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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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44
FiniteSets/representations/T.v
View File

@@ -7,7 +7,7 @@ Module Export T.
Private Inductive T (B : Type) : Type :=
| b : T B
| c : T B.
| c : T B.
Axiom p : A -> b A = c A.
Axiom setT : IsHSet (T A).
@@ -23,7 +23,7 @@ Module Export T.
Variable (bP : P (b A)).
Variable (cP : P (c A)).
Variable (pP : forall a : A, (p a) # bP = cP).
(* Induction principle *)
Fixpoint T_ind
(x : T A)
@@ -31,7 +31,7 @@ Module Export T.
: P x
:= (match x return _ -> _ -> P x with
| b => fun _ _ => bP
| c => fun _ _ => cP
| c => fun _ _ => cP
end) pP H.
Axiom T_ind_beta_p : forall (a : A),
@@ -68,7 +68,7 @@ Module Export T.
End T_recursion.
Instance T_recursion A : HitRecursion (T A)
:= {indTy := _; recTy := _;
:= {indTy := _; recTy := _;
H_inductor := T_ind A; H_recursor := T_rec A }.
End T.
@@ -119,44 +119,44 @@ Section merely_dec_lem.
Local Ltac f_prop := apply path_forall ; intro ; apply path_ishprop.
Lemma transport_code_b_x (a : A) :
Lemma transport_code_b_x (a : A) :
transport code_b (p a) = fun _ => a.
Proof.
f_prop.
Defined.
Lemma transport_code_c_x (a : A) :
Lemma transport_code_c_x (a : A) :
transport code_c (p a) = fun _ => tt.
Proof.
f_prop.
f_prop.
Defined.
Lemma transport_code_c_x_V (a : A) :
Lemma transport_code_c_x_V (a : A) :
transport code_c (p a)^ = fun _ => a.
Proof.
f_prop.
Proof.
f_prop.
Defined.
Lemma transport_code_x_b (a : A) :
Lemma transport_code_x_b (a : A) :
transport (fun x => code x (b A)) (p a) = fun _ => a.
Proof.
f_prop.
Defined.
Lemma transport_code_x_c (a : A) :
Lemma transport_code_x_c (a : A) :
transport (fun x => code x (c A)) (p a) = fun _ => tt.
Proof.
f_prop.
Defined.
Lemma transport_code_x_c_V (a : A) :
Lemma transport_code_x_c_V (a : A) :
transport (fun x => code x (c A)) (p a)^ = fun _ => a.
Proof.
f_prop.
Defined.
Lemma ap_diag {B : Type} {x y : B} (p : x = y) :
ap (fun x : B => (x, x)) p = path_prod' p p.
ap (fun x : B => (x, x)) p = path_prod' p p.
Proof.
by path_induction.
Defined.
@@ -217,7 +217,7 @@ Section merely_dec_lem.
refine (transport_arrow _ _ _ @ _).
refine (transport_paths_FlFr _ _ @ _).
rewrite transport_code_c_x_V.
hott_simpl.
hott_simpl.
Defined.
Lemma transport_paths_FlFr_trunc :
@@ -229,7 +229,7 @@ Section merely_dec_lem.
refine (ap tr _).
exact ((concat_1p r)^ @ (concat_p1 (1 @ r))^).
Defined.
Definition decode : forall (x y : T A), code x y -> x = y.
Proof.
simple refine (T_ind _ _ _ _ _ _); simpl.
@@ -248,7 +248,7 @@ Section merely_dec_lem.
f_ap.
refine (ap (fun x => (p x)) _).
apply path_ishprop.
+ intro.
+ intro.
rewrite transport_code_x_c_V.
hott_simpl.
+ intro b.
@@ -264,7 +264,7 @@ Section merely_dec_lem.
intros p. induction p.
simpl. revert u. simple refine (T_ind _ _ _ _ _ _).
+ simpl. reflexivity.
+ simpl. reflexivity.
+ simpl. reflexivity.
+ intro a.
apply set_path2.
Defined.
@@ -278,12 +278,12 @@ Section merely_dec_lem.
+ simpl. intro a. apply path_ishprop.
+ intro a. apply path_forall; intros ?. apply set_path2.
- simple refine (T_ind _ _ _ _ _ _).
+ simpl. intro a. apply path_ishprop.
+ simpl. apply path_ishprop.
+ simpl. intro a. apply path_ishprop.
+ simpl. apply path_ishprop.
+ intro a. apply path_forall; intros ?. apply set_path2.
- intro a. repeat (apply path_forall; intros ?). apply set_path2.
Defined.
Instance T_hprop (a : A) : IsHProp (b A = c A).
Proof.
@@ -307,7 +307,7 @@ Section merely_dec_lem.
rewrite ?decode_encode in H1.
apply H1.
Defined.
Lemma equiv_pathspace_T : (b A = c A) = A.
Proof.
apply path_iff_ishprop.

