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slide-1
SLIDE 1 ✂✁ ✄ ☎ ✆ ✝ ✞ ✟✡✠ ☛ ☞ ☛ ✌ ✍✏✎ ✑✓✒✔✒✖✕ ✑ ∗ ✗ ☎ ✍✘✎ ✍✚✙✜✛✣✢✥✤ ✗✧✦ ✍ ☞ ★✪✩✫✩✜✬ ∗

(*)DISI - Univ. of Genova, Italy,

✭✯✮✰✮✲✱✴✳✶✵✷✵✹✸✰✸✷✸✻✺✽✼✿✾❁❀❂✾❃✺❅❄✷❆❇✾❉❈❋❊✫✺●✾❍✮

(+)SCS - Univ. of Tech. Sydney, Australia,

✭✯✮✰✮✰✱✻✳✶✵✷✵✹✸✷✸✰✸✻✺■❀✲❏▲❑✲❀▼✺◆❄✷✮❋❀✪✺❖❊❁✼❁❄✴✺❖P✹❄
★❙❘❚✬❱❯ ✛ ❘❚✬✔❲✻✩ ✑❇❳❨✛ ◗ ❩ ✒✶✑▼❬ map rev ❬ L ❭ X → l ❭ X ❪❴❫❛❵❝❜❡❞❣❢✷❜✐❤✹❥❋❜❧❦♥♠✷♦♣♦rq s ❢❂t✉❤✈❦♣❥❋❜●❤✡t✇❤②①✇❜❧❦♣❥❝③ ❵④①⑤❤②①❉⑥
✑❛✒■✛ ❘ ✑❙⑧ ⑨ ★ ⑦✰⑩ ❬✴❶❸❷❡❹✻❺④❺❼❻❋❽❛❾✹❿➁➀❨❻❋❽❚➂✔❾❋❶➄➃ ❹✴➀❨❻❛➅❂❾➆➃ ➇➈❻❋➉✰➂✔❾❂❶➋➊➌❹➍❻❋➎✷➂✔❾ ❶➄➏ ❢❂❥ ❻❋➉➐❾✓➏ ❤➆❵ ❻❋➉✰➂✔❾❙❶➄➏ ♠❚q ❻❋➉ ♠ ➂
➑✻✛ ❘ ✬●➒ ➓☞ ☛ ❬❛➃ ➔ →

. . .

➣↕↔✇➙ ❲➜➛✿❘✲★ ⑦ ➒ ❾ ❜➝q➐❞➞❤②① ❾ ①✖➟➡➠❝❤✹s ♠✰① ❾ ❩ ★ ✒❱✢ ◗ ★ ⑦▲❩ ➑ ✬●➒ ◗ ➢ ❜●❤✡t✇s ① ❾ ❜➝q➐❞➞❤➓♠✰①➡①⑤❦➤③❋❥④s➥❤✹❥❋❜ ❾ t✉❤✲➦✿❵❇➟⑤❜❧❦♣❢❂❥
➑✻✛ ❘ ✛ ⑦ ✑ ➧☞ ☛ ↔✇➙ ❲➨➛➩❘✲★ ⑦ ➒ ❬ ➢ ➫ ❵➐❥✓➟⑤❜✉❢❂t✇① ➢ ➒ ➑✻✛ ❩ ✑✓✒✽✢ ↔✇➙ ❲➨➛➩❘✲★ ⑦ ➒ ❶➄➏ ♠❚q ❻❋➉ ❫ ➂ ➢ t✇❤✹❞❛t✉❤②①⑤❤✹❥❋❜➭♠✹❜❧❦♣❢❂❥④①❡❢ ➫ ①✖➠➐♠✷❞➞❤✹♦rq ➫ ❵❼❥✓➟⑤❜✉❢❂t✇①
  • ❩➜⑦
★ ❩ ✑ ⑦ ❘✲✬ ✑ ➒ ★ ↔ ➧☞ ☛ ❪✶♠✰①✚❤❍➯✓❞➞❤➆➟⑤❜✐❤✲➦➲⑥ ❶❸❷➳➏❋➃ ❻❂➵✯➂✔❬ ➢ ❜➸q➐❞➲❤✈❦♣❥ ➫ ❤✡t✇❤✹❥✓➟➺❤➻♠✷♦➤③❋❢❂t➡❦r❜➸➠❛s ➢ ①➭❵➼❫❼➽●❤➆➟❉❜➾t✉❤✲➦✿❵❇➟⑤❜❧❦♣❢❂❥ ❾➐➊ ➠➼❵❛t✉➟❍➠❝➚ ❿ ❢➪①➡①❉❤➆t➶❞❛t✉❢✲❞➞❤➆t✐❜➝q ➢ ①❧❜➡t✉❢❂❥❝③ ❥❛❢❂t✇s ♠✷♦♣❦➤➹❚♠✹❜❧❦♣❢❂❥ ❪❴❥❛❢➴➘➼➯❃⑥

