cmra.v 44 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 `````` 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. `````` Ralf Jung committed Jun 23, 2016 11 12 13 14 15 ``````(* The inclusion quantifies over [A], not [option A]. This means we do not get reflexivity. However, if we used [option A], the following would no longer hold: x ≼ y ↔ x.1 ≼ y.1 ∧ x.2 ≼ y.2 *) `````` Robbert Krebbers committed Feb 01, 2016 16 17 18 ``````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 19 ``````Hint Extern 0 (_ ≼ _) => reflexivity. `````` Robbert Krebbers committed Feb 01, 2016 20 21 ``````Instance: Params (@included) 3. `````` Robbert Krebbers committed Nov 11, 2015 22 23 ``````Class ValidN (A : Type) := validN : nat → A → Prop. Instance: Params (@validN) 3. `````` Robbert Krebbers committed Feb 11, 2016 24 ``````Notation "✓{ n } x" := (validN n x) `````` Robbert Krebbers committed Feb 19, 2016 25 `````` (at level 20, n at next level, format "✓{ n } x"). `````` Robbert Krebbers committed Nov 11, 2015 26 `````` `````` Robbert Krebbers committed Feb 01, 2016 27 28 ``````Class Valid (A : Type) := valid : A → Prop. Instance: Params (@valid) 2. `````` Robbert Krebbers committed Feb 11, 2016 29 ``````Notation "✓ x" := (valid x) (at level 20) : C_scope. `````` Robbert Krebbers committed Feb 01, 2016 30 `````` `````` Ralf Jung committed Feb 10, 2016 31 ``````Definition includedN `{Dist A, Op A} (n : nat) (x y : A) := ∃ z, y ≡{n}≡ x ⋅ z. `````` Robbert Krebbers committed Nov 20, 2015 32 ``````Notation "x ≼{ n } y" := (includedN n x y) `````` Robbert Krebbers committed Feb 19, 2016 33 `````` (at level 70, n at next level, format "x ≼{ n } y") : C_scope. `````` Robbert Krebbers committed Nov 20, 2015 34 ``````Instance: Params (@includedN) 4. `````` Robbert Krebbers committed Feb 13, 2016 35 ``````Hint Extern 0 (_ ≼{_} _) => reflexivity. `````` Robbert Krebbers committed Nov 20, 2015 36 `````` `````` Robbert Krebbers committed May 28, 2016 37 ``````Record CMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A} := { `````` Robbert Krebbers committed Nov 11, 2015 38 `````` (* setoids *) `````` Robbert Krebbers committed Jan 14, 2016 39 `````` mixin_cmra_op_ne n (x : A) : Proper (dist n ==> dist n) (op x); `````` Robbert Krebbers committed May 28, 2016 40 41 `````` 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 42 `````` mixin_cmra_validN_ne n : Proper (dist n ==> impl) (validN n); `````` Robbert Krebbers committed Nov 11, 2015 43 `````` (* valid *) `````` Robbert Krebbers committed Feb 24, 2016 44 `````` mixin_cmra_valid_validN x : ✓ x ↔ ∀ n, ✓{n} x; `````` Robbert Krebbers committed Feb 01, 2016 45 `````` mixin_cmra_validN_S n x : ✓{S n} x → ✓{n} x; `````` Robbert Krebbers committed Nov 11, 2015 46 `````` (* monoid *) `````` Robbert Krebbers committed Feb 11, 2016 47 48 `````` mixin_cmra_assoc : Assoc (≡) (⋅); mixin_cmra_comm : Comm (≡) (⋅); `````` Robbert Krebbers committed May 28, 2016 49 50 51 52 `````` 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 53 `````` mixin_cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x; `````` Robbert Krebbers committed Feb 24, 2016 54 55 56 `````` 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 57 ``````}. `````` Robbert Krebbers committed Nov 22, 2015 58 `````` `````` Robbert Krebbers committed Nov 11, 2015 59 ``````(** Bundeled version *) `````` Robbert Krebbers committed Jun 15, 2016 60 ``````Structure cmraT := CMRAT' { `````` Robbert Krebbers committed Nov 11, 2015 61 62 63 64 `````` cmra_car :> Type; cmra_equiv : Equiv cmra_car; cmra_dist : Dist cmra_car; cmra_compl : Compl cmra_car; `````` Robbert Krebbers committed May 28, 2016 65 `````` cmra_pcore : PCore cmra_car; `````` Robbert Krebbers committed Nov 11, 2015 66 `````` cmra_op : Op cmra_car; `````` Robbert Krebbers committed Feb 24, 2016 67 `````` cmra_valid : Valid cmra_car; `````` Robbert Krebbers committed Nov 11, 2015 68 `````` cmra_validN : ValidN cmra_car; `````` Robbert Krebbers committed Jan 14, 2016 69 `````` cmra_cofe_mixin : CofeMixin cmra_car; `````` Robbert Krebbers committed Jun 15, 2016 70 `````` cmra_mixin : CMRAMixin cmra_car; `````` Robbert Krebbers committed Jun 15, 2016 71 `````` _ : Type `````` Robbert Krebbers committed Nov 11, 2015 72 ``````}. `````` Robbert Krebbers committed Jun 15, 2016 73 74 ``````Arguments CMRAT' _ {_ _ _ _ _ _ _} _ _ _. Notation CMRAT A m m' := (CMRAT' A m m' A). `````` Robbert Krebbers committed Jan 14, 2016 75 76 77 78 ``````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 79 ``````Arguments cmra_pcore : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 80 ``````Arguments cmra_op : simpl never. `````` Robbert Krebbers committed Feb 24, 2016 81 ``````Arguments cmra_valid : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 82 83 84 ``````Arguments cmra_validN : simpl never. Arguments cmra_cofe_mixin : simpl never. Arguments cmra_mixin : simpl never. `````` Robbert Krebbers committed Nov 11, 2015 85 ``````Add Printing Constructor cmraT. `````` Robbert Krebbers committed Jun 14, 2016 86 87 88 89 ``````Hint Extern 0 (PCore _) => eapply (@cmra_pcore _) : typeclass_instances. Hint Extern 0 (Op _) => eapply (@cmra_op _) : typeclass_instances. Hint Extern 0 (Valid _) => eapply (@cmra_valid _) : typeclass_instances. Hint Extern 0 (ValidN _) => eapply (@cmra_validN _) : typeclass_instances. `````` Robbert Krebbers committed May 25, 2016 90 ``````Coercion cmra_cofeC (A : cmraT) : cofeT := CofeT A (cmra_cofe_mixin A). `````` Robbert Krebbers committed Nov 11, 2015 91 92 ``````Canonical Structure cmra_cofeC. `````` Robbert Krebbers committed Jan 14, 2016 93 94 95 96 97 98 ``````(** 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 99 100 101 `````` 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 102 103 `````` 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 104 105 `````` 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 106 107 `````` 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 108 109 110 111 `````` 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 112 113 114 115 116 117 118 `````` 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 119 120 `````` 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 121 `````` Lemma cmra_extend n x y1 y2 : `````` Ralf Jung committed Feb 10, 2016 122 123 `````` ✓{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 124 `````` Proof. apply (mixin_cmra_extend _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Jan 14, 2016 125 126 ``````End cmra_mixin. `````` Robbert Krebbers committed May 28, 2016 127 128 129 130 131 132 133 134 ``````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 {_} _ {_}. `````` Jacques-Henri Jourdan committed May 31, 2016 135 ``````(** * Exclusive elements (i.e., elements that cannot have a frame). *) `````` Robbert Krebbers committed Jun 16, 2016 136 137 ``````Class Exclusive {A : cmraT} (x : A) := exclusive0_l y : ✓{0} (x ⋅ y) → False. Arguments exclusive0_l {_} _ {_} _ _. `````` Jacques-Henri Jourdan committed May 31, 2016 138 `````` `````` Robbert Krebbers committed May 28, 2016 139 140 141 142 143 144 145 146 147 148 149 ``````(** * 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 150 ``````(** * CMRAs with a unit element *) `````` Robbert Krebbers committed Feb 01, 2016 151 ``````(** We use the notation ∅ because for most instances (maps, sets, etc) the `````` Ralf Jung committed Mar 08, 2016 152 ```````empty' element is the unit. *) `````` Robbert Krebbers committed May 28, 2016 153 ``````Record UCMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, Empty A} := { `````` Robbert Krebbers committed May 27, 2016 154 155 `````` mixin_ucmra_unit_valid : ✓ ∅; mixin_ucmra_unit_left_id : LeftId (≡) ∅ (⋅); `````` Robbert Krebbers committed May 28, 2016 156 157 `````` mixin_ucmra_unit_timeless : Timeless ∅; mixin_ucmra_pcore_unit : pcore ∅ ≡ Some ∅ `````` Robbert Krebbers committed Feb 01, 2016 158 ``````}. `````` Robbert Krebbers committed May 27, 2016 159 `````` `````` Robbert Krebbers committed Jun 15, 2016 160 ``````Structure ucmraT := UCMRAT' { `````` Robbert Krebbers committed May 27, 2016 161 162 163 164 `````` ucmra_car :> Type; ucmra_equiv : Equiv ucmra_car; ucmra_dist : Dist ucmra_car; ucmra_compl : Compl ucmra_car; `````` Robbert Krebbers committed May 28, 2016 165 `````` ucmra_pcore : PCore ucmra_car; `````` Robbert Krebbers committed May 27, 2016 166 167 168 169 170 171 `````` 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; `````` Robbert Krebbers committed Jun 15, 2016 172 `````` ucmra_mixin : UCMRAMixin ucmra_car; `````` Robbert Krebbers committed Jun 15, 2016 173 `````` _ : Type; `````` Robbert Krebbers committed May 27, 2016 174 ``````}. `````` Robbert Krebbers committed Jun 15, 2016 175 176 ``````Arguments UCMRAT' _ {_ _ _ _ _ _ _ _} _ _ _ _. Notation UCMRAT A m m' m'' := (UCMRAT' A m m' m'' A). `````` Robbert Krebbers committed May 27, 2016 177 178 179 180 ``````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 181 ``````Arguments ucmra_pcore : simpl never. `````` Robbert Krebbers committed May 27, 2016 182 183 184 185 186 187 188 ``````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. `````` Robbert Krebbers committed Jun 14, 2016 189 ``````Hint Extern 0 (Empty _) => eapply (@ucmra_empty _) : typeclass_instances. `````` Robbert Krebbers committed May 27, 2016 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 ``````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 206 207 `````` Lemma ucmra_pcore_unit : pcore (∅:A) ≡ Some ∅. Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin A)). Qed. `````` Robbert Krebbers committed May 27, 2016 208 ``````End ucmra_mixin. `````` Robbert Krebbers committed Jan 14, 2016 209 `````` `````` Robbert Krebbers committed Feb 24, 2016 210 ``````(** * Discrete CMRAs *) `````` Robbert Krebbers committed Feb 26, 2016 211 ``````Class CMRADiscrete (A : cmraT) := { `````` Robbert Krebbers committed Feb 24, 2016 212 213 214 215 `````` cmra_discrete :> Discrete A; cmra_discrete_valid (x : A) : ✓{0} x → ✓ x }. `````` Robbert Krebbers committed Jan 16, 2016 216 ``````(** * Morphisms *) `````` Robbert Krebbers committed Jan 14, 2016 217 ``````Class CMRAMonotone {A B : cmraT} (f : A → B) := { `````` Robbert Krebbers committed Feb 26, 2016 218 219 220 `````` 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 221 ``````}. `````` Robbert Krebbers committed Feb 26, 2016 222 223 ``````Arguments validN_preserving {_ _} _ {_} _ _ _. Arguments included_preserving {_ _} _ {_} _ _ _. `````` Robbert Krebbers committed Jan 14, 2016 224 `````` `````` Robbert Krebbers committed Jan 16, 2016 225 ``````(** * Properties **) `````` Robbert Krebbers committed Nov 11, 2015 226 ``````Section cmra. `````` Robbert Krebbers committed Jan 14, 2016 227 ``````Context {A : cmraT}. `````` Robbert Krebbers committed Nov 11, 2015 228 ``````Implicit Types x y z : A. `````` Robbert Krebbers committed Feb 01, 2016 229 ``````Implicit Types xs ys zs : list A. `````` Robbert Krebbers committed Nov 11, 2015 230 `````` `````` Robbert Krebbers committed Feb 01, 2016 231 ``````(** ** Setoids *) `````` Robbert Krebbers committed May 28, 2016 232 233 234 235 236 237 238 239 240 ``````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 241 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 242 243 244 `````` 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 245 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 246 247 248 249 ``````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 250 251 252 253 254 255 256 257 ``````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 258 259 260 261 ``````Proof. intros x y Hxy; rewrite !cmra_valid_validN. by split=> ? n; [rewrite -Hxy|rewrite Hxy]. Qed. `````` Robbert Krebbers committed Feb 01, 2016 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 ``````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 280 281 282 283 ``````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 01, 2016 284 `````` `````` Robbert Krebbers committed May 28, 2016 285 286 287 288 ``````(** ** 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 289 ``````(** ** Validity *) `````` Robbert Krebbers committed Feb 18, 2016 290 ``````Lemma cmra_validN_le n n' x : ✓{n} x → n' ≤ n → ✓{n'} x. `````` Robbert Krebbers committed Feb 01, 2016 291 292 293 ``````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 294 ``````Lemma cmra_validN_op_r n x y : ✓{n} (x ⋅ y) → ✓{n} y. `````` Robbert Krebbers committed Feb 11, 2016 295 ``````Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 296 297 298 ``````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 299 ``````(** ** Core *) `````` Robbert Krebbers committed May 28, 2016 300 301 302 303 304 305 306 307 ``````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. `````` Robbert Krebbers committed May 30, 2016 308 309 310 311 ``````Lemma cmra_pcore_dup x cx : pcore x = Some cx → cx ≡ cx ⋅ cx. Proof. intros; symmetry; eauto using cmra_pcore_r', cmra_pcore_idemp. Qed. Lemma cmra_pcore_dup' x cx : pcore x ≡ Some cx → cx ≡ cx ⋅ cx. Proof. intros; symmetry; eauto using cmra_pcore_r', cmra_pcore_idemp'. Qed. `````` Robbert Krebbers committed May 28, 2016 312 313 314 315 316 317 318 319 ``````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 320 `````` `````` Robbert Krebbers committed May 30, 2016 321 322 323 324 ``````(** ** Persistent elements *) Lemma persistent_dup x `{!Persistent x} : x ≡ x ⋅ x. Proof. by apply cmra_pcore_dup' with x. Qed. `````` Jacques-Henri Jourdan committed May 31, 2016 325 ``````(** ** Exclusive elements *) `````` Robbert Krebbers committed Jun 16, 2016 326 327 328 329 330 331 332 333 ``````Lemma exclusiveN_l n x `{!Exclusive x} y : ✓{n} (x ⋅ y) → False. Proof. intros ?