cmra.v 47.9 KB
 Robbert Krebbers committed Mar 10, 2016 1 ``````From iris.algebra Require Export cofe. `````` Robbert Krebbers committed Feb 01, 2016 2 `````` `````` Robbert Krebbers committed May 28, 2016 3 4 ``````Class PCore (A : Type) := pcore : A → option A. Instance: Params (@pcore) 2. `````` Robbert Krebbers committed Feb 01, 2016 5 6 7 8 9 10 11 12 13 `````` Class Op (A : Type) := op : A → A → A. Instance: Params (@op) 2. Infix "⋅" := op (at level 50, left associativity) : C_scope. Notation "(⋅)" := op (only parsing) : C_scope. Definition included `{Equiv A, Op A} (x y : A) := ∃ z, y ≡ x ⋅ z. Infix "≼" := included (at level 70) : C_scope. Notation "(≼)" := included (only parsing) : C_scope. `````` Robbert Krebbers committed Feb 13, 2016 14 ``````Hint Extern 0 (_ ≼ _) => reflexivity. `````` Robbert Krebbers committed Feb 01, 2016 15 16 ``````Instance: Params (@included) 3. `````` Robbert Krebbers committed Nov 11, 2015 17 18 ``````Class ValidN (A : Type) := validN : nat → A → Prop. Instance: Params (@validN) 3. `````` Robbert Krebbers committed Feb 11, 2016 19 ``````Notation "✓{ n } x" := (validN n x) `````` Robbert Krebbers committed Feb 19, 2016 20 `````` (at level 20, n at next level, format "✓{ n } x"). `````` Robbert Krebbers committed Nov 11, 2015 21 `````` `````` Robbert Krebbers committed Feb 01, 2016 22 23 ``````Class Valid (A : Type) := valid : A → Prop. Instance: Params (@valid) 2. `````` Robbert Krebbers committed Feb 11, 2016 24 ``````Notation "✓ x" := (valid x) (at level 20) : C_scope. `````` Robbert Krebbers committed Feb 01, 2016 25 `````` `````` Ralf Jung committed Feb 10, 2016 26 ``````Definition includedN `{Dist A, Op A} (n : nat) (x y : A) := ∃ z, y ≡{n}≡ x ⋅ z. `````` Robbert Krebbers committed Nov 20, 2015 27 ``````Notation "x ≼{ n } y" := (includedN n x y) `````` Robbert Krebbers committed Feb 19, 2016 28 `````` (at level 70, n at next level, format "x ≼{ n } y") : C_scope. `````` Robbert Krebbers committed Nov 20, 2015 29 ``````Instance: Params (@includedN) 4. `````` Robbert Krebbers committed Feb 13, 2016 30 ``````Hint Extern 0 (_ ≼{_} _) => reflexivity. `````` Robbert Krebbers committed Nov 20, 2015 31 `````` `````` Robbert Krebbers committed May 28, 2016 32 ``````Record CMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A} := { `````` Robbert Krebbers committed Nov 11, 2015 33 `````` (* setoids *) `````` Robbert Krebbers committed Jan 14, 2016 34 `````` mixin_cmra_op_ne n (x : A) : Proper (dist n ==> dist n) (op x); `````` Robbert Krebbers committed May 28, 2016 35 36 `````` mixin_cmra_pcore_ne n x y cx : x ≡{n}≡ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≡{n}≡ cy; `````` Robbert Krebbers committed Feb 11, 2016 37 `````` mixin_cmra_validN_ne n : Proper (dist n ==> impl) (validN n); `````` Robbert Krebbers committed Nov 11, 2015 38 `````` (* valid *) `````` Robbert Krebbers committed Feb 24, 2016 39 `````` mixin_cmra_valid_validN x : ✓ x ↔ ∀ n, ✓{n} x; `````` Robbert Krebbers committed Feb 01, 2016 40 `````` mixin_cmra_validN_S n x : ✓{S n} x → ✓{n} x; `````` Robbert Krebbers committed Nov 11, 2015 41 `````` (* monoid *) `````` Robbert Krebbers committed Feb 11, 2016 42 43 `````` mixin_cmra_assoc : Assoc (≡) (⋅); mixin_cmra_comm : Comm (≡) (⋅); `````` Robbert Krebbers committed May 28, 2016 44 45 46 47 `````` mixin_cmra_pcore_l x cx : pcore x = Some cx → cx ⋅ x ≡ x; mixin_cmra_pcore_idemp x cx : pcore x = Some cx → pcore cx ≡ Some cx; mixin_cmra_pcore_preserving x y cx : x ≼ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy; `````` Robbert Krebbers committed Feb 01, 2016 48 `````` mixin_cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x; `````` Robbert Krebbers committed Feb 24, 2016 49 50 51 `````` mixin_cmra_extend n x y1 y2 : ✓{n} x → x ≡{n}≡ y1 ⋅ y2 → { z | x ≡ z.1 ⋅ z.2 ∧ z.1 ≡{n}≡ y1 ∧ z.2 ≡{n}≡ y2 } `````` Robbert Krebbers committed Nov 11, 2015 52 ``````}. `````` Robbert Krebbers committed Nov 22, 2015 53 `````` `````` Robbert Krebbers committed Nov 11, 2015 54 55 56 57 58 59 ``````(** Bundeled version *) Structure cmraT := CMRAT { cmra_car :> Type; cmra_equiv : Equiv cmra_car; cmra_dist : Dist cmra_car; cmra_compl : Compl cmra_car; `````` Robbert Krebbers committed May 28, 2016 60 `````` cmra_pcore : PCore cmra_car; `````` Robbert Krebbers committed Nov 11, 2015 61 `````` cmra_op : Op cmra_car; `````` Robbert Krebbers committed Feb 24, 2016 62 `````` cmra_valid : Valid cmra_car; `````` Robbert Krebbers committed Nov 11, 2015 63 `````` cmra_validN : ValidN cmra_car; `````` Robbert Krebbers committed Jan 14, 2016 64 `````` cmra_cofe_mixin : CofeMixin cmra_car; `````` Robbert Krebbers committed Feb 24, 2016 65 `````` cmra_mixin : CMRAMixin cmra_car `````` Robbert Krebbers committed Nov 11, 2015 66 ``````}. `````` Robbert Krebbers committed May 25, 2016 67 ``````Arguments CMRAT _ {_ _ _ _ _ _ _} _ _. `````` Robbert Krebbers committed Jan 14, 2016 68 69 70 71 ``````Arguments cmra_car : simpl never. Arguments cmra_equiv : simpl never. Arguments cmra_dist : simpl never. Arguments cmra_compl : simpl never. `````` Robbert Krebbers committed May 28, 2016 72 ``````Arguments cmra_pcore : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 73 ``````Arguments cmra_op : simpl never. `````` Robbert Krebbers committed Feb 24, 2016 74 ``````Arguments cmra_valid : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 75 76 77 ``````Arguments cmra_validN : simpl never. Arguments cmra_cofe_mixin : simpl never. Arguments cmra_mixin : simpl never. `````` Robbert Krebbers committed Nov 11, 2015 78 ``````Add Printing Constructor cmraT. `````` Robbert Krebbers committed May 28, 2016 79 ``````Existing Instances cmra_pcore cmra_op cmra_valid cmra_validN. `````` Robbert Krebbers committed May 25, 2016 80 ``````Coercion cmra_cofeC (A : cmraT) : cofeT := CofeT A (cmra_cofe_mixin A). `````` Robbert Krebbers committed Nov 11, 2015 81 82 ``````Canonical Structure cmra_cofeC. `````` Robbert Krebbers committed Jan 14, 2016 83 84 85 86 87 88 ``````(** Lifting properties from the mixin *) Section cmra_mixin. Context {A : cmraT}. Implicit Types x y : A. Global Instance cmra_op_ne n (x : A) : Proper (dist n ==> dist n) (op x). Proof. apply (mixin_cmra_op_ne _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed May 28, 2016 89 90 91 `````` Lemma cmra_pcore_ne n x y cx : x ≡{n}≡ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≡{n}≡ cy. Proof. apply (mixin_cmra_pcore_ne _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Feb 01, 2016 