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Jul 21, 2023 223 likes •634 views

t rt t s r r rs t

slide-1
SLIDE 1

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪

❋♦r♠❛❧❧② ❝♦♠♣❛r✐♥❣ ❛♣♣r♦❛❝❤❡s t♦ ❞❛t❛t②♣❡✲❣❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣✱ ✉s✐♥❣ ❆❣❞❛

❏♦sé P❡❞r♦ ▼❛❣❛❧❤ã❡s

❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆♥❞r❡s ▲ö❤

❯tr❡❝❤t ❯♥✐✈❡rs✐t② ✫ ❲❡❧❧✲❚②♣❡❞ ▲▲P ❤tt♣✿✴✴❞r❡✐①❡❧✳♥❡t

❆✉❣✉st ✷✼✱ ✷✵✶✶

slide-2
SLIDE 2

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❘❡❣✉❧❛r P♦❧②P ■♥❞❡①❡❞ ❢✉♥❝t♦rs ■♥st❛♥t ❣❡♥❡r✐❝s ❈♦♥✈❡rs✐♦♥s ❈♦♥❝❧✉s✐♦♥

slide-3
SLIDE 3

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✸

❙❡tt✐♥❣

◮ ❚❤❡r❡ ❛r❡ ♠❛♥② ❧✐❜r❛r✐❡s ❢♦r ❣❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ✐♥ ❍❛s❦❡❧❧ ◮ ❉✐✛❡r❡♥t ❛♣♣r♦❛❝❤❡s ✈❛r② ✇✐❞❡❧② ✐♥ ✇❤❛t ❞❛t❛t②♣❡s t❤❡②

❝❛♥ ❡♥❝♦❞❡ ✭✉♥✐✈❡rs❡ s✐③❡✮ ❛♥❞ ✐♥ ✇❤❛t ❢✉♥❝t✐♦♥❛❧✐t② t❤❡② ❝❛♥ ♦✛❡r ✭❡①♣r❡ss✐✈❡♥❡ss✮

◮ ❚❤❡r❡ ✐s ❛ ❧♦t ♦❢ ❞✉♣❧✐❝❛t❡❞ ❝♦❞❡ ❛❝r♦ss ❞✐✛❡r❡♥t ❧✐❜r❛r✐❡s ◮ ◆❡✇❝♦♠❡rs t♦ t❤❡ ✜❡❧❞ ♥❡✈❡r ❦♥♦✇ ✇❤❛t ❧✐❜r❛r② t♦ ✉s❡ ◮ ■♥❢♦r♠❛❧ ❝♦♠♣❛r✐s♦♥s ❡①✐st✱ ❜✉t t❤❡r❡ ❛r❡ ♥♦ ❡♠❜❡❞❞✐♥❣s✱

♥♦r ❢♦r♠❛❧✐s❡❞ st❛t❡♠❡♥ts ❲❡ ✐♥t❡♥❞ t♦ ❝❤❛♥❣❡ t❤✐s✳

slide-4
SLIDE 4

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✸

❙❡tt✐♥❣

◮ ❚❤❡r❡ ❛r❡ ♠❛♥② ❧✐❜r❛r✐❡s ❢♦r ❣❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ✐♥ ❍❛s❦❡❧❧ ◮ ❉✐✛❡r❡♥t ❛♣♣r♦❛❝❤❡s ✈❛r② ✇✐❞❡❧② ✐♥ ✇❤❛t ❞❛t❛t②♣❡s t❤❡②

❝❛♥ ❡♥❝♦❞❡ ✭✉♥✐✈❡rs❡ s✐③❡✮ ❛♥❞ ✐♥ ✇❤❛t ❢✉♥❝t✐♦♥❛❧✐t② t❤❡② ❝❛♥ ♦✛❡r ✭❡①♣r❡ss✐✈❡♥❡ss✮

◮ ❚❤❡r❡ ✐s ❛ ❧♦t ♦❢ ❞✉♣❧✐❝❛t❡❞ ❝♦❞❡ ❛❝r♦ss ❞✐✛❡r❡♥t ❧✐❜r❛r✐❡s ◮ ◆❡✇❝♦♠❡rs t♦ t❤❡ ✜❡❧❞ ♥❡✈❡r ❦♥♦✇ ✇❤❛t ❧✐❜r❛r② t♦ ✉s❡ ◮ ■♥❢♦r♠❛❧ ❝♦♠♣❛r✐s♦♥s ❡①✐st✱ ❜✉t t❤❡r❡ ❛r❡ ♥♦ ❡♠❜❡❞❞✐♥❣s✱

♥♦r ❢♦r♠❛❧✐s❡❞ st❛t❡♠❡♥ts ❲❡ ✐♥t❡♥❞ t♦ ❝❤❛♥❣❡ t❤✐s✳

slide-5
SLIDE 5

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✹

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❲❡ ❤❛✈❡ ❧♦♦❦❡❞ ❛t ✜✈❡ ❧✐❜r❛r✐❡s✿

◮ r❡❣✉❧❛r✿ s✐♠♣❧❡ ❧✐❜r❛r②✱ ♦♥❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥✱ ♥♦

