mirror of https://github.com/nmvdw/HITs-Examples
Merge hrecursion into master
This commit is contained in:
Dan Frumin
commit
25d1e1c969
645
FinSets.v
645
FinSets.v
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@ -1,31 +1,39 @@
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Require Import HoTT.
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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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@ -128,200 +136,397 @@ 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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|
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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))
|
||||
(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))
|
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=
|
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(uP (U x y) z (uP x y px py) pz))
|
||||
(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) (px: P x),
|
||||
nl x # uP E x eP px = px)
|
||||
(nrP : forall (x : FSet) (px: P x),
|
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nr x # uP x E px eP = px)
|
||||
(idemP : forall (x : A),
|
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idem x # uP (L x) (L x) (lP x) (lP x) = lP x)
|
||||
(x : FSet),
|
||||
apD (FSet_ind P H eP lP uP assocP commP nlP nrP idemP) (nr x)
|
||||
=
|
||||
(nrP x (FSet_ind P H eP lP uP assocP commP nlP nrP idemP x)
|
||||
).
|
||||
|
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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))
|
||||
(eP : P E)
|
||||
(lP : forall a: A, P (L a))
|
||||
(uP : forall (x y: FSet), P x -> P y -> P (U x y))
|
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(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) (px: P x) (py: P y),
|
||||
comm x y #
|
||||
uP x y px py = uP y x py 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 H eP lP uP assocP commP nlP nrP idemP) (idem x)
|
||||
=
|
||||
idemP x.
|
||||
|
||||
End FSet.
|
||||
|
||||
Instance FSet_recursion A : HitRecursion (FSet A) := {
|
||||
indTy := _; recTy := _;
|
||||
H_inductor := FSet_ind A; H_recursor := FSet_rec A }.
|
||||
|
||||
Arguments E {_}.
|
||||
Arguments U {_} _ _.
|
||||
Arguments L {_} _.
|
||||
|
||||
End FinSet.
|
||||
|
||||
Definition isIn : forall
|
||||
(A : Type)
|
||||
(eq : A -> A -> Bool),
|
||||
A -> FinSets A -> Bool.
|
||||
Section functions.
|
||||
Parameter A : Type.
|
||||
Context {A_eqdec: DecidablePaths A}.
|
||||
|
||||
Notation "{| x |}" := (L x).
|
||||
Infix "∪" := U (at level 8, right associativity).
|
||||
Notation "∅" := E.
|
||||
|
||||
(** Properties of union *)
|
||||
Lemma union_idem : forall (X : FSet A), U X X = X.
|
||||
Proof.
|
||||
intro A.
|
||||
intro eq.
|
||||
intro a.
|
||||
refine (FinSets_rec A _ _ _ _ _ _ _ _ _).
|
||||
Unshelve.
|
||||
hinduction; try (intros; apply set_path2).
|
||||
- apply nr.
|
||||
- intros. apply idem.
|
||||
- intros X Y HX HY. etransitivity.
|
||||
rewrite assoc. rewrite (comm _ X Y). rewrite <- (assoc _ Y X X).
|
||||
rewrite comm.
|
||||
rewrite assoc. rewrite HX. rewrite HY. reflexivity.
|
||||
rewrite comm. reflexivity.
|
||||
Defined.
|
||||
|
||||
Focus 6.
|
||||
apply false.
|
||||
(** Membership predicate *)
|
||||
Definition isIn : A -> FSet A -> Bool.
|
||||
Proof.
|
||||
intros a.
|
||||
hrecursion.
|
||||
- exact false.
|
||||
- intro a'. destruct (dec (a = a')); [exact true | exact false].
|
||||
- apply orb.
|
||||
- intros x y z. compute. destruct x; reflexivity.
|
||||
- intros x y. compute. destruct x, y; reflexivity.
|
||||
- intros x. compute. reflexivity.
|
||||
- intros x. compute. destruct x; reflexivity.
|
||||
- intros a'. compute. destruct (A_eqdec a a'); reflexivity.
|
||||
Defined.
