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r t r r t t rt ts r

slide-1
SLIDE 1

❉❡r✐✈✐♥❣ t❤❡ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ❢r♦♠ t❤❡ ❙♠❛❧❧✲❙t❡♣ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ◆♦r♠❛❧ ❖r❞❡r

➪❧✈❛r♦ ●❛r❝í❛✲Pér❡③ P❛❜❧♦ ◆♦❣✉❡✐r❛ ❏✉❛♥ ❏♦sé ▼♦r❡♥♦✲◆❛✈❛rr♦ ▼❛❞r✐❞✱ ❙❡♣t❡♠❜❡r ✷✵✶✸

✶ ✴ ✶✺

slide-2
SLIDE 2

❆❜str❛❝t ▼❛❝❤✐♥❡s ✭✳ ✳ ✳ ✮ st♦♣ ✇❤❡♥ t❤❡ ✇❡❛❦ ❤❡❛❞ ♥♦r♠❛❧ ❢♦r♠ ♦❢ t❤❡✐r ❛r❣✉♠❡♥t ✐s r❡❛❝❤❡❞✳ ❆❧t❤♦✉❣❤✱ ✐t ✐s ❡♥♦✉❣❤ ✐♥ ♠♦st ❝❛s❡s ✭✳ ✳ ✳ ✮✱ ✐t ❝❛♥ ❜❡ s♦♠❡t✐♠❡s ❝♦♥✈❡♥✐❡♥t t♦ ❣♦ ❢✉rt❤❡r✳ ✳ ✳ ❬❈ré❣✉t✱ ✶✾✾✵❪ ❬▼❝●♦✇❛♥✱ ✶✾✼✵❪ ❬❈ré❣✉t✱ ✶✾✾✵❪ ❬P❛♦❧✐♥✐ ❛♥❞ ❉❡❧❧❛ ❘♦❝❝❛✱ ✶✾✾✾❪ ❬●r❡❣♦✐r❡ ❛♥❞ ▲❡r♦②✱ ✷✵✵✷❪ ❬❙❡st♦❢t✱ ✷✵✵✷❪ ❬❆❣❡r✱ ❇✐❡r♥❛❝❦✐✱ ❉❛♥✈② ❛♥❞ ▼✐❞t❣❛❛r❞✱ ✷✵✵✸❪ ❬❉❡❧❧❛ ❘♦❝❝❛ ❛♥❞ P❛♦❧✐♥✐✱ ✷✵✵✹❪ ❬❈ré❣✉t✱ ✷✵✵✼❪ ❬●❛r❝í❛ ❛♥❞ ◆♦❣✉❡✐r❛✱ ✷✵✶✸❪ ❬❉❛♥✈②✱ ▼✐❧❧✐❦✐♥ ❛♥❞ ▼✉♥❦✱ ✷✵✶✸❪

✷ ✴ ✶✺

slide-3
SLIDE 3

❆❜str❛❝t ▼❛❝❤✐♥❡s ✭✳ ✳ ✳ ✮ st♦♣ ✇❤❡♥ t❤❡ ✇❡❛❦ ❤❡❛❞ ♥♦r♠❛❧ ❢♦r♠ ♦❢ t❤❡✐r ❛r❣✉♠❡♥t ✐s r❡❛❝❤❡❞✳ ❆❧t❤♦✉❣❤✱ ✐t ✐s ❡♥♦✉❣❤ ✐♥ ♠♦st ❝❛s❡s ✭✳ ✳ ✳ ✮✱ ✐t ❝❛♥ ❜❡ s♦♠❡t✐♠❡s ❝♦♥✈❡♥✐❡♥t t♦ ❣♦ ❢✉rt❤❡r✳ ✳ ✳ ❬❈ré❣✉t✱ ✶✾✾✵❪ ❬▼❝●♦✇❛♥✱ ✶✾✼✵❪ ❬❈ré❣✉t✱ ✶✾✾✵❪ ❬P❛♦❧✐♥✐ ❛♥❞ ❉❡❧❧❛ ❘♦❝❝❛✱ ✶✾✾✾❪ ❬●r❡❣♦✐r❡ ❛♥❞ ▲❡r♦②✱ ✷✵✵✷❪ ❬❙❡st♦❢t✱ ✷✵✵✷❪ ❬❆❣❡r✱ ❇✐❡r♥❛❝❦✐✱ ❉❛♥✈② ❛♥❞ ▼✐❞t❣❛❛r❞✱ ✷✵✵✸❪ ❬❉❡❧❧❛ ❘♦❝❝❛ ❛♥❞ P❛♦❧✐♥✐✱ ✷✵✵✹❪ ❬❈ré❣✉t✱ ✷✵✵✼❪ ❬●❛r❝í❛ ❛♥❞ ◆♦❣✉❡✐r❛✱ ✷✵✶✸❪ ❬❉❛♥✈②✱ ▼✐❧❧✐❦✐♥ ❛♥❞ ▼✉♥❦✱ ✷✵✶✸❪

