#+Title: LARCS
#+Subtitle: The "Lisp Automated Rewrite and Calculation System"
#+AUTHOR: Samuel W. Flint
#+EMAIL: swflint@flintfam.org
#+DATE: \today
#+INFOJS_OPT: view:info toc:nil path:http://flintfam.org/org-info.js
#+OPTIONS: toc:nil H:5 ':t *:t todo:nil stat:nil d:nil
#+PROPERTY: header-args :noweb tangle :comments noweb
#+LATEX_HEADER: \usepackage[margins=0.75in]{geometry}
#+LATEX_HEADER: \parskip=5pt
#+LATEX_HEADER: \parindent=0pt
#+LATEX_HEADER: \lstset{texcl=true,breaklines=true,columns=fullflexible,basicstyle=\ttfamily,frame=lines,literate={<=}{$\leq$}1 {>=}{$\geq$}1}
#+LATEX_CLASS_OPTIONS: [10pt,twoside]
#+LATEX_HEADER: \pagestyle{headings}

* Export                                                           :noexport:
:PROPERTIES:
:CREATED:  <2016-06-09 Thu 12:49>
:END:

#+Caption: Export Document
#+Name: export-document
#+BEGIN_SRC emacs-lisp :exports none :results none
  (save-buffer)
  (let ((org-confirm-babel-evaluate
         (lambda (lang body)
           (declare (ignorable lang body))
           nil)))
    (org-latex-export-to-pdf))
#+END_SRC

* Tangle                                                           :noexport:
:PROPERTIES:
:CREATED:  <2016-06-09 Thu 12:50>
:END:

#+Caption: Tangle Document
#+Name: tangle-document
#+BEGIN_SRC emacs-lisp :exports none :results none
  (save-buffer)
  (let ((python-indent-offset 4))
    (org-babel-tangle))
#+END_SRC

* WORKING Introduction
:PROPERTIES:
:CREATED:  <2016-06-09 Thu 09:19>
:UNNUMBERED: t
:END:

It's a bold move to do what this does, building a Computer Algebra System from scratch, but I'm doing it anyway.  I've chosen to do this because I wanted to understand how most CASs work, and that can be accomplished by either reading thhe source code for one, or by building one.  While there are several very good CASs, the majority of them are non-free, and thus I'm not able to learn how exactly they work.  Those that are free software are either not complete, or are too complex to be able to learn from easily.

This is my Computer Algebra System, and it contains the following components:

 - [[id:f153a0fe-ec04-47b1-bdc5-290cc62bc985][Common Functionality]]
 - [[id:c6921b1e-d269-4243-acff-5a77685c331e][Expression Typing]]
 - [[id:b2c1fd45-b631-48f9-a093-66e1a0faa77f][Algebraic Manipulation]]
# - [[id:ffa664b6-e147-471c-9e97-f49c4619fc99][Symbolic Solver]]
# - [[id:b0259b8c-e285-48ca-bda0-4f04e3bc6ef2][Symbolic Trigonometry]]
 - [[id:552f402a-a25d-4f28-94af-17934c38a529][Symbolic Differentiation]]
# - [[id:61add971-a129-40d0-be13-24f8a4afc522][Symbolic Integration]]
 - [[id:75f65e8a-0cc9-477f-b5e9-3c563fe7ef5c][Symbolic To Typeset Form]]
 - [[id:89370949-8f58-41cf-8c4f-92f81d48ac23][Library Assembly]]

* DONE What's in a name?
CLOSED: [2016-06-09 Thu 12:48]
:PROPERTIES:
:CREATED:  <2016-06-09 Thu 12:37>
:UNNUMBERED: t
:END:

The CAS contained in this is called LARCS, or the Lisp Automated Rewrite and Calculation System.  This describes the system as follows:

 - Lisp :: The CAS is written in Lisp.  This is not novel, as other CAS have been written in Lisp before (Macsyma/Maxima), but it is unusual in that most new ones have been written in other languages.
 - Automated :: The CAS will perform rewrites and calculations automatically.
 - Rewrite :: The system is built on the concept of a rewrite system.  This workse because to perform many actions in the algebra, you rewrite an equation in one way or another.
 - Calculation :: The ability to go from a symbolic equation, something like $3x^3 + x^2 + 10x - 3$ (~(+ (* 3 (expt x 3)) (expt x 2) (* 10 x) -3)~), to the result where $x \gets 4$ (245).
 - System :: A complete library and application for symbolic algebra.

* TOC                                                                :ignore:
:PROPERTIES:
:CREATED:  <2016-06-09 Thu 09:19>
:END:

#+TOC: headlines 3
#+TOC: listings

* DONE Common Functionality [3/3]
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 22:23>
:ID:       f153a0fe-ec04-47b1-bdc5-290cc62bc985
:END:

There are several bits of common functions or variables that are required for use.  This primarily includes functions that some of the macros rely on, or things that are required for use in other parts of the system, but don't present as specific functionality.

** DONE Generate an Args List
CLOSED: [2016-07-30 Sat 16:08]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 17:19>
:ID:       49596957-2fc6-4458-ad85-99cbcf337b42
:END:

For some macros, an arguments list must be generated.  This is done by generating a list of variables starting with the word ~expression-~ followed by a letter from the alphabet, in turn.

#+Caption: Generate an Args List
#+Name: common-generate-an-args-list
#+BEGIN_SRC lisp
  (defun gen-args-list (count)
    (let ((letters '(a b c d e f g h i j k l m n o p q r s t u v w x y z)))
      (let ((variables-list '()))
        (dotimes (i count)
          (pushnew (symbolicate 'expression- (nth i letters)) variables-list))
        (reverse variables-list))))
#+END_SRC

** DONE Constants and Greeks
CLOSED: [2016-08-05 Fri 21:32]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 20:57>
:ID:       907fcf64-51eb-4a2c-a8bc-29e4f75f1dd3
:END:

This is a mapping between the names of constants and the way that they are correctly displayed in TeX.  Besides defining the mapping, which is in the form of an alist, it also collects the names of all of the constants, and exports the names themselves.

#+Caption: Constants and Greeks
#+Name: constants-and-greeks
#+BEGIN_SRC lisp
  (defvar *special-symbols-to-sequences*
    '((alpha . "\\alpha")
      (beta . "\\beta")
      (gamma . "\\gamma")
      (delta . "\\delta")
      (epsilon . "\\epsilon")
      (varepsilon . "\\varepsilon")
      (zeta . "\\zeta")
      (eta . "\\eta")
      (theta . "\\theta")
      (vartheta . "\\vartheta")
      (gamma . "\\gamma") (kappa . "\\kappa")
      (lambda . "\\lambda")
      (mu . "\\mu")
      (nu . "\\nu")
      (xi . "\\xi")
      (omicron . "\\o")
      (pi . "\\pi")
      (varpi . "\\varpi")
      (rho . "\\rho")
      (varrho . "\\varrho")
      (sigma . "\\sigma")
      (varsigm . "\\varsigm")
      (tau . "\\tau")
      (upsilon . "\\upsilon")
      (phi . "\\phi")
      (varphi . "\\varphi")
      (chi . "\\chi")
      (psi . "\\psi")
      (omega . "\\omega")
      (big-gamma . "\\Gamma")
      (big-delta . "\\Delta")
      (big-theta . "\\Theta")
      (big-lambda . "\\Lambda")
      (big-xi . "\\Xi")
      (big-pi . "\\Pi")
      (big-sigma . "\\Sigma")
      (big-upsilon . "\\Upsilon")
      (big-phi . "\\Phi")
      (big-psi . "\\Psi")
      (big-omega . "\\Omega")))

  (defvar *constant-names*
    (mapcar #'car *special-symbols-to-sequences*))

  (mapcar #'export *constant-names*)
#+END_SRC

** DONE Assembly
CLOSED: [2016-07-30 Sat 15:43]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 17:20>
:ID:       d583d5e4-a2c9-432c-9486-cc6baa4239f4
:END:

This is where the common functions and constants are assembled into their own package.  Almost all of the functions and variables are exported and available for everything else.

#+Caption: Assemble Common Functions
#+Name: assemble-common-functions
#+BEGIN_SRC lisp :tangle "larcs-common.lisp"
  (in-package #:larcs.common)

  <<common-generate-an-args-list>>

  <<constants-and-greeks>>
#+END_SRC

* DONE Expression Typing [8/8]
:PROPERTIES:
:CREATED:  <2016-04-30 Sat 23:15>
:ID:       c6921b1e-d269-4243-acff-5a77685c331e
:END:

To be able to provide various forms of matching and manipulation, the type of an expression must be determined.  This is done by analyzing the contents of the expression.  To accomplish this, there must be a way to define a classifier, store all possible classifiers, check a classifier and produce a classification.  To provide more flexibility in programming, there is also a special version of case, called ~classification-case~ and a when-pattern macro called ~when-classified-as~.

** DONE Define Classification
CLOSED: [2016-06-14 Tue 23:00]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 13:56>
:ID:       d8826a51-50b8-467a-9e52-158502bd4138
:END:

Classifications are defined as ~define-classification~.  This macro takes a ~name~, which is the name of the classification, and a body, which is classified within a function.  Inside the function, the following are bound: ~expression~, the expression to be classified; and, ~length~, which is the length of the expression if it's a list, otherwise, 0 if it's atomic.  A cons cell containing the name of the classification and the name of the classifier is pushed onto classification storage, and the classifier name is exported.

#+Caption: Define Classification
#+Name: et-define-classification
#+BEGIN_SRC lisp
  (defmacro define-classification (name &body body)
    (check-type name symbol)
    (let ((classifier-name (symbolicate name '-classifier)))
      `(progn
         (defun ,classifier-name (expression &aux (length (if (listp expression) (length expression) 0)))
           (declare (ignorable length))
           ,@body)
         (pushnew '(,name . ,classifier-name) *classifications*)
         (export ',name)
         ',name)))
#+END_SRC

** DONE Check Classification
CLOSED: [2016-06-14 Tue 23:10]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 13:56>
:ID:       6505b0b1-ffd8-4dd6-b81a-3e49483d8437
:END:

To classify an expression, the expression and name of the possible classification is passed in.  If the given name of the classification is ~*~, then ~t~ is returned, as this is a catch all; otherwise the classification is retrieved by name, and the expression is passed to the classifier, which will return either ~t~ or ~nil~.

#+Caption: Check Classification
#+Name: et-check-classification
#+BEGIN_SRC lisp
  (defun classified-as-p (expression classification)
    (if (eq '* classification)
        t
        (funcall (cdr (assoc classification *classifications*))
                 expression)))
#+END_SRC

** DONE Classify Expression
CLOSED: [2016-06-14 Tue 23:23]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 14:09>
:ID:       82d75d54-1d33-400b-86a3-7d16af938ac8
:END:

While being able to check if an expression is given a specific classification is vital, for some things, being able to see what all possible classifications for an expression are can be quite useful.  To do this, an expression is passed in, and for each possible classification in the classification storage, it is checked to see whether or not the classification is possible.  If it is, the classification is pushed on to a list of valid classifications.  When the possible classifications are exhausted, the list of valid classifications is reversed and returned.

