excl.v 6.89 KB
Newer Older
1
Require Export modures.cmra.
Robbert Krebbers's avatar
Robbert Krebbers committed
2
3
Local Arguments validN _ _ _ !_ /.
Local Arguments valid _ _  !_ /.
Robbert Krebbers's avatar
Robbert Krebbers committed
4
5
6

Inductive excl (A : Type) :=
  | Excl : A  excl A
Robbert Krebbers's avatar
Robbert Krebbers committed
7
  | ExclUnit : excl A
Robbert Krebbers's avatar
Robbert Krebbers committed
8
9
10
11
  | ExclBot : excl A.
Arguments Excl {_} _.
Arguments ExclUnit {_}.
Arguments ExclBot {_}.
Robbert Krebbers's avatar
Robbert Krebbers committed
12
13
Instance maybe_Excl {A} : Maybe (@Excl A) := λ x,
  match x with Excl a => Some a | _ => None end.
Robbert Krebbers's avatar
Robbert Krebbers committed
14

15
16
17
Section excl.
Context {A : cofeT}.

Robbert Krebbers's avatar
Robbert Krebbers committed
18
(* Cofe *)
19
Inductive excl_equiv : Equiv (excl A) :=
Robbert Krebbers's avatar
Robbert Krebbers committed
20
  | Excl_equiv (x y : A) : x  y  Excl x  Excl y
Robbert Krebbers's avatar
Robbert Krebbers committed
21
  | ExclUnit_equiv : ExclUnit  ExclUnit
Robbert Krebbers's avatar
Robbert Krebbers committed
22
23
  | ExclBot_equiv : ExclBot  ExclBot.
Existing Instance excl_equiv.
Robbert Krebbers's avatar
Robbert Krebbers committed
24
25
26
27
28
29
Inductive excl_dist `{Dist A} : Dist (excl A) :=
  | excl_dist_0 (x y : excl A) : x ={0}= y
  | Excl_dist (x y : A) n : x ={n}= y  Excl x ={n}= Excl y
  | ExclUnit_dist n : ExclUnit ={n}= ExclUnit
  | ExclBot_dist n : ExclBot ={n}= ExclBot.
Existing Instance excl_dist.
30
Program Definition excl_chain
Robbert Krebbers's avatar
Robbert Krebbers committed
31
32
33
    (c : chain (excl A)) (x : A) (H : maybe Excl (c 1) = Some x) : chain A :=
  {| chain_car n := match c n return _ with Excl y => y | _ => x end |}.
Next Obligation.
34
  intros c x ? n i ?; simpl; destruct (c 1) eqn:?; simplify_equality'.
Robbert Krebbers's avatar
Robbert Krebbers committed
35
36
37
38
39
  destruct (decide (i = 0)) as [->|].
  { by replace n with 0 by lia. }
  feed inversion (chain_cauchy c 1 i); auto with lia congruence.
  feed inversion (chain_cauchy c n i); simpl; auto with lia congruence.
Qed.
40
Instance excl_compl : Compl (excl A) := λ c,
Robbert Krebbers's avatar
Robbert Krebbers committed
41
42
43
  match Some_dec (maybe Excl (c 1)) with
  | inleft (exist x H) => Excl (compl (excl_chain c x H)) | inright _ => c 1
  end.
44
Definition excl_cofe_mixin : CofeMixin (excl A).
Robbert Krebbers's avatar
Robbert Krebbers committed
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
Proof.
  split.
  * intros mx my; split; [by destruct 1; constructor; apply equiv_dist|].
    intros Hxy; feed inversion (Hxy 1); subst; constructor; apply equiv_dist.
    by intros n; feed inversion (Hxy n).
  * intros n; split.
    + by intros [x| |]; constructor.
    + by destruct 1; constructor.
    + destruct 1; inversion_clear 1; constructor; etransitivity; eauto.
  * by inversion_clear 1; constructor; apply dist_S.
  * constructor.
  * intros c n; unfold compl, excl_compl.
    destruct (decide (n = 0)) as [->|]; [constructor|].
    destruct (Some_dec (maybe Excl (c 1))) as [[x Hx]|].
    { assert (c 1 = Excl x) by (by destruct (c 1); simplify_equality').
      assert ( y, c n = Excl y) as [y Hy].
      { feed inversion (chain_cauchy c 1 n); try congruence; eauto with lia. }
      rewrite Hy; constructor.
      by rewrite (conv_compl (excl_chain c x Hx) n); simpl; rewrite Hy. }
    feed inversion (chain_cauchy c 1 n); auto with lia; constructor.
    destruct (c 1); simplify_equality'.
Qed.
67
68
Canonical Structure exclC : cofeT := CofeT excl_cofe_mixin.