♣❛r❛♠❡t❡rs ♣♦❧②♣✿ ❤✐st♦r✐❝❛❧ ❛♣♣r♦❛❝❤✱ ♦♥❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥✱ ♦♥❡ ♣❛r❛♠❡t❡r✱ ❛♥❞ ❝♦♠♣♦s✐t✐♦♥ ♠✉❧t✐r❡❝✿ ♠✉❧t✐♣❧❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥s✱ ♥♦ ♣❛r❛♠❡t❡rs ✭♦♠✐tt❡❞ ❢r♦♠ t❤✐s t❛❧❦✮ ✐♥❞❡①❡❞✿ ♠✉❧t✐♣❧❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥s✱ ♠✉❧t✐♣❧❡ ♣❛r❛♠❡t❡rs✱ ❝♦♠♣♦s✐t✐♦♥✱ ❛♥❞ ✜①❡❞ ♣♦✐♥ts ✇✐t❤✐♥ t❤❡ ✉♥✐✈❡rs❡ ✐♥st❛♥t✲❣❡♥❡r✐❝s✿ ❝♦✐♥❞✉❝t✐✈❡ ❛♣♣r♦❛❝❤ ✇✐t❤ r❡❝✉rs✐✈❡ ❝♦❞❡s

slide-6
SLIDE 6

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✹

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❲❡ ❤❛✈❡ ❧♦♦❦❡❞ ❛t ✜✈❡ ❧✐❜r❛r✐❡s✿

◮ r❡❣✉❧❛r✿ s✐♠♣❧❡ ❧✐❜r❛r②✱ ♦♥❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥✱ ♥♦

♣❛r❛♠❡t❡rs

◮ ♣♦❧②♣✿ ❤✐st♦r✐❝❛❧ ❛♣♣r♦❛❝❤✱ ♦♥❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥✱ ♦♥❡

♣❛r❛♠❡t❡r✱ ❛♥❞ ❝♦♠♣♦s✐t✐♦♥ ♠✉❧t✐r❡❝✿ ♠✉❧t✐♣❧❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥s✱ ♥♦ ♣❛r❛♠❡t❡rs ✭♦♠✐tt❡❞ ❢r♦♠ t❤✐s t❛❧❦✮ ✐♥❞❡①❡❞✿ ♠✉❧t✐♣❧❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥s✱ ♠✉❧t✐♣❧❡ ♣❛r❛♠❡t❡rs✱ ❝♦♠♣♦s✐t✐♦♥✱ ❛♥❞ ✜①❡❞ ♣♦✐♥ts ✇✐t❤✐♥ t❤❡ ✉♥✐✈❡rs❡ ✐♥st❛♥t✲❣❡♥❡r✐❝s✿ ❝♦✐♥❞✉❝t✐✈❡ ❛♣♣r♦❛❝❤ ✇✐t❤ r❡❝✉rs✐✈❡ ❝♦❞❡s

slide-7
SLIDE 7

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✹

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❲❡ ❤❛✈❡ ❧♦♦❦❡❞ ❛t ✜✈❡ ❧✐❜r❛r✐❡s✿

◮ r❡❣✉❧❛r✿ s✐♠♣❧❡ ❧✐❜r❛r②✱ ♦♥❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥✱ ♥♦

♣❛r❛♠❡t❡rs

◮ ♣♦❧②♣✿ ❤✐st♦r✐❝❛❧ ❛♣♣r♦❛❝❤✱ ♦♥❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥✱ ♦♥❡

♣❛r❛♠❡t❡r✱ ❛♥❞ ❝♦♠♣♦s✐t✐♦♥

◮ ♠✉❧t✐r❡❝✿ ♠✉❧t✐♣❧❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥s✱ ♥♦ ♣❛r❛♠❡t❡rs

✭♦♠✐tt❡❞ ❢r♦♠ t❤✐s t❛❧❦✮ ✐♥❞❡①❡❞✿ ♠✉❧t✐♣❧❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥s✱ ♠✉❧t✐♣❧❡ ♣❛r❛♠❡t❡rs✱ ❝♦♠♣♦s✐t✐♦♥✱ ❛♥❞ ✜①❡❞ ♣♦✐♥ts ✇✐t❤✐♥ t❤❡ ✉♥✐✈❡rs❡ ✐♥st❛♥t✲❣❡♥❡r✐❝s✿ ❝♦✐♥❞✉❝t✐✈❡ ❛♣♣r♦❛❝❤ ✇✐t❤ r❡❝✉rs✐✈❡ ❝♦❞❡s

slide-8
SLIDE 8

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✹

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❲❡ ❤❛✈❡ ❧♦♦❦❡❞ ❛t ✜✈❡ ❧✐❜r❛r✐❡s✿

◮ r❡❣✉❧❛r✿ s✐♠♣❧❡ ❧✐❜r❛r②✱ ♦♥❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥✱ ♥♦

♣❛r❛♠❡t❡rs

◮ ♣♦❧②♣✿ ❤✐st♦r✐❝❛❧ ❛♣♣r♦❛❝❤✱ ♦♥❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥✱ ♦♥❡

♣❛r❛♠❡t❡r✱ ❛♥❞ ❝♦♠♣♦s✐t✐♦♥

◮ ♠✉❧t✐r❡❝✿ ♠✉❧t✐♣❧❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥s✱ ♥♦ ♣❛r❛♠❡t❡rs

✭♦♠✐tt❡❞ ❢r♦♠ t❤✐s t❛❧❦✮

◮ ✐♥❞❡①❡❞✿ ♠✉❧t✐♣❧❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥s✱ ♠✉❧t✐♣❧❡ ♣❛r❛♠❡t❡rs✱

❝♦♠♣♦s✐t✐♦♥✱ ❛♥❞ ✜①❡❞ ♣♦✐♥ts ✇✐t❤✐♥ t❤❡ ✉♥✐✈❡rs❡ ✐♥st❛♥t✲❣❡♥❡r✐❝s✿ ❝♦✐♥❞✉❝t✐✈❡ ❛♣♣r♦❛❝❤ ✇✐t❤ r❡❝✉rs✐✈❡ ❝♦❞❡s

slide-9
SLIDE 9

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✹

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❲❡ ❤❛✈❡ ❧♦♦❦❡❞ ❛t ✜✈❡ ❧✐❜r❛r✐❡s✿

◮ r❡❣✉❧❛r✿ s✐♠♣❧❡ ❧✐❜r❛r②✱ ♦♥❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥✱ ♥♦

♣❛r❛♠❡t❡rs

◮ ♣♦❧②♣✿ ❤✐st♦r✐❝❛❧ ❛♣♣r♦❛❝❤✱ ♦♥❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥✱ ♦♥❡

♣❛r❛♠❡t❡r✱ ❛♥❞ ❝♦♠♣♦s✐t✐♦♥

◮ ♠✉❧t✐r❡❝✿ ♠✉❧t✐♣❧❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥s✱ ♥♦ ♣❛r❛♠❡t❡rs

✭♦♠✐tt❡❞ ❢r♦♠ t❤✐s t❛❧❦✮

◮ ✐♥❞❡①❡❞✿ ♠✉❧t✐♣❧❡ r❡❝✉rs✐✈❡ ♣♦s✐t✐♦♥s✱ ♠✉❧t✐♣❧❡ ♣❛r❛♠❡t❡rs✱

❝♦♠♣♦s✐t✐♦♥✱ ❛♥❞ ✜①❡❞ ♣♦✐♥ts ✇✐t❤✐♥ t❤❡ ✉♥✐✈❡rs❡

◮ ✐♥st❛♥t✲❣❡♥❡r✐❝s✿ ❝♦✐♥❞✉❝t✐✈❡ ❛♣♣r♦❛❝❤ ✇✐t❤ r❡❝✉rs✐✈❡

❝♦❞❡s

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SLIDE 10

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✺

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❘❡❣✉❧❛r P♦❧②P ■♥❞❡①❡❞ ❢✉♥❝t♦rs ■♥st❛♥t ❣❡♥❡r✐❝s ❈♦♥✈❡rs✐♦♥s ❈♦♥❝❧✉s✐♦♥

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SLIDE 11

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✻

❘❡❣✉❧❛r✖✉♥✐✈❡rs❡

♠♦❞✉❧❡ ❘❡❣✉❧❛r ✇❤❡r❡ ❞❛t❛ Code : Set ✇❤❡r❡ U : Code I : Code K : (❳ : Set) → Code ❴⊕❴ : (❋ ● : Code) → Code ❴⊗❴ : (❋ ● : Code) → Code

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SLIDE 12

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✼

❘❡❣✉❧❛r✖✐♥t❡r♣r❡t❛t✐♦♥

❴ : Code → (Set → Set) U ❆ = ⊤ I ❆ = ❆ K ❳ ❆ = ❳ ❋ ⊕ ● ❆ = ❋ ❆ ⊎ ● ❆ ❋ ⊗ ● ❆ = ❋ ❆ × ● ❆ ❞❛t❛ µ (❋ : Code) : Set ✇❤❡r❡ ❴ : ❋ (µ ❋) → µ ❋

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SLIDE 13

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✽

❘❡❣✉❧❛r✖♠❛♣

♠❛♣ : {❆ ❇ : Set} (❋ : Code) → (❆ → ❇) → ❋ ❆ → ❋ ❇ ♠❛♣ U ❢ ❴ = tt ♠❛♣ I ❢ ① = ❢ ① ♠❛♣ (K ❳) ❢ ① = ① ♠❛♣ (❋ ⊕ ●) ❢ (inj1 ①) = inj1 (♠❛♣ ❋ ❢ ①) ♠❛♣ (❋ ⊕ ●) ❢ (inj2 ①) = inj2 (♠❛♣ ● ❢ ①) ♠❛♣ (❋ ⊗ ●) ❢ (①,②) = ♠❛♣ ❋ ❢ ①,♠❛♣ ● ❢ ②

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SLIDE 14

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✾

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❘❡❣✉❧❛r P♦❧②P ■♥❞❡①❡❞ ❢✉♥❝t♦rs ■♥st❛♥t ❣❡♥❡r✐❝s ❈♦♥✈❡rs✐♦♥s ❈♦♥❝❧✉s✐♦♥

