mirror of
https://github.com/nmvdw/HITs-Examples
synced 2025-10-31 05:33:51 +01:00
Separate the lattice properties proofs, get rid of the admits and general cleanup
This commit is contained in:
Dan Frumin
@@ -60,15 +60,16 @@ End Lattice.
|
|||||||
|
|
||||||
Arguments Lattice {_} _ _ _.
|
Arguments Lattice {_} _ _ _.
|
||||||
|
|
||||||
Ltac solve :=
|
|
||||||
let x := fresh in
|
|
||||||
repeat (intro x ; destruct x)
|
|
||||||
; compute
|
|
||||||
; auto
|
|
||||||
; try contradiction.
|
|
||||||
|
|
||||||
Section BoolLattice.
|
Section BoolLattice.
|
||||||
|
|
||||||
|
Local Ltac solve :=
|
||||||
|
let x := fresh in
|
||||||
|
repeat (intro x ; destruct x)
|
||||||
|
; compute
|
||||||
|
; auto
|
||||||
|
; try contradiction.
|
||||||
|
|
||||||
Instance min_com : Commutative orb.
|
Instance min_com : Commutative orb.
|
||||||
Proof.
|
Proof.
|
||||||
solve.
|
solve.
|
||||||
@@ -134,98 +135,7 @@ Section BoolLattice.
|
|||||||
|
|
||||||
End BoolLattice.
|
End BoolLattice.
|
||||||
|
|
||||||
Require Import definition.
|
|
||||||
Require Import properties.
|
|
||||||
|
|
||||||
Hint Resolve
|
Hint Resolve
|
||||||
min_com max_com min_assoc max_assoc min_idem max_idem min_nl min_nr
|
min_com max_com min_assoc max_assoc min_idem max_idem min_nl min_nr
|
||||||
bool_absorption_min_max bool_absorption_max_min
|
bool_absorption_min_max bool_absorption_max_min
|
||||||
: bool_lattice_hints.
|
: bool_lattice_hints.
|
||||||
|
|
||||||
Section SetLattice.
|
|
||||||
|
|
||||||
Context {A : Type}.
|
|
||||||
Context {A_deceq : DecidablePaths A}.
|
|
||||||
Context `{Funext}.
|
|
||||||
|
|
||||||
Lemma ext `{Funext} : forall S T, (forall a, isIn a S = isIn a T) -> S = T.
|
|
||||||
Proof.
|
|
||||||
intros.
|
|
||||||
destruct (fset_ext S T).
|
|
||||||
destruct equiv_isequiv.
|
|
||||||
apply equiv_inv.
|
|
||||||
apply X.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Ltac simplify_isIn :=
|
|
||||||
repeat (rewrite ?intersection_isIn ;
|
|
||||||
rewrite ?union_isIn).
|
|
||||||
|
|
||||||
Ltac toBool := try (intro) ;
|
|
||||||
intros ; apply ext ; intros ; simplify_isIn ; eauto with bool_lattice_hints.
|
|
||||||
|
|
||||||
Instance union_com : Commutative (@U A).
|
|
||||||
Proof.
|
|
||||||
toBool.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Instance intersection_com : Commutative intersection.
|
|
||||||
Proof.
|
|
||||||
toBool.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Instance union_assoc : Associative (@U A).
|
|
||||||
Proof.
|
|
||||||
toBool.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Instance intersection_assoc : Associative intersection.
|
|
||||||
Proof.
|
|
||||||
toBool.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Instance union_idem : Idempotent (@U A).
|
|
||||||
Proof.
|
|
||||||
toBool.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Instance intersection_idem : Idempotent intersection.
|
|
||||||
Proof.
|
|
||||||
toBool.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Instance union_nl : NeutralL (@U A) (@E A).
|
|
||||||
Proof.
|
|
||||||
toBool.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Instance union_nr : NeutralR (@U A) (@E A).
|
|
||||||
Proof.
|
|
||||||
toBool.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Instance set_absorption_intersection_union : Absorption (@U A) intersection.
|
|
||||||
Proof.
