I read in a few tutorials that if a then b else c stands for match a with true => b | false => c end. However the former very strangely does not check the type of a, while the latter of course makes sure that a is a boolean. For instance,
Coq < Check if nil then 1 else 2.
if nil then 1 else 2
: nat
where
?A : [ |- Type]
Coq < Check match nil with true => 1 | false => 2 end.
Toplevel input, characters 33-38:
> Check match nil with true => 1 | false => 2 end.
> ^^^^^
Error: Found a constructor of inductive type bool while
a constructor of list is expected.
Why is if ... then ... else ... allowing its first argument to be anything else than a non-boolean? Is there some overloading going on? (Locate "if". gives no result.)
Let me quote the Coq Reference manual:
For inductive types with exactly two constructors and for pattern-matchings expressions which do not depend on the arguments of the constructors, it is possible to use a if ... then ... else ... notation. More generally, for an inductive type with constructors C1 and C2, we have the following equivalence:
if term [dep_ret_type] then term1 else term2
is equivalent to
match term [dep_ret_type] with
| C1 _ ... _ => term1 (* we cannot bind the arguments *)
| C2 _ ... _ => term2
end
As you can see, the first constructor is treated as true value. Here is an example:
Definition is_empty {A : Type} (xs : list A) : bool :=
if xs then true else false.
Related
Why am I getting Warning this expression should have type unit with this code? although it does what it should do.
let matchInf42 list =
let a = ref 0 in
let lstLength = List.length list in
let rec matchInf4242 list =
match list with
|[]->[]
|m::body->
begin
if (m < 42) then a := !a + 1;
matchInf4242 body
end
in matchInf4242 list;
if(!a = lstLength) then -1 else 0
Warning:
ocamlopt match.ml -o m
File "match.ml", line 14, characters 7-24:
14 | in matchInf4242 list;
^^^^^^^^^^^^^^^^^
Warning 10: this expression should have type unit.
TL;DR: The error you obtain is a typical type error (mostly)
"Mostly" because, admittedly, it is not an "error" but a mere "warning" here, yet it appears this kind of warning (Warning 10: non-unit-statement) is always worth being addressed (i.e., avoided).
It is actually an instance of the following pattern:
42; print_string "…" ;;
(* or more generally *)
any_value_not_having_type_unit; any_value_having_type_unit ;;
(* which would raise *)
> Characters 0-2:
> 42; print_string "…";;
> ^^
> Warning 10: this expression should have type unit.
> …- : unit = ()
Further details
Just to recall, unit is a singleton type (which only has the value ()) and is typically chosen to assign a return type to functions that "return no specific value" but produce some side-effect.
Actually, the sequence operator is "a bit more general / more flexible" as what we might expect:
# let semicolon i j = i; j ;;
val semicolon : 'a -> 'b -> 'b = <fun>
That is to say, it's not semicolon : unit -> 'b -> 'b, and thereby the message we got for the code i; j when i does not have the type unit, is a mere warning, not a type error.
Fixes/workarounds
Two strategies to avoid this warning:
Either ignore it by relying on the ignore function
# ignore;;
- : 'a -> unit = <fun>
# ignore 42; print_string "…";;
…- : unit = ()
Or change/fix the way you compute the left-hand side of the sequence (so its type is unit).
In the particular case of your question example, it would suffice to write this (the only change being indicated with a symbol §):
let matchInf42 list =
let a = ref 0 in
let lstLength = List.length list in
let rec matchInf4242 list =
match list with
|[] -> () (*←§*)
|m::body->
begin
if (m < 42) then a := !a + 1;
matchInf4242 body
end
in matchInf4242 list;
if(!a = lstLength) then -1 else 0
Extra remark
Finally for completeness (even though it was not explicitly part of your question), note that the example function you considered could also be implemented in a more "functional" style (without references nor sequences, avoiding also the need for calling the List.length function beforehand):
let matchInf42 l =
if List.for_all (fun m -> m < 42) l
then -1 else 0
(* or *)
let matchInf42 l =
if List.fold_left (fun r e -> r && e < 42) true l
then -1 else 0
Change
matchInf4242 list
to either
ignore (matchInf4242 list)
or
matchInf4242 list in ().
This makes the statement return () (i.e. a unit), which is what ocaml expects.
