It’s here! Clash 1.0 is finally released! And it has a binary release for the very first time!
Check out the install instructions instructions on how to get it!

The team at QBayLogic would like to thank all of the Clash supporters and users for making this happen.
Read all about the new features of Clash 1.0 on the clash website.

In my previous post I discussed a solver plugin for KnownNat constraints, and with regards to user-defined type-level operations, I said:

i.e. it cannot solve constraints involving subtraction (-) or user-specified type-level operations on type-level naturals […] For user-defined type-level natural operations we are currently out of luck: unlike for the standard operations in GHC.TypeLits, we cannot encode the corresponding _dictionary functions_ beforehand.

well… it turns our we do not have to know these corresponding dictionary functions beforehand. So, I’m hereby announcing version 0.2 of the ghc-typelits-knownnat plugin, which can now also solve KnownNat constraints over types consisting of:

• Subtractions (GHC.TypeLits.-)
• User-defined type-level functions

Supporting user-defined operations

So what is the “trick” for handling user-defined operations? Well, in my previous post, we defined a class, and corresponding instance for every type-level operation. For example, for addition (GHC.TypeLits.+), we defined:

And for multiplication (GHC.TypeLits.*), we defined:

However, what we would want, is a single type class for functions of a certain arity, and subsequently define multiple instances for this single class. We can then export these type-classes, and developers can add an instance for their own type-level operations, and the solver plugin can then just lookup these instances.

Using some singletons trickery we can define such a type class as:

As it says in the comments, the first argument of this class is a type-level Symbol corresponding to the fully qualified name of the type-level operation. We need to do this, because, unlike normal type constructors (.i.e. Maybe), type families cannot be partially applied. This means we cannot write a class and corresponding instance of the form:

So, instead, for the first argument of the KnownNat2, we use a Symbol that corresponds to the name of the type-level operation. The solver plugin can then use the name of type-level operation to find the corresponding KnownNat2 instance.

Now, again due to the restriction on the partial application of type families, the associated type family KnownNatF2 cannot be of kind:

Instead, we must use a defunctionalised form of our type-level operations, which are provided by the singletons package:

Where the difference is that we use ~> (tilde-right_angle) instead of -> (dash-right_angle).

We can now define our addition and multiplication instances as:

And subtraction as:

which gets an extra constraint that KnownNat (a - b) only holds when b <= a.

Adding operations to the solver

So now that we have a type-class for constraint-level arithmetic, what steps should a developer take so that her type-level functions are supported by the ghc-typelits-knownnat solver? We will give a short example.

1.

Enable some language extensions

2.

Import the required modules

3.

Define the type-level operation

One restriction is that such a type-level operation must have at least two equations. This restriction is due GHC treating single-equation type families as type synonyms, which are expanded at the Core level.

That is, had we written our Max operation as a single-equation operation:

then a KnownNat (Max x y) constraint would show up as KnownNat (If (x <=? y) y x) inside the solver plugin. And, consequently, the solver wouldn’t be able to look up the KnownNat2 instance of Max.

4.

Use Template Haskell to generate the defunctionalised form of the type-level operation:

This will create the following definitions:

where we need MaxSym0.

5.

And finally specify the KnownNat2 instance:

where we use the nameToSymbol (exported by GHC.TypeLits.KnownNat) function to converts a Template Haskell Name to a Symbol.

Version 1.0?

The ghc-typelits-knownnat solver plugin is now at version 0.2; and it supports:

• All the type-level arithmetic functions of GHC.TypeLits
• User-defined type-level operations

Which might make you wonder: if it basically supports all type-level operations on Nat, why isn’t this version 1.0?

Well, I probably should have called it 1.0, and perhaps I will call it that in a next bugfix release. However, there are still some aspects that would be nice to add/fix:

• Add a KnownBool and KnownOrdering class and add support for the comparison operators of GHC.TypeLits.
• Find a way to index the KnownNat classes that is less error-prone than a Symbol corresponding to the fully qualified name of the type-level operation.

If you have ever worked with GHC.TypeLits then you have probably encountered an error that is very similar to:

In our case, the above error originates in the following file:

The problem is that, even though we have a KnownNat n (a type-level natural n which we can reflect to a term-level Integer), GHC cannot infer KnownNat (n+2). You would think that, in 2016, given a natural number n, an advanced compiler such as GHC should surely be able to infer the value corresponding to the natural number n + 2: you simply add 2 to already given n. Sadly, GHC can not do this… and, surely, complaints were made.

