cmra.v 57.7 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 *) `````` Ralf Jung committed Jan 27, 2017 40 `````` mixin_cmra_op_ne (x : A) : NonExpansive (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 Feb 09, 2017 74 75 76 77 78 ``````(* Given [m : CMRAMixin A], the notation [CMRAT A m] provides a smart constructor, which uses [ofe_mixin_of A] to infer the canonical OFE mixin of the type [A], so that it does not have to be given manually. *) Notation CMRAT A m := (CMRAT' A (ofe_mixin_of A%type) m A) (only parsing). `````` Robbert Krebbers committed Jan 14, 2016 79 80 81 ``````Arguments cmra_car : simpl never. Arguments cmra_equiv : simpl never. Arguments cmra_dist : simpl never. `````` Robbert Krebbers committed May 28, 2016 82 ``````Arguments cmra_pcore : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 83 ``````Arguments cmra_op : simpl never. `````` Robbert Krebbers committed Feb 24, 2016 84 ``````Arguments cmra_valid : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 85 ``````Arguments cmra_validN : simpl never. `````` Ralf Jung committed Nov 22, 2016 86 ``````Arguments cmra_ofe_mixin : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 87 ``````Arguments cmra_mixin : simpl never. `````` Robbert Krebbers committed Nov 11, 2015 88 ``````Add Printing Constructor cmraT. `````` Robbert Krebbers committed Jun 14, 2016 89 90 91 92 ``````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 93 94 ``````Coercion cmra_ofeC (A : cmraT) : ofeT := OfeT A (cmra_ofe_mixin A). Canonical Structure cmra_ofeC. `````` Robbert Krebbers committed Nov 11, 2015 95 `````` `````` Robbert Krebbers committed Feb 09, 2017 96 97 98 99 ``````Definition cmra_mixin_of' A {Ac : cmraT} (f : Ac → A) : CMRAMixin Ac := cmra_mixin Ac. Notation cmra_mixin_of A := ltac:(let H := eval hnf in (cmra_mixin_of' A id) in exact H) (only parsing). `````` Robbert Krebbers committed Jan 14, 2016 100 101 102 103 ``````(** Lifting properties from the mixin *) Section cmra_mixin. Context {A : cmraT}. Implicit Types x y : A. `````` Ralf Jung committed Jan 27, 2017 104 `````` Global Instance cmra_op_ne (x : A) : NonExpansive (op x). `````` Robbert Krebbers committed Jan 14, 2016 105 `````` Proof. apply (mixin_cmra_op_ne _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed May 28, 2016 106 107 108 `````` 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 109 110 `````` 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 111 112 `````` 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 113 114 `````` 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 115 116 117 118 `````` 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 119 120 121 122 `````` 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 123 `````` Lemma cmra_pcore_mono x y cx : `````` Robbert Krebbers committed May 28, 2016 124 `````` x ≼ y → pcore x = Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼ cy. `````` Ralf Jung committed Jul 25, 2016 125 `````` Proof. apply (mixin_cmra_pcore_mono _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Feb 01, 2016 126 127 `````` 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 128 `````` Lemma cmra_extend n x y1 y2 : `````` Ralf Jung committed Feb 10, 2016 129 `````` ✓{n} x → x ≡{n}≡ y1 ⋅ y2 → `````` Robbert Krebbers committed Aug 14, 2016 130 `````` ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2. `````` Robbert Krebbers committed Feb 24, 2016 131 `````` Proof. apply (mixin_cmra_extend _ (cmra_mixin A)). Qed. `````` Robbert Krebbers committed Jan 14, 2016 132 133 ``````End cmra_mixin. `````` Robbert Krebbers committed May 28, 2016 134 135 136 137 138 139 140 ``````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 {_} _ {_}. `````` Robbert Krebbers committed Jan 22, 2017 141 ``````Hint Mode Persistent + ! : typeclass_instances. `````` Robbert Krebbers committed