cmra.v 57.4 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 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 ``````(** Lifting properties from the mixin *) Section cmra_mixin. Context {A : cmraT}. Implicit Types x y : A. `````` Ralf Jung committed Jan 27, 2017 96 `````` Global Instance cmra_op_ne (x : A) : NonExpansive (op x). `````` Robbert Krebbers committed Jan 14, 2016 97 `````` 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 ``````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 133 ``````Hint Mode Persistent + ! : typeclass_instances. `````` Robbert Krebbers committed May 28, 2016 134 `````` `````` Jacques-Henri Jourdan committed May 31, 2016 135 ``````(** * Exclusive elements (i.e., elements that cannot have a frame). *) `````` Robbert Krebbers committed Jun 16, 2016 136 137 ``````Class Exclusive {A : cmraT} (x : A) := exclusive0_l y : ✓{0} (x ⋅ y) → False. Arguments exclusive0_l {_} _ {_} _ _. `````` Robbert Krebbers committed Jan 22, 2017 138 ``````Hint Mode Exclusive + ! : typeclass_instances. `````` Jacques-Henri Jourdan committed May 31, 2016 139 `````` `````` Jacques-Henri Jourdan committed Feb 01, 2017 140 141 142 143 144 145 146 147 148 149 150 151 ``````(** * 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 152 153 154 155 156 157 158 159 160 161 162 ``````(** * 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 163 ``````(** * CMRAs with a unit element *) `````` Robbert Krebbers committed Feb 01, 2016 164 ``````(** We use the notation ∅ because for most instances (maps, sets, etc) the `````` Ralf Jung committed Mar 08, 2016 165 ```````empty' element is the unit. *) `````` Robbert Krebbers committed May 28, 2016 166 ``````Record UCMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, Empty A} := { `````` Robbert Krebbers committed May 27, 2016 167 168 `````` mixin_ucmra_unit_valid : ✓ ∅; mixin_ucmra_unit_left_id : LeftId (≡) ∅ (⋅); `````` Robbert Krebbers committed May 28, 2016 169 `````` mixin_ucmra_pcore_unit : pcore ∅ ≡ Some ∅ `````` Robbert Krebbers committed Feb 01, 2016 170 ``````}. `````` Robbert Krebbers committed May 27, 2016 171 `````` `````` Robbert Krebbers committed Jun 15, 2016 172 ``````Structure ucmraT := UCMRAT' { `````` Robbert Krebbers committed May 27, 2016 173 174 175 `````` ucmra_car :> Type; ucmra_equiv : Equiv ucmra_car; ucmra_dist : Dist ucmra_car; `````` Robbert Krebbers committed May 28, 2016 176 `````` ucmra_pcore : PCore ucmra_car; `````` Robbert Krebbers committed May 27, 2016 177 178 179 180 `````` 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 181 `````` ucmra_ofe_mixin : OfeMixin ucmra_car; `````` Robbert Krebbers committed May 27, 2016 182 `````` ucmra_cmra_mixin : CMRAMixin ucmra_car; `````` Robbert Krebbers committed Jun 15, 2016 183 `````` ucmra_mixin : UCMRAMixin ucmra_car; `````` Robbert Krebbers committed Jun 15, 2016 184 `````` _ : Type; `````` Robbert Krebbers committed May 27, 2016 185 ``````}. `````` Ralf Jung committed Nov 22, 2016 186 ``````Arguments UCMRAT' _ {_ _ _ _ _ _ _} _ _ _ _. `````` Robbert Krebbers committed Jun 15, 2016 187 ``````Notation UCMRAT A m m' m'' := (UCMRAT' A m m' m'' A). `````` Robbert Krebbers committed May 27, 2016 188 189 190 ``````Arguments ucmra_car : simpl never. Arguments ucmra_equiv : simpl never. Arguments ucmra_dist : simpl never. `````` Robbert Krebbers committed May 28, 2016 191 ``````Arguments ucmra_pcore : simpl never. `````` Robbert Krebbers committed May 27, 2016 192 193 194 ``````Arguments ucmra_op : simpl never. Arguments ucmra_valid : simpl never. Arguments ucmra_validN : simpl never. `````` Ralf Jung committed Nov 22, 2016 195 ``````Arguments ucmra_ofe_mixin : simpl never. `````` Robbert Krebbers committed May 27, 2016 196 197 198 ``````Arguments ucmra_cmra_mixin : simpl never. Arguments ucmra_mixin : simpl never. Add Printing