assert | Thin assertion library for use in libraries

Dart SDK officially supports ARM64 and as of now, 2.14.2 is the latest (stable) Dart SDK that has support for ARM64. Though it was the same version that was bundled in my Flutter setup, it seemed to run on Intel architecture (Activity monitor shows dart processes running on Intel).

I manually tried replacing the dart SDK on my flutter installation bu replacing flutter-directory/bin/cache/dart-sdk/ with the contents of a zip file of the Dart SDK made for ARM64, downloaded from dart.dev archive. But trying to run an app on an Android emulator (which runs on ARM64 and was working on my old Flutter setup), throws this error:

I have a dockerfile that currently only installs pip-tools

When I run my project on web the Exception message and stack trace was

For some, simple thread related code, i.e:

Based on the documentation for Raku's lookaround assertions, I read the regex / / as saying "starting from the left, match but do not not consume one character that is a, b, or c and, once you have found a match, match and consume one alphabetic character."

Thus, this output makes sense:

I created an extension method to add all JSON configuration files to the IConfigurationBuilder

We are working on a project which allows us to record some sounds from a microphone with a 5k Hz sample rate with some Low-Pass filter & HighPass filter.

What we are using

We are using AvaudioEngine for this purpose.

We are using AVAudioConverter for downgrading the sample rate.

We are using AVAudioUnitEQ for the LowPass & HighPass filter.

Code

Consider:

(Disclaimer: I'm not 100% sure how codatatype works, especially when not referring to terminal algebras).

Consider the "category of types", something like Hask but with whatever adjustment that fits the discussion. Within such a category, it is said that (1) the initial algebras define datatypes, and (2) terminal algebras define codatatypes.

I'm struggling to convince myself of (2).

Consider the functor T(t) = 1 + a * t. I agree that the initial T-algebra is well-defined and indeed defines [a], the list of a. By definition, the initial T-algebra is a type X together with a function f :: 1+a*X -> X, such that for any other type Y and function g :: 1+a*Y -> Y, there is exactly one function m :: X -> Y such that m . f = g . T(m) (where . denotes the function combination operator as in Haskell). With f interpreted as the list constructor(s), g the initial value and the step function, and T(m) the recursion operation, the equation essentially asserts the unique existance of the function m given any initial value and any step function defined in g, which necessitates an underlying well-behaved fold together with the underlying type, the list of a.

For example, g :: Unit + (a, Nat) -> Nat could be () -> 0 | (_,n) -> n+1, in which case m defines the length function, or g could be () -> 0 | (_,n) -> 0, then m defines a constant zero function. An important fact here is that, for whatever g, m can always be uniquely defined, just as fold does not impose any contraint on its arguments and always produce a unique well-defined result.

This does not seem to hold for terminal algebras.

Consider the same functor T defined above. The definition of the terminal T-algebra is the same as the initial one, except that m is now of type X -> Y and the equation now becomes m . g = f . T(m). It is said that this should define a potentially infinite list.

I agree that this is sometimes true. For example, when g :: Unit + (Unit, Int) -> Int is defined as () -> 0 | (_,n) -> n+1 like before, m then behaves such that m(0) = () and m(n+1) = Cons () m(n). For non-negative n, m(n) should be a finite list of units. For any negative n, m(n) should be of infinite length. It can be verified that the equation above holds for such g and m.

With any of the two following modified definition of g, however, I don't see any well-defined m anymore.

First, when g is again () -> 0 | (_,n) -> n+1 but is of type g :: Unit + (Bool, Int) -> Int, m must satisfy that m(g((b,i))) = Cons b m(g(i)), which means that the result depends on b. But this is impossible, because m(g((b,i))) is really just m(i+1) which has no mentioning of b whatsoever, so the equation is not well-defined.

Second, when g is again of type g :: Unit + (Unit, Int) -> Int but is defined as the constant zero function g _ = 0, m must satisfy that m(g(())) = Nil and m(g(((),i))) = Cons () m(g(i)), which are contradictory because their left hand sides are the same, both being m(0), while the right hand sides are never the same.

In summary, there are T-algebras that have no morphism into the supposed terminal T-algebra, which implies that the terminal T-algebra does not exist. The theoretical modeling of the codatatype Stream (or infinite list), if any, cannot be based on the nonexistant terminal algebra of the functor T(t) = 1 + a * t.

Many thanks to any hint of any flaw in the story above.