1

slide-2
SLIDE 2 ✠ ◗ ★❙❘❚✬❱❯ ✛ ❘❚✬✔❲✻✩ ✑❇❳❨✛ ◗ ❩ ✒✶✑
  • map
❬ ∀F ❬②➅ .∀X, Y. ❪ X → Y ⑥ → FX → FY ➢ ➦④♠✹❜➭♠➴❞❣❢✯♦rq➐s ❢❂t✇❞❛➠❛❦➷①✖s ❬ F ➘➼➯✓❤✲➦ ❾ X, Y ➬ ♠✷t●q ➣➻①➭➠➐♠✷❞➲❤✈❞❣❢✯♦rq➐s ❢❂t✉❞❛➠❛❦➷①✖s ❬ F ➬ ♠✷t❍❦♣❤②① ❾ X, Y ➘➼➯✓❤✲➦
  • L
❬❁➅ ❜➸➠❝❤✈♦♣❦➮①❧❜ ➫ ❵❼❥✓➟⑤❜✉❢❂t
  • rev
❬ ∀X.LX → LX t✉❤ ➬ ❤✡t❧①⑤❦♣❥❝③ ♠↕♦♣❦➮①✇❜ ➱ ➑✻✛ ❘ ➛ ✛ ❲ ✃ ✑ ✬✔❲ ↔ ✑ ⑦✯⑦ ✑❙⑧✏❐
❵④♦♣♦rq ❤➡➯✓❞➐♦♣❦♣➟➺❦r❜

map L

❒ LX LX ❪ rev X ⑥ ❬ L ❭ X → L ❭ X

map L L

❒ X L ❒ X ❪ rev LX ⑥ ❬ L ❭ X → L ❭ X
  • ❜➝q➐❞➞❤❮❦♣s
❞➐♦♣❦❰➟✹❦r❜

map L

❒ rev ❬ L ❭ X → L ❭ X

map L rev

❬ L ❭ X → L ❭ X ➣ ➫ ❵④♦♣♦rq ❦♣s ❞❼♦♣❦♣➟➺❦r❜

map rev

❬ L ❭ X → L ❭ X ✠ ❩✻❩➨⑦ ★ ✛ ➛ ➑➍✑ ➒
➠➐♠✷t➡❦r❜➝q ❬ F s↕❵④①✇❜➳❫➞❤✂❤❍➯✓❞❼♦♣❦♣➟➺❦r❜
♠✰①✖Ï➪❤✹♦♣♦➞❜➸q➐❞➲❤❮➟✹♦♥♠✰①➡①⑤❤②① ❬ ➟✹❢✿➦❇❤ ➫ ❢❂t map ①✖❵❛❞❝❞❼♦♣❦♣❤✲➦ ❫➼q ❵➐①❉❤➆t
  • ❹✥➃
➔✥❬ F ❪ GX ⑥ Ð ❪ F ◦ G ⑥➺❪ X ⑥ ❾ ❢❂❥❛♦rq ❦➷①✖❢❂s ❢❂t✇❞❛➠❛❦♣➟▲Ñ

