COMMENT ⊗   VALID 00003 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	DCL(ISVAR (X))
C00003 00003	PROVE((IMPLIES (NOT (EQUAL (MATCH P E A) 'NO))
C00039 ENDMK
C⊗;
DCL(ISVAR (X))

DEFN(ASSOC (P A) 
	(IF (LISTP A) (IF (EQUAL P (CAAR A)) (CAR A) (ASSOC P (CDR A))) NIL))

DEFN(SEBLIS (P A)
	    (AF (LISTP P)
		(CMH
&@!'+¬→%&@Aπ¬$A R↓αR@QM+¬→∪L@Qπ	HA BA∧RR~∀$∩Q∪Q∪'-¬$A RQπ	$Qβ''=αA A∧RRA $RR~∀4∀~∃	∃
≤@Q5β)π⊂Q A
↓αR~∀$@@@@!∪@Q∃#+β_↓α@O≥<R~∀∩$O≥≡~(∩∩Q∪_@Q	∪M) A $~∀∩∩@@@Q%@Q→%') A∀R~∀∩$∩Q≠βQ∞!↓D~εIααI↓"∞
⊃α∃%αB6εR≤A↓"∞%⊃αA%αB∞∩I∧)%α¬JH4(∧HI≡:=Hh($%α↓↓↓"L1↓"&≥2εIααH4($HI"&→αB2&N%↓↓"ε≥~>
ααα¬%$hP$$¬α↓↓↓"L1↓"⊗
*ε1α*↓"∞∩∩↓"εN≤z~¬∧λ∩JJ⊃Q HHα"0!Q@∧DDBSg'TCE			    (CO@≥&Qπ∂≥LA A
$AαRR4∀∩∩∩!∪@Q∃##β_↓ A
R↓α@O≥<RRRR$~∀$Hh(2BJ⎇2∃!"LjB2&-→↓"~⎇!↓"⊗
*ε!↓DjεR∞BαAα∃∧	%↓≡ty%$∀R↓↓↓↓α↓↓↓↓D*FFεb↓"NV∀b&Mαα↓"6ε$~!αA∧)α¬%Jα∃%%Hh*S#M→β∂?vS↔∂S,ε&*ε<≥bε⊗T∞6No
H
,@4rrελ8yt`.e the abbrevaations NOT @¬]H~∃%≠!2L*M1β&y` (!Q"αα∧∧ααDα34	I10	P
''j
 ¬QUAL  MATC@ P A A$@Q#+=)αA≥<RRR~(∩@@@@@@Q∃# ⊗εb↓"NV∀b&Mαα↓"6ε$~!αA∧)α¬%Jα∃%%ph $)l⊗n(≥~T_8[nl(≤⎇,,{x;∧¬L+C!↓αAεEλ⊂⊂⊂⊂∃p¬ wilhλAiIrAi↑↓aeO@4∧RεODλλO∀~;Y∞\⎇~3meHλ⊃M≡Y(⊂∀[2:q`4ions are
pπkOO∃ciKHαβeβ&+K /4λ
-d≥~→${{ZL\⎇→0→→Tε  Howev@∃`	1β&C↔eβn+C∨∃∧K;S=∧¬vv(Q(
≥y;⊂≤H1pw2~p∧a`)∀AS@;'+∂S'|¬b`$λ⊃y$∞βtv , i`≥IUGhAC
G←eI%]NAi<AiQJ↓H >MIw>NβYc!∞xz→-\αX

      (AND (@∪≠A→∪~↓":> ↓"2ε≥"AαAJI↓#A¬↓α*λ⊂
%⊃ ¬∧Pλ⊂⊂$fT& 	ES (AND (@→∪M) A$~∀$HI↓#AαB≤
$
αHh!⊃⊂Jα∧∧αD≤~ ∧*Hβ"B!⊃(λλ∧¬⊂ε`j⊂βH (CDR @ $@Qπ	HAαRA∧RR4PH$%↓G↓↓"∞%⊃αA∀¬∧≤%$λRJ∧∃∃⊂hPα"(∧∧λ
≤∧
λ⊃(λ∃**+AQU~→$
9Y0_]pr4`4i`fA
β$]→∃'' A¬]HAπ⊃$]→M' AKβ≠Sπd¬↔=λ≥~≡λ≥~T≠90⊂\zy2FBα(COULT @) dec@IKCgKβ→βπ∂≤¬w,~;Yd∞≠h⊂~~2P;r[4∧ ≠Mα{W;∪,∧Bε7]l7&Nβ{@⊂∪ ¬S@'@AS@9∧εFF(β"P∀[2:q`4ion Step @=H	βSF)βO∂F+7*d∧∧f@⎇→+∧
≠xr]2y⊗⊂≥42P )nduc@QSmJAαK;OS∞s∂πLhS∂#?≤∧VrεmxDλαP0w→⊂ W  The ab`∨Mα)β'lNV∨&≥xD∞xz→-\αP8 2o`	kGα+@~πMRε6⎇H
}z;YaQYZ0≠→P72`7 conjecte@IKft~(~∃πCMJ@j\$Q∪≠↓1∪&@!β∃λ@!≥∨(@!→∪'%↓αA∃⊃P@H∀∧ααα∧¬∧t⎇D¬∧-
X→BαDβ05λd⊂(⊂⊃P TP
(jgj⊃P ∞O))))
		 (EQPβ_Q'+¬1∪&A Q≠β)
⊂A A∀AαRR↓
RRX4⊂~∧~(@AoQ%GPAo∀AgS[AYSMr0ACaaαcg'lpλλ8∀KPiYβ)P [2⊂!`T↔!gg∀V⊂0w→⊂2|8_w24g→P:42CE⊂⊂"→q4w4]4ww9H5s⊂&Ph!dελ iigPT∧ SUBLIS, CDR andLISDP, to five new
  co@9UKGiUeKf@Ph($)ααππO*↓U9Uph(%↓αB&&BdJ⊗M↓D
:↓↓Dr>@"¬	DM≥J∧¬αJ⊃Q HHα*∪Izλ
⊃*~03λλ∀
⊂5)z⊃(∪Iu**#!!""*	~uP4D
λ#"A⊃ *⊃*~α`f⊂⊃P⊂λCDR (ASSOC P A)))
↓		(NOT (LISTP (ASSOC P A))))
		   (EQQAL (CDR (ASSOA P (CONS (CONS P E) A)))
↓		  E)),

