Library UniMath.MoreFoundations.QuotientSet

Require Import
        UniMath.Foundations.Sets
        UniMath.MoreFoundations.PartA.

Definition iscomprelfun2' {X Y Z} (RX:hrel X) (RY:hrel Y)
           (f:X ->Y ->Z) : Type
  := (∏ x x', RX x x' -> ∏ y , f x y = f x' y ) ×
     (∏ y y', RY y y' -> ∏ x , f x y = f x y').

Definition iscomprelrelfun2' {X Y Z} (RX:hrel X) (RY:hrel Y) (RZ:eqrel Z)
           (f:X ->Y ->Z) : Type
  := (∏ x x' y , RX x x' -> RZ (f x y) (f x' y)) ×
     (∏ x y y', RY y y' -> RZ (f x y) (f x y')).

Lemma setquotuniv_equal { X : UU } ( R : hrel X ) ( Y : hSet )
      ( f f' : X -> Y ) (p : f = f')
      ( is : iscomprelfun R f ) ( is' : iscomprelfun R f' )
: setquotuniv R Y f is = setquotuniv R Y f' is'.
Proof.
  intros. destruct p. apply funextsec; intro c.
  assert(ip : isaprop (iscomprelfun R f)).
  { apply impred; intro x; apply impred; intro x'.
    apply impred; intro p. apply setproperty. }
  assert( q : is = is' ).
  { apply ip. }
  destruct q. reflexivity.

Definition setquotuniv2' {X Y} (RX:hrel X) (RY:hrel Y)
{Z:hSet} (f:X ->Y ->Z) (is:iscomprelfun2' RX RY f) :
setquot RX -> setquot RY -> Z.
Show proof.

  intros x''.
  simple refine (setquotuniv RX (funset (setquot RY) Z) _ _ _).
  { simpl. intro x. apply (setquotuniv RY Z (f x)).
    intros y y' e. unfold iscomprelfun2 in is.
    apply (pr2 is). assumption. }
  { intros x x' e.
    assert( p : f x = f x' ).
    { apply funextsec; intro y. apply (pr1 is). assumption. }
    apply setquotuniv_equal. assumption. } assumption.

Definition setquotfun2' {X Y Z} {RX:hrel X} {RY:hrel Y} {RZ:eqrel Z}
(f:X ->Y ->Z) (is:iscomprelrelfun2' RX RY RZ f) :
setquot RX -> setquot RY -> setquot RZ.
Show proof.

  set (f' := λ x y , setquotpr RZ (f x y) : setquotinset RZ).
  apply (setquotuniv2' RX RY f'). split.
  { intros ? ? p ?. apply iscompsetquotpr. exact (pr1 is x x' y p). }
  { intros ? ? p ?. apply iscompsetquotpr. exact (pr2 is x y y' p). }

Lemma setquotfun2_equal {X Y Z} (RX:eqrel X) (RY:eqrel Y) (RZ:eqrel Z)
           (f:X ->Y ->Z) (is:iscomprelrelfun2' RX RY RZ f)
           (x:X) (y:Y) :
  setquotfun2' f is (setquotpr RX x) (setquotpr RY y) =
  setquotpr RZ (f x y).
Show proof.

reflexivity.