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❆ ♣r♦♦❢ ✇✐t❤ s✐❞❡ ❡✛❡❝ts ♦❢ ●ö❞❡❧✬s ❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠ s✉✐t❛❜❧❡ ❢♦r s❡♠❛♥t✐❝ ♥♦r♠❛❧✐s❛t✐♦♥

❍✉❣♦ ❍❡r❜❡❧✐♥

❚❨P❊❙ ✶✾ ▼❛② ✷✵✶✺ ❚❛❧❧✐♥♥

✶

Pr❡❧✐♠✐♥❛r②✿ ♣r♦✈✐♥❣ ✇✐t❤ s✐❞❡ ❡✛❡❝ts

✲ ❈❧❛ss✐❝❛❧ ❧♦❣✐❝ s❡❡♥ ❛s ❛ s✐❞❡ ❡✛❡❝t✿ ✲ ❉✐r❡❝t st②❧❡ ❂ ❛ ❝♦♥tr♦❧ ♦♣❡r❛t♦r ✭❡✳❣✳ ❝❛❧❧❝❝ ♦❢ t②♣❡ P❡✐r❝❡✬s ❧❛✇✮ ❬●r✐✣♥ ✾✵❪ ✲ ■♥❞✐r❡❝t st②❧❡ ❂ ❝♦♥t✐♥✉❛t✐♦♥✲♣❛ss✐♥❣✲st②❧❡✴❞♦✉❜❧❡✲♥❡❣❛t✐♦♥ tr❛♥s❧❛t✐♦♥ ✇✐t❤✐♥ ✐♥t✉✐t✐♦♥✲ ✐st✐❝ ❧♦❣✐❝ ✲ ❚❤✐s t❛❧❦✿ ✲ ■♥t❡r♣r❡t✐♥❣ ❑r✐♣❦❡ ❢♦r❝✐♥❣ tr❛♥s❧❛t✐♦♥ ❛s ✐♥❞✐r❡❝t st②❧❡ ❢♦r ✇❤❛t ✐s ✐♥ ❞✐r❡❝t st②❧❡ ❛ ♠♦♥♦t♦♥✐❝ ♠❡♠♦r② ✉♣❞❛t❡ ✲ ❆♣♣❧②✐♥❣ t❤✐s t♦ ♦❜t❛✐♥ ❛ ♣r♦♦❢ ✇✐t❤ s✐❞❡✲❡✛❡❝t ♦❢ ●ö❞❡❧✬s ❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠ ❛s ❞✐r❡❝t✲st②❧❡ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ ♣r♦♦❢ ♦❢ ❝♦♠♣❧❡t❡♥❡ss ✇✳r✳t✳ ❑r✐♣❦❡ s❡♠❛♥t✐❝s

✷

❑r✐♣❦❡ ❢♦r❝✐♥❣ tr❛♥s❧❛t✐♦♥ ❛s ❛♥ ❡♥✈✐r♦♥♠❡♥t ♠♦♥❛❞

▲❡t ≥ ❜❡ ❛ ♣❛rt✐❛❧ ♦r❞❡r✳ ❆ ❦❡② ❝❧❛✉s❡ ♦❢ ❑r✐♣❦❡ ❢♦r❝✐♥❣ ✐s t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ✐♠♣❧✐✲ ❝❛t✐♦♥✿ w A → B ∀w′ ≥ w [(w′ A) → (w′ B)] ❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ wA(w) ∀w′ ≥ w A(w′) ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ❞❡♣❡♥❞❡♥t ❡♥✈✐r♦♥♠❡♥t ♠♦♥❛❞✱ ✐✳❡✳ ❛s ✐♥❞✐r❡❝t st②❧❡ ❢♦r ❛ ♠♦♥♦t♦♥✐❝ ♠❡♠♦r② ✉♣❞❛t❡ ❡✛❡❝t✳

