mirror of https://github.com/nmvdw/HITs-Examples
Changed lattice
This commit is contained in:
Niels
parent
77a449e68b
commit
7d74b45fc3
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@ -16,8 +16,8 @@ fsets/properties_decidable.v
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fsets/length.v
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fsets/monad.v
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FSets.v
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Sub.v
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implementations/lists.v
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variations/enumerated.v
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variations/k_finite.v
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#empty_set.v
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#ordered.v
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@ -2,7 +2,7 @@
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Require Import HoTT.
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Require Import lattice.
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Definition lor (X Y : hProp) : hProp := BuildhProp (Trunc (-1) (sum X Y)).
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Instance lor : maximum hProp := fun X Y => BuildhProp (Trunc (-1) (sum X Y)).
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Delimit Scope logic_scope with L.
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Notation "A ∨ B" := (lor A B) (at level 20, right associativity) : logic_scope.
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@ -75,7 +75,8 @@ Section lor_props.
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End lor_props.
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Definition land (X Y : hProp) : hProp := BuildhProp (prod X Y).
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Instance land : minimum hProp := fun X Y => BuildhProp (prod X Y).
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Instance lfalse : bottom hProp := False_hp.
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Notation "A ∧ B" := (land A B) (at level 20, right associativity) : logic_scope.
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Arguments land _%L _%L.
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@ -84,7 +85,7 @@ Open Scope logic_scope.
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Section hPropLattice.
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Context `{Univalence}.
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Global Instance lor_commutative : Commutative lor.
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Instance lor_commutative : Commutative lor.
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Proof.
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unfold Commutative.
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intros.
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@ -131,21 +132,21 @@ Section hPropLattice.
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- intros a ; apply (pair a a).
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Defined.
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Instance lor_neutrall : NeutralL lor False_hp.
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Instance lor_neutrall : NeutralL lor lfalse.
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Proof.
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unfold NeutralL.
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intros.
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apply lor_nl.
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Defined.
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Instance lor_neutralr : NeutralR lor False_hp.
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Instance lor_neutralr : NeutralR lor lfalse.
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Proof.
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unfold NeutralR.
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intros.
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apply lor_nr.
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Defined.
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Instance bool_absorption_orb_andb : Absorption lor land.
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Instance absorption_orb_andb : Absorption lor land.
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Proof.
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unfold Absorption.
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intros.
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@ -156,7 +157,7 @@ Section hPropLattice.
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apply (tr (inl X)).
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Defined.
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Instance bool_absorption_andb_orb : Absorption land lor.
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Instance absorption_andb_orb : Absorption land lor.
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Proof.
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unfold Absorption.
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intros.
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@ -169,15 +170,15 @@ Section hPropLattice.
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* apply (tr (inl X)).
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Defined.
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Global Instance lattice_hprop : Lattice land lor False_hp :=
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Global Instance lattice_hprop : Lattice hProp :=
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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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neutralL_max := _ ;
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neutralR_max := _ ;
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absorption_min_max := _ ;
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absorption_max_min := _
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}.
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@ -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)).
|
||||
Defined.
|
||||
Lemma L_isIn_b_true (a b : A) (p : a = b) :
|
||||
isIn_b a {|b|} = true.
|
||||
Proof.
|
||||
unfold isIn_b, dec.
|
||||
destruct (isIn_decidable a {|b|}) as [n | n] .
|
||||
- reflexivity.
|
||||
- simpl in n.
|
||||
contradiction (n (tr p)).
|
||||
Defined.
|
||||
|
||||
Lemma L_isIn_b_aa (a : A) :
|
||||
isIn_b a {|a|} = true.
|
||||
Proof.
|
||||
apply L_isIn_b_true ; reflexivity.
|
||||
Defined.
|
||||
Lemma L_isIn_b_aa (a : A) :
|
||||
isIn_b a {|a|} = true.
|
||||
Proof.