2

slide-3
SLIDE 3 Ò ✑❇✢ ✬ ⑧✻✑➲✛
  • F
❪ GX ⑥ Ð ❪ F ◦ G ⑥➺❪ X ⑥ ❾ ❢❂❥❛♦rq ❦➷①✖❢❂s ❢❂t✇❞❛➠❛❦♣➟▲Ñ
  • F
❪ GX ⑥ ❬ F ➫ ❵❼❥✓➟⑤❜✉❢❂t ❾ GX ➦④♠✹❜➭♠
  • FG
❪ X ⑥ ❬ ❪ F ◦ G ⑥ ➫ ❵➐❥✓➟⑤❜✉❢❂t ❾ X ➦④♠✹❜➭♠
  • Ó②Ô➸Õ✹Ö❝×✐Ø●Ù✓Ú❋Û❁Ü
❶❸❷ ❤✹❥ ➵➼Ý❚➂ ❢ ➫④➫ ❵➐❥❛➟❉❜✉❢❂t❧① ❬ ➟➺❢❂s ❞❃❢➪①❍❦r❜❧❦♣❢❂❥➥❵❛❞Þ❜➭❢ß❦➮①✖❢à❪✶♠✰① ①✖❢✷♦♣❵❝❜✐❦♣❢❂❥④①✏❜✉❢à➦④❢❂s ♠✷❦♣❥ ❤➆á✿❵➲♠✹❜❧❦♣❢❂❥④①⑤⑥ãâ
  • ➟✡♠✷❥❛❢❂❥❛❦❰➟❁♠✷♦▼❦➷①✖❢➪①➌♠✰①❡➦❛♠✹❜➭♠➴➦④❦➮①❧❜➡t➡❦♣❫❛❵✿❜❧❦♣❢❂❥
❶➄➏ ♠❚q ❻❋➉ ➟ ➂

F

❪ GX ⑥

FG

❪ X ⑥

dex

L

❪ L ❪ LX ⑥✉⑥ map rev

✲ L

❪ L ❪ LX ⑥✉⑥

L

❒ ❪ LX ⑥

dex

map rev

✲ L

❒ ❪ LX ⑥

dex

H

❪ X ⑥

map f

✲ H

❪ Y ⑥

3

slide-4
SLIDE 4
❲➨➛➩❘✲★ ⑦ ➒ ✗ ❘ ✢ ❩ ✑ ➒ ✗ ➒✰➛ ➑➍✑ ◗ ✛ ➣ ➫ ❵❼❥✓➟⑤❜✉❢❂t✇①➳ä

⊢ F

❬ m ❢ ➫ ♠✷t➡❦r❜➝q m

F, G

❬②❬ Ð

X | C |

å

m i | FGn | µmF

  • ❜➝q➐❞➞❤②①❡ä

⊢ τ

❬✖æ

τ

❬②❬ Ð

X | τ

ç → τ ❒ | . . . | F ❪ τ ⑥
  • ①➭➟❍➠❝❤✹s
♠èä

⊢ σ σ

❬②❬ Ð

τ | ∀X.σ | ∀X

❬ m.σ é ➠❝❤➆t✉❤✂ä ❬❁❬ Ð

|

ä , X ❬ m | ä , X ❬➭æ ❜➝q➐❞➞❤✂➟✹❢❂❥➪❜✐❤➡➯➐❜❚â
❲➜➛✿❘✲★ ⑦ ➒❼ê ✛ ⑦ ✬✽❘ ✢ ✛ ❲ ⑧ ë ◗ ✑➲✛ ❲➜✬■❲✻✩✜ì
  • ➟➺❢❂❥④①✇❜✉♠✷❥❋❜
➫ ❵➐❥✓➟⑤❜✉❢❂t✇① ❬ ➢ ➅▲❬❍❽ ❵➐❥❛❦r❜✏❜➝q➐❞➞❤ ➢

