The Broken Weight Problem

Some elderly academics were sitting next to me at the cafe today and they were discussing some mathematical puzzle, "The Broken Weight Problem." I'm not the best eavesdropper so I gathered a few search terms and popped them into Google, and viola.

Basically the idea is this. A merchant accepts goods in 40 pound increments. He uses a simple balance scale and a 40 pound weight to weigh the goods. One day, he accidentally dropped the weight and it broke into four pieces. After a bit of consideration, the merchant realized that, quite amazingly, each piece was an integer number of pounds in weight and, using his broken pieces of the weight and his balance scale, he was able to measure out any integer value from 1 to 40. The question is, what are the weights of the pieces? Hints and solutions after the break

---

Okay, a hint in case you are really stuck. You do realize that with a balance scale, the merchant is able to place some pieces on the other side of the scale which will act as a counter balance? This means that a five pound and a three pound piece can also measure two pounds, for example.

---

Now for solutions. Since we probably have a Common Lisp REPL close by, the easiest thing to do in this day and age is to just plug it in and have a computer solve it. But how do we do this? Well, it is simple enough using Screamer.

I start by grabbing a few libraries I will use, Iterate and, as mentioned, Screamer.

(ql:quickload '(:screamer :iterate))
(use-package :iterate)

First we define a test to see if a given set of piece weights can build an integer:

(defun check-integer (i w x y z)
  (iter (for s1 from (- w) to w by w)
    (finding
     s1 such-that
     (iter (for s2 from (- x) to x by x)
       (finding
        s2 such-that 
        (iter (for s3 from (- y) to y by y)
          (finding
           s3 such-that 
           (iter (for s4 from (- z) to z by z)
             (finding s4 such-that (= i (+ s1 s2 s3 s4)))))))))))

This function is pretty self explanatory. Given a set of weights, each weight can be in one of three places, on the right side of the scale, on the left side of the scale, or not on the scale at all. When applied to the balance of the scale (where zero means we have a balance) these states correspond to a negative value of the weight, a positive value of the weight, and a zero value for the weight. This is exactly what we are doing with these loops. Since there are only three states for each, it is simple to see that this will only require checking $3^{4}$ different possibilities at worst case.

Then we delve into the magic world of nondeterminism.

(screamer::defun stone-split (total-weight)
  (let* ((w (screamer:an-integer-between 1 (1+ total-weight)))
         (x (screamer:an-integer-between w (1+ total-weight)))
         (y (screamer:an-integer-between x (1+ total-weight)))
         (z (- total-weight w x y)))
    (if (and (> z 0)
             (iter (for i from 1 to total-weight)
               (let ((works (check-integer i w x y z)))
                 (always works))))
        (list w x y z)
        (screamer:fail))))

Here we are defining our function that will find zero, one, or several solutions to this problem depending on how we call it and if solutions exist. I left a free parameter total-weight so we can play with it, though it proves a bit boring to play with only that value. Things to note:

1. We need to use the defun from within the screamer package. This version of defun is special as it allows for simulated nondeterministic computation (via CPS I believe, but I'm not the guy to ask about that). The library suggests that you use screamer:define-screamer-package to make a package with the proper defun interned, but I am lazy.
2. The forms using screamer:an-integer-between tell Screamer that the solutions to this problem are some integer in that range. Screamer will handle figuring out which.
3. The screamer:fail form informs Screamer that it has chosen the wrong value. This will force Screamer to backtrack to the last integer range with values left and pick a new one.

We need to execute this function from within a "Nondeterministic Environment," which is commonly either inside a screamer:one-value or screamer:all-values form. So we invoke it like this ( fixed package namespace thanks to Derrell):

(screamer:one-value (stone-split 40))

…and it finds the correct result. Admittedly, this could be solved very simply without the use of Screamer, but I would argue that it requires less mental overhead when and produces much more flexible code using the Screamer route. What is the correct result?

