46 lines
1.5 KiB
Text
46 lines
1.5 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
|
|
|
|
Hedberg's Theorem: every type with decidable equality is a hset
|
|
-/
|
|
prelude
|
|
import init.nat init.trunc
|
|
open eq eq.ops nat truncation sigma.ops
|
|
|
|
-- TODO(Leo): move const coll and path_coll to a different file?
|
|
private definition const {A B : Type} (f : A → B) := ∀ x y, f x = f y
|
|
private definition coll (A : Type) := Σ f : A → A, const f
|
|
private definition path_coll (A : Type) := ∀ x y : A, coll (x = y)
|
|
|
|
context
|
|
parameter {A : Type}
|
|
hypothesis (h : decidable_eq A)
|
|
variables {x y : A}
|
|
|
|
private definition pc : path_coll A :=
|
|
λ a b, decidable.rec_on (h a b)
|
|
(λ p : a = b, ⟨(λ q, p), λ q r, rfl⟩)
|
|
(λ np : ¬ a = b, ⟨(λ q, q), λ q r, absurd q np⟩)
|
|
|
|
private definition f : x = y → x = y :=
|
|
sigma.rec_on (pc x y) (λ f c, f)
|
|
|
|
private definition f_const (p q : x = y) : f p = f q :=
|
|
sigma.rec_on (pc x y) (λ f c, c p q)
|
|
|
|
private definition aux (p : x = y) : p = (f (refl x))⁻¹ ⬝ (f p) :=
|
|
have aux : refl x = (f (refl x))⁻¹ ⬝ (f (refl x)), from
|
|
eq.rec_on (f (refl x)) rfl,
|
|
eq.rec_on p aux
|
|
|
|
definition is_hset_of_decidable_eq : is_hset A :=
|
|
is_hset.mk A (λ x y p q, calc
|
|
p = (f (refl x))⁻¹ ⬝ (f p) : aux
|
|
... = (f (refl x))⁻¹ ⬝ (f q) : f_const
|
|
... = q : aux)
|
|
end
|
|
|
|
instance [persistent] is_hset_of_decidable_eq
|