|
||||
apply L_isIn_b_true ; reflexivity.
|
||||
Defined.
|
||||
|
||||
Lemma L_isIn_b_false (a b : A) (p : a <> b) :
|
||||
isIn_b a {|b|} = false.
|
||||
Proof.
|
||||
unfold isIn_b, dec.
|
||||
destruct (isIn_decidable a {|b|}).
|
||||
- simpl in t.
|
||||
strip_truncations.
|
||||
contradiction.
|
||||
- reflexivity.
|
||||
Defined.
|
||||
|
||||
(* Union and membership *)
|
||||
Lemma union_isIn (X Y : FSet A) (a : A) :
|
||||
isIn_b a (U X Y) = orb (isIn_b a X) (isIn_b a Y).
|
||||
Proof.
|
||||
unfold isIn_b ; unfold dec.
|
||||
simpl.
|
||||
destruct (isIn_decidable a X) ; destruct (isIn_decidable a Y) ; reflexivity.
|
||||
Defined.
|
||||
Lemma L_isIn_b_false (a b : A) (p : a <> b) :
|
||||
isIn_b a {|b|} = false.
|
||||
Proof.
|
||||
unfold isIn_b, dec.
|
||||
destruct (isIn_decidable a {|b|}).
|
||||
- simpl in t.
|
||||
strip_truncations.
|
||||
contradiction.
|
||||
- reflexivity.
|
||||
Defined.
|
||||
|
||||
(* Union and membership *)
|
||||
Lemma union_isIn (X Y : FSet A) (a : A) :
|
||||
isIn_b a (U X Y) = orb (isIn_b a X) (isIn_b a Y).
|
||||
Proof.
|
||||
unfold isIn_b ; unfold dec.
|
||||
simpl.
|
||||
destruct (isIn_decidable a X) ; destruct (isIn_decidable a Y) ; reflexivity.
|
||||
Defined.
|
||||
|
||||
Lemma intersection_isIn (X Y: FSet A) (a : A) :
|
||||
Lemma intersection_isIn (X Y: FSet A) (a : A) :
|
||||
isIn_b a (intersection X Y) = andb (isIn_b a X) (isIn_b a Y).
|
||||
Proof.
|
||||
hinduction X; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intro b.
|
||||
destruct (dec (a = b)).
|
||||
* rewrite p.
|
||||
destruct (isIn_b b Y) ; symmetry ; eauto with bool_lattice_hints.
|
||||
* destruct (isIn_b b Y) ; destruct (isIn_b a Y) ; symmetry ; eauto with bool_lattice_hints.
|
||||
Proof.
|
||||
hinduction X; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intro b.
|
||||
destruct (dec (a = b)).
|
||||
* rewrite p.
|
||||
destruct (isIn_b b Y) ; symmetry ; eauto with bool_lattice_hints.
|
||||
* destruct (isIn_b b Y) ; destruct (isIn_b a Y) ; symmetry ; eauto with bool_lattice_hints.
|
||||
+ rewrite and_false.
|
||||
symmetry.
|
||||
apply (L_isIn_b_false a b n).
|
||||
+ rewrite and_true.
|
||||
apply (L_isIn_b_false a b n).
|
||||
- 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.
|
||||
- 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.
|
||||
|
|
|
|||
|
|
@ -1,7 +1,24 @@
|
|||
(* Typeclass for lattices *)
|
||||
Require Import HoTT.
|
||||
|
||||
Definition operation (A : Type) := A -> A -> A.
|
||||
Section binary_operation.
|
||||
Variable A : Type.
|
||||
|
||||
Definition operation := A -> A -> A.
|
||||
|
||||
Class maximum :=
|
||||
max_L : operation.
|
||||
|
||||
Class minimum :=
|
||||
min_L : operation.
|
||||
|
||||
Class bottom :=
|
||||
empty : A.
|
||||
End binary_operation.