%cmra_validN_le%exclusive0_l; auto with arith. Qed. Lemma exclusiveN_r n x `{!Exclusive x} y : ✓{n} (y ⋅ x) → False. Proof. rewrite comm. by apply exclusiveN_l. Qed. Lemma exclusive_l x `{!Exclusive x} y : ✓ (x ⋅ y) → False. Proof. by move /cmra_valid_validN /(_ 0) /exclusive0_l. Qed. Lemma exclusive_r x `{!Exclusive x} y : ✓ (y ⋅ x) → False. Proof. rewrite comm. by apply exclusive_l. Qed. `````` Robbert Krebbers committed Jun 16, 2016 334 335 ``````Lemma exclusiveN_opM n x `{!Exclusive x} my : ✓{n} (x ⋅? my) → my = None. Proof. destruct my. move=> /(exclusiveN_l _ x) []. done. Qed. `````` Jacques-Henri Jourdan committed May 31, 2016 336 `````` `````` Robbert Krebbers committed Feb 01, 2016 337 ``````(** ** Order *) `````` Robbert Krebbers committed Mar 11, 2016 338 339 ``````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 340 ``````Global Instance cmra_includedN_trans n : Transitive (@includedN A _ _ n). `````` Robbert Krebbers committed Feb 01, 2016 341 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 342 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 343 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 344 ``````Global Instance cmra_included_trans: Transitive (@included A _ _). `````` Robbert Krebbers committed Feb 01, 2016 345 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 346 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 347 ``````Qed. `````` Robbert Krebbers committed Feb 18, 2016 348 ``````Lemma cmra_validN_includedN n x y : ✓{n} y → x ≼{n} y → ✓{n} x. `````` Robbert Krebbers committed Feb 01, 2016 349 ``````Proof. intros Hyv [z ?]; cofe_subst y; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 18, 2016 350 ``````Lemma cmra_validN_included n x y : ✓{n} y → x ≼ y → ✓{n} x. `````` Robbert Krebbers committed Mar 11, 2016 351 ``````Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 352 `````` `````` Robbert Krebbers committed Feb 18, 2016 353 ``````Lemma cmra_includedN_S n x y : x ≼{S n} y → x ≼{n} y. `````` Robbert Krebbers committed Feb 01, 2016 354 ``````Proof. by intros [z Hz]; exists z; apply dist_S. Qed. `````` Robbert Krebbers committed Feb 18, 2016 355 ``````Lemma cmra_includedN_le n n' x y : x ≼{n} y → n' ≤ n → x ≼{n'} y. `````` Robbert Krebbers committed Feb 01, 2016 356 357 358 359 360 361 362 ``````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 363 ``````Proof. rewrite (comm op); apply cmra_includedN_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 364 ``````Lemma cmra_included_r x y : y ≼ x ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 365 ``````Proof. rewrite (comm op); apply cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 20, 2015 366 `````` `````` Robbert Krebbers committed May 28, 2016 367 368 369 370 371 372 373 374 375 ``````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 376 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 377 378 379 380 381 382 383 `````` 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 384 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 385 386 ``````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 387 ``````Lemma cmra_preservingN_l n x y z : x ≼{n} y → z ⋅ x ≼{n} z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 388 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Robbert Krebbers committed Feb 01, 2016 389 ``````Lemma cmra_preserving_l x y z : x ≼ y → z ⋅ x ≼ z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 390 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Robbert Krebbers committed Feb 11, 2016 391 ``````Lemma cmra_preservingN_r n x y z : x ≼{n} y → x ⋅ z ≼{n} y ⋅ z. `````` Robbert Krebbers committed Feb 11, 2016 392 ``````Proof. by intros; rewrite -!(comm _ z); apply cmra_preservingN_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 393 ``````Lemma cmra_preserving_r x y z : x ≼ y → x ⋅ z ≼ y ⋅ z. `````` Robbert Krebbers committed Feb 11, 2016 394 ``````Proof. by intros; rewrite -!