92 93 `````` Global Instance cmra_validN_ne n : Proper (dist n ==> impl) (@validN A _ n). Proof. apply (mixin_cmra_validN_ne _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Feb 24, 2016 94 95 `````` Lemma cmra_valid_validN x : ✓ x ↔ ∀ n, ✓{n} x. Proof. apply (mixin_cmra_valid_validN _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Feb 01, 2016 96 97 `````` Lemma cmra_validN_S n x : ✓{S n} x → ✓{n} x. Proof. apply (mixin_cmra_validN_S _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Feb 11, 2016 98 99 100 101 `````` Global Instance cmra_assoc : Assoc (≡) (@op A _). Proof. apply (mixin_cmra_assoc _ (cmra_mixin A)). Qed. Global Instance cmra_comm : Comm (≡) (@op A _). Proof. apply (mixin_cmra_comm _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed May 28, 2016 102 103 104 105 106 107 108 `````` Lemma cmra_pcore_l x cx : pcore x = Some cx → cx ⋅ x ≡ x. Proof. apply (mixin_cmra_pcore_l _ (cmra_mixin A)). Qed. Lemma cmra_pcore_idemp x cx : pcore x = Some cx → pcore cx ≡ Some cx. Proof. apply (mixin_cmra_pcore_idemp _ (cmra_mixin A)). Qed. Lemma cmra_pcore_preserving x y cx : x ≼ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy. Proof. apply (mixin_cmra_pcore_preserving _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Feb 01, 2016 109 110 `````` Lemma cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x. Proof. apply (mixin_cmra_validN_op_l _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Feb 24, 2016 111 `````` Lemma cmra_extend n x y1 y2 : `````` Ralf Jung committed Feb 10, 2016 112 113 `````` ✓{n} x → x ≡{n}≡ y1 ⋅ y2 → { z | x ≡ z.1 ⋅ z.2 ∧ z.1 ≡{n}≡ y1 ∧ z.2 ≡{n}≡ y2 }. `````` Robbert Krebbers committed Feb 24, 2016 114 `````` Proof. apply (mixin_cmra_extend _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Jan 14, 2016 115 116 ``````End cmra_mixin. `````` Robbert Krebbers committed May 28, 2016 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 ``````Definition opM {A : cmraT} (x : A) (my : option A) := match my with Some y => x ⋅ y | None => x end. Infix "⋅?" := opM (at level 50, left associativity) : C_scope. (** * Persistent elements *) Class Persistent {A : cmraT} (x : A) := persistent : pcore x ≡ Some x. Arguments persistent {_} _ {_}. (** * CMRAs whose core is total *) (** The function [core] may return a dummy when used on CMRAs without total core. *) Class CMRATotal (A : cmraT) := cmra_total (x : A) : is_Some (pcore x). Class Core (A : Type) := core : A → A. Instance: Params (@core) 2. Instance core' `{PCore A} : Core A := λ x, from_option id x (pcore x). Arguments core' _ _ _ /. `````` Ralf Jung committed Mar 08, 2016 136 ``````(** * CMRAs with a unit element *) `````` Robbert Krebbers committed Feb 01, 2016 137 ``````(** We use the notation ∅ because for most instances (maps, sets, etc) the `````` Ralf Jung committed Mar 08, 2016 138 ```````empty' element is the unit. *) `````` Robbert Krebbers committed May 28, 2016 139 ``````Record UCMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, Empty A} := { `````` Robbert Krebbers committed May 27, 2016 140 141 `````` mixin_ucmra_unit_valid : ✓ ∅; mixin_ucmra_unit_left_id : LeftId (≡) ∅ (⋅); `````` Robbert Krebbers committed May 28, 2016 142 143 `````` mixin_ucmra_unit_timeless : Timeless ∅; mixin_ucmra_pcore_unit : pcore ∅ ≡ Some ∅ `````` Robbert Krebbers committed Feb 01, 2016 144 ``````}. `````` Robbert Krebbers committed May 27, 2016 145 146 147 148 149 150 `````` Structure ucmraT := UCMRAT { ucmra_car :> Type; ucmra_equiv : Equiv ucmra_car; ucmra_dist : Dist ucmra_car; ucmra_compl : Compl ucmra_car; `````` Robbert Krebbers committed May 28, 2016 151 `````` ucmra_pcore : PCore ucmra_car; `````` Robbert Krebbers committed May 27, 2016 152 153 154 155 156 157 158 159 160 161 162 163 164 `````` ucmra_op : Op ucmra_car; ucmra_valid : Valid ucmra_car; ucmra_validN : ValidN ucmra_car; ucmra_empty : Empty ucmra_car; ucmra_cofe_mixin : CofeMixin ucmra_car; ucmra_cmra_mixin : CMRAMixin ucmra_car; ucmra_mixin : UCMRAMixin ucmra_car }. Arguments UCMRAT _ {_ _ _ _ _ _ _ _} _ _ _. Arguments ucmra_car : simpl never. Arguments ucmra_equiv : simpl never. Arguments ucmra_dist : simpl never. Arguments ucmra_compl : simpl never. `````` Robbert Krebbers committed May 28, 2016 165 ``````Arguments ucmra_pcore : simpl never. `````` Robbert Krebbers committed May 27, 2016 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 ``````Arguments ucmra_op : simpl never. Arguments ucmra_valid : simpl never. Arguments ucmra_validN : simpl never. Arguments ucmra_cofe_mixin : simpl never. Arguments ucmra_cmra_mixin : simpl never. Arguments ucmra_mixin : simpl never. Add Printing Constructor ucmraT. Existing Instances ucmra_empty. Coercion ucmra_cofeC (A : ucmraT) : cofeT := CofeT A (ucmra_cofe_mixin A). Canonical Structure ucmra_cofeC. Coercion ucmra_cmraR (A : ucmraT) : cmraT := CMRAT A (ucmra_cofe_mixin A) (ucmra_cmra_mixin A). Canonical Structure ucmra_cmraR. (** Lifting properties from the mixin *) Section ucmra_mixin. Context {A : ucmraT}. Implicit Types x y : A. Lemma ucmra_unit_valid : ✓ (∅ : A). Proof. apply (mixin_ucmra_unit_valid _ (ucmra_mixin A)). Qed. Global Instance ucmra_unit_left_id : LeftId (≡) ∅ (@op A _). Proof. apply (mixin_ucmra_unit_left_id _ (ucmra_mixin A)). Qed. Global Instance ucmra_unit_timeless : Timeless (∅ : A). Proof. apply (mixin_ucmra_unit_timeless _ (ucmra_mixin A)). Qed. `````` Robbert Krebbers committed May 28, 2016 190 191 `````` Lemma ucmra_pcore_unit : pcore (∅:A) ≡ Some ∅. Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin A)). Qed. `````` Robbert Krebbers committed May 27, 2016 192 ``````End ucmra_mixin. `````` Robbert Krebbers committed Jan 14, 2016 193 `````` `````` Robbert Krebbers committed Feb 24, 2016 194 ``````(** * Discrete CMRAs *) `````` Robbert Krebbers committed Feb 26, 2016 195 ``````Class CMRADiscrete (A : cmraT) := { `````` Robbert Krebbers committed Feb 24, 2016 196 197 198 199 `````` cmra_discrete :> Discrete A; cmra_discrete_valid (x : A) : ✓{0} x → ✓ x }. `````` Robbert Krebbers committed Jan 16, 2016 200 ``````(** * Morphisms *) `````` Robbert Krebbers committed Jan 14, 2016 201 ``````Class CMRAMonotone {A B : cmraT} (f : A → B) := { `````` Robbert Krebbers committed Feb 26, 2016 202 203 204 `````` cmra_monotone_ne n :> Proper (dist n ==> dist n) f; validN_preserving n x : ✓{n} x → ✓{n} f x; included_preserving x y : x ≼ y → f x ≼ f y `````` Robbert Krebbers committed Jan 14, 2016 205 ``````}. `````` Robbert Krebbers committed