cmra.v 47.7 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 `````` 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; `````` Ralf Jung committed Jul 25, 2016 51 `````` mixin_cmra_pcore_mono x y cx : `````` Robbert Krebbers committed May 28, 2016 52 `````` 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 `````` mixin_cmra_extend n x y1 y2 : ✓{n} x → x ≡{n}≡ y1 ⋅ y2 → `````` Robbert Krebbers committed Aug 14, 2016 56 `````` ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{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 `````` 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. `````` Ralf Jung committed Jul 25, 2016 116 `````` Lemma cmra_pcore_mono x y cx : `````` Robbert Krebbers committed May 28, 2016 117 `````` x ≼ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy. `````` Ralf Jung committed Jul 25, 2016 118 `````` Proof. apply (mixin_cmra_pcore_mono _ (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 `````` ✓{n} x → x ≡{n}≡ y1 ⋅ y2 → `````` Robbert Krebbers committed Aug 14, 2016 123 `````` ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{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 `````` mixin_ucmra_pcore_unit : pcore ∅ ≡ Some ∅ `````` Robbert Krebbers committed Feb 01, 2016 157 ``````}. `````` Robbert Krebbers committed May 27, 2016 158 `````` `````` Robbert Krebbers committed Jun 15, 2016 159 ``````Structure ucmraT := UCMRAT' { `````` Robbert Krebbers committed May 27, 2016 160 161 162 163 `````` ucmra_car :> Type; ucmra_equiv : Equiv ucmra_car; ucmra_dist : Dist ucmra_car; ucmra_compl : Compl ucmra_car; `````` Robbert Krebbers committed May 28, 2016 164 `````` ucmra_pcore : PCore ucmra_car; `````` Robbert Krebbers committed May 27, 2016 165 166 167 168 169 170 `````` 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 171 `````` ucmra_mixin : UCMRAMixin ucmra_car; `````` Robbert Krebbers committed Jun 15, 2016 172 `````` _ : Type; `````` Robbert Krebbers committed May 27, 2016 173 ``````}. `````` Robbert Krebbers committed Jun 15, 2016 174 175 ``````Arguments UCMRAT' _ {_ _ _ _ _ _ _ _} _ _ _ _. Notation UCMRAT A m m' m'' := (UCMRAT' A m m' m'' A). `````` Robbert Krebbers committed May 27, 2016 176 177 178 179 ``````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 180 ``````Arguments ucmra_pcore : simpl never. `````` Robbert Krebbers committed May 27, 2016 181 182 183 184 185 186 187 ``````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 188 ``````Hint Extern 0 (Empty _) => eapply (@ucmra_empty _) : typeclass_instances. `````` Robbert Krebbers committed May 27, 2016 189 190 191 192 193 194 195 196 197 198 199 200 201 202 ``````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. `````` Robbert Krebbers committed May 28, 2016 203 204 `````` Lemma ucmra_pcore_unit : pcore (∅:A) ≡ Some ∅. Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin A)). Qed. `````` Robbert Krebbers committed May 27, 2016 205 ``````End ucmra_mixin. `````` Robbert Krebbers committed Jan 14, 2016 206 `````` `````` Robbert Krebbers committed Feb 24, 2016 207 ``````(** * Discrete CMRAs *) `````` Robbert Krebbers committed Feb 26, 2016 208 ``````Class CMRADiscrete (A : cmraT) := { `````` Robbert Krebbers committed Feb 24, 2016 209 210 211 212 `````` cmra_discrete :> Discrete A; cmra_discrete_valid (x : A) : ✓{0} x → ✓ x }. `````` Robbert Krebbers committed Jan 16, 2016 213 ``````(** * Morphisms *) `````` Robbert Krebbers committed Jan 14, 2016 214 ``````Class CMRAMonotone {A B : cmraT} (f : A → B) := { `````` Robbert Krebbers committed Feb 26, 2016 215 216 `````` cmra_monotone_ne n :> Proper (dist n ==> dist n) f; validN_preserving n x : ✓{n} x → ✓{n} f x; `````` Ralf Jung committed Jul 25, 2016 217 `````` cmra_monotone x y : x ≼ y → f x ≼ f y `````` Robbert Krebbers committed Jan 14, 2016 218 ``````}. `````` Robbert Krebbers committed Feb 26, 2016 219 ``````Arguments validN_preserving {_ _} _ {_} _ _ _. `````` Ralf Jung committed Jul 25, 2016 220 ``````Arguments cmra_monotone {_ _} _ {_} _ _ _. `````` Robbert Krebbers committed Jan 14, 2016 221 `````` `````` Robbert Krebbers committed Jan 16, 2016 222 ``````(** * Properties **) `````` Robbert Krebbers committed Nov 11, 2015 223 ``````Section cmra. `````` Robbert Krebbers committed Jan 14, 2016 224 ``````Context {A : cmraT}. `````` Robbert Krebbers committed Nov 11, 2015 225 ``````Implicit Types x y z : A. `````` Robbert Krebbers committed Feb 01, 2016 226 ``````Implicit Types xs ys zs : list A. `````` Robbert Krebbers committed Nov 11, 2015 227 `````` `````` Robbert Krebbers committed Feb 01, 2016 228 ``````(** ** Setoids *) `````` Robbert Krebbers committed May 28, 2016 229 230 231 232 233 234 235 236 237 ``````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 238 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 239 240 241 `````` 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 242 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 243 244 245 246 ``````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 247 248 249 250 251 252 253 254 ``````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 255 256 257 258 ``````Proof. intros x y Hxy; rewrite !cmra_valid_validN. by split=> ? n; [rewrite -Hxy|rewrite Hxy]. Qed. `````` Robbert Krebbers committed Feb 01, 2016 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 ``````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 277 278 279 280 ``````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 281 `````` `````` Robbert Krebbers committed May 28, 2016 282 283 284 285 ``````(** ** 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 286 ``````(** ** Validity *) `````` Robbert Krebbers committed Feb 18, 2016 287 ``````Lemma cmra_validN_le n n' x : ✓{n} x → n' ≤ n → ✓{n'} x. `````` Robbert Krebbers committed Feb 01, 2016 288 289 290 ``````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 291 ``````Lemma cmra_validN_op_r n x y : ✓{n} (x ⋅ y) → ✓{n} y. `````` Robbert Krebbers committed Feb 11, 2016 292 ``````Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 293 294 295 ``````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 296 ``````(** ** Core *) `````` Robbert Krebbers committed May 28, 2016 297 298 299 300 301 302 303 304 ``````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 305 306 307 308 ``````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 309 310 311 312 313 314 315 316 ``````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 317 `````` `````` Robbert Krebbers committed May 30, 2016 318 319 320 321 ``````(** ** 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 322 ``````(** ** Exclusive elements *) `````` Robbert Krebbers committed Jun 16, 2016 323 ``````Lemma exclusiveN_l n x `{!Exclusive x} y : ✓{n} (x ⋅ y) → False. `````` Robbert Krebbers committed Aug 30, 2016 324 ``````Proof. intros. eapply (exclusive0_l x y), cmra_validN_le; eauto with lia. Qed. `````` Robbert Krebbers committed Jun 16, 2016 325 326 327 328 329 330 ``````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 331 ``````Lemma exclusiveN_opM n x `{!Exclusive x} my : ✓{n} (x ⋅? my) → my = None. `````` Robbert Krebbers committed Aug 30, 2016 332 ``````Proof. destruct my as [y|]. move=> /(exclusiveN_l _ x) []. done. Qed. `````` Jacques-Henri Jourdan committed May 31, 2016 333 `````` `````` Robbert Krebbers committed Feb 01, 2016 334 ``````(** ** Order *) `````` Robbert Krebbers committed Mar 11, 2016 335 336 ``````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 337 ``````Global Instance cmra_includedN_trans n : Transitive (@includedN A _ _ n). `````` Robbert Krebbers committed Feb 01, 2016 338 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 339 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 340 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 341 ``````Global Instance cmra_included_trans: Transitive (@included A _ _). `````` Robbert Krebbers committed Feb 01, 2016 342 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 343 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 344 ``````Qed. `````` Robbert Krebbers committed Sep 09, 2016 345 346 ``````Lemma cmra_valid_included x y : ✓ y → x ≼ y → ✓ x. Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_valid_op_l. 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 `````` `````` Ralf Jung committed Jul 25, 2016 366 ``````Lemma cmra_pcore_mono' x y cx : `````` Robbert Krebbers committed May 28, 2016 367 368 369 `````` x ≼ y → pcore x ≡ Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy. Proof. intros ? (cx'&?&Hcx)%equiv_Some_inv_r'. `````` Ralf Jung committed Jul 25, 2016 370 `````` destruct (cmra_pcore_mono x y cx') as (cy&->&?); auto. `````` Robbert Krebbers committed May 28, 2016 371 372 `````` exists cy; by rewrite Hcx. Qed. `````` Ralf Jung committed Jul 25, 2016 373 ``````Lemma cmra_pcore_monoN' n x y cx : `````` Robbert Krebbers committed May 28, 2016 374 `````` 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 `````` intros [z Hy] (cx'&?&Hcx)%dist_Some_inv_r'. `````` Ralf Jung committed Jul 25, 2016 377 `````` destruct (cmra_pcore_mono x (x ⋅ z) cx') `````` Robbert Krebbers committed May 28, 2016 378 379 380 381 382 `````` 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 Sep 27, 2016 386 `````` `````` Ralf Jung committed Jul 25, 2016 387 ``````Lemma cmra_monoN_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. `````` Ralf Jung committed Jul 25, 2016 389 ``````Lemma cmra_mono_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. `````` Ralf Jung committed Jul 25, 2016 391 392 393 394 ``````Lemma cmra_monoN_r n x y z : x ≼{n} y → x ⋅ z ≼{n} y ⋅ z. Proof. by intros; rewrite -!(comm _ z); apply cmra_monoN_l. Qed. Lemma cmra_mono_r x y z : x ≼ y → x ⋅ z ≼ y ⋅ z. Proof. by intros; rewrite -!(comm _ z); apply cmra_mono_l. Qed. `````` Robbert Krebbers committed Sep 27, 2016 395 396 397 398 ``````Lemma cmra_monoN n x1 x2 y1 y2 : x1 ≼{n} y1 → x2 ≼{n} y2 → x1 ⋅ x2 ≼{n} y1 ⋅ y2. Proof. intros; etrans; eauto using cmra_monoN_l, cmra_monoN_r. Qed. Lemma cmra_mono x1 x2 y1 y2 : x1 ≼ y1 → x2 ≼ y2 → x1 ⋅ x2 ≼ y1 ⋅ y2. Proof. intros; etrans; eauto using cmra_mono_l, cmra_mono_r. Qed. `````` Robbert Krebbers committed Feb 01, 2016 399 `````` `````` Robbert Krebbers committed Feb 18, 2016 400 ``````Lemma cmra_included_dist_l n x1 x2 x1' : `````` Ralf Jung committed Feb 10, 2016 401 `````` x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2. `````` Robbert Krebbers committed Nov 11, 2015 402 ``````Proof. `````` Robbert Krebbers committed Feb 01, 2016 403 404 `````` intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using cmra_included_l. by rewrite Hx1 Hx2. `````` Robbert Krebbers committed Nov 11, 2015 405 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2016 406 `````` `````` Robbert Krebbers committed May 28, 2016 407 408 409 410 411 412 413 414 415 416 417 418 ``````(** ** 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. `````` Ralf Jung committed Jul 25, 2016 419 `````` Lemma cmra_core_mono x y : x ≼ y → core x ≼ core y. `````` Robbert Krebbers committed May 28, 2016 420 421 `````` Proof. intros; destruct (cmra_total x) as [cx Hcx]. `````` Ralf Jung committed Jul 25, 2016 422 `````` destruct (cmra_pcore_mono x y cx) as (cy&Hcy&?); auto. `````` Robbert Krebbers committed May 28, 2016 423 424 425 426 427 428 429 430 431 432 433 434 435 `````` 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 436 437 `````` 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 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 `````` 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. `````` Ralf Jung committed Jul 25, 2016 468 `````` Lemma cmra_core_monoN n x y : x ≼{n} y → core x ≼{n} core y. `````` Robbert Krebbers committed May 28, 2016 469 470 `````` Proof. intros [z ->]. `````` Ralf Jung committed Jul 25, 2016 471 `````` apply cmra_included_includedN, cmra_core_mono, cmra_included_l. `````` Robbert Krebbers committed May 28, 2016 472 473 474 `````` Qed. End total_core. `````` Robbert Krebbers committed Jan 16, 2016 475 ``````(** ** Timeless *) `````` Robbert Krebbers committed Feb 10, 2016 476 ``````Lemma cmra_timeless_included_l x y : Timeless x → ✓{0} y → x ≼{0} y → x ≼ y. `````` Robbert Krebbers committed Dec 11, 2015 477 478 ``````Proof. intros ?? [x' ?]. `````` Robbert Krebbers committed Aug 14, 2016 479 `````` destruct (cmra_extend 0 y x x') as (z&z'&Hy&Hz&Hz'); auto; simpl in *. `````` Robbert Krebbers committed Jan 13, 2016 480 `````` by exists z'; rewrite Hy (timeless x z). `````` Robbert Krebbers committed Dec 11, 2015 481 ``````Qed. `````` Robbert Krebbers committed Aug 30, 2016 482 483 ``````Lemma cmra_timeless_included_r x y : Timeless y → x ≼{0} y → x ≼ y. Proof. intros ? [x' ?]. exists x'. by apply (timeless y). Qed. `````` Robbert Krebbers committed Jan 14, 2016 484 ``````Lemma cmra_op_timeless x1 x2 : `````` Robbert Krebbers committed Dec 11, 2015 485 `````` ✓ (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2). `````` Robbert Krebbers committed Nov 18, 2015 486 487 ``````Proof. intros ??? z Hz. `````` Robbert Krebbers committed Aug 14, 2016 488 `````` destruct (cmra_extend 0 z x1 x2) as (y1&y2&Hz'&?&?); auto; simpl in *. `````` Robbert Krebbers committed Feb 24, 2016 489 `````` { rewrite -?Hz. by apply cmra_valid_validN. } `````` Robbert Krebbers committed Jan 13, 2016 490 `````` by rewrite Hz' (timeless x1 y1) // (timeless x2 y2). `````` Robbert Krebbers committed Nov 18, 2015 491 ``````Qed. `````` Robbert Krebbers committed Nov 20, 2015 492 `````` `````` Robbert Krebbers committed Feb 24, 2016 493 494 495 496 497 498 499 500 ``````(** ** 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 501 `````` split; first by apply cmra_included_includedN. `````` Robbert Krebbers committed Feb 24, 2016 502 503 `````` intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 11, 2015 504 505 ``````End cmra. `````` Robbert Krebbers committed May 27, 2016 506 507 ``````(** * Properties about CMRAs with a unit element **) Section ucmra. `````` Robbert Krebbers committed May 28, 2016 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 `````` Context {A : ucmraT}. Implicit Types x y z : A. 