cmra.v 59.7 KB
Newer Older
1
From iris.algebra Require Export ofe monoid.
2
Set Default Proof Using "Type".
3

Robbert Krebbers's avatar
Robbert Krebbers committed
4
Class PCore (A : Type) := pcore : A  option A.
5
Hint Mode PCore ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
6
Instance: Params (@pcore) 2.
7 8

Class Op (A : Type) := op : A  A  A.
9
Hint Mode Op ! : typeclass_instances.
10 11 12 13
Instance: Params (@op) 2.
Infix "⋅" := op (at level 50, left associativity) : C_scope.
Notation "(⋅)" := op (only parsing) : C_scope.

14 15 16 17 18
(* 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
*)
19 20 21
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.
22
Hint Extern 0 (_  _) => reflexivity.
23 24
Instance: Params (@included) 3.

Robbert Krebbers's avatar
Robbert Krebbers committed
25
Class ValidN (A : Type) := validN : nat  A  Prop.
26
Hint Mode ValidN ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
27
Instance: Params (@validN) 3.
28
Notation "✓{ n } x" := (validN n x)
29
  (at level 20, n at next level, format "✓{ n }  x").
Robbert Krebbers's avatar
Robbert Krebbers committed
30

31
Class Valid (A : Type) := valid : A  Prop.
32
Hint Mode Valid ! : typeclass_instances.
33
Instance: Params (@valid) 2.
34
Notation "✓ x" := (valid x) (at level 20) : C_scope.
35

36
Definition includedN `{Dist A, Op A} (n : nat) (x y : A) :=  z, y {n} x  z.
Robbert Krebbers's avatar
Robbert Krebbers committed
37
Notation "x ≼{ n } y" := (includedN n x y)
38
  (at level 70, n at next level, format "x  ≼{ n }  y") : C_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
39
Instance: Params (@includedN) 4.
40
Hint Extern 0 (_ {_} _) => reflexivity.
Robbert Krebbers's avatar
Robbert Krebbers committed
41

42
Set Primitive Projections.
Robbert Krebbers's avatar
Robbert Krebbers committed
43
Record CMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A} := {
Robbert Krebbers's avatar
Robbert Krebbers committed
44
  (* setoids *)
45
  mixin_cmra_op_ne (x : A) : NonExpansive (op x);
Robbert Krebbers's avatar
Robbert Krebbers committed
46 47
  mixin_cmra_pcore_ne n x y cx :
    x {n} y  pcore x = Some cx   cy, pcore y = Some cy  cx {n} cy;
48
  mixin_cmra_validN_ne n : Proper (dist n ==> impl) (validN n);
Robbert Krebbers's avatar
Robbert Krebbers committed
49
  (* valid *)
50
  mixin_cmra_valid_validN x :  x   n, {n} x;
51
  mixin_cmra_validN_S n x : {S n} x  {n} x;
Robbert Krebbers's avatar
Robbert Krebbers committed
52
  (* monoid *)
53 54
  mixin_cmra_assoc : Assoc () ();
  mixin_cmra_comm : Comm () ();
Robbert Krebbers's avatar
Robbert Krebbers committed
55 56
  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;
57
  mixin_cmra_pcore_mono x y cx :
Robbert Krebbers's avatar
Robbert Krebbers committed
58
    x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy;
59
  mixin_cmra_validN_op_l n x y : {n} (x  y)  {n} x;
60 61
  mixin_cmra_extend n x y1 y2 :
    {n} x  x {n} y1  y2 
62
     z1 z2, x  z1  z2  z1 {n} y1  z2 {n} y2
Robbert Krebbers's avatar
Robbert Krebbers committed
63
}.
64
Unset Primitive Projections.
Robbert Krebbers's avatar
Robbert Krebbers committed
65