|
|
||||||
toBool.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Instance set_absorption_union_intersection : Absorption intersection (@U A).
|
|
||||||
Proof.
|
|
||||||
toBool.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Instance lattice_set : Lattice (@U A) intersection (@E A) :=
|
|
||||||
{
|
|
||||||
commutative_min := _ ;
|
|
||||||
commutative_max := _ ;
|
|
||||||
associative_min := _ ;
|
|
||||||
associative_max := _ ;
|
|
||||||
idempotent_min := _ ;
|
|
||||||
idempotent_max := _ ;
|
|
||||||
neutralL_min := _ ;
|
|
||||||
neutralR_min := _ ;
|
|
||||||
absorption_min_max := _ ;
|
|
||||||
absorption_max_min := _
|
|
||||||
}.
|
|
||||||
|
|
||||||
End SetLattice.
|
|
||||||
|
|||||||
@@ -1,43 +1,65 @@
|
|||||||
Require Import HoTT HitTactics.
|
Require Import HoTT HitTactics.
|
||||||
Require Export definition operations Ext.
|
Require Export definition operations Ext Lattice.
|
||||||
|
|
||||||
Section properties.
|
(* Lemmas relating operations to the membership predicate *)
|
||||||
|
Section operations_isIn.
|
||||||
Context {A : Type}.
|
Context {A : Type} `{DecidablePaths A}.
|
||||||
Context {A_deceq : DecidablePaths A}.
|
Lemma ext `{Funext} : forall (S T : FSet A), (forall a, isIn a S = isIn a T) -> S = T.
|
||||||
|
|
||||||
(** isIn properties *)
|
|
||||||
Lemma isIn_singleton_eq (a b: A) : isIn a (L b) = true -> a = b.
|
|
||||||
Proof. unfold isIn. simpl.
|
|
||||||
destruct (dec (a = b)). intro. apply p.
|
|
||||||
intro X.
|
|
||||||
contradiction (false_ne_true X).
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Lemma isIn_empty_false (a: A) : isIn a E = true -> Empty.
|
|
||||||
Proof.
|
|
||||||
cbv. intro X.
|
|
||||||
contradiction (false_ne_true X).
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Lemma isIn_union (a: A) (X Y: FSet A) :
|
|
||||||
isIn a (U X Y) = (isIn a X || isIn a Y)%Bool.
|
|
||||||
Proof. reflexivity. Qed.
|
|
||||||
|
|
||||||
(** comprehension properties *)
|
|
||||||
Lemma comprehension_false Y : comprehension (fun a => isIn a E) Y = E.
|
|
||||||
Proof.
|
Proof.
|
||||||
hrecursion Y; try (intros; apply set_path2).
|
apply fset_ext.
|
||||||
- cbn. reflexivity.
|
Defined.
|
||||||
- cbn. reflexivity.
|
|
||||||
- intros x y IHa IHb.
|
(* Union and membership *)
|
||||||
|
Theorem union_isIn (X Y : FSet A) (a : A) :
|
||||||
|
isIn a (U X Y) = orb (isIn a X) (isIn a Y).
|
||||||
|
Proof.
|
||||||
|
reflexivity.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
(* Intersection and membership. We need a bunch of supporting lemmas *)
|
||||||
|
Lemma intersection_0l: forall X: FSet A, intersection E X = E.
|
||||||
|
Proof.
|
||||||
|
hinduction;
|
||||||
|
try (intros ; apply set_path2).
|
||||||
|
- reflexivity.
|
||||||
|
- intro a.
|
||||||
|
reflexivity.
|
||||||
|
- intros x y P Q.
|
||||||
cbn.
|
cbn.
|
||||||
rewrite IHa.
|
rewrite P.
|
||||||
rewrite IHb.
|
rewrite Q.
|
||||||
apply union_idem.
|
apply union_idem.
|
||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
Theorem comprehension_or : forall ϕ ψ (x: FSet A),
|
Lemma intersection_0r (X : FSet A) : intersection X E = E.
|
||||||
|
Proof. exact idpath. Defined.
