TTHB+iBp2 AMi2`b2+iBQM hvT2b
sm sm21 - "`mMQ *X /X aX PHBp2B`1 - M/ LBM;MBM; sB22
1

lMBp2`bBiv Q7 >QM; EQM;- *?BMlMBp2`bBiv Q7 *K#`B/;2- lE

2#bi`+iX *H+mHB rBi? BMi2`b2+iBQM ivT2b ?p2 #22M mb2/ Qp2` i?2 v2`b
iQ KQ/2H p`BQmb 72im`2b- BM+Hm/BM;, Qp2`HQ/BM;- 2ti2MbB#H2 `2+Q`/b M/KQ`2 `2+2MiHv- M2bi2/ +QKTQbBiBQM M/ `2im`M ivT2 Qp2`HQ/BM;X L2p2`@
i?2H2bb MQ T`2pBQmb +H+mHmb bmTTQ`ib HH i?Qb2 72im`2b i QM+2X AM i?Bb
TT2` r2 bim/v 2tT`2bbBp2 +H+mHB rBi? BMi2`b2+iBQM ivT2b M/  K2`;2
QT2`iQ`X Pm` }`bi +H+mHmb bmTTQ`ib M mM`2bi`B+i2/ K2`;2 QT2`iQ`r?B+? Bb #H2 iQ bmTTQ`i HH i?2 72im`2b- M/ Bb T`Qp2M iQ #2 ivT2 bQmM/X>Qr2p2`- i?2 b2KMiB+b Bb MQM@/2i2`KBMBbiB+X AM i?2 b2+QM/ +H+mHmb r2
2KTHQv  T`2pBQmbHv T`QTQb2/ /BbDQBMiM2bb `2bi`B+iBQM- iQ KF2 i?2 b2@
KMiB+b /2i2`KBMBbiB+X aQK2 7Q`Kb Q7 Qp2`HQ/BM; `2 7Q`#B//2M- #mi HH
Qi?2` 72im`2b `2 bmTTQ`i2/X h?2 KBM +?HH2M;2 BM i?2 /2bB;M Bb `2Hi2/
iQ i?2 b2KMiB+b Q7 TTHB+iBQMb M/ `2+Q`/ T`QD2+iBQMbX q2 T`QTQb2 M
TTHB+iBp2 bm#ivTBM; `2HiBQM i?i 2M#H2b i?2 BM72`2M+2 Q7 `2bmHi ivT2b
7Q` TTHB+iBQMb M/ T`QD2+iBQMbX *Q``2bTQM/BM;Hv- i?2`2 Bb M TTHB+@iBp2 /BbTi+?BM; `2HiBQM i?i Bb mb2/ 7Q` i?2 /vMKB+ b2KMiB+bX h?2
irQ +H+mHB M/ i?2B` T`QQ7b `2 7Q`KHBx2/ BM i?2 *Q[ i?2Q`2K T`Qp2`X

R AMi`Q/m+iBQM*H+mHB rBi? BMi2`b2+iBQM ivT2b (j-d-RN-kk) ?p2  HQM; ?BbiQ`v BM T`Q;`KKBM;
HM;m;2bX _2vMQH/b (kR) rb i?2 }`bi iQ T`QKQi2 i?2 mb2 Q7 BMi2`b2+iBQM ivT2b BMT`+iB+H T`Q;`KKBM;X >2 BMi`Q/m+2/  K2`;2 QT2`iQ` i?i 2M#H2b #mBH/BM;pHm2b rBi? KmHiBTH2 ivT2b- r?2`2 i?2 KmHiBTH2 ivT2b `2 KQ/2HH2/ b BMi2`b2+iBQM
ivT2bX .mM}2H/ (N) `2}M2/ i?2 K2`;2 QT2`iQ`- iQ // bB;MB}+Mi //BiBQMH 2t@
T`2bbBp2 TQr2` Qp2` i?2 Q`B;BMH 7Q`KmHiBQM #v _2vMQH/bX Pp2` i?2 v2`b i?2`2?p2 #22M b2p2`H +H+mHB rBi? BMi2`b2+iBQM ivT2b 2[mBTT2/ rBi?  K2`;2 QT@
2`iQ`- M/ 2M#HBM; /Bz2`2Mi 72im`2b, Qp2`HQ/2/ 7mM+iBQMb (e-N)- `2im`M ivT2
Qp2`HQ/BM; (Re)- 2ti2MbB#H2 `2+Q`/b (N-kR) M/ M2bi2/ +QKTQbBiBQM (9-R9)X
L2p2`i?2H2bb- MQ T`2pBQmb +H+mHmb bmTTQ`ib HH 7Qm` 72im`2b iQ;2i?2`X aQK2
+H+mHB 2M#H2 7mM+iBQM Qp2`HQ/BM; (e)- #mi T`2+Hm/2 `2im`M ivT2 Qp2`HQ/BM;M/ M2bi2/ +QKTQbBiBQMX PM i?2 Qi?2` ?M/- +H+mHB rBi? /BbDQBMi BMi2`b2+iBQM
ivT2b (9-R9) bmTTQ`i `2im`M ivT2 Qp2`HQ/BM; M/ M2bi2/ +QKTQbBiBQM- #mi /BbH@
HQr +QMp2MiBQMH 7mM+iBQMH Qp2`HQ/BM;X .mM}2H/Ƕb +H+mHmb (N) bmTTQ`ib i?2 }`bii?`22 72im`2b- #mi MQi M2bi2/ +QKTQbBiBQMX h?Qb2 72im`2b `2 MQi +QKTH2i2Hv
Q`i?Q;QMH M/ i?2 BMi2`+iBQMb #2ir22M i?2K `2 BMi2`2biBM;- HHQrBM; 7Q` M2r
TTHB+iBQMbX >Qr2p2`- i?2 BMi2`+iBQMb HbQ TQb2 M2r i2+?MB+H +?HH2M;2bX
h?Bb TT2` bim/B2b 2tT`2bbBp2 +H+mHB rBi? BMi2`b2+iBQM ivT2b M/  K2`;2
QT2`iQ`X Pm` ;QH Bb iQ /2bB;M +H+mHB i?i /2H rBi? HH 7Qm` 72im`2b i QM+2- M/ksX sm2 2i HX

bim/v i?2 BMi2`+iBQM #2ir22M i?2b2 72im`2bX Pm` irQ KBM 7Q+mb2b `2 QM ivT2
BM72`2M+2 7Q` TTHB+iBQMb M/ `2+Q`/ T`QD2+iBQM- M/ i?2 /2bB;M Q7 i?2 QT2`iBQMH
b2KMiB+b 7Q` bm+? +H+mHBX hQ 2M#H2 HH i?2 72im`2b r2 BMi`Q/m+2  bT2+BHBx2/
7Q`K Q7 bm#ivTBM;- +HH2/ TTHB+iBp2 bm#ivTBM;- iQ /2H rBi? i?2 ~2tB#H2 7Q`Kb Q7
TTHB+iBQMb M/ `2+Q`/ T`QD2+iBQM HHQr2/ #v i?2 +H+mHBX *Q``2bTQM/BM;Hv- i?2`2
Bb M TTHB+iBp2 /BbTi+?BM; `2HiBQM i?i Bb mb2/ 7Q` i?2 /vMKB+ b2KMiB+bX AM
//BiBQM- r2 2tTHQ`2 i?2 BMi2`+iBQMb #2ir22M 72im`2bX AM T`iB+mH`- Qp2`HQ/BM;M/ M2bi2/ +QKTQbBiBQM 2M#H2 +m``B2/ Qp2`HQ/2/ 7mM+iBQMb- r?BH2 KQbi T`2pBQmb
rQ`F (R8-e-N-8) QMHv +QMbB/2`b mM+m``B2/ Qp2`HQ/2/ 7mM+iBQMbX
Pm` }`bi +H+mHmb bmTTQ`ib M mM`2bi`B+i2/ K2`;2 QT2`iQ`X h?Bb +H+mHmb Bb
#H2 iQ bmTTQ`i HH 7Qm` 72im`2b- M/ Bb T`Qp2M iQ #2 ivT2 bQmM/X >Qr2p2`- i?2
b2KMiB+b Bb MQM@/2i2`KBMBbiB+X AM T`+iB+2- BM M BKTH2K2MiiBQM Q7 i?Bb +H+mHmb
r2 +M 2KTHQv  #Bb2/ Q` bvKK2i`B+ K2`;2 QT2`iQ`- r?B+? ;Bp2b  U#Bb2/V
T`272`2M+2 iQ pHm2b QM i?2 H27i Q` `B;?i bB/2 Q7 K2`;2bX h?Bb TT`Q+? Bb bBKBH`iQ i?2 TT`Q+? iF2M #v .mM}2H/ BM ?2` +H+mHmb (N)- M/ bvKK2i`B+ K2`;2
UQ` +QM+i2MiBQMV QT2`iQ`b `2 HbQ /QTi2/ BM b2p2`H +H+mHB rBi? 2ti2MbB#H2`2+Q`/b (kd-ky)X AM i?2 b2+QM/ +H+mHmb r2 2KTHQv  T`2pBQmbHv T`QTQb2/ /BbDQBMi@
M2bb `2bi`B+iBQM (Rd)- iQ KF2 i?2 b2KMiB+b /2i2`KBMBbiB+X .BbDQBMiM2bb 2M#H2b
 bvKK2i`B+ K2`;2 QT2`iQ`- bBM+2 +QM~B+ib BM  K2`;2 `2 biiB+HHv `2D2+i2/`i?2` i?M `2bQHp2/ rBi?  #Bb2/ b2KMiB+bX AM i?2 b2+QM/ +H+mHmb bQK2 7Q`Kb
Q7 Qp2`HQ/BM; `2 7Q`#B//2M- #mi HH Qi?2` 72im`2b `2 bmTTQ`i2/X h?2 irQ +H+mHB
M/ i?2B` T`QQ7b `2 7Q`KHBx2/ BM i?2 *Q[ i?2Q`2K T`Qp2`X
AM bmKK`v- i?2 +QMi`B#miBQMb Q7 i?Bb TT2` `2,
Ĝ *H+mHB bmTTQ`iBM; Qp2`HQ/BM;- 2ti2MbB#H2 `2+Q`/b M/ M2bi2/ +QK@
TQbBiBQMX q2 T`QTQb2 +H+mHB rBi? BMi2`b2+iBQM ivT2b M/  K2`;2 QT2`iQ`r?B+? +M bmTTQ`i p`BQmb 72im`2b iQ;2i?2`- mMHBF2 T`2pBQmb +H+mHB r?2`2
QMHv bQK2 72im`2b r2`2 bmTTQ`i2/X
Ĝ TTHB+iBp2 bm#ivTBM; M/ /BbTi+?BM;X q2 /2p2HQT  bT2+BHBx2/ T@
THB+iBp2 bm#ivTBM; `2HiBQM iQ /2H rBi? i?2 T`Q#H2K Q7 BM72``BM; `2bmHi ivT2b
7Q` TTHB+iBQMb M/ `2+Q`/ T`QD2+iBQMbX AM //BiBQM- i?2 /vMKB+ b2KMiB+bbmTTQ`ib  +Q``2bTQM/BM; TTHB+iBp2 /BbTi+?BM; `2HiBQMX
Ĝ 6B`bi@+Hbb- +m``B2/ Qp2`HQ/BM;, q2 b?Qr i?i i?2 BMi2`+iBQM #2ir22M
Qp2`HQ/BM; M/ M2bi2/ +QKTQbBiBQM 2M#H2b Qp2`HQ/2/ 7mM+iBQMb iQ #2 }`bi@+Hbb- r?B+? HHQrb i?2 /2}MBiBQM Q7 +m``B2/ Qp2`HQ/2/ 7mM+iBQMbX
Ĝ J2+?MB+H 7Q`KHBxiBQM M/ BKTH2K2MiiBQM, HH i?2 +H+mHB M/
T`QQ7b `2 7Q`KHBx2/ BM i?2 *Q[ i?2Q`2K T`Qp2`X h?2 7Q`KHBxiBQM Bb pBH@
#H2 BM i?2 `iB7+i (R) M/  T`QiQivT2 BKTH2K2MiiBQM +M #2 7QmM/ i,?iiTb,ff;Bi?m#X+QKfDmMBQ`ttm2fTTHB+iBp2@BMi2`b2+iBQM

k Pp2`pB2r
h?Bb b2+iBQM ;Bp2b M Qp2`pB2r Q7 Qm` rQ`F M/ BMi`Q/m+2b i?2 F2v i2+?MB+H B/2bX
q2 i?2M BHHmbi`i2 bQK2 T`Q#H2Kb- +?HH2M;2b M/ bQHmiBQMb r?2M /2bB;MBM; ivT2
bvbi2Kb 7Q` bm+? +H+mHB M/ 72im`2bX

