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Setup for finite sets library.
This commit is contained in:
Leon Gondelman
5
FiniteSets/_CoqProject
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5
FiniteSets/_CoqProject
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-R . "" COQC = hoqc COQDEP = hoqdep
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definition.v
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operations.v
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properties.v
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207
FiniteSets/definition.v
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207
FiniteSets/definition.v
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Require Import HoTT.
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Require Export HoTT.
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Module Export definition.
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Section FSet.
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Variable A : Type.
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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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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 assoc : forall (x y z : FSet ),
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x ∪ (y ∪ z) = (x ∪ y) ∪ z.
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Axiom comm : forall (x y : FSet),
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x ∪ y = y ∪ x.
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Axiom nl : forall (x : FSet),
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∅ ∪ x = x.
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Axiom nr : forall (x : FSet),
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x ∪ ∅ = x.
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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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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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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 : 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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| 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 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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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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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 : 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 (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 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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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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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 : 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 (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 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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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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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 : 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 (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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).
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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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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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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 : 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 (FSet_rec P H e l u assocP commP nlP nrP idemP x)
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).
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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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(assocP : forall (x y z : P), u x (u y z) = u (u x y) z)
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(commP : forall (x y : P), u x y = u y x)
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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 : A),
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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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End FSet.
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Section FSet_induction.
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Arguments E {_}.
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Arguments U {_} _ _.
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Arguments L {_} _.
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Arguments assoc {_} _ _ _.
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Arguments comm {_} _ _.
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Arguments nl {_} _.
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Arguments nr {_} _.
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Arguments idem {_} _.
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Variable A: Type.
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Variable (P : FSet A -> Type).
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Variable (H : forall a : FSet A, IsHSet (P a)).
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Variable (eP : P E).
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Variable (lP : forall a: A, P (L a)).
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Variable (uP : forall (x y: FSet A), P x -> P y -> P (U x y)).
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Variable (assocP : forall (x y z : FSet A)
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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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Variable (commP : forall (x y: FSet A) (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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Variable (nlP : forall (x : FSet A) (px: P x),
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nl x # uP E x eP px = px).
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Variable (nrP : forall (x : FSet A) (px: P x),
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nr x # uP x E px eP = px).
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Variable (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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(* Induction principle *)
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Fixpoint FSet_ind
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(x : FSet A)
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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 y)
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(FSet_ind z)
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end) H assocP commP nlP nrP idemP.
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Axiom FSet_ind_beta_assoc : forall (x y z : FSet A),
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apD FSet_ind (assoc x y z) =
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(assocP x y z (FSet_ind x) (FSet_ind y) (FSet_ind z)).
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Axiom FSet_ind_beta_comm : forall (x y : FSet A),
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apD FSet_ind (comm x y) = (commP x y (FSet_ind x) (FSet_ind y)).
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Axiom FSet_ind_beta_nl : forall (x : FSet A),
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apD FSet_ind (nl x) = (nlP x (FSet_ind x)).
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Axiom FSet_ind_beta_nr : forall (x : FSet A),
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apD FSet_ind (nr x) = (nrP x (FSet_ind x)).
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Axiom FSet_ind_beta_idem : forall (x : A), apD FSet_ind (idem x) = idemP x.
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End FSet_induction.
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End definition.
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63
FiniteSets/operations.v
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63
FiniteSets/operations.v
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@@ -0,0 +1,63 @@
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Require Import HoTT.
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Require Export HoTT.
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Require Import definition.
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(*Set Implicit Arguments.*)
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Arguments E {_}.
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Arguments U {_} _ _.
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Arguments L {_} _.
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Arguments assoc {_} _ _ _.
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Arguments comm {_} _ _.
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Arguments nl {_} _.
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Arguments nr {_} _.
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Arguments idem {_} _.
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Section operations.
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Variable A : Type.
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Parameter A_eqdec : forall (x y : A), Decidable (x = y).
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Definition deceq (x y : A) :=
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if dec (x = y) then true else false.
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Definition isIn : A -> FSet A -> Bool.
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Proof.
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intros a.
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simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
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- exact false.
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- intro a'. apply (deceq a a').
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- apply orb.
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- intros x y z. destruct x; reflexivity.
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- intros x y. destruct x, y; reflexivity.
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- intros x. reflexivity.
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- intros x. destruct x; reflexivity.
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- intros a'. destruct (deceq a a'); reflexivity.
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Defined.
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Infix "∈" := isIn (at level 9, right associativity).
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Definition comprehension :
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(A -> Bool) -> FSet A -> FSet A.
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Proof.
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intros P.
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simple refine (FSet_rec A _ _ _ _ _ _ _ _ _ _).
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- apply E.
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- intro a.
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refine (if (P a) then L a else E).
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- apply U.
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- intros. cbv. apply assoc.
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- intros. cbv. apply comm.
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- intros. cbv. apply nl.
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- intros. cbv. apply nr.
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- intros. cbv.