I am trying to implement a program that takes a string and a list and returns NONE if it doesn't find any match and the list without the element if it does.
fun all_except_option ("string",["he","she","string"]) = SOME["he","she"]
I have managed to make it working, but without the options type and i have no idea how to make it return SOME list instead a normal list.
fun all_except_option(str,lst)=
case lst of
[] => []
| x::lst' => if same_string(x,str) = false
then let fun append (word, list) = word::list
in append(x,[]) :: all_except_option(str,lst')
end
else all_except_option(str,lst')
Thank you. I managed to make it working, but i still don't understand the "else case" and how is my programm handling it. Here is the working code. I would be glad if you can explain me the "else case all_except_option(str,list') of".
fun all_except_option(str,list)=
case list of
[] => NONE
| x::list' => if same_string(x,str) then
SOME( list')
else case all_except_option(str,list') of
NONE=>NONE
| SOME list'=>SOME(x::list')
implement a program that takes a string and a list and returns NONE if it doesn't find any match and the list without the element if it does.
all_except_option ("string",["he","she","string"]) = SOME ["he","she"]
How is SOME [] different from NONE? As in, if this function returned just a list, it would be possible to say that removing occurrences of "string" results in no other strings: Either the list was empty already, or it contained only occurrences of "string". I am not sure why NONE vs. SOME [] is warranted in one case over the other.
So the better function is one that simply returns a plain list:
fun except (x, ys) = List.filter (fn y => x <> y)
When is it useful to return 'a option?
For example when the return type does not have a way to indicate no result already:
fun lookup k1 [] = NONE
| lookup k1 ((k2,v)::pairs) =
if k1 = k2
then SOME v
else lookup k1 pairs
This function returns 0 or 1 thing. But it's also a simple function because it never aggregates results over its recursion. Recursive functions become complicated when they return composite data types like 'a option when it needs to unpack the result of the recursion.
A good example is an eval function that sometimes fails:
datatype expr
= Add of expr * expr
| Sub of expr * expr
| Mul of expr * expr
| Div of expr * expr
| Int of int
fun eval (Int n) = SOME n
| eval (Add (e1, e2)) = evalHelper ( op+ ) (e1, e2)
| eval (Sub (e1, e2)) = evalHelper ( op- ) (e1, e2)
| eval (Mul (e1, e2)) = evalHelper ( op* ) (e1, e2)
| eval (Div (e1, e2)) =
case eval e1 of
NONE => NONE
| SOME x => case eval e2 of
NONE => NONE
| SOME 0 => NONE
| SOME y => SOME (x div y)
and evalHelper binop (e1, e2) =
case eval e1 of
NONE => NONE
| SOME x => case eval e2 of
NONE => NONE
| SOME y => SOME (binop (x, y))
Here the return type is int option, which means that you most often return an int, but if you ever divide by zero, that results in "no value", so rather than raise an exception, we return NONE, which necessitates us to return SOME n when there is a result, so that the type fits in both cases.
A quick demonstration:
- eval (Div (Int 5, Int 2));
> val it = SOME 2 : int option
- eval (Div (Int 5, Int 0));
> val it = NONE : int option
- eval (Div (Int 2, Sub (Int 3, Int 3)));
> val it = NONE : int option
- eval (Div (Int 0, Int 1));
> val it = SOME 0 : int option
Here SOME 0 actually means "the result is 0", which is not the same as "cannot divide by zero."
How can I write a function in Ocaml, that would accept two boolean values (for instance a and b, that would represent logical values 0 and 1) and a character, that would determine the operation between these two logical values? The function would then return true or false, depending on the corresponding logical value input.
There is both a semantic, as well as a syntax error in this code; well, maybe there are more, but the compiler only made a complaint about this one so far:
line 2, characters 27-30:
Error: This expression has type char but an expression was expected of type
bool
This is what the compiler has stated about the code and here's what I've wrote:
let logic (a, b) operation = match operation with
| true -> if (operation == 'A') then match (a,b) with
| (true, true) -> true
| _ -> false
else if (operation == '0') then match (a,b) with
| (false,false) -> false
| _ -> true
else if (operation == 'X') then match (a,b) with
| (true,true) -> false
| (false,false) -> false
| _ -> true
else if (operation == 'I') then match (a,b) with
| (true, false) -> false
| _ -> true
else then match (a,b) with
| _ -> false
| false -> end;;
logic (a,b) 'A';;
--------------------------------------------------------------------------
OK, I've made some progress,at least it compiles now. However I still get an error, when I try to call the "logic" function with this line:
logic (true, false) 'A';;
The error states this:
Error: This expression has type bool
This is not a function; it cannot be applied.
