lang.v

From program_logic Require Export language.
From prelude Require Export stringmap.
From prelude Require Import gmap.

Module heap_lang.
Open Scope Z_scope.

(** Expressions and vals. *)
Definition loc := positive. (* Really, any countable type. *)

Inductive base_lit : Set :=
  | LitInt (n : Z) | LitBool (b : bool) | LitUnit.
Inductive un_op : Set :=
  | NegOp | MinusUnOp.
Inductive bin_op : Set :=
  | PlusOp | MinusOp | LeOp | LtOp | EqOp.

Inductive binder := BAnon | BNamed : string → binder.

Delimit Scope binder_scope with binder.
Bind Scope binder_scope with binder.

Inductive expr :=
  (* Base lambda calculus *)
  | Var (x : string)
  | Rec (f x : binder) (e : expr)
  | App (e1 e2 : expr)
  (* Base types and their operations *)
  | Lit (l : base_lit)
  | UnOp (op : un_op) (e : expr)
  | BinOp (op : bin_op) (e1 e2 : expr)
  | If (e0 e1 e2 : expr)
  (* Products *)
  | Pair (e1 e2 : expr)
  | Fst (e : expr)
  | Snd (e : expr)
  (* Sums *)
  | InjL (e : expr)
  | InjR (e : expr)
  | Case (e0 : expr) (e1 : expr) (e2 : expr)
  (* Concurrency *)
  | Fork (e : expr)
  (* Heap *)
  | Loc (l : loc)
  | Alloc (e : expr)
  | Load (e : expr)
  | Store (e1 : expr) (e2 : expr)
  | Cas (e0 : expr) (e1 : expr) (e2 : expr).

Bind Scope expr_scope with expr.
Delimit Scope expr_scope with E.

Inductive val :=
  | RecV (f x : binder) (e : expr) (* e should be closed *)
  | LitV (l : base_lit)
  | PairV (v1 v2 : val)
  | InjLV (v : val)
  | InjRV (v : val)
  | LocV (l : loc).

Bind Scope val_scope with val.
Delimit Scope val_scope with V.

Global Instance base_lit_dec_eq (l1 l2 : base_lit) : Decision (l1 = l2).
Proof. solve_decision. Defined.
Global Instance un_op_dec_eq (op1 op2 : un_op) : Decision (op1 = op2).
Proof. solve_decision. Defined.
Global Instance bin_op_dec_eq (op1 op2 : bin_op) : Decision (op1 = op2).
Proof. solve_decision. Defined.
Global Instance binder_dec_eq (x1 x2 : binder) : Decision (x1 = x2).
Proof. solve_decision. Defined.
Global Instance expr_dec_eq (e1 e2 : expr) : Decision (e1 = e2).
Proof. solve_decision. Defined.
Global Instance val_dec_eq (v1 v2 : val) : Decision (v1 = v2).
Proof. solve_decision. Defined.

Fixpoint of_val (v : val) : expr :=
  match v with
  | RecV f x e => Rec f x e
  | LitV l => Lit l
  | PairV v1 v2 => Pair (of_val v1) (of_val v2)
  | InjLV v => InjL (of_val v)
  | InjRV v => InjR (of_val v)
  | LocV l => Loc l
  end.
Fixpoint to_val (e : expr) : option val :=
  match e with
  | Rec f x e => Some (RecV f x e)
  | Lit l => Some (LitV l)
  | Pair e1 e2 => v1 ← to_val e1; v2 ← to_val e2; Some (PairV v1 v2)
  | InjL e => InjLV <$> to_val e
  | InjR e => InjRV <$> to_val e
  | Loc l => Some (LocV l)
  | _ => None
  end.

(** The state: heaps of vals. *)
Definition state := gmap loc val.

