cmra.v 53 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 `````` `````` Robbert Krebbers committed May 28, 2016 140 141 142 143 144 145 146 147 148 149 150 ``````(** * 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 151 ``````(** * CMRAs with a unit element *) `````` Robbert Krebbers committed Feb 01, 2016 152 ``````(** We use the notation ∅ because for most instances (maps, sets, etc) the `````` Ralf Jung committed Mar 08, 2016 153 ```````empty' element is the unit. *) `````` Robbert Krebbers committed May 28, 2016 154 ``````Record UCMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, Empty A} := { `````` Robbert Krebbers committed May 27, 2016 155 156 `````` mixin_ucmra_unit_valid : ✓ ∅; mixin_ucmra_unit_left_id : LeftId (≡) ∅ (⋅); `````` Robbert Krebbers committed May 28, 2016 157 `````` mixin_ucmra_pcore_unit : pcore ∅ ≡ Some ∅ `````` Robbert Krebbers committed Feb 01, 2016 158 ``````}. `````` Robbert Krebbers committed May 27, 2016 159 `````` `````` Robbert Krebbers committed Jun 15, 2016 160 ``````Structure ucmraT := UCMRAT' { `````` Robbert Krebbers committed May 27, 2016 161 162 163 `````` ucmra_car :> Type; ucmra_equiv : Equiv ucmra_car; ucmra_dist : Dist ucmra_car; `````` Robbert Krebbers committed May 28, 2016 164 `````` ucmra_pcore : PCore ucmra_car; `````` Robbert Krebbers committed May 27, 2016 165 166 167 168 `````` 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 169 `````` ucmra_ofe_mixin : OfeMixin ucmra_car; `````` Robbert Krebbers committed May 27, 2016 170 `````` ucmra_cmra_mixin : CMRAMixin ucmra_car; `````` Robbert Krebbers committed Jun 15, 2016 171 `````` ucmra_mixin : UCMRAMixin ucmra_car; `````` Robbert Krebbers committed Jun 15, 2016 172 `````` _ : Type; `````` Robbert Krebbers committed May 27, 2016 173 ``````}. `````` Ralf Jung committed Nov 22, 2016 174 ``````Arguments UCMRAT' _ {_ _ _ _ _ _ _} _ _ _ _. `````` Robbert Krebbers committed Jun 15, 2016 175 ``````Notation UCMRAT A m m' m'' := (UCMRAT' A m m' m'' A). `````` Robbert Krebbers committed May 27, 2016 176 177 178 ``````Arguments ucmra_car : simpl never. Arguments ucmra_equiv : simpl never. Arguments ucmra_dist : simpl never. `````` Robbert Krebbers committed May 28, 2016 179 ``````Arguments ucmra_pcore : simpl never. `````` Robbert Krebbers committed May 27, 2016 180 181 182 ``````Arguments ucmra_op : simpl never. Arguments ucmra_valid : simpl never. Arguments ucmra_validN : simpl never. `````` Ralf Jung committed Nov 22, 2016 183 ``````Arguments ucmra_ofe_mixin : simpl never. `````` Robbert Krebbers committed May 27, 2016 184 185 186 ``````Arguments ucmra_cmra_mixin : simpl never. Arguments ucmra_mixin : simpl never. Add Printing Constructor ucmraT. `````` Robbert Krebbers committed Jun 14, 2016 187 ``````Hint Extern 0 (Empty _) => eapply (@ucmra_empty _) : typeclass_instances. `````` Ralf Jung committed Nov 22, 2016 188 189 ``````Coercion ucmra_ofeC (A : ucmraT) : ofeT := OfeT A (ucmra_ofe_mixin A). Canonical Structure ucmra_ofeC. `````` Robbert Krebbers committed May 27, 2016 190 ``````Coercion ucmra_cmraR (A : ucmraT) : cmraT := `````` Ralf Jung committed Nov 22, 2016 191 `````` CMRAT A (ucmra_ofe_mixin A) (ucmra_cmra_mixin A). `````` Robbert Krebbers committed May 27, 2016 192 193 194 195 196 197 198 199 200 201 ``````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 202 203 `````` Lemma ucmra_pcore_unit : pcore (∅:A) ≡ Some ∅. Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin A)). Qed. `````` Robbert Krebbers committed May 27, 2016 204 ``````End ucmra_mixin. `````` Robbert Krebbers committed Jan 14, 2016 205 `````` `````` Robbert Krebbers committed Feb 24, 2016 206 ``````(** * Discrete CMRAs *) `````` Robbert Krebbers committed Feb 26, 2016 207 ``````Class CMRADiscrete (A : cmraT) := { `````` Robbert Krebbers committed Feb 24, 2016 208 209 210 211 `````` cmra_discrete :> Discrete A; cmra_discrete_valid (x : A) : ✓{0} x → ✓ x }. `````` Robbert Krebbers committed Jan 16, 2016 212 ``````(** * Morphisms *) `````` Robbert Krebbers committed Jan 14, 2016 213 ``````Class CMRAMonotone {A B : cmraT} (f : A → B) := { `````` Ralf Jung committed Jan 27, 2017 214 `````` cmra_monotone_ne :> NonExpansive f; `````` Robbert Krebbers committed Sep 28, 2016 215 `````` cmra_monotone_validN n x : ✓{n} x → ✓{n} f x; `````` Ralf Jung committed Jul 25, 2016 216 `````` cmra_monotone x y : x ≼ y → f x ≼ f y `````` Robbert Krebbers committed Jan 14, 2016 217 ``````}. `````` Robbert Krebbers committed Sep 28, 2016 218 ``````Arguments cmra_monotone_validN {_ _} _ {_} _ _ _. `````` Ralf Jung committed Jul 25, 2016 219 ``````Arguments cmra_monotone {_ _} _ {_} _ _ _. `````` Robbert Krebbers committed Jan 14, 2016 220 `````` `````` Robbert Krebbers committed Sep 28, 2016 221 222 223 ``````(* 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 224 `````` cmra_homomorphism_ne :> NonExpansive f; `````` Robbert Krebbers committed Sep 28, 2016 225 226 227 228 229 230 231 232 233 234 `````` 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 235 ``````(** * Properties **) `````` Robbert Krebbers committed Nov 11, 2015 236 ``````Section cmra. `````` Robbert Krebbers committed Jan 14, 2016 237 ``````Context {A : cmraT}. `````` Robbert Krebbers committed Nov 11, 2015 238 ``````Implicit Types x y z : A. `````` Robbert Krebbers committed Feb 01, 2016 239 ``````Implicit Types xs ys zs : list A. `````` Robbert Krebbers committed Nov 11, 2015 240 `````` `````` Robbert Krebbers committed Feb 01, 2016 241 ``````(** ** Setoids *) `````` Ralf Jung committed Jan 27, 2017 242 ``````Global Instance cmra_pcore_ne' : NonExpansive (@pcore A _). `````` Robbert Krebbers committed May 28, 2016 243 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 244 `````` intros n x y Hxy. destruct (pcore x) as [cx|] eqn:?. `````` Robbert Krebbers committed May 28, 2016 245 246 247 248 249 250 `````` { 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 251 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 252 253 254 `````` 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 255 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 256 257 ``````Global Instance cmra_pcore_proper' : Proper ((≡) ==> (≡)) (@pcore A _). Proof. apply (ne_proper _). Qed. `````` Ralf Jung committed Jan 27, 2017 258 259 ``````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 260 ``````Global Instance cmra_op_proper' : Proper ((≡) ==> (≡) ==> (≡)) (@op A _). `````` Robbert Krebbers committed Feb 01, 2016 261 262 263 264 265 266 267 ``````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 268 269 270 271 ``````Proof. intros x y Hxy; rewrite !cmra_valid_validN. by split=> ? n; [rewrite -Hxy|rewrite Hxy]. Qed. `````` Robbert Krebbers committed Feb 01, 2016 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 ``````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 290 ``````Global Instance cmra_opM_ne : NonExpansive2 (@opM A). `````` Robbert Krebbers committed May 28, 2016 291 292 293 ``````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 294 `````` `````` Robbert Krebbers committed May 28, 2016 295 296 297 298 ``````(** ** 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 299 ``````(** ** Validity *) `````` Robbert Krebbers committed Feb 18, 2016 300 ``````Lemma cmra_validN_le n n' x : ✓{n} x → n' ≤ n → ✓{n'} x. `````` Robbert Krebbers committed Feb 01, 2016 301 302 303 ``````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 304 ``````Lemma cmra_validN_op_r n x y : ✓{n} (x ⋅ y) → ✓{n} y. `````` Robbert Krebbers committed Feb 11, 2016 305 ``````Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 306 307 308 ``````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 309 ``````(** ** Core *) `````` Robbert Krebbers committed May 28, 2016 310 311 312 313 314 315 316 317 ``````Lemma cmra_pcore_l' x cx : pcore x ≡ Some cx → cx ⋅ x ≡ x. Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_l. Qed. Lemma cmra_pcore_r x cx : pcore x = Some cx → x ⋅ cx ≡ x. Proof. intros. rewrite comm. by apply cmra_pcore_l. Qed. Lemma cmra_pcore_r' x cx : pcore x ≡ Some cx → x ⋅ cx ≡ x. Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_r. Qed. Lemma cmra_pcore_idemp' x cx : pcore x ≡ Some cx → pcore cx ≡ Some cx. Proof. intros (cx'&?&->)%equiv_Some_inv_r'. eauto using cmra_pcore_idemp. Qed. `````` Robbert Krebbers committed May 30, 2016 318 319 320 321 ``````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 322 323 324 325 326 327 328 329 ``````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 330 `````` `````` Robbert Krebbers committed May 30, 2016 331 332 333 334 ``````(** ** 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 335 ``````(** ** Exclusive elements *) `````` Robbert Krebbers committed Jun 16, 2016 336 ``````Lemma exclusiveN_l n x `{!Exclusive x} y : ✓{n} (x ⋅ y) → False. `````` Robbert Krebbers committed Aug 30, 2016 337 ``````Proof. intros. eapply (exclusive0_l x y), cmra_validN_le; eauto with lia. Qed. `````` Robbert Krebbers committed Jun 16, 2016 338 339 340 341 342 343 ``````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 344 ``````Lemma exclusiveN_opM n x `{!Exclusive x} my : ✓{n} (x ⋅? my) → my = None. `````` Robbert Krebbers committed Aug 30, 2016 345 ``````Proof. destruct my as [y|]. move=> /(exclusiveN_l _ x) []. done. Qed. `````` Robbert Krebbers committed Oct 02, 2016 346 347 348 349 ``````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 350 `````` `````` Robbert Krebbers committed Feb 01, 2016 351 ``````(** ** Order *) `````` Robbert Krebbers committed Mar 11, 2016 352 353 ``````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 354 ``````Global Instance cmra_includedN_trans n : Transitive (@includedN A _ _ n). `````` Robbert Krebbers committed Feb 01, 2016 355 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 356 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 357 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 358 ``````Global Instance cmra_included_trans: Transitive (@included A _ _). `````` Robbert Krebbers committed Feb 01, 2016 359 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 360 `````` intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2). by rewrite assoc -Hy -Hz. `````` Robbert Krebbers committed Feb 01, 2016 361 ``````Qed. `````` Robbert Krebbers committed Sep 09, 2016 362 363 ``````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 364 ``````Lemma cmra_validN_includedN n x y : ✓{n} y → x ≼{n} y → ✓{n} x. `````` Robbert Krebbers committed Feb 01, 2016 365 ``````Proof. intros Hyv [z ?]; cofe_subst y; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 18, 2016 366 ``````Lemma cmra_validN_included n x y : ✓{n} y → x ≼ y → ✓{n} x. `````` Robbert Krebbers committed Mar 11, 2016 367 ``````Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_validN_op_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 368 `````` `````` Robbert Krebbers committed Feb 18, 2016 369 ``````Lemma cmra_includedN_S n x y : x ≼{S n} y → x ≼{n} y. `````` Robbert Krebbers committed Feb 01, 2016 370 ``````Proof. by intros [z Hz]; exists z; apply dist_S. Qed. `````` Robbert Krebbers committed Feb 18, 2016 371 ``````Lemma cmra_includedN_le n n' x y : x ≼{n} y → n' ≤ n → x ≼{n'} y. `````` Robbert Krebbers committed Feb 01, 2016 372 373 374 375 376 377 378 ``````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 379 ``````Proof. rewrite (comm op); apply cmra_includedN_l. Qed. `````` Robbert Krebbers committed Feb 01, 2016 380 ``````Lemma cmra_included_r x y : y ≼ x ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 381 ``````Proof. rewrite (comm op); apply cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 20, 2015 382 `````` `````` Ralf Jung committed Jul 25, 2016 