Constructor ucmraT. `````` Robbert Krebbers committed Jun 14, 2016 199 ``````Hint Extern 0 (Empty _) => eapply (@ucmra_empty _) : typeclass_instances. `````` Ralf Jung committed Nov 22, 2016 200 201 ``````Coercion ucmra_ofeC (A : ucmraT) : ofeT := OfeT A (ucmra_ofe_mixin A). Canonical Structure ucmra_ofeC. `````` Robbert Krebbers committed May 27, 2016 202 ``````Coercion ucmra_cmraR (A : ucmraT) : cmraT := `````` Ralf Jung committed Nov 22, 2016 203 `````` CMRAT A (ucmra_ofe_mixin A) (ucmra_cmra_mixin A). `````` Robbert Krebbers committed May 27, 2016 204 205 206 207 208 209 210 211 212 213 ``````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 214 215 `````` Lemma ucmra_pcore_unit : pcore (∅:A) ≡ Some ∅. Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin A)). Qed. `````` Robbert Krebbers committed May 27, 2016 216 ``````End ucmra_mixin. `````` Robbert Krebbers committed Jan 14, 2016 217 `````` `````` Robbert Krebbers committed Feb 24, 2016 218 ``````(** * Discrete CMRAs *) `````` Robbert Krebbers committed Feb 26, 2016 219 ``````Class CMRADiscrete (A : cmraT) := { `````` Robbert Krebbers committed Feb 24, 2016 220 221 222 223 `````` cmra_discrete :> Discrete A; cmra_discrete_valid (x : A) : ✓{0} x → ✓ x }. `````` Robbert Krebbers committed Jan 16, 2016 224 ``````(** * Morphisms *) `````` Robbert Krebbers committed Jan 14, 2016 225 ``````Class CMRAMonotone {A B : cmraT} (f : A → B) := { `````` Ralf Jung committed Jan 27, 2017 226 `````` cmra_monotone_ne :> NonExpansive f; `````` Robbert Krebbers committed Sep 28, 2016 227 `````` cmra_monotone_validN n x : ✓{n} x → ✓{n} f x; `````` Ralf Jung committed Jul 25, 2016 228 `````` cmra_monotone x y : x ≼ y → f x ≼ f y `````` Robbert Krebbers committed Jan 14, 2016 229 ``````}. `````` Robbert Krebbers committed Sep 28, 2016 230 ``````Arguments cmra_monotone_validN {_ _} _ {_} _ _ _. `````` Ralf Jung committed Jul 25, 2016 231 ``````Arguments cmra_monotone {_ _} _ {_} _ _ _. `````` Robbert Krebbers committed Jan 14, 2016 232 `````` `````` Robbert Krebbers committed Sep 28, 2016 233 234 235 ``````(* Not all intended homomorphisms preserve validity, in particular it does not hold for the [ownM] and [own] connectives. *) Class CMRAHomomorphism {A B : cmraT} (f : A → B) := { `````` Ralf Jung committed Jan 27, 2017 236 `````` cmra_homomorphism_ne :> NonExpansive f; `````` Robbert Krebbers committed Sep 28, 2016 237 238 239 240 241 242 243 244 245 246 `````` 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 247 ``````(** * Properties **) `````` Robbert Krebbers committed Nov 11, 2015 248 ``````Section cmra. `````` Robbert Krebbers committed Jan 14, 2016 249 ``````Context {A : cmraT}. `````` Robbert Krebbers committed Nov 11, 2015 250 ``````Implicit Types x y z : A. `````` Robbert Krebbers committed Feb 01, 2016 251 ``````Implicit Types xs ys zs : list A. `````` Robbert Krebbers committed Nov 11, 2015 252 `````` `````` Robbert Krebbers committed Feb 01, 2016 253 ``````(** ** Setoids *) `````` Ralf Jung committed Jan 27, 2017 254 ``````Global Instance cmra_pcore_ne' : NonExpansive (@pcore A _). `````` Robbert Krebbers committed May 28, 2016 255 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 256 `````` intros n x y Hxy. destruct (pcore x) as [cx|] eqn:?. `````` Robbert Krebbers committed May 28, 2016 257 258 259 260 261 262 `````` { 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 263 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 264 265 266 `````` 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 267 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 268 269 ``````Global Instance cmra_pcore_proper' : Proper ((≡) ==> (≡)) (@pcore A _). Proof. apply (ne_proper _). Qed. `````` Ralf Jung committed Jan 27, 2017 270 271 ``````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 272 ``````Global