TTHB+iBp2 AMi2`b2+iBQM hvT2b

jkXR "+F;`QmM/
AMi2`b2+iBQM hvT2b M/ J2`;2 PT2`iQ` AMi2`b2+iBQM ivT2b /2b+`B#2 2t@
T`2bbBQMb i?i +M ?p2 KmHiBTH2 ivT2bX h?2 BMi2`b2+iBQM ivT2 A  B Bb BM?#Bi2/#v i2`Kb i?i ?p2 #Qi? ivT2 A M/ ivT2 BX h?2 K2`;2 QT2`iQ` U/2MQi2/ b, ,V ?b #22M BMi`Q/m+2/ #v _2vMQH/b (kR)- M/ Hi2` `2}M2/ #v .mM}2H/ (N)- iQ+`2i2 i2`Kb rBi? BMi2`b2+iBQM ivT2b i i?2 i2`K H2p2HX M BKTQ`iMi 72im`2 Q7.mM}2H/Ƕb +H+mHmb Bb i?i Bi +QMiBMb  +QKTH2i2Hv mM`2bi`B+i2/ K2`;2 QT2`iQ`r?B+? 2M#H2b KQbi Q7 i?2 TTHB+iBQMb i?i r2 rBHH /Bb+mbb BM i?Bb TT2`- 2t+2Ti
7Q` M2bi2/ +QKTQbBiBQMX >Qr2p2`- i?Bb 2tT`2bbBp2 TQr2` +QK2b i  +QbiX h?2
b2KMiB+b Q7 i?2 +H+mHmb Bb K#B;mQmbX 6Q` 2tKTH2- R--k , AMi +M 2H#Q`i2
iQ #Qi? R M/ kX LQi2 i?i BMi2`b2+iBQM ivT2b BM i?2 T`2b2M+2 Q7 i?2 K2`;2 QT@
2`iQ` ?p2  /Bz2`2Mi BMi2`T`2iiBQM 7`QK i?2 Q`B;BMH K2MBM; (d)- r?2`2 ivT2
BMi2`b2+iBQMb   " `2 QMHv BM?#Bi2/ #v i?2 BMi2`b2+iBQM Q7 i?2 b2ib Q7 pHm2b
Q7  M/ "X AM ;2M2`H- rBi? i?2 K2`;2 QT2`iQ`- r2 +M Hrvb }M/  i2`K 7Q`
Mv BMi2`b2+iBQM ivT2- 2p2M r?2M i?2 irQ ivT2b BM i?2 BMi2`b2+iBQM `2 /BbDQBMi
UBX2X r?2M i?2 b2ib Q7 pHm2b /2MQi2/ #v i?2 irQ ivT2b `2 /BbDQBMiVX 6Q` 2tKTH2R--i`m2 ?b i?2 ivT2 AMi  "QQHX AM KMv +HbbB+H BMi2`b2+iBQM ivT2 bvbi2Kb
rBi?Qmi i?2 K2`;2 QT2`iQ`- bm+? ivT2 rQmH/ MQi #2 BM?#Bi2/ (RN)X h?mb- i?2
mb2 Q7 i?2 i2`K ǳBMi2`b2+iBQMǴ Bb K2`2Hv  ?BbiQ`B+H H2;+vX h?2 K2`;2 QT2`iQ`
//b 2tT`2bbBp2 TQr2` iQ +H+mHB rBi? BMi2`b2+iBQM ivT2bX b r2 b?HH b22- i?Bb
//2/ 2tT`2bbBp2 TQr2` Bb mb27mH iQ KQ/2H b2p2`H 72im`2b Q7 T`+iB+H BMi2`2bi
7Q` T`Q;`KKBM; HM;m;2bX
.BbDQBMi BMi2`b2+iBQM ivT2b PHBp2B` 2i HX (Rd) bQHp2/ i?2 K#B;mBiv T`Q#H2K#v BKTQbBM;  /BbDQBMiM2bb `2bi`B+iBQM QM K2`;2bX PMHv ivT2b i?i `2 /BbDQBMi+M #2 K2`;2/X 6Q` 2tKTH2- AMi M/ "QQH `2 /BbDQBMi- bQ i?2 ivT2 AMi  "QQH Bb
r2HH@7Q`K2/ M/ R--i`m2 Bb  pHB/ i2`KX >mM; 2i HX (R9) BKT`Qp2/ i?Bb +H+mHmb#v BMi`Q/m+BM;  ivT2@/B`2+i2/ QT2`iBQMH b2KMiB+b r?2`2 ivT2b `2 mb2/ iQ
bbBbi `2/m+iBQM M/ T`Qp2/ Bib ivT2 bQmM/M2bb M/ /2i2`KBMBbKX lM7Q`imMi2Hvi?2 `2bi`B+iBQM iQ /BbDQBMi BMi2`b2+iBQM ivT2b- r?BH2 HHQrBM; KMv Q7 i?2 Q`B;BMH
TTHB+iBQMb- `mH2b Qmi i`/BiBQMH 7mM+iBQM Qp2`HQ/BM; Ub22 a2+iBQM 8 7Q` KQ`2/2iBHbVX
kXk TTHB+iBQMb Q7 i?2 J2`;2 PT2`iQ`
hQ b?Qr i?i i?2 K2`;2 QT2`iQ` Bb mb27mH- r2 MQr +Qp2` 7Qm` TTHB+iBQMb Q7
i?2 K2`;2 QT2`iQ` i?i ?p2 TT2`2/ BM i?2 HBi2`im`2, `2+Q`/b M/ `2+Q`/
T`QD2+iBQMb- 7mM+iBQM Qp2`HQ/BM;- `2im`M ivT2 7mM+iBQM Qp2`HQ/BM; M/ M2bi2/+QKTQbBiBQMX HH TTHB+iBQMb +M #2 2M+Q/2/ #v Qm` +H+mHmb BM a2+iBQM 9X_2+Q`/b M/ _2+Q`/ S`QD2+iBQMb h?2 B/2 Q7 mbBM; i?2 K2`;2 QT2`iQ` iQ
KQ/2H `2+Q`/ +QM+i2MiBQM }`biHv TT2`b BM _2vMQH/Ƕb rQ`F (kj)X _2+Q`/b BM Qm`+H+mHB `2 KQ/2HH2/ b K2`;2b Q7 KmHiBTH2 bBM;H2@}2H/ `2+Q`/bX JmHiB@}2H/ `2+Q`/b+M #2 pB2r2/ b bvMi+iB+ bm;` M/ &t4]?2HHQ]- v4]rQ`H/]' Bb bBKTHv &t 4]?2HHQ]'--&v 4 ]rQ`H/]'X h?2 #2?pBQm` Q7 `2+Q`/ T`QD2+iBQM Bb KQbiHv biM/`/BM Qm` +H+mHBX 7i2` #2BM; T`QD2+i2/ #v  H#2H- i?2 K2`;2/ `2+Q`/b rBHH `2im`M
i?2 bbQ+Bi2/ i2`KbX 6Q` BMbiM+2 U!→ /2MQi2b `2/m+2 iQVX

9sX sm2 2i HX

U&t 4 ]?2HHQ ]'--&v 4 ]rQ`H/ ]'VXt !→ ]?2HHQ ]6mM+iBQM Qp2`HQ/BM; 6mM+iBQM Qp2`HQ/BM; Bb  7Q`K Q7 TQHvKQ`T?BbK r?2`2
i?2 BKTH2K2MiiBQM Q7 7mM+iBQMb +M p`v /2T2M/BM; QM /Bz2`2Mi ivT2b Q7 `;m@
K2Mib i?i `2 TTHB2/ #v 7mM+iBQMbX h?2`2 `2 KMv rvb iQ `2T`2b2Mi ivT2b Q7
Qp2`HQ/2/ 7mM+iBQMbX 6Q` 2tKTH2- bmTTQb2 b?Qr Bb M Qp2`HQ/2/ 7mM+iBQM i?i+M #2 TTHB2/ iQ 2Bi?2` BMi2;2`b Q` #QQH2MbX >bF2HH miBHBb2b ivT2 +Hbb2b (k8) iQ
bbB;M i?2 ivT2 a?Qr  ⇒  → ai`BM; iQ b?Qr rBi? BMbiM+2b /2}M2/X
qBi? BMi2`b2+iBQM ivT2b- r2 +M 2KTHQv i?2 K2`;2 QT2`iQ` (e-kR) iQ /2}M2 
bBKTHB}2/ p2`bBQM Q7 i?2 Qp2`HQ/2/ b?Qr 7mM+iBQMX 6Q` BMbiM+2- i?2 b?Qr 7mM+iBQM#2HQr ?b ivT2 UAMi → ai`BM;V  U"QQH → ai`BM;VX
b?Qr , UAMi → ai`BM; V  U"QQH → ai`BM; V 4 b?QrAMi -- b?Qr"QQH

h?2 #2?pBQm` Q7 b?Qr Bb biM/`/- +iBM; b  MQ`KH 7mM+iBQM, Bi +M #2
TTHB2/ iQ `;mK2Mib M/ i?2 +Q``2+i BKTH2K2MiiBQM Bb b2H2+i2/ #b2/ QM ivT2bX
b?Qr R !→ ]R]

b?Qr i`m2 !→ ]i`m2]_2im`M ivT2 Qp2`HQ/BM; PM2 +QKKQM 2tKTH2 Q7 `2im`M ivT2 Qp2`HQ/BM;
Bb i?2 `2/ ,, _2/  ⇒ ai`BM; →  7mM+iBQM BM >bF2HH- r?B+? Bb i?2 `2p2`b2
QT2`iBQM Q7 b?Qr M/ T`b2b  bi`BM; BMiQ bQK2 Qi?2` 7Q`K Q7 /iX GBF2 b?Qrr2 +M /2}M2  bBKTHB}2/ p2`bBQM Q7 `2/ mbBM; i?2 K2`;2 QT2`iQ`,
`2/ , U ai`BM; → AMiV  U ai`BM; → "QQHV 4 `2/AMi -- `2/"QQH

