Given this context:
open import IO
open import Data.String
open import Data.Unit
open import Coinduction
postulate
A : Set
f : String → A
g₁ g₂ : A → String
let's say I want to implement something like
foo : IO ⊤
foo = ♯ readFiniteFile "input.txt" >>= \s →
♯ (♯ putStrLn (g₁ (f s)) >>
♯ putStrLn (g₂ (f s)))
by let-binding the intermediate result f s. I was hoping this would work:
foo₁ : IO ⊤
foo₁ = ♯ readFiniteFile "input.txt" >>= \s →
let x = f s in
♯ (♯ putStrLn (g₁ x) >>
♯ putStrLn (g₂ x))
but this fails with
Not implemented: coinductive constructor in the scope of a
let-bound variable
so I tried moving out the ♯:
foo₂ : IO ⊤
foo₂ = ♯ readFiniteFile "input.txt" >>= \s →
♯ (let x = f s in
♯ putStrLn (g₁ x) >>
♯ putStrLn (g₂ x))
same problem as before.
I managed to get around this by just lifting out the ♯ening step:
_>>′_ : ∀ {a} {A B : Set a} → IO A → IO B → IO B
m >>′ m′ = ♯ m >> ♯ m′
foo₂ : IO ⊤
foo₂ = ♯ readFiniteFile "input.txt" >>= \s →
♯ let x = f s in
putStrLn (g₁ x) >>′ putStrLn (g₂ x)
but why does that work if the "inlined" version doesn't?
FYI, this problem was a long standing one and as I was having a look at it today, I noticed that it had been fixed just a few days before you asked your question. The fix is part of the currently released version (version 2.5.1.1).
Related
I have defined infinite streams as follows:
record Stream (A : Set) : Set where
coinductive
field head : A
field tail : Stream A
and an inductive type which shows that some element in a stream eventually satisfies a predicate:
data Eventually {A} (P : A -> Set) (xs : Stream A) : Set where
here : P (head xs) -> Eventually P xs
there : Eventually P (tail xs) -> Eventually P xs
I would like to write a function which skips over elements of the stream until the head of the stream satisfies a predicate. To ensure termination, we must know that an element eventually satisfies the predicate, else we could loop forever. Hence, the definition of Eventually must be passed as an argument. Furthermore, the function should not computationally depend on the Eventually predicate, as it is just there to prove termination, so I would like it to be an erased argument.
dropUntil : {A : Set} {P : A -> Set} (decide : ∀ x → Dec (P x)) → (xs : Stream A) → #0 Eventually P xs → Stream A
dropUntil decide xs ev with decide (head xs)
... | yes prf = xs
... | no contra = dropUntil decide (tail xs) ?
Here is the problem - I would like to fill in the hole in the definition. From contra in scope, we know that the head of the stream does not satisfy P, and hence by definition of eventually, some element in the tail of the stream must satisfy P. If Eventually wasn't erased in this context, we could simply pattern match on the predicate, and prove the here case impossible. Normally in these scenarios I would write an erased auxiliary function, on the lines of:
#0 eventuallyInv : ∀ {A} {P : A → Set} {xs : Stream A} → Eventually P xs → ¬ P (head xs) → Eventually P (tail xs)
eventuallyInv (here x) contra with contra x
... | ()
eventuallyInv (there ev) contra = ev
The problem with this approach is that the Eventually proof is the structurally recursive argument in dropUntil, and calling this auxiliary function does not pass the termination checker as Agda does not "look inside" the function definition.
Another approach I tried is inlining the above erased function into the definition of dropUntil. Unfortunately, I had no luck with this approach either - using the definition of case ... of like described here https://agda.readthedocs.io/en/v2.5.2/language/with-abstraction.html did not pass the termination checker either.
I have written an equivalent program in Coq which is accepted (using Prop rather than erased types), so I am confident that my reasoning is correct. The main reason why Coq accepted the definition and Agda doesn't is that Coq's termination checker expands function definitions, and hence the "auxiliary erased function" approach succeeds.
