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(** More generative ADT example from "State-Dependent
Represenation Independence" by A. Ahmed, D. Dreyer, A. Rossberg. *)
From iris.proofmode Require Import tactics.
From iris.algebra Require Import auth gset frac.
From iris.base_logic.lib Require Import auth.
From iris_logrel Require Import logrel examples.counter examples.lock examples.various.

(** * 5.4 Cell class *)
Definition cellτ : type :=
  TForall (TExists (TProd (TProd (TArrow (TVar 1) (TVar 0))
                                 (TArrow (TVar 0) (TVar 1)))
                                 (TArrow (TVar 0) (TArrow (TVar 1) TUnit)))).
Definition cell1 : val :=
  Λ: Pack (λ: "x", ref "x", λ: "r", !"r", λ: "r" "x", "r" <- "x").
Lemma cell1_typed Γ :
  typed Γ cell1 cellτ.
Proof.
  unfold cellτ. unlock cell1.
  solve_typed.
Qed.

Definition cell2 : val :=
  Λ: let: "l" := newlock #() in
    Pack ( λ: "x", acquire "l";; let: "v" := (ref #false, ref "x", ref "x") in
                   release "l";; "v"
        ,  λ: "r", acquire "l";;
                   let: "v" :=
                      if: !(Fst (Fst "r"))
                      then !(Snd "r")
                      else !(Snd (Fst "r")) in
                   release "l";;
                   "v"
         , λ: "r" "x", acquire "l";;
                       (if: !(Fst (Fst "r"))
                        then (Snd (Fst "r")) <- "x";; (Fst (Fst "r")) <- #false
                        else (Snd "r") <- "x";; (Fst (Fst "r")) <- #true);;
                       release "l").
Lemma cell2_typed Γ :
  typed Γ cell2 cellτ.
Proof.
  unfold cellτ. unlock cell2.
  solve_typed.
  econstructor.
  econstructor. 2: solve_typed.
  econstructor.
  eapply TPack with (TProd (TProd (Tref TBool) (Tref (TVar 0))) (Tref (TVar 0))).
  asimpl.
  econstructor; solve_typed.
  econstructor; solve_typed.
Qed.

Definition refPool := gset ((loc * loc * loc) * loc).
Definition refPoolR := gsetUR ((loc * loc * loc) * loc).

Class refPoolG Σ := RefPoolG
{ refPool_inG :> authG Σ refPoolR }.

Section cell_refinement.
  Context `{logrelG Σ, lockG Σ, refPoolG Σ}.
  Notation D := (prodC valC valC -n> iProp Σ).

  Definition cellInt (R : D) (r1 r2 r3 r : loc) : iProp Σ :=
    ( (a b c : val), r ↦ₛ a  r2 ↦ᵢ b  r3 ↦ᵢ c 
    ( (r1 ↦ᵢ #true  R (c, a))
     (r1 ↦ᵢ #false  R (b, a))))%I.

  Definition cellRegInv (P : refPool) (R : D) : iProp Σ :=
    ([ set] rs  P, match rs with
                      | ((r1, r2, r3), r) => cellInt R r1 r2 r3 r
                     end)%I.

  Definition cellInv γ R : iProp Σ :=
    ( (P : refPool), own γ ( P)  cellRegInv P R)%I.

  Program Definition cellR (γ : gname) : D := λne vv,
    ( r1 r2 r3 r : loc, vv.1 = (#r1, #r2, #r3)%V  vv.2 = #r
    own γ ( {[(r1, r2, r3), r]}))%I.
  Next Obligation. solve_proper. Qed.

  Instance cellR_persistent γ ww : PersistentP (cellR γ ww).
  Proof. apply _. Qed.

  Lemma cellRegInv_excl (P : refPool) R (r1 r2 r3 r : loc) (v : val) :
    cellRegInv P R - r1 ↦ᵢ v - (r1, r2, r3, r)  P.
  Proof.
    rewrite /cellRegInv.
    iIntros "HP Hr1".
    iAssert ((r1, r2, r3, r)  P  False)%I as %Hbaz.
    {
      iIntros (HP).
      rewrite (big_sepS_elem_of _ P _ HP).
      iDestruct "HP" as (a b c) "(Hr & Hr2 & Hr3 & Hrs)".
      iDestruct "Hrs" as "[[Hr1' ?]|[Hr1' ?]]";
      iDestruct (mapsto_valid_2 r1 with "Hr1' Hr1") as %Hfoo;
      compute in Hfoo; contradiction.
    }
    iPureIntro. eauto.
  Qed.

