cmra.v 52 KB
 Ralf Jung committed Nov 22, 2016 1 ``````From iris.algebra Require Export ofe. `````` Ralf Jung committed Jan 05, 2017 2 ``````Set Default Proof Using "Type". `````` Robbert Krebbers committed Feb 01, 2016 3 `````` `````` Robbert Krebbers committed May 28, 2016 4 5 ``````Class PCore (A : Type) := pcore : A → option A. Instance: Params (@pcore) 2. `````` Robbert Krebbers committed Feb 01, 2016 6 7 8 9 10 11 `````` 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 12 13 14 15 16 ``````(* 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 17 18 19 ``````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 20 ``````Hint Extern 0 (_ ≼ _) => reflexivity. `````` Robbert Krebbers committed Feb 01, 2016 21 22 ``````Instance: Params (@included) 3. `````` Robbert Krebbers committed Nov 11, 2015 23 24 ``````Class ValidN (A : Type) := validN : nat → A → Prop. Instance: Params (@validN) 3. `````` Robbert Krebbers committed Feb 11, 2016 25 ``````Notation "✓{ n } x" := (validN n x) `````` Robbert Krebbers committed Feb 19, 2016 26 `````` (at level 20, n at next level, format "✓{ n } x"). `````` Robbert Krebbers committed Nov 11, 2015 27 `````` `````` Robbert Krebbers committed Feb 01, 2016 28 29 ``````Class Valid (A : Type) := valid : A → Prop. Instance: Params (@valid) 2. `````` Robbert Krebbers committed Feb 11, 2016 30 ``````Notation "✓ x" := (valid x) (at level 20) : C_scope. `````` Robbert Krebbers committed Feb 01, 2016 31 `````` `````` Ralf Jung committed Feb 10, 2016 32 ``````Definition includedN `{Dist A, Op A} (n : nat) (x y : A) := ∃ z, y ≡{n}≡ x ⋅ z. `````` Robbert Krebbers committed Nov 20, 2015 33 ``````Notation "x ≼{ n } y" := (includedN n x y) `````` Robbert Krebbers committed Feb 19, 2016 34 `````` (at level 70, n at next level, format "x ≼{ n } y") : C_scope. `````` Robbert Krebbers committed Nov 20, 2015 35 ``````Instance: Params (@includedN) 4. `````` Robbert Krebbers committed Feb 13, 2016 36 ``````Hint Extern 0 (_ ≼{_} _) => reflexivity. `````` Robbert Krebbers committed Nov 20, 2015 37 `````` `````` Robbert Krebbers committed May 28, 2016 38 ``````Record CMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A} := { `````` Robbert Krebbers committed Nov 11, 2015 39 `````` (* setoids *) `````` Robbert Krebbers committed Jan 14, 2016 40 `````` mixin_cmra_op_ne n (x : A) : Proper (dist n ==> dist n) (op x); `````` Robbert Krebbers committed May 28, 2016 41 42 `````` 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 43 `````` mixin_cmra_validN_ne n : Proper (dist n ==> impl) (validN n); `````` Robbert Krebbers committed Nov 11, 2015 44 `````` (* valid *) `````` Robbert Krebbers committed Feb 24, 2016 45 `````` mixin_cmra_valid_validN x : ✓ x ↔ ∀ n, ✓{n} x; `````` Robbert Krebbers committed Feb 01, 2016 46 `````` mixin_cmra_validN_S n x : ✓{S n} x → ✓{n} x; `````` Robbert Krebbers committed Nov 11, 2015 47 `````` (* monoid *) `````` Robbert Krebbers committed Feb 11, 2016 48 49 `````` mixin_cmra_assoc : Assoc (≡) (⋅); mixin_cmra_comm : Comm (≡) (⋅); `````` Robbert Krebbers committed May 28, 2016 50 51 `````` 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 52 `````` mixin_cmra_pcore_mono x y cx : `````` Robbert Krebbers committed May 28, 2016 53 `````` x ≼ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy; `````` Robbert Krebbers committed Feb 01, 2016 54 `````` mixin_cmra_validN_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x; `````` Robbert Krebbers committed Feb 24, 2016 55 56 `````` mixin_cmra_extend n x y1 y2 : ✓{n} x → x ≡{n}≡ y1 ⋅ y2 → `````` Robbert Krebbers committed Aug 14, 2016 57 `````` ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2 `````` Robbert Krebbers committed Nov 11, 2015 58 ``````}. `````` Robbert Krebbers committed Nov 22, 2015 59 `````` `````` Robbert Krebbers committed Nov 11, 2015 60 ``````(** Bundeled version *) `````` Robbert Krebbers committed Jun 15, 2016 61 ``````Structure cmraT := CMRAT' { `````` Robbert Krebbers committed Nov 11, 2015 62 63 64 `````` cmra_car :> Type; cmra_equiv : Equiv cmra_car; cmra_dist : Dist 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; `````` Ralf Jung committed Nov 22, 2016 69 `````` cmra_ofe_mixin : OfeMixin 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 ``````}. `````` Ralf Jung committed Nov 22, 2016 73 ``````Arguments CMRAT' _ {_ _ _ _ _ _} _ _ _. `````` Robbert Krebbers committed Jun 15, 2016 74 ``````Notation CMRAT A m m' := (CMRAT' A m m' A). `````` Robbert Krebbers committed Jan 14, 2016 75 76 77 ``````Arguments cmra_car : simpl never. Arguments cmra_equiv : simpl never. Arguments cmra_dist : simpl never. `````` Robbert Krebbers committed May 28, 2016 78 ``````Arguments cmra_pcore : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 79 ``````Arguments cmra_op : simpl never. `````` Robbert Krebbers committed Feb 24, 2016 80 ``````Arguments cmra_valid : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 81 ``````Arguments cmra_validN : simpl never. `````` Ralf Jung committed Nov 22, 2016 82 ``````Arguments cmra_ofe_mixin : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 83 ``````Arguments cmra_mixin : simpl never. `````` Robbert Krebbers committed Nov 11, 2015 84 ``````Add Printing Constructor cmraT. `````` Robbert Krebbers committed Jun 14, 2016 85 86 87 88 ``````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. `````` Ralf Jung committed Nov 22, 2016 89 90 ``````Coercion cmra_ofeC (A : cmraT) : ofeT := OfeT A (cmra_ofe_mixin A). Canonical Structure cmra_ofeC. `````` Robbert Krebbers committed Nov 11, 2015 91 `````` `````` Robbert Krebbers committed Jan 14, 2016 92 93 94 95 96 97 ``````(** 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 98 99 100 `````` 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 101 102 `````` 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 103 104 `````` 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 105 106 `````` 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 107 108 109 110 `````` 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 111 112 113 114 `````` 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 115 `````` Lemma cmra_pcore_mono x y cx : `````` Robbert Krebbers committed May 28, 2016 116 `````` x ≼ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy. `````` Ralf Jung committed Jul 25, 2016 117 `````` Proof. apply (mixin_cmra_pcore_mono _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Feb 01, 2016 118 119 `````` 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 120 `````` Lemma cmra_extend n x y1 y2 : `````` Ralf Jung committed Feb 10, 2016 121 `````` ✓{n} x → x ≡{n}≡ y1 ⋅ y2 → `````` Robbert Krebbers committed Aug 14, 2016 122 `````` ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2. `````` Robbert Krebbers committed Feb 24, 2016 123 `````` Proof. apply (mixin_cmra_extend _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Jan 14, 2016 124 125 ``````End cmra_mixin. `````` Robbert Krebbers committed May 28, 2016 126 127 128 129 130 131 132 133 ``````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 134 ``````(** * Exclusive elements (i.e., elements that cannot have a frame). *) `````` Robbert Krebbers committed Jun 16, 2016 135 136 ``````Class Exclusive {A : cmraT} (x : A) := exclusive0_l y : ✓{0} (x ⋅ y) → False. Arguments exclusive0_l {_} _ {_} _ _. `````` Jacques-Henri Jourdan committed May 31, 2016 137 `````` `````` Robbert Krebbers committed May 28, 2016 138 139 140 141 142 143 144 145 146 147 148 ``````(** * 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 149 ``````(** * CMRAs with a unit element *) `````` Robbert Krebbers committed Feb 01, 2016 150 ``````(** We use the notation ∅ because for most instances (maps, sets, etc) the `````` Ralf Jung committed Mar 08, 2016 151 ```````empty' element is the unit. *) `````` Robbert Krebbers committed May 28, 2016 152 ``````Record UCMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, Empty A} := { `````` Robbert Krebbers committed May 27, 2016 153 154 `````` mixin_ucmra_unit_valid : ✓ ∅; mixin_ucmra_unit_left_id : LeftId (≡) ∅ (⋅); `````` Robbert Krebbers committed May 28, 2016 155 `````` mixin_ucmra_pcore_unit : pcore ∅ ≡ Some ∅ `````` Robbert Krebbers committed Feb 01, 2016 156 ``````}. `````` Robbert Krebbers committed May 27, 2016 157 `````` `````` Robbert Krebbers committed Jun 15, 2016 158 ``````Structure ucmraT := UCMRAT' { `````` Robbert Krebbers committed May 27, 2016 159 160 161 `````` ucmra_car :> Type; ucmra_equiv : Equiv ucmra_car; ucmra_dist : Dist ucmra_car; `````` Robbert Krebbers committed May 28, 2016 162 `````` ucmra_pcore : PCore ucmra_car; `````` Robbert Krebbers committed May 27, 2016 163 164 165 166 `````` ucmra_op : Op ucmra_car; ucmra_valid : Valid ucmra_car; ucmra_validN : ValidN ucmra_car; ucmra_empty : Empty ucmra_car; `````` Ralf Jung committed Nov 22, 2016 167 `````` ucmra_ofe_mixin : OfeMixin ucmra_car; `````` Robbert Krebbers committed May 27, 2016 168 `````` ucmra_cmra_mixin : CMRAMixin ucmra_car; `````` Robbert Krebbers committed Jun 15, 2016 169 `````` ucmra_mixin : UCMRAMixin ucmra_car; `````` Robbert Krebbers committed Jun 15, 2016 170 `````` _ : Type; `````` Robbert Krebbers committed May 27, 2016 171 ``````}. `````` Ralf Jung committed Nov 22, 2016 172 ``````Arguments UCMRAT' _ {_ _ _ _ _ _ _} _ _ _ _. `````` Robbert Krebbers committed Jun 15, 2016 173 ``````Notation UCMRAT A m m' m'' := (UCMRAT' A m m' m'' A). `````` Robbert Krebbers committed May 27, 2016 174 175 176 ``````Arguments ucmra_car : simpl never. Arguments ucmra_equiv : simpl never. Arguments ucmra_dist : simpl never. `````` Robbert Krebbers committed May 28, 2016 177 ``````Arguments ucmra_pcore : simpl never. `````` Robbert Krebbers committed May 27, 2016 178 179 180 ``````Arguments ucmra_op : simpl never. Arguments ucmra_valid : simpl never. Arguments ucmra_validN : simpl never. `````` Ralf Jung committed Nov 22, 2016 181 ``````Arguments ucmra_ofe_mixin : simpl never. `````` Robbert Krebbers committed May 27, 2016 182 183 184 ``````Arguments ucmra_cmra_mixin : simpl never. Arguments ucmra_mixin : simpl never. Add Printing Constructor ucmraT. `````` Robbert Krebbers committed Jun 14, 2016 185 ``````Hint Extern 0 (Empty _) => eapply (@ucmra_empty _) : typeclass_instances. `````` Ralf Jung committed Nov 22, 2016 186 187 ``````Coercion ucmra_ofeC (A : ucmraT) : ofeT := OfeT A (ucmra_ofe_mixin A). Canonical Structure ucmra_ofeC. `````` Robbert Krebbers committed May 27, 2016 188 ``````Coercion ucmra_cmraR (A : ucmraT) : cmraT := `````` Ralf Jung committed Nov 22, 2016 189 `````` CMRAT A (ucmra_ofe_mixin A) (ucmra_cmra_mixin A). `````` Robbert Krebbers committed May 27, 2016 190 191 192 193 194 195 196 197 198 199 ``````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 200 201 `````` Lemma ucmra_pcore_unit : pcore (∅:A) ≡ Some ∅. Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin A)). Qed. `````` Robbert Krebbers committed May 27, 2016 202 ``````End ucmra_mixin. `````` Robbert Krebbers committed Jan 14, 2016 203 `````` `````` Robbert Krebbers committed Feb 24, 2016 204 ``````(** * Discrete CMRAs *) `````` Robbert Krebbers committed Feb 26, 2016 205 ``````Class CMRADiscrete (A : cmraT) := { `````` Robbert Krebbers committed Feb 24, 2016 206 207 208 209 `````` cmra_discrete :> Discrete A; cmra_discrete_valid (x : A) : ✓{0} x → ✓ x }. `````` Robbert Krebbers committed Jan 16, 2016 210 ``````(** * Morphisms *) `````` Robbert Krebbers committed Jan 14, 2016 211 ``````Class CMRAMonotone {A B : cmraT} (f : A → B) := { `````` Robbert Krebbers committed Feb 26, 2016 212 `````` cmra_monotone_ne n :> Proper (dist n ==> dist n) f; `````` Robbert Krebbers committed Sep 28, 2016 213 `````` cmra_monotone_validN n x : ✓{n} x → ✓{n} f x; `````` Ralf Jung committed Jul 25, 2016 214 `````` cmra_monotone x y : x ≼ y → f x ≼ f y `````` Robbert Krebbers committed Jan 14, 2016 215 ``````}. `````` Robbert Krebbers committed Sep 28, 2016 216 ``````Arguments cmra_monotone_validN {_ _} _ {_} _ _ _. `````` Ralf Jung committed Jul 25, 2016 217 ``````Arguments cmra_monotone {_ _} _ {_} _ _ _. `````` Robbert Krebbers committed Jan 14, 2016 218 `````` `````` Robbert Krebbers committed Sep 28, 2016 219 220 221 222 223 224 225 226 227 228 229 230 231 232 ``````(* 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 233 ``````(** * Properties **) `````` Robbert Krebbers committed Nov 11, 2015 234 ``````Section cmra. `````` Robbert Krebbers committed Jan 14, 2016 235 ``````Context {A : cmraT}. `````` Robbert Krebbers committed Nov 11, 2015 236 ``````Implicit Types x y z : A. `````` Robbert Krebbers committed Feb 01, 2016 237 ``````Implicit Types xs ys zs : list A. `````` Robbert Krebbers committed Nov 11, 2015 238 `````` `````` Robbert Krebbers committed Feb 01, 2016 239 ``````(** ** Setoids *) `````` Robbert Krebbers committed May 28, 2016 240 241 242 243 244 245 246 247 248 ``````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 249 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 250 251 252 `````` 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 253 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 254 255 256 257 ``````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 258 ``````Global Instance cmra_op_proper' : Proper ((≡) ==> (≡) ==> (≡)) (@op A _). `````` Robbert Krebbers committed Feb 01, 2016 259 260 261 262 263 264 265 ``````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 266 267 268 269 ``````Proof. intros x y Hxy; rewrite !cmra_valid_validN. by split=> ? n; [rewrite -Hxy|rewrite Hxy]. Qed. `````` Robbert Krebbers committed Feb 01, 2016 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 ``````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 288 289 290 291 ``````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 292 `````` `````` Robbert Krebbers committed May 28, 2016 293 294 295 296 ``````(** ** 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 297 ``````(** ** Validity *) `````` Robbert Krebbers committed Feb 18, 2016 298 ``````Lemma cmra_validN_le n n' x : ✓{n} x → n' ≤ n → ✓{n'} x. `````` Robbert Krebbers committed Feb 01, 2016 299 300 301 ``````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 302 ``````Lemma cmra_validN_op_r n x y : ✓{n} (x ⋅ y) → ✓{n} y. `````` Robbert Krebbers committed Feb 11, 2016 303 ``````Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 304 305 306 ``````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 307 ``````(** ** Core *) `````` Robbert Krebbers committed May 28, 2016 308 309 310 311 312 313 314 315 ``````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 316 317 318 319 ``````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 320 321 322 323 324 325 326 327 ``````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 328 `````` `````` Robbert Krebbers committed May 30, 2016 329 330 331 332 ``````(** ** 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 333 ``````(** ** Exclusive elements *) `````` Robbert Krebbers committed Jun 16, 2016 334 ``````Lemma exclusiveN_l n x `{!Exclusive x} y : ✓{n} (x ⋅ y) → False. `````` Robbert Krebbers committed Aug 30, 2016 335 ``````Proof. intros. eapply (exclusive0_l x y), cmra_validN_le; eauto with lia. Qed. `````` Robbert Krebbers committed Jun 16, 2016 336 337 338 339 340 341 ``````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 342 ``````Lemma exclusiveN_opM n x `{!Exclusive x} my : ✓{n} (x ⋅? my) → my = None. `````` Robbert Krebbers committed Aug 30, 2016 343 ``````Proof. destruct my as [y|]. move=> /(exclusiveN_l _ x) []. done. Qed. `````` Robbert Krebbers committed Oct 02, 2016 344 345 346 347 ``````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 348 `````` `````` Robbert Krebbers committed Feb 01, 2016 349 ``````(** ** Order *) `````` Robbert Krebbers committed Mar 11, 2016 350 351 ``````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 352 ``````Global Instance cmra_includedN_trans n : Transitive (@includedN A _ _ n). `````` Robbert Krebbers committed Feb 01, 2016 353 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 354 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 355 