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

Require Import HoTT.Basics HoTT.Types.
[Loading ML file number_string_notation_plugin.cmxs (using legacy method) ... done]

Local Open Scope path_scope.

Generalizable Variables A B.

(** ** Alternate characterization of hprops. *)

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

Theorem equiv_hprop_inhabited_contr `{Funext} {A}
  : IsHProp A <~> (A -> Contr A).
H: Funext
A: Type
IsHProp A <~> (A -> Contr A)
Proof.
H: Funext
A: Type
IsHProp A <~> (A -> Contr A)
  apply (equiv_adjointify (@contr_inhabited_hprop A) (@hprop_inhabited_contr A)).
H: Funext
A: Type
(fun x : A -> Contr A => contr_inhabited_hprop A) ==
idmap
H: Funext
A: Type
(fun x : IsHProp A =>
 hprop_inhabited_contr A (contr_inhabited_hprop A)) ==
idmap
  -
H: Funext
A: Type
(fun x : A -> Contr A => contr_inhabited_hprop A) ==
idmap intro ic.
H: Funext
A: Type
ic: A -> Contr A
contr_inhabited_hprop A = ic by_extensionality x.
H: Funext
A: Type
ic: A -> Contr A
x: A
contr_inhabited_hprop A x = ic x
    apply @path_contr.
H: Funext
A: Type
ic: A -> Contr A
x: A
Contr (Contr A) apply contr_istrunc.
H: Funext
A: Type
ic: A -> Contr A
x: A
Contr A exact (ic x).
  -
H: Funext
A: Type
(fun x : IsHProp A =>
 hprop_inhabited_contr A (contr_inhabited_hprop A)) ==
idmap intro hp.
H: Funext
A: Type
hp: IsHProp A
hprop_inhabited_contr A (contr_inhabited_hprop A) = 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.
A: Type
H: IsEquiv (fun a : A => (a, a))
IsHProp A
Proof.
A: Type
H: IsEquiv (fun a : A => (a, a))
IsHProp A
  apply hprop_allpath; intros x y.
A: Type
H: IsEquiv (fun a : A => (a, a))
x, y: A
x = y
  set (d := fun (a:A) => (a,a)) in *.
A: Type
d:= fun a : A => (a, a): A -> A * A
H: IsEquiv d
x, y: A
x = y
  transitivity (fst (d (d^-1 (x,y)))).
A: Type
d:= fun a : A => (a, a): A -> A * A
H: IsEquiv d
x, y: A
x = fst (d (d^-1 (x, y)))
A: Type
d:= fun a : A => (a, a): A -> A * A
H: IsEquiv d
x, y: A
fst (d (d^-1 (x, y))) = y
  -
A: Type
d:= fun a : A => (a, a): A -> A * A
H: IsEquiv d
x, y: A
x = fst (d (d^-1 (x, y))) exact (ap fst (eisretr d (x,y))^).
  -
A: Type
d:= fun a : A => (a, a): A -> A * A
H: IsEquiv d
x, y: A
fst (d (d^-1 (x, y))) = y transitivity (snd (d (d^-1 (x,y)))).
A: Type
d:= fun a : A => (a, a): A -> A * A
H: IsEquiv d
x, y: A
fst (d (d^-1 (x, y))) = snd (d (d^-1 (x, y)))
A: Type
d:= fun a : A => (a, a): A -> A * A
H: IsEquiv d
x, y: A
snd (d (d^-1 (x, y))) = y
    +
A: Type
d:= fun a : A => (a, a): A -> A * A
H: IsEquiv d
x, y: A
fst (d (d^-1 (x, y))) = snd (d (d^-1 (x, y))) unfold d; reflexivity.
    +
A: Type
d:= fun a : A => (a, a): A -> A * A
H: IsEquiv d
x, y: A
snd (d (d^-1 (x, y))) = y exact (ap snd (eisretr d (x,y))).
Defined.