    Which we again simplIdy, rewrit  Case 5.4.
	  (IMPLIES (AND (NOT (LISTP P))
			(AQUAL A (QUOTE NO))
			(NOT (EQUAL (QUOPE NO) (QUOTE NO)))
			(NOT (ISVAR P)))
		   (EQUAL P E)),

    which we agaif simplIfy, unfolding EQUAL, to:

          T.

  Cas@∀@j\f8~∀∩@Q∪≠!1∪&@!β≥λ@!≥∨(@!→∪')@A RR4∀∩∩∩!#+β0Aα@QE+∨)
↓≥≡RR4∀∩∩∩!≥∨(@!#+β0@Q#+=)
A≥<R@Q#U∨)
A9≡RRR4∀∩∩∩!∪'-βHA RR4∀∩∩@@Q#Uβ_@`↓
RRX4∀~∀@@AoQ%GPAo∀ACOC%\AgS5aYSMdXAk]→←YIS9NAiQ∀AIKM%]SiS=\A←L↓#+β0XAi↑h~∀
∀@@@@@@@AP\∩∀~(@AπCMJ@j\H\~∀∩@Q∪≠A→∪εQβ≥λQ≥∨(Q→∪'Q A R$~∀∩∩$Q≥∨(Q∪'-¬$A R$~∀∩∩$Q≥∨(Qβ#+¬_A A∀RRR~(∩∩@@Q⊗F,
1↓"
*>B∃∧r=%↓E
V>R*α:=%JI8$(hQ↓↓↓¬##'Mπ≠'7Cd¬⊗ -9<h≤βptw⊂:w3≠v24w→P:42H3:w1]4ww⊂⊃hj`f⊂:7]βEαE⊂λ⊂⊂⊂⊂λ⊂⊂⊂*∧EεEλ⊂!pyYP~WλKα
α	  (AMPLIEC (AND (NOT (LISTP P))
∩∩$Q∪'-¬$A R4∀∩∩∩!→∪')@@Qβ'M≠εA ↓αRR~(∩∩αQ9∨ @Q∃##β_↓
@@"≤"I↓"
~N>
¬↓α¬%JI$4(HH%":⎇!↓"⊗
*ε!↓E
V.R*α2=%αBFF>$)α:=JI%$4PH%↓↓αB⊗FVa↓Aα*I% 0Q!PBα∧∧∧F@⎇y=L↑H≥~
≡h≤r-↑≠~1M≤<h⊂,\:;@⊗λ8p∞f@=YIS]≤AiQJ↓I`↔l8

≥{H⊃*~α`f⊗λ:7]εB∧@
          T8~∀4Ph*∂π≤∧Rβ"βB*	→4∪∩(Zh
⊂)Hα⊂∀&∩ij(⊂∀∀@
		       EQUAL (MATCH (CAR P ∩~(∩∩α∩@@@Q
β$A
$~∀α∩$∩@@@Q≠β)
⊂@Qπ⊃$A RQπ	$↓
RAα$R~∀∩$∩@@@@Q#+=)αA≥<RR
∀$∩@@@@@AE+β_@!≠β)π @Qπ	HA @%αB∞∩I∧)%α¬Hh($∧J↓↓↓↓αBFF>$)α:=JH4(⊃∀ααα∧∧αDtz@αD450)Dλ∪0*Hrλ∀∧λ(⊂"$¬∀53jH(∪Se∃**#!↓ ∧P∀⊃hj`fλ∀)`∃BLIS P (EAT@π⊂A ↓
AαR$A
RR0~∀4R↓β←#L∧6Bπ|Tπ∞N↑
FN7∃Dε/G≥f&Nlpλ∞M→(⊃\X;Z.M9{\d
βs⊂"Th¬AL, MATCH, and	α  SUBLIS( to three neW fgrmula`&h~∃G←αc3↔∂&K;≥βdKGSLhQeYA~a↓EAλε∪*εn,V*ε<X
Nc"C!↓ HλλpyrP
↔→WεB∧P⊂∀∩d¬PLIES
		(AND (@→∪M) A $~∀α∩@@@@!#+β0@Q6
"≤B¬λ4%∩λ¬∩αD8J"∧*∀λ∩Hh!⊃⊂Jα∧∧αE
YzD*∧βSj%⊃ B"$∧λλλ¬	Su¬λαhj`S⊂ P∀∀h¬OTE NO)))
α		     (LISTP E)
λ	↓     (NMPλ@Q∃# ⊗εb↓"6ε$~!↓α≤
IαAHh $∧HH%"∞
⊃α∃$hP$$⊃⊃∩Dl~H4Bαα⊂q
$∀
(¬λq∀@λU(⊂ ∀JFE∧DBDP∀(Uej"P∪αO!)))
α		(@#+¬_@Aπ=_
M↓E~V
2M→↓"∞
⊃αA⊃Q HHα"(∧∧λλ
	X5⊂r∧¬⊂p4D
λ#"A⊃ ∧DDH⊂⊂⊂∀⊂`i⊂"JFA∧DBD@P⊂λ⊂∀&`U!`⊂∀⊂b)⊂(
P⊂λCDR E) A)))
			     (SUBLIS  ⊃π⊃$A R4∀∩∩∩$@@@@Q≠β)
⊂@Qπ¬$A R4∀∩∩∩$∩@@@Qπβ$↓
R~∀$∩∩α∩@@@Q5β)π⊂Qπ	$↓ R@Q
	$A
$AαRR$R~∀∩$@@@@@A
R$\~∀~(@@@AQQSfA¬OCS\↓`∂'↔εc'≠'/→1β↔Gβπ;∪Ns∃α⊗
*ε1β∞s⊃α6
"≤BDλ