✷ ✴ ✶✺

slide-4
SLIDE 4

❆❜str❛❝t ▼❛❝❤✐♥❡s ✭✳ ✳ ✳ ✮ st♦♣ ✇❤❡♥ t❤❡ ✇❡❛❦ ❤❡❛❞ ♥♦r♠❛❧ ❢♦r♠ ♦❢ t❤❡✐r ❛r❣✉♠❡♥t ✐s r❡❛❝❤❡❞✳ ❆❧t❤♦✉❣❤✱ ✐t ✐s ❡♥♦✉❣❤ ✐♥ ♠♦st ❝❛s❡s ✭✳ ✳ ✳ ✮✱ ✐t ❝❛♥ ❜❡ s♦♠❡t✐♠❡s ❝♦♥✈❡♥✐❡♥t t♦ ❣♦ ❢✉rt❤❡r✳ ✳ ✳ ❬❈ré❣✉t✱ ✶✾✾✵❪ ❬▼❝●♦✇❛♥✱ ✶✾✼✵❪ ❬❈ré❣✉t✱ ✶✾✾✵❪ ❬P❛♦❧✐♥✐ ❛♥❞ ❉❡❧❧❛ ❘♦❝❝❛✱ ✶✾✾✾❪ ❬●r❡❣♦✐r❡ ❛♥❞ ▲❡r♦②✱ ✷✵✵✷❪ ❬❙❡st♦❢t✱ ✷✵✵✷❪ ❬❆❣❡r✱ ❇✐❡r♥❛❝❦✐✱ ❉❛♥✈② ❛♥❞ ▼✐❞t❣❛❛r❞✱ ✷✵✵✸❪ ❬❉❡❧❧❛ ❘♦❝❝❛ ❛♥❞ P❛♦❧✐♥✐✱ ✷✵✵✹❪ ❬❈ré❣✉t✱ ✷✵✵✼❪ ❬●❛r❝í❛ ❛♥❞ ◆♦❣✉❡✐r❛✱ ✷✵✶✸❪ ❬❉❛♥✈②✱ ▼✐❧❧✐❦✐♥ ❛♥❞ ▼✉♥❦✱ ✷✵✶✸❪

✷ ✴ ✶✺

slide-5
SLIDE 5

❉❡r✐✈❛t✐♦♥❛❧ ❉✐❛❣r❛♠

λ λ

ρ

λ∗

ρ

❡①♣✳ ❝tr❧✳ λ∗

ρ

r❡❞✳✲❜❛s❡❞ ♥♦r♠❛❧✐s❡r r❡❞✳✲❜❛s❡❞ ♥♦r♠❛❧✐s❡r ♣✉s❤✴❡♥t❡r ❑◆ ❡✈❛❧✴❛♣♣❧② ❛❜s✳ ♠❛❝❤✳ ❡✈❛❧✴❛♣♣❧② ❛❜s✳ ♠❛❝❤✳ ❡✈❛❧✴❛♣♣❧② ❛❜s✳ ♠❛❝❤✳ r❡❞✳✲❢r❡❡ ♥♦r♠❛❧✐s❡r ♣r❡♣♦♥❡❞ r❡❞✳✲❢r❡❡ ♥♦r♠❛❧✐s❡r ♣r❡♣♦♥❡❞ r❡❞✳✲❢r❡❡ ♥♦r♠❛❧✐s❡r ♣r❡♣♦♥❡❞ r❡❞✳✲❢r❡❡ ♥♦r♠❛❧✐s❡r

r❡❢♦❝✉s ✰ ▲❲❋ ✰ ✐♥❧✐♥❡ r❡❢✉♥❝✳ ✰ ❈P❙−✶ ♣r❡♣♦♥❡ σ❈ s❤♦rt❝✉t ❡①♣✳ ❝tr❧✳ ❈P❙ ✰ ❞❡❢✉♥❝ ♣r♦tr✉❞❡ ✰ ✐♥❧✐♥❡ r♠✳ ❝tr❧✳

✸ ✴ ✶✺

slide-6
SLIDE 6

❉❡r✐✈❛t✐♦♥❛❧ ❉✐❛❣r❛♠

λ λ

ρ

λ∗

ρ

❡①♣✳ ❝tr❧✳ λ∗

ρ

r❡❞✳✲❜❛s❡❞ ♥♦r♠❛❧✐s❡r r❡❞✳✲❜❛s❡❞ ♥♦r♠❛❧✐s❡r ♣✉s❤✴❡♥t❡r ❑◆ ❡✈❛❧✴❛♣♣❧② ❛❜s✳ ♠❛❝❤✳ ❡✈❛❧✴❛♣♣❧② ❛❜s✳ ♠❛❝❤✳ ❡✈❛❧✴❛♣♣❧② ❛❜s✳ ♠❛❝❤✳ r❡❞✳✲❢r❡❡ ♥♦r♠❛❧✐s❡r ♣r❡♣♦♥❡❞ r❡❞✳✲❢r❡❡ ♥♦r♠❛❧✐s❡r ♣r❡♣♦♥❡❞ r❡❞✳✲❢r❡❡ ♥♦r♠❛❧✐s❡r ♣r❡♣♦♥❡❞ r❡❞✳✲❢r❡❡ ♥♦r♠❛❧✐s❡r

r❡❢♦❝✉s ✰ ▲❲❋ ✰ ✐♥❧✐♥❡ r❡❢✉♥❝✳ ✰ ❈P❙−✶ ♣r❡♣♦♥❡ σ❈ s❤♦rt❝✉t ❡①♣✳ ❝tr❧✳ ❈P❙ ✰ ❞❡❢✉♥❝ ♣r♦tr✉❞❡ ✰ ✐♥❧✐♥❡ r♠✳ ❝tr❧✳

✸ ✴ ✶✺

slide-7
SLIDE 7

P✉r❡ ❯♥t②♣❡❞ ▲❛♠❜❞❛ ❈❛❧❝✉❧✉s ✭λ✮

Λ ::= ♥ | λ.Λ | Λ Λ

✹ ✴ ✶✺

slide-8
SLIDE 8

❊①❛♠♣❧❡

λ.✵((λ.✵)✵)

