mirror of https://github.com/nmvdw/HITs-Examples
Work on finite sets
This commit is contained in:
Niels
parent
b85976a96d
commit
6340f4cd77
652
FSet.v
652
FSet.v
|
|
@ -1,19 +1,20 @@
|
|||
Require Import HoTT.
|
||||
Require Export HoTT.
|
||||
Require Import FunextAxiom.
|
||||
|
||||
Module Export FinSet.
|
||||
|
||||
Set Implicit Arguments.
|
||||
|
||||
Parameter A: Type.
|
||||
Section FSet.
|
||||
Variable A : Type.
|
||||
|
||||
Private Inductive FSet : Type :=
|
||||
| empty : FSet
|
||||
| E : FSet
|
||||
| L : A -> FSet
|
||||
| U : FSet -> FSet -> FSet.
|
||||
Infix "∪" := U (at level 8, right associativity).
|
||||
Notation "∅" := empty.
|
||||
|
||||
Notation "{| x |}" := (L x).
|
||||
Infix "∪" := U (at level 8, right associativity).
|
||||
Notation "∅" := E.
|
||||
|
||||
Axiom assoc : forall (x y z : FSet ),
|
||||
x ∪ (y ∪ z) = (x ∪ y) ∪ z.
|
||||
|
|
@ -21,14 +22,14 @@ Axiom assoc : forall (x y z : FSet),
|
|||
Axiom comm : forall (x y : FSet),
|
||||
x ∪ y = y ∪ x.
|
||||
|
||||
Axiom neutl : forall (x : FSet),
|
||||
Axiom nl : forall (x : FSet),
|
||||
∅ ∪ x = x.
|
||||
|
||||
Axiom neutr : forall (x : FSet),
|
||||
Axiom nr : forall (x : FSet),
|
||||
x ∪ ∅ = x.
|
||||
|
||||
Axiom idem : forall (x : A),
|
||||
(L x) ∪ (L x) = L x.
|
||||
{| x |} ∪ {|x|} = {|x|}.
|
||||
|
||||
Axiom trunc : IsHSet FSet.
|
||||
|
||||
|
|
@ -47,14 +48,13 @@ Fixpoint FSet_rec
|
|||
{struct x}
|
||||
: P
|
||||
:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P with
|
||||
| empty => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => e
|
||||
| E => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => e
|
||||
| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => l a
|
||||
| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => u
|
||||
(FSet_rec H e l u assocP commP nlP nrP idemP y)
|
||||
(FSet_rec H e l u assocP commP nlP nrP idemP z)
|
||||
(FSet_rec P H e l u assocP commP nlP nrP idemP y)
|
||||
(FSet_rec P H e l u assocP commP nlP nrP idemP z)
|
||||
end) assocP commP nlP nrP idemP H.
|
||||
|
||||
|
||||
Axiom FSet_rec_beta_assoc : forall
|
||||
(P : Type)
|
||||
(H: IsHSet P)
|
||||
|
|
@ -67,11 +67,11 @@ Axiom FSet_rec_beta_assoc : forall
|
|||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x y z : FSet),
|
||||
ap (FSet_rec H e l u assocP commP nlP nrP idemP) (assoc x y z)
|
||||
ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (assoc x y z)
|
||||
=
|
||||
(assocP (FSet_rec H e l u assocP commP nlP nrP idemP x)
|
||||
(FSet_rec H e l u assocP commP nlP nrP idemP y)
|
||||
(FSet_rec H e l u assocP commP nlP nrP idemP z)
|
||||
(assocP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
|
||||
(FSet_rec P H e l u assocP commP nlP nrP idemP y)
|
||||
(FSet_rec P H e l u assocP commP nlP nrP idemP z)
|
||||
).