✸

❉✐r❡❝t✲st②❧❡ ❢♦r ❑r✐♣❦❡ ❢♦r❝✐♥❣

❆ r✉❧❡ ❢♦r ✐♥✐t✐❛❧✐s✐♥❣ t❤❡ ✉s❡ ♦❢ ❑r✐♣❦❡ ❢♦r❝✐♥❣✿ Γ, [b : x ≥ t] ⊢ q : T(x) Γ ⊢ r : refl ≥ Γ ⊢ s : trans ≥ x ❢r❡s❤ ✐♥ Γ ❛♥❞ T(t) Γ ⊢ s❡t x := t ❛s b/(r,s) ✐♥ q : T(t) s❡t❡❢❢ ❆ r✉❧❡ ❢♦r ✉♣❞❛t✐♥❣✿ Γ, [b : x ≥ t(x′)] ⊢ q : T(x) Γ ⊢ r : t(x′) ≥ x′ [x ≥ u] ∈ Γ ❢♦r s♦♠❡ u x′ ❢r❡s❤ ✐♥ Γ Γ ⊢ ✉♣❞❛t❡ x:=t(x) ♦❢ x′ ❛s b ❜② r ✐♥ q : T(t(x)) ✉♣❞❛t❡ ✇❤❡r❡ ✇❡ ✇r♦t❡ T✱ U ❢♦r ˙ → ✲ ˙ ∀✲❢r❡❡ ❢♦r♠✉❧❛s ✭❂ ✐♥t✉✐t✐✈❡❧② Σ0

1✲❢♦r♠✉❧❛s ❂ ❜❛s❡ t②♣❡s✮

✹

• ö❞❡❧✬s ❝♦♠♣❧❡t❡♥❡ss

✺

❖❜❥❡❝t ❧❛♥❣✉❛❣❡

❲❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ♥❡❣❛t✐✈❡ ❢r❛❣♠❡♥t ♦❢ ♣r❡❞✐❝❛t❡ ❧♦❣✐❝ ❛s ❛♥ ♦❜❥❡❝t ❧❛♥❣✉❛❣❡ ✭✇❡ ❝♦♥s✐❞❡r ˙ ⊥ t♦ ❜❡ ❛♥ ❛r❜✐tr❛r② ❛t♦♠ ❛♥❞ ❛❜❜r❡✈✐❛t❡ ˙ ¬A A ˙ → ˙ ⊥✮✳ t x | f(t1, ..., tn) F, G ˙ ⊥ | ˙ P(t1, ..., tn) | F ˙ → G | ˙ ∀x F Γ ǫ | Γ, F ❲❡ t❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❢❡r❡♥❝❡ r✉❧❡s✿ ˙ Ax

Γ,F,Γ′

: (Γ, F ⊂ Γ′) → (Γ′ ⊢ F) ˙ App

Γ,F,G →

: (Γ ⊢ F ˙ → G) → (Γ ⊢ F) → (Γ ⊢ G) ˙ Abs

Γ,F,G →

: (Γ, F ⊢ G) → (Γ ⊢ F ˙ → G) ˙ Abs

Γ,x,F ∀

: (Γ ⊢ F) → (x ∈ FV (Γ)) → (Γ ⊢ ˙ ∀x F(x)) ˙ App

Γ,x,t,F ∀

: (Γ ⊢ ˙ ∀x F) → (Γ ⊢ F[t/x]) ▼♦r❡♦✈❡r✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛❞♠✐ss✐❜❧❡✿ ˙ weak

Γ′ Γ,F

: (Γ ⊂ Γ′) → (Γ ⊢ F) → (Γ′ ⊢ F) ❲❡ s❤❛❧❧ ❛❧s♦ ✇r✐t❡ rΓ

F ❢♦r ❛ ♣r♦♦❢ ♦❢ Γ ⊂ (Γ, F)✱

✻

❚❛rs❦✐❛♥ ♠♦❞❡❧s

❆ ❚❛rs❦✐❛♥ ♠♦❞❡❧ M ✐s ♠❛❞❡ ♦❢ ❛ ❞♦♠❛✐♥ DM ❢♦r ✐♥t❡r♣r❡t✐♥❣ t❡r♠s✱ ♦❢ ❛♥ ✐♥t❡r✲ ♣r❡t❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥ s②♠❜♦❧s FM(f) : Daf → D ❛♥❞ ♦❢ ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❛t♦♠s PM( ˙ P) ⊂ Da ˙

