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$( ql.mm - Version of 11-Apr-2012
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Metamath source file for logic, set theory, numbers, and Hilbert space
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~~ PUBLIC DOMAIN ~~
This work is waived of all rights, including copyright, according to the CC0
Public Domain Dedication. http://creativecommons.org/publicdomain/zero/1.0/
Norman Megill - email: nm(at)alum(dot)mit(dot)edu - http://metamath.org
$)
$( placeholder
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AUQL - Algebraic Unified Quantum Logic of M. Pavicic
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$)
$(
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Ortholattices
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$)
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Basic syntax and axioms
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
$( Declare the primitive constant symbols. $)
$c ( $. $( Left parenthesis $)
$c ) $. $( Right parenthesis $)
$c = $. $( Equality (read: 'equals') $)
$c == $. $( Biconditional (read: 'equivalent') $)
$c v $. $( Disjunction (read: 'or') $)
$c ^ $. $( Conjuction (read: 'and') $)
$c 1 $. $( True constant (upside down ' ) (read: 'true') $)
$c 0 $. $( False constant ( ' ) (read: 'false') $)
$c ' $. $( Orthocomplement $)
$c wff $. $( Well-formed formula symbol (read: 'the following symbol
sequence is a wff') $)
$c term $. $( Term $)
$c |- $. $( Turnstile (read: 'the following symbol sequence is provable' or
'a proof exists for') $)
$( Relations as operations $)
$c C $. $( Commutes relation or commutator operation $)
$c =< $. $( Less-than-or-equal-to $)
$c =<2 $. $( Less-than-or-equal-to analogue for terms $)
$c ->0 $. $( Right arrow (read: 'implies') $)
$c ->1 $. $( Right arrow (read: 'implies') $)
$c ->2 $. $( Right arrow (read: 'implies') $)
$c ->3 $. $( Right arrow (read: 'implies') $)
$c ->4 $. $( Right arrow (read: 'implies') $)
$c ->5 $. $( Right arrow (read: 'implies') $)
$c ==0 $. $( Classical identity $)
$c ==1 $. $( Asymmetrical identity $)
$c ==2 $. $( Asymmetrical identity $)
$c ==3 $. $( Asymmetrical identity $)
$c ==4 $. $( Asymmetrical identity $)
$c ==5 $. $( Asymmetrical identity $)
$c ==OA $. $( Orthoarguesian identity $)
$c , $. $( Comma $)
$c <->3 $. $( Biconditional (read: 'equivalent') $)
$c <->1 $. $( Biconditional (read: 'equivalent') $)
$c u3 $. $( Disjunction (read: 'or') $)
$c ^3 $. $( Conjuction (read: 'and') $)
$( Introduce some variable names we will use to terms. $)
$v a $.
$v b $.
$v c $.
$v d $.
$v e $.
$v f $.
$v g $.
$v h $.
$v j $.
$v k $.
$v l $.
$v i $.
$v m $.
$v n $.
$v p $.
$v q $.
$v r $.
$v t $.
$v u $.
$v w $.
$v x $.
$v y $.
$v z $.
$v a0 a1 a2 b0 b1 b2 c0 c1 c2 p0 p1 p2 $.
$(
Specify some variables that we will use to represent terms.
The fact that a variable represents a wff is relevant only to a theorem
referring to that variable, so we may use $f hypotheses. The symbol
` term ` specifies that the variable that follows it represents a term.
$)
$( Let variable ` a ` be a term. $)
wva $f term a $.
$( Let variable ` b ` be a term. $)
wvb $f term b $.
$( Let variable ` c ` be a term. $)
wvc $f term c $.
$( Let variable ` d ` be a term. $)
wvd $f term d $.
$( Let variable ` e ` be a term. $)
wve $f term e $.
$( Let variable ` f ` be a term. $)
wvf $f term f $.
$( Let variable ` g ` be a term. $)
wvg $f term g $.
$( Let variable ` h ` be a term. $)
wvh $f term h $.
$( Let variable ` j ` be a term. $)
wvj $f term j $.
$( Let variable ` k ` be a term. $)
wvk $f term k $.
$( Let variable ` l ` be a term. $)
wvl $f term l $.
$( Let variable ` i ` be a term. $)
wvi $f term i $.
$( Let variable ` m ` be a term. $)
wvm $f term m $.
$( Let variable ` n ` be a term. $)
wvn $f term n $.
$( Let variable ` p ` be a term. $)
wvp $f term p $.
$( Let variable ` q ` be a term. $)
wvq $f term q $.
$( Let variable ` r ` be a term. $)
wvr $f term r $.
$( Let variable ` t ` be a term. $)
wvt $f term t $.
$( Let variable ` u ` be a term. $)
wvu $f term u $.
$( Let variable ` w ` be a term. $)
wvw $f term w $.
$( Let variable ` x ` be a term. $)
wvx $f term x $.
$( Let variable ` y ` be a term. $)
wvy $f term y $.
$( Let variable ` z ` be a term. $)
wvz $f term z $.
$( Let variable ` a0 ` be a term. $)
wva0 $f term a0 $.
$( Let variable ` a1 ` be a term. $)
wva1 $f term a1 $.
$( Let variable ` a2 ` be a term. $)
wva2 $f term a2 $.
$( Let variable ` b0 ` be a term. $)
wvb0 $f term b0 $.
$( Let variable ` b1 ` be a term. $)
wvb1 $f term b1 $.
$( Let variable ` b2 ` be a term. $)
wvb2 $f term b2 $.
$( Let variable ` c0 ` be a term. $)
wvc0 $f term c0 $.
$( Let variable ` c1 ` be a term. $)
wvc1 $f term c1 $.
$( Let variable ` c2 ` be a term. $)
wvc2 $f term c2 $.
$( Let variable ` p0 ` be a term. $)
wvp0 $f term p0 $.
$( Let variable ` p1 ` be a term. $)
wvp1 $f term p1 $.
$( Let variable ` p2 ` be a term. $)
wvp2 $f term p2 $.
$(
Recursively define terms and wffs.
$)
$( If ` a ` and ` b ` are terms, ` a = b ` is a wff. $)
wb $a wff a = b $.
$( If ` a ` and ` b ` are terms, ` a =< b ` is a wff. $)
wle $a wff a =< b $.
$( If ` a ` and ` b ` are terms, ` a C b ` is a wff. $)
wc $a wff a C b $.
$( If ` a ` is a term, so is ` a ' ` . $)
wn $a term a ' $.
$( If ` a ` and ` b ` are terms, so is ` ( a == b ) ` . $)
tb $a term ( a == b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a v b ) ` . $)
wo $a term ( a v b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ^ b ) ` . $)
wa $a term ( a ^ b ) $.
$(
@( If ` a ` and ` b ` are terms, so is ` ( a ' b ) ` . @)
wp @a term ( a ' b ) @.
$)
$( The logical true constant is a term. $)
wt $a term 1 $.
$( The logical false constant is a term. $)
wf $a term 0 $.
$( If ` a ` and ` b ` are terms, so is ` ( a =<2 b ) ` . $)
wle2 $a term ( a =<2 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ->0 b ) ` . $)
wi0 $a term ( a ->0 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ->1 b ) ` . $)
wi1 $a term ( a ->1 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ->2 b ) ` . $)
wi2 $a term ( a ->2 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ->3 b ) ` . $)
wi3 $a term ( a ->3 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ->4 b ) ` . $)
wi4 $a term ( a ->4 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ->5 b ) ` . $)
wi5 $a term ( a ->5 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ==0 b ) ` . $)
wid0 $a term ( a ==0 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ==1 b ) ` . $)
wid1 $a term ( a ==1 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ==2 b ) ` . $)
wid2 $a term ( a ==2 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ==3 b ) ` . $)
wid3 $a term ( a ==3 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ==4 b ) ` . $)
wid4 $a term ( a ==4 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ==5 b ) ` . $)
wid5 $a term ( a ==5 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a <->3 b ) ` . $)
wb3 $a term ( a <->3 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a <->3 b ) ` . $)
wb1 $a term ( a <->1 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a u3 b ) ` . $)
wo3 $a term ( a u3 b ) $.
