Yes, thank you!

The "should it merge?" function expects 3 arguments:

Shouldn't this say 1 argument?

The model solutions returns False for is_factorable([7, 6, 3, 0]), yet

$x^7 + x^6 + x^3 + 1 = (x + 1)(x^6 + x^2 - x + 1)$.

My fault, I think I have corrected the solution now.

Thank you! I've modified the test to have fun g (f, x) = f (x, x).

In sample tests "shadowing" you have this test:

invalid(`
  fun f (a, b) = a
  fun g (f, g) = f (a, b)

  out f`, "The parameter identifier 'f' overrides the global function 'f'");

However, this is also invalid due to a and b being unbound variables. This means the test can be passed without actually handling the shadowing case properly. I suggest a slight modification:

fun g (f, g) = f (f, g)

I'm not sure why the test case $x^{1000000} + 1$ gives false. It can be factored as $(y + 1)(y^{125624} - y^{125623} + y^{125622} - ... + 1)$, where $y = x^{64}$, can't it?

Thank you for the additional tests: I think that your first program never terminates though, so I modified it to

fun f(a, b) = b + if a < 0 then 0 else if a > 2 then a + b + f(a - 2, b + 1) else 3 + f(a - 1, b + 3)
out f

I've added that modified version and the other two tests you suggested.

A very enjoyable kata! One thing which I found challenging is handling conditional expressions. However, the current tests are not comprehensive enough. I suggest adding the following test cases:

1. Calls inside conditional branches which are not in tail positions:

fun f(a, b) = b + if a > 2 then f(a - 2, b + 1) else 3 + f(a - 1, b + 3)
out f

2. Calls inside the condition expression:

fun f(a) = if hd [g(a), 3] = a then a + 3 else f(a - 1)
fun g(x) = if x < 4 then x else x + 1
out f

3. A Fibonacci variation (the sequence starts from 1, 1) which combines 1 and 2:

fun fib (n) =
    (if n > hd [1, if n = 0 then 3 else fib(n - 1)] then fib(n - 2) else 0) + 
    (if n > 0 then fib(n - 1) else 1)

out fib

Yes :)

At least for sample tests, whe invalid is expected, is it possible to provide an error message stating why it should be invalid?

The syntax you described isn't strictly correct EBNF, but I appreciate the effort-it's still comprehensive and clearly presented.

Yes - some of the input programs, for example gcd, already has all of its calls in tail position, and a lot of tests are assert.doesNotThrow for checking that you haven't missed any cases for constructs in your parsing code.

Is it expected around 25% of tests pass if I just submit the initial code?