|
||||
|
||||
Arguments max_L {_} _ _.
|
||||
Arguments min_L {_} _ _.
|
||||
Arguments empty {_}.
|
||||
|
||||
Section Defs.
|
||||
Variable A : Type.
|
||||
|
|
@ -19,7 +36,7 @@ Section Defs.
|
|||
Variable g : operation A.
|
||||
|
||||
Class Absorption :=
|
||||
absrob : forall x y, f x (g x y) = x.
|
||||
absorb : forall x y, f x (g x y) = x.
|
||||
|
||||
Variable n : A.
|
||||
|
||||
|
|
@ -40,27 +57,32 @@ Arguments Absorption {_} _ _.
|
|||
|
||||
Section Lattice.
|
||||
Variable A : Type.
|
||||
Variable min max : operation A.
|
||||
Variable empty : A.
|
||||
Context {max_L : maximum A} {min_L : minimum A} {empty_L : bottom A}.
|
||||
|
||||
Class Lattice :=
|
||||
{
|
||||
commutative_min :> Commutative min ;
|
||||
commutative_max :> Commutative max ;
|
||||
associative_min :> Associative min ;
|
||||
associative_max :> Associative max ;
|
||||
idempotent_min :> Idempotent min ;
|
||||
idempotent_max :> Idempotent max ;
|
||||
neutralL_max :> NeutralL max empty ;
|
||||
neutralR_max :> NeutralR max empty ;
|
||||
absorption_min_max :> Absorption min max ;
|
||||
absorption_max_min :> Absorption max min
|
||||
commutative_min :> Commutative min_L ;
|
||||
commutative_max :> Commutative max_L ;
|
||||
associative_min :> Associative min_L ;
|
||||
associative_max :> Associative max_L ;
|
||||
idempotent_min :> Idempotent min_L ;
|
||||
idempotent_max :> Idempotent max_L ;
|
||||
neutralL_max :> NeutralL max_L empty_L ;
|
||||
neutralR_max :> NeutralR max_L empty_L ;
|
||||
absorption_min_max :> Absorption min_L max_L ;
|
||||
absorption_max_min :> Absorption max_L min_L
|
||||
}.
|
||||
|
||||
End Lattice.
|
||||
|
||||
Arguments Lattice {_} _ _ _.
|
||||
Arguments Lattice _ {_} {_} {_}.
|
||||
|
||||
Create HintDb lattice_hints.
|
||||
Hint Resolve associativity : lattice_hints.
|
||||
Hint Resolve commutative : lattice_hints.
|
||||
Hint Resolve absorb : lattice_hints.
|
||||
Hint Resolve idempotency : lattice_hints.
|
||||
Hint Resolve neutralityL : lattice_hints.
|
||||
Hint Resolve neutralityR : lattice_hints.
|
||||
|
||||
Section BoolLattice.
|
||||
|
||||
|
|
@ -71,69 +93,15 @@ Section BoolLattice.
|
|||
; auto
|
||||
; try contradiction.
|
||||
|
||||
Instance orb_com : Commutative orb.
|
||||
Instance maximum_bool : maximum Bool := orb.
|
||||
Instance minimum_bool : minimum Bool := andb.
|
||||
Instance bottom_bool : bottom Bool := false.
|
||||
|
||||
Global Instance lattice_bool : Lattice Bool.
|
||||
Proof.
|
||||
solve_bool.
|
||||
Defined.
|
||||
|
||||
Instance andb_com : Commutative andb.
|
||||
Proof.
|
||||
solve_bool.
|
||||
Defined.
|
||||
|
||||
Instance orb_assoc : Associative orb.
|
||||
Proof.
|
||||
solve_bool.
|
||||
Defined.
|
||||
|
||||
Instance andb_assoc : Associative andb.
|
||||
Proof.
|
||||
solve_bool.
|
||||
Defined.
|
||||
|
||||
Instance orb_idem : Idempotent orb.