AM >bF2HH- #2+mb2 i?2 `2im`M ivT2  +MMQi #2 /2i2`KBM2/ #v i?2 `;mK2Mi`2/ 2Bi?2` `2[mB`2b T`Q;`KK2`b iQ ;Bp2 M 2tTHB+Bi ivT2 MMQiiBQM- Q` M22/biQ miQKiB+HHv BM72` i?2 `2im`M ivT2 7`QK i?2 +QMi2tiX Pm` +H+mHB rQ`F BM bBKBH` KMM2`X amTTQb2 i?i bm++ Bb i?2 bm++2bbQ` 7mM+iBQM QM BMi2;2`b M/ MQi
Bb i?2 M2;iBQM 7mM+iBQM QM #QQH2Mb- i?2M r2 +M r`Bi2,
bm++ U`2/ ]R]V !→ k

MQi U`2/ ]i`m2 ]V !→ 7Hb2

L2bi2/ *QKTQbBiBQM aBKTHv bii2/- M2bi2/ +QKTQbBiBQM `2~2+ib /Bbi`B#miBpBiv
T`QT2`iB2b Q7 BMi2`b2+iBQM ivT2b i i?2 i2`K H2p2HX q?2M 2HBKBMiBM; i2`Kb +`2i2/#v i?2 K2`;2 QT2`iQ` UmbmHHv 7mM+iBQMb M/ `2+Q`/bV- i?2 `2bmHib 2ti`+i2/
7`QK M2bi2/ i2`Kb rBHH #2 +QKTQb2/X AM i?2 +QMi2ti Q7 `2+Q`/b- i?2 /Bbi`B#miBp2
bm#ivTBM; `mH2 2M#HBM; i?Bb #2?pBQm` Bb {l : A}  {l : B} <: {l : A  B}X qBi? i?Bb
`mH2 r2 +M ?p2 i?2 7QHHQrBM; 2tT`2bbBQM,
U&t 4 ]?2HHQ ]'--&t 4 R'VXt !→ ] ?2HHQ ]--R

LQi2 i?i ?2`2 r2 HHQr `2T2i2/ }2H/b rBi? i?2 bK2 MK2X PM2 Kv rQ``v
#Qmi K#B;mBiB2b #mi- rBi?  /BbDQBMiM2bb `2bi`B+iBQM- r2 +M QMHv ++2Ti }2H/b
rBi? i?2 bK2 H#2Hb B7 i?2 ivT2b Q7 i?2 }2H/b `2 /BbDQBMiX L2bi2/ +QKTQbBiBQM Bb 
F2v 72im`2 BM +QKTQbBiBQMH T`Q;`KKBM; (kN)- r?B+? mb2b Bi iQ bQHp2 +?HH2M;BM;
KQ/mH`Biv T`Q#H2Kb bm+? b i?2 1tT`2bbBQM S`Q#H2K (ke)- M/ iQ KQ/2H 7Q`Kb
Q7 7KBHv TQHvKQ`T?BbK (RR)X q2 `272` BMi2`2bi2/ `2/2`b iQ i?2 rQ`F Q7 w?M;
2i HX (kN) 7Q` /2iBHbX

TTHB+iBp2 AMi2`b2+iBQM hvT2b8

L2bi2/ +QKTQbBiBQM +M HbQ Q++m` rBi? 7mM+iBQMH BMi2`b2+iBQMb- mbBM; i?2
bm#ivTBM; `mH2 (A → B)  (A → C) <: A → (B  C)X qBi? i?Bb `mH2- r2 +M- 7Q`
2tKTH2- r`Bi2 i?2 7QHHQrBM; T`Q;`K,
Ubm++ -- BMihQ.B;Bi V 8 !→ e--^8^r?B+? TTHB2b irQ 7mM+iBQMb iQ i?2 BMi2;2` 8X LQi2 i?i ?2`2 BMihQ.B;Bi iF2b
M BMi2;2` M/ `2im`Mb  +Q``2bTQM/BM; +?`+i2`X q2 rBHH HbQ b22 i?i M2bi2/+QKTQbBiBQM 2M#H2b Qp2`HQ/2/ 7mM+iBQMb iQ #2 +m``B2/X
kXj *?HH2M;2b BM i?2 .2bB;M Q7 i?2 a2KMiB+bh?2 ;QH Q7 i?Bb rQ`F Bb iQ /2bB;M M miQKiB+ ivT2 BM72`2M+2 H;Q`Bi?K 7Q`
TTHB+iBQMb M/ `2+Q`/ T`QD2+iBQM M/ i?2 +Q``2bTQM/BM; /vMKB+ b2KMiB+bbQ i?i i?2 bvbi2K bmTTQ`ib HH TTHB+iBQMb T`2b2Mi2/ BM i?2 T`2pBQmb b2+iBQMX
lM7Q`imMi2Hv- /2bB;MBM; i?2 b2KMiB+b Q7 i?2 K2`;2 QT2`iQ` TQb2b bB;MB}+Mi+?HH2M;2b- r?B+? r2 2tTHBM BM i?2 `2bi Q7 i?Bb b2+iBQMX
AM72`2M+2 Q7 S`QD2+iBQMb M/ TTHB+iBQMb AM i`/BiBQMH ivT2 bvbi2Kb- BM
TTHB+iBQMb e1 e2 Q` T`QD2+iBQMb e.l- e1 Bb 2tT2+i2/ iQ ?p2 M ``Qr ivT2 M/
e Bb 2tT2+i2/ iQ ?p2  `2+Q`/ ivT2X am+? +QMp2MiBQM- ?Qr2p2`- +MMQi TTHv
iQ Qm` bvbi2K #2+mb2 +2`iBM 7Q`Kb Q7 BMi2`b2+iBQM ivT2b +M HbQ THv i?2 `QH2
Q7 ``Qr Q` `2+Q`/ ivT2bX AM T`iB+mH`- bm+? mb2 +b2b Q7 BMi2`b2+iBQM ivT2b `2?2HT7mH 7Q` KQ/2HHBM; Qp2`HQ/2/ 7mM+iBQMb M/ KmHiB@}2H/ `2+Q`/bX 6Q` 2tKTH2r2 FMQr i?i b?QrAMi Bb QM2 #`M+? Q7 b?Qr rBi? i?2 bm#ivTBM; bii2K2Mi,
UAMi → ai`BM; V  U"QQH → ai`BM; V I, AMi → ai`BM;

6`QK i?Bb 2tKTH2 r2 +M b22 i?i i?2 /vMKB+ b2KMiB+b Kmbi bQK2?Qr #2
ivT2@/2T2M/2MiX AM Qm` rQ`F r2 7QHHQr i?2 ivT2@/B`2+i2/ QT2`iBQMH b2KMiB+b
Uh.PaV (R9) TT`Q+?- r?B+? +?QQb2b #2ir22M K2`;2/ 7mM+iBQMb ++Q`/BM; iQ
ivT2 BM7Q`KiBQM /m`BM; `mMiBK2X >Qr2p2`- 2tBbiBM; h.Pa TT`Q+?2b /Q MQi
bmTTQ`i Qp2`HQ/BM; 7Q` irQ `2bQMbX 6B`biHv- h.Pa `2[mB`2b K2`;2/ 7mM+iBQMb iQ#2 /BbDQBMi rBi? 2+? Qi?2`- #mi BM i?Bb +b2 i?2 K2`;2/ 7mM+iBQMb `2 MQi /BbDQBMi
UBX2X- AMi → ai`BM; Bb MQi /BbDQBMi rBi? "QQH → ai`BM; #2+mb2 Q7 i?2 +QKKQM`2im`M ivT2 ai`BM;VX a2+QM/Hv- 2p2M B7 r2 rQmH/ bBKTHv B;MQ`2 i?2 /BbDQBMiM2bb`2bi`B+iBQM- r2 rQmH/ biBHH M22/ iQ Tmi M 2tTHB+Bi ivT2 MMQiiBQM AMi → ai`BM;
M/ r`Bi2 i?2 T`Q;`K b Ub?Qr , AMi → ai`BM;V R iQ b2H2+i i?2 +Q``2+i BKTH2@
K2MiiBQM iQ TTHv 7`QK i?2 Qp2`HQ/2/ b?Qr 7mM+iBQMX h?Bb Bb #2+mb2 T`2pBQmbh.Pa +H+mHB ?p2 `2bi`B+i2/ TTHB+iBQM `mH2b i?i +MMQi ++QKKQ/i2 i`/B@
iBQMH Qp2`HQ/BM;X *H2`Hv- BM  b2iiBM; rBi? Qp2`HQ/BM;- ?pBM; iQ r`Bi2 bm+?
2tTHB+Bi MMQiiBQMb rQmH/ #2 mMbiBb7vBM;X h?2`27Q`2 r2 rBb? iQ ?p2 M T@
T`Q+? r?2`2 r2 +M r`Bi2 Qp2`HQ/2/ 7mM+iBQMb Mim`HHvX
 bBKBH` T`Q#H2K Q++m`b mbBM; `2+Q`/ T`QD2+iBQMb BM 2tBbiBM; h.Pa +H+mHBX
6Q` BMbiM+2- i?2 ivT2 bvbi2K Q7 λi+ (R9) `2[mB`2b 2tTHB+Bi MMQiiBQMb 7Q` T`QD2+@iBQMb Q7 KmHiB@}2H/ `2+Q`/b rBi? /BbiBM+i H#2Hb- bm+? b U&t 4 R'--&v 4 i`m2', &t , AMi'VXtX h?Bb Bb Q7 +Qm`b2- [mBi2 mMMim`H iQ r`Bi2X Hi?Qm;? bQm`+2
HM;m;2b i`;2iiBM; i?2 h.Pa +H+mHB +M 2HBKBMi2 i?2 2tTHB+Bi mb2 Q7 bm+? M@
MQiiBQMb i i?2 bQm`+2 H2p2H- Bi rQmH/ #2 #2ii2` iQ //`2bb i?Bb T`Q#H2K /B`2+iHv
BM i?2 h.PaX

esX sm2 2i HX.vMKB+ a2KMiB+b :BpBM;  /B`2+i b2KMiB+b iQ Qp2`HQ/2/ TTHB+iBQMb Bb 
MQM@i`BpBH T`Q#H2KX h?MFb iQ i?2 K2`;2 QT2`iQ` M/ i?2 +HH@#v@pHm2 bi`i@2;v- Qm` Qp2`HQ/2/ 7mM+iBQMb `2 2tT2+i2/ iQ #2 BM i?2 7Q`K Q7 M2bi2/ K2`;2b
++Q`/BM; iQ i?2 bi`m+im`2 Q7 ivT2bX aQ r2 +M `2bQM #Qmi i?2 /vMKB+ b2KM@
iB+b b r2 /2H rBi? i?2 ivT2bX lM7Q`imMi2Hv- i?2 /Bbi`B#miBpBiv Q7 bm#ivTBM;+QKTHB+i2b i?2 biQ`vX h?2 +?HH2M;2 +QK2b 7`QK i?2 7+i i?i BM Qm` b2iiBM;
Qp2`HQ/2/ 7mM+iBQMb `2 }`bi@+HbbX h?i Bb- i?2v +M #2 iF2M b `;mK2Mib Q``2im`M2/ b `2bmHibX 6Q` BMbiM+2- r2 +M ?p2,
Tb?Qr , lMBi → UAMi → ai`BM; V  U"QQH → ai`BM; V
Tb?Qr 4 λtX b?Qr