(** Evaluation contexts *)
Inductive ectx_item :=
  | AppLCtx (e2 : expr)
  | AppRCtx (v1 : val)
  | UnOpCtx (op : un_op)
  | BinOpLCtx (op : bin_op) (e2 : expr)
  | BinOpRCtx (op : bin_op) (v1 : val)
  | IfCtx (e1 e2 : expr)
  | PairLCtx (e2 : expr)
  | PairRCtx (v1 : val)
  | FstCtx
  | SndCtx
  | InjLCtx
  | InjRCtx
  | CaseCtx (e1 : expr) (e2 : expr)
  | AllocCtx
  | LoadCtx
  | StoreLCtx (e2 : expr)
  | StoreRCtx (v1 : val)
  | CasLCtx (e1 : expr)  (e2 : expr)
  | CasMCtx (v0 : val) (e2 : expr)
  | CasRCtx (v0 : val) (v1 : val).

Notation ectx := (list ectx_item).

Definition fill_item (Ki : ectx_item) (e : expr) : expr :=
  match Ki with
  | AppLCtx e2 => App e e2
  | AppRCtx v1 => App (of_val v1) e
  | UnOpCtx op => UnOp op e
  | BinOpLCtx op e2 => BinOp op e e2
  | BinOpRCtx op v1 => BinOp op (of_val v1) e
  | IfCtx e1 e2 => If e e1 e2
  | PairLCtx e2 => Pair e e2
  | PairRCtx v1 => Pair (of_val v1) e
  | FstCtx => Fst e
  | SndCtx => Snd e
  | InjLCtx => InjL e
  | InjRCtx => InjR e
  | CaseCtx e1 e2 => Case e e1 e2
  | AllocCtx => Alloc e
  | LoadCtx => Load e
  | StoreLCtx e2 => Store e e2
  | StoreRCtx v1 => Store (of_val v1) e
  | CasLCtx e1 e2 => Cas e e1 e2
  | CasMCtx v0 e2 => Cas (of_val v0) e e2
  | CasRCtx v0 v1 => Cas (of_val v0) (of_val v1) e
  end.
Definition fill (K : ectx) (e : expr) : expr := fold_right fill_item e K.

(** Substitution *)
(** We have [subst' e BAnon v = e] to deal with anonymous binders *)
Fixpoint subst (e : expr) (x : string) (v : val) : expr :=
  match e with
  | Var y => if decide (x = y) then of_val v else Var y
  | Rec f y e =>
     Rec f y (if decide (BNamed x ≠ f ∧ BNamed x ≠ y) then subst e x v else e)
  | App e1 e2 => App (subst e1 x v) (subst e2 x v)
  | Lit l => Lit l
  | UnOp op e => UnOp op (subst e x v)
  | BinOp op e1 e2 => BinOp op (subst e1 x v) (subst e2 x v)
  | If e0 e1 e2 => If (subst e0 x v) (subst e1 x v) (subst e2 x v)
  | Pair e1 e2 => Pair (subst e1 x v) (subst e2 x v)
  | Fst e => Fst (subst e x v)
  | Snd e => Snd (subst e x v)
  | InjL e => InjL (subst e x v)
  | InjR e => InjR (subst e x v)
  | Case e0 e1 e2 =>
     Case (subst e0 x v) (subst e1 x v) (subst e2 x v)
  | Fork e => Fork (subst e x v)
  | Loc l => Loc l
  | Alloc e => Alloc (subst e x v)
  | Load e => Load (subst e x v)
  | Store e1 e2 => Store (subst e1 x v) (subst e2 x v)
  | Cas e0 e1 e2 => Cas (subst e0 x v) (subst e1 x v) (subst e2 x v)
  end.
Definition subst' (e : expr) (mx : binder) (v : val) : expr :=
  match mx with BNamed x => subst e x v | BAnon => e end.

(** The stepping relation *)
Definition un_op_eval (op : un_op) (l : base_lit) : option base_lit :=
  match op, l with
  | NegOp, LitBool b => Some (LitBool (negb b))
  | MinusUnOp, LitInt n => Some (LitInt (- n))
  | _, _ => None
  end.

