46 lines
No EOL
1.6 KiB
Text
46 lines
No EOL
1.6 KiB
Text
------------------------------------------------------------------------------------------------------
|
||
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||
-- Author: Leonardo de Moura
|
||
----------------------------------------------------------------------------------------------------
|
||
|
||
import logic.classes.inhabited logic.connectives.eq
|
||
|
||
namespace pair
|
||
inductive pair (A : Type) (B : Type) : Type :=
|
||
| mk_pair : A → B → pair A B
|
||
|
||
section
|
||
parameter {A : Type}
|
||
parameter {B : Type}
|
||
|
||
definition fst [inline] (p : pair A B) := pair_rec (λ x y, x) p
|
||
definition snd [inline] (p : pair A B) := pair_rec (λ x y, y) p
|
||
|
||
theorem pair_inhabited (H1 : inhabited A) (H2 : inhabited B) : inhabited (pair A B) :=
|
||
inhabited_elim H1 (λ a, inhabited_elim H2 (λ b, inhabited_intro (mk_pair a b)))
|
||
|
||
theorem fst_mk_pair (a : A) (b : B) : fst (mk_pair a b) = a :=
|
||
refl a
|
||
|
||
theorem snd_mk_pair (a : A) (b : B) : snd (mk_pair a b) = b :=
|
||
refl b
|
||
|
||
theorem pair_ext (p : pair A B) : mk_pair (fst p) (snd p) = p :=
|
||
pair_rec (λ x y, refl (mk_pair x y)) p
|
||
|
||
theorem pair_ext_eq {p1 p2 : pair A B} (H1 : fst p1 = fst p2) (H2 : snd p1 = snd p2) : p1 = p2 :=
|
||
calc p1 = mk_pair (fst p1) (snd p1) : symm (pair_ext p1)
|
||
... = mk_pair (fst p2) (snd p1) : {H1}
|
||
... = mk_pair (fst p2) (snd p2) : {H2}
|
||
... = p2 : pair_ext p2
|
||
end
|
||
|
||
instance pair_inhabited
|
||
|
||
precedence `×`:30
|
||
infixr × := pair
|
||
|
||
-- notation for n-ary tuples
|
||
notation `(` h `,` t:(foldl `,` (e r, mk_pair r e) h) `)` := t
|
||
end |