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SLIDE 15

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✶✵

P♦❧②P✖✉♥✐✈❡rs❡

♠♦❞✉❧❡ P♦❧②P ✇❤❡r❡ ❞❛t❛ Code : Set ✇❤❡r❡ U : Code I : Code P : Code K : (❳ : Set) → Code ❴⊕❴ : (❋ ● : Code) → Code ❴⊗❴ : (❋ ● : Code) → Code ❴⊚❴ : (❋ ● : Code) → Code

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SLIDE 16

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✶✶

P♦❧②P✖✐♥t❡r♣r❡t❛t✐♦♥

♠✉t✉❛❧ ❴ : Code → (Set → Set → Set) U ❆ ❘ = ⊤ I ❆ ❘ = ❘ P ❆ ❘ = ❆ K ❳ ❆ ❘ = ❳ ❋ ⊕ ● ❆ ❘ = ❋ ❆ ❘ ⊎ ● ❆ ❘ ❋ ⊗ ● ❆ ❘ = ❋ ❆ ❘ × ● ❆ ❘ ❋ ⊚ ● ❆ ❘ = µ ❋ ( ● ❆ ❘) ❞❛t❛ µ (❋ : Code) (❆ : Set) : Set ✇❤❡r❡ ❴ : ❋ ❆ (µ ❋ ❆) → µ ❋ ❆

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SLIDE 17

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✶✷

P♦❧②P✖♠❛♣

♠✉t✉❛❧ ♠❛♣ : {❆ ❇ ❘ ❙ : Set} (❋ : Code) → (❆ → ❇) → (❘ → ❙) → ❋ ❆ ❘ → ❋ ❇ ❙ ♠❛♣ U ❢ ❣ ❴ = tt ♠❛♣ I ❢ ❣ ① = ❣ ① ♠❛♣ P ❢ ❣ ① = ❢ ① ♠❛♣ (K ❳) ❢ ❣ ① = ① ♠❛♣ (❋ ⊕ ●) ❢ ❣ (inj1 ①) = inj1 (♠❛♣ ❋ ❢ ❣ ①) ♠❛♣ (❋ ⊕ ●) ❢ ❣ (inj2 ①) = inj2 (♠❛♣ ● ❢ ❣ ①) ♠❛♣ (❋ ⊗ ●) ❢ ❣ (①,②) = ♠❛♣ ❋ ❢ ❣ ①,♠❛♣ ● ❢ ❣ ② ♠❛♣ (❋ ⊚ ●) ❢ ❣ ① = ♠❛♣ ❋ (♠❛♣ ● ❢ ❣) (♠❛♣ (❋ ⊚ ●) ❢ ❣) ① ♣♠❛♣ : {❆ ❇ : Set} (❋ : Code) → (❆ → ❇) → µ ❋ ❆ → µ ❋ ❇ ♣♠❛♣ ❋ ❢ ① = ♠❛♣ ❋ ❢ (♣♠❛♣ ❋ ❢) ①

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SLIDE 18

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✶✸

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❘❡❣✉❧❛r P♦❧②P ■♥❞❡①❡❞ ❢✉♥❝t♦rs ■♥st❛♥t ❣❡♥❡r✐❝s ❈♦♥✈❡rs✐♦♥s ❈♦♥❝❧✉s✐♦♥

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SLIDE 19

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✶✹

■♥❞❡①❡❞ ❢✉♥❝t♦rs✖✉♥✐✈❡rs❡

♠♦❞✉❧❡ ■♥❞❡①❡❞ ✇❤❡r❡ ❞❛t❛ Code (■① : Set) (❖① : Set) : Set ✇❤❡r❡ U : Code ■① ❖① I : ■① → Code ■① ❖① ! : ❖① → Code ■① ❖① ❴⊕❴ : (❋ ● : Code ■① ❖①) → Code ■① ❖① ❴⊗❴ : (❋ ● : Code ■① ❖①) → Code ■① ❖① ❴⊚❴ : {▼① : Set} → (❋ : Code ▼① ❖①) → (● : Code ■① ▼①) → Code ■① ❖① Fix : (❋ : Code (■① ⊎ ❖①) ❖①) → Code ■① ❖①

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SLIDE 20

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✶✺

■♥❞❡①❡❞ ❢✉♥❝t♦rs✖✐♥t❡r♣r❡t❛t✐♦♥

■♥❞❡①❡❞ : Set → Set ■♥❞❡①❡❞ ■ = ■ → Set ♠✉t✉❛❧ ❴ : {■ ❖ : Set} → Code ■ ❖ → ■♥❞❡①❡❞ ■ → ■♥❞❡①❡❞ ❖ U r ✐ = ⊤ I ❥ r ✐ = r ❥ ! ❥ r ✐ = ✐ ≡ ❥ ❋ ⊕ ● r ✐ = ❋ r ✐ ⊎ ● r ✐ ❋ ⊗ ● r ✐ = ❋ r ✐ × ● r ✐ ❋ ⊚ ● r ✐ = ❋ ( ● r) ✐ Fix ❋ r ✐ = µ ❋ r ✐ ❞❛t❛ µ {■ ❖ : Set} (❋ : Code (■ ⊎ ❖) ❖) (r : ■♥❞❡①❡❞ ■) (♦ : ❖) : Set ✇❤❡r❡ ❴ : ❋ [r,µ ❋ r] ♦ → µ ❋ r ♦