---

I looked at it this way. You have three values a piece of weight $w$ can take, $-w$, 0, or $+w$. Therefore it seems to me that we have a set of equations:

\[ \alpha_i w + \beta_i x + \gamma_i y + \delta_i z = i ~~~~~\forall i \in {\mathbb N}_{1}^{40} \]

Where I use ${\mathbb N}_{1}^{40}$ to denote the natural numbers (starting from 1) less than or equal to 40. We know from the problem description that $w,x,y,z, \in {\mathbb N}_{1}^{40}$ and that $(\alpha, \beta, \gamma, \delta)+1 \in {\mathbb N}_{0}^{2}$. You could go about trying to solve these very over determined equations (edit: Actually it's underdetermined, hence we get multiple solutions. Limiting the values to integers is what makes it hard), however, this all might start looking familiar. This looks very much how you would calculate the value of a number in some (possibly irregular) base where the Latin symbols denote the value of each digit and the Greek symbols denote the value in that digit place. In this case, we are limited to three different values in our digit places, so it seems prudent to use base three. Thus the values of each digit place should be powers of three. Thus the answer should be 1, 3, 9, and 27.

How do we know we will have a solution? We need our trinary representation at least reach 40. Since half of our numbers are negative, we actually need to reach 80 in the positive only encoding. The largest value for four digits of trinary is (+ (* 2 27) (* 2 9) (* 2 3) (* 2 1)) or exactly 80. Also, it is a happy coincidence that our numbers also add up to 40, everything works just fine. If we look at how many unique solutions exist for other total weights we see that it seems that 40 is the largest weight we can get to with four pieces. This is a direct consequence of 80 being the largest 4 digit trinary number.

Total Weight	Number of unique solutions
4	1
5	1
6	2
7	3
8	4
9	5
10	7
11	7
12	9
13	10
14	10
15	11
16	12
17	11
18	12
19	12
20	11
21	11
22	12
23	9
24	9
25	9
26	7
27	7
28	7
29	5
30	5
31	5
32	3
33	3
34	3
35	2
36	2
37	2
38	1
39	1
40	1
41	0
42	0
43	0
44	0
45	0

On LaTeX and Microsoft Word

Sorry, this is a ranty post. Adjust your desire to read appropriately.

I was busy writing my APS abstract this last week. When we got near the end I was having a heck of a time getting Bibtex to work with Latex and insert my references properly. Later on I realized that APS doesn't really allow for citation ala Latex and Bibtex, but whatever. I was struggling with this, and mentioned to my advisor, jokingly, "Man, I hate Latex." He says, also jokingly, "you could use Microsoft Word." I say sure, despite the fact that the APS website says they prefer Latex and that Latex is actually easier for the submitter, MS Word is no problem. In fact, that is kind of the point. I wrote my abstract using Org-Mode precisely because it made little to no assumptions on the end result. I can export to Latex or one of the many other formats or just grab my text and paste it into MS Word with no problems. I can also send it to any human being in the world with a computer and that person can open it and understand what they are seeing. Hell, in a handful of keystrokes I could post it to this blog.

Later, when printing, my advisor changes from his joking suggestion to insist that I use MS Word (presumably so that I can export to a Rich Text Format document that the APS accepts). This will make it easier, right… So, I copy/paste it over to a document and save it as an RTF file. Low and behold, of course, the equations are not interpreted by word. It would take at least 15 minutes and perhaps up to an hour to figure out how to get the proper symbols, fonts, and kerning into the document for submission. The last kick to the nuts is that the APS provides an RTF template, but in order to use it, you must actually type the document into it yourself, by hand. If you copy and paste it into the template something, apparently, will get screwed up. I will repeat that, the American Physical Society requires you to actually, physically, press the keys on your keyboard while their template is open in order to submit an abstract (though I imagine something like xdotool might serve me well here). This is what happens with you use things you don't understand people. You get black magic crap like this.