Robbert Krebbers's avatar
Robbert Krebbers committed
66
(** Bundeled version *)
67
Structure cmraT := CMRAT' {
Robbert Krebbers's avatar
Robbert Krebbers committed
68 69 70
  cmra_car :> Type;
  cmra_equiv : Equiv cmra_car;
  cmra_dist : Dist cmra_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
71
  cmra_pcore : PCore cmra_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
72
  cmra_op : Op cmra_car;
73
  cmra_valid : Valid cmra_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
74
  cmra_validN : ValidN cmra_car;
75
  cmra_ofe_mixin : OfeMixin cmra_car;
76
  cmra_mixin : CMRAMixin cmra_car;
77
  _ : Type
Robbert Krebbers's avatar
Robbert Krebbers committed
78
}.
79
Arguments CMRAT' _ {_ _ _ _ _ _} _ _ _.
80 81 82 83 84
(* Given [m : CMRAMixin A], the notation [CMRAT A m] provides a smart
constructor, which uses [ofe_mixin_of A] to infer the canonical OFE mixin of
the type [A], so that it does not have to be given manually. *)
Notation CMRAT A m := (CMRAT' A (ofe_mixin_of A%type) m A) (only parsing).

85 86 87
Arguments cmra_car : simpl never.
Arguments cmra_equiv : simpl never.
Arguments cmra_dist : simpl never.
Robbert Krebbers's avatar
Robbert Krebbers committed
88
Arguments cmra_pcore : simpl never.
89
Arguments cmra_op : simpl never.
90
Arguments cmra_valid : simpl never.
91
Arguments cmra_validN : simpl never.
92
Arguments cmra_ofe_mixin : simpl never.
93
Arguments cmra_mixin : simpl never.
Robbert Krebbers's avatar
Robbert Krebbers committed
94
Add Printing Constructor cmraT.
95 96 97 98
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.
99 100
Coercion cmra_ofeC (A : cmraT) : ofeT := OfeT A (cmra_ofe_mixin A).
Canonical Structure cmra_ofeC.
Robbert Krebbers's avatar
Robbert Krebbers committed
101

102 103 104 105
Definition cmra_mixin_of' A {Ac : cmraT} (f : Ac  A) : CMRAMixin Ac := cmra_mixin Ac.
Notation cmra_mixin_of A :=
  ltac:(let H := eval hnf in (cmra_mixin_of' A id) in exact H) (only parsing).

106 107 108 109
(** Lifting properties from the mixin *)
Section cmra_mixin.
  Context {A : cmraT}.
  Implicit Types x y : A.
110
  Global Instance cmra_op_ne (x : A) : NonExpansive (op x).
111
  Proof. apply (mixin_cmra_op_ne _ (cmra_mixin A)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
112 113 114
  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.
115 116
  Global Instance cmra_validN_ne n : Proper (dist n ==> impl) (@validN A _ n).
  Proof. apply (mixin_cmra_validN_ne _ (cmra_mixin A)). Qed.
117 118
  Lemma cmra_valid_validN x :  x   n, {n} x.
  Proof. apply (mixin_cmra_valid_validN _ (cmra_mixin A)). Qed.
119 120
  Lemma cmra_validN_S n x : {S n} x  {n} x.
  Proof. apply (mixin_cmra_validN_S _ (cmra_mixin A)). Qed.
121 122 123 124
  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's avatar
Robbert Krebbers committed
125 126 127 128
  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.
129
  Lemma cmra_pcore_mono x y cx :
Robbert Krebbers's avatar
Robbert Krebbers committed
130
    x  y  pcore x = Some cx   cy, pcore y = Some cy  cx  cy.
131
  Proof. apply (mixin_cmra_pcore_mono _ (cmra_mixin A)). Qed.
132 133
  Lemma cmra_validN_op_l n x y : {n} (x  y)  {n} x.
  Proof. apply (mixin_cmra_validN_op_l _ (cmra_mixin A)). Qed.
134
  Lemma cmra_extend n x y1 y2 :
135
    {n} x  x {n} y1  y2 
136
     z1 z2, x  z1  z2  z1 {n} y1  z2 {n} y2.
137
  Proof. apply (mixin_cmra_extend _ (cmra_mixin A)). Qed.
138 139
End cmra_mixin.

