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

✭✇♦r❦ ✐♥ ♣r♦❣r❡ss✮ ❍✉❣♦ ❍❡r❜❡❧✐♥ ✸ ❋❡❜r✉❛r② ✷✵✶✷ ❋❆❚P❆ ✷✵✶✷ ❇❡❧❣r❛❞❡ ✭r❡✈✐s❡❞ ✺✴✷✴✶✷✮

✶

❚❤❡ ♣r♦♦❢✲❛s✲♣r♦❣r❛♠s ❝♦rr❡s♣♦♥❞❡♥❝❡

❆♥ ✐♥t✉✐t✐♦♥✐st✐❝ ♣r♦♦❢ ✐s ❛ ♣✉r❡❧② ❢✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠ ❬❈✉rr②✱ ✶✾✺✽✱ ❍♦✇❛r❞✱ ✶✾✻✾❪ ✲ ❛ ♣r♦♦❢ ♦❢ A → B ✐s ❛ ♣r♦❣r❛♠ ♦❢ t❤❡ ❢♦r♠ x => p ✇❤❡r❡ x ❤❛s t②♣❡ A ❛♥❞ p ❤❛s t②♣❡ B ✲ ❛ ♣r♦♦❢ ♦❢ A ∧ B ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♣❛✐r ♦❢ ❛♥ ❡❧❡♠❡♥t ♦❢ A ❛♥❞ ♦❢ ❛♥ ❡❧❡♠❡♥t ♦❢ B ✲ ❞❡r✐✈✐♥❣ B ❢r♦♠ A → B ❛♥❞ A ✐s t❤❡ s❛♠❡ ❛s ❛♣♣❧②✐♥❣ t❤❡ ❣✐✈❡♥ ♣r♦♦❢✲❢✉♥❝t✐♦♥ ♦❢ t②♣❡ A → B t♦ t❤❡ ❣✐✈❡♥ ❡❧❡♠❡♥t ♦❢ t②♣❡ A ✲ ❡t❝

✷

❚❤❡ ♣r♦♦❢✲❛s✲♣r♦❣r❛♠ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝

❆ ❝❧❛ss✐❝❛❧ ♣r♦♦❢ ✐s ❛ ♣r♦❣r❛♠ ✇✐t❤ ❝♦♥tr♦❧ ❬●r✐✣♥✱ ✶✾✾✵❪ ❝❛❧❧❝❝✴t❤r♦✇ ✭▼▲ ❧❛♥❣✉❛❣❡s✮ ♦r ❝❛❧❧✲✇✐t❤✲❝✉rr❡♥t✲❝♦♥t✐♥✉❛t✐♦♥ ✭❙❝❤❡♠❡✮ ❝❛♥ ❜❡ ✉s❡❞ t♦ ♣r♦✈❡ ¬¬A → A ❊①❝❡♣t✐♦♥ ♠❡❝❤❛♥✐s♠s ✭▼▲ ❧❛♥❣✉❛❣❡s✱ ❏❛✈❛✱ ✳✳✳✮✱ ♦r ❡✈❡♥ s❡t❥✉♠♣✴❧♦♥❣❥✉♠♣ ✭✐♥ ❈✮ ♣r♦✈✐❞❡ ❛ ✇❡❛❦ ❢♦r♠ ♦❢ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✿ t❤❡② ❝❛♥ ❜❡ ✉s❡❞ t♦ ♣r♦✈❡ ¬¬A → A ✇❤❡♥ A ✐s ❛ ❢♦r♠✉❧❛ ✇✐t❤♦✉t ✐♠♣❧✐❝❛t✐♦♥ ♥♦r ✉♥✐✈❡rs❛❧ q✉❛♥t✐✜❝❛t✐♦♥ t②♣❡ ✭t❤✐s ❧❛tt❡r ♣r✐♥❝✐♣❧❡ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ✈❛r✐❛♥t ♦❢ ▼❛r❦♦✈✬s ♣r✐♥❝✐♣❧❡ ❢♦r ♣r❡❞✐❝❛t❡ ❧♦❣✐❝✮ ❬❍❍✱ ✷✵✶✵❪