×,

→í❬⑤➎ ❫❛❦♣❥➐♠✷t●q ❞❛t✇❢❝➦✿❵❇➟⑤❜➈♠✷❥❇➦ ①✖❵❼s ➣Þ❞❛t✇❢✯➽●❤➆➟⑤❜❧❦♣❢❂❥ å

m i

❬ m ❾ é ➠❝❤➆t✉❤ i ∈ m å

m i

❪ X ⑥ ∼ Ð

Xi

➣↕➟➺❢❂s ❞❣❢❋①❍❦r❜❧❦♣❢❂❥ FGn ❬ n ❾ ❦ ➫ F ❬ m ♠✷❥❇➦ Gi∈m ❬ n

FGn

❪ X ⑥ ∼ Ð

F

❪ Gi ❪ X ⑥

i∈m

  • ❦♣❥❇➦✿❵❇➟⑤❜❧❦
➬ ❤ µmF ❬ m ❾ ❦ ➫ F ❬ m → ➅

µmF

❪ X ⑥ ∼ Ð

µY.F

❪ X, Y ⑥

4

slide-5
SLIDE 5 ➱ ➑✻✛ ❘ ✛ ⑦ ✑ ➓☞ ☛ ↔✇➙ ❲➜➛✿❘✲★ ⑦ ➒ ❐ ➣è♠➴t✇❤❉➘➐❥❝❤➆sà❤➆❥➪❜➾❢ ➫ ❜➝q➐❞➞❤❉➚■➟✹❢❂❥➐①✇❜➡t✇❵❇➟❉❜✉❢❂t❧① ❬ é ❦r❜➸➠ î Ú❋ï Ø ♠✯➦❛➦④❦r❜✐❦♣❢❂❥➐♠✷♦❃①✇❜➡t✇❵❇➟⑤❜❧❵❛t✉❤②ð✰❞❛t✇❢✲❞➲❤➆t✐❜✐❦♣❤②① ❤❂âñ③➞â ➫ ❵➐❥❛➟❉❜✉❢❂t❧① ❾ ①✖➠➐♠✷❞➞❤✹♦rq ➫ ❵❼❥✓➟⑤❜✉❢❂t✇① ❾ . . .
  • ❲❨★▼❘
➃ ➔ ➫ ❵❼❥✓➟⑤❜✉❢❂t✇① ❬ ❫❛❵✿❜➳➟➺❢❂❵❛♦➷➦ ❫➞❤ Ø②òóÕ✹Ú✯ô➐Ø✹ô ♠✰① ➃ ➔ ①✇❜➡t✇❵❇➟⑤❜❧❵❛t✉❤②①

F ∗

❬✡õ❃ö❂÷ ø é ➠❝❤✹❥ F ❬ m ❾ é ➠❝❤➆t✉❤ õ❃ö❂÷✣ù✜ú û õ❃ö❂÷ ü✓ý❛þ▼ÿ ✂✁ ✄ ☎ ✆✞✝ ✟✡✠ ÿ☞☛❼ü✓ý❛þ▼ÿ ✝✍✌ ✎✑✏✓✒ ✔ ✕ ✆✖✂✁✑✗✙✘✚✜✛✢✌✣✗✙✘✚✂✁ ✄✍✗✙✘✚✜✛ ✄ ☎ ✤✙✤✙✤ ÿ✦✥✓✧★☎ ❫❛❵✿❜ ❲ ★ ❦♣❥ ➬ ❤✡t❧①❉❤ ô④Ø✹Õ✹Ú✯ô➼Ô❱ò❛Ù â
  • ❲❨★▼❘
➇ ♠✰①✖Ï➪❤✹♦♣♦✘ð✪✩➓❢ ➫ ❤➆t✏❜➝q➐❞➞❤✈➟➺♦♥♠✰①➡①⑤❤②① ❬ ❫❛❵✿❜✈➟✹❢❂❵④♦➷➦ ❫➞❤ Ø❁òóÕ➆Ú✷ô④Ø➺ô ♠✰①❮❤✹♦♣❤✹s➥❤✹❥❋❜✇①✈❢ ➫ ♠➥❜➸q➐❞➲❤↕➟➺♦♥♠✰①➡①➥❪ ①⑤♠✷s➥❤ ❦➷➦❝❤✲♠í♠✰①➾♠❁❫❃❢ ➬ ❤②⑥ãâ ✫✭✬✯✮✱✰✳✲✵✴ ✶✵✷✹✸ ✺ ✻✽✼✿✾❁❀❃❂❅❄❇❆❉❈❊✻❋❄❇❆❍● ■✡✾ ✲❑❏▼▲❖◆✍P✓✲✵◗❘◆ ❙❁■❚❂❯■❚❂✜❱❁❲❨❳ ❄✡✻❩❂✂❱❁❲❨❳ ❀❬❄❇✾❭❈❪❂❃❆❍✼✑❀❃❂❅❄❇❆❉❈❴❫