Robbert Krebbers's avatar
Robbert Krebbers committed
69
70
71
72
Global Instance Excl_inj : Injective () () (@Excl A).
Proof. by inversion_clear 1. Qed.
Global Instance Excl_dist_inj n : Injective (dist n) (dist n) (@Excl A).
Proof. by inversion_clear 1. Qed.
73
Global Instance Excl_timeless (x : A) : Timeless x  Timeless (Excl x).
Robbert Krebbers's avatar
Robbert Krebbers committed
74
Proof. by inversion_clear 2; constructor; apply (timeless _). Qed.
75
Global Instance ExclUnit_timeless : Timeless (@ExclUnit A).
Robbert Krebbers's avatar
Robbert Krebbers committed
76
Proof. by inversion_clear 1; constructor. Qed.
77
Global Instance ExclBot_timeless : Timeless (@ExclBot A).
Robbert Krebbers's avatar
Robbert Krebbers committed
78
Proof. by inversion_clear 1; constructor. Qed.
79
Global Instance excl_timeless :
Robbert Krebbers's avatar
Robbert Krebbers committed
80
81
  ( x : A, Timeless x)   x : excl A, Timeless x.
Proof. intros ? []; apply _. Qed.
82
83
Global Instance excl_leibniz : LeibnizEquiv A  LeibnizEquiv (excl A).
Proof. by destruct 2; f_equal; apply leibniz_equiv. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
84
85

(* CMRA *)
86
Instance excl_validN : ValidN (excl A) := λ n x,
Robbert Krebbers's avatar
Robbert Krebbers committed
87
  match x with Excl _ | ExclUnit => True | ExclBot => n = 0 end.
88
89
90
Global Instance excl_empty : Empty (excl A) := ExclUnit.
Instance excl_unit : Unit (excl A) := λ _, .
Instance excl_op : Op (excl A) := λ x y,
Robbert Krebbers's avatar
Robbert Krebbers committed
91
92
93
94
95
  match x, y with
  | Excl x, ExclUnit | ExclUnit, Excl x => Excl x
  | ExclUnit, ExclUnit => ExclUnit
  | _, _=> ExclBot
  end.
96
Instance excl_minus : Minus (excl A) := λ x y,
Robbert Krebbers's avatar
Robbert Krebbers committed
97
98
99
100
101
  match x, y with
  | _, ExclUnit => x
  | Excl _, Excl _ => ExclUnit
  | _, _ => ExclBot
  end.
102
Definition excl_cmra_mixin : CMRAMixin (excl A).
Robbert Krebbers's avatar
Robbert Krebbers committed
103
104
Proof.
  split.
Robbert Krebbers's avatar
Robbert Krebbers committed
105
  * by intros n []; destruct 1; constructor.
Robbert Krebbers's avatar
Robbert Krebbers committed
106
  * constructor.
Robbert Krebbers's avatar
Robbert Krebbers committed
107
108
109
110
  * by destruct 1 as [? []| | |]; intros ?.
  * by destruct 1; inversion_clear 1; constructor.
  * by intros [].
  * intros n [?| |]; simpl; auto with lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
111
112
113
114
  * by intros [?| |] [?| |] [?| |]; constructor.
  * by intros [?| |] [?| |]; constructor.
  * by intros [?| |]; constructor.
  * constructor.
Robbert Krebbers's avatar
Robbert Krebbers committed
115
116
117
  * by intros n [?| |] [?| |]; exists .
  * by intros n [?| |] [?| |].
  * by intros n [?| |] [?| |] [[?| |] Hz]; inversion_clear Hz; constructor.
Robbert Krebbers's avatar
Robbert Krebbers committed
118
Qed.
119
Definition excl_cmra_extend_mixin : CMRAExtendMixin (excl A).
Robbert Krebbers's avatar
Robbert Krebbers committed
120
121
122
123
124
125
126
127
Proof.
  intros [|n] x y1 y2 ? Hx; [by exists (x,); destruct x|].
  by exists match y1, y2 with
    | Excl a1, Excl a2 => (Excl a1, Excl a2)
    | ExclBot, _ => (ExclBot, y2) | _, ExclBot => (y1, ExclBot)
    | ExclUnit, _ => (ExclUnit, x) | _, ExclUnit => (x, ExclUnit)
    end; destruct y1, y2; inversion_clear Hx; repeat constructor.
Qed.
128
129
Canonical Structure exclRA : cmraT :=
  CMRAT excl_cofe_mixin excl_cmra_mixin excl_cmra_extend_mixin.
130
131
Global Instance excl_cmra_identity : CMRAIdentity exclRA.
Proof. split. done. by intros []. apply _. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
132
133
134
135
Lemma excl_validN_inv_l n x y : {S n} (Excl x  y)  y = .
Proof. by destruct y. Qed.
Lemma excl_validN_inv_r n x y : {S n} (x  Excl y)  x = .
Proof. by destruct x. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
136
137
138
139
140
Lemma Excl_includedN n x y : {n} y  Excl x {n} y  y ={n}= Excl x.
Proof.
  intros Hvalid; split; [destruct n as [|n]; [done|]|by intros ->].
  by intros [z ?]; cofe_subst; rewrite (excl_validN_inv_l n x z).
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
141
142