(comm _ z); apply cmra_preserving_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 395 `````` `````` Robbert Krebbers committed Feb 18, 2016 396 ``````Lemma cmra_included_dist_l n x1 x2 x1' : `````` Ralf Jung committed Feb 10, 2016 397 `````` x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2. `````` Robbert Krebbers committed Nov 11, 2015 398 ``````Proof. `````` Robbert Krebbers committed Feb 01, 2016 399 400 `````` intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using cmra_included_l. by rewrite Hx1 Hx2. `````` Robbert Krebbers committed Nov 11, 2015 401 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2016 402 `````` `````` Robbert Krebbers committed May 28, 2016 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 ``````(** ** 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. `````` Robbert Krebbers committed May 30, 2016 432 433 `````` Lemma cmra_core_dup x : core x ≡ core x ⋅ core x. Proof. by rewrite -{3}(cmra_core_idemp x) cmra_core_r. Qed. `````` Robbert Krebbers committed May 28, 2016 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 `````` 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 471 ``````(** ** Timeless *) `````` Robbert Krebbers committed Feb 10, 2016 472 ``````Lemma cmra_timeless_included_l x y : Timeless x → ✓{0} y → x ≼{0} y → x ≼ y. `````` Robbert Krebbers committed Dec 11, 2015 473 474 ``````Proof. intros ?? [x' ?]. `````` Robbert Krebbers committed Feb 24, 2016 475 `````` destruct (cmra_extend 0 y x x') as ([z z']&Hy&Hz&Hz'); auto; simpl in *. `````` Robbert Krebbers committed Jan 13, 2016 476 `````` by exists z'; rewrite Hy (timeless x z). `````` Robbert Krebbers committed Dec 11, 2015 477 ``````Qed. `````` Robbert Krebbers committed Feb 10, 2016 478 ``````Lemma cmra_timeless_included_r n x y : Timeless y → x ≼{0} y → x ≼{n} y. `````` Robbert Krebbers committed Dec 11, 2015 479 ``````Proof. intros ? [x' ?]. exists x'. by apply equiv_dist, (timeless y). Qed. `````` Robbert Krebbers committed Jan 14, 2016 480 ``````Lemma cmra_op_timeless x1 x2 : `````` Robbert Krebbers committed Dec 11, 2015 481 `````` ✓ (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2). `````` Robbert Krebbers committed Nov 18, 2015 482 483 ``````Proof. intros ??? z Hz. `````` Robbert Krebbers committed Feb 24, 2016 484 `````` destruct (cmra_extend 0 z x1 x2) as ([y1 y2]&Hz'&?&?); auto; simpl in *. `````` Robbert Krebbers committed Feb 24, 2016 485 `````` { rewrite -?Hz. by apply cmra_valid_validN. } `````` Robbert Krebbers committed Jan 13, 2016 486 `````` by rewrite Hz' (timeless x1 y1) // (timeless x2 y2). `````` Robbert Krebbers committed Nov 18, 2015 487 ``````Qed. `````` Robbert Krebbers committed Nov 20, 2015 488 `````` `````` Robbert Krebbers committed Feb 24, 2016 489 490 491 492 493 494 495 496 ``````(** ** 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 497 `````` split; first by apply cmra_included_includedN. `````` Robbert Krebbers committed Feb 24, 2016 498 499 `````` intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 11, 2015 500 501 ``````End cmra. `````` Robbert Krebbers committed May 27, 2016 502 503 ``````(** * Properties about CMRAs with a unit element **) Section ucmra. `````` Robbert Krebbers committed May 28, 2016 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 `````` 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 525 ``````End ucmra. `````` Robbert Krebbers committed May 28, 2016 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 ``````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 558 `````` `````` Robbert Krebbers committed Feb 01, 2016 559 ``````(** * Properties about monotone functions *) `````` Robbert Krebbers committed Jan 14, 2016 560 ``````Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A). `````` Robbert Krebbers committed Feb 26, 2016 561 ``````Proof. repeat split; by try apply _. Qed. `````` Robbert Krebbers committed Feb 01, 2016 562 563 ``````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 564 565 ``````Proof. split. `````` Robbert Krebbers committed Feb 26, 2016 566 `````` - apply _. `````` Robbert Krebbers committed Feb 17, 2016 567 `````` - move=> n x Hx /=. by apply validN_preserving, validN_preserving. `````` Robbert Krebbers committed Feb 26, 2016 568 `````` - move=> x y Hxy /=. by apply included_preserving, included_preserving. `````` Robbert Krebbers committed Nov 20, 2015 569 ``````Qed. `````` Robbert Krebbers committed Nov 16, 2015 570 `````` `````` Robbert Krebbers committed Feb 01, 2016 571 572 ``````Section cmra_monotone. Context {A B : cmraT} (f : A → B) `{!CMRAMonotone f}. `````` Robbert Krebbers committed Feb 26, 2016 573 574 `````` 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 575 `````` Proof. `````` Robbert Krebbers committed Feb 26, 2016 576 `````` intros [z ->]. `````` Robbert Krebbers committed Feb 26, 2016 577 `````` apply cmra_included_includedN, (included_preserving f), cmra_included_l. `````` Robbert Krebbers committed Feb 01, 2016 578 `````` Qed. `````` Robbert Krebbers committed Feb 11, 2016 579 `````` Lemma valid_preserving x : ✓ x → ✓ f x. `````` Robbert Krebbers