Feb 26, 2016 206 207 ``````Arguments validN_preserving {_ _} _ {_} _ _ _. Arguments included_preserving {_ _} _ {_} _ _ _. `````` Robbert Krebbers committed Jan 14, 2016 208 `````` `````` Robbert Krebbers committed Feb 11, 2016 209 ``````(** * Local updates *) `````` Ralf Jung committed Feb 13, 2016 210 211 ``````(** The idea is that lemams taking this class will usually have L explicit, and leave Lv implicit - it will be inferred by the typeclass machinery. *) `````` Ralf Jung committed Feb 11, 2016 212 213 214 ``````Class LocalUpdate {A : cmraT} (Lv : A → Prop) (L : A → A) := { local_update_ne n :> Proper (dist n ==> dist n) L; local_updateN n x y : Lv x → ✓{n} (x ⋅ y) → L (x ⋅ y) ≡{n}≡ L x ⋅ y `````` Robbert Krebbers committed Feb 11, 2016 215 216 217 ``````}. Arguments local_updateN {_ _} _ {_} _ _ _ _ _. `````` Robbert Krebbers committed Feb 01, 2016 218 ``````(** * Frame preserving updates *) `````` Robbert Krebbers committed May 28, 2016 219 220 ``````Definition cmra_updateP {A : cmraT} (x : A) (P : A → Prop) := ∀ n mz, ✓{n} (x ⋅? mz) → ∃ y, P y ∧ ✓{n} (y ⋅? mz). `````` Robbert Krebbers committed Feb 02, 2016 221 ``````Instance: Params (@cmra_updateP) 1. `````` Ralf Jung committed Feb 03, 2016 222 ``````Infix "~~>:" := cmra_updateP (at level 70). `````` Robbert Krebbers committed May 28, 2016 223 224 225 `````` Definition cmra_update {A : cmraT} (x y : A) := ∀ n mz, ✓{n} (x ⋅? mz) → ✓{n} (y ⋅? mz). `````` Ralf Jung committed Feb 03, 2016 226 ``````Infix "~~>" := cmra_update (at level 70). `````` Robbert Krebbers committed Feb 02, 2016 227 ``````Instance: Params (@cmra_update) 1. `````` Robbert Krebbers committed Nov 22, 2015 228 `````` `````` Robbert Krebbers committed Jan 16, 2016 229 ``````(** * Properties **) `````` Robbert Krebbers committed Nov 11, 2015 230 ``````Section cmra. `````` Robbert Krebbers committed Jan 14, 2016 231 ``````Context {A : cmraT}. `````` Robbert Krebbers committed Nov 11, 2015 232 ``````Implicit Types x y z : A. `````` Robbert Krebbers committed Feb 01, 2016 233 ``````Implicit Types xs ys zs : list A. `````` Robbert Krebbers committed Nov 11, 2015 234 `````` `````` Robbert Krebbers committed Feb 01, 2016 235 ``````(** ** Setoids *) `````` Robbert Krebbers committed May 28, 2016 236 237 238 239 240 241 242 243 244 ``````Global Instance cmra_pcore_ne' n : Proper (dist n ==> dist n) (@pcore A _). Proof. intros x y Hxy. destruct (pcore x) as [cx|] eqn:?. { destruct (cmra_pcore_ne n x y cx) as (cy&->&->); auto. } destruct (pcore y) as [cy|] eqn:?; auto. destruct (cmra_pcore_ne n y x cy) as (cx&?&->); simplify_eq/=; auto. Qed. Lemma cmra_pcore_proper x y cx : x ≡ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≡ cy. `````` Robbert Krebbers committed Feb 01, 2016 245 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 246 247 248 `````` intros. destruct (cmra_pcore_ne 0 x y cx) as (cy&?&?); auto. exists cy; split; [done|apply equiv_dist=> n]. destruct (cmra_pcore_ne n x y cx) as (cy'&?&?); naive_solver. `````` Robbert Krebbers committed Feb 01, 2016 249 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 250 251 252 253 ``````Global Instance cmra_pcore_proper' : Proper ((≡) ==> (≡)) (@pcore A _). Proof. apply (ne_proper _). Qed. Global Instance cmra_op_ne' n : Proper (dist n ==> dist n ==> dist n) (@op A _). Proof. intros x1 x2 Hx y1 y2 Hy. by rewrite Hy (comm _ x1) Hx (comm _ y2). Qed. `````` Robbert Krebbers committed Feb 01, 2016 254 255 256 257 258 259 260 261 ``````Global Instance ra_op_proper' : Proper ((≡) ==> (≡) ==> (≡)) (@op A _). Proof. apply (ne_proper_2 _). Qed. Global Instance cmra_validN_ne' : Proper (dist n ==> iff) (@validN A _ n) | 1. Proof. by split; apply cmra_validN_ne. Qed. Global Instance cmra_validN_proper : Proper ((≡) ==> iff) (@validN A _ n) | 1. Proof. by intros n x1 x2 Hx; apply cmra_validN_ne', equiv_dist. Qed. Global Instance cmra_valid_proper : Proper ((≡) ==> iff) (@valid A _). `````` Robbert Krebbers committed Feb 24, 2016 262 263 264 265 ``````Proof. intros x y Hxy; rewrite !cmra_valid_validN. by split=> ? n; [rewrite -Hxy|rewrite Hxy]. Qed. `````` Robbert Krebbers committed Feb 01, 2016 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 ``````Global Instance cmra_includedN_ne n : Proper (dist n ==> dist n ==> iff) (@includedN A _ _ n) | 1. Proof. intros x x' Hx y y' Hy. by split; intros [z ?]; exists z; [rewrite -Hx -Hy|rewrite Hx Hy]. Qed. Global Instance cmra_includedN_proper n : Proper ((≡) ==> (≡) ==> iff) (@includedN A _ _ n) | 1. Proof. intros x x' Hx y y' Hy; revert Hx Hy; rewrite !equiv_dist=> Hx Hy. by rewrite (Hx n) (Hy n). Qed. Global Instance cmra_included_proper : Proper ((≡) ==> (≡) ==> iff) (@included A _ _) | 1. Proof. intros x x' Hx y y' Hy. by split; intros [z ?]; exists z; [rewrite -Hx -Hy|rewrite Hx Hy]. Qed. `````` Robbert Krebbers committed May 28, 2016 284 285 286 287 ``````Global Instance cmra_opM_ne n : Proper (dist n ==> dist n ==> dist n) (@opM A). Proof. destruct 2; by cofe_subst. Qed. Global Instance cmra_opM_proper : Proper ((≡) ==> (≡) ==> (≡)) (@opM A). Proof. destruct 2; by setoid_subst. Qed. `````` Robbert Krebbers committed Feb 02, 2016 288 289 290 ``````Global Instance cmra_updateP_proper : Proper ((≡) ==> pointwise_relation _ iff ==> iff) (@cmra_updateP A). Proof. `````` Robbert Krebbers committed May 28, 2016 291 292 293 294 295 296 297 `````` rewrite /pointwise_relation /cmra_updateP=> x x' Hx P P' HP; split=> ? n mz; setoid_subst; naive_solver. Qed. Global Instance cmra_update_proper : Proper ((≡) ==> (≡) ==> iff) (@cmra_update A). Proof. rewrite /cmra_update=> x x' Hx y y' Hy; split=> ? n mz ?; setoid_subst; auto. `````` Robbert Krebbers committed Feb 02, 2016 298 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2016 299 `````` `````` Robbert Krebbers committed May 28, 2016 300 301 302 303 ``````(** ** Op *) Lemma cmra_opM_assoc x y mz : (x ⋅ y) ⋅? mz ≡ x ⋅ (y ⋅? mz). Proof. destruct mz; by rewrite /= -?assoc. Qed. `````` Robbert Krebbers committed Feb 01, 2016 304 ``````(** ** Validity *) `````` Robbert Krebbers committed Feb 18, 2016 305 ``````Lemma cmra_validN_le n n' x : ✓{n} x → n' ≤ n → ✓{n'} x. `````` Robbert Krebbers committed Feb 01, 2016 306 307 308 ``````Proof. induction 2; eauto using cmra_validN_S. Qed. Lemma cmra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x. Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 18, 2016 309 ``````Lemma cmra_validN_op_r n x y : ✓{n} (x ⋅ y) → ✓{n} y. `````` Robbert Krebbers committed Feb 11, 2016 310 ``````Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 311 312 313 ``````Lemma cmra_valid_op_r x y : ✓ (x ⋅ y) → ✓ y. Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_r. Qed. `````` Ralf Jung committed Mar 08, 2016 314 ``````(** ** Core *) `````` Robbert Krebbers committed May 28, 2016 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 ``````Lemma cmra_pcore_l' x cx : pcore x ≡ Some cx → cx ⋅ x ≡ x. Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_l. Qed. Lemma cmra_pcore_r x cx : pcore x = Some cx → x ⋅ cx ≡ x. Proof. intros. rewrite comm. by apply cmra_pcore_l. Qed. Lemma cmra_pcore_r' x cx : pcore x ≡ Some cx → x ⋅ cx ≡ x. Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_r. Qed. Lemma cmra_pcore_idemp' x cx : pcore x ≡ Some cx → pcore cx ≡ Some cx. Proof. intros (cx'&?&->)%equiv_Some_inv_r'. eauto using cmra_pcore_idemp. Qed. Lemma cmra_pcore_pcore x cx : pcore x = Some cx → cx ⋅ cx ≡ cx. Proof. eauto using cmra_pcore_r', cmra_pcore_idemp. Qed. Lemma cmra_pcore_pcore' x cx : pcore x ≡ Some cx → cx ⋅ cx ≡ cx. Proof. eauto using cmra_pcore_r', cmra_pcore_idemp'. Qed. Lemma cmra_pcore_validN n x cx : ✓{n} x → pcore x = Some cx → ✓{n} cx. Proof. intros Hvx Hx%cmra_pcore_l. move: Hvx; rewrite -Hx. apply cmra_validN_op_l. Qed. Lemma cmra_pcore_valid x cx : ✓ x → pcore x = Some cx → ✓ cx. Proof. intros Hv Hx%cmra_pcore_l. move: Hv; rewrite -Hx. apply cmra_valid_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 335 336 `````` (** ** Order *) `````` Robbert Krebbers committed Mar 11, 2016 337 338 ``````Lemma cmra_included_includedN n x y : x ≼ y → x ≼{n} y. Proof. intros [z ->]. by exists z. Qed. `````` Robbert Krebbers committed May 28, 2016 339 ``````Global Instance cmra_includedN_trans n : Transitive (@includedN A _ _ n). `````` Robbert Krebbers committed Feb 01, 2016 340 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 341 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 342 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 343 ``````Global Instance cmra_included_trans: Transitive (@included A _ _). `````` Robbert Krebbers committed Feb 01, 2016 344 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 345 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 346 ``````Qed. `````` Robbert Krebbers committed Feb 18, 2016 347 ``````Lemma cmra_validN_includedN n x y : ✓{n} y → x ≼{n} y → ✓{n} x. `````` Robbert Krebbers committed Feb 01, 2016 348 ``````Proof. intros Hyv [z ?]; cofe_subst y; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 18, 2016 349 ``````Lemma cmra_validN_included n x y : ✓{n} y → x ≼ y → ✓{n} x. `````` Robbert Krebbers committed Mar 11, 2016 350 ``````Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 351 `````` `````` Robbert Krebbers committed Feb 18, 2016 352 ``````Lemma cmra_includedN_S n x y : x ≼{S n} y → x ≼{n} y. `````` Robbert Krebbers committed Feb 01, 2016 353 ``````Proof. by intros [z Hz]; exists z; apply dist_S. Qed. `````` Robbert Krebbers committed Feb 18, 2016 354 ``````Lemma cmra_includedN_le n n' x y : x ≼{n} y → n' ≤ n → x ≼{n'} y. `````` Robbert Krebbers committed Feb 01, 2016 355 356 357 358 359 360 361 ``````Proof. induction 2; auto using cmra_includedN_S. Qed. Lemma cmra_includedN_l n x y : x ≼{n} x ⋅ y. Proof. by exists y. Qed. Lemma cmra_included_l x y : x ≼ x ⋅ y. Proof. by exists y. Qed. Lemma cmra_includedN_r n x y : y ≼{n} x ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 362 ``````Proof. rewrite (comm op); apply cmra_includedN_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 363 ``````Lemma cmra_included_r x y : y ≼ x ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 364 ``````Proof. rewrite (comm op); apply cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 20, 2015 365 `````` `````` Robbert Krebbers committed May 28, 2016 366 367 368 369 370 371 372 373 374 ``````Lemma cmra_pcore_preserving' x y cx : x ≼ y → pcore x ≡ Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy. Proof. intros ? (cx'&?&Hcx)%equiv_Some_inv_r'. destruct (cmra_pcore_preserving x y cx') as (cy&->&?); auto. exists cy; by rewrite Hcx. Qed. Lemma cmra_pcore_preservingN' n x y cx : x ≼{n} y → pcore x ≡{n}≡ Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼{n} cy. `````` Robbert Krebbers committed Feb 26, 2016 375 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 376 377 378 379 380 381 382 `````` intros [z Hy] (cx'&?&Hcx)%dist_Some_inv_r'. destruct (cmra_pcore_preserving x (x ⋅ z) cx') as (cy&Hxy&?); auto using cmra_included_l. assert (pcore y ≡{n}≡ Some cy) as (cy'&?&Hcy')%dist_Some_inv_r'. { by rewrite Hy Hxy. } exists cy'; split; first done. rewrite Hcx -Hcy'; auto using cmra_included_includedN. `````` Robbert Krebbers committed Feb 26, 2016 383 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 384 385 ``````Lemma cmra_included_pcore x cx : pcore x = Some cx → cx ≼ x. Proof. exists x. by rewrite cmra_pcore_l. Qed. `````` Robbert Krebbers committed Feb 11, 2016 386 ``````Lemma cmra_preservingN_l n x y z : x ≼{n} y → z ⋅ x ≼{n} z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 387 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Robbert Krebbers committed Feb 01, 2016 388 ``````Lemma cmra_preserving_l x y z : x ≼ y → z ⋅ x ≼ z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 389 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Robbert Krebbers committed Feb 11, 2016 390 ``````Lemma cmra_preservingN_r n x y z : x ≼{n} y → x ⋅ z ≼{n} y ⋅ z. `````` Robbert Krebbers committed Feb 11, 2016 391 ``````Proof. by intros; rewrite -!(comm _ z); apply cmra_preservingN_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 392 ``````Lemma cmra_preserving_r x y z : x ≼ y → x ⋅ z ≼ y ⋅ z. `````` Robbert Krebbers committed Feb 11, 2016 393 ``````Proof. by intros; rewrite -!