Robbert Krebbers's avatar
Robbert Krebbers committed
140 141 142 143 144 145 146
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 {_} _ {_}.
147
Hint Mode Persistent + ! : typeclass_instances.
148
Instance: Params (@Persistent) 1.
Robbert Krebbers's avatar
Robbert Krebbers committed
149

150
(** * Exclusive elements (i.e., elements that cannot have a frame). *)
151 152
Class Exclusive {A : cmraT} (x : A) := exclusive0_l y : {0} (x  y)  False.
Arguments exclusive0_l {_} _ {_} _ _.
153
Hint Mode Exclusive + ! : typeclass_instances.
154
Instance: Params (@Exclusive) 1.
155

156 157 158 159 160
(** * 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.
161
Instance: Params (@Cancelable) 1.
162 163 164 165 166 167

(** * 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.
168
Instance: Params (@IdFree) 1.
169

Robbert Krebbers's avatar
Robbert Krebbers committed
170 171 172 173
(** * 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).
174
Hint Mode CMRATotal ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
175 176

Class Core (A : Type) := core : A  A.
177
Hint Mode Core ! : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
178 179 180 181 182
Instance: Params (@core) 2.

Instance core' `{PCore A} : Core A := λ x, from_option id x (pcore x).
Arguments core' _ _ _ /.

Ralf Jung's avatar
Ralf Jung committed
183
(** * CMRAs with a unit element *)
Robbert Krebbers's avatar
Robbert Krebbers committed
184 185 186 187 188 189 190
Class Unit (A : Type) := ε : A.
Arguments ε {_ _}.

Record UCMRAMixin A `{Dist A, Equiv A, PCore A, Op A, Valid A, Unit A} := {
  mixin_ucmra_unit_valid :  ε;
  mixin_ucmra_unit_left_id : LeftId () ε ();
  mixin_ucmra_pcore_unit : pcore ε  Some ε
191
}.
192

193
Structure ucmraT := UCMRAT' {
194 195 196
  ucmra_car :> Type;
  ucmra_equiv : Equiv ucmra_car;
  ucmra_dist : Dist ucmra_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
197
  ucmra_pcore : PCore ucmra_car;
198 199 200
  ucmra_op : Op ucmra_car;
  ucmra_valid : Valid ucmra_car;
  ucmra_validN : ValidN ucmra_car;
Robbert Krebbers's avatar
Robbert Krebbers committed
201
  ucmra_unit : Unit ucmra_car;
202
  ucmra_ofe_mixin : OfeMixin ucmra_car;
203
  ucmra_cmra_mixin : CMRAMixin ucmra_car;
204
  ucmra_mixin : UCMRAMixin ucmra_car;
205
  _ : Type;
206
}.
207
Arguments UCMRAT' _ {_ _ _ _ _ _ _} _ _ _ _.
208 209
Notation UCMRAT A m :=
  (UCMRAT' A (ofe_mixin_of A%type) (cmra_mixin_of A%type) m A) (only parsing).
210 211 212
Arguments ucmra_car : simpl never.
Arguments ucmra_equiv : simpl never.
Arguments ucmra_dist : simpl never.
Robbert Krebbers's avatar
Robbert Krebbers committed
213
Arguments ucmra_pcore : simpl never.
214 215 216
Arguments ucmra_op : simpl never.
Arguments ucmra_valid : simpl never.
Arguments ucmra_validN : simpl never.
217
Arguments ucmra_ofe_mixin : simpl never.
218 219 220
Arguments ucmra_cmra_mixin : simpl never.
Arguments ucmra_mixin : simpl never.
Add Printing Constructor ucmraT.
Robbert Krebbers's avatar
Robbert Krebbers committed
221
Hint Extern 0 (Unit _) => eapply (@ucmra_unit _) : typeclass_instances.
222 223
Coercion ucmra_ofeC (A : ucmraT) : ofeT := OfeT A (ucmra_ofe_mixin A).
Canonical Structure ucmra_ofeC.
224
Coercion ucmra_cmraR (A : ucmraT) : cmraT :=
225
  CMRAT' A (ucmra_ofe_mixin A) (ucmra_cmra_mixin A) A.
226 227 228 229 230 231
Canonical Structure ucmra_cmraR.

