 ) ACEZBCEZLMEZABEZCEZLPNABCFGO
|
|
ACABDHIJK $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Contrapositive for l.e. $)
|
|
lecon $p |- b ' =< a ' $=
|
|
( wn wa wo ax-a2 oran df-le2 3tr2 con3 df2le1 ) BDZADZMNEZBBAFABFODBBAGBA
|
|
HABCIJKL $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
lecon1.1 $e |- a ' =< b ' $.
|
|
$( Contrapositive for l.e. $)
|
|
lecon1 $p |- b =< a $=
|
|
( wn lecon ax-a1 le3tr1 ) BDZDADZDBAIHCEBFAFG $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
lecon2.1 $e |- a ' =< b $.
|
|
$( Contrapositive for l.e. $)
|
|
lecon2 $p |- b ' =< a $=
|
|
( wn ax-a1 lbtr lecon1 ) ABDZADBHDCBEFG $.
|
|
$( [19-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
lecon3.1 $e |- a =< b ' $.
|
|
$( Contrapositive for l.e. $)
|
|
lecon3 $p |- b =< a ' $=
|
|
( wn lecon lecon2 lecon1 ) ADZBBDZHAICEFG $.
|
|
$( [19-Dec-98] $)
|
|
$}
|
|
|
|
$( L.e. absorption. $)
|
|
leo $p |- a =< ( a v b ) $=
|
|
( wo anabs df2le1 ) AABCABDE $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( L.e. absorption. $)
|
|
leor $p |- a =< ( b v a ) $=
|
|
( wo leo ax-a2 lbtr ) AABCBACABDABEF $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( L.e. absorption. $)
|
|
lea $p |- ( a ^ b ) =< a $=
|
|
( wa wo ax-a2 orabs ax-r2 df-le1 ) ABCZAIADAIDAIAEABFGH $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( L.e. absorption. $)
|
|
lear $p |- ( a ^ b ) =< b $=
|
|
( wa ancom lea bltr ) ABCBACBABDBAEF $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( L.e. absorption. $)
|
|
leao1 $p |- ( a ^ b ) =< ( a v c ) $=
|
|
( wa wo lea leo letr ) ABDAACEABFACGH $.
|
|
$( [8-Jul-00] $)
|
|
|
|
$( L.e. absorption. $)
|
|
leao2 $p |- ( b ^ a ) =< ( a v c ) $=
|
|
( wa wo lear leo letr ) BADAACEBAFACGH $.
|
|
$( [8-Jul-00] $)
|
|
|
|
$( L.e. absorption. $)
|
|
leao3 $p |- ( a ^ b ) =< ( c v a ) $=
|
|
( wa wo lea leor letr ) ABDACAEABFACGH $.
|
|
$( [8-Jul-00] $)
|
|
|
|
$( L.e. absorption. $)
|
|
leao4 $p |- ( b ^ a ) =< ( c v a ) $=
|
|
( wa wo lear leor letr ) BADACAEBAFACGH $.
|
|
$( [8-Jul-00] $)
|
|
|
|
${
|
|
lel.1 $e |- a =< b $.
|
|
$( Add disjunct to left of both sides. $)
|
|
lelor $p |- ( c v a ) =< ( c v b ) $=
|
|
( wo leror ax-a2 le3tr1 ) ACEBCECAECBEABCDFCAGCBGH $.
|
|
$( [25-Oct-97] $)
|
|
|
|
$( Add conjunct to left of both sides. $)
|
|
lelan $p |- ( c ^ a ) =< ( c ^ b ) $=
|
|
( wa leran ancom le3tr1 ) ACEBCECAECBEABCDFCAGCBGH $.
|
|
$( [25-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
le2.1 $e |- a =< b $.
|
|
le2.2 $e |- c =< d $.
|
|
$( Disjunction of 2 l.e.'s. $)
|
|
le2or $p |- ( a v c ) =< ( b v d ) $=
|
|
( wo leror lelor letr ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$( [25-Oct-97] $)
|
|
|
|
$( Conjunction of 2 l.e.'s. $)
|
|
le2an $p |- ( a ^ c ) =< ( b ^ d ) $=
|
|
( wa leran lelan letr ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$( [25-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
lel2.1 $e |- a =< b $.
|
|
lel2.2 $e |- c =< b $.
|
|
$( Disjunction of 2 l.e.'s. $)
|
|
lel2or $p |- ( a v c ) =< b $=
|
|
( wo le2or oridm lbtr ) ACFBBFBABCBDEGBHI $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( Conjunction of 2 l.e.'s. $)
|
|
lel2an $p |- ( a ^ c ) =< b $=
|
|
( wa le2an anidm lbtr ) ACFBBFBABCBDEGBHI $.
|
|
$( [11-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
ler2.1 $e |- a =< b $.
|
|
ler2.2 $e |- a =< c $.
|
|
$( Disjunction of 2 l.e.'s. $)
|
|
ler2or $p |- a =< ( b v c ) $=
|
|
( wo oridm ax-r1 le2or bltr ) AAAFZBCFKAAGHABACDEIJ $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( Conjunction of 2 l.e.'s. $)
|
|
ler2an $p |- a =< ( b ^ c ) $=
|
|
( wa anidm ax-r1 le2an bltr ) AAAFZBCFKAAGHABACDEIJ $.
|
|
$( [11-Nov-97] $)
|
|
$}
|
|
|
|
$( Half of distributive law. $)
|
|
ledi $p |- ( ( a ^ b ) v ( a ^ c ) ) =< ( a ^ ( b v c ) ) $=
|
|
( wa wo anidm ax-r1 lea le2or oridm lbtr ancom bltr le2an ) ABDZACDZEZQQDZA
|
|
BCEZDRQQFGQAQSQAAEAOAPAABHACHIAJKOBPCOBADBABLBAHMPCADCACLCAHMINM $.
|
|
$( [28-Aug-97] $)
|
|
|
|
$( Half of distributive law. $)
|
|
ledir $p |- ( ( b ^ a ) v ( c ^ a ) ) =< ( ( b v c ) ^ a ) $=
|
|
( wa wo ledi ancom 2or le3tr1 ) ABDZACDZEABCEZDBADZCADZELADABCFMJNKBAGCAGHL
|
|
AGI $.
|
|
$( [30-Nov-98] $)
|
|
|
|
$( Half of distributive law. $)
|
|
ledio $p |- ( a v ( b ^ c ) ) =< ( ( a v b ) ^ ( a v c ) ) $=
|
|
( wa wo anidm ax-r1 leo le2an bltr ax-a2 lbtr le2or oridm ) ABCDZEABEZACEZD
|
|
ZRERARORAAADZRSAAFGAPAQABHACHIJBPCQBBAEPBAHBAKLCCAEQCAHCAKLIMRNL $.
|
|
$( [28-Aug-97] $)
|
|
|
|
$( Half of distributive law. $)
|
|
ledior $p |- ( ( b ^ c ) v a ) =< ( ( b v a ) ^ ( c v a ) ) $=
|
|
( wa wo ledio ax-a2 2an le3tr1 ) ABCDZEABEZACEZDJAEBAEZCAEZDABCFJAGMKNLBAGC
|
|
AGHI $.
|
|
$( [30-Nov-98] $)
|
|
|
|
$( Commutation with 0. Kalmbach 83 p. 20. $)
|
|
comm0 $p |- a C 0 $=
|
|
( wf wo wa wn ax-a2 or0 ax-r2 ax-r1 an0 wt df-f con2 lan an1 2or df-c1 ) AB
|
|
ABACZABDZABEZDZCZRARABCABAFAGHIUBRSBUAAAJUAAKDATKABKLMNAOHPIHQ $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Commutation with 1. Kalmbach 83 p. 20. $)
|
|
comm1 $p |- 1 C a $=
|
|
( wt wn wo wa df-t ancom an1 ax-r2 2or ax-r1 df-c1 ) BABAACZDZBAEZBMEZDZAFQ
|
|
NOAPMOABEABAGAHIPMBEMBMGMHIJKIL $.
|
|
$( [27-Aug-97] $)
|
|
|
|
${
|
|
lecom.1 $e |- a =< b $.
|
|
$( Comparable elements commute. Beran 84 2.3(iii) p. 40. $)
|
|
lecom $p |- a C b $=
|
|
( wn wa wo orabs ax-r1 df2le2 ax-r5 ax-r2 df-c1 ) ABAAABDZEZFZABEZNFOAAMG
|
|
HAPNPAABCIHJKL $.
|
|
$( [30-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
bctr.1 $e |- a = b $.
|
|
bctr.2 $e |- b C c $.
|
|
$( Transitive inference. $)
|
|
bctr $p |- a C c $=
|
|
( wa wn wo df-c2 ran 2or 3tr1 df-c1 ) ACBBCFZBCGZFZHAACFZAOFZHBCEIDQNRPAB
|
|
CDJABODJKLM $.
|
|
$( [30-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
cbtr.1 $e |- a C b $.
|
|
cbtr.2 $e |- b = c $.
|
|
$( Transitive inference. $)
|
|
cbtr $p |- a C c $=
|
|
( wa wn wo df-c2 lan ax-r4 2or ax-r2 df-c1 ) ACAABFZABGZFZHACFZACGZFZHABD
|
|
IORQTBCAEJPSABCEKJLMN $.
|
|
$( [30-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
comcom2.1 $e |- a C b $.
|
|
$( Commutation equivalence. Kalmbach 83 p. 23. Does not use OML. $)
|
|
comcom2 $p |- a C b ' $=
|
|
( wn wa wo df-c2 ax-a1 lan ax-r5 ax-r2 ax-a2 df-c1 ) ABDZAANDZEZANEZFZQPF
|
|
AABEZQFRABCGSPQBOABHIJKPQLKM $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
$( Commutation law. Does not use OML. $)
|
|
comorr $p |- a C ( a v b ) $=
|
|
( wo leo lecom ) AABCABDE $.
|
|
$( [30-Aug-97] $)
|
|
|
|
$( Commutation law. Does not use OML. $)
|
|
coman1 $p |- ( a ^ b ) C a $=
|
|
( wa lea lecom ) ABCAABDE $.
|
|
$( [30-Aug-97] $)
|
|
|
|
$( Commutation law. Does not use OML. $)
|
|
coman2 $p |- ( a ^ b ) C b $=
|
|
( wa ancom coman1 bctr ) ABCBACBABDBAEF $.
|
|
$( [9-Nov-97] $)
|
|
|
|
$( Identity law for commutation. Does not use OML. $)
|
|
comid $p |- a C a $=
|
|
( wo comorr oridm cbtr ) AAABAAACADE $.
|
|
$( [9-Nov-97] $)
|
|
|
|
${
|
|
distlem.1 $e |- ( a ^ ( b v c ) ) =< b $.
|
|
$( Distributive law inference (uses OL only). $)
|
|
distlem $p |- ( a ^ ( b v c ) ) = ( ( a ^ b ) v ( a ^ c ) ) $=
|
|
( wo wa lea ler2an leo letr ledi lebi ) ABCEZFZABFZACFZEZNOQNABAMGDHOPIJA
|
|
BCKL $.
|
|
$( [17-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
str.1 $e |- a =< ( b v c ) $.
|
|
str.2 $e |- ( a ^ ( b v c ) ) =< b $.
|
|
$( Strengthening rule. $)
|
|
str $p |- a =< b $=
|
|
( wo wa id bile ler2an letr ) AABCFZGBAALAAAHIDJEK $.
|
|
$( [18-Nov-98] $)
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Commutator (ortholattice theorems)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Commutative law for commutator. $)
|
|
cmtrcom $p |- C ( a , b ) = C ( b , a ) $=
|
|
( wa wn wo wcmtr ancom 2or or4 ax-r2 df-cmtr 3tr1 ) ABCZABDZCZEZADZBCZQNCZE
|
|
ZEZBACZBQCZENACZNQCZEEZABFBAFUAUBUDEZUCUEEZEUFPUGTUHMUBOUDABGANGHRUCSUEQBGQ
|
|
NGHHUBUDUCUEIJABKBAKL $.
|
|
$( [24-Jan-99] $)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Weak "orthomodular law" in ortholattices.
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
|
|
All theorems here do not require R3 and
|
|
are true in all ortholattices.
|
|
$)
|
|
|
|
$( Weak A1. $)
|
|
wa1 $p |- ( a == a ' ' ) = 1 $=
|
|
( wn ax-a1 bi1 ) AABBACD $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( Weak A2. $)
|
|
wa2 $p |- ( ( a v b ) == ( b v a ) ) = 1 $=
|
|
( wo ax-a2 bi1 ) ABCBACABDE $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( Weak A3. $)
|
|
wa3 $p |- ( ( ( a v b ) v c ) == ( a v ( b v c ) ) ) = 1 $=
|
|
( wo ax-a3 bi1 ) ABDCDABCDDABCEF $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( Weak A4. $)
|
|
wa4 $p |- ( ( a v ( b v b ' ) ) == ( b v b ' ) ) = 1 $=
|
|
( wn wo ax-a4 bi1 ) ABBCDZDGABEF $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( Weak A5. $)
|
|
wa5 $p |- ( ( a v ( a ' v b ' ) ' ) == a ) = 1 $=
|
|
( wn wo ax-a5 bi1 ) AACBCZDCDAAGEF $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( Weak A6. $)
|
|
wa6 $p |- ( ( a == b ) == ( ( a ' v b ' ) ' v ( a v b ) ' ) )
|
|
= 1 $=
|
|
( tb wn wo df-b bi1 ) ABCADBDEDABEDEABFG $.
|
|
$( [12-Jul-98] $)
|
|
|
|
${
|
|
wr1.1 $e |- ( a == b ) = 1 $.
|
|
$( Weak R1. $)
|
|
wr1 $p |- ( b == a ) = 1 $=
|
|
( tb wt bicom ax-r2 ) BADABDEBAFCG $.
|
|
$( [2-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wr3.1 $e |- ( 1 == a ) = 1 $.
|
|
$( Weak R3. $)
|
|
wr3 $p |- a = 1 $=
|
|
( wt tb 1b ax-r1 ax-r2 ) ACADZCHAAEFBG $.
|
|
$( [2-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wr4.1 $e |- ( a == b ) = 1 $.
|
|
$( Weak R4. $)
|
|
wr4 $p |- ( a ' == b ' ) = 1 $=
|
|
( wn tb wt conb ax-r1 ax-r2 ) ADBDEZABEZFKJABGHCI $.
|
|
$( [2-Sep-97] $)
|
|
$}
|
|
|
|
$( Absorption law. $)
|
|
wa5b $p |- ( ( a v ( a ^ b ) ) == a ) = 1 $=
|
|
( wa wo orabs bi1 ) AABCDAABEF $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( Absorption law. $)
|
|
wa5c $p |- ( ( a ^ ( a v b ) ) == a ) = 1 $=
|
|
( wo wa anabs bi1 ) AABCDAABEF $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( Contraposition law. $)
|
|
wcon $p |- ( ( a == b ) == ( a ' == b ' ) ) = 1 $=
|
|
( tb wn conb bi1 ) ABCADBDCABEF $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( Commutative law. $)
|
|
wancom $p |- ( ( a ^ b ) == ( b ^ a ) ) = 1 $=
|
|
( wa ancom bi1 ) ABCBACABDE $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( Associative law. $)
|
|
wanass $p |- ( ( ( a ^ b ) ^ c ) == ( a ^ ( b ^ c ) ) ) = 1 $=
|
|
( wa anass bi1 ) ABDCDABCDDABCEF $.
|
|
$( [27-Sep-97] $)
|
|
|
|
${
|
|
wwbmp.1 $e |- a = 1 $.
|
|
wwbmp.2 $e |- ( a == b ) = 1 $.
|
|
$( Weak weak equivalential detachment (WBMP). $)
|
|
wwbmp $p |- b = 1 $=
|
|
( wt tb rbi ax-r1 ax-r2 wr3 ) BEBFZABFZELKAEBCGHDIJ $.
|
|
$( [2-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wwbmpr.1 $e |- b = 1 $.
|
|
wwbmpr.2 $e |- ( a == b ) = 1 $.
|
|
$( Weak weak equivalential detachment (WBMP). $)
|
|
wwbmpr $p |- a = 1 $=
|
|
( wr1 wwbmp ) BACABDEF $.
|
|
$( [24-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wcon1.1 $e |- ( a ' == b ' ) = 1 $.
|
|
$( Weak contraposition. $)
|
|
wcon1 $p |- ( a == b ) = 1 $=
|
|
( tb wn wt conb ax-r2 ) ABDAEBEDFABGCH $.
|
|
$( [24-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wcon2.1 $e |- ( a == b ' ) = 1 $.
|
|
$( Weak contraposition. $)
|
|
wcon2 $p |- ( a ' == b ) = 1 $=
|
|
( wn tb wt conb ax-a1 rbi ax-r1 ax-r2 ) ADZBEZABDZEZFMLDZNEZOLBGOQAPNAHIJ
|
|
KCK $.
|
|
$( [24-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wcon3.1 $e |- ( a ' == b ) = 1 $.
|
|
$( Weak contraposition. $)
|
|
wcon3 $p |- ( a == b ' ) = 1 $=
|
|
( wn tb wt ax-a1 ax-r1 lbi ax-r2 wcon1 ) ABDZADZLDZEMBEFNBMBNBGHICJK $.
|
|
$( [24-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wlem3.1.1 $e |- ( a v b ) = b $.
|
|
wlem3.1.2 $e |- ( b ' v a ) = 1 $.
|
|
$( Weak analogue to lemma used in proof of Th. 3.1 of Pavicic 1993. $)
|
|
wlem3.1 $p |- ( a == b ) = 1 $=
|
|
( tb wn wo wt wa dfb leoa oran ax-r1 ax-r2 con3 2or ax-a2 ) ABEZBFZAGZHRA
|
|
BIZAFSIZGZTABJUCASGTUAAUBSABBCKUBBUBFZABGZBUEUDABLMCNOPASQNNDN $.
|
|
$( [2-Sep-97] $)
|
|
$}
|
|
|
|
$( Theorem structurally similar to orthomodular law but does not require
|
|
R3. $)
|
|
woml $p |- ( ( a v ( a ' ^ ( a v b ) ) ) == ( a v b ) ) = 1 $=
|
|
( wn wo wa omlem1 omlem2 wlem3.1 ) AACABDZEDIABFABGH $.
|
|
$( [2-Sep-97] $)
|
|
|
|
${
|
|
wwoml2.1 $e |- a =< b $.
|
|
$( Weak orthomodular law. $)
|
|
wwoml2 $p |- ( ( a v ( a ' ^ b ) ) == b ) = 1 $=
|
|
( wn wa wo tb wt df-le2 ax-r1 lan lor rbi lbi woml 3tr2 ) AADZBEZFZABFZGA
|
|
QTEZFZTGSBGHSUBTRUAABTQTBABCIZJKLMTBSUCNABOP $.
|
|
$( [2-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wwoml3.1 $e |- a =< b $.
|
|
wwoml3.2 $e |- ( b ^ a ' ) = 0 $.
|
|
$( Weak orthomodular law. $)
|
|
wwoml3 $p |- ( a == b ) = 1 $=
|
|
( wf wo tb wn wa wt ax-r1 ancom ax-r2 lor rbi or0 wwoml2 3tr2 ) AEFZBGAAH
|
|
ZBIZFZBGABGJSUBBEUAAEBTIZUAUCEDKBTLMNOSABAPOABCQR $.
|
|
$( [2-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wwcomd.1 $e |- a ' C b $.
|
|
$( Commutation dual (weak). Kalmbach 83 p. 23. $)
|
|
wwcomd $p |- a = ( ( a v b ) ^ ( a v b ' ) ) $=
|
|
( wo wn wa df-c2 oran ax-a2 anor2 ax-r1 con3 2an ax-r4 3tr1 ax-r2 con1 )
|
|
AABDZABEZDZFZAEZUBBFZUBSFZDZUAEZUBBCGUDUCDUDEZUCEZFZEUEUFUDUCHUCUDIUAUIRU
|
|
GTUHABHTUCUCTEABJKLMNOPQ $.
|
|
$( [2-Sep-97] $)
|
|
|
|
$}
|
|
|
|
${
|
|
wwcom3ii.1 $e |- b ' C a $.
|
|
$( Lemma 3(ii) (weak) of Kalmbach 83 p. 23. $)
|
|
wwcom3ii $p |- ( a ^ ( a ' v b ) ) = ( a ^ b ) $=
|
|
( wa wn wo wwcomd lan anass ax-r1 ax-a2 anabs ax-r2 2an ) ABDZAAEZBFZDZOA
|
|
BAFZBPFZDZDZRBUAABACGHUBASDZTDZRUDUBASTIJUCATQUCAABFZDASUEABAKHABLMBPKNMM
|
|
J $.
|
|
$( [2-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wwfh.1 $e |- b C a $.
|
|
wwfh.2 $e |- c C a $.
|
|
$( Foulis-Holland Theorem (weak). $)
|
|
wwfh1 $p |- ( ( a ^ ( b v c ) ) == ( ( a ^ b ) v ( a ^ c ) ) )
|
|
= 1 $=
|
|
( wo wa tb wn wf df-a ax-r1 con3 ax-r2 2an ax-a1 bctr wwcom3ii anandi lan
|
|
wt bicom ledi ancom 2or con2 anass 3tr1 an12 oran dff an0 wwoml3 ) ABCFZG
|
|
ZABGZACGZFZHURUOHUAUOURUBURUOABCUCUOURIZGZAUNBIZCIZGZGZGZJUTUNAGZAIZVAFZV
|
|
GVBFZGZGZVEUOVFUSVJAUNUDURVJURVHIZVIIZFZVJIUPVLUQVMABKACKUEVNVJVJVNIVHVIK
|
|
LMNUFOVKUNAVCGZGZVEVKUNAVJGZGVPUNAVJUGVQVOUNAVHGZAVIGZGAVAGZAVBGZGVQVOVRV
|
|
TVSWAAVAVAIZBABWBBPLDQRAVBVBIZCACWCCPLEQROAVHVISAVAVBSUHTNUNAVCUINNVEAJGJ
|
|
VDJAVDUNUNIZGZJVCWDUNVCUNUNVCIBCUJLMTJWEUNUKLNTAULNNUMN $.
|
|
$( [3-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wwfh2.1 $e |- a C b $.
|
|
wwfh2.2 $e |- c ' C a $.
|
|
$( Foulis-Holland Theorem (weak). $)
|
|
wwfh2 $p |- ( ( b ^ ( a v c ) ) == ( ( b ^ a ) v ( b ^ c ) ) )
|
|
= 1 $=
|
|
( wo wa tb wt bicom wn wf con2 ran ax-r2 lan an4 ax-r1 wwcom3ii anass dff
|
|
ledi oran df-a ax-r4 ax-a1 bctr ancom ax-r5 comcom2 an12 3tr1 an0 wwoml3
|
|
) BACFZGZBAGZBCGZFZHUSUPHIUPUSJUSUPBACUBUPUSKZGZAKZCBURKZGZGZGZLVAVBCGZVD
|
|
GZVFVAVBUOGZVDGZVHVAVBBGZUOVCGZGZVJVAUPBKVBFZVCGZGZVMUTVOUPUSVOUSUQKZVCGZ
|
|
KVOKUQURUCVRVOVQVNVCUQVNBAUDMNUEOMPVPBVNGZVLGVMBUOVNVCQVSVKVLVSBVBGVKBVBV
|
|
BKZABAVTAUFZRDUGSBVBUHONOOVBBUOVCQOVIVGVDVIVBVTCFZGVGUOWBVBAVTCWAUIPVBCCK
|
|
AEUJSONOVBCVDTOVFVBLGLVELVBBCVCGGZURVCGZVELWDWCBCVCTRCBVCUKURUAULPVBUMOOU
|
|
NO $.
|
|
$( [3-Sep-97] $)
|
|
|
|
$}
|
|
|
|
${
|
|
wwfh3.1 $e |- b ' C a $.
|
|
wwfh3.2 $e |- c ' C a $.
|
|
$( Foulis-Holland Theorem (weak). $)
|
|
wwfh3 $p |- ( ( a v ( b ^ c ) ) == ( ( a v b ) ^ ( a v c ) ) )
|
|
= 1 $=
|
|
( wa wo tb wn wt conb oran df-a con2 lan ax-r4 ax-r2 2or 2bi comcom2
|
|
wwfh1 ) ABCFZGZABGZACGZFZHZAIZBIZCIZGZFZUHUIFZUHUJFZGZHZJUGUCIZUFIZHUPUCU
|
|
FKUQULURUOUCULUCUHUBIZFZIULIAUBLUTULUSUKUHUBUKBCMNOPQNUFUOUFUDIZUEIZGZIUO
|
|
IUDUEMVCUOVAUMVBUNUDUMABLNUEUNACLNRPQNSQUHUIUJUIADTUJAETUAQ $.
|
|
$( [3-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wwfh4.1 $e |- a ' C b $.
|
|
wwfh4.2 $e |- c C a $.
|
|
$( Foulis-Holland Theorem (weak). $)
|
|
wwfh4 $p |- ( ( b v ( a ^ c ) ) == ( ( b v a ) ^ ( b v c ) ) )
|
|
= 1 $=
|
|
( wa wo tb wn wt conb oran df-a con2 lan ax-r4 ax-r2 2or 2bi comcom2 bctr
|
|
ax-a1 ax-r1 wwfh2 ) BACFZGZBAGZBCGZFZHZBIZAIZCIZGZFZUKULFZUKUMFZGZHZJUJUF
|
|
IZUIIZHUSUFUIKUTUOVAURUFUOUFUKUEIZFZIUOIBUELVCUOVBUNUKUEUNACMNOPQNUIURUIU
|
|
GIZUHIZGZIURIUGUHMVFURVDUPVEUQUGUPBALNUHUQBCLNRPQNSQULUKUMULBDTUMIZAVGCAC
|
|
VGCUBUCEUATUDQ $.
|
|
$( [3-Sep-97] $)
|
|
$}
|
|
|
|
$( Weak OM-like absorption law for ortholattices. $)
|
|
womao $p |- ( a ' ^ ( a v ( a ' ^ ( a v b ) ) ) ) =
|
|
( a ' ^ ( a v b ) ) $=
|
|
( wn wo wa lea lear leo lel2or letr ler2an leor lebi ) ACZANABDZEZDZEZPRNON
|
|
QFRQONQGAOPABHNOGIJKPNQNOFPALKM $.
|
|
$( [8-Nov-98] $)
|
|
|
|
$( Weak OM-like absorption law for ortholattices. $)
|
|
womaon $p |- ( a ^ ( a ' v ( a ^ ( a ' v b ) ) ) ) =
|
|
( a ^ ( a ' v b ) ) $=
|
|
( wn wo wa lea lear leo lel2or letr ler2an leor lebi ) AACZANBDZEZDZEZPRAOA
|
|
QFRQOAQGNOPNBHAOGIJKPAQAOFPNLKM $.
|
|
$( [8-Nov-98] $)
|
|
|
|
$( Weak OM-like absorption law for ortholattices. $)
|
|
womaa $p |- ( a ' v ( a ^ ( a ' v ( a ^ b ) ) ) ) =
|
|
( a ' v ( a ^ b ) ) $=
|
|
( wn wa wo leo lear lel2or lea leor ler2an letr lebi ) ACZANABDZEZDZEZPNPQN
|
|
OFAPGHNRONQFOQROAPABIONJKQNJLHM $.
|
|
$( [8-Nov-98] $)
|
|
|
|
$( Weak OM-like absorption law for ortholattices. $)
|
|
womaan $p |- ( a v ( a ' ^ ( a v ( a ' ^ b ) ) ) ) =
|
|
( a v ( a ' ^ b ) ) $=
|
|
( wn wa wo leo lear lel2or lea leor ler2an letr lebi ) AACZANBDZEZDZEZPAPQA
|
|
OFNPGHAROAQFOQRONPNBIOAJKQAJLHM $.
|
|
$( [8-Nov-98] $)
|
|
|
|
$( Absorption law for ortholattices. $)
|
|
anorabs2 $p |- ( a ^ ( b v ( a ^ ( b v c ) ) ) ) =
|
|
( a ^ ( b v c ) ) $=
|
|
( wo wa lea lear leo lel2or letr ler2an leor lebi ) ABABCDZEZDZEZOQANAPFQPN
|
|
APGBNOBCHANGIJKOAPANFOBLKM $.
|
|
$( [13-Nov-98] $)
|
|
|
|
$( Absorption law for ortholattices. $)
|
|
anorabs $p |- ( a ' ^ ( b v ( a ' ^ ( a v b ) ) ) ) =
|
|
( a ' ^ ( a v b ) ) $=
|
|
( wn wo wa anorabs2 ax-a2 lan lor 3tr1 ) ACZBKBADZEZDZEMKBKABDZEZDZEPKBAFQN
|
|
KPMBOLKABGHZIHRJ $.
|
|
$( [8-Nov-98] $)
|
|
|
|
$( Axiom KA2a in Pavicic and Megill, 1998 $)
|
|
ska2a $p |- ( ( ( a v c ) == ( b v c ) ) ==
|
|
( ( c v a ) == ( c v b ) ) ) = 1 $=
|
|
( wo tb ax-a2 2bi bi1 ) ACDZBCDZECADZCBDZEIKJLACFBCFGH $.
|
|
$( [9-Nov-98] $)
|
|
|
|
$( Axiom KA2b in Pavicic and Megill, 1998 $)
|
|
ska2b $p |- ( ( ( a v c ) == ( b v c ) ) ==
|
|
( ( a ' ^ c ' ) ' == ( b ' ^ c ' ) ' ) ) = 1 $=
|
|
( wo tb wn wa oran 2bi bi1 ) ACDZBCDZEAFCFZGFZBFMGFZEKNLOACHBCHIJ $.
|
|
$( [9-Nov-98] $)
|
|
|
|
$( Lemma for KA4 soundness (OR version) - uses OL only. $)
|
|
ka4lemo $p |- ( ( a v b ) v ( ( a v c ) == ( b v c ) ) ) = 1 $=
|
|
( wo tb wt le1 wn df-t wa ax-a2 lbtr lelor leror oran con2 ax-r1 ax-r2 bltr
|
|
2an leo ax-a3 ledio le3tr1 dfb anor1 anandir ax-r5 ax-r4 3tr1 lor letr lebi
|
|
) ABDZACDZBCDZEZDZFURGFUNCDZUSHZDZURUSIVAUNABJZCDZDZUTDZURUSVDUTCVCUNCCVBDZ
|
|
VCCVBUACVBKLMNVEUNVCUTDZDURUNVCUTUBVGUQUNVGUOUPJZUTDZUQVCVHUTVFCADZCBDZJVCV
|
|
HCABUCVBCKUOVJUPVKACKBCKTUDNUQVIUQVHUOHZUPHZJZDVIUOUPUEVNUTVHVNAHZCHZJZBHZV
|
|
PJZJZUTVLVQVMVSUOVQACOPUPVSBCOPTVOVRJZVPJZWAHZCDZHVTUTWACUFWBVTVOVRVPUGQUSW
|
|
DUNWCCABOUHUIUJRUKRQLMSULSUM $.
|
|
$( [25-Oct-97] $)
|
|
|
|
$( Lemma for KA4 soundness (AND version) - uses OL only. $)
|
|
ka4lem $p |- ( ( a ^ b ) ' v ( ( a ^ c ) == ( b ^ c ) ) ) = 1 $=
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( wa wn tb wo wt df-a con2 2bi conb ax-r1 ax-r2 2or ka4lemo ) ABDZEZACDZBCD
|
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ZFZGAEZBEZGZUBCEZGZUCUEGZFZGHRUDUAUHQUDABIJUAUFEZUGEZFZUHSUITUJACIBCIKUHUKU
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FUGLMNOUBUCUEPN $.
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$( [25-Oct-97] $)
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$(
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
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Kalmbach axioms (soundness proofs) that are true in all ortholattices
|
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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$)
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${
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sklem.1 $e |- a =< b $.
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$( Soundness lemma. $)
|
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sklem $p |- ( a ' v b ) = 1 $=
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( wn wo wt or12 df-t ax-r5 ax-r1 ax-a3 ax-a2 3tr2 ax-r2 df-le2 lor or1 )
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ADZABEZEZBFEZRBEZFTAUBEZUARABGAREZBEZFBEZUCUAUFUEFUDBAHIJARBKFBLMNSBRABCO
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PBQM $.
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$( [30-Aug-97] $)
|
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$}
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$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA1. $)
|
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ska1 $p |- ( a == a ) = 1 $=
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( biid ) AB $.
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$( [30-Aug-97] $)
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$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA3. $)
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ska3 $p |- ( ( a == b ) ' v ( a ' == b ' ) ) = 1 $=
|
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( wn tb wo wt conb ax-r4 lor ax-a2 df-t 3tr1 ) ACBCDZABDZCZEMMCZEOMEFOPMNMA
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BGHIOMJMKL $.
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$( [30-Aug-97] $)
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$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA5. $)
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ska5 $p |- ( ( a ^ b ) == ( b ^ a ) ) = 1 $=
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( wa ancom bi1 ) ABCBACABDE $.
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$( [30-Aug-97] $)
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$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA6. $)
|
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ska6 $p |- ( ( a ^ ( b ^ c ) ) == ( ( a ^ b ) ^ c ) ) = 1 $=
|
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( wa anass ax-r1 bi1 ) ABCDDZABDCDZIHABCEFG $.
|
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$( [30-Aug-97] $)
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$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA7. $)
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ska7 $p |- ( ( a ^ ( a v b ) ) == a ) = 1 $=
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( wo wa anabs bi1 ) AABCDAABEF $.
|
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$( [30-Aug-97] $)
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$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA8. $)
|
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ska8 $p |- ( ( a ' ^ a ) == ( ( a ' ^ a ) ^ b ) ) = 1 $=
|
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( wn wa wf an0 ax-r1 ancom ax-r2 dff ran 3tr2 bi1 ) ACZADZOBDZEEBDZOPEBEDZQ
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REBFGBEHIEANDOAJANHIZEOBSKLM $.
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$( [30-Aug-97] $)
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|
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$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA9. $)
|
|
ska9 $p |- ( a == a ' ' ) = 1 $=
|
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( wn ax-a1 bi1 ) AABBACD $.
|
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$( [30-Aug-97] $)
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|
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$( Soundness theorem for Kalmbach's quantum propositional logic axiom
|
|
KA10. $)
|
|
ska10 $p |- ( ( a v b ) ' == ( a ' ^ b ' ) ) = 1 $=
|
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( wo wn wa oran con2 bi1 ) ABCZDADBDEZIJABFGH $.
|
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$( [30-Aug-97] $)
|
|
|
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$( Soundness theorem for Kalmbach's quantum propositional logic axiom
|
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KA11. $)
|
|
ska11 $p |- ( ( a v ( a ' ^ ( a v b ) ) ) == ( a v b ) ) = 1 $=
|
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( woml ) ABC $.
|
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$( [2-Sep-97] $)
|
|
$( [30-Aug-97] $)
|
|
|
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$( Soundness theorem for Kalmbach's quantum propositional logic axiom
|
|
KA12. $)
|
|
ska12 $p |- ( ( a == b ) == ( b == a ) ) = 1 $=
|
|
( tb bicom bi1 ) ABCBACABDE $.
|
|
$( [30-Aug-97] $)
|
|
|
|
$( Soundness theorem for Kalmbach's quantum propositional logic axiom
|
|
KA13. $)
|
|
ska13 $p |- ( ( a == b ) ' v ( a ' v b ) ) = 1 $=
|
|
( tb wn wo wa ledio lea letr ancom bltr leror dfb ax-a2 le3tr1 sklem ) ABCZ
|
|
ADZBEZABFZRBDZFEZBREZQSUBTREZUCUBUDTUAEZFUDTRUAGUDUEHITBRTBAFBABJBAHKLIABMR
|
|
BNOP $.
|
|
$( [30-Aug-97] $)
|
|
|
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${
|
|
skr0.1 $e |- a = 1 $.
|
|
skr0.2 $e |- ( a ' v b ) = 1 $.
|
|
$( Soundness theorem for Kalmbach's quantum propositional logic axiom
|
|
KR0. $)
|
|
skr0 $p |- b = 1 $=
|
|
( wn wo wt wf ax-a2 or0 ax-r1 ax-r4 df-f ax-r2 ax-r5 3tr1 ) BAEZBFZGBHFZH
|
|
BFBRBHISBBJKQHBQGEZHAGCLHTMKNOPDN $.
|
|
$( [30-Aug-97] $)
|
|
$}
|
|
|
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$( Lemma for 2-variable WOML proof. $)
|
|
wlem1 $p |- ( ( a == b ) ' v ( ( a ->1 b ) ^ ( b ->1 a ) ) ) = 1 $=
|
|
( tb wn wi1 wa wo wt le1 df-t ax-a2 ax-r2 dfb ledio df-i1 ancom ax-r5 ax-r1
|
|
2an bltr lbtr lelor lebi ) ABCZDZABEZBAEZFZGZHUIIHUEUDGZUIHUDUEGUJUDJUDUEKL
|
|
UDUHUEUDABFZADZBDZFGZUHABMUNUKULGZUKUMGZFZUHUKULUMNUHUQUFUOUGUPUFULUKGUOABO
|
|
ULUKKLUGUMBAFZGZUPBAOUSURUMGUPUMURKURUKUMBAPQLLSRUATUBTUC $.
|
|
$( [11-Nov-98] $)
|
|
|
|
$( Soundness theorem for Kalmbach's quantum propositional logic axiom
|
|
KA15. $)
|
|
ska15 $p |- ( ( a ->3 b ) ' v ( a ' v b ) ) = 1 $=
|
|
( wi3 wn wo wa df-i3 ax-a2 lea lear le2or bltr oridm lbtr sklem ) ABCZADZBE
|
|
ZPQBFZQBDZFZEZARFZEZRABGUDRRERUBRUCRUBUASERSUAHUAQSBQTIQBJKLARJKRMNLO $.
|
|
$( [2-Nov-97] $)
|
|
|
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${
|
|
skmp3.1 $e |- a = 1 $.
|
|
skmp3.2 $e |- ( a ->3 b ) = 1 $.
|
|
$( Soundness proof for KMP3. $)
|
|
skmp3 $p |- b = 1 $=
|
|
( wi3 wn wo ska15 skr0 ) ABCABEAFBGDABHII $.
|
|
$( [2-Nov-97] $)
|
|
$}
|
|
|
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${
|
|
lei3.1 $e |- a =< b $.
|
|
$( L.e. to Kalmbach implication. $)
|
|
lei3 $p |- ( a ->3 b ) = 1 $=
|
|
( wn wa wo wi3 wt ax-a3 ax-a2 ancom lecon df2le2 ax-r2 sklem lan an1 3tr1
|
|
2or anor2 con2 lor df-i3 df-t ) ADZBEZUEBDZEZFAUEBFZEZFZUFUFDZFZABGHUKUFU
|
|
HUJFZFUMUFUHUJIUNULUFUGAFAUGFZUNULUGAJUHUGUJAUHUGUEEUGUEUGKUGUEABCLMNUJAH
|
|
EAUIHAABCOPAQNSUFUOABTUARUBNABUCUFUDR $.
|
|
$( [3-Nov-97] $)
|
|
$}
|
|
|
|
$( E2 - OL theorem proved by EQP $)
|
|
mccune2 $p |- ( a v ( ( a ' ^ ( ( a v b ' ) ^ ( a v b ) ) ) v (
|
|
a ' ^ ( ( a ' ^ b ) v ( a ' ^ b ' ) ) ) ) ) = 1 $=
|
|
( wn wo wa wt ax-a3 ax-r1 anor2 lear lel2or id bile ler2an lebi anor3 oran3
|
|
lea 2or ax-r2 ax-a2 lor df-t 3tr1 ) AABCZDZABDZEZCZAUIDZCZDZDZUJUKDZAACZUHE
|
|
ZUOUOBEZUOUEEZDZEZDZDFUNUMAUIUKGHVAULAVAUKUIDULUPUKUTUIAUHIUTUSUIUTUSUOUSJU
|
|
SUOUSUQUOURUOBRUOUERKUSUSUSLMNOUSUFCZUGCZDUIUQVBURVCABIABPSUFUGQTTSUKUIUATU
|
|
BUJUCUD $.
|
|
$( [14-Nov-98] $)
|
|
|
|
$( E3 - OL theorem proved by EQP $)
|
|
mccune3 $p |- ( ( ( ( a ' ^ b ) v ( a ' ^ b ' ) ) v ( a ^ ( a '
|
|
v b ) ) ) ' v ( a ' v b ) ) = 1 $=
|
|
( wn wa wo wi3 wt df-i3 ax-r1 ax-r4 ax-r5 ska15 ax-r2 ) ACZBDNBCDEANBEZDEZC
|
|
ZOEABFZCZOEGQSOPRRPABHIJKABLM $.
|
|
$( [14-Nov-98] $)
|
|
|
|
$( Equivalence for Kalmbach implication. $)
|
|
i3n1 $p |- ( a ' ->3 b ' ) = ( ( ( a ^ b ' ) v ( a ^ b ) ) v
|
|
( a ' ^ ( a v b ' ) ) ) $=
|
|
( wn wi3 wa wo df-i3 ax-a1 ran 2an 2or ax-r5 lan ax-r1 ax-r2 ) ACZBCZDPCZQE
|
|
ZRQCZEZFZPRQFZEZFZAQEZABEZFZPAQFZEZFZPQGUKUEUHUBUJUDUFSUGUAARQAHZIARBTULBHJ
|
|
KUIUCPARQULLMKNO $.
|
|
$( [9-Nov-97] $)
|
|
|
|
$( Equivalence for Kalmbach implication. $)
|
|
ni31 $p |- ( a ->3 b ) ' = ( ( ( a v b ' ) ^ ( a v b ) ) ^
|
|
( a ' v ( a ^ b ' ) ) ) $=
|
|
( wi3 wn wo wa df-i3 oran anor2 con2 ax-r1 2an ax-r4 ax-r2 df-a anor1 lor )
|
|
ABCZABDZEZABEZFZADZASFZEZFZRUCBFZUCSFZEZAUCBEZFZEZUFDZABGULUIDZUKDZFZDUMUIU
|
|
KHUPUFUNUBUOUEUIUBUIUGDZUHDZFZDUBDUGUHHUSUBUQTURUAUGTABIJUAURABHKLMNJUKUEUK
|
|
UCUJDZEZDUEDAUJOVAUEUTUDUCUDUTABPKQMNJLMNNJ $.
|
|
$( [9-Nov-97] $)
|
|
|
|
$( Identity for Kalmbach implication. $)
|
|
i3id $p |- ( a ->3 a ) = 1 $=
|
|
( wn wa wo wi3 wt wf ancom dff ax-r1 ax-r2 anidm 2or ax-a2 or0 df-t lan an1
|
|
df-i3 3tr1 ) ABZACZUAUACZDZAUAADZCZDZAUADZAAEFUGUEUHUDUAUFAUDUAGDZUAUDGUADU
|
|
IUBGUCUAUBAUACZGUAAHGUJAIJKUALMGUANKUAOKUFAFCAUEFAUEUHFUAANZFUHAPZJKQARKMUK
|
|
KAASULT $.
|
|
$( [2-Nov-97] $)
|
|
|
|
${
|
|
li3.1 $e |- a = b $.
|
|
$( Introduce Kalmbach implication to the left. $)
|
|
li3 $p |- ( c ->3 a ) = ( c ->3 b ) $=
|
|
( wn wa wo wi3 lan ax-r4 2or lor df-i3 3tr1 ) CEZAFZOAEZFZGZCOAGZFZGOBFZO
|
|
BEZFZGZCOBGZFZGCAHCBHSUEUAUGPUBRUDABODIQUCOABDJIKTUFCABODLIKCAMCBMN $.
|
|
$( [2-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
ri3.1 $e |- a = b $.
|
|
$( Introduce Kalmbach implication to the right. $)
|
|
ri3 $p |- ( a ->3 c ) = ( b ->3 c ) $=
|
|
( wn wa wo wi3 ax-r4 ran 2or ax-r5 2an df-i3 3tr1 ) AEZCFZPCEZFZGZAPCGZFZ
|
|
GBEZCFZUCRFZGZBUCCGZFZGACHBCHTUFUBUHQUDSUEPUCCABDIZJPUCRUIJKABUAUGDPUCCUI
|
|
LMKACNBCNO $.
|
|
$( [2-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
2i3.1 $e |- a = b $.
|
|
2i3.2 $e |- c = d $.
|
|
$( Join both sides with Kalmbach implication. $)
|
|
2i3 $p |- ( a ->3 c ) = ( b ->3 d ) $=
|
|
( wi3 li3 ri3 ax-r2 ) ACGADGBDGCDAFHABDEIJ $.
|
|
$( [2-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
ud1lem0a.1 $e |- a = b $.
|
|
$( Introduce ` ->1 ` to the left. $)
|
|
ud1lem0a $p |- ( c ->1 a ) = ( c ->1 b ) $=
|
|
( wn wa wo wi1 lan lor df-i1 3tr1 ) CEZCAFZGMCBFZGCAHCBHNOMABCDIJCAKCBKL
|
|
$.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Introduce ` ->1 ` to the right. $)
|
|
ud1lem0b $p |- ( a ->1 c ) = ( b ->1 c ) $=
|
|
( wn wa wo wi1 ax-r4 ran 2or df-i1 3tr1 ) AEZACFZGBEZBCFZGACHBCHNPOQABDIA
|
|
BCDJKACLBCLM $.
|
|
$( [23-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
ud1lem0ab.1 $e |- a = b $.
|
|
ud1lem0ab.2 $e |- c = d $.
|
|
$( Join both sides of hypotheses with ` ->1 ` . $)
|
|
ud1lem0ab $p |- ( a ->1 c ) = ( b ->1 d ) $=
|
|
( wi1 ud1lem0b ud1lem0a ax-r2 ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$( [19-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
ud2lem0a.1 $e |- a = b $.
|
|
$( Introduce ` ->2 ` to the left. $)
|
|
ud2lem0a $p |- ( c ->2 a ) = ( c ->2 b ) $=
|
|
( wn wa wo wi2 ax-r4 lan 2or df-i2 3tr1 ) ACEZAEZFZGBNBEZFZGCAHCBHABPRDOQ
|
|
NABDIJKCALCBLM $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Introduce ` ->2 ` to the right. $)
|
|
ud2lem0b $p |- ( a ->2 c ) = ( b ->2 c ) $=
|
|
( wn wa wo wi2 ax-r4 ran lor df-i2 3tr1 ) CAEZCEZFZGCBEZOFZGACHBCHPRCNQOA
|
|
BDIJKACLBCLM $.
|
|
$( [23-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
ud3lem0a.1 $e |- a = b $.
|
|
$( Introduce Kalmbach implication to the left. $)
|
|
ud3lem0a $p |- ( c ->3 a ) = ( c ->3 b ) $=
|
|
( li3 ) ABCDE $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Introduce Kalmbach implication to the right. $)
|
|
ud3lem0b $p |- ( a ->3 c ) = ( b ->3 c ) $=
|
|
( ri3 ) ABCDE $.
|
|
$( [23-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
ud4lem0a.1 $e |- a = b $.
|
|
$( Introduce ` ->4 ` to the left. $)
|
|
ud4lem0a $p |- ( c ->4 a ) = ( c ->4 b ) $=
|
|
( wa wn wo wi4 lan 2or lor ax-r4 2an df-i4 3tr1 ) CAEZCFZAEZGZQAGZAFZEZGC
|
|
BEZQBEZGZQBGZBFZEZGCAHCBHSUEUBUHPUCRUDABCDIABQDIJTUFUAUGABQDKABDLMJCANCBN
|
|
O $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Introduce ` ->4 ` to the right. $)
|
|
ud4lem0b $p |- ( a ->4 c ) = ( b ->4 c ) $=
|
|
( wa wn wo wi4 ran ax-r4 2or ax-r5 df-i4 3tr1 ) ACEZAFZCEZGZPCGZCFZEZGBCE
|
|
ZBFZCEZGZUCCGZTEZGACHBCHRUEUAUGOUBQUDABCDIPUCCABDJZIKSUFTPUCCUHLIKACMBCMN
|
|
$.
|
|
$( [23-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
ud5lem0a.1 $e |- a = b $.
|
|
$( Introduce ` ->5 ` to the left. $)
|
|
ud5lem0a $p |- ( c ->5 a ) = ( c ->5 b ) $=
|
|
( wa wn wo wi5 lan 2or ax-r4 df-i5 3tr1 ) CAEZCFZAEZGZOAFZEZGCBEZOBEZGZOB
|
|
FZEZGCAHCBHQUBSUDNTPUAABCDIABODIJRUCOABDKIJCALCBLM $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Introduce ` ->5 ` to the right. $)
|
|
ud5lem0b $p |- ( a ->5 c ) = ( b ->5 c ) $=
|
|
( wa wn wo wi5 ran ax-r4 2or df-i5 3tr1 ) ACEZAFZCEZGZOCFZEZGBCEZBFZCEZGZ
|
|
UAREZGACHBCHQUCSUDNTPUBABCDIOUACABDJZIKOUARUEIKACLBCLM $.
|
|
$( [23-Nov-97] $)
|
|
$}
|
|
|
|
$( Correspondence between Sasaki and Dishkant conditionals. $)
|
|
i1i2 $p |- ( a ->1 b ) = ( b ' ->2 a ' ) $=
|
|
( wn wa wo wi1 wi2 ax-a1 2an ancom ax-r2 lor df-i1 df-i2 3tr1 ) ACZABDZEPBC
|
|
ZCZPCZDZEABFRPGQUAPQTSDUAATBSAHBHITSJKLABMRPNO $.
|
|
$( [25-Nov-98] $)
|
|
|
|
$( Correspondence between Sasaki and Dishkant conditionals. $)
|
|
i2i1 $p |- ( a ->2 b ) = ( b ' ->1 a ' ) $=
|
|
( wn wi2 wi1 ax-a1 ud2lem0b ud2lem0a i1i2 3tr1 ) ABCZCZDACZCZLDABDKMEANLAFG
|
|
BLABFHKMIJ $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Correspondence between Sasaki and Dishkant conditionals. $)
|
|
i1i2con1 $p |- ( a ->1 b ' ) = ( b ->2 a ' ) $=
|
|
( wn wi1 wi2 i1i2 ax-a1 ax-r1 ud2lem0b ax-r2 ) ABCZDKCZACZEBMEAKFLBMBLBGHIJ
|
|
$.
|
|
$( [28-Feb-99] $)
|
|
|
|
$( Correspondence between Sasaki and Dishkant conditionals. $)
|
|
i1i2con2 $p |- ( a ' ->1 b ) = ( b ' ->2 a ) $=
|
|
( wn wi1 wi2 i1i2 ax-a1 ax-r1 ud2lem0a ax-r2 ) ACZBDBCZKCZELAEKBFMALAMAGHIJ
|
|
$.
|
|
$( [28-Feb-99] $)
|
|
|
|
$( Correspondence between Kalmbach and non-tollens conditionals. $)
|
|
i3i4 $p |- ( a ->3 b ) = ( b ' ->4 a ' ) $=
|
|
( wn wa wi3 wi4 ax-a2 ancom ax-a1 ran ax-r2 2or ax-r5 2an df-i3 df-i4 3tr1
|
|
wo ) ACZBDZSBCZDZRZASBRZDZRUASDZUACZSDZRZUGSRZSCZDZRABEUASFUCUIUEULUCUBTRUI
|
|
TUBGUBUFTUHSUAHTBSDUHSBHBUGSBIZJKLKUEUDADULAUDHUDUJAUKUDBSRUJSBGBUGSUMMKAIN
|
|
KLABOUASPQ $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Correspondence between Kalmbach and non-tollens conditionals. $)
|
|
i4i3 $p |- ( a ->4 b ) = ( b ' ->3 a ' ) $=
|
|
( wi4 wn wi3 ax-a1 ud4lem0a ud4lem0b ax-r2 i3i4 ax-r1 ) ABCZADZDZBDZDZCZOME
|
|
ZLAPCQBPABFGANPAFHIRQOMJKI $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Converse of ` ->5 ` . $)
|
|
i5con $p |- ( a ->5 b ) = ( b ' ->5 a ' ) $=
|
|
( wa wn wo wi5 ancom ax-a2 ax-a1 ran ax-r2 2an 2or ax-a3 3tr1 df-i5 ) ABCZA
|
|
DZBCZEZRBDZCZEZUARCZUADZRCZEUERDZCZEZABFUARFUBTEUDUFUHEZEUCUIUBUDTUJRUAGTSQ
|
|
EUJQSHSUFQUHSBRCUFRBGBUERBIZJKQBACUHABGBUEAUGUKAILKMKMTUBHUDUFUHNOABPUARPO
|
|
$.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Antecedent of 0 on Sasaki conditional. $)
|
|
0i1 $p |- ( 0 ->1 a ) = 1 $=
|
|
( wf wi1 wn wa wo wt df-i1 ax-a2 df-f con2 lor ax-r2 or1 3tr ) BACBDZBAEZFZ
|
|
QGFZGBAHRQPFSPQIPGQBGJKLMQNO $.
|
|
$( [24-Dec-98] $)
|
|
|
|
$( Antecedent of 1 on Sasaki conditional. $)
|
|
1i1 $p |- ( 1 ->1 a ) = a $=
|
|
( wt wi1 wn wa wo df-i1 wf df-f ax-r1 ancom an1 ax-r2 2or ax-a2 or0 ) BACBD
|
|
ZBAEZFZABAGSHAFZAQHRAHQIJRABEABAKALMNTAHFAHAOAPMMM $.
|
|
$( [24-Dec-98] $)
|
|
|
|
$( Identity law for Sasaki conditional. $)
|
|
i1id $p |- ( a ->1 a ) = 1 $=
|
|
( wi1 wn wa wo wt df-i1 ax-a2 anidm lor df-t 3tr1 ax-r2 ) AABACZAADZEZFAAGN
|
|
AEANEPFNAHOANAIJAKLM $.
|
|
$( [25-Dec-98] $)
|
|
|
|
$( Identity law for Dishkant conditional. $)
|
|
i2id $p |- ( a ->2 a ) = 1 $=
|
|
( wi2 wn wa wo wt df-i2 anidm lor df-t ax-r1 ax-r2 ) AABAACZMDZEZFAAGOAMEZF
|
|
NMAMHIFPAJKLL $.
|
|
$( [26-Jun-03] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud1lem0c $p |- ( a ->1 b ) ' = ( a ^ ( a ' v b ' ) ) $=
|
|
( wi1 wn wo wa df-i1 df-a ax-r1 lor ax-r4 ax-r2 con3 con2 ) ABCZAADZBDEZFZO
|
|
PABFZEZRDABGTRRTDZRPQDZEZDUAAQHUCTUBSPSUBABHIJKLIMLN $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud2lem0c $p |- ( a ->2 b ) ' = ( b ' ^ ( a v b ) ) $=
|
|
( wi2 wn wo wa df-i2 oran ax-r1 lan ax-r4 ax-r2 con2 ) ABCZBDZABEZFZNBADOFZ
|
|
EZQDZABGSORDZFZDTBRHUBQUAPOPUAABHIJKLLM $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem0c $p |- ( a ->3 b ) ' = ( ( ( a v b ' ) ^ ( a v b ) ) ^
|
|
( a ' v ( a ^ b ' ) ) ) $=
|
|
( ni31 ) ABC $.
|
|
$( [22-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud4lem0c $p |- ( a ->4 b ) ' = ( ( ( a ' v b ' ) ^ ( a v b ' ) ) ^
|
|
( ( a ^ b ' ) v b ) ) $=
|
|
( wi4 wn wo wa df-i4 oran df-a con2 anor2 2an ax-r4 ax-r2 anor1 ax-r1 ax-r5
|
|
) ABCZADZBDZEZATEZFZATFZBEZFZRABFZSBFZEZSBEZTFZEZUFDZABGULUIDZUKDZFZDUMUIUK
|
|
HUPUFUNUCUOUEUIUCUIUGDZUHDZFZDUCDUGUHHUSUCUQUAURUBUGUAABIJUHUBABKJLMNJUKUEU
|
|
KUJDZBEZDUEDUJBOVAUEUTUDBUDUTABOPQMNJLMNNJ $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud5lem0c $p |- ( a ->5 b ) ' = ( ( ( a ' v b ' ) ^ ( a v b ' ) ) ^
|
|
( a v b ) ) $=
|
|
( wi5 wn wo wa df-i5 oran df-a con2 anor2 2an ax-r4 ax-r2 ax-r1 ) ABCZADZBD
|
|
ZEZAREZFZABEZFZPABFZQBFZEZQRFZEZUCDZABGUHUFDZUGDZFZDUIUFUGHULUCUJUAUKUBUFUA
|
|
UFUDDZUEDZFZDUADUDUEHUOUAUMSUNTUDSABIJUETABKJLMNJUBUKABHOLMNNJ $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Pavicic binary logic ax-a1 analog. $)
|
|
bina1 $p |- ( a ->3 a ' ' ) = 1 $=
|
|
( wi3 wn i3id ax-a1 li3 bi1 wwbmp ) AABZAACCZBZADIKAJAAEFGH $.
|
|
$( [5-Nov-97] $)
|
|
|
|
$( Pavicic binary logic ax-a2 analog. $)
|
|
bina2 $p |- ( a ' ' ->3 a ) = 1 $=
|
|
( wi3 wn i3id ax-a1 ri3 bi1 wwbmp ) AABZACCZABZADIKAJAAEFGH $.
|
|
$( [5-Nov-97] $)
|
|
|
|
$( Pavicic binary logic ax-a3 analog. $)
|
|
bina3 $p |- ( a ->3 ( a v b ) ) = 1 $=
|
|
( wo leo lei3 ) AABCABDE $.
|
|
$( [5-Nov-97] $)
|
|
|
|
$( Pavicic binary logic ax-a4 analog. $)
|
|
bina4 $p |- ( b ->3 ( a v b ) ) = 1 $=
|
|
( wo leo ax-a2 lbtr lei3 ) BABCZBBACHBADBAEFG $.
|
|
$( [5-Nov-97] $)
|
|
|
|
$( Pavicic binary logic ax-a5 analog. $)
|
|
bina5 $p |- ( b ->3 ( a v a ' ) ) = 1 $=
|
|
( wn wo wt le1 df-t lbtr lei3 ) BAACDZBEJBFAGHI $.
|
|
$( [5-Nov-97] $)
|
|
|
|
${
|
|
wql1lem.1 $e |- ( a ->1 b ) = 1 $.
|
|
$( Classical implication inferred from Sakaki implication. $)
|
|
wql1lem $p |- ( a ' v b ) = 1 $=
|
|
( wn wo wt le1 wi1 ax-r1 wa df-i1 lear lelor bltr lebi ) ADZBEZFQGFABHZQR
|
|
FCIRPABJZEQABKSBPABLMNNO $.
|
|
$( [5-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
wql2lem.1 $e |- ( a ->2 b ) = 1 $.
|
|
$( Classical implication inferred from Dishkant implication. $)
|
|
wql2lem $p |- ( a ' v b ) = 1 $=
|
|
( wn wo wt le1 wa wi2 df-i2 ax-a2 3tr2 lea leror bltr lebi ) ADZBEZFRGFQB
|
|
DZHZBEZRABIBTEFUAABJCBTKLTQBQSMNOP $.
|
|
$( [6-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
wql2lem2.1 $e |- ( ( a v c ) ->2 ( b v c ) ) = 1 $.
|
|
$( Lemma for ` ->2 ` WQL axiom. $)
|
|
wql2lem2 $p |- ( ( a v ( b v c ) ) ' v ( b v c ) ) = 1 $=
|
|
( wo wn wi2 wt df-i2 anor3 ax-a3 ax-r1 orordir ax-r2 ax-r4 lor ax-a2 3tr
|
|
wa ) ABCEZEZFZTEZACEZTGZHUEUCUETUDFTFSZETUBEUCUDTIUFUBTUFUDTEZFZUBUDTJUBU
|
|
HUAUGUAABECEZUGUIUAABCKLABCMNOLNPTUBQRLDN $.
|
|
$( [6-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
wql2lem3.1 $e |- ( a ->2 b ) = 1 $.
|
|
$( Lemma for ` ->2 ` WQL axiom. $)
|
|
wql2lem3 $p |- ( ( a ^ b ' ) ->2 a ' ) = 1 $=
|
|
( wn wa wi2 wo wt df-i2 oran2 ax-r1 ran ancom lor wql2lem omlem2 skr0 3tr
|
|
ax-r2 ) ABDEZADZFUATDZUADZEZGUAUCUABGZEZGZHTUAIUDUFUAUDUEUCEUFUBUEUCUEUBA
|
|
BJKLUEUCMSNUEUGABCOUABPQR $.
|
|
$( [6-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
wql2lem4.1 $e |- ( ( ( a ^ b ' ) v ( a ^ b ) ) ->2
|
|
( a ' v ( a ^ b ) ) ) = 1 $.
|
|
wql2lem4.2 $e |- ( ( a ->1 b ) v ( a ^ b ' ) ) = 1 $.
|
|
$( Lemma for ` ->2 ` WQL axiom. $)
|
|
wql2lem4 $p |- ( a ->1 b ) = 1 $=
|
|
( wi1 wn wa wo wt df-i1 id ax-a2 ax-r5 ax-r1 3tr wql2lem2 skr0 ) ABEZAFZA
|
|
BGZHZUAIABJZUAKABFGZUAHZUAUDUAUCHZRUCHZIUCUALUFUERUAUCUBMNDOUCSTCPQO $.
|
|
$( [6-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
wql2lem5.1 $e |- ( a ->2 b ) = 1 $.
|
|
$( Lemma for ` ->2 ` WQL axiom. $)
|
|
wql2lem5 $p |- ( ( b ' ^ ( a v b ) ) ->2 a ' ) = 1 $=
|
|
( wn wo wa wi2 wt anor3 oran3 ud2lem0c ax-r5 ran ancom an1 3tr ax-r4 3tr2
|
|
ax-r2 lor df-i2 df-t 3tr1 ) ADZBDABEFZDUDDZFZEUDUFEUEUDGHUGUFUDUGUEUDEZDU
|
|
FUEUDIUHUDABGZDZUDEUIAFZDUHUDUIAJUJUEUDABKLUKAUKHAFAHFAUIHACMHANAOPQRQSTU
|
|
EUDUAUDUBUC $.
|
|
$( [6-Dec-98] $)
|
|
$}
|
|
|
|
|
|
${
|
|
wql1.1 $e |- ( a ->1 b ) = 1 $.
|
|
wql1.2 $e |- ( ( a v c ) ->1 ( b v c ) ) = 1 $.
|
|
wql1.3 $e |- c = b $.
|
|
$( The 2nd hypothesis is the first ` ->1 ` WQL axiom. We show it implies
|
|
the WOM law. $)
|
|
wql1 $p |- ( a ->2 b ) = 1 $=
|
|
( wi2 wn wa wo wt df-i2 anor3 lor ax-a2 wi1 oridm ax-r2 ud1lem0a ax-r1
|
|
ud1lem0b 3tr2 wql1lem 3tr ) ABGBAHBHIZJBABJZHZJZKABLUEUGBABMNUHUGBJKBUGOU
|
|
FBACJZBPZUIBCJZPZUFBPKULUJUKBUIUKBBJBCBBFNBQRSTUIUFBCBAFNUAEUBUCRUD $.
|
|
$( [5-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oaidlem1.1 $e |- ( a ^ b ) =< c $.
|
|
$( Lemma for OA identity-like law. $)
|
|
oaidlem1 $p |- ( a ' v ( b ->1 c ) ) = 1 $=
|
|
( wn wi1 wo wa df-i1 lor oran3 ax-r5 ax-a3 lear ler2an sklem 3tr2 ax-r2
|
|
wt ) AEZBCFZGTBEZBCHZGZGZSUAUDTBCIJTUBGZUCGABHZEZUCGUESUFUHUCABKLTUBUCMUG
|
|
UCUGBCABNDOPQR $.
|
|
$( [22-Jan-99] $)
|
|
$}
|
|
|
|
|
|
${
|
|
womle2a.1 $e |- ( a ^ ( a ->2 b ) ) =<
|
|
( ( a ->2 b ) ' v ( a ->1 b ) ) $.
|
|
$( An equivalent to the WOM law. $)
|
|
womle2a $p |- ( ( a ->2 b ) ' v ( a ->1 b ) ) = 1 $=
|
|
( wi2 wn wi1 wo wa wt or4 oridm df-i1 ax-r5 or32 3tr1 ax-r2 2or ax-a2 lor
|
|
oran3 3tr2 le1 df-t leror bltr lebi ) ABDZEZABFZGZUJAUGHZEZGZIUHUHGZUIAEZ
|
|
GZGUJUHUOGZGUJUMUHUHUIUOJUNUHUPUIUHKUPUOABHZGZUOGZUIUIUSUOABLZMUOUOGZURGU
|
|
SUTUIVBUOURUOKMUOURUONVAOPQUQULUJUQUOUHGULUHUORAUGTPSUAUMIUMUBIUKULGUMUKU
|
|
CUKUJULCUDUEUFP $.
|
|
$( [24-Jan-99] $)
|
|
$}
|
|
|
|
${
|
|
womle2b.1 $e |- ( ( a ->2 b ) ' v ( a ->1 b ) ) = 1 $.
|
|
$( An equivalent to the WOM law. $)
|
|
womle2b $p |- ( a ^ ( a ->2 b ) ) =<
|
|
( ( a ->2 b ) ' v ( a ->1 b ) ) $=
|
|
( wi2 wa wt wn wi1 wo le1 ax-r1 lbtr ) AABDZEZFMGABHIZNJOFCKL $.
|
|
$( [24-Jan-99] $)
|
|
$}
|
|
|
|
${
|
|
womle3b.1 $e |- ( ( a ->1 b ) ' v ( a ->2 b ) ) = 1 $.
|
|
$( Implied by the WOM law. $)
|
|
womle3b $p |- ( a ^ ( a ->1 b ) ) =<
|
|
( ( a ->1 b ) ' v ( a ->2 b ) ) $=
|
|
( wi1 wa wt wn wi2 wo le1 ax-r1 lbtr ) AABDZEZFMGABHIZNJOFCKL $.
|
|
$( [27-Jan-99] $)
|
|
$}
|
|
|
|
${
|
|
womle.1 $e |- ( a ^ ( a ->1 b ) ) = ( a ^ ( a ->2 b ) ) $.
|
|
$( An equality implying the WOM law. $)
|
|
womle $p |- ( ( a ->2 b ) ' v ( a ->1 b ) ) = 1 $=
|
|
( wi2 wa wi1 wn wo ax-r1 lear bltr leor letr womle2a ) ABAABDZEZABFZOGZQH
|
|
PAQEZQSPCIAQJKQRLMN $.
|
|
$( [24-Jan-99] $)
|
|
$}
|
|
|
|
$( Lemma for "Non-Orthomodular Models..." paper. $)
|
|
nomb41 $p |- ( a ==4 b ) = ( b ==1 a ) $=
|
|
( wn wo wa wid4 wid1 ax-a2 ancom lor 2an df-id4 df-id1 3tr1 ) ACZBDZBCZABEZ
|
|
DZEBODZQBAEZDZEABFBAGPTSUBOBHRUAQABIJKABLBAMN $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Lemma for "Non-Orthomodular Models..." paper. $)
|
|
nomb32 $p |- ( a ==3 b ) = ( b ==2 a ) $=
|
|
( wn wo wa wid3 wid2 ax-a2 ancom lor 2an df-id3 df-id2 3tr1 ) ACZBDZAOBCZEZ
|
|
DZEBODZAQOEZDZEABFBAGPTSUBOBHRUAAOQIJKABLBAMN $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Lemma for "Non-Orthomodular Models..." paper. $)
|
|
nomcon0 $p |- ( a ==0 b ) = ( b ' ==0 a ' ) $=
|
|
( wn wo wa wid0 ax-a2 ax-a1 ax-r5 ax-r2 2an df-id0 3tr1 ) ACZBDZBCZADZEPCZN
|
|
DZNCZPDZEABFPNFOSQUAOBNDSNBGBRNBHIJQAPDUAPAGATPAHIJKABLPNLM $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Lemma for "Non-Orthomodular Models..." paper. $)
|
|
nomcon1 $p |- ( a ==1 b ) = ( b ' ==2 a ' ) $=
|
|
( wn wo wa wid1 wid2 ax-a2 ax-a1 lor ax-r2 ancom 2an df-id1 df-id2 3tr1 ) A
|
|
BCZDZACZABEZDZEQSCZDZSQCZUBEZDZEABFQSGRUCUAUFRQADUCAQHAUBQAIZJKTUESTBAEUEAB
|
|
LBUDAUBBIUGMKJMABNQSOP $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Lemma for "Non-Orthomodular Models..." paper. $)
|
|
nomcon2 $p |- ( a ==2 b ) = ( b ' ==1 a ' ) $=
|
|
( wn wo wa wid2 wid1 ax-a2 ax-a1 lor ax-r2 ancom 2or 2an df-id2 df-id1 3tr1
|
|
) ABCZDZBACZREZDZERTCZDZRCZRTEZDZEABFRTGSUDUBUGSRADUDARHAUCRAIJKBUEUAUFBITR
|
|
LMNABORTPQ $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Lemma for "Non-Orthomodular Models..." paper. $)
|
|
nomcon3 $p |- ( a ==3 b ) = ( b ' ==4 a ' ) $=
|
|
( wid2 wn wid1 wid3 wid4 nomcon2 nomb32 nomb41 3tr1 ) BACADZBDZEABFMLGBAHAB
|
|
IMLJK $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Lemma for "Non-Orthomodular Models..." paper. $)
|
|
nomcon4 $p |- ( a ==4 b ) = ( b ' ==3 a ' ) $=
|
|
( wid1 wn wid2 wid4 wid3 nomcon1 nomb41 nomb32 3tr1 ) BACADZBDZEABFMLGBAHAB
|
|
IMLJK $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Lemma for "Non-Orthomodular Models..." paper. $)
|
|
nomcon5 $p |- ( a == b ) = ( b ' == a ' ) $=
|
|
( tb wn bicom conb ax-r2 ) ABCBACBDADCABEBAFG $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom10 $p |- ( a ->0 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wn wa wo wi0 wi1 id df-i0 df-i1 3tr1 ) ACABDZEZMALFABGMHALIABJK $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom11 $p |- ( a ->1 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wn wa wo wi1 anass ax-r1 anidm ran ax-r2 lor df-i1 3tr1 ) ACZAABDZDZEOPEA
|
|
PFABFQPOQAADZBDZPSQAABGHRABAIJKLAPMABMN $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom12 $p |- ( a ->2 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wa wn wo wi2 wi1 oran ax-r1 orabs ax-r2 con3 lor ax-a2 df-i2 df-i1 3tr1 )
|
|
ABCZADZRDCZEZSREZARFABGUARSEUBTSRTATDZAREZAUDUCARHIABJKLMRSNKAROABPQ $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom13 $p |- ( a ->3 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wn wa wo wi3 wi1 oran ax-r1 orabs ax-r2 con3 lor df-le2 ax-r5 womaa df-i3
|
|
lea df-i1 3tr1 ) ACZABDZDZUAUBCDZEZAUAUBEZDZEZUFAUBFABGUHUAUGEUFUEUAUGUEUCU
|
|
AEUAUDUAUCUDAUDCZAUBEZAUJUIAUBHIABJKLMUCUAUAUBRNKOABPKAUBQABST $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom14 $p |- ( a ->4 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wa wn wi4 wi1 ax-a2 anass ax-r1 anidm ran ax-r2 lor lear df-le2 3tr ax-r5
|
|
wo leo lea lbtr lel2or lecon ler2an lelor lebi df-i4 df-i1 3tr1 ) AABCZCZAD
|
|
ZUJCZRZULUJRZUJDZCZRZUOAUJEABFURUJUQRZUJULRZUOUNUJUQUNUMUKRUMUJRUJUKUMGUKUJ
|
|
UMUKAACZBCZUJVBUKAABHIVAABAJKLMUMUJULUJNOPQUSUTUJUTUQUJULSUQUOUTUOUPTULUJGU
|
|
AUBULUQUJULUOUPULUJSUJAABTUCUDUEUFUJULGPAUJUGABUHUI $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom15 $p |- ( a ->5 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wa wn wo wi5 wi1 anass ax-r1 anidm ran ax-r2 ax-r5 ax-a2 df-le2 3tr oran3
|
|
lear lan anabs 2or df-i5 df-i1 3tr1 ) AABCZCZADZUECZEZUGUEDZCZEZUGUEEZAUEFA
|
|
BGULUEUGEUMUIUEUKUGUIUEUHEUHUEEUEUFUEUHUFAACZBCZUEUOUFAABHIUNABAJKLMUEUHNUH
|
|
UEUGUEROPUKUGUGBDZEZCZUGURUKUQUJUGABQSIUGUPTLUAUEUGNLAUEUBABUCUD $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom20 $p |- ( a ==0 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wn wa wo wid0 wi1 lea leor letr lelor ax-a3 ax-r1 oran3 ax-r5 lbtr df2le2
|
|
ax-r2 df-id0 df-i1 3tr1 ) ACZABDZEZUCCZAEZDUDAUCFABGUDUFUDUBBCZAEZEZUFUCUHU
|
|
BUCAUHABHAUGIJKUIUBUGEZAEZUFUKUIUBUGALMUJUEAABNORPQAUCSABTUA $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom21 $p |- ( a ==1 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wa wn wo wid1 wi1 ancom oran3 lor ax-r2 anidm ran ax-r1 anass 2an lea leo
|
|
or12 letr lelor df2le2 3tr2 df-id1 df-i1 3tr1 ) AABCZDZEZADZAUGCZEZCZUJUGEZ
|
|
AUGFABGUJABDZEZEZUNCUNUQCUMUNUQUNHUQUIUNULUQAUJUOEZEUIUJAUOSURUHAABIJKUGUKU
|
|
JUGAACZBCZUKUTUGUSABALMNAABOKJPUNUQUGUPUJUGAUPABQAUORTUAUBUCAUGUDABUEUF $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom22 $p |- ( a ==2 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wa wn wid2 wi1 oran3 lor ax-r1 or12 ax-r2 ax-a2 lan anabs ax-r5 2an ancom
|
|
wo lea leo letr lelor df2le2 3tr df-id2 df-i1 3tr1 ) AABCZDZRZUHADZUICZRZCZ
|
|
UKUHRZAUHEABFUNUKABDZRZRZUOCUOURCUOUJURUMUOUJAUKUPRZRZURUTUJUSUIAABGZHIAUKU
|
|
PJKUMULUHRUOUHULLULUKUHULUKUSCZUKVBULUSUIUKVAMIUKUPNKOKPURUOQUOURUHUQUKUHAU
|
|
QABSAUPTUAUBUCUDAUHUEABUFUG $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom23 $p |- ( a ==3 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wn wa wo wid3 wi1 wt le1 df-t anabs ax-r1 oran3 lan ax-r2 lor lbtr df2le2
|
|
df-id3 df-i1 3tr1 ) ACZABDZEZAUBUCCZDZEZDUDAUCFABGUDUGUDHUGUDIHAUBEUGAJUBUF
|
|
AUBUBUBBCZEZDZUFUJUBUBUHKLUIUEUBABMNOPOQRAUCSABTUA $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom24 $p |- ( a ==4 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wn wa wo wid4 wi1 leo leror oran3 anidm ran ax-r1 anass ax-r2 lbtr df2le2
|
|
2or df-id4 df-i1 3tr1 ) ACZABDZEZUCCZAUCDZEZDUDAUCFABGUDUGUDUBBCZEZUCEUGUBU
|
|
IUCUBUHHIUIUEUCUFABJUCAADZBDZUFUKUCUJABAKLMAABNORPQAUCSABTUA $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom25 $p |- ( a == ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( wa wn wo tb wi1 anass ax-r1 anidm ran ax-r2 oran3 lan anabs 2or ax-a2 dfb
|
|
df-i1 3tr1 ) AABCZCZADZUADZCZEZUCUAEZAUAFABGUFUAUCEUGUBUAUEUCUBAACZBCZUAUIU
|
|
BAABHIUHABAJKLUEUCUCBDZEZCZUCULUEUKUDUCABMNIUCUJOLPUAUCQLAUARABST $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom30 $p |- ( ( a ^ b ) ==0 a ) = ( a ->1 b ) $=
|
|
( wa wid0 wi1 wn wo ancom df-id0 3tr1 nom20 ax-r2 ) ABCZADZAMDZABEMFAGZAFMG
|
|
ZCQPCNOPQHMAIAMIJABKL $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom31 $p |- ( ( a ^ b ) ==1 a ) = ( a ->1 b ) $=
|
|
( wa wid1 wid4 wi1 nomb41 ax-r1 nom24 ax-r2 ) ABCZADZAKEZABFMLAKGHABIJ $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom32 $p |- ( ( a ^ b ) ==2 a ) = ( a ->1 b ) $=
|
|
( wa wid2 wid3 wi1 nomb32 ax-r1 nom23 ax-r2 ) ABCZADZAKEZABFMLAKGHABIJ $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom33 $p |- ( ( a ^ b ) ==3 a ) = ( a ->1 b ) $=
|
|
( wa wid3 wid2 wi1 nomb32 nom22 ax-r2 ) ABCZADAJEABFJAGABHI $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom34 $p |- ( ( a ^ b ) ==4 a ) = ( a ->1 b ) $=
|
|
( wa wid4 wid1 wi1 nomb41 nom21 ax-r2 ) ABCZADAJEABFJAGABHI $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. $)
|
|
nom35 $p |- ( ( a ^ b ) == a ) = ( a ->1 b ) $=
|
|
( wa tb wi1 bicom nom25 ax-r2 ) ABCZADAIDABEIAFABGH $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom40 $p |- ( ( a v b ) ->0 b ) = ( a ->2 b ) $=
|
|
( wn wa wi0 wi1 wo wi2 nom10 ax-a2 ax-a1 ancom anor3 ax-r2 ax-r1 df-i0 3tr1
|
|
2or i2i1 ) BCZTACZDZEZTUAFABGZBEZABHTUAIUDCZBGZTCZUBGZUEUCUGBUFGUIUFBJBUHUF
|
|
UBBKUBUFUBUATDUFTUALABMNORNUDBPTUBPQABSQ $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom41 $p |- ( ( a v b ) ->1 b ) = ( a ->2 b ) $=
|
|
( wn wo wi2 wi1 wa ancom anor3 ax-r2 ud2lem0a ax-r1 nom12 i1i2 i2i1 3tr1 )
|
|
BCZABDZCZEZQACZFZRBFABETQQUAGZEZUBUDTUCSQUCUAQGSQUAHABIJKLQUAMJRBNABOP $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom42 $p |- ( ( a v b ) ->2 b ) = ( a ->2 b ) $=
|
|
( wn wo wi1 wi2 wa ancom anor3 ax-r2 ud1lem0a ax-r1 nom11 i2i1 3tr1 ) BCZAB
|
|
DZCZEZPACZEZQBFABFSPPTGZEZUAUCSUBRPUBTPGRPTHABIJKLPTMJQBNABNO $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom43 $p |- ( ( a v b ) ->3 b ) = ( a ->2 b ) $=
|
|
( wn wo wi4 wi1 wi3 wi2 wa ancom anor3 ax-r2 ud4lem0a ax-r1 nom14 i3i4 i2i1
|
|
3tr1 ) BCZABDZCZEZSACZFZTBGABHUBSSUCIZEZUDUFUBUEUASUEUCSIUASUCJABKLMNSUCOLT
|
|
BPABQR $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom44 $p |- ( ( a v b ) ->4 b ) = ( a ->2 b ) $=
|
|
( wn wo wi3 wi1 wi4 wi2 wa ancom anor3 ax-r2 ud3lem0a ax-r1 nom13 i4i3 i2i1
|
|
3tr1 ) BCZABDZCZEZSACZFZTBGABHUBSSUCIZEZUDUFUBUEUASUEUCSIUASUCJABKLMNSUCOLT
|
|
BPABQR $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom45 $p |- ( ( a v b ) ->5 b ) = ( a ->2 b ) $=
|
|
( wn wo wi5 wi1 wi2 ancom anor3 ax-r2 ud5lem0a ax-r1 nom15 i5con i2i1 3tr1
|
|
wa ) BCZABDZCZEZRACZFZSBEABGUARRUBQZEZUCUEUAUDTRUDUBRQTRUBHABIJKLRUBMJSBNAB
|
|
OP $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom50 $p |- ( ( a v b ) ==0 b ) = ( a ->2 b ) $=
|
|
( wn wo wid0 wi1 wi2 wa ancom anor3 ax-r2 lor ax-r4 ax-r5 ax-r1 df-id0 3tr1
|
|
2an nom20 nomcon0 i2i1 ) BCZABDZCZEZUBACZFZUCBEABGUEUBUBUFHZEZUGUBCZUDDZUDC
|
|
ZUBDZHZUJUHDZUHCZUBDZHZUEUIURUNUOUKUQUMUHUDUJUHUFUBHUDUBUFIABJKZLUPULUBUHUD
|
|
USMNROUBUDPUBUHPQUBUFSKUCBTABUAQ $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom51 $p |- ( ( a v b ) ==1 b ) = ( a ->2 b ) $=
|
|
( wn wo wid2 wi1 wid1 wi2 wa ancom anor3 ax-r2 ax-r1 lor lan 2or 2an df-id2
|
|
ax-r4 3tr1 nom22 nomcon1 i2i1 ) BCZABDZCZEZUDACZFZUEBGABHUGUDUDUHIZEZUIUDUF
|
|
CZDZUFUDCZULIZDZIUDUJCZDZUJUNUQIZDZIUGUKUMURUPUTULUQUDUFUJUJUFUJUHUDIZUFUDU
|
|
HJABKZLMZSNUFUJUOUSVCULUQUNUFUJUFVAUJVAUFVBMUHUDJLSOPQUDUFRUDUJRTUDUHUALUEB
|
|
UBABUCT $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom52 $p |- ( ( a v b ) ==2 b ) = ( a ->2 b ) $=
|
|
( wn wo wid1 wi1 wid2 wi2 wa ancom anor3 ax-r2 ax-r1 ax-r4 lor lan 2an 3tr1
|
|
df-id1 nom21 nomcon2 i2i1 ) BCZABDZCZEZUCACZFZUDBGABHUFUCUCUGIZEZUHUCUECZDZ
|
|
UCCZUCUEIZDZIUCUICZDZUMUCUIIZDZIUFUJULUQUOUSUKUPUCUEUIUIUEUIUGUCIUEUCUGJABK
|
|
LMZNOUNURUMUEUIUCUTPOQUCUESUCUISRUCUGTLUDBUAABUBR $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom53 $p |- ( ( a v b ) ==3 b ) = ( a ->2 b ) $=
|
|
( wn wo wid4 wi1 wid3 wi2 wa ancom anor3 ax-r2 ax-r1 lor lan 2or 2an df-id4
|
|
ax-r4 3tr1 nom24 nomcon3 i2i1 ) BCZABDZCZEZUDACZFZUEBGABHUGUDUDUHIZEZUIUDCZ
|
|
UFDZUFCZUDUFIZDZIULUJDZUJCZUDUJIZDZIUGUKUMUQUPUTUFUJULUJUFUJUHUDIUFUDUHJABK
|
|
LMZNUNURUOUSUFUJVASUFUJUDVAOPQUDUFRUDUJRTUDUHUALUEBUBABUCT $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom54 $p |- ( ( a v b ) ==4 b ) = ( a ->2 b ) $=
|
|
( wn wo wid3 wi1 wid4 wi2 wa ancom anor3 ax-r2 lor ax-r4 lan 2an 3tr1 ax-r1
|
|
df-id3 nom23 nomcon4 i2i1 ) BCZABDZCZEZUCACZFZUDBGABHUFUCUCUGIZEZUHUJUFUCCZ
|
|
UIDZUCUKUICZIZDZIUKUEDZUCUKUECZIZDZIUJUFULUPUOUSUIUEUKUIUGUCIUEUCUGJABKLZMU
|
|
NURUCUMUQUKUIUEUTNOMPUCUISUCUESQRUCUGTLUDBUAABUBQ $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom55 $p |- ( ( a v b ) == b ) = ( a ->2 b ) $=
|
|
( wn wa tb wi1 wo wi2 nom25 conb bicom ancom anor3 ax-r2 ax-r1 lbi 3tr i2i1
|
|
3tr1 ) BCZTACZDZEZTUAFABGZBEZABHTUAIUEUDCZTETUFEUCUDBJUFTKUFUBTUBUFUBUATDUF
|
|
TUALABMNOPQABRS $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom60 $p |- ( b ==0 ( a v b ) ) = ( a ->2 b ) $=
|
|
( wo wid0 wi2 wn wa ancom df-id0 3tr1 nom50 ax-r2 ) BABCZDZMBDZABEBFMCZMFBC
|
|
ZGQPGNOPQHBMIMBIJABKL $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom61 $p |- ( b ==1 ( a v b ) ) = ( a ->2 b ) $=
|
|
( wo wid1 wid4 wi2 nomb41 ax-r1 nom54 ax-r2 ) BABCZDZKBEZABFMLKBGHABIJ $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom62 $p |- ( b ==2 ( a v b ) ) = ( a ->2 b ) $=
|
|
( wo wid2 wid3 wi2 nomb32 ax-r1 nom53 ax-r2 ) BABCZDZKBEZABFMLKBGHABIJ $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom63 $p |- ( b ==3 ( a v b ) ) = ( a ->2 b ) $=
|
|
( wo wid3 wid2 wi2 nomb32 nom52 ax-r2 ) BABCZDJBEABFBJGABHI $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom64 $p |- ( b ==4 ( a v b ) ) = ( a ->2 b ) $=
|
|
( wo wid4 wid1 wi2 nomb41 nom51 ax-r2 ) BABCZDJBEABFBJGABHI $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. $)
|
|
nom65 $p |- ( b == ( a v b ) ) = ( a ->2 b ) $=
|
|
( wo tb wi2 bicom nom55 ax-r2 ) BABCZDIBDABEBIFABGH $.
|
|
$( [7-Feb-99] $)
|
|
|
|
$( Lemma for proof of Mayet 8-variable "full" equation from 4-variable
|
|
Godowski equation. $)
|
|
go1 $p |- ( ( a ^ b ) ^ ( a ->1 b ' ) ) = 0 $=
|
|
( wa wn wi1 wo wf df-i1 lan lear lelor lelan oran3 dff ax-r1 ax-r2 lbtr le0
|
|
lebi ) ABCZABDZEZCTADZAUACZFZCZGUBUETAUAHIUFGUFTUCUAFZCZGUEUGTUDUAUCAUAJKLU
|
|
HTTDZCZGUGUITABMIGUJTNOPQUFRSP $.
|
|
$( [19-Nov-99] $)
|
|
|
|
$( Lemma for disjunction of ` ->2 ` . $)
|
|
i2or $p |- ( ( a ->2 c ) v ( b ->2 c ) ) =< ( ( a ^ b ) ->2 c ) $=
|
|
( wi2 wo wa wn df-i2 lea lecon leran lelor bltr lear lel2or ax-r1 lbtr ) AC
|
|
DZBCDZECABFZGZCGZFZEZTCDZRUDSRCAGZUBFZEUDACHUGUCCUFUAUBTAABIJKLMSCBGZUBFZEU
|
|
DBCHUIUCCUHUAUBTBABNJKLMOUEUDTCHPQ $.
|
|
$( [5-Jul-00] $)
|
|
|
|
$( Lemma for disjunction of ` ->1 ` . $)
|
|
i1or $p |- ( ( c ->1 a ) v ( c ->1 b ) ) =< ( c ->1 ( a v b ) ) $=
|
|
( wi1 wo wn wa df-i1 leo lelan lelor bltr leor lel2or ax-r1 lbtr ) CADZCBDZ
|
|
ECFZCABEZGZEZCTDZQUBRQSCAGZEUBCAHUDUASATCABIJKLRSCBGZEUBCBHUEUASBTCBAMJKLNU
|
|
CUBCTHOP $.
|
|
$( [5-Jul-00] $)
|
|
|
|
$( "Less than" analogue is equal to ` ->2 ` implication. $)
|
|
lei2 $p |- ( a =<2 b ) = ( a ->2 b ) $=
|
|
( wo tb wn wa wle2 wi2 mi df-le df-i2 3tr1 ) ABCBDBAEBEFCABGABHABIABJABKL
|
|
$.
|
|
$( [28-Jan-02] $)
|
|
|
|
$( Relevance implication is l.e. Sasaki implication. $)
|
|
i5lei1 $p |- ( a ->5 b ) =< ( a ->1 b ) $=
|
|
( wa wn wi5 wi1 ax-a3 ax-a2 ax-r2 lea lel2or leror bltr df-i5 df-i1 le3tr1
|
|
wo ) ABCZADZBCZQSBDZCZQZSRQZABEABFUCTUBQZRQZUDUCRUEQUFRTUBGRUEHIUESRTSUBSBJ
|
|
SUAJKLMABNABOP $.
|
|
$( [26-Jun-03] $)
|
|
|
|
$( Relevance implication is l.e. Dishkant implication. $)
|
|
i5lei2 $p |- ( a ->5 b ) =< ( a ->2 b ) $=
|
|
( wa wn wo wi5 wi2 lear lel2or leror df-i5 df-i2 le3tr1 ) ABCZADZBCZEZOBDCZ
|
|
EBREABFABGQBRNBPABHOBHIJABKABLM $.
|
|
$( [26-Jun-03] $)
|
|
|
|
$( Relevance implication is l.e. Kalmbach implication. $)
|
|
i5lei3 $p |- ( a ->5 b ) =< ( a ->3 b ) $=
|
|
( wa wn wo wi5 wi3 leor lelan leror df-i5 ax-a3 ax-r2 df-i3 ax-a2 le3tr1 )
|
|
ABCZADZBCZRBDCZEZEZARBEZCZUAEZABFZABGZQUDUABUCABRHIJUFQSETEUBABKQSTLMUGUAUD
|
|
EUEABNUAUDOMP $.
|
|
$( [26-Jun-03] $)
|
|
|
|
$( Relevance implication is l.e. non-tollens implication. $)
|
|
i5lei4 $p |- ( a ->5 b ) =< ( a ->4 b ) $=
|
|
( wa wn wo wi5 wi4 leo leran lelor df-i5 df-i4 le3tr1 ) ABCADZBCEZNBDZCZEON
|
|
BEZPCZEABFABGQSONRPNBHIJABKABLM $.
|
|
$( [26-Jun-03] $)
|
|
|
|
$( Quantum identity is less than classical identity. $)
|
|
id5leid0 $p |- ( a == b ) =< ( a ==0 b ) $=
|
|
( wa wn wo tb wid0 ax-a2 lea lear le2or ler2an bltr dfb df-id0 le3tr1 ) ABC
|
|
ZADZBDZCZEZRBEZSAEZCZABFABGUATQEZUDQTHUEUBUCTRQBRSIABJKTSQARSJABIKLMABNABOP
|
|
$.
|
|
$( [4-Mar-06] $)
|
|
|
|
${
|
|
id5id0.1 $e |- ( a == b ) = 1 $.
|
|
$( Show that classical identity follows from quantum identity in OL. $)
|
|
id5id0 $p |- ( a ==0 b ) = 1 $=
|
|
( tb wid0 id5leid0 sklem skr0 ) ABDZABEZCIJABFGH $.
|
|
$( [4-Mar-06] $)
|
|
$}
|
|
|
|
${
|
|
k1-6.1 $e |- x = ( ( x ^ c ) v ( x ^ c ' ) ) $.
|
|
$( Statement (6) in proof of Theorem 1 of Kalmbach, _Orthomodular
|
|
Lattices_, p. 21. $)
|
|
k1-6 $p |- ( x ' ^ c ) = ( ( x ' v c ' ) ^ c ) $=
|
|
( wn wa wo anor3 cm con4 oran3 oran2 2an 3tr1 ran anass ancom ax-a2 anabs
|
|
lan 3tr ) BDZAEUAADZFZUAAFZEZAEUCUDAEZEUCAEUAUEABAEZBUBEZFZDZUGDZUHDZEZUA
|
|
UEUMUJUGUHGHBUICIUCUKUDULBAJBAKLMNUCUDAOUFAUCUFAUDEAAUAFZEAUDAPUDUNAUAAQS
|
|
AUARTST $.
|
|
$( [26-May-2008] $)
|
|
$}
|
|
|
|
${
|
|
k1-7.1 $e |- x = ( ( x ^ c ) v ( x ^ c ' ) ) $.
|
|
$( Statement (7) in proof of Theorem 1 of Kalmbach, _Orthomodular
|
|
Lattices_, p. 21. $)
|
|
k1-7 $p |- ( x ' ^ c ' ) = ( ( x ' v c ) ^ c ' ) $=
|
|
( wn wa wo anor3 cm ax-a1 lan ror orcom 3tr con4 oran3 oran2 2an 3tr1 ran
|
|
lor anass tr ancom ax-a2 anabs ) BDZADZEUFUGDZFZUFUGFZEZUGEZUFAFZUJUGEZEZ
|
|
UMUGEUFUKUGBUGEZBUHEZFZDZUPDZUQDZEZUFUKVBUSUPUQGHBURBBAEZUPFUQUPFURCVCUQU
|
|
PAUHBAIZJKUQUPLMNUIUTUJVABUGOBUGPQRSULUMUJEZUGEZUOVFULVEUKUGUMUIUJAUHUFVD
|
|
TSSHUMUJUGUAUBUNUGUMUNUGUJEUGUGUFFZEUGUJUGUCUJVGUGUFUGUDJUGUFUEMJM $.
|
|
$( [26-May-2008] $)
|
|
$}
|
|
|
|
${
|
|
k1-8a.1 $e |- x ' = ( ( x ' ^ c ) v ( x ' ^ c ' ) ) $.
|
|
k1-8a.2 $e |- x =< c $.
|
|
k1-8a.3 $e |- y =< c ' $.
|
|
$( First part of statement (8) in proof of Theorem 1 of Kalmbach,
|
|
_Orthomodular Lattices_, p. 21. $)
|
|
k1-8a $p |- x = ( ( x v y ) ^ c ) $=
|
|
( wo wa leo ler2an wn lelor leran ax-a1 ror ran k1-6 tr cm df2le2 lbtr
|
|
3tr lebi ) BBCGZAHZBUDABCIEJUEBAKZGZAHZBUDUGACUFBFLMUHBKZKZUFGZAHZBAHZBUG
|
|
UKABUJUFBNZOPUMULUMUJAHULBUJAUNPAUIDQRSBAETUBUAUC $.
|
|
$( [27-May-2008] $)
|
|
$}
|
|
|
|
${
|
|
k1-8b.1 $e |- y ' = ( ( y ' ^ c ) v ( y ' ^ c ' ) ) $.
|
|
k1-8b.2 $e |- x =< c $.
|
|
k1-8b.3 $e |- y =< c ' $.
|
|
$( Second part of statement (8) in proof of Theorem 1 of Kalmbach,
|
|
_Orthomodular Lattices_, p. 21. $)
|
|
k1-8b $p |- y = ( ( x v y ) ^ c ' ) $=
|
|
( wo wn wa ax-a1 lan ror orcom 3tr lbtr k1-8a ran tr ) CCBGZAHZIBCGZTITCB
|
|
CHZUBAIZUBTIZGUBTHZIZUDGUDUFGDUCUFUDAUEUBAJZKLUFUDMNFBAUEEUGOPSUATCBMQR
|
|
$.
|
|
$( [27-May-2008] $)
|
|
$}
|
|
|
|
${
|
|
k1-2.1 $e |- x = ( ( x ^ c ) v ( x ^ c ' ) ) $.
|
|
k1-2.2 $e |- y = ( ( y ^ c ) v ( y ^ c ' ) ) $.
|
|
k1-2.3 $e |- ( ( x ^ c ) v ( y ^ c ) ) ' = ( ( ( ( x ^ c )
|
|
v ( y ^ c ) ) ' ^ c ) v ( ( ( x ^ c ) v ( y ^ c ) ) ' ^ c ' ) ) $.
|
|
$( Statement (2) in proof of Theorem 1 of Kalmbach, _Orthomodular
|
|
Lattices_, p. 21. $)
|
|
k1-2 $p |- ( ( x v y ) ^ c ) = ( ( x ^ c ) v ( y ^ c ) ) $=
|
|
( wo wa wn 2or or4 ax-r2 ran lear lel2or k1-8a ax-r1 tr ) BCGZAHBAHZCAHZG
|
|
ZBAIZHZCUCHZGZGZAHZUBSUGASTUDGZUAUEGZGUGBUICUJDEJTUDUAUEKLMUBUHAUBUFFTAUA
|
|
BANCANOUDUCUEBUCNCUCNOPQR $.
|
|
$( [27-May-2008] $)
|
|
$}
|
|
|
|
${
|
|
k1-3.1 $e |- x = ( ( x ^ c ) v ( x ^ c ' ) ) $.
|
|
k1-3.2 $e |- y = ( ( y ^ c ) v ( y ^ c ' ) ) $.
|
|
k1-3.3 $e |- ( ( x ^ c ' ) v ( y ^ c ' ) ) ' = ( ( ( ( x ^ c ' )
|
|
v ( y ^ c ' ) ) ' ^ c ) v ( ( ( x ^ c ' ) v ( y ^ c ' ) ) ' ^ c ' ) ) $.
|
|
$( Statement (3) in proof of Theorem 1 of Kalmbach, _Orthomodular
|
|
Lattices_, p. 21. $)
|
|
k1-3 $p |- ( ( x v y ) ^ c ' ) = ( ( x ^ c ' ) v ( y ^ c ' ) ) $=
|
|
( wo wn wa 2or or4 ax-r2 ran lear lel2or k1-8b ax-r1 tr ) BCGZAHZIBAIZCAI
|
|
ZGZBTIZCTIZGZGZTIZUFSUGTSUAUDGZUBUEGZGUGBUICUJDEJUAUDUBUEKLMUFUHAUCUFFUAA
|
|
UBBANCANOUDTUEBTNCTNOPQR $.
|
|
$( [27-May-2008] $)
|
|
$}
|
|
|
|
${
|
|
k1-4.1 $e |- ( x ' ^ ( x v c ' ) ) =
|
|
( ( ( x ' ^ ( x v c ' ) ) ^ c ) v ( ( x ' ^ ( x v c ' ) ) ^ c ' ) ) $.
|
|
k1-4.2 $e |- x =< c $.
|
|
$( Statement (4) in proof of Theorem 1 of Kalmbach, _Orthomodular
|
|
Lattices_, p. 21. $)
|
|
k1-4 $p |- ( x v ( x ' ^ c ) ) = c $=
|
|
( wn wa wo oran1 lan cm anor3 an32 dff 3tr1 leao4 df2le2 df-le2 ax-r4 3tr
|
|
wf tr 2or or0r 3tr2 con1 ) BBEZAFZGZAUFUGEZFZUFBAEZGZFZUHEUKUMUJULUIUFBAH
|
|
ZIJBUGKUMUMAFZUMUKFZGTUKGUKCUOTUPUKUGULFUGUIFUOTULUIUGUNIUFULALUGMNUPUFUK
|
|
FZULFUQUKUFULUKLUQULUKUFBOPUQBAGZEUKBAKURABADQRUASUBUKUCSUDUE $.
|
|
$( [27-May-2008] $)
|
|
$}
|
|
|
|
${
|
|
k1-5.1 $e |- ( x ' ^ ( x v c ) ) =
|
|
( ( ( x ' ^ ( x v c ) ) ^ c ) v ( ( x ' ^ ( x v c ) ) ^ c ' ) ) $.
|
|
k1-5.2 $e |- x =< c ' $.
|
|
$( Statement (5) in proof of Theorem 1 of Kalmbach, _Orthomodular
|
|
Lattices_, p. 21. $)
|
|
k1-5 $p |- ( x v ( x ' ^ c ' ) ) = c ' $=
|
|
( wn wo wa ax-a1 lor lan orcom ran 2an 2or tr 3tr2 k1-4 ) AEZBBEZBAFZGZUA
|
|
AGZUARGZFZSBREZFZGZUGRGZUGUEGZFZCTUFSAUEBAHZIJZUDUCUBFUJUBUCKUCUHUBUIUAUG
|
|
RULLUAUGAUEULUKMNOPDQ $.
|
|
$( [27-May-2008] $)
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Weakly orthomodular lattices
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Weak orthomodular law
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
ax-wom.1 $e |- ( a ' v ( a ^ b ) ) = 1 $.
|
|
$( 2-variable WOML rule. $)
|
|
ax-wom $a |- ( b v ( a ' ^ b ' ) ) = 1 $.
|
|
$}
|
|
|
|
${
|
|
2vwomr2.1 $e |- ( b v ( a ' ^ b ' ) ) = 1 $.
|
|
$( 2-variable WOML rule. $)
|
|
2vwomr2 $p |- ( a ' v ( a ^ b ) ) = 1 $=
|
|
( wn wa wo wt ancom ax-a1 2an ax-r2 lor 2or ax-r1 ax-wom ) ADZABEZFPBDZDZ
|
|
PDZEZFGQUAPQBAEUAABHBSATBIZAIJKLRPSRPEZFZBPREZFZGUFUDBSUEUCUBPRHMNCKOK $.
|
|
$( [13-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
2vwomr1a.1 $e |- ( a ->1 b ) = 1 $.
|
|
$( 2-variable WOML rule. $)
|
|
2vwomr1a $p |- ( a ->2 b ) = 1 $=
|
|
( wi2 wn wa wo wt df-i2 wi1 df-i1 ax-r1 ax-r2 ax-wom ) ABDBAEZBEFGHABIABO
|
|
ABFGZABJZHQPABKLCMNM $.
|
|
$( [13-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
2vwomr2a.1 $e |- ( a ->2 b ) = 1 $.
|
|
$( 2-variable WOML rule. $)
|
|
2vwomr2a $p |- ( a ->1 b ) = 1 $=
|
|
( wi1 wn wa wo wt df-i1 wi2 df-i2 ax-r1 ax-r2 2vwomr2 ) ABDAEZABFGHABIABB
|
|
OBEFGZABJZHQPABKLCMNM $.
|
|
$( [13-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
2vwomlem.1 $e |- ( a ->2 b ) = 1 $.
|
|
2vwomlem.2 $e |- ( b ->2 a ) = 1 $.
|
|
$( Lemma from 2-variable WOML rule. $)
|
|
2vwomlem $p |- ( a == b ) = 1 $=
|
|
( tb wa wn wo wt dfb wf df-f ax-r1 wi2 anor3 ax-r2 lor df-i2 3tr 3tr2 ran
|
|
anor2 ancom ax-r4 anabs anass oran 2an lan or0 le1 2vwomr2 lea leo ler2an
|
|
oran3 lelor bltr lebi ax-wom ) ABEABFZAGZBGZFZHZIABJVEKHVEVBVEGZFZHVEIKVG
|
|
VEKIGZVGLAABHZGZHZGZVBVIFZVHVGVMVLAVIUBMVKIVKAVCVBFZHZBANZIVJVNAVJVDVNVDV
|
|
JABOMVBVCUCPQVPVOBARMDSUDVMVBVBVCHZFZVIFVBVQVIFZFVGVBVRVIVRVBVBVCUEMUAVBV
|
|
QVIUFVSVFVBVSVAGZVDGZFVFVQVTVIWAABUPABUGUHVAVDOPUISTPQVEUJAVEVBAVEFZHZIWC
|
|
UKIVBVAHZWCWDIABBVDHZABNZIWFWEABRMCPULMVAWBVBVAAVEABUMVAVDUNUOUQURUSUTTP
|
|
$.
|
|
$( [13-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
wr5-2v.1 $e |- ( a == b ) = 1 $.
|
|
$( WOML derived from 2-variable axioms. $)
|
|
wr5-2v $p |- ( ( a v c ) == ( b v c ) ) = 1 $=
|
|
( wo wi2 wn wa wt df-i2 ax-r1 anandir anass ax-r2 3tr2 wi1 df-i1 bltr le1
|
|
lebi anor3 lan 2an lor tb wlem1 skr0 lea leo lelan 2vwomr1a lear 2vwomlem
|
|
lelor ) ACEZBCEZUOUPFUPUOGZUPGZHZEZIUOUPJUPAGZURHZEZAUPFZUTIVDVCAUPJKVBUS
|
|
UPVABGZHCGZHZVAVFHZVEVFHZHVBUSVAVEVFLVGVAVIHVBVAVEVFMVIURVABCUAZUBNVHUQVI
|
|
URACUAZVJUCOUDAUPAUPPVAAUPHZEZIAUPQIVMIVMIVAABHZEZVMIABPZVOIVPIVPBAPZHZVP
|
|
VRIABUEVRDABUFUGKZVPVQUHRVPSTABQNVNVLVABUPABCUIUJUNRVMSTKNUKONUPUOFUOURUQ
|
|
HZEZIUPUOJUOVEUQHZEZBUOFZWAIWDWCBUOJKWBVTUOVEVAHVFHZVIVHHWBVTVEVAVFLWEVEV
|
|
HHWBVEVAVFMVHUQVEVKUBNVIURVHUQVJVKUCOUDBUOBUOPVEBUOHZEZIBUOQIWGIWGIVEBAHZ
|
|
EZWGIVQWIIVQIVRVQVSVPVQULRVQSTBAQNWHWFVEAUOBACUIUJUNRWGSTKNUKONUM $.
|
|
$( [11-Nov-98] $)
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Weakly orthomodular lattices
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
wom3.1 $e |- ( a == b ) = 1 $.
|
|
$( Weak orthomodular law for study of weakly orthomodular lattices. $)
|
|
wom3 $p |- a =< ( ( a v c ) == ( b v c ) ) $=
|
|
( wt wo tb le1 wr5-2v ax-r1 bile letr ) AEACFBCFGZAHEMMEABCDIJKL $.
|
|
$( [13-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
wlor.1 $e |- ( a == b ) = 1 $.
|
|
$( Weak orthomodular law. $)
|
|
wlor $p |- ( ( c v a ) == ( c v b ) ) = 1 $=
|
|
( wo tb wt ax-a2 2bi wr5-2v ax-r2 ) CAEZCBEZFACEZBCEZFGLNMOCAHCBHIABCDJK
|
|
$.
|
|
$( [24-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wran.1 $e |- ( a == b ) = 1 $.
|
|
$( Weak orthomodular law. $)
|
|
wran $p |- ( ( a ^ c ) == ( b ^ c ) ) = 1 $=
|
|
( wa tb wn wo wt df-a 2bi wr4 wr5-2v ax-r2 ) ACEZBCEZFAGZCGZHZGZBGZRHZGZF
|
|
IOTPUCACJBCJKSUBQUARABDLMLN $.
|
|
$( [24-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wlan.1 $e |- ( a == b ) = 1 $.
|
|
$( Weak orthomodular law. $)
|
|
wlan $p |- ( ( c ^ a ) == ( c ^ b ) ) = 1 $=
|
|
( wa tb wt ancom 2bi wran ax-r2 ) CAEZCBEZFACEZBCEZFGLNMOCAHCBHIABCDJK $.
|
|
$( [24-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wr2.1 $e |- ( a == b ) = 1 $.
|
|
wr2.2 $e |- ( b == c ) = 1 $.
|
|
$( Inference rule of AUQL. $)
|
|
wr2 $p |- ( a == c ) = 1 $=
|
|
( tb wa wn wo wt dfb rbi wr1 wran wr5-2v ax-r2 wwbmp wr4 wlor wwbmpr ) AC
|
|
FZACGZBHZCHZGZIZBCFZUFEUGUFFBCGZUEIZUFFJUGUIUFBCKLUHUBUEBACABDMNOPQUAUFFU
|
|
BAHZUDGZIZUFFJUAULUFACKLUKUEUBUJUCUDABDRNSPT $.
|
|
$( [24-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
w2or.1 $e |- ( a == b ) = 1 $.
|
|
w2or.2 $e |- ( c == d ) = 1 $.
|
|
$( Join both sides with disjunction. $)
|
|
w2or $p |- ( ( a v c ) == ( b v d ) ) = 1 $=
|
|
( wo wlor wr5-2v wr2 ) ACGADGBDGCDAFHABDEIJ $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
w2an.1 $e |- ( a == b ) = 1 $.
|
|
w2an.2 $e |- ( c == d ) = 1 $.
|
|
$( Join both sides with conjunction. $)
|
|
w2an $p |- ( ( a ^ c ) == ( b ^ d ) ) = 1 $=
|
|
( wa wlan wran wr2 ) ACGADGBDGCDAFHABDEIJ $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
w3tr1.1 $e |- ( a == b ) = 1 $.
|
|
w3tr1.2 $e |- ( c == a ) = 1 $.
|
|
w3tr1.3 $e |- ( d == b ) = 1 $.
|
|
$( Transitive inference useful for introducing definitions. $)
|
|
w3tr1 $p |- ( c == d ) = 1 $=
|
|
( wr1 wr2 ) CADFABDEDBGHII $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
w3tr2.1 $e |- ( a == b ) = 1 $.
|
|
w3tr2.2 $e |- ( a == c ) = 1 $.
|
|
w3tr2.3 $e |- ( b == d ) = 1 $.
|
|
$( Transitive inference useful for eliminating definitions. $)
|
|
w3tr2 $p |- ( c == d ) = 1 $=
|
|
( wr1 w3tr1 ) ABCDEACFHBDGHI $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Relationship analogues (ordering; commutation) in WOML
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
wleoa.1 $e |- ( ( a v c ) == b ) = 1 $.
|
|
$( Relation between two methods of expressing "less than or equal to". $)
|
|
wleoa $p |- ( ( a ^ b ) == a ) = 1 $=
|
|
( wa wo wr1 wlan wa5c wr2 ) ABEAACFZEABKAKBDGHACIJ $.
|
|
$( [27-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wleao.1 $e |- ( ( c ^ b ) == a ) = 1 $.
|
|
$( Relation between two methods of expressing "less than or equal to". $)
|
|
wleao $p |- ( ( a v b ) == b ) = 1 $=
|
|
( wo wa wa2 wr1 wancom wr2 wlor wa5b ) ABEZBBCFZEZBMBAEOABGANBACBFZNPADHN
|
|
PBCIHJKJBCLJ $.
|
|
$( [27-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wdf-le1.1 $e |- ( ( a v b ) == b ) = 1 $.
|
|
$( Define 'less than or equal to' analogue for ` == ` analogue of
|
|
` = ` . $)
|
|
wdf-le1 $p |- ( a =<2 b ) = 1 $=
|
|
( wle2 wo tb wt df-le ax-r2 ) ABDABEBFGABHCI $.
|
|
$( [27-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wdf-le2.1 $e |- ( a =<2 b ) = 1 $.
|
|
$( Define 'less than or equal to' analogue for ` == ` analogue of
|
|
` = ` . $)
|
|
wdf-le2 $p |- ( ( a v b ) == b ) = 1 $=
|
|
( wo tb wle2 wt df-le ax-r1 ax-r2 ) ABDBEZABFZGLKABHICJ $.
|
|
$( [27-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wom4.1 $e |- ( a =<2 b ) = 1 $.
|
|
$( Orthomodular law. Kalmbach 83 p. 22. $)
|
|
wom4 $p |- ( ( a v ( a ' ^ b ) ) == b ) = 1 $=
|
|
( wn wo wa woml wdf-le2 wlan wlor w3tr2 ) AADZABEZFZEMALBFZEBABGNOAMBLABC
|
|
HZIJPK $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wom5.1 $e |- ( a =<2 b ) = 1 $.
|
|
wom5.2 $e |- ( ( b ^ a ' ) == 0 ) = 1 $.
|
|
$( Orthomodular law. Kalmbach 83 p. 22. $)
|
|
wom5 $p |- ( a == b ) = 1 $=
|
|
( wf wo wn wa wr1 ancom bi1 wr2 wlor or0 wom4 w3tr2 ) AEFZAAGZBHZFABESAEB
|
|
RHZSTEDITSBRJKLMQAANKABCOP $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wcomlem.1 $e |- ( a == ( ( a ^ b ) v ( a ^ b ' ) ) ) = 1 $.
|
|
$( Analogue of commutation is symmetric. Similar to Kalmbach 83 p. 22. $)
|
|
wcomlem $p |- ( b == ( ( b ^ a ) v ( b ^ a ' ) ) ) = 1 $=
|
|
( wa wn ax-a2 bi1 wran ancom wr2 anabs wlan df-a anor1 w2or wr4 wr1 anass
|
|
wo wlor wcon2 w3tr1 orabs wdf-le1 wom4 w3tr2 ) ABDZUGEZBDZSZUGAEZBDZSZBBA
|
|
DZBUKDZSUMUJULUIUGUKBEZSZUKBSZBDZDZUQBDULUIUSBUQUSBBUKSZDZBUSVABDZVBURVAB
|
|
URVAUKBFGHVCVBVABIGJVBBBUKKGJLULUQURDZBDZUTUKVDBUKUQEZUREZSZEZVDAVHAUGAUP
|
|
DZSVHCUGVFVJVGUGVFABMGZVJVGABNGOJPVDVIVDVIUQURMGQJHVEUTUQURBRGJUHUQBUGUQV
|
|
KUAHUBTQUGBUGBUGBSZBUGSZBVLVMUGBFGVMBUNSZBUGUNBUGUNABIGZTVNBBAUCGJJUDUEUG
|
|
UNULUOVOULUOUKBIGOUF $.
|
|
$( [27-Jan-02] $)
|
|
$}
|
|
|
|
${
|
|
wdf-c1.1 $e |- ( a == ( ( a ^ b ) v ( a ^ b ' ) ) ) = 1 $.
|
|
$( Show that commutator is a 'commutes' analogue for ` == ` analogue of
|
|
` = ` . $)
|
|
wdf-c1 $p |- C ( a , b ) = 1 $=
|
|
( wcmtr wa wn wo cmtrcom df-cmtr df-t bi1 wcomlem ax-a1 ax-r5 ax-a2 ax-r2
|
|
wt lan wr2 w2or wr3 3tr ) ABDBADBAEBAFZEGZBFZAEUEUCEGZGZQABHBAIUGQBUEGZUG
|
|
QUHBJKBUDUEUFABCLAUEAABEZAUEEZGZUJAUEFZEZGZCUKUNUKUMUJGUNUIUMUJBULABMRNUM
|
|
UJOPKSLTSUAUB $.
|
|
$( [27-Jan-02] $)
|
|
$}
|
|
|
|
${
|
|
wdf-c2.1 $e |- C ( a , b ) = 1 $.
|
|
$( Show that commutator is a 'commutes' analogue for ` == ` analogue of
|
|
` = ` . $)
|
|
wdf-c2 $p |- ( a == ( ( a ^ b ) v ( a ^ b ' ) ) ) = 1 $=
|
|
( wa wn wo tb wt le1 lea lel2or lelor wcmtr ax-r1 df-cmtr ax-r2 dfb ancom
|
|
2an anabs df2le2 anandi oran3 oran2 anor3 lan anidm 3tr2 2or le3tr1 lebi
|
|
) AABDZABEZDZFZGZHUPIUOAEZBDZUQUMDZFZFZUOUQFZHUPUTUQUOURUQUSUQBJUQUMJKLHA
|
|
BMZVAVCHCNABOPUPAUODZUQUOEZDZFVBAUOQVDUOVFUQVDUOADUOAUORUOAULAUNABJAUMJKU
|
|
APUQUQUMFZUQBFZDZDUQVGDZUQVHDZDZVFUQUQVGVHUBVIVEUQVIULEZUNEZDVEVGVMVHVNAB
|
|
UCABUDSULUNUEPUFVLUQUQDUQVJUQVKUQUQUMTUQBTSUQUGPUHUIPUJUK $.
|
|
$( [27-Jan-02] $)
|
|
$}
|
|
|
|
${
|
|
wdf2le1.1 $e |- ( ( a ^ b ) == a ) = 1 $.
|
|
$( Alternate definition of 'less than or equal to'. $)
|
|
wdf2le1 $p |- ( a =<2 b ) = 1 $=
|
|
( wleao wdf-le1 ) ABABACDE $.
|
|
$( [27-Sep-97] $)
|
|
$}
|
|
|
|
${
|
|
wdf2le2.1 $e |- ( a =<2 b ) = 1 $.
|
|
$( Alternate definition of 'less than or equal to'. $)
|
|
wdf2le2 $p |- ( ( a ^ b ) == a ) = 1 $=
|
|
( wdf-le2 wleoa ) ABBABCDE $.
|
|
$( [27-Sep-97] $)
|
|
$}
|
|
|
|
$( L.e. absorption. $)
|
|
wleo $p |- ( a =<2 ( a v b ) ) = 1 $=
|
|
( wo wa5c wdf2le1 ) AABCABDE $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( L.e. absorption. $)
|
|
wlea $p |- ( ( a ^ b ) =<2 a ) = 1 $=
|
|
( wa wo wa2 wa5b wr2 wdf-le1 ) ABCZAIADAIDAIAEABFGH $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( Anything is l.e. 1. $)
|
|
wle1 $p |- ( a =<2 1 ) = 1 $=
|
|
( wt wo or1 bi1 wdf-le1 ) ABABCBADEF $.
|
|
$( [27-Sep-97] $)
|
|
|
|
$( 0 is l.e. anything. $)
|
|
wle0 $p |- ( 0 =<2 a ) = 1 $=
|
|
( wf wle2 wo tb wt df-le ax-a2 or0 ax-r2 bi1 ) BACBADZAEFBAGLALABDABAHAIJKJ
|
|
$.
|
|
$( [11-Oct-97] $)
|
|
|
|
${
|
|
wle.1 $e |- ( a =<2 b ) = 1 $.