A type-checker plugin for solving KnownNat constraints

As I’ve blogged about in another post, GHC supports so-called type-checker plugins since version 7.10.1. Type checker plugins let us extend GHC’s constraint solver, i.e., extend the range of programs GHC can type check.

So, given that GHC 7.10.1 was released on March 27 2015, surely somebody must have already written a type-checker plugin that can fix our annoying KnownNat issue, right? Well, even if someone did, such a plugin has never been released on hackage, until today that is: https://hackage.haskell.org/package/ghc-typelits-knownnat

By just adding the following line:

{-# OPTIONS_GHC -fplugin GHC.TypeLits.KnownNat.Solver #-}

to our program:

we can finally see that “all is right with the world (of KnownNat)”:

Creating this type-checker plugin

Now, there are probably a multitude of reasons as to why a plugin for solving KnownNat constraints wasn’t build/released before now; one of the likely reasons being that both the GHC API, and type-checker plugins, are somewhat obscure. Then again, I, the author of this post, have written several type-checker plugins, and I am (at least somewhat) familiar with those parts of the GHC API that are involved with constraint solving. So what are the reasons that I didn’t build such a plugin (much) earlier:

1. To be honest, I was personally never too bothered with typing in the extra KnownNat constraints. That is, until I wanted to simplify the API of one of the libraries that I work on.
2. I thought it couldn’t be done.

I will now focus on point 2, the reason I thought it couldn’t be done.

Evidence for KnownNat constraints

Basically, what Type checker plugins must do is provide evidence for the constraints GHC wants them to solve; where this evidence takes on the form of EvTerm:

Where, for example, the evidence for a KnownNat 8 constraint, would be encoded by EvLit (EvNum 8). OK, so what’s the problem with EvTerm? Well, let me first explain what we are trying to do:

• Given e.g. evidence for KnownNat n and KnownNat 2
• We want to add the two values of the evidence to create evidence for KnownNat (n+2).

And this is where we run into problems: EvTerm does not have a constructor for adding two KnownNat evidence terms. Actually, the more general problem is that it doesn’t allow us to create (arbitrary) Core expressions which will perform the addition. This problem is discussed in more detail at: https://ghc.haskell.org/trac/ghc/wiki/Plugins/TypeChecker#Underdiscussion:EmbeddingCoreExprinEvTerm.

So, this made me think that it was simply not possible to create a KnownNat solver given the current (version 8.0.1) state of the GHC API. And then, yesterday (August 9th, 2016), just before bed, after staring for some time at the EvTerm definition, I realised I could use the EvDFunApp constructor to do what I want!

Building a dictionary (function)

As some of you might already know, class constraints in Haskell are translated to so-called dictionaries by GHC. A dictionary is record, where the fields correspond to the methods (and super-classes) of the class. So, e.g. for the Eq class:

GHC creates the following dictionary:

Now, what perhaps fewer people know, is that for instances with class constraints, GHC creates dictionary functions: functions that create dictionaries. i.e. given the instance:

GHC creates a dictionary function:

Now that we know what dictionary functions are, we can come back to the EvDFunApp constructor of EvTerm.
EvDFunApp applies dictionaries to a given dictionary function. That is, where dictionary functions are our evidence-level lambdas, EvDFunApp is our evidence-level application. This means that all we need to do is figure out how to write an instance that gives us our evidence-level arithmetic operations, and we would be in business.

Evidence-level addition

Below follows the code for our specially crafted class, and corresponding instance, so that we get an _evidence-level_ addition operation:

For the above code, GHC will generate the desired _dictionary function_ to do _evidence-level_ addition:

OK, so we can now build a DKnownNatAdd dictionary, however, this only gives us _evidence_ for a KnownNatAdd constraint, but not for a KnownNat constraint. And this is where another lesser-known aspect of GHCs translation of classes comes in:

• Single method classes are represented by newtype dictionaries in GHC.

And, we can safe cast a newtype to (and from) its underlying representation using coercions.
Looking at the code for our evidence-level addition, we can now see that:

• KnownNatAdd is nothing more than a newtype wrapper around SNatKn
• SNatKn is a newtype wrapper around Integer.

