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Make everything work with the new notation

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This commit is contained in:
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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12
FiniteSets/variations/enumerated.v
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@@ -223,7 +223,7 @@ Section enumerated_fset.
end.
Lemma list_to_fset_ext (ls : list A) (a : A):
listExt ls a -> isIn a (list_to_fset ls).
listExt ls a -> a ∈ (list_to_fset ls).
Proof.
induction ls as [|x xs]; simpl.
- apply idmap.
@@ -269,7 +269,7 @@ Section fset_dec_enumerated.
- intros a X Hls.
strip_truncations. apply tr.
destruct Hls as [ls Hls].
exists (cons a ls). intros b. simpl.
exists (cons a ls). intros b. cbn.
f_ap.
- intros. apply path_ishprop.
- intros. apply path_ishprop.
@@ -294,16 +294,16 @@ Section subobjects.
Definition enumeratedS (P : Sub A) : hProp :=
enumerated (sigT P).
Lemma enumeratedS_empty : enumeratedS empty_sub.
Lemma enumeratedS_empty : closedEmpty enumeratedS.
Proof.
unfold enumeratedS.
apply tr. exists nil. simpl.
intros [a Ha]. assumption.
Defined.
Lemma enumeratedS_singleton (x : A) : enumeratedS (singleton x).
Lemma enumeratedS_singleton : closedSingleton enumeratedS.
Proof.
apply tr. simpl.
intros x. apply tr. simpl.
exists (cons (x;tr idpath) nil).
intros [y Hxy]. simpl.
strip_truncations. apply tr.
@@ -417,7 +417,7 @@ Section subobjects.
end.
Lemma list_weaken_to_fset_ext (P : Sub A) (ls : list (sigT P)) (a : A) (Ha : P a):
listExt ls (a;Ha) -> isIn a (list_weaken_to_fset P ls).
listExt ls (a;Ha) -> a ∈ (list_weaken_to_fset P ls).
Proof.
induction ls as [|[x Hx] xs]; simpl.
- apply idmap.

22
FiniteSets/variations/k_finite.v
View File

@@ -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.