coq_tactics.v 37.9 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1 2 3
From iris.algebra Require Export upred.
From iris.algebra Require Import upred_big_op upred_tactics.
From iris.proofmode Require Export environments.
4
From iris.prelude Require Import stringmap hlist.
Robbert Krebbers's avatar
Robbert Krebbers committed
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Import uPred.

Local Notation "Γ !! j" := (env_lookup j Γ).
Local Notation "x ← y ; z" := (match y with Some x => z | None => None end).
Local Notation "' ( x1 , x2 ) ← y ; z" :=
  (match y with Some (x1,x2) => z | None => None end).
Local Notation "' ( x1 , x2 , x3 ) ← y ; z" :=
  (match y with Some (x1,x2,x3) => z | None => None end).

Record envs (M : cmraT) :=
  Envs { env_persistent : env (uPred M); env_spatial : env (uPred M) }.
Add Printing Constructor envs.
Arguments Envs {_} _ _.
Arguments env_persistent {_} _.
Arguments env_spatial {_} _.

Record envs_wf {M} (Δ : envs M) := {
  env_persistent_valid : env_wf (env_persistent Δ);
  env_spatial_valid : env_wf (env_spatial Δ);
  envs_disjoint i : env_persistent Δ !! i = None  env_spatial Δ !! i = None
}.

Coercion of_envs {M} (Δ : envs M) : uPred M :=
  ( envs_wf Δ   Π env_persistent Δ  Π★ env_spatial Δ)%I.
29 30 31 32 33 34
Instance: Params (@of_envs) 1.

Record envs_Forall2 {M} (R : relation (uPred M)) (Δ1 Δ2 : envs M) : Prop := {
  env_persistent_Forall2 : env_Forall2 R (env_persistent Δ1) (env_persistent Δ2);
  env_spatial_Forall2 : env_Forall2 R (env_spatial Δ1) (env_spatial Δ2)
}.
Robbert Krebbers's avatar
Robbert Krebbers committed
35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

Instance envs_dom {M} : Dom (envs M) stringset := λ Δ,
  dom stringset (env_persistent Δ)  dom stringset (env_spatial Δ).

Definition envs_lookup {M} (i : string) (Δ : envs M) : option (bool * uPred M) :=
  let (Γp,Γs) := Δ in
  match env_lookup i Γp with
  | Some P => Some (true, P) | None => P  env_lookup i Γs; Some (false, P)
  end.

Definition envs_delete {M} (i : string) (p : bool) (Δ : envs M) : envs M :=
  let (Γp,Γs) := Δ in
  match p with
  | true => Envs (env_delete i Γp) Γs | false => Envs Γp (env_delete i Γs)
  end.

Definition envs_lookup_delete {M} (i : string)
    (Δ : envs M) : option (bool * uPred M * envs M) :=
  let (Γp,Γs) := Δ in
  match env_lookup_delete i Γp with
  | Some (P,Γp') => Some (true, P, Envs Γp' Γs)
  | None => '(P,Γs')  env_lookup_delete i Γs; Some (false, P, Envs Γp Γs')
  end.

Definition envs_app {M} (p : bool)
    (Γ : env (uPred M)) (Δ : envs M) : option (envs M) :=
  let (Γp,Γs) := Δ in
  match p with
  | true => _  env_app Γ Γs; Γp'  env_app Γ Γp; Some (Envs Γp' Γs)
  | false => _  env_app Γ Γp; Γs'  env_app Γ Γs; Some (Envs Γp Γs')
  end.

Definition envs_simple_replace {M} (i : string) (p : bool) (Γ : env (uPred M))
    (Δ : envs M) : option (envs M) :=
  let (Γp,Γs) := Δ in
  match p with
  | true => _  env_app Γ Γs; Γp'  env_replace i Γ Γp; Some (Envs Γp' Γs)
  | false => _  env_app Γ Γp; Γs'  env_replace i Γ Γs; Some (Envs Γp Γs')
  end.

Definition envs_replace {M} (i : string) (p q : bool) (Γ : env (uPred M))
    (Δ : envs M) : option (envs M) :=
  if eqb p q then envs_simple_replace i p Γ Δ
  else envs_app q Γ (envs_delete i p Δ).

80 81
(* if [lr = false] then [result = (hyps named js, remaining hyps)],
   if [lr = true] then [result = (remaining hyps, hyps named js)] *)
Robbert Krebbers's avatar
Robbert Krebbers committed
82 83 84 85 86 87 88 89 90 91 92 93
Definition envs_split {M}
    (lr : bool) (js : list string) (Δ : envs M) : option (envs M * envs M) :=
  let (Γp,Γs) := Δ in
  '(Γs1,Γs2)  env_split js Γs;
  match lr with
  | false  => Some (Envs Γp Γs1, Envs Γp Γs2)
  | true => Some (Envs Γp Γs2, Envs Γp Γs1)
  end.

Definition envs_persistent {M} (Δ : envs M) :=
  if env_spatial Δ is Enil then true else false.

94 95 96
Definition envs_clear_spatial {M} (Δ : envs M) : envs M :=
  Envs (env_persistent Δ) Enil.

Robbert Krebbers's avatar
Robbert Krebbers committed
97 98 99 100 101 102 103
(* Coq versions of the tactics *)
Section tactics.
Context {M : cmraT}.
Implicit Types Γ : env (uPred M).
Implicit Types Δ : envs M.
Implicit Types P Q : uPred M.

104 105 106 107
Lemma of_envs_def Δ :
  of_envs Δ = ( envs_wf Δ   Π env_persistent Δ  Π★ env_spatial Δ)%I.
Proof. done. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
108 109 110 111 112 113 114 115 116 117
Lemma envs_lookup_delete_Some Δ Δ' i p P :
  envs_lookup_delete i Δ = Some (p,P,Δ')
   envs_lookup i Δ = Some (p,P)  Δ' = envs_delete i p Δ.
Proof.
  rewrite /envs_lookup /envs_delete /envs_lookup_delete.
  destruct Δ as [Γp Γs]; rewrite /= !env_lookup_delete_correct.
  destruct (Γp !! i), (Γs !! i); naive_solver.
Qed.

Lemma envs_lookup_sound Δ i p P :
118
  envs_lookup i Δ = Some (p,P)  Δ  (?p P  envs_delete i p Δ).
Robbert Krebbers's avatar
Robbert Krebbers committed
119 120 121 122 123 124 125 126 127 128 129 130 131 132 133
Proof.
  rewrite /envs_lookup /envs_delete /of_envs=>?; apply const_elim_sep_l=> Hwf.
  destruct Δ as [Γp Γs], (Γp !! i) eqn:?; simplify_eq/=.
  - rewrite (env_lookup_perm Γp) //= always_and_sep always_sep.
    ecancel [ Π _;  P; Π★ _]%I; apply const_intro.
    destruct Hwf; constructor;
      naive_solver eauto using env_delete_wf, env_delete_fresh.
  - destruct (Γs !! i) eqn:?; simplify_eq/=.
    rewrite (env_lookup_perm Γs) //=.
    ecancel [ Π _; P; Π★ _]%I; apply const_intro.
    destruct Hwf; constructor;
      naive_solver eauto using env_delete_wf, env_delete_fresh.
Qed.
Lemma envs_lookup_sound' Δ i p P :
  envs_lookup i Δ = Some (p,P)  Δ  (P  envs_delete i p Δ).
134
Proof. intros. rewrite envs_lookup_sound //. by rewrite always_if_elim. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
135 136 137 138 139 140 141
Lemma envs_lookup_persistent_sound Δ i P :
  envs_lookup i Δ = Some (true,P)  Δ  ( P  Δ).
Proof.
  intros. apply: always_entails_l. by rewrite envs_lookup_sound // sep_elim_l.
Qed.

