mirror of https://github.com/nmvdw/HITs-Examples
Some modifications to the finset module
This commit is contained in:
Dan Frumin
parent
399fab467b
commit
5e594d10f0
344
FinSets.v
344
FinSets.v
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@ -2,30 +2,38 @@ Require Export HoTT.
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Require Import HitTactics.
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Module Export FinSet.
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Section FSet.
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Variable A : Type.
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Private Inductive FinSets (A : Type) : Type :=
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| empty : FinSets A
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| L : A -> FinSets A
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| U : FinSets A -> FinSets A -> FinSets A.
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Private Inductive FSet : Type :=
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| E : FSet
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| L : A -> FSet
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| U : FSet -> FSet -> FSet.
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Axiom assoc : forall (A : Type) (x y z : FinSets A),
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U A x (U A y z) = U A (U A x y) z.
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Notation "{| x |}" := (L x).
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Infix "∪" := U (at level 8, right associativity).
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Notation "∅" := E.
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Axiom comm : forall (A : Type) (x y : FinSets A),
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U A x y = U A y x.
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Axiom assoc : forall (x y z : FSet ),
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x ∪ (y ∪ z) = (x ∪ y) ∪ z.
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Axiom nl : forall (A : Type) (x : FinSets A),
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U A (empty A) x = x.
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Axiom comm : forall (x y : FSet),
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x ∪ y = y ∪ x.
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Axiom nr : forall (A : Type) (x : FinSets A),
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U A x (empty A) = x.
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Axiom nl : forall (x : FSet),
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∅ ∪ x = x.
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Axiom idem : forall (A : Type) (x : A),
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U A (L A x) (L A x) = L A x.
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Axiom nr : forall (x : FSet),
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x ∪ ∅ = x.
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Fixpoint FinSets_rec
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(A : Type)
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Axiom idem : forall (x : A),
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{| x |} ∪ {|x|} = {|x|}.
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Axiom trunc : IsHSet FSet.
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Fixpoint FSet_rec
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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@ -34,20 +42,20 @@ Fixpoint FinSets_rec
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FinSets A)
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(x : FSet)
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{struct x}
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: P
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:= (match x return _ -> _ -> _ -> _ -> _ -> P with
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| empty => fun _ => fun _ => fun _ => fun _ => fun _ => e
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| L a => fun _ => fun _ => fun _ => fun _ => fun _ => l a
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| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => u
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(FinSets_rec A P e l u assocP commP nlP nrP idemP y)
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(FinSets_rec A P e l u assocP commP nlP nrP idemP z)
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end) assocP commP nlP nrP idemP.
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:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P with
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| E => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => e
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| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => l a
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| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => u
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(FSet_rec P H e l u assocP commP nlP nrP idemP y)
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(FSet_rec P H e l u assocP commP nlP nrP idemP z)
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end) assocP commP nlP nrP idemP H.
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Axiom FinSets_beta_assoc : forall
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(A : Type)
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Axiom FSet_rec_beta_assoc : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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@ -56,17 +64,17 @@ Axiom FinSets_beta_assoc : forall
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x y z : FinSets A),
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ap (FinSets_rec A P e l u assocP commP nlP nrP idemP) (assoc A x y z)
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(x y z : FSet),
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (assoc x y z)
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=
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(assocP (FinSets_rec A P e l u assocP commP nlP nrP idemP x)
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(FinSets_rec A P e l u assocP commP nlP nrP idemP y)
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(FinSets_rec A P e l u assocP commP nlP nrP idemP z)
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(assocP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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(FSet_rec P H e l u assocP commP nlP nrP idemP y)
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(FSet_rec P H e l u assocP commP nlP nrP idemP z)
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).
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Axiom FinSets_beta_comm : forall
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(A : Type)
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Axiom FSet_rec_beta_comm : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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@ -75,16 +83,16 @@ Axiom FinSets_beta_comm : forall
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x y : FinSets A),
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ap (FinSets_rec A P e l u assocP commP nlP nrP idemP) (comm A x y)
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(x y : FSet),
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (comm x y)
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=
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(commP (FinSets_rec A P e l u assocP commP nlP nrP idemP x)
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(FinSets_rec A P e l u assocP commP nlP nrP idemP y)
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(commP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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(FSet_rec P H e l u assocP commP nlP nrP idemP y)
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).
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Axiom FinSets_beta_nl : forall
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(A : Type)
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Axiom FSet_rec_beta_nl : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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@ -93,15 +101,15 @@ Axiom FinSets_beta_nl : forall
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FinSets A),
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ap (FinSets_rec A P e l u assocP commP nlP nrP idemP) (nl A x)
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(x : FSet),
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (nl x)
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=
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(nlP (FinSets_rec A P e l u assocP commP nlP nrP idemP x)
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(nlP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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).
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Axiom FinSets_beta_nr : forall
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(A : Type)
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Axiom FSet_rec_beta_nr : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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@ -110,15 +118,15 @@ Axiom FinSets_beta_nr : forall
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(nlP : forall (x : P), u e x = x)
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : FinSets A),
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ap (FinSets_rec A P e l u assocP commP nlP nrP idemP) (nr A x)
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(x : FSet),
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (nr x)
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=
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(nrP (FinSets_rec A P e l u assocP commP nlP nrP idemP x)
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(nrP (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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).