|
||||
|
||||
Axiom FSet_rec_beta_comm : forall
|
||||
|
|
@ -86,10 +86,10 @@ Axiom FSet_rec_beta_comm : forall
|
|||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x y : FSet),
|
||||
ap (FSet_rec H e l u assocP commP nlP nrP idemP) (comm x y)
|
||||
ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (comm x y)
|
||||
=
|
||||
(commP (FSet_rec H e l u assocP commP nlP nrP idemP x)
|
||||
(FSet_rec H e l u assocP commP nlP nrP idemP y)
|
||||
(commP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
|
||||
(FSet_rec P H e l u assocP commP nlP nrP idemP y)
|
||||
).
|
||||
|
||||
Axiom FSet_rec_beta_nl : forall
|
||||
|
|
@ -104,9 +104,9 @@ Axiom FSet_rec_beta_nl : forall
|
|||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x : FSet),
|
||||
ap (FSet_rec H e l u assocP commP nlP nrP idemP) (neutl x)
|
||||
ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (nl x)
|
||||
=
|
||||
(nlP (FSet_rec H e l u assocP commP nlP nrP idemP x)
|
||||
(nlP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
|
||||
).
|
||||
|
||||
Axiom FSet_rec_beta_nr : forall
|
||||
|
|
@ -121,9 +121,9 @@ Axiom FSet_rec_beta_nr : forall
|
|||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x : FSet),
|
||||
ap (FSet_rec H e l u assocP commP nlP nrP idemP) (neutr x)
|
||||
ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (nr x)
|
||||
=
|
||||
(nrP (FSet_rec H e l u assocP commP nlP nrP idemP x)
|
||||
(nrP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
|
||||
).
|
||||
|
||||
Axiom FSet_rec_beta_idem : forall
|
||||
|
|
@ -138,14 +138,16 @@ Axiom FSet_rec_beta_idem : forall
|
|||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x : A),
|
||||
ap (FSet_rec H e l u assocP commP nlP nrP idemP) (idem x)
|
||||
ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (idem x)
|
||||
=
|
||||
idemP x.
|
||||
|
||||
|
||||
(* Induction principle *)
|
||||
Fixpoint FSet_ind
|
||||
(P : FSet -> Type)
|
||||
(H : forall a : FSet, IsHSet (P a))
|
||||
(eP : P empty)
|
||||
(eP : P E)
|
||||
(lP : forall a: A, P (L a))
|
||||
(uP : forall (x y: FSet), P x -> P y -> P (U x y))
|
||||
(assocP : forall (x y z : FSet)
|
||||
|
|
@ -158,16 +160,16 @@ Fixpoint FSet_ind
|
|||
comm x y #
|
||||
uP x y px py = uP y x py px)
|
||||
(nlP : forall (x : FSet) (px: P x),
|
||||
neutl x # uP empty x eP px = px)
|
||||
nl x # uP E x eP px = px)
|
||||
(nrP : forall (x : FSet) (px: P x),
|
||||
neutr x # uP x empty px eP = px)
|
||||
nr x # uP x E px eP = px)
|
||||
(idemP : forall (x : A),
|
||||
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
||||
(x : FSet)
|
||||
{struct x}
|
||||
: P x
|
||||
:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P x with
|
||||
| empty => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => eP
|
||||
| E => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => eP
|
||||
| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => lP a
|
||||
| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => uP y z
|
||||
(FSet_ind P H eP lP uP assocP commP nlP nrP idemP y)
|
||||
|
|
@ -176,517 +178,279 @@ Fixpoint FSet_ind
|
|||
|
||||
|
||||
Axiom FSet_ind_beta_assoc : forall
|
||||
(A : Type)
|
||||
(P : FSet A -> Type)
|
||||
(eP : P (empty A))
|
||||
(P : FSet -> Type)
|
||||
