mirror of https://github.com/nmvdw/HITs-Examples
Proofs of the lattice properties (via extensionality)
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Niels
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@ -0,0 +1,225 @@
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Require Import HoTT.
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Definition operation (A : Type) := A -> A -> A.
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Section Defs.
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Variable A : Type.
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Variable f : A -> A -> A.
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Class Commutative :=
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commutative : forall x y, f x y = f y x.
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Class Associative :=
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associativity : forall x y z, f (f x y) z = f x (f y z).
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Class Idempotent :=
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idempotency : forall x, f x x = x.
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Variable g : operation A.
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Class Absorption :=
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absrob : forall x y, f x (g x y) = x.
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Variable n : A.
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Class NeutralL :=
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neutralityL : forall x, f x n = x.
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Class NeutralR :=
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neutralityR : forall x, f n x = x.
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End Defs.
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Arguments Commutative {_} _.
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Arguments Associative {_} _.
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Arguments Idempotent {_} _.
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Arguments NeutralL {_} _ _.
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Arguments NeutralR {_} _ _.
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Arguments Absorption {_} _ _.
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Section Lattice.
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Variable A : Type.
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Variable min max : operation A.
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Variable empty : A.
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Class Lattice :=
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{
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commutative_min :> Commutative min ;
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commutative_max :> Commutative max ;
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associative_min :> Associative min ;
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associative_max :> Associative max ;
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idempotent_min :> Idempotent min ;
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idempotent_max :> Idempotent max ;
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neutralL_min :> NeutralL min empty ;
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neutralR_min :> NeutralR min empty ;
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absorption_min_max :> Absorption min max ;
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absorption_max_min :> Absorption max min
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}.
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End Lattice.
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Arguments Lattice {_} _ _ _.
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Ltac solve :=
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repeat (intro x ; destruct x)
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; compute
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; auto
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; try contradiction.
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Section BoolLattice.
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Instance min_com : Commutative orb.
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Proof.
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solve.
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Defined.
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Instance max_com : Commutative andb.
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Proof.
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solve.
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Defined.
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Instance min_assoc : Associative orb.
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Proof.
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solve.
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Defined.
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Instance max_assoc : Associative andb.
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Proof.
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solve.
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Defined.
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Instance min_idem : Idempotent orb.
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Proof.
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solve.
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Defined.
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Instance max_idem : Idempotent andb.
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Proof.
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solve.
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Defined.
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Instance min_nl : NeutralL orb false.
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Proof.
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solve.
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Defined.
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Instance min_nr : NeutralR orb false.
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Proof.
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solve.
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Defined.
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Instance bool_absorption_min_max : Absorption orb andb.
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Proof.
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solve.
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Defined.
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Instance bool_absorption_max_min : Absorption andb orb.
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Proof.
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solve.
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Defined.
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Global Instance lattice_bool : Lattice orb andb false :=
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{ commutative_min := _ ;
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commutative_max := _ ;
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associative_min := _ ;
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associative_max := _ ;
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idempotent_min := _ ;
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idempotent_max := _ ;
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neutralL_min := _ ;
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neutralR_min := _ ;
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absorption_min_max := _ ;
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absorption_max_min := _
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}.
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End BoolLattice.
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Require Import definition.
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Require Import properties.
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Section SetLattice.
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Context {A : Type}.
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Context {A_deceq : DecidablePaths A}.
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Context `{Funext}.
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Lemma ext `{Funext} : forall S T, (forall a, isIn a S = isIn a T) -> S = T.
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Proof.
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intros.
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destruct (fset_ext S T).
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destruct equiv_isequiv.
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apply equiv_inv.
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apply X.
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Defined.
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Ltac simplify_isIn :=
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repeat (rewrite ?intersection_isIn ;
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rewrite ?union_isIn).
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Ltac toBool :=
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intros ; apply ext ; intros ; simplify_isIn.
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Instance union_com : Commutative (@U A).
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Proof.
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unfold Commutative ; toBool ; apply min_com.
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Defined.
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Instance intersection_com : Commutative intersection.
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Proof.
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unfold Commutative ; toBool ; apply max_com.
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Defined.
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Instance union_assoc : Associative (@U A).
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Proof.
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unfold Associative ; toBool ; apply min_assoc.
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Defined.
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Instance intersection_assoc : Associative intersection.
