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rs ts t r ts Prs

  1. ❆ ❙❝♦♣❡✲❛♥❞✲❚②♣❡ ❙❛❢❡ ❯♥✐✈❡rs❡ ♦❢ ❙②♥t❛①❡s ✇✐t❤ ❇✐♥❞✐♥❣✱ ❚❤❡✐r ❙❡♠❛♥t✐❝s ❛♥❞ Pr♦♦❢s ●✉✐❧❧❛✉♠❡ ❆❧❧❛✐s✱ ❘♦❜❡rt ❆t❦❡②✱ ❏❛♠❡s ❈❤❛♣♠❛♥✱ ❈♦♥♦r ▼❝❇r✐❞❡✱ ❏❛♠❡s ▼❝❑✐♥♥❛ ■❈❋P✬✶✽✱ ❙t ▲♦✉✐s✱ ▼❖ ❙❡♣t❡♠❜❡r ✷✹✱ ✷✵✶✽

  2. ❘♦❛❞✲♠❛♣ ▼♦t✐✈❛t✐♦♥ ❆ Pr♦❣r❛♠ ❚r❛♥s❢♦r♠❛t✐♦♥ ❆ ❙♦✉♥❞♥❡ss ▲❡♠♠❛ ❆ ❯♥✐✈❡rs❡ ♦❢ ❙②♥t❛①❡s ✇✐t❤ ❇✐♥❞✐♥❣ ❆♥❛t♦♠② ♦❢ ❛ ▲❛♥❣✉❛❣❡✬s ❙②♥t❛① ❈♦❞❡s ❢♦r ❙②♥t❛①❡s ❙❝♦♣❡✲❛♥❞✲❑✐♥❞ ❆✇❛r❡ ❚r❛✈❡rs❛❧s ❆ ●❡♥❡r✐❝ ◆♦t✐♦♥ ♦❢ ❙❡♠❛♥t✐❝s ❆ ❈❛t❛❧♦❣✉❡ ♦❢ ❙❝♦♣❡✲❛♥❞✲❑✐♥❞ Pr❡s❡r✈✐♥❣ Pr♦❣r❛♠s Pr♦♦❢ ❋r❛♠❡✇♦r❦s

  3. ❙ ❙ ❙ ❙ ❧❡t ❙ ✐♥ ❙ Pr♦❜❧❡♠ ❲r✐t❡ ❛ ♣r♦❣r❛♠ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ ❙ t♦ ❚ ✐♥❧✐♥✐♥❣ ❧❡t✳✳✐♥✳✳ Pr♦✈❡ ❛ s✐♠✉❧❛t✐♦♥ ❧❡♠♠❛ ❢♦r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t ❚ ::= x | ❚ ❚ | λ x . ❚

  4. ❙ ❙ ❙ ❙ ❧❡t ❙ ✐♥ ❙ Pr♦❜❧❡♠ ❲r✐t❡ ❛ ♣r♦❣r❛♠ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ ❙ t♦ ❚ ✐♥❧✐♥✐♥❣ ❧❡t✳✳✐♥✳✳ Pr♦✈❡ ❛ s✐♠✉❧❛t✐♦♥ ❧❡♠♠❛ ❢♦r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t ❚ ::= x | ❚ ❚ | λ x . ❚

  5. ❙ ❙ ❙ ❙ ❙ ✐♥ ❙ ❧❡t Pr♦❜❧❡♠ ❲r✐t❡ ❛ ♣r♦❣r❛♠ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ ❙ t♦ ❚ ✐♥❧✐♥✐♥❣ ❧❡t✳✳✐♥✳✳ Pr♦✈❡ ❛ s✐♠✉❧❛t✐♦♥ ❧❡♠♠❛ ❢♦r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t ❚ ::= x | ❚ ❚ | λ x . ❚

  6. ❙ ❙ ❙ ❙ ❙ ✐♥ ❙ ❧❡t Pr♦❜❧❡♠ ❲r✐t❡ ❛ ♣r♦❣r❛♠ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ ❙ t♦ ❚ ✐♥❧✐♥✐♥❣ ❧❡t✳✳✐♥✳✳ Pr♦✈❡ ❛ s✐♠✉❧❛t✐♦♥ ❧❡♠♠❛ ❢♦r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t ❚ ::= x | ❚ ❚ | λ x . ❚

  7. Pr♦❜❧❡♠ ❲r✐t❡ ❛ ♣r♦❣r❛♠ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ ❙ t♦ ❚ ✐♥❧✐♥✐♥❣ ❧❡t✳✳✐♥✳✳ Pr♦✈❡ ❛ s✐♠✉❧❛t✐♦♥ ❧❡♠♠❛ ❢♦r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t ❙ ::= x | ❙ ❙ | λ x . ❙ | ❧❡t x = ❙ ✐♥ ❙ ❚ ::= x | ❚ ❚ | λ x . ❚