✺ ✴ ✶✺

slide-9
SLIDE 9

❊①❛♠♣❧❡

λ.✵((λ.✵)✵)

✺ ✴ ✶✺

slide-10
SLIDE 10

❊①❛♠♣❧❡

λ.✵((λ.✵)✵)

✺ ✴ ✶✺

slide-11
SLIDE 11

❊①❛♠♣❧❡

λ.✵ ✵()

✺ ✴ ✶✺

slide-12
SLIDE 12

❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λρ✮ ❬❈ré❣✉t✱ ✷✵✵✼❪

❈ ::= Λ[ρ] | ♥ | ⌊Λ, ❧⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ, ❧⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✻ ✴ ✶✺

slide-13
SLIDE 13

❊①❛♠♣❧❡

λ.✵((λ.✵)✵)

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-14
SLIDE 14

❊①❛♠♣❧❡

λ.✵((λ.✵)✵)

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-15
SLIDE 15

❊①❛♠♣❧❡

( λ.✵((λ.✵)✵)[ǫ] , ǫ , ✵ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-16
SLIDE 16

❊①❛♠♣❧❡

( λ.✵((λ.✵)✵)[ǫ] , ǫ , ✵ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-17
SLIDE 17

❊①❛♠♣❧❡

( ✵((λ.✵)✵)[✶] , λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-18
SLIDE 18

❊①❛♠♣❧❡

( ✵((λ.✵)✵)[✶] , λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-19
SLIDE 19

❊①❛♠♣❧❡

( ✵[✶] , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-20
SLIDE 20

❊①❛♠♣❧❡

( ✵[✶] , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-21
SLIDE 21

❊①❛♠♣❧❡

( ✶ , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-22
SLIDE 22

❊①❛♠♣❧❡

( ✶ , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-23
SLIDE 23

❊①❛♠♣❧❡

( ⌊✵, ✶⌋ , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-24
SLIDE 24

❊①❛♠♣❧❡

( ⌊✵, ✶⌋ , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-25
SLIDE 25

❊①❛♠♣❧❡

( ((λ.✵)✵)[✶] , ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-26
SLIDE 26

❊①❛♠♣❧❡

( ((λ.✵)✵)[✶] , ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-27
SLIDE 27

❊①❛♠♣❧❡

( (λ.✵)[✶] , ✵[✶] : ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-28
SLIDE 28

❊①❛♠♣❧❡

( (λ.✵)[✶] , ✵[✶] : ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-29
SLIDE 29

❊①❛♠♣❧❡

( ✵[✵[✶] : ✶] , ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-30
SLIDE 30

❊①❛♠♣❧❡

( ✵[✵[✶] : ✶] , ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-31
SLIDE 31

❊①❛♠♣❧❡

( ✵[✶] , ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-32
SLIDE 32

❊①❛♠♣❧❡

( ✵[✶] , ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-33
SLIDE 33

❊①❛♠♣❧❡

( ✶ , ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-34
SLIDE 34

❊①❛♠♣❧❡

( ✶ , ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-35
SLIDE 35

❊①❛♠♣❧❡

( ⌊✵, ✶⌋ , ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-36
SLIDE 36

❊①❛♠♣❧❡

( ⌊✵, ✶⌋ , ⌊✵, ✶⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-37
SLIDE 37

❊①❛♠♣❧❡

( ⌊✵ ✵, ✶⌋ , λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-38
SLIDE 38

❊①❛♠♣❧❡

( ⌊✵ ✵, ✶⌋ , λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-39
SLIDE 39

❊①❛♠♣❧❡

( ⌊λ.✵ ✵, ✶⌋ , ǫ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-40
SLIDE 40

❊①❛♠♣❧❡

( ⌊λ.✵ ✵, ✶⌋ , ǫ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-41
SLIDE 41

❊①❛♠♣❧❡

λ.✵ ✵

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚

✼ ✴ ✶✺

slide-42
SLIDE 42

❈❛❧❝✉❧✉s ♦❢ ❈❧♦s✉r❡s ✭λ

ρ✮ ❬❇✐❡r♥❛❝❦❛ ❛♥❞ ❉❛♥✈②✱ ✷✵✵✼❪ ❈ ::= Λ[ρ] | ♥ | ⌊♥⌋ | λ λ.❈ | ❈ · ❈ ρ ::= ǫ | ❈ : ρ ❊ ♥ ❊ ❊ ❊

✽ ✴ ✶✺

slide-43
SLIDE 43

❈❛❧❝✉❧✉s ♦❢ ❈❧♦s✉r❡s ✭λ

ρ✮ ❬❈✉r✐❡♥✱ ✶✾✾✶❪ ❈ ::= Λ[ρ] | ♥ | ⌊♥⌋ | λ λ.❈ | ❈ · ❈ ρ ::= ǫ | ❈ : ρ ❊ ♥ ❊ ❊ ❊

✽ ✴ ✶✺

slide-44
SLIDE 44

❈❛❧❝✉❧✉s ♦❢ ❈❧♦s✉r❡s ✭λ

ρ✮ ❬❇✐❡r♥❛❝❦❛ ❛♥❞ ❉❛♥✈②✱ ✷✵✵✼❪ ❈ ::= Λ[ρ] | ♥ | ⌊♥⌋ | λ λ.❈ | ❈ · ❈ ρ ::= ǫ | ❈ : ρ ❊ ♥ ❊ ❊ ❊