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. `````` Ralf Jung committed Jul 25, 2016 524 `````` intros x. destruct (cmra_pcore_mono' ∅ x ∅) as (cx&->&?); `````` Robbert Krebbers committed May 28, 2016 525 526 `````` eauto using ucmra_unit_least, (persistent ∅). Qed. `````` Robbert Krebbers committed May 27, 2016 527 ``````End ucmra. `````` Robbert Krebbers committed May 28, 2016 528 529 ``````Hint Immediate cmra_unit_total. `````` Robbert Krebbers committed Sep 01, 2016 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 `````` (** * Properties about CMRAs with Leibniz equality *) Section cmra_leibniz. Context {A : cmraT} `{!LeibnizEquiv A}. Implicit Types x y : A. Global Instance cmra_assoc_L : Assoc (=) (@op A _). Proof. intros x y z. unfold_leibniz. by rewrite assoc. Qed. Global Instance cmra_comm_L : Comm (=) (@op A _). Proof. intros x y. unfold_leibniz. by rewrite comm. Qed. Lemma cmra_pcore_l_L x cx : pcore x = Some cx → cx ⋅ x = x. Proof. unfold_leibniz. apply cmra_pcore_l'. Qed. Lemma cmra_pcore_idemp_L x cx : pcore x = Some cx → pcore cx = Some cx. Proof. unfold_leibniz. apply cmra_pcore_idemp'. Qed. Lemma cmra_opM_assoc_L x y mz : (x ⋅ y) ⋅? mz = x ⋅ (y ⋅? mz). Proof. unfold_leibniz. apply cmra_opM_assoc. Qed. (** ** Core *) Lemma cmra_pcore_r_L x cx : pcore x = Some cx → x ⋅ cx = x. Proof. unfold_leibniz. apply cmra_pcore_r'. Qed. Lemma cmra_pcore_dup_L x cx : pcore x = Some cx → cx = cx ⋅ cx. Proof. unfold_leibniz. apply cmra_pcore_dup'. Qed. (** ** Persistent elements *) Lemma persistent_dup_L x `{!Persistent x} : x ≡ x ⋅ x. Proof. unfold_leibniz. by apply persistent_dup. Qed. (** ** Total core *) Section total_core. Context `{CMRATotal A}. Lemma cmra_core_r_L x : x ⋅ core x = x. Proof. unfold_leibniz. apply cmra_core_r. Qed. Lemma cmra_core_l_L x : core x ⋅ x = x. Proof. unfold_leibniz. apply cmra_core_l. Qed. Lemma cmra_core_idemp_L x : core (core x) = core x. Proof. unfold_leibniz. apply cmra_core_idemp. Qed. Lemma cmra_core_dup_L x : core x = core x ⋅ core x. Proof. unfold_leibniz. apply cmra_core_dup. Qed. Lemma persistent_total_L x : Persistent x ↔ core x = x. Proof. unfold_leibniz. apply persistent_total. Qed. Lemma persistent_core_L x `{!Persistent x} : core x = x. Proof. by apply persistent_total_L. Qed. End total_core. End cmra_leibniz. Section ucmra_leibniz. Context {A : ucmraT} `{!LeibnizEquiv A}. Implicit Types x y z : A. Global Instance ucmra_unit_left_id_L : LeftId (=) ∅ (@op A _). Proof. intros x. unfold_leibniz. by rewrite left_id. Qed. Global Instance ucmra_unit_right_id_L : RightId (=) ∅ (@op A _). Proof. intros x. unfold_leibniz. by rewrite right_id. Qed. End ucmra_leibniz. `````` Robbert Krebbers committed May 28, 2016 588 589 590 591 592 593 594 595 596 597 598 599 600 ``````(** * 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). `````` Ralf Jung committed Jul 25, 2016 601 `````` Context (core_mono : ∀ x y : A, x ≼ y → core x ≼ core y). `````` Robbert Krebbers committed May 28, 2016 602 603 604 `````` 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 → `````` Robbert Krebbers committed Aug 14, 2016 605 `````` ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2). `````` Robbert Krebbers committed May 28, 2016 606 607 608 609 610 611 612 613 `````` 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. `````` Ralf Jung committed Jul 25, 2016 614 `````` - intros x y cx Hxy%core_mono Hx. move: Hxy. `````` Robbert Krebbers committed May 28, 2016 615 616 617 `````` rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto. Qed. End cmra_total. `````` Robbert Krebbers committed May 27, 2016 618 `````` `````` Robbert Krebbers committed Feb 01, 2016 619 ``````(** * Properties about monotone functions *) `````` Robbert Krebbers committed Jan 14, 2016 620 ``````Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A). `````` Robbert Krebbers committed Feb 26, 2016 621 ``````Proof. repeat