And, when we take a look at the (hidden) definition of KnownNat:

we see that KnownNat is, ultimately, also nothing more than a newtype wrapper around Integer. This means we can safely cast our DKnownNatAdd dictionary to the DKnownNat dictionary (the code for these coercions can be found here):

1. KnownNatAdd a b ~ SNatKn (a + b)
2. SNatKn (a + b) ~ Integer
3. Integer ~ SNat (a + b)
4. SNat (a + b) ~ KnownNat (a + b)

And that’s it:

1. We have our DKnownNatAdd dictionary function for _evidence-level_ addition.
2. A way to safely cast a DKnownNatAdd dictionary to a DKnownNat dictionary, the evidence for KnownNat constraints.

Closing thoughts

So now finally, on August 10th 2016, using the ghc-typelits-knownnat type checker plugin, GHC can finally solve “complex” KnownNat constraints from other simple, single variable, KnownNat constraints. So does that mean it can infer all kinds of “complex” KnownNat constraints? Well… no, it can (currently) only infer constraints involving:

• Type-level natural numbers
• Type variables
• Applications of the following type-level operations: +, *, and ^.

i.e. it cannot solve constraints involving subtraction (-) or user-specified type-level operations on type-level naturals, such as:

Where the problem with subtraction is that it is a partial function on natural numbers. Let’s say someone writes:

then we should not just automatically solve the KnownNat (n-4) constraint from the given KnownNat n constraint, because for 0 <= n < 4, n-4 is not a natural number. In this case, we should only automatically infer KnownNat (n-4) when an additional 4 <= n constraint is given:

I think I know how to check for these additional constraints, so a future version of the plugin will automatically solve KnownNat constraints involving subtractions. For user-defined type-level natural operations we are currently out of luck: unlike for the standard operations in GHC.TypeLits, we cannot encode the corresponding dictionary functions beforehand. Perhaps, once arbitrary Core expressions can be used as evidence (discussed here), we can reflect type-level operations to evidence-level operations, and then also automatically solve KnownNat constraints over user-defined type-level operations.

This version of the tutorial is outdated, please follow the new version.

In this tutorial we’ll be programming the Terasic DE0-Nano FPGA development board using the functional hardware description language CλaSH. The end result of this tutorial is demonstrated in the video below:

This tutorial is not a general introduction to CλaSH, nor to programming FPGAs. It is meant to demonstrate how to use of TopEntity annotations (added in version 0.5.5 of CλaSH) to configure your CλaSH designs for an FPGA, without writing a single line of VHDL or (System)Verilog. Even then, this tutorial is already almost too long for single blog post, but here goes:

This tutorial used to be specified in terms of .topentity files, a method which is now superseded by TopEntity annotions. You can still see the old version on github.

Blinker circuit

We start with some general information about the DE0-Nano board:

• It has a 50 MHz crystal connected to a clock pin of FPGA, which we will connect to one of the PLLs that is on the FPGA to create a stable 50 MHz clock signal.
• It has 8 green LEDs that turn on when we put a 1 on the FPGA pin connected to the LED.
• It has two buttons which are already properly debounced by a Schmitt trigger. The buttons are 0 when they are pressed, and 1 when they are not.

The circuit that we are making will repeatedly do one of two things:

• Invert the state of the LEDs
• Rotate the state of the LEDs

We switch between these two modes when the KEY1 button on the board is pressed and subsequently released. We reset the circuit by pressing the KEY0 button.

CλaSH circuit specification

The Blinker.hs file given below implements the behaviour of circuit (sans reset behaviour) we just described. As the clock is running at 50 MHz, we shouldn’t update the state of the LEDs every clock cycle, as they will just seem to to all be turned on all the time to the human eye. We hence use a counter that counts up to a certain amount of clock cycles, and only update the state of the LEDs when the desired number of cycles is reached and the count is reset.

blinker.hs:

TopEntity annotations

Now that we’ve created our CλaSH design, it’s time to move on to the important part of this tutorial, elaboration of the TopEntity annotation. The TopEntity is a Haskell data type that guides the CλaSH compiler in generating port names and setting up clocks. These annotations are applied using the ANN pragma:

{-# ANN foo (TopEntity {t_name = .., .., ..}) #-}

In our example, we extend the minimalist defTop annotation with:

• t_name: the name our component should have. In our case: blinker
• t_inputs: a list of names our inputs should have. In our case: KEY1
• t_outputs: a list of names our outputs should have. In our case: LED
• t_extraIn: a list of extra inputs that do not correspond to the arguments of our topEntity function.
  These extra inputs are always bit vectors.
  Every item in the extra_in list is a tuple (encoded as a 2-element list) of a name, and the number of bits for that input.
  In our case we add an extra 1-bit input CLOCK_50 which will correspond to the pin to which the 50MHz crystal is attached, and a 1-bit input KEY0 which is the button we will use as a reset.
• t_clocks: a list of clock sources.