✸

■s ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ❛ s✐❞❡ ❡✛❡❝t❄

❋r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ ♣✉r❡ ❢✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ✭❍❛s❦❡❧❧✱ ♦r t❤❡ ❝♦r❡ ♦❢ ❋★✮✱ ❝♦♥tr♦❧ ♦♣❡r❛t♦rs ✐♠♣❧❡♠❡♥t s✐❞❡ ❡✛❡❝ts ❚♦ ❞❡❛❧ ✇✐t❤ s✐❞❡ ❡✛❡❝ts ✐♥ ❍❛s❦❡❧❧✱ t❤❡ ❝♦♠♠♦♥ ❛♣♣r♦❛❝❤ ✐s t♦ r❡❛s♦♥ ✐♥ ❛ ✏♠♦♥❛❞✑ ❛♥❞ t♦ ❡①tr❛❝t ❛❢t❡r✇❛r❞s ♣✉r❡ ❝♦♥t❡♥ts ❜② ✏r✉♥♥✐♥❣✑ t❤❡ ♠♦♥❛❞ ❈❛♥ ✇❡ ✐♠❛❣✐♥❡ ❛❞❞✐♥❣ ♦t❤❡r s✐❞❡ ❡✛❡❝ts t♦ ❧♦❣✐❝ t❤❛t ❝♦✉❧❞ ❜❡ ❝❧❡❛r❡❞ ❜② ✏r✉♥♥✐♥❣✑ t❤❡ ♠♦♥❛❞❄

✹

❊①❛♠♣❧❡ ♦❢ ♠♦♥❛❞✐❝ ♣r♦❣r❛♠♠✐♥❣✿ t❤❡ st❛t❡ ♠♦♥❛❞

❚♦ s✐♠✉❧❛t❡ t❤❡ ❛❜✐❧✐t② t♦ r❡❛❞ ❛♥❞ ✇r✐t❡ ❛ ❣❧♦❜❛❧ ♠❡♠♦r② ♦❢ t②♣❡ S ✇✐t❤♦✉t ✉s✐♥❣ ❛♥② ♠❡♠♦r② ❛t ❛❧❧✱ ♦♥❡ s❡ts Tst(A) S → S × A ❛♥❞✱ ✇❤❡♥❡✈❡r ♦♥❡ ✇❛♥ts t♦ ✇r✐t❡ ❛ ♣r♦❣r❛♠ ♦❢ t②♣❡ A✱ ✇❡ ❛❝t✉❛❧❧② ✇r✐t❡ ❛ ♣r♦❣r❛♠ ♦❢ t②♣❡ Tst(A) ❚♦ ❜❡ ❛❜❧❡ t♦ ✇r✐t❡ ❛♥② ❢✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠ ✐♥ ❛ ✏♠♦♥❛❞✑ T✱ ✇❡ ♥❡❡❞ t✇♦ ♦♣❡r❛t✐♦♥s✶ η : A → T(A) t♦ ❡♥t❡r t❤❡ ♠♦♥❛❞

∗

: (A → T(B)) → T(A) → T(B) t♦ ❛♣♣❧② ❢✉♥❝t✐♦♥s run : T(base) → base t♦ ✏r✉♥✑ t❤❡ ♠♦♥❛❞ ♦♥ ❜❛s❡ t②♣❡s ✇❤✐❝❤✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ st❛t❡ ♠♦♥❛❞✱ ❛r❡ ✐♠♣❧❡♠❡♥t❡❞ ❜②✿ η x λs.(s, x) f ∗ x λs.let (s′, x′) = xs in f x′ s′ run x let (❴, x′) = xs0 in x′ ✇❤❡r❡ s0 ✐s t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ t❤❡ ♠❡♠♦r②✳

✶❚❤❡s❡ ♦♣❡r❛t✐♦♥s ❤❛✈❡ t♦ s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥s η∗x = x✱ f ∗(ηx) = fx✱ g∗(f ∗x) = (g∗ ◦ f)∗x✳

✺

❲❤❛t ❞✐❞ ✇❡ ❣❛✐♥❄

❲❤❛t ✇❡ ❣❛✐♥❡❞ ✐s t❤❛t ✐♥ t❤❡ st❛t❡ ♠♦♥❛❞✱ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ♥❡✇ ♦♣❡r❛t✐♦♥s ❛r❡ ❛✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡✦ read : unit → Tst(S) write : S → Tst(unit) ✇❤✐❝❤ ❛r❡ ✐♠♣❧❡♠❡♥t❡❞ ❜②✿ read () λs.(s, s) write s′ λ❴.(s′, ())