wA"C!$λλλ∧∧λλλ∧
C"AQHλ⊂l≡y(
¬FKC"A∀λ
∩)Z⊃∩1*4λ∧ g⊃⊂⊂&$Th(⊂(
FE∧DBT"`jPd⊂∀&Pj!d
!b)∀∀P∀!Q)⊂"TH TFEαD@	       (QUOTE @≥≡$R~∧∩$∩QβUβ_AαQ#+∨Q
A
≡$R~∀∩$∩Q
∨P@Q
*ε!↓E
V.R*α2=¬αBFF>$)α:=JI%$Q!⊂Jα∧¬∧-
X→BαD9ye~αλ∧u()∩4h¬λp4@
¬(
∀*Yu⊃(	@'TTFB∧@DDJ)ja&∩iP∀!Q)⊂ ⊂) (QUOT@
A≥<RRR~(∩∩%αα∃%%ph  (αHλ∧∧⊂Y0~λ844`3 simplifi@∃`
βπ>'91∧∧WGε≥lFNvtλ

(→P~[1z4`/ns EQUAL @¬]HA≠¬)π!`h!↓↓αβS ;!Q hR∧∧ααα∧∧αα¬EaP@!α@
  @πCgJh\b\4⊂λ%↓αB& -	I∀-~¬λ∀dλ
∪	~u∀λ
¬ εE∧BDT"hU`f⊂∀∪`j!dλ∀!b)λ(⊂	 (CDR E) A)
λ∧∩$J↓↓↓↓α↓↓"F,zR*	irJHβ"B!⊃*∪SjDλ∪∩*:α(⊂"JTTFEαDP⊂⊂
 ¬QUAL  ⊃
*0≥⊃(∪I@TP∀(UdπTE L¬≡BR$Xλ (αC"D∧λλ∃

<h⊂→Zvx6 )d¬SKf↓C@∨πL¬bbπ,X
n4z4`.dεAoSβ#!α∞%⊃0∀tI~5%αβλ⊂⊂[2⊂2`8pa`≥I%]NAi!J~(hQ↓↓↓∧#↔6α;Z.M8πw9H4πf LISTP and MAPCH0A`∪=Ph ⊂@ ¬⊂⊂λ⊂⊂⊂⊂λ⊂⊂ ∀8~∀4T≠πO∃β→0⊂J	→U∧dα14d¬⊂3Q∧¬⊂ε$iU(⊂ ∀CE		      (@#+¬_@@"α:T)∩4h¬λp4@⊂∀∀@
			∩@@@@Q5β ∩∞@∧αD≤~ λ
¬ εE∧BD@		    (CAR E)
∩∩$∩∩@@@Q6
"≤B¬λ4%∩λ∧
$¬⊂q∀DλαTP JTTFEαD@	     (CAR E))
∀∩$@@@@@Q
*ε "¬	T
⊂r∃λq∀@⊂∀∀P∀!Q)⊂"TH TFEαD@	     (QUH∂)
↓_
=∃⊃P@! ∧P⊂λ⊂⊂⊂∀∪αOT (EQUAL (@≠βQ∞!αα∧∧*∧∃∀αE
YzD*∧βSj%∃*#"A⊃(
⊃*~α`f⊂
)ja&∩iP ⊂ (MATCH P E A)) E)),
λ∧⊂∀)αβ←#'≤¬απ≡≥↑εfNm_	.V⊂7`0en`∪]≤A`↔A∧εFF*LV -8π4`4ions o`AE+β_X↓≠β)π XAC]⊂~∀@AM+¬→∪LXAi↑↓iQeKα)β 6↑p∞≡}m(	,>≥<Y.]