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destruct (P x); simpl.
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+ apply idem.
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+ apply nl.
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Defined.
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Definition intersection :
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FSet A -> FSet A -> FSet A.
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Proof.
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intros X Y.
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apply (comprehension (fun (a : A) => isIn a X) Y).
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Defined.
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End operations.
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271
FiniteSets/properties.v
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271
FiniteSets/properties.v
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@@ -0,0 +1,271 @@
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Require Import HoTT.
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Require Export HoTT.
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Require Import definition.
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Require Import operations.
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Section properties.
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Arguments E {_}.
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Arguments U {_} _ _.
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Arguments L {_} _.
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Arguments assoc {_} _ _ _.
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Arguments comm {_} _ _.
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Arguments nl {_} _.
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Arguments nr {_} _.
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Arguments idem {_} _.
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Arguments isIn {_} _ _.
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Arguments comprehension {_} _ _.
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Arguments intersection {_} _ _.
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Variable A : Type.
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Parameter A_eqdec : forall (x y : A), Decidable (x = y).
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Definition deceq (x y : A) :=
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if dec (x = y) then true else false.
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Theorem deceq_sym : forall x y, deceq x y = deceq y x.
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Proof.
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intros x y.
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unfold deceq.
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destruct (dec (x = y)) ; destruct (dec (y = x)) ; cbn.
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- reflexivity.
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- pose (n (p^)) ; contradiction.
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- pose (n (p^)) ; contradiction.
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- reflexivity.
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Defined.
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Lemma comprehension_false: forall Y: FSet A,
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comprehension (fun a => isIn a E) Y = E.
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Proof.
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simple refine (FSet_ind _ _ _ _ _ _ _ _ _ _ _);
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try (intros; apply set_path2).
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- cbn. reflexivity.
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- cbn. reflexivity.
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- intros x y IHa IHb.
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cbn.
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rewrite IHa.
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rewrite IHb.
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rewrite nl.
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reflexivity.
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Defined.
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Lemma isIn_singleton_eq (a b: A) : isIn a (L b) = true -> a = b.
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Proof. unfold isIn. simpl. unfold operations.deceq.
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destruct (dec (a = b)). intro. apply p.
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intro X.
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contradiction (false_ne_true X).
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Defined.
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Lemma isIn_empty_false (a: A) : isIn a E = true -> Empty.
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Proof.
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cbv. intro X.
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contradiction (false_ne_true X).
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Defined.
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Lemma isIn_union (a: A) (X Y: FSet A) :
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isIn a (U X Y) = (isIn a X || isIn a Y)%Bool.
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Proof. reflexivity. Qed.
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Theorem comprehension_or : forall ϕ ψ (x: FSet A),
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comprehension (fun a => orb (ϕ a) (ψ a)) x = U (comprehension ϕ x)
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(comprehension ψ x).
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Proof.
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intros ϕ ψ.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
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- cbn. symmetry ; apply nl.
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- cbn. intros.
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destruct (ϕ a) ; destruct (ψ a) ; symmetry.
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* apply idem.
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* apply nr.
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* apply nl.
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* apply nl.
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- simpl. intros x y P Q.
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cbn.
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rewrite P.
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rewrite Q.
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rewrite <- assoc.
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rewrite (assoc (comprehension ψ x)).
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rewrite (comm (comprehension ψ x) (comprehension ϕ y)).
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rewrite <- assoc.
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rewrite <- assoc.
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reflexivity.
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Defined.
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Theorem intersection_isIn : forall a (x y: FSet A),
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isIn a (intersection x y) = andb (isIn a x) (isIn a y).
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Proof.
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Admitted.
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(*
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intros a.
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simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; cbn.
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- intros y.
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rewrite intersection_E.
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reflexivity.
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- intros b y.
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rewrite intersection_La.
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unfold deceq.
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destruct (dec (a = b)) ; cbn.
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* rewrite p.
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destruct (isIn b y).
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+ cbn.
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unfold deceq.
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destruct (dec (b = b)).
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{ reflexivity. }
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{ pose (n idpath). contradiction. }
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+ reflexivity.
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* destruct (isIn b y).
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+ cbn.
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unfold deceq.
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destruct (dec (a = b)).
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{ pose (n p). contradiction. }
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{ reflexivity. }
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+ reflexivity.
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- intros x y P Q z.
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enough (intersection (U x y) z = U (intersection x z) (intersection y z)).
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rewrite X.
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cbn.
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rewrite P.
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rewrite Q.
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destruct (isIn a x) ; destruct (isIn a y) ; destruct (isIn a z) ; reflexivity.
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admit.
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Admitted.
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||||
*)
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||||
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||||
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Theorem union_idem : forall x: FSet A, U x x = x.
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Proof.
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||||
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ;
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||||
try (intros ; apply set_path2) ; cbn.
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||||
- apply nl.
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||||
- apply idem.
|
||||
- intros x y P Q.