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Proof.
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unfold Associative ; toBool ; apply max_assoc.
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Defined.
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Instance union_idem : Idempotent (@U A).
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Proof.
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unfold Idempotent ; toBool ; apply min_idem.
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Defined.
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Instance intersection_idem : Idempotent intersection.
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Proof.
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unfold Idempotent ; toBool ; apply max_idem.
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Defined.
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Instance union_nl : NeutralL (@U A) (@E A).
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Proof.
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unfold NeutralL ; toBool ; apply min_nl.
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Defined.
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Instance union_nr : NeutralR (@U A) (@E A).
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Proof.
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unfold NeutralR ; toBool ; apply min_nr.
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Defined.
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Instance set_absorption_intersection_union : Absorption (@U A) intersection.
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Proof.
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unfold Absorption ; toBool ; apply bool_absorption_min_max.
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Defined.
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Instance set_absorption_union_intersection : Absorption intersection (@U A).
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Proof.
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unfold Absorption ; toBool ; apply bool_absorption_max_min.
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Defined.
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Instance lattice_set : Lattice (@U A) intersection (@E A) :=
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{
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commutative_min := _ ;
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commutative_max := _ ;
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associative_min := _ ;
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associative_max := _ ;
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idempotent_min := _ ;
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idempotent_max := _ ;
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neutralL_min := _ ;
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neutralR_min := _ ;
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absorption_min_max := _ ;
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absorption_max_min := _
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}.
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End SetLattice.
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@ -6,3 +6,4 @@ properties.v
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empty_set.v
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ordered.v
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cons_repr.v
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Lattice.v
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@ -49,9 +49,11 @@ hrecursion Y; try (intros; apply set_path2).
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- cbn. reflexivity.
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- cbn. reflexivity.
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- intros x y IHa IHb.
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cbn.
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rewrite IHa.
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rewrite IHb.
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apply union_idem.
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rewrite nl.
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reflexivity.
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Defined.
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Theorem comprehension_or : forall ϕ ψ (x: FSet A),
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@ -60,7 +62,7 @@ Theorem comprehension_or : forall ϕ ψ (x: FSet A),
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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. symmetry ; apply nl.
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- cbn. intros.
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destruct (ϕ a) ; destruct (ψ a) ; symmetry.
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* apply idem.
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@ -84,7 +86,7 @@ Theorem comprehension_subset : forall ϕ (X : FSet A),
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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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- apply nl.
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- intro a.
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destruct (ϕ a).
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* apply union_idem.
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@ -108,7 +110,9 @@ try (intros ; apply set_path2).
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- reflexivity.
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- intro a.
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reflexivity.
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- intros x y P Q.
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- unfold intersection.
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intros x y P Q.
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cbn.
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rewrite P.
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rewrite Q.
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apply nl.
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@ -146,26 +150,27 @@ Defined.
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Lemma intersection_comm X Y: intersection X Y = intersection Y X.
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Proof.
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hrecursion X; try (intros; apply set_path2).
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- apply intersection_0l.
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- intro a.
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- cbn. unfold intersection. apply comprehension_false.
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- cbn. unfold intersection. intros a.
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hrecursion Y; try (intros; apply set_path2).
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+ reflexivity.
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+ intros b.
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+ cbn. reflexivity.
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+ cbn. intros b.
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destruct (dec (a = b)) as [pa|npa].
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* rewrite pa.
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destruct (dec (b = b)) as [|nb]; [reflexivity|].
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by contradiction nb.
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* destruct (dec (b = a)) as [pb|]; [|reflexivity].
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by contradiction npa.
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+ intros Y1 Y2 IH1 IH2.
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+ cbn -[isIn]. intros Y1 Y2 IH1 IH2.
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rewrite IH1.
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rewrite IH2.
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symmetry.
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apply (comprehension_or (fun a => isIn a Y1) (fun a => isIn a Y2) (L a)).
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- intros X1 X2 IH1 IH2.
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cbn.
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unfold intersection in *.
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rewrite <- IH1.
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rewrite <- IH2.
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unfold intersection. simpl.
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apply comprehension_or.
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Defined.
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@ -437,7 +442,6 @@ Proof.
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reflexivity.
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Defined.
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(* Proof of the extensionality axiom *)
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(* Properties about subset relation. *)
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Lemma subset_union `{Funext} (X Y : FSet A) :
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subset X Y = true -> U X Y = Y.