#+Caption: Classify Expression
#+Name: et-classify-expression
#+BEGIN_SRC lisp
  (defun classify (expression)
    (let ((classifications '()))
      (dolist (possible
                ,*classifications*
               (reverse classifications))
        (let ((name (car possible))
              (checker (cdr possible)))
          (when (funcall checker expression)
            (push name classifications))))))
#+END_SRC

** DONE Classification Case
CLOSED: [2016-06-14 Tue 23:34]
:PROPERTIES:
:CREATED:  <2016-05-20 Fri 14:15>
:ID:       19a4e467-baa0-47eb-9267-93ff3801b1fd
:END:

Because case is such a useful tool, and because it provides a way to ensure that an expression doesn't fall through when acting on it, I've written the ~classification-case~ macro.  It takes an expression, named ~var~ and a list of cases, in the form of ~(classification body-form-1 body-form-2 body-form-n)~.  It transforms the cases, converting them to the form ~((classified-as-p expression 'type) body-form-1 body-form-2 body-form-n)~.  It finally expands to a ~cond~ in which ~the-classification~ is bound to the full and complete classification of the passed expression.

#+Caption: Classification Case
#+Name: et-classification-case
#+BEGIN_SRC lisp
  (defmacro classification-case (var &rest cases)
    (let ((conditions (map 'list #'(lambda (case)
                                     (destructuring-bind (type &body body) case
                                       (if (eq type 't)
                                           `((classified-as-p ,var '*) ,@body)
                                           `((classified-as-p ,var ',type) ,@body))))
                           cases)))
      `(let ((the-classification (classify ,var)))
         (declare (ignorable the-classification))
         (cond
           ,@conditions))))
#+END_SRC

** DONE When Classified
CLOSED: [2016-06-14 Tue 23:44]
:PROPERTIES:
:CREATED:  <2016-05-30 Mon 18:31>
:ID:       5c7c3e0b-9170-48e9-a414-6ac4528f9ac3
:END:

Another utility macro is ~when-classified-as~, which takes a ~classification~, an expressiond named ~variable~ and a body.  It expands fairly simply to a ~when~ form, with the predicate taking the following form ~(classified-as-p variable 'classification)~, wrapping around the passed in body.

#+Caption: When Classified
#+Name: et-when-classified
#+BEGIN_SRC lisp
  (defmacro when-classified-as (classification variable &body body)
    `(when (classified-as-p ,variable ',classification)
       ,@body))
#+END_SRC

** DONE Classifications [13/13]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 13:56>
:ID:       dcce4a6b-1b2d-4638-a82b-0c4917b0698a
:END:

I define the following classifications:

 - Numerics :: All numbers
 - Variables :: Any symbols
 - Non-atomics :: Anything that isn't simply a number or a variable
 - Additives :: Expressions that are adding multiple terms
 - Subtractives :: Expressions subtracting multiple terms
 - Powers :: Expressions of the form $x^n$, where $x$ is a variable, and $n$ is a numeric.
 - Exponentials :: Expressions of the form $x^y$ or $e^y$, where $x$ and $y$ are generic expressions, and $e$ is Euler's constant.
 - Logarithmics :: Expressions of the form of $\ln x$ or $\log_b x$, where $x$ and $b$ are generic expressions.
 - Rationals :: Expressions of the form $\frac{f(x)}{g(x)}$.
 - Polynomial Terms :: Any integers, multiplicatives of the form $nx^m$ or powers of the form $x^m$, where $x$ is a variable and $n$ and $m$ are numerics.
 - Polynomials :: Additives or Subtractives consisting solely of Polynomial Terms.
 - Trigonometrics :: The trig functions: $\sin$, $\cos$, $\tan$, $\csc$, $\sec$ and $\cot$.

#+Caption: Possible Classifications
#+Name: et-possible-classifications
#+BEGIN_SRC lisp
  <<et-classify-numbers>>
  <<et-classify-variables>>
  <<et-classify-non-atomics>>
  <<et-classify-additives>>
  <<et-classify-subtractives>>
  <<et-classify-powers>>
  <<et-classify-exponentials>>
  <<et-classify-multiplicatives>>
  <<et-classify-logarithmics>>
  <<et-classify-rationals>>
  <<et-classify-polynomial-term>>
  <<et-classify-polynomials>>
  <<et-classify-trigonometrics>>
#+END_SRC

*** DONE Numbers
CLOSED: [2016-06-14 Tue 23:58]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 14:26>
:ID:       42081153-7cc5-42ff-a17f-53e171c6d1a7
:END:

A number is defined as anything that satisfies the built-in ~numberp~.  This includes integers, rationals, floats and complex numbers.

#+Caption: Classify Numbers
#+Name: et-classify-numbers
#+BEGIN_SRC lisp
  (define-classification numeric
    (numberp expression))
#+END_SRC

*** DONE Variables
CLOSED: [2016-06-15 Wed 00:00]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 14:26>
:ID:       4c676754-ef9a-485f-91a2-8f1bd83c7659
:END:

Variables are defined as anything that satisfies the Common Lisp predicate, ~symbolp~.

#+Caption: Classify Variables
#+Name: et-classify-variables
#+BEGIN_SRC lisp
  (define-classification variable
    (symbolp expression))
#+END_SRC

*** DONE Non Atomics
CLOSED: [2016-06-15 Wed 00:02]
:PROPERTIES:
:CREATED:  <2016-05-04 Wed 19:52>
:ID:       414df063-0be1-4849-8b9f-d71aa828be2a
:END:

Non-atomic is a classification for anything other than numerics and variables.  It is defined as anything that satisfies the predicate ~listp~.

#+Caption: Classify Non-Atomics
#+Name: et-classify-non-atomics
#+BEGIN_SRC lisp
  (define-classification non-atomic
    (listp expression))
#+END_SRC

*** DONE Additives
CLOSED: [2016-06-15 Wed 00:03]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 14:26>
:ID:       736d79dc-f34c-4247-b592-690d7f2fddd9
:END:

When an expression is non-atomic, and the first element is the symbol ~+~, it is classified as an additive expression.

#+Caption: Classify Additives
#+Name: et-classify-additives
#+BEGIN_SRC lisp
  (define-classification additive
    (when-classified-as non-atomic expression
      (eq '+ (first expression))))
#+END_SRC

*** DONE Subtractive
CLOSED: [2016-06-15 Wed 00:06]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 14:26>
:ID:       c59d086f-2f49-485a-8f96-57d85e774f60
:END:

A non-atomic expression for which the first element is the symbol ~-~ is a subtractive expression.

#+Caption: Classify Subtractives
#+Name: et-classify-subtractives
#+BEGIN_SRC lisp
  (define-classification subtractive
    (when-classified-as non-atomic expression
      (eq '- (first expression))))
#+END_SRC

*** DONE Powers
CLOSED: [2016-06-15 Wed 00:07]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 14:27>
:ID:       cc15dd10-7cc0-4370-9e69-daf903b30ad5
:END:

A power is any expression that is non-atomic, the first element is the symbol ~expt~, the second is a variable and the third is a numeric.

#+Caption: Classify Powers
#+Name: et-classify-powers
#+BEGIN_SRC lisp
  (define-classification power
    (when-classified-as non-atomic expression
      (and (eq 'expt (first expression))
         (classified-as-p (second expression) 'variable)
         (classified-as-p (third expression) 'numeric))))
#+END_SRC

*** DONE Exponentials
CLOSED: [2016-06-15 Wed 00:11]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 15:04>
:ID:       a11fdd94-d56c-4749-bb22-dca75159dbcb
:END:

There are two types of exponentials, natural and non-natural.  Natural exponentials are defined as being non-atomic, two elements long, and the first element being ~exp~.  Non-natural exponentials are defined similarly, but are three elements long, and the first of which is the symbol ~expt~.

#+Caption: Classify Exponentials
#+Name: et-classify-exponentials
#+BEGIN_SRC lisp
  (define-classification natural-exponential
    (when-classified-as non-atomic expression
      (and (= 2 length)
         (eq 'exp (first expression)))))

  (define-classification exponential
    (when-classified-as non-atomic expression
      (and (= 3 length)
         (eq 'expt (first expression)))))
#+END_SRC

*** DONE Multiplicatives
CLOSED: [2016-06-15 Wed 00:12]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 14:27>
:ID:       feb85a20-93e3-45a1-be01-9893ecc07c53
:END:

A multiplicative expression is non-atomic, of any length, and the first element is the symbol ~*~.

#+Caption: Classify Multiplicatives
#+Name: et-classify-multiplicatives
#+BEGIN_SRC lisp
  (define-classification multiplicative
    (when-classified-as non-atomic expression
      (eq '* (first expression))))
#+END_SRC

*** DONE Logarithmics
CLOSED: [2016-06-15 Wed 00:14]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 14:27>
:ID:       0b733d75-e1ab-413f-8f8a-6a8a47db409c
:END:

There are two types of logarithmic classifications, natural and non-natural.  Natural logarithmics are non-atomic, two elements long, and the first element is the symbol ~log~.  Natural logarithmics are also non-atomic, but they are three elements long, starting with the symbol ~log~.

#+Caption: Classify Lograthmics
#+Name: et-classify-logarithmics
#+BEGIN_SRC lisp
  (define-classification natural-logarithmic
    (when-classified-as non-atomic expression
      (and (= 2 length)
         (eq 'log (first expression)))))

  (define-classification logarithmic
    (when-classified-as non-atomic expression
      (and (= 3 length)
         (eq 'log (first expression)))))
#+END_SRC

*** DONE Rationals
CLOSED: [2016-06-15 Wed 00:15]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 14:28>
:ID:       a4505a66-c249-4438-a6df-81e21718e23e
:END:

Rationals are non-atomic, three elements long, and the first element is the symbol ~/~.

#+Caption: Classify Rationals
#+Name: et-classify-rationals
#+BEGIN_SRC lisp
  (define-classification rational
    (when-classified-as non-atomic expression
      (and (= 3 length)
         (eq '/ (first expression)))))
#+END_SRC

*** DONE Polynomial Terms
CLOSED: [2016-06-15 Wed 00:17]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 14:28>
:ID:       37da52b7-98a0-4a16-8a17-a62fcff2ba59
:END:

Polynomials are a compound classification:
 - Numerics
 - Variables
 - Powers
 - Multiplicatives that are a numeric and a variable
 - Multiplicatives that are a numeric and a power

#+Caption: Classify Polynomial Term
#+Name: et-classify-polynomial-term
#+BEGIN_SRC lisp
  (define-classification polynomial-term
    (or (classified-as-p expression 'numeric)
       (classified-as-p expression 'variable)
       (classified-as-p expression 'power)
       (and (classified-as-p expression 'multiplicative)
          (= (length (rest expression)) 2)
          (or (and (classified-as-p (second expression) 'numeric)
                (or (classified-as-p (third expression) 'power)
                   (classified-as-p (third expression) 'variable)))
             (and (classified-as-p (third expression) 'numeric)
                (or (classified-as-p (second expression) 'power)
                   (classified-as-p (second expression) 'variable)))))))
#+END_SRC

*** DONE Polynomials
CLOSED: [2016-06-15 Wed 00:19]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 14:28>
:ID:       8cd9045b-81dd-4571-930a-a852f81969c9
:END:

Polynomials are compound classifications that are defined as expressions which are either additive or subtrative, for which each term is a polynomial term.

