(define-struct Not (arg))
(define-struct And (left right))
(define-struct Or (left right))
(define-struct Implies (left right))
(define-struct If (test conseq alt))

parse: bool-SchemeExp -> boolExp
unparse: boolExp -> bool-SchemeExp

(make-Not  X)   	=>	(make-If  X  false true)
(make-And  X  Y)	=>	(make-If  X  Y  false)
(make-Or  X  Y)		=>	(make-If  X  true  Y)
(make-Implies  X  Y)	=>	(make-If  X  Y  true)

(check-expect  (convert-to-if (make-Or (make-And 'x 'y) 'z))    (make-If (make-If 'x 'y false) true 'z))
(check-expect  (convert-to-if (make-Implies 'x (make-Not 'y))   (make-If 'x (make-If 'y false  true) true))

a {{boolExp}} is either:
   *  a boolean value {{true}} and {{false}};
   *  a symbol {{s}} representing a boolean variable;
   *  {{(make-Not M)}} where {{M}} is a {{boolExp}};
   *  {{(make-And M N)}} where {{M}} and {{N}} are {{boolExps}};
   *  {{(make-Or M N)}} where {{M}} and {{N}} are {{boolExps}};
   *  =(make-Implies M N) where =M= and =N= are =boolExps=; or
   *  =(make-If M N P)= where =M=, =N=, and =P= are =boolExps=.

The provided function {{parse: input -> boolExp}} takes a Scheme expression and returns the corresponding {{boolExp}}.

----+Normalization

An =ifExp= is _normalized_ iff every sub-expression in =test= position is either
a variable (symbol) or a constant (=true= or =false=).  We call this type =norm-ifExp=.

For example, the =ifExp=

is not a =norm-ifExp= because it has an =If= construction in test
position.
In contrast, the equivalent =ifExp=

is normalized and hence is an =norm-ifExp=.

The normalization process, implemented by the function
=normalize: ifExp -> norm-ifExp= eliminates all =if= constructions that
appear in the _test_ position of other =if= constructions.
We perform this transformation by
repeatedly applying the following rewrite rule
(to any portion of the expression) until it is inapplicable:

This transformation always terminates and yields a unique answer
independent of the order in which rewrites are performed.
The proof of this fact is left as an optional exercise.

In the =normalize= function, it is critically important not to duplicate any work, so
the order in which reductions are made really matters.  Do *NOT* apply
the normalization rule above unless =U= and =V= are already
normalized, because the rule duplicates both =U= and =V=.  If you
reduce the _consequent_ and the _alternative_ (=U= and =V= in the left
hand side of the rule above) before reducing the _test_, =normalize=
runs in linear time (in the number of nodes in the input); if done in
the wrong order it runs in exponential time in the worst case.  (And
some of our test cases will exhibit this worst case behavior.)

Hint: define a sub-function head-normalize that takes three =norm-ifExp=s =X=, =Y=, and =Z=
and constructs a =norm-ifExp= equivalent to =(makeIf X Y Z)=.  This help function processes
=X= because the =test= position must be a variable or a constant, yet =X= can be an
arbitrary =norm-ifExp=. In contrast, =(head-normalize X Y Z)= never even inspects =Y=
and =Z= because they are already normalized and the normalizing transformations performed
in =head-normalize= never place these expressions in =test= position.

The following examples illustrate how the =normalize= and =head-normalize= functions behave:

Once a large formula has been normalized, do not try to print it unless you know that the formula is small!
The printed form can be exponentially larger than the internal representation (because the internal representation can share subtrees).

Before you start writing =normalize=, write the template corresponding to the inductive data definition of =norm-ifExp=.

----+Symbolic Evaluation

The symbolic evaluation process, implemented by the function =eval: norm-if-form environment -> norm-if-form=, reduces a =norm-if-form=
to simple form.  In particular, it reduces all tautologies
(expressions that are always true) to =true= and
all contradictions (expressions that are always false) to =false=.
<p>
Symbolic evaluation applies the following rewrite rules to
an expression until
none is applicable (with one exception
discussed below):

The notation =M[X <- N]= means =M= with all occurrences
of the symbol =X= replaced by the expression =N=.  It is very costly to actually
perform these subtitutions on =norm-if-form= data.  To avoid this computational
expense, we simply maintain a list of bindings which are pairs consisting
of symbols (variable names) and boolean values { =true=, =false= }.  The
following data definition definition formally defines the =binding= type.

A =binding= is a pair
(make-binding s v)
where s is a symbol (a variable) and v is a boolean value (an element of { =true=, =false= }.

An =environment= is a =(listOf binding)=.

When the =eval= function encounters a variable (symbol), it looks up the symbol in the environment and replaces the symbol it's boolean value if it exists.
<p>
These rewrite rules do not have the Church-Rosser property.  The
last two rewrite rules are the spoilers; the relative order
in which they are applied can affect the result in some cases.
However, the rewrite rules
do have the Church-Rosser property on expressions which are
tautologies or contradictions.
<p>
If the final rule is applied only when
=X= actually occurs in either
=Y= or =Z=,
then the symbolic evaluation process
is guaranteed to terminate.  In this case, every rule
either reduces the size of the expression or the number of variable
occurrences in it.

We recommend applying the rules in the order shown from the top down
until no more reductions
are possible (using the constraint on the final rule).  Note that the
last rule should only be applied once to a given sub-expression.

----+Conversion to Boolean Form

The final phase converts an expression in (not necessarily reduced) =If= form
to an equivalent expression constructed from
variables and { =true=, =false=, =And=, =Or=, =Not=,
=Implies=, =If=}.
This process
eliminates every expression of the form

where one of the arguments {=X=, =Y=, =Z=} is a constant
{ =true=, =false=}.

Use the following set of reduction rules to perform this conversion