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 356 ``````Global Instance cmra_included_trans: Transitive (@included A _ _). `````` Robbert Krebbers committed Feb 01, 2016 357 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 358 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 359 ``````Qed. `````` Robbert Krebbers committed Sep 09, 2016 360 361 ``````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 362 ``````Lemma cmra_validN_includedN n x y : ✓{n} y → x ≼{n} y → ✓{n} x. `````` Robbert Krebbers committed Feb 01, 2016 363 ``````Proof. intros Hyv [z ?]; cofe_subst y; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 18, 2016 364 ``````Lemma cmra_validN_included n x y : ✓{n} y → x ≼ y → ✓{n} x. `````` Robbert Krebbers committed Mar 11, 2016 365 ``````Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 366 `````` `````` Robbert Krebbers committed Feb 18, 2016 367 ``````Lemma cmra_includedN_S n x y : x ≼{S n} y → x ≼{n} y. `````` Robbert Krebbers committed Feb 01, 2016 368 ``````Proof. by intros [z Hz]; exists z; apply dist_S. Qed. `````` Robbert Krebbers committed Feb 18, 2016 369 ``````Lemma cmra_includedN_le n n' x y : x ≼{n} y → n' ≤ n → x ≼{n'} y. `````` Robbert Krebbers committed Feb 01, 2016 370 371 372 373 374 375 376 ``````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 377 ``````Proof. rewrite (comm op); apply cmra_includedN_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 378 ``````Lemma cmra_included_r x y : y ≼ x ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 379 ``````Proof. rewrite (comm op); apply cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 20, 2015 380 `````` `````` Ralf Jung committed Jul 25, 2016 381 ``````Lemma cmra_pcore_mono' x y cx : `````` Robbert Krebbers committed May 28, 2016 382 383 384 `````` 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 385 `````` destruct (cmra_pcore_mono x y cx') as (cy&->&?); auto. `````` Robbert Krebbers committed May 28, 2016 386 387 `````` exists cy; by rewrite Hcx. Qed. `````` Ralf Jung committed Jul 25, 2016 388 ``````Lemma cmra_pcore_monoN' n x y cx : `````` Robbert Krebbers committed May 28, 2016 389 `````` x ≼{n} y → pcore x ≡{n}≡ Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼{n} cy. `````` Robbert Krebbers committed Feb 26, 2016 390 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 391 `````` intros [z Hy] (cx'&?&Hcx)%dist_Some_inv_r'. `````` Ralf Jung committed Jul 25, 2016 392 `````` destruct (cmra_pcore_mono x (x ⋅ z) cx') `````` Robbert Krebbers committed May 28, 2016 393 394 395 396 397 `````` 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 398 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 399 400 ``````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 401 `````` `````` Ralf Jung committed Jul 25, 2016 402 ``````Lemma cmra_monoN_l n x y z : x ≼{n} y → z ⋅ x ≼{n} z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 403 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Ralf Jung committed Jul 25, 2016 404 ``````Lemma cmra_mono_l x y z : x ≼ y → z ⋅ x ≼ z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 405 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Ralf Jung committed Jul 25, 2016 406 407 408 409 ``````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 410 411 412 413 ``````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 414 `````` `````` Robbert Krebbers committed Sep 28, 2016 415 416 417 418 419 420 421 ``````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 422 ``````Lemma cmra_included_dist_l n x1 x2 x1' : `````` Ralf Jung committed Feb 10, 2016 423 `````` x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2. `````` Robbert Krebbers committed Nov 11, 2015 424 ``````Proof. `````` Robbert Krebbers committed Feb 01, 2016 425 426 `````` intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using cmra_included_l. by rewrite Hx1 Hx2. `````` Robbert Krebbers committed Nov 11, 2015 427 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2016 428 `````` `````` Robbert Krebbers