Here's what I wrote now:
let logic (a, b) operation = match operation with
| 'A' -> (match (a,b) with
| (true, true) -> true
| _ -> false
)
| '0' -> (match (a,b) with
| (false,false) -> false
| _ -> true
)
| 'X' -> (match (a,b) with
| (true,true) -> false
| (false,false) -> false
| _ -> true)
| 'I' -> (match (a,b) with
| (true, false) -> false
| _ -> true
)
| _ -> (match (a,b) with
| _ -> false
)
To expand on the comment of #Elan-Hamburger a little bit.
You have this:
match operation with
| true -> ...
But your operation is a character. So this won't work. You can only match operation against specific characters.
It's especially strange since you later compare operator to various characters using ==. But the match will do that for you.
In other words you can have something like this:
match operation with
| 'A' -> ...
| '0' -> ...
| 'X' -> ...
| 'I' -> ...
| _ -> ...
There are many other problems with your code. Here are some comments:
You have nested match statements, which requires parentheses to work out right. If you try to nest without parentheses there's no way to tell when the inner match is over and further alternatives (starting with |) of the outer match are given.
Nested match statements look like this:
match expr with
| X ab ->
(match ab with
| A -> 4
| B -> 2
)
| Y _ -> 0
(You can also use begin/end instead of parentheses if you prefer the way that looks.)
You're using == to compare values. The ordinary equality comparison operator is =. The special operator == should only be used when you have a specific reason for it.
You have code that reads like this:
else then match ...
This can't be syntactically correct. Possibly the then is left over from an edit.
You can simplify this expression
match (a, b) with
| _ -> false
to this simpler expression:
false
Update
I can't reproduce your new reported problem. In fact your new code works OK for me in two quick tests.
. . .
val logic : bool * bool -> char -> bool = <fun>
# logic (true, false) 'A';;
- : bool = false
# logic (true, true) 'A';;
- : bool = true
Possibly there was some extraneous input (or extraneous definitions) in your top-level session. I suggest just trying again with a fresh session.
Write a function remove_option, which takes a string and a string list. Return NONE if the string is not in the list, else return SOME xs where xs is identical to the argument list except the string is not in it. You may assume the string is in the list at most once. Use same_string, provided to you, to compare strings. Sample solution is around 8 lines.
The function type should be fn : string * string list -> string list option.Here is my code
fun same_string(s1 : string, s2 : string) =
s1 = s2
fun remove_option (str: string ,str_list : string list) =
case str_list of
[] => NONE
| x::xs => if same_string(x,str)
then SOME xs
else x :: remove_option( str,xs)
and the error report
hw2provided.sml:10.5-15.37 Error: right-hand-side of clause doesn't agree with f
unction result type [tycon mismatch]
expression: _ option
result type: string list
in declaration:
remove_option =
(fn (<pat> : string,<pat> : string list) =>
(case str_list
of <pat> => <exp>
| <pat> => <exp>))
uncaught exception Error
raised at: ../compiler/TopLevel/interact/evalloop.sml:66.19-66.27
../compiler/TopLevel/interact/evalloop.sml:44.55
../compiler/TopLevel/interact/evalloop.sml:292.17-292.20
So where is the bug ?
The problem is that you want to return a string list option but the line
else x :: remove_option( str,xs)
makes it seem that you want to return a string list
What you need to do with the return value of remove_option( str,xs) is
1) decide what to do if it is NONE
2) extract the string list strings (or whatever you want to call it) if it is of the form SOME strings, tack x onto the front of the list, and repackage it with SOME before returning it.
You seem comfortable with the use of case, so you could use it here.
Since John showed where the bug is, here are some extra comments:
Since the function same_string is not injected, it is superfluous. You might as well use =.
Recursive functions that return 'a option are kind of tricky, since you need to unpack the result:
fun remove_option (s1, []) = NONE
| remove_option (s1, s2::ss) =
if s1 = s2
then SOME ss
else case remove_option (s1, ss) of
NONE => NONE
| SOME ss' => SOME (s2::ss')
Generally, when you see the pattern
case x_opt of
NONE => NONE
| SOME x => SOME (f x))
this can be refactored into using e.g. Option.map : ('a -> 'b) -> 'a option -> 'b option:
Option.map f x_opt
In this case,
fun curry f x y = f (x, y)
fun remove_option (s1, []) = NONE
| remove_option (s1, s2::ss) =
if s1 = s2
then SOME ss
else Option.map (curry op:: s2) (remove_option (s1, ss))
where curry op:: s2, the function that puts s2 in front of a list.