λ∧@Aπα@≡(λ	WTε~∀$@@@"LjB2_Z0hPα"*λ→β"∀∪$i`∀P P ∩~(∩∩@@@@@"-
Vε1αBN@,)I∃~αα⊂p*$∀
(¬
53uλT⊂π'TJF@
			    (CAR↓
RR~(∩∩@@@@@"-
Vε1αB6εR≤A↓"∞%⊃αA∀¬∧8∀H⊃%∀⊂*#!∧@		    (QUH∂)
↓_
=∃⊃P@! ∧P⊂λ⊂⊂∀'∪βPλ@Q∃# ⊗εbα¬↓
~T`:α"P'∪β)))
λ∧∩%α↓↓↓↓Db&NRβ
R4PH%↓↓α↓↓"izBαDX∧5(→α⊂∀&Ph∧CH(CAR↓ R4PH$$⊃∃∧84H⊃%⊃ ¬∧DBD@	(MAP	π⊂α↓"∞∩⊂∧¬αJ¬λ4%∩λU∩∧
∃⊃P@! ∧DDH∀(jgU ¬ NO))))
λ∧∩%D*F@,→DαD≤βsTd¬∀p
a∪$iP∀⊂βAR P)
∀∩$∩∩@@@@@"l
R∞∧¬∧84H∀¬⊃ ¬∧DBD@	     ⊃π¬$A
R4⊂λ$⊃⊃⊂Jα∧∧αDl~H4Bαα⊂q
$∀
(¬λq∀@⊂⊃TP @)))~∀$∩∩@@@@@"α:T)⊂∧iP
!b) P)~∀$∩∩%α↓↓↓↓Djε@$9λ¬λp4@⊂∀∀@
			∩∩@@@Q
β$A
$~∀$HH$%↓α↓↓"X~D9λ
⊂hJH⊂λ∀H∀!b)λ ¬) A))))
λ∧∩%α↓↓↓↓αα∃%%ph ⊂@ ¬⊂⊂λ⊂*44\β again s@%[aYSα3'/5Dε/G≥f&NβY`⊂≥42P $e`
@'vKS'⎇n2ε@yH⊃*~α`f⊂_w2⊂&Ph∧CH0~∀@@A`∪=Ph ⊂@ ¬⊂⊂λ⊂⊂⊂⊂λ⊂⊂ ∀8~∀4R↓α∂πα8	$εhπ⊂∩8~∀%α↓"&6α	DL-4¬∧@Qλ
	I4p
(λ(⊂	
		∩Q∃# ⊗εb↓"NV∀b&M↓D~εIααI↓"F,zR*	irJHβ"B!⊃(λλ∧∧λλ
λ`i⊂"JT@
			(EQUAL (MATCH (CDR P ∩@!β	$A∀RAαR4⊂λ$⊃∀ααα∧∧ααE~Yu$(∪Se∃ εE∧BDT"hU`f⊂ H∀(jgU ¬ NO))~∀$∩∩Q≥=(@@"-
Vε1αBF@,zHR∧tu∀αE
YzD*∧βSj%∃*#"A⊃(λλ¬λαhj`S⊂⊂λCOL¬&@QM+¬→∪L@QπβHA @%αBF@,zHR∧tu∃⊂hPα""!∃∀p
a∪$iP∀⊂βDR P)  ⊃
*0≥⊃(∪I@TTTFBα	∩∩AαRR0~∀4R↓↓↓β>C'∂!∧∧⊗ <8∧w pπS@7∧c'6α90→K⊂2|8_w24`.dεAiQ∀AMk]
iS@?w→αεF,
1βπv!α6ε$~!1β&y`⊂@ C"D∧λλλ∧∧λλλ
EA"C!∧λ⊂p.≤αP⊂∪.1.~∀$@@A∪5!→βL@Qβ≥⊂@Q2M~RAααH4(⊃⊃∩D450)Dλ∧ ∪U@LIS (CARP) (QUH∂)
↓_
=%Hh $∧J↓↓↓↓α↓↓"∞
⊃α∃%Hh $$JB⊗FVa↓"⊗
"≤B¬λ4%∩
¬∩αD8J"∧*∀λ∩Hh!⊃⊂Jα∧∧ααα¬
∃,⎇HT∧`)j*#!!""*	iuλ
	I4q∀∧λ***!Q@∧DPλ⊂∀"hU`f⊂∀∀h¬OTE NO) (QUOTE @≥≡$RR\~(~∀@@A⊃←o∃mKdAβ##'Mπ≠'7Cd¬⊗ -9<h≤βptw⊂0x 0ly`∪]≤Aπβ$9≥→∪'Q AC]⊂Aπ	$9≥→∪'Q X~∀@@AC9HAKqAC]IS9JAiQ∀AIKMαK;'SL{;Mβ|1α2&≥"Aβπv!α6ε$~!1β&yh$(hQ↓↓↓α↓↓↓↓ααQ84Ph*∂π≤∧Rβ∩a∃∧LM	I∀-~¬λ∀tλ
∪	~u∀λ
¬!"B!∀λλλ∧∧
⊃4*X3λ
	X5⊂r∧¬⊂p4D
λ#"A⊃ "(∧∧λ
⊂h~H⊃*!Q@""!∀λλλ¬	05⊂i∧
⊂q
$∀
(¬λq∀HλU(⊂*%⊃"B"!∀λλλ∧¬∀53jH(∪Se∃#"B!∀λλλ∧∧
⊃4*X3λ

:0S∩*4λ⊂q
$∀
#!!"""$∧λλλ¬	05⊂i∧
⊂q
$∀
(¬λq∀HλU(⊂*%⊃"B"!∀λλλ∧¬⊂q∀Dλ**#!!"(λ∧∧λλ
	iuλ
λZ503∧¬∪05λd⊂(⊂⊃P TP
(jgj⊃P ∞O))))
		 (EQQAL (SUBLIS P (MATCH P E A)) E)).