Global Instance isequiv_diag_ishprop {A} `{IsHProp A}
: IsEquiv (fun (a:A) => (a,a)).
A: Type
IsHProp0: IsHProp A
IsEquiv (fun a : A => (a, a))
Proof.
A: Type
IsHProp0: IsHProp A
IsEquiv (fun a : A => (a, a))
  refine (isequiv_adjointify _ fst _ _).
A: Type
IsHProp0: IsHProp A
(fun x : A * A => (fst x, fst x)) == idmap
A: Type
IsHProp0: IsHProp A
(fun x : A => fst (x, x)) == idmap
  -
A: Type
IsHProp0: IsHProp A
(fun x : A * A => (fst x, fst x)) == idmap intros [x y].
A: Type
IsHProp0: IsHProp A
x, y: A
(fst (x, y), fst (x, y)) = (x, y)
    apply path_prod; simpl.
A: Type
IsHProp0: IsHProp A
x, y: A
x = x
A: Type
IsHProp0: IsHProp A
x, y: A
x = y
    +
A: Type
IsHProp0: IsHProp A
x, y: A
x = x reflexivity.
    +
A: Type
IsHProp0: IsHProp A
x, y: A
x = y apply path_ishprop.
  -
A: Type
IsHProp0: IsHProp A
(fun x : A => fst (x, x)) == idmap intros a; simpl.
A: Type
IsHProp0: IsHProp A
a: A
a = a
    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 : forall x1 x2, (f x1 = f x2) -> (x1 = x2))
           (H : forall x1 x2, (@ap X Y f x1 x2) o (s x1 x2) == idmap)
  : IsEmbedding f.
X, Y: Type
f: X -> Y
s: forall x1 x2 : X, f x1 = f x2 -> x1 = x2
H: forall x1 x2 : X, ap f o s x1 x2 == idmap
IsEmbedding f
Proof.
X, Y: Type
f: X -> Y
s: forall x1 x2 : X, f x1 = f x2 -> x1 = x2
H: forall x1 x2 : X, ap f o s x1 x2 == idmap
IsEmbedding f
  intros y.
X, Y: Type
f: X -> Y
s: forall x1 x2 : X, f x1 = f x2 -> x1 = x2
H: forall x1 x2 : X, ap f o s x1 x2 == idmap
y: Y
IsHProp (hfiber f y)
  apply hprop_allpath.
X, Y: Type
f: X -> Y
s: forall x1 x2 : X, f x1 = f x2 -> x1 = x2
H: forall x1 x2 : X, ap f o s x1 x2 == idmap
y: Y
forall x y0 : hfiber f y, x = y0
  intros [x1 p1] [x2 p2].
X, Y: Type
f: X -> Y
s: forall x1 x2 : X, f x1 = f x2 -> x1 = x2
H: forall x1 x2 : X, ap f o s x1 x2 == idmap
y: Y
x1: X
p1: f x1 = y
x2: X
p2: f x2 = y
(x1; p1) = (x2; p2)
  apply path_sigma with (s x1 x2 (p1 @ p2^)).
X, Y: Type
f: X -> Y
s: forall x1 x2 : X, f x1 = f x2 -> x1 = x2
H: forall x1 x2 : X, ap f o s x1 x2 == idmap
y: Y
x1: X
p1: f x1 = y
x2: X
p2: f x2 = y
transport (fun x : X => f x = y) (s x1 x2 (p1 @ p2^))
  (x1; p1).2 = (x2; p2).2
  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.
H: Funext
A: Type
Contr A <~> A * IsHProp A
Proof.
H: Funext
A: Type
Contr A <~> A * IsHProp A
  assert (f : Contr A -> A * IsHProp A).
H: Funext
A: Type
Contr A -> A * IsHProp A
H: Funext
A: Type
f: Contr A -> A * IsHProp A
Contr A <~> A * IsHProp A
  -
H: Funext
A: Type
Contr A -> A * IsHProp A intro P.
H: Funext
A: Type
P: Contr A
A * IsHProp A split.
H: Funext
A: Type
P: Contr A
A
H: Funext
A: Type
P: Contr A
IsHProp A
    +
H: Funext
A: Type
P: Contr A
A exact (@center _ P).
    +
H: Funext
A: Type
P: Contr A
IsHProp A apply @istrunc_succ.
H: Funext
A: Type
P: Contr A
Contr A exact P.
  -
H: Funext
A: Type
f: Contr A -> A * IsHProp A
Contr A <~> A * IsHProp A assert (g : A * IsHProp A -> Contr A).
H: Funext
A: Type
f: Contr A -> A * IsHProp A
A * IsHProp A -> Contr A
H: Funext
A: Type
f: Contr A -> A * IsHProp A
g: A * IsHProp A -> Contr A
Contr A <~> A * IsHProp A
    +
H: Funext
A: Type
f: Contr A -> A * IsHProp A
A * IsHProp A -> Contr A intros [a P].
H: Funext
A: Type
f: Contr A -> A * IsHProp A
a: A
P: IsHProp A
Contr A apply (@contr_inhabited_hprop _ P a).
    +
H: Funext
A: Type
f: Contr A -> A * IsHProp A
g: A * IsHProp A -> Contr A
Contr A <~> A * IsHProp A refine (@equiv_iff_hprop _ _ _ _ f g).
H: Funext
A: Type
f: Contr A -> A * IsHProp A
g: A * IsHProp A -> Contr A
IsHProp (A * IsHProp A)
      apply hprop_inhabited_contr; intro p.
H: Funext
A: Type
f: Contr A -> A * IsHProp A
g: A * IsHProp A -> Contr A
p: A * IsHProp A
Contr (A * IsHProp A)
      apply @contr_prod.
H: Funext
A: Type
f: Contr A -> A * IsHProp A
g: A * IsHProp A -> Contr A
p: A * IsHProp A
Contr A
H: Funext
A: Type
f: Contr A -> A * IsHProp A
g: A * IsHProp A -> Contr A
p: A * IsHProp A
Contr (IsHProp A)
      *
H: Funext
A: Type
f: Contr A -> A * IsHProp A
g: A * IsHProp A -> Contr A
p: A * IsHProp A
Contr A exact (g p).
      *
H: Funext
A: Type
f: Contr A -> A * IsHProp A
g: A * IsHProp A -> Contr A
p: A * IsHProp A
Contr (IsHProp A) apply (@contr_inhabited_hprop _ _ (snd p)).
Defined.

