### begriffs

I am experimenting with layered code tests. Each layer is more specific and strict. The layers test continuously, watching the behavior of the code. Beware, the tests are relentless and can hurt your feelings.

Let me illustrate in the Coffeescript interpreter with an example from algebra. You can grab the code here and play along. Let’s define some sets and operations. First we’ll include the Coffeecup structures library.

``````coffee> s = require './Structure.coffee'
{ Set: [Function: Set],
Magma: [Function: Magma],
Monoid:
{ [Function: Monoid]
__super__: { op: [Function], pow: [Function], cross: [Function] } } }``````

Sets add extra type checks to the language and are the first layer of the onion. Let’s create the sets of all integers and odd integers.

``````coffee> Z = new s.Set (x) -> x == (Math.floor x) and x < Infinity
{ char: [Function] }
coffee> Odd = Z.where (x) -> x%2 in [1,-1]
{ char: [Function] }``````

Note that the “[1,-1]” thing corrects a bug in the underlying JavaScript modulo operator. Here is what these sets report.

``````coffee> Odd.contains -1
true
coffee> Odd.contains 2
false
coffee> Odd.contains 3.14
false``````

Now let’s add a layer to the onion. We’ll associate a binary operation with Z and Odd and assert that they form magmas. This means the sets are closed under the operation. We will use ordinary addition as our operation.

``````coffee> Z = new s.Magma Z, ((x,y) -> x+y)
{ s: { char: [Function] }, o: [Function] }
coffee> Odd = new s.Magma Odd, ((x,y) -> x+y)
{ s: { char: [Function] }, o: [Function] }``````

Take them for a test drive.

``````coffee> Z.op 2, 2
4
coffee> Odd.op 2, 2
AssertionError: values outside of magma``````

The magma adds a check to protect us from elements outside its domain. Furthermore,

``````coffee> Odd.op 1, 3
AssertionError: magma operation not closed``````

This response highlights that the operation we chose (simple addition) is not suitable for making a magma out of odd numbers. Onion testing always watches what we do and compares it to what we assert.

Now let’s see how onion testing can have a memory. We want to assert that strings form a monoid under concatenation. This means, in part, that concatenation is associative. A failure of associativity only manifests after several uses of an operation so this layer of the onion must compare previous operation results to uncover it.

We will define two operations on strings, normal concatenation and weird concatenation. We will assert they are both monoids. Monoids add another layer of testing to magmas.

``````coffee> Str = new s.Monoid (new s.Set (x) -> typeof(x) == 'string'), ((x,y) -> (x+y)), ''
{ s: { char: [Function] },
o: [Function],
i: '',
log: [] }
coffee> WeirdStr = new s.Monoid (new s.Set (x) -> typeof(x) == 'string'), ((x,y) -> (x+x+y)), ''
{ s: { char: [Function] },
o: [Function],
i: '',
log: [] }``````

First try using Str.

``````coffee> Str.op 'a', 'b'
'ab'
coffee> Str.op 'ab', 'c'
'abc'``````

Now try the same on WeirdStr.

``````coffee> WeirdStr.op 'a', 'b'
'aab'``````

So far so good in its own weird way. But is it associative?

``````coffee> WeirdStr.op 'aab', 'c'
AssertionError: monoid not left-associative``````

No. The testing uncovered the fact that in this weird concatenation `('a' + 'b') + 'c' = 'aabaabc'` whereas `'a' + ('b' + 'c') = 'aabbc'`. It tried things both ways behind the scenes. Written more concisely, these are the errors we get from WeirdStr.

``````coffee> WeirdStr.op (WeirdStr.op 'a', 'b'), 'c'
AssertionError: monoid not left-associative
coffee> WeirdStr.op 'a', (WeirdStr.op 'b', 'c')
AssertionError: monoid not right-associative``````

I’ll be adding more structures to Coffeecup and will continue developing this layered testing approach.