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 Robbert Krebbers committed Mar 21, 2016 1 ``````From iris.algebra Require Export upred list. `````` Robbert Krebbers committed Mar 10, 2016 2 ``````From iris.prelude Require Import gmap fin_collections. `````` Ralf Jung committed Feb 17, 2016 3 ``````Import uPred. `````` Robbert Krebbers committed Feb 14, 2016 4 `````` `````` Robbert Krebbers committed Feb 16, 2016 5 6 ``````(** * Big ops over lists *) (* These are the basic building blocks for other big ops *) `````` Robbert Krebbers committed Feb 16, 2016 7 8 9 10 11 12 13 14 ``````Fixpoint uPred_big_and {M} (Ps : list (uPred M)) : uPred M:= match Ps with [] => True | P :: Ps => P ∧ uPred_big_and Ps end%I. Instance: Params (@uPred_big_and) 1. Notation "'Π∧' Ps" := (uPred_big_and Ps) (at level 20) : uPred_scope. Fixpoint uPred_big_sep {M} (Ps : list (uPred M)) : uPred M := match Ps with [] => True | P :: Ps => P ★ uPred_big_sep Ps end%I. Instance: Params (@uPred_big_sep) 1. Notation "'Π★' Ps" := (uPred_big_sep Ps) (at level 20) : uPred_scope. `````` Robbert Krebbers committed Feb 14, 2016 15 `````` `````` Robbert Krebbers committed Feb 16, 2016 16 17 ``````(** * Other big ops *) (** We use a type class to obtain overloaded notations *) `````` Robbert Krebbers committed Feb 17, 2016 18 ``````Definition uPred_big_sepM {M} `{Countable K} {A} `````` Robbert Krebbers committed Feb 18, 2016 19 20 `````` (m : gmap K A) (Φ : K → A → uPred M) : uPred M := uPred_big_sep (curry Φ <\$> map_to_list m). `````` Robbert Krebbers committed Feb 17, 2016 21 ``````Instance: Params (@uPred_big_sepM) 6. `````` Robbert Krebbers committed Feb 18, 2016 22 23 ``````Notation "'Π★{map' m } Φ" := (uPred_big_sepM m Φ) (at level 20, m at level 10, format "Π★{map m } Φ") : uPred_scope. `````` Robbert Krebbers committed Feb 14, 2016 24 `````` `````` Robbert Krebbers committed Feb 17, 2016 25 ``````Definition uPred_big_sepS {M} `{Countable A} `````` Robbert Krebbers committed Feb 18, 2016 26 `````` (X : gset A) (Φ : A → uPred M) : uPred M := uPred_big_sep (Φ <\$> elements X). `````` Robbert Krebbers committed Feb 17, 2016 27 ``````Instance: Params (@uPred_big_sepS) 5. `````` Robbert Krebbers committed Feb 18, 2016 28 29 ``````Notation "'Π★{set' X } Φ" := (uPred_big_sepS X Φ) (at level 20, X at level 10, format "Π★{set X } Φ") : uPred_scope. `````` Robbert Krebbers committed Feb 16, 2016 30 31 `````` (** * Always stability for lists *) `````` Robbert Krebbers committed Mar 11, 2016 32 ``````Class PersistentL {M} (Ps : list (uPred M)) := `````` Robbert Krebbers committed Mar 15, 2016 33 `````` persistentL : Forall PersistentP Ps. `````` Robbert Krebbers committed Mar 11, 2016 34 ``````Arguments persistentL {_} _ {_}. `````` Robbert Krebbers committed Feb 14, 2016 35 36 37 38 39 40 41 `````` Section big_op. Context {M : cmraT}. Implicit Types Ps Qs : list (uPred M). Implicit Types A : Type. (* Big ops *) `````` Ralf Jung committed Mar 10, 2016 42 ``````Global Instance big_and_proper : Proper ((≡) ==> (⊣⊢)) (@uPred_big_and M). `````` Robbert Krebbers