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||||
rewrite assoc.
|
||||
rewrite (comm x y).
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||||
rewrite <- (assoc y x x).
|
||||
rewrite P.
|
||||
rewrite (comm y x).
|
||||
rewrite <- (assoc x y y).
|
||||
rewrite Q.
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||||
reflexivity.
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||||
Defined.
|
||||
|
||||
Lemma intersection_0l: forall X: FSet A, intersection E X = E.
|
||||
Proof.
|
||||
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.
|
||||
|
||||
Definition intersection_0r (X: FSet A): intersection X E = E := idpath.
|
||||
|
||||
|
||||
|
||||
|
||||
Lemma intersection_comm (X Y: FSet A): intersection X Y = intersection Y X.
|
||||
Proof.
|
||||
(*
|
||||
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _ X) ;
|
||||
try (intros; apply set_path2).
|
||||
- cbn. unfold intersection. apply comprehension_false.
|
||||
- cbn. unfold intersection. intros a.
|
||||
hrecursion Y; try (intros; apply set_path2).
|
||||
+ cbn. reflexivity.
|
||||
+ cbn. intros.
|
||||
destruct (dec (a0 = a)).
|
||||
rewrite p. destruct (dec (a=a)).
|
||||
reflexivity.
|
||||
contradiction n.
|
||||
reflexivity.
|
||||
destruct (dec (a = a0)).
|
||||
contradiction n. apply p^. reflexivity.
|
||||
+ cbn -[isIn]. intros Y1 Y2 IH1 IH2.
|
||||
rewrite IH1.
|
||||
rewrite IH2.
|
||||
apply (comprehension_union (L a)).
|
||||
- intros X1 X2 IH1 IH2.
|
||||
cbn.
|
||||
unfold intersection in *.
|
||||
rewrite <- IH1.
|
||||
rewrite <- IH2. symmetry.
|
||||
apply comprehension_union.
|
||||
Defined.*)
|
||||
Admitted.
|
||||
|
||||
Lemma comprehension_union (X Y Z: FSet A) :
|
||||
U (comprehension (fun a => isIn a Y) X)
|
||||
(comprehension (fun a => isIn a Z) X) =
|
||||
comprehension (fun a => isIn a (U Y Z)) X.
|
||||
Proof.
|
||||
Admitted.
|
||||
(*
|
||||
hrecursion X; try (intros; apply set_path2).
|
||||
- cbn. apply nl.
|
||||
- cbn. intro a.
|
||||
destruct (isIn a Y); simpl;
|
||||
destruct (isIn a Z); simpl.
|
||||
apply idem.
|
||||
apply nr.
|
||||
apply nl.
|
||||
apply nl.
|
||||
- cbn. intros X1 X2 IH1 IH2.
|
||||
rewrite assoc.
|
||||
rewrite (comm _ (comprehension (fun a : A => isIn a Y) X1)
|
||||
(comprehension (fun a : A => isIn a Y) X2)).
|
||||
rewrite <- (assoc _
|
||||
(comprehension (fun a : A => isIn a Y) X2)
|
||||
(comprehension (fun a : A => isIn a Y) X1)
|
||||
(comprehension (fun a : A => isIn a Z) X1)
|
||||
).
|
||||
rewrite IH1.
|
||||
rewrite comm.
|
||||
rewrite assoc.
|
||||
rewrite (comm _ (comprehension (fun a : A => isIn a Z) X2) _).
|
||||
rewrite IH2.
|
||||
apply comm.
|
||||
Defined.*)
|
||||
|
||||
Theorem intersection_assoc : forall (x y z: FSet A),
|
||||
intersection x (intersection y z) = intersection (intersection x y) z.
|
||||
Admitted.
|
||||
(*
|
||||
simple refine (FSet_ind A _ _ _ _ _ _ _ _ _ _) ; try (intros ; apply set_path2).
|
||||
- intros y z.
|
||||
cbn.
|
||||
rewrite intersection_E.
|
||||
rewrite intersection_E.
|
||||
rewrite intersection_E.
|
||||
reflexivity.
|
||||
- intros a y z.
|
||||
cbn.
|
||||
rewrite intersection_La.
|
||||
rewrite intersection_La.
|
||||
rewrite intersection_isIn.
|
||||
destruct (isIn a y).
|
||||
* rewrite intersection_La.
|
||||
destruct (isIn a z).
|
||||
+ reflexivity.
|
||||
+ reflexivity.
|
||||
* rewrite intersection_E.
|
||||
reflexivity.
|
||||
- unfold intersection. cbn.
|
||||
intros x y P Q z z'.
|
||||
rewrite comprehension_or.
|
||||
rewrite P.
|
||||
rewrite Q.
|
||||
rewrite comprehension_or.
|
||||
cbn.
|
||||
rewrite comprehension_or.
|
||||
reflexivity.
|
||||
*)
|
||||
|
||||
End properties.
|
||||
|
||||
Reference in New Issue
Block a user