mirror of
https://github.com/nmvdw/HITs-Examples
synced 2025-10-31 13:43:51 +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.
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Arguments Lattice {_} _ _ _.
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Ltac solve :=
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Section BoolLattice.
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Local Ltac solve :=
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let x := fresh in
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repeat (intro x ; destruct x)
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; compute
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; auto
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; try contradiction.
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Section BoolLattice.
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Instance min_com : Commutative orb.
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Proof.
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solve.
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@@ -134,98 +135,7 @@ Section BoolLattice.
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End BoolLattice.
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Require Import definition.
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Require Import properties.
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Hint Resolve
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min_com max_com min_assoc max_assoc min_idem max_idem min_nl min_nr
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bool_absorption_min_max bool_absorption_max_min
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: bool_lattice_hints.
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Section SetLattice.
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Context {A : Type}.
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Context {A_deceq : DecidablePaths A}.
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Context `{Funext}.
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Lemma ext `{Funext} : forall S T, (forall a, isIn a S = isIn a T) -> S = T.
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Proof.
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intros.
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destruct (fset_ext S T).
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destruct equiv_isequiv.
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apply equiv_inv.
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apply X.
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Defined.
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Ltac simplify_isIn :=
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repeat (rewrite ?intersection_isIn ;
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rewrite ?union_isIn).
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Ltac toBool := try (intro) ;
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intros ; apply ext ; intros ; simplify_isIn ; eauto with bool_lattice_hints.
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Instance union_com : Commutative (@U A).
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Proof.
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toBool.
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Defined.
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Instance intersection_com : Commutative intersection.
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Proof.
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toBool.
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Defined.
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Instance union_assoc : Associative (@U A).
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Proof.
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toBool.
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Defined.
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Instance intersection_assoc : Associative intersection.
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Proof.
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toBool.
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Defined.
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Instance union_idem : Idempotent (@U A).
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Proof.
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toBool.
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Defined.
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Instance intersection_idem : Idempotent intersection.
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Proof.
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toBool.
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Defined.
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Instance union_nl : NeutralL (@U A) (@E A).
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Proof.
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toBool.
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Defined.
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Instance union_nr : NeutralR (@U A) (@E A).
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Proof.
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toBool.
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Defined.
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Instance set_absorption_intersection_union : Absorption (@U A) intersection.
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Proof.
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toBool.
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Defined.
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Instance set_absorption_union_intersection : Absorption intersection (@U A).
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Proof.
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toBool.
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Defined.
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Instance lattice_set : Lattice (@U A) intersection (@E A) :=
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{
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commutative_min := _ ;
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commutative_max := _ ;
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associative_min := _ ;
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associative_max := _ ;
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idempotent_min := _ ;
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idempotent_max := _ ;
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neutralL_min := _ ;
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neutralR_min := _ ;
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absorption_min_max := _ ;
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absorption_max_min := _
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}.
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End SetLattice.
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@@ -1,6 +1,237 @@
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Require Import HoTT HitTactics.
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Require Export definition operations Ext.
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Require Export definition operations Ext Lattice.
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(* Lemmas relating operations to the membership predicate *)
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Section operations_isIn.
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Context {A : Type} `{DecidablePaths A}.
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Lemma ext `{Funext} : forall (S T : FSet A), (forall a, isIn a S = isIn a T) -> S = T.
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Proof.
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apply fset_ext.
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Defined.
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(* Union and membership *)
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Theorem union_isIn (X Y : FSet A) (a : A) :
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isIn a (U X Y) = orb (isIn a X) (isIn a Y).
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Proof.
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reflexivity.
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Defined.
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(* Intersection and membership. We need a bunch of supporting lemmas *)
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Lemma intersection_0l: forall X: FSet A, intersection E X = E.
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Proof.
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hinduction;
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try (intros ; apply set_path2).
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- reflexivity.
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- intro a.
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reflexivity.
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- intros x y P Q.
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cbn.
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rewrite P.
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rewrite Q.
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apply union_idem.
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Defined.
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Lemma intersection_0r (X : FSet A) : intersection X E = E.
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Proof. exact idpath. Defined.
