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Code Issues Releases Wiki Activity

Clean up trailing whitespaces and an unused definition.

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This commit is contained in:
Dan Frumin
2017-08-09 18:05:58 +02:00
parent 31e08af1d1
commit 33808928db
15 changed files with 137 additions and 152 deletions
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10
FiniteSets/fsets/extensionality.v
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@@ -6,7 +6,7 @@ Section ext.
Context {A : Type}.
Context `{Univalence}.
Lemma subset_union (X Y : FSet A) :
Lemma subset_union (X Y : FSet A) :
X ⊆ Y -> X ∪ Y = Y.
Proof.
hinduction X ; try (intros; apply path_forall; intro; apply set_path2).
@@ -17,7 +17,7 @@ Section ext.
contradiction.
+ intro a0.
simple refine (Trunc_ind _ _).
intro p ; simpl.
intro p ; simpl.
rewrite p; apply idem.
+ intros X1 X2 IH1 IH2.
simple refine (Trunc_ind _ _).
@@ -112,8 +112,8 @@ Section ext.
unshelve esplit.
{ intros [H1 H2]. etransitivity. apply H1^.
rewrite comm. apply H2. }
intro; apply path_prod; apply set_path2.
all: intro; apply set_path2.
intro; apply path_prod; apply set_path2.
all: intro; apply set_path2.
Defined.
Lemma eq_subset (X Y : FSet A) :
@@ -149,4 +149,4 @@ Section ext.
split ; apply subset_isIn.
Defined.
End ext.
End ext.

8
FiniteSets/fsets/isomorphism.v
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@@ -14,7 +14,7 @@ Section Iso.
- apply E.
- intros a x.
apply ({|a|} ∪ x).
- intros. cbn.
- intros. cbn.
etransitivity. apply assoc.
apply (ap (∪ x)).
apply idem.
@@ -22,7 +22,7 @@ Section Iso.
etransitivity. apply assoc.
etransitivity. refine (ap (∪ x) _ ).
apply FSet.comm.
symmetry.
symmetry.
apply assoc.
Defined.
@@ -39,10 +39,10 @@ Section Iso.
- apply singleton_idem.
Defined.
Lemma append_union: forall (x y: FSetC A),
Lemma append_union: forall (x y: FSetC A),
FSetC_to_FSet (x ∪ y) = (FSetC_to_FSet x) ∪ (FSetC_to_FSet y).
Proof.
intros x.
intros x.
hrecursion x; try (intros; apply path_forall; intro; apply set_path2).
- intros. symmetry. apply nl.
- intros a x HR y. unfold union, fsetc_union in *. rewrite HR. apply assoc.

27
FiniteSets/fsets/operations.v
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@@ -34,7 +34,7 @@ Section operations.
- apply comm.
- apply nl.
- apply nr.
- intros; simpl.
- intros; simpl.
destruct (P x).
+ apply idem.
+ apply nl.
@@ -61,7 +61,7 @@ Section operations.
- intros ; apply nr.
- intros ; apply idem.
Defined.
Definition product {A B : Type} : FSet A -> FSet B -> FSet (A * B).
Proof.
intros X Y.
@@ -76,7 +76,7 @@ Section operations.
- intros ; apply nr.
- intros ; apply union_idem.
Defined.
Global Instance fset_subset : forall A, hasSubset (FSet A).
Proof.
intros A X Y.
@@ -103,21 +103,6 @@ Section operations.
split ; apply p.
Defined.
Definition map (A B : Type) (f : A -> B) : FSet A -> FSet B.
Proof.
hrecursion.
- apply ∅.
- intro a.
apply {|f a|}.
- apply (∪).
- apply assoc.
- apply comm.
- apply nl.
- apply nr.
- intros.
apply idem.
Defined.
Local Ltac remove_transport
:= intros ; apply path_forall ; intro Z ; rewrite transport_arrow
; rewrite transport_const ; simpl.
@@ -155,6 +140,6 @@ Section operations.
- remove_transport.
rewrite <- (idem (Z (x; tr 1%path))).
pointwise.
Defined.
End operations.
Defined.
End operations.