committed Feb 14, 2016 43 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Ralf Jung committed Mar 10, 2016 44 ``````Global Instance big_sep_proper : Proper ((≡) ==> (⊣⊢)) (@uPred_big_sep M). `````` Robbert Krebbers committed Feb 14, 2016 45 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Robbert Krebbers committed Feb 17, 2016 46 `````` `````` Robbert Krebbers committed Mar 21, 2016 47 ``````Global Instance big_and_ne n : Proper (dist n ==> dist n) (@uPred_big_and M). `````` Robbert Krebbers committed Feb 17, 2016 48 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Robbert Krebbers committed Mar 21, 2016 49 ``````Global Instance big_sep_ne n : Proper (dist n ==> dist n) (@uPred_big_sep M). `````` Robbert Krebbers committed Feb 17, 2016 50 51 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Ralf Jung committed Mar 10, 2016 52 ``````Global Instance big_and_mono' : Proper (Forall2 (⊢) ==> (⊢)) (@uPred_big_and M). `````` Robbert Krebbers committed Feb 17, 2016 53 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Ralf Jung committed Mar 10, 2016 54 ``````Global Instance big_sep_mono' : Proper (Forall2 (⊢) ==> (⊢)) (@uPred_big_sep M). `````` Robbert Krebbers committed Feb 17, 2016 55 56 ``````Proof. by induction 1 as [|P Q Ps Qs HPQ ? IH]; rewrite /= ?HPQ ?IH. Qed. `````` Ralf Jung committed Mar 10, 2016 57 ``````Global Instance big_and_perm : Proper ((≡ₚ) ==> (⊣⊢)) (@uPred_big_and M). `````` Robbert Krebbers committed Feb 14, 2016 58 59 ``````Proof. induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto. `````` Robbert Krebbers committed Feb 17, 2016 60 61 `````` - by rewrite IH. - by rewrite !assoc (comm _ P). `````` Ralf Jung committed Feb 20, 2016 62 `````` - etrans; eauto. `````` Robbert Krebbers committed Feb 14, 2016 63 ``````Qed. `````` Ralf Jung committed Mar 10, 2016 64 ``````Global Instance big_sep_perm : Proper ((≡ₚ) ==> (⊣⊢)) (@uPred_big_sep M). `````` Robbert Krebbers committed Feb 14, 2016 65 66 ``````Proof. induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto. `````` Robbert Krebbers committed Feb 17, 2016 67 68 `````` - by rewrite IH. - by rewrite !assoc (comm _ P). `````` Ralf Jung committed Feb 20, 2016 69 `````` - etrans; eauto. `````` Robbert Krebbers committed Feb 14, 2016 70 ``````Qed. `````` Robbert Krebbers committed Feb 17, 2016 71 `````` `````` Robbert Krebbers committed Mar 21, 2016 72 ``````Lemma big_and_app Ps Qs : Π∧ (Ps ++ Qs) ⊣⊢ (Π∧ Ps ∧ Π∧ Qs). `````` Robbert Krebbers committed Feb 14, 2016 73 ``````Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed. `````` Robbert Krebbers committed Mar 21, 2016 74 ``````Lemma big_sep_app Ps Qs : Π★ (Ps ++ Qs) ⊣⊢ (Π★ Ps ★ Π★ Qs). `````` Robbert Krebbers committed Feb 14, 2016 75 ``````Proof. by induction Ps as [|?? IH]; rewrite /= ?left_id -?assoc ?IH. Qed. `````` Robbert Krebbers committed Feb 17, 2016 76 `````` `````` Robbert Krebbers committed Mar 21, 2016 77 ``````Lemma big_and_contains Ps Qs : Qs `contains` Ps → Π∧ Ps ⊢ Π∧ Qs. `````` Robbert Krebbers committed Feb 17, 2016 78 ``````Proof. `````` Ralf Jung committed Feb 