mirror of
https://github.com/nmvdw/HITs-Examples
synced 2025-11-03 15:13:51 +01:00
Changed lattice
This commit is contained in:
Niels
@@ -75,11 +75,11 @@ Proof.
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- apply true.
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- apply (fun _ => false).
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- apply andb.
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- intros. symmetry. eauto with bool_lattice_hints.
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- eauto with bool_lattice_hints.
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- eauto with bool_lattice_hints.
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- eauto with bool_lattice_hints.
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- eauto with bool_lattice_hints.
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- intros. symmetry. eauto with lattice_hints typeclass_instances.
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- eauto with bool_lattice_hints typeclass_instances.
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- eauto with bool_lattice_hints typeclass_instances.
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- eauto with bool_lattice_hints typeclass_instances.
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- eauto with bool_lattice_hints typeclass_instances.
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Defined.
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End operations.
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@@ -12,7 +12,14 @@ Section decidable_A.
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intros a.
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hinduction ; try (intros ; apply path_ishprop).
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- apply _.
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- intros. apply _.
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- intros.
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unfold Decidable.
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destruct (dec (a = a0)) as [p | np].
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* apply (inl (tr p)).
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* right.
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intro p.
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strip_truncations.
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contradiction.
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- intros. apply _.
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Defined.
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@@ -4,153 +4,151 @@ Require Export representations.definition disjunction fsets.operations.
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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}.
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Context `{Univalence}.
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Context {A : Type}.
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Context `{Univalence}.
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Lemma union_idem : forall x: FSet A, U x x = x.
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Proof.
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hinduction;
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try (intros ; apply set_path2).
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- apply nl.
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- apply idem.
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- intros x y P Q.
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rewrite assoc.
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rewrite (comm x y).
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rewrite <- (assoc y x x).
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rewrite P.
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rewrite (comm y x).
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rewrite <- (assoc x y y).
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f_ap.
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Qed.
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(** ** Properties about subset relation. *)
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Lemma subset_union (X Y : FSet A) :
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subset X Y -> U X Y = Y.
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Proof.
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hinduction X; try (intros; apply path_forall; intro; apply set_path2).
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- intros. apply nl.
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- intros a. hinduction Y;
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try (intros; apply path_forall; intro; apply set_path2).
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+ intro.
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contradiction.
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+ intro a0.
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simple refine (Trunc_ind _ _).
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intro p ; simpl.
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rewrite p; apply idem.
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+ intros X1 X2 IH1 IH2.
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simple refine (Trunc_ind _ _).
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intros [e1 | e2].
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++ rewrite assoc.
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rewrite (IH1 e1).
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reflexivity.
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++ rewrite comm.
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rewrite <- assoc.
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rewrite (comm X2).
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rewrite (IH2 e2).
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reflexivity.
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- intros X1 X2 IH1 IH2 [G1 G2].
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rewrite <- assoc.
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rewrite (IH2 G2).
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apply (IH1 G1).
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Defined.
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Lemma union_idem : forall x: FSet A, U x x = x.
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Proof.
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hinduction;
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try (intros ; apply set_path2) ; cbn.
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- apply nl.
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- apply idem.
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- intros x y P Q.
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rewrite assoc.
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rewrite (comm x y).
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rewrite <- (assoc y x x).
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rewrite P.
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rewrite (comm y x).
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rewrite <- (assoc x y y).
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f_ap.
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Defined.
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Lemma subset_union_l (X : FSet A) :
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forall Y, subset X (U X Y).
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Proof.
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hinduction X ;
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try (repeat (intro; intros; apply path_forall); intro; apply equiv_hprop_allpath ; apply _).
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- apply (fun _ => tt).
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- intros a Y.
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apply tr ; left ; apply tr ; reflexivity.
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- intros X1 X2 HX1 HX2 Y.
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split.
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* rewrite <- assoc. apply HX1.
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* rewrite (comm X1 X2). rewrite <- assoc. apply HX2.
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Defined.
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(** ** Properties about subset relation. *)
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Lemma subset_union (X Y : FSet A) :
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subset X Y -> U X Y = Y.
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Proof.
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hinduction X; try (intros; apply path_forall; intro; apply set_path2).
