lean2/hott/init/hedberg.hlean

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/-
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.trunc
open eq eq.ops nat truncation sigma
-- 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 [reducible] : 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 [reducible] : 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
attribute is_hset_of_decidable_eq [instance]