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SLIDE 21

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✶✻

■♥❞❡①❡❞ ❢✉♥❝t♦rs✖♠❛♣

❴⇒❴ : {■ : Set} → ■♥❞❡①❡❞ ■ → ■♥❞❡①❡❞ ■ → Set ❘ ⇒ ❙ = (✐ : ❴) → ❘ ✐ → ❙ ✐ ❴❴ : {■ ❏ : Set} {❆ ❈ : ■♥❞❡①❡❞ ■} {❇ ❉ : ■♥❞❡①❡❞ ❏} → ❆ ⇒ ❈ → ❇ ⇒ ❉ → [❆,❇] ⇒ [❈,❉] ♠❛♣ : {■ ❖ : Set} {❘ ❙ : ■♥❞❡①❡❞ ■} (❋ : Code ■ ❖) → ❘ ⇒ ❙ → ❋ ❘ ⇒ ❋ ❙ ♠❛♣ U ❢ ✐ ❴ = tt ♠❛♣ (I ❥) ❢ ✐ ① = ❢ ❥ ① ♠❛♣ (! ❥) ❢ ✐ ① = ① ♠❛♣ (❋ ⊕ ●) ❢ ✐ (inj1 ①) = inj1 (♠❛♣ ❋ ❢ ✐ ①) ♠❛♣ (❋ ⊕ ●) ❢ ✐ (inj2 ①) = inj2 (♠❛♣ ● ❢ ✐ ①) ♠❛♣ (❋ ⊗ ●) ❢ ✐ (①,②) = ♠❛♣ ❋ ❢ ✐ ①,♠❛♣ ● ❢ ✐ ② ♠❛♣ (❋ ⊚ ●) ❢ ✐ ① = ♠❛♣ ❋ (♠❛♣ ● ❢) ✐ ① ♠❛♣ (Fix ❋) ❢ ✐ ① = ♠❛♣ ❋ (❢ ♠❛♣ (Fix ❋) ❢) ✐ ①

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SLIDE 22

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✶✼

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❘❡❣✉❧❛r P♦❧②P ■♥❞❡①❡❞ ❢✉♥❝t♦rs ■♥st❛♥t ❣❡♥❡r✐❝s ❈♦♥✈❡rs✐♦♥s ❈♦♥❝❧✉s✐♦♥

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SLIDE 23

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✶✽

■♥st❛♥t ❣❡♥❡r✐❝s✖✉♥✐✈❡rs❡

♠♦❞✉❧❡ ■♥st❛♥t●❡♥❡r✐❝s ✇❤❡r❡ ♠✉t✉❛❧ ❞❛t❛ Code : Set ✇❤❡r❡ Z : Code U : Code K : Set → Code R : ∞ Code → Code ❴⊕❴ : ∞ Code → ∞ Code → Code ❴⊗❴ : ∞ Code → ∞ Code → Code

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SLIDE 24

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✶✾

■♥st❛♥t ❣❡♥❡r✐❝s✖✐♥t❡r♣r❡t❛t✐♦♥

❞❛t❛ ❴ : Code → Set ✇❤❡r❡ tt : U

  • k

: {❆ : Set} → ❆ → K ❆

  • rec

: {❈ : ∞ Code} → ♭ ❈ → R ❈

  • inl

: {❈ ❉ : ∞ Code} → ♭ ❈ → ❈ ⊕ ❉ inr : {❈ ❉ : ∞ Code} → ♭ ❉ → ❈ ⊕ ❉ ❴ ,❴ : {❈ ❉ : ∞ Code} → ♭ ❈ → ♭ ❉ → ❈ ⊗ ❉

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SLIDE 25

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✵

■♥st❛♥t ❣❡♥❡r✐❝s✖s❛♠♣❧❡ ❣❡♥❡r✐❝ ❢✉♥❝t✐♦♥

❈♦✐♥❞✉❝t✐✈❡ ❝♦❞❡s ❞♦ ♥♦t ♥❛t✉r❛❧❧② ❞❡✜♥❡ ❢✉♥❝t♦rs✳ ❲❡ s❤♦✇ ❛♥ ❡①❛♠♣❧❡ ❣❡♥❡r✐❝ ❢✉♥❝t✐♦♥✿ s✐③❡ : (❆ : Code) → ❆ → N s✐③❡ Z () s✐③❡ U ① = ✶ s✐③❡ (K ❆) (k ①) = ✶ s✐③❡ (R ❈) (rec ①) = ✶ + s✐③❡ (♭ ❈) ① s✐③❡ (❆ ⊕ ❇) (inl ①) = s✐③❡ (♭ ❆) ① s✐③❡ (❆ ⊕ ❇) (inr ①) = s✐③❡ (♭ ❇) ① s✐③❡ (❆ ⊗ ❇) (①,②) = s✐③❡ (♭ ❆) ① + s✐③❡ (♭ ❇) ②