✻

▼♦♥❛❞✐❝✲st②❧❡ ✈s ❞✐r❡❝t✲st②❧❡✿ t❤❡ ❝❛s❡ ♦❢ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝

❘❡♠❛r❦✿ ❞♦✉❜❧❡✲♥❡❣❛t✐♦♥ tr❛♥s❧❛t✐♦♥ Tcont(A) ¬¬A ✐s t❤❡ ✏❝♦♥t✐♥✉❛t✐♦♥✑ ♠♦♥❛❞ t❤❛t ❛❧❧♦✇s t♦ r❡❛s♦♥ ❝❧❛ss✐❝❛❧❧② ✭✐✳❡✳ ✉s✐♥❣ ¬¬A → A✮ ✐♥s✐❞❡ ✐♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝✳ ❈♦♥✈❡rs❡❧②✱ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ❝❛♥ ❜❡ s❡❡♥ ❛s r❡❛s♦♥✐♥❣ ✐♥t✉✐t✐♦♥✐st✐❝❛❧❧② ✇✐t❤✐♥ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♠♦♥❛❞✱ ❜✉t ❞♦✐♥❣ ✐t ✐♥ ❞✐r❡❝t✲st②❧❡✳

✼

▼♦♥❛❞✐❝✲st②❧❡ ✈s ❞✐r❡❝t✲st②❧❡✿ t❤❡ ❝❛s❡ ♦❢ ▼❛r❦♦✈✬s ♣r✐♥❝✐♣❧❡

❚❤❡r❡ ✐s ❛ ✇❡❛❦ ❢♦r♠ ♦❢ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♠♦♥❛❞✱ ♥❛♠❡❧② t❤❡ ✏❡①❝❡♣t✐♦♥✑ ♠♦♥❛❞ Texc(A) A∨E ❢♦r s✉♣♣♦rt✐♥❣ ❤❛♥❞✐♥❣ ❛♥❞ r❛✐s✐♥❣ ❡①❝❡♣t✐♦♥s ✐♥ s♦♠❡ t②♣❡ E✳ ❚❤✐s ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈✐❞❡ ❛ ♣r❡❞✐❝❛t❡ ❧♦❣✐❝ ✈❛r✐❛♥t ♦❢ ▼❛r❦♦✈✬s ♣r✐♥❝✐♣❧❡ ✭✐✳❡✳ ¬¬A → A ❢♦r A ✇✐t❤♦✉t ✐♠♣❧✐❝❛t✐♦♥ ♥♦r ✉♥✐✈❡rs❛❧ q✉❛♥t✐✜❝❛t✐♦♥✮✳

✽

▼♦♥❛❞✐❝✲st②❧❡ ✈s ❞✐r❡❝t✲st②❧❡

■t t✉r♥s ♦✉t t❤❛t s♦♠❡ ❢♦r♠ ♦❢ ♠❡♠♦r② ❛ss✐❣♥♠❡♥t ✐s ✉s❡❞ ✐♥ ❧♦❣✐❝✿ ❜♦t❤ ♦❢ ✲ ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❜❛s❡❞ tr❛♥s❧❛t✐♦♥s ✲ ❈♦❤❡♥✬s ❢♦r❝✐♥❣ tr❛♥s❧❛t✐♦♥ ❝♦rr❡s♣♦♥❞ t♦ ♣r♦✈✐❞✐♥❣ ♠♦♥♦t♦♥✐❝❛❧❧② ♠❡♠♦r② ❛ss✐❣♥♠❡♥t ✐♥ ♠♦♥❛❞✐❝ st②❧❡✳ ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❜❛s❡❞ tr❛♥s❧❛t✐♦♥s ✭✐✳❡✳ Tkr(A)(x) ∀x′ ≥ x A(x′) ❢♦r x r❛♥❣✐♥❣ ♦✈❡r W✮ ✐s ❛ ❞❡♣❡♥❞❡♥t ❢♦r♠ ♦❢ t❤❡ ❡♥✈✐r♦♥♠❡♥t ♠♦♥❛❞ ✭Tenv(A) W → A✮ ❦♥♦✇♥ t♦ ♣r♦✈✐❞❡ ▲✐s♣✲st②❧❡ ❞②♥❛♠✐❝ ❜✐♥❞✐♥❣s✳ ❈♦❤❡♥✬s ❢♦r❝✐♥❣ tr❛♥s❧❛t✐♦♥ ✭Tforc(A)(x) ∀x′ ≥ x∃x′′ ≥ x′ A(x′′) ❢♦r x r❛♥❣✐♥❣ ♦✈❡r s♦♠❡ ❞♦♠❛✐♥ S✮ ✐s ❛ ❞❡♣❡♥❞❡♥t ❢♦r♠ ♦❢ t❤❡ st❛t❡ ♠♦♥❛❞ ✭Tst(A) S → S × A✮✳ ❲❤❛t ✐❢ ✇❡ tr② t♦ ❞❡s✐❣♥ ❛ ❧♦❣✐❝❛❧ s②st❡♠ t❤❛t ♣r♦✈✐❞❡s s✉❝❤ ❦✐♥❞s ♦❢ ♠❡♠♦r② ❛ss✐❣♥♠❡♥t ✐♥ ❞✐r❡❝t✲st②❧❡❄