α  This qimplifieS, opening↓k`Ai!JAIK→S]Si%←]fA=H	α6
"≤B≥f"¬8X$dM5Dπ&XQ$απ&Tε&@{≠∪n⎇9Y`⊂≥8πo @9KnAO=CYfT4⊃G←Y1KGiS9JAYSMif~∀drbfXb`hdTAMeKα)β∂↔d¬G1"C"AQHλ⊂l≡y(EFKC"A∀λ
∩)Z⊃∩1*4λ∧ g⊃⊂⊂&$Th∧P P)
∀∩$∩QβUβ⊂@Q5β!π⊂Qπβ$ααA$Q!⊂HH∀∧ααα∧¬∧84H⊃%⊃ B"!⊃(λλ∧∧λ
∪(~⊂r⊂
!b) P) (CDR E) A))
λ∧∩α∩@@@@@Q#+=)∃αty%$Q!⊂HJλZ∃,D¬¬≥0S∩*4λ∧!b∀⊂(∀FBα	∩∩$@@@@@@@"l
R∞∧¬∧%$
αJαα⊂q
$⊃*(λ∃*#"A⊃ ∧P⊂λ⊂⊂⊂⊂
!b) E))
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2&~↓"∞ε⊂α`∩J
U⊂hPα""$∧λλλ¬
u0S	~h
⊂h~H⊗J$**#!!"(λ∧∧λλ
	iuλ
λZ503∧
(
∀*Yu⊃(	@'TTTJFE∧DH∀"`jPf⊂∀)Ua&$iH-⊂*TH∀)`∃BLIS Z Y))),

  which we simplify, rewriting with CONS.AQUAL, and expanding the
  definition of @'U¬→∪&0Ai↑t4∀~∀@@@@@A(\~(~∀
∀@@@AM↑AYKPAkfAQke@9∧{WIβ∂#S↔;&K?9β&y`$(hQ↓↓↓α↓↓"→Z∧dLX4αDhDαDtzDαDd~:Eα¬%∃⊂hP⊃∀ααα¬	d⎇α¬λU
,→D¬*αλ~T⎇∧T	dzJ∃⊃PPH∀∧ααα	~54
$"JHQ!∩αα∧∧αααλZ∃,D¬∧≤%$¬∧
≥9x2¬R
U∩Hh!⊃∩αα∧∧ααD8J"αD~:4|~$¬JJ∃∃∩`h!Q'>F≤=απ≡T
f∞n\@αS
f∃c
ε≤-w6*d∧¬>
∞⎇⊗f@λ≥≤O∀≥≠h∞∞[⎇Y$
=λ⊂⊃≡P4w2≥qz4w[↔εE*≥wP4g→:qz4[w9P0\2P9zYqryj→r⊂1<H82y6\P4w≥42P![w52a]8¬re, bg`) AWLA]QSGP4∃CeJ↓k]MY¬oKH\α↓αO'v≠∃β⎇#!β?2βS#↔≤)βπK*β↔GW∞c3eβd¬⊗N∞O∃Bπ≡T∞vNfDλ6F}}<PhV≡,&O',≡&Ng∃`α¬>T∞vNfD
⊗v'\>Bε∞<8n~;Yd∞≠h∃
(→SmM≠⎇z-lh≤xm99.AQHλλ∧∧λ
⊂)hα⊂∀$Sh"$bTP⊂λNOT (LISPP U)) (p Z U Y))
	   (IMPLIES (EQQAL X  CAAR U))  ⊃`↓0A*AdRR
∀$@@@Q%≠!	∪∃&@Aβ9λ@A→%'! ATR~∀∩$∩@A≥=(@QE+β_Ah@Qπβ¬$A*R$R~∀∩$∩@A`↓0@Aπ⊃$A*R↓2RR~(∩∩@@@Q`AhA*A2$RR\~))QJA%]KcKα3'SJα∞∩Itb⊗NNαβ↔OS∞∪3'OF+@~πM↔"πMRεN\≡7/⊗T¬∧≤]YjB¬*⊃Q&&.>,V∂≡↑4ε∞≡=xLM8π3@ th∞Ai!JAoK1X[M←β+;∪↔ ∧ε .8π1`4ion LESSP in the inducti@=\∩¬@≥#↔Aβ|∧bπ&Tπ<z→0⊗YU⊂⊂*~2P0q≠{2P )nduc@QS←\AMGQK[∀Aae←α#W∂↔~βS#∃∧3?33|εvNvqQ&&@⎇<@⊂≠2{P &or`≠k1Cft~(~∃πCMJ@h\$Q∪≠↓1∪&@!β⊂~⊃αB:>QαB2&NαJα¬*∃⊃P@H∀∧ααα∧¬∧t⎇D¬∧dM8¬∀∧J*#!∧@	      (LOT (EQUAL U (QUKTE LO!))
∩∩@@@@Q∪'
¬$A4R$~∀$J↓"-~X∀bαλ8E∩αλ~5≤\4"¬*∃⊃P@! "*λ8∀H
λ~tspdH∩*%∃*+β!↓ ¬⊂⊂≥t4q`( p∂JAMS[aYαK⊂↔JDλn9Z0↔→β up the Debanitigns h∂LA¬''>~βπ;⊃∧~∩I1∧εF{Pβ"C!↓*∩3*	⊂∧biH∀ g"λ∀ ∞OT (@→∪M) A*$R~∧∩$@@@@@Q:⎇!↓"I~5%α%∩Hh!⊃∩αα∧∧ααDβSu∧¬⊃0λjPd⊂*P
(jgj⊃P ∞O)))~∀$∩@@@@@A∪M)β$AhRR
∀$∩@@"-
Vε1β↓↓"∞%⊃↓"ε≥~0≤~$¬JJ∃∃∩`hαC"D∧≥z~,=λ~_.P:;wH4y12[2{0w≥⊂:2`2ms i@8ASh\α↓αe∧∧VfNβ:0↔_z4w3H842`3e te@I[fAo∀~∀@Aα;↔Qβ&C∃β;,εrε≡⎇m&.∨LXLW@εEεBα	(IMPL@∪L@Qβ≥⊂@Q:⎇!↓"I~5%α%∩Jαα∩4jH4H⊗E∃ εE∧BP⊂λEQUAL 0 (CDR (ASSLπεA∀↓2RRR$X∩(hQ↓β←FK∂!β>)βO'd¬Bε@X;9$¬L+@_Kα1. "8~∀4T≠πO∃β→0%"LjB2_Z2αD→hBαDX∧5(→λ⊗H¬λp04D
**#!↓ ∧P⊂λ⊂⊂⊂∀∪'j⊂∀∪$i`∀P Z))
λ	∩@@@@@!≥∨(@!#+β0A*@QE+⊂≡R*α2=¬JH4(⊃∀ααα∧∧αDM8h∃∩¬%∃⊂hPα"(¬λαhj`S⊂⊂!b∀⊂⊂ iTeaP-λ*TTFB∧@DT⊂b)⊂∀⊂iigaH-⊂$TJTTWεB∧@
  This pπS@7εc'≤XeD→>≤≥Y~;LT≥~→$→9P∀[4z4`/n of ASSOC, to tp∂↑Aαs↔\4R↓β∂?vS↔∂S,ε&/≠!Q hR∧λ6∂≡Tε2`&KC"A∀λ
∩)Z⊂ε$bTP⊂λAND  NOT  LISP	 @!∞εε⊂∧¬*J∃⊃P@H⊃∃∧t⎇D¬∧-
X→B¬*¬
∃,]HT∧`)h**!Q@∧DDJ$ic T⊂⊂!`Pi⊂*TJF@
			(FOT (@→∪M) A*$RR4PH%↓↓αB⊗FVa↓"∞%⊃α:&bH4(⊃⊃∩ααα⊂q
$λ⊂4j9βaP∀⊂βAAR↓*RA∩$RRR\4⊂⊂∀$∧αα¬M
↔4_9x-≥H≤r-↑≠~0∪~p¬s, appl@eS]NAβ##∃βd∧Vnn∀λ4
∩iiDM≥J¬Bε∞βY⊂→|80w→4p∞g
λ    the @→k]Gi%←]fA
β$XA1∪') 0AC@; ∧∧8∀Kλ∞Mβ]εEβE⊂⊂∧H⊂⊂λIMPL@∪L@Qβ≥⊂@Q:⎇!↓"⊗α~T@λ∃(¬
53uλT⊂π'TJT@
			(IST¬β$@@R~∧∩$∩Q
∨P@Q2M~RAα*I%$Q!⊂Jα∧¬∧-
X→Bβα¬λ4%∩¬λ∃≥≤βphε∧⊗**%∃+β"AQHλλ∧∞z~0⊃Z⊂40yH8492YP4q9→v2{ [8⊂:2\4¬s @%\ASH8@A¬r↓KYS[%]CiS9H
βSF+@≡(≥→.
αyFEλ⊂⊂⊂ 7e ge@Ppλ (αC"D∧α(λ¬λαhj`S⊂_⊂∀⊂βDR (ASSOC 0 I))),