|
||||
Proof.
|
||||
solve_bool.
|
||||
Defined.
|
||||
|
||||
Instance andb_idem : Idempotent andb.
|
||||
Proof.
|
||||
solve_bool.
|
||||
Defined.
|
||||
|
||||
Instance orb_nl : NeutralL orb false.
|
||||
Proof.
|
||||
solve_bool.
|
||||
Defined.
|
||||
|
||||
Instance orb_nr : NeutralR orb false.
|
||||
Proof.
|
||||
solve_bool.
|
||||
Defined.
|
||||
|
||||
Instance bool_absorption_orb_andb : Absorption orb andb.
|
||||
Proof.
|
||||
solve_bool.
|
||||
Defined.
|
||||
|
||||
Instance bool_absorption_andb_orb : Absorption andb orb.
|
||||
Proof.
|
||||
solve_bool.
|
||||
split ; solve_bool.
|
||||
Defined.
|
||||
|
||||
Global Instance lattice_bool : Lattice andb orb false :=
|
||||
{ commutative_min := _ ;
|
||||
commutative_max := _ ;
|
||||
associative_min := _ ;
|
||||
associative_max := _ ;
|
||||
idempotent_min := _ ;
|
||||
idempotent_max := _ ;
|
||||
neutralL_max := _ ;
|
||||
neutralR_max := _ ;
|
||||
absorption_min_max := _ ;
|
||||
absorption_max_min := _
|
||||
}.
|
||||
|
||||
Definition and_true : forall b, andb b true = b.
|
||||
Proof.
|
||||
solve_bool.
|
||||
|
|
@ -165,123 +133,52 @@ Section BoolLattice.
|
|||
End BoolLattice.
|
||||
|
||||
Section fun_lattice.
|
||||
Context {A B : Type} {maxB minB : B -> B -> B} {botB : B}.
|
||||
Context `{Lattice B minB maxB botB}.
|
||||
Context {A B : Type}.
|
||||
Context `{Lattice B}.
|
||||
Context `{Funext}.
|
||||
|
||||
Definition max_fun (f g : (A -> B)) (a : A) : B
|
||||
:= maxB (f a) (g a).
|
||||
Global Instance max_fun : maximum (A -> B) :=
|
||||
fun (f g : A -> B) (a : A) => max_L0 (f a) (g a).
|
||||
|
||||
Definition min_fun (f g : (A -> B)) (a : A) : B
|
||||
:= minB (f a) (g a).
|
||||
Global Instance min_fun : minimum (A -> B) :=
|
||||
fun (f g : A -> B) (a : A) => min_L0 (f a) (g a).
|
||||
|
||||
Definition bot_fun (a : A) : B
|
||||
:= botB.
|
||||
Global Instance bot_fun : bottom (A -> B)
|
||||
:= fun _ => empty_L.
|
||||
|
||||
Hint Unfold max_fun min_fun bot_fun.
|
||||
Ltac solve_fun :=
|
||||
compute ; intros ; apply path_forall ; intro ;
|
||||
eauto with lattice_hints typeclass_instances.
|
||||
|
||||
Ltac solve_fun := compute ; intros ; apply path_forall ; intro.
|
||||
|
||||
Instance max_fun_com : Commutative max_fun.
|
||||
Global Instance lattice_fun : Lattice (A -> B).
|
||||
Proof.
|
||||
solve_fun.
|
||||
refine (commutative_max _ _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance min_fun_com : Commutative min_fun.
|
||||
Proof.
|
||||
solve_fun.
|
||||
refine (commutative_min _ _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance max_fun_assoc : Associative max_fun.
|
||||
Proof.
|
||||
solve_fun.
|
||||
refine (associative_max _ _ _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance min_fun_assoc : Associative min_fun.
|
||||
Proof.
|
||||
solve_fun.
