%% Name: ***
%% Date: ***
%% Tabs: ***

Package derivative

==============================================================
%% This package defines type Expression, and 
%% some functions on expressions, including
%% 
%%    taking the derivative of an expression
%%   
%%    simplifying an expression
%%   
%% The simplification is rudimentary.
==============================================================

==============================================================
			export
==============================================================

==============================================================
%%			Expression
==============================================================
%% The expression type is defined in the body of this package
%% so that its representation is hidden from view.  Here,
%% we only advertise that the type exists.
==============================================================

%: An expression involves constants, an independent variable
%: called xx and a collection of operations.  The operations
%: that are supported are
%:
%:   xx	                The independent variable
%:   constant(c)        A constant expression having value c
%:   A+B                The sum of expressions A and B

Type Expression interface

  with ==, $
%Type

==============================================================
%%                derivativeX
==============================================================

Exception derivativeX(Expression) 
  "Unable to take the derivative of an expression"
%Exception

==============================================================
%%			Functions
==============================================================
%% Here are the exported functions for expressions.  Only 
%% these are public.
==============================================================

Expect
  constant	: Real -> Expression
  
  		%: constant(r) is an expression that is the
		%: real number r.  For example, constant(2.5)
		%: is the real number 2.5, as an expression.
		;
		
  xx		: Expression
  
  		%: xx is the independent variable.
		;
		
  +		: (Expression, Expression) -> Expression
  
  		%: a + b is the expression that is the sum of
		%: expressions a and b.
		;
		
  derivative	: Expression -> Expression
  
  		%: deriv(e) is the derivative of e with
		%: respect to variable xx.
		;
  
  simplify	: Expression -> Expression
  
  		%: simplify(e) is a (possibly) simplified
		%: form of expression e.
                ;

%Expect


========================================================
			implementation
========================================================

========================================================
%%		   Type Expression
========================================================
%% Here, type Expression type and its constructors
%% and unconstructors are defined.  We do not want the
%% default definition of $, and instead define our own.
========================================================

Type Expression =
  
  --------------------------------------------------
  %% An expression can be a real constant.
  --------------------------------------------------
    
  | constant(Real)
    
  --------------------------------------------------
  %% An expression can be the independent variable xx.
  --------------------------------------------------
    
  | xx
    
  --------------------------------------------------
  %% An expression can be the sum of two expressions.
  --------------------------------------------------
    
  | Expression + Expression

  with ==
    
%Type
========================================================

========================================================
%%		Expression Simplification
========================================================

========================================================
%%			simplifySum
========================================================
%% simplifySum(e) is a (possibly) simplified form
%% of expression e, which is presumed to be a sum.
%% The following simplifications are employed.
%%
%%    0 + a = a         a + 0 = a
%%
%%    Computation with constants.  We replace 
%%    constant(m) + constant(n) by constant(m+n).
%%
%% simplifySum only looks at e to see if it
%% is exactly one of these forms.  It does not
%% look inside e for subexpressions that have these
%% forms.
========================================================

Define
  
  ----------------------------------------------------
  %% constant arithmetic
  ----------------------------------------------------

  case simplifySum(constant(m) + constant(n)) = constant(m+n)

  ----------------------------------------------------
  %% 0 + a = a
  ----------------------------------------------------

  case simplifySum(constant(0) + a)  = a

  ----------------------------------------------------
  %% a + 0 = a
  ----------------------------------------------------

  case simplifySum(a + constant(0))  = a

  ----------------------------------------------------
  %% Otherwise, don't simplify.  Return a as is.
  ----------------------------------------------------

  else simplifySum(a)  = a
%Define

========================================================
%%			simplify
========================================================
%% Function simplify does a full simplification, going into
%% the structure of an expression looking for subexpressions
%% that can be simplified.
========================================================

Define
  case simplify(constant(r)) = constant(r)
  case simplify(=xx)         = xx
  case simplify(a + b)       = simplifySum(simplify(a) + simplify(b))
%Define


========================================================
%%			Computing Derivatives
========================================================

========================================================
%%			deriv
========================================================
%% deriv(e) is the derivative of expression e with
%% respect to variable xx, but it does not do any
%% simplifications.  It employs the rules for
%% derivatives directly.
========================================================

Define 
    
  ----------------------------------------------------
  %% c' = 0.
  ----------------------------------------------------

  case deriv(constant(?)) = constant(0)

  ----------------------------------------------------
  %% xx' = 1.
  ----------------------------------------------------

  case deriv(=xx) = constant(1)

  ----------------------------------------------------
  %% (a+b)' = a' + b'.
  ----------------------------------------------------

  case deriv(a + b) = deriv(a) + deriv(b)
%Define

========================================================
%%			derivative
=======================================================

Define derivative(a) = simplify(deriv(a)).

========================================================
%%			$ for Expressions
========================================================
%% Function $ is used to convert an expression to a string.
%% It tries to make the expression look readable.
========================================================

Define
  case $(constant(r)) = $(r)
  case $(=xx)         = "x"
  case $(a + b)       = concat["(", $(a), "+", $(b), ")"]
%Define

%Package