Library UniMath.CategoryTheory.Inductives.Trees
Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Propositions.
Require Import UniMath.Foundations.Sets.
Require Import UniMath.Foundations.NaturalNumbers.
Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.limits.graphs.colimits.
Require Import UniMath.CategoryTheory.categories.HSET.Core.
Require Import UniMath.CategoryTheory.categories.HSET.Limits.
Require Import UniMath.CategoryTheory.categories.HSET.Colimits.
Require Import UniMath.CategoryTheory.limits.initial.
Require Import UniMath.CategoryTheory.FunctorAlgebras.
Require Import UniMath.CategoryTheory.limits.binproducts.
Require Import UniMath.CategoryTheory.limits.terminal.
Require Import UniMath.CategoryTheory.Chains.All.
Require Import UniMath.CategoryTheory.exponentials.
Require Import UniMath.CategoryTheory.limits.bincoproducts.
Require Import UniMath.CategoryTheory.Inductives.Lists.
Local Open Scope cat.
The tree functor: F(X) = 1 + A * X * X
Definition treeOmegaFunctor : omega_cocont_functor HSET HSET := '1 + 'A × (Id × Id).
Let treeFunctor : functor HSET HSET := pr1 treeOmegaFunctor.
Let is_omega_cocont_treeFunctor : is_omega_cocont treeFunctor := pr2 treeOmegaFunctor.
Lemma treeFunctor_Initial :
Initial (precategory_FunctorAlg treeFunctor has_homsets_HSET).
Proof.
apply (colimAlgInitial _ InitialHSET is_omega_cocont_treeFunctor (ColimCoconeHSET _ _)).
Defined.
Let treeFunctor : functor HSET HSET := pr1 treeOmegaFunctor.
Let is_omega_cocont_treeFunctor : is_omega_cocont treeFunctor := pr2 treeOmegaFunctor.
Lemma treeFunctor_Initial :
Initial (precategory_FunctorAlg treeFunctor has_homsets_HSET).
Proof.
apply (colimAlgInitial _ InitialHSET is_omega_cocont_treeFunctor (ColimCoconeHSET _ _)).
Defined.
The type of binary trees
Definition Tree : HSET :=
alg_carrier _ (InitialObject treeFunctor_Initial).
Let Tree_mor : HSET⟦treeFunctor Tree,Tree⟧ :=
alg_map _ (InitialObject treeFunctor_Initial).
Let Tree_alg : algebra_ob treeFunctor :=
InitialObject treeFunctor_Initial.
Definition leaf_map : HSET⟦unitHSET,Tree⟧.
Proof.
simpl; intro x.
use Tree_mor.
apply inl, x.
Defined.
Definition leaf : pr1 Tree := leaf_map tt.
Definition node_map : HSET⟦(A × (Tree × Tree))%set,Tree⟧.
Proof.
intros xs.
use Tree_mor.
exact (inr xs).
Defined.
Definition node : pr1 A × (pr1 Tree × pr1 Tree) → pr1 Tree := node_map.
alg_carrier _ (InitialObject treeFunctor_Initial).
Let Tree_mor : HSET⟦treeFunctor Tree,Tree⟧ :=
alg_map _ (InitialObject treeFunctor_Initial).
Let Tree_alg : algebra_ob treeFunctor :=
InitialObject treeFunctor_Initial.
Definition leaf_map : HSET⟦unitHSET,Tree⟧.
Proof.
simpl; intro x.
use Tree_mor.
apply inl, x.
Defined.
Definition leaf : pr1 Tree := leaf_map tt.
Definition node_map : HSET⟦(A × (Tree × Tree))%set,Tree⟧.
Proof.
intros xs.
use Tree_mor.
exact (inr xs).
Defined.
Definition node : pr1 A × (pr1 Tree × pr1 Tree) → pr1 Tree := node_map.
Get recursion/iteration scheme:
x : X f : A × X × X -> X
------------------------------------
foldr x f : Tree A -> X
Definition make_treeAlgebra (X : HSET) (x : pr1 X)
(f : HSET⟦(A × X × X)%set,X⟧) : algebra_ob treeFunctor.
Proof.
set (x' := λ (_ : unit), x).
apply (tpair _ X (sumofmaps x' f) : algebra_ob treeFunctor).
Defined.