(H : forall a : FSet, IsHSet (P a))
|
||||
(eP : P E)
|
||||
(lP : forall a: A, P (L a))
|
||||
(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
|
||||
(assocP : forall (x y z : FSet A)
|
||||
(uP : forall (x y: FSet), P x -> P y -> P (U x y))
|
||||
(assocP : forall (x y z : FSet)
|
||||
(px: P x) (py: P y) (pz: P z),
|
||||
assoc x y z #
|
||||
(uP x (U y z) px (uP y z py pz))
|
||||
=
|
||||
(uP (U x y) z (uP x y px py) pz))
|
||||
(commP : forall (x y: FSet A) (px: P x) (py: P y),
|
||||
(commP : forall (x y: FSet) (px: P x) (py: P y),
|
||||
comm x y #
|
||||
uP x y px py = uP y x py px)
|
||||
(nlP : forall (x : FSet A) (px: P x),
|
||||
nl x # uP (empty A) x eP px = px)
|
||||
(nrP : forall (x : FSet A) (px: P x),
|
||||
nr x # uP x (empty A) px eP = px)
|
||||
(nlP : forall (x : FSet) (px: P x),
|
||||
nl x # uP E x eP px = px)
|
||||
(nrP : forall (x : FSet) (px: P x),
|
||||
nr x # uP x E px eP = px)
|
||||
(idemP : forall (x : A),
|
||||
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
||||
(x y z : FSet A),
|
||||
apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP)
|
||||
(x y z : FSet),
|
||||
apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP)
|
||||
(assoc x y z)
|
||||
=
|
||||
(assocP x y z
|
||||
(FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
|
||||
(FSet_ind P eP lP uP assocP commP nlP nrP idemP y)
|
||||
(FSet_ind P eP lP uP assocP commP nlP nrP idemP z)
|
||||
(FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
|
||||
(FSet_ind P H eP lP uP assocP commP nlP nrP idemP y)
|
||||
(FSet_ind P H eP lP uP assocP commP nlP nrP idemP z)
|
||||
).
|
||||
|
||||
|
||||
|
||||
Axiom FSet_ind_beta_comm : forall
|
||||
(A : Type)
|
||||
(P : FSet A -> Type)
|
||||
(eP : P (empty A))
|
||||
(P : FSet -> Type)
|
||||
(H : forall a : FSet, IsHSet (P a))
|
||||
(eP : P E)
|
||||
(lP : forall a: A, P (L a))
|
||||
(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
|
||||
(assocP : forall (x y z : FSet A)
|
||||
(uP : forall (x y: FSet), P x -> P y -> P (U x y))
|
||||
(assocP : forall (x y z : FSet)
|
||||
(px: P x) (py: P y) (pz: P z),
|
||||
assoc x y z #
|
||||
(uP x (U y z) px (uP y z py pz))
|
||||
=
|
||||
(uP (U x y) z (uP x y px py) pz))
|
||||
(commP : forall (x y: FSet A) (px: P x) (py: P y),
|
||||
(commP : forall (x y : FSet) (px: P x) (py: P y),
|
||||
comm x y #
|
||||
uP x y px py = uP y x py px)
|
||||
(nlP : forall (x : FSet A) (px: P x),
|
||||
nl x # uP (empty A) x eP px = px)
|
||||
(nrP : forall (x : FSet A) (px: P x),
|
||||
nr x # uP x (empty A) px eP = px)
|
||||
(nlP : forall (x : FSet) (px: P x),
|
||||
nl x # uP E x eP px = px)
|
||||
(nrP : forall (x : FSet) (px: P x),
|
||||
nr x # uP x E px eP = px)
|
||||
(idemP : forall (x : A),
|
||||
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
||||
(x y : FSet A),
|
||||
apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (comm x y)
|
||||
(x y : FSet),
|
||||
apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (comm x y)
|
||||
=
|
||||
(commP x y
|
||||
(FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
|
||||
(FSet_ind P eP lP uP assocP commP nlP nrP idemP y)
|
||||
(FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
|
||||
(FSet_ind P H eP lP uP assocP commP nlP nrP idemP y)
|
||||
).