|
||||
refine (associative_min _ _ _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance max_fun_idem : Idempotent max_fun.
|
||||
Proof.
|
||||
solve_fun.
|
||||
refine (idempotent_max _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance min_fun_idem : Idempotent min_fun.
|
||||
Proof.
|
||||
solve_fun.
|
||||
refine (idempotent_min _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance max_fun_nl : NeutralL max_fun bot_fun.
|
||||
Proof.
|
||||
solve_fun.
|
||||
simple refine (neutralL_max _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance max_fun_nr : NeutralR max_fun bot_fun.
|
||||
Proof.
|
||||
solve_fun.
|
||||
simple refine (neutralR_max _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance absorption_max_fun_min_fun : Absorption max_fun min_fun.
|
||||
Proof.
|
||||
solve_fun.
|
||||
simple refine (absorption_max_min _ _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance absorption_min_fun_max_fun : Absorption min_fun max_fun.
|
||||
Proof.
|
||||
solve_fun.
|
||||
simple refine (absorption_min_max _ _ _ _ _ _).
|
||||
split ; solve_fun.
|
||||
Defined.
|
||||
|
||||
Global Instance lattice_fun : Lattice min_fun max_fun bot_fun :=
|
||||
{ commutative_min := _ ;
|
||||
commutative_max := _ ;
|
||||
associative_min := _ ;
|
||||
associative_max := _ ;
|
||||
idempotent_min := _ ;
|
||||
idempotent_max := _ ;
|
||||
neutralL_max := _ ;
|
||||
neutralR_max := _ ;
|
||||
absorption_min_max := _ ;
|
||||
absorption_max_min := _
|
||||
}.
|
||||
End fun_lattice.
|
||||
|
||||
Section sub_lattice.
|
||||
Context {A : Type} {P : A -> hProp} {maxA minA : A -> A -> A} {botA : A}.
|
||||
Context {Hmax : forall x y, P x -> P y -> P (maxA x y)}.
|
||||
Context {Hmin : forall x y, P x -> P y -> P (minA x y)}.
|
||||
Context {Hbot : P botA}.
|
||||
Context `{Lattice A minA maxA botA}.
|
||||
Context {A : Type} {P : A -> hProp}.
|
||||
Context `{Lattice A}.
|
||||
Context {Hmax : forall x y, P x -> P y -> P (max_L0 x y)}.
|
||||
Context {Hmin : forall x y, P x -> P y -> P (min_L0 x y)}.
|
||||
Context {Hbot : P empty_L}.
|
||||
|
||||
Definition AP : Type := sig P.
|
||||
|
||||
Definition botAP : AP := (botA ; Hbot).
|
||||
Instance botAP : bottom AP := (empty_L ; Hbot).
|
||||
|
||||
Definition maxAP (x y : AP) : AP :=
|
||||
match x with
|
||||
| (a ; pa) => match y with
|
||||
| (b ; pb) => (maxA a b ; Hmax a b pa pb)
|
||||
end
|
||||
end.
|
||||
Instance maxAP : maximum AP :=
|
||||
fun x y =>
|
||||
match x, y with
|
||||
| (a ; pa), (b ; pb) => (max_L0 a b ; Hmax a b pa pb)
|
||||
end.
|
||||
|
||||
Definition minAP (x y : AP) : AP :=
|
||||
match x with
|
||||
| (a ; pa) => match y with
|
||||
| (b ; pb) => (minA a b ; Hmin a b pa pb)
|
||||
end
|
||||
end.
|
||||
|
||||
Hint Unfold maxAP minAP botAP.
|
||||
Instance minAP : minimum AP :=
|
||||
fun x y =>
|
||||
match x, y with
|
||||
| (a ; pa), (b ; pb) => (min_L0 a b ; Hmin a b pa pb)
|
||||
end.
|
||||
|
||||
Instance hprop_sub : forall c, IsHProp (P c).
|
||||
Proof.