5

slide-6
SLIDE 6 ❵ ★ ◗ ✑ ➒ ✑ ◗ ✛ ❲✜❘❚✬✔➛❼➒ ↔ ★ ⑦ F ❬❚➅ ✬✔❲ ❵✻✑ ❘
  • F
❬❛❵ ✑ ❘ → ❵✻✑ ❘ ➫ ❵❼❥✓➟⑤❜➭❢❂t ➟➺❢❂s ❞❣❢❋①❍❦r❜❧❦♣❢❂❥ ♠✷❥❇➦ ❞❛t✇❢✯➽●❤➆➟⑤❜❧❦♣❢❂❥④①❡❵❛❞ ❜✉❢à❤➆á✿❵➲♠✷♦♣❦r❜➸q ❥❛❢à❦♣s ❞❼♦♣❦♣➟➺❦r❜➳①✖➠➐♠✷❞➞❤✈❞❣❢✯♦rq➐s ❢❂t✇❞❛➠❛❦➷①✖s ❪❴❥❛❢❂t µmF ⑥
  • F
❬❛❵ ✑ ❘ → ❵✻✑ ❘ ➫ ❵❼❥✓➟⑤❜➭❢❂t é ❦r❜➸➠ t✉❤✹♦♥♠✹❜❧❦♣❢❂❥➐♠✷♦✫♠✯➟⑤❜❧❦♣❢❂❥

X

ç

f

✲ Y

ç

X

R g

✲ Y

S

❦♣s ❞➐♦♣❦♣❤②①

FX

ç

Ff

✲ FY

ç

FX

FR Fg

✲ FY

FS

  • F
❬❛❵ ✑ ❘ → ❵✻✑ ❘ ①✖➠➐♠✷❞➞❤✹♦rq ➫ ❵❼❥✓➟⑤❜➭❢❂t ❾ ❦❱â ❤❂â

F

❪ X ⑥

dataX

✲ L

❪ X ⑥

S ∼

Ð

F

❪ ➅ ⑥

F

❪❧Ñ➄⑥

❜ Ð

data

ç

✲ L

❪ ➅ ⑥ ∼ Ð

N

L

❪✇Ñ➄⑥

F

❪ X ⑥ ∼ Ð ❝ s ❬ S.X ❞ s ❥➐♠✹❜❧❵❛t➡♠✷♦♣♦rq ❦♣❥ X ➣ F t✉❤✹❞❛t➆â S ∈ ❵✻✑ ❘ , ❜ ❬ S → N ❢ ➫ ①➭➠➐♠✷❞➲❤✹♦rq ➫ ❵➐❥❛➟❉❜✉❢❂t

F

❪ X ⑥ Ð➻Ð ❝ s ❬ S.X ❞ s å ç ❡ Ð➻Ð
  • ➅ , λ
❬②➅ . ➅ ➟➺❢❂s ❞❣❢❋①❍❦r❜❧❦♣❢❂❥ ♠✷❥❇➦ ❞❛t✇❢✯➽●❤➆➟⑤❜❧❦♣❢❂❥④①❡❢❂❥❛♦rq ❵❛❞ ❜➭❢è❦➮①➭❢ ❦♣s ❞➐♦♣❦♣➟➺❦r❜❡①✖➠➐♠✷❞➞❤✈❞❃❢✷♦rq➐s ❢❂t✉❞④➠❛❦➮①➭s ❪✶♠✷❥❇➦ µmF ⑥

6

slide-7
SLIDE 7 ✆ ✑ ⑦➪◗ ➒ ✛ ✒■✛ ☎ ➑ ➙ ⑦ ➛ ➑
  • ❜●❤✡t✇s
①❡ä ❢❍❣ ⊢ t ❬ σ

t

❬❁❬ Ð

x | c | λx.t | t

ç t ❒ | ♦♣❤❉❜ x Ð

t

ç ❦♣❥ t ❒ é ➠❝❤➆t✉❤❤❣ ❬②❬ Ð

|

❣ , x ❬ σ ❪✽❦ ➫ ä

⊢ σ

⑥ ❜●❤✡t✇s ➟✹❢❂❥➪❜✐❤➡➯➐❜❚â ✞ ➙ ✒❖✑ ➒ ↔ ★ ⑦ ❘ ✢ ❩ ✑ ✛ ➒❂➒➪✬❖✩ ❲ ◗ ✑ ❲✜❘ ❪ c ⑥ ä ❢❍❣ ⊢ ä ❢❍❣ ⊢ c ❬ σc ❪ x ⑥ ä ❢❍❣ ⊢ ä ❢❍❣ ⊢ x ❬ σ σ Ð ❣❨❪ x ⑥ ❪■➣❥✐ ❞❝❞

