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Simplified no union for Bishops

This commit is contained in:
Niels van der Weide 2017-10-03 14:45:00 +02:00
parent c9e6b35949
commit fffdb87b4f
1 changed files with 22 additions and 24 deletions
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46
FiniteSets/subobjects/b_finite.v
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@ -64,9 +64,9 @@ Section finite_hott.
Lemma no_union `{IsHSet A}
(f : forall (X Y : Sub A),
Bfin X -> Bfin Y -> Bfin (X ∪ Y))
(a b : A) :
hor (a = b) (a = b -> Empty).
: MerelyDecidablePaths A.
Proof.
intros a b.
specialize (f {|a|} {|b|} (singleton a) (singleton b)).
unfold Bfin in f.
destruct f as [n pn].
@ -78,7 +78,7 @@ Section finite_hott.
exists a. apply (tr(inl(tr idpath))).
- destruct n as [|n].
+ (* If the size of the union is 1, then (a = b) *)
refine (tr (inl _)).
refine (inl (tr _)).
pose (s1 := (a;tr(inl(tr idpath)))
: {c : A & Trunc (-1) (Trunc (-1) (c = a) + Trunc (-1) (c = b))}).
pose (s2 := (b;tr(inr(tr idpath)))
@ -88,8 +88,8 @@ Section finite_hott.
{ by apply path_ishprop. }
refine (ap (fun x => (g x).1) fs_eq).
+ (* Otherwise, ¬(a = b) *)
refine (tr (inr _)).
intros p.
refine (inr (fun p => _)).
strip_truncations.
pose (s1 := inl (inr tt) : Fin n + Unit + Unit).
pose (s2 := inr tt : Fin n + Unit + Unit).
pose (gs1 := g s1).
@ -99,37 +99,35 @@ Section finite_hott.
assert (Hgs1 : gs1 = c) by reflexivity.
assert (Hgs2 : gs2 = d) by reflexivity.
destruct c as [x px'].
destruct d as [y py'].
simple refine (Trunc_ind _ _ px') ; intros px.
simple refine (Trunc_ind _ _ py') ; intros py.
simpl.
destruct d as [y py'].
simple refine (Trunc_ind _ _ px') ; intros px
; simple refine (Trunc_ind _ _ py') ; intros py ; simpl.
cut (x = y).
{
enough (s1 = s2) as X.
{
intros.
unfold s1, s2 in X.
refine (not_is_inl_and_inr' (inl(inr tt)) _ _).
+ apply tt.
+ rewrite X ; apply tt.
contradiction (inl_ne_inr _ _ X).
}
transitivity (f gs1).
{ apply (fg s1)^. }
symmetry ; transitivity (f gs2).
{ apply (fg s2)^. }
unfold gs1, gs2 in *.
refine ((fg s1)^ @ _ @ fg s2).
rewrite Hgs1, Hgs2.
f_ap.
simple refine (path_sigma _ _ _ _ _); [ | apply path_ishprop ]; simpl.
destruct px as [p1 | p1] ; destruct py as [p2 | p2] ; strip_truncations.
* apply (p2 @ p1^).
* refine (p2 @ _^ @ p1^). auto.
* refine (p2 @ _ @ p1^). auto.
* apply (p2 @ p1^).
destruct px as [px | px] ; destruct py as [py | py]
; refine (Trunc_rec _ px) ; clear px ; intro px
; refine (Trunc_rec _ py) ; clear py ; intro py.
* apply (px @ py^).
* refine (px @ _ @ py^). auto.
* refine (px @ _^ @ py^). auto.
* apply (px @ py^).
}
destruct px as [px | px] ; destruct py as [py | py]; strip_truncations.
destruct px as [px | px] ; destruct py as [py | py]
; refine (Trunc_rec _ px) ; clear px ; intro px
; refine (Trunc_rec _ py) ; clear py ; intro py.
** apply (px @ py^).
** refine (px @ _ @ py^). auto.
** refine (px @ _ @ py^). symmetry. auto.
** refine (px @ _^ @ py^). auto.
** apply (px @ py^).
Defined.
End finite_hott.