committed May 28, 2016 429 430 ``````(** ** Total core *) Section total_core. `````` Ralf Jung committed Jan 05, 2017 431 `````` Set Default Proof Using "Type*". `````` Robbert Krebbers committed May 28, 2016 432 433 434 435 436 437 438 439 440 441 `````` 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 442 `````` Lemma cmra_core_mono x y : x ≼ y → core x ≼ core y. `````` Robbert Krebbers committed May 28, 2016 443 444 `````` Proof. intros; destruct (cmra_total x) as [cx Hcx]. `````` Ralf Jung committed Jul 25, 2016 445 `````` destruct (cmra_pcore_mono x y cx) as (cy&Hcy&?); auto. `````` Robbert Krebbers committed May 28, 2016 446 447 448 449 450 451 452 453 454 455 456 457 458 `````` 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 459 460 `````` 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 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 `````` 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 491 `````` Lemma cmra_core_monoN n x y : x ≼{n} y → core x ≼{n} core y. `````` Robbert Krebbers committed May 28, 2016 492 493 `````` Proof. intros [z ->]. `````` Ralf Jung committed Jul 25, 2016 494 `````` apply cmra_included_includedN, cmra_core_mono, cmra_included_l. `````` Robbert Krebbers committed May 28, 2016 495 496 497 `````` Qed. End total_core. `````` Robbert Krebbers committed Jan 16, 2016 498 ``````(** ** Timeless *) `````` Robbert Krebbers committed Feb 10, 2016 499 ``````Lemma cmra_timeless_included_l x y : Timeless x → ✓{0} y → x ≼{0} y → x ≼ y. `````` Robbert Krebbers committed Dec 11, 2015 500 501 ``````Proof. intros ?? [x' ?]. `````` Robbert Krebbers committed Aug 14, 2016 502 `````` destruct (cmra_extend 0 y x x') as (z&z'&Hy&Hz&Hz'); auto; simpl in *. `````` Robbert Krebbers committed Jan 13, 2016 503 `````` by exists z'; rewrite Hy (timeless x z). `````` Robbert Krebbers committed Dec 11, 2015 504 ``````Qed. `````` Robbert Krebbers committed Aug 30, 2016 505 506 ``````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 507 ``````Lemma cmra_op_timeless x1 x2 : `````` Robbert Krebbers committed Dec 11, 2015 508 `````` ✓ (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2). `````` Robbert Krebbers committed Nov 18, 2015 509 510 ``````Proof. intros ??? z Hz. `````` Robbert Krebbers committed Aug 14, 2016 511 `````` destruct (cmra_extend 0 z x1 x2) as (y1&y2&Hz'&?&?); auto; simpl in *. `````` Robbert Krebbers committed Feb 24, 2016 512 `````` { rewrite -?Hz. by apply cmra_valid_validN. } `````` Robbert Krebbers committed Jan 13, 2016 513 `````` by rewrite Hz' (timeless x1 y1) // (timeless x2 y2). `````` Robbert Krebbers committed Nov 18, 2015 514 ``````Qed. `````` Robbert Krebbers committed Nov 20, 2015 515 `````` `````` Robbert Krebbers committed Feb 24, 2016 516 517 518 519 520 521 522 523 ``````(** ** 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 524 `````` split; first by apply cmra_included_includedN. `````` Robbert Krebbers committed Feb 24, 2016 525 526 `````` intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 11, 2015 527 528 ``````End cmra. `````` Robbert Krebbers committed May 27, 2016 529 530 ``````(** * Properties about CMRAs with a unit element **) Section ucmra. `````` Robbert Krebbers committed May 28, 2016 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 `````` 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 547 `````` intros x. destruct (cmra_pcore_mono' ∅ x ∅) as (cx&->&?); `````` Robbert Krebbers committed May 28, 2016 548 549 `````` eauto using ucmra_unit_least, (persistent ∅). Qed. `````` Robbert Krebbers committed May 27, 2016 550 ``````End ucmra. `````` Robbert Krebbers committed May 28, 2016 551 552 ``````Hint Immediate cmra_unit_total. `````` Robbert Krebbers committed Sep 01, 2016 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 `````` (** * 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 *) `````` Robbert Krebbers committed Jan 04, 2017 579 `````` Lemma persistent_dup_L x `{!Persistent x} : x = x ⋅ x. `````` Robbert Krebbers committed Sep 01, 2016 