May 28, 2016 142 `````` `````` Jacques-Henri Jourdan committed May 31, 2016 143 ``````(** * Exclusive elements (i.e., elements that cannot have a frame). *) `````` Robbert Krebbers committed Jun 16, 2016 144 145 ``````Class Exclusive {A : cmraT} (x : A) := exclusive0_l y : ✓{0} (x ⋅ y) → False. Arguments exclusive0_l {_} _ {_} _ _. `````` Robbert Krebbers committed Jan 22, 2017 146 ``````Hint Mode Exclusive + ! : typeclass_instances. `````` Jacques-Henri Jourdan committed May 31, 2016 147 `````` `````` Jacques-Henri Jourdan committed Feb 01, 2017 148 149 150 151 152 153 154 155 156 157 158 159 ``````(** * Cancelable elements. *) Class Cancelable {A : cmraT} (x : A) := cancelableN n y z : ✓{n}(x ⋅ y) → x ⋅ y ≡{n}≡ x ⋅ z → y ≡{n}≡ z. Arguments cancelableN {_} _ {_} _ _ _ _. Hint Mode Cancelable + ! : typeclass_instances. (** * Identity-free elements. *) Class IdFree {A : cmraT} (x : A) := id_free0_r y : ✓{0}x → x ⋅ y ≡{0}≡ x → False. Arguments id_free0_r {_} _ {_} _ _. Hint Mode IdFree + ! : typeclass_instances. `````` Robbert Krebbers committed May 28, 2016 160 161 162 163 164 165 166 167 168 169 170 ``````(** * 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 171 ``````(** * CMRAs with a unit element *) `````` Robbert Krebbers committed Feb 01, 2016 172 ``````(** We use the notation ∅ because for most instances (maps, sets, etc) the `````` Ralf Jung committed Mar 08, 2016 173 ```````empty' element is the unit. *) `````` Robbert Krebbers committed May 28, 2016 174 ``````Record UCMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, Empty A} := { `````` Robbert Krebbers committed May 27, 2016 175 176 `````` mixin_ucmra_unit_valid : ✓ ∅; mixin_ucmra_unit_left_id : LeftId (≡) ∅ (⋅); `````` Robbert Krebbers committed May 28, 2016 177 `````` mixin_ucmra_pcore_unit : pcore ∅ ≡ Some ∅ `````` Robbert Krebbers committed Feb 01, 2016 178 ``````}. `````` Robbert Krebbers committed May 27, 2016 179 `````` `````` Robbert Krebbers committed Jun 15, 2016 180 ``````Structure ucmraT := UCMRAT' { `````` Robbert Krebbers committed May 27, 2016 181 182 183 `````` ucmra_car :> Type; ucmra_equiv : Equiv ucmra_car; ucmra_dist : Dist ucmra_car; `````` Robbert Krebbers committed May 28, 2016 184 `````` ucmra_pcore : PCore ucmra_car; `````` Robbert Krebbers committed May 27, 2016 185 186 187 188 `````` 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 189 `````` ucmra_ofe_mixin : OfeMixin ucmra_car; `````` Robbert Krebbers committed May 27, 2016 190 `````` ucmra_cmra_mixin : CMRAMixin ucmra_car; `````` Robbert Krebbers committed Jun 15, 2016 191 `````` ucmra_mixin : UCMRAMixin ucmra_car; `````` Robbert Krebbers committed Jun 15, 2016 192 `````` _ : Type; `````` Robbert Krebbers committed May 27, 2016 193 ``````}. `````` Ralf Jung committed Nov 22, 2016 194 ``````Arguments UCMRAT' _ {_ _ _ _ _ _ _} _ _ _ _. `````` Robbert Krebbers committed Feb 09, 2017 195 196 ``````Notation UCMRAT A m := (UCMRAT' A (ofe_mixin_of A%type) (cmra_mixin_of A%type) m A) (only parsing). `````` Robbert Krebbers committed May 27, 2016 197 198 199 ``````Arguments ucmra_car : simpl never. Arguments ucmra_equiv : simpl never. Arguments ucmra_dist : simpl never. `````` Robbert Krebbers committed May 28, 2016 200 ``````Arguments ucmra_pcore : simpl never. `````` Robbert Krebbers committed May 27, 2016 201 202 203 ``````Arguments ucmra_op : simpl never. Arguments ucmra_valid : simpl never. Arguments ucmra_validN : simpl never. `````` Ralf Jung committed Nov 22, 2016 204 ``````Arguments ucmra_ofe_mixin : simpl never. `````` Robbert Krebbers committed May 27, 2016 205 206 207 ``````Arguments ucmra_cmra_mixin : simpl never. Arguments ucmra_mixin : simpl never. Add Printing Constructor ucmraT. `````` Robbert Krebbers committed Jun 14, 2016 208 ``````Hint Extern 0 (Empty _) => eapply (@ucmra_empty _) : typeclass_instances. `````` Ralf