upred.v 52.2 KB
Newer Older
1
From algebra Require Export cmra.
2 3
Local Hint Extern 1 (_  _) => etrans; [eassumption|].
Local Hint Extern 1 (_  _) => etrans; [|eassumption].
Robbert Krebbers's avatar
Robbert Krebbers committed
4 5
Local Hint Extern 10 (_  _) => omega.

Robbert Krebbers's avatar
Robbert Krebbers committed
6
Record uPred (M : cmraT) : Type := IProp {
Robbert Krebbers's avatar
Robbert Krebbers committed
7
  uPred_holds :> nat  M  Prop;
Robbert Krebbers's avatar
Robbert Krebbers committed
8 9
  uPred_ne n x1 x2 : uPred_holds n x1  x1 {n} x2  uPred_holds n x2;
  uPred_weaken  n1 n2 x1 x2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
10
    uPred_holds n1 x1  x1  x2  n2  n1  {n2} x2  uPred_holds n2 x2
Robbert Krebbers's avatar
Robbert Krebbers committed
11
}.
12
Arguments uPred_holds {_} _ _ _ : simpl never.
Robbert Krebbers's avatar
Robbert Krebbers committed
13 14
Add Printing Constructor uPred.
Instance: Params (@uPred_holds) 3.
Robbert Krebbers's avatar
Robbert Krebbers committed
15

16 17 18 19
Delimit Scope uPred_scope with I.
Bind Scope uPred_scope with uPred.
Arguments uPred_holds {_} _%I _ _.

20 21
Section cofe.
  Context {M : cmraT}.
22 23 24 25 26 27 28

  Inductive uPred_equiv' (P Q : uPred M) : Prop :=
    { uPred_in_equiv :  n x, {n} x  P n x  Q n x }.
  Instance uPred_equiv : Equiv (uPred M) := uPred_equiv'.
  Inductive uPred_dist' (n : nat) (P Q : uPred M) : Prop :=
    { uPred_in_dist :  n' x, n'  n  {n'} x  P n' x  Q n' x }.
  Instance uPred_dist : Dist (uPred M) := uPred_dist'.
29
  Program Instance uPred_compl : Compl (uPred M) := λ c,
30
    {| uPred_holds n x := c n n x |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
31
  Next Obligation. by intros c n x y ??; simpl in *; apply uPred_ne with x. Qed.
32
  Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
33
    intros c n1 n2 x1 x2 ????; simpl in *.
34
    apply (chain_cauchy c n2 n1); eauto using uPred_weaken.
35 36 37 38
  Qed.
  Definition uPred_cofe_mixin : CofeMixin (uPred M).
  Proof.
    split.
39 40 41
    - intros P Q; split.
      + by intros HPQ n; split=> i x ??; apply HPQ.
      + intros HPQ; split=> n x ?; apply HPQ with n; auto.
42
    - intros n; split.
43 44 45 46 47
      + by intros P; split=> x i.
      + by intros P Q HPQ; split=> x i ??; symmetry; apply HPQ.
      + intros P Q Q' HP HQ; split=> i x ??.
        by trans (Q i x);[apply HP|apply HQ].
    - intros n P Q HPQ; split=> i x ??; apply HPQ; auto.
48
    - intros n c; split=>i x ??; symmetry; apply (chain_cauchy c i n); auto.
49 50 51 52 53
  Qed.
  Canonical Structure uPredC : cofeT := CofeT uPred_cofe_mixin.
End cofe.
Arguments uPredC : clear implicits.

54
Instance uPred_ne' {M} (P : uPred M) n : Proper (dist n ==> iff) (P n).
Robbert Krebbers's avatar
Robbert Krebbers committed
55
Proof. intros x1 x2 Hx; split; eauto using uPred_ne. Qed.
56 57 58 59
Instance uPred_proper {M} (P : uPred M) n : Proper (() ==> iff) (P n).
Proof. by intros x1 x2 Hx; apply uPred_ne', equiv_dist. Qed.

Lemma uPred_holds_ne {M} (P1 P2 : uPred M) n x :
60
  P1 {n} P2  {n} x  P1 n x  P2 n x.
61
Proof. intros HP ?; apply HP; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
62
Lemma uPred_weaken' {M} (P : uPred M) n1 n2 x1 x2 :
63
  x1  x2  n2  n1  {n2} x2  P n1 x1  P n2 x2.
64
Proof. eauto using uPred_weaken. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
65 66

