weakestpre.v

Require Export program_logic.pviewshifts.
Require Import program_logic.wsat.
Local Hint Extern 10 (_ ≤ _) => omega.
Local Hint Extern 100 (@eq coPset _ _) => eassumption || solve_elem_of.
Local Hint Extern 100 (_ ∉ _) => solve_elem_of.
Local Hint Extern 100 (@subseteq coPset _ _ _) => solve_elem_of.
Local Hint Extern 10 (✓{_} _) =>
  repeat match goal with H : wsat _ _ _ _ |- _ => apply wsat_valid in H end;
  solve_validN.

Record wp_go {Λ Σ} (E : coPset) (Q Qfork : expr Λ → nat → iRes Λ Σ → Prop)
    (k : nat) (rf : iRes Λ Σ) (e1 : expr Λ) (σ1 : state Λ) := {
  wf_safe : reducible e1 σ1;
  wp_step e2 σ2 ef :
    prim_step e1 σ1 e2 σ2 ef →
    ∃ r2 r2',
      wsat k E σ2 (r2 ⋅ r2' ⋅ rf) ∧
      Q e2 k r2 ∧
      ∀ e', ef = Some e' → Qfork e' k r2'
}.
CoInductive wp_pre {Λ Σ} (E : coPset)
     (Q : val Λ → iProp Λ Σ) : expr Λ → nat → iRes Λ Σ → Prop :=
  | wp_pre_value n r v : pvs E E (Q v) n r → wp_pre E Q (of_val v) n r
  | wp_pre_step n r1 e1 :
     to_val e1 = None →
     (∀ rf k Ef σ1,
       1 < k < n → E ∩ Ef = ∅ →
       wsat (S k) (E ∪ Ef) σ1 (r1 ⋅ rf) →
       wp_go (E ∪ Ef) (wp_pre E Q)
                      (wp_pre coPset_all (λ _, True%I)) k rf e1 σ1) →
     wp_pre E Q e1 n r1.
Program Definition wp {Λ Σ} (E : coPset) (e : expr Λ)
  (Q : val Λ → iProp Λ Σ) : iProp Λ Σ := {| uPred_holds := wp_pre E Q e |}.
Next Obligation.
  intros Λ Σ E e Q r1 r2 n Hwp Hr.
  destruct Hwp as [|n r1 e2 ? Hgo]; constructor; rewrite -?Hr; auto.
  intros rf k Ef σ1 ?; rewrite -(dist_le _ _ _ _ Hr); naive_solver.
Qed.
Next Obligation.
  intros Λ Σ E e Q r; destruct (to_val e) as [v|] eqn:?.
  * by rewrite -(of_to_val e v) //; constructor.
  * constructor; auto with lia.
Qed.
Next Obligation.
  intros Λ Σ E e Q r1 r2 n1; revert Q E e r1 r2.
  induction n1 as [n1 IH] using lt_wf_ind; intros Q E e r1 r1' n2.
  destruct 1 as [|n1 r1 e1 ? Hgo].
  * constructor; eauto using uPred_weaken.
  * intros [rf' Hr] ??; constructor; [done|intros rf k Ef σ1 ???].
    destruct (Hgo (rf' ⋅ rf) k Ef σ1) as [Hsafe Hstep];
      rewrite ?associative -?Hr; auto; constructor; [done|].
    intros e2 σ2 ef ?; destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
    exists r2, (r2' ⋅ rf'); split_ands; eauto 10 using (IH k), cmra_included_l.
    by rewrite -!associative (associative _ r2).
Qed.
Instance: Params (@wp) 4.

Section wp.
Context {Λ : language} {Σ : iFunctor}.
Implicit Types P : iProp Λ Σ.
Implicit Types Q : val Λ → iProp Λ Σ.
Implicit Types v : val Λ.
Implicit Types e : expr Λ.
Transparent uPred_holds.

Global Instance wp_ne E e n :
  Proper (pointwise_relation _ (dist n) ==> dist n) (@wp Λ Σ E e).
Proof.
  cut (∀ Q1 Q2, (∀ v, Q1 v ={n}= Q2 v) →
    ∀ r n', n' ≤ n → ✓{n'} r → wp E e Q1 n' r → wp E e Q2 n' r).
  { by intros help Q Q' HQ; split; apply help. }
  intros Q1 Q2 HQ r n'; revert e r.
  induction n' as [n' IH] using lt_wf_ind=> e r.
  destruct 3 as [n' r v HpvsQ|n' r e1 ? Hgo].
  { constructor. by eapply pvs_ne, HpvsQ; eauto. }
  constructor; [done|]=> rf k Ef σ1 ???.
  destruct (Hgo rf k Ef σ1) as [Hsafe Hstep]; auto.
  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
  exists r2, r2'; split_ands; [|eapply IH|]; eauto.
Qed.
Global Instance wp_proper E e :
  Proper (pointwise_relation _ (≡) ==> (≡)) (@wp Λ Σ E e).
Proof.
  by intros Q Q' ?; apply equiv_dist=>n; apply wp_ne=>v; apply equiv_dist.
Qed.
Lemma wp_mask_frame_mono E1 E2 e Q1 Q2 :
  E1 ⊆ E2 → (∀ v, Q1 v ⊑ Q2 v) → wp E1 e Q1 ⊑ wp E2 e Q2.
Proof.
  intros HE HQ r n; revert e r; induction n as [n IH] using lt_wf_ind=> e r.
  destruct 2 as [n' r v HpvsQ|n' r e1 ? Hgo].
  { constructor; eapply pvs_mask_frame_mono, HpvsQ; eauto. }
  constructor; [done|]=> rf k Ef σ1 ???.
  assert (E2 ∪ Ef = E1 ∪ (E2 ∖ E1 ∪ Ef)) as HE'.
  { by rewrite associative_L -union_difference_L. }
  destruct (Hgo rf k ((E2 ∖ E1) ∪ Ef) σ1) as [Hsafe Hstep]; rewrite -?HE'; auto.
  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
  exists r2, r2'; split_ands; [rewrite HE'|eapply IH|]; eauto.
Qed.