Instance cmra_op_proper' : Proper ((≡) ==> (≡) ==> (≡)) (@op A _). `````` Robbert Krebbers committed Feb 01, 2016 273 274 275 276 277 278 279 ``````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 280 281 282 283 ``````Proof. intros x y Hxy; rewrite !cmra_valid_validN. by split=> ? n; [rewrite -Hxy|rewrite Hxy]. Qed. `````` Robbert Krebbers committed Feb 01, 2016 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 ``````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 302 ``````Global Instance cmra_opM_ne : NonExpansive2 (@opM A). `````` Robbert Krebbers committed May 28, 2016 303 304 305 ``````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 306 `````` `````` Robbert Krebbers committed May 28, 2016 307 308 309 310 ``````(** ** 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 311 ``````(** ** Validity *) `````` Robbert Krebbers committed Feb 18, 2016 312 ``````Lemma cmra_validN_le n n' x : ✓{n} x → n' ≤ n → ✓{n'} x. `````` Robbert Krebbers committed Feb 01, 2016 313 314 315 ``````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 316 ``````Lemma cmra_validN_op_r n x y : ✓{n} (x ⋅ y) → ✓{n} y. `````` Robbert Krebbers committed Feb 11, 2016 317 ``````Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 318 319 320 ``````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 321 ``````(** ** Core *) `````` Robbert Krebbers committed May 28, 2016 322 323 324 ``````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 325 ``````Proof. intros. rewrite comm. by apply cmra_pcore_l. Qed. `````` Robbert Krebbers committed May 28, 2016 326 ``````Lemma cmra_pcore_r' x cx : pcore x ≡ Some cx → x ⋅ cx ≡ x. `````` Jacques-Henri Jourdan committed Feb 01, 2017 327 ``````Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_r. Qed. `````` Robbert Krebbers committed May 28, 2016 328 ``````Lemma cmra_pcore_idemp' x cx : pcore x ≡ Some cx → pcore cx ≡ Some cx. `````` Jacques-Henri Jourdan committed Feb 01, 2017 329 ``````Proof. intros (cx'&?&->)%equiv_Some_inv_r'. eauto using cmra_pcore_idemp. Qed. `````` Robbert Krebbers committed May 30, 2016 330 331 332 333 ``````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 334 335 336 337 338 339 340 341 ``````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 342 `````` `````` Robbert Krebbers committed May 30, 2016 343 344 345 346 ``````(** ** 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 347 ``````(** ** Exclusive elements *) `````` Robbert Krebbers committed Jun 16, 2016 348 ``````Lemma exclusiveN_l n x `{!Exclusive x} y : ✓{n} (x ⋅ y) → False. `````` Robbert Krebbers committed Aug 30, 2016 349 ``````Proof. intros. eapply (exclusive0_l x y), cmra_validN_le; eauto with lia. Qed. `````` Robbert Krebbers committed Jun 16, 2016 350 351 352 353 354 355 ``````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 356 ``````Lemma exclusiveN_opM n x `{!Exclusive x} my : ✓{n} (x ⋅? my) → my = None. `````` Robbert Krebbers committed Aug 30, 2016 357 ``````Proof. destruct my as [y|]. move=> /(exclusiveN_l _ x) []. done. Qed. `````` Robbert Krebbers committed Oct 02, 2016 358 359 360 361 ``````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 362 `````` `````` Robbert Krebbers committed Feb 01, 2016 363 ``````(** ** Order *) `````` Robbert Krebbers committed Mar 11, 2016 364 365 ``````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 366 ``````Global Instance cmra_includedN_trans n : Transitive (@includedN A _ _ n). `````` Robbert Krebbers committed Feb 01, 2016 367 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 368 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 369 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 370 ``````Global Instance cmra_included_trans: Transitive (@included A _ _). `````` Robbert Krebbers committed Feb 01, 2016 371 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 