Jung committed Nov 22, 2016 209 210 ``````Coercion ucmra_ofeC (A : ucmraT) : ofeT := OfeT A (ucmra_ofe_mixin A). Canonical Structure ucmra_ofeC. `````` Robbert Krebbers committed May 27, 2016 211 ``````Coercion ucmra_cmraR (A : ucmraT) : cmraT := `````` Robbert Krebbers committed Feb 09, 2017 212 `````` CMRAT' A (ucmra_ofe_mixin A) (ucmra_cmra_mixin A) A. `````` Robbert Krebbers committed May 27, 2016 213 214 215 216 217 218 219 220 221 222 ``````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 223 224 `````` Lemma ucmra_pcore_unit : pcore (∅:A) ≡ Some ∅. Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin A)). Qed. `````` Robbert Krebbers committed May 27, 2016 225 ``````End ucmra_mixin. `````` Robbert Krebbers committed Jan 14, 2016 226 `````` `````` Robbert Krebbers committed Feb 24, 2016 227 ``````(** * Discrete CMRAs *) `````` Robbert Krebbers committed Feb 26, 2016 228 ``````Class CMRADiscrete (A : cmraT) := { `````` Robbert Krebbers committed Feb 24, 2016 229 230 231 232 `````` cmra_discrete :> Discrete A; cmra_discrete_valid (x : A) : ✓{0} x → ✓ x }. `````` Robbert Krebbers committed Jan 16, 2016 233 ``````(** * Morphisms *) `````` Robbert Krebbers committed Jan 14, 2016 234 ``````Class CMRAMonotone {A B : cmraT} (f : A → B) := { `````` Ralf Jung committed Jan 27, 2017 235 `````` cmra_monotone_ne :> NonExpansive f; `````` Robbert Krebbers committed Sep 28, 2016 236 `````` cmra_monotone_validN n x : ✓{n} x → ✓{n} f x; `````` Ralf Jung committed Jul 25, 2016 237 `````` cmra_monotone x y : x ≼ y → f x ≼ f y `````` Robbert Krebbers committed Jan 14, 2016 238 ``````}. `````` Robbert Krebbers committed Sep 28, 2016 239 ``````Arguments cmra_monotone_validN {_ _} _ {_} _ _ _. `````` Ralf Jung committed Jul 25, 2016 240 ``````Arguments cmra_monotone {_ _} _ {_} _ _ _. `````` Robbert Krebbers committed Jan 14, 2016 241 `````` `````` Robbert Krebbers committed Sep 28, 2016 242 243 244 ``````(* 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) := { `````` Ralf Jung committed Jan 27, 2017 245 `````` cmra_homomorphism_ne :> NonExpansive f; `````` Robbert Krebbers committed Sep 28, 2016 246 247 248 249 250 251 252 253 254 255 `````` 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 256 ``````(** * Properties **) `````` Robbert Krebbers committed Nov 11, 2015 257 ``````Section cmra. `````` Robbert Krebbers committed Jan 14, 2016 258 ``````Context {A : cmraT}. `````` Robbert Krebbers committed Nov 11, 2015 259 ``````Implicit Types x y z : A. `````` Robbert Krebbers committed Feb 01, 2016 260 ``````Implicit Types xs ys zs : list A. `````` Robbert Krebbers committed Nov 11, 2015 261 `````` `````` Robbert Krebbers committed Feb 01, 2016 262 ``````(** ** Setoids *) `````` Ralf Jung committed Jan 27, 2017 263 ``````Global Instance cmra_pcore_ne' : NonExpansive (@pcore A _). `````` Robbert Krebbers committed May 28, 2016 264 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 265 `````` intros n x y Hxy. destruct (pcore x) as [cx|] eqn:?. `````` Robbert Krebbers committed May 28, 2016 266 267 268 269 270 271 `````` { 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 272 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 273 274 275 `````` 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 276 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 277 278 ``````Global Instance cmra_pcore_proper' : Proper ((≡) ==> (≡)) (@pcore A _). Proof. apply (ne_proper _). Qed. `````` Ralf Jung committed Jan 27, 2017 279 280 ``````Global Instance cmra_op_ne' : NonExpansive2 (@op A _). Proof. intros n x1 x2 Hx y1 y2 Hy. by rewrite Hy (comm _ x1) Hx (comm _ y2). Qed. `````` Robbert Krebbers committed Sep 28, 2016 281 ``````Global Instance cmra_op_proper' : Proper ((≡) ==> (≡) ==> (≡)) (@op A _). `````` Robbert Krebbers committed Feb 01, 2016 282 283 284 285 286 