# Library UniMath.MoreFoundations.NullHomotopies

## Null homotopies, an aid for proving things about propositional truncation

Require Export UniMath.Foundations.PartD.
Require Import UniMath.Foundations.UnivalenceAxiom.
Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.MoreFoundations.Notations.

Local Open Scope transport.

Definition nullHomotopyTo {X Y} (f:XY) (y:Y) := x:X, f x = y.
Definition NullHomotopyTo {X Y} (f:XY) := total2 (nullHomotopyTo f).
Definition NullHomotopyTo_center {X Y} (f:XY) : NullHomotopyTo f Y := pr1.
Definition NullHomotopyTo_path {X Y} {f:XY} (r:NullHomotopyTo f) := pr2 r.

Definition nullHomotopyFrom {X Y} (f:XY) (y:Y) := x:X, y = f x.
Definition NullHomotopyFrom {X Y} (f:XY) := total2 (nullHomotopyFrom f).
Definition NullHomotopyFrom_center {X Y} (f:XY) : NullHomotopyFrom f Y := pr1.
Definition NullHomotopyFrom_path {X Y} {f:XY} (r:NullHomotopyFrom f) := pr2 r.

Definition nullHomotopyTo_transport {X Y} {f:XY} {y:Y} (h : nullHomotopyTo f y)
{y':Y} (p:y = y') (x:X) : (p # h) x = h x @ p.
Proof.
intros. induction p. apply pathsinv0. apply pathscomp0rid.
Defined.

Lemma isaset_NullHomotopyTo {X Y} (i:isaset Y) (f:XY) : isaset (NullHomotopyTo f).
Proof.
intros. apply (isofhleveltotal2 2).
{ apply i. }
intros y. apply impred; intros x. apply isasetaprop. apply i.
Defined.

Lemma isaprop_nullHomotopyTo {X Y} (is:isaset Y) (f:XY) (y:Y) :
isaprop (nullHomotopyTo f y).
Proof.
apply impred; intros x. apply is.
Defined.

Lemma isaprop_NullHomotopyTo {X} {Y} (is:isaset Y) (f:XY) :
X isaprop (NullHomotopyTo f).
Proof.
apply factor_through_squash.
apply isapropisaprop.
intros x. apply invproofirrelevance. intros [r i] [s j].
apply subtypePairEquality.
- intros n. apply (isaprop_nullHomotopyTo is).
- exact (!i x @ j x).
Defined.

Definition cone_squash_map {X Y} (f:XY) (y:Y) :
nullHomotopyTo f y X Y.
Proof.
intros e h. exact (point_from (h (paths_to_prop y) (λ x, f x,,e x))).
Defined.

Goal X Y (y:Y) (f:XY) (e: m:X, f m = y),
f = cone_squash_map f y e squash_element.
Proof.
reflexivity.
Qed.