Definition foldr_map (X : HSET) (x : pr1 X) (f : HSET⟦(A × X × X)%set,X⟧) :
algebra_mor _ Tree_alg (make_treeAlgebra X x f).
Proof.
apply (InitialArrow treeFunctor_Initial (make_treeAlgebra X x f)).
Defined.
Definition foldr (X : HSET) (x : pr1 X)
(f : pr1 A × pr1 X × pr1 X → pr1 X) : pr1 Tree → pr1 X.
Proof.
apply (foldr_map _ x f).
Defined.
Lemma foldr_leaf (X : hSet) (x : X) (f : pr1 A × X × X → X) : foldr X x f leaf = x.
Proof.
assert (F := maponpaths (λ x, BinCoproductIn1 _ (BinCoproductsHSET _ _) · x)
(algebra_mor_commutes _ _ _ (foldr_map X x f))).
apply (toforallpaths _ _ _ F tt).
Qed.
Lemma foldr_node (X : hSet) (x : X) (f : pr1 A × X × X → X)
(a : pr1 A) (l1 l2 : pr1 Tree) :
foldr X x f (node (a,,l1,,l2)) = f (a,,foldr X x f l1,,foldr X x f l2).
Proof.
assert (F := maponpaths (λ x, BinCoproductIn2 _ (BinCoproductsHSET _ _)· x)
(algebra_mor_commutes _ _ _ (foldr_map X x f))).
assert (Fal := toforallpaths _ _ _ F (a,,l1,,l2)).
clear F.
unfold compose in Fal.
simpl in Fal.
apply Fal.
Opaque foldr_map.
Qed. Transparent foldr_map.
(f : HSET⟦(A × X × X)%set,X⟧) : algebra_ob treeFunctor.
Proof.
set (x' := λ (_ : unit), x).
apply (tpair _ X (sumofmaps x' f) : algebra_ob treeFunctor).
Defined.
Definition foldr_map (X : HSET) (x : pr1 X) (f : HSET⟦(A × X × X)%set,X⟧) :
algebra_mor _ Tree_alg (make_treeAlgebra X x f).
Proof.
apply (InitialArrow treeFunctor_Initial (make_treeAlgebra X x f)).
Defined.
Definition foldr (X : HSET) (x : pr1 X)
(f : pr1 A × pr1 X × pr1 X → pr1 X) : pr1 Tree → pr1 X.
Proof.
apply (foldr_map _ x f).
Defined.
Lemma foldr_leaf (X : hSet) (x : X) (f : pr1 A × X × X → X) : foldr X x f leaf = x.
Proof.
assert (F := maponpaths (λ x, BinCoproductIn1 _ (BinCoproductsHSET _ _) · x)
(algebra_mor_commutes _ _ _ (foldr_map X x f))).
apply (toforallpaths _ _ _ F tt).
Qed.
Lemma foldr_node (X : hSet) (x : X) (f : pr1 A × X × X → X)
(a : pr1 A) (l1 l2 : pr1 Tree) :
foldr X x f (node (a,,l1,,l2)) = f (a,,foldr X x f l1,,foldr X x f l2).
Proof.
assert (F := maponpaths (λ x, BinCoproductIn2 _ (BinCoproductsHSET _ _)· x)
(algebra_mor_commutes _ _ _ (foldr_map X x f))).
assert (Fal := toforallpaths _ _ _ F (a,,l1,,l2)).
clear F.
unfold compose in Fal.
simpl in Fal.
apply Fal.
Opaque foldr_map.
Qed. Transparent foldr_map.
This defines the induction principle for trees using foldr
Section tree_induction.
Variables (P : pr1 Tree → UU) (PhSet : ∏ l, isaset (P l)).
Variables (P0 : P leaf)
(Pc : ∏ (a : pr1 A) (l1 l2 : pr1 Tree), P l1 → P l2 → P (node (a,,l1,,l2))).
Let P' : UU := ∑ l, P l.
Let P0' : P' := (leaf,, P0).
Let Pc' : pr1 A × P' × P' → P'.
Proof.
intros ap.
apply (tpair _ (node (pr1 ap,,pr1 (pr1 (pr2 ap)),,pr1 (pr2 (pr2 ap))))).
apply (Pc _ _ _ (pr2 (pr1 (pr2 ap))) (pr2 (pr2 (pr2 ap)))).
Defined.
Definition P'HSET : HSET.
