(* Copyright (c) 2012, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects definitions and theorems on finite collections. Most
importantly, it implements a fold and size function and some useful induction
principles on finite collections . *)
Require Import Permutation ars.
Require Export collections numbers listset.
Instance collection_size `{Elements A C} : Size C := length ∘ elements.
Definition collection_fold `{Elements A C} {B}
(f : A → B → B) (b : B) : C → B := foldr f b ∘ elements.
Section fin_collection.
Context `{FinCollection A C}.
Global Instance elements_proper: Proper ((≡) ==> Permutation) elements.
Proof.
intros ?? E. apply NoDup_Permutation.
* apply elements_nodup.
* apply elements_nodup.
* intros. by rewrite <-!elements_spec, E.
Qed.
Global Instance collection_size_proper: Proper ((≡) ==> (=)) size.
Proof. intros ?? E. apply Permutation_length. by rewrite E. Qed.
Lemma size_empty : size (∅ : C) = 0.
Proof.
unfold size, collection_size. simpl.
rewrite (elem_of_nil_inv (elements ∅)).
* done.
* intro. rewrite <-elements_spec. solve_elem_of.
Qed.
Lemma size_empty_inv (X : C) : size X = 0 → X ≡ ∅.
Proof.
intros. apply equiv_empty. intro. rewrite elements_spec.
rewrite (nil_length (elements X)). by rewrite elem_of_nil. done.
Qed.
Lemma size_empty_iff (X : C) : size X = 0 ↔ X ≡ ∅.
Proof. split. apply size_empty_inv. intros E. by rewrite E, size_empty. Qed.
Lemma size_non_empty_iff (X : C) : size X ≠ 0 ↔ X ≢ ∅.
Proof. by rewrite size_empty_iff. Qed.
Lemma size_singleton (x : A) : size {[ x ]} = 1.
Proof.
change (length (elements {[ x ]}) = length [x]).
apply Permutation_length, NoDup_Permutation.
* apply elements_nodup.
* apply NoDup_singleton.
* intros.
by rewrite <-elements_spec, elem_of_singleton, elem_of_list_singleton.
Qed.
Lemma size_singleton_inv X x y : size X = 1 → x ∈ X → y ∈ X → x = y.
Proof.
unfold size, collection_size. simpl. rewrite !elements_spec.
generalize (elements X). intros [|? l].
* done.
* injection 1. intro. rewrite (nil_length l) by done.
simpl. rewrite !elem_of_list_singleton. congruence.
Qed.
Lemma elem_of_or_empty X : (∃ x, x ∈ X) ∨ X ≡ ∅.
Proof.
destruct (elements X) as [|x xs] eqn:E.
* right. apply equiv_empty. intros x Ex.
by rewrite elements_spec, E, elem_of_nil in Ex.
* left. exists x. rewrite elements_spec, E.
by constructor.
Qed.
Lemma choose X : X ≢ ∅ → ∃ x, x ∈ X.
Proof.
destruct (elem_of_or_empty X) as [[x ?]|?].
* by exists x.
* done.
Qed.
Lemma size_pos_choose X : 0 < size X → ∃ x, x ∈ X.
Proof.
intros E1. apply choose.
intros E2. rewrite E2, size_empty in E1.
by apply (Lt.lt_n_0 0).
Qed.
Lemma size_1_choose X : size X = 1 → ∃ x, X ≡ {[ x ]}.
Proof.
intros E. destruct (size_pos_choose X).
* rewrite E. auto with arith.
* exists x. apply elem_of_equiv. split.
+ intro. rewrite elem_of_singleton.
eauto using size_singleton_inv.
+ solve_elem_of.
Qed.
Lemma size_union X Y : X ∩ Y ≡ ∅ → size (X ∪ Y) = size X + size Y.
Proof.
intros [E _]. unfold size, collection_size. simpl. rewrite <-app_length.
apply Permutation_length, NoDup_Permutation.
* apply elements_nodup.
* apply NoDup_app; repeat split; try apply elements_nodup.
intros x. rewrite <-!elements_spec. esolve_elem_of.
* intros. rewrite elem_of_app, <-!elements_spec. solve_elem_of.
Qed.
Instance elem_of_dec_slow (x : A) (X : C) : Decision (x ∈ X) | 100.
Proof.
refine (cast_if (decide_rel (∈) x (elements X)));
by rewrite (elements_spec _).
Defined.