P ✭❢♦r af✱ a ˙

P t❤❡ ❛r✐t② ♦❢ f✱ ˙

P r❡s♣✳✮✳ ❚r✉t❤ ✐s ❞❡✜♥❡❞ ❜② [ [x] ]σ

M

σ(x) [ [ft1 . . . taf] ]σ

M

FM(f)([ [t1] ]σ

M, . . . , [

[taf] ]σ

M)

σ

M ˙

P(t1, . . . , ta ˙

P) PM( ˙

P)([ [t1] ]σ

M, . . . , [

[ta ˙

P]

]σ

M)

σ

M ˙

⊥ PM( ˙ ⊥) σ

M F ˙

→ G σ

M F → σ M G

σ

M ˙

∀x F ∀t ∈ MD σ[x←t]

M

F

✼

❈♦♠♣❧❡t❡♥❡ss ✇✳r✳t ❚❛rs❦✐❛♥ ♠♦❞❡❧s

▲❡t Clas ❜❡ t❤❡ t❤❡♦r② ❝♦♥t❛✐♥✐♥❣ ˙ ¬ ˙ ¬F ˙ → F ❢♦r ❛❧❧ ❢♦r♠✉❧❛s F ✭❛t♦♠s ❛r❡ ❡♥♦✉❣❤✮✳ ❲❡ ❞❡✜♥❡ ⊢C F t♦ ❜❡ Clas ⊢M F ✐♥ ♠✐♥✐♠❛❧ ❧♦❣✐❝✳ ❆ ❚❛rs❦✐❛♥ ♠♦❞❡❧ M ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ✐s ❛ ❚❛rs❦✐❛♥ ♠♦❞❡❧ ✇❤✐❝❤ s❛t✐s✜❡s M Clas ✭✐♥ ❛ ❝❧❛ss✐❝❛❧ ♠❡t❛✲❧❛♥❣✉❛❣❡✱ ❛❧❧ ❚❛rs❦✐❛♥ ♠♦❞❡❧s ❛r❡ ❝❧❛ss✐❝❛❧✱ ❜✉t ♥♦t ✐♥ ❛♥ ✐♥t✉✐t✐♦♥✐st✐❝ ♠❡t❛✲❧❛♥❣✉❛❣❡✮✳ ❚❤❡ st❛t❡♠❡♥t ♦❢ ❝♦♠♣❧❡t❡♥❡ss ✇✳r✳t ❚❛rs❦✐❛♥ ♠♦❞❡❧s ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ✐s✿ [∀M ∀σ (σ

M Clas → σ M F)] → Clas ⊢M F

❚❤❡ ✉s✉❛❧ ♣r♦♦❢ ✐s ❜② ❝♦♥tr❛❞✐❝t✐♦♥✱ ❜✉✐❧❞✐♥❣ ❛ s❛t✉r❛t❡❞ ❝♦✉♥t❡r✲♠♦❞❡❧ ❜② ❡♥✉♠❡r❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠✉❧❛s✳ ❚❤❡ ♣r♦♦❢ ✇✐t❤ ❡✛❡❝ts ✇❡ s❤❛❧❧ ❝♦♥s✐❞❡r ❛❝t✉❛❧❧② ✇♦r❦s ❢♦r ❛r❜✐tr❛r② t❤❡♦r✐❡s✱ s♦ t❤❛t ✇❡ s❤❛❧❧ ❝♦♥s✐❞❡r ✐♥st❡❛❞ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥t✿ (∀M ∀σ σ