split; by try apply _. Qed. `````` Robbert Krebbers committed Feb 01, 2016 622 623 ``````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 624 625 ``````Proof. split. `````` Robbert Krebbers committed Feb 26, 2016 626 `````` - apply _. `````` Robbert Krebbers committed Feb 17, 2016 627 `````` - move=> n x Hx /=. by apply validN_preserving, validN_preserving. `````` Ralf Jung committed Jul 25, 2016 628 `````` - move=> x y Hxy /=. by apply cmra_monotone, cmra_monotone. `````` Robbert Krebbers committed Nov 20, 2015 629 ``````Qed. `````` Robbert Krebbers committed Nov 16, 2015 630 `````` `````` Robbert Krebbers committed Feb 01, 2016 631 632 ``````Section cmra_monotone. Context {A B : cmraT} (f : A → B) `{!CMRAMonotone f}. `````` Robbert Krebbers committed Feb 26, 2016 633 `````` Global Instance cmra_monotone_proper : Proper ((≡) ==> (≡)) f := ne_proper _. `````` Ralf Jung committed Jul 25, 2016 634 `````` Lemma cmra_monotoneN n x y : x ≼{n} y → f x ≼{n} f y. `````` Robbert Krebbers committed Feb 01, 2016 635 `````` Proof. `````` Robbert Krebbers committed Feb 26, 2016 636 `````` intros [z ->]. `````` Ralf Jung committed Jul 25, 2016 637 `````` apply cmra_included_includedN, (cmra_monotone f), cmra_included_l. `````` Robbert Krebbers committed Feb 01, 2016 638 `````` Qed. `````` Robbert Krebbers committed Feb 11, 2016 639 `````` Lemma valid_preserving x : ✓ x → ✓ f x. `````` Robbert Krebbers committed Feb 01, 2016 640 641 642 `````` Proof. rewrite !cmra_valid_validN; eauto using validN_preserving. Qed. End cmra_monotone. `````` Robbert Krebbers committed May 25, 2016 643 644 ``````(** Functors *) Structure rFunctor := RFunctor { `````` Robbert Krebbers committed May 27, 2016 645 `````` rFunctor_car : cofeT → cofeT → cmraT; `````` Robbert Krebbers committed May 25, 2016 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 `````` 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 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 ``````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 702 703 704 705 706 707 708 709 710 711 712 713 714 ``````(** * 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 715 `````` Lemma cmra_transport_core x : T (core x) = core (T x). `````` Robbert Krebbers committed Feb 08, 2016 716 `````` Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 717 `````` Lemma cmra_transport_validN n x : ✓{n} T x ↔ ✓{n} x. `````` Robbert Krebbers committed Feb 08, 2016 718 `````` Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 719 `````` Lemma cmra_transport_valid x : ✓ T x ↔ ✓ x. `````` Robbert Krebbers committed Feb 08, 2016 720 721 722 `````` 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 723 724 `````` Global Instance cmra_transport_persistent x : Persistent x → Persistent (T x). Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 08, 2016 725 726 ``````End cmra_transport. `````` Robbert Krebbers committed Feb 01, 2016 727 728 ``````(** * Instances *) (** ** Discrete CMRA *) `````` Robbert Krebbers committed May 28, 2016 729 ``````Record RAMixin A `{Equiv A, PCore A, Op A, Valid A} := { `````` Robbert Krebbers committed Feb 01, 2016 730 `````` (* setoids *) `````` Robbert Krebbers committed May 28, 2016 731 732 733 734 `````` 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 735 `````` (* monoid *) `````` Robbert Krebbers committed May 25, 2016 736 737 `````` ra_assoc : Assoc (≡) (⋅); ra_comm : Comm (≡) (⋅); `````` Robbert Krebbers committed May 28, 2016 738 739 `````` 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; `````` Ralf Jung committed Jul 25, 2016 740 `````` ra_pcore_mono x y cx : `````` Robbert