mirror of https://github.com/nmvdw/HITs-Examples
Removed View_rec2
This commit is contained in:
Niels
parent
ae60ac0146
commit
8ff54def39
|
|
@ -130,6 +130,8 @@ Section quotient_surjection.
|
|||
|
||||
End quotient_surjection.
|
||||
|
||||
Arguments quotient_iso {_} {_} _ {_} {_} {_}.
|
||||
|
||||
Ltac reduce T :=
|
||||
intros ;
|
||||
repeat (rewrite (f_empty T _)
|
||||
|
|
@ -137,12 +139,6 @@ Ltac reduce T :=
|
|||
|| rewrite (f_union T _)
|
||||
|| rewrite (f_filter T _)
|
||||
|| rewrite (f_member T _)).
|
||||
Ltac simplify T := intros ; autounfold in * ; apply reflect_eq ; reduce T.
|
||||
Ltac reflect_equality T := simplify T ; eauto with lattice_hints typeclass_instances.
|
||||
Ltac reflect_eq T := autounfold
|
||||
; repeat (hinduction ; try (intros ; apply path_ishprop) ; intro)
|
||||
; apply related_classes_eq
|
||||
; reflect_equality T.
|
||||
|
||||
Section quotient_properties.
|
||||
Variable (T : Type -> Type).
|
||||
|
|
@ -152,45 +148,73 @@ Section quotient_properties.
|
|||
Definition set_eq A := f_eq (T A) (FSet A) (f A).
|
||||
Definition View A : Type := quotientB (T A) (FSet A) (f A).
|
||||
|
||||
Definition View_rec2 {A} (P : Type) (HP : IsHSet P) (u : T A -> T A -> P)
|
||||
(Hresp : forall (x x' y y': T A), set_eq A x x' -> set_eq A y y' -> u x y = u x' y')
|
||||
: forall (x y : View A), P.
|
||||
Instance f_is_surjective A : IsSurjection (f A).
|
||||
Proof.
|
||||
unfold View ; hrecursion.
|
||||
- intros a.
|
||||
hrecursion.
|
||||
+ apply (u a).
|
||||
+ intros b b' Hbb'.
|
||||
apply Hresp.
|
||||
++ reflexivity.
|
||||
++ assumption.
|
||||
- intros ; simpl.
|
||||
apply path_forall.
|
||||
red.
|
||||
hinduction ; try (intros ; apply path_ishprop).
|
||||
intros b.
|
||||
apply Hresp.
|
||||
++ assumption.
|
||||
++ reflexivity.
|
||||
apply (f_surjective T f A).
|
||||
Defined.
|
||||
|
||||
Definition well_defined_union (A : Type) (X1 X2 Y1 Y2 : T A) :
|
||||
set_eq A X1 Y1 -> set_eq A X2 Y2 -> set_eq A (union X1 X2) (union Y1 Y2).
|
||||
Global Instance view_union (A : Type) : hasUnion (View A).
|
||||
Proof.
|
||||
intros HXY1 HXY2.
|
||||
simplify T.
|
||||
by rewrite HXY1, HXY2.
|
||||
intros X Y.
|
||||
apply (quotient_iso _)^-1.
|
||||
simple refine (union _ _).
|
||||
- simple refine (quotient_iso (f A) X).
|
||||
- simple refine (quotient_iso (f A) Y).
|
||||
Defined.
|
||||
|
||||
Definition well_defined_filter (A : Type) (ϕ : A -> Bool) (X Y : T A) :
|
||||
set_eq A X Y -> set_eq A (filter ϕ X) (filter ϕ Y).
|
||||
Definition well_defined_union (A : Type) (X Y : T A) :
|
||||
(class_of _ X) ∪ (class_of _ Y) = class_of _ (X ∪ Y).
|
||||
Proof.
|
||||
intros HXY.
|
||||
simplify T.
|
||||
by rewrite HXY.
|
||||
rewrite <- (eissect (quotient_iso _)).
|
||||
simpl.
|
||||
rewrite (f_union T _).
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Global Instance View_member A: hasMembership (View A) A.
|
||||
Global Instance view_comprehension (A : Type) : hasComprehension (View A) A.
|
||||
Proof.
|
||||
intros ϕ X.
|
||||
apply (quotient_iso _)^-1.
|
||||
simple refine ({|_ & ϕ|}).
|
||||
apply (quotient_iso (f A) X).
|
||||
Defined.
|
||||
|
||||
Definition well_defined_filter (A : Type) (ϕ : A -> Bool) (X : T A) :
|
||||
{|class_of _ X & ϕ|} = class_of _ {|X & ϕ|}.
|
||||
Proof.
|
||||
rewrite <- (eissect (quotient_iso _)).
|
||||
simpl.
|
||||
rewrite (f_filter T _).
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Global Instance View_empty A : hasEmpty (View A).
|
||||
Proof.
|
||||
apply ((quotient_iso _)^-1 ∅).
|
||||
Defined.
|
||||
|
||||
Definition well_defined_empty A : ∅ = class_of (set_eq A) ∅.
|
||||
Proof.
|
||||
rewrite <- (eissect (quotient_iso _)).
|
||||
simpl.
|
||||
rewrite (f_empty T _).