M F) → ⊢M F

✽

❈♦♠♣❧❡t❡♥❡ss ✇✳r✳t✳ ❑r✐♣❦❡ ♠♦❞❡❧s

✾

❑r✐♣❦❡ ♠♦❞❡❧s

❆ ❑r✐♣❦❡ ♠♦❞❡❧ K ✐s ❛♥ ✐♥❝r❡❛s✐♥❣ ❢❛♠✐❧② ♦❢ ❚❛rs❦✐❛♥ ♠♦❞❡❧s ✐♥❞❡①❡❞ ♦✈❡r ❛ s❡t ♦❢ ✇♦r❧❞s WK ♦r❞❡r❡❞ ❜② ≥K✳ ■♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ∨ ❛♥❞ ∃✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❛❦❡ DK ❝♦♥st❛♥t✳ ❚r✉t❤ r❡❧❛t✐✈❡❧② t♦ K ❛t ✇♦r❧❞ w ✐s ❞❡✜♥❡❞ ❜②✿ [ [x] ]σ

K

σ(x) [ [ft1 . . . taf] ]σ

K

FK(f)([ [t1] ]σ

K, . . . , [

[taf] ]σ

K)

w σ

K ˙

P(t1 . . . ta ˙

P) PK( ˙

P)w([ [t1] ]σ

K, . . . , [

[ta ˙

P]

]σ

K)

w σ

K ˙

⊥ PK( ˙ ⊥)w w σ

K F ˙

→ G ∀w′ ≥K w (w′ σ

K F → w′ σ K G)

w σ

K ∀x F

∀t ∈ KD w σ[x←t]

K

F ❚❤❡ st❛t❡♠❡♥t ♦❢ ❝♦♠♣❧❡t❡♥❡ss ✇✳r✳t✳ ❑r✐♣❦❡ ♠♦❞❡❧s ✐s✿ (∀K ∀σ ∀w ∈ WK w σ

K F) → ⊢M F

✶✵

❈♦♠♣❧❡t❡♥❡ss ✇✳r✳t ❑r✐♣❦❡ ♠♦❞❡❧s

❚❤❡ ✏st❛♥❞❛r❞✑ ♣r♦♦❢ ✇♦r❦s ❜② ❜✉✐❧❞✐♥❣ t❤❡ ❝❛♥♦♥✐❝❛❧ ♠♦❞❡❧ K0 ❞❡✜♥❡❞ ❜② t❛❦✐♥❣ WK0 t♦ ❜❡ t❤❡ t②♣✐♥❣ ❝♦♥t❡①ts ♦r❞❡r❡❞ ❜② ✐♥❝❧✉s✐♦♥✱ DK0 t♦ ❜❡ t❤❡ t❡r♠s✱ KF(f) t♦ ❜❡ t❤❡ s②♥t❛❝t✐❝ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ f✱ ❛♥❞ KP( ˙ P)(Γ)(t1, ..., ta ˙

P) t♦ ❜❡ Γ ⊢M ˙

P(t1, ..., ta ˙

P)

❚❤❡ ♠❛✐♥ ❧❡♠♠❛ ♣r♦✈❡s Γ ⊢M F ↔ Γ K0 F ❜② ✐♥❞✉❝t✐♦♥ ♦♥ F ↑Γ

F

Γ ⊢M F − → Γ K0 F ↑Γ

˙ P( t)

p p ↑Γ

F ˙ → G

p Γ′ → h → m → ↑Γ′

G

˙ App

Γ′,F,G →

( ˙ weak

Γ′ Γ,F(h, p), ↓Γ′ F m)

↑Γ

˙ ∀x F

p t → ↑Γ

F[t/x]

˙ App

Γ,x,F ∀

(p, t) ↓Γ

F

Γ K0 F − → Γ ⊢M F ↓Γ

˙ P( t)

m m ↓Γ

F ˙ → G

m

• ˙

Abs

Γ,F,G →

(↓Γ,F

G

(m (Γ, F) rΓ

F (↑Γ,F F

˙ Ax

Γ1,F,,Γ(bF))))

