mirror of https://github.com/nmvdw/HITs-Examples
Add a general hit recursion tactic.
The tactic resolves the induction principle for a HIT using Canonical Structures, following the suggestion of @gallais
This commit is contained in:
Dan Frumin
parent
e81986a920
commit
44da34d72f
180
FinSets.v
180
FinSets.v
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@ -1,5 +1,5 @@
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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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@ -45,6 +45,10 @@ Fixpoint FinSets_rec
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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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Definition FinSetsCL A : HitRec.class (FinSets A) _ :=
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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).
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Canonical Structure FinSetsTy A : HitRec.type := HitRec.Pack _ _ (FinSetsCL A).
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Axiom FinSets_beta_assoc : forall
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(A : Type)
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(P : Type)
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@ -133,129 +137,66 @@ Axiom FinSets_beta_idem : forall
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idemP x.
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End FinSet.
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Definition isIn : forall
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(A : Type)
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(eq : A -> A -> Bool),
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A -> FinSets A -> Bool.
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Section functions.
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Parameter A : Type.
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Parameter eq : A -> A -> Bool.
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Definition isIn : A -> FinSets A -> Bool.
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Proof.
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intro A.
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intro eq.
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intro a.
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refine (FinSets_rec A _ _ _ _ _ _ _ _ _).
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Unshelve.
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Focus 6.
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apply false.
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Focus 6.
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intro a'.
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apply (eq a a').
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Focus 6.
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intro b.
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intro b'.
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apply (orb b b').
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Focus 3.
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intros.
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compute.
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reflexivity.
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Focus 1.
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intros.
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compute.
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destruct x.
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reflexivity.
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destruct y.
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reflexivity.
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reflexivity.
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Focus 1.
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intros.
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compute.
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destruct x.
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destruct y.
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reflexivity.
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reflexivity.
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destruct y.
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reflexivity.
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reflexivity.
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Focus 1.
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intros.
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compute.
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destruct x.
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reflexivity.
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reflexivity.
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intros.
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compute.
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destruct (eq a x).
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reflexivity.
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reflexivity.
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intros a X.
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hrecursion X.
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- exact false.
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- intro a'. apply (eq a a').
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- apply orb.
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- intros x y z. compute. destruct x; reflexivity.
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- intros x y. compute. destruct x, y; reflexivity.
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- intros x. compute. reflexivity.
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- intros x. compute. destruct x; reflexivity.
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- intros a'. compute. destruct (eq a a'); reflexivity.
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Defined.
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Definition comprehension : forall
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(A : Type)
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(eq : A -> A -> Bool),
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Definition comprehension :
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(A -> Bool) -> FinSets A -> FinSets A.
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Proof.
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intro A.
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intro eq.
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intro phi.
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refine (FinSets_rec A _ _ _ _ _ _ _ _ _).
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Unshelve.
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Focus 6.
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apply empty.
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Focus 6.
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intro a.
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apply (if (phi a) then L A a else (empty A)).
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Focus 6.
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intro x.
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intro y.
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apply (U A x y).
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Focus 3.
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intros.
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compute.
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apply nl.
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Focus 1.
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intros.
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compute.
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apply assoc.
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Focus 1.
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intros.
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compute.
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apply comm.
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Focus 1.
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intros.
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compute.
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apply nr.
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intros.
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compute.
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destruct (phi x).
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apply idem.
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apply nl.
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intros P X.
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hrecursion X.
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- apply empty.
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- intro a.
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refine (if (P a) then L A a else empty A).
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- intros X Y.
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apply (U A X Y).
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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 : forall (A : Type) (eq : A -> A -> Bool),
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Definition intersection :
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FinSets A -> FinSets A -> FinSets A.
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Proof.
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intro A.
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intro eq.
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intro x.
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intro y.
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apply (comprehension A eq (fun (a : A) => isIn A eq a x) y).
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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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Definition subset (A : Type) (eq : A -> A -> Bool) (x : FinSets A) (y : FinSets A) : Bool.