We create a single clock source by instantiating the _default_ template for the Altera PPL component altpll for the Cyclone IV. The first argument of this function is the name, the second an expression corresponding to the clock pin to connect, and the third the expression corresponding to the reset pin to connect. So to elaborate the arguments in order:

• name: the name of the component generating our clock. In our case "altpll50", an instantiated PLL component we will create later on in this tutorial.
• clock port: Now, I lied a bit that we didn’t have to write any VHDL or SystemVerilog. As you can see, we connect the clock port to the expression CLOCK_50(0), which is a VHDL expression. The altpll50s clock port expects a single bit, but our CLOCK_50 port is a 1-bit vector. Hence we need to select the first (and only) bit: "CLOCK_50(0)".
• reset port: Again, we have to write some VHDL: not KEY0(0). Remember that our extra inputs are bit vectors, but the reset port expects a single bit. Hence we need to select the first (and only) bit: KEY0(0). Additionally, remember that the keys are 0 when they are pressed, but the reset port expects an active-high signal. Hence we need to negate the signal: "not KEY0(0)".

VHDL generation

Now it’s time to generate some VHDL, yay! Make sure that the Blinker.hs file is in your current working directory, and then run the CλaSH compiler to generate VHDL:

clash --vhdl Blinker.hs

This will create a ./vhdl/Blinker/ directory, of which, for this tutorial, the most interesting file is within that directory is:

blinker.vhdl:

Port/Pin mappings

It’s almost time to start programming our FPGA. But first we need to setup how the component ports are connected to the right pins on the FPGA.
For this we will use the DE0_Nano_SystemBuilder which you can find on the DE0-Nano CD-ROM. This will create both a pin/port map file, and a Quartus II project file. Note that I work on OS X, so both DE0_Nano_SystemBuilder and Quartus II are running inside a VirtualBox VM running Windows 8.

In DE0_Nano_SystemBuilder, call the project blinker, and ensure that only CLOCK, Button x 2, and LED x 8 are selected.
Then press the Generate button, this will open a save dialog window: select a location where you want to create the project and hit the save button.
You can now close DE0_Nano_SystemBuilder.

In the directory to which you saved the project you will find the files:

• blinker.htm
• blinker.qpf
• blinker.qsf
• blinker.sdc
• blinker.v

You can delete blinker.v as we will be using our blinker.vhdl. The port/pin mappings that are generated by DE0_Nano_SystemBuilder do not completely correspond to our port names. We will hence need to update the mapping file blinker.qsf:

• Replace every occurrence of CLOCK_50 by CLOCK_50[0]
• Replace every occurrence of KEY[0] by KEY0[0]
• Replace every occurrence of KEY[1] by KEY1[0]

The final blinker.qsf file will look like:

Also, since we will be using the PLL for clocks, you can replace the entire contents of blinker.sdc by:

derive_pll_clocks

Quartus II

We now open the blinker.qpf file in Quartus II. For this tutorial, I will be using version 14.1 of the Quartus II software.
The first thing we will do is add the VHDL files generated by CλaSH to our design. In the menu bar, go to, Project -> Add/Remove Files in Project..., press the ... button next to File name, in the load dialog, select all the files generated by clash, and press Open. Then press OK to close the Settings dialog.

Next we will create a PLL component. On the right of the Quartus II main window you will see the IP Catalog. Go to Installed IP > Library > Basic Functions > Clocks; PLLs and Resets > PLL, and double click on ALTPLL.

This opens a dialog called Save IP Variation, at the end of the line enter altpll50 and press OK.

This opens the MegaWizard dialog for the PLL. Change What is the frequency of the inclk0 input? to 50.000.

Then press Finish twice. This closes the MegaWizard dialog, and opens a new dialog asking if you want to add this IP block to your project. We want this, so select ‘Yes’.

