Library HoTT.Constant

(* -*- mode: coq; mode: visual-line -*-  *)
Require Import HoTT.Basics HoTT.Types.
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 ∀ 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).
Proof.
  apply hprop_inhabited_contr; intros [x0 p0].
  refine (contr_equiv' {x:X & f x0 = x} _); unfold FixedBy.
  apply equiv_functor_sigma_id. intros x.
  apply equiv_concat_l.
  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.
Proof.
  apply equiv_iff_hprop.
  - intros [x p]; exact (tr x).
  - apply Trunc_rec; intros 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.
Proof.
  refine (_ o (equiv_fix_merely collapse)^-1).
  apply pr1.
Defined.

Definition collapsible_splitsupp {X} (s : merely X → X)
: Collapsible X.
Proof.
  refine (Build_Collapsible _ (s o tr) _); intros x 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).
Proof.
  ∃ (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).
Proof.
  refine (_ oE (equiv_sigma_symm _)).
  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.
Proof.
  pose (g' := Trunc_rec g : merely X → P).
  ∃ (h o g'); intros 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.
Proof.
  assert (merely X → IsHProp P).
  { apply Trunc_rec.
Uses funext
    intros x; pose (Pc x); apply istrunc_succ. }
  pose (g' := Trunc_ind (fun _ ⇒ P) g : merely X → P).
  ∃ (h o g'); intros 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.
Proof.
  ∃ (f o splitsupp_collapsible); intros x.
  unfold splitsupp_collapsible; simpl.
  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.
Proof.
  assert (Ys' : merely X → IsHSet Y).
  { apply Trunc_rec. intros x; exact (Ys x). }
  simple refine (cconst_factors_hprop f (image (-1) f) _ _ _).
  - apply hprop_allpath; intros [y1 p1] [y2 p2].
    apply path_sigma_hprop; simpl.
    pose proof (Ys' (Trunc_functor (-1) pr1 p1)).
    strip_truncations.
    destruct p1 as [x1 q1], p2 as [x2 q2].
    exact (q1^ @ wconst x1 x2 @ q2).
  - apply factor1.
  - apply factor2.
  - 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).
Proof.
  assert (Ys' : merely X → IsHSet Y).
  { apply Trunc_rec. intros x; exact (Ys x). }
  simple refine (equiv_adjointify
            (fun fc ⇒ @merely_rec_hset _ _ _ fc.1 _ fc.2)
            (fun g ⇒ (g o tr ; _)) _ _); try exact _.
  - intros x y; apply (ap g), path_ishprop.
  - intros g; apply path_arrow; intros mx.
    pose proof (Ys' mx).
    strip_truncations; reflexivity.
  - intros [f ?].
    refine (path_sigma_hprop _ _ _).
    + intros f'; apply hprop_allpath; intros w1 w2.
      apply path_forall; intros x; apply path_forall; intros y.
      pose (Ys x); apply path_ishprop.
    + apply path_arrow; intros x; reflexivity.
Defined.