$( If ` a ` and ` b ` are terms, so is ` ( a ^3 b ) ` . $)
wan3 $a term ( a ^3 b ) $.
$( If ` a ` , ` b ` , and ` c ` are terms, so is ` ( a == c ==OA b ) ` . $)
wid3oa $a term ( a == c ==OA b ) $.
$( If ` a ` , ` b ` , ` c ` , and ` d ` are terms, so is
` ( a == c , d ==OA b ) ` . $)
wid4oa $a term ( a == c , d ==OA b ) $.
$( If ` a ` and ` b ` are terms, so is ` C ( a , b ) ` . $)
wcmtr $a term C ( a , b ) $.
$( Axiom for ortholattices. $)
ax-a1 $a |- a = a ' ' $.
$( Axiom for ortholattices. $)
ax-a2 $a |- ( a v b ) = ( b v a ) $.
$( Axiom for ortholattices. $)
ax-a3 $a |- ( ( a v b ) v c ) = ( a v ( b v c ) ) $.
$( Axiom for ortholattices. $)
ax-a4 $a |- ( a v ( b v b ' ) ) = ( b v b ' ) $.
$(
ax-a5 $a |- ( a v ( a ' v b ' ) ' ) = a $.
$)
$( Axiom for ortholattices. $)
ax-a5 $a |- ( a v ( a ' v b ) ' ) = a $.
$(
df-b $a |- ( a == b ) =
( ( a ' ' v b ' ' ) ' v ( a ' v b ' ) ' ) $.
$)
${
r1.1 $e |- a = b $.
$( Inference rule for ortholattices. $)
ax-r1 $a |- b = a $.
$}
${
r2.1 $e |- a = b $.
r2.2 $e |- b = c $.
$( Inference rule for ortholattices. $)
ax-r2 $a |- a = c $.
$}
$( Axiom ~ax-r3 is the orthomodular axiom and will be introduced
when we start to use it. $)
${
r4.1 $e |- a = b $.
$( Inference rule for ortholattices. $)
ax-r4 $a |- a ' = b ' $.
$}
${
r5.1 $e |- a = b $.
$( Inference rule for ortholattices. $)
ax-r5 $a |- ( a v c ) = ( b v c ) $.
$}
$( Define biconditional. $)
df-b $a |- ( a == b ) = ( ( a ' v b ' ) ' v ( a v b ) ' ) $.
$( Define conjunction. $)
df-a $a |- ( a ^ b ) = ( a ' v b ' ) ' $.
$( Define true. $)
df-t $a |- 1 = ( a v a ' ) $.
$( Define false. $)
df-f $a |- 0 = 1 ' $.
$( Define classical conditional. $)
df-i0 $a |- ( a ->0 b ) = ( a ' v b ) $.
$( Define Sasaki (Mittelstaedt) conditional. $)
df-i1 $a |- ( a ->1 b ) = ( a ' v ( a ^ b ) ) $.
$( Define Dishkant conditional. $)
df-i2 $a |- ( a ->2 b ) = ( b v ( a ' ^ b ' ) ) $.
$( Define Kalmbach conditional. $)
df-i3 $a |- ( a ->3 b ) = ( ( ( a ' ^ b ) v ( a ' ^ b ' ) ) v
( a ^ ( a ' v b ) ) ) $.
$( Define non-tollens conditional. $)
df-i4 $a |- ( a ->4 b ) = ( ( ( a ^ b ) v ( a ' ^ b ) ) v
( ( a ' v b ) ^ b ' ) ) $.
$( Define relevance conditional. $)
df-i5 $a |- ( a ->5 b ) = ( ( ( a ^ b ) v ( a ' ^ b ) ) v
( a ' ^ b ' ) ) $.
$( Define classical identity. $)
df-id0 $a |- ( a ==0 b ) = ( ( a ' v b ) ^ ( b ' v a ) ) $.
$( Define asymmetrical identity (for "Non-Orthomodular Models..." paper). $)
df-id1 $a |- ( a ==1 b ) = ( ( a v b ' ) ^ ( a ' v ( a ^ b ) ) ) $.
$( Define asymmetrical identity (for "Non-Orthomodular Models..." paper). $)
df-id2 $a |- ( a ==2 b ) = ( ( a v b ' ) ^ ( b v ( a ' ^ b ' ) ) ) $.
$( Define asymmetrical identity (for "Non-Orthomodular Models..." paper). $)
df-id3 $a |- ( a ==3 b ) = ( ( a ' v b ) ^ ( a v ( a ' ^ b ' ) ) ) $.
$( Define asymmetrical identity (for "Non-Orthomodular Models..." paper). $)
df-id4 $a |- ( a ==4 b ) = ( ( a ' v b ) ^ ( b ' v ( a ^ b ) ) ) $.
$( Defined disjunction. $)
df-o3 $a |- ( a u3 b ) = ( a ' ->3 ( a ' ->3 b ) ) $.
$( Defined conjunction. $)
df-a3 $a |- ( a ^3 b ) = ( a ' u3 b ' ) ' $.
$( Defined biconditional. $)
df-b3 $a |- ( a <->3 b ) = ( ( a ->3 b ) ^ ( b ->3 a ) ) $.
$( The 3-variable orthoarguesian identity term. $)
df-id3oa $a |- ( a == c ==OA b ) = ( ( ( a ->1 c ) ^ ( b ->1 c ) )
v ( ( a ' ->1 c ) ^ ( b ' ->1 c ) ) ) $.
$( The 4-variable orthoarguesian identity term. $)
df-id4oa $a |- ( a == c , d ==OA b ) = ( ( a == d ==OA b ) v
( ( a == d ==OA c ) ^ ( b == d ==OA c ) ) ) $.
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Basic lemmas
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
$( Identity law. $)
id $p |- a = a $=
( wn ax-a1 ax-r1 ax-r2 ) AABBZAACZAFGDE $.
$( [9-Aug-97] $)
$( Justification of definition ~df-t of true ( ` 1 ` ). This shows that the
definition is independent of the variable used to define it. $)
tt $p |- ( a v a ' ) = ( b v b ' ) $=
( wn wo ax-a4 ax-r1 ax-a2 ax-r2 ) AACDZIBBCDZDZJIJIDZKLIJAEFJIGHIBEH $.
$( [9-Aug-97] $)
${
cm.1 $e |- a = b $.
$( Commutative inference rule for ortholattices. $)
cm $p |- b = a $=
( ax-r1 ) ABCD $.
$( [26-May-2008] $) $( [26-May-2008] $)
$}
${
tr.1 $e |- a = b $.
tr.2 $e |- b = c $.
$( Transitive inference rule for ortholattices. $)
tr $p |- a = c $=
( ax-r2 ) ABCDEF $.
$( [26-May-2008] $) $( [26-May-2008] $)
$}
${
3tr1.1 $e |- a = b $.
3tr1.2 $e |- c = a $.
3tr1.3 $e |- d = b $.
$( Transitive inference useful for introducing definitions. $)
3tr1 $p |- c = d $=
( ax-r1 ax-r2 ) CADFABDEDBGHII $.
$( [10-Aug-97] $)
$}
${
3tr2.1 $e |- a = b $.
3tr2.2 $e |- a = c $.
3tr2.3 $e |- b = d $.
$( 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 $=
( wa wn tb wo wt df-a con2 2bi conb ax-r1 ax-r2 2or ka4lemo ) ABDZEZACDZBCD
ZFZGAEZBEZGZUBCEZGZUCUEGZFZGHRUDUAUHQUDABIJUAUFEZUGEZFZUHSUITUJACIBCIKUHUKU
FUGLMNOUBUCUEPN $.
$( [25-Oct-97] $)
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Kalmbach axioms (soundness proofs) that are true in all ortholattices
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
${
sklem.1 $e |- a =< b $.
$( Soundness lemma. $)
sklem $p |- ( a ' v b ) = 1 $=
( wn wo wt or12 df-t ax-r5 ax-r1 ax-a3 ax-a2 3tr2 ax-r2 df-le2 lor or1 )
ADZABEZEZBFEZRBEZFTAUBEZUARABGAREZBEZFBEZUCUAUFUEFUDBAHIJARBKFBLMNSBRABCO
PBQM $.