Lemma envs_lookup_split Δ i p P :
142
  envs_lookup i Δ = Some (p,P)  Δ  (?p P  (?p P - Δ)).
Robbert Krebbers's avatar
Robbert Krebbers committed
143 144 145 146 147 148 149 150 151 152 153 154
Proof.
  rewrite /envs_lookup /of_envs=>?; apply const_elim_sep_l=> Hwf.
  destruct Δ as [Γp Γs], (Γp !! i) eqn:?; simplify_eq/=.
  - rewrite (env_lookup_perm Γp) //= always_and_sep always_sep.
    rewrite const_equiv // left_id.
    cancel [ P]%I. apply wand_intro_l. solve_sep_entails.
  - destruct (Γs !! i) eqn:?; simplify_eq/=.
    rewrite (env_lookup_perm Γs) //=. rewrite const_equiv // left_id.
    cancel [P]. apply wand_intro_l. solve_sep_entails.
Qed.

Lemma envs_lookup_delete_sound Δ Δ' i p P :
155
  envs_lookup_delete i Δ = Some (p,P,Δ')  Δ  (?p P  Δ')%I.
Robbert Krebbers's avatar
Robbert Krebbers committed
156 157 158 159 160
Proof. intros [? ->]%envs_lookup_delete_Some. by apply envs_lookup_sound. Qed.
Lemma envs_lookup_delete_sound' Δ Δ' i p P :
  envs_lookup_delete i Δ = Some (p,P,Δ')  Δ  (P  Δ')%I.
Proof. intros [? ->]%envs_lookup_delete_Some. by apply envs_lookup_sound'. Qed.

161
Lemma envs_app_sound Δ Δ' p Γ : envs_app p Γ Δ = Some Δ'  Δ  (?p Π★ Γ - Δ').
Robbert Krebbers's avatar
Robbert Krebbers committed
162 163 164 165 166 167 168 169 170 171
Proof.
  rewrite /of_envs /envs_app=> ?; apply const_elim_sep_l=> Hwf.
  destruct Δ as [Γp Γs], p; simplify_eq/=.
  - destruct (env_app Γ Γs) eqn:Happ,
      (env_app Γ Γp) as [Γp'|] eqn:?; simplify_eq/=.
    apply wand_intro_l, sep_intro_True_l; [apply const_intro|].
    + destruct Hwf; constructor; simpl; eauto using env_app_wf.
      intros j. apply (env_app_disjoint _ _ _ j) in Happ.
      naive_solver eauto using env_app_fresh.
    + rewrite (env_app_perm _ _ Γp') //.
172 173
      rewrite big_and_app always_and_sep always_sep (big_sep_and Γ).
      solve_sep_entails.
Robbert Krebbers's avatar
Robbert Krebbers committed
174 175 176 177 178 179 180 181 182 183 184
  - destruct (env_app Γ Γp) eqn:Happ,
      (env_app Γ Γs) as [Γs'|] eqn:?; simplify_eq/=.
    apply wand_intro_l, sep_intro_True_l; [apply const_intro|].
    + destruct Hwf; constructor; simpl; eauto using env_app_wf.
      intros j. apply (env_app_disjoint _ _ _ j) in Happ.
      naive_solver eauto using env_app_fresh.
    + rewrite (env_app_perm _ _ Γs') // big_sep_app. solve_sep_entails.
Qed.

Lemma envs_simple_replace_sound' Δ Δ' i p Γ :
  envs_simple_replace i p Γ Δ = Some Δ' 
185
  envs_delete i p Δ  (?p Π★ Γ - Δ')%I.
Robbert Krebbers's avatar
Robbert Krebbers committed
186 187 188 189 190 191 192 193 194 195
Proof.
  rewrite /envs_simple_replace /envs_delete /of_envs=> ?.
  apply const_elim_sep_l=> Hwf. destruct Δ as [Γp Γs], p; simplify_eq/=.
  - destruct (env_app Γ Γs) eqn:Happ,
      (env_replace i Γ Γp) as [Γp'|] eqn:?; simplify_eq/=.
    apply wand_intro_l, sep_intro_True_l; [apply const_intro|].
    + destruct Hwf; constructor; simpl; eauto using env_replace_wf.
      intros j. apply (env_app_disjoint _ _ _ j) in Happ.
      destruct (decide (i = j)); try naive_solver eauto using env_replace_fresh.
    + rewrite (env_replace_perm _ _ Γp') //.
196 197
      rewrite big_and_app always_and_sep always_sep (big_sep_and Γ).
      solve_sep_entails.
Robbert Krebbers's avatar
Robbert Krebbers committed
198 199 200 201 202 203 204 205 206 207 208
  - destruct (env_app Γ Γp) eqn:Happ,
      (env_replace i Γ Γs) as [Γs'|] eqn:?; simplify_eq/=.
    apply wand_intro_l, sep_intro_True_l; [apply const_intro|].
    + destruct Hwf; constructor; simpl; eauto using env_replace_wf.
      intros j. apply (env_app_disjoint _ _ _ j) in Happ.
      destruct (decide (i = j)); try naive_solver eauto using env_replace_fresh.
    + rewrite (env_replace_perm _ _ Γs') // big_sep_app. solve_sep_entails.
Qed.

Lemma envs_simple_replace_sound Δ Δ' i p P Γ :
  envs_lookup i Δ = Some (p,P)  envs_simple_replace i p Γ Δ = Some Δ' 
209
  Δ  (?p P  (?p Π★ Γ - Δ'))%I.
Robbert Krebbers's avatar
Robbert Krebbers committed
210 211 212
Proof. intros. by rewrite envs_lookup_sound// envs_simple_replace_sound'//. Qed.

Lemma envs_replace_sound' Δ Δ' i p q Γ :
213
  envs_replace i p q Γ Δ = Some Δ'  envs_delete i p Δ  (?q Π★ Γ - Δ')%I.
Robbert Krebbers's avatar
Robbert Krebbers committed
214 215 216 217 218 219 220 221
Proof.
  rewrite /envs_replace; destruct (eqb _ _) eqn:Hpq.
  - apply eqb_prop in Hpq as ->. apply envs_simple_replace_sound'.
  - apply envs_app_sound.
Qed.

Lemma envs_replace_sound Δ Δ' i p q P Γ :
  envs_lookup i Δ = Some (p,P)  envs_replace i p q Γ Δ = Some Δ' 
222
  Δ  (?p P  (?q Π★ Γ - Δ'))%I.
Robbert Krebbers's avatar
Robbert Krebbers committed
223 224 225 226 227 228 229 230 231 232 233 234 235 236
Proof. intros. by rewrite envs_lookup_sound// envs_replace_sound'//. Qed.

Lemma envs_split_sound Δ lr js Δ1 Δ2 :
  envs_split lr js Δ = Some (Δ1,Δ2)  Δ  (Δ1  Δ2).
Proof.
  rewrite /envs_split /of_envs=> ?; apply const_elim_sep_l=> Hwf.
  destruct Δ as [Γp Γs], (env_split js _) as [[Γs1 Γs2]|] eqn:?; simplify_eq/=.
  rewrite (env_split_perm Γs) // big_sep_app {1}always_sep_dup'.
  destruct lr; simplify_eq/=; cancel [ Π Γp;  Π Γp; Π★ Γs1; Π★ Γs2]%I;
    destruct Hwf; apply sep_intro_True_l; apply const_intro; constructor;
      naive_solver eauto using env_split_wf_1, env_split_wf_2,
      env_split_fresh_1, env_split_fresh_2.
Qed.