committed Feb 01, 2016 580 581 582 `````` Proof. rewrite !cmra_valid_validN; eauto using validN_preserving. Qed. End cmra_monotone. `````` Robbert Krebbers committed May 25, 2016 583 584 ``````(** Functors *) Structure rFunctor := RFunctor { `````` Robbert Krebbers committed May 27, 2016 585 `````` rFunctor_car : cofeT → cofeT → cmraT; `````` Robbert Krebbers committed May 25, 2016 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 `````` 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 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 ``````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 642 643 644 645 646 647 648 649 650 651 652 653 654 ``````(** * 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 655 `````` Lemma cmra_transport_core x : T (core x) = core (T x). `````` Robbert Krebbers committed Feb 08, 2016 656 `````` Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 657 `````` Lemma cmra_transport_validN n x : ✓{n} T x ↔ ✓{n} x. `````` Robbert Krebbers committed Feb 08, 2016 658 `````` Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 659 `````` Lemma cmra_transport_valid x : ✓ T x ↔ ✓ x. `````` Robbert Krebbers committed Feb 08, 2016 660 661 662 `````` 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 663 664 `````` Global Instance cmra_transport_persistent x : Persistent x → Persistent (T x). Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 08, 2016 665 666 ``````End cmra_transport. `````` Robbert Krebbers committed Feb 01, 2016 667 668 ``````(** * Instances *) (** ** Discrete CMRA *) `````` Robbert Krebbers committed May 28, 2016 669 ``````Record RAMixin A `{Equiv A, PCore A, Op A, Valid A} := { `````` Robbert Krebbers committed Feb 01, 2016 670 `````` (* setoids *) `````` Robbert Krebbers committed May 28, 2016 671 672 673 674 `````` 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 675 `````` (* monoid *) `````` Robbert Krebbers committed May 25, 2016 676 677 `````` ra_assoc : Assoc (≡) (⋅); ra_comm : Comm (≡) (⋅); `````` Robbert Krebbers committed May 28, 2016 678 679 680 681 `````` 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 682 `````` ra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x `````` Robbert Krebbers committed Feb 01, 2016 683 684 ``````}. `````` Robbert Krebbers committed Nov 16, 2015 685 ``````Section discrete. `````` Robbert Krebbers committed May 28, 2016 686 `````` Context `{Equiv A, PCore A, Op A, Valid A, @Equivalence A (≡)}. `````` Robbert Krebbers committed May 25, 2016 687 688 `````` Context (ra_mix : RAMixin A). Existing Instances discrete_dist discrete_compl. `````` Robbert Krebbers committed Feb 01, 2016 689 `````` `````` Robbert Krebbers committed Feb 10, 2016 690 `````` Instance discrete_validN : ValidN A := λ n x, ✓ x. `````` Robbert Krebbers committed Jan 14, 2016 691 `````` Definition discrete_cmra_mixin : CMRAMixin A. `````` Robbert Krebbers committed Nov 16, 2015 692 `````` Proof. `````` Robbert Krebbers committed May 25, 2016 693 `````` destruct ra_mix; split; try done. `````` Robbert Krebbers committed Feb 24, 2016 694 `````` - intros x; split; first done. by move=> /(_ 0). `````` Robbert Krebbers committed May 25, 2016 695 `````` - intros n x y1 y2 ??; by exists (y1,y2). `````` Robbert Krebbers committed Nov 16, 2015 696 697 698 `````` Qed. End discrete. `````` Robbert Krebbers committed May 25, 2016 699 700 ``````Notation discreteR A ra_mix := (CMRAT A discrete_cofe_mixin (discrete_cmra_mixin ra_mix)). `````` Robbert Krebbers committed Jun 30, 2016 701 702 ``````Notation discreteUR A ra_mix ucmra_mix := (UCMRAT A discrete_cofe_mixin (discrete_cmra_mixin ra_mix) ucmra_mix). `````` Robbert Krebbers committed May 25, 2016 703 `````` `````` Robbert Krebbers committed May 28, 2016 704 ``````Global Instance discrete_cmra_discrete `{Equiv A, PCore A, Op A, Valid A, `````` Robbert Krebbers committed May 25, 2016 705 706 707 `````` @Equivalence A (≡)} (ra_mix : RAMixin A) : CMRADiscrete (discreteR A ra_mix). Proof. split. apply _. done. Qed. `````` Robbert Krebbers committed May 28, 2016 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 ``````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 733 734 735 ``````(** ** CMRA for the unit type *) Section unit. Instance unit_valid : Valid () := λ x, True. `````` Robbert Krebbers committed May 25, 2016 736 `````` Instance unit_validN : ValidN () := λ n x, True. `````` Robbert Krebbers committed May 28, 2016 737 `````` Instance unit_pcore : PCore () := λ x, Some x. `````` Robbert Krebbers committed Feb 01, 2016 738 `````` Instance