|
||||||
|
|
||||||
|
Lemma intersection_La (X : FSet A) (a : A) :
|
||||||
|
intersection (L a) X = if isIn a X then L a else E.
|
||||||
|
Proof.
|
||||||
|
hinduction X; try (intros ; apply set_path2).
|
||||||
|
- reflexivity.
|
||||||
|
- intro b.
|
||||||
|
cbn.
|
||||||
|
destruct (dec (a = b)) as [p|np].
|
||||||
|
* rewrite p.
|
||||||
|
destruct (dec (b = b)) as [|nb]; [reflexivity|].
|
||||||
|
by contradiction nb.
|
||||||
|
* destruct (dec (b = a)); [|reflexivity].
|
||||||
|
by contradiction np.
|
||||||
|
- unfold intersection.
|
||||||
|
intros X Y P Q.
|
||||||
|
cbn.
|
||||||
|
rewrite P.
|
||||||
|
rewrite Q.
|
||||||
|
destruct (isIn a X) ; destruct (isIn a Y).
|
||||||
|
* apply union_idem.
|
||||||
|
* apply nr.
|
||||||
|
* apply nl.
|
||||||
|
* apply union_idem.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Lemma comprehension_or : forall ϕ ψ (x: FSet A),
|
||||||
comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x)
|
comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x)
|
||||||
(comprehension ψ x).
|
(comprehension ψ x).
|
||||||
Proof.
|
Proof.
|
||||||
@@ -61,6 +83,190 @@ hinduction; try (intros; apply set_path2).
|
|||||||
reflexivity.
|
reflexivity.
|
||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
|
Lemma distributive_La (z : FSet A) (a : A) : forall Y : FSet A,
|
||||||
|
intersection (U (L a) z) Y = U (intersection (L a) Y) (intersection z Y).
|
||||||
|
Proof.
|
||||||
|
hinduction; try (intros ; apply set_path2) ; cbn.
|
||||||
|
- symmetry ; apply nl.
|
||||||
|
- intros b.
|
||||||
|
destruct (dec (b = a)) ; cbn.
|
||||||
|
* destruct (isIn b z).
|
||||||
|
+ rewrite union_idem.
|
||||||
|
reflexivity.
|
||||||
|
+ rewrite nr.
|
||||||
|
reflexivity.
|
||||||
|
* rewrite nl ; reflexivity.
|
||||||
|
- intros X1 X2 P Q.
|
||||||
|
rewrite P. rewrite Q.
|
||||||
|
rewrite <- assoc.
|
||||||
|
rewrite (comm (comprehension (fun a0 : A => isIn a0 z) X1)).
|
||||||
|
rewrite <- assoc.
|
||||||
|
rewrite assoc.
|
||||||
|
rewrite (comm (comprehension (fun a0 : A => isIn a0 z) X2)).
|
||||||
|
reflexivity.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Lemma distributive_intersection_U (X1 X2 Y : FSet A) :
|
||||||
|
intersection (U X1 X2) Y = U (intersection X1 Y) (intersection X2 Y).
|
||||||
|
Proof.
|
||||||
|
hinduction X1; try (intros ; apply set_path2) ; cbn.
|
||||||
|
- rewrite intersection_0l.
|
||||||
|
rewrite nl.
|
||||||
|
rewrite nl.
|
||||||
|
reflexivity.
|
||||||
|
- intro a.
|
||||||
|
rewrite intersection_La.
|
||||||
|
rewrite distributive_La.
|
||||||
|
rewrite intersection_La.
|
||||||
|
reflexivity.
|
||||||
|
- intros Z1 Z2 P Q.
|
||||||
|
unfold intersection in *. simpl in *.
|
||||||
|
apply comprehension_or.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Theorem intersection_isIn (X Y: FSet A) (a : A) :
|
||||||
|
isIn a (intersection X Y) = andb (isIn a X) (isIn a Y).
|
||||||
|
Proof.
|
||||||
|
hinduction X; try (intros ; apply set_path2) ; cbn.