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Lemma intersection_La (X : FSet A) (a : A) :
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intersection (L a) X = if isIn a X then L a else E.
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Proof.
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hinduction X; try (intros ; apply set_path2).
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- reflexivity.
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- intro b.
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cbn.
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destruct (dec (a = b)) as [p|np].
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* rewrite p.
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destruct (dec (b = b)) as [|nb]; [reflexivity|].
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by contradiction nb.
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* destruct (dec (b = a)); [|reflexivity].
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by contradiction np.
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- unfold intersection.
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intros X Y P Q.
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cbn.
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rewrite P.
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rewrite Q.
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destruct (isIn a X) ; destruct (isIn a Y).
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* apply union_idem.
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* apply nr.
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* apply nl.
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* apply union_idem.
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Defined.
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Lemma comprehension_or : forall ϕ ψ (x: FSet A),
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comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x)
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(comprehension ψ x).
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Proof.
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intros ϕ ψ.
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hinduction; try (intros; apply set_path2).
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- cbn. apply (union_idem _)^.
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- cbn. intros.
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destruct (ϕ a) ; destruct (ψ a) ; symmetry.
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* apply union_idem.
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* apply nr.
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* apply nl.
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* apply union_idem.
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- simpl. intros x y P Q.
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rewrite P.
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rewrite Q.
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rewrite <- assoc.
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rewrite (assoc (comprehension ψ x)).
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rewrite (comm (comprehension ψ x) (comprehension ϕ y)).
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rewrite <- assoc.
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rewrite <- assoc.
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reflexivity.
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Defined.
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Lemma distributive_La (z : FSet A) (a : A) : forall Y : FSet A,
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intersection (U (L a) z) Y = U (intersection (L a) Y) (intersection z Y).
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Proof.
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hinduction; try (intros ; apply set_path2) ; cbn.
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- symmetry ; apply nl.
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- intros b.
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destruct (dec (b = a)) ; cbn.
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* destruct (isIn b z).
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+ rewrite union_idem.
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reflexivity.
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+ rewrite nr.
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reflexivity.
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* rewrite nl ; reflexivity.
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- intros X1 X2 P Q.
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rewrite P. rewrite Q.
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rewrite <- assoc.
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rewrite (comm (comprehension (fun a0 : A => isIn a0 z) X1)).
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rewrite <- assoc.
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rewrite assoc.
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rewrite (comm (comprehension (fun a0 : A => isIn a0 z) X2)).
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reflexivity.
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Defined.
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Lemma distributive_intersection_U (X1 X2 Y : FSet A) :
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intersection (U X1 X2) Y = U (intersection X1 Y) (intersection X2 Y).
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Proof.
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hinduction X1; try (intros ; apply set_path2) ; cbn.
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- rewrite intersection_0l.
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rewrite nl.
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rewrite nl.
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reflexivity.
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- intro a.
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rewrite intersection_La.
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rewrite distributive_La.
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rewrite intersection_La.
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reflexivity.
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- intros Z1 Z2 P Q.
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unfold intersection in *. simpl in *.
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apply comprehension_or.
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Defined.
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Theorem intersection_isIn (X Y: FSet A) (a : A) :
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isIn a (intersection X Y) = andb (isIn a X) (isIn a Y).
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Proof.
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hinduction X; try (intros ; apply set_path2) ; cbn.
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- rewrite intersection_0l.
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reflexivity.
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- intro b.
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rewrite intersection_La.
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destruct (dec (a = b)) ; cbn.
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* rewrite p.
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destruct (isIn b Y).
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+ cbn.
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destruct (dec (b = b)); [reflexivity|].
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by contradiction n.
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+ reflexivity.
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* destruct (isIn b Y).
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+ cbn.
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destruct (dec (a = b)); [|reflexivity].
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by contradiction n.
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+ reflexivity.
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- intros X1 X2 P Q.
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rewrite distributive_intersection_U. simpl.
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rewrite P.
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rewrite Q.
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destruct (isIn a X1) ; destruct (isIn a X2) ; destruct (isIn a Y) ;
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reflexivity.
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Defined.
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End operations_isIn.
