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Some cleaning in notation

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Niels
2017-08-07 16:49:46 +02:00
parent 1bab2206a3
commit 1e373364b2
9 changed files with 136 additions and 137 deletions
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12
FiniteSets/implementations/lists.v
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@@ -47,7 +47,7 @@ Section ListToSet.
Context `{Univalence}.
Lemma member_isIn (a : A) (l : list A) :
member a l = isIn a (list_to_set A l).
member a l = a ∈ (list_to_set A l).
Proof.
induction l ; unfold member in * ; simpl in *.
- reflexivity.
@@ -60,7 +60,7 @@ Section ListToSet.
* apply (tr (inr z2)).
Defined.
Definition empty_empty : list_to_set A empty = E := idpath.
Definition empty_empty : list_to_set A empty = ∅ := idpath.
Lemma filter_comprehension (ϕ : A -> Bool) (l : list A) :
list_to_set A (filter ϕ l) = comprehension ϕ (list_to_set A l).
@@ -68,17 +68,17 @@ Section ListToSet.
induction l ; cbn in *.
- reflexivity.
- destruct (ϕ a) ; cbn in * ; unfold list_to_set in IHl.
* refine (ap (fun y => U {|a|} y) _).
* refine (ap (fun y => {|a|} ∪ y) _).
apply IHl.
* rewrite nl.
apply IHl.
Defined.
Definition singleton_single (a : A) : list_to_set A (singleton a) = L a :=
nr (L a).
Definition singleton_single (a : A) : list_to_set A (singleton a) = {|a|} :=
nr {|a|}.
Lemma append_union (l1 l2 : list A) :
list_to_set A (union l1 l2) = U (list_to_set A l1) (list_to_set A l2).
list_to_set A (union l1 l2) = (list_to_set A l1) ∪ (list_to_set A l2).
Proof.
induction l1 ; induction l2 ; cbn.
- apply (union_idem _)^.