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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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24
FiniteSets/fsets/properties.v
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@@ -7,7 +7,7 @@ Section operations_isIn.
Context {A : Type}.
Context `{Univalence}.
Lemma union_idem : forall x: FSet A, U x x = x.
Lemma union_idem : forall x: FSet A, x ∪ x = x.
Proof.
hinduction ; try (intros ; apply set_path2).
- apply nl.
@@ -24,7 +24,7 @@ Section operations_isIn.
(** ** Properties about subset relation. *)
Lemma subset_union (X Y : FSet A) :
subset X Y -> U X Y = Y.
X ⊆ Y -> X ∪ Y = Y.
Proof.
hinduction X ; try (intros; apply path_forall; intro; apply set_path2).
- intros. apply nl.
@@ -54,7 +54,7 @@ Section operations_isIn.
Defined.
Lemma subset_union_l (X : FSet A) :
forall Y, subset X (U X Y).
forall Y, X ⊆ X ∪ Y.
Proof.
hinduction X ; try (repeat (intro; intros; apply path_forall);
intro ; apply equiv_hprop_allpath ; apply _).
@@ -69,8 +69,8 @@ Section operations_isIn.
(* simplify it using extensionality *)
Lemma comprehension_or : forall ϕ ψ (x: FSet A),
comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x)
(comprehension ψ x).
comprehension (fun a => orb (ϕ a) (ψ a)) x
= (comprehension ϕ x) ∪ (comprehension ψ x).
Proof.
intros ϕ ψ.
hinduction ; try (intros; apply set_path2).
@@ -101,15 +101,15 @@ Section properties.
Context `{Univalence}.
(** isIn properties *)
Definition empty_isIn (a: A) : isIn a E -> Empty := idmap.
Definition empty_isIn (a: A) : a ∈ E -> Empty := idmap.
Definition singleton_isIn (a b: A) : isIn a (L b) -> Trunc (-1) (a = b) := idmap.
Definition singleton_isIn (a b: A) : a ∈ {|b|} -> Trunc (-1) (a = b) := idmap.
Definition union_isIn (X Y : FSet A) (a : A)
: isIn a (U X Y) = isIn a X ∨ isIn a Y := idpath.
: a ∈ X ∪ Y = a ∈ X ∨ a ∈ Y := idpath.
Lemma comprehension_isIn (ϕ : A -> Bool) (a : A) : forall X : FSet A,
isIn a (comprehension ϕ X) = if ϕ a then isIn a X else False_hp.
a ∈ (comprehension ϕ X) = if ϕ a then a ∈ X else False_hp.
Proof.
hinduction ; try (intros ; apply set_path2) ; cbn.
- destruct (ϕ a) ; reflexivity.
@@ -139,7 +139,7 @@ Section properties.
(* The proof can be simplified using extensionality *)
(** comprehension properties *)
Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = E.
Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = ∅.
Proof.
hrecursion Y; try (intros; apply set_path2).
- reflexivity.
@@ -151,7 +151,7 @@ Section properties.
(* Can be simplified using extensionality *)
Lemma comprehension_subset : forall ϕ (X : FSet A),
U (comprehension ϕ X) X = X.
(comprehension ϕ X) ∪ X = X.
Proof.
intros ϕ.
hrecursion; try (intros ; apply set_path2) ; cbn.
@@ -171,7 +171,7 @@ Section properties.
reflexivity.
Defined.
Lemma merely_choice : forall X : FSet A, hor (X = E) (hexists (fun a => isIn a X)).
Lemma merely_choice : forall X : FSet A, hor (X = ∅) (hexists (fun a => a ∈ X)).
Proof.
hinduction; try (intros; apply equiv_hprop_allpath ; apply _).
- apply (tr (inl idpath)).