Trying to understand the SML option structure - functional-programming

Okay so I started learning SML for a class, and I'm stuck with the option structure.
What I have so far for this example:
datatype suit = spades|hearts|clubs|diamonds;
datatype rank = ace|two|three|...|j|q|k|joker;
type card = suit*rank;
My lecturer was trying to explain the use of the option structure by saying that not all cards necessarily have a suit; jokers don't have a suit associated with them.
So when designing a function getsuit to get the suit of a card, we have the following:
datatype 'a option = NONE | SOME of 'a;
fun getsuit ((joker,_):card):suit option = NONE
| getsuit ((_,s):card):suit option = SOME s;
But using emacs, I get two errors, one saying how the pattern and constraint don't agree,
pattern: rank * ?.suit
constraint: rank * suit
and the other saying how the expression type and the resulting types don't agree.
expression: ?.suit option
result type: suit option
This was the code provided by the lecturer so clearly they're not of much help if it results in errors.
What's the meaning of "?." and why does it show up? How would I correctly define this function?

Not really a problem with option as you've defined it.
You've got the order of suit and rank in your card patterns the wrong way:
Try:
datatype 'a option = NONE | SOME of 'a;
fun getsuit ((_, joker):card):suit option = NONE
| getsuit ((s, _):card):suit option = SOME s;
My version of ML probably prints errors differently so I'm not sure how to explain the the meaning of ?. etc. But it's simple enough if you take it bit by bit:
Try
(clubs, ace);
The interpreter (or emacs if that's what you're using) tells you the type is a product of a suit * rank. That's ML's type inference at work, but you can specify the type (you expect) like this:
(clubs, ace): suit*rank;
Or
(clubs, ace): card; (* Works, since card is defined as (suit*rank) *)
And you won't have any complaints. But obviously you would if you did
(clubs, ace): rank*suit;
Or
(clubs, ace): card; (* card is defined as (rank*) *)
You placed a constraint on the type of getsuit's argument (that it must be a card, or the compatible (suit*rank) product), but the pattern is of type (rank*?) or (?*rank), neither of which are compatible with (suit*rank).

Related

SProxy in purescript?

What's the use of Sproxy in purescript?
In Pursuit, it's written as
data SProxy (sym :: Symbol)
--| A value-level proxy for a type-level symbol.
and what is meant by Symbol in purescipt?
First, please note that PureScript now has polykinds since version 0.14 and most functions now use Proxy instead of SProxy. Proxy is basically a generalisation of SProxy.
About Symbols and Strings
PureScript knows value level strings (known as String) and type level strings (known as Symbol).
A String can have any string value at runtime. The compiler does not track the value of the string.
A Symbol is different, it can only have one value (but remember, it is on the type level). The compiler keeps track of this string. This allows the compiler to type check certain expressions.
Symbols in Practice
The most prominent use of Symbols is in records. The difference between a Record and a String-Map is that the compiler knows about the keys at compile time and can typecheck lookups.
Now, sometimes we need to bridge the gap between these two worlds: The type level and the value level world. Maybe you know that PureScript records are implemented as JavaScript objects in the official compiler. This means we need to somehow receive a string value from our symbol. The magical function reflectSymbol allows us to turn a symbol into a string. But a symbol is on the type level. This means we can only write a symbol where we can write types (so for example in type definition after ::). This is where the Proxy hack comes in. The SProxy is a simple value that "stores" the type by applying it.
For example the get function from purescript-records allows us to get a value at a property from a record.
get :: forall proxy r r' l a. IsSymbol l => Cons l a r' r => proxy l -> Record r -> a
If we apply the first paramerter we get:
get (Proxy :: Proxy "x") :: forall r a. { x :: a | r } -> a
Now you could argue that you can get the same function by simply writing:
_.x :: forall r a. { x :: a | r } -> a
It has exactly the same type. This leads to one last question:
But why?
Well, there are certain meta programming szenarios, where you don't programm for a specific symbol, but rather for any symbol. Imagine you want to write a JSON serialiser for any record. You might want to "iterate" over every property of the record, get the value, turn the value itself into JSON and then concatinate the key value pair with all the other keys and values.
An example for such an implementation can be found here
This is maybe not the most technical explanation of it all, but this is how I understand it.