AM i?Bb bBimiBQM- M Qp2`HQ/2/ 7mM+iBQM Bb r`TT2/ rBi?  HK#/ #bi`+@
iBQM- r?BH2 Bi b?QmH/ HbQ #2 pB2r2/ b M Qp2`HQ/2/ 7mM+iBQMX 6Q` 2tKTH2- r2
2tT2+i i?2 7QHHQrBM; iQ ?QH/,
Tb?Qr mMBi R !→ ]R]

Tb?Qr mMBi i`m2 !→ ]i`m2]

AM i?2 Hbi irQ +b2b- rBi?  i`/BiBQMH TT`Q+? iQ TTHB+iBQMb- Tb?Qr Bb 2t@
T2+i2/ iQ ?p2 ivT2 lMBi → AMi → ai`BM; M/ lMBi → "QQH → ai`BM; `2bT2+@
iBp2HvX 6`QK i?2 T2`bT2+iBp2 Q7 BMi2`b2+iBQM Qp2`HQ/BM;- Tb?Qr b?QmH/ #2 Q7 ivT2
UlMBi → AMi → ai`BM;V  UlMBi → "QQH → ai`BM;V- r?B+?- ?Qr2p2`- Bb /Bz2`@
2Mi 7`QK i?2 ;Bp2M ivT2 MMQiiBQMX h?Bb Hi2`MiBp2 pB2r Q7 ivT2b M/ 7mM+iBQMb
TQb2b +?HH2M;2b iQ i?2 /2bB;M Q7 i?2 biiB+ b r2HH b i?2 /vMKB+ b2KMiB+bX
K#B;mBiB2b QM i?2 BMTmi ivT2b AM HM;m;2b HBF2 *YY M/ Cp- Qp2`HQ/@BM; +MMQi #2 /2}M2/ QM `2im`M ivT2b- M/ K#B;mBiB2b `2 /2i2+i2/ r?2M i?2
BMTmi ivT2b Q7 Qp2`HQ/2/ 7mM+iBQMb Qp2`HTX h?Bb Bb HbQ  `2bQM r?v KMv
rQ`Fb KQ/2H i?2 BMTmib Q7 Qp2`HQ/2/ 7mM+iBQMb b T`Q/m+i ivT2b (R8-e-N)X h?2
/pMi;2b `2 Q#pBQmb, Bi Bb 2bB2` iQ `2bQHp2 i?2 +Q``2+i #`M+? #v QMHv +QK@
T`BM; i?2 T`Q/m+i ivT2b M/ ivT2b Q7 `;mK2MibX h?2 /`r#+F Q7 i?Bb KQ/2H
Bb i?i T`Q/m+i ivT2b rBHH T`2p2Mi Qp2`HQ/2/ 7mM+iBQMb iQ #2 +m``B2/X h?Bb Bb #2@+mb2 Qp2`HQ/2/ 7mM+iBQMb #b2/ QM T`Q/m+i ivT2b 2tT2+i  imTH2 +QMiBMBM; HHi?2 `;mK2Mib M/ `2D2+i T`iBH TTHB+iBQMbX h?2 +?HH2M;2 Q7 KQ/2HHBM; Qp2`@
HQ/2/ +m``B2/ 7mM+iBQMb Bb i?i T`iBH TTHB+iBQMb Kv #2 BMbm{+B2Mi iQ 7mHHv/2i2`KBM2 i?2 BKTH2K2MiiBQM iQ iF2 7`QK i?2 Qp2`HQ/2/ 7mM+iBQMX h?2b2 TBMb+M #2 HH2pBi2/ mbBM; BMi2`b2+iBQM ivT2b- i?2 K2`;2 QT2`iQ` M/ i?2 72im`2 Q7
M2bi2/ +QKTQbBiBQMX
7 , AMi → AMi → AMi; , AMi → "QQH → "QQH

6Q` 2tKTH2- rBi? 7--;- r2 +M bBKTHv `2bQM i?i i?2 `2bmHi Q7 U7--;V R i`m2 Bb; R i`m2X h?2 T`Q#H2K Q++m`b BM i?2 T`iBHHv TTHB2/ i2`K U7--;V R- 7Q` r?B+?
i?2`2 `2 irQ TQbbB#H2 /2bB;M +?QB+2bX h?2 }`bi +?QB+2 Bb iQ `2D2+i i?Bb TTHB+iBQM
i2`K bBM+2 r2 +MMQi b2H2+i #2ir22M Qp2`HQ/b- i?mb 7Q`#B//BM; KMv mb2 +b2b
HBF2 i?BbX MQi?2` +?QB+2 Bb iQ TTHv 7 M/ ; BM T`HH2H iQ R- `2bmHiBM; BM U7
RV--U; RV- r?B+? ?b i?2 ivT2 UAMi → AMiV  U"QQH → "QQHVXTTHB+iBp2 AMi2`b2+iBQM hvT2bd

kX9 E2v A/2b M/ _2bmHib
TTHB+iBp2 bm#ivTBM; hQ ?2HT rBi? i?2 BM72`2M+2 Q7 i?2 `2bmHi ivT2b 7Q` T@
THB+iBQMb M/ T`QD2+iBQMb- r2 T`QTQb2  M2r bT2+BHBx2/ bm#ivTBM; H;Q`Bi?K
7Q` TTHB+iBQMb M/ T`QD2+iBQMbX aT2+B}+HHv- +QMp2MiBQMH bm#ivTBM; H;Q`Bi?KbiF2 irQ ivT2b b BMTmib- M/ `2im`M  #QQH2M BM/B+iBM; r?2i?2` irQ ivT2b `2
BM  bm#ivTBM; `2HiBQM Q` MQiX q2 T`2b2Mi M TTHB+iBp2 bm#ivTBM; H;Q`Bi?Kr?Qb2 BMimBiBQM Bb bBKTH2, ;Bp2M  7mM+iBQMH ivT2 A Ur?B+? Kv #2 M BMi2`b2+@
iBQM Q7 7mM+iBQMbV- M/ i?2 ivT2 Q7 M `;mK2Mi B- Bi i2HHb r?2i?2` i?Bb 7mM+iBQM+M #2 TTHB2/ iQ i?Bb `;mK2Mi M/- B7 v2b- Bi +QKTmi2b i?2 QmiTmi ivT2X aBK@
BH`Hv- ;Bp2M  `2+Q`/ ivT2 A- M/  H#2H l- TTHB+iBp2 bm#ivTBM; i2HHb r?2i?2`
i?Bb `2+Q`/ +M #2 T`QD2+i2/ #v i?Bb H#2H- M/ B7 v2b- Bi +QKTmi2b i?2 `2bmHi ivT2
bbQ+Bi2/ rBi? i?Bb H#2HX "bB+HHv- r2 i`v iQ bQHp2 i?2 T`Q#H2K BM i?2 7QHHQrBM;
bm#ivTBM; 7Q`K U/2MQi2/ b <:V- r?2`2 r2 BM72` i?2 ivT2 \ ;Bp2M i?2 `;mK2Mi
ivT2 AMi,
UAMi → ai`BM; V  U"QQH → ai`BM; V I, AMi → \h?Bb T`Q#H2K +M #2 bTHBi BMiQ irQ bi2Tb, }`bi- +?2+F r?2i?2` i?2 TTHB+iBQM
Bb r2HH@ivT2/- M/ B7 bQ- /2i2`KBM2 Bib QmiTmi ivT2X 6Q` i?2 #Qp2 2tKTH2- \ Bb
2tT2+i2/ iQ #2 ai`BM;- b AMi Bb M `;mK2Mi iQ AMi → ai`BM;X _2+Q`/ T`QD2+iBQM
rQ`Fb bBKBH`HvX ai`BM;  AMi b?QmH/ #2 /2`Bp2/ b i?2 `2bmHi ivT2 7Q` T`QD2+iBQM
U&t 4 ]?2HHQ]'--&t 4 R'VXtX&t , ai`BM; '  &t , AMi' I, &t , \'

TTHB+iBp2 bm#ivTBM; Bb mb2/ r?2M ivTBM; TTHB+iBQMb M/ T`QD2+iBQMbX Pm`
H;Q`Bi?K /QTib i?2 MQiBQM Q7 b2H2+iQ`b S i?i #bi`+i i?2 `;mK2Mib Ub  ivT2
7Q` TTHB+iBQMb- Q`  H#2H 7Q` T`QD2+iBQMbVX h?2 #2?pBQm` Q7 TTHB+iBp2 bm#ivT@BM; 7Q` BMi2`b2+iBQM ivT2b Bb +Tim`2/ #v  bBKTH2 +QKTQbBiBQM QT2`iQ` ! r?B+?
BbQHi2b T`iB+mH` /2bB;M +?QB+2bX AM TTHB+iBp2 bm#ivTBM;-  TQbbB#H2 `2bmHi Bb
i?i i?2 TTHB+iBQM 7BHbX q2 /2MQi2 7BHm`2 rBi?  . bvK#QHX q2 BHHmbi`i2 i?2`2bmHib Q7 TTHB+iBp2 bm#ivTBM; U/2MQi2/ b #V 7Q` i?2 #Qp2 2tKTH2b M2tiX
UAMi → ai`BM; V  U"QQH → ai`BM; V # AMi
U ai`BM; → AMiV  U ai`BM; → "QQHV # ai`BM;&t , ai`BM; V &v , ai`BM; V# t&t , ai`BM; V &t ,AMiV# t

4 ai`BM; ! X
4 AMi! "QQH
4 ai`BM; ! X
4 ai`BM; ! AMi

h?2`2 `2 irQ BKTQ`iMi i?BM;b iQ MQiB+2 ?2`2X 6B`biHv- b /Bb+mbb2/ #Qp2- i?2
b2+QM/ `;mK2Mi Q7 # Bb Dmbi i?2 7mM+iBQM `;mK2Mi ivT2 Q`  H#2H- BMbi2/ Q7 i?2+QKTH2i2 ivT2 i?i rQmH/ #2 MQ`KHHv mb2/ BM  bm#ivTBM; +QKT`BbQMX a2+QM/Hvi?2 `2bmHib `2 ;Bp2M BM i2`Kb Q7 i?2 +QKTQbBiBQM QT2`iQ` !X h?2 +QKTQbBiBQM
QT2`iQ` #bi`+ib #2?pBQm` i?i Bb bT2+B}+ iQ T`iB+mH` bm#ivTBM; `2HiBQMbX
q?2M /2bB;MBM; TTHB+iBp2 bm#ivTBM;-  /2bB`#H2 T`QT2`iv Bb i?i Bi b?QmH/ #2
bQmM/ M/ +QKTH2i2 rBi? `2bT2+i iQ bm#ivTBM;X h?2 bQmM/M2bb M/ +QKTH2i2M2bb
T`QT2`iB2b +M #2 bii2/ b 7QHHQrb U?2`2 r2 b?Qr i?2 +b2 7Q` 7mM+iBQMbV,G2KK R UaQmM/M2bbVX A7 A # B = C- i?2M A <: B → CXG2KK k U*QKTH2i2M2bbVX A7 A <: B → C- i?2M ∃D, A # B = D ∧ D <: CX

3

sX sm2 2i HX

.!.

=.

A1 ! .