Definition bin_op_eval (op : bin_op) (l1 l2 : base_lit) : option base_lit :=
  match op, l1, l2 with
  | PlusOp, LitInt n1, LitInt n2 => Some $ LitInt (n1 + n2)
  | MinusOp, LitInt n1, LitInt n2 => Some $ LitInt (n1 - n2)
  | LeOp, LitInt n1, LitInt n2 => Some $ LitBool $ bool_decide (n1 ≤ n2)
  | LtOp, LitInt n1, LitInt n2 => Some $ LitBool $ bool_decide (n1 < n2)
  | EqOp, LitInt n1, LitInt n2 => Some $ LitBool $ bool_decide (n1 = n2)
  | _, _, _ => None
  end.

Inductive head_step : expr → state → expr → state → option expr → Prop :=
  | BetaS f x e1 e2 v2 σ :
     to_val e2 = Some v2 →
     head_step (App (Rec f x e1) e2) σ
       (subst' (subst' e1 f (RecV f x e1)) x v2) σ None
  | UnOpS op l l' σ :
     un_op_eval op l = Some l' → 
     head_step (UnOp op (Lit l)) σ (Lit l') σ None
  | BinOpS op l1 l2 l' σ :
     bin_op_eval op l1 l2 = Some l' → 
     head_step (BinOp op (Lit l1) (Lit l2)) σ (Lit l') σ None
  | IfTrueS e1 e2 σ :
     head_step (If (Lit $ LitBool true) e1 e2) σ e1 σ None
  | IfFalseS e1 e2 σ :
     head_step (If (Lit $ LitBool false) e1 e2) σ e2 σ None
  | FstS e1 v1 e2 v2 σ :
     to_val e1 = Some v1 → to_val e2 = Some v2 →
     head_step (Fst (Pair e1 e2)) σ e1 σ None
  | SndS e1 v1 e2 v2 σ :
     to_val e1 = Some v1 → to_val e2 = Some v2 →
     head_step (Snd (Pair e1 e2)) σ e2 σ None
  | CaseLS e0 v0 e1 e2 σ :
     to_val e0 = Some v0 →
     head_step (Case (InjL e0) e1 e2) σ (App e1 e0) σ None
  | CaseRS e0 v0 e1 e2 σ :
     to_val e0 = Some v0 →
     head_step (Case (InjR e0) e1 e2) σ (App e2 e0) σ None
  | ForkS e σ:
     head_step (Fork e) σ (Lit LitUnit) σ (Some e)
  | AllocS e v σ l :
     to_val e = Some v → σ !! l = None →
     head_step (Alloc e) σ (Loc l) (<[l:=v]>σ) None
  | LoadS l v σ :
     σ !! l = Some v →
     head_step (Load (Loc l)) σ (of_val v) σ None
  | StoreS l e v σ :
     to_val e = Some v → is_Some (σ !! l) →
     head_step (Store (Loc l) e) σ (Lit LitUnit) (<[l:=v]>σ) None
  | CasFailS l e1 v1 e2 v2 vl σ :
     to_val e1 = Some v1 → to_val e2 = Some v2 →
     σ !! l = Some vl → vl ≠ v1 →
     head_step (Cas (Loc l) e1 e2) σ (Lit $ LitBool false) σ None
  | CasSucS l e1 v1 e2 v2 σ :
     to_val e1 = Some v1 → to_val e2 = Some v2 →
     σ !! l = Some v1 →
     head_step (Cas (Loc l) e1 e2) σ (Lit $ LitBool true) (<[l:=v2]>σ) None.

(** Atomic expressions *)
Definition atomic (e: expr) : Prop :=
  match e with
  | Alloc e => is_Some (to_val e)
  | Load e => is_Some (to_val e)
  | Store e1 e2 => is_Some (to_val e1) ∧ is_Some (to_val e2)
  | Cas e0 e1 e2 => is_Some (to_val e0) ∧ is_Some (to_val e1) ∧ is_Some (to_val e2)
  (* Make "skip" atomic *)
  | App (Rec _ _ (Lit _)) (Lit _) => True
  | _ => False
  end.