|
||||||
|
- rewrite intersection_0l.
|
||||||
|
reflexivity.
|
||||||
|
- intro b.
|
||||||
|
rewrite intersection_La.
|
||||||
|
destruct (dec (a = b)) ; cbn.
|
||||||
|
* rewrite p.
|
||||||
|
destruct (isIn b Y).
|
||||||
|
+ cbn.
|
||||||
|
destruct (dec (b = b)); [reflexivity|].
|
||||||
|
by contradiction n.
|
||||||
|
+ reflexivity.
|
||||||
|
* destruct (isIn b Y).
|
||||||
|
+ cbn.
|
||||||
|
destruct (dec (a = b)); [|reflexivity].
|
||||||
|
by contradiction n.
|
||||||
|
+ reflexivity.
|
||||||
|
- intros X1 X2 P Q.
|
||||||
|
rewrite distributive_intersection_U. simpl.
|
||||||
|
rewrite P.
|
||||||
|
rewrite Q.
|
||||||
|
destruct (isIn a X1) ; destruct (isIn a X2) ; destruct (isIn a Y) ;
|
||||||
|
reflexivity.
|
||||||
|
Defined.
|
||||||
|
End operations_isIn.
|
||||||
|
|
||||||
|
(* Some suporting tactics *)
|
||||||
|
Ltac simplify_isIn :=
|
||||||
|
repeat (rewrite ?intersection_isIn ;
|
||||||
|
rewrite ?union_isIn).
|
||||||
|
|
||||||
|
Ltac toBool := try (intro) ;
|
||||||
|
intros ; apply ext ; intros ; simplify_isIn ; eauto with bool_lattice_hints.
|
||||||
|
|
||||||
|
Section SetLattice.
|
||||||
|
|
||||||
|
Context {A : Type}.
|
||||||
|
Context {A_deceq : DecidablePaths A}.
|
||||||
|
Context `{Funext}.
|
||||||
|
|
||||||
|
Instance fset_union_com : Commutative (@U A).
|
||||||
|
Proof.
|
||||||
|
toBool.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Instance fset_intersection_com : Commutative intersection.
|
||||||
|
Proof.
|
||||||
|
toBool.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Instance fset_union_assoc : Associative (@U A).
|
||||||
|
Proof.
|
||||||
|
toBool.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Instance fset_intersection_assoc : Associative intersection.
|
||||||
|
Proof.
|
||||||
|
toBool.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Instance fset_union_idem : Idempotent (@U A).
|
||||||
|
Proof. exact union_idem. Defined.
|
||||||
|
|
||||||
|
Instance fset_intersection_idem : Idempotent intersection.
|
||||||
|
Proof.
|
||||||
|
toBool.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Instance fset_union_nl : NeutralL (@U A) (@E A).
|
||||||
|
Proof.
|
||||||
|
toBool.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Instance fset_union_nr : NeutralR (@U A) (@E A).
|
||||||
|
Proof.
|
||||||
|
toBool.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Instance fset_absorption_intersection_union : Absorption (@U A) intersection.
|
||||||
|
Proof.
|
||||||
|
toBool.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Instance fset_absorption_union_intersection : Absorption intersection (@U A).
|
||||||
|
Proof.
|
||||||
|
toBool.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Instance lattice_fset : Lattice (@U A) intersection (@E A) :=
|
||||||
|
{
|
||||||
|
commutative_min := _ ;
|
||||||
|
commutative_max := _ ;
|
||||||
|
associative_min := _ ;
|
||||||
|
associative_max := _ ;
|
||||||
|
idempotent_min := _ ;
|
||||||
|
idempotent_max := _ ;
|
||||||
|
neutralL_min := _ ;
|
||||||
|
neutralR_min := _ ;
|
||||||
|
absorption_min_max := _ ;
|
||||||
|
absorption_max_min := _
|
||||||
|
}.
|
||||||
|
|
||||||
|
End SetLattice.
|
||||||
|
|
||||||
|
(* Other properties *)
|
||||||
|
Section properties.
|
||||||
|
|
||||||
|
Context {A : Type}.
|
||||||
|
Context {A_deceq : DecidablePaths A}.
|
||||||
|
|
||||||
|
(** isIn properties *)
|
||||||
|
Lemma isIn_singleton_eq (a b: A) : isIn a (L b) = true -> a = b.
|
||||||
|
Proof. unfold isIn. simpl.
|
||||||
|
destruct (dec (a = b)). intro. apply p.
|
||||||
|
intro X.
|
||||||
|
contradiction (false_ne_true X).