Proof.
apply (tpair _ P').
abstract (apply (isofhleveltotal2 2); [ apply setproperty | intro x; apply PhSet ]).
Defined.
Opaque is_omega_cocont_treeFunctor.
Lemma isalghom_pr1foldr :
is_algebra_mor _ Tree_alg Tree_alg (λ l, pr1 (foldr P'HSET P0' Pc' l)).
Proof.
apply BinCoproductArrow_eq_cor.
- apply funextfun; intro x; destruct x.
apply (maponpaths pr1 (foldr_leaf P'HSET P0' Pc')).
- apply funextfun; intro x; destruct x as [a [l1 l2]].
apply (maponpaths pr1 (foldr_node P'HSET P0' Pc' a l1 l2)).
Qed.
Transparent is_omega_cocont_treeFunctor.
Definition pr1foldr_algmor : algebra_mor _ Tree_alg Tree_alg :=
tpair _ _ isalghom_pr1foldr.
Lemma pr1foldr_algmor_identity : identity _ = pr1foldr_algmor.
Proof.
now rewrite (@InitialEndo_is_identity _ treeFunctor_Initial pr1foldr_algmor).
Qed.
Lemma treeInd l : P l.
Proof.
assert (H : pr1 (foldr P'HSET P0' Pc' l) = l).
apply (toforallpaths _ _ _ (!pr1foldr_algmor_identity) l).
rewrite <- H.
apply (pr2 (foldr P'HSET P0' Pc' l)).
Defined.
End tree_induction.
Lemma treeIndProp (P : pr1 Tree → UU) (HP : ∏ l, isaprop (P l)) :
P leaf → (∏ a l1 l2, P l1 → P l2 → P (node (a,,l1,,l2))) → ∏ l, P l.
Proof.
intros Pnil Pcons.
apply treeInd; try assumption.
intro l; apply isasetaprop, HP.
Defined.
End bintrees.
Variables (P : pr1 Tree → UU) (PhSet : ∏ l, isaset (P l)).
Variables (P0 : P leaf)
(Pc : ∏ (a : pr1 A) (l1 l2 : pr1 Tree), P l1 → P l2 → P (node (a,,l1,,l2))).
Let P' : UU := ∑ l, P l.
Let P0' : P' := (leaf,, P0).
Let Pc' : pr1 A × P' × P' → P'.
Proof.
intros ap.
apply (tpair _ (node (pr1 ap,,pr1 (pr1 (pr2 ap)),,pr1 (pr2 (pr2 ap))))).
apply (Pc _ _ _ (pr2 (pr1 (pr2 ap))) (pr2 (pr2 (pr2 ap)))).
Defined.
Definition P'HSET : HSET.
Proof.
apply (tpair _ P').
abstract (apply (isofhleveltotal2 2); [ apply setproperty | intro x; apply PhSet ]).
Defined.
Opaque is_omega_cocont_treeFunctor.
Lemma isalghom_pr1foldr :
is_algebra_mor _ Tree_alg Tree_alg (λ l, pr1 (foldr P'HSET P0' Pc' l)).
Proof.
apply BinCoproductArrow_eq_cor.
- apply funextfun; intro x; destruct x.
apply (maponpaths pr1 (foldr_leaf P'HSET P0' Pc')).
- apply funextfun; intro x; destruct x as [a [l1 l2]].
apply (maponpaths pr1 (foldr_node P'HSET P0' Pc' a l1 l2)).
Qed.
Transparent is_omega_cocont_treeFunctor.
Definition pr1foldr_algmor : algebra_mor _ Tree_alg Tree_alg :=
tpair _ _ isalghom_pr1foldr.
Lemma pr1foldr_algmor_identity : identity _ = pr1foldr_algmor.
Proof.
now rewrite (@InitialEndo_is_identity _ treeFunctor_Initial pr1foldr_algmor).
Qed.
Lemma treeInd l : P l.
Proof.
assert (H : pr1 (foldr P'HSET P0' Pc' l) = l).
apply (toforallpaths _ _ _ (!pr1foldr_algmor_identity) l).
rewrite <- H.
apply (pr2 (foldr P'HSET P0' Pc' l)).
Defined.
End tree_induction.
Lemma treeIndProp (P : pr1 Tree → UU) (HP : ∏ l, isaprop (P l)) :
P leaf → (∏ a l1 l2, P l1 → P l2 → P (node (a,,l1,,l2))) → ∏ l, P l.