Point is, I miss my previous colleagues and advisor, nay entire department, nay entire institution where they held the opinion that any individual in the sciences should be using Latex as their format for correspondence. Also, bravo to the APS for encouraging people to use Latex for submitting abstracts.

Time Lapse Fun with FFMPEG and Mencoder

I was in the office early this morning and saw the sun rising. I captured some video of it (my phone can do 30 minutes of 720p, though it doesn't look that good). However, the thirty minutes of video is rather boring to watch in real time. So I decided to speed up the video.

Speeding up the video was actually much more difficult that I thought it would be (and I am somewhat acclimated to the troublesome nature of FFMPEG and Mencoder). Anyway, in order to perform the speed up I used the FFMPEG filter select to only pass every n^th frame. The large the n, the faster the speed up. At the same time I chose to convert to WebM (the phone puts out H.264+AAC and placed in an MPEG4 container format). This produces a video file with only every n^th frame, but the video just hangs for the time it would have been displaying the other frames. To fix this, I then passed it through another program, mencoder. This program fairly similar to ffmpeg in purpose. The mencoder program has a special option, -speed, that will speed up the video stream by a factor you specify. This means that if we use -speed n then we should get the desired video frame rate. Refer to the man pages and FFMPEG manual for more insight for what is happening here.

ffmpeg -i VID_20111025_071454.m4v -vf select='not(mod(n\,50))' -an temp.webm
mencoder -ovc copy -oac copy -speed 50 temp.webm -o faster.webm

This would have been all it would take, however, since I am using Maverick Meerkat, the version of FFMPEG included doesn't have support for filters. This means that, ugh, you can't use the first command. Not unless you install a newer version. I avoid compiling libraries from source like the plague. Instead I added this PPA to my software sources and updated my software. This actually gave a lot of scary errors and almost tricked me into a dist-upgrade to Natty, (shame on you apt, I wouldn't do that to my worst enemy). But in the end, those commands did work (as evidenced by the videos).

The end result, we have a video where only every 50th frame show up in the output. This resulted in a pretty good effect. I didn't actually point the camera at the sunrise as that would cause problems with the contrast and I couldn't balance the camera on the window sill pointing that direction from my office. So it's not as dramatic as an actual sunrise.

I also captured another 30 minutes later that morning as the clouds forming as the air passed off the flatirons was a pretty cool dynamic effect.

Of course, all of this is kind of stupid (neat, to me, but stupid), as time-lapse is usually done by taking individual frames spaced seconds to minutes (or more) apart. Also, there are several options available on the Android Market. Though I wouldn't install them without researching the application and developers.

Fixed Point Emacs Completion

One of those annoying things about GNU Emacs is that when a window is popped up containing completions, it derails what you were doing.

First, completion windows take up half the frame. I can't actually figure out how to change this easily. Think of how you use the completions window. One primary way I use it is: I try to complete a symbol, realize that it doesn't have enough information, then type more and try again. In this case, having so many options doesn't help at all. I only need to know if there the symbol can be completed or if there are multiple possibilities.

Second, the position of the cursor on the screen is almost always changed when a window is popped up, and interestingly enough, not even in a predictable way. This means that if I try to perform a completion which I expect to have only one completion, thus not popping up a window, but it actually has several possibilities, I have an annoying situation. This could happen as a result of a symbol I didn't know about making the completion ambiguous resulting in a pop-up and a partial completion or it could be due to a typo. In either case, I need to examine what is at the command prompt in order to find out the proper course of action. This is unavoidable. What is avoidable, however, is the need to move the command prompt around the screen forcing the user to go searching for the new location.

In addition to these pretty legitimate claims, I also just find the entire process to be jarring. I often find myself kicked out of my current train of thought as I search for where my cursor has gone. Here is an example of how completion tends to work in Emacs.