unit_op : Op () := λ x y, (). `````` Robbert Krebbers committed May 27, 2016 739 `````` Lemma unit_cmra_mixin : CMRAMixin (). `````` Robbert Krebbers committed Jun 14, 2016 740 `````` Proof. apply discrete_cmra_mixin, ra_total_mixin; by eauto. Qed. `````` Robbert Krebbers committed May 25, 2016 741 `````` Canonical Structure unitR : cmraT := CMRAT () unit_cofe_mixin unit_cmra_mixin. `````` Robbert Krebbers committed May 27, 2016 742 743 744 745 746 747 748 `````` 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 749 `````` Global Instance unit_cmra_discrete : CMRADiscrete unitR. `````` Robbert Krebbers committed May 25, 2016 750 `````` Proof. done. Qed. `````` Robbert Krebbers committed Mar 15, 2016 751 `````` Global Instance unit_persistent (x : ()) : Persistent x. `````` Robbert Krebbers committed May 28, 2016 752 `````` Proof. by constructor. Qed. `````` Robbert Krebbers committed Feb 01, 2016 753 ``````End unit. `````` Ralf Jung committed Jan 19, 2016 754 `````` `````` Robbert Krebbers committed Jun 14, 2016 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 ``````(** ** Natural numbers *) Section nat. Instance nat_valid : Valid nat := λ x, True. Instance nat_validN : ValidN nat := λ n x, True. Instance nat_pcore : PCore nat := λ x, Some 0. Instance nat_op : Op nat := plus. Lemma nat_included (x y : nat) : x ≼ y ↔ x ≤ y. Proof. split. - intros [z ->]; unfold op, nat_op; lia. - exists (y - x). by apply le_plus_minus. Qed. Lemma nat_cmra_mixin : CMRAMixin nat. Proof. apply discrete_cmra_mixin, ra_total_mixin; try by eauto. - solve_proper. - intros x y z. apply Nat.add_assoc. - intros x y. apply Nat.add_comm. - by exists 0. Qed. Canonical Structure natR : cmraT := CMRAT nat (@discrete_cofe_mixin _ equivL _) nat_cmra_mixin. Instance nat_empty : Empty nat := 0. Lemma nat_ucmra_mixin : UCMRAMixin nat. Proof. split; apply _ || done. Qed. Canonical Structure natUR : ucmraT := UCMRAT nat (@discrete_cofe_mixin _ equivL _) nat_cmra_mixin nat_ucmra_mixin. Global Instance nat_cmra_discrete : CMRADiscrete natR. Proof. constructor; apply _ || done. Qed. Global Instance nat_persistent (x : ()) : Persistent x. Proof. by constructor. Qed. End nat. `````` Robbert Krebbers committed Feb 01, 2016 790 ``````(** ** Product *) `````` Robbert Krebbers committed Jan 14, 2016 791 792 ``````Section prod. Context {A B : cmraT}. `````` Robbert Krebbers committed May 28, 2016 793 794 795 `````` Local Arguments pcore _ _ !_ /. Local Arguments cmra_pcore _ !_/. `````` Robbert Krebbers committed Jan 14, 2016 796 `````` Instance prod_op : Op (A * B) := λ x y, (x.1 ⋅ y.1, x.2 ⋅ y.2). `````` Robbert Krebbers committed May 28, 2016 797 798 799 `````` 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 800 `````` Instance prod_valid : Valid (A * B) := λ x, ✓ x.1 ∧ ✓ x.2. `````` Robbert Krebbers committed Feb 11, 2016 801 `````` Instance prod_validN : ValidN (A * B) := λ n x, ✓{n} x.1 ∧ ✓{n} x.2. `````` Robbert Krebbers committed May 27, 2016 802 `````` `````` Robbert Krebbers committed May 28, 2016 803 804 805 806 807 808 809 810 811 812 813 814 `````` Lemma prod_pcore_Some (x cx : A * B) : pcore x = Some cx ↔ pcore (x.1) = Some (cx.1) ∧ pcore (x.2) = Some (cx.2). Proof. destruct x, cx; by intuition simplify_option_eq. Qed. Lemma prod_pcore_Some' (x cx : A * B) : pcore x ≡ Some cx ↔ pcore (x.1) ≡ Some (cx.1) ∧ pcore (x.2) ≡ Some (cx.2). Proof. split; [by intros (cx'&[-> ->]%prod_pcore_Some&->)%equiv_Some_inv_r'|]. rewrite {3}/pcore /prod_pcore. (* TODO: use setoid rewrite *) intros [Hx1 Hx2]; inversion_clear Hx1; simpl; inversion_clear Hx2. by constructor. Qed. `````` Robbert Krebbers committed Jan 14, 2016 815 816 817 818 819 820 821 822 823 824 `````` Lemma prod_included (x y : A * B) : x ≼ y ↔ x.1 ≼ y.1 ∧ x.2 ≼ y.2. Proof. split; [intros [z Hz]; split; [exists (z.1)|exists (z.2)]; apply Hz|]. intros [[z1 Hz1] [z2 Hz2]]; exists (z1,z2); split; auto. Qed. Lemma prod_includedN (x y : A * B) n : x ≼{n} y ↔ x.1 ≼{n} y.1 ∧ x.2 ≼{n} y.2. Proof. split; [intros [z Hz]; split; [exists (z.1)|exists (z.2)]; apply Hz|]. intros [[z1 Hz1] [z2 Hz2]]; exists (z1,z2); split; auto. Qed. `````` Robbert Krebbers committed May 27, 2016 825 `````` `````` Robbert Krebbers committed Jan 14, 2016 826 827 828 `````` Definition prod_cmra_mixin : CMRAMixin (A * B). Proof. split; try apply _. `````` Robbert Krebbers committed Feb 17, 2016 829 `````` - by intros n x y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2. `````` Robbert Krebbers committed May 28, 2016 830 831 832 833 `````` - intros n x y cx; setoid_rewrite prod_pcore_Some=> -[??] [??]