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Definition subset (x : FinSets A) (y : FinSets A) : Bool.
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Proof.
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hrecursion x.
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- apply true.
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- intro a. apply (isIn a y).
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- apply andb.
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- intros a b c. compute. destruct a; reflexivity.
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- intros a b. compute. destruct a, b; reflexivity.
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- intros x. compute. reflexivity.
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- intros x. compute. destruct x;reflexivity.
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- intros a. simpl.
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destruct (isIn a y); reflexivity.
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Defined.
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Definition subset' (x : FinSets A) (y : FinSets A) : Bool.
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Proof.
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refine (FinSets_rec A _ _ _ _ _ _ _ _ _ _).
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Unshelve.
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@ -268,7 +209,7 @@ refine (FinSets_rec A _ _ _ _ _ _ _ _ _ _).
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Focus 6.
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intro a.
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apply (isIn A eq a y).
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apply (isIn a y).
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Focus 6.
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intro b.
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@ -308,15 +249,16 @@ refine (FinSets_rec A _ _ _ _ _ _ _ _ _ _).
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reflexivity.
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intros.
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destruct (isIn A eq x0 y).
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destruct (isIn x0 y).
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compute.
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reflexivity.
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compute.
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reflexivity.
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Defined.
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(* TODO: subset = subset' *)
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Definition equal_set (A : Type) (eq : A -> A -> Bool) (x : FinSets A) (y : FinSets A) : Bool
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:= andb (subset A eq x y) (subset A eq y x).
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Definition equal_set (x : FinSets A) (y : FinSets A) : Bool
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:= andb (subset x y) (subset y x).
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Fixpoint eq_nat n m : Bool :=
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match n, m with
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@ -324,4 +266,6 @@ Fixpoint eq_nat n m : Bool :=
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| O, S _ => false
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| S _, O => false
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| S n1, S m1 => eq_nat n1 m1
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end.
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end.
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End functions.
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@ -0,0 +1,72 @@
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Module HitRec.
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Record class (H : Type) (* The HIT type itself *)
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(recTy : H -> Type) (* The type of the recursion principle *)
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:= Class { recursion_term : forall (x : H), recTy x }.
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Structure type := Pack { obj : Type ; rec : obj -> Type ; class_of : class obj rec }.
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Definition hrecursion (e : type) : forall x, rec e x :=
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let 'Pack _ _ (Class r) := e
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in r.
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Arguments hrecursion {e} x : simpl never.
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Arguments Class H {recTy} recursion_term.
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End HitRec.
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Ltac hrecursion_ target :=
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generalize dependent target;
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match goal with
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| [ |- forall target, ?P] => intro target;
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match type of (HitRec.hrecursion target) with
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| ?Q =>
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match (eval simpl in Q) with
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| forall Y , Y target =>
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simple refine (HitRec.hrecursion target (fun target => P) _)
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| forall Y _, Y target =>
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simple refine (HitRec.hrecursion target (fun target => P) _)
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| forall Y _ _, Y target =>
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simple refine (HitRec.hrecursion target (fun target => P) _ _)
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| forall Y _ _ _, Y target =>
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let hrec:=(eval hnf in (HitRec.hrecursion target)) in
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simple refine (hrec (fun target => P) _ _ _)
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| forall Y _ _ _ _, Y target =>
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simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _)
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| forall Y _ _ _ _ _, Y target =>
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simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _)
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| forall Y _ _ _ _ _ _, Y target =>
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simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _ _)
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| forall Y _ _ _ _ _ _ _, Y target =>
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simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _ _ _)
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| forall Y _ _ _ _ _ _ _, Y target =>
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simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _ _ _)
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| forall Y _ _ _ _ _ _ _ _, Y target =>
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simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _ _ _ _)
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| forall Y _ _ _ _ _ _ _ _ _, Y target =>
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simple refine (HitRec.hrecursion target (fun target => P) _ _ _ _ _ _ _ _ _)