287 288 ``````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 289 290 291 292 ``````Proof. intros x y Hxy; rewrite !cmra_valid_validN. by split=> ? n; [rewrite -Hxy|rewrite Hxy]. Qed. `````` Robbert Krebbers committed Feb 01, 2016 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 ``````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. `````` Ralf Jung committed Jan 27, 2017 311 ``````Global Instance cmra_opM_ne : NonExpansive2 (@opM A). `````` Robbert Krebbers committed Feb 09, 2017 312 ``````Proof. destruct 2; by ofe_subst. Qed. `````` Robbert Krebbers committed May 28, 2016 313 314 ``````Global Instance cmra_opM_proper : Proper ((≡) ==> (≡) ==> (≡)) (@opM A). Proof. destruct 2; by setoid_subst. Qed. `````` Robbert Krebbers committed Feb 01, 2016 315 `````` `````` Robbert Krebbers committed May 28, 2016 316 317 318 319 ``````(** ** 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 320 ``````(** ** Validity *) `````` Robbert Krebbers committed Feb 18, 2016 321 ``````Lemma cmra_validN_le n n' x : ✓{n} x → n' ≤ n → ✓{n'} x. `````` Robbert Krebbers committed Feb 01, 2016 322 323 324 ``````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 325 ``````Lemma cmra_validN_op_r n x y : ✓{n} (x ⋅ y) → ✓{n} y. `````` Robbert Krebbers committed Feb 11, 2016 326 ``````Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 327 328 329 ``````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 330 ``````(** ** Core *) `````` Robbert Krebbers committed May 28, 2016 331 332 333 ``````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. `````` Jacques-Henri Jourdan committed Feb 01, 2017 334 ``````Proof. intros. rewrite comm. by apply cmra_pcore_l. Qed. `````` Robbert Krebbers committed May 28, 2016 335 ``````Lemma cmra_pcore_r' x cx : pcore x ≡ Some cx → x ⋅ cx ≡ x. `````` Jacques-Henri Jourdan committed Feb 01, 2017 336 ``````Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_r. Qed. `````` Robbert Krebbers committed May 28, 2016 337 ``````Lemma cmra_pcore_idemp' x cx : pcore x ≡ Some cx → pcore cx ≡ Some cx. `````` Jacques-Henri Jourdan committed Feb 01, 2017 338 ``````Proof. intros (cx'&?&->)%equiv_Some_inv_r'. eauto using cmra_pcore_idemp. Qed. `````` Robbert Krebbers committed May 30, 2016 339 340 341 342 ``````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 343 344 345 346 347 348 349 350 ``````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 351 `````` `````` Robbert Krebbers committed May 30, 2016 352 353 354 355 ``````(** ** 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 356 ``````(** ** Exclusive elements *) `````` Robbert Krebbers committed Jun 16, 2016 357 ``````Lemma exclusiveN_l n x `{!Exclusive x} y : ✓{n} (x ⋅ y) → False. `````` Robbert Krebbers committed Aug 30, 2016 358 ``````Proof. intros. eapply (exclusive0_l x y), cmra_validN_le; eauto with lia. Qed. `````` Robbert Krebbers committed Jun 16, 2016 359 360 361 362 363 364 ``````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 365 ``````Lemma exclusiveN_opM n x `{!Exclusive x} my : ✓{n} (x ⋅? my) → my = None. `````` Robbert Krebbers committed Aug 30, 2016 366 ``````Proof. destruct my as [y|]. move=> /(exclusiveN_l _ x) []. done. Qed. `````` Robbert Krebbers committed Oct 02, 2016 367 368 369 370 ``````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 371 `````` `````` Robbert Krebbers committed Feb 01, 2016 372 ``````(** ** Order *) `````` Robbert Krebbers committed Mar 11, 2016 373 374 ``````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 375 ``````Global Instance cmra_includedN_trans n : Transitive (@includedN A _ _ n). `````` Robbert Krebbers committed Feb 01, 2016 376 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 377 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 378 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 379 ``````Global Instance