383 ``````Lemma cmra_pcore_mono' x y cx : `````` Robbert Krebbers committed May 28, 2016 384 385 386 `````` 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 387 `````` destruct (cmra_pcore_mono x y cx') as (cy&->&?); auto. `````` Robbert Krebbers committed May 28, 2016 388 389 `````` exists cy; by rewrite Hcx. Qed. `````` Ralf Jung committed Jul 25, 2016 390 ``````Lemma cmra_pcore_monoN' n x y cx : `````` Robbert Krebbers committed May 28, 2016 391 `````` x ≼{n} y → pcore x ≡{n}≡ Some cx → ∃ cy, pcore y = Some cy ∧ cx ≼{n} cy. `````` Robbert Krebbers committed Feb 26, 2016 392 ``````Proof. `````` Robbert Krebbers committed May 28, 2016 393 `````` intros [z Hy] (cx'&?&Hcx)%dist_Some_inv_r'. `````` Ralf Jung committed Jul 25, 2016 394 `````` destruct (cmra_pcore_mono x (x ⋅ z) cx') `````` Robbert Krebbers committed May 28, 2016 395 396 397 398 399 `````` 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 400 ``````Qed. `````` Robbert Krebbers committed May 28, 2016 401 402 ``````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 403 `````` `````` Ralf Jung committed Jul 25, 2016 404 ``````Lemma cmra_monoN_l n x y z : x ≼{n} y → z ⋅ x ≼{n} z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 405 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Ralf Jung committed Jul 25, 2016 406 ``````Lemma cmra_mono_l x y z : x ≼ y → z ⋅ x ≼ z ⋅ y. `````` Robbert Krebbers committed Feb 11, 2016 407 ``````Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed. `````` Ralf Jung committed Jul 25, 2016 408 409 410 411 ``````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 412 413 414 415 ``````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 416 `````` `````` Robbert Krebbers committed Sep 28, 2016 417 418 419 420 421 422 423 ``````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 424 ``````Lemma cmra_included_dist_l n x1 x2 x1' : `````` Ralf Jung committed Feb 10, 2016 425 `````` x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2. `````` Robbert Krebbers committed Nov 11, 2015 426 ``````Proof. `````` Robbert Krebbers committed Feb 01, 2016 427 428 `````` intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using cmra_included_l. by rewrite Hx1 Hx2. `````` Robbert Krebbers committed Nov 11, 2015 429 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2016 430 `````` `````` Robbert Krebbers committed May 28, 2016 431 432 ``````(** ** Total core *) Section total_core. `````` Ralf Jung committed Jan 25, 2017 433 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed May 28, 2016 434 435 436 437 438 439 440 441 442 443 `````` Context `{CMRATotal A}. Lemma cmra_core_l x : core x ⋅ x ≡ x. Proof. destruct (cmra_total x) as [cx Hcx]. by rewrite /core /= Hcx cmra_pcore_l. Qed. Lemma cmra_core_idemp x : core (core x) ≡ core x. Proof. destruct (cmra_total x) as [cx Hcx]. by rewrite /core /= Hcx cmra_pcore_idemp. Qed. `````` Ralf Jung committed Jul 25, 2016 444 `````` Lemma cmra_core_mono x y : x ≼ y → core x ≼ core y. `````` Robbert Krebbers committed May 28, 2016 445 446 `````` Proof. intros; destruct (cmra_total x) as [cx Hcx]. `````` Ralf Jung committed Jul 25, 2016 447 `````` destruct (cmra_pcore_mono x y cx) as (cy&Hcy&?); auto. `````` Robbert Krebbers committed May 28, 2016 448 449 450 `````` by rewrite /core /= Hcx Hcy. Qed. `````` Ralf Jung committed Jan 27, 2017 451 `````` Global Instance cmra_core_ne : NonExpansive (@core A _). `````` Robbert Krebbers committed May 28, 2016 452 `````` Proof. `````` Ralf Jung committed Jan 27, 2017 453 `````` intros n x y Hxy. destruct (cmra_total x) as [cx Hcx]. `````` Robbert Krebbers committed May 28, 2016 454 455 456 457 458 459 460 `````` by rewrite /core /= -Hxy Hcx. Qed. Global Instance cmra_core_proper : Proper ((≡) ==> (≡)) (@core A _). Proof. apply (ne_proper _). Qed. Lemma cmra_core_r x : x ⋅ core x ≡ x. Proof. by rewrite (comm _ x) cmra_core_l. Qed. `````` Robbert