#+Caption: Classify Polynomials
#+Name: et-classify-polynomials
#+BEGIN_SRC lisp
  (define-classification polynomial
    (when-classified-as non-atomic expression
      (and (or (eq '- (first expression))
            (eq '+ (first expression)))
         (reduce #'(lambda (a b)
                     (and a b))
                 (map 'list
                   #'(lambda (the-expression)
                       (classified-as-p the-expression 'polynomial-term))
                   (rest expression))))))
#+END_SRC

*** DONE Trigonometrics
CLOSED: [2016-06-15 Wed 00:22]
:PROPERTIES:
:CREATED:  <2016-05-04 Wed 13:38>
:ID:       6f433cad-4b81-4a6f-ab65-981f4a924812
:END:

Trigonometrics are defined as non atomic expressions that are two elements long, for which the first element of the expression is either ~sin~, ~cos~, ~tan~, ~csc~, ~sec~, or ~cot~.  For each of these there is a classification seperate from the generic ~trigonometric~ classification.

#+Caption: Classify Trigonometrics
#+Name: et-classify-trigonometrics
#+BEGIN_SRC lisp
  (define-classification trigonometric
    (when-classified-as non-atomic expression
      (member (first expression) '(sin cos tan csc sec cot))))

  (define-classification sin
    (when-classified-as non-atomic expression
      (eq 'sin (first expression))))

  (define-classification cos
    (when-classified-as non-atomic expression
      (eq 'cos (first expression))))

  (define-classification tan
    (when-classified-as non-atomic expression
      (eq 'tan (first expression))))

  (define-classification csc
    (when-classified-as non-atomic expression
      (eq 'csc (first expression))))

  (define-classification sec
    (when-classified-as non-atomic expression
      (eq 'sec (first expression))))

  (define-classification cot
    (when (classified-as-p expression 'non-atomic)
      (eq 'cot (first expression))))
#+END_SRC

** DONE Classification Storage
CLOSED: [2016-06-14 Tue 23:48]
:PROPERTIES:
:CREATED:  <2016-05-02 Mon 13:55>
:ID:       ff35cd33-3c10-4a45-a2c5-32bc3fdc1acc
:END:

Classifications are stored in an alist, with the key being the name of the classification, and the value being the classifier itself.  These cons cells are stored in the ~*classifications*~ variable.

#+Caption: Classification Storage
#+Name: et-classification-storage
#+BEGIN_SRC lisp
  (defvar *classifications* '())
#+END_SRC

** DONE Assembly
CLOSED: [2016-06-15 Wed 00:26]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 16:59>
:ID:       bb1d3eb5-b9bf-4378-9716-87ab57dcc8a3
:END:

This assembles the classification library, which in the ~#:larcs.classify~ package.  It correctly resolves the order of the code, taking it from simple blocks to a complete file.

#+Caption: Expression Typing Assembly
#+Name: et-assembly
#+BEGIN_SRC lisp :tangle "larcs-classify.lisp"
  (in-package #:larcs.classify)
  <<et-classification-storage>>
  <<et-define-classification>>
  <<et-check-classification>>
  <<et-classify-expression>>
  <<et-classification-case>>
  <<et-when-classified>>
  <<et-possible-classifications>>
#+END_SRC

* WORKING Polynomial Related Functions [3/10]
:PROPERTIES:
:CREATED:  <2016-05-01 Sun 12:29>
:ID:       984d0f52-4c52-4bfa-a150-f3289d25bdf1
:END:

Polynomials, and polynomial terms, are a somewhat common occurence.  Because of this, it is important to provide a set of functions that manipulate these structures.  These functions include a coefficient retriever, a variable retriever, a power retriever and a couple of order comparators.

** DONE Collect Variables
CLOSED: [2016-06-21 Tue 22:10]
:PROPERTIES:
:CREATED:  <2016-05-20 Fri 15:15>
:ID:       6333322c-e12f-4ef6-8394-2fe219a72836
:END:

The task of collecting all variables in a given expression is fairly important to the task of algebraic manipulation.  This is accomplished using a fairly simple recursive algorithm, collecting the elements that are classified as variables.

An expression is passed in, and if a variable, it is collected, if non-atomic, all but the first element are passed again to ~collect-variables~, and it happens all over again, this time, with those variables being added to the list, and when all is said and done, a list of all variables in a given expression is returned.  See figure [[fig:variable-collection]] for a graphical representation.

#+Caption: Variable Collection
#+Name: variable-collection
#+BEGIN_SRC dot :file "imgs/variable-collection.png" :export results :cache yes
  digraph {
          start [label = "Start"];
          stop [label = "Stop"];
          collect [label = "Collect"];
          if_var [label = "If Variable", shape = rectangle];
          recurse_collect [label = "Iterate, Recurse and Collect Results"];

          start -> if_var;
          if_var -> collect [label = "True"];
          collect -> stop;

          if_var -> recurse_collect [label = "Non-atomic"];
          recurse_collect -> start;
  }
#+END_SRC

#+Caption: Variable Collection Algorithm
#+Name: fig:variable-collection
#+ATTR_LATEX: :width 8cm
#+RESULTS[e1586dc50921f7ba260f125e7221a978d489bd34]: variable-collection
[[file:imgs/variable-collection.png]]

#+Caption: Collect Variables
#+Name: poly-collect-variables
#+BEGIN_SRC lisp
  (defun collect-variables (expression)
    (let ((variables '()))
      (flet ((merge-variables (variable)
               (pushnew variable variables)))
        (classification-case expression
                             (variable (merge-variables expression))
                             (non-atomic (map 'list #'(lambda (expr)
                                                        (dolist (variable (collect-variables expr))
                                                          (merge-variables variable)))
                                              (rest expression)))))
      (reverse variables)))
#+END_SRC

** DONE Get Coefficient
CLOSED: [2016-06-26 Sun 19:50]
:PROPERTIES:
:CREATED:  <2016-05-31 Tue 19:08>
:ID:       cbc927fc-ae5e-46bf-a028-2872b5c31831
:END:

To get the coefficient of a polynomial term there are three possibilities

 - Numerics :: The term itself
 - Multiplicatives :: The second element in the term
 - All else :: 1

#+Caption: Get Coefficient
#+Name: poly-get-coefficient
#+BEGIN_SRC lisp
  (defun coefficient (term)
    (when (classified-as-p term 'polynomial-term)
      (classification-case term
                           (numeric term)
                           (multiplicative (second term))
                           (* 1))))
#+END_SRC

** DONE Get Term Variables
CLOSED: [2016-06-27 Mon 18:40]
:PROPERTIES:
:CREATED:  <2016-05-31 Tue 19:08>
:ID:       55729698-bd51-48af-ab42-197871c54dbb
:END:

The ability to retrieve tha variable in a polynomial term is important.  This is accomplished by collecting the variables in the term and returning the first.  If this is simply a numeric expression, ~nil~ is returned as there are no variables.

#+Caption: Get Term Variable
#+Name: poly-get-term-variable
#+BEGIN_SRC lisp
  (defun term-variable (term)
    (when (classified-as-p term 'polynomial-term)
      (first (collect-variables term))))
#+END_SRC

** TODO Get Power
:PROPERTIES:
:CREATED:  <2016-05-31 Tue 19:08>
:ID:       7d5a10da-bb30-496f-b285-470057a46db0
:END:

#+Caption: Get Power
#+Name: poly-get-power
#+BEGIN_SRC lisp
  (defun get-power (term)
    (classification-case term
                         (numeric 0)
                         (variable 1)
                         (power (third term))
                         (multiplicative
                          (if (listp (third term))
                              (third (third term))
                              1))
                         (* 0)))
#+END_SRC

** TODO Term Order Less Than
:PROPERTIES:
:CREATED:  <2016-06-24 Fri 20:57>
:ID:       b7fcb056-2494-4142-aad5-7619629b1980
:END:

#+Caption: Term Order Less Than
#+Name: poly-term-order-less-than
#+BEGIN_SRC lisp
  (defun term-order-< (a b)
    (< (get-power a)
       (get-power b)))
#+END_SRC

** TODO Same Order
:PROPERTIES:
:CREATED:  <2016-05-31 Tue 19:08>
:ID:       c56a1496-f4c2-4693-9448-5043570a752f
:END:

#+Caption: Same Order
#+Name: poly-same-order
#+BEGIN_SRC lisp
  (defun term-order-= (term-a term-b)
    (= (get-power term-a)
       (get-power term-b)))
#+END_SRC

** TODO Term Order Greater Than
:PROPERTIES:
:CREATED:  <2016-06-24 Fri 20:57>
:ID:       27f836aa-8522-4a4c-88b3-6f19a505916f
:END:

#+Caption: Term Order Greater Than
#+Name: poly-term-order-greater-than
#+BEGIN_SRC lisp
  (defun term-order-> (a b)
    (> (get-power a)
       (get-power b)))
#+END_SRC

** TODO Same Variable
:PROPERTIES:
:CREATED:  <2016-05-31 Tue 19:08>
:ID:       3806c97a-12fa-4488-b38c-d9ff3570c139
:END:

#+Caption: Same Variable
#+Name: poly-same-variable
#+BEGIN_SRC lisp
  (defun same-variable-p (term-a term-b)
    (eq (term-variable term-a)
        (term-variable term-b)))
#+END_SRC

** TODO Is Combinable
:PROPERTIES:
:CREATED:  <2016-05-31 Tue 19:08>
:ID:       db0410aa-bb12-4933-9be7-1a50d70ae90f
:END:

#+Caption: Is Combinable
#+Name: poly-is-combinable
#+BEGIN_SRC lisp
  (defun single-term-combinable-p (term-a term-b)
    (and (term-order-= term-a term-b)
       (same-variable-p term-a term-b)))
#+END_SRC

** TODO Assembly
:PROPERTIES:
:CREATED:  <2016-08-21 Sun 11:03>
:ID:       0c64cf6f-818d-4d39-9dc3-ab9353a6d66e
:END:

#+Caption: Polynomial Related Functions
#+Name: poly-assembly
#+BEGIN_SRC lisp :tangle "larcs-polynomials.lisp"
  (in-package #:larcs.polynomials)
  <<poly-collect-variables>>
  <<poly-get-coefficient>>
  <<poly-get-term-variable>>
  <<poly-get-power>>
  <<poly-term-order-less-than>>
  <<poly-same-order>>
  <<poly-term-order-greater-than>>
  <<poly-same-variable>>
  <<poly-is-combinable>>
#+END_SRC

* WORKING Algebraic Manipulation [2/3]
:PROPERTIES:
:CREATED:  <2016-06-09 Thu 09:20>
:ID:       b2c1fd45-b631-48f9-a093-66e1a0faa77f
:END:

One of the most important parts of this system is the "algebraic manipulator", a sub-system that provides utilities for symbolic arithmetic, that is to say addition, subtraction, multiplication and division, along with trigonometric functions and exponential/logarithmic functions.  These function, as many other portions of this system, using rewrite rules, implementing a form of specialized generic programming.

