option.v 8.62 KB
 Robbert Krebbers committed Feb 13, 2016 1 ``````From algebra Require Export cmra. `````` Robbert Krebbers committed Feb 13, 2016 2 ``````From algebra Require Import functor upred. `````` Robbert Krebbers committed Dec 15, 2015 3 4 `````` (* COFE *) `````` Robbert Krebbers committed Jan 14, 2016 5 6 7 ``````Section cofe. Context {A : cofeT}. Inductive option_dist : Dist (option A) := `````` Ralf Jung committed Feb 10, 2016 8 9 `````` | Some_dist n x y : x ≡{n}≡ y → Some x ≡{n}≡ Some y | None_dist n : None ≡{n}≡ None. `````` Robbert Krebbers committed Dec 15, 2015 10 ``````Existing Instance option_dist. `````` Robbert Krebbers committed Jan 14, 2016 11 ``````Program Definition option_chain `````` Robbert Krebbers committed Dec 15, 2015 12 13 14 `````` (c : chain (option A)) (x : A) (H : c 1 = Some x) : chain A := {| chain_car n := from_option x (c n) |}. Next Obligation. `````` Robbert Krebbers committed Feb 10, 2016 15 `````` intros c x ? n [|i] ?; [omega|]; simpl. `````` Robbert Krebbers committed Feb 17, 2016 16 `````` destruct (c 1) eqn:?; simplify_eq/=. `````` Robbert Krebbers committed Feb 10, 2016 17 `````` by feed inversion (chain_cauchy c n (S i)). `````` Robbert Krebbers committed Dec 15, 2015 18 ``````Qed. `````` Robbert Krebbers committed Jan 14, 2016 19 ``````Instance option_compl : Compl (option A) := λ c, `````` Robbert Krebbers committed Dec 15, 2015 20 21 22 `````` match Some_dec (c 1) with | inleft (exist x H) => Some (compl (option_chain c x H)) | inright _ => None end. `````` Robbert Krebbers committed Jan 14, 2016 23 ``````Definition option_cofe_mixin : CofeMixin (option A). `````` Robbert Krebbers committed Dec 15, 2015 24 25 ``````Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 26 `````` - intros mx my; split; [by destruct 1; constructor; apply equiv_dist|]. `````` Robbert Krebbers committed Dec 15, 2015 27 28 `````` intros Hxy; feed inversion (Hxy 1); subst; constructor; apply equiv_dist. by intros n; feed inversion (Hxy n). `````` Robbert Krebbers committed Feb 17, 2016 29 `````` - intros n; split. `````` Robbert Krebbers committed Dec 15, 2015 30 31 `````` + by intros [x|]; constructor. + by destruct 1; constructor. `````` Ralf Jung committed Feb 20, 2016 32 `````` + destruct 1; inversion_clear 1; constructor; etrans; eauto. `````` Robbert Krebbers committed Feb 17, 2016 33 `````` - by inversion_clear 1; constructor; apply dist_S. `````` Robbert Krebbers committed Feb 18, 2016 34 `````` - intros n c; unfold compl, option_compl. `````` Robbert Krebbers committed Dec 15, 2015 35 `````` destruct (Some_dec (c 1)) as [[x Hx]|]. `````` Robbert Krebbers committed Feb 10, 2016 36 37 `````` { assert (is_Some (c (S n))) as [y Hy]. { feed inversion (chain_cauchy c 0 (S n)); eauto with lia congruence. } `````` Robbert Krebbers committed Dec 15, 2015 38 `````` rewrite Hy; constructor. `````` Robbert Krebbers committed Feb 18, 2016 39 `````` by rewrite (conv_compl n (option_chain c x Hx)) /= Hy. } `````` Robbert Krebbers committed Feb 10, 2016 40 41 `````` feed inversion (chain_cauchy c 0 (S n)); eauto with lia congruence. constructor. `````` Robbert Krebbers committed Dec 15, 2015 42 ``````Qed. `````` Robbert Krebbers committed Jan 14, 2016 43 ``````Canonical Structure optionC := CofeT option_cofe_mixin. `````` Robbert Krebbers committed Feb 24, 2016 44 45 46 ``````Global Instance option_discrete : Discrete A → Discrete optionC. Proof. inversion_clear 2; constructor; by apply (timeless _). Qed. `````` Robbert Krebbers committed Jan 14, 2016 47 ``````Global Instance Some_ne : Proper (dist n ==> dist n) (@Some A). `````` Robbert Krebbers committed Dec 15, 2015 48 ``````Proof. by constructor. Qed. `````` Robbert Krebbers committed Feb 10, 2016 49 ``````Global Instance is_Some_ne n : Proper (dist n ==> iff) (@is_Some A). `````` Robbert Krebbers committed Jan 16, 2016 50 ``````Proof. inversion_clear 1; split; eauto. Qed. `````` Robbert Krebbers committed Feb 11, 2016 51 ``````Global Instance Some_dist_inj : Inj (dist n) (dist n) (@Some A). `````` Robbert Krebbers committed Jan 16, 2016 52 ``````Proof. by inversion_clear 1. Qed. `````` Robbert Krebbers committed Jan 14, 2016 53 ``````Global Instance None_timeless : Timeless (@None A). `````` Robbert Krebbers committed Dec 15, 2015 54 ``````Proof. inversion_clear 1; constructor. Qed. `````` Robbert Krebbers committed Jan 14, 2016 55 ``````Global Instance Some_timeless x : Timeless x → Timeless (Some x). `````` Robbert Krebbers committed Dec 15, 2015 56 ``````Proof. by intros ?; inversion_clear 1; constructor; apply timeless. Qed. `````` Robbert Krebbers committed Jan 14, 2016 57 58 59 60 ``````End cofe. Arguments optionC : clear implicits. `````` Robbert Krebbers committed Dec 15, 2015 61 ``````(* CMRA *) `````` Robbert Krebbers committed Jan 14, 2016 62 63 64 ``````Section cmra. Context {A : cmraT}. `````` Robbert Krebbers committed Feb 24, 2016 65 66 ``````Instance option_valid : Valid (option A) := λ mx, match mx with Some x => ✓ x | None => True end. `````` Robbert Krebbers committed Jan 14, 2016 67 ``````Instance option_validN : ValidN (option A) := λ n mx, `````` Robbert Krebbers committed Dec 15, 2015 68 `````` match mx with Some x => ✓{n} x | None => True end. `````` Robbert Krebbers committed Feb 04, 2016 69 ``````Global Instance option_empty : Empty (option A) := None. `````` Robbert Krebbers committed Jan 14, 2016 70 71 72 ``````Instance option_unit : Unit (option A) := fmap unit. Instance option_op : Op (option A) := union_with (λ x y, Some (x ⋅ y)). Instance option_minus : Minus (option A) := `````` Robbert Krebbers committed Dec 15, 2015 73 `````` difference_with (λ x y, Some (x ⩪ y)). `````` Robbert Krebbers committed Jan 14, 2016 74 ``````Lemma option_includedN n (mx my : option A) : `````` Robbert Krebbers committed Feb 10, 2016 75 `````` mx ≼{n} my ↔ mx = None ∨ ∃ x y, mx = Some x ∧ my = Some y ∧ x ≼{n} y. `````` Robbert Krebbers committed Dec 15, 2015 76 77 ``````Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 78 `````` - intros [mz Hmz]. `````` Robbert Krebbers committed Dec 15, 2015 79 80 `````` destruct mx as [x|]; [right|by left]. destruct my as [y|]; [exists x, y|destruct mz; inversion_clear Hmz]. `````` Robbert Krebbers committed Feb 19, 2016 81 `````` destruct mz as [z|]; inversion_clear Hmz; split_and?; auto; `````` Robbert Krebbers committed Feb 01, 2016 82 `````` cofe_subst; eauto using cmra_includedN_l. `````` Robbert Krebbers committed Feb 17, 2016 83 `````` - intros [->|(x&y&->&->&z&Hz)]; try (by exists my; destruct my; constructor). `````` Robbert Krebbers committed Dec 15, 2015 84 85 `````` by exists (Some z); constructor. Qed. `````` Robbert Krebbers committed Jan 16, 2016 86 87 88 89 ``````Lemma None_includedN n (mx : option A) : None ≼{n} mx. Proof. rewrite option_includedN; auto. Qed. Lemma Some_Some_includedN n (x y : A) : x ≼{n} y → Some x ≼{n} Some y. Proof. rewrite option_includedN; eauto 10. Qed. `````` 90 ``````Definition Some_op a b : Some (a ⋅ b) = Some a ⋅ Some b := eq_refl. `````` Robbert Krebbers committed Jan 16, 2016 91 `````` `````` Robbert Krebbers committed Jan 14, 2016 92 ``````Definition option_cmra_mixin : CMRAMixin (option A). `````` Robbert Krebbers committed Dec 15, 2015 93 94 ``````Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 95 96 97 98 `````` - by intros n [x|]; destruct 1; constructor; cofe_subst. - by destruct 1; constructor; cofe_subst. - by destruct 1; rewrite /validN /option_validN //=; cofe_subst. - by destruct 1; inversion_clear 1; constructor; cofe_subst. `````` Robbert Krebbers