Lemma wp_value_inv E Q v n r : wp E (of_val v) Q n r → pvs E E (Q v) n r.
Proof.
  by inversion 1 as [|??? He]; [|rewrite ?to_of_val in He]; simplify_equality.
Qed.
Lemma wp_step_inv E Ef Q e k n σ r rf :
  to_val e = None → 1 < k < n → E ∩ Ef = ∅ →
  wp E e Q n r → wsat (S k) (E ∪ Ef) σ (r ⋅ rf) →
  wp_go (E ∪ Ef) (λ e, wp E e Q) (λ e, wp coPset_all e (λ _, True%I)) k rf e σ.
Proof. intros He; destruct 3; [by rewrite ?to_of_val in He|eauto]. Qed.

Lemma wp_value E Q v : Q v ⊑ wp E (of_val v) Q.
Proof. by constructor; apply pvs_intro. Qed.
Lemma pvs_wp E e Q : pvs E E (wp E e Q) ⊑ wp E e Q.
Proof.
  intros r [|n] ?; [done|]; intros Hvs.
  destruct (to_val e) as [v|] eqn:He; [apply of_to_val in He; subst|].
  { constructor; eapply pvs_trans', pvs_mono, Hvs; eauto.
    intros ???; apply wp_value_inv. }
  constructor; [done|]=> rf k Ef σ1 ???.
  destruct (Hvs rf (S k) Ef σ1) as (r'&Hwp&?); auto.
  eapply wp_step_inv with (S k) r'; eauto.
Qed.
Lemma wp_pvs E e Q : wp E e (λ v, pvs E E (Q v)) ⊑ wp E e Q.
Proof.
  intros r n; revert e r; induction n as [n IH] using lt_wf_ind=> e r Hr HQ.
  destruct (to_val e) as [v|] eqn:He; [apply of_to_val in He; subst|].
  { constructor; apply pvs_trans', (wp_value_inv _ (pvs E E ∘ Q)); auto. }
  constructor; [done|]=> rf k Ef σ1 ???.
  destruct (wp_step_inv E Ef (pvs E E ∘ Q) e k n σ1 r rf) as [? Hstep]; auto.
  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&Hwp'&?); auto.
  exists r2, r2'; split_ands; [|apply (IH k)|]; auto.
Qed.
Lemma wp_atomic E1 E2 e Q :
  E2 ⊆ E1 → atomic e → pvs E1 E2 (wp E2 e (λ v, pvs E2 E1 (Q v))) ⊑ wp E1 e Q.
Proof.
  intros ? He r n ? Hvs; constructor; eauto using atomic_not_val.
  intros rf k Ef σ1 ???.
  destruct (Hvs rf (S k) Ef σ1) as (r'&Hwp&?); auto.
  destruct (wp_step_inv E2 Ef (pvs E2 E1 ∘ Q) e k (S k) σ1 r' rf)
    as [Hsafe Hstep]; auto using atomic_not_val.
  split; [done|]=> e2 σ2 ef ?.
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&Hwp'&?); clear Hsafe Hstep; auto.
  destruct Hwp' as [k r2 v Hvs'|k r2 e2 Hgo];
    [|destruct (atomic_step e σ1 e2 σ2 ef); naive_solver].
  apply pvs_trans in Hvs'; auto.
  destruct (Hvs' (r2' ⋅ rf) k Ef σ2) as (r3&[]); rewrite ?(associative _); auto.
  exists r3, r2'; split_ands; last done.
  * by rewrite -(associative _).
  * constructor; apply pvs_intro; auto.
Qed.
Lemma wp_frame_r E e Q R : (wp E e Q ★ R) ⊑ wp E e (λ v, Q v ★ R).
Proof.
  intros r' n Hvalid (r&rR&Hr&Hwp&?); revert Hvalid.
  rewrite Hr; clear Hr; revert e r Hwp.
  induction n as [n IH] using lt_wf_ind; intros e r1.
  destruct 1 as [|n r e ? Hgo]=>?.
  { constructor; apply pvs_frame_r; auto. exists r, rR; eauto. }
  constructor; [done|]=> rf k Ef σ1 ???.
  destruct (Hgo (rR⋅rf) k Ef σ1) as [Hsafe Hstep]; auto.
  { by rewrite (associative _). }
  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
  exists (r2 ⋅ rR), r2'; split_ands; auto.
  * by rewrite -(associative _ r2)
      (commutative _ rR) !associative -(associative _ _ rR).
  * apply IH; eauto using uPred_weaken.
Qed.
Lemma wp_frame_later_r E e Q R :
  to_val e = None → (wp E e Q ★ ▷ R) ⊑ wp E e (λ v, Q v ★ R).
Proof.
  intros He r' n Hvalid (r&rR&Hr&Hwp&?); revert Hvalid; rewrite Hr; clear Hr.
  destruct Hwp as [|[|n] r e ? Hgo]; [by rewrite to_of_val in He|done|].
  constructor; [done|intros rf k Ef σ1 ???].
  destruct (Hgo (rR⋅rf) k Ef σ1) as [Hsafe Hstep];rewrite ?(associative _);auto.
  split; [done|intros e2 σ2 ef ?].
  destruct (Hstep e2 σ2 ef) as (r2&r2'&?&?&?); auto.
  exists (r2 ⋅ rR), r2'; split_ands; auto.
  * by rewrite -(associative _ r2)
      (commutative _ rR) !associative -(associative _ _ rR).
  * apply wp_frame_r; [auto|exists r2, rR; split_ands; auto].
    eapply uPred_weaken with rR n; eauto.
Qed.
Lemma wp_bind `{LanguageCtx Λ K} E e Q :
  wp E e (λ v, wp E (K (of_val v)) Q) ⊑ wp E (K e) Q.
Proof.
  intros r n; revert e r; induction n as [n IH] using lt_wf_ind=> e r ?.
  destruct 1 as [|n r e ? Hgo]; [by apply pvs_wp|].
  constructor; auto using fill_not_val=> rf k Ef σ1 ???.
  destruct (Hgo rf k Ef σ1) as [Hsafe Hstep]; auto.
  split.
  { destruct Hsafe as (e2&σ2&ef&?).
    by exists (K e2), σ2, ef; apply fill_step. }
  intros e2 σ2 ef ?.
  destruct (fill_step_inv e σ1 e2 σ2 ef) as (e2'&->&?); auto.
  destruct (Hstep e2' σ2 ef) as (r2&r2'&?&?&?); auto.
  exists r2, r2'; split_ands; try eapply IH; eauto.
Qed.