Krebbers committed May 30, 2016 461 462 `````` Lemma cmra_core_dup x : core x ≡ core x ⋅ core x. Proof. by rewrite -{3}(cmra_core_idemp x) cmra_core_r. Qed. `````` Robbert Krebbers committed May 28, 2016 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 `````` Lemma cmra_core_validN n x : ✓{n} x → ✓{n} core x. Proof. rewrite -{1}(cmra_core_l x); apply cmra_validN_op_l. Qed. Lemma cmra_core_valid x : ✓ x → ✓ core x. Proof. rewrite -{1}(cmra_core_l x); apply cmra_valid_op_l. Qed. Lemma persistent_total x : Persistent x ↔ core x ≡ x. Proof. split; [intros; by rewrite /core /= (persistent x)|]. rewrite /Persistent /core /=. destruct (cmra_total x) as [? ->]. by constructor. Qed. Lemma persistent_core x `{!Persistent x} : core x ≡ x. Proof. by apply persistent_total. Qed. Global Instance cmra_core_persistent x : Persistent (core x). Proof. destruct (cmra_total x) as [cx Hcx]. rewrite /Persistent /core /= Hcx /=. eauto using cmra_pcore_idemp. Qed. Lemma cmra_included_core x : core x ≼ x. Proof. by exists x; rewrite cmra_core_l. Qed. Global Instance cmra_includedN_preorder n : PreOrder (@includedN A _ _ n). Proof. split; [|apply _]. by intros x; exists (core x); rewrite cmra_core_r. Qed. Global Instance cmra_included_preorder : PreOrder (@included A _ _). Proof. split; [|apply _]. by intros x; exists (core x); rewrite cmra_core_r. Qed. `````` Ralf Jung committed Jul 25, 2016 493 `````` Lemma cmra_core_monoN n x y : x ≼{n} y → core x ≼{n} core y. `````` Robbert Krebbers committed May 28, 2016 494 495 `````` Proof. intros [z ->]. `````` Ralf Jung committed Jul 25, 2016 496 `````` apply cmra_included_includedN, cmra_core_mono, cmra_included_l. `````` Robbert Krebbers committed May 28, 2016 497 498 499 `````` Qed. End total_core. `````` Robbert Krebbers committed Jan 16, 2016 500 ``````(** ** Timeless *) `````` Robbert Krebbers committed Feb 10, 2016 501 ``````Lemma cmra_timeless_included_l x y : Timeless x → ✓{0} y → x ≼{0} y → x ≼ y. `````` Robbert Krebbers committed Dec 11, 2015 502 503 ``````Proof. intros ?? [x' ?]. `````` Robbert Krebbers committed Aug 14, 2016 504 `````` destruct (cmra_extend 0 y x x') as (z&z'&Hy&Hz&Hz'); auto; simpl in *. `````` Robbert Krebbers committed Jan 13, 2016 505 `````` by exists z'; rewrite Hy (timeless x z). `````` Robbert Krebbers committed Dec 11, 2015 506 ``````Qed. `````` Robbert Krebbers committed Aug 30, 2016 507 508 ``````Lemma cmra_timeless_included_r x y : Timeless y → x ≼{0} y → x ≼ y. Proof. intros ? [x' ?]. exists x'. by apply (timeless y). Qed. `````` Robbert Krebbers committed Jan 14, 2016 509 ``````Lemma cmra_op_timeless x1 x2 : `````` Robbert Krebbers committed Dec 11, 2015 510 `````` ✓ (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2). `````` Robbert Krebbers committed Nov 18, 2015 511 512 ``````Proof. intros ??? z Hz. `````` Robbert Krebbers committed Aug 14, 2016 513 `````` destruct (cmra_extend 0 z x1 x2) as (y1&y2&Hz'&?&?); auto; simpl in *. `````` Robbert Krebbers committed Feb 24, 2016 514 `````` { rewrite -?Hz. by apply cmra_valid_validN. } `````` Robbert Krebbers committed Jan 13, 2016 515 `````` by rewrite Hz' (timeless x1 y1) // (timeless x2 y2). `````` Robbert Krebbers committed Nov 18, 2015 516 ``````Qed. `````` Robbert Krebbers committed Nov 20, 2015 517 `````` `````` Robbert Krebbers committed Feb 24, 2016 518 519 520 521 522 523 524 525 ``````(** ** Discrete *) Lemma cmra_discrete_valid_iff `{CMRADiscrete A} n x : ✓ x ↔ ✓{n} x. Proof. split; first by rewrite cmra_valid_validN. eauto using cmra_discrete_valid, cmra_validN_le with lia. Qed. Lemma cmra_discrete_included_iff `{Discrete A} n x y : x ≼ y ↔ x ≼{n} y. Proof. `````` Robbert Krebbers committed Mar 11, 2016 526 `````` split; first by apply cmra_included_includedN. `````` Robbert Krebbers committed Feb 24, 2016 527 528 `````` intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l. Qed. `````` Robbert