|
|
$( Add disjunct to right of l.e. $)
|
|
wler $p |- ( a =<2 ( b v c ) ) = 1 $=
|
|
( wo wle2 tb wt df-le ax-a3 ax-r1 rbi ax-r2 wr5-2v ) ABCEZFAOEZOGZHAOIQAB
|
|
EZCEZOGHPSOSPABCJKLRBCRBGZABFZHUATABIKDMNMM $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Add conjunct to left of l.e. $)
|
|
wlel $p |- ( ( a ^ c ) =<2 b ) = 1 $=
|
|
( wa an32 bi1 wdf2le2 wran wr2 wdf2le1 ) ACEZBLBEZABEZCEZLMOACBFGNACABDHI
|
|
JK $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Add disjunct to right of both sides. $)
|
|
wleror $p |- ( ( a v c ) =<2 ( b v c ) ) = 1 $=
|
|
( wo orordir bi1 wr1 wdf-le2 wr5-2v wr2 wdf-le1 ) ACEZBCEZMNEZABEZCEZNQOQ
|
|
OABCFGHPBCABDIJKL $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Add conjunct to right of both sides. $)
|
|
wleran $p |- ( ( a ^ c ) =<2 ( b ^ c ) ) = 1 $=
|
|
( wa anandir bi1 wr1 wdf2le2 wran wr2 wdf2le1 ) ACEZBCEZMNEZABEZCEZMQOQOA
|
|
BCFGHPACABDIJKL $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Contrapositive for l.e. $)
|
|
wlecon $p |- ( b ' =<2 a ' ) = 1 $=
|
|
( wn wa wo ax-a2 bi1 oran wdf-le2 w3tr2 wcon3 wdf2le1 ) BDZADZNOEZBBAFZAB
|
|
FZPDZBQRBAGHQSBAIHABCJKLM $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$}
|
|
|
|
${
|
|
wletr.1 $e |- ( a =<2 b ) = 1 $.
|
|
wletr.2 $e |- ( b =<2 c ) = 1 $.
|
|
$( Transitive law for l.e. $)
|
|
wletr $p |- ( a =<2 c ) = 1 $=
|
|
( wa wo wdf-le2 wr5-2v wr1 ax-a3 bi1 w3tr2 wlan anabs wr2 wdf2le1 ) ACACF
|
|
AABCGZGZFZACSARABGZCGZCSUBRUABCABDHIJBCEHUBSABCKLMNTAAROLPQ $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wbltr.1 $e |- ( a == b ) = 1 $.
|
|
wbltr.2 $e |- ( b =<2 c ) = 1 $.
|
|
$( Transitive inference. $)
|
|
wbltr $p |- ( a =<2 c ) = 1 $=
|
|
( wo wr5-2v wdf-le2 wr2 wdf-le1 ) ACACFBCFCABCDGBCEHIJ $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wlbtr.1 $e |- ( a =<2 b ) = 1 $.
|
|
wlbtr.2 $e |- ( b == c ) = 1 $.
|
|
$( Transitive inference. $)
|
|
wlbtr $p |- ( a =<2 c ) = 1 $=
|
|
( wa wr1 wlan wdf2le2 wr2 wdf2le1 ) ACACFABFACBABCEGHABDIJK $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wle3tr1.1 $e |- ( a =<2 b ) = 1 $.
|
|
wle3tr1.2 $e |- ( c == a ) = 1 $.
|
|
wle3tr1.3 $e |- ( d == b ) = 1 $.
|
|
$( Transitive inference useful for introducing definitions. $)
|
|
wle3tr1 $p |- ( c =<2 d ) = 1 $=
|
|
( wbltr wr1 wlbtr ) CBDCABFEHDBGIJ $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wle3tr2.1 $e |- ( a =<2 b ) = 1 $.
|
|
wle3tr2.2 $e |- ( a == c ) = 1 $.
|
|
wle3tr2.3 $e |- ( b == d ) = 1 $.
|
|
$( Transitive inference useful for eliminating definitions. $)
|
|
wle3tr2 $p |- ( c =<2 d ) = 1 $=
|
|
( wr1 wle3tr1 ) ABCDEACFHBDGHI $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wbile.1 $e |- ( a == b ) = 1 $.
|
|
$( Biconditional to l.e. $)
|
|
wbile $p |- ( a =<2 b ) = 1 $=
|
|
( wo wr5-2v oridm bi1 wr2 wdf-le1 ) ABABDBBDZBABBCEJBBFGHI $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wlebi.1 $e |- ( a =<2 b ) = 1 $.
|
|
wlebi.2 $e |- ( b =<2 a ) = 1 $.
|
|
$( L.e. to biconditional. $)
|
|
wlebi $p |- ( a == b ) = 1 $=
|
|
( wo wdf-le2 wr1 ax-a2 bi1 wr2 ) AABEZBABAEZKLABADFGLKBAHIJABCFJ $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wle2.1 $e |- ( a =<2 b ) = 1 $.
|
|
wle2.2 $e |- ( c =<2 d ) = 1 $.
|
|
$( Disjunction of 2 l.e.'s. $)
|
|
wle2or $p |- ( ( a v c ) =<2 ( b v d ) ) = 1 $=
|
|
( wo wleror ax-a2 bi1 wle3tr1 wletr ) ACGBCGZBDGZABCEHCBGZDBGZMNCDBFHMOBC
|
|
IJNPBDIJKL $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Conjunction of 2 l.e.'s. $)
|
|
wle2an $p |- ( ( a ^ c ) =<2 ( b ^ d ) ) = 1 $=
|
|
( wa wleran ancom bi1 wle3tr1 wletr ) ACGBCGZBDGZABCEHCBGZDBGZMNCDBFHMOBC
|
|
IJNPBDIJKL $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
$( Half of distributive law. $)
|
|
wledi $p |- ( ( ( a ^ b ) v ( a ^ c ) ) =<2
|
|
( a ^ ( b v c ) ) ) = 1 $=
|
|
( wa wo anidm bi1 wr1 wlea wle2or oridm wlbtr ancom wbltr wle2an ) ABDZACDZ
|
|
EZRRDZABCEZDSRSRRFGHRARTRAAEZAPAQAABIACIJUAAAKGLPBQCPBADZBPUBABMGBAINQCADZC
|
|
QUCACMGCAINJON $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Half of distributive law. $)
|
|
wledio $p |- ( ( a v ( b ^ c ) ) =<2
|
|
( ( a v b ) ^ ( a v c ) ) ) = 1 $=
|
|
( wa wo anidm bi1 wr1 wleo wle2an wbltr ax-a2 wlbtr wle2or oridm ) ABCDZEAB
|
|
EZACEZDZSEZSASPSAAADZSUAAUAAAFGHAQARABIACIJKBQCRBBAEZQBAIUBQBALGMCCAEZRCAIU
|
|
CRCALGMJNTSSOGM $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Commutation with 0. Kalmbach 83 p. 20. $)
|
|
wcom0 $p |- C ( a , 0 ) = 1 $=
|
|
( wf wa wn wo comm0 df-c2 bi1 wdf-c1 ) ABAABCABDCEABAFGHI $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Commutation with 1. Kalmbach 83 p. 20. $)
|
|
wcom1 $p |- C ( 1 , a ) = 1 $=
|
|
( wt wa wn wo comm1 df-c2 bi1 wdf-c1 ) BABBACBADCEBAAFGHI $.
|
|
$( [13-Oct-97] $)
|
|
|
|
${
|
|
wlecom.1 $e |- ( a =<2 b ) = 1 $.
|
|
$( Comparable elements commute. Beran 84 2.3(iii) p. 40. $)
|
|
wlecom $p |- C ( a , b ) = 1 $=
|
|
( wn wa wo orabs bi1 wr1 wdf2le2 wr5-2v wr2 wdf-c1 ) ABAAABDZEZFZABEZOFPA
|
|
PAANGHIAQOQAABCJIKLM $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wbctr.1 $e |- ( a == b ) = 1 $.
|
|
wbctr.2 $e |- C ( b , c ) = 1 $.
|
|
$( Transitive inference. $)
|
|
wbctr $p |- C ( a , c ) = 1 $=
|
|
( wa wn wo wdf-c2 wran w2or w3tr1 wdf-c1 ) ACBBCFZBCGZFZHAACFZAOFZHBCEIDQ
|
|
NRPABCDJABODJKLM $.
|
|
$( [13-Oct-97] $)
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wcbtr.1 $e |- C ( a , b ) = 1 $.
|
|
wcbtr.2 $e |- ( b == c ) = 1 $.
|
|
$( Transitive inference. $)
|
|
wcbtr $p |- C ( a , c ) = 1 $=
|
|
( wa wn wo wdf-c2 wlan wr4 w2or wr2 wdf-c1 ) ACAABFZABGZFZHACFZACGZFZHABD
|
|
IORQTBCAEJPSABCEKJLMN $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
$( Weak commutation law. $)
|
|
wcomorr $p |- C ( a , ( a v b ) ) = 1 $=
|
|
( wo wleo wlecom ) AABCABDE $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Weak commutation law. $)
|
|
wcoman1 $p |- C ( ( a ^ b ) , a ) = 1 $=
|
|
( wa wlea wlecom ) ABCAABDE $.
|
|
$( [13-Oct-97] $)
|
|
|
|
${
|
|
wcomcom.1 $e |- C ( a , b ) = 1 $.
|
|
$( Commutation is symmetric. Kalmbach 83 p. 22. $)
|
|
wcomcom $p |- C ( b , a ) = 1 $=
|
|
( wcmtr wt cmtrcom ax-r2 ) BADABDEBAFCG $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Commutation equivalence. Kalmbach 83 p. 23. $)
|
|
wcomcom2 $p |- C ( a , b ' ) = 1 $=
|
|
( wn wa wo wdf-c2 ax-a1 bi1 wlan wr5-2v wr2 ax-a2 wdf-c1 ) ABDZAAODZEZAOE
|
|
ZFZRQFZAABEZRFSABCGUAQRBPABPBHIJKLSTQRMILN $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Commutation equivalence. Kalmbach 83 p. 23. $)
|
|
wcomcom3 $p |- C ( a ' , b ) = 1 $=
|
|
( wn wcomcom wcomcom2 ) BADBAABCEFE $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Commutation equivalence. Kalmbach 83 p. 23. $)
|
|
wcomcom4 $p |- C ( a ' , b ' ) = 1 $=
|
|
( wn wcomcom3 wcomcom2 ) ADBABCEF $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Commutation dual. Kalmbach 83 p. 23. $)
|
|
wcomd $p |- ( a == ( ( a v b ) ^ ( a v b ' ) ) ) = 1 $=
|
|
( wn wa wo wcomcom4 wdf-c2 wcon3 oran bi1 wcon2 w2an wr1 wr2 ) AADZBDZEZP
|
|
QDEZFZDZABFZAQFZEZATPQABCGHIUARDZSDZEZUDTUGTUGDRSJKLUDUGUBUEUCUFUBUEABJKU
|
|
CUFAQJKMNOO $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Lemma 3(ii) of Kalmbach 83 p. 23. $)
|
|
wcom3ii $p |- ( ( a ^ ( a ' v b ) ) == ( a ^ b ) ) = 1 $=
|
|
( wa wn wo wcomcom wcomd wlan anass bi1 wr1 ax-a2 anabs wr2 w2an ) ABDZAA
|
|
EZBFZDZQABAFZBRFZDZDZTBUCABAABCGHIUDAUADZUBDZTUFUDUFUDAUAUBJKLUEAUBSUEAAB
|
|
FZDZAUAUGAUAUGBAMKIUHAABNKOUBSBRMKPOOL $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wcomcom5.1 $e |- C ( a ' , b ' ) = 1 $.
|
|
$( Commutation equivalence. Kalmbach 83 p. 23. $)
|
|
wcomcom5 $p |- C ( a , b ) = 1 $=
|
|
( wn wa wo wcomcom4 wdf-c2 ax-a1 bi1 w2an w2or w3tr1 wdf-c1 ) ABADZDZPBDZ
|
|
DZEZPRDZEZFAABEZAQEZFPROQCGHAPAIJZUBSUCUAAPBRUDBRBIJKAPQTUDQTQIJKLMN $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wcomdr.1 $e |- ( a == ( ( a v b ) ^ ( a v b ' ) ) ) = 1 $.
|
|
$( Commutation dual. Kalmbach 83 p. 23. $)
|
|
wcomdr $p |- C ( a , b ) = 1 $=
|
|
( wn wa wo df-a bi1 oran wcon2 w2or wr4 wr2 wdf-c1 wcomcom5 ) ABADZBDZAPQ
|
|
EZPQDEZFZAABFZAQFZEZTDZCUCUADZUBDZFZDZUDUCUHUAUBGHUGTUERUFSUARUARDABIHJUB
|
|
SUBSDAQIHJKLMMJNO $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wcom3i.1 $e |- ( ( a ^ ( a ' v b ) ) == ( a ^ b ) ) = 1 $.
|
|
$( Lemma 3(i) of Kalmbach 83 p. 23. $)
|
|
wcom3i $p |- C ( a , b ) = 1 $=
|
|
( wn wa anor1 bi1 wcon2 wran ancom wr2 wlor wlea wom4 ax-a2 w3tr2 wdf-c1
|
|
wo ) ABABDZEZTDZAEZRTABEZRZAUCTRZUBUCTUBAADBRZEZUCUBUFAEZUGUAUFATUFTUFDAB
|
|
FGHIUHUGUFAJGKCKLTAASMNUDUETUCOGPQ $.
|
|
$( [13-Oct-97] $)
|
|
$}
|
|
|
|
${
|
|
wfh.1 $e |- C ( a , b ) = 1 $.
|
|
wfh.2 $e |- C ( a , c ) = 1 $.
|
|
$( Weak structural analog of Foulis-Holland Theorem. $)
|
|
wfh1 $p |- ( ( a ^ ( b v c ) ) ==
|
|
( ( a ^ b ) v ( a ^ c ) ) ) = 1 $=
|
|
( wa wo wledi wn bi1 df-a wr1 wcon3 wr2 w2an wcomcom2 wcom3ii anandi wlan
|
|
wf ancom w2or wcon2 anass w3tr1 an12 oran dff an0 wom5 ) ABFZACFZGZABCGZF
|
|
ZUMUOABCHUOUMIZFZAUNBIZCIZFZFZFZTUQUNAFZAIZURGZVDUSGZFZFZVBUOVCUPVGUOVCAU
|
|
NUAJUMVGUMVEIZVFIZGZVGIUKVIULVJUKVIABKJULVJACKJUBVKVGVGVKIZVGVLVEVFKJLMNU
|
|
COVHUNAUTFZFZVBVHUNAVGFZFZVNVHVPUNAVGUDJVOVMUNAVEFZAVFFZFZAURFZAUSFZFZVOV
|
|
MVQVTVRWAAURABDPQAUSACEPQOVOVSAVEVFRJVMWBAURUSRJUESNVNVBUNAUTUFJNNVBATFZT
|
|
VATAVAUNUNIZFZTUTWDUNUTUNUNUTIZUNWFBCUGJLMSTWETWEUNUHJLNSWCTAUIJNNUJL $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Weak structural analog of Foulis-Holland Theorem. $)
|
|
wfh2 $p |- ( ( b ^ ( a v c ) ) ==
|
|
( ( b ^ a ) v ( b ^ c ) ) ) = 1 $=
|
|
( wa wo wledi wn wf oran bi1 wcon2 wran wr2 wlan an4 wcom3ii anass wr1
|
|
df-a wr4 wcomcom wcomcom2 ancom ax-a1 wr5-2v wcomcom3 an12 dff w3tr1 wom5
|
|
an0 ) BAFZBCFZGZBACGZFZUPURBACHURUPIZFZAIZCBUOIZFZFZFZJUTVACFZVCFZVEUTVAU
|
|
QFZVCFZVGUTVABFZUQVBFZFZVIUTURBIVAGZVBFZFZVLUSVNURUPVNUPUNIZVBFZIZVNIUPVR
|
|
UNUOKLVQVNVPVMVBUNVMUNVMIBAUALMNUBOMPVOBVMFZVKFZVLVOVTBUQVMVBQLVSVJVKVSBV
|
|
AFZVJBVABAABDUCUDRWAVJBVAUELONOOVLVIVABUQVBQLOVHVFVCVHVAVAIZCGZFVFUQWCVAA
|
|
WBCAWBAUFLUGPVACACEUHRONOVGVEVACVCSLOVEVAJFZJVDJVABCVBFFZUOVBFZVDJWFWEWFW
|
|
EBCVBSLTVDWECBVBUILJWFUOUJLUKPWDJVAUMLOOULT $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Weak structural analog of Foulis-Holland Theorem. $)
|
|
wfh3 $p |- ( ( a v ( b ^ c ) ) ==
|
|
( ( a v b ) ^ ( a v c ) ) ) = 1 $=
|
|
( wa wo wn wcomcom4 wfh1 anor2 bi1 df-a wr1 wlor wr4 wr2 oran w2an w3tr2
|
|
wcon1 ) ABCFZGZABGZACGZFZAHZBHZCHZGZFZUGUHFZUGUIFZGZUCHZUFHZUGUHUIABDIACE
|
|
IJUKAUJHZGZHZUOUKUSAUJKLURUCUQUBAUBUQUBUQBCMLNOPQUNULHZUMHZFZHZUPUNVCULUM
|
|
RLVBUFUFVBUDUTUEVAUDUTABRLUEVAACRLSNPQTUA $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Weak structural analog of Foulis-Holland Theorem. $)
|
|
wfh4 $p |- ( ( b v ( a ^ c ) ) ==
|
|
( ( b v a ) ^ ( b v c ) ) ) = 1 $=
|
|
( wa wo wn wcomcom4 wfh2 anor2 bi1 df-a wr1 wlor wr4 wr2 oran w2an w3tr2
|
|
wcon1 ) BACFZGZBAGZBCGZFZBHZAHZCHZGZFZUGUHFZUGUIFZGZUCHZUFHZUHUGUIABDIACE
|
|
IJUKBUJHZGZHZUOUKUSBUJKLURUCUQUBBUBUQUBUQACMLNOPQUNULHZUMHZFZHZUPUNVCULUM
|
|
RLVBUFUFVBUDUTUEVAUDUTBARLUEVABCRLSNPQTUA $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Th. 4.2 Beran p. 49. $)
|
|
wcom2or $p |- C ( a , ( b v c ) ) = 1 $=
|
|
( wo wa wn wcomcom wdf-c2 ancom 2or bi1 wr2 w2or or4 wfh1 wcomcom3 wdf-c1
|
|
wr1 ) BCFZAUAAUAABGZACGZFZAHZBGZUECGZFZFZUAAGZUAUEGZFZUAUBUFFZUCUGFZFZUIB
|
|
UMCUNBBAGZBUEGZFZUMBAABDIJURUMUPUBUQUFBAKBUEKLMNCCAGZCUEGZFZUNCAACEIJVAUN
|
|
USUCUTUGCAKCUEKLMNOUOUIUBUFUCUGPMNULUIUJUDUKUHUJAUAGZUDUJVBUAAKMABCDEQNUK
|
|
UEUAGZUHUKVCUAUEKMUEBCABDRACERQNOTNSI $.
|
|
$( [10-Nov-98] $)
|
|
|
|
$( Th. 4.2 Beran p. 49. $)
|
|
wcom2an $p |- C ( a , ( b ^ c ) ) = 1 $=
|
|
( wa wn wo wcomcom4 wcom2or df-a con2 ax-r1 bi1 wcbtr wcomcom5 ) ABCFZAGZ
|
|
BGZCGZHZQGZRSTABDIACEIJUAUBUBUAQUABCKLMNOP $.
|
|
$( [10-Nov-98] $)
|
|
|
|
$}
|
|
|
|
$( Negated biconditional (distributive form) $)
|
|
wnbdi $p |- ( ( a == b ) ' ==
|
|
( ( ( a v b ) ^ a ' ) v ( ( a v b ) ^ b ' ) ) ) = 1 $=
|
|
( tb wn wo wa dfnb bi1 wcomorr wcomcom wcomcom2 ax-a2 wcbtr wfh1 wr2 ) ABCD
|
|
ZABEZADZBDZEFZQRFQSFEPTABGHQRSQAAQABIJKQBBQBBAEZQBAIUAQBALHMJKNO $.
|
|
$( [13-Oct-97] $)
|
|
|
|
$( Lemma for KA14 soundness. $)
|
|
wlem14 $p |- ( ( ( a ^ b ' ) v a ' ) ' v
|
|
( ( a ^ b ' ) v ( ( a ' ^ ( ( a v b ' ) ^ ( a v b ) ) )
|
|
v ( a ' ^ ( ( a v b ' ) ^ ( a v b ) ) ' ) ) ) ) = 1 $=
|
|
( wn wa wo wt df-t ax-r1 ax-a2 bi1 wwbmpr wlan anidm wr1 wleo wle2an wlecom
|
|
wbltr wcomcom3 wlor wcomcom4 wfh1 an1 w3tr2 ) ABCZDZACZEZCZUFUGAUEEZABEZDZD
|
|
UGULCZDEZEZEUIUHEZUPUHUIEZFUQUHGHUPUQUIUHIJKUOUHUIUNUGUFUGULUMEZDUGFDZUNUGU
|
|
RFUGURFFURULGHJLUGULUMAULAULAAADZULUTAUTAAMJNAUJAUKAUEOABOPRQZSAULVAUAUBUSU
|
|
GUGUCJUDTTK $.
|
|
$( [25-Oct-97] $)
|
|
|
|
${
|
|
wr5.1 $e |- ( a == b ) = 1 $.
|
|
$( Proof of weak orthomodular law from weaker-looking equivalent, ~ wom3 ,
|
|
which in turn is derived from ~ ax-wom . $)
|
|
wr5 $p |- ( ( a v c ) == ( b v c ) ) = 1 $=
|
|
( wr5-2v ) ABCDE $.
|
|
$( [25-Oct-97] $)
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Kalmbach axioms (soundness proofs) that require WOML
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( >>>Remove "id" when bug is fixed. $)
|
|
$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA2. $)
|
|
ska2 $p |- ( ( a == b ) ' v ( ( b == c ) ' v ( a == c ) ) ) = 1 $=
|
|
( wn wo wa wt ax-a3 ax-r1 ax-a2 or12 lor ax-r2 wcomorr bi1 wcomcom wcomcom2
|
|
bltr ancom wr2 tb dfnb dfb 2or id le1 df-t oran3 leor le2or lelor letr lebi
|
|
orordi wcbtr wfh4 or1 ran an1 or32 w2or wlor orordir anor3 wcom2or oran leo
|
|
wwbmpr wr5-2v wcomcom3 wfh1 wwbmp ax-r5 ledi leror ) ABUADZBCUADZACUAZEZEAB
|
|
EZADZBDZEFZBCEZWBCDZEZFZACFZWAWEFZEZEZEZGVPWCVSWKABUBVQWGVRWJBCUBACUCUDUDWL
|
|
WCWGEZWJEZGWNWLWCWGWJHIWNWNGWNUEWNGWNUFGVTWAFZWBVTFZEZWBWDFZWDWEFZEZEZWJEZW
|
|
NGWOWPWREZWSEZEZWJEZXBXFGWOWBVTWDEZFZWSEZEZWJEZXFXKWJXJEZGXJWJJXLWOWJXIEZEZ
|
|
GWJWOXIKXNWOWHWIWBEZWSEZEZEZXRWHWOXPEZEZGWOWHXPKXTWHWIWBWOEZWBWSEZEZEZEZGXS
|
|
YDWHXSXOWOWSEZEZYDWOXOWSKYGWIWBYFEZEYDWIWBYFHYHYCWIWBWOWSUNLMMLYEWHWIWBWAEZ
|
|
WFEZEZEZYLWIWHYJEZEZGWHWIYJKYNGYNUFGYMYNGWHWAWEEZEZYMGWHWHDZEZYPWHUGYPYRYOY
|
|
QWHACUHLIMYOYJWHWAYIWEWFWAWBUIWEWBUIUJUKRYMWIUIULUMMYDYKWHYCYJWIYAYIYBWFYAW
|
|
BVTEZYIFZYIVTWBWAVTBBVTBBAEZVTBANUUAVTBAJOUOZPQVTAAVTABNPQUPYTYIYTGYIFZYIYS
|
|
GYIYSAWBBEZEZGWBABKUUEAGEGUUDGAUUDBWBEZGWBBJGUUFBUGIZMLAUQMMURUUCYIGFYIGYIS
|
|
YIUSMMOTYBWBWDEZWFFZWFWDWBWEWDBBWDBCNZPQWDCCWDCCBEZWDCBNUUKWDCBJZOUOPQUPUUI
|
|
WFUUIGWFFZWFUUHGWFUUHWDWBEZGWBWDJUUNUUFCEZGBCWBUTUUOCUUFEZGUUFCJUUPCGEGUUFG
|
|
CUUGLCUQMMMMURUUMWFGFWFGWFSWFUSMMOTVAVBVBVHMMXMXQWOXMWHWIXIEZEZXQXMUURWHWIX
|
|
IHOUUQXPWHUUQWIXHEZWSEZXPUUQUUTUUTUUQWIXHWSHIOUUSXOWSUUSWIXGWBFZEZXOUUSUVBX
|
|
HUVAWIWBXGSLOUVBWIXGEZXOFZXOXGWIWBXGACEZDZWIXGUVEUVEXGUVEUVEBEZXGUVEBNUVGXG
|
|
UVGVTUUKEXGACBVCUUKWDVTUULLMOUOPQUVFWIWIUVFACVDIOUOXGBBXGBVTWDUUBUUJVEPQUPU
|
|
VDXOUVDGXOFZXOUVCGXOUVCGUVCUFGWIUVEEZUVCGWIWIDZEUVIWIUGUVJUVEWIUVEUVJACVFIL
|
|
MUVEXGWIAVTCWDABVGCBUIUJUKRUMURUVHXOGFXOGXOSXOUSMMOTTVITVBTVBVHMMXJXEWJXIXD
|
|
WOXHXCWSWBVTWDBVTUUBVJBWDUUJVJVKVIVBVIVLIXEXAWJXEWQWREZWSEZXAXEWOXCEZWSEZUV
|
|
LUVNXEWOXCWSHIUVMUVKWSUVKUVMWOWPWRHIVMMWQWRWSHMVMMXAWMWJWQWCWTWGWQWOVTWBFZE
|
|
WCWPUVOWOWBVTSLVTWAWBVNRWTWDWBFZWSEWGWRUVPWSWBWDSVMWDWBWEVNRUJVORUMMMM $.
|
|
$( [10-Nov-98] $)
|
|
|
|
$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA4. $)
|
|
ska4 $p |- ( ( a == b ) ' v ( ( a ^ c ) == ( b ^ c ) ) ) = 1 $=
|
|
( tb wn wa wo wt 2or ax-a2 le1 df-t lor ax-r1 ax-r2 lea lecon leror wcomcom
|
|
wcomcom2 dfnb dfb ax-a3 oran le2an bltr lebi ran ancom an1 3tr anandir lear
|
|
oran3 ax-r5 ler2an lelor wlea wleo wletr wlecom wlbtr wcom2an wcomorr wcbtr
|
|
bi1 wcom2or wfh4 wlor wwbmpr ) ABDEZACFZBCFZDZGABGZAEZBEZGZFZVLVMFZVLEZVMEZ
|
|
FZGZGWDVSGZHVKVSVNWDABUAVLVMUBIVSWDJWEVTWCVSGZGZHVTWCVSUCWGVTWCVOGZWCVRGZFZ
|
|
GZWKVTWIGZHWJWIVTWJHWIFWIHFWIWHHWIWHHWHKHVPVQFZVOGZWHHWMWMEZGZWNWMLWNWPVOWO
|
|
WMABUDMNOWMWCVOVPWAVQWBVLAACPQVMBBCPQUERUFUGUHHWIUIWIUJUKMWLHWLKHVTCEZVRGZG
|
|
ZWLHABFZCFZXAEZGWSXALXAVTXBWRABCULXBVRWQGZWRXCXBXCWTEZWQGXBVRXDWQABUNUOWTCU
|
|
NONVRWQJOIOWRWIVTWQWCVRWQWAWBVLCACUMQVMCBCUMQUPRUQUFUGOWFWJVTVOWCVRVOWAWBVO
|
|
VLVLVOVLVOVLAVOACURABUSUTVASTVOVMVMVOVMVOVMBVOBCURBBAGZVOBAUSXEVOBAJVFZVBUT
|
|
VASTVCVOVPVQVOAAVOABVDSTVOBBVOBXEVOBAVDXFVESTVGVHVIVJOUK $.
|
|
$( [9-Nov-98] $)
|
|
|
|
$( Weak orthomodular law for study of weakly orthomodular lattices. $)
|
|
wom2 $p |- a =< ( ( a == b ) ' v ( ( a v c ) == ( b v c ) ) ) $=
|
|
( wt tb wn wo le1 wa conb ax-r4 oran 2bi ax-r1 ax-r2 2or ska4 lbtr ) ADABEZ
|
|
FZACGZBCGZEZGZAHUDDUDAFZBFZEZFZUECFZIZUFUIIZEZGDTUHUCULSUGABJKUCUJFZUKFZEZU
|
|
LUAUMUBUNACLBCLMULUOUJUKJNOPUEUFUIQONR $.
|
|
$( [13-Nov-98] $)
|
|
|
|
$( 3-variable version of weakly orthomodular law. It is proved from a
|
|
weaker-looking equivalent, ~ wom2 , which in turn is proved from
|
|
~ ax-wom . $)
|
|
ka4ot $p |- ( ( a == b ) ' v ( ( a v c ) == ( b v c ) ) ) = 1 $=
|
|
( tb wn wo wt le1 wom2 bicom ax-r4 2or lbtr le2or oridm leror ka4lemo ax-a3
|
|
lor ax-r2 le3tr2 lebi ) ABDZEZACFZBCFZDZFZGUHHABFZUGFUHUGFZGUHUIUHUGUIUHUHF
|
|
UHAUHBUHABCIBBADZEZUFUEDZFUHBACIULUDUMUGUKUCBAJKUFUEJLMNUHOMPABCQUJUDUGUGFZ
|
|
FUHUDUGUGRUNUGUDUGOSTUAUB $.
|
|
$( [25-Oct-97] $)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Weak orthomodular law variants
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Variant of weakly orthomodular law. $)
|
|
woml6 $p |- ( ( a ->1 b ) ' v ( a ->2 b ) ) = 1 $=
|
|
( wn wo wa wt df-a lor ax-r2 ax-r1 2or ax-a2 ancom wcomorr wcomcom wcomcom3
|
|
bi1 wcomcom5 df-t 3tr wi1 wi2 df-i1 ax-r4 df-i2 ax-r5 ax-a3 tb wcbtr wr5-2v
|
|
1b wfh4 or12 or1 ran an1 anor3 wr2 wr1 3tr2 ) ABUAZCZABUBZDAACZBCZDZEZBVDVE
|
|
EZDZDZFVBVGVCVIVBVDVFCZDZCZVGVAVLVAVDABEZDVLABUCVNVKVDABGHIUDVGVMAVFGJIABUE
|
|
KVGBDZVHDBVFAEZDZVHDZVJFVOVQVHVOBVGDVQVGBLVGVPBAVFMHIUFVGBVHUGVRFVRUHZFVSVR
|
|
VRUKJVRFVRBVFDZBADZEZVHDZFVQWBVHVFBAVFBVFVEVEVFVEVEVDDZVFVEVDNWDVFVEVDLQUIO
|
|
PRVFAVFVDVDVFVDVENOPRULUJWCFWCABDZWECZDZFWBWEVHWFWBWAFEZWAWEWBFWAEWHVTFWAVT
|
|
VDBVEDZDZVDFDZFBVDVEUMWKWJFWIVDBSHJVDUNTUOFWAMIWAUPBALTABUQKFWGWESJIQURUSIU
|
|
TI $.
|
|
$( [14-Nov-98] $)
|
|
|
|
$( Variant of weakly orthomodular law. $)
|
|
woml7 $p |- ( ( ( a ->2 b ) ^ ( b ->2 a ) ) ' v ( a == b ) ) = 1 $=
|
|
( wi2 wa wn tb wo wt df-i2 ax-a2 ax-r2 ancom ax-r5 3tr 2an wcoman1 wcomcom3
|
|
bi1 wcomcom5 wr2 ax-r4 id dfb 2or 1b ax-r1 df-t wa2 wbctr wfh3 wr4 wr5-2v )
|
|
ABCZBACZDZEZABFZGAEZBEZDZAGZUTBGZDZEZABDZUTGZGZHVGFZHUPVDUQVFUPVDVDUOVCUOVB
|
|
VADVCUMVBUNVAUMBUTGVBABIBUTJKUNAUSURDZGVIAGVABAIAVIJVIUTAUSURLMNOVBVALKUAVD
|
|
UBKABUCUDVHVGVGUEUFHVFEZVFGZVGHVKHVFVJGVKVFUGVFVJJKRVJVDVFVFVCVFUTVEGVCVEUT
|
|
UHUTABUTAUTURURUSPQSUTBUTUSUTVIUSUTVIURUSLRUSURPUIQSUJTUKULTN $.
|
|
$( [14-Nov-98] $)
|
|
|
|
${
|
|
ortha.1 $e |- a =< b ' $.
|
|
$( Property of orthogonality. $)
|
|
ortha $p |- ( a ^ b ) = 0 $=
|
|
( wa wf wn lecon3 lelan dff ax-r1 lbtr le0 lebi ) ABDZENAAFZDZEBOAABCGHEP
|
|
AIJKNLM $.
|
|
$( [10-Mar-02] $)
|
|
$}
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Orthomodular lattices
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Orthomodular law
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
r3.1 $e |- ( c v c ' ) = ( ( a ' v b ' ) ' v ( a v b ) ' ) $.
|
|
$( Orthomodular law - when added to an ortholattice, it makes the
|
|
ortholattice an orthomodular lattice. See ~ r3a for a more compact
|
|
version. $)
|
|
ax-r3 $a |- a = b $.
|
|
$}
|
|
|
|
${
|
|
r3a.1 $e |- 1 = ( a == b ) $.
|
|
$( Orthomodular law restated. $)
|
|
r3a $p |- a = b $=
|
|
( wt tb wn wo df-t df-b 3tr2 ax-r3 ) ABADABEAAFZGLBFGFABGFGCAHABIJK $.
|
|
$( [12-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
wed.1 $e |- a = b $.
|
|
wed.2 $e |- ( a == b ) = ( c == d ) $.
|
|
$( Weak equivalential detachment (WBMP). $)
|
|
wed $p |- c = d $=
|
|
( wt tb 1bi ax-r2 r3a ) CDGABHCDHABEIFJK $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
r3b.1 $e |- ( c v c ' ) = ( a == b ) $.
|
|
$( Orthomodular law from weak equivalential detachment (WBMP). $)
|
|
r3b $p |- a = b $=
|
|
( wt tb wn wo df-t ax-r2 1b wed ) EABFZABECCGHMCIDJMKL $.
|
|
$( [10-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
lem3.1.1 $e |- ( a v b ) = b $.
|
|
lem3.1.2 $e |- ( b ' v a ) = 1 $.
|
|
$( Lemma used in proof of Th. 3.1 of Pavicic 1993. $)
|
|
lem3.1 $p |- a = b $=
|
|
( tb wt wlem3.1 ax-r1 r3a ) ABABEFABCDGHI $.
|
|
$( [12-Aug-97] $)
|
|
|
|
$( Lemma used in proof of Th. 3.1 of Pavicic 1993. $)
|
|
lem3a.1 $p |- ( a v b ) = a $=
|
|
( wo lem3.1 ax-r1 lor oridm ax-r2 ) ABEAAEABAAABABCDFGHAIJ $.
|
|
$( [12-Aug-97] $)
|
|
$}
|
|
|
|
$( Orthomodular law. Compare Th. 1 of Pavicic 1987. $)
|
|
oml $p |- ( a v ( a ' ^ ( a v b ) ) ) = ( a v b ) $=
|
|
( wn wo wa omlem1 omlem2 lem3.1 ) AACABDZEDIABFABGH $.
|
|
$( [12-Aug-97] $)
|
|
|
|
$( Orthomodular law. $)
|
|
omln $p |- ( a ' v ( a ^ ( a ' v b ) ) ) = ( a ' v b ) $=
|
|
( wn wo wa ax-a1 ran lor oml ax-r2 ) ACZAKBDZEZDKKCZLEZDLMOKANLAFGHKBIJ $.
|
|
$( [2-Nov-97] $)
|
|
|
|
$( Orthomodular law. $)
|
|
omla $p |- ( a ^ ( a ' v ( a ^ b ) ) ) = ( a ^ b ) $=
|
|
( wn wa wo df-a ax-r1 lor ax-r4 ax-r2 omln con2 3tr1 con1 ) AACZABDZEZDZPOQ
|
|
CZEZOBCZEZRCPCTOAUBDZEUBSUCOUCSUCOUBCZEZCSAUBFUEQUDPOPUDABFZGHIJGHAUAKJRTAQ
|
|
FLPUBUFLMN $.
|
|
$( [7-Nov-97] $)
|
|
|
|
$( Orthomodular law. $)
|
|
omlan $p |- ( a ' ^ ( a v ( a ' ^ b ) ) ) = ( a ' ^ b ) $=
|
|
( wn wa wo ax-a1 ax-r5 lan omla ax-r2 ) ACZAKBDZEZDKKCZLEZDLMOKANLAFGHKBIJ
|
|
$.
|
|
$( [7-Nov-97] $)
|
|
|
|
$( Orthomodular law. $)
|
|
oml5 $p |- ( ( a ^ b ) v ( ( a ^ b ) ' ^ ( b v c ) ) )
|
|
= ( b v c ) $=
|
|
( wa wn wo oml ax-a3 ancom lor orabs ax-r2 ax-r5 or12 3tr2 lan 3tr1 ) ABDZR
|
|
EZBCFZDZFZBRFZCFZTRSRTFZDZFUEUBUDRTGUAUFRTUESUDBRCFFZTUEBRCHZUCBCUCBBADZFBR
|
|
UIBABIJBAKLMZBRCNZOPJUDUGUEUHUKLQUJL $.
|
|
$( [16-Nov-97] $)
|
|
|
|
$( Orthomodular law. $)
|
|
oml5a $p |- ( ( a v b ) ^ ( ( a v b ) ' v ( b ^ c ) ) )
|
|
= ( b ^ c ) $=
|
|
( wo wn wa omla anass ax-a2 lan anabs ax-r2 ran an12 3tr2 lor 3tr1 ) ABDZRE
|
|
ZBCFZDZFZBRFZCFZTRSRTFZDZFUEUBUDRTGUAUFRTUESUDBRCFFZTUEBRCHZUCBCUCBBADZFBRU
|
|
IBABIJBAKLMZBRCNZOPJUDUGUEUHUKLQUJL $.
|
|
$( [16-Nov-97] $)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Relationship analogues using OML (ordering; commutation)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
oml2.1 $e |- a =< b $.
|
|
$( Orthomodular law. Kalmbach 83 p. 22. $)
|
|
oml2 $p |- ( a v ( a ' ^ b ) ) = b $=
|
|
( wn wo wa oml df-le2 lan lor 3tr2 ) AADZABEZFZEMALBFZEBABGNOAMBLABCHZIJP
|
|
K $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
oml3.1 $e |- a =< b $.
|
|
oml3.2 $e |- ( b ^ a ' ) = 0 $.
|
|
$( Orthomodular law. Kalmbach 83 p. 22. $)
|
|
oml3 $p |- a = b $=
|
|
( wf wo wn wa ax-r1 ancom ax-r2 lor or0 oml2 3tr2 ) AEFAAGZBHZFABEQAEBPHZ
|
|
QREDIBPJKLAMABCNO $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
comcom.1 $e |- a C b $.
|
|
$( Commutation is symmetric. Kalmbach 83 p. 22. $)
|
|
comcom $p |- b C a $=
|
|
( wa wn wo ax-a2 ran ancom ax-r2 anabs df-c2 df-a anor1 ax-r4 ax-r1 anass
|
|
lan 2or lor con2 3tr1 orabs df-le1 oml2 3tr2 df-c1 ) BAABDZUHEZBDZFZUHAEZ
|
|
BDZFZBBADZBULDZFUNUKUMUJUHULBEZFZULBFZBDZDZURBDUMUJUTBURUTBBULFZDZBUTVBBD
|
|
VCUSVBBULBGHVBBIJBULKJRUMURUSDZBDVAULVDBULUREZUSEZFZEZVDAVGAUHAUQDZFVGABC
|
|
LUHVEVIVFABMZABNSJOVDVHURUSMPJHURUSBQJUIURBUHURVJUAHUBTPUHBUHBUHBFBUHFZBU
|
|
HBGVKBUOFBUHUOBABIZTBAUCJJUDUEUHUOUMUPVLULBISUFUG $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Commutation equivalence. Kalmbach 83 p. 23. $)
|
|
comcom3 $p |- a ' C b $=
|
|
( wn comcom comcom2 ) BADBAABCEFE $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Commutation equivalence. Kalmbach 83 p. 23. $)
|
|
comcom4 $p |- a ' C b ' $=
|
|
( wn comcom3 comcom2 ) ADBABCEF $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Commutation dual. Kalmbach 83 p. 23. $)
|
|
comd $p |- a = ( ( a v b ) ^ ( a v b ' ) ) $=
|
|
( wn wa wo comcom4 df-c2 con3 oran con2 2an ax-r1 ax-r2 ) AADZBDZEZOPDEZF
|
|
ZDZABFZAPFZEZASOPABCGHITQDZRDZEZUCSUFQRJKUCUFUAUDUBUEABJAPJLMNN $.
|
|
$( [27-Aug-97] $)
|
|
|
|
$( Lemma 3(ii) of Kalmbach 83 p. 23. $)
|
|
com3ii $p |- ( a ^ ( a ' v b ) ) = ( a ^ b ) $=
|
|
( wa wn wo comcom comd lan anass ax-r1 ax-a2 anabs ax-r2 2an ) ABDZAAEZBF
|
|
ZDZPABAFZBQFZDZDZSBUBABAABCGHIUCATDZUADZSUEUCATUAJKUDAUARUDAABFZDATUFABAL
|
|
IABMNBQLONNK $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
comcom5.1 $e |- a ' C b ' $.
|
|
$( Commutation equivalence. Kalmbach 83 p. 23. $)
|
|
comcom5 $p |- a C b $=
|
|
( wn wa wo comcom4 df-c2 ax-a1 2an 2or 3tr1 df-c1 ) ABADZDZOBDZDZEZOQDZEZ
|
|
FAABEZAPEZFOQNPCGHAIZUARUBTAOBQUCBIJAOPSUCPIJKLM $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
comcom6.1 $e |- a ' C b $.
|
|
$( Commutation equivalence. Kalmbach 83 p. 23. $)
|
|
comcom6 $p |- a C b $=
|
|
( wn comcom2 comcom5 ) ABADBCEF $.
|
|
$( [26-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
comcom7.1 $e |- a C b ' $.
|
|
$( Commutation equivalence. Kalmbach 83 p. 23. $)
|
|
comcom7 $p |- a C b $=
|
|
( wn comcom3 comcom5 ) ABABDCEF $.
|
|
$( [26-Nov-97] $)
|
|
$}
|
|
|
|
$( Commutation law. $)
|
|
comor1 $p |- ( a v b ) C a $=
|
|
( wo comorr comcom ) AABCABDE $.
|
|
$( [9-Nov-97] $)
|
|
|
|
$( Commutation law. $)
|
|
comor2 $p |- ( a v b ) C b $=
|
|
( wo ax-a2 comor1 bctr ) ABCBACBABDBAEF $.
|
|
$( [9-Nov-97] $)
|
|
|
|
$( Commutation law. $)
|
|
comorr2 $p |- b C ( a v b ) $=
|
|
( wo comor2 comcom ) ABCBABDE $.
|
|
$( [26-Nov-97] $)
|
|
|
|
$( Commutation law. $)
|
|
comanr1 $p |- a C ( a ^ b ) $=
|
|
( wa coman1 comcom ) ABCAABDE $.
|
|
$( [26-Nov-97] $)
|
|
|
|
$( Commutation law. $)
|
|
comanr2 $p |- b C ( a ^ b ) $=
|
|
( wa coman2 comcom ) ABCBABDE $.
|
|
$( [26-Nov-97] $)
|
|
|
|
${
|
|
comdr.1 $e |- a = ( ( a v b ) ^ ( a v b ' ) ) $.
|
|
$( Commutation dual. Kalmbach 83 p. 23. $)
|
|
comdr $p |- a C b $=
|
|
( wn wa wo df-a oran con2 2or ax-r4 ax-r2 df-c1 comcom5 ) ABADZBDZAOPEZOP
|
|
DEZFZAABFZAPFZEZSDZCUBTDZUADZFZDUCTUAGUFSUDQUERTQABHIUARAPHIJKLLIMN $.
|
|
$( [27-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
com3i.1 $e |- ( a ^ ( a ' v b ) ) = ( a ^ b ) $.
|
|
$( Lemma 3(i) of Kalmbach 83 p. 23. $)
|
|
com3i $p |- a C b $=
|
|
( wn wa wo anor1 con2 ran ancom ax-r2 lor lea oml2 ax-a2 3tr2 df-c1 ) ABA
|
|
BDZEZSDZAEZFSABEZFAUBSFUAUBSUAAADBFZEZUBUAUCAEUDTUCASUCABGHIUCAJKCKLSAARM
|
|
NSUBOPQ $.
|
|
$( [28-Aug-97] $)
|
|
$}
|
|
|
|
${
|
|
df2c1.1 $e |- a = ( ( a v b ) ^ ( a v b ' ) ) $.
|
|
$( Dual 'commutes' analogue for ` == ` analogue of ` = ` . $)
|
|
df2c1 $p |- a C b $=
|
|
( wn wa wo df-a anor3 2or ax-r1 ax-r4 ax-r2 con2 df-c1 comcom5 ) ABADZBDZ
|
|
APQEZPQDEZFZAABFZAQFZEZTDZCUCUADZUBDZFZDUDUAUBGUGTTUGRUESUFABHAQHIJKLLMNO
|
|
$.
|
|
$( [20-Sep-98] $)
|
|
$}
|
|
|
|
${
|
|
fh.1 $e |- a C b $.
|
|
fh.2 $e |- a C c $.
|
|
$( Foulis-Holland Theorem. $)
|
|
fh1 $p |- ( a ^ ( b v c ) ) = ( ( a ^ b ) v ( a ^ c ) ) $=
|
|
( wa wo ledi wn ancom df-a ax-r1 con3 ax-r2 2an comcom2 com3ii anandi lan
|
|
wf 2or con2 anass 3tr1 an12 oran dff an0 oml3 ) ABFZACFZGZABCGZFZULUNABCH
|
|
UNULIZFZAUMBIZCIZFZFZFZTUPUMAFZAIZUQGZVCURGZFZFZVAUNVBUOVFAUMJULVFULVDIZV
|
|
EIZGZVFIUJVHUKVIABKACKUAVJVFVFVJIVDVEKLMNUBOVGUMAUSFZFZVAVGUMAVFFZFVLUMAV
|
|
FUCVMVKUMAVDFZAVEFZFAUQFZAURFZFVMVKVNVPVOVQAUQABDPQAURACEPQOAVDVERAUQURRU
|
|
DSNUMAUSUENNVAATFTUTTAUTUMUMIZFZTUSVRUMUSUMUMUSIBCUFLMSTVSUMUGLNSAUHNNUIL
|
|
$.
|
|
$( [29-Aug-97] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh2 $p |- ( b ^ ( a v c ) ) = ( ( b ^ a ) v ( b ^ c ) ) $=
|
|
( wa wo ledi wn wf oran df-a con2 ran ax-r2 lan an4 com3ii anass ax-r1
|
|
ax-r4 comcom comcom2 ancom ax-a1 ax-r5 comcom3 an12 dff 3tr1 an0 oml3 ) B
|
|
AFZBCFZGZBACGZFZUOUQBACHUQUOIZFZAIZCBUNIZFZFZFZJUSUTCFZVBFZVDUSUTUPFZVBFZ
|
|
VFUSUTBFZUPVAFZFZVHUSUQBIUTGZVAFZFZVKURVMUQUOVMUOUMIZVAFZIVMIUMUNKVPVMVOV
|
|
LVAUMVLBALMNUAOMPVNBVLFZVJFVKBUPVLVAQVQVIVJVQBUTFVIBUTBAABDUBUCRBUTUDONOO
|
|
UTBUPVAQOVGVEVBVGUTUTIZCGZFVEUPVSUTAVRCAUEUFPUTCACEUGRONOUTCVBSOVDUTJFJVC
|
|
JUTBCVAFFZUNVAFZVCJWAVTBCVASTCBVAUHUNUIUJPUTUKOOULT $.
|
|
$( [29-Aug-97] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh3 $p |- ( a v ( b ^ c ) ) = ( ( a v b ) ^ ( a v c ) ) $=
|
|
( wa wo comcom4 fh1 anor2 df-a ax-r1 lor ax-r4 ax-r2 oran 2an 3tr2 con1
|
|
wn ) ABCFZGZABGZACGZFZATZBTZCTZGZFZUFUGFZUFUHFZGZUBTZUETZUFUGUHABDHACEHIU
|
|
JAUITZGZTUNAUIJUQUBUPUAAUAUPBCKLMNOUMUKTZULTZFZTUOUKULPUTUEUEUTUCURUDUSAB
|
|
PACPQLNORS $.
|
|
$( [29-Aug-97] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh4 $p |- ( b v ( a ^ c ) ) = ( ( b v a ) ^ ( b v c ) ) $=
|
|
( wa wo comcom4 fh2 anor2 df-a ax-r1 lor ax-r4 ax-r2 oran 2an 3tr2 con1
|
|
wn ) BACFZGZBAGZBCGZFZBTZATZCTZGZFZUFUGFZUFUHFZGZUBTZUETZUGUFUHABDHACEHIU
|
|
JBUITZGZTUNBUIJUQUBUPUABUAUPACKLMNOUMUKTZULTZFZTUOUKULPUTUEUEUTUCURUDUSBA
|
|
PBCPQLNORS $.
|
|
$( [29-Aug-97] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh1r $p |- ( ( b v c ) ^ a ) = ( ( b ^ a ) v ( c ^ a ) ) $=
|
|
( wo wa fh1 ancom 2or 3tr1 ) ABCFZGABGZACGZFLAGBAGZCAGZFABCDEHLAIOMPNBAIC
|
|
AIJK $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh2r $p |- ( ( a v c ) ^ b ) = ( ( a ^ b ) v ( c ^ b ) ) $=
|
|
( wo wa fh2 ancom 2or 3tr1 ) BACFZGBAGZBCGZFLBGABGZCBGZFABCDEHLBIOMPNABIC
|
|
BIJK $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh3r $p |- ( ( b ^ c ) v a ) = ( ( b v a ) ^ ( c v a ) ) $=
|
|
( wa wo fh3 ax-a2 2an 3tr1 ) ABCFZGABGZACGZFLAGBAGZCAGZFABCDEHLAIOMPNBAIC
|
|
AIJK $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh4r $p |- ( ( a ^ c ) v b ) = ( ( a v b ) ^ ( c v b ) ) $=
|
|
( wa wo fh4 ax-a2 2an 3tr1 ) BACFZGBAGZBCGZFLBGABGZCBGZFABCDEHLBIOMPNABIC
|
|
BIJK $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh2c $p |- ( b ^ ( c v a ) ) = ( ( b ^ c ) v ( b ^ a ) ) $=
|
|
( wo wa fh2 ax-a2 lan 3tr1 ) BACFZGBAGZBCGZFBCAFZGNMFABCDEHOLBCAIJNMIK $.
|
|
$( [20-Sep-98] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh4c $p |- ( b v ( c ^ a ) ) = ( ( b v c ) ^ ( b v a ) ) $=
|
|
( wa wo fh4 ancom lor 3tr1 ) BACFZGBAGZBCGZFBCAFZGNMFABCDEHOLBCAIJNMIK $.
|
|
$( [20-Sep-98] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh1rc $p |- ( ( c v b ) ^ a ) = ( ( c ^ a ) v ( b ^ a ) ) $=
|
|
( wo wa fh1r ax-a2 ran 3tr1 ) BCFZAGBAGZCAGZFCBFZAGNMFABCDEHOLACBIJNMIK
|
|
$.
|
|
$( [10-Mar-02] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh2rc $p |- ( ( c v a ) ^ b ) = ( ( c ^ b ) v ( a ^ b ) ) $=
|
|
( wo wa fh2r ax-a2 ran 3tr1 ) ACFZBGABGZCBGZFCAFZBGNMFABCDEHOLBCAIJNMIK
|
|
$.
|
|
$( [20-Sep-98] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh3rc $p |- ( ( c ^ b ) v a ) = ( ( c v a ) ^ ( b v a ) ) $=
|
|
( wa wo fh3r ancom ax-r5 3tr1 ) BCFZAGBAGZCAGZFCBFZAGNMFABCDEHOLACBIJNMIK
|
|
$.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Foulis-Holland Theorem. $)
|
|
fh4rc $p |- ( ( c ^ a ) v b ) = ( ( c v b ) ^ ( a v b ) ) $=
|
|
( wa wo fh4r ancom ax-r5 3tr1 ) ACFZBGABGZCBGZFCAFZBGNMFABCDEHOLBCAIJNMIK
|
|
$.
|
|
$( [20-Sep-98] $)
|
|
|
|
$( Th. 4.2 Beran p. 49. $)
|
|
com2or $p |- a C ( b v c ) $=
|
|
( wo wa wn comcom df-c2 ancom 2or ax-r2 or4 fh1 comcom3 ax-r1 df-c1 ) BCF
|
|
ZASASABGZACGZFZAHZBGZUCCGZFZFZSAGZSUCGZFZSTUDFZUAUEFZFUGBUKCULBBAGZBUCGZF
|
|
UKBAABDIJUMTUNUDBAKBUCKLMCCAGZCUCGZFULCAACEIJUOUAUPUECAKCUCKLMLTUDUAUENMU
|
|
JUGUHUBUIUFUHASGUBSAKABCDEOMUIUCSGUFSUCKUCBCABDPACEPOMLQMRI $.
|
|
$( [7-Nov-97] $)
|
|
|
|
$( Th. 4.2 Beran p. 49. $)
|
|
com2an $p |- a C ( b ^ c ) $=
|
|
( wa wn wo comcom4 com2or df-a con2 ax-r1 cbtr comcom5 ) ABCFZAGZBGZCGZHZ
|
|
PGZQRSABDIACEIJUATPTBCKLMNO $.
|
|
$( [7-Nov-97] $)
|
|
|
|
$}
|
|
|
|
$( Commutation theorem for Sasaki implication. $)
|
|
combi $p |- a C ( a == b ) $=
|
|
( wa wn wo tb comanr1 comcom6 com2or dfb ax-r1 cbtr ) AABCZADZBDZCZEZABFZAM
|
|
PABGAPNOGHIRQABJKL $.
|
|
$( [25-Oct-98] $)
|
|
|
|
$( Negated biconditional (distributive form) $)
|
|
nbdi $p |- ( a == b ) ' =
|
|
( ( ( a v b ) ^ a ' ) v ( ( a v b ) ^ b ' ) ) $=
|
|
( tb wn wo wa dfnb comorr comcom comcom2 ax-a2 cbtr fh1 ax-r2 ) ABCDABEZADZ
|
|
BDZEFOPFOQFEABGOPQOAAOABHIJOBBOBBAEOBAHBAKLIJMN $.
|
|
$( [30-Aug-97] $)
|
|
|
|
$( Orthomodular law. $)
|
|
oml4 $p |- ( ( a == b ) ^ a ) =< b $=
|
|
( tb wa ancom wn wo dfb lan coman1 comcom comcom2 comcom5 fh1 or0 ran anass
|
|
wf ax-r2 3tr2 anidm ax-r1 an0 dff 2or lea bltr ) ABCZADZBADZBUIAUHDZUJUHAEU
|
|
KAABDZAFZBFZDZGZDZUJUHUPAABHIUQAULDZAUODZGZUJAULUOULAABJKAUOUMUOUOUMUMUNJKL
|
|
MNULRGULUTUJULOULURRUSULAADZBDZURVBULVAABAUAPUBAABQSRAUMDZUNDZUSUNRDRUNDRVD
|
|
UNREUNUCRVCUNAUDPTAUMUNQSUEABETSSSBAUFUG $.
|
|
$( [25-Oct-97] $)
|
|
|
|
$( Orthomodular law. $)
|
|
oml6 $p |- ( a v ( b ^ ( a ' v b ' ) ) ) = ( a v b ) $=
|
|
( wn wo wa comor1 comcom7 comor2 fh4c df-t ax-r5 ax-a2 or1 ax-r2 ax-a3 3tr2
|
|
wt ax-r1 lan an1 3tr ) ABACZBCZDZEDABDZAUDDZEUEQEUEUDABUDAUBUCFGUDBUBUCHGIU
|
|
FQUEQUFQUCDZAUBDZUCDQUFQUHUCAJKUGUCQDQQUCLUCMNAUBUCOPRSUETUA $.
|
|
$( [3-Jan-99] $)
|
|
|
|
${
|
|
gsth.1 $e |- a C b $.
|
|
gsth.2 $e |- b C c $.
|
|
gsth.3 $e |- a C ( b ^ c ) $.
|
|
$( Gudder-Schelp's Theorem. Beran, p. 262, Th. 4.1. $)
|
|
gsth $p |- ( a ^ b ) C c $=
|
|
( wa wo wn comcom fh4rc comcom2 lan fh1r ran lea ancom wf ax-r1 3tr lecom
|
|
2an an4 an32 comd leo letr coman2 com2or df2le2 fh1 anass dff an0 lor or0
|
|
cbtr ax-r2 2or ax-a2 lelan bltr df-le2 3tr2 df2c1 ) ABGZCVFCHZVFCIZHZGZVF
|
|
VJACHZBCHZGZAVHHZBVHHZGZGVKVNGZVLVOGZGZVFVGVMVIVPBCAEABDJZKBVHABCELVTKUBV
|
|
KVLVNVOUCVQBGVKBGZVNGZVSVFVKVNBUDBVRVQBCEUEMWBVFCBGZHZVNGVFVNGZWCVNGZHZVF
|
|
WAWDVNBACVTENOVNVFWCVFVNVFVNVFAVNABPAVHUFUGZUAJVNBCGZWCWIVNWIAVHAWIFJZWIC
|
|
BCUHLZUIJBCQUQNWGVFWIAGZHWLVFHVFWEVFWFWLVFVNWHUJWFWIVNGWLWIVHGZHZWLWCWIVN
|
|
CBQOWIAVHWJWKUKWNWLRHWLWMRWLWMBCVHGZGBRGRBCVHULWORBRWOCUMSMBUNTUOWLUPURTU
|
|
SVFWLUTWLVFWLAWIGVFWIAQWIBABCPVAVBVCTTVDTSVE $.
|
|
$( [20-Sep-98] $)
|
|
$}
|
|
|
|
${
|
|
gsth2.1 $e |- b C c $.
|
|
gsth2.2 $e |- a C ( b ^ c ) $.
|
|
$( Stronger version of Gudder-Schelp's Theorem. Beran, p. 263, Th. 4.2. $)
|
|
gsth2 $p |- ( a ^ b ) C c $=
|
|
( wa wn comcom ancom ax-a2 ran ax-r2 comor2 comcom7 comcom2 coman1 com2or
|
|
wo df-a cbtr gsth bctr lor ax-r4 ax-r1 com2an omla ) CABFZCBBGZBAFZRZFZUH
|
|
CBUKBCDHCUKCBUIAGZRZFZUKGZUOCUOUMUIRZBFZCUOUNBFURBUNIUNUQBUIUMJKLUQBCUQBU
|
|
MUIMNDBCFZUQUSUMUIUSAAUSEHOUSBBCPOQHUAUBHUOUIUNGZRZGZUPBUNSUPVBUKVAUJUTUI
|
|
BASUCUDUELTNUFULUJUHBAUGBAILTH $.
|
|
$( [20-Sep-98] $)
|
|
$}
|
|
|
|
${
|
|
gstho.1 $e |- b C c $.
|
|
gstho.2 $e |- a C ( b v c ) $.
|
|
$( "OR" version of Gudder-Schelp's Theorem. $)
|
|
gstho $p |- ( a v b ) C c $=
|
|
( wo wn wa anor3 ax-r1 comcom4 cbtr gsth2 bctr comcom5 ) ABFZCPGZAGZBGZHZ
|
|
CGZTQABIJRSUABCDKRBCFZGZSUAHZAUBEKUDUCBCIJLMNO $.
|
|
$( [19-Oct-98] $)
|
|
$}
|
|
|
|
${
|
|
gt1.1 $e |- a = ( b v c ) $.
|
|
gt1.2 $e |- b =< d $.
|
|
gt1.3 $e |- c =< d ' $.
|
|
$( Part of Lemma 1 from Gaisi Takeuti, "Quantum Set Theory". $)
|
|
gt1 $p |- a C d $=
|
|
( wo lecom comcom wn comcom7 com2or bctr ) ABCHZDEDODBCBDBDFIJCDCDCDKGILJ
|
|
MJN $.
|
|
$( [2-Dec-98] $)
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Commutator (orthomodular lattice theorems)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
cmtr1com.1 $e |- C ( a , b ) = 1 $.
|
|
$( Commutator equal to 1 commutes. Theorem 2.11 of Beran, p. 86. $)
|
|
cmtr1com $p |- a C b $=
|
|
( wa wn wo lea lel2or df-le2 le1 wcmtr df-cmtr ax-a2 3tr2 leror bltr lebi
|
|
wt lem3.1 ax-r1 df-c1 ) ABABDZABEZDZFZAUEAUEAUBAUDABGAUCGHIAEZUEFZRUGJRUF
|
|
BDZUFUCDZFZUEFZUGABKUEUJFRUKABLCUEUJMNUJUFUEUHUFUIUFBGUFUCGHOPQSTUA $.
|
|
$( [24-Jan-99] $)
|
|
$}
|
|
|
|
${
|
|
comcmtr1.1 $e |- a C b $.
|
|
$( Commutation implies commutator equal to 1. Theorem 2.11 of Beran,
|
|
p. 86. $)
|
|
comcmtr1 $p |- C ( a , b ) = 1 $=
|
|
( wa wn wo wcmtr wt df-c2 comcom3 2or ax-r1 df-cmtr df-t 3tr1 ) ABDABEZDF
|
|
ZAEZBDRPDFZFZARFZABGHUATAQRSABCIRBABCJIKLABMANO $.
|
|
$( [24-Jan-99] $)
|
|
$}
|
|
|
|
${
|
|
i0cmtrcom.1 $e |- ( a ->0 C ( a , b ) ) = 1 $.
|
|
$( Commutator element ` ->0 ` commutator implies commutation. $)
|
|
i0cmtrcom $p |- a C b $=
|
|
( wa wn wo lea lel2or df-le2 wcmtr wi0 df-cmtr lor ax-r1 ax-a2 ax-r2 or12
|
|
wt 3tr df-i0 3tr1 lem3.1 df-c1 ) ABABDZABEZDZFZAUGAUGAUDAUFABGAUEGHIAEZUG
|
|
FZAABJZKZRUHUGUHBDZUHUEDZFZFZFZUHUJFZUIUKUQUPUJUOUHABLMNUIUGUHFZUGUHUNFZF
|
|
ZUPUHUGOUTURUSUHUGUSUNUHFUHUHUNOUNUHULUHUMUHBGUHUEGHIPMNUGUHUNQSAUJTUACPU
|
|
BNUC $.
|
|
$( [24-Jan-99] $)
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Kalmbach conditional
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Kalmbach implication and biconditional. $)
|
|
i3bi $p |- ( ( a ->3 b ) ^ ( b ->3 a ) ) = ( a == b ) $=
|
|
( wn wa wo lea leo ax-a2 letr ancom lecom comcom2 comcom bctr wf ax-r2 bltr
|
|
ax-r1 lan 2or wi3 tb anor2 lbtr le3tr1 le2or oridm fh2 cbtr fh1 ran an4 dff
|
|
anor1 2an anidm an12 con2 an0 anandi coman1 an32 or0 lor oran con3 fh3 3tr2
|
|
anass df-i3 or32 dfb 3tr1 ) ACZBCZDZVNBDZAVNBEZDZEZEZVPVOADZBVOAEZDZEZEZDZA
|
|
BDZVPEZABUAZBAUAZDABUBVPVTWEDZEVPWHEWGWIWLWHVPWLVTWBDZVTWDDZEZWHWBVTWDWBBVN
|
|
EZCZVTBAUCZVTWQVTWPVTWPVTWPWPEWPVQWPVSWPVQVNWPVNBFVNVRWPVNBGVNBHUDIVRADZVRV
|
|
SWPVRAFZAVRJZBVNHZUEUFWPUGUDKLMNWBWQWDWRWDWQWDWPWDWPWDBWPBWCFZBVNGIKLMNUHWO
|
|
OWHEZWHWMOWNWHWMWBVTDZOVTWBJXEWBVQDZWBVSDZEZOWBVQVSWBWCCZVQWBWCWBWCWBVOWCVO
|
|
AFVOAGIKLVQXIVQBVNDZXIVNBJZBAUNPZRZUIWBVRCZVSWBAVODZXNVOAJZABUNPVSXNVSVRVSV
|
|
RVSWSVRXAWTQKLMNUJXHOOEOXFOXGOXFXOVQDZOWBXOVQXPUKXQAVNDZVOBDZDZOAVOVNBULXTO
|
|
ODZOYAXTOXROXSAUMZOBVODZXSBUMZBVOJPUOROUPPPPXGAWBVRDZDZOWBAVRUQYFAODZOYGYFO
|
|
YEAOWBWBCZDYEWBUMYHVRWBYHWPVRWBWPWRURXBPSPSRAUSZPPTOUGPPPWNWDVTDZWHVTWDJYJW
|
|
DVQDZWDVSDZEZWHVQWDVSVQXIWDXLWDXIWDWCWDWCWDWCBDWCBWCJWCBFQKLMNVQAVOEZCZVSAB
|
|
UCVSYOVSYNVSYNVSAYNAVRFZAVOGIKLMNUHYMYLYKEZWHYKYLHYQWHOEZWHYLWHYKOYLBADZWCV
|
|
RDDZWHBWCAVRULYTYSWCDZYSVRDZDZWHYSWCVRUTUUCYSYSDZWHUUAYSUUBYSUUAYSVODZYSADZ
|
|
EZYSYSVOAYSBBAVAZLYSWHABAJZABVANZUJUUGYSOEZYSUUGOYSEZUUKUUEOUUFYSUUEYCADZOB
|
|
AVOVBUUMAYCDZOYCAJUUNYGOYGUUNOYCAYDSRYIPPPUUFBAADZDYSBAAVIUUOABAUPSPTOYSHZP
|
|
YSVCZPPUUBYSVNDZYSBDZEZYSYSVNBYSAUUJLUUHUJUUTUUKYSUUTUULUUKUUROUUSYSUURBXRD
|
|
ZOBAVNVIUVABODZOUVBUVAOXRBYBSRBUSZPPUUSBBDZADYSBABVBUVDBABUPUKPTUUPPUUQPPUO
|
|
UUDYSWHYSUPUUIPPPPYKUVBOYKBWCVQDZDZUVBBWCVQVIUVBUVFOUVEBOWCXIDUVEWCUMXIVQWC
|
|
XMSPSRPUVCPTWHVCZPPPPTXDYRWHOWHHUVGPPPVDVPVTWEVPBAEZCZVTVPVOVNDZUVIVNVOJUVJ
|
|
UVHUVHUVJCBAVERVFPVTUVIVTUVHVTUVHVQBVSAVQXJBXKBVNFQYPUFKLMNVPABEZCZWEVPUVKU
|
|
VKVPCABVERVFWEUVLWEUVKWEUVKWBAWDBWBXOAXPAVOFQXCUFKLMNVGVPWHHVHWJWAWKWFWJVQV
|
|
PEVSEZWAABVJUVMVTVPEWAVQVPVSVKVTVPHPPWKWBUVJEWDEZWFBAVJUVNWEUVJEZWFWBUVJWDV
|
|
KUVOWEVPEWFUVJVPWEVOVNJVDWEVPHPPPUOABVLVM $.
|
|
$( [5-Nov-97] $)
|
|
|
|
$( Kalmbach implication OR builder. $)
|
|
i3or $p |- ( ( a == b ) ' v ( ( a v c ) ->3 ( b v c ) ) ) = 1 $=
|
|
( tb wn wo wi3 wt le1 ka4ot ax-r1 wa i3bi lea bltr lelor lebi ) ABDEZACFZBC
|
|
FZGZFZHUBIHRSTDZFZUBUDHABCJKUCUARUCUATSGZLZUAUFUCSTMKUAUENOPOQ $.
|
|
$( [26-Dec-97] $)
|
|
|
|
$( Alternate definition for Kalmbach implication. $)
|
|
df2i3 $p |- ( a ->3 b ) = ( ( a ' ^ b ' ) v ( ( a ' v b ) ^
|
|
( a v ( a ' ^ b ) ) ) ) $=
|
|
( wi3 wn wa wo df-i3 ax-a3 coman1 comcom comcom2 comcom5 comorr fh4 lea leo
|
|
or12 letr lan ax-r2 df-le2 ancom ax-a2 lor ) ABCADZBEZUEBDEZFAUEBFZEZFZUGUH
|
|
AUFFZEZFZABGUJUFUGUIFFZUMUFUGUIHUNUGUFUIFZFUMUFUGUIQUOULUGUOUFAFZUFUHFZEZUL
|
|
AUFUHAUFUEUFUFUEUEBIJKLAUHUEUHUEBMKLNURUPUHEZULUQUHUPUFUHUFUEUHUEBOUEBPRUAS
|
|
USUHUPEULUPUHUBUPUKUHUFAUCSTTTUDTTT $.
|
|
$( [7-Nov-97] $)
|
|
|
|
$( Alternate Kalmbach conditional. $)
|
|
dfi3b $p |- ( a ->3 b ) =
|
|
( ( a ' v b ) ^ ( ( a v ( a ' ^ b ' ) ) v ( a ' ^ b ) ) ) $=
|
|
( wn wa wo wi3 ax-a2 ax-a3 oridm ax-r1 anidm ran anass ax-r2 lan 2or com2an
|
|
ancom fh1 3tr1 an12 lea leo letr df2le2 comor1 comcom7 comor2 coman1 coman2
|
|
comcom2 fh1r df-i3 com2or ) ACZBDZUOBCZDZEAUOBEZDZEZUSAUREZDZUSUPDZEZABFUSV
|
|
BUPEDUOUPDZBUPDZEZUSADZUSURDZEZEZVKVHEVAVEVHVKGVAUPURUTEZEVLUPURUTHUPVHVMVK
|
|
UPUPUPEZVHVNUPUPIJUPVFUPVGUPUOUODZBDVFUOVOBVOUOUOKJLUOUOBMNUPUOBBDZDVGBVPUO
|
|
VPBBKJOUOBBUANPNVMVJVIEVKURVJUTVIURURUSDZVJVQURURUSURUOUSUOUQUBUOBUCUDUEJUR
|
|
USRNAUSRPVJVIGNPNVCVKVDVHUSAURUSAUOBUFZUGZUSUOUQVRUSBUOBUHZUKQZSUPUOBUOBUIU
|
|
OBUJULPTABUMUSVBUPUSAURVSWAUNUSUOBVRVTQST $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Alternate non-tollens conditional. $)
|
|
dfi4b $p |- ( a ->4 b ) =
|
|
( ( a ' v b ) ^ ( ( b ' v ( b ^ a ' ) ) v ( b ^ a ) ) ) $=
|
|
( wi4 wn wi3 wo wa i4i3 dfi3b ax-a2 ax-a1 ax-r5 ax-r2 ran lor 2an 2or ax-r1
|
|
or32 ) ABCBDZADZEZUABFZTBUAGZFZBAGZFZGZABHUBTDZUAFZTUIUADZGZFUIUAGZFZGZUHTU
|
|
AIUHUOUCUJUGUNUCBUAFUJUABJBUIUABKZLMUGTUMFZULFUNUEUQUFULUDUMTBUIUAUPNOBUIAU
|
|
KUPAKPQTUMULSMPRMM $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Equivalence for Kalmbach implication. $)
|
|
i3n2 $p |- ( a ' ->3 b ' ) = ( ( a ^ b ) v ( ( a v b ' ) ^
|
|
( a ' v ( a ^ b ' ) ) ) ) $=
|
|
( wn wi3 wa wo df2i3 ax-a1 2an ax-r5 ran lor 2or ax-r1 ax-r2 ) ACZBCZDPCZQC
|
|
ZEZRQFZPRQEZFZEZFZABEZAQFZPAQEZFZEZFZPQGUKUEUFTUJUDARBSAHZBHIUGUAUIUCARQULJ
|
|
UHUBPARQULKLIMNO $.
|
|
$( [9-Nov-97] $)
|
|
|
|
$( Equivalence for Kalmbach implication. $)
|
|
ni32 $p |- ( a ->3 b ) ' = ( ( a v b ) ^ ( ( a ^ b ' ) v
|
|
( a ' ^ ( a v b ' ) ) ) ) $=
|
|
( wi3 wo wn wa df2i3 oran anor1 con2 ax-r1 anor2 lan ax-r4 ax-r2 2an ) ABCZ
|
|
ABDZABEZFZAEZASDZFZDZFZQUASFZUABDZAUABFZDZFZDZUEEZABGUKUFEZUJEZFZEULUFUJHUO
|
|
UEUEUORUMUDUNABHUDTEZUCEZFZEUNTUCHURUJUJURUGUPUIUQUPUGTUGABIJKUIUAUHEZFZEUQ
|
|
AUHHUTUCUSUBUAUHUBABLJMNOPKNOPKNOOJ $.
|
|
$( [9-Nov-97] $)
|
|
|
|
$( Theorem for Kalmbach implication. $)
|
|
oi3ai3 $p |- ( ( a ^ b ) v ( a ->3 b ) ' ) =
|
|
( ( a v b ) ^ ( a ' ->3 b ' ) ) $=
|
|
( wa wo wn wi3 lea leo letr lecom coman1 ancom comcom2 com2an com2or df-le2
|
|
bctr fh3 ax-a3 ax-r2 ax-r1 ax-a2 ax-r5 2an ni32 lor i3n1 lan 3tr1 ) ABCZABD
|
|
ZABEZCZAEZAULDZCZDZCZDZUKUMUJDZUPDZCZUJABFEZDUKUNULFZCUSUJUKDZUJUQDZCVBUJUK
|
|
UQUJUKUJAUKABGABHIZJUJUMUPUJAULABKZUJBUJBACBABLBAKQMZNUJUNUOUJAVHMUJAULVHVI
|
|
ONORVEUKVFVAUJUKVGPVFUJUMDZUPDZVAVKVFUJUMUPSUAVJUTUPUJUMUBUCTUDTVCURUJABUEU
|
|
FVDVAUKABUGUHUI $.
|
|
$( [9-Nov-97] $)
|
|
|
|
${
|
|
i3lem.1 $e |- ( a ->3 b ) = 1 $.
|
|
$( Lemma for Kalmbach implication. $)
|
|
i3lem1 $p |- ( ( a ' ^ b ) v ( a ' ^ b ' ) ) = a ' $=
|
|
( wn wa wo wt coman1 comcom comorr comcom3 com2an anass ax-r1 anidm ax-r2
|
|
fh1 ran anabs omlan 2or ax-a2 wi3 df2i3 lan an1 ) ADZBEZUGBDZEZFZUGGEZUGU
|
|
KUGUJUGBFZAUHFZEZFZEZULUQUKUQUGUJEZUGUOEZFZUKUGUJUOUJUGUGUIHIUGUMUNUGBJAU
|
|
NAUHJKLQUTUJUHFUKURUJUSUHURUGUGEZUIEZUJVBURUGUGUIMNVAUGUIUGORPUSUGUMEZUNE
|
|
ZUHVDUSUGUMUNMNVDUGUNEUHVCUGUNUGBSRABTPPUAUJUHUBPPNUPGUGUPABUCZGVEUPABUDN
|
|
CPUEPUGUFP $.
|
|
$( [7-Nov-97] $)
|
|
|
|
$( Lemma for Kalmbach implication. $)
|
|
i3lem2 $p |- a C b $=
|
|
( wn wa wo i3lem1 ax-r1 df-c1 comcom2 comcom5 ) ABADZBLBLBELBDEFLABCGHIJK
|
|
$.
|
|
$( [7-Nov-97] $)
|
|
|
|
$( Lemma for Kalmbach implication. $)
|
|
i3lem3 $p |- ( ( a ' v b ) ^ b ' ) = ( a ' ^ b ' ) $=
|
|
( wn wa omlan ancom ax-a2 ax-a3 ax-r1 i3lem1 lor orabs ax-r2 2or 3tr2 lan
|
|
wo 3tr1 ) BDZBTADZEZRZEZUBUABRZTEZUATEZBUAFUFTUEEUDUETGUEUCTUEBUARZUCUABH
|
|
BUABEZUGRZRZBUIRZUGRZUHUCUMUKBUIUGIJUJUABABCKLULBUGUBULBBUAEZRBUIUNBUABGL
|
|
BUAMNUATGZOPNQNUOS $.
|
|
$( [7-Nov-97] $)
|
|
|
|
$( Lemma for Kalmbach implication. $)
|
|
i3lem4 $p |- ( a ' v b ) = 1 $=
|
|
( wn wo wa wt i3lem1 ax-r5 ax-r1 omln wi3 df-i3 ax-r2 3tr2 ) ADZAPBEZFZEZ
|
|
PBFPBDFEZREZQGUASTPRABCHIJABKUAABLZGUBUAABMJCNO $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
$( Commutation theorem. $)
|
|
comi31 $p |- a C ( a ->3 b ) $=
|
|
( wn wa wo wi3 coman1 comcom comcom2 comcom5 com2or df-i3 ax-r1 cbtr ) AACZ
|
|
BDZOBCZDZEZAOBEZDZEZABFZASUAAPRAPOPPOOBGHIJARORROOQGHIJKUAAATGHKUCUBABLMN
|
|
$.
|
|
$( [9-Nov-97] $)
|
|
|
|
${
|
|
com2i3.1 $e |- a C b $.
|
|
com2i3.2 $e |- a C c $.
|
|
$( Commutation theorem. $)
|
|
com2i3 $p |- a C ( b ->3 c ) $=
|
|
( wn wa wo wi3 comcom2 com2an com2or df-i3 ax-r1 cbtr ) ABFZCGZPCFZGZHZBP
|
|
CHZGZHZBCIZATUBAQSAPCABDJZEKAPRUEACEJKLABUADAPCUEELKLUDUCBCMNO $.
|
|
$( [9-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
comi32.1 $e |- a C b $.
|
|
$( Commutation theorem. $)
|
|
comi32 $p |- a C ( b ->3 a ) $=
|
|
( comid com2i3 ) ABACADE $.
|
|
$( [9-Nov-97] $)
|
|
$}
|
|
|
|
$( Lemma 4 of Kalmbach p. 240. $)
|
|
lem4 $p |- ( a ->3 ( a ->3 b ) ) = ( a ' v b ) $=
|
|
( wi3 wn wa wo df-i3 lan oridm lecom comcom wf ancom ax-r2 ax-r1 3tr2 orabs
|
|
lea lor 2or le2or lbtr comcom3 fh1 anass dff df2le2 orordi or32 ax-r5 ax-r4
|
|
an0 or0 oran con2 oml2 ax-a3 omln ) AABCZCADZUSEZUTUSDEZFZAUTUSFZEZFZUTBFZA
|
|
USGVFUTAVGEZFZVGVCUTVEVHVCUTBEZUTBDZEZFZVMDZUTEZFUTVAVMVBVOVAUTVMVHFZEZVMUS
|
|
VPUTABGZHVQUTVMEZUTVHEZFZVMUTVMVHVMUTVMUTVMUTUTFZUTVJUTVLUTUTBRUTVKRUAUTIZU
|
|
BZJKAVHVHAVHAAVGRJKUCUDWAVSLFZVMVTLVSUTAEZVGEVGWFEZVTLWFVGMUTAVGUEWGVGLEZLW
|
|
HWGLWFVGLAUTEWFAUFAUTMNHOVGULNPSWEVSVMVSUMVSVMUTEVMUTVMMVMUTWDUGNNNNNAUSFZD
|
|
AVMFZDZVBVOWIWJWIAVPFZWJUSVPAVRSWLWJAVHFZFZWJAVMVHUHWNWJAFZWJWMAWJAVGQSWOAA
|
|
FZVMFWJAVMAUIWPAVMAIUJNNNNUKWIVBAUSUNUOWKUTVNEZVOWJWQAVMUNUOUTVNMNPTVMUTWDU
|
|
PNVDVGAVDVIVGVDUTVPFZVIUSVPUTVRSWRUTVMFZVHFZVIWTWRUTVMVHUQOWSUTVHWSUTVJFZUT
|
|
VLFZFZUTUTVJVLUHXCWBUTXAUTXBUTUTBQUTVKQTWCNNUJNNABURZNHTXDNN $.
|
|
$( [5-Nov-97] $)
|
|
|
|
${
|
|
i0i3.1 $e |- ( a ' v b ) = 1 $.
|
|
$( Translation to Kalmbach implication. $)
|
|
i0i3 $p |- ( a ->3 ( a ->3 b ) ) = 1 $=
|
|
( wi3 wn wo wt lem4 ax-r2 ) AABDDAEBFGABHCI $.
|
|
$( [9-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i3i0.1 $e |- ( a ->3 ( a ->3 b ) ) = 1 $.
|
|
$( Translation from Kalmbach implication. $)
|
|
i3i0 $p |- ( a ' v b ) = 1 $=
|
|
( wn wo wi3 wt lem4 ax-r1 ax-r2 ) ADBEZAABFFZGLKABHICJ $.
|
|
$( [9-Nov-97] $)
|
|
$}
|
|
|
|
$( Soundness proof for KA14. $)
|
|
ska14 $p |- ( ( a ' v b ) ->3 ( a ->3 ( a ->3 b ) ) ) = 1 $=
|
|
( wn wo wi3 wt lem4 ax-r1 ri3 i3id ax-r2 ) ACBDZAABEEZEMMEFLMMMLABGHIMJK $.
|
|
$( [3-Nov-97] $)
|
|
|
|
${
|
|
i3le.1 $e |- ( a ->3 b ) = 1 $.
|
|
$( L.e. to Kalmbach implication. $)
|
|
i3le $p |- a =< b $=
|
|
( wn wt wa ancom wo i3lem3 i3lem4 ran 3tr2 an1 df2le1 lecon1 ) BABDZADZEP
|
|
FZPEFPQFZPEPGQBHZPFQPFRSABCITEPABCJKQPGLPMLNO $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
$( Biconditional implies Kalmbach implication. $)
|
|
bii3 $p |- ( ( a == b ) ->3 ( a ->3 b ) ) = 1 $=
|
|
( tb wi3 wa i3bi ax-r1 lea bltr lei3 ) ABCZABDZKLBADZEZLNKABFGLMHIJ $.
|
|
$( [9-Nov-97] $)
|
|
|
|
${
|
|
binr1.1 $e |- ( a ->3 b ) = 1 $.
|
|
$( Pavicic binary logic ax-r1 analog. $)
|
|
binr1 $p |- ( b ' ->3 a ' ) = 1 $=
|
|
( wn i3le lecon lei3 ) BDADABABCEFG $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
binr2.1 $e |- ( a ->3 b ) = 1 $.
|
|
binr2.2 $e |- ( b ->3 c ) = 1 $.
|
|
$( Pavicic binary logic ax-r2 analog. $)
|
|
binr2 $p |- ( a ->3 c ) = 1 $=
|
|
( i3le letr lei3 ) ACABCABDFBCEFGH $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
binr3.1 $e |- ( a ->3 c ) = 1 $.
|
|
binr3.2 $e |- ( b ->3 c ) = 1 $.
|
|
$( Pavicic binary logic axr3 analog. $)
|
|
binr3 $p |- ( ( a v b ) ->3 c ) = 1 $=
|
|
( wo i3le le2or oridm lbtr lei3 ) ABFZCLCCFCACBCACDGBCEGHCIJK $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
$( Theorem for Kalmbach implication. $)
|
|
i31 $p |- ( a ->3 1 ) = 1 $=
|
|
( wt wi3 wn wo df-t li3 bina3 ax-r2 ) ABCAAADZEZCBBKAAFGAJHI $.
|
|
$( [7-Nov-97] $)
|
|
|
|
${
|
|
i3aa.1 $e |- a = 1 $.
|
|
$( Add antecedent. $)
|
|
i3aa $p |- ( b ->3 a ) = 1 $=
|
|
( wi3 wt i31 li3 bi1 wwbmpr ) BADZBEDZBFJKAEBCGHI $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
$( Antecedent absorption. $)
|
|
i3abs1 $p |- ( a ->3 ( a ->3 ( a ->3 b ) ) ) = ( a ->3 ( a ->3 b ) ) $=
|
|
( wn wa wo wi3 orordi orabs 2or oridm ax-r2 ax-r5 ax-a3 omln 3tr2 df-i3 lor
|
|
lem4 3tr1 ) ACZTBDZTBCZDZEZATBEZDZEZEZUEAAABFZFZFZUJTUDEZUFETUFEUHUEULTUFUL
|
|
TUAEZTUCEZEZTTUAUCGUOTTETUMTUNTTBHTUBHITJKKLTUDUFMABNOUKTUIEUHAUIRUIUGTABPQ
|
|
KABRS $.
|
|
$( [16-Nov-97] $)
|
|
|
|
${
|
|
i3abs2.1 $e |- ( a ->3 ( a ->3 ( a ->3 b ) ) ) = 1 $.
|
|
$( Antecedent absorption. $)
|
|
i3abs2 $p |- ( a ->3 ( a ->3 b ) ) = 1 $=
|
|
( wi3 i3abs1 bi1 wwbmp ) AAABDDZDZHCIHABEFG $.
|
|
$( [9-Nov-97] $)
|
|
$}
|
|
|
|
$( Antecedent absorption. $)
|
|
i3abs3 $p |- ( ( a ->3 b ) ->3 ( ( a ->3 b ) ->3 a ) ) =
|
|
( ( a ->3 b ) ->3 a ) $=
|
|
( wi3 wn wo wa wt df-t lan an1 comi31 comcom comcom3 comcom4 fh1 3tr2 ax-r1
|
|
wf ax-a2 ax-r2 comid comcom2 dff ax-r5 or0 2or fh4 ancom lem4 df-i3 3tr1
|
|
ran ) ABCZDZAEZUNAFUNADZFEZUMUOFZEZUMUMACZCUTUSUOUSUNUMAFZEZUOUQUNURVAUNUQU
|
|
NGFUNAUPEZFUNUQGVCUNAHIUNJUNAUPUMAAUMABKLZMUMAVDNOPQURUMUNFZVAEZVAUMUNAUMUM
|
|
UMUAUBZVDORVAEVAREVFVARVASRVEVAUMUCUDVAUEPTUFVBUNUMEZUOFZUOUMUNAVGVDUGVIUOG
|
|
FZUOVIGUOFVJVHGUOVHUMUNEZGUNUMSGVKUMHQTULGUOUHTUOJTTTQUMAUIUMAUJUK $.
|
|
$( [19-Nov-97] $)
|
|
|
|
$( Commutative law for conjunction with Kalmbach implication. $)
|
|
i3orcom $p |- ( ( a v b ) ->3 ( b v a ) ) = 1 $=
|
|
( wo wi3 i3id ax-a2 ri3 bi1 wwbmp ) BACZJDZABCZJDZJEKMJLJBAFGHI $.
|
|
$( [7-Nov-97] $)
|
|
|
|
$( Commutative law for disjunction with Kalmbach implication. $)
|
|
i3ancom $p |- ( ( a ^ b ) ->3 ( b ^ a ) ) = 1 $=
|
|
( wa wi3 i3id ancom ri3 bi1 wwbmp ) BACZJDZABCZJDZJEKMJLJBAFGHI $.
|
|
$( [7-Nov-97] $)
|
|
|
|
${
|
|
bi3tr.1 $e |- a = b $.
|
|
bi3tr.2 $e |- ( b ->3 c ) = 1 $.
|
|
$( Transitive inference. $)
|
|
bi3tr $p |- ( a ->3 c ) = 1 $=
|
|
( wi3 ri3 bi1 wwbmpr ) ACFZBCFZEJKABCDGHI $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i3btr.1 $e |- ( a ->3 b ) = 1 $.
|
|
i3btr.2 $e |- b = c $.
|
|
$( Transitive inference. $)
|
|
i3btr $p |- ( a ->3 c ) = 1 $=
|
|
( wi3 li3 bi1 wwbmp ) ABFZACFZDJKBCAEGHI $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i33tr1.1 $e |- ( a ->3 b ) = 1 $.
|
|
i33tr1.2 $e |- c = a $.
|
|
i33tr1.3 $e |- d = b $.
|
|
$( Transitive inference useful for introducing definitions. $)
|
|
i33tr1 $p |- ( c ->3 d ) = 1 $=
|
|
( bi3tr ax-r1 i3btr ) CBDCABFEHDBGIJ $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i33tr2.1 $e |- ( a ->3 b ) = 1 $.
|
|
i33tr2.2 $e |- a = c $.
|
|
i33tr2.3 $e |- b = d $.
|
|
$( Transitive inference useful for eliminating definitions. $)
|
|
i33tr2 $p |- ( c ->3 d ) = 1 $=
|
|
( ax-r1 i33tr1 ) ABCDEACFHBDGHI $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i3con1.1 $e |- ( a ' ->3 b ' ) = 1 $.
|
|
$( Contrapositive. $)
|
|
i3con1 $p |- ( b ->3 a ) = 1 $=
|
|
( wn binr1 ax-a1 i33tr1 ) BDZDADZDBAIHCEBFAFG $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i3ror.1 $e |- ( a ->3 b ) = 1 $.
|
|
$( WQL (Weak Quantum Logic) rule. $)
|
|
i3ror $p |- ( ( a v c ) ->3 ( b v c ) ) = 1 $=
|
|
( wo bina3 binr2 bina4 binr3 ) ACBCEZABJDBCFGBCHI $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i3lor.1 $e |- ( a ->3 b ) = 1 $.
|
|
$( WQL (Weak Quantum Logic) rule. $)
|
|
i3lor $p |- ( ( c v a ) ->3 ( c v b ) ) = 1 $=
|
|
( wo i3orcom i3ror binr2 ) CAEACEZCBEZCAFIBCEJABCDGBCFHH $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i32or.1 $e |- ( a ->3 b ) = 1 $.
|
|
i32or.2 $e |- ( c ->3 d ) = 1 $.
|
|
$( WQL (Weak Quantum Logic) rule. $)
|
|
i32or $p |- ( ( a v c ) ->3 ( b v d ) ) = 1 $=
|
|
( wo i3ror i3lor binr2 ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i3ran.1 $e |- ( a ->3 b ) = 1 $.
|
|
$( WQL (Weak Quantum Logic) rule. $)
|
|
i3ran $p |- ( ( a ^ c ) ->3 ( b ^ c ) ) = 1 $=
|
|
( wn wo wa binr1 i3ror df-a i33tr1 ) AEZCEZFZEBEZMFZEACGBCGPNOLMABDHIHACJ
|
|
BCJK $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i3lan.1 $e |- ( a ->3 b ) = 1 $.
|
|
$( WQL (Weak Quantum Logic) rule. $)
|
|
i3lan $p |- ( ( c ^ a ) ->3 ( c ^ b ) ) = 1 $=
|
|
( wa i3ran ancom i33tr1 ) ACEBCECAECBEABCDFCAGCBGH $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i32an.1 $e |- ( a ->3 b ) = 1 $.
|
|
i32an.2 $e |- ( c ->3 d ) = 1 $.
|
|
$( WQL (Weak Quantum Logic) rule. $)
|
|
i32an $p |- ( ( a ^ c ) ->3 ( b ^ d ) ) = 1 $=
|
|
( wa i3ran i3lan binr2 ) ACGBCGBDGABCEHCDBFIJ $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i3ri3.1 $e |- ( a ->3 b ) = 1 $.
|
|
i3ri3.2 $e |- ( b ->3 a ) = 1 $.
|
|
$( WQL (Weak Quantum Logic) rule. $)
|
|
i3ri3 $p |- ( ( a ->3 c ) ->3 ( b ->3 c ) ) = 1 $=
|
|
( wi3 i3le lebi ri3 bile lei3 ) ACFZBCFZLMABCABABDGBAEGHIJK $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i3li3.1 $e |- ( a ->3 b ) = 1 $.
|
|
i3li3.2 $e |- ( b ->3 a ) = 1 $.
|
|
$( WQL (Weak Quantum Logic) rule. $)
|
|
i3li3 $p |- ( ( c ->3 a ) ->3 ( c ->3 b ) ) = 1 $=
|
|
( wi3 i3le lebi li3 bile lei3 ) CAFZCBFZLMABCABABDGBAEGHIJK $.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i32i3.1 $e |- ( a ->3 b ) = 1 $.
|
|
i32i3.2 $e |- ( b ->3 a ) = 1 $.
|
|
i32i3.3 $e |- ( c ->3 d ) = 1 $.
|
|
i32i3.4 $e |- ( d ->3 c ) = 1 $.
|
|
$( WQL (Weak Quantum Logic) rule. $)
|
|
i32i3 $p |- ( ( a ->3 c ) ->3 ( b ->3 d ) ) = 1 $=
|
|
( wi3 i3le lebi 2i3 bile lei3 ) ACIZBDIZOPABCDABABEJBAFJKCDCDGJDCHJKLMN
|
|
$.
|
|
$( [7-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i0i3tr.1 $e |- ( a ->3 ( a ->3 b ) ) = 1 $.
|
|
i0i3tr.2 $e |- ( b ->3 c ) = 1 $.
|
|
$( Transitive inference. $)
|
|
i0i3tr $p |- ( a ->3 ( a ->3 c ) ) = 1 $=
|
|
( wn wo i3i0 i3lor skmp3 i0i3 ) ACAFZBGLCGABDHBCLEIJK $.
|
|
$( [9-Nov-97] $)
|
|
$}
|
|
|
|
${
|
|
i3i0tr.1 $e |- ( a ->3 b ) = 1 $.
|
|
i3i0tr.2 $e |- ( b ->3 ( b ->3 c ) ) = 1 $.
|
|
$( Transitive inference. $)
|
|
i3i0tr $p |- ( a ->3 ( a ->3 c ) ) = 1 $=
|
|
( wn wo i3i0 binr1 i3ror skmp3 i0i3 ) ACBFZCGAFZCGBCEHMNCABDIJKL $.
|
|
$( [9-Nov-97] $)
|
|
$}
|
|
|
|
$( Theorem for Kalmbach implication. $)
|
|
i3th1 $p |- ( a ->3 ( a ->3 ( b ->3 a ) ) ) = 1 $=
|
|
( wn wi3 wo wa wt df2i3 lor ax-a3 anor1 ax-a2 anor2 ax-r1 ax-r2 ancom orabs
|
|
lem4 ax-r5 3tr1 con2 2an oml5 3tr2 df-t ) ACZBADZEUFBCZUFFZUHAEZBUHAFZEZFZE
|
|
ZEZAAUGDDGUGUNUFBAHIAUGRGUFUIEZUMEZUOUFBEZURCZEZUFUMEZGUQURAUHFZEUFBVBEZEZU
|
|
TVAUFBVBJVBUSURABKIVDUFUFBFZEZUMEZVAVGVDVGUFVEUMEZEVDUFVEUMJVHVCUFVHVEVECZV
|
|
CFZEVCUMVJVEUJVIULVCUJAUHEZVIUHALVIVKVEVKABMUANOUKVBBUHAPIUBIUFBVBUCOIONVFU
|
|
FUMUFBQSOUDURUEUPUFUMUPUFUFUHFZEUFUIVLUFUHUFPIUFUHQOSTUFUIUMJOT $.
|
|
$( [16-Nov-97] $)
|
|
|
|
$( Theorem for Kalmbach implication. $)
|
|
i3th2 $p |- ( a ->3 ( b ->3 ( b ->3 a ) ) ) = 1 $=
|
|
( wi3 wn wo wt lem4 li3 bina4 ax-r2 ) ABBACCZCABDZAEZCFKMABAGHLAIJ $.
|
|
$( [7-Nov-97] $)
|
|
|
|
$( Theorem for Kalmbach implication. $)
|
|
i3th3 $p |- ( a ' ->3 ( a ->3 ( a ->3 b ) ) ) = 1 $=
|
|
( wn wi3 wo wt lem4 li3 bina3 ax-r2 ) ACZAABDDZDKKBEZDFLMKABGHKBIJ $.
|
|
$( [7-Nov-97] $)
|
|
|
|
$( Theorem for Kalmbach implication. $)
|
|
i3th4 $p |- ( a ->3 ( b ->3 b ) ) = 1 $=
|
|
( wt wi3 i31 i3id ax-r1 li3 rbi wed ) ACDZCABBDZDZCAEKMCCLALCBFGHIJ $.
|
|
$( [7-Nov-97] $)
|
|
|
|
$( Theorem for Kalmbach implication. $)
|
|
i3th5 $p |- ( ( a ->3 b ) ->3 ( a ->3 ( a ->3 b ) ) ) = 1 $=
|
|
( wi3 wn wa wo ax-a2 lea lear le2or bltr oridm lbtr df-i3 lem4 le3tr1 lei3
|
|
) ABCZARCZADZBEZTBDZEZFZATBFZEZFZUERSUGUEUEFUEUDUEUFUEUDUCUAFUEUAUCGUCTUABT
|
|
UBHTBIJKAUEIJUELMABNABOPQ $.
|
|
$( [16-Nov-97] $)
|
|
|
|
$( Theorem for Kalmbach implication. $)
|
|
i3th6 $p |- ( ( a ->3 ( a ->3 ( a ->3 b ) ) ) ->3 ( a ->3 ( a ->3 b ) ) )
|
|
= 1 $=
|
|
( wi3 tb i3abs1 bi1 bii3 skmp3 ) AAABCCZCZIDJICJIABEFJIGH $.
|
|
$( [16-Nov-97] $)
|
|
|
|
$( Theorem for Kalmbach implication. $)
|
|
i3th7 $p |- ( a ->3 ( ( a ->3 b ) ->3 a ) ) = 1 $=
|
|
( wi3 wn wo leor lem4 ax-r1 i3abs3 ax-r2 lbtr lei3 ) AABCZACZAMDZAEZNAOFPMN
|
|
CZNQPMAGHABIJKL $.
|
|
$( [19-Nov-97] $)
|
|
|
|
$( Theorem for Kalmbach implication. $)
|
|
i3th8 $p |- ( ( a ->3 b ) ' ->3 ( ( a ->3 b ) ->3 a ) ) = 1 $=
|
|
( wi3 wn wo leo lem4 ax-r1 i3abs3 ax-r2 lbtr lei3 ) ABCZDZMACZNNAEZONAFPMOC
|
|
ZOQPMAGHABIJKL $.
|
|
$( [19-Nov-97] $)
|
|
|
|
$( Theorem for Kalmbach implication. $)
|
|
i3con $p |- ( ( a ->3 b ) ->3 ( ( a ->3 b ) ->3 ( b ' ->3 a ' ) ) )
|
|
= 1 $=
|
|
( wn wo wt ax-a2 com2an com2or fh4 ax-a3 ancom lor orabs ax-r2 comcom3 df-t
|
|
wa comcom ax-r1 2an wi3 ni32 i3n1 2or comor2 comcom2 or12 lea bltr leo letr
|
|
comor1 df-le2 comorr or1 ax-r5 or4 coman2 anor1 con2 coman1 anor2 df-a 3tr1
|
|
an1 i0i3 ) ABUAZBCZACZUAZVGCZVJDZEEQZEVLABDZAVHQZVIAVHDZQZDZQZBVIQZBAQZDZVH
|
|
BVIDZQZDZDZVMVKVSVJWEABUBBAUCUDWFWEVSDZVMVSWEFWGWEVNDZWEVRDZQVMVNWEVRVNWBWD
|
|
VNVTWAVNBVIABUEZVNAABULZUFZGVNBAWJWKGHVNVHWCVNBWJUFZVNBVIWJWLHGHVNVOVQVNAVH
|
|
WKWMGVNVIVPWLVNAVHWKWMHGHIWHEWIEWHWBWDVNDZDZEWBWDVNJWOWDWBVNDZDZEWBWDVNUGWQ
|
|
WNEWPVNWDWPVTWAVNDZDZVNVTWAVNJWSVTVNDZVNWRVNVTWAVNWAAVNWAABQABAKABUHUIABUJU
|
|
KUMLWTVNVTDZVNVTVNFXAABVTDZDVNABVTJXBBABVIMLNNNNLWNVNWDDZEWDVNFXCVNVHDZVNWC
|
|
DZQZEVHVNWCBVNVNBWJROBWCBVIUNOZIXFVMEXDEXEEXDABVHDZDZEABVHJXIAEDEXHEAEXHBPS
|
|
LAUONNXEBADZWCDZEVNXJWCABFUPXKBBDZAVIDZDZEBABVIUQXNXLEDEXMEXLEXMAPSLXLUONNN
|
|
TEVEZNNNNNNWIWBWDVRDZDZEWBWDVRJXQWAVIBQZDZVHVQDZDZEWBXSXPXTWBWAVTDXSVTWAFVT
|
|
XRWABVIKLNXPWDVODZVQDZXTYCXPWDVOVQJSYBVHVQYBVOWDDZVHWDVOFYDVOVHDZVOWCDZQZVH
|
|
VHVOWCVOVHAVHURRXGIYGVHEQVHYEVHYFEYEVHVODZVHVOVHFYHVHVHAQZDVHVOYIVHAVHKLVHA
|
|
MNNYFVOVOCZDZEWCYJVOWCVIBDZYJBVIFYJYLVOYLABUSUTSNLEYKVOPSNTVHVENNNUPNUDYAWA
|
|
VHDZXRVQDZDZEWAXRVHVQUQYOYMVIDZEYNVIYMYNXRVIDZXRVPDZQZVIVIXRVPXRVIVIBVARAVP
|
|
AVHUNOIYSVIEQVIYQVIYREYQVIXRDVIXRVIFVIBMNYRXRXRCZDZEVPYTXRYTVPXRVPABVBUTSLE
|
|
UUAXRPSNTVIVENNLWAVHVIDZDWAWACZDYPEUUBUUCWAUUCUUBWAUUBBAVCUTSLWAVHVIJWAPVDN
|
|
NNNTNNNXONVF $.
|
|
$( [9-Nov-97] $)
|
|
|
|
$( Lemma for Kalmbach implication OR builder. $)
|
|
i3orlem1 $p |- ( ( a v c ) ^ ( ( a v c ) ' v ( b v c ) ) ) =<
|
|
( ( a v c ) ->3 ( b v c ) ) $=
|
|
( wo wn wa wi3 leor df-i3 ax-r1 lbtr ) ACDZLEZBCDZDFZMNFMNEFDZODZLNGZOPHRQL
|
|
NIJK $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( Lemma for Kalmbach implication OR builder. $)
|
|
i3orlem2 $p |- ( a ^ b ) =< ( ( a v c ) ->3 ( b v c ) ) $=
|
|
( wa wo wi3 leo le2an wn leor ledi letr i3orlem1 ) ABDACEZBCEZDZNOFZANBOACG
|
|
BCGHPNNIZOEDZQPNRDZPESPTJNROKLABCMLL $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( Lemma for Kalmbach implication OR builder. $)
|
|
i3orlem3 $p |- c =< ( ( a v c ) ->3 ( b v c ) ) $=
|
|
( wo wn wi3 ax-a2 lan anabs ax-r2 ax-r1 leor lelor le2an bltr i3orlem1 letr
|
|
wa ) CACDZSEZBCDZDZRZSUAFCCTCDZRZUCUECUECCTDZRCUDUFCTCGHCTIJKCSUDUBCALCUATC
|
|
BLMNOABCPQ $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( Lemma for Kalmbach implication OR builder. $)
|
|
i3orlem4 $p |- ( ( a v c ) ' ^ ( b v c ) ) =<
|
|
( ( a v c ) ->3 ( b v c ) ) $=
|
|
( wo wn wa wi3 leo ler df-i3 ax-r1 lbtr ) ACDZEZBCDZFZPNOEFZDZMNODFZDZMOGZP
|
|
RSPQHIUATMOJKL $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( Lemma for Kalmbach implication OR builder. $)
|
|
i3orlem5 $p |- ( ( a ' ^ b ' ) ^ c ' ) =<
|
|
( ( a v c ) ->3 ( b v c ) ) $=
|
|
( wo wn wa wi3 leo anandir oran con2 ax-r1 2an ax-r2 df2i3 le3tr1 ) ACDZEZB
|
|
CDZEZFZUARSDQRSFDFZDAEZBEZFCEZFZQSGUAUBHUFUCUEFZUDUEFZFUAUCUDUEIUGRUHTRUGQU
|
|
GACJKLTUHSUHBCJKLMNQSOP $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( Lemma for Kalmbach implication OR builder. $)
|
|
i3orlem6 $p |- ( ( a ->3 b ) ' v ( ( a v c ) ->3 ( b v c ) ) ) =
|
|
( ( ( a v b ) ^ ( a ' ->3 b ' ) ) v ( ( a v c ) ->3 ( b v c ) ) ) $=
|
|
( wa wi3 wn wo ax-a3 ax-r1 i3orlem2 lerr df-le2 oi3ai3 ax-r5 3tr2 ) ABDZABE
|
|
FZACGBCGEZGZGZPQGZRGZSABGAFBFEDZRGUBTPQRHIPSPRQABCJKLUAUCRABMNO $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( Lemma for Kalmbach implication OR builder. $)
|
|
i3orlem7 $p |- ( a ^ b ' ) =<
|
|
( ( a ->3 b ) ' v ( ( a v c ) ->3 ( b v c ) ) ) $=
|
|
( wn wa wo wi3 lea leo letr ler2an ler i3n1 lan comor1 comcom2 com2an ax-r1
|
|
com2or lbtr comor2 fh1 ax-r2 i3orlem6 ) ABDZEZABFZADZUEGZEZACFBCFGZFZABGDUK
|
|
FZUFUJUKUFUGUFABEZFZEZUGUHAUEFZEZEZFZUJUFUPUSUFUGUOUFAUGAUEHABIJUFUNIKLUJUT
|
|
UJUGUOURFZEUTUIVAUGABMNUGUOURUGUFUNUGAUEABOZUGBABUAZPZQUGABVBVCQSUGUHUQUGAV
|
|
BPUGAUEVBVDSQUBUCRTLUMULABCUDRT $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$( Lemma for Kalmbach implication OR builder. $)
|
|
i3orlem8 $p |- ( ( ( a v b ) ^ ( a v b ' ) ) ^ a ' ) =<
|
|
( ( a ->3 b ) ' v ( ( a v c ) ->3 ( b v c ) ) ) $=
|
|
( wo wn wa wi3 anass ancom lan ax-r2 leor bltr comor1 comcom2 com2an com2or
|
|
i3n1 ax-r1 lbtr comor2 fh1 ler i3orlem6 ) ABDZABEZDZFAEZFZUEUHUFGZFZACDBCDG
|
|
ZDZABGEULDZUIUKULUIUEAUFFZABFZDZFZUEUHUGFZFZDZUKUIUTVAUIUEUGUHFZFUTUEUGUHHV
|
|
BUSUEUGUHIJKUTURLMUKVAUKUEUQUSDZFVAUJVCUEABRJUEUQUSUEUOUPUEAUFABNZUEBABUAZO
|
|
ZPUEABVDVEPQUEUHUGUEAVDOUEAUFVDVFQPUBKSTUCUNUMABCUDST $.
|
|
$( [11-Nov-97] $)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Unified disjunction
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud1lem1 $p |- ( ( a ->1 b ) ->1 ( b ->1 a ) ) =
|
|
( a v ( a ' ^ b ' ) ) $=
|
|
( wi1 wn wa df-i1 ud1lem0c 2an 2or ancom lor lan coman1 comcom2 coman2 fh3r
|
|
wo ax-r1 ax-r2 wt or12 comcom comorr comcom5 fh4r orabs df-a df-t an1 ax-a2
|
|
) ABCZBACZCUKDZUKULEZQZAADZBDZEZQZUKULFUOAUPUQQZEZUPABEZQZUQBAEZQZEZQZUSUMV
|
|
AUNVFABGUKVCULVEABFBAFHIVGVAURVBQZQZUSVFVHVAVFVCUQVBQZEZVHVEVJVCVDVBUQBAJKL
|
|
VHVKVBUPUQVBAABMZNVBBABONPRSKVIURVAVBQZQZUSVAURVBUAVNURAVBQZUTVBQZEZQZUSVMV
|
|
QURAVBUTVBAVLUBAUTUPUTUPUQUCNUDUEKVRURAQUSVQAURVQATEAVOAVPTABUFVPUTUTDZQZTV
|
|
BVSUTABUGKTVTUTUHRSHAUISKURAUJSSSSSS $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud1lem2 $p |- ( ( a v ( a ' ^ b ' ) ) ->1 a ) = ( a v b ) $=
|
|
( wn wa wo wi1 df-i1 comid comcom3 comor1 fh3 wt ancom ax-a2 df-t ax-r1 lan
|
|
ax-r2 oran 3tr an1 ax-r4 con2 ax-r5 oml ) AACZBCDZEZAFUHCZUHADEUIUHEZUIAEZD
|
|
ZABEZUHAGUIUHAUHUHUHHIUHAAUGJIKULUKUJDUKLDZUMUJUKMUJLUKUJUHUIEZLUIUHNLUOUHO
|
|
PRQUNUKUFUMDZAEZUMUKUAUIUPAUHUPUHUFUGCZDZCUPCAUGSUSUPURUMUFUMURABSPQUBRUCUD
|
|
UQAUPEUMUPANABUERTTT $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud1lem3 $p |- ( ( a ->1 b ) ->1 ( a v b ) ) = ( a v b ) $=
|
|
( wi1 wo wn wa df-i1 ud1lem0c con3 ran 2or comid comcom2 df-t ax-r1 lan an1
|
|
wt comorr ax-r2 comor1 comor2 com2or com2an comcom ancom comcom5 fh4r ax-a2
|
|
fh3 or4 lor or1 ax-a3 oridm ax-r5 ) ABCZABDZCUQEZUQURFZDZURUQURGVAAAEZBEZDZ
|
|
FZVEEZURFZDZURUSVEUTVGABHZUQVFURUQVEVIIJKVHVEVFDZVEURDZFZURVEVFURVEVEVELMUR
|
|
VEURAVDABUAZURVBVCURAVMMURBABUBMUCUDUEUJVLVKVJFZURVJVKUFVNVKRFZURVJRVKRVJVE
|
|
NOPVOVKURVKQVKAURDZVDURDZFZURAURVDABSAVDVBVDVBVCSMUGUHVRVPRFZURVQRVPVQURVDD
|
|
ZRVDURUIVTAVBDZBVCDZDZRABVBVCUKWCWARDRWBRWARWBBNOULWAUMTTTPVSVPURVPQVPAADZB
|
|
DZURWEVPAABUNOWDABAUOUPTTTTTTTTTT $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud2lem1 $p |- ( ( a ->2 b ) ->2 ( b ->2 a ) ) =
|
|
( a v ( a ' ^ b ' ) ) $=
|
|
( wi2 wn wa wo df-i2 ud2lem0c 2an 2or wf ancom lor dff oran ax-r1 lan ax-r2
|
|
anandir ax-a2 ran or0 ) ABCZBACZCUDUCDZUDDZEZFZAADZBDZEZFZUCUDGUHAUJUIEZFZU
|
|
JABFZEZUIBAFZEZEZFZULUDUNUGUSBAGUEUPUFURABHBAHIJUTULKFULUNULUSKUMUKAUJUILMK
|
|
USKUMUQEZUSKUMUMDZEVAUMNVBUQUMUQVBBAOPQRVAUJUQEZUREUSUJUIUQSVCUPURUQUOUJBAT
|
|
QUARRPJULUBRRR $.
