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Make everything work with the new notation

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Dan Frumin
2017-08-08 17:44:27 +02:00
parent 80dabe3162
commit dad6686c4c
4 changed files with 37 additions and 40 deletions
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22
FiniteSets/variations/k_finite.v
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@@ -6,7 +6,7 @@ Section k_finite.
Context (A : Type).
Context `{Univalence}.
Definition map (X : FSet A) : Sub A := fun a => isIn a X.
Definition map (X : FSet A) : Sub A := fun a => a ∈ X.
Global Instance map_injective : IsEmbedding map.
Proof.
@@ -69,37 +69,35 @@ Section structure_k_finite.
Context (A : Type).
Context `{Univalence}.
Lemma map_union : forall X Y : FSet A, map (U X Y) = max_fun (map X) (map Y).
Lemma map_union : forall X Y : FSet A, map (X ∪ Y) = max_fun (map X) (map Y).
Proof.
intros.
unfold map, max_fun.
reflexivity.
Defined.
Lemma k_finite_union : hasUnion (Kf_sub A).
Lemma k_finite_union : closedUnion (Kf_sub A).
Proof.
unfold hasUnion, Kf_sub, Kf_sub_intern.
unfold closedUnion, Kf_sub, Kf_sub_intern.
intros.
destruct X0 as [SX XP].
destruct X1 as [SY YP].
exists (U SX SY).
exists (SX ∪ SY).
rewrite map_union.
rewrite XP, YP.
reflexivity.
Defined.
Lemma k_finite_empty : hasEmpty (Kf_sub A).
Lemma k_finite_empty : closedEmpty (Kf_sub A).
Proof.
unfold hasEmpty, Kf_sub, Kf_sub_intern, map, empty_sub.
exists E.
exists ∅.
reflexivity.
Defined.
Lemma k_finite_singleton : hasSingleton (Kf_sub A).
Lemma k_finite_singleton : closedSingleton (Kf_sub A).
Proof.
unfold hasSingleton, Kf_sub, Kf_sub_intern, map, singleton.
intro.
exists (L a).
exists {|a|}.
cbn.
apply path_forall.
intro z.
@@ -108,7 +106,7 @@ Section structure_k_finite.
Lemma k_finite_hasDecidableEmpty : hasDecidableEmpty (Kf_sub A).
Proof.
unfold hasDecidableEmpty, hasEmpty, Kf_sub, Kf_sub_intern, map.
unfold hasDecidableEmpty, closedEmpty, Kf_sub, Kf_sub_intern, map.
intros.
destruct X0 as [SX EX].
rewrite EX.