(* -*- mode: coq; mode: visual-line -*-  *)
Require Import HoTT.Basics HoTT.Types.
[Loading ML file number_string_notation_plugin.cmxs (using legacy method) ... done]
Require Import Extensions Factorization.
Require Import Truncations.Core Modalities.Modality.

Local Open Scope path_scope.
Local Open Scope trunc_scope.

(** * Varieties of constant function *)

(** Recall that a function [f : X -> Y] is *weakly constant*, [WeaklyConstant f], if [forall x y, f x = f y].  We show, following Kraus, Escardo, Coquand, and Altenkirch, that the type of fixed points of a weakly constant endofunction is an hprop.  However, to avoid potential confusion with [Coq.Init.Wf.Fix], instead of their notation [Fix], we denote this type by [FixedBy]. *)

Definition FixedBy {X : Type} (f : X -> X) := {x : X & f x = x}.

Global Instance ishprop_fix_wconst {X : Type} (f : X -> X)
       `{WeaklyConstant _ _ f}
: IsHProp (FixedBy f).
X: Type
f: X -> X
H: WeaklyConstant f
IsHProp (FixedBy f)
Proof.
X: Type
f: X -> X
H: WeaklyConstant f
IsHProp (FixedBy f)
  apply hprop_inhabited_contr; intros [x0 p0].
X: Type
f: X -> X
H: WeaklyConstant f
x0: X
p0: f x0 = x0
Contr (FixedBy f)
  refine (contr_equiv' {x:X & f x0 = x} _); unfold FixedBy.
X: Type
f: X -> X
H: WeaklyConstant f
x0: X
p0: f x0 = x0
{x : X & f x0 = x} <~> {x : X & f x = x}
  apply equiv_functor_sigma_id.
X: Type
f: X -> X
H: WeaklyConstant f
x0: X
p0: f x0 = x0
forall a : X, f x0 = a <~> f a = a intros x.
X: Type
f: X -> X
H: WeaklyConstant f
x0: X
p0: f x0 = x0
x: X
f x0 = x <~> f x = x
  apply equiv_concat_l.
X: Type
f: X -> X
H: WeaklyConstant f
x0: X
p0: f x0 = x0
x: X
f x = f x0
  apply wconst.
Defined.

(** It follows that if a type [X] admits a weakly constant endofunction [f], then [FixedBy f] is equivalent to [merely X]. *)
Definition equiv_fix_merely {X : Type} (f : X -> X)
           `{WeaklyConstant _ _ f}
: FixedBy f <~> merely X.
X: Type
f: X -> X
H: WeaklyConstant f
FixedBy f <~> merely X
Proof.
X: Type
f: X -> X
H: WeaklyConstant f
FixedBy f <~> merely X
  apply equiv_iff_hprop.
X: Type
f: X -> X
H: WeaklyConstant f
FixedBy f -> merely X
X: Type
f: X -> X
H: WeaklyConstant f
merely X -> FixedBy f
  -
X: Type
f: X -> X
H: WeaklyConstant f
FixedBy f -> merely X intros [x p]; exact (tr x).
  -
X: Type
f: X -> X
H: WeaklyConstant f
merely X -> FixedBy f apply Trunc_rec; intros x.
X: Type
f: X -> X
H: WeaklyConstant f
x: X
FixedBy f
    exists (f x).
X: Type
f: X -> X
H: WeaklyConstant f
x: X
f (f x) = f x
    apply wconst.
Defined.

(** Therefore, a type is collapsible (admits a weakly constant endomap) if and only if [merely X -> X] (it has "split support"). *)
Definition splitsupp_collapsible {X} `{Collapsible X}
: merely X -> X.
X: Type
H: Collapsible X
merely X -> X
Proof.
X: Type
H: Collapsible X
merely X -> X
  refine (_ o (equiv_fix_merely collapse)^-1).
X: Type
H: Collapsible X
FixedBy collapse -> X
  apply pr1.
Defined.

Definition collapsible_splitsupp {X} (s : merely X -> X)
: Collapsible X.
X: Type
s: merely X -> X
Collapsible X
Proof.
X: Type
s: merely X -> X
Collapsible X
  refine (Build_Collapsible _ (s o tr) _); intros x y.
X: Type
s: merely X -> X
x, y: X
s (tr x) = s (tr y)
  apply (ap s), path_ishprop.
Defined.

(**  We say that [f] is *conditionally constant* if it factors through the propositional truncation [merely X], and *constant* if it factors through [Unit]. *)

Definition ConditionallyConstant {X Y : Type} (f : X -> Y)
  := ExtensionAlong (@tr (-1) X) (fun _ => Y) f.