✽ ✴ ✶✺

slide-45
SLIDE 45

❈❛❧❝✉❧✉s ♦❢ ❈❧♦s✉r❡s ✭λ

ρ✮ ❬❇✐❡r♥❛❝❦❛ ❛♥❞ ❉❛♥✈②✱ ✷✵✵✼❪ ❈ ::= Λ[ρ] | ♥ | ⌊♥⌋ | λ λ.❈ | ❈ · ❈ ρ ::= ǫ | ❈ : ρ ❊ ♥ ❊ ❊ ❊

✽ ✴ ✶✺

slide-46
SLIDE 46

❈❛❧❝✉❧✉s ♦❢ ❈❧♦s✉r❡s ✭λ

ρ✮ ❬❇✐❡r♥❛❝❦❛ ❛♥❞ ❉❛♥✈②✱ ✷✵✵✼❪ ❈ ::= Λ[ρ] | ♥ | ⌊♥⌋ | λ λ.❈ | ❈ · ❈ ρ ::= ǫ | ❈ : ρ ❊ ::= ⌊♥⌋ | λ λ.❊ | ❊ · ❊

✽ ✴ ✶✺

slide-47
SLIDE 47

❈❛❧❝✉❧✉s ♦❢ ❈❧♦s✉r❡s ✭λ

ρ✮ ❬❇✐❡r♥❛❝❦❛ ❛♥❞ ❉❛♥✈②✱ ✷✵✵✼❪ ❈ ::= Λ[ρ] | ♥ | ⌊♥⌋ | λ λ.❈ | ❈ · ❈ ρ ::= ǫ | ❈ : ρ ❊ ::= ⌊♥⌋ | λ λ.❊ | ❊ · ❊ (≃ Λ)

✽ ✴ ✶✺

slide-48
SLIDE 48

❊①❛♠♣❧❡

λ.✵((λ.✵)✵)

✾ ✴ ✶✺

slide-49
SLIDE 49

❊①❛♠♣❧❡

  (λ.✵((λ.✵)✵))[ǫ]  

✵

✾ ✴ ✶✺

slide-50
SLIDE 50

❊①❛♠♣❧❡

  λ λ.(✵((λ.✵)✵)[ǫ])  

✵

✾ ✴ ✶✺

slide-51
SLIDE 51

❊①❛♠♣❧❡

  λ λ.(✵((λ.✵)✵)[ǫ])  

✵

✾ ✴ ✶✺

slide-52
SLIDE 52

❊①❛♠♣❧❡

  λ λ.(✵((λ.✵)✵)[ǫ])  

✵

✾ ✴ ✶✺

slide-53
SLIDE 53

❊①❛♠♣❧❡

  λ λ.(✵((λ.✵)✵)[✶])  

✵

✾ ✴ ✶✺

slide-54
SLIDE 54

❊①❛♠♣❧❡

λ λ.(   (✵(λ.✵)✵)[✶]  

✶

)

✾ ✴ ✶✺

slide-55
SLIDE 55

❊①❛♠♣❧❡

λ λ.(   ✵[✶] · ((λ.✵)✵)[✶]  

✶

)

✾ ✴ ✶✺

slide-56
SLIDE 56

❊①❛♠♣❧❡

λ λ.(   ✵[✶]  

✶

· ((λ.✵)✵)[✶])

✾ ✴ ✶✺

slide-57
SLIDE 57

❊①❛♠♣❧❡

λ λ.(   ✵[✶]  

✶

· ((λ.✵)✵)[✶])

✾ ✴ ✶✺

slide-58
SLIDE 58

❊①❛♠♣❧❡

λ λ.(   ✶  

✶

· ((λ.✵)✵)[✶])

✾ ✴ ✶✺

slide-59
SLIDE 59

❊①❛♠♣❧❡

λ λ.(   ⌊✶ − ✶⌋  

✶

· ((λ.✵)✵)[✶])

✾ ✴ ✶✺

slide-60
SLIDE 60

❊①❛♠♣❧❡

λ λ.(   ⌊✵⌋  

✶

· ((λ.✵)✵)[✶])

✾ ✴ ✶✺

slide-61
SLIDE 61

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   ((λ.✵)✵)[✶]  

✶

)

✾ ✴ ✶✺

slide-62
SLIDE 62

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   (λ.✵)[✶] · ✵[✶]  

✶

)

✾ ✴ ✶✺

slide-63
SLIDE 63

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   (λ.✵)[✶]  

✶

· ✵[✶])

✾ ✴ ✶✺

slide-64
SLIDE 64

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   λ λ.(✵[✶])  

✶

· ✵[✶])

✾ ✴ ✶✺

slide-65
SLIDE 65

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   λ λ.(✵[✶])  

✶

· ✵[✶])

✾ ✴ ✶✺

slide-66
SLIDE 66

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   λ λ.(✵[✷ : ✶])  

✶

· ✵[✶])

✾ ✴ ✶✺

slide-67
SLIDE 67

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   λ λ.(✵[✷ : ✶]) · ✵[✶]  

✶

)

✾ ✴ ✶✺

slide-68
SLIDE 68

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   λ λ.(✵[✷ : ✶]) · ✵[✶]  

✶

)

✾ ✴ ✶✺

slide-69
SLIDE 69

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   ✵[✵[✶] : ✶]  

✶

)

✾ ✴ ✶✺

slide-70
SLIDE 70

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   ✵[✵[✶] : ✶]  

✶

)

✾ ✴ ✶✺

slide-71
SLIDE 71

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   ✵[✶]  

✶

)

✾ ✴ ✶✺

slide-72
SLIDE 72

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   ✵[✶]  

✶

)

✾ ✴ ✶✺

slide-73
SLIDE 73

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   ✶  

✶

)

