(** These two definitions specify the _abstract syntax_ of
    arithmetic and boolean expressions. *)

Inductive aexp : Type :=
  | ANum : nat -> aexp
  | APlus : aexp -> aexp -> aexp
  | AMinus : aexp -> aexp -> aexp
  | AMult : aexp -> aexp -> aexp.(** These two definitions specify the _abstract syntax_ of
    arithmetic and boolean expressions. *)

Inductive aexp : Type :=
  | ANum : nat -> aexp
  | APlus : aexp -> aexp -> aexp
  | AMinus : aexp -> aexp -> aexp
  | AMult : aexp -> aexp -> aexp.(** These two definitions specify the _abstract syntax_ of
    arithmetic and boolean expressions. *)

Inductive aexp : Type :=
  | ANum : nat -> aexp
  | APlus : aexp -> aexp -> aexp
  | AMinus : aexp -> aexp -> aexp
  | AMult : aexp -> aexp -> aexp.(** These two definitions specify the _abstract syntax_ of
    arithmetic and boolean expressions. *)

Inductive aexp : Type :=
  | ANum : nat -> aexp
  | APlus : aexp -> aexp -> aexp
  | AMinus : aexp -> aexp -> aexp
  | AMult : aexp -> aexp -> aexp.(** These two definitions specify the _abstract syntax_ of
    arithmetic and boolean expressions. *)

Inductive aexp : Type :=
  | ANum : nat -> aexp
  | APlus : aexp -> aexp -> aexp
  | AMinus : aexp -> aexp -> aexp
  | AMult : aexp -> aexp -> aexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.Inductive bexp : Type :=
  | BTrue : bexp
  | BFalse : bexp
  | BEq : aexp -> aexp -> bexp
  | BLe : aexp -> aexp -> bexp
  | BNot : bexp -> bexp
  | BAnd : bexp -> bexp -> bexp.(** In this chapter, we'll elide the translation from the
    concrete syntax that a programmer would actually write to these
    abstract syntax trees -- the process that, for example, would
    translate the string ["1+2*3"] to the AST

      APlus (ANum 1) (AMult (ANum 2) (ANum 3)).

    The optional chapter [ImpParser] develops a simple
    implementation of a lexical analyzer and parser that can perform
    this translation.  You do _not_ need to understand that chapter to
    understand this one, but if you haven't taken a course where these
    techniques are covered (e.g., a compilers course) you may want to
    skim it. *)

(** For comparison, here's a conventional BNF (Backus-Naur Form)
    grammar defining the same abstract syntax:

    a ::= nat
        | a + a
        | a - a
        | a * a

    b ::= true
        | false
        | a = a
        | a <= a
        | not b
        | b and b
*)

(** Compared to the Coq version above...

       - The BNF is more informal -- for example, it gives some
         suggestions about the surface syntax of expressions (like the
         fact that the addition operation is written [+] and is an
         infix symbol) while leaving other aspects of lexical analysis
         and parsing (like the relative precedence of [+], [-], and
         [*], the use of parens to explicitly group subexpressions,
         etc.) unspecified.  Some additional information (and human
         intelligence) would be required to turn this description into
         a formal definition, for example when implementing a
         compiler.

         The Coq version consistently omits all this information and
         concentrates on the abstract syntax only.

       - On the other hand, the BNF version is lighter and easier to
         read.  Its informality makes it flexible, a big advantage in
         situations like discussions at the blackboard, where
         conveying general ideas is more important than getting every
         detail nailed down precisely.

         Indeed, there are dozens of BNF-like notations and people
         switch freely among them, usually without bothering to say
         which form of BNF they're using because there is no need to:
         a rough-and-ready informal understanding is all that's
         important.

    It's good to be comfortable with both sorts of notations: informal
    ones for communicating between humans and formal ones for carrying
    out implementations and proofs. *)

(* ================================================================= *)
(** ** Evaluation *)

(** _Evaluating_ an arithmetic expression produces a number. *)

Fixpoint aeval (a : aexp) : nat :=
  match a with
  | ANum n => n
  | APlus a1 a2 => (aeval a1) + (aeval a2)
  | AMinus a1 a2  => (aeval a1) - (aeval a2)
  | AMult a1 a2 => (aeval a1) * (aeval a2)
  end.Example test_aeval1:
  aeval (APlus (ANum 2) (ANum 2)) = 4.(** Similarly, evaluating a boolean expression yields a boolean. *)

Fixpoint beval (b : bexp) : bool :=
  match b with
  | BTrue       => true
  | BFalse      => false
  | BEq a1 a2   => beq_nat (aeval a1) (aeval a2)
  | BLe a1 a2   => leb (aeval a1) (aeval a2)
  | BNot b1     => negb (beval b1)
  | BAnd b1 b2  => andb (beval b1) (beval b2)
  end.(* ================================================================= *)
(** ** Optimization *)

(** We haven't defined very much yet, but we can already get
    some mileage out of the definitions.  Suppose we define a function
    that takes an arithmetic expression and slightly simplifies it,
    changing every occurrence of [0+e] (i.e., [(APlus (ANum 0) e])
    into just [e]. *)

Fixpoint optimize_0plus (a:aexp) : aexp :=
  match a with
  | ANum n =>
      ANum n
  | APlus (ANum 0) e2 =>
      optimize_0plus e2
  | APlus e1 e2 =>
      APlus (optimize_0plus e1) (optimize_0plus e2)
  | AMinus e1 e2 =>
      AMinus (optimize_0plus e1) (optimize_0plus e2)
  | AMult e1 e2 =>
      AMult (optimize_0plus e1) (optimize_0plus e2)
  end.(** To make sure our optimization is doing the right thing we
    can test it on some examples and see if the output looks OK. *)

Example test_optimize_0plus:
  optimize_0plus (APlus (ANum 2)
                        (APlus (ANum 0)
                               (APlus (ANum 0) (ANum 1))))
  = APlus (ANum 2) (ANum 1).

Theorem optimize_0plus_sound'': forall a, aeval (optimize_0plus a) = aeval a.