237 238 239 240 241 242 243 244 245 246 247 248 249 250 251
Lemma envs_clear_spatial_sound Δ :
  Δ  (envs_clear_spatial Δ  Π★ env_spatial Δ)%I.
Proof.
  rewrite /of_envs /envs_clear_spatial /=; apply const_elim_sep_l=> Hwf.
  rewrite right_id -assoc; apply sep_intro_True_l; [apply const_intro|done].
  destruct Hwf; constructor; simpl; auto using Enil_wf.
Qed.

Lemma env_fold_wand Γ Q : env_fold uPred_wand Q Γ  (Π★ Γ - Q).
Proof.
  revert Q; induction Γ as [|Γ IH i P]=> Q /=; [by rewrite wand_True|].
  by rewrite IH wand_curry (comm uPred_sep).
Qed.

Lemma envs_persistent_persistent Δ : envs_persistent Δ = true  PersistentP Δ.
Robbert Krebbers's avatar
Robbert Krebbers committed
252
Proof. intros; destruct Δ as [? []]; simplify_eq/=; apply _. Qed.
253
Hint Immediate envs_persistent_persistent : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
254

255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287
Global Instance envs_Forall2_refl (R : relation (uPred M)) :
  Reflexive R  Reflexive (envs_Forall2 R).
Proof. by constructor. Qed.
Global Instance envs_Forall2_sym (R : relation (uPred M)) :
  Symmetric R  Symmetric (envs_Forall2 R).
Proof. intros ??? [??]; by constructor. Qed.
Global Instance envs_Forall2_trans (R : relation (uPred M)) :
  Transitive R  Transitive (envs_Forall2 R).
Proof. intros ??? [??] [??] [??]; constructor; etrans; eauto. Qed.
Global Instance envs_Forall2_antisymm (R R' : relation (uPred M)) :
  AntiSymm R R'  AntiSymm (envs_Forall2 R) (envs_Forall2 R').
Proof. intros ??? [??] [??]; constructor; by eapply (anti_symm _). Qed.
Lemma envs_Forall2_impl (R R' : relation (uPred M)) Δ1 Δ2 :
  envs_Forall2 R Δ1 Δ2  ( P Q, R P Q  R' P Q)  envs_Forall2 R' Δ1 Δ2.
Proof. intros [??] ?; constructor; eauto using env_Forall2_impl. Qed.

Global Instance of_envs_mono : Proper (envs_Forall2 () ==> ()) (@of_envs M).
Proof.
  intros [Γp1 Γs1] [Γp2 Γs2] [Hp Hs]; unfold of_envs; simpl in *.
  apply const_elim_sep_l=>Hwf. apply sep_intro_True_l.
  - destruct Hwf; apply const_intro; constructor;
      naive_solver eauto using env_Forall2_wf, env_Forall2_fresh.
  - by repeat f_equiv.
Qed.
Global Instance of_envs_proper : Proper (envs_Forall2 () ==> ()) (@of_envs M).
Proof.
  intros Δ1 Δ2 ?; apply (anti_symm ()); apply of_envs_mono;
    eapply envs_Forall2_impl; [| |symmetry|]; eauto using equiv_entails.
Qed.
Global Instance Envs_mono (R : relation (uPred M)) :
  Proper (env_Forall2 R ==> env_Forall2 R ==> envs_Forall2 R) (@Envs M).
Proof. by constructor. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
288 289 290 291 292 293 294 295
(** * Adequacy *)
Lemma tac_adequate P : Envs Enil Enil  P  True  P.
Proof.
  intros <-. rewrite /of_envs /= always_const !right_id.
  apply const_intro; repeat constructor.
Qed.

(** * Basic rules *)
296
Class ToAssumption (p : bool) (P Q : uPred M) := to_assumption : ?p P  Q.
297
Arguments to_assumption _ _ _ {_}.
298
Global Instance to_assumption_exact p P : ToAssumption p P P.
299
Proof. destruct p; by rewrite /ToAssumption /= ?always_elim. Qed.
300 301 302 303
Global Instance to_assumption_always_l p P Q :
  ToAssumption p P Q  ToAssumption p ( P) Q.
Proof. rewrite /ToAssumption=><-. by rewrite always_elim. Qed.
Global Instance to_assumption_always_r P Q :
304 305 306 307 308 309
  ToAssumption true P Q  ToAssumption true P ( Q).
Proof. rewrite /ToAssumption=><-. by rewrite always_always. Qed.

Lemma tac_assumption Δ i p P Q :
  envs_lookup i Δ = Some (p,P)  ToAssumption p P Q  Δ  Q.
Proof. intros. by rewrite envs_lookup_sound // sep_elim_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
310 311 312 313 314 315 316 317 318 319 320 321

Lemma tac_rename Δ Δ' i j p P Q :
  envs_lookup i Δ = Some (p,P) 
  envs_simple_replace i p (Esnoc Enil j P) Δ = Some Δ' 
  Δ'  Q  Δ  Q.
Proof.
  intros. rewrite envs_simple_replace_sound //.
  destruct p; simpl; by rewrite right_id wand_elim_r.
Qed.
Lemma tac_clear Δ Δ' i p P Q :
  envs_lookup_delete i Δ = Some (p,P,Δ')  Δ'  Q  Δ  Q.
Proof. intros. by rewrite envs_lookup_delete_sound // sep_elim_r. Qed.
322 323 324
Lemma tac_clear_spatial Δ Δ' Q :
  envs_clear_spatial Δ = Δ'  Δ'  Q  Δ  Q.
Proof. intros <- ?. by rewrite envs_clear_spatial_sound // sep_elim_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
325 326 327 328 329 330

(** * False *)
Lemma tac_ex_falso Δ Q : Δ  False  Δ  Q.
Proof. by rewrite -(False_elim Q). Qed.

(** * Pure *)
331
Class ToPure (P : uPred M) (φ : Prop) := to_pure : P   φ.
Robbert Krebbers's avatar
Robbert Krebbers committed
332 333 334 335 336 337 338 339 340
Arguments to_pure : clear implicits.
Global Instance to_pure_const φ : ToPure ( φ) φ.
Proof. done. Qed.
Global Instance to_pure_eq {A : cofeT} (a b : A) :
  Timeless a  ToPure (a  b) (a  b).
Proof. intros; red. by rewrite timeless_eq. Qed.
Global Instance to_pure_valid `{CMRADiscrete A} (a : A) : ToPure ( a) ( a).
Proof. intros; red. by rewrite discrete_valid. Qed.

341 342 343
Lemma tac_pure_intro Δ Q (φ : Prop) : ToPure Q φ  φ  Δ  Q.
Proof. intros ->. apply const_intro. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387
Lemma tac_pure Δ Δ' i p P φ Q :
  envs_lookup_delete i Δ = Some (p, P, Δ')  ToPure P φ 
  (φ  Δ'  Q)  Δ  Q.
Proof.
  intros ?? HQ. rewrite envs_lookup_delete_sound' //; simpl.
  rewrite (to_pure P); by apply const_elim_sep_l.
Qed.