= A1. ! A2 = A2

A1 ! A2 = A1  A2
UV L2bi2/ +QKTQbBiBQM b2KMiB+bX

.!.

=.

A1 ! .

= A1. ! A2
Amb ! O

= A2
= Amb

O ! Amb = Amb
A1 ! A2= Amb

U#V Pp2`HQ/BM; b2KMiB+bX

6B;X R, hrQ TQbbB#H2 +QKTQbBiBQM QT2`iQ`b- 7Q` +H+mHB rBi? /Bbi`B#miBp2 bm#ivT@BM; UQM i?2 H27iV- M/ rBi?Qmi /Bbi`B#miBp2 bm#ivTBM; UQM i?2 `B;?iVX h?2 MQiiBQM
Amb /2MQi2b K#B;mBiv- r?B+? `2T`2b2Mib  ivT2@2``Q`- r?BH2 i?2 K2i@p`B#H2
O /2MQi2b Mv QmiTmi ivT2 UBX2X 2Bi?2`  ivT2- 7BHm`2 Q` K#B;mBivVX.2T2M/BM; QM i?2 2tT`2bbBp2 TQr2` Q7 i?2 bm#ivTBM; `2HiBQM r2 M22/ /Bz2`2Mi
BKTH2K2MiiBQMb Q7 ! iQ biBb7v bQmM/M2bbf+QKTH2i2M2bbX AM T`iB+mH`- r?2i?2`
Q` MQi i?2 bm#ivTBM; `2HiBQM BM+Hm/2b /Bbi`B#miBp2 bm#ivTBM; `mH2b z2+ib i?2 /27@
BMBiBQM Q7 !X 6B;m`2 R BHHmbi`i2b i?Bb /Bz2`2M+2X AM  `2HiBQM rBi? /Bbi`B#miBp2
bm#ivTBM; `mH2b Ubm+? b (A → B)  (A → C) <: A → (B  C)V- i?2 +QKTQbBiBQM
QT2`iQ` QM i?2 H27i UV H2/b iQ  bQmM/ M/ +QKTH2i2 /2}MBiBQM Q7 TTHB+iBp2
bm#ivTBM;X h?Bb QT2`iQ`- r?B+? Bb [mBi2 bBKTH2- HHQrb +QK#BMBM; `2bmHib i?i`Bb2 7`QK KmHiBTH2 #`M+?2bX q2 bv i?i i?Bb +QKTQbBiBQM QT2`iQ` BKTH2K2Mib
 M2bi2/ +QKTQbBiBQM b2KMiB+b- bBM+2 Bi HHQrb +QK#BMBM; KmHiBTH2 `2bmHibX 6Q`
BMbiM+2- AMi ! "QQH bBKTHv /2MQi2b AMi  "QQH- K2MBM; i?i M TTHB+iBp2
bm#ivTBM; bii2K2Mi Uai`BM; → AMiV  Uai`BM; → "QQHV # ai`BM; bm++22/b+QKTmiBM; i?2 QmiTmi ivT2 AMi  "QQHX qBi?Qmi i?2 /Bbi`B#miBp2 bm#ivTBM; `mH2
i?2 +QKTQbBiBQM QT2`iQ` QM i?2 H27i UV Bb MQi bQmM/ rBi? `2bT2+i iQ bm#ivTBM;X
AMbi2/ r2 b?QmH/ mb2 i?2 BKTH2K2MiiBQM QM i?2 `B;?i U#V- r?B+? rBHH `2D2+i+b2b HBF2 AMi ! "QQH- bBM+2 bm+? +b2b /2MQi2  7Q`K Q7 K#B;mBivX q2 bv
i?i i?2 +QKTQbBiBQM QT2`iQ` QM i?2 `B;?i BKTH2K2Mib M Qp2`HQ/BM; b2KM@
iB+b- bBM+2 B7 KmHiBTH2 BKTH2K2MiiBQMb BM M Qp2`HQ/2/ /2}MBiBQM Ki+? rBi?
M `;mK2Mi UQ`  H#2HV- r2 `2D2+i i?2 TTHB+iBQMX h?Bb Bb bBKBH` iQ i`/BiBQMH
Qp2`HQ/BM; K2+?MBbKb- r?B+? `2D2+i bm+? +b2b b  7Q`K Q7 K#B;mBivX AM Qi?2`
rQ`/b- BM i?2 Qp2`HQ/BM; b2KMiB+b- QMHv QM2 BKTH2K2MiiBQM +M #2 b2H2+i2/
7`QK M Qp2`HQ/2/ /2}MBiBQMX LQi2 i?i i?2 Qp2`HQ/BM; b2KMiB+b BKTH2K2M@
iiBQM +M HbQ #2 mb2/ BM  +H+mHmb rBi? /Bbi`B#miBpBiv- #mi i?Bb rQmH/ HQb2
i?2 +QKTH2i2M2bb T`QT2`ivX PM2 +QmMi2` 2tKTH2 Bb UAMi → ai`BM;V  UAMi →AMiV I, AMi → Uai`BM;  AMiV- r?B+? ?QH/b ++Q`/BM; iQ i?2 /Bbi`B#miBpBiv `mH2#mi i?2 TTHB+iBp2 bm#ivTBM; #b2/ QM i?2 Qp2`HQ/BM; b2KMiB+b rBHH /2`Bp2 M
K#B;mBiv K# 2``Q`X
h.Pa 7Q` Pp2`HQ/BM; 6Q` i?2 b2KMiB+b- r2 7QHHQr mT i?2 B/2 Q7 ivT2/@/B`2+i2/ QT2`iBQMH b2KMiB+b (R9) M/ /2}M2  M2r Dm/;K2Mi i?i T2`7Q`Kb
TTHB+iBp2 /BbTi+?BM; iQ bmTTQ`i Qp2`HQ/BM;X i  ?B;? H2p2H- TTHB+iBp2 /Bb@Ti+? `2~2+ib TTHB+iBp2 bm#ivTBM; BM i?2 /vMKB+ b2KMiB+bX b r2 MHvx2/

TTHB+iBp2 AMi2`b2+iBQM hvT2bN

#Qp2- /Bbi`B#miBpBiv 7Q`#B/b Qp2`HQ/2/ 7mM+iBQMb iQ #2 2t+i M2bi2/ K2`;2b- i?mb
 +MQMB+H 7Q`K Q7 Qp2`HQ/2/ 7mM+iBQM b?QmH/ #2 b2iiH2/X hQ bQHp2 i?Bb T`Q#H2K
r2 mb2 M 2tTHB+Bi K2`;2 rBi? 2ti` MMQiiBQMb i?i THv  `QH2 Q7 ǳ`mMiBK2
ivT2bǴ- r?B+? `2 mb2/ #v TTHB+iBp2 /BbTi+?BM; iQ b2H2+i i?2 +Q``2+i #`M+?
/m`BM; `mMiBK2X
hrQ *H+mHB q2 T`2b2Mi irQ +H+mHB iQ /2KQMbi`i2 i?2 TTHB+iBp2 bm#ivT@BM; M/ TTHB+iBp2 /BbTi+?BM;X "Qi? +H+mHB miBHBx2 i?2 +QKTQbBiBQM QT2`iQ`
U6B;m`2 RV bBM+2 r2 rMi iQ HHQr M2bi2/ +QKTQbBiBQM- M/ 2tTHQ`2 +m``B2/ M/}`bi@+Hbb Qp2`HQ/2/ 7mM+iBQMbX h?2 }`bi +H+mHmb 2K#`+2b  bBKTH2 /2bB;M M/
/QTib M mM`2bi`B+i2/ K2`;2 QT2`iQ`X HH 72im`2b K2MiBQM2/ #Qp2 +M #2 2M@+Q/2/ BM i?Bb +H+mHmb- #mi i?2 +H+mHmb rBHH ?p2  MQM@/2i2`KBMBbiB+ b2KMiB+b/m2 iQ K#B;mBiB2bX 6Q` K#B;mBiB2b- r2 T`2b2Mi  b2+QM/ +H+mHmb- r?B+? /QTib `2bi`B+i2/ K2`;2 QT2`iQ`, QMHv i2`Kb rBi? /BbDQBMi ivT2b +M #2 K2`;2/X h?Bb
+H+mHmb Bb /2i2`KBMBbiB+ #mi 2t+Hm/2b +2`iBM 7Q`Kb Q7 Qp2`HQ/BM;- HBF2 i?2 b?Qr
7mM+iBQMX aBM+2 AMi → ai`BM; Bb MQi /BbDQBMi rBi? "QQH → ai`BM;- bm+? K2`;2brBHH #2 `2D2+i2/X

j TTHB+iBp2 am#ivTBM;
AM i?Bb b2+iBQM- r2 }`bi T`2b2Mi i?2 MQ`KH bm#ivTBM; H;Q`Bi?K 7Q` BMi2`b2+iBQM
ivT2b M/ i?2M T`2b2Mi TTHB+iBp2 bm#ivTBM;X
jXR hvT2b M/ am#ivTBM;
h?2 ivT2b i?i r2 +QMbB/2` BM i?Bb rQ`F `2,hvT2bP`/BM`v hvT2b

A, B ::= Int | T op | A → B | A  B | {l : A}o

A , Bo ::= Int | T op | A → Bo | {l : Ao }A M/ B `2 K2ip`B#H2b r?B+? `M;2 Qp2` ivT2bX Int M/ T op `2 #b2 ivT2b
M/ T op Bb i?2 bmT2`ivT2 Q7 HH ivT2bX *QKTQmM/ ivT2b `2 7mM+iBQM ivT2b A → BBMi2`b2+iBQM ivT2b A  B- M/ `2+Q`/ ivT2b {l : A}X P`/BM`v ivT2b (3-R9) `2 2b@b2MiBHHv ivT2b rBi?Qmi BMi2`b2+iBQM ivT2b- 2t+2Ti 7Q` 7mM+iBQMb r?2`2 BMi2`b2+iBQM
ivT2b +M TT2` BM `;mK2Mi ivT2bX
H;Q`Bi?KB+ "*. bm#ivTBM; am#ivTBM; `2HiBQMb 7Q` BMi2`b2+iBQM ivT2b +M
p`v BM r?2i?2` /Bbi`B#miBpBiv `mH2b `2 BM+Hm/2/ Q` MQiX 6Q` +H+mHB rBi? BMi2`@b2+iBQM ivT2b-  +QKKQM `mH2 HHQrb i?2 BMi2`b2+iBQM Q7 ``Qr ivT2b iQ /Bbi`B#mi2
Qp2` ``QrbX PM2 r2HH@FMQrM bm#ivTBM; `2HiBQM rBi? bm+? /Bbi`B#miBpBiv `mH2 Bb"*. bm#ivTBM; (j)X >mM; 2i HX (R9) T`QpB/2  bQmM/ M/ +QKTH2i2 H;Q`Bi?K 7Q`"*. bm#ivTBM; #v 2HBKBMiBM; i?2 i`MbBiBpBiv `mH2- M/ 2KTHQvBM; i?2 MQiBQMb
Q7 Q`/BM`v ivT2b M/ bTHBii#H2 ivT2bX q2 T`2b2Mi i?i bm#ivTBM; `2HiBQM BM
6B;X kX aTHBii#H2 ivT2b /2b+`B#2 i?i ivT2b +M #2 bTHBi BMiQ irQ bBKTH2` ivT2b
M/ Q`/BM`v ivT2b `2 i?Qb2 r?B+? +MMQi #2 bTHBiX _mH2 am#@M/ Bb i?2 KQbi
BMi2`2biBM; `mH2 b Bi +Tim`2b i?2 /Bbi`B#miBpBiv Q7 BMi2`b2+iBQM ivT2b Qp2` 7mM+@
iBQM ivT2b M/ `2+Q`/ ivT2bX h?Bb `mH2 bTHBib i?2 ivT2 B BMiQ irQ ivT2b B1 M/
B2 M/ T`Q+22/b #v i2biBM; r?2i?2` A Bb  bm#ivT2 Q7 #Qi? B1 M/ B2 X