When these two effects combine, I got so frustrated I started leaving my frames split in half so that completions will show up in the other window without effecting the window I was editing in at all. In a recent Reddit conversation with Stassats, he states that he also always maintains a completion window. To me, this is a big waste of screen space, but at least the point doesn't jump.

There are other solutions to this problem. My guess is that many people struggle with this and each produce their own solution. For instance, here is a post of how to designate a particular window for completions in someones (rather complicated for my taste) completion window setup.

My solution

The code is here, though I can't vouch for it being bug free.

My solution is to have windows pop-up with completions but have those windows be smaller than half the screen. An important distinction is that when a completion window pops up, it will not alter the screen location of the point. In order for this to be the case, we need to pop-up a window in an area of the frame that the cursor isn't utilizing. So, we handle this by identifying where in the frame the point is, then popping up a smaller completion window in either the top or bottom of the frame depending on its location.

To be precise:

1. See if there already is a visible completions window. If one exists, use it.
2. See if the current frame has only one window (when not counting the minibuffer). If it does, we know that a popup is desired.
   1. If the current window is actually the minibuffer, create the popup at the bottom of the frame.
   2. If the point is above the halfway point in the frame, create the popup window at the bottom of the frame.
   3. If below the halfway point, create it at the top of the frame.
3. If it matches none of this, let Emacs do as it would have.

Here is the result:

This works pretty well for me, but I imagine that is just because I might use Emacs in a quirky way. For instance, I usually have one desktop where all of my Emacs windows are. If they are on different desktops, perhaps the "use completions window if 'visible'" won't work very well. I don't know how Emacs handles this. What about terminal emacsclient instances and how does completion there interact with open X windows. All of these things should be configurable.

Anyhow, if anybody else finds this useful and wants to push patches back to me, please do. If anybody wants to grab this code and take over the project, or include it in your project, again, please do.

Other completion extension libraries

While I haven't found anything out there that does exactly what Fixed-Point-Completions does, there are a few libraries that also tweak completion. Most notably, popwin and icicles extend the completions functionality. Icicles is a bit heavy handed for my taste, changing many things in how Emacs functions. Popwin does a lot towards fixing my completion issues, but not enough. The thing which is really missing from both of these is that they still shift the visual location of the point at times.

Modf support on ABCL

No posts in a long time, sorry for that.  Probably going to be dead around here until PhD is taken care of.    I take solace in the fact that nobody reads this blog ;)

One good piece of news that crossed my RSS feed today: ABCL 0.26.2 was released.  Normally this isn't something I would really care that much about, but the change log says they fixed the apply/setf problem, which is a current failing of Modf on ABCL.  This means that, while Modf will still fail the test suite (due to no MOP, which is needed to expand class accessors if the user didn't specify an expander), it will work much more closely to how it does on other Lisps.  I think it is correct to say that Modf will work as long as all expanders are actually specified (i.e. not expanded at run time).  This is great news for any ABCL+Modf users of which I'd bet there are approximately zero.  Maybe now there will be some.

This should just work without any changes to Modf, but I'll try to find time to test it sometime soon.

Emacs Pinky

Whelp, it has happened to me.  After switching to Emacs 2-3 years ago, I have finally developed a repetitive stress injury due to my programming.  I don't think this would have happened with Vim (my old editor), but I have been coding much more these days.  Whatever.

I am resistant to using Viper mode as I think it will mess with the key bindings in the 20 some odd major modes I use every day.  So I am going to try something else.  I am deleting the Ctrl and Shift modifier keys from the X key bindings on the left side of my keyboard and will force myself to use Shift and Ctrl on the right hand side for a while.  Spreading the wear out over both hands should help things a bit.  Of course things are complicated a bit by the fact that this Mac keyboard doesn't have a left control at all.  No biggie with xmodmap.  Just use xev to figure out what keys send what code and make a xmodmap input file like this.

clear control
clear mod1
clear mod4
clear shift

keycode 50 = Shift_L
keycode 62 = Shift_R
keycode 37 = Control_L
keycode 108 = Control_R
keycode 64 = Super_L
keycode 134 = Alt_R
keycode 133 = Alt_L

add shift = Shift_R
add control = Alt_R
add mod1 = Alt_L
add mod4 = Super_L

Oh yeah, if you are using Ubuntu or probably any mainstream distribution, changes to xmodmap are often times intercepted on startup and different key mapping utility is used. in short, make this file as xmodmap in your home space, apply it, then restart the X server. You will probably get a dialog asking you if you want to include these key mappings or not.