Lemma tac_pure_revert Δ φ Q : Δ  ( φ  Q)  (φ  Δ  Q).
Proof. intros HΔ ?. by rewrite HΔ const_equiv // left_id. Qed.

(** * Later *)
Class StripLaterEnv (Γ1 Γ2 : env (uPred M)) :=
  strip_later_env : env_Forall2 StripLaterR Γ1 Γ2.
Global Instance strip_later_env_nil : StripLaterEnv Enil Enil.
Proof. constructor. Qed.
Global Instance strip_later_env_snoc Γ1 Γ2 i P Q :
  StripLaterEnv Γ1 Γ2  StripLaterR P Q 
  StripLaterEnv (Esnoc Γ1 i P) (Esnoc Γ2 i Q).
Proof. by constructor. Qed.

Class StripLaterEnvs (Δ1 Δ2 : envs M) := {
  strip_later_persistent: StripLaterEnv (env_persistent Δ1) (env_persistent Δ2);
  strip_later_spatial: StripLaterEnv (env_spatial Δ1) (env_spatial Δ2)
}.
Global Instance strip_later_envs Γp1 Γp2 Γs1 Γs2 :
  StripLaterEnv Γp1 Γp2  StripLaterEnv Γs1 Γs2 
  StripLaterEnvs (Envs Γp1 Γs1) (Envs Γp2 Γs2).
Proof. by split. Qed.
Lemma strip_later_env_sound Δ1 Δ2 : StripLaterEnvs Δ1 Δ2  Δ1   Δ2.
Proof.
  intros [Hp Hs]; rewrite /of_envs /= !later_sep -always_later.
  repeat apply sep_mono; try apply always_mono.
  - rewrite -later_intro; apply const_mono; destruct 1; constructor;
      naive_solver eauto using env_Forall2_wf, env_Forall2_fresh.
  - induction Hp; rewrite /= ?later_and; auto using and_mono, later_intro.
  - induction Hs; rewrite /= ?later_sep; auto using sep_mono, later_intro.
Qed.

Lemma tac_next Δ Δ' Q Q' :
  StripLaterEnvs Δ Δ'  StripLaterL Q Q'  Δ'  Q'  Δ  Q.
Proof. intros ?? HQ. by rewrite -(strip_later_l Q) strip_later_env_sound HQ. Qed.

Lemma tac_löb Δ Δ' i Q :
388 389
  envs_persistent Δ = true 
  envs_app true (Esnoc Enil i ( Q)%I) Δ = Some Δ' 
Robbert Krebbers's avatar
Robbert Krebbers committed
390 391
  Δ'  Q  Δ  Q.
Proof.
392 393 394 395
  intros ?? HQ. rewrite -(always_elim Q) -(löb ( Q)) -always_later.
  apply impl_intro_l, (always_intro _ _).
  rewrite envs_app_sound //; simpl.
  by rewrite right_id always_and_sep_l' wand_elim_r HQ.
Robbert Krebbers's avatar
Robbert Krebbers committed
396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413
Qed.

(** * Always *)
Lemma tac_always_intro Δ Q : envs_persistent Δ = true  Δ  Q  Δ   Q.
Proof. intros. by apply: always_intro. Qed.

Class ToPersistentP (P Q : uPred M) := to_persistentP : P   Q.
Arguments to_persistentP : clear implicits.
Global Instance to_persistentP_always_trans P Q :
  ToPersistentP P Q  ToPersistentP ( P) Q | 0.
Proof. rewrite /ToPersistentP=> ->. by rewrite always_always. Qed.
Global Instance to_persistentP_always P : ToPersistentP ( P) P | 1.
Proof. done. Qed.
Global Instance to_persistentP_persistent P :
  PersistentP P  ToPersistentP P P | 100.
Proof. done. Qed.

Lemma tac_persistent Δ Δ' i p P P' Q :
414
  envs_lookup i Δ = Some (p, P)  ToPersistentP P P' 
Robbert Krebbers's avatar
Robbert Krebbers committed
415 416 417
  envs_replace i p true (Esnoc Enil i P') Δ = Some Δ' 
  Δ'  Q  Δ  Q.
Proof.
418 419
  intros ??? <-. rewrite envs_replace_sound //; simpl.
  by rewrite right_id (to_persistentP P) always_if_always wand_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446
Qed.

(** * Implication and wand *)
Lemma tac_impl_intro Δ Δ' i P Q :
  envs_persistent Δ = true 
  envs_app false (Esnoc Enil i P) Δ = Some Δ' 
  Δ'  Q  Δ  (P  Q).
Proof.
  intros ?? HQ. rewrite (persistentP Δ) envs_app_sound //; simpl.
  by rewrite right_id always_wand_impl always_elim HQ.
Qed.
Lemma tac_impl_intro_persistent Δ Δ' i P P' Q :
  ToPersistentP P P' 
  envs_app true (Esnoc Enil i P') Δ = Some Δ' 
  Δ'  Q  Δ  (P  Q).
Proof.
  intros ?? HQ. rewrite envs_app_sound //; simpl. apply impl_intro_l.
  by rewrite right_id {1}(to_persistentP P) always_and_sep_l wand_elim_r.
Qed.
Lemma tac_impl_intro_pure Δ P φ Q : ToPure P φ  (φ  Δ  Q)  Δ  (P  Q).
Proof.
  intros. by apply impl_intro_l; rewrite (to_pure P); apply const_elim_l.
Qed.

Lemma tac_wand_intro Δ Δ' i P Q :
  envs_app false (Esnoc Enil i P) Δ = Some Δ'  Δ'  Q  Δ  (P - Q).
Proof.
447
  intros ? HQ. rewrite envs_app_sound //; simpl. by rewrite right_id HQ.
Robbert Krebbers's avatar
Robbert Krebbers committed
448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471
Qed.
Lemma tac_wand_intro_persistent Δ Δ' i P P' Q :
  ToPersistentP P P' 
  envs_app true (Esnoc Enil i P') Δ = Some Δ' 
  Δ'  Q  Δ  (P - Q).
Proof.
  intros. rewrite envs_app_sound //; simpl.
  rewrite right_id. by apply wand_mono.
Qed.
Lemma tac_wand_intro_pure Δ P φ Q : ToPure P φ  (φ  Δ  Q)  Δ  (P - Q).
Proof.
  intros. by apply wand_intro_l; rewrite (to_pure P); apply const_elim_sep_l.
Qed.