(** Close reduction under evaluation contexts.
We could potentially make this a generic construction. *)
Inductive prim_step
    (e1 : expr) (σ1 : state) (e2 : expr) (σ2: state) (ef: option expr) : Prop :=
  Ectx_step K e1' e2' :
    e1 = fill K e1' → e2 = fill K e2' →
    head_step e1' σ1 e2' σ2 ef → prim_step e1 σ1 e2 σ2 ef.

(** Basic properties about the language *)
Lemma to_of_val v : to_val (of_val v) = Some v.
Proof. by induction v; simplify_option_eq. Qed.

Lemma of_to_val e v : to_val e = Some v → of_val v = e.
Proof.
  revert v; induction e; intros; simplify_option_eq; auto with f_equal.
Qed.

Instance: Inj (=) (=) of_val.
Proof. by intros ?? Hv; apply (inj Some); rewrite -!to_of_val Hv. Qed.

Instance fill_item_inj Ki : Inj (=) (=) (fill_item Ki).
Proof. destruct Ki; intros ???; simplify_eq/=; auto with f_equal. Qed.

Instance ectx_fill_inj K : Inj (=) (=) (fill K).
Proof. red; induction K as [|Ki K IH]; naive_solver. Qed.

Lemma fill_app K1 K2 e : fill (K1 ++ K2) e = fill K1 (fill K2 e).
Proof. revert e; induction K1; simpl; auto with f_equal. Qed.

Lemma fill_val K e : is_Some (to_val (fill K e)) → is_Some (to_val e).
Proof.
  intros [v' Hv']; revert v' Hv'.
  induction K as [|[]]; intros; simplify_option_eq; eauto.
Qed.

Lemma fill_not_val K e : to_val e = None → to_val (fill K e) = None.
Proof. rewrite !eq_None_not_Some; eauto using fill_val. Qed.

Lemma val_head_stuck e1 σ1 e2 σ2 ef :
  head_step e1 σ1 e2 σ2 ef → to_val e1 = None.
Proof. destruct 1; naive_solver. Qed.

Lemma val_stuck e1 σ1 e2 σ2 ef : prim_step e1 σ1 e2 σ2 ef → to_val e1 = None.
Proof. intros [??? -> -> ?]; eauto using fill_not_val, val_head_stuck. Qed.

Lemma atomic_not_val e : atomic e → to_val e = None.
Proof. destruct e; naive_solver. Qed.

Lemma atomic_fill_item Ki e : atomic (fill_item Ki e) → is_Some (to_val e).
Proof.
  intros. destruct Ki; simplify_eq/=; destruct_and?;
    repeat (case_match || contradiction); eauto.
Qed.

Lemma atomic_fill K e : atomic (fill K e) → to_val e = None → K = [].
Proof.
  destruct K as [|Ki K]; [done|].
  rewrite eq_None_not_Some=> /= ? []; eauto using atomic_fill_item, fill_val.
Qed.

Lemma atomic_head_step e1 σ1 e2 σ2 ef :
  atomic e1 → head_step e1 σ1 e2 σ2 ef → is_Some (to_val e2).
Proof.
  destruct 2; simpl; rewrite ?to_of_val; try by eauto.
  unfold subst'; repeat (case_match || contradiction || simplify_eq/=); eauto.
Qed.

Lemma atomic_step e1 σ1 e2 σ2 ef :
  atomic e1 → prim_step e1 σ1 e2 σ2 ef → is_Some (to_val e2).
Proof.
  intros Hatomic [K e1' e2' -> -> Hstep].
  assert (K = []) as -> by eauto 10 using atomic_fill, val_head_stuck.
  naive_solver eauto using atomic_head_step.
Qed.

Lemma head_ctx_step_val Ki e σ1 e2 σ2 ef :
  head_step (fill_item Ki e) σ1 e2 σ2 ef → is_Some (to_val e).
Proof. destruct Ki; inversion_clear 1; simplify_option_eq; eauto. Qed.