(** functor *)
67 68 69 70
Program Definition uPred_map {M1 M2 : cmraT} (f : M2 -n> M1)
  `{!CMRAMonotone f} (P : uPred M1) :
  uPred M2 := {| uPred_holds n x := P n (f x) |}.
Next Obligation. by intros M1 M2 f ? P y1 y2 n ? Hy; rewrite /= -Hy. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
71
Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
72
  naive_solver eauto using uPred_weaken, included_preserving, validN_preserving.
Robbert Krebbers's avatar
Robbert Krebbers committed
73
Qed.
74
Instance uPred_map_ne {M1 M2 : cmraT} (f : M2 -n> M1)
Robbert Krebbers's avatar
Robbert Krebbers committed
75
  `{!CMRAMonotone f} n : Proper (dist n ==> dist n) (uPred_map f).
Robbert Krebbers's avatar
Robbert Krebbers committed
76
Proof.
77 78
  intros x1 x2 Hx; split=> n' y ??.
  split; apply Hx; auto using validN_preserving.
Robbert Krebbers's avatar
Robbert Krebbers committed
79
Qed.
80
Lemma uPred_map_id {M : cmraT} (P : uPred M): uPred_map cid P  P.
81
Proof. by split=> n x ?. Qed.
82
Lemma uPred_map_compose {M1 M2 M3 : cmraT} (f : M1 -n> M2) (g : M2 -n> M3)
Robbert Krebbers's avatar
Robbert Krebbers committed
83
    `{!CMRAMonotone f, !CMRAMonotone g} (P : uPred M3):
84
  uPred_map (g  f) P  uPred_map f (uPred_map g P).
85
Proof. by split=> n x Hx. Qed.
86
Lemma uPred_map_ext {M1 M2 : cmraT} (f g : M1 -n> M2)
87
      `{!CMRAMonotone f} `{!CMRAMonotone g}:
88
  ( x, f x  g x)   x, uPred_map f x  uPred_map g x.
89
Proof. intros Hf P; split=> n x Hx /=; by rewrite /uPred_holds /= Hf. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
90
Definition uPredC_map {M1 M2 : cmraT} (f : M2 -n> M1) `{!CMRAMonotone f} :
Robbert Krebbers's avatar
Robbert Krebbers committed
91
  uPredC M1 -n> uPredC M2 := CofeMor (uPred_map f : uPredC M1  uPredC M2).
92
Lemma uPredC_map_ne {M1 M2 : cmraT} (f g : M2 -n> M1)
Robbert Krebbers's avatar
Robbert Krebbers committed
93
    `{!CMRAMonotone f, !CMRAMonotone g} n :
94
  f {n} g  uPredC_map f {n} uPredC_map g.
Robbert Krebbers's avatar
Robbert Krebbers committed
95
Proof.
96
  by intros Hfg P; split=> n' y ??;
97
    rewrite /uPred_holds /= (dist_le _ _ _ _(Hfg y)); last lia.
Robbert Krebbers's avatar
Robbert Krebbers committed
98
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
99

100 101 102 103
Program Definition uPredCF (F : rFunctor) : cFunctor := {|
  cFunctor_car A B := uPredC (rFunctor_car F B A);
  cFunctor_map A1 A2 B1 B2 fg := uPredC_map (rFunctor_map F (fg.2, fg.1))
|}.
104 105 106 107
Next Obligation.
  intros F A1 A2 B1 B2 n P Q HPQ.
  apply uPredC_map_ne, rFunctor_ne; split; by apply HPQ.
Qed.
108 109 110 111 112 113 114 115 116
Next Obligation.
  intros F A B P; simpl. rewrite -{2}(uPred_map_id P).
  apply uPred_map_ext=>y. by rewrite rFunctor_id.
Qed.
Next Obligation.
  intros F A1 A2 A3 B1 B2 B3 f g f' g' P; simpl. rewrite -uPred_map_compose.
  apply uPred_map_ext=>y; apply rFunctor_compose.
Qed.

117 118 119 120 121 122 123
Instance uPredCF_contractive F :
  rFunctorContractive F  cFunctorContractive (uPredCF F).
Proof.
  intros ? A1 A2 B1 B2 n P Q HPQ.
  apply uPredC_map_ne, rFunctor_contractive=> i ?; split; by apply HPQ.
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
124
(** logical entailement *)
125 126
Inductive uPred_entails {M} (P Q : uPred M) : Prop :=
  { uPred_in_entails :  n x, {n} x  P n x  Q n x }.
127
Hint Extern 0 (uPred_entails _ _) => reflexivity.
128
Instance uPred_entails_rewrite_relation M : RewriteRelation (@uPred_entails M).
Robbert Krebbers's avatar
Robbert Krebbers committed
129 130

(** logical connectives *)
131
Program Definition uPred_const_def {M} (φ : Prop) : uPred M :=
132
  {| uPred_holds n x := φ |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
133
Solve Obligations with done.
134 135 136 137 138
Definition uPred_const_aux : { x | x = @uPred_const_def }. by eexists. Qed.
Definition uPred_const {M} := proj1_sig uPred_const_aux M.
Definition uPred_const_eq :
  @uPred_const = @uPred_const_def := proj2_sig uPred_const_aux.

139
Instance uPred_inhabited M : Inhabited (uPred M) := populate (uPred_const True).
Robbert Krebbers's avatar
Robbert Krebbers committed
140

141
Program Definition uPred_and_def {M} (P Q : uPred M) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
142 143
  {| uPred_holds n x := P n x  Q n x |}.
Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
144 145 146 147 148
Definition uPred_and_aux : { x | x = @uPred_and_def }. by eexists. Qed.
Definition uPred_and {M} := proj1_sig uPred_and_aux M.
Definition uPred_and_eq: @uPred_and = @uPred_and_def := proj2_sig uPred_and_aux.