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@ -473,9 +477,20 @@ hinduction X; try (intros; apply path_forall; intro; apply set_path2).
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* contradiction (false_ne_true).
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Defined.
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Lemma eq1 (X Y : FSet A) : X = Y <~> (U Y X = X) * (U X Y = Y).
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Proof.
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unshelve eapply BuildEquiv.
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{ intro H. rewrite H. split; apply union_idem. }
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unshelve esplit.
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{ intros [H1 H2]. etransitivity. apply H1^.
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rewrite comm. apply H2. }
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intro; apply path_prod; apply set_path2.
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all: intro; apply set_path2.
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Defined.
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Lemma subset_union_l `{Funext} X :
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forall Y, subset X (U X Y) = true.
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Proof.
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hinduction X;
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try (intros; apply path_forall; intro; apply set_path2).
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- reflexivity.
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@ -483,45 +498,37 @@ hinduction X;
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* reflexivity.
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* by contradiction n.
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- intros X1 X2 HX1 HX2 Y.
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refine (ap (fun z => (X1 ⊆ z && X2 ⊆ (X1 ∪ X2) ∪ Y))%Bool (assoc X1 X2 Y)^ @ _).
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refine (ap (fun z => (X1 ⊆ _ && X2 ⊆ z ∪ Y))%Bool (comm _ _) @ _).
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refine (ap (fun z => (X1 ⊆ _ && X2 ⊆ z))%Bool (assoc _ _ _)^ @ _).
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rewrite HX1. simpl. apply HX2.
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enough (subset X1 (U (U X1 X2) Y) = true).
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enough (subset X2 (U (U X1 X2) Y) = true).
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rewrite X. rewrite X0. reflexivity.
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{ rewrite (comm X1 X2).
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rewrite <- (assoc X2 X1 Y).
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apply (HX2 (U X1 Y)). }
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{ rewrite <- (assoc X1 X2 Y). apply (HX1 (U X2 Y)). }
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Defined.
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Lemma subset_union_equiv `{Funext}
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: forall X Y : FSet A, subset X Y = true <~> U X Y = Y.
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Proof.
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intros X Y.
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eapply equiv_iff_hprop_uncurried.
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split.
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- apply subset_union.
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- intros HXY. etransitivity.
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apply (ap _ HXY^).
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apply subset_union_l.
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Defined.
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Lemma eq_subset' (X Y : FSet A) : X = Y <~> (U Y X = X) * (U X Y = Y).
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Proof.
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eapply equiv_iff_hprop_uncurried.
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split.
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- intro H. split.
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apply (comm Y X @ ap (U X) H^ @ union_idem X).
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apply (ap (U X) H^ @ union_idem X @ H).
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- intros [H1 H2]. etransitivity. apply H1^.
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apply (comm Y X @ H2).
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unshelve eapply BuildEquiv.
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apply subset_union.
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unshelve esplit.
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{ intros HXY. rewrite <- HXY. clear HXY.
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apply subset_union_l. }
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all: intro; apply set_path2.
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Defined.
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Lemma eq_subset `{Funext} (X Y : FSet A) :
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X = Y <~> ((subset Y X = true) * (subset X Y = true)).
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Proof.
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transitivity ((U Y X = X) * (U X Y = Y)).
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apply eq_subset'.
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apply eq1.
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symmetry.
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eapply equiv_functor_prod'; apply subset_union_equiv.
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Defined.
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Lemma subset_isIn `{FE :Funext} (X Y : FSet A) :
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Lemma subset_isIn `{FE : Funext} (X Y : FSet A) :
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(forall (a : A), isIn a X = true -> isIn a Y = true)
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<-> (subset X Y = true).
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Proof.
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@ -555,26 +562,44 @@ Proof.
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destruct (dec (a = b)).
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+ intros ; rewrite p ; apply H.
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+ intros X ; contradiction (false_ne_true X).
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* intros X1 X2.
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intros IH1 IH2 H1 a H2.
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destruct (subset X1 Y) ; destruct (subset X2 Y);
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cbv in H1; try by contradiction false_ne_true.
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specialize (IH1 idpath a). specialize (IH2 idpath a).
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destruct (isIn a X1); destruct (isIn a X2);
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cbv in H2; try by contradiction false_ne_true.
