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Strengthened mere choice

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Niels
2017-08-14 12:43:15 +02:00
parent bf0b9f8771
commit 0f6e98a377
2 changed files with 31 additions and 15 deletions
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43
FiniteSets/fsets/properties.v
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@@ -177,30 +177,47 @@ Section properties.
toHProp.
Defined.
Lemma merely_choice : forall X : FSet A, hor (X = ∅) (hexists (fun a => a ∈ X)).
Local Ltac solve_mc :=
repeat (let x := fresh in intro x ; try(destruct x))
; simpl
; rewrite transport_sum
; try (f_ap ; apply path_ishprop)
; try (f_ap ; apply set_path2).
Lemma merely_choice : forall X : FSet A, (X = ∅) + (hexists (fun a => a ∈ X)).
Proof.
hinduction; try (intros; apply equiv_hprop_allpath ; apply _).
- apply (tr (inl idpath)).
hinduction.
- apply (inl idpath).
- intro a.
refine (tr (inr (tr (a ; tr idpath)))).
refine (inr (tr (a ; tr idpath))).
- intros X Y TX TY.
strip_truncations.
destruct TX as [XE | HX] ; destruct TY as [YE | HY] ; try(strip_truncations ; apply tr).
* refine (tr (inl _)).
destruct TX as [XE | HX] ; destruct TY as [YE | HY].
* refine (inl _).
rewrite XE, YE.
apply (union_idem ∅).
* destruct HY as [a Ya].
refine (inr (tr _)).
* right.
strip_truncations.
destruct HY as [a Ya].
refine (tr _).
exists a.
apply (tr (inr Ya)).
* destruct HX as [a Xa].
refine (inr (tr _)).
* right.
strip_truncations.
destruct HX as [a Xa].
refine (tr _).
exists a.
apply (tr (inl Xa)).
* destruct (HX, HY) as [[a Xa] [b Yb]].
refine (inr (tr _)).
* right.
strip_truncations.
destruct (HX, HY) as [[a Xa] [b Yb]].
refine (tr _).
exists a.
apply (tr (inl Xa)).
- solve_mc.
- solve_mc.
- solve_mc.
- solve_mc.
- solve_mc.
Defined.
Lemma separation_isIn (B : Type) : forall (X : FSet A) (f : {a | a ∈ X} -> B) (b : B),