Program Definition uPred_or_def {M} (P Q : uPred M) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
149 150
  {| uPred_holds n x := P n x  Q n x |}.
Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
151 152 153 154 155
Definition uPred_or_aux : { x | x = @uPred_or_def }. by eexists. Qed.
Definition uPred_or {M} := proj1_sig uPred_or_aux M.
Definition uPred_or_eq: @uPred_or = @uPred_or_def := proj2_sig uPred_or_aux.

Program Definition uPred_impl_def {M} (P Q : uPred M) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
156
  {| uPred_holds n x :=  n' x',
Robbert Krebbers's avatar
Robbert Krebbers committed
157
       x  x'  n'  n  {n'} x'  P n' x'  Q n' x' |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
158
Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
159 160
  intros M P Q n1 x1' x1 HPQ Hx1 n2 x2 ????.
  destruct (cmra_included_dist_l n1 x1 x2 x1') as (x2'&?&Hx2); auto.
161
  assert (x2' {n2} x2) as Hx2' by (by apply dist_le with n1).
162
  assert ({n2} x2') by (by rewrite Hx2'); rewrite -Hx2'.
Robbert Krebbers's avatar
Robbert Krebbers committed
163
  eauto using uPred_weaken, uPred_ne.
Robbert Krebbers's avatar
Robbert Krebbers committed
164
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
165
Next Obligation. intros M P Q [|n] x1 x2; auto with lia. Qed.
166 167 168 169
Definition uPred_impl_aux : { x | x = @uPred_impl_def }. by eexists. Qed.
Definition uPred_impl {M} := proj1_sig uPred_impl_aux M.
Definition uPred_impl_eq :
  @uPred_impl = @uPred_impl_def := proj2_sig uPred_impl_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
170

171
Program Definition uPred_forall_def {M A} (Ψ : A  uPred M) : uPred M :=
172
  {| uPred_holds n x :=  a, Ψ a n x |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
173
Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
174 175 176 177 178 179
Definition uPred_forall_aux : { x | x = @uPred_forall_def }. by eexists. Qed.
Definition uPred_forall {M A} := proj1_sig uPred_forall_aux M A.
Definition uPred_forall_eq :
  @uPred_forall = @uPred_forall_def := proj2_sig uPred_forall_aux.

Program Definition uPred_exist_def {M A} (Ψ : A  uPred M) : uPred M :=
180
  {| uPred_holds n x :=  a, Ψ a n x |}.
181
Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
182 183 184
Definition uPred_exist_aux : { x | x = @uPred_exist_def }. by eexists. Qed.
Definition uPred_exist {M A} := proj1_sig uPred_exist_aux M A.
Definition uPred_exist_eq: @uPred_exist = @uPred_exist_def := proj2_sig uPred_exist_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
185

186
Program Definition uPred_eq_def {M} {A : cofeT} (a1 a2 : A) : uPred M :=
187
  {| uPred_holds n x := a1 {n} a2 |}.
188
Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)).
189 190 191
Definition uPred_eq_aux : { x | x = @uPred_eq_def }. by eexists. Qed.
Definition uPred_eq {M A} := proj1_sig uPred_eq_aux M A.
Definition uPred_eq_eq: @uPred_eq = @uPred_eq_def := proj2_sig uPred_eq_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
192

193
Program Definition uPred_sep_def {M} (P Q : uPred M) : uPred M :=
194
  {| uPred_holds n x :=  x1 x2, x {n} x1  x2  P n x1  Q n x2 |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
195
Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
196
  by intros M P Q n x y (x1&x2&?&?&?) Hxy; exists x1, x2; rewrite -Hxy.
Robbert Krebbers's avatar
Robbert Krebbers committed
197 198
Qed.
Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
199
  intros M P Q n1 n2 x y (x1&x2&Hx&?&?) Hxy ??.
200
  assert ( x2', y {n2} x1  x2'  x2  x2') as (x2'&Hy&?).
201
  { destruct Hxy as [z Hy]; exists (x2  z); split; eauto using cmra_included_l.
202
    apply dist_le with n1; auto. by rewrite (assoc op) -Hx Hy. }
203
  clear Hxy; cofe_subst y; exists x1, x2'; split_and?; [done| |].
Robbert Krebbers's avatar
Robbert Krebbers committed
204 205
  - apply uPred_weaken with n1 x1; eauto using cmra_validN_op_l.
  - apply uPred_weaken with n1 x2; eauto using cmra_validN_op_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
206
Qed.
207 208 209
Definition uPred_sep_aux : { x | x = @uPred_sep_def }. by eexists. Qed.
Definition uPred_sep {M} := proj1_sig uPred_sep_aux M.
Definition uPred_sep_eq: @uPred_sep = @uPred_sep_def := proj2_sig uPred_sep_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
210

211
Program Definition uPred_wand_def {M} (P Q : uPred M) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
212
  {| uPred_holds n x :=  n' x',
Robbert Krebbers's avatar
Robbert Krebbers committed
213
       n'  n  {n'} (x  x')  P n' x'  Q n' (x  x') |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
214
Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
215
  intros M P Q n1 x1 x2 HPQ Hx n2 x3 ???; simpl in *.
Robbert Krebbers's avatar
Robbert Krebbers committed
216
  rewrite -(dist_le _ _ _ _ Hx) //; apply HPQ; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
217
  by rewrite (dist_le _ _ _ _ Hx).
Robbert Krebbers's avatar
Robbert Krebbers committed
218 219
Qed.
Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
220 221
  intros M P Q n1 n2 x1 x2 HPQ ??? n3 x3 ???; simpl in *.
  apply uPred_weaken with n3 (x1  x3);
222
    eauto using cmra_validN_included, cmra_preserving_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
223
Qed.
224 225 226 227
Definition uPred_wand_aux : { x | x = @uPred_wand_def }. by eexists. Qed.
Definition uPred_wand {M} := proj1_sig uPred_wand_aux M.
Definition uPred_wand_eq :
  @uPred_wand = @uPred_wand_def := proj2_sig uPred_wand_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
228

