feat(library/standard): add inhabited, and files option and unit

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2014-06-29 19:30:38 -07:00
parent 360e9b9486
commit e1d807a077
3 changed files with 42 additions and 2 deletions

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@ -13,6 +13,9 @@ definition not (a : Bool) := a → false
precedence `¬`:40
notation `¬` a := not a
notation `assume` binders `,` r:(scoped f, f) := r
notation `take` binders `,` r:(scoped f, f) := r
theorem not_intro {a : Bool} (H : a → false) : ¬ a
:= H
@ -23,10 +26,10 @@ theorem absurd {a : Bool} (H1 : a) (H2 : ¬ a) : false
:= H2 H1
theorem mt {a b : Bool} (H1 : a → b) (H2 : ¬ b) : ¬ a
:= λ Ha : a, absurd (H1 Ha) H2
:= assume Ha : a, absurd (H1 Ha) H2
theorem contrapos {a b : Bool} (H : a → b) : ¬ b → ¬ a
:= λ Hnb : ¬ b, mt H Hnb
:= assume Hnb : ¬ b, mt H Hnb
theorem absurd_elim {a : Bool} (b : Bool) (H1 : a) (H2 : ¬ a) : b
:= false_elim b (absurd H1 H2)
@ -72,3 +75,25 @@ theorem congr1 {A B : Type} {f g : A → B} (H : f = g) (a : A) : f a = g a
theorem congr2 {A B : Type} {a b : A} (f : A → B) (H : a = b) : f a = f b
:= subst H (refl (f a))
inductive Exists {A : Type} (P : A → Bool) : Bool :=
| exists_intro : ∀ (a : A), P a → Exists P
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 inhabited (A : Type) := ∃ x : A, true
theorem inhabited_intro {A : Type} (a : A) : inhabited A
:= exists_intro a trivial
theorem inhabited_elim {A : Type} {B : Bool} (H1 : inhabited A) (H2 : A → B) : B
:= exists_elim H1 (λ (a : A) (H : true), H2 a)
theorem inhabited_Bool : inhabited Bool
:= inhabited_intro true
theorem inhabited_fun (A : Type) {B : Type} (H : inhabited B) : inhabited (A → B)
:= inhabited_elim H (take (b : B), inhabited_intro (λ a : A, b))

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@ -0,0 +1,8 @@
import logic
inductive option (A : Type) : Type :=
| none {} : option A
| some : A → option A
theorem inhabited_option (A : Type) : inhabited (option A)
:= inhabited_intro none

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@ -0,0 +1,7 @@
import logic
inductive unit : Type :=
| tt : unit
theorem inhabited_unit : inhabited unit
:= inhabited_intro tt