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Lemma isIn_empty_false (a: A) : isIn a E = true -> Empty.
|
||||||
|
Proof.
|
||||||
|
cbv. intro X.
|
||||||
|
contradiction (false_ne_true X).
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Lemma isIn_union (a: A) (X Y: FSet A) :
|
||||||
|
isIn a (U X Y) = (isIn a X || isIn a Y)%Bool.
|
||||||
|
Proof. reflexivity. Qed.
|
||||||
|
|
||||||
|
(** comprehension properties *)
|
||||||
|
Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = E.
|
||||||
|
Proof.
|
||||||
|
hrecursion Y; try (intros; apply set_path2).
|
||||||
|
- reflexivity.
|
||||||
|
- reflexivity.
|
||||||
|
- intros x y IHa IHb.
|
||||||
|
rewrite IHa.
|
||||||
|
rewrite IHb.
|
||||||
|
apply union_idem.
|
||||||
|
Defined.
|
||||||
|
|
||||||
Theorem comprehension_subset : forall ϕ (X : FSet A),
|
Theorem comprehension_subset : forall ϕ (X : FSet A),
|
||||||
U (comprehension ϕ X) X = X.
|
U (comprehension ϕ X) X = X.
|
||||||
Proof.
|
Proof.
|
||||||
@@ -83,48 +289,7 @@ hrecursion; try (intros ; apply set_path2) ; cbn.
|
|||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
(** intersection properties *)
|
(** intersection properties *)
|
||||||
Lemma intersection_0l: forall X: FSet A, intersection E X = E.
|
|
||||||
Proof.
|
|
||||||
hinduction;
|
|
||||||
try (intros ; apply set_path2).
|
|
||||||
- reflexivity.
|
|
||||||
- intro a.
|
|
||||||
reflexivity.
|
|
||||||
- intros x y P Q.
|
|
||||||
cbn.
|
|
||||||
rewrite P.
|
|
||||||
rewrite Q.
|
|
||||||
apply union_idem.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Lemma intersection_0r (X: FSet A): intersection X E = E.
|
|
||||||
Proof. exact idpath. Defined.
|
|
||||||
|
|
||||||
Theorem intersection_La : forall (a : A) (X : FSet A),
|
|
||||||
intersection (L a) X = if isIn a X then L a else E.
|
|
||||||
Proof.
|
|
||||||
intro a.
|
|
||||||
hinduction; try (intros ; apply set_path2).
|
|
||||||
- reflexivity.
|
|
||||||
- intro b.
|
|
||||||
cbn.
|
|
||||||
destruct (dec (a = b)) as [p|np].
|
|
||||||
* rewrite p.
|
|
||||||
destruct (dec (b = b)) as [|nb]; [reflexivity|].
|
|
||||||
by contradiction nb.
|
|
||||||
* destruct (dec (b = a)); [|reflexivity].
|
|
||||||
by contradiction np.
|
|
||||||
- unfold intersection.
|
|
||||||
intros x y P Q.
|
|
||||||
cbn.
|
|
||||||
rewrite P.
|
|
||||||
rewrite Q.
|
|
||||||
destruct (isIn a x) ; destruct (isIn a y).
|
|
||||||
* apply idem.
|
|
||||||
* apply nr.
|
|
||||||
* apply nl.
|
|
||||||
* apply nl.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Lemma intersection_comm X Y: intersection X Y = intersection Y X.
|
Lemma intersection_comm X Y: intersection X Y = intersection Y X.
|
||||||
Proof.
|
Proof.