229
Program Definition uPred_always_def {M} (P : uPred M) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
230
  {| uPred_holds n x := P n (unit x) |}.
231
Next Obligation. by intros M P x1 x2 n ? Hx; rewrite /= -Hx. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
232
Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
233
  intros M P n1 n2 x1 x2 ????; eapply uPred_weaken with n1 (unit x1);
234
    eauto using cmra_unit_preserving, cmra_unit_validN.
Robbert Krebbers's avatar
Robbert Krebbers committed
235
Qed.
236 237 238 239 240 241
Definition uPred_always_aux : { x | x = @uPred_always_def }. by eexists. Qed.
Definition uPred_always {M} := proj1_sig uPred_always_aux M.
Definition uPred_always_eq :
  @uPred_always = @uPred_always_def := proj2_sig uPred_always_aux.

Program Definition uPred_later_def {M} (P : uPred M) : uPred M :=
242 243 244 245 246
  {| uPred_holds n x := match n return _ with 0 => True | S n' => P n' x end |}.
Next Obligation. intros M P [|n] ??; eauto using uPred_ne,(dist_le (A:=M)). Qed.
Next Obligation.
  intros M P [|n1] [|n2] x1 x2; eauto using uPred_weaken,cmra_validN_S; try lia.
Qed.
247 248 249 250
Definition uPred_later_aux : { x | x = @uPred_later_def }. by eexists. Qed.
Definition uPred_later {M} := proj1_sig uPred_later_aux M.
Definition uPred_later_eq :
  @uPred_later = @uPred_later_def := proj2_sig uPred_later_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
251

252
Program Definition uPred_ownM_def {M : cmraT} (a : M) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
253
  {| uPred_holds n x := a {n} x |}.
Robbert Krebbers's avatar
Robbert Krebbers committed
254
Next Obligation. by intros M a n x1 x2 [a' ?] Hx; exists a'; rewrite -Hx. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
255
Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
256
  intros M a n1 n2 x1 x [a' Hx1] [x2 Hx] ??.
257
  exists (a'  x2). by rewrite (assoc op) -(dist_le _ _ _ _ Hx1) // Hx.
Robbert Krebbers's avatar
Robbert Krebbers committed
258
Qed.
259 260 261 262 263 264
Definition uPred_ownM_aux : { x | x = @uPred_ownM_def }. by eexists. Qed.
Definition uPred_ownM {M} := proj1_sig uPred_ownM_aux M.
Definition uPred_ownM_eq :
  @uPred_ownM = @uPred_ownM_def := proj2_sig uPred_ownM_aux.

Program Definition uPred_valid_def {M A : cmraT} (a : A) : uPred M :=
Robbert Krebbers's avatar
Robbert Krebbers committed
265
  {| uPred_holds n x := {n} a |}.
266
Solve Obligations with naive_solver eauto 2 using cmra_validN_le.
267 268 269 270
Definition uPred_valid_aux : { x | x = @uPred_valid_def }. by eexists. Qed.
Definition uPred_valid {M A} := proj1_sig uPred_valid_aux M A.
Definition uPred_valid_eq :
  @uPred_valid = @uPred_valid_def := proj2_sig uPred_valid_aux.
Robbert Krebbers's avatar
Robbert Krebbers committed
271

272 273
Notation "P ⊑ Q" := (uPred_entails P%I Q%I) (at level 70) : C_scope.
Notation "(⊑)" := uPred_entails (only parsing) : C_scope.
274 275
Notation "■ φ" := (uPred_const φ%C%type)
  (at level 20, right associativity) : uPred_scope.
Ralf Jung's avatar
Ralf Jung committed
276
Notation "x = y" := (uPred_const (x%C%type = y%C%type)) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
277 278 279
Notation "'False'" := (uPred_const False) : uPred_scope.
Notation "'True'" := (uPred_const True) : uPred_scope.
Infix "∧" := uPred_and : uPred_scope.
280
Notation "(∧)" := uPred_and (only parsing) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
281
Infix "∨" := uPred_or : uPred_scope.
282
Notation "(∨)" := uPred_or (only parsing) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
283 284
Infix "→" := uPred_impl : uPred_scope.
Infix "★" := uPred_sep (at level 80, right associativity) : uPred_scope.
285
Notation "(★)" := uPred_sep (only parsing) : uPred_scope.
286
Notation "P -★ Q" := (uPred_wand P Q)
287
  (at level 199, Q at level 200, right associativity) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
288
Notation "∀ x .. y , P" :=
289
  (uPred_forall (λ x, .. (uPred_forall (λ y, P)) ..)%I) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
290
Notation "∃ x .. y , P" :=
291
  (uPred_exist (λ x, .. (uPred_exist (λ y, P)) ..)%I) : uPred_scope.
292 293
Notation "□ P" := (uPred_always P)
  (at level 20, right associativity) : uPred_scope.
294 295
Notation "▷ P" := (uPred_later P)
  (at level 20, right associativity) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
296
Infix "≡" := uPred_eq : uPred_scope.
297
Notation "✓ x" := (uPred_valid x) (at level 20) : uPred_scope.
Robbert Krebbers's avatar
Robbert Krebbers committed
298

299 300 301
Definition uPred_iff {M} (P Q : uPred M) : uPred M := ((P  Q)  (Q  P))%I.
Infix "↔" := uPred_iff : uPred_scope.