n

⑥ ä ❢❪❣ ⊢ t ❬ ∀X ❬ n.σ ä

⊢ F

❬ n ä ❢❍❣ ⊢ t ❬ σ{F/X} ❪■➣❥❦

n

⑥ ä , Y ❬ n ❢❪❣ ⊢ t ❬ σ{Y/X} ä ❢❉❣ ⊢ t ❬ ∀X ❬ n.σ

Y ∈

❧♥♠ ❪❱ä ⑥ ❪♦✐ ❞❝❞❣⑥ ä ❢❉❣ ⊢ t ❬ ∀X.σ ❣ ⊢ τ ä ❢❍❣ ⊢ t ❬ σ{τ/X} ❪♦❦➾⑥ ä , Y ❬➭æ ❢❉❣ ⊢ t ❬ σ{Y/X} ä ❢❉❣ ⊢ t ❬ ∀X.σ

Y ∈

❧♥♠ ❪❱ä ⑥ ❪✶♠✷❞❝❞➲⑥ ä ❢❪❣ ⊢ t ❬ τ ç → τ ❒ ä ❢❍❣ ⊢ t ç ❬ τ ç ä ❢❍❣ ⊢ ❪ t t ç ⑥ ❬ τ ❒ ❪ λ ⑥ ä ❢❍❣ , y ❬ τ ç ⊢ t{y/x} ❬ τ ❒ ä ❢❍❣ ⊢ ❪ λx.t ⑥ ❬ τ ç → τ ❒

y ∈

❧♥♠ ❪✽❣ ⑥ ❪✽♦♣❤❉❜❉⑥ ä ❢❪❣ ⊢ t ç ❬ σ ç ä ❢❪❣ , y ❬ σ ç ⊢ t ❒ {y/x} ❬ σ ❒ ä ❢❉❣ ⊢ ♦♣❤❉❜ x Ð

t

ç ❦♣❥ t ❒ ❬ σ ❒

y ∈

❧♥♠ ❪✽❣ ⑥

7

slide-8
SLIDE 8 ☎ ★ ❲➜➒✲❘ ✛ ❲✜❘✯➒ ✛ ❲ ⑧ ❘ ➑ ✑ ✬ ⑦ ➒✰➛ ➑➍✑ ◗ ✛ s ♠✷❞

m

❬ ∀F ❬ m.∀X, Y . ❪ Xi → Yi ⑥

i∈m → F

❪ X ⑥ → F ❪ Y ⑥ ❞➞❤➡➯

m,i

❬ ∀X.Xi → å

m i

❪ X ⑥ ❞❛❦♣❥

m,i

❬ ∀X. å

m i

❪ X ⑥ → Xi ➦❝❤➡➯

m,n

❬ ∀F ❬ m.∀G ❬ n.∀X.F ❪ Gi ❪ X ⑥

i∈m

⑥ → FGn ❪ X ⑥ ➦④❦♣❥

m,n

❬ ∀F ❬ m.∀G ❬ n.∀X.FGn ❪ X ⑥ → F ❪ Gi ❪ X ⑥

i∈m

⑥ ❦♣❥❋❜✖t✉❢

m

❬ ∀F ❬ m → ➅ .∀X.F ❪ X, µmF ❪ X ⑥✉⑥ → µmF ❪ X ⑥ ➫ ❢✯♦➷➦

m

❬ ∀F ❬ m → ➅ .∀X, Y. ❪ F ❪ X, Y ⑥ → Y ⑥ → µmF ❪ X ⑥ → Y ❵➐❥ ❬í➅ ❞➲♠✷❦♣t ❬ ∀X ❡ , X ç .X ❡ → X ç → X ❡ × X ç ❞④❦

j

❬ ∀X ❡ , X ç .X ❡ × X ç → Xj ❦♣❥

j

❬ ∀X ❡ , X ç .Xj → X ❡ →

X

ç ➟❁♠✰①⑤❤ ❬ ∀X ❡ , X ç , Y. ❪ X ❡ → Y ⑥ → ❪ X ç → Y ⑥ → X ❡ →

X

ç → Y ➘➼➯ ❬ ∀X. ❪ X → X ⑥ → X
✛ ✬■❲ ⑦ ✑ ➒ ➙ ✒ ❘ ❬ ➃ ➔ ➚■♦♣❦♣Ï➪❤➁❜➝q➐❞➞❤❮❦♣❥ ➫ ❤➆t✉❤✹❥✓➟➺❤➻♠✷♦➤③❋❢❂t➡❦r❜➝➠④s â

