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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✐♦♥ � w A ( 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 ) s❡t❡❢❢ Γ ⊢ s❡t x := t ❛s b/ ( r,s ) ✐♥ q : T ( 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 ✉♣❞❛t❡ Γ ⊢ ✉♣❞❛t❡ x := t ( x ) ♦❢ x ′ ❛s b ❜② r ✐♥ q : T ( t ( x )) → ✲ ˙ ∀ ✲❢r❡❡ ❢♦r♠✉❧❛s ✭❂ ✐♥t✉✐t✐✈❡❧② Σ 0 ✇❤❡r❡ ✇❡ ✇r♦t❡ T ✱ U ❢♦r ˙ 1 ✲❢♦r♠✉❧❛s ❂ ❜❛s❡ t②♣❡s✮ ✹

●ö❞❡❧✬s ❝♦♠♣❧❡t❡♥❡ss ✺

❖❜❥❡❝t ❧❛♥❣✉❛❣❡ ❲❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ♥❡❣❛t✐✈❡ ❢r❛❣♠❡♥t ♦❢ ♣r❡❞✐❝❛t❡ ❧♦❣✐❝ ❛s ❛♥ ♦❜❥❡❝t ❧❛♥❣✉❛❣❡ ✭✇❡ ❝♦♥s✐❞❡r ˙ → ˙ ¬ A � A ˙ ⊥ t♦ ❜❡ ❛♥ ❛r❜✐tr❛r② ❛t♦♠ ❛♥❞ ❛❜❜r❡✈✐❛t❡ ˙ ⊥ ✮✳ � x | f ( t 1 , ..., t n ) t F, G � ˙ ⊥ | ˙ → G | ˙ P ( t 1 , ..., t n ) | F ˙ ∀ x F � ǫ | Γ , F Γ ❲❡ t❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❢❡r❡♥❝❡ r✉❧❡s✿ Γ ,F, Γ ′ : (Γ , F ⊂ Γ ′ ) → (Γ ′ ⊢ F ) ˙ Ax Γ ,F,G ˙ : (Γ ⊢ F ˙ → G ) → (Γ ⊢ F ) → (Γ ⊢ G ) App → Γ ,F,G ˙ : (Γ , F ⊢ G ) → (Γ ⊢ F ˙ → G ) Abs → Γ ,x,F : (Γ ⊢ F ) → ( x �∈ FV (Γ)) → (Γ ⊢ ˙ ˙ ∀ x F ( x )) Abs ∀ Γ ,x,t,F : (Γ ⊢ ˙ ˙ ∀ x F ) → (Γ ⊢ F [ t/x ]) App ∀ ▼♦r❡♦✈❡r✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛❞♠✐ss✐❜❧❡✿ Γ ′ : (Γ ⊂ Γ ′ ) → (Γ ⊢ F ) → (Γ ′ ⊢ F ) ˙ weak Γ ,F ❲❡ s❤❛❧❧ ❛❧s♦ ✇r✐t❡ r Γ F ❢♦r ❛ ♣r♦♦❢ ♦❢ Γ ⊂ (Γ , F ) ✱ ✻

❚❛rs❦✐❛♥ ♠♦❞❡❧s ❆ ❚❛rs❦✐❛♥ ♠♦❞❡❧ M ✐s ♠❛❞❡ ♦❢ ❛ ❞♦♠❛✐♥ D M ❢♦r ✐♥t❡r♣r❡t✐♥❣ t❡r♠s✱ ♦❢ ❛♥ ✐♥t❡r✲ ♣r❡t❛t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥ s②♠❜♦❧s F M ( f ) : D a f → D ❛♥❞ ♦❢ ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❛t♦♠s P M ( ˙ P t❤❡ ❛r✐t② ♦❢ f ✱ ˙ P ) ⊂ D a ˙ P ✭❢♦r a f ✱ a ˙ P r❡s♣✳✮✳ ❚r✉t❤ ✐s ❞❡✜♥❡❞ ❜② ] σ � σ ( x ) [ [ x ] M ] σ ] σ ] σ � F M ( f )([ [ [ ft 1 . . . t a f ] [ t 1 ] M , . . . , [ [ t a f ] M ) M M ˙ P ) � P M ( ˙ � σ ] σ ] σ P ( t 1 , . . . , t a ˙ P )([ [ t 1 ] M , . . . , [ [ t a ˙ P ] M ) M ˙ � P M ( ˙ � σ ⊥ ⊥ ) � σ � � σ M F → � σ M F ˙ → G M G � ∀ t ∈ M D � σ [ x ← t ] M ˙ � σ ∀ x F F M ✼

❈♦♠♣❧❡t❡♥❡ss ✇✳r✳t ❚❛rs❦✐❛♥ ♠♦❞❡❧s ▲❡t C las ❜❡ t❤❡ t❤❡♦r② ❝♦♥t❛✐♥✐♥❣ ˙ ¬ ˙ ¬ F ˙ → F ❢♦r ❛❧❧ ❢♦r♠✉❧❛s F ✭❛t♦♠s ❛r❡ ❡♥♦✉❣❤✮✳ ❲❡ ❞❡✜♥❡ ⊢ C F t♦ ❜❡ C las ⊢ M F ✐♥ ♠✐♥✐♠❛❧ ❧♦❣✐❝✳ ❆ ❚❛rs❦✐❛♥ ♠♦❞❡❧ M ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ✐s ❛ ❚❛rs❦✐❛♥ ♠♦❞❡❧ ✇❤✐❝❤ s❛t✐s✜❡s � M C las ✭✐♥ ❛ ❝❧❛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 C las → � σ M F )] → C las ⊢ 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 W K ♦r❞❡r❡❞ ❜② ≥ K ✳ ■♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ ∨ ❛♥❞ ∃ ✱ ✐t ✐s ❡♥♦✉❣❤ t♦ t❛❦❡ D K ❝♦♥st❛♥t✳ ❚r✉t❤ r❡❧❛t✐✈❡❧② t♦ K ❛t ✇♦r❧❞ w ✐s ❞❡✜♥❡❞ ❜②✿ ] σ � σ ( x ) [ [ x ] K ] σ ] σ ] σ � F K ( f )([ [ [ ft 1 . . . t a f ] [ t 1 ] K , . . . , [ [ t a f ] K ) K K ˙ P ) � P K ( ˙ w � σ ] σ ] σ P ( t 1 . . . t a ˙ P ) w ([ [ t 1 ] K , . . . , [ [ t a ˙ P ] K ) K ˙ � P K ( ˙ w � σ ⊥ ⊥ ) w � ∀ w ′ ≥ K w ( w ′ � σ K F → w ′ � σ w � σ K F ˙ → G K G ) � ∀ t ∈ K D w � σ [ x ← t ] w � σ K ∀ x F F K ❚❤❡ st❛t❡♠❡♥t ♦❢ ❝♦♠♣❧❡t❡♥❡ss ✇✳r✳t✳ ❑r✐♣❦❡ ♠♦❞❡❧s ✐s✿ ( ∀K ∀ σ ∀ w ∈ W K w � σ K F ) → ⊢ M F ✶✵