Isabelle tactic definition

I was trying to copy a tactic file that might help proving my theorem, however, there seems to be having some problems.
The original tactic is like this:
fun typechk_step_tac tyrls =
FIRSTGOAL (test_assume_tac ORELSE' filt_resolve_tac tyrls 4);
(* Output:
Value or constructor (filt_resolve_tac) has not been declared
*)
I tried to find this tactic in the internet, but there is not much explanation of this. I saw that some theory file in 2009 use this method, and for 2020 one, a theory file use a similar method called filt_resolve_from_net_tac, which I think the types of them are different, so I am not sure about how to use them.
Other than the filt_resolve_tac, the tactic file used a function called ref like this:
val basictypeinfo = ref([]:thm list);
(* Output:
Value or constructor (ref) has not been declared
*)
However, the Isabelle 2020 seems know something about the ref, since when I changed something:
val basictypeinfo = [];
fun addT typelist = (basictypeinfo := (typelist #(!basictypeinfo)));
(* It shows error:
Type error in function application.
Function: ! : 'a ref -> 'a
Argument: basictypeinfo : 'a list
Reason: Can't unify 'a ref (*In Basis*) with 'a list (*In Basis*) (Different type constructors)
*)
It clearly shows that ref is like a type, and it is defined in the Isabelle right?
Therefore the ref in ref([]:the list) should be similar to a casting function, and I found that there is a thing called Unsynchronized.ref which solves the type problem, may I know are they the same thing in this context?
In the later part of the files, there are also some tactic and rule set seems to be not defined, for example:
etac: Value or constructor (etac) has not been declared
(*
I saw Prof. Paulson had shown this tactic in his Isabelle lecture,
but I couldn't find it in the Isabelle manual or the implementation manual,
is the name of it changed?
*)
ZF_typechecks: I couldn't find any rule sets that has this name in the whole ZF directory.
Sorry to have so many questions, it seems that the tactic file is no longer really well-supported by the new Isabelle, are there still people using Isar/ML to define new tactic? Or most people are doing this with the method declaration in the Isabelle? And which part of the Isabelle/Isar reference manual should I read to master this skill? Thank you very much indeed.

Find the instantiation of a class for a type

In Isabelle/HOL, how do I find where a given type was instantiated for a given class? For the sake of this post for example, where real was instantiated as a conditionally_complete_linorder. To justify the question: I might want to know this for inspiration for a similar instantiation, for showing it to someone(s), for Isabelle/HOL practice reading, for curiosity, and so on. My process at the moment:
First, check it actually is: type instantiation real :: conditionally_complete_linorder begin end and see if I get the error message "No parameters and no pending instance proof obligations in instantiation."
Next, ideally before where I'd need to know how i.e. whether it was direct, or implicit via classes C_1[, C_2, C_3, etc]. Then, I would need to find where those instantiations are, either an explicit instantiation real :: conditionally_complete_linorder or the implicit ones for the C_i (same process for either case ofc). I don't know how to find out how, so I have to check for an explicit instantiation, then all possible implicit instantiations.
For explicit, I can do a grep -Ern ~/.local/src/Isabelle2019 -e 'instantiation real :: conditionally_complete_linorder' (and hope the whitespace isn't weird, or do a more robust search :)). Repeat for AFP location. Alternatively, to stay within the jEdit window:
I can find where the class itself was defined by typing term "x::'a::conditionally_complete_linorder" then Ctrl-clicking the class name, and then check if real is directly instantiated in that file with Ctrl-f.
I could then check if it's instantiated where the type real is defined by typing term "x::real" and Ctrl-clicking real, then Ctrl-f for conditionally_complete_linorder in that file.
(If it is in either place it'll be whichever is further down in the import hierarchy, but I find just going through those two steps simpler.) However, if neither two places turn it up then either, for whatever reason, it is explicitly instantiated somewhere else or is implicitly instantiated. For that reason grep is more robust.
If explicit turns nothing up then I check implicit. Looking at the class_deps graph I can see that conditionally_complete_linorder can follow from either complete_linorder or linear_continuum. I can then continue the search by seeing if real is instantiated as either of them (disregarding any I happen to know real can't be instantiated as). I can also check to see if it's instantiated as both conditioanlly_complete_lattice and linorder, which is what I can see conditionally_complete_linorder is a simple (no additional assumptions) combination of*. Repeat for all of these classes recursively until the instantiations are found. In this case, I can see that linear_continuum_topology implies linear_continuum, so kill two birds with one stone with grep -Ern ~/.local/src/Isabelle2019 -e "instantiation.*real" | grep continuum and find /path/to/.local/src/Isabelle2019/src/HOL/Real.thy:897:instantiation real :: linear_continuum.
This process is quite tedious. Less but still quite tedious** would be to get the class_deps graph up and Ctrl-f for "instantiation real" in Real.thy and look for instantiations of: the original class, the superclasses of it, or the classes which imply it. Then in the files each those classes are defined search for "instantiation real". Do this recursively till done. In this case I would have found what I needed in Real.thy.
Is there an easier way? Hope I just missed something obvious.
* I can't Ctrl-click in Conditionally_Complete_Lattices.thy to jump to linorder directly, I guess because of something to do with it being pre-built, so I have to do the term "x::'a::linorder" thing again.
** And also less robust, as it is minus grep-ing which can turn up weirder instantiation locations, then again I'm not sure if this ever comes up in practice.
Thanks
You can import the theory in the code listing below and then use the command find_instantiations. I will leave the code without further explanation, but please feel free to ask further questions in the comments if you need further details or suspect that something is not quite right.
section ‹Auxiliary commands›
theory aux_cmd
imports Complex_Main
keywords "find_instantiations" :: thy_decl
begin
subsection ‹Commands›
ML ‹
fun find_instantiations ctxt c =
let
val {classes, ...} = ctxt |> Proof_Context.tsig_of |> Type.rep_tsig;
val algebra = classes |> #2
val arities = algebra |> Sorts.arities_of;
in
Symtab.lookup arities c
|> the
|> map #1
|> Sorts.minimize_sort algebra
end
fun find_instantiations_cmd tc st =
let
val ctxt = Toplevel.context_of st;
val _ = tc
|> Syntax.parse_typ ctxt
|> dest_Type
|> fst
|> find_instantiations ctxt
|> map Pretty.str
|> Pretty.writeln_chunks
in () end
val q = Outer_Syntax.command
\<^command_keyword>‹find_instantiations›
"find all instantiations of a given type constructor"
(Parse.type_const >> (fn tc => Toplevel.keep (find_instantiations_cmd tc)));
›
subsection ‹Examples›
find_instantiations filter
find_instantiations nat
find_instantiations real
end
Remarks
I would be happy to provide amendments if you find any problems with it, but do expect a reasonable delay in further replies.
The command finds both explicit and implicit instantiations, i.e. it also finds the ones that were achieved by means other than the use of the commands instance or instantiation, e.g. inside an ML function.
Unfortunately, the command does not give you the location of the file where the instantiation was performed - this is something that would be more difficult to achieve, especially, given that instantiations can also be performed programmatically. Nevertheless, given a list of all instantiations, I believe, it is nearly always easy to use the in-built search functionality on the imported theories to narrow down the exact place where the instantiation was performed.

