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✐♥❣ ❘✉❧❡

px1 ...xn ¬py1 ...yn x1 = y1 | ... | xn = yn

▼❛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

■♥tr♦❞✉❝t✐♦♥ t♦ ❏✐tPr♦ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦q ❊①❛♠♣❧❡✿ ❈❤❛♥❣✐♥❣ t❤❡ ♦r❞❡r ♦❢ q✉❛♥t✐✜❡rs ❊①❛♠♣❧❡✿ ❑❛♠✐♥s❦✐ ❊q✉❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥

❯s❛❜✐❧✐t②

❏✐tPr♦✿

❈❤❡❝❦s r✉❧❡ ❛♣♣❧✐❝❛❜✐❧✐t② Pr♦♦❢ ❜② ❝❧✐❝❦✐♥❣

❈♦q✿

❘❡✉s❛❜❧❡ ♣r♦♦❢ s❝r✐♣ts

▼❛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✐♦♥

❊①t❡♥❞❛❜✐❧✐t② ❛♥❞ ❆✉t♦♠❛t✐③❛t✐♦♥

❏✐tPr♦✿

◆❛t✐✈❡ ❏❛✈❛s❝r✐♣t ❢✉♥❝t✐♦♥s

❈♦q✿

❉❡✈❡❧♦♣✐♥❣ ❛♥❞ ♣r♦✈✐♥❣ ✐♥ s❛♠❡ ❡♥✈✐r♦♥♠❡♥t

• ✉❛r❛♥t❡❡❞ s♦✉♥❞♥❡ss

▼❛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✐♦♥

■✳ ❇❡rt♦t✱ P✳ ❈❛stér❛♥✱ ■♥t❡r❛❝t✐✈❡ ❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❛♥❞ Pr♦❣r❛♠ ❉❡✈❡❧♦♣♠❡♥t✱ ❙♣r✐♥❣❡r ❱❡r❧❛❣✱ ✷✵✵✹ ❚❤❡ ❈♦q Pr♦♦❢ ❆ss✐st❛♥t ❘❡❢❡r❡♥❝❡ ▼❛♥✉❛❧ ✭❤tt♣✿✴✴❝♦q✳✐♥r✐❛✳❢r✴❱✽✳✶♣❧✸✴r❡❢♠❛♥✴✐♥❞❡①✳❤t♠❧✮

• ✳ ❙♠♦❧❦❛✱ ❈✳ ❊✳ ❇r♦✇♥✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❈♦♠♣✉t❛t✐♦♥❛❧ ▲♦❣✐❝

✷✵✵✽ ▲❡❝t✉r❡ ◆♦t❡s ✭❤tt♣✿✴✴✇✇✇✳♣s✳✉♥✐✲s❜✳❞❡✴❝♦✉rs❡s✴❝❧✲ss✵✽✴s❝r✐♣t✴✐❝❧✳♣❞❢✮

• ✳ ❙♠♦❧❦❛✱ ❈✳ ❊✳ ❇r♦✇♥✱ ❚❡r♠✐♥❛t✐♥❣ ❚❛❜❧❡❛✉① ❢♦r t❤❡ ❇❛s✐❝

❋r❛❣♠❡♥t ♦❢ ❙✐♠♣❧❡ ❚②♣❡ ❚❤❡♦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✐♦♥

❚❤❛♥❦ ❨♦✉

▼❛tt❤✐❛s ❍ös❝❤❡❧❡ ❆❞✈✐s♦r✿ ❉r✳ ❈❤❛❞ ❊✳ ❇r♦✇♥ ❆❞✈✐s♦r✿ Pr♦❢✳ ●❡rt ❙♠♦❧❦❛ ❈♦♠♣❛r✐♥❣ ❏✐tPr♦ ❛♥❞ ❈♦q