✾ ✴ ✶✺

slide-74
SLIDE 74

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   ⌊✶ − ✶⌋  

✶

)

✾ ✴ ✶✺

slide-75
SLIDE 75

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ ·   ⌊✵⌋  

✶

)

✾ ✴ ✶✺

slide-76
SLIDE 76

❊①❛♠♣❧❡

λ λ.(⌊✵⌋ · ⌊✵⌋)

✾ ✴ ✶✺

slide-77
SLIDE 77

❊①❛♠♣❧❡

λ.✵ ✵

✾ ✴ ✶✺

slide-78
SLIDE 78

❙tr✉❝t✉r❛❧ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ✭λ

ρ✮ ♥ < |ρ| ♥[ρ], ❧ → ♥t❤(ρ), ❧

✭❱❛r✮

(▼ ◆)[ρ], ❧ → ▼[ρ] · ◆[ρ], ❧

✭❆♣♣✮

♥, ❧ → ⌊❧ − ♥⌋, ❧

✭P❛r✮

♥ ≥ |ρ| ♥[ρ], ❧ → ⌊♥ − (|ρ| − ❧)⌋, ❧

✭❋r❡✮

(λ.❇)[ρ], ❧ → λ λ.❇[❧ + ✶ : ρ], ❧

✭▲❛♠✮

(λ λ.❇[♥ : ρ]) · ◆, ❧ → ❇[◆ : ρ], ❧

✭β✮

▼ ∈ ❲❍◆❋❈ ▼, ❧ → ▼′, ❧ ▼ · ◆, ❧ → ▼′ · ◆, ❧

✭µ✶✮

▼ ∈ ❲❍◆❋❈ ▼ ≡ λ λ.❇ ▼, ❧ → ▼′, ❧ ▼ · ◆, ❧ → ▼′ · ◆, ❧

✭µ✷✮

▼ ∈ ◆❋❈ ▼ ≡ λ λ.❇ ◆, ❧ → ◆′, ❧ ▼ · ◆, ❧ → ▼ · ◆′, ❧

✭ν✮

❇, ❧ + ✶ → ❇′, ❧ + ✶ λ λ.❇, ❧ → λ λ.❇′, ❧

✭ξ✮

❲❍◆❋❈ ::= λ λ.❈ | ⌊♥⌋{·❈}∗ ◆❋❈ ::= λ λ.◆❋❈ | ⌊♥⌋{·◆❋❈}∗ ◆❋

✶✵ ✴ ✶✺

slide-79
SLIDE 79

❙tr✉❝t✉r❛❧ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ✭λ

ρ✮ ♥ < |ρ| ♥[ρ], ❧ → ♥t❤(ρ), ❧

✭❱❛r✮

(▼ ◆)[ρ], ❧ → ▼[ρ] · ◆[ρ], ❧

✭❆♣♣✮

♥, ❧ → ⌊❧ − ♥⌋, ❧

✭P❛r✮

♥ ≥ |ρ| ♥[ρ], ❧ → ⌊♥ − (|ρ| − ❧)⌋, ❧

✭❋r❡✮

(λ.❇)[ρ], ❧ → λ λ.❇[❧ + ✶ : ρ], ❧

✭▲❛♠✮

(λ λ.❇[♥ : ρ]) · ◆, ❧ → ❇[◆ : ρ], ❧

✭β✮

▼ ∈ ❲❍◆❋❈ ▼, ❧ → ▼′, ❧ ▼ · ◆, ❧ → ▼′ · ◆, ❧

✭µ✶✮

▼ ∈ ❲❍◆❋❈ ▼ ≡ λ λ.❇ ▼, ❧ → ▼′, ❧ ▼ · ◆, ❧ → ▼′ · ◆, ❧

✭µ✷✮

▼ ∈ ◆❋❈ ▼ ≡ λ λ.❇ ◆, ❧ → ◆′, ❧ ▼ · ◆, ❧ → ▼ · ◆′, ❧

✭ν✮

❇, ❧ + ✶ → ❇′, ❧ + ✶ λ λ.❇, ❧ → λ λ.❇′, ❧

✭ξ✮

❲❍◆❋❈ ::= λ λ.❈ | ⌊♥⌋{·❈}∗ ◆❋❈ ::= λ λ.◆❋❈ | ⌊♥⌋{·◆❋❈}∗ (≃ ◆❋)

✶✵ ✴ ✶✺

slide-80
SLIDE 80

◆❛t✉r❛❧ ❙❡♠❛♥t✐❝s ✇✐t❤ ❊①♣❧✐❝✐t ❈♦♥tr♦❧ ✭λ∗

ρ✮ ❝ ::= ♥ | ✇ ♥ < |ρ| ♥t❤(ρ), ❧, ❝ ⇓ ◆ ♥[ρ], ❧, ❝ ⇓ ◆

✭❱❛r✮

♥, ❧, ❝ ⇓ ⌊❧ − ♥⌋

✭P❛r✮

♥ ≥ |ρ| ♥[ρ], ❧, ❝ ⇓ ⌊♥ − (|ρ| − ❧)⌋

✭❋r❡✮

(λ.❇)[ρ], ❧, ✇ ⇓ ❇[❧ + ✶ : ρ]

