mirror of https://github.com/nmvdw/HITs-Examples
64 lines
1.7 KiB
Coq
64 lines
1.7 KiB
Coq
(** Some general prerequisities in homotopy type theory. *)
|
|
Require Import HoTT.
|
|
|
|
Lemma ap_inl_path_sum_inl {A B} (x y : A) (p : inl x = inl y) :
|
|
ap inl (path_sum_inl B p) = p.
|
|
Proof.
|
|
transitivity (@path_sum _ B (inl x) (inl y) (path_sum_inl B p));
|
|
[ | apply (eisretr_path_sum _) ].
|
|
destruct (path_sum_inl B p).
|
|
reflexivity.
|
|
Defined.
|
|
|
|
Lemma ap_equiv {A B} (f : A <~> B) {x y : A} (p : x = y) :
|
|
ap (f^-1 o f) p = eissect f x @ p @ (eissect f y)^.
|
|
Proof.
|
|
destruct p.
|
|
hott_simpl.
|
|
Defined.
|
|
|
|
Global Instance hprop_lem `{Univalence} (T : Type) (Ttrunc : IsHProp T) : IsHProp (T + ~T).
|
|
Proof.
|
|
apply (equiv_hprop_allpath _)^-1.
|
|
intros [x | nx] [y | ny] ; try f_ap ; try (apply Ttrunc) ; try contradiction.
|
|
- apply equiv_hprop_allpath. apply _.
|
|
Defined.
|
|
|
|
Class MerelyDecidablePaths A :=
|
|
m_dec_path : forall (a b : A), Decidable(Trunc (-1) (a = b)).
|
|
|
|
Global Instance DecidableToMerely A (H : DecidablePaths A) : MerelyDecidablePaths A.
|
|
Proof.
|
|
intros x y.
|
|
destruct (dec (x = y)).
|
|
- apply (inl(tr p)).
|
|
- refine (inr(fun p => _)).
|
|
strip_truncations.
|
|
apply (n p).
|
|
Defined.
|
|
|
|
Global Instance merely_decidable_paths_prod (A B : Type)
|
|
`{MerelyDecidablePaths A} `{MerelyDecidablePaths B}
|
|
: MerelyDecidablePaths(A * B).
|
|
Proof.
|
|
intros x y.
|
|
destruct (m_dec_path (fst x) (fst y)) as [p1 | n1],
|
|
(m_dec_path (snd x) (snd y)) as [p2 | n2].
|
|
- apply inl.
|
|
strip_truncations.
|
|
apply tr.
|
|
apply path_prod ; assumption.
|
|
- apply inr.
|
|
intros pp.
|
|
strip_truncations.
|
|
apply (n2 (tr (ap snd pp))).
|
|
- apply inr.
|
|
intros pp.
|
|
strip_truncations.
|
|
apply (n1 (tr (ap fst pp))).
|
|
- apply inr.
|
|
intros pp.
|
|
strip_truncations.
|
|
apply (n1 (tr (ap fst pp))).
|
|
Defined.
|