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SLIDE 26

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✶

❊♥❝♦❞✐♥❣ ❞❛t❛t②♣❡s

♠♦❞✉❧❡ ❊①❛♠♣❧❡ ✇❤❡r❡ ❞❛t❛ List (❆ : Set) : Set ✇❤❡r❡ nil : List ❆ cons : ❆ → List ❆ → List ❆ ▲✐st❈♣ : Code♣ ▲✐st❈♣ = U♣ ⊕♣ P♣ ⊗♣ I♣ ▲✐st❈✐ : Code✐ ⊤ ⊤ ▲✐st❈✐ = Fix✐ (U✐ ⊕✐ (I✐ (inj1 tt))⊗✐ (I✐ (inj2 tt))) ▲✐st❈✐❣ : ∞ Code✐❣ → Code✐❣ ▲✐st❈✐❣ ❆ = (♯ U✐❣)⊕✐❣ (♯ (❆⊗✐❣ (♯ (R✐❣ (♯ (▲✐st❈✐❣ ❆))))))

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SLIDE 27

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✷

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❘❡❣✉❧❛r P♦❧②P ■♥❞❡①❡❞ ❢✉♥❝t♦rs ■♥st❛♥t ❣❡♥❡r✐❝s ❈♦♥✈❡rs✐♦♥s ❈♦♥❝❧✉s✐♦♥

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SLIDE 28

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✸

❈♦♥✈❡rs✐♦♥s

❘❡❣✉❧❛r

I✶ ⊕ ⊗

P♦❧②P

I✶ P✶ ⊕ ⊗ ⊚

■♥st❛♥t●❡♥❡r✐❝s

R ⊕ ⊗

▼✉❧t✐r❡❝

I♥ ⊕ ⊗

■♥❞❡①❡❞

I♥ P♥ ⊕ ⊗ ⊚ Fix

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SLIDE 29

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✹

❘❡❣✉❧❛r t♦ P♦❧②P ■

♠♦❞✉❧❡ ❘❡❣✉❧❛rP♦❧②P ✇❤❡r❡

r

⇑♣ : Coder → Code♣

r

⇑♣ Ur = U♣

r

⇑♣ Ir = I♣

r

⇑♣ (Kr ❳) = K♣ ❳

r

⇑♣ (❋⊕r ●) = (r ⇑♣ ❋)⊕♣ (r ⇑♣ ●)

r

⇑♣ (❋⊗r ●) = (r ⇑♣ ❋)⊗♣ (r ⇑♣ ●)

r

↑♣ : {❆ ❘ : Set} → (❈ : Coder) → ❈r ❘ ≡ r ⇑♣ ❈♣ ❆ ❘

r

↑♣ (Ur) = refl

r

↑♣ (Ir) = refl

r

↑♣ (Kr ❳) = refl

r

↑♣ (❋⊕r ●) = ❝♦♥❣2 ❴⊎❴ (r ↑♣ ❋) (r ↑♣ ●)

r

↑♣ (❋⊗r ●) = ❝♦♥❣2 ❴×❴ (r ↑♣ ❋) (r ↑♣ ●)

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SLIDE 30

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✺

❘❡❣✉❧❛r t♦ P♦❧②P ■■

❢r♦♠r : {❆ ❘ : Set} (❈ : Coder) → ❈r ❘ → r ⇑♣ ❈♣ ❆ ❘ ❢r♦♠r Ur = ✐❞ ❢r♦♠r Ir = ✐❞ ❢r♦♠r (Kr ❳) = ✐❞ ❢r♦♠r (❋⊕r ●) = [inj1 ◦ ❢r♦♠r ❋,inj2 ◦ ❢r♦♠r ●] ❢r♦♠r (❋⊗r ●) = ❁ ❢r♦♠r ❋ ◦ ♣r♦❥1 ,❢r♦♠r ● ◦ ♣r♦❥2 ❃ ❢r♦♠µr : {❆ : Set} (❈ : Coder) → µr ❈ → µ♣ (r ⇑♣ ❈) ❆ ❢r♦♠µr ❈ ①r = ❢r♦♠r ❈ (♠❛♣r ❈ (❢r♦♠µr ❈) ①)♣

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SLIDE 31

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✻

❘❡❣✉❧❛r t♦ P♦❧②P ■■■

✐s♦1 : {❆ ❘ : Set} (❈ : Coder) {① : ❴r ❈ ❘} → t♦r {❆} ❈ (❢r♦♠r ❈ ①) ≡ ① ✐s♦1 Ur = refl ✐s♦1 Ir = refl ✐s♦1 (Kr ❳) = refl ✐s♦1 (❋⊕r ●) {inj1 ❴} = ❝♦♥❣ inj1 (✐s♦1 ❋) ✐s♦1 (❋⊕r ●) {inj2 ❴} = ❝♦♥❣ inj2 (✐s♦1 ●) ✐s♦1 (❋⊗r ●) {❴ ,❴} = ❝♦♥❣2 ❴ ,❴ (✐s♦1 ❋) (✐s♦1 ●)