|
||||
|
||||
Axiom FSet_ind_beta_nl : forall
|
||||
(A : Type)
|
||||
(P : FSet A -> Type)
|
||||
(eP : P (empty A))
|
||||
(P : FSet -> Type)
|
||||
(H : forall a : FSet, IsHSet (P a))
|
||||
(eP : P E)
|
||||
(lP : forall a: A, P (L a))
|
||||
(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
|
||||
(assocP : forall (x y z : FSet A)
|
||||
(uP : forall (x y: FSet), P x -> P y -> P (U x y))
|
||||
(assocP : forall (x y z : FSet)
|
||||
(px: P x) (py: P y) (pz: P z),
|
||||
assoc x y z #
|
||||
(uP x (U y z) px (uP y z py pz))
|
||||
=
|
||||
(uP (U x y) z (uP x y px py) pz))
|
||||
(commP : forall (x y: FSet A) (px: P x) (py: P y),
|
||||
(commP : forall (x y : FSet) (px: P x) (py: P y),
|
||||
comm x y #
|
||||
uP x y px py = uP y x py px)
|
||||
(nlP : forall (x : FSet A) (px: P x),
|
||||
nl x # uP (empty A) x eP px = px)
|
||||
(nrP : forall (x : FSet A) (px: P x),
|
||||
nr x # uP x (empty A) px eP = px)
|
||||
(nlP : forall (x : FSet) (px: P x),
|
||||
nl x # uP E x eP px = px)
|
||||
(nrP : forall (x : FSet) (px: P x),
|
||||
nr x # uP x E px eP = px)
|
||||
(idemP : forall (x : A),
|
||||
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
||||
(x : FSet A),
|
||||
apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (nl x)
|
||||
(x : FSet),
|
||||
apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (nl x)
|
||||
=
|
||||
(nlP x (FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
|
||||
(nlP x (FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
|
||||
).
|
||||
|
||||
Axiom FSet_ind_beta_nr : forall
|
||||
(A : Type)
|
||||
(P : FSet A -> Type)
|
||||
(eP : P (empty A))
|
||||
(P : FSet -> Type)
|
||||
(H : forall a : FSet, IsHSet (P a))
|
||||
(eP : P E)
|
||||
(lP : forall a: A, P (L a))
|
||||
(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
|
||||
(assocP : forall (x y z : FSet A)
|
||||
(uP : forall (x y: FSet), P x -> P y -> P (U x y))
|
||||
(assocP : forall (x y z : FSet)
|
||||
(px: P x) (py: P y) (pz: P z),
|
||||
assoc x y z #
|
||||
(uP x (U y z) px (uP y z py pz))
|
||||
=
|
||||
(uP (U x y) z (uP x y px py) pz))
|
||||
(commP : forall (x y: FSet A) (px: P x) (py: P y),
|
||||
(commP : forall (x y : FSet) (px: P x) (py: P y),
|
||||
comm x y #
|
||||
uP x y px py = uP y x py px)
|
||||
(nlP : forall (x : FSet A) (px: P x),
|
||||
nl x # uP (empty A) x eP px = px)
|
||||
(nrP : forall (x : FSet A) (px: P x),
|
||||
nr x # uP x (empty A) px eP = px)
|
||||
(nlP : forall (x : FSet) (px: P x),
|
||||
nl x # uP E x eP px = px)
|
||||
(nrP : forall (x : FSet) (px: P x),
|
||||
nr x # uP x E px eP = px)
|
||||
(idemP : forall (x : A),
|
||||
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
||||
(x : FSet A),
|
||||
apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (nr x)
|
||||
(x : FSet),
|
||||
apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (nr x)
|
||||
=
|
||||
(nrP x (FSet_ind P eP lP uP assocP commP nlP nrP idemP x)
|
||||
(nrP x (FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
|
||||
).