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Global Instance View_singleton A: hasSingleton (View A) A.
|
||||
Proof.
|
||||
intro a ; apply ((quotient_iso _)^-1 {|a|}).
|
||||
Defined.
|
||||
|
||||
Definition well_defined_sungleton A (a : A) : {|a|} = class_of _ {|a|}.
|
||||
Proof.
|
||||
rewrite <- (eissect (quotient_iso _)).
|
||||
simpl.
|
||||
rewrite (f_singleton T _).
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Global Instance View_member A : hasMembership (View A) A.
|
||||
Proof.
|
||||
intros a ; unfold View.
|
||||
hrecursion.
|
||||
|
|
@ -200,54 +224,30 @@ Section quotient_properties.
|
|||
apply (ap _ HXY).
|
||||
Defined.
|
||||
|
||||
Global Instance View_empty A: hasEmpty (View A).
|
||||
Proof.
|
||||
apply (class_of _ ∅).
|
||||
Defined.
|
||||
|
||||
Global Instance View_singleton A: hasSingleton (View A) A.
|
||||
Proof.
|
||||
intros a.
|
||||
apply (class_of _ {|a|}).
|
||||
Defined.
|
||||
|
||||
Instance View_max A : maximum (View A).
|
||||
Proof.
|
||||
simple refine (View_rec2 _ _ _ _).
|
||||
- intros a b.
|
||||
apply (class_of _ (union a b)).
|
||||
- intros x x' Hxx' y y' Hyy' ; simpl.
|
||||
apply related_classes_eq.
|
||||
eapply well_defined_union ; eauto.
|
||||
apply view_union.
|
||||
Defined.
|
||||
|
||||
Global Instance View_union A: hasUnion (View A).
|
||||
Proof.
|
||||
apply max_L.
|
||||
Defined.
|
||||
|
||||
Global Instance View_comprehension A: hasComprehension (View A) A.
|
||||
Proof.
|
||||
intros ϕ ; unfold View.
|
||||
hrecursion.
|
||||
- intros X.
|
||||
apply (class_of _ (filter ϕ X)).
|
||||
- intros X X' HXX' ; simpl.
|
||||
apply related_classes_eq.
|
||||
eapply well_defined_filter ; eauto.
|
||||
Defined.
|
||||
|
||||
Hint Unfold Commutative Associative Idempotent NeutralL NeutralR.
|
||||
Hint Unfold Commutative Associative Idempotent NeutralL NeutralR View_max view_union.
|
||||
|
||||
Instance bottom_view A : bottom (View A).
|
||||
Proof.
|
||||
unfold bottom.
|
||||
apply ∅.
|
||||
apply View_empty.
|
||||
Defined.
|
||||
|
||||
Ltac sq1 := autounfold ; intros ; try f_ap
|
||||
; rewrite ?(eisretr (quotient_iso _))
|
||||
; eauto with lattice_hints typeclass_instances.
|
||||
|
||||
Ltac sq2 := autounfold ; intros
|
||||
; rewrite <- (eissect (quotient_iso _)), ?(eisretr (quotient_iso _))
|
||||
; f_ap ; simpl
|
||||
; reduce T ; eauto with lattice_hints typeclass_instances.
|
||||
|
||||
Global Instance view_lattice A : JoinSemiLattice (View A).
|
||||
Proof.
|
||||
split ; reflect_eq T.
|
||||
split ; try sq1 ; try sq2.
|
||||
Defined.
|
||||
|
||||
End quotient_properties.
|
||||
|
|
@ -294,20 +294,8 @@ Section properties.
|
|||
refine (same_class _ _ _ _ _ _) ; assumption.
|
||||
Defined.
|
||||
|
||||
Lemma class_union (A : Type) (X Y : T A) :
|
||||
class_of (set_eq f) (X ∪ Y) = (class_of (set_eq f) X) ∪ (class_of (set_eq f) Y).
|
||||
Proof.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Lemma class_filter (A : Type) (X : T A) (ϕ : A -> Bool) :
|
||||
class_of (set_eq f) ({|X & ϕ|}) = {|(class_of (set_eq f) X) & ϕ|}.
|
||||
Proof.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Ltac via_quotient := intros ; apply reflect_f_eq
|
||||
; rewrite ?class_union, ?class_filter
|
||||
Ltac via_quotient := intros ; apply reflect_f_eq ; simpl
|
||||
; rewrite <- ?(well_defined_union T _), <- ?(well_defined_empty T _)
|
||||
; eauto with lattice_hints typeclass_instances.
|
||||
|
||||
Lemma union_comm : forall A (X Y : T A),
|
||||
|
|
@ -328,10 +316,16 @@ Section properties.
|
|||
via_quotient.
|
||||
Defined.
|
||||
|
||||
Lemma union_neutral : forall A (X : T A),
|
||||
Lemma union_neutralL : forall A (X : T A),
|
||||
set_eq f (∅ ∪ X) X.
|
||||
Proof.
|
||||
via_quotient.
|
||||
Defined.
|
||||
|
||||
Lemma union_neutralR : forall A (X : T A),
|
||||
set_eq f (X ∪ ∅) X.
|
||||
Proof.
|
||||
via_quotient.
|
||||
Defined.
|
||||
|
||||
End properties.
|
||||
Loading…
Reference in New Issue