✭▲❛♠✶✮

❇[❧ + ✶ : ρ], ❧ + ✶, ♥ ⇓ ⌊❇′⌋ (λ.❇)[ρ], ❧, ♥ ⇓ ⌊λ.❇′⌋

✭▲❛♠✷✮

▼[ρ], ❧, ✇ ⇓ ▼′ ▼′ ≡ ❇[♥ : ρ] ❇[◆[ρ] : ρ], ❧, ❝ ⇓ ❇′ (▼ ◆)[ρ], ❧, ❝ ⇓ ❇′

✭❘❡❞✮

▼[ρ], ❧, ✇ ⇓ ▼′ ▼′ ≡ ⌊▼′⌋ ◆[ρ], ❧, ♥ ⇓ ⌊◆′⌋ (▼ ◆)[ρ], ❧, ❝ ⇓ ⌊▼′ ◆′⌋

✭◆❡✉✮

✶✶ ✴ ✶✺

slide-81
SLIDE 81

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-82
SLIDE 82

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-83
SLIDE 83

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-84
SLIDE 84

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-85
SLIDE 85

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ♥ ❙ ❧ ♥ ❧ ❙ ❧ ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-86
SLIDE 86

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-87
SLIDE 87

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-88
SLIDE 88

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-89
SLIDE 89

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚ ♥(λ.❇)◆

✶✷ ✴ ✶✺

slide-90
SLIDE 90

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ, ❧⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ, ❧⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-91
SLIDE 91

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ, ❧⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ, ❧⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-92
SLIDE 92

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ, ❧⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ, ❧⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-93
SLIDE 93

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ, ❧⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ, ❧⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-94
SLIDE 94

❖♣t✐♠✐s❡❞ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ ✭λ∗

ρ✮ ❈ ::= Λ[ρ] | ♥ | ⌊Λ, ❧⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ, ❧⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ✵ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-95
SLIDE 95

❖r✐❣✐♥❛❧ ❋✉❧❧✲❘❡❞✉❝✐♥❣ ❑r✐✈✐♥❡ ▼❛❝❤✐♥❡ λρ

❈ ::= Λ[ρ] | ♥ | ⌊Λ, ❧⌋ ρ ::= ǫ | ❈ : ρ ❙ ::= ǫ | Λ[ρ] : ❙ | λ λ : ❙ | ⌊Λ, ❧⌋ : ❙ ❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥, ❧⌋ , ❙ , ❧ ) ( ⌊▼, ❧′⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼, ❧′⌋ : ❙ , ❧′ ) ( ⌊❇, ❧′⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇, ❧′⌋ , ❙ , ❧ ) ( ⌊◆, ❧′⌋ , ⌊▼, ❧′′⌋ : ❙ , ❧ ) → ( ⌊▼ ◆, ❧′′⌋ , ❙ , ❧ ) ( ⌊❚, ❧′⌋ , ǫ , ❧ ) → ❚ ♥ ❇ ◆

✶✷ ✴ ✶✺

slide-96
SLIDE 96

❙✉♠♠❛r②

◮ ❙tr✉❝t✉r❛❧ ♦♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ✭s♠❛❧❧✲st❡♣✮✿ ❡①♣♦s❡s

❡♣❤❡♠❡r❛❧ ❝♦♥str✉❝ts✱ ❞❡❇r✉✐❥♥ ❧❡✈❡❧s r❡♣r❡s❡♥t✐♥❣ ♣❛r❛♠❡t❡rs ✭♥✮✳ ◆❛t✉r❛❧ s❡♠❛♥t✐❝s ✭❜✐❣✲st❡♣✮✿ ❝♦♥tr♦❧ ❝❤❛r❛❝t❡r ❞✐st✐♥❣✉✐s❤❡s ❜❡t✇❡❡♥ ✇❡❛❦ ✭✇✮ ❛♥❞ ❢✉❧❧ ✭♥✮ ❡✈❛❧✉❛t✐♦♥✳ ❆❜str❛❝t ♠❛❝❤✐♥❡ ✭✜rst✲♦r❞❡r st❛t❡✲tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✮✿ t❤❡ ❞✐st✐♥❝t✐♦♥ ✐s ❞♦♥❡ ❧♦♦❦✐♥❣ ❛t t❤❡ ❝♦♥tr♦❧ st❛❝❦✳

✶✸ ✴ ✶✺

slide-97
SLIDE 97

❙✉♠♠❛r②

◮ ❙tr✉❝t✉r❛❧ ♦♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ✭s♠❛❧❧✲st❡♣✮✿ ❡①♣♦s❡s

❡♣❤❡♠❡r❛❧ ❝♦♥str✉❝ts✱ ❞❡❇r✉✐❥♥ ❧❡✈❡❧s r❡♣r❡s❡♥t✐♥❣ ♣❛r❛♠❡t❡rs ✭♥✮✳

◮ ◆❛t✉r❛❧ s❡♠❛♥t✐❝s ✭❜✐❣✲st❡♣✮✿ ❝♦♥tr♦❧ ❝❤❛r❛❝t❡r ❞✐st✐♥❣✉✐s❤❡s

❜❡t✇❡❡♥ ✇❡❛❦ ✭✇✮ ❛♥❞ ❢✉❧❧ ✭♥✮ ❡✈❛❧✉❛t✐♦♥✳ ❆❜str❛❝t ♠❛❝❤✐♥❡ ✭✜rst✲♦r❞❡r st❛t❡✲tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✮✿ t❤❡ ❞✐st✐♥❝t✐♦♥ ✐s ❞♦♥❡ ❧♦♦❦✐♥❣ ❛t t❤❡ ❝♦♥tr♦❧ st❛❝❦✳

✶✸ ✴ ✶✺

slide-98
SLIDE 98

❙✉♠♠❛r②

◮ ❙tr✉❝t✉r❛❧ ♦♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ✭s♠❛❧❧✲st❡♣✮✿ ❡①♣♦s❡s

