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Decidable emptiness improved

This commit is contained in:
Niels 2017-09-07 17:22:14 +02:00
parent 7a636c7035
commit f8234375c8
1 changed files with 36 additions and 9 deletions
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43
FiniteSets/kuratowski/operations.v
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@ -1,6 +1,6 @@
(** Operations on the [FSet A] for an arbitrary [A] *) (** Operations on the [FSet A] for an arbitrary [A] *)
Require Import HoTT HitTactics. Require Import HoTT HitTactics.
Require Import kuratowski_sets monad_interface. Require Import kuratowski_sets monad_interface extensionality.
Section operations. Section operations.
(** Monad operations. *) (** Monad operations. *)
@ -58,13 +58,6 @@ Section operations.
+ apply nl. + apply nl.
Defined. Defined.
Definition isEmpty (A : Type) :
FSet A -> Bool.
Proof.
simple refine (FSet_rec _ _ _ true (fun _ => false) andb _ _ _ _ _)
; try eauto with bool_lattice_hints typeclass_instances.
Defined.
Definition single_product {A B : Type} (a : A) : FSet B -> FSet (A * B). Definition single_product {A B : Type} (a : A) : FSet B -> FSet (A * B).
Proof. Proof.
hrecursion. hrecursion.
@ -95,6 +88,9 @@ Section operations.
- intros ; apply union_idem. - intros ; apply union_idem.
Defined. Defined.
Definition disjoint_sum {A B : Type} (X : FSet A) (Y : FSet B) : FSet (A + B) :=
(fset_fmap inl X) ∪ (fset_fmap inr Y).
Local Ltac remove_transport Local Ltac remove_transport
:= intros ; apply path_forall ; intro Z ; rewrite transport_arrow := intros ; apply path_forall ; intro Z ; rewrite transport_arrow
; rewrite transport_const ; simpl. ; rewrite transport_const ; simpl.
@ -136,6 +132,37 @@ Section operations.
rewrite <- (idem (Z (x; tr 1%path))). rewrite <- (idem (Z (x; tr 1%path))).
pointwise. pointwise.
Defined. Defined.
(** [FSet A] has decidable emptiness. *)
Definition isEmpty {A : Type} (X : FSet A) : Decidable (X = ∅).
Proof.
hinduction X ; try (intros ; apply path_ishprop).
- apply (inl idpath).
- intros.
refine (inr (fun p => _)).
refine (transport (fun Z : hProp => Z) (ap (fun z => a ∈ z) p) _).
apply (tr idpath).
- intros X Y H1 H2.
destruct H1 as [p1 | p1], H2 as [p2 | p2].
* left.
rewrite p1, p2.
apply nl.
* right.
rewrite p1, nl.
apply p2.
* right.
rewrite p2, nr.
apply p1.
* right.
intros p.
apply p1.
apply fset_ext.
intros.
apply path_iff_hprop ; try contradiction.
intro H1.
refine (transport (fun z => a ∈ z) p _).
apply (tr(inl H1)).
Defined.
End operations. End operations.
Section operations_decidable. Section operations_decidable.