EDIT:
This is my attempt using sized types, however it does not pass the termination checker and I can't figure out why.
record Stream (A : Set) : Set where
coinductive
field
head : A
tail : Stream A
open Stream
data Eventually {A} (P : A → Set) (xs : Stream A) : Size → Set where
here : ∀ {i} → P (head xs) → Eventually P xs (↑ i)
there : ∀ {i} → Eventually P (tail xs) i → Eventually P xs (↑ i)
#0 eventuallyInv : ∀ {A P i} {xs : Stream A} → Eventually P xs (↑ i) → ¬ P (head xs) → Eventually P (tail xs) i
eventuallyInv (here p) ¬p with ¬p p
... | ()
eventuallyInv (there ev) ¬p = ev
dropUntil : ∀ {A P i} → (∀ x → Dec (P x)) → (xs : Stream A) → #0 Eventually P xs (↑ i) → Stream A
dropUntil decide xs ev with decide (head xs)
... | yes p = xs
... | no ¬p = dropUntil decide (tail xs) (eventuallyInv ev ¬p)
In your case you can work with a weaker notion of Eventually which matches what dropUntil actually needs to know. It's also single constructor so you can match on it even when erased.
data Eventually' {A} (P : A -> Set) (xs : Stream A) : Set where
next : (¬ P (head xs) → Eventually' P (tail xs)) → Eventually' P xs
eventuallyInv : ∀ {A} {P : A → Set} {xs : Stream A} → (ev : Eventually P xs) → Eventually' P xs
eventuallyInv (here p) = next \ np → ⊥-elim (np p)
eventuallyInv (there ev) = next \ np → eventuallyInv ev
dropUntil' : {A : Set} {P : A -> Set} (decide : ∀ x → Dec (P x)) → (xs : Stream A) → #0 Eventually' P xs → Stream A
dropUntil' decide xs (next ev) with decide (head xs)
... | yes prf = xs
... | no contra = dropUntil' decide (tail xs) (ev contra)
dropUntil : {A : Set} {P : A -> Set} (decide : ∀ x → Dec (P x)) → (xs : Stream A) → #0 Eventually P xs → Stream A
dropUntil decide xs ev = dropUntil' decide xs (eventuallyInv ev)
Given a list, a "property" (a function to Bools), and a proof any f xs ≡ true, I want to get a value of Any (λ x → T (f x)) ns, i.e. I want a to convert a proof from any to a proof of Any.
So far, I have been able to sketch the proof, but I am stuck at a seemingly simple statement (please see the second to last line below):
open import Data.Bool
open import Data.Bool.Properties
open import Data.List
open import Data.List.Relation.Unary.Any using (Any; here; there)
open import Data.Unit
open import Relation.Binary.PropositionalEquality
≡→T : ∀ {b : Bool} → b ≡ true → T b
≡→T refl = tt
any-val : ∀ {a} {A : Set a} (f) (ns : List A)
→ any f ns ≡ true
→ Any (λ x → T (f x)) ns
any-val f [] ()
any-val f (n ∷ ns) any-f-⟨n∷ns⟩≡true with f n
... | true = here (≡→T ?)
... | false = there (any-val f ns any-f-⟨n∷ns⟩≡true)
Apparently, I need to prove that f n ≡ true, which should be trivial to prove because we are in the true branch of the with statement f ≡ true.
What is supposed to go in there? What am I understanding incorrectly?
The key is to use the inspect idiom, which I already described thoroughly in another answer here:
`with f x` matches `false`, but cannot construct `f x == false`
This gives the following code, which I cleaned up, including imports:
open import Data.Bool
open import Data.List
open import Data.List.Relation.Unary.Any using (Any; here; there)
open import Data.Unit
open import Relation.Binary.PropositionalEquality
open import Function
≡→T : ∀ {b : Bool} → b ≡ true → T b
≡→T refl = tt
any-val : ∀ {a} {A : Set a}
f (ns : List A) → any f ns ≡ true → Any (T ∘ f) ns
any-val f (x ∷ l) p with f x | inspect f x
... | false | _ = there (any-val f l p)
... | true | [ fx≡true ] = here (≡→T fx≡true)
On a side note, be sure that you have Agda updated to the last version so that you don't have to handle the empty case any-val f [] (), as in your code.
