/-
Copyright (c) 2014 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Jakob von Raumer

Categories of (hprop value) ordered sets.
-/
import ..category algebra.order types.fin

open algebra category is_trunc is_equiv equiv iso

namespace category

section
universe variable l
parameters (A : Type.{l}) [HA : is_set A] [OA : weak_order.{l} A] 
  [Hle : Π a b : A, is_prop (a ≤ b)]
include A HA OA Hle

definition precategory_order [constructor] : precategory.{l l} A :=
begin
  fconstructor,
  { intro a b, exact a ≤ b },
  { intro a b c, exact ge.trans },
  { intro a, apply le.refl },
  do 5 (intros; apply is_prop.elim),
  { intros, apply is_trunc_succ }
end

local attribute [instance] precategory_order

definition category_order : category.{l l} A :=
begin
  fapply category.mk precategory_order,
  intros a b, fapply adjointify,
  { intro f, apply le.antisymm, apply iso.to_hom f, apply iso.to_inv f },
  { intro f, fapply iso_eq, esimp[precategory_order], apply is_prop.elim },
  { intro p, apply is_prop.elim }
end

end

end category