↓Γ

˙ ∀x F

m

• ˙

Abs

Γ,x,F ∀

( ˙ y, ↓Γ

F[z/x] (m ˙

y)) ˙ y ❢r❡s❤ ✐♥ Γ ❆♥❞ ✜♥❛❧❧②✿ ❝♦♠♣❧ v →↓ǫ

A (v K0 ∅ ǫ) : (∀K ∀σ ∀w ∈ WK w σ K F) → ⊢M F

✶✶

❈♦♠♣❧❡t❡♥❡ss ✇✳r✳t✳ ❑r✐♣❦❡ ♠♦❞❡❧s ✐♥ ❞✐r❡❝t✲st②❧❡

✶✷

❑r✐♣❦❡ ❢♦r❝✐♥❣ tr❛♥s❧❛t✐♦♥ ❢♦r s❡❝♦♥❞✲♦r❞❡r ❛r✐t❤♠❡t✐❝

❲❡ ❝♦♥s✐❞❡r ❛ s❡❝♦♥❞✲♦r❞❡r ❛r✐t❤♠❡t✐❝ ♠✉❧t✐✲s♦rt❡❞ ♦✈❡r ✜rst✲♦r❞❡r ❞❛t❛t②♣❡s s✉❝❤ ❛s N✱ ❧✐sts✱ ❢♦r♠✉❧❛s✱ ❡t❝✳✱ ❛♥❞ ✇✐t❤ ♣r✐♠✐t✐✈❡ r❡❝✉rs✐✈❡ ❛t♦♠s ✇r✐tt❡♥ P(t1, ..., taP)✳ A, B X(t1, ..., taX) | P(t1, ..., taP) | A ∧ B | A → B | ∀x A | ∀X A ▲❡t ≥ ❜❡ ❛ ♣❛rt✐❛❧ ♦r❞❡r✳ ❲❡ ❡①t❡♥❞ ❑r✐♣❦❡ ❢♦r❝✐♥❣ t♦ s❡❝♦♥❞ ♦r❞❡r q✉❛♥t✐✜❝❛t✐♦♥✳ w X(t1, ..., taX) X(w, t1, ..., taX) w P(t1, ..., taP) P(t1, ..., taP) w A ∧ B (w A) ∧ (w B) w A → B ∀w′ ≥ w [(w′ A) → (w′ B)] w ∀x A ∀x w A w ∀X A ∀X (mon(X) → w A) ✇❤❡r❡ mon(X) ∀w ∀w′ ≥ w (X(w, t1, ..., taX) → X(w′, t1, ..., taX))

✶✸

❘❡❧❛t✐♥❣ ❝♦♠♣❧❡t❡♥❡ss ✇✳r✳t ❚❛rs❦✐❛♥ ♠♦❞❡❧s t♦ ❝♦♠♣❧❡t❡♥❡ss ✇✳r✳t✳ ❑r✐♣❦❡ ♠♦❞❡❧s

❲❡ ❣❡t ❛ str♦♥❣❡r st❛t❡♠❡♥t ♦❢ ❝♦♠♣❧❡t❡♥❡ss ❜② ❝♦♥s✐❞❡r✐♥❣ ❝♦♠♣❧❡t❡♥❡ss ✇✳r✳t ❑r✐♣❦❡ ♠♦❞❡❧s ❜② s♣❡❝✐✜❝❛❧❧② ✐♥st❛♥t✐❛t✐♥❣ WK t♦ ❜❡ t❤❡ t②♣✐♥❣ ❝♦♥t❡①ts ❛♥❞ ≥ t♦ ❜❡ ✐♥❝❧✉s✐♦♥ ♦❢ ❝♦♥t❡①ts✳ (∀(DK, FK, PK) ∀σ [ǫ σ

(WK,DK,KF,PK) F]) → ⊢M F

◆♦✇✱ ❛♣♣❧②✐♥❣ ❢♦r❝✐♥❣ s❤♦✇s t❤❛t ǫ x (∀(DM, FM, PM)∀σ (DM,FM,PM) F) ✐s ❡q✉✐✈❛❧❡♥t t♦ ∀(DK, FK, PK) ∀σ (ǫ (WK,DK,KF,PK) F) ❛♥❞ ❤❡♥❝❡ t❤❛t ❢♦r❝✐♥❣ ♦✈❡r t❤❡ st❛t❡♠❡♥t ♦❢ ❝♦♠♣❧❡t❡♥❡ss ✇✳r✳t✳ ❚❛rs❦✐❛♥ ♠♦❞❡❧s ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ✐♥st❛♥t✐❛t✐♦♥ ♦❢ t❤❡ s❡t ♦❢ ✇♦r❧❞s t♦ t②♣✐♥❣ ❝♦♥t❡①ts ♦❢ ❝♦♠♣❧❡t❡♥❡ss ✇✳r✳t✳ ❑r✐♣❦❡ ♠♦❞❡❧s