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by apply IH1.
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by apply IH1.
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by apply IH2.
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* repeat (intro; intros; apply path_forall).
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intros; intro; intros; apply set_path2.
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* repeat (intro; intros; apply path_forall).
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intros; intro; intros; apply set_path2.
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* repeat (intro; intros; apply path_forall).
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intros; intro; intros; apply set_path2.
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* repeat (intro; intros; apply path_forall).
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intros; intro; intros; apply set_path2.
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* repeat (intro; intros; apply path_forall);
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intros; intro; intros; apply set_path2.
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* intros X1 X2.
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intros IH1 IH2 H1 a H2.
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destruct (subset X1 Y) ; destruct (subset X2 Y);
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cbv in H1; try by contradiction false_ne_true.
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specialize (IH1 idpath a). specialize (IH2 idpath a).
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destruct (isIn a X1); destruct (isIn a X2);
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cbv in H2; try by contradiction false_ne_true.
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by apply IH1.
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by apply IH1.
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by apply IH2.
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* repeat (intro; intros; apply path_forall).
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intros; intro; intros; apply set_path2.
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* repeat (intro; intros; apply path_forall).
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intros; intro; intros; apply set_path2.
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* repeat (intro; intros; apply path_forall).
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intros; intro; intros; apply set_path2.
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* repeat (intro; intros; apply path_forall).
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intros; intro; intros; apply set_path2.
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* repeat (intro; intros; apply path_forall);
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intros; intro; intros; apply set_path2.
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Defined.
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Lemma HPropEquiv (X Y : Type) (P : IsHProp X) (Q : IsHProp Y) :
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(X <-> Y) -> (X <~> Y).
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Proof.
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intros [f g].
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simple refine (BuildEquiv _ _ _ _).
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apply f.
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simple refine (BuildIsEquiv _ _ _ _ _ _ _).
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- apply g.
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- unfold Sect.
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intro x.
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apply Q.
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- unfold Sect.
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intro x.
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apply P.
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- intros.
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apply set_path2.
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Defined.
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Theorem fset_ext `{Funext} (X Y : FSet A) :
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@ -584,10 +609,18 @@ Proof.
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transitivity
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((forall a, isIn a Y = true -> isIn a X = true)
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*(forall a, isIn a X = true -> isIn a Y = true)).
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- eapply equiv_functor_prod';
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apply equiv_iff_hprop_uncurried;
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- eapply equiv_functor_prod'.
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apply HPropEquiv.
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exact _.
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exact _.
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split ; apply subset_isIn.
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- apply equiv_iff_hprop_uncurried.
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apply HPropEquiv.
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exact _.
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exact _.
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split ; apply subset_isIn.
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- apply HPropEquiv.
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exact _.
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exact _.
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split.
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* intros [H1 H2 a].
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specialize (H1 a) ; specialize (H2 a).
|
||||
|
|
@ -604,4 +637,37 @@ Proof.
|
|||
apply H2.
|
||||
Defined.
|
||||
|
||||
Ltac solve :=
|
||||
repeat (intro x ; destruct x)
|
||||
; compute
|
||||
; auto
|
||||
; try contradiction.
|
||||
|
||||
Ltac simplify_isIn :=
|
||||
repeat (rewrite ?intersection_isIn ;
|
||||
rewrite ?union_isIn).
|
||||
|
||||
Lemma ext `{Funext} : forall S T, (forall a, isIn a S = isIn a T) -> S = T.
|
||||
Proof.
|
||||
intros.
|
||||
destruct (fset_ext S T).
|
||||
destruct equiv_isequiv.
|
||||
apply equiv_inv.
|
||||
apply X.
|
||||
Defined.
|
||||
|
||||
Ltac toBool :=
|
||||
intros ; apply ext ; intros ; simplify_isIn.
|
||||
|
||||
Lemma andb_comm : forall x y, andb x y = andb y x.
|
||||
Proof.
|
||||
solve.
|
||||
Defined.
|
||||
|
||||
Lemma intersectioncomm `{Funext} : forall x y, intersection x y = intersection y x.
|
||||
Proof.
|
||||
toBool.
|
||||
destruct_all bool. apply andb_comm.
|
||||
Defined.
|
||||
|
||||
End properties.
|
||||
Loading…
Reference in New Issue