Why can't a range of char be collected?

I'm trying to generate a vector containing lowercase ASCII characters. This more convoluted approach works:
let ascii_lowercase = (b'a'..=b'z').map(|b| b as char).collect::<Vec<char>>();
But this more straightforward one, which I came up with in the first place, does not:
let ascii_lowercase = ('a'..='z').collect::<Vec<char>>();
The error is:
error[E0599]: no method named `collect` found for type `std::ops::RangeInclusive<char>` in the current scope
--> src/main.rs:2:39
|
2 | let ascii_lowercase = ('a'..='z').collect::<Vec<char>>();
| ^^^^^^^
|
= note: the method `collect` exists but the following trait bounds were not satisfied:
`std::ops::RangeInclusive<char> : std::iter::Iterator`
`&mut std::ops::RangeInclusive<char> : std::iter::Iterator`
Which is weird, because as far as I understand, there is a blanket implementation of Iterator for RangeInclusive.
Is it impossible to use a range of chars as an iterator? If so, why? If not, what am I doing wrong? I'm using stable Rust 2018 1.31.1.
EDIT 2020-07-17: since Rust 1.45.0, the trait Step is implemented for char, making Range<char> (and some other char ranges) work as an iterator. The code in the question now compiles without problem!
Old answer below.
The expression b'a'..=b'z' has the type RangeInclusive<u8> (see on Playground) because the expression b'a' has the type u8: that's what the b in front of the character literal is for. On the other hand, the expression 'a'..='z' (without the bs) has the type RangeInclusive<char>.
[...] there is a blanket implementation of Iterator for RangeInclusive.
For one, this is not what we call "blanket implementation" (this is when the impl block is for T or for &T (or similar) with T being a generic type). But yes, there is an impl. But let's take a closer look:
impl<A> Iterator for RangeInclusive<A>
where
A: Step, // <--- important
The A: Step bound is important. As you can see in the documentation for Step, this trait is implemented for all primitive integer types, but not for char. This means that there is no clear "add one" operation on characters. Yes, you could define it to be the next valid Unicode codepoint, but the Rust developers probably decided against that for a good reason.
As a consequence, RangeInclusive<char> does not implement Iterator.
So your solution is already a good one. I would probably write this:
(b'a'..=b'z').map(char::from).collect::<Vec<_>>()
The only real advantage is that in this version, char doesn't appear twice.
The problem was that the iteration capabilities of range types depend on the Step trait (see extended answer).
However, starting from Rust 1.45, char also implements Step (PR 72413), which means that the code in the question now works!
let ascii_lowercase: Vec<char> = ('a'..='h').collect();
assert_eq!(
ascii_lowercase,
vec!['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h'],
);

How to call two functions and use their results as arguments for each other?

I have code:
let join a ~with':b ~by:key =
let rec a' = link a ~to':b' ~by:key
and b' = link b ~to':a' ~by:(Key.opposite key) in
a'
and compilation result for it is:
Error: This kind of expression is not allowed as right-hand side of
`let rec' build complete
I can rewrite it to:
let join a ~with':b ~by:key =
let rec a'() = link a ~to':(b'()) ~by:key
and b'() = link b ~to':(a'()) ~by:(Key.opposite key) in
a'()
It is compilable variant, but implemented function is infinitely recursive and it is not what I need.
My questions: Why is first implementation invalid? How to call two functions and use their results as arguments for each other?
My compiler version = 4.01.0
The answer to your first question is given in Section 7.3 of the OCaml manual. Here's what it says:
Informally, the class of accepted definitions consists of those definitions where the defined names occur only inside function bodies or as argument to a data constructor.
Your names appear as function arguments, which isn't supported.
I suspect the reason is that you can't assign a semantics otherwise. It seems to me the infinite computation that you see is impossible to avoid in general.

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