I've found myself repeating a pattern over and over again, and I'd like to abstract it. I'm fairly confident that coq is sufficiently expressive to capture the pattern, but I'm having a bit of trouble figuring out how to do so. I'm defining a programming language, which has mutually recursive inductive datatypes representing the syntactic terms:
Inductive Expr : Set :=
| eLambda (x:TermVar) (e:Expr)
| eVar (x:TermVar)
| eAscribe (e:Expr) (t:IFType)
| ePlus (e1:Expr) (e2:Expr)
| ... many other forms ...
with DType : Set :=
| tArrow (x:TermVar) (t:DType) (c:Constraint) (t':DType)
| tInt
| ... many other forms ...
with Constraint : Set :=
| cEq (e1:Expr) (e2:Expr)
| ...
Now, there are a number of functions that I need to define over these types. For example, I'd like a function to find all of the free variables, a function to perform substitution, and a function to pull out the set of all constraints. These functions all have the following form:
Fixpoint doExpr (e:Expr) := match e with
(* one or two Interesting cases *)
| ...
(* lots and lots of boring cases,
** all of which just recurse on the subterms
** and then combine the results in the same way
*)
| ....
with doIFType (t:IFType) := match t with
(* same structure as above *)
with doConstraint (c:Constraint) := match c with
(* ditto *)
For example, to find free variables, I need to do something interesting in the variable cases and the cases that do binding, but for everything else I just recursively find all of the free variables of the subexpressions and then union those lists together. Similarly for the function that produces a list of all of the constraints. The substitution case is a little bit more tricky, because the result types of the three functions are different, and the constructors used to combine the subexpressions are also different:
Variable x:TermVar, v:Expr.
Fixpoint substInExpr (e:Expr) : **Expr** := match e with
(* interesting cases *)
| eLambda y e' =>
if x = y then eLambda y e' else eLambda y (substInExpr e')
| eVar y =>
if x = y then v else y
(* boring cases *)
| eAscribe e' t => **eAscribe** (substInExpr e') (substInType t)
| ePlus e1 e2 => **ePlus** (substInExpr e1) (substInExpr e2)
| ...
with substInType (t:Type) : **Type** := match t with ...
with substInConstraint (c:Constraint) : **Constraint** := ...
.
Writing these functions is tedious and error prone, because I have to write out all of the uninteresting cases for each function, and I need to make sure I recurse on all of the subterms. What I would like to write is something like the following:
Fixpoint freeVars X:syntax := match X with
| syntaxExpr eVar x => [x]
| syntaxExpr eLambda x e => remove x (freeVars e)
| syntaxType tArrow x t1 c t2 => remove x (freeVars t1)++(freeVars c)++(freeVars t2)
| _ _ args => fold (++) (map freeVars args)
end.
Variable x:TermVar, v:Expr.
Fixpoint subst X:syntax := match X with
| syntaxExpr eVar y => if y = x then v else eVar y
| syntaxExpr eLambda y e => eLambda y (if y = x then e else (subst e))
| syntaxType tArrow ...
| _ cons args => cons (map subst args)
end.
The key to this idea is the ability to generally apply a constructor to some number of arguments, and to have some kind of "map" that that preserves the type and number of arguments.
Clearly this pseudocode doesn't work, because the _ cases just aren't right. So my question is, is it possible to write code that is organized this way, or am I doomed to just manually listing out all of the boring cases?
Here's another way, though it's not everyone's cup of tea.
The idea is to move recursion out of the types and the evaluators, parameterizing it instead, and turning your expression values into folds. This offers convenience in some ways, but more effort in others -- it's really a question of where you end up spending the most time. The nice aspect is that evaluators can be easy to write, and you won't have to deal with mutually recursive definitions. However, some things that are simpler the other way can become brain-twisters in this style.
Require Import Ssreflect.ssreflect.
Require Import Ssreflect.ssrbool.
Require Import Ssreflect.eqtype.
Require Import Ssreflect.seq.
Require Import Ssreflect.ssrnat.
Inductive ExprF (d : (Type -> Type) -> Type -> Type)
(c : Type -> Type) (e : Type) : Type :=
| eLambda (x:nat) (e':e)
| eVar (x:nat)
| eAscribe (e':e) (t:d c e)
| ePlus (e1:e) (e2:e).
Inductive DTypeF (c : Type -> Type) (e : Type) : Type :=
| tArrow (x:nat) (t:e) (c':c e) (t':e)
| tInt.
