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❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t PP❙✴ πr 2 ✷✶st ▼❛r❝❤ ✷✵✶✹ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶ ✴ ✹✶

❲❡ ❞♦ ♥♦t ✜❣❤t ❛❧✐❡♥❛t✐♦♥ ✇✐t❤ ❛♥ ❛❧✐❡♥❛t❡❞ ❧♦❣✐❝✳ ❏✉st✐❢②✐♥❣ ❛r✐t❤♠❡t✐❝ ❞✐✛❡r❡♥t❧② ✳✳✳ ■♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝✦ ❉♦✉❜❧❡✲♥❡❣❛t✐♦♥ tr❛♥s❧❛t✐♦♥ ✭✶✾✸✸✮ ❉✐❛❧❡❝t✐❝❛ ✭✸✵✬s✱ ♣✉❜❧✐s❤❡❞ ✐♥ ✶✾✺✽✮ ❖♥❝❡ ✉♣♦♥ ❛ t✐♠❡✳✳✳ ❈❛t❛❝❧②s♠✿ ●ö❞❡❧✬s ✐♥❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠ ✭✶✾✸✶✮ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷ ✴ ✹✶

❏✉st✐❢②✐♥❣ ❛r✐t❤♠❡t✐❝ ❞✐✛❡r❡♥t❧② ✳✳✳ ■♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝✦ ❉♦✉❜❧❡✲♥❡❣❛t✐♦♥ tr❛♥s❧❛t✐♦♥ ✭✶✾✸✸✮ ❉✐❛❧❡❝t✐❝❛ ✭✸✵✬s✱ ♣✉❜❧✐s❤❡❞ ✐♥ ✶✾✺✽✮ ❖♥❝❡ ✉♣♦♥ ❛ t✐♠❡✳✳✳ ❈❛t❛❝❧②s♠✿ ●ö❞❡❧✬s ✐♥❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠ ✭✶✾✸✶✮ ❲❡ ❞♦ ♥♦t ✜❣❤t ❛❧✐❡♥❛t✐♦♥ ✇✐t❤ ❛♥ ❛❧✐❡♥❛t❡❞ ❧♦❣✐❝✳ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷ ✴ ✹✶

❖♥❝❡ ✉♣♦♥ ❛ t✐♠❡✳✳✳ ❈❛t❛❝❧②s♠✿ ●ö❞❡❧✬s ✐♥❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠ ✭✶✾✸✶✮ ❲❡ ❞♦ ♥♦t ✜❣❤t ❛❧✐❡♥❛t✐♦♥ ✇✐t❤ ❛♥ ❛❧✐❡♥❛t❡❞ ❧♦❣✐❝✳ ❏✉st✐❢②✐♥❣ ❛r✐t❤♠❡t✐❝ ❞✐✛❡r❡♥t❧② ✳✳✳ ■♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝✦ ❉♦✉❜❧❡✲♥❡❣❛t✐♦♥ tr❛♥s❧❛t✐♦♥ ✭✶✾✸✸✮ ❉✐❛❧❡❝t✐❝❛ ✭✸✵✬s✱ ♣✉❜❧✐s❤❡❞ ✐♥ ✶✾✺✽✮ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷ ✴ ✹✶

P❧❛♥ ❍✐st♦r✐❝❛❧ ♣r❡s❡♥t❛t✐♦♥ ✶ ❆ st❡♣ ✐♥t♦ ♠♦❞❡r♥✐t② ✷ ❊♥t❡rs ▲✐♥❡❛r ▲♦❣✐❝ ✸ ❆ s②♥t❛❝t✐❝ ♣r❡s❡♥t❛t✐♦♥ ✹ ❚♦✇❛r❞s CC ω ✺ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✸ ✴ ✹✶