|
|
$( [22-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud2lem2 $p |- ( ( a v ( a ' ^ b ' ) ) ->2 a ) = ( a v b ) $=
|
|
( wn wa wi2 df-i2 oran con2 ax-r1 lor anor2 con3 ax-r2 ran an32 anidm oml
|
|
wo ) AACZBCDZRZAEAUACZSDZRZABRZUAAFUDASUEDZRUEUCUFAUCUAARZCZUFUHUCUGUCUAAGH
|
|
ZIUHUCUFUIUCUFSDZUFUBUFSUAUFUAAUECZRZUFCTUKAUKTUETABGHIJULUFUFULCAUEKILMHNU
|
|
JSSDZUEDUFSUESOUMSUESPNMMMMJABQMM $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud2lem3 $p |- ( ( a ->2 b ) ->2 ( a v b ) ) = ( a v b ) $=
|
|
( wi2 wo wn wa df-i2 ud2lem0c ran lor coman2 comcom comid comcom2 fh3 ancom
|
|
wt df-t ax-r1 ax-r2 2an an1 orabs ) ABCZABDZCUEUDEZUEEZFZDZUEUDUEGUIUEBEZUE
|
|
FZUGFZDZUEUHULUEUFUKUGABHIJUMUEUKDZUEUGDZFZUEUEUKUGUKUEUJUEKLUEUEUEMNOUPUEU
|
|
EUJFZDZQFZUEUNURUOQUKUQUEUJUEPJQUOUERSUAUSURUEURUBUEUJUCTTTTT $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem1a $p |- ( ( a ->3 b ) ' ^ ( b ->3 a ) ) = ( a ^ b ' ) $=
|
|
( wn wa wo 2an comor2 comor1 com2an comcom2 com2or comcom anass ancom ax-r2
|
|
wf lan ax-r1 dff an0 ud3lem0c comanr2 comcom3 coman2 coman1 comanr1 comcom6
|
|
wi3 df-i3 fh2 comcom7 ax-a2 anabs lea leo ler2an letr df2le2 an32 oran con3
|
|
ran an0r 2or or0 fh2r or0r anor1 ) ABUHCZBAUHZDABCZEZABEZDZACZAVKDZEZDZVKAD
|
|
ZVKVODZEZBVKAEZDZEZDZVPVIVRVJWDABUABAUIFWEVRWADZVRWCDZEZVPWAVRWCWAVNVQWAVLV
|
|
MVLWAVLVSVTVLVKAAVKGZAVKHZIVLVKVOWIVLAWJJIKLVMWAVMVSVTVMVKAVMBABGJZABHZIVMV
|
|
KVOWKVMAWLJIKLIWAVOVPVOWAVOVSVTAVSVKAUBUCVKVOUBKLVPWAVPVSVTVPVKAAVKUDZAVKUE
|
|
ZIVPVKVOWMVPAWNJZIKLKIWABWBBWABVSVTBVSVKAUFUGBVTVKVOUFUGKLWBWAWBVSVTWBVKAVK
|
|
AHZVKAGZIWBVKVOWPWBAWQJIKLIUJWHVPPEZVPWFVPWGPWFVRVSDZVRVTDZEZVPVSVRVTVSVNVQ
|
|
VSVLVMVSAVKVKAUDZVKAUEZKVSABXBVSBXCUKKIVSVOVPVSAXBJZVSAVKXBXCIKIVSVKVOXCXDI
|
|
UJXAWRVPWSVPWTPWSVNVQVSDZDZVPVNVQVSMXFVNVPDZVPXEVPVNXEVSVQDZVPVQVSNXHVPVPVO
|
|
EZDVPVSVPVQXIVKANVOVPULZFVPVOUMOOQXGVPVNDVPVNVPNVPVNVPAVNAVKUNAVLVMAVKUOABU
|
|
OUPUQUROOOWTVNVTDZVQDZPVNVQVTUSXLPVQDPXKPVQXKVLVMVTDZDZPVLVMVTMXNVLPDPXMPVL
|
|
XMBAEZXOCZDZPVMXOVTXPABULVTXOXOVTCBAUTRVAFPXQXOSROQVLTOOVBVQVCOOVDVPVEZOOWG
|
|
VNVQWCDZDZPVNVQWCMXTVNPDPXSPVNXSVQBDZWBDZPYBXSVQBWBMRYBBVODZWBDZPYAYCWBYAXI
|
|
BDZYCVQXIBXJVBYEVPBDZVOBDZEZYCVPBVOVPBWMUKWOVFYHPYCEYCYFPYGYCYFAVKBDZDZPAVK
|
|
BMYJAPDPYIPAYIBVKDZPVKBNPYKBSROQATOOVOBNVDYCVGOOOVBYDYCYCCZDZPWBYLYCWBYCYCW
|
|
BCBAVHRVAQPYMYCSROOOQVNTOOVDXROOO $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem1b $p |- ( ( a ->3 b ) ' ^ ( b ->3 a ) ' ) = 0 $=
|
|
( wi3 wn wa wo ud3lem0c 2an an32 comor2 comcom7 ancom ax-a2 lan ax-r2 ax-r1
|
|
wf dff anass ran an12 comor1 comcom2 com2an fh1 anabs anor1 anidm fh1r oran
|
|
2or or0 ) ABCDZBACDZEABDZFZABFZEZADZAUOEZFZEZBUSFZBAFZEZUOBUSEZFZEZEZQUMVBU
|
|
NVHABGBAGHVIURVHEZVAEZQURVAVHIVKUOVDEZVCEZVAEZQVJVMVAVJUPVHEZUQEZVMUPUQVHIV
|
|
PUOVCEZVDEZUQEZVMVOVRUQVOVEUPVGEZEZVRUPVEVGUAWAVEUOEZVRVTUOVEVTUPUOEZUPVFEZ
|
|
FZUOUPUOVFAUOJZUPBUSUPBWFKUPAAUOUBUCUDUEWEUOQFUOWCUOWDQWCUOUOAFZEZUOWCUOUPE
|
|
WHUPUOLUPWGUOAUOMZNOUOAUFOWDWGWGDZEZQUPWGVFWJWIBAUGHQWKWGRPOUKUOULOONWBUOVE
|
|
EZVRVEUOLVRWLUOVCVDSPOOOTVSVRVDEZVMUQVDVRABMNWMVQVDVDEZEZVMVQVDVDSWOVRVMWNV
|
|
DVQVDUHNUOVCVDIOOOOOTVNVLVCVAEZEZQVLVCVASWQVLUSEZQWPUSVLWPUSBFZVAEZUSVCWSVA
|
|
BUSMTWTVAWSEZUSWSVALXAUSWSEZUTWSEZFZUSWSUSUTUSBUBZWSAUOWSAXEKWSBUSBJUCUDUIX
|
|
DUSQFUSXBUSXCQUSBUFXCWSUTEZQUTWSLXFWSWSDZEZQUTXGWSABUGNQXHWSRPOOUKUSULOOOON
|
|
WRUOUSEZVDEZQUOVDUSIXJXIXIDZEZQVDXKXIBAUJNQXLXIRPOOOOOOO $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem1c $p |- ( ( a ->3 b ) ' v ( b ->3 a ) ) = ( a v b ' ) $=
|
|
( wn wo wa 2or coman2 coman1 com2or comcom7 com2an comcom ax-a2 ax-r2 ax-a3
|
|
wt ax-r1 df-t lor or1 ud3lem0c comorr2 comcom6 comor2 comor1 comorr comcom3
|
|
wi3 df-i3 fh4r comcom2 lea lel2or leor letr lear lbtr or12 ancom oran ax-r5
|
|
df-le2 or1r 2an an1 fh4 ran an1r anor1 ) ABUHCZBAUHZDABCZDZABDZEZACZAVLEZDZ
|
|
EZVLAEZVLVPEZDZBVLADZEZDZDZVMVJVSVKWEABUABAUIFWFVOWEDZVRWEDZEZVMVOWEVRVOWBW
|
|
DVOVTWAVTVOVTVMVNVTAVLVLAGZVLAHZIVTABWJVTBWKJIKLWAVOWAVMVNWAAVLWAAVLVPGJZVL
|
|
VPHZIWAABWLWABWMJIKLIVOBWCBVOBVMVNBVMAVLUBUCABUBKLWCVOWCVMVNWCAVLVLAUDZVLAU
|
|
EZIWCABWNWCBWOJZIKLKIVOVPVQVPVOVPVMVNAVMAVLUFUGAVNABUFUGKLVQVOVQVMVNVQAVLAV
|
|
LHZAVLGZIVQABWQVQBWRJIKLIUJWIVMPEZVMWGVMWHPWGVMWEDZVNWEDZEZVMVMWEVNVMWBWDVM
|
|
VTWAVMVLAAVLUDZAVLUEZKVMVLVPXCVMAXDUKKIVMBWCVMBXCJZVMVLAXCXDIKIVMABXDXEIUJX
|
|
BWSVMWTVMXAPWTWEVMDVMVMWEMWEVMWBVMWDWBVLVMVTVLWAVLAULVLVPULUMVLAUNUOWDWCVMB
|
|
WCUPVLAMUQUMVBNXAWBVNWDDDZPVNWBWDURXFWBVNDZWDDZPXHXFWBVNWDOQXHPWDDPXGPWDXGV
|
|
TWAVNDZDZPVTWAVNOXJVTPDPXIPVTXIVPVLEZXKCZDZPWAXKVNXLVLVPUSABUTFPXMXKRQNSVTT
|
|
NNVAWDVCNNNVDVMVEZNNWHWBVRWDDZDZPVRWBWDURXPWBPDPXOPWBXOVQVPDZWCBEZDZPVRXQWD
|
|
XRVPVQMBWCUSFXSVQVPXRDZDZPVQVPXROYAVQVPBDZDZPXTYBVQXTVPWCDZYBEZYBWCVPBWCAWN
|
|
UKWPVFYEPYBEYBYDPYBYDWCVPDZPVPWCMYFVLAVPDZDZPVLAVPOYHVLPDPYGPVLPYGARQSVLTNN
|
|
NVGYBVHNNSYCYBVQDZPVQYBMYIYBYBCZDZPVQYJYBABVISPYKYBRQNNNNNSWBTNNVDXNNNN $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem1d $p |- ( ( a ->3 b ) ^ ( ( a ->3 b ) ' v ( b ->3 a ) ) ) =
|
|
( ( a ' ^ b ' ) v ( a ^ ( a ' v b ) ) ) $=
|
|
( wi3 wn wo wa df-i3 ud3lem1c 2an comor1 comcom2 comor2 comcom7 com2an fh1r
|
|
com2or an32 ax-r2 2or wf anabs ran ancom anor2 lan dff ax-r1 lear leor letr
|
|
df2le2 or0r ax-r5 ) ABCZUNDBACEZFADZBFZUPBDZFZEZAUPBEZFZEZAUREZFZUSVBEZUNVC
|
|
UOVDABGABHIVEUTVDFZVBVDFZEZVFVDUTVBVDUQUSVDUPBVDAAURJZKZVDBAURLZMZNZVDUPURV
|
|
KVLNZPVDAVAVJVDUPBVKVMPNOVIUQVDFZUSVDFZEZVBEVFVGVRVHVBVDUQUSVNVOOVHAVDFZVAF
|
|
VBAVAVDQVSAVAAURUAUBRSVRUSVBVRTUSEUSVPTVQUSVPVDUQFZTUQVDUCVTVDVDDZFZTUQWAVD
|
|
ABUDUETWBVDUFUGRRUSVDUSURVDUPURUHURAUIUJUKSUSULRUMRRR $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem1 $p |- ( ( a ->3 b ) ->3 ( b ->3 a ) ) =
|
|
( a v ( a ' ^ b ' ) ) $=
|
|
( wi3 wn wa wo df-i3 wf ud3lem1a ud3lem1b 2or ax-r2 ud3lem1d coman1 comcom2
|
|
or0 coman2 wt ax-a2 lor comcom7 com2or fh3 orabs anor1 df-t ax-r1 or12 3tr1
|
|
2an an1 ) ABCZBACZCULDZUMEZUNUMDEZFZULUNUMFEZFZAADZBDZEZFZULUMGUSAVAEZVBAUT
|
|
BFZEZFZFZVCUQVDURVGUQVDHFVDUOVDUPHABIABJKVDPLABMKVBVDVFFZFVBAFVHVCVIAVBVIVD
|
|
AFZVDVEFZEZAVDAVEAVANZVDUTBVDAVMOVDBAVAQUAUBUCVLAREAVJAVKRVJAVDFAVDASAVAUDL
|
|
VKVEVDFZRVDVESVNVEVEDZFZRVDVOVEABUETRVPVEUFUGLLUJAUKLLTVDVBVFUHAVBSUILL $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem2 $p |- ( ( a v ( a ' ^ b ' ) ) ->3 a ) = ( a v b ) $=
|
|
( wn wa wo wi3 oran ax-r1 con3 lor anor2 ax-r2 ax-a2 wf ran lan dff 2or or0
|
|
ancom ud3lem0b df-i3 ax-a3 ax-a1 an32 anidm ax-r5 2an oml comorr fh2r anabs
|
|
comcom2 anass an0 ) AACZBCDZEZAFUPABEZDZCZAFZUSURVAAURAUSCZEZVAUQVCAUQUSUSU
|
|
QCABGHIJVDUTUTVDCAUSKHIZLUAVBVACZADZVFUPDZEVAVFAEZDZEZUSVAAUBVKVGVHVJEZEZUS
|
|
VGVHVJUCVMVLVGEZUSVGVLMVNUSNEUSVLUSVGNVLAUTEZUSVLUTAEZVOVHUTVJAVHUTUPDZUTVQ
|
|
VHUTVFUPUTUDZOHVQUPUPDZUSDUTUPUSUPUEVSUPUSUPUFOLLVJVDVPDZAVTVJVDVAVPVIVEUTV
|
|
FAVRUGUHHVTVDUSDZAVPUSVDVPVOUSUTAMZABUIZLPWAAUSDZVCUSDZEZAAUSVCABUJZAUSWGUM
|
|
UKWFANEAWDAWENABULWEUSVCDZNVCUSTNWHUSQHLRASLLLLRWBLWCLVGAVFDZNVFATWIAUTDZNW
|
|
JWIUTVFAVRPHWJAUPDZUSDZNWLWJAUPUSUNHWLUSWKDZNWKUSTWMUSNDZNWNWMNWKUSAQPHUSUO
|
|
LLLLLRUSSLLLLL $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem3a $p |- ( ( a ->3 b ) ' ^ ( a v b ) ) = ( a ->3 b ) ' $=
|
|
( wi3 wn wo wa ud3lem0c lea lear letr bltr df2le2 ) ABCDZABEZMABDZEZNFZADAO
|
|
FEZFZNABGSQNQRHPNIJKL $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem3b $p |- ( ( a ->3 b ) ' ^ ( a v b ) ' ) = 0 $=
|
|
( wi3 wn wo wa wf ud3lem0c ran an32 anass dff ax-r1 lan an0 ax-r2 an0r ) AB
|
|
CDZABEZDZFABDZEZSFZADAUAFEZFZTFZGRUETABHIUFUCTFZUDFZGUCUDTJUHGUDFGUGGUDUGUB
|
|
STFZFZGUBSTKUJUBGFGUIGUBGUISLMNUBOPPIUDQPPP $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem3c $p |- ( ( a ->3 b ) ' v ( a v b ) ) = ( a v b ) $=
|
|
( wi3 wn wo wa ud3lem0c an32 ancom ax-r2 ax-r5 ax-a2 orabs ) ABCDZABEZEOABD
|
|
ZEZADAPFEZFZFZOEZONTONQOFRFZTABGUBSOFTQORHSOIJJKUAOTEOTOLOSMJJ $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem3d $p |- ( ( a ->3 b ) ^ ( ( a ->3 b ) ' v ( a v b ) ) ) =
|
|
( ( a ' ^ b ) v ( a ^ ( a ' v b ) ) ) $=
|
|
( wi3 wn wo wa ud3lem3c 2an comor1 comcom2 comor2 com2an com2or fh1r coman1
|
|
df-i3 wf letr df2le2 ax-r2 comcom7 coman2 fh2r lear leor oran lan dff ax-r1
|
|
2or or0 ax-r5 lea leo lor ) ABCZUPDABEZEZFADZBFZUSBDZFZEZAUSBEZFZEZUQFZUTVE
|
|
EZUPVFURUQABPABGHVGVCUQFZVEUQFZEZVHUQVCVEUQUTVBUQUSBUQAABIZJZABKZLUQUSVAVMU
|
|
QBVNJLMUQAVDVLUQUSBVMVNMLNVKUTVJEVHVIUTVJVIUTUQFZVBUQFZEZUTUTUQVBUTABUTAUSB
|
|
OZUAUSBUBZMUTUSVAVRUTBVSJLUCVQUTQEUTVOUTVPQUTUQUTBUQUSBUDBAUERSVPVBVBDZFZQU
|
|
QVTVBABUFUGQWAVBUHUITUJUTUKTTULVJVEUTVEUQVEAUQAVDUMABUNRSUOTTT $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud3lem3 $p |- ( ( a ->3 b ) ->3 ( a v b ) ) = ( a v b ) $=
|
|
( wi3 wo wn wa ax-r2 2or coman1 comcom7 coman2 comcom2 com2or com2an comcom
|
|
wf comorr wt ax-r1 lor df-i3 ud3lem3a ud3lem0c ud3lem3b or0 ud3lem3d comor1
|
|
comor2 comcom3 fh4r ax-a3 anor2 df-t ax-r5 or1r ax-a2 lear leor letr lel2or
|
|
lea leo df-le2 2an an1r or12 df-a anor1 ax-r4 or1 an1 ) ABCZABDZCVLEZVMFZVN
|
|
VMEFZDZVLVNVMDFZDZVMVLVMUAVSABEZDZVMFZAEZAVTFZDZFZWCBFZAWCBDZFZDZDZVMVQWFVR
|
|
WJVQWFPDWFVOWFVPPVOVNWFABUBABUCGABUDHWFUEGABUFHWKWBWJDZWEWJDZFZVMWBWJWEWBWG
|
|
WIWGWBWGWAVMWGAVTWGAWCBIJZWGBWCBKZLMWGABWOWPMNOWBAWHAWBAWAVMAVTQZABQZNOWHWB
|
|
WHWAVMWHAVTWHAWCBUGJZWHBWCBUHZLMWHABWSWTMNONMWBWCWDWCWBWCWAVMAWAWQUIAVMWRUI
|
|
NOWDWBWDWAVMWDAVTAVTIZAVTKZMWDABXAWDBXBJMNOMUJWNVMRFVMWLVMWMRWLWAWJDZVMWJDZ
|
|
FZVMWAWJVMWAWGWIWAWCBWAAAVTUGZLZWABAVTUHJZNWAAWHXFWAWCBXGXHMNMWAABXFXHMUJXE
|
|
RVMFVMXCRXDVMXCWAWGDZWIDZRXJXCWAWGWIUKSXJRWIDRXIRWIXIWAWAEZDZRWGXKWAABULTRX
|
|
LWAUMSGUNWIUOGGXDWJVMDVMVMWJUPWJVMWGVMWIWGBVMWCBUQBAURUSWIAVMAWHVAABVBUSUTV
|
|
CGVDVMVEGGWMWGWEWIDZDZRWEWGWIVFXNWGRDRXMRWGXMWEWEEZDZRWIXOWEWIWCWHEZDZEXOAW
|
|
HVGXRWEXQWDWCWDXQABVHSTVIGTRXPWEUMSGTWGVJGGVDVMVKGGGG $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud4lem1a $p |- ( ( a ->4 b ) ^ ( b ->4 a ) ) =
|
|
( ( a ^ b ) v ( a ' ^ b ' ) ) $=
|
|
( wa wn wo coman2 comcom com2or coman1 comcom2 com2an comcom3 ancom 2or lan
|
|
wf dff ax-r1 an0 ax-r2 wi4 df-i4 2an comcom5 fh2r ax-a2 ran fh1 an4 lor fh3
|
|
or0 3tr2 an12 an32 anass anor2 con3 fh2 lecon lelan oran anor1 ax-r4 le3tr1
|
|
lea con2 le0 lebi leo le2an df2le2 ) ABUAZBAUAZCABCZADZBCZEZVPBEZBDZCZEZBAC
|
|
ZVTACZEZVTAEZVPCZEZCZVOVPVTCZEZVMWBVNWHABUBBAUBUCWIVRWHCZWAWHCZEWKVRWHWAVRW
|
|
EWGVRWCWDVRBABVRBVOVQVOBABFZGZVQBVPBFGZHGZAVRAVOVQVOAABIZGZAVQVPVQVQVPVPBIG
|
|
ZJUDHGZKVRVTAVTVRVTVOVQBVOWOLBVQWPLHGZXAKHVRWFVPVRVTAXBXAHVPVRVPVOVQAVOWSLW
|
|
THGZKHVRVSVTVRVPBXCWQHVRBWQJZKUEWLVOWMWJWLVOPEZVOWLVRVOAVTCZEZAVTEZVPCZEZCZ
|
|
XEWHXJVRWEXGWGXIWCVOWDXFBAMVTAMZNWFXHVPVTAUFUGNOXKVRXGCZVRXICZEXEVRXGXIVRVO
|
|
XFVRABXAWQKVRAVTXAXBKHVRXHVPVRAVTXAXDHXCKUHXMVOXNPVOVQXFCZEXEXMVOXOPVOXOVPA
|
|
CZBVTCZCZPVPBAVTUIXRXPPCPXQPXPPXQBQROXPSTTUJVOVQXFVOVPBVOAWRJZWNKZVOAVTWRVO
|
|
BWNJZKUKVOULZUMXNVOXICZVQXICZEZPVOXIVQVOXHVPVOAVTWRYAHXSKXTUEYEPPEPYCPYDPYC
|
|
XHVOVPCZCZPVOXHVPUNYGXHPCPYFPXHYFAVPCZBCZPABVPUOYIBYHCZPYHBMYJBPCPYHPBPYHAQ
|
|
ROBSTTTOXHSTTVQXHCZVPCVPYKCZYDPYKVPMVQXHVPUPYLVPPCPYKPVPYKVQVQDZCZPXHYMVQXH
|
|
VQVQXHDABUQRUROPYNVQQRTOVPSTUMNPULTTNTTYBTWMWAWECZWAWGCZEZWJWEWAWGWEVSVTWEV
|
|
PBVPWEAWEAWCWDWCABAFGZWDAVTAFGZHZLGBWEBWCWDWCBBAIGZBWDVTWDWDVTVTAIGZJUDHGHV
|
|
TWEVTWCWDBWCUUALUUBHGZKWEWFVPWEVTAUUCAWEYTGHVPWEVPWCWDAWCYRLAWDYSLHGKUSYQPW
|
|
JEZWJYOPYPWJYOPWAVSWCDZCZDZCWAWADZCYOPUUGUUHWAWAUUFVTUUEVSWCBBAVFUTVAUTVAWE
|
|
UUGWAWEWDWCEZUUGWCWDUFUUIWDDZUUECZDUUGWDWCVBUUKUUFUUJVSUUEWDVSWDXFVSDXLABVC
|
|
TVGUGVDTTOWAQVEYOVHVIYPVSWFCZVTVPCZCZWJVSVTWFVPUIUUNUUMWJUUNUUMUULCZUUMUULU
|
|
UMMUUOUUMWFVSCZCUUMUULUUPUUMVSWFMOUUMUUPVTWFVPVSVTAVJVPBVJVKVLTTVTVPMTTNUUD
|
|
WJPEWJPWJUFWJULTTTNTT $.
|
|
$( [24-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud4lem1b $p |- ( ( a ->4 b ) ' ^ ( b ->4 a ) ) =
|
|
( a ^ b ' ) $=
|
|
( wn wa wo coman2 comcom2 coman1 com2or comcom com2an comcom5 wf an32 ax-r2
|
|
dff ancom an0 2or lan wi4 ud4lem0c df-i4 2an comor2 comor1 comcom3 fh2 df-a
|
|
ax-a2 ax-r1 ran lea leor letr lear leo ler2an bltr 3tr1 or0 anass fh2r an12
|
|
df2le2 anor1 ) ABUACZBAUAZDACZBCZEZAVJEZDZAVJDZBEZDZBADZVJADZEZVJAEZVIDZEZD
|
|
ZVNVGVPVHWBABUBBAUCUDWCVPVSDZVPWADZEZVNVSVPWAVSVMVOVSVKVLVKVSVKVQVRVQVKVQVI
|
|
VJVQABAFZGVQBBAHZGZIZJVRVKVRVIVJVRAVJAFZGVJAHZIJIJVLVSVLVQVRVQVLVQAVJWGWIIZ
|
|
JVLVJAAVJUEAVJUFKIJKVSVNBVNVSVNVQVRVNBAVNBVNVJAVJFZUGLZAVJHZKVNVJAWNWPKIJBV
|
|
SBVQVRVQBWHJBVRVJVRVRVJWLJGLIJIKVSVTVIVTVSVTVQVRVTBAVTBVTVJVJAUFZUGLVJAUEZK
|
|
VTVJAWQWRKIJVSAAVSAVQVRVQAWGJVRAWKJIJGKUHWFVNMEZVNWDVNWEMWDVPVQDZVPVRDZEZVN
|
|
VQVPVRVQVMVOVQVKVLWJWMKVQVNBVQAVJWGWIKWHIKVQVJAWIWGKUHXBWSVNXBMVNEWSWTMXAVN
|
|
WTVMVQDZVODZMVMVOVQNXDMVODZMXCMVOXCVKVQDZVLDZMVKVLVQNXGMVLDZMXFMVLXFVJVIEZX
|
|
ICZDZMVKXIVQXJVIVJUJBAUIUDMXKXIPUKOULXHVLMDMMVLQVLROOOULXEVOMDMMVOQVOROOOVR
|
|
VPDVRXAVNVRVPVRVMVOVRVKVLVRVJVKVJAUMVJVIUNUOVRAVLVJAUPAVJUQUOURVRVNVOVJAQVN
|
|
BUQUSURVEVPVRQAVJQUTSMVNUJOVNVAZOOWEVMVOWADZDZMVMVOWAVBXNVMMDMXMMVMXMVNWADZ
|
|
BWADZEZMVNWABVNVTVIVNVJAWNWPIVNAWPGKWOVCXQMMEMXOMXPMXOVTVNVIDZDZMVNVTVIVDXS
|
|
VTMDMXRMVTXRAVIDZVJDZMAVJVINYAVJXTDZMXTVJQYBVJMDMXTMVJMXTAPUKTVJROOOTVTROOV
|
|
TBVIDZDVTVTCZDXPMYCYDVTBAVFTBVTVIVDVTPUTSMVAOOTVMROOSXLOOO $.
|
|
$( [25-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud4lem1c $p |- ( ( a ->4 b ) ' v ( b ->4 a ) ) =
|
|
( a v b ' ) $=
|
|
( wn wo wa comor2 comcom3 comcom5 comor1 com2an com2or comcom coman1 coman2
|
|
comcom2 wt ax-a2 ax-a3 ax-r2 or1 ud4lem0c df-i4 comorr fh4r df-a df-t ax-r1
|
|
wi4 2or lor 3tr2 ax-r5 lear lel2or leo letr lea lbtr 2an ancom an1 or32 or4
|
|
df-le2 fh4 anor2 con2 3tr1 ) ABUHCZBAUHZDACZBCZDZAVLDZEZAVLEZBDZEZBAEZVLAEZ
|
|
DZVLADZVKEZDZDZVNVIVRVJWDABUABAUBUIWEVOWDDZVQWDDZEZVNVOWDVQVOWAWCWAVOWAVMVN
|
|
VMWAVMVSVTVMBAVMBVMVLVKVLFZGHVMAVMVKVKVLIZGHZJVMVLAWIWKJKLVNWAVNVSVTVNBAVNB
|
|
VNVLAVLFZGHAVLIZJVNVLAWLWMJKLJLVOWBVKWBVOWBVMVNWBVKVLWBAVLAFOZVLAIZKZWBAVLW
|
|
BAWBVKWNGHWOKJLVKVOVKVMVNVKVLUCAVNAVLUCGJLJKVOVPBVPVOVPVMVNVPVKVLVPAAVLMZOA
|
|
VLNZKVPAVLWQWRKJLVOBVOVLVLVOVLVMVNVMVLWILVNVLWLLJLGHKUDWHVNPEZVNWFVNWGPWFVM
|
|
WDDZVNWDDZEZVNVMWDVNVMWAWCVMVSVTVSVMVSVKVLVSABANOVSBBAMOKLVTVMVTVKVLVTAVLAN
|
|
OVLAMKLKVMWBVKWBVMWPLWJJKVNVMVNVKVLVNAWMOWLKLUDXBPVNEZVNWTPXAVNVMWADZWCDPWC
|
|
DZWTPXDPWCVMVSDZVTDVTXFDZXDPXFVTQVMVSVTRXGVTPDPXFPVTXFVLVKDZXHCZDZPVMXHVSXI
|
|
VKVLQBAUEUIPXJXHUFUGSUJVTTSUKULVMWAWCRXEWCPDPPWCQWCTSUKXAWDVNDVNVNWDQWDVNWA
|
|
VNWCWAAVNVSAVTBAUMVLAUMUNAVLUOUPWCWBVNWBVKUQVLAQURUNVDSUSXCWSVNPVNUTVNVAZSS
|
|
SWGWDVQDZPVQWDQXLVSVPDZPDZPXLWAVQDZWCDZXNWAWCVQVBXPXMVTBDZDZWCDZXNXOXRWCVSV
|
|
TVPBVCULXSXMXQWCDZDXNXMXQWCRXTPXMVTBWCDZDVTVTCZDXTPYAYBVTYABWBDZBVKDZEZYBWB
|
|
BVKWBBWBVLWOGHWNVEYEPYBEZYBYCPYDYBBVLDZADPADZYCPYGPAPYGBUFUGULBVLARYHAPDPPA
|
|
QATSUKYBYDVTYDBAVFVGUGUSYFYBPEYBPYBUTYBVASSSUJVTBWCRVTUFVHUJSSSXMTSSUSXKSSS
|
|
$.
|
|
$( [25-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud4lem1d $p |- ( ( ( a ->4 b ) ' v ( b ->4 a ) ) ^ ( b ->4 a ) ' ) =
|
|
( ( ( a ' v b ' ) ^ ( a ' v b ) ) ^ a ) $=
|
|
( wi4 wn wo ud4lem1c ud4lem0c 2an an12 ax-a2 comor2 comcom3 comcom5 comcom2
|
|
wa comor1 com2an fh1 wf ax-r2 anor1 dff ax-r1 ancom anabs 2or or0 ) ABCDBAC
|
|
ZEZUHDZOABDZEZUKADZEZBUMEZOZBUMOZAEZOZOZUMUKEZUMBEZOZAOZUIULUJUSABFBAGHUTUP
|
|
ULUROZOVDULUPURIUPVCVEAUNVAUOVBUKUMJBUMJHVEULUQOZULAOZEZAULUQAULBUMULBULUKA
|
|
UKKLMULAAUKPZNQVIRVHSAEZAVFSVGAVFUKAEZVKDZOZSULVKUQVLAUKJBAUAHSVMVKUBUCTVGA
|
|
ULOAULAUDAUKUETUFVJASEASAJAUGTTTHTT $.
|
|
$( [25-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud4lem1 $p |- ( ( a ->4 b ) ->4 ( b ->4 a ) ) =
|
|
( a v ( a ' ^ b ' ) ) $=
|
|
( wi4 wa 2or lor coman1 comcom comcom3 com2or comcom2 comcom5 comorr com2an
|
|
wn wo ax-r2 wt ax-a2 or1 df-i4 ud4lem1a ud4lem1b ud4lem1d ancom fh4 or4 lea
|
|
ax-a3 lel2or leor letr df-le2 coman2 comor1 or32 df-a con2 ax-r1 df-t ax-r5
|
|
comor2 anor1 3tr1 2an an1 ) ABCZBACZCVGVHDZVGOZVHDZPZVJVHPVHODZPZAAOZBOZDZP
|
|
ZVGVHUAVNABDZVQPZAVPDZPZVOVPPZVOBPZDZADZPZVRVLWBVMWFVIVTVKWAABUBABUCEABUDEW
|
|
GWBAPZWBWEPZDZVRWGWBAWEDZPWJWFWKWBWEAUEFAWBWEAWBVOWBVOVTWAVOVSVQAVSVSAABGZH
|
|
IVQVOVOVPGHJAWAWAAAVPGHIJKLAWEVOWEVOWCWDVOVPMVOBMNKLUFQWJVRRDVRWHVRWIRWHVQA
|
|
PZVRWHVTWAAPPZWMVTWAAUIWNVSWAPZWMPWMVSVQWAAUGWOWMWOAWMVSAWAABUHAVPUHUJAVQUK
|
|
ULUMQQVQASQWIRRDZRWIWBWCPZWBWDPZDWPWCWBWDWCVTWAWCVSVQVSWCVSVOVPVSAWLKVSBABU
|
|
NKJHWCVOVPVOVPUOZVOVPVBZNJWCAVPWCAWCVOWSILWTNJWCVOBWSWCBWCVPWTILJUFWQRWRRWQ
|
|
VTWCPZWAPZRVTWAWCUPXBRWAPZRXARWAXAVSWCPZVQPZRVSVQWCUPXERVQPZRXDRVQXDVSVSOZP
|
|
ZRWCXGVSXGWCVSWCABUQURUSFRXHVSUTUSQVAXFVQRPRRVQSVQTQQQVAXCWARPRRWASWATQQQWR
|
|
VTWAWDPZPZRVTWAWDUIXJVTRPRXIRVTWDWAPWDWDOZPXIRWAXKWDABVCFWAWDSWDUTVDFVTTQQV
|
|
EQRVFQVEVRVFQQQQ $.
|
|
$( [25-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud4lem2 $p |- ( ( a v ( a ' ^ b ' ) ) ->4 a ) = ( a v b ) $=
|
|
( wn wa wo wi4 df-i4 wf ancom anabs ax-r2 oran con2 ran ax-r1 lan 2or ax-r5
|
|
con3 wt anass dff an0 or0 lor anor2 comid comorr fh3r or32 oridm df-t ax-a2
|
|
comcom2 2an an1 oml ) AACZBCDZEZAFUTADZUTCZADZEZVBAEZURDZEZABEZUTAGVGAURVHD
|
|
ZEVHVDAVFVIVDAHEAVAAVCHVAAUTDAUTAIAUSJKVCURUSCZDZADZHVBVKAUTVKAUSLMNVLAVKDZ
|
|
HVKAIVMAURDZVJDZHVOVMAURVJUAOVOVJVNDZHVNVJIVPVJHDZHVQVPHVNVJAUBPOVJUCKKKKKQ
|
|
AUDKVFURVEDVIVEURIVEVHURVEVIAEZVHVBVIAUTVIUTAVHCZEZVICUSVSAUSVHVHVJABLOSUEV
|
|
TVIVIVTCAVHUFOSKMRVRURAEZVHAEZDZVHAURVHAAAUGUNABUHUIWCVHTDZVHWCWBWADWDWAWBI
|
|
WBVHWATWBAAEZBEVHABAUJWEABAUKRKTWATAUREWAAULAURUMKOUOKVHUPKKKPKQABUQKK $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud4lem3a $p |- ( ( a ->4 b ) ' ^ ( a v b ) ) = ( a ->4 b ) ' $=
|
|
( wn wo wa wi4 anass lea leror df2le2 lan ax-r2 ud4lem0c ran 3tr1 ) ACBCZDA
|
|
PDEZAPEZBDZEZABDZEZTABFCZUAEUCUBQSUAEZETQSUAGUDSQSUARABAPHIJKLUCTUAABMZNUEO
|
|
$.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud4lem3b $p |- ( ( a ->4 b ) ' v ( a v b ) ) = ( a v b ) $=
|
|
( wi4 wn wo wa ud4lem0c comcom2 com2or com2an fh3r wt ax-a2 or4 ax-r1 ax-r2
|
|
lor or1 2an an1 ax-r5 comor1 comor2 df-t lea leror df-le2 ancom ) ABCDZABEZ
|
|
EADZBDZEZAULEZFZAULFZBEZFZUJEZUJUIURUJABGUAUSUOUJEZUQUJEZFZUJUJUOUQUJUMUNUJ
|
|
UKULUJAABUBZHUJBABUCZHZIZUJAULVCVEIZJUJUPBUJAULVCVEJVDIKVBLUJFZUJUTLVAUJUTL
|
|
LFZLUTUMUJEZUNUJEZFVIUJUMUNVFVGKVJLVKLVJUJUMEZLUMUJMVLAUKEZBULEZEZLABUKULNV
|
|
OVMLEZLVPVOLVNVMBUDZQOVMRPPPVKUJUNEZLUNUJMVRAAEZVNEZLABAULNVTVSLEZLWAVTLVNV
|
|
SVQQOVSRPPPSPLTPUQUJUPABAULUEUFUGSVHUJLFUJLUJUHUJTPPPP $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud4lem3 $p |- ( ( a ->4 b ) ->4 ( a v b ) ) = ( a v b ) $=
|
|
( wi4 wo wa wn df-i4 ud4lem3a lor comid comcom2 comor1 comor2 com2an com2or
|
|
wf comcom wt ax-r2 ax-r1 bctr fh4r ancom ax-a2 ud4lem3b 2an an1 ran dff 2or
|
|
df-t or0 ) ABCZABDZCUMUNEZUMFZUNEZDZUPUNDZUNFZEZDZUNUMUNGVBUNPDUNURUNVAPURU
|
|
OUPDZUNUQUPUOABHIVCUMUPDZUNUPDZEZUNUMUPUNUMUMUMJKUMABEZAFZBEZDZVHBDZBFZEZDZ
|
|
UNABGUNVNUNVJVMUNVGVIUNABABLZABMZNUNVHBUNAVOKZVPNOUNVKVLUNVHBVQVPOUNBVPKNOQ
|
|
UAUBVFVEVDEZUNVDVEUCVRUNREUNVEUNVDRVEUSUNUNUPUDABUEZSRVDUMUKTUFUNUGSSSSVAUN
|
|
UTEZPUSUNUTVSUHPVTUNUITSUJUNULSS $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud5lem1a $p |- ( ( a ->5 b ) ^ ( b ->5 a ) ) =
|
|
( ( a ^ b ) v ( a ' ^ b ' ) ) $=
|
|
( wa wo lan coman2 comcom2 coman1 com2an comcom com2or wf anass ax-r1 ancom
|
|
fh1r an0 ax-r2 2or or0 wi5 wn df-i5 2an ax-a2 fh2 comcom3 comcom5 dff anidm
|
|
an12 lor ran ) ABUAZBAUAZCABCZAUBZBCZDZUQBUBZCZDZBACZUTACZDZUTUQCZDZCZUPVAD
|
|
ZUNVBUOVGABUCBAUCUDVHVBVFVEDZCZVIVGVJVBVEVFUEEVKVBVFCZVBVECZDZVIVFVBVEVFUSV
|
|
AVFUPURUPVFUPUTUQUPBABFGZUPAABHZGIJZURVFURUTUQURBUQBFGUQBHIJZKVFUQUTUTUQFZU
|
|
TUQHZIZKVFVCVDVCVFVCUTUQVCBBAHZGZVCABAFZGZIJVDVFVDUTUQUTAHZVDAUTAFZGZIJKUFV
|
|
NVAUPDVIVLVAVMUPVLUSVFCZVAVFCZDZVAVFUSVAVFUPURVFABVFAVFUQVSUGUHVFBVFUTVTUGU
|
|
HZIVFUQBVSWLIKWAPWKLVADZVAWILWJVAWIUPVFCZURVFCZDZLVFUPURVQVRPWPLLDZLWNLWOLW
|
|
NABVFCZCZLABVFMWSABUTCZUQCZCZLWRXAAXAWRBUTUQMNZEXBALCZLXALAXAUQWTCZLWTUQOXE
|
|
UQLCZLWTLUQLWTBUINZEUQQZRRZEAQZRRRWOUQWRCZLUQBVFMXKUQXACZLWRXAUQXCEXLXFLXAL
|
|
UQXIEXHRRRSLTZRRWJVAVACVAVFVAVAUTUQOEVAUJRSWMVALDVALVAUEVATRRRVMVBVCCZVBVDC
|
|
ZDZUPVCVBVDVCUSVAVCUPURVCABWDWBIZVCUQBWEWBIZKZVCUQUTWEWCIZKVCUTAWCWDIUFXPUS
|
|
VCCZUSVDCZDZUPXNYAXOYBXNYAVAVCCZDZYAVCUSVAXSXTPYEYALDZYAYDLYAYDUQUTVCCZCZLU
|
|
QUTVCMYHXFLYGLUQYGBVDCZLYIYGBUTAUKNYIWTACZLYJYIBUTAMNYJAWTCZLWTAOYKXDLWTLAX
|
|
GEXJRRRZREXHRRULYATZRRXOYBVAVDCZDZYBVDUSVAVDUPURVDABWGVDBVDUTWFUGUHZIVDUQBW
|
|
HYPIZKVDUQUTWHWFIPYOYBLDYBYNLYBYNVDVACZLVAVDOYRUTAVACZCZLUTAVAMYTUTLCZLYSLU
|
|
TYSAUQCZUTCZLUUCYSAUQUTMNUUCUTUUBCZLUUBUTOUUDUUALUUBLUTLUUBAUINZEUTQZRRREUU
|
|
FRRRULYBTRRSYCYFUPYBLYAYBUPVDCZURVDCZDZLVDUPURUPVDUPUTAVOVPIJYQPUUIWQLUUGLU
|
|
UHLUUGAYICZLABVDMUUJXDLYILAYLEXJRRUUHUQYICZLUQBVDMUUKXFLYILUQYLEXHRRSXMRRUL
|
|
YFYAUPYMYAUPVCCZURVCCZDZUPVCUPURXQXRPUUNUPLDUPUULUPUUMLUULUPUPCUPVCUPUPBAOE
|
|
UPUJRUUMVCURCZLURVCOUUOBAURCZCZLBAURMUUQBLCZLUUPLBUUPUURLUUPUUBBCZUURUUSUUP
|
|
AUQBMNUUSLBCUURUUBLBUUEUMLBORRBQZREUUTRRRSUPTRRRRRRSVAUPUERRRR $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud5lem1b $p |- ( ( a ->5 b ) ' ^ ( b ->5 a ) ) = ( a ^ b ' ) $=
|
|
( wi5 wn wa wo ax-a2 ax-r2 2an coman2 coman1 com2or comcom7 com2an wf ax-r1
|
|
fh2 dff comcom2 an32 ud5lem0c df-i5 anass oran con3 lan an0 df-a ran ler2an
|
|
an0r lea leor letr lear leo df2le2 ancom 3tr1 2or or0r ) ABCDZBACZEADZBDZFZ
|
|
AVEFZEZABFZEZVEVDEZBAEZVEAEZFZFZEZAVEEZVBVJVCVOABUAVCVNVKFVOBAUBVNVKGHIVPVJ
|
|
VKEZVJVNEZFZVQVKVJVNVKVHVIVKVFVGVKVDVEVEVDJZVEVDKZLVKAVEVKAWAMZWBLNVKABWCVK
|
|
BWBMZLNVKVLVMVKBAWDWCNVKVEAWBWCNLQVTOVQFZVQVROVSVQVRVHVIVKEZEZOVHVIVKUCWGVH
|
|
OEOWFOVHWFBAFZWHDZEZOVIWHVKWIABGVKWHWHVKDBAUDPUEIOWJWHRPHUFVHUGHHVSVJVLEZVJ
|
|
VMEZFZVQVLVJVMVLVHVIVLVFVGVLVDVEVLABAJZSVLBBAKZSZLVLAVEWNWPLNVLABWNWOLNVLVE
|
|
AWPWNNQWMWEVQWKOWLVQWKVHVLEZVIEZOVHVIVLTWROVIEOWQOVIWQVFVLEZVGEZOVFVGVLTWTO
|
|
VGEOWSOVGWSVEVDFZXADZEZOVFXAVLXBVDVEGBAUHIOXCXARPHUIVGUKHHUIVIUKHHVMVJEVMWL
|
|
VQVMVJVMVHVIVMVEVHVEAULVEVFVGVEVDUMVEAUMUJUNVMAVIVEAUOABUPUNUJUQVJVMURAVEUR
|
|
USUTVQVAZHHUTXDHHH $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud5lem1c $p |- ( ( a ->5 b ) ' ^ ( b ->5 a ) ' ) =
|
|
( ( ( a v b ) ^ ( a v b ' ) ) ^
|
|
( ( a ' v b ) ^ ( a ' v b ' ) ) ) $=
|
|
( wi5 wn wa wo ud5lem0c ax-a2 2an ax-r2 an4 ancom anidm ran anass ax-r1 ) A
|
|
BCDZBACDZEADZBDZFZATFZEZABFZEZUASBFZEZUDEZEZUDUBEUFUAEZEZQUERUHABGRTSFZBSFZ
|
|
EZBAFZEUHBAGUNUGUOUDULUAUMUFTSHBSHIBAHIJIUIUCUGEZUDUDEZEZUKUCUDUGUDKURUQUPE
|
|
ZUKUPUQLUSUDUBUJEZEZUKUQUDUPUTUDMUPUAUAEZUBUFEZEZUTUAUBUAUFKVDVCUAEZUTVDUAV
|
|
CEVEVBUAVCUAMNUAVCLJUBUFUAOJJIUKVAUDUBUJOPJJJJ $.
|
|
$( [26-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud5lem1 $p |- ( ( a ->5 b ) ->5 ( b ->5 a ) ) =
|
|
( a v b ' ) $=
|
|
( wa wn wo coman1 coman2 com2or comcom2 com2an comcom comcom7 comor1 comor2
|
|
wi5 fh4 wt lor ax-r1 ax-r2 df-i5 ud5lem1a ud5lem1b ud5lem1c or32 ax-a3 oran
|
|
2or df-t or1 ax-r5 or1r lea leo letr lear leor lel2or df-le2 2an an1r anor1
|
|
con3 df-a an1 ) ABOZBAOZOVFVGCZVFDZVGCZEZVIVGDCZEZABDZEZVFVGUAVMABCZADZVNCZ
|
|
EZAVNCZEZABEZVOCZVQBEZVQVNEZCZCZEZVOVKWAVLWGVHVSVJVTABUBABUCUHABUDUHWHWAWCE
|
|
ZWAWFEZCZVOWCWAWFWCVSVTWCVPVRVPWCVPWBVOVPABABFZABGZHVPAVNWLVPBWMIHJKVRWCVRW
|
|
BVOVRABVRAVQVNFLZVRBVQVNGZLHVRAVNWNWOHJKHVTWCVTWBVOVTABAVNFZVTBAVNGZLHVTAVN
|
|
WPWQHJKHWCWDWEWDWCWDWBVOWDABWDAVQBMZLZVQBNZHWDAVNWSWDBWTIZHJKWEWCWEWBVOWEAB
|
|
WEAVQVNMLZWEBVQVNNZLHWEAVNXBXCHJKJPWKVOQCVOWIVOWJQWIWAWBEZWAVOEZCZVOWBWAVOW
|
|
BVSVTWBVPVRWBABABMZABNZJWBVQVNWBAXGIWBBXHIZJHWBAVNXGXIJHWBAVNXGXIHPXFQVOCVO
|
|
XDQXEVOXDVSWBEZVTEZQVSVTWBUEXKQVTEZQXJQVTXJVPVRWBEZEZQVPVRWBUFXNVPQEQXMQVPX
|
|
MVRVRDZEZQWBXOVRABUGRQXPVRUISTRVPUJTTUKVTULZTTWAVOVSVOVTVPVOVRVPAVOABUMAVNU
|
|
NZUOVRVNVOVQVNUPVNAUQUOURVTAVOAVNUMXRUOURUSUTVOVATTWJWAWDEZWAWEEZCZQWDWAWEW
|
|
DVSVTWDVPVRWDABWSWTJWDVQVNWRXAJHWDAVNWSXAJHWDVQVNWRXAHPYAQQCQXSQXTQXSVSVTWD
|
|
EZEZQVSVTWDUFYCVSQEQYBQVSYBVTVTDZEZQWDYDVTWDVTVTWDDABVBSVCRQYEVTUISTRVSUJTT
|
|
XTVSWEEZVTEZQVSVTWEUEYGXLQYFQVTYFVPWEEZVREZQVPVRWEUEYIQVREQYHQVRYHVPVPDZEZQ
|
|
WEYJVPWEVPVPWEDABVDSVCRQYKVPUISTUKVRULTTUKXQTTUTQVETTUTVOVETTTT $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud5lem2 $p |- ( ( a v b ' ) ->5 a ) = ( a v ( a ' ^ b ) ) $=
|
|
( wn wo wi5 wa df-i5 ax-a3 ancom anabs ax-r2 ax-a2 wf anor2 ax-r1 ran anidm
|
|
an32 dff 2or lan an0 or0 ) ABCZDZAEUEAFZUECZAFZDUGACZFZDZAUIBFZDZUEAGUKUFUH
|
|
UJDZDUMUFUHUJHUFAUNULUFAUEFAUEAIAUDJKUNUJUHDZULUHUJLUOULMDULUJULUHMUJULUIFZ
|
|
ULUGULUIULUGABNOZPUPUIUIFZBFULUIBUIRURUIBUIQPKKUHULAFZMUGULAUQPUSUIAFZBFZMU
|
|
IBARVABUTFZMUTBIVBBMFMUTMBUTAUIFZMUIAIMVCASOKUABUBKKKKTULUCKKTKK $.
|
|
$( [10-Apr-2012] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud5lem3a $p |- ( ( a ->5 b ) ^ ( a v ( a ' ^ b ) ) ) =
|
|
( ( a ^ b ) v ( a ' ^ b ) ) $=
|
|
( wn wa wo ran comanr1 comcom6 com2or fh1r ax-r2 ancom anass ax-r1 dff an0r
|
|
wf 2or or0 com2an wi5 df-i5 fh2 an32 anidm coman1 comcom7 comcom2 ax-a2 lan
|
|
coman2 anabs an4 an0 lor ) ABUAZAACZBDZEZDABDZUREZUQBCZDZEZUSDZVAUPVDUSABUB
|
|
FVEVDADZVDURDZEZVAAVDURAVAVCAUTURABGZAURUQBGHZIZAVCUQVBGHZIVJUCVHUTVAURDZVC
|
|
URDZEZEVAVFUTVGVOVFVAADZVCADZEZUTAVAVCVKVLJVRUTQEZUTVPUTVQQVPUTADZURADZEZUT
|
|
AUTURVIVJJWBVSUTVTUTWAQVTAADZBDUTABAUDWCABAUEFKWAAURDZQURALWDAUQDZBDZQWFWDA
|
|
UQBMNWFQBDQWEQBQWEAONZFBPKKKRUTSZKKVQAVCDZQVCALWIWEVBDZQWJWIAUQVBMNWJQVBDQW
|
|
EQVBWGFVBPKKKRWHKKURVAVCURUTURURABURAUQBUFZUGUQBUKZTURUQBWKWLTIURUQVBWKURBW
|
|
LUHTJRVOURUTVOURQEURVMURVNQVMURVADZURVAURLWMURURUTEZDURVAWNURUTURUIUJURUTUL
|
|
KKVNUQUQDZVBBDZDZQUQVBUQBUMWQWOQDQWPQWOWPBVBDZQVBBLQWRBONKUJWOUNKKRURSKUOKK
|
|
K $.
|
|
$( [27-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud5lem3b $p |- ( ( a ->5 b ) ' ^ ( a v ( a ' ^ b ) ) ) =
|
|
( a ^ ( a ' v b ' ) ) $=
|
|
( wi5 wn wa wo ud5lem0c ran comorr comcom6 com2an comanr1 anass ancom anabs
|
|
fh2 wf ax-r2 lan an32 anor2 dff ax-r1 an0 an0r 2or or0 ) ABCDZAADZBEZFZEUIB
|
|
DZFZAULFZEZABFZEZUKEZAUMEZUHUQUKABGHURUQAEZUQUJEZFZUSAUQUJAUOUPAUMUNAUMUIUL
|
|
IJAULIKABIKAUJUIBLJPVBUSQFUSUTUSVAQUTUOUPAEZEZUSUOUPAMVDUOAEZUSVCAUOVCAUPEA
|
|
UPANABORSVEUMUNAEZEZUSUMUNAMVGUMAEUSVFAUMVFAUNEAUNANAULORSUMANRRRRVAUOUJEZU
|
|
PEZQUOUPUJTVIQUPEQVHQUPVHUMUNUJEZEZQUMUNUJMVKUMQEQVJQUMVJUNUNDZEZQUJVLUNABU
|
|
ASQVMUNUBUCRSUMUDRRHUPUERRUFUSUGRRR $.
|
|
$( [26-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud5lem3c $p |- ( ( a ->5 b ) ' ^ ( a v ( a ' ^ b ) ) ' ) =
|
|
( ( ( a v b ) ^ ( a v b ' ) ) ^ a ' ) $=
|
|
( wi5 wn wa wo ud5lem0c oran con2 anor2 lan ax-r2 2an an4 ancom anabs anidm
|
|
an32 ran anass ax-r1 ) ABCDZAADZBEZFZDZEUCBDZFZAUGFZEZABFZEZUCUIEZEZUKUIEUC
|
|
EZUBULUFUMABGUFUCUDDZEZUMUEUQAUDHIUPUIUCUDUIABJIKLMUNUJUMEZUKEZUOUJUKUMRUSU
|
|
KUIUCEZEZUOUSUTUKEVAURUTUKURUHUCEZUIUIEZEZUTUHUIUCUINVDUMUTVBUCVCUIVBUCUHEU
|
|
CUHUCOUCUGPLUIQMUCUIOLLSUTUKOLUOVAUKUIUCTUALLL $.
|
|
$( [26-Nov-97] $)
|
|
|
|
$( Lemma for unified disjunction. $)
|
|
ud5lem3 $p |- ( ( a ->5 b ) ->5 ( a v ( a ' ^ b ) ) ) = ( a v b ) $=
|
|
( wi5 wn wa wo 2or fh4 ax-a2 orabs ax-r2 ax-r1 con3 lor df-t 2an an1 com2or
|
|
wt comcom2 df-i5 ud5lem3a ud5lem3b ud5lem3c or4 comanr1 comorr comcom6 df-a
|
|
ax-a3 coman1 comcom7 coman2 com2an fh3 comor1 comor2 fh3r oridm ancom anabs
|
|
or12 anor2 oml ) ABCZAADZBEZFZCVEVHEZVEDZVHEZFZVJVHDEZFZABFZVEVHUAVNABEZVGF
|
|
ZAVFBDZFZEZFZVOAVRFZEZVFEZFZVOVLWAVMWDVIVQVKVTABUBABUCGABUDGWEVQVTWDFFZVOVQ
|
|
VTWDUJWFVPVTFZVGWDFZFZVOVPVGVTWDUEWIAVFVOEZFVOWGAWHWJWGVPAFZVPVSFZEZAAVPVSA
|
|
BUFAVSVFVRUGUHHWMASEAWKAWLSWKAVPFAVPAIABJKWLVPVPDZFZSVSWNVPVSVPVPVSDABUILMN
|
|
SWOVPOLKPAQKKWHVGWCFZVGVFFZEZWJVGWCVFVGVOWBVGABVGAVFBUKZULZVFBUMZRVGAVRWTVG
|
|
BXATRUNWSUOWRVOVFEWJWPVOWQVFWPVGVOFZVGWBFZEZVOVOVGWBVOVFBVOAABUPZTZABUQZUNV
|
|
OAVRXEVOBXGTRHXDVOSEVOXBVOXCSXBVFVOFZBVOFZEZVOVOVFBXFXGURXJVOVFFZVOEZVOXHXK
|
|
XIVOVFVOIXIABBFZFVOBABVBXMBABUSNKPXLVOXKEVOXKVOUTVOVFVAKKKXCVGVGDZFZSWBXNVG
|
|
WBVGVGWBDABVCLMNSXOVGOLKPVOQKKWQVFVGFVFVGVFIVFBJKPVOVFUTKKGABVDKKKKK $.
|
|
$( [26-Nov-97] $)
|
|
|
|
$( Unified disjunction for Sasaki implication. $)
|
|
ud1 $p |- ( a v b ) =
|
|
( ( a ->1 b ) ->1 ( ( ( a ->1 b ) ->1 ( b ->1 a ) ) ->1 a ) ) $=
|
|
( wi1 wo wn wa ud1lem1 ud1lem0b ud1lem2 ax-r2 ud1lem0a ud1lem3 ax-r1 ) ABCZ
|
|
NBACCZACZCZABDZQNRCRPRNPAAEBEFDZACROSAABGHABIJKABLJM $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Unified disjunction for Dishkant implication. $)
|
|
ud2 $p |- ( a v b ) =
|
|
( ( a ->2 b ) ->2 ( ( ( a ->2 b ) ->2 ( b ->2 a ) ) ->2 a ) ) $=
|
|
( wi2 wo wn wa ud2lem1 ud2lem0b ud2lem2 ax-r2 ud2lem0a ud2lem3 ax-r1 ) ABCZ
|
|
NBACCZACZCZABDZQNRCRPRNPAAEBEFDZACROSAABGHABIJKABLJM $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Unified disjunction for Kalmbach implication. $)
|
|
ud3 $p |- ( a v b ) =
|
|
( ( a ->3 b ) ->3 ( ( ( a ->3 b ) ->3 ( b ->3 a ) ) ->3 a ) ) $=
|
|
( wi3 wo wn wa ud3lem1 ud3lem0b ud3lem2 ax-r2 ud3lem0a ud3lem3 ax-r1 ) ABCZ
|
|
NBACCZACZCZABDZQNRCRPRNPAAEBEFDZACROSAABGHABIJKABLJM $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Unified disjunction for non-tollens implication. $)
|
|
ud4 $p |- ( a v b ) =
|
|
( ( a ->4 b ) ->4 ( ( ( a ->4 b ) ->4 ( b ->4 a ) ) ->4 a ) ) $=
|
|
( wi4 wo wn wa ud4lem1 ud4lem0b ud4lem2 ax-r2 ud4lem0a ud4lem3 ax-r1 ) ABCZ
|
|
NBACCZACZCZABDZQNRCRPRNPAAEBEFDZACROSAABGHABIJKABLJM $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$( Unified disjunction for relevance implication. $)
|
|
ud5 $p |- ( a v b ) =
|
|
( ( a ->5 b ) ->5 ( ( ( a ->5 b ) ->5 ( b ->5 a ) ) ->5 a ) ) $=
|
|
( wi5 wo wn wa ud5lem1 ud5lem0b ud5lem2 ax-r2 ud5lem0a ud5lem3 ax-r1 ) ABCZ
|
|
NBACCZACZCZABDZQNAAEBFDZCRPSNPABEDZACSOTAABGHABIJKABLJM $.
|
|
$( [23-Nov-97] $)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Lemmas for unified implication study
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Lemma for Sasaki implication study. Equation 4.10 of [MegPav2000] p. 23.
|
|
This is the second part of the equation. $)
|
|
u1lemaa $p |- ( ( a ->1 b ) ^ a ) = ( a ^ b ) $=
|
|
( wi1 wa wn wo df-i1 ran comid comcom2 comanr1 fh1r ax-a2 anidm ax-r2 ancom
|
|
wf an32 dff ax-r1 2or or0 ) ABCZADAEZABDZFZADZUEUCUFAABGHUGUDADZUEADZFZUEAU
|
|
DUEAAAIJABKLUJUEQFZUEUJUIUHFUKUHUIMUIUEUHQUIAADZBDUEABARULABANHOUHAUDDZQUDA
|
|
PQUMASTOUAOUEUBOOO $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemaa $p |- ( ( a ->2 b ) ^ a ) = ( a ^ b ) $=
|
|
( wi2 wa wn wo df-i2 ran ax-a2 coman1 comcom7 coman2 fh2r ancom anass ax-r1
|
|
wf dff lan ax-r2 an0 2or or0 ) ABCZADBAEZBEZDZFZADZABDZUDUHAABGHUIUGBFZADZU
|
|
JUHUKABUGIHULUGADZBADZFZUJUGABUGAUEUFJKUGBUEUFLKMUOUNUMFZUJUMUNIUPUJQFUJUNU
|
|
JUMQBANUMAUGDZQUGANUQAUEDZUFDZQUSUQAUEUFOPUSUFURDZQURUFNUTUFQDQURQUFQURARPS
|
|
UFUATTTTUBUJUCTTTTT $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemaa $p |- ( ( a ->3 b ) ^ a ) = ( a ^ ( a ' v b ) ) $=
|
|
( wi3 wa wn wo df-i3 ran comanr1 comcom6 wf ancom anass ax-r1 lan an0 ax-r2
|
|
fh1r 2or or0 com2or comid comorr com2an dff an32 anidm ax-a2 ) ABCZADAEZBDZ
|
|
UJBEZDZFZAUJBFZDZFZADZUPUIUQAABGHURUNADZUPADZFZUPAUNUPAUKUMAUKUJBIJZAUMUJUL
|
|
IJZUAAAUOAUBAUOUJBUCJUDRVAKUPFZUPUSKUTUPUSUKADZUMADZFZKAUKUMVBVCRVGKKFKVEKV
|
|
FKVEAUKDZKUKALVHAUJDZBDZKVJVHAUJBMNVJBVIDZKVIBLVKBKDKVIKBKVIAUENZOBPQQQQVFA
|
|
UMDZKUMALVMVIULDZKVNVMAUJULMNVNULVIDZKVIULLVOULKDKVIKULVLOULPQQQQSKTQQUTAAD
|
|
ZUODUPAUOAUFVPAUOAUGHQSVDUPKFUPKUPUHUPTQQQQ $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemaa $p |- ( ( a ->4 b ) ^ a ) = ( a ^ b ) $=
|
|
( wi4 wa wn wo df-i4 ran comanr1 com2or comcom comanr2 wf ax-r2 ancom anass
|
|
ax-r1 dff lan 2or comcom6 comcom3 comcom2 com2an fh2r fh1r an32 anidm anor1
|
|
an0 or0 ) ABCZADABDZAEZBDZFZUNBFZBEZDZFZADZUMULUTAABGHVAUPADZUSADZFZUMUPAUS
|
|
AUPAUMUOABIZAUOUNBIZUAZJKUPUQURUPUNBUNUPUNUMUOAUMVEUBVFJKBUPBUMUOABLUNBLJKZ
|
|
JUPBVHUCUDUEVDUMMFZUMVBUMVCMVBUMADZUOADZFZUMAUMUOVEVGUFVLVIUMVJUMVKMVJAADZB
|
|
DUMABAUGVMABAUHHNVKAUODZMUOAOVNAUNDZBDZMVPVNAUNBPQVPBVODZMVOBOVQBMDMVOMBMVO
|
|
ARQSBUJNNNNTUMUKZNNVCUQURADZDZMUQURAPVTUQUQEZDZMVSWAUQVSAURDWAURAOABUINSMWB
|
|
UQRQNNTVRNNN $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemaa $p |- ( ( a ->5 b ) ^ a ) = ( a ^ b ) $=
|
|
( wi5 wa wn wo df-i5 ran comanr1 comcom6 fh1r wf an32 ax-r2 ancom ax-r1 lan
|
|
an0 2or anass com2or anidm dff or0 fh4 ax-a2 orabs fh1 ) ABCZADABDZAEZBDZFZ
|
|
UKBEZDZFZADZUJUIUPAABGHUQUMADZUOADZFZUJAUMUOAUJULABIZAULUKBIJZUAAUOUKUNIJZK
|
|
UTUJAUODZFZUJURUJUSVDURUJADZULADZFZUJAUJULVAVBKVHUJLFZUJVFUJVGLVFAADZBDZUJA
|
|
BAMVJABAUBHZNVGUKADZBDZLUKBAMVNBVMDZLVMBOVOBLDLVMLBVMAUKDZLVPVMAUKOPLVPAUCZ
|
|
PNQBRNNNSUJUDZNNUOAOSVEUJAFZUJUOFZDZUJAUJUOVAVCUEWAAVTDZUJVSAVTVSAUJFAUJAUF
|
|
ABUGNHWBAUJDZVDFZUJAUJUOVAVCUHWDVIUJWCUJVDLWCVKUJVKWCAABTPVLNVDVPUNDZLWEVDA
|
|
UKUNTPWEUNVPDZLVPUNOWFUNLDZLWGWFLVPUNVQQPUNRNNNSVRNNNNNNN $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemana $p |- ( ( a ->1 b ) ^ a ' ) = a ' $=
|
|
( wi1 wn wa wo df-i1 ran ancom anabs ax-r2 ) ABCZADZEMABEZFZMEZMLOMABGHPMOE
|
|
MOMIMNJKK $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemana $p |- ( ( a ->2 b ) ^ a ' ) =
|
|
( ( a ' ^ b ) v ( a ' ^ b ' ) ) $=
|
|
( wi2 wn wa wo df-i2 ran ax-a2 coman1 coman2 comcom7 fh2r anidm ax-r2 ancom
|
|
an32 2or ) ABCZADZEBTBDZEZFZTEZTBEZUBFZSUCTABGHUDUBBFZTEZUFUCUGTBUBIHUHUBTE
|
|
ZBTEZFZUFUBTBTUAJUBBTUAKLMUKUBUEFUFUIUBUJUEUITTEZUAEUBTUATQULTUATNHOBTPRUBU
|
|
EIOOOO $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemana $p |- ( ( a ->3 b ) ^ a ' ) =
|
|
( ( a ' ^ b ) v ( a ' ^ b ' ) ) $=
|
|
( wi3 wn wa wo df-i3 ran comanr1 com2or comid comcom3 comorr com2an fh1r wf
|
|
lea lel2or df2le2 ax-r2 an32 ancom dff ax-r1 lan an0 2or or0 ) ABCZADZEUJBE
|
|
ZUJBDZEZFZAUJBFZEZFZUJEZUNUIUQUJABGHURUNUJEZUPUJEZFZUNUJUNUPUJUKUMUJBIUJULI
|
|
JUJAUOAAAKLUJBMNOVAUNPFUNUSUNUTPUNUJUKUJUMUJBQUJULQRSUTAUJEZUOEZPAUOUJUAVCU
|
|
OVBEZPVBUOUBVDUOPEPVBPUOPVBAUCUDUEUOUFTTTUGUNUHTTT $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemana $p |- ( ( a ->4 b ) ^ a ' ) =
|
|
( ( a ' ^ b ) v ( a ' ^ b ' ) ) $=
|
|
( wi4 wn wa wo df-i4 comanr1 comcom3 com2or comcom comor1 com2an comanr2 wf
|
|
ran an32 ancom ax-r2 2or comcom7 comor2 fh2r fh1r dff ax-r1 lan anidm ax-a2
|
|
an0 or0 leo df2le2 id ) ABCZADZEABEZUPBEZFZUPBFZBDZEZFZUPEZURUPVAEZFZUOVCUP
|
|
ABGPVDUSUPEZVBUPEZFZVFUSUPVBUPUSUPUQURAUQABHIZUPBHZJKUSUTVAUTUSUTUQURUTABUT
|
|
AUPBLZUAUPBUBZMUTUPBVLVMMJKVAUSVAUQURBUQABNIBURUPBNIJKMUCVIVFVFVGURVHVEVGUQ
|
|
UPEZURUPEZFZURUPUQURVJVKUDVPOURFZURVNOVOURVNAUPEZBEZOABUPQVSBVREZOVRBRVTBOE
|
|
OVROBOVRAUEUFUGBUJSSSVOUPUPEZBEURUPBUPQWAUPBUPUHPSTVQUROFUROURUIURUKSSSVHUT
|
|
UPEZVAEVEUTVAUPQWBUPVAWBUPUTEUPUTUPRUPUTUPBULUMSPSTVFUNSSS $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemana $p |- ( ( a ->5 b ) ^ a ' ) =
|
|
( ( a ' ^ b ) v ( a ' ^ b ' ) ) $=
|
|
( wi5 wn wa wo df-i5 ran comanr1 comcom3 com2or fh1r ax-a2 an32 anidm ax-r2
|
|
wf ancom dff 2or lan ax-r1 an0 or0 ) ABCZADZEABEZUFBEZFZUFBDZEZFZUFEZUHUKFZ
|
|
UEULUFABGHUMUIUFEZUKUFEZFUNUFUIUKUFUGUHAUGABIJZUFBIZKUFUJILUOUHUPUKUOUGUFEZ
|
|
UHUFEZFZUHUFUGUHUQURLVAUTUSFZUHUSUTMVBUHQFUHUTUHUSQUTUFUFEZBEUHUFBUFNVCUFBU
|
|
FOZHPUSAUFEZBEZQABUFNVFBVEEZQVEBRVGBQEZQVHVGQVEBASUAUBBUCPPPTUHUDPPPUPVCUJE
|
|
UKUFUJUFNVCUFUJVDHPTPP $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. Equation 4.10 of [MegPav2000] p. 23.
|
|
This is the second part of the equation. $)
|
|
u1lemab $p |- ( ( a ->1 b ) ^ b ) = ( ( a ^ b ) v ( a ' ^ b ) ) $=
|
|
( wi1 wa wn wo df-i1 ran ax-a2 coman2 coman1 comcom2 fh2r ax-r2 anass anidm
|
|
lan ax-r5 ) ABCZBDAEZABDZFZBDZUATBDZFZSUBBABGHUCUABDZUDFZUEUCUATFZBDUGUBUHB
|
|
TUAIHUABTABJUAAABKLMNUFUAUDUFABBDZDUAABBOUIBABPQNRNN $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemab $p |- ( ( a ->2 b ) ^ b ) = b $=
|
|
( wi2 wa wn wo df-i2 ran ancom anabs ax-r2 ) ABCZBDBAEBEDZFZBDZBLNBABGHOBND
|
|
BNBIBMJKK $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemab $p |- ( ( a ->3 b ) ^ b ) = ( ( a ^ b ) v ( a ' ^ b ) ) $=
|
|
( wi3 wa wn wo df-i3 comanr2 com2or comcom coman1 comcom7 coman2 com2an lan
|
|
wf anass ax-r2 2or ax-a2 ran comcom6 fh2r fh1r anidm an32 dff ax-r1 an0 or0
|
|
ancom anabs ) ABCZBDAEZBDZUNBEZDZFZAUNBFZDZFZBDZABDZUOFZUMVABABGUAVBURBDZUT
|
|
BDZFZVDURBUTBURBUOUQUNBHZBUQUNUPHUBZIJUTURUTUOUQUOUTUOAUSUOAUNBKZLUOUNBVJUN
|
|
BMINJUQUTUQAUSUQAUNUPKZLUQUNBVKUQBUNUPMLINJIJUCVGUOVCFVDVEUOVFVCVEUOBDZUQBD
|
|
ZFZUOBUOUQVHVIUDVNUOPFUOVLUOVMPVLUNBBDZDUOUNBBQVOBUNBUEORVMUOUPDZPUNUPBUFVP
|
|
UNBUPDZDZPUNBUPQVRUNPDPVQPUNPVQBUGUHOUNUIRRRSUOUJRRVFAUSBDZDVCAUSBQVSBAVSBU
|
|
SDZBUSBUKVTBBUNFZDBUSWABUNBTOBUNULRRORSUOVCTRRR $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemab $p |- ( ( a ->4 b ) ^ b ) = ( ( a ^ b ) v ( a ' ^ b ) ) $=
|
|
( wi4 wa wn wo df-i4 comanr2 com2or comcom6 fh1r wf lear lel2or df2le2 an32
|
|
ran anass dff ax-r2 lan ax-r1 an0 2or or0 ) ABCZBDABDZAEZBDZFZUHBFZBEZDZFZB
|
|
DZUJUFUNBABGQUOUJBDZUMBDZFZUJBUJUMBUGUIABHUHBHIBUMUKULHJKURUJLFUJUPUJUQLUJB
|
|
UGBUIABMUHBMNOUQUKBDULDZLUKULBPUSUKBULDZDZLUKBULRVAUKLDZLVBVALUTUKBSUAUBUKU
|
|
CTTTUDUJUETTT $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemab $p |- ( ( a ->5 b ) ^ b ) = ( ( a ^ b ) v ( a ' ^ b ) ) $=
|
|
( wi5 wa wn wo df-i5 comanr2 com2or comcom6 fh1r wf lear lel2or df2le2 an32
|
|
ran anass dff ax-r2 lan ax-r1 an0 2or or0 ) ABCZBDABDZAEZBDZFZUHBEZDZFZBDZU
|
|
JUFUMBABGQUNUJBDZULBDZFZUJBUJULBUGUIABHUHBHIBULUHUKHJKUQUJLFUJUOUJUPLUJBUGB
|
|
UIABMUHBMNOUPUIUKDZLUHUKBPURUHBUKDZDZLUHBUKRUTUHLDZLVAUTLUSUHBSUAUBUHUCTTTU
|
|
DUJUETTT $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemanb $p |- ( ( a ->1 b ) ^ b ' ) = ( a ' ^ b ' ) $=
|
|
( wi1 wn wa wo df-i1 ran ax-a2 coman2 comcom2 coman1 wf anass dff lan ax-r1
|
|
fh2r an0 ax-r2 lor or0 ) ABCZBDZEADZABEZFZUDEZUEUDEZUCUGUDABGHUHUFUEFZUDEZU
|
|
IUGUJUDUEUFIHUKUFUDEZUIFZUIUFUDUEUFBABJKUFAABLKRUMUIULFZUIULUIIUNUIMFUIULMU
|
|
IULABUDEZEZMABUDNUPAMEZMUQUPMUOABOPQASTTUAUIUBTTTTT $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemanb $p |- ( ( a ->2 b ) ^ b ' ) = ( a ' ^ b ' ) $=
|
|
( wi2 wn wa wo df-i2 ran comid comcom3 comanr2 fh1r ax-a2 anass anidm ax-r2
|
|
wf lan dff ax-r1 2or or0 ) ABCZBDZEBADZUDEZFZUDEZUFUCUGUDABGHUHBUDEZUFUDEZF
|
|
ZUFUDBUFBBBIJUEUDKLUKUJUIFZUFUIUJMULUFQFUFUJUFUIQUJUEUDUDEZEUFUEUDUDNUMUDUE
|
|
UDORPQUIBSTUAUFUBPPPP $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemanb $p |- ( ( a ->3 b ) ^ b ' ) = ( a ' ^ b ' ) $=
|
|
( wn wa wo comanr2 com2or comcom coman1 comcom7 coman2 com2an fh2r wf anass
|
|
lan ax-r2 dff ax-r1 2or wi3 df-i3 ran comcom3 comcom2 ax-a2 anidm an0 ancom
|
|
or0 an32 anor1 ) ABUAZBCZDACZBDZUOUNDZEZAUOBEZDZEZUNDZUQUMVAUNABUBUCVBURUND
|
|
ZUTUNDZEZUQURUNUTUNURUNUPUQBUPUOBFUDUOUNFGHUTURUTUPUQUPUTUPAUSUPAUOBIZJUPUO
|
|
BVFUOBKZGLHUQUTUQAUSUQAUOUNIZJUQUOBVHUQBUOUNKJGLHGHMVEUQNEZUQVCUQVDNVCUPUND
|
|
ZUQUNDZEZUQUPUNUQUPBVGUEZUPUOUNVFVMLMVLVKVJEZUQVJVKUFVNVIUQVKUQVJNVKUOUNUND
|
|
ZDUQUOUNUNOVOUNUOUNUGPQVJUOBUNDZDZNUOBUNOVQUONDZNVRVQNVPUOBRPSUOUHQQTUQUJZQ
|
|
QQVDAUNDZUSDZNAUSUNUKWAUSVTDZNVTUSUIWBUSUSCZDZNVTWCUSABULPNWDUSRSQQQTVSQQQ
|
|
$.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemanb $p |- ( ( a ->4 b ) ^ b ' ) = ( ( a ' v b ) ^ b ' ) $=
|
|
( wi4 wn wa wo df-i4 ran comanr2 comcom3 com2or fh1r wf anass lan ax-r2 an0
|
|
ax-r1 2or or0 comorr2 comid com2an ax-a2 anidm dff ) ABCZBDZEABEZADZBEZFZUJ
|
|
BFZUHEZFZUHEZUNUGUOUHABGHUPULUHEZUNUHEZFZUNUHULUNUHUIUKBUIABIJZBUKUJBIJZKUH
|
|
UMUHBUMUJBUAJUHUBUCLUSURUQFZUNUQURUDVBUNMFUNURUNUQMURUMUHUHEZEUNUMUHUHNVCUH
|
|
UMUHUEOPUQUIUHEZUKUHEZFZMUHUIUKUTVALVFMMFMVDMVEMVDABUHEZEZMABUHNVHAMEZMVIVH
|
|
MVGABUFZORAQPPVEUJVGEZMUJBUHNVKUJMEZMVLVKMVGUJVJORUJQPPSMTPPSUNTPPPP $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemanb $p |- ( ( a ->5 b ) ^ b ' ) = ( a ' ^ b ' ) $=
|
|
( wi5 wn wa wo df-i5 ran comanr2 comcom3 com2or fh1r wf anass lan ax-r2 an0
|
|
ax-r1 2or or0 ax-a2 anidm dff ) ABCZBDZEABEZADZBEZFZUGUEEZFZUEEZUJUDUKUEABG
|
|
HULUIUEEZUJUEEZFZUJUEUIUJUEUFUHBUFABIJZBUHUGBIJZKUGUEILUOUNUMFZUJUMUNUAURUJ
|
|
MFUJUNUJUMMUNUGUEUEEZEUJUGUEUENUSUEUGUEUBOPUMUFUEEZUHUEEZFZMUEUFUHUPUQLVBMM
|
|
FMUTMVAMUTABUEEZEZMABUENVDAMEZMVEVDMVCABUCZORAQPPVAUGVCEZMUGBUENVGUGMEZMVHV
|
|
GMVCUGVFORUGQPPSMTPPSUJTPPPP $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemoa $p |- ( ( a ->1 b ) v a ) = 1 $=
|
|
( wi1 wo wn wa wt df-i1 ax-r5 ax-a2 ax-a3 ax-r1 df-t lor or1 ax-r2 ) ABCZAD
|
|
AEZABFZDZADZGQTAABHIUAATDZGTAJUBARDZSDZGUDUBARSKLUDSUCDZGUCSJUESGDZGUFUEGUC
|
|
SAMNLSOPPPPP $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemoa $p |- ( ( a ->2 b ) v a ) = 1 $=
|
|
( wi2 wo wn wa wt df-i2 ax-r5 ax-a2 ax-a3 ax-r1 oran lor df-t ax-r2 ) ABCZA
|
|
DBAEBEFZDZADZGQSAABHITASDZGSAJUAABDZRDZGUCUAABRKLUCRUBDZGUBRJUDRREZDZGUBUER
|
|
ABMNGUFROLPPPPP $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemoa $p |- ( ( a ->3 b ) v a ) =
|
|
( a v ( ( a ' ^ b ) v ( a ' ^ b ' ) ) ) $=
|
|
( wi3 wo wn wa df-i3 ax-r5 ax-a3 lea df-le2 lor ax-a2 ax-r2 ) ABCZADAEZBFPB
|
|
EFDZAPBDZFZDZADZAQDZOTAABGHUAQSADZDZUBQSAIUDQADUBUCAQSAARJKLQAMNNN $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemoa $p |- ( ( a ->4 b ) v a ) = 1 $=
|
|
( wi4 wo wa wn df-i4 ax-r5 ax-a3 comor1 comcom7 comor2 ax-a2 df-t ax-r2 lor
|
|
wt ax-r1 or1 ancom comcom2 fh4r or32 ran an1 anor1 ) ABCZADABEZAFZBEZDZUIBD
|
|
ZBFZEZDZADZQUGUOAABGHUPUKUNADZDZQUKUNAIURUKUMADZDZQUQUSUKUQULADZUSEZUSULAUM
|
|
ULAUIBJKULBUIBLUAUBVBQUSEZUSVAQUSVAUIADZBDZQUIBAUCVEBVDDZQVDBMVFBQDZQVGVFQV
|
|
DBQAUIDVDANAUIMOPRBSOOOUDVCUSQEUSQUSTUSUEOOOPUTUHUJUSDZDZQUHUJUSIVIUHQDQVHQ
|
|
UHVHUSUJDZQUJUSMVJUSUSFZDZQUJVKUSUJBUIEVKUIBTBAUFOPQVLUSNROOPUHSOOOOO $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemoa $p |- ( ( a ->5 b ) v a ) =
|
|
( a v ( ( a ' ^ b ) v ( a ' ^ b ' ) ) ) $=
|
|
( wi5 wo wa wn df-i5 ax-r5 ax-a2 ax-a3 lor ax-r1 orabs ax-r2 ) ABCZADABEZAF
|
|
ZBEZDQBFEZDZADZARSDZDZOTAABGHUAATDZUCTAIUDAPUBDZDZUCTUEAPRSJKUFAPDZUBDZUCUH
|
|
UFAPUBJLUGAUBABMHNNNN $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemona $p |- ( ( a ->1 b ) v a ' ) = ( a ' v ( a ^ b ) ) $=
|
|
( wi1 wn wo wa df-i1 ax-r5 or32 oridm ax-r2 ) ABCZADZEMABFZEZMEZOLOMABGHPMM
|
|
EZNEOMNMIQMNMJHKK $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemona $p |- ( ( a ->2 b ) v a ' ) = ( a ' v b ) $=
|
|
( wi2 wn wo wa df-i2 ax-r5 ax-a3 ax-a2 lea df-le2 ax-r2 ) ABCZADZEBOBDZFZEZ
|
|
OEZOBEZNROABGHSBQOEZEZTBQOIUBUABETBUAJUAOBQOOPKLHMMM $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemona $p |- ( ( a ->3 b ) v a ' ) = ( a ' v b ) $=
|
|
( wi3 wn wo wa df-i3 ax-r5 or32 lea lel2or df-le2 omln ax-r2 ) ABCZADZEPBFZ
|
|
PBDZFZEZAPBEZFZEZPEZUAOUCPABGHUDTPEZUBEZUATUBPIUFPUBEUAUEPUBTPQPSPBJPRJKLHA
|
|
BMNNN $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemona $p |- ( ( a ->4 b ) v a ' ) = ( a ' v b ) $=
|
|
( wi4 wn wo wa df-i4 ax-r5 ax-a3 lea df-le2 lor ax-r2 comor1 comcom7 comor2
|
|
or32 com2an wt ax-r1 com2or comcom2 fh4 lear leor letr leo lel2or df-a con3
|
|
df-t 2an an1 ) ABCZADZEABFZUOBFZEZUOBEZBDZFZEZUOEZUSUNVBUOABGHVCURUOEZVAEZU
|
|
SURVAUOQVEUPUOEZVAEZUSVDVFVAVDUPUQUOEZEVFUPUQUOIVHUOUPUQUOUOBJKLMHVGVFUSEZV
|
|
FUTEZFZUSUSVFUTUSUPUOUSABUSAUOBNZOUOBPZRVLUAUSBVMUBUCVKUSSFUSVIUSVJSVFUSUPU
|
|
SUOUPBUSABUDBUOUEUFUOBUGUHKVJUPUOUTEZEZSUPUOUTIVOUPUPDZEZSVNVPUPVNUPUPVNDAB
|
|
UITUJLSVQUPUKTMMULUSUMMMMMM $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemona $p |- ( ( a ->5 b ) v a ' ) = ( a ' v ( a ^ b ) ) $=
|
|
( wi5 wn wo wa df-i5 ax-r5 ax-a3 lea lel2or df-le2 lor ax-a2 ax-r2 ) ABCZAD
|
|
ZEABFZQBFZEQBDZFZEZQEZQREZPUBQABGHUCRSUAEZEZQEZUDUBUFQRSUAIHUGRUEQEZEZUDRUE
|
|
QIUIRQEUDUHQRUEQSQUAQBJQTJKLMRQNOOOO $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemob $p |- ( ( a ->1 b ) v b ) = ( a ' v b ) $=
|
|
( wi1 wo wn wa df-i1 ax-r5 or32 ax-a2 lear leor letr df-le2 ax-r2 ) ABCZBDA
|
|
EZABFZDZBDZQBDZPSBABGHTUARDZUAQRBIUBRUADUAUARJRUARBUAABKBQLMNOOO $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemob $p |- ( ( a ->2 b ) v b ) = ( ( a ' ^ b ' ) v b ) $=
|
|
( wi2 wo wn wa df-i2 ax-r5 or32 ax-a2 oridm lor ax-r2 ) ABCZBDBAEBEFZDZBDZO
|
|
BDZNPBABGHQBBDZODZRBOBITOSDRSOJSBOBKLMMM $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemob $p |- ( ( a ->3 b ) v b ) = ( a ' v b ) $=
|
|
( wi3 wo wn wa df-i3 ax-r5 or32 lear df-le2 ax-r2 2or comor2 comor1 comcom2
|
|
ancom wt lor ax-r1 com2an com2or comcom7 fh4 or12 oridm ax-a2 lea letr oran
|
|
leo con2 df-t 2an an1 ) ABCZBDAEZBFZUQBEZFZDZAUQBDZFZDZBDZVBUPVDBABGHVEVABD
|
|
ZVCDZVBVAVCBIVGBUTDZVBAFZDZVBVFVHVCVIVFURBDZUTDVHURUTBIVKBUTURBUQBJKHLAVBQM
|
|
VJVHVBDZVHADZFZVBVBVHAVBBUTUQBNZVBUQUSUQBOZVBBVOPUAUBVBAVPUCUDVNVBRFVBVLVBV
|
|
MRVLBVBDZUTDZVBBUTVBIVRVBUTDZVBVQVBUTVQUQBBDZDVBBUQBUEVTBUQBUFSLHVSUTVBDVBV
|
|
BUTUGUTVBUTUQVBUQUSUHUQBUKUIKLLLVMBADZUTDZRBUTAIWBWAWAEZDZRUTWCWAUTUSUQFZWC
|
|
UQUSQWCWEWAWEBAUJULTLSRWDWAUMTLLUNVBUOLLLLL $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemob $p |- ( ( a ->4 b ) v b ) = ( a ' v b ) $=
|
|
( wi4 wo wa wn df-i4 ax-r5 or32 lear lel2or df-le2 comorr2 comid comcom2 wt
|
|
fh3 or12 oridm ax-r2 lor df-t ax-r1 2an an1 ) ABCZBDABEZAFZBEZDZUHBDZBFZEZD
|
|
ZBDZUKUFUNBABGHUOUJBDZUMDZUKUJUMBIUQBUMDZUKUPBUMUJBUGBUIABJUHBJKLHURBUKDZBU
|
|
LDZEZUKBUKULUHBMBBBNOQVAUKPEUKUSUKUTPUSUHBBDZDUKBUHBRVBBUHBSUATPUTBUBUCUDUK
|
|
UETTTTT $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemob $p |- ( ( a ->5 b ) v b ) = ( ( a ' ^ b ' ) v b ) $=
|
|
( wi5 wo wa wn df-i5 ax-r5 ax-a3 lear lel2or leor letr df-le2 ax-r2 ) ABCZB
|
|
DABEZAFZBEZDZRBFEZDZBDZUABDZPUBBABGHUCTUDDUDTUABITUDTBUDQBSABJRBJKBUALMNOO
|
|
$.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemonb $p |- ( ( a ->1 b ) v b ' ) = 1 $=
|
|
( wi1 wn wo wa wt df-i1 ax-r5 or32 df-a lor df-t ax-r1 ax-r2 ) ABCZBDZEADZA
|
|
BFZEZQEZGPTQABHIUARQEZSEZGRSQJUCUBUBDZEZGSUDUBABKLGUEUBMNOOO $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemonb $p |- ( ( a ->2 b ) v b ' ) = 1 $=
|
|
( wi2 wn wo wa wt df-i2 ax-r5 or32 ax-a2 df-t lor ax-r1 or1 ax-r2 ) ABCZBDZ
|
|
EBADRFZEZREZGQTRABHIUABREZSEZGBSRJUCSUBEZGUBSKUDSGEZGUEUDGUBSBLMNSOPPPP $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemonb $p |- ( ( a ->3 b ) v b ' ) = 1 $=
|
|
( wi3 wn wo wa df-i3 ax-r5 or32 ax-a3 lear df-le2 lor ax-r2 ancom 2or ax-r1
|
|
wt df-t or1 comor1 comor2 com2an comcom2 com2or comcom7 fh4 ax-a2 anor1 2an
|
|
con2 an1 ) ABCZBDZEADZBFZUOUNFZEZAUOBEZFZEZUNEZRUMVAUNABGHVBURUNEZUTEZRURUT
|
|
UNIVDUPUNEZUSAFZEZRVCVEUTVFVCUPUQUNEZEVEUPUQUNJVHUNUPUQUNUOUNKLMNAUSOPVGVEU
|
|
SEZVEAEZFZRUSVEAUSUPUNUSUOBUOBUAZUOBUBZUCUSBVMUDUEUSAVLUFUGVKRRFRVIRVJRVIUP
|
|
UNUSEZEZRUPUNUSJVOUPRERVNRUPVNUSUNEZRUNUSUHVPUOBUNEZEZRUOBUNJVRUORERVQRUORV
|
|
QBSQMUOTNNNMUPTNNVJUPUNAEZEZRUPUNAJVTUPUPDZEZRVSWAUPWAVSUPVSUPBUOFVSDUOBOBA
|
|
UINUKQMRWBUPSQNNUJRULNNNNN $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemonb $p |- ( ( a ->4 b ) v b ' ) =
|
|
( ( ( a ^ b ) v ( a ' ^ b ) ) v b ' ) $=
|
|
( wi4 wn wo wa df-i4 ax-r5 ax-a3 lear df-le2 lor ax-r2 ) ABCZBDZEABFADZBFEZ
|
|
PBEZOFZEZOEZQOEZNTOABGHUAQSOEZEUBQSOIUCOQSOROJKLMM $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemonb $p |- ( ( a ->5 b ) v b ' ) =
|
|
( ( ( a ^ b ) v ( a ' ^ b ) ) v b ' ) $=
|
|
( wi5 wn wo wa df-i5 ax-r5 ax-a3 lear df-le2 lor ax-r2 ) ABCZBDZEABFADZBFEZ
|
|
POFZEZOEZQOEZNSOABGHTQROEZEUAQROIUBOQROPOJKLMM $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemnaa $p |- ( ( a ->1 b ) ' ^ a ) = ( a ^ ( a ' v b ' ) ) $=
|
|
( wi1 wn wa wo anor2 u1lemona ax-r4 df-a lor ax-r1 ax-r2 ) ABCZDAENADZFZDZA
|
|
OBDFZEZNAGQOABEZFZDZSPUAABHISUBSORDZFZDZUBARJUBUEUAUDTUCOABJKILMLMM $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemnaa $p |- ( ( a ->2 b ) ' ^ a ) = ( a ^ b ' ) $=
|
|
( wi2 wn wa wo anor2 u2lemona ax-r4 ax-r2 anor1 ax-r1 ) ABCZDAEZADZBFZDZABD
|
|
EZNMOFZDQMAGSPABHIJRQABKLJ $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemnaa $p |- ( ( a ->3 b ) ' ^ a ) = ( a ^ b ' ) $=
|
|
( wi3 wn wa wo anor2 anor1 u3lemona ax-r4 ax-r1 ax-r2 ) ABCZDAEMADZFZDZABDE
|
|
ZMAGQPQNBFZDZPABHPSORABIJKLKL $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemnaa $p |- ( ( a ->4 b ) ' ^ a ) = ( a ^ b ' ) $=
|
|
( wi4 wn wa wo anor2 u4lemona ax-r4 anor1 ax-r1 ax-r2 ) ABCZDAEMADZFZDZABDE
|
|
ZMAGPNBFZDZQORABHIQSABJKLL $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemnaa $p |- ( ( a ->5 b ) ' ^ a ) = ( a ^ ( a ' v b ' ) ) $=
|
|
( wi5 wn wa wo anor2 u5lemona ax-r4 anor1 ax-r1 df-a con2 lan ax-r2 ) ABCZD
|
|
AEPADZFZDZAQBDFZEZPAGSQABEZFZDZUARUCABHIUDAUBDZEZUAUFUDAUBJKUETAUBTABLMNOOO
|
|
$.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemnana $p |- ( ( a ->1 b ) ' ^ a ' ) = 0 $=
|
|
( wi1 wn wa wt wf wo anor3 u1lemoa ax-r4 ax-r2 df-f ax-r1 ) ABCZDADEZFDZGPO
|
|
AHZDQOAIRFABJKLGQMNL $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemnana $p |- ( ( a ->2 b ) ' ^ a ' ) = 0 $=
|
|
( wi2 wn wa wt wf wo anor3 u2lemoa ax-r4 ax-r2 df-f ax-r1 ) ABCZDADEZFDZGPO
|
|
AHZDQOAIRFABJKLGQMNL $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemnana $p |- ( ( a ->3 b ) ' ^ a ' ) =
|
|
( a ' ^ ( ( a v b ) ^ ( a v b ' ) ) ) $=
|
|
( wi3 wn wa wo u3lemoa ax-a2 anor3 anor2 2or oran3 ax-r2 lor oran 3tr2 con1
|
|
oran1 ) ABCZDADZEZTABFZABDZFZEZEZSAFZAUEDZFZUADUFDUGATBEZTUCEZFZFUIABGULUHA
|
|
ULUKUJFZUHUJUKHUMUBDZUDDZFUHUKUNUJUOABIABJKUBUDLMMNMSAOAUERPQ $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemnana $p |- ( ( a ->4 b ) ' ^ a ' ) = 0 $=
|
|
( wi4 wn wa wt wf wo anor3 u4lemoa ax-r4 ax-r2 df-f ax-r1 ) ABCZDADEZFDZGPO
|
|
AHZDQOAIRFABJKLGQMNL $.
|
|
$( [15-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemnana $p |- ( ( a ->5 b ) ' ^ a ' ) =
|
|
( a ' ^ ( ( a v b ) ^ ( a v b ' ) ) ) $=
|
|
( wi5 wn wa wo u5lemoa ax-a2 anor3 anor2 2or oran3 ax-r2 lor oran 3tr2 con1
|
|
oran1 ) ABCZDADZEZTABFZABDZFZEZEZSAFZAUEDZFZUADUFDUGATBEZTUCEZFZFUIABGULUHA
|
|
ULUKUJFZUHUJUKHUMUBDZUDDZFUHUKUNUJUOABIABJKUBUDLMMNMSAOAUERPQ $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemnab $p |- ( ( a ->1 b ) ' ^ b ) = 0 $=
|
|
( wi1 wn wa wf wo wt u1lemonb oran1 df-f con2 ax-r1 3tr2 con1 ) ABCZDBEZFPB
|
|
DGHQDFDZABIPBJRHFHKLMNO $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemnab $p |- ( ( a ->2 b ) ' ^ b ) = 0 $=
|
|
( wi2 wn wa wf wo wt u2lemonb oran1 df-f con2 ax-r1 3tr2 con1 ) ABCZDBEZFPB
|
|
DGHQDFDZABIPBJRHFHKLMNO $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemnab $p |- ( ( a ->3 b ) ' ^ b ) = 0 $=
|
|
( wi3 wn wa wf wo wt u3lemonb oran1 df-f con2 ax-r1 3tr2 con1 ) ABCZDBEZFPB
|
|
DGHQDFDZABIPBJRHFHKLMNO $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemnab $p |- ( ( a ->4 b ) ' ^ b ) =
|
|
( ( ( a v b ' ) ^ ( a ' v b ' ) ) ^ b ) $=
|
|
( wi4 wn wa u4lemonb ax-a2 anor2 df-a 2or oran3 ax-r2 ax-r5 oran1 3tr2 con1
|
|
wo ) ABCZDBEZABDZQZADZTQZEZBEZRTQZUDDZTQZSDUEDUFABEZUBBEZQZTQUHABFUKUGTUKUJ
|
|
UIQZUGUIUJGULUADZUCDZQUGUJUMUIUNABHABIJUAUCKLLMLRBNUDBKOP $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemnab $p |- ( ( a ->5 b ) ' ^ b ) =
|
|
( ( ( a v b ' ) ^ ( a ' v b ' ) ) ^ b ) $=
|
|
( wi5 wn wa u5lemonb ax-a2 anor2 df-a 2or oran3 ax-r2 ax-r5 oran1 3tr2 con1
|
|
wo ) ABCZDBEZABDZQZADZTQZEZBEZRTQZUDDZTQZSDUEDUFABEZUBBEZQZTQUHABFUKUGTUKUJ
|
|
UIQZUGUIUJGULUADZUCDZQUGUJUMUIUNABHABIJUAUCKLLMLRBNUDBKOP $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemnanb $p |- ( ( a ->1 b ) ' ^ b ' ) = ( a ^ b ' ) $=
|
|
( wi1 wn wa wo u1lemob oran oran2 3tr2 con1 ) ABCZDBDZEZAMEZLBFADBFNDODABGL
|
|
BHABIJK $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemnanb $p |- ( ( a ->2 b ) ' ^ b ' ) = ( ( a v b ) ^ b ' ) $=
|
|
( wi2 wn wa wo u2lemob anor3 ax-r5 ax-r2 oran oran2 3tr2 con1 ) ABCZDBDZEZA
|
|
BFZPEZOBFZRDZBFZQDSDTADPEZBFUBABGUCUABABHIJOBKRBLMN $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemnanb $p |- ( ( a ->3 b ) ' ^ b ' ) = ( a ^ b ' ) $=
|
|
( wi3 wn wa wo u3lemob oran oran2 3tr2 con1 ) ABCZDBDZEZAMEZLBFADBFNDODABGL
|
|
BHABIJK $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemnanb $p |- ( ( a ->4 b ) ' ^ b ' ) = ( a ^ b ' ) $=
|
|
( wi4 wn wa wo u4lemob oran oran2 3tr2 con1 ) ABCZDBDZEZAMEZLBFADBFNDODABGL
|
|
BHABIJK $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemnanb $p |- ( ( a ->5 b ) ' ^ b ' ) = ( ( a v b ) ^ b ' ) $=
|
|
( wi5 wn wa wo u5lemob anor3 ax-r5 ax-r2 oran oran2 3tr2 con1 ) ABCZDBDZEZA
|
|
BFZPEZOBFZRDZBFZQDSDTADPEZBFUBABGUCUABABHIJOBKRBLMN $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemnoa $p |- ( ( a ->1 b ) ' v a ) = a $=
|
|
( wi1 wn wo wa anor1 ax-r1 u1lemana ax-r2 con1 ) ABCZDAEZAMDZLADZFZOPNLAGHA
|
|
BIJK $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemnoa $p |- ( ( a ->2 b ) ' v a ) = ( ( a v b ) ^ ( a v b ' ) ) $=
|
|
( wi2 wn wo wa u2lemana ax-a2 anor3 anor2 2or ax-r2 anor1 oran3 3tr2 con1 )
|
|
ABCZDAEZABEZABDZEZFZQADZFZSDZUADZEZRDUBDUDUCBFZUCTFZEZUGABGUJUIUHEUGUHUIHUI
|
|
UEUHUFABIABJKLLQAMSUANOP $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemnoa $p |- ( ( a ->3 b ) ' v a ) = ( ( a v b ) ^ ( a v b ' ) ) $=
|
|
( wi3 wn wo wa u3lemana ax-a2 anor3 anor2 2or ax-r2 anor1 oran3 3tr2 con1 )
|
|
ABCZDAEZABEZABDZEZFZQADZFZSDZUADZEZRDUBDUDUCBFZUCTFZEZUGABGUJUIUHEUGUHUIHUI
|
|
UEUHUFABIABJKLLQAMSUANOP $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemnoa $p |- ( ( a ->4 b ) ' v a ) = ( ( a v b ) ^ ( a v b ' ) ) $=
|
|
( wi4 wn wo wa u4lemana ax-a2 anor3 anor2 2or ax-r2 anor1 oran3 3tr2 con1 )
|
|
ABCZDAEZABEZABDZEZFZQADZFZSDZUADZEZRDUBDUDUCBFZUCTFZEZUGABGUJUIUHEUGUHUIHUI
|
|
UEUHUFABIABJKLLQAMSUANOP $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemnoa $p |- ( ( a ->5 b ) ' v a ) = ( ( a v b ) ^ ( a v b ' ) ) $=
|
|
( wi5 wn wo wa u5lemana ax-a2 anor3 anor2 2or ax-r2 anor1 oran3 3tr2 con1 )
|
|
ABCZDAEZABEZABDZEZFZQADZFZSDZUADZEZRDUBDUDUCBFZUCTFZEZUGABGUJUIUHEUGUHUIHUI
|
|
UEUHUFABIABJKLLQAMSUANOP $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemnona $p |- ( ( a ->1 b ) ' v a ' ) = ( a ' v b ' ) $=
|
|
( wi1 wn wo wa u1lemaa df-a 3tr2 con1 ) ABCZDADZEZLBDEZKAFABFMDNDABGKAHABHI
|
|
J $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemnona $p |- ( ( a ->2 b ) ' v a ' ) = ( a ' v b ' ) $=
|
|
( wi2 wn wo wa u2lemaa df-a 3tr2 con1 ) ABCZDADZEZLBDEZKAFABFMDNDABGKAHABHI
|
|
J $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemnona $p |- ( ( a ->3 b ) ' v a ' ) = ( a ' v ( a ^ b ' ) ) $=
|
|
( wi3 wn wo wa u3lemaa oran2 lan ax-r2 df-a anor1 3tr2 con1 ) ABCZDADZEZPAB
|
|
DFZEZOAFZARDZFZQDSDTAPBEZFUBABGUCUAAABHIJOAKARLMN $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemnona $p |- ( ( a ->4 b ) ' v a ' ) = ( a ' v b ' ) $=
|
|
( wi4 wn wo wa u4lemaa df-a 3tr2 con1 ) ABCZDADZEZLBDEZKAFABFMDNDABGKAHABHI
|
|
J $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemnona $p |- ( ( a ->5 b ) ' v a ' ) = ( a ' v b ' ) $=
|
|
( wi5 wn wo wa u5lemaa df-a 3tr2 con1 ) ABCZDADZEZLBDEZKAFABFMDNDABGKAHABHI
|
|
J $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemnob $p |- ( ( a ->1 b ) ' v b ) = ( a v b ) $=
|
|
( wi1 wn wo wa u1lemanb anor1 anor3 3tr2 con1 ) ABCZDBEZABEZLBDZFADOFMDNDAB
|
|
GLBHABIJK $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemnob $p |- ( ( a ->2 b ) ' v b ) = ( a v b ) $=
|
|
( wi2 wn wo wa u2lemanb anor1 anor3 3tr2 con1 ) ABCZDBEZABEZLBDZFADOFMDNDAB
|
|
GLBHABIJK $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemnob $p |- ( ( a ->3 b ) ' v b ) = ( a v b ) $=
|
|
( wi3 wn wo wa u3lemanb anor1 anor3 3tr2 con1 ) ABCZDBEZABEZLBDZFADOFMDNDAB
|
|
GLBHABIJK $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemnob $p |- ( ( a ->4 b ) ' v b ) = ( ( a ^ b ' ) v b ) $=
|
|
( wi4 wn wo wa u4lemanb oran2 ran ax-r2 anor1 anor3 3tr2 con1 ) ABCZDBEZABD
|
|
ZFZBEZOQFZRDZQFZPDSDTADBEZQFUBABGUCUAQABHIJOBKRBLMN $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemnob $p |- ( ( a ->5 b ) ' v b ) = ( a v b ) $=
|
|
( wi5 wn wo wa u5lemanb anor1 anor3 3tr2 con1 ) ABCZDBEZABEZLBDZFADOFMDNDAB
|
|
GLBHABIJK $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemnonb $p |- ( ( a ->1 b ) ' v b ' ) =
|
|
( ( a v b ' ) ^ ( a ' v b ' ) ) $=
|
|
( wi1 wn wo wa u1lemab ax-a2 anor2 df-a 2or ax-r2 oran3 3tr2 con1 ) ABCZDBD
|
|
ZEZAQEZADZQEZFZPBFZSDZUADZEZRDUBDUCABFZTBFZEZUFABGUIUHUGEUFUGUHHUHUDUGUEABI
|
|
ABJKLLPBJSUAMNO $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemnonb $p |- ( ( a ->2 b ) ' v b ' ) = b ' $=
|
|
( wi2 wn wo wa df-a ax-r1 u2lemab ax-r2 con3 ) ABCZDBDEZBMDZLBFZBONLBGHABIJ
|
|
K $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemnonb $p |- ( ( a ->3 b ) ' v b ' ) =
|
|
( ( a v b ' ) ^ ( a ' v b ' ) ) $=
|
|
( wi3 wn wo wa u3lemab ax-a2 anor2 df-a 2or ax-r2 oran3 3tr2 con1 ) ABCZDBD
|
|
ZEZAQEZADZQEZFZPBFZSDZUADZEZRDUBDUCABFZTBFZEZUFABGUIUHUGEUFUGUHHUHUDUGUEABI
|
|
ABJKLLPBJSUAMNO $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemnonb $p |- ( ( a ->4 b ) ' v b ' ) =
|
|
( ( a v b ' ) ^ ( a ' v b ' ) ) $=
|
|
( wi4 wn wo wa u4lemab ax-a2 anor2 df-a 2or ax-r2 oran3 3tr2 con1 ) ABCZDBD
|
|
ZEZAQEZADZQEZFZPBFZSDZUADZEZRDUBDUCABFZTBFZEZUFABGUIUHUGEUFUGUHHUHUDUGUEABI
|
|
ABJKLLPBJSUAMNO $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemnonb $p |- ( ( a ->5 b ) ' v b ' ) =
|
|
( ( a v b ' ) ^ ( a ' v b ' ) ) $=
|
|
( wi5 wn wo wa u5lemab ax-a2 anor2 df-a 2or ax-r2 oran3 3tr2 con1 ) ABCZDBD
|
|
ZEZAQEZADZQEZFZPBFZSDZUADZEZRDUBDUCABFZTBFZEZUFABGUIUHUGEUFUGUHHUHUDUGUEABI
|
|
ABJKLLPBJSUAMNO $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Commutation theorem for Sasaki implication. $)
|
|
u1lemc1 $p |- a C ( a ->1 b ) $=
|
|
( wn wa wo wi1 comid comcom2 comanr1 com2or df-i1 ax-r1 cbtr ) AACZABDZEZAB
|
|
FZANOAAAGHABIJQPABKLM $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for Dishkant implication. $)
|
|
u2lemc1 $p |- b C ( a ->2 b ) $=
|
|
( wn wa wo wi2 comid comanr2 comcom6 com2or df-i2 ax-r1 cbtr ) BBACZBCZDZEZ
|
|
ABFZBBPBGBPNOHIJRQABKLM $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for Kalmbach implication. $)
|
|
u3lemc1 $p |- a C ( a ->3 b ) $=
|
|
( comi31 ) ABC $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for non-tollens implication. $)
|
|
u4lemc1 $p |- b C ( a ->4 b ) $=
|
|
( wa wn wo wi4 comanr2 com2or comorr2 comid comcom2 com2an df-i4 ax-r1 cbtr
|
|
) BABCZADZBCZEZQBEZBDZCZEZABFZBSUBBPRABGQBGHBTUAQBIBBBJKLHUDUCABMNO $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for relevance implication. $)
|
|
u5lemc1 $p |- a C ( a ->5 b ) $=
|
|
( wa wn wo wi5 comanr1 comcom6 com2or df-i5 ax-r1 cbtr ) AABCZADZBCZEZNBDZC
|
|
ZEZABFZAPRAMOABGAONBGHIARNQGHITSABJKL $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for relevance implication. $)
|
|
u5lemc1b $p |- b C ( a ->5 b ) $=
|
|
( wa wn wo wi5 comanr2 com2or comcom6 df-i5 ax-r1 cbtr ) BABCZADZBCZEZNBDZC
|
|
ZEZABFZBPRBMOABGNBGHBRNQGIHTSABJKL $.
|
|
$( [14-Dec-97] $)
|
|
|
|
${
|
|
ulemc2.1 $e |- a C b $.
|
|
ulemc2.2 $e |- a C c $.
|
|
$( Commutation theorem for Sasaki implication. $)
|
|
u1lemc2 $p |- a C ( b ->1 c ) $=
|
|
( wn wa wo wi1 comcom2 com2an com2or df-i1 ax-r1 cbtr ) ABFZBCGZHZBCIZAPQ
|
|
ABDJABCDEKLSRBCMNO $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for Dishkant implication. $)
|
|
u2lemc2 $p |- a C ( b ->2 c ) $=
|
|
( wn wa wo wi2 comcom2 com2an com2or df-i2 ax-r1 cbtr ) ACBFZCFZGZHZBCIZA
|
|
CREAPQABDJACEJKLTSBCMNO $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for Kalmbach implication. $)
|
|
u3lemc2 $p |- a C ( b ->3 c ) $=
|
|
( com2i3 ) ABCDEF $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for non-tollens implication. $)
|
|
u4lemc2 $p |- a C ( b ->4 c ) $=
|
|
( wa wn wo wi4 com2an comcom2 com2or df-i4 ax-r1 cbtr ) ABCFZBGZCFZHZQCHZ
|
|
CGZFZHZBCIZASUBAPRABCDEJAQCABDKZEJLATUAAQCUEELACEKJLUDUCBCMNO $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for relevance implication. $)
|
|
u5lemc2 $p |- a C ( b ->5 c ) $=
|
|
( wa wn wo wi5 com2an comcom2 com2or df-i5 ax-r1 cbtr ) ABCFZBGZCFZHZQCGZ
|
|
FZHZBCIZASUAAPRABCDEJAQCABDKZEJLAQTUDACEKJLUCUBBCMNO $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$}
|
|
|
|
${
|
|
ulemc3.1 $e |- a C b $.
|
|
$( Commutation theorem for Sasaki implication. $)
|
|
u1lemc3 $p |- a C ( b ->1 a ) $=
|
|
( comid u1lemc2 ) ABACADE $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for Dishkant implication. $)
|
|
u2lemc3 $p |- a C ( b ->2 a ) $=
|
|
( u2lemc1 ) BAD $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for Kalmbach implication. $)
|
|
u3lemc3 $p |- a C ( b ->3 a ) $=
|
|
( comi32 ) ABCD $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for non-tollens implication. $)
|
|
u4lemc3 $p |- a C ( b ->4 a ) $=
|
|
( u4lemc1 ) BAD $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for relevance implication. $)
|
|
u5lemc3 $p |- a C ( b ->5 a ) $=
|
|
( u5lemc1b ) BAD $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Commutation theorem for Sasaki implication. $)
|
|
u1lemc5 $p |- a C ( a ->1 b ) $=
|
|
( u1lemc1 ) ABD $.
|
|
$( [11-Jan-98] $)
|
|
|
|
$( Commutation theorem for Dishkant implication. $)
|
|
u2lemc5 $p |- a C ( a ->2 b ) $=
|
|
( comid u2lemc2 ) AABADCE $.
|
|
$( [11-Jan-98] $)
|
|
|
|
$( Commutation theorem for Kalmbach implication. $)
|
|
u3lemc5 $p |- a C ( a ->3 b ) $=
|
|
( comi31 ) ABD $.
|
|
$( [11-Jan-98] $)
|
|
|
|
$( Commutation theorem for non-tollens implication. $)
|
|
u4lemc5 $p |- a C ( a ->4 b ) $=
|
|
( comid u4lemc2 ) AABADCE $.
|
|
$( [11-Jan-98] $)
|
|
|
|
$( Commutation theorem for relevance implication. $)
|
|
u5lemc5 $p |- a C ( a ->5 b ) $=
|
|
( u5lemc1 ) ABD $.
|
|
$( [11-Jan-98] $)
|
|
|
|
$( Lemma for Sasaki implication study. $)
|
|
u1lemc4 $p |- ( a ->1 b ) = ( a ' v b ) $=
|
|
( wi1 wn wa wo df-i1 comid comcom2 fh4 ancom wt ax-a2 ax-r1 ax-r2 lan an1
|
|
df-t ) ABDAEZABFGZTBGZABHUATAGZUBFZUBATBAAAIJCKUDUBUCFZUBUCUBLUEUBMFUBUCM
|
|
UBUCATGZMTANMUFASOPQUBRPPPP $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for Dishkant implication study. $)
|
|
u2lemc4 $p |- ( a ->2 b ) = ( a ' v b ) $=
|
|
( wi2 wn wa wo df-i2 comcom3 comcom4 fh4 ax-a2 df-t ax-r1 2an an1 ax-r2
|
|
wt ) ABDBAEZBEZFGZSBGZABHUABSGZBTGZFZUBSBTABCIABCJKUEUBRFUBUCUBUDRBSLRUDB
|
|
MNOUBPQQQ $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for Kalmbach implication study. $)
|
|
u3lemc4 $p |- ( a ->3 b ) = ( a ' v b ) $=
|
|
( wi3 wn wa wo df-i3 comcom3 comcom4 fh1 ax-r1 df-t lan ax-r2 comid ax-a2
|
|
wt an1 wf comcom2 dff lor or0 2or fh4 ancom ) ABDAEZBFUHBEZFGZAUHBGZFZGZU
|
|
KABHUMUHABFZGZUKUJUHULUNUJUHBUIGZFZUHUQUJUHBUIABCIABCJKLUQUHRFUHUPRUHRUPB
|
|
MLNUHSOOULAUHFZUNGZUNAUHBAAAPUAZCKUSUNURGZUNURUNQVAUNTGUNURTUNTURAUBLUCUN
|
|
UDOOOUEUOUHAGZUKFZUKAUHBUTCUFVCUKVBFZUKVBUKUGVDUKRFUKVBRUKVBAUHGZRUHAQRVE
|
|
AMLONUKSOOOOO $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for non-tollens implication study. $)
|
|
u4lemc4 $p |- ( a ->4 b ) = ( a ' v b ) $=
|
|
( wi4 wa wn wo df-i4 comid comcom2 fh2r ax-r1 ancom wt df-t lan an1 ax-r2
|
|
comcom4 wf comcom3 dff lor or0 2or fh4 ax-a2 2an ) ABDABEAFZBEGZUIBGZBFZE
|
|
ZGZUKABHUNBUIULEZGZUKUJBUMUOUJAUIGZBEZBURUJABUICAAAIJKLURBUQEZBUQBMUSBNEB
|
|
UQNBNUQAOLPBQRRRUMUOBULEZGZUOUIULBABCSZABCUAZKVAUOTGUOUTTUOTUTBUBLUCUOUDR
|
|
RUEUPBUIGZBULGZEZUKUIBULVCVBUFVFUKNEUKVDUKVENBUIUGNVEBOLUHUKQRRRR $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for relevance implication study. $)
|
|
u5lemc4 $p |- ( a ->5 b ) = ( a ' v b ) $=
|
|
( wi5 wa wn wo df-i5 comid comcom2 fh2r ax-r1 ancom wt df-t lan an1 ax-r2
|
|
ax-r5 comcom3 comcom4 fh4 ax-a2 2an ) ABDABEAFZBEGZUEBFZEZGZUEBGZABHUIBUH
|
|
GZUJUFBUHUFAUEGZBEZBUMUFABUECAAAIJKLUMBULEZBULBMUNBNEBULNBNULAOLPBQRRRSUK
|
|
BUEGZBUGGZEZUJUEBUGABCTABCUAUBUQUJNEUJUOUJUPNBUEUCNUPBOLUDUJQRRRR $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$}
|
|
|
|
$( Commutation theorem for Sasaki implication. $)
|
|
u1lemc6 $p |- ( a ->1 b ) C ( a ' ->1 b ) $=
|
|
( wi1 wn wo wa lea ax-a1 lbtr leo letr ud1lem0c df-i1 le3tr1 lecom comcom6
|
|
) ABCZADZBCZQDZSARBDEZFZRDZRBFZEZTSUBUCUEUBAUCAUAGAHIUCUDJKABLRBMNOP $.
|
|
$( [19-Mar-99] $)
|
|
|
|
$( Commutation theorem for ` ->1 ` and ` ->2 ` . $)
|
|
comi12 $p |- ( a ->1 b ) C ( c ->2 a ) $=
|
|
( wi1 wn wa wo wi2 df-i1 lea leo letr lecom comcom anor3 cbtr comcom7 df-i2
|
|
ax-r1 bctr ) ABDAEZABFZGZCAHZABIUCACEUAFZGZUDUCUFUCUAUEEZFZUFEUHUCUHUCUHUAU
|
|
CUAUGJUAUBKLMNAUEOPQUDUFCARSPT $.
|
|
$( [5-Jul-00] $)
|
|
|
|
${
|
|
i1com.1 $e |- b =< ( a ->1 b ) $.
|
|
$( Commutation expressed with ` ->1 ` . $)
|
|
i1com $p |- a C b $=
|
|
( wi1 wa wn wo ancom df2le2 u1lemab 2or ax-r2 3tr2 df-c1 comcom ) BABABAB
|
|
DZEPBEZBBAEZBAFZEZGZBPHBPCIQABEZSBEZGUAABJUBRUCTABHSBHKLMNO $.
|
|
$( [1-Dec-99] $)
|
|
$}
|
|
|
|
${
|
|
comi1.1 $e |- a C b $.
|
|
$( Commutation expressed with ` ->1 ` . $)
|
|
comi1 $p |- b =< ( a ->1 b ) $=
|
|
( wa wn wo wi1 ancom ax-r5 ax-a2 ax-r2 lear leror bltr comcom df-c2 df-i1
|
|
le3tr1 ) BADZBAEZDZFZTABDZFZBABGUBUAUCFZUDUBUCUAFUESUCUABAHIUCUAJKUATUCBT
|
|
LMNBAABCOPABQR $.
|
|
$( [1-Dec-99] $)
|
|
$}
|
|
|
|
${
|
|
ulemle1.1 $e |- a =< b $.
|
|
$( L.e. to Sasaki implication. $)
|
|
u1lemle1 $p |- ( a ->1 b ) = 1 $=
|
|
( wi1 wn wo wt lecom u1lemc4 sklem ax-r2 ) ABDAEBFGABABCHIABCJK $.
|
|
$( [11-Jan-98] $)
|
|
|
|
$( L.e. to Dishkant implication. $)
|
|
u2lemle1 $p |- ( a ->2 b ) = 1 $=
|
|
( wi2 wn wo wt lecom u2lemc4 sklem ax-r2 ) ABDAEBFGABABCHIABCJK $.
|
|
$( [11-Jan-98] $)
|
|
|
|
$( L.e. to Kalmbach implication. $)
|
|
u3lemle1 $p |- ( a ->3 b ) = 1 $=
|
|
( wi3 wn wo wt lecom u3lemc4 sklem ax-r2 ) ABDAEBFGABABCHIABCJK $.
|
|
$( [11-Jan-98] $)
|
|
|
|
$( L.e. to non-tollens implication. $)
|
|
u4lemle1 $p |- ( a ->4 b ) = 1 $=
|
|
( wi4 wn wo wt lecom u4lemc4 sklem ax-r2 ) ABDAEBFGABABCHIABCJK $.
|
|
$( [11-Jan-98] $)
|
|
|
|
$( L.e. to relevance implication. $)
|
|
u5lemle1 $p |- ( a ->5 b ) = 1 $=
|
|
( wi5 wn wo wt lecom u5lemc4 sklem ax-r2 ) ABDAEBFGABABCHIABCJK $.
|
|
$( [11-Jan-98] $)
|
|
|
|
$}
|
|
|
|
${
|
|
u1lemle2.1 $e |- ( a ->1 b ) = 1 $.
|
|
$( Sasaki implication to l.e. $)
|
|
u1lemle2 $p |- a =< b $=
|
|
( wa wf wo wt wn anidm ran ax-r1 anass ax-r2 dff 2or ax-a2 coman1 comcom2
|
|
lan fh2 wi1 df-i1 or0 an1 3tr2 df2le1 ) ABABDZEFZAGDZUGAUHAAHZUGFZDZUIUHA
|
|
UGDZAUJDZFZULUGUMEUNUGAADZBDZUMUQUGUPABAIJKAABLMANOULUOULAUGUJFZDUOUKURAU
|
|
JUGPSUGAUJABQZUGAUSRTMKMUKGAUKABUAZGUTUKABUBKCMSMUGUCAUDUEUF $.
|
|
$( [11-Jan-98] $)
|
|
$}
|
|
|
|
${
|
|
u2lemle2.1 $e |- ( a ->2 b ) = 1 $.
|
|
$( Dishkant implication to l.e. $)
|
|
u2lemle2 $p |- a =< b $=
|
|
( wa wf wo wt ax-a2 lan coman1 comcom7 coman2 fh2 ancom anass ax-r1 ax-r2
|
|
wn dff 3tr2 an0 ax-r5 wi2 df-i2 or0 an1 df2le1 ) ABABDZEFZAGDZUHAUIABARZB
|
|
RZDZFZDZUJUOUIUOAUMBFZDZUIUNUPABUMHIUQAUMDZUHFZUIUMABUMAUKULJKUMBUKULLKMU
|
|
SEUHFUIUREUHAUKDZULDULUTDZUREUTULNAUKULOVAULEDEUTEULEUTASPIULUAQTUBEUHHQQ
|
|
QPUNGAUNABUCZGVBUNABUDPCQIQUHUEAUFTUG $.
|
|
$( [11-Jan-98] $)
|
|
$}
|
|
|
|
${
|
|
u3lemle2.1 $e |- ( a ->3 b ) = 1 $.
|
|
$( Kalmbach implication to l.e. $)
|
|
u3lemle2 $p |- a =< b $=
|
|
( i3le ) ABCD $.
|
|
$( [11-Jan-98] $)
|
|
$}
|
|
|
|
${
|
|
u4lemle2.1 $e |- ( a ->4 b ) = 1 $.
|
|
$( Non-tollens implication to l.e. $)
|
|
u4lemle2 $p |- a =< b $=
|
|
( wa wn wo wt ax-r1 ax-r2 comanr1 com2or comcom com2an comanr2 comcom3 wf
|
|
lan anass dff 3tr2 wi4 df-i4 comcom6 comor1 comcom7 fh2 fh1 anidm ran an0
|
|
comor2 ancom 2or or0 anor1 an12 3tr1 an1 df2le1 ) ABAABDZAEZBDZFZVABFZBEZ
|
|
DZFZDZAGDUTAVGGAVGABUAZGVIVGABUBHCIQVHAVCDZAVFDZFZUTVCAVFAVCAUTVBABJZAVBV
|
|
ABJUCZKLVCVDVEVDVCVDUTVBVDABVDAVABUDZUEVABUKZMVDVABVOVPMKLVEVCVEUTVBBUTAB
|
|
NOBVBVABNOKLMUFVLUTPFZUTVJUTVKPVJAUTDZAVBDZFZUTAUTVBVMVNUGVTVQUTVQVTUTVRP
|
|
VSUTAADZBDZVRWBUTWAABAUHUIHAABRIPAVADZBDZVSBPDBWCDPWDPWCBASQBUJBWCULTAVAB
|
|
RIUMHUTUNZIIVDAVEDZDVDVDEZDVKPWFWGVDABUOQAVDVEUPVDSUQUMWEIIAURTUS $.
|
|
$( [11-Jan-98] $)
|
|
$}
|
|
|
|
${
|
|
u5lemle2.1 $e |- ( a ->5 b ) = 1 $.