372 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 373 ``````Qed. `````` Robbert Krebbers committed Sep 09, 2016 374 375 ``````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 376 ``````Lemma cmra_validN_includedN n x y : ✓{n} y → x ≼{n} y → ✓{n} x. `````` Robbert Krebbers committed Feb 01, 2016 377 ``````Proof. intros Hyv [z ?]; cofe_subst y; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 18, 2016 378 ``````Lemma cmra_validN_included n x y : ✓{n} y → x ≼ y → ✓{n} x. `````` Robbert Krebbers committed Mar 11, 2016 379 ``````Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 380 `````` `````` Robbert Krebbers committed Feb 18, 2016 381 ``````Lemma cmra_includedN_S n x y : x ≼{S n} y → x ≼{n} y. `````` Robbert Krebbers committed Feb 01, 2016 382 ``````Proof. by intros [z Hz]; exists z; apply dist_S. Qed. `````` Robbert Krebbers committed Feb 18, 2016 383 ``````Lemma cmra_includedN_le n n' x y : x ≼{n} y → n' ≤ n → x ≼{n'} y. `````` Robbert Krebbers committed Feb 01, 2016 384 385 386 387 388 389 390 ``````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 391 ``````Proof. rewrite (comm op); apply cmra_includedN_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 392 ``````Lemma cmra_included_r x y : y ≼ x ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 393 ``````Proof. rewrite (comm op); apply cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 20, 2015 394 `````` `````` Ralf Jung committed Jul 25, 2016 395 ``````Lemma cmra_pcore_mono' x y cx : `````` Robbert Krebbers committed May 28, 2016 396 397 398 `````` 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 399 `````` destruct (cmra_pcore_mono x y cx') as (cy&->&?); auto. `````` Robbert Krebbers committed May 28, 2016 400 401 `````` exists cy; by rewrite Hcx. Qed. `````` Ralf Jung committed Jul 25, 2016 402 ``````Lemma cmra_pcore_monoN' n x y cx : `````` Robbert Krebbers committed May 28, 2016 403 `````` x ≼{n} y → pcore x ≡{n}≡ Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼{n} cy. `````` Robbert Krebbers committed Feb 26, 2016 404 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 405 `````` intros [z Hy] (cx'&?&Hcx)%dist_Some_inv_r'. `````` Ralf Jung committed Jul 25, 2016 406 `````` destruct (cmra_pcore_mono x (x ⋅ z) cx') `````` Robbert Krebbers committed May 28, 2016 407 408 409 410 411 `````` 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 412 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 413 414 ``````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 415 `````` `````` Ralf Jung committed Jul 25, 2016 416 ``````Lemma cmra_monoN_l n x y z : x ≼{n} y → z ⋅ x ≼{n} z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 417 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Ralf Jung committed Jul 25, 2016 418 ``````Lemma cmra_mono_l x y z : x ≼ y → z ⋅ x ≼ z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 419 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Ralf Jung committed Jul 25, 2016 420 421 422 423 ``````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 424 425 426 427 ``````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 428 `````` `````` Robbert Krebbers committed Sep 28, 2016 429 430 431 432 433 434 435 ``````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 436 ``````Lemma cmra_included_dist_l n x1 x2 x1' : `````` Ralf Jung committed Feb 10, 2016 437 `````` x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2. `````` Robbert Krebbers committed Nov 11, 2015 438 ``````Proof. `````` Robbert Krebbers committed Feb 01, 2016 439 440 `````` intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using cmra_included_l. by rewrite Hx1 Hx2. `````` Robbert Krebbers committed Nov 11, 2015 441 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2016 442 `````` `````` Robbert Krebbers committed May 28, 2016 443 444 ``````(** ** Total core *) Section total_core. `````` Ralf Jung committed Jan 25, 2017 445 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed May 28, 2016 446 447 448 449 450 451 452 453 454 455 `````` 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 456 `````` Lemma cmra_core_mono x y : x ≼ y → core x ≼ core y. `````` Robbert Krebbers committed May 28, 2016 457 458 `````` Proof. intros; destruct (cmra_total x) as [cx Hcx]. `````` Ralf Jung committed Jul 25, 2016 459 `````` destruct (cmra_pcore_mono x y cx) as (cy&Hcy&?); auto. `````` Robbert Krebbers committed May 28, 2016 460 461 462 `````` by rewrite /core /= Hcx Hcy. Qed. `````` Ralf Jung committed Jan 27, 2017 463 `````` Global Instance cmra_core_ne : NonExpansive (@core A _). `````` Robbert Krebbers committed May 28, 2016 464 `````` Proof. `````` Ralf Jung committed Jan 27, 2017 465 `````` intros n x y Hxy. destruct (cmra_total x) as [cx Hcx]. `````` Robbert Krebbers committed May 28, 2016 466 467 468 469 470 471 472 `````` 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 473 474 `````` 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 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 `````` 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 505 `````` Lemma cmra_core_monoN n x y : x ≼{n} y → core x ≼{n} core y. `````` Robbert Krebbers committed May 28, 2016 506 507 `````` Proof. intros [z ->]. `````` Ralf Jung committed Jul 25, 2016 508 `````` apply cmra_included_includedN, cmra_core_mono, cmra_included_l. `````` Robbert Krebbers committed May 28, 2016 509 510 511 `````` Qed. End total_core. `````` Robbert Krebbers committed Jan 16, 2016 512 ``````(** ** Timeless *) `````` Robbert Krebbers committed Feb 10, 2016 513 ``````Lemma cmra_timeless_included_l x y : Timeless x → ✓{0} y → x ≼{0} y → x ≼ y. `````` Robbert Krebbers committed Dec 11, 2015 514 515 ``````Proof. intros ?? [x' ?]. `````` Robbert Krebbers committed Aug 14, 2016 516 `````` destruct (cmra_extend 0 y x x') as (z&z'&Hy&Hz&Hz'); auto; simpl in *. `````` Robbert Krebbers committed Jan 13, 2016 517 `````` by exists z'; rewrite Hy (timeless x z). `````` Robbert Krebbers committed Dec 11, 2015 518 ``````Qed. `````` Robbert Krebbers committed Aug 30, 2016 519 520 ``````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 521 ``````Lemma cmra_op_timeless x1 x2 : `````` Robbert Krebbers committed Dec 11, 2015 522 `````` ✓ (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2). `````` Robbert Krebbers committed Nov 18, 2015 523 524 ``````Proof. intros ??? z Hz. `````` Robbert Krebbers committed Aug 14, 2016 525 `````` destruct (cmra_extend 0 z x1 x2) as (y1&y2&Hz'&?&?); auto; simpl in *. `````` Robbert Krebbers committed Feb 24, 2016 526 `````` { rewrite -?Hz. by apply cmra_valid_validN. } `````` Robbert Krebbers committed Jan 13, 2016 527 `````` by rewrite Hz' (timeless x1 y1) // (timeless x2 y2). `````` Robbert Krebbers committed Nov 18, 2015 528 ``````Qed. `````` Robbert Krebbers committed Nov 20, 2015 529 `````` `````` Robbert Krebbers committed Feb 24, 2016 530 531 532 533 534 535 536 537 ``````(** ** 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 538 `````` split; first by apply cmra_included_includedN. `````` Robbert Krebbers committed Feb 24, 2016 539 540 `````` intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l. Qed. `````` Jacques-Henri Jourdan committed Feb 01, 2017 541 542 543 `````` (** Cancelable elements *) Global Instance cancelable_proper : Proper (equiv ==> iff) (@Cancelable A). `````` Robbert Krebbers committed Feb 03, 2017 544 545 ``````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 546 547 548 549 550 551 552 ``````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 553 `````` intros ?? n z z' ??. apply (cancelableN y), (cancelableN x). `````` Jacques-Henri Jourdan committed Feb 