(** Lifting properties from the mixin *)
Section ucmra_mixin.
  Context {A : ucmraT}.
  Implicit Types x y : A.
Robbert Krebbers's avatar
Robbert Krebbers committed
232
  Lemma ucmra_unit_valid :  (ε : A).
233
  Proof. apply (mixin_ucmra_unit_valid _ (ucmra_mixin A)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
234
  Global Instance ucmra_unit_left_id : LeftId () ε (@op A _).
235
  Proof. apply (mixin_ucmra_unit_left_id _ (ucmra_mixin A)). Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
236
  Lemma ucmra_pcore_unit : pcore (ε:A)  Some ε.
Robbert Krebbers's avatar
Robbert Krebbers committed
237
  Proof. apply (mixin_ucmra_pcore_unit _ (ucmra_mixin A)). Qed.
238
End ucmra_mixin.
239

240
(** * Discrete CMRAs *)
241
Class CMRADiscrete (A : cmraT) := {
242 243 244
  cmra_discrete :> Discrete A;
  cmra_discrete_valid (x : A) : {0} x   x
}.
245
Hint Mode CMRADiscrete ! : typeclass_instances.
246

Robbert Krebbers's avatar
Robbert Krebbers committed
247
(** * Morphisms *)
248 249 250 251 252
Class CMRAMorphism {A B : cmraT} (f : A  B) := {
  cmra_morphism_ne :> NonExpansive f;
  cmra_morphism_validN n x : {n} x  {n} f x;
  cmra_morphism_pcore x : pcore (f x)  f <$> pcore x;
  cmra_morphism_op x y : f x  f y  f (x  y)
253
}.
254 255 256
Arguments cmra_morphism_validN {_ _} _ {_} _ _ _.
Arguments cmra_morphism_pcore {_ _} _ {_} _.
Arguments cmra_morphism_op {_ _} _ {_} _ _.
257

Robbert Krebbers's avatar
Robbert Krebbers committed
258
(** * Properties **)
Robbert Krebbers's avatar
Robbert Krebbers committed
259
Section cmra.
260
Context {A : cmraT}.
Robbert Krebbers's avatar
Robbert Krebbers committed
261
Implicit Types x y z : A.
262
Implicit Types xs ys zs : list A.
Robbert Krebbers's avatar
Robbert Krebbers committed
263

264
(** ** Setoids *)
265
Global Instance cmra_pcore_ne' : NonExpansive (@pcore A _).
Robbert Krebbers's avatar
Robbert Krebbers committed
266
Proof.
267
  intros n x y Hxy. destruct (pcore x) as [cx|] eqn:?.
Robbert Krebbers's avatar
Robbert Krebbers committed
268 269 270 271 272 273
  { 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.
274
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
275 276 277
  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.
278
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
279 280
Global Instance cmra_pcore_proper' : Proper (() ==> ()) (@pcore A _).
Proof. apply (ne_proper _). Qed.
281 282
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.
283
Global Instance cmra_op_proper' : Proper (() ==> () ==> ()) (@op A _).
284 285 286 287 288 289 290
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 _).
291 292 293 294
Proof.
  intros x y Hxy; rewrite !cmra_valid_validN.
  by split=> ? n; [rewrite -Hxy|rewrite Hxy].
Qed.
295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312
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.
313
Global Instance cmra_opM_ne : NonExpansive2 (@opM A).
314
Proof. destruct 2; by ofe_subst. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
315 316
Global Instance cmra_opM_proper : Proper (() ==> () ==> ()) (@opM A).
Proof. destruct 2; by setoid_subst. Qed.
317