The code is:
filter-pos : {A : Set} → 𝕃 A → (ℕ → 𝔹) → 𝕃 A
filter-pos = {!!}
filter-pos-test : filter-pos ('a' :: 'b' :: 'c' :: 'd' :: 'e' :: 'f' :: []) is-even ≡ 'a' :: 'c' :: 'e' :: []
filter-pos-test = refl
My thought is to use nth to output nth index, use map function to make them into a list, and if it's a even number n, it will return. However, that is not working correctly.
Can someone let me know how I should go about solving this? I think it will be helpful to write a help function to solve the problem.
I'll be using the standard library.
One dumb solution is to zip a list with indices of elements, use the usual filter and then remove the indices. E.g.
'a' :: 'b' :: 'c' :: 'd' :: 'e' :: 'f' :: []
becomes
('a', 0) :: ('b', 1) :: ('c', 2) :: ('d', 3) :: ('e', 4) :: ('f', 5) :: []
filter (is-even ∘ snd) returns
('a', 0) :: ('c', 2) :: ('e', 4) :: []
and map fst results in
'a' :: 'c' :: 'e' :: []
A more natural solution is to traverse a list and increment a counter on each recursive call:
filter-pos : {A : Set} → List A → (ℕ → Bool) → List A
filter-pos {A} xs p = go 0 xs where
go : ℕ -> List A -> List A
go i [] = []
go i (x ∷ xs) = if p i then x ∷ r else r where
r = go (suc i) xs
Here i is the index of an element. However now whenever you need to prove something about filter-pos, you'll need to prove a lemma about go first, because it's go does the actual job, while filter-pos is just a wrapper around it. An idiomatic solution looks like this:
filter-pos : {A : Set} → List A → (ℕ → Bool) → List A
filter-pos [] p = []
filter-pos (x ∷ xs) p = if p 0 then x ∷ r else r where
r = filter-pos xs (p ∘ suc)
Here instead of incrementing a counter we adjust a predicate and compose it with suc. So on a first element we check whether p 0 is true, on a second element we check whether (p ∘ suc) 0 (which immediately reduces to p 1) is true, on a third element we check whether (p ∘ suc ∘ suc) 0 (which immediately reduces to p 2) is true and so on. I.e. this is the same solution as with a counter, but uses only one function.
The last version also can be tuned to work with Fin instead of ℕ
filter-pos-fin : {A : Set} → (xs : List A) → (Fin (length xs) → Bool) → List A
filter-pos-fin [] p = []
filter-pos-fin (x ∷ xs) p = if p zero then x ∷ r else r where
r = filter-pos-fin xs (p ∘ suc)
I am trying to parse a string with natural numbers in Agda.
e.g., the result of stringListToℕ "1,2,3" should be Just (1 ∷ 2 ∷ 3 ∷ [])
My current code is not quite right or by any means nice, but it works.
However it returns the type:
Maybe (List (Maybe ℕ))
The Question is:
How to implement the function stringListToℕ in a nice way (compared to my code);
it should have the type Maybe (List ℕ)
(optional, not important) How can I convert the type Maybe (List (Maybe ℕ)) to Maybe (List ℕ)?
