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CayleyDickson: some cleanups, remove funext #1801

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CayleyDickson: some cleanups, remove funext
Signed-off-by: Ali Caglayan <[email protected]>
Alizter committed Jan 8, 2024

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commit b68c3ab18a0247d1815869a3b27c8789ddc7ae48
58 changes: 28 additions & 30 deletions theories/Homotopy/CayleyDickson.v
Original file line number Diff line number Diff line change
@@ -185,7 +185,7 @@ Defined.
Section ImaginaroidHSpace.

(* Let A be a Cayley-Dickson imaginaroid with associative H-space multiplication on Susp A *)
Context `{Funext} {A} `(CayleyDicksonImaginaroid A)
Context {A} `(CayleyDicksonImaginaroid A)
`(!Associative hspace_op).

(** Declaring these as local instances so that they can be found *)
@@ -249,26 +249,29 @@ Section ImaginaroidHSpace.
: SgOp (pjoin (psusp A) (psusp A)).
Proof.
unfold psusp, pjoin; cbn.
intros x y; revert x.
srapply Join_rec; hnf.
{ intro a.
revert y.
srapply Join_rec; hnf.
- intro c.
exact (joinl (hspace_op a c)).
exact (joinl (a * c)).
- intro d.
exact (joinr (hspace_op (conj a) d)).
exact (joinr (conj a * d)).
- intros x y.
apply jglue. }
{ intro b.
revert y.
srapply Join_rec; hnf.
- intro c.
exact (joinr (hspace_op c b)).
exact (joinr (c * b)).
- intro d.
exact (joinl (hspace_op (-d) (conj b))).
exact (joinl ((-d) * conj b)).
- intros x y.
symmetry.
apply jglue. }
intros a b.
apply path_forall.
revert y.
srapply Join_ind_dp.
1: intro; apply jglue.
1: intro; cbn; symmetry; apply jglue.
@@ -301,37 +304,32 @@ Section ImaginaroidHSpace.
Global Instance cd_op_left_identity
: LeftIdentity cd_op pt.
Proof.
srapply Join_ind_dp.
{ unfold cd_op, Join_rec, Pushout.Pushout_rec, Pushout.Pushout_ind; simpl;
intro a; apply ap.
srapply hspace_left_identity. }
{ unfold cd_op, Join_rec, Pushout.Pushout_rec, Pushout.Pushout_ind.
simpl; intro b; apply ap.
srapply hspace_left_identity. }
snrapply Join_ind_FFlr.
1,2: exact (fun _ => ap _ (hspace_left_identity _)).
intros a b.
simpl. apply sq_dp^-1.
rewrite ap_idmap, Join_rec_beta_jglue.
apply join_natsq_v.
refine (whiskerR _ _ @ _).
{ refine (ap _ (ap_idmap _) @ _).
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exact (Join_rec_beta_jglue
(fun c => joinl (pt * c))
(fun d => joinr (conj pt * d))
(fun x y => jglue (pt * x) (conj pt * y))
a b). }
symmetry.
apply join_natsq.
Defined.

Global Instance cd_op_right_identity
: RightIdentity cd_op pt.
Proof.
srapply Join_ind_dp.
{ unfold cd_op, Join_rec, Pushout.Pushout_rec, Pushout.Pushout_ind; simpl.
intro a; apply ap.
srapply hspace_right_identity. }
{ unfold cd_op, Join_rec, Pushout.Pushout_rec, Pushout.Pushout_ind; simpl.
intro b; apply ap.
srapply hspace_left_identity. }
snrapply Join_ind_FFlr.
1: exact (fun _ => ap joinl (hspace_right_identity _)).
1: exact (fun _ => ap joinr (hspace_left_identity _)).
intros a b.
apply sq_dp^-1.
rewrite ap_idmap.
rewrite (ap_compose _ (fun f => f _)).
rewrite Join_rec_beta_jglue.
rewrite ap_apply_l.
rewrite eisretr.
apply join_natsq_v.
refine (whiskerR _ _ @ _).
{ refine (ap _ (ap_idmap _) @ _).
simpl; rapply Join_rec_beta_jglue. }
symmetry.
apply join_natsq.
Defined.

Global Instance hspace_cdi_susp_assoc