17, 2016 79 `````` intros [Ps' ->]%contains_Permutation. by rewrite big_and_app and_elim_l. `````` Robbert Krebbers committed Feb 17, 2016 80 ``````Qed. `````` Robbert Krebbers committed Mar 21, 2016 81 ``````Lemma big_sep_contains Ps Qs : Qs `contains` Ps → Π★ Ps ⊢ Π★ Qs. `````` Robbert Krebbers committed Feb 17, 2016 82 ``````Proof. `````` Ralf Jung committed Feb 17, 2016 83 `````` intros [Ps' ->]%contains_Permutation. by rewrite big_sep_app sep_elim_l. `````` Robbert Krebbers committed Feb 17, 2016 84 85 ``````Qed. `````` Robbert Krebbers committed Mar 21, 2016 86 ``````Lemma big_sep_and Ps : Π★ Ps ⊢ Π∧ Ps. `````` Robbert Krebbers committed Feb 14, 2016 87 ``````Proof. by induction Ps as [|P Ps IH]; simpl; auto with I. Qed. `````` Robbert Krebbers committed Feb 17, 2016 88 `````` `````` Robbert Krebbers committed Mar 21, 2016 89 ``````Lemma big_and_elem_of Ps P : P ∈ Ps → Π∧ Ps ⊢ P. `````` Robbert Krebbers committed Feb 14, 2016 90 ``````Proof. induction 1; simpl; auto with I. Qed. `````` Robbert Krebbers committed Mar 21, 2016 91 ``````Lemma big_sep_elem_of Ps P : P ∈ Ps → Π★ Ps ⊢ P. `````` Robbert Krebbers committed Feb 14, 2016 92 93 ``````Proof. induction 1; simpl; auto with I. Qed. `````` Robbert Krebbers committed Feb 14, 2016 94 ``````(* Big ops over finite maps *) `````` Robbert Krebbers committed Feb 17, 2016 95 96 97 ``````Section gmap. Context `{Countable K} {A : Type}. Implicit Types m : gmap K A. `````` Robbert Krebbers committed Feb 18, 2016 98 `````` Implicit Types Φ Ψ : K → A → uPred M. `````` Robbert Krebbers committed Feb 14, 2016 99 `````` `````` Robbert Krebbers committed Feb 18, 2016 100 `````` Lemma big_sepM_mono Φ Ψ m1 m2 : `````` Ralf Jung committed Mar 10, 2016 101 `````` m2 ⊆ m1 → (∀ x k, m2 !! k = Some x → Φ k x ⊢ Ψ k x) → `````` Robbert Krebbers committed Mar 21, 2016 102 `````` Π★{map m1} Φ ⊢ Π★{map m2} Ψ. `````` Robbert Krebbers committed Feb 16, 2016 103 `````` Proof. `````` Ralf Jung committed Feb 20, 2016 104 `````` intros HX HΦ. trans (Π★{map m2} Φ)%I. `````` Robbert Krebbers committed Feb 17, 2016 105 `````` - by apply big_sep_contains, fmap_contains, map_to_list_contains. `````` Robbert Krebbers committed Mar 21, 2016 106 `````` - apply big_sep_mono', Forall2_fmap, Forall_Forall2. `````` Robbert Krebbers committed Feb 18, 2016 107 `````` apply Forall_forall=> -[i x] ? /=. by apply HΦ, elem_of_map_to_list. `````` Robbert Krebbers committed Feb 16, 2016 108 `````` Qed. `````` Robbert Krebbers committed Feb 17, 2016 109 110 111 112 113 `````` Global Instance big_sepM_ne m n : Proper (pointwise_relation _ (pointwise_relation _ (dist n)) ==> (dist n)) (uPred_big_sepM (M:=M) m). Proof. `````` Robbert Krebbers committed Feb 18, 2016 114 `````` intros Φ1 Φ2 HΦ. apply big_sep_ne, Forall2_fmap. `````` Robbert Krebbers committed Mar 21, 2016 115 `````` apply Forall_Forall2, Forall_true=> -[i x]; apply HΦ. `````` Robbert Krebbers committed Feb 17, 2016 116 117 `````` Qed. Global Instance big_sepM_proper m : `````` Ralf Jung committed Mar 10, 2016 118 `````` Proper (pointwise_relation _ (pointwise_relation _ (⊣⊢)) ==> (⊣⊢)) `````` Robbert Krebbers committed Feb 17, 2016 119 120 `````` (uPred_big_sepM (M:=M) m). Proof. `````` Robbert Krebbers committed Feb 18, 2016 121 122 `````` intros Φ1 Φ2 HΦ; apply equiv_dist=> n. apply big_sepM_ne=> k x; apply equiv_dist, HΦ. `````` Robbert Krebbers committed Feb 17, 2016 123 124 `````` Qed. Global Instance big_sepM_mono' m : `````` Ralf Jung committed Mar 10, 2016 125 `````` Proper (pointwise_relation _ (pointwise_relation _ (⊢)) ==> (⊢)) `````` Robbert Krebbers committed Feb 17, 2016 126 `````` (uPred_big_sepM (M:=M) m). `````` Robbert Krebbers committed Feb 18, 2016 127 `````` Proof. intros Φ1 Φ2 HΦ. apply big_sepM_mono; intros; [done|apply HΦ]. Qed. `````` Robbert Krebbers committed Feb 17, 2016 128 `````` `````` Robbert Krebbers committed Mar 21, 2016 129 `````` Lemma big_sepM_empty Φ : Π★{map ∅} Φ ⊣⊢ True. `````` Robbert Krebbers committed Feb 17, 2016 130 `````` Proof. by rewrite /uPred_big_sepM map_to_list_empty. Qed. `````` Robbert Krebbers committed Feb 18, 2016 131 `````` Lemma big_sepM_insert Φ (m : gmap K A) i x : `````` Robbert Krebbers committed Mar 21, 2016 132 `````` m !! i = None → Π★{map <[i:=x]> m} Φ ⊣⊢ (Φ i x ★ Π★{map m} Φ). `````` Robbert Krebbers committed Feb 17, 2016 133 `````` Proof. intros ?; by rewrite /uPred_big_sepM map_to_list_insert. Qed. `````` Robbert Krebbers committed Mar 21, 2016 134 `````` Lemma big_sepM_singleton Φ i x : Π★{map {[i := x]}} Φ ⊣⊢ (Φ i x). `````` Robbert Krebbers committed Feb 14, 2016 135 136 137 138 `````` Proof. rewrite -insert_empty big_sepM_insert/=; last auto using lookup_empty. by rewrite big_sepM_empty right_id. Qed. `````` Ralf Jung committed Feb 17, 2016 139 `````` `````` Robbert Krebbers committed Feb 18, 2016 140 `````` Lemma big_sepM_sepM Φ Ψ m : `````` Robbert Krebbers committed Mar 21, 2016 141 `````` Π★{map m} (λ i x, Φ i x ★ Ψ i x) ⊣⊢ (Π★{map m} Φ ★ Π★{map m} Ψ). `````` Ralf Jung committed Feb 17, 2016 142 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 143 144 `````` rewrite /uPred_big_sepM. induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?right_id //. `````` Robbert Krebbers committed Feb 18, 2016 145 `````` by rewrite IH -!assoc (assoc _ (Ψ _ _)) [(Ψ _ _ ★ _)%I]comm -!assoc. `````` Ralf Jung committed Feb 17, 2016 146 `````` Qed. `````` Robbert Krebbers committed Mar 21, 2016 147 `````` Lemma big_sepM_later Φ m : ▷ Π★{map m} Φ ⊣⊢ Π★{map m} (λ i x, ▷ Φ i x). `````` Ralf Jung committed Feb 17, 2016 148 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 149 150 151 `````` rewrite /uPred_big_sepM. induction (map_to_list m) as [|[i x] l IH]; csimpl; rewrite ?later_True //. by rewrite later_sep IH. `````` Ralf Jung committed Feb 17, 2016 152 `````` Qed. `````` Robbert Krebbers committed Feb 17, 2016 153 154 155 156 157 158 ``````End gmap. (* Big ops over finite sets *) Section gset. Context `{Countable A}. Implicit Types X : gset A. `````` Robbert Krebbers committed Feb 18, 2016 159 `````` Implicit Types Φ : A → uPred M. `````` Robbert Krebbers committed Feb 17, 2016 160 `````` `````` Robbert