(comm _ z); apply cmra_preserving_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 394 `````` `````` Robbert Krebbers committed Feb 18, 2016 395 ``````Lemma cmra_included_dist_l n x1 x2 x1' : `````` Ralf Jung committed Feb 10, 2016 396 `````` x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2. `````` Robbert Krebbers committed Nov 11, 2015 397 ``````Proof. `````` Robbert Krebbers committed Feb 01, 2016 398 399 `````` intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using cmra_included_l. by rewrite Hx1 Hx2. `````` Robbert Krebbers committed Nov 11, 2015 400 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2016 401 `````` `````` Robbert Krebbers committed May 28, 2016 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 ``````(** ** Total core *) Section total_core. Context `{CMRATotal A}. Lemma cmra_core_l x : core x ⋅ x ≡ x. Proof. destruct (cmra_total x) as [cx Hcx]. by rewrite /core /= Hcx cmra_pcore_l. Qed. Lemma cmra_core_idemp x : core (core x) ≡ core x. Proof. destruct (cmra_total x) as [cx Hcx]. by rewrite /core /= Hcx cmra_pcore_idemp. Qed. Lemma cmra_core_preserving x y : x ≼ y → core x ≼ core y. Proof. intros; destruct (cmra_total x) as [cx Hcx]. destruct (cmra_pcore_preserving x y cx) as (cy&Hcy&?); auto. by rewrite /core /= Hcx Hcy. Qed. Global Instance cmra_core_ne n : Proper (dist n ==> dist n) (@core A _). Proof. intros x y Hxy. destruct (cmra_total x) as [cx Hcx]. by rewrite /core /= -Hxy Hcx. Qed. Global Instance cmra_core_proper : Proper ((≡) ==> (≡)) (@core A _). Proof. apply (ne_proper _). Qed. Lemma cmra_core_r x : x ⋅ core x ≡ x. Proof. by rewrite (comm _ x) cmra_core_l. Qed. Lemma cmra_core_core x : core x ⋅ core x ≡ core x. Proof. by rewrite -{2}(cmra_core_idemp x) cmra_core_r. Qed. Lemma cmra_core_validN n x : ✓{n} x → ✓{n} core x. Proof. rewrite -{1}(cmra_core_l x); apply cmra_validN_op_l. Qed. Lemma cmra_core_valid x : ✓ x → ✓ core x. Proof. rewrite -{1}(cmra_core_l x); apply cmra_valid_op_l. Qed. Lemma persistent_total x : Persistent x ↔ core x ≡ x. Proof. split; [intros; by rewrite /core /= (persistent x)|]. rewrite /Persistent /core /=. destruct (cmra_total x) as [? ->]. by constructor. Qed. Lemma persistent_core x `{!Persistent x} : core x ≡ x. Proof. by apply persistent_total. Qed. Global Instance cmra_core_persistent x : Persistent (core x). Proof. destruct (cmra_total x) as [cx Hcx]. rewrite /Persistent /core /= Hcx /=. eauto using cmra_pcore_idemp. Qed. Lemma cmra_included_core x : core x ≼ x. Proof. by exists x; rewrite cmra_core_l. Qed. Global Instance cmra_includedN_preorder n : PreOrder (@includedN A _ _ n). Proof. split; [|apply _]. by intros x; exists (core x); rewrite cmra_core_r. Qed. Global Instance cmra_included_preorder : PreOrder (@included A _ _). Proof. split; [|apply _]. by intros x; exists (core x); rewrite cmra_core_r. Qed. Lemma cmra_core_preservingN n x y : x ≼{n} y → core x ≼{n} core y. Proof. intros [z ->]. apply cmra_included_includedN, cmra_core_preserving, cmra_included_l. Qed. End total_core. `````` Robbert Krebbers committed Jan 16, 2016 470 ``````(** ** Timeless *) `````` Robbert Krebbers committed Feb 10, 2016 471 ``````Lemma cmra_timeless_included_l x y : Timeless x → ✓{0} y → x ≼{0} y → x ≼ y. `````` Robbert Krebbers committed Dec 11, 2015 472 473 ``````Proof. intros ?? [x' ?]. `````` Robbert Krebbers committed Feb 24, 2016 474 `````` destruct (cmra_extend 0 y x x') as ([z z']&Hy&Hz&Hz'); auto; simpl in *. `````` Robbert Krebbers committed Jan 13, 2016 475 `````` by exists z'; rewrite Hy (timeless x z). `````` Robbert Krebbers committed Dec 11, 2015 476 ``````Qed. `````` Robbert Krebbers committed Feb 10, 2016 477 ``````Lemma cmra_timeless_included_r n x y : Timeless y → x ≼{0} y → x ≼{n} y. `````` Robbert Krebbers committed Dec 11, 2015 478 ``````Proof. intros ? [x' ?]. exists x'. by apply equiv_dist, (timeless y). Qed. `````` Robbert Krebbers committed Jan 14, 2016 479 ``````Lemma cmra_op_timeless x1 x2 : `````` Robbert Krebbers committed Dec 11, 2015 480 `````` ✓ (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2). `````` Robbert Krebbers committed Nov 18, 2015 481 482 ``````Proof. intros ??? z Hz. `````` Robbert Krebbers committed Feb 24, 2016 483 `````` destruct (cmra_extend 0 z x1 x2) as ([y1 y2]&Hz'&?&?); auto; simpl in *. `````` Robbert Krebbers committed Feb 24, 2016 484 `````` { rewrite -?Hz. by apply cmra_valid_validN. } `````` Robbert Krebbers committed Jan 13, 2016 485 `````` by rewrite Hz' (timeless x1 y1) // (timeless x2 y2). `````` Robbert Krebbers committed Nov 18, 2015 486 ``````Qed. `````` Robbert Krebbers committed Nov 20, 2015 487 `````` `````` Robbert Krebbers committed Feb 24, 2016 488 489 490 491 492 493 494 495 ``````(** ** Discrete *) Lemma cmra_discrete_valid_iff `{CMRADiscrete A} n x : ✓ x ↔ ✓{n} x. Proof. split; first by rewrite cmra_valid_validN. eauto using cmra_discrete_valid, cmra_validN_le with lia. Qed. Lemma cmra_discrete_included_iff `{Discrete A} n x y : x ≼ y ↔ x ≼{n} y. Proof. `````` Robbert Krebbers committed Mar 11, 2016 496 `````` split; first by apply cmra_included_includedN. `````` Robbert Krebbers committed Feb 24, 2016 497 498 499 `````` intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l. Qed. `````` Robbert Krebbers committed Feb 11, 2016 500 ``````(** ** Local updates *) `````` Ralf Jung committed Feb 11, 2016 501 502 ``````Global Instance local_update_proper Lv (L : A → A) : LocalUpdate Lv L → Proper ((≡) ==> (≡)) L. `````` Robbert Krebbers committed Feb 11, 2016 503 504 ``````Proof. intros; apply (ne_proper _). Qed. `````` Ralf Jung committed Feb 11, 2016 505 506 ``````Lemma local_update L `{!LocalUpdate Lv L} x y : Lv x → ✓ (x ⋅ y) → L (x ⋅ y) ≡ L x ⋅ y. `````` Robbert Krebbers committed Feb 24, 2016 507 508 509 ``````Proof. by rewrite cmra_valid_validN equiv_dist=>?? n; apply (local_updateN L). Qed. `````` Robbert Krebbers committed Feb 11, 2016 510 511 `````` Global Instance local_update_op x : LocalUpdate (λ _, True) (op x). `````` Robbert Krebbers committed Feb 11, 2016 512 ``````Proof. split. apply _. by intros n y1 y2 _ _; rewrite assoc. Qed. `````` Robbert Krebbers committed Feb 11, 2016 513 `````` `````` Ralf Jung committed Feb 13, 2016 514 515 516 ``````Global Instance local_update_id : LocalUpdate (λ _, True) (@id A). Proof. split; auto with typeclass_instances. Qed. `````` Robbert Krebbers committed Feb 01, 2016 517 ``````(** ** Updates *) `````` Ralf Jung committed Feb 03, 2016 518 ``````Lemma cmra_update_updateP x y : x ~~> y ↔ x ~~>: (y =). `````` Robbert Krebbers committed May 28, 2016 519 ``````Proof. split=> Hup n z ?; eauto. destruct (Hup n z) as (?&<-&?); auto. Qed. `````` Ralf Jung committed Feb 03, 2016 520 ``````Lemma cmra_updateP_id (P : A → Prop) x : P x → x ~~>: P. `````` Robbert Krebbers committed May 28, 2016 521 ``````Proof. intros ? n mz ?; eauto. Qed. `````` Robbert Krebbers committed Feb 02, 2016 522 ``````Lemma cmra_updateP_compose (P Q : A → Prop) x : `````` Ralf Jung committed Feb 03, 2016 523 `````` x ~~>: P → (∀ y, P y → y ~~>: Q) → x ~~>: Q. `````` Robbert Krebbers committed May 28, 2016 524 ``````Proof. intros Hx Hy n mz ?. destruct (Hx n mz) as (y&?