$( [30-Aug-97] $)
$}
$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA1. $)
ska1 $p |- ( a == a ) = 1 $=
( biid ) AB $.
$( [30-Aug-97] $)
$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA3. $)
ska3 $p |- ( ( a == b ) ' v ( a ' == b ' ) ) = 1 $=
( wn tb wo wt conb ax-r4 lor ax-a2 df-t 3tr1 ) ACBCDZABDZCZEMMCZEOMEFOPMNMA
BGHIOMJMKL $.
$( [30-Aug-97] $)
$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA5. $)
ska5 $p |- ( ( a ^ b ) == ( b ^ a ) ) = 1 $=
( wa ancom bi1 ) ABCBACABDE $.
$( [30-Aug-97] $)
$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA6. $)
ska6 $p |- ( ( a ^ ( b ^ c ) ) == ( ( a ^ b ) ^ c ) ) = 1 $=
( wa anass ax-r1 bi1 ) ABCDDZABDCDZIHABCEFG $.
$( [30-Aug-97] $)
$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA7. $)
ska7 $p |- ( ( a ^ ( a v b ) ) == a ) = 1 $=
( wo wa anabs bi1 ) AABCDAABEF $.
$( [30-Aug-97] $)
$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA8. $)
ska8 $p |- ( ( a ' ^ a ) == ( ( a ' ^ a ) ^ b ) ) = 1 $=
( wn wa wf an0 ax-r1 ancom ax-r2 dff ran 3tr2 bi1 ) ACZADZOBDZEEBDZOPEBEDZQ
REBFGBEHIEANDOAJANHIZEOBSKLM $.
$( [30-Aug-97] $)
$( Soundness theorem for Kalmbach's quantum propositional logic axiom KA9. $)
ska9 $p |- ( a == a ' ' ) = 1 $=
( wn ax-a1 bi1 ) AABBACD $.
$( [30-Aug-97] $)
$( Soundness theorem for Kalmbach's quantum propositional logic axiom
KA10. $)
ska10 $p |- ( ( a v b ) ' == ( a ' ^ b ' ) ) = 1 $=
( wo wn wa oran con2 bi1 ) ABCZDADBDEZIJABFGH $.
$( [30-Aug-97] $)
$( Soundness theorem for Kalmbach's quantum propositional logic axiom
KA11. $)
ska11 $p |- ( ( a v ( a ' ^ ( a v b ) ) ) == ( a v b ) ) = 1 $=
( woml ) ABC $.
$( [2-Sep-97] $)
$( [30-Aug-97] $)
$( 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] $)
${
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] $)
$}
$( 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] $)
${
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] $)
$}
${
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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 $.
$( [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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 $.
$( [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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 $.
$( [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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 $.
$( [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
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
$(
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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 $.
$( [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 ) $=
( tb wo wa wn ax-r2 lbtr ran 2or ax-r1 anass comcom7 wf biao bile wi2 wi1
bicom orbile imp3 le2an 2bi ax-r4 lan 2an lea 3tr1 rbi ler2an coman1 bctr
ancom an32 coman2 com2an com2or fh2c anor3 comanr1 fh2rc leao1 df2le2 dff
comcom3 oran an0r 3tr2 or0 3tr an4 anidm or0r dfb lor mlaoml bltr letr
bi3 ) ABIZACIZBCIZJZKABJZABKZIZWKLZCLZKZCWJKZJZKZWGWFWLWIWQWFWLWFWKWJIZWL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$.
$( [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] $)
${
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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 $.
$( [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] $)
$}
${
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] $)
$}
${
gomaex3lem2.5 $e |- e =< f ' $.
$( Lemma for Godowski 6-var -> Mayet Example 3. $)
gomaex3lem2 $p |- ( ( e v f ) ' v f ) = e ' $=
( wo wn wt lecon3 lecom comid comcom2 fh3r anor3 ax-r5 ax-r1 anabs df2le1
wa leid lel2or ax-r2 lebi df-t ax-a2 2an 3tr1 an1 ) ABDEZBDZAEZFQZUIUIBEZ
QZBDZUIBDZUKBDZQUHUJBUIUKBUIABCGZHBBBIJKUMUHULUGBABLMNUIUNFUOUIUNUIUNUIBO
PUIUIBUIRUPSUAFBUKDUOBUBBUKUCTUDUEUIUFT $.
$( [29-Nov-99] $)
$}
$( Lemma for Godowski 6-var -> Mayet Example 3. $)
gomaex3lem3 $p |- ( ( p ' ->1 q ) ' v ( p ' ^ q ) ) = p ' $=
( wn wi1 wa wo anor1 ax-r1 df-i1 ax-r4 3tr1 ax-r5 coman1 comid comcom2 fh3r
id wt orabs ax-r2 ax-a2 df-t 2an an1 3tr ) ACZBDZCZUFBEZFUFUICZEZUIFUFUIFZU
JUIFZEZUFUHUKUIUFCUIFZCZUKUHUKUKUPUFUIGHUGUOUFBIJUKQKLUIUFUJUFBMUIUIUINOPUN
UFREUFULUFUMRUFBSUMUIUJFZRUJUIUARUQUIUBHTUCUFUDTUE $.
$( [29-Nov-99] $)
${
gomaex3lem4.9 $e |- p = ( ( a v b ) ->1 ( d v e ) ' ) ' $.
$( Lemma for Godowski 6-var -> Mayet Example 3. $)
gomaex3lem4 $p |- ( ( a v b ) ^ ( d v e ) ' ) =< p ' $=
( wo wn wa leor wi1 ax-a1 df-i1 ax-r1 ax-r4 3tr1 lbtr ) ABGZCDGHZIZRHZTGZ
EHZTUAJRSKZUDHZHUBUCUDLUDUBRSMNEUEFOPQ $.
$( [29-Nov-99] $)
$}
${
gomaex3lem5.1 $e |- a =< b ' $.
gomaex3lem5.2 $e |- b =< c ' $.
gomaex3lem5.3 $e |- c =< d ' $.
gomaex3lem5.5 $e |- e =< f ' $.
gomaex3lem5.6 $e |- f =< a ' $.
gomaex3lem5.8 $e |- ( ( ( i ->2 g ) ^ ( g ->2 y ) ) ^
( ( ( y ->2 w ) ^ ( w ->2 n ) ) ^
( ( n ->2 k ) ^ ( k ->2 i ) ) ) ) =< ( g ->2 i ) $.
gomaex3lem5.9 $e |- p = ( ( a v b ) ->1 ( d v e ) ' ) ' $.
gomaex3lem5.10 $e |- q = ( ( e v f ) ->1 ( b v c ) ' ) ' $.
gomaex3lem5.11 $e |- r = ( ( p ' ->1 q ) ' ^ ( c v d ) ) $.
gomaex3lem5.12 $e |- g = a $.
gomaex3lem5.13 $e |- h = b $.
gomaex3lem5.14 $e |- i = c $.
gomaex3lem5.15 $e |- j = ( c v d ) ' $.
gomaex3lem5.16 $e |- k = r $.
gomaex3lem5.17 $e |- m = ( p ' ->1 q ) $.
gomaex3lem5.18 $e |- n = ( p ' ->1 q ) ' $.
gomaex3lem5.19 $e |- u = ( p ' ^ q ) $.
gomaex3lem5.20 $e |- w = q ' $.
gomaex3lem5.21 $e |- x = q $.
gomaex3lem5.22 $e |- y = ( e v f ) ' $.
gomaex3lem5.23 $e |- z = f $.
$( Lemma for Godowski 6-var -> Mayet Example 3. $)
gomaex3lem5 $p |- ( ( ( g v h ) ^ ( i v j ) ) ^
( ( ( k v m ) ^ ( n v u ) ) ^
( ( w v x ) ^ ( y v z ) ) ) ) =< ( h v i ) $=
( gomaex3h1 gomaex3h2 gomaex3h3 gomaex3h4 gomaex3h5 gomaex3h10 gomaex3h11
gomaex3h6 gomaex3h7 gomaex3h8 gomaex3h9 gomaex3h12 go2n6 ) GHIJKLMQRSTUAA
BGHUBUKULVCBCHKUCULUMVDCDIKUMUNVECDIJNOPUJUNUOVFCDJLNOPUJUOUPVGLMNOUPUQVJ
MNOQUQURVKNOQRURUSVLORSUSUTVMBCEFOSTUIUTVAVHEFTUAVAVBVIAFGUAUFUKVBVNUGVO
$.