Ry

sX sm2 2i HXUaTHBii#H2 hvT2bV

AR ! A " AkaT@``

aT@M/A ! AB " B

aT@_+/
AR ! A " Ak

BR ! B " Bk

A → BR ! A → B " A → Bk

{H : AR } ! {H : A} " {H : Ak }Uam#ivTBM;V

A <: B
am#@AMiam#@hQT

am#@``
C <: AB <: Do

am#@_+/
A <: BoAMi <: AMi

A <: hQT

A → B <: C → Do

{H : A} <: {H : Bo }am#@M/
BR ! B " Bk

A <: BRA <: Bk

A <: Bam#@M/@G
A <: Coam#@M/@_
B <: CoA  B <: CoA  B <: Co

6B;X k, aTHBii#H2 hvT2b M/ H;Q`Bi?KB+ am#ivTBM;
when B <: A1

URVA1 → A2 # B = .

when ¬(B <: A1 )UkV

{l1 = A} # l2 = .

when l1 %= l2

U9Votherwise

UeV

A1 → A2 # B = A2

UjV{l = A} # l = A

A1  A2 # S = (A1 # S) ! (A2 # S)
A#S =.

U8V

6B;X j, TTHB+iBp2 am#ivTBM;
jXk TTHB+iBp2 am#ivTBM;
TTHB+iBp2 bm#ivTBM; miBHBb2b i?2 MQiBQM Q7 b2H2+iQ`b iQ }M/ i?2 +Q``2+i QmiTmi
ivT2 7`QK TTHB+#H2 ivT2bX q2 +QMbB/2` TTHB+#H2 ivT2b iQ #2 7mM+iBQM Q` `2+Q`/
ivT2bX h?Bb `2HiBQM 2M#H2b i?2 ivT2 bvbi2K iQ BM72` i?2 ivT2 Q7 TTHB+iBQMb M/`2+Q`/ T`QD2+iBQMb- b b?QrM BM a2+iBQM 9X q2 KQ/2H i?2 ivT2b Q7 `;mK2Mib M/
H#2Hb Q7 `2+Q`/ T`QD2+iBQMb b b2H2+iQ`b- M/ i?2 QmiTmib b #2BM; 2Bi?2`  ivT2
Q` MQi?BM; U/2MQiBM; i?2 7BHm`2 iQ }M/  bmBi#H2 QmiTmi ivT2VX
a2H2+iQ`b

S ::= A | l

PmiTmib

O ::= . | A

h?2 /2}MBiBQM Q7 TTHB+iBp2 bm#ivTBM; Bb ;Bp2M BM 6B;X jX a2H2+iQ`b `2 mb2/ b
i?2 b2+QM/ T`K2i2` M/ T`QT;i2 i?`Qm;? i?2 bm#ivTBM; +?2+Fb- mMiBH r2`2+? ``Qr Q` `2+Q`/ ivT2bX 6Q` ``Qr ivT2b- BM `mH2b URV UkV- r2 +?2+F i?2+QMi`p`BMi bm#ivTBM; #2ir22M BMTmi ivT2 A1 M/ `;mK2Mi BX A7 bm++2bb7mHi?2 QmiTmi ivT2 A2 Bb `2im`M2/- Qi?2`rBb2 r2 7BHX 6Q` `2+Q`/ ivT2b UjV U9V- r2+?2+F i?2 2[mHBiv #2ir22M H#2HbX A7 i?2 H#2Hb `2 2[mH- r2 `2im`M i?2 QmiTmi
ivT2 A- Qi?2`rBb2 r2 7BHX 6Q` i?2 +b2 Q7 i?2 BMi2`b2+iBQM ivT2b A1  A2 U8V- r2
BMi`Q/m+2  +QKTQbBiBQM QT2`iQ` ! iQ +QK#BM2 irQ `2bmHib r?B+? `2 /2`Bp2/
7`QK TTHvBM; A1 M/ A2 rBi? i?2 bK2 b2H2+iQ` BX _mH2 UeV +Qp2`b  MmK#2` Q7

TTHB+iBp2 AMi2`b2+iBQM hvT2b

RR

KBbbBM; +b2b Ubm+? b Int # SV r?B+? rBHH HH 7BHX 6Q` bBKTHB+Biv Q7 T`2b2MiiBQM
r2 r`Bi2 i?Qb2 `mH2b b  bBM;H2 `mH2 UeVX
h?2 +QKTQbBiBQM QT2`iQ` ++2Tib QmiTmi `2bmHib M/ `2im`Mb  M2r QmiTmi`2bmHiX 6Q` bvbi2Kb rBi? "*. bm#ivTBM;- r?B+? BM+Hm/2 /Bbi`B#miBpBiv `mH2b- r2
mb2 i?2 +QKTQbBiBQM QT2`iQ` BKTH2K2MiBM; i?2 M2bi2/ +QKTQbBiBQM b2KMiB+b BM
6B;X RX
jXj J2ii?2Q`v
q2 T`Qp2/ i?2 bQmM/M2bb M/ +QKTH2i2M2bb Q7 Qm` TTHB+iBp2 bm#ivTBM; rBi?
`2bT2+i iQ i?2 MQ`KH bm#ivTBM;X h?2 /2+B/#BHBiv Q7 TTHB+iBp2 bm#ivTBM; Bbbi`B;?i7Q`r`/ bBM+2 Bi Bb KQ/2HH2/ b  bi`m+im`HHv `2+m`bBp2 7mM+iBQMX q2 ?p2
irQ p2`bBQMb Q7 bQmM/M2bb M/ +QKTH2i2M2bb H2KKbX h?2 }`bi p2`bBQM TTHB2b iQ
i?2 +b2 r?2`2 i?2 bmT2`ivT2 Bb  7mM+iBQM,
G2KK j UaQmM/M2bb U6mM+iBQMVVX A7 A # B = C- i?2M A <: B → CXG2KK 9 U*QKTH2i2M2bb U6mM+iBQMVVX A7 A <: B → C- i?2M ∃D, A # B =D ∧ D <: CX
h?2 bQmM/M2bb H2KK Bb BMimBiBp2X A7 i?2 `2bmHi Q7 +?2+FBM; TTHB+iBp2 bm#ivTBM;
rBi?  bm#ivT2 A M/ M/ BMTmi ivT2 B +QKTmi2b  ivT2 C i?2M Bi b?QmH/ #2 i?2+b2 i?i A <: B → CX 6Q` +QKTH2i2M2bb r2 rBb? iQ b?Qr i?i B7 A Bb  bm#ivT2
Q7  7mM+iBQM ivT2 B → C i?2M TTHB+iBp2 bm#ivTBM; rBHH Hrvb #2 #H2 iQ }M/bQK2 QmiTmi ivT2 D r?B+? Bb  bm#ivT2 Q7 CX
h?2 b2+QM/ p2`bBQM Q7 i?2 H2KK- r?B+? TTHB2b iQ i?2 +b2 r?2`2 i?2 bmT2`@
ivT2 Bb  `2+Q`/- Bb /2}M2/ BM  bBKBH` KMM2`X
G2KK 8 UaQmM/M2bb U_2+Q`/VVX A7 A # l = B- i?2M A <: {l : B}X

G2KK e U*QKTH2i2M2bb U_2+Q`/VVX A7 A <: {l : B}- i?2M ∃C, A # l =C ∧ C <: BX_2K`F LQi2 i?i- B7 r2 rQmH/ /`QT i?2 /Bbi`B#miBpBiv Q7 BMi2`b2+iBQMb Qp2` Qi?2`+QMbi`m+ib #v `2KQpBM; i?2 `mH2b aT@`` M/ aT@_+/- i?2M iQ ?p2 bQmM/M2bb
M/ +QKTH2i2M2bb r2 M22/ iQ 2KTHQv i?2 +QKTQbBiBQM QT2`iQ` BKTH2K2MiBM;
i?2 Qp2`HQ/BM; b2KMiB+b iQ i?2 `B;?i Q7 6B;X RX q?2M mbBM; i?i +QKTQbBiBQM
QT2`iQ`- i?2 bQmM/M2bb H2KKb `2KBM i?2 bK2- #mi r2 M22/ iQ /Dmbi i?2+QKTH2i2M2bb H2KKb iQ +QMbB/2` i?2 K#B;mQmb +b2bX 6Q` BMbiM+2- i?2 +QK@
TH2i2M2bb 7Q` i?2 +b2 Q7  7mM+iBQM bmT2`ivT2 rQmH/ #2+QK2,
G2KK d U*QKTH2i2M2bbVX A7 A <: B → C- i?2M (∃D, A # B = D ∧ D <:C) ∨ A # B = AmbX

9  *H+mHmb rBi? M lM`2bi`B+i2/ J2`;2 PT2`iQ`
h?Bb b2+iBQM T`2b2Mib  ivT2 bQmM/ +H+mHmb i?i bmTTQ`ib #Qi? BMi2`b2+iBQM ivT2b
M/  K2`;2 QT2`iQ`X h?Bb +H+mHmb +M #2 pB2r2/ b  p`BMi Q7 .mM}2H/Ƕb +H@
+mHmb UrBi?Qmi mMBQM ivT2bV (N)X Pm` +H+mHmb 2KTHQvb  ivT2@/B`2+i2/ QT2`iBQMH
b2KMiB+b (R9) BMbi2/ Q7 mbBM; 2H#Q`iBQM b2KMiB+b b T`QTQb2/ #v .mM}2H/
M/ /QTib TTHB+iBp2 bm#ivTBM; M/ /Bbi`B#miBp2 bm#ivTBM;XRksX sm2 2i HXΓ &e⇔A
h@GBi
Γ & B ⇒ AMih@TTΓ & eR ⇒ A
h@J`;Γ & eR ⇒ A