Global Program Instance collection_subseteq_dec_slow (X Y : C) :
Decision (X ⊆ Y) | 100 :=
match decide_rel (=) (size (X ∖ Y)) 0 with
| left E1 => left _
| right E1 => right _
end.
Next Obligation.
intros x Ex; apply dec_stable; intro.
destruct (proj1 (elem_of_empty x)).
apply (size_empty_inv _ E1).
by rewrite elem_of_difference.
Qed.
Next Obligation.
intros E2. destruct E1.
apply size_empty_iff, equiv_empty. intros x.
rewrite elem_of_difference. intros [E3 ?].
by apply E2 in E3.
Qed.
Lemma size_union_alt X Y : size (X ∪ Y) = size X + size (Y ∖ X).
Proof.
rewrite <-size_union by solve_elem_of.
setoid_replace (Y ∖ X) with ((Y ∪ X) ∖ X) by esolve_elem_of.
rewrite <-union_difference, (commutative (∪)); solve_elem_of.
Qed.
Lemma subseteq_size X Y : X ⊆ Y → size X ≤ size Y.
Proof.
intros. rewrite (union_difference X Y), size_union_alt by done. lia.
Qed.
Lemma subset_size X Y : X ⊂ Y → size X < size Y.
Proof.
intros. rewrite (union_difference X Y) by solve_elem_of.
rewrite size_union_alt, difference_twice.
cut (size (Y ∖ X) ≠ 0); [lia |].
by apply size_non_empty_iff, non_empty_difference.
Qed.
Lemma collection_wf : wf (@subset C _).
Proof. apply well_founded_lt_compat with size, subset_size. Qed.
Lemma collection_ind (P : C → Prop) :
Proper ((≡) ==> iff) P →
P ∅ →
(∀ x X, x ∉ X → P X → P ({[ x ]} ∪ X)) →
∀ X, P X.
Proof.
intros ? Hemp Hadd. apply well_founded_induction with (⊂).
{ apply collection_wf. }
intros X IH. destruct (elem_of_or_empty X) as [[x ?]|HX].
* rewrite (union_difference {[ x ]} X) by solve_elem_of.
apply Hadd. solve_elem_of. apply IH. esolve_elem_of.
* by rewrite HX.
Qed.
Lemma collection_fold_ind {B} (P : B → C → Prop) (f : A → B → B) (b : B) :
Proper ((=) ==> (≡) ==> iff) P →
P b ∅ →
(∀ x X r, x ∉ X → P r X → P (f x r) ({[ x ]} ∪ X)) →
∀ X, P (collection_fold f b X) X.
Proof.
intros ? Hemp Hadd.
cut (∀ l, NoDup l → ∀ X, (∀ x, x ∈ X ↔ x ∈ l) → P (foldr f b l) X).
{ intros help ?. apply help. apply elements_nodup. apply elements_spec. }
induction 1 as [|x l ?? IH]; simpl.
* intros X HX. setoid_rewrite elem_of_nil in HX.
rewrite equiv_empty. done. esolve_elem_of.
* intros X HX. setoid_rewrite elem_of_cons in HX.
rewrite (union_difference {[ x ]} X) by esolve_elem_of.
apply Hadd. solve_elem_of. apply IH. esolve_elem_of.
Qed.
Lemma collection_fold_proper {B} (R : relation B)
`{!Equivalence R}
(f : A → B → B) (b : B)
`{!Proper ((=) ==> R ==> R) f}
(Hf : ∀ a1 a2 b, R (f a1 (f a2 b)) (f a2 (f a1 b))) :
Proper ((≡) ==> R) (collection_fold f b).
Proof.
intros ?? E. apply (foldr_permutation R f b).
* auto.
* by rewrite E.
Qed.
Global Instance cforall_dec `(P : A → Prop)
`{∀ x, Decision (P x)} X : Decision (cforall P X) | 100.
Proof.
refine (cast_if (decide (Forall P (elements X))));
abstract (unfold cforall; setoid_rewrite elements_spec;
by rewrite <-Forall_forall).
Defined.
Global Instance cexists_dec `(P : A → Prop) `{∀ x, Decision (P x)} X :
Decision (cexists P X) | 100.
Proof.
refine (cast_if (decide (Exists P (elements X))));
abstract (unfold cexists; setoid_rewrite elements_spec;
by rewrite <-Exists_exists).
Defined.
Global Instance rel_elem_of_dec `{∀ x y, Decision (R x y)} x X :
Decision (elem_of_upto R x X) | 100 := decide (cexists (R x) X).
End fin_collection.