|
||||
|
||||
Axiom FSet_ind_beta_idem : forall
|
||||
(A : Type)
|
||||
(P : FSet A -> Type)
|
||||
(eP : P (empty A))
|
||||
(P : FSet -> Type)
|
||||
(H : forall a : FSet, IsHSet (P a))
|
||||
(eP : P E)
|
||||
(lP : forall a: A, P (L a))
|
||||
(uP : forall (x y: FSet A), P x -> P y -> P (U x y))
|
||||
(assocP : forall (x y z : FSet A)
|
||||
(uP : forall (x y: FSet), P x -> P y -> P (U x y))
|
||||
(assocP : forall (x y z : FSet)
|
||||
(px: P x) (py: P y) (pz: P z),
|
||||
assoc x y z #
|
||||
(uP x (U y z) px (uP y z py pz))
|
||||
=
|
||||
(uP (U x y) z (uP x y px py) pz))
|
||||
(commP : forall (x y: FSet A) (px: P x) (py: P y),
|
||||
(commP : forall (x y : FSet) (px: P x) (py: P y),
|
||||
comm x y #
|
||||
uP x y px py = uP y x py px)
|
||||
(nlP : forall (x : FSet A) (px: P x),
|
||||
nl x # uP (empty A) x eP px = px)
|
||||
(nrP : forall (x : FSet A) (px: P x),
|
||||
nr x # uP x (empty A) px eP = px)
|
||||
(nlP : forall (x : FSet) (px: P x),
|
||||
nl x # uP E x eP px = px)
|
||||
(nrP : forall (x : FSet) (px: P x),
|
||||
nr x # uP x E px eP = px)
|
||||
(idemP : forall (x : A),
|
||||
idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
||||
(x : A),
|
||||
apD (FSet_ind P eP lP uP assocP commP nlP nrP idemP) (idem x)
|
||||
apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (idem x)
|
||||
=
|
||||
idemP x.
|
||||
|
||||
End FSet.
|
||||
|
||||
Parameter A : Type.
|
||||
Parameter eq : A -> A -> Bool.
|
||||
Parameter eq_refl: forall a:A, eq a a = true.
|
||||
|
||||
Theorem idemSet :
|
||||
forall x: FSet A, U x x = x.
|
||||
Proof.
|
||||
simple refine (FSet_ind _ _ _ _ _ _ _ _ _).
|
||||
- cbn.
|
||||
apply nl.
|
||||
- cbn.
|
||||
apply idem.
|
||||
- cbn.
|
||||
intros.
|
||||
rewrite assoc.
|
||||
rewrite (comm (U x y)).
|
||||
rewrite assoc.
|
||||
rewrite X.
|
||||
rewrite <- assoc.
|
||||
rewrite X0.
|
||||
reflexivity.
|
||||
- cbn.
|
||||
Arguments E {_}.
|
||||
Arguments U {_} _ _.
|
||||
Arguments L {_} _.
|
||||
|
||||
(* todo optimisation *)
|
||||
Theorem FSetRec (A : Type)
|
||||
(P : Type)
|
||||
(e : P)
|
||||
(l : A -> P)
|
||||
(u : P -> P -> P)
|
||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
||||
(commP : forall (x y : P), u x y = u y x)
|
||||
(nlP : forall (x : P), u e x = x)
|
||||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x : FSet A) : P.
|
||||
Definition isIn : A -> FSet A -> Bool.
|
||||
Proof.
|
||||
simple refine (FSet_ind _ _ _ _ _ _ _ _ _ x).
|
||||
- apply e.
|
||||
- apply l.
|
||||
- apply (fun _ => fun _ => u).
|
||||
- cbn.
|
||||
intros.
|
||||
transitivity (u px (u py pz)).
|
||||
apply transport_const.
|
||||
apply assocP.
|
||||
- cbn.
|
||||
intros.
|
||||
transitivity (u px py).
|
||||
apply transport_const.
|
||||
apply commP.
|
||||
- cbn.
|
||||
intros.
|
||||
transitivity (u e px).
|
||||
apply transport_const.
|
||||
apply nlP.
|
||||
- cbn.
|
||||
intros.
|
||||
transitivity (u px e).
|
||||
apply transport_const.
|
||||
apply nrP.
|
||||
- cbn.
|
||||
intros.
|
||||
transitivity (u (l x0) (l x0)).
|
||||
apply transport_const.