22
FiniteSets/representations/cons_repr.v
View File

@@ -3,7 +3,7 @@ Require Import HoTT HitTactics.
Require Export notation.
Module Export FSetC.
Section FSetC.
Private Inductive FSetC (A : Type) : Type :=
| Nil : FSetC A
@@ -14,9 +14,9 @@ Module Export FSetC.
Variable A : Type.
Arguments Cns {_} _ _.
Infix ";;" := Cns (at level 8, right associativity).
Axiom dupl : forall (a : A) (x : FSetC A),
a ;; a ;; x = a ;; x.
a ;; a ;; x = a ;; x.
Axiom comm : forall (a b : A) (x : FSetC A),
a ;; b ;; x = b ;; a ;; x.
@@ -41,9 +41,9 @@ Module Export FSetC.
Variable (duplP : forall (a: A) (x: FSetC A) (px : P x),
dupl a x # cnsP a (a;;x) (cnsP a x px) = cnsP a x px).
Variable (commP : forall (a b: A) (x: FSetC A) (px: P x),
comm a b x # cnsP a (b;;x) (cnsP b x px) =
comm a b x # cnsP a (b;;x) (cnsP b x px) =
cnsP b (a;;x) (cnsP a x px)).
(* Induction principle *)
Fixpoint FSetC_ind
(x : FSetC A)
@@ -76,18 +76,18 @@ Module Export FSetC.
(* Recursion principle *)
Definition FSetC_rec : FSetC A -> P.
Proof.
simple refine (FSetC_ind _ _ _ _ _ _ _ );
simple refine (FSetC_ind _ _ _ _ _ _ _ );
try (intros; simple refine ((transport_const _ _) @ _ )); cbn.
- apply nil.
- apply (fun a => fun _ => cns a).
- apply (fun a => fun _ => cns a).
- apply duplP.
- apply commP.
- apply commP.
Defined.
Definition FSetC_rec_beta_dupl : forall (a: A) (x : FSetC A),
ap FSetC_rec (dupl a x) = duplP a (FSetC_rec x).
Proof.
intros.
intros.
eapply (cancelL (transport_const (dupl a x) _)).
simple refine ((apD_const _ _)^ @ _).
apply FSetC_ind_beta_dupl.
@@ -96,7 +96,7 @@ Module Export FSetC.
Definition FSetC_rec_beta_comm : forall (a b: A) (x : FSetC A),
ap FSetC_rec (comm a b x) = commP a b (FSetC_rec x).
Proof.
intros.
intros.
eapply (cancelL (transport_const (comm a b x) _)).
simple refine ((apD_const _ _)^ @ _).
apply FSetC_ind_beta_comm.
@@ -107,7 +107,7 @@ Module Export FSetC.
Instance FSetC_recursion A : HitRecursion (FSetC A) :=
{
indTy := _; recTy := _;
indTy := _; recTy := _;
H_inductor := FSetC_ind A; H_recursor := FSetC_rec A
}.