Krebbers committed Feb 18, 2016 161 `````` Lemma big_sepS_mono Φ Ψ X Y : `````` Robbert Krebbers committed Mar 21, 2016 162 `````` Y ⊆ X → (∀ x, x ∈ Y → Φ x ⊢ Ψ x) → Π★{set X} Φ ⊢ Π★{set Y} Ψ. `````` Robbert Krebbers committed Feb 17, 2016 163 `````` Proof. `````` Ralf Jung committed Feb 20, 2016 164 `````` intros HX HΦ. trans (Π★{set Y} Φ)%I. `````` Robbert Krebbers committed Feb 17, 2016 165 `````` - by apply big_sep_contains, fmap_contains, elements_contains. `````` Robbert Krebbers committed Mar 21, 2016 166 `````` - apply big_sep_mono', Forall2_fmap, Forall_Forall2. `````` Robbert Krebbers committed Feb 18, 2016 167 `````` apply Forall_forall=> x ? /=. by apply HΦ, elem_of_elements. `````` Robbert Krebbers committed Feb 17, 2016 168 169 170 171 172 `````` Qed. Lemma big_sepS_ne X n : Proper (pointwise_relation _ (dist n) ==> dist n) (uPred_big_sepS (M:=M) X). Proof. `````` Robbert Krebbers committed Feb 18, 2016 173 `````` intros Φ1 Φ2 HΦ. apply big_sep_ne, Forall2_fmap. `````` Robbert Krebbers committed Mar 21, 2016 174 `````` apply Forall_Forall2, Forall_true=> x; apply HΦ. `````` Robbert Krebbers committed Feb 17, 2016 175 176 `````` Qed. Lemma big_sepS_proper X : `````` Ralf Jung committed Mar 10, 2016 177 `````` Proper (pointwise_relation _ (⊣⊢) ==> (⊣⊢)) (uPred_big_sepS (M:=M) X). `````` Robbert Krebbers committed Feb 17, 2016 178 `````` Proof. `````` Robbert Krebbers committed Feb 18, 2016 179 180 `````` intros Φ1 Φ2 HΦ; apply equiv_dist=> n. apply big_sepS_ne=> x; apply equiv_dist, HΦ. `````` Robbert Krebbers committed Feb 17, 2016 181 182 `````` Qed. Lemma big_sepS_mono' X : `````` Ralf Jung committed Mar 10, 2016 183 `````` Proper (pointwise_relation _ (⊢) ==> (⊢)) (uPred_big_sepS (M:=M) X). `````` Robbert Krebbers committed Feb 18, 2016 184 `````` Proof. intros Φ1 Φ2 HΦ. apply big_sepS_mono; naive_solver. Qed. `````` Robbert Krebbers committed Feb 17, 2016 185 `````` `````` Robbert Krebbers committed Mar 21, 2016 186 `````` Lemma big_sepS_empty Φ : Π★{set ∅} Φ ⊣⊢ True. `````` Robbert Krebbers committed Feb 17, 2016 187 `````` Proof. by rewrite /uPred_big_sepS elements_empty. Qed. `````` Robbert Krebbers committed Feb 18, 2016 188 `````` Lemma big_sepS_insert Φ X x : `````` Robbert Krebbers committed Mar 21, 2016 189 `````` x ∉ X → Π★{set {[ x ]} ∪ X} Φ ⊣⊢ (Φ x ★ Π★{set X} Φ). `````` Robbert Krebbers committed Feb 17, 2016 190 `````` Proof. intros. by rewrite /uPred_big_sepS elements_union_singleton. Qed. `````` Robbert Krebbers committed Feb 18, 2016 191 `````` Lemma big_sepS_delete Φ X x : `````` Robbert Krebbers committed Mar 21, 2016 192 `````` x ∈ X → Π★{set X} Φ ⊣⊢ (Φ x ★ Π★{set X ∖ {[ x ]}} Φ). `````` Robbert Krebbers committed Feb 17, 2016 193 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 194 195 `````` intros. rewrite -big_sepS_insert; last set_solver. by rewrite -union_difference_L; last set_solver. `````` Robbert Krebbers committed Feb 17, 2016 196 `````` Qed. `````` Robbert Krebbers committed Mar 21, 2016 197 `````` Lemma big_sepS_singleton Φ x : Π★{set {[ x ]}} Φ ⊣⊢ (Φ x). `````` Robbert