18
FiniteSets/fsets/properties.v
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@@ -10,7 +10,7 @@ Section characterize_isIn.
(** isIn properties *)
Definition empty_isIn (a: A) : a ∈ ∅ -> Empty := idmap.
Definition singleton_isIn (a b: A) : a ∈ {|b|} -> Trunc (-1) (a = b) := idmap.
Definition union_isIn (X Y : FSet A) (a : A)
@@ -20,7 +20,7 @@ Section characterize_isIn.
{|X ∪ Y & ϕ|} = {|X & ϕ|} ∪ {|Y & ϕ|}.
Proof.
reflexivity.
Defined.
Defined.
Lemma comprehension_isIn (ϕ : A -> Bool) (a : A) : forall X : FSet A,
a ∈ {|X & ϕ|} = if ϕ a then a ∈ X else False_hp.
@@ -37,7 +37,7 @@ Section characterize_isIn.
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 (Hd^ @ ap ϕ X^ @ Hc).
* apply (Hc^ @ ap ϕ X @ Hd).
}
apply (X (ϕ a) (ϕ b) idpath idpath).
@@ -57,7 +57,7 @@ End characterize_isIn.
Section product_isIn.
Context {A B : Type}.
Context `{Univalence}.
Lemma isIn_singleproduct (a : A) (b : B) (c : A) : forall (Y : FSet B),
(a, b) ∈ (single_product c Y) = land (BuildhProp (Trunc (-1) (a = c))) (b ∈ Y).
Proof.
@@ -65,7 +65,7 @@ Section product_isIn.
- apply path_hprop ; symmetry ; apply prod_empty_r.
- intros d.
apply path_iff_hprop.
* intros.
* intros.
strip_truncations.
split ; apply tr ; try (apply (ap fst X)) ; try (apply (ap snd X)).
* intros [Z1 Z2].
@@ -93,7 +93,7 @@ Section product_isIn.
** right.
split ; try (apply (tr H1)) ; try (apply Hb2).
Defined.
Definition isIn_product (a : A) (b : B) (X : FSet A) (Y : FSet B) :
(a,b) ∈ (product X Y) = land (a ∈ X) (b ∈ Y).
Proof.
@@ -147,7 +147,7 @@ Section properties.
Global Instance joinsemilattice_fset : JoinSemiLattice (FSet A).
Proof.
split ; toHProp.
Defined.
Defined.
(** comprehension properties *)
Lemma comprehension_false : forall (X : FSet A), {|X & fun _ => false|} = ∅.
@@ -176,7 +176,7 @@ Section properties.
Proof.
toHProp.
Defined.
Lemma merely_choice : forall X : FSet A, hor (X = ∅) (hexists (fun a => a ∈ X)).
Proof.
hinduction; try (intros; apply equiv_hprop_allpath ; apply _).
@@ -295,5 +295,5 @@ Section properties.
repeat f_ap.
apply path_ishprop.
Defined.
End properties.

24
FiniteSets/fsets/properties_cons_repr.v
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@@ -7,7 +7,7 @@ Section properties.
Context {A : Type}.
Definition append_nl : forall (x: FSetC A), ∅ ∪ x = x
:= fun _ => idpath.
:= fun _ => idpath.
Lemma append_nr : forall (x: FSetC A), x ∪ ∅ = x.
Proof.
@@ -15,20 +15,20 @@ Section properties.
- reflexivity.
- intros. apply (ap (fun y => a;;y) X).
Defined.
Lemma append_assoc {H: Funext}:
Lemma append_assoc {H: Funext}:
forall (x y z: FSetC A), x ∪ (y ∪ z) = (x ∪ y) ∪ z.
Proof.
hinduction
; try (intros ; apply path_forall ; intro
; apply path_forall ; intro ; apply set_path2).
- reflexivity.
- intros a x HR y z.
- intros a x HR y z.
specialize (HR y z).
apply (ap (fun y => a;;y) HR).
apply (ap (fun y => a;;y) HR).
Defined.
Lemma append_singleton: forall (a: A) (x: FSetC A),
Lemma append_singleton: forall (a: A) (x: FSetC A),
a ;; x = x ∪ (a ;; ∅).
Proof.
intro a. hinduction; try (intros; apply set_path2).
@@ -38,22 +38,22 @@ Section properties.
+ apply (ap (fun y => b ;; y) HR).
Defined.
Lemma append_comm {H: Funext}:
Lemma append_comm {H: Funext}:
forall (x1 x2: FSetC A), x1 ∪ x2 = x2 ∪ x1.
Proof.
hinduction ; try (intros ; apply path_forall ; intro ; apply set_path2).
- intros. symmetry. apply append_nr.
- intros. symmetry. apply append_nr.
- intros a x1 HR x2.
etransitivity.
apply (ap (fun y => a;;y) (HR x2)).
apply (ap (fun y => a;;y) (HR x2)).
transitivity ((x2 ∪ x1) ∪ (a;;∅)).
+ apply append_singleton.
+ apply append_singleton.
+ etransitivity.
* symmetry. apply append_assoc.
* simple refine (ap (x2 ∪) (append_singleton _ _)^).
Defined.
Lemma singleton_idem: forall (a: A),
Lemma singleton_idem: forall (a: A),
{|a|} ∪ {|a|} = {|a|}.
Proof.
unfold singleton.

12
FiniteSets/fsets/properties_decidable.v
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@@ -60,7 +60,7 @@ Section operations_isIn.
contradiction.
- reflexivity.
Defined.
(* Union and membership *)
Lemma union_isIn_b (X Y : FSet A) (a : A) :
a ∈_d (X ∪ Y) = orb (a ∈_d X) (a ∈_d Y).
@@ -111,24 +111,24 @@ Section SetLattice.
intros x y.
apply (x ∪ y).
Defined.
Instance fset_min : minimum (FSet A).
Proof.
intros x y.
apply (x ∩ y).
Defined.
Instance fset_bot : bottom (FSet A).
Proof.
unfold bottom.
apply ∅.
Defined.
Instance lattice_fset : Lattice (FSet A).
Proof.
split; toBool.
Defined.
End SetLattice.
(* With extensionality we can prove decidable equality *)
@@ -142,4 +142,4 @@ Section dec_eq.
apply decidable_prod.
Defined.
End dec_eq.
End dec_eq.