  Lemma cellRegInv_open (P : refPool) R (r1 r2 r3 r : loc) :
    (r1, r2, r3, r)  P 
    cellRegInv P R -
    (cellInt R r1 r2 r3 r)  (cellInt R r1 r2 r3 r - cellRegInv P R).
  Proof.
    iIntros (Hrin) "Hreg".
    rewrite /cellRegInv.
    iDestruct (big_sepS_elem_of_acc _ P (r1, r2, r3, r) with "Hreg") as "[Hrs Hreg]"; first assumption.
    by iFrame.
  Qed.

  Lemma refPool_alloc γ (P : refPool) (r1 r2 r3 r : loc) :
    (r1, r2, r3, r)  P 
    own γ ( P)
    == own γ ( ({[(r1, r2, r3, r)]}  P))
       own γ ( ({[(r1, r2, r3, r)]})).
  Proof.
    iIntros (Hin) "HP".
    iMod (own_update with "HP") as "[HP Hfrag]".
    { eapply auth_update_alloc.
      eapply (gset_local_update _ _ ({[(r1, r2, r3, r)]}  P)).
      apply union_subseteq_r. }
    iFrame.
    rewrite -gset_op_union auth_frag_op.
    by iDestruct "Hfrag" as "[$ _]".
  Qed.

  Lemma refPool_lookup γ (P : refPool) (r1 r2 r3 r : loc) :
    own γ ( P) -
    own γ ( {[(r1, r2), r3, r]}) -
    (r1, r2, r3, r)  P.
  Proof.
    iIntros "Ho Hrs".
    iDestruct (own_valid_2 with "Ho Hrs") as %Hfoo.
    iPureIntro.
    apply auth_valid_discrete in Hfoo.
    simpl in Hfoo. destruct Hfoo as [Hfoo _].
    revert Hfoo. rewrite left_id.
    by rewrite gset_included elem_of_subseteq_singleton.
  Qed.

