Library HoTT.HProp

(* -*- mode: coq; mode: visual-line -*-  *)

HPropositions


Require Import HoTT.Basics HoTT.Types.

Local Open Scope path_scope.

Generalizable Variables A B.

Alternate characterization of hprops.


Theorem equiv_hprop_allpath `{Funext} (A : Type)
  : IsHProp A <~> (∀ (x y : A), x = y).
Proof.
  rapply (equiv_iff_hprop (@path_ishprop A) (@hprop_allpath A)).
  apply hprop_allpath; intros f g.
  funext x y.
  pose (C := Build_Contr A x (f x)).
  apply path_contr.
Defined.

Theorem equiv_hprop_inhabited_contr `{Funext} {A}
  : IsHProp A <~> (A → Contr A).
Proof.
  apply (equiv_adjointify (@contr_inhabited_hprop A) (@hprop_inhabited_contr A)).
  - intro ic. by_extensionality x.
    apply @path_contr. apply contr_istrunc. exact (ic x).
  - intro hp.
    apply path_ishprop.
Defined.

Being an hprop is also equivalent to the diagonal being an equivalence.
Definition ishprop_isequiv_diag {A} `{IsEquiv _ _ (fun (a:A) ⇒ (a,a))}
: IsHProp A.
Proof.
  apply hprop_allpath; intros x y.
  set (d := fun (a:A) ⇒ (a,a)) in ×.
  transitivity (fst (d (d^-1 (x,y)))).
  - exact (ap fst (eisretr d (x,y))^).
  - transitivity (snd (d (d^-1 (x,y)))).
    + unfold d; reflexivity.
    + exact (ap snd (eisretr d (x,y))).
Defined.

Global Instance isequiv_diag_ishprop {A} `{IsHProp A}
: IsEquiv (fun (a:A) ⇒ (a,a)).
Proof.
  refine (isequiv_adjointify _ fst _ _).
  - intros [x y].
    apply path_prod; simpl.
    + reflexivity.
    + apply path_ishprop.
  - intros a; simpl.
    reflexivity.
Defined.

A map is an embedding as soon as its ap's have sections.


Definition isembedding_sect_ap {X Y} (f : X → Y)
           (s : ∀ x1 x2, (f x1 = f x2) → (x1 = x2))
           (H : ∀ x1 x2, (@ap X Y f x1 x2) o (s x1 x2) == idmap)
  : IsEmbedding f.
Proof.
  intros y.
  apply hprop_allpath.
  intros [x1 p1] [x2 p2].
  apply path_sigma with (s x1 x2 (p1 @ p2^)).
  abstract (rewrite transport_paths_Fl; cbn;
            rewrite (H x1 x2 (p1 @ p2^));
            rewrite inv_pp, inv_V; apply concat_pV_p).
Defined.

Alternate characterizations of contractibility.


Theorem equiv_contr_inhabited_hprop `{Funext} {A}
  : Contr A <~> A × IsHProp A.
Proof.
  assert (f : Contr A → A × IsHProp A).
  - intro P. split.
    + exact (@center _ P).
    + apply @istrunc_succ. exact P.
  - assert (g : A × IsHProp A → Contr A).
    + intros [a P]. apply (@contr_inhabited_hprop _ P a).
    + refine (@equiv_iff_hprop _ _ _ _ f g).
      apply hprop_inhabited_contr; intro p.
      apply @contr_prod.
      × exact (g p).
      × apply (@contr_inhabited_hprop _ _ (snd p)).
Defined.

Theorem equiv_contr_inhabited_allpath `{Funext} {A}
  : Contr A <~> A × ∀ (x y : A), x = y.
Proof.
  transitivity (A × IsHProp A).
  - apply equiv_contr_inhabited_hprop.
  - exact (1 ×E equiv_hprop_allpath _).
Defined.

Logical equivalence of hprops

Logical equivalence of hprops is not just logically equivalent to equivalence, it is equivalent to it.
Global Instance isequiv_equiv_iff_hprop_uncurried
       `{Funext} {A B} `{IsHProp A} `{IsHProp B}
: IsEquiv (@equiv_iff_hprop_uncurried A _ B _) | 0.
Proof.
  pose (@istrunc_equiv).
  refine (isequiv_adjointify
            equiv_iff_hprop_uncurried
            (fun e ⇒ (@equiv_fun _ _ e, @equiv_inv _ _ e _))
            _ _);
    intro;
      by apply path_ishprop.
Defined.

Definition equiv_equiv_iff_hprop
       `{Funext} (A B : Type) `{IsHProp A} `{IsHProp B}
  : (A ↔ B) <~> (A <~> B)
  := Build_Equiv _ _ (@equiv_iff_hprop_uncurried A _ B _) _.

Inhabited and uninhabited hprops

If an hprop is inhabited, then it is equivalent to Unit.
Lemma if_hprop_then_equiv_Unit (hprop : Type) `{IsHProp hprop} : hprop → hprop <~> Unit.
Proof.
  intro p.
  apply equiv_iff_hprop.
  - exact (fun _ ⇒ tt).
  - exact (fun _ ⇒ p).
Defined.

If an hprop is not inhabited, then it is equivalent to Empty.
Lemma if_not_hprop_then_equiv_Empty (hprop : Type) `{IsHProp hprop} : ¬hprop → hprop <~> Empty.
Proof.
  intro np.
  exact (Build_Equiv _ _ np _).
Defined.

Thus, a decidable hprop is either equivalent to Unit or Empty.
Definition equiv_decidable_hprop (hprop : Type)
           `{IsHProp hprop} `{Decidable hprop}
: (hprop <~> Unit) + (hprop <~> Empty).
Proof.
  destruct (dec hprop) as [x|nx].
  - exact (inl (if_hprop_then_equiv_Unit hprop x)).
  - exact (inr (if_not_hprop_then_equiv_Empty hprop nx)).
Defined.