|
||||
|
||||
Focus 6.
|
||||
intro a'.
|
||||
apply (eq a a').
|
||||
Infix "∈" := isIn (at level 9, right associativity).
|
||||
|
||||
Focus 6.
|
||||
intro b.
|
||||
intro b'.
|
||||
apply (orb b b').
|
||||
Lemma isIn_singleton_eq a b : a ∈ {| b |} = true -> a = b.
|
||||
Proof. cbv.
|
||||
destruct (A_eqdec a b). intro. apply p.
|
||||
intro X.
|
||||
contradiction (false_ne_true X).
|
||||
Defined.
|
||||
|
||||
Focus 3.
|
||||
intros.
|
||||
compute.
|
||||
reflexivity.
|
||||
Lemma isIn_empty_false a : a ∈ ∅ = true -> Empty.
|
||||
Proof.
|
||||
cbv. intro X.
|
||||
contradiction (false_ne_true X).
|
||||
Defined.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x.
|
||||
reflexivity.
|
||||
destruct y.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
Lemma isIn_union a X Y : a ∈ (X ∪ Y) = (a ∈ X || a ∈ Y)%Bool.
|
||||
Proof. reflexivity. Qed.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x.
|
||||
destruct y.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
destruct y.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
(** Set comprehension *)
|
||||
Definition comprehension :
|
||||
(A -> Bool) -> FSet A -> FSet A.
|
||||
Proof.
|
||||
intros P.
|
||||
hrecursion.
|
||||
- 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.
|
||||
|
||||
Focus 1.
|
||||
intros.
|
||||
compute.
|
||||
destruct x.
|
||||
reflexivity.
|
||||
reflexivity.
|
||||
|
||||
intros.
|
||||
compute.
|
||||
destruct (eq a x).
|
||||
reflexivity.
|
||||
Lemma comprehension_false Y : comprehension (fun a => a ∈ ∅) Y = E.
|
||||
Proof.
|
||||
hrecursion Y; try (intros; apply set_path2).
|
||||
- cbn. reflexivity.
|
||||
- cbn. reflexivity.
|
||||
- intros x y IHa IHb.
|
||||
cbn.
|
||||
rewrite IHa.
|
||||
rewrite IHb.
|
||||
rewrite nl.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Definition comprehension : forall
|
||||
(A : Type)
|
||||
(eq : A -> A -> Bool),
|
||||
(A -> Bool) -> FinSets A -> FinSets A.
|
||||
Lemma comprehension_union X Y Z :
|
||||
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.
|
||||
intro A.
|
||||
intro eq.
|
||||
intro phi.
|
||||
refine (FinSets_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).
|
||||
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.
|
||||
|
||||
Definition intersection : forall (A : Type) (eq : A -> A -> Bool),
|
||||
FinSets A -> FinSets A -> FinSets A.
|
||||
|
||||
Lemma comprehension_idem' `{Funext}:
|
||||
forall (X:FSet A), forall Y, comprehension (fun x => x ∈ (U X Y)) X = X.
|
||||
Proof.
|
||||
intro A.
|
||||
intro eq.
|
||||
intro x.
|
||||
intro y.
|
||||
apply (comprehension A eq (fun (a : A) => isIn A eq a x) y).
|
||||
hinduction.
|
||||
all: try (intros; apply path_forall; intro; apply set_path2).
|
||||
- intro Y. cbv. reflexivity.
|
||||
- intros a Y. cbn.
|
||||
destruct (dec (a = a)); simpl.
|
||||
+ reflexivity.
|
||||
+ contradiction n. reflexivity.
|
||||
- intros X1 X2 IH1 IH2 Y.
|
||||
cbn -[isIn].
|
||||
f_ap.
|
||||
+ rewrite <- assoc. apply (IH1 (U X2 Y)).
|
||||
+ rewrite (comm _ X1 X2).
|
||||
rewrite <- (assoc _ X2 X1 Y).
|
||||
apply (IH2 (U X1 Y)).
|
||||
Defined.