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- intros. apply nl.
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- intros a. hinduction Y;
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try (intros; apply path_forall; intro; apply set_path2).
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+ intro.
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contradiction.
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+ intro a0.
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simple refine (Trunc_ind _ _).
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intro p ; cbn.
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rewrite p; apply idem.
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+ intros X1 X2 IH1 IH2.
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simple refine (Trunc_ind _ _).
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intros [e1 | e2].
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++ rewrite assoc.
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rewrite (IH1 e1).
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reflexivity.
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++ rewrite comm.
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rewrite <- assoc.
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rewrite (comm X2).
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rewrite (IH2 e2).
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reflexivity.
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- intros X1 X2 IH1 IH2 [G1 G2].
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rewrite <- assoc.
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rewrite (IH2 G2).
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apply (IH1 G1).
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Defined.
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(* Union and membership *)
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Lemma union_isIn (X Y : FSet A) (a : A) :
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isIn a (U X Y) = 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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Lemma subset_union_l (X : FSet A) :
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forall Y, subset X (U X Y).
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Proof.
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hinduction X ;
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try (repeat (intro; intros; apply path_forall); intro; apply equiv_hprop_allpath ; apply _).
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- apply (fun _ => tt).
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- intros a Y.
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apply tr ; left ; apply tr ; reflexivity.
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- intros X1 X2 HX1 HX2 Y.
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split.
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* rewrite <- assoc. apply HX1.
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* rewrite (comm X1 X2). rewrite <- assoc. apply HX2.
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Defined.
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(* Union and membership *)
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Lemma union_isIn (X Y : FSet A) (a : A) :
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isIn a (U X Y) = 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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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 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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- apply (union_idem _)^.
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- 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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End operations_isIn.
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(* Other properties *)
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Section properties.
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Context {A : Type}.
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Context `{Univalence}.
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Context {A : Type}.
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Context `{Univalence}.
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(** isIn properties *)
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Lemma singleton_isIn (a b: A) : isIn a (L b) -> Trunc (-1) (a = b).
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Proof.
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apply idmap.
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Defined.
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(** isIn properties *)
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Lemma singleton_isIn (a b: A) : isIn a (L b) -> Trunc (-1) (a = b).
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Proof.
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apply idmap.
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Defined.
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Lemma empty_isIn (a: A) : isIn a E -> Empty.
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Proof.
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apply idmap.
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Defined.
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Lemma empty_isIn (a: A) : isIn a E -> Empty.
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Proof.
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apply idmap.
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Defined.
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(** comprehension properties *)
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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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- reflexivity.
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- reflexivity.
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- intros x y IHa IHb.
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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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(** comprehension properties *)
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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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- reflexivity.
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- reflexivity.
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- intros x y IHa IHb.
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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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Lemma 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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intros ϕ.
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hrecursion; try (intros ; apply set_path2) ; cbn.
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- apply union_idem.
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- intro a.
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destruct (ϕ a).
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* apply union_idem.
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* apply nl.
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- intros X Y P Q.
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rewrite assoc.
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rewrite <- (assoc (comprehension ϕ X) (comprehension ϕ Y) X).
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rewrite (comm (comprehension ϕ Y) X).
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rewrite assoc.
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rewrite P.
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rewrite <- assoc.
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rewrite Q.
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reflexivity.
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Defined.
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Lemma 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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intros ϕ.
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hrecursion; try (intros ; apply set_path2) ; cbn.
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- apply union_idem.
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- intro a.
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destruct (ϕ a).
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* apply union_idem.
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* apply nl.
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- intros X Y P Q.
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rewrite assoc.
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rewrite <- (assoc (comprehension ϕ X) (comprehension ϕ Y) X).
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rewrite (comm (comprehension ϕ Y) X).
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rewrite assoc.
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rewrite P.
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rewrite <- assoc.
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rewrite Q.
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reflexivity.
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Defined.
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End properties.
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@@ -6,143 +6,150 @@ Require Export 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} `{Univalence}.
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Context {A : Type} `{DecidablePaths A} `{Univalence}.
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Lemma ext : forall (S T : FSet A), (forall a, isIn_b a S = isIn_b a T) -> S = T.
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Proof.
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intros X Y H2.