✶✹

❊①❝❡r♣t ♦❢ ♦✉r ♠❡t❛✲❧❛♥❣✉❛❣❡ ✇✐t❤ ❡✛❡❝ts

Γ ⊢ p : A(y) y ❢r❡s❤ ✐♥ Γ Γ ⊢ λy.p : ∀y A(y) ∀I Γ ⊢ p : ∀x A(x) t ✉♣❞❛t❛❜❧❡✲✈❛r✐❛❜❧❡✲❢r❡❡ ♦r t ❛♥ ✉♣❞❛t❛❜❧❡ ✈❛r✐❛❜❧❡ ❛♥❞ A(x) ♦❢ t②♣❡ ✶ Γ ⊢ pt : A(t) ∀E Γ ⊢ p : A(X) X ❢r❡s❤ ✐♥ Γ Γ ⊢ p : ∀X A(X) ∀2

I

Γ ⊢ p : ∀X A(X) Γ ⊢ q : monΓ B( y) Γ ⊢ p : A(X)[B( y)/X( y)] ∀2

E

Γ, [b : x ≥ t] ⊢ q : T(x) Γ ⊢ r : refl ≥ Γ ⊢ s : trans ≥ x ❢r❡s❤ ✐♥ Γ ❛♥❞ T(t) Γ ⊢ s❡t x := t ❛s b/(r,s) ✐♥ q : T(t) s❡t❡❢❢ Γ, [b : x ≥ t(x′)] ⊢ q : T(x) Γ ⊢ r : t(x′) ≥ x′ [x ≥ u] ∈ Γ ❢♦r s♦♠❡ u x′ ❢r❡s❤ ✐♥ Γ Γ ⊢ ✉♣❞❛t❡ x:=t(x) ♦❢ x′ ❛s b ❜② r ✐♥ q : T(t(x)) ✉♣❞❛t❡ ✇❤❡r❡ C ♦❢ t②♣❡ ✶ ♠❡❛♥s ✐♥ t❤❡ ❣r❛♠♠❛r C ::= P(t1, ..., taP) | P(t1, ..., taP) → C | ∀x C ❛♥❞ monΓ B ♠❡❛♥s B ♠♦♥♦t♦♥✐❝ ❢♦r ❛❧❧ ✉♣❞❛t❛❜❧❡ ✈❛r✐❛❜❧❡s ✐♥ Γ

✶✺

❚❤❡ ❝♦♠♣❧❡t❡♥❡ss ♣r♦♦❢ ✐♥ ❞✐r❡❝t✲st②❧❡

■♥ ❞✐r❡❝t st②❧❡✱ K0 ✐s t❤❡ ♠♦❞❡❧ M0 ❞❡✜♥❡❞ ❜② PM( ˙ P)(t1, ..., ta ˙

P) Γ ⊢ ˙

P(t1, ..., ta ˙

P)

❢♦r Γ ❛ ❣✐✈❡♥ ✉♣❞❛t❛❜❧❡ ✈❛r✐❛❜❧❡ ↑F Γ ⊢M F − → M0 F ↑P(

t)

g g ↑F ˙

→ G

g m → ↑G ˙ App

Γ,F,G →

(g, ↓F m) ↑˙

∀x F

g t → ↑F[t/x] ˙ App

Γ,x,F ∀

(g, t) ↓F M0 F − → Γ ⊢M F ↓P(

t)

m m ↓F ˙

→ G

m

• ˙

Abs

Γ,F,G →

(✉♣❞❛t❡ Γ:=(Γ, F) ♦❢ Γ1 ❛s bF ❜② rΓ

F ✐♥ ↓G (m (↑F ˙

Ax

Γ1,F,,Γ(bF))))