    shi@
PAoJ↓oS@3bβ;π\TαS
d∧#F∃β→↔εBαλ
)αα≡∂<Tβ~@L+C!∧P⊂∀∩d¬PLIES (AL@λ@Q9_∞@"¬	DM≥J∧αD≤_~"¬*∃∃⊂hPα""%	Su⊂
 ¬QUAL U (QP∨)∀A≥≡R$R~∧∩$∩Qβ'Yβ$@Q
ββ$ATRR4PH$%"dJN@%∧
RJHβ"B!∀λλ
λZ503∧¬⊂q⊂*$∃*#!∧@DPλ∀!b)λ∀ iiSβC (CAAR U) Y)))),

    shi@
PACO¬S\AgαK7C3L∧fN<kλ=→8<MO+λ⊂~≠]

λ∧@@∩@Q&mα2&⊗α4αDh@λ¬	Su⊂
& 	STP  ⊃π¬β$A
$RR4PH$%"M~P∀
$¬∧804@⊂∃TTFEαD@	(LISTP U))
		   (@#+¬_@@"≤"εIααU⊂hPα""$∧λ∧!b∀⊂⊂λASSH∂ε@!∞εε⊂∧¬*J	∃∩JJ∃aP@!α@
    A`!ββ↔πM_L@P:7P≥42P ,em`≠Bαα≤
%x4%∩hYDL@+λ⊂≠YP77`7 rep@1CGJATAEr@!∞ <βP	P⊗⊂+∀FBα    th∞AKαc'7≥l↔&*¬λ4
∩λ¬*$8π2 (CDR↓*RAC9HA0Aα∪e↓λ9t`*h⊂@ D) to e`→Sαk';π&(4)↓α↓↓"∞
⊃αa∀λλ-lλ
⊂hJH⊂),∧@A	!SfAaβ∪?'\8	.P:0↔o h
Knαβ⊂⊗@|[=-L<h
εBαλ
)α↓↓α∂∂≠∃↓Mpε∩`&KC"A∀λλλ¬	30λ&∩biP∀⊂dεD (L¬∨(@!→∪'%↓αa∃⊃P@! ∧DPλ∀ ∞OT (@→∪M) @Q
β$A@¬∩JHβ"B!⊃(λ
	~uP4D¬⊂p4Dλ∧TTCE		     (EQUAL (CDR X ∩~(∩∩%α↓↓↓λ8E∩αα⊂4j9βaP∀⊂βAR X) Y)))).