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apply fset_ext.
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intro a.
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specialize (H2 a).
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unfold isIn_b, dec in H2.
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destruct (isIn_decidable a X) ; destruct (isIn_decidable a Y).
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- apply path_iff_hprop ; intro ; assumption.
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- contradiction (true_ne_false).
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- contradiction (true_ne_false) ; apply H2^.
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- apply path_iff_hprop ; intro ; contradiction.
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Defined.
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Lemma ext : forall (S T : FSet A), (forall a, isIn_b a S = isIn_b a T) -> S = T.
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Proof.
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intros X Y H2.
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apply fset_ext.
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intro a.
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specialize (H2 a).
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unfold isIn_b, dec in H2.
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destruct (isIn_decidable a X) ; destruct (isIn_decidable a Y).
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- apply path_iff_hprop ; intro ; assumption.
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- contradiction (true_ne_false).
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- contradiction (true_ne_false) ; apply H2^.
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- apply path_iff_hprop ; intro ; contradiction.
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Defined.
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Lemma empty_isIn (a : A) :
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isIn_b a ∅ = false.
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Proof.
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reflexivity.
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Defined.
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Lemma empty_isIn (a : A) :
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isIn_b a ∅ = false.
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Proof.
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reflexivity.
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Defined.
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Lemma L_isIn (a b : A) :
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isIn a {|b|} -> a = b.
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Proof.
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intros. strip_truncations. assumption.
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Defined.
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Lemma L_isIn (a b : A) :
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isIn a {|b|} -> a = b.
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Proof.
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intros. strip_truncations. assumption.
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Defined.
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Lemma L_isIn_b_true (a b : A) (p : a = b) :
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isIn_b a {|b|} = true.
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Proof.
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unfold isIn_b, dec.
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destruct (isIn_decidable a {|b|}) as [n | n] .
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- reflexivity.
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- simpl in n.
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contradiction (n (tr p)).
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Defined.
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Lemma L_isIn_b_true (a b : A) (p : a = b) :
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isIn_b a {|b|} = true.
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Proof.
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unfold isIn_b, dec.
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destruct (isIn_decidable a {|b|}) as [n | n] .
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- reflexivity.
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- simpl in n.
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contradiction (n (tr p)).
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Defined.
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Lemma L_isIn_b_aa (a : A) :
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isIn_b a {|a|} = true.
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Proof.
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apply L_isIn_b_true ; reflexivity.
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Defined.
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Lemma L_isIn_b_aa (a : A) :
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isIn_b a {|a|} = true.
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Proof.
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apply L_isIn_b_true ; reflexivity.
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Defined.
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Lemma L_isIn_b_false (a b : A) (p : a <> b) :
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isIn_b a {|b|} = false.
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Proof.
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unfold isIn_b, dec.
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destruct (isIn_decidable a {|b|}).
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- simpl in t.
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strip_truncations.
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contradiction.
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- reflexivity.
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Defined.
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(* Union and membership *)
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Lemma union_isIn (X Y : FSet A) (a : A) :
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isIn_b a (U X Y) = orb (isIn_b a X) (isIn_b a Y).
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Proof.
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unfold isIn_b ; unfold dec.
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simpl.
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destruct (isIn_decidable a X) ; destruct (isIn_decidable a Y) ; reflexivity.
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Defined.
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Lemma L_isIn_b_false (a b : A) (p : a <> b) :
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isIn_b a {|b|} = false.
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Proof.
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unfold isIn_b, dec.
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destruct (isIn_decidable a {|b|}).
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- simpl in t.
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strip_truncations.
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contradiction.
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- reflexivity.
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Defined.
|
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|
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(* Union and membership *)
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Lemma union_isIn (X Y : FSet A) (a : A) :
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isIn_b a (U X Y) = orb (isIn_b a X) (isIn_b a Y).
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Proof.
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unfold isIn_b ; unfold dec.
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simpl.
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destruct (isIn_decidable a X) ; destruct (isIn_decidable a Y) ; reflexivity.
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||||
Defined.
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||||
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Lemma intersection_isIn (X Y: FSet A) (a : A) :
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Lemma intersection_isIn (X Y: FSet A) (a : A) :
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isIn_b a (intersection X Y) = andb (isIn_b a X) (isIn_b a Y).
