70 lines
2.6 KiB
Text
70 lines
2.6 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, Jeremy Avigad
|
|
|
|
import logic.core.prop logic.classes.inhabited logic.classes.decidable
|
|
|
|
using inhabited decidable
|
|
|
|
namespace sum
|
|
|
|
-- TODO: take this outside the namespace when the inductive package handles it better
|
|
inductive sum (A B : Type) : Type :=
|
|
inl : A → sum A B,
|
|
inr : B → sum A B
|
|
|
|
infixr `+`:25 := sum
|
|
|
|
abbreviation rec_on {A B : Type} {C : (A + B) → Type} (s : A + B)
|
|
(H1 : ∀a : A, C (inl B a)) (H2 : ∀b : B, C (inr A b)) : C s :=
|
|
sum_rec H1 H2 s
|
|
|
|
|
|
theorem inl_inj {A B : Type} {a1 a2 : A} (H : inl B a1 = inl B a2) : a1 = a2 :=
|
|
let f := λs, rec_on s (λa, a1 = a) (λb, false) in
|
|
have H1 : f (inl B a1), from rfl,
|
|
have H2 : f (inl B a2), from subst H H1,
|
|
H2
|
|
|
|
abbreviation cases_on {A B : Type} {P : (A + B) → Prop} (s : A + B)
|
|
(H1 : ∀a : A, P (inl B a)) (H2 : ∀b : B, P (inr A b)) : P s :=
|
|
sum_rec H1 H2 s
|
|
|
|
theorem inl_neq_inr {A B : Type} {a : A} {b : B} (H : inl B a = inr A b) : false :=
|
|
let f := λs, rec_on s (λa', a = a') (λb, false) in
|
|
have H1 : f (inl B a), from rfl,
|
|
have H2 : f (inr A b), from subst H H1,
|
|
H2
|
|
|
|
theorem inr_inj {A B : Type} {b1 b2 : B} (H : inr A b1 = inr A b2) : b1 = b2 :=
|
|
let f := λs, rec_on s (λa, false) (λb, b1 = b) in
|
|
have H1 : f (inr A b1), from rfl,
|
|
have H2 : f (inr A b2), from subst H H1,
|
|
H2
|
|
|
|
theorem sum_inhabited_left [instance] {A B : Type} (H : inhabited A) : inhabited (A + B) :=
|
|
inhabited_mk (inl B (default A))
|
|
|
|
theorem sum_inhabited_right [instance] {A B : Type} (H : inhabited B) : inhabited (A + B) :=
|
|
inhabited_mk (inr A (default B))
|
|
|
|
theorem sum_eq_decidable [instance] {A B : Type} (s1 s2 : A + B)
|
|
(H1 : ∀a1 a2 : A, decidable (inl B a1 = inl B a2))
|
|
(H2 : ∀b1 b2 : B, decidable (inr A b1 = inr A b2)) : decidable (s1 = s2) :=
|
|
rec_on s1
|
|
(take a1, show decidable (inl B a1 = s2), from
|
|
rec_on s2
|
|
(take a2, show decidable (inl B a1 = inl B a2), from H1 a1 a2)
|
|
(take b2,
|
|
have H3 : (inl B a1 = inr A b2) ↔ false,
|
|
from iff_intro inl_neq_inr (assume H4, false_elim H4),
|
|
show decidable (inl B a1 = inr A b2), from decidable_iff_equiv _ (iff_symm H3)))
|
|
(take b1, show decidable (inr A b1 = s2), from
|
|
rec_on s2
|
|
(take a2,
|
|
have H3 : (inr A b1 = inl B a2) ↔ false,
|
|
from iff_intro (assume H4, inl_neq_inr (symm H4)) (assume H4, false_elim H4),
|
|
show decidable (inr A b1 = inl B a2), from decidable_iff_equiv _ (iff_symm H3))
|
|
(take b2, show decidable (inr A b1 = inr A b2), from H2 b1 b2))
|
|
|
|
end sum
|