36
FiniteSets/representations/definition.v
View File

@@ -14,7 +14,7 @@ Module Export FSet.
Global Instance fset_union : forall A, hasUnion (FSet A) := U.
Variable A : Type.
Axiom assoc : forall (x y z : FSet A),
x ∪ (y ∪ z) = (x ∪ y) ∪ z.
@@ -38,7 +38,7 @@ Module Export FSet.
Arguments comm {_} _ _.
Arguments nl {_} _.
Arguments nr {_} _.
Arguments idem {_} _.
Arguments idem {_} _.
Section FSet_induction.
Variable A: Type.
@@ -47,22 +47,22 @@ Module Export FSet.
Variable (eP : P ∅).
Variable (lP : forall a: A, P {|a|}).
Variable (uP : forall (x y: FSet A), P x -> P y -> P (x ∪ y)).
Variable (assocP : forall (x y z : FSet A)
Variable (assocP : forall (x y z : FSet A)
(px: P x) (py: P y) (pz: P z),
assoc x y z #
(uP x (y ∪ z) px (uP y z py pz))
=
(uP x (y ∪ z) px (uP y z py pz))
=
(uP (x ∪ y) z (uP x y px py) pz)).
Variable (commP : forall (x y: FSet A) (px: P x) (py: P y),
comm x y #
uP x y px py = uP y x py px).
Variable (nlP : forall (x : FSet A) (px: P x),
Variable (nlP : forall (x : FSet A) (px: P x),
nl x # uP ∅ x eP px = px).
Variable (nrP : forall (x : FSet A) (px: P x),
Variable (nrP : forall (x : FSet A) (px: P x),
nr x # uP x ∅ px eP = px).
Variable (idemP : forall (x : A),
Variable (idemP : forall (x : A),
idem x # uP {|x|} {|x|} (lP x) (lP x) = lP x).
(* Induction principle *)
Fixpoint FSet_ind
(x : FSet A)
@@ -119,7 +119,7 @@ Module Export FSet.
Defined.
Definition FSet_rec_beta_assoc : forall (x y z : FSet A),
ap FSet_rec (assoc x y z)
ap FSet_rec (assoc x y z)
=
assocP (FSet_rec x) (FSet_rec y) (FSet_rec z).
Proof.
@@ -131,7 +131,7 @@ Module Export FSet.
Defined.
Definition FSet_rec_beta_comm : forall (x y : FSet A),
ap FSet_rec (comm x y)
ap FSet_rec (comm x y)
=
commP (FSet_rec x) (FSet_rec y).
Proof.
@@ -143,7 +143,7 @@ Module Export FSet.
Defined.
Definition FSet_rec_beta_nl : forall (x : FSet A),
ap FSet_rec (nl x)
ap FSet_rec (nl x)
=
nlP (FSet_rec x).
Proof.
@@ -155,7 +155,7 @@ Module Export FSet.
Defined.
Definition FSet_rec_beta_nr : forall (x : FSet A),
ap FSet_rec (nr x)
ap FSet_rec (nr x)
=
nrP (FSet_rec x).
Proof.
@@ -167,7 +167,7 @@ Module Export FSet.
Defined.
Definition FSet_rec_beta_idem : forall (a : A),
ap FSet_rec (idem a)
ap FSet_rec (idem a)
=
idemP a.
Proof.
@@ -177,12 +177,12 @@ Module Export FSet.
simple refine ((apD_const _ _)^ @ _).
apply FSet_ind_beta_idem.
Defined.
End FSet_recursion.
Instance FSet_recursion A : HitRecursion (FSet A) :=
{
indTy := _; recTy := _;
indTy := _; recTy := _;
H_inductor := FSet_ind A; H_recursor := FSet_rec A
}.
@@ -200,5 +200,5 @@ Proof.
rewrite P.
rewrite (comm y x).
rewrite <- (assoc x y y).
f_ap.
Defined.
f_ap.
Defined.