Synthesis time

Finally, we’re finished with configuration, and we can start creating the configuration file for the FPGA. In the menu bar, click: ‘Processing -> Start Compilation’. In my extremely slow VM this compilation/synthesis process will take a while, if but might be much faster on your machine.

Programming time

After synthesis has finished, it is time to program our FPGA board. Connect the FPGA board to a USB port, and start the programmer from the menu bar: Tools -> Programmer. Press the Start button on the left to program your FPGA and wait until the progress bar says 100% (Successful).

Time to play

Your FPGA should now be fully configured with the CλaSH generated design for you to play with. So yeah, go press those buttons!

This ends the tutorial on the use of TopEntity annotations to program your FPGA devices with CλaSH designs. I hope you got the gist of it. If/When I get my hands on a Xilinx FPGA board, I hope to make an alternate version of this tutorial, but then use Xilinx’ ISE/Vivado tool-flow instead.

Since version 7.10.2, GHC supports so-called type-checker plugins which let us extend GHC’s constraint solver, i.e. extend the the range of programs GHC can type check. Several plugins have already been released, the ones I know are:

• Adam Gundry’s uom-plugin: for supporting units of measure.
• Iavor Diatchki’s type-nat-solver: which allows you to hook up SMT solvers to solve numeric constraints.
• My own plugins:
  • ghc-typelits-natnormalise: which solves numeric constraints using a custom normalisation procedure as opposed to Iavor’s approach, which uses SMT solvers.
  • ghc-typelits-extra: which adds a type-level greatest common denominator (GCD) and (ceiling of) logarithm operator for GHCs type-level Natural numbers.

This post will be about writing our own type-checker plugin: we will build a plugin that (just like ghc-typelits-extra) allows GHC to perform GCD on types of kind Nat.

Why a type-checker plugin?

So the first question that might pop up: why do we even need to write a type-checker plugin to support GCD over Nat? The “problem” is that Nat is _not_ inductively defined like so:

, however, it is simply defined as an Integer (see CoreSyn.hs).

For an inductively defined Nats, it is fairly simple to write a (closed) type family symbolising an (arithmetic) operation, e.g.:

However, an inductively defined Nats is not what GHC gives us, GHC gives us Integers. Having only Integers, we might try to start with:

where we have no real way to define the part indicated by “???”. Indeed, all the currently defined type-level operations on types of kind Nat as defined in GHC.TypeLits, are all implemented by compiler magic. Using type-checker plugins, that kind of magic is, however, no longer just reserved for the wizards over at GHC HQ!

As to why GHCs native type-level natural numbers are implemented using Integer, instead of an inductively defined datatype, I can only guess:

• Unary encodings (such as the one seen earlier) are “expensive”, in terms of memory, for large numbers. I’m sure there must be a way to internally use a more efficient representation for what is, from the user’s point of view, a unary encoding. To my my knowledge, no efforts have been made to explore this option in GHC.
• Base-2 encodings, though low-cost in terms of memory, have harder to define arithmetic operations.

With the advent of type-checker plugins, the Integer-representation aspect of Nat has not been a problem any longer for most of my projects.

UPDATE:
@Bodigrim on reddit showed that we can just implement GCD with the tools already in GHC.TypeLits:

I guess the lesson learned here is that type-checker plugins should always be a last resort measure.

Type-level GCD

OK, enough about why we need to implement GCD using a type-checker plugin. Let’s get on with implementing it. Before we implement the plugin, we first need to define our GCD operation:

The reason I’ve chosen a closed type-family (as opposed to an open one) is that I don’t want our users to pull any shenanigans such as adding additional bogus equations. For GHC 7.10 we need to add at least one equation (see GHC issue #9840) to our closed type family; this should no longer be needed for GHC 8+. Although an empty closed type family is possible in GHC 8+, the current single equation version simplifies the design of the type-checker plugin, as we will see at the end of this blog post.

Type-checker plugin

I guess the reason there are not too many blog posts on GHC type checker plugins is because you need to be sort-of familiar with the GHC API. I’ve been using the GHC API for over 6 years now, so I’ve taught myself how to browse its documentation (the comments in the source code); however, it’s lack of haddock documentation makes it somewhat intimidating for the uninitiated. I try to keep the GHC API code to a minimum, but it can sadly not be avoided.