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| forall Y, Y =>
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simple refine (HitRec.hrecursion target P)
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| forall Y _, Y =>
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simple refine (HitRec.hrecursion target P _)
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| forall Y _ _, Y =>
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simple refine (HitRec.hrecursion target P _ _)
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| forall Y _ _ _, Y =>
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simple refine (HitRec.hrecursion target P _ _ _)
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| forall Y _ _ _ _, Y =>
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simple refine (HitRec.hrecursion target P _ _ _ _)
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| forall Y _ _ _ _ _, Y =>
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simple refine (HitRec.hrecursion target P _ _ _ _ _)
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| forall Y _ _ _ _ _ _, Y =>
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simple refine (HitRec.hrecursion target P _ _ _ _ _ _)
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| forall Y _ _ _ _ _ _ _, Y =>
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simple refine (HitRec.hrecursion target P _ _ _ _ _ _ _)
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| forall Y _ _ _ _ _ _ _ _, Y =>
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simple refine (HitRec.hrecursion target P _ _ _ _ _ _ _ _)
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| forall Y _ _ _ _ _ _ _ _ _, Y =>
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simple refine (HitRec.hrecursion target P _ _ _ _ _ _ _ _ _)
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| forall Y _ _ _ _ _ _ _ _ _ _, Y =>
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simple refine (HitRec.hrecursion target P _ _ _ _ _ _ _ _ _ _)
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| _ => fail "Cannot handle the recursion principle (too many parameters?) :("
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end
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end
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end.
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Ltac hrecursion x := hrecursion_ x; simpl; try (clear x).
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60
Mod2.v
60
Mod2.v
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@ -1,6 +1,6 @@
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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 modulo.
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Private Inductive Mod2 : Type0 :=
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@ -51,44 +51,38 @@ Axiom Mod2_rec_beta_mod : forall
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(mod' : a = s (s a))
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, ap (Mod2_rec P a s mod') mod = mod'.
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Definition Mod2CL : HitRec.class Mod2 _ :=
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HitRec.Class Mod2 (fun x P a s p => Mod2_ind P a s p x).
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Canonical Structure Mod2ty : HitRec.type := HitRec.Pack Mod2 _ Mod2CL.
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End modulo.
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Definition negate : Mod2 -> Mod2.
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Proof.
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refine (Mod2_ind _ _ _ _).
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Unshelve.
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Focus 2.
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apply (succ Z).
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Focus 2.
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intros.
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apply (succ H).
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simpl.
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rewrite transport_const.
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rewrite <- mod.
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reflexivity.
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Defined.
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Theorem modulo2 : forall n : Mod2, n = succ(succ n).
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Proof.
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refine (Mod2_ind _ _ _ _).
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Unshelve.
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Focus 2.
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apply mod.
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intro n.
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hrecursion n.
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- apply mod.
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- intros n p.
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apply (ap succ p).
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- simpl.
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etransitivity.
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eapply (@transport_paths_FlFr _ _ idmap (fun n => succ (succ n))).
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hott_simpl.
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apply ap_compose.
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Defined.
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Focus 2.
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intro n.
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intro p.
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apply (ap succ p).
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simpl.
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rewrite @HoTT.Types.Paths.transport_paths_FlFr.
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rewrite ap_idmap.
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rewrite concat_Vp.
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rewrite concat_1p.
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rewrite ap_compose.
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reflexivity.
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Definition negate : Mod2 -> Mod2.
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Proof.
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intro x.
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hrecursion x.
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- apply Z.
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- intros. apply (succ H).
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- simpl.
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etransitivity.
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eapply transport_const.
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eapply modulo2.
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Defined.
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Definition plus : Mod2 -> Mod2 -> Mod2.
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@ -1,5 +1,6 @@
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-R . "" COQC = hoqc COQDEP = hoqdep
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HitTactics.v
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Mod2.v
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FinSets.v
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Expressions.v
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