Lemma fill_item_no_val_inj Ki1 Ki2 e1 e2 :
  to_val e1 = None → to_val e2 = None →
  fill_item Ki1 e1 = fill_item Ki2 e2 → Ki1 = Ki2.
Proof.
  destruct Ki1, Ki2; intros; try discriminate; simplify_eq/=;
    repeat match goal with
    | H : to_val (of_val _) = None |- _ => by rewrite to_of_val in H
    end; auto.
Qed.

(* When something does a step, and another decomposition of the same expression
has a non-val [e] in the hole, then [K] is a left sub-context of [K'] - in
other words, [e] also contains the reducible expression *)
Lemma step_by_val K K' e1 e1' σ1 e2 σ2 ef :
  fill K e1 = fill K' e1' → to_val e1 = None → head_step e1' σ1 e2 σ2 ef →
  K `prefix_of` K'.
Proof.
  intros Hfill Hred Hnval; revert K' Hfill.
  induction K as [|Ki K IH]; simpl; intros K' Hfill; auto using prefix_of_nil.
  destruct K' as [|Ki' K']; simplify_eq/=.
  { exfalso; apply (eq_None_not_Some (to_val (fill K e1)));
      eauto using fill_not_val, head_ctx_step_val. }
  cut (Ki = Ki'); [naive_solver eauto using prefix_of_cons|].
  eauto using fill_item_no_val_inj, val_head_stuck, fill_not_val.
Qed.

Lemma alloc_fresh e v σ :
  let l := fresh (dom _ σ) in
  to_val e = Some v → head_step (Alloc e) σ (Loc l) (<[l:=v]>σ) None.
Proof. by intros; apply AllocS, (not_elem_of_dom (D:=gset _)), is_fresh. Qed.

(** Closed expressions *)
Definition of_binder (mx : binder) : stringset :=
  match mx with BAnon => ∅ | BNamed x => {[ x ]} end.
Lemma elem_of_of_binder x mx: x ∈ of_binder mx ↔ mx = BNamed x.
Proof. destruct mx; set_solver. Qed.
Global Instance set_unfold_of_binder (mx : binder) x :
  SetUnfold (x ∈ of_binder mx) (mx = BNamed x).
Proof. constructor; destruct mx; set_solver. Qed.

Fixpoint is_closed (X : stringset) (e : expr) : bool :=
  match e with
  | Var x => bool_decide (x ∈ X)
  | Rec f y e => is_closed (of_binder f ∪ of_binder y ∪ X) e
  | App e1 e2 => is_closed X e1 && is_closed X e2
  | Lit l => true
  | UnOp _ e => is_closed X e
  | BinOp _ e1 e2 => is_closed X e1 && is_closed X e2
  | If e0 e1 e2 => is_closed X e0 && is_closed X e1 && is_closed X e2
  | Pair e1 e2 => is_closed X e1 && is_closed X e2
  | Fst e => is_closed X e
  | Snd e => is_closed X e
  | InjL e => is_closed X e
  | InjR e => is_closed X e
  | Case e0 e1 e2 => is_closed X e0 && is_closed X e1 && is_closed X e2
  | Fork e => is_closed X e
  | Loc l => true
  | Alloc e => is_closed X e
  | Load e => is_closed X e
  | Store e1 e2 => is_closed X e1 && is_closed X e2
  | Cas e0 e1 e2 => is_closed X e0 && is_closed X e1 && is_closed X e2
  end.
Class Closed (e : expr) := closed : ∀ x v, subst e x v = e.

Lemma is_closed_subst_same X e x v : is_closed X e → x ∉ X → subst e x v = e.
Proof.
  revert X. induction e; simpl;
    repeat case_decide; naive_solver eauto with f_equal set_solver.
Qed.
Lemma is_closed_Closed e : is_closed ∅ e → Closed e.
Proof. intros ? x v. by eapply is_closed_subst_same, not_elem_of_empty. Qed.