  8. Pr♦❜❧❡♠ ❲r✐t❡ ❛ ♣r♦❣r❛♠ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ ❙ t♦ ❚ ✐♥❧✐♥✐♥❣ ❧❡t✳✳✐♥✳✳ Pr♦✈❡ ❛ s✐♠✉❧❛t✐♦♥ ❧❡♠♠❛ ❢♦r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t ❙ ::= x | ❙ ❙ | λ x . ❙ | ❧❡t x = ❙ ✐♥ ❙ ❚ ::= x | ❚ ❚ | λ x . ❚

  9. Pr♦❜❧❡♠ ❲r✐t❡ ❛ ♣r♦❣r❛♠ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ ❙ t♦ ❚ ✐♥❧✐♥✐♥❣ ❧❡t✳✳✐♥✳✳ Pr♦✈❡ ❛ s✐♠✉❧❛t✐♦♥ ❧❡♠♠❛ ❢♦r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t ❙ ::= x | ❙ ❙ | λ x . ❙ | ❧❡t x = ❙ ✐♥ ❙ ❚ ::= x | ❚ ❚ | λ x . ❚

  10. Pr♦❜❧❡♠ ❙t❛t❡♠❡♥t ❙ ::= x | ❙ ❙ | λ x . ❙ | ❧❡t x = ❙ ✐♥ ❙ ❚ ::= x | ❚ ❚ | λ x . ❚ Pr♦❜❧❡♠ ◮ ❲r✐t❡ ❛ ♣r♦❣r❛♠ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠ ❙ t♦ ❚ ✐♥❧✐♥✐♥❣ ❧❡t✳✳✐♥✳✳ ◮ Pr♦✈❡ ❛ s✐♠✉❧❛t✐♦♥ ❧❡♠♠❛ ❢♦r t❤✐s tr❛♥s❢♦r♠❛t✐♦♥

  11. ❧❡t ✐♥ ❍♦♥❡st② t❛① ✭❾✶✮✿ ❚ ❛❞♠✐ts ✇❡❛❦❡♥✐♥❣ ▲❡t✲❡❧❛❜♦r❛t✐♦♥✿ ❢r♦♠ ❙ t♦ ❚ � · � · : ❙ → ( ❱❛r ⇒ ❚ ) → ❚

  12. ❧❡t ✐♥ ❍♦♥❡st② t❛① ✭❾✶✮✿ ❚ ❛❞♠✐ts ✇❡❛❦❡♥✐♥❣ ▲❡t✲❡❧❛❜♦r❛t✐♦♥✿ ❢r♦♠ ❙ t♦ ❚ � · � · : ❙ → ( ❱❛r ⇒ ❚ ) → ❚

  13. ❧❡t ✐♥ ❍♦♥❡st② t❛① ✭❾✶✮✿ ❚ ❛❞♠✐ts ✇❡❛❦❡♥✐♥❣ ▲❡t✲❡❧❛❜♦r❛t✐♦♥✿ ❢r♦♠ ❙ t♦ ❚ � · � · : ❙ → ( ❱❛r ⇒ ❚ ) → ❚

  14. ❧❡t ✐♥ ❍♦♥❡st② t❛① ✭❾✶✮✿ ❚ ❛❞♠✐ts ✇❡❛❦❡♥✐♥❣ ▲❡t✲❡❧❛❜♦r❛t✐♦♥✿ ❢r♦♠ ❙ t♦ ❚ � · � · : ❙ → ( ❱❛r ⇒ ❚ ) → ❚

  15. ❧❡t ✐♥ ❍♦♥❡st② t❛① ✭❾✶✮✿ ❚ ❛❞♠✐ts ✇❡❛❦❡♥✐♥❣ ▲❡t✲❡❧❛❜♦r❛t✐♦♥✿ ❢r♦♠ ❙ t♦ ❚ � · � · : ❙ → ( ❱❛r ⇒ ❚ ) → ❚ � x � ρ = ρ ( x )

  16. ❧❡t ✐♥ ❍♦♥❡st② t❛① ✭❾✶✮✿ ❚ ❛❞♠✐ts ✇❡❛❦❡♥✐♥❣ ▲❡t✲❡❧❛❜♦r❛t✐♦♥✿ ❢r♦♠ ❙ t♦ ❚ � · � · : ❙ → ( ❱❛r ⇒ ❚ ) → ❚ � x � ρ = ρ ( x ) � f t � ρ = ( � f � ρ ) ( � t � ρ )

  17. ❧❡t ✐♥ ❍♦♥❡st② t❛① ✭❾✶✮✿ ❚ ❛❞♠✐ts ✇❡❛❦❡♥✐♥❣ ▲❡t✲❡❧❛❜♦r❛t✐♦♥✿ ❢r♦♠ ❙ t♦ ❚ � · � · : ❙ → ( ❱❛r ⇒ ❚ ) → ❚ � x � ρ = ρ ( x ) � f t � ρ = ( � f � ρ ) ( � t � ρ ) � λ x . b � ρ = λ x . ( � b � ( ρ · x ))

  18. ❧❡t ✐♥ ❍♦♥❡st② t❛① ✭❾✶✮✿ ❚ ❛❞♠✐ts ✇❡❛❦❡♥✐♥❣ ▲❡t✲❡❧❛❜♦r❛t✐♦♥✿ ❢r♦♠ ❙ t♦ ❚ � · � · : ❙ → ( ❱❛r ⇒ ❚ ) → ❚ � x � ρ = ρ ( x ) � f t � ρ = ( � f � ρ ) ( � t � ρ ) � λ x . b � ρ = λ x . ( � b � ( ρ · x ) )

