fin_collections.v 7.5 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 ``````From Coq Require Import Permutation. `````` Robbert Krebbers committed Mar 10, 2016 7 8 ``````From iris.prelude Require Import relations listset. From iris.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 Jul 22, 2016 20 21 22 23 24 25 26 27 `````` 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. (** * The [elements] operation *) `````` Robbert Krebbers committed Nov 18, 2015 28 ``````Global Instance elements_proper: Proper ((≡) ==> (≡ₚ)) (elements (C:=C)). `````` Robbert Krebbers committed Nov 11, 2015 29 30 ``````Proof. intros ?? E. apply NoDup_Permutation. `````` Robbert Krebbers committed Feb 17, 2016 31 32 33 `````` - apply NoDup_elements. - apply NoDup_elements. - intros. by rewrite !elem_of_elements, E. `````` Robbert Krebbers committed Nov 11, 2015 34 ``````Qed. `````` Robbert Krebbers committed Jul 22, 2016 35 `````` `````` Robbert Krebbers committed Feb 17, 2016 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 ``````Lemma elements_empty : elements (∅ : C) = []. Proof. apply elem_of_nil_inv; intros x. rewrite elem_of_elements, elem_of_empty; tauto. Qed. Lemma elements_union_singleton (X : C) x : x ∉ X → elements ({[ x ]} ∪ X) ≡ₚ x :: elements X. Proof. intros ?; apply NoDup_Permutation. { apply NoDup_elements. } { by constructor; rewrite ?elem_of_elements; try apply NoDup_elements. } intros y; rewrite elem_of_elements, elem_of_union, elem_of_singleton. by rewrite elem_of_cons, elem_of_elements. Qed. Lemma elements_singleton x : elements {[ x ]} = [x]. Proof. apply Permutation_singleton. by rewrite <-(right_id ∅ (∪) {[x]}), `````` Robbert Krebbers committed Feb 17, 2016 53 `````` elements_union_singleton, elements_empty by set_solver. `````` Robbert Krebbers committed Feb 17, 2016 54 55 56 57 58 59 60 ``````Qed. Lemma elements_contains X Y : X ⊆ Y → elements X `contains` elements Y. Proof. intros; apply NoDup_contains; auto using NoDup_elements. intros x. rewrite !elem_of_elements; auto. Qed. `````` Robbert Krebbers committed Jul 22, 2016 61 ``````(** * The [size] operation *) `````` Robbert Krebbers committed Nov 18, 2015 62 ``````Global Instance collection_size_proper: Proper ((≡) ==> (=)) (@size C _). `````` Robbert Krebbers committed Nov 11, 2015 63 ``````Proof. intros ?? E. apply Permutation_length. by rewrite E. Qed. `````` Robbert Krebbers committed Jul 22, 2016 64 `````` `````` Robbert Krebbers committed Nov 11, 2015 65 ``````Lemma size_empty : size (∅ : C) = 0. `````` Robbert Krebbers committed Feb 17, 2016 66 ``````Proof. unfold size, collection_size. simpl. by rewrite elements_empty. Qed. `````` Robbert Krebbers committed Nov 11, 2015 67 68 ``````Lemma size_empty_inv (X : C) : size X = 0 → X ≡ ∅. Proof. `````` Robbert Krebbers committed Dec 11, 2015 69 70 `````` 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 71 72 ``````Qed. Lemma size_empty_iff (X : C) : size X = 0 ↔ X ≡ ∅. `````` Robbert Krebbers committed Dec 11, 2015 73 ``````Proof. split. apply size_empty_inv. by intros ->; rewrite size_empty. Qed. `````` Robbert Krebbers committed Nov 11, 2015 74 75 ``````Lemma size_non_empty_iff (X : C) : size X ≠ 0 ↔ X ≢ ∅. Proof. by rewrite size_empty_iff. Qed. `````` Robbert Krebbers committed Jul 22, 2016 76 `````` `````` Robbert Krebbers committed Nov 11, 2015 77 78 79 ``````Lemma collection_choose_or_empty X : (∃ x, x ∈ X) ∨ X ≡ ∅. Proof. destruct (elements X) as [|x l] eqn:HX; [right|left]. `````` Robbert Krebbers committed Feb 17, 2016 80 81 `````` - apply equiv_empty; intros x. by rewrite <-elem_of_elements, HX, elem_of_nil. - exists x. rewrite <-elem_of_elements, HX. by left. `````` Robbert Krebbers committed Nov 11, 2015 82 83 84 85 86 87 88 89 90 91 ``````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. `````` Robbert Krebbers committed Jul 22, 2016 92 93 94 95 96 97 98 99 100 `````` Lemma size_singleton (x : A) : size {[ x ]} = 1. Proof. unfold size, collection_size. simpl. by rewrite elements_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 <-!elem_of_elements. generalize (elements X). intros [|? l]; intro; simplify_eq/=. rewrite (nil_length_inv l), !elem_of_list_singleton by done; congruence. Qed. `````` Robbert Krebbers committed Nov 11, 2015 101 102 103 104 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 105 `````` - rewrite elem_of_singleton. eauto using size_singleton_inv. `````` Robbert Krebbers committed Feb 17, 2016 106 `````` - set_solver. `````` Robbert Krebbers committed Nov 11, 2015 107 ``````Qed. `````` Robbert Krebbers committed Jul 22, 2016 108 `````` `````` Robbert Krebbers committed Mar 23, 2016 109 ``````Lemma size_union X Y : X ⊥ Y → size (X ∪ Y) = size X + size Y. `````` Robbert Krebbers committed Nov 11, 2015 110 ``````Proof. `````` Robbert Krebbers committed Mar 23, 2016 111 `````` intros. unfold size, collection_size. simpl. rewrite <-app_length. `````` Robbert Krebbers committed Nov 11, 2015 112 `````` apply Permutation_length, NoDup_Permutation. `````` Robbert Krebbers committed Feb 17, 2016 113 114 `````` - apply NoDup_elements. - apply NoDup_app; repeat split; try apply NoDup_elements. `````` Robbert Krebbers committed Feb 17, 2016 115 `````` intros x; rewrite !elem_of_elements; set_solver. `````` Robbert Krebbers committed Feb 17, 2016 116 `````` - intros. by rewrite elem_of_app, !elem_of_elements, elem_of_union. `````` Robbert Krebbers committed Nov 11, 2015 117 118 119 ``````Qed. Lemma size_union_alt X Y : size (X ∪ Y) = size X + size (Y ∖ X). Proof. `````` Robbert Krebbers committed Feb 17, 2016 120 121 122 `````` rewrite <-size_union by set_solver. setoid_replace (Y ∖ X) with ((Y ∪ X) ∖ X) by set_solver. rewrite <-union_difference, (comm (∪)); set_solver. `````` Robbert Krebbers committed Nov 11, 2015 123 ``````Qed. `````` Robbert Krebbers committed Jul 22, 2016 124 `````` `````` Robbert Krebbers committed Nov 11, 2015 125 126 127 128 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 129 `````` intros. rewrite (union_difference X Y) by set_solver. `````` Robbert Krebbers committed Nov 11, 2015 130 131 132 133 `````` rewrite size_union_alt, difference_twice. cut (size (Y ∖ X) ≠ 0); [lia |]. by apply size_non_empty_iff, non_empty_difference. Qed. `````` Robbert Krebbers committed Jul 22, 2016 134 135 `````` (** * Induction principles *) `````` Robbert Krebbers committed Nov 11, 2015 136 137 138 139 140 141 142 143 144 ``````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]. `````` Robbert Krebbers committed Feb 17, 2016 145 146 `````` - rewrite (union_difference {[ x ]} X) by set_solver. apply Hadd. set_solver. apply IH; set_solver. `````` Robbert Krebbers committed Feb 17, 2016 147 `````` - by rewrite HX. `````` Robbert Krebbers committed Nov 11, 2015 148 ``````Qed. `````` Robbert Krebbers committed Jul 22, 2016 149 150 `````` (** * The [collection_fold] operation *) `````` Robbert Krebbers committed Nov 11, 2015 151 152 153 154 155 156 157 158 159 160 ``````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. `````` Robbert Krebbers committed Feb 17, 2016 161 `````` - intros X HX. setoid_rewrite elem_of_nil in HX. `````` Robbert Krebbers committed Feb 17, 2016 162 `````` rewrite equiv_empty. done. set_solver. `````` Robbert Krebbers committed Feb 17, 2016 163 `````` - intros X HX. setoid_rewrite elem_of_cons in HX. `````` Robbert Krebbers committed Feb 17, 2016 164 165 `````` rewrite (union_difference {[ x ]} X) by set_solver. apply Hadd. set_solver. apply IH. set_solver. `````` Robbert Krebbers committed Nov 11, 2015 166 167 168 169 ``````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 170 `````` Proper ((≡) ==> R) (collection_fold f b : C → B). `````` Robbert Krebbers committed Nov 11, 2015 171 ``````Proof. intros ?? E. apply (foldr_permutation R f b); auto. by rewrite E. Qed. `````` Robbert Krebbers committed Jul 22, 2016 172 173 `````` (** * Decision procedures *) `````` Robbert Krebbers committed Nov 11, 2015 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 ``````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. End fin_collection.``````