(* Updates *)
143
Lemma excl_update (x : A) y :  y  Excl x  y.
Robbert Krebbers's avatar
Robbert Krebbers committed
144
Proof. by destruct y; intros ? [?| |]. Qed.
145
146
Lemma excl_updateP (P : excl A  Prop) x y :  y  P y  Excl x : P.
Proof. intros ?? z n ?; exists y. by destruct y, z as [?| |]. Qed.
147
148
149
150
End excl.

Arguments exclC : clear implicits.
Arguments exclRA : clear implicits.
Robbert Krebbers's avatar
Robbert Krebbers committed
151

Robbert Krebbers's avatar
Robbert Krebbers committed
152
153
154
155
156
(* Functor *)
Instance excl_fmap : FMap excl := λ A B f x,
  match x with
  | Excl a => Excl (f a) | ExclUnit => ExclUnit | ExclBot => ExclBot
  end.
157
158
159
160
161
Lemma excl_fmap_id {A} (x : excl A) : id <$> x = x.
Proof. by destruct x. Qed.
Lemma excl_fmap_compose {A B C} (f : A  B) (g : B  C) (x : excl A) :
  g  f <$> x = g <$> f <$> x.
Proof. by destruct x. Qed.
162
Instance excl_fmap_cmra_ne {A B : cofeT} n :
Robbert Krebbers's avatar
Robbert Krebbers committed
163
164
  Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@fmap excl _ A B).
Proof. by intros f f' Hf; destruct 1; constructor; apply Hf. Qed.
165
Instance excl_fmap_cmra_monotone {A B : cofeT} :
Robbert Krebbers's avatar
Robbert Krebbers committed
166
167
168
169
170
171
172
  ( n, Proper (dist n ==> dist n) f)  CMRAMonotone (fmap f : excl A  excl B).
Proof.
  split.
  * intros n x y [z Hy]; exists (f <$> z); rewrite Hy.
    by destruct x, z; constructor.
  * by intros n [a| |].
Qed.
173
Definition exclRA_map {A B} (f : A -n> B) : exclRA A -n> exclRA B :=
Robbert Krebbers's avatar
Robbert Krebbers committed
174
175
176
  CofeMor (fmap f : exclRA A  exclRA B).
Lemma exclRA_map_ne A B n : Proper (dist n ==> dist n) (@exclRA_map A B).
Proof. by intros f f' Hf []; constructor; apply Hf. Qed.