318 319 320 321 322 323 324 325 326
Global Instance Persistent_proper : Proper (() ==> iff) (@Persistent A).
Proof. solve_proper. Qed.
Global Instance Exclusive_proper : Proper (() ==> iff) (@Exclusive A).
Proof. intros x y Hxy. rewrite /Exclusive. by setoid_rewrite Hxy. Qed.
Global Instance Cancelable_proper : Proper (() ==> iff) (@Cancelable A).
Proof. intros x y Hxy. rewrite /Cancelable. by setoid_rewrite Hxy. Qed.
Global Instance IdFree_proper : Proper (() ==> iff) (@IdFree A).
Proof. intros x y Hxy. rewrite /IdFree. by setoid_rewrite Hxy. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
327 328 329 330
(** ** Op *)
Lemma cmra_opM_assoc x y mz : (x  y) ? mz  x  (y ? mz).
Proof. destruct mz; by rewrite /= -?assoc. Qed.

331
(** ** Validity *)
Robbert Krebbers's avatar
Robbert Krebbers committed
332
Lemma cmra_validN_le n n' x : {n} x  n'  n  {n'} x.
333 334 335
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's avatar
Robbert Krebbers committed
336
Lemma cmra_validN_op_r n x y : {n} (x  y)  {n} y.
337
Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed.
338 339 340
Lemma cmra_valid_op_r x y :  (x  y)   y.
Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_r. Qed.

Ralf Jung's avatar
Ralf Jung committed
341
(** ** Core *)
Robbert Krebbers's avatar
Robbert Krebbers committed
342 343 344
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.
345
Proof. intros. rewrite comm. by apply cmra_pcore_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
346
Lemma cmra_pcore_r' x cx : pcore x  Some cx  x  cx  x.
347
Proof. intros (cx'&?&->)%equiv_Some_inv_r'. by apply cmra_pcore_r. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
348
Lemma cmra_pcore_idemp' x cx : pcore x  Some cx  pcore cx  Some cx.
349
Proof. intros (cx'&?&->)%equiv_Some_inv_r'. eauto using cmra_pcore_idemp. Qed.
350 351 352 353
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's avatar
Robbert Krebbers committed
354 355 356 357 358 359 360 361
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.
362

363 364 365 366
(** ** Persistent elements *)
Lemma persistent_dup x `{!Persistent x} : x  x  x.
Proof. by apply cmra_pcore_dup' with x. Qed.

367
(** ** Exclusive elements *)
368
Lemma exclusiveN_l n x `{!Exclusive x} y : {n} (x  y)  False.
369
Proof. intros. eapply (exclusive0_l x y), cmra_validN_le; eauto with lia. Qed.
370 371 372 373 374 375
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.
376
Lemma exclusiveN_opM n x `{!Exclusive x} my : {n} (x ? my)  my = None.
377
Proof. destruct my as [y|]. move=> /(exclusiveN_l _ x) []. done. Qed.
378 379 380 381
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.
382

383
(** ** Order *)
Robbert Krebbers's avatar
Robbert Krebbers committed
384 385
Lemma cmra_included_includedN n x y : x  y  x {n} y.
Proof. intros [z ->]. by exists z. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
386
Global Instance cmra_includedN_trans n : Transitive (@includedN A _ _ n).
387
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
388
  intros x y z [z1 Hy] [z2 Hz]; exists (z1  z2). by rewrite assoc -Hy -Hz.
389
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
390
Global Instance cmra_included_trans: Transitive (@included A _ _).
391
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
392
  intros x y z [z1 Hy] [z2 Hz]; exists (z1  z2). by rewrite assoc -Hy -Hz.
393
Qed.
394 395
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's avatar
Robbert Krebbers committed
396
Lemma cmra_validN_includedN n x y : {n} y  x {n} y  {n} x.
397
Proof. intros Hyv [z ?]; ofe_subst y; eauto using cmra_validN_op_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
398
Lemma cmra_validN_included n x y : {n} y  x  y  {n} x.
Robbert Krebbers's avatar
Robbert Krebbers committed
399
Proof. intros Hyv [z ?]; setoid_subst; eauto using cmra_validN_op_l. Qed.
400