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SLIDE 32

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✼

❘❡❣✉❧❛r t♦ P♦❧②P ■❱

✐s♦µ1 : {❆ : Set} (❈ : Coder) (① : µr ❈) → t♦µr {❆} ❈ (❢r♦♠µr ❈ ①) ≡ ① ✐s♦µ1 {❆} ❈ ①r = ❝♦♥❣ ❴r $ ❜❡❣✐♥ t♦r {❆} ❈ (♠❛♣♣ (r ⇑♣ ❈) ✐❞ (t♦µr ❈) (❢r♦♠r ❈ (♠❛♣r ❈ (❢r♦♠µr ❈) ①))) ≡ ♠❝ {❆} ❈ ♠❛♣r ❈ (t♦µr {❆} ❈) (t♦r {❆} ❈ (❢r♦♠r ❈ (♠❛♣r ❈ (❢r♦♠µr ❈) ①))) ≡ ❝♦♥❣ (♠❛♣r ❈ (t♦µr {❆} ❈)) (✐s♦1 ❈) ♠❛♣r ❈ (t♦µr {❆} ❈) (♠❛♣r ❈ (❢r♦♠µr {❆} ❈) ①) ♠❝ ❆ ❘ ❘ ① ❴ ❢ ❘ ❘ ❈

r

t♦r ❆ ❘ ❈ ♠❛♣♣

r ♣ ❈ ✐❞ ❢ ①

♠❛♣r ❈ ❢ t♦r ❈ ①

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SLIDE 33

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✼

❘❡❣✉❧❛r t♦ P♦❧②P ■❱

✐s♦µ1 : {❆ : Set} (❈ : Coder) (① : µr ❈) → t♦µr {❆} ❈ (❢r♦♠µr ❈ ①) ≡ ① ✐s♦µ1 {❆} ❈ ①r = ❝♦♥❣ ❴r $ ❜❡❣✐♥ t♦r {❆} ❈ (♠❛♣♣ (r ⇑♣ ❈) ✐❞ (t♦µr ❈) (❢r♦♠r ❈ (♠❛♣r ❈ (❢r♦♠µr ❈) ①))) ≡ ♠❝ {❆} ❈ ♠❛♣r ❈ (t♦µr {❆} ❈) (t♦r {❆} ❈ (❢r♦♠r ❈ (♠❛♣r ❈ (❢r♦♠µr ❈) ①))) ≡ ❝♦♥❣ (♠❛♣r ❈ (t♦µr {❆} ❈)) (✐s♦1 ❈) ♠❛♣r ❈ (t♦µr {❆} ❈) (♠❛♣r ❈ (❢r♦♠µr {❆} ❈) ①) ♠❝ : {❆ ❘1 ❘2 : Set} {① : ❴} {❢ : ❘1 → ❘2} (❈ : Coder) → t♦r {❆} {❘2} ❈ (♠❛♣♣ (r ⇑♣ ❈) ✐❞ ❢ ①) ≡ ♠❛♣r ❈ ❢ (t♦r ❈ ①)

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SLIDE 34

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✽

❘❡❣✉❧❛r t♦ P♦❧②P ❱

♠❛♣r ❈ (t♦µr {❆} ❈) (♠❛♣r ❈ (❢r♦♠µr {❆} ❈) ①) ≡ ♠❛♣◦

r ❈

♠❛♣r ❈ (t♦µr ❈ ◦ ❢r♦♠µr ❈) ① ≡ ♠❛♣∀

r ❈ (t♦µr ❈ ◦ ❢r♦♠µr ❈) ✐❞ (✐s♦µ1 ❈) ①

♠❛♣r ❈ ✐❞ ① ≡ ♠❛♣✐❞

r ❈

① ♠❛♣r ❆ ❇ ❈

r

❢ ❣ ❆ ❇ ① ❢ ① ❣ ① ① ♠❛♣r ❈ ❢ ① ♠❛♣r ❈ ❣ ①

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SLIDE 35

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✽

❘❡❣✉❧❛r t♦ P♦❧②P ❱

♠❛♣r ❈ (t♦µr {❆} ❈) (♠❛♣r ❈ (❢r♦♠µr {❆} ❈) ①) ≡ ♠❛♣◦

r ❈

♠❛♣r ❈ (t♦µr ❈ ◦ ❢r♦♠µr ❈) ① ≡ ♠❛♣∀

r ❈ (t♦µr ❈ ◦ ❢r♦♠µr ❈) ✐❞ (✐s♦µ1 ❈) ①

♠❛♣r ❈ ✐❞ ① ≡ ♠❛♣✐❞

r ❈

① ♠❛♣∀

r : {❆ ❇ : Set} (❈ : Coder)

→ (❢ ❣ : ❆ → ❇) → (∀ ① → ❢ ① ≡ ❣ ①) → (∀ ① → ♠❛♣r ❈ ❢ ① ≡ ♠❛♣r ❈ ❣ ①)

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SLIDE 36

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✷✾

P♦❧②P t♦ ■♥st❛♥t●❡♥❡r✐❝s

♠♦❞✉❧❡ P♦❧②P■♥st❛♥t●❡♥❡r✐❝s ✇❤❡r❡

♣

⇑✐❣ : Code♣ → Set → Code✐❣

♣

⇑✐❣ ❈ ❆ = ♣ ⇑✐❣. ❈ ❈ ❆ ✇❤❡r❡

♣

⇑✐❣. : Code♣ → Code♣ → Set → Code✐❣

♣

⇑✐❣. ❈ U♣ ❆ = U✐❣

♣

⇑✐❣. ❈ I♣ ❆ = R✐❣ (♯ ♣ ⇑✐❣. ❈ ❈ ❆)