U"B/B`2+iBQMH hvTBM;Vh@o`
x:A ∈ ΓΓ &t⇒Ah@_+/Γ &e⇒Ah@GK
Γ, t : A & e ⇐ B

Γ & λt. e : A → B ⇒ A → B

Γ & ek ⇒ B

Γ & eR ek ⇒ C
Γ & ek ⇒ B

Γ & eR , , ek ⇒ A  B

A#B=C
h@MM
Γ &e⇐AΓ &e:A⇒A

Γ & {H = e} ⇒ {H : A}

h@S`QDΓ &e⇒A

A#H=B

Γ & e.H ⇒ B

h@am#Γ &e⇒A

A <: BΓ &e⇐B

6B;X 9, "B@/B`2+iBQMH ivTBM;X h?2 #B/B`2+iBQMH KQ/2 bvMit Bb ⇔ ::= ⇐|⇒X
9XR avMith?2 bvMit Q7 i?Bb +H+mHmb Bb,
1tT`2bbBQMb_r oHm2boHm2b
*QMi2tibe ::= x | i | e : A | e1 e2 | λx .e : A → B | e1 , , e2 | {l = e} | e.l

p ::= i | λx .e : A → B

v ::= p : Ao | v1 , , v2 | {l = v}

Γ ::= · | Γ, x : A

JQbi 2tT`2bbBQMb `2 biM/`/X GK#/ 2tT`2bbBQMb λx. e : A → B `2 7mHHv MMQ@ii2/ #2+mb2 i?2 QT2`iBQMH b2KMiB+b Bb ivT2@/B`2+i2/X h?2 2tT`2bbBQM e1 , , e2+`2i2b  K2`;2 Q7 irQ 2tT`2bbBQMb e1 M/ e2 X h?2 2tT`2bbBQM {l = e} /2MQi2b
 bBM;H2@}2H/ `2+Q`/ rBi? H#2H l M/ }2H/ eX h?2 T`QD2+iBQM Q7 `2+Q`/b Bb `2T`2@
b2Mi2/ #v e.lX _r pHm2b BM+Hm/2 BMi2;2`b M/ HK#/b- M/ pHm2b `2 /2}M2/
QM `r pHm2b MMQii2/ rBi? Q`/BM`v ivT2b- K2`;2b Q7 pHm2b M/ `2+Q`/b r?Qb2}2H/b `2 pHm2bX q2 bi`iB7v `r pHm2b M/ pHm2b #2+mb2 r2 M22/ iQ miBHBb2
MMQiiBQMb iQ /QTi /BbTi+?BM; BM i?2 b2KMiB+bX h?2 Q`/BM`v `2bi`B+iBQM QM
pHm2b 2M7Q`+2b  +MQMB+H 7Q`K 7Q` Qp2`HQ/2/ 7mM+iBQMbX Pp2`HQ/2/ 7mM+iBQMb
rBHH #2 `2/m+2/ iQ 2tTHB+Bi M2bi2/ K2`;2b- 2p2M BM b2iiBM;b rBi? /Bbi`B#mBpBivX
9Xk hvTBM;
6B;X 9 b?Qrb Qm` #B@/B`2+iBQMH ivT2 bvbi2KX JQbi Q7 i?2 `mH2b `2 /Ti2/ 7`QK
i`/BiBQMH #B@/B`2+iBQMH ivTBM; (Ry)X h?2 MQp2H `mH2b `2 `mH2b h@TT M/ h@
S`QD- r?Qb2 BM72``2/ ivT2 Bb /2`Bp2/ 7`QK TTHB+iBp2 bm#ivTBM;X
hvTBM; Q7 TTHB+iBQM M/ S`QD2+iBQM Pm` TT`Q+? iQ ivT2 TTHB+iBQMb (k3)
Bb iQ BM72` i?2 ivT2 Q7 7mM+iBQMb M/ `;mK2Mib i i?2 bK2 iBK2- Tbb i?2B` ivT2b
BMiQ TTHB+iBp2 bm#ivTBM; UA M/ B BM `mH2 h@TTV- M/ bbB;M i?2 +QKTmi2/`2bmHi C iQ TTHB+iBQMbX h?Bb Bb #2+mb2 r2 HHQr BMi2`b2+iBQM ivT2b iQ /Bbi`B#mi2
Qp2` ``Qr ivT2b- i?mb i?2 ivT2 Q7 i?2 7mM+iBQM +M #2 M ``Qr ivT2 Q` M BM@
i2`b2+iBQM ivT2X q2 +MMQi bBKTHv 2ti`+i i?2 BMTmi ivT2 Q7  7mM+iBQMX aBM+2

TTHB+iBp2 AMi2`b2+iBQM hvT2b

Rj

KmHiB@}2H/ `2+Q`/b `2 HbQ BMi2`b2+iBQM ivT2b BM Qm` bvbi2K- i?2 ivTBM; 7Q` T`Q@
D2+iBQMb U`mH2 h@S`QDV mb2b  bBKBH` B/2 iQ TTHB+iBQMbX
1tKTH2b q2 b?Qr M 2tKTH2 Q7 ?Qr i?2 `mH2 h@TT rQ`FbX amTTQb2 i?i
r2 ?p2 Γ = f : I → I → I, g : I → B → BX UI M/ B biM/ 7Q` Int M/ BoolV
Γ & (f, , g) ⇒ (I → I → I)  (I → B → B)

Γ & (f, , g) 2 ⇒ (I → I)  (B → B)
Γ & (f, , g) 2 true ⇒ BΓ &2⇒Ih@TT
Γ & true ⇒ B

h@TT

LQi2 i?i 7Q` bT+2 `2bQMb r2 QKBi i?2 TTHB+iBp2 bm#ivTBM; /2`BpiBQMb ?2`2r?B+? `2 bi`B;?i7Q`r`/X hQ BM72` i?2 ivT2 Q7 (f, , g) 2 true- r2 }`bi BM72` #Qi?
i?2 ivT2 Q7 (f, , g) 2 M/ trueX h?2 ivT2 Q7 (f, , g) 2 Bb (Int → Int)  (Bool →Bool)X h?Bb `2bmHi Bb +QKTmi2/ 7`QK TTHB+iBp2 bm#ivTBM; rBi? irQ BMTmib, ivT2
Q7 7mM+iBQM K2`;2b f, , g M/ ivT2 Q7 2X Gi2` r2 mb2 i?2 +QKTmi2/ `2bmHi Q7(f, , g) 2 iQ /2`Bp2 Qm` }MH ivT2 BoolX
9Xj a2KMiB+bh?Bb +H+mHmb /QTib  ivT2@/B`2+i2/ QT2`iBQMH b2KMiB+b (R9)- r?2`2 ivT2 M@
MQiiBQMb `2 mb2/ iQ +bi i2`Kb BMbi2/ Q7 #2BM; 2`b2/ 7i2` ivT2 +?2+FBM;X*biBM; q2 BMi`Q/m+2 i?2 +biBM; Dm/;K2Mi BM 6B;X 8X Cm/;K2Mi v *−→A v !
/2b+`B#2b i?i pHm2 v Bb +bi iQ pHm2 v ! #v ivT2 A- i?mb 7Q`+BM; i?2 pHm2 iQKi+? i?2 ivT2 bi`m+im`2 Q7 AX h?2 +biBM; `mH2b `2 2bb2MiBHHv i?2 bK2 b i?2`mH2b T`QTQb2/ #v >mM; 2i HX (R9)X _mH2b *i@J`;@G M/ *i@J`;@_ bii2
i?i K2`;2b rBHH #2 +bi iQ QM2 `2bmHi #v Q`/BM`v ivT2bX 6Q` 2tKTH2- b?Qr rBHH
#2 +bi iQ b?QrAMi #v ivT2 AMi → ai`BM;X
TTHB+iBp2 .BbTi+?BM; q2 BMi`Q/m+2  M2r Dm/;2K2Mi +HH2/ TTHB+iBp2
/BbTi+?BM; U6B;X 8V- r?B+? 2ti2M/b >mM; 2i HǶb (R9) T`HH2H TTHB+iBQM Dm/;2@K2MiX AM +QMi`bi iQ T`HH2H TTHB+iBQM- r2 Kmbi HbQ /2H rBi? Qp2`HQ/BM;X
Cm/;K2Mi (v • vl) !→ e /2b+`B#2b i?i pHm2 v Bb TTHB2/ iQ pHm2 Q` H#2H vl
UBX2X vl ::= v | lV M/ i?2M `2/m+2/ iQ  i2`K eX _mH2 TT@GK T2`7Q`Kb #2i@`2/m+iBQM M/ TT2M/b M 2ti` MMQiiBQM D iQ 2M7Q`+2 i?2 QmiTmi ivT2 Q7 i?2
TTHB+iBQMX _mH2 TT@S`QD bBKTHv 2ti`+ib i?2 pHm2 7`QK i?2 bBM;H2 `2+Q`/}2H/X h?2 BMi2`2biBM; T`i Bb i?2 `2KBMBM; i?`22 `mH2b 7Q` K2`;2bX h?2 7mM+iBQM,vl- bBKTHv 2ti`+ib Qmi i?2 ivT2 Q7  pHm2- iQ T`QpB/2 i?2 ivT2b iQ #2 +QK@
T`2/ rBi? TTHB+iBp2 bm#ivTBM;X hQ /2H rBi? Qp2`HQ/BM; r2 M22/ iQ BMi`Q/m+2`mH2b TT@J`;@G M/ TT@J`;@_- r?B+? HHQrb  K2`;2 iQ #2 TTHB2/ r?2M
QMHv QM2 Q7 i?2 pHm2b Bb TTHB+#H2X h?2 Hbi `mH2- `mH2 TT@J`;@S /2Hb rBi?
i?2 T`HH2H TTHB+iBQM- r?2`2 #Qi? pHm2b BM i?2 K2`;2 +M #2 TTHB2/X
PT2`iBQMH a2KMiB+b q2 T`2b2Mi Qm` bKHH@bi2T `2/m+iBQM `mH2b BM 6B;X eX_mH2b ai2T@AMi@MM M/ ai2T@``@MM TT2M/ 2ti` MMQiiBQMb iQ i?2 T`@iBH pHm2- BM Q`/2` iQ T`2b2`p2 i?2 T`2+Bb2 ivT2b i `mMiBK2X _mH2 ai2T@Sp@aTHBi
rBHH bTHBi i2`Kb ++Q`/BM; iQ bTHBii#H2 ivT2b- 7Q`+BM; i?2 ivT2 Q7 2+? #`M+? BM
K2`;2b iQ #2 Q`/BM`vX _mH2b ai2T@TT M/ ai2T@S`D /B`2+iHv +HH TTHB+iBp2
/BbTi+?BM;X _mH2 ai2T@oH@MM i`B;;2`b +biBM;, v Bb +bi iQ v ! #v ivT2 AX

R9sX sm2 2i HX
U*biBM;Vv *−→A v !
*i@AMi*i@hQT

B : A *−→AMi B : AMiv *−→hQT . : hQT*i@``*i@_+/
v *−→Ao v !

{H = v} *−→{H:Ao } {H = v ! }*i@J`;@GvR *−→Ao vR!

E <: C → Do

vR , , vk *−→Ao vR!

(λt. e : A → B) : E *−→(C→Do ) (λt. e : A → Do ) : (C → Do )*i@J`;@_
vk *−→Ao vk!

*i@M/
AR ! A " AkvR , , vk −* →Ao vk!

v *−→AR vRv *−→Ak vkv *−→A vR , , vk

UTTHB+iBp2 .BbTi+?BM;V(v • vl) !→ e
TT@GK

TT@S`QD

v *−→A v !

({H = v} • H) !→ v((λt. e : A → B) : C → D • v) !→ e[t *→ v ! ] : D
TT@J`;@G,v2 - # ,vl- = .

(vR • vl) !→ e

TT@J`;@S,v1 - # ,vl- %= .

,v2 - # ,vl- %= .

((vR , , vk ) • vl) !→ e

TT@J`;@_,v1 - # ,vl- = .