|
|
$( Relevance implication to l.e. $)
|
|
u5lemle2 $p |- a =< b $=
|
|
( wa wn wo wt wi5 ax-r1 ax-r2 lan comanr1 comcom6 fh1 wf anass ancom 3tr2
|
|
an0 2or df-i5 com2or anidm ran dff or0 an1 df2le1 ) ABAABDZAEZBDZFZUJBEZD
|
|
ZFZDZAGDUIAUOGAUOABHZGUQUOABUAICJKUPAULDZAUNDZFZUIAULUNAUIUKABLZAUKUJBLMZ
|
|
UBAUNUJUMLMNUTUIOFZUIURUIUSOURAUIDZAUKDZFZUIAUIUKVAVBNVFVCUIVDUIVEOVDAADZ
|
|
BDZUIVHVDAABPIVGABAUCUDJAUJDZBDBVIDZVEOVIBQAUJBPVJBODOVIOBOVIAUEZIKBSJRTU
|
|
IUFZJJVIUMDUMVIDZUSOVIUMQAUJUMPVMUMODZOVNVMOVIUMVKKIUMSJRTVLJJAUGRUH $.
|
|
$( [11-Jan-98] $)
|
|
$}
|
|
|
|
$( Sasaki implication and biconditional. $)
|
|
u1lembi $p |- ( ( a ->1 b ) ^ ( b ->1 a ) ) = ( a == b ) $=
|
|
( wn wa wo wi1 tb ax-a2 2an coman1 comcom2 coman2 fh3 ax-r1 ax-r2 df-i1 lor
|
|
ancom dfb 3tr1 ) ACZABDZEZBCZUBEZDZUBUAUDDEZABFZBAFZDABGUFUBUAEZUBUDEZDZUGU
|
|
CUJUEUKUAUBHUDUBHIUGULUBUAUDUBAABJKUBBABLKMNOUHUCUIUEABPUIUDBADZEUEBAPUMUBU
|
|
DBARQOIABST $.
|
|
$( [17-Jan-98] $)
|
|
|
|
$( Dishkant implication and biconditional. $)
|
|
u2lembi $p |- ( ( a ->2 b ) ^ ( b ->2 a ) ) = ( a == b ) $=
|
|
( wn wa wo wi2 tb ancom coman1 comcom7 coman2 ax-r1 ax-r2 df-i2 lor 2an dfb
|
|
fh3r 3tr1 ) BACZBCZDZEZAUBEZDZABDUBEZABFZBAFZDABGUEUDUCDZUFUCUDHUFUIUBABUBA
|
|
TUAIJUBBTUAKJRLMUGUCUHUDABNUHAUATDZEUDBANUJUBAUATHOMPABQS $.
|
|
$( [17-Jan-98] $)
|
|
|
|
$( Dishkant implication expressed with biconditional. $)
|
|
i2bi $p |- ( a ->2 b ) = ( b v ( a == b ) ) $=
|
|
( wi2 tb wo wn wa leor lelor df-i2 dfb lor le3tr1 leo lbtr u2lembi lea bltr
|
|
ax-r1 lel2or lebi ) ABCZBABDZEZBAFBFGZEZBABGZUEEZEUBUDUEUHBUEUGHIABJZUCUHBA
|
|
BKLMBUBUCBUFUBBUENUBUFUISOUCUBBACZGZUBUKUCABPSUBUJQRTUA $.
|
|
$( [20-Nov-98] $)
|
|
|
|
$( Kalmbach implication and biconditional. $)
|
|
u3lembi $p |- ( ( a ->3 b ) ^ ( b ->3 a ) ) = ( a == b ) $=
|
|
( i3bi ) ABC $.
|
|
$( [17-Jan-98] $)
|
|
|
|
$( Non-tollens implication and biconditional. $)
|
|
u4lembi $p |- ( ( a ->4 b ) ^ ( b ->4 a ) ) = ( a == b ) $=
|
|
( wi4 wa wn wo tb ud4lem1a dfb ax-r1 ax-r2 ) ABCBACDABDAEBEDFZABGZABHMLABIJ
|
|
K $.
|
|
$( [17-Jan-98] $)
|
|
|
|
$( Relevance implication and biconditional. $)
|
|
u5lembi $p |- ( ( a ->5 b ) ^ ( b ->5 a ) ) = ( a == b ) $=
|
|
( wi5 wa wn wo tb u5lemc1b comcom com2an comcom2 wf ancom df-i5 ax-r2 anabs
|
|
fh1 2an lan 2or u5lemc1 com2or ax-a3 u5lemanb u5lemaa an4 dff ax-r1 an0 or0
|
|
anandi ax-a2 id dfb 3tr1 ) ABCZBADZBEZADZFZURAEZDZFZDZABDZVAURDZFZUPBACZDAB
|
|
GVDUPUTDZUPVBDZFZVGUPUTVBUPUQUSUPBABUPABHIZAUPABUAIZJZUPURAUPBVLKZVMJZUBUPU
|
|
RVAVOUPAVMKJQVKVGVGVIVEVJVFVIUPUQDZUPUSDZFZVEUPUQUSVNVPQVSVELFVEVQVEVRLVQUQ
|
|
UPDZVEUPUQMVTVEVEVABDZVFFZFZDVEUQVEUPWCBAMUPVEWAFZVFFZWCABNZVEWAVFUCORVEWBP
|
|
OOVRUPURDZUPADZDZLUPURAUKWIVFVEDZLWGVFWHVEABUDABUERWJVEVFDZLVFVEMWKAVADZBUR
|
|
DZDZLABVAURUFWNWLLDLWMLWLLWMBUGUHSWLUIOOOOOTVEUJOOVJVBUPDZVFUPVBMWOVFVFWDFZ
|
|
DVFVBVFUPWPURVAMUPWEWPWFWDVFULORVFWDPOOTVGUMOOVHVCUPBANSABUNUO $.
|
|
$( [17-Jan-98] $)
|
|
|
|
$( Sasaki/Dishkant implication and biconditional. Equation 4.14 of
|
|
[MegPav2000] p. 23. The variable i in the paper is set to 1, and j is set
|
|
to 2. $)
|
|
u12lembi $p |- ( ( a ->1 b ) ^ ( b ->2 a ) ) = ( a == b ) $=
|
|
( wi1 wn wa wo wi2 tb u1lemc1 comcom lear leo df-i1 ax-r1 lbtr letr u1lemaa
|
|
lecom fh1 lan an12 u1lemana ancom 3tr 2or ax-r2 df-i2 dfb 3tr1 ) ABCZABDZAD
|
|
ZEZFZEZABEZULUKEZFZUJBAGZEABHUOUJAEZUJUMEZFURUJAUMAUJABIJUMUJUMUJUMULUJUKUL
|
|
KULULUPFZUJULUPLUJVBABMNOPRJSUTUPVAUQABQVAUKUJULEZEUMUQUJUKULUAVCULUKABUBTU
|
|
KULUCUDUEUFUSUNUJBAUGTABUHUI $.
|
|
$( [2-Mar-00] $)
|
|
|
|
$( Dishkant/Sasaki implication and biconditional. $)
|
|
u21lembi $p |- ( ( a ->2 b ) ^ ( b ->1 a ) ) = ( a == b ) $=
|
|
( wi2 wn wa wo wi1 u2lemc1 comcom3 comanr1 fh2 u2lemanb u2lemab anass ancom
|
|
tb ran 3tr2 2or ax-a2 3tr df-i1 lan dfb 3tr1 ) ABCZBDZBAEZFZEZABEZADUGEZFZU
|
|
FBAGZEABPUJUFUGEZUFUHEZFULUKFUMUGUFUHBUFABHIBUHBAJIKUOULUPUKABLUFBEZAEUHUPU
|
|
KUQBAABMQUFBANBAORSULUKTUAUNUIUFBAUBUCABUDUE $.
|
|
$( [3-Mar-00] $)
|
|
|
|
$( Commutation theorem for biimplication. $)
|
|
ublemc1 $p |- a C ( a == b ) $=
|
|
( combi ) ABC $.
|
|
$( [19-Sep-98] $)
|
|
|
|
$( Commutation theorem for biimplication. $)
|
|
ublemc2 $p |- b C ( a == b ) $=
|
|
( tb ublemc1 bicom cbtr ) BBACABCBADBAEF $.
|
|
$( [19-Sep-98] $)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Some proofs contributed by Josiah Burroughs
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( This theorem continues the line of proofs such as ~ u1lemnaa ,
|
|
~ ud1lem0b , ~ u1lemnanb , etc. (Contributed by Josiah Burroughs
|
|
26-May-04.) $)
|
|
u1lemn1b $p |- ( a ->1 b ) = ( ( a ->1 b ) ' ->1 b ) $=
|
|
( wi1 wf wo wn wa ax-a1 u1lemnab ax-r1 2or or0 df-i1 3tr1 ) ABCZDEZOFZFZQBG
|
|
ZEOQBCORDSOHSDABIJKPOOLJQBMN $.
|
|
$( [26-May-04] $)
|
|
|
|
$( A 3-variable formula. (Contributed by Josiah Burroughs 26-May-04.) $)
|
|
u1lem3var1 $p |- ( ( ( a ->1 c ) ^ ( b ->1 c ) ) ' v
|
|
( ( ( a ->1 c ) ' ->1 c ) ^ ( ( b ->1 c ) ' ->1 c ) ) ) = 1 $=
|
|
( wi1 wa wn wo wt ax-a2 u1lemn1b 2an ax-r1 lor df-t 3tr1 ) ACDZBCDZEZFZRGRS
|
|
GSPFCDZQFCDZEZGHSRIUBRSRUBPTQUAACJBCJKLMRNO $.
|
|
$( [26-May-04] $)
|
|
|
|
${
|
|
oi3oa3lem1.1 $e |- 1 = ( b == a ) $.
|
|
$( An attempt at the OA3 conjecture, which is true if ` ( a == b ) = 1 ` .
|
|
(Contributed by Josiah Burroughs 27-May-04.) $)
|
|
oi3oa3lem1 $p |- ( ( ( a ->1 c ) ^ ( b ->1 c ) ) v ( a ^ b ) ) = 1 $=
|
|
( wi1 wa wo wt r3a ud1lem0b lan 2or anidm u1lemoa 3tr ) ACEZBCEZFZABFZGPP
|
|
FZAAFZGPAGHRTSUAQPPBACBADIZJKBAAUBKLTPUAAPMAMLACNO $.
|
|
$( [27-May-04] $)
|
|
$}
|
|
|
|
${
|
|
oi3oa3.1 $e |- 1 = ( b == a ) $.
|
|
$( An attempt at the OA3 conjecture, which is true if ` ( a == b ) = 1 ` .
|
|
(Contributed by Josiah Burroughs 27-May-04.) $)
|
|
oi3oa3 $p |- ( ( ( a ->1 c ) ^ ( b ->1 c ) ) v
|
|
( ( ( ( a ->1 c ) ^
|
|
( ( ( a ->1 c ) ^ ( b ->1 c ) ) v ( a ^ b ) ) ) ->1 c ) ^
|
|
( ( ( b ->1 c ) ^
|
|
( ( ( a ->1 c ) ^ ( b ->1 c ) ) v ( a ^ b ) ) ) ->1 c )
|
|
) ) = 1 $=
|
|
( wi1 wa wo oi3oa3lem1 lan an1 ax-r2 ud1lem0b 2an lor ax-a2 r3a 1bi 3tr
|
|
wt ) ACEZBCEZFZTUBABFGZFZCEZUAUCFZCEZFZGUBTCEZUACEZFZGUKUBGSUHUKUBUEUIUGU
|
|
JUDTCUDTSFTUCSTABCDHZITJKLUFUACUFUASFUAUCSUAULIUAJKLMNUBUKOTUACUATBACBADP
|
|
LQHR $.
|
|
$( [27-May-04] $)
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
More lemmas for unified implication
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
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$(
|
|
u1lem0 $p |- ( b ' ->1 a ' ) = ( a ->2 b ) $=
|
|
?$.
|
|
|
|
u2lem0 $p |- ( b ' ->2 a ' ) = ( a ->1 b ) $=
|
|
?$.
|
|
|
|
u3lem0 $p |- ( b ' ->3 a ' ) = ( a ->4 b ) $=
|
|
?$.
|
|
|
|
u4lem0 $p |- ( b ' ->4 a ' ) = ( a ->3 b ) $=
|
|
?$.
|
|
|
|
u5lem0 $p |- ( b ' ->5 a ' ) = ( a ->5 b ) $=
|
|
?$.
|
|
$)
|
|
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u1lem1 $p |- ( ( a ->1 b ) ->1 a ) = a $=
|
|
( wi1 wn wo u1lemc1 comcom u1lemc4 u1lemnoa ax-r2 ) ABCZACKDAEAKAAKABFGHABI
|
|
J $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u2lem1 $p |- ( ( a ->2 b ) ->2 a ) = a $=
|
|
( wi2 wn wa wo df-i2 wf ud2lem0c ran an32 ax-a2 ax-r2 lan dff ax-r1 lor or0
|
|
oran ) ABCZACATDZADZEZFZATAGUDAHFAUCHAUCBDZABFZEZUBEZHUAUGUBABIJUHUEUBEZUFE
|
|
ZHUEUFUBKUJUIUIDZEZHUFUKUIUFBAFUKABLBASMNHULUIOPMMMQARMM $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem1 $p |- ( ( a ->3 b ) ->3 a ) = ( ( a v b ) ^ ( a v b ' ) ) $=
|
|
( wi3 wn wo wa comi31 comcom u3lemc4 u3lemnoa ax-r2 ) ABCZACLDAEABEABDEFLAA
|
|
LABGHIABJK $.
|
|
$( [14-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u4lem1 $p |- ( ( a ->4 b ) ->4 a ) =
|
|
( ( ( ( a ^ b ) v ( a ^ b ' ) ) v a ' ) ^
|
|
( ( a v b ) ^ ( a v b ' ) ) ) $=
|
|
( wi4 wa wn wo u4lemaa 2or comanr1 com2or comcom3 comorr com2an fh4 lea leo
|
|
df-i4 letr df-le2 ax-r2 u4lemnaa ran ancom lor comor1 comor2 comcom2 lel2or
|
|
u4lemnoa 2an lan id ) ABCZACUMADZUMEZADZFZUOAFZAEZDZFZABDZABEZDZFZUSFZABFZA
|
|
VCFZDZDZUMAQVAVEVIUSDZFZVJUQVEUTVKUNVBUPVDABGABUAHURVIUSABUIUBHVLVEUSVIDZFZ
|
|
VJVKVMVEVIUSUCUDVNVFVEVIFZDZVJUSVEVIAVEAVBVDABIAVCIJKAVIAVGVHABLAVCLMKNVPVJ
|
|
VJVOVIVFVOVEVGFZVEVHFZDVIVGVEVHVGVBVDVGABABUEZABUFZMVGAVCVSVGBVTUGZMJVGAVCV
|
|
SWAJNVQVGVRVHVEVGVEAVGVBAVDABOAVCOUHZABPRSVEVHVEAVHWBAVCPRSUJTUKVJULTTTTT
|
|
$.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u5lem1 $p |- ( ( a ->5 b ) ->5 a ) = ( ( a v b ) ^ ( a v b ' ) ) $=
|
|
( wi5 wn wo wa u5lemc1 comcom u5lemc4 u5lemnoa ax-r2 ) ABCZACLDAEABEABDEFLA
|
|
ALABGHIABJK $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u1lem1n $p |- ( ( a ->1 b ) ->1 a ) ' = a ' $=
|
|
( wi1 u1lem1 ax-r4 ) ABCACAABDE $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u2lem1n $p |- ( ( a ->2 b ) ->2 a ) ' = a ' $=
|
|
( wi2 u2lem1 ax-r4 ) ABCACAABDE $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem1n $p |- ( ( a ->3 b ) ->3 a ) ' =
|
|
( ( a ' ^ b ) v ( a ' ^ b ' ) ) $=
|
|
( wi3 wn wa wo u3lem1 ancom df-a anor2 anor3 2or ax-r4 ax-r1 ax-r2 con2 ) A
|
|
BCACZADZBEZRBDZEZFZQABFZATFZEZUBDZABGUEUDUCEZUFUCUDHUGUDDZUCDZFZDZUFUDUCIUF
|
|
UKUBUJSUHUAUIABJABKLMNOOOP $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u4lem1n $p |- ( ( a ->4 b ) ->4 a ) ' =
|
|
( ( ( ( a ' v b ) ^ ( a ' v b ' ) ) ^ a ) v
|
|
( ( a ' ^ b ) v ( a ' ^ b ' ) ) ) $=
|
|
( wa wn wo wi4 oran1 df-a anor1 2or ax-r4 ax-r1 ax-r2 ancom ran anor2 anor3
|
|
2an u4lem1 oran 3tr1 ) ABCZABDZCZEZADZEZABEZAUCEZCZCZDUFBEZUFUCEZCZACZDZUFB
|
|
CZUFUCCZEZDZCZDABFAFZDUOUSEUKVAUGUPUJUTUGUEDZACZDUPUEAGVDUOVCUNAVCUMULCZUNV
|
|
CUMDZULDZEZDZVEUEVHUBVFUDVGABHABIJKVEVIUMULHLMUMULNMOKMUJUIUHCZUTUHUINVJUID
|
|
ZUHDZEZDZUTUIUHHUTVNUSVMUQVKURVLABPABQJKLMMRKVBUKABSKUOUSTUA $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u5lem1n $p |- ( ( a ->5 b ) ->5 a ) ' =
|
|
( ( a ' ^ b ) v ( a ' ^ b ' ) ) $=
|
|
( wi5 wn wa wo u5lem1 ancom df-a anor2 anor3 2or ax-r4 ax-r1 ax-r2 con2 ) A
|
|
BCACZADZBEZRBDZEZFZQABFZATFZEZUBDZABGUEUDUCEZUFUCUDHUGUDDZUCDZFZDZUFUDUCIUF
|
|
UKUBUJSUHUAUIABJABKLMNOOOP $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u1lem2 $p |- ( ( ( a ->1 b ) ->1 a ) ->1 a ) = 1 $=
|
|
( wi1 wn wa wo wt df-i1 u1lem1n u1lem1 ran anidm ax-r2 2or ax-a2 df-t ax-r1
|
|
) ABCACZACRDZRAEZFZGRAHUAADZAFZGSUBTAABITAAEARAAABJKALMNUCAUBFZGUBAOGUDAPQM
|
|
MM $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u2lem2 $p |- ( ( ( a ->2 b ) ->2 a ) ->2 a ) = 1 $=
|
|
( wi2 wn wa wo wt df-i2 u2lem1n ran anidm ax-r2 lor df-t ax-r1 ) ABCACZACAP
|
|
DZADZEZFZGPAHTARFZGSRASRRERQRRABIJRKLMGUAANOLL $.
|
|
$( [16-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem2 $p |- ( ( ( a ->3 b ) ->3 a ) ->3 a ) =
|
|
( a v ( ( a ' ^ b ) v ( a ' ^ b ' ) ) ) $=
|
|
( wi3 wn wo comi31 comid u3lemc2 comcom u3lemc4 u3lem1n ax-r5 ax-a2 ax-r2
|
|
wa ) ABCZACZACQDZAEZAADZBOTBDOEZEZQAAQAPAABFAGHIJSUAAEUBRUAAABKLUAAMNN $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u4lem2 $p |- ( ( ( a ->4 b ) ->4 a ) ->4 a ) =
|
|
( a v ( ( a ' ^ b ) v ( a ' ^ b ' ) ) ) $=
|
|
( wn wo wa u4lemc1 comcom u4lemc4 u4lem1n ax-r5 ax-a3 lear leor letr df-le2
|
|
wi4 ax-a2 ax-r2 ) ABPZAPZAPTCZADZAACZBEUCBCZEDZDZTAATSAFGHUBUCBDUCUDDEZAEZU
|
|
EDZADZUFUAUIAABIJUJUHUEADZDZUFUHUEAKULUKUFUHUKUHAUKUGALAUEMNOUEAQRRRR $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u5lem2 $p |- ( ( ( a ->5 b ) ->5 a ) ->5 a ) =
|
|
( a v ( ( a ' ^ b ) v ( a ' ^ b ' ) ) ) $=
|
|
( wi5 wn wo wa u5lemc1b comcom u5lemc4 u5lem1n ax-r5 ax-a2 ax-r2 ) ABCZACZA
|
|
CODZAEZAADZBFRBDFEZEZOAAONAGHIQSAETPSAABJKSALMM $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u1lem3 $p |- ( a ->1 ( b ->1 a ) ) =
|
|
( a ' v ( ( a ^ b ) v ( a ^ b ' ) ) ) $=
|
|
( wi1 wn wa wo df-i1 ancom 2or u1lemab ax-r1 ax-r2 lor id ) ABACZCADZAOEZFZ
|
|
PABEZABDZEZFZFZAOGRUCUCQUBPUBQUBOAEZQUBBAEZTAEZFZUDSUEUAUFABHATHIUDUGBAJKLO
|
|
AHLKMUCNLL $.
|
|
$( [17-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u2lem3 $p |- ( a ->2 ( b ->2 a ) ) = 1 $=
|
|
( wi2 wn wa wo wt df-i2 u2lemc1 comcom3 comcom4 fh4 u2lemonb df-t ax-r1 2an
|
|
an1 ax-r2 ) ABACZCSADZSDZEFZGASHUBSTFZSUAFZEZGTSUAASBAIZJASUFKLUEGGEGUCGUDG
|
|
BAMGUDSNOPGQRRR $.
|
|
$( [17-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem3 $p |- ( a ->3 ( b ->3 a ) ) =
|
|
( a v ( ( a ' ^ b ) v ( a ' ^ b ' ) ) ) $=
|
|
( wi3 wn wa df-i3 ancom u3lemanb ax-r2 u3lemnanb 2or ax-a2 u3lemonb lan an1
|
|
wo wt ) ABACZCADZREZSRDZEZPZASRPZEZPZASBEZSBDZEZPZPZARFUFUJAPUKUCUJUEAUCUHS
|
|
EZBSEZPZUJTULUBUMTRSEULSRGBAHIUBUASEUMSUAGBAJIKUNUIUGPUJULUIUMUGUHSGBSGKUIU
|
|
GLIIUEAQEAUDQAUDRSPQSRLBAMINAOIKUJALII $.
|
|
$( [17-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u4lem3 $p |- ( a ->4 ( b ->4 a ) ) =
|
|
( a ' v ( ( a ^ b ) v ( a ^ b ' ) ) ) $=
|
|
( wi4 wn wo wa u4lemc1 u4lemc4 ax-a2 u4lemonb ancom 2or ax-r5 ax-r2 ) ABACZ
|
|
CADZOEZPABFZABDZFZEZEZAOBAGHQOPEZUBPOIUCBAFZSAFZEZPEZUBBAJUGUAPEUBUFUAPUDRU
|
|
ETBAKSAKLMUAPINNNN $.
|
|
$( [17-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u5lem3 $p |- ( a ->5 ( b ->5 a ) ) =
|
|
( a ' v ( ( a ^ b ) v ( a ^ b ' ) ) ) $=
|
|
( wi5 wn wo wa u5lemc1b u5lemc4 ax-a2 u5lemonb ancom 2or ax-r5 ax-r2 ) ABAC
|
|
ZCADZOEZPABFZABDZFZEZEZAOBAGHQOPEZUBPOIUCBAFZSAFZEZPEZUBBAJUGUAPEUBUFUAPUDR
|
|
UETBAKSAKLMUAPINNNN $.
|
|
$( [17-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem3n $p |- ( a ->3 ( b ->3 a ) ) ' =
|
|
( a ' ^ ( ( a v b ) ^ ( a v b ' ) ) ) $=
|
|
( wi3 wn wo wa u3lem3 ax-a2 anor3 anor2 2or oran3 ax-r2 lor oran1 con2 ) AB
|
|
ACCZADZABEZABDZEZFZFZQARBFZRTFZEZEZUCDZABGUGAUBDZEUHUFUIAUFUEUDEZUIUDUEHUJS
|
|
DZUADZEUIUEUKUDULABIABJKSUALMMNAUBOMMP $.
|
|
$( [17-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u4lem3n $p |- ( a ->4 ( b ->4 a ) ) ' =
|
|
( a ^ ( ( a ' v b ) ^ ( a ' v b ' ) ) ) $=
|
|
( wi4 wn wo wa u4lem3 ax-a2 anor1 df-a 2or oran3 ax-r2 lor con2 ) ABACCZAAD
|
|
ZBEZQBDZEZFZFZPQABFZASFZEZEZUBDZABGUFQUADZEUGUEUHQUEUDUCEZUHUCUDHUIRDZTDZEU
|
|
HUDUJUCUKABIABJKRTLMMNAUALMMO $.
|
|
$( [17-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u5lem3n $p |- ( a ->5 ( b ->5 a ) ) ' =
|
|
( a ^ ( ( a ' v b ) ^ ( a ' v b ' ) ) ) $=
|
|
( wi5 wn wo wa u5lem3 ax-a2 anor1 df-a 2or oran3 ax-r2 lor con2 ) ABACCZAAD
|
|
ZBEZQBDZEZFZFZPQABFZASFZEZEZUBDZABGUFQUADZEUGUEUHQUEUDUCEZUHUCUDHUIRDZTDZEU
|
|
HUDUJUCUKABIABJKRTLMMNAUALMMO $.
|
|
$( [17-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u1lem4 $p |- ( a ->1 ( a ->1 ( b ->1 a ) ) ) = ( a ->1 ( b ->1 a ) ) $=
|
|
( wi1 wn wa wo df-i1 comid comcom2 u1lemc1 fh4 wt ax-a2 df-t ax-r1 u1lemona
|
|
ax-r2 ancom lor lan u1lem3 coman1 coman2 fh2 anass anidm ran ax-r5 2an an1
|
|
) AABACZCZCADZAULEFZULAULGUNUMAFZUMULFZEZULAUMULAAAHIAUKJKUQLULEZULUOLUPULU
|
|
OAUMFZLUMAMLUSANOQUPULUMFZULUMULMUTUMAUKEZFZULAUKPVBUMABDZABEZFZEZFZULVAVFU
|
|
MUKVEAUKVCBAEZFVEBAGVHVDVCBARSQTSULVGULUMVDAVCEZFZFVGABUAVJVFUMVFVJVFAVDVCF
|
|
ZEZVJVEVKAVCVDMTVLAVDEZVIFVJVDAVCABUBVDBABUCIUDVMVDVIVMAAEZBEZVDVOVMAABUEOV
|
|
NABAUFUGQUHQQOSQOQQQUIURULLEULLULRULUJQQQQ $.
|
|
$( [11-Jan-98] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem4 $p |- ( a ->3 ( a ->3 ( b ->3 a ) ) ) = 1 $=
|
|
( wi3 wn wo wt lem4 ax-a2 u3lemonb ax-r2 ) AABACZCCADZKEZFAKGMKLEFLKHBAIJJ
|
|
$.
|
|
$( [21-Jan-98] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u4lem4 $p |- ( a ->4 ( a ->4 ( b ->4 a ) ) ) = ( a ->4 ( b ->4 a ) ) $=
|
|
( wi4 wa wn wo df-i4 comid comcom2 comanr1 com2or comcom ax-r1 df-t lan an1
|
|
wt ax-r2 wf ax-r5 u4lem3 bctr fh2r comcom4 comcom3 fh1r dff lor or0 2or fh3
|
|
ancom or32 oridm ) AABACCZCAUODAEZUODFZUPUOFUOEZDZFZUOAUOGUTUOUPURDZFZUOUQU
|
|
OUSVAUQAUPFZUODZUOVDUQAUOUPUOAUOUPABDZABEZDZFZFZAABUAZAVIAUPVHAAAHIZAVEVGAB
|
|
JAVFJKKLUBZLVKUCMVDUOVCDZUOVCUOULVMUOQDUOVCQUOQVCANMOUOPRRRUSVAUOURDZFZVAUR
|
|
UPUOUOAVLUDUOUOUOHZUEUFVOVASFVAVNSVASVNUOUGMUHVAUIRRUJVBUOUPFZUOVBVQUOURFZD
|
|
ZVQUOUPURUOAVLIUOUOVPIUKVSVQQDVQVRQVQQVRUONMOVQPRRVQVIUPFZUOUOVIUPVJTVTUPUP
|
|
FZVHFZUOUPVHUPUMWBVIUOWAUPVHUPUNTUOVIVJMRRRRRR $.
|
|
$( [18-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u5lem4 $p |- ( a ->5 ( a ->5 ( b ->5 a ) ) ) = ( a ->5 ( b ->5 a ) ) $=
|
|
( wi5 wn wo u5lemc1 u5lemc4 wa u5lem3 lor ax-a3 ax-r1 oridm ax-r5 ax-r2 ) A
|
|
ABACZCZCADZQEZQAQAPFGSRRABHABDHEZEZEZQQUARABIZJUBRREZTEZQUEUBRRTKLUEUAQUDRT
|
|
RMNQUAUCLOOOO $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u1lem5 $p |- ( a ->1 ( a ->1 b ) ) = ( a ->1 b ) $=
|
|
( wi1 wn wa wo df-i1 ancom u1lemaa ax-r2 lor ax-r1 ) AABCZCADZAMEZFZMAMGPNA
|
|
BEZFZMOQNOMAEQAMHABIJKMRABGLJJ $.
|
|
$( [20-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u2lem5 $p |- ( a ->2 ( a ->2 b ) ) = ( a ->2 b ) $=
|
|
( wi2 wn wa wo df-i2 wf ancom u2lemnana ax-r2 lor or0 ) AABCZCNADZNDZEZFZNA
|
|
NGRNHFNQHNQPOEHOPIABJKLNMKK $.
|
|
$( [20-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem5 $p |- ( a ->3 ( a ->3 b ) ) = ( a ' v b ) $=
|
|
( wi3 wn wo comi31 u3lemc4 ax-a2 u3lemona ax-r2 ) AABCZCADZKEZLBEZAKABFGMKL
|
|
ENLKHABIJJ $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u4lem5 $p |- ( a ->4 ( a ->4 b ) ) = ( ( a ' ^ b ' ) v b ) $=
|
|
( wi4 wa wn wo ancom ax-r2 2or ax-a3 ax-r1 ax-a2 2an comcom7 com2an comanr2
|
|
wf com2or wt lor df-i4 u4lemaa u4lemana u4lemona ud4lem0c anass comor1 fh1r
|
|
comor2 comcom2 leor df2le2 lan dff or0 comcom6 comorr2 fh4 or32 lear lel2or
|
|
oran2 df-le2 ax-r5 or4 oran3 df-t or1 oran1 an1 ) AABCZCAVKDZAEZVKDZFZVMVKF
|
|
ZVKEZDZFZVMBEZDZBFZAVKUAVSABDZVMBDZFZWAFZBVMVTFZAVTFZDZDZFZWBVOWFVRWJVOWCWD
|
|
WAFZFZWFVLWCVNWLVLVKADWCAVKGABUBHVNVKVMDWLVMVKGABUCHIWFWMWCWDWAJKHVRVMBFZWI
|
|
AVTDZBFZDZDZWJVPWNVQWQVPVKVMFWNVMVKLABUDHABUEMWRWQWNDZWJWNWQGWSWIWPWNDZDZWJ
|
|
WIWPWNUFXAWIBDWJWTBWIWTWOWNDZBWNDZFZBWNWOBWNAVTWNAVMBUGNWNBVMBUIZUJOXEUHXDX
|
|
CXBFZBXBXCLXFBQFBXCBXBQBWNBVMUKULXBWOWOEZDZQWNXGWOABVBUMQXHWOUNKHIBUOHHHUMW
|
|
IBGHHHHIWKWFBFZWFWIFZDZWBBWFWIBWEWABWCWDABPVMBPRBWAVMVTPUPRBWIVTWGWHVMVTUQA
|
|
VTUQOUPURXKBWAFZSDZWBXIXLXJSXIWEBFZWAFXLWEWABUSXNBWAWEBWCBWDABUTVMBUTVAVCVD
|
|
HXJWFWGFZWFWHFZDZSWGWFWHWGWEWAWGWCWDWGABWGAVMVTUGZNZWGBVMVTUIZNZOWGVMBXRYAO
|
|
RWGVMVTXRXTORWGAVTXSXTRURXQSSDSXOSXPSXOWEWGFWAFZSWEWAWGUSYBWEWGWAFFZSWEWGWA
|
|
JYCWCWGFZWLFZSWCWDWGWAVEYEWLYDFZSYDWLLYFWLSFSYDSWLYDWCWCEZFZSWGYGWCABVFTSYH
|
|
WCVGKHTWLVHHHHHHXPWEWAWHFFZSWEWAWHJYIWCWAFZWDWHFZFZSWCWDWAWHVEYLYJSFSYKSYJY
|
|
KWDWDEZFZSWHYMWDABVITSYNWDVGKHTYJVHHHHMSVJHHMXMXLWBXLVJBWALHHHHH $.
|
|
$( [26-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u5lem5 $p |- ( a ->5 ( a ->5 b ) ) = ( a ' v ( a ^ b ) ) $=
|
|
( wi5 wa wn wo df-i5 u5lemc1 comcom comcom2 fh1r ax-r1 ancom df-t lan ax-r2
|
|
wt an1 ax-r5 comcom3 comcom4 fh4 u5lemona ) AABCZCAUDDAEZUDDFZUEUDEZDZFZUEA
|
|
BDFZAUDGUIUDUHFZUJUFUDUHUFAUEFZUDDZUDUMUFUDAUEAUDABHZIZUDAUOJKLUMUDULDZUDUL
|
|
UDMUPUDQDUDULQUDQULANLOUDRPPPSUKUDUEFZUDUGFZDZUJUEUDUGAUDUNTAUDUNUAUBUSUQQD
|
|
ZUJURQUQQURUDNLOUTUQUJUQRABUCPPPPP $.
|
|
$( [20-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u4lem5n $p |- ( a ->4 ( a ->4 b ) ) ' = ( ( a v b ) ^ b ' ) $=
|
|
( wi4 wo wn wa u4lem5 anor3 ax-r5 ax-r2 oran2 con2 ) AABCCZABDZBEZFZMNEZBDZ
|
|
PEMAEOFZBDRABGSQBABHIJNBKJL $.
|
|
$( [20-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem6 $p |- ( a ->3 ( a ->3 ( a ->3 b ) ) ) = ( a ->3 ( a ->3 b ) ) $=
|
|
( wi3 wn wo comi31 u3lemc4 u3lem5 lor ax-a3 ax-r1 oridm ax-r5 ax-r2 ) AAABC
|
|
ZCZCADZPEZPAPAOFGRQQBEZEZPPSQABHZITQQEZBEZPUCTQQBJKUCSPUBQBQLMPSUAKNNNN $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u4lem6 $p |- ( a ->4 ( a ->4 ( a ->4 b ) ) ) = ( a ->4 b ) $=
|
|
( wi4 wa wn wo lan comcom7 fh2 ax-a2 ancom ax-r1 ax-r2 lor ax-r5 2an com2an
|
|
wf wt com2or df-i4 u4lem5 coman1 coman2 anass dff an0 3tr2 or0 anidm ran id
|
|
2or or12 comor1 comcom2 fh3r ax-a3 oridm df-t or1 an1 u4lem5n fh4 lear leor
|
|
comor2 letr lea lel2or leo df-le2 or32 anor3 comorr2 comcom3 comanr2 df2le2
|
|
3tr1 ) AAABCZCZCAWADZAEZWADZFZWCWAFZWAEZDZFZVTAWAUAWIABDZWCBEZDZWCBDZFZFZWC
|
|
BFZABFZWKDZDZFZVTWEWOWHWSWEWOWOWBWJWDWNWBAWLBFZDZWJWAXAAABUBZGXBAWLDZWJFZWJ
|
|
WLABWLAWCWKUCZHWLBWCWKUDHZIXEWJXDFZWJXDWJJXHWJRFWJXDRWJAWCDZWKDWKXIDZXDRXIW
|
|
KKAWCWKUEXJWKRDRXIRWKRXIAUFLGWKUGMUHNWJUIMMMMWDWCXADZWNWAXAWCXCGXKWCWLDZWMF
|
|
WNWLWCBXFXGIXLWLWMXLWCWCDZWKDZWLXNXLWCWCWKUELXMWCWKWCUJUKMOMMUMWOULMWHWSWSW
|
|
FWPWGWRWFWCXAFZWPWAXAWCXCNXOWLWPFZWPWCWLBUNXPWCWPFZWKWPFZDZWPWPWCWKWCBUOZWP
|
|
BWCBVGZUPZUQXSWPSDWPXQWPXRSXQWCWCFZBFZWPYDXQWCWCBURLYCWCBWCUSOMXRWCWKBFZFZS
|
|
WKWCBUNYFWCSFSYESWCYEBWKFZSWKBJSYGBUTLMNWCVAMMPWPVBMMMMABVCPWSULMUMWTWOWPFZ
|
|
WOWRFZDZVTWPWOWRWPWJWNWPABWPAXTHZYAQWPWLWMWPWCWKXTYBQWPWCBXTYAQTTWPWQWKWPAB
|
|
YKYATYBQVDYJWPWKWJWMFZFZDZVTYHWPYIYMWOWPWJWPWNWJBWPABVEZBWCVFZVHWNWCWPWLWCW
|
|
MWCWKVIWCBVIVJWCBVKVHVJVLYIWOWQFZWOWKFZDZYMWQWOWKWQWJWNWQABABUOZABVGZQWQWLW
|
|
MWQWCWKWQAYTUPZWQBUUAUPZQWQWCBUUBUUAQTTUUCVDYSSYMDZYMYQSYRYMYQWJWQFZWNFZSWJ
|
|
WNWQVMUUEWLFZWMFSWMFZUUFSUUGSWMUUGWJWQWLFZFZSWJWQWLURUUJWJSFSUUISWJUUIWQWQE
|
|
ZFZSWLUUKWQABVNNSUULWQUTLMNWJVAMMOUUEWLWMURUUHWMSFSSWMJWMVAMUHMYRWLYLFZWKFZ
|
|
YMWOUUMWKWJWLWMUNOWLWKFZYLFYMUUNYMUUOWKYLWLWKWCWKVEVLOWLYLWKVMYMULVSMPUUDYM
|
|
SDYMSYMKYMVBMMMPYNWPWKDZWPYLDZFZVTWKWPYLBWPWCBVOVPBYLBWJWMABVQWCBVQTVPIUUPY
|
|
LFYLUUPFUURVTUUPYLJUUQYLUUPUUQYLWPDYLWPYLKYLWPYLBWPWJBWMYOWCBVEVJYPVHVRMNAB
|
|
UAVSMMMMM $.
|
|
$( [26-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u5lem6 $p |- ( a ->5 ( a ->5 ( a ->5 b ) ) ) = ( a ->5 ( a ->5 b ) ) $=
|
|
( wi5 wa wn wo df-i5 ancom u5lemc1 comcom comcom2 fh1r df-t ax-r1 lan ax-r2
|
|
wt an1 3tr2 ax-r5 comcom3 comcom4 fh4 u5lem5 oridm or32 3tr1 ) AAABCZCZCAUI
|
|
DAEZUIDFZUJUIEZDZFZUIAUIGUNUIUMFZUIUKUIUMAUJFZUIDUIUPDZUKUIUPUIHUIAUJAUIAUH
|
|
IZJZUIAUSKLUQUIQDUIUPQUIQUPAMNOUIRPSTUOUIUJFZUIULFZDZUIUJUIULAUIURUAAUIURUB
|
|
UCVBUTQDZUIVAQUTQVAUIMNOVCUTUIUTRUTUJABDZFZUJFZUIUIVEUJABUDZTUJUJFZVDFVEVFU
|
|
IVHUJVDUJUETUJVDUJUFVGUGPPPPPP $.
|
|
$( [20-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u24lem $p |- ( ( a ->2 b ) ^ ( a ->4 b ) ) = ( a ->5 b ) $=
|
|
( wi2 wi4 wa wn wo wi5 df-i2 u4lemc1 comanr2 comcom6 fh2r ancom ax-r2 anass
|
|
ran ax-r1 2or id u4lemanb lan anabs comanr1 com2or fh1 u4lemab ax-r5 df2le2
|
|
fh4r leor ax-a3 lear df-le2 lor df-i5 ) ABCZABDZEBAFZBFZEZGZUREZABHZUQVBURA
|
|
BIQVCBUREZVAUREZGZVDBURVAABJZBVAUSUTKLMVGVEUTUSEZGZVDVEVEVFVIVEURBEZVEBURNZ
|
|
URBNOVFUSUTUREZEZVIUSUTURPVNUSUSBGZUTEZEZVIVMVPUSVMURUTEVPUTURNABUAOUBVQUSV
|
|
OEZUTEZVIVSVQUSVOUTPRVSVAVIVRUSUTUSBUCQUSUTNOOOOSVJBVIGURVIGZEZVDBVIURBVIUT
|
|
USUDLZVHUJWABVTEZVIVTEZGZVDBVTVIBURVIVHWBUEWBMWEABEUSBEGZBVIEZGZVIGZVDWCWHW
|
|
DVIWCVEWGGZWHBURVIVHWBUFWJWHWHVEWFWGVEVKWFVLABUGOUHWHTOOVIVTVIURUKUISWIWFWG
|
|
VIGZGZVDWFWGVIULWLWFVAGZVDWKVAWFWKVIVAWGVIBVIUMUNUTUSNOUOWMVDVDVDWMABUPRVDT
|
|
OOOOOOOOO $.
|
|
$( [20-Dec-97] $)
|
|
|
|
$( Implication lemma. $)
|
|
u12lem $p |- ( ( a ->1 b ) v ( a ->2 b ) ) = ( a ->0 b ) $=
|
|
( wi1 wn wa wo wi2 wi0 orordi u1lemob df-i1 ax-r5 or32 orabs ax-r2 2or bile
|
|
id lear lelor lel2or leo lebi df-i2 lor df-i0 3tr1 ) ABCZBADZBDZEZFZFZUIBFZ
|
|
UHABGZFABHUMUHBFZUHUKFZFZUNUHBUKIURUNUIABEZFZFZUNUPUNUQUTABJUQUTUKFZUTUHUTU
|
|
KABKLVBUIUKFZUSFUTUIUSUKMVCUIUSUIUJNLOOPVAUNUNUNUTUNUNUNRQUSBUIABSTUAUNUTUB
|
|
UCOOUOULUHABUDUEABUFUG $.
|
|
$( [17-Nov-98] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u1lem7 $p |- ( a ->1 ( a ' ->1 b ) ) = 1 $=
|
|
( wn wi1 wa wo wt df-i1 ax-a1 ran ancom u1lemana ax-r2 lor df-t ax-r1 ) AAC
|
|
ZBDZDQAREZFZGARHTQQCZFZGSUAQSUAREZUAAUARAIJUCRUAEUAUARKQBLMMNGUBQOPMM $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u2lem7 $p |- ( a ->2 ( a ' ->2 b ) ) =
|
|
( ( ( a ^ b ' ) v ( a ' ^ b ' ) ) v b ) $=
|
|
( wn wi2 wa df-i2 ax-a1 ax-r1 ran lor ax-r2 ancom u2lemnaa 2or ax-a3 ax-a2
|
|
wo ) AACZBDZDSRSCZEZQZABCZEZRUCEZQZBQZASFUBBUDQZUEQZUGSUHUAUESBRCZUCEZQUHRB
|
|
FUKUDBUJAUCAUJAGHIJKUATREUERTLRBMKNUIBUFQUGBUDUEOBUFPKKK $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem7 $p |- ( a ->3 ( a ' ->3 b ) ) =
|
|
( a ' v ( ( a ^ b ) v ( a ^ b ' ) ) ) $=
|
|
( wn wi3 wo comi31 comcom6 u3lemc4 df-i3 lor or12 ax-a1 ran 2or ax-r1 orabs
|
|
wa ax-a2 ax-r2 ) AACZBDZDTUAEZTABQZABCZQZEZEZAUAAUATBFGHUBTTCZBQZUHUDQZEZTU
|
|
HBEZQZEZEZUGUAUNTTBIJUOUKTUMEZEZUGTUKUMKUQUFTEUGUKUFUPTUFUKUCUIUEUJAUHBALZM
|
|
AUHUDURMNOTULPNUFTRSSSS $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u2lem7n $p |- ( a ->2 ( a ' ->2 b ) ) ' =
|
|
( ( ( a v b ) ^ ( a ' v b ) ) ^ b ' ) $=
|
|
( wn wi2 wo wa u2lem7 ax-a2 anor3 anor1 2or ax-r2 oran3 ax-r5 oran2 con2 )
|
|
AACZBDDZABEZQBEZFZBCZFZRAUBFZQUBFZEZBEZUCCZABGUGUACZBEUHUFUIBUFSCZTCZEZUIUF
|
|
UEUDEULUDUEHUEUJUDUKABIABJKLSTMLNUABOLLP $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma used in study of orthoarguesian law. $)
|
|
u1lem8 $p |- ( ( a ->1 b ) ^ ( a ' ->1 b ) ) =
|
|
( ( a ^ b ) v ( a ' ^ b ) ) $=
|
|
( wi1 wn wa df-i1 ax-a1 ax-r5 ax-r1 2an comor1 comcom2 coman1 coman2 com2an
|
|
wo ax-r2 com2or comcom fh1r omlan lea leo letr df2le2 2or ax-a2 3tr ) ABCZA
|
|
DZBCZEUJABEZPZAUJBEZPZEUJUOEZULUOEZPZULUNPZUIUMUKUOABFUKUJDZUNPZUOUJBFUOVAA
|
|
UTUNAGHIQJUOUJULUOAAUNKLULUOULAUNABMZULUJBULAVBLABNORSTURUNULPUSUPUNUQULABU
|
|
AULUOULAUOABUBAUNUCUDUEUFUNULUGQUH $.
|
|
$( [27-Dec-98] $)
|
|
|
|
$( Lemma used in study of orthoarguesian law. Equation 4.11 of [MegPav2000]
|
|
p. 23. This is the first part of the inequality. $)
|
|
u1lem9a $p |- ( a ' ->1 b ) ' =< a ' $=
|
|
( wn wi1 wa wo df-i1 ax-r4 anor1 ax-r1 ax-r2 lea bltr ) ACZBDZCZNNBEZCZEZNP
|
|
NCQFZCZSOTNBGHSUANQIJKNRLM $.
|
|
$( [28-Dec-98] $)
|
|
|
|
$( Lemma used in study of orthoarguesian law. Equation 4.11 of [MegPav2000]
|
|
p. 23. This is the second part of the inequality. $)
|
|
u1lem9b $p |- a ' =< ( a ->1 b ) $=
|
|
( wn wa wo wi1 leo df-i1 ax-r1 lbtr ) ACZKABDZEZABFZKLGNMABHIJ $.
|
|
$( [27-Dec-98] $)
|
|
|
|
$( Lemma used in study of orthoarguesian law. $)
|
|
u1lem9ab $p |- ( a ' ->1 b ) ' =< ( a ->1 b ) $=
|
|
( wn wi1 u1lem9a u1lem9b letr ) ACZBDCHABDABEABFG $.
|
|
$( [27-Dec-98] $)
|
|
|
|
$( Lemma used in study of orthoarguesian law. $)
|
|
u1lem11 $p |- ( ( a ' ->1 b ) ->1 b ) = ( a ->1 b ) $=
|
|
( wn wi1 wa ud1lem0c ax-a1 ax-r1 ax-r5 lan 3tr comanr1 com2or comcom com2an
|
|
wo ax-r2 wt lor df-i1 u1lemab ran 2or comcom3 comor1 comor2 comcom7 comcom2
|
|
ax-a2 fh3r or32 ax-a3 orabs 3tr2 or12 anor2 df-t or1 2an an1 3tr1 ) ACZBDZC
|
|
ZVCBEZPZVBABEZPZVCBDABDVFVBABCZPZEZVGVBBEZPZPVBVMPZVJVMPZEZVHVDVKVEVMVDVBVB
|
|
CZVIPZEVKVBBFVRVJVBVQAVIAVQAGZHIJQVEVLVQBEZPVTVLPZVMVBBUAVLVTUIVMWAVGVTVLAV
|
|
QBVSUBIHKUCVMVBVJVBVMVBVGVLAVGABLUDVBBLMNVJVMVJVGVLVJABAVIUEZVJBAVIUFUGZOVJ
|
|
VBBVJAWBUHWCOMNUJVPVHREVHVNVHVORVHVLPVBVLPZVGPVNVHVBVGVLUKVBVGVLULWDVBVGVBB
|
|
UMIUNVOVGVJVLPZPVGRPRVJVGVLUOWERVGWEVJVJCZPZRVLWFVJABUPSRWGVJUQHQSVGURKUSVH
|
|
UTQKVCBTABTVA $.
|
|
$( [28-Dec-98] $)
|
|
|
|
$( Lemma used in study of orthoarguesian law. Equation 4.12 of [MegPav2000]
|
|
p. 23. $)
|
|
u1lem12 $p |- ( ( a ->1 b ) ->1 b ) = ( a ' ->1 b ) $=
|
|
( wi1 wn ax-a1 ud1lem0b u1lem11 ax-r2 ) ABCZBCADZDZBCZBCJBCILBAKBAEFFJBGH
|
|
$.
|
|
$( [28-Dec-98] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u2lem8 $p |- ( a ' ->2 ( a ->2 ( a ' ->2 b ) ) ) =
|
|
( a ->2 ( a ' ->2 b ) ) $=
|
|
( wn wi2 wa wo df-i2 wf u2lem7 ax-a1 ax-r1 u2lem7n 2an an12 anass anor1 lan
|
|
dff ax-r2 an0 2or or0 ) ACZAUCBDDZDUDUCCZUDCZEZFZUDUCUDGUHABCZEZUCUIEFBFZHF
|
|
ZUDUDUKUGHABIZUGAABFZUCBFZEZUIEZEZHUEAUFUQAUEAJKABLMURUPUJEZHAUPUINUSUNUOUJ
|
|
EZEZHUNUOUJOVAUNHEHUTHUNUTUOUOCZEZHUJVBUOABPQHVCUORKSQUNTSSSSUAULUKUDUKUBUD
|
|
UKUMKSSS $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem8 $p |- ( a ' ->3 ( a ->3 ( a ' ->3 b ) ) ) = 1 $=
|
|
( wn wi3 wo wt comi31 comcom3 u3lemc4 wa ax-a1 ax-r1 u3lem7 2or ax-a3 ax-a2
|
|
df-t lor or1 ax-r2 ) ACZAUABDZDZDUACZUCEZFUAUCAUCAUBGHIUEAUAABJABCJEZEZEZFU
|
|
DAUCUGAUDAKLABMNUHAUAEZUFEZFUJUHAUAUFOLUJUFUIEZFUIUFPUKUFFEFUIFUFFUIAQLRUFS
|
|
TTTTT $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem9 $p |- ( a ->3 ( a ->3 ( a ' ->3 b ) ) ) =
|
|
( a ->3 ( a ' ->3 b ) ) $=
|
|
( wn wi3 wo comi31 u3lemc4 wa u3lem7 lor ax-a3 ax-r1 oridm ax-r5 ax-r2 ) AA
|
|
ACZBDZDZDPREZRARAQFGSPPABHABCHEZEZEZRRUAPABIZJUBPPEZTEZRUEUBPPTKLUEUARUDPTP
|
|
MNRUAUCLOOOO $.
|
|
$( [24-Dec-97] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem10 $p |- ( a ->3 ( a ' ^ ( a v b ) ) ) = a ' $=
|
|
( wn wo wi3 df-i3 anass ax-r1 anidm ran ax-r2 anor3 lor oran1 lan omlan 2or
|
|
wa wt orabs comanr1 comorr comcom3 fh4r df-t 2an an1 ancom ) AACZABDZRZEUIU
|
|
KRZUIUKCZRZDZAUIUKDZRZDZUIAUKFURUIUIARZDUIUOUIUQUSUOUKUIBCZRZDZUIULUKUNVAUL
|
|
UIUIRZUJRZUKVDULUIUIUJGHVCUIUJUIIJKUNUIAVADZRVAUMVEUIVEUMVEAUJCZDUMVAVFAABL
|
|
ZMAUJNKHOAUTPKQVBUIVADZUJVADZRZUIUIVAUJUIUTUAAUJABUBUCUDVJUISRUIVHUIVISUIUT
|
|
TVIUJVFDZSVAVFUJVGMSVKUJUEHKUFUIUGKKKUQAUIRUSUPUIAUIUJTOAUIUHKQUIATKK $.
|
|
$( [17-Jan-98] $)
|
|
|
|
$(
|
|
u3lem10a $p |- ( a ->3 ( ( a ->3 b ) ->3 ( b ->3 a ) ) ' ) = a ' $=
|
|
?$.
|
|
$)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem11 $p |- ( a ->3 ( b ' ^ ( a v b ) ) ) = ( a ->3 b ' ) $=
|
|
( wn wo wa wi3 df-i3 lan lor ax-r5 wf anass ax-r1 ax-a2 ax-r2 ran 2or ancom
|
|
3tr1 wt ax-a1 oran dff anor3 oran1 coman1 coman2 comcom7 fh2 anidm or0 df-t
|
|
ax-a3 or1 3tr2 an1 comor1 comcom2 comor2 fh4 id ) ABCZABDZEZFACZVDEZVEVDCZE
|
|
ZDZAVEVDDZEZDZAVBFZAVDGVEVBEZVEBEZDZAVEVBDZEZDVNVEVBCZEZDZVRDVLVMVPWAVRVOVT
|
|
VNBVSVEBUAHIJVIVPVKVRVIKVPDZVPVFKVHVPVNVCEZVNVNCZEVFKVCWDVNABUBHWCVFVEVBVCL
|
|
MVNUCSVHVEVNBDZEZVPVGWEVEWEVGWEBVCCZDZVGWEWGBDWHVNWGBABUDJWGBNOBVCUEOMHWFVE
|
|
VNEZVODVPVNVEBVEVBUFVNBVEVBUGUHUIWIVNVOWIVEVEEZVBEZVNWKWIVEVEVBLMWJVEVBVEUJ
|
|
POJOOQWBVPKDVPKVPNVPUKOOAVEVCDZVQEZEVRVKVRWMVQAWMTVQEZVQWLTVQVEADZBDTBDZWLT
|
|
WOTBWOAVEDZTVEANTWQAULMOJVEABUMWPBTDTTBNBUNOUOPWNVQTEVQTVQRVQUPOOHVJWMAVJVE
|
|
VCVBEZDWMVDWRVEVBVCRIVCVEVBVCAABUQURVCBABUSURUTOHVRVASQAVBGSO $.
|
|
$( [18-Jan-98] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem11a $p |- ( a ->3 ( ( b ->3 a ) ->3 ( a ->3 b ) ) ' ) =
|
|
( a ->3 b ' ) $=
|
|
( wi3 wn wo wa ud3lem1 ancom anor3 ax-r2 lor oran1 con2 ud3lem0a u3lem11 )
|
|
ABACABCCZDZCABDZABEZFZCARCQTAPTPBRADZFZEZTDZBAGUCBSDZEUDUBUEBUBUARFUERUAHAB
|
|
IJKBSLJJMNABOJ $.
|
|
$( [18-Jan-98] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem12 $p |- ( a ->3 ( a ->3 b ' ) ) ' = ( a ^ b ) $=
|
|
( wn wi3 wo wa lem4 ax-r4 df-a ax-r1 ax-r2 ) AABCZDDZCACLEZCZABFZMNALGHPOAB
|
|
IJK $.
|
|
$( [18-Jan-98] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem13a $p |- ( a ->3 ( a ->3 b ' ) ' ) = ( a ->1 b ) $=
|
|
( wn wi3 wa wo ancom ax-r2 ax-a1 ax-r1 lan 2or comanr1 comorr ax-a2 lea lor
|
|
wt comcom2 wf wi1 df-i3 u3lemnana u3lemana com2or com2an fh4r lel2or df-le2
|
|
comcom3 anor2 anor3 oran3 df-t 2an an1 comid comi31 fh1 dff u3lemnaa df-i1
|
|
or0 ) AABCZDZCZDACZVFEZVGVFCZEZFZAVGVFFEZFZABUAZAVFUBVMVGAVDCZEZFZVNVKVGVLV
|
|
PVKVGAVDFZAVOFZEZEZVGVDEZVGVOEZFZFZVGVHWAVJWDVHVFVGEWAVGVFGAVDUCHVJVGVEEZWD
|
|
VIVEVGVEVIVEIJKWFVEVGEWDVGVEGAVDUDHHLWEVGWDFZVTWDFZEZVGVGWDVTVGWBWCVGVDMVGV
|
|
OMUEAVTAVRVSAVDNAVONUFUJUGWIVGREVGWGVGWHRWGWDVGFVGVGWDOWDVGWBVGWCVGVDPVGVOP
|
|
UHUIHWHVTVTCZFZRWDWJVTWDVRCZVSCZFZWJWDWMWLFWNWBWMWCWLAVDUKAVDULLWMWLOHVRVSU
|
|
MHQRWKVTUNJHUOVGUPHHHVLAVGEZAVFEZFZVPAVGVFAAAUQSAVEAVDURSUSWQTVPFZVPWOTWPVP
|
|
TWOAUTJWPVFAEVPAVFGAVDVAHLWRVPTFVPTVPOVPVCHHHLVQVGABEZFZVNVPWSVGVOBABVOBIJK
|
|
QVNWTABVBJHHH $.
|
|
$( [18-Jan-98] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem13b $p |- ( ( a ->3 b ' ) ->3 a ' ) = ( a ->1 b ) $=
|
|
( wn wa wo ax-r1 lan ax-r2 2or comanr1 comcom3 com2an com2or ax-a2 lea letr
|
|
leo wf comcom wt wi3 wi1 df-i3 u3lemnana u3lemnaa comorr fh4r coman1 coman2
|
|
ax-a1 comcom7 fh3r df-le2 2an u3lemnona comi31 fh2 u3lemana anandi u3lemanb
|
|
id u3lemaa an4 ancom dff an0 or0 comanr2 comcom2 comorr2 lel2or anor3 anor2
|
|
oran3 lor df-t an1 df-i1 ud1lem0a ) ABCZUAZACZUAWACZWBDZWCWBCZDZEZWAWCWBEZD
|
|
ZEZABUBZWAWBUCWJWBAVTCZDZEZAVTEZAWLEZDZDZWBVTDZWBWLDZEZEZWKWGWRWIXAWGWBWQDZ
|
|
WMEZWRWDXCWFWMAVTUDWFWCADWMWEAWCAWEAUJFGAVTUEHIXDWNWQWMEZDZWRWBWMWQAWMAWLJK
|
|
ZAWQAWOWPAVTUFAWLUFLZKUGXFWNWOWMEZWPWMEZDZDZWRXEXKWNWMWOWPWMAVTAWLUHZWMVTAW
|
|
LUIZUKMWMAWLXMXNMULGXLWRWRXKWQWNXKWQWQXIWOXJWPXIWMWOEWOWOWMNWMWOWMAWOAWLOZA
|
|
VTQPUMHXJWMWPEWPWPWMNWMWPWMAWPXOAWLQPUMHUNWQVAHGWRVAHHHHWIWAWNDZXAWHWNWAAVT
|
|
UOGXPWAWBDZWAWMDZEZXAWBWAWMAWAAVTUPKXGUQXSXAREXAXQXAXRRAVTURXRWAADZWAWLDZDZ
|
|
RWAAWLUSYBAWBVTEZDZWTDZRXTYDYAWTAVTVBAVTUTUNYEAWBDZYCWLDZDZRAYCWBWLVCYHYGYF
|
|
DZRYFYGVDYIYGRDRYFRYGRYFAVEFGYGVFHHHHHIXAVGHHHIXBWNXAEZWQXAEZDZWKWNXAWQXAWN
|
|
XAWBWMWBXAWBWSWTWBVTJWBWLJMSZXAAWLXAAYMUKWLXAWLWSWTVTWSWBVTVHKWBWLVHMSLMSWQ
|
|
WNWQWBWMWQAAWQXHSZVIWQAWLYNWLWQWLWOWPVTWOAVTVJKAWLVJLSLMSUGYLWNTDZWKYJWNYKT
|
|
YJXAWNEZWNWNXANYPWNWNXAWNXAWBWNWSWBWTWBVTOWBWLOVKWBWMQPUMWNVAHHYKWQWQCZEZTX
|
|
AYQWQXAWTWSEZYQWSWTNYSWOCZWPCZEYQWTYTWSUUAAVTVLAVTVMIWOWPVNHHVOTYRWQVPFHUNY
|
|
OWNWKWNVQWNAWLUBZWKUUBWNAWLVRFWLBABWLBUJFVSHHHHHH $.
|
|
$( [19-Jan-98] $)
|
|
|
|
$( Lemma for unified implication study. $)
|
|
u3lem14a $p |- ( a ->3 ( ( b ->3 a ' ) ->3 b ' ) ) =
|
|
( a ->3 ( b ->3 a ) ) $=
|
|
( wn wi3 u3lem13b ud3lem0a wa wo df-i3 ancom u1lemanb ax-r2 u1lemnanb ax-a2
|
|
wi1 2or wt u1lemonb lan an1 u3lem3 ax-r1 id ) ABACZDBCZDZDABAOZDZABADDZUFUG
|
|
ABAEFUHUDUGGZUDUGCZGZHZAUDUGHZGZHZUIAUGIUPUDBGZUDUEGZHZAHZUIUMUSUOAUMUEUDGZ
|
|
BUDGZHZUSUJVAULVBUJUGUDGVAUDUGJBAKLULUKUDGVBUDUKJBAMLPVCVBVAHUSVAVBNVBUQVAU
|
|
RBUDJUEUDJPLLUOAQGAUNQAUNUGUDHQUDUGNBARLSATLPUTAUSHZUIUSANVDUIUIUIVDABUAUBU
|
|
IUCLLLLL $.
|
|
$( [19-Jan-98] $)
|
|
|
|
$( Used to prove ` ->1 ` "add antecedent" rule in ` ->3 ` system. $)
|
|
u3lem14aa $p |- ( a ->3 ( a ->3 ( ( b ->3 a ' ) ->3 b ' ) ) ) = 1 $=
|
|
( wn wi3 wt u3lem14a ud3lem0a i3th1 ax-r2 ) AABACDBCDDZDAABADDZDEJKAABFGABH
|
|
I $.
|
|
$( [19-Jan-98] $)
|
|
|
|
$( Used to prove ` ->1 ` "add antecedent" rule in ` ->3 ` system. $)
|
|
u3lem14aa2 $p |- ( a ->3 ( a ->3 ( b ->3 ( b ->3 a ' ) ' ) ) ) = 1 $=
|
|
( wn wi3 wt wi1 u3lem13a u3lem13b ax-r1 ax-r2 ud3lem0a u3lem14aa ) AABBACDZ
|
|
CDZDZDAAMBCDZDZDEOQANPANBAFZPBAGPRBAHIJKKABLJ $.
|
|
$( [19-Jan-98] $)
|
|
|
|
$( Used to prove ` ->1 ` modus ponens rule in ` ->3 ` system. $)
|
|
u3lem14mp $p |- ( ( a ->3 b ' ) ' ->3 ( a ->3 ( a ->3 b ) ) ) = 1 $=
|
|
( wn wo wa lear ax-a1 ax-r1 lbtr lelor letr ud3lem0c u3lem5 le3tr1 u3lemle1
|
|
wi3 ) ABCZPCZAABPPZAQCZDAQDEZACZATEZDZEZUBBDZRSUEUDUFUAUDFUCBUBUCTBATFBTBGH
|
|
IJKAQLABMNO $.
|
|
$( [19-Jan-98] $)
|
|
|
|
$( Lemma for Kalmbach implication. $)
|
|
u3lem15 $p |- ( ( a ->3 b ) ^ ( a v b ) )
|
|
= ( ( a ' v b ) ^ ( a v ( a ' ^ b ) ) ) $=
|
|
( wi3 wo wa wn dfi3b ran anass comor1 comcom2 comor2 com2an com2or fh1r lan
|
|
wf ax-r2 2or 3tr leao4 lecom comcom anabs oran dff ax-r1 or0 df2le2 ) ABCZA
|
|
BDZEAFZBDZAULBFZEZDZULBEZDZEZUKEUMURUKEZEUMAUQDZEUJUSUKABGHUMURUKIUTVAUMUTU
|
|
PUKEZUQUKEZDVAUKUPUQUKAUOABJZUKULUNUKAVDKUKBABLKMZNUQUKUQUKBULAUAZUBUCOVBAV
|
|
CUQVBAUKEZUOUKEZDAQDAUKAUOVDVEOVGAVHQABUDVHUOUOFZEZQUKVIUOABUEPQVJUOUFUGRSA
|
|
UHTUQUKVFUISRPT $.
|
|
$( [7-Aug-01] $)
|
|
|
|
$( Possible axiom for Kalmbach implication system. $)
|
|
u3lemax4 $p |- ( ( a ->3 b ) ->3 ( ( a ->3 b ) ->3 ( ( b ->3 a )
|
|
->3 ( ( b ->3 a )
|
|
->3 ( ( c ->3 ( c ->3 a ) ) ->3 ( c ->3 ( c ->3 b ) ) ) ) ) ) ) = 1 $=
|
|
( wi3 wn wo wt lem4 2i3 lor ax-r2 tb wa u3lembi ax-r4 ax-r1 conb ancom bltr
|
|
anor1 oran3 ax-r5 ax-a3 le1 ska4 2bi 2or lea lelor lebi 3tr2 ) ABDZULBADZUM
|
|
CCADDZCCBDDZDZDDZDDULEZUQFZGULUQHUSURUMEZCEZAFZVABFZDZFZFZGUQVEURUQUTUPFVEU
|
|
MUPHUPVDUTUNVBUOVCCAHCBHIJKJURUTFZVDFABLZEZVDFZVFGVGVIVDVGULUMMZEVIULUMUAVK
|
|
VHABNOKUBURUTVDUCVJGVJUDGVIVBVCLZFZVJGAEZBEZLZEZVNCMZVOCMZLZFZVMWAGVNVOCUEP
|
|
VMWAVIVQVLVTVHVPABQOVLVBEZVCEZLZVTVBVCQVTWDVRWBVSWCVRCVNMWBVNCRCATKVSCVOMWC
|
|
VOCRCBTKUFPKUGPKVLVDVIVLVDVCVBDZMZVDWFVLVBVCNPVDWEUHSUISUJUKKK $.
|
|
$( [21-Jan-98] $)
|
|
|
|
|
|
$( Possible axiom for Kalmbach implication system. $)
|
|
u3lemax5 $p |- ( ( a ->3 b ) ->3 ( ( a ->3 b )
|
|
->3 ( ( b ->3 a ) ->3 ( ( b ->3 a )
|
|
->3 ( ( b ->3 c ) ->3 ( ( b ->3 c )
|
|
->3 ( ( c ->3 b ) ->3 ( ( c ->3 b )
|
|
->3 ( a ->3 c ) ) ) ) ) ) ) ) ) = 1 $=
|
|
( wi3 wn wo wt lem4 tb lor ax-a3 ax-r1 oran3 u3lembi ax-r4 ax-r2 ax-r5 bltr
|
|
wa lelor le1 ska2 lea lebi ) ABDZUEBADZUFBCDZUGCBDZUHACDZDDZDDZDDZDDUEEZULF
|
|
ZGUEULHUNUMUFEZBCIZEZUIFZFZFZGULUSUMULUOUKFUSUFUKHUKURUOUKUGEZUJFZURUGUJHVB
|
|
VAUHEZUIFZFZURUJVDVAUHUIHJVEVAVCFZUIFZURVGVEVAVCUIKLVFUQUIVFUGUHSZEUQUGUHMV
|
|
HUPBCNOPQPPPJPJUTUMUOFZURFZGVJUTUMUOURKLVJABIZEZURFZGVIVLURVIUEUFSZEVLUEUFM
|
|
VNVKABNOPQVMGVMUAGVLUQACIZFZFZVMVQGABCUBLVPURVLVOUIUQVOUICADZSZUIVSVOACNLUI
|
|
VRUCRTTRUDPPPP $.
|
|
$( [23-Jan-98] $)
|
|
|
|
$( Equivalence to biconditional. $)
|
|
bi1o1a $p |- ( a == b ) =
|
|
( ( a ->1 ( a ^ b ) ) ^ ( ( a v b ) ->1 a ) ) $=
|
|
( wn wa wo tb wi1 lea leo letr ax-r1 leid ler2an lear lebi ax-r2 3tr1 df-i1
|
|
wf 2or lecom comcom comor1 comcom7 fh1 dfb ax-a2 dff ancom ax-r5 or0r comid
|
|
df2le2 comcom2 comanr1 fh1r 3tr lor anor3 2an ) ACZABDZEZVABCZDZDZVCADZEZVC
|
|
VEAEZDZABFZAVBGZABEZAGZDVJVHVCVEAVEVCVEVCVEVAVCVAVDHVAVBIJZUAUBVCAVAVBUCUDU
|
|
EKVKVBVEEVEVBEVHABUFVBVEUGVEVFVBVGVEVFVEVCVEVOVELMVCVENOSVBADZEZVAADZVPEVBV
|
|
GSVRVPSAVADVRAUHAVAUIPUJVBVPVQVPVBVBAABHZUMKVQVPVPUKKPAVAVBAAAULUNABUOUPQTU
|
|
QVLVCVNVIVLVAAVBDZEVCAVBRVTVBVAVTVBAVBNVBAVBVSVBLMOURPVNVMCZVMADZEVIVMARWAV
|
|
EWBAVEWAABUSKWBAVMANAVMAABIALMOTPUTQ $.
|
|
$( [5-Jul-00] $)
|
|
|
|
$( Equivalence to biconditional. $)
|
|
biao $p |- ( a == b ) = ( ( a ^ b ) == ( a v b ) ) $=
|
|
( wa wn wo tb leao1 df2le2 ax-r1 anor3 lecon df-le1 ler2an lear df-le2 lebi
|
|
oridm ax-r2 2or dfb 3tr1 ) ABCZADBDCZEUBABEZCZUBDZUDDZCZEABFUBUDFUBUEUCUHUE
|
|
UBUBUDABBGZHIUCUGUHABJUGUHUGUFUGUBUDUIKUGUGUGQLMUHUGUHUGUFUGNOLPRSABTUBUDTU
|
|
A $.
|
|
$( [8-Jul-00] $)
|
|
|
|
$( Equivalence to ` ->2 ` . $)
|
|
i2i1i1 $p |- ( a ->2 b ) =
|
|
( ( a ->1 ( a v b ) ) ^ ( ( a v b ) ->1 b ) ) $=
|
|
( wn wa wo wi2 wi1 an1r ax-r1 df-i2 anabs ax-a2 ax-r2 df-i1 df-t 3tr1 anor3
|
|
wt lor leor leid ler2an lear lebi 2or 3tr 2an ) BACZBCDZEZRUJDZABFAABEZGZUL
|
|
BGZDUKUJUJHIABJUMRUNUJUHAULDZEZAUHEZUMRUPUHAEUQUOAUHABKSUHALMAULNAOPUNULCZU
|
|
LBDZEZUIBEZUJULBNVAUTUIURBUSABQBUSBULBBATBUAUBULBUCUDUEIUIBLUFUGP $.
|
|
$( [5-Jul-00] $)
|
|
|
|
$( An absorption law for ` ->1 ` . $)
|
|
i1abs $p |- ( ( a ->1 b ) ' v ( a ^ b ) ) = a $=
|
|
( wi1 wn wa wo ud1lem0c ax-r5 comanr1 comorr comcom6 fh4r wt orabs df-a lor
|
|
df-t ax-r1 ax-r2 2an an1 3tr ) ABCDZABEZFAADZBDZFZEZUDFAUDFZUGUDFZEZAUCUHUD
|
|
ABGHAUDUGABIAUGUEUFJKLUKAMEAUIAUJMABNUJUGUGDZFZMUDULUGABOPMUMUGQRSTAUASUB
|
|
$.
|
|
$( [21-Feb-02] $)
|
|
|
|
$( Part of an attempt to crack a potential Kalmbach axiom. $)
|
|
test $p |- ( ( ( c v ( a ' v b ' ) ) ^ ( c ' ^ ( c v ( a ^ b ) ) ) )
|
|
v ( ( c ' ^ ( a ^ b ) ) v ( c ^ ( c ' v ( a ^ b ) ) ) ) )
|
|
= ( ( c v ( a ^ b ) ) ^ ( c ' v ( a ^
|
|
b ) ) ) $=
|
|
( wn wo wa oran3 lor ax-r5 comor1 comor2 com2an com2or wt ax-a3 ax-r1 ax-a2
|
|
comcom7 ax-r2 2an ran comcom2 fh4r anor2 df-t or1 leor df-le2 coman1 comcom
|
|
lelan fh3 oml or12 orabs ancom an1 ) CADBDEZEZCDZCABFZEZFZFZUTVAFZCUTVAEZFZ
|
|
EZECVADZEZVCFZVHEZVBVFFZVDVKVHUSVJVCURVICABGHUAIVLVJVHEZVCVHEZFZVMVJVHVCVJV
|
|
EVGVJUTVAVJCCVIJZUBZVJVACVIKRZLVJCVFVQVJUTVAVRVSMLMVJUTVBVRVJCVAVQVSMLUCVPN
|
|
VMFZVMVNNVOVMVNVJVEEZVGEZNWBVNVJVEVGOPWBVGWAEZNWAVGQWCVGNENWANVGWAVJVJDZEZN
|
|
VEWDVJCVAUDHNWEVJUEPSHVGUFSSSVOVCVEEZVGEZVMWGVOVCVEVGOPWGVCVGEZVMWFVCVGWFVE
|
|
VCEVCVCVEQVEVCVAVBUTVACUGUKUHSIWHVCCEZVCVFEZFVMVCCVFVCCUTVBUIRVFVCVFUTVBUTV
|
|
AJZVFCVAVFCWKRUTVAKMLUJULWIVBWJVFWICVCEVBVCCQCVAUMSWJUTVCVAEEZVFVCUTVAUNWLU
|
|
TVCEZVAEZVFWNWLUTVCVAOPWMUTVAUTVBUOISSTSSSTVTVMNFVMNVMUPVMUQSSSS $.
|
|
$( [29-Dec-97] $)
|
|
|
|
$( Part of an attempt to crack a potential Kalmbach axiom. $)
|
|
test2 $p |- ( a v b ) =<
|
|
( ( a == b ) ' v ( ( c v ( a ^ b ) ) ^ ( c ' v ( a ^
|
|
b ) ) ) ) $=
|
|
( wo tb wn wa dfnb anidm 2or comor1 comor2 com2an comcom2 com2or fh4r ax-r2
|
|
wt ax-r1 leor ax-a2 lea leo letr df-le2 df-a lor df-t 2an le2an lelor bltr
|
|
an1 ) ABDZABEFZABGZUPGZDZUOCUPDZCFZUPDZGZDURUNURUNAFZBFZDZGZUPDZUNUOVFUQUPA
|
|
BHUPIJVGUNUPDZVEUPDZGZUNUNUPVEUNABABKZABLZMUNVCVDUNAVKNUNBVLNOPVJUNRGUNVHUN
|
|
VIRVHUPUNDUNUNUPUAUPUNUPAUNABUBABUCUDUEQVIVEVEFZDZRUPVMVEABUFUGRVNVEUHSQUIU
|
|
NUMQQQSUQVBUOUPUSUPVAUPCTUPUTTUJUKUL $.
|
|
$( [29-Dec-97] $)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Some 3-variable theorems
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( A 3-variable theorem. Equivalent to OML. $)
|
|
3vth1 $p |- ( ( a ->2 b ) ^ ( b v c ) ' ) =< ( a ->2 c ) $=
|
|
( wn wa wo wi2 anor3 lan ax-r1 anass ax-r2 ancom omlan lear bltr leran leor
|
|
letr df-i2 lor ran le3tr1 ) BBDZADZEZFZBCFDZEZCUECDZEZFZABGZUHEACGUIUKULUIU
|
|
GUDEZUJEZUKUIUGUDUJEZEZUOUQUIUPUHUGBCHIJUOUQUGUDUJKJLUNUEUJUNUFUEUNUDUGEUFU
|
|
GUDMBUENLUDUEOPQPUKCRSUMUGUHUMBUEUDEZFUGABTURUFBUEUDMUALUBACTUC $.
|
|
$( [18-Oct-98] $)
|
|
|
|
$( A 3-variable theorem. Equivalent to OML. $)
|
|
3vth2 $p |- ( ( a ->2 b ) ^ ( b v c ) ' ) =
|
|
( ( a ->2 c ) ^ ( b v c ) ' ) $=
|
|
( wi2 wo wn wa 3vth1 lear ler2an ax-a2 ax-r4 lan bltr lebi ) ABDZBCEZFZGZAC
|
|
DZRGZSTRABCHPRIJUAPRUATCBEZFZGPRUCTQUBBCKLMACBHNTRIJO $.
|
|
$( [18-Oct-98] $)
|
|
|
|
$( A 3-variable theorem. Equivalent to OML. $)
|
|
3vth3 $p |- ( ( a ->2 c ) v ( ( a ->2 b ) ^ ( b v c ) ' ) ) =
|
|
( a ->2 c ) $=
|
|
( wi2 wo wn wa ax-a2 3vth1 df-le2 ax-r2 ) ACDZABDBCEFGZEMLELLMHMLABCIJK $.
|
|
$( [18-Oct-98] $)
|
|
|
|
$( A 3-variable theorem. $)
|
|
3vth4 $p |- ( ( a ->2 b ) ' ->2 ( b v c ) ) =
|
|
( ( a ->2 c ) ' ->2 ( b v c ) ) $=
|
|
( wo wi2 wn wa 3vth2 ax-a1 ran 3tr2 lor df-i2 3tr1 ) BCDZABEZFZFZOFZGZDOACE
|
|
ZFZFZSGZDQOEUBOETUDOPSGUASGTUDABCHPRSPIJUAUCSUAIJKLQOMUBOMN $.
|
|
$( [18-Oct-98] $)
|
|
|
|
$( A 3-variable theorem. $)
|
|
3vth5 $p |- ( ( a ->2 b ) ' ->2 ( b v c ) ) =
|
|
( c v ( ( a ->2 b ) ^ ( c ->2 b ) ) ) $=
|
|
( wo wn wi2 ax-a3 or12 comorr comcom2 fh3 ax-r1 oridm ax-r5 ax-r2 ancom lor
|
|
wa 2an df-i2 anor3 ax-a1 ran 3tr1 ) BCDZBAEBEZRZDZUEEZRZDZCUHBCEZUFRZDZRZDZ
|
|
ABFZEZUEFZCUQCBFZRZDUKBCUJDDZUPBCUJGVBCBUJDZDUPBCUJHVCUOCVCBUHDZBUIDZRUOBUH
|
|
UIBUGIBUEBCIJKVDUHVEUNVDBBDZUGDZUHVGVDBBUGGLVFBUGBMNOUIUMBUMUIUMUFULRUIULUF
|
|
PBCUAOLQSOQOOUSUEUREZUIRZDZUKURUETUKVJUJVIUEUHVHUIUHUQVHUQUHABTZLUQUBOUCQLO
|
|
VAUOCUQUHUTUNVKCBTSQUD $.
|
|
$( [18-Oct-98] $)
|
|
|
|
$( A 3-variable theorem. $)
|
|
3vth6 $p |- ( ( a ->2 b ) ' ->2 ( b v c ) ) =
|
|
( ( ( a ->2 b ) ^ ( c ->2 b ) ) v
|
|
( ( a ->2 c ) ^ ( b ->2 c ) ) ) $=
|
|
( wi2 wn wo wa oridm ax-r1 3vth4 3vth5 ax-a2 ax-r2 2or or4 leo df-i2 ler2an
|
|
lbtr df-le2 lor ud2lem0a ax-r5 ) ABDZEBCFZDZUFUFFZUDCBDZGZACDZBCDZGZFZUGUFU
|
|
FHIUGUFUJEZUEDZFZUMUFUOUFABCJUAUPCUIFZBULFZFZUMUFUQUOURABCKUOUNCBFZDURUEUTU
|
|
NBCLUBACBKMNUSUTUMFZUMCUIBULOVAUEUMFZUMUTUEUMCBLUCVBBUIFZCULFZFUMBCUIULOVCU
|
|
IVDULBUIBUDUHBBAEZBEZGZFZUDBVGPUDVHABQISBBCEZVFGZFZUHBVJPUHVKCBQISRTCULCUJU
|
|
KCCVEVIGZFZUJCVLPUJVMACQISCCVFVIGZFZUKCVNPUKVOBCQISRTNMMMMMM $.
|
|
$( [18-Oct-98] $)
|
|
|
|
|
|
$( A 3-variable theorem. $)
|
|
3vth7 $p |- ( ( a ->2 b ) ' ->2 ( b v c ) ) =
|
|
( a ->2 ( b v c ) ) $=
|
|
( wi2 wa wo wn df-i2 2an anass ax-r1 anor3 lan an32 3tr lor comanr2 comcom6
|
|
3tr2 ax-r2 anidm an4 fh3 3vth5 ax-a3 or12 3tr1 ) CABDZCBDZEZFCBAGZBCFZGZEZF
|
|
ZFZUHGULDAULDZUJUOCUJBUKBGZEZFZBCGZUREZFZEZUOUHUTUIVCABHCBHIUOVDUOBUSVBEZFV
|
|
DUNVEBUNUKVAEZUREZVFURUREZEZVEUKURVAEZEZUSVAEZUNVGVLVKUKURVAJKVJUMUKBCLMUKU
|
|
RVANSVIVGVHURVFURUAMKUKVAURURUBOPBUSVBBUSUKURQRBVBVAURQRUCTKTPABCUDUQULUNFB
|
|
CUNFFUPAULHBCUNUEBCUNUFOUG $.
|
|
$( [18-Oct-98] $)
|
|
|
|
$( A 3-variable theorem. $)
|
|
3vth8 $p |- ( a ->2 ( b v c ) ) =
|
|
( ( ( a ->2 b ) ^ ( c ->2 b ) ) v
|
|
( ( a ->2 c ) ^ ( b ->2 c ) ) ) $=
|
|
( wo wi2 wn wa 3vth7 ax-r1 3vth6 ax-r2 ) ABCDZEZABEZFLEZNCBEGACEBCEGDOMABCH
|
|
IABCJK $.
|
|
$( [18-Oct-98] $)
|
|
|
|
$( A 3-variable theorem. $)
|
|
3vth9 $p |- ( ( a v b ) ->1 ( c ->2 b ) ) =
|
|
( ( b v c ) ->2 ( a ->2 b ) ) $=
|
|
( wo wn wi2 wa wi1 anor3 ax-r1 df-i2 lan 2or df-i1 ud2lem0c 2an ax-r2 ancom
|
|
anandi lor anass or32 comanr1 comcom6 comorr2 or12 oridm ax-r5 ax-a2 3tr1
|
|
fh3 ) ABDZEZULCBFZGZDAEBEZGZULBCEZUPGZDZGZDZULUNHBCDZABFZFZUMUQUOVAUQUMABIJ
|
|
UNUTULCBKLMULUNNVEBUQDZUPURGZULGZDZVBVEVDVCEZVDEZGZDVIVCVDKVDVFVLVHABKVLVGU
|
|
PULGZGZVHVJVGVKVMVGVJBCIJABOPVNUPURULGGZVHVOVNUPURULSJVHVOUPURULUAJQQMQVIBV
|
|
HDZUQDZVBBUQVHUBVQVAUQDVBVPVAUQVPBVGDZBULDZGZVABVGULBVGUPURUCUDABUEUKVTUTUL
|
|
GVAVRUTVSULVGUSBUPURRTVSABBDZDULBABUFWABABUGTQPUTULRQQUHVAUQUIQQQUJ $.
|
|
$( [16-Nov-98] $)
|
|
|
|
$( 3-variable commutation theorem. $)
|
|
3vcom $p |- ( ( a ->1 c ) v ( b ->1 c ) ) C
|
|
( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) $=
|
|
( wn wi1 wa wo oran3 ax-r1 u1lem9ab le2or lecom bctr comcom6 comcom ) ADCEZ
|
|
BDCEZFZACEZBCEZGZRUARDZPDZQDZGZUAUEUBPQHIUEUAUCSUDTACJBCJKLMNO $.
|
|
$( [19-Mar-99] $)
|
|
|
|
${
|
|
3vded11.1 $e |- b =< ( c ->1 ( b ->1 a ) ) $.
|
|
$( A 3-variable theorem. Experiment with weak deduction theorem. $)
|
|
3vded11 $p |- c =< ( b ->1 a ) $=
|
|
( wi1 wt le1 wn wa df-t ancom anor2 ax-r2 lor ax-r1 ax-a3 3tr df-i1 lbtr
|
|
wo leo lelan lelor lel2or bltr lebi u1lemle2 ) CBAEZCUHEZFUIGFBCHZCBHZIZT
|
|
ZTZUIFBUJTZUOHZTZUOULTZUNUOJURUQULUPUOULUKCIUPCUKKBCLMNOBUJULPQBUIUMDUMUJ
|
|
CUHIZTZUIULUSUJUKUHCUKUKBAIZTZUHUKVAUAUHVBBAROSUBUCUIUTCUHROSUDUEUFUG $.
|
|
$( [25-Oct-98] $)
|
|
$}
|
|
|
|
${
|
|
3vded12.1 $e |- ( a ^ ( c ->1 a ) ) =< ( c ->1 ( b ->1 a ) ) $.
|
|
3vded12.2 $e |- c =< a $.
|
|
$( A 3-variable theorem. Experiment with weak deduction theorem. $)
|
|
3vded12 $p |- c =< ( b ->1 a ) $=
|
|
( wi1 wt le1 wn wo df-t wa an1 ax-r1 u1lemle1 lan ax-r2 bltr lecon leo
|
|
df-i1 lbtr letr lel2or lebi u1lemle2 ) CBAFZCUGFZGUHHGAAIZJUHAKAUHUIAACAF
|
|
ZLZUHAAGLZUKULAAMNUKULUJGACAEOPNQDRUICIZUHCAESUMUMCUGLZJZUHUMUNTUHUOCUGUA
|
|
NUBUCUDRUEUF $.
|
|
$( [25-Oct-98] $)
|
|
$}
|
|
|
|
${
|
|
3vded13.1 $e |- ( b ^ ( c ->1 a ) ) =< ( c ->1 ( b ->1 a ) ) $.
|
|
3vded13.2 $e |- c =< a $.
|
|
$( A 3-variable theorem. Experiment with weak deduction theorem. $)
|
|
3vded13 $p |- c =< ( b ->1 a ) $=
|
|
( wi1 wa wt an1 ax-r1 u1lemle1 lan ax-r2 bltr 3vded11 ) ABCBBCAFZGZCBAFFB
|
|
BHGZQRBBIJHPBPHCAEKJLMDNO $.
|
|
$( [25-Oct-98] $)
|
|
$}
|
|
|
|
${
|
|
3vded21.1 $e |- c =< ( ( a ->0 b ) ->0 ( c ->2 b ) ) $.
|
|
3vded21.2 $e |- c =< ( a ->0 b ) $.
|
|
$( A 3-variable theorem. Experiment with weak deduction theorem. $)
|
|
3vded21 $p |- c =< b $=
|
|
( wf wo wa wn wi0 df-i0 lbtr lor ax-r2 2or ax-a2 3tr comor2 comcom2 anabs
|
|
wi2 ax-r4 df-i2 anor3 ler2an leror ax-a3 oridm lecom comcom comid fh1 or0
|
|
com2or ax-r1 dff ran ancom ax-r5 3tr2 leran com2an fh1r an32 anass le3tr2
|
|
lan an0 ) CBFGZBCCBGZHBAIZBGZVJIZHZGZVJHZCVICVOVJCVLBVMGZVLIZGZHZVOCVLVSC
|
|
ABJZVLEABKZLZCWACBUAZJZVSDWEWAIZWDGVRVQGVSWAWDKWFVRWDVQWAVLWBUBWDBCIBIHZG
|
|
VQCBUCWGVMBCBUDMNOVRVQPQLUEVTVLVQHZVLVRHZGZVOVLVQVRVLBVMVKBRZVLVJVJVLVJVL
|
|
VJVLBGZVLCVLBWCUFWLVKBBGZGVLVKBBUGWMBVKBUHMNLUIZUJSZUNVLVLVLUKSULVLBHZVNG
|
|
ZFGWQWJVOWQUMWQWHFWIWHWQVLBVMWKWOULUOVLUPOWPBVNWPBVKGZBHBWRHBVLWRBVKBPUQW
|
|
RBURBVKTQUSUTNLVACBTVPBVJHZVNVJHZGVIVJBVNCBRVJVLVMWNVJVJVJUKSVBVCWSBWTFWS
|
|
BBCGZHBVJXABCBPVGBCTNWTVLVJHVMHVLVJVMHZHZFVLVMVJVDVLVJVMVEXCVLFHZFXDXCFXB
|
|
VLVJUPVGUOVLVHNQONVFBUML $.
|
|
$( [31-Oct-98] $)
|
|
$}
|
|
|
|
${
|
|
3vded22.1 $e |- c =< ( C ( a , b ) v C ( c , b ) ) $.
|
|
3vded22.2 $e |- c =< a $.
|
|
3vded22.3 $e |- c =< ( a ->0 b ) $.
|
|
$( A 3-variable theorem. Experiment with weak deduction theorem. $)
|
|
3vded22 $p |- c =< b $=
|
|
( wn wa wo wi0 wcmtr df-cmtr or4 ax-r2 lear lel2or leran le2or bltr df-i0
|
|
wi2 lecon lelor leror letr or12 ax-r4 anor1 ax-r1 df-i2 2or oridm 3vded21
|
|
3tr1 lbtr ) ABCCBABGZHZCGZUPHZIZIZVAIZABJZCBUAZJZCABKZCBKZIVBDVFVAVGVAVFA
|
|
BHZAGZBHZIZUQVIUPHZIZIZVAVFVHUQIVJVLIIVNABLVHUQVJVLMNVKBVMUTVHBVJABOVIBOP
|
|
VLUSUQVIURUPCAEUBQUCRSVGCBHZURBHZIZCUPHZUSIZIZVAVGVOVRIVPUSIIVTCBLVOVRVPU
|
|
SMNVQBVSUTVOBVPCBOURBOPVRUQUSCAUPEQUDRSRUEVEVBVEVCGZVDIZVBVCVDTUQBUSIZIVA
|
|
WBVBUQBUSUFWAUQVDWCWAVIBIZGZUQVCWDABTUGUQWEABUHUINCBUJUKVAULUNNUIUOFUM $.
|
|
$( [31-Oct-98] $)
|
|
$}
|
|
|
|
${
|
|
3vded3.1 $e |- ( c ->0 C ( a , c ) ) = 1 $.
|
|
3vded3.2 $e |- ( c ->0 a ) = 1 $.
|
|
3vded3.3 $e |- ( c ->0 ( a ->0 b ) ) = 1 $.
|
|
$( A 3-variable theorem. Experiment with weak deduction theorem. $)
|
|
3vded3 $p |- ( c ->0 b ) = 1 $=
|
|
( wi0 wn wo wt df-i0 wa wcmtr lor 3tr1 ax-r2 ax-r1 wf ancom 3tr2 cmtrcom
|
|
ax-a3 i0cmtrcom comcom4 comid comcom3 fh1 lan dff or0 an1 orabs ax-r5 3tr
|
|
comcom ) CBGCHZBIZCABGZGZJCBKUPAHZIZBIZUPUTBIZIZUQUSUPUTBUBVBUQVAUPBVAUPU
|
|
PUTLZIUPUTVEUPUTJLZUTUPLZUTVEUTUPAIZLVGUTALZIZVFVGUTUPAUPUTCACACCAMZGZCAC
|
|
MZGZJUPVKIUPVMIVLVNVKVMUPCAUANCVKKCVMKODPUCUDUOAAAUEUFUGVHJUTVHCAGZJVOVHC
|
|
AKQEPUHVJVGRIZVGVPVJRVIVGRAUTLVIAUIAUTSPNQVGUJPTUTUKUTUPSTNUPUTULPUMQUSUP
|
|
URIVDCURKURVCUPABKNPOFUN $.
|
|
$( [24-Jan-99] $)
|
|
$}
|
|
|
|
$( Orthoarguesian-like law with ` ->1 ` instead of ` ->0 ` but true in all
|
|
OMLs. $)
|
|
1oa $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ->1 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) )
|
|
=< ( a ->2 c ) $=
|
|
( wn wa wo wi2 lear an12 lerr lan ax-r1 coman1 bctr comcom2 df-i2 2an anass
|
|
wf ax-r2 wi1 bltr leid letr df-i1 fh2c anor3 comid comcom3 comanr2 fh1r dff
|
|
lel2or anidm 2or ax-a2 or0 3tr ran 3tr2 lea lecom fh3 coman2 oran cbtr 3tr1
|
|
com2an le3tr1 ) ADZBDZCDZEZEZBCFZFZVNBVJVKEZFZFZEZVNCVJVLEZFZFZEZWBABGZVOWE
|
|
ACGZEZUAZEZWFWDWCWBVTWCHVNWBWBVNVKWAEZWBVJVKVLIZWJWACVKWAHJZUBWBUCUMUDWIWEV
|
|
ODZVOWGEZFZEWEWMEZWEWNEZFZWDWHWOWEVOWGUEKWNWEWMWNWEVOWFEZEZWEWTWNWEVOWFILWE
|
|
WSMNWNVOVOWGMOUFWRVNVOVRWBEZEZFZWDWPVNWQXBWPVRVMEZVNWEVRWMVMABPZVMWMBCUGLQV
|
|
RVKEZVLEVQVLEZXDVNXFVQVLXFBVKEZVQVKEZFSVQFZVQVKBVQBBBUHUIVJVKUJUKXHSXIVQSXH
|
|
BULLXIVJVKVKEZEVQVJVKVKRXKVKVJVKUNKTUOXJVQSFVQSVQUPVQUQTURUSVRVKVLRVJVKVLRZ
|
|
UTTWQVOWEWGEZEXBWEVOWGIXMXAVOXMWEWEEZWFEZXAXOXMWEWEWFRLXNVRWFWBXNWEVRWEUNXE
|
|
TACPZQTKTUOVPVNXAFZEVPVSWCEZEXCWDXQXRVPVNVRWBVNXGVRXGVNXLLXGVRXGVQBVQVLVAJV
|
|
BNZVNWJWBWKWJWBWLVBNZVCKVNVOXAVNVMDZVOVNVMVJVMVDOVOYABCVELVFVNVRWBXSXTVHVCV
|
|
PVSWCRVGTURXPVI $.
|
|
$( [1-Nov-98] $)
|
|
|
|
$( Orthoarguesian-like OM law. $)
|
|
1oai1 $p |- ( ( a ->1 c ) ^
|
|
( ( a ^ b ) ' ->1 ( ( a ->1 c ) ^ ( b ->1 c ) ) ) )
|
|
=< ( b ->1 c ) $=
|
|
( wn wi2 wo wa wi1 1oa i1i2 oran3 ax-r1 2an ud1lem0ab le3tr1 ) CDZADZEZQBDZ
|
|
FZRPSEZGZHZGUAACHZABGDZUDBCHZGZHZGUFPQSIUDRUHUCACJZUETUGUBTUEABKLUDRUFUAUIB
|
|
CJZMNMUJO $.
|
|
$( [30-Dec-98] $)
|
|
|
|
$( Orthoarguesian-like OM law. $)
|
|
2oai1u $p |- ( ( a ->1 c ) ^
|
|
( ( ( a ->1 c ) ^ ( b ->1 c ) ) ' ->2 ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ) )
|
|
=< ( b ->1 c ) $=
|
|
( wn wi1 wa wi2 1oai1 u1lem11 2an ax-r1 ud1lem0a i1i2con2 ax-r2 le3tr2 ) AD
|
|
CEZCEZPBDCEZFZDZQRCEZFZEZFUAACEZUDBCEZFZDSGZFUEPRCHQUDUCUGACIZUCTUFEZUGUIUC
|
|
UFUBTUBUFQUDUAUEUHBCIZJKLKSUFMNJUJO $.
|
|
$( [28-Feb-99] $)
|
|
|
|
$( OML analog to orthoarguesian law of Godowski/Greechie, Eq. III with
|
|
` ->1 ` instead of ` ->0 ` . $)
|
|
1oaiii $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ->1 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) =
|
|
( ( a ->2 c ) ^ ( ( b v c ) ->1 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) $=
|
|
( wi2 wo wa wi1 anass anidm lan ax-r2 ax-r1 leran bltr ancom ud1lem0a ax-a2
|
|
1oa ud1lem0b ran lebi ) ABDZBCEZUBACDZFZGZFZUDUFFZUGUGUFFZUHUIUGUIUBUFUFFZF
|
|
UGUBUFUFHUJUFUBUFIZJKLUGUDUFABCRMNUHUDCBEZUDUBFZGZFZUFFZUGUPUHUPUDUNUFFZFUH
|
|
UDUNUFHUQUFUDUQUJUFUNUFUFUNULUEGUFUMUEULUDUBOPULUCUECBQSKTUKKJKLUOUBUFACBRM
|
|
NUA $.
|
|
$( [1-Nov-98] $)
|
|
|
|
$( OML analog to orthoarguesian law of Godowski/Greechie, Eq. II with
|
|
` ->1 ` instead of ` ->0 ` . $)
|
|
1oaii $p |- ( b ' ^ ( ( a ->2 b ) v ( ( a ->2 c ) ^ ( ( b v c )
|
|
->1 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) ) =< a ' $=
|
|
( wn wi2 wo wi1 orabs 1oaiii lor df-i2 ancom ax-r2 3tr2 lan omlan lear bltr
|
|
wa ) BDZABEZACEZBCFUAUBSGZSZFZSZTADZSZUGUFTBUHFZSUHUEUITUAUAUCSZFUAUEUIUAUC
|
|
HUJUDUAABCIJUABUGTSZFUIABKUKUHBUGTLJMNOBUGPMTUGQR $.
|
|
$( [1-Nov-98] $)
|
|
|
|
$( Lemma for OA-like stuff with ` ->2 ` instead of ` ->0 ` . $)
|
|
2oalem1 $p |- ( ( a ->2 b ) ' v ( ( b v c ) v ( ( a ->2 b ) ^
|
|
( a ->2 c ) ) ) ) = 1 $=
|
|
( wi2 wn wo wa wt or12 df-i2 2an lor or32 ax-a2 lan ax-r5 ax-r2 anor3 ax-r1
|
|
3tr ud2lem0c 2or oml 3tr1 ax-a3 oran lear bltr leo letr lecom comcom le3tr2
|
|
comcom6 fh3 df-t or1 anidm 3tr2 ) ABDZEZBCFZUTACDZGZFFVBVAVDFZFVBBEZABFZGZB
|
|
AEZVFGZFZCVICEZGZFZGZFZFZHVAVBVDIVEVPVBVAVHVDVOABUAUTVKVCVNABJACJKUBLVBVHFZ
|
|
VOFVGCFZVOFZVQHVRVSVOVRBVHFZCFVSBCVHMWAVGCBVFBAFZGZFWBWAVGBAUCVHWCBVGWBVFAB
|
|
NZOLWDUDPQPVBVHVOUEVTVSVKFZVSVNFZGHHGHVSVKVNVKVSVKVSVKEZVSWGVGVSWGVHVGVHWGV
|
|
HVFVJEZGWGVGWHVFABUFOBVJRQSVFVGUGUHVGCUIUJUKUNULVNVSVNVSVNEZVSVLACFZGZWJBFZ
|
|
WIVSWKWJWLVLWJUGWJBUIUJWKVLVMEZGWIWJWMVLACUFOCVMRQACBMUMUKUNULUOWEHWFHWEBVS
|
|
VJFZFBHFZHVSBVJIWNHBWNVGVJFZCFCWPFZHVGCVJMWPCNWQCHFZHWPHCWPVGVGEZFZHVJWSVGA
|
|
BRLHWTVGUPSQLCUQZQTLBUQZTWFCVSVMFZFWRHVSCVMIXCHCXCBWJFZVMFBWJVMFZFZHVSXDVMV
|
|
SWLXDABCMWJBNQPBWJVMUEXFWOHXEHBXEWJWJEZFZHVMXGWJACRLHXHWJUPSQLXBQTLXATKHURT
|
|
UST $.
|
|
$( [15-Nov-98] $)
|
|
|
|
$( OA-like theorem with ` ->2 ` instead of ` ->0 ` . $)
|
|
2oath1 $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) =
|
|
( ( a ->2 b ) ^ ( a ->2 c ) ) $=
|
|
( wi2 wo wa wn df-i2 lan coman1 comorr2 comcom2 anor3 ax-r1 fh2 anass ax-r2
|
|
cbtr wf wt anidm ran oran lor 2oalem1 ax-r4 df-a df-f 3tr1 2or or0 3tr ) AB
|
|
DZBCEZUMACDZFZDZFUMUPUNGUPGFZEZFUMUPFZUMURFZEZUPUQUSUMUNUPHIUPUMURUMUOJUPUN
|
|
UPEZGZURUPVCUNUPKLURVDUNUPMNROVBUPSEUPUTUPVASUTUMUMFZUOFZUPVFUTUMUMUOPNVEUM
|
|
UOUMUAUBQUMGZURGZEZGTGVASVITVIVGVCEZTVJVIVCVHVGUNUPUCUDNABCUEQUFUMURUGUHUIU
|
|
JUPUKQUL $.
|
|
$( [15-Nov-98] $)
|
|
|
|
$( Orthoarguesian-like OM law. $)
|
|
2oath1i1 $p |- ( ( a ->1 c ) ^
|
|
( ( a ^ b ) ' ->2 ( ( a ->1 c ) ^ ( b ->1 c ) ) ) )
|
|
= ( ( a ->1 c ) ^ ( b ->1 c ) ) $=
|
|
( wn wi2 wo wa wi1 2oath1 i1i2 2an ud2lem0a oran3 ax-r1 ud2lem0b ax-r2 3tr1
|
|
) CDZADZEZSBDZFZTRUAEZGZEZGUDACHZABGDZUFBCHZGZEZGUIRSUAIUFTUJUEACJZUJUGUDEU
|
|
EUIUDUGUFTUHUCUKBCJKZLUGUBUDUBUGABMNOPKULQ $.
|
|
$( [30-Dec-98] $)
|
|
|
|
$( Orthoarguesian-like OM law. $)
|
|
1oath1i1u $p |- ( ( a ->1 c ) ^
|
|
( ( ( a ->1 c ) ^ ( b ->1 c ) ) ' ->1 ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ) )
|
|
= ( ( a ->1 c ) ^ ( b ->1 c ) ) $=
|
|
( wn wi1 wa wi2 2oath1i1 u1lem11 2an ud2lem0a i1i2con2 ax-r1 ax-r2 3tr2 ) A
|
|
DCEZCEZPBDCEZFZDZQRCEZFZGZFUBACEZUDBCEZFZDSEZFUFPRCHQUDUCUGACIZUCTUFGZUGUBU
|
|
FTQUDUAUEUHBCIJZKUGUIUFSLMNJUJO $.
|
|
$( [28-Feb-99] $)
|
|
|
|
$( Relation for studying OA. $)
|
|
oale $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ' ) =< ( a ->2 c ) $=
|
|
( wi2 wo wa wn df-i2 lan coman1 comanr2 comcom6 fh2 anass ax-r1 anidm ax-r2
|
|
ran anor3 2or ax-a2 3tr 2oath1 df-le1 lear letr ) ABDZBCEZUGACDZFZEGZFZUJUI
|
|
ULUJULUJEZUGUHUJDZFZUJUOUMUOUGUJUHGZUJGZFZEZFUGUJFZUGURFZEZUMUNUSUGUHUJHIUJ
|
|
UGURUGUIJUJURUPUQKLMVBUJULEUMUTUJVAULUTUGUGFZUIFZUJVDUTUGUGUINOVCUGUIUGPRQU
|
|
RUKUGUHUJSITUJULUAQUBOABCUCQUDUGUIUEUF $.
|
|
$( [18-Nov-98] $)
|
|
|
|
${
|
|
oaeqv.1 $e |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ' v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) )
|
|
=< ( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
$( Weakened OA implies OA). $)
|
|
oaeqv $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ' v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) )
|
|
=< ( a ->2 c ) $=
|
|
( wi2 wo wn wa lea ler2an 2oath1 lbtr lear letr ) ABEZBCFZGOACEZHZFZHZRQT
|
|
OPREZHRTOUAOSIDJABCKLOQMN $.
|
|
$( [16-Nov-98] $)
|
|
$}
|
|
|
|
|
|
${
|
|
3vroa.1 $e |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) = 1 $.
|
|
$( OA-like inference rule (requires OM only). $)
|
|
3vroa $p |- ( a ->2 c ) = 1 $=
|
|
( wi2 wn wa wo wt df-i2 or12 oridm lor le1 wi0 ax-r1 lea bltr lebi ax-r2
|
|
ran ancom an1 3tr lear df-i0 anor3 ax-r5 le3tr2 u2lemle2 lecon leran 3tr2
|
|
leror ) ACEZCAFZCFZGZHZIACJZCURURHZHURUSHZUSICURURKVAURCURLMVBURABEZUOGZH
|
|
ZIVEVBVDUSURVDUOIGZUOUSVDIUOGVFVCIUOVCIVCNIVCBCHZVDOZGZVCVIIDPVCVHQRSZUAI
|
|
UOUBTUOUCUTUDMPVEIVENIBFZUQGZVDHZVEIVMVIVHIVMVCVHUEDVHVGFZVDHZVMVGVDUFVMV
|
|
OVLVNVDBCUGUHPTUIVMNSVLURVDVKUPUQABABVJUJUKULUNRSTUMT $.
|
|
$( [13-Nov-98] $)
|
|
$}
|
|
|
|
$( Lemma for Mladen's OML. $)
|
|
mlalem $p |- ( ( a == b ) ^ ( b ->1 c ) ) =< ( a ->1 c ) $=
|
|
( wa wn wo tb wi1 comcom3 anass ax-r1 3tr bltr ax-r2 lear letr lel2or df-i1
|
|
wf leo comanr2 comanr1 fh2 dff lan an0 le0 an12 an4 leor lea dfb 2an coman1
|
|
lecom coman2 com2or oran3 cbtr comcom7 fh2rc le3tr1 ) ABDZBEZBCDZFZDZAEZVDD
|
|
ZVFDZFZVHACDZFZABGZBCHZDZACHVGVMVJVGVCVDDZVCVEDZFVMVDVCVEBVCABUAIBVEBCUBIUC
|
|
VQVMVRVQSVMVQABVDDZDASDSABVDJVSSASVSBUDKUEAUFLVMUGMVRBBDZVLDZVMVRABVEDDBAVE
|
|
DDZWAABVEJABVEUHWBBADVEDZWAWCWBBAVEJKBABCUINLWAVLVMVTVLOVLVHUJPMQMVJVHVDVFD
|
|
ZDZVMVHVDVFJWEVHVMVHWDUKVHVLTPMQVPVCVIFZVFDVKVNWFVOVFABULBCRUMVIVFVCVIVFVIV
|
|
DVFVHVDOVDVETPUOVIVCVIVHVDFVCEVIVHVDVHVDUNVHVDUPUQABURUSUTVANACRVB $.
|
|
$( [4-Nov-98] $)
|
|
|
|
|
|
$( Mladen's OML. $)
|
|
mlaoml $p |- ( ( a == b ) ^ ( b == c ) ) =< ( a == c ) $=
|
|
( wi1 wa tb u1lembi ran mlalem bltr ancom an32 3tr le2an an12 id 3tr1 anass
|
|
anandi anandir 3tr2 2an le3tr2 ) ABDZBADZEZBCDZEZUECBDZEZUGEZEZACDZCADZEABF
|
|
ZBCFZEZACFUHUMUKUNUHUOUGEUMUFUOUGABGZHABCIJUKCBFZUEEZUNUKUIUEEZUGEUIUGEZUEE
|
|
UTUJVAUGUEUIKHUIUEUGLVBUSUECBGHMCBAIJNULUHUIEZUFUGUIEZEUQUFUJEZUGEUFUIEZUGE
|
|
ULVCVEVFUGUEUDUIEEZUDUJEVEVFUEUDUIOVEUEUDEZUJEVEVGUFVHUJUDUEKHVEPUEUDUISQUD
|
|
UEUIRQHUFUJUGTUFUIUGLUAUFUGUIRUFUOVDUPURBCGUBMACGUC $.
|
|
$( [4-Nov-98] $)
|
|
|
|
$( 4-variable transitive law for equivalence. $)
|
|
eqtr4 $p |- ( ( ( a == b ) ^ ( b == c ) ) ^ ( c == d ) ) =< ( a == d ) $=
|
|
( tb wa mlaoml leran letr ) ABEBCEFZCDEZFACEZKFADEJLKABCGHACDGI $.
|
|
$( [26-Jun-03] $)
|
|
|
|
${
|
|
sac.1 $e |- ( a ->1 c ) = ( b ->1 c ) $.
|
|
$( Theorem showing "Sasaki complement" is an operation. $)
|
|
sac $p |- ( a ' ->1 c ) = ( b ' ->1 c ) $=
|
|
( wi1 wn ud1lem0b u1lem12 3tr2 ) ACEZCEBCEZCEAFCEBFCEJKCDGACHBCHI $.
|
|
$( [3-Jan-99] $)
|
|
$}
|
|
|
|
${
|
|
sa5.1 $e |- ( a ->1 c ) =< ( b ->1 c ) $.
|
|
$( Possible axiom for a "Sasaki algebra" for orthoarguesian lattices. $)
|
|
sa5 $p |- ( b ' ->1 c ) =< ( ( a ' ->1 c ) v c ) $=
|
|
( wn wa wo wi1 leor ax-a2 lan ax-r5 oml6 ax-r1 ud1lem0c le3tr2 letr ax-a1
|
|
3tr df-i1 lecon lea leror bltr orabs ancom 3tr2 ax-a3 ax-r2 lel2or le3tr1
|
|
2or lear ) BEZEZUNCFZGAEZEZUQCFZGZCGZUNCHUQCHZCGUOVAUPBACGZUOVABCBGZVCBCI
|
|
VDBUNCEZGZFZCGZVCVHVDVHBVEUNGZFZCGCVJGVDVGVJCVFVIBUNVEJKLVJCJCBMSNVGACVGA
|
|
UQVEGZFZABCHZEACHZEVGVLVNVMDUABCOACOPAVKUBQUCUDQBRVCURUSCGZGZVAAURCVOARCC
|
|
UQFZGVQCGCVOCVQJCUQUEVQUSCCUQUFLUGULVAVPURUSCUHNUIPUPCVAUNCUMCUTIQUJUNCTV
|
|
BUTCUQCTLUK $.
|
|
$( [3-Jan-99] $)
|
|
$}
|
|
|
|
|
|
$(
|
|
lattice (((-xIy)vy)Iy)=(x2y)
|
|
lattice "((xIw)v(yIw))<((((-xIw)^(-yIw))Iw)vw)"
|
|
lattice "((((-xIw)vw)Iw)^(((-yIw)vw)Iw))<((((-xIw)v(-yIw))Iw)vw)"
|
|
lattice "(((-xIw)^(-yIw))Iw)<((xIw)v(yIw))"
|
|
lattice "(((-xIw)v(-yIw))Iw)<(((xIw)^(yIw))vw)"
|
|
a' v b' =< (a ^ b)' v 0
|
|
(a v 0)' ^ (b v 0)' =< (a ^ b)' v 0
|
|
(a ^ b)' =< a' v b'
|
|
(a v b)' =< (a' ^ b') v 0
|
|
$)
|
|
$( Lemma for attempt at Sasaki algebra. $)
|
|
salem1 $p |- ( ( ( a ' ->1 b ) v b ) ->1 b ) = ( a ->2 b ) $=
|
|
( wn wi1 wo wi2 u1lemob ax-r4 anor1 ax-r1 ax-r2 ran ax-a2 ancom anabs df-i1
|
|
wa 3tr 2or df-i2 3tr1 ) ACZBDBEZCZUCBQZEZBUBBCQZEZUCBDABFUFUGBEUHUDUGUEBUDU
|
|
BCZBEZCZUGUCUJUBBGZHUGUKUBBIJKUEUJBQZBBUIEZQZBUCUJBULLUMUNBQUOUJUNBUIBMLUNB
|
|
NKBUIORSUGBMKUCBPABTUA $.
|
|
$( [4-Jan-99] $)
|
|
|
|
|
|
$( Weak DeMorgan's law for attempt at Sasaki algebra. $)
|
|
sadm3 $p |- ( ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ->1 c ) =<
|
|
( ( a ->1 c ) v ( b ->1 c ) ) $=
|
|
( wn wi1 wa wo oran3 ax-r1 u1lem9a bltr an32 lea leo or32 lbtr u1lemab letr
|
|
le2or df-i1 ax-a1 bile leran lel2or lelor 2or le3tr1 ) ADZCEZBDZCEZFZDZULCF
|
|
ZGZUHACFZGZUJBCFZGZGZULCEACEZBCEZGUOUQUJGZUTUOUHUJGZUICFZGVCUMVDUNVEUMUIDZU
|
|
KDZGZVDVHUMUIUKHIVFUHVGUJACJBCJSKUNVEUKFVEUIUKCLVEUKMKSVDVCVEVDVDUPGVCVDUPN
|
|
UHUJUPOPVEUQVCVEUHCFZUHDZCFZGUQUHCQVIUHVKUPUHCMVJACVJAAVJAUAIUBUCSKUQUJNRUD
|
|
RUJUSUQUJURNUERULCTVAUQVBUSACTBCTUFUG $.
|
|
$( [4-Jan-99] $)
|
|
|
|
$( Weak DeMorgan's law for attempt at Sasaki algebra. $)
|
|
$(
|
|
sadm1 $p |- ( ( a ->1 c ) v ( b ->1 c ) ) =<
|
|
( ( ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ->1 c ) v c ) $=
|
|
?$.
|
|
$)
|
|
|
|
$( Weak DeMorgan's law for attempt at Sasaki algebra. $)
|
|
$(
|
|
sadm2 $p |- ( ( ( ( a ' ->1 c ) v c ) ->1 c ) ^
|
|
( ( ( b ' ->1 c ) v c ) ->1 c ) ) =<
|
|
( ( ( ( a ' ->1 c ) v ( b ' ->1 c ) ) ->1 c ) v c ) $=
|
|
?$.
|
|
$)
|
|
|
|
$( Weak DeMorgan's law for attempt at Sasaki algebra. $)
|
|
$(
|
|
sadm4 $p |- ( ( ( a ' ->1 c ) v ( b ' ->1 c ) ) ->1 c ) =<
|
|
( ( ( a ->1 c ) ^ ( b ->1 c ) ) v c ) $=
|
|
?$.
|
|
$)
|
|
|
|
$( Chained biconditional. $)
|
|
bi3 $p |- ( ( a == b ) ^ ( b == c ) ) =
|
|
( ( ( a ^ b ) ^ c ) v ( ( a ' ^ b ' ) ^ c ' ) ) $=
|
|
( tb wa wn wo ax-r1 lan leo letr lecom comcom7 wf anass 3tr ax-r2 ran ancom
|
|
2or wi1 wi2 dfb u12lembi 2an df-i1 lear coman1 coman2 fh2rc comanr2 comcom3
|
|
com2an comanr1 fh2 dff an0 anidm or0r comcom an4 an0r 3tr2 or0 df-i2 le3tr1
|
|
an32 lea bltr oran lbtr fh2r u2lemab u2lemanb an12 ) ABDZBCDZEABEZAFZBFZEZG
|
|
ZBCUAZCBUBZEZEZVRCEZWACFZEZGZVPWBVQWEABUCWEVQBCUDHUEWBWCEZWDEWGWAGZWDEZWFWJ
|
|
WKWLWDWKWBVTBCEZGZEVRWOEZWAWOEZGWLWCWOWBBCUFIWAWOVRWAWOWAVTWOVSVTUGVTWNJKLW
|
|
AABWAAVSVTUHMWABVSVTUIZMUMUJWPWGWQWAWPVRVTEZVRWNEZGNWGGWGVTVRWNBVRABUKULBWN
|
|
BCUNULZUOWSNWTWGWSABVTEZEZANEZNABVTOXDXCNXBABUPZIHAUQPWTVRBEZCEZWGXGWTVRBCO
|
|
HXFVRCXFABBEZEVRABBOXHBABURIQRQTWGUSPWQWAVTEZWAWNEZGWANGWAVTWAWNWAVTWRUTXAU
|
|
OXIWAXJNXIVSVTVTEZEWAVSVTVTOXKVTVSVTURIQXJVSBEVTCEZEVSBXLEZEZNVSVTBCVAVSBXL
|
|
OXNVSNENXMNVSXBCEZNCEZXMNXPXONXBCXERHBVTCOCVBVCIVSUQQPTWAVDPTPRWBWCWDOWMWGW
|
|
DEZWAWDEZGWJWGWDWAWGWDACEZBEZBWHVTEZGZWGWDXTBYBXSBUGBYAJKABCVGCBVEVFLWGWAWG
|
|
WAFZWGABGZYCWGAYDWGAWNEAABCOAWNVHVIABJKABVJVKLMVLXQWGXRWIXQVRCWDEEXSBWDEZEZ
|
|
WGVRCWDOABCWDVAYFXTWGYEBXSYEWDBEBBWDSCBVMQIACBVGQPXRVSVTWDEZEVSYAEZWIVSVTWD
|
|
OYGYAVSYGWDVTEYAVTWDSCBVNQIYHWHWAEWIVSWHVTVOWHWASQPTQVCQ $.
