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ReflectiveSubuniverse: add a couple of things and simplify one proof #1822

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Jan 21, 2024
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ReflectiveSubuniverse: add a couple of things and simplify one proof #1822

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e67b773
ReflectiveSubuniverse: add a couple of things and simplify one proof
jdchristensen Jan 20, 2024
ce70eab
ReflectiveSubuniverse: cleaner fix for typeclass oddness
jdchristensen Jan 21, 2024
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27 changes: 19 additions & 8 deletions theories/Modalities/ReflectiveSubuniverse.v
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Original file line number Diff line number Diff line change
Expand Up @@ -298,7 +298,7 @@ Section Reflective_Subuniverse.
Defined.

(** If [T] is in the subuniverse, then [to O T] is an equivalence. *)
Definition isequiv_to_O_inO (T : Type) `{In O T} : IsEquiv (to O T).
Global Instance isequiv_to_O_inO (T : Type) `{In O T} : IsEquiv (to O T).
Proof.
pose (g := O_rec idmap : O T -> T).
refine (isequiv_adjointify (to O T) g _ _).
Expand All @@ -309,7 +309,6 @@ Section Reflective_Subuniverse.
- intros x.
apply O_rec_beta.
Defined.
Global Existing Instance isequiv_to_O_inO.

Definition equiv_to_O (T : Type) `{In O T} : T <~> O T
:= Build_Equiv T (O T) (to O T) _.
Expand Down Expand Up @@ -509,6 +508,11 @@ Section Reflective_Subuniverse.
- symmetry. exact (O_rec_beta (g o f) x).
Defined.

(** In particular, we have: *)
Definition O_rec_postcompose_to_O {A B : Type} (f : A -> B) `{In O B}
: to O B o O_rec f == O_functor f
:= O_rec_postcompose f (to O B).

End Functor.

Section Replete.
Expand Down Expand Up @@ -575,19 +579,26 @@ Section Reflective_Subuniverse.
{A B : Type} `{In O B} (f : A -> B) `{O_inverts f}
: IsEquiv (O_rec (O := O) f).
Proof.
apply isequiv_O_inverts.
nrefine (cancelR_isequiv (O_functor (to O A))); [ exact _ | ].
nrefine (isequiv_homotopic (O_functor (O_rec f o to O A))
(O_functor_compose _ _)).
refine (isequiv_homotopic (O_functor f)
(O_functor_homotopy _ _ (fun x => (O_rec_beta f x)^))).
(* Not sure why we need [C:=O B] on the next line to get Coq to use two typeclass instances. *)
rapply (cancelL_isequiv (C:=O B) (to O B)).
rapply (isequiv_homotopic (O_functor f) (fun x => (O_rec_postcompose_to_O f x)^)).
Defined.

Definition equiv_O_rec_O_inverts
{A B : Type} `{In O B} (f : A -> B) `{O_inverts f}
: O A <~> B
:= Build_Equiv _ _ _ (isequiv_O_rec_O_inverts f).

Definition isequiv_to_O_O_inverts
{A B : Type} `{In O A} (f : A -> B) `{O_inverts f}
: IsEquiv (to O B o f)
:= isequiv_homotopic (O_functor f o to O A) (to_O_natural f).

Definition equiv_to_O_O_inverts
{A B : Type} `{In O A} (f : A -> B) `{O_inverts f}
: A <~> O B
:= Build_Equiv _ _ _ (isequiv_to_O_O_inverts f).

(** If [f] is inverted by [O], then mapping out of it into any modal type is an equivalence. First we prove a version not requiring funext. For use in [O_inverts_O_leq] below, we allow the types [A], [B], and [Z] to perhaps live in smaller universes than the one [i] on which our subuniverse lives. This the first half of Lemma 1.23 of RSS. *)
Definition ooextendable_O_inverts@{a b z i}
{A : Type@{a}} {B : Type@{b}} (f : A -> B) `{O_inverts f}
Expand Down
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