Library iris.program_logic.language

From iris.algebra Require Export ofe.
Set Default Proof Using "Type".

Section language_mixin.
  Context {expr val state observation : Type}.
  Context (of_val : val expr).
  Context (to_val : expr option val).
We annotate the reduction relation with observations κ, which we will use in the definition of weakest preconditions to predict future observations and assert correctness of the predictions.
  Context (prim_step : expr state list observation expr state list expr Prop).

  Record LanguageMixin := {
    mixin_to_of_val v : to_val (of_val v) = Some v;
    mixin_of_to_val e v : to_val e = Some v of_val v = e;
    mixin_val_stuck e σ κ e' σ' efs : prim_step e σ κ e' σ' efs to_val e = None
  }.
End language_mixin.

Structure language := Language {
  expr : Type;
  val : Type;
  state : Type;
  observation : Type;
  of_val : val expr;
  to_val : expr option val;
  prim_step : expr state list observation expr state list expr Prop;
  language_mixin : LanguageMixin of_val to_val prim_step
}.
Delimit Scope expr_scope with E.
Delimit Scope val_scope with V.
Bind Scope expr_scope with expr.
Bind Scope val_scope with val.

Arguments Language {_ _ _ _ _ _ _} _.
Arguments of_val {_} _.
Arguments to_val {_} _.
Arguments prim_step {_} _ _ _ _ _ _.

Canonical Structure stateO Λ := leibnizO (state Λ).
Canonical Structure valO Λ := leibnizO (val Λ).
Canonical Structure exprO Λ := leibnizO (expr Λ).

Definition cfg (Λ : language) := (list (expr Λ) × state Λ)%type.

Class LanguageCtx {Λ : language} (K : expr Λ expr Λ) := {
  fill_not_val e :
    to_val e = None to_val (K e) = None;
  fill_step e1 σ1 κ e2 σ2 efs :
    prim_step e1 σ1 κ e2 σ2 efs
    prim_step (K e1) σ1 κ (K e2) σ2 efs;
  fill_step_inv e1' σ1 κ e2 σ2 efs :
    to_val e1' = None prim_step (K e1') σ1 κ e2 σ2 efs
     e2', e2 = K e2' prim_step e1' σ1 κ e2' σ2 efs
}.

Instance language_ctx_id Λ : LanguageCtx (@id (expr Λ)).
Proof. constructor; naive_solver. Qed.

Inductive atomicity := StronglyAtomic | WeaklyAtomic.

Section language.
  Context {Λ : language}.
  Implicit Types v : val Λ.
  Implicit Types e : expr Λ.

  Lemma to_of_val v : to_val (of_val v) = Some v.
  Proof. apply language_mixin. Qed.
  Lemma of_to_val e v : to_val e = Some v of_val v = e.
  Proof. apply language_mixin. Qed.
  Lemma val_stuck e σ κ e' σ' efs : prim_step e σ κ e' σ' efs to_val e = None.
  Proof. apply language_mixin. Qed.

  Definition reducible (e : expr Λ) (σ : state Λ) :=
     κ e' σ' efs, prim_step e σ κ e' σ' efs.
  Definition reducible_no_obs (e : expr Λ) (σ : state Λ) :=
     e' σ' efs, prim_step e σ [] e' σ' efs.
  Definition irreducible (e : expr Λ) (σ : state Λ) :=
     κ e' σ' efs, ¬prim_step e σ κ e' σ' efs.
  Definition stuck (e : expr Λ) (σ : state Λ) :=
    to_val e = None irreducible e σ.
  Definition not_stuck (e : expr Λ) (σ : state Λ) :=
    is_Some (to_val e) reducible e σ.

  Class Atomic (a : atomicity) (e : expr Λ) : Prop :=
    atomic σ e' κ σ' efs :
      prim_step e σ κ e' σ' efs
      if a is WeaklyAtomic then irreducible e' σ' else is_Some (to_val e').