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(* Some suporting tactics *)
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Ltac simplify_isIn :=
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repeat (rewrite ?intersection_isIn ;
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rewrite ?union_isIn).
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Ltac toBool := try (intro) ;
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intros ; apply ext ; intros ; simplify_isIn ; eauto with bool_lattice_hints.
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Section SetLattice.
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Context {A : Type}.
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Context {A_deceq : DecidablePaths A}.
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Context `{Funext}.
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Instance fset_union_com : Commutative (@U A).
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Proof.
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toBool.
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Defined.
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Instance fset_intersection_com : Commutative intersection.
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Proof.
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toBool.
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Defined.
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Instance fset_union_assoc : Associative (@U A).
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Proof.
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toBool.
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Defined.
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Instance fset_intersection_assoc : Associative intersection.
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Proof.
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toBool.
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Defined.
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Instance fset_union_idem : Idempotent (@U A).
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Proof. exact union_idem. Defined.
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Instance fset_intersection_idem : Idempotent intersection.
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Proof.
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toBool.
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Defined.
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Instance fset_union_nl : NeutralL (@U A) (@E A).
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Proof.
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toBool.
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Defined.
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Instance fset_union_nr : NeutralR (@U A) (@E A).
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Proof.
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toBool.
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Defined.
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Instance fset_absorption_intersection_union : Absorption (@U A) intersection.
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Proof.
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toBool.
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Defined.
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Instance fset_absorption_union_intersection : Absorption intersection (@U A).
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Proof.
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toBool.
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Defined.
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Instance lattice_fset : Lattice (@U A) intersection (@E A) :=
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{
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commutative_min := _ ;
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commutative_max := _ ;
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associative_min := _ ;
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associative_max := _ ;
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idempotent_min := _ ;
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idempotent_max := _ ;
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neutralL_min := _ ;
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neutralR_min := _ ;
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absorption_min_max := _ ;
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absorption_max_min := _
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}.
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End SetLattice.
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(* Other properties *)
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Section properties.
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Context {A : Type}.
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@@ -25,42 +256,17 @@ Lemma isIn_union (a: A) (X Y: FSet A) :
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Proof. reflexivity. Qed.
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(** comprehension properties *)
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Lemma comprehension_false Y : comprehension (fun a => isIn a E) Y = E.
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Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = E.
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Proof.
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hrecursion Y; try (intros; apply set_path2).
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- cbn. reflexivity.
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- cbn. reflexivity.
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- reflexivity.
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- reflexivity.
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- intros x y IHa IHb.
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cbn.
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rewrite IHa.
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rewrite IHb.
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apply union_idem.
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Defined.
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Theorem comprehension_or : forall ϕ ψ (x: FSet A),
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comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x)
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(comprehension ψ x).
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Proof.
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intros ϕ ψ.
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hinduction; try (intros; apply set_path2).
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- cbn. apply (union_idem _)^.
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- cbn. intros.
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destruct (ϕ a) ; destruct (ψ a) ; symmetry.
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* apply union_idem.
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* apply nr.
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* apply nl.
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* apply union_idem.
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- simpl. intros x y P Q.
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rewrite P.
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rewrite Q.
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rewrite <- assoc.
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rewrite (assoc (comprehension ψ x)).
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rewrite (comm (comprehension ψ x) (comprehension ϕ y)).
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rewrite <- assoc.
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rewrite <- assoc.
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reflexivity.
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Defined.
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Theorem comprehension_subset : forall ϕ (X : FSet A),
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U (comprehension ϕ X) X = X.
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Proof.
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@@ -83,48 +289,7 @@ hrecursion; try (intros ; apply set_path2) ; cbn.
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Defined.
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(** intersection properties *)
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Lemma intersection_0l: forall X: FSet A, intersection E X = E.
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Proof.
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hinduction;
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try (intros ; apply set_path2).
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- reflexivity.
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- intro a.
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reflexivity.
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- intros x y P Q.
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cbn.
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rewrite P.
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rewrite Q.
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apply union_idem.
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Defined.
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Lemma intersection_0r (X: FSet A): intersection X E = E.
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Proof. exact idpath. Defined.