|
||||
|
||||
Definition subset (A : Type) (eq : A -> A -> Bool) (x : FinSets A) (y : FinSets A) : Bool.
|
||||
Lemma comprehension_idem `{Funext}:
|
||||
forall (X:FSet A), comprehension (fun x => x ∈ X) X = X.
|
||||
Proof.
|
||||
refine (FinSets_rec A _ _ _ _ _ _ _ _ _ _).
|
||||
Unshelve.
|
||||
|
||||
Focus 6.
|
||||
apply x.
|
||||
|
||||
Focus 6.
|
||||
apply true.
|
||||
|
||||
Focus 6.
|
||||
intro a.
|
||||
apply (isIn A eq a y).
|
||||
|
||||
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.
|
||||
intros X.
|
||||
enough (comprehension (fun x : A => isIn x (U X X)) X = X).
|
||||
rewrite (union_idem) in X0. assumption.
|
||||
apply comprehension_idem'.
|
||||
Defined.
|
||||
|
||||
Definition equal_set (A : Type) (eq : A -> A -> Bool) (x : FinSets A) (y : FinSets A) : Bool
|
||||
:= andb (subset A eq x y) (subset A eq y x).
|
||||
(** Set intersection *)
|
||||
Definition intersection :
|
||||
FSet A -> FSet A -> FSet A.
|
||||
Proof.
|
||||
intros X Y.
|
||||
apply (comprehension (fun (a : A) => isIn a X) Y).
|
||||
Defined.
|
||||
|
||||
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.
|
||||
Lemma intersection_comm X Y: intersection X Y = intersection Y X.
|
||||
Proof.
|
||||
hrecursion 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.
|
||||
|
||||
|
||||
(** Subset ordering *)
|
||||
Definition subset (x : FSet A) (y : FSet A) : Bool.
|
||||
Proof.
|
||||
hrecursion x.
|
||||
- apply true.
|
||||
- intro a. apply (isIn a y).
|
||||
- intros a b. apply (andb a b).
|
||||
- intros a b c. compute. destruct a; reflexivity.
|
||||
- intros a b. compute. destruct a, b; reflexivity.
|
||||
- intros x. compute. reflexivity.
|
||||
- intros x. compute. destruct x;reflexivity.
|
||||
- intros a. simpl.
|
||||
destruct (isIn a y); reflexivity.
|
||||
Defined.
|
||||
|
||||
Infix "⊆" := subset (at level 8, right associativity).
|
||||
|
||||
End functions.
|
||||
|
|
@ -0,0 +1,85 @@
|
|||
Class HitRecursion (H : Type) := {
|
||||
indTy : Type;
|
||||
recTy : Type;
|
||||
H_inductor : indTy;
|
||||
H_recursor : recTy
|
||||
}.
|
||||
|
||||
Definition hrecursion (H : Type) {HR : HitRecursion H} : @recTy H HR :=
|
||||
@H_recursor H HR.
|
||||
|
||||
Definition hinduction (H : Type) {HR : HitRecursion H} : @indTy H HR :=
|
||||
@H_inductor H HR.