  Inductive step (ρ1 : cfg Λ) (κ : list (observation Λ)) (ρ2 : cfg Λ) : Prop :=
    | step_atomic e1 σ1 e2 σ2 efs t1 t2 :
       ρ1 = (t1 ++ e1 :: t2, σ1)
       ρ2 = (t1 ++ e2 :: t2 ++ efs, σ2)
       prim_step e1 σ1 κ e2 σ2 efs
       step ρ1 κ ρ2.
  Hint Constructors step : core.

  Inductive nsteps : nat cfg Λ list (observation Λ) cfg Λ Prop :=
    | nsteps_refl ρ :
       nsteps 0 ρ [] ρ
    | nsteps_l n ρ1 ρ2 ρ3 κ κs :
       step ρ1 κ ρ2
       nsteps n ρ2 κs ρ3
       nsteps (S n) ρ1 (κ ++ κs) ρ3.
  Hint Constructors nsteps : core.

  Definition erased_step (ρ1 ρ2 : cfg Λ) := κ, step ρ1 κ ρ2.

rtc erased_step and nsteps encode the same thing, just packaged in a different way.
  Lemma erased_steps_nsteps ρ1 ρ2 :
    rtc erased_step ρ1 ρ2 n κs, nsteps n ρ1 κs ρ2.
  Proof.
    split.
    - induction 1; firstorder eauto.     - intros (n & κs & Hsteps). unfold erased_step.
      induction Hsteps; eauto using rtc_refl, rtc_l.
  Qed.

  Lemma of_to_val_flip v e : of_val v = e to_val e = Some v.
  Proof. intros <-. by rewrite to_of_val. Qed.

  Lemma not_reducible e σ : ¬reducible e σ irreducible e σ.
  Proof. unfold reducible, irreducible. naive_solver. Qed.
  Lemma reducible_not_val e σ : reducible e σ to_val e = None.
  Proof. intros (?&?&?&?&?); eauto using val_stuck. Qed.
  Lemma reducible_no_obs_reducible e σ : reducible_no_obs e σ reducible e σ.
  Proof. intros (?&?&?&?); eexists; eauto. Qed.
  Lemma val_irreducible e σ : is_Some (to_val e) irreducible e σ.
  Proof. intros [??] ???? ?%val_stuck. by destruct (to_val e). Qed.
  Global Instance of_val_inj : Inj (=) (=) (@of_val Λ).
  Proof. by intros v v' Hv; apply (inj Some); rewrite -!to_of_val Hv. Qed.
  Lemma not_not_stuck e σ : ¬not_stuck e σ stuck e σ.
  Proof.
    rewrite /stuck /not_stuck -not_eq_None_Some -not_reducible.
    destruct (decide (to_val e = None)); naive_solver.
  Qed.

  Lemma strongly_atomic_atomic e a :
    Atomic StronglyAtomic e Atomic a e.
  Proof. unfold Atomic. destruct a; eauto using val_irreducible. Qed.

  Lemma reducible_fill `{!@LanguageCtx Λ K} e σ :
    reducible e σ reducible (K e) σ.
  Proof. unfold reducible in ×. naive_solver eauto using fill_step. Qed.
  Lemma reducible_fill_inv `{!@LanguageCtx Λ K} e σ :
    to_val e = None reducible (K e) σ reducible e σ.
  Proof.
    intros ? (e'&σ'&k&efs&Hstep); unfold reducible.
    apply fill_step_inv in Hstep as (e2' & _ & Hstep); eauto.
  Qed.
  Lemma reducible_no_obs_fill `{!@LanguageCtx Λ K} e σ :
    reducible_no_obs e σ reducible_no_obs (K e) σ.
  Proof. unfold reducible_no_obs in ×. naive_solver eauto using fill_step. Qed.
  Lemma reducible_no_obs_fill_inv `{!@LanguageCtx Λ K} e σ :
    to_val e = None reducible_no_obs (K e) σ reducible_no_obs e σ.
  Proof.
    intros ? (e'&σ'&efs&Hstep); unfold reducible_no_obs.
    apply fill_step_inv in Hstep as (e2' & _ & Hstep); eauto.
  Qed.
  Lemma irreducible_fill `{!@LanguageCtx Λ K} e σ :
    to_val e = None irreducible e σ irreducible (K e) σ.
  Proof. rewrite -!not_reducible. naive_solver eauto using reducible_fill_inv. Qed.
  Lemma irreducible_fill_inv `{!@LanguageCtx Λ K} e σ :
    irreducible (K e) σ irreducible e σ.
  Proof. rewrite -!not_reducible. naive_solver eauto using reducible_fill. Qed.