cmra_included_trans: Transitive (@included A _ _). `````` Robbert Krebbers committed Feb 01, 2016 380 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 381 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 382 ``````Qed. `````` Robbert Krebbers committed Sep 09, 2016 383 384 ``````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 385 ``````Lemma cmra_validN_includedN n x y : ✓{n} y → x ≼{n} y → ✓{n} x. `````` Robbert Krebbers committed Feb 09, 2017 386 ``````Proof. intros Hyv [z ?]; ofe_subst y; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 18, 2016 387 ``````Lemma cmra_validN_included n x y : ✓{n} y → x ≼ y → ✓{n} x. `````` Robbert Krebbers committed Mar 11, 2016 388 ``````Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 389 `````` `````` Robbert Krebbers committed Feb 18, 2016 390 ``````Lemma cmra_includedN_S n x y : x ≼{S n} y → x ≼{n} y. `````` Robbert Krebbers committed Feb 01, 2016 391 ``````Proof. by intros [z Hz]; exists z; apply dist_S. Qed. `````` Robbert Krebbers committed Feb 18, 2016 392 ``````Lemma cmra_includedN_le n n' x y : x ≼{n} y → n' ≤ n → x ≼{n'} y. `````` Robbert Krebbers committed Feb 01, 2016 393 394 395 396 397 398 399 ``````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 400 ``````Proof. rewrite (comm op); apply cmra_includedN_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 401 ``````Lemma cmra_included_r x y : y ≼ x ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 402 ``````Proof. rewrite (comm op); apply cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 20, 2015 403 `````` `````` Ralf Jung committed Jul 25, 2016 404 ``````Lemma cmra_pcore_mono' x y cx : `````` Robbert Krebbers committed May 28, 2016 405 406 407 `````` 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 408 `````` destruct (cmra_pcore_mono x y cx') as (cy&->&?); auto. `````` Robbert Krebbers committed May 28, 2016 409 410 `````` exists cy; by rewrite Hcx. Qed. `````` Ralf Jung committed Jul 25, 2016 411 ``````Lemma cmra_pcore_monoN' n x y cx : `````` Robbert Krebbers committed May 28, 2016 412 `````` x ≼{n} y → pcore x ≡{n}≡ Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼{n} cy. `````` Robbert Krebbers committed Feb 26, 2016 413 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 414 `````` intros [z Hy] (cx'&?&Hcx)%dist_Some_inv_r'. `````` Ralf Jung committed Jul 25, 2016 415 `````` destruct (cmra_pcore_mono x (x ⋅ z) cx') `````` Robbert Krebbers committed May 28, 2016 416 417 418 419 420 `````` 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 421 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 422 423 ``````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 424 `````` `````` Ralf Jung committed Jul 25, 2016 425 ``````Lemma cmra_monoN_l n x y z : x ≼{n} y → z ⋅ x ≼{n} z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 426 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Ralf Jung committed Jul 25, 2016 427 ``````Lemma cmra_mono_l x y z : x ≼ y → z ⋅ x ≼ z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 428 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Ralf Jung committed Jul 25, 2016 429 430 431 432 ``````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 433 434 435 436 ``````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 437 `````` `````` Robbert Krebbers committed Sep 28, 2016 438 439 440 441 442 443 444 ``````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 445 ``````Lemma cmra_included_dist_l n x1 x2 x1' : `````` Ralf Jung committed Feb 10, 2016 446 `````` x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2. `````` Robbert Krebbers committed Nov 11, 2015 447 ``````Proof. `````` Robbert Krebbers committed Feb 01, 2016 448 449 `````` intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using cmra_included_l. by rewrite Hx1 Hx2. `````` Robbert Krebbers committed