Krebbers committed Feb 17, 2016 198 `````` Proof. intros. by rewrite /uPred_big_sepS elements_singleton /= right_id. Qed. `````` Ralf Jung committed Feb 17, 2016 199 `````` `````` Robbert Krebbers committed Feb 18, 2016 200 `````` Lemma big_sepS_sepS Φ Ψ X : `````` Robbert Krebbers committed Mar 21, 2016 201 `````` Π★{set X} (λ x, Φ x ★ Ψ x) ⊣⊢ (Π★{set X} Φ ★ Π★{set X} Ψ). `````` Ralf Jung committed Feb 17, 2016 202 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 203 204 `````` rewrite /uPred_big_sepS. induction (elements X) as [|x l IH]; csimpl; first by rewrite ?right_id. `````` Robbert Krebbers committed Feb 18, 2016 205 `````` by rewrite IH -!assoc (assoc _ (Ψ _)) [(Ψ _ ★ _)%I]comm -!assoc. `````` Ralf Jung committed Feb 17, 2016 206 207 `````` Qed. `````` Robbert Krebbers committed Mar 21, 2016 208 `````` Lemma big_sepS_later Φ X : ▷ Π★{set X} Φ ⊣⊢ Π★{set X} (λ x, ▷ Φ x). `````` Ralf Jung committed Feb 17, 2016 209 `````` Proof. `````` Robbert Krebbers committed Feb 17, 2016 210 211 212 `````` rewrite /uPred_big_sepS. induction (elements X) as [|x l IH]; csimpl; first by rewrite ?later_True. by rewrite later_sep IH. `````` Ralf Jung committed Feb 17, 2016 213 `````` Qed. `````` Robbert Krebbers committed Feb 17, 2016 214 ``````End gset. `````` Robbert Krebbers committed Feb 14, 2016 215 `````` `````` Robbert Krebbers committed Feb 14, 2016 216 ``````(* Always stable *) `````` Robbert Krebbers committed Mar 15, 2016 217 ``````Global Instance big_and_persistent Ps : PersistentL Ps → PersistentP (Π∧ Ps). `````` Robbert Krebbers committed Feb 14, 2016 218 ``````Proof. induction 1; apply _. Qed. `````` Robbert Krebbers committed Mar 15, 2016 219 ``````Global Instance big_sep_persistent Ps : PersistentL Ps → PersistentP (Π★ Ps). `````` Robbert Krebbers committed Feb 14, 2016 220 221 ``````Proof. induction 1; apply _. Qed. `````` Robbert Krebbers committed Mar 11, 2016 222 ``````Global Instance nil_persistent : PersistentL (@nil (uPred M)). `````` Robbert Krebbers committed Feb 14, 2016 223 ``````Proof. constructor. Qed. `````` Robbert Krebbers committed Mar 11, 2016 224 ``````Global Instance cons_persistent P Ps : `````` Robbert Krebbers committed Mar 15, 2016 225 `````` PersistentP P → PersistentL Ps → PersistentL (P :: Ps). `````` Robbert Krebbers committed Feb 14, 2016 226 ``````Proof. by constructor. Qed. `````` Robbert Krebbers committed Mar 11, 2016 227 228 ``````Global Instance app_persistent Ps Ps' : PersistentL Ps → PersistentL Ps' → PersistentL (Ps ++ Ps'). `````` Robbert Krebbers committed Feb 14, 2016 229 ``````Proof. apply Forall_app_2. Qed. `````` Robbert Krebbers committed Mar 11, 2016 230 ``````Global Instance zip_with_persistent {A B} (f : A → B → uPred M) xs ys : `````` Robbert Krebbers committed Mar 15, 2016 231 `````` (∀ x y, PersistentP (f x y)) → PersistentL (zip_with f xs ys). `````` Robbert Krebbers committed Mar 11, 2016 232 233 234 ``````Proof. unfold PersistentL=> ?; revert ys; induction xs=> -[|??]; constructor; auto. Qed. `````` Robbert Krebbers committed Feb 16, 2016 235 ``End big_op.``