Proof.
intros Pnil Pcons.
apply treeInd; try assumption.
intro l; apply isasetaprop, HP.
Defined.
End bintrees.
Some tests
Section nat_examples.
Local Open Scope nat_scope.
Definition size : pr1 (Tree natHSET) → nat :=
foldr natHSET natHSET 0 (λ x, S (pr1 (pr2 x) + pr2 (pr2 x))).
Lemma size_node a l1 l2 : size (node natHSET (a,,l1,,l2)) = 1 + size l1 + size l2.
Proof.
unfold size.
now rewrite foldr_node.
Qed.
Definition map (f : nat → nat) (l : pr1 (Tree natHSET)) : pr1 (Tree natHSET) :=
foldr natHSET (Tree natHSET) (leaf natHSET)
(λ a, node natHSET (f (pr1 a),, pr1 (pr2 a),, pr2 (pr2 a))) l.
Lemma size_map (f : nat → nat) : ∏ l, size (map f l) = size l.
Proof.
apply treeIndProp.
- intros l. apply isasetnat.
- now unfold map; rewrite foldr_leaf.
- intros a l1 l2 ih1 ih2; unfold map.
now rewrite foldr_node, !size_node, <- ih1, <- ih2.
Qed.
Definition sum : pr1 (Tree natHSET) → nat :=
foldr natHSET natHSET 0 (λ x, pr1 x + pr1 (pr2 x) + pr2 (pr2 x)).
Definition testtree : pr1 (Tree natHSET).
Proof.
use node_map; repeat split.
- apply 5.
- use node_map; repeat split.
+ apply 6.
+ exact (leaf_map _ tt).
+ exact (leaf_map _ tt).
- exact (leaf_map _ tt).
Defined.
End nat_examples.
Local Open Scope nat_scope.
Definition size : pr1 (Tree natHSET) → nat :=
foldr natHSET natHSET 0 (λ x, S (pr1 (pr2 x) + pr2 (pr2 x))).
Lemma size_node a l1 l2 : size (node natHSET (a,,l1,,l2)) = 1 + size l1 + size l2.
Proof.
unfold size.
now rewrite foldr_node.
Qed.
Definition map (f : nat → nat) (l : pr1 (Tree natHSET)) : pr1 (Tree natHSET) :=
foldr natHSET (Tree natHSET) (leaf natHSET)
(λ a, node natHSET (f (pr1 a),, pr1 (pr2 a),, pr2 (pr2 a))) l.
Lemma size_map (f : nat → nat) : ∏ l, size (map f l) = size l.
Proof.
apply treeIndProp.
- intros l. apply isasetnat.
- now unfold map; rewrite foldr_leaf.
- intros a l1 l2 ih1 ih2; unfold map.
now rewrite foldr_node, !size_node, <- ih1, <- ih2.
Qed.
Definition sum : pr1 (Tree natHSET) → nat :=
foldr natHSET natHSET 0 (λ x, pr1 x + pr1 (pr2 x) + pr2 (pr2 x)).
Definition testtree : pr1 (Tree natHSET).
Proof.
use node_map; repeat split.
- apply 5.
- use node_map; repeat split.
+ apply 6.
+ exact (leaf_map _ tt).
+ exact (leaf_map _ tt).
- exact (leaf_map _ tt).
Defined.
End nat_examples.
Local Notation "a :: b" := (cons _ a b).
Definition flatten (A : HSET) : pr1 (Tree A) → List A.
Proof.
intro t.
use (foldr A).
- apply nil.
- intro all'.
set (a := pr1 all').
set (l := pr1 (pr2 all')).
set (l' := pr2 (pr2 all')). cbn beta in l'.
exact (concatenate _ l (concatenate _ (a :: nil _ ) l')).
- exact t.
Defined.
Goal Lists.sum (flatten _ testtree) = sum testtree. reflexivity. Qed.
Definition flatten (A : HSET) : pr1 (Tree A) → List A.
Proof.
intro t.
use (foldr A).
- apply nil.
- intro all'.
set (a := pr1 all').
set (l := pr1 (pr2 all')).
set (l' := pr2 (pr2 all')). cbn beta in l'.
exact (concatenate _ l (concatenate _ (a :: nil _ ) l')).
- exact t.
Defined.
Goal Lists.sum (flatten _ testtree) = sum testtree. reflexivity. Qed.