$( [29-Nov-99] $)
$( Lemma for Godowski 6-var -> Mayet Example 3. $)
gomaex3lem6 $p |- ( ( ( a v b ) ^ ( c v ( c v d ) ' ) ) ^
( ( ( r v ( p ' ->1 q ) ) ^ ( ( p ' ->1 q ) '
v ( p ' ^ q ) ) ) ^
( ( q ' v q ) ^ ( ( e v f ) ' v f ) ) ) ) =< ( b v c ) $=
( wo wa wn wi1 gomaex3lem5 2or 2an le3tr2 ) GHVCZKIVCZVDZJLVCZMQVCZVDZRSV
CZTUAVCZVDZVDZVDHKVCABVCZCCDVCVEZVCZVDZPNVEZOVFZVCZWFVEZWEOVDZVCZVDZOVEZO
VCZEFVCVEZFVCZVDZVDZVDBCVCABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMU
NUOUPUQURUSUTVAVBVGVMWDVTWQVKWAVLWCGAHBUKULVHKCIWBUMUNVHVIVPWKVSWPVNWGVOW
JJPLWFUOUPVHMWHQWIUQURVHVIVQWMVRWORWLSOUSUTVHTWNUAFVAVBVHVIVIVIHBKCULUMVH
VJ $.
$( [29-Nov-99] $)
$( Lemma for Godowski 6-var -> Mayet Example 3. $)
gomaex3lem7 $p |- ( ( ( a v b ) ^ d ' ) ^
( ( ( r v ( p ' ->1 q ) ) ^ p ' ) ^ e ' ) ) =< ( b v c ) $=
( wo wn wa wi1 gomaex3lem1 gomaex3lem3 ancom gomaex3lem2 ax-a2 df-t ax-r1
lan wt ax-r2 2an an1 3tr gomaex3lem6 bltr ) ABVCZDVDZVEZPNVDZOVFZVCZWEVEZ
EVDZVEZVEZWBCCDVCVDVCZVEZWGWFVDWEOVEVCZVEZOVDZOVCZEFVCVDFVCZVEZVEZVEZBCVC
XAWKWMWDWTWJWLWCWBCDUDVGVNWOWHWSWIWNWEWGNOVHVNWSWRWQVEWIVOVEWIWQWRVIWRWIW
QVOEFUEVJWQOWPVCZVOWPOVKVOXBOVLVMVPVQWIVRVSVQVQVMABCDEFGHIJKLMNOPQRSTUAUB
UCUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBVTWA $.
$( [29-Nov-99] $)
$( Lemma for Godowski 6-var -> Mayet Example 3. $)
gomaex3lem8 $p |- ( ( ( a v b ) ^ ( d v e ) ' ) ^
( ( r v ( p ' ->1 q ) ) ^ p ' ) ) =< ( b v c ) $=
( wo wn wa wi1 an32 anor3 lan ran an4 3tr2 gomaex3lem7 bltr ) ABVCZDEVCVD
ZVEZPNVDZOVFVCVRVEZVEZVODVDZVEVSEVDZVEVEZBCVCVOWAWBVEZVEZVSVEVOVSVEWDVEVT
WCVOWDVSVGWEVQVSWDVPVODEVHVIVJVOVSWAWBVKVLABCDEFGHIJKLMNOPQRSTUAUBUCUDUEU
FUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBVMVN $.
$( [29-Nov-99] $)
$( Lemma for Godowski 6-var -> Mayet Example 3. $)
gomaex3lem9 $p |- ( ( ( a v b ) ^ ( d v e ) ' ) ^
( r v ( p ' ->1 q ) ) ) =< ( b v c ) $=
( wo wn wi1 ancom gomaex3lem4 df2le2 ax-r1 lan an12 3tr gomaex3lem8 bltr
wa ) ABVCDEVCVDVOZPNVDZOVEVCZVOZVPVRVQVOVOZBCVCVSVRVPVOVRVPVQVOZVOVTVPVRV
FVPWAVRWAVPVPVQABDENUHVGVHVIVJVRVPVQVKVLABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFU
GUHUIUJUKULUMUNUOUPUQURUSUTVAVBVMVN $.
$( [29-Nov-99] $)
$( Lemma for Godowski 6-var -> Mayet Example 3. $)
gomaex3lem10 $p |- ( ( ( a v b ) ^ ( d v e ) ' ) ^
( r v ( p ' ->1 q ) ) ) =< ( ( b v c ) v ( e v f ) ' ) $=
( wo wn wa wi1 gomaex3lem9 leo letr ) ABVCDEVCVDVEPNVDOVFVCVEBCVCZVJEFVCV
DZVCABCDEFGHIJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMUNUOUPUQURUSUTVAVBVGVJV
KVHVI $.
$( [29-Nov-99] $)
$}
${
gomaex3.1 $e |- a =< b ' $.
gomaex3.2 $e |- b =< c ' $.
gomaex3.3 $e |- c =< d ' $.
gomaex3.5 $e |- e =< f ' $.
gomaex3.6 $e |- f =< a ' $.
gomaex3.8 $e |- ( ( ( i ->2 g ) ^ ( g ->2 y ) ) ^
( ( ( y ->2 w ) ^ ( w ->2 n ) ) ^
( ( n ->2 k ) ^ ( k ->2 i ) ) ) ) =< ( g ->2 i ) $.
gomaex3.9 $e |- p = ( ( a v b ) ->1 ( d v e ) ' ) ' $.
gomaex3.10 $e |- q = ( ( e v f ) ->1 ( b v c ) ' ) ' $.
gomaex3.11 $e |- r = ( ( p ' ->1 q ) ' ^ ( c v d ) ) $.
gomaex3.12 $e |- g = a $.
gomaex3.14 $e |- i = c $.
gomaex3.16 $e |- k = r $.
gomaex3.18 $e |- n = ( p ' ->1 q ) ' $.
gomaex3.20 $e |- w = q ' $.
gomaex3.22 $e |- y = ( e v f ) ' $.
$( 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. $)
gomaex3 $p |- ( ( ( a v b ) ^ ( d v e ) ' ) ^
( ( ( ( a v b ) ->1 ( d v e ) ' ) ->1
( ( e v f ) ->1 ( b v c ) ' ) ' ) ' ->1 ( c v d ) ) )
=< ( ( b v c ) v ( e v f ) ' ) $=
( wo wn wa wi1 df-i1 ax-a2 con2 ud1lem0ab ax-a1 ax-r2 ax-r4 ran 2or ax-r1
lan id gomaex3lem10 bltr ) ABUKZDEUKULZUMZVIVJUNZEFUKZBCUKZULUNULZUNZULZC
DUKZUNZUMVKMKULZLUNZUKZUMVNVMULUKVSWBVKVSVQULZVQVRUMZUKZWBVQVRUOWBWEWBWAM
UKWEMWAUPWAWCMWDWAVPWCVTVLLVOKVLUBUQUCURZVPUSUTMWAULZVRUMWDUDWGVQVRWAVPWF
VAVBUTVCUTVDUTVEABCDEFGBVRULZHIWAJKLMVTLUMZNLOFPQRSTUAUBUCUDUEBVFUFWHVFUG
WAVFUHWIVFUILVFUJFVFVGVH $.
$( [27-May-00] $)
$}
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
OML Lemmas for studying orthoarguesian laws
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
${
oas.1 $e |- ( a ' ^ ( a v b ) ) =< c $.