01, 2017 554 555 556 557 558 `````` - 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 559 ``````Proof. intros ? n z z' []%(exclusiveN_l _ x). Qed. `````` Jacques-Henri Jourdan committed Feb 01, 2017 560 561 `````` (** Id-free elements *) `````` Robbert Krebbers committed Feb 03, 2017 562 ``````Global Instance id_free_ne n : Proper (dist n ==> iff) (@IdFree A). `````` Jacques-Henri Jourdan committed Feb 01, 2017 563 ``````Proof. `````` Robbert Krebbers committed Feb 03, 2017 564 565 `````` 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 566 567 ``````Qed. Global Instance id_free_proper : Proper (equiv ==> iff) (@IdFree A). `````` Robbert Krebbers committed Feb 03, 2017 568 ``````Proof. by move=> P Q /equiv_dist /(_ 0)=> ->. Qed. `````` Jacques-Henri Jourdan committed Feb 01, 2017 569 570 571 572 573 574 575 576 577 ``````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 578 `````` (∀ y, ✓ x → x ⋅ y ≡ x → False) → IdFree x. `````` Jacques-Henri Jourdan committed Feb 01, 2017 579 ``````Proof. repeat intro. eauto using cmra_discrete_valid, cmra_discrete, timeless. Qed. `````` Robbert Krebbers committed Feb 03, 2017 580 ``````Global Instance id_free_op_r x y : IdFree y → Cancelable x → IdFree (x ⋅ y). `````` Jacques-Henri Jourdan committed Feb 01, 2017 581 ``````Proof. `````` Robbert Krebbers committed Feb 03, 2017 582 `````` intros ?? z ? Hid%symmetry. revert Hid. rewrite -assoc=>/(cancelableN x) ?. `````` Jacques-Henri Jourdan committed Feb 01, 2017 583 584 `````` eapply (id_free0_r _); [by eapply cmra_validN_op_r |symmetry; eauto]. Qed. `````` Robbert Krebbers committed Feb 03, 2017 585 ``````Global Instance id_free_op_l x y : IdFree x → Cancelable y → IdFree (x ⋅ y). `````` Jacques-Henri Jourdan committed Feb 01, 2017 586 587 588 ``````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 589 590 ``````End cmra. `````` Robbert Krebbers committed May 27, 2016 591 592 ``````(** * Properties about CMRAs with a unit element **) Section ucmra. `````` Robbert Krebbers committed May 28, 2016 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 `````` 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 609 `````` intros x. destruct (cmra_pcore_mono' ∅ x ∅) as (cx&->&?); `````` Robbert Krebbers committed Jan 22, 2017 610 `````` eauto using ucmra_unit_least, (persistent (∅:A)). `````` Robbert Krebbers committed May 28, 2016 611 `````` Qed. `````` Jacques-Henri Jourdan committed Feb 01, 2017 612 613 `````` Global Instance empty_cancelable : Cancelable (∅:A). Proof. intros ???. by rewrite !left_id. Qed. `````` Robbert Krebbers committed May 27, 2016 614 ``````End ucmra. `````` Robbert Krebbers committed May 28, 2016 615 616 ``````Hint Immediate cmra_unit_total. `````` Robbert Krebbers committed Sep 01, 2016 617 618 619 `````` (** * Properties about CMRAs with Leibniz equality *) Section cmra_leibniz. `````` Ralf Jung committed Jan 25, 2017 620 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Sep 01, 2016 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 `````` 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 644 `````` Lemma persistent_dup_L x `{!Persistent x} : x = x ⋅ x. `````` Robbert Krebbers committed Sep 01, 2016 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 `````` 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 667 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Sep 01, 2016 668 669 670 671 672 673 674 675 676 `````` 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 677 678 679 680 ``````(** * 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 681 682 `````` Context (op_ne : ∀ (x : A), NonExpansive (op x)). Context (core_ne : NonExpansive (@core A _)). `````` Robbert Krebbers committed May 28, 2016 683 684 685 686 687 688 689 `````` 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 690 `````` Context (core_mono : ∀ x y : A, x ≼ y → core x ≼ core y). `````` Robbert Krebbers committed May 28, 2016 691 692 693 `````` 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 694 `````` ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2). `````` Robbert Krebbers committed May 28, 2016 695 `````` Lemma cmra_total_mixin : CMRAMixin A. `````` Ralf Jung committed Jan 25, 2017 696 `````` Proof using Type*. `````` Robbert Krebbers committed May 28, 2016 697 698 699 700 701 702 `````` 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 703 `````` - intros x y cx Hxy%core_mono Hx. move: Hxy. `````` Robbert Krebbers committed May 28, 2016 704 705 706 `````` rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto. Qed. End cmra_total. `````` Robbert Krebbers committed May 27, 2016 707 `````` `````` Robbert Krebbers committed Feb 01, 2016 708 ``````(** * Properties about monotone functions *) `````` Robbert Krebbers committed Jan 14, 2016 709 ``````Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A). `````` Robbert Krebbers committed Feb 26, 2016 710 ``````Proof. repeat split; by try apply _. Qed. `````` Robbert Krebbers committed Feb 01, 2016 711 712 ``````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 713 714 ``````Proof. split. `````` Jacques-Henri Jourdan committed Feb 01, 2017 715 `````` - apply _. `````` Robbert Krebbers committed Sep 28, 2016 716 `````` - move=> n x Hx /=. by apply cmra_monotone_validN, cmra_monotone_validN. `````` Ralf Jung committed Jul 25, 2016 717 `````` - move=> x y Hxy /=. by apply cmra_monotone, cmra_monotone. `````` Robbert Krebbers committed Nov 20, 2015 718 ``````Qed. `````` Robbert Krebbers committed Nov 16, 2015 719 `````` `````` Robbert Krebbers committed Feb 01, 2016 720 ``````Section cmra_monotone. `````` Ralf Jung committed Jan 25, 2017 721 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Feb 01, 2016 722 `````` Context {A B : cmraT} (f : A → B) `{!CMRAMonotone f}. `````` Robbert Krebbers committed Feb 26, 2016 723 `````` Global Instance cmra_monotone_proper : Proper ((≡) ==> (≡)) f := ne_proper _. `````` Ralf Jung committed Jul 25, 2016 724 `````` Lemma cmra_monotoneN n x y : x ≼{n} y → f x ≼{n} f y. `````` Robbert Krebbers committed Feb 01, 2016 725 `````` Proof. `````` Robbert Krebbers committed Feb 26, 2016 726 `````` intros [z ->]. `````` Ralf Jung committed Jul 25, 2016 727 `````` apply cmra_included_includedN, (cmra_monotone f), cmra_included_l. `````` Robbert Krebbers committed Feb 01, 2016 728 `````` Qed. `````` Robbert Krebbers committed Sep 28, 2016 729 730 `````` 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 731 732 ``````End cmra_monotone. `````` Robbert Krebbers committed Sep 28, 2016 733 734 735 736 737 738 ``````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. `````` Jacques-Henri Jourdan committed Feb 01, 2017 739 `````` - apply _. `````` Robbert Krebbers committed Sep 28, 2016 740 741 742 743 744 745 746 747 748 749 750 751 752 `````` - 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 753 754 ``````(** Functors *) Structure rFunctor := RFunctor { `````` Ralf Jung committed Nov 22, 2016 755 `````` rFunctor_car : ofeT → ofeT → cmraT; `````` Robbert Krebbers committed May 25, 2016 756 757 `````` rFunctor_map {A1 A2 B1 B2} : ((A2 -n> A1) * (B1 -n> B2)) → rFunctor_car A1 B1 -n> rFunctor_car A2 B2; `````` Ralf Jung committed Jan 27, 2017 758 759 `````` rFunctor_ne A1 A2 B1 B2 : NonExpansive (@rFunctor_map A1 A2 B1 B2); `````` Robbert Krebbers committed May 25, 2016 760 761 762 763 764 `````` 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)) : `````` Jacques-Henri Jourdan committed Feb 01, 2017 765 `````` CMRAMonotone (rFunctor_map fg) `````` Robbert Krebbers committed May 25, 2016 766 767 768 769 ``````}. Existing Instances rFunctor_ne rFunctor_mono. Instance: Params (@rFunctor_map) 5. ``````