Library UniMath.OrderTheory.DCPOs.Core.IntrinsicApartness

Require Import UniMath.MoreFoundations.All.
Require Import UniMath.OrderTheory.DCPOs.Core.DirectedSets.
Require Import UniMath.OrderTheory.DCPOs.Core.Basics.
Require Import UniMath.OrderTheory.DCPOs.Core.WayBelow.
Require Import UniMath.OrderTheory.DCPOs.Core.ScottTopology.
Require Import UniMath.OrderTheory.DCPOs.Basis.Continuous.

Local Open Scope dcpo.

1. The specialization preorder
Definition dcpo_specialization
           {X : dcpo}
           (x y : X)
  : hProp
  := (∀ (P : X → hProp), is_scott_open P ⇒ P x ⇒ P y)%logic.

Notation "x ⊑s y" := (dcpo_specialization x y) (at level 70) : dcpo.

2. Properties of the specialization preorder
Section PropertiesSpecialization.
  Context {X : dcpo}.

  Proposition refl_dcpo_specialization
              (x : X)
    : x ⊑s x.
  Show proof.
    intros P HP Px.
    exact Px.

  Proposition trans_dcpo_specialization
              {x y z : X}
              (p : x ⊑s y)
              (q : y ⊑s z)
    : x ⊑s z.
  Show proof.
    intros P HP Px.
    apply (q P HP).
    apply (p P HP Px).

  Proposition le_dcpo_specialization
              {x y : X}
              (p : x ⊑ y)
    : x ⊑s y.
  Show proof.
    intros P HP Px.
    exact (is_scott_open_upper_set HP Px p).

  Proposition le_dcpo_specialization_equiv
              (CX : continuous_dcpo_struct X)
              (x y : X)
    : x ⊑ y ≃ x ⊑s y.
  Show proof.
    use weqimplimpl.
    - exact le_dcpo_specialization.
    - intros p.
      use (invmap (continuous_dcpo_struct_le_via_approximation CX x y)).
      intro z.
      intro q.
      exact (p _ (upper_set_is_scott_open CX z) q).
    - apply propproperty.
    - apply propproperty.
End PropertiesSpecialization.

3. The 'does not specialize' relation
Definition dcpo_not_specialization
           {X : dcpo}
           (x y : X)
  : hProp
  := (∃ (P : X → hProp), is_scott_open P ∧ P x ∧ ¬(P y))%logic.

Notation "x ⊄ y" := (dcpo_not_specialization x y) (at level 70) : dcpo.
4. Properties of the 'does not specialize' relation
Section PropertiesNotSpecialization.
  Context {X : dcpo}.

  Proposition irrefl_dcpo_not_specialization
              (x : X)
    : ¬(x ⊄ x).
  Show proof.
    use factor_through_squash.
    {
      apply isapropempty.
    }
    intro P.
    induction P as [ P [ HP [ p n ]]].
    exact (n p).

  Proposition dcpo_not_specialization_le
              {x y : X}
              (p : x ⊄ y)
    : ¬(x ⊑ y).
  Show proof.
    intro q.
    revert p.
    use factor_through_squash.
    {
      apply isapropempty.
    }
    intros P.
    induction P as [ P [ HP [ Px Py ]]].
    refine (Py _).
    exact (is_scott_open_upper_set HP Px q).

  Proposition continuous_not_specialization_weq
              (CX : continuous_dcpo_struct X)
              (x y : X)
    : x ⊄ y ≃ (∃ (b : X), b ≪ x ∧ ¬(b ⊑ y))%logic.
  Show proof.
    use weqimplimpl.
    - use factor_through_squash.
      {
        apply propproperty.
      }
      intros P.
      induction P as [ P [ HP [ p₁ p₂ ]]] ; cbn in p₂.
      pose (D := approximating_family CX x).
      assert (HD : P (⨆ D)).
      {
        unfold D.
        rewrite approximating_family_lub.
        exact p₁.
      }
      assert (H := is_scott_open_lub_inaccessible HP D HD).
      revert H.
      use factor_through_squash.
      {
        apply propproperty.
      }
      intro i.
      induction i as [ i Pi ].
      refine (hinhpr (D i ,, _ ,, _)).
      + apply approximating_family_way_below.
      + intro n.
        apply p₂.
        apply (is_scott_open_upper_set HP Pi n).
    - use factor_through_squash.
      {
        apply propproperty.
      }
      intro b.
      induction b as [ b [ p₁ p₂ ]] ; cbn in p₂.
      apply hinhpr.
      refine (_ ,, upper_set_is_scott_open CX b ,, p₁ ,, _).
      intro p.
      apply p₂.
      apply way_below_to_le.
      exact p.
    - apply propproperty.
    - apply propproperty.
End PropertiesNotSpecialization.

5. The intrinsic apartness relation
Definition dcpo_intrinsic_apartness
           {X : dcpo}
           (x y : X)
  : hProp
  := dcpo_not_specialization x y ∨ dcpo_not_specialization y x.

Notation "x # y" := (dcpo_intrinsic_apartness x y).

6. Properties of the intrinsic apartness relation
Section PropertiesApartness.
  Context {X : dcpo}.

  Proposition irrefl_intrinsic_apartness
              (x : X)
    : ¬(x # x).
  Show proof.
    use factor_through_squash.
    {
      apply isapropempty.
    }
    intros p.
    induction p as [ p | p ].
    - apply (irrefl_dcpo_not_specialization x).
      exact p.
    - apply (irrefl_dcpo_not_specialization x).
      exact p.

  Proposition symmetric_intrinsic_apartness
              {x y : X}
              (p : x # y)
    : y # x.
  Show proof.
    revert p.
    use factor_through_squash.
    {
      apply propproperty.
    }
    intros p.
    induction p as [ p | p ].
    - exact (hinhpr (inr p)).
    - exact (hinhpr (inl p)).

  Proposition intrinsic_apartness_not_eq
              {x y : X}
              (p : x # y)
    : ¬(x = y).
  Show proof.
    intro q.
    induction q.
    exact (irrefl_intrinsic_apartness x p).
End PropertiesApartness.