|
|
$( [2-Mar-00] $)
|
|
|
|
$( Chained biconditional. $)
|
|
bi4 $p |- ( ( ( a == b ) ^ ( b == c ) ) ^ ( c == d ) ) =
|
|
( ( ( ( a ^ b ) ^ c ) ^ d ) v
|
|
( ( ( a ' ^ b ' ) ^ c ' ) ^ d ' ) ) $=
|
|
( tb wa wn wo ax-r1 lan lecom leao4 lbtr wf anass 3tr ax-r2 ran 2or ancom
|
|
wi1 wi2 bi3 u12lembi df-i1 leao2 oran2 comcom comcom6 fh2rc comanr2 comcom3
|
|
2an comanr1 fh2 dff an0 anidm or0r an4 an0r 3tr2 or0 u2lemab df2le1 comcom7
|
|
an32 bltr fh2r u2lemanb an12 ) ABEBCEFZCDEZFABFZCFZAGBGFZCGZFZHZCDUAZDCUBZF
|
|
ZFZVODFZVRDGZFZHZVLVSVMWBABCUCWBVMCDUDIUMVSVTFZWAFWDVRHZWAFZWCWGWHWIWAWHVSV
|
|
QCDFZHZFVOWLFZVRWLFZHWIVTWLVSCDUEJVRWLVOVRWLVQVPWKUFKVRVOVOVRGZVOWOVOVPGZCH
|
|
ZWOCVNWPLVPCUGZMKUHUIUJWMWDWNVRWMVOVQFZVOWKFZHNWDHWDVQVOWKCVOVNCUKULCWKCDUN
|
|
ULZUOWSNWTWDWSVNCVQFZFZVNNFZNVNCVQOXDXCNXBVNCUPZJIVNUQPWTVOCFZDFZWDXGWTVOCD
|
|
OIXFVODXFVNCCFZFVOVNCCOXHCVNCURJQRQSWDUSPWNVRVQFZVRWKFZHVRNHVRVQVRWKVPVQUKX
|
|
AUOXIVRXJNXIVPVQVQFZFVRVPVQVQOXKVQVPVQURJQXJVPCFVQDFZFVPCXLFZFZNVPVQCDUTVPC
|
|
XLOXNVPNFNXMNVPXBDFZNDFZXMNXPXONXBDXERICVQDODVAVBJVPUQQPSVRVCPSPRVSVTWAOWJW
|
|
DWAFZVRWAFZHWGWDWAVRWDWAWDWAXQVODWAFFVNDFZCWAFZFZWDVODWAOVNCDWAUTYAXSCFZWDX
|
|
TCXSXTWACFCCWATDCVDQJVNDCVGQPZVEKWDVRWDWOWDWQWOWDYBWQVNCDVGCXSWPLVHWRMKVFVI
|
|
XQWDXRWFYCXRVPVQWAFZFVPWEVQFZFZWFVPVQWAOYDYEVPYDWAVQFYEVQWATDCVJQJYFWEVRFWF
|
|
VPWEVQVKWEVRTQPSQVBQ $.
|
|
$( [25-Jun-03] $)
|
|
|
|
$( Implicational product with 3 variables. Theorem 3.20 of "Equations,
|
|
states, and lattices..." paper. $)
|
|
imp3 $p |- ( ( a ->2 b ) ^ ( b ->1 c ) ) =
|
|
( ( a ' ^ b ' ) v ( b ^ c ) ) $=
|
|
( wi2 wi1 wa wn wo df-i1 lan u2lemc1 comcom3 comanr1 fh2 u2lemanb ancom lea
|
|
u2lem3 u2lemle2 letr df2le2 ax-r2 2or 3tr ) ABDZBCEZFUEBGZBCFZHZFUEUGFZUEUH
|
|
FZHAGUGFZUHHUFUIUEBCIJUGUEUHBUEABKLBUHBCMLNUJULUKUHABOUKUHUEFUHUEUHPUHUEUHB
|
|
UEBCQBUEBARSTUAUBUCUD $.
|
|
$( [3-Mar-00] $)
|
|
|
|
$( Disjunction of biconditionals. $)
|
|
orbi $p |- ( ( a == c ) v ( b == c ) ) =
|
|
( ( ( a ->2 c ) v ( b ->2 c ) ) ^ ( ( c ->1 a ) v ( c ->1 b ) ) ) $=
|
|
( tb wo wa wn wi2 2or ax-a2 ax-a3 lor ax-r2 ax-r5 leo letr lecom comcom 3tr
|
|
bctr wi1 dfb ancom imp3 ax-r1 df-i1 lear comi12 fh4rc df-le2 lan 3tr2 df-i2
|
|
lea anor1 cbtr comcom7 fh4 orordi 3tr1 or12 2an ) ACDZBCDZEACFZAGZCGZFZEZBC
|
|
FZBGVGFZEZEVLVIEZACHZBCHZEZCAUAZCBUAZEZFZVCVIVDVLACUBBCUBIVIVLJVMVJVKVIEZEV
|
|
JVOVHEZVGCAFZEZFZEZVTVJVKVIKWAWEVJVKVEEZVHEVOVQFZVHEZWAWEWGWHVHWGVKWCEZWHVE
|
|
WCVKACUCLWHWJBCAUDUEMNVKVEVHKWIWBVQVHEZFWEVQVHVOVQWDVHCAUFZVHWDVHWDVHVGWDVF
|
|
VGUGVGWCOPZQRTCABUHUIWKWDWBWKWDVHEVHWDEWDVQWDVHWLNWDVHJVHWDWMUJSUKMULLWFVJW
|
|
BEZVJWDEZFVTWBVJWDWBCVKVHEZEZVJWBCVKEZVHEZWQVOWRVHBCUMZNCVKVHKZMZVJWQVJWQVJ
|
|
CWQBCUGZCWPOZPQRTWBWQWDXBWQWDWQCWCGZFZWDGXFWQXFWQXFCWQCXEUNXDPQRCWCUOUPUQTU
|
|
RWNVPWOVSVJVOEZVHEWSWNVPXGWRVHXGVJWREWRVOWRVJWTLVJWRVJCWRXCCVKOPUJMNVJVOVHK
|
|
WSWQVPXAWRCVHEZEXHWREWQVPWRXHJCVKVHUSVNXHVOWRACUMWTIUTMULWOVGVJWCEEZVSVJVGW
|
|
CVAVGVJEZWDEWDXJEXIVSXJWDJVGVJWCUSVQWDVRXJWLVRVGCBFZEXJCBUFXKVJVGCBUCLMIUTM
|
|
VBMSS $.
|
|
$( [5-Jul-00] $)
|
|
|
|
$( Disjunction of biconditionals. $)
|
|
orbile $p |- ( ( a == c ) v ( b == c ) ) =<
|
|
( ( ( a ^ b ) ->2 c ) ^ ( c ->1 ( a v b ) ) ) $=
|
|
( tb wo wi2 wi1 wa orbi i2or i1or le2an bltr ) ACDBCDEACFBCFEZCAGCBGEZHABHC
|
|
FZCABEGZHABCINPOQABCJABCKLM $.
|
|
$( [5-Jul-00] $)
|
|
|
|
${
|
|
mlaconj4.1 $e |- ( ( d == e ) ^ ( ( e ' ^ c ' ) v ( d ^ c ) ) ) =<
|
|
( d == c ) $.
|
|
mlaconj4.2 $e |- d = ( a v b ) $.
|
|
mlaconj4.3 $e |- e = ( a ^ b ) $.
|
|
$( For 4GO proof of Mladen's conjecture, that it follows from Eq. (3.30)
|
|
in OA-GO paper. $)
|
|
mlaconj4 $p |- ( ( a == b ) ^ ( ( a == c ) v ( b == c ) ) ) =<
|
|
( a == c ) $=
|
|
( tb wo wa wn ax-r2 lbtr ran 2or ax-r1 anass comcom7 wf biao bile wi2 wi1
|
|
bicom orbile imp3 le2an 2bi ax-r4 lan 2an lea 3tr1 rbi ler2an coman1 bctr
|
|
ancom an32 coman2 com2an com2or fh2c anor3 comanr1 fh2rc leao1 df2le2 dff
|
|
comcom3 oran an0r 3tr2 or0 3tr an4 anidm or0r dfb lor mlaoml bltr letr
|
|
bi3 ) ABIZACIZBCIZJZKABJZABKZIZWKLZCLZKZCWJKZJZKZWGWFWLWIWQWFWLWFWKWJIZWL
|
|
ABUAZWKWJUEMUBWIWKCUCCWJUDKWQABCUFWKCWJUGNUHWRDEIZELZWNKZDCKZJZKZWGXFWRXA
|
|
WLXEWQDWJEWKGHUIZXCWOXDWPXBWMWNEWKHUJOXDCDKWPDCUSDWJCGUKMPULQXFWFWJCIZKZW
|
|
GXFWFXHXFXAWFXAXEUMWLWSXAWFWJWKUEXGWTUNNXFDCIXHFDWJCGUONUPXIWFWHKZWGWKALZ
|
|
BLZKZJZWJCKZXMWNKZJZKZWKCKZXPJZXIXJXRXNXOKZXNXPKZJXTXPXNXOXPWKXMXPABXPXKX
|
|
LWNKZKZAXKXLWNRYDAXKYCUQSURZXPXKWNKZXLKZBXKXLWNUTYGBYFXLVASURZVBXMWNUQVCX
|
|
PWJCXPABYEYHVCXPCXMWNVASVBVDYAXSYBXPYAWKXOKZXMXOKZJXSTJXSXMXOWKXMWJLZXOAB
|
|
VEZWJXOWJCVFVKURXMABXMAXKXLUQSXMBXKXLVASVBZVGYIXSYJTYIWKWJKZCKZXSYOYIWKWJ
|
|
CRQYNWKCWKWJABBVHVIOMXMWJKZCKZTCKZYJTYRYQTYPCTXMXMLZKZYPXMVJYPYTWJYSXMABV
|
|
LUKQMOQXMWJCRCVMVNPXSVOVPYBWKXPKZXMXPKZJTXPJXPXMXPWKXMWNVFYMVGUUATUUBXPUU
|
|
AAXMKZBWNKZKZTUUDKZTABXMWNVQUUFUUETUUCUUDTXLKAXKKZXLKTUUCTUUGXLAVJOXLVMAX
|
|
KXLRVNOQUUDVMVPUUBXMXMKZWNKZXPUUIUUBXMXMWNRQUUHXMWNXMVROMPXPVSVPPMWFXNXHX
|
|
QABVTXHXOYKWNKZJZXQWJCVTXQUUKXPUUJXOXMYKWNYLOWAQMULABCWEUNABCWBWCWDWCWD
|
|
$.
|
|
$( [8-Jul-00] $)
|
|
$}
|
|
|
|
|
|
$( For 5GO proof of Mladen's conjecture. $)
|
|
mlaconj $p |- ( ( a == b ) ^ ( ( a == c ) v ( b == c ) ) ) =<
|
|
( ( ( ( a ->1 ( a ^ b ) ) ^ ( ( a ^ b ) ->1 ( ( a ^ b ) v c ) ) ) ^
|
|
( ( ( ( a ^ b ) v c ) ->1 c ) ^ ( c ->1 ( a v b ) ) ) ) ^
|
|
( ( a v b ) ->1 a ) ) $=
|
|
( tb wo wa wi2 wi1 orbile lelan ancom ran anass ax-r2 3tr lan bi1o1a i2i1i1
|
|
id 3tr1 2an lbtr ) ABDZACDBCDEZFUCABFZCGZCABEZHZFZFZAUEHZUEUECEZHZFZULCHZUH
|
|
FZFUGAHZFZUDUIUCABCIJUKUQFZUMUOFZUHFZFZUNUPUQFZFZUJURUKUQVAFZFUKUMVCFZFVBVD
|
|
VEVFUKVEVAUQFUMUPFZUQFVFUQVAKVAVGUQVAVAVGUTUTUHUTSLUMUOUHMNLUMUPUQMOPUKUQVA
|
|
MUKUMVCMTUCUSUIVAABQUFUTUHUECRLUAUNUPUQMTUB $.
|
|
$( [20-Jan-02] $)
|
|
|
|
${
|
|
mlaconj2.1 $e |- ( ( ( ( a ->1 ( a ^ b ) ) ^
|
|
( ( a ^ b ) ->1 ( ( a ^ b ) v c ) ) ) ^
|
|
( ( ( ( a ^ b ) v c ) ->1 c ) ^ ( c ->1 ( a v b ) ) ) ) ^
|
|
( ( a v b ) ->1 a ) ) =< ( a == c ) $.
|
|
$( For 5GO proof of Mladen's conjecture. Hypothesis is 5GO law
|
|
consequence. $)
|
|
mlaconj2 $p |- ( ( a == b ) ^ ( ( a == c ) v ( b == c ) ) ) =<
|
|
( a == c ) $=
|
|
( tb wo wa wi1 mlaconj letr ) ABEACEZBCEFGAABGZHLLCFZHGMCHCABFZHGGNAHGKAB
|
|
CIDJ $.
|
|
$( [6-Jul-00] $)
|
|
$}
|
|
|
|
$( Equivalence to chained biconditional. $)
|
|
$( [Appears not to be a theorem.]
|
|
bi3eq $p |- ( ( a == b ) ^ ( ( a ^ c ) v ( b ' ^ c ' ) ) ) =
|
|
( ( a == c ) ^ ( b == c ) ) $=
|
|
( u1lembi ran ax-r1 anass wn lea wa leo df-i1 lbtr letr lecom lear lecon
|
|
comcom7 fh2c u1lemab id ax-r2 u1lemana 2or lan bi3 bicom ) ??????????DEF???
|
|
???GZ?????????????????BHZCHZI???UIBAJK????LFMNO???????UIUJP??ACPQNORS??????
|
|
??????UHFZ????????TE?UAZUBUB???UK????????UCEULUBUBUDULUBUBUE???????????UB??
|
|
???UFFUB?????UGUEUBUBUBUB $.
|
|
$)
|
|
$( [3-Mar-00] $)
|
|
|
|
$( Complemented antecedent lemma. $)
|
|
i1orni1 $p |- ( ( a ->1 b ) v ( a ' ->1 b ) ) = 1 $=
|
|
( wi1 wn wo wa wt df-i1 ax-a1 ax-r5 ax-r1 ax-r2 lor orordi u1lemoa or1r ) A
|
|
BCZADZBCZEQARBFZEZEZGSUAQSRDZTEZUARBHUAUDAUCTAIJKLMUBQAEZQTEZEZGQATNUGGUFEG
|
|
UEGUFABOJUFPLLL $.
|
|
$( [6-Aug-01] $)
|
|
|
|
${
|
|
negant.1 $e |- ( a ->1 c ) = ( b ->1 c ) $.
|
|
$( Lemma for negated antecedent identity. $)
|
|
negantlem1 $p |- a C ( b ->1 c ) $=
|
|
( wi1 wn wa wo leo df-i1 ax-r1 ax-r2 lbtr lecom comcom6 ) ABCEZAFZPQQACGZ
|
|
HZPQRISACEZPTSACJKDLMNO $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Lemma for negated antecedent identity. $)
|
|
negantlem2 $p |- a =< ( b ' ->1 c ) $=
|
|
( wn wi1 wo leo wa wt i1orni1 lan ax-r1 an1 u1lemc6 negantlem1 ancom lear
|
|
bltr letr comcom fh4rc 3tr1 u1lemaa 3tr2 ler2an ax-a1 leror u1lemab ax-r2
|
|
lea lbtr df-i1 le3tr1 leid lel2or ) AABEZCFZGZURAURHUSABCFZIZURGZURUSJIZU
|
|
SUTURGZIZUSVBVEVCVDJUSBCKLMVCUSUSNMUTURABCOAUTABCDPUAUBUCVAURURVACUTIZURV
|
|
ACUTVAACIZCAACFZIVHAIVAVGAVHQVHUTADLACUDUEACRSAUTRUFBCIZUQCIZGZUQEZVJGVFU
|
|
RVIVLVJVIBVLBCUKBUGULUHVFUTCIVKCUTQBCUIUJUQCUMUNTURUOUPST $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Lemma for negated antecedent identity. $)
|
|
negantlem3 $p |- ( a ' ^ c ) =< ( b ' ->1 c ) $=
|
|
( wn wa wi1 wo leo df-i1 ax-r1 ax-r2 lbtr leran leror u1lemab ax-a1 ax-r5
|
|
lea le3tr1 letr ) AEZCFBCGZCFZBEZCGZUBUCCUBUBACFZHZUCUBUGIUHACGZUCUIUHACJ
|
|
KDLMNBCFZUECFZHBUKHZUDUFUJBUKBCSOBCPUFUEEZUKHZULUECJULUNBUMUKBQRKLTUA $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Lemma for negated antecedent identity. $)
|
|
negantlem4 $p |- ( a ' ->1 c ) =< ( b ' ->1 c ) $=
|
|
( wn wi1 wa wo df-i1 ax-a1 ax-r5 ax-r1 ax-r2 negantlem2 negantlem3 lel2or
|
|
bltr ) AEZCFZARCGZHZBECFZSREZTHZUARCIUAUDAUCTAJKLMAUBTABCDNABCDOPQ $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Negated antecedent identity. $)
|
|
negant $p |- ( a ' ->1 c ) = ( b ' ->1 c ) $=
|
|
( wn wi1 negantlem4 ax-r1 lebi ) AECFBECFABCDGBACACFBCFDHGI $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Negated antecedent identity. $)
|
|
negantlem5 $p |- ( a ' ^ c ' ) = ( b ' ^ c ' ) $=
|
|
( wi1 wn wa ran u1lemanb 3tr2 ) ACEZCFZGBCEZLGAFLGBFLGKMLDHACIBCIJ $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Negated antecedent identity. $)
|
|
negantlem6 $p |- ( a ^ c ' ) = ( b ^ c ' ) $=
|
|
( wn wa negant negantlem5 ax-a1 ran 3tr1 ) AEZEZCEZFBEZEZNFANFBNFLOCABCDG
|
|
HAMNAIJBPNBIJK $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Negated antecedent identity. $)
|
|
negantlem7 $p |- ( a v c ) = ( b v c ) $=
|
|
( wo wn wa negantlem5 anor3 3tr2 con1 ) ACEZBCEZAFCFZGBFNGLFMFABCDHACIBCI
|
|
JK $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Negated antecedent identity. $)
|
|
negantlem8 $p |- ( a ' v c ) = ( b ' v c ) $=
|
|
( wn wa wo negantlem6 ax-r4 oran2 3tr1 ) ACEZFZEBLFZEAECGBECGMNABCDHIACJB
|
|
CJK $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Negated antecedent identity. $)
|
|
negant0 $p |- ( a ' ->0 c ) = ( b ' ->0 c ) $=
|
|
( wn wo wi0 negantlem7 ax-a1 ax-r5 3tr2 df-i0 3tr1 ) AEZEZCFZBEZEZCFZNCGQ
|
|
CGACFBCFPSABCDHAOCAIJBRCBIJKNCLQCLM $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Negated antecedent identity. $)
|
|
negant2 $p |- ( a ' ->2 c ) = ( b ' ->2 c ) $=
|
|
( wn wa wo wi2 negantlem6 ax-a1 ran 3tr2 lor df-i2 3tr1 ) CAEZEZCEZFZGCBE
|
|
ZEZRFZGPCHTCHSUBCARFBRFSUBABCDIAQRAJKBUARBJKLMPCNTCNO $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Negated antecedent identity. $)
|
|
negantlem9 $p |- ( a ->3 c ) =< ( b ->3 c ) $=
|
|
( wn wa wo wi3 leor wi1 df-i1 ax-a1 ax-r5 ax-r1 leo bltr letr ler2an lbtr
|
|
ax-r2 leao4 sac 3tr2 leror leao1 negantlem8 negantlem5 ler lear lel df-i3
|
|
lel2or dfi3b le3tr1 ) AEZCFZUOCEZFZGZAUOCGZFZGBEZCGZBVBUQFZGZVBCFZGZFZACH
|
|
BCHUSVHVAUPVHURUPVCVGCUOVBUAUPAUPGZVGUPAIVIBVFGZVGUOCJZVBCJZVIVJABCDUBVKU
|
|
OEZUPGZVIUOCKVIVNAVMUPALMNTVLVBEZVFGZVJVBCKVJVPBVOVFBLMNTUCZBVEVFBVDOUDZP
|
|
QRURVCVGURUTVCUOUQCUEABCDUFZSURVDVGABCDUGVDVEVFVDBIUHPRULVAVCVGVAUTVCAUTU
|
|
IVSSAVGUTAVJVGAVIVJAUPOVQSVRQUJRULACUKBCUMUN $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Negated antecedent identity. $)
|
|
negant3 $p |- ( a ' ->3 c ) = ( b ' ->3 c ) $=
|
|
( wn wi3 sac negantlem9 wi1 ax-r1 lebi ) AEZCFBEZCFLMCABCDGZHMLCLCIMCINJH
|
|
K $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Lemma for negated antecedent identity. $)
|
|
negantlem10 $p |- ( a ->4 c ) =< ( b ->4 c ) $=
|
|
( wa wn wo wi4 leao4 wi1 leor df-i1 ax-r1 lbtr lear ler2an ran ancom bltr
|
|
ax-r2 u1lemab 2or ax-a2 lor ax-a3 letr negant ax-a1 lel2or lea negantlem8
|
|
leao2 ler df-i4 dfi4b le3tr1 ) ACEZAFZCEZGZURCGZCFZEZGBFZCGZVBCVDEZGZCBEZ
|
|
GZEZACHBCHUTVJVCUQVJUSUQVEVICAVDIUQACJZCEZVIUQVKCUQURUQGZVKUQURKVKVMACLMN
|
|
ACOPVLBCJZCEZVIVKVNCDQVOBCEZVDCEZGZVIBCUAVRVBVRGZVIVRVBKVSVBVFVHGZGZVIVRV
|
|
TVBVRVHVFGVTVPVHVQVFBCRVDCRUBVHVFUCTUDVIWAVBVFVHUEMZTNSSUFPUSVEVICURVDIUS
|
|
URCJZCEZVIUSWCCUSURFZUSGZWCUSWEKWCWFURCLMNURCOPWDVDCJZCEZVIWCWGCABCDUGQWH
|
|
VQVDFZCEZGZVIVDCUAWKVBWKGZVIWKVBKWLWAVIWAWLVTWKVBVFVQVHWJCVDRVHVPWJCBRBWI
|
|
CBUHQTUBUDMWBTNSSUFPUIVCVEVIVCVAVEVAVBUJABCDUKNVCVGVHVBVAVFULUMPUIACUNBCU
|
|
OUP $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Negated antecedent identity. $)
|
|
negant4 $p |- ( a ' ->4 c ) = ( b ' ->4 c ) $=
|
|
( wn wi4 sac negantlem10 wi1 ax-r1 lebi ) AEZCFBEZCFLMCABCDGZHMLCLCIMCINJ
|
|
HK $.
|
|
$( [6-Aug-01] $)
|
|
|
|
$( Negated antecedent identity. $)
|
|
negant5 $p |- ( a ' ->5 c ) = ( b ' ->5 c ) $=
|
|
( wn wi2 wi4 wa wi5 negant2 negant4 2an u24lem 3tr2 ) AEZCFZOCGZHBEZCFZRC
|
|
GZHOCIRCIPSQTABCDJABCDKLOCMRCMN $.
|
|
$( [6-Aug-01] $)
|
|
$}
|
|
|
|
${
|
|
neg3ant.1 $e |- ( a ->3 c ) = ( b ->3 c ) $.
|
|
$( Lemma for negated antecedent identity. $)
|
|
neg3antlem1 $p |- ( a ^ c ) =< ( b ->1 c ) $=
|
|
( wa wi1 wn wo leo wi3 ran u3lemab 3tr2 u1lemab ax-r1 ax-r2 lbtr lea letr
|
|
) ACEZBCFZCEZUATTAGCEZHZUBTUCIUDBCEBGCEHZUBACJZCEBCJZCEUDUEUFUGCDKACLBCLM
|
|
UBUEBCNOPQUACRS $.
|
|
$( [7-Aug-01] $)
|
|
|
|
$( Lemma for negated antecedent identity. $)
|
|
neg3antlem2 $p |- a ' =< ( b ->1 c ) $=
|
|
( wn wa wo leor wi3 u3lemab 3tr2 lbtr leao1 lel2or letr ax-r2 ax-r1 wf wt
|
|
ran wi1 df-i3 u3lemanb anor3 con1 ler2an u3lem15 lear oran2 lan anor1 lor
|
|
anor2 oran1 le3tr2 lecon1 leo ax-r5 u3lemob comor1 comcom7 comor2 comcom2
|
|
lel com2an fh1r anabs dff 2or or0 3tr ler id ax-a2 orabs 3tr1 df-t coman1
|
|
2an an1 coman2 com2or fh3 df-i1 le3tr1 ) AEZCFZWFACEZGZFZGZBEZBCFZGZWFBCU
|
|
AWGWNWJWGWMWLCFZGZWNWGACFZWGGZWPWGWQHACIZCFBCIZCFWRWPWSWTCDTACJBCJKLWMWNW
|
|
OWMWLHWLCWMMNOWJWLWMWJWLBWHFZGZWLCGZFZWLWJXBXCXBWJBXCFZAWGGZXBEZWJEZXEWFC
|
|
GZXFFZXFXEWSACGZFXJXEWSXKXEWOWLWHFZGZXEGZWSXEXMHWSXNWSWTXNDBCUBPQLXEBCGZX
|
|
KBXCCMXKXOXKXOWFWHFZXLXKEXOEWSWHFWTWHFXPXLWSWTWHDTACUCBCUCKACUDBCUDKUEQLU
|
|
FACUGLXIXFUHOXEBXAEZFXGXCXQBBCUIZUJBXAUKPXFAWIEZGXHWGXSAACUMULAWIUNPUOUPW
|
|
FXCWIWFXIXCWFCUQWSCGWTCGXIXCWSWTCDURACUSBCUSKLVDUFXDWLXCFZXAXCFZGWLRGWLXC
|
|
WLXAWLCUTZXCBWHXCBYBVAXCCWLCVBVCVEVFXTWLYARWLCVGYAXAXQFZRXCXQXAXRUJRYCXAV
|
|
HQPVIWLVJVKLVLNWFSFZWGWFGZWGWIGZFWFWKWFYESYFWFWFWFYEWFVMZYGYEWFWGGWFWGWFV
|
|
NWFCVOPVPSWGWGEZGZYFWGVQYFYIWIYHWGACUNULQPVSYDWFWFVTQWGWFWIWFCVRZWGAWHWGA
|
|
YJVAWGCWFCWAVCWBWCVPBCWDWE $.
|
|
$( [7-Aug-01] $)
|
|
|
|
$( Lemma for negated antecedent identity. $)
|
|
neg3ant1 $p |- ( a ->1 c ) = ( b ->1 c ) $=
|
|
( wn wa wi1 neg3antlem2 neg3antlem1 lel2or df-i1 lbtr wi3 ax-r1 lebi 3tr1
|
|
wo ) AEZACFZQZBEZBCFZQZACGZBCGZTUCTUEUCRUESABCDHABCDIJBCKZLUCUDTUAUDUBBAC
|
|
ACMBCMDNZHBACUGIJACKZLOUHUFP $.
|
|
$( [7-Aug-01] $)
|
|
$}
|
|
|
|
${
|
|
elimcons.1 $e |- ( a ->1 c ) = ( b ->1 c ) $.
|
|
elimcons.2 $e |- ( a ^ c ) =< ( b v c ' ) $.
|
|
$( Lemma for consequent elimination law. $)
|
|
elimconslem $p |- a =< ( b v c ' ) $=
|
|
( wn wo wa wt df-t lecon oran3 ax-r1 lbtr bltr df-a wi1 df-i1 3tr2 lor
|
|
lelor lelan an1 comor1 comcom7 lecom comcom6 fh2c le3tr2 ax-r4 3tr1 leror
|
|
lear letr ax-a2 leao1 df-le2 ax-r2 ) ABCFZGZBBFZUSGZHZGZUTAAUTHZVCGZVDAVE
|
|
AAFZUSGZHZGZVFAIHAUTVHGZHAVJIVKAIUTUTFZGVKUTJVLVHUTVLACHZFZVHVMUTEKVHVNAC
|
|
LMNUAOUBAUCVHAUTVHAVGUSUDUEVHUTVHFZUTVOVMUTVMVOACPZMEOUFUGUHUIVIVCVEVGVOG
|
|
ZFVAVBFZGZFVIVCVQVSVGVMGZVABCHZGZVQVSACQBCQVTWBDACRBCRSVMVOVGVPTWAVRVABCP
|
|
TSUJAVHPBVBPUKTNVEUTVCAUTUMULUNVDVCUTGUTUTVCUOVCUTBVBUSUPUQURN $.
|
|
$( [3-Mar-02] $)
|
|
|
|
$( Consequent elimination law. $)
|
|
elimcons $p |- a =< b $=
|
|
( wn wo wa df-t elimconslem leror bltr wi1 df-i1 3tr2 anor2 lor df-a lbtr
|
|
wt lelan an1 comor1 comcom2 lecom comcom3 comcom le3tr2 negant ax-r1 3tr1
|
|
fh2 ax-r4 ax-r5 lear lelor letr lea df-le2 lecon1 ) BABFZAFZACFZGZHZVBGZV
|
|
BVAVEVAVBHZGZVFVAVABVCGZHZVGGZVHVATHVAVIVBGZHVAVKTVLVATAVBGVLAIAVIVBABCDE
|
|
JZKLUAVAUBVIVAVBVIBBVCUCUDVBVIAVIAVIVMUEUFUGULUHVJVEVGVAFZVIFZGZFVBFZVDFZ
|
|
GZFVJVEVPVSVSVPVQVBCHZGZVNVACHZGZVSVPVBCMVACMWAWCABCDUIVBCNVACNOVTVRVQACP
|
|
QWBVOVNBCPQOUJUMVAVIRVBVDRUKUNSVGVBVEVAVBUOUPUQVEVBVBVDURUSSUT $.
|
|
$( [3-Mar-02] $)
|
|
$}
|
|
|
|
${
|
|
elimcons2.1 $e |- ( a ->1 c ) = ( b ->1 c ) $.
|
|
elimcons2.2 $e |- ( a ^ ( c ^ ( b ->1 c ) ) ) =<
|
|
( b v ( c ' v ( a ->1 c ) ' ) ) $.
|
|
$( Consequent elimination law. $)
|
|
elimcons2 $p |- a =< b $=
|
|
( wi1 wa wn ax-r1 df-i1 ax-r2 lan anass leor df2le2 3tr ax-r4 lor ax-a2
|
|
wo ud1lem0c ax-a3 lea df-le2 ax-r5 le3tr2 elimcons ) ABCDACBCFZGZGZBCHZAC
|
|
FZHZTZTZACGZBUKTZEUJACAHZUPTZGZGZUPUSGZUPUIUTAUHUSCUHULUSULUHDIACJKLLVBVA
|
|
ACUSMIUPUSUPURNOPUOBBBHUKTZGZUKTZTZBVDTZUKTZUQUNVEBUNUKVDTVEUMVDUKUMUHHVD
|
|
ULUHDQBCUAKRUKVDSKRVHVFBVDUKUBIVGBUKVGVDBTBBVDSVDBBVCUCUDKUEPUFUG $.
|
|
$( [12-Mar-02] $)
|
|
$}
|
|
|
|
$( Lemma for biconditional commutation law. $)
|
|
comanblem1 $p |- ( ( a == c ) ^ ( b == c ) ) =
|
|
( ( ( a v c ) ' v ( ( a ^ b ) ^ c ) ) ^ ( b ->1 c ) ) $=
|
|
( wi1 wa tb wo wn u1lembi 2an df-i1 comanr1 comcom3 ax-r1 ax-r2 lan ran 3tr
|
|
ancom wf an4 an32 fh3 lea leor bltr letr lecom com2an comcom coman2 comcom2
|
|
fh2c coman1 fh2rc anass dff an0 lor or0 anor3 bctr anandi leran df2le2 lear
|
|
2or df-le2 3tr2 ) ACDZCADZEZBCDZCBDZEZEVJVMEVKVNEZEZACFZBCFZEACGHZABEZCEZGZ
|
|
VMEZVJVKVMVNUAVLVRVOVSACIBCIJVQVJVPEZVMEWDVJVMVPUBWEWCVMWEVJCHZCAEZCBEZEZGZ
|
|
EAHZACEZGZWJEZWCVPWJVJVPWFWGGZWFWHGZEZWJVKWOVNWPCAKCBKJWJWQWFWGWHCWGCALMZCW
|
|
HCBLMZUCNOPVJWMWJACKQWNWMWFEZWMWIEZGWCWIWMWFWIWMWIWGWMWGWHUDWGWLWMCASWLWKUE
|
|
UFUGUHWFWIWFWGWHWRWSUIUJUMWTVTXAWBWTWKWFEZWLWFEZGXBTGZVTWLWFWKWLCACUKULWLAA
|
|
CUNULZUOXCTXBXCACWFEZEZATEZTACWFUPXHXGTXFACUQPNAURRUSXDXBVTXBUTACVAORXAWKWI
|
|
EZWLWIEZGWKWBEZWBGWBWLWIWKWLWGWIACSWGWHLVBXEUOXIXKXJWBWIWBWKWICWAEZWBXLWICA
|
|
BVCNCWASOZPXJWLWBEWBWLEWBWIWBWLXMPWLWBSWBWLWAACABUDVDVERVGXKWBWKWBVFVHRVGOR
|
|
QOVI $.
|
|
$( [1-Dec-99] $)
|
|
|
|
$( Lemma for biconditional commutation law. $)
|
|
comanblem2 $p |- ( ( a ^ b ) ^ ( ( a == c ) ^ ( b == c ) ) ) =
|
|
( ( a ^ b ) ^ c ) $=
|
|
( wa tb wn wo dfb 2an wf comanr1 comcom6 fh1 anass ax-r1 anidm ran dff 3tr2
|
|
ax-r2 lan an0r 2or or0 3tr an4 anandir 3tr1 ) ABDZACEZBCEZDZDUIACDZAFZCFZDZ
|
|
GZBCDZBFZUODZGZDZDZUICDZULVBUIUJUQUKVAACHBCHIUAAUQDZBVADZDUMURDVCVDVEUMVFUR
|
|
VEAUMDZAUPDZGUMJGUMAUMUPACKAUPUNUOKLMVGUMVHJVGAADZCDZUMVJVGAACNOVIACAPQTAUN
|
|
DZUODZJUODZVHJVMVLJVKUOARQOAUNUONUOUBZSUCUMUDUEVFBURDZBUTDZGURJGURBURUTBCKB
|
|
UTUSUOKLMVOURVPJVOBBDZCDZURVRVOBBCNOVQBCBPQTBUSDZUODZVMVPJVMVTJVSUOBRQOBUSU
|
|
ONVNSUCURUDUEIABUQVAUFABCUGUHT $.
|
|
$( [1-Dec-99] $)
|
|
|
|
$( Biconditional commutation law. $)
|
|
comanb $p |- ( a ^ b ) C ( ( a == c ) ^ ( b == c ) ) $=
|
|
( wa tb wo wn wi1 lea leo lecon leror comanblem1 df-i1 comanblem2 lor ax-r2
|
|
letr le3tr1 i1com ) ABDZACEBCEDZACFZGZUACDZFZBCHZDZUAGZUEFZUBUAUBHZUHUFUJUF
|
|
UGIUDUIUEUAUCUAAUCABIACJRKLRABCMUKUIUAUBDZFUJUAUBNULUEUIABCOPQST $.
|
|
$( [1-Dec-99] $)
|
|
|
|
$( Biconditional commutation law. $)
|
|
comanbn $p |- ( a ' ^ b ' ) C ( ( a == c ) ^ ( b == c ) ) $=
|
|
( wn wa tb comanb conb 2an ax-r1 cbtr ) ADZBDZELCDZFZMNFZEZACFZBCFZEZLMNGTQ
|
|
ROSPACHBCHIJK $.
|
|
$( [1-Dec-99] $)
|
|
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${
|
|
mhlem.1 $e |- ( a v b ) =< ( c v d ) ' $.
|
|
$( Lemma for Lemma 7.1 of Kalmbach, p. 91. $)
|
|
mhlemlem1 $p |- ( ( ( a v b ) v c ) ^ ( a v ( c v d ) ) ) = ( a v c ) $=
|
|
( wo wa leo ler lecom wn letr comcom7 fh2 ancom ax-a3 anabs 3tr wf 2or
|
|
lan comor1 lecon3 fh1rc ortha or0r ax-r2 ) ABFZCFZACDFZFGUIAGZUIUJGZFACFA
|
|
UIUJAUIAUHCABHZIJAUJAUJKZAUHUNUMELJMNUKAULCUKAUIGAABCFZFZGAUIAOUIUPAABCPU
|
|
AAUOQRULUHUJGZCUJGZFSCFCUJCUHCDUBUJUHUJUHKUHUJEUCJMUDUQSURCUHUJEUECDQTCUF
|
|
RTUG $.
|
|
$( [10-Mar-02] $)
|
|
|
|
$( Lemma for Lemma 7.1 of Kalmbach, p. 91. $)
|
|
mhlemlem2 $p |- ( ( ( a v b ) v d ) ^ ( b v ( c v d ) ) ) = ( b v d ) $=
|
|
( wo wa ax-a2 ax-r5 lor 2an wn ax-r4 le3tr1 mhlemlem1 ax-r2 ) ABFZDFZBCDF
|
|
ZFZGBAFZDFZBDCFZFZGBDFRUBTUDQUADABHISUCBCDHJKBADCQSLUAUCLEBAHUCSDCHMNOP
|
|
$.
|
|
$( [10-Mar-02] $)
|
|
|
|
$( Lemma 7.1 of Kalmbach, p. 91. $)
|
|
mhlem $p |- ( ( a v c ) ^ ( b v d ) ) = ( ( a ^ b ) v ( c ^ d ) ) $=
|
|
( wo wa comor1 comor2 com2an wn lecom comcom7 leao1 letr comcom 3tr ax-r2
|
|
3tr1 wf fh1r fh2rc 2or lerr fh3 id mhlemlem1 mhlemlem2 ancom ax-a2 df-le2
|
|
2an an4 lor ax-r1 or12 lan leor fh3r lecon3 com2or ax-a3 ax-r5 le2an lbtr
|
|
leo fh2 ortha or0 df2le2 lear leid ler2an lebi ) ACFZBDFZGZABGZABFZGZCDFZ
|
|
VSGZVRCDGZGZFZFZWAWCGZFZVRWCFVQVTWBFZWDFZWGFZWHVRWAFZVSGZWLWCGZFZWIWDWGFZ
|
|
FVQWKWMWIWNWPVSVRWAVSABABHABIJZVSWAVSWAKZELMZUAWAWCVRWACDCDHCDIJVRWAVRWAV
|
|
RWRVRVSWRABBNZEOLMZPUBUCVQWLVSWCFZGZWOVRWAWCVSFZGFZWLVRXDFZGZVQXCVRWAXDXA
|
|
VRXDVRVSWCWTUDLUEZVQVSCFZAWAFZGZVSDFZBWAFZGZGZXEVQVQVQXOVQUFZXPXKVOXNVPAB
|
|
CDEUGABCDEUHULSXOXIXLGZXJXMGZGXRXQGZXEXIXJXLXMUMXQXRUIXCXGXSXEXBXFWLXBXDW
|
|
CVRVSFZFZXFVSWCUJYAXDXTVSWCVRVSWTUKUNUOWCVRVSUPQUQZXCXSWLXRXBXQWAABAWAAWA
|
|
AWRAVSWRABVFEOLMPBWABWABWRBVSWRBAUREOLMPUSVSCDCVSCVSCVSKZCWAYCCDVFVSWAEUT
|
|
ZOLMPDVSDVSDYCDWAYCDCURYDOLMPUEULUOXHSQRYBSVSWLWCVSVRWAWQWSVAWCVSWCVSWCYC
|
|
WCWAYCCDDNZYDOLMPVGRWIWDWGVBSWJWFWGVTWBWDVBVCRWFVRWGWCWFVTTFVTVRWETVTWEVS
|
|
WAGZTWDWBFWBWEYFWDWBWDYFWBVRVSWCWAWTYEVDVSWAUIZVEUKWBWDUJYGSVSWAEVHRUNVTV
|
|
IVRVSWTVJQWGWCWAWCVKWCWAWCYEWCVLVMVNUCR $.
|
|
$( [10-Mar-02] $)
|
|
$}
|
|
|
|
${
|
|
mhlem1.1 $e |- a C b $.
|
|
mhlem1.2 $e |- c C b $.
|
|
$( Lemma for Marsden-Herman distributive law. $)
|
|
mhlem1 $p |- ( ( a v b ) ^ ( b ' v c ) ) = ( ( a ^ b ' ) v ( b ^ c ) ) $=
|
|
( wo wn wa wt lan comcom2 fh1 ax-a2 wf comcom lor ax-r1 3tr comcom6 ax-r5
|
|
df-t an1 comor2 comid comcom3 fh1r dff or0 ancom anabs 2or comorr comanr2
|
|
3tr2 ran fh2rc leao2 df2le2 ax-r2 or0r ) ABFZBGZCFZHAVBHZBFZVCHZVDBVCHZFZ
|
|
VDBCHZFVAVEVCVAIHVABVBFZHZVAVEIVJVABUAJVAUBVKVABHZVAVBHZFVMVLFVEVABVBABUC
|
|
ZVABVNKLVLVMMVMVDVLBVMVDBVBHZFZVDNFZVDVBABAVBABDKOBBBUDZUEUFVQVPNVOVDBUGZ
|
|
PQVDUHRVLBVAHBBAFZHBVABUIVAVTBABMJBAUJRUKRUNUOVFVDVCHZVGFVHBVCVDBVCVBCULS
|
|
BVDAVBUMSUPWAVDVGVDVCVBACUQURTUSVGVIVDVGVOVIFZNVIFZVIBVBCBBVRKCBEOLWCWBNV
|
|
OVIVSTQVIUTRPR $.
|
|
$( [10-Mar-02] $)
|
|
$}
|
|
|
|
${
|
|
mh.1 $e |- a C c $.
|
|
mh.2 $e |- a C d $.
|
|
mh.3 $e |- b C c $.
|
|
mh.4 $e |- b C d $.
|
|
$( Lemma for Marsden-Herman distributive law. $)
|
|
mhlem2 $p |- ( ( ( a v c ) ^ ( c ' v b ' ) ) ^
|
|
( ( b v d ) ^ ( a ' v d ' ) ) ) =
|
|
( ( ( a ^ c ' ) ^ ( b ^ d ' ) ) v
|
|
( ( c ^ b ' ) ^ ( d ^ a ' ) ) ) $=
|
|
( wo wn wa comcom3 mhlem1 ax-a2 ax-r2 2an leao2 leao3 ler2an oran2 lel2or
|
|
lan anor3 lbtr mhlem ) ACICJZBJZIKZBDIZAJZDJZIZKZKAUFKZCUGKZIZBUKKZDUJKZI
|
|
ZKUNUQKUOURKIUHUPUMUSACUGEBCGLMUMUIUKUJIZKUSULUTUIUJUKNUBBDUJHADFLMOPUNUQ
|
|
UOURUNUQIUFBIZUKAIZKZUOURIJZUNVCUQUNVAVBUFABQAUFUKRSUQVAVBBUKUFRUKBAQSUAV
|
|
CUOJZURJZKVDVAVEVBVFCBTDATPUOURUCOUDUEO $.
|
|
$( [10-Mar-02] $)
|
|
|
|
$( Marsden-Herman distributive law. Lemma 7.2 of Kalmbach, p. 91. $)
|
|
mh $p |- ( ( a v c ) ^ ( b v d ) )
|
|
= ( ( ( a ^ b ) v ( a ^ d ) ) v ( ( c ^ b ) v ( c ^ d ) ) ) $=
|
|
( wa wo leao1 leao2 ler2an leao4 lel2or wn ax-r1 ax-r2 lea ax-a3 leao3 wf
|
|
anass an4 mhlem2 le2an leo letr leor bltr leran anor3 ax-a2 or12 3tr 3tr1
|
|
lor ax-r4 oran3 2an ran lan dff le3tr1 le0 lebi oml3 ) ABIZADIZJZCBIZCDIZ
|
|
JZJZACJZBDJZIZVNVQVJVQVMVHVQVIVHVOVPABCKBADLMVIVOVPADCKDABNMOVKVQVLVKVOVP
|
|
CBAUABCDLMVLVOVPCDAUADCBNMOOVQVNPZIZUBVQCPZBPZJZAPZDPZJZIZVHVLJZPZIZIZWGW
|
|
HIZVSUBWJVQWFIZWHIZWKWMWJVQWFWHUCQWLWGWHWLAVTIZBWDIZIZCWAIZDWCIZIZJZWGWLV
|
|
OWBIVPWEIIWTVOVPWBWEUDABCDEFGHUERWPWGWSWPVHWGWNAWOBAVTSBWDSUFVHVLUGUHWSVL
|
|
WGWQCWRDCWASDWCSUFVLVHUIUHOUJUKUJVRWIVQVKVIJZWGJZPZXAPZWHIZVRWIXEXCXAWGUL
|
|
QVNXBVNVMVJJZXBVJVMUMVKVLVJJZJVKVIWGJZJXFXBXGXHVKXGVHVLVIJJZWGVIJZXHVLVHV
|
|
IUNXJXIVHVLVITQWGVIUMUOUQVKVLVJTVKVIWGTUPRURWFXDWHWFVKPZVIPZIXDWBXKWEXLCB
|
|
USADUSUTVKVIULRVAUPVBWGVCVDVSVEVFVGQ $.
|
|
$( [10-Mar-02] $)
|
|
$}
|
|
|
|
${
|
|
marsden.1 $e |- a C b $.
|
|
marsden.2 $e |- b C c $.
|
|
marsden.3 $e |- c C d $.
|
|
marsden.4 $e |- d C a $.
|
|
$( Lemma for Marsden-Herman distributive law. $)
|
|
marsdenlem1 $p |- ( ( a v b ) ^ ( a ' v d ' ) )
|
|
= ( ( a ' ^ ( a v b ) ) v ( d ' ^ ( a v b ) ) ) $=
|
|
( wo wn wa ancom comorr comcom3 comcom4 comcom fh2r ax-r2 ) ABIZAJZDJZIZK
|
|
UBSKTSKUASKISUBLTSUAASABMNUATDAHOPQR $.
|
|
$( [26-Feb-02] $)
|
|
|
|
$( Lemma for Marsden-Herman distributive law. $)
|
|
marsdenlem2 $p |- ( ( c v d ) ^ ( b ' v c ' ) )
|
|
= ( ( ( b ' ^ c ) v ( c ' ^ d ) ) v ( b ' ^ d ) ) $=
|
|
( wo wn wa ancom comorr comcom3 comcom4 comcom fh2 wf ax-r2 3tr fh2rc dff
|
|
comcom6 comid comcom2 ax-r5 ax-r1 or0r 2or or32 ) CDIZBJZCJZIZKUNUKKULUKK
|
|
ZUMUKKZIZULCKZUMDKZIULDKZIZUKUNLUMUKULCUKCDMNULUMBCFOPZUAUQURUTIZUSIVAUOV
|
|
CUPUSCULDCULVBUCGQUPUMCKZUSIZRUSIZUSCUMDCCCUDUEGQVFVERVDUSRCUMKVDCUBCUMLS
|
|
UFUGUSUHTUIURUTUSUJST $.
|
|
$( [26-Feb-02] $)
|
|
|
|
$( Lemma for Marsden-Herman distributive law. $)
|
|
marsdenlem3 $p |- ( ( ( b ' ^ c ) v ( c ' ^ d ) ) ^ ( b ^ d ' ) ) = 0 $=
|
|
( wn wa wo wf lea lecom comcom7 comcom an4 dff ax-r1 3tr lecon lear oran2
|
|
lel lerr lbtr fh1r ancom ax-r2 ran an0r lan an0 2or or0 ) BIZCJZCIZDJZKBD
|
|
IZJZJUQVAJZUSVAJZKLLKLVAUQUSUQVAUQVAUQVAIZUPVDCVABBUTMUAUDNOPUSVAUSVAUSVD
|
|
USUPDKVDUSDUPURDUBUEBDUCUFNOPUGVBLVCLVBUPBJZCUTJZJLVFJLUPCBUTQVELVFVEBUPJ
|
|
ZLUPBUHLVGBRSUIUJVFUKTVCURBJZDUTJZJVHLJLURDBUTQVILVHLVIDRSULVHUMTUNLUOT
|
|
$.
|
|
$( [26-Feb-02] $)
|
|
|
|
$( Lemma for Marsden-Herman distributive law. $)
|
|
marsdenlem4 $p |- ( ( ( a ' ^ b ) v ( a ^ d ' ) ) ^ ( b ' ^ d ) ) = 0 $=
|
|
( wn wa wo wf lbtr lecom comcom7 ancom lan an4 dff 3tr leao3 fh1r an0 2or
|
|
oran1 leao4 oran2 ax-r1 ax-r2 or0 ) AIZBJZADIZJZKBIZDJZJULUPJZUNUPJZKLLKL
|
|
UPULUNUPULUPULIZUPAUOKUSUODAUAABUEMNOUPUNUPUNIZUPUKDKUTDUOUKUFADUGMNOUBUQ
|
|
LURLUQULDUOJZJUKDJZBUOJZJZLUPVAULUODPQUKBDUORVDVBLJZLVEVDLVCVBBSQUHVBUCUI
|
|
TURAUOJZUMDJZJVFLJLAUMUODRVGLVFVGDUMJZLUMDPLVHDSUHUIQVFUCTUDLUJT $.
|
|
$( [26-Feb-02] $)
|
|
|
|
$( Marsden-Herman distributive law. Corollary 3.3 of Beran, p. 259. $)
|
|
mh2 $p |- ( ( a v b ) ^ ( c v d ) )
|
|
= ( ( ( a ^ c ) v ( a ^ d ) ) v ( ( b ^ c ) v ( b ^ d ) ) ) $=
|
|
( comcom mh ) ACBDEDAHIBCFIGJ $.
|
|
$( [10-Mar-02] $)
|
|
$}
|
|
|
|
$( Lemma for OML proof of Mladen's conjecture, $)
|
|
mlaconjolem $p |- ( ( a == c ) v ( b == c ) ) =<
|
|
( ( c ^ ( a v b ) ) v ( c ' ^ ( a ' v b ' ) ) ) $=
|
|
( tb wo wa wi2 wi1 wn orbile df-i2 oran3 ran lor ax-r1 ax-r2 df-i1 2an 3tr
|
|
ancom comor1 comcom2 leao1 lecom comcom fh1 omlan df2le2 2or ax-a2 lbtr ) A
|
|
CDBCDEABFZCGZCABEZHZFZCUNFZCIZAIBIEZFZEZABCJUPCUSURFZEZURUQEZFVCURFZVCUQFZE
|
|
ZVAUMVCUOVDUMCULIZURFZEZVCULCKVCVJVBVICUSVHURABLMNOPCUNQRVCURUQVCCCVBUAUBUQ
|
|
VCUQVCCUNVBUCZUDUEUFVGUTUQEVAVEUTVFUQVECUTEZURFURVLFUTVCVLURVBUTCUSURTNMVLU
|
|
RTCUSUGSVFUQVCFUQVCUQTUQVCVKUHPUIUTUQUJPSUK $.
|
|
$( [10-Mar-02] $)
|
|
|
|
$( OML proof of Mladen's conjecture. $)
|
|
mlaconjo $p |- ( ( a == b ) ^ ( ( a == c ) v ( b == c ) ) ) =<
|
|
( a == c ) $=
|
|
( tb wo wa wn dfb le2an lea leao1 lbtr lecom comcom7 lor ax-r2 an12 lan dff
|
|
wf bile mlaconjolem le2or oran leor df-a oran1 lear oran3 ax-r1 an0 3tr or0
|
|
mh ax-r5 or0r 2or le3tr1 letr ) ABDZACDZBCDEZFABFZAGZBGZFZEZCABEZFZCGZVDVEE
|
|
ZFZEZFZVAUTVGVBVMUTVGABHUAABCUBIVCVIFZVFVLFZEZACFZVDVJFZEVNVAVOVRVPVSVCAVIC
|
|
ABJCVHJIVFVDVLVJVDVEJVJVKJIUCVNVOVCVLFZEZVFVIFZVPEZEVQVCVIVFVLVCVFVCVFGZVCV
|
|
HWDABBKABUDZLMNVCVLVCVLGZVCCVCEZWFVCCUEWGCVKGZEZWFVCWHCABUFOCVKUGZPLMNVIVFV
|
|
IWDVIVHWDCVHUHWELMNVIVLVIWFVIWIWFCVHWHKWJLMNUNWAVOWCVPWAVOTEVOVTTVOVTVJVCVK
|
|
FZFVJTFTVCVJVKQWKTVJWKVCVCGZFZTVKWLVCABUIRTWMVCSUJPRVJUKULOVOUMPWCTVPEVPWBT
|
|
VPWBCVFVHFZFCTFTVFCVHQWNTCWNVFWDFZTVHWDVFWERTWOVFSUJPRCUKULUOVPUPPUQPACHURU
|
|
S $.
|
|
$( [10-Mar-02] $)
|
|
|
|
$( Distributive law for identity. $)
|
|
distid $p |- ( ( a == b ) ^ ( ( a == c ) v ( b == c ) ) ) =
|
|
( ( ( a == b ) ^ ( a == c ) ) v ( ( a == b ) ^ ( b == c ) ) ) $=
|
|
( tb wo wa lea mlaconjo ler2an bicom ax-a2 2an bltr ler2or ledi lebi ) ABDZ
|
|
ACDZBCDZEZFZQRFZQSFZEUAUBUCUAQRQTGZABCHIUAQSUDUABADZSREZFSQUETUFABJRSKLBACH
|
|
MINQRSOP $.
|
|
$( [17-Mar-02] $)
|
|
|
|
$( Corollary of Marsden-Herman Lemma. $)
|
|
mhcor1 $p |- ( ( ( ( a ->1 b ) ^ ( b ->2 c ) ) ^
|
|
( c ->1 d ) ) ^ ( d ->2 a ) ) =
|
|
( ( ( a == b ) ^ ( b == c ) ) ^ ( c == d ) ) $=
|
|
( wa wn wo tb anass ancom ax-r2 lbtr lecom comcom7 wf ran lan 3tr ax-r1 2or
|
|
wi2 wi1 imp3 2an leao3 oran comcom leao2 mh2 an4 3tr1 dff an0r 3tr2 an0 or0
|
|
or0r ax-a2 bi4 ) BCUAZCDUBZEZABUBZEZDAUAZEZABEZCEDEZAFZBFZEZCFZEDFZEZGZVCUT
|
|
EVAEZVEEABHBCHECDHEVFVBVCVEEZEVJVLEZCDEZGZVMVIEZVGGZEZVOVBVCVEIVBVTVQWBBCDU
|
|
CVQVEVCEWBVCVEJDABUCKUDWCVRWAEZVRVGEZGZVSWAEZVSVGEZGZGVNVHGVOVRVSWAVGVSVRVS
|
|
VRVSVRFZVSBCGZWJCDBUEBCUFZLMNUGVSWAVSWAFZVSDAGZWMDCAUHDAUFZLMNVGWAVGWAVGWMV
|
|
GWNWMABDUEWOLMNUGVGVRVGWJVGWKWJBACUHWLLMNUIWFVNWIVHWFVNOGVNWDVNWEOVLVMEZVKE
|
|
ZVKWPEWDVNWPVKJWDVLVJEZWAEWPVJVIEZEWQVRWRWAVJVLJPVLVJVMVIUJWSVKWPVJVIJQRVKV
|
|
LVMIUKWEVGVREABVREZEZOVRVGJABVRIXAAOEOWTOABVJEZVLEZOVLEZWTOXDXCOXBVLBULPSBV
|
|
JVLIVLUMUNQAUOKRTVNUPKWIOVHGVHWGOWHVHWGCDWAEZECOEOCDWAIXEOCXEDVMEZVIEZOXGXE
|
|
DVMVIISOXGOOVIEZXGXHOVIUMSOXFVIDULPKSKQCUORWHVGVSEZVHVSVGJVHXIVGCDISKTVHUQK
|
|
TVNVHURRRVPVDVEVPVCVBEVDVCUTVAIVCVBJKPABCDUSUK $.
|
|
$( [26-Jun-03] $)
|
|
|
|
$( Equation (3.29) of "Equations, states, and lattices..." paper. This shows
|
|
that it holds in all OMLs, not just 4GO. $)
|
|
oago3.29 $p |- ( ( a ->1 b ) ^ ( ( b ->2 c ) ^ ( c ->1 a ) ) )
|
|
=< ( a == c ) $=
|
|
( wi1 wi2 wa tb anass i2id 2an ax-r1 an1 mhcor1 3tr2 lear bicom lbtr bltr
|
|
wt ) ABDZBCEZCADZFFZABGBCGFZCAGZFZACGZUCSFZTUAFUBFZAAEZFZUCUFUKUHUIUCUJSTUA
|
|
UBHAIJKUCLABCAMNUFUEUGUDUEOCAPQR $.
|
|
$( [22-Jun-03] $)
|
|
|
|
$( 4-variable extension of Equation (3.21) of "Equations, states, and
|
|
lattices..." paper. This shows that it holds in all OMLs, not just
|
|
4GO. $)
|
|
oago3.21x $p |- ( ( ( ( a ->5 b ) ^ ( b ->5 c ) ) ^
|
|
( c ->5 d ) ) ^ ( d ->5 a ) ) =
|
|
( ( ( a == b ) ^ ( b == c ) ) ^ ( c == d ) ) $=
|
|
( wi5 wa tb wi1 wi2 i5lei1 i5lei2 le2an mhcor1 lbtr eqtr4 u5lembi ax-r1 lea
|
|
leid bltr bicom ler2an letr lebi ) ABEZBCEZFZCDEZFZDAEZFZABGZBCGZFZCDGZFZUK
|
|
ABHZBCIZFZCDHZFZDAIZFUPUIVAUJVBUGUSUHUTUEUQUFURABJBCKLCDJLDAKLABCDMNUPUPDAG
|
|
ZFUKUPUPVCUPSUPADGVCABCDOADUANUBUPUIVCUJUNUGUOUHULUEUMUFULUEBAEZFZUEVEULABP
|
|
QUEVDRTUMUFCBEZFZUFVGUMBCPQUFVFRTLUOUHDCEZFZUHVIUOCDPQUHVHRTLVCUJADEZFZUJVK
|
|
VCDAPQUJVJRTLUCUD $.
|
|
$( [26-Jun-03] $)
|
|
|
|
${
|
|
cancel.1 $e |- ( ( d v ( a ->1 c ) ) ->1 c ) = ( ( d v ( b ->1 c ) ) ->1 c
|
|
) $.
|
|
$( Lemma for cancellation law eliminating ` ->1 ` consequent. $)
|
|
cancellem $p |- ( d v ( a ->1 c ) ) =< ( d v ( b ->1 c ) ) $=
|
|
( wi1 wo wn i1abs ax-r1 leo df-i1 ax-r2 lbtr lecon2 ran 3tr lel2or bltr
|
|
wa leor lear ler2an coman2 coman1 comcom2 fh2rc 3tr1 leao4 lerr lor ax-r4
|
|
id an12 anor1 lan anor3 ancom anass le3tr1 lea lel letr ) DACFGZVDCFZHZVD
|
|
CTZGZDBCFZGZVHVDVDCIJVFVJVGVJVEVJHZVKVJCTZGZVEVKVLKVEVMVEVJCFZVMEVJCLZMJN
|
|
OVGVNCTZVJVGVNCVGVDHZVGGZVNVGVQUAVRVEVNVEVRVDCLJEMNVDCUBUCVPVKCTZVLCTZGZV
|
|
JVMCTWAVPWAVLCVKVJCUDVLVJVJCUEUFUGVNVMCVOPWAUMUHVSVJVTDHZBCTZHZTZWCTZDBHZ
|
|
WCGZGZVSVJWFWHDWCWEWGUIUJVSWEBTZCTWFVKWJCVKWIHZBWETZWJVJWIVIWHDBCLUKZULWL
|
|
WKWLWBBWDTZTWBWHHZTWKBWBWDUNWNWOWBBWCUOUPDWHUQQJBWEURQPWEBCUSMWMUTVLVJCVJ
|
|
CVAVBRSVCRS $.
|
|
$( [21-Feb-02] $)
|
|
|
|
$( Cancellation law eliminating ` ->1 ` consequent. $)
|
|
cancel $p |- ( d v ( a ->1 c ) ) = ( d v ( b ->1 c ) ) $=
|
|
( wi1 wo cancellem ax-r1 lebi ) DACFGZDBCFGZABCDEHBACDKCFLCFEIHJ $.
|
|
$( [21-Feb-02] $)
|
|
$}
|
|
|
|
${
|
|
kb10iii.1 $e |- b ' =< ( a ->1 c ) $.
|
|
$( Exercise 10(iii) of Kalmbach p. 30 (in a rewritten form). $)
|
|
kb10iii $p |- c ' =< ( a ->1 b ) $=
|
|
( wi1 wn wo wa ud1lem0c omln u1lem9b lel2or bltr lelan ancom lbtr u1lemaa
|
|
womaon le3tr2 lear letr lecon2 ) ABEZCUCFAAFZBFZGZHZCABIUGACHZCAUDUGGZHZA
|
|
CEZAHZUGUHUJAUKHULUIUKAUIUFUKAUEJUDUKUEACKDLMNAUKOPAUERACQSACTUAMUB $.
|
|
$( [9-Jan-04] $)
|
|
$}
|
|
|
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${
|
|
e2ast2.1 $e |- a =< b ' $.
|
|
e2ast2.2 $e |- c =< d ' $.
|
|
e2ast2.3 $e |- a =< c ' $.
|
|
$( Show that the E*_2 derivative on p. 23 of Mayet, "Equations holding in
|
|
Hilbert lattices" IJTP 2006, holds in all OMLs. $)
|
|
e2ast2 $p |- ( ( a v b ) ^ ( c v d ) ) =< ( ( b v d ) v ( a v c ) ' ) $=
|
|
( wo wa wn leror lecon3 lecom comcom df-le2 ax-r2 ax-r1 lor ax-a3 ax-r5
|
|
le2an comcom2 fh4c lan anor3 leao4 com2or fh4 or32 lear 3tr2 df2le2 ax-a2
|
|
2an ancom 3tr 3tr1 lbtr ) ABHZCDHZICJZBHZAJZDHZIZBDHACHJZHZUSVBUTVDAVABGK
|
|
CVCDACGLKUABDVCHZVAIZHZBDVFHZHVEVGVIVKBVIDVCVAIZHZVKVMVIVMVHDVAHZIVIVADVC
|
|
DVADVACDFLZMNZVAAAVAAVAGMNUBZUCVNVAVHDVAVOOUDPQVLVFDACUERPRVEBVCIZVIHZVJV
|
|
SVEVSVRVHHZVRVAHZIVDVBIVEVHVRVAVRVHVRVHVCBDUFMNVAVHVADVCVPVQUGNUHVTVDWAVB
|
|
VRDHVCHVRVCHZDHVTVDVRDVCUIVRDVCSWBVCDVRVCBVCUJOTUKWABVAHVBVRBVABVCABELULZ
|
|
TBVAUMPUNVDVBUOUPQVRBVIWCTPBDVFSUQUR $.
|
|
$( [24-Jun-2006] $)
|
|
$}
|
|
|
|
${
|
|
e2ast.1 $e |- a =< b ' $.
|
|
e2ast.2 $e |- c =< d ' $.
|
|
e2ast.3 $e |- r =< a ' $.
|
|
e2ast.4 $e |- a =< c ' $.
|
|
e2ast.5 $e |- c =< r ' $.
|
|
$( Lemma towards a possible proof that E*_2 on p. 23 of Mayet, "Equations
|
|
holding in Hilbert lattices" IJTP 2006, holds in all OMLs. $)
|
|
e2astlem1 $p |- ( ( ( a v b ) ^ ( c v d ) ) ^ ( ( a v c ) v r ) ) =
|
|
( ( a v ( b ^ ( c v r ) ) ) ^ ( c v ( d ^ ( a v r ) ) ) ) $=
|
|
( wo wa ler lecom wn comcom7 fh2r df2le2 wf ax-r2 leo ax-a3 comcom com2or
|
|
anandir lan fh2 lecon3 ortha ax-r5 or0r 3tr 2or leor or32 fh2c lor or0
|
|
2an ) ABKZCDKZLACKZEKZLUTVCLZVAVCLZLABCEKZLZKZCDAEKZLZKZLUTVAVCUEVDVHVEVK
|
|
VDAVCLZBVCLZKVHAVCBAVCAVBEACUAMZNABABOFNPZQVLAVMVGAVCVNRVMBAVFKZLZSVGKZVG
|
|
VCVPBACEUBUFVQBALZVGKVRABVFVOACEACACOINPZEAEAEAOHNPUCUDUGVSSVGBAABFUHUIUJ
|
|
TVGUKULUMTVECVCLZDVCLZKVKCVCDCVCCVBECAUNMZNCDCDOGNPZQWACWBVJCVCWCRWBDVICK
|
|
ZLVJDCLZKZVJVCWEDACEUOUFCDVIWDCAEACVTUCCECEOJNPUDUPWGVJSKVJWFSVJDCCDGUHUI
|
|
UQVJURTULUMTUST $.
|
|
$( [25-Jun-2006] $)
|
|
|
|
$( Show that E*_2 on p. 23 of Mayet, "Equations holding in Hilbert
|
|
lattices" IJTP 2006, holds in all OMLs. $)
|
|
$(
|
|
e2ast $p |- ( ( ( a v b ) ^ ( c v d ) ) ^ ( ( a v c ) v r ) ) =<
|
|
( ( b v d ) v r ) $=
|
|
( wo wa wn ax-a3 comor1 bctr comcom3 comcom7 comorr2 comcom6 com2an mh2
|
|
ax-r2 lbtr wf anass ax-r1 anor3 ran ancom dff lan an0 an0r le0 bltr
|
|
lel2or letr ) ABKCDKLACKEKLZBEKZCKZMZDEBMZKZLZLZVBCEAKMZLZLZKZBVELZBVHLZK
|
|
ZKZBDKEKZUS?VN??VBBKVEVHKLVN?VBBVEVHVABVABECKZKBBECNBVPOPQBDVDBD?RBVDEVCS
|
|
TUA??UBUCUDVJVOVMVFVOVI?VIUEVOVIVBCLZVGLZUEVRVIVBCVGUFUGVRUEVGLUEVQUEVGVQ
|
|
UTMZCMZLZCLZUEWBVQWAVBCUTCUHUIUGWBVSVTCLZLZUEVSVTCUFWDVSUELUEWCUEVSWCCVTL
|
|
ZUEVTCUJUEWECUKUGUCULVSUMUCUCUCUIVGUNUCUCVOUOUPUQVKVOVL??UQUQUR $.
|
|
$)
|
|
$( [25-Jun-2006] $)
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
OML Lemmas for studying Godowski equations.
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
govar.1 $e |- a =< b ' $.
|
|
govar.2 $e |- b =< c ' $.
|
|
$( Lemma for converting n-variable Godowski equations to 2n-variable
|
|
equations. $)
|
|
govar $p |- ( ( a v b ) ^ ( a ->2 c ) ) =< ( b v c ) $=
|
|
( wo wi2 wa wn df-i2 lan ax-a2 ran lecom comcom7 comcom comcom2 lor 3tr
|
|
wf com2an com2or fh2r ax-r2 coman1 fh2c dff ax-r1 anass an0r 3tr2 or0 lea
|
|
coman2 lear le2or bltr ) ABFZACGZHZBCAIZCIZHZFZHZACHZFZBCFUTURVDHZVEAVDHZ
|
|
FZVGUSVDURACJKVHBAFZVDHVJURVKVDABLMBVDABCVCBCBVBENZOBVAVBBAABABABIDNOPZQV
|
|
LUAUBVMUCUDVIVFVEVIVFAVCHZFVFTFVFVCACVCAVAVBUEOVCCVAVBUNOUFVNTVFAVAHZVBHZ
|
|
TVBHZVNTVQVPTVOVBAUGMUHAVAVBUIVBUJUKRVFULSRSVEBVFCBVDUMACUOUPUQ $.
|
|
$( [19-Nov-99] $)
|
|
|
|
$( Lemma for converting n-variable to 2n-variable Godowski equations. $)
|
|
govar2 $p |- ( a v b ) =< ( c ->2 a ) $=
|
|
( wo wn wa wi2 lecon3 ler2an lelor df-i2 ax-r1 lbtr ) ABFACGZAGZHZFZCAIZB
|
|
RABPQEABDJKLTSCAMNO $.
|
|
$( [19-Nov-99] $)
|
|
|
|
${
|
|
gon2n.3 $e |- ( ( c ->2 a ) ^ d ) =< ( a ->2 c ) $.
|
|
gon2n.4 $e |- e =< d $.
|
|
$( Lemma for converting n-variable to 2n-variable Godowski equations. $)
|
|
gon2n $p |- ( ( a v b ) ^ e ) =< ( b v c ) $=
|
|
( wo wa wi2 lea govar2 le2an letr ler2an govar ) ABJZEKZSACLZKBCJTSUASE
|
|
MTCALZDKUASUBEDABCFGNIOHPQABCFGRP $.
|
|
$( [19-Nov-99] $)
|
|
$}
|
|
$}
|
|
|
|
${
|
|
go2n4.1 $e |- a =< b ' $.
|
|
go2n4.2 $e |- b =< c ' $.
|
|
go2n4.3 $e |- c =< d ' $.
|
|
go2n4.4 $e |- d =< e ' $.
|
|
go2n4.5 $e |- e =< f ' $.
|
|
go2n4.6 $e |- f =< g ' $.
|
|
go2n4.7 $e |- g =< h ' $.
|
|
go2n4.8 $e |- h =< a ' $.
|
|
${
|
|
go2n4.9 $e |- ( ( ( c ->2 a ) ^ ( a ->2 g ) ) ^
|
|
( ( g ->2 e ) ^ ( e ->2 c ) ) ) =< ( a ->2 c ) $.
|
|
$( 8-variable Godowski equation derived from 4-variable one. The last
|
|
hypothesis is the 4-variable Godowski equation. $)
|
|
go2n4 $p |- ( ( ( a v b ) ^ ( c v d ) ) ^
|
|
( ( e v f ) ^ ( g v h ) ) ) =< ( b v c ) $=
|
|
( wo wa wi2 anass ancom lan ax-r2 an32 ax-r1 bltr govar2 le2an gon2n )
|
|
ABRZCDRZSEFRZGHRZSZSZUKUOULSZSZBCRUPUKULUOSZSURUKULUOUAUSUQUKULUOUBUCUD
|
|
ABCGETZAGTZSZECTZSZUQIJCATZVDSZVEVASUTVCSZSZACTVHVFVHVEVAVGSZSVFVEVAVGU
|
|
AVIVDVEVIVGVASVDVAVGUBUTVCVAUEUDUCUDUFQUGUOVBULVCUMUTUNVAEFGMNUHGHAOPUH
|
|
UICDEKLUHUIUJUG $.
|
|
$( [19-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex4.9 $e |- ( ( ( a ->2 g ) ^ ( g ->2 e ) ) ^ ( (
|
|
e ->2 c ) ^ ( c ->2 a ) ) ) =< ( g ->2 a ) $.
|
|
gomaex4.10 $e |- ( ( ( e ->2 c ) ^ ( c ->2 a ) ) ^ ( (
|
|
a ->2 g ) ^ ( g ->2 e ) ) ) =< ( c ->2 e ) $.
|
|
$( Proof of Mayet Example 4 from 4-variable Godowski equation. R. Mayet,
|
|
"Equational bases for some varieties of orthomodular lattices related
|
|
to states," Algebra Universalis 23 (1986), 167-195. $)
|
|
gomaex4 $p |- ( ( ( ( a v b ) ^ ( c v d ) ) ^
|
|
( ( e v f ) ^ ( g v h ) ) ) ^ ( ( a v h ) ->1 ( d v e ) ' ) ) = 0 $=
|
|
( wo wa wn wi1 wf go2n4 an4 ancom ran ax-r2 3tr ax-a2 le3tr1 lan ler2an
|
|
2an bltr leran go1 lbtr le0 lebi ) ABSZCDSZTZEFSZGHSZTZTZAHSZDESZUAUBZT
|
|
ZUCVKVHVITZVJTUCVGVLVJVGVHVIVEVATZVBVDTZTZHASVGVHGHABCDEFOPIJKLMNQUDVGV
|
|
AVDTZVBVETZTZVEVBTZVPTZVOVAVBVDVEUEVRVQVPTVTVPVQUFVQVSVPVBVEUFUGUHVEVBV
|
|
AVDUEUIAHUJUKVGVNVMTZVIVGVAVETZVDVBTZTZVOWAVGVCVEVDTZTWBVNTWDVFWEVCVDVE
|
|
UFULVAVBVEVDUEVNWCWBVBVDUFULUIWBVMWCVNVAVEUFVDVBUFUNVMVNUFUICDEFGHABKLM
|
|
NOPIJRUDUOUMUPVHVIUQURVKUSUT $.
|
|
$( [19-Nov-99] $)
|
|
$}
|
|
$}
|
|
|
|
${
|
|
go2n6.1 $e |- g =< h ' $.
|
|
go2n6.2 $e |- h =< i ' $.
|
|
go2n6.3 $e |- i =< j ' $.
|
|
go2n6.4 $e |- j =< k ' $.
|
|
go2n6.5 $e |- k =< m ' $.
|
|
go2n6.6 $e |- m =< n ' $.
|
|
go2n6.7 $e |- n =< u ' $.
|
|
go2n6.8 $e |- u =< w ' $.
|
|
go2n6.9 $e |- w =< x ' $.
|
|
go2n6.10 $e |- x =< y ' $.
|
|
go2n6.11 $e |- y =< z ' $.
|
|
go2n6.12 $e |- z =< g ' $.
|
|
go2n6.13 $e |- ( ( ( i ->2 g ) ^ ( g ->2 y ) ) ^
|
|
( ( ( y ->2 w ) ^ ( w ->2 n ) ) ^
|
|
( ( n ->2 k ) ^ ( k ->2 i ) ) ) ) =< ( g ->2 i ) $.
|
|
$( 12-variable Godowski equation derived from 6-variable one. The last
|
|
hypothesis is the 6-variable Godowski equation. $)
|
|
go2n6 $p |- ( ( ( g v h ) ^ ( i v j ) ) ^
|
|
( ( ( k v m ) ^ ( n v u ) ) ^
|
|
( ( w v x ) ^ ( y v z ) ) ) ) =< ( h v i ) $=
|
|
( wo anass ancom lan 3tr ran ax-r2 ax-r1 3tr2 3tr1 wi2 govar2 le2an gon2n
|
|
wa bltr ) ABUFZECUFZUTDFUFZGHUFZUTZIJUFZKLUFZUTZUTZUTZVBVHVGVEUTZVDVCUTZU
|
|
TZUTZUTZBEUFVKVBVHUTVNUTZVPVBVCVJUTZUTVPVKVQVRVOVBVCVFUTZVIUTZVNVHUTZVRVO
|
|
WAVTWAVSVGUTZVHUTVTVNWBVHVNVGVEVMUTZUTVGVSUTWBVGVEVMUGWCVSVGWCVEVCVDUTZUT
|
|
WDVEUTVSVMWDVEVDVCUHUIVEWDUHVCVDVEUGUJUIVGVSUHUJUKVSVGVHUGULUMVCVFVIUGVNV
|
|
HUHUNUIVBVCVJUGVBVHVNUGZUOWEULABEAKUPZKIUPZIGUPZUTZGDUPZDEUPZUTZUTZUTZVOM
|
|
NEAUPZWNUTZWOWFUTWMUTZAEUPWQWPWOWFWMUGUMUEVAVHWFVNWMKLAUCUDUQVLWIVMWLVGWG
|
|
VEWHIJKUAUBUQGHISTUQURVDWJVCWKDFGQRUQECDOPUQURURURUSVA $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h1.1 $e |- a =< b ' $.
|
|
gomaex3h1.12 $e |- g = a $.
|
|
gomaex3h1.13 $e |- h = b $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h1 $p |- g =< h ' $=
|
|
( wn ax-r4 le3tr1 ) ABHCDHEFDBGIJ $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h2.2 $e |- b =< c ' $.
|
|
gomaex3h2.13 $e |- h = b $.
|
|
gomaex3h2.14 $e |- i = c $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h2 $p |- h =< i ' $=
|
|
( wn ax-r4 le3tr1 ) ABHCDHEFDBGIJ $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h3.14 $e |- i = c $.
|
|
gomaex3h3.15 $e |- j = ( c v d ) ' $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h3 $p |- i =< j ' $=
|
|
( wo wn leo ax-a1 lbtr ax-r4 le3tr1 ) AABGZHZHZDCHANPABINJKECOFLM $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h4.11 $e |- r = ( ( p ' ->1 q ) ' ^ ( c v d ) ) $.
|
|
gomaex3h4.15 $e |- j = ( c v d ) ' $.
|
|
gomaex3h4.16 $e |- k = r $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h4 $p |- j =< k ' $=
|
|
( wo wn wi1 wa lear bltr lecon ax-r4 le3tr1 ) ABKZLGLCDLGTGELFMLZTNTHUATO
|
|
PQIDGJRS $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h5.11 $e |- r = ( ( p ' ->1 q ) ' ^ ( c v d ) ) $.
|
|
gomaex3h5.16 $e |- k = r $.
|
|
gomaex3h5.17 $e |- m = ( p ' ->1 q ) $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h5 $p |- k =< m ' $=
|
|
( wn wi1 wo wa lea bltr ax-r4 le3tr1 ) GEKFLZKZCDKGTABMZNTHTUAOPIDSJQR $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h6.17 $e |- m = ( p ' ->1 q ) $.
|
|
gomaex3h6.18 $e |- n = ( p ' ->1 q ) ' $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h6 $p |- m =< n ' $=
|
|
( wn wi1 leid ax-a1 lbtr ax-r4 le3tr1 ) CGDHZNGZGZABGNNPNINJKEBOFLM $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h7.18 $e |- n = ( p ' ->1 q ) ' $.
|
|
gomaex3h7.19 $e |- u = ( p ' ^ q ) $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h7 $p |- n =< u ' $=
|
|
( wn wi1 wa wo leor df-i1 ax-r1 lbtr lecon ax-r4 le3tr1 ) BGZCHZGRCIZGADG
|
|
TSTRGZTJZSTUAKSUBRCLMNOEDTFPQ $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h8.19 $e |- u = ( p ' ^ q ) $.
|
|
gomaex3h8.20 $e |- w = q ' $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h8 $p |- u =< w ' $=
|
|
( wn wa lear ax-a1 lbtr ax-r4 le3tr1 ) AGZBHZBGZGZCDGOBQNBIBJKEDPFLM $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h9.20 $e |- w = q ' $.
|
|
gomaex3h9.21 $e |- x = q $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h9 $p |- w =< x ' $=
|
|
( wn leid ax-r4 le3tr1 ) AFZJBCFJGDCAEHI $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h10.10 $e |- q = ( ( e v f ) ->1 ( b v c ) ' ) ' $.
|
|
gomaex3h10.21 $e |- x = q $.
|
|
gomaex3h10.22 $e |- y = ( e v f ) ' $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h10 $p |- x =< y ' $=
|
|
( wo wn wa lea wi1 df-i1 ax-r4 ax-r1 ax-r2 le3tr1 anor1 ax-a1 ) ECDKZLZLZ
|
|
FGLUCUCABKLZMZLZMZUCEUEUCUHNEUCUFOZLZUIHUKUDUGKZLZUIUJULUCUFPQUIUMUCUGUAR
|
|
SSUCUEUCUBRTIGUDJQT $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h11.22 $e |- y = ( e v f ) ' $.
|
|
gomaex3h11.23 $e |- z = f $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h11 $p |- y =< z ' $=
|
|
( wo wn leor lecon ax-r4 le3tr1 ) ABGZHBHCDHBMBAIJEDBFKL $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3h12.6 $e |- f =< a ' $.
|
|
gomaex3h12.12 $e |- g = a $.
|
|
gomaex3h12.23 $e |- z = f $.
|
|
$( Hypothesis for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3h12 $p |- z =< g ' $=
|
|
( wn ax-r4 le3tr1 ) BAHDCHEGCAFIJ $.
|
|
$( [29-Nov-99] $)
|
|
$}
|
|
|
|
${
|
|
gomaex3lem1.3 $e |- c =< d ' $.
|
|
$( Lemma for Godowski 6-var -> Mayet Example 3. $)
|
|
gomaex3lem1 $p |- ( c v ( c v d ) ' ) = d ' $=
|
|
( wn wa wo comid comcom2 lecom fh3 anor3 lor wt ancom df-le2 df-t 2an an1
|
|
ax-r1 3tr 3tr2 ) AADZBDZEZFAUBFZAUCFZEZAABFDZFUCAUBUCAAAGHAUCCIJUDUHAABKL
|
|
UGUFUEEZUCMEZUCUEUFNUJUIUCUFMUEUFUCAUCCOSAPQSUCRTUA $.
|
|
$( [29-Nov-99] $)
|
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$}
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${
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gomaex3lem2.5 $e |- e =< f ' $.
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$( Lemma for Godowski 6-var -> Mayet Example 3. $)
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gomaex3lem2 $p |- ( ( e v f ) ' v f ) = e ' $=
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( wo wn wt lecon3 lecom comid comcom2 fh3r anor3 ax-r5 ax-r1 anabs df2le1
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wa leid lel2or ax-r2 lebi df-t ax-a2 2an 3tr1 an1 ) ABDEZBDZAEZFQZUIUIBEZ
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QZBDZUIBDZUKBDZQUHUJBUIUKBUIABCGZHBBBIJKUMUHULUGBABLMNUIUNFUOUIUNUIUNUIBO
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PUIUIBUIRUPSUAFBUKDUOBUBBUKUCTUDUEUIUFT $.
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$( [29-Nov-99] $)
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$}
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$( Lemma for Godowski 6-var -> Mayet Example 3. $)
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gomaex3lem3 $p |- ( ( p ' ->1 q ) ' v ( p ' ^ q ) ) = p ' $=
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( wn wi1 wa wo anor1 ax-r1 df-i1 ax-r4 3tr1 ax-r5 coman1 comid comcom2 fh3r
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id wt orabs ax-r2 ax-a2 df-t 2an an1 3tr ) ACZBDZCZUFBEZFUFUICZEZUIFUFUIFZU
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JUIFZEZUFUHUKUIUFCUIFZCZUKUHUKUKUPUFUIGHUGUOUFBIJUKQKLUIUFUJUFBMUIUIUINOPUN
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UFREUFULUFUMRUFBSUMUIUJFZRUJUIUARUQUIUBHTUCUFUDTUE $.
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$( [29-Nov-99] $)
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${
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gomaex3lem4.9 $e |- p = ( ( a v b ) ->1 ( d v e ) ' ) ' $.
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$( Lemma for Godowski 6-var -> Mayet Example 3. $)
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gomaex3lem4 $p |- ( ( a v b ) ^ ( d v e ) ' ) =< p ' $=
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( wo wn wa leor wi1 ax-a1 df-i1 ax-r1 ax-r4 3tr1 lbtr ) ABGZCDGHZIZRHZTGZ
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EHZTUAJRSKZUDHZHUBUCUDLUDUBRSMNEUEFOPQ $.
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$( [29-Nov-99] $)
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$}
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${
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gomaex3lem5.1 $e |- a =< b ' $.
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gomaex3lem5.2 $e |- b =< c ' $.
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gomaex3lem5.3 $e |- c =< d ' $.
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gomaex3lem5.5 $e |- e =< f ' $.
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gomaex3lem5.6 $e |- f =< a ' $.
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gomaex3lem5.8 $e |- ( ( ( i ->2 g ) ^ ( g ->2 y ) ) ^
|
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( ( ( y ->2 w ) ^ ( w ->2 n ) ) ^
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( ( n ->2 k ) ^ ( k ->2 i ) ) ) ) =< ( g ->2 i ) $.
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gomaex3lem5.9 $e |- p = ( ( a v b ) ->1 ( d v e ) ' ) ' $.
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gomaex3lem5.10 $e |- q = ( ( e v f ) ->1 ( b v c ) ' ) ' $.
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gomaex3lem5.11 $e |- r = ( ( p ' ->1 q ) ' ^ ( c v d ) ) $.
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gomaex3lem5.12 $e |- g = a $.
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gomaex3lem5.13 $e |- h = b $.
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gomaex3lem5.14 $e |- i = c $.
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gomaex3lem5.15 $e |- j = ( c v d ) ' $.
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gomaex3lem5.16 $e |- k = r $.
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gomaex3lem5.17 $e |- m = ( p ' ->1 q ) $.
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gomaex3lem5.18 $e |- n = ( p ' ->1 q ) ' $.
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gomaex3lem5.19 $e |- u = ( p ' ^ q ) $.
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gomaex3lem5.20 $e |- w = q ' $.
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gomaex3lem5.21 $e |- x = q $.
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gomaex3lem5.22 $e |- y = ( e v f ) ' $.
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gomaex3lem5.23 $e |- z = f $.
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$( Lemma for Godowski 6-var -> Mayet Example 3. $)
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gomaex3lem5 $p |- ( ( ( g v h ) ^ ( i v j ) ) ^
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( ( ( k v m ) ^ ( n v u ) ) ^
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( ( w v x ) ^ ( y v z ) ) ) ) =< ( h v i ) $=
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( gomaex3h1 gomaex3h2 gomaex3h3 gomaex3h4 gomaex3h5 gomaex3h10 gomaex3h11
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gomaex3h6 gomaex3h7 gomaex3h8 gomaex3h9 gomaex3h12 go2n6 ) GHIJKLMQRSTUAA
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BGHUBUKULVCBCHKUCULUMVDCDIKUMUNVECDIJNOPUJUNUOVFCDJLNOPUJUOUPVGLMNOUPUQVJ
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MNOQUQURVKNOQRURUSVLORSUSUTVMBCEFOSTUIUTVAVHEFTUAVAVBVIAFGUAUFUKVBVNUGVO
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$.
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$( [29-Nov-99] $)
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$( Lemma for Godowski 6-var -> Mayet Example 3. $)
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gomaex3lem6 $p |- ( ( ( a v b ) ^ ( c v ( c v d ) ' ) ) ^
|
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( ( ( r v ( p ' ->1 q ) ) ^ ( ( p ' ->1 q ) '
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v ( p ' ^ q ) ) ) ^
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( ( q ' v q ) ^ ( ( e v f ) ' v f ) ) ) ) =< ( b v c ) $=
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( wo wa wn wi1 gomaex3lem5 2or 2an le3tr2 ) GHVCZKIVCZVDZJLVCZMQVCZVDZRSV
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CZTUAVCZVDZVDZVDHKVCABVCZCCDVCVEZVCZVDZPNVEZOVFZVCZWFVEZWEOVDZVCZVDZOVEZO
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VCZEFVCVEZFVCZVDZVDZVDBCVCABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMU
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NUOUPUQURUSUTVAVBVGVMWDVTWQVKWAVLWCGAHBUKULVHKCIWBUMUNVHVIVPWKVSWPVNWGVOW
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JJPLWFUOUPVHMWHQWIUQURVHVIVQWMVRWORWLSOUSUTVHTWNUAFVAVBVHVIVIVIHBKCULUMVH
|
|
VJ $.
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$( [29-Nov-99] $)
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$( Lemma for Godowski 6-var -> Mayet Example 3. $)
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gomaex3lem7 $p |- ( ( ( a v b ) ^ d ' ) ^
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( ( ( r v ( p ' ->1 q ) ) ^ p ' ) ^ e ' ) ) =< ( b v c ) $=
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( wo wn wa wi1 gomaex3lem1 gomaex3lem3 ancom gomaex3lem2 ax-a2 df-t ax-r1
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lan wt ax-r2 2an an1 3tr gomaex3lem6 bltr ) ABVCZDVDZVEZPNVDZOVFZVCZWEVEZ
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EVDZVEZVEZWBCCDVCVDVCZVEZWGWFVDWEOVEVCZVEZOVDZOVCZEFVCVDFVCZVEZVEZVEZBCVC
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XAWKWMWDWTWJWLWCWBCDUDVGVNWOWHWSWIWNWEWGNOVHVNWSWRWQVEWIVOVEWIWQWRVIWRWIW
|
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QVOEFUEVJWQOWPVCZVOWPOVKVOXBOVLVMVPVQWIVRVSVQVQVMABCDEFGHIJKLMNOPQRSTUAUB
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UCUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBVTWA $.
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$( [29-Nov-99] $)
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$( Lemma for Godowski 6-var -> Mayet Example 3. $)
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gomaex3lem8 $p |- ( ( ( a v b ) ^ ( d v e ) ' ) ^
|
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( ( r v ( p ' ->1 q ) ) ^ p ' ) ) =< ( b v c ) $=
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( wo wn wa wi1 an32 anor3 lan ran an4 3tr2 gomaex3lem7 bltr ) ABVCZDEVCVD
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ZVEZPNVDZOVFVCVRVEZVEZVODVDZVEVSEVDZVEVEZBCVCVOWAWBVEZVEZVSVEVOVSVEWDVEVT
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WCVOWDVSVGWEVQVSWDVPVODEVHVIVJVOVSWAWBVKVLABCDEFGHIJKLMNOPQRSTUAUBUCUDUEU
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FUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBVMVN $.
|
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$( [29-Nov-99] $)
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|
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$( Lemma for Godowski 6-var -> Mayet Example 3. $)
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gomaex3lem9 $p |- ( ( ( a v b ) ^ ( d v e ) ' ) ^
|
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( r v ( p ' ->1 q ) ) ) =< ( b v c ) $=
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( wo wn wi1 ancom gomaex3lem4 df2le2 ax-r1 lan an12 3tr gomaex3lem8 bltr
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wa ) ABVCDEVCVDVOZPNVDZOVEVCZVOZVPVRVQVOVOZBCVCVSVRVPVOVRVPVQVOZVOVTVPVRV
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FVPWAVRWAVPVPVQABDENUHVGVHVIVJVRVPVQVKVLABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFU
|
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GUHUIUJUKULUMUNUOUPUQURUSUTVAVBVMVN $.
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$( [29-Nov-99] $)
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|
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$( Lemma for Godowski 6-var -> Mayet Example 3. $)
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gomaex3lem10 $p |- ( ( ( a v b ) ^ ( d v e ) ' ) ^
|
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( r v ( p ' ->1 q ) ) ) =< ( ( b v c ) v ( e v f ) ' ) $=
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( wo wn wa wi1 gomaex3lem9 leo letr ) ABVCDEVCVDVEPNVDOVFVCVEBCVCZVJEFVCV
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DZVCABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBVGVJV
|
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KVHVI $.
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$( [29-Nov-99] $)
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$}
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${
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gomaex3.1 $e |- a =< b ' $.
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gomaex3.2 $e |- b =< c ' $.
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gomaex3.3 $e |- c =< d ' $.
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gomaex3.5 $e |- e =< f ' $.
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gomaex3.6 $e |- f =< a ' $.
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gomaex3.8 $e |- ( ( ( i ->2 g ) ^ ( g ->2 y ) ) ^
|
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( ( ( y ->2 w ) ^ ( w ->2 n ) ) ^
|
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( ( n ->2 k ) ^ ( k ->2 i ) ) ) ) =< ( g ->2 i ) $.
|
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gomaex3.9 $e |- p = ( ( a v b ) ->1 ( d v e ) ' ) ' $.
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gomaex3.10 $e |- q = ( ( e v f ) ->1 ( b v c ) ' ) ' $.
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gomaex3.11 $e |- r = ( ( p ' ->1 q ) ' ^ ( c v d ) ) $.
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gomaex3.12 $e |- g = a $.
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gomaex3.14 $e |- i = c $.
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gomaex3.16 $e |- k = r $.
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gomaex3.18 $e |- n = ( p ' ->1 q ) ' $.
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gomaex3.20 $e |- w = q ' $.
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gomaex3.22 $e |- y = ( e v f ) ' $.
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$( Proof of Mayet Example 3 from 6-variable Godowski equation. R. Mayet,
|
|
"Equational bases for some varieties of orthomodular lattices related to
|
|
states," Algebra Universalis 23 (1986), 167-195. $)
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gomaex3 $p |- ( ( ( a v b ) ^ ( d v e ) ' ) ^
|
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( ( ( ( a v b ) ->1 ( d v e ) ' ) ->1
|
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( ( e v f ) ->1 ( b v c ) ' ) ' ) ' ->1 ( c v d ) ) )
|
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=< ( ( b v c ) v ( e v f ) ' ) $=
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( wo wn wa wi1 df-i1 ax-a2 con2 ud1lem0ab ax-a1 ax-r2 ax-r4 ran 2or ax-r1
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lan id gomaex3lem10 bltr ) ABUKZDEUKULZUMZVIVJUNZEFUKZBCUKZULUNULZUNZULZC
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DUKZUNZUMVKMKULZLUNZUKZUMVNVMULUKVSWBVKVSVQULZVQVRUMZUKZWBVQVRUOWBWEWBWAM
|
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UKWEMWAUPWAWCMWDWAVPWCVTVLLVOKVLUBUQUCURZVPUSUTMWAULZVRUMWDUDWGVQVRWAVPWF
|
|
VAVBUTVCUTVDUTVEABCDEFGBVRULZHIWAJKLMVTLUMZNLOFPQRSTUAUBUCUDUEBVFUFWHVFUG
|
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WAVFUHWIVFUILVFUJFVFVGVH $.
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$( [27-May-00] $)
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$}
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$(
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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|
OML Lemmas for studying orthoarguesian laws
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=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
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$)
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${
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oas.1 $e |- ( a ' ^ ( a v b ) ) =< c $.
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$( "Strengthening" lemma for studying the orthoarguesian law. $)
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oas $p |- ( ( a ->1 c ) ^ ( a v b ) ) =< c $=
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( wi1 wo wa oml ax-r1 lea ler2an lelor bltr lelan u1lemc1 lbtr letr ax-r2
|
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wn lear comanr1 comcom6 fh2 u1lemaa ancom leo df-i1 df2le2 2or lel2or ) A
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CEZABFZGZACGZASZCGZFZCUMUKAUPFZGZUQULURUKULAUOULGZFZURVAULABHIUTUPAUTUOCU
|
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OULJDKLMNUSUKAGZUKUPGZFUQAUKUPACOAUPUOCUAUBUCVBUNVCUPACUDVCUPUKGUPUKUPUEU
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PUKUPUOUKUOCJUOUOUNFZUKUOUNUFUKVDACUGIPQUHRUIRPUNCUPACTUOCTUJQ $.
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$( [25-Dec-98] $)
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$}
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${
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oasr.1 $e |- ( ( a ->1 c ) ^ ( a v b ) ) =< c $.
|
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$( Reverse of ~ oas lemma for studying the orthoarguesian law. $)
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oasr $p |- ( a ' ^ ( a v b ) ) =< c $=
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( wn wo wa wi1 u1lem9b leran letr ) AEZABFZGACHZMGCLNMACIJDK $.
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$( [28-Dec-98] $)
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$}
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${
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oat.1 $e |- ( a ' ^ ( a v b ) ) =< c $.
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$( Transformation lemma for studying the orthoarguesian law. $)
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oat $p |- b =< ( a ' ->1 c ) $=
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( wn wa wo wi1 leor oml ax-r1 lea lelor bltr letr ax-a1 ax-r5 df-i1 ax-r2
|
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ler2an lbtr ) BAAEZCFZGZUBCHZBABGZUDBAIUFAUBUFFZGZUDUHUFABJKUGUCAUGUBCUBU
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FLDTMNOUDUBEZUCGZUEAUIUCAPQUEUJUBCRKSUA $.
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$( [26-Dec-98] $)
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$}
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${
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oatr.1 $e |- b =< ( a ' ->1 c ) $.
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$( Reverse transformation lemma for studying the orthoarguesian law. $)
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oatr $p |- ( a ' ^ ( a v b ) ) =< c $=
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( wn wo wa leo df-i1 ax-a1 ax-r5 ax-r1 ax-r2 lbtr lel2or lelan omlan lear
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wi1 letr ) AEZABFZGZUACGZCUCUAAUDFZGUDUBUEUAAUEBAUDHBUACSZUEDUFUAEZUDFZUE
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UACIUEUHAUGUDAJKLMNOPACQNUACRT $.
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$( [26-Dec-98] $)
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$}
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${
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oau.1 $e |- ( a ^ ( ( a ->1 c ) v b ) ) =< c $.
|
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$( Transformation lemma for studying the orthoarguesian law. $)
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oau $p |- b =< ( a ->1 c ) $=
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( wi1 wo ax-a2 wa lea ler2an u1lemaa ax-r1 lelor wt u1lemc1 comcom comorr
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lbtr fh3 ax-r2 u1lemoa ax-a3 oridm ax-r5 2an ancom an1 3tr orabs leo lebi
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le3tr2 df-le1 ) BACEZBUNFUNBFZUNBUNGUOUNUNAUOHZFZUNUNAHZFUOUNUPURUNUPACHZ
|
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URUPACAUOIDJURUSACKLRMUQUNAFZUNUOFZHNUOHZUOUNAUOAUNACOPUNBQSUTNVAUOACUAVA
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UNUNFZBFZUOVDVAUNUNBUBLVCUNBUNUCUDTUEVBUONHUONUOUFUOUGTUHUNAUIULUNBUJUKTU
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M $.
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$( [28-Dec-98] $)
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$}
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${
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oaur.1 $e |- b =< ( a ->1 c ) $.
|
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$( Transformation lemma for studying the orthoarguesian law. $)
|
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oaur $p |- ( a ^ ( ( a ->1 c ) v b ) ) =< c $=
|
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( wi1 wo wa leid lel2or lelan ancom u1lemaa ax-r2 lbtr lear letr ) AACEZB
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FZGZACGZCSAQGZTRQAQQBQHDIJUAQAGTAQKACLMNACOP $.
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$( [28-Dec-98] $)
|
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$}
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${
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oaidlem2.1 $e |- ( ( d v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) ' v
|
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( ( a ->1 c ) ->1 ( b ->1 c ) ) ) = 1 $.
|
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$( Lemma for identity-like OA law. $)
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oaidlem2 $p |- ( ( a ->1 c ) ^
|
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( d v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) )
|
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=< ( b ->1 c ) $=
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( wi1 wa wo anidm ax-r1 ran anass ax-r2 leor lelan bltr df-le2 wn ax-a3
|
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wt ax-a2 oran3 ax-r5 df-i1 lor 3tr2 lem3.1 bile lear letr ) ACFZDUKBCFZGZ
|
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HZGZUMULUOUMUMUOUMUOUMUOUMUKUMGZUOUMUKUKGZULGUPUKUQULUQUKUKIJKUKUKULLMUMU
|
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NUKUMDNOPQUNRZUKRZHZUMHURUSUMHZHZUORZUMHTURUSUMSUTVCUMUTUSURHVCURUSUAUKUN
|
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UBMUCVBURUKULFZHZTVEVBVDVAURUKULUDUEJEMUFUGJUHUKULUIUJ $.
|
|
$( [22-Jan-99] $)
|
|
$}
|
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${
|
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oaidlem2g.1 $e |- ( ( c v ( a ^ b ) ) ' v
|
|
( a ->1 b ) ) = 1 $.
|
|
$( Lemma for identity-like OA law (generalized). $)
|
|
oaidlem2g $p |- ( a ^
|
|
( c v ( a ^ b ) ) )
|
|
=< b $=
|
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( wa wo anidm ax-r1 ran anass ax-r2 leor lelan bltr df-le2 wn ax-a3 ax-a2
|
|
wt oran3 ax-r5 wi1 df-i1 lor 3tr2 lem3.1 bile lear letr ) ACABEZFZEZUJBUL
|
|
UJUJULUJULUJULUJAUJEZULUJAAEZBEUMAUNBUNAAGHIAABJKUJUKAUJCLMNOUKPZAPZFZUJF
|
|
UOUPUJFZFZULPZUJFSUOUPUJQUQUTUJUQUPUOFUTUOUPRAUKTKUAUSUOABUBZFZSVBUSVAURU
|
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OABUCUDHDKUEUFHUGABUHUI $.
|
|
$( [18-Feb-02] $)
|
|
$}
|
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|
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${
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oa6v4v.1 $e |- ( ( ( a v b ) ^ ( c v d ) ) ^ ( e v f ) ) =<
|
|
( b v ( a ^ ( c v ( ( ( a v c ) ^ ( b v d ) ) ^
|
|
( ( ( a v e ) ^ ( b v f ) ) v ( ( c v e ) ^ ( d v f ) ) ) ) ) ) ) $.
|
|
oa6v4v.2 $e |- e = 0 $.
|
|
oa6v4v.3 $e |- f = 1 $.
|
|
$( 6-variable OA to 4-variable OA. $)
|
|
oa6v4v $p |- ( ( a v b ) ^ ( c v d ) ) =< ( b v ( a ^ ( c v
|
|
( ( a v c ) ^ ( b v d ) ) ) ) ) $=
|
|
( wo wa wt wf 2or ax-r2 lan an1 lor or0 or1 or0r 2an an32 anidm le3tr2
|
|
ran ) ABJCDJKZEFJZKZBACACJZBDJZKZAEJZBFJZKZCEJZDFJZKZJZKZJZKZJUGBACULJZKZ
|
|
JGUIUGLKUGUHLUGUHMLJLEMFLHINLUAOPUGQOVBVDBVAVCAUTULCUTULUJKZULUSUJULUOAUR
|
|
CUOALKAUMAUNLUMAMJAEMAHRASOUNBLJLFLBIRBTOUBAQOURCLKCUPCUQLUPCMJCEMCHRCSOU
|
|
QDLJLFLDIRDTOUBCQONPVEUJUJKZUKKULUJUKUJUCVFUJUKUJUDUFOORPRUE $.
|
|
$( [29-Nov-98] $)
|
|
$}
|
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|
|
${
|
|
oa4v3v.1 $e |- d =< b ' $.
|
|
oa4v3v.2 $e |- e =< c ' $.
|
|
oa4v3v.3 $e |- ( ( d v b ) ^ ( e v c ) ) =< ( b v ( d ^ ( e v
|
|
( ( d v e ) ^ ( b v c ) ) ) ) ) $.
|
|
oa4v3v.4 $e |- d = ( a ->2 b ) ' $.
|
|
oa4v3v.5 $e |- e = ( a ->2 c ) ' $.
|
|
$( 4-variable OA to 3-variable OA (Godowski/Greechie Eq. IV). $)
|
|
oa4v3v $p |- ( b ' ^ ( ( a ->2 b ) v ( ( a ->2 c ) ^ ( ( b v c ) '
|
|
v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) ) =<
|
|
( ( b ' ^ ( a ->2 b ) ) v ( c ' ^ ( a ->2 c ) ) ) $=
|
|
( wn wi2 wa wo ax-a2 lor oran1 3tr 2an ax-r2 anor3 ancom 2or oran3 le3tr2
|
|
lan anor1 lecon1 ) BKZABLZMZCKACLZMZNZUIUJULBCNZKUJULMZNZMZNZMZDBNZECNZMZ
|
|
BDEDENZUOMZNZMZNZUNKZUTKZHVCUKKZUMKZMVIVAVKVBVLVABDNBUJKZNVKDBODVMBIPBUJQ
|
|
RVBCENCULKZNVLECOEVNCJPCULQRSUKUMUATVHBUSKZNVJVGVOBVGVMURKZMVODVMVFVPIVFV
|
|
NUQKZNVPEVNVEVQJVEUOVDMUOUPKZMVQVDUOUBVDVRUOVDVMVNNVRDVMEVNIJUCUJULUDTUFU
|
|
OUPUGRUCULUQUDTSUJURUATPBUSQTUEUH $.
|
|
$( [28-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oal42.1 $e |- ( b ' ^ ( ( a ->2 b ) v ( ( a ->2 c ) ^ ( ( b v c ) '
|
|
v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) ) =<
|
|
( ( b ' ^ ( a ->2 b ) ) v ( c ' ^ ( a ->2 c ) ) ) $.
|
|
$( Derivation of Godowski/Greechie Eq. II from Eq. IV. $)
|
|
oal42 $p |- ( b ' ^ ( ( a ->2 b ) v ( ( a ->2 c ) ^ ( ( b v c ) '
|
|
v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) ) =< a ' $=
|
|
( wn wi2 wo wa ancom u2lemanb ax-r2 2or lbtr lea lel2or letr ) BEZABFZACF
|
|
ZBCGERSHGHGHZAEZQHZUACEZHZGZUATQRHZUCSHZGUEDUFUBUGUDUFRQHUBQRIABJKUGSUCHU
|
|
DUCSIACJKLMUBUAUDUAQNUAUCNOP $.
|
|
$( [25-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oa23.1 $e |- ( c ' ^ ( ( a ->2 c ) v ( ( a ->2 b ) ^ ( ( c v b ) '
|
|
v ( ( a ->2 c ) ^ ( a ->2 b ) ) ) ) ) ) =< a ' $.
|
|
$( Derivation of OA from Godowski/Greechie Eq. II. $)
|
|
oa23 $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ' v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) )
|
|
=< ( a ->2 c ) $=
|
|
( wi2 wo wn wa ax-a2 ax-r4 ancom 2or lan ax-r5 wt ax-a3 ax-r1 oridm ax-r2
|
|
u2lemonb 2an an1 comorr u2lemc1 comcom comcom2 fh3 3tr1 lea ler2an le3tr1
|
|
u2lemanb lelor orabs lbtr bltr leo lebi 3tr df-le1 ) ABEZBCFZGZVAACEZHZFZ
|
|
HZVDVGVDFVACBFZGZVDVAHZFZHZVDFVDVLFZVDVGVLVDVFVKVAVCVIVEVJVBVHBCIJVAVDKLM
|
|
NVLVDIVMVDVMVDVMCGZHZFZVDVMOHZVDVMFZVDVNFZHZVMVPVTVQVRVMVSOVRVDVDFZVLFZVM
|
|
WBVRVDVDVLPQWAVDVLVDRNSACTUAQVQVMVMUBQVDVMVNVDVLUCVDCCVDACUDUEUFUGUHVPVDV
|
|
DVNHZFVDVOWCVDVNVMHZAGZVNHVOWCWDWEVNDVNVMUIUJVMVNKACULUKUMVDVNUNUOUPVDVLU
|
|
QURUSUT $.
|
|
$( [25-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oa4lem1.1 $e |- a =< b ' $.
|
|
oa4lem1.2 $e |- c =< d ' $.
|
|
$( Lemma for 3-var to 4-var OA. $)
|
|
oa4lem1 $p |- ( a v b ) =< ( ( a v c ) ' ->2 b ) $=
|
|
( wo wn wa wi2 leo ax-a1 lbtr ler2an lelor ax-a2 df-i2 le3tr1 ) BAGBACGZH
|
|
ZHZBHZIZGABGTBJAUCBAUAUBASUAACKSLMENOABPTBQR $.
|
|
$( [27-Nov-98] $)
|
|
|
|
$( Lemma for 3-var to 4-var OA. $)
|
|
oa4lem2 $p |- ( c v d ) =< ( ( a v c ) ' ->2 d ) $=
|
|
( wo wn wa wi2 leor ax-a1 lbtr ler2an lelor ax-a2 df-i2 le3tr1 ) DCGDACGZ
|
|
HZHZDHZIZGCDGTDJCUCDCUAUBCSUACAKSLMFNOCDPTDQR $.
|
|
$( [27-Nov-98] $)
|
|
|
|
$( Lemma for 3-var to 4-var OA. $)
|
|
oa4lem3 $p |- ( ( a v b ) ^ ( c v d ) ) =< ( ( b v d ) ' v
|
|
( ( ( a v c ) ' ->2 b ) ^ ( ( a v c ) ' ->2 d ) ) ) $=
|
|
( wo wa wn wi2 oa4lem1 oa4lem2 le2an leor letr ) ABGZCDGZHACGIZBJZRDJZHZB
|
|
DGIZUAGPSQTABCDEFKABCDEFLMUAUBNO $.
|
|
$( [27-Nov-98] $)
|
|
$}
|
|
|
|
|
|
${
|
|
$( Substitutions into OA distributive law. $)
|
|
distoa.1 $e |- d = ( a ->2 b ) $.
|
|
distoa.2 $e |- e = ( ( b v c ) ->1 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
distoa.3 $e |- f = ( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
$( Satisfaction of distributive law hypothesis. $)
|
|
distoah1 $p |- d =< ( a ->2 b ) $=
|
|
( wi2 bile ) DABJGK $.
|
|
$( [29-Nov-98] $)
|
|
|
|
$( Satisfaction of distributive law hypothesis. $)
|
|
distoah2 $p |- e =< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $=
|
|
( wo wi2 wa wi1 wi0 leo ax-r1 u12lem le3tr2 ) BCJZABKACKLZMZUASTKZJESTNUA
|
|
UBOEUAHPSTQR $.
|
|
$( [29-Nov-98] $)
|
|
|
|
$( Satisfaction of distributive law hypothesis. $)
|
|
distoah3 $p |- f =< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $=
|
|
( wo wi2 wa wi1 wi0 leor ax-r1 u12lem le3tr2 ) BCJZABKACKLZKZSTMZUAJFSTNU
|
|
AUBOFUAIPSTQR $.
|
|
$( [29-Nov-98] $)
|
|
|
|
$( Satisfaction of distributive law hypothesis. $)
|
|
distoah4 $p |- ( d ^ ( a ->2 c ) ) =< f $=
|
|
( wi2 wa wo wn leo ran df-i2 ax-r2 le3tr1 ) ABJZACJZKZUABCLZMUAMKZLZDTKFU
|
|
AUCNDSTGOFUBUAJUDIUBUAPQR $.
|
|
$( [29-Nov-98] $)
|
|
|
|
${
|
|
|
|
$( OA distributive law as hypothesis. $)
|
|
distoa.4 $e |- ( d ^ ( e v f ) ) = ( ( d ^ e ) v ( d ^ f ) ) $.
|
|
$( Derivation in OM of OA, assuming OA distributive law ~ oadistd . $)
|
|
distoa $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ' v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) )
|
|
=< ( a ->2 c ) $=
|
|
( wi2 wo wa wi1 wn 1oa 2oath1 2or 2an ax-r2 lear bltr le2or 3tr2 u12lem
|
|
ax-r1 wi0 df-i0 lan oridm le3tr2 ) ABKZBCLZULACKZMZNZMZULUMUOKZMZLZUNUN
|
|
LULUMOUOLZMZUNUQUNUSUNABCPUSUOUNABCQULUNUAUBUCUTULUPURLZMZVBVDUTDEFLZMD
|
|
EMZDFMZLVDUTJDULVEVCGEUPFURHIRSVFUQVGUSDULEUPGHSDULFURGISRUDUFVCVAULVCU
|
|
MUOUGVAUMUOUEUMUOUHTUITUNUJUK $.
|
|
$( [29-Nov-98] $)
|
|
$}
|
|
$}
|
|
|
|
|
|
${
|
|
oa3to4lem.1 $e |- a ' =< b $.
|
|
oa3to4lem.2 $e |- c ' =< d $.
|
|
oa3to4lem.3 $e |- g = ( ( a ^ b ) v ( c ^ d ) ) $.
|
|
$( Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable
|
|
proof). $)
|
|
oa3to4lem1 $p |- b =< ( a ->1 g ) $=
|
|
( wn wa wo wi1 leor comid comcom3 wt ax-r2 ran ax-r1 lbtr lecom fh3 ancom
|
|
df-t ax-a2 an1 3tr2 anidm anass lor leo lelan lelor letr ud1lem0a df-i1 )
|
|
BAIZAABJZCDJZKZJZKZAELZBUQAURJZKZVBBUQBKZVEBUQMVFUQURKZVEVGVFVGUQAKZVFJZV
|
|
FUQABAAANOUQBFUAUBPVFJVFPJVIVFPVFUCPVHVFPAUQKVHAUDAUQUEQRVFUFUGQSURVDUQUR
|
|
AAJZBJZVDVKURVJABAUHRSAABUIQUJQTVDVAUQURUTAURUSUKULUMUNVCVBVCAUTLVBEUTAHU
|
|
OAUTUPQST $.
|
|
$( [19-Dec-98] $)
|
|
|
|
$( Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable
|
|
proof). $)
|
|
oa3to4lem2 $p |- d =< ( c ->1 g ) $=
|
|
( wn wa wo wi1 leor comid comcom3 wt ax-r2 ran ax-r1 lbtr lecom fh3 ancom
|
|
df-t ax-a2 an1 3tr2 anidm anass lor lelan lelor letr ud1lem0a df-i1 ) DCI
|
|
ZCABJZCDJZKZJZKZCELZDUPCURJZKZVADUPDKZVDDUPMVEUPURKZVDVFVEVFUPCKZVEJZVEUP
|
|
CDCCCNOUPDGUAUBPVEJVEPJVHVEPVEUCPVGVEPCUPKVGCUDCUPUEQRVEUFUGQSURVCUPURCCJ
|
|
ZDJZVCVJURVICDCUHRSCCDUIQUJQTVCUTUPURUSCURUQMUKULUMVBVAVBCUSLVAEUSCHUNCUS
|
|
UOQST $.
|
|
$( [19-Dec-98] $)
|
|
|
|
$( Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable
|
|
proof). $)
|
|
oa3to4lem3 $p |- ( a ^ ( b v ( d ^ ( ( a ^ c ) v ( b ^ d ) ) ) ) )
|
|
=< ( a ^ ( ( a ->1 g ) v ( ( c ->1 g ) ^ ( ( a ^ c ) v
|
|
( ( a ->1 g ) ^ ( c ->1 g ) ) ) ) ) ) $=
|
|
( wa wo wi1 oa3to4lem1 oa3to4lem2 le2an lelor le2or lelan ) BDACIZBDIZJZI
|
|
ZJAEKZCEKZRUBUCIZJZIZJABUBUAUFABCDEFGHLZDUCTUEABCDEFGHMZSUDRBUBDUCUGUHNON
|
|
PQ $.
|
|
$( [19-Dec-98] $)
|
|
|
|
${
|
|
$( Godowski/Greechie 3-variable OA as hypothesis $)
|
|
oa3to4lem.oa3 $e |- ( a ^ ( ( a ->1 g ) v ( ( c ->1 g ) ^ ( ( a ^ c ) v
|
|
( ( a ->1 g ) ^ ( c ->1 g ) ) ) ) ) )
|
|
=< ( ( a ^ g ) v ( c ^ g ) ) $.
|
|
$( Lemma for orthoarguesian law (Godowski/Greechie 3-variable to
|
|
4-variable proof). $)
|
|
oa3to4lem4 $p |- ( a ^ ( b v ( d ^ ( ( a ^ c ) v ( b ^ d ) ) ) ) )
|
|
=< g $=
|
|
( wa wo wi1 oa3to4lem3 lear lel2or letr ) ABDACJZBDJKJKJAAELZCELZQRSJKJ
|
|
KJZEABCDEFGHMTAEJZCEJZKEIUAEUBAENCENOPP $.
|
|
$( [19-Dec-98] $)
|
|
$}
|
|
$}
|
|
|
|
${
|
|
oa3to4lem5.1 $e |- ( ( a v b ) ^ ( c v d ) ) =< ( a v ( b ^ ( d v
|
|
( ( a v c ) ^ ( b v d ) ) ) ) ) $.
|
|
$( Lemma for orthoarguesian law (Godowski/Greechie 3-variable to 4-variable
|
|
proof). $)
|
|
oa3to4lem5 $p |- ( ( b v a ) ^ ( d v c ) ) =< ( a v ( b ^ ( d v
|
|
( ( b v d ) ^ ( a v c ) ) ) ) ) $=
|
|
( wo wa ax-a2 2an ancom lor lan le3tr1 ) ABFZCDFZGABDACFZBDFZGZFZGZFBAFZD
|
|
CFZGABDQPGZFZGZFEUANUBOBAHDCHIUETAUDSBUCRDQPJKLKM $.
|
|
$( [19-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa3to4lem6.oa4.1 $e |- a =< b ' $.
|
|
oa3to4lem6.oa4.2 $e |- c =< d ' $.
|
|
$( Variable substitutions to make into the 4-variable OA. $)
|
|
oa3to4lem6.3 $e |- g = ( ( a ' ^ b ' ) v ( c ' ^ d ' ) ) $.
|
|
oa3to4lem6.4 $e |- e = a ' $.
|
|
oa3to4lem6.5 $e |- f = c ' $.
|
|
$( Godowski/Greechie 3-variable OA as hypothesis $)
|
|
oa3to4lem6.oa3 $e |- ( e ^ ( ( e ->1 g ) v ( ( f ->1 g ) ^ ( ( e ^ f ) v
|
|
( ( e ->1 g ) ^ ( f ->1 g ) ) ) ) ) )
|
|
=< ( ( e ^ g ) v ( f ^ g ) ) $.
|
|
$( Orthoarguesian law (Godowski/Greechie 3-variable to 4-variable). The
|
|
first 2 hypotheses are those for 4-OA. The next 3 are variable
|
|
substitutions into 3-OA. The last is the 3-OA. The proof uses OM logic
|
|
only. $)
|
|
oa3to4lem6 $p |- ( ( a v b ) ^ ( c v d ) ) =< ( a v ( b ^ ( d v
|
|
( ( a v c ) ^ ( b v d ) ) ) ) ) $=
|
|
( wo wa wn 2an 2or anor3 ax-r2 lecon3 lecon id wi1 ud1lem0ab le3tr2 oran3
|
|
oa3to4lem4 lan lor lecon1 ) ABDACNZBDNZOZNZOZNZABNZCDNZOZAPZBPZDPZVACPZOZ
|
|
VBVCOZNZOZNZOZVAVBOZVDVCOZNZUQPZUTPZVAVBVDVCVMBVAABHUAUBDVDCDIUAUBVMUCEEG
|
|
UDZFGUDZEFOZVPVQOZNZOZNZOEGOZFGOZNVAVAVMUDZVDVMUDZVEWEWFOZNZOZNZOVAVMOZVD
|
|
VMOZNMEVAWBWJKVPWEWAWIEVAGVMKJUEZVQWFVTWHFVDGVMLJUEZVRVEVSWGEVAFVDKLQVPWE
|
|
VQWFWMWNQRQRQWCWKWDWLEVAGVMKJQFVDGVMLJQRUFUHVJVAUPPZOVNVIWOVAVIVBUOPZNWOV
|
|
HWPVBVHVCUNPZOWPVGWQVCVGULPZUMPZNWQVEWRVFWSACSBDSRULUMUGTUIDUNSTUJBUOUGTU
|
|
IAUPSTVMURPZUSPZNVOVKWTVLXAABSCDSRURUSUGTUFUK $.
|
|
$( [19-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa3to4.oa4.1 $e |- a =< b ' $.
|
|
oa3to4.oa4.2 $e |- c =< d ' $.
|
|
$( Variable substitutions to make into the 4-variable OA. $)
|
|
oa3to4.3 $e |- g = ( ( b ' ^ a ' ) v ( d ' ^ c ' ) ) $.
|
|
oa3to4.4 $e |- e = b ' $.
|
|
oa3to4.5 $e |- f = d ' $.
|
|
$( Godowski/Greechie 3-variable OA as hypothesis $)
|
|
oa3to4.oa3 $e |- ( e ^ ( ( e ->1 g ) v ( ( f ->1 g ) ^ ( ( e ^ f ) v
|
|
( ( e ->1 g ) ^ ( f ->1 g ) ) ) ) ) )
|
|
=< ( ( e ^ g ) v ( f ^ g ) ) $.
|
|
$( Orthoarguesian law (Godowski/Greechie 3-variable to 4-variable). The
|
|
first 2 hypotheses are those for 4-OA. The next 3 are variable
|
|
substitutions into 3-OA. The last is the 3-OA. The proof uses OM logic
|
|
only. $)
|
|
oa3to4 $p |- ( ( a v b ) ^ ( c v d ) ) =< ( b v ( a ^ ( c v
|
|
( ( a v c ) ^ ( b v d ) ) ) ) ) $=
|
|
( lecon3 oa3to4lem6 oa3to4lem5 ) BADCBADCEFGABHNCDINJKLMOP $.
|
|
$( [19-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa6todual.1 $e |- ( ( ( a ' v b ' ) ^ ( c ' v d ' ) ) ^ ( e ' v f ' ) )
|
|
=< ( b ' v ( a ' ^ ( c ' v ( ( ( a ' v c ' ) ^ ( b ' v d ' )
|
|
) ^ ( ( ( a ' v e ' ) ^ ( b ' v f ' ) ) v ( ( c ' v e ' ) ^
|
|
( d ' v f ' ) ) ) ) ) ) ) $.
|
|
$( Conventional to dual 6-variable OA law. $)
|
|
oa6todual $p |- ( b ^ ( a v ( c ^ ( ( ( a ^ c ) v ( b ^ d ) ) v
|
|
( ( ( a ^ e ) v ( b ^ f ) ) ^ ( ( c ^ e ) v ( d ^ f ) ) ) ) ) ) )
|
|
=< ( ( ( a ^ b ) v ( c ^ d ) ) v ( e ^ f ) ) $=
|
|
( wn wo wa lecon ax-a1 df-a 2or oran3 ax-r2 2an anor3 le3tr1 ) BHZAHZCHZU
|
|
AUBIZTDHZIZJZUAEHZIZTFHZIZJZUBUGIZUDUIIZJZIZJZIZJZIZHZUATIZUBUDIZJZUGUIIZ
|
|
JZHZBACACJZBDJZIZAEJZBFJZIZCEJZDFJZIZJZIZJZIZJZABJZCDJZIZEFJZIZVEUSGKVTTH
|
|
ZURHZJUTBWFVSWGBLVSUAHZUQHZIWGAWHVRWIALVRUBHZUPHZJWICWJVQWKCLVQUFHZUOHZIW
|
|
KVIWLVPWMVIUCHZUEHZIWLVGWNVHWOACMBDMNUCUEOPVPUKHZUNHZJWMVLWPVOWQVLUHHZUJH
|
|
ZIWPVJWRVKWSAEMBFMNUHUJOPVOULHZUMHZIWQVMWTVNXACEMDFMNULUMOPQUKUNRPNUFUOOP
|
|
QUBUPRPNUAUQOPQTURRPWEVCHZVDHZIVFWCXBWDXCWCVAHZVBHZIXBWAXDWBXEABMCDMNVAVB
|
|
OPEFMNVCVDOPS $.
|
|
$( [22-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa6fromdual.1 $e |- ( b ' ^ ( a ' v ( c ' ^ ( ( ( a ' ^ c ' ) v ( b '
|
|
^ d ' ) ) v ( ( ( a ' ^ e ' ) v ( b ' ^ f ' ) ) ^ (
|
|
( c ' ^ e ' ) v ( d ' ^ f ' ) ) ) ) ) ) )
|
|
=< ( ( ( a ' ^ b ' ) v ( c ' ^ d ' ) ) v ( e ' ^ f ' ) ) $.
|
|
$( Dual to conventional 6-variable OA law. $)
|
|
oa6fromdual $p |- ( ( ( a v b ) ^ ( c v d ) ) ^ ( e v f ) ) =<
|
|
( b v ( a ^ ( c v ( ( ( a v c ) ^ ( b v d ) ) ^
|
|
( ( ( a v e ) ^ ( b v f ) ) v ( ( c v e ) ^ ( d v f ) ) ) ) ) ) ) $=
|
|
( wn wa wo lecon oran 2an anor3 ax-r2 ax-a1 2or oran3 le3tr1 ) AHZBHZIZCH
|
|
ZDHZIZJZEHZFHZIZJZHZUATUCTUCIZUAUDIZJZTUGIZUAUHIZJZUCUGIZUDUHIZJZIZJZIZJZ
|
|
IZHZABJZCDJZIZEFJZIZBACACJZBDJZIZAEJZBFJZIZCEJZDFJZIZJZIZJZIZJZVEUJGKVKUF
|
|
HZUIHZIUKVIWFVJWGVIUBHZUEHZIWFVGWHVHWIABLCDLMUBUENOEFLMUFUINOWEUAHZVDHZJV
|
|
FBWJWDWKBPWDTHZVCHZIWKAWLWCWMAPWCUCHZVBHZJWMCWNWBWOCPWBUNHZVAHZIWOVNWPWAW
|
|
QVNULHZUMHZIWPVLWRVMWSACLBDLMULUMNOWAUQHZUTHZJWQVQWTVTXAVQUOHZUPHZIWTVOXB
|
|
VPXCAELBFLMUOUPNOVTURHZUSHZIXAVRXDVSXECELDFLMURUSNOQUQUTROMUNVANOQUCVBROM
|
|
TVCNOQUAVDROS $.
|
|
$( [22-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa6fromdualn.1 $e |- ( b ^ ( a v ( c ^ ( ( ( a ^ c ) v ( b ^ d ) ) v
|
|
( ( ( a ^ e ) v ( b ^ f ) ) ^ ( ( c ^ e ) v ( d ^ f ) ) ) ) ) ) )
|
|
=< ( ( ( a ^ b ) v ( c ^ d ) ) v ( e ^ f ) ) $.
|
|
$( Dual to conventional 6-variable OA law. $)
|
|
oa6fromdualn $p |- ( ( ( a ' v b ' ) ^ ( c ' v d ' ) ) ^ ( e ' v f ' ) )
|
|
=< ( b ' v ( a ' ^ ( c ' v ( ( ( a ' v c ' ) ^ ( b ' v d ' )
|
|
) ^ ( ( ( a ' v e ' ) ^ ( b ' v f ' ) ) v ( ( c ' v e ' ) ^
|
|
( d ' v f ' ) ) ) ) ) ) ) $=
|
|
( wn wa wo ax-a1 2an 2or le3tr2 oa6fromdual ) AHZBHZCHZDHZEHZFHZBACACIZBD
|
|
IZJZAEIZBFIZJZCEIZDFIZJZIZJZIZJZIABIZCDIZJZEFIZJQHZPHZRHZUTVAIZUSSHZIZJZU
|
|
TTHZIZUSUAHZIZJZVAVFIZVCVHIZJZIZJZIZJZIUTUSIZVAVCIZJZVFVHIZJGBUSUNVQBKZAU
|
|
TUMVPAKZCVAULVOCKZUDVEUKVNUBVBUCVDAUTCVAWCWDLBUSDVCWBDKZLMUGVJUJVMUEVGUFV
|
|
IAUTEVFWCEKZLBUSFVHWBFKZLMUHVKUIVLCVAEVFWDWFLDVCFVHWEWGLMLMLMLUQVTURWAUOV
|
|
RUPVSAUTBUSWCWBLCVADVCWDWELMEVFFVHWFWGLMNO $.
|
|
$( [24-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
$( Substitutions into 6-variable OA law. $)
|
|
oa6to4.1 $e |- b ' = ( a ->1 g ) ' $.
|
|
oa6to4.2 $e |- d ' = ( c ->1 g ) ' $.
|
|
oa6to4.3 $e |- f ' = ( e ->1 g ) ' $.
|
|
$( Satisfaction of 6-variable OA law hypothesis. $)
|
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oa6to4h1 $p |- a ' =< b ' ' $=
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( wn wa wo leo wi1 df-i1 ax-r4 ax-r2 ax-r1 con3 lbtr ) AKZUBAGLZMZBKZKUBU
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CNUDUEUEUDKZUEAGOZKUFHUGUDAGPQRSTUA $.