❆ tr❛♥s❧❛t✐♦♥ ❢r♦♠ ❍❆ ✐♥t♦ ❍❆ ❚❤❛t ♣r❡s❡r✈❡s ✐♥t✉✐t✐♦♥✐st✐❝ ❝♦♥t❡♥t ❇✉t ♦✛❡rs t✇♦ s❡♠✐✲❝❧❛ss✐❝❛❧ ♣r✐♥❝✐♣❧❡s✿ ▼P ■P ❖✈❡r✈✐❡✇ ❲❤❛t ✐s ❉✐❛❧❡❝t✐❝❛❄ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✹ ✴ ✹✶

❇✉t ♦✛❡rs t✇♦ s❡♠✐✲❝❧❛ss✐❝❛❧ ♣r✐♥❝✐♣❧❡s✿ ▼P ■P ❖✈❡r✈✐❡✇ ❲❤❛t ✐s ❉✐❛❧❡❝t✐❝❛❄ ❆ tr❛♥s❧❛t✐♦♥ ❢r♦♠ ❍❆ ✐♥t♦ ❍❆ ω ❚❤❛t ♣r❡s❡r✈❡s ✐♥t✉✐t✐♦♥✐st✐❝ ❝♦♥t❡♥t P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✹ ✴ ✹✶

❖✈❡r✈✐❡✇ ❲❤❛t ✐s ❉✐❛❧❡❝t✐❝❛❄ ❆ tr❛♥s❧❛t✐♦♥ ❢r♦♠ ❍❆ ✐♥t♦ ❍❆ ω ❚❤❛t ♣r❡s❡r✈❡s ✐♥t✉✐t✐♦♥✐st✐❝ ❝♦♥t❡♥t ❇✉t ♦✛❡rs t✇♦ s❡♠✐✲❝❧❛ss✐❝❛❧ ♣r✐♥❝✐♣❧❡s✿ ¬ ( ∀ n ∈ N . ¬ P n ) ( ∀ n ∈ N . P n ) → ∃ m ∈ N . Q m ■P ▼P ∃ n ∈ N . P n ∃ m ∈ N . ( ∀ n ∈ N . P n ) → Q m P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✹ ✴ ✹✶

❲❛r♥✐♥❣✦ ❉✉st② ❧♦❣✐❝ ✐♥s✐❞❡ ❚r❛♥s❧❛t✐♦♥ ❛❝t✐♥❣ ♦♥ ❢♦r♠✉❧æ Pr❡✈❛❧❡♥❝❡ ♦❢ ♥❡❣❛t✐✈❡ ❝♦♥♥❡❝t✐✈❡s ❋✐rst✲♦r❞❡r ❧♦❣✐❝ ▲♦ts ♦❢ ❛r✐t❤♠❡t✐❝ ❡♥❝♦❞✐♥❣ ❉♦❡s ♥♦t ♣r❡s❡r✈❡ ✲r❡❞✉❝t✐♦♥ P❛r❡♥t❛❧ ❛❞✈✐s♦r② r❡q✉✐r❡❞ ❋♦r t❤❡ s❛❦❡ ♦❢ ❡①❤❛✉st✐✈✐t②✱ ✇❡✬❧❧ t❛❦❡ ❛ ❣❧✐♠♣s❡ ❛t t❤❡ ❤✐st♦r✐❝❛❧ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❉✐❛❧❡❝t✐❝❛✳ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✺ ✴ ✹✶

P❛r❡♥t❛❧ ❛❞✈✐s♦r② r❡q✉✐r❡❞ ❋♦r t❤❡ s❛❦❡ ♦❢ ❡①❤❛✉st✐✈✐t②✱ ✇❡✬❧❧ t❛❦❡ ❛ ❣❧✐♠♣s❡ ❛t t❤❡ ❤✐st♦r✐❝❛❧ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❉✐❛❧❡❝t✐❝❛✳ ❲❛r♥✐♥❣✦ ❉✉st② ❧♦❣✐❝ ✐♥s✐❞❡ ❚r❛♥s❧❛t✐♦♥ ❛❝t✐♥❣ ♦♥ ❢♦r♠✉❧æ Pr❡✈❛❧❡♥❝❡ ♦❢ ♥❡❣❛t✐✈❡ ❝♦♥♥❡❝t✐✈❡s ❋✐rst✲♦r❞❡r ❧♦❣✐❝ ▲♦ts ♦❢ ❛r✐t❤♠❡t✐❝ ❡♥❝♦❞✐♥❣ ❉♦❡s ♥♦t ♣r❡s❡r✈❡ β ✲r❡❞✉❝t✐♦♥ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✺ ✴ ✹✶