(** We don't yet have a need for a predicate [Constant] on functions; we do already have the operation [const] which constructs the constant function at a given point.  Every such constant function is, of course, conditionally constant. *)
Definition cconst_const {X Y} (y : Y)
: ConditionallyConstant (@const X Y y).
X, Y: Type
y: Y
ConditionallyConstant (const y)
Proof.
X, Y: Type
y: Y
ConditionallyConstant (const y)
  exists (const y); intros x; reflexivity.
Defined.

(** The type of conditionally constant functions is equivalent to [merely X -> Y]. *)
Definition equiv_cconst_from_merely `{Funext} (X Y : Type)
: { f : X -> Y & ConditionallyConstant f } <~> (merely X -> Y).
H: Funext
X, Y: Type
{f : X -> Y & ConditionallyConstant f} <~>
(merely X -> Y)
Proof.
H: Funext
X, Y: Type
{f : X -> Y & ConditionallyConstant f} <~>
(merely X -> Y)
  refine (_ oE (equiv_sigma_symm _)).
H: Funext
X, Y: Type
{b : Trunc (-1) X -> Y &
{a : X -> Y & forall x : X, b (tr x) = a x}} <~>
(merely X -> Y)
  refine (equiv_sigma_contr _).
Defined.

(** If a function factors through any hprop, it is conditionally constant. *)
Definition cconst_factors_hprop {X Y : Type} (f : X -> Y)
           (P : Type) `{IsHProp P}
           (g : X -> P) (h : P -> Y) (p : h o g == f)
: ConditionallyConstant f.
X, Y: Type
f: X -> Y
P: Type
IsHProp0: IsHProp P
g: X -> P
h: P -> Y
p: h o g == f
ConditionallyConstant f
Proof.
X, Y: Type
f: X -> Y
P: Type
IsHProp0: IsHProp P
g: X -> P
h: P -> Y
p: h o g == f
ConditionallyConstant f
  pose (g' := Trunc_rec g : merely X -> P).
X, Y: Type
f: X -> Y
P: Type
IsHProp0: IsHProp P
g: X -> P
h: P -> Y
p: h o g == f
g':= Trunc_rec g : merely X -> P: merely X -> P
ConditionallyConstant f
  exists (h o g'); intros x.
X, Y: Type
f: X -> Y
P: Type
IsHProp0: IsHProp P
g: X -> P
h: P -> Y
p: h o g == f
g':= Trunc_rec g : merely X -> P: merely X -> P
x: X
h (g' (tr x)) = f x
  apply p.
Defined.

(** Thus, if it factors through a type that [X] implies is contractible, then it is also conditionally constant. *)
Definition cconst_factors_contr `{Funext}  {X Y : Type} (f : X -> Y)
           (P : Type) `{Pc : X -> Contr P}
           (g : X -> P) (h : P -> Y) (p : h o g == f)
: ConditionallyConstant f.
H: Funext
X, Y: Type
f: X -> Y
P: Type
Pc: X -> Contr P
g: X -> P
h: P -> Y
p: h o g == f
ConditionallyConstant f
Proof.
H: Funext
X, Y: Type
f: X -> Y
P: Type
Pc: X -> Contr P
g: X -> P
h: P -> Y
p: h o g == f
ConditionallyConstant f
  assert (merely X -> IsHProp P).
H: Funext
X, Y: Type
f: X -> Y
P: Type
Pc: X -> Contr P
g: X -> P
h: P -> Y
p: h o g == f
merely X -> IsHProp P
H: Funext
X, Y: Type
f: X -> Y
P: Type
Pc: X -> Contr P
g: X -> P
h: P -> Y
p: h o g == f
X0: merely X -> IsHProp P
ConditionallyConstant f
  {
H: Funext
X, Y: Type
f: X -> Y
P: Type
Pc: X -> Contr P
g: X -> P
h: P -> Y
p: h o g == f
merely X -> IsHProp P apply Trunc_rec.            (** Uses funext *)
H: Funext
X, Y: Type
f: X -> Y
P: Type
Pc: X -> Contr P
g: X -> P
h: P -> Y
p: h o g == f
X -> IsHProp P
    intros x; pose (Pc x); apply istrunc_succ. }
H: Funext
X, Y: Type
f: X -> Y
P: Type
Pc: X -> Contr P
g: X -> P
h: P -> Y
p: h o g == f
X0: merely X -> IsHProp P
ConditionallyConstant f
  pose (g' := Trunc_ind (fun _ => P) g : merely X -> P).
H: Funext
X, Y: Type
f: X -> Y
P: Type
Pc: X -> Contr P
g: X -> P
h: P -> Y
p: h o g == f
X0: merely X -> IsHProp P
g':= Trunc_ind (fun _ : Trunc (-1) X => P) g
:
merely X -> P: merely X -> P
ConditionallyConstant f
  exists (h o g'); intros x.
H: Funext
X, Y: Type
f: X -> Y
P: Type
Pc: X -> Contr P
g: X -> P
h: P -> Y
p: h o g == f
X0: merely X -> IsHProp P
g':= Trunc_ind (fun _ : Trunc (-1) X => P) g
:
merely X -> P: merely X -> P
x: X
h (g' (tr x)) = f x
  apply p.
Defined.