Just run xmodmap <input-file> to effect the changes.  Also, I am going to try and mimic the key placement of the older space cadet keyboards, which were made for Emacs.  The Wikipedia page states that a big difference was that the space cadet keyboards had modifier keys in a "thumbable" position.  So I am leaving my left Alt as Alt and setting my right Alt as a Ctrl.  This means that in a few weeks time I should be using thumbs for most Emacs work (and everything work, as xmodmap changes are X global).

Now, the only thing I use my left pinky for is typing and Tab, which seems fair.  I'll see if things get better.  Maybe I will end up in Viper mode.

Update: Things did get better, but just barely. I do believe that the numbness is caused primarily by just a few common Emacs commands, like C-c C-c, C-x C-s, or any command that uses the C-c or C-x prefixes. In fact, my opinion is that the left hand is way over used in Emacs. I had to re-enable the left shift key for the sake of my typing speed. This has also revealed that Apple cut a few corners when it came to their keyboard hardware (everybody does). The left and right alt keys, when pressed at the same time, will stop any keys on the "zxcv" row from even producing key codes, so that's a pain right now. As of testing just a few minutes ago, the numbness comes back with just a few minutes of typing on a normal keyboard layout.

Update (after a few months): This has basically been resolved.  A few thoughts: mapping to thumbs seemed like a good idea, but thumbs get worn out too.  After a few months, my thumbs were pretty ache-y.  In the end, the two things I found that really did make a difference were: 1. binding caps-lock as a control and 2. switching to a Dvorak layout, more specifically the Programmers Dvorak layout, which I think is pretty smart.  I can't say how much each of these contributed individually, as I changed them simultaneously.  I plan to write a post on Programmers Dvorak soon but suffice it to say, switching layouts isn't a decision to take lightly.  The caps-lock thing, on the other hand, is dead simple and helps tremendously; I very much suggest it.  I only wish I had another key right beside the caps-lock (or in the pinky position on the right hand) that I could bind to Meta/Alt.

Introducting Index-Mapped-Arrays

I was writing a blog post regarding my release of a library I have been using for a while now but have recently cleaned up, but it got too wordy. I will post it as well (perhaps already have), but I wanted to have a short post showing the capabilities of this library. That way people might actually be interested enough to download the library or read the other post. So without further ado, I present Index-Mapped-Arrays:

Index-Mapped-Arrays provides a generic interface, imref, to Common Lisp types.

(imref '(a b c d e f) 2)
;; C

(imref '(((a b) (c d)) ((e f) (g h))) 1 1 1)
;; H

(imref '((1 2 3) (4 5 6) (7 8 9)) 1 1)
;; 5

(imref #2A((1 2 3) (4 5 6) (7 8 9)) 1 1)
;; 5

;;; As well as setting

(let ((ima '((1 2 3) (4 5 6) (7 8 9))))
  (setf (imref ima 1 1) 'set)
  ima )
;; ((1 2 3) (4 SET 6) (7 8 9))

(let ((ima #2A((1 2 3) (4 5 6) (7 8 9))))
  (setf (imref ima 1 1) 'set)
  ima )
;; #2A((1 2 3) (4 SET 6) (7 8 9))

As the name implies, you can map indices. Mapping indices allows you to do things like pull vectors out of matrices and grab sub-regions of arrays.