❉✉st② ❧♦❣✐❝s ❉✐❛❧❡❝t✐❝❛✱ ❉❛✇♥ ♦❢ ❈✉rr②✲❍♦✇❛r❞✿ ⊢ A D ≡ ∃ � ⊢ A �→ u. ∀ � x. A D [ � u, � x ] A ∧ B ∃ � ∀ � x ] ∧ B D [ � u� v. x � y. A D [ � u, � v, � y ] A ∨ B ∃ � ∀ � ( b = 0 ∧ A D [ � x ]) ∨ ( b = 1 ∧ B D [ � u� v b. x � y. u, � v, � y ]) ϕ � u, � A → B ∃ � ∀ � y )] → B D [ � ψ. u � y. A D [ � ψ ( � u, � ϕ ( � u ) , � y ] ∀ n. A [ n ] ∃ � ∀ � A D [ � ϕ ( n ) , � x, n ] ϕ. x n. ∃ n. A [ n ] ∃ � ∀ � A D [ � x ] u n. x. u, n, � ❙♦✉♥❞ tr❛♥s❧❛t✐♦♥✱ ❜❧❛❤ ❜❧❛❤ ❜❧❛❤✳ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✻ ✴ ✹✶

❆ st❡♣ ✐♥t♦ ♠♦❞❡r♥✐t② ▲❡t ✉s ❢♦r❣❡t t❤❡ ✺✵✬s✱ ❛♥❞ r❛t❤❡r ❥✉♠♣ ❞✐r❡❝t❧② t♦ t❤❡ ✾✵✬s✳ ❚❛❦❡ s❡r✐♦✉s❧② t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♥t❡♥t ❉✐❛❧❡❝t✐❝❛ ❛s ❛ t②♣❡❞ ♦❜❥❡❝t ❲♦r❦s ♦❢ ❉❡ P❛✐✈❛✱ ❍②❧❛♥❞✱ ❡t❝✳ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✼ ✴ ✹✶

●ö❞❡❧✬s ❛♥❛t♦♠② P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✽ ✴ ✹✶

■❢ ✇❡ ✇✐s❤ t♦ ♣✉t ♠♦r❡ t②♣❡s ✐♥ t❤❡r❡✿ ❚❤❡ s❛♠❡✱ ✇✐t❤ t②♣❡s ❆ ♣r♦♦❢ ⊢ u : A ✐s ❛ t❡r♠ ⊢ u : W ( A ) s✉❝❤ t❤❛t✿ ∀ x : C ( A ) . u ⊥ A x P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✾ ✴ ✹✶

❚❤❡ s❛♠❡✱ ✇✐t❤ t②♣❡s ❆ ♣r♦♦❢ ⊢ u : A ✐s ❛ t❡r♠ ⊢ u : W ( A ) s✉❝❤ t❤❛t✿ ∀ x : C ( A ) . u ⊥ A x ■❢ ✇❡ ✇✐s❤ t♦ ♣✉t ♠♦r❡ t②♣❡s ✐♥ t❤❡r❡✿ W C A ∧ B ∃ � u� v. ∀ � x � y. A × B W ( A ) × W ( B ) C ( A ) × C ( B ) A ∨ B ∃ b � ∀ � u� v. x � y. bool × W ( A ) × W ( B ) C ( A ) × C ( B ) A + B ϕ � A → B ∃ � ∀ � ψ. u � y. � W ( A ) → W ( B ) A → B W ( A ) × C ( B ) C ( B ) → W ( A ) → C ( A ) P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴ πr 2 ✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✾ ✴ ✹✶