|
||||
|
|
@ -291,85 +188,28 @@ Section sub_lattice.
|
|||
Ltac solve_sub :=
|
||||
let x := fresh in
|
||||
repeat (intro x ; destruct x)
|
||||
; simple refine (path_sigma _ _ _ _ _) ; try (apply hprop_sub).
|
||||
; simple refine (path_sigma _ _ _ _ _)
|
||||
; simpl
|
||||
; try (apply hprop_sub)
|
||||
; eauto 3 with lattice_hints typeclass_instances.
|
||||
|
||||
Instance max_sub_com : Commutative maxAP.
|
||||
Global Instance lattice_sub : Lattice AP.
|
||||
Proof.
|
||||
solve_sub.
|
||||
refine (commutative_max _ _ _ _ _ _).
|
||||
split ; solve_sub.
|
||||
Defined.
|
||||
|
||||
Instance min_sub_com : Commutative minAP.
|
||||
Proof.
|
||||
solve_sub.
|
||||
refine (commutative_min _ _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance max_sub_assoc : Associative maxAP.
|
||||
Proof.
|
||||
solve_sub.
|
||||
refine (associative_max _ _ _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance min_sub_assoc : Associative minAP.
|
||||
Proof.
|
||||
solve_sub.
|
||||
refine (associative_min _ _ _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance max_sub_idem : Idempotent maxAP.
|
||||
Proof.
|
||||
solve_sub.
|
||||
refine (idempotent_max _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance min_sub_idem : Idempotent minAP.
|
||||
Proof.
|
||||
solve_sub.
|
||||
refine (idempotent_min _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance max_sub_nl : NeutralL maxAP botAP.
|
||||
Proof.
|
||||
solve_sub.
|
||||
simple refine (neutralL_max _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance max_sub_nr : NeutralR maxAP botAP.
|
||||
Proof.
|
||||
solve_sub.
|
||||
simple refine (neutralR_max _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance absorption_max_sub_min_sub : Absorption maxAP minAP.
|
||||
Proof.
|
||||
solve_sub.
|
||||
simple refine (absorption_max_min _ _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Instance absorption_min_sub_max_sub : Absorption minAP maxAP.
|
||||
Proof.
|
||||
solve_sub.
|
||||
simple refine (absorption_min_max _ _ _ _ _ _).
|
||||
Defined.
|
||||
|
||||
Global Instance lattice_sub : Lattice minAP maxAP botAP :=
|
||||
{ commutative_min := _ ;
|
||||
commutative_max := _ ;
|
||||
associative_min := _ ;
|
||||
associative_max := _ ;
|
||||
idempotent_min := _ ;
|
||||
idempotent_max := _ ;
|
||||
neutralL_max := _ ;
|
||||
neutralR_max := _ ;
|
||||
absorption_min_max := _ ;
|
||||
absorption_max_min := _
|
||||
}.
|
||||
|
||||
End sub_lattice.
|
||||
|
||||
Create HintDb bool_lattice_hints.
|
||||
Hint Resolve associativity : bool_lattice_hints.
|
||||
Hint Resolve commutative : bool_lattice_hints.
|
||||
Hint Resolve absorb : bool_lattice_hints.
|
||||
Hint Resolve idempotency : bool_lattice_hints.
|
||||
Hint Resolve neutralityL : bool_lattice_hints.
|
||||
Hint Resolve neutralityR : bool_lattice_hints.
|
||||
|
||||
Hint Resolve
|
||||
orb_com andb_com orb_assoc andb_assoc orb_idem andb_idem orb_nl orb_nr
|
||||
bool_absorption_orb_andb bool_absorption_andb_orb and_true and_false
|
||||
associativity
|
||||
and_true and_false
|
||||
dist₁ dist₂ max_min
|
||||
: bool_lattice_hints.
|
||||
|
|
|
|||
Loading…
Reference in New Issue