302
Class TimelessP {M} (P : uPred M) := timelessP :  P  (P   False).
303
Arguments timelessP {_} _ {_}.
304
Class AlwaysStable {M} (P : uPred M) := always_stable : P   P.
305
Arguments always_stable {_} _ {_}.
Robbert Krebbers's avatar
Robbert Krebbers committed
306

307 308 309 310 311 312 313 314
Module uPred.
Definition unseal :=
  (uPred_const_eq, uPred_and_eq, uPred_or_eq, uPred_impl_eq, uPred_forall_eq,
  uPred_exist_eq, uPred_eq_eq, uPred_sep_eq, uPred_wand_eq, uPred_always_eq,
  uPred_later_eq, uPred_ownM_eq, uPred_valid_eq).
Ltac unseal := rewrite !unseal.

Section uPred_logic.
315
Context {M : cmraT}.
316
Implicit Types φ : Prop.
Robbert Krebbers's avatar
Robbert Krebbers committed
317
Implicit Types P Q : uPred M.
318
Implicit Types A : Type.
319
Notation "P ⊑ Q" := (@uPred_entails M P%I Q%I). (* Force implicit argument M *)
320
Arguments uPred_holds {_} !_ _ _ /.
321
Hint Immediate uPred_in_entails.
Robbert Krebbers's avatar
Robbert Krebbers committed
322

323
Global Instance: PreOrder (@uPred_entails M).
324 325 326 327 328
Proof.
  split.
  * by intros P; split=> x i.
  * by intros P Q Q' HP HQ; split=> x i ??; apply HQ, HP.
Qed.
329
Global Instance: AntiSymm () (@uPred_entails M).
330
Proof. intros P Q HPQ HQP; split=> x n; by split; [apply HPQ|apply HQP]. Qed.
331
Lemma equiv_spec P Q : P  Q  P  Q  Q  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
332
Proof.
333
  split; [|by intros [??]; apply (anti_symm ())].
334
  intros HPQ; split; split=> x i; apply HPQ.
Robbert Krebbers's avatar
Robbert Krebbers committed
335
Qed.
336 337 338 339
Lemma equiv_entails P Q : P  Q  P  Q.
Proof. apply equiv_spec. Qed.
Lemma equiv_entails_sym P Q : Q  P  P  Q.
Proof. apply equiv_spec. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
340
Global Instance entails_proper :
341
  Proper (() ==> () ==> iff) (() : relation (uPred M)).
Robbert Krebbers's avatar
Robbert Krebbers committed
342
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
343
  move => P1 P2 /equiv_spec [HP1 HP2] Q1 Q2 /equiv_spec [HQ1 HQ2]; split; intros.
344 345
  - by trans P1; [|trans Q1].
  - by trans P2; [|trans Q2].
Robbert Krebbers's avatar
Robbert Krebbers committed
346
Qed.
347 348 349 350
Lemma entails_equiv_l (P Q R : uPred M) : P  Q  Q  R  P  R.
Proof. by intros ->. Qed.
Lemma entails_equiv_r (P Q R : uPred M) : P  Q  Q  R  P  R.
Proof. by intros ? <-. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
351