Krebbers committed Nov 11, 2015 529 530 ``````End cmra. `````` Robbert Krebbers committed May 27, 2016 531 532 ``````(** * Properties about CMRAs with a unit element **) Section ucmra. `````` Robbert Krebbers committed May 28, 2016 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 `````` Context {A : ucmraT}. Implicit Types x y z : A. Lemma ucmra_unit_validN n : ✓{n} (∅:A). Proof. apply cmra_valid_validN, ucmra_unit_valid. Qed. Lemma ucmra_unit_leastN n x : ∅ ≼{n} x. Proof. by exists x; rewrite left_id. Qed. Lemma ucmra_unit_least x : ∅ ≼ x. Proof. by exists x; rewrite left_id. Qed. Global Instance ucmra_unit_right_id : RightId (≡) ∅ (@op A _). Proof. by intros x; rewrite (comm op) left_id. Qed. Global Instance ucmra_unit_persistent : Persistent (∅:A). Proof. apply ucmra_pcore_unit. Qed. Global Instance cmra_unit_total : CMRATotal A. Proof. `````` Ralf Jung committed Jul 25, 2016 549 `````` intros x. destruct (cmra_pcore_mono' ∅ x ∅) as (cx&->&?); `````` Robbert Krebbers committed Jan 22, 2017 550 `````` eauto using ucmra_unit_least, (persistent (∅:A)). `````` Robbert Krebbers committed May 28, 2016 551 `````` Qed. `````` Robbert Krebbers committed May 27, 2016 552 ``````End ucmra. `````` Robbert Krebbers committed May 28, 2016 553 554 ``````Hint Immediate cmra_unit_total. `````` Robbert Krebbers committed Sep 01, 2016 555 556 557 `````` (** * Properties about CMRAs with Leibniz equality *) Section cmra_leibniz. `````` Ralf Jung committed Jan 25, 2017 558 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Sep 01, 2016 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 `````` 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 582 `````` Lemma persistent_dup_L x `{!Persistent x} : x = x ⋅ x. `````` Robbert Krebbers committed Sep 01, 2016 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 `````` 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 605 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Sep 01, 2016 606 607 608 609 610 611 612 613 614 `````` 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 615 616 617 618 ``````(** * 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 619 620 `````` Context (op_ne : ∀ (x : A), NonExpansive (op x)). Context (core_ne : NonExpansive (@core A _)). `````` Robbert Krebbers committed May 28, 2016 621 622 623 624 625 626 627 `````` 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 628 `````` Context (core_mono : ∀ x y : A, x ≼ y → core x ≼ core y). `````` Robbert Krebbers committed May 28, 2016 629 630 631 `````` 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 632 `````` ∃ z1 z2, x ≡ z1 ⋅ z2 ∧ z1 ≡{n}≡ y1 ∧ z2 ≡{n}≡ y2). `````` Robbert Krebbers committed May 28, 2016 633 `````` Lemma cmra_total_mixin : CMRAMixin A. `````` Ralf Jung committed Jan 25, 2017 634 `````` Proof using Type*. `````` Robbert Krebbers committed May 28, 2016 635 636 637 638 639 640 `````` 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 641 `````` - intros x y cx Hxy%core_mono Hx. move: Hxy. `````` Robbert Krebbers committed May 28, 2016 642 643 644 `````` rewrite /core /= Hx /=. case (total y)=> [cy ->]; eauto. Qed. End cmra_total. `````` Robbert Krebbers committed May 27, 2016 645 `````` `````` Robbert Krebbers committed Feb 01, 2016 646 ``````(** * Properties about monotone functions *) `````` Robbert Krebbers committed Jan 14, 2016 647 ``````Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A). `````` Robbert Krebbers committed Feb 26, 2016 648 ``````Proof. repeat split; by try apply _. Qed. `````` Robbert Krebbers committed Feb 01, 2016 649 650 ``````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 651 652 ``````Proof. split. `````` Robbert Krebbers committed Feb 26, 2016 653 `````` - apply _. `````` Robbert Krebbers committed Sep 28, 2016 654 `````` - move=> n x Hx /=. by apply cmra_monotone_validN, cmra_monotone_validN. `````` Ralf Jung committed Jul 25, 2016 655 `````` - move=> x y Hxy /=. by apply cmra_monotone, cmra_monotone. `````` Robbert Krebbers committed Nov 20, 2015 656 ``````Qed. `````` Robbert Krebbers committed Nov 16, 2015 657 `````` `````` Robbert Krebbers committed Feb 01, 2016 658 ``````Section cmra_monotone. `````` Ralf Jung committed Jan 25, 2017 659 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Feb 01, 2016 660 `````` Context {A B : cmraT} (f : A → B) `{!CMRAMonotone f}. `````` Robbert Krebbers committed Feb 26, 2016 661 `````` Global Instance cmra_monotone_proper : Proper ((≡) ==> (≡)) f := ne_proper _. `````` Ralf Jung committed Jul 25, 2016 662 `````` Lemma cmra_monotoneN n x y : x ≼{n} y → f x ≼{n} f y. `````` Robbert Krebbers committed Feb 01, 2016 663 `````` Proof. `````` Robbert Krebbers committed Feb 26, 2016 664 `````` intros [z ->]. `````` Ralf Jung committed Jul 25, 2016 665 `````` apply cmra_included_includedN, (cmra_monotone f), cmra_included_l. `````` Robbert Krebbers committed Feb 01, 2016 666 `````` Qed. `````` Robbert Krebbers committed Sep 28, 2016 667 668 `````` 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 669 670 ``````End cmra_monotone. `````` Robbert Krebbers committed Sep 28, 2016 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 ``````Instance cmra_homomorphism_id {A : cmraT} : CMRAHomomorphism (@id A). Proof. repeat split; by try apply _. Qed. Instance cmra_homomorphism_compose {A B C : cmraT} (f : A → B) (g : B → C) : CMRAHomomorphism f → CMRAHomomorphism g → CMRAHomomorphism (g ∘ f). Proof. split. - apply _. - move=> x y /=. rewrite -(cmra_homomorphism g). by apply (ne_proper _), cmra_homomorphism. Qed. Instance cmra_homomorphism_proper {A B : cmraT} (f : A → B) : CMRAHomomorphism f → Proper ((≡) ==> (≡)) f := λ _, ne_proper _. Instance ucmra_homomorphism_id {A : ucmraT} : UCMRAHomomorphism (@id A). Proof. repeat split; by try apply _. Qed. Instance ucmra_homomorphism_compose {A B C : ucmraT} (f : A → B) (g : B → C) : UCMRAHomomorphism f → UCMRAHomomorphism g → UCMRAHomomorphism (g ∘ f). Proof. split. apply _. by rewrite /= !ucmra_homomorphism_unit. Qed. `````` Robbert Krebbers committed May 25, 2016 691 692 ``````(** Functors *) Structure rFunctor := RFunctor { `````` Ralf Jung committed Nov 22, 2016 693 `````` rFunctor_car : ofeT → ofeT → cmraT; `````` Robbert Krebbers committed May 25, 2016 694 695 `````` 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 696 697 `````` rFunctor_ne A1 A2 B1 B2 : NonExpansive (@rFunctor_map A1 A2 B1 B2); `````` Robbert Krebbers committed May 25, 2016 698 699 700 701 702 703 704 705 706 707 `````` rFunctor_id {A B} (x : rFunctor_car A B) : rFunctor_map (cid,cid) x ≡ x; rFunctor_compose {A1 A2 A3 B1 B2 B3} (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x : rFunctor_map (f◎g, g'◎f') x ≡ rFunctor_map (g,g') (rFunctor_map (f,f') x); rFunctor_mono {A1 A2 B1 B2} (fg : (A2 -n> A1) * (B1 -n> B2)) : CMRAMonotone (rFunctor_map fg) }. Existing Instances rFunctor_ne rFunctor_mono. Instance: Params (@rFunctor_map) 5. `````` Ralf Jung committed Jan 06, 2017 708 709 710 ``````Delimit Scope rFunctor_scope with RF. Bind Scope rFunctor_scope with rFunctor. `````` Robbert Krebbers committed May 25, 2016 711 712 713 ``````Class rFunctorContractive (F : rFunctor) := rFunctor_contractive A1 A2 B1 B2 :> Contractive (@rFunctor_map F A1 A2 B1 B2). `````` Ralf Jung committed Nov 22, 2016 714 ``````Definition rFunctor_diag (F: rFunctor) (A: ofeT) : cmraT := rFunctor_car F A A. `````` Robbert Krebbers committed May 25, 2016 715 716 717 718 719 ``````Coercion rFunctor_diag : rFunctor >-> Funclass. Program Definition constRF (B : cmraT) : rFunctor := {| rFunctor_car A1 A2 := B; rFunctor_map A1 A2 B1 B2 f := cid |}. Solve Obligations with done. `````` Ralf Jung committed Jan 06, 2017 720 ``````Coercion constRF : cmraT >-> rFunctor. `````` Robbert Krebbers committed May 25, 2016 721 722 723 724 `````` Instance