(** Any weakly constant function with collapsible domain is conditionally constant. *)
Definition cconst_wconst_collapsible {X Y : Type} (f : X -> Y)
           `{Collapsible X} `{WeaklyConstant _ _ f}
: ConditionallyConstant f.
X, Y: Type
f: X -> Y
H: Collapsible X
H0: WeaklyConstant f
ConditionallyConstant f
Proof.
X, Y: Type
f: X -> Y
H: Collapsible X
H0: WeaklyConstant f
ConditionallyConstant f
  exists (f o splitsupp_collapsible); intros x.
X, Y: Type
f: X -> Y
H: Collapsible X
H0: WeaklyConstant f
x: X
f (splitsupp_collapsible (tr x)) = f x
  unfold splitsupp_collapsible; simpl.
X, Y: Type
f: X -> Y
H: Collapsible X
H0: WeaklyConstant f
x: X
f (collapse x) = f x
  apply wconst.
Defined.

(** Any weakly constant function with hset codomain is conditionally constant. *)
Definition cconst_wconst_hset `{Funext} {X Y : Type} (f : X -> Y)
           `{Ys : X -> IsHSet Y} `{WeaklyConstant _ _ f}
: ConditionallyConstant f.
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
ConditionallyConstant f
Proof.
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
ConditionallyConstant f
  assert (Ys' : merely X -> IsHSet Y).
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
merely X -> IsHSet Y
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
ConditionallyConstant f
  {
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
merely X -> IsHSet Y apply Trunc_rec.
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
X -> IsHSet Y intros x; exact (Ys x). }
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
ConditionallyConstant f
  simple refine (cconst_factors_hprop f (image (-1) f) _ _ _).
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
IsHProp (image (Tr (-1)) f)
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
X -> image (Tr (-1)) f
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
image (Tr (-1)) f -> Y
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
?h o ?g == f
  -
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
IsHProp (image (Tr (-1)) f) apply hprop_allpath; intros [y1 p1] [y2 p2].
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
y1: Y
p1: Tr (-1) (hfiber f y1)
y2: Y
p2: Tr (-1) (hfiber f y2)
(y1; p1) = (y2; p2)
    apply path_sigma_hprop; simpl.
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
y1: Y
p1: Tr (-1) (hfiber f y1)
y2: Y
p2: Tr (-1) (hfiber f y2)
y1 = y2
    pose proof (Ys' (Trunc_functor (-1) pr1 p1)).
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
y1: Y
p1: Tr (-1) (hfiber f y1)
y2: Y
p2: Tr (-1) (hfiber f y2)
X0: IsHSet Y
y1 = y2
    strip_truncations.
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
y1, y2: Y
X0: IsHSet Y
p2: hfiber f y2
p1: hfiber f y1
y1 = y2
    destruct p1 as [x1 q1], p2 as [x2 q2].
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
y1, y2: Y
X0: IsHSet Y
x2: X
q2: f x2 = y2
x1: X
q1: f x1 = y1
y1 = y2
    exact (q1^ @ wconst x1 x2 @ q2).
  -
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
X -> image (Tr (-1)) f apply factor1.
  -
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
image (Tr (-1)) f -> Y apply factor2.
  -
H: Funext
X, Y: Type
f: X -> Y
Ys: X -> IsHSet Y
H0: WeaklyConstant f
Ys': merely X -> IsHSet Y
factor2 (image (Tr (-1)) f)
o factor1 (image (Tr (-1)) f) == f apply fact_factors.
Defined.

(** We can decompose this into an "induction principle" and its computation rule. *)
Definition merely_rec_hset `{Funext} {X Y : Type} (f : X -> Y)
           `{Ys : X -> IsHSet Y} `{WeaklyConstant _ _ f}
: merely X -> Y
  := (cconst_wconst_hset f).1.

Definition merely_rec_hset_beta `{Funext} {X Y : Type} (f : X -> Y)
           `{Ys : X -> IsHSet Y} `{WeaklyConstant _ _ f}
           (x : X)
: merely_rec_hset f (tr x) = f x
  := (cconst_wconst_hset f).2 x.