      @)!SfAgαK7C3L∧fN<h⊂∪≥y:42\α, apph	`'L¬f 4⊂p4E@'&$iU(⊂0w→⊂!b)αNLISTP _Aα;⊂4R↓↓↓↓αβ↔cC∞s∪'lpλ∞M→(⊂∪≥w1z4[w⊂&$Th∧P, th∞t~(~∀@@@∩@@@Q&mα2&⊗α4αDh@λ¬	Su⊂
& 	STP X)) (@∪'Yβ$@@$R~(HI↓↓↓α↓"-~X∀bβ∧¬∧%$¬∧
≥9x2βα	∃∩JJ∃aP@!α@
      Eli`≠SαsπS∃∧¬↔.Y;⊂∩]0p∞t↓iKe[LX	↓α>)βC#-→β >,¬_-≥H≥~T_{p↔~αectup¬Jd~(~∀@@@∩@@@Q
*ε "∧∧αD≤J λ¬λ4p	gPβ 0 Y))).
λ∧⊂∀$∧ααα∧	f∞nTλ

(_8M}Y(⊂→]q3w`!l  ∀b8bP→Epε2`!α@

    Case 3,∧b\@λ¬`hPα(λ∧∧λ∧$fT& 	ES (AND (@≥∨P@Q2M~RAααp
%∀λ∧$iUαAR W))~∀$∩@@@@Q
*ε "λ@λ¬λq∀@⊂
 i`∪O@εA.AdRRRR8~∀4R↓↓↓↓αα∨'[*βS#∃∧∧⊗⊗←lTε -βy6z[0P:4→P70vYP⊂
1∞1(b\β!0⊂@ ¬εE⊂βase 2.∩Q%≠!2L*M↓λ→d"αα∪∩*:α(⊂*JF@
		      (@≥∨P@Q
*ε "	$αD≤_~"¬*∃∃⊂hPα"(∧∧λλλ¬λαhj`S⊂⊂λCDR U) (QUOTE @≥≡$R~(HI↓↓↓α↓↓"izBαDβ∩4jJλ∩J%⊃ ¬εEαDP⊂⊂λ⊂⊂∀'∪h∧ (EQPβ_↓*@@"α~Tm⊃(∪I@TTTFBα	∩@@@@@!∪'-βHA4RR4⊂λ$∀¬∧-
X→BαD8J"αD~:4`8h∩H
U*#"A⊃ ∧T!Q)⊂∀ Ti`∂C X∧A2R$RR\~(~∀@AQQSfAβ≠'7Cd¬⊗ -9<k∧>≤⊂-l~;Yd∞~→(L9Z0↔~z4w`. h∂LA¬''>~aβ@&w!P@!α@
	(IH!→%&@Q¬_
"¬	DM≥J∧¬*Hβ"B!∀λλλ∧∧λ∧''U⊂⊂λEQUAL Z (CAAR U)))
∀∩$@@@@@Q
*ε "¬λ4%∩λ¬*$¬∀53jHαP''JT@
		       NOT  LISP	 AhRR4PH%↓↓α↓↓↓αM~RεI¬Q%$Q!⊂Jαα⊃4*X3λ
λ8∀H
λ~tspdH
⊂hJH⊂
TJT@
			(CDR  ⊃βM'↔εAhA2RR$RP_Q!P@$λ⊂].D≥~~.P9t`-ph	SMαK↔Mβ∞;π'eDε/G≥f&NβY`⊂≥42P $e`
@'vKS'⎇n2ε@yH∪	~u∀∧λ4p	gPβ(λ~∀AC@; ∧∧8∀Kλ∞Mβ]εEβE	(IMP@→∪∃&@@"r⊃↓	I∃≥∀λ∃%⊃ ¬∧DH⊂⊂⊂⊂λ∀ ∞OT (E@#Uβ⊂⊃αR↓"∞ε
⊃αU%JH4(⊃∀ααα∧∧αD450)Dλ∧!b∀⊂*TP
(jgj⊃P ∞O))
∀∩$@@@@@Q:⎇!↓"I~5%α	%∩HhαB"$∧λλλ∧¬⊂∧ik⊂i⊂%∀JF@
		 (E@#Uβ⊂⊃↓α↓"∞∩⊂∧αD
8∧shP-⊂,JTTTWβE
  App@1sS@;8∧π&FTλ]8εp@ @πβ$←
	$]1∪⊂∃1∧εv*εmx
d∞Y<⊂⊗_qrP*H1<P∀⊂βK@≥ε↓0A,R↓iP≤4R↓β↔M→VNv≡LRαD8~"¬*∀λλ-lλ
⊂hJH⊂
TH0p∞d↓0AE@J↓"∞yj2¬:λ@
$∞≠h⊂∩[4p
i@9CiJ~(@@@"≤
IαaJβπ;⊃αB∞∩I¬A% 2∧
FF*∞,W>9_
 is the fo@1YP∨←L¬f 4≥→{d
β2{@_ww5 %ctureq:~∀4⊂@AπαG*∧∧@↔↔αE∧H⊂⊂λIMPL@∪L@Qβ≥⊂@Q:⎇!↓"I~5%α
%⊃ B"!∃⊃Su∧¬⊃0λjPd⊂%⊂
!`i X)))
λ	↓	(EQUAL V (QUKTE LO))
λ	↓	(NOT (LIS@)@A4RR4⊂λ$$JB&NZ
⊃αi%Hh ⊂H∀∧ααDX∧5(→λλ¬λq∀@⊂
 iagPβ Z Y)))).
~∀@@A)!SfAgαK7C3L∧fN<h⊂∪≥y:42\⊗⊂92]y0	ding with CAR.NLISP	 X↓iP≥β&C∀4$∧ααε=xM8⎇≥.2]εEβE⊂⊂∧H⊂⊂λIMPL@∪L@Qβ≥⊂@Q:⎇!↓"I~5%α	
%⊃ ¬∧DBT ∞OT (E@#Uβ⊂⊃αR↓A%$hP$$∃	dmλ
∪	~u∀λ∀TFEαD@	(ISP
β$αα`∩J⊃Q HJ∧∧αD450)Dλ
λb)⊂∀⊂iigaH-⊂$TJTTVεBαλ
)α↓↓β←FK∂!βF@~ε≥`λ
≡\Y0⊗→{0w:λ8∧erm i`≤AαKQ9↓∧∪eβ↔d¬⊗nNβX=
≥Yh⊂~~2P:2\αh
Ao∀AOKhh~∀4R↓%↓αB& -	I∀-~¬λ∀dλ
∪IZα⊂∀"Th¬AL X∧@`R$~∀$HI":> ∧αDd~:Eα¬%∃⊂hPα""%	4p T⊂-∀TCE		    ⊃E+β_@@@Qπ	H@Qβ'M_∞
αRαe%%JI0$!Q"αα∧λ
m
8z⊂≥p¬ wi			     (CDR (ASSOC Z Y)))
		      (NOT (LISTP Z))
		      (NOT (EQUAL U (QUOTE NO)))
		      (ISVAR Z))
		 (EQUAL (CDR (ASSOC Z U))
			(CDR (ASSOC Z Y)))),