Theorem equiv_contr_inhabited_allpath `{Funext} {A}
  : Contr A <~> A * forall (x y : A), x = y.
H: Funext
A: Type
Contr A <~> A * (forall x y : A, x = y)
Proof.
H: Funext
A: Type
Contr A <~> A * (forall x y : A, x = y)
  transitivity (A * IsHProp A).
H: Funext
A: Type
Contr A <~> A * IsHProp A
H: Funext
A: Type
A * IsHProp A <~> A * (forall x y : A, x = y)
  -
H: Funext
A: Type
Contr A <~> A * IsHProp A apply equiv_contr_inhabited_hprop.
  -
H: Funext
A: Type
A * IsHProp A <~> A * (forall x y : A, x = y) 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.
H: Funext
A, B: Type
IsHProp0: IsHProp A
IsHProp1: IsHProp B
IsEquiv equiv_iff_hprop_uncurried
Proof.
H: Funext
A, B: Type
IsHProp0: IsHProp A
IsHProp1: IsHProp B
IsEquiv equiv_iff_hprop_uncurried
  pose (@istrunc_equiv).
H: Funext
A, B: Type
IsHProp0: IsHProp A
IsHProp1: IsHProp B
i:= @istrunc_equiv: Funext ->
forall (n : trunc_index) 
(A B : Type),
IsTrunc n.+1 B -> IsTrunc n.+1 (A <~> B)
IsEquiv equiv_iff_hprop_uncurried
  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.
hprop: Type
IsHProp0: IsHProp hprop
hprop -> hprop <~> Unit
Proof.
hprop: Type
IsHProp0: IsHProp hprop
hprop -> hprop <~> Unit
  intro p.
hprop: Type
IsHProp0: IsHProp hprop
p: hprop
hprop <~> Unit
  apply equiv_iff_hprop.
hprop: Type
IsHProp0: IsHProp hprop
p: hprop
hprop -> Unit
hprop: Type
IsHProp0: IsHProp hprop
p: hprop
Unit -> hprop
  -
hprop: Type
IsHProp0: IsHProp hprop
p: hprop
hprop -> Unit exact (fun _ => tt).
  -
hprop: Type
IsHProp0: IsHProp hprop
p: hprop
Unit -> hprop 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.
hprop: Type
IsHProp0: IsHProp hprop
~ hprop -> hprop <~> Empty
Proof.
hprop: Type
IsHProp0: IsHProp hprop
~ hprop -> hprop <~> Empty
  intro np.
hprop: Type
IsHProp0: IsHProp hprop
np: ~ hprop
hprop <~> Empty
  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).
hprop: Type
IsHProp0: IsHProp hprop
H: Decidable hprop
(hprop <~> Unit) + (hprop <~> Empty)
Proof.
hprop: Type
IsHProp0: IsHProp hprop
H: Decidable hprop
(hprop <~> Unit) + (hprop <~> Empty)
  destruct (dec hprop) as [x|nx].
hprop: Type
IsHProp0: IsHProp hprop
H: Decidable hprop
x: hprop
(hprop <~> Unit) + (hprop <~> Empty)
hprop: Type
IsHProp0: IsHProp hprop
H: Decidable hprop
nx: ~ hprop
(hprop <~> Unit) + (hprop <~> Empty)
  -
hprop: Type
IsHProp0: IsHProp hprop
H: Decidable hprop
x: hprop
(hprop <~> Unit) + (hprop <~> Empty) exact (inl (if_hprop_then_equiv_Unit hprop x)).
  -
hprop: Type
IsHProp0: IsHProp hprop
H: Decidable hprop
nx: ~ hprop
(hprop <~> Unit) + (hprop <~> Empty) exact (inr (if_not_hprop_then_equiv_Empty hprop nx)).
Defined.