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 `````` 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 611 612 613 614 615 616 617 618 619 620 621 622 623 ``````(** * 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 624 `````` Context (core_mono : ∀ x y : A, x ≼ y → core x ≼ core y). `````` Robbert Krebbers committed May 28, 2016 625 626 627 `````` 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 628 `````` ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2). `````` Robbert Krebbers committed May 28, 2016 629 630 631 632 633 634 635 636 `````` 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 637 `````` - intros x y cx Hxy%core_mono Hx. move: Hxy. `````` Robbert Krebbers committed May 28, 2016 638 639 640 `````` rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto. Qed. End cmra_total. `````` Robbert Krebbers committed May 27, 2016 641 `````` `````` Robbert Krebbers committed Feb 01, 2016 642 ``````(** * Properties about monotone functions *) `````` Robbert Krebbers committed Jan 14, 2016 643 ``````Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A). `````` Robbert Krebbers committed Feb 26, 2016 644 ``````Proof. repeat split; by try apply _. Qed. `````` Robbert Krebbers committed Feb 01, 2016 645 646 ``````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 647 648 ``````Proof. split. `````` Robbert Krebbers committed Feb 26, 2016 649 `````` - apply _. `````` Robbert Krebbers committed Sep 28, 2016 650 `````` - move=> n x Hx /=. by apply cmra_monotone_validN, cmra_monotone_validN. `````` Ralf Jung committed Jul 25, 2016 651 `````` - move=> x y Hxy /=. by apply cmra_monotone, cmra_monotone. `````` Robbert Krebbers committed Nov 20, 2015 652 ``````Qed. `````` Robbert Krebbers committed Nov 16, 2015 653 `````` `````` Robbert Krebbers committed Feb 01, 2016 654 655 ``````Section cmra_monotone. Context {A B : cmraT} (f : A → B) `{!CMRAMonotone f}. `````` Robbert Krebbers committed Feb 26, 2016 656 `````` Global Instance cmra_monotone_proper : Proper ((≡) ==> (≡)) f := ne_proper _. `````` Ralf Jung committed Jul 25, 2016 657 `````` Lemma cmra_monotoneN n x y : x ≼{n} y → f x ≼{n} f y. `````` Robbert Krebbers committed Feb 01, 2016 658 `````` Proof. `````` Robbert Krebbers committed Feb 26, 2016 659 `````` intros [z ->]. `````` Ralf Jung committed Jul 25, 2016 660 `````` apply cmra_included_includedN, (cmra_monotone f), cmra_included_l. `````` Robbert Krebbers committed Feb 01, 2016 661 `````` Qed. `````` Robbert Krebbers committed Sep 28, 2016 662 663 `````` 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 664 665 ``````End cmra_monotone. `````` Robbert Krebbers committed Sep 28, 2016 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 ``````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 686 687 ``````(** Functors *) Structure rFunctor := RFunctor { `````` Ralf Jung committed Nov 22, 2016 688 `````` rFunctor_car : ofeT → ofeT → cmraT; `````` Robbert Krebbers committed May 25, 2016 689 690 691 692 693 694 695 696 697 698 699 700 701 702 `````` 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. `````` Ralf Jung committed Jan 06, 2017 703 704 705 ``````Delimit Scope rFunctor_scope with RF. Bind Scope rFunctor_scope with rFunctor. `````` Robbert Krebbers committed May 25, 2016 706 707 708 ``````Class rFunctorContractive (F : rFunctor) := rFunctor_contractive A1 A2 B1 B2 :> Contractive (@rFunctor_map F A1 A2 B1 B2). `````` Ralf Jung committed Nov 22, 2016 709 ``````Definition rFunctor_diag (F: rFunctor) (A: ofeT) : cmraT := rFunctor_car F A A. `````` Robbert Krebbers committed May 25, 2016 710 711 712 713 714 ``````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. `````` Ralf Jung committed Jan 06, 2017 715 ``````Coercion constRF : cmraT >-> rFunctor. `````` Robbert Krebbers committed May 25, 2016 716 717 718 719 `````` Instance constRF_contractive B : rFunctorContractive (constRF B). Proof. rewrite /rFunctorContractive; apply _. Qed. `````` Robbert Krebbers committed May 27, 2016 720 ``````Structure urFunctor := URFunctor { `````` Ralf Jung committed Nov 22, 2016 721 `````` urFunctor_car : ofeT → ofeT → ucmraT; `````` Robbert Krebbers committed May 27, 2016 722 723 724 725 726 727 728 729 730 731 732 733 734 735 `````` 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. `````` Ralf Jung committed Jan 06, 2017 736 737 738 ``````Delimit Scope urFunctor_scope with URF. Bind Scope urFunctor_scope with urFunctor. `````` Robbert Krebbers committed May 27, 2016 739 740 741 ``````Class urFunctorContractive (F : urFunctor) := urFunctor_contractive A1 A2 B1 B2 :> Contractive (@urFunctor_map F A1 A2 B1 B2). `````` Ralf Jung committed Nov 22, 2016 742 ``````Definition urFunctor_diag (F: urFunctor) (A: ofeT) : ucmraT := urFunctor_car F A A. `````` Robbert Krebbers committed May 27, 2016 743 744 745 746 747 ``````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. `````` Ralf Jung committed Jan 06, 2017 748 ``````Coercion constURF : ucmraT >-> urFunctor. `````` Robbert Krebbers committed May 27, 2016 749 750 751 752 `````` Instance constURF_contractive B : urFunctorContractive (constURF B). Proof. rewrite /urFunctorContractive; apply _. Qed. `````` Robbert Krebbers committed Feb 08, 2016 753 754 755 756 757 758 759 760 761 762 763 764 765 ``````(** * 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 766 `````` Lemma cmra_transport_core x : T (core x) = core (T x). `````` Robbert Krebbers committed Feb 08, 2016 767 `````` Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 768 `````` Lemma cmra_transport_validN n x : ✓{n} T x ↔ ✓{n} x. `````` Robbert Krebbers committed Feb 08, 2016 769 `````` Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 770 `````` Lemma cmra_transport_valid x : ✓ T x ↔ ✓ x. `````` Robbert Krebbers committed Feb 08, 2016 771 772 773 `````` 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 774 775 `````` Global Instance cmra_transport_persistent x : Persistent x → Persistent (T x). Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 08, 2016 776 777 ``````End cmra_transport. `````` Robbert Krebbers committed Feb 01, 2016 778 779 ``````(** * Instances *) (** ** Discrete CMRA *) `````` Robbert Krebbers committed May 28, 2016 780 ``````Record RAMixin A `{Equiv A, PCore A, Op A, Valid A} := { `````` Robbert Krebbers committed Feb 01, 2016 781 `````` (* setoids *) `````` Robbert Krebbers committed May 28, 2016 782 783 784 785 `````` 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 786 `````` (* monoid *) `````` Robbert Krebbers committed May 25, 2016 787 788 `````` ra_assoc : Assoc (≡) (⋅); ra_comm : Comm (≡) (⋅); `````` Robbert Krebbers committed May 28, 2016 789 790 `````` 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 791 `````` ra_pcore_mono x y cx : `````` Robbert Krebbers committed May 28, 2016 792 `````` x ≼ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy; `````` Robbert Krebbers committed Mar 11, 2016 793 `````` ra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x `````` Robbert Krebbers committed Feb 01, 2016 794 795 ``````}. `````` Robbert Krebbers committed Nov 16, 2015 796 ``````Section discrete. `````` Robbert Krebbers committed May 28, 2016 797 `````` Context `{Equiv A, PCore A, Op A, Valid A, @Equivalence A (≡)}. `````` Robbert Krebbers committed May 25, 2016 798 `````` Context (ra_mix : RAMixin A). `````` Ralf Jung committed Nov 22, 2016 799 `````` Existing Instances discrete_dist. `````` Robbert Krebbers committed Feb 01, 2016 800 `````` `````` Robbert Krebbers committed Feb 10, 2016 801 `````` Instance discrete_validN : ValidN A := λ n x, ✓ x. `````` Robbert Krebbers committed Jan 14, 2016 802 `````` Definition discrete_cmra_mixin : CMRAMixin A. `````` Robbert Krebbers committed Nov 16, 2015 803 `````` Proof. `````` Robbert Krebbers committed May 25, 2016 804 `````` destruct ra_mix; split; try done. `````` Robbert Krebbers committed Feb 24, 2016 805 `````` - intros x; split; first done. by move=> /(_ 0). `````` Robbert Krebbers committed Aug 14, 2016 806 `````` - intros n x y1 y2 ??; by exists y1, y2. ``````