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$( [22-Dec-98] $)
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$( Satisfaction of 6-variable OA law hypothesis. $)
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oa6to4h2 $p |- c ' =< d ' ' $=
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( wn wa wo leo wi1 df-i1 ax-r4 ax-r2 ax-r1 con3 lbtr ) CKZUBCGLZMZDKZKUBU
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CNUDUEUEUDKZUECGOZKUFIUGUDCGPQRSTUA $.
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$( [22-Dec-98] $)
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$( Satisfaction of 6-variable OA law hypothesis. $)
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oa6to4h3 $p |- e ' =< f ' ' $=
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( wn wa wo leo wi1 df-i1 ax-r4 ax-r2 ax-r1 con3 lbtr ) EKZUBEGLZMZFKZKUBU
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CNUDUEUEUDKZUEEGOZKUFJUGUDEGPQRSTUA $.
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$( [22-Dec-98] $)
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${
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$( 6-variable OA law as hypothesis. $)
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oa6to4.oa6 $e |- ( ( ( a ' v b ' ) ^ ( c ' v d ' ) )
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^ ( e ' v f ' ) )
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=< ( b ' v ( a ' ^ ( c ' v ( ( ( a ' v c ' ) ^ ( b ' v d ' )
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) ^ ( ( ( a ' v e ' ) ^ ( b ' v f ' ) ) v ( ( c ' v e ' ) ^
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( d ' v f ' ) ) ) ) ) ) ) $.
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$( Derivation of 4-variable proper OA law, assuming 6-variable OA law. $)
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oa6to4 $p |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
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( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
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( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
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( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
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=< ( ( ( a ^ g ) v ( c ^ g ) ) v ( e ^ g ) ) $=
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( wa wo wi1 con1 2an lor 2or lan ancom oa6todual u1lemaa 3tr le3tr2 ) B
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ACACLZBDLZMZAELZBFLZMZCELZDFLZMZLZMZLZMZLABLZCDLZMZEFLZMAGNZACUEVBCGNZL
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ZMZUHVBEGNZLZMZUKVCVFLZMZLZMZLZMZLAGLZCGLZMZEGLZMABCDEFKUABVBUQVNBVBHOZ
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UPVMAUOVLCUGVEUNVKUFVDUEBVBDVCVSDVCIOZPQUJVHUMVJUIVGUHBVBFVFVSFVFJOZPQU
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LVIUKDVCFVFVTWAPQPRSQPUTVQVAVRURVOUSVPURAVBLVBALVOBVBAVSSAVBTAGUBUCUSCV
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CLVCCLVPDVCCVTSCVCTCGUBUCRVAEVFLVFELVRFVFEWASEVFTEGUBUCRUD $.
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$( [22-Dec-98] $)
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$}
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$}
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${
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oa4b.1 $e |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
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( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
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( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
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( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
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=< ( ( ( a ^ g ) v ( c ^ g ) ) v ( e ^ g ) ) $.
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$( Derivation of 4-OA law variant. $)
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oa4b $p |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
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( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
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( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
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( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
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=< g $=
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( wi1 wa wo lear lel2or letr ) ADFZABABGLBDFZGHACGLCDFZGHBCGMNGHGHGHGADGZ
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BDGZHZCDGZHDEQDRODPADIBDIJCDIJK $.
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$( [22-Dec-98] $)
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$}
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${
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oa4to6lem.1 $e |- a ' =< b $.
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oa4to6lem.2 $e |- c ' =< d $.
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oa4to6lem.3 $e |- e ' =< f $.
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oa4to6lem.4 $e |- g = ( ( ( a ^ b ) v ( c ^ d ) ) v ( e ^ f ) ) $.
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$( Lemma for orthoarguesian law (4-variable to 6-variable proof). $)
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oa4to6lem1 $p |- b =< ( a ->1 g ) $=
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( wn wa wo wi1 wt ax-r2 ran ax-r1 lbtr leor comid comcom3 lecom fh3 ancom
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df-t ax-a2 an1 3tr2 anidm anass lor ax-a3 lelan lelor letr ud1lem0a df-i1
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leo ) BALZAABMZCDMZNEFMZNZMZNZAGOZBVAAVBMZNZVGBVABNZVJBVAUAVKVAVBNZVJVLVK
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VLVAANZVKMZVKVAABAAAUBUCVABHUDUEPVKMVKPMVNVKPVKUFPVMVKPAVANVMAUGAVAUHQRVK
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UIUJQSVBVIVAVBAAMZBMZVIVPVBVOABAUKRSAABULQUMQTVIVFVAVBVEAVBVBVCVDNZNZVEVB
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VQUTVEVRVBVCVDUNSTUOUPUQVHVGVHAVEOVGGVEAKURAVEUSQST $.
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$( [18-Dec-98] $)
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$( Lemma for orthoarguesian law (4-variable to 6-variable proof). $)
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oa4to6lem2 $p |- d =< ( c ->1 g ) $=
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( wa wo wi1 leor wt ax-r2 ran ax-r1 lbtr wn comid comcom3 lecom fh3 ancom
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df-t ax-a2 an1 3tr2 anidm anass lor or32 lelan lelor letr ud1lem0a df-i1
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) DCUAZCABLZCDLZMEFLZMZLZMZCGNZDUTCVBLZMZVFDUTDMZVIDUTOVJUTVBMZVIVKVJVKUT
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CMZVJLZVJUTCDCCCUBUCUTDIUDUEPVJLVJPLVMVJPVJUFPVLVJPCUTMVLCUGCUTUHQRVJUIUJ
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QSVBVHUTVBCCLZDLZVHVOVBVNCDCUKRSCCDULQUMQTVHVEUTVBVDCVBVAVCMZVBMVDVBVPOVA
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VCVBUNTUOUPUQVGVFVGCVDNVFGVDCKURCVDUSQST $.
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$( [18-Dec-98] $)
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$( Lemma for orthoarguesian law (4-variable to 6-variable proof). $)
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oa4to6lem3 $p |- f =< ( e ->1 g ) $=
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( wa wo wi1 leor wt ax-r2 ran ax-r1 lbtr wn comid comcom3 lecom fh3 ancom
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df-t ax-a2 an1 3tr2 anidm anass lor lelan lelor letr ud1lem0a df-i1 ) FEU
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AZEABLCDLMZEFLZMZLZMZEGNZFUSEVALZMZVDFUSFMZVGFUSOVHUSVAMZVGVIVHVIUSEMZVHL
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ZVHUSEFEEEUBUCUSFJUDUEPVHLVHPLVKVHPVHUFPVJVHPEUSMVJEUGEUSUHQRVHUIUJQSVAVF
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USVAEELZFLZVFVMVAVLEFEUKRSEEFULQUMQTVFVCUSVAVBEVAUTOUNUOUPVEVDVEEVBNVDGVB
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EKUQEVBURQST $.
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$( [18-Dec-98] $)
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$( Lemma for orthoarguesian law (4-variable to 6-variable proof). $)
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oa4to6lem4 $p |- ( b ^ ( a v ( c ^ ( ( ( a ^ c ) v ( b ^ d ) ) v
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( ( ( a ^ e ) v ( b ^ f ) ) ^ ( ( c ^ e ) v ( d ^ f ) ) ) ) ) ) )
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=< ( ( a ->1 g ) ^ ( a v ( c ^ (
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( ( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
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( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
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( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) ) $=
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( wi1 wa wo oa4to6lem1 oa4to6lem2 le2an lelor oa4to6lem3 le2or lelan ) BA
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GLZACACMZBDMZNZAEMZBFMZNZCEMZDFMZNZMZNZMZNACUCUBCGLZMZNZUFUBEGLZMZNZUIUOU
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RMZNZMZNZMZNABCDEFGHIJKOZUNVEAUMVDCUEUQULVCUDUPUCBUBDUOVFABCDEFGHIJKPZQRU
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HUTUKVBUGUSUFBUBFURVFABCDEFGHIJKSZQRUJVAUIDUOFURVGVHQRQTUARQ $.
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$( [18-Dec-98] $)
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${
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$( Proper 4-variable OA as hypothesis $)
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oa4to6lem.oa4 $e |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
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( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
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( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
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( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
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=< g $.
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$( Lemma for orthoarguesian law (4-variable to 6-variable proof). $)
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oa4to6dual $p |- ( b ^ ( a v ( c ^ ( ( ( a ^ c ) v ( b ^ d ) ) v
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( ( ( a ^ e ) v ( b ^ f ) ) ^ ( ( c ^ e ) v ( d ^ f ) ) ) ) ) ) )
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=< g $=
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( wa wo wi1 oa4to6lem4 letr ) BACACMZBDMNAEMZBFMNCEMZDFMNMNMNMAGOZACRUA
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CGOZMNSUAEGOZMNTUBUCMNMNMNMGABCDEFGHIJKPLQ $.
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$( [19-Dec-98] $)
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$}
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$}
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${
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oa4to6.oa6.1 $e |- a =< b ' $.
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oa4to6.oa6.2 $e |- c =< d ' $.
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oa4to6.oa6.3 $e |- e =< f ' $.
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$( Variable substitutions to make into the 4-variable OA. $)
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oa4to6.4 $e |- g =
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( ( ( a ' ^ b ' ) v ( c ' ^ d ' ) ) v ( e ' ^ f ' ) ) $.
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oa4to6.5 $e |- h = a ' $.
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oa4to6.6 $e |- j = c ' $.
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oa4to6.7 $e |- k = e ' $.
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$( Proper 4-variable OA as hypothesis $)
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oa4to6.oa4 $e |- ( ( h ->1 g ) ^ ( h v ( j ^ ( (
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( h ^ j ) v ( ( h ->1 g ) ^ ( j ->1 g ) ) ) v
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( ( ( h ^ k ) v ( ( h ->1 g ) ^ ( k ->1 g ) ) ) ^
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( ( j ^ k ) v ( ( j ->1 g ) ^ ( k ->1 g ) ) ) ) ) ) ) )
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=< g $.
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$( Orthoarguesian law (4-variable to 6-variable proof). The first 3
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|
hypotheses are those for 6-OA. The next 4 are variable substitutions
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into 4-OA. The last is the 4-OA. The proof uses OM logic only. $)
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oa4to6 $p |- ( ( ( a v b ) ^ ( c v d ) ) ^ ( e v f ) ) =<
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( b v ( a ^ ( c v ( ( ( a v c ) ^ ( b v d ) ) ^
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( ( ( a v e ) ^ ( b v f ) ) v ( ( c v e ) ^ ( d v f ) ) ) ) ) ) ) $=
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( wa wo lecon3 lecon wi1 ud1lem0ab 2an 2or le3tr2 oa4to6dual oa6fromdual
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wn id ) ABCDEFAUJZBUJZCUJZDUJZEUJZFUJZULUMSUNUOSTUPUQSTZBULABKUAUBDUNCDLU
|
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AUBFUPEFMUAUBURUKHGUCZHIHISZUSIGUCZSZTZHJSZUSJGUCZSZTZIJSZVAVESZTZSZTZSZT
|
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ZSGULURUCZULUNULUNSZVOUNURUCZSZTZULUPSZVOUPURUCZSZTZUNUPSZVQWASZTZSZTZSZT
|
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ZSURRUSVOVNWJHULGURONUDZHULVMWIOIUNVLWHPVCVSVKWGUTVPVBVRHULIUNOPUEUSVOVAV
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QWKIUNGURPNUDZUEUFVGWCVJWFVDVTVFWBHULJUPOQUEUSVOVEWAWKJUPGURQNUDZUEUFVHWD
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VIWEIUNJUPPQUEVAVQVEWAWLWMUEUFUEUFUEUFUENUGUHUI $.
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$( [19-Dec-98] $)
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$}
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${
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oa4btoc.1 $e |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
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( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
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( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
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( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
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=< g $.
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$( Derivation of 4-OA law variant. $)
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oa4btoc $p |- ( a ' ^ ( a v ( c ^ ( (
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( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
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( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
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( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
|
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=< g $=
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( wn wa wi1 wo leo df-i1 ax-r1 lbtr leid lelor lelan le2an letr ) AFZABAB
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GADHZBDHZGIZACGTCDHZGIBCGUAUCGIGZIZGZIZGTUGGDSTUGUGSSADGZIZTSUHJTUIADKLMU
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FUFAUEUEBUDUDUBUDNOPOQER $.
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$( [22-Dec-98] $)
|
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$}
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${
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oa4ctob.1 $e |- ( a ' ^ ( a v ( c ^ ( (
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( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
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( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
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( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
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=< g $.
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$( Derivation of 4-OA law variant. $)
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oa4ctob $p |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
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( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
|
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( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
|
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( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
|
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=< g $=
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( wa wi1 wo oas ) ABABFADGZBDGZFHACFJCDGZFHBCFKLFHFHFDEI $.
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$( [22-Dec-98] $)
|
|
$}
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|
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${
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oa4ctod.1 $e |- ( a ' ^ ( a v ( b ^ ( (
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( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v
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( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
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( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) ) ) ) )
|
|
=< d $.
|
|
$( Derivation of 4-OA law variant. $)
|
|
oa4ctod $p |- ( b ^ ( (
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( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v
|
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( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
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( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) ) )
|
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=< ( a ' ->1 d ) $=
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( wa wi1 wo oat ) ABABFADGZBDGZFHACFJCDGZFHBCFKLFHFHFDEI $.
|
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$( [24-Dec-98] $)
|
|
$}
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${
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oa4dtoc.1 $e |- ( b ^ ( (
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( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v
|
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( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
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( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) ) )
|
|
=< ( a ' ->1 d ) $.
|
|
$( Derivation of 4-OA law variant. $)
|
|
oa4dtoc $p |- ( a ' ^ ( a v ( b ^ ( (
|
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( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v
|
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( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
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( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) ) ) ) )
|
|
=< d $=
|
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( wa wi1 wo oatr ) ABABFADGZBDGZFHACFJCDGZFHBCFKLFHFHFDEI $.
|
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$( [24-Dec-98] $)
|
|
$}
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$( Lemma commuting terms. $)
|
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oa4dcom $p |- ( b ^ ( (
|
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( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v
|
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( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) ) )
|
|
= ( b ^ ( (
|
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( b ^ a ) v ( ( b ->1 d ) ^ ( a ->1 d ) ) ) v
|
|
( ( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ) ) ) $=
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( wa wi1 wo ancom 2or lan ) ABEZADFZBDFZEZGZACELCDFZEGZBCEMPEGZEZGBAEZMLEZG
|
|
ZRQEZGBOUBSUCKTNUAABHLMHIQRHIJ $.
|
|
$( [24-Dec-98] $)
|
|
|
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|
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$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
5OA law
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
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${
|
|
oa8to5.1 $e |- ( ( ( a ' v b ' ) ^ ( c ' v d ' ) ) ^
|
|
( ( e ' v f ' ) ^ ( g ' v h ' ) ) ) =< ( b ' v ( a ' ^ ( c ' v
|
|
( ( ( ( a ' v c ' ) ^ ( b ' v d ' ) ) ^
|
|
( ( ( a ' v g ' ) ^ ( b ' v h ' ) ) v
|
|
( ( c ' v g ' ) ^ ( d ' v h ' ) ) ) )
|
|
^ ( ( ( ( a ' v e ' ) ^ ( b ' v f ' ) ) ^
|
|
( ( ( a ' v g ' ) ^ ( b ' v h ' ) ) v
|
|
( ( e ' v g ' ) ^ ( f ' v h ' ) ) ) )
|
|
v ( ( ( c ' v e ' ) ^ ( d ' v f ' ) ) ^
|
|
( ( ( c ' v g ' ) ^ ( d ' v h ' ) ) v
|
|
( ( e ' v g ' ) ^ ( f ' v h ' ) ) ) ) ) ) ) ) ) $.
|
|
$( Conventional to dual 8-variable OA law. $)
|
|
oa8todual $p |- ( b ^ ( a v ( c ^
|
|
( ( ( ( a ^ c ) v ( b ^ d ) ) v
|
|
( ( ( a ^ g ) v ( b ^ h ) ) ^
|
|
( ( c ^ g ) v ( d ^ h ) ) ) )
|
|
v ( ( ( ( a ^ e ) v ( b ^ f ) ) v
|
|
( ( ( a ^ g ) v ( b ^ h ) ) ^
|
|
( ( e ^ g ) v ( f ^ h ) ) ) )
|
|
^ ( ( ( c ^ e ) v ( d ^ f ) ) v
|
|
( ( ( c ^ g ) v ( d ^ h ) ) ^
|
|
( ( e ^ g ) v ( f ^ h ) ) ) ) ) ) ) ) )
|
|
=< ( ( ( a ^ b ) v ( c ^ d ) ) v
|
|
( ( e ^ f ) v ( g ^ h ) ) ) $=
|
|
( wn wo wa lecon ax-a1 df-a 2or oran3 ax-r2 2an anor3 le3tr1 ) BJZAJZCJZU
|
|
CUDKZUBDJZKZLZUCGJZKZUBHJZKZLZUDUIKZUFUKKZLZKZLZUCEJZKZUBFJZKZLZUMUSUIKZV
|
|
AUKKZLZKZLZUDUSKZUFVAKZLZUPVFKZLZKZLZKZLZKZJZUCUBKZUDUFKZLZUSVAKZUIUKKZLZ
|
|
LZJZBACACLZBDLZKZAGLZBHLZKZCGLZDHLZKZLZKZAELZBFLZKZWMEGLZFHLZKZLZKZCELZDF
|
|
LZKZWPXDLZKZLZKZLZKZLZABLZCDLZKZEFLZGHLZKZKZWFVRIMXPUBJZVQJZLVSBYDXOYEBNX
|
|
OUCJZVPJZKYEAYFXNYGANXNUDJZVOJZLYGCYHXMYICNXMURJZVNJZKYIWRYJXLYKWRUHJZUQJ
|
|
ZKYJWJYLWQYMWJUEJZUGJZKYLWHYNWIYOACOBDOPUEUGQRWQUMJZUPJZLYMWMYPWPYQWMUJJZ
|
|
ULJZKYPWKYRWLYSAGOBHOPUJULQRZWPUNJZUOJZKYQWNUUAWOUUBCGODHOPUNUOQRZSUMUPTR
|
|
PUHUQQRXLVHJZVMJZLYKXFUUDXKUUEXFVCJZVGJZKUUDXAUUFXEUUGXAUTJZVBJZKUUFWSUUH
|
|
WTUUIAEOBFOPUTVBQRXEYPVFJZLUUGWMYPXDUUJYTXDVDJZVEJZKUUJXBUUKXCUULEGOFHOPV
|
|
DVEQRZSUMVFTRPVCVGQRXKVKJZVLJZKUUEXIUUNXJUUOXIVIJZVJJZKUUNXGUUPXHUUQCEODF
|
|
OPVIVJQRXJYQUUJLUUOWPYQXDUUJUUCUUMSUPVFTRPVKVLQRSVHVMTRPURVNQRSUDVOTRPUCV
|
|
PQRSUBVQTRYCWBJZWEJZKWGXSUURYBUUSXSVTJZWAJZKUURXQUUTXRUVAABOCDOPVTWAQRYBW
|
|
CJZWDJZKUUSXTUVBYAUVCEFOGHOPWCWDQRPWBWEQRUA $.
|
|
$( [8-May-00] $)
|
|
|
|
${
|
|
$( Substitutions into 8-variable 5OA law. $)
|
|
oa8to5.2 $e |- b ' = ( a ->1 j ) ' $.
|
|
oa8to5.3 $e |- d ' = ( c ->1 j ) ' $.
|
|
oa8to5.4 $e |- f ' = ( e ->1 j ) ' $.
|
|
oa8to5.5 $e |- h ' = ( g ->1 j ) ' $.
|
|
$( Orthoarguesian law 5OA converted from 8 to 5 variables. $)
|
|
oa8to5 $p |- ( ( a ->1 j ) ^ ( a v ( c ^ (
|
|
( ( ( a ^ c ) v ( ( a ->1 j ) ^ ( c ->1 j ) ) ) v
|
|
( ( ( a ^ g ) v ( ( a ->1 j ) ^ ( g ->1 j ) ) )
|
|
^ ( ( c ^ g ) v ( ( c ->1 j ) ^ ( g ->1 j ) ) ) ) )
|
|
v (
|
|
( ( ( a ^ e ) v ( ( a ->1 j ) ^ ( e ->1 j ) ) ) v
|
|
( ( ( a ^ g ) v ( ( a ->1 j ) ^ ( g ->1 j ) ) )
|
|
^ ( ( e ^ g ) v ( ( e ->1 j ) ^ ( g ->1 j ) ) ) ) )
|
|
^
|
|
( ( ( c ^ e ) v ( ( c ->1 j ) ^ ( e ->1 j ) ) ) v
|
|
( ( ( c ^ g ) v ( ( c ->1 j ) ^ ( g ->1 j ) ) )
|
|
^ ( ( e ^ g ) v ( ( e ->1 j ) ^ ( g ->1 j ) ) ) ) ) ) ) )
|
|
) )
|
|
=< ( ( ( a ^ j ) v ( c ^ j ) ) v
|
|
( ( e ^ j ) v ( g ^ j ) ) ) $=
|
|
( wa wo 2an lor 2or lan wi1 oa8todual con1 ancom u1lemaa 3tr le3tr2 ) B
|
|
ACACOZBDOZPZAGOZBHOZPZCGOZDHOZPZOZPZAEOZBFOZPZUMEGOZFHOZPZOZPZCEOZDFOZP
|
|
ZUPVDOZPZOZPZOZPZOABOZCDOZPZEFOZGHOZPZPAIUAZACUHWBCIUAZOZPZUKWBGIUAZOZP
|
|
ZUNWCWFOZPZOZPZUSWBEIUAZOZPZWHVBWMWFOZPZOZPZVGWCWMOZPZWJWQOZPZOZPZOZPZO
|
|
AIOZCIOZPZEIOZGIOZPZPABCDEFGHJUBBWBVOXGBWBKUCZVNXFAVMXECURWLVLXDUJWEUQW
|
|
KUIWDUHBWBDWCXNDWCLUCZQRUMWHUPWJULWGUKBWBHWFXNHWFNUCZQRZUOWIUNDWCHWFXOX
|
|
PQRZQSVFWSVKXCVAWOVEWRUTWNUSBWBFWMXNFWMMUCZQRUMWHVDWQXQVCWPVBFWMHWFXSXP
|
|
QRZQSVIXAVJXBVHWTVGDWCFWMXOXSQRUPWJVDWQXRXTQSQSTRQVRXJWAXMVPXHVQXIVPAWB
|
|
OWBAOXHBWBAXNTAWBUDAIUEUFVQCWCOWCCOXIDWCCXOTCWCUDCIUEUFSVSXKVTXLVSEWMOW
|
|
MEOXKFWMEXSTEWMUDEIUEUFVTGWFOWFGOXLHWFGXPTGWFUDGIUEUFSSUG $.
|
|
$( [8-May-00] $)
|
|
$}
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
"Godowski/Greechie" form of proper 4-OA
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
oa4to4u.1 $e |- ( ( e ->1 d ) ^ ( e v ( f ^ ( (
|
|
( e ^ f ) v ( ( e ->1 d ) ^ ( f ->1 d ) ) ) v
|
|
( ( ( e ^ g ) v ( ( e ->1 d ) ^ ( g ->1 d ) ) ) ^
|
|
( ( f ^ g ) v ( ( f ->1 d ) ^ ( g ->1 d ) ) ) ) ) ) ) )
|
|
=< ( ( ( e ^ d ) v ( f ^ d ) ) v ( g ^ d ) ) $.
|
|
$( Substitutions into 4-variable OA law. $)
|
|
oa4to4u.2 $e |- e = ( a ' ->1 d ) $.
|
|
oa4to4u3 $e |- f = ( b ' ->1 d ) $.
|
|
oa4to4u.4 $e |- g = ( c ' ->1 d ) $.
|
|
$( A "universal" 4-OA. The hypotheses are the standard proper 4-OA and
|
|
substitutions into it. $)
|
|
oa4to4u $p |- ( ( a ->1 d ) ^ ( ( a ' ->1 d ) v ( ( b ' ->1 d ) ^ ( (
|
|
( ( a ->1 d ) ^ ( b ->1 d ) ) v ( ( a ' ->1 d ) ^ ( b ' ->1 d ) ) ) v
|
|
( ( ( ( a ->1 d ) ^ ( c ->1 d ) ) v ( ( a ' ->1 d ) ^ ( c ' ->1 d ) ) ) ^
|
|
( ( ( b ->1 d ) ^ ( c ->1 d ) ) v ( ( b ' ->1 d ) ^ ( c ' ->1 d ) ) ) )
|
|
) ) ) ) =< ( ( ( ( a ->1 d ) ^ ( a ' ->1 d ) ) v
|
|
( ( b ->1 d ) ^ ( b ' ->1 d ) ) ) v
|
|
( ( c ->1 d ) ^ ( c ' ->1 d ) ) ) $=
|
|
( wn wi1 wa wo 2an 2or ran ax-a2 ax-r2 ud1lem0b u1lem11 ax-r5 lan u1lemab
|
|
le3tr2 lor u1lem8 ax-a1 3tr ax-r1 ) ALZDMZDMZUMBLZDMZUMUPNZUNUPDMZNZOZUMC
|
|
LZDMZNZUNVBDMZNZOZUPVBNZURVDNZOZNZOZNZOZNZUMDNZUPDNZOZVBDNZOZADMZUMUPVTBD
|
|
MZNZUQOZVTCDMZNZVCOZWAWDNZVGOZNZOZNZOZNVTUMNZWAUPNZOZWDVBNZOEDMZEFEFNZWQF
|
|
DMZNZOZEGNZWQGDMZNZOZFGNZWSXCNZOZNZOZNZOZNEDNZFDNZOZGDNZOVNVSHWQUNXLVMEUM
|
|
DIUAZEUMXKVLIFUPXJVKJXAUTXIVJWRUQWTUSEUMFUPIJPWQUNWSURXQFUPDJUAZPQXEVFXHV
|
|
IXBVCXDVEEUMGVBIKPWQUNXCVDXQGVBDKUAZPQXFVGXGVHFUPGVBJKPWSURXCVDXRXSPQPQPQ
|
|
PXOVQXPVRXMVOXNVPEUMDIRFUPDJRQGVBDKRQUFUNVTVMWLADUBZVLWKUMVKWJUPUTWCVJWIU
|
|
TUSUQOWCUQUSSUSWBUQUNVTURWAXTBDUBZPUCTVFWFVIWHVFVEVCOWFVCVESVEWEVCUNVTVDW
|
|
DXTCDUBZPUCTVIVHVGOWHVGVHSVHWGVGURWAVDWDYAYBPUCTPQUDUGPVQWOVRWPVOWMVPWNVO
|
|
ULDNZULLZDNZOZWMULDUEWMYFWMADNZYCOYCYGOYFADUHYGYCSYGYEYCAYDDAUIRUGUJUKTVP
|
|
UODNZUOLZDNZOZWNUODUEWNYKWNBDNZYHOYHYLOYKBDUHYLYHSYLYJYHBYIDBUIRUGUJUKTQV
|
|
RVADNZVALZDNZOZWPVADUEWPYPWPCDNZYMOYMYQOYPCDUHYQYMSYQYOYMCYNDCUIRUGUJUKTQ
|
|
UF $.
|
|
$( [28-Dec-98] $)
|
|
|
|
$( A weaker-looking "universal" proper 4-OA. $)
|
|
oa4to4u2 $p |- ( ( a ->1 d ) ^ ( ( a ' ->1 d ) v ( ( b ' ->1 d ) ^ ( (
|
|
( ( a ->1 d ) ^ ( b ->1 d ) ) v ( ( a ' ->1 d ) ^ ( b ' ->1 d ) ) ) v
|
|
( ( ( ( a ->1 d ) ^ ( c ->1 d ) ) v ( ( a ' ->1 d ) ^ ( c ' ->1 d ) ) ) ^
|
|
( ( ( b ->1 d ) ^ ( c ->1 d ) ) v ( ( b ' ->1 d ) ^ ( c ' ->1 d ) ) ) )
|
|
) ) ) ) =< d $=
|
|
( wi1 wn wa wo oa4to4u u1lem8 lear lel2or bltr letr ) ADLZAMZDLZBMZDLZUBB
|
|
DLZNUDUFNOUBCDLZNUDCMZDLZNOUGUHNUFUJNONONONUBUDNZUGUFNZOZUHUJNZODABCDEFGH
|
|
IJKPUMDUNUKDULUKADNZUCDNZODADQUODUPADRUCDRSTULBDNZUEDNZODBDQUQDURBDRUEDRS
|
|
TSUNCDNZUIDNZODCDQUSDUTCDRUIDRSTSUA $.
|
|
$( [29-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa4uto4g.1 $e |- ( ( b ' ->1 d ) ^ ( ( b ' ' ->1 d ) v
|
|
( ( a ' ' ->1 d ) ^ ( (
|
|
( ( b ' ->1 d ) ^ ( a ' ->1 d ) ) v ( ( b ' ' ->1 d ) ^
|
|
( a ' ' ->1 d ) ) ) v
|
|
( ( ( ( b ' ->1 d ) ^ ( c ' ->1 d ) ) v ( ( b ' ' ->1 d ) ^
|
|
( c ' ' ->1 d ) ) ) ^
|
|
( ( ( a ' ->1 d ) ^ ( c ' ->1 d ) ) v ( ( a ' ' ->1 d ) ^
|
|
( c ' ' ->1 d ) ) ) ) ) ) ) ) =< d $.
|
|
|
|
$( Expression involving 4th variable. $)
|
|
oa4uto4g.4 $e |- h =
|
|
( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) $.
|
|
$( Derivation of "Godowski/Greechie" 4-variable proper OA law variant from
|
|
"universal" variant ~ oa4to4u2 . $)
|
|
oa4uto4g $p |- ( ( a ->1 d ) ^ ( (
|
|
( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v h ) )
|
|
=< ( b ->1 d ) $=
|
|
( wi1 wa wo ancom 2or lan lor wn u1lem9a lecon1 le2an leror 2an bltr letr
|
|
ax-r5 le2or lelan lelor ax-a1 ud1lem0b ax-r2 oau ) BADHZABIZUKBDHZIZJZEJZ
|
|
IZDBUMUQJZIBUMUKBAIZUMUKIZJZEJZIZJZIZDURVDBUQVCUMUPVBUKUOVAEULUSUNUTABKUK
|
|
UMKLUCMNMVEBOZDHZUMUKVGAOZDHZIZUTJZVICOZDHZIZUKCDHZIZJZVGVMIZUMVOIZJZIZJZ
|
|
IZJZIZDBVGVDWDVGBBDPQZVCWCUMVBWBUKVAVKEWAUSVJUTBVGAVIWFVIAADPQZRSEACIZVPJ
|
|
ZBCIZVSJZIWAGWIVQWKVTWHVNVPAVICVMWGVMCCDPQZRSWJVRVSBVGCVMWFWLRSRUAUDUEUFR
|
|
WEVGVFOZDHZVHOZDHZVJWNWPIZJZVRWNVLOZDHZIZJZVNWPWTIZJZIZJZIZJZIDWDXHVGUMWN
|
|
WCXGBWMDBUGUHZUKWPWBXFAWODAUGUHZVKWRWAXEUTWQVJUMWNUKWPXIXJTNWAVTVQIXEVQVT
|
|
KVTXBVQXDVSXAVRUMWNVOWTXICWSDCUGUHZTNVPXCVNUKWPVOWTXJXKTNTUILTLMFUAUBUAUJ
|
|
$.
|
|
$( [28-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa4gto4u.1 $e |- ( ( e ->1 d ) ^ ( (
|
|
( e ^ f ) v ( ( e ->1 d ) ^ ( f ->1 d ) ) ) v
|
|
( ( ( e ^ g ) v ( ( e ->1 d ) ^ ( g ->1 d ) ) ) ^
|
|
( ( f ^ g ) v ( ( f ->1 d ) ^ ( g ->1 d ) ) ) ) ) )
|
|
=< ( f ->1 d ) $.
|
|
$( Substitutions into 4-variable OA law. $)
|
|
oa4gto4u.2 $e |- f = ( a ->1 d ) $.
|
|
oa4gto4u3 $e |- e = ( b ->1 d ) $.
|
|
oa4gto4u.4 $e |- g = ( c ->1 d ) $.
|
|
$( A "universal" 4-OA derived from the Godowski/Greechie form. The
|
|
hypotheses are the Godowski/Greechie form of the proper 4-OA and
|
|
substitutions into it. $)
|
|
oa4gto4u $p |- ( ( a ->1 d ) ^ ( ( a ' ->1 d ) v ( ( b ' ->1 d ) ^ ( (
|
|
( ( a ->1 d ) ^ ( b ->1 d ) ) v ( ( a ' ->1 d ) ^ ( b ' ->1 d ) ) ) v
|
|
( ( ( ( a ->1 d ) ^ ( c ->1 d ) ) v ( ( a ' ->1 d ) ^ ( c ' ->1 d ) ) ) ^
|
|
( ( ( b ->1 d ) ^ ( c ->1 d ) ) v ( ( b ' ->1 d ) ^ ( c ' ->1 d ) ) ) )
|
|
) ) ) ) =< d $=
|
|
( wi1 wn wa wo ud1lem0b u1lem12 ax-r2 2an 2or ancom ax-r1 oaur bltr ) ADL
|
|
ZAMDLZBMDLZUEBDLZNZUFUGNZOZUECDLZNZUFCMDLZNZOZUHULNZUGUNNZOZNZOZNZOZNZFFD
|
|
LZEDLZEFNZVFVENZOZEGNZVFGDLZNZOZFGNZVEVKNZOZNZOZNZOZNZDWAVDFUEVTVCIVEUFVS
|
|
VBVEUEDLUFFUEDIPADQRZVFUGVRVAVFUHDLUGEUHDJPBDQRZVIUKVQUTVGUIVHUJVGFENUIEF
|
|
UAFUEEUHIJSRVHVEVFNUJVFVEUAVEUFVFUGWBWCSRTVQVPVMNUTVMVPUAVPUPVMUSVNUMVOUO
|
|
FUEGULIKSVEUFVKUNWBVKULDLUNGULDKPCDQRZSTVJUQVLUREUHGULJKSVFUGVKUNWCWDSTSR
|
|
TSTSUBFVSDHUCUD $.
|
|
$( [30-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa4uto4.1 $e |- ( ( a ->1 d ) ^ ( ( a ' ->1 d ) v ( ( b ' ->1 d ) ^ ( (
|
|
( ( a ->1 d ) ^ ( b ->1 d ) ) v ( ( a ' ->1 d ) ^ ( b ' ->1 d ) ) ) v
|
|
( ( ( ( a ->1 d ) ^ ( c ->1 d ) ) v ( ( a ' ->1 d ) ^ ( c ' ->1 d ) ) ) ^
|
|
( ( ( b ->1 d ) ^ ( c ->1 d ) ) v ( ( b ' ->1 d ) ^ ( c ' ->1 d ) ) ) )
|
|
) ) ) ) =< d $.
|
|
$( Derivation of standard 4-variable proper OA law from "universal" variant
|
|
~ oa4to4u2 . $)
|
|
oa4uto4 $p |- ( ( a ->1 d ) ^ ( a v ( b ^ ( (
|
|
( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v
|
|
( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) ) ) ) )
|
|
=< d $=
|
|
( wi1 wa wo wn u1lem9a lecon1 ax-a2 le2an lelor bltr le2or lelan letr ) A
|
|
DFZABABGZSBDFZGZHZACGZSCDFZGZHZBCGZUAUEGZHZGZHZGZHZGSAIDFZBIDFZUBUOUPGZHZ
|
|
UFUOCIDFZGZHZUIUPUSGZHZGZHZGZHZGDUNVGSAUOUMVFUOAADJKZBUPULVEUPBBDJKZUCURU
|
|
KVDUCUBTHURTUBLTUQUBAUOBUPVHVIMNOUGVAUJVCUGUFUDHVAUDUFLUDUTUFAUOCUSVHUSCC
|
|
DJKZMNOUJUIUHHVCUHUILUHVBUIBUPCUSVIVJMNOMPMPQER $.
|
|
$( [30-Dec-98] $)
|
|
$}
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Some 3-OA inferences (derived under OM)
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
|
|
$( Lemma for 3-OA(2). Equivalence with substitution into 4-OA. $)
|
|
oa3-2lema $p |- ( ( a ->1 c ) ^ ( a v ( b ^ ( (
|
|
( a ^ b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) v
|
|
( ( ( a ^ 0 ) v ( ( a ->1 c ) ^ ( 0 ->1 c ) ) ) ^
|
|
( ( b ^ 0 ) v ( ( b ->1 c ) ^ ( 0 ->1 c ) ) ) ) ) ) ) )
|
|
= ( ( a ->1 c ) ^ ( a v ( b ^ ( ( a ^ b ) v (
|
|
( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) $=
|
|
( wa wi1 wo wf ax-a3 an0 ax-r5 ax-a2 wt or0 0i1 lan an1 3tr 2an lor ax-r2
|
|
oridm ) ABABDZACEZBCEZDZFZAGDZUCGCEZDZFZBGDZUDUHDZFZDZFZDZFABUFDZFUCUPUQAUO
|
|
UFBUOUBUEUNFZFUFUBUEUNHURUEUBURUEUEFUEUNUEUEUJUCUMUDUJGUIFUIGFZUCUGGUIAIJGU
|
|
IKUSUIUCLDUCUIMUHLUCCNZOUCPQQUMGULFULGFZUDUKGULBIJGULKVAULUDLDUDULMUHLUDUTO
|
|
UDPQQRSUEUATSTOSO $.
|
|
$( [24-Dec-98] $)
|
|
|
|
$( Lemma for 3-OA(2). Equivalence with substitution into 4-OA. $)
|
|
oa3-2lemb $p |- ( ( a ->1 c ) ^ ( a v ( b ^ ( (
|
|
( a ^ b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) v
|
|
( ( ( a ^ c ) v ( ( a ->1 c ) ^ ( c ->1 c ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 c ) ^ ( c ->1 c ) ) ) ) ) ) ) )
|
|
= ( ( a ->1 c ) ^ ( a v ( b ^ ( ( a ^ b ) v (
|
|
( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) $=
|
|
( wa wi1 wo ax-a3 wt i1id lan an1 ax-r2 lor wn or12 oridm df-i1 3tr1 2an )
|
|
ABABDZACEZBCEZDZFZACDZUACCEZDZFZBCDZUBUFDZFZDZFZDZFABUDDZFUAUNUOAUMUDBUMTUC
|
|
ULFZFUDTUCULGUPUCTUPUCUCFUCULUCUCUHUAUKUBUHUEUAFZUAUGUAUEUGUAHDUAUFHUACIZJU
|
|
AKLMUEANZUEFZFZUTUQUAVAUSUEUEFZFUTUEUSUEOVBUEUSUEPMLUAUTUEACQZMVCRLUKUIUBFZ
|
|
UBUJUBUIUJUBHDUBUFHUBURJUBKLMUIBNZUIFZFZVFVDUBVGVEUIUIFZFVFUIVEUIOVHUIVEUIP
|
|
MLUBVFUIBCQZMVIRLSMUCPLMLJMJ $.
|
|
$( [24-Dec-98] $)
|
|
|
|
$( Lemma for 3-OA(6). Equivalence with substitution into 4-OA. $)
|
|
oa3-6lem $p |- ( ( a ->1 c ) ^ ( a v ( b ^ ( (
|
|
( a ^ b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) v
|
|
( ( ( a ^ 1 ) v ( ( a ->1 c ) ^ ( 1 ->1 c ) ) ) ^
|
|
( ( b ^ 1 ) v ( ( b ->1 c ) ^ ( 1 ->1 c ) ) ) ) ) ) ) )
|
|
= ( ( a ->1 c ) ^ ( a v ( b ^ ( (
|
|
( a ' ->1 c ) ^ ( b ' ->1 c ) ) v (
|
|
( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) $=
|
|
( wa wi1 wo wt wn an1 lan u1lemab ax-r2 2or ax-a3 ax-r1 orabs ax-r5 3tr 2an
|
|
lor 1i1 or32 leo le2an df-le2 ax-a1 df-i1 ) ABABDZACEZBCEZDZFZAGDZUIGCEZDZF
|
|
ZBGDZUJUNDZFZDZFZDZFABAHZCEZBHZCEZDZUKFZDZFUIVBVIAVAVHBVAULAVCCDZFZBVECDZFZ
|
|
DZFUHVNFZUKFVHUTVNULUPVKUSVMUPAACDZVJFZFZAVPFZVJFZVKUMAUOVQAIUOUICDVQUNCUIC
|
|
UAZJACKLMVTVRAVPVJNOVSAVJACPQRUSBBCDZVLFZFZBWBFZVLFZVMUQBURWCBIURUJCDWCUNCU
|
|
JWAJBCKLMWFWDBWBVLNOWEBVLBCPQRSTUHUKVNUBVOVGUKVOVNVGUHVNAVKBVMAVJUCBVLUCUDU
|
|
EVKVDVMVFVKVCHZVJFZVDAWGVJAUFQVDWHVCCUGOLVMVEHZVLFZVFBWIVLBUFQVFWJVECUGOLSL
|
|
QRJTJ $.
|
|
$( [24-Dec-98] $)
|
|
|
|
$( Lemma for 3-OA(3). Equivalence with substitution into 6-OA dual. $)
|
|
oa3-3lem $p |- ( a ' ^ ( a v ( b ^ ( ( ( a ^ b ) v ( a '
|
|
^ b ' ) ) v ( ( ( a ^ 1 ) v ( a ' ^ c ) ) ^
|
|
( ( b ^ 1 ) v ( b ' ^ c ) ) ) ) ) ) ) =
|
|
( a ' ^ ( a v ( b ^
|
|
( ( a == b ) v ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ) ) ) ) $=
|
|
( wa wn wo wt tb wi1 dfb ax-r1 an1 ax-a1 ax-r2 ax-r5 df-i1 2an 2or lan lor
|
|
) ABABDAEZBEZDFZAGDZUACDZFZBGDZUBCDZFZDZFZDZFABABHZUACIZUBCIZDZFZDZFUAULURA
|
|
UKUQBUCUMUJUPUMUCABJKUFUNUIUOUFUAEZUEFZUNUDUSUEUDAUSALAMNOUNUTUACPKNUIUBEZU
|
|
HFZUOUGVAUHUGBVABLBMNOUOVBUBCPKNQRSTS $.
|
|
$( [24-Dec-98] $)
|
|
|
|
$( Lemma for 3-OA(1). Equivalence with substitution into 6-OA dual. $)
|
|
oa3-1lem $p |- ( 1 ^ ( 0 v ( a ^ ( ( ( 0 ^ a ) v ( 1 ^ ( a ->1 c ) )
|
|
) v ( ( ( 0 ^ b ) v ( 1 ^ ( b ->1 c ) ) ) ^ ( ( a ^ b ) v
|
|
( ( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) ) )
|
|
= ( a ^ ( ( a ->1 c ) v ( ( b ->1 c ) ^
|
|
( ( a ^ b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) $=
|
|
( wt wf wa wi1 wo ancom an1 ax-a2 or0 an0 ax-r2 2or 3tr ax-r5 ran lor lan )
|
|
DEAEAFZDACGZFZHZEBFZDBCGZFZHZABFUBUFFHZFZHZFZHZFUMDFUMAUBUFUIFZHZFZDUMIUMJU
|
|
MULEHULUPEULKULLUKUOAUKUBUJHUOUDUBUJUDEUBHUBEHUBUAEUCUBUAAEFEEAIAMNUCUBDFUB
|
|
DUBIUBJNOEUBKUBLPQUJUNUBUHUFUIUHUGUEHUFEHUFUEUGKUGUFUEEUGUFDFUFDUFIUFJNUEBE
|
|
FEEBIBMNOUFLPRSNTPP $.
|
|
$( [25-Dec-98] $)
|
|
|
|
$( Lemma for 3-OA(4). Equivalence with substitution into 6-OA dual. $)
|
|
oa3-4lem $p |- ( a ' ^ ( a v ( b ^ ( ( ( a ^ b ) v ( a '
|
|
^ b ' ) ) v ( ( ( a ^ c ) v ( a ' ^ 1 ) ) ^
|
|
( ( b ^ c ) v ( b ' ^ 1 ) ) ) ) ) ) ) =
|
|
( a ' ^ ( a v ( b ^
|
|
( ( a == b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) $=
|
|
( wa wn wo wt tb wi1 dfb ax-a2 df-i1 an1 lor 3tr1 2an 2or ax-r1 lan ) ABABD
|
|
AEZBEZDFZACDZTGDZFZBCDZUAGDZFZDZFZDZFABABHZACIZBCIZDZFZDZFTUKUQAUJUPBUPUJUL
|
|
UBUOUIABJUMUEUNUHTUCFUCTFUMUETUCKACLUDTUCTMNOUAUFFUFUAFUNUHUAUFKBCLUGUAUFUA
|
|
MNOPQRSNS $.
|
|
$( [25-Dec-98] $)
|
|
|
|
$( Lemma for 3-OA(5). Equivalence with substitution into 6-OA dual. $)
|
|
oa3-5lem $p |- ( ( a ->1 c ) ^ ( a v ( c ^ ( ( ( a ^ c ) v (
|
|
( a ->1 c ) ^ 1 ) ) v ( ( ( a ^ b ) v ( ( a ->1 c ) ^
|
|
( b ->1 c ) ) ) ^ ( ( c ^ b ) v ( 1 ^ ( b ->1 c ) ) ) ) ) ) ) ) =
|
|
( ( a ->1 c ) ^ ( a v ( c ^ ( ( a ->1 c ) v ( ( b ->1 c ) ^
|
|
( ( a ^ b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) ) ) $=
|
|
( wa wi1 wt wo or12 oridm lor ax-r2 an1 df-i1 3tr1 ancom ax-r5 3tr lan 2or
|
|
wn ) ACACDZACEZFDZGZABDUBBCEZDGZCBDZFUEDZGZDZGZDZGACUBUEUFDZGZDZGUBULUOAUKU
|
|
NCUDUBUJUMUAATZUAGZGZUQUDUBURUPUAUAGZGUQUAUPUAHUSUAUPUAIJKUCUQUAUCUBUQUBLAC
|
|
MZKJUTNUJUFUEDUMUIUEUFUGBTZBCDZGZGZVCUIUEVDVAUGVBGZGVCUGVAVBHVEVBVAVEVBVBGV
|
|
BUGVBVBCBOPVBIKJKUHVCUGUHUEFDUEVCFUEOUELBCMZQJVFNRUFUEOKSRJR $.
|
|
$( [25-Dec-98] $)
|
|
|
|
$( Lemma for a "universal" 3-OA. Equivalence with substitution into 6-OA
|
|
dual. $)
|
|
oa3-u1lem $p |- ( 1 ^ ( c v ( ( a ' ->1 c ) ^ ( ( ( c ^ ( a ' ->1 c ) )
|
|
v ( 1 ^ ( a ->1 c ) ) ) v ( ( ( c ^ ( b ' ->1 c ) ) v ( 1
|
|
^ ( b ->1 c ) ) ) ^ ( ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) v
|
|
( ( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) ) ) =
|
|
( c v ( ( a ' ->1 c ) ^ ( ( a ->1 c ) v ( ( b ->1 c ) ^ (
|
|
( ( a ->1 c ) ^ ( b ->1 c ) ) v ( ( a ' ->1 c ) ^ ( b ' ->1
|
|
c ) ) ) ) ) ) ) $=
|
|
( wt wn wi1 wa wo ancom an1 lea leo letr leor lel2or df-le2 u1lemab lor 3tr
|
|
2or ax-a1 ax-r1 ran df-i1 3tr1 ax-a2 2an lan ) DCAEZCFZCUJGZDACFZGZHZCBEZCF
|
|
ZGZDBCFZGZHZUJUPGZULURGZHZGZHZGZHZGVGDGVGCUJULURVBVAHZGZHZGZHDVGIVGJVFVKCVE
|
|
VJUJUNULVDVIUICGZACGZHZUIVMHZHVOUNULVNVOVLVOVMVLUIVOUICKUIVMLMVMUINOPUKVNUM
|
|
VOUKUJCGVLUIEZCGZHVNCUJIUICQVQVMVLVPACAVPAUAUBUCRSUMULDGULVODULIULJACUDZSTV
|
|
RUEUTURVCVHUOCGZBCGZHZUOVTHZHWBUTURWAWBVSWBVTVSUOWBUOCKUOVTLMVTUONOPUQWAUSW
|
|
BUQUPCGVSUOEZCGZHWACUPIUOCQWDVTVSWCBCBWCBUAUBUCRSUSURDGURWBDURIURJBCUDZSTWE
|
|
UEVAVBUFUGTUHRS $.
|
|
$( [26-Dec-98] $)
|
|
|
|
$( Lemma for a "universal" 3-OA. Equivalence with substitution into 6-OA
|
|
dual. $)
|
|
oa3-u2lem $p |- ( ( a ->1 c ) ^ ( ( a ' ->1 c ) v ( c ^ (
|
|
( ( ( a ' ->1 c ) ^ c ) v ( ( a ->1 c ) ^ 1 ) ) v (
|
|
( ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) v ( ( a ->1 c ) ^
|
|
( b ->1 c ) ) ) ^ ( ( c ^ ( b ' ->1 c ) ) v
|
|
( 1 ^ ( b ->1 c ) ) ) ) ) ) ) ) =
|
|
( ( a ->1 c ) ^ ( ( a ' ->1 c ) v ( c ^ ( ( a ->1 c ) v
|
|
( ( b ->1 c ) ^ ( ( ( a ->1 c ) ^ ( b ->1 c ) ) v
|
|
( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ) ) ) ) ) ) $=
|
|
( wn wi1 wa wt u1lemab an1 2or lea ax-a1 ax-r1 leid leran le2or ax-r2 ancom
|
|
wo bltr df-i1 lbtr df-le2 ax-a2 2an lan lor ) ADZCEZCUICFZACEZGFZSZUIBDZCEZ
|
|
FZUKBCEZFZSZCUOFZGUQFZSZFZSZFZSUICUKUQURUPSZFZSZFZSUKVEVIUIVDVHCUMUKVCVGUMU
|
|
HCFZUHDZCFZSZUKSUKUJVMULUKUHCHUKIJVMUKVMUHACFZSZUKVJUHVLVNUHCKVKACVKAAAVKAL
|
|
MANTOPUKVOACUAMUBUCQVCVBUSFVGUSVBRVBUQUSVFVBUNCFZUNDZCFZSZUQSUQUTVSVAUQUTUO
|
|
CFVSCUORUNCHQVAUQGFUQGUQRUQIQJVSUQVSUNBCFZSZUQVPUNVRVTUNCKVQBCVQBBBVQBLMBNT
|
|
OPUQWABCUAMUBUCQUPURUDUEQJUFUGUF $.
|
|
$( [27-Dec-98] $)
|
|
|
|
${
|
|
oa3-6to3.1 $e |- ( ( a ->1 c ) ^ ( a v ( b ^ ( (
|
|
( a ' ->1 c ) ^ ( b ' ->1 c ) ) v (
|
|
( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) =< c $.
|
|
$( Derivation of 3-OA variant (3) from (6). $)
|
|
oa3-6to3 $p |- ( a ' ^ ( a v ( b ^
|
|
( ( a == b ) v ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ) ) ) )
|
|
=< c $=
|
|
( wn tb wi1 wa wo wt oa3-3lem ax-r1 leid wf df-f bltr ax-r2 dff 2or or0
|
|
le0 ancom an1 ax-a2 oa3-6lem oa4to6dual ) AEZABABFUGCGBEZCGHZIHIHZUGABABH
|
|
ZUGUHHIAJHZUGCHIBJHZUHCHIHIHIHZCUNUJABCKLAUGBUHJCCUGMUHMJEZNCNUOOLCUAPCJC
|
|
HZAUGHZBUHHZIZIZUSUPIUTCUTCNICUPCUSNUPCJHCJCUBCUCQUSNNIZNVAUSNUQNURARBRSL
|
|
NTQSCTQLUPUSUDQACGZABUKVBBCGZHZIULVBJCGZHIUMVCVEHIHIHIHVBABUIVDIHIHCABCUE
|
|
DPUFP $.
|
|
$( [24-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa3-2to4.1 $e |- ( ( a ->1 c ) ^ ( a v ( b ^ ( ( a ^ b ) v (
|
|
( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) =< c $.
|
|
$( Derivation of 3-OA variant (4) from (2). $)
|
|
oa3-2to4 $p |- ( a ' ^ ( a v ( b ^
|
|
( ( a == b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) )
|
|
=< c $=
|
|
( wn tb wi1 wa wo wt oa3-4lem ax-r1 leid le1 wf dff 2or or0 ax-r2 bltr
|
|
an1 ax-a2 oa3-2lemb oa4to6dual ) AEZABABFACGZBCGZHZIHIHZUEABABHZUEBEZHIAC
|
|
HZUEJHIBCHZUKJHIHIHIHZCUNUIABCKLAUEBUKCJCUEMUKMCENCCJHZAUEHZBUKHZIZIZURUO
|
|
IUSCUSCOICUOCUROCUAUROOIZOUTUROUPOUQAPBPQLORSQCRSLUOURUBSUFABUJUHIZULUFCC
|
|
GZHIUMUGVBHIHIHIHUFABVAHIHCABCUCDTUDT $.
|
|
$( [24-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa3-2wto2.1 $e |- ( a ' ^ ( a v ( b ^ ( ( a ^ b ) v (
|
|
( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) =< c $.
|
|
$( Derivation of 3-OA variant from weaker version. $)
|
|
oa3-2wto2 $p |- ( ( a ->1 c ) ^ ( a v ( b ^ ( ( a ^ b ) v (
|
|
( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) =< c $=
|
|
( wa wi1 wo oas ) ABABEACFBCFEGECDH $.
|
|
$( [25-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa3-2to2s.1 $e |- ( ( a ->1 d ) ^ ( a v ( b ^ ( ( a ^ b ) v (
|
|
( a ->1 d ) ^ ( b ->1 d ) ) ) ) ) ) =< d $.
|
|
$( Substitution into weaker version. $)
|
|
oa3-2to2s.2 $e |- d = ( ( a ^ c ) v ( b ^ c ) ) $.
|
|
$( Derivation of 3-OA variant from weaker version. $)
|
|
oa3-2to2s $p |- ( ( a ->1 c ) ^ ( a v ( b ^ ( ( a ^ b ) v (
|
|
( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) ) =<
|
|
( ( a ^ c ) v ( b ^ c ) ) $=
|
|
( wi1 wa wo wf wn id leo df-i1 ax-r1 ax-a1 ax-r2 lbtr 2an wt or0 lan omla
|
|
2or an1 0i1 oa3-2lema bltr oa4to6 oa6to4 ancom an0 lor le3tr2 ) ACGZABABH
|
|
ZUOBCGZHIZAJHZUOJCGZHIBJHZUQUTHIHIHIHACHZBCHZIZJCHZIZUOABURHIHVDAUOBUQJUT
|
|
CUOKZLUQKZLUTKZLAKZVGBKZVHJKZVIDABJVJVJVBIZVGKZVJVBMVMUOVNUOVMACNZOUOPZQR
|
|
VKVKVCIZVHKZVKVCMVQUQVRUQVQBCNZOUQPZQRVLVLVEIZVIKZVLVEMWAUTWBUTWAJCNOUTPZ
|
|
QRDDJIZVJKZVNHZVKKZVRHZIZVLKZWBHZIWDDDUAODWIJWKDVDWIFVBWFVCWHVBAUOHZWFWLV
|
|
BWLAVMHVBUOVMAVOUBACUCQOAWEUOVNAPZVPSQVCBUQHZWHWNVCWNBVQHVCUQVQBVSUBBCUCQ
|
|
OBWGUQVRBPZVTSQUDQJJTHZWKWPJJUEOJWJTWBJPZTUTWBUTTCUFOWCQSQUDQWMWOWQADGZAB
|
|
UPWRBDGZHIZUSWRJDGZHIVAWSXAHIHIHIHWRABWTHIHDABDUGEUHUIUJABCUGVFVDJIVDVEJV
|
|
DVECJHJJCUKCULQUMVDUAQUN $.
|
|
$( [25-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa3-u1.1 $e |- ( ( c ->1 c ) ^ ( c v ( ( a ' ->1 c ) ^
|
|
( ( ( c ^ ( a ' ->1 c ) ) v ( ( c ->1 c ) ^
|
|
( ( a ' ->1 c ) ->1 c ) ) ) v ( ( ( c ^ ( b ' ->1 c ) )
|
|
v ( ( c ->1 c ) ^ ( ( b ' ->1 c ) ->1 c ) ) ) ^ ( (
|
|
( a ' ->1 c ) ^ ( b ' ->1 c ) ) v
|
|
( ( ( a ' ->1 c ) ->1 c ) ^ ( ( b ' ->1 c ) ->1 c ) ) )
|
|
) ) ) ) ) =< c $.
|
|
$( Derivation of a "universal" 3-OA. The hypothesis is a substitution
|
|
instance of the proper 4-OA. $)
|
|
oa3-u1 $p |- ( c v ( ( a ' ->1 c ) ^ ( ( a ->1 c ) v
|
|
( ( b ->1 c ) ^ ( ( ( a ->1 c ) ^ ( b ->1
|
|
c ) ) v ( ( a ' ->1 c ) ^ ( b ' ->1
|
|
c ) ) ) ) ) ) ) =< c $=
|
|
( wn wi1 wa wo wt oa3-u1lem ax-r1 u1lem9ab ax-a2 lear lel2or df-le2 ax-r2
|
|
ancom u1lem8 2or le1 an1 3tr oa4to6dual leid letr bltr ) CAEZCFZACFZBCFZU
|
|
JUKGZUIBEZCFZGZHGHGHZICUICUIGIUJGHCUNGIUKGHUOULHGHGHGZCUQUPABCJKUQCCCIUIU
|
|
JUNUKCCEUAACLBCLCCBCGZUMCGZHZHZCIGZUIUJGZHZUNUKGZHZVACVAUTCHCCUTMUTCURCUS
|
|
BCNUMCNOPQKVFVAVDCVEUTVDCACGZUHCGZHZHVICHCVBCVCVICUBVCUJUIGVIUIUJRACSQTCV
|
|
IMVICVGCVHACNUHCNOPUCVEUKUNGUTUNUKRBCSQTKQDUDCUEUFUG $.
|
|
$( [27-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa3-u2.1 $e |- ( ( ( a ' ->1 c ) ->1 c ) ^ ( ( a ' ->1 c
|
|
) v ( c ^ ( ( ( ( a ' ->1 c ) ^ c ) v ( ( ( a ' ->1 c ) ->1 c ) ^ ( c ->1
|
|
c ) ) ) v ( ( ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) v ( ( ( a ' ->1 c ) ->1 c )
|
|
^ ( ( b ' ->1 c ) ->1 c ) ) ) ^ ( ( c ^ ( b ' ->1 c ) ) v ( ( c ->1 c ) ^
|
|
( ( b ' ->1 c ) ->1 c ) ) ) ) ) ) ) ) =< c $.
|
|
$( Derivation of a "universal" 3-OA. The hypothesis is a substitution
|
|
instance of the proper 4-OA. $)
|
|
oa3-u2 $p |- ( ( a ->1 c ) ^ ( ( a ' ->1 c ) v ( c ^ ( ( a ->1 c ) v
|
|
( ( b ->1 c ) ^ ( ( ( a ->1 c ) ^ ( b ->1 c ) ) v
|
|
( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ) ) ) ) ) ) =< c $=
|
|
( wi1 wn wa wo wt oa3-u2lem ax-r1 u1lem9ab le1 or32 ancom u1lem8 2or lear
|
|
ax-r2 lel2or an1 df-le2 3tr oa4to6dual bltr ) ACEZAFZCEZCUFBCEZUFUIGZUHBF
|
|
ZCEZGZHGHGHGZUFUHCUHCGUFIGHUMUJHCULGIUIGHGHGHGZCUOUNABCJKUHUFCIULUICACLCF
|
|
MBCLUHUFGZCIGZHULUIGZHZCUSUPURHZUQHACGZUGCGZHZBCGZUKCGZHZHZCHCUPUQURNUTVG
|
|
UQCUPVCURVFUPUFUHGVCUHUFOACPSURUIULGVFULUIOBCPSQCUAQVGCVCCVFVACVBACRUGCRT
|
|
VDCVEBCRUKCRTTUBUCKDUDUE $.
|
|
$( [27-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
oa3-1to5.1 $e |- ( ( a ->1 c ) ^
|
|
( ( a ^ b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) )
|
|
=< ( b ->1 c ) $.
|
|
$( Derivation of an equivalent of the second "universal" 3-OA U2 from an
|
|
equivalent of the first "universal" 3-OA U1. This shows that U2 is
|
|
redundant in a system containg U1. The hypothesis is theorem
|
|
~ oal1 . $)
|
|
oa3-1to5 $p |- ( c ^ ( ( b ->1 c ) v ( ( a ->1 c ) ^
|
|
( ( a ^ b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) ) ) )
|
|
=< ( b ' ->1 c ) $=
|
|
( wi1 wa wo wn leid lel2or lelan ax-a1 ran ax-r5 ax-a2 ax-r2 u1lemab 3tr1
|
|
ancom lbtr lear letr ) CBCEZACEZABFUDUCFGFZGZFZCBHZCEZFZUIUGCUCFZUJUFUCCU
|
|
CUCUEUCIDJKUCCFZUICFZUKUJBCFZUHCFZGZUOUHHZCFZGZULUMUPURUOGUSUNURUOBUQCBLM
|
|
NURUOOPBCQUHCQRCUCSCUISRTCUIUAUB $.
|
|
$( [1-Jan-99] $)
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Derivation of 4-variable proper OA from OA distributive law
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$( In this section, we postulate a temporary axiom (intended not to
|
|
be used outside of this section) for the OA distributive law, and derive
|
|
from it the proper 4-OA. This shows that the OA distributive law
|
|
implies the proper 4-OA (and therefore the 6-OA). $)
|
|
|
|
${
|
|
oad.1 $e |- e =
|
|
( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) $.
|
|
oad.2 $e |- f =
|
|
( ( ( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v e ) $.
|
|
oad.3 $e |- h =< ( a ->1 d ) $.
|
|
oad.4 $e |- j =< f $.
|
|
oad.5 $e |- k =< f $.
|
|
oad.6 $e |- ( h ^ ( b ->1 d ) ) =< k $.
|
|
$( OA Distributive law. In this section, we postulate this temporary axiom
|
|
(intended not to be used outside of this section) for the OA
|
|
distributive law, and derive from it the 6-OA, in theorem ~ d6oa . This
|
|
together with the derivation of the distributive law in theorem
|
|
~ 4oadist shows that the OA distributive law is equivalent to the
|
|
6-OA. $)
|
|
ax-oadist $a |- ( h ^ ( j v k ) ) = ( ( h ^ j ) v ( h ^ k ) ) $.
|
|
$}
|
|
|
|
${
|
|
d3oa.1 $e |- f =
|
|
( ( a ^ b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) $.
|
|
$( Derivation of 3-OA from OA distributive law. $)
|
|
d3oa $p |- ( ( a ->1 c ) ^ f ) =< ( b ->1 c ) $=
|
|
( wi1 wa wn wi2 wo lear bltr le2or id leid ax-r1 leo letr ax-r2 lbtr bile
|
|
1oai1 2oath1i1 df-i1 ax-a1 df-i2 ax-a2 lea ax-oadist wi0 u12lem df-i0 lan
|
|
ax-r5 oridm le3tr2 ) ACFZABGZHZUQBCFZGZFZGZUQUSVAIZGZJZUTUTJUQDGZUTVCUTVE
|
|
UTABCUBVEVAUTABCUCUQUTKLMVFUQVBVDJZGZVGVIVFABACAAGUQUQGJBAGUTUQGJGZURVAJZ
|
|
VJJZUQVBVDVJNVLNUQOVBVKVLVBUSHZUSVAGZJVKUSVAUDVMURVNVAVMURURVMURUEZPZUAUS
|
|
VAKMLVKVJQZRVDVKVLVDVMVAHZGZVAJZVKVDVAVSJZVTUSVAUFZVAVSUGSVSURVAVAVSVMURV
|
|
MVRUHVPTVAOMLVQRVAWAVDVAVSQVDWAWBPTUIPVHDUQVHVKDVHVMVAJZVKVHUSVAUJWCUSVAU
|
|
KUSVAULSVKWCURVMVAVOUNPSDVKEPSUMSUTUOUP $.
|
|
$( [30-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
d4oa.2 $e |- e =
|
|
( ( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) $.
|
|
d4oa.1 $e |- f =
|
|
( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) $.
|
|
$( Variant of proper 4-OA proved from OA distributive law. $)
|
|
d4oa $p |- ( ( a ->1 d ) ^ ( e v f ) ) =< ( b ->1 d ) $=
|
|
( wi1 wo wa lan id 2or leor ax-r1 ax-r2 d3oa bltr ancom ax-a2 anass leran
|
|
leid leo lbtr ax-oadist letr lel2or ) ADIZEFJZKZUJFKZUJEKZJZBDIZULUJFEJZK
|
|
UOUKUQUJEFUALABCDACKUJCDIZKJZBCKZUPURKZJZKZUKUJFEVCMEABKZUJUPKZJZFVCGHNUJ
|
|
UDFEOEFUEVEVFEVEVDOEVFGPUFUGQUMUPUNUMURVBKZUPUMUJUSKZVBKZVGUMUJVCKZVIFVCU
|
|
JHLVIVJUJUSVBUBPQVHURVBACDUSUSMRUCSCBDVBUTCBKVAURUPKBCTUPURTNRUHABDEGRUIS
|
|
$.
|
|
$( [30-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
d6oa.1 $e |- a =< b ' $.
|
|
d6oa.2 $e |- c =< d ' $.
|
|
d6oa.3 $e |- e =< f ' $.
|
|
$( Derivation of 6-variable orthoarguesian law from OA distributive law. $)
|
|
d6oa $p |- ( ( ( a v b ) ^ ( c v d ) ) ^ ( e v f ) ) =<
|
|
( b v ( a ^ ( c v ( ( ( a v c ) ^ ( b v d ) ) ^
|
|
( ( ( a v e ) ^ ( b v f ) ) v ( ( c v e ) ^ ( d v f ) ) ) ) ) ) ) $=
|
|
( wn wa wo id wi1 d4oa oa4gto4u oa4uto4 oa4to6 ) ABCDEFAJZBJKCJZDJKLEJZFJ
|
|
KLZSTUAGHIUBMSMTMUAMSTUAUBSTUAUBTUBNZSUBNZUAUBNZUCUDUEUBUCUDKUCUBNZUDUBNZ
|
|
KLZUCUEKUFUEUBNZKLUDUEKUGUIKLKZUHMUJMOUDMUCMUEMPQR $.
|
|
$( [30-Dec-98] $)
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Orthoarguesian laws
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$( R. Godowski and R. Greechie, Demonstratio Mathematica 17, 241 (1984) $)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
3-variable orthoarguesian law
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( 3-variable consequence of the orthoarguesion law. $)
|
|
ax-3oa $a |- ( ( a ->1 c ) ^
|
|
( ( a ^ b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) )
|
|
=< ( b ->1 c ) $.
|
|
|
|
$( Orthoarguesian law - ` ->2 ` version. $)
|
|
oal2 $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ' v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) )
|
|
=< ( a ->2 c ) $=
|
|
( wn wi1 wa wo wi2 ax-3oa i2i1 anor3 ax-r1 2an 2or le3tr1 ) BDZADZEZPCDZFZR
|
|
SQEZFZGZFUAABHZBCGDZUDACHZFZGZFUFPSQIUDRUHUCABJZUETUGUBTUEBCKLUDRUFUAUIACJZ
|
|
MNMUJO $.
|
|
$( [20-Jul-99] $)
|
|
|
|
$( Orthoarguesian law - ` ->1 ` version derived from ` ->1 ` version. $)
|
|
oal1 $p |- ( ( a ->1 c ) ^
|
|
( ( a ^ b ) v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) )
|
|
=< ( b ->1 c ) $=
|
|
( wn wi2 wo wa wi1 oal2 i1i2 df-a 2an 2or le3tr1 ) CDZADZEZPBDZFDZQOREZGZFZ
|
|
GTACHZABGZUCBCHZGZFZGUEOPRIUCQUGUBACJZUDSUFUAABKUCQUETUHBCJZLMLUIN $.
|
|
$( [25-Nov-98] $)
|
|
|
|
$( Orthoarguesian law. Godowski/Greechie, Eq. III. $)
|
|
oaliii $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ' v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) =
|
|
( ( a ->2 c ) ^ ( ( b v c ) ' v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) $=
|
|
( wi2 wo wn wa anass anidm lan ax-r2 ax-r1 oal2 leran ax-a2 ax-r4 ancom 2or
|
|
bltr ran lebi ) ABDZBCEZFZUBACDZGZEZGZUEUGGZUHUHUGGZUIUJUHUJUBUGUGGZGUHUBUG
|
|
UGHUKUGUBUGIZJKLUHUEUGABCMNSUIUECBEZFZUEUBGZEZGZUGGZUHURUIURUEUPUGGZGUIUEUP
|
|
UGHUSUGUEUSUKUGUPUGUGUNUDUOUFUMUCCBOPUEUBQRTULKJKLUQUBUGACBMNSUA $.
|
|
$( [22-Sep-98] $)
|
|
|
|
$( Orthoarguesian law. Godowski/Greechie, Eq. II. This proof references
|
|
~ oaliii only. $)
|
|
oalii $p |- ( b ' ^ ( ( a ->2 b ) v ( ( a ->2 c ) ^ ( ( b v c ) '
|
|
v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) ) =< a ' $=
|
|
( wn wi2 wo wa orabs oaliii lor df-i2 ancom ax-r2 3tr2 lan omlan lear bltr
|
|
) BDZABEZACEZBCFDTUAGFZGZFZGZSADZGZUFUESBUGFZGUGUDUHSTTUBGZFTUDUHTUBHUIUCTA
|
|
BCIJTBUFSGZFUHABKUJUGBUFSLJMNOBUFPMSUFQR $.
|
|
$( [22-Sep-98] $)
|
|
|
|
$( Orthoarguesian law. Godowski/Greechie, Eq. IV. $)
|
|
oaliv $p |- ( b ' ^ ( ( a ->2 b ) v ( ( a ->2 c ) ^ ( ( b v c ) '
|
|
v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) ) =<
|
|
( ( b ' ^ ( a ->2 b ) ) v ( c ' ^ ( a ->2 c ) ) ) $=
|
|
( wn wi2 wo lea oalii ler2an df-i2 ancom lor ax-r2 lan omlan ax-r1 lbtr leo
|
|
wa letr ) BDZABEZACEZBCFDUBUCSFSFZSZUAUBSZUFCDUCSZFUEUAADZSZUFUEUAUHUAUDGAB
|
|
CHIUFUIUFUABUIFZSUIUBUJUAUBBUHUASZFUJABJUKUIBUHUAKLMNBUHOMPQUFUGRT $.
|
|
$( [25-Nov-98] $)
|
|
|
|
$( OA theorem. $)
|
|
oath1 $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ' v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) =
|
|
( ( a ->2 b ) ^ ( a ->2 c ) ) $=
|
|
( wi2 wo wn wa oaliii lan anidm ax-r1 anandir 3tr1 ax-a2 anabs 3tr ) ABDZBC
|
|
EFZQACDZGZEZGZTUAGZTTREZGTUBUBGZUBSUAGZGUBUCUBUFUBABCHIUEUBUBJKQSUALMUAUDTR
|
|
TNITROP $.
|
|
$( [12-Oct-98] $)
|
|
|
|
$( Lemma. $)
|
|
oalem1 $p |- ( ( b v c ) v ( ( b v c ) ' ^ ( ( a ->2 b )
|
|
v ( ( a ->2 c ) ^ ( ( b v c ) '
|
|
v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) ) ) =<
|
|
( a ->2 ( b v c ) ) $=
|
|
( wo wn wi2 wa anidm ran ax-r1 anor3 an32 ax-r2 3tr2 anass oalii lelan bltr
|
|
ancom lbtr lelor df-i2 ) BCDZUCEZABFZACFZUDUEUFGDGDZGZDUCAEZUDGZDZAUCFZUHUJ
|
|
UCUHUDUIGZUJUHUDBEZGZUGGZUMUDUOUGUNCEZGZUNUNGZUQGZUDUOUTURUSUNUQUNHIJBCKZUT
|
|
URUNGUOUNUNUQLURUDUNVAIMNIUPUDUNUGGZGUMUDUNUGOVBUIUDABCPQRRUDUISTUAULUKAUCU
|
|
BJT $.
|
|
$( [16-Oct-98] $)
|
|
|
|
$( Lemma. $)
|
|
oalem2 $p |- ( ( a ->2 b )
|
|
v ( ( a ->2 c ) ^ ( ( b v c ) '
|
|
v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) =
|
|
( a ->2 b ) $=
|
|
( wi2 wo wn wa ax-a2 ax-r4 ancom 2or lan oath1 ax-r2 lor orabs 3tr ) ABDZAC
|
|
DZBCEZFZRSGZEZGZERSRGZERUBERUDUERUDSCBEZFZUEEZGUEUCUHSUAUGUBUETUFBCHIRSJKLA
|
|
CBMNOUEUBRSRJORSPQ $.
|
|
$( [16-Oct-98] $)
|
|
|
|
${
|
|
oadist2a.1 $e |- ( d v ( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) )
|
|
=< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
$( Distributive inference derived from OA. $)
|
|
oadist2a $p |- ( ( a ->2 b ) ^
|
|
( d v ( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) =
|
|
( ( ( a ->2 b ) ^ d ) v
|
|
( ( a ->2 b ) ^ ( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) $=
|
|
( wi2 wo wa ax-a2 lan wi0 bltr lelan wn df-i0 oath1 ax-r2 leo df-i2 ax-r1
|
|
lbtr letr distlem ) ABFZDBCGZUDACFHZFZGZHUDUGDGZHZUDDHZUDUGHZGZUHUIUDDUGI
|
|
JUJULUKGUMUDUGDUJUDUEUFKZHZUGUIUNUDUIUHUNUGDIELMUOUFUGUOUDUENZUFGZHUFUNUQ
|
|
UDUEUFOJABCPQUFUFUPUFNHZGZUGUFURRUGUSUEUFSTUALUBUCULUKIQQ $.
|
|
$( [17-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oadist2b.1 $e |- d =<
|
|
( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
$( Distributive inference derived from OA. $)
|
|
oadist2b $p |- ( ( a ->2 b ) ^
|
|
( d v ( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) =
|
|
( ( ( a ->2 b ) ^ d ) v
|
|
( ( a ->2 b ) ^ ( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) $=
|
|
( wo wi2 wa wi1 wi0 u12lem ax-r1 lbtr leor lel2or oadist2a ) ABCDDBCFZABG
|
|
ACGHZGZFQRIZSFZQRJZDUASDUBUAEUAUBQRKZLMSTNOUCMP $.
|
|
$( [17-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oadist2.1 $e |- ( d v ( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) )
|
|
= ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
$( Distributive inference derived from OA. $)
|
|
oadist2 $p |- ( ( a ->2 b ) ^
|
|
( d v ( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) =
|
|
( ( ( a ->2 b ) ^ d ) v
|
|
( ( a ->2 b ) ^ ( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) $=
|
|
( wo wi2 wa wi0 bile oadist2a ) ABCDDBCFZABGACGHZGFLMIEJK $.
|
|
$( [17-Nov-98] $)
|
|
$}
|
|
|
|
$( Distributive law derived from OA. $)
|
|
oadist12 $p |- ( ( a ->2 b ) ^
|
|
( ( ( b v c ) ->1 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) v
|
|
( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) =
|
|
( ( ( a ->2 b ) ^ ( ( b v c ) ->1 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) v
|
|
( ( a ->2 b ) ^ ( ( b v c ) ->2 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) ) $=
|
|
( wo wi2 wa wi1 u12lem oadist2 ) ABCBCDZABEACEFZGJKHI $.
|
|
$( [17-Nov-98] $)
|
|
|
|
${
|
|
oacom.1 $e |- d C ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
oacom.2 $e |- ( d ^ ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) )
|
|
C ( a ->2 b ) $.
|
|
$( Commutation law requiring OA. $)
|
|
oacom $p |- d C ( ( a ->2 b ) ^ ( a ->2 c ) ) $=
|
|
( wi2 wo wa wi0 comcom ancom bctr gsth2 wn df-i0 lan oath1 ax-r2 cbtr ) D
|
|
ABGZBCHZUAACGIZJZIZUCUEDUAUDDDUDEKUDDIZUAUFDUDIUAUDDLFMKNKUEUAUBOUCHZIUCU
|
|
DUGUAUBUCPQABCRST $.
|
|
$( [19-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oacom2.1 $e |- d =<
|
|
( ( a ->2 b ) ^ ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) $.
|
|
$( Commutation law requiring OA. $)
|
|
oacom2 $p |- d C ( ( a ->2 b ) ^ ( a ->2 c ) ) $=
|
|
( wo wi2 wa wi0 lear letr lecom lea oacom ) ABCDDBCFABGZACGHIZDOPHZPEOPJK
|
|
LDPHZORDODPMDQOEOPMKKLN $.
|
|
$( [19-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oacom3.1 $e |- ( d ^ ( a ->2 b ) )
|
|
C ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
oacom3.2 $e |- d C ( a ->2 b ) $.
|
|
$( Commutation law requiring OA. $)
|
|
oacom3 $p |- d C ( ( a ->2 b ) ^ ( a ->2 c ) ) $=
|
|
( wo wi2 wa wi0 comcom ancom bctr gsth2 wn df-i0 ran oath1 3tr cbtr ) DBC
|
|
GZABHZACHIZJZUBIZUCUEDUDUBDDUBFKUBDIZUDUFDUBIUDUBDLEMKNKUEUAOUCGZUBIUBUGI
|
|
UCUDUGUBUAUCPQUGUBLABCRST $.
|
|
$( [19-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oagen1.1 $e |- d =< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
$( "Generalized" OA. $)
|
|
oagen1 $p |- ( ( a ->2 b ) ^
|
|
( d v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) =
|
|
( ( a ->2 b ) ^ ( a ->2 c ) ) $=
|
|
( wi2 wa wo wn wi0 df-i0 lbtr leror ax-a3 oridm lor ax-r2 lelan oath1 lea
|
|
leor ler2an lebi ) ABFZDUDACFZGZHZGZUFUHUDBCHZIZUFHZGUFUGUKUDUGUKUFHZUKDU
|
|
KUFDUIUFJUKEUIUFKLMULUJUFUFHZHUKUJUFUFNUMUFUJUFOPQLRABCSLUFUDUGUDUETUFDUA
|
|
UBUC $.
|
|
$( [19-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oagen1b.1 $e |- d =< ( a ->2 b ) $.
|
|
oagen1b.2 $e |- e =< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
$( "Generalized" OA. $)
|
|
oagen1b $p |- ( d ^ ( e v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) =
|
|
( d ^ ( a ->2 c ) ) $=
|
|
( wi2 wa wo oagen1 lan anass ax-r1 df2le2 ran ax-r2 3tr2 ) DABHZESACHZIZJ
|
|
ZIZIZDUAIZDUBIZDTIZUCUADABCEGKLUDDSIZUBIZUFUIUDDSUBMNUHDUBDSFOZPQUEUHTIZU
|
|
GUKUEDSTMNUHDTUJPQR $.
|
|
$( [21-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oagen2.1 $e |- d =< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
$( "Generalized" OA. $)
|
|
oagen2 $p |- ( ( a ->2 b ) ^ d ) =< ( a ->2 c ) $=
|
|
( wi2 wa wo wn wi0 df-i0 lbtr lelan oal2 letr ) ABFZDGPBCHZIPACFZGZHZGRDT
|
|
PDQSJTEQSKLMABCNO $.
|
|
$( [19-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oagen2b.1 $e |- d =< ( a ->2 b ) $.
|
|
oagen2b.2 $e |- e =< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
$( "Generalized" OA. $)
|
|
oagen2b $p |- ( d ^ e ) =< ( a ->2 c ) $=
|
|
( wa wi2 leran oagen2 letr ) DEHABIZEHACIDMEFJABCEGKL $.
|
|
$( [21-Nov-98] $)
|
|
$}
|
|
|
|
$( Mladen's OA $)
|
|
mloa $p |- ( ( a == b ) ^ ( ( b == c ) v ( ( b v ( a == b ) )
|
|
^ ( c v ( a == c ) ) ) ) ) =< ( c v ( a == c ) ) $=
|
|
( wi2 wa wn wo tb lea ax-a3 or12 anor3 ax-r2 leo df-i2 ax-r1 lbtr le2an 2an
|
|
i2bi ax-r5 id bile lel2or lelor bltr oal2 letr u2lembi dfb 2or le3tr2 ) ABD
|
|
ZBADZEZBCEZBFZCFZEZGZUMACDZEZGZEZVAABHZBCHZBVEGZCACHGZEZGZEVHVDUMBCGFZVBGZE
|
|
VAUOUMVCVLUMUNIVCVKUPVBGZGZVLVCUPUSVBGGZVNUPUSVBJVOUSVMGVNUPUSVBKUSVKVMBCLU
|
|
AMMVMVBVKUPVBVBBUMCVABBAFZUQEZGZUMBVQNUMVRABOPQCCVPUREZGZVACVSNVAVTACOPQRVB
|
|
VBVBUBUCUDUEUFRABCUGUHUOVEVCVJABUIUTVFVBVIVFUTBCUJPUMVGVAVHABTACTZSUKSWAUL
|
|
$.
|
|
$( [20-Nov-98] $)
|
|
|
|
|
|
${
|
|
oadist.1 $e |- d =< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
$( Distributive law derived from OAL. $)
|
|
oadist $p |- ( ( a ->2 b ) ^
|
|
( d v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) =
|
|
( ( ( a ->2 b ) ^ d ) v ( ( a ->2 b ) ^
|
|
( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) $=
|
|
( wi2 wa oagen1 bile anidm ax-r1 ran anass ax-r2 leor bltr letr ledi lebi
|
|
wo ) ABFZDUAACFZGZTGZUADGZUAUCGZTZUDUCUGUDUCABCDEHIUCUFUGUCUAUAGZUBGUFUAU
|
|
HUBUHUAUAJKLUAUAUBMNUFUEOPQUADUCRS $.
|
|
$( [20-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oadistb.2 $e |- d =< ( a ->2 b ) $.
|
|
oadistb.1 $e |- e =< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
$( Distributive law derived from OAL. $)
|
|
oadistb $p |- ( d ^
|
|
( e v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) =
|
|
( ( d ^ e ) v ( d ^
|
|
( ( a ->2 b ) ^ ( a ->2 c ) ) ) ) $=
|
|
( wi2 wa wo df2le2 ran ax-r1 anass oagen1 lan ax-r2 leor bltr ledi lebi )
|
|
DEABHZACHIZJZIZDEIZDUCIZJZUEUGUHUEDUBIZUDIZUGUJUEUIDUDDUBFKLMUJDUBUDIZIUG
|
|
DUBUDNUKUCDABCEGOPQQUGUFRSDEUCTUA $.
|
|
$( [20-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oadistc0.1 $e |- d =< ( ( a ->2 b ) ^ ( a ->2 c ) ) $.
|
|
$( Note: inference of 2nd hyp. from 1st may be an OM theorem. $)
|
|
oadistc0.2 $e |- ( ( a ->2 c ) ^
|
|
( ( a ->2 b ) ^ ( ( b v c ) ' v d ) ) ) =<
|
|
( ( ( a ->2 b ) ^ ( b v c ) ' ) v d ) $.
|
|
$( Pre-distributive law. $)
|
|
oadistc0 $p |- ( ( a ->2 b ) ^ ( ( b v c ) ' v d ) ) =
|
|
( ( ( a ->2 b ) ^ ( b v c ) ' ) v d ) $=
|
|
( wi2 wo wn wa ancom lelor lelan oal2 letr df2le2 ax-r2 ax-r1 bltr ledior
|
|
ax-a2 lea df-le2 ran lbtr lebi ) ABGZBCHIZDHZJZUGUHJDHZUJACGZUJJZUKUMUJUM
|
|
UJULJUJULUJKUJULUJUGUHUGULJZHZJULUIUOUGDUNUHELMABCNOPQRFSUKUGDHZUIJUJDUGU
|
|
HTUPUGUIUPDUGHUGUGDUADUGDUNUGEUGULUBOUCQUDUEUF $.
|
|
$( [30-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oadistc.1 $e |- d =< ( ( a ->2 b ) ^ ( a ->2 c ) ) $.
|
|
oadistc.2 $e |- ( ( a ->2 b ) ^ ( ( b v c ) ' v d ) ) =<
|
|
( ( ( a ->2 b ) ^ ( b v c ) ' ) v d ) $.
|
|
$( Distributive law. $)
|
|
oadistc $p |- ( ( a ->2 b ) ^ ( ( b v c ) ' v d ) ) =
|
|
( ( ( a ->2 b ) ^ ( b v c ) ' ) v ( ( a ->2 b ) ^ d ) ) $=
|
|
( wi2 wo wn wa lea letr df2le2 ax-r1 ancom ax-r2 lor lbtr ledi lebi ) ABG
|
|
ZBCHIZDHJZUAUBJZUADJZHZUCUDDHUFFDUEUDDDUAJZUEUGDDUADUAACGZJUAEUAUHKLMNDUA
|
|
OPQRUAUBDST $.
|
|
$( [21-Nov-98] $)
|
|
$}
|
|
|
|
${
|
|
oadistd.1 $e |- d =< ( a ->2 b ) $.
|
|
oadistd.2 $e |- e =< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
oadistd.3 $e |- f =< ( ( b v c ) ->0 ( ( a ->2 b ) ^ ( a ->2 c ) ) ) $.
|
|
oadistd.4 $e |- ( d ^ ( a ->2 c ) ) =< f $.
|
|
$( OA distributive law. $)
|
|
oadistd $p |- ( d ^ ( e v f ) ) = ( ( d ^ e ) v ( d ^ f ) ) $=
|
|
( wo wa wi2 lbtr df2le2 ax-r1 lan ax-r2 bltr letr le2or oridm lelan df-i0
|
|
wi0 wn leo oagen1b lear an32 lea leor ledi lebi ) DEFKZLZDELZDFLZKZUPURUS
|
|
UPUPDACMZLZLZURUPUPDBCKZABMUTLZUEZLZLZVBVGUPUPVFUOVEDUOVEVEKVEEVEFVEHIUAV
|
|
EUBNUCOPVFVAUPVFDVCUFZVDKZLVAVEVIDVCVDUDZQABCDVHGVHVIVEVHVDUGVEVIVJPNUHRQ
|
|
RVBVAURUPVAUIVAURUTLZURVAVAFLZVKVLVAVAFJOPDUTFUJRURUTUKSTSURUQULTDEFUMUN
|
|
$.
|
|
$( [21-Nov-98] $)
|
|
$}
|
|
|
|
$( Alternate form for the 3-variable orthoarguesion law. $)
|
|
3oa2 $p |- ( ( a ->1 c ) ^
|
|
( ( ( a ->1 c ) ^ ( b ->1 c ) ) v
|
|
( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ) )
|
|
=< ( b ->1 c ) $=
|
|
( wn wi1 wa wo ax-3oa u1lem11 ax-a2 2an ax-r5 ax-r2 le3tr2 ) ADCEZCEZOBDCEZ
|
|
FZPQCEZFZGZFSACEZUBBCEZFZRGZFUCOQCHPUBUAUEACIZUATRGUERTJTUDRPUBSUCUFBCIZKLM
|
|
KUGN $.
|
|
$( [27-May-04] $)
|
|
|
|
$( 3-variable orthoarguesion law expressed with the 3OA identity
|
|
abbreviation. $)
|
|
3oa3 $p |- ( ( a ->1 c ) ^ ( a == c ==OA b ) ) =< ( b ->1 c ) $=
|
|
( wi1 wid3oa wa wn wo df-id3oa lan 3oa2 bltr ) ACDZABCEZFMMBCDZFAGCDBGCDFHZ
|
|
FONPMABCIJABCKL $.
|
|
$( [27-May-04] $)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
4-variable orthoarguesian law
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
oal4.1 $e |- a =< b ' $.
|
|
oal4.2 $e |- c =< d ' $.
|
|
$( Orthoarguesian law (4-variable version). $)
|
|
ax-oal4 $a |- ( ( a v b ) ^ ( c v d ) ) =< ( b v ( a ^ ( c v
|
|
( ( a v c ) ^ ( b v d ) ) ) ) ) $.
|
|
$}
|
|
|
|
$( 4-variable OA closed equational form) $)
|
|
oa4cl $p |- ( ( a v ( b ^ a ' ) ) ^ ( c v ( d ^ c ' ) ) ) =<
|
|
( ( b ^ a ' ) v ( a ^ ( c v
|
|
( ( a v c ) ^ ( ( b ^ a ' ) v ( d ^ c ' ) ) ) ) ) ) $=
|
|
( wn wa wo leor oran2 lbtr ax-oal4 ) ABAEFZCDCEFZABEZAGLEANHBAIJCDEZCGMECOH
|
|
DCIJK $.
|
|
$( [1-Dec-98] $)
|
|
|
|
$( Derivation of 3-variable OA from 4-variable OA. $)
|
|
oa43v $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ' v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) )
|
|
=< ( a ->2 c ) $=
|
|
( wi2 wn wo wa ud2lem0c lea bltr ax-oal4 id oa4v3v oal42 oa23 ) ABCACBACBAC
|
|
DEZABDEZPCEZACFZGRACHRSIJZQBEZABFZGUAABHUAUBIJZPCQBTUCKPLQLMNO $.
|
|
$( [28-Nov-98] $)
|
|
|
|
|
|
${
|
|
oa3moa3.1 $e |- a =< b ' $.
|
|
oa3moa3.2 $e |- c =< d ' $.
|
|
oa3moa3.3 $e |- d =< e ' $.
|
|
oa3moa3.4 $e |- e =< c ' $.
|
|
$( 4-variable 3OA to 5-variable Mayet's 3OA. $)
|
|
oa3moa3 $p |- ( ( a v b ) ^ ( ( c v d ) v e ) ) =< ( a v ( ( ( b ^ ( c v
|
|
( ( b v c ) ^ ( ( a v d ) v e ) ) ) ) ^ ( d v ( ( b v d ) ^ ( ( a v
|
|
c ) v e ) ) ) ) ^ ( e v ( ( b v e ) ^ ( ( a v c ) v d ) ) ) ) ) $=
|
|
( wo wa lecon3 wn lel2or lan lor lel lecom comcom7 comcom ax-a2 ax-a3 2an
|
|
ax-oal4 orass le3tr1 ror tr ler2an fh3 cm anandi lbtr ax-r1 anass 3tr1 )
|
|
ABJZCDJZEJZKZABCBCJZADJEJZKZJZKZJZABDBDJZACJZEJZKZJZEBEJZVHDJZKZJZKZKZJZK
|
|
ZAVEVKKVOKZJZUTVFVRBAJZCDEJZJZKABCVAAWCJZKZJZKZJUTVFBACWCABFLZWCCDCMECDGL
|
|
INLUDUQWBUSWDABUAZCDEUBUCVEWHAVDWGBVCWFCVBWEVAADEUEOPOPUFUTABVKKZJZABVOKZ
|
|
JZKZVRUTWLWNWBDCEJZJZKABDVGAWPJZKZJZKZJUTWLBADWPWIWPDCDMEGDEHLNLUDUQWBUSW
|
|
QWJUSDCJZEJWQURXBECDUAUGDCEUEUHUCWKXAAVKWTBVJWSDVIWRVGACEUEOPOPUFWBEURJZK
|
|
ABEVLAURJZKZJZKZJUTWNBAEURWIURECEMDECILHNLUDUQWBUSXCWJUREUAUCWMXGAVOXFBVN
|
|
XEEVMXDVLACDUBOPOPUFUIWOAWKWMKZJZVRXIWOAWKWMWKAWKAWKAMZBXJVKWIQRSTWMAWMAW
|
|
MXJBXJVOWIQRSTUJUKXHVQAVQXHBVKVOULUKPUHUMUIVSAVEVQKZJZWAXLVSAVEVQVEAVEAVE
|
|
XJBXJVDWIQRSTVQAVQAVQXJBXJVPWIQRSTUJUNXKVTABVDVPKKZVEVPKZXKVTXNXMBVDVPUOU
|
|
KXMXKBVDVPULUNVEVKVOUOUPPUHUM $.
|
|
$( [31-Mar-2011] $) $( [3-Apr-2009] $)
|
|
$}
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
6-variable orthoarguesian law
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
oal6.1 $e |- a =< b ' $.
|
|
oal6.2 $e |- c =< d ' $.
|
|
oal6.3 $e |- e =< f ' $.
|
|
$( Orthoarguesian law (6-variable version). $)
|
|
ax-oa6 $a |- ( ( ( a v b ) ^ ( c v d ) ) ^ ( e v f ) ) =<
|
|
( b v ( a ^ ( c v ( ( ( a v c ) ^ ( b v d ) ) ^
|
|
( ( ( a v e ) ^ ( b v f ) ) v ( ( c v e ) ^ ( d v f ) ) ) ) ) ) ) $.
|
|
$}
|
|
|
|
|
|
${
|
|
oa64v.1 $e |- a =< b ' $.
|
|
oa64v.2 $e |- c =< d ' $.
|
|
$( Derivation of 4-variable OA from 6-variable OA. $)
|
|
oa64v $p |- ( ( a v b ) ^ ( c v d ) ) =< ( b v ( a ^ ( c v
|
|
( ( a v c ) ^ ( b v d ) ) ) ) ) $=
|
|
( wf wt wn le0 ax-oa6 id oa6v4v ) ABCDGHABCDGHEFHIJKGLHLM $.
|
|
$( [29-Nov-98] $)
|
|
$}
|
|
|
|
|
|
$( Derivation of 3-variable OA from 6-variable OA. $)
|
|
oa63v $p |- ( ( a ->2 b ) ^
|
|
( ( b v c ) ' v ( ( a ->2 b ) ^ ( a ->2 c ) ) ) )
|
|
=< ( a ->2 c ) $=
|
|
( wi2 wn wo wa ud2lem0c lea bltr oa64v id oa4v3v oal42 oa23 ) ABCACBACBACDE
|
|
ZABDEZPCEZACFZGRACHRSIJZQBEZABFZGUAABHUAUBIJZPCQBTUCKPLQLMNO $.
|
|
$( [28-Nov-98] $)
|
|
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
The proper 4-variable orthoarguesian law
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The proper 4-variable OA law. $)
|
|
ax-4oa $a |- ( ( a ->1 d ) ^ ( (
|
|
( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v
|
|
( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) ) )
|
|
=< ( b ->1 d ) $.
|
|
|
|
$( The proper 4-variable OA law. $)
|
|
axoa4 $p |- ( a ' ^ ( a v ( b ^
|
|
( ( ( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v (
|
|
( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) ) ) ) ) =<
|
|
d $=
|
|
( wn wa wi1 wo u1lem9b leran ax-4oa id oa4gto4u oa4uto4 letr ) AEZABABFADGZ
|
|
BDGZFHACFQCDGZFHBCFRSFHFHFHZFQTFDPQTADIJABCDABCDRQSRQSDKQLRLSLMNO $.
|
|
$( [20-Jul-99] $)
|
|
|
|
$( Proper 4-variable OA law variant. $)
|
|
axoa4b $p |- ( ( a ->1 d ) ^ ( a v ( b ^
|
|
( ( ( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v (
|
|
( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) ) ) ) ) =<
|
|
d $=
|
|
( axoa4 oa4ctob ) ABCDABCDEF $.
|
|
$( [22-Dec-98] $)
|
|
|
|
${
|
|
oa6.1 $e |- a =< b ' $.
|
|
oa6.2 $e |- c =< d ' $.
|
|
oa6.3 $e |- e =< f ' $.
|
|
$( Derivation of 6-variable orthoarguesian law from 4-variable version. $)
|
|
oa6 $p |- ( ( ( a v b ) ^ ( c v d ) ) ^ ( e v f ) ) =<
|
|
( b v ( a ^ ( c v ( ( ( a v c ) ^ ( b v d ) ) ^
|
|
( ( ( a v e ) ^ ( b v f ) ) v ( ( c v e ) ^ ( d v f ) ) ) ) ) ) ) $=
|
|
( wn wa wo id axoa4b oa4to6 ) ABCDEFAJZBJKCJZDJKLEJZFJKLZPQRGHISMPMQMRMPQ
|
|
RSNO $.
|
|
$( [18-Dec-98] $)
|
|
$}
|
|
|
|
$( Proper 4-variable OA law variant. $)
|
|
axoa4a $p |- ( ( a ->1 d ) ^ ( a v ( b ^
|
|
( ( ( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v (
|
|
( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) ) ) ) ) =<
|
|
( ( ( a ^ d ) v ( b ^ d ) ) v ( c ^ d ) ) $=
|
|
( wi1 wn id wa wo leo df-i1 ax-r1 ax-a1 ax-r2 lbtr oa6 oa6to4 ) AADEZBBDEZC
|
|
CDEZDRFZGSFZGTFZGAFZUABFZUBCFZUCUDUDADHZIZUAFZUDUGJUHRUIRUHADKLRMNOUEUEBDHZ
|
|
IZUBFZUEUJJUKSULSUKBDKLSMNOUFUFCDHZIZUCFZUFUMJUNTUOTUNCDKLTMNOPQ $.
|
|
$( [22-Dec-98] $)
|
|
|
|
$( Proper 4-variable OA law variant. $)
|
|
axoa4d $p |- ( a ^ ( (
|
|
( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v
|
|
( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) ) )
|
|
=< ( b ' ->1 d ) $=
|
|
( wa wi1 wo wn oa4dcom ax-r1 axoa4 oa4ctod bltr ) AABEADFZBDFZEGACENCDFZEGZ
|
|
BCEOPEGZEGEZABAEONEGRQEGEZBHDFTSBACDIJBACDBACDKLM $.
|
|
$( [24-Dec-98] $)
|
|
|
|
${
|
|
4oa.1 $e |- e =
|
|
( ( ( a ^ c ) v ( ( a ->1 d ) ^ ( c ->1 d ) ) ) ^
|
|
( ( b ^ c ) v ( ( b ->1 d ) ^ ( c ->1 d ) ) ) ) $.
|
|
$( Generalized "alpha" expression. $)
|
|
4oa.2 $e |- f =
|
|
( ( ( a ^ b ) v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) v e ) $.
|
|
$( Variant of proper 4-OA. $)
|
|
4oa $p |- ( ( a ->1 d ) ^ f ) =< ( b ->1 d ) $=
|
|
( wi1 wa wo lan wn axoa4a id oa4to4u2 oa4uto4g bltr ) ADIZFJSABJSBDIZJKEK
|
|
ZJTFUASHLABCDEBMZAMZCMZDUBMDIZUCMDIZUDMDIZUEUFUGDNUEOUFOUGOPGQR $.
|
|
$( [29-Dec-98] $)
|
|
|
|
$( Proper OA analog to Godowski/Greechie, Eq. III. $)
|
|
4oaiii $p |- ( ( a ->1 d ) ^ f ) = ( ( b ->1 d ) ^ f ) $=
|
|
( wi1 wa 4oa lear ler2an wo ancom ax-r2 2or ax-r5 lebi ) ADIZFJZBDIZFJZUA
|
|
UBFABCDEFGHKTFLMUCTFBACDEFEACJTCDIZJNZBCJUBUDJNZJUFUEJGUEUFOPFABJZTUBJZNZ
|
|
ENBAJZUBTJZNZENHUIULEUGUJUHUKABOTUBOQRPKUBFLMS $.
|
|
$( [29-Dec-98] $)
|
|
|
|
$( Proper 4-OA theorem. $)
|
|
4oath1 $p |- ( ( a ->1 d ) ^ f ) = ( ( a ->1 d ) ^ ( b ->1 d ) ) $=
|
|
( wi1 wa wo 4oaiii lan or32 ax-r2 2an anidm ax-r1 anandir 3tr1 ax-a2 3tr
|
|
anabs ) ADIZFJZUDBDIZJZABJZEKZUGKZJZUGUGUIKZJUGUEUEJZUDUJJZUFUJJZJZUEUKUM
|
|
UEUFFJZJUPUEUQUEABCDEFGHLMUEUNUQUOFUJUDFUHUGKEKUJHUHUGENOZMFUJUFURMPOUMUE
|
|
UEQRUDUFUJSTUJULUGUIUGUAMUGUIUCUB $.
|
|
$( [29-Dec-98] $)
|
|
|
|
${
|
|
4oagen1.1 $e |- g =< f $.
|
|
$( "Generalized" 4-OA. $)
|
|
4oagen1 $p |- ( ( a ->1 d ) ^
|
|
( g v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) ) =
|
|
( ( a ->1 d ) ^ ( b ->1 d ) ) $=
|
|
( wi1 wa wo or32 ax-r2 lbtr leror ax-a3 oridm lor ax-r1 4oath1 lea leor
|
|
lelan ler2an lebi ) ADKZGUHBDKZLZMZLZUJULUHFLUJUKFUHUKABLZEMZUJMZUJMZFG
|
|
UOUJGFUOJFUMUJMEMUOIUMUJENOZPQUPUNUJUJMZMZFUNUJUJRUSUOFURUJUNUJSTFUOUQU
|
|
AOOPUEABCDEFHIUBPUJUHUKUHUIUCUJGUDUFUG $.
|
|
$( [29-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
4oagen1b.1 $e |- g =< f $.
|
|
4oagen1b.2 $e |- h =< ( a ->1 d ) $.
|
|
$( "Generalized" OA. $)
|
|
4oagen1b $p |- ( h ^ ( g v ( ( a ->1 d ) ^ ( b ->1 d ) ) ) ) =
|
|
( h ^ ( b ->1 d ) ) $=
|
|
( wi1 wa wo 4oagen1 anass ax-r1 ran ax-r2 lan df2le2 3tr2 ) HADMZGUDBDM
|
|
ZNZOZNZNZHUFNZHUGNZHUENZUHUFHABCDEFGIJKPUAUIHUDNZUGNZUKUNUIHUDUGQRUMHUG
|
|
HUDLUBZSTUJUMUENZULUPUJHUDUEQRUMHUEUOSTUC $.
|
|
$( [29-Dec-98] $)
|
|
$}
|
|
|
|
${
|
|
4oadist.1 $e |- h =< ( a ->1 d ) $.
|
|
4oadist.2 $e |- j =< f $.
|
|
4oadist.3 $e |- k =< f $.
|
|
4oadist.4 $e |- ( h ^ ( b ->1 d ) ) =< k $.
|
|
$( OA Distributive law. This is equivalent to the 6-variable OA law, as
|
|
shown by theorem ~ d6oa . $)
|
|
4oadist $p |- ( h ^ ( j v k ) ) = ( ( h ^ j ) v ( h ^ k ) ) $=
|
|
( wo wa wi1 ax-r1 ax-r2 le2or oridm lbtr lelan df2le2 or32 lan 4oagen1b
|
|
leo lear an32 lea bltr letr leor ledi lebi ) GHIPZQZGHQZGIQZPZUSVAVBUSU
|
|
SGBDRZQZQZVAUSUSGFQZQZVEVGUSUSVFURFGURFFPFHFIFMNUAFUBUCUDUESVFVDUSVFGAB
|
|
QZEPZADRVCQZPZQVDFVKGFVHVJPEPVKKVHVJEUFTZUGABCDEFVIGJKVIVKFVIVJUIFVKVLS
|
|
UCLUHTUGTVEVDVAUSVDUJVDVAVCQZVAVDVDIQZVMVNVDVDIOUESGVCIUKTVAVCULUMUNUMV
|
|
AUTUOUNGHIUPUQ $.
|
|
$( [29-Dec-98] $)
|
|
$}
|
|
|
|
$}
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Other stronger-than-OML laws
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
New state-related equation
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( New equation that holds in Hilbert space, discovered by Pavicic and Megill
|
|
(unpublished). $)
|
|
ax-newstateeq $a |- ( ( ( a ->1 b ) ->1 ( c ->1 b ) ) ^
|
|
( ( a ->1 c ) ^ ( b ->1 a ) ) ) =< ( c ->1 a ) $.