$( "Strengthening" lemma for studying the orthoarguesian law. $)
oas $p |- ( ( a ->1 c ) ^ ( a v b ) ) =< c $=
( wi1 wo wa oml ax-r1 lea ler2an lelor bltr lelan u1lemc1 lbtr letr ax-r2
wn lear comanr1 comcom6 fh2 u1lemaa ancom leo df-i1 df2le2 2or lel2or ) A
CEZABFZGZACGZASZCGZFZCUMUKAUPFZGZUQULURUKULAUOULGZFZURVAULABHIUTUPAUTUOCU
OULJDKLMNUSUKAGZUKUPGZFUQAUKUPACOAUPUOCUAUBUCVBUNVCUPACUDVCUPUKGUPUKUPUEU
PUKUPUOUKUOCJUOUOUNFZUKUOUNUFUKVDACUGIPQUHRUIRPUNCUPACTUOCTUJQ $.
$( [25-Dec-98] $)
$}
${
oasr.1 $e |- ( ( a ->1 c ) ^ ( a v b ) ) =< c $.
$( Reverse of ~ oas lemma for studying the orthoarguesian law. $)
oasr $p |- ( a ' ^ ( a v b ) ) =< c $=
( wn wo wa wi1 u1lem9b leran letr ) AEZABFZGACHZMGCLNMACIJDK $.
$( [28-Dec-98] $)
$}
${
oat.1 $e |- ( a ' ^ ( a v b ) ) =< c $.
$( Transformation lemma for studying the orthoarguesian law. $)
oat $p |- b =< ( a ' ->1 c ) $=
( wn wa wo wi1 leor oml ax-r1 lea lelor bltr letr ax-a1 ax-r5 df-i1 ax-r2
ler2an lbtr ) BAAEZCFZGZUBCHZBABGZUDBAIUFAUBUFFZGZUDUHUFABJKUGUCAUGUBCUBU
FLDTMNOUDUBEZUCGZUEAUIUCAPQUEUJUBCRKSUA $.
$( [26-Dec-98] $)
$}
${
oatr.1 $e |- b =< ( a ' ->1 c ) $.
$( Reverse transformation lemma for studying the orthoarguesian law. $)
oatr $p |- ( a ' ^ ( a v b ) ) =< c $=
( wn wo wa leo df-i1 ax-a1 ax-r5 ax-r1 ax-r2 lbtr lel2or lelan omlan lear
wi1 letr ) AEZABFZGZUACGZCUCUAAUDFZGUDUBUEUAAUEBAUDHBUACSZUEDUFUAEZUDFZUE
UACIUEUHAUGUDAJKLMNOPACQNUACRT $.
$( [26-Dec-98] $)
$}
${
oau.1 $e |- ( a ^ ( ( a ->1 c ) v b ) ) =< c $.
$( Transformation lemma for studying the orthoarguesian law. $)
oau $p |- b =< ( a ->1 c ) $=
( wi1 wo ax-a2 wa lea ler2an u1lemaa ax-r1 lelor wt u1lemc1 comcom comorr
lbtr fh3 ax-r2 u1lemoa ax-a3 oridm ax-r5 2an ancom an1 3tr orabs leo lebi
le3tr2 df-le1 ) BACEZBUNFUNBFZUNBUNGUOUNUNAUOHZFZUNUNAHZFUOUNUPURUNUPACHZ
URUPACAUOIDJURUSACKLRMUQUNAFZUNUOFZHNUOHZUOUNAUOAUNACOPUNBQSUTNVAUOACUAVA
UNUNFZBFZUOVDVAUNUNBUBLVCUNBUNUCUDTUEVBUONHUONUOUFUOUGTUHUNAUIULUNBUJUKTU
M $.
$( [28-Dec-98] $)
$}
${
oaur.1 $e |- b =< ( a ->1 c ) $.
$( Transformation lemma for studying the orthoarguesian law. $)
oaur $p |- ( a ^ ( ( a ->1 c ) v b ) ) =< c $=
( wi1 wo wa leid lel2or lelan ancom u1lemaa ax-r2 lbtr lear letr ) AACEZB
FZGZACGZCSAQGZTRQAQQBQHDIJUAQAGTAQKACLMNACOP $.
$( [28-Dec-98] $)
$}
${
oaidlem2.1 $e |- ( ( d v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) ' v
( ( a ->1 c ) ->1 ( b ->1 c ) ) ) = 1 $.
$( Lemma for identity-like OA law. $)
oaidlem2 $p |- ( ( a ->1 c ) ^
( d v ( ( a ->1 c ) ^ ( b ->1 c ) ) ) )
=< ( b ->1 c ) $=
( wi1 wa wo anidm ax-r1 ran anass ax-r2 leor lelan bltr df-le2 wn ax-a3
wt ax-a2 oran3 ax-r5 df-i1 lor 3tr2 lem3.1 bile lear letr ) ACFZDUKBCFZGZ
HZGZUMULUOUMUMUOUMUOUMUOUMUKUMGZUOUMUKUKGZULGUPUKUQULUQUKUKIJKUKUKULLMUMU
NUKUMDNOPQUNRZUKRZHZUMHURUSUMHZHZUORZUMHTURUSUMSUTVCUMUTUSURHVCURUSUAUKUN
UBMUCVBURUKULFZHZTVEVBVDVAURUKULUDUEJEMUFUGJUHUKULUIUJ $.
$( [22-Jan-99] $)
$}
${
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 $=
( 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
OABUCUDHDKUEUFHUGABUHUI $.
$( [18-Feb-02] $)
$}
${
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] $)
$}
${
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. $)
oa6to4h1 $p |- a ' =< b ' ' $=
( wn wa wo leo wi1 df-i1 ax-r4 ax-r2 ax-r1 con3 lbtr ) AKZUBAGLZMZBKZKUBU
CNUDUEUEUDKZUEAGOZKUFHUGUDAGPQRSTUA $.
$( [22-Dec-98] $)
$( Satisfaction of 6-variable OA law hypothesis. $)
oa6to4h2 $p |- c ' =< d ' ' $=
( wn wa wo leo wi1 df-i1 ax-r4 ax-r2 ax-r1 con3 lbtr ) CKZUBCGLZMZDKZKUBU
CNUDUEUEUDKZUECGOZKUFIUGUDCGPQRSTUA $.
$( [22-Dec-98] $)
$( Satisfaction of 6-variable OA law hypothesis. $)
oa6to4h3 $p |- e ' =< f ' ' $=
( wn wa wo leo wi1 df-i1 ax-r4 ax-r2 ax-r1 con3 lbtr ) EKZUBEGLZMZFKZKUBU
CNUDUEUEUDKZUEEGOZKUFJUGUDEGPQRSTUA $.
$( [22-Dec-98] $)
${
$( 6-variable OA law as hypothesis. $)
oa6to4.oa6 $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 ' ) ) ) ) ) ) ) $.
$( Derivation of 4-variable proper OA law, assuming 6-variable OA law. $)
oa6to4 $p |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
=< ( ( ( a ^ g ) v ( c ^ g ) ) v ( e ^ g ) ) $=
( wa wo wi1 con1 2an lor 2or lan ancom oa6todual u1lemaa 3tr le3tr2 ) B
ACACLZBDLZMZAELZBFLZMZCELZDFLZMZLZMZLZMZLABLZCDLZMZEFLZMAGNZACUEVBCGNZL
ZMZUHVBEGNZLZMZUKVCVFLZMZLZMZLZMZLAGLZCGLZMZEGLZMABCDEFKUABVBUQVNBVBHOZ
UPVMAUOVLCUGVEUNVKUFVDUEBVBDVCVSDVCIOZPQUJVHUMVJUIVGUHBVBFVFVSFVFJOZPQU
LVIUKDVCFVFVTWAPQPRSQPUTVQVAVRURVOUSVPURAVBLVBALVOBVBAVSSAVBTAGUBUCUSCV
CLVCCLVPDVCCVTSCVCTCGUBUCRVAEVFLVFELVRFVFEWASEVFTEGUBUCRUD $.
$( [22-Dec-98] $)
$}
$}
${
oa4b.1 $e |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
=< ( ( ( a ^ g ) v ( c ^ g ) ) v ( e ^ g ) ) $.
$( Derivation of 4-OA law variant. $)
oa4b $p |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
=< g $=
( wi1 wa wo lear lel2or letr ) ADFZABABGLBDFZGHACGLCDFZGHBCGMNGHGHGHGADGZ
BDGZHZCDGZHDEQDRODPADIBDIJCDIJK $.
