(** **** Exercise: 3 stars, advanced, optional (not_congr)  *)
(** We've shown that the [cequiv] relation is both an equivalence and
    a congruence on commands.  Can you think of a relation on commands
    that is an equivalence but _not_ a congruence? *)

(* FILL IN HERE *)
(** [] *)

(* ################################################################# *)
(** * Program Transformations *)

(** A _program transformation_ is a function that takes a program as
    input and produces some variant of the program as output.
    Compiler optimizations such as constant folding are a canonical
    example, but there are many others. *)

(** A program transformation is _sound_ if it preserves the
    behavior of the original program. *)

Definition atrans_sound (atrans : aexp -> aexp) : Prop :=
  forall (a : aexp),
    aequiv a (atrans a).Definition btrans_sound (btrans : bexp -> bexp) : Prop :=
  forall (b : bexp),
    bequiv b (btrans b).Definition ctrans_sound (ctrans : com -> com) : Prop :=
  forall (c : com),
    cequiv c (ctrans c).(* ================================================================= *)
(** ** The Constant-Folding Transformation *)

(** An expression is _constant_ when it contains no variable
    references.

    Constant folding is an optimization that finds constant
    expressions and replaces them by their values. *)

Fixpoint fold_constants_aexp (a : aexp) : aexp :=
  match a with
  | ANum n       => ANum n
  | AId i        => AId i
  | APlus a1 a2  =>
    match (fold_constants_aexp a1, fold_constants_aexp a2)
    with
    | (ANum n1, ANum n2) => ANum (n1 + n2)
    | (a1', a2') => APlus a1' a2'
    end
  | AMinus a1 a2 =>
    match (fold_constants_aexp a1, fold_constants_aexp a2)
    with
    | (ANum n1, ANum n2) => ANum (n1 - n2)
    | (a1', a2') => AMinus a1' a2'
    end
  | AMult a1 a2  =>
    match (fold_constants_aexp a1, fold_constants_aexp a2)
    with
    | (ANum n1, ANum n2) => ANum (n1 * n2)
    | (a1', a2') => AMult a1' a2'
    end
  end.Example fold_aexp_ex1 :
    fold_constants_aexp
      (AMult (APlus (ANum 1) (ANum 2)) (AId X))
  = AMult (ANum 3) (AId X).(** Note that this version of constant folding doesn't eliminate
    trivial additions, etc. -- we are focusing attention on a single
    optimization for the sake of simplicity.  It is not hard to
    incorporate other ways of simplifying expressions; the definitions
    and proofs just get longer. *)

Example fold_aexp_ex2 :
    fold_constants_aexp
      (AMinus (AId X) (APlus (AMult (ANum 0) (ANum 6))
                             (AId Y)))
  = AMinus (AId X) (APlus (ANum 0) (AId Y)).(** Not only can we lift [fold_constants_aexp] to [bexp]s (in the
    [BEq] and [BLe] cases); we can also look for constant _boolean_
    expressions and evaluate them in-place. *)

Fixpoint fold_constants_bexp (b : bexp) : bexp :=
  match b with
  | BTrue        => BTrue
  | BFalse       => BFalse
  | BEq a1 a2  =>
      match (fold_constants_aexp a1, fold_constants_aexp a2) with
      | (ANum n1, ANum n2) =>
          if beq_nat n1 n2 then BTrue else BFalse
      | (a1', a2') =>
          BEq a1' a2'
      end
  | BLe a1 a2  =>
      match (fold_constants_aexp a1, fold_constants_aexp a2) with
      | (ANum n1, ANum n2) =>
          if leb n1 n2 then BTrue else BFalse
      | (a1', a2') =>
          BLe a1' a2'
      end
  | BNot b1  =>
      match (fold_constants_bexp b1) with
      | BTrue => BFalse
      | BFalse => BTrue
      | b1' => BNot b1'
      end
  | BAnd b1 b2  =>
      match (fold_constants_bexp b1, fold_constants_bexp b2) with
      | (BTrue, BTrue) => BTrue
      | (BTrue, BFalse) => BFalse
      | (BFalse, BTrue) => BFalse
      | (BFalse, BFalse) => BFalse
      | (b1', b2') => BAnd b1' b2'
      end
  end.Example fold_bexp_ex1 :
    fold_constants_bexp (BAnd BTrue (BNot (BAnd BFalse BTrue)))
  = BTrue.Example fold_bexp_ex2 :
    fold_constants_bexp
      (BAnd (BEq (AId X) (AId Y))
            (BEq (ANum 0)
                 (AMinus (ANum 2) (APlus (ANum 1)
                                         (ANum 1)))))
  = BAnd (BEq (AId X) (AId Y)) BTrue.

Theorem fold_constants_bexp_sound: btrans_sound fold_constants_bexp.