8

slide-9
SLIDE 9 ✆ ✢ ❩ ✑q♣ ↔ ⑦ ✑④✑ ✞ ✑❙⑧ ➙ ➛➩❘✲✬❖★✜❲ ❪ λx.t ❒ ⑥ t ç > t ❒ {t ç /x} ♦♣❤❉❜ x Ð

t

ç ❦♣❥ t ❒

> t

❒ {t ç /x} ❞❛❦

j

❪✽❞➞♠✷❦♣t t ❡ t ç ⑥ > tj ➟✡♠✰①❉❤ f ❡ f ç ❪✽❦♣❥

j t

⑥ > fj t ❞❛❦♣❥

m,i

❪✽❞➞❤➡➯

m,i t

⑥ > t ➦④❦♣❥

m,n

❪❱➦❝❤➡➯

m,n t

⑥ > t s ♠✷❞

m fk∈m

❪✽❞➞❤➡➯

m,i t

⑥ > ❞➞❤➡➯

m,i

❪ fi t ⑥ s ♠✷❞

n fk∈n

❪❱➦❝❤➡➯

m,n t

⑥ > ➦❝❤➡➯

m,n

❪✽s ♠✷❞

m

❪✽s ♠✷❞

n f

i∈m t

⑥ s ♠✷❞

m fi∈m

❪✽❦♣❥❋❜✖t✉❢

m t

⑥ > ❦♣❥❋❜✖t✉❢

m

❪✽s ♠✷❞

m

✤ ç f ❪❴s ♠✷❞

m f

⑥ t ⑥ s ♠✷❞ ❒ f ❡ f ç ❪✽❞➞♠✷❦♣t t ❡ t ç ⑥ > ❞➲♠✷❦♣t➓❪ f ❡ t ❡ ⑥➁❪ f ç t ç ⑥ s ♠✷❞ ❒ f ❡ f ç ❪✽❦♣❥

j t

⑥ > ❦♣❥

j

❪ fj t ⑥ s ♠✷❞ ❡ ❵➐❥

>

❵❼❥ ➫ ❢✷♦➷➦

m f

❪✽❦♣❥❋❜✖t✉❢

m t

⑥ > f ❪✽s ♠✷❞

m

✤ ç ❪ λx.x ⑥

i∈m

❪ ➫ ❢✯♦➷➦

m f

⑥ t ⑥ ◗ ✛ ✬✔❲ ⑦ ✑ ➒ ➙ ✒ ❘✯➒
➠❋❵❛t➭➟➡➠❇➚ ❿ ❢❋①➡①⑤❤✡t✏❞❛t✇❢✲❞➲❤➆t✐❜➸q✿❢
❵❋❫➐➽➝❤✡➟⑤❜➾t✉❤✲➦✿❵❇➟⑤❜❧❦♣❢❂❥❭❢
❜✖t✉❢❂❥❝③í❥❛❢❂t✇s ♠✷♦♣❦➤➹❚♠✹❜❧❦♣❢❂❥ ❪ é ➠❝❤✹❥ ❥❛❢➴➘➼➯❃⑥ãâ

9

slide-10
SLIDE 10 ❵✥✛ ◗ ❩ ✒❖✑ ⑦ ✑✣⑧ ➙ ➛➩❘❚✬✶★✜❲➨➒
❢✷♦➷➦ ❡ f ❪✽❦♣❥❋❜✖t✉❢ ❡ t ⑥ > f ❪❴s ♠✷❞ ç ❪ ➫ ❢✯♦➷➦ ❡ f ⑥ t ⑥

