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From iris.proofmode Require Import tactics.
From iris.algebra Require Import auth.
From iris.base_logic Require Import lib.auth.
From iris_logrel.F_mu_ref_conc Require Export examples.lock.
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From iris_logrel.F_mu_ref_conc Require Import tactics rel_tactics soundness_binary relational_properties.
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From iris.program_logic Require Import adequacy.

From iris_logrel.F_mu_ref_conc Require Import hax.

Definition rand : val := λ: <>,
  let: "y" := (ref (# false))
  in Fork ("y" <- # true);;
     !"y".
Definition lateChoice : val := λ: "x",
  "x" <- #n 0;; rand #().
Definition earlyChoice : val := λ: "x",
  let: "r" := rand #() in "x" <- #n 0;; "r".

Section Refinement.
  Context `{heapIG Σ, cfgSG Σ}.

  Definition choiceN : namespace := nroot .@ "choice".

  Definition choice_inv y y' : iProp Σ :=
    ( f, y ↦ᵢ (#v f)  y' ↦ₛ (#v f))%I.

  Lemma wp_rand :
    (WP rand #() {{ v, v = #v true  v = #v false}})%I.
  Proof.
    iStartProof.
    unfold rand. unlock.
    iApply wp_rec; eauto. solve_closed. iNext. simpl.
    wp_bind (Alloc _). iApply wp_alloc; auto. iNext. iIntros (y) "Hy".
    iMod (inv_alloc choiceN _ (y ↦ᵢ (#v false)  y ↦ᵢ (#v true))%I with "[Hy]") as "#Hinv"; eauto.
    iApply wp_rec; eauto. solve_closed. iNext. simpl.
    wp_bind (Fork _). iApply wp_fork. iNext.
    iSplitL.
    - iModIntro. iApply wp_rec; eauto. solve_closed. iNext; simpl.
      iInv choiceN as "[Hy | Hy]" "Hcl"; iApply (wp_load with "Hy"); eauto; iNext;
        iIntros "Hy"; iMod ("Hcl" with "[Hy]"); eauto.
    - iInv choiceN as "[Hy | Hy]" "Hcl"; iApply (wp_store with "Hy"); eauto; iNext;
        iIntros "Hy"; iMod ("Hcl" with "[Hy]"); eauto.
  Qed.

  Lemma rand_l Γ E1 K ρ t τ :
    choiceN  E1 
    spec_ctx ρ - ( b, {E1,E1;Γ}  fill K (# b) log t : τ)
    - {E1,E1;Γ}  fill K (rand #())%E log t : τ.
  Proof.
    iIntros (?) "#Hs Hlog".
    unfold rand at 1. unlock. simpl.
    iApply (bin_log_related_rec_l Γ E1 K BAnon BAnon _ #()%E); first done.
    iNext. simpl.
    rel_bind_l (Alloc _).
    iApply bin_log_related_alloc_l'; first eauto. iIntros (y) "Hy". simpl.
    iApply (bin_log_related_rec_l _ _ K); first eauto. iNext. simpl.
    iMod (inv_alloc choiceN _ ( b, y ↦ᵢ (#v b))%I with "[Hy]") as "#Hinv".
    { iNext. eauto. }
    rel_bind_l (Fork _).
    iApply bin_log_related_fork_l. iModIntro.
    iSplitR.
    - iNext.
      iInv choiceN as (b) "Hy" "Hcl".
      iApply (wp_store with "Hy"); eauto. iNext. iIntros "Hy".
      iMod ("Hcl" with "[Hy]").
      { iNext. iExists true. by iFrame. }
      done.
    - simpl.
      iApply (bin_log_related_rec_l _ _ K); first eauto. iNext. simpl.
      iApply (bin_log_related_load_l _ _ _ K).
      iInv choiceN as (b) "Hy" "Hcl". iModIntro.
      iExists (#v b). iFrame. iIntros "Hy".
      iMod ("Hcl" with "[Hy]").
      { iNext. iExists b. iFrame. }
      done. 
  Qed.
 
  Lemma lateChoice_l Γ x v ρ t :
    spec_ctx ρ - x ↦ᵢ v -
    (x ↦ᵢ (#nv 0) -  b, Γ  (# b) log t : TBool) -
    Γ  lateChoice #x log t : TBool.
  Proof.  
    iIntros "#Hs Hx Hlog".
    unfold lateChoice. unlock.
    iApply (bin_log_related_rec_l _ _ []); eauto. iNext. simpl. rewrite !Closed_subst_id.
    rel_bind_l (#x <- _)%E.
    iApply (bin_log_related_store_l' with "Hx"); eauto. iIntros "Hx".
    simpl.
    iApply (bin_log_related_rec_l _ _ []); eauto. iNext. simpl.
    
    unfold rand at 1. unlock.
    iApply (bin_log_related_rec_l _ _ []); eauto. iNext. simpl.
    rel_bind_l (Alloc _).
    iApply bin_log_related_alloc_l'; eauto. iIntros (y) "Hy". simpl.
    iApply (bin_log_related_rec_l _ _ []); eauto. iNext. simpl.
    
    iMod (inv_alloc choiceN _ ( b, y ↦ᵢ (#v b))%I with "[Hy]") as "#Hinv".
    { iNext. eauto. }
    rel_bind_l (Fork _).
    iApply bin_log_related_fork_l. iModIntro.
    iSplitR.
    - iNext.
      iInv choiceN as (b) "Hy" "Hcl".
      iApply (wp_store with "Hy"); eauto. iNext. iIntros "Hy".
      iMod ("Hcl" with "[Hy]").
      { iNext. iExists true. by iFrame. }
      done.
    - simpl.
      iApply (bin_log_related_rec_l _ _ []); eauto. iNext. simpl.
      iApply (bin_log_related_load_l _ _ _ []).
      iInv choiceN as (b) "Hy" "Hcl". iModIntro.
      iExists (#v b). iFrame. iIntros "Hy".
      iMod ("Hcl" with "[Hy]").
      { iNext. iExists b. iFrame. }
      simpl. iApply ("Hlog" with "Hx").
  Qed.
  