|
||||||
@@ -175,79 +340,6 @@ hinduction; try (intros ; apply set_path2).
|
|||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
(** assorted lattice laws *)
|
(** assorted lattice laws *)
|
||||||
Lemma distributive_La (z : FSet A) (a : A) : forall Y : FSet A,
|
|
||||||
intersection (U (L a) z) Y = U (intersection (L a) Y) (intersection z Y).
|
|
||||||
Proof.
|
|
||||||
hinduction; try (intros ; apply set_path2) ; cbn.
|
|
||||||
- symmetry ; apply nl.
|
|
||||||
- intros b.
|
|
||||||
destruct (dec (b = a)) ; cbn.
|
|
||||||
* destruct (isIn b z).
|
|
||||||
+ rewrite union_idem.
|
|
||||||
reflexivity.
|
|
||||||
+ rewrite nr.
|
|
||||||
reflexivity.
|
|
||||||
* rewrite nl ; reflexivity.
|
|
||||||
- intros X1 X2 P Q.
|
|
||||||
rewrite P. rewrite Q.
|
|
||||||
rewrite <- assoc.
|
|
||||||
rewrite (comm (comprehension (fun a0 : A => isIn a0 z) X1)).
|
|
||||||
rewrite <- assoc.
|
|
||||||
rewrite assoc.
|
|
||||||
rewrite (comm (comprehension (fun a0 : A => isIn a0 z) X2)).
|
|
||||||
reflexivity.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Theorem distributive_intersection_U (X1 X2 Y : FSet A) :
|
|
||||||
intersection (U X1 X2) Y = U (intersection X1 Y) (intersection X2 Y).
|
|
||||||
Proof.
|
|
||||||
hinduction X1; try (intros ; apply set_path2) ; cbn.
|
|
||||||
- rewrite intersection_0l.
|
|
||||||
rewrite nl.
|
|
||||||
rewrite nl.
|
|
||||||
reflexivity.
|
|
||||||
- intro a.
|
|
||||||
rewrite intersection_La.
|
|
||||||
rewrite distributive_La.
|
|
||||||
rewrite intersection_La.
|
|
||||||
reflexivity.
|
|
||||||
- intros Z1 Z2 P Q.
|
|
||||||
unfold intersection in *.
|
|
||||||
cbn.
|
|
||||||
rewrite comprehension_or.
|
|
||||||
rewrite comprehension_or.
|
|
||||||
reflexivity.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Theorem intersection_isIn : forall a (x y: FSet A),
|
|
||||||
isIn a (intersection x y) = andb (isIn a x) (isIn a y).
|
|
||||||
Proof.
|
|
||||||
intros a x y.
|
|
||||||
hinduction x; try (intros ; apply set_path2) ; cbn.
|
|
||||||
- rewrite intersection_0l.
|
|
||||||
reflexivity.
|
|
||||||
- intro b.
|
|
||||||
rewrite intersection_La.
|
|
||||||
destruct (dec (a = b)) ; cbn.
|
|
||||||
* rewrite p.
|
|
||||||
destruct (isIn b y).
|
|
||||||
+ cbn.
|
|
||||||
destruct (dec (b = b)); [reflexivity|].
|
|
||||||
by contradiction n.
|
|
||||||
+ reflexivity.
|
|
||||||
* destruct (isIn b y).
|
|
||||||
+ cbn.
|
|
||||||
destruct (dec (a = b)); [|reflexivity].
|
|
||||||
by contradiction n.
|
|
||||||
+ reflexivity.
|
|
||||||
- intros X1 X2 P Q.
|
|
||||||
rewrite distributive_intersection_U.
|
|
||||||
cbn.
|
|
||||||
rewrite P.
|
|
||||||
rewrite Q.
|
|
||||||
destruct (isIn a X1) ; destruct (isIn a X2) ; destruct (isIn a y) ;
|
|
||||||
reflexivity.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Theorem intersection_assoc (X Y Z: FSet A) :
|
Theorem intersection_assoc (X Y Z: FSet A) :
|
||||||
intersection X (intersection Y Z) = intersection (intersection X Y) Z.