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Contributions of Roy Longton
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Roy's first section
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
lem3.3.2.1 $e |- a = 1 $.
|
|
lem3.3.2.2 $e |- ( a ->0 b ) = 1 $.
|
|
$( Equation 3.2 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem3.3.2 $p |- b = 1 $=
|
|
( wn wo wi0 wt df-i0 ax-r1 ax-r2 skr0 ) ABCAEBFZABGZHNMABIJDKL $.
|
|
$( [3-Jul-05] $) $( [27-Jun-05] $)
|
|
$}
|
|
|
|
$( Define asymmetrical identity (for "Non-Orthomodular Models..." paper). $)
|
|
df-id5 $a |- ( a ==5 b ) = ( ( a ^ b ) v ( a ' ^ b ' ) ) $.
|
|
|
|
$( Define biconditional for ` ->1 ` . $)
|
|
df-b1 $a |- ( a <->1 b ) = ( ( a ->1 b ) ^ ( b ->1 a ) ) $.
|
|
|
|
$( Lemma for ~ lem3.3.3 . $)
|
|
lem3.3.3lem1 $p |- ( a ==5 b ) =< ( a ->1 b ) $=
|
|
( wa wn wo wid5 wi1 ax-a2 lea leror bltr df-id5 df-i1 le3tr1 ) ABCZADZBDZCZ
|
|
EZPOEZABFABGSROETORHRPOPQIJKABLABMN $.
|
|
$( [3-Jul-05] $) $( [27-Jun-05] $)
|
|
|
|
$( Lemma for ~ lem3.3.3 . $)
|
|
lem3.3.3lem2 $p |- ( a ==5 b ) =< ( b ->1 a ) $=
|
|
( wa wn wo wid5 wi1 lear leror ax-a2 ancom lor le3tr1 df-id5 df-i1 ) ABCZAD
|
|
ZBDZCZEZRBACZEZABFBAGSPERPETUBSRPQRHIPSJUAPRBAKLMABNBAOM $.
|
|
$( [3-Jul-05] $) $( [27-Jun-05] $)
|
|
|
|
$( Lemma for ~ lem3.3.3 . $)
|
|
lem3.3.3lem3 $p |- ( a ==5 b ) =< ( ( a ->1 b ) ^ ( b ->1 a ) ) $=
|
|
( wid5 wi1 lem3.3.3lem1 lem3.3.3lem2 ler2an ) ABCABDBADABEABFG $.
|
|
$( [3-Jul-05] $) $( [27-Jun-05] $)
|
|
|
|
$( Equation 3.3 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem3.3.3 $p |- ( ( a ==5 b ) ->0 ( a <->1 b ) ) = 1 $=
|
|
( wid5 wb1 wi0 wn wo wi1 wa wt df-i0 df-b1 lor lem3.3.3lem3 sklem 3tr ) ABC
|
|
ZABDZEQFZRGSABHBAHIZGJQRKRTSABLMQTABNOP $.
|
|
$( [3-Jul-05] $) $( [27-Jun-05] $)
|
|
|
|
${
|
|
lem3.3.4.1 $e |- ( b ->2 a ) = 1 $.
|
|
$( Equation 3.4 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem3.3.4 $p |- ( a ->2 ( a ==5 b ) ) = ( a ==5 b ) $=
|
|
( wid5 wi2 wn wa wo df-i2 df-id5 ax-r4 lan anor3 ax-r1 ax-r2 lor 3tr1 3tr
|
|
wf wt oran3 oran 2an anabs ran anass con2 ancom oran1 con3 df-f 3tr2 or0r
|
|
ax-a2 ) AABDZEUOAFZUOFZGZHZSUOHZUOAUOIUSUOSHUTURSUOURUPABGZUPBFZGZHZFZGUP
|
|
UPVBHZABHZGZGZSUQVEUPUOVDABJKLVEVHUPVEVAFZVCFZGZVHVLVEVAVCMNVJVFVKVGVFVJA
|
|
BUANVGVKABUBZNUCOLUPVFGZVGGUPVGGZVISVNUPVGUPVBUDUEUPVFVGUFBAEZFTFVOSVPTCK
|
|
VOVPAVGFZHZAVBUPGZHVOFZVPVQVSAVQVCVSVGVCVMUGUPVBUHOPVRVTAVGUINBAIQUJUKQUL
|
|
RPUOSUNOUOUMR $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
$}
|
|
|
|
${
|
|
lem3.3.5lem.1 $e |- 1 =< a $.
|
|
$( A fundamental property in quantum logic. Lemma for ~ lem3.3.5 . $)
|
|
lem3.3.5lem $p |- a = 1 $=
|
|
( wt le1 lebi ) ACADBE $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
$}
|
|
|
|
${
|
|
lem3.3.5.1 $e |- ( a ==5 b ) = 1 $.
|
|
$( Equation 3.5 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem3.3.5 $p |- ( a ->1 ( b v c ) ) = 1 $=
|
|
( wo wi1 wb1 wn wa wt df-b1 lea bltr df-i1 lbtr leo lelan lelor letr wid5
|
|
lem3.3.3 lem3.3.2 ax-r1 le3tr1 lem3.3.5lem ) ABCEZFZABGZAHZAUFIZEZJUGUHUI
|
|
ABIZEZUKUHABFZUMUHUNBAFZIUNABKUNUOLMABNOULUJUIBUFABCPQRSUHJABTUHDABUAUBUC
|
|
AUFNUDUE $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
$}
|
|
|
|
$( Equation 3.6 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem3.3.6 $p |- ( a ->2 ( b v c ) ) = ( ( a v c ) ->2 ( b v c ) ) $=
|
|
( wo wn wa wi2 anor3 ax-r1 lan anandir anass 2an 3tr2 ax-r2 lor df-i2 3tr1
|
|
) BCDZAEZSEZFZDSACDZEZUAFZDASGUCSGUBUESUBTBEZCEZFZFZUEUAUHTUHUABCHZIJTUFFUG
|
|
FTUGFZUHFUIUETUFUGKTUFUGLUKUDUHUAACHUJMNOPASQUCSQR $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
0, and this is the first part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i0e1 $p |- ( a ->0 ( a ^ b ) ) = ( a ==0 ( a ^ b ) ) $=
|
|
( wn wa wo wi0 wid0 or1 ax-r1 lan an1 df-t lor 3tr2 ax-a2 ax-a3 ax-r5 oran3
|
|
wt 3tr df-i0 df-id0 3tr1 ) ACZABDZEZUFUECZAEZDZAUEFAUEGUFUFBCZUDEZAEZDZUFUD
|
|
UJEZAEZDUIUFUFUJAUDEZEZDZUFUJUDAEZEZDUMUFSDUFUJSEZDUFURSVAUFVASUJHIJUFKVAUQ
|
|
UFSUPUJALMJNUQUTUFUPUSUJAUDOMJUTULUFULUTUJUDAPIJTULUOUFUKUNAUJUDOQJUOUHUFUN
|
|
UGAABRQJTAUEUAAUEUBUC $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
0, and this is the second part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i0e2 $p |- ( a ==0 ( a ^ b ) ) = ( ( a ^ b ) ==0 a ) $=
|
|
( wn wa wo wid0 ancom df-id0 3tr1 ) ACABDZEZJCAEZDLKDAJFJAFKLGAJHJAHI $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
0, and this is the third part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i0e3 $p |- ( a ->0 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( nom10 ) ABC $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
1, and this is the first part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i1e1 $p |- ( a ->1 ( a ^ b ) ) = ( a ==1 ( a ^ b ) ) $=
|
|
( wn wa wo wi1 wid1 or1r ax-r1 ran an1r df-t ax-r5 3tr2 ax-a3 oran3 lor 3tr
|
|
wt df-i1 df-id1 3tr1 ) ACZAABDZDEZAUDCZEZUEDZAUDFAUDGUEAUCEZBCZEZUEDZAUCUJE
|
|
ZEZUEDUHSUEDSUJEZUEDUEULSUOUEUOSUJHIJUEKUOUKUESUIUJALMJNUKUNUEAUCUJOJUNUGUE
|
|
UMUFAABPQJRAUDTAUDUAUB $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
1, and this is the second part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i1e2 $p |- ( a ==1 ( a ^ b ) ) = ( ( a ^ b ) ==1 a ) $=
|
|
( wa wn wo wid1 oran3 ax-r1 lor ran ax-a3 wt df-t ax-r5 anass ax-a2 lan 3tr
|
|
or1r df-id1 an1r anidm an1 ancom 3tr1 ) AABCZDZEZADZAUFCZEZCZUFUIEZUGUFACZE
|
|
ZCZAUFFUFAFULAUIBDZEZEZUKCAUIEZUQEZUKCZUPUHUSUKUGURAURUGABGHIJUSVAUKVAUSAUI
|
|
UQKHJVBLUQEZUKCLUKCZUPVAVCUKUTLUQLUTAMHNJVCLUKUQSJVDUKUIAACZBCZEZUPUKUAUJVF
|
|
UIVFUJAABOZHIVGUIUFEZUMUGABACZCZEZCZUPVFUFUIVEABAUBZJIVIUMUGVFEZCZUMUGUJEZC
|
|
VMVIUMUFUGEZCZUMUGUFEZCVPVIUMUMLCZVSUIUFPWAUMUMUCHLVRUMUFMQRVRVTUMUFUGPQVTV
|
|
OUMUFVFUGAVEBVEAVNHJIQRVOVQUMVFUJUGVHIQVQVLUMUJVKUGUFVJAABUDQIQRVLUOUMVKUNU
|
|
GUNVKABAOHIQRRRRAUFTUFATUE $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
1, and this is the third part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i1e3 $p |- ( a ->1 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( nom11 ) ABC $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
2, and this is the first part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i2e1 $p |- ( a ->2 ( a ^ b ) ) = ( a ==2 ( a ^ b ) ) $=
|
|
( wa wn wo wi2 wid2 or1r ax-r1 ran an1r df-t ax-r5 3tr2 ax-a3 oran3 lor 3tr
|
|
wt df-i2 df-id2 3tr1 ) ABCZADZUCDZCEZAUEEZUFCZAUCFAUCGUFAUDEZBDZEZUFCZAUDUJ
|
|
EZEZUFCUHSUFCSUJEZUFCUFULSUOUFUOSUJHIJUFKUOUKUFSUIUJALMJNUKUNUFAUDUJOJUNUGU
|
|
FUMUEAABPQJRAUCTAUCUAUB $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
2, and this is the second part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i2e2 $p |- ( a ==2 ( a ^ b ) ) = ( ( a ^ b ) ==2 a ) $=
|
|
( wa wn wo wid2 oran3 ax-r1 lor ran ax-a3 wt df-t ax-r5 anor3 ax-r4 lan 3tr
|
|
ax-r2 df-id2 or1r an1r orabs an1 lea df-le2 3tr1 ) AABCZDZEZUHADZUICZEZCZUH
|
|
UKEZAUIUKCZEZCZAUHFUHAFUNAUKBDZEZEZUMCAUKEZUSEZUMCZURUJVAUMUIUTAUTUIABGHIJV
|
|
AVCUMVCVAAUKUSKHJVDLUSEZUMCLUMCZURVCVEUMVBLUSLVBAMZHNJVELUMUSUAJVFUMUHAUHEZ
|
|
DZEZURUMUBULVIUHAUHOIVJUOURVIUKUHVHAABUCPIUOUOVBCZUOAUHAEZDZEZCURUOUOLCZVKV
|
|
OUOUOUDHLVBUOVGQSVBVNUOUKVMAAVLVLAUHAABUEUFHPIQVNUQUOVMUPAUPVMUHAOHIQRSRRRA
|
|
UHTUHATUG $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
2, and this is the third part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i2e3 $p |- ( a ->2 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( nom12 ) ABC $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
3, and this is the first part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i3e1 $p |- ( a ->3 ( a ^ b ) ) = ( a ==3 ( a ^ b ) ) $=
|
|
( wn wa wo wi3 wid3 anass ax-r1 ax-r5 ancom ran wf dff ax-r4 wt lan 3tr lor
|
|
an0r or0r anor3 orabs womaa an1 df-t ax-r2 df-i3 df-id3 3tr1 ) ACZABDZDZUKU
|
|
LCDZEZAUKULEZDZEZUPAUNEZDZAULFAULGURUKADZBDZUNEZUQEAUKDZBDZUNEZUQEZUTUOVCUQ
|
|
UMVBUNVBUMUKABHIJJVCVFUQVBVEUNVAVDBUKAKLJJVGMBDZUNEZUQEMUNEZUQEZUTVFVIUQVEV
|
|
HUNVDMBMVDANILJJVIVJUQVHMUNBTJJVKUNUQEZUTVJUNUQUNUAJVLAULEZCZUQEUKUQEZUTUNV
|
|
NUQAULUBZJVNUKUQVMAABUCZOJVOUPAUKEZDZUPAVNEZDUTVOUPUPPDZVSABUDWAUPUPUEIPVRU
|
|
PAUFQRVRVTUPUKVNAAVMVMAVQIOSQVTUSUPVNUNAUNVNVPISQRRUGRRAULUHAULUIUJ $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
3, and this is the second part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i3e2 $p |- ( a ==3 ( a ^ b ) ) = ( ( a ^ b ) ==3 a ) $=
|
|
( wn wa wo wid3 wt anor3 lor lan orabs ax-r4 df-t ax-r1 an1 ax-a2 3tr ax-r5
|
|
ran df-id3 lea df-le2 an1r ax-r2 or1 ax-a3 oran3 3tr1 ) ACZABDZEZAUIUJCZDZE
|
|
ZDZULAEZUJULUIDZEZDZAUJFUJAFUOBCZUIEZAEZURDZUIUTEZAEZURDUSUOUTAUIEZEZURDZUT
|
|
UIAEZEZURDVCUOGURDZUTGEZURDVHUOURVKUOUJUIEZUJUJAEZCZEURUOUKAAUJEZCZEZDUKVFD
|
|
ZVMUNVRUKUMVQAAUJHIJVRVFUKVQUIAVPAABKLIJVSUKGDUKVMVFGUKGVFAMZNJUKOUIUJPQQUI
|
|
VOUJAVNVNAUJAABUAUBNLIVOUQUJUQVOUJAHNIQVKURURUCNUDGVLURVLGUTUENSVLVGURGVFUT
|
|
VTISQVGVJURVFVIUTAUIPISVJVBURVBVJUTUIAUFNSQVBVEURVAVDAUTUIPRSVEUPURVDULAABU
|
|
GRSQAUJTUJATUH $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
3, and this is the third part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i3e3 $p |- ( a ->3 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( nom13 ) ABC $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
4, and this is the first part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i4e1 $p |- ( a ->4 ( a ^ b ) ) = ( a ==4 ( a ^ b ) ) $=
|
|
( wa wn wo wi4 wid4 lear lea ler2an lebi ax-r5 wt wf lor lel2or leo 3tr lan
|
|
ax-r1 leid lecon ortha or0 leor lerr an1 sklem df-i4 df-id4 3tr1 ) AABCZCZA
|
|
DZULCZEZUNULEZULDZCZEZUQURUMEZCZAULFAULGUTULUOEZUSEZUQURULEZCZVBUPVCUSUMULU
|
|
OUMULAULHZULAULABIZULUAZJZKLLVDUQUQMCZVFVDULNEZUSEULUSEZUQVCVLUSUONULUNULUL
|
|
AVHUBZUCOLVLULUSULUDLVMUQULUQUSULUNUEUQURIPUNVMULUNUSULUNUQURUNULQVNJUFULUS
|
|
QPKRVKUQUQUGTMVEUQVEMULULVIUHTSRVEVAUQULUMURULUMVJVGKOSRAULUIAULUJUK $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
4, and this is the second part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i4e2 $p |- ( a ==4 ( a ^ b ) ) = ( ( a ^ b ) ==4 a ) $=
|
|
( wn wa wo wid4 wt lear lea leid ler2an lebi lor lan sklem an1 df2le2 ax-r1
|
|
3tr df-id4 an1r ax-r2 ran 3tr1 ) ACZABDZEZUFCZAUFDZEZDZUHAEZUEUFADZEZDZAUFF
|
|
UFAFUKUGUHUFEZDUGGDZUOUJUPUGUIUFUHUIUFAUFHUFAUFABIZUFJZKLMNUPGUGUFUFUSONUQU
|
|
GGUNDZUOUGPUGUNUTUFUMUEUMUFUFAURQRMUTUNUNUARUBGULUNULGUFAURORUCSSAUFTUFATUD
|
|
$.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
4, and this is the third part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i4e3 $p |- ( a ->4 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( nom14 ) ABC $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
5, and this is the first part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i5e1 $p |- ( a ->5 ( a ^ b ) ) = ( a ==5 ( a ^ b ) ) $=
|
|
( wa wn wo wi5 wid5 wf lear lea leid ler2an lebi lecon ortha 2or or0 df2le2
|
|
ax-r5 ax-r1 3tr df-i5 df-id5 3tr1 ) AABCZCZADZUECZEZUGUEDZCZEZUFUKEZAUEFAUE
|
|
GULUEHEZUKEUEUGEUMUIUNUKUFUEUHHUFUEAUEIZUEAUEABJZUEKLZMUGUEUEAUPNZOPSUNUEUK
|
|
UGUEQUGUJURRZPUEUFUGUKUEUFUQUOMUKUGUSTPUAAUEUBAUEUCUD $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
5, and this is the second part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i5e2 $p |- ( a ==5 ( a ^ b ) ) = ( ( a ^ b ) ==5 a ) $=
|
|
( wa wn wo wid5 ancom 2or ax-r1 df-id5 3tr1 ) AABCZCZADZLDZCZEZLACZONCZEZAL
|
|
FLAFTQRMSPLAGONGHIALJLAJK $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( Equation 3.7 of [PavMeg1999] p. 9. The variable i in the paper is set to
|
|
5, and this is the third part of the equation. (Contributed by Roy F.
|
|
Longton, 3-Jul-05.) $)
|
|
lem3.3.7i5e3 $p |- ( a ->5 ( a ^ b ) ) = ( a ->1 b ) $=
|
|
( nom15 ) ABC $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$(
|
|
lem3.3.8i0e1 $p |- ( ( a v b ) ->0 b ) = ( ( a v b ) ==0 b ) $= ? $.
|
|
|
|
lem3.3.8i0e2 $p |- ( ( a v b ) ==0 b ) = ( b ==0 ( a v b ) ) $= ? $.
|
|
|
|
lem3.3.8i0e3 $p |- ( ( a v b ) ->0 b ) = ( a ->2 b ) $=
|
|
wva wvb nom40 $.
|
|
|
|
lem3.3.8i1e1 $p |- ( ( a v b ) ->1 b ) = ( ( a v b ) ==1 b ) $= ? $.
|
|
|
|
lem3.3.8i1e2 $p |- ( ( a v b ) ==1 b ) = ( b ==1 ( a v b ) ) $= ? $.
|
|
|
|
lem3.3.8i1e3 $p |- ( ( a v b ) ->1 b ) = ( a ->2 b ) $=
|
|
wva wvb nom41 $.
|
|
|
|
lem3.3.8i2e1 $p |- ( ( a v b ) ->2 b ) = ( ( a v b ) ==2 b ) $= ? $.
|
|
|
|
lem3.3.8i2e2 $p |- ( ( a v b ) ==2 b ) = ( b ==2 ( a v b ) ) $= ? $.
|
|
|
|
lem3.3.8i2e3 $p |- ( ( a v b ) ->2 b ) = ( a ->2 b ) $=
|
|
wva wvb nom42 $.
|
|
|
|
lem3.3.8i3e1 $p |- ( ( a v b ) ->3 b ) = ( ( a v b ) ==3 b ) $= ? $.
|
|
|
|
lem3.3.8i3e2 $p |- ( ( a v b ) ==3 b ) = ( b ==3 ( a v b ) ) $= ? $.
|
|
|
|
lem3.3.8i3e3 $p |- ( ( a v b ) ->3 b ) = ( a ->2 b ) $= ? $.
|
|
|
|
lem3.3.8i4e1 $p |- ( ( a v b ) ->4 b ) = ( ( a v b ) ==4 b ) $= ? $.
|
|
|
|
lem3.3.8i4e2 $p |- ( ( a v b ) ==4 b ) = ( b ==4 ( a v b ) ) $= ? $.
|
|
|
|
lem3.3.8i4e3 $p |- ( ( a v b ) ->4 b ) = ( a ->2 b ) $= ? $.
|
|
|
|
lem3.3.8i5e1 $p |- ( ( a v b ) ->5 b ) = ( ( a v b ) ==5 b ) $= ? $.
|
|
|
|
lem3.3.8i5e2 $p |- ( ( a v b ) ==5 b ) = ( b ==5 ( a v b ) ) $= ? $.
|
|
|
|
lem3.3.8i5e3 $p |- ( ( a v b ) ->5 b ) = ( a ->2 b ) $= ? $.
|
|
$)
|
|
$( [28-Jun-05] $)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Roy's second section
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( Equation 3.9 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem3.4.1 $p |- ( ( a ->1 b ) ->0 ( a ->2 b ) ) = 1 $=
|
|
( wi1 wi2 wi0 wn wo wt df-i0 woml6 ax-r2 ) ABCZABDZELFMGHLMIABJK $.
|
|
$( [3-Jul-05] $) $( [28-Jun-05] $)
|
|
|
|
$( lem3.4.2 is 2vwomr1a and 2vwomr2a $)
|
|
|
|
${
|
|
lem3.4.3.1 $e |- ( a ->2 b ) = 1 $.
|
|
$( Equation 3.11 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem3.4.3 $p |- ( a ->2 ( a ==5 b ) ) = 1 $=
|
|
( wid5 wi1 wt 2vwomr2a ax-r1 wn wa wo anidm ran lea lel leran ler2an bltr
|
|
ler df-i1 df-id5 lan lbtr lelor le3tr1 lem3.3.5lem 2vwomr1a ) AABDZAUHEZF
|
|
ABEZUIUJFABCGHAIZABJZKUKAUHJZKUJUIULUMUKULAULUKBIJZKZJZUMULAAJZBJZUPAUQBU
|
|
QAALHMURAUOUQABAANZOURULUNUQABUSPSQRUOUHAUHUOABUAHUBUCUDABTAUHTUERUFUG $.
|
|
$( [3-Jul-05] $) $( [29-Jun-05] $)
|
|
$}
|
|
|
|
${
|
|
lem3.4.4.1 $e |- ( a ->2 b ) = 1 $.
|
|
lem3.4.4.2 $e |- ( b ->2 a ) = 1 $.
|
|
$( Equation 3.12 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem3.4.4 $p |- ( a ==5 b ) = 1 $=
|
|
( wid5 wi2 wt lem3.3.4 ax-r1 lem3.4.3 ax-r2 ) ABEZALFZGMLABDHIABCJK $.
|
|
$( [3-Jul-05] $) $( [29-Jun-05] $)
|
|
$}
|
|
|
|
${
|
|
lem3.4.5.1 $e |- ( a ==5 b ) = 1 $.
|
|
$( Equation 3.13 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem3.4.5 $p |- ( a ->2 ( b v c ) ) = 1 $=
|
|
( wo lem3.3.5 2vwomr1a ) ABCEABCDFG $.
|
|
$( [3-Jul-05] $) $( [29-Jun-05] $)
|
|
$}
|
|
|
|
${
|
|
lem3.4.6.1 $e |- ( a ==5 b ) = 1 $.
|
|
$( Equation 3.14 of [PavMeg1999] p. 9. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem3.4.6 $p |- ( ( a v c ) ==5 ( b v c ) ) = 1 $=
|
|
( wo wi2 wt lem3.3.6 ax-r1 lem3.4.5 ax-r2 wid5 wa wn df-id5 ancom 2or 3tr
|
|
lem3.4.4 ) ACEZBCEZTUAFZAUAFZGUCUBABCHIABCDJKUATFZBTFZGUEUDBACHIBACBALBAM
|
|
ZBNZANZMZEZGBAOUJABMZUHUGMZEZABLZGUFUKUIULBAPUGUHPQUNUMABOIDRKJKS $.
|
|
$( [3-Jul-05] $) $( [29-Jun-05] $)
|
|
$}
|
|
|
|
$(
|
|
@( Lemma intended for ~ thm3.8i1 . @)
|
|
thm3.8i1lem @p |- ( a ==1 b ) = ( ( b ->0 a ) ^ ( a ->1 b ) ) @=
|
|
wva wvb wn wo wva wn wva wvb wa wo wa wvb wn wva wo wva wn wva wvb wa wo wa
|
|
wva wvb wid1 wvb wva wi0 wva wvb wi1 wa wva wvb wn wo wvb wn wva wo wva wn
|
|
wva wvb wa wo wva wvb wn ax-a2 ran wva wvb df-id1 wvb wva wi0 wvb wn wva wo
|
|
wva wvb wi1 wva wn wva wvb wa wo wvb wva df-i0 wva wvb df-i1 2an 3tr1 @.
|
|
@( [31-Mar-2011] @) @( [30-Jun-05] @)
|
|
|
|
@{
|
|
thm3.8i1.1 @e |- ( a ==1 b ) = 1 @.
|
|
thm3.8i1 @p |- ( ( a v c ) ==1 ( b v c ) ) = 1 @= ? @.
|
|
@}
|
|
|
|
@{
|
|
thm3.8i2.1 @e |- ( a ==2 b ) = 1 @.
|
|
thm3.8i2 @p |- ( ( a v c ) ==2 ( b v c ) ) = 1 @= ? @.
|
|
@}
|
|
|
|
@{
|
|
thm3.8i3.1 @e |- ( a ==3 b ) = 1 @.
|
|
thm3.8i3 @p |- ( ( a v c ) ==3 ( b v c ) ) = 1 @= ? @.
|
|
@}
|
|
|
|
@{
|
|
thm3.8i4.1 @e |- ( a ==4 b ) = 1 @.
|
|
thm3.8i4 @p |- ( ( a v c ) ==4 ( b v c ) ) = 1 @= ? @.
|
|
@}
|
|
|
|
@{
|
|
thm3.8i5.1 @e |- ( a ==5 b ) = 1 @.
|
|
thm3.8i5 @p |- ( ( a v c ) ==5 ( b v c ) ) = 1 @=
|
|
wva wvb wvc thm3.8i5.1 lem3.4.6 @.
|
|
@( [31-Mar-2011] @) @( [29-Jun-05] @)
|
|
@}
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Roy's third section
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( lem4.6.1 is u1lemaa $)
|
|
|
|
$( Equation 4.10 of [MegPav2000] p. 23. This is the first part of the
|
|
equation. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.2e1 $p |- ( ( a ->1 b ) ^ ( a ' ->1 b ) ) = ( ( a ->1 b ) ^ b ) $=
|
|
( wi1 wn wa wo df-i1 2an ax-a1 ax-r1 ax-r5 lan comcom fh1 lor coman1 coman2
|
|
ancom ran 3tr comorr comcom6 leao1 lecom comcom7 com2an anass anidm comcom2
|
|
omla orabs fh3 ax-a2 lear df-le2 ) ABCZADZBCZEUQABEZFZUQDZUQBEZFZEZUTBEZUPB
|
|
EUPUTURVCABGZUQBGHVDUTAVBFZEUTAEZUTVBEZFZVEVCVGUTVAAVBAVAAIJKLUTAVBAUTAUTUQ
|
|
USUAUBMVBUTVBUTUQBUSUCUDMNVJAUTEZVIFUSVIFZVEVHVKVIUTARKVKUSVIABUJKVLUSVBUTE
|
|
ZFUSVBUQEZVBUSEZFZFZVEVIVMUSUTVBROVMVPUSVBUQUSUQBPZVBABVBAVRUEUQBQUFNOVQUSB
|
|
UQEZUQEZVOFZFUSBUQUQEZEZVOFZFZVEVPWAUSVNVTVOVBVSUQUQBRSKOWAWDUSVTWCVOBUQUQU
|
|
GKOWEUSVSVOFZFUSVBVOFZFZVEWDWFUSWCVSVOWBUQBUQUHLKOWFWGUSVSVBVOBUQRKOWHUSVBF
|
|
USUQFZUSBFZEVEWGVBUSVBUSUKOUSUQBUSAABPUIABQULWIUTWJBUSUQUMUSBABUNUOHTTTTTTU
|
|
TUPBUPUTVFJST $.
|
|
$( [3-Jul-05] $) $( [29-Jun-05] $)
|
|
|
|
$( Equation 4.10 of [MegPav2000] p. 23. This is the second part of the
|
|
equation. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.2e2 $p |- ( ( a ->1 b ) ^ b ) = ( ( a ^ b ) v ( a ' ^ b ) ) $=
|
|
( u1lemab ) ABC $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.11 of [MegPav2000] p. 23. This is the first part of the
|
|
equation. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.3le1 $p |- ( a ' ->1 b ) ' =< a ' $=
|
|
( u1lem9a ) ABC $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.11 of [MegPav2000] p. 23. This is the second part of the
|
|
equation. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.3le2 $p |- a ' =< ( a ->1 b ) $=
|
|
( u1lem9b ) ABC $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.12 of [MegPav2000] p. 23. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem4.6.4 $p |- ( ( a ->1 b ) ->1 b ) = ( a ' ->1 b ) $=
|
|
( u1lem12 ) ABC $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.13 of [MegPav2000] p. 23. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem4.6.5 $p |- ( ( a ->1 b ) ' ->1 b ) = ( a ->1 b ) $=
|
|
( wi1 wn u1lemn1b ax-r1 ) ABCZGDBCABEF $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 0, and j is set to 1. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i0j1 $p |- ( ( a ->0 b ) v ( a ->1 b ) ) = ( a ->0 b ) $=
|
|
( wn wo wa wi0 wi1 leid lear lelor lel2or leo lebi df-i0 df-i1 2or 3tr1 ) A
|
|
CZBDZRABEZDZDZSABFZABGZDUCUBSSSUASHTBRABIJKSUALMUCSUDUAABNZABOPUEQ $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 0, and j is set to 2. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i0j2 $p |- ( ( a ->0 b ) v ( a ->2 b ) ) = ( a ->0 b ) $=
|
|
( wn wo wa wi0 wi2 leid leor leao1 lel2or leo lebi df-i0 df-i2 2or 3tr1 ) A
|
|
CZBDZBRBCZEZDZDZSABFZABGZDUDUCSSSUBSHBSUABRIRTBJKKSUBLMUDSUEUBABNZABOPUFQ
|
|
$.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 0, and j is set to 3. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i0j3 $p |- ( ( a ->0 b ) v ( a ->3 b ) ) = ( a ->0 b ) $=
|
|
( wn wo wa wi0 wi3 leid leao1 lel2or lear leo lebi df-i0 df-i3 2or 3tr1 ) A
|
|
CZBDZRBEZRBCZEZDZASEZDZDZSABFZABGZDUGUFSSSUESHUCSUDTSUBRBBIRUABIJASKJJSUELM
|
|
UGSUHUEABNZABOPUIQ $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 0, and j is set to 4. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i0j4 $p |- ( ( a ->0 b ) v ( a ->4 b ) ) = ( a ->0 b ) $=
|
|
( wn wo wi0 wi4 leid leao4 leao1 lel2or lea leo lebi df-i0 df-i4 2or 3tr1
|
|
wa ) ACZBDZABRZSBRZDZTBCZRZDZDZTABEZABFZDUHUGTTTUFTGUCTUEUATUBBASHSBBIJTUDK
|
|
JJTUFLMUHTUIUFABNZABOPUJQ $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 1, and j is set to 0. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i1j0 $p |- ( ( a ->1 b ) v ( a ->0 b ) ) = ( a ->0 b ) $=
|
|
( wn wa wo wi1 wi0 lear lelor df-le2 df-i1 df-i0 2or 3tr1 ) ACZABDZEZOBEZER
|
|
ABFZABGZETQRPBOABHIJSQTRABKABLZMUAN $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 1, and j is set to 2. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i1j2 $p |- ( ( a ->1 b ) v ( a ->2 b ) ) = ( a ->0 b ) $=
|
|
( u12lem ) ABC $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 1, and j is set to 3. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i1j3 $p |- ( ( a ->1 b ) v ( a ->3 b ) ) = ( a ->0 b ) $=
|
|
( wn wa wo wi1 wi3 ler lecom lea lel2or ax-a3 ax-a2 ran ax-r1 wt lor df-le2
|
|
ax-r5 3tr wi0 leo comcom6 comcom lear lelor ax-a4 df-le1 lem3.3.5lem orordi
|
|
fh3 an1r ax-r2 3tr2 df-i1 df-i3 2or df-i0 3tr1 ) ACZABDZEZUTBDZUTBCZDZEZAUT
|
|
BEZDZEZEZVGABFZABGZEABUAVBVFEZVHEVMAEZVMVGEZDZVJVGVMAVGAVMAVMUTVMUTVBVFUTVA
|
|
UBHIUCUDVMVGVBVGVFVABUTABUEZUFVFUTBVCUTVEUTBJUTVDJKHZKIUKVBVFVHLVPAVMEZVODA
|
|
VBEZVFEZVODZVGVNVSVOVMAMNVSWAVOWAVSAVBVFLONWBAUTEZVAEZVFEZVODPVODZVGWAWEVOV
|
|
TWDVFWDVTAUTVALOSNWEPVOWEPWDVFPWCVAPWCPAUGUHHHUINWFVOVFVBEZVGEZVGVOULVMWGVG
|
|
VBVFMSWHVFVBVGEZEZVGVFVBVGLWJVFUTVABEZEZEVFVGEVGWIWLVFWLWIUTVABUJOQWLVGVFWK
|
|
BUTVABVQRQQVFVGVRRTUMTTTUNVKVBVLVIABUOABUPUQABURUS $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 2, and j is set to 0. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i2j0 $p |- ( ( a ->2 b ) v ( a ->0 b ) ) = ( a ->0 b ) $=
|
|
( wn wa wo wi2 wi0 leor leao1 lel2or df-le2 df-i2 df-i0 2or 3tr1 ) BACZBCZD
|
|
ZEZPBEZETABFZABGZEUBSTBTRBPHPQBIJKUASUBTABLABMZNUCO $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 2, and j is set to 1. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i2j1 $p |- ( ( a ->2 b ) v ( a ->1 b ) ) = ( a ->0 b ) $=
|
|
( wn wa wo wi2 wi1 wi0 leor leao1 lel2or lear lelor leo lerr ler lebi df-i2
|
|
df-i1 2or df-i0 3tr1 ) BACZBCZDZEZUCABDZEZEZUCBEZABFZABGZEABHUIUJUFUJUHBUJU
|
|
EBUCIUCUDBJKUGBUCABLMKUCUIBUCUHUFUCUGNOBUFUHBUENPKQUKUFULUHABRABSTABUAUB $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 2, and j is set to 4. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i2j4 $p |- ( ( a ->2 b ) v ( a ->4 b ) ) = ( a ->0 b ) $=
|
|
( wn wa wo wi2 wi4 wi0 ax-a2 ax-r5 ax-a3 ax-r1 lor ancom lan oml 3tr lel2or
|
|
ax-r2 leao1 leao4 leid leor lerr lebi df-i2 df-i4 2or df-i0 3tr1 ) BACZBCZD
|
|
ZEZABDZUKBDZEZUKBEZULDZEZEZURABFZABGZEABHVAUMBEZUTEUMBUTEZEZURUNVDUTBUMIJUM
|
|
BUTKVFUMBUQEZUSEZEUMUQBEZUSEZEZURVEVHUMVHVEBUQUSKLMVHVJUMVGVIUSBUQIJMVKUMUQ
|
|
BUSEZEZEUMUQUREZEZURVJVMUMUQBUSKMVMVNUMVLURUQVLBULURDZEBULBUKEZDZEZURUSVPBU
|
|
RULNMVPVRBURVQULUKBIOMVSVQURBUKPBUKISQMMVOURUMURVNUKULBTUQURURUOURUPBAUKUAU
|
|
KBBTRURUBRRURVNUMURUQUCUDUEQQQVBUNVCUTABUFABUGUHABUIUJ $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 3, and j is set to 0. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i3j0 $p |- ( ( a ->3 b ) v ( a ->0 b ) ) = ( a ->0 b ) $=
|
|
( wn wa wi3 wi0 ax-a3 ax-r1 lor ax-a2 omln ax-r2 ax-r5 leid leor lel2or leo
|
|
wo leao1 3tr lebi df-le2 df-i3 df-i0 2or 3tr1 ) ACZBDZUGBCZDZRZAUGBRZDZRZUL
|
|
RZULABEZABFZRUQUOUKUMULRZRUKUMUGRZBRZRZULUKUMULGURUTUKUTURUMUGBGHIVAUKULBRZ
|
|
RUKULRULUTVBUKUSULBUSUGUMRULUMUGJABKLMIVBULUKVBULULULBULNBUGOPULBQUAIUKULUH
|
|
ULUJUGBBSUGUIBSPUBTTUPUNUQULABUCABUDZUEVCUF $.
|
|
$( [3-Jul-05] $) $( [1-Jul-05] $)
|
|
|
|
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
|
|
to 3, and j is set to 1. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
|
lem4.6.6i3j1 $p |- ( ( a ->3 b ) v ( a ->1 b ) ) = ( a ->0 b ) $=
|
|
( wn wa wo wi3 wi1 wi0 ax-a3 ax-r1 ax-a2 omln ax-r2 ax-r5 leao1 lel2or leid
|
|
lor leao4 leo lerr lebi 3tr df-i3 df-i1 2or df-i0 3tr1 ) ACZBDZUIBCZDZEZAUI
|
|
BEZDZEZUIABDZEZEZUNABFZABGZEABHUSUMUOUREZEUMUOUIEZUQEZEZUNUMUOURIVBVDUMVDVB
|
|
UOUIUQIJRVEUMUNUQEZEZUNVDVFUMVCUNUQVCUIUOEUNUOUIKABLMNRVGUNUMUNVFUJUNULUIBB
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OUIUKBOPUNUNUQUNQBAUISPPUNVFUMUNUQTUAUBMUCUTUPVAURABUDABUEUFABUGUH $.
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$( [3-Jul-05] $) $( [1-Jul-05] $)
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$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
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|
to 4, and j is set to 0. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
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lem4.6.6i4j0 $p |- ( ( a ->4 b ) v ( a ->0 b ) ) = ( a ->0 b ) $=
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( wa wn wo wi4 wi0 leao4 leao1 lel2or lea df-le2 df-i4 df-i0 2or 3tr1 ) ABC
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ZADZBCZEZRBEZBDZCZEZUAEUAABFZABGZEUFUDUATUAUCQUASBARHRBBIJUAUBKJLUEUDUFUAAB
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MABNZOUGP $.
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$( [3-Jul-05] $) $( [2-Jul-05] $)
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$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
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|
to 4, and j is set to 2. (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
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lem4.6.6i4j2 $p |- ( ( a ->4 b ) v ( a ->2 b ) ) = ( a ->0 b ) $=
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( wa wn wi4 wi2 wi0 ax-a3 ax-r1 ax-a2 ancom lor leor oml2 ax-r5 ax-r2 leao1
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wo 3tr lel2or leao4 leid leo lerr lebi df-i4 df-i2 2or df-i0 3tr1 ) ABCZADZ
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BCZRZULBRZBDZCZRZBULUPCZRZRZUOABEZABFZRABGVAUNUQUTRZRUNUOUSRZRZUOUNUQUTHVDV
|
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EUNVDUQBRZUSRZVEVHVDUQBUSHIVGUOUSVGBUQRBUPUOCZRUOUQBJUQVIBUOUPKLBUOBULMNSOP
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LVFUOUNUOVEUKUOUMBAULUAULBBQTUOUOUSUOUBULUPBQTTUOVEUNUOUSUCUDUESVBURVCUTABU
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FABUGUHABUIUJ $.
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$( [3-Jul-05] $) $( [2-Jul-05] $)
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${
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com3iia.1 $e |- a C b $.
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$( The dual of ~ com3ii . (Contributed by Roy F. Longton, 3-Jul-05.) $)
|
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com3iia $p |- ( a v ( a ' ^ b ) ) = ( a v b ) $=
|
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( wn wa wo comid comcom2 fh3 lear ax-a4 df-le1 leid ler2an lebi ax-r2 ) A
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ADZBEFAQFZABFZEZSAQBAAAGHCITSRSJSRSSRSAKLSMNOP $.
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$( [3-Jul-05] $) $( [2-Jul-05] $)
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$}
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$(
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@( Note: This theorem is unfinished. This is the progress that I was able
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to make. @)
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lem4.6.6i4j3 @p |- ( ( a ->4 b ) v ( a ->3 b ) ) = ( a ->0 b ) @=
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wva wvb wa wva wn wvb wa wo wva wn wvb wo wvb wn wa wo wva wn wvb wa wva wn
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wvb wn wa wo wva wva wn wvb wo wa wo wo wva wn wvb wo wva wvb wi4 wva wvb
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wi3 wo wva wvb wi0 wva wvb wa wva wn wvb wa wo wva wn wvb wo wvb wn wa wo
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wva wn wvb wa wva wn wvb wn wa wo wva wva wn wvb wo wa wo wo wva wvb wa wva
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wn wvb wo wva wn wvb wa wvb wn wo wa wo wva wn wvb wn wa wva wn wvb wa wva
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wo wva wn wvb wo wa wo wo wva wn wvb wo wva wvb wa wva wn wvb wa wo wva wn
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wvb wo wvb wn wa wo wva wvb wa wva wn wvb wo wva wn wvb wa wvb wn wo wa wo
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wva wn wvb wa wva wn wvb wn wa wo wva wva wn wvb wo wa wo wva wn wvb wn wa
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wva wn wvb wa wva wo wva wn wvb wo wa wo wva wvb wa wva wn wvb wa wo wva wn
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wvb wo wvb wn wa wo wva wvb wa wva wn wvb wa wva wn wvb wo wvb wn wa wo wo
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wva wvb wa wva wn wvb wa wva wn wvb wo wo wva wn wvb wa wvb wn wo wa wo wva
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wvb wa wva wn wvb wo wva wn wvb wa wvb wn wo wa wo wva wvb wa wva wn wvb wa
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wva wn wvb wo wvb wn wa ax-a3 wva wn wvb wa wva wn wvb wo wvb wn wa wo wva
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wn wvb wa wva wn wvb wo wo wva wn wvb wa wvb wn wo wa wva wvb wa wva wn wvb
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wa wva wn wvb wo wvb wn wva wn wvb wa wva wn wvb wo wva wn wvb wvb leao1
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lecom wva wn wvb wa wvb wva wn wvb coman2 comcom2 fh3 lor wva wn wvb wa wva
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wn wvb wo wo wva wn wvb wa wvb wn wo wa wva wn wvb wo wva wn wvb wa wvb wn
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wo wa wva wvb wa wva wn wvb wa wva wn wvb wo wo wva wn wvb wo wva wn wvb wa
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wvb wn wo wva wn wvb wa wva wn wvb wo wva wn wvb wvb leao1 df-le2 ran lor
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3tr wva wn wvb wa wva wn wvb wn wa wo wva wva wn wvb wo wa wo wva wn wvb wn
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wa wva wn wvb wa wo wva wva wn wvb wo wa wo wva wn wvb wn wa wva wn wvb wa
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wva wva wn wvb wo wa wo wo wva wn wvb wn wa wva wn wvb wa wva wo wva wn wvb
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wo wa wo wva wn wvb wa wva wn wvb wn wa wo wva wn wvb wn wa wva wn wvb wa
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wo wva wva wn wvb wo wa wva wn wvb wa wva wn wvb wn wa ax-a2 ax-r5 wva wn
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wvb wn wa wva wn wvb wa wva wva wn wvb wo wa ax-a3 wva wn wvb wa wva wva wn
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wvb wo wa wo wva wn wvb wa wva wo wva wn wvb wo wa wva wn wvb wn wa wva wn
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wvb wa wva wva wn wvb wo wa wo wva wn wvb wa wva wo wva wn wvb wa wva wn
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wvb wo wo wa wva wn wvb wa wva wo wva wn wvb wo wa wva wn wvb wa wva wva wn
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wvb wo wva wva wn wvb wa wva wva wn wvb wa wva wn wvb comanr1 comcom6
|
|
comcom wva wn wvb wa wva wn wvb wo wva wn wvb wvb leao1 lecom fh3 wva wn
|
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wvb wa wva wn wvb wo wo wva wn wvb wo wva wn wvb wa wva wo wva wn wvb wa
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|
wva wn wvb wo wva wn wvb wvb leao1 df-le2 lan ax-r2 lor 3tr 2or wva wvb wa
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|
wva wn wvb wo wva wn wvb wa wvb wn wo wa wo wva wn wvb wn wa wva wn wvb wa
|
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wva wo wva wn wvb wo wa wo wo wva wvb wa wva wn wvb wo wva wn wvb wa wvb wn
|
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wo wa wo wva wn wvb wo wva wn wvb wa wva wo wa wva wn wvb wn wa wo wo wva
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wn wvb wo wva wn wvb wn wa wva wn wvb wa wva wo wva wn wvb wo wa wo wva wn
|
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wvb wo wva wn wvb wa wva wo wa wva wn wvb wn wa wo wva wvb wa wva wn wvb wo
|
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wva wn wvb wa wvb wn wo wa wo wva wn wvb wn wa wva wn wvb wa wva wo wva wn
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wvb wo wa wo wva wn wvb wn wa wva wn wvb wo wva wn wvb wa wva wo wa wo wva
|
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wn wvb wo wva wn wvb wa wva wo wa wva wn wvb wn wa wo wva wn wvb wa wva wo
|
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wva wn wvb wo wa wva wn wvb wo wva wn wvb wa wva wo wa wva wn wvb wn wa wva
|
|
wn wvb wa wva wo wva wn wvb wo ancom lor wva wn wvb wn wa wva wn wvb wo wva
|
|
wn wvb wa wva wo wa ax-a2 ax-r2 lor wva wvb wa wva wn wvb wo wva wn wvb wa
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|
wvb wn wo wa wo wva wn wvb wo wva wn wvb wa wva wo wa wva wn wvb wn wa wo
|
|
wo wva wvb wa wva wn wvb wo wva wn wvb wa wvb wn wo wa wva wn wvb wo wva wn
|
|
wvb wa wva wo wa wva wn wvb wn wa wo wo wo wva wn wvb wo wva wvb wa wva wn
|
|
wvb wo wva wn wvb wa wvb wn wo wa wva wn wvb wo wva wn wvb wa wva wo wa wva
|
|
wn wvb wn wa wo ax-a3 wva wvb wa wva wn wvb wo wva wn wvb wa wvb wn wo wa
|
|
wva wn wvb wo wva wn wvb wa wva wo wa wva wn wvb wn wa wo wo wo wva wvb wa
|
|
wva wn wvb wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo wo wa wva wn wvb
|
|
wn wa wo wo wva wn wvb wo wva wn wvb wo wva wn wvb wa wvb wn wo wa wva wn
|
|
wvb wo wva wn wvb wa wva wo wa wva wn wvb wn wa wo wo wva wn wvb wo wva wn
|
|
wvb wa wvb wn wo wva wn wvb wa wva wo wo wa wva wn wvb wn wa wo wva wvb wa
|
|
wva wn wvb wo wva wn wvb wa wvb wn wo wa wva wn wvb wo wva wn wvb wa wva wo
|
|
wa wva wn wvb wn wa wo wo wva wn wvb wo wva wn wvb wa wvb wn wo wa wva wn
|
|
wvb wo wva wn wvb wa wva wo wa wo wva wn wvb wn wa wo wva wn wvb wo wva wn
|
|
wvb wa wvb wn wo wva wn wvb wa wva wo wo wa wva wn wvb wn wa wo wva wn wvb
|
|
wo wva wn wvb wa wvb wn wo wa wva wn wvb wo wva wn wvb wa wva wo wa wo wva
|
|
wn wvb wn wa wo wva wn wvb wo wva wn wvb wa wvb wn wo wa wva wn wvb wo wva
|
|
wn wvb wa wva wo wa wva wn wvb wn wa wo wo wva wn wvb wo wva wn wvb wa wvb
|
|
wn wo wa wva wn wvb wo wva wn wvb wa wva wo wa wva wn wvb wn wa ax-a3 ax-r1
|
|
wva wn wvb wo wva wn wvb wa wvb wn wo wa wva wn wvb wo wva wn wvb wa wva wo
|
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wa wo wva wn wvb wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo wo wa wva
|
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wn wvb wn wa wva wn wvb wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo wo
|
|
wa wva wn wvb wo wva wn wvb wa wvb wn wo wa wva wn wvb wo wva wn wvb wa wva
|
|
wo wa wo wva wn wvb wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo wva wn
|
|
wvb wo wva wn wvb wa wvb wn wva wn wvb wa wva wn wvb wo wva wn wvb wa wva
|
|
wn wvb wo wva wn wvb wvb leao1 lecom comcom wva wn wvb wo wvb wva wn wvb
|
|
comor2 comcom2 com2or wva wn wvb wo wva wn wvb wa wva wva wn wvb wa wva wn
|
|
wvb wo wva wn wvb wa wva wn wvb wo wva wn wvb wvb leao1 lecom comcom wva wn
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|
wvb wo wva wva wn wvb comor1 comcom7 com2or fh1 ax-r1 ax-r5 ax-r2 lor wva
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|
wvb wa wva wn wvb wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo wo wa wva
|
|
wn wvb wn wa wo wo wva wvb wa wva wn wvb wn wa wva wn wvb wo wva wn wvb wa
|
|
wvb wn wo wva wn wvb wa wva wo wo wa wo wo wva wn wvb wo wva wn wvb wo wva
|
|
wn wvb wa wvb wn wo wva wn wvb wa wva wo wo wa wva wn wvb wn wa wo wva wn
|
|
wvb wn wa wva wn wvb wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo wo wa
|
|
wo wva wvb wa wva wn wvb wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo wo
|
|
wa wva wn wvb wn wa ax-a2 lor wva wvb wa wva wn wvb wn wa wva wn wvb wo wva
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|
wn wvb wa wvb wn wo wva wn wvb wa wva wo wo wa wo wo wva wvb wa wva wn wvb
|
|
wn wa wo wva wn wvb wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo wo wa
|
|
wo wva wn wvb wo wva wvb wa wva wn wvb wn wa wo wva wn wvb wo wva wn wvb wa
|
|
wvb wn wo wva wn wvb wa wva wo wo wa wo wva wvb wa wva wn wvb wn wa wva wn
|
|
wvb wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo wo wa wo wo wva wvb wa
|
|
wva wn wvb wn wa wva wn wvb wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo
|
|
wo wa ax-a3 ax-r1 wva wvb wa wva wn wvb wn wa wo wva wn wvb wo wva wn wvb
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|
wa wvb wn wo wva wn wvb wa wva wo wo wa wo wva wvb wa wva wn wvb wn wa wo
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wva wn wvb wo wva wn wvb wa wvb wn wva wo wo wa wo wva wn wvb wo wva wn wvb
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wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo wo wa wva wn wvb wo wva wn
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wvb wa wvb wn wva wo wo wa wva wvb wa wva wn wvb wn wa wo wva wn wvb wa wvb
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wn wo wva wn wvb wa wva wo wo wva wn wvb wa wvb wn wva wo wo wva wn wvb wo
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|
wva wn wvb wa wvb wn wva wo wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo
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|
wo wva wn wvb wa wvb wn wva orordi ax-r1 lan lor wva wvb wa wva wn wvb wn
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|
wa wo wva wn wvb wo wva wn wvb wa wvb wn wva wo wo wa wo wva wvb wa wva wn
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wvb wn wa wo wva wn wvb wo wva wn wvb wa wva wvb wn wo wo wa wo wva wn wvb
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wo wva wn wvb wo wva wn wvb wa wvb wn wva wo wo wa wva wn wvb wo wva wn wvb
|
|
wa wva wvb wn wo wo wa wva wvb wa wva wn wvb wn wa wo wva wn wvb wa wvb wn
|
|
wva wo wo wva wn wvb wa wva wvb wn wo wo wva wn wvb wo wvb wn wva wo wva
|
|
wvb wn wo wva wn wvb wa wvb wn wva ax-a2 lor lan lor ? ax-r2 ax-r2 ax-r2
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|
ax-r2 ax-r2 ax-r2 ax-r2 ax-r2 wva wvb wi4 wva wvb wa wva wn wvb wa wo wva
|
|
wn wvb wo wvb wn wa wo wva wvb wi3 wva wn wvb wa wva wn wvb wn wa wo wva
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|
wva wn wvb wo wa wo wva wvb df-i4 wva wvb df-i3 2or wva wvb df-i0 3tr1 @.
|
|
@( [31-Mar-2011] @) @( [2-Jul-05] @)
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lem4.6.6i1j4 @p |- ( ( a ->1 b ) v ( a ->4 b ) ) = ( a ->0 b ) @= ? @.
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|
|
lem4.6.6i2j3 @p |- ( ( a ->2 b ) v ( a ->3 b ) ) = ( a ->0 b ) @= ? @.
|
|
|
|
lem4.6.6i3j2 @p |- ( ( a ->3 b ) v ( a ->2 b ) ) = ( a ->0 b ) @= ? @.
|
|
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|
lem4.6.6i3j4 @p |- ( ( a ->3 b ) v ( a ->4 b ) ) = ( a ->0 b ) @= ? @.
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|
|
|
lem4.6.6i4j1 @p |- ( ( a ->4 b ) v ( a ->1 b ) ) = ( a ->0 b ) @= ? @.
|
|
$)
|
|
|
|
${
|
|
lem4.6.7.1 $e |- a ' =< b $.
|
|
$( Equation 4.15 of [MegPav2000] p. 23. (Contributed by Roy F. Longton,
|
|
3-Jul-05.) $)
|
|
lem4.6.7 $p |- b =< ( a ->1 b ) $=
|
|
( wn wa wo wi1 wt leid sklem ax-r1 df-le2 ax-a3 ler2an lel2or leran leao2
|
|
2an le1 ler lebi ax-r2 comid comcom3 lecom fh3 3tr1 df-le1 df-i1 lbtr ) B
|
|
ADZABEZFZABGZBUMHBEZUKAFZUKBFZEBUMFZUMHUPBUQUPHAAAIJKUQBUKBCLKRURBUKFZULF
|
|
ZUOUTURBUKULMKUTUOUSUOULBUOUKBHBBSBINUKHBUKSCNOAHBASPOUOUSULBHUKQTUAUBUKA
|
|
BAAAUCUDUKBCUEUFUGUHUNUMABUIKUJ $.
|
|
$( [3-Jul-05] $) $( [3-Jul-05] $)
|
|
$}
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$( $t
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/* The '$t' token indicates the beginning of the typesetting definition
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|
section, embedded in a Metamath comment. There may only be one per
|
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source file, and the typesetting section ends with the end of the
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Metamath comment. The typesetting section uses C-style comment
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delimiters. */
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/* These are the LaTeX and HTML definitions in the order the tokens are
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introduced in $c or $v statements. See HELP TEX or HELP HTML in the
|
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Metamath program. */
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/* Definitions for LaTeX output of various Metamath commands */
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/* (LaTeX definitions have not been written for this file.) */
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/* Definitions for HTML output of various Metamath commands. */
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/* Title */
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htmltitle "Quantum Logic Explorer";
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|
|
|
/* Home page link */
|
|
htmlhome '<A HREF="mmql.html"><FONT SIZE=-2 FACE=sans-serif>' +
|
|
'<IMG SRC="l46-7icon.gif" BORDER=0 ALT=' +
|
|
'"[Lattice L46-7]Home Page" HEIGHT=32 WIDTH=32 ALIGN=MIDDLE>' +
|
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'Home</FONT></A>';
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|
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/* Optional file where bibliographic references are kept */
|
|
htmlbibliography "mmql.html";
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|
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/* Variable color key */
|
|
htmlvarcolor '<FONT COLOR="#CC4400">term</FONT>';
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/* GIF and Symbol Font HTML directories */
|
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htmldir "../qlegif/";
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|
althtmldir "../qleuni/";
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/* Symbol definitions */
|
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htmldef "a" as "<IMG SRC='_ba.gif' WIDTH=9 HEIGHT=19 ALT='a' ALIGN=TOP>";
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htmldef "b" as "<IMG SRC='_bb.gif' WIDTH=8 HEIGHT=19 ALT='b' ALIGN=TOP>";
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htmldef "c" as "<IMG SRC='_bc.gif' WIDTH=7 HEIGHT=19 ALT='c' ALIGN=TOP>";
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htmldef "d" as "<IMG SRC='_bd.gif' WIDTH=9 HEIGHT=19 ALT='d' ALIGN=TOP>";
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htmldef "e" as "<IMG SRC='_be.gif' WIDTH=8 HEIGHT=19 ALT='e' ALIGN=TOP>";
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htmldef "f" as "<IMG SRC='_bf.gif' WIDTH=9 HEIGHT=19 ALT='f' ALIGN=TOP>";
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htmldef "g" as "<IMG SRC='_bg.gif' WIDTH=9 HEIGHT=19 ALT='g' ALIGN=TOP>";
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htmldef "h" as "<IMG SRC='_bh.gif' WIDTH=10 HEIGHT=19 ALT='h' ALIGN=TOP>";
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htmldef "i" as "<IMG SRC='_browni.gif' WIDTH=6 HEIGHT=19 ALT='i' ALIGN=TOP>";
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htmldef "j" as "<IMG SRC='_bj.gif' WIDTH=7 HEIGHT=19 ALT='j' ALIGN=TOP>";
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htmldef "k" as "<IMG SRC='_bk.gif' WIDTH=9 HEIGHT=19 ALT='k' ALIGN=TOP>";
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htmldef "l" as "<IMG SRC='_bl.gif' WIDTH=6 HEIGHT=19 ALT='l' ALIGN=TOP>";
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htmldef "m" as "<IMG SRC='_bm.gif' WIDTH=14 HEIGHT=19 ALT='m' ALIGN=TOP>";
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htmldef "n" as "<IMG SRC='_bn.gif' WIDTH=10 HEIGHT=19 ALT='n' ALIGN=TOP>";
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htmldef "p" as "<IMG SRC='_bp.gif' WIDTH=10 HEIGHT=19 ALT='p' ALIGN=TOP>";
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htmldef "q" as "<IMG SRC='_bq.gif' WIDTH=8 HEIGHT=19 ALT='q' ALIGN=TOP>";
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htmldef "r" as "<IMG SRC='_br.gif' WIDTH=8 HEIGHT=19 ALT='r' ALIGN=TOP>";
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htmldef "t" as "<IMG SRC='_bt.gif' WIDTH=7 HEIGHT=19 ALT='t' ALIGN=TOP>";
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htmldef "u" as "<IMG SRC='_bu.gif' WIDTH=10 HEIGHT=19 ALT='u' ALIGN=TOP>";
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htmldef "w" as "<IMG SRC='_bw.gif' WIDTH=12 HEIGHT=19 ALT='w' ALIGN=TOP>";
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htmldef "x" as "<IMG SRC='_bx.gif' WIDTH=10 HEIGHT=19 ALT='x' ALIGN=TOP>";
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htmldef "y" as "<IMG SRC='_by.gif' WIDTH=9 HEIGHT=19 ALT='y' ALIGN=TOP>";
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htmldef "z" as "<IMG SRC='_bz.gif' WIDTH=9 HEIGHT=19 ALT='z' ALIGN=TOP>";
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htmldef "(" as "<IMG SRC='lp.gif' WIDTH=5 HEIGHT=19 ALT='(' ALIGN=TOP>";
|
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htmldef ")" as "<IMG SRC='rp.gif' WIDTH=5 HEIGHT=19 ALT=')' ALIGN=TOP>";
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|
htmldef "=" as " <IMG SRC='eq.gif' WIDTH=12 HEIGHT=19 ALT='=' ALIGN=TOP> ";
|
|
htmldef "==" as " <IMG SRC='equiv.gif' WIDTH=12 HEIGHT=19 ALT='=='" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "v" as " <IMG SRC='cup.gif' WIDTH=10 HEIGHT=19 ALT='v' ALIGN=TOP> ";
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htmldef "^" as " <IMG SRC='cap.gif' WIDTH=10 HEIGHT=19 ALT='^' ALIGN=TOP> ";
|
|
htmldef "0" as "<IMG SRC='0.gif' WIDTH=8 HEIGHT=19 ALT='0' ALIGN=TOP>";
|
|
htmldef "1" as "<IMG SRC='1.gif' WIDTH=7 HEIGHT=19 ALT='1' ALIGN=TOP>";
|
|
/* htmldef "-" as "<IMG SRC='shortminus.gif' WIDTH=8 HEIGHT=19 ALT='-'" +
|
|
" ALIGN=TOP>"; */
|
|
/* htmldef "_|_" as " <IMG SRC='perp.gif' WIDTH=11 HEIGHT=19 ALT='_|_'" +
|
|
" ALIGN=TOP> "; */
|
|
htmldef "'" as "<IMG SRC='supperp.gif' WIDTH=9 HEIGHT=19 ALT=" +
|
|
'"' + "'" + '"' + " ALIGN=TOP>";
|
|
htmldef "wff" as "<IMG SRC='_wff.gif' WIDTH=24 HEIGHT=19 ALT='wff'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "term" as "<IMG SRC='_term.gif' WIDTH=32 HEIGHT=19 ALT='term'" +
|
|
" ALIGN=TOP> ";
|
|
/* Mladen wants the turnstile to go away 2/9/02 */
|
|
/*htmldef "|-" as "<IMG SRC='_vdash.gif' WIDTH=10 HEIGHT=19 ALT='|-'" +
|
|
" ALIGN=TOP> ";*/
|
|
htmldef "|-" as "";
|
|
htmldef "C" as " <IMG SRC='cc.gif' WIDTH=12 HEIGHT=19 ALT='C' ALIGN=TOP> ";
|
|
htmldef "," as "<IMG SRC='comma.gif' WIDTH=4 HEIGHT=19 ALT=',' ALIGN=TOP> ";
|
|
htmldef "=<" as " <IMG SRC='le.gif' WIDTH=11 HEIGHT=19 ALT='=<'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "=<2" as " <IMG SRC='_le2.gif' WIDTH=17 HEIGHT=19 ALT='=<2'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "->0" as " <IMG SRC='_to0.gif' WIDTH=21 HEIGHT=19 ALT='->0'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "->1" as " <IMG SRC='_to1.gif' WIDTH=19 HEIGHT=19 ALT='->1'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "->2" as " <IMG SRC='_to2.gif' WIDTH=21 HEIGHT=19 ALT='->2'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "->3" as " <IMG SRC='_to3.gif' WIDTH=21 HEIGHT=19 ALT='->3'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "->4" as " <IMG SRC='_to4.gif' WIDTH=20 HEIGHT=19 ALT='->4'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "->5" as " <IMG SRC='_to5.gif' WIDTH=20 HEIGHT=19 ALT='->5'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "<->1" as " <IMG SRC='_bi1.gif' WIDTH=19 HEIGHT=19" +
|
|
" ALT='<->1' ALIGN=TOP> ";
|
|
htmldef "<->3" as " <IMG SRC='_bi3.gif' WIDTH=21 HEIGHT=19" +
|
|
" ALT='<->3' ALIGN=TOP> ";
|
|
htmldef "u3" as " <IMG SRC='_cup3.gif' WIDTH=16 HEIGHT=19 ALT='u3'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "^3" as " <IMG SRC='_cap3.gif' WIDTH=16 HEIGHT=19 ALT='^3'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "==0" as " <IMG SRC='_equiv0.gif' WIDTH=18 HEIGHT=19 ALT='==0'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "==1" as " <IMG SRC='_equiv1.gif' WIDTH=16 HEIGHT=19 ALT='==1'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "==2" as " <IMG SRC='_equiv2.gif' WIDTH=18 HEIGHT=19 ALT='==2'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "==3" as " <IMG SRC='_equiv3.gif' WIDTH=18 HEIGHT=19 ALT='==3'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "==4" as " <IMG SRC='_equiv4.gif' WIDTH=18 HEIGHT=19 ALT='==4'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "==5" as " <IMG SRC='_equiv5.gif' WIDTH=18 HEIGHT=19 ALT='==5'" +
|
|
" ALIGN=TOP> ";
|
|
htmldef "==OA" as " <IMG SRC='_oa.gif' WIDTH=26 HEIGHT=19 ALT='==OA'" +
|
|
" ALIGN=TOP> ";
|
|
/*
|
|
htmldef "==u" as '<FONT FACE="Symbol"> º</FONT ><I><SUB>u</SUB> </I>';
|
|
htmldef "u.u" as '<FONT FACE="Symbol"> Ú</FONT ><I><SUB>u</SUB> </I>';
|
|
htmldef "^u" as '<FONT FACE="Symbol"> Ù</FONT ><I><SUB>u</SUB> </I>';
|
|
htmldef "-u" as '<FONT FACE="Symbol"> Ø</FONT ><I><SUB>u</SUB> </I>';
|
|
htmldef "=<u" as '<FONT FACE="Symbol"> £</FONT ><I><SUB>u</SUB> </I>';
|
|
htmldef "=" as '<FONT FACE="Symbol"> = </FONT>';
|
|
*/
|
|
|
|
|
|
/* Definitions for Unicode version */
|
|
althtmldef "a" as '<I><FONT COLOR="#CC4400">a</FONT></I>';
|
|
althtmldef "b" as '<I><FONT COLOR="#CC4400">b</FONT></I>';
|
|
althtmldef "c" as '<I><FONT COLOR="#CC4400">c</FONT></I>';
|
|
althtmldef "d" as '<I><FONT COLOR="#CC4400">d</FONT></I>';
|
|
althtmldef "e" as '<I><FONT COLOR="#CC4400">e</FONT></I>';
|
|
althtmldef "f" as '<I><FONT COLOR="#CC4400">f</FONT></I>';
|
|
althtmldef "g" as '<I><FONT COLOR="#CC4400">g</FONT></I>';
|
|
althtmldef "h" as '<I><FONT COLOR="#CC4400">h</FONT></I>';
|
|
althtmldef "i" as '<I><FONT COLOR="#CC4400">i</FONT></I>';
|
|
althtmldef "j" as '<I><FONT COLOR="#CC4400">j</FONT></I>';
|
|
althtmldef "k" as '<I><FONT COLOR="#CC4400">k</FONT></I>';
|
|
althtmldef "l" as '<I><FONT COLOR="#CC4400">l</FONT></I>';
|
|
althtmldef "m" as '<I><FONT COLOR="#CC4400">m</FONT></I>';
|
|
althtmldef "n" as '<I><FONT COLOR="#CC4400">n</FONT></I>';
|
|
althtmldef "p" as '<I><FONT COLOR="#CC4400">p</FONT></I>';
|
|
althtmldef "q" as '<I><FONT COLOR="#CC4400">q</FONT></I>';
|
|
althtmldef "r" as '<I><FONT COLOR="#CC4400">r</FONT></I>';
|
|
althtmldef "t" as '<I><FONT COLOR="#CC4400">t</FONT></I>';
|
|
althtmldef "u" as '<I><FONT COLOR="#CC4400">u</FONT></I>';
|
|
althtmldef "w" as '<I><FONT COLOR="#CC4400">w</FONT></I>';
|
|
althtmldef "x" as '<I><FONT COLOR="#CC4400">x</FONT></I>';
|
|
althtmldef "y" as '<I><FONT COLOR="#CC4400">y</FONT></I>';
|
|
althtmldef "z" as '<I><FONT COLOR="#CC4400">z</FONT></I>';
|
|
althtmldef "a0" as '<I><FONT COLOR="#CC4400">a<SUB>0</SUB></FONT></I>';
|
|
althtmldef "a1" as '<I><FONT COLOR="#CC4400">a<SUB>1</SUB></FONT></I>';
|
|
althtmldef "a2" as '<I><FONT COLOR="#CC4400">a<SUB>2</SUB></FONT></I>';
|
|
althtmldef "b0" as '<I><FONT COLOR="#CC4400">b<SUB>0</SUB></FONT></I>';
|
|
althtmldef "b1" as '<I><FONT COLOR="#CC4400">b<SUB>1</SUB></FONT></I>';
|
|
althtmldef "b2" as '<I><FONT COLOR="#CC4400">b<SUB>2</SUB></FONT></I>';
|
|
althtmldef "c0" as '<I><FONT COLOR="#CC4400">c<SUB>0</SUB></FONT></I>';
|
|
althtmldef "c1" as '<I><FONT COLOR="#CC4400">c<SUB>1</SUB></FONT></I>';
|
|
althtmldef "c2" as '<I><FONT COLOR="#CC4400">c<SUB>2</SUB></FONT></I>';
|
|
althtmldef "p0" as '<I><FONT COLOR="#CC4400">p<SUB>0</SUB></FONT></I>';
|
|
althtmldef "p1" as '<I><FONT COLOR="#CC4400">p<SUB>1</SUB></FONT></I>';
|
|
althtmldef "p2" as '<I><FONT COLOR="#CC4400">p<SUB>2</SUB></FONT></I>';
|
|
htmldef "a0" as '<I><FONT COLOR="#CC4400">a<SUB>0</SUB></FONT></I>';
|
|
htmldef "a1" as '<I><FONT COLOR="#CC4400">a<SUB>1</SUB></FONT></I>';
|
|
htmldef "a2" as '<I><FONT COLOR="#CC4400">a<SUB>2</SUB></FONT></I>';
|
|
htmldef "b0" as '<I><FONT COLOR="#CC4400">b<SUB>0</SUB></FONT></I>';
|
|
htmldef "b1" as '<I><FONT COLOR="#CC4400">b<SUB>1</SUB></FONT></I>';
|
|
htmldef "b2" as '<I><FONT COLOR="#CC4400">b<SUB>2</SUB></FONT></I>';
|
|
htmldef "c0" as '<I><FONT COLOR="#CC4400">c<SUB>0</SUB></FONT></I>';
|
|
htmldef "c1" as '<I><FONT COLOR="#CC4400">c<SUB>1</SUB></FONT></I>';
|
|
htmldef "c2" as '<I><FONT COLOR="#CC4400">c<SUB>2</SUB></FONT></I>';
|
|
htmldef "p0" as '<I><FONT COLOR="#CC4400">p<SUB>0</SUB></FONT></I>';
|
|
htmldef "p1" as '<I><FONT COLOR="#CC4400">p<SUB>1</SUB></FONT></I>';
|
|
htmldef "p2" as '<I><FONT COLOR="#CC4400">p<SUB>2</SUB></FONT></I>';
|
|
althtmldef "(" as '(';
|
|
althtmldef ")" as ')';
|
|
althtmldef "=" as ' = '; /* = */
|
|
althtmldef "==" as ' ≡ ';
|
|
althtmldef "v" as ' ∪ ';
|
|
althtmldef "^" as ' ∩ ';
|
|
althtmldef "1" as '1';
|
|
althtmldef "0" as '0';
|
|
/* althtmldef "-" as ' - '; */
|
|
/* althtmldef "'" as '⊥'; */
|
|
althtmldef "'" as '<SUP>⊥</SUP> ';
|
|
althtmldef "wff" as '<FONT COLOR="#00CC00">wff </FONT>';
|
|
althtmldef "term" as '<FONT COLOR="#00CC00">term </FONT>';
|
|
/* Mladen wants the turnstile to go away 2/9/02 */
|
|
/*althtmldef "|-" as '<FONT COLOR="#00CC00">|- </FONT>';*/
|
|
althtmldef "|-" as '';
|
|
althtmldef "C" as '<I> C </I>';
|
|
althtmldef "," as ', ';
|
|
althtmldef "=<" as ' ≤ ';
|
|
althtmldef "=<2" as ' ≤<SUB>2 </SUB>';
|
|
althtmldef "->0" as ' →<SUB>0 </SUB>';
|
|
althtmldef "->1" as ' →<SUB>1 </SUB>';
|
|
althtmldef "->2" as ' →<SUB>2 </SUB>';
|
|
althtmldef "->3" as ' →<SUB>3 </SUB>';
|
|
althtmldef "->4" as ' →<SUB>4 </SUB>';
|
|
althtmldef "->5" as ' →<SUB>5 </SUB>';
|
|
althtmldef "<->1" as ' ↔<SUB>1 </SUB> ';
|
|
althtmldef "<->3" as ' ↔<SUB>3 </SUB> ';
|
|
althtmldef "u3" as ' ∪<SUB>3 </SUB> ';
|
|
althtmldef "^3" as ' ∩<SUB>3 </SUB> ';
|
|
althtmldef "==0" as ' ≡<SUB>0 </SUB> ';
|
|
althtmldef "==1" as ' ≡<SUB>1 </SUB>';
|
|
althtmldef "==2" as ' ≡<SUB>2 </SUB>';
|
|
althtmldef "==3" as ' ≡<SUB>3 </SUB>';
|
|
althtmldef "==4" as ' ≡<SUB>4 </SUB>';
|
|
althtmldef "==5" as ' ≡<SUB>5 </SUB>';
|
|
althtmldef "==OA" as ' ≡<SUB>OA </SUB>';
|
|
/*
|
|
althtmldef "==u" as ' ≡<I><SUB>u</SUB> </I>';
|
|
althtmldef "u.u" as ' ·<I><SUB>u</SUB> </I>';
|
|
althtmldef "^u" as ' ∩<I><SUB>u</SUB> </I>';
|
|
althtmldef "-u" as ' −<I><SUB>u</SUB> </I>';
|
|
althtmldef "=<u" as ' ≤<I><SUB>u</SUB> </I>';
|
|
althtmldef "=" as ' = ';
|
|
*/
|
|
/* End of Unicode defintions */
|
|
|
|
|
|
latexdef "a" as "a";
|
|
latexdef "b" as "b";
|
|
latexdef "c" as "c";
|
|
latexdef "d" as "d";
|
|
latexdef "e" as "e";
|
|
latexdef "f" as "f";
|
|
latexdef "g" as "g";
|
|
latexdef "h" as "h";
|
|
latexdef "i" as "i";
|
|
latexdef "j" as "j";
|
|
latexdef "k" as "k";
|
|
latexdef "l" as "l";
|
|
latexdef "m" as "m";
|
|
latexdef "n" as "n";
|
|
latexdef "p" as "p";
|
|
latexdef "q" as "q";
|
|
latexdef "r" as "r";
|
|
latexdef "t" as "t";
|
|
latexdef "u" as "u";
|
|
latexdef "w" as "w";
|
|
latexdef "x" as "x";
|
|
latexdef "y" as "y";
|
|
latexdef "z" as "z";
|
|
latexdef "(" as "(";
|
|
latexdef ")" as ")";
|
|
latexdef "=" as "=";
|
|
latexdef "==" as "\equiv ";
|
|
latexdef "v" as "\vee ";
|
|
latexdef "^" as "\wedge ";
|
|
latexdef "0" as "0";
|
|
latexdef "1" as "1";
|
|
latexdef "'" as "'";
|
|
latexdef "wff" as "{\rm wff}";
|
|
latexdef "term" as "{\rm term}";
|
|
latexdef "|-" as "";
|
|
latexdef "C" as "C";
|
|
latexdef "," as ",";
|
|
latexdef "=<" as "\le ";
|
|
latexdef "=<2" as "\le_2";
|
|
latexdef "->0" as "\to_0";
|
|
latexdef "->1" as "\to_1";
|
|
latexdef "->2" as "\to_2";
|
|
latexdef "->3" as "\to_3";
|
|
latexdef "->4" as "\to_4";
|
|
latexdef "->5" as "\to_5";
|
|
latexdef "<->1" as "\leftrightarrow_1";
|
|
latexdef "<->3" as "\leftrightarrow_3";
|
|
latexdef "u3" as "\vee_3";
|
|
latexdef "^3" as "\wedge_3";
|
|
latexdef "==0" as "\equiv_0";
|
|
latexdef "==1" as "\equiv_1";
|
|
latexdef "==2" as "\equiv_2";
|
|
latexdef "==3" as "\equiv_3";
|
|
latexdef "==4" as "\equiv_4";
|
|
latexdef "==5" as "\equiv_5";
|
|
latexdef "==OA" as "\equiv_{\mathrm{OA}}";
|
|
|
|
/* End of typesetting definition section */
|
|
$)
|
|
|
|
$( 456789012345 (79-character line to adjust text window width) 567890123456 $)
|
|
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Weakly distributive ortholattices (WDOL)
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
WDOL law
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The WDOL (weakly distributive ortholattice) axiom. $)
|
|
ax-wdol $a |- ( ( a == b ) v ( a == b ' ) ) = 1 $.
|
|
|
|
$( Any two variables (weakly) commute in a WDOL. $)
|
|
wdcom $p |- C ( a , b ) = 1 $=
|
|
( wcmtr wa wn wo wt df-cmtr or42 tb dfb ax-a1 lan ax-r1 lor 2or ax-wdol 3tr
|
|
ax-r2 ) ABCABDZABEZDZFAEZBDZUCUADZFFTUEFZUBUDFZFZGABHTUBUDUEIUHABJZAUAJZFZG
|
|
UKUHUIUFUJUGABKUJUBUCUAEZDZFUGAUAKUMUDUBUDUMBULUCBLMNOSPNABQSR $.
|
|
$( [4-Mar-06] $)
|
|
|
|
${
|
|
wdwom.1 $e |- ( a ' v ( a ^ b ) ) = 1 $.
|
|
$( Prove 2-variable WOML rule in WDOL. This will make all WOML theorems
|
|
available to us. The proof does not use ~ ax-r3 or ~ ax-wom . Since
|
|
this is the same as ~ ax-wom , from here on we will freely use those
|
|
theorems invoking ~ ax-wom . $)
|
|
wdwom $p |- ( b v ( a ' ^ b ' ) ) = 1 $=
|
|
( wn wa wo wi2 wt df-i2 ax-r1 le1 wi5 df-i5 wi1 df-i1 ax-r2 wql1lem wcmtr
|
|
or4 anor1 lor ax-r5 or12 df-cmtr 3tr1 wdcom skr0 i5lei2 bltr lebi ) BADZB
|
|
DZEZFZABGZHUOUNABIJUOHUOKHABLZUOUPHUPABEZUKBEZFZUMFZHABMUKBFZUTABABNUKUQF
|
|
HABOCPQVADZUTFZABRZHUSVBUMFFZUQAULEZFZURUMFZFZVCVDVEUQVBFZVHFVIUQURVBUMSV
|
|
JVGVHVBVFUQVFVBABTJUAUBPVBUSUMUCABUDUEABUFPUGPJABUHUIUJP $.
|
|
$( [4-Mar-06] $)
|
|
$}
|
|
|
|
$( Prove the weak distributive law in WDOL. This is our first WDOL theorem
|
|
making use of ~ ax-wom , which is justified by ~ wdwom . $)
|
|
wddi1 $p |- ( ( a ^ ( b v c ) ) == ( ( a ^ b ) v ( a ^ c ) ) ) = 1 $=
|
|
( wdcom wfh1 ) ABCABDACDE $.
|
|
$( [4-Mar-06] $)
|
|
|
|
|
|
$( The weak distributive law in WDOL. $)
|
|
wddi2 $p |- ( ( ( a v b ) ^ c ) == ( ( a ^ c ) v ( b ^ c ) ) ) = 1 $=
|
|
( wo wa wancom wddi1 w2or wr2 ) ABDZCECJEZACEZBCEZDZJCFKCAEZCBEZDNCABGOLPMC
|
|
AFCBFHII $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$( The weak distributive law in WDOL. $)
|
|
wddi3 $p |- ( ( a v ( b ^ c ) ) ==
|
|
( ( a v b ) ^ ( a v c ) ) ) = 1 $=
|
|
( wdcom wfh3 ) ABCABDACDE $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$( The weak distributive law in WDOL. $)
|
|
wddi4 $p |- ( ( ( a ^ b ) v c ) ==
|
|
( ( a v c ) ^ ( b v c ) ) ) = 1 $=
|
|
( wa wo wa2 wddi3 w2an wr2 ) ABDZCECJEZACEZBCEZDZJCFKCAEZCBEZDNCABGOLPMCAFC
|
|
BFHII $.
|
|
$( [5-Mar-06] $)
|
|
|
|
|
|
${
|
|
wdid0id5.1 $e |- ( a ==0 b ) = 1 $.
|
|
$( Show that quantum identity follows from classical identity in a WDOL. $)
|
|
wdid0id5 $p |- ( a == b ) = 1 $=
|
|
( tb wa wn wo wt dfb wid0 df-id0 ax-r1 ax-r2 wa4 wleoa wancom wddi3 w3tr1
|
|
wr1 wa2 wr2 w2an wddi4 wwbmp ) ABDABEAFZBFZEZGZHABIUEBGZUFAGZEZUHUKABJZHU
|
|
LUKABKLCMUJUIEAUGGZBUGGZEUKUHUJUMUIUNAUFGZAUEGZUOEZUJUMUOUOUPEZUQURUOUOUP
|
|
UPUOANOSUOUPPUAUFATAUEUFQRBUEGZUSBUFGZEZUIUNVAUSUSUTUTUSBNOSUEBTBUEUFQRUB
|
|
UIUJPABUGUCRUDM $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$( Show a quantum identity that follows from classical identity in a
|
|
WDOL. $)
|
|
wdid0id1 $p |- ( a ==1 b ) = 1 $=
|
|
( wid1 wn wo wa wt df-id1 wid0 df-id0 ax-r1 ax-r2 wancom wa2 wlan wa4 wr2
|
|
wleoa wr1 wddi3 w2an biid w3tr1 wwbmp ) ABDABEZFZAEZABGFZGZHABIUHBFZUFAFZ
|
|
GZUJUMABJZHUNUMABKLCMUMUIUGGUMUJUKUIULUGUKUHAFZUKGZUIUPUKUPUKUOGZUKUOUKNU
|
|
QUKAUHFZGUKUOURUKUHAOPUKURURUKAQSRRTUIUPUHABUATRUFAOUBUMUCUGUINUDUEM $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$( Show a quantum identity that follows from classical identity in a
|
|
WDOL. $)
|
|
wdid0id2 $p |- ( a ==2 b ) = 1 $=
|
|
( wid2 wn wo wa df-id2 wid0 df-id0 ax-r1 ax-r2 wancom wa2 wa4 wleoa wddi3
|
|
wt wr1 w3tr1 w2an wr2 wwbmp ) ABDABEZFZBAEZUDGFZGZRABHUFBFZUDAFZGZUHUKABI
|
|
ZRULUKABJKCLUKUJUIGUHUIUJMUJUEUIUGUDANBUFFZUMBUDFZGZUIUGUOUMUMUNUNUMBOPSU
|
|
FBNBUFUDQTUAUBUCL $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$( Show a quantum identity that follows from classical identity in a
|
|
WDOL. $)
|
|
wdid0id3 $p |- ( a ==3 b ) = 1 $=
|
|
( wid3 wn wo wa wt df-id3 df-id0 ax-r1 ax-r2 wa4 wleoa wr1 wancom wr2 wa2
|
|
wid0 wddi3 w3tr1 wlan wwbmp ) ABDAEZBFZAUDBEZGFZGZHABIUEUFAFZGZUHUJABSZHU
|
|
KUJABJKCLUIUGUEAUFFZAUDFZULGZUIUGULULUMGZUNUOULULUMUMULAMNOULUMPQUFARAUDU
|
|
FTUAUBUCL $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$( Show a quantum identity that follows from classical identity in a
|
|
WDOL. $)
|
|
wdid0id4 $p |- ( a ==4 b ) = 1 $=
|
|
( wid4 wn wo wa wt df-id4 wid0 df-id0 ax-r1 ax-r2 wddi3 wa2 wa4 wleoa wr2
|
|
wlan wr1 wwbmp ) ABDAEBFZBEZABGFZGZHABIUBUCAFZGZUEUGABJZHUHUGABKLCMUFUDUB
|
|
UDUFUDUFUCBFZGZUFUCABNUJUFBUCFZGUFUIUKUFUCBOSUFUKUKUFBPQRRTSUAM $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$( Show WDOL analog of WOM law. $)
|
|
wdka4o $p |- ( ( a v c ) ==0 ( b v c ) ) = 1 $=
|
|
( wo wdid0id5 wr5 id5id0 ) ACEBCEABCABDFGH $.
|
|
$( [5-Mar-06] $)
|
|
$}
|
|
|
|
$( The weak distributive law in WDOL. $)
|
|
wddi-0 $p |- ( ( a ^ ( b v c ) ) ==0 ( ( a ^ b ) v ( a ^ c ) ) ) = 1 $=
|
|
( wo wa wddi1 id5id0 ) ABCDEABEACEDABCFG $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$( The weak distributive law in WDOL. $)
|
|
wddi-1 $p |- ( ( a ^ ( b v c ) ) ==1 ( ( a ^ b ) v ( a ^ c ) ) ) = 1 $=
|
|
( wo wa wddi-0 wdid0id1 ) ABCDEABEACEDABCFG $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$( The weak distributive law in WDOL. $)
|
|
wddi-2 $p |- ( ( a ^ ( b v c ) ) ==2 ( ( a ^ b ) v ( a ^ c ) ) ) = 1 $=
|
|
( wo wa wddi-0 wdid0id2 ) ABCDEABEACEDABCFG $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$( The weak distributive law in WDOL. $)
|
|
wddi-3 $p |- ( ( a ^ ( b v c ) ) ==3 ( ( a ^ b ) v ( a ^ c ) ) ) = 1 $=
|
|
( wo wa wddi-0 wdid0id3 ) ABCDEABEACEDABCFG $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$( The weak distributive law in WDOL. $)
|
|
wddi-4 $p |- ( ( a ^ ( b v c ) ) ==4 ( ( a ^ b ) v ( a ^ c ) ) ) = 1 $=
|
|
( wo wa wddi-0 wdid0id4 ) ABCDEABEACEDABCFG $.
|
|
$( [5-Mar-06] $)
|
|
|
|
$(
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
Modular ortholattices (MOL)
|
|
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Modular law
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
$( The modular law axiom. $)
|
|
ax-ml $a |- ( ( a v b ) ^ ( a v c ) ) =< ( a v ( b ^ ( a v c ) ) ) $.
|
|
|
|
$( Modular law in equational form. $)
|
|
ml $p |- ( a v ( b ^ ( a v c ) ) ) = ( ( a v b ) ^ ( a v c ) ) $=
|
|
( wo wa leo ler2an leor leran lel2or ax-ml lebi ) ABACDZEZDABDZMEZAPNAOMABF
|
|
ACFGBOMBAHIJABCKL $.
|
|
$( [31-Mar-2011] $) $( [15-Mar-2010] $)
|
|
|
|
$( Dual of modular law. $)
|
|
mldual $p |- ( a ^ ( b v ( a ^ c ) ) ) = ( ( a ^ b ) v ( a ^ c ) ) $=
|
|
( wa wo wn anor3 cm oran3 lan ax-r1 tr lor ml 2an 3tr 3tr2 con1 ) ABACDZEZD
|
|
ZABDZSEZAFZTFZEZUBFZSFZDZUAFUCFUFUDBFZUDCFZEZDZEUDUJEZULDUIUEUMUDUEUJUHDZUM
|
|
UOUEBSGHUMUOULUHUJACIZJKLMUDUJUKNUNUGULUHABIUPOPATIUBSGQR $.
|
|
$( [31-Mar-2011] $) $( [15-Mar-2010] $)
|
|
|
|
${
|
|
mli.1 $e |- c =< a $.
|
|
$( Inference version of modular law. $)
|
|
ml2i $p |- ( c v ( b ^ a ) ) = ( ( c v b ) ^ a ) $=
|
|
( wo wa ml df-le2 lan lor 3tr2 ) CBCAEZFZECBEZLFCBAFZENAFCBAGMOCLABCADHZI
|
|
JLANPIK $.
|
|
$( [1-Apr-2012] $)
|
|
|
|
$( Inference version of modular law. $)
|
|
mli $p |- ( ( a ^ b ) v c ) = ( a ^ ( b v c ) ) $=
|
|
( wa wo ancom ror orcom ml2i 3tr ran ) ABEZCFZCBFZAEZBCFZAEAQENBAEZCFCRFP
|
|
MRCABGHRCIABCDJKOQACBILQAGK $.
|
|
$( [1-Apr-2012] $)
|
|
|
|
$}
|
|
|
|
${
|
|
mlduali.1 $e |- a =< c $.
|
|
$( Inference version of dual of modular law. $)
|
|
mldual2i $p |- ( c ^ ( b v a ) ) = ( ( c ^ b ) v a ) $=
|
|
( wa wo mldual lear leid ler2an lebi lor lan 3tr2 ) CBCAEZFZECBEZOFCBAFZE
|
|
QAFCBAGPRCOABOACAHACADAIJKZLMOAQSLN $.
|
|
$( [1-Apr-2012] $)
|
|
|
|
$( Inference version of dual of modular law. $)
|
|
mlduali $p |- ( ( a v b ) ^ c ) = ( a v ( b ^ c ) ) $=
|
|
( wo wa ax-a2 ran ancom mldual2i 3tr ror orcom ) ABEZCFZCBFZAEZBCFZAEAREO
|
|
BAEZCFCSFQNSCABGHSCIABCDJKPRACBILRAMK $.
|
|
$( [1-Apr-2012] $)
|
|
$}
|
|
|
|
$( Form of modular law that swaps two terms. $)
|
|
ml3le $p |- ( a v ( b ^ ( c v a ) ) ) =< ( a v ( c ^ ( b v a ) ) ) $=
|
|
( wo wa lear lelor or12 oridm lor orcom 3tr lbtr leor lel2or ler2an mlduali
|
|
leao1 ) ABCADZEZDZACDZBADZEACUCEDUAUBUCUAASDZUBTSABSFGUDCAADZDSUBACAHUEACAI
|
|
JCAKLMAUCTABNZBSAROPACUCUFQM $.
|
|
$( [1-Apr-2012] $)
|
|
|
|
$( Form of modular law that swaps two terms. $)
|
|
ml3 $p |- ( a v ( b ^ ( c v a ) ) ) = ( a v ( c ^ ( b v a ) ) ) $=
|
|
( wo wa ml3le lebi ) ABCADEDACBADEDABCFACBFG $.
|
|
$( [1-Apr-2012] $)
|
|
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem1 $p |- ( ( ( x v y ) v u ) ^ w )
|
|
= ( ( ( x v y ) v u ) ^ ( ( u v w ) ^ w ) ) $=
|
|
( wo wa leor leid ler2an lear lebi lan ) BABEZBFZCDEAEBNBMBBAGBHIMBJKL $.
|
|
$( [31-Mar-2011] $) $( [15-Mar-2010] $)
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem2 $p |- ( ( ( x v y ) v u ) ^ ( ( u v w ) ^ w ) )
|
|
= ( ( ( ( x v y ) ^ ( u v w ) ) v u ) ^ w ) $=
|
|
( wo wa anass cm ax-a2 ran ml orcom 3tr tr ) CDEZAEZABEZBFFZPQFZBFZOQFZAEZB
|
|
FTRPQBGHSUBBSAOEZQFZAUAEZUBPUCQOAIJUEUDAOBKHAUALMJN $.
|
|
$( [31-Mar-2011] $) $( [15-Mar-2010] $)
|
|
|
|
${
|
|
vneulem3.1 $e |- ( ( x v y ) ^ ( u v w ) ) = 0 $.
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem3 $p |- ( ( ( ( x v y ) ^ ( u v w ) ) v u ) ^ w ) = ( u ^ w ) $=
|
|
( wo wa wf ror or0r tr ran ) CDFABFGZAFZABNHAFAMHAEIAJKL $.
|
|
$( [31-Mar-2011] $) $( [15-Mar-2010] $)
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem4 $p |- ( ( ( x v y ) v u ) ^ w ) = ( u ^ w ) $=
|
|
( wo wa vneulem1 vneulem2 vneulem3 3tr ) CDFZAFZBGMABFZBGGLNGAFBGABGABCDH
|
|
ABCDIABCDEJK $.
|
|
$( [31-Mar-2011] $) $( [15-Mar-2010] $)
|
|
$}
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem5 $p |- ( ( ( x v y ) v u ) ^ ( ( x v y ) v w ) )
|
|
= ( ( x v y ) v ( ( ( x v y ) v u ) ^ w ) ) $=
|
|
( wo wa ancom ml cm lor 3tr ) CDEZAEZLBEZFNMFZLBMFZEZLMBFZEMNGQOLBAHIPRLBMG
|
|
JK $.
|
|
$( [31-Mar-2011] $) $( [15-Mar-2010] $)
|
|
|
|
${
|
|
vneulem6.1 $e |- ( ( a v b ) ^ ( c v d ) ) = 0 $.
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem6 $p |- ( ( ( a v b ) v d ) ^ ( ( b v c ) v d ) )
|
|
= ( ( c ^ a ) v ( b v d ) ) $=
|
|
( wo wa orcom ror or32 tr 2an vneulem5 leor ax-a2 leao3 bltr lel2or leror
|
|
ler ax-r2 ran wf vneulem4 lerr leao2 leo ler2an lebi ) ABFZDFZBCFZDFZGZCA
|
|
GZBDFZFZUNUPUPAFZCGZFZUQUNURUPCFZGUTUKURUMVAUKBAFZDFZURUJVBDABHIBADJKBCDJ
|
|
LACBDMUAUPUQUSUPUONUSDCGZUQUSVCCGVDURVCCBDAJUBDCBAVBDCFZGUJCDFZGUCVBUJVEV
|
|
FBAODCOLEKUDKVDUPUODCBPUEQRQUQUKUMUOUKUPUOUJDACBUFTBUJDBANSRUOUMUPUOULDCA
|
|
BPTBULDBCUGSRUHUI $.
|
|
$( [31-Mar-2011] $) $( [15-Mar-2010] $)
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem7 $p |- ( ( c ^ a ) v ( b v d ) ) = ( b v d ) $=
|
|
( wa wo wf leao2 leao1 ler2an lbtr le0 lebi ror or0r tr ) CAFZBDGZGHSGSRH
|
|
SRHRABGZCDGZFHRTUAACBICADJKELRMNOSPQ $.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem8 $p |- ( ( ( a v b ) v d ) ^ ( ( b v c ) v d ) ) = ( b v d ) $=
|
|
( wo wa vneulem6 vneulem7 tr ) ABFDFBCFDFGCAGBDFZFKABCDEHABCDEIJ $.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem9 $p |- ( ( ( a v b ) v d ) ^ ( ( a v b ) v c ) )
|
|
= ( ( c ^ d ) v ( a v b ) ) $=
|
|
( wo wa ancom vneulem5 ax-r2 orcom vneulem4 ror 3tr ) ABFZDFZOCFZGZOQDGZF
|
|
ZSOFCDGZOFRQPGTPQHCDABIJOSKSUAOCDABELMN $.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem10 $p |- ( ( ( a v b ) v c ) ^ ( ( a v c ) v d ) ) = ( a v c ) $=
|
|
( wo wa ax-a2 ax-r5 or32 2an wf orcom tr vneulem8 ) ABFZCFZACFZDFZGBAFZCF
|
|
ZADFCFZGRQUASUBPTCABHIACDJKBADCTDCFZGPCDFZGLTPUCUDBAMDCMKENON $.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem11 $p |- ( ( ( b v c ) v d ) ^ ( ( a v c ) v d ) )
|
|
= ( ( c v d ) v ( a ^ b ) ) $=
|
|
( wo wa ax-a3 orcom tr ax-a2 ror or32 2an wf ancom vneulem9 3tr ) BCFDFZA
|
|
CFZDFZGCDFZBFZUBAFZGABGZUBFUBUEFSUCUAUDSBUBFUCBCDHBUBIJUACAFZDFUDTUFDACKL
|
|
CADMJNCDABUBABFZGUGUBGOUBUGPEJQUEUBIR $.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
$}
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem12 $p |- ( ( ( c ^ d ) v ( a v b ) ) ^ ( ( c v d ) v ( a ^ b ) ) )
|
|
= ( ( c ^ d ) v ( ( a v b ) ^ ( ( c v d ) v ( a ^ b ) ) ) ) $=
|
|
( wa wo ml cm orass leao1 df-le2 ror tr lan lor 3tr2 ) CDEZABFZFZQCDFZABEZF
|
|
ZFZEZQRUCEZFZSUBEQRUBEZFUFUDQRUBGHUCUBSUCQTFZUAFZUBUIUCQTUAIHUHTUAQTCDDJKLM
|
|
ZNUEUGQUCUBRUJNOP $.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
|
|
${
|
|
vneulem13.1 $e |- ( ( a v b ) ^ ( c v d ) ) = 0 $.
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem13 $p |- ( ( c ^ d ) v ( ( a v b ) ^ ( ( c v d ) v ( a ^ b ) ) ) )
|
|
= ( ( c ^ d ) v ( a ^ b ) ) $=
|
|
( wo wa leao1 leid ler2an lear lebi lor lan mldual wf 2or or0r tr 3tr ) A
|
|
BFZCDFZABGZFZGZUCCDGUEUAUBUAUCGZFZGUAUBGZUFFZUCUDUGUAUCUFUBUCUFUCUAUCABBH
|
|
UCIJZUAUCKZLMNUAUBUCOUIPUCFUCUHPUFUCEUFUCUKUJLQUCRSTM $.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem14 $p |- ( ( ( c ^ d ) v ( a v b ) ) ^ ( ( c v d ) v ( a ^ b ) ) )
|
|
= ( ( c ^ d ) v ( a ^ b ) ) $=
|
|
( wa wo vneulem12 vneulem13 tr ) CDFZABGZGCDGABFZGZFKLNFGKMGABCDHABCDEIJ
|
|
$.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem15 $p |- ( ( a v c ) ^ ( b v d ) )
|
|
= ( ( ( ( a v b ) v c ) ^ ( ( a v c ) v d ) )
|
|
^ ( ( ( a v b ) v d ) ^ ( ( b v c ) v d ) ) ) $=
|
|
( wo wa vneulem10 vneulem8 2an cm ) ABFZCFACFZDFGZLDFBCFDFGZGMBDFZGNMOPAB
|
|
CDEHABCDEIJK $.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
|
|
$( Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem16 $p |- ( ( ( ( a v b ) v c ) ^ ( ( a v c ) v d ) )
|
|
^ ( ( ( a v b ) v d ) ^ ( ( b v c ) v d ) ) )
|
|
= ( ( a ^ b ) v ( c ^ d ) ) $=
|
|
( wo wa ancom an4 vneulem9 vneulem11 2an tr vneulem14 orcom 3tr ) ABFZCFZ
|
|
ACFDFZGZQDFZBCFDFZGZGUCTGZCDGZQFZCDFABGZFZGZUGUEFZTUCHUDUARGZUBSGZGUIUAUB
|
|
RSIUKUFULUHABCDEJABCDEKLMUIUEUGFUJABCDENUEUGOMP $.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
$}
|
|
|
|
${
|
|
vneulem.1 $e |- ( ( a v b ) ^ ( c v d ) ) = 0 $.
|
|
$( von Neumann's modular law lemma. Lemma 9, Kalmbach p. 96 $)
|
|
vneulem $p |- ( ( a v c ) ^ ( b v d ) ) = ( ( a ^ b ) v ( c ^ d ) ) $=
|
|
( wo wa vneulem15 vneulem16 tr ) ACFZBDFGABFZCFKDFGLDFBCFDFGGABGCDGFABCDE
|
|
HABCDEIJ $.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
$}
|
|
|
|
${
|
|
vneulemexp.1 $e |- ( ( a v b ) ^ ( c v d ) ) = 0 $.
|
|
$( Expanded version of ~ vneulem . $)
|
|
vneulemexp $p |- ( ( a v c ) ^ ( b v d ) ) = ( ( a ^ b ) v ( c ^ d ) ) $=
|
|
( wo wa or32 2an orcom ror tr ancom ml cm 3tr ran ler2an lebi wf lor leor
|
|
ax-a2 ax-r5 ax-r2 leid lear anass or0r leao3 lerr bltr lel2or leao2 leror
|
|
lan ler leo leao1 lbtr le0 an4 ax-a3 orass df-le2 3tr2 mldual 2or ) ACFZB
|
|
DFZGZABFZCFZVIDFZGZVLDFZBCFZDFZGZGZABGZCDGZFZVTVKVOVIVSVJVOBAFZCFZADFZCFZ
|
|
GZVIVMWEVNWGVLWDCABUCUDACDHIWHDBGZVIFZVIWHWJWHVIVIBFZDGZFZWJWHWKVNGZWMWEW
|
|
KWGVNWEVMWKWDVLCBAJZKABCHLADCHIWNVNWKGZVIDWKGZFZWMWKVNMWRWPVIDBNOWQWLVIDW
|
|
KMUAPUEVIWJWLVIWIUBWLWBWJWLVMDGZWBWKVMDACBHQWSVMCDFZDGZGZVLWTGZCFZDGZWBDX
|
|
AVMDXADWTDDCUBDUFRWTDUGSUPXBVMWTGZDGZXEXGXBVMWTDUHOXFXDDXFCVLFZWTGZCXCFZX
|
|
DVMXHWTVLCUCQXJXICVLDNOCXCJPQLXDCDXDTCFCXCTCEKCUILQPZLWBVIWICDAUJUKULUMUL
|
|
WJWEWGWIWEVIWIWDCBDAUNZUQAWDCABUBUOUMWIWGVIWIWFCDBAUJUQAWFCADURUOUMRSWJTV
|
|
IFVIWITVIWITWIWDDCFZGZTWIWDXMXLDBCUSRXNXCTWDVLXMWTWODCJIELZUTWIVASKVIUILL
|
|
LVSCAGZVJFZVJVSXQVSVJVJAFZCGZFZXQVSXRVJCFZGZXTVPXRVRYAVPWDDFZXRVLWDDABJKB
|
|
ADHLBCDHIYBYAXRGZVJCXRGZFZXTXRYAMYFYDVJCANOYEXSVJCXRMUAPUEVJXQXSVJXPUBXSD
|
|
CGZXQXSYCCGZYGXRYCCBDAHQYHYCXMCGZGZXNDFZCGZYGCYIYCCYICXMCCDUBCUFRXMCUGSUP
|
|
YJYCXMGZCGZYLYNYJYCXMCUHOYMYKCYMDWDFZXMGZDXNFZYKYCYOXMWDDUCQYQYPDWDCNODXN
|
|
JPQLYKDCYKTDFDXNTDXOKDUILQPLYGVJXPDCBUJUKULUMULXQVPVRXPVPVJXPVLDACBUNZUQB
|
|
VLDBAUBZUOUMXPVRVJXPVQDCABUJUQBVQDBCURUOUMRSXQTVJFVJXPTVJXPTXPXCTXPVLWTYR
|
|
CADUSREUTXPVASKVJUILLIOVTVSVOGZWBVLFZWTWAFZGZWCVOVSMYTVPVMGZVRVNGZGUUCVPV
|
|
RVMVNVBUUDUUAUUEUUBUUDVLWSFZWSVLFUUAUUDVMVPGZUUFVPVMMUUGUUDVLDVMGZFZUUFVM
|
|
VPMUUIUUDVLDCNOUUHWSVLDVMMUAPUEVLWSJWSWBVLXKKPUUEWTBFZWTAFZGZWAWTFZUUBVRU
|
|
UJVNUUKVRBWTFUUJBCDVCBWTJLVNCAFZDFUUKVIUUNDACUCKCADHLIUULWTUUKBGZFZUUOWTF
|
|
UUMUULUUKUUJGZUUPUUJUUKMUUQUULWTBUUKGZFZUUPUUKUUJMUUSUULWTBANOUURUUOWTBUU
|
|
KMUAPUEWTUUOJUUOWAWTUUOUUKVLBGZGZWTVLGZAFZBGZWABUUTUUKBUUTBVLBYSBUFRVLBUG
|
|
SUPUVAUUKVLGZBGZUVDUVFUVAUUKVLBUHOUVEUVCBUVEAWTFZVLGZAUVBFZUVCUUKUVGVLWTA
|
|
UCQUVIUVHAWTBNOAUVBJPQLUVCABUVCTAFAUVBTAUVBXCTWTVLMELKAUILQPKPWAWTJPILUUC
|
|
WBWAFZWCUUCWBVLUUBGZFZUVJUUAWBUUBFZGZWBVLUVMGZFZUUCUVLUVPUVNWBVLUUBNOUVMU
|
|
UBUUAUVMWBWTFZWAFZUUBUVRUVMWBWTWAVDOUVQWTWAWBWTCDDUSVEKLZUPUVOUVKWBUVMUUB
|
|
VLUVSUPUAVFUVKWAWBUVKVLWTVLWAGZFZGXCUVTFZWAUUBUWAVLWAUVTWTWAUVTWAVLWAABBU
|
|
SWAUFRZVLWAUGZSUAUPVLWTWAVGUWBTWAFWAXCTUVTWAEUVTWAUWDUWCSVHWAUILPUALWBWAJ
|
|
LPL $.
|
|
$( [31-Mar-2011] $) $( [31-Mar-2011] $)
|
|
$}
|
|
|
|
$( Lemma for ~ l42mod .. $)
|
|
l42modlem1 $p |- ( ( ( a v b ) v d ) ^ ( ( a v b ) v e ) ) =
|
|
( ( a v b ) v ( ( a v d ) ^ ( b v e ) ) ) $=
|
|
( wo wa leo ml2i ancom tr lor cm orass or12 2an lerr 3tr 3tr1 ) ABDEZBACEZE
|
|
ZFZEZABTSFZEZEZABEZCEZUGDEZFZUGUDEUFUCUEUBAUEUASFUBSTBBDGHUASIJKLUJUAASEZFU
|
|
KUAFZUCUHUAUIUKUHABCEEUAABCMABCNJABDMOUAUKIUCULUASAATBACGPHLQABUDMR $.
|
|
$( [8-Apr-2012] $)
|
|
|
|
$( Lemma for ~ l42mod .. $)
|
|
l42modlem2 $p |- ( ( ( ( a v b ) ^ c ) v d ) ^ e ) =<
|
|
( ( ( a v b ) v d ) ^ ( ( a v b ) v e ) ) $=
|
|
( wo wa lea leror leor le2an ) ABFZCGZDFLDFELEFMLDLCHIELJK $.
|
|
$( [8-Apr-2012] $)
|
|
|
|
$( An equation that fails in OML L42 when converted to a Hilbert space
|
|
equation. $)
|
|
l42mod $p |- ( ( ( ( a v b ) ^ c ) v d ) ^ e )
|
|
=< ( ( a v b ) v ( ( a v d ) ^ ( b v e ) ) ) $=
|
|
( wo wa l42modlem2 l42modlem1 lbtr ) ABFZCGDFEGKDFKEFGKADFBEFGFABCDEHABDEIJ
|
|
$.
|
|
$( [8-Apr-2012] $)
|
|
|
|
$( Expansion by modular law. $)
|
|
modexp $p |- ( a ^ ( b v c ) ) = ( a ^ ( b v ( c ^ ( a v b ) ) ) ) $=
|
|
( wo wa anass anabs ran ancom leor mlduali tr lan 3tr2 ) AABDZEZBCDZEAOQEZE
|
|
AQEABCOEDZEAOQFPAQABGHRSARQOESOQIBCOBAJKLMN $.
|
|
$( [10-Apr-2012] $)
|
|
|
|
$( Experimental expansion of l42mod.
|
|
l42modexp $p |- ( ( ( a v b ) v d ) ^ ( ( a v b ) v e ) ) =
|
|
( ( a v b ) v ( ( a v d ) ^ ( b v e ) ) ) $=
|
|
( l42modlem1 modexp id tr lor lan cm ) ???????E???????????????FZ?????????L?
|
|
????????L?????????L?????????L?????????L?????????L?????????L?GHIJHIJHIJHIJHI
|
|
JHIJHIJHIKHH $.
|
|
$)
|
|
|
|
$(
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
Arguesian law
|
|
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
|
|
$)
|
|
|
|
${
|
|
arg.1 $e |- ( ( a0 v b0 ) ^ ( a1 v b1 ) ) =< ( a2 v b2 ) $.
|
|
$( The Arguesian law as an axiom. $)
|
|
ax-arg $a |- ( ( a0 v a1 ) ^ ( b0 v b1 ) )
|
|
=< ( ( ( a0 v a2 ) ^ ( b0 v b2 ) ) v ( ( a1 v a2 ) ^ ( b1 v b2 ) ) ) $.
|
|
$}
|
|
|
|
${
|
|
dp15lema.1 $e |- d = ( a2 v ( a0 ^ ( a1 v b1 ) ) ) $.
|
|
dp15lema.2 $e |- p0 = ( ( a1 v b1 ) ^ ( a2 v b2 ) ) $.
|
|
dp15lema.3 $e |- e = ( b0 ^ ( a0 v p0 ) ) $.
|
|
$( Part of proof (1)=>(5) in Day/Pickering 1982. $)
|
|
dp15lema $p |- ( ( a0 v e ) ^ ( a1 v b1 ) ) =< ( d v b2 ) $=
|
|
( wo wa lor tr ran wt leran cm lan le1 lelor an1r orass oridm ror 3tr lea
|
|
orcom mlduali lear leror bltr or32 lbtr letr ) CBMZDGMZNCFCUSEHMZNZMZNZMZ
|
|
USNZAHMZURVDUSBVCCBFCIMZNVCLVGVBFIVACKOUAPOQVECRVBNZMZUSNZVFVDVIUSVCVHCFR
|
|
VBFUBSUCSVJUTCUSNZMZVFVJVAVKMZVLVJVACMZUSNVMVIVNUSVICVBMZCCMZVAMZVNVHVBCV
|
|
BUDOVQVOCCVAUETVQVBVNVPCVACUFUGCVAUJPUHQVACUSUSUTUIUKPVAUTVKUSUTULUMUNVLE
|
|
VKMZHMZVFEHVKUOVFVSAVRHJUGTPUPUQUN $.
|
|
$( [1-Apr-2012] $)
|
|
|
|
$( Part of proof (1)=>(5) in Day/Pickering 1982. $)
|
|
dp15lemb $p |- ( ( a0 v a1 ) ^ ( e v b1 ) )
|
|
=< ( ( ( a0 v d ) ^ ( e v b2 ) ) v ( ( a1 v d ) ^ ( b1 v b2 ) ) ) $=
|
|
( dp15lema ax-arg ) CDABGHABCDEFGHIJKLMN $.
|
|
$( [1-Apr-2012] $)
|
|
|
|
$( Part of proof (1)=>(5) in Day/Pickering 1982. $)
|
|
dp15lemc $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
=< ( ( ( a0 v ( a2 v ( a0 ^ ( a1 v b1 ) ) ) )
|
|
^ ( ( b0 ^ ( a0 v p0 ) ) v b2 ) )
|
|
v ( ( a1 v ( a2 v ( a0 ^ ( a1 v b1 ) ) ) ) ^ ( b1 v b2 ) ) ) $=
|
|
( wo wa dp15lemb ror lan lor 2an ran 2or le3tr2 ) CDMZBGMZNCAMZBHMZNZDAMZ
|
|
GHMZNZMUCFCIMNZGMZNCECDGMNMZMZUKHMZNZDUMMZUINZMABCDEFGHIJKLOUDULUCBUKGLPQ
|
|
UGUPUJURUEUNUFUOAUMCJRBUKHLPSUHUQUIAUMDJRTUAUB $.
|
|
$( [10-Apr-2012] $)
|
|
|
|
$( Part of proof (1)=>(5) in Day/Pickering 1982. $)
|
|
dp15lemd $p |- ( ( ( a0 v ( a2 v ( a0 ^ ( a1 v b1 ) ) ) )
|
|
^ ( ( b0 ^ ( a0 v p0 ) ) v b2 ) )
|
|
v ( ( a1 v ( a2 v ( a0 ^ ( a1 v b1 ) ) ) ) ^ ( b1 v b2 ) ) )
|
|
= ( ( ( a0 v a2 )
|
|
^ ( ( b0 ^ ( a0 v p0 ) ) v b2 ) )
|
|
v ( ( ( a1 v a2 ) v ( a0 ^ ( a1 v b1 ) ) ) ^ ( b1 v b2 ) ) ) $=
|
|
( wo wa or12 orabs lor orcom 3tr ran orass cm 2or ) CECDGMZNZMZMZFCIMNHMZ
|
|
NCEMZUHNDUFMZGHMZNZDEMUEMZUKNZUGUIUHUGECUEMZMECMUICEUEOUOCECUDPQECRSTUNUL
|
|
UMUJUKDEUEUATUBUC $.
|
|
$( [1-Apr-2012] $)
|
|
|
|
$( Part of proof (1)=>(5) in Day/Pickering 1982. $)
|
|
dp15leme $p |- ( ( ( a0 v a2 )
|
|
^ ( ( b0 ^ ( a0 v p0 ) ) v b2 ) )
|
|
v ( ( ( a1 v a2 ) v ( a0 ^ ( a1 v b1 ) ) ) ^ ( b1 v b2 ) ) )
|
|
=< ( ( ( a0 v a2 )
|
|
^ ( ( b0 ^ ( a0 v p0 ) ) v b2 ) )
|
|
v ( ( ( a1 v a2 ) v ( b1 ^ ( a0 v a1 ) ) ) ^ ( b1 v b2 ) ) ) $=
|
|
( wo wa ax-a2 lan 2or orass tr lelor ml3le bltr cm ror lbtr leran ) DEMZC
|
|
DGMZNZMZGHMZNUGGCDMNZMZUKNCEMFCIMNHMNUJUMUKUJEDULMZMZUMUJEDCGDMZNZMZMZUOU
|
|
JEDMZUQMUSUGUTUIUQDEOUHUPCDGOPQEDUQRSURUNEDCGUATUBUOUTULMZUMVAUOEDULRUCUT
|
|
UGULEDOUDSUEUFT $.
|
|
$( [1-Apr-2012] $)
|
|
|
|
$( Part of proof (1)=>(5) in Day/Pickering 1982. $)
|
|
dp15lemf $p |- ( ( ( a0 v a2 )
|
|
^ ( ( b0 ^ ( a0 v p0 ) ) v b2 ) )
|
|
v ( ( ( a1 v a2 ) v ( b1 ^ ( a0 v a1 ) ) ) ^ ( b1 v b2 ) ) )
|
|
=< ( ( ( a1 v a2 )
|
|
^ ( b1 v b2 ) )
|
|
v ( ( ( a0 v a2 ) ^ ( b0 v b2 ) ) v ( b1 ^ ( a0 v a1 ) ) ) ) $=
|
|
( wo wa lea leror lelan leao1 mldual2i ancom 3tr2 bile le2or or12 lbtr
|
|
ror ) CEMZFCIMZNZHMZNZDEMZGCDMZNZMZGHMZNZMUGFHMZNZULUPNZUNMZMUTUSUNMMUKUS
|
|
UQVAUJURUGUIFHFUHOPQUQVAUPUONUPULNZUNMUQVAUNULUPGUMHRSUPUOTVBUTUNUPULTUFU
|
|
AUBUCUSUTUNUDUE $.
|
|
$( [1-Apr-2012] $)
|
|
|
|
dp15lemg.4 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
dp15lemg.5 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
$( Part of proof (1)=>(5) in Day/Pickering 1982. $)
|
|
dp15lemg $p |- ( ( ( a1 v a2 )
|
|
^ ( b1 v b2 ) )
|
|
v ( ( ( a0 v a2 ) ^ ( b0 v b2 ) ) v ( b1 ^ ( a0 v a1 ) ) ) )
|
|
= ( ( c0 v c1 ) v ( b1 ^ ( a0 v a1 ) ) ) $=
|
|
( wo wa ror cm 2or orass tr ) DEQGHQRZCEQFHQRZGCDQRZQZQZIJUFQZQZIJQUFQZUJ
|
|
UHIUDUIUGOJUEUFPSUATUKUJIJUFUBTUC $.
|
|
$( [1-Apr-2012] $)
|
|
|
|
$( Part of proof (1)=>(5) in Day/Pickering 1982. $)
|
|
dp15lemh $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
=< ( ( c0 v c1 ) v ( b1 ^ ( a0 v a1 ) ) ) $=
|
|
( wo wa lbtr letr dp15lemc dp15lemd dp15leme dp15lemf dp15lemg ) CDQZFCKQ
|
|
RZGQRZDEQZGHQZRCEQZFHQRGUFRZQQZIJQULQUHUKUGHQZRZUIULQUJRQZUMUHUOUICDGQRZQ
|
|
UJRQZUPUHCEUQQZQUNRDUSQUJRQURABCDEFGHKLMNUAABCDEFGHKLMNUBSABCDEFGHKLMNUCT
|
|
ABCDEFGHKLMNUDTABCDEFGHIJKLMNOPUES $.
|
|
$( [2-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
dp15.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
dp15.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
dp15.3 $e |- p0 = ( ( a1 v b1 ) ^ ( a2 v b2 ) ) $.
|
|
$( Part of theorem from Alan Day and Doug Pickering, "A note on the
|
|
Arguesian lattice identity," Studia Sci. Math. Hungar. 19:303-305
|
|
(1982). (1)=>(5) $)
|
|
dp15 $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
=< ( ( c0 v c1 ) v ( b1 ^ ( a0 v a1 ) ) ) $=
|
|
( wo wa id dp15lemh ) CABEMNMZDAIMNZABCDEFGHIQOLROJKP $.
|
|
$( [1-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
dp53lem.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
dp53lem.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
dp53lem.3 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
dp53lem.4 $e |- p0 = ( ( a1 v b1 ) ^ ( a2 v b2 ) ) $.
|
|
dp53lem.5 $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
$( Part of proof (5)=>(3) in Day/Pickering 1982. $)
|
|
dp53lema $p |- ( b1 v ( b0 ^ ( a0 v p0 ) ) )
|
|
=< ( b1 v ( ( a0 v a1 ) ^ ( c0 v c1 ) ) ) $=
|
|
( wo wa lbtr letr leo lor lan lear lea lelor cm bltr ler2an leor mldual2i
|
|
ax-a3 ancom ror tr dp15 orcom leid lel2or ) FFBCQZHIQZRZQZEBKQZRZFVBUAZVE
|
|
UTVEFQZRZVCQZVCVEVHFQZVIVEVGUTFQZRZVJVEVGVKVEFUAVEEBCFQZDGQZRZQZRZVKVDVPE
|
|
KVOBOUBUCVQVPVKEVPUDVPBVMQZVKVOVMBVMVNUEUFVKVRBCFULUGSTUHUIVLVGUTRZFQVJFU
|
|
TVGFVEUJUKVSVHFVGUTUMUNUOSFVCVHVFUFTVHVCVCVHVBFQZVCVHVBFUTRZQZVTVHUTVAWAQ
|
|
ZRWBVHUTWCUTVGUEBCDEFGHIKLMOUPUIWAVAUTFUTUDUKSWAFVBFUTUEUFTVBFUQSVCURUSTU
|
|
S $.
|
|
$( [2-Apr-2012] $)
|
|
|
|
$( Part of proof (5)=>(3) in Day/Pickering 1982. $)
|
|
dp53lemb $p |- ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) )
|
|
= ( b0 ^ ( b1 v ( ( a0 v a1 ) ^ ( c0 v c1 ) ) ) ) $=
|
|
( wo wa ran 3tr an32 tr lor leor ml2i ancom lan anass cm anabs ) EFJHIQZR
|
|
ZQZREEFQZFBCQZUKRZQZRZRZEUNRZUQRZEUQRUMUREUMFUPUNRZQUQUNRURULVBFULUOUNRZU
|
|
KRVBJVCUKNSUOUNUKUAUBUCUNUPFFEUDUEUQUNUFTUGVAUSEUNUQUHUIUTEUQEFUJST $.
|
|
$( [2-Apr-2012] $)
|
|
|
|
$( Part of proof (5)=>(3) in Day/Pickering 1982. $)
|
|
dp53lemc $p |- ( b0 ^ ( ( ( a0 ^ b0 ) v b1 ) v ( c2 ^ ( c0 v c1 ) ) ) )
|
|
= ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) $=
|
|
( wa wo leo le2an or32 orcom cm lbtr lerr ler2an df-le2 lor 3tr lan ) BEQ
|
|
ZFRJHIRZQZRZFUMRZEUNUKUMRZFRFUPRUOUKFUMUAUPFUBUPUMFUKUMUKJULUKBCRZEFRZQZJ
|
|
BUQEURBCSEFSTJUSNUCUDUKIHUKBDRZEGRZQZIBUTEVABDSEGSTIVBMUCUDUEUFUGUHUIUJ
|
|
$.
|
|
$( [2-Apr-2012] $)
|
|
|
|
$( Part of proof (5)=>(3) in Day/Pickering 1982. $)
|
|
dp53lemd $p |- ( b0 ^ ( a0 v p0 ) )
|
|
=< ( b0 ^ ( ( ( a0 ^ b0 ) v b1 ) v ( c2 ^ ( c0 v c1 ) ) ) ) $=
|
|
( wo wa lea leor dp53lema letr ler2an dp53lemc dp53lemb tr cm lbtr ) EBKQ
|
|
ZRZEFBCQHIQZRQZRZEBERFQJUKRZQRZUJEULEUISUJFUJQULUJFTABCDEFGHIJKLMNOPUAUBU
|
|
CUOUMUOEFUNQRUMABCDEFGHIJKLMNOPUDABCDEFGHIJKLMNOPUEUFUGUH $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (5)=>(3) in Day/Pickering 1982. $)
|
|
dp53leme $p |- ( b0 ^ ( a0 v p0 ) )
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa dp53lemd orcom orass tr lan lear mldual2i 3tr lea leror bltr letr
|
|
) EBKQREBERZFQJHIQRZQZRZBEFULQZRZQZABCDEFGHIJKLMNOPSUNUKUPQZUQUNEUOUKQZRU
|
|
PUKQURUMUSEUMUKUOQUSUKFULUAUKUOTUBUCUKUOEBEUDUEUPUKTUFUKBUPBEUGUHUIUJ $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (5)=>(3) in Day/Pickering 1982. $)
|
|
dp53lemf $p |- ( a0 v p )
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa leo lbtr anass tr lan cm leao4 bltr lea orcom ler2an mldual2i ror
|
|
ancom lelor letr dp53leme df-le2 lel2or ) BBEFJHIQRQRZQZABURSZAEBKQZRZUSQ
|
|
ZUSAVBBQZVCAVAEBQZRZVDABEQZCFQZDGQZRZRZVFAVGVHRVIRVKPVGVHVIUAUBVKVAVEVKVG
|
|
KRZVAVLVKKVJVGOUCUDKVGBUEUFVKVGVEVGVJUGBEUHTUIUFVFVAERZBQVDBEVABKSUJVMVBB
|
|
VAEULUKUBTBUSVBUTUMUNVBUSABCDEFGHIJKLMNOPUOUPTUQ $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (5)=>(3) in Day/Pickering 1982. $)
|
|
dp53lemg $p |- p
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa leor dp53lemf letr ) ABAQBEFJHIQRQRQABSABCDEFGHIJKLMNOPTUA $.
|
|
$( [2-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
dp53.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
dp53.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
dp53.3 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
dp53.4 $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
$( Part of theorem from Alan Day and Doug Pickering, "A note on the
|
|
Arguesian lattice identity," Studia Sci. Math. Hungar. 19:303-305
|
|
(1982). (5)=>(3) $)
|
|
dp53 $p |- p =< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa id dp53lemg ) ABCDEFGHIJCFODGOPZKLMSQNR $.
|
|
$( [2-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
dp35lem.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
dp35lem.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
dp35lem.3 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
dp35lem.4 $e |- p0 = ( ( a1 v b1 ) ^ ( a2 v b2 ) ) $.
|
|
dp35lem.5 $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
dp35lemg $p |- p
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( dp53 ) ABCDEFGHIJLMNPQ $.
|
|
$( [12-Apr-2012] $)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
dp35lemf $p |- ( a0 v p )
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa leo dp35lemg lel2or ) BBEFJHIQRQRZQABUBSABCDEFGHIJKLMNOPTUA $.
|
|
$( [12-Apr-2012] $)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
dp35leme $p |- ( b0 ^ ( a0 v p0 ) )
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa lor ancom leor bile le2an anass cm leo mlduali 3tr1 dp35lemf bltr
|
|
tr letr ) EBKQZRBEQZBCFQZDGQZRZQZRZBEFJHIQRQRQZEUNUMUREBUAUMURKUQBOSUBUCU
|
|
SBAQZUTBUQUNRZQZBUNUORUPRZQUSVAVBVDBVBUNUQRZVDUQUNTVDVEUNUOUPUDUEUKSUSURU
|
|
NRVCUNURTBUQUNBEUFUGUKAVDBPSUHABCDEFGHIJKLMNOPUIUJUL $.
|
|
$( [12-Apr-2012] $)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
dp35lemd $p |- ( b0 ^ ( a0 v p0 ) )
|
|
=< ( b0 ^ ( ( ( a0 ^ b0 ) v b1 ) v ( c2 ^ ( c0 v c1 ) ) ) ) $=
|
|
( wo wa lea ler2an dp35leme mldual2i lel2or ancom bile lear le2or cm lbtr
|
|
orass bltr letr ) EBKQZRZEBEFJHIQRZQZRZQZRZEBERZFQUOQZRZUNEUREUMSABCDEFGH
|
|
IJKLMNOPUATUSEBRZUQQZVBUQBEEUPSZUBVDEVAVCEUQEBSVEUCVDUTUPQZVAVCUTUQUPVCUT
|
|
EBUDUEEUPUFUGVAVFUTFUOUJUHUITUKUL $.
|
|
$( [12-Apr-2012] $)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
dp35lemc $p |- ( b0 ^ ( ( ( a0 ^ b0 ) v b1 ) v ( c2 ^ ( c0 v c1 ) ) ) )
|
|
= ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) $=
|
|
( wa wo leo le2an or32 orcom cm lbtr lerr ler2an df-le2 lor 3tr lan ) BEQ
|
|
ZFRJHIRZQZRZFUMRZEUNUKUMRZFRFUPRUOUKFUMUAUPFUBUPUMFUKUMUKJULUKBCRZEFRZQZJ
|
|
BUQEURBCSEFSTJUSNUCUDUKIHUKBDRZEGRZQZIBUTEVABDSEGSTIVBMUCUDUEUFUGUHUIUJ
|
|
$.
|
|
$( [2-Apr-2012] $)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
dp35lemb $p |- ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) )
|
|
= ( b0 ^ ( b1 v ( ( a0 v a1 ) ^ ( c0 v c1 ) ) ) ) $=
|
|
( wo wa ran 3tr an32 tr lor leor ml2i ancom lan anass cm anabs ) EFJHIQZR
|
|
ZQZREEFQZFBCQZUKRZQZRZRZEUNRZUQRZEUQRUMUREUMFUPUNRZQUQUNRURULVBFULUOUNRZU
|
|
KRVBJVCUKNSUOUNUKUAUBUCUNUPFFEUDUEUQUNUFTUGVAUSEUNUQUHUIUTEUQEFUJST $.
|
|
$( [2-Apr-2012] $)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
dp35lembb $p |- ( b0 ^ ( a0 v p0 ) )
|
|
=< ( b0 ^ ( b1 v ( ( a0 v a1 ) ^ ( c0 v c1 ) ) ) ) $=
|
|
( wo wa dp35lemd dp35lemc dp35lemb tr lbtr ) EBKQREBERFQJHIQZRZQRZEFBCQUD
|
|
RQRZABCDEFGHIJKLMNOPSUFEFUEQRUGABCDEFGHIJKLMNOPTABCDEFGHIJKLMNOPUAUBUC $.
|
|
$( [12-Apr-2012] $)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
dp35lema $p |- ( b1 v ( b0 ^ ( a0 v p0 ) ) )
|
|
=< ( b1 v ( ( a0 v a1 ) ^ ( c0 v c1 ) ) ) $=
|
|
( wo wa leo dp35lembb lear letr lel2or ) FFBCQHIQRZQZEBKQRZFUDSUFEUERUEAB
|
|
CDEFGHIJKLMNOPTEUEUAUBUC $.
|
|
$( [12-Apr-2012] $)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
dp35lem0 $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
=< ( ( c0 v c1 ) v ( b1 ^ ( a0 v a1 ) ) ) $=
|
|
( wo wa orcom letr leid bltr dp35lema lelan id lea mldual2i tr ancom lbtr
|
|
ror lear lelor ) BCQZEBKQRZFQZRZFUNRZUNHIQZRZQZUSURQZUQUNFUTQZRZVAUPVCUNU
|
|
PFUOQZVCUPVEVEUOFSVEUAUBABCDEFGHIJKLMNOPUCTUDVDUNFRZUTQZVAVDVDVGVDUEUTFUN
|
|
UNUSUFUGUHVFURUTUNFUIUKUHUJVAURUSQVBUTUSURUNUSULUMURUSSUJT $.
|
|
$( [12-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
dp35.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
dp35.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
dp35.3 $e |- p0 = ( ( a1 v b1 ) ^ ( a2 v b2 ) ) $.
|
|
$( Part of theorem from Alan Day and Doug Pickering, "A note on the
|
|
Arguesian lattice identity," Studia Sci. Math. Hungar. 19:303-305
|
|
(1982). (3)=>(5) $)
|
|
dp35 $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
=< ( ( c0 v c1 ) v ( b1 ^ ( a0 v a1 ) ) ) $=
|
|
( wo wa id dp35lem0 ) ADMBEMNCFMNZABCDEFGHABMDEMNZIJKROLQOP $.
|
|
$( [12-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
dp34.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
dp34.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
dp34.3 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
dp34.4 $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
$( Part of theorem from Alan Day and Doug Pickering, "A note on the
|
|
Arguesian lattice identity," Studia Sci. Math. Hungar. 19:303-305
|
|
(1982). (3)=>(4) $)
|
|
dp34 $p |- p =< ( ( a0 v b1 ) v ( c2 ^ ( c0 v c1 ) ) ) $=
|
|
( wo wa dp53 lear lelor letr orass cm lbtr ) ABFJHIOPZOZOZBFOUDOZABEUEPZO
|
|
UFABCDEFGHIJKLMNQUHUEBEUERSTUGUFBFUDUAUBUC $.
|
|
$( [3-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
dp41lem.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
dp41lem.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
dp41lem.3 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
dp41lem.4 $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
dp41lem.5 $e |- p2 = ( ( a0 v b0 ) ^ ( a1 v b1 ) ) $.
|
|
dp41lem.6 $e |- p2 =< ( a2 v b2 ) $.
