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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} Lemma no_union `{IsHSet A}
(f : forall (X Y : Sub A), (f : forall (X Y : Sub A),
Bfin X -> Bfin Y -> Bfin (X ∪ Y)) Bfin X -> Bfin Y -> Bfin (X ∪ Y))
(a b : A) : : MerelyDecidablePaths A.
hor (a = b) (a = b -> Empty).
Proof. Proof.
intros a b.
specialize (f {|a|} {|b|} (singleton a) (singleton b)). specialize (f {|a|} {|b|} (singleton a) (singleton b)).
unfold Bfin in f. unfold Bfin in f.
destruct f as [n pn]. destruct f as [n pn].
@ -78,7 +78,7 @@ Section finite_hott.
exists a. apply (tr(inl(tr idpath))). exists a. apply (tr(inl(tr idpath))).
- destruct n as [|n]. - destruct n as [|n].
+ (* If the size of the union is 1, then (a = b) *) + (* If the size of the union is 1, then (a = b) *)
refine (tr (inl _)). refine (inl (tr _)).
pose (s1 := (a;tr(inl(tr idpath))) pose (s1 := (a;tr(inl(tr idpath)))
: {c : A & Trunc (-1) (Trunc (-1) (c = a) + Trunc (-1) (c = b))}). : {c : A & Trunc (-1) (Trunc (-1) (c = a) + Trunc (-1) (c = b))}).
pose (s2 := (b;tr(inr(tr idpath))) pose (s2 := (b;tr(inr(tr idpath)))
@ -88,8 +88,8 @@ Section finite_hott.
{ by apply path_ishprop. } { by apply path_ishprop. }
refine (ap (fun x => (g x).1) fs_eq). refine (ap (fun x => (g x).1) fs_eq).
+ (* Otherwise, ¬(a = b) *) + (* Otherwise, ¬(a = b) *)
refine (tr (inr _)). refine (inr (fun p => _)).
intros p. strip_truncations.
pose (s1 := inl (inr tt) : Fin n + Unit + Unit). pose (s1 := inl (inr tt) : Fin n + Unit + Unit).
pose (s2 := inr tt : Fin n + Unit + Unit). pose (s2 := inr tt : Fin n + Unit + Unit).
pose (gs1 := g s1). pose (gs1 := g s1).
@ -99,37 +99,35 @@ Section finite_hott.
assert (Hgs1 : gs1 = c) by reflexivity. assert (Hgs1 : gs1 = c) by reflexivity.
assert (Hgs2 : gs2 = d) by reflexivity. assert (Hgs2 : gs2 = d) by reflexivity.
destruct c as [x px']. destruct c as [x px'].
destruct d as [y py']. destruct d as [y py'].
simple refine (Trunc_ind _ _ px') ; intros px. simple refine (Trunc_ind _ _ px') ; intros px
simple refine (Trunc_ind _ _ py') ; intros py. ; simple refine (Trunc_ind _ _ py') ; intros py ; simpl.
simpl.
cut (x = y). cut (x = y).
{ {
enough (s1 = s2) as X. enough (s1 = s2) as X.
{ {
intros.
unfold s1, s2 in X. unfold s1, s2 in X.
refine (not_is_inl_and_inr' (inl(inr tt)) _ _). contradiction (inl_ne_inr _ _ X).
+ apply tt.
+ rewrite X ; apply tt.
} }
transitivity (f gs1). unfold gs1, gs2 in *.
{ apply (fg s1)^. } refine ((fg s1)^ @ _ @ fg s2).
symmetry ; transitivity (f gs2).
{ apply (fg s2)^. }
rewrite Hgs1, Hgs2. rewrite Hgs1, Hgs2.
f_ap. f_ap.
simple refine (path_sigma _ _ _ _ _); [ | apply path_ishprop ]; simpl. simple refine (path_sigma _ _ _ _ _); [ | apply path_ishprop ]; simpl.
destruct px as [p1 | p1] ; destruct py as [p2 | p2] ; strip_truncations. destruct px as [px | px] ; destruct py as [py | py]
* apply (p2 @ p1^). ; refine (Trunc_rec _ px) ; clear px ; intro px
* refine (p2 @ _^ @ p1^). auto. ; refine (Trunc_rec _ py) ; clear py ; intro py.
* refine (p2 @ _ @ p1^). auto. * apply (px @ py^).
* apply (p2 @ p1^). * 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^). ** apply (px @ py^).
** refine (px @ _ @ py^). auto. ** refine (px @ _ @ py^). auto.
** refine (px @ _ @ py^). symmetry. auto. ** refine (px @ _^ @ py^). auto.
** apply (px @ py^). ** apply (px @ py^).
Defined. Defined.
End finite_hott. End finite_hott.