Class ToWand (R P Q : uPred M) := to_wand : R  (P - Q).
Arguments to_wand : clear implicits.
Global Instance to_wand_wand P Q : ToWand (P - Q) P Q.
Proof. done. Qed.
Global Instance to_wand_impl P Q : ToWand (P  Q) P Q.
Proof. apply impl_wand. Qed.
Global Instance to_wand_iff_l P Q : ToWand (P  Q) P Q.
Proof. by apply and_elim_l', impl_wand. Qed.
Global Instance to_wand_iff_r P Q : ToWand (P  Q) Q P.
Proof. apply and_elim_r', impl_wand. Qed.
472 473
Global Instance to_wand_always R P Q : ToWand R P Q  ToWand ( R) P Q.
+Proof. rewrite /ToWand=> ->. apply always_elim. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
474 475 476 477 478

(* This is pretty much [tac_specialize_assert] with [js:=[j]] and [tac_exact],
but it is doing some work to keep the order of hypotheses preserved. *)
Lemma tac_specialize Δ Δ' Δ'' i p j q P1 P2 R Q :
  envs_lookup_delete i Δ = Some (p, P1, Δ') 
479
  envs_lookup j (if p then Δ else Δ') = Some (q, R) 
Robbert Krebbers's avatar
Robbert Krebbers committed
480 481 482 483 484 485 486 487 488 489
  ToWand R P1 P2 
  match p with
  | true  => envs_simple_replace j q (Esnoc Enil j P2) Δ
  | false => envs_replace j q false (Esnoc Enil j P2) Δ'
             (* remove [i] and make [j] spatial *)
  end = Some Δ'' 
  Δ''  Q  Δ  Q.
Proof.
  intros [? ->]%envs_lookup_delete_Some ??? <-. destruct p.
  - rewrite envs_lookup_persistent_sound // envs_simple_replace_sound //; simpl.
490 491
    rewrite assoc (to_wand R) (always_elim_if q) -always_if_sep wand_elim_r.
    by rewrite right_id wand_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
492 493
  - rewrite envs_lookup_sound //; simpl.
    rewrite envs_lookup_sound // (envs_replace_sound' _ Δ'') //; simpl.
494
    by rewrite right_id assoc (to_wand R) always_if_elim wand_elim_r wand_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
495 496
Qed.

497 498 499 500 501 502 503
Class ToAssert (P : uPred M) (Q : uPred M) (R : uPred M) :=
  to_assert : (R  (P - Q))  Q.
Global Arguments to_assert _ _ _ {_}.
Lemma to_assert_fallthrough P Q : ToAssert P Q P.
Proof. by rewrite /ToAssert wand_elim_r. Qed.

Lemma tac_specialize_assert Δ Δ' Δ1 Δ2' j q lr js R P1 P2 P1' Q :
504
  envs_lookup_delete j Δ = Some (q, R, Δ') 
505
  ToWand R P1 P2  ToAssert P1 Q P1' 
Robbert Krebbers's avatar
Robbert Krebbers committed
506
  ('(Δ1,Δ2)  envs_split lr js Δ';
507
    Δ2'  envs_app false (Esnoc Enil j P2) Δ2;
Robbert Krebbers's avatar
Robbert Krebbers committed
508
    Some (Δ1,Δ2')) = Some (Δ1,Δ2')  (* does not preserve position of [j] *)
509
  Δ1  P1'  Δ2'  Q  Δ  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
510
Proof.
511
  intros [? ->]%envs_lookup_delete_Some ??? HP1 HQ.
Robbert Krebbers's avatar
Robbert Krebbers committed
512 513 514 515
  destruct (envs_split _ _ _) as [[? Δ2]|] eqn:?; simplify_eq/=;
    destruct (envs_app _ _ _) eqn:?; simplify_eq/=.
  rewrite envs_lookup_sound // envs_split_sound //.
  rewrite (envs_app_sound Δ2) //; simpl.
516 517 518
  rewrite right_id (to_wand R) HP1 assoc -(comm _ P1') -assoc.
  rewrite -(to_assert P1 Q); apply sep_mono_r, wand_intro_l.
  by rewrite always_if_elim assoc !wand_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
519 520
Qed.

521 522 523 524 525
Lemma tac_specialize_pure Δ Δ' j q R P1 P2 φ Q :
  envs_lookup j Δ = Some (q, R) 
  ToWand R P1 P2  ToPure P1 φ 
  envs_simple_replace j q (Esnoc Enil j P2) Δ = Some Δ' 
  φ  Δ'  Q  Δ  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
526
Proof.
527 528
  intros. rewrite envs_simple_replace_sound //; simpl.
  by rewrite right_id (to_wand R) (to_pure P1) const_equiv // wand_True wand_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
529 530
Qed.

531 532 533 534 535 536
Lemma tac_specialize_persistent Δ Δ' Δ'' j q P1 P2 R Q :
  envs_lookup_delete j Δ = Some (q, R, Δ') 
  ToWand R P1 P2 
  envs_simple_replace j q (Esnoc Enil j P2) Δ = Some Δ'' 
  Δ'  P1  (PersistentP P1  PersistentP P2) 
  Δ''  Q  Δ  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
537
Proof.
538 539 540 541 542 543 544 545 546 547 548
  intros [? ->]%envs_lookup_delete_Some ?? HP1 [?|?] <-.
  - rewrite envs_lookup_sound //.
    rewrite -(idemp uPred_and (envs_delete _ _ _)).
    rewrite {1}HP1 (persistentP P1) always_and_sep_l assoc.
    rewrite envs_simple_replace_sound' //; simpl.
    rewrite right_id (to_wand R) (always_elim_if q) -always_if_sep wand_elim_l.
    by rewrite wand_elim_r.
  - rewrite -(idemp uPred_and Δ) {1}envs_lookup_sound //; simpl; rewrite HP1.
    rewrite envs_simple_replace_sound //; simpl.
    rewrite (sep_elim_r _ (_ - _)) right_id (to_wand R) always_if_elim.
    by rewrite wand_elim_l always_and_sep_l -{1}(always_if_always q P2) wand_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
549 550 551 552 553 554 555 556 557 558 559
Qed.

Lemma tac_revert Δ Δ' i p P Q :
  envs_lookup_delete i Δ = Some (p,P,Δ') 
  Δ'  (if p then  P  Q else P - Q)  Δ  Q.
Proof.
  intros ? HQ. rewrite envs_lookup_delete_sound //; simpl. destruct p.
  - by rewrite HQ -always_and_sep_l impl_elim_r.
  - by rewrite HQ wand_elim_r.
Qed.

560 561 562 563 564 565
Lemma tac_revert_spatial Δ Q :
  envs_clear_spatial Δ  (env_fold uPred_wand Q (env_spatial Δ))  Δ  Q.
Proof.
  intros HΔ. by rewrite envs_clear_spatial_sound HΔ env_fold_wand wand_elim_l.
Qed.

566 567
Lemma tac_assert Δ Δ1 Δ2 Δ2' lr js j P Q R :
  ToAssert P Q R 
Robbert Krebbers's avatar
Robbert Krebbers committed
568
  envs_split lr js Δ = Some (Δ1,Δ2) 
569
  envs_app false (Esnoc Enil j P) Δ2 = Some Δ2' 
570
  Δ1  R  Δ2'  Q  Δ  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
571
Proof.
572
  intros ??? HP HQ. rewrite envs_split_sound //.
573
  rewrite (envs_app_sound Δ2) //; simpl.
574
  by rewrite right_id HP HQ.
Robbert Krebbers's avatar
Robbert Krebbers committed
575 576 577 578
Qed.

Lemma tac_assert_persistent Δ Δ' j P Q :
  envs_app true (Esnoc Enil j P) Δ = Some Δ' 
579
  Δ  P  PersistentP P  Δ'  Q  Δ  Q.
Robbert Krebbers's avatar
Robbert Krebbers committed
580
Proof.
581
  intros ? HP ??.
Robbert Krebbers's avatar
Robbert Krebbers committed
582 583 584 585
  rewrite -(idemp uPred_and Δ) {1}HP envs_app_sound //; simpl.
  by rewrite right_id {1}(persistentP P) always_and_sep_l wand_elim_r.
Qed.