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Axiom FinSets_beta_idem : forall
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(A : Type)
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Axiom FSet_rec_beta_idem : forall
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(P : Type)
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(H: IsHSet P)
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(e : P)
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(l : A -> P)
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(u : P -> P -> P)
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@ -128,21 +136,206 @@ Axiom FinSets_beta_idem : forall
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(nrP : forall (x : P), u x e = x)
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(idemP : forall (x : A), u (l x) (l x) = l x)
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(x : A),
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ap (FinSets_rec A P e l u assocP commP nlP nrP idemP) (idem A x)
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ap (FSet_rec P H e l u assocP commP nlP nrP idemP) (idem x)
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=
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idemP x.
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(* Induction principle *)
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Fixpoint FSet_ind
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(P : FSet -> Type)
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(H : forall a : FSet, IsHSet (P a))
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(eP : P E)
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y: FSet) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet) (px: P x),
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nl x # uP E x eP px = px)
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(nrP : forall (x : FSet) (px: P x),
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nr x # uP x E px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x : FSet)
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{struct x}
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: P x
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:= (match x return _ -> _ -> _ -> _ -> _ -> _ -> P x with
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| E => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => eP
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| L a => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => lP a
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| U y z => fun _ => fun _ => fun _ => fun _ => fun _ => fun _ => uP y z
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP y)
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP z)
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end) H assocP commP nlP nrP idemP.
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Axiom FSet_ind_beta_assoc : forall
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(P : FSet -> Type)
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(H : forall a : FSet, IsHSet (P a))
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(eP : P E)
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y: FSet) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet) (px: P x),
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nl x # uP E x eP px = px)
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(nrP : forall (x : FSet) (px: P x),
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nr x # uP x E px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x y z : FSet),
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apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP)
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(assoc x y z)
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=
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(assocP x y z
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP y)
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP z)
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).
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Axiom FSet_ind_beta_comm : forall
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(P : FSet -> Type)
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(H : forall a : FSet, IsHSet (P a))
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(eP : P E)
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y : FSet) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet) (px: P x),
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nl x # uP E x eP px = px)
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(nrP : forall (x : FSet) (px: P x),
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nr x # uP x E px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x y : FSet),
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apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (comm x y)
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=
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(commP x y
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
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(FSet_ind P H eP lP uP assocP commP nlP nrP idemP y)
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).
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Axiom FSet_ind_beta_nl : forall
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(P : FSet -> Type)
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(H : forall a : FSet, IsHSet (P a))
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(eP : P E)
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y : FSet) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet) (px: P x),
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nl x # uP E x eP px = px)
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(nrP : forall (x : FSet) (px: P x),
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nr x # uP x E px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x : FSet),
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apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (nl x)
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=
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(nlP x (FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
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).
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Axiom FSet_ind_beta_nr : forall
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(P : FSet -> Type)
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(H : forall a : FSet, IsHSet (P a))
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(eP : P E)
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y : FSet) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet) (px: P x),
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nl x # uP E x eP px = px)
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(nrP : forall (x : FSet) (px: P x),
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nr x # uP x E px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x : FSet),
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apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (nr x)
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=
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(nrP x (FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
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).
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Axiom FSet_ind_beta_idem : forall
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(P : FSet -> Type)
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(H : forall a : FSet, IsHSet (P a))
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(eP : P E)
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(lP : forall a: A, P (L a))
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(uP : forall (x y: FSet), P x -> P y -> P (U x y))
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(assocP : forall (x y z : FSet)
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(px: P x) (py: P y) (pz: P z),
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assoc x y z #
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(uP x (U y z) px (uP y z py pz))
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=
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(uP (U x y) z (uP x y px py) pz))
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(commP : forall (x y : FSet) (px: P x) (py: P y),
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comm x y #
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uP x y px py = uP y x py px)
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(nlP : forall (x : FSet) (px: P x),
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nl x # uP E x eP px = px)
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(nrP : forall (x : FSet) (px: P x),
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nr x # uP x E px eP = px)
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(idemP : forall (x : A),
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
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(x : A),
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apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (idem x)
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=
|
||||
idemP x.
|
||||
|
||||
End FSet.
|
||||
|
||||
(* TODO: add an induction principle *)
|
||||
Definition FinSetsCL A : HitRec.class (FinSets A) _ _ :=
|
||||
HitRec.Class (FinSets A) (fun x P e l u aP cP lP rP iP => FinSets_rec A P e l u aP cP lP rP iP x) (fun x P e l u aP cP lP rP iP => FinSets_rec A P e l u aP cP lP rP iP x).
|
||||
Canonical Structure FinSetsTy A : HitRec.type := HitRec.Pack _ _ _ (FinSetsCL A).
