mirror of https://github.com/nmvdw/HITs-Examples
144 lines
3.5 KiB
Coq
144 lines
3.5 KiB
Coq
(* Properties of [FSet A] where [A] has decidable equality *)
|
||
Require Import HoTT HitTactics.
|
||
From fsets Require Export properties extensionality operations_decidable.
|
||
Require Export lattice.
|
||
|
||
(* Lemmas relating operations to the membership predicate *)
|
||
Section operations_isIn.
|
||
|
||
Context {A : Type} `{DecidablePaths A} `{Univalence}.
|
||
|
||
Lemma ext : forall (S T : FSet A), (forall a, isIn_b a S = isIn_b a T) -> S = T.
|
||
Proof.
|
||
intros X Y H2.
|
||
apply fset_ext.
|
||
intro a.
|
||
specialize (H2 a).
|
||
unfold isIn_b, dec in H2.
|
||
destruct (isIn_decidable a X) ; destruct (isIn_decidable a Y).
|
||
- apply path_iff_hprop ; intro ; assumption.
|
||
- contradiction (true_ne_false).
|
||
- contradiction (true_ne_false) ; apply H2^.
|
||
- apply path_iff_hprop ; intro ; contradiction.
|
||
Defined.
|
||
|
||
Lemma empty_isIn (a : A) :
|
||
isIn_b a ∅ = false.
|
||
Proof.
|
||
reflexivity.
|
||
Defined.
|
||
|
||
Lemma L_isIn (a b : A) :
|
||
a ∈ {|b|} -> a = b.
|
||
Proof.
|
||
intros. strip_truncations. assumption.
|
||
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_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_b (X Y : FSet A) (a : A) :
|
||
isIn_b a (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 comprehension_isIn_b (Y : FSet A) (ϕ : A -> Bool) (a : A) :
|
||
isIn_b a {|Y & ϕ|} = andb (isIn_b a Y) (ϕ a).
|
||
Proof.
|
||
unfold isIn_b, dec ; simpl.
|
||
destruct (isIn_decidable a {|Y & ϕ|}) as [t | t]
|
||
; destruct (isIn_decidable a Y) as [n | n] ; rewrite comprehension_isIn in t
|
||
; destruct (ϕ a) ; try reflexivity ; try contradiction.
|
||
Defined.
|
||
|
||
Lemma intersection_isIn_b (X Y: FSet A) (a : A) :
|
||
isIn_b a (intersection X Y) = andb (isIn_b a X) (isIn_b a Y).
|
||
Proof.
|
||
apply comprehension_isIn_b.
|
||
Defined.
|
||
|
||
End operations_isIn.
|
||
|
||
(* Some suporting tactics *)
|
||
Ltac simplify_isIn_b :=
|
||
repeat (rewrite union_isIn_b
|
||
|| rewrite L_isIn_b_aa
|
||
|| rewrite intersection_isIn_b
|
||
|| rewrite comprehension_isIn_b).
|
||
|
||
Ltac toBool :=
|
||
repeat intro;
|
||
apply ext ; intros ;
|
||
simplify_isIn_b ; eauto with bool_lattice_hints typeclass_instances.
|
||
|
||
Section SetLattice.
|
||
|
||
Context {A : Type}.
|
||
Context {A_deceq : DecidablePaths A}.
|
||
Context `{Univalence}.
|
||
|
||
Instance fset_max : maximum (FSet A).
|
||
Proof.
|
||
intros x y.
|
||
apply (x ∪ y).
|
||
Defined.
|
||
|
||
Instance fset_min : minimum (FSet A).
|
||
Proof.
|
||
intros x y.
|
||
apply (intersection x y).
|
||
Defined.
|
||
|
||
Instance fset_bot : bottom (FSet A).
|
||
Proof.
|
||
unfold bottom.
|
||
apply ∅.
|
||
Defined.
|
||
|
||
Instance lattice_fset : Lattice (FSet A).
|
||
Proof.
|
||
split; toBool.
|
||
Defined.
|
||
|
||
End SetLattice.
|
||
|
||
(* With extensionality we can prove decidable equality *)
|
||
Section dec_eq.
|
||
Context (A : Type) `{DecidablePaths A} `{Univalence}.
|
||
|
||
Instance fsets_dec_eq : DecidablePaths (FSet A).
|
||
Proof.
|
||
intros X Y.
|
||
apply (decidable_equiv' ((Y ⊆ X) * (X ⊆ Y)) (eq_subset X Y)^-1).
|
||
apply decidable_prod.
|
||
Defined.
|
||
|
||
End dec_eq. |