** DONE Term Collection
CLOSED: [2016-06-24 Fri 20:57]
:PROPERTIES:
:CREATED:  <2016-04-30 Sat 22:59>
:ID:       c1856735-914b-4f73-8825-3e5a062113d2
:END:

To aid in the design and implementation of various sub-systems, from simplification to the basics of algebraic manipulators, the ability to collect terms is extremely important. It is accomplished as follows:

 1. Lists for each of the types are initialized as empty.
 2. For each term in the given expression, put it into the given list.
 3. Return an alist containing the names of the types and the given lists, with the conses removed if the CDR is null.

#+Caption: Collect Terms
#+Name: am-collect-terms
#+BEGIN_SRC lisp
  (defun collect-terms (expression &aux (terms (rest expression)))
    (let ((numerics '())
          (variables '())
          (additives '())
          (subtractives '())
          (multiplicatives '())
          (polynomial-terms '())
          (rationals '())
          (powers '())
          (natural-exponentials '())
          (exponentials '())
          (natural-logarithmics '())
          (trigonometrics '()))
      (dolist (term terms)
        (classification-case term
                             (numeric (pushnew term numerics))
                             (variable (pushnew term variables))
                             (power (pushnew term powers))
                             (additive (pushnew term additives))
                             (subtractive (pushnew term subtractives))
                             (polynomial-term (pushnew term polynomial-terms))
                             (multiplicative (pushnew term multiplicatives))
                             (rational (pushnew term rationals))
                             (power (pushnew term powers))
                             (natural-exponential (pushnew term natural-exponentials))
                             (exponential (pushnew term exponentials))
                             (natural-logarithmic (pushnew term natural-logarithmics))
                             (trigonometric (pushnew term trigonometrics))))
      (remove-if #'(lambda (expr) (null (cdr expr)))
                 (list (cons :numerics numerics)
                       (cons :variables variables)
                       (cons :powers powers)
                       (cons :additives additives)
                       (cons :subtractives subtractives)
                       (cons :multiplicatives multiplicatives)
                       (cons :polynomial-terms polynomial-terms)
                       (cons :rationals rationals)
                       (cons :powers powers)
                       (cons :natural-exponentials natural-exponentials)
                       (cons :exponentials exponentials)
                       (cons :natural-logarithmics natural-logarithmics)
                       (cons :trigonometrics trigonometrics)))))
#+END_SRC

** WORKING Expression Manipulators [0/2]
:PROPERTIES:
:CREATED:  <2016-04-30 Sat 22:58>
:ID:       4fe60cc1-be66-4d5e-8922-590554d99004
:END:

Foo

#+Caption: Expression Manipulation
#+Name: am-expression-manipulation
#+BEGIN_SRC lisp
  <<am-define-expression-manipulator>>
  <<am-external-manipulator>>
#+END_SRC

*** WORKING Define Expression Manipulator
:PROPERTIES:
:CREATED:  <2016-04-30 Sat 22:57>
:ID:       63909972-428d-47f3-9dc3-3e1fb213aa70
:END:

#+Caption: Define Expression Manipulator
#+Name: am-define-expression-manipulator
#+BEGIN_SRC lisp
  (defvar *manipulator-map* '())

  (defmacro define-operation (name arity short)
    (check-type name symbol)
    (check-type arity (integer 1 26))
    (check-type short symbol)
    (let* ((args (gen-args-list arity))
           (expression-types (map 'list #'(lambda (x)
                                            (symbolicate x '-type)) args))
           (rules-name (symbolicate '*manipulators- name '*))
           (base-manipulator-name (symbolicate name '-manipulator-))
           (manipulator-define-name (symbolicate 'define- name '-manipulator))
           (is-applicable-name (symbolicate name '-is-applicable-p))
           (get-operations-name (symbolicate 'get- name '-manipulators))
           (type-check-list (let ((i 0))
                              (loop for arg in args
                                 collect (prog1
                                             `(classified-as-p ,arg (nth ,i types))
                                           (incf i))))))
      `(progn
         (push '(,short . ,name) *manipulator-map*)
         (defvar ,rules-name '())
         (defun ,is-applicable-name (types ,@args)
           (and ,@type-check-list))
         (defun ,get-operations-name (,@args)
           (remove-if #'null
                      (map 'list #'(lambda (option)
                                     (let ((types (car option))
                                           (name (cdr option)))
                                       (if (,is-applicable-name types ,@args)
                                           name)))
                           ,rules-name)))
         (defun ,name (,@args)
           (funcall (first (,get-operations-name ,@args))
                    ,@args))
         (defmacro ,manipulator-define-name ((,@expression-types) &body body)
           (let ((manipulator-name (symbolicate ',base-manipulator-name ,@expression-types)))
             `(progn
                (setf ,',rules-name (append ,',rules-name '(((,,@expression-types) . ,manipulator-name))))
                (defun ,manipulator-name ,',args
                  ,@body)))))))
#+END_SRC

#+Caption: Manipulation Example
#+Name: am-ex-manip-example
#+BEGIN_SRC lisp :results output raw :exports results :cache yes
  (ql:quickload :larcs-lib :silent t)

  (in-package #:larcs.manipulate)

  (format t "#+Caption: Expression Manipulator Expansion~%#+Name: am-ex-manip-expansion~%#+BEGIN_SRC lisp :exports code~%~a~%#+END_SRC"
          (macroexpand-1 '(define-operation frobnicate 2 frob)))
#+END_SRC

#+RESULTS[468c43c82d25c9776f1e482487e63f458252edcb]: am-ex-manip-example
#+Caption: Expression Manipulator Expansion
#+Name: am-ex-manip-expansion
#+BEGIN_SRC lisp :exports code
(PROGN
 (PUSH '(FROB . FROBNICATE) *MANIPULATOR-MAP*)
 (DEFVAR *MANIPULATORS-FROBNICATE* 'NIL)
 (DEFUN FROBNICATE-IS-APPLICABLE-P (TYPES EXPRESSION-A EXPRESSION-B)
   (AND (CLASSIFIED-AS-P EXPRESSION-A (NTH 0 TYPES))
        (CLASSIFIED-AS-P EXPRESSION-B (NTH 1 TYPES))))
 (DEFUN GET-FROBNICATE-MANIPULATORS (EXPRESSION-A EXPRESSION-B)
   (REMOVE-IF #'NULL
              (MAP 'LIST
                   #'(LAMBDA (OPTION)
                       (LET ((TYPES (CAR OPTION)) (NAME (CDR OPTION)))
                         (IF (FROBNICATE-IS-APPLICABLE-P TYPES EXPRESSION-A
                              EXPRESSION-B)
                             NAME)))
                   *MANIPULATORS-FROBNICATE*)))
 (DEFUN FROBNICATE (EXPRESSION-A EXPRESSION-B)
   (FUNCALL (FIRST (GET-FROBNICATE-MANIPULATORS EXPRESSION-A EXPRESSION-B))
            EXPRESSION-A EXPRESSION-B))
 (DEFMACRO DEFINE-FROBNICATE-MANIPULATOR
           ((EXPRESSION-A-TYPE EXPRESSION-B-TYPE) &BODY BODY)
   (LET ((MANIPULATOR-NAME
          (SYMBOLICATE 'FROBNICATE-MANIPULATOR- EXPRESSION-A-TYPE
                       EXPRESSION-B-TYPE)))
     `(PROGN
       (SETF ,'*MANIPULATORS-FROBNICATE*
               (APPEND ,'*MANIPULATORS-FROBNICATE*
                       '(((,EXPRESSION-A-TYPE ,EXPRESSION-B-TYPE)
                          ,@MANIPULATOR-NAME))))
       (DEFUN ,MANIPULATOR-NAME ,'(EXPRESSION-A EXPRESSION-B) ,@BODY)))))
#+END_SRC

*** TODO External Manipulator
:PROPERTIES:
:CREATED:  <2016-05-01 Sun 14:33>
:ID:       6419490c-3cb0-47e4-840a-c20af4bfb3d7
:END:

The Expression Manipulators should not be touched outside of this package, as they are not designed to be used outside of it.  Instead, they should be used through this simple function.  It takes an action and a list of expressions.  The function used to perform the action correctly is determined, and used to reduce the expressions.

#+Caption: External Manipulator
#+Name: am-external-manipulator
#+BEGIN_SRC lisp
  (defun manipulate (action &rest expressions)
    (let ((the-manipulator (cdr (assoc action *manipulator-map*))))
      (reduce the-manipulator
              expressions)))
#+END_SRC

** DONE Assembly
CLOSED: [2016-06-18 Sat 13:38]
:PROPERTIES:
:CREATED:  <2016-04-30 Sat 23:07>
:ID:       d487ed31-295b-4274-aef2-b45e4fa7bec2
:END:

This is the assembly of the ~#:larcs.manipulate~ package.  It includes, in correct order, all bits of functionality.  It places all of this in the ~larcs-manipulation.lisp~ file.

#+Caption: Packaging
#+Name: am-packaging
#+BEGIN_SRC lisp :tangle "larcs-manipulation.lisp"
  (in-package #:larcs.manipulate)
  <<am-determine-expression-type>>
  <<am-collect-variables>>
  <<am-collect-terms>>
  <<am-expression-manipulation>>
#+END_SRC

* TODO Manipulators for Addition
:PROPERTIES:
:CREATED:  <2016-04-30 Sat 23:08>
:ID:       b794486c-e493-408f-b80c-a440edae1bc8
:END:

Foo

#+Caption: Addition Manipulator
#+Name: am-addition-manipulator
#+BEGIN_SRC lisp :tangle "larcs-manipulators-addition.lisp""
  (in-package #:larcs.manipulate)

  (define-operation add 2 +)

  (define-add-manipulator (numeric numeric)
    (+ expression-a expression-b))

  (define-add-manipulator (numeric additive)
    (let ((total expression-a)
          (remainder (rest expression-b))
          (non-numeric '()))
      (dolist (element remainder)
        (if (classified-as-p element 'numeric)
            (incf total element)
            (push element non-numeric)))
      (cond
        ((null non-numeric)
         total)
        ((= 0 total)
         `(+ ,@non-numeric))
        (t
         `(+ ,total ,@non-numeric)))))

  (define-add-manipulator (additive additive)
    (let ((total 0)
          (elements (append (rest expression-a)
                            (rest expression-b)))
          (non-numeric '()))
      (dolist (element elements)
        (if (classified-as-p element 'numeric)
            (incf total element)
            (push element non-numeric)))
      (cond
        ((null non-numeric)
         total)
        ((= 0 total)
         `(+ ,@non-numeric))
        (t
         `(+ ,total ,@non-numeric)))))

  (define-add-manipulator (numeric subtractive)
    (let ((total expression-a)
          (the-other (rest expression-b))
          (non-numeric '()))
      (dolist (element the-other)
        (if (classified-as-p element 'numeric)
            (decf total element)
            (push element non-numeric)))
      (cond
        ((null non-numeric)
         total)
        ((= 0 total)
         `(+ ,@non-numeric))
        (t
         `(+ ,total (-,@non-numeric))))))