352
(** Non-expansiveness and setoid morphisms *)
Robbert Krebbers's avatar
Robbert Krebbers committed
353
Global Instance const_proper : Proper (iff ==> ()) (@uPred_const M).
354
Proof. intros φ1 φ2 Hφ. by unseal; split=> -[|n] ?; try apply Hφ. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
355
Global Instance and_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_and M).
Robbert Krebbers's avatar
Robbert Krebbers committed
356
Proof.
357
  intros P P' HP Q Q' HQ; unseal; split=> x n' ??.
358
  split; (intros [??]; split; [by apply HP|by apply HQ]).
Robbert Krebbers's avatar
Robbert Krebbers committed
359
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
360
Global Instance and_proper :
Robbert Krebbers's avatar
Robbert Krebbers committed
361
  Proper (() ==> () ==> ()) (@uPred_and M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
362
Global Instance or_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_or M).
Robbert Krebbers's avatar
Robbert Krebbers committed
363
Proof.
364
  intros P P' HP Q Q' HQ; split=> x n' ??.
365
  unseal; split; (intros [?|?]; [left; by apply HP|right; by apply HQ]).
Robbert Krebbers's avatar
Robbert Krebbers committed
366
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
367
Global Instance or_proper :
Robbert Krebbers's avatar
Robbert Krebbers committed
368
  Proper (() ==> () ==> ()) (@uPred_or M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
369
Global Instance impl_ne n :
Robbert Krebbers's avatar
Robbert Krebbers committed
370
  Proper (dist n ==> dist n ==> dist n) (@uPred_impl M).
Robbert Krebbers's avatar
Robbert Krebbers committed
371
Proof.
372
  intros P P' HP Q Q' HQ; split=> x n' ??.
373
  unseal; split; intros HPQ x' n'' ????; apply HQ, HPQ, HP; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
374
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
375
Global Instance impl_proper :
Robbert Krebbers's avatar
Robbert Krebbers committed
376
  Proper (() ==> () ==> ()) (@uPred_impl M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
377
Global Instance sep_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_sep M).
Robbert Krebbers's avatar
Robbert Krebbers committed
378
Proof.
379
  intros P P' HP Q Q' HQ; split=> n' x ??.
380
  unseal; split; intros (x1&x2&?&?&?); cofe_subst x;
381
    exists x1, x2; split_and!; try (apply HP || apply HQ);
382
    eauto using cmra_validN_op_l, cmra_validN_op_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
383
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
384
Global Instance sep_proper :
Robbert Krebbers's avatar
Robbert Krebbers committed
385
  Proper (() ==> () ==> ()) (@uPred_sep M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
386
Global Instance wand_ne n :
Robbert Krebbers's avatar
Robbert Krebbers committed
387
  Proper (dist n ==> dist n ==> dist n) (@uPred_wand M).
Robbert Krebbers's avatar
Robbert Krebbers committed
388
Proof.
389
  intros P P' HP Q Q' HQ; split=> n' x ??; unseal; split; intros HPQ x' n'' ???;
390
    apply HQ, HPQ, HP; eauto using cmra_validN_op_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
391
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
392
Global Instance wand_proper :
Robbert Krebbers's avatar
Robbert Krebbers committed
393
  Proper (() ==> () ==> ()) (@uPred_wand M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
394
Global Instance eq_ne (A : cofeT) n :
Robbert Krebbers's avatar
Robbert Krebbers committed
395
  Proper (dist n ==> dist n ==> dist n) (@uPred_eq M A).
Robbert Krebbers's avatar
Robbert Krebbers committed
396
Proof.
397
  intros x x' Hx y y' Hy; split=> n' z; unseal; split; intros; simpl in *.
398 399
  * by rewrite -(dist_le _ _ _ _ Hx) -?(dist_le _ _ _ _ Hy); auto.
  * by rewrite (dist_le _ _ _ _ Hx) ?(dist_le _ _ _ _ Hy); auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
400
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
401
Global Instance eq_proper (A : cofeT) :
Robbert Krebbers's avatar
Robbert Krebbers committed
402
  Proper (() ==> () ==> ()) (@uPred_eq M A) := ne_proper_2 _.
403
Global Instance forall_ne A n :
Robbert Krebbers's avatar
Robbert Krebbers committed
404
  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_forall M A).
405 406 407
Proof.
  by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
408
Global Instance forall_proper A :
Robbert Krebbers's avatar
Robbert Krebbers committed
409
  Proper (pointwise_relation _ () ==> ()) (@uPred_forall M A).
410 411 412
Proof.
  by intros Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ.
Qed.
413
Global Instance exist_ne A n :
Robbert Krebbers's avatar
Robbert Krebbers committed
414
  Proper (pointwise_relation _ (dist n) ==> dist n) (@uPred_exist M A).
415
Proof.
416 417
  intros Ψ1 Ψ2 HΨ.
  unseal; split=> n' x ??; split; intros [a ?]; exists a; by apply HΨ.
418
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
419
Global Instance exist_proper A :
Robbert Krebbers's avatar
Robbert Krebbers committed
420
  Proper (pointwise_relation _ () ==> ()) (@uPred_exist M A).
421
Proof.
422 423
  intros Ψ1 Ψ2 HΨ.
  unseal; split=> n' x ?; split; intros [a ?]; exists a; by apply HΨ.
424
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
425
Global Instance later_contractive : Contractive (@uPred_later M).
Robbert Krebbers's avatar
Robbert Krebbers committed
426
Proof.
427
  intros n P Q HPQ; unseal; split=> -[|n'] x ??; simpl; [done|].
428
  apply (HPQ n'); eauto using cmra_validN_S.
Robbert Krebbers's avatar
Robbert Krebbers committed
429
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
430
Global Instance later_proper :
Robbert Krebbers's avatar
Robbert Krebbers committed
431
  Proper (() ==> ()) (@uPred_later M) := ne_proper _.
432 433
Global Instance always_ne n : Proper (dist n ==> dist n) (@uPred_always M).
Proof.
434 435
  intros P1 P2 HP.
  unseal; split=> n' x; split; apply HP; eauto using cmra_unit_validN.
436
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
437
Global Instance always_proper :
Robbert Krebbers's avatar
Robbert Krebbers committed
438
  Proper (() ==> ()) (@uPred_always M) := ne_proper _.
439
Global Instance ownM_ne n : Proper (dist n ==> dist n) (@uPred_ownM M).
440
Proof.
441 442
  intros a b Ha.
  unseal; split=> n' x ? /=. by rewrite (dist_le _ _ _ _ Ha); last lia.
443
Qed.
444 445
Global Instance ownM_proper: Proper (() ==> ()) (@uPred_ownM M) := ne_proper _.
Global Instance valid_ne {A : cmraT} n :
446 447
Proper (dist n ==> dist n) (@uPred_valid M A).
Proof.
448 449
  intros a b Ha; unseal; split=> n' x ? /=.
  by rewrite (dist_le _ _ _ _ Ha); last lia.
450
Qed.
451 452
Global Instance valid_proper {A : cmraT} :
  Proper (() ==> ()) (@uPred_valid M A) := ne_proper _.
Robbert Krebbers's avatar
Robbert Krebbers committed
453
Global Instance iff_ne n : Proper (dist n ==> dist n ==> dist n) (@uPred_iff M).
454
Proof. unfold uPred_iff; solve_proper. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
455
Global Instance iff_proper :
456
  Proper (() ==> () ==> ()) (@uPred_iff M) := ne_proper_2 _.
Robbert Krebbers's avatar
Robbert Krebbers committed
457 458