Robbert Krebbers's avatar
Robbert Krebbers committed
401
Lemma cmra_includedN_S n x y : x {S n} y  x {n} y.
402
Proof. by intros [z Hz]; exists z; apply dist_S. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
403
Lemma cmra_includedN_le n n' x y : x {n} y  n'  n  x {n'} y.
404 405 406 407 408 409 410
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.
411
Proof. rewrite (comm op); apply cmra_includedN_l. Qed.
412
Lemma cmra_included_r x y : y  x  y.
413
Proof. rewrite (comm op); apply cmra_included_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
414

415
Lemma cmra_pcore_mono' x y cx :
Robbert Krebbers's avatar
Robbert Krebbers committed
416 417 418
  x  y  pcore x  Some cx   cy, pcore y = Some cy  cx  cy.
Proof.
  intros ? (cx'&?&Hcx)%equiv_Some_inv_r'.
419
  destruct (cmra_pcore_mono x y cx') as (cy&->&?); auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
420 421
  exists cy; by rewrite Hcx.
Qed.
422
Lemma cmra_pcore_monoN' n x y cx :
Robbert Krebbers's avatar
Robbert Krebbers committed
423
  x {n} y  pcore x {n} Some cx   cy, pcore y = Some cy  cx {n} cy.
Robbert Krebbers's avatar
Robbert Krebbers committed
424
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
425
  intros [z Hy] (cx'&?&Hcx)%dist_Some_inv_r'.
426
  destruct (cmra_pcore_mono x (x  z) cx')
Robbert Krebbers's avatar
Robbert Krebbers committed
427 428 429 430 431
    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's avatar
Robbert Krebbers committed
432
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
433 434
Lemma cmra_included_pcore x cx : pcore x = Some cx  cx  x.
Proof. exists x. by rewrite cmra_pcore_l. Qed.
435

436
Lemma cmra_monoN_l n x y z : x {n} y  z  x {n} z  y.
437
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed.
438
Lemma cmra_mono_l x y z : x  y  z  x  z  y.
439
Proof. by intros [z1 Hz1]; exists z1; rewrite Hz1 (assoc op). Qed.
440 441 442 443
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.
444 445 446 447
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.
448

449 450 451 452 453 454 455
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's avatar
Robbert Krebbers committed
456
Lemma cmra_included_dist_l n x1 x2 x1' :
457
  x1  x2  x1' {n} x1   x2', x1'  x2'  x2' {n} x2.
Robbert Krebbers's avatar
Robbert Krebbers committed
458
Proof.
459 460
  intros [z Hx2] Hx1; exists (x1'  z); split; auto using cmra_included_l.
  by rewrite Hx1 Hx2.
Robbert Krebbers's avatar
Robbert Krebbers committed
461
Qed.
462

Robbert Krebbers's avatar
Robbert Krebbers committed
463 464
(** ** Total core *)
Section total_core.
465
  Local Set Default Proof Using "Type*".
Robbert Krebbers's avatar
Robbert Krebbers committed
466 467 468 469 470 471 472 473 474 475
  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.
476
  Lemma cmra_core_mono x y : x  y  core x  core y.
Robbert Krebbers's avatar
Robbert Krebbers committed
477 478
  Proof.
    intros; destruct (cmra_total x) as [cx Hcx].
479
    destruct (cmra_pcore_mono x y cx) as (cy&Hcy&?); auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
480 481 482
    by rewrite /core /= Hcx Hcy.
  Qed.