committed Feb 24, 2016 99 `````` - intros [x|]; [apply cmra_valid_validN|done]. `````` Robbert Krebbers committed Feb 17, 2016 100 101 102 103 104 105 `````` - intros n [x|]; unfold validN, option_validN; eauto using cmra_validN_S. - intros [x|] [y|] [z|]; constructor; rewrite ?assoc; auto. - intros [x|] [y|]; constructor; rewrite 1?comm; auto. - by intros [x|]; constructor; rewrite cmra_unit_l. - by intros [x|]; constructor; rewrite cmra_unit_idemp. - intros n mx my; rewrite !option_includedN;intros [->|(x&y&->&->&?)]; auto. `````` Robbert Krebbers committed Feb 10, 2016 106 `````` right; exists (unit x), (unit y); eauto using cmra_unit_preservingN. `````` Robbert Krebbers committed Feb 17, 2016 107 `````` - intros n [x|] [y|]; rewrite /validN /option_validN /=; `````` Robbert Krebbers committed Feb 01, 2016 108 `````` eauto using cmra_validN_op_l. `````` Robbert Krebbers committed Feb 17, 2016 109 `````` - intros n mx my; rewrite option_includedN. `````` Robbert Krebbers committed Feb 10, 2016 110 `````` intros [->|(x&y&->&->&?)]; [by destruct my|]. `````` Robbert Krebbers committed Dec 15, 2015 111 `````` by constructor; apply cmra_op_minus. `````` Robbert Krebbers committed Feb 24, 2016 112 113 114 115 116 117 118 119 120 `````` - intros n mx my1 my2. destruct mx as [x|], my1 as [y1|], my2 as [y2|]; intros Hx Hx'; try (by exfalso; inversion Hx'; auto). + destruct (cmra_extend n x y1 y2) as ([z1 z2]&?&?&?); auto. { by inversion_clear Hx'. } by exists (Some z1, Some z2); repeat constructor. + by exists (Some x,None); inversion Hx'; repeat constructor. + by exists (None,Some x); inversion Hx'; repeat constructor. + exists (None,None); repeat constructor. `````` Robbert Krebbers committed Dec 15, 2015 121 ``````Qed. `````` Robbert Krebbers committed Feb 24, 2016 122 ``````Canonical Structure optionRA := CMRAT option_cofe_mixin option_cmra_mixin. `````` Robbert Krebbers committed Feb 04, 2016 123 124 ``````Global Instance option_cmra_identity : CMRAIdentity optionRA. Proof. split. done. by intros []. by inversion_clear 1. Qed. `````` Robbert Krebbers committed Feb 24, 2016 125 126 ``````Global Instance option_cmra_discrete : CMRADiscrete A → CMRADiscrete optionRA. Proof. split; [apply _|]. by intros [x|]; [apply (cmra_discrete_valid x)|]. Qed. `````` Robbert Krebbers committed Jan 14, 2016 127 `````` `````` Robbert Krebbers committed Feb 13, 2016 128 ``````(** Misc *) `````` Robbert Krebbers committed Feb 02, 2016 129 130 131 132 ``````Lemma op_is_Some mx my : is_Some (mx ⋅ my) ↔ is_Some mx ∨ is_Some my. Proof. destruct mx, my; rewrite /op /option_op /= -!not_eq_None_Some; naive_solver. Qed. `````` Ralf Jung committed Feb 10, 2016 133 ``````Lemma option_op_positive_dist_l n mx my : mx ⋅ my ≡{n}≡ None → mx ≡{n}≡ None. `````` Robbert Krebbers committed Feb 02, 2016 134 ``````Proof. by destruct mx, my; inversion_clear 1. Qed. `````` Ralf Jung committed Feb 10, 2016 135 ``````Lemma option_op_positive_dist_r n mx my : mx ⋅ my ≡{n}≡ None → my ≡{n}≡ None. `````` Robbert Krebbers committed Feb 02, 2016 136 137 ``````Proof. by destruct mx, my; inversion_clear 1. Qed. `````` Robbert Krebbers committed Feb 13, 2016 138 139 140 141 142 ``````(** Internalized properties *) Lemma option_equivI {M} (x y : option A) : (x ≡ y)%I ≡ (match x, y with | Some a, Some b => a ≡ b | None, None => True | _, _ => False end : uPred M)%I. `````` Robbert Krebbers committed Feb 25, 2016 143 144 145 ``````Proof. uPred.unseal. do 2 split. by destruct 1. by destruct x, y; try constructor. Qed. `````` Robbert Krebbers committed Feb 13, 2016 146 147 ``````Lemma option_validI {M} (x : option A) : (✓ x)%I ≡ (match x with Some a => ✓ a | None => True end : uPred M)%I. `````` Robbert Krebbers committed Feb 25, 2016 148 ``````Proof. uPred.unseal. by destruct x. Qed. `````` Robbert Krebbers committed Feb 13, 2016 149 150 `````` (** Updates *) `````` Robbert Krebbers committed Feb 02, 2016 151 ``````Lemma option_updateP (P : A → Prop) (Q : option A → Prop) x : `````` Ralf Jung committed Feb 03, 2016 152 `````` x ~~>: P → (∀ y, P y → Q (Some y)) → Some x ~~>: Q. `````` Robbert Krebbers committed Feb 02, 2016 153 ``````Proof. `````` Robbert Krebbers committed Feb 18, 2016 154 155 156 `````` intros Hx Hy n [y|] ?. { destruct (Hx n y) as (y'&?