(** Introduction and elimination rules *)
459
Lemma const_intro φ P : φ  P   φ.
460
Proof. by intros ?; unseal; split. Qed.
461
Lemma const_elim φ Q R : Q   φ  (φ  Q  R)  Q  R.
462 463 464
Proof.
  unseal; intros HQP HQR; split=> n x ??; apply HQR; first eapply HQP; eauto.
Qed.
465
Lemma False_elim P : False  P.
466
Proof. by unseal; split=> n x ?. Qed.
467
Lemma and_elim_l P Q : (P  Q)  P.
468
Proof. by unseal; split=> n x ? [??]. Qed.
469
Lemma and_elim_r P Q : (P  Q)  Q.
470
Proof. by unseal; split=> n x ? [??]. Qed.
471
Lemma and_intro P Q R : P  Q  P  R  P  (Q  R).
472
Proof. intros HQ HR; unseal; split=> n x ??; by split; [apply HQ|apply HR]. Qed.
473
Lemma or_intro_l P Q : P  (P  Q).
474
Proof. unseal; split=> n x ??; left; auto. Qed.
475
Lemma or_intro_r P Q : Q  (P  Q).
476
Proof. unseal; split=> n x ??; right; auto. Qed.
477
Lemma or_elim P Q R : P  R  Q  R  (P  Q)  R.
478
Proof. intros HP HQ; unseal; split=> n x ? [?|?]. by apply HP. by apply HQ. Qed.
479
Lemma impl_intro_r P Q R : (P  Q)  R  P  (Q  R).
Robbert Krebbers's avatar
Robbert Krebbers committed
480
Proof.
481
  unseal; intros HQ; split=> n x ?? n' x' ????.
482
  apply HQ; naive_solver eauto using uPred_weaken.
Robbert Krebbers's avatar
Robbert Krebbers committed
483
Qed.
484
Lemma impl_elim P Q R : P  (Q  R)  P  Q  P  R.
485
Proof. by unseal; intros HP HP'; split=> n x ??; apply HP with n x, HP'. Qed.
486
Lemma forall_intro {A} P (Ψ : A  uPred M): ( a, P  Ψ a)  P  ( a, Ψ a).
487
Proof. unseal; intros HPΨ; split=> n x ?? a; by apply HPΨ. Qed.
488
Lemma forall_elim {A} {Ψ : A  uPred M} a : ( a, Ψ a)  Ψ a.
489
Proof. unseal; split=> n x ? HP; apply HP. Qed.
490
Lemma exist_intro {A} {Ψ : A  uPred M} a : Ψ a  ( a, Ψ a).
491
Proof. unseal; split=> n x ??; by exists a. Qed.
492
Lemma exist_elim {A} (Φ : A  uPred M) Q : ( a, Φ a  Q)  ( a, Φ a)  Q.
493
Proof. unseal; intros HΦΨ; split=> n x ? [a ?]; by apply HΦΨ with a. Qed.
494
Lemma eq_refl {A : cofeT} (a : A) P : P  (a  a).
495
Proof. unseal; by split=> n x ??; simpl. Qed.
496 497
Lemma eq_rewrite {A : cofeT} a b (Ψ : A  uPred M) P
  `{HΨ :  n, Proper (dist n ==> dist n) Ψ} : P  (a  b)  P  Ψ a  P  Ψ b.
498
Proof.
499
  unseal; intros Hab Ha; split=> n x ??.
500
  apply HΨ with n a; auto. by symmetry; apply Hab with x. by apply Ha.
501
Qed.
502
Lemma eq_equiv `{Empty M, !CMRAIdentity M} {A : cofeT} (a b : A) :
503
  True  (a  b)  a  b.
504
Proof.
505
  unseal=> Hab; apply equiv_dist; intros n; apply Hab with ; last done.
506
  apply cmra_valid_validN, cmra_empty_valid.
507
Qed.
508
Lemma iff_equiv P Q : True  (P  Q)  P  Q.
509 510 511 512
Proof.
  rewrite /uPred_iff; unseal=> HPQ.
  split=> n x ?; split; intros; by apply HPQ with n x.
Qed.
513 514