constRF_contractive B : rFunctorContractive (constRF B). Proof. rewrite /rFunctorContractive; apply _. Qed. `````` Robbert Krebbers committed May 27, 2016 725 ``````Structure urFunctor := URFunctor { `````` Ralf Jung committed Nov 22, 2016 726 `````` urFunctor_car : ofeT → ofeT → ucmraT; `````` Robbert Krebbers committed May 27, 2016 727 728 `````` urFunctor_map {A1 A2 B1 B2} : ((A2 -n> A1) * (B1 -n> B2)) → urFunctor_car A1 B1 -n> urFunctor_car A2 B2; `````` Ralf Jung committed Jan 27, 2017 729 730 `````` urFunctor_ne A1 A2 B1 B2 : NonExpansive (@urFunctor_map A1 A2 B1 B2); `````` Robbert Krebbers committed May 27, 2016 731 732 733 734 735 736 737 738 739 740 `````` urFunctor_id {A B} (x : urFunctor_car A B) : urFunctor_map (cid,cid) x ≡ x; urFunctor_compose {A1 A2 A3 B1 B2 B3} (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x : urFunctor_map (f◎g, g'◎f') x ≡ urFunctor_map (g,g') (urFunctor_map (f,f') x); urFunctor_mono {A1 A2 B1 B2} (fg : (A2 -n> A1) * (B1 -n> B2)) : CMRAMonotone (urFunctor_map fg) }. Existing Instances urFunctor_ne urFunctor_mono. Instance: Params (@urFunctor_map) 5. `````` Ralf Jung committed Jan 06, 2017 741 742 743 ``````Delimit Scope urFunctor_scope with URF. Bind Scope urFunctor_scope with urFunctor. `````` Robbert Krebbers committed May 27, 2016 744 745 746 ``````Class urFunctorContractive (F : urFunctor) := urFunctor_contractive A1 A2 B1 B2 :> Contractive (@urFunctor_map F A1 A2 B1 B2). `````` Ralf Jung committed Nov 22, 2016 747 ``````Definition urFunctor_diag (F: urFunctor) (A: ofeT) : ucmraT := urFunctor_car F A A. `````` Robbert Krebbers committed May 27, 2016 748 749 750 751 752 ``````Coercion urFunctor_diag : urFunctor >-> Funclass. Program Definition constURF (B : ucmraT) : urFunctor := {| urFunctor_car A1 A2 := B; urFunctor_map A1 A2 B1 B2 f := cid |}. Solve Obligations with done. `````` Ralf Jung committed Jan 06, 2017 753 ``````Coercion constURF : ucmraT >-> urFunctor. `````` Robbert Krebbers committed May 27, 2016 754 755 756 757 `````` Instance constURF_contractive B : urFunctorContractive (constURF B). Proof. rewrite /urFunctorContractive; apply _. Qed. `````` Robbert Krebbers committed Feb 08, 2016 758 759 760 761 762 763 764 ``````(** * Transporting a CMRA equality *) Definition cmra_transport {A B : cmraT} (H : A = B) (x : A) : B := eq_rect A id x _ H. Section cmra_transport. Context {A B : cmraT} (H : A = B). Notation T := (cmra_transport H). `````` Ralf Jung committed Jan 27, 2017 765 `````` Global Instance cmra_transport_ne : NonExpansive T. `````` Robbert Krebbers committed Feb 08, 2016 766 767 768 769 770 `````` Proof. by intros ???; destruct H. Qed. Global Instance cmra_transport_proper : Proper ((≡) ==> (≡)) T. Proof. by intros ???; destruct H. Qed. Lemma cmra_transport_op x y : T (x ⋅ y) = T x ⋅ T y. Proof. by destruct H. Qed. `````` Ralf Jung committed Mar 08, 2016 771 `````` Lemma cmra_transport_core x : T (core x) = core (T x). `````` Robbert Krebbers committed Feb 08, 2016 772 `````` Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 773 `````` Lemma cmra_transport_validN n x : ✓{n} T x ↔ ✓{n} x. `````` Robbert Krebbers committed Feb 08, 2016 774 `````` Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 11, 2016 775 `````` Lemma cmra_transport_valid x : ✓ T x ↔ ✓ x. `````` Robbert Krebbers committed Feb 08, 2016 776 777 778 `````` Proof. by destruct H. Qed. Global Instance cmra_transport_timeless x : Timeless x → Timeless (T x). Proof. by destruct H. Qed. `````` Robbert Krebbers committed Mar 15, 2016 779 780 `````` Global Instance cmra_transport_persistent x : Persistent x → Persistent (T x). Proof. by destruct H. Qed. `````` Robbert Krebbers committed Feb 08, 2016 781 782 ``````End cmra_transport. `````` Robbert Krebbers committed Feb 01, 2016 783 784 ``````(** * Instances *) (** ** Discrete CMRA *) `````` Robbert Krebbers committed May 28, 2016 785 ``Record RAMixin A ``