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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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destruct (dec (a = b)).
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* rewrite p.
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destruct (isIn_b b Y) ; symmetry ; eauto with bool_lattice_hints.
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* destruct (isIn_b b Y) ; destruct (isIn_b a Y) ; symmetry ; eauto with bool_lattice_hints.
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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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destruct (dec (a = b)).
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* rewrite p.
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destruct (isIn_b b Y) ; symmetry ; eauto with bool_lattice_hints.
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* destruct (isIn_b b Y) ; destruct (isIn_b a Y) ; symmetry ; eauto with bool_lattice_hints.
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+ rewrite and_false.
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symmetry.
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apply (L_isIn_b_false a b n).
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+ rewrite and_true.
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apply (L_isIn_b_false a b n).
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||||
- intros X1 X2 P Q.
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rewrite union_isIn ; rewrite union_isIn.
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||||
rewrite P.
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rewrite Q.
|
||||
unfold isIn_b, dec.
|
||||
destruct (isIn_decidable a X1)
|
||||
; destruct (isIn_decidable a X2)
|
||||
; destruct (isIn_decidable a Y)
|
||||
; reflexivity.
|
||||
Defined.
|
||||
- intros X1 X2 P Q.
|
||||
rewrite union_isIn ; rewrite union_isIn.
|
||||
rewrite P.
|
||||
rewrite Q.
|
||||
unfold isIn_b, dec.
|
||||
destruct (isIn_decidable a X1)
|
||||
; destruct (isIn_decidable a X2)
|
||||
; destruct (isIn_decidable a Y)
|
||||
; reflexivity.
|
||||
Defined.
|
||||
|
||||
Lemma comprehension_isIn (Y : FSet A) (ϕ : A -> Bool) (a : A) :
|
||||
isIn_b a (comprehension ϕ Y) = andb (isIn_b a Y) (ϕ a).
|
||||
Proof.
|
||||
hinduction Y ; try (intros; apply set_path2).
|
||||
- apply empty_isIn.
|
||||
- intro b.
|
||||
destruct (isIn_decidable a {|b|}).
|
||||
* simpl in t.
|
||||
strip_truncations.
|
||||
rewrite t.
|
||||
destruct (ϕ b).
|
||||
** rewrite (L_isIn_b_true _ _ idpath).
|
||||
eauto with bool_lattice_hints.
|
||||
** rewrite empty_isIn ; rewrite (L_isIn_b_true _ _ idpath).
|
||||
eauto with bool_lattice_hints.
|
||||
* destruct (ϕ b).
|
||||
** rewrite L_isIn_b_false.
|
||||
*** eauto with bool_lattice_hints.
|
||||
*** intro.
|
||||
apply (n (tr X)).
|
||||
** rewrite empty_isIn.
|
||||
rewrite L_isIn_b_false.
|
||||
*** eauto with bool_lattice_hints.
|
||||
*** intro.
|
||||
apply (n (tr X)).
|
||||
- intros.
|
||||
Opaque isIn_b.
|
||||
rewrite ?union_isIn.
|
||||
rewrite X.
|
||||
rewrite X0.
|
||||
assert (forall b1 b2 b3,
|
||||
(b1 && b2 || b3 && b2)%Bool = ((b1 || b3) && b2)%Bool).
|
||||
{ intros ; destruct b1, b2, b3 ; reflexivity. }
|
||||
apply X1.
|
||||
Defined.
|
||||
Lemma comprehension_isIn (Y : FSet A) (ϕ : A -> Bool) (a : A) :
|
||||
isIn_b a (comprehension ϕ Y) = andb (isIn_b a Y) (ϕ a).
|
||||
Proof.
|
||||
hinduction Y ; try (intros; apply set_path2).
|
||||
- apply empty_isIn.
|
||||
- intro b.
|
||||
destruct (isIn_decidable a {|b|}).
|
||||
* simpl in t.
|
||||
strip_truncations.
|
||||
rewrite t.
|
||||
destruct (ϕ b).
|
||||
** rewrite (L_isIn_b_true _ _ idpath).
|
||||
eauto with bool_lattice_hints.
