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Adding compatibility with HoTT for 8.19 #600

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16 changes: 8 additions & 8 deletions theories/HoTT/EqDec.v
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Original file line number Diff line number Diff line change
Expand Up @@ -12,6 +12,8 @@ Require Import Equations.Init.
Require Import Equations.HoTT.Logic.
Require Import Equations.HoTT.Classes.

From HoTT Require Import Basics.Trunc.

(** Decidable equality.

We redevelop the derivation of [K] from decidable equality on [A] making
Expand Down Expand Up @@ -201,8 +203,9 @@ Qed.
#[export]
Instance eqdec_hset (A : Type) `(UIP A) : IsHSet A.
Proof.
red. red. intros *. exists (uip x0 y0). intros e.
destruct x0. apply uip.
apply equiv_istrunc_unfold ; cbn ; intros ??.
apply hprop_allpath.
now apply H.
Defined.

Lemma sigma_eq@{i} (A : Type@{i}) (P : A -> Type@{i}) (x y : sigma P) :
Expand All @@ -215,7 +218,7 @@ Defined.

Lemma is_hset {A} `{H : IsHSet A} {x y : A} (p q : x = y) : p = q.
Proof.
apply H.
apply path_ishprop.
Defined.

Theorem inj_sigma_r@{i} {A : Type@{i}} `{H : IsHSet A} {P : A -> Type@{i}} {x} {y y':P x} :
Expand All @@ -230,8 +233,6 @@ Definition apd_eq {A} {x y : A} (p : x = y) {z} (q : z = x) :
transport (@paths A z) p q = q @ p.
Proof. now destruct p, q. Defined.

Require Import HoTT.Basics.Trunc.

Lemma hprop_hset {A} (h : IsHProp A) : IsHSet A.
Proof.
apply istrunc_hprop.
Expand All @@ -240,9 +241,8 @@ Defined.
(** Proof that equality proofs in 0-truncated types are connected *)
Lemma hset_pi {A} `{H : IsHSet A} (x y : A) (p q : x = y) (r : p = q) : is_hset p q = r.
Proof.
red in H.
pose (hprop_hset (H x y)).
apply i.
enough (Contr (p = q)) by now apply path_ishprop.
apply (H _ _ _).
Defined.

Lemma is_hset_refl {A} `{H : IsHSet A} (x : A) : is_hset (@idpath _ x) 1 = 1%path.
Expand Down