  (define-add-manipulator (numeric polynomial-term)
    `(+ ,expression-a ,expression-b))

  (define-add-manipulator (polynomial-term polynomial-term)
    (if (single-term-combinable-p expression-a expression-b)
        (let ((new-coefficient (+ (coefficient expression-a)
                                  (coefficient expression-b)))
              (variable (term-variable expression-a))
              (power (get-power expression-a)))
          `(* ,new-coefficient (expt ,variable ,power)))
        `(+ ,expression-a ,expression-b)))

  (define-add-manipulator (* numeric)
    (add expression-b expression-a))
#+END_SRC

* TODO Manipulators for Subtraction
:PROPERTIES:
:CREATED:  <2016-04-30 Sat 23:08>
:ID:       f675fd81-e995-41ee-9570-cc78261d9dc1
:END:

Foo

#+Caption: Subtraction Manipulator
#+Name: am-subtraction-manipulator
#+BEGIN_SRC lisp :tangle "larcs-manipulators-subtraction.lisp"
  (in-package #:larcs.manipulate)

  (define-operation subtract 2 -)

  (define-subtract-manipulator (numeric numeric)
    (- expression-a expression-b))

  (define-subtract-manipulator (numeric subtractive)
    (let ((total expression-a)
          (elements (rest expression-b))
          (non-numeric '()))
      (dolist (element elements)
        (if (classified-as-p element 'numeric)
            (decf total element)
            (push element non-numeric)))
      (cond
        ((null non-numeric)
         total)
        ((= 0 total)
         `(- ,@(reverse non-numeric)))
        (t
         `(- ,total ,@(reverse non-numeric))))))

  (define-subtract-manipulator (* numeric)
    (subtract expression-b expression-a))
#+END_SRC

* TODO Manipulators for Multiplication
:PROPERTIES:
:CREATED:  <2016-04-30 Sat 23:08>
:ID:       cddffdaa-10dd-425f-9697-3f0617162953
:END:

Foo

#+Caption: Multiplication Manipulators
#+Name: am-multiplication-manipulators
#+BEGIN_SRC lisp :tangle "larcs-manipulators-multiplication.lisp"
  (in-package #:larcs.manipulate)

  (define-operation multiply 2 *)

  (define-multiply-manipulator (numeric numeric)
      (* expression-a expression-b))

  (define-multiply-manipulator (numeric polynomial-term)
    (let ((new-coefficient (* expression-a (coefficient expression-b)))
          (variable (term-variable expression-b))
          (power (get-power expression-b)))
      (if (= 1 power)
          `(* ,new-coefficient ,variable)
          `(* ,new-coefficient (expt ,variable ,power)))))

  (define-multiply-manipulator (polynomial-term polynomial-term)
    (let ((new-coefficient (* (coefficient expression-a)
                              (coefficient expression-b)))
          (variable (term-variable expression-b))
          (power (+ (get-power expression-a)
                    (get-power expression-b))))
      `(* ,new-coefficient (expt ,variable ,power))))
#+END_SRC

* TODO Manipulators for Division
:PROPERTIES:
:CREATED:  <2016-04-30 Sat 23:09>
:ID:       4c4f7034-555a-46b0-85b9-56a08cf48f9b
:END:

Foo

#+Caption: Division Manipulators
#+Name: am-division-manipulators
#+BEGIN_SRC lisp :tangle "larcs-manipulators-division.lisp"
  (in-package #:larcs.manipulate)

  (define-operation division 2 /)

  (define-division-manipulator (numeric numeric)
      (/ expression-a expression-b))

  (define-division-manipulator (polynomial-term polynomial-term)
    (let ((new-coefficient (/ (coefficient expression-a)
                              (coefficient expression-b)))
          (variable (term-variable expression-b))
          (power (- (get-power expression-a)
                    (get-power expression-b))))
      `(* ,new-coefficient (expt ,variable ,power))))
#+END_SRC

* WORKING Manipulators for Trigonometric [0/6]
:PROPERTIES:
:CREATED:  <2016-04-30 Sat 23:09>
:ID:       ba4acf37-9074-429b-a2c8-a23094e1c86b
:END:

Foo

#+Caption: Trigonometric Manipulators
#+Name: am-trigonometric-manipulators
#+BEGIN_SRC lisp :tangle "larcs-manipulators-trig.lisp"
  (in-package #:larcs.manipulate)

  <<am-sine-manipulators>>
  <<am-cosine-manipulators>>
  <<am-tangent-manipulators>>
  <<am-cosecant-manipulators>>
  <<am-secant-manipulators>>
  <<am-cotangent-manipulators>>
#+END_SRC

** WORKING Sine
:PROPERTIES:
:CREATED:  <2016-05-08 Sun 16:22>
:ID:       c733c6b3-a44a-488f-8b6e-38346830b257
:END:

Foo

#+Caption: Sine Manipulators
#+Name: am-sine-manipulators
#+BEGIN_SRC lisp
  (define-operation sine 1 sin)

  (define-sine-manipulator (numeric)
    (sin expression-a))
#+END_SRC

** WORKING Cosine
:PROPERTIES:
:CREATED:  <2016-05-08 Sun 16:22>
:ID:       c2fbd362-6932-4483-8270-e3ad72a308fd
:END:

Foo

#+Caption: Cosine Manipulators
#+Name: am-cosine-manipulators
#+BEGIN_SRC lisp
  (define-operation cosine 1 cos)

  (define-cosine-manipulator (numeric)
    (cosine expression-a))
#+END_SRC

** WORKING Tangent
:PROPERTIES:
:CREATED:  <2016-05-08 Sun 16:22>
:ID:       07222206-1c22-411e-a8ab-13e1a627e9ef
:END:

Foo

#+Caption: Tangent Manipulators
#+Name: am-tangent-manipulators
#+BEGIN_SRC lisp
  (define-operation tangent 1 tan)

  (define-tangent-manipulator (numeric)
    (tan expression-a))
#+END_SRC

** WORKING Cosecant
:PROPERTIES:
:CREATED:  <2016-05-08 Sun 16:22>
:ID:       e77c0317-7281-45ff-b86b-8d66fb8c38ef
:END:

Foo

#+Caption: Cosecant Manipulators
#+Name: am-cosecant-manipulators
#+BEGIN_SRC lisp
  (define-operation cosecant 1 csc)
#+END_SRC

** WORKING Secant
:PROPERTIES:
:CREATED:  <2016-05-08 Sun 16:23>
:ID:       6c377c7d-ec84-4fcf-be94-db89b832c2d8
:END:

Foo

#+Caption: Secant Manipulators
#+Name: am-secant-manipulators
#+BEGIN_SRC lisp
  (define-operation secant 1 sec)
#+END_SRC

** WORKING Cotangent
:PROPERTIES:
:CREATED:  <2016-05-08 Sun 16:23>
:ID:       70a9fb76-7ca7-4c7d-b25b-0fa94d390b6c
:END:

Foo

#+Caption: Cotangent Manipulators
#+Name: am-cotangent-manipulators
#+BEGIN_SRC lisp
  (define-operation cotangent 1 cot)
#+END_SRC

* DONE Symbolic Differentiation [4/4]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 22:45>
:ID:       552f402a-a25d-4f28-94af-17934c38a529
:END:

While this isn't exactly /algebra/, differentiation is important mathematically.  This is done rather simply using rules to rewrite an initial expression forming the derivative.

** DONE Rule Definition [3/3]
CLOSED: [2016-08-16 Tue 22:10]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 22:51>
:END:

Rule definition is how this system works.  This is done rather simply, and is comprised of definition, retrieval and storage portions.

*** DONE Definition
CLOSED: [2016-07-19 Tue 20:48]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 22:51>
:ID:       de915ee7-47bd-4f7f-ad06-39f0201a4651
:END:

Rules are defined using the ~define-derivative~ macro, which takes an expression type, an arguments list, and a body.  If the expression type is not already in the expansions map, it pushes the expression type and expansion name onto the the mapping.  Following that, it defines a function for the expansion, using the given arguments list as the lambda-list and the given body for the body for the function.

#+Caption: Rule Definition
#+Name: sd-rule-definition
#+BEGIN_SRC lisp
  (defmacro define-derivative (expression-type (&rest arguments-list) &body body)
    (let ((expansion-name (symbolicate expression-type '-expansion)))
      `(progn
         (when (not (member ',expression-type (mapcar #'car *rules*)))
           (setq *rules* (append *rules* '((,expression-type . ,expansion-name)))))
         (defun ,expansion-name (,@arguments-list)
           ,@body))))
#+END_SRC

*** DONE Retrieval
CLOSED: [2016-08-16 Tue 22:00]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 23:08>
:ID:       97d8b24e-dd75-4919-a953-cba8035cb691
:END:

Rule retrieval works by matching a rewrite rule to an expression by classification.  This is done by iterating through the possible classification-rewrite rule pairs, and if the expression matches the classification, returing the rewrite rule to be used.

#+Caption: Rule Retrieval
#+Name: sd-rule-retrieval
#+BEGIN_SRC lisp
  (defun get-rule (expression)
    (cdr (first (remove-if #'(lambda (pair)
                               (let ((type (first pair)))
                                 (not (classified-as-p expression type))))
                           ,*rules*))))
#+END_SRC

*** DONE Storage
CLOSED: [2016-08-16 Tue 22:05]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 22:52>
:ID: 372dc2d7-ee67-4eba-a9f7-3633eaf0996e
:END:

Rules are stored rather simply, in a list of cons cells, with the ~CAR~ being the classification, and the ~CDR~ being the actualy rewrite function.  They are found in the variable ~*rules*~.

#+Caption: Rule Storage
#+Name: sd-rule-storage
#+BEGIN_SRC lisp
  (defvar *rules* '())
#+END_SRC

** DONE Rules [9/9]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 22:52>
:ID:       fdcebadd-b53d-4f59-99a4-4a3782e017a2
:END:

The process of symbolic derivation is carried out through rules.  These are defined in the following sections, taking apart the equations and putting them back together again as their derivatives.

#+Caption: Rules
#+Name: sd-rules
#+BEGIN_SRC lisp
  <<sd-numbers>>
  <<sd-variables>>
  <<sd-polynomial-terms>>
  <<sd-multiplicatives>>
  <<sd-rationals>>
  <<sd-additives>>
  <<sd-subtractives>>
  <<sd-exponentials-and-logarithmics>>
#+END_SRC

*** DONE Numbers
CLOSED: [2016-08-19 Fri 17:07]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 23:18>
:ID:       bb1f9175-2e86-43a3-94b3-9467d233539c
:END:

Numbers are perhaps one of the simplest rules to define, ignore everything and simply return 0, which is, by definition, the derivative of any bare number.

#+Caption: Numbers
#+Name: sd-numbers
#+BEGIN_SRC lisp
  (define-derivative numeric (&rest junk)
    (declare (ignorable junk))
    0)
#+END_SRC

*** DONE Variables
CLOSED: [2016-08-19 Fri 17:19]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 23:19>
:ID:       ecc17ca3-2989-4908-aded-4b6e20b1855c
:END:

As with Numbers, Variables are just as simple, if something is simply a bare variable, the derivative of such is 1.