&?); auto. exists (Some y'); auto. } destruct (Hx n (unit x)) as (y'&?&?); rewrite ?cmra_unit_r; auto. `````` Robbert Krebbers committed Feb 02, 2016 157 158 `````` by exists (Some y'); split; [auto|apply cmra_validN_op_l with (unit x)]. Qed. `````` Robbert Krebbers committed Feb 02, 2016 159 ``````Lemma option_updateP' (P : A → Prop) x : `````` Ralf Jung committed Feb 03, 2016 160 `````` x ~~>: P → Some x ~~>: λ y, default False y P. `````` Robbert Krebbers committed Feb 02, 2016 161 ``````Proof. eauto using option_updateP. Qed. `````` Ralf Jung committed Feb 03, 2016 162 ``````Lemma option_update x y : x ~~> y → Some x ~~> Some y. `````` Robbert Krebbers committed Jan 16, 2016 163 ``````Proof. `````` Robbert Krebbers committed Feb 02, 2016 164 `````` rewrite !cmra_update_updateP; eauto using option_updateP with congruence. `````` Robbert Krebbers committed Jan 16, 2016 165 ``````Qed. `````` Robbert Krebbers committed Feb 08, 2016 166 167 ``````Lemma option_update_None `{Empty A, !CMRAIdentity A} : ∅ ~~> Some ∅. Proof. `````` Robbert Krebbers committed Feb 18, 2016 168 `````` intros n [x|] ?; rewrite /op /cmra_op /validN /cmra_validN /= ?left_id; `````` Robbert Krebbers committed Feb 24, 2016 169 `````` auto using cmra_empty_validN. `````` Robbert Krebbers committed Feb 08, 2016 170 ``````Qed. `````` Robbert Krebbers committed Jan 14, 2016 171 172 173 ``````End cmra. Arguments optionRA : clear implicits. `````` Robbert Krebbers committed Feb 04, 2016 174 175 176 177 ``````(** Functor *) Instance option_fmap_ne {A B : cofeT} (f : A → B) n: Proper (dist n ==> dist n) f → Proper (dist n==>dist n) (fmap (M:=option) f). Proof. by intros Hf; destruct 1; constructor; apply Hf. Qed. `````` Robbert Krebbers committed Jan 14, 2016 178 179 ``````Instance option_fmap_cmra_monotone {A B : cmraT} (f: A → B) `{!CMRAMonotone f} : CMRAMonotone (fmap f : option A → option B). `````` Robbert Krebbers committed Dec 15, 2015 180 181 ``````Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 182 `````` - intros n mx my; rewrite !option_includedN. `````` Robbert Krebbers committed Feb 10, 2016 183 `````` intros [->|(x&y&->&->&?)]; simpl; eauto 10 using @includedN_preserving. `````` Robbert Krebbers committed Feb 17, 2016 184 `````` - by intros n [x|] ?; rewrite /cmra_validN /=; try apply validN_preserving. `````` Robbert Krebbers committed Jan 14, 2016 185 ``````Qed. `````` Robbert Krebbers committed Feb 04, 2016 186 187 188 189 ``````Definition optionC_map {A B} (f : A -n> B) : optionC A -n> optionC B := CofeMor (fmap f : optionC A → optionC B). Instance optionC_map_ne A B n : Proper (dist n ==> dist n) (@optionC_map A B). Proof. by intros f f' Hf []; constructor; apply Hf. Qed. `````` Ralf Jung committed Feb 05, 2016 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 `````` Program Definition optionF (Σ : iFunctor) : iFunctor := {| ifunctor_car := optionRA ∘ Σ; ifunctor_map A B := optionC_map ∘ ifunctor_map Σ |}. Next Obligation. by intros Σ A B n f g Hfg; apply optionC_map_ne, ifunctor_map_ne. Qed. Next Obligation. intros Σ A x. rewrite /= -{2}(option_fmap_id x). apply option_fmap_setoid_ext=>y; apply ifunctor_map_id. Qed. Next Obligation. intros Σ A B C f g x. rewrite /= -option_fmap_compose. apply option_fmap_setoid_ext=>y; apply ifunctor_map_compose. Qed.``````