We first start with some imports:

Aside from the GHC API, and some Prelude modules, we also import some functions from my ghc-tcplugins-extra package, GHC.TcPluginM.Extra: it provides some type-checker plugin utility functions with the intention to provide a stable API spanning multiple GHC releases. As those of us who use the GHC API often know, the GHC API is fairly unstable, hence my personal need for ghc-tcplugins-extra. Currently, ghc-tcplugins-extra supports GHC 7.10 and GHC 8.0.

Next up, setting up the plugin:

We use defaultPlugin to create a plugin that does nothing at all, and add our own gcdPlugin type-checker plugin. Plugin’s can be given command line options, but we just ignore them.

The TcPlugin record has three fields:

• * tcPluginInit: Initialises the state used by our type-checker plugin. We will use it to look up the TyCon (type constructor) of the GCD type family. One can also perform IO operations in tcPluginInit, and have the state contain IORefs.
  * tcPluginSolve: The main solver function, this is were we will actually perform the _gcd_ calculation.
  * tcPluginStop: Cleaning up of the state. We have no use for it, but it can for example be used to remove files that are used as a communication medium with an SMT solver.

Our type-checker plugin solves equations involving our GCD type family, so we need to know that the types that we are dealing with actually involve GCD. To do that, we need GHC’s internal representation of GCD, which is a TyCon. We do the lookup of our GCD TyCon in the initialisation of our type-checker plugin, so that it becomes our plugin’s internal state:

The name of our package is going to be “ghc-typelits-gcd”, and the module in which we defined GCD is “GHC.TypeLits.GCD”. In the above code, we first look up the GHC.TypeLits.GCD module in the ghc-typelits-gcd package, we then look up the full Name of the thing we call “GCD”, and finally look up the actual GCD TyCon.

Actually solving something

So we’ve done all the setting up for our type-checker plugin, and are now ready to implement the business end of it. But what does a type-checker plugin actually do? Well, it solves Constraints, the types that in our Haskell code usually appear on the left of the => arrow.

In our case, we will be solving equality constraints, that is, given the following haskell code:

we have to say whether the equality GCD 6 8 ~ 2 can be proven. So let’s start with the type signature of our solver:

The first argument is the plugin’s state, which in our case is just the (immutable) TyCon of our GCD type family. Next come three categories of constraints that GHC’s solver pipeline is giving to us:

• First are the given constraints, these are the proven facts about types available to the compiler.
• Second are the derived constraints… to be honest, I never really understood what they do, apparently they help with improving error messages, but I never investigated them in depth.
• And finally, there are the wanted constraints.

These are the constraints that the compiler wants us to either:

• Prove, so that the valid program type checks, or,
• Disprove, so that it can give a good error message.

That is, unsolved constraints also result in errors, but constraints that are outright rejected result in better error messages. The result of our plugin is:

where we can either return TcPluginContradictions cts, where cts are all the wanted constraints that we deemed insolvable (think 2 + 3 ~ 4 for ordinary natural numbers). Or we can return TcPluginOk proven newWanted, where the proven are all the wanted constraints (Ct) that we solved together with with their proofs (EvTerm), and newWanted are all the new wanted constraints that must additionally be solved.

An example of the latter would be: given the constraint x + GCD 6 8 ~ 2 + x, my [ghc-typelits-natnormalise](http://hackage.haskell.org/package/ghc-typelits-natnormalise) plugin knows that addition is commutative, but doesn’t know anything about GCD, so it would return: TcPluginOK [("Believe me",x + GCD 6 8 ~ 2 + x)] [GCD 6 8 ~ 2]. The new wanted constraint, GCD 6 8 ~ 2, would then be picked up by the plugin that we are currently describing in this blog post.

Next, the body of our solveGCD function:

The first clause basically says that, if there are no wanted constraints for us to solve we are done. The thing is, our type-checker plugin is used inside two phases of GHC’s solver pipeline. In the first phase, only given (and probably derived) constraints are passed through the pipeline. A type-checker plugin could then extend GHCs knowledge about the constraints by creating more given constraints, which could then help in solving future wanted constraints. In the second phase, we’ll be constantly given new wanted constraints that pop up during iterations of the solver, until all wanted constraints have passed trough the pipeline at least once.