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  (* Lemma lateChoice_l Γ E1 E2 K x t τ R : *)
  (*    (|={E1,E2}=>  n, x ↦ᵢ (#nv n)  R(n)  *)
  (*      (( n, x ↦ᵢ (#nv n)  R(n)) ={E2,E1}= True)  *)
  (*       (x ↦ᵢ (#nv 0)  R(0) - *)
  (*            b, {E2,E1;Γ}  fill K (# b) log t : τ)) *)
  (*   - {E1,E1;Γ}  fill K (lateChoice #x)%E log t : τ. *)
  (* Proof. *)
  (*   iIntros "#H". *)
  (*   unfold lateChoice. unlock. *)
  (*   iApply (bin_log_related_rec_l _ _ K); eauto. iNext. simpl. rewrite !Closed_subst_id. *)
  (*   rel_bind_l (#x <- _)%E. *)
  (*   iApply (bin_log_related_store_l); eauto.  *)
  (*     iMod "H" as (n) "[Hx [HR Hfin]]". *)
  (*     iModIntro. iExists (#nv n). iFrame. *)
  (*     iIntros "Hx". *)
  (*   simpl. *)
  (*   rewrite ->uPred.and_elim_l. *)
   
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  Lemma prerefinement Γ x x' n ρ :
    (spec_ctx ρ - x ↦ᵢ (#nv n) - x' ↦ₛ (#nv n) -
      Γ  lateChoice #x log earlyChoice #x' : TBool)%I.
  Proof.
    iIntros "#Hspec Hx Hx'".
    unfold lateChoice, earlyChoice. unlock.
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    rel_rec_l. rel_rec_r.
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    rel_bind_l (#x <- _)%E.
    iApply (bin_log_related_store_l' with "Hx"); eauto. iIntros "Hx".
    simpl.
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    rel_rec_l.
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    unfold rand at 1. unlock.
    iApply (bin_log_related_rec_l _ _ []); eauto. iNext. simpl.
    rel_bind_l (Alloc _).
    iApply bin_log_related_alloc_l'; eauto. iIntros (y) "Hy". simpl.
    iApply (bin_log_related_rec_l _ _ []); eauto. iNext. simpl.

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    unfold rand. unlock.
    rel_rec_r.
    rel_alloc_r as y' "Hy'".
    rel_rec_r.
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    iAssert (choice_inv y y') with "[Hy Hy']" as "Hinv".
    { iExists false. by iFrame. }
    iMod (inv_alloc choiceN with "[Hinv]") as "#Hinv".
    { iNext. iApply "Hinv". }
    rel_bind_r (Fork _).
    iApply bin_log_related_fork_r; eauto. iIntros (i) "Hi".

    rel_bind_l (Fork _).
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    iApply bin_log_related_fork_l;simpl. iModIntro.
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    iSplitL "Hi".
    - iNext.
      iInv choiceN as (f) "[Hy Hy']" "Hcl".
      iApply (wp_store with "Hy"); eauto. iNext. iIntros "Hy".
      tp_store i.
      iMod ("Hcl" with "[Hy Hy']").
      { iNext. iExists true. by iFrame. }
      done.
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    - rel_rec_l.
      rel_rec_r.
      
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      iApply (bin_log_related_load_l _ _ _ []).
      iInv choiceN as (f) "[Hy >Hy']" "Hcl". iModIntro.
      iExists (#v f). iFrame. iIntros "Hy".
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      rel_load_r.
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      iMod ("Hcl" with "[Hy Hy']").
      { iNext. iExists f. iFrame. }

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      rel_rec_r.
      rel_store_r. simpl.
      rel_seq_r.
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      iApply bin_log_related_val; eauto.
      { iIntros (Δ). iModIntro. simpl. eauto. }
  Qed.

  Lemma refinement Γ ρ :
    (spec_ctx ρ -
      Γ  lateChoice log earlyChoice : TArrow (Tref TNat) TBool)%I.
  Proof.
    iIntros "#Hspec".
    unfold lateChoice in *. unfold earlyChoice in *. unlock.
    iApply bin_log_related_arrow.
    iAlways. iIntros (Δ (l,l')) "Hxx'". simpl.
    iDestruct "Hxx'" as ([x x']) "[% #Hxx']". inversion H1; subst. simpl.    
    replace (λ: "x", "x" <- Nat 0 ;; rand #())%E
      with (of_val lateChoice); last first.
    { unfold lateChoice. unlock. reflexivity. }
    replace (λ: "x", (λ: "r", "x" <- Nat 0 ;; "r") (rand #()))%E
      with (of_val earlyChoice); last first.
    { unfold earlyChoice. unlock. reflexivity. }
    Abort.
    (* iApply prerefinement; eauto. *)
    
End Refinement.