Ltac solve_closed := apply is_closed_Closed; vm_compute; exact I.

Lemma is_closed_weaken X Y e : is_closed X e → X ⊆ Y → is_closed Y e.
Proof.
  revert X Y.
  induction e; simpl; intros; naive_solver eauto with f_equal set_solver.
Qed.

Lemma is_closed_fill_item X Ki e : is_closed X (fill_item Ki e) → is_closed X e.
Proof. destruct Ki; rewrite /= ?andb_True; tauto. Qed.
Lemma is_closed_fill X K e : is_closed X (fill K e) → is_closed X e.
Proof. induction K; simpl; eauto using is_closed_fill_item. Qed.

Lemma is_closed_subst X e x v :
  is_closed ({[x]} ∪ X) e → is_closed ∅ (of_val v) → is_closed X (subst e x v).
Proof.
  revert X. elim: e; simpl; try by intros; destruct_and?; split_and?; auto.
  - intros y X; rewrite bool_decide_spec=>??.
    case_decide; subst; last set_solver.
    eauto using is_closed_weaken with set_solver.
  - intros f y e IH X ??. case_decide as Hx.
    + apply IH; clear IH; eauto using is_closed_weaken with set_solver.
    + clear IH; apply not_and_l in Hx;
        destruct Hx as [Hx|Hx]; apply dec_stable in Hx; subst;
        eauto using is_closed_weaken with set_solver.
Qed.
Lemma is_closed_subst' X e x v :
  is_closed (of_binder x ∪ X) e → is_closed ∅ (of_val v) →
  is_closed X (subst' e x v).
Proof. destruct x; rewrite /= ?left_id_L; auto using is_closed_subst. Qed.
Lemma is_closed_head_step e1 σ1 e2 σ2 ef :
  (* for load, the state should be closed *)
  match e1 with Load _ => False | _ => True end →
  head_step e1 σ1 e2 σ2 ef → is_closed ∅ e1 → is_closed ∅ e2.
Proof.
  destruct 2; rewrite /= ?andb_True; try
    match goal with H : to_val _ = _ |- _ => apply of_to_val in H; subst end;
    try tauto.
  intros [??]. repeat apply is_closed_subst'; rewrite /= ?assoc_L; auto.
Qed.
End heap_lang.

(** Language *)
Program Canonical Structure heap_lang : language := {|
  expr := heap_lang.expr; val := heap_lang.val; state := heap_lang.state;
  of_val := heap_lang.of_val; to_val := heap_lang.to_val;
  atomic := heap_lang.atomic; prim_step := heap_lang.prim_step;
|}.
Solve Obligations with eauto using heap_lang.to_of_val, heap_lang.of_to_val,
  heap_lang.val_stuck, heap_lang.atomic_not_val, heap_lang.atomic_step.

Global Instance heap_lang_ctx K : LanguageCtx heap_lang (heap_lang.fill K).
Proof.
  split.
  - eauto using heap_lang.fill_not_val.
  - intros ????? [K' e1' e2' Heq1 Heq2 Hstep].
    by exists (K ++ K') e1' e2'; rewrite ?heap_lang.fill_app ?Heq1 ?Heq2.
  - intros e1 σ1 e2 σ2 ? Hnval [K'' e1'' e2'' Heq1 -> Hstep].
    destruct (heap_lang.step_by_val
      K K'' e1 e1'' σ1 e2'' σ2 ef) as [K' ->]; eauto.
    rewrite heap_lang.fill_app in Heq1; apply (inj _) in Heq1.
    exists (heap_lang.fill K' e2''); rewrite heap_lang.fill_app; split; auto.
    econstructor; eauto.
Qed.

Global Instance heap_lang_ctx_item Ki :
  LanguageCtx heap_lang (heap_lang.fill_item Ki).
Proof. change (LanguageCtx heap_lang (heap_lang.fill [Ki])). apply _. Qed.