|
||||
apply idemP.
|
||||
intros a.
|
||||
simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
|
||||
- exact false.
|
||||
- intro a'. apply (eq a a').
|
||||
- apply orb.
|
||||
- intros x y z. destruct x; reflexivity.
|
||||
- intros x y. destruct x, y; reflexivity.
|
||||
- intros x. reflexivity.
|
||||
- intros x. destruct x; reflexivity.
|
||||
- intros a'. destruct (eq a a'); reflexivity.
|
||||
Defined.
|
||||
|
||||
Set Implicit Arguments.
|
||||
|
||||
|
||||
|
||||
Theorem FSet_rec_beta_assocT : forall
|
||||
(A : Type)
|
||||
(P : Type)
|
||||
(e : P)
|
||||
(l : A -> P)
|
||||
(u : P -> P -> P)
|
||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
||||
(commP : forall (x y : P), u x y = u y x)
|
||||
(nlP : forall (x : P), u e x = x)
|
||||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x y z : FSet A),
|
||||
ap (FSetRec e l u assocP commP nlP nrP idemP) (assoc x y z)
|
||||
=
|
||||
(assocP (FSetRec e l u assocP commP nlP nrP idemP x)
|
||||
(FSetRec e l u assocP commP nlP nrP idemP y)
|
||||
(FSetRec e l u assocP commP nlP nrP idemP z)
|
||||
).
|
||||
Proof.
|
||||
intros.
|
||||
eapply (cancelL (transport_const (assoc x y z) _ ) ).
|
||||
simple refine
|
||||
((apD_const
|
||||
(FSetRec e l u assocP commP nlP nrP idemP)
|
||||
(assoc x y z))^ @ _).
|
||||
apply FSet_ind_beta_assoc.
|
||||
|
||||
|
||||
Theorem FSet_rec_beta_commT : forall
|
||||
(A : Type)
|
||||
(P : Type)
|
||||
(e : P)
|
||||
(l : A -> P)
|
||||
(u : P -> P -> P)
|
||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
||||
(commP : forall (x y : P), u x y = u y x)
|
||||
(nlP : forall (x : P), u e x = x)
|
||||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x y : FSet A),
|
||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (comm x y)
|
||||
=
|
||||
(commP (FSet_rec e l u assocP commP nlP nrP idemP x)
|
||||
(FSet_rec e l u assocP commP nlP nrP idemP y)
|
||||
).
|
||||
|
||||
Axiom FSet_rec_beta_nl : forall
|
||||
(A : Type)
|
||||
(P : Type)
|
||||
(e : P)
|
||||
(l : A -> P)
|
||||
(u : P -> P -> P)
|
||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
||||
(commP : forall (x y : P), u x y = u y x)
|
||||
(nlP : forall (x : P), u e x = x)
|
||||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x : FSet A),
|
||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (nl x)
|
||||
=
|
||||
(nlP (FSet_rec e l u assocP commP nlP nrP idemP x)
|
||||
).
|
||||
|
||||
Axiom FSet_rec_beta_nr : forall
|
||||
(A : Type)
|
||||
(P : Type)
|
||||
(e : P)
|
||||
(l : A -> P)
|
||||
(u : P -> P -> P)
|
||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
||||
(commP : forall (x y : P), u x y = u y x)
|
||||
(nlP : forall (x : P), u e x = x)
|
||||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x : FSet A),
|
||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (nr x)
|
||||
=
|
||||
(nrP (FSet_rec e l u assocP commP nlP nrP idemP x)
|
||||
).
|
||||
|
||||
Axiom FSet_rec_beta_idem : forall
|
||||
(A : Type)
|
||||
(P : Type)
|
||||
(e : P)
|
||||
(l : A -> P)
|
||||
(u : P -> P -> P)
|
||||
(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
|
||||
(commP : forall (x y : P), u x y = u y x)
|
||||
(nlP : forall (x : P), u e x = x)
|
||||
(nrP : forall (x : P), u x e = x)
|
||||
(idemP : forall (x : A), u (l x) (l x) = l x)
|
||||
(x : A),
|
||||
ap (FSet_rec e l u assocP commP nlP nrP idemP) (idem x)
|
||||
=
|
||||
idemP x.