(* Derived rules *)
Opaque uPred_holds.
Import uPred.
Lemma wp_mono E e Q1 Q2 : (∀ v, Q1 v ⊑ Q2 v) → wp E e Q1 ⊑ wp E e Q2.
Proof. by apply wp_mask_frame_mono. Qed.
Global Instance wp_mono' E e :
  Proper (pointwise_relation _ (⊑) ==> (⊑)) (@wp Λ Σ E e).
Proof. by intros Q Q' ?; apply wp_mono. Qed.
Lemma wp_value' E Q e v : to_val e = Some v → Q v ⊑ wp E e Q.
Proof. intros; rewrite -(of_to_val e v) //; by apply wp_value. Qed.
Lemma wp_frame_l E e Q R : (R ★ wp E e Q) ⊑ wp E e (λ v, R ★ Q v).
Proof. setoid_rewrite (commutative _ R); apply wp_frame_r. Qed.
Lemma wp_frame_later_l E e Q R :
  to_val e = None → (▷ R ★ wp E e Q) ⊑ wp E e (λ v, R ★ Q v).
Proof.
  rewrite (commutative _ (▷ R)%I); setoid_rewrite (commutative _ R).
  apply wp_frame_later_r.
Qed.
Lemma wp_always_l E e Q R `{!AlwaysStable R} :
  (R ∧ wp E e Q) ⊑ wp E e (λ v, R ∧ Q v).
Proof. by setoid_rewrite (always_and_sep_l' _ _); rewrite wp_frame_l. Qed.
Lemma wp_always_r E e Q R `{!AlwaysStable R} :
  (wp E e Q ∧ R) ⊑ wp E e (λ v, Q v ∧ R).
Proof. by setoid_rewrite (always_and_sep_r' _ _); rewrite wp_frame_r. Qed.
Lemma wp_impl_l E e Q1 Q2 : ((□ ∀ v, Q1 v → Q2 v) ∧ wp E e Q1) ⊑ wp E e Q2.
Proof.
  rewrite wp_always_l; apply wp_mono=> // v.
  by rewrite always_elim (forall_elim v) impl_elim_l.
Qed.
Lemma wp_impl_r E e Q1 Q2 : (wp E e Q1 ∧ □ ∀ v, Q1 v → Q2 v) ⊑ wp E e Q2.
Proof. by rewrite commutative wp_impl_l. Qed.
End wp.