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Added `S1` has merely decidable equality

This commit is contained in:
Niels van der Weide 2017-09-25 13:03:51 +02:00
parent 617451da28
commit 7281bfc0bf
2 changed files with 50 additions and 1 deletions
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49
FiniteSets/prelude.v
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@ -57,6 +57,55 @@ Defined.
Class MerelyDecidablePaths A := Class MerelyDecidablePaths A :=
m_dec_path : forall (a b : A), Decidable(Trunc (-1) (a = b)). m_dec_path : forall (a b : A), Decidable(Trunc (-1) (a = b)).
Definition S1_merely_decidable_equality `{Univalence} : MerelyDecidablePaths S1.
Proof.
simple refine (S1_ind _ _ _) ; simpl.
- simple refine (S1_ind _ _ _) ; simpl.
* apply (inl (tr idpath)).
* apply path_ishprop.
- apply path_forall.
intro z.
rewrite transport_forall, transport_sum.
destruct
(transportD (fun _ : S1 => S1)
(fun x y : S1 => Decidable (Trunc (-1) (x = y)))
loop
(transport (fun _ : S1 => S1) loop^ z)
(S1_ind
(fun x : S1 => Decidable (Trunc (-1) (base = x)))
(inl (tr 1%path))
(transport_sum loop (inl (tr 1))
@
ap inl
(path_ishprop
(transport
(fun a : S1 => Trunc (-1) (base = a))
loop
(tr 1))
(tr 1)
)
)
(transport (fun _ : S1 => S1) loop^ z)
)
)
as [t | n].
** revert t.
revert z.
simple refine (S1_ind (fun _ => _) _ _) ; simpl.
*** intros.
apply path_ishprop.
*** apply path_forall.
intro z.
rewrite transport_forall, transport_paths_FlFr, ap_const.
hott_simpl.
apply set_path2.
** contradiction n.
rewrite ?transport_const.
simple refine (S1_ind (fun q => Trunc (-1) (base = q)) _ _ z) ; simpl.
*** apply (tr idpath).
*** apply path_ishprop.
Defined.
Global Instance DecidableToMerely A (H : DecidablePaths A) : MerelyDecidablePaths A. Global Instance DecidableToMerely A (H : DecidablePaths A) : MerelyDecidablePaths A.
Proof. Proof.
intros x y. intros x y.