$( [22-Dec-98] $)
$}
${
oa4to6lem.1 $e |- a ' =< b $.
oa4to6lem.2 $e |- c ' =< d $.
oa4to6lem.3 $e |- e ' =< f $.
oa4to6lem.4 $e |- g = ( ( ( a ^ b ) v ( c ^ d ) ) v ( e ^ f ) ) $.
$( Lemma for orthoarguesian law (4-variable to 6-variable proof). $)
oa4to6lem1 $p |- b =< ( a ->1 g ) $=
( wn wa wo wi1 wt ax-r2 ran ax-r1 lbtr leor comid comcom3 lecom fh3 ancom
df-t ax-a2 an1 3tr2 anidm anass lor ax-a3 lelan lelor letr ud1lem0a df-i1
leo ) BALZAABMZCDMZNEFMZNZMZNZAGOZBVAAVBMZNZVGBVABNZVJBVAUAVKVAVBNZVJVLVK
VLVAANZVKMZVKVAABAAAUBUCVABHUDUEPVKMVKPMVNVKPVKUFPVMVKPAVANVMAUGAVAUHQRVK
UIUJQSVBVIVAVBAAMZBMZVIVPVBVOABAUKRSAABULQUMQTVIVFVAVBVEAVBVBVCVDNZNZVEVB
VQUTVEVRVBVCVDUNSTUOUPUQVHVGVHAVEOVGGVEAKURAVEUSQST $.
$( [18-Dec-98] $)
$( Lemma for orthoarguesian law (4-variable to 6-variable proof). $)
oa4to6lem2 $p |- d =< ( c ->1 g ) $=
( wa wo wi1 leor wt ax-r2 ran ax-r1 lbtr wn comid comcom3 lecom fh3 ancom
df-t ax-a2 an1 3tr2 anidm anass lor or32 lelan lelor letr ud1lem0a df-i1
) DCUAZCABLZCDLZMEFLZMZLZMZCGNZDUTCVBLZMZVFDUTDMZVIDUTOVJUTVBMZVIVKVJVKUT
CMZVJLZVJUTCDCCCUBUCUTDIUDUEPVJLVJPLVMVJPVJUFPVLVJPCUTMVLCUGCUTUHQRVJUIUJ
QSVBVHUTVBCCLZDLZVHVOVBVNCDCUKRSCCDULQUMQTVHVEUTVBVDCVBVAVCMZVBMVDVBVPOVA
VCVBUNTUOUPUQVGVFVGCVDNVFGVDCKURCVDUSQST $.
$( [18-Dec-98] $)
$( Lemma for orthoarguesian law (4-variable to 6-variable proof). $)
oa4to6lem3 $p |- f =< ( e ->1 g ) $=
( wa wo wi1 leor wt ax-r2 ran ax-r1 lbtr wn comid comcom3 lecom fh3 ancom
df-t ax-a2 an1 3tr2 anidm anass lor lelan lelor letr ud1lem0a df-i1 ) FEU
AZEABLCDLMZEFLZMZLZMZEGNZFUSEVALZMZVDFUSFMZVGFUSOVHUSVAMZVGVIVHVIUSEMZVHL
ZVHUSEFEEEUBUCUSFJUDUEPVHLVHPLVKVHPVHUFPVJVHPEUSMVJEUGEUSUHQRVHUIUJQSVAVF
USVAEELZFLZVFVMVAVLEFEUKRSEEFULQUMQTVFVCUSVAVBEVAUTOUNUOUPVEVDVEEVBNVDGVB
EKUQEVBURQST $.
$( [18-Dec-98] $)
$( Lemma for orthoarguesian law (4-variable to 6-variable proof). $)
oa4to6lem4 $p |- ( b ^ ( a v ( c ^ ( ( ( a ^ c ) v ( b ^ d ) ) v
( ( ( a ^ e ) v ( b ^ f ) ) ^ ( ( c ^ e ) v ( d ^ f ) ) ) ) ) ) )
=< ( ( a ->1 g ) ^ ( a v ( c ^ (
( ( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) ) $=
( wi1 wa wo oa4to6lem1 oa4to6lem2 le2an lelor oa4to6lem3 le2or lelan ) BA
GLZACACMZBDMZNZAEMZBFMZNZCEMZDFMZNZMZNZMZNACUCUBCGLZMZNZUFUBEGLZMZNZUIUOU
RMZNZMZNZMZNABCDEFGHIJKOZUNVEAUMVDCUEUQULVCUDUPUCBUBDUOVFABCDEFGHIJKPZQRU
HUTUKVBUGUSUFBUBFURVFABCDEFGHIJKSZQRUJVAUIDUOFURVGVHQRQTUARQ $.
$( [18-Dec-98] $)
${
$( Proper 4-variable OA as hypothesis $)
oa4to6lem.oa4 $e |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
=< g $.
$( Lemma for orthoarguesian law (4-variable to 6-variable proof). $)
oa4to6dual $p |- ( b ^ ( a v ( c ^ ( ( ( a ^ c ) v ( b ^ d ) ) v
( ( ( a ^ e ) v ( b ^ f ) ) ^ ( ( c ^ e ) v ( d ^ f ) ) ) ) ) ) )
=< g $=
( wa wo wi1 oa4to6lem4 letr ) BACACMZBDMNAEMZBFMNCEMZDFMNMNMNMAGOZACRUA
CGOZMNSUAEGOZMNTUBUCMNMNMNMGABCDEFGHIJKPLQ $.
$( [19-Dec-98] $)
$}
$}
${
oa4to6.oa6.1 $e |- a =< b ' $.
oa4to6.oa6.2 $e |- c =< d ' $.
oa4to6.oa6.3 $e |- e =< f ' $.
$( Variable substitutions to make into the 4-variable OA. $)
oa4to6.4 $e |- g =
( ( ( a ' ^ b ' ) v ( c ' ^ d ' ) ) v ( e ' ^ f ' ) ) $.
oa4to6.5 $e |- h = a ' $.
oa4to6.6 $e |- j = c ' $.
oa4to6.7 $e |- k = e ' $.
$( Proper 4-variable OA as hypothesis $)
oa4to6.oa4 $e |- ( ( h ->1 g ) ^ ( h v ( j ^ ( (
( h ^ j ) v ( ( h ->1 g ) ^ ( j ->1 g ) ) ) v
( ( ( h ^ k ) v ( ( h ->1 g ) ^ ( k ->1 g ) ) ) ^
( ( j ^ k ) v ( ( j ->1 g ) ^ ( k ->1 g ) ) ) ) ) ) ) )
=< g $.
$( Orthoarguesian law (4-variable to 6-variable proof). The first 3
hypotheses are those for 6-OA. The next 4 are variable substitutions
into 4-OA. The last is the 4-OA. The proof uses OM logic only. $)
oa4to6 $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 ) ) ) ) ) ) ) $=
( wa wo lecon3 lecon wi1 ud1lem0ab 2an 2or le3tr2 oa4to6dual oa6fromdual
wn id ) ABCDEFAUJZBUJZCUJZDUJZEUJZFUJZULUMSUNUOSTUPUQSTZBULABKUAUBDUNCDLU
AUBFUPEFMUAUBURUKHGUCZHIHISZUSIGUCZSZTZHJSZUSJGUCZSZTZIJSZVAVESZTZSZTZSZT
ZSGULURUCZULUNULUNSZVOUNURUCZSZTZULUPSZVOUPURUCZSZTZUNUPSZVQWASZTZSZTZSZT
ZSURRUSVOVNWJHULGURONUDZHULVMWIOIUNVLWHPVCVSVKWGUTVPVBVRHULIUNOPUEUSVOVAV
QWKIUNGURPNUDZUEUFVGWCVJWFVDVTVFWBHULJUPOQUEUSVOVEWAWKJUPGURQNUDZUEUFVHWD
VIWEIUNJUPPQUEVAVQVEWAWLWMUEUFUEUFUEUFUENUGUHUI $.
$( [19-Dec-98] $)
$}
${
oa4btoc.1 $e |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
=< g $.
