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feat(library/standard/logic): add exists unique

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
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Leonardo de Moura 2014-06-30 18:22:28 -07:00
parent ad12abcdb4
commit d4b29cd132
1 changed files with 10 additions and 0 deletions
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library/standard/logic.lean
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@ -178,6 +178,16 @@ notation `∃` binders `,` r:(scoped P, Exists P) := r
theorem exists_elim {A : Type} {P : A → Bool} {B : Bool} (H1 : ∃ x : A, P x) (H2 : ∀ (a : A) (H : P a), B) : B
:= Exists_rec H2 H1
definition exists_unique {A : Type} (p : A → Bool) := ∃ x, p x ∧ ∀ y, y ≠ x → ¬ p y
notation `∃!` binders `,` r:(scoped P, exists_unique P) := r
theorem exists_unique_intro {A : Type} {p : A → Bool} (w : A) (H1 : p w) (H2 : ∀ y, y ≠ w → ¬ p y) : ∃! x, p x
:= exists_intro w (and_intro H1 H2)
theorem exists_unique_elim {A : Type} {p : A → Bool} {b : Bool} (H2 : ∃! x, p x) (H1 : ∀ x, p x → (∀ y, y ≠ x → ¬ p y) → b) : b
:= exists_elim H2 (λ w Hw, H1 w (and_elim_left Hw) (and_elim_right Hw))
definition inhabited (A : Type) := ∃ x : A, true
theorem inhabited_intro {A : Type} (a : A) : inhabited A