|
intersection X (intersection Y Z) = intersection (intersection X Y) Z.
|
||||||
@@ -300,77 +392,12 @@ hinduction; try (intros ; apply set_path2).
|
|||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
|
|
||||||
Theorem distributive_U_int (X1 X2 Y : FSet A) :
|
Theorem distributive_U_int `{Funext} (X1 X2 Y : FSet A) :
|
||||||
U (intersection X1 X2) Y = intersection (U X1 Y) (U X2 Y).
|
U (intersection X1 X2) Y = intersection (U X1 Y) (U X2 Y).
|
||||||
Proof.
|
Proof.
|
||||||
hinduction X1; try (intros ; apply set_path2) ; cbn.
|
toBool.
|
||||||
- rewrite intersection_0l.
|
destruct (a ∈ X1), (a ∈ X2), (a ∈ Y); eauto.
|
||||||
rewrite nl.
|
Defined.
|
||||||
unfold intersection.
|
|
||||||
rewrite comprehension_all.
|
|
||||||
pose (intersection_comm Y X2).
|
|
||||||
unfold intersection in p.
|
|
||||||
rewrite p.
|
|
||||||
rewrite comprehension_subset.
|
|
||||||
reflexivity.
|
|
||||||
- intros.
|
|
||||||
assert (Y = intersection (U (L a) Y) Y) as HY.
|
|
||||||
{ unfold intersection. symmetry.
|
|
||||||
transitivity (U (comprehension (fun x => isIn x (L a)) Y) (comprehension (fun x => isIn x Y) Y)).
|
|
||||||
apply comprehension_or.
|
|
||||||
rewrite comprehension_all.
|
|
||||||
apply comprehension_subset. }
|
|
||||||
rewrite <- HY.
|
|
||||||
admit.
|
|
||||||
- unfold intersection.
|
|
||||||
intros Z1 Z2 P Q.
|
|
||||||
rewrite comprehension_or.
|
|
||||||
assert (U (U (comprehension (fun a : A => isIn a Z1) X2)
|
|
||||||
(comprehension (fun a : A => isIn a Z2) X2))
|
|
||||||
Y = U (U (comprehension (fun a : A => isIn a Z1) X2)
|
|
||||||
(comprehension (fun a : A => isIn a Z2) X2))
|
|
||||||
(U Y Y)).
|
|
||||||
rewrite (union_idem Y).
|
|
||||||
reflexivity.
|
|
||||||
rewrite X.
|
|
||||||
rewrite <- assoc.
|
|
||||||
rewrite (assoc (comprehension (fun a : A => isIn a Z2) X2)).
|
|
||||||
rewrite Q.
|
|
||||||
cbn.
|
|
||||||
rewrite
|
|
||||||
(comm (U (comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) X2)
|
|
||||||
(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) Y)) Y).
|
|
||||||
rewrite assoc.
|
|
||||||
rewrite P.
|
|
||||||
rewrite <- assoc. cbn.
|
|
||||||
rewrite (assoc (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) Y)).
|
|
||||||
rewrite (comm (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) Y)).
|
|
||||||
rewrite <- assoc.
|
|
||||||
rewrite assoc.
|
|
||||||
enough (C : (U (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) X2)
|
|
||||||
(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) X2))
|
|
||||||
= (comprehension (fun a : A => (isIn a Z1 || isIn a Z2 || isIn a Y)%Bool) X2)).
|
|
||||||
rewrite C.
|
|
||||||
enough (D : (U (comprehension (fun a : A => (isIn a Z1 || isIn a Y)%Bool) Y)
|
|
||||||
(comprehension (fun a : A => (isIn a Z2 || isIn a Y)%Bool) Y))
|
|
||||||
= (comprehension (fun a : A => (isIn a Z1 || isIn a Z2 || isIn a Y)%Bool) Y)).
|
|
||||||
rewrite D.
|
|
||||||
reflexivity.
|
|
||||||
* repeat (rewrite comprehension_or).