✾

❚♦✇❛r❞s ❛ ❧♦❣✐❝ ✇✐t❤ s✐❞❡ ❡✛❡❝ts

❲❡ s❤❛❧❧ ♥♦✇ ❞❡s❝r✐❜❡ ❛ ✭s♦✉♥❞✮ ❡①t❡♥s✐♦♥ ♦❢ s❡❝♦♥❞✲♦r❞❡r ✐♥t✉✐t✐♦♥✐st✐❝ ❛r✐t❤♠❡t✐❝ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❡✛❡❝ts✿ ✲ ❡①❝❡♣t✐♦♥s ✭✐✳❡✳ ❛ ✇❡❛❦ ❢♦r♠ ♦❢ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✮ ✲ ♠♦♥♦t♦♥✐❝❛❧❧② ✉♣❞❛t❛❜❧❡ ♠❡♠♦r② ◆♦t❡✿ ❙♦✉♥❞♥❡ss ❝♦♠❡s ❜② ❡♠❜❡❞❞✐♥❣ ✐♥t♦ s❡❝♦♥❞✲♦r❞❡r ✐♥t✉✐t✐♦♥✐st✐❝ ❛r✐t❤♠❡t✐❝ ✉s✐♥❣ ❛ ❝♦♠✲ ❜✐♥❛t✐♦♥ ♦❢ ❑r✐♣❦❡✲st②❧❡ ✭❞❡♣❡♥❞❡♥t✮ ❡♥✈✐r♦♥♠❡♥t ♠♦♥❛❞✱ ❋r✐❡❞♠❛♥✲st②❧❡ ❡①❝❡♣t✐♦♥ ♠♦♥❛❞✱ ❛♥❞ ❈♦q✉❛♥❞✲❍♦❢♠❛♥♥✬s tr❛♥s❧❛t✐♦♥ ❬✶✾✾✾❪✳

✶✵

▲♦❣✐❝❛❧ r✉❧❡s ♣r♦✈✐❞✐♥❣ ❞❡❧✐♠✐t❡❞ ❞✐r❡❝t✲st②❧❡ ❡①❝❡♣t✐♦♥s ❛♥❞ ♠♦♥♦t♦♥❡ ♠❡♠♦r② ✉♣❞❛t❡s

❆ r✉❧❡ t♦ s✐♠✉❧t❛♥❡♦✉s❧② ❞❡❝❧❛r❡ ❛ ♠❡♠♦r② x ✇✐t❤ ✐♥✐t✐❛❧ ✈❛❧✉❡ t ❛♥❞ ♠♦♥♦t♦♥✐❝❛❧❧② ✉♣❞❛t❛❜❧❡ ❛❧♦♥❣ ❛ ♣r❡♦r❞❡r ≥ ❛♥❞ ❛♥ ❤❛♥❞❧❡r ♦❢ ❡①❝❡♣t✐♦♥s ♦❢ ♥❛♠❡ ˆ α ✐♥ t②♣❡ U ✭t❤✐s ❞❡❧✐♠✐ts t❤❡ ❞✐r❡❝t✲st②❧❡ ✉s❡ ♦❢ t❤❡ ♠❡♠♦r② ✉♣❞❛t❡ ❛♥❞ ❡①❝❡♣t✐♦♥ ❡✛❡❝ts✮✿ Γ, ˆ α : ¬U(x), b : x ≥ t ⊢ q : U(x) Γ ⊢ r : preorder ≥ Γ ⊢ s : monotone≥ U(x) Γ ⊢ s❡t x := t ❛s b ✉s✐♥❣ (r, s) ✐♥ #ˆ