Nov 11, 2015 450 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2016 451 `````` `````` Robbert Krebbers committed May 28, 2016 452 453 ``````(** ** Total core *) Section total_core. `````` Ralf Jung committed Jan 25, 2017 454 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed May 28, 2016 455 456 457 458 459 460 461 462 463 464 `````` 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 465 `````` Lemma cmra_core_mono x y : x ≼ y → core x ≼ core y. `````` Robbert Krebbers committed May 28, 2016 466 467 `````` Proof. intros; destruct (cmra_total x) as [cx Hcx]. `````` Ralf Jung committed Jul 25, 2016 468 `````` destruct (cmra_pcore_mono x y cx) as (cy&Hcy&?); auto. `````` Robbert Krebbers committed May 28, 2016 469 470 471 `````` by rewrite /core /= Hcx Hcy. Qed. `````` Ralf Jung committed Jan 27, 2017 472 `````` Global Instance cmra_core_ne : NonExpansive (@core A _). `````` Robbert Krebbers committed May 28, 2016 473 `````` Proof. `````` Ralf Jung committed Jan 27, 2017 474 `````` intros n x y Hxy. destruct (cmra_total x) as [cx Hcx]. `````` Robbert Krebbers committed May 28, 2016 475 476 477 478 479 480 481 `````` 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 482 483 `````` 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 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 `````` 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 514 `````` Lemma cmra_core_monoN n x y : x ≼{n} y → core x ≼{n} core y. `````` Robbert Krebbers committed May 28, 2016 515 516 `````` Proof. intros [z ->]. `````` Ralf Jung committed Jul 25, 2016 517 `````` apply cmra_included_includedN, cmra_core_mono, cmra_included_l. `````` Robbert Krebbers committed May 28, 2016 518 519 520 `````` Qed. End total_core. `````` Robbert Krebbers committed Jan 16, 2016 521 ``````(** ** Timeless *) `````` Robbert Krebbers committed Feb 10, 2016 522 ``````Lemma cmra_timeless_included_l x y : Timeless x → ✓{0} y → x ≼{0} y → x ≼ y. `````` Robbert Krebbers committed Dec 11, 2015 523 524 ``````Proof. intros ?? [x' ?]. `````` Robbert Krebbers committed Aug 14, 2016 525 `````` destruct (cmra_extend 0 y x x') as (z&z'&Hy&Hz&Hz'); auto; simpl in *. `````` Robbert Krebbers committed Jan 13, 2016 526 `````` by exists z'; rewrite Hy (timeless x z). `````` Robbert Krebbers committed Dec 11, 2015 527 ``````Qed. `````` Robbert Krebbers committed Aug 30, 2016 528 529 ``````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 530 ``````Lemma cmra_op_timeless x1 x2 : `````` Robbert Krebbers committed Dec 11, 2015 531 `````` ✓ (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2). `````` Robbert Krebbers committed Nov 18, 2015 532 533 ``````Proof. intros ??? z Hz. `````` Robbert Krebbers committed Aug 14, 2016 534 `````` destruct (cmra_extend 0 z x1 x2) as (y1&y2&Hz'&?&?); auto; simpl in *. `````` Robbert Krebbers committed Feb 24, 2016 535 `````` { rewrite -?Hz. by apply cmra_valid_validN. } `````` Robbert Krebbers committed Jan 13, 2016 536 `````` by rewrite Hz' (timeless x1 y1) // (timeless x2 y2). `````` Robbert Krebbers committed Nov 18, 2015 537 ``````Qed. `````` Robbert Krebbers committed Nov 20, 2015 538 `````` `````` Robbert Krebbers committed Feb 24, 2016 539 540 541 542 543 544 545 546 ``````(** ** 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 547 `````` split; first by apply cmra_included_includedN. `````` Robbert Krebbers committed Feb 24, 2016 548 549 `````` intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l. Qed. `````` Jacques-Henri Jourdan committed Feb 01, 2017 550 551 552 `````` (** Cancelable elements *) Global Instance cancelable_proper : Proper (equiv ==> iff) (@Cancelable A). `````` Robbert Krebbers committed Feb 03, 2017 553 554 ``````Proof. unfold Cancelable. intros x x' EQ. by setoid_rewrite EQ. Qed. Lemma cancelable x `{!Cancelable x} y z : ✓(x ⋅ y) → x ⋅ y ≡ x ⋅ z → y ≡ z. `````` Jacques-Henri Jourdan committed Feb 01, 2017 555 556 557 558 559 560 561 ``````Proof. rewrite !equiv_dist cmra_valid_validN. intros. by apply (cancelableN x). Qed. Lemma discrete_cancelable x `{CMRADiscrete A}: (∀ y z, ✓(x ⋅ y) → x ⋅ y ≡ x ⋅ z → y ≡ z) → Cancelable x. Proof. intros ????. rewrite -!timeless_iff -cmra_discrete_valid_iff. auto. Qed. Global Instance cancelable_op x y : Cancelable x → Cancelable y → Cancelable (x ⋅ y). Proof. `````` Robbert Krebbers committed Feb 03, 2017 562 `````` intros ?? n z z' ??. apply (cancelableN y), (cancelableN x). `````` Jacques-Henri Jourdan committed Feb 01, 2017 563 564 565 566 567 `````` - eapply cmra_validN_op_r. by rewrite assoc. - by rewrite assoc. - by rewrite !assoc. Qed. Global Instance exclusive_cancelable (x : A) : Exclusive x → Cancelable x. `````` Robbert Krebbers committed Feb 03, 2017 568 ``````Proof. intros ? n z z' []%(exclusiveN_l _ x). Qed. `````` Jacques-Henri Jourdan committed Feb 01, 2017 569 570 `````` (** Id-free elements *) `````` Robbert Krebbers committed Feb 03, 2017 571 ``````Global Instance id_free_ne n : Proper (dist n ==> iff) (@IdFree A). `````` Jacques-Henri Jourdan committed Feb 01, 2017 572 ``````Proof. `````` Robbert Krebbers committed Feb 03, 2017 573 574 `````` intros x x' EQ%(dist_le _ 0); last lia. rewrite /IdFree. split=> y ?; (rewrite -EQ || rewrite EQ); eauto. `````` Jacques-Henri Jourdan committed Feb 01, 2017 575 576 ``````Qed. Global Instance id_free_proper : Proper (equiv ==> iff) (@IdFree A). `````` Robbert Krebbers committed Feb 03, 2017 577 ``````Proof. by move=> P Q /equiv_dist /(_ 0)=> ->. Qed. `````` Jacques-Henri Jourdan committed Feb 01, 2017 578 579 580 581 582 583 584 585 586 ``````Lemma id_freeN_r n n' x `{!IdFree x} y : ✓{n}x → x ⋅ y ≡{n'}≡ x → False. Proof. eauto using cmra_validN_le, dist_le with lia. Qed. Lemma id_freeN_l n n' x `{!IdFree x} y : ✓{n}x → y ⋅ x ≡{n'}≡ x → False. Proof. rewrite comm. eauto using id_freeN_r. Qed. Lemma id_free_r x `{!IdFree x} y : ✓x → x ⋅ y ≡ x → False. Proof. move=> /cmra_valid_validN ? /equiv_dist. eauto. Qed. Lemma id_free_l x `{!IdFree x} y : ✓x → y ⋅ x ≡ x → False. Proof. rewrite comm. eauto using id_free_r. Qed. Lemma discrete_id_free x `{CMRADiscrete A}: `````` Robbert Krebbers committed Feb 03, 2017 587 `````` (∀ y, ✓ x → x ⋅ y ≡ x → False) → IdFree x. `````` Jacques-Henri Jourdan committed Feb 01, 2017 588 ``````Proof. repeat intro. eauto using cmra_discrete_valid, cmra_discrete, timeless. Qed. `````` Robbert Krebbers committed Feb 03, 2017 589 ``````Global Instance id_free_op_r x y : IdFree y → Cancelable x → IdFree (x ⋅ y). `````` Jacques-Henri Jourdan committed Feb 01, 2017 590 ``````Proof. `````` Robbert Krebbers committed Feb 03, 2017 591 `````` intros ?? z ? Hid%symmetry. revert Hid. rewrite -assoc=>/(cancelableN x) ?. `````` Jacques-Henri Jourdan committed Feb 01, 2017 592 593 `````` eapply (id_free0_r _); [by eapply cmra_validN_op_r |symmetry; eauto]. Qed. `````` Robbert Krebbers committed Feb 03, 2017 594 ``````Global Instance id_free_op_l x y : IdFree x → Cancelable y → IdFree (x ⋅ y). `````` Jacques-Henri Jourdan committed Feb 01, 2017 595 596 597 ``````Proof. intros. rewrite comm. apply _. Qed. Global Instance exclusive_id_free x : Exclusive x → IdFree x. Proof. intros ? z ? Hid. apply (exclusiveN_l 0 x z). by rewrite Hid. Qed. `````` Robbert Krebbers committed Nov 11, 2015 598 599 ``````End cmra. `````` Robbert Krebbers committed May 27, 2016 600 601 ``````(** * Properties about CMRAs with a unit element **) Section ucmra. `````` Robbert Krebbers committed May 28, 2016 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 `````` 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 618 `````` intros x. destruct (cmra_pcore_mono' ∅ x ∅) as (cx&->&?); `````` Robbert Krebbers committed Jan 22, 