(* Derived logical stuff *)
Robbert Krebbers's avatar
Robbert Krebbers committed
515 516
Lemma True_intro P : P  True.
Proof. by apply const_intro. Qed.
517
Lemma and_elim_l' P Q R : P  R  (P  Q)  R.
518
Proof. by rewrite and_elim_l. Qed.
519
Lemma and_elim_r' P Q R : Q  R  (P  Q)  R.
520
Proof. by rewrite and_elim_r. Qed.
521
Lemma or_intro_l' P Q R : P  Q  P  (Q  R).
522
Proof. intros ->; apply or_intro_l. Qed.
523
Lemma or_intro_r' P Q R : P  R  P  (Q  R).
524
Proof. intros ->; apply or_intro_r. Qed.
525
Lemma exist_intro' {A} P (Ψ : A  uPred M) a : P  Ψ a  P  ( a, Ψ a).
526
Proof. intros ->; apply exist_intro. Qed.
Ralf Jung's avatar
Ralf Jung committed
527
Lemma forall_elim' {A} P (Ψ : A  uPred M) : P  ( a, Ψ a)  ( a, P  Ψ a).
528
Proof. move=> HP a. by rewrite HP forall_elim. Qed.
529

530
Hint Resolve or_elim or_intro_l' or_intro_r'.
Robbert Krebbers's avatar
Robbert Krebbers committed
531 532
Hint Resolve and_intro and_elim_l' and_elim_r'.
Hint Immediate True_intro False_elim.
533

534 535
Lemma impl_intro_l P Q R : (Q  P)  R  P  (Q  R).
Proof. intros HR; apply impl_intro_r; rewrite -HR; auto. Qed.
536 537 538 539 540 541 542 543
Lemma impl_elim_l P Q : ((P  Q)  P)  Q.
Proof. apply impl_elim with P; auto. Qed.
Lemma impl_elim_r P Q : (P  (P  Q))  Q.
Proof. apply impl_elim with P; auto. Qed.
Lemma impl_elim_l' P Q R : P  (Q  R)  (P  Q)  R.
Proof. intros; apply impl_elim with Q; auto. Qed.
Lemma impl_elim_r' P Q R : Q  (P  R)  (P  Q)  R.
Proof. intros; apply impl_elim with P; auto. Qed.
544
Lemma impl_entails P Q : True  (P  Q)  P  Q.
545
Proof. intros HPQ; apply impl_elim with P; rewrite -?HPQ; auto. Qed.
546
Lemma entails_impl P Q : (P  Q)  True  (P  Q).
547
Proof. auto using impl_intro_l. Qed.
548

549 550
Lemma const_mono φ1 φ2 : (φ1  φ2)   φ1   φ2.
Proof. intros; apply const_elim with φ1; eauto using const_intro. Qed.
551
Lemma and_mono P P' Q Q' : P  Q  P'  Q'  (P  P')  (Q  Q').
552
Proof. auto. Qed.
553 554 555 556
Lemma and_mono_l P P' Q : P  Q  (P  P')  (Q  P').
Proof. by intros; apply and_mono. Qed.
Lemma and_mono_r P P' Q' : P'  Q'  (P  P')  (P  Q').
Proof. by apply and_mono. Qed.
557
Lemma or_mono P P' Q Q' : P  Q  P'  Q'  (P  P')  (Q  Q').
558
Proof. auto. Qed.
559 560 561 562
Lemma or_mono_l P P' Q : P  Q  (P  P')  (Q  P').
Proof. by intros; apply or_mono. Qed.
Lemma or_mono_r P P' Q' : P'  Q'  (P  P')  (P  Q').
Proof. by apply or_mono. Qed.
563
Lemma impl_mono P P' Q Q' : Q  P  P'  Q'  (P  P')  (Q  Q').
564
Proof.
565
  intros HP HQ'; apply impl_intro_l; rewrite -HQ'.
566
  apply impl_elim with P; eauto.
567
Qed.
568 569
Lemma forall_mono {A} (Φ Ψ : A  uPred M) :
  ( a, Φ a  Ψ a)  ( a, Φ a)  ( a, Ψ a).
570
Proof.
571
  intros HP. apply forall_intro=> a; rewrite -(HP a); apply forall_elim.
572
Qed.
573 574 575
Lemma exist_mono {A} (Φ Ψ : A  uPred M) :
  ( a, Φ a  Ψ a)  ( a, Φ a)  ( a, Ψ a).
Proof. intros HΦ. apply exist_elim=> a; rewrite (HΦ a); apply exist_intro. Qed.
576
Global Instance const_mono' : Proper (impl ==> ()) (@uPred_const M).
577
Proof. intros φ1 φ2; apply const_mono. Qed.
578
Global Instance and_mono' : Proper (() ==> () ==> ()) (@uPred_and M).
Robbert Krebbers's avatar
Robbert Krebbers committed
579
Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
580 581 582
Global Instance and_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@uPred_and M).
Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed.
583
Global Instance or_mono' : Proper (() ==> () ==> ()) (@uPred_or M).
Robbert Krebbers's avatar
Robbert Krebbers committed
584
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
585 586 587
Global Instance or_flip_mono' :
  Proper (flip () ==> flip () ==> flip ()) (@uPred_or M).
Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
588
Global Instance impl_mono' :
589
  Proper (flip () ==> () ==> ()) (@uPred_impl M).