α q : U(t)

s❡t❡❢❢ ❆ r✉❧❡ t♦ ♠♦♥♦t♦♥✐❝❛❧❧② ✉♣❞❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♠❡♠♦r② x ✭t❤✐s ♣r♦✈✐❞❡s ✐♥ ❞✐r❡❝t✲st②❧❡ ✇❤❛t ❑r✐♣❦❡✲st②❧❡ tr❛♥s❧❛t✐♦♥ Tkr ♣r♦✈✐❞❡s ✐♥ ♠♦♥❛❞✐❝ st②❧❡✮✿ Γ, b : x ≥ u(x′) ⊢ q : U(x) Γ ⊢ r : u(x) ≥ x (ˆ α : ¬U(x)) ∈ Γ x′ ❢r❡s❤ Γ ⊢ ✉♣❞❛t❡ x := u(x) ❛s (x′, b) ❜② r ✐♥ #ˆ

α q : U(u(x))

✉♣❞❛t❡ ❆ r✉❧❡ t♦ r❛✐s❡ ❛♥ ❡①❝❡♣t✐♦♥ ♦❢ ♥❛♠❡ ˆ α ✐♥ t②♣❡ U ✭t❤✐s ♣r♦✈✐❞❡s ✐♥ ❞✐r❡❝t✲st②❧❡ ✇❤❛t t❤❡ ❡①❝❡♣t✐♦♥ ♠♦♥❛❞ Texc ♣r♦✈✐❞❡s ✐♥ ♠♦♥❛❞✐❝ st②❧❡✮✿ Γ ⊢ p : U(x) (ˆ α : ¬U(x)) ∈ Γ Γ ⊢ raiseˆ

α p : B

❛❜♦rt

✶✶

❆♣♣❧✐❝❛t✐♦♥✿ ❛♥ ❡♥✉♠❡r❛t✐♦♥✲❢r❡❡ ♣r♦♦❢ ♦❢ ●ö❞❡❧✬s ❝♦♠♣❧❡t❡♥❡ss

✭❲❡❛❦✮ ❝♦♠♣❧❡t❡♥❡ss✿ ✐❢ A ✐s tr✉❡ ✐♥ ❛❧❧ ♠♦❞❡❧s M✱ t❤❡♥ A ✐s ♣r♦✈❛❜❧❡ ▲❡t C0 ❜❡ ❛ ❢♦r♠✉❧❛ ❛♥❞ ♣r♦✈❡ ˙ ¬C0 ✳ ⊢ ˙ ⊥✳ ❚❤❡ ✐❞❡❛ ✐s t♦ ❝♦♥s✐❞❡r ❛♥ ✉♣❞❛t❛❜❧❡ ✈❛r✐❛❜❧❡ Γ ✐♥✐t✐❛❧✐③❡❞ t♦ ˙ ¬C0 ✇✐t❤ Γ ✳ ⊢ ˙ ⊥ ❛s ♦❜❥❡❝t✐✈❡ ❛♥❞ t♦ t❛❦❡ t❤❡ s②♥t❛❝t✐❝ ♠♦❞❡❧ M0 ❞❡✜♥❡❞ ❜② A ∈ M0 ✐✛ Γ ✳ ⊢ A✳ ❲❡ ♥♦✇ ❤❛✈❡ t♦ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ (A ˙ →B) ∈ M0 ✐✛ A ∈ M0 → B ∈ M0 ( ˙ ∀x A) ∈ M0 ✐✛ ∀t A[t/x] ∈ M0 ˙ ⊥ ∈ M0 ✐✛ ⊥ ˙ ¬ ˙ ¬A ∈ M0 ✐✛ A ∈ M0 ✐✳❡✳ Γ ✳ ⊢ A ˙ →B ✐✛ Γ ✳ ⊢ A → Γ ✳ ⊢ B Γ ✳ ⊢ ( ˙ ∀x A) ✐✛ ∀t Γ ✳ ⊢ A[t/x] Γ ✳ ⊢ ˙ ⊥ ✐✛ ⊥ Γ ✳ ⊢ ˙ ¬ ˙ ¬A ✐✛ Γ ✳ ⊢ A ✇❤❡r❡ Γ ✐s ❛ ♠♦♥♦t♦♥✐❝❛❧❧② ✉♣❞❛t❛❜❧❡ ✈❛r✐❛❜❧❡✳