&?); naive_solver. Qed. `````` Robbert Krebbers committed Feb 08, 2016 525 526 527 ``````Lemma cmra_updateP_compose_l (Q : A → Prop) x y : x ~~> y → y ~~>: Q → x ~~>: Q. Proof. rewrite cmra_update_updateP. `````` Robbert Krebbers committed May 28, 2016 528 `````` intros; apply cmra_updateP_compose with (y =); naive_solver. `````` Robbert Krebbers committed Feb 08, 2016 529 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 530 531 ``````Lemma cmra_updateP_weaken (P Q : A → Prop) x : x ~~>: P → (∀ y, P y → Q y) → x ~~>: Q. `````` Robbert Krebbers committed Feb 02, 2016 532 ``````Proof. eauto using cmra_updateP_compose, cmra_updateP_id. Qed. `````` Robbert Krebbers committed May 28, 2016 533 534 535 536 537 538 539 ``````Global Instance cmra_update_preorder : PreOrder (@cmra_update A). Proof. split. - intros x. by apply cmra_update_updateP, cmra_updateP_id. - intros x y z. rewrite !cmra_update_updateP. eauto using cmra_updateP_compose with subst. Qed. `````` Robbert Krebbers committed Feb 02, 2016 540 `````` `````` Robbert Krebbers committed Feb 02, 2016 541 ``````Lemma cmra_updateP_op (P1 P2 Q : A → Prop) x1 x2 : `````` Robbert Krebbers committed May 28, 2016 542 543 `````` x1 ~~>: P1 → x2 ~~>: P2 → (∀ y1 y2, P1 y1 → P2 y2 → Q (y1 ⋅ y2)) → x1 ⋅ x2 ~~>: Q. `````` Robbert Krebbers committed Feb 02, 2016 544 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 545 546 547 548 549 550 `````` intros Hx1 Hx2 Hy n mz ?. destruct (Hx1 n (Some (x2 ⋅? mz))) as (y1&?&?). { by rewrite /= -cmra_opM_assoc. } destruct (Hx2 n (Some (y1 ⋅? mz))) as (y2&?&?). { by rewrite /= -cmra_opM_assoc (comm _ x2) cmra_opM_assoc. } exists (y1 ⋅ y2); split; last rewrite (comm _ y1) cmra_opM_assoc; auto. `````` Robbert Krebbers committed Feb 02, 2016 551 ``````Qed. `````` Robbert Krebbers committed Feb 02, 2016 552 ``````Lemma cmra_updateP_op' (P1 P2 : A → Prop) x1 x2 : `````` Robbert Krebbers committed May 28, 2016 553 554 `````` x1 ~~>: P1 → x2 ~~>: P2 → x1 ⋅ x2 ~~>: λ y, ∃ y1 y2, y = y1 ⋅ y2 ∧ P1 y1 ∧ P2 y2. `````` Robbert Krebbers committed Feb 02, 2016 555 ``````Proof. eauto 10 using cmra_updateP_op. Qed. `````` Ralf Jung committed Feb 03, 2016 556 ``````Lemma cmra_update_op x1 x2 y1 y2 : x1 ~~> y1 → x2 ~~> y2 → x1 ⋅ x2 ~~> y1 ⋅ y2. `````` Robbert Krebbers committed Feb 02, 2016 557 ``````Proof. `````` Robbert Krebbers committed Feb 02, 2016 558 `````` rewrite !cmra_update_updateP; eauto using cmra_updateP_op with congruence. `````` Robbert Krebbers committed Feb 02, 2016 559 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 `````` Section total_updates. Context `{CMRATotal A}. Lemma cmra_total_updateP x (P : A → Prop) : x ~~>: P ↔ ∀ n z, ✓{n} (x ⋅ z) → ∃ y, P y ∧ ✓{n} (y ⋅ z). Proof. split=> Hup; [intros n z; apply (Hup n (Some z))|]. intros n [z|] ?; simpl; [by apply Hup|]. destruct (Hup n (core x)) as (y&?&?); first by rewrite cmra_core_r. eauto using cmra_validN_op_l. Qed. Lemma cmra_total_update x y : x ~~> y ↔ ∀ n z, ✓{n} (x ⋅ z) → ✓{n} (y ⋅ z). Proof. rewrite cmra_update_updateP cmra_total_updateP. naive_solver. Qed. Context `{CMRADiscrete A}. Lemma cmra_discrete_updateP (x : A) (P : A → Prop) : x ~~>: P ↔ ∀ z, ✓ (x ⋅ z) → ∃ y, P y ∧ ✓ (y ⋅ z). Proof. rewrite cmra_total_updateP; setoid_rewrite <-cmra_discrete_valid_iff. naive_solver eauto using 0. Qed. Lemma cmra_discrete_update `{CMRADiscrete A} (x y : A) : x ~~> y ↔ ∀ z, ✓ (x ⋅ z) → ✓ (y ⋅ z). Proof. rewrite cmra_total_update; setoid_rewrite <-cmra_discrete_valid_iff. naive_solver eauto using 0. Qed. End total_updates. `````` Robbert Krebbers committed Nov 11, 2015 590 591 ``````End cmra. `````` Robbert Krebbers committed May 27, 2016 592 593 ``````(** * Properties about CMRAs with a unit element **) Section ucmra. `````` Robbert Krebbers committed May 28, 2016 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 `````` Context {A : ucmraT}. Implicit Types x y z : A. Global Instance ucmra_unit_inhabited : Inhabited A := populate ∅. Lemma ucmra_unit_validN n : ✓{n} (∅:A). Proof. apply cmra_valid_validN, ucmra_unit_valid. Qed. Lemma ucmra_unit_leastN n x : ∅ ≼{n} x. Proof. by exists x; rewrite left_id. Qed. Lemma ucmra_unit_least x : ∅ ≼ x. Proof. by exists x; rewrite left_id. Qed. Global Instance ucmra_unit_right_id : RightId (≡) ∅ (@op A _). Proof. by intros x; rewrite (comm op) left_id. Qed. Global Instance ucmra_unit_persistent : Persistent (∅:A). Proof. apply ucmra_pcore_unit. Qed. Global Instance cmra_unit_total : CMRATotal A. Proof. intros x. destruct (cmra_pcore_preserving' ∅ x ∅) as (cx&->&?); eauto using ucmra_unit_least, (persistent ∅). Qed. `````` Robbert Krebbers committed May 27, 2016 615 `````` `````` Robbert Krebbers committed May 28, 2016 616 617 618 619 620 621 `````` Lemma ucmra_update_unit x : x ~~> ∅. Proof. apply cmra_total_update=> n z. rewrite left_id; apply cmra_validN_op_r. Qed. Lemma ucmra_update_unit_alt y : ∅ ~~> y ↔ ∀ x, x ~~> y. Proof. split; [intros; trans ∅|]; auto using ucmra_update_unit. Qed. `````` Robbert Krebbers committed May 27, 2016 622 ``````End ucmra. `````` Robbert Krebbers committed May 28, 2016 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 ``````Hint Immediate cmra_unit_total. (** * Constructing a CMRA with total core *) Section cmra_total. Context A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A}. Context (total : ∀ x, is_Some (pcore x)). Context (op_ne : ∀ n (x : A), Proper (dist n ==> dist n) (op x)). Context (core_ne : ∀ n, Proper (dist n ==> dist n) (@core A _)). Context (validN_ne : ∀ n, Proper (dist n ==> impl) (@validN A _ n)). Context (valid_validN : ∀ (x : A), ✓ x ↔ ∀ n, ✓{n} x). Context (validN_S : ∀ n (x : A), ✓{S n} x → ✓{n} x). Context (op_assoc : Assoc (≡) (@op A _)). Context (op_comm : Comm (≡) (@op A _)). Context (core_l : ∀ x : A, core x ⋅ x ≡ x). Context (core_idemp : ∀ x : A, core (core x) ≡ core x). Context (core_preserving : ∀ x y : A, x ≼ y → core x ≼ core y). Context (validN_op_l : ∀ n (x y : A), ✓{n} (x ⋅ y) → ✓{n} x). Context (extend : ∀ n (x y1 y2 : A), ✓{n} x → x ≡{n}≡ y1 ⋅ y2 → { z | x ≡ z.1 ⋅ z.2 ∧ z.1 ≡{n}≡ y1 ∧ z.2 ≡{n}≡ y2 }). Lemma cmra_total_mixin : CMRAMixin A. Proof. split; auto. - intros n x y ? Hcx%core_ne Hx; move: Hcx. rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto. - intros x cx Hcx. move: (core_l x). by rewrite /core /= Hcx. - intros x cx Hcx. move: (core_idemp x). rewrite /core /= Hcx /=. case (total cx)=>[ccx ->]; by constructor. - intros x y cx Hxy%core_preserving Hx. move: Hxy. rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto. Qed. End