|
||||
|
||||
Ltac hrecursion_ :=
|
||||
lazymatch goal with
|
||||
| [ |- ?T -> ?P ] =>
|
||||
let hrec1 := eval cbv[hrecursion H_recursor] in (hrecursion T) in
|
||||
let hrec := eval simpl in hrec1 in
|
||||
match type of hrec with
|
||||
| ?Q =>
|
||||
match (eval simpl in Q) with
|
||||
| forall Y, T -> Y =>
|
||||
simple refine (hrec P)
|
||||
| forall Y _, T -> Y =>
|
||||
simple refine (hrec P _)
|
||||
| forall Y _ _, T -> Y =>
|
||||
simple refine (hrec P _ _)
|
||||
| forall Y _ _ _, T -> Y =>
|
||||
simple refine (hrec P _ _ _)
|
||||
| forall Y _ _ _ _, T -> Y =>
|
||||
simple refine (hrec P _ _ _ _)
|
||||
| forall Y _ _ _ _ _, T -> Y =>
|
||||
simple refine (hrec P _ _ _ _ _)
|
||||
| forall Y _ _ _ _ _ _, T -> Y =>
|
||||
simple refine (hrec P _ _ _ _ _ _)
|
||||
| forall Y _ _ _ _ _ _ _, T -> Y =>
|
||||
simple refine (hrec P _ _ _ _ _ _ _)
|
||||
| forall Y _ _ _ _ _ _ _ _, T -> Y =>
|
||||
simple refine (hrec P _ _ _ _ _ _ _ _)
|
||||
| forall Y _ _ _ _ _ _ _ _ _, T -> Y =>
|
||||
simple refine (hrec P _ _ _ _ _ _ _ _ _)
|
||||
| forall Y _ _ _ _ _ _ _ _ _ _, T -> Y =>
|
||||
simple refine (hrec P _ _ _ _ _ _ _ _ _ _)
|
||||
| _ => fail "Cannot handle the recursion principle (too many parameters?) :("
|
||||
end
|
||||
end
|
||||
| [ |- forall (target:?T), ?P] =>
|
||||
let hind1 := eval cbv[hinduction H_inductor] in (hinduction T) in
|
||||
let hind := eval simpl in hind1 in
|
||||
match type of hind with
|
||||
| ?Q =>
|
||||
match (eval simpl in Q) with
|
||||
| forall Y , (forall x, Y x) =>
|
||||
simple refine (hind (fun target => P) _)
|
||||
| forall Y _, (forall x, Y x) =>
|
||||
simple refine (hind (fun target => P) _)
|
||||
| forall Y _ _, (forall x, Y x) =>
|
||||
simple refine (hind (fun target => P) _ _)
|
||||
| forall Y _ _ _, (forall x, Y x) =>
|
||||
simple refine (hind (fun target => P) _ _ _)
|
||||
| forall Y _ _ _ _, (forall x, Y x) =>
|
||||
simple refine (hind (fun target => P) _ _ _ _)
|
||||
| forall Y _ _ _ _ _, (forall x, Y x) =>
|
||||
simple refine (hind (fun target => P) _ _ _ _ _)
|
||||
| forall Y _ _ _ _ _ _, (forall x, Y x) =>
|
||||
simple refine (hind (fun target => P) _ _ _ _ _ _)
|
||||
| forall Y _ _ _ _ _ _ _, (forall x, Y x) =>
|
||||
simple refine (hind (fun target => P) _ _ _ _ _ _ _)
|
||||
| forall Y _ _ _ _ _ _ _, (forall x, Y x) =>
|
||||
simple refine (hind (fun target => P) _ _ _ _ _ _ _)
|
||||
| forall Y _ _ _ _ _ _ _ _, (forall x, Y x) =>
|
||||
simple refine (hind (fun target => P) _ _ _ _ _ _ _ _)
|
||||
| forall Y _ _ _ _ _ _ _ _ _, (forall x, Y x) =>
|
||||
simple refine (hind (fun target => P) _ _ _ _ _ _ _ _ _)
|
||||
| _ => fail "Cannot handle the induction principle (too many parameters?) :("
|
||||
end
|
||||
end
|
||||
(*| _ => fail "I am sorry, but something went wrong!"*)
|
||||
end.
|
||||
|
||||
Tactic Notation "hrecursion" := hrecursion_; simpl.
|
||||
Tactic Notation "hrecursion" ident(x) := revert x; hrecursion.
|
||||
Tactic Notation "hinduction" := hrecursion_; simpl.
|
||||
Tactic Notation "hinduction" ident(x) := revert x; hrecursion.
|
||||
|
||||
168
Mod2.v
168
Mod2.v
|
|
@ -1,5 +1,5 @@
|
|||
Require Import HoTT.
|
||||
Require Export HoTT.
|
||||
Require Import HitTactics.
|
||||
|
||||
Module Export modulo.
|
||||
|
||||
|
|
@ -51,166 +51,98 @@ Axiom Mod2_rec_beta_mod : forall
|
|||
(mod' : a = s (s a))
|
||||
, ap (Mod2_rec P a s mod') mod = mod'.