✶✷

❆♣♣❧✐❝❛t✐♦♥✿ ❛♥ ❡♥✉♠❡r❛t✐♦♥✲❢r❡❡ ♣r♦♦❢ ♦❢ ●ö❞❡❧✬s ❝♦♠♣❧❡t❡♥❡ss

❚❤❡ t✇♦ st❛t❡♠❡♥ts t❤❛t ✉s❡ ❡✛❡❝ts ❛r❡✿ ⇐ ˙

→ : (Γ

✳ ⊢ A → Γ ✳ ⊢ B) → Γ ✳ ⊢ A ˙ →B ⇒ ˙

⊥ : Γ

✳ ⊢ ˙ ⊥ → ⊥ ❚❤❡ ♣r♦♦❢s ❛r❡✿ ⇐ ˙

→ f

˙ IMPIΓ,A,B ˙ DN(Γ,A),B ✉♣❞❛t❡ Γ := (Γ, A, ˙ ¬B) ❛s (Γ′, b) ❜② r0 ✐♥ #ˆ

α

˙ IMPEΓ,B, ˙

⊥( ˙

AX(Γ′,A), ˙

¬B,Γ b, f ( ˙

AXΓ′,A,Γ φ(b))) ⇒ ˙

⊥ p raiseˆ α p

✇❤❡r❡ r0 ♣r♦✈❡s Γ ⊂ (Γ, A, ˙ ¬B) ✇❤✐❧❡ b ♦❢ t②♣❡ (Γ′, A, ˙ ¬B) ⊂ Γ ❛♥❞ φ(b)✱ ♦❜t❛✐♥❡❞ ❢r♦♠ b ❛♥❞ ♦❢ t②♣❡ (Γ′, A) ⊂ Γ✱ r❡s♣❡❝t✐✈❡❧② ❥✉st✐❢② t❤❡ ❝♦rr❡❝t♥❡ss ♦❢ t❤❡ ❛①✐♦♠ r✉❧❡s✳ ❚❤❡ r❡❧❡✈❛♥t ❡①❝❡r♣t ♦❢ ✐♥❢❡r❡♥❝❡ r✉❧❡s ♦❢ t❤❡ ♦❜❥❡❝t ❧❛♥❣✉❛❣❡ ✐s✿ p : (∆, A ✳ ⊢ B) ˙ IMPI∆,A,B p : (∆ ✳ ⊢ A ˙ →B) p : ((∆′, A) ⊂ ∆) ˙ AX∆′,A,∆ p : (∆ ✳ ⊢ A) p : (∆, ˙ ¬A ✳ ⊢ ⊥) ˙ DN∆,A p : (∆ ✳ ⊢ A) p : (∆ ✳ ⊢ A ˙ →B) q : (∆ ✳ ⊢ A) ˙ IMPE∆,A,B (p, q) : (∆ ✳ ⊢ B)

✶✸

❆♣♣❧✐❝❛t✐♦♥✿ ❛♥ ❡♥✉♠❡r❛t✐♦♥✲❢r❡❡ ♣r♦♦❢ ♦❢ ●ö❞❡❧✬s ❝♦♠♣❧❡t❡♥❡ss

■♥ s❤♦rt✱ ✐❢ ✇❡ ❝❛❧❧ H t❤❡ ♣r♦♦❢ ♦❢ ∀M ♠♦❞❡❧✭M✮ → C0 ∈ M✱ o0 ❛♥❞ s0 ♣r♦♦❢s r❡s♣❡❝t✐✈❡❧② ♦❢ t❤❡ ♦r❞❡r✐♥❣ ♦❢ ⊂ ❛♥❞ ♦❢ (Γ ⊂ Γ′) → (Γ ✳ ⊢ A) → (Γ′ ✳ ⊢ A) ❛♥❞ ok0 t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ⇐ ˙

