The Foundations of Dawn article came up on Lobsters recently. In this article, the author of Dawn defines a core calculus for the language, and provides its semantics. The core calculus is called the untyped concatenative calculus, or UCC. The definitions in the semantics seemed so clean and straightforward that I wanted to try my hand at translating them into machinechecked code. I am most familiar with Coq, and that’s what I reached for when making this attempt.
Defining the Syntax
Expressions and Intrinsics
This is mostly the easy part. A UCC expression is one of three things:
 An “intrinsic”, written $i$, which is akin to a builtin function or command.
 A “quote”, written $[e]$, which takes a UCC expression $e$ and moves it onto the stack (UCC is stackbased).
 A composition of several expressions, written $e_1\ e_2\ \ldots\ e_n$, which effectively evaluates them in order.
This is straightforward to define in Coq, but I’m going to make a little simplifying change.
Instead of making “composition of $n$ expressions” a core language feature, I’ll only
allow “composition of $e_1$ and $e_2$”, written $e_1\ e_2$. This change does not
in any way reduce the power of the language; we can still
[note:
The same expression can, of course, be written as $e_1\ \ldots\ (e_{n1}\ e_n)$.
So, which way should we really use when translating the manyexpression composition
from the Dawn article into the twoexpression composition I am using here? Well, the answer is,
it doesn't matter; expression composition is associative, so both ways effectively mean
the same thing.
This is quite similar to what we do in algebra: the regular old addition operator, $+$ is formally
only defined for pairs of numbers, like $a+b$. However, no one really bats an eye when we
write $1+2+3$, because we can just insert parentheses any way we like, and get the same result:
$(1+2)+3$ is the same as $1+(2+3)$.
]
With that in mind, we can translate each of the three types of expressions in UCC into cases
of an inductive data type in Coq.


Why do we need e_int
? We do because a token like $\text{swap}$ can be viewed
as belonging to the set of intrinsics $i$, or the set of expressions, $e$. While writing
down the rules in mathematical notation, what exactly the token means is inferred from context  clearly
$\text{swap}\ \text{drop}$ is an expression built from two other expressions. In staticallytyped
functional languages like Coq or Haskell, however, the same expression can’t belong to two different,
arbitrary types. Thus, to turn an intrinsic into an expression, we need to wrap it up in a constructor,
which we called e_int
here. Other than that, e_quote
accepts as argument another expression, e
(the
thing being quoted), and e_comp
accepts two expressions, e1
and e2
(the two subexpressions being composed).
The definition for intrinsics themselves is even simpler:


We simply define a constructor for each of the six intrinsics. Since none of the intrinsic names are reserved in Coq, we can just call our constructors exactly the same as their names in the written formalization.
Values and Value Stacks
Values are up next. My initial thought was to define a value much like I defined an intrinsic expression: by wrapping an expression in a constructor for a new data type. Something like:
Inductive value :=
 v_quot (e : expr).
Then, v_quot (e_int swap)
would be the Coq translation of the expression $[\text{swap}]$.
However, I didn’t decide on this approach for two reasons:
 There are now two ways to write a quoted expression: either
v_quote e
to represent a quoted expression that is a value, ore_quote e
to represent a quoted expression that is just an expression. In the extreme case, the value $[[e]]$ would be represented byv_quote (e_quote e)
 two different constructors for the same concept, in the same expression!  When formalizing the lambda calculus, Programming Language Foundations uses an inductivelydefined property to indicate values. In the simply typed lambda calculus, much like in UCC, values are a subset of expressions.
I took instead the approach from Programming Language Foundations: a value is merely an expression
for which some predicate, IsValue
, holds. We will define this such that IsValue (e_quote e)
is provable,
but also such that here is no way to prove IsValue (e_int swap)
, since that expression is not
a value. But what does “provable” mean, here?
By the CurryHoward correspondence, a predicate is just a function that takes something and returns a type. Thus, if $\text{Even}$ is a predicate, then $\text{Even}\ 3$ is actually a type. Since $\text{Even}$ takes numbers in, it is a predicate on numbers. Our $\text{IsValue}$ predicate will be a predicate on expressions, instead. In Coq, we can write this as:


You might be thinking,
Huh,
Prop
? But you just said that predicates return types!
This is a good observation; In Coq, Prop
is a special sort of type that corresponds to logical
propositions. It’s special for a few reasons, but those reasons are beyond the scope of this post;
for our purposes, it’s sufficient to think of IsValue e
as a type.
Alright, so what good is this new IsValue e
type? Well, we will define IsValue
such that
this type is only inhabited if e
is a value according to the UCC specification. A type
is inhabited if and only if we can find a value of that type. For instance, the type of natural
numbers, nat
, is inhabited, because any number, like 0
, has this type. Uninhabited types
are harder to come by, but take as an example the type 3 = 4
, the type of proofs that three is equal
to four. Three is not equal to four, so we can never find a proof of equality, and thus, 3 = 4
is
uninhabited. As I said, IsValue e
will only be inhabited if e
is a value per the formal
specification of UCC; specifically, this means that e
is a quoted expression, like e_quote e'
.
To this end, we define IsValue
as follows:


Now, IsValue
is a new data type with only only constructor, ValQuote
. For any expression e
,
this constructor creates a value of type IsValue (e_quote e)
. Two things are true here:
 Since
Val_quote
accepts any expressione
to be put insidee_quote
, we can useVal_quote
to create anIsValue
instance for any quoted expression.  Because
Val_quote
is the only constructor, and because it always returnsIsValue (e_quote e)
, there’s no way to getIsValue (e_int i)
, or anything else.
Thus, IsValue e
is inhabited if and only if e
is a UCC value, as we intended.
Just one more thing. A value is just an expression, but Coq only knows about this as long
as there’s an IsValue
instance around to vouch for it. To be able to reason about values, then,
we will need both the expression and its IsValue
proof. Thus, we define the type value
to mean
a pair of two things: an expression v
and a proof that it’s a value, IsValue v
:


A value stack is just a list of values:


Semantics
Remember our IsValue
predicate? Well, it’s not just any predicate, it’s a unary predicate.
Unary means that it’s a predicate that only takes one argument, an expression in our case. However,
this is far from the only type of predicate. Here are some examples:
 Equality,
=
, is a binary predicate in Coq. It takes two arguments, sayx
andy
, and builds a typex = y
that is only inhabited ifx
andy
are equal.  The mathematical “less than” relation is also a binary predicate, and it’s called
le
in Coq. It takes two numbersn
andm
and returns a typele n m
that is only inhabited ifn
is less than or equal tom
.  The evaluation relation in UCC is a ternary predicate. It takes two stacks,
vs
andvs'
, and an expression,e
, and creates a type that’s inhabited if and only if evaluatinge
starting at a stackvs
results in the stackvs'
.
Binary predicates are just functions of two inputs that return types. For instance, here’s what Coq has
to say about the type of eq
:
eq : ?A > ?A > Prop
By a similar logic, ternary predicates, much like UCC’s evaluation relation, are functions of three inputs. We can thus write the type of our evaluation relation as follows:


We define the constructors just like we did in our IsValue
predicate. For each evaluation
rule in UCC, such as:
We introduce a constructor. For the swap
rule mentioned above, the constructor looks like this:


Although the stacks are written in reverse order (which is just a consequence of Coq’s list notation), I hope that the correspondence is fairly clear. If it’s not, try reading this rule out loud:
The rule
Sem_swap
says that for every two valuesv
andv'
, and for any stackvs
, evaluatingswap
in the original stackv' :: v :: vs
, aka $\langle V, v, v'\rangle$, results in a final stackv :: v' :: vs
, aka $\langle V, v', v\rangle$.
With that in mind, here’s a definition of a predicate Sem_int
, the evaluation predicate
for intrinsics:


Hey, what’s all this with v_quote
and projT1
? It’s just a little bit of bookkeeping.
Given a value – a pair of an expression e
and a proof IsValue e
– the function projT1
just returns the expression e
. That is, it’s basically a way of converting a value back into
an expression. The function v_quote
takes us in the other direction: given an expression $e$,
it constructs a quoted expression $[e]$, and combines it with a proof that the newly constructed
quote is a value.
The above two function in combination help us define the quote
intrinsic, which
wraps a value on the stack in an additional layer of quotes. When we create a new quote, we
need to push it onto the value stack, so it needs to be a value; we thus use v_quote
. However,
v_quote
needs an expression to wrap in a quote, so we use projT1
to extract the expression from
the value on top of the stack.
In addition to intrinsics, we also define the evaluation relation for actual expressions.


Here, we may as well go through the three constructors to explain what they mean:

Sem_e_int
says that if the expression being evaluated is an intrinsic, and if the intrinsic has an effect on the stack as described bySem_int
above, then the effect of the expression itself is the same. 
Sem_e_quote
says that if the expression is a quote, then a corresponding quoted value is placed on top of the stack. 
Sem_e_comp
says that if one expressione1
changes the stack fromvs1
tovs2
, and if another expressione2
takes this new stackvs2
and changes it intovs3
, then running the two expressions one after another (i.e. composing them) means starting at stackvs1
and ending in stackvs3
.
$\text{true}$, $\text{false}$, $\text{or}$ and Proofs
Now it’s time for some fun! The UCC language specification starts by defining two values: true and false. Why don’t we do the same thing?
UCC Spec  Coq encoding  

$\text{false}$=$[\text{drop}]$ 
From Dawn.v, lines 41 through 42


$\text{true}$=$[\text{swap} \ \text{drop}]$ 
From Dawn.v, lines 44 through 45

Let’s try prove that these two work as intended.


This is the first real proof in this article. Rather than getting into the technical details, I invite you to take a look at the “shape” of the proof:
 After the initial use of
intros
, which brings the variablesv
,v
, andvs
into scope, we start by applyingSem_e_comp
. Intuitively, this makes sense  at the top level, our expression, $\text{false}\ \text{apply}$, is a composition of two other expressions, $\text{false}$ and $\text{apply}$. Because of this, we need to use the rule from our semantics that corresponds to composition.  The composition rule requires that we describe the individual effects on the stack of the
two constituent expressions (recall that the first expression takes us from the initial stack
v1
to some intermediate stackv2
, and the second expression takes us from that stackv2
to the final stackv3
). Thus, we have two “bullet points”: The first expression, $\text{false}$, is just a quoted expression. Thus, the rule
Sem_e_quote
applies, and the contents of the quote are puhsed onto the stack.  The second expression, $\text{apply}$, is an intrinsic, so we need to use the rule
Sem_e_int
, which handles the intrinsic case. This, in turn, requires that we show the effect of the intrinsic itself; theapply
intrinsic evaluates the quoted expression on the stack. The quoted expression contains the body of false, or $\text{drop}$. This is once again an intrinsic, so we useSem_e_int
; the intrinsic in question is $\text{drop}$, so theSem_drop
rule takes care of that.
 The first expression, $\text{false}$, is just a quoted expression. Thus, the rule
Following these steps, we arrive at the fact that evaluating false
on the stack simply drops the top
element, as specified. The proof for $\text{true}$ is very similar in spirit:


We can also formalize the $\text{or}$ operator:
UCC Spec  Coq encoding  

$\text{or}$=$\text{clone}\ \text{apply}$ 
From Dawn.v, line 65

We can write two toplevel proofs about how this works: the first says that $\text{or}$, when the first argument is $\text{false}$, just returns the second argument (this is in agreement with the truth table, since $\text{false}$ is the identity element of $\text{or}$). The proof proceeds much like before:


To shorten the proof a little bit, I used the Proof with
construct from Coq, which runs
an additional tactic (like apply
) whenever ...
is used.
Because of this, in this proof writing apply Sem_apply...
is the same
as apply Sem_apply. apply Sem_e_int
. Since the Sem_e_int
rule is used a lot, this makes for a
very convenient shorthand.
Similarly, we prove that $\text{or}$ applied to $\text{true}$ always returns $\text{true}$.