586
(** Whenever posing [lem : True ⊢ Q] as [H] we want it to appear as [H : Q] and
587 588 589 590 591 592 593 594 595 596 597 598
not as [H : True -★ Q]. The class [ToPosedProof] is used to strip off the
[True]. Note that [to_posed_proof_True] is declared using a [Hint Extern] to
make sure it is not used while posing [lem : ?P ⊢ Q] with [?P] an evar. *)
Class ToPosedProof (P1 P2 R : uPred M) := to_pose_proof : P1  P2  True  R.
Arguments to_pose_proof : clear implicits.
Instance to_posed_proof_True P : ToPosedProof True P P.
Proof. by rewrite /ToPosedProof. Qed.
Global Instance to_posed_proof_wand P Q : ToPosedProof P Q (P - Q).
Proof. rewrite /ToPosedProof. apply entails_wand. Qed.

Lemma tac_pose_proof Δ Δ' j P1 P2 R Q :
  P1  P2  ToPosedProof P1 P2 R  envs_app true (Esnoc Enil j R) Δ = Some Δ' 
Robbert Krebbers's avatar
Robbert Krebbers committed
599 600
  Δ'  Q  Δ  Q.
Proof.
601 602
  intros HP ?? <-. rewrite envs_app_sound //; simpl.
  by rewrite right_id -(to_pose_proof P1 P2 R) // always_const wand_True.
Robbert Krebbers's avatar
Robbert Krebbers committed
603 604
Qed.

605 606 607 608 609 610 611 612 613 614 615 616
Lemma tac_pose_proof_hyp Δ Δ' Δ'' i p j P Q :
  envs_lookup_delete i Δ = Some (p, P, Δ') 
  envs_app p (Esnoc Enil j P) (if p then Δ else Δ') = Some Δ'' 
  Δ''  Q  Δ  Q.
Proof.
  intros [? ->]%envs_lookup_delete_Some ? <-. destruct p.
  - rewrite envs_lookup_persistent_sound // envs_app_sound //; simpl.
    by rewrite right_id wand_elim_r.
  - rewrite envs_lookup_sound // envs_app_sound //; simpl.
    by rewrite right_id wand_elim_r.
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
617
Lemma tac_apply Δ Δ' i p R P1 P2 :
618
  envs_lookup_delete i Δ = Some (p, R, Δ')  ToWand R P1 P2 
Robbert Krebbers's avatar
Robbert Krebbers committed
619 620 621 622 623 624 625 626 627
  Δ'  P1  Δ  P2.
Proof.
  intros ?? HP1. rewrite envs_lookup_delete_sound' //.
  by rewrite (to_wand R) HP1 wand_elim_l.
Qed.

(** * Rewriting *)
Lemma tac_rewrite Δ i p Pxy (lr : bool) Q :
  envs_lookup i Δ = Some (p, Pxy) 
Ralf Jung's avatar
Ralf Jung committed
628
   {A : cofeT} (x y : A) (Φ : A  uPred M),
629
    Pxy  (x  y) 
Robbert Krebbers's avatar
Robbert Krebbers committed
630 631 632 633 634 635 636 637 638 639 640
    Q  Φ (if lr then y else x) 
    ( n, Proper (dist n ==> dist n) Φ) 
    Δ  Φ (if lr then x else y)  Δ  Q.
Proof.
  intros ? A x y ? HPxy -> ?; apply eq_rewrite; auto.
  rewrite {1}envs_lookup_sound' //; rewrite sep_elim_l HPxy.
  destruct lr; auto using eq_sym.
Qed.

Lemma tac_rewrite_in Δ i p Pxy j q P (lr : bool) Q :
  envs_lookup i Δ = Some (p, Pxy) 
641
  envs_lookup j Δ = Some (q, P) 
Robbert Krebbers's avatar
Robbert Krebbers committed
642
   {A : cofeT} Δ' x y (Φ : A  uPred M),
643
    Pxy  (x  y) 
Robbert Krebbers's avatar
Robbert Krebbers committed
644 645 646 647 648 649 650 651
    P  Φ (if lr then y else x) 
    ( n, Proper (dist n ==> dist n) Φ) 
    envs_simple_replace j q (Esnoc Enil j (Φ (if lr then x else y))) Δ = Some Δ' 
    Δ'  Q  Δ  Q.
Proof.
  intros ?? A Δ' x y Φ HPxy HP ?? <-.
  rewrite -(idemp uPred_and Δ) {2}(envs_lookup_sound' _ i) //.
  rewrite sep_elim_l HPxy always_and_sep_r.
652 653 654 655 656
  rewrite (envs_simple_replace_sound _ _ j) //; simpl.
  rewrite HP right_id -assoc; apply wand_elim_r'. destruct lr.
  - apply (eq_rewrite x y (λ y, ?q Φ y - Δ')%I); eauto with I. solve_proper.
  - apply (eq_rewrite y x (λ y, ?q Φ y - Δ')%I); eauto using eq_sym with I.
    solve_proper.
Robbert Krebbers's avatar
Robbert Krebbers committed
657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717
Qed.

(** * Conjunction splitting *)
Class AndSplit (P Q1 Q2 : uPred M) := and_split : (Q1  Q2)  P.
Arguments and_split : clear implicits.

Global Instance and_split_and P1 P2 : AndSplit (P1  P2) P1 P2.
Proof. done. Qed.
Global Instance and_split_sep_persistent_l P1 P2 :
  PersistentP P1  AndSplit (P1  P2) P1 P2.
Proof. intros. by rewrite /AndSplit always_and_sep_l. Qed.
Global Instance and_split_sep_persistent_r P1 P2 :
  PersistentP P2  AndSplit (P1  P2) P1 P2.
Proof. intros. by rewrite /AndSplit always_and_sep_r. Qed.

Lemma tac_and_split Δ P Q1 Q2 : AndSplit P Q1 Q2  Δ  Q1  Δ  Q2  Δ  P.
Proof. intros. rewrite -(and_split P). by apply and_intro. Qed.

(** * Separating conjunction splitting *)
Class SepSplit (P Q1 Q2 : uPred M) := sep_split : (Q1  Q2)  P.
Arguments sep_split : clear implicits.

Global Instance sep_split_sep P1 P2 : SepSplit (P1  P2) P1 P2 | 100.
Proof. done. Qed.
Global Instance sep_split_ownM (a b : M) :
  SepSplit (uPred_ownM (a  b)) (uPred_ownM a) (uPred_ownM b) | 99.
Proof. by rewrite /SepSplit ownM_op. Qed.

Lemma tac_sep_split Δ Δ1 Δ2 lr js P Q1 Q2 :
  SepSplit P Q1 Q2 
  envs_split lr js Δ = Some (Δ1,Δ2) 
  Δ1  Q1  Δ2  Q2  Δ  P.
Proof.
  intros. rewrite envs_split_sound // -(sep_split P). by apply sep_mono.
Qed.

(** * Combining *)
Lemma tac_combine Δ1 Δ2 Δ3 Δ4 i1 p P1 i2 q P2 j P Q :
  envs_lookup_delete i1 Δ1 = Some (p,P1,Δ2) 
  envs_lookup_delete i2 (if p then Δ1 else Δ2) = Some (q,P2,Δ3) 
  SepSplit P P1 P2 
  envs_app (p && q) (Esnoc Enil j P)
    (if q then (if p then Δ1 else Δ2) else Δ3) = Some Δ4 
  Δ4  Q  Δ1  Q.
Proof.
  intros [? ->]%envs_lookup_delete_Some [? ->]%envs_lookup_delete_Some ?? <-.
  destruct p.
  - rewrite envs_lookup_persistent_sound //. destruct q.
    + rewrite envs_lookup_persistent_sound // envs_app_sound //; simpl.
      by rewrite right_id assoc -always_sep (sep_split P) wand_elim_r.
    + rewrite envs_lookup_sound // envs_app_sound //; simpl.
      by rewrite right_id assoc always_elim (sep_split P) wand_elim_r.
  - rewrite envs_lookup_sound //; simpl. destruct q.
    + rewrite envs_lookup_persistent_sound // envs_app_sound //; simpl.
      by rewrite right_id assoc always_elim (sep_split P) wand_elim_r.
    + rewrite envs_lookup_sound // envs_app_sound //; simpl.
      by rewrite right_id assoc (sep_split P) wand_elim_r.
Qed.