#+Caption: Variables
#+Name: sd-variables
#+BEGIN_SRC lisp
  (define-derivative variable (&rest junk)
    (declare (ignorable junk))
    1)
#+END_SRC

*** DONE Polynomial Terms
CLOSED: [2016-08-19 Fri 17:31]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 23:33>
:ID:       6ca719d7-b584-4ae6-ae44-23bed186c6e9
:END:

A Polynomial Term is a bit more complex than the previous two, the rewrite rule has to be able to get the variable, coefficient and current power of the polynomial term.  Given this information, there are three possible cases:

 - The power is equal to 1. :: The coefficient is returned.
 - The power is equal to 2. :: The coefficient times the variable (in symbolic form) is returned.
 - The power is greater than 2. :: This comes in the form of $(nm)x^{^}{(n-1)}$.

#+Caption: Polynomial Terms
#+Name: sd-polynomial-terms
#+BEGIN_SRC lisp
  (define-derivative polynomial-term (&rest term)
    (let* ((coefficient (coefficient term))
           (variable (term-variable term))
           (power (get-power term)))
      (cond
        ((= 1 power)
         coefficient)
        ((= 2 power)
         `(* ,(* coefficient power) ,variable))
        (t
         `(* ,(* coefficient power) (expt ,variable ,(1- power)))))))
#+END_SRC

*** DONE Multiplicatives
CLOSED: [2016-08-19 Fri 20:21]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 09:57>
:ID:       161906a4-5c14-4a84-bf1d-7fae9e20b14f
:END:

Differentiation of Multiplicative equations are performed by the product rule.  This is defined as $\frac{\mathrm{d}}{\mathrm{d}x} f(x) \cdot g(x) = f(x) \cdot g^{\prime}(x) + f^{\prime}(x) \cdot g(x)$.  There are some minor exceptions, if $f(x)$ and $g(x)$ are numeric, then the result is the product of the two; if either $f(x)$ or $g(x)$ is numeric and the other is not, then the numeric is placed in front of the other derivative of the remainder.

#+Caption: Multiplicatives
#+Name: sd-multiplicatives
#+BEGIN_SRC lisp
  (define-derivative multiplicative (function first &rest rest)
    (declare (ignore function))
    (if (= 1 (length rest))
        (let ((second (first rest)))
          (cond
            ((and (classified-as-p first 'numeric)
                (classified-as-p second 'numeric))
             (* first second))
            ((classified-as-p first 'numeric)
             `(* ,first ,(differentiate second)))
            ((classified-as-p second 'numeric)
             `(* ,second ,(differentiate first)))
            (t
             `(+ (* ,first ,(differentiate second))
                 (* ,second ,(differentiate first))))))
        (differentiate `(* ,first (* ,@rest)))))
#+END_SRC

*** DONE Rationals
CLOSED: [2016-08-20 Sat 13:28]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 10:21>
:ID:       cd681a61-a143-4e02-a6a9-e7b8f9b9c77d
:END:

This follows the quotient rule, which is defined as $\frac{\mathrm{d}}{\mathrm{d}x} \frac{f(x)}{g(x)} = \frac{f^{\prime}(x)g(x) - f(x)g^{\prime}(x)}{g(x)^2}$.

#+Caption: Rational Derivatives
#+Name: sd-rationals
#+BEGIN_SRC lisp
  (define-derivative rational (function numerator denominator)
    (declare (ignore function))
    `(/ (- (* ,numerator ,(differentiate denominator))
           (* ,denominator ,(differentiate numerator)))
        (expt ,denominator 2)))
#+END_SRC

*** DONE Additives
CLOSED: [2016-08-20 Sat 11:20]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 10:30>
:ID:       d3a07d51-977c-4b1e-9a63-0eb415977f46
:END:

This is quite simple, differentiate each term and add them together.  This is accomplished using ~map~.

#+Caption: Additives
#+Name: sd-additives
#+BEGIN_SRC lisp
  (define-derivative additive (function &rest terms)
    (declare (ignore function))
    `(+ ,@(map 'list #'(lambda (term) (differentiate term)) terms)))
#+END_SRC

*** DONE Subtractives
CLOSED: [2016-08-20 Sat 13:13]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 10:30>
:ID:       063f61ee-6fd9-4286-9008-9c80ef0985a5
:END:

Following the same pattern as for additives, subtractives map over, deriving each term and prepending the ~-~ symbol to render the derivative of the originally passed subtractive equation.

#+Caption: Subtractives
#+Name: sd-subtractives
#+BEGIN_SRC lisp
  (define-derivative subtractive (function &rest terms)
    (declare (ignore function))
    `(- ,@(map 'list #'(lambda (term) (differentiate term)) terms)))
#+END_SRC

*** DONE Exponentials and Logarithmics
CLOSED: [2016-08-20 Sat 13:39]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 10:37>
:END:

There are four types of functions that are supported:

 - Natural Exponentials :: Effectively this returns itself.
 - Exponentials :: The original function multiplied by the log of the base.
 - Natural Logarithmics :: The derivative of the function being passed to $\ln$ over the same function.
 - Logarithmics :: The derivative of the natural log of the expression over the log of the base times the expression.

#+Caption: Exponentials and Logarithms
#+Name: sd-exponentials-and-logarithms
#+BEGIN_SRC lisp
  (define-derivative natural-exponential (function expression)
    (declare (ignore function))
    `(exp ,expression))

  (define-derivative exponential (function base power)
    (declare (ignore function))
    (if (numberp power)
        (if (listp base)
            `(* ,power (expt ,base ,(1- power)) ,(differentiate base))
            `(* ,power (expt ,base ,(1- power))))
        `(* (expt ,base ,power) (log ,base))))

  (define-derivative natural-logarithmic (function expression)
    (declare (ignore function))
    `(/ ,(differentiate expression) ,expression))

  (define-derivative logarithmic (function number base)
    (declare (ignore function))
    `(/ ,(differentiate (cons 'log number)) (* (log ,base) ,number)))
#+END_SRC

*** DONE Trigonometric Functions
CLOSED: [2016-08-19 Fri 20:36]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 10:45>
:END:

The following trig functions are supported:

 - Sine :: $f^{\prime}(x) \cdot \cos(f(x))$
 - Cosine :: $^{}f^{\prime}(x) \cdot -\sin(f(x))$
 - Tangent :: $f^{\prime}(x) \cdot {\sec(f(x))}^2$
 - Cosecant :: $f^{\prime}(x) \cdot \left( \csc(f(x)) - \cot(f(x)) \right)$
 - Cotangent :: $f^{\prime}(x) \cdot -{\csc(f(x))}^2$

#+Caption: Trigonometric Functions
#+Name: sd-trigonometric-functions
#+BEGIN_SRC lisp
  (define-derivative sin (function expression)
    (declare (ignore function))
    `(* ,(differentiate expression) (cos ,expression)))

  (define-derivative cos (function expression)
    (declare (ignore function))
    `(* ,(differentiate expression) (- (sin ,expression))))

  (define-derivative tan (function expression)
    (declare (ignore function))
    `(* ,(differentiate expression) (expt (sec ,expression) 2)))

  (define-derivative csc (function expression)
    (declare (ignore function))
    `(* ,(differentiate expression) (- (csc ,expression)) (cot ,expression)))

  (define-derivative cot (function expression)
    (declare (ignore function))
    `(* ,(differentiate expression) (- (expt (csc ,expression) 2))))
#+END_SRC

** DONE Driver
CLOSED: [2016-07-18 Mon 18:31]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 22:59>
:ID:       b40ed5ad-2eb7-43b1-bab7-39592894e5be
:END:

This is the derivative driver, ~differentiate~, which in the end is called recursively.  It takes an expression (called function), finds a rule using the ~get-rule~ function, and applies the rule to the function, ensuring that the function is passed as a list.

#+Caption: Derivative Driver
#+Name: sd-derivative-driver
#+BEGIN_SRC lisp
  (defun differentiate (function)
    (let ((rule (get-rule function)))
      (when rule
        (apply rule (ensure-list function)))))
#+END_SRC

** DONE Assembly
CLOSED: [2016-07-19 Tue 20:36]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 22:46>
:ID:       d87d49e3-8245-4ff0-aaf0-57b9e19edeba
:END:

This assembles the package, placing the contents in the correct order and puts them in the file ~larcs-differentiate.lisp~.

#+Caption: Symbolic Differentiation
#+Name: sd-symbolic-differentiation
#+BEGIN_SRC lisp :tangle "larcs-differentiate.lisp"
  (in-package #:larcs.differentiate)
  <<sd-rule-storage>>
  <<sd-rule-definition>>
  <<sd-rule-retrieval>>
  <<sd-rules>>
  <<sd-derivative-driver>>
#+END_SRC

* WORKING Symbolic To Typeset Form [3/5]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:13>
:ID:       75f65e8a-0cc9-477f-b5e9-3c563fe7ef5c
:END:

One of the less important parts of this system is the format converter, which converts between the internal symbolic form and a format that is capable of being typeset using TeX.  This is done using a variant of the common rewrite system, but instead of going between variants of the symbolic format, it converts from a symbolic format to string-based format.

** DONE Rule Management [2/2]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:17>
:END:

To accomplish the task of conversion from symbolic form to typeset form, rules are necessary.  It is done using three main things, rule definition, rule retrieval and rule storage.

*** DONE Define Rules
CLOSED: [2016-06-24 Fri 22:28]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:18>
:ID:       ec6fdb0d-546e-41fc-a7b7-5fbbfe4b7931
:END:

Rule definitions are built using the ~define-converter~ macro, which takes an expression type, a lambda list and a body.  It creates a function using the body and the given arguments list, and if it hasn't been pushed onto the storage system, the converter function is pushed into storage.

#+Caption: Rule Definition
#+Name: stf-define-rule
#+BEGIN_SRC lisp
  (defvar *rules* '())

  (defmacro define-converter (expression-type (&rest arguments-list) &body body)
    (let ((expansion-name (symbolicate expression-type '-conversion)))
      `(progn
         (when (not (member ',expression-type (mapcar #'car *rules*)))
           (setq *rules* (append *rules* '((,expression-type . ,expansion-name)))))
         (defun ,expansion-name (,@arguments-list)
           ,@body))))
#+END_SRC

*** DONE Rule Retrieval
CLOSED: [2016-06-24 Fri 22:36]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:18>
:ID:       0c34c744-7847-46c2-bdef-228feee7c84e
:END:

Rule retrieval is done by taking an expression, comparing it against given classifications, and from the first classification, returning the second element of the ~(classification . converter)~ pair.