cmra_total. `````` Robbert Krebbers committed May 27, 2016 655 `````` `````` Robbert Krebbers committed Feb 01, 2016 656 ``````(** * Properties about monotone functions *) `````` Robbert Krebbers committed Jan 14, 2016 657 ``````Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A). `````` Robbert Krebbers committed Feb 26, 2016 658 ``````Proof. repeat split; by try apply _. Qed. `````` Robbert Krebbers committed Feb 01, 2016 659 660 ``````Instance cmra_monotone_compose {A B C : cmraT} (f : A → B) (g : B → C) : CMRAMonotone f → CMRAMonotone g → CMRAMonotone (g ∘ f). `````` Robbert Krebbers committed Nov 20, 2015 661 662 ``````Proof. split. `````` Robbert Krebbers committed Feb 26, 2016 663 `````` - apply _. `````` Robbert Krebbers committed Feb 17, 2016 664 `````` - move=> n x Hx /=. by apply validN_preserving, validN_preserving. `````` Robbert Krebbers committed Feb 26, 2016 665 `````` - move=> x y Hxy /=. by apply included_preserving, included_preserving. `````` Robbert Krebbers committed Nov 20, 2015 666 ``````Qed. `````` Robbert Krebbers committed Nov 16, 2015 667 `````` `````` Robbert Krebbers committed Feb 01, 2016 668 669 ``````Section cmra_monotone. Context {A B : cmraT} (f : A → B) `{!CMRAMonotone f}. `````` Robbert Krebbers committed Feb 26, 2016 670 671 `````` Global Instance cmra_monotone_proper : Proper ((≡) ==> (≡)) f := ne_proper _. Lemma includedN_preserving n x y : x ≼{n} y → f x ≼{n} f y. `````` Robbert Krebbers committed Feb 01, 2016 672 `````` Proof. `````` Robbert Krebbers committed Feb 26, 2016 673 `````` intros [z ->]. `````` Robbert Krebbers committed Feb 26, 2016 674 `````` apply cmra_included_includedN, (included_preserving f), cmra_included_l. `````` Robbert Krebbers committed Feb 01, 2016 675 `````` Qed. `````` Robbert Krebbers committed Feb 11, 2016 676 `````` Lemma valid_preserving x : ✓ x → ✓ f x. `````` Robbert Krebbers committed Feb 01, 2016 677 678 679 `````` Proof. rewrite !cmra_valid_validN; eauto using validN_preserving. Qed. End cmra_monotone. `````` Robbert Krebbers committed May 25, 2016 680 681 ``````(** Functors *) Structure rFunctor := RFunctor { `````` Robbert Krebbers committed May 27, 2016 682 `````` rFunctor_car : cofeT → cofeT → cmraT; `````` Robbert Krebbers committed May 25, 2016 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 `````` rFunctor_map {A1 A2 B1 B2} : ((A2 -n> A1) * (B1 -n> B2)) → rFunctor_car A1 B1 -n> rFunctor_car A2 B2; rFunctor_ne A1 A2 B1 B2 n : Proper (dist n ==> dist n) (@rFunctor_map A1 A2 B1 B2); rFunctor_id {A B} (x : rFunctor_car A B) : rFunctor_map (cid,cid) x ≡ x; rFunctor_compose {A1 A2 A3 B1 B2 B3} (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x : rFunctor_map (f◎g, g'◎f') x ≡ rFunctor_map (g,g') (rFunctor_map (f,f') x); rFunctor_mono {A1 A2 B1 B2} (fg : (A2 -n> A1) * (B1 -n> B2)) : CMRAMonotone (rFunctor_map fg) }. Existing Instances rFunctor_ne rFunctor_mono. Instance: Params (@rFunctor_map) 5. Class rFunctorContractive (F : rFunctor) := rFunctor_contractive A1 A2 B1 B2 :> Contractive (@rFunctor_map F A1 A2 B1 B2). Definition rFunctor_diag (F: rFunctor) (A: cofeT) : cmraT := rFunctor_car F A A. Coercion rFunctor_diag : rFunctor >-> Funclass. Program Definition constRF (B : cmraT) : rFunctor := {| rFunctor_car A1 A2 := B; rFunctor_map A1 A2 B1 B2 f := cid |}. Solve Obligations with done. Instance constRF_contractive B : rFunctorContractive (constRF B). Proof. rewrite /rFunctorContractive; apply _. Qed. `````` Robbert Krebbers committed May 27, 2016 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 ``````Structure urFunctor := URFunctor { urFunctor_car : cofeT → cofeT → ucmraT; urFunctor_map {A1 A2 B1 B2} : ((A2 -n> A1) * (B1 -n> B2)) → urFunctor_car A1 B1 -n> urFunctor_car A2 B2; urFunctor_ne A1 A2 B1 B2 n : Proper (dist n ==> dist n) (@urFunctor_map A1 A2 B1 B2); urFunctor_id {A B} (x : urFunctor_car A B) : urFunctor_map (cid,cid) x ≡ x; urFunctor_compose {A1 A2 A3 B1 B2 B3} (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x : urFunctor_map (f◎g, g'◎f') x ≡ urFunctor_map (g,g') (urFunctor_map (f,f') x); urFunctor_mono {A1 A2 B1 B2} (fg : (A2 -n> A1) * (B1 -n> B2)) : CMRAMonotone (urFunctor_map fg) }. Existing Instances urFunctor_ne urFunctor_mono. Instance: Params (@urFunctor_map) 5. Class urFunctorContractive (F : urFunctor) := urFunctor_contractive A1 A2 B1 B2 :> Contractive (@urFunctor_map F A1 A2 B1 B2). Definition urFunctor_diag (F: urFunctor) (A: cofeT) : ucmraT := urFunctor_car F A A. Coercion urFunctor_diag : urFunctor >-> Funclass. Program Definition constURF (B : ucmraT) : urFunctor := {| urFunctor_car A1 A2 := B; urFunctor_map A1 A2 B1 B2 f := cid |}. Solve Obligations with done. Instance constURF_contractive B : urFunctorContractive (constURF B). Proof. rewrite /urFunctorContractive; apply _. Qed. `````` Robbert Krebbers committed Feb 08, 2016 739 740 741 742 743 744 745 746 747 748 749 750 751 ``````(** * Transporting a CMRA equality *) Definition cmra_transport {A B : cmraT} (H : A = B) (x : A) : B := eq_rect A id x _ H. Section cmra_transport. Context {A B : cmraT} (H : A = B). Notation T := (cmra_transport H). Global Instance cmra_transport_ne n : Proper (dist n ==> dist n) T. Proof. by intros ???; destruct H. Qed. Global Instance cmra_transport_proper : Proper ((≡) ==> (≡)) T. Proof. by intros ???; destruct H. Qed. Lemma cmra_transport_op x y : T (x ⋅ y) = T x ⋅ T y. Proof. by destruct H. Qed. `````` Ralf Jung committed Mar 08, 2016 752 `````` Lemma cmra_transport_core x : T (core x) = core (T x). `````` Robbert Krebbers committed Feb 08, 2016 753 `````` Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 754 `````` Lemma cmra_transport_validN n x : ✓{n} T x ↔ ✓{n} x. `````` Robbert Krebbers committed Feb 08, 2016 755 `````` Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 756 `````` Lemma cmra_transport_valid x : ✓ T x ↔ ✓ x. `````` Robbert Krebbers committed Feb 08, 2016 757 758 759 `````` Proof. by destruct H. Qed. Global Instance cmra_transport_timeless x : Timeless x → Timeless (T x). Proof. by destruct H. Qed. `````` Robbert Krebbers committed Mar 15, 2016 760 761 `````` Global Instance cmra_transport_persistent x : Persistent x → Persistent (T x). Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 08, 2016 762 763 764 765 766 767 768 769 `````` Lemma cmra_transport_updateP (P : A → Prop) (Q : B → Prop) x : x ~~>: P → (∀ y, P y → Q (T y)) → T x ~~>: Q. Proof. destruct H; eauto using cmra_updateP_weaken. Qed. Lemma cmra_transport_updateP' (P : A → Prop) x : x ~~>: P → T x ~~>: λ y, ∃ y', y = cmra_transport H y' ∧ P y'. Proof. eauto using cmra_transport_updateP. Qed. End cmra_transport. `````` Robbert Krebbers committed Feb 01, 2016 770 771 ``````(** * Instances *) (** ** Discrete CMRA *) `````` Robbert Krebbers committed May 28, 2016 772 ``````Record RAMixin A `{Equiv A, PCore A, Op A, Valid A} := { `````` Robbert Krebbers committed Feb 01, 2016 773 `````` (* setoids *) `````` Robbert Krebbers committed May 28, 2016 774 775 776 777 `````` ra_op_proper (x : A) : Proper ((≡) ==> (≡)) (op x); ra_core_proper x y cx : x ≡ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≡ cy; ra_validN_proper : Proper ((≡) ==> impl) valid; `````` Robbert Krebbers committed Feb 01, 2016 778 `````` (* monoid *) `````` Robbert Krebbers committed May 25, 2016 779 780 `````` ra_assoc : Assoc (≡) (⋅); ra_comm : Comm (≡) (⋅); `````` Robbert Krebbers committed May 28, 2016 781 782 783 784 `````` ra_pcore_l x cx : pcore x = Some cx → cx ⋅ x ≡ x; ra_pcore_idemp x cx : pcore x = Some cx → pcore cx ≡ Some cx; ra_pcore_preserving x y cx : x ≼ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy; `````` Robbert Krebbers committed Mar 11, 2016 785 `````` ra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x `````` Robbert Krebbers committed Feb 01, 2016 786 787 ``````}. `````` Robbert Krebbers committed Nov 16, 2015 788 ``````Section discrete. `````` Robbert Krebbers committed May 28, 2016 789 `````` Context `{Equiv A, PCore A, Op A, Valid A, @Equivalence A (≡)}. `````` Robbert Krebbers committed May 25, 2016 790 791 `````` Context (ra_mix : RAMixin A). Existing Instances discrete_dist discrete_compl. `````` Robbert Krebbers committed Feb 01, 2016 792 `````` `````` Robbert Krebbers committed Feb 10, 2016 793 `````` Instance discrete_validN : ValidN A := λ n x, ✓ x. `````` Robbert Krebbers committed Jan 14, 2016 794 `````` Definition discrete_cmra_mixin : CMRAMixin A. `````` Robbert Krebbers committed Nov 16, 2015 795 `````` Proof. `````` Robbert Krebbers committed May 25, 2016 796 `````` destruct ra_mix; split; try done. `````` Robbert Krebbers committed Feb 24, 2016 797 `````` - intros x; split; first done. by move=> /(_ 0). `````` Robbert Krebbers committed May 25, 2016 798 `````` - intros n x y1 y2 ??; by exists (y1,y2). `````` Robbert Krebbers committed Nov 16, 2015 799 800 801 `````` Qed. End discrete. `````` Robbert Krebbers committed May 25, 2016 802 803 804 805 806 ``````Notation discreteR A ra_mix := (CMRAT A discrete_cofe_mixin (discrete_cmra_mixin ra_mix)). Notation discreteLeibnizR A ra_mix := (CMRAT A (@discrete_cofe_mixin _ equivL _) (discrete_cmra_mixin ra_mix)). `````` Robbert Krebbers committed May 28, 2016 807 ``````Global Instance discrete_cmra_discrete `{Equiv A, PCore A, Op A, Valid A, `````` Robbert Krebbers committed May 25, 2016 808 809 810 `````` @Equivalence A (≡)} (ra_mix : RAMixin A) : CMRADiscrete (discreteR A ra_mix). Proof. split. apply _. done. Qed. `````` Robbert Krebbers committed May 28, 2016 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 ``````Section ra_total. Context A `{Equiv A, PCore A, Op A, Valid A}. Context (total : ∀ x, is_Some (pcore x)). Context (op_proper : ∀ (x : A), Proper ((≡) ==> (≡)) (op x)). Context (core_proper: Proper ((≡) ==> (≡)) (@core A _)). Context (valid_proper : Proper ((≡) ==> impl) (@valid A _)). Context (op_assoc : Assoc (≡) (@op A _)). Context (op_comm : Comm (≡) (@op A _)). Context (core_l : ∀ x : A, core x ⋅ x ≡ x). Context (core_idemp : ∀ x : A, core (core x) ≡ core x). Context (core_preserving : ∀ x y : A, x ≼ y → core x ≼ core y). Context (valid_op_l : ∀ x y : A, ✓ (x ⋅ y) → ✓ x). Lemma ra_total_mixin : RAMixin A. Proof. split; auto. - intros x y ? Hcx%core_proper Hx; move: Hcx. rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto. - intros x cx Hcx. move: (core_l x). by rewrite /core /= Hcx. - intros x cx Hcx. move: (core_idemp x). rewrite /core /= Hcx /=. case (total cx)=>[ccx ->]; by constructor. - intros x y cx Hxy%core_preserving Hx. move: Hxy. rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto. Qed. End ra_total. `````` Robbert Krebbers committed Feb 01, 2016 836 837 838 ``````(** ** CMRA for the unit type *) Section unit. Instance unit_valid : Valid () := λ x, True. `````` Robbert Krebbers committed May 25, 2016 839 `````` Instance unit_validN : ValidN () := λ n x, True. `````` Robbert Krebbers committed May 28, 2016 840 `````` Instance unit_pcore : PCore () := λ x, Some x. `````` Robbert Krebbers committed Feb 01, 2016 841 `````` Instance unit_op : Op () := λ x y, (). `````` Robbert Krebbers committed May 27, 2016 842 `````` Lemma unit_cmra_mixin : CMRAMixin (). `````` Robbert Krebbers committed May 28, 2016 843 `````` Proof. apply cmra_total_mixin; try done. eauto. by exists ((),()). Qed. `````` Robbert Krebbers committed May 25, 2016 844 `````` Canonical Structure unitR : cmraT := CMRAT () unit_cofe_mixin unit_cmra_mixin. `````` Robbert Krebbers committed May 27, 2016 845 846 847 848 849 850 851 `````` Instance unit_empty : Empty () := (). Lemma unit_ucmra_mixin : UCMRAMixin (). Proof. done. Qed. Canonical Structure unitUR : ucmraT := UCMRAT () unit_cofe_mixin unit_cmra_mixin unit_ucmra_mixin. `````` Robbert Krebbers committed Mar 01, 2016 852 `````` Global Instance unit_cmra_discrete : CMRADiscrete unitR. `````` Robbert Krebbers committed May 25, 2016 853 `````` Proof. done. Qed. `````` Robbert Krebbers committed Mar 15, 2016 854 `````` Global Instance unit_persistent (x : ()) : Persistent x. `````` Robbert Krebbers committed May 28, 2016 855 `````` Proof. by constructor. Qed. `````` Robbert Krebbers committed Feb 01, 2016 856 ``````End unit. `````` Ralf Jung committed Jan 19, 2016 857 `````` `````` Robbert Krebbers committed Feb 01, 2016 858 ``````(** ** Product *) `````` Robbert Krebbers committed Jan 14, 2016 859 860 ``````Section prod. Context {A B : cmraT}. `````` Robbert Krebbers committed May 28, 2016 861 862 863 `````` Local Arguments pcore _ _ !_ /. Local Arguments cmra_pcore _ !_/. `````` Robbert Krebbers committed Jan 14, 2016 864 `````` Instance prod_op : Op (A * B) := λ x y, (x.1 ⋅ y.1, x.2 ⋅ y.2). `````` Robbert Krebbers committed May 28, 2016 865 866 867 `````` Instance prod_pcore : PCore (A * B) := λ x, c1 ← pcore (x.1); c2 ← pcore (x.2); Some (c1, c2). Arguments prod_pcore !_ /. `````` Robbert Krebbers committed Feb 24, 2016 868 `` Instance prod_valid : Valid (A * B) := λ x, ✓ x.1 <``