|
||||
|
||||
|
||||
End FinSet.
|
||||
|
||||
Definition isIn : forall
|
||||
(A : Type)
|
||||
(eq : A -> A -> Bool),
|
||||
A -> FSet A -> Bool.
|
||||
Proof.
|
||||
intro A.
|
||||
intro eq.
|
||||
intro a.
|
||||
refine (FSet_rec A _ _ _ _ _ _ _ _ _).
|
||||
Unshelve.
|
||||
|
||||
Focus 6.
|
||||
apply false.
|
||||
|
||||
Focus 6.
|
||||
intro a'.
|
||||
apply (eq a a').
|
||||
|
||||
Focus 6.
|
||||
intro b.
|
||||
intro b'.
|
||||
apply (orb b b').
|
||||
|
||||
Focus 3.
|
||||
intros.
|
||||
compute.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x.
|
||||
reflexivity.
|
||||
destruct y.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x.
|
||||
destruct y.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
destruct y.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
intros.
|
||||
compute.
|
||||
destruct (eq a x).
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Definition comprehension : forall
|
||||
(A : Type)
|
||||
(eq : A -> A -> Bool),
|
||||
Definition comprehension :
|
||||
(A -> Bool) -> FSet A -> FSet A.
|
||||
Proof.
|
||||
intro A.
|
||||
intro eq.
|
||||
intro phi.
|
||||
refine (FSet_rec A _ _ _ _ _ _ _ _ _).
|
||||
Unshelve.
|
||||
|
||||
Focus 6.
|
||||
apply empty.
|
||||
|
||||
Focus 6.
|
||||
intro a.
|
||||
apply (if (phi a) then L A a else (empty A)).
|
||||
|
||||
Focus 6.
|
||||
intro x.
|
||||
intro y.
|
||||
apply (U A x y).
|
||||
|
||||
Focus 3.
|
||||
intros.
|
||||
compute.
|
||||
apply nl.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
apply assoc.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
apply comm.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
apply nr.
|
||||
|
||||
intros.
|
||||
compute.
|
||||
destruct (phi x).
|
||||
apply idem.
|
||||
apply nl.
|
||||
intros P.
|
||||
simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
|
||||
- apply E.
|
||||
- intro a.
|
||||
refine (if (P a) then L a else E).
|
||||
- apply U.
|
||||
- intros. cbv. apply assoc.
|
||||
- intros. cbv. apply comm.
|
||||
- intros. cbv. apply nl.
|
||||
- intros. cbv. apply nr.
|
||||
- intros. cbv.
|
||||
destruct (P x); simpl.
|
||||
+ apply idem.
|
||||
+ apply nl.
|
||||
Defined.
|
||||
|
||||
Definition intersection : forall (A : Type) (eq : A -> A -> Bool),
|
||||
Definition intersection :
|
||||
FSet A -> FSet A -> FSet A.
|
||||
Proof.
|
||||
intro A.
|
||||
intro eq.
|
||||
intro x.
|
||||
intro y.
|
||||
apply (comprehension A eq (fun (a : A) => isIn A eq a x) y).
|
||||
intros X Y.
|
||||
apply (comprehension (fun (a : A) => isIn a X) Y).
|
||||
Defined.
|
||||
|
||||
Definition subset (A : Type) (eq : A -> A -> Bool) (x : FSet A) (y : FSet A) : Bool.
|
||||
Lemma intersection_E : forall x,
|
||||
intersection E x = E.
|
||||
Proof.
|
||||
refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
|
||||
Unshelve.
|
||||
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intro a.
|
||||
reflexivity.
|
||||
- unfold intersection.
|
||||
intros x y P Q.
|
||||
cbn.
|
||||
rewrite P.
|
||||
rewrite Q.
