r tPr q tts s - PowerPoint PPT Presentation

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❋❡❜r✉❛r② ✶✸✱ ✷✵✵✾ ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐t♣r♦ ❏ ❛✈❛❙❝r✐♣t ■ ♥t❡r❛❝t✐✈❡ ❍✐❣❤❡r✲❖r❞❡r ❚ ❛❜❧❡❛✉ Pr♦ ✈❡r ❙✐♠♣❧② t②♣❡❞ ❘❡❢✉t❛t✐♦♥ ❝❛❧❝✉❧✉s ❇✉✐❧t✲✐♥ ❈❧❛ss✐❝❛❧ ▲♦❣✐❝ Pr♦♣♦s✐t✐♦♥s ♦❢ t②♣❡ ❇♦♦❧ ❘✉❧❡ ❛♣♣❧✐❝❛t✐♦♥ ❜② t❤❡ ❝❧✐❝❦ ♦❢ ❛ ❜✉tt♦♥ ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❆ s♠❛❧❧ ❊①❛♠♣❧❡✿ ∀ A , B : B . A ∧ B → B ∧ A ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ●❆▲▲■◆❆ s♣❡❝✐✜❝❛t✐♦♥ ❧❛♥❣✉❛❣❡ ❈❛❧❝✉❧✉s ♦❢ ■♥❞✉❝t✐✈❡ ❈♦♥str✉❝t✐♦♥s P♦❧②♠♦r♣❤✐❝ Pr♦♦❢s ✐♥ ❈♦q s✐♠✐❧❛r t♦ Pr♦♦❢s ✐♥ ◆❉ ❇✉✐❧❞✐♥❣ ♣r♦♦❢s ❜② ❛♣♣❧✐❝❛t✐♦♥s ♦❢ t❛❝t✐❝s ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ✸ ❜❛s✐❝ s♦rts✿ Pr♦♣ ❙❡t ❚②♣❡ ❊✈❡r②t❤✐♥❣ ✐s ❛ t❡r♠ ❈✉rr②✲❍♦✇❛r❞ ■s♦♠♦r♣❤✐s♠ Pr♦♦❢ ❝❤❡❝❦✐♥❣ ❜② t②♣❡ ❝❤❡❝❦✐♥❣ ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❆✉t♦♠❛t✐③❛t✐♦♥ ❋❡❛t✉r❡s ❯s❡r✲❞❡✜♥❡❞ t❛❝t✐❝s ❆✉t♦ t❛❝t✐❝ ❍✐♥t ❞❛t❛❜❛s❡s ❆❞❞ ❛r❜✐tr❛r② t❤❡♦r❡♠s✴t❛❝t✐❝s t♦ ❞❛t❛❜❛s❡ ▼❛t❝❤❡s ❝✉rr❡♥t ❣♦❛❧ ✇✐t❤ ❤✐♥t ❞❛t❛❜❛s❡ ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❆❧♠♦st t❤❡ s❛♠❡ s♠❛❧❧ ❊①❛♠♣❧❡✿ ∀ A , B : Prop . A ∧ B → B ∧ A ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs✿ ∀ P : A → A → B . ( ∃ x : A ∀ y : A , Pxy ) ⇒ ∀ y : A ∃ x : A . Pxy Pr♦♦❢ ✐♥ ❏✐tPr♦ ✳✳✳ ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ✭♣♦❧②♠♦r♣❤✐❝✮✿ ∀ A : Type . ∀ P : A → A → Prop . ( ∃ x : A ∀ y : A , Pxy ) ⇒ ∀ y : A ∃ x : A . Pxy Pr♦♦❢ ✐♥ ❈♦q ✳✳✳ ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ∀ f : B → B . ∀ x : B . f ( f ( f x )) = f x ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ▼❛t✐♥❣ ❘✉❧❡ ¬ py 1 ... y n px 1 ... x n x 1 � = y 1 | | x n � = y n ... ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ▼❛t✐♥❣ ❚❛❝t✐❝ Theorem mating : forall (P:(bool -> bool)) (a b:bool), P a = true -> P b <> true->(a<>b). Ltac t_mate f a b P1 P2 := assert(a <> b); [exact (mating f a b P1 P2) | idtac]. ▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q