$( Derivation of 4-OA law variant. $)
oa4btoc $p |- ( a ' ^ ( a v ( c ^ ( (
( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
=< g $=
( wn wa wi1 wo leo df-i1 ax-r1 lbtr leid lelor lelan le2an letr ) AFZABAB
GADHZBDHZGIZACGTCDHZGIBCGUAUCGIGZIZGZIZGTUGGDSTUGUGSSADGZIZTSUHJTUIADKLMU
FUFAUEUEBUDUDUBUDNOPOQER $.
$( [22-Dec-98] $)
$}
${
oa4ctob.1 $e |- ( a ' ^ ( a v ( c ^ ( (
( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
=< g $.
$( Derivation of 4-OA law variant. $)
oa4ctob $p |- ( ( a ->1 g ) ^ ( a v ( c ^ ( (
( a ^ c ) v ( ( a ->1 g ) ^ ( c ->1 g ) ) ) v
( ( ( a ^ e ) v ( ( a ->1 g ) ^ ( e ->1 g ) ) ) ^
( ( c ^ e ) v ( ( c ->1 g ) ^ ( e ->1 g ) ) ) ) ) ) ) )
=< g $=
( wa wi1 wo oas ) ABABFADGZBDGZFHACFJCDGZFHBCFKLFHFHFDEI $.
$( [22-Dec-98] $)
$}
${
oa4ctod.1 $e |- ( 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 $.
$( Derivation of 4-OA law variant. $)
oa4ctod $p |- ( 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 ' ->1 d ) $=
( wa wi1 wo oat ) ABABFADGZBDGZFHACFJCDGZFHBCFKLFHFHFDEI $.
$( [24-Dec-98] $)
$}
${
oa4dtoc.1 $e |- ( 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 ' ->1 d ) $.
$( Derivation of 4-OA law variant. $)
oa4dtoc $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 $=
( wa wi1 wo oatr ) ABABFADGZBDGZFHACFJCDGZFHBCFKLFHFHFDEI $.
$( [24-Dec-98] $)
$}
$( Lemma commuting terms. $)
oa4dcom $p |- ( 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 ) ) ) ) ) )
= ( b ^ ( (
( 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 ) ) ) ) ) ) $=
( wa wi1 wo ancom 2or lan ) ABEZADFZBDFZEZGZACELCDFZEGZBCEMPEGZEZGBAEZMLEZG
ZRQEZGBOUBSUCKTNUAABHLMHIQRHIJ $.
$( [24-Dec-98] $)
$(
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
5OA law
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$)
${
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
OUIUKBOPUNUNUQUNQBAUISPPUNVFUMUNUQTUAUBMUCUTUPVAURABUDABUEUFABUGUH $.
$( [3-Jul-05] $) $( [1-Jul-05] $)
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
to 4, and j is set to 0. (Contributed by Roy F. Longton, 3-Jul-05.) $)
lem4.6.6i4j0 $p |- ( ( a ->4 b ) v ( a ->0 b ) ) = ( a ->0 b ) $=
( wa wn wo wi4 wi0 leao4 leao1 lel2or lea df-le2 df-i4 df-i0 2or 3tr1 ) ABC
ZADZBCZEZRBEZBDZCZEZUAEUAABFZABGZEUFUDUATUAUCQUASBARHRBBIJUAUBKJLUEUDUFUAAB
MABNZOUGP $.
$( [3-Jul-05] $) $( [2-Jul-05] $)
$( Equation 4.14 of [MegPav2000] p. 23. The variable i in the paper is set
to 4, and j is set to 2. (Contributed by Roy F. Longton, 3-Jul-05.) $)
lem4.6.6i4j2 $p |- ( ( a ->4 b ) v ( a ->2 b ) ) = ( a ->0 b ) $=
( wa wn wi4 wi2 wi0 ax-a3 ax-r1 ax-a2 ancom lor leor oml2 ax-r5 ax-r2 leao1
wo 3tr lel2or leao4 leid leo lerr lebi df-i4 df-i2 2or df-i0 3tr1 ) ABCZADZ
BCZRZULBRZBDZCZRZBULUPCZRZRZUOABEZABFZRABGVAUNUQUTRZRUNUOUSRZRZUOUNUQUTHVDV
EUNVDUQBRZUSRZVEVHVDUQBUSHIVGUOUSVGBUQRBUPUOCZRUOUQBJUQVIBUOUPKLBUOBULMNSOP
LVFUOUNUOVEUKUOUMBAULUAULBBQTUOUOUSUOUBULUPBQTTUOVEUNUOUSUCUDUESVBURVCUTABU
FABUGUHABUIUJ $.
$( [3-Jul-05] $) $( [2-Jul-05] $)
${
com3iia.1 $e |- a C b $.
$( The dual of ~ com3ii . (Contributed by Roy F. Longton, 3-Jul-05.) $)
com3iia $p |- ( a v ( a ' ^ b ) ) = ( a v b ) $=
( wn wa wo comid comcom2 fh3 lear ax-a4 df-le1 leid ler2an lebi ax-r2 ) A
ADZBEFAQFZABFZEZSAQBAAAGHCITSRSJSRSSRSAKLSMNOP $.
$( [3-Jul-05] $) $( [2-Jul-05] $)
$}
$(
@( Note: This theorem is unfinished. This is the progress that I was able
to make. @)
lem4.6.6i4j3 @p |- ( ( a ->4 b ) v ( a ->3 b ) ) = ( a ->0 b ) @=
wva wvb wa wva wn wvb wa wo wva wn wvb wo wvb wn wa wo wva wn wvb wa wva wn
wvb wn wa wo wva wva wn wvb wo wa wo wo wva wn wvb wo wva wvb wi4 wva wvb
wi3 wo wva wvb wi0 wva wvb wa wva wn wvb wa wo wva wn wvb wo wvb wn wa wo
wva wn wvb wa wva wn wvb wn wa wo wva wva wn wvb wo wa wo wo wva wvb wa wva
wn wvb wo wva wn wvb wa wvb wn wo wa wo wva wn wvb wn wa wva wn wvb wa wva
wo wva wn wvb wo wa wo wo wva wn wvb wo wva wvb wa wva wn wvb wa wo wva wn
wvb wo wvb wn wa wo wva wvb wa wva wn wvb wo wva wn wvb wa wvb wn wo wa wo
wva wn wvb wa wva wn wvb wn wa wo wva wva wn wvb wo wa wo wva wn wvb wn wa
wva wn wvb wa wva wo wva wn wvb wo wa wo wva wvb wa wva wn wvb wa wo wva wn
wvb wo wvb wn wa wo wva wvb wa wva wn wvb wa wva wn wvb wo wvb wn wa wo wo
wva wvb wa wva wn wvb wa wva wn wvb wo wo wva wn wvb wa wvb wn wo wa wo wva
wvb wa wva wn wvb wo wva wn wvb wa wvb wn wo wa wo wva wvb wa wva wn wvb wa
wva wn wvb wo wvb wn wa ax-a3 wva wn wvb wa wva wn wvb wo wvb wn wa wo wva
wn wvb wa wva wn wvb wo wo wva wn wvb wa wvb wn wo wa wva wvb wa wva wn wvb
wa wva wn wvb wo wvb wn wva wn wvb wa wva wn wvb wo wva wn wvb wvb leao1
lecom wva wn wvb wa wvb wva wn wvb coman2 comcom2 fh3 lor wva wn wvb wa wva
wn wvb wo wo wva wn wvb wa wvb wn wo wa wva wn wvb wo wva wn wvb wa wvb wn
wo wa wva wvb wa wva wn wvb wa wva wn wvb wo wo wva wn wvb wo wva wn wvb wa
wvb wn wo wva wn wvb wa wva wn wvb wo wva wn wvb wvb leao1 df-le2 ran lor
3tr wva wn wvb wa wva wn wvb wn wa wo wva wva wn wvb wo wa wo wva wn wvb wn
wa wva wn wvb wa wo wva wva wn wvb wo wa wo wva wn wvb wn wa wva wn wvb wa
wva wva wn wvb wo wa wo wo wva wn wvb wn wa wva wn wvb wa wva wo wva wn wvb
wo wa wo wva wn wvb wa wva wn wvb wn wa wo wva wn wvb wn wa wva wn wvb wa
wo wva wva wn wvb wo wa wva wn wvb wa wva wn wvb wn wa ax-a2 ax-r5 wva wn
wvb wn wa wva wn wvb wa wva wva wn wvb wo wa ax-a3 wva wn wvb wa wva wva wn
wvb wo wa wo wva wn wvb wa wva wo wva wn wvb wo wa wva wn wvb wn wa wva wn
wvb wa wva wva wn wvb wo wa wo wva wn wvb wa wva wo wva wn wvb wa wva wn
wvb wo wo wa wva wn wvb wa wva wo wva wn wvb wo wa wva wn wvb wa wva wva wn
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
wvb wa wva wn wvb wo wo wva wn wvb wo wva wn wvb wa wva wo wva wn wvb wa
wva wn wvb wo wva wn wvb wvb leao1 df-le2 lan ax-r2 lor 3tr 2or wva wvb wa
wva wn wvb wo wva wn wvb wa wvb wn wo wa wo wva wn wvb wn wa wva wn wvb wa
wva wo wva wn wvb wo wa wo wo wva wvb wa wva wn wvb wo wva wn wvb wa wvb wn
wo wa wo 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 wn wa wva wn wvb wa wva wo wva wn wvb wo wa wo wva wn
wvb wo wva wn wvb wa wva wo wa wva wn wvb wn wa wo wva wvb wa wva wn wvb wo
wva wn wvb wa wvb wn wo wa wo wva wn wvb wn wa wva wn wvb wa wva wo wva wn
wvb wo wa wo wva wn wvb wn wa wva wn wvb wo wva wn wvb wa wva wo wa wo wva
wn wvb wo wva wn wvb wa wva wo wa wva wn wvb wn wa wo wva wn wvb wa wva wo
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
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
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 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