↓˙

∀x F

m

• ˙

Abs

Γ,x,F ∀

( ˙ y, ↓F[z/x] (m ˙ y)) ❝♦♠♣❧ v → s❡t Γ := ǫ ❛s b/(r,s) ✐♥ ↓ǫ

F (v M0 ∅)

❖❜✈✐♦✉s❧②✱ t❤❡ r❡s✉❧t✐♥❣ ♣r♦♦❢ ✐♥ t❤❡ ♦❜❥❡❝t ❧❛♥❣✉❛❣❡ ✐s ❛ r❡✐✜❝❛t✐♦♥ ♦❢ t❤❡ ♣r♦♦❢ ♦❢ ✈❛✲ ❧✐❞✐t② ❛s ✐♥ ◆♦r♠❛❧✐s❛t✐♦♥✲❜②✲❊✈❛❧✉❛t✐♦♥ ✴ s❡♠❛♥t✐❝ ♥♦r♠❛❧✐s❛t✐♦♥ ❬❈✳ ❈♦q✉❛♥❞ ✾✸✱ ❉❛♥✈② ✾✻✱ ❆❧t❡♥❦✐r❝❤✲❍♦❢♠❛♥♥ ✾✻✱ ❖❦❛❞❛ ✾✾✱ ✳✳✳❪

✶✻

✬❡✳

Pr♦♣❡rt✐❡s ♦❢ t❤❡ ♠❡t❛✲❧❛♥❣✉❛❣❡ ✇✐t❤ ✉♣❞❛t❡ ❡✛❡❝t

✲ ❈❛♥ ❜❡ ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ r❡❞✉❝t✐♦♥ s❡♠❛♥t✐❝s ✭❞❡r✐✈❡❞ ❢r♦♠ t❤❡ ❢♦r❝✐♥❣ ✐♥t❡r♣r❡t❛t✐♦♥✮ ✲ ❈♦♥s✐st❡♥t ❜❡❝❛✉s❡ ✐♥t❡r♣r❡t❛❜❧❡ ❜② ❢♦r❝✐♥❣ ✐♥ ♣✉r❡ ✐♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝ ✲ ■♥❝♦♥s✐st❡♥t ✇✐t❤ ❛♥② ♥♦♥✲✐♥t✉✐t✐♦♥✐st✐❝ ❛ss✉♠♣t✐♦♥ ✭❧✐❦❡ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✮ ✲ ❍♦✇❡✈❡r✱ ✈❛r✐❛♥ts ✭✉♥❞❡r ✐♥✈❡st✐❣❛t✐♦♥✮ ❛r❡ ♣♦ss✐❜❧❡✿ ✲ ▲♦❝❛❧ ✉s❡ ♦❢ ❝❧❛ss✐❝❛❧ r❡❛s♦♥✐♥❣ ♣r♦✈✐❞✐♥❣ ▼❛r❦♦✈✬s ♣r✐♥❝✐♣❧❡ ❛♥❞ ❉♦✉❜❧❡ ◆❡❣❛t✐♦♥ ❙❤✐❢t ❛r❡ ♣♦ss✐❜❧❡ ✉s✐♥❣ ■❧✐❦✬s ✈❛r✐❛♥t ♦❢ ❑r✐♣❦❡ ❢♦r❝✐♥❣ ✲ ❋✉❧❧ ❝♦♠♣❛t✐❜✐❧✐t② ✇✐t❤ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ✉s✐♥❣ ❈♦❤❡♥ ❢♦r❝✐♥❣ t♦ ❜❡ ✐♥✈❡st✐❣❛t❡❞ ✲ Pr❡s❡r✈❛t✐♦♥ ♦❢ ❝♦♥s✐st❡♥❝② ✇❤❡♥ ♠✐①✐♥❣ s❡✈❡r❛❧ ✉s❡s ♦❢ ❢♦r❝✐♥❣ ♦♥ ❢✉♥❝t✐♦♥❛❧ ✏❝♦♥❞✐t✐♦♥s✑ t♦ ❜❡ ✐♥✈❡st✐❣❛t❡❞

✶✼