61901cff81
also add more definitions in types.pi, types.path, algebra.precategory the (pre)category library still needs cleanup authors of this commit: @avigad, @javra, @fpvandoorn
46 lines
1.5 KiB
Text
46 lines
1.5 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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Hedberg's Theorem: every type with decidable equality is a hset
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-/
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prelude
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import init.trunc
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open eq eq.ops nat is_trunc sigma
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-- TODO(Leo): move const coll and path_coll to a different file?
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private definition const {A B : Type} (f : A → B) := ∀ x y, f x = f y
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private definition coll (A : Type) := Σ f : A → A, const f
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private definition path_coll (A : Type) := ∀ x y : A, coll (x = y)
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context
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parameter {A : Type}
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hypothesis (h : decidable_eq A)
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variables {x y : A}
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private definition pc [reducible] : path_coll A :=
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λ a b, decidable.rec_on (h a b)
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(λ p : a = b, ⟨(λ q, p), λ q r, rfl⟩)
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(λ np : ¬ a = b, ⟨(λ q, q), λ q r, absurd q np⟩)
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private definition f [reducible] : x = y → x = y :=
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sigma.rec_on (pc x y) (λ f c, f)
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private definition f_const (p q : x = y) : f p = f q :=
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sigma.rec_on (pc x y) (λ f c, c p q)
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private definition aux (p : x = y) : p = (f (refl x))⁻¹ ⬝ (f p) :=
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have aux : refl x = (f (refl x))⁻¹ ⬝ (f (refl x)), from
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eq.rec_on (f (refl x)) rfl,
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eq.rec_on p aux
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definition is_hset_of_decidable_eq : is_hset A :=
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is_hset.mk A (λ x y p q, calc
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p = (f (refl x))⁻¹ ⬝ (f p) : aux
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... = (f (refl x))⁻¹ ⬝ (f q) : f_const
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... = q : aux)
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end
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attribute is_hset_of_decidable_eq [instance]
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