|
||||
|
||||
Instance: HitRecursion Mod2 := {
|
||||
indTy := _; recTy := _;
|
||||
H_inductor := Mod2_ind;
|
||||
H_recursor := Mod2_rec }.
|
||||
|
||||
End modulo.
|
||||
|
||||
Definition negate : Mod2 -> Mod2.
|
||||
Proof.
|
||||
refine (Mod2_ind _ _ _ _).
|
||||
Unshelve.
|
||||
Focus 2.
|
||||
apply (succ Z).
|
||||
|
||||
Focus 2.
|
||||
intros.
|
||||
apply (succ H).
|
||||
|
||||
simpl.
|
||||
rewrite transport_const.
|
||||
rewrite <- mod.
|
||||
reflexivity.
|
||||
Defined.
|
||||
|
||||
Theorem modulo2 : forall n : Mod2, n = succ(succ n).
|
||||
Proof.
|
||||
refine (Mod2_ind _ _ _ _).
|
||||
Unshelve.
|
||||
Focus 2.
|
||||
apply mod.
|
||||
|
||||
Focus 2.
|
||||
intro n.
|
||||
intro p.
|
||||
intro n.
|
||||
hinduction n.
|
||||
- apply mod.
|
||||
- intros n p.
|
||||
apply (ap succ p).
|
||||
- simpl.
|
||||
etransitivity.
|
||||
eapply (@transport_paths_FlFr _ _ idmap (fun n => succ (succ n))).
|
||||
hott_simpl.
|
||||
apply ap_compose.
|
||||
Defined.
|
||||
|
||||
simpl.
|
||||
rewrite @HoTT.Types.Paths.transport_paths_FlFr.
|
||||
rewrite ap_idmap.
|
||||
rewrite concat_Vp.
|
||||
rewrite concat_1p.
|
||||
rewrite ap_compose.
|
||||
reflexivity.
|
||||
Definition negate : Mod2 -> Mod2.
|
||||
Proof.
|
||||
hrecursion.
|
||||
- apply Z.
|
||||
- intros. apply (succ H).
|
||||
- simpl. apply mod.
|
||||
Defined.
|
||||
|
||||
Definition plus : Mod2 -> Mod2 -> Mod2.
|
||||
Proof.
|
||||
intro n.
|
||||
refine (Mod2_ind _ _ _ _).
|
||||
Unshelve.
|
||||
|
||||
Focus 2.
|
||||
apply n.
|
||||
|
||||
Focus 2.
|
||||
intro m.
|
||||
intro k.
|
||||
apply (succ k).
|
||||
|
||||
simpl.
|
||||
rewrite transport_const.
|
||||
apply modulo2.
|
||||
intros n m.
|
||||
hrecursion m.
|
||||
- exact n.
|
||||
- apply succ.
|
||||
- apply modulo2.
|
||||
Defined.
|
||||
|
||||
Definition Bool_to_Mod2 : Bool -> Mod2.
|
||||
Proof.
|
||||
intro b.
|
||||
destruct b.
|
||||
apply (succ Z).
|
||||
|
||||
apply Z.
|
||||
+ apply (succ Z).
|
||||
+ apply Z.
|
||||
Defined.
|
||||
|
||||
Definition Mod2_to_Bool : Mod2 -> Bool.
|
||||
Proof.
|
||||
refine (Mod2_ind _ _ _ _).
|
||||
Unshelve.
|
||||
Focus 2.
|
||||
apply false.
|
||||
|
||||
Focus 2.
|
||||
intro n.
|
||||
apply negb.
|
||||
|
||||
Focus 1.
|
||||
simpl.
|
||||
apply transport_const.
|
||||
intro x.
|
||||
hrecursion x.
|
||||
- exact false.
|
||||
- exact negb.
|
||||
- simpl. reflexivity.
|
||||
Defined.
|
||||
|
||||
Theorem eq1 : forall n : Bool, Mod2_to_Bool (Bool_to_Mod2 n) = n.