  Lemma not_stuck_fill_inv K `{!@LanguageCtx Λ K} e σ :
    not_stuck (K e) σ not_stuck e σ.
  Proof.
    rewrite /not_stuck -!not_eq_None_Some. intros [?|?].
    - auto using fill_not_val.
    - destruct (decide (to_val e = None)); auto using reducible_fill_inv.
  Qed.

  Lemma stuck_fill `{!@LanguageCtx Λ K} e σ :
    stuck e σ stuck (K e) σ.
  Proof. rewrite -!not_not_stuck. eauto using not_stuck_fill_inv. Qed.

  Lemma step_Permutation (t1 t1' t2 : list (expr Λ)) κ σ1 σ2 :
    t1 ≡ₚ t1' step (t1,σ1) κ (t2,σ2) t2', t2 ≡ₚ t2' step (t1',σ1) κ (t2',σ2).
  Proof.
    intros Ht [e1 σ1' e2 σ2' efs tl tr ?? Hstep]; simplify_eq/=.
    move: Ht; rewrite -Permutation_middle (symmetry_iff (≡ₚ)).
    intros (tl'&tr'&->&Ht)%Permutation_cons_inv.
     (tl' ++ e2 :: tr' ++ efs); split; [|by econstructor].
    by rewrite -!Permutation_middle !assoc_L Ht.
  Qed.

  Lemma step_insert i t2 σ2 e κ e' σ3 efs :
    t2 !! i = Some e
    prim_step e σ2 κ e' σ3 efs
    step (t2, σ2) κ (<[i:=e']>t2 ++ efs, σ3).
  Proof.
    intros.
    edestruct (elem_of_list_split_length t2) as (t21&t22&?&?);
      first (by eauto using elem_of_list_lookup_2); simplify_eq.
    econstructor; eauto.
    by rewrite insert_app_r_alt // Nat.sub_diag /= -assoc_L.
  Qed.

  Lemma erased_step_Permutation (t1 t1' t2 : list (expr Λ)) σ1 σ2 :
    t1 ≡ₚ t1' erased_step (t1,σ1) (t2,σ2) t2', t2 ≡ₚ t2' erased_step (t1',σ1) (t2',σ2).
  Proof.
    intros Heq [? Hs]. pose proof (step_Permutation _ _ _ _ _ _ Heq Hs). firstorder.
  Qed.

  Record pure_step (e1 e2 : expr Λ) := {
    pure_step_safe σ1 : reducible_no_obs e1 σ1;
    pure_step_det σ1 κ e2' σ2 efs :
      prim_step e1 σ1 κ e2' σ2 efs κ = [] σ2 = σ1 e2' = e2 efs = []
  }.

  Notation pure_steps_tp := (Forall2 (rtc pure_step)).

  Class PureExec (φ : Prop) (n : nat) (e1 e2 : expr Λ) :=
    pure_exec : φ relations.nsteps pure_step n e1 e2.

  Lemma pure_step_ctx K `{!@LanguageCtx Λ K} e1 e2 :
    pure_step e1 e2
    pure_step (K e1) (K e2).
  Proof.
    intros [Hred Hstep]. split.
    - unfold reducible_no_obs in ×. naive_solver eauto using fill_step.
    - intros σ1 κ e2' σ2 efs Hpstep.
      destruct (fill_step_inv e1 σ1 κ e2' σ2 efs) as (e2'' & → & ?); [|exact Hpstep|].
      + destruct (Hred σ1) as (? & ? & ? & ?); eauto using val_stuck.
      + edestruct (Hstep σ1 κ e2'' σ2 efs) as (? & → & → & ->); auto.
  Qed.

