idris

问题 In Agda, there is Set n . As I understand, Set n extends the Haskell-style value-type-kind hierarchy to infinite levels. That is, Set 0 is the universe of normal types, Set 1 is the universe of normal kinds, Set 2 is the universe of normal sorts, etc. In contrast, Idris has the so called "cumulative hierarchy of universes". It seems that for a < b , Type a: Type b , and universe levels are inferred. But what does it mean in real world programs? Can't we define something that only operate on

问题 Most proof assistants are functional programming languages with dependent types. They can proof programs/algorithms. I'm interested, instead, in proof assistant suitable best for mathematics and only (calculus for instance). Can you recommend one? I heard about Mizar but I don’t like that the source code is closed, but if it is best for math I will use it. How well the new languages such as Agda and Idris are suited for mathematical proofs? 回答1: Coq has extensive libraries covering real

问题 I've been studying dependent types and I understand the following: Why universal quantification is represented as a dependent function type. ∀(x:A).B(x) means “for all x of type A there is a value of type B(x) ” . Hence it's represented as a function which when given any value x of type A returns a value of type B(x) . Why existential quantification is represented as a dependent pair type. ∃(x:A).B(x) means “there exists an x of type A for which there is a value of type B(x) ” . Hence it's

问题 I am trying to model a simple survey form in Idris and currently struggling with validating user input, which comes as a string, w.r.t. to the type of questions asked. Currently I have the following types: data Question : Type where QCM : {numOptions : Nat} -> (question : String) -> (qcmOptions : Vect numOptions String) -> (expected : Fin numOptions) -> Question data Answer : (q : Question) -> Type where AnswerQCM : (option : Fin n) -> Answer (QCM {numOptions = n } q opts exp) total

问题 I am new to dependent types and am confused about the difference between the two. It seems people usually say a type is parameterized by another type and indexed by some value . But isn't there no distinction between types and terms in a dependently typed language? Is the distinction between parameters and indices fundamental? Can you show me examples showing difference in their meanings in both programming and theorem proving? 回答1: When you see a family of types, you may wonder whether each

问题 I want to prove that if b (Integer) divides a (Integer), then b also divides a * c (where c is an Integer). First, I need to reformulate the problem into a computer understandable problem, here was an attempt: -- If a is divisible by b, then there exists an integer such that a = b * n divisibleBy : (a, b : Integer) -> (n : Integer ** (a = b * n)) -- If b | a then b | ac. alsoDividesMultiples : (a, b, c : Integer) -> (divisibleBy a b) -> (divisibleBy (a * c) b) However, I am getting

问题 I attempted to compare a String and String , expecting True . Idris> String == String Can't find implementation for Eq Type Then I expected False when comparing a String to a Bool . Idris> String /= Bool Can't find implementation for Eq Type Am I missing an import ? 回答1: You can't as it would break parametricity, which we have in Idris. We can't pattern match on types. But this would be necessary to write the Eq implementation, for example: {- Doesn't work! eqNat : Type -> Bool eqNat Nat =

问题 The function convert below has the type signature : SUnit fromUnit-> SUnit toUnit ->Value fromUnit -> Value toUnit , which has redundancy , because the same information could be expressed by : Value fromUnit -> Value toUnit . 1) Is there a way to get rid of the first two arguments ( SUnit fromUnit-> SUnit toUnit ) ? 2) Is there some other way by which this simple dependently typed program could be written more elegantly ? 3) How would this program look like in Idris ? {-# LANGUAGE GADTs

问题 I'm trying to compute parity together with the floor of the half, over natural numbers: data IsEven : Nat -> Nat -> Type where Times2 : (n : Nat) -> IsEven (n + n) n data IsOdd : Nat -> Nat -> Type where Times2Plus1 : (n : Nat) -> IsOdd (S (n + n)) n parity : (n : Nat) -> Either (Exists (IsEven n)) (Exists (IsOdd n)) I tried going with the obvious implementation of parity : parity Z = Left $ Evidence _ $ Times2 0 parity (S Z) = Right $ Evidence _ $ Times2Plus1 0 parity (S (S n)) with (parity

问题 I'm learning Idris and I got stuck on a very simple lemma that shows that some specific index is impossible for a data type. I've tried to use impossible patterns but Idris refuses them with the following error message: RegExp.idr line 32 col 13: hasEmptyZero prf is a valid case The complete code piece is available in the following gist: https://gist.github.com/rodrigogribeiro/f27f1f035e5a98f506ee Any help is appreciated. 回答1: I've talk with people at freenode Idris community and they