|
||||
Definition FSetCL A : HitRec.class (FSet A) _ _ :=
|
||||
HitRec.Class (FSet A) (fun x P H e l u aP cP lP rP iP => FSet_rec A P H e l u aP cP lP rP iP x) (fun x P H e l u aP cP lP rP iP => FSet_ind A P H e l u aP cP lP rP iP x).
|
||||
Canonical Structure FSetTy A : HitRec.type := HitRec.Pack _ _ _ (FSetCL A).
|
||||
|
||||
Arguments E {_}.
|
||||
Arguments U {_} _ _.
|
||||
Arguments L {_} _.
|
||||
|
||||
End FinSet.
|
||||
|
||||
Section functions.
|
||||
Parameter A : Type.
|
||||
Parameter eq : A -> A -> Bool.
|
||||
Definition isIn : A -> FinSets A -> Bool.
|
||||
Parameter eq_refl: forall a:A, eq a a = true.
|
||||
|
||||
Definition isIn : A -> FSet A -> Bool.
|
||||
Proof.
|
||||
intros a X.
|
||||
hrecursion X.
|
||||
|
|
@ -156,17 +349,15 @@ hrecursion X.
|
|||
- intros a'. compute. destruct (eq a a'); reflexivity.
|
||||
Defined.
|
||||
|
||||
|
||||
Definition comprehension :
|
||||
(A -> Bool) -> FinSets A -> FinSets A.
|
||||
(A -> Bool) -> FSet A -> FSet A.
|
||||
Proof.
|
||||
intros P X.
|
||||
hrecursion X.
|
||||
- apply empty.
|
||||
- apply E.
|
||||
- intro a.
|
||||
refine (if (P a) then L A a else empty A).
|
||||
- intros X Y.
|
||||
apply (U A X Y).
|
||||
refine (if (P a) then L a else E).
|
||||
- apply U.
|
||||
- intros. cbv. apply assoc.
|
||||
- intros. cbv. apply comm.
|
||||
- intros. cbv. apply nl.
|
||||
|
|
@ -176,15 +367,37 @@ hrecursion X.
|
|||
+ apply idem.
|
||||
+ apply nl.
|
||||
Defined.
|
||||
Require Import FunextAxiom.
|
||||
Lemma comprehension_idem:
|
||||
forall (X:FSet A), forall Y, comprehension (fun x => isIn x (U X Y)) X = X.
|
||||
Proof.
|
||||
simple refine (FSet_ind _ _ _ _ _ _ _ _ _ _ _); simpl.
|
||||
- intro Y. cbv. reflexivity.
|
||||
- intros a Y. unfold comprehension. unfold HitRec.hrecursion. simpl.
|
||||
enough (isIn a (U (L a) Y) = true).
|
||||
+ rewrite X. reflexivity.
|
||||
+ unfold isIn. unfold HitRec.hrecursion. simpl.
|
||||
rewrite eq_refl. auto.
|
||||
- intros X1 X2 IH1 IH2 Y.
|
||||
unfold comprehension. unfold HitRec.hrecursion. simpl.
|
||||
rewrite <- (assoc _ X1 X2 Y).
|
||||
f_ap.
|
||||
+ apply (IH1 (U X2 Y)).
|
||||
+ rewrite (assoc _ X1 X2 Y).
|
||||
rewrite (comm _ X1 X2).
|
||||
rewrite <- (assoc _ X2 X1 Y).
|
||||
apply (IH2 (U X1 Y)).
|
||||
Admitted.
|
||||
|
||||
|
||||
Definition intersection :
|
||||
FinSets A -> FinSets A -> FinSets A.
|
||||
FSet A -> FSet A -> FSet A.
|
||||
Proof.
|
||||
intros X Y.
|
||||
apply (comprehension (fun (a : A) => isIn a X) Y).
|
||||
Defined.
|
||||
|
||||
Definition subset (x : FinSets A) (y : FinSets A) : Bool.
|
||||
Definition subset (x : FSet A) (y : FSet A) : Bool.
|
||||
Proof.
|
||||
hrecursion x.
|
||||
- apply true.
|
||||
|
|
@ -198,9 +411,10 @@ hrecursion x.
|
|||
destruct (isIn a y); reflexivity.
|
||||
Defined.
|
||||
|
||||
Definition subset' (x : FinSets A) (y : FinSets A) : Bool.
|
||||
|
||||
Definition subset' (x : FSet A) (y : FSet A) : Bool.
|
||||
Proof.
|
||||
refine (FinSets_rec A _ _ _ _ _ _ _ _ _ _).
|
||||
refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
|
||||
Unshelve.
|
||||
|
||||
Focus 6.
|
||||
|
|
@ -259,7 +473,7 @@ refine (FinSets_rec A _ _ _ _ _ _ _ _ _ _).
|
|||
Defined.
|
||||
(* TODO: subset = subset' *)
|
||||
|
||||
Definition equal_set (x : FinSets A) (y : FinSets A) : Bool
|
||||
Definition equal_set (x : FSet A) (y : FSet A) : Bool
|
||||
:= andb (subset x y) (subset y x).
|
||||
|
||||
Fixpoint eq_nat n m : Bool :=
|
||||
|
|
|
|||
Loading…
Reference in New Issue