(** More generally, the type of weakly constant functions [X -> Y], when [Y] is a set, is equivalent to [merely X -> Y]. *)
Definition equiv_merely_rec_hset `{Funext} (X Y : Type)
           `{Ys : X -> IsHSet Y}
: { f : X -> Y & WeaklyConstant f } <~> (merely X -> Y).
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
{f : X -> Y & WeaklyConstant f} <~> (merely X -> Y)
Proof.
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
{f : X -> Y & WeaklyConstant f} <~> (merely X -> Y)
  assert (Ys' : merely X -> IsHSet Y).
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
merely X -> IsHSet Y
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
{f : X -> Y & WeaklyConstant f} <~> (merely X -> Y)
  {
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
merely X -> IsHSet Y apply Trunc_rec.
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
X -> IsHSet Y intros x; exact (Ys x). }
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
{f : X -> Y & WeaklyConstant f} <~> (merely X -> Y)
  simple refine (equiv_adjointify
            (fun fc => @merely_rec_hset _ _ _ fc.1 _ fc.2)
            (fun g => (g o tr ; _)) _ _); try exact _.
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
g: merely X -> Y
WeaklyConstant (g o tr)
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
(fun fc : {x : _ & WeaklyConstant x} =>
 merely_rec_hset fc.1)
o (fun g : merely X -> Y => (g o tr; ?proj2)) == idmap
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
(fun g : merely X -> Y => (g o tr; ?proj2))
o (fun fc : {x : _ & WeaklyConstant x} =>
   merely_rec_hset fc.1) == idmap
  -
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
g: merely X -> Y
WeaklyConstant (g o tr) intros x y; apply (ap g), path_ishprop.
  -
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
(fun fc : {x : _ & WeaklyConstant x} =>
 merely_rec_hset fc.1)
o (fun g : merely X -> Y =>
   (g o tr;
   (fun x y : X => ap g (path_ishprop (tr x) (tr y)))
   :
   WeaklyConstant (g o tr))) == idmap intros g; apply path_arrow; intros mx.
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
g: merely X -> Y
mx: merely X
merely_rec_hset
  (fun x : X => g (tr x);
  fun x y : X => ap g (path_ishprop (tr x) (tr y))).1
  mx = g mx
    pose proof (Ys' mx).
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
g: merely X -> Y
mx: merely X
X0: IsHSet Y
merely_rec_hset
  (fun x : X => g (tr x);
  fun x y : X => ap g (path_ishprop (tr x) (tr y))).1
  mx = g mx
    strip_truncations; reflexivity.
  -
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
(fun g : merely X -> Y =>
 (g o tr;
 (fun x y : X => ap g (path_ishprop (tr x) (tr y)))
 :
 WeaklyConstant (g o tr)))
o (fun fc : {x : _ & WeaklyConstant x} =>
   merely_rec_hset fc.1) == idmap intros [f ?].
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
f: X -> Y
proj2: WeaklyConstant f
(fun x : X => merely_rec_hset (f; proj2).1 (tr x);
fun x y : X =>
ap (merely_rec_hset (f; proj2).1)
  (path_ishprop (tr x) (tr y))) = (f; proj2)
    refine (path_sigma_hprop _ _ _).
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
f: X -> Y
proj2: WeaklyConstant f
forall x : X -> Y, IsHProp (WeaklyConstant x)
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
f: X -> Y
proj2: WeaklyConstant f
(fun x : X => merely_rec_hset (f; proj2).1 (tr x);
fun x y : X =>
ap (merely_rec_hset (f; proj2).1)
  (path_ishprop (tr x) (tr y))).1 = 
(f; proj2).1
    +
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
f: X -> Y
proj2: WeaklyConstant f
forall x : X -> Y, IsHProp (WeaklyConstant x) intros f'; apply hprop_allpath; intros w1 w2.
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
f: X -> Y
proj2: WeaklyConstant f
f': X -> Y
w1, w2: WeaklyConstant f'
w1 = w2
      apply path_forall; intros x; apply path_forall; intros y.
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
f: X -> Y
proj2: WeaklyConstant f
f': X -> Y
w1, w2: WeaklyConstant f'
x, y: X
w1 x y = w2 x y
      pose (Ys x); apply path_ishprop.
    +
H: Funext
X, Y: Type
Ys: X -> IsHSet Y
Ys': merely X -> IsHSet Y
f: X -> Y
proj2: WeaklyConstant f
(fun x : X => merely_rec_hset (f; proj2).1 (tr x);
fun x y : X =>
ap (merely_rec_hset (f; proj2).1)
  (path_ishprop (tr x) (tr y))).1 = 
(f; proj2).1 apply path_arrow; intros x; reflexivity.
Defined.