My Code:
charToℕ : Char → Maybe ℕ
charToℕ '0' = just 0
charToℕ '1' = just 1
charToℕ '2' = just 2
charToℕ '3' = just 3
charToℕ '4' = just 4
charToℕ '5' = just 5
charToℕ '6' = just 6
charToℕ '7' = just 7
charToℕ '8' = just 8
charToℕ '9' = just 9
charToℕ _ = nothing
stringToℕ' : List Char → (acc : ℕ) → Maybe ℕ
stringToℕ' [] acc = just acc
stringToℕ' (x ∷ xs) acc = charToℕ x >>= λ n → stringToℕ' xs ( 10 * acc + n )
stringToℕ : String → Maybe ℕ
stringToℕ s = stringToℕ' (toList s) 0
isComma : Char → Bool
isComma h = h Ch.== ','
notComma : Char → Bool
notComma ',' = false
notComma _ = true
{-# NO_TERMINATION_CHECK #-}
split : List Char → List (List Char)
split [] = []
split s = l ∷ split (drop (length(l) + 1) s)
where l : List Char
l = takeWhile notComma s
isNothing' : Maybe ℕ → Bool
isNothing' nothing = true
isNothing' _ = false
isNothing : List (Maybe ℕ) → Bool
isNothing l = any isNothing' l
-- wrong type, should be String -> Maybe (List N)
stringListToℕ : String → Maybe (List (Maybe ℕ))
stringListToℕ s = if (isNothing res) then nothing else just res
where res : List (Maybe ℕ)
res = map stringToℕ (map fromList( split (Data.String.toList s)))
test1 = stringListToℕ "1,2,3"
-- => just (just 1 ∷ just 2 ∷ just 3 ∷ [])
EDIT
I tried to write a conversion function using from-just, but this gives a error when type checking:
conv : Maybe (List (Maybe ℕ)) → Maybe (List ℕ)
conv (just xs) = map from-just xs
conv _ = nothing
the error is:
Cannot instantiate the metavariable _143 to solution
(Data.Maybe.From-just (_145 xs) x) since it contains the variable x
which is not in scope of the metavariable or irrelevant in the
metavariable but relevant in the solution
when checking that the expression from-just has type
Maybe (_145 xs) → _143 xs
I took the liberty of rewriting your split function into something more general which also works with the termination check:
open import Data.List
open import Data.Product
open import Function
splitBy : ∀ {a} {A : Set a} → (A → Bool) → List A → List (List A)
splitBy {A = A} p = uncurry′ _∷_ ∘ foldr step ([] , [])
where
step : A → List A × List (List A) → List A × List (List A)
step x (cur , acc) with p x
... | true = x ∷ cur , acc
... | false = [] , cur ∷ acc
Also, stringToℕ "" should most likely be nothing, unless you really want:
stringListToℕ "1,,2" ≡ just (1 ∷ 0 ∷ 2 ∷ [])
Let's rewrite it a bit (note that helper is your original stringToℕ function):
stringToℕ : List Char → Maybe ℕ
stringToℕ [] = nothing
stringToℕ list = helper list 0
where {- ... -}
And now we can put it all together. For simplicity I'm using List Char everywhere, sprinkle with fromList/toList as necessary):
let x1 = s : List Char -- start
let x2 = splitBy notComma x1 : List (List Char) -- split at commas
let x3 = map stringToℕ x2 : List (Maybe ℕ) -- map our ℕ-conversion
let x4 = sequence x3 : Maybe (List ℕ) -- turn Maybe inside out
You can find sequence in Data.List; we also have to specify which monad instance we want to use. Data.Maybe exports its monad instance under the name monad. Final code:
open import Data.Char
open import Data.List
open import Data.Maybe
open import Data.Nat
open import Function
stringListToℕ : List Char → Maybe (List ℕ)
stringListToℕ = sequence Data.Maybe.monad ∘ map stringToℕ ∘ splitBy notComma
And a small test:
open import Relation.Binary.PropositionalEquality
test : stringListToℕ ('1' ∷ '2' ∷ ',' ∷ '3' ∷ []) ≡ just (12 ∷ 3 ∷ [])
test = refl
Considering your second question: there are many ways to turn a Maybe (List (Maybe ℕ)) into a Maybe (List ℕ), for example:
silly : Maybe (List (Maybe ℕ)) → Maybe (List ℕ)
silly _ = nothing
Right, this doesn't do much. We'd like the conversion to preserve the elements if they are all just. isNothing already does this part of checking but it cannot get rid of the inner Maybe layer.
from-just could work since we know that when we use it, all elements of the List must be just x for some x. The problem is that conv in its current form is just wrong - from-just works as a function of type Maybe A → A only when the Maybe value is just x! We could very well do something like this:
test₂ : Maybe (List ℕ)
test₂ = conv ∘ just $ nothing ∷ just 1 ∷ []
And since from-list behaves as a Maybe A → ⊤ when given nothing, we are esentially trying to construct a heterogeneous list with elements of type both ⊤ and ℕ.
