RTLPar Generation

#[local] Open Scope positive.
#[local] Open Scope forest.
#[local] Open Scope pred_op.

Abstract Computations

Define the abstract computation using the update function, which will set each register to its symbolic value. First we need to define a few helper functions to correctly translate the predicates.

Fixpoint list_translation (l : list reg) (f : forest) {struct l} : list pred_expr :=
  match l with
  | nil => nil
  | i :: l => (f # (Reg i)) :: (list_translation l f)

Fixpoint replicate {A} (n: nat) (l: A) :=
  match n with
  | O => nil
  | S n => l :: replicate n l

Definition merge''' x y :=
  match x, y with
  | Some p1, Some p2 => Some (Pand p1 p2)
  | Some p, None | None, Some p => Some p
  | None, None => None

Definition merge'' x :=
  match x with
  | ((a, e), (b, el)) => (merge''' a b, Econs e el)

Definition map_pred_op {A B} (pf: option pred_op * (A -> B)) (pa: option pred_op * A): option pred_op * B :=
  match pa, pf with
  | (p, a), (p', f) => (merge''' p p', f a)

Definition predicated_prod {A B: Type} (p1: predicated A) (p2: predicated B) := (fun x => match x with ((a, b), (c, d)) => (Pand a c, (b, d)) end)
         (NE.non_empty_prod p1 p2).

Definition predicated_map {A B: Type} (f: A -> B) (p: predicated A): predicated B := (fun x => (fst x, f (snd x))) p.

Definition merge' (pel: pred_expr) (tpel: predicated expression_list) :=
  predicated_map (uncurry Econs) (predicated_prod pel tpel).

Fixpoint merge (pel: list pred_expr): predicated expression_list :=
  match pel with
  | nil => NE.singleton (T, Enil)
  | a :: b => merge' a (merge b)

Definition map_predicated {A B} (pf: predicated (A -> B)) (pa: predicated A): predicated B :=
  predicated_map (fun x => (fst x) (snd x)) (predicated_prod pf pa).

Definition predicated_apply1 {A B} (pf: predicated (A -> B)) (pa: A): predicated B := (fun x => (fst x, (snd x) pa)) pf.

Definition predicated_apply2 {A B C} (pf: predicated (A -> B -> C)) (pa: A) (pb: B): predicated C := (fun x => (fst x, (snd x) pa pb)) pf.

Definition predicated_apply3 {A B C D} (pf: predicated (A -> B -> C -> D)) (pa: A) (pb: B) (pc: C): predicated D := (fun x => (fst x, (snd x) pa pb pc)) pf.

Definition predicated_from_opt {A: Type} (p: option pred_op) (a: A) :=
  match p with
  | Some p' => NE.singleton (p', a)
  | None => NE.singleton (T, a)

#[local] Open Scope non_empty_scope.
#[local] Open Scope pred_op.

Fixpoint NEfold_left {A B} (f: A -> B -> A) (l: NE.non_empty B) (a: A) : A :=
  match l with
  | NE.singleton a' => f a a'
  | a' ::| b => NEfold_left f b (f a a')

Fixpoint NEapp {A} (l m: NE.non_empty A) :=
  match l with
  | NE.singleton a => a ::| m
  | a ::| b => a ::| NEapp b m

Definition app_predicated' {A: Type} (a b: predicated A) :=
  let negation := ¬ (NEfold_left (fun a b => a (fst b)) b ) in
  NEapp ( (fun x => (negation fst x, snd x)) a) b.

Definition app_predicated {A: Type} (p: option pred_op) (a b: predicated A) :=
  match p with
  | Some p' => NEapp ( (fun x => (¬ p' fst x, snd x)) a)
                     ( (fun x => (p' fst x, snd x)) b)
  | None => b

Definition pred_ret {A: Type} (a: A) : predicated A :=
  NE.singleton (T, a).

Update Function

The update function will generate a new forest given an existing forest and a new instruction, so that it can evaluate a symbolic expression by folding over a list of instructions. The main problem is that predicates need to be merged as well, so that:
1. The predicates are *independent*. 2. The expression assigned to the register should still be correct.
This is done by multiplying the predicates together, and assigning the negation of the expression to the other predicates.

Definition update (f : forest) (i : instr) : forest :=
  match i with
  | RBnop => f
  | RBop p op rl r =>
    f # (Reg r) <-
    (app_predicated p
       (f # (Reg r))
       (map_predicated (pred_ret (Eop op)) (merge (list_translation rl f))))
  | RBload p chunk addr rl r =>
    f # (Reg r) <-
      (app_predicated p
         (f # (Reg r))
            (map_predicated (pred_ret (Eload chunk addr)) (merge (list_translation rl f)))
            (f # Mem)))
  | RBstore p chunk addr rl r =>
    f # Mem <-
      (app_predicated p
         (f # Mem)
               (predicated_apply2 (map_predicated (pred_ret Estore) (f # (Reg r))) chunk addr)
               (merge (list_translation rl f))) (f # Mem)))
  | RBsetpred p' c args p =>
    f # (Pred p) <-
    (app_predicated p'
       (f # (Pred p))
       (map_predicated (pred_ret (Esetpred c)) (merge (list_translation args f))))

Implementing which are necessary to show the correctness of the translation validation by showing that there aren't any more effects in the resultant RTLPar code than in the RTLBlock code.
Get a sequence from the basic block.

Fixpoint abstract_sequence (f : forest) (b : list instr) : forest :=
  match b with
  | nil => f
  | i :: l => abstract_sequence (update f i) l

Check equivalence of control flow instructions. As none of the basic blocks should have been moved, none of the labels should be different, meaning the control-flow instructions should match exactly.

Definition check_control_flow_instr (c1 c2: cf_instr) : bool :=
  if cf_instr_eq c1 c2 then true else false.

We define the top-level oracle that will check if two basic blocks are equivalent after a scheduling transformation.

Definition empty_trees (bb: (bbt: : bool :=
  match bb with
  | nil =>
    match bbt with
    | nil => true
    | _ => false
  | _ => true

Definition schedule_oracle (bb: RTLBlock.bblock) (bbt: RTLPar.bblock) : bool :=
  check (abstract_sequence empty (bb_body bb))
        (abstract_sequence empty (concat (concat (bb_body bbt)))) &&
  check_control_flow_instr (bb_exit bb) (bb_exit bbt) &&
  empty_trees (bb_body bb) (bb_body bbt).

Definition check_scheduled_trees := beq2 schedule_oracle.

Lemma check_scheduled_trees_correct:
  forall f1 f2 x y1,
    check_scheduled_trees f1 f2 = true ->
    PTree.get x f1 = Some y1 ->
    exists y2, PTree.get x f2 = Some y2 /\ schedule_oracle y1 y2 = true.

Lemma check_scheduled_trees_correct2:
  forall f1 f2 x,
    check_scheduled_trees f1 f2 = true ->
    PTree.get x f1 = None ->
    PTree.get x f2 = None.

Top-level Functions

Parameter schedule : RTLBlock.function -> RTLPar.function.

Definition transl_function (f: RTLBlock.function) : Errors.res RTLPar.function :=
  let tfcode := fn_code (schedule f) in
  if check_scheduled_trees f.(fn_code) tfcode then
    Errors.OK (mkfunction f.(fn_sig)
    Errors.Error (Errors.msg "RTLPargen: Could not prove the blocks equivalent.").

Definition transl_fundef := transf_partial_fundef transl_function.

Definition transl_program (p : RTLBlock.program) : Errors.res RTLPar.program :=
  transform_partial_program transl_fundef p.