Library UniMath.MoreFoundations.Sets
Contents
- (More entries need to be added here...)
- Other universal properties for setquot
- The equivalence relation of being in the same fiber
- Subsets
Local Open Scope set.
Definition isconst {X:UU} {Y:hSet} (f : X → Y) : hProp := ∀ x x', f x = f x'.
Definition squash_to_hSet {X : UU} {Y : hSet} (f : X → Y) : isconst f → ∥ X ∥ → Y.
Proof.
apply squash_to_set, setproperty.
Defined.
Definition isconst_2 {X Y:UU} {Z:hSet} (f : X → Y → Z) : hProp :=
(∀ x x' y y', f x y = f x' y')%set.
Definition squash_to_hSet_2 {X Y : UU} {Z : hSet} (f : X → Y → Z) :
isconst_2 f → ∥ X ∥ → ∥ Y ∥ → Z.
Proof.
intros c. use squash_to_set.
{ apply isaset_forall_hSet. }
{ intros x. use squash_to_hSet. exact (f x). intros y y'. exact (c x x y y'). }
{ intros x x'. apply funextfun; intros yn.
apply (squash_to_prop yn).
{ apply setproperty. }
intros y. assert (e : hinhpr y = yn).
{ apply propproperty. }
induction e. change ( f x y = f x' y ). exact (c x x' y y). }
Defined.
Definition isconst_2' {X Y:UU} {Z:hSet} (f : X → Y → Z) : hProp :=
(
(∀ x x' y, f x y = f x' y)
∧
(∀ x y y', f x y = f x y')
)%set.
Definition squash_to_hSet_2' {X Y : UU} {Z : hSet} (f : X → Y → Z) :
isconst_2' f → ∥ X ∥ → ∥ Y ∥ → Z.
Proof.
intros [c d]. use squash_to_set.
{ apply isaset_forall_hSet. }
{ intros x. use squash_to_hSet. exact (f x). intros y y'. exact (d x y y'). }
{ intros x x'. apply funextfun; intros yn.
apply (squash_to_prop yn).
{ apply setproperty. }
intros y. assert (e : hinhpr y = yn).
{ apply propproperty. }
induction e. change ( f x y = f x' y ). exact (c x x' y). }
Defined.
Definition eqset_to_path {X:hSet} (x y:X) : eqset x y → x = y := λ e, e.
Lemma isapropiscomprelfun {X : UU} {Y : hSet} (R : hrel X) (f : X → Y) : isaprop (iscomprelfun R f).
Proof.
apply impred. intro x. apply impred. intro x'. apply impred. intro r. apply Y.
Defined.
Lemma iscomprelfun_funcomp {X Y Z : UU} {R : hrel X} {S : hrel Y}
{f : X → Y} {g : Y → Z} (Hf : iscomprelrelfun R S f) (Hg : iscomprelfun S g) :
iscomprelfun R (g ∘ f).
Proof.
intros x x' r. exact (Hg _ _ (Hf x x' r)).
Defined.
Other universal properties for setquot
Theorem setquotunivprop' {X : UU} {R : eqrel X} (P : setquot (pr1 R) → UU)
(H : ∏ x, isaprop (P x)) (ps : ∏ x : X, P (setquotpr R x)) : ∏ c : setquot (pr1 R), P c.
Proof.
exact (setquotunivprop R (λ x, hProppair (P x) (H x)) ps).
Defined.
Theorem setquotuniv2prop' {X : UU} {R : eqrel X} (P : setquot (pr1 R) → setquot (pr1 R) → UU)
(H : ∏ x1 x2, isaprop (P x1 x2))
(ps : ∏ x1 x2, P (setquotpr R x1) (setquotpr R x2)) : ∏ c1 c2 : setquot (pr1 R), P c1 c2.
Proof.
exact (setquotuniv2prop R (λ x1 x2, hProppair (P x1 x2) (H x1 x2)) ps).
Defined.
Theorem setquotuniv3prop' {X : UU} {R : eqrel X}
(P : setquot (pr1 R) → setquot (pr1 R) → setquot (pr1 R) → UU)
(H : ∏ x1 x2 x3, isaprop (P x1 x2 x3))
(ps : ∏ x1 x2 x3, P (setquotpr R x1) (setquotpr R x2) (setquotpr R x3)) :
∏ c1 c2 c3 : setquot (pr1 R), P c1 c2 c3.
Proof.
exact (setquotuniv3prop R (λ x1 x2 x3, hProppair (P x1 x2 x3) (H x1 x2 x3)) ps).
Defined.
Theorem setquotuniv4prop' {X : UU} {R : eqrel X}
(P : setquot (pr1 R) → setquot (pr1 R) → setquot (pr1 R) → setquot (pr1 R) → UU)
(H : ∏ x1 x2 x3 x4, isaprop (P x1 x2 x3 x4))
(ps : ∏ x1 x2 x3 x4,
P (setquotpr R x1) (setquotpr R x2) (setquotpr R x3) (setquotpr R x4)) :
∏ c1 c2 c3 c4 : setquot (pr1 R), P c1 c2 c3 c4.
Proof.
exact (setquotuniv4prop R (λ x1 x2 x3 x4, hProppair (P x1 x2 x3 x4) (H x1 x2 x3 x4)) ps).
Defined.
Definition setcoprod (X Y : hSet) : hSet :=
hSetpair (X ⨿ Y) (isasetcoprod X Y (pr2 X) (pr2 Y)).