(defparameter *arr* '((1 2 3) (4 5 6) (7 8 9)))

;; Note, it returns a list
(row-vector *arr* 1)
;; (4 5 6)

;; We can't easily get a column vector, so we return an IMA object.
(column-vector *arr* 1)
;; #1D-IMA(2 5 8)

;; Print readably
(let ((*print-readably* t))
  (print (column-vector *arr* 1)) )
;; #(2 5 8) ; Printed as a normal array
;; #1D-IMA(2 5 8)

;; Same sort of thing with a submatrix
(submatrix *arr* 0 1 3 2)
;; #2D-IMA((2 3) (5 6) (8 9))

(row-vector (submatrix *arr* 0 1 3 2) 1)
;; #1D-IMA(5 6)

Note that IMA tries its best to give you a native data structure back, but if it can't, it gives to an instance of class index-mapped-array which emulates the proper mapping.

But that's not all. Index maps are themselves setfable places.

(setf (column-vector *arr* 0) '(a b c))
;; (A B C)

*arr*
;; ((A 2 3) (B 5 6) (C 8 9))

(defparameter *subarr* (submatrix *arr* 1 1 2 2))
;; *SUBARR*

(setf (row-vector *subarr* 0) '(d e))
;; (D E)

*subarr*
;; #2D-IMA((D E) (8 9))

*arr*
;; ((A 2 3) (B D E) (C 8 9))

And to get all of this for any data structure that can be accessed via a list of integer indices, all you have to do is specialize the methods, ima-dimensions, ima-dimension, imref, and (setf imref) (or if your data structure is immutable immod). Do this, and everything above just works, and it's portable so it should work everywhere the spec is obeyed (I test on SBCL, CMUCL, CCL, CLISP, ECL, and ABCL. All but ABCL work).

And don't think that this is only useful for blocks of data. It's a bit more flexible than that. Observe:

(defclass point () ((x-value :initarg :x-value :initform 0 :accessor x-of)
                    (y-value :initarg :y-value :initform 0 :accessor y-of)
                    (z-value :initarg :z-value :initform 0 :accessor z-of) ))

(defmethod print-object ((point point) str)
  (format str
          "#<Point: x=~A y=~A z=~A>"
          (x-of point)
          (y-of point)
          (z-of point) ))

(make-instance 'point :x-value 1 :y-value 2 :z-value 3)
;; #<Point: x=1 y=2 z=3>

;; Define the IMA interface
(defmethod ima-dimensions ((ima point))
  (list 3) )
(defmethod ima-dimension ((ima point) n)
  3 )

(defmethod imref ((point point) &rest i)
  (case (first i)
    (0 (x-of point))
    (1 (y-of point))
    (2 (z-of point))
    (otherwise (error "Index ~S out of bounds" i)) ))
(defmethod (setf imref) (new-value (point point) &rest i)
  (case (first i)
    (0 (setf (x-of point) new-value))
    (1 (setf (y-of point) new-value))
    (2 (setf (z-of point) new-value))
    (otherwise (error "Index ~S out of bounds" i)) )
  new-value )

(let ((point (make-instance 'point :x-value 1 :y-value 2 :z-value 3)))
  (iter (for i below 3)
    (collect (imref point i)) ))
;; (1 2 3)

For extra functionality, and you do want it, you can use the def-unmapper macro and specialize the make-ima-like method which will tell IMA how to change other IMAs into this kind of IMA, and how to make an another IMA like this one so we can fill it with data, respectively.