(** * Conjunction/separating conjunction elimination *)
Class SepDestruct (p : bool) (P Q1 Q2 : uPred M) :=
718
  sep_destruct : ?p P  ?p (Q1  Q2).
Robbert Krebbers's avatar
Robbert Krebbers committed
719 720
Arguments sep_destruct : clear implicits.
Lemma sep_destruct_spatial p P Q1 Q2 : P  (Q1  Q2)  SepDestruct p P Q1 Q2.
721
Proof. rewrite /SepDestruct=> ->. by rewrite always_if_sep. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
722 723 724 725 726 727 728 729

Global Instance sep_destruct_sep p P Q : SepDestruct p (P  Q) P Q.
Proof. by apply sep_destruct_spatial. Qed.
Global Instance sep_destruct_ownM p (a b : M) :
  SepDestruct p (uPred_ownM (a  b)) (uPred_ownM a) (uPred_ownM b).
Proof. apply sep_destruct_spatial. by rewrite ownM_op. Qed.

Global Instance sep_destruct_and P Q : SepDestruct true (P  Q) P Q.
730
Proof. by rewrite /SepDestruct /= always_and_sep. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
731 732
Global Instance sep_destruct_and_persistent_l P Q :
  PersistentP P  SepDestruct false (P  Q) P Q.
733
Proof. intros; by rewrite /SepDestruct /= always_and_sep_l. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
734 735
Global Instance sep_destruct_and_persistent_r P Q :
  PersistentP Q  SepDestruct false (P  Q) P Q.
736
Proof. intros; by rewrite /SepDestruct /= always_and_sep_r. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
737 738 739

Global Instance sep_destruct_later p P Q1 Q2 :
  SepDestruct p P Q1 Q2  SepDestruct p ( P) ( Q1) ( Q2).
740
Proof. by rewrite /SepDestruct -later_sep !always_if_later=> ->. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
741 742

Lemma tac_sep_destruct Δ Δ' i p j1 j2 P P1 P2 Q :
743
  envs_lookup i Δ = Some (p, P)  SepDestruct p P P1 P2 
Robbert Krebbers's avatar
Robbert Krebbers committed
744 745 746
  envs_simple_replace i p (Esnoc (Esnoc Enil j1 P1) j2 P2) Δ = Some Δ' 
  Δ'  Q  Δ  Q.
Proof.
747 748
  intros. rewrite envs_simple_replace_sound //; simpl.
  by rewrite (sep_destruct p P) right_id (comm uPred_sep P1) wand_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
749 750 751
Qed.

(** * Framing *)
Robbert Krebbers's avatar
Robbert Krebbers committed
752 753 754 755
(** The [option] is to account for formulas that can be framed entirely, so
we do not end up with [True]s everywhere. *)
Class Frame (R P : uPred M) (mQ : option (uPred M)) :=
  frame : (R  from_option True mQ)  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
756 757
Arguments frame : clear implicits.

758
Global Instance frame_here R : Frame R R None.
Robbert Krebbers's avatar
Robbert Krebbers committed
759
Proof. by rewrite /Frame right_id. Qed.
760
Global Instance frame_sep_l R P1 P2 mQ :
Robbert Krebbers's avatar
Robbert Krebbers committed
761
  Frame R P1 mQ 
762
  Frame R (P1  P2) (Some $ if mQ is Some Q then Q  P2 else P2)%I | 9.
Robbert Krebbers's avatar
Robbert Krebbers committed
763
Proof. rewrite /Frame => <-. destruct mQ; simpl; solve_sep_entails. Qed.
764
Global Instance frame_sep_r R P1 P2 mQ :
Robbert Krebbers's avatar
Robbert Krebbers committed
765
  Frame R P2 mQ 
766
  Frame R (P1  P2) (Some $ if mQ is Some Q then P1  Q else P1)%I | 10.
Robbert Krebbers's avatar
Robbert Krebbers committed
767
Proof. rewrite /Frame => <-. destruct mQ; simpl; solve_sep_entails. Qed.
768
Global Instance frame_and_l R P1 P2 mQ :
Robbert Krebbers's avatar
Robbert Krebbers committed
769
  Frame R P1 mQ 
770
  Frame R (P1  P2) (Some $ if mQ is Some Q then Q  P2 else P2)%I | 9.
Robbert Krebbers's avatar
Robbert Krebbers committed
771
Proof. rewrite /Frame => <-. destruct mQ; simpl; eauto 10 with I. Qed.
772
Global Instance frame_and_r R P1 P2 mQ :
Robbert Krebbers's avatar
Robbert Krebbers committed
773
  Frame R P2 mQ 
774
  Frame R (P1  P2) (Some $ if mQ is Some Q then P1  Q else P1)%I | 10.
Robbert Krebbers's avatar
Robbert Krebbers committed
775
Proof. rewrite /Frame => <-. destruct mQ; simpl; eauto 10 with I. Qed.
776
Global Instance frame_or R P1 P2 mQ1 mQ2 :
Robbert Krebbers's avatar
Robbert Krebbers committed
777 778 779 780 781 782 783 784 785
  Frame R P1 mQ1  Frame R P2 mQ2 
  Frame R (P1  P2) (match mQ1, mQ2 with
                     | Some Q1, Some Q2 => Some (Q1  Q2)%I | _, _ => None
                     end).
Proof.
  rewrite /Frame=> <- <-.
  destruct mQ1 as [Q1|], mQ2 as [Q2|]; simpl; auto with I.
  by rewrite -sep_or_l.
Qed.
786
Global Instance frame_later R P mQ :
Robbert Krebbers's avatar
Robbert Krebbers committed
787 788 789 790 791
  Frame R P mQ  Frame R ( P) (if mQ is Some Q then Some ( Q) else None)%I.
Proof.
  rewrite /Frame=><-.
  by destruct mQ; rewrite /= later_sep -(later_intro R) ?later_True.
Qed.
792
Global Instance frame_exist {A} R (Φ : A  uPred M) mΨ :
Robbert Krebbers's avatar
Robbert Krebbers committed
793 794
  ( a, Frame R (Φ a) (mΨ a)) 
  Frame R ( x, Φ x) (Some ( x, if mΨ x is Some Q then Q else True))%I.
Robbert Krebbers's avatar
Robbert Krebbers committed
795
Proof. rewrite /Frame=> ?. by rewrite sep_exist_l; apply exist_mono. Qed.
796
Global Instance frame_forall {A} R (Φ : A  uPred M) mΨ :
Robbert Krebbers's avatar
Robbert Krebbers committed
797 798
  ( a, Frame R (Φ a) (mΨ a)) 
  Frame R ( x, Φ x) (Some ( x, if mΨ x is Some Q then Q else True))%I.
Robbert Krebbers's avatar
Robbert Krebbers committed
799 800
Proof. rewrite /Frame=> ?. by rewrite sep_forall_l; apply forall_mono. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
801
Lemma tac_frame Δ Δ' i p R P mQ :
802
  envs_lookup_delete i Δ = Some (p, R, Δ')  Frame R P mQ 
Robbert Krebbers's avatar
Robbert Krebbers committed
803 804
  (if mQ is Some Q then (if p then Δ else Δ')  Q else True) 
  Δ  P.
Robbert Krebbers's avatar
Robbert Krebbers committed
805
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
806 807 808 809 810
  intros [? ->]%envs_lookup_delete_Some ? HQ. destruct p.
  - rewrite envs_lookup_persistent_sound // always_elim.
    rewrite -(frame R P). destruct mQ as [Q|]; rewrite ?HQ /=; auto with I.
  - rewrite envs_lookup_sound //; simpl.
    rewrite -(frame R P). destruct mQ as [Q|]; rewrite ?HQ /=; auto with I.
Robbert Krebbers's avatar
Robbert Krebbers committed
811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832
Qed.