|
||||
** rewrite empty_isIn ; rewrite (L_isIn_b_true _ _ idpath).
|
||||
eauto with bool_lattice_hints.
|
||||
* destruct (ϕ b).
|
||||
** rewrite L_isIn_b_false.
|
||||
*** eauto with bool_lattice_hints.
|
||||
*** intro.
|
||||
apply (n (tr X)).
|
||||
** rewrite empty_isIn.
|
||||
rewrite L_isIn_b_false.
|
||||
*** eauto with bool_lattice_hints.
|
||||
*** intro.
|
||||
apply (n (tr X)).
|
||||
- intros.
|
||||
Opaque isIn_b.
|
||||
rewrite ?union_isIn.
|
||||
rewrite X.
|
||||
rewrite X0.
|
||||
assert (forall b1 b2 b3,
|
||||
(b1 && b2 || b3 && b2)%Bool = ((b1 || b3) && b2)%Bool).
|
||||
{ intros ; destruct b1, b2, b3 ; reflexivity. }
|
||||
apply X1.
|
||||
Defined.
|
||||
End operations_isIn.
|
||||
|
||||
Global Opaque isIn_b.
|
||||
(* Some suporting tactics *)
|
||||
(*
|
||||
Ltac simplify_isIn :=
|
||||
repeat (rewrite union_isIn
|
||||
|| rewrite L_isIn_b_aa
|
||||
|| rewrite intersection_isIn
|
||||
|| rewrite comprehension_isIn).
|
||||
*)
|
||||
|
||||
Ltac simplify_isIn :=
|
||||
repeat (rewrite union_isIn
|
||||
|| rewrite L_isIn_b_aa
|
||||
|| rewrite intersection_isIn
|
||||
|| rewrite comprehension_isIn).
|
||||
|
||||
|
||||
Ltac toBool := try (intro) ;
|
||||
intros ; apply ext ; intros ; simplify_isIn ; eauto with bool_lattice_hints.
|
||||
intros ; apply ext ; intros ; simplify_isIn ; eauto with bool_lattice_hints typeclass_instances.
|
||||
|
||||
Section SetLattice.
|
||||
|
||||
@@ -150,70 +157,15 @@ Section SetLattice.
|
||||
Context {A_deceq : DecidablePaths A}.
|
||||
Context `{Univalence}.
|
||||
|
||||
Instance fset_union_com : Commutative (@U A).
|
||||
Proof.
|
||||
toBool.
|
||||
Defined.
|
||||
Instance fset_max : maximum (FSet A) := U.
|
||||
Instance fset_min : minimum (FSet A) := intersection.
|
||||
Instance fset_bot : bottom (FSet A) := E.
|
||||
|
||||
Instance fset_intersection_com : Commutative intersection.
|
||||
Instance lattice_fset : Lattice (FSet A).
|
||||
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.
|
||||
split ; toBool.
|
||||
Defined.
|
||||
|
||||
Instance lattice_fset : Lattice intersection (@U A) (@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.
|
||||
|
||||
(* Comprehension properties *)
|
||||
@@ -236,16 +188,16 @@ Section comprehension_properties.
|
||||
Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = E.
|
||||
Proof.
|
||||
toBool.
|
||||
Defined.
|
||||
Qed.
|
||||
|
||||
Lemma comprehension_all : forall (X : FSet A),
|
||||
comprehension (fun a => isIn_b a X) X = X.
|
||||
Proof.
|
||||
toBool.
|
||||
Defined.
|
||||
Qed.
|
||||
|
||||
Lemma comprehension_subset : forall ϕ (X : FSet A),
|
||||
U (comprehension ϕ X) X = X.
|
||||
U (comprehension ϕ X) X = X.
|
||||
Proof.
|
||||
toBool.
|
||||
Defined.
|
||||
@@ -266,7 +218,7 @@ Section dec_eq.
|
||||
- right. intros [p1 p2]. contradiction.
|
||||
- right. intros [p1 p2]. contradiction.
|
||||
Defined.
|
||||
|
||||
|
||||
Instance fsets_dec_eq : DecidablePaths (FSet A).
|
||||
Proof.
|
||||
intros X Y.
|
||||
|
||||
Reference in New Issue
Block a user