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41lema $p |- ( ( a0 v b0 ) ^ ( a1 v b1 ) )
|
|
=< ( ( a0 v b1 ) v ( c2 ^ ( c0 v c1 ) ) ) $=
|
|
( wo wa cm bltr df2le2 tr dp34 ) BERCFRSZABFRJHIRSRUEUEDGRZSZAUGUEUEUFUEK
|
|
UFKUEPTQUAUBTAUGOTUCABCDEFGHIJLMNOUDUA $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41lemb $p |- c2
|
|
= ( ( c2 ^ ( ( a0 v b0 ) v b1 ) ) ^ ( ( a0 v a1 ) v b1 ) ) $=
|
|
( wo wa tr ancom leor leror leo le2an bltr df2le2 cm anass ) JJBERZFRZBCR
|
|
ZFRZSZSZJUKSUMSZUOJJUNJEFRZULSZUNJULUQSURNULUQUATUQUKULUMEUJFEBUBUCULFUDU
|
|
EUFUGUHUPUOJUKUMUIUHT $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41lemc0 $p |- ( ( ( a0 v b0 ) v b1 ) ^ ( ( a0 v a1 ) v b1 ) )
|
|
= ( ( a0 v b1 ) v ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ) $=
|
|
( wo wa tr ax-a2 ror or32 lan ancom leor ler mldual2i leo 3tr orass orcom
|
|
) BERZFRZBCRZFRZSZUMCFRZSZBRZFRZUSBFRZRVBUSRUQURBRZUNSZVCUMSZFRVAUQUNVCSV
|
|
DUPVCUNUPCBRZFRVCUOVFFBCUAUBCBFUCTUDUNVCUETFUMVCFURBFCUFUGUHVEUTFVEUMVCSU
|
|
TVCUMUEBURUMBEUIUHTUBUJUSBFUKUSVBULUJ $.
|
|
$( [4-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41lemc $p |- ( ( c2 ^ ( ( a0 v b0 ) v b1 ) ) ^ ( ( a0 v a1 ) v b1 ) )
|
|
=< ( c2 ^ ( ( a0 v b1 ) v ( c2 ^ ( c0 v c1 ) ) ) ) $=
|
|
( wo wa bltr anass dp41lemc0 leo dp41lema lel2or lelan ) JBERZFRZSBCRFRZS
|
|
JUHUISZSJBFRZJHIRSZRZSJUHUIUAUJUMJUJUKUGCFRSZRUMABCDEFGHIJKLMNOPQUBUKUMUN
|
|
UKULUCABCDEFGHIJKLMNOPQUDUETUFT $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41lemd $p |- ( c2 ^ ( ( a0 v b1 ) v ( c2 ^ ( c0 v c1 ) ) ) )
|
|
= ( c2 ^ ( ( c0 v c1 ) v ( c2 ^ ( a0 v b1 ) ) ) ) $=
|
|
( wo wa ancom mldual lor lea ml2i ax-a2 lan 3tr ) JBFRZJHIRZSZRSJUHSZUJRU
|
|
KUIJSZRZJUIUKRZSZJUHUIUAUJULUKJUITUBUMUKUIRZJSJUPSUOJUIUKJUHUCUDUPJTUPUNJ
|
|
UKUIUEUFUGUG $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41leme $p |- ( c2 ^ ( ( c0 v c1 ) v ( c2 ^ ( a0 v b1 ) ) ) )
|
|
=< ( ( c0 v c1 ) v ( ( a0 ^ ( b0 v b1 ) ) v ( b1 ^ ( a0 v a1 ) ) ) ) $=
|
|
( wo wa lor mldual ran anass leor mldual2i orcom ancom 3tr lan leao1 lear
|
|
tr leror bltr ) JHIRZJBFRZSZRSZJUOSZBEFRZSZFBCRZSZRZRZUOVDRURUSUQRVEJUOUP
|
|
UAUQVDUSUQVBUTSZUPSVBUTUPSZSZVDJVFUPNUBVBUTUPUCVHVBFVARZSVBFSZVARZVDVGVIV
|
|
BVGUTBSZFRFVLRVIFBUTFEUDUEVLFUFVLVAFUTBUGTUHUIVAFVBBUTCUJUEVKVAVJRVDVJVAU
|
|
FVJVCVAVBFUGTULUHUHTULUSUOVDJUOUKUMUN $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41lemf $p |- ( ( c0 v c1 ) v
|
|
( ( a0 ^ ( b0 v b1 ) ) v ( b1 ^ ( a0 v a1 ) ) ) )
|
|
= ( ( ( b1 v b2 ) ^ ( ( a1 v a2 ) v ( b1 ^ ( a0 v a1 ) ) ) )
|
|
v ( ( a0 v a2 ) ^ ( ( b0 v b2 ) v ( a0 ^ ( b0 v b1 ) ) ) ) ) $=
|
|
( wo wa tr orcom lor or4 ancom ror 2or leao1 mli 3tr ) HIRZBEFRZSZFBCRZSZ
|
|
RZRUJUNULRZRZFGRZCDRZSZUNRZBDRZEGRZSZULRZRZURUSUNRSZVBVCULRSZRUOUPUJULUNU
|
|
AUBUQHUNRZIULRZRVFHIUNULUCVIVAVJVEHUTUNHUSURSUTLUSURUDTUEIVDULMUEUFTVAVGV
|
|
EVHURUSUNFUMGUGUHVBVCULBUKDUGUHUFUI $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41lemg $p |- ( ( ( b1 v b2 ) ^ ( ( a1 v a2 ) v ( b1 ^ ( a0 v a1 ) ) ) )
|
|
v ( ( a0 v a2 ) ^ ( ( b0 v b2 ) v ( a0 ^ ( b0 v b1 ) ) ) ) )
|
|
= ( ( ( b1 v b2 ) ^ ( ( a1 v a2 ) v ( a0 ^ ( a1 v b1 ) ) ) )
|
|
v ( ( a0 v a2 ) ^ ( ( b0 v b2 ) v ( b1 ^ ( a0 v b0 ) ) ) ) ) $=
|
|
( wo wa or32 ml3 orcom lan lor tr ror 3tr 2or ) FGRZCDRZFBCRSZRZSUIUJBCFR
|
|
ZSZRZSBDRZEGRZBEFRZSZRZSUPUQFBERSZRZSULUOUIULCUKRZDRCUNRZDRUOCDUKTVCVDDVC
|
|
CBFCRZSZRVDCFBUAVFUNCVEUMBFCUBUCUDUEUFCUNDTUGUCUTVBUPUTEUSRZGREVARZGRVBEG
|
|
USTVGVHGVGEBFERZSZRVHUSVJEURVIBEFUBUCUDEBFUAUEUFEVAGTUGUCUH $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. "By CP(a,b)". $)
|
|
dp41lemh $p |- ( ( ( b1 v b2 ) ^ ( ( a1 v a2 ) v ( a0 ^ ( a1 v b1 ) ) ) )
|
|
v ( ( a0 v a2 ) ^ ( ( b0 v b2 ) v ( b1 ^ ( a0 v b0 ) ) ) ) )
|
|
=< ( ( ( b1 v b2 ) ^ ( ( a1 v a2 ) v ( a0 ^ ( a2 v b2 ) ) ) )
|
|
v ( ( a0 v a2 ) ^ ( ( b0 v b2 ) v ( b1 ^ ( a2 v b2 ) ) ) ) ) $=
|
|
( wo wa ler2an lea leo leran cm bltr letr lelor lelan lear leao3 le2or )
|
|
FGRZCDRZBCFRZSZRZSULUMBDGRZSZRZSBDRZEGRZFBERZSZRZSUTVAFUQSZRZSUPUSULUOURU
|
|
MUOBUQBUNUAUOVBUNSZUQBVBUNBEUBUCVGKUQKVGPUDQUEZUFTUGUHVDVFUTVCVEVAVCFUQFV
|
|
BUAVCVGUQVCVBUNFVBUIFVBCUJTVHUFTUGUHUK $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41lemj $p |- ( ( ( b1 v b2 ) ^ ( ( a1 v a2 ) v ( a0 ^ ( a2 v b2 ) ) ) )
|
|
v ( ( a0 v a2 ) ^ ( ( b0 v b2 ) v ( b1 ^ ( a2 v b2 ) ) ) ) )
|
|
= ( ( ( b1 v b2 ) ^ ( ( a1 v a2 ) v ( b2 ^ ( a0 v a2 ) ) ) )
|
|
v ( ( a0 v a2 ) ^ ( ( b0 v b2 ) v ( a2 ^ ( b1 v b2 ) ) ) ) ) $=
|
|
( wo wa orass ax-a2 lan lor ml3 tr 3tr1 2or ) FGRZCDRZBDGRZSZRZSUHUIGBDRZ
|
|
SZRZSUMEGRZFUJSZRZSUMUPDUHSZRZSULUOUHCDUKRZRCDUNRZRULUOVAVBCVADBGDRZSZRVB
|
|
UKVDDUJVCBDGUAUBUCDBGUDUEUCCDUKTCDUNTUFUBURUTUMEGUQRZREGUSRZRURUTVEVFEGFD
|
|
UDUCEGUQTEGUSTUFUBUG $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41lemk $p |- ( ( ( b1 v b2 ) ^ ( ( a1 v a2 ) v ( b2 ^ ( a0 v a2 ) ) ) )
|
|
v ( ( a0 v a2 ) ^ ( ( b0 v b2 ) v ( a2 ^ ( b1 v b2 ) ) ) ) )
|
|
= ( ( c0 v ( b2 ^ ( a0 v a2 ) ) ) v ( c1 v ( a2 ^ ( b1 v b2 ) ) ) ) $=
|
|
( wo wa tr leao3 mldual2i ancom ror cm 2or ) FGRZCDRZGBDRZSZRSZHUJRZUIEGR
|
|
ZDUGSZRSZIUNRZUKUGUHSZUJRZULUJUHUGGUIFUAUBULURHUQUJHUHUGSUQLUHUGUCTUDUETU
|
|
OUIUMSZUNRZUPUNUMUIDUGBUAUBUPUTIUSUNMUDUETUF $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41leml $p |- ( ( c0 v ( b2 ^ ( a0 v a2 ) ) )
|
|
v ( c1 v ( a2 ^ ( b1 v b2 ) ) ) )
|
|
= ( c0 v c1 ) $=
|
|
( wo wa orcom or4 ancom leor lelan bltr leran le2or 2or cm tr lbtr df-le2
|
|
3tr ) HGBDRZSZRIDFGRZSZRRHIRZUOUQRZRUSURRURHUOIUQUAURUSTUSURUSUNEGRZSZCDR
|
|
ZUPSZRZURUOVAUQVCUOUNGSVAGUNUBGUTUNGEUCUDUEDVBUPDCUCUFUGVDIHRZURVEVDIVAHV
|
|
CMLUHUIIHTUJUKULUM $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
dp41lemm $p |- c2 =< ( c0 v c1 ) $=
|
|
( wo wa lbtr dp41lemb dp41lemc dp41lemd dp41leme dp41lemf dp41lemg tr 3tr
|
|
bltr letr dp41lemh dp41lemj dp41lemk dp41leml ) JFGRZCDRZBDGRZSRSBDRZEGRZ
|
|
FUQSRSRZHIRZJUOUPBCFRSRSURUSFBERZSRSRZUTJVABEFRSZFBCRZSZRRZVCJJVAJBFRZSRS
|
|
ZVGJJVHJVASRSZVIJJVBFRSVEFRSVJABCDEFGHIJKLMNOPQUAABCDEFGHIJKLMNOPQUBUIABC
|
|
DEFGHIJKLMNOPQUCTABCDEFGHIJKLMNOPQUDUJVGUOUPVFRSURUSVDRSRVCABCDEFGHIJKLMN
|
|
OPQUEABCDEFGHIJKLMNOPQUFUGTABCDEFGHIJKLMNOPQUKUJUTUOUPGURSZRSURUSDUOSZRSR
|
|
HVKRIVLRRVAABCDEFGHIJKLMNOPQULABCDEFGHIJKLMNOPQUMABCDEFGHIJKLMNOPQUNUHT
|
|
$.
|
|
$( [3-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
dp41.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
dp41.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
dp41.3 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
dp41.4 $e |- p2 = ( ( a0 v b0 ) ^ ( a1 v b1 ) ) $.
|
|
dp41.5 $e |- p2 =< ( a2 v b2 ) $.
|
|
$( Part of theorem from Alan Day and Doug Pickering, "A note on the
|
|
Arguesian lattice identity," Studia Sci. Math. Hungar. 19:303-305
|
|
(1982). (4)=>(1) $)
|
|
dp41 $p |- c2 =< ( c0 v c1 ) $=
|
|
( wo wa id dp41lemm ) ADPBEPQCFPQZABCDEFGHIJKLMTRNOS $.
|
|
$( [3-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
dp32.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
dp32.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
dp32.3 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
dp32.4 $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
$( Part of theorem from Alan Day and Doug Pickering, "A note on the
|
|
Arguesian lattice identity," Studia Sci. Math. Hungar. 19:303-305
|
|
(1982). (3)=>(2) $)
|
|
dp32 $p |- p =< ( ( a0 ^ ( a1 v ( c2 ^ ( c0 v c1 ) ) ) )
|
|
v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa ancom tr orcom ler2an dp53 2an leao1 mldual2i mldual cm lbtr lerr
|
|
leao2 ml2i lea df-le2 ran 3tr ror ) ABEFJHIOZPZOZPZOZEBCUQOZPZOZPZVBUSOZA
|
|
UTVCABCDEFGHIJKLMNUAAEFGBCDHIJHCDOZFGOZPVGVFPKVFVGQRIBDOZEGOZPZVIVHPLVHVI
|
|
QRJBCOZEFOZPZVLVKPMVKVLQRABEOZCFOZPZDGOZPEBOZFCOZPZGDOZPNVPVTVQWAVNVRVOVS
|
|
BESCFSUBDGSUBRUATVDUTEPZVBOUSVBOVEVBEUTBVAUSUCUDWBUSVBWBEUTPEBPZUSOZUSUTE
|
|
QEBURUEWDWCEOZURPUSUREWCWCUQFWCJUPWCVMJWCVKVLBECUIEBFUCTJVMMUFUGWCIHWCVJI
|
|
WCVHVIBEDUIEBGUCTIVJLUFUGUHTUHUJWEEURWCEEBUKULUMRUNUOUSVBSUNUG $.
|
|
$( [4-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
dp23.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
dp23.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
dp23.3 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
dp23.4 $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
$( Part of theorem from Alan Day and Doug Pickering, "A note on the
|
|
Arguesian lattice identity," Studia Sci. Math. Hungar. 19:303-305
|
|
(1982). (2)=>(3) $)
|
|
dp23 $p |- p =< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa dp32 lea leror letr ) ABCJHIOPZOZPZEFUAOPZOBUDOABCDEFGHIJKLMNQUCB
|
|
UDBUBRST $.
|
|
$( [4-Apr-2012] $)
|
|
$}
|
|
|
|
|
|
${
|
|
xdp41.c0 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
xdp41.c1 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
xdp41.c2 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
xdp41.p $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
xdp41.p2 $e |- p2 = ( ( a0 v b0 ) ^ ( a1 v b1 ) ) $.
|
|
xdp41.1 $e |- p2 =< ( a2 v b2 ) $.
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
xdp41 $p |- c2 =< ( c0 v c1 ) $=
|
|
( wo wa tr ancom leor leror leo le2an bltr df2le2 cm anass ax-a2 ror or32
|
|
lan ler mldual2i 3tr orass orcom dp34 lel2or mldual lor lea ml2i lbtr ran
|
|
lelan leao1 lear letr or4 2or mli ml3 leran ler2an lelor leao3 le2or 3tr1
|
|
df-le2 ) JFGRZCDRZBDGRZSZRZSZBDRZEGRZFWDSZRZSZRZHIRZJWBWCBCFRZSZRZSZWHWIF
|
|
BERZSZRZSZRZWMJWNBEFRZSZFBCRZSZRZRZXCJJWNJBFRZSZRZSZXIJJXJJWNSZRZSZXMJJWS
|
|
FRZSXFFRZSZXPJJXQXRSZSZXSYAJJXTJXDXFSZXTJXFXDSZYBNXFXDUATXDXQXFXREWSFEBUB
|
|
UCXFFUDUEUFUGUHXSYAJXQXRUIZUHTXSYAXPYDXTXOJXTXJWSWOSZRZXOXTYEBRZFRZYEXJRY
|
|
FXTWOBRZXQSZYIWSSZFRYHXTXQYISYJXRYIXQXRCBRZFRYIXFYLFBCUJUKCBFULTUMXQYIUAT
|
|
FWSYIFWOBFCUBUNUOYKYGFYKWSYISYGYIWSUABWOWSBEUDZUOTUKUPYEBFUQYEXJURUPXJXOY
|
|
EXJXNUDYEAXOYEYEWDSZAYNYEYEWDYEKWDKYEPUHQUFZUGUHAYNOUHTABCDEFGHIJLMNOUSUF
|
|
UTUFVGUFUFXPXKXNRXKWNJSZRZXMJXJWNVAXNYPXKJWNUAVBYQXKWNRZJSJYRSXMJWNXKJXJV
|
|
CVDYRJUAYRXLJXKWNUJUMUPUPVEXMXNXHRZXIXMXNXKRYSJWNXJVAXKXHXNXKYCXJSXFXDXJS
|
|
ZSZXHJYCXJNVFXFXDXJUIUUAXFFXERZSXFFSZXERZXHYTUUBXFYTXDBSZFRFUUERUUBFBXDFE
|
|
UBUOUUEFURUUEXEFXDBUAVBUPUMXEFXFBXDCVHUOUUDXEUUCRXHUUCXEURUUCXGXEXFFUAVBT
|
|
UPUPVBTXNWNXHJWNVIUCUFVJXIWBWCXGRZSZWHWIXERZSZRZXCXIWNXGXERZRZWBWCSZXGRZW
|
|
HWISZXERZRZUUJXHUUKWNXEXGURVBUULHXGRZIXERZRUUQHIXGXEVKUURUUNUUSUUPHUUMXGH
|
|
WCWBSZUUMLWCWBUATZUKIUUOXEMUKVLTUUNUUGUUPUUIWBWCXGFXFGVHVMWHWIXEBXDDVHVMV
|
|
LUPUUGWRUUIXBUUFWQWBUUFCXGRZDRCWPRZDRWQCDXGULUVBUVCDUVBCBFCRZSZRUVCCFBVNU
|
|
VEWPCUVDWOBFCURUMVBTUKCWPDULUPUMUUHXAWHUUHEXERZGREWTRZGRXAEGXEULUVFUVGGUV
|
|
FEBFERZSZRUVGXEUVIEXDUVHBEFURUMVBEBFVNTUKEWTGULUPUMVLTVEWRWGXBWLWQWFWBWPW
|
|
EWCWPBWDBWOVCWPYEWDBWSWOYMVOYOVJVPVQVGXAWKWHWTWJWIWTFWDFWSVCWTYEWDWTWSWOF
|
|
WSVIFWSCVRVPYOVJVPVQVGVSVJWMWBWCGWHSZRZSZWHWIDWBSZRZSZRHUVJRZIUVMRZRZWNWG
|
|
UVLWLUVOWFUVKWBCDWERZRCDUVJRZRWFUVKUVSUVTCUVSDBGDRZSZRUVTWEUWBDWDUWABDGUJ
|
|
UMVBDBGVNTVBCDWEUQCDUVJUQVTUMWKUVNWHEGWJRZREGUVMRZRWKUVNUWCUWDEGFDVNVBEGW
|
|
JUQEGUVMUQVTUMVLUVLUVPUVOUVQUVLUUMUVJRZUVPUVJWCWBGWHFVRUOUVPUWEHUUMUVJUVA
|
|
UKUHTUVOUUOUVMRZUVQUVMWIWHDWBBVRUOUVQUWFIUUOUVMMUKUHTVLUVRWNUVJUVMRZRUWGW
|
|
NRWNHUVJIUVMVKWNUWGURUWGWNUWGUUOUUTRZWNUVJUUOUVMUUTUVJWHGSUUOGWHUAGWIWHGE
|
|
UBVGUFDWCWBDCUBVOVSUWHIHRZWNUWIUWHIUUOHUUTMLVLUHIHURTVEWAUPUPVE $.
|
|
$( [3-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
xdp15.d $e |- d = ( a2 v ( a0 ^ ( a1 v b1 ) ) ) $.
|
|
xdp15.p0 $e |- p0 = ( ( a1 v b1 ) ^ ( a2 v b2 ) ) $.
|
|
xdp15.e $e |- e = ( b0 ^ ( a0 v p0 ) ) $.
|
|
xdp15.c0 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
xdp15.c1 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
$( Part of proof (1)=>(5) in Day/Pickering 1982. $)
|
|
xdp15 $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
=< ( ( c0 v c1 ) v ( b1 ^ ( a0 v a1 ) ) ) $=
|
|
( wo wa tr ror lor lan ran wt le1 leran lelor an1r orass cm oridm 3tr lea
|
|
orcom mlduali lear leror bltr or32 lbtr letr ax-arg 2an le3tr2 or12 orabs
|
|
2or ax-a2 ml3le lelan leao1 mldual2i ancom 3tr2 bile le2or ) CDQZFCKQZRZG
|
|
QZRZDEQZGHQZRZCEQZFHQZRZGVQRZQZQZIJQWHQZWAWEVSHQZRZWBWHQZWCRZQZWJWAWMWBCD
|
|
GQZRZQZWCRZQZWPWACEWRQZQZWLRZDXBQZWCRZQZXAVQBGQZRCAQZBHQZRZDAQZWCRZQWAXGC
|
|
DABGHCBQZWQRCFCWQEHQZRZQZRZQZWQRZAHQZXNXSWQBXRCBVSXRNVRXQFKXPCMUAUBSUAUCX
|
|
TCUDXQRZQZWQRZYAXSYCWQXRYBCFUDXQFUEUFUGUFYDXOWRQZYAYDXPWRQZYEYDXPCQZWQRYF
|
|
YCYGWQYCCXQQZCCQZXPQZYGYBXQCXQUHUAYJYHCCXPUIUJYJXQYGYICXPCUKTCXPUNSULUCXP
|
|
CWQWQXOUMUOSXPXOWRWQXOUPUQURYEXBHQZYAEHWRUSYAYKAXBHLTUJSUTVAURVBXHVTVQBVS
|
|
GNTUBXKXDXMXFXIXCXJWLAXBCLUABVSHNTVCXLXEWCAXBDLUAUCVGVDXDWMXFWTXCWEWLXCEC
|
|
WRQZQECQWECEWRVEYLCECWQVFUAECUNULUCWTXFWSXEWCDEWRUIUCUJVGUTWTWOWMWSWNWCWS
|
|
EDWHQZQZWNWSEDCGDQZRZQZQZYNWSEDQZYPQYRWBYSWRYPDEVHWQYOCDGVHUBVGEDYPUISYQY
|
|
MEDCGVIUGURYNYSWHQZWNYTYNEDWHUIUJYSWBWHEDVHTSUTUFUGVAWPWGWDWHQZQWJWMWGWOU
|
|
UAWLWFWEVSFHFVRUMUQVJWOUUAWCWNRWCWBRZWHQWOUUAWHWBWCGVQHVKVLWCWNVMUUBWDWHW
|
|
CWBVMTVNVOVPWGWDWHVEUTVAWJIJWHQZQZWKUUDWJIWDUUCWIOJWGWHPTVGUJWKUUDIJWHUIU
|
|
JSUT $.
|
|
$( [11-Apr-2012] $)
|
|
$}
|
|
|
|
${
|
|
xdp53.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
xdp53.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
xdp53.3 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
xdp53.4 $e |- p0 = ( ( a1 v b1 ) ^ ( a2 v b2 ) ) $.
|
|
xdp53.5 $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
$( Part of proof (5)=>(3) in Day/Pickering 1982. $)
|
|
xdp53 $p |- p
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa leo lbtr leor anass tr lan cm leao4 bltr lea orcom mldual2i ancom
|
|
ler2an ror lelor letr lor lear ax-a3 dp15 leid lel2or or32 le2an lerr 3tr
|
|
df-le2 ran an32 ml2i anabs orass leror ) ABAQBEFJHIQZRZQZRZQZABUABVQABVPS
|
|
ZAEBKQZRZVQQZVQAVTBQZWAAVSEBQZRZWBABEQZCFQZDGQZRZRZWDAWEWFRWGRWIPWEWFWGUB
|
|
UCWIVSWCWIWEKRZVSWJWIKWHWEOUDUEKWEBUFUGWIWEWCWEWHUHBEUITULUGWDVSERZBQWBBE
|
|
VSBKSUJWKVTBVSEUKUMUCTBVQVTVRUNUOVTVQVTEBERZFQVNQZRZVQVTEFBCQZVMRZQZRZWNV
|
|
TEWQEVSUHVTFVTQWQVTFUAFWQVTFWPSZVTWOVTFQZRZWQQZWQVTXAFQZXBVTWTWOFQZRZXCVT
|
|
WTXDVTFSVTEBWHQZRZXDVSXFEKWHBOUPUDXGXFXDEXFUQXFBWFQZXDWHWFBWFWGUHUNXDXHBC
|
|
FURUETUOUGULXEWTWORZFQXCFWOWTFVTUAUJXIXAFWTWOUKUMUCTFWQXAWSUNUOXAWQWQXAWP
|
|
FQZWQXAWPFWORZQZXJXAWOVMXKQZRXLXAWOXMWOWTUHBCDEFGHIKLMOUSULXKVMWOFWOUQUJT
|
|
XKFWPFWOUHUNUOWPFUITWQUTVAUOVAUOULWNWRWNVPWRWMVOEWMWLVNQZFQFXNQVOWLFVNVBX
|
|
NFUIXNVNFWLVNWLJVMWLWOEFQZRZJBWOEXOBCSEFSVCJXPNUETWLIHWLBDQZEGQZRZIBXQEXR
|
|
BDSEGSVCIXSMUETVDULVFUPVEUDVPEXOWQRZRZEXORZWQRZWRVOXTEVOFWPXORZQWQXORXTVN
|
|
YDFVNXPVMRYDJXPVMNVGWOXOVMVHUCUPXOWPFFEUAVIWQXOUKVEUDYCYAEXOWQUBUEYBEWQEF
|
|
VJVGVEUCUETWNWLVPQZVQWNEVOWLQZRVPWLQYEWMYFEWMWLVOQYFWLFVNVKWLVOUIUCUDWLVO
|
|
EBEUQUJVPWLUIVEWLBVPBEUHVLUGUOVFTVAUO $.
|
|
$( [11-Apr-2012] $)
|
|
$}
|
|
|
|
|
|
${
|
|
xxdp.1 $e |- p2 =< ( a2 v b2 ) $.
|
|
xxdp.c0 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
xxdp.c1 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
xxdp.c2 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
xxdp.d $e |- d = ( a2 v ( a0 ^ ( a1 v b1 ) ) ) $.
|
|
xxdp.e $e |- e = ( b0 ^ ( a0 v p0 ) ) $.
|
|
xxdp.p $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
xxdp.p0 $e |- p0 = ( ( a1 v b1 ) ^ ( a2 v b2 ) ) $.
|
|
xxdp.p2 $e |- p2 = ( ( a0 v b0 ) ^ ( a1 v b1 ) ) $.
|
|
$( Part of proof (4)=>(1) in Day/Pickering 1982. $)
|
|
xxdp41 $p |- c2 =< ( c0 v c1 ) $=
|
|
( wo wa ancom tr leor leror leo le2an bltr df2le2 cm anass ax-a2 ror or32
|
|
lan ler mldual2i 3tr orass orcom dp34 lel2or mldual lor lea ml2i lbtr ran
|
|
lelan leao1 lear letr or4 2or mli ml3 leran ler2an lelor leao3 le2or 3tr1
|
|
df-le2 ) LHIUDZEFUDZDFIUDZUEZUDZUEZDFUDZGIUDZHWJUEZUDZUEZUDZJKUDZLWHWIDEH
|
|
UDZUEZUDZUEZWNWOHDGUDZUEZUDZUEZUDZWSLWTDGHUDZUEZHDEUDZUEZUDZUDZXILLWTLDHU
|
|
DZUEZUDZUEZXOLLXPLWTUEZUDZUEZXSLLXEHUDZUEXLHUDZUEZYBLLYCYDUEZUEZYEYGLLYFL
|
|
XJXLUEZYFLXLXJUEZYHRXLXJUFUGXJYCXLYDGXEHGDUHUIXLHUJUKULUMUNYEYGLYCYDUOZUN
|
|
UGYEYGYBYJYFYALYFXPXEXAUEZUDZYAYFYKDUDZHUDZYKXPUDYLYFXADUDZYCUEZYOXEUEZHU
|
|
DYNYFYCYOUEYPYDYOYCYDEDUDZHUDYOXLYRHDEUPUQEDHURUGUSYCYOUFUGHXEYOHXADHEUHU
|
|
TVAYQYMHYQXEYOUEYMYOXEUFDXAXEDGUJZVAUGUQVBYKDHVCYKXPVDVBXPYAYKXPXTUJYKCYA
|
|
YKYKWJUEZCYTYKYKWJYKNWJNYKUCUNOULZUMUNCYTUAUNUGCDEFGHIJKLPQRUAVEULVFULVMU
|
|
LULYBXQXTUDXQWTLUEZUDZXSLXPWTVGXTUUBXQLWTUFVHUUCXQWTUDZLUELUUDUEXSLWTXQLX
|
|
PVIVJUUDLUFUUDXRLXQWTUPUSVBVBVKXSXTXNUDZXOXSXTXQUDUUELWTXPVGXQXNXTXQYIXPU
|
|
EXLXJXPUEZUEZXNLYIXPRVLXLXJXPUOUUGXLHXKUDZUEXLHUEZXKUDZXNUUFUUHXLUUFXJDUE
|
|
ZHUDHUUKUDUUHHDXJHGUHVAUUKHVDUUKXKHXJDUFVHVBUSXKHXLDXJEVNVAUUJXKUUIUDXNUU
|
|
IXKVDUUIXMXKXLHUFVHUGVBVBVHUGXTWTXNLWTVOUIULVPXOWHWIXMUDZUEZWNWOXKUDZUEZU
|
|
DZXIXOWTXMXKUDZUDZWHWIUEZXMUDZWNWOUEZXKUDZUDZUUPXNUUQWTXKXMVDVHUURJXMUDZK
|
|
XKUDZUDUVCJKXMXKVQUVDUUTUVEUVBJUUSXMJWIWHUEZUUSPWIWHUFUGZUQKUVAXKQUQVRUGU
|
|
UTUUMUVBUUOWHWIXMHXLIVNVSWNWOXKDXJFVNVSVRVBUUMXDUUOXHUULXCWHUULEXMUDZFUDE
|
|
XBUDZFUDXCEFXMURUVHUVIFUVHEDHEUDZUEZUDUVIEHDVTUVKXBEUVJXADHEVDUSVHUGUQEXB
|
|
FURVBUSUUNXGWNUUNGXKUDZIUDGXFUDZIUDXGGIXKURUVLUVMIUVLGDHGUDZUEZUDUVMXKUVO
|
|
GXJUVNDGHVDUSVHGDHVTUGUQGXFIURVBUSVRUGVKXDWMXHWRXCWLWHXBWKWIXBDWJDXAVIXBY
|
|
KWJDXEXAYSWAUUAVPWBWCVMXGWQWNXFWPWOXFHWJHXEVIXFYKWJXFXEXAHXEVOHXEEWDWBUUA
|
|
VPWBWCVMWEVPWSWHWIIWNUEZUDZUEZWNWOFWHUEZUDZUEZUDJUVPUDZKUVSUDZUDZWTWMUVRW
|
|
RUWAWLUVQWHEFWKUDZUDEFUVPUDZUDWLUVQUWEUWFEUWEFDIFUDZUEZUDUWFWKUWHFWJUWGDF
|
|
IUPUSVHFDIVTUGVHEFWKVCEFUVPVCWFUSWQUVTWNGIWPUDZUDGIUVSUDZUDWQUVTUWIUWJGIH
|
|
FVTVHGIWPVCGIUVSVCWFUSVRUVRUWBUWAUWCUVRUUSUVPUDZUWBUVPWIWHIWNHWDVAUWBUWKJ
|
|
UUSUVPUVGUQUNUGUWAUVAUVSUDZUWCUVSWOWNFWHDWDVAUWCUWLKUVAUVSQUQUNUGVRUWDWTU
|
|
VPUVSUDZUDUWMWTUDWTJUVPKUVSVQWTUWMVDUWMWTUWMUVAUVFUDZWTUVPUVAUVSUVFUVPWNI
|
|
UEUVAIWNUFIWOWNIGUHVMULFWIWHFEUHWAWEUWNKJUDZWTUWOUWNKUVAJUVFQPVRUNKJVDUGV
|
|
KWGVBVBVK $.
|
|
$( [3-Apr-2012] $)
|
|
|
|
$( Part of proof (1)=>(5) in Day/Pickering 1982. $)
|
|
xxdp15 $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
=< ( ( c0 v c1 ) v ( b1 ^ ( a0 v a1 ) ) ) $=
|
|
( wo wa lor lan tr ran wt le1 leran lelor an1r orass cm oridm ror 3tr lea
|
|
orcom mlduali lear leror bltr or32 lbtr letr ax-arg 2an le3tr2 or12 orabs
|
|
2or ax-a2 ml3le lelan leao1 mldual2i ancom 3tr2 bile le2or ) DEUDZGDMUDZU
|
|
EZHUDZUEZEFUDZHIUDZUEZDFUDZGIUDZUEZHWDUEZUDZUDZJKUDWOUDZWHWLWFIUDZUEZWIWO
|
|
UDZWJUEZUDZWQWHWTWIDEHUDZUEZUDZWJUEZUDZXCWHDFXEUDZUDZWSUEZEXIUDZWJUEZUDZX
|
|
HWDBHUDZUEDAUDZBIUDZUEZEAUDZWJUEZUDWHXNDEABHIDBUDZXDUEDGDXDFIUDZUEZUDZUEZ
|
|
UDZXDUEZAIUDZYAYFXDBYEDBWFYETWEYDGMYCDUBUFUGUHUFUIYGDUJYDUEZUDZXDUEZYHYFY
|
|
JXDYEYIDGUJYDGUKULUMULYKYBXEUDZYHYKYCXEUDZYLYKYCDUDZXDUEYMYJYNXDYJDYDUDZD
|
|
DUDZYCUDZYNYIYDDYDUNUFYQYODDYCUOUPYQYDYNYPDYCDUQURDYCVAUHUSUIYCDXDXDYBUTV
|
|
BUHYCYBXEXDYBVCVDVEYLXIIUDZYHFIXEVFYHYRAXIISURUPUHVGVHVEVIXOWGWDBWFHTURUG
|
|
XRXKXTXMXPXJXQWSAXIDSUFBWFITURVJXSXLWJAXIESUFUIVNVKXKWTXMXGXJWLWSXJFDXEUD
|
|
ZUDFDUDWLDFXEVLYSDFDXDVMUFFDVAUSUIXGXMXFXLWJEFXEUOUIUPVNVGXGXBWTXFXAWJXFF
|
|
EWOUDZUDZXAXFFEDHEUDZUEZUDZUDZUUAXFFEUDZUUCUDUUEWIUUFXEUUCEFVOXDUUBDEHVOU
|
|
GVNFEUUCUOUHUUDYTFEDHVPUMVEUUAUUFWOUDZXAUUGUUAFEWOUOUPUUFWIWOFEVOURUHVGUL
|
|
UMVHXCWNWKWOUDZUDWQWTWNXBUUHWSWMWLWFGIGWEUTVDVQXBUUHWJXAUEWJWIUEZWOUDXBUU
|
|
HWOWIWJHWDIVRVSWJXAVTUUIWKWOWJWIVTURWAWBWCWNWKWOVLVGVHWQJKWOUDZUDZWRUUKWQ
|
|
JWKUUJWPPKWNWOQURVNUPWRUUKJKWOUOUPUHVG $.
|
|
$( [11-Apr-2012] $)
|
|
|
|
$( Part of proof (5)=>(3) in Day/Pickering 1982. $)
|
|
xxdp53 $p |- p
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa leor leo anass tr lan cm leao4 bltr lea orcom lbtr mldual2i ancom
|
|
ler2an ror lelor letr lor lear ax-a3 dp15 leid lel2or or32 le2an lerr 3tr
|
|
df-le2 ran an32 ml2i anabs orass leror ) CDCUDDGHLJKUDZUEZUDZUEZUDZCDUFDW
|
|
DCDWCUGZCGDMUDZUEZWDUDZWDCWGDUDZWHCWFGDUDZUEZWICDGUDZEHUDZFIUDZUEZUEZWKCW
|
|
LWMUEWNUEWPUAWLWMWNUHUIWPWFWJWPWLMUEZWFWQWPMWOWLUBUJUKMWLDULUMWPWLWJWLWOU
|
|
NDGUOUPUSUMWKWFGUEZDUDWIDGWFDMUGUQWRWGDWFGURUTUIUPDWDWGWEVAVBWGWDWGGDGUEZ
|
|
HUDWAUDZUEZWDWGGHDEUDZVTUEZUDZUEZXAWGGXDGWFUNWGHWGUDXDWGHUFHXDWGHXCUGZWGX
|
|
BWGHUDZUEZXDUDZXDWGXHHUDZXIWGXGXBHUDZUEZXJWGXGXKWGHUGWGGDWOUDZUEZXKWFXMGM
|
|
WODUBVCUJXNXMXKGXMVDXMDWMUDZXKWOWMDWMWNUNVAXKXODEHVEUKUPVBUMUSXLXGXBUEZHU
|
|
DXJHXBXGHWGUFUQXPXHHXGXBURUTUIUPHXDXHXFVAVBXHXDXDXHXCHUDZXDXHXCHXBUEZUDZX
|
|
QXHXBVTXRUDZUEXSXHXBXTXBXGUNDEFGHIJKMPQUBVFUSXRVTXBHXBVDUQUPXRHXCHXBUNVAV
|
|
BXCHUOUPXDVGVHVBVHVBUSXAXEXAWCXEWTWBGWTWSWAUDZHUDHYAUDWBWSHWAVIYAHUOYAWAH
|
|
WSWAWSLVTWSXBGHUDZUEZLDXBGYBDEUGGHUGVJLYCRUKUPWSKJWSDFUDZGIUDZUEZKDYDGYED
|
|
FUGGIUGVJKYFQUKUPVKUSVMVCVLUJWCGYBXDUEZUEZGYBUEZXDUEZXEWBYGGWBHXCYBUEZUDX
|
|
DYBUEYGWAYKHWAYCVTUEYKLYCVTRVNXBYBVTVOUIVCYBXCHHGUFVPXDYBURVLUJYJYHGYBXDU
|
|
HUKYIGXDGHVQVNVLUIUKUPXAWSWCUDZWDXAGWBWSUDZUEWCWSUDYLWTYMGWTWSWBUDYMWSHWA
|
|
VRWSWBUOUIUJWSWBGDGVDUQWCWSUOVLWSDWCDGUNVSUMVBVMUPVHVB $.
|
|
$( [11-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(5) in Day/Pickering 1982. $)
|
|
xdp45lem $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
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|
|
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|
|
ror lan ler mldual2i 3tr orass orcom lor ran wt le1 leran lelor oridm lea
|
|
an1r mlduali lear lbtr letr id dp34 lel2or lelan mldual leao1 or4 2or mli
|
|
ml2i ml3 ler2an leao3 le2or 3tr1 df-le2 le3tr2 or12 orabs ml3le 3tr2 bile
|
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) DEUDZGDMUDZUEZHUDZUEZEFUDZHIUDZUEZDFUDZGIUDZUEZHXAUEZUDZUDZJKUDXLUDZXEX
|
|
IXCIUDZUEZXFXLUDZXGUEZUDZXNXEXQXFDEHUDZUEZUDZXGUEZUDZXTXEDFYBUDZUDZXPUEZE
|
|
YFUDZXGUEZUDZYEXABHUDZUEZDAUDZBIUDZUEZEAUDZXGUEZUDZXEYKYMYOYNEAIUDZUEZUDZ
|
|
UEZYQXGBYTUEZUDZUEZUDZYSYMYOYNEDBUDZUEZUDZUEZYQXGBYAUEZUDZUEZUDZUUGYMYSEH
|
|
BUDZUEZBEDUDZUEZUDZUDZUUOYMYMYSYMEBUDZUEZUDZUEZUVAYMYMUVBYMYSUEZUDZUEZUVE
|
|
YMYMYABUDZUEUURBUDZUEZUVHYMYMUVIUVJUEZUEZUVKUVMYMYMUVLYMUUPUURUEZUVLYMUUR
|
|
UUPUEZUVNXAUURYLUUPDEUFBHUFUGZUURUUPUHUIUUPUVIUURUVJHYABHEUJUKUURBULUMUNU
|
|
OUPUVKUVMYMUVIUVJUQZUPUIUVKUVMUVHUVQUVLUVGYMUVLUVBYAUUHUEZUDZUVGUVLUVREUD
|
|
ZBUDZUVRUVBUDUVSUVLUUHEUDZUVIUEZUWBYAUEZBUDUWAUVLUVIUWBUEUWCUVJUWBUVIUVJX
|
|
ABUDUWBUURXABEDUFUSDEBURUIUTUVIUWBUHUIBYAUWBBUUHEBDUJVAVBUWDUVTBUWDYAUWBU
|
|
EUVTUWBYAUHEUUHYAEHULZVBUIUSVCUVREBVDUVRUVBVEVCUVBUVGUVRUVBUVFULUVRUVRYTU
|
|
EZUVGUVRUWFUWFUWFUVRUVRYTUVRUUHYAUEZYTUWGUVRUUHYAUHUPUWGDGDYAFIUDZUEZUDZU
|
|
EZUDZYAUEZYTUUHUWLYABUWKDBXCUWKTXBUWJGMUWIDUBVFUTUIVFVGUWMDVHUWJUEZUDZYAU
|
|
EZYTUWLUWOYAUWKUWNDGVHUWJGVIVJVKVJUWPUWHYBUDZYTUWPUWIYBUDZUWQUWPUWIDUDZYA
|
|
UEUWRUWOUWSYAUWODUWJUDZDDUDZUWIUDZUWSUWNUWJDUWJVNVFUXBUWTDDUWIVDUPUXBUWJU
|
|
WSUXADUWIDVLUSDUWIVEUIVCVGUWIDYAYAUWHVMVOUIUWIUWHYBYAUWHVPUKUNUWQYFIUDZYT
|
|
FIYBURYTUXCAYFISUSUPUIVQVRUNUNZUOUPUWFUWFUWFVSZUPUIUWFEDAHBIYPYRYMYPVSZYR
|
|
VSZXAUURYLUUPDEVEBHVEUGUXEVTUNWAUNWBUNUNUVHUVCUVFUDUVCYSYMUEZUDZUVEYMUVBY
|
|
SWCUVFUXHUVCYMYSUHVFUXIUVCYSUDZYMUEYMUXJUEUVEYMYSUVCYMUVBVMWHUXJYMUHUXJUV
|
|
DYMUVCYSUFUTVCVCVQUVEUVFUUTUDZUVAUVEUVFUVCUDUXKYMYSUVBWCUVCUUTUVFUVCUVOUV
|
|
BUEUURUUPUVBUEZUEZUUTYMUVOUVBUVPVGUURUUPUVBUQUXMUURBUUQUDZUEUURBUEZUUQUDZ
|
|
UUTUXLUXNUURUXLUUPEUEZBUDBUXQUDUXNBEUUPBHUJVBUXQBVEUXQUUQBUUPEUHVFVCUTUUQ
|
|
BUUREUUPDWDVBUXPUUQUXOUDUUTUXOUUQVEUXOUUSUUQUURBUHVFUIVCVCVFUIUVFYSUUTYMY
|
|
SVPUKUNVRUVAYOYNUUSUDZUEZYQXGUUQUDZUEZUDZUUOUVAYSUUSUUQUDZUDZYOYNUEZUUSUD
|
|
ZYRUUQUDZUDZUYBUUTUYCYSUUQUUSVEVFUYDYPUUSUDZUYGUDUYHYPYRUUSUUQWEUYIUYFUYG
|
|
UYGYPUYEUUSYPYPUYEUXFYNYOUHUIZUSYRYRUUQUXGUSWFUIUYFUXSUYGUYAYOYNUUSBUURIW
|
|
DWGYQXGUUQEUUPAWDWGWFVCUXSUUKUYAUUNUXRUUJYOUXRDUUSUDZAUDDUUIUDZAUDUUJDAUU
|
|
SURUYKUYLAUYKDEBDUDZUEZUDUYLDBEWIUYNUUIDUYMUUHEBDVEUTVFUIUSDUUIAURVCUTUXT
|
|
UUMYQUXTHUUQUDZIUDHUULUDZIUDUUMHIUUQURUYOUYPIUYOHEYLUEZUDUYPUUQUYQHUUPYLE
|
|
HBVEUTVFHEBWIUIUSHUULIURVCUTWFUIVQUUKUUCUUNUUFUUJUUBYOUUIUUAYNUUIEYTEUUHV
|
|
MUUIUVRYTEYAUUHUWEVJUXDVRWJVKWBUUMUUEYQUULUUDXGUULBYTBYAVMUULUVRYTUULYAUU
|
|
HBYAVPBYADWKWJUXDVRWJVKWBWLVRUUGYOYNIYQUEZUDZUEZYQXGAYOUEZUDZUEZUDYPUYRUD
|
|
ZYRVUAUDZUDZYSUUCUYTUUFVUCUUBUYSYODAUUAUDZUDDAUYRUDZUDUUBUYSVUGVUHDVUGAEI
|
|
AUDZUEZUDVUHUUAVUJAYTVUIEAIUFUTVFAEIWIUIVFDAUUAVDDAUYRVDWMUTUUEVUBYQHIUUD
|
|
UDZUDHIVUAUDZUDUUEVUBVUKVULHIBAWIVFHIUUDVDHIVUAVDWMUTWFUYTVUDVUCVUEUYTUYE
|
|
UYRUDZVUDUYRYNYOIYQBWKVBVUDVUMYPUYEUYRUYJUSUPUIVUCVUEVUEVUAXGYQAYOEWKVBVU
|
|
EVUEYRYRVUAUXGUSUPUIWFVUFYSUYRVUAUDZUDVUNYSUDYSYPUYRYRVUAWEYSVUNVEVUNYSVU
|
|
NYRYPUDZYSUYRYRVUAYPUYRYQIUEYRIYQUHIXGYQIHUJWBUNAYNYOADUJVJWLVUOVUOYSVUOV
|
|
UOYRYRYPYPUXGUXFWFUPYRYPVEUIVQWNVCVCVQYLXDXABXCHTUSUTYPYHYRYJYNYGYOXPAYFD
|
|
SVFBXCITUSUGYQYIXGAYFESVFVGWFWOYHXQYJYDYGXIXPYGFDYBUDZUDFDUDXIDFYBWPVUPDF
|
|
DYAWQVFFDVEVCVGYDYJYCYIXGEFYBVDVGUPWFVQYDXSXQYCXRXGYCFEXLUDZUDZXRYCFEDHEU
|
|
DZUEZUDZUDZVURYCFEUDZVUTUDVVBXFVVCYBVUTEFUFYAVUSDEHUFUTWFFEVUTVDUIVVAVUQF
|
|
EDHWRVKUNVURVVCXLUDZXRVVDVURFEXLVDUPVVCXFXLFEUFUSUIVQVJVKVRXTXKXHXLUDZUDX
|
|
NXQXKXSVVEXPXJXIXCGIGXBVMUKWBXSVVEXGXRUEXGXFUEZXLUDXSVVEXLXFXGHXAIWDVBXGX
|
|
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|
|
FUPXOVVHJKXLVDUPUIVQ $.
|
|
$( [11-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(5) in Day/Pickering 1982. Proof before putting in
|
|
id's, ancom/orcom/2an (why?) $)
|
|
$(
|
|
xdp45lemtest $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
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|
|
( wo wa ancom tr leor leror leo le2an bltr df2le2 cm anass ax-a2 ror
|
|
or32 lan ler mldual2i 3tr orass orcom lor ran le1 leran lelor an1r oridm
|
|
lea mlduali lear lbtr letr dp34 lel2or lelan mldual ml2i leao1 or4 2or
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|
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|
|
DZXCXGXAIUDZUEZXDXJUDZXEUEZUDZXLXCXOXDDEHUDZUEZUDZXEUEZUDZXRXCDFXTUDZUDZX
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|
|
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|
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|
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|
|
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|
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|
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|
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|
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|
|
HTUQUSYMYFYOYHYKYEYLXNAYDDSVEBXAITUQWLYNYGXEAYDESVEVFWDWMYFXOYHYBYEXGXNYE
|
|
FDXTUDZUDFDUDXGDFXTWNUVBDFDXSWOVEFDVDVBVFYBYHYAYGXEEFXTVCVFUNWDVOYBXQXOYA
|
|
XPXEYAFEXJUDZUDZXPYAFEDHEUDZUEZUDZUDZUVDYAFEUDZUVFUDUVHXDUVIXTUVFEFUPXSUV
|
|
EDEHUPUSWDFEUVFVCUGUVGUVCFEDHWPVIULUVDUVIXJUDZXPUVJUVDFEXJVCUNUVIXDXJFEUP
|
|
UQUGVOVHVIVPXRXIXFXJUDZUDXLXOXIXQUVKXNXHXGXAGIGWTVLUIVSXQUVKXEXPUEXEXDUEZ
|
|
XJUDXQUVKXJXDXEHWSIWBVAXEXPUFUVLXFXJXEXDUFUQWQWRWIXIXFXJWNVOVPXLJKXJUDZUD
|
|
ZXMUVNXLJXFUVMXKPKXIXJQUQWDUNXMUVNJKXJVCUNUGVO $.
|
|
$)
|
|
$( [11-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(3) in Day/Pickering 1982. $)
|
|
xdp43lem $p |- p
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa leor leo anass tr lan cm leao4 bltr lea orcom lbtr mldual2i ancom
|
|
ler2an ror lelor letr lor lear ax-a3 ax-a2 2an leror le2an df2le2 ler 3tr
|
|
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|
|
lelan leao1 or4 2or mli ml3 leao3 le2or 3tr1 df-le2 or12 orabs ml3le 3tr2
|
|
le3tr2 bile leid lerr an32 anabs ) CDCUDDGHLJKUDZUEZUDZUEZUDZCDUFDXKCDXJU
|
|
GZCGDMUDZUEZXKUDZXKCXNDUDZXOCXMGDUDZUEZXPCDGUDZEHUDZFIUDZUEZUEZXRCXSXTUEY
|
|
AUEYCUAXSXTYAUHUIYCXMXQYCXSMUEZXMYDYCMYBXSUBUJUKMXSDULUMYCXSXQXSYBUNDGUOU
|
|
PUSUMXRXMGUEZDUDXPDGXMDMUGUQYEXNDXMGURUTUIUPDXKXNXLVAVBXNXKXNGDGUEZHUDXHU
|
|
DZUEZXKXNGHDEUDZXGUEZUDZUEZYHXNGYKGXMUNZXNHXNUDYKXNHUFHYKXNHYJUGZXNYIXNHU
|
|
DZUEZYKUDZYKXNYPHUDZYQXNYOYIHUDZUEZYRXNYOYSXNHUGXNGDYBUDZUEZYSXMUUAGMYBDU
|
|
BVCUJZUUBUUAYSGUUAVDUUADXTUDZYSYBXTDXTYAUNZVAYSUUDDEHVEUKUPVBUMUSYTYOYIUE
|
|
ZHUDYRHYIYOHXNUFUQUUFYPHYOYIURUTUIUPHYKYPYNVAVBYPYKYKYPYJHUDZYKYPYJHYIUEZ
|
|
UDZUUGYPYIXGUUHUDZUEUUIYPYIUUJYIYOUNYPEFUDZHIUDZUEZDFUDZGIUDZUEZUUHUDZUDZ
|
|
UUJYPUUNXNIUDZUEZUUKUUHUDZUULUEZUDZUURYPUUTUUKDXTUEZUDZUULUEZUDZUVCYPDFUV
|
|
DUDZUDZUUSUEZEUVHUDZUULUEZUDZUVGYIBHUDZUEZDAUDZBIUDZUEZEAUDZUULUEZUDZYPUV
|
|
MUVOUVQUVPEAIUDZUEZUDZUEZUVSUULBUWBUEZUDZUEZUDZUWAUVOUVQUVPEDBUDZUEZUDZUE
|
|
ZUVSUULBXTUEZUDZUEZUDZUWIUVOUWAEHBUDZUEZBEDUDZUEZUDZUDZUWQUVOUVOUWAUVOEBU
|
|
DZUEZUDZUEZUXCUVOUVOUXDUVOUWAUEZUDZUEZUXGUVOUVOXTBUDZUEUWTBUDZUEZUXJUVOUV
|
|
OUXKUXLUEZUEZUXMUXOUVOUVOUXNUVOUWRUWTUEZUXNUVOUWTUWRUEZUXPYIUWTUVNUWRDEVF
|
|
BHVFVGZUWTUWRURUIUWRUXKUWTUXLHXTBHEUFVHUWTBUGVIUMVJUKUXMUXOUVOUXKUXLUHZUK
|
|
UIUXMUXOUXJUXSUXNUXIUVOUXNUXDXTUWJUEZUDZUXIUXNUXTEUDZBUDZUXTUXDUDUYAUXNUW
|
|
JEUDZUXKUEZUYDXTUEZBUDUYCUXNUXKUYDUEUYEUXLUYDUXKUXLYIBUDUYDUWTYIBEDVFUTDE
|
|
BVMUIUJUXKUYDURUIBXTUYDBUWJEBDUFVKUQUYFUYBBUYFXTUYDUEUYBUYDXTUREUWJXTEHUG
|
|
ZUQUIUTVLUXTEBVNUXTUXDUOVLUXDUXIUXTUXDUXHUGUXTUXTUWBUEZUXIUXTUYHUYHUYHUXT
|
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UXTUWBUXTUWJXTUEZUWBUYIUXTUWJXTURUKUYIDUUBUDZXTUEZUWBUWJUYJXTBUUBDBXNUUBT
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UUCUIVCVOUYKDVPUUAUEZUDZXTUEZUWBUYJUYMXTUUBUYLDGVPUUAGVQVRVAVRUYNYAUVDUDZ
|
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UWBUYNYBUVDUDZUYOUYNYBDUDZXTUEUYPUYMUYQXTUYMDUUAUDZDDUDZYBUDZUYQUYLUUADUU
|
|
AVSVCUYTUYRDDYBVNUKUYTUUAUYQUYSDYBDVTUTDYBUOUIVLVOYBDXTUUEWAUIYBYAUVDXTYA
|
|
VDVHUMUYOUVHIUDZUWBFIUVDVMUWBVUAAUVHISUTUKUIUPVBUMUMZVJUKUYHUYHUYHWBZUKUI
|
|
UYHEDAHBIUVRUVTUVOUVRWBZUVTWBZYIUWTUVNUWRDEUOBHUOVGVUCWCUMWDUMWGUMUMUXJUX
|
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EUXHUDUXEUWAUVOUEZUDZUXGUVOUXDUWAWEUXHVUFUXEUVOUWAURVCVUGUXEUWAUDZUVOUEUV
|
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OVUHUEUXGUVOUWAUXEUVOUXDUNWFVUHUVOURVUHUXFUVOUXEUWAVFUJVLVLUPUXGUXHUXBUDZ
|
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UXCUXGUXHUXEUDVUIUVOUWAUXDWEUXEUXBUXHUXEUXQUXDUEUWTUWRUXDUEZUEZUXBUVOUXQU
|
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XDUXRVOUWTUWRUXDUHVUKUWTBUWSUDZUEUWTBUEZUWSUDZUXBVUJVULUWTVUJUWREUEZBUDBV
|
|
UOUDVULBEUWRBHUFUQVUOBUOVUOUWSBUWREURVCVLUJUWSBUWTEUWRDWHUQVUNUWSVUMUDUXB
|
|
VUMUWSUOVUMUXAUWSUWTBURVCUIVLVLVCUIUXHUWAUXBUVOUWAVDVHUMVBUXCUVQUVPUXAUDZ
|
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UEZUVSUULUWSUDZUEZUDZUWQUXCUWAUXAUWSUDZUDZUVQUVPUEZUXAUDZUVTUWSUDZUDZVUTU
|
|
XBVVAUWAUWSUXAUOVCVVBUVRUXAUDZVVEUDVVFUVRUVTUXAUWSWIVVGVVDVVEVVEUVRVVCUXA
|
|
UVRUVRVVCVUDUVPUVQURUIZUTUVTUVTUWSVUEUTWJUIVVDVUQVVEVUSUVQUVPUXABUWTIWHWK
|
|
UVSUULUWSEUWRAWHWKWJVLVUQUWMVUSUWPVUPUWLUVQVUPDUXAUDZAUDDUWKUDZAUDUWLDAUX
|
|
AVMVVIVVJAVVIDEBDUDZUEZUDVVJDBEWLVVLUWKDVVKUWJEBDUOUJVCUIUTDUWKAVMVLUJVUR
|
|
UWOUVSVURHUWSUDZIUDHUWNUDZIUDUWOHIUWSVMVVMVVNIVVMHEUVNUEZUDVVNUWSVVOHUWRU
|
|
VNEHBUOUJVCHEBWLUIUTHUWNIVMVLUJWJUIUPUWMUWEUWPUWHUWLUWDUVQUWKUWCUVPUWKEUW
|
|
BEUWJUNUWKUXTUWBEXTUWJUYGVRVUBVBUSVAWGUWOUWGUVSUWNUWFUULUWNBUWBBXTUNUWNUX
|
|
TUWBUWNXTUWJBXTVDBXTDWMUSVUBVBUSVAWGWNVBUWIUVQUVPIUVSUEZUDZUEZUVSUULAUVQU
|
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EZUDZUEZUDUVRVVPUDZUVTVVSUDZUDZUWAUWEVVRUWHVWAUWDVVQUVQDAUWCUDZUDDAVVPUDZ
|
|
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|
|
VVPVNWOUJUWGVVTUVSHIUWFUDZUDHIVVSUDZUDUWGVVTVWIVWJHIBAWLVCHIUWFVNHIVVSVNW
|
|
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|
|
IVWAVWCVWCVVSUULUVSAUVQEWMUQVWCVWCUVTUVTVVSVUEUTUKUIWJVWDUWAVVPVVSUDZUDVW
|
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|
|
UVTIUVSURIUULUVSIHUFWGUMAUVPUVQADUFVRWNVWMVWMUWAVWMVWMUVTUVTUVRUVRVUEVUDW
|
|
JUKUVTUVRUOUIUPWPVLVLUPUVNYOYIBXNHTUTUJUVRUVJUVTUVLUVPUVIUVQUUSAUVHDSVCBX
|
|
NITUTVGUVSUVKUULAUVHESVCVOWJXAUVJUUTUVLUVFUVIUUNUUSUVIFDUVDUDZUDFDUDUUNDF
|
|
UVDWQVWNDFDXTWRVCFDUOVLVOUVFUVLUVEUVKUULEFUVDVNVOUKWJUPUVFUVBUUTUVEUVAUUL
|
|
UVEFEUUHUDZUDZUVAUVEFEDHEUDZUEZUDZUDZVWPUVEFEUDZVWRUDVWTUUKVXAUVDVWREFVFX
|
|
TVWQDEHVFUJWJFEVWRVNUIVWSVWOFEDHWSVAUMVWPVXAUUHUDZUVAVXBVWPFEUUHVNUKVXAUU
|
|
KUUHFEVFUTUIUPVRVAVBUVCUUPUUMUUHUDZUDUURUUTUUPUVBVXCUUSUUOUUNXNGIYMVHWGUV
|
|
BVXCUULUVAUEUULUUKUEZUUHUDUVBVXCUUHUUKUULHYIIWHUQUULUVAURVXDUUMUUHUULUUKU
|
|
RUTWTXBWNUUPUUMUUHWQUPVBUURJKUUHUDZUDZUUJVXFUURJUUMVXEUUQPKUUPUUHQUTWJUKU
|
|
UJVXFJKUUHVNUKUIUPUSUUHXGYIHYIVDUQUPUUHHYJHYIUNVAVBYJHUOUPYKXCWDVBWDVBUSY
|
|
HYLYHXJYLYGXIGYGYFXHUDZHUDHVXGUDXIYFHXHVMVXGHUOVXGXHHYFXHYFLXGYFYIGHUDZUE
|
|
ZLDYIGVXHDEUGGHUGVILVXIRUKUPYFKJYFUUPKDUUNGUUODFUGGIUGVIKUUPQUKUPXDUSWPVC
|
|
VLUJXJGVXHYKUEZUEZGVXHUEZYKUEZYLXIVXJGXIHYJVXHUEZUDYKVXHUEVXJXHVXNHXHVXIX
|
|
GUEVXNLVXIXGRVOYIVXHXGXEUIVCVXHYJHHGUFWFYKVXHURVLUJVXMVXKGVXHYKUHUKVXLGYK
|
|
GHXFVOVLUIUKUPYHYFXJUDZXKYHGXIYFUDZUEXJYFUDVXOYGVXPGYGYFXIUDVXPYFHXHVNYFX
|
|
IUOUIUJYFXIGDGVDUQXJYFUOVLYFDXJDGUNVHUMVBWPUPWDVB $.
|
|
$( [11-Apr-2012] $)
|
|
$}
|
|
|
|
|
|
${
|
|
xxxdp.c0 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
xxxdp.c1 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
xxxdp.c2 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
xxxdp.d $e |- d = ( a2 v ( a0 ^ ( a1 v b1 ) ) ) $.
|
|
xxxdp.e $e |- e = ( b0 ^ ( a0 v p0 ) ) $.
|
|
xxxdp.p $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
xxxdp.p0 $e |- p0 = ( ( a1 v b1 ) ^ ( a2 v b2 ) ) $.
|
|
xxxdp.p2 $e |- p2 = ( ( a0 v b0 ) ^ ( a1 v b1 ) ) $.
|
|
$( Part of proof (4)=>(5) in Day/Pickering 1982. $)
|
|
xdp45 $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
=< ( ( c0 v c1 ) v ( b1 ^ ( a0 v a1 ) ) ) $=
|
|
( wo wa ax-a2 2an ancom tr leor leror leo le2an bltr df2le2 cm anass or32
|
|
ror lan ler mldual2i 3tr orass orcom lor ran wt le1 leran lelor oridm lea
|
|
an1r mlduali lear lbtr letr id dp34 lel2or lelan mldual leao1 or4 2or mli
|
|
ml2i ml3 ler2an leao3 le2or 3tr1 df-le2 le3tr2 or12 orabs ml3le 3tr2 bile
|
|
) DEUCZGDMUCZUDZHUCZUDZEFUCZHIUCZUDZDFUCZGIUCZUDZHWTUDZUCZUCZJKUCXKUCZXDX
|
|
HXBIUCZUDZXEXKUCZXFUDZUCZXMXDXPXEDEHUCZUDZUCZXFUDZUCZXSXDDFYAUCZUCZXOUDZE
|
|
YEUCZXFUDZUCZYDWTBHUCZUDZDAUCZBIUCZUDZEAUCZXFUDZUCZXDYJYLYNYMEAIUCZUDZUCZ
|
|
UDZYPXFBYSUDZUCZUDZUCZYRYLYNYMEDBUCZUDZUCZUDZYPXFBXTUDZUCZUDZUCZUUFYLYREH
|
|
BUCZUDZBEDUCZUDZUCZUCZUUNYLYLYRYLEBUCZUDZUCZUDZUUTYLYLUVAYLYRUDZUCZUDZUVD
|
|
YLYLXTBUCZUDUUQBUCZUDZUVGYLYLUVHUVIUDZUDZUVJUVLYLYLUVKYLUUOUUQUDZUVKYLUUQ
|
|
UUOUDZUVMWTUUQYKUUODEUEBHUEUFZUUQUUOUGUHUUOUVHUUQUVIHXTBHEUIUJUUQBUKULUMU
|
|
NUOUVJUVLYLUVHUVIUPZUOUHUVJUVLUVGUVPUVKUVFYLUVKUVAXTUUGUDZUCZUVFUVKUVQEUC
|
|
ZBUCZUVQUVAUCUVRUVKUUGEUCZUVHUDZUWAXTUDZBUCUVTUVKUVHUWAUDUWBUVIUWAUVHUVIW
|
|
TBUCUWAUUQWTBEDUEURDEBUQUHUSUVHUWAUGUHBXTUWABUUGEBDUIUTVAUWCUVSBUWCXTUWAU
|
|
DUVSUWAXTUGEUUGXTEHUKZVAUHURVBUVQEBVCUVQUVAVDVBUVAUVFUVQUVAUVEUKUVQUVQYSU
|
|
DZUVFUVQUWEUWEUWEUVQUVQYSUVQUUGXTUDZYSUWFUVQUUGXTUGUOUWFDGDXTFIUCZUDZUCZU
|
|
DZUCZXTUDZYSUUGUWKXTBUWJDBXBUWJSXAUWIGMUWHDUAVEUSUHVEVFUWLDVGUWIUDZUCZXTU
|
|
DZYSUWKUWNXTUWJUWMDGVGUWIGVHVIVJVIUWOUWGYAUCZYSUWOUWHYAUCZUWPUWOUWHDUCZXT
|
|
UDUWQUWNUWRXTUWNDUWIUCZDDUCZUWHUCZUWRUWMUWIDUWIVMVEUXAUWSDDUWHVCUOUXAUWIU
|
|
WRUWTDUWHDVKURDUWHVDUHVBVFUWHDXTXTUWGVLVNUHUWHUWGYAXTUWGVOUJUMUWPYEIUCZYS
|
|
FIYAUQYSUXBAYEIRURUOUHVPVQUMUMZUNUOUWEUWEUWEVRZUOUHUWEEDAHBIYOYQYLYOVRZYQ
|
|
VRZWTUUQYKUUODEVDBHVDUFUXDVSUMVTUMWAUMUMUVGUVBUVEUCUVBYRYLUDZUCZUVDYLUVAY
|
|
RWBUVEUXGUVBYLYRUGVEUXHUVBYRUCZYLUDYLUXIUDUVDYLYRUVBYLUVAVLWGUXIYLUGUXIUV
|
|
CYLUVBYRUEUSVBVBVPUVDUVEUUSUCZUUTUVDUVEUVBUCUXJYLYRUVAWBUVBUUSUVEUVBUVNUV
|
|
AUDUUQUUOUVAUDZUDZUUSYLUVNUVAUVOVFUUQUUOUVAUPUXLUUQBUUPUCZUDUUQBUDZUUPUCZ
|
|
UUSUXKUXMUUQUXKUUOEUDZBUCBUXPUCUXMBEUUOBHUIVAUXPBVDUXPUUPBUUOEUGVEVBUSUUP
|
|
BUUQEUUODWCVAUXOUUPUXNUCUUSUXNUUPVDUXNUURUUPUUQBUGVEUHVBVBVEUHUVEYRUUSYLY
|
|
RVOUJUMVQUUTYNYMUURUCZUDZYPXFUUPUCZUDZUCZUUNUUTYRUURUUPUCZUCZYNYMUDZUURUC
|
|
ZYQUUPUCZUCZUYAUUSUYBYRUUPUURVDVEUYCYOUURUCZUYFUCUYGYOYQUURUUPWDUYHUYEUYF
|
|
UYFYOUYDUURYOYOUYDUXEYMYNUGUHZURYQYQUUPUXFURWEUHUYEUXRUYFUXTYNYMUURBUUQIW
|
|
CWFYPXFUUPEUUOAWCWFWEVBUXRUUJUXTUUMUXQUUIYNUXQDUURUCZAUCDUUHUCZAUCUUIDAUU
|
|
RUQUYJUYKAUYJDEBDUCZUDZUCUYKDBEWHUYMUUHDUYLUUGEBDVDUSVEUHURDUUHAUQVBUSUXS
|
|
UULYPUXSHUUPUCZIUCHUUKUCZIUCUULHIUUPUQUYNUYOIUYNHEYKUDZUCUYOUUPUYPHUUOYKE
|
|
HBVDUSVEHEBWHUHURHUUKIUQVBUSWEUHVPUUJUUBUUMUUEUUIUUAYNUUHYTYMUUHEYSEUUGVL
|
|
UUHUVQYSEXTUUGUWDVIUXCVQWIVJWAUULUUDYPUUKUUCXFUUKBYSBXTVLUUKUVQYSUUKXTUUG
|
|
BXTVOBXTDWJWIUXCVQWIVJWAWKVQUUFYNYMIYPUDZUCZUDZYPXFAYNUDZUCZUDZUCYOUYQUCZ
|
|
YQUYTUCZUCZYRUUBUYSUUEVUBUUAUYRYNDAYTUCZUCDAUYQUCZUCUUAUYRVUFVUGDVUFAEIAU
|
|
CZUDZUCVUGYTVUIAYSVUHEAIUEUSVEAEIWHUHVEDAYTVCDAUYQVCWLUSUUDVUAYPHIUUCUCZU
|
|
CHIUYTUCZUCUUDVUAVUJVUKHIBAWHVEHIUUCVCHIUYTVCWLUSWEUYSVUCVUBVUDUYSUYDUYQU
|
|
CZVUCUYQYMYNIYPBWJVAVUCVULYOUYDUYQUYIURUOUHVUBVUDVUDUYTXFYPAYNEWJVAVUDVUD
|
|
YQYQUYTUXFURUOUHWEVUEYRUYQUYTUCZUCVUMYRUCYRYOUYQYQUYTWDYRVUMVDVUMYRVUMYQY
|
|
OUCZYRUYQYQUYTYOUYQYPIUDYQIYPUGIXFYPIHUIWAUMAYMYNADUIVIWKVUNVUNYRVUNVUNYQ
|
|
YQYOYOUXFUXEWEUOYQYOVDUHVPWMVBVBVPYKXCWTBXBHSURUSYOYGYQYIYMYFYNXOAYEDRVEB
|
|
XBISURUFYPYHXFAYEERVEVFWEWNYGXPYIYCYFXHXOYFFDYAUCZUCFDUCXHDFYAWOVUODFDXTW
|
|
PVEFDVDVBVFYCYIYBYHXFEFYAVCVFUOWEVPYCXRXPYBXQXFYBFEXKUCZUCZXQYBFEDHEUCZUD
|
|
ZUCZUCZVUQYBFEUCZVUSUCVVAXEVVBYAVUSEFUEXTVURDEHUEUSWEFEVUSVCUHVUTVUPFEDHW
|
|
QVJUMVUQVVBXKUCZXQVVCVUQFEXKVCUOVVBXEXKFEUEURUHVPVIVJVQXSXJXGXKUCZUCXMXPX
|
|
JXRVVDXOXIXHXBGIGXAVLUJWAXRVVDXFXQUDXFXEUDZXKUCXRVVDXKXEXFHWTIWCVAXFXQUGV
|
|
VEXGXKXFXEUGURWRWSWKXJXGXKWOVPVQXMJKXKUCZUCZXNVVGXMJXGVVFXLOKXJXKPURWEUOX
|
|
NVVGJKXKVCUOUHVP $.
|
|
$( [11-Apr-2012] $)
|
|
|
|
$( Part of proof (4)=>(3) in Day/Pickering 1982. $)
|
|
xdp43 $p |- p
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa leor leo anass tr lan cm leao4 bltr lea orcom lbtr mldual2i ancom
|
|
ler2an ror lelor letr lor lear ax-a3 ax-a2 2an leror le2an df2le2 ler 3tr
|
|
or32 orass ran wt le1 leran an1r oridm mlduali id dp34 lel2or mldual ml2i
|
|
lelan leao1 or4 2or mli ml3 leao3 le2or 3tr1 df-le2 or12 orabs ml3le 3tr2
|
|
le3tr2 bile leid lerr an32 anabs ) CDCUCDGHLJKUCZUDZUCZUDZUCZCDUEDXJCDXIU
|
|
FZCGDMUCZUDZXJUCZXJCXMDUCZXNCXLGDUCZUDZXOCDGUCZEHUCZFIUCZUDZUDZXQCXRXSUDX
|
|
TUDYBTXRXSXTUGUHYBXLXPYBXRMUDZXLYCYBMYAXRUAUIUJMXRDUKULYBXRXPXRYAUMDGUNUO
|
|
URULXQXLGUDZDUCXODGXLDMUFUPYDXMDXLGUQUSUHUODXJXMXKUTVAXMXJXMGDGUDZHUCXGUC
|
|
ZUDZXJXMGHDEUCZXFUDZUCZUDZYGXMGYJGXLUMZXMHXMUCYJXMHUEHYJXMHYIUFZXMYHXMHUC
|
|
ZUDZYJUCZYJXMYOHUCZYPXMYNYHHUCZUDZYQXMYNYRXMHUFXMGDYAUCZUDZYRXLYTGMYADUAV
|
|
BUIZUUAYTYRGYTVCYTDXSUCZYRYAXSDXSXTUMZUTYRUUCDEHVDUJUOVAULURYSYNYHUDZHUCY
|
|
QHYHYNHXMUEUPUUEYOHYNYHUQUSUHUOHYJYOYMUTVAYOYJYJYOYIHUCZYJYOYIHYHUDZUCZUU
|
|
FYOYHXFUUGUCZUDUUHYOYHUUIYHYNUMYOEFUCZHIUCZUDZDFUCZGIUCZUDZUUGUCZUCZUUIYO
|
|
UUMXMIUCZUDZUUJUUGUCZUUKUDZUCZUUQYOUUSUUJDXSUDZUCZUUKUDZUCZUVBYODFUVCUCZU
|
|
CZUURUDZEUVGUCZUUKUDZUCZUVFYHBHUCZUDZDAUCZBIUCZUDZEAUCZUUKUDZUCZYOUVLUVNU
|
|
VPUVOEAIUCZUDZUCZUDZUVRUUKBUWAUDZUCZUDZUCZUVTUVNUVPUVOEDBUCZUDZUCZUDZUVRU
|
|
UKBXSUDZUCZUDZUCZUWHUVNUVTEHBUCZUDZBEDUCZUDZUCZUCZUWPUVNUVNUVTUVNEBUCZUDZ
|
|
UCZUDZUXBUVNUVNUXCUVNUVTUDZUCZUDZUXFUVNUVNXSBUCZUDUWSBUCZUDZUXIUVNUVNUXJU
|
|
XKUDZUDZUXLUXNUVNUVNUXMUVNUWQUWSUDZUXMUVNUWSUWQUDZUXOYHUWSUVMUWQDEVEBHVEV
|
|
FZUWSUWQUQUHUWQUXJUWSUXKHXSBHEUEVGUWSBUFVHULVIUJUXLUXNUVNUXJUXKUGZUJUHUXL
|
|
UXNUXIUXRUXMUXHUVNUXMUXCXSUWIUDZUCZUXHUXMUXSEUCZBUCZUXSUXCUCUXTUXMUWIEUCZ
|
|
UXJUDZUYCXSUDZBUCUYBUXMUXJUYCUDUYDUXKUYCUXJUXKYHBUCUYCUWSYHBEDVEUSDEBVLUH
|
|
UIUXJUYCUQUHBXSUYCBUWIEBDUEVJUPUYEUYABUYEXSUYCUDUYAUYCXSUQEUWIXSEHUFZUPUH
|
|
USVKUXSEBVMUXSUXCUNVKUXCUXHUXSUXCUXGUFUXSUXSUWAUDZUXHUXSUYGUYGUYGUXSUXSUW
|
|
AUXSUWIXSUDZUWAUYHUXSUWIXSUQUJUYHDUUAUCZXSUDZUWAUWIUYIXSBUUADBXMUUASUUBUH
|
|
VBVNUYJDVOYTUDZUCZXSUDZUWAUYIUYLXSUUAUYKDGVOYTGVPVQUTVQUYMXTUVCUCZUWAUYMY
|
|
AUVCUCZUYNUYMYADUCZXSUDUYOUYLUYPXSUYLDYTUCZDDUCZYAUCZUYPUYKYTDYTVRVBUYSUY
|
|
QDDYAVMUJUYSYTUYPUYRDYADVSUSDYAUNUHVKVNYADXSUUDVTUHYAXTUVCXSXTVCVGULUYNUV
|
|
GIUCZUWAFIUVCVLUWAUYTAUVGIRUSUJUHUOVAULULZVIUJUYGUYGUYGWAZUJUHUYGEDAHBIUV
|
|
QUVSUVNUVQWAZUVSWAZYHUWSUVMUWQDEUNBHUNVFVUBWBULWCULWFULULUXIUXDUXGUCUXDUV
|
|
TUVNUDZUCZUXFUVNUXCUVTWDUXGVUEUXDUVNUVTUQVBVUFUXDUVTUCZUVNUDUVNVUGUDUXFUV
|
|
NUVTUXDUVNUXCUMWEVUGUVNUQVUGUXEUVNUXDUVTVEUIVKVKUOUXFUXGUXAUCZUXBUXFUXGUX
|
|
DUCVUHUVNUVTUXCWDUXDUXAUXGUXDUXPUXCUDUWSUWQUXCUDZUDZUXAUVNUXPUXCUXQVNUWSU
|
|
WQUXCUGVUJUWSBUWRUCZUDUWSBUDZUWRUCZUXAVUIVUKUWSVUIUWQEUDZBUCBVUNUCVUKBEUW
|
|
QBHUEUPVUNBUNVUNUWRBUWQEUQVBVKUIUWRBUWSEUWQDWGUPVUMUWRVULUCUXAVULUWRUNVUL
|
|
UWTUWRUWSBUQVBUHVKVKVBUHUXGUVTUXAUVNUVTVCVGULVAUXBUVPUVOUWTUCZUDZUVRUUKUW
|
|
RUCZUDZUCZUWPUXBUVTUWTUWRUCZUCZUVPUVOUDZUWTUCZUVSUWRUCZUCZVUSUXAVUTUVTUWR
|
|
UWTUNVBVVAUVQUWTUCZVVDUCVVEUVQUVSUWTUWRWHVVFVVCVVDVVDUVQVVBUWTUVQUVQVVBVU
|
|
CUVOUVPUQUHZUSUVSUVSUWRVUDUSWIUHVVCVUPVVDVURUVPUVOUWTBUWSIWGWJUVRUUKUWREU
|
|
WQAWGWJWIVKVUPUWLVURUWOVUOUWKUVPVUODUWTUCZAUCDUWJUCZAUCUWKDAUWTVLVVHVVIAV
|
|
VHDEBDUCZUDZUCVVIDBEWKVVKUWJDVVJUWIEBDUNUIVBUHUSDUWJAVLVKUIVUQUWNUVRVUQHU
|
|
WRUCZIUCHUWMUCZIUCUWNHIUWRVLVVLVVMIVVLHEUVMUDZUCVVMUWRVVNHUWQUVMEHBUNUIVB
|
|
HEBWKUHUSHUWMIVLVKUIWIUHUOUWLUWDUWOUWGUWKUWCUVPUWJUWBUVOUWJEUWAEUWIUMUWJU
|
|
XSUWAEXSUWIUYFVQVUAVAURUTWFUWNUWFUVRUWMUWEUUKUWMBUWABXSUMUWMUXSUWAUWMXSUW
|
|
IBXSVCBXSDWLURVUAVAURUTWFWMVAUWHUVPUVOIUVRUDZUCZUDZUVRUUKAUVPUDZUCZUDZUCU
|
|
VQVVOUCZUVSVVRUCZUCZUVTUWDVVQUWGVVTUWCVVPUVPDAUWBUCZUCDAVVOUCZUCUWCVVPVWD
|
|
VWEDVWDAEIAUCZUDZUCVWEUWBVWGAUWAVWFEAIVEUIVBAEIWKUHVBDAUWBVMDAVVOVMWNUIUW
|
|
FVVSUVRHIUWEUCZUCHIVVRUCZUCUWFVVSVWHVWIHIBAWKVBHIUWEVMHIVVRVMWNUIWIVVQVWA
|
|
VVTVWBVVQVVBVVOUCZVWAVVOUVOUVPIUVRBWLUPVWAVWJUVQVVBVVOVVGUSUJUHVVTVWBVWBV
|
|
VRUUKUVRAUVPEWLUPVWBVWBUVSUVSVVRVUDUSUJUHWIVWCUVTVVOVVRUCZUCVWKUVTUCUVTUV
|
|
QVVOUVSVVRWHUVTVWKUNVWKUVTVWKUVSUVQUCZUVTVVOUVSVVRUVQVVOUVRIUDUVSIUVRUQIU
|
|
UKUVRIHUEWFULAUVOUVPADUEVQWMVWLVWLUVTVWLVWLUVSUVSUVQUVQVUDVUCWIUJUVSUVQUN
|
|
UHUOWOVKVKUOUVMYNYHBXMHSUSUIUVQUVIUVSUVKUVOUVHUVPUURAUVGDRVBBXMISUSVFUVRU
|
|
VJUUKAUVGERVBVNWIWTUVIUUSUVKUVEUVHUUMUURUVHFDUVCUCZUCFDUCUUMDFUVCWPVWMDFD
|
|
XSWQVBFDUNVKVNUVEUVKUVDUVJUUKEFUVCVMVNUJWIUOUVEUVAUUSUVDUUTUUKUVDFEUUGUCZ
|
|
UCZUUTUVDFEDHEUCZUDZUCZUCZVWOUVDFEUCZVWQUCVWSUUJVWTUVCVWQEFVEXSVWPDEHVEUI
|
|
WIFEVWQVMUHVWRVWNFEDHWRUTULVWOVWTUUGUCZUUTVXAVWOFEUUGVMUJVWTUUJUUGFEVEUSU
|
|
HUOVQUTVAUVBUUOUULUUGUCZUCUUQUUSUUOUVAVXBUURUUNUUMXMGIYLVGWFUVAVXBUUKUUTU
|
|
DUUKUUJUDZUUGUCUVAVXBUUGUUJUUKHYHIWGUPUUKUUTUQVXCUULUUGUUKUUJUQUSWSXAWMUU
|
|
OUULUUGWPUOVAUUQJKUUGUCZUCZUUIVXEUUQJUULVXDUUPOKUUOUUGPUSWIUJUUIVXEJKUUGV
|
|
MUJUHUOURUUGXFYHHYHVCUPUOUUGHYIHYHUMUTVAYIHUNUOYJXBWCVAWCVAURYGYKYGXIYKYF
|
|
XHGYFYEXGUCZHUCHVXFUCXHYEHXGVLVXFHUNVXFXGHYEXGYELXFYEYHGHUCZUDZLDYHGVXGDE
|
|
UFGHUFVHLVXHQUJUOYEKJYEUUOKDUUMGUUNDFUFGIUFVHKUUOPUJUOXCURWOVBVKUIXIGVXGY
|
|
JUDZUDZGVXGUDZYJUDZYKXHVXIGXHHYIVXGUDZUCYJVXGUDVXIXGVXMHXGVXHXFUDVXMLVXHX
|
|
FQVNYHVXGXFXDUHVBVXGYIHHGUEWEYJVXGUQVKUIVXLVXJGVXGYJUGUJVXKGYJGHXEVNVKUHU
|
|
JUOYGYEXIUCZXJYGGXHYEUCZUDXIYEUCVXNYFVXOGYFYEXHUCVXOYEHXGVMYEXHUNUHUIYEXH
|
|
GDGVCUPXIYEUNVKYEDXIDGUMVGULVAWOUOWCVA $.
|
|
$( [11-Apr-2012] $)
|
|
$}
|
|
|
|
|
|
${
|
|
3dp.c0 $e |- c0 = ( ( a1 v a1 ) ^ ( b1 v b1 ) ) $.
|
|
3dp.c1 $e |- c1 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
3dp.c2 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
3dp.d $e |- d = ( a1 v ( a0 ^ ( a1 v b1 ) ) ) $.
|
|
3dp.e $e |- e = ( b0 ^ ( a0 v p0 ) ) $.
|
|
3dp.p $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a1 v b1 ) ) $.
|
|
3dp.p0 $e |- p0 = ( ( a1 v b1 ) ^ ( a1 v b1 ) ) $.
|
|
3dp.p2 $e |- p2 = ( ( a0 v b0 ) ^ ( a1 v b1 ) ) $.
|
|
$( "3OA" version of ~ xdp43 . Changed ` a2 ` to ` a1 ` and ` b2 ` to
|
|
` b1 ` . $)
|
|
3dp43 $p |- p
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa leor leo anass tr lan cm leao4 bltr lea orcom lbtr mldual2i ancom
|
|
ler2an ror lelor letr lor lear ax-a3 ax-a2 2an leror le2an df2le2 ler 3tr
|
|
or32 orass ran wt le1 leran an1r oridm mlduali id dp34 lel2or mldual ml2i
|
|
lelan leao1 or4 2or mli ml3 leao3 le2or 3tr1 df-le2 or12 orabs ml3le 3tr2
|
|
le3tr2 bile leid lerr an32 anabs ) CDCUADFGJHIUAZUBZUAZUBZUAZCDUCDXHCDXGU
|
|
DZCFDKUAZUBZXHUAZXHCXKDUAZXLCXJFDUAZUBZXMCDFUAZEGUAZXQUBZUBZXOCXPXQUBXQUB
|
|
XSRXPXQXQUEUFXSXJXNXSXPKUBZXJXTXSKXRXPSUGUHKXPDUIUJXSXPXNXPXRUKDFULUMUPUJ
|
|
XOXJFUBZDUAXMDFXJDKUDUNYAXKDXJFUOUQUFUMDXHXKXIURUSXKXHXKFDFUBZGUAXEUAZUBZ
|
|
XHXKFGDEUAZXDUBZUAZUBZYDXKFYGFXJUKZXKGXKUAYGXKGUCGYGXKGYFUDZXKYEXKGUAZUBZ
|
|
YGUAZYGXKYLGUAZYMXKYKYEGUAZUBZYNXKYKYOXKGUDXKFDXRUAZUBZYOXJYQFKXRDSUTUGZY
|
|
RYQYOFYQVAYQDXQUAZYOXRXQDXQXQUKZURYOYTDEGVBUHUMUSUJUPYPYKYEUBZGUAYNGYEYKG
|
|
XKUCUNUUBYLGYKYEUOUQUFUMGYGYLYJURUSYLYGYGYLYFGUAZYGYLYFGYEUBZUAZUUCYLYEXD
|
|
UUDUAZUBUUEYLYEUUFYEYKUKYLEEUAZGGUAZUBZYEFGUAZUBZUUDUAZUAZUUFYLYLUUGUUDUA
|
|
ZUUHUBZUAZUUMYLYLUUGDXQUBZUAZUUHUBZUAZUUPYLDEUUQUAZUAZYKUBZEUVAUAZUUHUBZU
|
|
AZUUTYEBGUAZUBZDAUAZUVGUBZEAUAZUUHUBZUAZYLUVFUVHUVGUVIEAGUAZUBZUAZUBZUVKU
|
|
UHBUVNUBZUAZUBZUAZUVMUVHUVGUVIEDBUAZUBZUAZUBZUVKUUHBXQUBZUAZUBZUAZUWAUVHU
|
|
VMEGBUAZUBZBEDUAZUBZUAZUAZUWIUVHUVHUVMUVHEBUAZUBZUAZUBZUWOUVHUVHUWPUVHUVM
|
|
UBZUAZUBZUWSUVHUVHXQBUAZUBUWLBUAZUBZUXBUVHUVHUXCUXDUBZUBZUXEUXGUVHUVHUXFU
|
|
VHUWJUWLUBZUXFUVHUWLUWJUBZUXHYEUWLUVGUWJDEVCBGVCVDZUWLUWJUOUFUWJUXCUWLUXD
|
|
GXQBGEUCVEUWLBUDVFUJVGUHUXEUXGUVHUXCUXDUEZUHUFUXEUXGUXBUXKUXFUXAUVHUXFUWP
|
|
XQUWBUBZUAZUXAUXFUXLEUAZBUAZUXLUWPUAUXMUXFUWBEUAZUXCUBZUXPXQUBZBUAUXOUXFU
|
|
XCUXPUBUXQUXDUXPUXCUXDYEBUAUXPUWLYEBEDVCUQDEBVJUFUGUXCUXPUOUFBXQUXPBUWBEB
|
|
DUCVHUNUXRUXNBUXRXQUXPUBUXNUXPXQUOEUWBXQEGUDZUNUFUQVIUXLEBVKUXLUWPULVIUWP
|
|
UXAUXLUWPUWTUDUXLUXLUVNUBZUXAUXLUXTUXTUXTUXLUXLUVNUXLUWBXQUBZUVNUYAUXLUWB
|
|
XQUOUHUYADYRUAZXQUBZUVNUWBUYBXQBYRDBXKYRQYSUFUTVLUYCDVMYQUBZUAZXQUBZUVNUY
|
|
BUYEXQYRUYDDFVMYQFVNVOURVOUYFXQUUQUAZUVNUYFXRUUQUAZUYGUYFXRDUAZXQUBUYHUYE
|
|
UYIXQUYEDYQUAZDDUAZXRUAZUYIUYDYQDYQVPUTUYLUYJDDXRVKUHUYLYQUYIUYKDXRDVQUQD
|
|
XRULUFVIVLXRDXQUUAVRUFXRXQUUQXQXQVAVEUJUYGUVAGUAZUVNEGUUQVJUVNUYMAUVAGPUQ
|
|
UHUFUMUSUJUJZVGUHUXTUXTUXTVSZUHUFUXTEDAGBGUVJUVLUVHUVJVSZUVLVSZYEUWLUVGUW
|
|
JDEULBGULVDUYOVTUJWAUJWDUJUJUXBUWQUWTUAUWQUVMUVHUBZUAZUWSUVHUWPUVMWBUWTUY
|
|
RUWQUVHUVMUOUTUYSUWQUVMUAZUVHUBUVHUYTUBUWSUVHUVMUWQUVHUWPUKWCUYTUVHUOUYTU
|
|
WRUVHUWQUVMVCUGVIVIUMUWSUWTUWNUAZUWOUWSUWTUWQUAVUAUVHUVMUWPWBUWQUWNUWTUWQ
|
|
UXIUWPUBUWLUWJUWPUBZUBZUWNUVHUXIUWPUXJVLUWLUWJUWPUEVUCUWLBUWKUAZUBUWLBUBZ
|
|
UWKUAZUWNVUBVUDUWLVUBUWJEUBZBUABVUGUAVUDBEUWJBGUCUNVUGBULVUGUWKBUWJEUOUTV
|
|
IUGUWKBUWLEUWJDWEUNVUFUWKVUEUAUWNVUEUWKULVUEUWMUWKUWLBUOUTUFVIVIUTUFUWTUV
|
|
MUWNUVHUVMVAVEUJUSUWOUVGUVIUWMUAZUBZUVKUUHUWKUAZUBZUAZUWIUWOUVMUWMUWKUAZU
|
|
AZUVGUVIUBZUWMUAZUVLUWKUAZUAZVULUWNVUMUVMUWKUWMULUTVUNUVJUWMUAZVUQUAVURUV
|
|
JUVLUWMUWKWFVUSVUPVUQVUQUVJVUOUWMUVJUVJVUOUYPUVIUVGUOUFZUQUVLUVLUWKUYQUQW
|
|
GUFVUPVUIVUQVUKUVGUVIUWMBUWLGWEWHUVKUUHUWKEUWJAWEWHWGVIVUIUWEVUKUWHVUHUWD
|
|
UVGVUHDUWMUAZAUADUWCUAZAUAUWDDAUWMVJVVAVVBAVVADEBDUAZUBZUAVVBDBEWIVVDUWCD
|
|
VVCUWBEBDULUGUTUFUQDUWCAVJVIUGVUJUWGUVKVUJGUWKUAZGUAGUWFUAZGUAUWGGGUWKVJV
|
|
VEVVFGVVEGEUVGUBZUAVVFUWKVVGGUWJUVGEGBULUGUTGEBWIUFUQGUWFGVJVIUGWGUFUMUWE
|
|
UVQUWHUVTUWDUVPUVGUWCUVOUVIUWCEUVNEUWBUKUWCUXLUVNEXQUWBUXSVOUYNUSUPURWDUW
|
|
GUVSUVKUWFUVRUUHUWFBUVNBXQUKUWFUXLUVNUWFXQUWBBXQVABXQDWJUPUYNUSUPURWDWKUS
|
|
UWAUVGUVIGUVKUBZUAZUBZUVKUUHAUVGUBZUAZUBZUAUVJVVHUAZUVLVVKUAZUAZUVMUVQVVJ
|
|
UVTVVMUVPVVIUVGDAUVOUAZUADAVVHUAZUAUVPVVIVVQVVRDVVQAEGAUAZUBZUAVVRUVOVVTA
|
|
UVNVVSEAGVCUGUTAEGWIUFUTDAUVOVKDAVVHVKWLUGUVSVVLUVKGGUVRUAZUAGGVVKUAZUAUV
|
|
SVVLVWAVWBGGBAWIUTGGUVRVKGGVVKVKWLUGWGVVJVVNVVMVVOVVJVUOVVHUAZVVNVVHUVIUV
|
|
GGUVKBWJUNVVNVWCUVJVUOVVHVUTUQUHUFVVMVVOVVOVVKUUHUVKAUVGEWJUNVVOVVOUVLUVL
|
|
VVKUYQUQUHUFWGVVPUVMVVHVVKUAZUAVWDUVMUAUVMUVJVVHUVLVVKWFUVMVWDULVWDUVMVWD
|
|
UVLUVJUAZUVMVVHUVLVVKUVJVVHUVKGUBUVLGUVKUOGUUHUVKGGUCWDUJAUVIUVGADUCVOWKV
|
|
WEVWEUVMVWEVWEUVLUVLUVJUVJUYQUYPWGUHUVLUVJULUFUMWMVIVIUMUVGYKYEBXKGQUQZUG
|
|
UVJUVCUVLUVEUVIUVBUVGYKAUVADPUTVWFVDUVKUVDUUHAUVAEPUTVLWGWRUVCYLUVEUUSUVB
|
|
YEYKUVBEDUUQUAZUAUWLYEDEUUQWNVWGDEDXQWOUTEDULVIVLUUSUVEUURUVDUUHEEUUQVKVL
|
|
UHWGUMUUSUUOYLUURUUNUUHUUREEUUDUAZUAZUUNUUREEDGEUAZUBZUAZUAZVWIUURUUGVWKU
|
|
AVWMUUGUUGUUQVWKEEVCZXQVWJDEGVCUGWGEEVWKVKUFVWLVWHEEDGWPURUJVWIUUNUUNUUNV
|
|
WIEEUUDVKUHUUGUUGUUDVWNUQUFUMVOURUSUUPUUKUUIUUDUAZUAUUMYLUUKUUOVWOYKUUJYE
|
|
XKFGYIVEWDUUOVWOUUHUUNUBUUHUUGUBZUUDUAUUOVWOUUDUUGUUHGYEGWEUNUUHUUNUOVWPU
|
|
UIUUDUUHUUGUOUQWQWSWKUUKUUIUUDWNUMUSUUMHIUUDUAZUAZUUFVWRUUMHUUIVWQUULMIUU
|
|
KUUDNUQWGUHUUFVWRHIUUDVKUHUFUMUPUUDXDYEGYEVAUNUMUUDGYFGYEUKURUSYFGULUMYGW
|
|
TWAUSWAUSUPYDYHYDXGYHYCXFFYCYBXEUAZGUAGVWSUAXFYBGXEVJVWSGULVWSXEGYBXEYBJX
|
|
DYBUUKJDYEFUUJDEUDFGUDVFZJUUKOUHUMYBIHYBUUKIVWTIUUKNUHUMXAUPWMUTVIUGXGFUU
|
|
JYGUBZUBZFUUJUBZYGUBZYHXFVXAFXFGYFUUJUBZUAYGUUJUBVXAXEVXEGXEUUKXDUBVXEJUU
|
|
KXDOVLYEUUJXDXBUFUTUUJYFGGFUCWCYGUUJUOVIUGVXDVXBFUUJYGUEUHVXCFYGFGXCVLVIU
|
|
FUHUMYDYBXGUAZXHYDFXFYBUAZUBXGYBUAVXFYCVXGFYCYBXFUAVXGYBGXEVKYBXFULUFUGYB
|
|
XFFDFVAUNXGYBULVIYBDXGDFUKVEUJUSWMUMWAUS $.
|
|
$( [11-Apr-2012] $)
|
|
$}
|
|
|
|
|
|
${
|
|
oadp35lem.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
oadp35lem.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
oadp35lem.3 $e |- c2 = ( ( a0 v a1 ) ^ ( b0 v b1 ) ) $.
|
|
oadp35lem.4 $e |- p0 = ( ( a1 v b1 ) ^ ( a2 v b2 ) ) $.
|
|
oadp35lem.5 $e |- p = ( ( ( a0 v b0 ) ^ ( a1 v b1 ) ) ^ ( a2 v b2 ) ) $.
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
oadp35lemg $p |- p
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( dp53 ) ABCDEFGHIJLMNPQ $.
|
|
$( [12-Jul-2015] $)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
oadp35lemf $p |- ( a0 v p )
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
( wo wa leo oadp35lemg lel2or ) BBEFJHIQRQRZQABUBSABCDEFGHIJKLMNOPTUA $.
|
|
$( [12-Jul-2015] $)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
$(
|
|
oadp35leme $p |- ( b0 ^ ( a0 v p0 ) )
|
|
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
|
|
? $.
|
|
$)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
$(
|
|
oadp35lemd $p |- ( b0 ^ ( a0 v p0 ) )
|
|
=< ( b0 ^ ( ( ( a0 ^ b0 ) v b1 ) v ( c2 ^ ( c0 v c1 ) ) ) ) $=
|
|
? $.
|
|
$)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
oadp35lemc $p |- ( b0 ^ ( ( ( a0 ^ b0 ) v b1 ) v ( c2 ^ ( c0 v c1 ) ) ) )
|
|
= ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) $=
|
|
( wa wo leo le2an or32 orcom cm lbtr lerr ler2an df-le2 lor 3tr lan ) BEQ
|
|
ZFRJHIRZQZRZFUMRZEUNUKUMRZFRFUPRUOUKFUMUAUPFUBUPUMFUKUMUKJULUKBCRZEFRZQZJ
|
|
BUQEURBCSEFSTJUSNUCUDUKIHUKBDRZEGRZQZIBUTEVABDSEGSTIVBMUCUDUEUFUGUHUIUJ
|
|
$.
|
|
$( [12-Jul-2015] $)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
$(
|
|
oadp35lemb $p |- ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) )
|
|
= ( b0 ^ ( b1 v ( ( a0 v a1 ) ^ ( c0 v c1 ) ) ) ) $=
|
|
? $.
|
|
$)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
$(
|
|
oadp35lembb $p |- ( b0 ^ ( a0 v p0 ) )
|
|
=< ( b0 ^ ( b1 v ( ( a0 v a1 ) ^ ( c0 v c1 ) ) ) ) $=
|
|
( wo wa oadp35lemd oadp35lemc oadp35lemb tr lbtr )
|
|
EBKQREBERFQJHIQZRZQRZEFBCQUD
|
|
RQRZABCDEFGHIJKLMNOPSUFEFUEQRUGABCDEFGHIJKLMNOPTABCDEFGHIJKLMNOPUAUBUC $.
|
|
$)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
$(
|
|
oadp35lema $p |- ( b1 v ( b0 ^ ( a0 v p0 ) ) )
|
|
=< ( b1 v ( ( a0 v a1 ) ^ ( c0 v c1 ) ) ) $=
|
|
( wo wa leo oadp35lembb lear letr lel2or ) FFBCQHIQRZQZEBKQRZFUDSUFEUERUEAB
|
|
CDEFGHIJKLMNOPTEUEUAUBUC $.
|
|
$)
|
|
|
|
$( Part of proof (3)=>(5) in Day/Pickering 1982. $)
|
|
$(
|
|
oadp35lem0 $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
=< ( ( c0 v c1 ) v ( b1 ^ ( a0 v a1 ) ) ) $=
|
|
? $.
|
|
$)
|
|
$}
|
|
|
|
|
|
${
|
|
oadp35.1 $e |- c0 = ( ( a1 v a2 ) ^ ( b1 v b2 ) ) $.
|
|
oadp35.2 $e |- c1 = ( ( a0 v a2 ) ^ ( b0 v b2 ) ) $.
|
|
oadp35.3 $e |- p0 = ( ( a1 v b1 ) ^ ( a2 v b2 ) ) $.
|
|
$( Part of theorem from Alan Day and Doug Pickering, "A note on the
|
|
Arguesian lattice identity," Studia Sci. Math. Hungar. 19:303-305
|
|
(1982). (3)=>(5) $)
|
|
oadp35 $p |- ( ( a0 v a1 ) ^ ( ( b0 ^ ( a0 v p0 ) ) v b1 ) )
|
|
=< ( ( c0 v c1 ) v ( b1 ^ ( a0 v a1 ) ) ) $=
|
|
( wo wa id dp35lem0 ) ADMBEMNCFMNZABCDEFGHABMDEMNZIJKROLQOP $.
|
|
$( [12-Apr-2012] $)
|
|
$}
|
|
|
|
$( A modular law experiment. $)
|
|
testmod $p |- ( ( ( c v a ) v ( ( b v c ) ^ ( d v a ) ) )
|
|
^ ( a v ( b ^ ( d v ( ( a v c ) ^ ( b v d ) ) ) ) ) )
|
|
= ( ( b ^ ( ( ( ( a v c ) v ( ( b v c
|
|
) ^ ( d v a ) ) ) ^ d ) v ( ( a v c ) ^ ( b v d ) ) ) ) v a ) $=
|
|
( wo wa leao1 mli orass ran tr lan ror an12 leo orcom or32 2an 3tr cm ) BAC
|
|
EZBCEDAEFZEZDFUABDEZFZEZFZAEZCAEUBEZABDUEEZFZEZFZUHACUBEZEZUKFZAEZUOUKAEZFU
|
|
MUHBUOUJFZFZAEUQUGUTAUFUSBUFUCUJFUSUCDUEUAUDUBGHUCUOUJACUBIJKLMUTUPABUOUJNM
|
|
KUOUKAAUNOHUOUIURULUOUNAEUIAUNPCUBAQKUKAPRST $.
|
|
$( [21-Apr-2012] $)
|
|
|
|
$( A modular law experiment. $)
|
|
testmod1 $p |- ( ( ( c v a ) v ( ( b v c ) ^ ( d v a ) ) )
|
|
^ ( a v ( b ^ ( d v ( ( a v c ) ^ ( b v d ) ) ) ) ) )
|
|
= ( a v ( b ^ ( ( ( a v c ) ^ ( b v d ) )
|
|
v ( d ^ ( ( a v c ) v ( ( b v c ) ^ ( d v a ) ) ) ) ) ) ) $=
|
|
( wo wa testmod orcom ancom lor tr lan ) CAEBCEDAEFZEABDACEZBDEFZEFEFBNMEZD
|
|
FZOEZFZAEZABODPFZEZFZEZABCDGTASEUDSAHSUCARUBBROQEUBQOHQUAOPDIJKLJKK $.
|
|
$( [21-Apr-2012] $)
|
|
|
|
$( A modular law experiment. $)
|
|
testmod2 $p |- ( ( a v b ) ^ ( a v ( c v d ) ) )
|
|
= ( a v ( b ^ ( ( ( a v c ) ^ ( b v d ) )
|
|
v ( d ^ ( ( a v c ) v ( ( b v c ) ^ ( d v a ) ) ) ) ) ) ) $=
|
|
( wo wa orass lan cm leo ler mlduali leor df2le2 ran anass ancom orcom lor
|
|
tr ler2an an32 mldual2i ror lea leror l42modlem1 2an leao1 ) ABEZACDEEZFZAB
|
|
ACEZDEZFZEZABUMBDEZFZDUMBCEZDAEZFZEZFEZFZEULUJUNFZUPVEULUNUKUJACDGHIABUNAUM
|
|
DACJKLTUOVDAUOBUQUMBEZFZUNFZFZVDUOBVGFZUNFZVIVKUOVJBUNBVGBUQVFBDJBUMMUANOIB
|
|
VGUNPTVHVCBVHURDEZVBFZVCVHVLUNVFFZFZVMVHVLUNFZVFFZVOVHVLVFFZVQVHUQUNFZVFFVR
|
|
UQVFUNUBVSVLVFVSUQUMFZDEVLDUMUQDBMUCVTURDUQUMQUDTOTVQVRVPVLVFVLUNURUMDUMUQU
|
|
EUFNOITVLUNVFPTVNVBVLVNUMADEZCBEZFZEVBACDBUGWCVAUMWCUTUSFVAWAUTWBUSADRCBRUH
|
|
UTUSQTSTHTURDVBUMUQVAUILTHTST $.
|
|
$( [21-Apr-2012] $)
|
|
|
|
$( A modular law experiment. $)
|
|
testmod2expanded $p |- ( ( a v b ) ^ ( a v ( c v d ) ) )
|
|
= ( a v ( b ^ ( ( ( a v c ) ^ ( b v d ) )
|
|
v ( d ^ ( ( a v c ) v ( ( b v c ) ^ ( d v a ) ) ) ) ) ) ) $=
|
|
( wo wa orass lan cm leo ler mlduali leor df2le2 ran lor anass ancom orcom
|
|
tr ler2an an32 mldual2i ror lea leror l42modlem1 2an leao1 ) ABEZACDEEZFZAB
|
|
ACEZBDEZFZDEZUMBCEZDAEZFZEZFZFZEZABUODUTFEZFZEULABUPUMADEZCBEZFZEZFZFZEZVCU
|
|
LABUPUMDEZUMBEZFZFZFZEZVLULABUPVNFZFZEZVRULABUNUMFZDEZVNFZFZEZWAULABUNVMFZV
|
|
NFZFZEZWFULABUNVNFZVMFZFZEZWJULABWKFZVMFZEZWNULABVMFZEZWQULUJVMFZWSWTULVMUK
|
|
UJACDGHIABVMAUMDACJKLTWRWPAWPWRWOBVMBWKBUNVNBDJBUMMUANOIPTWPWMABWKVMQPTWMWI
|
|
AWLWHBUNVNVMUBHPTWIWEAWHWDBWGWCVNDUMUNDBMUCOHPTWEVTAWDVSBWCUPVNWBUODUNUMRUD
|
|
OHPTVTVQAVSVPBVSUPVMFZVNFZVPXBVSXAUPVNUPVMUOUMDUMUNUEUFNOIUPVMVNQTHPTVQVKAV
|
|
PVJBVOVIUPACDBUGHHPTVKVBAVJVABVIUTUPVHUSUMVHURUQFUSVFURVGUQADSCBSUHURUQRTPH
|
|
HPTVBVEAVAVDBUODUTUMUNUSUILHPT $.
|
|
$( [21-Apr-2012] $)
|
|
|
|
$( A modular law experiment. $)
|
|
testmod3 $p |- ( ( ( c v a ) v ( ( b v c ) ^ ( d v a ) ) )
|
|
^ ( a v ( b ^ ( d v ( ( a v c ) ^ ( b v d ) ) ) ) ) )
|
|
= ( a v ( ( ( c v a ) v ( ( b v c ) ^ ( d v
|
|
a ) ) ) ^ ( b ^ ( d v ( ( a v c ) ^ ( b v d ) ) ) ) ) ) $=
|
|
( wo wa orcom leor ler mli tr lan cm ) ACAEZBCEDAEFZEZBDACEBDEFEFZFZEZPAQEZ
|
|
FZSPQAEZFZUASRAEUCARGPQAANOACHIJKUBTPQAGLKM $.
|
|
$( [21-Apr-2012] $)
|
|
|
|
$( A modular law experiment. $)
|
|
$(
|
|
testmod4 $p |- ( ( ( c v a ) v ( ( b v c ) ^ ( d v a ) ) )
|
|
^ ( a v ( b ^ ( d v ( ( a v c ) ^ ( b v d ) ) ) ) ) )
|
|
= ( a v ( ( ( c v a ) v ( ( b v c ) ^ ( d v
|
|
a ) ) ) ^ ( b ^ ( d v ( ( a v c ) ^ ( b v d ) ) ) ) ) ) $=
|
|
( wvx wvr wvy wvq wvp wo wa leo id lor lan lear lea lelor ax-a3 cm lbtr
|
|
letr bltr ler2an leor mldual2i ancom ror tr orcom leid lel2or lebi ) CAJBCJ
|
|
DAJKJZABDACJBDJKJKZJKEFAGJZHKZJZKZAUNUOKJZ?UTUSUTGFAJZIKZJZUS?VCUS?USVAUSGJ
|
|
ZKZVCJZVCUSVEGJZVFUSVDVAGJZKZVGUSVDVHUSGLUSUSVHURUREUQUQFUQMNOUSURVHEURPURF
|
|
UPJZVHUQUPFUPHQRVHVJFAGSTUAUBUCUDVIVDVAKZGJVGGVAVDGUSUEUFVKVEGVDVAUGUHUIUAG
|
|
VCVEGVBLRUBVEVCVCVEVBGJZVCVEVBGVAKZJZVLVEVAIVMJZKVNVEVAVOVAVDQ?UDVMIVAGVAPU
|
|
FUAVMGVBGVAQRUBVBGUJUAVCUKULUBUMUITUI $.
|
|
$)
|
|
$( [22-Apr-2012] $)
|