Timings for SpanPushout.v
Require Import HoTT.Basics HoTT.Colimits.Pushout.
(** * Pushouts of "dependent spans". *)
Context {X Y : Type} (Q : X -> Y -> Type).
Definition SPushout := @Pushout@{up _ _ up} (sig@{up _} (fun (xy : X * Y) => Q (fst xy) (snd xy))) X Y
(fst o pr1) (snd o pr1).
Definition spushl : X -> SPushout := pushl.
Definition spushr : Y -> SPushout := pushr.
Definition spglue {x:X} {y:Y} : Q x y -> spushl x = spushr y
:= fun q => pglue ((x,y) ; q).
Definition SPushout_rec (R : Type)
(spushl' : X -> R) (spushr' : Y -> R)
(sglue' : forall x y (q : Q x y), spushl' x = spushr' y)
: SPushout -> R.
srapply (@Pushout_rec {xy:X * Y & Q (fst xy) (snd xy)} X Y
(fst o pr1) (snd o pr1) R spushl' spushr').
intros [[x y] q]; cbn in *.
apply sglue'; assumption.
Definition SPushout_ind (R : SPushout -> Type)
(spushl' : forall x, R (spushl x))
(spushr' : forall y, R (spushr y))
(sglue' : forall x y (q : Q x y),
transport R (spglue q) (spushl' x) = (spushr' y))
: forall p, R p.
srapply (@Pushout_ind {xy:X * Y & Q (fst xy) (snd xy)} X Y
(fst o pr1) (snd o pr1) R spushl' spushr').
intros [[x y] q]; cbn in *.
apply sglue'; assumption.
Definition spushout_ind_beta_sglue (R : SPushout -> Type)
(spushl' : forall x, R (spushl x))
(spushr' : forall y, R (spushr y))
(spglue' : forall x y (q : Q x y),
transport R (spglue q) (spushl' x) = (spushr' y))
(x:X) (y:Y) (q:Q x y)
: apD (SPushout_ind R spushl' spushr' spglue') (spglue q) = spglue' x y q
:= Pushout_ind_beta_pglue _ _ _ _ ((x,y);q).