  19. ❍♦♥❡st② t❛① ✭❾✶✮✿ ❚ ❛❞♠✐ts ✇❡❛❦❡♥✐♥❣ ▲❡t✲❡❧❛❜♦r❛t✐♦♥✿ ❢r♦♠ ❙ t♦ ❚ � · � · : ❙ → ( ❱❛r ⇒ ❚ ) → ❚ � x � ρ = ρ ( x ) � f t � ρ = ( � f � ρ ) ( � t � ρ ) � λ x . b � ρ = λ x . ( � b � ( ρ · x )) � ❧❡t x = e ✐♥ b � ρ = � b � ( ρ · � e � ρ )

  20. ▲❡t✲❡❧❛❜♦r❛t✐♦♥✿ ❢r♦♠ ❙ t♦ ❚ � · � · : ❙ → ( ❱❛r ⇒ ❚ ) → ❚ � x � ρ = ρ ( x ) � f t � ρ = ( � f � ρ ) ( � t � ρ ) � λ x . b � ρ = λ x . ( � b � ( ρ · x )) � ❧❡t x = e ✐♥ b � ρ = � b � ( ρ · � e � ρ ) ❍♦♥❡st② t❛① ✭❾✶✮✿ ❚ ❛❞♠✐ts ✇❡❛❦❡♥✐♥❣

  21. ✭❾✶✷✰✮ ♠❡❛♥s ❳ ✐s st❛❜❧❡ ✉♥❞❡r s✉❜st✐t✉t✐♦♥ ❚♦ ❞❡✜♥❡ s✉❜st✐t✉t✐♦♥ ✇❡ ♥❡❡❞ t♦ ❞❡✜♥❡ ✇❡❛❦❡♥✐♥❣ ❋✉s✐♦♥ ❧❡♠♠❛s ❊①t❡♥s✐♦♥❛❧✐t② ❧❡♠♠❛s ■❞❡♥t✐t② ❧❡♠♠❛s s ❛♥❞ s✬ s✳t✳ s s✬ ❛♥❞ ✬ s✳t✳ ❍♦♥❡st② t❛① ✿ ❙t❛t❡♠❡♥t Pr♦♦❢ ❆ ❙♦✉♥❞♥❡ss ▲❡♠♠❛ ▲❡♠♠❛ ✭❙✐♠✉❧❛t✐♦♥✮ ●✐✈❡♥✿ ❲❡ ❝❛♥ ♣r♦✈❡ t❤❛t✿

  22. ✭❾✶✷✰✮ ♠❡❛♥s ❳ ✐s st❛❜❧❡ ✉♥❞❡r s✉❜st✐t✉t✐♦♥ ❚♦ ❞❡✜♥❡ s✉❜st✐t✉t✐♦♥ ✇❡ ♥❡❡❞ t♦ ❞❡✜♥❡ ✇❡❛❦❡♥✐♥❣ ❋✉s✐♦♥ ❧❡♠♠❛s ❊①t❡♥s✐♦♥❛❧✐t② ❧❡♠♠❛s ■❞❡♥t✐t② ❧❡♠♠❛s ❛♥❞ ✬ s✳t✳ ❍♦♥❡st② t❛① ✿ ❙t❛t❡♠❡♥t Pr♦♦❢ ❆ ❙♦✉♥❞♥❡ss ▲❡♠♠❛ ▲❡♠♠❛ ✭❙✐♠✉❧❛t✐♦♥✮ ●✐✈❡♥✿ ◮ s ❛♥❞ s✬ s✳t✳ s � S s✬ ❲❡ ❝❛♥ ♣r♦✈❡ t❤❛t✿

  23. ✭❾✶✷✰✮ ♠❡❛♥s ❳ ✐s st❛❜❧❡ ✉♥❞❡r s✉❜st✐t✉t✐♦♥ ❚♦ ❞❡✜♥❡ s✉❜st✐t✉t✐♦♥ ✇❡ ♥❡❡❞ t♦ ❞❡✜♥❡ ✇❡❛❦❡♥✐♥❣ ❋✉s✐♦♥ ❧❡♠♠❛s ❊①t❡♥s✐♦♥❛❧✐t② ❧❡♠♠❛s ■❞❡♥t✐t② ❧❡♠♠❛s ❍♦♥❡st② t❛① ✿ ❙t❛t❡♠❡♥t Pr♦♦❢ ❆ ❙♦✉♥❞♥❡ss ▲❡♠♠❛ ▲❡♠♠❛ ✭❙✐♠✉❧❛t✐♦♥✮ ●✐✈❡♥✿ ◮ s ❛♥❞ s✬ s✳t✳ s � S s✬ ◮ ρ ❛♥❞ ρ ✬ s✳t✳ ∀ x .ρ ( x ) � ⋆ T ρ ′ ( x ) ❲❡ ❝❛♥ ♣r♦✈❡ t❤❛t✿

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