♣

⇑✐❣. ❈ P♣ ❆ = K✐❣ ❆

♣

⇑✐❣. ❈ (K♣ ❳) ❆ = K✐❣ ❳

♣

⇑✐❣. ❈ (❋⊕♣ ●) ❆ = (♯ ♣ ⇑✐❣. ❈ ❋ ❆)⊕✐❣ (♯ ♣ ⇑✐❣. ❈ ● ❆)

♣

⇑✐❣. ❈ (❋⊗♣ ●) ❆ = (♯ ♣ ⇑✐❣. ❈ ❋ ❆)⊗✐❣ (♯ ♣ ⇑✐❣. ❈ ● ❆)

♣

⇑✐❣. ❈ (❋⊚♣ ●) ❆ = R✐❣ (♯ ♣ ⇑✐❣. ❋ ❋ (♣ ⇑✐❣. ❈ ● ❆)✐❣)

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SLIDE 37

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✸✵

■♥❞❡①❡❞ t♦ ■♥st❛♥t●❡♥❡r✐❝s

♠♦❞✉❧❡ ■♥❞❡①❡❞■♥st❛♥t●❡♥❡r✐❝s ✇❤❡r❡ ✐✷❝′

❝ : {■ ❖ : Set}

→ Code✐ ■ ❖ → (■ → Code✐❣) → (❖ → Code✐❣) ✐✷❝′

❝ U✐

r ♦ = U✐❣ ✐✷❝′

❝ (I✐ ✐)

r ♦ = r ✐ ✐✷❝′

❝ (!✐ ✐)

r ♦ = U✐❣ ✐✷❝′

❝ (❋⊕✐ ●) r ♦ = (♯ ✐✷❝′ ❝ ❋ r ♦)⊕✐❣ (♯ ✐✷❝′ ❝ ● r ♦)

✐✷❝′

❝ (❋⊗✐ ●) r ♦ = (♯ ✐✷❝′ ❝ ❋ r ♦)⊗✐❣ (♯ ✐✷❝′ ❝ ● r ♦)

✐✷❝′

❝ (❋⊚✐ ●) r ♦ = R✐❣ (♯ ✐✷❝′ ❝ ❋ (✐✷❝′ ❝ ● r) ♦)

✐✷❝′

❝ (Fix✐ ❋) r ♦ = R✐❣ (♯ ✐✷❝′ ❝ ❋ [r,✐✷❝′ ❝ (Fix✐ ❋) r] ♦)

✐✷❝❝ : {■ ❖ : Set} → Code✐ ■ ❖ → (■ → Set) → (❖ → Code✐❣) ✐✷❝❝ ❈ r ♦ = ✐✷❝′

❝ ❈ (K✐❣ ◦ r) ♦

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SLIDE 38

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✸✶

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥

  • ❡♥❡r✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧✐❜r❛r✐❡s✱ ✐♥ ❆❣❞❛

❘❡❣✉❧❛r P♦❧②P ■♥❞❡①❡❞ ❢✉♥❝t♦rs ■♥st❛♥t ❣❡♥❡r✐❝s ❈♦♥✈❡rs✐♦♥s ❈♦♥❝❧✉s✐♦♥

slide-39
SLIDE 39

❬❋❛❝✉❧t② ♦❢ ❙❝✐❡♥❝❡ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣ ❙❝✐❡♥❝❡s❪ ✸✷

❈♦♥❝❧✉s✐♦♥

◮ ❆♥❛❧②③❡❞ ✜✈❡ ❧✐❜r❛r✐❡s✱ ✇✐t❤ ❛♥ ❡♥❝♦❞✐♥❣ ✐♥ ❆❣❞❛ ❢♦r ❡❛❝❤ ◮ ❘❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ✜①❡❞✲♣♦✐♥t ❛♣♣r♦❛❝❤❡s ♠❛❞❡

✭❢♦r♠❛❧❧②✮ ❝❧❡❛r

◮ ❚❤❡ s❛♠❡ ❣❡♥❡r✐❝ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ✉s❡❞ ✐♥ ❞✐✛❡r❡♥t ❧✐❜r❛r✐❡s ◮ ■♥st❛♥t●❡♥❡r✐❝s ❤❛s ❛ ✈❡r② ✢❡①✐❜❧❡ ✉♥✐✈❡rs❡ ◮ ❊♠❜❡❞❞✐♥❣ ✐♥t♦ ■♥st❛♥t●❡♥❡r✐❝s r❡q✉✐r❡s ❡①♣❛♥❞✐♥❣ ✜①❡❞

♣♦✐♥ts✱ ❤✐❣❤❧✐❣❤t✐♥❣ ❡✳❣✳ t❤❡ ✇❛② ❝♦♠♣♦s✐t✐♦♥ ✇♦r❦s ✐♥ P♦❧②P