|
||||
apply nl.
|
||||
Defined.
|
||||
|
||||
Focus 6.
|
||||
apply x.
|
||||
|
||||
Focus 6.
|
||||
apply true.
|
||||
|
||||
Focus 6.
|
||||
Theorem intersection_La : forall a x,
|
||||
intersection (L a) x = if isIn a x then L a else E.
|
||||
Proof.
|
||||
intro a.
|
||||
apply (isIn A eq a y).
|
||||
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
|
||||
- reflexivity.
|
||||
- intro b.
|
||||
admit.
|
||||
- unfold intersection.
|
||||
intros x y P Q.
|
||||
cbn.
|
||||
rewrite P.
|
||||
rewrite Q.
|
||||
destruct (isIn a x) ; destruct (isIn a y).
|
||||
* apply idem.
|
||||
* apply nr.
|
||||
* apply nl.
|
||||
* apply nl.
|
||||
Admitted.
|
||||
|
||||
Focus 6.
|
||||
intro b.
|
||||
intro b'.
|
||||
apply (andb b b').
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x0.
|
||||
destruct y0.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x0.
|
||||
destruct y0.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
destruct y0.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
reflexivity.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x0.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
intros.
|
||||
destruct (isIn A eq x0 y).
|
||||
compute.
|
||||
reflexivity.
|
||||
compute.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Definition equal_set (A : Type) (eq : A -> A -> Bool) (x : FSet A) (y : FSet A) : Bool
|
||||
:= andb (subset A eq x y) (subset A eq y x).
|
||||
|
||||
Fixpoint eq_nat n m : Bool :=
|
||||
match n, m with
|
||||
| O, O => true
|
||||
| O, S _ => false
|
||||
| S _, O => false
|
||||
| S n1, S m1 => eq_nat n1 m1
|
||||
end.
|
||||
|
||||
|
||||
|
||||
Theorem test : forall (A:Type) (eq : A -> A -> Bool)
|
||||
(u: FSet A), ~(u = empty _) -> exists (a: A) (v: FSet A),
|
||||
u = U _ (L _ a) v /\ (isIn _ eq a v) = False.
|
||||
Theorem comprehension_or : forall ϕ ψ x,
|
||||
comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x) (comprehension ψ x).
|
||||
Proof.
|
||||
intros A eq.
|
||||
i
|
||||
|
||||
intros ϕ ψ.
|
||||
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
|
||||
- cbn. symmetry ; apply nl.
|
||||
- cbn. intros.
|
||||
destruct (ϕ a) ; destruct (ψ a) ; symmetry.
|
||||
* apply idem.
|
||||
* apply nr.
|
||||
* apply nl.
|
||||
* apply nl.
|
||||
- simpl. intros x y P Q.
|
||||
cbn.
|
||||
rewrite P.
|
||||
rewrite Q.
|
||||
|
||||
|
||||
Theorem intersection_assoc : forall x y z,
|
||||
intersection x (intersection y z) = intersection (intersection x y) z.
|
||||
Proof.
|
||||
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
|
||||
- cbn.
|
||||
intros y z.
|
||||
rewrite intersection_E.
|
||||
rewrite intersection_E.
|
||||
rewrite intersection_E.
|
||||
reflexivity.
|
||||
- intro a.
|
||||
cbn.
|
||||
intros y z.
|
||||
(* simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _ y) ; try (intros ; apply set_path2). *)
|
||||
admit.
|
||||
- unfold intersection.
|
||||
intros x y P Q z z'.
|
||||
cbn.
|
||||
rewrite Q.
|
||||
|
||||
rewrite intersection_La.
|
||||
rewrite intersection_La.
|
||||
destruct (isIn a y).
|
||||
* rewrite intersection_La.
|
||||
destruct (isIn a (intersection y z)).
|
||||
+ reflexivity.
|
||||
+
|
||||
*
|
||||
destruct (isIn a (intersection y z)).
|
||||
|
||||
Loading…
Reference in New Issue