(vk • vl) !→ e((vR , , vk ) • vl) !→ e(vR • vl) !→ eR((vR , , vk ) • vl) !→ eR , , ek

(vk • vl) !→ ek

6B;X 8, *biBM; M/ TTHB+iBp2 .BbTi+?BM;_mH2 ai2T@MM Bb  +QM;`m2M+2 `mH2 rBi?  `2bi`B+iBQM i?i e +MMQi #2  `r
pHm2 pX h?2 `2KBMBM; `mH2b `2 MQ`KH +QM;`m2M+2 `mH2bX
9X9 hvT2 aQmM/M2bb
6Q` ivT2 bQmM/M2bb- r2 2KTHQv  T`QQ7 i2+?MB[m2 bBKBH` iQ i?2 QM2 #v 6M 2i
HX (Rk)X 6B`bi r2 M22/  MmK#2` Q7 `2bmHib #Qmi i?2 mtBHB`v `2HiBQMb mb2/ BM`2/m+iBQMX q2 b?Qr bQK2 Q7 i?2 KQ`2 BMi2`2biBM; H2KKb M2ti,
G2KK 3 US`2b2`piBQM UTTHB+iBQMb M/ S`QD2+iBQMbVVX
Ĝ A7 · & v1 v2 ⇒ A M/ v1 • v2 !→ e- i?2M · & e ⇐ AX
Ĝ A7 · & v.l ⇒ A M/ v • l !→ e- i?2M · & e ⇐ AX
G2KK N US`Q;`2bb UTTHB+iBQMb M/ S`QD2+iBQMbVVX
Ĝ A7 · & v1 v2 ⇒ A- i?2M ∃e, v1 • v2 !→ e
Ĝ A7 · & v.l ⇒ A- i?2M ∃e, v • l !→ eX

hvT2 bQmM/M2bb Bb T`Qp2M pB biM/`/ T`2b2`piBQM M/ T`Q;`2bb i?2Q`2KbX
h?2Q`2K R US`2b2`piBQMVX A7 · & e ⇔ A M/ e *−→ e ! - i?2M · & e ! ⇐ AX
h?2Q`2K k US`Q;`2bbVX A7 · & e ⇔ A- i?2M e Bb  pHm2 Q` ∃e ! , e *−→ e ! X

TTHB+iBp2 AMi2`b2+iBQM hvT2b

UaKHH@ai2T _2/m+iBQMV

e *−→ e !
ai2T@AMi@MM

ai2T@``@MM

B −* → B : AMi

λt. e : A → B *−→ (λt. e : A → B) : A → B

ai2T@Sp@aTHBi
AR ! A " Ak

ai2T@S`D(v • H) !→ v !
v.H *−→ v !

p : A *−→ p : AR , , p : Ak

ai2T@oH@MM
v *−→A v !
v : A *−→ v !

R8

ai2T@TT@G
eR *−→ eR!

eR ek *−→ eR! ek

ai2T@J`;@_ek *−→ ek!

vR , , ek *−→ vR , , ek!

ai2T@TT@_ek *−→ ek!

vR ek *−→ vR ek!

ai2T@_+/@_
e *−→ e !

{H = e} *−→ {H = e ! }ai2T@TT(vR • vk ) !→ evR vk *−→ eai2T@MM¬e ∈ pe *−→ e !
e : A *−→ e ! : A

ai2T@J`;@GeR *−→ eR!

eR , , ek *−→ eR! , , ek
ai2T@S`D@G
e *−→ e !
e.H *−→ e ! .H

6B;X e, PT2`iBQMH a2KMiB+b

8  *H+mHmb rBi?  .BbDQBMi J2`;2 PT2`iQ`
h?Bb b2+iBQM T`2b2Mib  b2+QM/ +H+mHmb rBi?  /BbDQBMiM2bb `2bi`B+iBQM QM K2`;2b (Rd)iQ `2+Qp2` /2i2`KBMBbKX h?Bb +H+mHmb 7Q`#B/b bQK2 +b2b Q7 +QMp2MiBQMH Qp2`@
HQ/BM;- #mi biBHH bmTTQ`ib i?2 Qi?2` 72im`2bX q2 7Q+mb QM i?2 F2v /Bz2`2M+2biQ i?2 T`2pBQmb +H+mHmb- bBM+2 KQbi `mH2b M/ `2HiBQMb `2 i?2 bK2X *QKT`2/
iQ T`2pBQmb +H+mHB rBi? /BbDQBMi BMi2`b2+iBQM ivT2b- i?2 KBM MQp2Hiv Bb i?2 mb2
Q7 i?2 TTHB+iBp2 bm#ivTBM; M/ /BbTi+?BM; `2HiBQMb- r?B+? 2M#H2b bmTTQ`i
7Q` `2+Q`/ T`QD2+iBQMb M/  `2bi`B+i2/ 7Q`K Q7 Qp2`HQ/BM; Mim`HHv UrBi?Qmi`2/mM/Mi ivT2 MMQiiBQMbVX
8XR .BbDQBMiM2bb
q2 2KTHQv i?2 /2}MBiBQM Q7 /BbDQBMiM2bb T`QTQb2/ #v PHBp2B` 2i HX (Rd)X AM7Q`@
KHHv- B7 HH +QKKQM bmT2`ivT2b Q7 irQ ivT2b `2 iQT@HBF2 ivT2b- r2 +M +QM+Hm/2
i?i i?2 irQ ivT2b `2 /BbDQBMiX hQT@HBF2 ivT2b `2 i?Qb2 r?Q `2 bmT2`ivT2b Q7
HH ivT2b U2X;X- T op- T op  T opV M/ /2}M2/ b,
hQT@HBF2 ivT2b/A0 ::= T op | /A0  /B0 | A →/B0 | {l :/A0}

LQi2 i?i i?2 BM+Hm/BM; Q7 bm+? ivT2b BMiQ iQT@HBF2 ivT2b Bb HbQ T`i Q7 i?2+HbbB+H "*. bm#ivTBM; `2HiBQM (j)X  7Q`KH bT2+B}+iBQM Q7 /BbDQBMiM2bb Bb;Bp2M #2HQrX h?2`2 Bb  bQmM/ M/ +QKTH2i2 b2i Q7 H;Q`Bi?KB+ /BbDQBMiM2bb `mH2bi?i +QM7Q`K iQ i?Bb bT2+B}+iBQMX h?2 BMi2`2bi2/ `2/2` +M +?2+F 2tBbiBM; rQ`F
7Q` i?2 H;Q`Bi?KB+ `mH2b (Rd-R9)X 6Q` bT+2 `2bQMb r2 QKBi i?2K ?2`2X.2}MBiBQM RX U.BbDQBMiM2bbV A∗B " ∀C B7 A <: C∧B <: C- i?2M C Bb iQT@HBF2X

Re

sX sm2 2i HX

AM Qm` +H+mHmb r2 HHQr K2`;2b Q7 /BbDQBMi 7mM+iBQMbX h?mb- ivT2b bm+? b Int →Int Q` Int → Bool `2 /BbDQBMiX hQ BM+Hm/2 7mM+iBQM ivT2b BMiQ Qm` /BbDQBMiM2bbivT2b HBF2 Int → T op b?QmH/ #2 iQT@HBF2 M/ bmT2`ivT2b Q7 HH ivT2b- bBM+2 Qi?2`@
rBb2 Int → Int M/ Int → Bool +MMQi #2 /BbDQBMi ++Q`/BM; iQ Qm` /2}MBiBQMX>Qr2p2`- i?Bb /BbDQBMiM2bb /2}MBiBQM T`2p2Mib bQK2 7Q`Kb Q7 Qp2`HQ/BM;X 6Q` 2t@
KTH2- i?2 ivT2 Q7 b?Qr Bb UAMi → ai`BM;V  U"QQH → ai`BM;V- r?B+? rBHH #2`2D2+i2/ #v i?2 /BbDQBMiM2bb +QM/BiBQM bBM+2 AMi → ai`BM; Bb MQi /BbDQBMi rBi?
"QQH → ai`BM;X 6Q` i?Qb2 irQ ivT2b r2 +M }M/  +QKKQM bmT2`ivT2 AMi "QQH → ai`BM;- r?B+? Bb MQi iQT@HBF2X hQ b22 r?v r2 b?QmH/ T`2p2Mi bm+? K2`;2b
bbmK2 i?i b?Qr Bb HHQr2/- i?2M b?Qr UR--i`m2V , ai`BM; Bb K#B;mQmb bBM+2
i?2 `2bmHi +M #2 2Bi?2` ]R] Q` ]i`m2]X LQi2 i?i bQK2 7Q`Kb Q7 Qp2`HQ/BM; `2
biBHH TQbbB#H2X 6Q` BMbiM+2 bm++--MQi rBHH #2 ++2Ti2/ bBM+2 AMi → AMi Bb /BbDQBMi
rBi? "QQH → "QQHX
q2 7QHHQr T`2pBQmb rQ`F QM /BbDQBMi BMi2`b2+iBQM ivT2b (9) M/ ;2M2`HBx2 Qm`
bm#ivTBM; `mH2 7Q` `mH2 a@hQT iQ #2 A <:/B0 r?2`2 /B0 K2Mb i?i B Bb  iQT@HBF2
ivT2X .BbDQBMiM2bb ?b BKTQ`iMi T`QT2`iB2b- r?B+? `2 ?2HT7mH 7Q` i?2 K2ii?2Q`v
Q7 i?2 +H+mHmbX AM T`iB+mH`- B7 irQ ivT2b `2 /BbDQBMi- i?2B` TTHB+iBp2 bm#ivTBM;`2bmHib mM/2` i?2 bK2 T`iBH ivT2b `2 HbQ /BbDQBMiXG2KK Ry UTTHB+iBp2 am#ivTBM; M/ .BbDQBMiM2bbVX A7 A ∗ B- A # S =C1 M/ B # S = C2 - i?2M C1 ∗ C2 X
aQmM/M2bb M/ +QKTH2i2M2bb Q7 TTHB+iBp2 bm#ivTBM;X qBi? i?2 KQ`2
;2M2`H bm#ivTBM; `mH2 7Q` iQT@HBF2 ivT2b- TTHB+iBp2 bm#ivTBM; `2KBMb bQmM/
UrBi? H2KKb j- 8 BM a2+iBQM jV rBi? `2bT2+i iQ bm#ivTBM;X >Qr2p2`- i?2 +QK@
TH2i2M2bb Q7 Qm` TTHB+iBp2 bm#ivTBM; M22/b iQ #2 bHB;?iHv /Ti2/X
G2KK RR U*QKTH2i2M2bb Q7 TTHB+iBp2 am#ivTBM;VX A7 A <: B → Ci?2M (∃D, A # B = D ∧ D <: C) ∨ T op <: CX
AM Qi?2` rQ`/b- TTHB+iBp2 bm#ivTBM; Bb +QKTH2i2 2t+2Ti 7Q` i?2 +b2 r?2`2 i?2
QmiTmi ivT2 Bb iQT@HBF2X AM bm+? +b2 TTHB+iBp2 bm#ivTBM; 7BHbX LQi2 i?Qm;? i?i
i?Bb 7BHm`2 T`2p2Mib bi`M;2 T`Q;`Kb 7`QK #2BM; ivT2@+?2+F2/X 6Q` 2tKTH2bm#ivTBM; ?b BMbiM+2b hQT I,  → hQT- HHQrBM; UR , hQTV k iQ #2 r2HH@ivT2/r?B+? rQmH/ `2[mB`2 bT2+BH i`2iK2Mi BM i?2 ivTBM; `mH2bX q2 `2D2+i bm+? +b2bKFBM; i?2 ivTBM; `mH2b bBKTH2`- M/ pQB/BM; ivT2@+?2+FBM; bm+? T`Q;`KbX
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