Inductive ConstraintF (e : Type) : Type :=
| cEq (e1:e) (e2:e).
Definition Mu (f : Type -> Type) := forall a, (f a -> a) -> a.
Definition Constraint := Mu ConstraintF.
Definition DType := Mu (DTypeF ConstraintF).
Definition Expr := Mu (ExprF DTypeF ConstraintF).
Definition substInExpr (x:nat) (v:Expr) (e':Expr) : Expr := fun a phi =>
e' a (fun e => match e return a with
(* interesting cases *)
| eLambda y e' =>
if (x == y) then e' else phi e
| eVar y =>
if (x == y) then v _ phi else phi e
(* boring cases *)
| _ => phi e
end).
Definition varNum (x:ExprF DTypeF ConstraintF nat) : nat :=
match x with
| eLambda _ e => e
| eVar y => y
| _ => 0
end.
Compute (substInExpr 2 (fun a psi => psi (eVar _ _ _ 3))
(fun _ phi =>
phi (eLambda _ _ _ 1 (phi (eVar _ _ _ 2)))))
nat varNum.
Compute (substInExpr 1 (fun a psi => psi (eVar _ _ _ 3))
(fun _ phi =>
phi (eLambda _ _ _ 1 (phi (eVar _ _ _ 2)))))
nat varNum.
Here is a way to go, but it does not give very readable code: use tactics.
Let's say I have a language with many constructors of various arity, and I want to apply a specific goal only to the case given by constructor aaa, and I want to traverse all the other constructors, to get down to the aaa's that may appear under them. I can do the following:
Say you want to define a function A -> B (A is the type of the language), you will need to keep track of what case you are in,
so you should define a phantom type over A, reducing to B.
Definition phant (x : A) : Type := B.
I suppose that the union function has type B -> B -> B and that you have a default value in B, called empty_B
Ltac generic_process f acc :=
match goal with
|- context [phan (aaa _)] => (* assume aaa has arith 1 *)
intros val_of_aaa_component; exact process_this_value val_of_aaa_component
| |- _ =>
(* This should be used when the next argument of the current
constructor is in type A, you want to process recursively
down this argument, using the function f, and keep this result
in the accumulator. *)
let v := fresh "val_in_A" in
intros v; generic_process f (union acc (f v))
(* This clause will fail if val_in_A is not in type A *)
| |- _ => let v := fresh "val_not_in_A" in
(* This should be used when the next argument of the current
constructor is not in type A, you want to ignore it *)
intros v; generic_process f acc
| |- phant _ =>
(* this rule should be used at the end, when all
the arguments of the constructor have been used. *)
exact acc
end.
Now, you define the function by a proof. Let's say the function is called process_aaa.
Definition process_aaa (x : A) : phant x.
fix process_aaa 1.
(* This adds process_add : forall x:A, phant x. in the context. *)
intros x; case x; generic_process process_aaa empty_B.
Defined.
Note that the definition of generic_process only mention one constructor by name, aaa, all others
are treated in a systematic way. We use the type information to detect those sub-components in which we want to perform a recursive descent. If you have several mutually inductive types, you can add arguments to the generic_process function to indicate which function will be used for each type and have more clauses, one for each argument of each type.
Here is a test of this idea, where the language has 4 constructors, values to be processed are the ones that appear in the constructor var and the type nat is also used in another constructor (c2). We use the type of lists of natural numbers as the type B, with nil as the empty and singleton lists as result when encountering variables. The function collects all occurrences of var.
Require Import List.
Inductive expr : Type :=
var : nat -> expr
| c1 : expr -> expr -> expr -> expr
| c2 : expr -> nat -> expr
| c3 : expr -> expr -> expr
| c4 : expr -> expr -> expr
.
Definition phant (x : expr) : Type := list nat.
Definition union := (#List.app nat).
Ltac generic_process f acc :=
match goal with
|- context[phant (var _)] => exact (fun y => y::nil)
| |- _ => let v := fresh "val_in_expr" in
intros v; generic_process f (union acc (f v))
| |- _ => let v := fresh "val_not_in_expr" in
intros v; generic_process f acc
| |- phant _ => exact acc
end.
Definition collect_vars : forall x : expr, phant x.
fix collect_vars 1.
intros x; case x; generic_process collect_vars (#nil nat).
Defined.
Compute collect_vars (c1 (var 0) (c2 (var 4) 1)
(c3 (var 2) (var 3))).
The last computation returns a list containing values 0 4 2 and 3 as expected, but not 1, which did not occur inside a var constructor.