(defmethod make-ima-like ((point point) &key &allow-other-keys)
  (make-instance 'point) )

(def-unmapper point (ima)
  (when (or (< 1 (length (ima-dimensions ima)))
            (< 3 (ima-dimension ima 0)) )
    (error "Can only map a vector with 3 or fewer elements to a point") )
  (let ((point (make-instance 'point)))
    (setf (a-of point) (imref ima 0)
          (b-of point) (imref ima 1)
          (c-of point) (imref ima 2) )
    point ))

;; Or, more interestingly
(let ((point1 (make-instance 'point :x-value 1 :y-value 2 :z-value 3))
      (point2 (make-instance 'point :x-value 3 :y-value 2 :z-value 1)) )
  (list (v:dot point1 point2)
        (v:cross point1 point2)
        (v:+ point1 point2)
        (v:- point1 point2) ))
(10 #<Point: x=-4 y=8 z=-4> #<Point: x=4 y=4 z=4> #<Point: x=-2 y=0 z=2>)

This works because my vector arithmetic library uses IMA to access the elements of its arguments, and uses make-ima-like to make the structures that it returns to the caller. The unmapper is used if you need to get some mapped data into a equivalent form but without any of the index mappings.

And if you thought that was all, I just added a new feature, a map simplifier. This means that in certain simple cases, multiple levels of mappings are reduced to a single map. For instance, taking a transpose of a transpose will return the original array. This means there is less to fear when it comes to using a mapping as part of long recursion or iteration.

(let ((ima '((1 2) (3 4))))
  (iter (for i below 1000)
    (setf ima (transpose ima)) )
  ima )
;; ((1 2) (3 4)) <- the original list

(let ((ima '((1 2) (3 4))))
  (iter (for i below 1001)
    (setf ima (transpose ima)) )
  ima )
;; #2D-IMA((1 3) (2 4)) <- Only one mapping here

;; Let's really test the simplifier
(defun count-items (item-to-count ima)
  (cond ((= 0 (ima-dimension ima 0))
         0 )
        ((eql item-to-count (imref ima 0))
         (+ 1 (count-items item-to-count (subvector ima 1))) )
        (t (count-items item-to-count (subvector ima 1)) )))

;; We have to make a kind of complicated vector structure because if the vector
;; is of type list or array, there are quick, native ways of getting at the
;; subvector we need, so the simplifier isn't used at all.
(let ((ima (column-vector
            (iter (for i below 10000)
              (collect (list 1 (alexandria:random-elt '(not-it it)))) )
            1 )))
  (time (count-items 'it ima)) )
;; Evaluation took:
;;   0.192 seconds of real time
;;   0.190000 seconds of total run time (0.190000 user, 0.000000 system)
;;   98.96% CPU
;;   458,864,121 processor cycles
;;   4,484,640 bytes consed
;; 4900

;; Compare to without the simplifier
(let ((ima (column-vector
            (iter (for i below 10000)
              (collect (list 1 (alexandria:random-elt '(not-it it)))))
            1 ))
      ;; *simplify* controls whether IMA attempts to simplify or not.  Don't try
      ;; *this at home.
      (ima::*simplify* nil) )
  (time (count-items 'it ima)) )
;; Evaluation took:
;;   20.112 seconds of real time
;;   20.030000 seconds of total run time (19.720000 user, 0.310000 system)
;;   [ Run times consist of 1.030 seconds GC time, and 19.0000 seconds non-GC time. ]
;;   99.59% CPU
;;   48,151,308,831 processor cycles
;;   8 page faults
;;   1,602,561,216 bytes consed
;; 5026

Also, if you want to exploit your data structures capabilities, you can by specializing the mapping methods. But you only need bother if you want to try and eke out some extra performance for your particular case.

It's even integrated with Iterate (although more can be done here).

;; One way to "flatten" an IMA
(iter (for el in-ima '((1 2) (3 4)))
  (collect el) )
;; (1 2 3 4)

;; These are pretty self explanatory
(iter
  (for col in-column-vectors-of '((1 2 3) (4 5 6) (7 8 9)))
  (for row in-row-vectors-of '((1 2 3) (4 5 6) (7 8 9)))
  (collect (v:dot col row)) )
;; (30 81 150)

Basically, I wanted to share this with the community. I've been using it for a couple years and I really like it. Maybe others out there will as well. The code is on my github page and is released under the Lesser Lisp GPL.