(** * Disjunction *)
Class OrSplit (P Q1 Q2 : uPred M) := or_split : (Q1  Q2)  P.
Arguments or_split : clear implicits.
Global Instance or_split_or P1 P2 : OrSplit (P1  P2) P1 P2.
Proof. done. Qed.

Lemma tac_or_l Δ P Q1 Q2 : OrSplit P Q1 Q2  Δ  Q1  Δ  P.
Proof. intros. rewrite -(or_split P). by apply or_intro_l'. Qed.
Lemma tac_or_r Δ P Q1 Q2 : OrSplit P Q1 Q2  Δ  Q2  Δ  P.
Proof. intros. rewrite -(or_split P). by apply or_intro_r'. Qed.

Class OrDestruct P Q1 Q2 := or_destruct : P  (Q1  Q2).
Arguments or_destruct : clear implicits.
Global Instance or_destruct_or P Q : OrDestruct (P  Q) P Q.
Proof. done. Qed.
Global Instance or_destruct_later P Q1 Q2 :
  OrDestruct P Q1 Q2  OrDestruct ( P) ( Q1) ( Q2).
Proof. rewrite /OrDestruct=>->. by rewrite later_or. Qed.

Lemma tac_or_destruct Δ Δ1 Δ2 i p j1 j2 P P1 P2 Q :
Ralf Jung's avatar
Ralf Jung committed
833
  envs_lookup i Δ = Some (p, P)  OrDestruct P P1 P2 
Robbert Krebbers's avatar
Robbert Krebbers committed
834 835 836 837
  envs_simple_replace i p (Esnoc Enil j1 P1) Δ = Some Δ1 
  envs_simple_replace i p (Esnoc Enil j2 P2) Δ = Some Δ2 
  Δ1  Q  Δ2  Q  Δ  Q.
Proof.
838 839 840 841 842 843
  intros ???? HP1 HP2. rewrite envs_lookup_sound //.
  rewrite (or_destruct P) always_if_or sep_or_r; apply or_elim.
  - rewrite (envs_simple_replace_sound' _ Δ1) //.
    by rewrite /= right_id wand_elim_r.
  - rewrite (envs_simple_replace_sound' _ Δ2) //.
    by rewrite /= right_id wand_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
844 845 846 847 848 849
Qed.

(** * Forall *)
Lemma tac_forall_intro {A} Δ (Φ : A  uPred M) : ( a, Δ  Φ a)  Δ  ( a, Φ a).
Proof. apply forall_intro. Qed.

850 851 852 853 854 855 856 857 858 859 860 861 862 863 864
Class ForallSpecialize {As} (xs : hlist As)
    (P : uPred M) (Φ : himpl As (uPred M)) :=
  forall_specialize : P  happly Φ xs.
Arguments forall_specialize {_} _ _ _ {_}.

Global Instance forall_specialize_nil P : ForallSpecialize hnil P P | 100.
Proof. done. Qed.
Global Instance forall_specialize_cons A As x xs Φ (Ψ : A  himpl As (uPred M)) :
  ( x, ForallSpecialize xs (Φ x) (Ψ x)) 
  ForallSpecialize (hcons x xs) ( x : A, Φ x) Ψ.
Proof. rewrite /ForallSpecialize /= => <-. by rewrite (forall_elim x). Qed.

Lemma tac_forall_specialize {As} Δ Δ' i p P (Φ : himpl As (uPred M)) Q xs :
  envs_lookup i Δ = Some (p, P)  ForallSpecialize xs P Φ 
  envs_simple_replace i p (Esnoc Enil i (happly Φ xs)) Δ = Some Δ' 
Robbert Krebbers's avatar
Robbert Krebbers committed
865 866
  Δ'  Q  Δ  Q.
Proof.
867 868
  intros. rewrite envs_simple_replace_sound //; simpl.
  by rewrite right_id (forall_specialize _ P) wand_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
869 870 871 872 873 874 875 876 877 878 879 880 881
Qed.

Lemma tac_forall_revert {A} Δ (Φ : A  uPred M) :
  Δ  ( a, Φ a)  ( a, Δ  Φ a).
Proof. intros HΔ a. by rewrite HΔ (forall_elim a). Qed.

(** * Existential *)
Class ExistSplit {A} (P : uPred M) (Φ : A  uPred M) :=
  exist_split : ( x, Φ x)  P.
Arguments exist_split {_} _ _ {_}.
Global Instance exist_split_exist {A} (Φ: A  uPred M): ExistSplit ( a, Φ a) Φ.
Proof. done. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
882 883 884
Lemma tac_exist {A} Δ P (Φ : A  uPred M) :
  ExistSplit P Φ  ( a, Δ  Φ a)  Δ  P.
Proof. intros ? [a ?]. rewrite -(exist_split P). eauto using exist_intro'. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
885 886 887 888 889 890 891 892

Class ExistDestruct {A} (P : uPred M) (Φ : A  uPred M) :=
  exist_destruct : P  ( x, Φ x).
Arguments exist_destruct {_} _ _ {_}.
Global Instance exist_destruct_exist {A} (Φ : A  uPred M) :
  ExistDestruct ( a, Φ a) Φ.
Proof. done. Qed.
Global Instance exist_destruct_later {A} P (Φ : A  uPred M) :
893 894
  ExistDestruct P Φ  Inhabited A  ExistDestruct ( P) (λ a,  (Φ a))%I.
Proof. rewrite /ExistDestruct=> HP ?. by rewrite HP later_exist. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
895 896

Lemma tac_exist_destruct {A} Δ i p j P (Φ : A  uPred M) Q :
897
  envs_lookup i Δ = Some (p, P)  ExistDestruct P Φ 
Robbert Krebbers's avatar
Robbert Krebbers committed
898 899 900 901
  ( a,  Δ',
    envs_simple_replace i p (Esnoc Enil j (Φ a)) Δ = Some Δ'  Δ'  Q) 
  Δ  Q.
Proof.
902 903 904 905
  intros ?? HΦ. rewrite envs_lookup_sound //.
  rewrite (exist_destruct P) always_if_exist sep_exist_r.
  apply exist_elim=> a; destruct (HΦ a) as (Δ'&?&?).
  rewrite envs_simple_replace_sound' //; simpl. by rewrite right_id wand_elim_r.
Robbert Krebbers's avatar
Robbert Krebbers committed
906 907
Qed.
End tactics.
Robbert Krebbers's avatar
Robbert Krebbers committed
908

909 910
Hint Extern 0 (ToPosedProof True _ _) =>
  class_apply @to_posed_proof_True : typeclass_instances.