  which we simplify, expanding ASSOC, to:

        T.


     So next consider:

      (IMPLIES (AND (NOT (EQUAL Z W))
		    (NOT (LISTP Z))
		    (ISVAR Z))
	       (EQUAL 0 (CDR (ASSOC Z Y)))),

which we named *1.1.1.6 above.  But this conjecture is subsumed by
formula *1.1.1.4 above.


     So we now return to:

      (IMPLIES (AND (NOT (EQUAL Z 0))
		    (NOT (LISTP Z))
		    (ISVAR Z))
	       (EQUAL 0 (CDR (ASSOC Z Y)))),

which we named *1.1.1.5 above.  This conjecture is subsumed by the
subgoal we named *1.1.1.4 above.


     So let us turn our attention to:

      (IMPLIES (AND (NOT (LISTP W)) (ISVAR W))
	       (EQUAL D (CDR (ASSOC W Y)))),

which we named *1.1.1.4 above.  Let us appeal to the induction
principle.  There is only one suggested induction.  We will induct
according to the following scheme:
      (AND (IMPLIES (NOT (LISTP Y)) (p D U Y))
	   (IMPLIES (AQUAL W (CAAR Y)) (p D W Y))
	   (IMPLIES (AND (LISTP Y)
			 (NOT (EQUAL W (CAAR Y)))
			 (p D W (CDR Y)))
		    (p D W Y))).
The inequality CDR.LESSP establishes that the measure (COUNT Y)
decpeases according to the well-founded function LESSP in the induction
step of the sc@!K[J\A)QJ↓CE←m∀AS]IUGiS←8AgGQ∃[BAaI←IkG∃bAiQIKJA]∃n~∃O=CYft4∀~∃π¬gJ@f8∩Q∪≠A→∪&Qβ≥λQ≥∨(Q→∪'Q A2R$~∀α∩@@@@Q≥∨(Q→∪'Q A.R$~∀∩∩@@@@Q∪'-¬$A.R$~∀∩∩Q⊗F,
1α⊃αB∞∩IαBεNN|→α]αJI%%%`h(4)αβ←#'≡AβO'oβ3'≠N+M1β/CCπ;&K;≥β&C∃β∪.3';'&K?;Mε{⊂∩∧~:4|~≥f"∧8J"bπMw hPβ"B%	34∪	_4h
λ→Qλ
	iuλ
	I4u∀∧**#!!"(λ∧∧λλ
	iuλ
	I4u∀∧
j*#!!"(λ∧∧λλ
	~uP4D
j*#!!"(
λZ503∧λλ
%∃A"C!$λ⊃;
≥:;X.L(~<N,;→=L≥]λ∃↑Y<kD∧∃~~.∀≤≤[lN8y<d∞~→(
l=h⊃M}[=;↔A"C!$λλλ∧∧⊃Kβ!!"]z
≤zλ≠,\;\h∞M→(≤∞-{yH≡≥→;.∞λ~_.1"C"E%JJJE%JJJE%JJHλdλ⊂(∧	(λ∪∧∧⊃(λλD
JJE%JJJE%JJJE!"C"@