This brings us to the second clause, which is the clause that handles this phase 2 part. It gets all the wanted constraints (involving GCD) for which there is some hope that we actually can solve them; these interesting wanted constraints are gcdWanteds. As we will see later on, the solving part basically consists of trying to reduce both sides of the equality relation to a number. So given the equality GCD 6 8 ~ 2, we reduce it to 2 ~ 2. Consequently, all the solved constraints are the constraints where both sides of the equality reduced to the same number, and failed constraints are the constraints where the sides of the equality reduced to a different number.

Finally, when there are no failed constraints, we return all our solved constraints, together with a proof that the constraints hold; we don’t return any new wanted constraints. The proofs for the solved constraints are constructed by evMagic (which we will discuss later). If there are any failed constraints, we return those in a TcPluginContradiction.

Next comes the code by which we find the constraints that are interesting for our plugin:

The toGCDEquality first looks at what type of constraint we are dealing with. We are only interested in the equality constraints; other constraints are things like class constraints and implicit parameters. If the constraint is an equality constraint (EqPred), we see if we can reduce both sides of the equality to a number using reduceGCD.

Finally, reduceGCD is the function that applies the actual gcd function! If reduceGCD finds a type-level natural, it’s done. If it finds an application of our GCD type family, it first tries to reduce the argument, and subsequently applies the gcd function. If it’s not a type-level natural, or the application GCD type family, we give up.

Burden of proof

The last part of our plugin is the evidence generation: the actual proof that the wanted equality relation holds. This is sadly the part where we take the easy way out (the easy way is often pragmatic :-)). We use the evByFiat function from ghc-tcplugins-extra, which, as its name implies, gives a proof by fiat. The evByFiat function basically uses the same “proof”, as the one that unsafeCoerce uses internally in GHC.

Perhaps these “proofs” by fiat are OK for the tiny type-checker plugin we have written here, perhaps it is not. Looking at the Agda code for GCD it seems possible to provide a real proof for GCD. Then again, Agda’s proof uses inductively defined natural numbers, which we didn’t have to begin with.

In the end, I use Haskell for actually building programs that run, or actually, building programs that get translated to digital circuits; and not to prove things about programs. I’m fairly confident that the plugin we have written does not allow for incorrect programs to be type-checked (I know, assumptions are the mother of all …), so I’m personally OK with these by fiat “proofs”.

Wrapping up

The complete code for this blog can be found on https://github.com/christiaanb/ghc-typelits-gcd. It contains the following, very minor, test suite:

which, using our type-checker plugin, passes the type-checker, YAY!

When you start writing “real” type-checker plugins, you’ll have to think _at least_ about the following aspects:

• How am I going to let given and wanted constraints interact?
• How do I ensure that I do not keep on generating new wanted constraints every time my plugin gets called?
  Because if you keep on generating new constraints, the GHCs solver will never finish, and compiling just becomes an exercise in watching a blinking cursor for all eternity.
• How do I interact with other type-checker plugins?
  Remember where I said that: given the constraint x + GCD 6 8 ~ 2 + x, our ghc-typelits-natnormalise plugin knows that addition is commutative, but doesn’t know anything about GCD, so it would return: TcPluginOK [("Believe me",x + GCD 6 8 ~ 2 + x)] [GCD 6 8 ~ 2]. And the the GCD plugin would do the rest. That was actually a white lie! The evidence it generates is not our simple evByFiat (i.e. “Believe me”), but a _conditional_ evidence. We need it to be conditional, because if GCD 6 8 ~ 2 turned out not to be true, then x + GCD 6 8 ~ 2 + x isn’t true either. I used to solve this problem with tricks/hacks, but GHC 8+ seems to allow me to do this in a more sanctioned way.

Remember where I said it was easier to have the single equation definition of the GCD type family as opposed to the zero equation definition. Our test case, test2, is the proof of that: our solver, as it is currently defined, is completely unable to handle test2. Our current solver only works when both arguments of GCD reduce to a natural number, and gives up otherwise.

Indeed, this blog post was written with the intention of demonstrating a “working” prototype of a type-checker plugin with a “minimal” amount of code, and I think it serves that purpose well. Given that there isn’t much material on writing type-checker plugins out there, I understand there aren’t too many type-checker plugins in the wild. I hope this blog post serves as that final push for some of you to start writing your own type-checker plugins and further explore its design space.