. destruct (cmra_pcore_ne n (x.1) (y.1) (cx.1)) as (z1&->&?); auto. destruct (cmra_pcore_ne n (x.2) (y.2) (cx.2)) as (z2&->&?); auto. exists (z1,z2); repeat constructor; auto. `````` Robbert Krebbers committed Feb 17, 2016 834 `````` - by intros n y1 y2 [Hy1 Hy2] [??]; split; rewrite /= -?Hy1 -?Hy2. `````` Robbert Krebbers committed Feb 24, 2016 835 836 837 `````` - intros x; split. + intros [??] n; split; by apply cmra_valid_validN. + intros Hxy; split; apply cmra_valid_validN=> n; apply Hxy. `````` Robbert Krebbers committed Feb 17, 2016 838 839 840 `````` - by intros n x [??]; split; apply cmra_validN_S. - by split; rewrite /= assoc. - by split; rewrite /= comm. `````` Robbert Krebbers committed May 28, 2016 841 842 843 844 845 846 847 848 `````` - intros x y [??]%prod_pcore_Some; constructor; simpl; eauto using cmra_pcore_l. - intros x y; rewrite prod_pcore_Some prod_pcore_Some'. naive_solver eauto using cmra_pcore_idemp. - intros x y cx; rewrite prod_included prod_pcore_Some=> -[??] [??]. destruct (cmra_pcore_preserving (x.1) (y.1) (cx.1)) as (z1&?&?); auto. destruct (cmra_pcore_preserving (x.2) (y.2) (cx.2)) as (z2&?&?); auto. exists (z1,z2). by rewrite prod_included prod_pcore_Some. `````` Robbert Krebbers committed Feb 17, 2016 849 `````` - intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l. `````` Robbert Krebbers committed Feb 24, 2016 850 851 852 853 `````` - intros n x y1 y2 [??] [??]; simpl in *. destruct (cmra_extend n (x.1) (y1.1) (y2.1)) as (z1&?&?&?); auto. destruct (cmra_extend n (x.2) (y1.2) (y2.2)) as (z2&?&?&?); auto. by exists ((z1.1,z2.1),(z1.2,z2.2)). `````` Robbert Krebbers committed Jan 14, 2016 854 `````` Qed. `````` Robbert Krebbers committed May 27, 2016 855 `````` Canonical Structure prodR := `````` Robbert Krebbers committed May 25, 2016 856 `````` CMRAT (A * B) prod_cofe_mixin prod_cmra_mixin. `````` Robbert Krebbers committed May 27, 2016 857 `````` `````` Robbert Krebbers committed Jun 14, 2016 858 `````` Lemma pair_op (a a' : A) (b b' : B) : (a, b) ⋅ (a', b') = (a ⋅ a', b ⋅ b'). `````` Jacques-Henri Jourdan committed Jun 01, 2016 859 860 `````` Proof. done. Qed. `````` Robbert Krebbers committed May 28, 2016 861 862 863 864 865 866 `````` Global Instance prod_cmra_total : CMRATotal A → CMRATotal B → CMRATotal prodR. Proof. intros H1 H2 [a b]. destruct (H1 a) as [ca ?], (H2 b) as [cb ?]. exists (ca,cb); by simplify_option_eq. Qed. `````` Robbert Krebbers committed Feb 24, 2016 867 `````` Global Instance prod_cmra_discrete : `````` Robbert Krebbers committed Mar 01, 2016 868 `````` CMRADiscrete A → CMRADiscrete B → CMRADiscrete prodR. `````` Robbert Krebbers committed Feb 24, 2016 869 870 `````` Proof. split. apply _. by intros ? []; split; apply cmra_discrete_valid. Qed. `````` Robbert Krebbers committed Mar 15, 2016 871 872 `````` Global Instance pair_persistent x y : Persistent x → Persistent y → Persistent (x,y). `````` Robbert Krebbers committed May 28, 2016 873 `````` Proof. by rewrite /Persistent prod_pcore_Some'. Qed. `````` Robbert Krebbers committed Mar 15, 2016 874 `````` `````` Robbert Krebbers committed Jun 16, 2016 875 `````` Global Instance pair_exclusive_l x y : Exclusive x → Exclusive (x,y). `````` Robbert Krebbers committed Jun 16, 2016 876 `````` Proof. by intros ?[][?%exclusive0_l]. Qed. `````` Robbert Krebbers committed Jun 16, 2016 877 `````` Global Instance pair_exclusive_r x y : Exclusive y → Exclusive (x,y). `````` Robbert Krebbers committed Jun 16, 2016 878 `````` Proof. by intros ?[][??%exclusive0_l]. Qed. `````` Robbert Krebbers committed Jan 14, 2016 879 ``````End prod. `````` Robbert Krebbers committed May 22, 2016 880 `````` `````` Robbert Krebbers committed Mar 01, 2016 881 ``````Arguments prodR : clear implicits. `````` Robbert Krebbers committed Jan 14, 2016 882 `````` `````` Robbert Krebbers committed May 27, 2016 883 884 885 886 887 888 889 890 891 892 ``````Section prod_unit. Context {A B : ucmraT}. Instance prod_empty `{Empty A, Empty B} : Empty (A * B) := (∅, ∅). Lemma prod_ucmra_mixin : UCMRAMixin (A * B). Proof. split. - split; apply ucmra_unit_valid. - by split; rewrite /=left_id. - by intros ? [??]; split; apply (timeless _). `````` Robbert Krebbers committed May 28, 2016 893 `````` - rewrite prod_pcore_Some'; split; apply (persistent _). `````` Robbert Krebbers committed May 27, 2016 894 895 896 897 898 `````` Qed. Canonical Structure prodUR := UCMRAT (A * B) prod_cofe_mixin prod_cmra_mixin prod_ucmra_mixin. Lemma pair_split (x : A) (y : B)``````