❡♣❤❡♠❡r❛❧ ❝♦♥str✉❝ts✱ ❞❡❇r✉✐❥♥ ❧❡✈❡❧s r❡♣r❡s❡♥t✐♥❣ ♣❛r❛♠❡t❡rs ✭♥✮✳

◮ ◆❛t✉r❛❧ s❡♠❛♥t✐❝s ✭❜✐❣✲st❡♣✮✿ ❝♦♥tr♦❧ ❝❤❛r❛❝t❡r ❞✐st✐♥❣✉✐s❤❡s

❜❡t✇❡❡♥ ✇❡❛❦ ✭✇✮ ❛♥❞ ❢✉❧❧ ✭♥✮ ❡✈❛❧✉❛t✐♦♥✳

◮ ❆❜str❛❝t ♠❛❝❤✐♥❡ ✭✜rst✲♦r❞❡r st❛t❡✲tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✮✿ t❤❡

❞✐st✐♥❝t✐♦♥ ✐s ❞♦♥❡ ❧♦♦❦✐♥❣ ❛t t❤❡ ❝♦♥tr♦❧ st❛❝❦✳

✶✸ ✴ ✶✺

slide-99
SLIDE 99

❈❛❧❝✉❧✐ ♦❢ ❈❧♦s✉r❡s ❍✐❡r❛r❝❤② λρ λ∗

ρ

λ

ρ

λ

ρ

❈ ::= Λ[ρ]

❬❈✉r✐❡♥✱ ✶✾✾✶❪ ❬❇✐❡r♥❛❝❦❛ ❛♥❞ ❉❛♥✈②✱ ✷✵✵✼❪ ❜❛s❡❞ ♦♥ ❬❈ré❣✉t✱ ✷✵✵✼❪

+♥ + ⌊Λ⌋ +❈ · ❈ ⌊Λ⌋≃⌊♥⌋ | λ λ.❈ | ❈ · ❈ +♥ + ⌊♥⌋ + λ λ.❈

❚❤❛♥❦s✦

✶✹ ✴ ✶✺

slide-100
SLIDE 100

❈❛❧❝✉❧✐ ♦❢ ❈❧♦s✉r❡s ❍✐❡r❛r❝❤② λρ λ∗

ρ

λ

ρ

λ

ρ

❈ ::= Λ[ρ]

❬❈✉r✐❡♥✱ ✶✾✾✶❪ ❬❇✐❡r♥❛❝❦❛ ❛♥❞ ❉❛♥✈②✱ ✷✵✵✼❪ ❜❛s❡❞ ♦♥ ❬❈ré❣✉t✱ ✷✵✵✼❪

+♥ + ⌊Λ⌋ +❈ · ❈ ⌊Λ⌋≃⌊♥⌋ | λ λ.❈ | ❈ · ❈ +♥ + ⌊♥⌋ + λ λ.❈

❚❤❛♥❦s✦

✶✹ ✴ ✶✺

slide-101
SLIDE 101

❈❧♦s✉r❡✲❇❛s❡❞ ❙✉❜st✐t✉t✐♦♥

σ❈(⌊♥⌋, ❧) = ♥ σ❈(♥[ρ], ❧) =

  • σ❈(♥t❤(ρ), ❧)

✐❢ ♥ < |ρ| ♥ − (|ρ| − ❧) ✐❢ ♥ ≥ |ρ| σ❈((λ.❇)[ρ], ❧) = σ❈(λ λ.❇[❧ + ✶ : ρ], ❧) σ❈(λ λ.❇, ❧) = λ.σ❈(❇, ❧ + ✶) σ❈(♥, ❧) = ❧ − ♥ σ❈((▼ ◆)[ρ], ❧) = σ❈(▼[ρ] · ◆[ρ], ❧) σ❈(▼ · ◆, ❧) = σ❈(▼, ❧) σ❈(◆, ❧)

✶✹ ✴ ✶✺

slide-102
SLIDE 102

❙t❡♣✲❜②✲❙t❡♣ ❙✐♠✉❧❛t✐♦♥ ❜❡t✇❡❡♥ λ

ρ ❛♥❞ λ ▼ ▼′ ▼✶ ▼′

✶

▼✷ ▼ ▼✶ ▼✷ ▼ ≡ ▼[ǫ]

ρ ∗ β ρ ∗ β σ σ

λ

ρ

→ρ ❛❧❧ t❤❡ r✉❧❡s ❜✉t β →β β ❛♥❞ ❛❧❧ t❤❡ ❝♦♠♣❛t✐❜✐❧✐t② r✉❧❡s →σ ❝❧♦s✉r❡✲❜❛s❡❞ s✉❜st✐t✉t✐♦♥ λ → s✉❜st✐t✉t✐♦♥✲❜❛s❡❞ ♥♦r♠❛❧ ♦r❞❡r

✶✹ ✴ ✶✺

slide-103
SLIDE 103

❍❡✐❣❤t ♦❢ ❛ ❈❧♦s✉r❡

❤(♥[ρ]) =

  • ❤(♥t❤(ρ))

✐❢ ♥ < |ρ| ✵ ✐❢ ♥ ≥ |ρ| ❤((λ.❇)[ρ]) = ✶ + ❤(❇[♥ : ρ]) ❤((▼ ◆)[ρ]) = ♠❛①{❤(▼[ρ]), ❤(◆[ρ])} ❤(♥) = ✵ ❤(⌊♥⌋) = ✵ ❤(λ λ.❇) = ✶ + ❤(❇) ❤(▼ · ◆) = ♠❛①{❤(▼), ❤(◆)}