2017 619 `````` eauto using ucmra_unit_least, (persistent (∅:A)). `````` Robbert Krebbers committed May 28, 2016 620 `````` Qed. `````` Jacques-Henri Jourdan committed Feb 01, 2017 621 622 `````` Global Instance empty_cancelable : Cancelable (∅:A). Proof. intros ???. by rewrite !left_id. Qed. `````` Robbert Krebbers committed May 27, 2016 623 ``````End ucmra. `````` Robbert Krebbers committed May 28, 2016 624 625 ``````Hint Immediate cmra_unit_total. `````` Robbert Krebbers committed Sep 01, 2016 626 627 628 `````` (** * Properties about CMRAs with Leibniz equality *) Section cmra_leibniz. `````` Ralf Jung committed Jan 25, 2017 629 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Sep 01, 2016 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 `````` 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 653 `````` Lemma persistent_dup_L x `{!Persistent x} : x = x ⋅ x. `````` Robbert Krebbers committed Sep 01, 2016 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 `````` 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. `````` Ralf Jung committed Jan 25, 2017 676 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Sep 01, 2016 677 678 679 680 681 682 683 684 685 `````` 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 686 687 688 689 ``````(** * 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)). `````` Ralf Jung committed Jan 27, 2017 690 691 `````` Context (op_ne : ∀ (x : A), NonExpansive (op x)). Context (core_ne : NonExpansive (@core A _)). `````` Robbert Krebbers committed May 28, 2016 692 693 694 695 696 697 698 `````` 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 699 `````` Context (core_mono : ∀ x y : A, x ≼ y → core x ≼ core y). `````` Robbert Krebbers committed May 28, 2016 700 701 702 `````` 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 703 `````` ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2). `````` Robbert Krebbers committed May 28, 2016 704 `````` Lemma cmra_total_mixin : CMRAMixin A. `````` Ralf Jung committed Jan 25, 2017 705 `````` Proof using Type*. `````` Robbert Krebbers committed May 28, 2016 706 707 708 709 710 711 `````` 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 712 `````` - intros x y cx Hxy%core_mono Hx. move: Hxy. `````` Robbert Krebbers committed May 28, 2016 713 714 715 `````` rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto. Qed. End cmra_total. `````` Robbert Krebbers committed May 27, 2016 716 `````` `````` Robbert Krebbers committed Feb 01, 2016 717 ``````(** * Properties about monotone functions *) `````` Robbert Krebbers committed Jan 14, 2016 718 ``````Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A). `````` Robbert Krebbers committed Feb 26, 2016 719 ``````Proof. repeat split; by try apply _. Qed. `````` Robbert Krebbers committed Feb 01, 2016 720 721 ``````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 722 723 ``````Proof. split. `````` Jacques-Henri Jourdan committed Feb 01, 2017 724 `````` - apply _. `````` Robbert Krebbers committed Sep 28, 2016 725 `````` - move=> n x Hx /=. by apply cmra_monotone_validN, cmra_monotone_validN. `````` Ralf Jung committed Jul 25, 2016 726 `````` - move=> x y Hxy /=. by apply cmra_monotone, cmra_monotone. `````` Robbert Krebbers committed Nov 20, 2015 727 ``````Qed. `````` Robbert Krebbers committed Nov 16, 2015 728 `````` `````` Robbert Krebbers committed Feb 01, 2016 729 ``````Section cmra_monotone. `````` Ralf Jung committed Jan 25, 2017 730 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Feb 01, 2016 731 `````` Context {A B : cmraT} (f : A → B) `{!CMRAMonotone f}. `````` Robbert Krebbers committed Feb 26, 2016 732 `````` Global Instance cmra_monotone_proper : Proper ((≡) ==> (≡)) f := ne_proper _. `````` Ralf Jung committed Jul 25, 2016 733 `````` Lemma cmra_monotoneN n x y : x ≼{n} y → f x ≼{n} f y. ``````