  Lemma cell2_cell1_refinement Γ :
    Γ  cell2 log cell1 : cellτ.
  Proof.
    iIntros (Δ).
    unlock cell2 cell1 cellτ.
    pose (N:=logrelN.@"locked").
    assert (Closed (dom _ Γ) (let: "l" := newlock #() in
        Pack
          (λ: "x", acquire "l";; let: "v" := (ref #false, ref "x", ref "x") in
                   release "l";; "v",
          λ: "r",
            acquire "l" ;;
            let: "v" := if: ! (Fst (Fst "r")) then
                        ! (Snd "r") else ! (Snd (Fst "r")) in
            release "l" ;; "v",
          λ: "r" "x",
            acquire "l" ;;
            (if: ! (Fst (Fst "r"))
             then Snd (Fst "r") <- "x" ;; Fst (Fst "r") <- #false
             else Snd "r" <- "x" ;;
                  Fst (Fst "r") <- #true) ;; release "l"))).
    { apply Closed_mono with ; eauto.
      set_solver. }
    assert (Closed (dom _ Γ) (Pack (λ: "x", ref "x", λ: "r", ! "r", λ: "r" "x", "r" <- "x"))).
    { apply Closed_mono with ; eauto.
      set_solver. }
    iApply bin_log_related_tlam; auto.
    iIntros (R HR) "!#".
    iApply fupd_logrel'.
    iMod (own_alloc ( ( : refPoolR))) as (γ) "Ho"; first done.
    iModIntro.
    rel_apply_l (bin_log_related_newlock_l N (cellInv γ R)%I with "[Ho]").
    { iExists _. iFrame. unfold cellRegInv.
      by rewrite big_sepS_empty. }
    iIntros (lk γl) "#Hlk".
    rel_let_l.
    iApply (bin_log_related_pack _ (cellR γ)).
    repeat iApply bin_log_related_pair.
    - (* New cell *)
      iApply bin_log_related_arrow_val; eauto.
      iAlways. iIntros (v1 v2) "/= #Hv".
      rel_let_l. rel_let_r.
      rel_apply_l (bin_log_related_acquire_l N _ lk with "Hlk"); first auto.
      iIntros "Hl".
      iDestruct 1 as (P) "[Ho HP]".
      rel_let_l.
      rel_alloc_l as r1 "Hr1".
      rel_alloc_l as r2 "Hr2".
      rel_alloc_l as r3 "Hr3".
      rel_alloc_r as r "Hr".
      rel_let_l.
      iDestruct (cellRegInv_excl with "HP Hr1") as %Hrs.
      iApply fupd_logrel'.
      iMod (refPool_alloc γ P r1 r2 r3 r with "Ho") as "[Ho #Hrs]"; first apply Hrs.
      iModIntro.
      rel_apply_l (bin_log_related_release_l with "Hlk Hl [-Hrs]"); first auto.
      { iExists _; iFrame. rewrite {2}/cellRegInv.
        rewrite big_sepS_insert; last assumption.
        iFrame. iExists _,_,_. iFrame. iRight; by iFrame. }
      rel_let_l.
      rel_vals. iModIntro. iAlways. iExists _,_,_,_. eauto.
    - (* Read cell *)
      iApply bin_log_related_arrow_val; eauto.
      iAlways. iIntros (v1 v2) "/=".
      iDestruct 1 as (r1 r2 r3 r) "[% [% #Hrs]]". simplify_eq.
      rel_let_l.
      rel_apply_l (bin_log_related_acquire_l N _ lk with "Hlk"); first auto.
      iIntros "Hl".
      iDestruct 1 as (P) "[Ho HP]".
      rel_let_l. repeat rel_proj_l.
      iDestruct (refPool_lookup with "Ho Hrs") as %Hrin.
      iDestruct (cellRegInv_open with "HP") as "[Hr HclP]"; first apply Hrin.
      iDestruct "Hr" as (a b c) "(Hr & Hr2 & Hr3 & [[Hr1 #HR] | [Hr1 #HR]])";
        rel_load_l; rel_if_l; repeat rel_proj_l; rel_load_l; rel_let_l;
        rel_let_r; rel_load_r.
      + rel_apply_l (bin_log_related_release_l with "Hlk Hl [-]"); auto.
        { iExists _; iFrame. iApply "HclP".
          iExists _,_,_; iFrame. iLeft; by iFrame. }
        rel_let_l. rel_vals; eauto.
      + rel_apply_l (bin_log_related_release_l with "Hlk Hl [-]"); auto.
        { iExists _; iFrame. iApply "HclP".
          iExists _,_,_; iFrame. iRight; by iFrame. }
        rel_let_l. rel_vals; eauto.
    - (* Insert cell *)
      iApply bin_log_related_arrow_val; eauto.
      iAlways. iIntros (v1 v2) "/=".
      iDestruct 1 as (r1 r2 r3 r) "[% [% #Hrs]]". simplify_eq.
      rel_let_l. rel_let_r.
      iApply bin_log_related_arrow_val; eauto.
      iAlways. iIntros (v1 v2) "/= #Hv".
      rel_let_l. rel_let_r.
      rel_apply_l (bin_log_related_acquire_l N _ lk with "Hlk"); first auto.
      iIntros "Hl".
      iDestruct 1 as (P) "[Ho HP]".
      rel_let_l. repeat rel_proj_l.
      iDestruct (refPool_lookup with "Ho Hrs") as %Hrin.
      iDestruct (cellRegInv_open with "HP") as "[Hr HclP]"; first apply Hrin.
      iDestruct "Hr" as (a b c) "(Hr & Hr2 & Hr3 & [[Hr1 #HR] | [Hr1 #HR]])";
        rel_load_l; rel_if_l; repeat rel_proj_l;
        rel_store_l; rel_let_l; repeat rel_proj_l;
        rel_store_l; rel_store_r; rel_let_l.
      + rel_apply_l (bin_log_related_release_l with "Hlk Hl [-]"); auto.
        { iExists _; iFrame. iApply "HclP".
          iExists _,_,_; iFrame. iRight; by iFrame. }
        iApply bin_log_related_unit.
      + rel_apply_l (bin_log_related_release_l with "Hlk Hl [-]"); auto.
        { iExists _; iFrame. iApply "HclP".
          iExists _,_,_; iFrame. iLeft; by iFrame. }
        iApply bin_log_related_unit.
    Qed.
End cell_refinement.