F

❪ µ ❡

F

⑥ ❦♣❥❋❜➡t✇❢ ❡

✲ µ

F F

❪ Y ⑥ s ♠✷❞ ç ❪ ➫ ❢✯♦➷➦ ❡ f ⑥

f

✲ Y ❄

➫ ❢✯♦➷➦ ❡ f
  • s
♠✷❞ ç f ❪❴❦♣❥❋❜➡t✇❢ ç t ⑥ > ❦♣❥❋❜✖t✉❢ ç ❪✽s ♠✷❞ ❒ f ❪✽s ♠✷❞ ç f ⑥ t ⑥

F

❪ X, µ ç F ❪ X ⑥✉⑥ ❦♣❥❋❜➡t✇❢ ç

✲ µ

ç F ❪ X ⑥

F

❪ Y, µ ç F ❪ Y ⑥✉⑥ s ♠✷❞ ❒ ❪ f, s ♠✷❞ ç f ⑥

❦♣❥❋❜➡t✇❢ ç

✲ µ

ç F ❪ Y ⑥

s ♠✷❞ ç f

10

slide-11
SLIDE 11 ☎ ★✜❲➨➛ ✒ ➙ ➒➪✬✶★✜❲➨➒
  • ❹✥➃
➔ ➫ ❵❼❥✓➟⑤❜✉❢❂t✇①➾♠✰①★✐ ❧ æ ➢ ①⑤❤ ➬ ❤✡t✖♠✷♦ Ô❱ï❤r❁s Ø②ï Ø②ò✪×➝Ö❇×❖Ô➝Ú❋ò î❡❞❣❢❋①➡①❍❦♣❫❼♦♣❤ ➢ ♠➴①➭s ❢✿❢✷❜➸➠ é ♠❚q ❜✉❢è❤❍➯➐❜●❤✹❥❇➦ ➃ ➔❘t ✉✇✈ ❺✓➔✥❾ ➟➺❢❂t✉❤⑤➚ ➃ ➔ ⊂ ❹✴➃ ➔ ➢ ❫④❵❝❜➳t➡♠✹❜➸➠❝❤✡t✏♦♣❢ é ➚■♦♣❤ ➬ ❤✹♦ ❾ ❤❂âñ③➞â ♦♣❦➷①✇❜✇① L Ð

µ

ç F é ➠❝❤✡t✇❤ F Ð → ➅ ❒ , × ❥④❦♣♦ Ð ❪❴❦♣❥❋❜➡t✇❢ ç ◦ ➦❝❤➡➯ ❒ , ❒ ◦ ❦♣❥ ❡ ◦ ➦❇❤➡➯ ❡ , ❒ ⑥➜❵➐❥ ❬ LX
  • ①✇❜✐❵✓➦❇q
❹✥➃ ➔ ❜✐❤➆t✉s ①➓♠ ♦♥♠ ➊ ❵❛t✇t✐q ❬ t➡❦♣➟➡➠❝❤✡t➁s ❢✿➦❇❤✹♦❰① ❾ ❫➲❤❉❜✔❜✐❤➆t❮❢✲❞❇➚ ❜❧❦♣s ❦➤➹❚♠➆❜✐❦♣❢❂❥④① ❶❸➇✂➃ ❻❋➉✲➂
  • ➟✡♠✷❥❛❢❂❥❛❦❰➟❁♠✷♦▼❦➷①✖❢➪①❡❦♣❥
➫ ❤✡t✇❤➆❥❛➟✹❤✂❞❛t✇❢❂❫➐♦♣❤✹s
  • ❹✥➃
➔ ♠✰① ❦♣❥❋❜✐❤➆t✉sà❤✲➦④❦♥♠✹❜✐❤ ♦♥♠✷❥❇③❋❵➞♠✲③❋❤ ➫ ❢❂tÞ♠✷❥➐♠✷♦rq❼①❍❦➷① ❢ ➫ ♠✯➟➆➟➺❤②①➡①➓❜✉❢ ➦④♠✹❜➭♠ ❪✽➟❁♠✷❥④❢❂❥❛❦♣➟✡♠✷♦▼❦➮①✖❢❂s ❢❂t✉❞❛➠④❦➮①✖s ①❉⑥ ❶❸➀➶➏✷❻❛➅❂❾❛➔ ❤✡t ❻❋➎❚➂

11

slide-12
SLIDE 12 ①✇②④③⑤②⑦⑥❉②⑦⑧✖⑨✞②❶⑩

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