|
||||
Proof.
|
||||
intro b.
|
||||
destruct b.
|
||||
Focus 1.
|
||||
compute.
|
||||
reflexivity.
|
||||
|
||||
compute.
|
||||
reflexivity.
|
||||
destruct b; compute; reflexivity.
|
||||
Qed.
|
||||
|
||||
Theorem Bool_to_Mod2_negb : forall x : Bool,
|
||||
succ (Bool_to_Mod2 x) = Bool_to_Mod2 (negb x).
|
||||
Proof.
|
||||
intros.
|
||||
destruct x.
|
||||
compute.
|
||||
apply mod^.
|
||||
|
||||
compute.
|
||||
apply reflexivity.
|
||||
destruct x; compute.
|
||||
+ apply mod^.
|
||||
+ apply reflexivity.
|
||||
Defined.
|
||||
|
||||
Theorem eq2 : forall n : Mod2, Bool_to_Mod2 (Mod2_to_Bool n) = n.
|
||||
Proof.
|
||||
refine (Mod2_ind _ _ _ _).
|
||||
Unshelve.
|
||||
Focus 2.
|
||||
compute.
|
||||
reflexivity.
|
||||
|
||||
Focus 2.
|
||||
intro n.
|
||||
intro IHn.
|
||||
symmetry.
|
||||
transitivity (succ (Bool_to_Mod2 (Mod2_to_Bool n))).
|
||||
|
||||
Focus 1.
|
||||
symmetry.
|
||||
apply (ap succ IHn).
|
||||
|
||||
transitivity (Bool_to_Mod2 (negb (Mod2_to_Bool n))).
|
||||
apply Bool_to_Mod2_negb.
|
||||
enough (negb (Mod2_to_Bool n) = Mod2_to_Bool (succ n)).
|
||||
apply (ap Bool_to_Mod2 X).
|
||||
|
||||
compute.
|
||||
reflexivity.
|
||||
simpl.
|
||||
rewrite concat_p1.
|
||||
rewrite concat_1p.
|
||||
rewrite @HoTT.Types.Paths.transport_paths_FlFr.
|
||||
rewrite concat_p1.
|
||||
rewrite ap_idmap.
|
||||
intro n.
|
||||
hinduction n.
|
||||
- reflexivity.
|
||||
- intros n IHn.
|
||||
symmetry. etransitivity. apply (ap succ IHn^).
|
||||
etransitivity. apply Bool_to_Mod2_negb.
|
||||
hott_simpl.
|
||||
- rewrite @HoTT.Types.Paths.transport_paths_FlFr.
|
||||
hott_simpl.
|
||||
rewrite ap_compose.
|
||||
|
||||
enough (ap Mod2_to_Bool mod = reflexivity false).
|
||||
rewrite X.
|
||||
simpl.
|
||||
rewrite concat_1p.
|
||||
rewrite inv_V.
|
||||
reflexivity.
|
||||
|
||||
enough (IsHSet Bool).
|
||||
apply axiomK_hset.
|
||||
apply X.
|
||||
apply hset_bool.
|
||||
enough (ap Mod2_to_Bool mod = idpath).
|
||||
+ rewrite X. hott_simpl.
|
||||
+ apply (Mod2_rec_beta_mod Bool false negb 1).
|
||||
Defined.
|
||||
|
||||
Theorem adj :
|
||||
forall x : Mod2, eq1 (Mod2_to_Bool x) = ap Mod2_to_Bool (eq2 x).
|
||||
Proof.
|
||||
intro x.
|
||||
enough (IsHSet Bool).
|
||||
apply set_path2.
|
||||
apply hset_bool.
|
||||
Defined.
|
||||
|
||||
|
|
|
|||
|
|
@ -1,5 +1,6 @@
|
|||
-R . "" COQC = hoqc COQDEP = hoqdep
|
||||
|
||||
HitTactics.v
|
||||
Mod2.v
|
||||
FinSets.v
|
||||
Expressions.v
|
||||
|
|
|
|||
Loading…
Reference in New Issue