  Lemma pure_step_nsteps_ctx K `{!@LanguageCtx Λ K} n e1 e2 :
    relations.nsteps pure_step n e1 e2
    relations.nsteps pure_step n (K e1) (K e2).
  Proof. eauto using nsteps_congruence, pure_step_ctx. Qed.

  Lemma rtc_pure_step_ctx K `{!@LanguageCtx Λ K} e1 e2 :
    rtc pure_step e1 e2 rtc pure_step (K e1) (K e2).
  Proof. eauto using rtc_congruence, pure_step_ctx. Qed.

  Lemma pure_exec_ctx K `{!@LanguageCtx Λ K} φ n e1 e2 :
    PureExec φ n e1 e2
    PureExec φ n (K e1) (K e2).
  Proof. rewrite /PureExec; eauto using pure_step_nsteps_ctx. Qed.

  Class IntoVal (e : expr Λ) (v : val Λ) :=
    into_val : of_val v = e.

  Class AsVal (e : expr Λ) := as_val : v, of_val v = e.
  Global Instance as_vals_of_val vs : TCForall AsVal (of_val <$> vs).
  Proof.
    apply TCForall_Forall, Forall_fmap, Forall_truev.
    rewrite /AsVal /=; eauto.
  Qed.

  Lemma as_val_is_Some e :
    ( v, of_val v = e) is_Some (to_val e).
  Proof. intros [v <-]. rewrite to_of_val. eauto. Qed.

  Lemma prim_step_not_stuck e σ κ e' σ' efs :
    prim_step e σ κ e' σ' efs not_stuck e σ.
  Proof. rewrite /not_stuck /reducible. eauto 10. Qed.

  Lemma rtc_pure_step_val `{!Inhabited (state Λ)} v e :
    rtc pure_step (of_val v) e to_val e = Some v.
  Proof.
    intros ?; rewrite <- to_of_val.
    f_equal; symmetry; eapply rtc_nf; first done.
    intros [e' [Hstep _]].
    destruct (Hstep inhabitant) as (?&?&?&Hval%val_stuck).
    by rewrite to_of_val in Hval.
  Qed.

Let thread pools t1 and t3 be such that each thread in t1 makes (zero or more) pure steps to the corresponding thread in t3. Furthermore, let t2 be a thread pool such that t1 under state σ1 makes a (single) step to thread pool t2 and state σ2. In this situation, either the step from t1 to t2 corresponds to one of the pure steps between t1 and t3, or, there is an i such that ith thread does not participate in the pure steps between t1 and t3 and t2 corresponds to taking a step in the ith thread starting from t1.
  Lemma erased_step_pure_step_tp t1 σ1 t2 σ2 t3 :
    erased_step (t1, σ1) (t2, σ2)
    pure_steps_tp t1 t3
    (σ1 = σ2 pure_steps_tp t2 t3)
    ( i e efs e' κ,
      t1 !! i = Some e t3 !! i = Some e
      t2 = <[i:=e']>t1 ++ efs
      prim_step e σ1 κ e' σ2 efs).
  Proof.
    intros [κ [e σ e' σ' ? t11 t12 ?? Hstep]] Hps; simplify_eq/=.
    apply Forall2_app_inv_l in Hps
      as (t31&?&Hpsteps&(e''&t32&Hps&?&->)%Forall2_cons_inv_l&->).
    destruct Hps as [e|e1 e2 e3 [_ Hprs]].
    - right.
       (length t11), e, efs, e', κ; split_and!; last done.
      + by rewrite lookup_app_r // Nat.sub_diag.
      + apply Forall2_length in Hpsteps.
        by rewrite lookup_app_r Hpsteps // Nat.sub_diag.
      + by rewrite insert_app_r_alt // Nat.sub_diag /= -assoc_L.
    - edestruct Hprs as (?&?&?&?); first done; simplify_eq.
      left; split; first done.
      rewrite right_id_L.
      eauto using Forall2_app.
  Qed.

End language.

Notation pure_steps_tp := (Forall2 (rtc pure_step)).