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Code Issues Releases Wiki Activity

Some cleaning

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This commit is contained in:
Niels
2017-08-07 16:22:55 +02:00
parent a0844f6be4
commit 1bab2206a3
2 changed files with 86 additions and 97 deletions
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121
FiniteSets/fsets/properties.v
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@@ -9,8 +9,7 @@ Section operations_isIn.
Lemma union_idem : forall x: FSet A, U x x = x.
Proof.
hinduction;
try (intros ; apply set_path2).
hinduction ; try (intros ; apply set_path2).
- apply nl.
- apply idem.
- intros x y P Q.
@@ -21,16 +20,16 @@ Section operations_isIn.
rewrite (comm y x).
rewrite <- (assoc x y y).
f_ap.
Qed.
Defined.
(** ** Properties about subset relation. *)
Lemma subset_union (X Y : FSet A) :
subset X Y -> U X Y = Y.
Proof.
hinduction X; try (intros; apply path_forall; intro; apply set_path2).
hinduction X ; try (intros; apply path_forall; intro; apply set_path2).
- intros. apply nl.
- intros a. hinduction Y;
try (intros; apply path_forall; intro; apply set_path2).
- intros a.
hinduction Y ; try (intros; apply path_forall; intro; apply set_path2).
+ intro.
contradiction.
+ intro a0.
@@ -53,34 +52,28 @@ Section operations_isIn.
rewrite (IH2 G2).
apply (IH1 G1).
Defined.
Lemma subset_union_l (X : FSet A) :
forall Y, subset X (U X Y).
Proof.
hinduction X ;
try (repeat (intro; intros; apply path_forall); intro; apply equiv_hprop_allpath ; apply _).
hinduction X ; try (repeat (intro; intros; apply path_forall);
intro ; apply equiv_hprop_allpath ; apply _).
- apply (fun _ => tt).
- intros a Y.
apply tr ; left ; apply tr ; reflexivity.
apply (tr(inl(tr idpath))).
- intros X1 X2 HX1 HX2 Y.
split.
* rewrite <- assoc. apply HX1.
* rewrite (comm X1 X2). rewrite <- assoc. apply HX2.
Defined.
(* Union and membership *)
Lemma union_isIn (X Y : FSet A) (a : A) :
isIn a (U X Y) = isIn a X ∨ isIn a Y.
Proof.
reflexivity.
Defined.
(* 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).
Proof.
intros ϕ ψ.
hinduction; try (intros; apply set_path2).
hinduction ; try (intros; apply set_path2).
- apply (union_idem _)^.
- intros.
destruct (ϕ a) ; destruct (ψ a) ; symmetry.
@@ -108,16 +101,43 @@ Section properties.
Context `{Univalence}.
(** isIn properties *)
Lemma singleton_isIn (a b: A) : isIn a (L b) -> Trunc (-1) (a = b).
Definition empty_isIn (a: A) : isIn a E -> Empty := idmap.
Definition singleton_isIn (a b: A) : isIn a (L 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.
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.
Proof.
apply idmap.
Defined.
Lemma empty_isIn (a: A) : isIn a E -> Empty.
Proof.
apply idmap.
hinduction ; try (intros ; apply set_path2) ; cbn.
- destruct (ϕ a) ; reflexivity.
- intros b.
assert (forall c d, ϕ a = c -> ϕ b = d ->
a ∈ (if ϕ b then {|b|} else ∅)
=
(if ϕ a then BuildhProp (Trunc (-1) (a = b)) else False_hp)) as X.
{
intros c d Hc Hd.
destruct c ; destruct d ; rewrite Hc, Hd ; try reflexivity
; apply path_iff_hprop ; try contradiction ; intros ; strip_truncations
; apply (false_ne_true).
* apply (Hd^ @ ap ϕ X^ @ Hc).
* apply (Hc^ @ ap ϕ X @ Hd).
}
apply (X (ϕ a) (ϕ b) idpath idpath).
- intros X Y H1 H2.
rewrite H1, H2.
destruct (ϕ a).
* reflexivity.
* apply path_iff_hprop.
** intros Z ; strip_truncations.
destruct Z ; assumption.
** intros ; apply tr ; right ; assumption.
Defined.
(* The proof can be simplified using extensionality *)
(** comprehension properties *)
Lemma comprehension_false Y : comprehension (fun (_ : A) => false) Y = E.
Proof.
@@ -125,11 +145,11 @@ Section properties.
- reflexivity.
- reflexivity.
- intros x y IHa IHb.
rewrite IHa.
rewrite IHb.
rewrite IHa, IHb.
apply union_idem.
Defined.
(* Can be simplified using extensionality *)
Lemma comprehension_subset : forall ϕ (X : FSet A),
U (comprehension ϕ X) X = X.
Proof.
@@ -160,54 +180,21 @@ Section properties.
- intros X Y TX TY.
strip_truncations.
destruct TX as [XE | HX] ; destruct TY as [YE | HY] ; try(strip_truncations ; apply tr).
* apply tr ; left.
* refine (tr (inl _)).
rewrite XE, YE.
apply (union_idem E).
* right.
destruct HY as [a Ya].
apply tr.
* destruct HY as [a Ya].
refine (inr (tr _)).
exists a.
apply (tr (inr Ya)).
* right.
destruct HX as [a Xa].
apply tr.
* destruct HX as [a Xa].
refine (inr (tr _)).
exists a.
apply (tr (inl Xa)).
* right.
destruct HX as [a Xa].
destruct HY as [b Yb].
apply tr.
* destruct (HX, HY) as [[a Xa] [b Yb]].
refine (inr (tr _)).
exists a.
apply (tr (inl Xa)).
Defined.
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.
Proof.
hinduction ; try (intros ; apply set_path2) ; cbn.
- destruct (ϕ a) ; reflexivity.
- intros b.
assert (forall c d, ϕ a = c -> ϕ b = d ->
a ∈ (if ϕ b then {|b|} else ∅)
=
(if ϕ a then BuildhProp (Trunc (-1) (a = b)) else False_hp)) as X.
{
intros c d Hc Hd.
destruct c ; destruct d ; rewrite Hc, Hd ; try reflexivity
; apply path_iff_hprop ; try contradiction ; intros ; strip_truncations
; apply (false_ne_true).
* apply (Hd^ @ ap ϕ X^ @ Hc).
* apply (Hc^ @ ap ϕ X @ Hd).
}
apply (X (ϕ a) (ϕ b) idpath idpath).
- intros X Y H1 H2.
rewrite H1, H2.
destruct (ϕ a).
* reflexivity.
* apply path_iff_hprop.
** intros Z ; strip_truncations.
destruct Z ; assumption.
** intros ; apply tr ; right ; assumption.
Defined.
End properties.