Finally, the specific facts (like $\text{false}\ \text{or}\ \text{false}$ evaluating to $\text{false}$)
can be expressed using our two new proofs, or_false_v
and or_true
.


Derived Expressions
Quotes
The UCC specification defines $\text{quote}_n$ to make it more convenient to quote multiple terms. For example, $\text{quote}_2$ composes and quotes the first two values on the stack. This is defined in terms of other UCC expressions as follows:
$\text{quote}_n = \text{quote}_{n1}\ \text{swap}\ \text{quote}\ \text{swap}\ \text{compose}$We can write this in Coq as follows:


This definition diverges slightly from the one given in the UCC specification; particularly,
UCC’s spec mentions that $\text{quote}_n$ is only defined for $n \geq 1$.However,
this means that in our code, we’d have to somehow handle the error that would arise if the
term $\text{quote}_0$ is used. Instead, I defined quote_n n
to simply mean
$\text{quote}_{n+1}$; thus, in Coq, no matter what n
we use, we will have a valid
expression, since quote_n 0
will simply correspond to $\text{quote}_1 = \text{quote}$.
We can now attempt to prove that this definition is correct by ensuring that the examples given in the specification are valid. We may thus write,


We used a new tactic here, repeat
, but overall, the structure of the proof is pretty straightforward:
the definition of quote_n
consists of many intrinsics, and we apply the corresponding rules
onebyone until we arrive at the final stack. Writing this proof was kind of boring, since
I just had to see which intrinsic is being used in each step, and then write a line of apply
code to handle that intrinsic. This gets worse for $\text{quote}_3$:


It’s so long! Instead, I decided to try out Coq’s Ltac2
mechanism to teach Coq how
to write proofs like this itself. Here’s what I came up with:


You don’t have to understand the details, but in brief, this checks what kind of proof
we’re asking Coq to do (for instance, if we’re trying to prove that a $\text{swap}$
instruction has a particular effect), and tries to apply a corresponding semantic rule.
Thus, it will try Sem_swap
if the expression is $\text{swap}$,
Sem_clone
if the expression is $\text{clone}$, and so on. Then, the two proofs become:


Rotations
There’s a little trick to formalizing rotations. Values have an important property: when a value is run against a stack, all it does is place itself on a stack. We can state this as follows:
$\langle V \rangle\ v = \langle V\ v \rangle$Or, in Coq,


This is the trick to how $\text{rotate}_n$ works: it creates a quote of $n$ reordered and composed
values on the stack, and then evaluates that quote. Since evaluating each value
just places it on the stack, these values end up back on the stack, in the same order that they
were in the quote. When writing the proof, solve_basic ()
gets us almost all the way to the
end (evaluating a list of values against a stack). Then, we simply apply the composition
rule over and over, following it up with eval_value
to prove that the each value is just being
placed back on the stack.


e_comp
is Associative
When composing three expressions, which way of inserting parentheses is correct? Is it $(e_1\ e_2)\ e_3$? Or is it $e_1\ (e_2\ e_3)$? Well, both! Expression composition is associative, which means that the order of the parentheses doesn’t matter. We state this in the following theorem, which says that the two ways of writing the composition, if they evaluate to anything, evaluate to the same thing.


Conclusion
That’s all I’ve got in me for today. However, we got pretty far! The UCC specification says:
One of my long term goals for UCC is to democratize formal software verification in order to make it much more feasible and realistic to write perfect software.
I think that UCC is definitely getting there: formally defining the semantics outlined on the page was quite straightforward. We can now have complete confidence in the behavior of $\text{true}$, $\text{false}$, $\text{or}$, $\text{quote}_n$ and $\text{rotate}_n$. The proof of associativity is also enough to possibly argue for simplifying the core calculus’ syntax even more. All of this we got from an official source, with only a little bit of tweaking to get from the written description of the language to code! I’m looking forward to reading the next post about the multistack concatenative calculus.