483
  Global Instance cmra_core_ne : NonExpansive (@core A _).
Robbert Krebbers's avatar
Robbert Krebbers committed
484
  Proof.
485
    intros n x y Hxy. destruct (cmra_total x) as [cx Hcx].
Robbert Krebbers's avatar
Robbert Krebbers committed
486 487 488 489 490 491 492
    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.
493 494
  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's avatar
Robbert Krebbers committed
495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524
  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.
525
  Lemma cmra_core_monoN n x y : x {n} y  core x {n} core y.
Robbert Krebbers's avatar
Robbert Krebbers committed
526 527
  Proof.
    intros [z ->].
528
    apply cmra_included_includedN, cmra_core_mono, cmra_included_l.
Robbert Krebbers's avatar
Robbert Krebbers committed
529 530 531
  Qed.
End total_core.

Robbert Krebbers's avatar
Robbert Krebbers committed
532
(** ** Timeless *)
533
Lemma cmra_timeless_included_l x y : Timeless x  {0} y  x {0} y  x  y.
Robbert Krebbers's avatar
Robbert Krebbers committed
534 535
Proof.
  intros ?? [x' ?].
536
  destruct (cmra_extend 0 y x x') as (z&z'&Hy&Hz&Hz'); auto; simpl in *.
Robbert Krebbers's avatar
Robbert Krebbers committed
537
  by exists z'; rewrite Hy (timeless x z).
Robbert Krebbers's avatar
Robbert Krebbers committed
538
Qed.
539 540
Lemma cmra_timeless_included_r x y : Timeless y  x {0} y  x  y.
Proof. intros ? [x' ?]. exists x'. by apply (timeless y). Qed.
541
Lemma cmra_op_timeless x1 x2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
542
   (x1  x2)  Timeless x1  Timeless x2  Timeless (x1  x2).
Robbert Krebbers's avatar
Robbert Krebbers committed
543 544
Proof.
  intros ??? z Hz.
545
  destruct (cmra_extend 0 z x1 x2) as (y1&y2&Hz'&?&?); auto; simpl in *.
546
  { rewrite -?Hz. by apply cmra_valid_validN. }
Robbert Krebbers's avatar
Robbert Krebbers committed
547
  by rewrite Hz' (timeless x1 y1) // (timeless x2 y2).
Robbert Krebbers's avatar
Robbert Krebbers committed
548
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
549

550 551 552 553 554 555 556 557
(** ** 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's avatar
Robbert Krebbers committed
558
  split; first by apply cmra_included_includedN.
559 560
  intros [z ->%(timeless_iff _ _)]; eauto using cmra_included_l.
Qed.
561 562 563

(** Cancelable elements  *)
Global Instance cancelable_proper : Proper (equiv ==> iff) (@Cancelable A).
564 565
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.
566 567 568 569 570 571 572
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.
573
  intros ?? n z z' ??. apply (cancelableN y), (cancelableN x).
574 575 576 577 578
  - 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.
579
Proof. intros ? n z z' []%(exclusiveN_l _ x). Qed.
580 581

(** Id-free elements  *)
582
Global Instance id_free_ne n : Proper (dist n ==> iff) (@IdFree A).
583
Proof.
584 585
  intros x x' EQ%(dist_le _ 0); last lia. rewrite /IdFree.
  split=> y ?; (rewrite -EQ || rewrite EQ); eauto.
586 587
Qed.
Global Instance id_free_proper : Proper (equiv ==> iff) (@IdFree A).
588
Proof. by move=> P Q /equiv_dist /(_ 0)=> ->. Qed.
589 590 591 592 593 594 595 596 597
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}:
598
  ( y,  x  x  y  x  False)  IdFree x.
599
Proof. repeat intro. eauto using cmra_discrete_valid, cmra_discrete, timeless. Qed.
600
Global Instance id_free_op_r x y : IdFree y  Cancelable x  IdFree (x  y).
601
Proof.
602
  intros ?? z ? Hid%symmetry. revert Hid. rewrite -assoc=>/(cancelableN x) ?.
603 604
  eapply (id_free0_r _); [by eapply cmra_validN_op_r |symmetry; eauto].
Qed.
605
Global Instance id_free_op_l x y : IdFree x  Cancelable y  IdFree (x  y).
606 607 608
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's avatar
Robbert Krebbers committed
609 610
End cmra.