Let's scrap this solution, I'll show a much simpler one (in fact, it should resemble the first part of this answer).
We are given a Maybe (List (Maybe ℕ)) and we gave two goals:
take the inner List (Maybe ℕ) (if any), check if all elements are just x and in this case put them all into a list wrapped in a just, otherwise return nothing
squash the doubled Maybe layer into one
Well, the second point sounds familiar - that's something monads can do! We get:
join : {A : Set} → Maybe (Maybe A) → Maybe A
join mm = mm >>= λ x → x
where
open RawMonad Data.Maybe.monad
This function could work with any monad but we'll be fine with Maybe.
And for the first part, we need a way to turn a List (Maybe ℕ) into a Maybe (List ℕ) - that is, we want to swap the layers while propagating the possible error (i.e. nothing) into the outer layer. Haskell has specialized typeclass for this kind of stuff (Traversable from Data.Traversable), this question has some excellent answers if you'd like to know more. Basically, it's all about rebuilding the structure while collecting the "side effects". We'll be fine with the version that works just for Lists and we're back at sequence again.
There's still one piece missing, let's look at what we have so far:
sequence-maybe : List (Maybe ℕ) → Maybe (List ℕ)
sequence-maybe = sequence Data.Maybe.monad
join : Maybe (Maybe (List ℕ)) → Maybe (List ℕ)
-- substituting A with List ℕ
We need to apply sequence-maybe inside one Maybe layer. That's where the Maybe functor instance comes into play (you could do it with a monad instance alone, but it's more convenient). With this functor instance, we can lift an ordinary function of type a → b into a function of type Maybe a → Maybe b. And finally:
open import Category.Functor
open import Data.Maybe
final : Maybe (List (Maybe ℕ)) → Maybe (List ℕ)
final mlm = join (sequence-maybe <$> mlm)
where
open RawFunctor functor
I had a go at it trying not to be clever and using simple recursive functions rather than stdlib magic. parse xs m ns parses xs by recording the (possibly empty) prefix already read in m while keeping the list of numbers already parsed in the accumulator ns.
If a parsing failure happens (non recognized character, two consecutive ,, etc.) everything is thrown away and we return nothing.
module parseList where
open import Data.Nat
open import Data.List
open import Data.Maybe
open import Data.Char
open import Data.String
isDigit : Char → Maybe ℕ
isDigit '0' = just 0
isDigit '1' = just 1
isDigit '2' = just 2
isDigit '3' = just 3
isDigit _ = nothing
attach : Maybe ℕ → ℕ → ℕ
attach nothing n = n
attach (just m) n = 10 * m + n
Quote : List Char → Maybe (List ℕ)
Quote xs = parse xs nothing []
where
parse : List Char → Maybe ℕ → List ℕ → Maybe (List ℕ)
parse [] nothing ns = just ns
parse [] (just n) ns = just (n ∷ ns)
parse (',' ∷ tl) (just n) ns = parse tl nothing (n ∷ ns)
parse (hd ∷ tl) m ns with isDigit hd
... | nothing = nothing
... | just n = parse tl (just (attach m n)) ns
stringListToℕ : String → Maybe (List ℕ)
stringListToℕ xs with Quote (toList xs)
... | nothing = nothing
... | just ns = just (reverse ns)
open import Relation.Binary.PropositionalEquality
test : stringListToℕ ("12,3") ≡ just (12 ∷ 3 ∷ [])
test = refl
Here is the Code from Vitus as a running example that uses the Agda Prelude
module Parse where
open import Prelude
-- Install Prelude
---- clone this git repo:
---- https://github.com/fkettelhoit/agda-prelude
-- Configure Prelude
--- press Meta/Alt and the letter X together
--- type "customize-group" (i.e. in the mini buffer)
--- type "agda2"
--- expand the Entry "Agda2 Include Dirs:"
--- add the directory
open import Data.Product using (uncurry′)