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Theorem intersection_La : forall (a : A) (X : FSet A),
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intersection (L a) X = if isIn a X then L a else E.
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Proof.
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intro a.
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hinduction; try (intros ; apply set_path2).
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- reflexivity.
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- intro b.
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cbn.
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destruct (dec (a = b)) as [p|np].
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* rewrite p.
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destruct (dec (b = b)) as [|nb]; [reflexivity|].
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by contradiction nb.
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* destruct (dec (b = a)); [|reflexivity].
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by contradiction np.
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- unfold intersection.
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intros x y P Q.
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cbn.
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rewrite P.
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rewrite Q.
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destruct (isIn a x) ; destruct (isIn a y).
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* apply idem.
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* apply nr.
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* apply nl.
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* apply nl.
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Defined.
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Lemma intersection_comm X Y: intersection X Y = intersection Y X.
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Proof.
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@@ -175,79 +340,6 @@ hinduction; try (intros ; apply set_path2).
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Defined.
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(** assorted lattice laws *)
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Lemma distributive_La (z : FSet A) (a : A) : forall Y : FSet A,
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intersection (U (L a) z) Y = U (intersection (L a) Y) (intersection z Y).
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Proof.
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hinduction; try (intros ; apply set_path2) ; cbn.
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- symmetry ; apply nl.
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- intros b.
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destruct (dec (b = a)) ; cbn.
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* destruct (isIn b z).
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+ rewrite union_idem.
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reflexivity.
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+ rewrite nr.
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reflexivity.
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* rewrite nl ; reflexivity.
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- intros X1 X2 P Q.
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rewrite P. rewrite Q.
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rewrite <- assoc.
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rewrite (comm (comprehension (fun a0 : A => isIn a0 z) X1)).
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rewrite <- assoc.
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rewrite assoc.
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rewrite (comm (comprehension (fun a0 : A => isIn a0 z) X2)).
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reflexivity.
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Defined.
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Theorem distributive_intersection_U (X1 X2 Y : FSet A) :
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intersection (U X1 X2) Y = U (intersection X1 Y) (intersection X2 Y).
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Proof.
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hinduction X1; try (intros ; apply set_path2) ; cbn.
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- rewrite intersection_0l.
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rewrite nl.
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rewrite nl.
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reflexivity.
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- intro a.
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rewrite intersection_La.
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rewrite distributive_La.
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rewrite intersection_La.
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reflexivity.
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- intros Z1 Z2 P Q.
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unfold intersection in *.
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cbn.
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rewrite comprehension_or.
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rewrite comprehension_or.
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reflexivity.
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Defined.
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Theorem intersection_isIn : forall a (x y: FSet A),
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isIn a (intersection x y) = andb (isIn a x) (isIn a y).
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Proof.
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intros a x y.
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hinduction x; try (intros ; apply set_path2) ; cbn.
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- rewrite intersection_0l.
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reflexivity.
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- intro b.
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rewrite intersection_La.
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destruct (dec (a = b)) ; cbn.
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* rewrite p.
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destruct (isIn b y).
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+ 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) :
|
||||
intersection X (intersection Y Z) = intersection (intersection X Y) Z.
|
||||
@@ -300,77 +392,12 @@ hinduction; try (intros ; apply set_path2).
|
||||
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).
|
||||
Proof.
|
||||
hinduction X1; try (intros ; apply set_path2) ; cbn.
|
||||
- rewrite intersection_0l.
|
||||
rewrite nl.
|
||||
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.
|
||||
toBool.
|
||||
destruct (a ∈ X1), (a ∈ X2), (a ∈ Y); eauto.
|
||||
Defined.
|
||||
|
||||
Theorem absorb_0 (X Y : FSet A) : U X (intersection X Y) = X.
|
||||
Proof.
|
||||
@@ -395,62 +422,9 @@ hinduction X; try (intros ; apply set_path2) ; cbn.
|
||||
reflexivity.
|
||||
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.
|
||||
hrecursion X; try (intros ; apply set_path2).
|
||||
- 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.
|
||||
|
||||
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.
|
||||
toBool.
|
||||
Defined.
|
||||
|
||||
End properties.
|
||||
|
||||
Reference in New Issue
Block a user