#+Caption: Rule Retrieval
#+Name: stf-rule-retrieval
#+BEGIN_SRC lisp
  (defun get-rule (expression)
    (cdr (first (remove-if #'(lambda (pair)
                               (let ((type (first pair)))
                                 (not (classified-as-p expression type))))
                           ,*rules*))))
#+END_SRC

** WORKING Rules [2/9]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:18>
:ID:       90accad9-81d0-4aaf-9c7f-2418e36e1f3c
:END:

The following contains all of the defined rules, which are as follows:

 - Numerics
 - Variables
 - Polynomial Terms
 - Multiplicatives
 - Rationals
 - Additives
 - Subtractives
 - Trigonometrics
 - Exponentials & Logarithmics

#+Caption: Rules
#+Name: stf-rules
#+BEGIN_SRC lisp
  <<stf-numerics>>
  <<stf-variables>>
  <<stf-polynomial-terms>>
  <<stf-multiplicatives>>
  <<stf-rationals>>
  <<stf-additives>>
  <<stf-subtractives>>
  <<stf-trigonometrics>>
  <<stf-exponentials-logarithmics>>
#+END_SRC

*** DONE Numbers
CLOSED: [2016-08-02 Tue 22:09]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:22>
:ID:       fbc3e5ac-3276-4f54-b53e-9d4cc0263405
:END:

Numbers are formatted fairly simply, as they are simply surrounded by curly braces, and formatted as to be normal read syntax, which is generally correct.

#+Caption: Numerics
#+Name: stf-numerics
#+BEGIN_SRC lisp
  (define-converter numeric (number)
    (with-tex-output
      (format nil "{~A}" number)))
#+END_SRC

*** DONE Variables
CLOSED: [2016-08-02 Tue 22:20]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:22>
:ID:       8ec363f7-da0f-4023-90bb-e08a85623f55
:END:

As with numbers, variables are a relatively simple thing to format.  If the variable passed is in the ~*constant-names*~ list, then it must be a formattable constant for which there is a known TeX command.  If there is, it is looked up in the ~*special-symbols-to-sequences*~ alist, otherwise, the given variable is downcased and output as a string.  Either way, they are surrounded by, as usual, curly braces.

#+Caption: Variables
#+Name: stf-variables
#+BEGIN_SRC lisp
  (define-converter variable (var)
    (if (member var *constant-names*)
        (with-tex-output
          (format nil "{~A}" (cdr (assoc var *special-symbols-to-sequences*))))
        (with-tex-output
          (format nil "{~A}" (string-downcase var)))))
#+END_SRC

*** TODO Polynomial Terms
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:23>
:ID:       ac2283d0-da70-4672-90cb-08511bd9105e
:END:

#+Caption: Polynomial Terms
#+Name: stf-polynomial-terms
#+BEGIN_SRC lisp
  (define-converter polynomial-term (&rest term)
    (let ((variable (term-variable term))
          (coefficient (coefficient term))
          (power (get-power term)))
      (cond
        ((= 1 power)
         (with-tex-output
           (format nil "{~A}{~A}"
                   (convert-for-display coefficient)
                   (convert-for-display power))))
        ((= 0 coefficient)
         (with-tex-output
           (format nil "{~A}^{~A}"
                   (convert-for-display variable)
                   (convert-for-display power))))
        (t
         (with-tex-output
           (format nil "{~A}{~A}^{~A}"
                   (convert-for-display coefficient)
                   (convert-for-display variable)
                   (convert-for-display power)))))))
#+END_SRC

*** TODO Multiplicatives
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:23>
:ID:       87a7e236-072e-4c19-9f09-c458e5b50397
:END:

#+Caption: Multiplicatives
#+Name: stf-multiplicatives
#+BEGIN_SRC lisp
  (define-converter multiplicative (op &rest elements)
    (declare (ignore op))
    (with-tex-output
      (format nil "{~{~A~^ \\cdot ~}}"
              (mapcar #'convert-for-display
                      elements))))
#+END_SRC

*** TODO Rationals
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:23>
:ID:       4f8f984f-e567-4efb-ba15-8a98e15915fe
:END:

#+Caption: Rationals
#+Name: stf-rationals
#+BEGIN_SRC lisp
  (define-converter rational (op numerator denominator)
    (declare (ignore op))
    (with-tex-output
      (format nil "{\\frac{~A}{~A}}"
              (convert-for-display numerator)
              (convert-for-display denominator))))
#+END_SRC

*** TODO Additives
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:23>
:ID:       10ec8596-094a-4900-aba0-22b958ffdc9a
:END:

#+Caption: Additives
#+Name: stf-additives
#+BEGIN_SRC lisp
  (define-converter additive (op &rest terms)
    (declare (ignore op))
    (with-tex-output
      (format nil "{~{~A~^ + ~}}"
              (mapcar #'convert-for-display terms))))
#+END_SRC

*** TODO Subtractives
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:23>
:ID:       1037cb8e-f127-4c87-9312-2817bc2cfc25
:END:

#+Caption: Subtractives
#+Name: stf-subtractives
#+BEGIN_SRC lisp
  (define-converter subtractive (op &rest terms)
    (declare (ignore op))
    (with-tex-output
      (format nil "{~{~A~^ - ~}}"
              (mapcar #'convert-for-display terms))))
#+END_SRC

*** TODO Trigonometrics
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 18:38>
:ID:       742d303a-dcea-4bb2-9553-19b968a70272
:END:

#+Caption: Trigonometrics
#+Name: stf-trigonometrics
#+BEGIN_SRC lisp
  (define-converter sin (op term)
    (declare (ignore op))
    (with-tex-output
      (format nil "{\\sin {~A}}" (convert-for-display term))))

  (define-converter cos (op term)
    (declare (ignore op))
    (with-tex-output
      (format nil "{\\cos {~A}}" (convert-for-display term))))

  (define-converter tan (op term)
    (declare (ignore op))
    (with-tex-output
      (format nil "{\\tan {~A}}" (convert-for-display term))))

  (define-converter csc (op term)
    (declare (ignore op))
    (with-tex-output
      (format nil "{\\csc {~A}}" (convert-for-display term))))

  (define-converter sec (op term)
    (declare (ignore op))
    (with-tex-output
      (format nil "{\\sec {~A}}" (convert-for-display term))))

  (define-converter cot (op term)
    (declare (ignore op))
    (with-tex-output
      (format nil "{\\cot {~A}}" (convert-for-display term))))
#+END_SRC

*** TODO Exponentials and Logarithmics
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:24>
:ID:       24dc527f-0b9d-44b0-ae0f-4515f0c1d119
:END:

#+Caption: Exponentials and Logarithmics
#+Name: stf-exponentials-logarithmics
#+BEGIN_SRC lisp
  (define-converter natural-exponential (op term)
    (declare (ignore op))
    (with-tex-output
      (format nil "{e^~A}" (convert-for-display term))))

  (define-converter exponential (op base power)
    (declare (ignore op))
    (with-tex-output
      (format nil "{~A^~A}"
              (convert-for-display base)
              (convert-for-display power))))

  (define-converter natural-logarithmic (op term)
    (declare (ignore op))
    (with-tex-output
      (format nil "{\\ln ~A}"
              (convert-for-display term))))

  (define-converter logarithmic (op term base)
    (declare (ignore op))
    (with-tex-output
      (format nil "{\\log_~a ~a}"
              (convert-for-display base)
              (convert-for-display term))))
#+END_SRC

** WORKING Converter [0/7]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:18>
:ID:       88d433ad-e381-4747-8a29-2d78bc759fbf
:END:

#+Caption: Conversion Driver
#+Name: stf-conversion-driver
#+BEGIN_SRC lisp
  (defun convert-for-display (function)
    (if (and (listp function)
           (member (first function) '(and or not = sum integrate parens)))
        (let ((operator (first function)))
          (cond
            ((eq operator 'and)
             <<stf-and-operator>>
             )
            ((eq operator 'or)
             <<stf-or-operator>>
             )
            ((eq operator 'not)
             <<stf-not-operator>>
             )
            ((eq operator '=)
             <<stf-equality-operator>>
             )
            ((eq operator 'sum)
             <<stf-summation>>
             )
            ((eq operator 'integrate)
             <<stf-integration>>
             )
            ((eq operator 'parens)
             <<stf-parenthesis>>
             )))
        (let ((rule (get-rule function)))
          (when rule
            (apply rule (ensure-list function))))))
#+END_SRC

*** TODO And
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:38>
:ID:       733b98a1-90f1-4d13-abe8-cb86a5608aee
:END:

Foo

#+Caption: And Operator
#+Name: stf-and-operator
#+BEGIN_SRC lisp
  (destructuring-bind (op &rest terms) function
    (declare (ignore op))
    (with-tex-output
      (format nil "{~{~A~^ \\wedge ~}}"
              (mapcar #'convert-for-display terms))))
#+END_SRC

*** TODO Or
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:38>
:ID:       276de305-32c4-4f79-96e7-d0a99ff24f78
:END:

Foo

#+Caption: Or Operator
#+Name: stf-or-operator
#+BEGIN_SRC lisp
  (destructuring-bind (op &rest terms) function
    (declare (ignore op))
    (with-tex-output
      (format nil "{~{~A~^ \\vee ~}}"
              (mapcar #'convert-for-display terms))))
#+END_SRC

*** TODO Not
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:38>
:ID:       1b0a28a4-744d-44d1-a328-7b2bb10bd0c7
:END:

Foo

#+Caption: Not Operator
#+Name: stf-not-operator
#+BEGIN_SRC lisp
  (destructuring-bind (op term) function
    (with-tex-output
      (format nil "{\\not ~A}"
              (convert-for-display term))))
#+END_SRC

*** TODO Equality
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:41>
:ID:       4ce4835c-e196-4494-ab4b-591690e4164c
:END:

Foo

#+Caption: Equality Operator
#+Name: stf-equality-operator
#+BEGIN_SRC lisp
  (destructuring-bind (op lhs rhs) function
    (declare (ignore op))
    (format nil "{~A = ~A}"
            (convert-for-display lhs)
            (convert-for-display rhs)))
#+END_SRC

*** TODO Summation
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:24>
:ID:       98404213-b8b8-410f-b660-23b701518cea
:END:

#+Caption: Summation
#+Name: stf-summation
#+BEGIN_SRC lisp
  (destructuring-bind (op start stop expression) function
    (declare (ignore op))
    (format nil "{\sum_~A^~A ~A}"
            (convert-for-display start)
            (convert-for-display stop)
            (convert-for-display expression)))
#+END_SRC

*** TODO Integration
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:39>
:ID:       60c16d30-2bb3-497c-aaa0-4529ecfc523c
:END:

#+Caption: Integration
#+Name: stf-integration
#+BEGIN_SRC lisp
  (destructuring-bind (op from to expression wrt) function
    (declare (ignore op))
    (with-tex-output
      (format nil "{\\int_~A^~A ~A\\,\\mathrm{d}~A}"
              (convert-for-display from)
              (convert-for-display to)
              (convert-for-display expression)
              (convert-for-display wrt))))
#+END_SRC

*** TODO Parenthesis
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:24>
:ID:       93d643d6-2219-4c49-bba5-190520a6ff29
:END:

#+Caption: Parenthesis
#+Name: stf-parenthesis
#+BEGIN_SRC lisp
  (destructuring-bind (op type expression) function
    (declare (ignore op))
    (let* ((types '((square . ("[" . "]"))
                    (curly . ("{" . "}"))
                    (smooth . ("(" . ")"))))
           (left (cadr (assoc type types)))
           (right (cddr (assoc type types))))
      (with-tex-output
        (format nil "{\\left~a {~a} \\right~a}"
                left
                (convert-for-display expression)
                right))))
#+END_SRC

** DONE Special Macros
CLOSED: [2016-06-25 Sat 16:27]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:20>
:ID:       56ca6afe-912a-4530-91e4-a63123dc6d9d
:END:

There is one specialty macro, ~with-tex-output~, which is used to ensure that an expression is wrapped to be a part of correct (La)TeX output.  It works by checking to see whether or not the variable ~*tex-outputp*~ is true, if so, it simply pass through the given body, and if not, it binds the variable to ~t~, and makes sure that the given body is wrapped in ~$~, allowing the expression to be typeset correctly.