611 612
(** * Properties about CMRAs with a unit element **)
Section ucmra.
Robbert Krebbers's avatar
Robbert Krebbers committed
613 614 615
  Context {A : ucmraT}.
  Implicit Types x y z : A.

Robbert Krebbers's avatar
Robbert Krebbers committed
616
  Lemma ucmra_unit_validN n : {n} (ε:A).
Robbert Krebbers's avatar
Robbert Krebbers committed
617
  Proof. apply cmra_valid_validN, ucmra_unit_valid. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
618
  Lemma ucmra_unit_leastN n x : ε {n} x.
Robbert Krebbers's avatar
Robbert Krebbers committed
619
  Proof. by exists x; rewrite left_id. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
620
  Lemma ucmra_unit_least x : ε  x.
Robbert Krebbers's avatar
Robbert Krebbers committed
621
  Proof. by exists x; rewrite left_id. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
622
  Global Instance ucmra_unit_right_id : RightId () ε (@op A _).
Robbert Krebbers's avatar
Robbert Krebbers committed
623
  Proof. by intros x; rewrite (comm op) left_id. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
624
  Global Instance ucmra_unit_persistent : Persistent (ε:A).
Robbert Krebbers's avatar
Robbert Krebbers committed
625 626 627 628
  Proof. apply ucmra_pcore_unit. Qed.

  Global Instance cmra_unit_total : CMRATotal A.
  Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
629 630
    intros x. destruct (cmra_pcore_mono' ε x ε) as (cx&->&?);
      eauto using ucmra_unit_least, (persistent (ε:A)).
Robbert Krebbers's avatar
Robbert Krebbers committed
631
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
632
  Global Instance empty_cancelable : Cancelable (ε:A).
633
  Proof. intros ???. by rewrite !left_id. Qed.
634 635

  (* For big ops *)
Robbert Krebbers's avatar
Robbert Krebbers committed
636
  Global Instance cmra_monoid : Monoid (@op A _) := {| monoid_unit := ε |}.
637
End ucmra.
Robbert Krebbers's avatar
Robbert Krebbers committed
638

639
Hint Immediate cmra_unit_total.
640 641 642

(** * Properties about CMRAs with Leibniz equality *)
Section cmra_leibniz.
643
  Local Set Default Proof Using "Type*".
644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666
  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's avatar
Robbert Krebbers committed
667
  Lemma persistent_dup_L x `{!Persistent x} : x = x  x.
668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689
  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.
690
  Local Set Default Proof Using "Type*".
691 692 693
  Context {A : ucmraT} `{!LeibnizEquiv A}.
  Implicit Types x y z : A.

Robbert Krebbers's avatar
Robbert Krebbers committed
694
  Global Instance ucmra_unit_left_id_L : LeftId (=) ε (@op A _).
695
  Proof. intros x. unfold_leibniz. by rewrite left_id. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
696
  Global Instance ucmra_unit_right_id_L : RightId (=) ε (@op A _).
697 698 699
  Proof. intros x. unfold_leibniz. by rewrite right_id. Qed.
End ucmra_leibniz.

Robbert Krebbers's avatar
Robbert Krebbers committed
700 701 702
(** * Constructing a CMRA with total core *)
Section cmra_total.
  Context A `{Dist A, Equiv A, PCore A, Op A, Valid A, ValidN A}.
703 704
  Context (total :  x : A, is_Some (pcore x)).
  Context (op_ne :  x : A, NonExpansive (op x)).
705
  Context (core_ne : NonExpansive (@core A _)).
Robbert Krebbers's avatar
Robbert Krebbers committed
706 707 708 709 710 711 712
  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 () (@