open import Data.Maybe using ()
open import Data.List using (sequence)
splitBy : ∀ {a} {A : Set a} → (A → Bool) → List A → List (List A)
splitBy {A = A} p = uncurry′ _∷_ ∘ foldr step ([] , [])
where
step : A → List A × List (List A) → List A × List (List A)
step x (cur , acc) with p x
... | true = x ∷ cur , acc
... | false = [] , cur ∷ acc
charsToℕ : List Char → Maybe ℕ
charsToℕ [] = nothing
charsToℕ list = stringToℕ (fromList list)
notComma : Char → Bool
notComma c = not (c == ',')
-- Finally:
charListToℕ : List Char → Maybe (List ℕ)
charListToℕ = Data.List.sequence Data.Maybe.monad ∘ map charsToℕ ∘ splitBy notComma
stringListToℕ : String → Maybe (List ℕ)
stringListToℕ = charListToℕ ∘ toList
-- Test
test1 : charListToℕ ('1' ∷ '2' ∷ ',' ∷ '3' ∷ []) ≡ just (12 ∷ 3 ∷ [])
test1 = refl
test2 : stringListToℕ "12,33" ≡ just (12 ∷ 33 ∷ [])
test2 = refl
test3 : stringListToℕ ",,," ≡ nothing
test3 = refl
test4 : stringListToℕ "abc,def" ≡ nothing
test4 = refl
I was proving some properties of filter and map, everything went quite good until I stumbled on this property: filter p (map f xs) ≡ map f (filter (p ∘ f) xs). Here's a part of the code that's relevant:
open import Relation.Binary.PropositionalEquality
open import Data.Bool
open import Data.List hiding (filter)
import Level
filter : ∀ {a} {A : Set a} → (A → Bool) → List A → List A
filter _ [] = []
filter p (x ∷ xs) with p x
... | true = x ∷ filter p xs
... | false = filter p xs
Now, because I love writing proofs using the ≡-Reasoning module, the first thing I tried was:
open ≡-Reasoning
open import Function
filter-map : ∀ {a b} {A : Set a} {B : Set b}
(xs : List A) (f : A → B) (p : B → Bool) →
filter p (map f xs) ≡ map f (filter (p ∘ f) xs)
filter-map [] _ _ = refl
filter-map (x ∷ xs) f p with p (f x)
... | true = begin
filter p (map f (x ∷ xs))
≡⟨ refl ⟩
f x ∷ filter p (map f xs)
-- ...
But alas, that didn't work. After trying for one hour, I finally gave up and proved it in this way:
filter-map (x ∷ xs) f p with p (f x)
... | true = cong (λ a → f x ∷ a) (filter-map xs f p)
... | false = filter-map xs f p
Still curious about why going through ≡-Reasoning didn't work, I tried something very trivial:
filter-map-def : ∀ {a b} {A : Set a} {B : Set b}
(x : A) xs (f : A → B) (p : B → Bool) → T (p (f x)) →
filter p (map f (x ∷ xs)) ≡ f x ∷ filter p (map f xs)
filter-map-def x xs f p _ with p (f x)
filter-map-def x xs f p () | false
filter-map-def x xs f p _ | true = -- not writing refl on purpose
begin
filter p (map f (x ∷ xs))
≡⟨ refl ⟩
f x ∷ filter p (map f xs)
∎
But typechecker doesn't agree with me. It would seem that the current goal remains filter p (f x ∷ map f xs) | p (f x) and even though I pattern matched on p (f x), filter just won't reduce to f x ∷ filter p (map f xs).
Is there a way to make this work with ≡-Reasoning?
Thanks!
The trouble with with-clauses is that Agda forgets the information it learned from pattern match unless you arrange beforehand for this information to be preserved.
More precisely, when Agda sees a with expression clause, it replaces all the occurences of expression in the current context and goal with a fresh variable w and then gives you that variable with updated context and goal into the with-clause, forgetting everything about its origin.
In your case, you write filter p (map f (x ∷ xs)) inside the with-block, so it goes into scope after Agda has performed the rewriting, so Agda has already forgotten the fact that p (f x) is true and does not reduce the term.
You can preserve the proof of equality by using one of the "Inspect"-patterns from the standard library, but I'm not sure how it can be useful in your case.