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Simplified proof of extensionalty and proofs in interface.v

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Niels
2017-08-14 21:38:50 +02:00
parent b274fcddfc
commit 06701dcdf8
4 changed files with 104 additions and 188 deletions
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30
FiniteSets/implementations/interface.v
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@@ -77,49 +77,46 @@ Section properties.
Ltac simplify := intros ; autounfold in * ; apply reflect_eq ; reduce.
Definition well_defined_union : forall (A : Type) (X1 X2 Y1 Y2 : T A),
Definition well_defined_union (A : Type) (X1 X2 Y1 Y2 : T A) :
set_eq A X1 Y1 -> set_eq A X2 Y2 -> set_eq A (union X1 X2) (union Y1 Y2).
Proof.
intros A X1 X2 Y1 Y2 HXY1 HXY2.
intros HXY1 HXY2.
simplify.
by rewrite HXY1, HXY2.
Defined.
Definition well_defined_filter : forall (A : Type) (ϕ : A -> Bool) (X Y : T A),
Definition well_defined_filter (A : Type) (ϕ : A -> Bool) (X Y : T A) :
set_eq A X Y -> set_eq A (filter ϕ X) (filter ϕ Y).
Proof.
intros A ϕ X Y HXY.
intros HXY.
simplify.
by rewrite HXY.
Defined.
Ltac reflect_equality := simplify ; eauto with lattice_hints typeclass_instances.
Lemma union_comm : forall A (X Y : T A),
set_eq A (X ∪ Y) (Y ∪ X).
set_eq A (X ∪ Y) (Y ∪ X).
Proof.
simplify.
apply comm.
reflect_equality.
Defined.
Lemma union_assoc : forall A (X Y Z : T A),
set_eq A ((X ∪ Y) ∪ Z) (X ∪ (Y ∪ Z)) .
set_eq A ((X ∪ Y) ∪ Z) (X ∪ (Y ∪ Z)).
Proof.
simplify.
symmetry.
apply assoc.
reflect_equality.
Defined.
Lemma union_idem : forall A (X : T A),
set_eq A (X ∪ X) X.
Proof.
simplify.
apply union_idem.
reflect_equality.
Defined.
Lemma union_neutral : forall A (X : T A),
set_eq A (∅ ∪ X) X.
Proof.
simplify.
apply nl.
reflect_equality.
Defined.
End properties.
@@ -258,8 +255,7 @@ Proof.
- intros. apply path_forall; intro. apply set_path2.
Defined.
Ltac buggeroff := intros;
(repeat (apply path_forall; intro)); apply set_path2.
Ltac buggeroff := intros; apply path_ishprop.
Instance View_max_assoc A: Associative (@max_L (View A) _).
Proof.