Krebbers committed May 28, 2016 741 `````` x ≼ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy; `````` Robbert Krebbers committed Mar 11, 2016 742 `````` ra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x `````` Robbert Krebbers committed Feb 01, 2016 743 744 ``````}. `````` Robbert Krebbers committed Nov 16, 2015 745 ``````Section discrete. `````` Robbert Krebbers committed May 28, 2016 746 `````` Context `{Equiv A, PCore A, Op A, Valid A, @Equivalence A (≡)}. `````` Robbert Krebbers committed May 25, 2016 747 748 `````` Context (ra_mix : RAMixin A). Existing Instances discrete_dist discrete_compl. `````` Robbert Krebbers committed Feb 01, 2016 749 `````` `````` Robbert Krebbers committed Feb 10, 2016 750 `````` Instance discrete_validN : ValidN A := λ n x, ✓ x. `````` Robbert Krebbers committed Jan 14, 2016 751 `````` Definition discrete_cmra_mixin : CMRAMixin A. `````` Robbert Krebbers committed Nov 16, 2015 752 `````` Proof. `````` Robbert Krebbers committed May 25, 2016 753 `````` destruct ra_mix; split; try done. `````` Robbert Krebbers committed Feb 24, 2016 754 `````` - intros x; split; first done. by move=> /(_ 0). `````` Robbert Krebbers committed Aug 14, 2016 755 `````` - intros n x y1 y2 ??; by exists y1, y2. `````` Robbert Krebbers committed Nov 16, 2015 756 757 758 `````` Qed. End discrete. `````` Robbert Krebbers committed May 25, 2016 759 760 ``````Notation discreteR A ra_mix := (CMRAT A discrete_cofe_mixin (discrete_cmra_mixin ra_mix)). `````` Robbert Krebbers committed Jun 30, 2016 761 762 ``````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 763 `````` `````` Robbert Krebbers committed May 28, 2016 764 ``````Global Instance discrete_cmra_discrete `{Equiv A, PCore A, Op A, Valid A, `````` Robbert Krebbers committed May 25, 2016 765 766 767 `````` @Equivalence A (≡)} (ra_mix : RAMixin A) : CMRADiscrete (discreteR A ra_mix). Proof. split. apply _. done. Qed. `````` Robbert Krebbers committed May 28, 2016 768 769 770 771 772 773 774 775 776 777 ``````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). `````` Ralf Jung committed Jul 25, 2016 778 `````` Context (core_mono : ∀ x y : A, x ≼ y → core x ≼ core y). `````` Robbert Krebbers committed May 28, 2016 779 780 781 782 783 784 785 786 787 `````` 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. `````` Ralf Jung committed Jul 25, 2016 788 `````` - intros x y cx Hxy%core_mono Hx. move: Hxy. `````` Robbert Krebbers committed May 28, 2016 789 790 791 792 `````` rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto. Qed. End ra_total. `````` Robbert Krebbers committed Feb 01, 2016 793 794 795 ``````(** ** CMRA for the unit type *) Section unit. Instance unit_valid : Valid () := λ x, True. `````` Robbert Krebbers committed May 25, 2016 796 `````` Instance unit_validN : ValidN () := λ n x, True. `````` Robbert Krebbers committed May 28, 2016 797 `````` Instance unit_pcore : PCore () := λ x, Some x. `````` Robbert Krebbers committed Feb 01, 2016 798 `````` Instance unit_op : Op () := λ x y, (). `````` Robbert Krebbers committed May 27, 2016 799 `````` Lemma unit_cmra_mixin : CMRAMixin (). `````` Robbert Krebbers committed Jun 14, 2016 800 `````` Proof. apply discrete_cmra_mixin, ra_total_mixin; by eauto. Qed. `````` Robbert Krebbers committed May 25, 2016 801 `````` Canonical Structure unitR : cmraT := CMRAT () unit_cofe_mixin unit_cmra_mixin. `````` Robbert Krebbers committed May 27, 2016 802 803 804 805 806 807 808 `````` 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 809 `````` Global Instance unit_cmra_discrete : CMRADiscrete unitR. ``````