fin_collections.v 7.39 KB
 Robbert Krebbers committed Nov 11, 2015 1 2 3 4 5 ``````(* Copyright (c) 2012-2015, 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 . *) `````` Robbert Krebbers committed Feb 13, 2016 6 7 8 ``````From Coq Require Import Permutation. From prelude Require Import relations listset. From prelude Require Export numbers collections. `````` Robbert Krebbers committed Nov 11, 2015 9 10 11 12 13 14 15 `````` 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}. `````` Robbert Krebbers committed Nov 18, 2015 16 ``````Implicit Types X Y : C. `````` Robbert Krebbers committed Nov 11, 2015 17 `````` `````` Robbert Krebbers committed Dec 11, 2015 18 19 ``````Lemma fin_collection_finite X : set_finite X. Proof. by exists (elements X); intros; rewrite elem_of_elements. Qed. `````` Robbert Krebbers committed Nov 18, 2015 20 ``````Global Instance elements_proper: Proper ((≡) ==> (≡ₚ)) (elements (C:=C)). `````` Robbert Krebbers committed Nov 11, 2015 21 22 23 24 25 26 ``````Proof. intros ?? E. apply NoDup_Permutation. * apply NoDup_elements. * apply NoDup_elements. * intros. by rewrite !elem_of_elements, E. Qed. `````` Robbert Krebbers committed Nov 18, 2015 27 ``````Global Instance collection_size_proper: Proper ((≡) ==> (=)) (@size C _). `````` Robbert Krebbers committed Nov 11, 2015 28 29 30 31 ``````Proof. intros ?? E. apply Permutation_length. by rewrite E. Qed. Lemma size_empty : size (∅ : C) = 0. Proof. unfold size, collection_size. simpl. `````` Robbert Krebbers committed Dec 11, 2015 32 33 `````` rewrite (elem_of_nil_inv (elements ∅)); [done|intro]. rewrite elem_of_elements, elem_of_empty; auto. `````` Robbert Krebbers committed Nov 11, 2015 34 35 36 ``````Qed. Lemma size_empty_inv (X : C) : size X = 0 → X ≡ ∅. Proof. `````` Robbert Krebbers committed Dec 11, 2015 37 38 `````` intros; apply equiv_empty; intros x; rewrite <-elem_of_elements. by rewrite (nil_length_inv (elements X)), ?elem_of_nil. `````` Robbert Krebbers committed Nov 11, 2015 39 40 ``````Qed. Lemma size_empty_iff (X : C) : size X = 0 ↔ X ≡ ∅. `````` Robbert Krebbers committed Dec 11, 2015 41 ``````Proof. split. apply size_empty_inv. by intros ->; rewrite size_empty. Qed. `````` Robbert Krebbers committed Nov 11, 2015 42 43 44 45 46 47 48 49 ``````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 NoDup_elements. * apply NoDup_singleton. `````` Robbert Krebbers committed Dec 11, 2015 50 51 `````` * intros y. by rewrite elem_of_elements, elem_of_singleton, elem_of_list_singleton. `````` Robbert Krebbers committed Nov 11, 2015 52 53 54 55 56 ``````Qed. Lemma size_singleton_inv X x y : size X = 1 → x ∈ X → y ∈ X → x = y. Proof. unfold size, collection_size. simpl. rewrite <-!elem_of_elements. generalize (elements X). intros [|? l]; intro; simplify_equality'. `````` Robbert Krebbers committed Dec 11, 2015 57 `````` rewrite (nil_length_inv l), !elem_of_list_singleton by done; congruence. `````` Robbert Krebbers committed Nov 11, 2015 58 59 60 61 ``````Qed. Lemma collection_choose_or_empty X : (∃ x, x ∈ X) ∨ X ≡ ∅. Proof. destruct (elements X) as [|x l] eqn:HX; [right|left]. `````` Robbert Krebbers committed Dec 11, 2015 62 `````` * apply equiv_empty; intros x. by rewrite <-elem_of_elements, HX, elem_of_nil. `````` Robbert Krebbers committed Nov 11, 2015 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 `````` * exists x. rewrite <-elem_of_elements, HX. by left. Qed. Lemma collection_choose X : X ≢ ∅ → ∃ x, x ∈ X. Proof. intros. by destruct (collection_choose_or_empty X). Qed. Lemma collection_choose_L `{!LeibnizEquiv C} X : X ≠ ∅ → ∃ x, x ∈ X. Proof. unfold_leibniz. apply collection_choose. Qed. Lemma size_pos_elem_of X : 0 < size X → ∃ x, x ∈ X. Proof. intros Hsz. destruct (collection_choose_or_empty X) as [|HX]; [done|]. contradict Hsz. rewrite HX, size_empty; lia. Qed. Lemma size_1_elem_of X : size X = 1 → ∃ x, X ≡ {[ x ]}. Proof. intros E. destruct (size_pos_elem_of X); auto with lia. exists x. apply elem_of_equiv. split. * 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 NoDup_elements. * apply NoDup_app; repeat split; try apply NoDup_elements. `````` Robbert Krebbers committed Jan 16, 2016 87 `````` intros x; rewrite !elem_of_elements; solve_elem_of. `````` Robbert Krebbers committed Dec 11, 2015 88 `````` * intros. by rewrite elem_of_app, !elem_of_elements, elem_of_union. `````` Robbert Krebbers committed Nov 11, 2015 89 90 91 92 93 94 95 96 ``````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 <-(elem_of_elements _). Defined. Global Program Instance collection_subseteq_dec_slow (X Y : C) : Decision (X ⊆ Y) | 100 := `````` Robbert Krebbers committed Jan 12, 2016 97 98 `````` match decide_rel (=) (size (X ∖ Y)) 0 return _ with | left _ => left _ | right _ => right _ `````` Robbert Krebbers committed Nov 11, 2015 99 100 `````` end. Next Obligation. `````` Robbert Krebbers committed Jan 12, 2016 101 `````` intros X Y E1 x ?; apply dec_stable; intro. destruct (proj1(elem_of_empty x)). `````` Robbert Krebbers committed Nov 11, 2015 102 103 104 `````` apply (size_empty_inv _ E1). by rewrite elem_of_difference. Qed. Next Obligation. `````` Robbert Krebbers committed Jan 12, 2016 105 `````` intros X Y E1 E2; destruct E1. apply size_empty_iff, equiv_empty. intros x. `````` Robbert Krebbers committed Nov 11, 2015 106 107 108 109 110 `````` 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. `````` Robbert Krebbers committed Jan 16, 2016 111 `````` setoid_replace (Y ∖ X) with ((Y ∪ X) ∖ X) by solve_elem_of. `````` Robbert Krebbers committed Feb 11, 2016 112 `````` rewrite <-union_difference, (comm (∪)); solve_elem_of. `````` Robbert Krebbers committed Nov 11, 2015 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 ``````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 (strict (@subseteq C _)). Proof. apply (wf_projected (<) size); auto using subset_size, lt_wf. 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 (collection_choose_or_empty X) as [[x ?]|HX]. * rewrite (union_difference {[ x ]} X) by solve_elem_of. `````` Robbert Krebbers committed Jan 16, 2016 133 `````` apply Hadd. solve_elem_of. apply IH; solve_elem_of. `````` Robbert Krebbers committed Nov 11, 2015 134 135 136 137 138 139 140 141 142 143 144 145 146 `````` * 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 NoDup_elements|]. symmetry. apply elem_of_elements. } induction 1 as [|x l ?? IH]; simpl. * intros X HX. setoid_rewrite elem_of_nil in HX. `````` Robbert Krebbers committed Jan 16, 2016 147 `````` rewrite equiv_empty. done. solve_elem_of. `````` Robbert Krebbers committed Nov 11, 2015 148 `````` * intros X HX. setoid_rewrite elem_of_cons in HX. `````` Robbert Krebbers committed Jan 16, 2016 149 150 `````` rewrite (union_difference {[ x ]} X) by solve_elem_of. apply Hadd. solve_elem_of. apply IH. solve_elem_of. `````` Robbert Krebbers committed Nov 11, 2015 151 152 153 154 ``````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))) : `````` Robbert Krebbers committed Nov 18, 2015 155 `````` Proper ((≡) ==> R) (collection_fold f b : C → B). `````` Robbert Krebbers committed Nov 11, 2015 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 ``````Proof. intros ?? E. apply (foldr_permutation R f b); auto. by rewrite E. Qed. Global Instance set_Forall_dec `(P : A → Prop) `{∀ x, Decision (P x)} X : Decision (set_Forall P X) | 100. Proof. refine (cast_if (decide (Forall P (elements X)))); abstract (unfold set_Forall; setoid_rewrite <-elem_of_elements; by rewrite <-Forall_forall). Defined. Global Instance set_Exists_dec `(P : A → Prop) `{∀ x, Decision (P x)} X : Decision (set_Exists P X) | 100. Proof. refine (cast_if (decide (Exists P (elements X)))); abstract (unfold set_Exists; setoid_rewrite <-elem_of_elements; 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 (set_Exists (R x) X). End fin_collection.``````