#+Caption: Special Macros
#+Name: stf-special-macros
#+BEGIN_SRC lisp
  (defvar *tex-outputp* nil)
  (declaim (special *tex-outputp*))

  (defmacro with-tex-output (&body body)
    `(if *tex-outputp*
         (progn
           ,@body)
         (let ((*tex-outputp* t))
           (format nil "$~a$"
                   (progn
                     ,@body)))))
#+END_SRC

** DONE Assembly
CLOSED: [2016-06-24 Fri 21:34]
:PROPERTIES:
:CREATED:  <2016-06-14 Tue 17:15>
:ID:       bbd15b88-8256-4b5b-abcc-4783fc096c29
:END:

The final assembly of this portion of the system is as simple as the rest, resolving dependencies and placing everything in a single file.  As normal, this is done using NoWeb syntax, with everything tangled to the file ~larcs-typeset.lisp~.

#+Caption: Assemble Symbolic to Typeset Form
#+Name: stf-assemble
#+BEGIN_SRC lisp :tangle "larcs-typeset.lisp"
  (in-package #:larcs.typeset)
  <<stf-special-macros>>
  <<stf-rule-retrieval>>
  <<stf-define-rule>>
  <<stf-conversion-driver>>
  <<stf-rules>>
#+END_SRC

* WORKING Symbolic Solver [0/3]                                    :noexport:
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 17:55>
:ID:       ffa664b6-e147-471c-9e97-f49c4619fc99
:END:

** TODO Techniques
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 17:55>
:END:

** TODO Rules
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 17:56>
:END:

** TODO Assembly
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 17:56>
:END:

* WORKING Symbolic Trigonometry [0/2]                              :noexport:
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 17:58>
:ID:       b0259b8c-e285-48ca-bda0-4f04e3bc6ef2
:END:

** TODO Rules
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 17:58>
:END:

** TODO Assembly
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 17:58>
:END:

* WORKING Symbolic Evaluation of Limits [0/2]                      :noexport:
:PROPERTIES:
:CREATED:  <2016-07-18 Mon 18:15>
:END:

** TODO Rules
:PROPERTIES:
:CREATED:  <2016-07-18 Mon 18:19>
:END:

** TODO Assembly
:PROPERTIES:
:CREATED:  <2016-07-18 Mon 18:20>
:END:

* WORKING Symbolic Integration [0/3]                               :noexport:
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 18:02>
:ID:       61add971-a129-40d0-be13-24f8a4afc522
:END:

** TODO Rules
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 18:02>
:END:

** TODO Techniques
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 18:02>
:END:

** TODO Assembly
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 18:02>
:END:

* DONE Library Assembly [2/2]
:PROPERTIES:
:CREATED:  <2016-06-11 Sat 22:30>
:ID:       89370949-8f58-41cf-8c4f-92f81d48ac23
:END:

LARCS is primarily a library, and as such, the formal interface for the library, and a way to load it must be provided.  This is done here by defining packages centrally and defining an ASDF system to load the library quickly and easily.

** DONE Package Definitions [6/6]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 15:00>
:ID:       573a8352-8cbe-408c-8c27-3cf0b66da885
:END:

As all of the packages are defined centrally, this makes resolving inter-package dependencies much easier and convenient.

#+Caption: LARCS Packages
#+Name: larcs-packages
#+BEGIN_SRC lisp :tangle "larcs-packages.lisp"
  <<common-package-def>>
  <<classification-package-def>>
  <<polynomial-package-def>>
  <<manipulation-package-def>>
  <<differentiation-package-def>>
  <<typesetting-package-def>>
#+END_SRC

*** DONE Common
CLOSED: [2016-08-21 Sun 11:53]
:PROPERTIES:
:CREATED:  <2016-08-05 Fri 22:07>
:ID:       84845974-c272-4007-a490-6bc328a8d22d
:END:

This defines the common package, which keeps a few macros and variables that are used by all other packages.  It exports a function to generate arguments lists, a list of special symbols to TeX sequences and a list of constant names.

#+Caption: Common Package Definition
#+Name: common-package-def
#+BEGIN_SRC lisp
  (defpackage #:larcs.common
    (:use #:cl)
    (:import-from #:alexandria
                  #:symbolicate)
    (:export #:gen-args-list
             #:*special-symbols-to-sequences*
             #:*constant-names*)
    (:nicknames #:common))
#+END_SRC

*** DONE Classification
CLOSED: [2016-08-21 Sun 12:03]
:PROPERTIES:
:CREATED:  <2016-08-05 Fri 22:07>
:ID:       f68d9d69-3cca-4767-8204-8077ae8ac2b2
:END:

Classification is another system that is used by just as many packages as the common packages.  In this case, it exports two functions (one to get classification and one to check whether something is classified as a given classification) and a macro (the case pattern for classification).

#+Caption: Classification Package Definition
#+Name: classification-package-def
#+BEGIN_SRC lisp
  (defpackage #:larcs.classify
    (:use #:cl
          #:larcs.common)
    (:import-from #:alexandria
                  #:symbolicate)
    (:export #:classify
             #:classified-as-p
             #:classification-case)
    (:nicknames #:classify))
#+END_SRC

*** DONE Polynomial Functions
CLOSED: [2016-08-21 Sun 12:09]
:PROPERTIES:
:CREATED:  <2016-08-21 Sun 10:58>
:ID:       c97b187c-22c8-4332-902e-3be2247b277e
:END:

Polynomial Functions are used in several of the other packages, and provide a way of comparing and retrieving information about specific terms, or providing a way to collect information about variables.

#+Caption: Polynomial Package Definition
#+Name: polynomial-package-def
#+BEGIN_SRC lisp 
  (defpackage #:larcs.polynomials
    (:use #:cl
          #:larcs.common
          #:larcs.classify)
    (:export #:collect-variables
             #:coefficient
             #:term-variable
             #:get-power
             #:term-order-<
             #:term-order-=
             #:term-order->
             #:save-variable-p
             #:single-term-combinable-p)
    (:nicknames #:polynomials))
#+END_SRC

*** DONE Manipulation
CLOSED: [2016-08-21 Sun 12:21]
:PROPERTIES:
:CREATED:  <2016-08-05 Fri 22:07>
:ID:       07a9d8fc-5bdc-4c03-9df3-3e3d46d1b704
:END:

This package provides for a method to manipulate and perform algebra symbolically.  In a way, this is the core of the library, upon which nearly everything else is built.

#+Caption: Manipulation Package Definition
#+Name: manipulation-package-def
#+BEGIN_SRC lisp
  (defpackage #:larcs.manipulate
    (:use #:cl
          #:larcs.common
          #:larcs.classify
          #:larcs.polynomials)
    (:import-from #:alexandria
                  #:symbolicate)
    (:export #:manipulate
             #:collect-terms)
    (:nicknames #:manipulate))
#+END_SRC

*** DONE Differentiation
CLOSED: [2016-08-21 Sun 13:19]
:PROPERTIES:
:CREATED:  <2016-08-05 Fri 22:07>
:ID:       53a94c2e-3fb1-4cc1-9b3a-0e53657c3b91
:END:

This is a rather simple package, providing a way to differentiate equations for use in a variety of functions.

#+Caption: Differentiation Package Definition
#+Name: differentiation-package-def
#+BEGIN_SRC lisp
  (defpackage #:larcs.differentiate
    (:use #:cl
          #:larcs.common
          #:larcs.classify
          #:larcs.manipulate
          #:larcs.polynomials)
    (:import-from #:alexandria
                  #:symbolicate)
    (:import-from #:com.informatimago.common-lisp.cesarum.list
                  #:aget
                  #:ensure-list)
    (:export :differentiate)
    (:nicknames :diff))
#+END_SRC

*** DONE Typesetting
CLOSED: [2016-08-21 Sun 13:35]
:PROPERTIES:
:CREATED:  <2016-08-05 Fri 22:07>
:ID:       f3ae4f48-7fc9-4f4c-a128-642cd3babaf1
:END:

Again, this is a relatively simple package definition, providing a way to convert symbolic equations to a form that can be more easily read by most humans.

#+Caption: Typesetting Package Definition
#+Name: typesetting-package-def
#+BEGIN_SRC lisp
  (defpackage #:larcs.typeset
    (:use #:cl
          #:larcs.common
          #:larcs.classify
          #:larcs.manipulate
          #:larcs.polynomials)
    (:import-from #:alexandria
                  #:symbolicate)
    (:import-from #:com.informatimago.common-lisp.cesarum.list
                  #:aget
                  #:ensure-list)
    (:export #:convert-for-display)
    (:nicknames #:typeset))
#+END_SRC

** DONE System Definition
CLOSED: [2016-08-21 Sun 11:23]
:PROPERTIES:
:CREATED:  <2016-06-13 Mon 15:00>
:ID:       35b2ec01-a933-4b5b-af73-b6b7f1c45cb6
:END:

This defines the LARCS Library system, allowing the functionality to be loaded by other systems or to provide a way to make development quickly and easily.

#+Caption: Library System Definition
#+Name: library-system-definition
#+BEGIN_SRC lisp :tangle "larcs-lib.asd"
  (asdf:defsystem #:larcs-lib
    :description "A CAS Library for use within Lisp Software."
    :author "Samuel Flint <swflint@flintfam.org>"
    :license "GNU GPLv3 or Later"
    :depends-on (#:alexandria
                 #:com.informatimago)
    :serial t
    :components ((:file "larcs-packages")
                 (:file "larcs-common")
                 (:file "larcs-classify")
                 (:file "larcs-polynomials")
                 (:file "larcs-manipulation")
                 (:file "larcs-manipulators-addition")
                 (:file "larcs-manipulators-subtraction")
                 (:file "larcs-manipulators-multiplication")
                 (:file "larcs-manipulators-division")
                 (:file "larcs-manipulators-trig")
                 (:file "larcs-differentiate")
                 (:file "larcs-typeset")))
#+END_SRC

* Push to bottom                                                     :ignore:
:PROPERTIES:
:CREATED:  <2016-07-17 Sun 13:58>
:END:

#+LATEX: \newpage

* Version Information
:PROPERTIES:
:CREATED:  <2016-07-17 Sun 13:58>
:UNNUMBERED: t
:END:

This document is version src_sh{git describe --always --long --dirty --abbrev=10 --tags}.