✶✺ ✴ ✶✺

slide-104
SLIDE 104

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

λ.✵((λ.✵)✵)

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-105
SLIDE 105

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

λ.✵((λ.✵)✵)

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-106
SLIDE 106

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( λ.✵((λ.✵)✵)[ǫ] , ǫ , ✵ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-107
SLIDE 107

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( λ.✵((λ.✵)✵)[ǫ] , ǫ , ✵ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-108
SLIDE 108

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✵((λ.✵)✵)[✶] , λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-109
SLIDE 109

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✵((λ.✵)✵)[✶] , λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-110
SLIDE 110

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✵[✶] , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-111
SLIDE 111

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✵[✶] , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-112
SLIDE 112

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✶ , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-113
SLIDE 113

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✶ , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-114
SLIDE 114

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ⌊✵⌋ , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-115
SLIDE 115

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ⌊✵⌋ , ((λ.✵)✵)[✶] : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-116
SLIDE 116

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ((λ.✵)✵)[✶] , ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-117
SLIDE 117

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ((λ.✵)✵)[✶] , ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-118
SLIDE 118

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( (λ.✵)[✶] , ✵[✶] : ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-119
SLIDE 119

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( (λ.✵)[✶] , ✵[✶] : ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-120
SLIDE 120

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✵[✵[✶] : ✶] , ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-121
SLIDE 121

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✵[✵[✶] : ✶] , ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-122
SLIDE 122

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✵[✶] , ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-123
SLIDE 123

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✵[✶] , ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-124
SLIDE 124

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✶ , ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-125
SLIDE 125

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ✶ , ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-126
SLIDE 126

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ⌊✵⌋ , ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-127
SLIDE 127

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ⌊✵⌋ , ⌊✵⌋ : λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-128
SLIDE 128

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ⌊✵ ✵⌋ , λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-129
SLIDE 129

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ⌊✵ ✵⌋ , λ λ , ✶ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-130
SLIDE 130

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ⌊λ.✵ ✵⌋ , ǫ , ✵ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-131
SLIDE 131

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

( ⌊λ.✵ ✵⌋ , ǫ , ✵ )

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-132
SLIDE 132

❊①❛♠♣❧❡ ✇✐t❤ ❖♣t✐♠✐s❡❞ ▼❛❝❤✐♥❡

λ.✵ ✵

❚ → ( ❚[ǫ] , ǫ , ✵ ) ( (♥ + ✶)[❈ : ρ] , ❙ , ❧ ) → ( ♥[ρ] , ❙ , ❧ ) ( ♥[ǫ] , ❙ , ❧ ) → ( ⌊♥ + ❧⌋ , ❙ , ❧ ) ( ✵[❈ : ρ] , ❙ , ❧ ) → ( ❈ , ❙ , ❧ ) ( (▼ ◆)[ρ] , ❙ , ❧ ) → ( ▼[ρ] , ◆[ρ] : ❙ , ❧ ) ( (λ.❇)[ρ] , ◆[ρ′] : ❙ , ❧ ) → ( ❇[◆[ρ′] : ρ] , ❙ , ❧ ) ( (λ.❇)[ρ] , ❙ , ❧ ) → ( ❇[❧ + ✶ : ρ] , λ λ : ❙ , ❧ + ✶ ) ( ♥ , ❙ , ❧ ) → ( ⌊❧ − ♥⌋ , ❙ , ❧ ) ( ⌊▼⌋ , ◆[ρ] : ❙ , ❧ ) → ( ◆[ρ] , ⌊▼⌋ : ❙ , ❧ ) ( ⌊❇⌋ , λ λ : ❙ , ❧ ) → ( ⌊λ.❇⌋ , ❙ , ❧ − ✶ ) ( ⌊◆⌋ , ⌊▼⌋ : ❙ , ❧ ) → ( ⌊▼ ◆⌋ , ❙ , ❧ ) ( ⌊❚⌋ , ǫ , ✵ ) → ❚

✶✺ ✴ ✶✺

slide-133
SLIDE 133

❘❡❢❡r❡♥❝❡s

❇✐❡r♥❛❝❦❛✱ ▼✳ ❛♥❞ ❉❛♥✈②✱ ❖✳ ✭✷✵✵✼✮✳ ❆ ❝♦♥❝r❡t❡ ❢r❛♠❡✇♦r❦ ❢♦r ❡♥✈✐r♦♥♠❡♥t ♠❛❝❤✐♥❡s✳ ❆❈▼ ❚r❛♥s✳ ❈♦♠♣✉t✳ ▲♦❣✱ ✾✭✶✮✿✻✿✶✕✻✿✷✾✳ ❈ré❣✉t✱ P✳ ✭✷✵✵✼✮✳ ❙tr♦♥❣❧② r❡❞✉❝✐♥❣ ✈❛r✐❛♥ts ♦❢ t❤❡ ❑r✐✈✐♥❡ ❛❜str❛❝t ♠❛❝❤✐♥❡✳ ❍✐❣❤❡r✲❖r❞❡r ❛♥❞ ❙②♠❜♦❧✐❝ ❈♦♠♣✉t❛t✐♦♥✱ ✷✵✭✸✮✿✷✵✾✕✷✸✵✳ ❈✉r✐❡♥✱ P✳✲▲✳ ✭✶✾✾✶✮✳ ❆♥ ❛❜str❛❝t ❢r❛♠❡✇♦r❦ ❢♦r ❡♥✈✐r♦♥♠❡♥t ♠❛❝❤✐♥❡s✳ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✽✷✭✷✮✿✸✽✾✕✹✵✷✳

✶✺ ✴ ✶✺