cmra.v 50.1 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 `````` cmra_monotone_ne n :> Proper (dist n ==> dist n) f; `````` Robbert Krebbers committed Sep 28, 2016 216 `````` cmra_monotone_validN 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 Sep 28, 2016 219 ``````Arguments cmra_monotone_validN {_ _} _ {_} _ _ _. `````` Ralf Jung committed Jul 25, 2016 220 ``````Arguments cmra_monotone {_ _} _ {_} _ _ _. `````` Robbert Krebbers committed Jan 14, 2016 221 `````` `````` Robbert Krebbers committed Sep 28, 2016 222 223 224 225 226 227 228 229 230 231 232 233 234 235 ``````(* Not all intended homomorphisms preserve validity, in particular it does not hold for the [ownM] and [own] connectives. *) Class CMRAHomomorphism {A B : cmraT} (f : A → B) := { cmra_homomorphism_ne n :> Proper (dist n ==> dist n) f; cmra_homomorphism x y : f (x ⋅ y) ≡ f x ⋅ f y }. Arguments cmra_homomorphism {_ _} _ _ _ _. Class UCMRAHomomorphism {A B : ucmraT} (f : A → B) := { ucmra_homomorphism :> CMRAHomomorphism f; ucmra_homomorphism_unit : f ∅ ≡ ∅ }. Arguments ucmra_homomorphism_unit {_ _} _ _. `````` Robbert Krebbers committed Jan 16, 2016 236 ``````(** * Properties **) `````` Robbert Krebbers committed Nov 11, 2015 237 ``````Section cmra. `````` Robbert Krebbers committed Jan 14, 2016 238 ``````Context {A : cmraT}. `````` Robbert Krebbers committed Nov 11, 2015 239 ``````Implicit Types x y z : A. `````` Robbert Krebbers committed Feb 01, 2016 240 ``````Implicit Types xs ys zs : list A. `````` Robbert Krebbers committed Nov 11, 2015 241 `````` `````` Robbert Krebbers committed Feb 01, 2016 242 ``````(** ** Setoids *) `````` Robbert Krebbers committed May 28, 2016 243 244 245 246 247 248 249 250 251 ``````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 252 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 253 254 255 `````` 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 256 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 257 258 259 260 ``````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 Sep 28, 2016 261 ``````Global Instance cmra_op_proper' : Proper ((≡) ==> (≡) ==> (≡)) (@op A _). `````` Robbert Krebbers committed Feb 01, 2016 262 263 264 265 266 267 268 ``````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 269 270 271 272 ``````Proof. intros x y Hxy; rewrite !cmra_valid_validN. by split=> ? n; [rewrite -Hxy|rewrite Hxy]. Qed. `````` Robbert Krebbers committed Feb 01, 2016 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 ``````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 291 292 293 294 ``````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 295 `````` `````` Robbert Krebbers committed May 28, 2016 296 297 298 299 ``````(** ** 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 300 ``````(** ** Validity *) `````` Robbert Krebbers committed Feb 18, 2016 301 ``````Lemma cmra_validN_le n n' x : ✓{n} x → n' ≤ n → ✓{n'} x. `````` Robbert Krebbers committed Feb 01, 2016 302 303 304 ``````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 305 ``````Lemma cmra_validN_op_r n x y : ✓{n} (x ⋅ y) → ✓{n} y. `````` Robbert Krebbers committed Feb 11, 2016 306 ``````Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 307 308 309 ``````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 310 ``````(** ** Core *) `````` Robbert Krebbers committed May 28, 2016 311 312 313 314 315 316 317 318 ``````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 319 320 321 322 ``````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 323 324 325 326 327 328 329 330 ``````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 331 `````` `````` Robbert Krebbers committed May 30, 2016 332 333 334 335 ``````(** ** 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 336 ``````(** ** Exclusive elements *) `````` Robbert Krebbers committed Jun 16, 2016 337 ``````Lemma exclusiveN_l n x `{!Exclusive x} y : ✓{n} (x ⋅ y) → False. `````` Robbert Krebbers committed Aug 30, 2016 338 ``````Proof. intros. eapply (exclusive0_l x y), cmra_validN_le; eauto with lia. Qed. `````` Robbert Krebbers committed Jun 16, 2016 339 340 341 342 343 344 ``````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 345 ``````Lemma exclusiveN_opM n x `{!Exclusive x} my : ✓{n} (x ⋅? my) → my = None. `````` Robbert Krebbers committed Aug 30, 2016 346 ``````Proof. destruct my as [y|]. move=> /(exclusiveN_l _ x) []. done. Qed. `````` Robbert Krebbers committed Oct 02, 2016 347 348 349 350 ``````Lemma exclusive_includedN n x `{!Exclusive x} y : x ≼{n} y → ✓{n} y → False. Proof. intros [? ->]. by apply exclusiveN_l. Qed. Lemma exclusive_included x `{!Exclusive x} y : x ≼ y → ✓ y → False. Proof. intros [? ->]. by apply exclusive_l. Qed. `````` Jacques-Henri Jourdan committed May 31, 2016 351 `````` `````` Robbert Krebbers committed Feb 01, 2016 352 ``````(** ** Order *) `````` Robbert Krebbers committed Mar 11, 2016 353 354 ``````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 355 ``````Global Instance cmra_includedN_trans n : Transitive (@includedN A _ _ n). `````` Robbert Krebbers committed Feb 01, 2016 356 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 357 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 358 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 359 ``````Global Instance cmra_included_trans: Transitive (@included A _ _). `````` Robbert Krebbers committed Feb 01, 2016 360 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 361 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 362 ``````Qed. `````` Robbert Krebbers committed Sep 09, 2016 363 364 ``````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 365 ``````Lemma cmra_validN_includedN n x y : ✓{n} y → x ≼{n} y → ✓{n} x. `````` Robbert Krebbers committed Feb 01, 2016 366 ``````Proof. intros Hyv [z ?]; cofe_subst y; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 18, 2016 367 ``````Lemma cmra_validN_included n x y : ✓{n} y → x ≼ y → ✓{n} x. `````` Robbert Krebbers committed Mar 11, 2016 368 ``````Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 369 `````` `````` Robbert Krebbers committed Feb 18, 2016 370 ``````Lemma cmra_includedN_S n x y : x ≼{S n} y → x ≼{n} y. `````` Robbert Krebbers committed Feb 01, 2016 371 ``````Proof. by intros [z Hz]; exists z; apply dist_S. Qed. `````` Robbert Krebbers committed Feb 18, 2016 372 ``````Lemma cmra_includedN_le n n' x y : x ≼{n} y → n' ≤ n → x ≼{n'} y. `````` Robbert Krebbers committed Feb 01, 2016 373 374 375 376 377 378 379 ``````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 380 ``````Proof. rewrite (comm op); apply cmra_includedN_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 381 ``````Lemma cmra_included_r x y : y ≼ x ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 382 ``````Proof. rewrite (comm op); apply cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 20, 2015 383 `````` `````` Ralf Jung committed Jul 25, 2016 384 ``````Lemma cmra_pcore_mono' x y cx : `````` Robbert Krebbers committed May 28, 2016 385 386 387 `````` 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 388 `````` destruct (cmra_pcore_mono x y cx') as (cy&->&?); auto. `````` Robbert Krebbers committed May 28, 2016 389 390 `````` exists cy; by rewrite Hcx. Qed. `````` Ralf Jung committed Jul 25, 2016 391 ``````Lemma cmra_pcore_monoN' n x y cx : `````` Robbert Krebbers committed May 28, 2016 392 `````` x ≼{n} y → pcore x ≡{n}≡ Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼{n} cy. `````` Robbert Krebbers committed Feb 26, 2016 393 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 394 `````` intros [z Hy] (cx'&?&Hcx)%dist_Some_inv_r'. `````` Ralf Jung committed Jul 25, 2016 395 `````` destruct (cmra_pcore_mono x (x ⋅ z) cx') `````` Robbert Krebbers committed May 28, 2016 396 397 398 399 400 `````` 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 401 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 402 403 ``````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 404 `````` `````` Ralf Jung committed Jul 25, 2016 405 ``````Lemma cmra_monoN_l n x y z : x ≼{n} y → z ⋅ x ≼{n} z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 406 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Ralf Jung committed Jul 25, 2016 407 ``````Lemma cmra_mono_l x y z : x ≼ y → z ⋅ x ≼ z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 408 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Ralf Jung committed Jul 25, 2016 409 410 411 412 ``````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 413 414 415 416 ``````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 417 `````` `````` Robbert Krebbers committed Sep 28, 2016 418 419 420 421 422 423 424 ``````Global Instance cmra_monoN' n : Proper (includedN n ==> includedN n ==> includedN n) (@op A _). Proof. intros x1 x2 Hx y1 y2 Hy. by apply cmra_monoN. Qed. Global Instance cmra_mono' : Proper (included ==> included ==> included) (@op A _). Proof. intros x1 x2 Hx y1 y2 Hy. by apply cmra_mono. Qed. `````` Robbert Krebbers committed Feb 18, 2016 425 ``````Lemma cmra_included_dist_l n x1 x2 x1' : `````` Ralf Jung committed Feb 10, 2016 426 `````` x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2. `````` Robbert Krebbers committed Nov 11, 2015 427 ``````Proof. `````` Robbert Krebbers committed Feb 01, 2016 428 429 `````` intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using cmra_included_l. by rewrite Hx1 Hx2. `````` Robbert Krebbers committed Nov 11, 2015 430 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2016 431 `````` `````` Robbert Krebbers committed May 28, 2016 432 433 434 435 436 437 438 439 440 441 442 443 ``````(** ** 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 444 `````` Lemma cmra_core_mono x y : x ≼ y → core x ≼ core y. `````` Robbert Krebbers committed May 28, 2016 445 446 `````` Proof. intros; destruct (cmra_total x) as [cx Hcx]. `````` Ralf Jung committed Jul 25, 2016 447 `````` destruct (cmra_pcore_mono x y cx) as (cy&Hcy&?); auto. `````` Robbert Krebbers committed May 28, 2016 448 449 450 451 452 453 454 455 456 457 458 459 460 `````` 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 461 462 `````` 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 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 `````` 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 493 `````` Lemma cmra_core_monoN n x y : x ≼{n} y → core x ≼{n} core y. `````` Robbert Krebbers committed May 28, 2016 494 495 `````` Proof. intros [z ->]. `````` Ralf Jung committed Jul 25, 2016 496 `````` apply cmra_included_includedN, cmra_core_mono, cmra_included_l. `````` Robbert Krebbers committed May 28, 2016 497 498 499 `````` Qed. End total_core. `````` Robbert Krebbers committed Jan 16, 2016 500 ``````(** ** Timeless *) `````` Robbert Krebbers committed Feb 10, 2016 501 ``````Lemma cmra_timeless_included_l x y : Timeless x → ✓{0} y → x ≼{0} y → x ≼ y. `````` Robbert Krebbers committed Dec 11, 2015 502 503 ``````Proof. intros ?? [x' ?]. `````` Robbert Krebbers committed Aug 14, 2016 504 `````` destruct (cmra_extend 0 y x x') as (z&z'&Hy&Hz&Hz'); auto; simpl in *. `````` Robbert Krebbers committed Jan 13, 2016 505 `````` by exists z'; rewrite Hy (timeless x z). `````` Robbert Krebbers committed Dec 11, 2015 506 ``````Qed. `````` Robbert Krebbers committed Aug 30, 2016 507 508 ``````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 509 ``````Lemma cmra_op_timeless x1 x2 : `````` Robbert Krebbers committed Dec 11, 2015 510 `````` ✓ (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2). `````` Robbert Krebbers committed Nov 18, 2015 511 512 ``````Proof. intros ??? z Hz. `````` Robbert Krebbers committed Aug 14, 2016 513 `````` destruct (cmra_extend 0 z x1 x2) as (y1&y2&Hz'&?&?); auto; simpl in *. `````` Robbert Krebbers committed Feb 24, 2016 514 `````` { rewrite -?Hz. by apply cmra_valid_validN. } `````` Robbert Krebbers committed Jan 13, 2016 515 `````` by rewrite Hz' (timeless x1 y1) // (timeless x2 y2). `````` Robbert Krebbers committed Nov 18, 2015 516 ``````Qed. `````` Robbert Krebbers committed Nov 20, 2015 517 `````` `````` Robbert Krebbers committed Feb 24, 2016 518 519 520 521 522 523 524 525 ``````(** ** 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 526 `````` split; first by apply cmra_included_includedN. `````` Robbert Krebbers committed Feb 24, 2016 527 528 `````` intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 11, 2015 529 530 ``````End cmra. `````` Robbert Krebbers committed May 27, 2016 531 532 ``````(** * Properties about CMRAs with a unit element **) Section ucmra. `````` Robbert Krebbers committed May 28, 2016 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 `````` 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 549 `````` intros x. destruct (cmra_pcore_mono' ∅ x ∅) as (cx&->&?); `````` Robbert Krebbers committed May 28, 2016 550 551 `````` eauto using ucmra_unit_least, (persistent ∅). Qed. `````` Robbert Krebbers committed May 27, 2016 552 ``````End ucmra. `````` Robbert Krebbers committed May 28, 2016 553 554 ``````Hint Immediate cmra_unit_total. `````` Robbert Krebbers committed Sep 01, 2016 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 `````` (** * 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 613 614 615 616 617 618 619 620 621 622 623 624 625 ``````(** * 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 626 `````` Context (core_mono : ∀ x y : A, x ≼ y → core x ≼ core y). `````` Robbert Krebbers committed May 28, 2016 627 628 629 `````` 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 630 `````` ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2). `````` Robbert Krebbers committed May 28, 2016 631 632 633 634 635 636 637 638 `````` 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 639 `````` - intros x y cx Hxy%core_mono Hx. move: Hxy. `````` Robbert Krebbers committed May 28, 2016 640 641 642 `````` rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto. Qed. End cmra_total. `````` Robbert Krebbers committed May 27, 2016 643 `````` `````` Robbert Krebbers committed Feb 01, 2016 644 ``````(** * Properties about monotone functions *) `````` Robbert Krebbers committed Jan 14, 2016 645 ``````Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A). `````` Robbert Krebbers committed Feb 26, 2016 646 ``````Proof. repeat split; by try apply _. Qed. `````` Robbert Krebbers committed Feb 01, 2016 647 648 ``````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 649 650 ``````Proof. split. `````` Robbert Krebbers committed Feb 26, 2016 651 `````` - apply _. `````` Robbert Krebbers committed Sep 28, 2016 652 `````` - move=> n x Hx /=. by apply cmra_monotone_validN, cmra_monotone_validN. `````` Ralf Jung committed Jul 25, 2016 653 `````` - move=> x y Hxy /=. by apply cmra_monotone, cmra_monotone. `````` Robbert Krebbers committed Nov 20, 2015 654 ``````Qed. `````` Robbert Krebbers committed Nov 16, 2015 655 `````` `````` Robbert Krebbers committed Feb 01, 2016 656 657 ``````Section cmra_monotone. Context {A B : cmraT} (f : A → B) `{!CMRAMonotone f}. `````` Robbert Krebbers committed Feb 26, 2016 658 `````` Global Instance cmra_monotone_proper : Proper ((≡) ==> (≡)) f := ne_proper _. `````` Ralf Jung committed Jul 25, 2016 659 `````` Lemma cmra_monotoneN n x y : x ≼{n} y → f x ≼{n} f y. `````` Robbert Krebbers committed Feb 01, 2016 660 `````` Proof. `````` Robbert Krebbers committed Feb 26, 2016 661 `````` intros [z ->]. `````` Ralf Jung committed Jul 25, 2016 662 `````` apply cmra_included_includedN, (cmra_monotone f), cmra_included_l. `````` Robbert Krebbers committed Feb 01, 2016 663 `````` Qed. `````` Robbert Krebbers committed Sep 28, 2016 664 665 `````` Lemma cmra_monotone_valid x : ✓ x → ✓ f x. Proof. rewrite !cmra_valid_validN; eauto using cmra_monotone_validN. Qed. `````` Robbert Krebbers committed Feb 01, 2016 666 667 ``````End cmra_monotone. `````` Robbert Krebbers committed Sep 28, 2016 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 ``````Instance cmra_homomorphism_id {A : cmraT} : CMRAHomomorphism (@id A). Proof. repeat split; by try apply _. Qed. Instance cmra_homomorphism_compose {A B C : cmraT} (f : A → B) (g : B → C) : CMRAHomomorphism f → CMRAHomomorphism g → CMRAHomomorphism (g ∘ f). Proof. split. - apply _. - move=> x y /=. rewrite -(cmra_homomorphism g). by apply (ne_proper _), cmra_homomorphism. Qed. Instance cmra_homomorphism_proper {A B : cmraT} (f : A → B) : CMRAHomomorphism f → Proper ((≡) ==> (≡)) f := λ _, ne_proper _. Instance ucmra_homomorphism_id {A : ucmraT} : UCMRAHomomorphism (@id A). Proof. repeat split; by try apply _. Qed. Instance ucmra_homomorphism_compose {A B C : ucmraT} (f : A → B) (g : B → C) : UCMRAHomomorphism f → UCMRAHomomorphism g → UCMRAHomomorphism (g ∘ f). Proof. split. apply _. by rewrite /= !ucmra_homomorphism_unit. Qed. `````` Robbert Krebbers committed May 25, 2016 688 689 ``````(** Functors *) Structure rFunctor := RFunctor { `````` Robbert Krebbers committed May 27, 2016 690 `````` rFunctor_car : cofeT → cofeT → cmraT; `````` Robbert Krebbers committed May 25, 2016 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 `````` 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 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 ``````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 747 748 749 750 751 752 753 754 755 756 757 758 759 ``````(** * 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 760 `````` Lemma cmra_transport_core x : T (core x) = core (T x). `````` Robbert Krebbers committed Feb 08, 2016 761