wvb wo wva wva wn wvb comor1 comcom7 com2or fh1 ax-r1 ax-r5 ax-r2 lor 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 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
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
wa wvb wn wo wva wn wvb wa wva wo wo wa wo wva wvb wa wva wn wvb wn wa wo
wva wn wvb wo wva wn wvb wa wvb wn wva wo wo wa 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 wo wva wn
wvb wa wvb wn wva wo wo wa wva wvb wa wva wn wvb wn wa wo wva wn wvb wa wvb
wn wo wva wn wvb wa wva wo wo wva wn wvb wa wvb wn wva wo wo wva wn wvb wo
wva wn wvb wa wvb wn wva wo wo wva wn wvb wa wvb wn wo wva wn wvb wa wva wo
wo wva wn wvb wa wvb wn wva orordi ax-r1 lan lor wva wvb wa wva wn wvb wn
wa wo wva wn wvb wo wva wn wvb wa wvb wn wva wo wo wa wo wva wvb wa wva wn
wvb wn wa wo wva wn wvb wo wva wn wvb wa wva wvb wn wo wo wa wo wva wn wvb
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
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
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] @)
lem4.6.6i1j4 @p |- ( ( a ->1 b ) v ( a ->4 b ) ) = ( a ->0 b ) @= ? @.
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 ) @= ? @.
lem4.6.6i3j4 @p |- ( ( a ->3 b ) v ( a ->4 b ) ) = ( a ->0 b ) @= ? @.
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] $)
$}
$( $t
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introduced in $c or $v statements. See HELP TEX or HELP HTML in the
Metamath program. */
/* Definitions for LaTeX output of various Metamath commands */
/* (LaTeX definitions have not been written for this file.) */
/* Definitions for HTML output of various Metamath commands. */
/* Title */
htmltitle "Quantum Logic Explorer";
/* 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>' +
'Home</FONT></A>';
/* Optional file where bibliographic references are kept */
htmlbibliography "mmql.html";
/* Variable color key */
htmlvarcolor '<FONT COLOR="#CC4400">term</FONT>';
/* GIF and Symbol Font HTML directories */
htmldir "../qlegif/";
althtmldir "../qleuni/";
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htmldef "term" as "<IMG SRC='_term.gif' WIDTH=32 HEIGHT=19 ALT='term'" +
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/* Mladen wants the turnstile to go away 2/9/02 */
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htmldef "=<" as " <IMG SRC='le.gif' WIDTH=11 HEIGHT=19 ALT='=<'" +
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althtmldef "b" as '<I><FONT COLOR="#CC4400">b</FONT></I>';
althtmldef "c" as '<I><FONT COLOR="#CC4400">c</FONT></I>';
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althtmldef "g" as '<I><FONT COLOR="#CC4400">g</FONT></I>';
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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>';
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althtmldef "m" as '<I><FONT COLOR="#CC4400">m</FONT></I>';
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althtmldef "c2" as '<I><FONT COLOR="#CC4400">c<SUB>2</SUB></FONT></I>';
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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>';
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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
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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
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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
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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
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FVTVHGIWPVCGIUVSVCWFUSVRUVRUWBUWAUWCUVRUUSUVPUDZUWBUVPWIWHIWNHWDVAUWBUWKJ
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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
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GVNFEUUCUOUHUUDYTFEDHVPUMVEUUAUUFWOUDZXAUUGUUAFEWOUOUPUUFWIWOFEVOURUHVGUL
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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
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VRWSWBUOUIUJWSWBGDGVDUQWCWSUOVLWSDWCDGUNVSUMVBVMUPVHVB $.
$( [11-Apr-2012] $)
$( Part of proof (4)=>(5) in Day/Pickering 1982. $)
xdp45lem $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
) DEUDZGDMUDZUEZHUDZUEZEFUDZHIUDZUEZDFUDZGIUDZUEZHXAUEZUDZUDZJKUDXLUDZXEX
IXCIUDZUEZXFXLUDZXGUEZUDZXNXEXQXFDEHUDZUEZUDZXGUEZUDZXTXEDFYBUDZUDZXPUEZE
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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 ) )
=< ( ( c0 v c1 ) v ( b1 ^ ( a0 v a1 ) ) ) $=
( wo wa ancom tr leor leror leo le2an bltr df2le2 cm anass ax-a2 ror
or32 lan ler mldual2i 3tr orass orcom lor ran le1 leran lelor an1r oridm
lea mlduali lear lbtr letr dp34 lel2or lelan mldual ml2i leao1 or4 2or
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????????????????????????UPZUQ???URZUGUSYPUG??????YRUTVA??????YP???YSVAUGU
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??VLZVMUG?????VNZUIZUL???UUB?????SUQUNUGVOVPULULZUMUNUUEUG??????????????V
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$)
$( [11-Apr-2012] $)
$( Part of proof (4)=>(3) in Day/Pickering 1982. $)
xdp43lem $p |- p
=< ( a0 v ( b0 ^ ( b1 v ( c2 ^ ( c0 v c1 ) ) ) ) ) $=
( wo wa leor leo anass tr lan cm leao4 bltr lea orcom lbtr mldual2i ancom
ler2an ror lelor letr lor lear ax-a3 ax-a2 2an leror le2an df2le2 ler 3tr
or32 orass ran wt le1 leran an1r oridm mlduali id dp34 lel2or mldual ml2i
lelan leao1 or4 2or mli ml3 leao3 le2or 3tr1 df-le2 or12 orabs ml3le 3tr2
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$( [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 ) ) ) $=
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ror lan ler mldual2i 3tr orass orcom lor ran wt le1 leran lelor oridm lea
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ml2i ml3 ler2an leao3 le2or 3tr1 df-le2 le3tr2 or12 orabs ml3le 3tr2 bile
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$( [11-Apr-2012] $)
$( Part of proof (4)=>(3) in Day/Pickering 1982. $)
xdp43 $p |- p
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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
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VVKUYQUQUHUFWGVVPUVMVVHVVKUAZUAVWDUVMUAUVMUVJVVHUVLVVKWFUVMVWDULVWDUVMVWD
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AVWMUUGUUGUUQVWKEEVCZXQVWJDEGVCUGWGEEVWKVKUFVWLVWHEEDGWPURUJVWIUUNUUNUUNV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 $.
$( [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] $)
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