|
|
||||||
rewrite <- assoc.
|
|
||||||
rewrite (comm (comprehension (fun a : A => isIn a Y) Y)).
|
|
||||||
rewrite <- (assoc (comprehension (fun a : A => isIn a Z2) Y)).
|
|
||||||
rewrite union_idem.
|
|
||||||
rewrite assoc.
|
|
||||||
reflexivity.
|
|
||||||
* repeat (rewrite comprehension_or).
|
|
||||||
rewrite <- assoc.
|
|
||||||
rewrite (comm (comprehension (fun a : A => isIn a Y) X2)).
|
|
||||||
rewrite <- (assoc (comprehension (fun a : A => isIn a Z2) X2)).
|
|
||||||
rewrite union_idem.
|
|
||||||
rewrite assoc.
|
|
||||||
reflexivity.
|
|
||||||
Admitted.
|
|
||||||
|
|
||||||
Theorem absorb_0 (X Y : FSet A) : U X (intersection X Y) = X.
|
Theorem absorb_0 (X Y : FSet A) : U X (intersection X Y) = X.
|
||||||
Proof.
|
Proof.
|
||||||
@@ -395,62 +422,9 @@ hinduction X; try (intros ; apply set_path2) ; cbn.
|
|||||||
reflexivity.
|
reflexivity.
|
||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
Theorem absorb_1 (X Y : FSet A) : intersection X (U X Y) = X.
|
Theorem absorb_1 `{Funext} (X Y : FSet A) : intersection X (U X Y) = X.
|
||||||
Proof.
|
Proof.
|
||||||
hrecursion X; try (intros ; apply set_path2).
|
toBool.
|
||||||
- cbn.
|
|
||||||
rewrite nl.
|
|
||||||
apply comprehension_false.
|
|
||||||
- intro a.
|
|
||||||
rewrite intersection_La.
|
|
||||||
destruct (dec (a = a)).
|
|
||||||
* destruct (isIn a Y).
|
|
||||||
+ apply union_idem.
|
|
||||||
+ apply nr.
|
|
||||||
* contradiction (n idpath).
|
|
||||||
- intros X1 X2 P Q.
|
|
||||||
cbn in *.
|
|
||||||
symmetry.
|
|
||||||
rewrite <- P.
|
|
||||||
rewrite <- Q.
|
|
||||||
Admitted.
|
|
||||||
|
|
||||||
Theorem union_isIn (X Y : FSet A) (a : A) : isIn a (U X Y) = orb (isIn a X) (isIn a Y).
|
|
||||||
Proof.
|
|
||||||
reflexivity.
|
|
||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
Ltac solve :=
|
|
||||||
let x := fresh in
|
|
||||||
repeat (intro x ; destruct x)
|
|
||||||
; compute
|
|
||||||
; auto
|
|
||||||
; try contradiction.
|
|
||||||
|
|
||||||
Ltac simplify_isIn :=
|
|
||||||
repeat (rewrite ?intersection_isIn ;
|
|
||||||
rewrite ?union_isIn).
|
|
||||||
|
|
||||||
Lemma ext `{Funext} : forall S T, (forall a, isIn a S = isIn a T) -> S = T.
|
|
||||||
Proof.
|
|
||||||
intros.
|
|
||||||
destruct (fset_ext S T).
|
|
||||||
destruct equiv_isequiv.
|
|
||||||
apply equiv_inv.
|
|
||||||
apply X.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Ltac toBool :=
|
|
||||||
intros ; apply ext ; intros ; simplify_isIn.
|
|
||||||
|
|
||||||
Lemma andb_comm : forall x y, andb x y = andb y x.
|
|
||||||
Proof.
|
|
||||||
solve.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
Lemma intersectioncomm `{Funext} : forall x y, intersection x y = intersection y x.
|
|
||||||
Proof.
|
|
||||||
toBool. apply andb_comm.
|
|
||||||
Defined.
|
|
||||||
|
|
||||||
End properties.
|
End properties.
|
||||||
|
|||||||
Reference in New Issue
Block a user