→✱

⇐˙

∀✱ ⇐ ˙ ⊥✱ ⇒ ˙ →✱ ⇒˙ ∀ ❛♥❞ ⇒ ˙ ⊥ t❤❛t s❤♦✇s t❤❛t M0 ✐s ❛ ♠♦❞❡❧✱ t❤❡♥ t❤❡ ♣r♦♦❢ ♦❢

✳ ⊢ C0 ✇r✐tt❡♥ ❛s ❛ ♣r♦❣r❛♠ ✐s compl H ˙ DN∅,C0 s❡t Γ := ˙ ¬C0 ❛s b ✉s✐♥❣ (o0, s0) ✐♥ #ˆ

α

˙ IMPEΓ,C0, ˙

⊥ ( ˙

AX∅, ˙

¬C0,Γ b, H M0 ok0)

❚❤✐s ♣r♦♦❢ ✐s ❝♦♥str✉❝t✐✈❡✳✳✳ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ✇✐t❤ ✐t ✉s✐♥❣ ❡①❝❡♣t✐♦♥s ❛♥❞ ❛ss✐❣♥♠❡♥t✦ ❚❤❡ ♣r♦♦❢ ✐s ❛❧s♦ ❡①t❡♥s✐❜❧❡ t♦ str♦♥❣ ❝♦♠♣❧❡t❡♥❡ss✿ ✐❢ A ✐s tr✉❡ ✐♥ ❛❧❧ ♠♦❞❡❧s M s❛t✐s❢②✐♥❣ t❤❡♦r② T ✱ t❤❡♥ A ✐s ♣r♦✈❛❜❧❡ ✐♥ s♦♠❡ ✜♥✐t❡ s✉❜s❡t ♦❢ T ✳

✶✹

❆♣♣❧✐❝❛t✐♦♥✿ ❛♥ ❡♥✉♠❡r❛t✐♦♥✲❢r❡❡ ♣r♦♦❢ ♦❢ ●ö❞❡❧✬s ❝♦♠♣❧❡t❡♥❡ss

❆❧t❡r♥❛t✐✈❡❧②✱ t❤❡ ♣r❡✈✐♦✉s ♣r♦♦❢ ❡①♣r❡ss❡s ✐♥ ❞✐r❡❝t✲st②❧❡ t❤❛t t❤❡ ❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠ ❤♦❧❞s ❢♦r ❛ ♠♦❞❡❧ ❞❡✜♥❡❞ ❢r♦♠ ✐ts ✈❛❧✉❡ ♦♥ ❛t♦♠s ❜② Γ M ˙ ⊥ ⊥ Γ M A ˙ →B ∀Γ′ ⊃ Γ Γ′ M A → Γ′ M B Γ M ∀x A ∀t ∈ Dom(M) Γ M A[t/x] ❛♥❞ s✉❝❤ t❤❛t Γ M ¬¬A ✐♠♣❧✐❡s Γ M A✳ ❲❡ r❡❝♦❣♥✐s❡ ❤❡r❡ t❤❛t t❤❡ ♣r♦♦❢ ✇✐t❤ ❡✛❡❝ts ✐s s✐♠✐❧❛r t♦ ❛ ❞✐r❡❝t✲st②❧❡ ❢♦r♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♣❧❡t❡♥❡ss ✇rt ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❢♦r t❤❡ ♥❡❣❛t✐✈❡ ❢r❛❣♠❡♥t ♦❢ ✐♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝✱ t❤✐s ❢r❛❣♠❡♥t ✇❤❡r❡ ✐♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝ ♣r❡❝✐s❡❧② ❝♦✐♥❝✐❞❡s ✇✐t❤ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ✭✐♥ ♣❛rt✐❝✉❧❛r ❛t♦♠s ❛r❡ t❛❦❡♥ ♣r❡✜①❡❞ ❜② ❛ ❞♦✉❜❧❡ ♥❡❣❛t✐♦♥✮✳

✶✺