# Library UniMath.MoreFoundations.Univalence

Funextsec and toforallpaths are mutually inverses
Lemma funextsec_toforallpaths {T : UU} {P : T UU} {f g : t : T, P t} :
(h : f = g), funextsec _ _ _ (toforallpaths _ _ _ h) = h.
Proof.
intro h; exact (!homotinvweqweq0 (weqtoforallpaths _ _ _) h).
Defined.

Lemma toforallpaths_funextsec {T : UU} {P : T UU} {f g : t : T, P t} :
(h : t : T, f t = g t), toforallpaths _ _ _ (funextsec _ _ _ h) = h.
Proof.
intro h; exact (homotweqinvweq (weqtoforallpaths _ _ _) h).
Defined.

Definition toforallpaths_funextsec_comp {T : UU} {P : T UU} (f g : t, P t) :
toforallpaths P f g funextsec P f g = idfun _.
Proof.
apply funextsec; intro; unfold funcomp.
apply toforallpaths_funextsec.
Defined.

Lemma maponpaths_funextsec {T : UU} {P : T UU}
(f g : t, P t) (t : T) (p : f ¬ g) :
maponpaths (λ h, h t) (funextsec _ f g p) = p t.
Proof.
intermediate_path (toforallpaths _ _ _ (funextsec _ f g p) t).
- generalize (funextsec _ f g p); intros q.
induction q.
reflexivity.
- apply (eqtohomot (eqtohomot (toforallpaths_funextsec_comp f g) p) t).
Qed.

Definition weqonsec {X Y} (P:XType) (Q:YType)
(f:X Y) (g: x, weq (P x) (Q (f x))) :
( x:X, P x) ( y:Y, Q y).
Proof.
intros.
exact (weqcomp (weqonsecfibers P (λ x, Q(f x)) g)
(invweq (weqonsecbase Q f))).
Defined.

Definition weq_transportf {X} (P:XType) {x y:X} (p:x = y) : (P x) (P y).
Proof.
intros. induction p. apply idweq.
Defined.

Definition weq_transportf_comp {X} (P:XType) {x y:X} (p:x = y) (f: x, P x) :
weq_transportf P p (f x) = f y.
Proof.
intros. induction p. reflexivity.
Defined.

Definition weqonpaths2 {X Y} (w:X Y) {x x':X} {y y':Y} :
w x = y w x' = y' (x = x') (y = y').
Proof.
intros p q. induction p,q. apply weqonpaths.
Defined.

Definition eqweqmap_ap {T} (P:TType) {t t':T} (e:t = t') (f: t:T, P t) :
eqweqmap (maponpaths P e) (f t) = f t'. Proof.
intros. induction e. reflexivity.
Defined.

Definition eqweqmap_ap' {T} (P:TType) {t t':T} (e:t = t') (f: t:T, P t) :
invmap (eqweqmap (maponpaths P e)) (f t') = f t. Proof.
intros. induction e. reflexivity.
Defined.

weak equivalences

Definition pr1_eqweqmap { X Y } ( e: X = Y ) : cast e = pr1 (eqweqmap e).
Proof.
intros. induction e. reflexivity.
Defined.

Definition path_to_fun {X Y} : X=Y XY.
Proof.
intros p. induction p. exact (idfun _).
Defined.

Definition pr1_eqweqmap2 { X Y } ( e: X = Y ) :
pr1 (eqweqmap e) = transportf (λ T:Type, T) e.
Proof.
intros. induction e. reflexivity.
Defined.

Definition weqpath_transport {X Y} (w : X Y) :
transportf (idfun UU) (weqtopaths w) = pr1 w.
Proof.
intros. exact (!pr1_eqweqmap2 _ @ maponpaths pr1 (weqpathsweq w)).
Defined.

Definition weqpath_cast {X Y} (w : X Y) : cast (weqtopaths w) = w.
Proof.
intros. exact (pr1_eqweqmap _ @ maponpaths pr1 (weqpathsweq w)).
Defined.

Definition switch_weq {X Y} (f:X Y) {x y} : y = f x invweq f y = x.
Proof.
intros p. exact (maponpaths (invweq f) p @ homotinvweqweq f x).
Defined.

Definition switch_weq' {X Y} (f:X Y) {x y} : invweq f y = x y = f x.
Proof.
intros p. exact (! homotweqinvweq f y @ maponpaths f p).
Defined.

Local Open Scope transport.

Definition weq_over_sections {S T} (w:S T)
{s0:S} {t0:T} (k:w s0 = t0)
{P:TType}
(p0:P t0) (pw0:P(w s0)) (l:k#pw0 = p0)
(H:( t, P t) UU)
(J:( s, P(w s)) UU)
(g: f:( t, P t), weq (H f) (J (maponsec1 P w f))) :
weq (hfiber (λ fh:total2 H, pr1 fh t0) p0 )
(hfiber (λ fh:total2 J, pr1 fh s0) pw0).
Proof.
intros. simple refine (weqbandf _ _ _ _).
{ simple refine (weqbandf _ _ _ _).
{ exact (weqonsecbase P w). }
{ unfold weqonsecbase; simpl. exact g. } }
{ intros [f h]. simpl. unfold maponsec1; simpl.
induction k, l; simpl. unfold transportf; simpl.
unfold idfun; simpl. apply idweq. }
Defined.